NFEM Complete

NONLINEAR FINITE ELEMENT METHODS

Carlos A. Felippa

Department of Aerospace Engineering Sciences andCenter for Space Structures and Controls

University of ColoradoBoulder, Colorado 80309-0429, USA

August 2001

Material assembled from lecture notes for the course Nonlinear Finite Elements Methods, offeredsince 1987 to date at the Aerospace Engineering Sciences Department of the University of Coloradoat Boulder.

Preface

This textbook presents an Introduction to the computer-based simulation of nonlinear structures bythe Finite Element Method (FEM). It assembles the still “unconverged” lecture notes of NonlinearFinite Element Methods or NFEM. This is an advanced graduate course offered in the AerospaceEngineering Sciences of the University of Colorado at Boulder.NFEM was first taught on the Spring Semester 1986 and has been repeated every two or three years.Unlike the Introduction to Finite Element Methods (IFEM), NFEM is not a core course. It istypicall taken by second year graduate students that are interested in the topic of nonlinear simulationof mechanical systems.Prerequisites for the course are an introductory course in finite elements such as IFEM, graduate-levelcalculus, linear algebra, knowledge of structural mechanics at the Mechanics of Materials level, andability to program in a higher level language such as Matlab or Mathematica.The course originally used Fortran 77 as computer implementation language. This has been graduallychanged toMathematica since 1995. Unlike IFEM the changeover is not yet complete since the coursehas been offered only twice since.

Book Objectives

(To be completed)

Book Organization

(To be completed)

Exercises

Each Chapter is followed by a list of homework exercises that pose problems of varying difficulty.Each exercise is labeled by a tag of the form

[type:rating]

The type is indicated by letters A, C, D or N for exercises to be answered primarily by analyticalwork, computer programming, descriptive narration, and numerical calculations, respectively. Someexercises involve a combination of these traits, in which case a combination of letters separatedby + is used, e.g., A+N. For some problems heavy analytical work may be helped by the use of acomputer-algebra system, in which case the type is identified as A/C.The rating is a number between 5 and 50 that estimates the degree of difficulty of an Exercise, in thefollowing “logarithmic” scale:5 A simple question that can be answered in seconds, or is already answered in the text if the

student has read and understood the material.10 A straightforward question that can be answered in minutes.15 A relatively simple question that requires some thinking, and may take on the order of half to

one hour to answer.

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20 Either a problem of moderate difficulty, or a straightforward one requiring lengthy computationsor some programming, normally taking one to six hours of work.

25 A scaled up version of the above, estimated to require six hours to one day of work.30 A problem of moderate difficulty that normally requires on the order of one or two days of work.

Arriving at the answer may involve a combination of techniques, some background or referencematerial, or lenghty but straightforward programming.

40 A difficult problem that may be solvable only by gifted and well prepared individual students,or a team. Difficulties may be due to the need of correct formulation, advanced mathematics,or high level programming. With the proper preparation, background and tools these problemsmay be solved in hours or days, while remaining inaccessible to unprepared or average students.

50 A research problem, worthy of publication if solved.Most Exercises have a rating of 15 or 20. Assigning three or four per week puts a load of roughly 5-10hours of solution work, plus the time needed to prepare the answer material. Assignments of difficulty25 or 30 are better handled by groups, or given in take-home exams. Assignments of difficulty over30 are never assigned in the course, but provided as a challenge for an elite group.Occasionally an Exercise has two or more distinct but related parts identified as items. In that case arating may be given for each item. For example: [A/C:15+20]. This does not mean that the exerciseas a whole has a difficulty of 35, because the scale is roughly logarithmic; the numbers simply ratethe expected effort per item.

Selecting Course Material

(To be completed)

Acknowledgements

Thanks are due to students and colleagues who have provided valuable feedback on the original courseNotes, and helped metamorphosis into a textbook. Two invigorating sabbaticals in 1993 and 2001provided blocks of time to develop, reformat and integrate material. The hospitality of Dr. P. G.Bergan of Det Norske Veritas at Oslo, Norway and Professor E. Onate of CIMNE/UPC at Barcelona,Spain, during those sabbaticals is gratefully acknowleged.

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Chapter ContentsSection1 . . . . . . . . . . . . . . . . . . 1-12 . . . . . . . . . . . . . . . . . . 2-13 . . . . . . . . . . . . . . . . . . 3-14 . . . . . . . . . . . . . . . . . . 4-15 . . . . . . . . . . . . . . . . . . 5-16 . . . . . . . . . . . . . . . . . . 6-17 . . . . . . . . . . . . . . . . . . 7-18 . . . . . . . . . . . . . . . . . . 8-19 . . . . . . . . . . . . . . . . . . 9-110 . . . . . . . . . . . . . . . . . . 10-111 . . . . . . . . . . . . . . . . . . 11-112 . . . . . . . . . . . . . . . . . . 12-113 . . . . . . . . . . . . . . . . . . 13-114 . . . . . . . . . . . . . . . . . . 14-115 . . . . . . . . . . . . . . . . . . 15-116 . . . . . . . . . . . . . . . . . . 16-117 . . . . . . . . . . . . . . . . . . 17-118 . . . . . . . . . . . . . . . . . . 18-119 . . . . . . . . . . . . . . . . . . 19-120 . . . . . . . . . . . . . . . . . . 20-121 . . . . . . . . . . . . . . . . . . 21-122 . . . . . . . . . . . . . . . . . . 22-123 . . . . . . . . . . . . . . . . . . 23-124 . . . . . . . . . . . . . . . . . . 24-123 . . . . . . . . . . . . . . . . . . 23-124 . . . . . . . . . . . . . . . . . . 24-125 . . . . . . . . . . . . . . . . . . 25-126 . . . . . . . . . . . . . . . . . . 26-127 . . . . . . . . . . . . . . . . . . 27-128 . . . . . . . . . . . . . . . . . . 28-1AppendicesA . . . . . . . . . . . . . . . . . . A-1B . . . . . . . . . . . . . . . . . . B-1C . . . . . . . . . . . . . . . . . . C-1D . . . . . . . . . . . . . . . . . . D-1H . . . . . . . . . . . . . . . . . . H-1

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1Overview

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Chapter 1: OVERVIEW 1–2

TABLE OF CONTENTS

Page§1.1. Book Scope 1–3§1.2. Where the Book Fits 1–3

§1.2.1. Top Level Classification . . . . . . . . . . . . . . 1–3§1.2.2. Computational Mechanics . . . . . . . . . . . . . 1–3§1.2.3. Statics versus Dynamics . . . . . . . . . . . . . . 1–5§1.2.4. Linear versus Nonlinear . . . . . . . . . . . . . 1–5§1.2.5. Discretization Methods . . . . . . . . . . . . . . 1–5§1.2.6. FEM Variants . . . . . . . . . . . . . . . . . 1–6

§1.3. The FEM Analysis Process 1–6§1.3.1. The Physical FEM . . . . . . . . . . . . . . . . 1–6§1.3.2. The Mathematical FEM . . . . . . . . . . . . . 1–7§1.3.3. Synergy of Physical and Mathematical FEM . . . . . . . 1–8§1.3.4. Streamlined Idealization and Discretization . . . . . . . 1–10

§1.4. Method Interpretations 1–10§1.4.1. Physical Interpretation . . . . . . . . . . . . . . 1–10§1.4.2. Mathematical Interpretation . . . . . . . . . . . . 1–11

§1.5. The Solution Morass 1–11§1.6. Historical Background 1–12

§1.6.1. Smooth Nonlinearities . . . . . . . . . . . . . . 1–13§1.6.2. Rough Nonlinearities . . . . . . . . . . . . . . 1–15§1.6.3. Hybrid Approach . . . . . . . . . . . . . . . . 1–15§1.6.4. Summary of Present Status . . . . . . . . . . . . 1–16

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1–3 §1.2 WHERE THE BOOK FITS

§1.1. Book ScopeThis is a textbook about nonlinear structural analysis using the Finite Element Method (FEM) as adiscretization tool. It is intended to support a course at the second-year level of graduate studies inAerospace, Mechanical, or Civil Engineering. The focus of the book is on geometrically nonlinearproblems as well as structural stability.Basic prerequisite to understanding the material covered here is an introductory FEM course at thegraduate level. Such course typically focuses on linear problems and assumes working knowledgeof matrix algebra, as well as that of structural and solid mechanics at the undergraduate level.This Chapter presents an overview of what the book covers. As noted above, it is assumed thatthe reader has a good idea of what finite elements are and what they are used for. Therefore thoseaspects are glossed over.

§1.2. Where the Book FitsThis Section outlines where the book material fits within the large scope of Mechanics. In theensuing multilevel classification, topics addressed in some depth are emphasized in bold typeface.

§1.2.1. Top Level ClassificationDefinitions of Mechanics in dictionaries usually state two flavors:• The branch of Physics that studies the effect of forces and energy on physical bodies.1

• The practical application of that science to the design, construction or operation of materialsystems or devices, such as machines, vehicles or structures.

These flavors are science and engineering oriented, respectively. But dictionaries are notoriouslyoutdated. For our objectives it will be convenient to distinguish four major areas:

Mechanics

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TheoreticalAppliedComputationalExperimental

(1.1)

Theoretical mechanics deals with fundamental laws and principles studied for their intrinsic sci-entific value. Applied mechanics transfers this theoretical knowledge to scientific and engineeringapplications, especially as regards the construction of mathematical models of physical phenomena.Computational mechanics solves specific problems by model-based simulation through numericalmethods implemented on digital computers. Experimental mechanics subjects the knowledge de-rived from theory, application and simulation to the ultimate test of observation.

Remark 1.1. Paraphrasing an old joke about mathematicians, one may define a computational mechanicianas a person who searches for solutions to given problems, an applied mechanician as a person who searchesfor problems that fit given solutions, and a theoretical mechanician as a person who can prove the existenceof problems and solutions. As regards experimentalists, make up your own joke.

1 Here the term “bodies” includes all forms of matter, whether solid, liquid or gaseous; as well as all physical scales, fromsubatomic through cosmic.

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§1.2.2. Computational MechanicsSeveral branches of computational mechanics can be distinguished according to the physical scaleof the focus of attention:

Computational Mechanics

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NanomechanicsMicromechanics

Continuum mechanics% Solids and StructuresFluidsMultiphysics

Systems

(1.2)

Nanomechanics deals with phenomena at the molecular and atomic levels. As such, it is closelyrelated to particle physics and chemistry. At the atomic scale it transitions to quantum mechanics.Micromechanics looks primarily at the crystallographic and granular levels of matter. Its maintechnological application is the design and fabrication of materials and microdevices.Continuummechanics studies bodies at themacroscopic level, using continuummodels inwhich themicrostructure is homogenized by phenomenological averaging. The two traditional areas of appli-cation are solid and fluid mechanics. Structural mechanics is a conjoint branch of solid mechanics,since structures, for obvious reasons, are fabricated with solids. Computational solid mechan-ics favors a applied-sciences approach, whereas computational structural mechanics emphasizestechnological applications to the analysis and design of structures.Computational fluid mechanics deals with problems that involve the equilibrium and motion ofliquid and gases. Well developed related subareas are hydrodynamics, aerodynamics, atmosphericphysics, propulsion, and combustion.Multiphysics is a more recent newcomer.2 This area is meant to include mechanical systems thattranscend the classical boundaries of solid and fluid mechanics. A key example is interactionbetween fluids and structures, which has important application subareas such as aeroelasticity andhydroelasticity. Phase change problems such as ice melting and metal solidification fit into thiscategory, as do the interaction of control, mechanical and electromagnetic systems.Finally, system identifies mechanical objects, whether natural or artificial, that perform a distin-guishable function. Examples of man-made systems are airplanes, building, bridges, engines, cars,microchips, radio telescopes, robots, roller skates and garden sprinklers. Biological systems, suchas a whale, amoeba, virus or pine tree are included if studied from the viewpoint of biomechanics.Ecological, astronomical and cosmological entities also form systems.3

In the progression of (1.2), system is the most general concept. Systems are studied by decompo-sition: its behavior is that of its components plus the interaction between the components. Com-ponents are broken down into subcomponents and so on. As this hierarchical process continues

2 This unifying term is in fact missing from most dictionaries, as it was introduced by computational mechanicians in the1970s. Some multiphysics problems, however, are older. For example, aircraft aeroelasticity emerged in the 1920s.

3 Except that their function may not be clear to us. “What is that breathes fire into the equations and makes a universefor them to describe? The usual approach of science of constructing a mathematical model cannot answer the questionsof why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?”(Stephen Hawking).

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1–5 §1.2 WHERE THE BOOK FITS

the individual components become simple enough to be treated by individual disciplines, but theirinteractions may get more complex. Thus there are tradeoff skills in deciding where to stop.4

§1.2.3. Statics versus Dynamics

Continuum mechanics problems may be subdivided according to whether inertial effects are takeninto account or not:

Continuum mechanics

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Statics&

Time InvariantQuasi-static

Dynamics(1.3)

In statics inertial forces are ignored or neglected. Static problems may be subclassified into timeinvariant and quasi-static. For the former time need not be considered explicitly; any historicaltime-like response-ordering parameter (should one is needed) will do. In quasi-static problems suchas foundation settlement, creep flow, rate-dependent plasticity or fatigue cycling, a more realisticestimation of time is required but inertial forces are neglected because motions are slow.In dynamics the time dependence is explicitly considered because the calculation of inertial (and/ordamping) forces requires derivatives respect to actual time to be taken.

§1.2.4. Linear versus Nonlinear

A classification of static problems that is particularly relevant to this book is

Statics&

LinearNonlinear

(1.4)

Linear static analysis deals with static problems in which the response is linear in the cause-and-effect sense. For example: if the applied forces are doubled, the displacements and internal stressesalso double. Problems outside this domain are classified as nonlinear.

§1.2.5. Discretization Methods

A final classification of computational solid and structural mechanics (CSSM) is based on thediscretization method by which the continuum mathematical model is discretized in space, i.e.,converted to a discrete model of finite number of degrees of freedom:

Computational solid and structuralmechanics spatial discretization

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Finite Element Method (FEM)Boundary Element Method (BEM)Finite Difference Method (FDM)Finite Volume Method (FVM)Spectral MethodMesh-Free Method

(1.5)

4 Thus in breaking down a car engine, say, the decomposition does not usually proceed beyond the components that maybe bought at a automotive shop.

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For linear problems finite element methods currently dominate the scene, with boundary elementmethods posting a strong second choice in selected application areas. For nonlinear problems thedominance of finite element methods is overwhelming.Classical finite difference methods in solid and structural mechanics have virtually disappearedfrom practical use. This statement is not true, however, for fluid mechanics, where finite differencediscretization methods are still dominant. Finite-volume methods, which focus on conservationlaws, are important in highly nonlinear problems of fluid mechanics. Spectral methods are basedon global transformations, based on eigendecomposition of the governing equations, that map thephysical computational domain to transform spaces where the problem can be efficiently solved.A recent newcomer to the scene are the mesh-free methods. These are finite different methods onarbitrary grids constructed using a subset of finite element techniques

§1.2.6. FEM Variants

The term Finite Element Method actually identifies a broad spectrum of techniques that share com-mon features, as outlined in introductory FEM textbooks. Two subclassifications thatfit applicationsto structural mechanics particularly well are

FEM Formulation

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DisplacementEquilibriumMixedHybrid

FEM Solution% StiffnessFlexibilityMixed (a.k.a. Combined)

(1.6)

Using the classification of (1.1) through (1.5) we can now state the book topic more precisely:

The continuum-model-based simulation of nonlinear static structures discretized by FEM(1.7)

Of the FEM variants listed in (1.6) emphasis will be placed on the displacement formulation andstiffness solution. This particular combination is called the Direct Stiffness Method or DSM.

§1.3. The FEM Analysis Process

Processes that use FEM involve carrying out a sequence of steps in some way. Those sequencestake two canonical configurations, depending on (i) the environment in which FEM is used and (ii)the main objective: model-based simulation of physical systems, or numerical approximation tomathematical problems. Both are reviewed below to introduce terminology used in the sequel.

§1.3.1. The Physical FEM

A canonical use of FEM is simulation of physical systems. This requires models of such systems.Consequenty the methodology is often called model-based simulation.The process is illustrated in Figure 1.1. The centerpiece is the physical system to be modeled.Accordingly, this configuration is called the Physical FEM. The processes of idealization anddiscretization are carried out concurrently to produce the discrete model. The solution step ishandled by an equation solver often customized to FEM, which delivers a discrete solution (orsolutions).

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1–7 §1.3 THE FEM ANALYSIS PROCESS

Physical system

simulation error= modeling + solution error

solution error

Discretemodel

Discretesolution

VALIDATION

VERIFICATION

FEM

CONTINUIFICATION

Ideal Mathematical

model

IDEALIZATION &DISCRETIZATION

SOLUTION

generallyirrelevant

Figure 1.1. The Physical FEM. The physical system (left) is the source of the simulation process.The ideal mathematical model (should one go to the trouble of constructing it) is inessential.

Figure 1.1 also shows an ideal mathematical model. This may be presented as a continuum limit or“continuification” of the discrete model. For some physical systems, notably those well modeledby continuum fields, this step is useful. For others, such as complex engineering systems (say, aflying aircraft) it makes no sense. Indeed Physical FEM discretizations may be constructed andadjusted without reference to mathematical models, simply from experimental measurements.The concept of error arises in the Physical FEM in two ways. These are known as verificationand validation, respectively. Verification is done by replacing the discrete solution into the discretemodel to get the solution error. This error is not generally important. Substitution in the idealmathematical model in principle provides the discretization error. This step is rarely useful incomplex engineering systems, however, because there is no reason to expect that the continuummodel exists, and even if it does, that it is more physically relevant than the discrete model.Validation tries to compare the discrete solution against observation by computing the simulationerror, which combines modeling and solution errors. As the latter is typically unimportant, thesimulation error in practice can be identified with the modeling error. In real-life applications thiserror overwhelms the others.5

§1.3.2. The Mathematical FEM

The other canonical way of using FEM focuses on themathematics. The process steps are illustratedin Figure 1.2. The spotlight now falls on the mathematical model. This is often an ordinarydifferential equation (ODE), or a partial differential equation (PDE) in space and time. A discretefinite element model is generated from a variational or weak form of the mathematical model.6This is the discretization step. The FEM equations are solved as described for the Physical FEM.

5 “All models are wrong; some are useful” (George Box)6 The distinction between strong, weak and variational forms is discussed in advanced FEM courses. In the present booksuch forms will be largely stated (and used) as recipes.

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Discretization + solution error

REALIZATIONIDEALIZATION

solution error

Discretemodel

Discretesolution

VERIFICATION

VERIFICATION

FEM

IDEALIZATION &DISCRETIZATION

SOLUTION

Idealphysical system

Mathematicalmodel

generally irrelevant

Figure 1.2. The Mathematical FEM. The mathematical model (top) is the source ofthe simulation process. Discrete model and solution follow from it. The ideal physical

system (should one go to the trouble of exhibiting it) is inessential.

On the left, Figure 1.2 shows an ideal physical system. This may be presented as a realization ofthe mathematical model. Conversely, the mathematical model is said to be an idealization of thissystem. E.g., if the mathematical model is the Poisson’s PDE, realizations may be heat conductionor an electrostatic charge-distribution problem. This step is inessential and may be left out. IndeedMathematical FEM discretizations may be constructed without any reference to physics.The concept of error arises when the discrete solution is substituted in the “model” boxes. Thisreplacement is generically called verification. As in the Physical FEM, the solution error is theamount by which the discrete solution fails to satisfy the discrete equations. This error is relativelyunimportant when using computers, and in particular direct linear equation solvers, for the solutionstep. More relevant is the discretization error, which is the amount by which the discrete solutionfails to satisfy themathematical model.7 Replacing into the ideal physical systemwould in principlequantify modeling errors. In the Mathematical FEM this is largely irrelevant, however, because theideal physical system is merely that: a figment of the imagination.

§1.3.3. Synergy of Physical and Mathematical FEM

The foregoing canonical sequences are not exclusive but complementary. This synergy8 is one ofthe reasons behind the power and acceptance of the method. Historically the Physical FEMwas thefirst one to be developed to model complex physical systems such as aircraft, as narrated in §1.7.The Mathematical FEM came later and, among other things, provided the necessary theoreticalunderpinnings to extend FEM beyond structural analysis.Aglance at the schematics of a commercial jet aircraftmakes obvious the reasons behind thePhysical

7 This error can be computed in several ways, the details of which are of no importance here.8 Such interplay is not exactly a new idea: “The men of experiment are like the ant, they only collect and use; the reasonersresemble spiders, who make cobwebs out of their own substance. But the bee takes the middle course: it gathers itsmaterial from the flowers of the garden and field, but transforms and digests it by a power of its own.” (Francis Bacon).

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1–9 §1.3 THE FEM ANALYSIS PROCESS

FEM Library

Component

discrete

model

Component

equations

Physical

system

System

discrete

model

Complete

solution

Mathematical

model

SYSTEM

LEVEL

COMPONENT

LEVEL

Figure 1.3. Combining physical and mathematical modeling through multilevel FEM. Onlytwo levels (system and component) are shown for simplicity.

FEM. There is no simple differential equation that captures, at a continuum mechanics level,9 thestructure, avionics, fuel, propulsion, cargo, and passengers eating dinner. There is no reason fordespair, however. The time honored divide and conquer strategy, coupled with abstraction, comesto the rescue.First, separate the structure out and view the rest as masses and forces. Second, consider the aircraftstructure as built up of substructures (a part of a structure devoted to a specific function): wings,fuselage, stabilizers, engines, landing gears, and so on.Take each substructure, and continue to break it down into components: rings, ribs, spars, coverplates, actuators, etc. Continue through as many levels as necessary. Eventually those componentsbecome sufficiently simple in geometry and connectivity that they can be reasonably well describedby the mathematical models provided, for instance, by Mechanics of Materials or the Theory ofElasticity. At that point, stop. The component level discrete equations are obtained from a FEMlibrary based on the mathematical model.The system model is obtained by going through the reverse process: from component equationsto substructure equations, and from those to the equations of the complete aircraft. This systemassembly process is governed by the classical principles of Newtonian mechanics, which providethe necessary inter-component “glue.” The multilevel decomposition process is diagramed inFigure 1.3, in which intermediate levels are omitted for simplicity

Remark 1.2. More intermediate decomposition levels are used in systems such as offshore and ship structures,

9 Of course at the (sub)atomic level quantum mechanics works for everything, from landing gears to passengers. Butit would be slightly impractical to represent the aircraft by, say, 1036 interacting particles modeled by the Schrodingerequations. More seriously, Truesdell and Toupin correctly note that “Newtonian mechanics, while not appropriate to thecorpuscles making up a body, agrees with experience when applied to the body as a whole, except for certain phenomenaof astronomical scale” [388, p. 228].

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joint

Physical System

support

member

IDEALIZATION

Figure 1.4. The idealization process for a simple structure. The physical system— here aroof truss— is directly idealized by the mathematical model: a pin-jointed bar assembly. For

this particular structure idealized and discrete models coalesce.

which are characterized by a modular fabrication process. In that case multilevel decomposition mimics theway the system is actually fabricated. The general technique, called superelements, is discussed in Chapter 11.

Remark 1.3. There is no point in practice in going beyond a certain component level while considering thecomplete system. The reason is that the level of detail can become overwhelming without adding relevantinformation. Usually that point is reached when uncertainty impedes further progress. Further refinementof specific components is done by the so-called global-local analysis technique outlined in Chapter 10. Thistechnique is an instance of multiscale analysis.

§1.3.4. Streamlined Idealization and Discretization

For sufficiently simple structures, passing to a discrete model is carried out in a single idealizationand discretization step, as illustrated for the truss roof structure shown in Figure 1.4. Other levelsare unnecessary in such cases. Of course the truss may be viewed as a substructure of the roof, andthe roof as a a substructure of a building. If so the multilevel process would be more appropriate.

§1.4. Method Interpretations

Just like there are two complementary ways of using the FEM, there are two complementaryinterpretations for explaining it, a choice that obviously impacts teaching. One interpretationstresses the physical significance and is aligned with the Physical FEM. The other focuses on themathematical context, and is aligned with the Mathematical FEM. They are outlined next.

§1.4.1. Physical Interpretation

The physical interpretation focuses on the flowchart of Figure 1.1. This interpretation has beenshaped by the discovery and extensive use of the method in the field of structural mechanics. Thehistorical connection is reflected in the use of structural terms such as “stiffness matrix”, “forcevector” and “degrees of freedom,” a terminology that carries over to non-structural applications.

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1–11 §1.5 THE SOLUTION MORASS

The basic concept in the physical interpretation is the breakdown (! disassembly, tearing, partition,separation, decomposition) of a complex mechanical system into simpler, disjoint componentscalled finite elements, or simply elements. The mechanical response of an element is characterizedin terms of a finite number of degrees of freedom. These degrees of freedoms are represented asthe values of the unknown functions as a set of node points. The element response is defined byalgebraic equations constructed from mathematical or experimental arguments. The response ofthe original system is considered to be approximated by that of the discrete model constructed byconnecting or assembling the collection of all elements.

The breakdown-assembly concept occurs naturally when an engineer considers many artificial andnatural systems. For example, it is easy and natural to visualize an engine, bridge, aircraft orskeleton as being fabricated from simpler parts.

As discussed in §1.3.3, the underlying theme is divide and conquer. If the behavior of a systemis too complex, the recipe is to divide it into more manageable subsystems. If these subsystemsare still too complex the subdivision process is continued until the behavior of each subsystem issimple enough to fit a mathematical model that represents well the knowledge level the analystis interested in. In the finite element method such “primitive pieces” are called elements. Thebehavior of the total system is that of the individual elements plus their interaction. A key factor inthe initial acceptance of the FEM was that the element interaction could be physically interpretedand understood in terms that were eminently familiar to structural engineers.

§1.4.2. Mathematical Interpretation

This interpretation is closely aligned with the flowchart of Figure 1.2. The FEM is viewed asa procedure for obtaining numerical approximations to the solution of boundary value problems(BVPs) posed over a domain !. This domain is replaced by the union " of disjoint subdomains!(e) called finite elements. In general the geometry of ! is only approximated by that of "!(e).

The unknown function (or functions) is locally approximated over each element by an interpolationformula expressed in terms of values taken by the function(s), and possibly their derivatives, at aset of node points generally located on the element boundaries. The states of the assumed unknownfunction(s) determined by unit node values are called shape functions. The union of shape functions“patched” over adjacent elements form a trial function basis for which the node values represent thegeneralized coordinates. The trial function space may be inserted into the governing equations andthe unknown node values determined by the Ritz method (if the solution extremizes a variationalprinciple) or by the Galerkin, least-squares or other weighted-residual minimization methods if theproblem cannot be expressed in a standard variational form.

Remark 1.4. In the mathematical interpretation the emphasis is on the concept of local (piecewise) approx-imation. The concept of element-by-element breakdown and assembly, while convenient in the computerimplementation, is not theoretically necessary. The mathematical interpretation permits a general approachto the questions of convergence, error bounds, trial and shape function requirements, etc., which the physicalapproach leaves unanswered. It also facilitates the application of FEM to classes of problems that are not soreadily amenable to physical visualization as structures; for example electromagnetics and heat conduction.

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§1.5. The Solution Morass

In nonlinear analysis the two FEM interpretations are not equal in importance. Nonlinear analysisdemands a persistent attention to the underlying physics to avoid getting astray as the “real world”is covered by layer upon layer of mathematics and numerics.Why is concern for physics of paramount importance? A key component of finite element nonlinearanalysis is the solution of the nonlinear algebraic systems of equations that arise upon discretization.

FACT

The numerical solution of nonlinear systems in “black box” mode is muchmore difficult than in the linear case.

The key difficulty is tied to the essentially obscure nature of general nonlinear systems, about whichvery little can be said in advance. And you can be sure that Murphy’s law10 works silently in thebackground.One particularly vexing aspect of dealing with nonlinear systems is the solution morass. A deter-minate system of 1, 1000, or 1000000 linear equations has, under mild conditions, one and onlyone solution. The computer effort to obtain this solution can be estimated fairly accurately if thesparseness (or denseness) of the coefficient matrix is known. Thus setting up linear equation solversas “black-box” stand-alone functions or modules is a perfectly sensible thing to do.By way of contrast, a system of 1000 cubic equations has 31000 # 10300 solutions in the complexplane. This is much, much larger than the number of atoms in the Universe, which is merely 1050give or take a few. Suppose just several billions or millions of these are real solutions. Whichsolution(s) have physical meaning? And how do you compute those solutions without wasting timeon the others?This combinatorial difficulty is overcome by the concept of continuation, which engineers also callincremental analysis. Briefly speaking, we start the analysis from an easily computable solution— for example, the linear solution— and then try to follow the behavior of the system as actionsapplied to it are changed by small steps called increments. The previous solution is used as a startingpoint for the iterative solution-search procedure. The underlying prescription: follow the physics.This technique is interwined with the concept of response explained in Chapter 2.

Remark 1.5. Not surprisingly, incremental analysiswas used by the aerospace engineers thatfirst used thefiniteelement method for geometrically nonlinear analysis in the late 1950s. Techniques have been considerablyrefined since then, but the underlying idea remains the same.

We conclude this overview with a historical perspective on nonlinear finite element methods insolid and structural mechanics, along with a succint bibliography.

10 If something can go wrong, it will go wrong.

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1–13 §1.6 HISTORICAL BACKGROUND

§1.6. Historical BackgroundIn the history of finite element methods the year 1960 stands out. The name “finite element method” appearsfor the first time in the open literature in an article by Clough [487]. And Turner, Dill, Martin andMelosh [620]publish a pioneering paper in nonlinear structural analysis. The then-five-year-old “direct stiffness method”(what we now call displacement-assumed finite element method) was applied to“problems involving nonuniform heating and/or large deflections . . . in a series of linearized steps. Stiffnessmatrices are revised at the beginning of each step to account for changes in internal loads, temperatures, andgeometric configuration.”

Thirty years and several thousand publications later, computerized nonlinear structural analysis has acquiredfull adult rights, but has not developed equally in all areas.The first fifteen years (1960-1975) were dominated by formulation concerns. For example, not until thelate 1960s were correct finite-deflection incremental forms for displacement models rigorously derived. Andinteraction of flow-like constitutive behavior with the spatial discretization (the so called “incompressibilitylocking” effects) led to important research into constitutive equations and element formulations.While the investigators of this period devoted much energy to obtaining correct and implementable nonlinearfinite-element equations, the art of solving such equations in a reliable and efficientmanner was understandablyneglected. This helps to explain the dominance of purely incremental methods. Corrective methods of Newtontype did not get much attention until the early 1970s, and then only for geometrically nonlinear problems. Atthe time of this writing, progress in numerical solution techniques has been uneven: well developed for certainproblems, largely a black art in others. To understand the difference, it pays to distinguish between smoothnonlinearities and rough nonlinearities.

§1.6.1. Smooth NonlinearitiesProblem with smooth nonlinearities are characterized by continuous, path-independent nonlinear relations atthe local level. Some examples:

1. Finite deflections (geometric nonlinearities). Nonlinear effects arise from strain-displacement equations,which are well behaved for all strain measures in practical use.

2. Nonlinear elasticity. Stresses are nonlinear but reversible functions of strains.

3. Follower forces (e.g., pressure loading). External forces are smooth nonlinear functions of displacements.

A unifying characteristic of this problem is that nonlinearities are of equality type, i.e., reversible, and theserelations are continuous at each point within the structure. Mathematicians call these smooth mappings.It is important to point out, however, that the overall structural behavior is not necessarily smooth; as witnessedby the phenomena of buckling, snapping and flutter. But at the local level everything is smooth: nonlinearstrain-displacement equations, nonlinear elasticity law, follower pressures.Methods for solving this class of problems are highly developed, and have received a great deal of attentionfrom the mathematical and numerical analysis community. This research has directly benefitted many areasof structural analysis.Let us consider finite deflection problems as prototype. Within the finite element community, these wereoriginally treated by purely incremental (step-by-step) techniques; but anomalies detected in the mid-1960’sprompted research into consistent linearizations. A good exposition of this early work is given in the book byOden [558]. Once formulation questions were settled, investigators had correct forms of the “residual” out-of-balance forces and tangent stiffness matrix, and incremental steps began to be augmented with corrective

1–13


iterations in the late 1960s. Conventional and modified Newton methods were used in the corrective phase.These were further extended through restricted step (safeguarded Newton) and, more recently, variants of thepowerful conjugate-gradient and quasi-Newton methods.But difficulties in detecting and traversing limit and bifurcation points still remained. Pressing engineeringrequirements for post-buckling and post-collapse analyses led to the development of displacement control,alternating load/displacement control, and finally arclength control. The resultant increment-control methodshave no difficulty in passing limit points. The problem of reliably traversing simple bifurcation points withoutguessing imperfections remains a research subject, while passing multiple or clustered bifurcation pointsremains a frontier subject. A concerted effort is underway, however, to subsume these final challenges.These reliable solution methods have been implemented into many special-purpose finite element programs,and incorporation into general-purpose programs is proceeding steadily.

Remark 1.6. As noted above, incremental methods were the first to be used in nonlinear structural analysis. Among thepre-1970 contributions along this line we may cite Argyris and coworkers [443,444], Felippa [497], Goldberg and Richard[517], Marcal, Hibbitt and coworkers [524,543,544], Oden [557], Turner, Martin and coworkers [545,620,621],

Remark 1.7. The earliest applications of Newton methods to finite element nonlinear analysis are by Oden [557], Malletand Marcal [542], and Murray and Wilson [552,553]. During the early 1970s Stricklin, Haisler and coworkers at TexasA&M implemented and evaluated self-corrective, pseudo-force, energy-search and Newton-type methods and presentedextensive comparisons; see Stricklin et. al. [608,611], Tillerson et. al. [618], and Haisler et. al. [520]. Almroth, Brogan,Bushnell and coworkers at Lockheed began using true and modified Newton methods in the late 1960s for energy-basedfinite-difference collapse analysis of shells; see Brogan and Almroth [469], Almroth and Felippa [439], Brush and Almroth[472], and Bushnell [473,474]. By the late 1970s Newton-like methods enjoyed widespread acceptance for geometricallynonlinear analysis.

Remark 1.8. Displacement control strategies forfinite element post-buckling and collapse analysis were presented byArgyris[445] and Felippa [497] in 1966, and generalized in different directions by Sharifi and Popov [600,601] (fictitious springs),Bergan et. al. [460,461], (current stiffness parameter), Powell and Simons [577] and Bergan and Simons [462] (multipledisplacement controls). A modification of Newton’s method to traverse bifurcation points was described by Thurston [617].Arclength control schemes for structural problems may be found in the following source papers: Wempner [627], Riks[589], Schmidt [595], Crisfield [483,484], Ramm [581], Felippa [500,501], Fried [508], Park [568], Padovan [565,566],Simo et.al. [602], Yang and McGuire [632], Bathe and Dvorkin [454]. Other articles of particular interest are Bathe andCimento [452], Batoz and Dhatt [456], Bushnell [474], Bergan [462], Geradin et al. [514,515]. Meek and Tan [547], Ramm[581,582], Riks [590,593], Sobel and Thomas [604], Zienkiewicz [633,634,636]. Several conferences have been devotedexclusively to nonlinear problems in structural mechanics, for example [446,463,451,560,612,613,631]. Finite elementtextbooks and monographs dealing rather extensively with nonlinear problems are by Oden [558], Bathe [453], Bushnell[475], White [629] and Zienkiewicz [635].

Remark 1.9. In the mathematical literature the concept of continuation (also called imbedding) can be traced back to the1930s. A survey of the work up to 1950 is given by Ficken [505]. The use of continuation by parameter differentiation asa numerical method is attributed to Davidenko [489]. Key papers of this early period are by Freudenstein and Roth [507],Deist and Sefor [493] andMeyer [549], as well as the survey byWasserstrom [624]. This early history is covered byWacker[623].

Remark 1.10. Arclength continuation methods in the mathematical literature are generally attributed to Haselgrove [522]and Klopfestein [534] although these papers remained largely unnoticed until the late 1970s. Important contributionsto the mathematical treatment are by Abbott [435], Anselone and Moore [442], Avila [448], Brent [466], Boggs [464],Branin [465], Broyden [470,471], Cassel [478], Chow et. al. [480], Crandall and Rabinowitz [482], Georg [512,513],Keller and coworkers [479,491,492,528,531], Matthies and Strang [546], Moore [550,551], Ponish [574,575], Rheinboldtand coworkers [494,548,584,585], Watson [625] and Werner and Spence [628]. Of these, key contributions in terms ofsubsequent influence are [480,529,584]. For surveys and edited proceedings see Allgower [436,437], Byrne and Hall [477],Kupper [539,540], Rall [580], Wacker [623], and references therein. Textbooks and monographs dealing with nonlinearequation solving include Chow and Hale [481], Dennis and Schnabel [496], Kubıcek and Hlavacek [537], Kubıcek andMarek [538], Ortega and Rheinboldt [563], Rabinowitz [578], Rall [579], Rheinboldt [588], and Seydel [599]. Of these, the

1–14


book by Ortega and Rheinboldt [563] remains a classic and an invaluable source to essentially all mathematically orientedwork done prior to 1970. The book by Seydel [599] contains material on treatment of conventional and Hopf bifurcationsnot readily available elsewhere. Nonlinear equation solving is interwined with the larger subject of optimization andmathematical programming; for the latter the textbooks by Gill, Murray and Wright [516] and Fletcher [506] are highlyrecommended.

§1.6.2. Rough NonlinearitiesRoughnonlinearities are characterizedbydiscontinuousfield relations, usually involving inequality constraints.Examples: flow-rule plasticity, contact, friction. The local response is nonsmooth.Solution techniques for these problems are in a less satisfactory state, and case-by-case consideration is calledfor. The local and overall responses are generally path-dependent, an attribute that forces the past responsehistory to be taken into account.The key difficulty is that conventional solution procedures based on Taylor expansions or similar differentialforms may fail, because such Taylor expansions need not exist! An encompassing mathematical treatmentis lacking, and consequently problem-dependent handling is presently the rule. For this class of problemsincremental methods, as opposed to incremental-iterative methods, still dominate.

Remark 1.11. Earliest publications on computational plasticity using finite element methods are by Gallagher et. al. [509],Argyris [443,444], Marcal [543], Pope [573] and Felippa [497]. By now there is an enormous literature on the numericaltreatment of inelastic processes, especially plasticity and creep. Fortunately the survey by Bushnell [475], although focusingon plastic buckling, contains over 300 references that collectively embody most of the English-speaking work prior to 1980.Other important surveys are by Armen [447] andWillam [630]. For contact problems, see Oden [561], Bathe and Chaudhary[453], Kikuchi and Oden [532,533], Simo et. al. [602] Stein et. al. [605], Nour-Omid and Wriggers [556], and referencestherein.

§1.6.3. Hybrid ApproachWhat does an analyst do when faced with an unfamiliar nonlinear problem? If the problem falls into thesmooth-nonlinear type, there is no need to panic. Robust and efficient methods are available. Even if thewhizziest methods are not implemented into one’s favorite computer program, there is a wealth of theory andpractice available for trouble-shooting.But what if the problem include rough nonlinearities? A time-honored general strategy is divide and conquer.More specifically, two powerful techniques are frequently available: splitting and nesting.Splitting can be used if the nonlinearities can be separated in an additive form:

Smooth + Rough

This separation is usually done at the force level. Then the smooth-nonlinear term is treated by conventionaltechniques whereas the rough-nonlinear term is treated by special techniques. This scheme can be particularlyeffective when the rough nonlinearity is localized, for example in contact and impact problems.Nesting may be used when a simple additive separation is not available. This is best illustrated by an actualexample. In the early 1970s, some authors argued that Newton’s method would be useless for finite-deflectionelastoplasticity, as no unique Jacobian exists in plastic regions on account of loading/unloading switches. Theargument was compelling but turned out to be a false alarm. The problem was eventually solved by “nesting”geometric nonlinearities within the material nonlinearity, as illustrated in Fig. 1.1.In the inner equilibrium loop thematerial law is “frozen”, whichmakes the highly effective Newton-typemeth-ods applicable. The non-conservative material behavior is treated in an outer loop where material propertiesand constitutive variables are updated in an incremental or sub-incremental manner.

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Another application of nesting comes in the global function approach (also called Rayleigh-Ritz or reduced-basis approach), which is presently pursued by several investigators. The key idea is to try to describe theoverall response behavior by a few parameters, which are amplitudes of globally defined functions. The smallnonlinear system for the global parameters is solved in an inner loop, while an external loop involving residualcalculations over the detailed finite element model is executed occasionally.Despite its inherent implementation complexity, the global function approach appears cost-effective for smooth,path-independent nonlinear systems. This is especially so when expensive parametric studies are involved, asin structural optimization under nonlinear stability constraints.

Remark 1.12. For geometric-material nesting and subincremental techniques see Bushnell [?], and references therein. Theglobal-function approach in its modern form was presented by Almroth, Stern and Brogan [441] and pursued by Noor andcoworkers under the name of reduced-basis technique; see Noor and Peters [554,555]. For perturbation techniques see thesurvey by Gallagher [510].

§1.6.4. Summary of Present StatusSolution techniques for smooth nonlinearities are in a fairly satisfactory state. Although further refinementsin the area of traversing bifurcation points can be expected, incremental-iterative methods implemented withgeneral increment control appear to be as reliable as an engineer user may reasonably expect.For rough nonlinearities, case-by-case handling is still necessary in view of the lack of general theories andimplementation procedures. Separation or nesting of nonlinearities, when applicable, can lead to signifi-cant gains in efficiency and reliability, but at the cost of programming complexity and problem-dependentimplementations.

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.

2A Tour ofNonlinearAnalysis

2–1

Chapter 2: A TOUR OF NONLINEAR ANALYSIS 2–2

TABLE OF CONTENTS

Page§2.1. Introduction 2–3§2.2. Equilibrium Path and Response Diagrams 2–3

§2.2.1. Load-deflection response . . . . . . . . . . . . . 2–3§2.2.2. Terminology . . . . . . . . . . . . . . . . . 2–3

§2.3. Special Equilibrium Points 2–5§2.3.1. Critical points . . . . . . . . . . . . . . . . . 2–5§2.3.2. Turning points . . . . . . . . . . . . . . . . 2–5§2.3.3. Failure points . . . . . . . . . . . . . . . . . 2–5

§2.4. Linear Response 2–6§2.5. Tangent Stiffness and Stability 2–6§2.6. Parametrized Response 2–7§2.7. Response Flavors 2–7§2.8. Engineering Applications 2–9§2.9. Sources of Nonlinearities 2–9§2.10. Geometric Nonlinearity 2–10§2.11. Material Nonlinearity 2–11§2.12. Force BC Nonlinearity 2–12§2.13. Displacement BC Nonlinearity 2–13§2. Exercises . . . . . . . . . . . . . . . . . . . . . . 2–14

2–2

2–3 §2.2 EQUILIBRIUM PATH AND RESPONSE DIAGRAMS

§2.1. Introduction

This chapter reviews nonlinear structural problems by looking at the manifestation and physicalsources of nonlinear behavior.We begin by introducing response as a pictorial characterization of nonlinearity of a structuralsystem. Response is a graphical representation of the fundamental concept of equilibrium path.This concept permeates the entire course because of both its intrinsic physical value and the factthat incremental solution methods (mentioned in Chapter 1) are based on it.Finally, nonlinearities are classified according to their source in the mathematical model of con-tinuum mechanics and correlated with the physical system. Examples of these nonlinearities inpractical engineering applications are given.

§2.2. Equilibrium Path and Response Diagrams

The concept of equilibrium path plays a central role in explaining the mysteries of nonlinearstructural analysis. This concept lends itself to graphical representation in the form of responsediagrams. The most widely used form of these pictures is the load-deflection response diagram.Through this representation many key concepts can be illustrated and interpreted in physical,mathematical or computational terms.

§2.2.1. Load-deflection response

The gross or overall static behavior of many structures can be characterized by a load-deflectionor force-displacement response. The response is usually drawn in two dimensions as a x-y plotas illustrated in Figure 2.1. In this figure a “representative” force quantity is plotted against a“representative” displacement quantity. If the response plot is nonlinear, the structure behavior isnonlinear.

Remark 2.1. We will see below that a response diagram generally depicts relationships between inputs andoutputs. Or, in more physical terms, between what is applied and what is measured. For structures the mostcommon inputs are forces and the most common outputs are displacements or deflections1

Remark 2.2. The qualifier “representative” implies a choice among many possible candidates. For relativelysimple structures the choice of load and deflection variables is often clear-cut from considerations such asthe availability of experimental data. For more complex structures the choice may not be obvious, and manypossibilities may exist. The load is not necessarily an applied force but may be an integrated quantity: forexample the weight of traffic on a bridge, or the total lift on an airplane wing.

Remark 2.3. This type of response should not be confused with what in structural dynamics is called theresponse time history. A response history involves time, which is the independent variable, plotted usuallyalong the horizontal axis, with either inputs or outputs plotted vertically.

1 A deflection is the magnitude or amplitude of a displacement. Displacements are vector quantities whereas deflectionsare scalars.

2–3


Representativeload

Representative deflection

Reference state Reference state

Representative load


(b)(a)

Equilibrium path

Critical point

Fundamental or primary path

Secondary path

Initial linear response

Figure 2.1. Response diagrams: (a) typical load-deflection diagram showing equilibrium path; (b)diagram distinguishing fundamental (a.k.a. primary) from secondary equilibrium path.

§2.2.2. Terminology

A smooth curve shown in a load-deflection diagram is called a path.2 Each point in the pathrepresents a possible configuration or state of the structure. If the path represents configurationsin static equilibrium it is called an equilibrium path. Each point in an equilibrium path is called anequilibrium point. An equilibrium point is the graphical representation of an equilibrium state orequilibrium configuration. See Figure 2.1(a).The origin of the response plot (zero load, zero deflection) is called the reference state because it isthe configuration from which loads and deflections are measured. It should be noted, however, thatthe reference state may be chosen rather arbitrarily; this freedom is exploited in some nonlinearformulations and solution methods, as we shall see later.For problems involving perfect structures3 the reference state is unstressed and undeformed, and isalso an equilibrium state. This means that an equilibrium path passes through the reference state,as in Figure 2.1(a).The path that crosses the reference state is called the fundamental equilibrium path, or fundamentalpath for short. (Many authors also call this a primary path.) The fundamental path extends fromthe reference state up to special states called critical points which are introduced in §2.3.1. Anypath that is not a fundamental path but connects with it at a critical point is called a secondary path.See Figure 2.1(b).Qualifiers “fundamental” and “secondary” are linked with the relative importance of these equi-librium paths in design. Most structures are designed to operate in the fundamental path when

2 The terms branch and trajectory are also found in the literature. “Branch” is commonly used in the treatment of bifurcationphenomena, in which multiple paths emanate from one equilibrium point. On the other hand, “trajectory” has temporalas well as history connotations, and is mostly used in the context of dynamic analysis.

3 A concept to be explained later in connectionwith stability analysis. A perfect structure involves some formof idealizationsuch as perfectly centered loads or perfect fabrication. An imperfect structure is one that deviates from that idealizationin measurable ways.

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2–5 §2.3 SPECIAL EQUILIBRIUM POINTS

in service, with some sort of safety factor against reaching a critical point. But knowledge ofsecondary paths may be important in some aspects of the design process, for example in the assess-ment of structural behavior under emergency scenarios (e.g., a vehicle crash or a building hit by anearthquake), which may directly or indirectly affect safety.

§2.3. Special Equilibrium Points

Certain points of an equilibrium path have special significance in the applications and thus receivespecial names. Of particular interest to our subject are critical, turning and failure points.

§2.3.1. Critical points

Critical points are characterized mathematically in later chapters. It is sufficient to mention herethat there are two types:

1. Limit points, at which the tangent to the equilibrium path is horizontal, i.e. parallel to thedeflection axis, and

2. Bifurcation points, at which two or more equilibrium paths cross.At critical points the relation between the given characteristic load and the associated deflectionis not unique. Physically, the structure becomes uncontrollable or marginally controllable there.This property endows such points with engineering significance from a design standpoint.

§2.3.2. Turning points

Points at which the tangent to the equilibrium path is vertical, i.e. parallel to the load axis, are calledturning points. These are not critical points and have less physical significance, although they are ofoccassionaly of interest in connection with the so-called “snap-back” phenomena. Turning pointsmay have computational significance, however, because they can affect the performance of certain“path following” solution methods.

§2.3.3. Failure points

Points at which a path suddenly stops or “breaks” because of physical failure are called failurepoints. The phenomenon of failure may be local or global in nature. In the first case (e.g, failure ofa noncritical structure component) the structuremay regain functional equilibrium after dynamically“jumping” to another equilibrium path. In the latter case the failure is catastrophic or destructiveand the structure does not regain functional equilibrium.In the present exposition, bifurcation, limit, turning and failure points are often identified by theletters B, L, T and F, respectively.Equilibrium points that are not critical are called regular.

2–5


Linear fundamental path

goes on forever

State parameter µ or u

Control parameter λ Representativeload


Reference state Reference state

(b)(a)

Equilibrium path

Figure 2.2. Two response diagram specializations: (a) linear response; (b) parametrized form.

§2.4. Linear Response

A linear structure is a mathematical model characterized by a linear fundamental equilibrium pathfor all possible choices of load and deflection variables. This is shown schematically in Figure 2.2.The consequences of such behavior are not difficult to foresee:

1. A linear structure can sustain any load level and undergo any displacement magnitude.2. There are no critical, turning or failure points.3. Response to different load systems can be obtained by superposition.4. Removing all loads returns the structure to the reference position.

The requirements for such a model to be applicable are:• Perfect linear elasticity for any deformation• Infinitesimal deformations• Infinite strengthThese assumptions are not only physically unrealistic but mutually contradictory. For example, ifthe deformations are to remain infinitesimal for any load, the body must be rigid rather than elastic,which contradicts the first assumption. Thus, there are necessarily limits placed on the validity ofthe linear model.Despite these obvious limitations, the linear model can be a good approximation of portions of thenonlinear response. In particular, the fundamental path response in the vicinity of the reference state.See for instance Figure 2.1(b). Because for many structures this segment represents the operationalor service range, the linear model is widely used in design calculations. The key advantage ofthis idealization is that the superposition-of-effects principle applies. Practical implications of thefailure of the superposition principle are further discussed in Chapter 3.

2–6

2–7 §2.7 RESPONSE FLAVORS

§2.5. Tangent Stiffness and Stability

The tangent to an equilibrium path may be informally viewed as the limit of the ratio

force incrementdisplacement increment

This is by definition a stiffness or, more precisely, the tangent stiffness associated with the repre-sentative force and displacement. The reciprocal ratio is called flexibility or compliance.The sign of the tangent stiffness is closely associated with the question of stability of an equilibriumstate. A negative stiffness is necessarily associated with unstable equilibrium. A positive stiffnessis necessary but not sufficient for stability.4

If the load and deflection quantities are conjugate in the virtual work sense, the area under aload-deflection diagram may be interpreted as work performed by the system.

§2.6. Parametrized Response

It is often useful to be able to parametrize the load-displacement curve of Figure 2.1 in the followingway. A control-state response involves two ingredients:1. A control parameter, called !, plotted along the vertical axis versus2. A state parameter, called u or µ, plotted along the horizontal axis.5

We shall see in following Chapters that ! and u (or µ) characterize in some way the actions appliedto the structure and the state of the structure, respectively.A diagram such as that shown in Figure 2.2(b) is called a control-state response. Throughout thisbook the abbreviated term response is often used in this particular sense. In practice the controlparameter is usually a load amplitude or load factor, whereas the state parameter is a displacementamplitude. Thus the usual load-deflection response is one form of the control-state response.

Remark 2.4. The interpretation of the tangent-to-the-path as stiffness discussed in §2.5 does not necessarilycarry over to more control-state diagrams. Similarly, the interpretations of the sign of the tangent and of theenclosed-area in terms of stability indicator and stored work, respectively, do not necessarily hold. This isbecause control and state are not necessarily conjugate in the virtual work sense.

§2.7. Response Flavors

The response diagrams in Figure 2.3 illustrate three “monotonic” types of response: linear, harden-ing, and softening. In these diagrams symbols F and L identify failure and limit points, respectively.The response shown in (a): linear until fracture, is characteristic of pure crystals, glassy, as well ascertain high strength composite materials that contain such materials as fibers.The response illustrated by (b) is typical of cable, netted and pneumatic (inflatable) structures,which may be collectively called tensile structures. The stiffening effect comes from geometry

4 These sign criteria would be sufficient for a one-degree-of-freedom system.5 We shall use the symbol µ primarily for dimensionless state quantities.

2–7


F F L

F

R R R

(a) (b) (c)

Figure 2.3. Basic flavors of nonlinear response: (a) Linear until brittle failure; (b) Stiffeningor hardening; (c) Softening.

“adaptation” to the applied loads. Some flat-plate assemblies also display this behavior initiallybecause of load redistribution as membrane stresses develop while the midsurface stretches.A response such as in (c) is more common for structure materials than the previous two. A linearresponse is followed by a softening regime that may occur suddenly (yield, slip) or gradually. More“softening flavors” are given in Figure 2.4The diagrams of Figure 2.4 illustrate a “combination of basic flavors” that can complicate theresponse as well as the task of the analyst. Here B and T denote bifurcation and turning points,respectively.

L

LT

TL

LF

F

FF

TL

BB

B

(d) (e) (f) (g)

R R R R

Figure 2.4. More complex response patterns: (d) snap-through, (e) snap-back, (f) bifurcation,(g) bifurcation combined with limit points and snap-back.

The snap-through response (d) combines softening with hardening following the second limit point.The response branch between the two limit points has a negative stiffness and is therefore unstable.(If the structure is subject to a prescribed constant load, the structure “takes off” dynamically whenthe first limit point is reached.) A response of this type is typical of slightly curved structures suchas shallow arches.The snap-back response (e) is an exaggerated snap-through, in which the response curve “turnsback” in itself with the consequent appearance of turning points. The equilibrium between the

2–8

2–9 §2.9 SOURCES OF NONLINEARITIES

two turning points may be stable and consequently physically realizable. This type of response isexhibited by trussed-dome, folded and thin-shell structures in which “moving arch” effects occurfollowing the first limit point; for example cylindrical shells with free edges and supported by enddiaphragms.In all previous diagrams the response was a unique curve. The presence of bifurcation (popularlyknown as “buckling” by structural engineers) points as in (f) and (g) introduces more features. Atsuch points more than one response path is possible. The structure takes the path that is dynamicallypreferred (in the sense of having a lower energy) over the others. Bifurcation points may occur inany sufficiently thin structure that experiences compressive stresses.Bifurcation, limit and turning points may occur in many combinations as illustrated in (g). Astriking example of such a complicated response is provided by thin cylindrical shells under axialcompression.

§2.8. Engineering Applications

Nonlinear Structural Analysis is the prediction of the response of nonlinear structures by model-based simulation. Simulation involves a combination of mathematical modeling, discretizationmethods and numerical techniques. As noted in Chapter 1, finite element methods dominate thediscretization step.Table 2.1 summarizes the most important applications of nonlinear structural analysis.

§2.9. Sources of Nonlinearities

A response diagram characterizes only the gross behavior of a structure, as it might be observedsimply by conducting an experiment on amechanical testingmachine. Further insight into the sourceof nonlinearity is required to capture such physical behavior with mathematical and computationalmodels for computer simulation.For structural analysis there are four sources of nonlinear behavior. The corresponding nonlineareffects are identified by the terms material, geometric, force B.C. and displacement B.C., in whichB.C. means “boundary conditions.” In this course we shall be primarily concerned with the lastthree types of nonlinearity, with emphasis on the geometric one.6

The four sources are discussed in more detail in following sections. To remember where the nonlin-ear terms appear in the governing equations, it is useful to recall the fields that continuummechanicsdeals with, and the relationships among these fields. For linear solid continuum mechanics infor-mation is presented in Figure 2.6 7

In linear solid mechanics or linear structural mechanics the connecting relationships shown inFigure 2.6 are linear, and so are the governing equations obtained by eliminating all fields but one.Any of these relations, however, may be nonlinear. Tracing this fact back to physics gives rise tothe types of nonlinearities depicted in Figure 2.7. Relations between body force and stress (theequilibrium equations) and between strains and displacements (the kinematic equations) are closely

6 The exclusion of constitutive or material nonlinearities does not imply that there are less important than the others. Quitethe contrary. But the topic is well covered in separate courses offered in the Civil Engineering department.

7 These are actually the so-called Tonti diagrams introduced in the IFEM course.

2–9


Table 2.1 Engineering Applications of Nonlinear Structural Analysis

Application Explanation

Strength analysis How much load can the structure support beforeglobal failure occurs?

Deflection analysis When deflection control is of primary importance

Stability analysis Finding critical points (limit points or bifurcationpoints) closest to operational range

Service configuration analysis Finding the “operational” equilibrium form of certainslender structures when the fabrication and serviceconfigurations are quite different (e.g. cables, inflat-able structures, helicoids)

Reserve strength analysis Finding the load carrying capacity beyond criticalpoints to assess safety under abnormal conditions.

Progressive failure analysis A variant of stability and strength analysis in whichprogressive deterioration (e.g. cracking) is consid-ered.

Envelope analysis A combination of previous analyses inwhichmultipleparameters are varied and the strength informationthus obtained is condensed into failure envelopes.

linked in a “duality” sense, and so the term geometric nonlinearities applies collectively to both setsof relations. The force BC nonlinearities couple displacements and applied forces (surface tractionsand/or body forces) and thus bring the additional links drawn in Figure 2.6.In the following sections these sources of nonlinearities are correlated to the physics in more detail.

§2.10. Geometric Nonlinearity

Physical source. Change in geometry as the structure deforms is taken into account in setting upthe strain-displacement and equilibrium equations.

Applications. Slender structures in aerospace, civil and mechanical engineering applications.Tensile structures such as cables and inflatable membranes. Metal and plastic forming. Stabilityanalysis of all types.

Mathematical model source. Strain-displacement equations, symbolically represented in operatorform as

e = D(u). (2.1)

2–10

2–11 §2.11 MATERIAL NONLINEARITY

Displacements

Strains

Body forces

Stresses

Prescribeddisplacements

Prescribedtractions or forces

Kinematicequations

Displacement BCs

Force(Traction)

BCsConstitutiveequations

Equilibriumequations

Figure 2.5. Fields in solid continuum mechanics and connecting relationships

u

e

b

σ

u = uu

in V on Stt

= E eσ ^

on Su

σn= t

e = D uin V

D + b = 0Tσin V

Figure 2.6. Same as Figure 2.7, with symbols and equations written down for the linear case.

The operator D is nonlinear when finite strains (as opposed to infinitesimal strains) are expressedin terms of displacements. Internal equilibrium equations:

b = !D"(!). (2.2)

In the classical linear theory of elasticity, (2.1) and (2.2) may be expressed as matrix-operator form,andD" = DT is then the formal adjoint ofD. That is not necessarily true if geometric nonlinearitiesare considered.

Remark 2.5. The term geometric nonlinerities models a myriad of physical problems:

Large strain. The strains themselves may be large, say over 5%. Examples: rubber structures (tires,membranes, air bags, polymer dampers), metal forming. These are frequently associated with material non-linearities.

Small strains but finite displacements and/or rotations. Slender structures undergoing finite displacementsand rotations although the deformational strains may be treated as infinitesimal. Example: cables, springs,arches, bars, thin plates.

Linearized prebucking. When both strains and displacements may be treated as infinitesimal before loss ofstability by buckling. These may be viewed as initially stressed members. Example: many civil engineeringstructures such as buildings and stiff (non-suspended) bridges.

2–11


Materialnonlinearities

Displacement B.C. nonlinearities

Geometricnonlinearities

Force B.C.nonlinearities

u

e

b

σ

u

t

Figure 2.7. Graphical depiction of sources of nonlinearities in solid and structural mechanics.

§2.11. Material Nonlinearity

Physical source. Material behavior depends on current deformation state and possibly past historyof the deformation. Other constitutive variables (prestress, temperature, time, moisture, electro-magnetic fields, etc.) may be involved.

Applications. Structures undergoing nonlinear elasticity, plasticity, viscoelasticity, creep, or in-elastic rate effects.

Mathematical model source. Constitutive equations that relate stresses and strains. For a linearelastic material

! = E e, or e = C !, (2.3)

in which the elasticity matrix E contains elastic moduli and the compliance matrix C = E!1 (ifE is nonsingular) contains compliance coefficients. If the material does not fit the elastic model,generalizations of this equation are necessary, and a whole branch of continuum mechanics isdevoted to the formulation, study and validation of constitutive equations.

Remark 2.6. The engineering significance of material nonlinearities varies greatly across disciplines. Theyseem to occur most often in civil engineering, that deals with inherently nonlinear materials such as concrete,soils and low-strength steel. In mechanical engineering creep and plasticity are most important, frequentlyoccurring in combination with strain-rate and thermal effects. In aerospace engineering material nonlinearitiesare less important and tend to be local in nature (for example, cracking and “localization” failures of compositematerials).

Remark 2.7. Material nonlinearities may give rise to very complex phenomena such as path dependence,hysteresis, localization, shakedown, fatigue, progressive failure. The detailed numerical simulation of thesephenomena in three dimensions is still beyond the capabilities of the most powerful computers. (This waswritten in the early 1990s.)

2–12

2–13 §2.13 DISPLACEMENT BC NONLINEARITY

§2.12. Force BC Nonlinearity

Physical Source. Applied forces depend on deformation.

Applications. Themost important engineering application concerns pressure loads of fluids. Theseinclude hydrostatic loads on submerged or container structures; aerodynamic and hydrodynamicloads caused by the motion of aeriform and hydroform fluids (wind loads, wave loads, drag forces).Of more mathematical interest are gyroscopic and non-conservative follower forces, but these areof interest only in a limited class of problems, particularly in aerospace engineering.

Mathematical model source. The applied forces (prescribed surface tractions!t and/or body forcesb) depend on the displacements:

!t =!t(u), b = b(u). (2.4)

The former dependence (of surface forces) in (2.4) is more important in practice.

§2.13. Displacement BC Nonlinearity

Physical source. Displacement boundary conditions depend on the deformation of the structure.

Applications. Themost important application is the contact problem,8 inwhich no-interpenetrationconditions are enforced on flexible bodies while the extent of the contact area is unknown. Non-structural applications of this problem pertain to the more general class of free boundary problems,for example: ice melting, phase changes, flow in porous media. The determination of the essentialboundary conditions is a key part of the solution process.

Mathematical model source. For the contact problem: prescribed displacements !d depend oninternal displacements u:

!d = !d(u), (2.5)

in which u is unknown. More complicated dependencies can occur in the free-boundary problemsmentioned in §2.12, in which finding the boundary extent is part of the problem.

8 Contact-impact in dynamics.

2–13


Homework Exercises for Chapter 2

A Tour of Nonlinear Analysis

EXERCISE 2.1 [D:10] Explain the difference, if any, between a load-deflection response and a control-stateresponse.

EXERCISE 2.2 [D:20] Can the following occur simultaneously: (a) a limit and a bifurcation point, (b) abifurcation and a turning point, (c) a limit and a turning point, (d) two bifurcation points coalescing into one.If you answer “yes” to an item, sketch a response diagram to justify that reply.

EXERCISE 2.3 [D:25] In §2.10–13, nonlinearities are classified according to physical source into geomet-ric, material, force boundary conditions, and displacement boundary conditions. For each of the followingmechanical systems indicate the source(s) of nonlinearity that you think are significant; note that there maybe more than one. (If you are not familiar with the underlying concepts, read those sections.)

(a) a long, slender elastic pipe bent under end couples while the pipe material stays elastic. See Figure E2.1.(b) an inflating balloon. See Figure E2.2.(c) a cable deflecting under action of wind forces while its material stays elastic. See Figure E2.3.(d) a forming process in which hot metal is extruded through a rigid die. See Figure E2.4.(e) a metal anchor is drilled into the soil to serve as a cable support; the hole is then filled with concrete.

See Figures E2.5 and E2.6. The question refers to the soil-drilling process, ignoring dynamics.(f) a hefty bird — say a condor — is sucked into an aircraft jet engine. Ignore dynamics; engine is the

structure, bird the load.

EXERCISE 2.4 [D:15] Can you think of a mechanical component that has the load-deflection responsediagram pictured in Figure E2.7? (Explain why). Hint: Think of a helicoidal spring.

2–14

2–15 Exercises

Slender tube bentby end couples applied by an eccentric force pair

Figure E2.1. Slender elastic pipe bent under end couples for Exercise 2.3(a).

Figure E2.2. Inflating balloon for Exercise 2.3(b).

Wind

Cable

wind load

Figure E2.3. Cable deflecting under wind forces for Exercise 2.3(c)

2–15


Hot metal

Die

Figure E2.4. Hot metal extruded trough a rigid die for Exercise 2.3(d).

Figure E2.5. Drill element of a cable anchor, for Exercise 2.3(e).

concrete grouting

hole

soil

(a) (b)

Figure E2.6. Configuration of cable anchor after drilling in the soil, for Exercise 2.3(e).

2–16

2–17 Exercises

Axial force

Axial deflection (shortening)

B

R

III

II

I

Figure E2.7. A “mystery” response diagram for Exercise 2.4.

2–17

.

3Residual

ForceEquations

3–1

Chapter 3: RESIDUAL FORCE EQUATIONS 3–2

TABLE OF CONTENTS

Page§3.1. Introduction 3–3§3.2. Residual Equations 3–3

§3.2.1. Control and State Variables . . . . . . . . . . . . . 3–3§3.2.2. Balanced Force Residual . . . . . . . . . . . . . 3–4§3.2.3. Accounting for Memory Effects . . . . . . . . . . . 3–4

§3.3. Stiffness and Control Matrix 3–5§3.4. Parametric Representations and Rate Forms 3–6§3.5. Reduction to Single Control Parameter by Staging 3–10

§3.5.1. Explicit Reduction . . . . . . . . . . . . . . . 3–10§3.5.2. *Implicit Reduction . . . . . . . . . . . . . . . 3–12

§3. Notes and Bibliography. . . . . . . . . . . . . . . . . . . . . . 3–13§3. Exercises . . . . . . . . . . . . . . . . . . . . . . 3–14

3–2

3–3 §3.2 RESIDUAL EQUATIONS

§3.1. IntroductionChapters 3 through 6 discuss basic properties of systems of algebraic nonlinear equations thatdepend on one or more control parameters. Algebraic means that these systems contain a finitenumber of equations and unknowns.Those systems result from the discretization of continuummodels of nonlinear structures. As notedin Chapter 1, the most widely used discretization method is the displacement-based Finite ElementMethod (FEM). The description of FEM discretization techniques is deferred until Chapter 7 andfollowing ones. For the moment it is assumed that the discretization has been carried out.Physically the algebraic systems represent force equilibrium at the discrete level. Specifically, fordiscrete models coming from the displacement FEM, the sum of external and internal node forceson each degree of freedom vanishes. These are collectively known as force residual equations orresidual equations for short. In the present Chapter, residual and differential forms of use in laterChapters are presented, and the key concept of staging, through which multiple control parametersare reduced to one, is introduced.

§3.2. Residual EquationsDiscrete equilibrium equations encountered in nonlinear static structural analysis formulated bythe displacement method may be presented in the compact total force residual form

r (u,!) = 0. (3.1)

Here u is the state vector that contains the degrees of freedom that uniquely characterize the stateof the structure, r is the residual vector that contains out-of-balance forces conjugate to u, and !is an array of assignable control parameters. Occasionally an indicial form of (3.1) is convenient:

ri (u j , !k) = 0. (3.2)

Here ri , u j , and !k are indexed entries of vectors r, u, and !, respectively, that range overappropriate dimensions. We will normally restrict consideration to problems where the dimensionsof r and u are the same; if that is the case, the range of indices i and j is identical.As noted above, the formulation of these discrete equations using FEM is treated in later Chapters.For the moment it is simply assumed that, given u and !, a computational procedure exists thatreturns r. This “black box view” is sketched in the block diagram of Figure 3.1(a). In addition,most solution methods may require residual derivative (rate) information, as discussed later.

§3.2.1. Control and State VariablesIn structural mechanics, control parameters are commonly mechanical load levels. They mayalso be, however, prescribed physical or generalized displacements, temperature variations, fluidvelocities, imperfection amplitudes, and even (in design and optimization) design variables such asgeometric dimensions or material properties.The degrees of freedom collected in u are usually physical or generalized unknown displacements.In this book the term state variables will be used to designate them in general terms.1

1 The names behavior and configuration variables for the entries of u are also found in the literature. However, “statevariable” is more precise, and in line with standard terminology in analytical dynamics. In a general mathematicalcontext, u and! are called the active and passive variables, respectively.

3–3


Control: Λ

State u Residual rResidualevaluation

(a) Control: Λ

State u Residual r

(b)

Ext forceevaluation

Int forceevaluation

−

f

p

Figure 3.1. Black box diagrams for residual evaluation: (a) total force residual form (3.1);(b) balanced force residual form (3.4).

The dependence of r on u and ! is assumed to be piecewise smooth so that first and secondderivatives exist except possibly at isolated critical points. If the system is conservative,2 r is thegradient of the total potential energy !(u,!) for fixed !:

r = "!

"u, or ri = "!

"ui. (3.3)

If (3.3) holds, (3.1) expresses that equilibrium is associated with a energy-stationarity condition.

§3.2.2. Balanced Force Residual

An alternative version of (3.1) that displays more physical meaning is the balanced force residualform:

p(u) = f(u,!). (3.4)

Here p denotes the configuration-dependent internal forces resisted by the structure whereas f arethe control-dependent external or applied forces, which may also be state dependent. This formsays that in static equilibrium, the internal forces p balance the external forces f. A black box viewof (3.4) is sketched in Figure 3.1(b).

The total force residual corresponding to (3.4) is either r = p ! f or r = f ! p, the two versionsbeing equivalent except for sign. In the sequel only the first form: r = p ! f, is used so as to agreewith the conventional definition of tangent stiffness matrix in §3.3.

If a total potential energy ! exists, the decomposition associated with (3.4) is

p = "U"u

, f = "W"u

, (3.5)

in whichU and W are the internal energy and external work potential components, respectively, of! = U ! W .

2 A property studied more carefully in Chapter 6.

3–4

3–5 §3.3 STIFFNESS AND CONTROL MATRIX

§3.2.3. Accounting for Memory Effects

Both the total force residual form (3.1) and its force-balance variant (3.4) are restricted in that noaccount for historical or memory effects is made. A more general form would be

r(u, !,!) = 0, (3.6)

in which! is a functional of the past history of deformation. This generalized form is needed for thetreatment of inelastic, path-dependent materials. (Think, for example, of loading versus unloadingpaths in plasticity). The path-independent forms (3.1) or (3.4) are sufficient, however, for the classof problems considered here. In addition to geometric and force B.C. nonlinearities, these formsare applicable to nonlinear elasticity (hyperelasticity), as well as several types of displacement B.C.nonlinearities.

Example 3.1. Consider the following residual equilibrium equations

r1 = 4 u1 ! u2 + u2 u3 ! 6"1 = 0,r2 = 6 u2 ! u1 + u1 u3 ! 3"2 = 0,r3 = 4 u3 + u1 u2 ! 3"2 = 0.

(3.7)

The vector form of (3.7) is

r =

!

r1(u,!)

r2(u,!)

r3(u,!)

"

=

! 4 u1 ! u2 + u2 u3 ! 6"16 u2 ! u1 + u1 u3 ! 3"24 u3 + u1 u2 ! 3"2

"

= 0, with u =

!

u1u2u3

"

, ! =#

"1"2

$

. (3.8)

The force-balance vector form of = (3.7) is p = f, in which

p =

!

p1(u)p2(u)p3(u)

"

=

! 4 u1 ! u2 + u2 u36 u2 ! u1 + u1 u34 u3 + u1 u2

"

, f =

!

f1(!)

f2(!)

f3(!)

"

= 3

! 2"1"2"2

"

. (3.9)

In this particular case f does not depend on u.

Remark 3.1. The function u(!) characterizes the equilibrium surface of the structure in the space spannedby u and !. In a general mathematical context the set of {u,!} pairs that satisfies (3.1) is called a solutionmanifold.

Remark 3.2. The usefulness of the residual equation (3.1) is not restricted to static problems. It is alsoapplicable to nonlinear dynamical systems

r(u(# ),!(# )) = 0, (3.10)

which have been discretized in time # by implicit methods.3 In this case a system of nonlinear equationsarises at each time station.

3 # is used throughout this book for real time in lieu of t , which is used more extensively to denote pseudotime.

3–5


§3.3. Stiffness and Control Matrix

Varying the vector rwith respect to the components of uwhile keeping! fixed yields the Jacobianmatrix K:

K = !r!u

, with entries Ki j = !ri!u j

. (3.11)

This is called the tangent stiffness matrix in structural mechanics applications. The inverse ofK, ifit exists, is denoted by F = K!1; a notation suggested by the name flexibility matrix used in linearstructural analysis for the reciprocal of the stiffness. If (3.1) derives from a potential, both K andF are symmetric matrices (see Exercise 3.5).Varying the negative of r with respect to ! while keeping u fixed yields

Q = ! !r!!

, with entries Qi j = ! !ri!" j

. (3.12)

There is no agreed upon name for Q in the literature. In the sequel it is called the control matrix,although the name loads matrix could be also appropriate. The specialization of Q to the usualincremental load vector q is discussed in Chapter 4.

Remark 3.3. Are the foregoing definitions compatible with linear FEM? The master stiffness equationsproduced by the linear Direct Stiffness Method (DSM) [120, Chapters 2-3] are

Ku = f, (3.13)

in which K and f are constant. The associated residual can be written as r = Ku! f. Evidently K = !r/!u,which is consistent with (3.11). For the control matrix the verification is less obvious, because the concept ofcontrol parameters is unnecessary in linear FEM analysis. But we can bring them in, somewhat artificially, byexpressing the force vector as a linear combination of the "i as

f = !T Q, (3.14)

in which the columns of Q contain the so-called load cases. Then r = Ku! !TQ, and Q = !!r/!! = Q.This is consistent with (3.12).

§3.4. Parametric Representations and Rate Forms

Parametric representations of the state variables u and control parameters ! are useful in thedescription of solution methods as pseudodynamical processes. The general form is

u = u(t), ! = !(t), (3.15)

in which t is a dimensionless time-like parameter. Derivatives with respect to t will be denoted bysuperposed dots, as in real dynamics. The first two t-derivatives of the residual in component formare (with summation convention implied):

ri = !ri!u j

u j + !ri!" j

" j , (3.16)

3–6

3–7 §3.4 PARAMETRIC REPRESENTATIONS AND RATE FORMS

A B

C

L L

k = β

C'

C'

(a) (b)

(c)

2L

P = λ EA

P = λ EA

A B

k

θ θ u

FBCFAC

Fs

E,A constant

EAL

Figure 3.2. Pin-jointed truss structure composed of two aligned elastic bars propped by anextensional spring: (a) unloaded and undeformed reference state; (b) deformed configuration

under load P; (c) FBD of midspan joint C ! in the deformed configuration.

ri = !ri

!u ju j +

! !2ri

!u j!ukuk + !2ri

!u j!"k"k

"

u j +!ri

!" j" j +

! !2ri

!" j!ukuk + !2ri

!" j!"k"k

"

" j . (3.17)

In matrix form,r = Ku"Q!, (3.18)

r = Ku+ Ku"Q! " Q!. (3.19)

Note that both K and Q are matrices. Their (i, j) entries are shown in square brackets in (3.17).On the other hand, terms such as !2ri/!u j!uk , are three-dimensional arrays that may be visualizedas “cubic matrices.” Matrices K and Q are projections of those arrays on the subspace spanned bythe directions u and !. Often these matrices can be more expediently formed by direct pseudotimedifferentiation:

K = dKdt

, Q = dQdt

. (3.20)

Example 3.2. The pin-jointed truss structure shown in Figure 3.2(a) consists of two aligned, identical elasticbars of length L , elastic modulus E and cross section area A. The bars are propped at the midspan joint Cby an extensional spring as shown (without the spring, the bars would form an infinitesimal mechanism). Theextensional spring stiffness is expressed as k = # E A/L for convenience, where # is dimensionless.A vertical point load P = $ E A is applied to the center pin hinge C as shown in Figure 3.2(b). The loadfactor $ = P/(E A) is a dimensionless control parameter defined in terms of E A for convenience in eventuallymaking all equations dimensionless.4 The structure deflects symmetrically as pictured in Figure 3.2(b). Thedeformed configuration is fully specified by either the deflection u, which is the movement CC’ of the loadpoint, or by the rise angle % defined in that figure. For convenience we introduce the dimensionless deflectionµ = u/L . Deflection and rise angle are related by tan % = u/L = µ. The deformed bar lengths are

4 This $ is labeled the stage parameter in §3.5.1, and further studied in the next Chapter.

3–7


−1 −0.5 0 0.5 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−1 −0.5 0 0.5 1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−1 −0.5 00

0

0.5 1

0.5

1

1.5

2

2.5

−1 −0.5 0 0.5 1

0.5

1

1.5

2

2.5

Dimensionless displacement µ = u/LDimensionless displacement µ = u/L

Dimensionless displacement µ = u/LDimensionless displacement µ = u/L Stiff

ness

coe

ffici

ent

K =

dr/

dµSt

iffne

ss c

oeffi

cien

t K

= d

r/dµ

Load

fact

or λ

= P

/EA

Load

fact

or λ

= P

/EA

Response usingengineeringstrain measure

Stiffness usingengineeringstrain measure

Response usingGreen-Lagrangestrain measure

Stiffness usingGreen-Lagrangestrain measure

β = 1 β = 1/10 β = 1/100, 1/1000 (indistinguishable at plot scale)


(a) (b)

(c) (d)

Figure 3.3. Results for problems worked out in Examples 3.2 and 3.3: (a,b) load-deflection response ! = !(µ)

and stiffness coefficient K = K (µ), respectively, using engineering strain measure; (c,d) load-deflection response! = !(µ) and stiffness coefficient K = K (µ), respectively, using GL strain measure.

Ld =!

L2 + u2 = L!

1 + µ2. The bar elongations are " = Ld " L = L(!

1 + µ2 " 1). We shall assumethat Hooke’s law holds for both bars in terms of the axial engineering strain

# = Ld " LL

= "

L=!

1 + µ2 " 1, (3.21)

so that the axial bar stress is $ = E #. Neglecting cross section change, the internal bar forces are Fb = FAC =FBC = A $ = E A #. The restoring force of the extensional spring is Fs = k u = % (E A/L) µL = % E A µ.To establish the total force residual equation in terms of µ, it is sufficient to draw the Free Body Diagram (FBD)of the center hinge in the deformed configuration C #. This FBD is depicted in Figure 3.2(c). Horizontal andmoment equilibrium are automatically verified, whereas vertical equilibrium requires 2Fb sin & + Fs = !P .Since sin & = tan &/

!

(1 + tan2 &) = µ/!

1 + µ2, replacing the sine and canceling out E A yields the forceresidual in dimensionless form as

r(µ, !) = 2"

!

1 + µ2 " 1#

$

µ

1 + µ2 + % µ " ! = µ

%

2 + % " 2!

1 + µ2

&

" ! = 0. (3.22)

Note that µ = ! = 0 give r = 0, so the undeformed configuration is in equilibrium. Equation (3.22) can beeasily solved for ! to get the response ! = !(µ) in terms of the deflection µ. On the other hand, obtaining aclosed form solution for µ = µ(!) is not feasible since it requires finding the roots of a sixth order polynomialin µ in terms of its symbolic coefficients. The associated tangent stiffness matrix is simply 1 $ 1 and its only

3–8

3–9 §3.4 PARAMETRIC REPRESENTATIONS AND RATE FORMS

entry is the dimensionless stiffness coefficient

K = !r!µ

= " + 2(1+ µ2)3/2 ! 1(1+ µ2)3/2

. (3.23)

Figures 3.3(a,b) display plots of # = #(µ) and K = K (µ) for µ " [!1, 1] and four extensional spring ratios:" = 1, 1/10, 1/100, and 1/1000. The #(µ) curves are of hardening type, a response flavor introduced in§2.7. Physical reason: as the deflection grows, both bars develop increasing tension forces, which stiffen thestructure. This effect is qualitatively displayed by the K (µ) plots. Note that if " # 0, the stiffness at µ = 0approaches zero, and the response curve has a zero-slope tangent at the inflexion point there.

Example 3.3. Consider again the problem defined in Figure 3.2. Would the response change appreciably ifanother bar strain measure is used? More specifically, assume now that the bar responds linearly in terms ofthe Green-Lagrange strain5

e = $ + 12 $2, (3.24)

in which $ is the axial engineering strain given in (3.21). The axial bar stress is now % = E e,6 and the axialforce Fb = FAC = FBC = A % = E A e. Repeating the previous derivations one finds, after simplification

r(µ, #) = µ

!

" + µ2"

1+ µ2

#

! # = 0, K = !r!µ

= " + µ2 (3+ 2µ2)

(1+ µ2)3/2. (3.25)

Observe that the total residual r is significantly simpler than (3.22) despite (3.24) being a more complicatedstrain measure than (3.21). Figures 3.3(c,d) display plots of # = #(µ) and K = K (µ) for µ " [!1, 1] andfour spring ratios: " = 1, 1/10, 1/100, and 1/1000. Eyeball comparison to Figures 3.3(a,b) show someminor discrepancies in the load response diagrams #(µ) as µ increases, and more significant ones as regardthe stiffness K (µ), since differentiation amplifies differences.

Example 3.4. This is a variation of the previous example that brings into focus the effect of initial forces (alsocalled prestress) on the nonlinear response. Suppose that in the reference state of Figure 3.2(a), the alignedbars are under an internal axial force F0, which is the same in both. This force may be positive or negative.The reference state µ = # = 0 is still in equilibrium since the initial forces cancel each other at C.We will again use the Green-Lagrange (GL) strain measure (3.24) for the bar elasticity to keep the residualrelatively simple. For convenience define the initial GL strain to be e0 = F0/(E A). The bar constitutiveequation becomes % = E(e + e0), and the bar axial forces are Fb = FAB = FBC = A% = E A(e + e0). Thetotal force residual and the associated stiffness coefficient are easily worked out to be

r(µ, #) = µ

!

" + µ2 + 2e0"

1+ µ2

#

! #, K = !r!µ

= " + 2µ4 + 3µ2 + 2e0(1+ µ2)3/2

. (3.26)

These reduce to (3.25) if e0 = 0, as may be expected. The reference state # = µ = 0 is still an equilibriumconfiguration for any e0. However the initial stiffness at µ = 0 is K0 = " + 2e0. If e0 is negative and |e0|exceeds "/2, K0 < 0, whence the initial slope of the load-deflection response is negative, and the structure isunstable in the reference state. It recovers stability after it deflects enough under P , so the bar compressiveforces are sufficiently reduced.Figures 3.3(c,d) display plots of # = #(µ) and K = K (µ) for µ " [!1, 1] the four spring ratios: " = 1,1/10, 1/100, and 1/1000, and a negative initial strain e0 = !1/5 = !0.2, so aligned bars are in compression.

5 A strain measure defined generally (for 3D) in Chapter 7. For now assume that (3.24) is correct.6 Strictly speaking the bar stress-strain equation should be s = E e, in which s is the second Piola-Kirchhoff (PK2) stressmeasure that is conjugate to e. For now we will not bother about such nitpicking details.

3–9


−1 −0.5 0 0.5 1

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−1 −0.5 0 0.5 1Dimensionless displacement µ = u/LDimensionless displacement µ = u/L St

iffne

ss c

oeffi

cien

t K

= d

r/dµ

Load

fact

or λ

= P

/EA

Response usingGreen-Lagrangestrain measureand e = −0.2


(a) (b)

0

0.5

−0.5

1

1.5

2

2.5

0

Stiffness usingGreen-Lagrangestrain measureand e = −0.20

Figure 3.4. Results for problem worked out in Example 3.4 with initial GL strain e0 = !0.2:(a) load-deflection response ! = !(µ); (b) stiffness coefficient K = K (µ).

(This is admittedly a very large initial strain, exaggerated for plot visualization convenience.) The structure isinitially unstable if " < |2e0| = 2/5 = 0.4. It recovers stability upon traversing a limit point at which K = 0.For instance, if " = 1/10 those limit points occur at µ = ±3/10.The benefits of presenting results in dimensionless form for the foregoing examples should be evident. Theload-deflection and stiffness-deflection solutions are valid for any E , A, L and k, once a bar strain measure isdecided upon. One plot chart for each of !(µ) and K (µ) suffices. Also notice that working out these simpleproblems does not require any use of finite elements: simple Mechanics of Materials does it.

§3.5. Reduction to Single Control Parameter by Staging

Multiple control parameters are quite common in real-life nonlinear problems. They are the analogof multiple load conditions in linear problems. But in the linear world, multiple load conditions canbe processed independently because any load combination is readily handled by superposition. Innonlinear problems, however, control parameters are not varied independently. This aspect deservesan explanation, as it is rarely mentioned in the literature.

§3.5.1. Explicit Reduction

Typically the analysis proceeds as follows. The user defines the control parameters to the computerprogram during a model preprocessing phase. To illustrate the process with a real problem, let usassume that for the analysis of a suspension bridge (Figure 3.5) there are six control parameters

! = [ #1 #2 #3 #4 #5 #6 ]T , (3.27)

in which parameter #1 is associated with own weight, #2 and #3 with live loads, #4 with temper-ature changes, #5 with foundation settlement and #6 with wind velocity.Suppose that #1 = 10 corresponds to full own weight. The first nonlinear analysis involves goingfrom the reference state at

!re f = [ 0 0 0 0 0 0 ]T (3.28)

to the full own-weight condition

!W = [ 10 0 0 0 0 0 ]T . (3.29)

3–10

3–11 §3.5 REDUCTION TO SINGLE CONTROL PARAMETER BY STAGING

Figure 3.5. TheBrooklynBridge in 1876, drawn by F.Hildebrand for the Library ofCongress.

Next, assume that the effect of a temperature drop of!20"C is to be investigated. If a unit incrementof !4 corresponds to 1"C, then the next nonlinear analysis corresponds to going from !W to

!T = [ 10 0 0 !20 0 0 ]T . (3.30)

Note that own weight obviously stays on as a fixed value of !1 = 10, once the bridge is assumedfinished and in service. Analysis for a live load parametrized by !2 = 20, combined with a 6"

temperature rise, would entails going from !W to

!LL = [ 10 20 0 6 0 0 ]T , (3.31)

and so on. Each of these processes is called an analysis stage or simply stage. A stage can bedefined as “advancing the solution” from

!A to !B . (3.32)

when the solution uA at !A is known. Furthermore, if we assume that the components of ! willvary proportionally, we can introduce a single control parameter " that varies from 0 through 1 asper the linear interpolation

! = (1! ")!A + "!B . (3.33)

This " is called the stage control parameter. The nonlinear equation to be solved in (A # B) is

r (u, ") = 0, (3.34)

with the initial condition u = uA at " = 0. The solution curve u = u(") is called the response ofthe structure in the (A # B) stage.The importance of staging in nonlinear static analysis arises from the inapplicability of the super-position principle of linear analysis. For example, the sequences

!A # !B # !C , !A # !C , (3.35)

3–11


or!A ! !B ! !A, !A ! !C ! !A, (3.36)

do not necessarily produce the same final solution. Of course, this phenomenon is especiallyimportant for path-dependent problems in which material failure may occur during a stage.

Remark 3.4. In mathematical circles, (3.33) receive names such as linear homotopy imbedding and piecewiselinear imbedding. The vector ure f for !re f = 0 is called the reference or initial configuration. Commonlyure f = 0, i.e., the reference state is the origin of displacements. This is not necessarily a physically attainableconfiguration. In the suspension bridge example, ure f is fictitious because bridges are not erected in zerogravity fields; on the other hand the own-weight configuration uW is physically relevant once the bridge iscontructed.

Remark 3.5. Many staged analysis sequences do start from the same configuration, for example the own-weight solution uW in the case of the suspension bridge. A comprehensive analysis system must thereforeprovides facilities for saving selected solutions on a permanent database, and the ability to restart the analysisfrom any previously saved solution.

Remark 3.6. The definition (3.33) of ! as a [0 ! 1] parametrization of the [A ! B] stage is somewhatartificial for the following reasons. First, there is no guarantee that a solution at !B exists, so ! = 1 may notbe in fact attainable. Second, in stability analysis (3.33) defines only a direction in control parameter space;in this case the user may want to know the smallest ! (or largest"!) at which a critical point of (3.34) occurs,and the analysis stops there.

Remark 3.7. As the number of control parameters grows, the number of possible analysis sequences increasescombinatorially. Given the substantial costs usually incurred in these analyses, the experience and ability ofthe engineer can play an important role in weeding out unproductive paths. Selective linearization can alsoreduce the number of cases substantially, because invoking the superposition principle “factors out” certainparameters. For example, if the bridge response to live loads is essentially linear about the own-weight solution,parameters "2 and "3 may be removed from the nonlinear analysis, and the dimension of ! is reduced from6 to 4.

§3.5.2. *Implicit ReductionThe most general reduction from multiple control parameters ! to a single parameter ! can beformulated as follows. Suppose that over a stage an implicit vector algebraic relation is established:

m(!, !) = 0, (3.37)

wherem is a vector of m equations that uniquely defines ! if ! is given. Augmenting the residual(3.1) with (3.37) we obtain the expanded system

!

r(u,!)

m(!, !)

"

=!

00

"

. (3.38)

This relation implicitly defines u as a function of !. (It would be explicit if the ! are eliminated,but that might not be possible or convenient.) Derivatives of r with respect to pseudotime t areobtained by the chain rule, for example

!

r(u,!)

m(!, !)

"

=!

Ku"Q!A! " aT !

"

=!

00

"

, (3.39)

3–12

3–13 §3. Notes and Bibliography

in which A = !m/!! and a = !!m/!". The explicit definition of " by the linear interpolation(3.33) is a special case of (3.38), for which

m(!, ") = ! ! (1! ")!A ! "!B = 0. (3.40)

In this case the elimination of! is trivial. For most applications use of the more general expression(3.38) is unnecessary. It may come handy, however, in complicated load parameter dependencies.For example, those associated with hydro or aerodynamic loads: beyond the laminar flow regimephysical loads on the structure are complicated nonlinear functions of relative fluid velocities.

Notes and Bibliography

The general treatment of nonlinear structural problems in terms of force residuals is a natural one, since itmerges physics visualization with standard techniques in numerical analysis and computational mathematics.In textbooks of that ilk, systems of nonlinear equations are typically presented as

f(x) = 0, (3.41)

and attention is directed to finding the roots, whether a few, billions or an infinite number. Filtering the rootmorass by attaching physical significance to roots requires the introduction of control parameters, as done in§3.2. No treatment of that nature can be found in FEM textbooks. One needs to go to the literature in systemcontrol to run into comparable formulations.The use of parametrized residual derivative forms is a useful tool that unifies continuation solution methods.It also provides a natural link to nonlinear dynamics. Again those rate forms are ignored in most FEMpresentations, reflecting the unease of many authors in dealing with rate equations in pseudotime. A shorthistorical account of rate forms is provided in the book by Ortega and Rheinboldt [563, pp. 234–235].NoFEMbook evenmentions staging, despite its crucial importance in design verification by nonlinear analysis.

3–13



Residual Force Equations

EXERCISE 3.1 [A:15] For the example system (3.7) find K and Q.

EXERCISE 3.2 [A:15] For the example system (3.7), and assuming the parametric representation (3.15) foru and !, write down r, K, Q and r in matrix-vector form.

EXERCISE 3.3 [A:15] Continuing the above exercise: if !1 = 2" and !2 = ", write down r(u, ") andr(u, ") in matrix-vector form.

EXERCISE 3.4 [A:20] Is the example system (3.7) derivable from a potential energy function # so that rcan be represented as r = $#/$u? Can you guess by inspection what the potential # is?

EXERCISE 3.5 [A:20] Show that both K and F = K!1 (assuming the latter exists) are symmetric if theresidual is derivable from a potential energy function.

EXERCISE 3.6 [A:20] For the example problem defined in Figure 3.2, derive equations for "(µ) and K (µ)

if the logarithmic (also called Hencky) strain eH = log(1 + %) is used for the bars, so that & = E eH andFb = FAC = FBC = A& = E A eH . Provide plots of "(µ) and K (µ) similar to those shown in Figure 3.3.Does the change in strain measure make much difference?

EXERCISE 3.7 [A:15] Show that the total potential energy # for the example problem in Figure 3.2, using(3.21) as bar strain measure, is given by

# = U ! W, U = E A L %2 + 12 k u

2, W = P u. (E3.1)

Then derive the total residual force equation directly as r = $#/$µ = 0. Does this method provide the sameresult as (3.22)?

EXERCISE 3.8 [D:10] The K (µ) curves plotted in Figures 3.3(b,d) for varying ' are identical except for atranslation along the vertical " axis. Explain the physical reason behind this feature.

EXERCISE 3.9 [A:20] Assume that the total force residual equation for a one-DOF system with stateparameterµ and control parameterµ is r(µ, "), and that the response satisfying r = 0 can be explicitly solvedas " = "(µ). The stiffness matrix is simply 1 " 1, and its only entry is K = $r/$µ. Discuss under whichcondition(s) the following relation holds true:

K = $r$µ

?= $"

$µ. (E3.2)

(Hint: differentiate both sides of r = 0 wrt µ and apply the chain rule.) Can (E3.2) be extended to the casewhere the response " = "(µ) satisfies r = rc, where rc is constant?

3–14

.

4One-Parameter

ResidualEquations

4–1

Chapter 4: ONE-PARAMETER RESIDUAL EQUATIONS 4–2

TABLE OF CONTENTS

Page§4.1. Introduction 4–3§4.2. Residual Equations For One Control Parameter 4–3

§4.2.1. Residual Derivatives . . . . . . . . . . . . . . . 4–3§4.2.2. Rate Forms . . . . . . . . . . . . . . . . . 4–4§4.2.3. Incremental Velocity . . . . . . . . . . . . . . . 4–4§4.2.4. Separable Residuals and Proportional Loading . . . . . . 4–4

§4.3. Response Visualization 4–5§4.3.1. Path Traversal Sense . . . . . . . . . . . . . . . 4–5§4.3.2. Incremental Flow Visualization . . . . . . . . . . . 4–5§4.3.3. Diagrams for Single-DOF System . . . . . . . . . . 4–7§4.3.4. Diagrams for Multiple-DOF System . . . . . . . . . 4–8

§4.4. Intrinsic Geometry of Incremental Flow 4–9§4.4.1. Tangent Vector . . . . . . . . . . . . . . . . . 4–9§4.4.2. Normal Hyperplane and Flow-Orthogonal Envelope . . . . 4–10§4.4.3. ArcLength Distance . . . . . . . . . . . . . . . 4–11

§4.5. *State Vector Scaling 4–11§4. Notes and Bibliography. . . . . . . . . . . . . . . . . . . . . . 4–12§4. Exercises . . . . . . . . . . . . . . . . . . . . . . 4–14

4–2

4–3 §4.2 RESIDUAL EQUATIONS FOR ONE CONTROL PARAMETER

§4.1. Introduction

This Chapter continues on the topic of residual equations introduced in Chapter 3. The generalresidual force equation presented there is specialized, through the concept of staging introduced in§3.5, to the one-parameter form in which r is a function of the state vector u and the single controlparameter !.1 Taken together these form the control-state space. The separable case, in which uand ! can be segregated to both sides of the balanced force residual form, is noted on account ofits frequent occurence.Further insight into the structural response may be achieved with the help of constant-residualincremental flows. This and related features are geometrically illustrated. Finally, the concepts oftangent and arclength in control-state space and scaling are discussed.

§4.2. Residual Equations For One Control Parameter

In this section we study further the one-parameter, total residual force form introduced in §3.5 as(3.34). This equation is reproduced below for convenience:

r(u, !) = 0. (4.1)

Here the stage parameter ! is now the only control parameter. In the sequel partial derivatives withrespect to ! will be often abbreviated with a prime. For example,

u! = "u"!

, r!! = "2r"!2

, !! = 1, !!! = 0. (4.2)

§4.2.1. Residual Derivatives

We recall here the parametric representations of the state variables in terms of pseudotime, asintroduced in §3.4 through (3.15). This definition is specialized to a single control parameter:

u = u(t), ! = !(t), (4.3)

in which t is again the pseudotime variable. The choice ! = t is often made in purely incrementalmethods; this is equivalent to “rewinding the clock” after each stage since then ! = t = 0 is resetat each stage start.Derivatives with respect to t will be denoted by superposed dots, as in real dynamics. The first twot-derivatives associated with the total residual force r(u, !) are

r = K u" q !, (4.4)

r = K u+ K u" q ! " q !, (4.5)

1 Why it is convenient to have only one control parameter at a time? Think of driving a car. To go anywhere along a roadmaze, it is easier to have one steering wheel than two or four. Mathematically, the reduction to one independent variableallows the use of well known techniques to handle the resulting ODE as an initial-value problem.

4–3


in whichK = !r

!u, q = ! !r

!"= !r". (4.6)

Here K is the tangent stiffness matrix introduced in §3.3 via equation (3.11), whereas q is theincremental load vector. The latter is the specialization of the control matrix Q defined in §3.3through equation (3.12), to the one-parameter case.Equations (4.1) through (4.6) will be used in the sequel instead of those given in §3.3 and §3.4 forthe multiple control parameter case, unless otherwise noted.

§4.2.2. Rate FormsRate forms of the total force residual equation (4.1) are obtained by equating the foregoing residualderivatives to zero:

r = 0, or K u = q", (4.7)

r = 0, or K u+ K u = q " + q ". (4.8)Equations r = 0 and r = 0 will be called the first order and second order rate incrementalequations, respectively. The qualifier incremental arises from their important role in incrementalsolution methods. Both (4.7) and (4.8) are ordinary differential equations (ODE) in t .

§4.2.3. Incremental VelocityAt regular points of the control-state space the tangent stiffness K is nonsingular. If so, we cansolve the first-order rate form (4.7) for u:

u = K!1 q " = u" "def= v ", (4.9)

in whichv = u" = K!1 q. (4.10)

This v is called the incremental velocity vector. It is an important component of all solutionmethodsbased on continuation. Note that u = v " in (4.9) can be most easily remembered using the chainrule: u = (!u/!")/(!"/!t); then replace !u/!" = u" by v, and !"/!t by ".

§4.2.4. Separable Residuals and Proportional LoadingThe balanced force residual equivalent of (3.4) in §3.2.2 for a one-parameter form is

p(u) = f(u, "), (4.11)

in which p and f denote internal and external forces, respectively. The total residual is r = p! f.If the external forces do not depend on the state variables u, that is

p(u) = f("), (4.12)

the residual eqilibrium equations are called separable. In this case the tangent stiffness matrixK = !r/!u = !p/!u depends only on u. Furthermore, if f is linear in " the loading is said to beproportional. If so,

q = ! !r!"

= !r" = !f!"

= f", (4.13)

is a constant vector.

4–4

4–5 §4.3 RESPONSE VISUALIZATION

u

λ

Response curve r = 0(primary equilibrium path)

!

Pseudo-time t decreasing

Pseudo-time t increasing

+

−

u

(b) !

Negative tangentvector

Positive tangentvector

u

(c)(a)

t−

t+P

Figure 4.1. Load-displacement response for single control parameter: (a) an equilibrium path in control-state space;(b) positive and negative traversal senses on equilibrium path; (c) positive and negative tangent vectors at point P .

Remark 4.1. The more general balanced force residual (3.4) that contains multiple control parameters collectedin array !, is said to be separable if

p(u) = f(!). (4.14)

In this case the loading is called proportional if f is linear in all control parameters, thus providing a constantcontrol matrix Q = !"r/"! = "f/"!.

Remark 4.2. If a separable system derives from a total potential energy # = U ! W , then the external workpotential W must be linear in the state variables ui . Furthermore, for the loading to be proportional, W mustalso be linear in all $ j in the multiple control parameter case; or linear in ! if there is only one parameter.

§4.3. Response Visualization

As discussed in §2.2, the solution of the one-parameter residual form

r(u, !) = 0, (4.15)

is often plotted on the u versus ! plane, where u (or a dimensionless version µ) is a representativecomponent (or some norm) of the state vector u. One such diagram is illustrated in Figure 4.1(a).If ! is a load amplitude, this is called a load-displacement response curve or simply a responsecurve. It was noted in §2.2.2 that it is common practice to make the curve pass through the originof the (u, !) plane. A response curve that satisfies r = 0 is called equilibrium path or equilibriumtrajectory. The equilibrium path that passes through the origin is called the primary or fundamentalpath, since it usually represents the operation of the structure under normal service conditions.

§4.3.1. Path Traversal Sense

Away from bifurcation points, a path can be traversed in two directions. These are identified aspositive or + sense, and negative or ! sense. As illustrated in Figure 4.1(b) we shall use theconvention that the positive sense is associated with increasing values of the pseudotime t whenthe path is parametrically described as in (4.3). At a regular point P of an equilibrium path theresponse curve has a unique tangent, but the presence of two directions mean that two oppositetangent vectors, called positive and negative, respectively, can be drawn. See Figure 4.1(c).

4–5


equilibrium path r = 0

t+

positive tangentvector at P

P(u, λ )

λ

u

t+

positive tangentvector at P

normal "hyperplane"

at P P(u, λ )

λ

u

equilibrium path r = 0

(a) (b)

Figure 4.2. Response visualization tools: (a) incremental flow as a family of constant-residualtrajectories; (b) incremental flow (full curves) and the flow-orthogonal envelope (dashed curves).

§4.3.2. Incremental Flow Visualization

A diagram such as that in Figure 4.1(a) gives of course only a partial picture of the structuralbehavior. For a better understanding of the way numerical solution procedures work (or fail to) itis instructive to “look around” the equilibrium path by considering the perturbed residual equation

r(u, !) = rc, (4.16)

in which rc is a constant vector. This is the general solution of r = 0 viewed as a first-order ODEin pseudotime t . Additional behavioral information can be conveyed by drawing the solutions of(4.16) for various values of the right-hand side near zero. This produces constant-residual paths asillustrated in Figure 4.2(a). Collectively these paths form the incremental flow whose differentialequation is either r = 0, or, if we take t ! ! :

r" = "r"!

= 0, assuming t ! !. (4.17)

Here the notational convention (4.2) for ! derivatives is used. If the residual is separable, (4.17)can also be presented as

r" = p" # r" = "p"u

u" # r" = K u" # q = 0. (4.18)

If K is nonsingular, solving (4.18) yields u" = K#1q = v. The incremental solution methodscovered later exploit these forms. This use explains the qualifier “incremental” applied to the flow.

4–6

4–7 §4.3 RESPONSE VISUALIZATION

−1 −0.5 0 0.5 1

−0.2

−0.1

0

0.1

0.2

−1 −0.5 0 0.5 1

−0.2

−0.1

0

0.1

0.2

Dimensionless displacement µ = u/L Dimensionless displacement µ = u/L

Load

fact

or λ

= P/EA

Load

fact

or λ

= P/EA

(a) (b)

0.5

0.05

0.01

0.1

0.25

−0.5

−0.05

−0.01

−0.1−0.25

0.5

0.05

0.01

0.10.25

−0.5

−0.05−0.01

−0.1−0.25

0 0

Figure 4.3. Incremental flow plots for the residual (4.19) with ! = 1/10, and initial straine0 = 0 for (a), and e0 = !0.2 for (b). Numbers annotated near curves are the values of rc .

Figures 4.2(a,b) also depicts the construction of the tangent vector t+ at an arbitrary point P(u, ")

for a one degree-of-freedom (DOF) case. This procedure is described more precisely in §4.4.1.Figure 4.2(b) pictures a set of curves whose trajectories are orthogonal to the incremental flow.This set is called the flow-orthogonal envelope. It will be explained in §4.4.2 that this set generallyconsists of a family of hypersurfaces. For a one-DOF system, however, the envelope reduces to afamily of curves, as in that figure. This graphical representation will be useful in Chapters 12ff inexplaining how incremental-iterative solution methods work.

Remark 4.3. The consideration of a “perturbed residual” equation such as (4.16), as well as the picture of theassociated incremental flow, will be also useful later when discussing effects of imperfections due to fabrication(e.g., lack of fit, crooked members), eccentric loading, or residual stresses.

§4.3.3. Diagrams for Single-DOF SystemFor simple one-DOF systems it is easy to produce the incremental flow using standard graphicpackages as long as the perturbed residual equation (4.16) can be solved explicitly for " as functionof rc and the single state parameter, say µ. As an example, consider the following total forceresidual derived in Example 3.4 of Chapter 3:

r(µ, ") = µ

!

! + µ2 + 2e0"1 + µ2

#

! ". (4.19)

Here the state parameter µ is a dimensionless deflection, " a load factor chosen as control parameter,! a scaled spring constant and e0 an initial strain. [The source problem of (4.19) is shown inFigure 3.2.] Inasmuch as " appears linearly in (4.19), it is easy to set r = rc and solve for ":

" = µ

!

! + µ2 + 2e0"1 + µ2

#

! rc. (4.20)

Setting ! = 1/10 and e0 = 0, Mathematica produced the plots shown in Figure 4.3(a) for rc = 0,rc = ±0.01, rc = ±0.05, rc = ±0.1, rc = ±0.25, and rc = ±0.5, using a cyclical color scheme

4–7


−1 −0.5 0 0.5 1

−0.2

−0.1

0

0.1

0.2

−1 −0.5 0 0.5 1

−0.2

−0.1

0

0.1

0.2

−1 −0.5 0 0.5 1

−0.2

−0.1

0

0.1

0.2

r=µ*(β+(2*e0+µ^2)/Sqrt[1+µ^2])-λ; r=Simplify[r/.{β->1/10,e0->-0.2}];ContourPlot[-r,{µ,-1,1},{λ,-0.25,.25},PlotPoints->101];ContourPlot[-Sqrt[Abs[r]],{µ,-1,1},{λ,-0.25,.25},PlotPoints->101];ContourPlot[-Sqrt[Abs[r]],{µ,-1,1},{λ,-0.25,.25},PlotPoints->301];

(d)

(a) (b) (c)

Figure 4.4. Incremental flow contour plots for the residual (4.19) with ! = 1/10 and e0 = !0.2, producedbyMathematica: (a) contour plot of!r ; (b) contour plot of!

"|r |with 101 points aver axis ranges; (c) contour

plot of !"

|r | with 301 points aver axis ranges; (d) Mathematica script that generated the three plots.

for rc #= 0. Changing the initial strain to e0 = !0.2 with ! = 1/10 and same list of rc values,generates the plot of Figure 4.3(b).If no explicit solution for " is possible, or it is procurable but too complicated, the incremental flowcan be generated through a contour plot of r(µ, "), using graphical software such as ContourPlotinMathematica. In such plots one varies " andµ along the vertical and horizonatl axes, respectively.Challenges posed by that kind of canned software is that it may be difficult to grade rc near zero,as was automatically done in the plots of Figure 4.3.To illustrate how to overcome those troubles, consider the plots shown in Figure 4.4(a,b,c). Thosewere produced for the residual (4.19) through the Mathematica script of Figure 4.4(d). The plotin Figure 4.4(a) directly contours !r (sign reversal produces better gray scale). This is too coarse,however, to display curves near the equilibriumpath becauseContourPlotusesfixed contour-valueincrements and there is no way to specify grading near r = 0 in the calling sequence.Plotting !

"|r | improves visibility since taking the square root “flattens” r near zero, as shown in

Figure 4.4(b). (It also gets rid of the tresidual sign, which usually is of no interest.) However, using101 points over each axis range is insufficient to clearly resolve contours near r = 0, and resultsin visible “smallpox” noise there. Increasing the resolution to 301 points produces a better display,as shown in Figure 4.4(c). This is comparable to Figure 4.3(b).In conclusion, the chief advantage of contouring an incremental flow is that one can directly plotthe residual, or function thereof, without having to explicitly solve for the perturbed load-deflectionresponses. Such solution may be difficult or impossible. The main drawback is the need toexperiment with plots (as well as “residual grader” functions) to get reasonable visualizations.

4–8

4–9 §4.4 INTRINSIC GEOMETRY OF INCREMENTAL FLOW

1u

2u 2u

λ

r = 0 r = 0

1u

λ

(b)(a)

Figure 4.5. Response diagram for two DOF: (a) incremental flow; only a few paths are shown toreduce clutter; (b) some members of the flow-orthogonal envelope, with only the primary equilibrium

path r = 0 shown to reduce clutter.

§4.3.4. Diagrams for Multiple-DOF System

If the number of degrees of freedom (DOF) increases to N > 1 the incremental flow still remains afamily of curves in the N +1-dimensional control-state space space (u, !). Visualization, however,is restricted to N = 2 as illustrated in Figure 4.5(a). For three or more DOF, only cross sectionsof the control-state space can be displayed, in which one or two representative state variables orfunctions of such are plotted. Such “projections” requires some ingenuity and experience.The flow-orthogonal envelope becomes a family of ordinary surfaces if N = 2, as illustrated inFigure 4.5(b). For three or more DOF, the envelope becomes a family of hypersurfaces.

§4.4. Intrinsic Geometry of Incremental Flow

This section focuses on some geometric objects that will be of interest later when describing certain“path following” solution methods.

§4.4.1. Tangent Vector

At a generic regular point P of coordinates (u, !), not necessarily on the equilibrium path, we canconstruct an unnormalized tangent vector t defined by

t =!

u!

!!

"=!

v1

", (4.21)

4–9


t+

t+

P

P

1u 1u

2u 2u

λ + sense of increasing t

TNormal hyperplanev u + λ = 0 at P. .

PQ

Normal hyperplane at P

!s

Positive tangent direction

Equilibrium pathEquilibrium path

λ

(b)(a)

r = 0 r = 0

Figure 4.6. Tangent and normal hyperplane and arclength illustrated for a two DOF case: (a) tangent vector andnormal hyperplane, here P is on the primary equilibrium path but P is generic; (b) concept of arclength distance!s from point P to point Q — note that the point order is important: the arclength distance from Q to P is not

generally the same as that from P to Q.

where v = K!1q is the incremental velocity vector. Tangent vectors are illustrated in Figures 4.2and 4.6 for one and two DOF, respectively.The tangent vector normalized to unit length is

tu =!

v/ f1/ f

", (4.22)

where f is the scaling factor

f = |t| = +#

||t||2 = +#

1 + vT v. (4.23)

The positive tangent direction and the positive unit tangent are defined as

t+ = ±!

v1

", t+u = t+

f= ±

!v/ f1/ f

". (4.24)

The positive tangent direction points along the positive sense of path traversal, as illustrated inFigures 4.1(b) and 4.6(a).

§4.4.2. Normal Hyperplane and Flow-Orthogonal EnvelopeThe hyperplane NP normal to the tangent vector t at P(u, ") has the equation

vT !u + !" = 0, (4.25)

where !u = u ! uP and !" = " ! "P are increments from P . Dividing these increments by !tand passing to the limit one obtains

vT u + " = 0. (4.26)

4–10

4–11 §4.5 *STATE VECTOR SCALING

For one DOF u the hyperplane reduce to a line in (u, !) space, as illustrated in Figure 4.2. Fortwo DOF the normal hyperplane is an ordinary plane in the 3D control-state space (u1, u2, !), asillustrated in Figure 4.6(b).For one DOF (4.26) is the differential equation of a flow orthogonal to the incremental flow, asillustrated in Figure 4.2; this flow is the envelope of the normals. For two DOF (4.26) represents afamily of surfaces, see Figure 4.5(b). For more degrees of freedom (4.26) is a family of hypersur-faces. The orthogonality property plays an important role in corrective solution methods such asNewton-Raphson and their variants.

§4.4.3. ArcLength DistanceThe left hand side of the hyperplane equation (4.25) normalized on dividing through by f

"s = 1f(vT "u+ "!), (4.27)

acquires the following geometric meaning: "s is the signed distance from the normal hyperplane atP to a point Q("u, "!). For small increments ("u,"!),"smaybe considered as an approximationto the arclength s of the path that passes through P because

ds = 1f(vT du+ d!). (4.28)

This important concept is illustrated in Figure 4.6(b).

Remark 4.4. At isolated limit points studied in Chapter 5, the normalization process (4.28) reduces the unittangent to

tu =! z0

", t+u =

!±z0

", (4.29)

where z is the unit length null eigenvector ofK, that is,Kz = 0. The sign ambiguity arises because+z and!zare both eigenvectors; one of them has to be chosen to satisfy the positive-traversal convention. At bifurcationpoints and non isolated limit points t is not unique.

Remark 4.5. From previous equations we note the formulas

duds

= vf,

d!

ds= 1

f. (4.30)

Remark 4.6. In the mathematical literature the incremental flow projected on the u state space is sometimescalled aDavidenko flow in honor of the father of continuationmethods, should! be interpreted as a continuationparameter.

Remark 4.7. An alternative to plotting (4.16) for response visualization, is to consider the use of the constant-residual-norm equation

||r(u, !)|| = C, (4.31)in which ||r|| denotes a vector norm such as, for instance, the Euclidean norm ||r||2 = rT r, and C is anonnegative numeric constant. This relation does not generally represent a family of curves but a familyof tube-like hypersurfaces that for sufficiently small C “wrap around” equilibrium paths, as illustrated inFigure 4.7. Because of the visual clutter evident in that figure, (4.31) is less suitable than (4.16) to study whathappens in the neighboorhood of equilibrium paths.

4–11


||r|| = C1

||r|| = C2 > C1

||r|| = C3 > C2

1u

2u

λ

r = 0

Figure 4.7. For Remark 4.7: illustrating that the constant residual-norm equation ||r|| = constgenerally represents a family of tube-like surfaces “wrapping around” equilibrium paths.

§4.5. *State Vector Scaling

In applying nonlinear equation solving techniques to structural mechanics (or, in general, to problems inengineering and physics) the issue of scaling often arises because of two aspects:1. The residual r has two types of arguments: u and !. Translational DOF collected in the state vector u

have physical dimensions of length (displacement) whereas ! is dimensionless.2. The DOF in u may have heterogeneous physical dimensions. For example, in the analysis of finite

element models that account for bending effects u may contain both translations and rotations.To reduce the sensitivity of solution procedures to these factors, it is often advisable to introduce a scaling ofthe state vector u to render it dimensionless and thus placed on an equal footing with !:

!u = Su. (4.32)

Here the scaling matrix S is diagonal, and a superposed tilde identifies a scaled quantity. If all entries of uhave homogeneous dimensions, one may take simply S = (1/u) I, where the scalar u has the dimension of u.The scaled versions of other quantities defined previously are

"!u = S "u, !q = S!1q, !K = S!1KS!1, (4.33)

!v = Sv, !t ="Sv

1

#="!v

1

#, !f =

$1 + vTS2v =

$1 +!vT!v, (4.34)

!tu = (1/!f )"!v

1

#, "!s = (!vT "!u + "!)/!f = (vTS2 "u + "!)/!f . (4.35)

4–12

4–13 §4. Notes and Bibliography


None of the material in this Chapter is available in comprehensive form in textbooks devoted to computationalnonlinear analysis. Portions of it are hidden under various names in the applied mathematics literature. Forexample, what is here called the tangent stiffness matrix often receives the more generic name of Jacobianmatrix, or of Hessian matrix if the residual derives from potentials. The incremental flow is occasionallylabeled Davidenko flow, in honor of the father of continuation methods. The name Branin sometimes appearsinstead of Davidenko in the optimization literature.

4–13



One-Parameter Residual Equations

EXERCISE 4.1 [A:5+15] Consider the residual force equations! r1r2

"=! u1 + 3u22 ! 2!1u2 + 6u1u2 ! !2

"=! 00

". (E4.1)

(a) Is this system of equations separable in the sense discussed in §4.2.4?(b) If so, can f and p be expressed as u- gradients of energy functions U and W , and what are those?

EXERCISE 4.2 [A:15+15] Suppose that (E4.1) is to be solved in two stages:

Stage 1. Start from !1 = !2 = 0 and go to !1 = 0 and !2 = 5. Parameter " varies from 0 to 1.

Stage 2. Start from !1 = 0, !2 = 5 and go to !1 = !2 = 10. Again " varies from 0 to 1.

(a) Express the residual in the one-parameter form (4.1) for each stage.(b) Find the expression of the incremental load vector q in each stage. Is the loading proportional?

EXERCISE 4.3 [A:20] Suppose the first residual force above is replaced by r1 = u1 + 3u22 ! 2!21.

(a) Is the system still separable?(b) For the same two stages of the previous exercise, is the loading proportional?

EXERCISE 4.4 [A:25] For stage 1 of Exercise 4.2, write down the analytical expressions of the incrementalvelocity, the tangent vectors t and tu , the normal hyperplane equation, and the differential equations of theflow-orthogonal envelope. Note: explicit inversion of K!1 may be done using the formulas to invert a 2" 2matrix.

EXERCISE 4.5 [A:25] Verify the assertion of Remark 4.7 by using the Euclidean norm ||r|| = rT r of theresidual vector.

EXERCISE 4.6 [A:25] Explainwhether the unnormalized tangent vector t introduced in §4.4.1may be definedas

t =! u

"

", (E4.2)

and whether this definition is more general than (4.21).

4–14

.

5Critical Points

5–1

Chapter 5: CRITICAL POINTS 5–2

TABLE OF CONTENTS

Page§5.1. Introduction 5–3§5.2. Critical Points 5–3

§5.2.1. Behavioral Assumptions . . . . . . . . . . . . . . 5–3§5.2.2. Stiffness Matrix Properties . . . . . . . . . . . . 5–3§5.2.3. Regular Versus Critical Points . . . . . . . . . . . . 5–4§5.2.4. Isolated Versus Multiple Critical Points . . . . . . . . 5–4§5.2.5. Limit Versus Bifurcation Points . . . . . . . . . . . 5–4

§5.3. Limit or Bifurcation Point? 5–5§5.4. Limit Point Sensors 5–7§5.5. Critical Point Computation Examples 5–8

§5.5.1. The Circle Game . . . . . . . . . . . . . . . . 5–8§5.5.2. Perfect Propped Rigid Cantilever Column . . . . . . . . 5–10§5.5.3. Imperfect Propped Rigid Cantilever Column . . . . . . 5–11

§5.6. *Turning Points 5–14§5. Notes and Bibliography. . . . . . . . . . . . . . . . . . . . . . 5–14§5. Exercises . . . . . . . . . . . . . . . . . . . . . . 5–15

5–2

5–3 §5.2 CRITICAL POINTS

§5.1. Introduction

This Chapter provides additional material on properties of the one-parameter force residual equa-tions. It begins with a study of critical points, which are classified into limit and bifurcation points.Limit point “sensors” and turning points are briefly described. Two worked out examples have beenadded in the present revision.

§5.2. Critical Points

This section deals with the classification and characterization of critical points. The determinationof such points is a key application of geometrically nonlinear analysis on account of the followingproperty:

Along a static equilibrium path of a conservative system, transition fromstability to instability can only occur at critical points. (5.1)

This property does not extend to nonconservative systems, which require a dynamic treatment. Italso does not apply to conservative systems away from equilibrium.

§5.2.1. Behavioral Assumptions

We shall restrict the class of systems considered here to those that satisfy the following assumptions:• There is only one control parameter: the staging parameter !.• The system is conservative: the total residual is the gradient of a real total-energy function":

r(u, !) = #"(u, !)

#u. (5.2)

Since " and the state vector entries are real, the entries of the residual vector are also real.

§5.2.2. Stiffness Matrix Properties

A consequence of the conservativeness assumption (5.2) is that the tangent stiffness matrix is theHessian of the total energy function:

K = #r(u, !)

#u= #"(u, !)

#u #u. (5.3)

Transposing both sides of (5.3) gives K = KT . Thus K is symmetric real. This guarantees twoimportant spectral properties:• All eigenvalues of K are real.• K has a full set of independent real eigenvectors that can be orthonormalized. Furthermore

left and right eigenvectors coalesce.

5–3


To state these properties more precisely, let the eigensystem of the N ! N tangent stiffness matrixbe

Kzi = !i zi , i = 1, 2, . . . N . (5.4)The eigenvalues !i of K = KT are real, and the orthonormalized eigenvectors satisfy zTi z j = "i j ,where "i j is the Kronecker delta.

Remark 5.1. If the system is nonconservative, K is generally unsymmetric, and the foregoing spectralproperties are lost. The major consequence is that purely static stability analysis is no longer possible becauseof the possible occurrence of growing oscillations in real time (flutter). Investigation of that possibility requiresa dynamic analysis, which substantially complicated the model as well as the analysis process. This case isrelegated to the final Chapters.

§5.2.3. Regular Versus Critical PointsEach point of an equilibrium path represents a (static) equilibrium state. These are classified asfollows according to whether the tangent stiffness matrix evaluated at that point is singular or not:Regular point: K is nonsingular.Critical point: K is singular. Also called singular or nonregular points.Recall the incremental velocity vector defined in §4.2.3 is

v = u" = K#1 q. (5.5)At a critical point v becomes undefined according to (5.5), since K#1 does not exist. Physicallythis means that the structural behavior cannot be controlled by the parameter #.Since the determinant of a singular matrix is zero, the foregoing classification can be stated as

The determinant of K vanishes at a critical point (5.6)

This rule provides a practical mean for locating critical points analytically in simple problems withclosed form solutions for the response. The procedure is illustrated in a later section.

§5.2.4. Isolated Versus Multiple Critical PointsWe shall denote by

ucr , #cr , Kcr , qcr , (5.7)the value of the state vector, control parameter, tangent stiffnessmatrix, and incremental load vector,respectively, evaluated at a critical point. SinceKcr is singular, at least one eigenvalue ofK is zero.The following subclassification takes into account the number of zero eigenvalues:Isolated critical point: Kcr has only one zero eigenvalue. Its rank deficiency is one.Multiple critical point: Kcr has two or more zero eigenvalues. Its rank deficiency is two or more.This distinction has importance from both computational and engineering viewpoints. A multiplecritical point is more difficult to “capture” and traverse numerically in a response computationprocess. Physically, a structure with a multiple critical point is more sensitive to imperfectionsin the vicinity of that critical state. It might be thought that critical point coalescence has a lowprobability of happening in a typical structure. However, such occurrence may be the unfortunateside effect of a design optimization process.

5–4

5–5 §5.3 LIMIT OR BIFURCATION POINT?

u

!

BL1

L2

1

! L1

L2

B2

1B

u1

(b)(a)

Figure 5.1. Critical points for a two degree of freedom system (u1, u2) shown on the u1 versus ! plane:(a) Limit point (“snap through” behavior) L1 occurs before bifurcation B1; (b) Bifurcation point B1 occursbefore limit point L1, in which case L1 is physically unreachable. Full lines represent physically “preferred”

paths. A more realistic three-dimensional view of this case is shown in Figure 5.2.

§5.2.5. Limit Versus Bifurcation Points

For simplicity we restrict attention to isolated critical points, at which Kcr has a single zero eigen-value and a rank deficiency of one. It is convenient to distinguish two types of critical pointsLimit points, at which the tangent (4.29) to the equilibrium path is unique but normal to the ! axisso v becomes infinitely large.1

Bifurcation point, also called branch point or branching point, from which two equilibrium pathbranches emanate and so there is no unique tangent.Since the tangent at a limit point is normal to !, it must correspond to a maximum, minimum orinflexion point with respect to !. In the case of a maximum or a minimum, the occurrence of alimit point is informally called snap through or snap buckling by structural engineers for reasonsexplained in a Remark below.The type classification a for multiple critical point is more complicated, and is discussed in theChapters dealing specifically with stability.Figures 5.1 illustrates two possible configurations of limit and bifurcation points (all of themisolated) for a two DOF system. It is assumed that the fundamental path occurs with u2 = 0, andso the response is shown on the {!, u1} plane for clarity. Limit points are identified as L1, L2, . . .

whereas bifurcation points are marked as B1, B2, . . ..

§5.3. Limit or Bifurcation Point?

We shall focus here on an isolated critical point, at which we have available the quantities listed in(5.7). How can we mathematically characterize its type? Consider the eigensystem (5.4), and letzcr be the right eigenvector of Kcr associated with the single zero eigenvalue:

Kcr zcr = 0! zcr = 0. (5.8)

1 Some authors, e.g. Seydel [599], call limit points “turning points.” That term is here used for a different type cf. §5.6.

5–5


This eigenvector will be called the null eigenvector. It spans the null space of Kcr . Transposingboth sides of (5.8) gives

zTcr KTcr = zTcr Kcr = 0T . (5.9)

That is, zcr is also a left null eigenvector. Recall the first-order rate equation of the equilibriumpath, stated in (4.7) as K u = q !. Evaluate this equation at the critical point: Kcr ucr = qcr !cr .Multiply through by dt to express it in terms of differentials:

Kcr du = qcr d!. (5.10)

in which du and d! denote the differentials at (ucr , !cr ). Premultiply both sides of (5.10) by zTcrand use (5.9) to get

zTcr qcr d! = 0. (5.11)This states that the product of two scalars: zTcr qcr , and d!, must vanish. So one of them must bezero. If

zTcr qcr != 0, (5.12)

then d! must vanish, and we have a limit point. On the other hand, if

zTcr qcr = 0, (5.13)

then d! is not necessarily zero, and we have a bifurcation or branching point. The quantity zTcr qcris called an critical point type indicator.The key physical characteristic of a bifurcation point is an abrupt transition from one deformationmode to another mode; the latter having been previously “concealed” by virtue of being orthogonalto the incremental load vector. This is plainly shown by (5.13).

Remark 5.2. IfK is not symmetric, several changes must bemade in the previous assumptions and derivations.These are explained in the Chapters that deal with nonconservative systems. In such systems the possible lossof stability may occur on account of growing dynamic oscillations, a phenomenom called flutter when theconconservative loads are due to aerodynamic effects. .

Remark 5.3. If ! is an applied load multiplier, a limit point associated with a maximum or a minimum, suchas L1 and L2 in Figure 5.1(a) is called a snap-through point by structural engineers. The reason is that, if theload is kept constant, the structure “snaps” dynamically to another equilibrium position. The term collapseapplies to critical points beyond which the structure becomes useless. .

Remark 5.4. As an isolated limit point is approached, the incremental velocity vector v tends to becomeparallel to zcr whereas its magnitude goes to". If v is normalized (for example, to length one), then

v|v|

# zcr . (5.14)

Therefore the normalized vmay be a good null eigenvector estimate ifK has been factored near the limit point.(This is just a restatement of the well known inverse iteration process [411] for finding eigenvectors.) .

Remark 5.5. The set of control parameters for which detK = 0 while r(u, !) = 0 is sometimes called thebifurcation set in the applied mathematics literature. The name is misleading, however, in that the set mayinclude limit points; the name critical set would be more appropriate. .

5–6

5–7 §5.4 LIMIT POINT SENSORS

! L1

L2

B2

1B

u1u2Figure 5.2. The equilibrium path of Figure 5.2 shown in the 3D space (u1, u2, !). This is the typeof response exhibited by a uniformly pressurized deep arch, for which u1 and u2 are amplitudes of the

symmetric and antisymmetric deformation mode, respectively, and ! is a pressure multiplier.

Remark 5.6. Showing bifurcation points of a 2-DOF system on the ! versus u1 plane as in Figure 5.1may be misleading, as it conceals the phenomenon of transition from one mode of deformation to another.Figure 5.2 provides a more realistic picture. This diagram shows the classical bifurcation behavior for asymmetrically loaded shallow arch. Here u1 and u2 measure amplitude of symmetric and antisymmetricdisplacement shapes, respectively. At B1 the arch, which had been deforming symmetrically, takes off alongan antisymmetric deformation mode; at B2 the latter disappears and the arch rejoins the symmetric path. .

Remark 5.7. Physically the distinction between the two types of critical points is not so marked, inasmuch asimperfect structures display limit-point behavior. A bifurcation point may be viewed as the limit of a sequenceof critical points of limit type, realized as the structure strives towards mathematical perfection. The exampleworked out in §5.5.3 clearly illustrates that point; see Figure 5.8. .

§5.4. Limit Point SensorsScalar estimates of the overall stiffness of the structure as the control parameter varies are usefulas limit points sensors. The following estimator is based on the Rayleigh quotient approximationto the fundamental eigenvalue of K:

kx = xTKxxT x

, (5.15)

where x is an arbitrary nonnull vector, and K is evaluated at an equilibrium position u(!). An“equilibrium-path stiffness” estimator is obtained by taking x to be v = K!1q, in which case

k = kv = qT vvT v

. (5.16)

This value of course depends on !. It is convenient in practice to work with the dimensionless ratio

" = k(!)/k(0), (5.17)

This ratio takes the value 1 at the start of an analysis stage, and goes to zero as a limit point isapproached. A stiffness estimator with this behavior (although computed in a different way) wasintroduced by Bergan and coworkers [460,461] under the name current stiffness parameter. Itshould be noted, however, that no estimator of this type can reliably predict the occurrence of abifurcation point. Sensors for such points are described later in the context of augmented equations.

5–7


−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

State parameter µ

Con

trol

par

amet

er λ

State parameter µ

Con

trol

par

amet

er λ

R

L1

B1

B2

T1

V2

V1

L2

T2

(a) (b)

0.04

0

0

0.080.15

0.15

0.30

0.30

0.50

0.50

1.00 2.00

−0.04−0.08

−0.15

−0.15

−0.30

−0.30

−0.50

−0.50

−1.00−2.00

Figure 5.3. The Circle Game example: (a) Equilibrium paths in control-state space displayinginteresting points; (b) Incremental flow; numbers annotated on paths are values of rc .

§5.5. Critical Point Computation Examples

This section goes over the computation of critical points and associated attributes, such as stabilitytransitions. Two problems with one degree of freedom (DOF) are used. The first example isartificial, only used to simultaneously illustrate all types of special points (except fracture). Thesecond example pertains to a real but highly idealized structure, and brings attention to the effect ofimperfections. All calculations are carried out in closed form. No FEM discretization is required.

§5.5.1. The Circle Game

The first example assumes the following total residual function

r(µ, !) = (! ! µ) (!2 + µ2 ! 1) = 0, (5.18)

in which µ is a dimensionless state parameter and ! a dimensionless control parameter. The non-separable residual (5.18) is wholly artificial: no real structure produces it.2 It is useful, however,in illustrating several types of special points using simple diagrams. Plainly the equilibrium pathsassociated with (5.18) are: (i) the fundamental path ! = µ that passes through the origin referencepoint, and (ii) the unit-radius circle !2 + µ2 = 1, which forms a secondary path. See Figure 5.3(a).The following interesting points are marked there:

One reference point R, at ! = µ = 0.Two bifurcation points B1 and B2, at ! = ±1/

"2, µ = ±1/

"2.

Two limit points L1 and L2, at ! = ±1, µ = 0.Two turning points T1 and T2, at ! = 0, µ = ±1.Two non-equilibrium vortex points V1 and V2, at ! = ±1/

"6, µ = #1/

"6.

The last type (vortex point) has not been introduced before.3 Those points are characterized below.Unlike the other seven points, V1 and V2 do not lie on an equilibrium path, but may be reached viaperturbed residuals rc = ±4/(3

"6) = ±0.544331.

2 To get a roughly similar response associated with a real structural system, at least two DOF are required.3 They are also called center points in the literature of nonlinear systems.

5–8

5–9 §5.5 CRITICAL POINT COMPUTATION EXAMPLES

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

State parameter µ

Con

trol

par

amet

er λ

R

L1

B1

B2

T1

L2

T2

(a)

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

State parameter µ

Con

trol

par

amet

er λ

R

L1B1

B2

T1

L2

T2

(b)

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

State parameter µ

Con

trol

par

amet

er λ

R

L1

S1

S2

B1

B2

T1

L2

T2

(c)

K >0Stable

K<0Unstable

K = 0Neutral

K = 0q = 0

q=0 & v=0

v=q/K=0/0

q > 0

q< 0

Figure 5.4. Circle Game example, with equilibrium paths shown as dashed lines: (a) Sign regions for K ; (b) Signregions for q; (c) Combination of (a) and (b) that displays both K = 0 and q = 0 ellipses to give sign of v = q/K .

Figure 5.3(b) shows the incremental flow r = rc drawn for several sample values of rc. Theequation r(µ, !) = rc is cubic in both µ and !, and its closed form solution for either µ or ! givesthree roots. For example, the solution ! = !(µ, rc) obtained by Mathematica can be stated as

A1 = 2µ2 ! 3, A2 = 2 3"2 A1, A3 = 27 rc ! 18µ + 20µ3,

A4 =!

4 A31 + (27 rc ! 18µ + 20µ3)2, A5 = 3"

A3 + A4,

B1 = 3"4 A5, B2 = A2/A5, C1 = 1+ j"3, C2 = 1! j

"3, with j =

"!1),

!1 = 16 (2µ ! B2 + B1), !2 = 1

12 (4µ + C1 B2 ! C2 B1), !3 = 112 (4µ + C2 B2 ! C1 B1).

(5.19)For specifiedµ and rc, (5.19) gives 1 or 3 real roots, as can be graphically observed in Figure 5.3(b).The vortex points V1 and V2 are characterized by neighboring closed orbits of constant rc.The tangent stiffness matrixK, incremental load vector q and incremental velocity vector v reduceto scalars K , q and v, respectively, which are obtained as

K = "r"µ

= 1! !2 + 2!µ ! 3µ2, q = ! "r"!

= 1! 3!2 + 2!µ ! µ2, v = qK

.

Signs taken by K , q , and v over the {!, µ} plane are shown in Figure 5.4(a,b,c), respectively.The {!, µ} region where K > 0 is called stable, whereas that where K < 0 is unstable.4 Thetransition locus K = 0, which lies on the ellipse !2 ! 2!µ + 3µ2 = 1 highlighted in Figure 5.4(a),is labelled neutrally stable or simply neutral. Note that transition from stability to instability alongan equilibrium path occurs at the critical points B1, B2, L1 and L2. As noted previously in (5.1),this is a general property of a conservative system in static equilibrium.Figure 5.4(b) displays sign regions for the incremental load q . This value vanishes over the ellipse3!2 ! 2!µ + µ2 = 1. This ellipse is tilted with respect to the zero-stiffness one. It passes throughthe bifurcation and turning points, but not through the limit points.

4 Those conclusions are justified in a later chapter that specifically deals with stability.

5–9


A

B

(a)k

A'

A

B

P = λ kL(b) u = L sin θA

rigid

L

θ

CC'

spring stayshorizontal ascolumn tilts L

L cos θ

Figure 5.5. Geometrically exact analysis of a perfect propped rigid cantilever (PRC) column with extensionalspring remaining horizontal: (a) untilted, unloaded column; (b) tilted column after buckling.

The ellipses of Figures 5.4(a,b) insersect at four points: the two bifurcation points B1 and B2,and the two vortex points V1 and V2, as pictured in Figure 5.4(c). The bifurcation points lie onequilibrium paths but the vortex points do not. At those four points both K and q vanish, whencethe incremental velocity v = q/K takes on the indeterminate form 0/0.The conditions stated in §5.3 to distingish limit and bifurcation points can be easily checked in thisexample. Since K is a scalar, stiffness singularity means K = 0. The null eigenvector, normalizedto unit length, has only one entry: zcr = 1. Thus the indicator zT

cr qcr reduces to qcr , which is qevaluated at a critical point. From inspection of Figure 5.4(b) it is plain that qcr = 0 at B1 and B2whereas qcr != 0 at L1 and L2. This corroborates the rules stated in (5.12) and (5.13)

§5.5.2. Perfect Propped Rigid Cantilever Column

The second example considers the configuration shown in Figure 5.5(a). A rigid strut AB of lengthL is hinged at B and supports a downward vertical load P at tip A. The load remains vertical asthe column tilts. The column is propped by an extensional spring of stiffness k attached to A. Thisconfiguration will be called a propped cantilevered rigid column; or PCR column for short.Since the column is rigid, there is only one DOF. Two convenient choices for it are: the tilt angle! , or the tip horizontal displacement: u A = L cos ! . We select the latter as state parameter, butrender it dimensionless on dividing by L: µ = u A/L = sin ! . For convenience, the single controlparameter " is defined from P = " k L , which also makes " dimensionless.For a geometrically exact analysis is it important to know what happens to the extensional spring asthe column tilts. The simplest assumption is that it remains horizontal, as pictured in Figure 5.5(b).The spring force k u A is then horizontal and points to the left if u A > 0. Doing a FBD at thedisplaced tip position A" and taking moments with respect to B yields the equilibrium condition

P L sin ! = k u A L(1 # cos !), (5.20)

which can be transformed to " k L2 sin ! = k L2 sin ! (1 # cos !). Cancelling out k L2, replacing

5–10


B

State parameter µState parameter µ

Con

trol

par

amet

er λ

Stiff

ness

coe

ffici

ent K

−1 −0.5 0 0.5 1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

B

R T1T2−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

1.2

1.4

Unstable Unstable

Stable

Unstable(a) (b)

λcr

Figure 5.6. Response of perfect PRC column: (a) Response plot ! = !(µ) showing two equilibriumpaths interfection at B; (b) Stiffness coefficient K = K (µ) for unstable secondary path.

µ = sin " , and converting to total residual yields5

r(!, µ) = µ ! !µ

!

1! µ2. (5.21)

This can be also be derived as the gradient r = #$/#µ of the total potential energy function

$(!, µ) = 12 µ2 + !

!

1! µ2. (5.22)

The two equilibrium solutions provided by r = 0 are

µ = 0 for any !, ! =!

1! µ2. (5.23)

These solutions yield the vertical (untilted) and tilted column equilibrium paths, respectively. Theseare the primary and secondary paths shown in Figure 5.6(a). The secondary path falls on the unitcircle !2 + µ2 = 1. For ! > 0 the two paths intersect at µ = 0 and ! = !cr = 1, which is abifurcation point. The secondary path exhibits two turning points at ! = 0, µ = ±1.As in the Circle Game example, the tangent stiffness matrix K, incremental load vector q andincremental velocity vector v reduce to scalars K , q and v, respectively, which are given by

K (µ) = #r#µ

= ! µ2

1! µ2, q(µ) = ! #r

#!= µ

!

1! µ2, v(µ) = q

K= !

!

1! µ2

µ. (5.24)

In terms of the tilt angle, K = ! tan2 " , q = tan " and v = ! cot " . The K and v given above,however, are only valid for the secondary path. On the primary path µ = 0, K " ±# andv " 0/0. The stiffness coefficient for the secondary branch is plotted in Figure 5.6(b). SinceK < 0 except at µ = 0, the entire path is unstable. In fact, of the four branches that emanatefrom B, only one (the µ = 0 primary path for ! < !cr ) is stable. After buckling the tilting columnsupports only a decreasing load, which vanishes at the turning points. Consequently this structuralconfiguration is poor from the standpoint of post-buckling safety.

5 It is important not to cancel out the sin " on both sides, as otherwise the primary path µ = sin " = 0 would be lost.

5–11


B

(a)k

A'AA

B

(b)

θ

CC'

εL

InitialImperfection

θ0θ0

A0A0

L

rigid

spring stayshorizontal ascolumn tilts

P = λ kLu = L sin θA

L cos θL

Figure 5.7. Geometrically exact analysis of a imperfect propped rigid cantilever (PRC) column with extensionalspring remaining horizontal: (a) untilted, unloaded column; (b) tilted loaded column.

§5.5.3. Imperfect Propped Rigid Cantilever Column

We now consider a variation of the previous problem, in which the PRC column exhibits an initialgeometric imperfection, as shown in Figure 5.7(a). When the column is unloaded, it tilts by anangle !0 or, equivalent, a tip horizontal displacement L " = L sin ! . We shall take " = sin ! asmeasure of initial imperfection. The state parameter µ = u A/L and control parameter # = P/(kL)

are defined as before. The total residual (5.21) changes to

r(#, µ, ") = µ ! " ! #µ

!

1 ! µ2. (5.25)

This is the gradient of the total potential energy function

$(#, µ, ") = 12 (µ ! ")2 + #

!

1 ! µ2. (5.26)

which reduces to (5.22) if " = 0. The equilibrium path satisfying r = 0 is given by

# = (1 ! "

µ)!

1 ! µ2. (5.27)

If " "= 0, this solution gives two branches separated by the # axis, as pictured in Figure 5.8(a).(The formal limit " # 0 gives only the secondary path # =

!

1 ! µ2 of the perfect column.) Theprevious formulas for K and v given in (5.24) change to

K (µ, ") = %r%µ

= " ! µ3

µ(1 ! µ2), v(µ, ") = q

K= !µ2

!

1 ! µ2

µ3 ! ", (5.28)

whereas q remains the same. Figure 5.8(a) plots the equilibrium paths given by (5.27) for selectedvalues of ". The resemblance of this picture to an incremental flow is not accidental. If the flow isproduced by setting the residual (5.21) of the perfect column to rc = ", the residual (5.25) of theimperfect column is obtained.

5–12


−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

1.2

1.4

−1 −0.5 0 0.5 1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

State parameter µState parameter µ

Con

trol

par

amet

er λ

Stiff

ness

coe

ffici

ent K

0.01

0.01

−0.01

−0.01

0.02

0.02

0.05

0.05

0.1

0.1

0.2

0.2

0.5

0.51.0

−0.02

−0.02

−0.05

−0.05

−0.1

−0.1

−0.2

−0.2

−0.5

−0.5−1.0

0.01

0.01

−0.01−0.01

0.02

0.02

0.05

0.05

0.1

0.1

0.2

0.20.5

−0.02

−0.02

−0.05

−0.05

−0.1

−0.1

−0.2

−0.2−0.5

(a) (b)

0 −1

Figure 5.8. Response of imperfect PRC column: (a) Load-deflection response for sample values of !; (b) Stiffnesscoefficient versus state parameter for those values. Sample values of ! are annotated near the corresponding curves.

−1 −0.5 00 0

0.5 1

0.2

0.4

0.6

0.8

1

1.2

1.4

State parameter µ Imperfection parameter ε

Con

trol

par

amet

er λ

Cri

tical

load

par

amet

er λ

Unstable Unstable

Stable

(a) (b)

−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1.2

1.4

−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

1.2

1.4

λ(µ)=(1−µ )2 3/2

λ(ε)=(1−ε )3/2 2/3

Figure 5.9. More detailed display of imperfection effects: (a) critical point locus separatingstable and unstable regions; (b) imperfection sensitivity diagram, which displays decreasing

load capacity as the imperfection parameter grows.

Inspection of Figure 5.8(a) shows that if ! != 0 the bifurcation point disappears. If so the responseconsists of two paths, which do not intersect:• A primary path that takes off from the unloaded but imperfect configuration " = 0 µ = !.

This path exhibits a limit point. Those points are marked by circles in Figure 5.8(a).• A secondary path located in the opposite half-plane. This path has no critical points.The two paths are separated by the perfect column equilibrium paths (5.23). The primary path limitpoints collectively represent a set of critical load values, which lie on the curve

"cr (µ) = (!

1 " µ2)3 = cos3/2 # . (5.29)

This is called a critical point locus, which separates the stable and unstable regions, as can besurmised from the stiffness coefficient plotted in Figure 5.8(b). This separation is shown in further

5–13


detail in Figure 5.9(a). Eliminating µ in (5.29) in favor of ! yields the imperfection sensitivitydiagram

"cr (!) = (1! !2/3)3/2 = 1! 32!2/3 + 3

8!4/3 + 1

16!2 + . . . (5.30)

This expression is plotted in Figure 5.9(b). From the Taylor series given above one can see that thiscurve has a vertical tangent as ! " 0. This highlights that this structural configuration exhibitshigh sensitivity of load capacity to small initial imperfections.

§5.6. *Turning PointsTurning points are regular points at which the tangent is parallel to the " axis so that v = 0. The unit tangenttakes the form

tu =! 0

±1

"

. (5.31)

Although these points generally do not have physical meaning, they can cause special problems in path-following solution procedures because of “turnback” effects.To detect the vicinity of a turning point one can check the two mathematical conditions: v becomes orthogonalto q and u tends to zero faster than q. For example:

| cos(v,q)| < #, |$| > $min, (5.32)

where $ is the current stiffness parameter. Typical values may be # = 0.01, $min = 100.


Overall there is a huge literature on critical points, and their application to stability. That pile is fractured,however, according to applications as well as communities.As regards structural stability, the classic reference isTimoshenko [378], first published in 1936. It collectsmosteverything known on the subject until then. Focus is on linearized bifurcation buckling. Nonlinear stabilityand limit points receive scant attention, as such calculations were considered too demanding given the absenceof computers. Timoshenko’s problem-by-problem, example-focused approach approach has influenced manybooks on structural stability since. Among them the textbook by Bazant and Cedolin [458] stands out by itscomprehensive coverage that includes plasticity, creep, dynamics, localization and fracture. The example-driven monograph by Panovko and Gubanova [637], translated from the Russian, is less ambitious but makesdelightful reading.Stronger from a computational viewpoint is the textbook by Brush and Almroth [472]. The monograph byBushnell [475] has more physics (e.g., temperature, plasticity and creep effects) and a wider selection ofpractical problems. The book by Seydel [599] has a nice description of computational methods although it isoriented to chemical engineering problems.Before computers came olong, perturbation methods were popular for treating complicated problems in elasticstability. The monographs by Thompson and Hunt [614,616] dwelve on the topic.The connection between potential-based structural stability and “catastrophe theory” is presented in a readablemanner by Poston and Steward [576] and Thompson [615]. That theory began as a serious effort to systematizesingular behavior of potential-driven systems and ended as a joke after outlandish claims of application to thenatural and social sciences. But, like chaos and fractals, it had its 15 minutes of fame.

5–14

5–15 Exercises

Homework Exercise for Chapter 5

Critical Points and Related Properties

EXERCISE 5.1 Given the one-parameter, two-degree-of-freedom residual-force system

r(u1, u2, !) =! r1r2

"

=! 6u1 ! 2u2 ! u21 ! 12!

!2u1 + 4u2 ! u22 + 2!

"

(E5.1)

Consider the point P(u1, u2, !) located at

u1 = 2, u2 = 1, ! = 12 , (E5.2)

(a) Show that P is on an equilibrium path,(b) Show that P is a critical point,(c) Determine whether it is a limit or a bifurcation point. [Compute the null eigenvector z ofK at that point].(d) Verify whether the limit point sensor " is zero at P .

EXERCISE 5.2 Show that all critical points of (E5.1) satisfy either of the equations

63! u1 ! 36u2 = 0, 5! 2u1 ! 3u2 + u1u2 = 0 (E5.3)

called critical point surfaces, and that the only intersection of these surfaces and the equilibrium path is at(E5.2).

EXERCISE 5.3 Show that the critical point surface defined by det(K) = 0 is independent of ! if the residualforce system is separable.

EXERCISE 5.4 Show that qT z is independent of ! if the residual force system is separable and the load isproportional.

EXERCISE 5.5 (Advanced, requires knowledge of matrix eigensystem theory). If K is not symmetric, thecritical point classification argument based on qT z fails. Explain why.

5–15

.

6Conservative

Systems

6–1

Chapter 6: CONSERVATIVE SYSTEMS 6–2

TABLE OF CONTENTS

Page§6.1. Introduction 6–3§6.2. Work and Work Functions 6–3

§6.2.1. Concept Of Work . . . . . . . . . . . . . . . . 6–3§6.2.2. Power . . . . . . . . . . . . . . . . . . . 6–4§6.2.3. Conservative Forces and Potential Energy . . . . . . . . 6–4§6.2.4. Work Under Multiple Forces . . . . . . . . . . . . 6–5§6.2.5. Work in a Force Field . . . . . . . . . . . . . . . 6–5§6.2.6. Work Associated With Multiple Points . . . . . . . . 6–6§6.2.7. Separation Into Internal and External . . . . . . . . . 6–6

§6.3. Force Residual of Conservative Systems 6–6§6.3.1. Advantages of Energy Derivation . . . . . . . . . . 6–7

§6.4. Construction of Work Functions 6–8§6.4.1. Point Loads . . . . . . . . . . . . . . . . . . 6–8§6.4.2. Distributed Dead Loads . . . . . . . . . . . . . 6–9§6.4.3. The Internal Energy: A Linear Spring . . . . . . . . . 6–10§6.4.4. The Internal Energy: How Geometric Nonlinearities Arise . . 6–10

§6.5. Internal Energy: Additivity Property 6–12§6.6. *Derivatives of Energy Functions 6–12§6.7. *Energy Increments 6–13§6. Exercises . . . . . . . . . . . . . . . . . . . . . . 6–15

6–2

6–3 §6.2 WORK AND WORK FUNCTIONS

§6.1. Introduction

This Chapter treats topics pertaining to energy methods for conservative systems in a more detailedmanner. The concepts of work and energy are introduced, along with work and potential functions.This is followed by several examples involving very simple conservative systems, which show, instep by step fashion, how to construct work functions. The last example of this series illustrateshow geometric nonlinearities naturally arise when large motions are considered.The Chapter concludes with some mathematical derivations that will be of use in Chapters devotedto incremental solution methods. Such derivations are considered advanced material. This meansthat are covered during the course, although key results (for example, amplification matrices) canbe quated and applied in later Chapters.

§6.2. Work and Work Functions

The following definition appears in scienceworld.wolfram.com. A conservative system is one inwhich work done by a force, or set of forces, is1. Independent of path.2. Equal to the difference between the final and initial values of an energy function.3. Completely reversible.These three definitions can be shown to be mathematically equivalent,1 and are elucidated in thefollowing subsections.

Remark 6.1. The two most notable conservative systems are gravitational and electrical fields. For example,in the case of a uniform gravity field, the gravitational potential energy acquired or lost by a mass dependsonly on the difference between heights, and not on the path taken to get from one state to the other.

§6.2.1. Concept Of Work

The concept of mechanical work can be brought up as follows. Mechanical force f, located atposition coordinate x, displaces an infinitesimal distance du, to x + du. See Figure 6.1(a). TheCartesian components of f are fx , fy, fz , whereas those of du are {dux , duy, duz}. The workdifferential is defined as the inner product

dW = f T du = fx dux + fy duy + fz duz, (6.1)

in which W denotes work.2 From the definition it is obvious that mechanical work is a scalarwith physical dimension of force times length. Since forces add vectorially, work differentials addalgebraically. If f is the resultant of, say, f1, f2, and f3, then

dW = f T du = (f1 + f2 + f3)T du = fT1 du+ fT2 du+ fT3 du = dW1 + dW2 + dW3. (6.2)

1 The Wikipedia definition is similar: “A force is conservative if the work done by a particle between two points isindependent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done is zero.” The Wikidefinition has a Physics flavor as it focus on one particle; for engineering systems like structures, replace “particle” by“system,” “travel” by “motion” and “point” by “state.”

2 In the differential-of-work definition (6.1), du could be substituted by dx. Use of the displacement, however, fits betterwithin the usual nomenclature of the Finite Element Method.

6–3


P(x)P(x)

A(x )A

BB(x )

(a)

A

(b)

x y

z

du =dududu

xyz du

xyf =fffz

f

Position vector x

Figure 6.1. Concept of work: (a) work of a force along a path joining two points A and B;(b) work along a closed path that returns to A.

a property used later in §6.2.4. Power is the rate of work, i.e., work spent per unit time (or pseudotimein this course). Consequently its differential is

dP = dWdt

= f T du. (6.3)

The work spent by moving force f between two spatial positions: initial at xA and final at xB =xA + uBA, can be calculated by the path integral

WBA =! xB

xA

dW =! xA+uBA

xA

dW, (6.4)

carried out over a path that joins points A and B, as pictured in Figure 6.1(a). Here the displacementuBA = xB ! xA is no longer infinitesimal. An important question is whether WBA can be expressedas the difference of values taken by a work potential function (or simply work potential) W (x) atthe initial and final positions:

WBA = W (xB) ! W (xA) = W (xA + uBA) ! W (xA), (6.5)

independently of the xA " xB path taken. A condition for this to occur is discussed in the nextsubsection.

§6.2.2. PowerPower is the rate of work, i.e., work spent per unit time (or pseudotime in this course). Consequentlyits differential is

dP = dWdt

= f T du. (6.6)

The total power spent is obtained by the line integral of P .

6–4

6–5 §6.2 WORK AND WORK FUNCTIONS

§6.2.3. Conservative Forces and Potential Energy

A force f = f(x) that is a function of position x, but independent of time, is called conservative ifit is the gradient of a work potential function W (x) ! W (u):

f = "W =! !W

!x!W!y

!W!z

"T =#

!W!ux

!W!uy

!W!uz

$T. (6.7)

in which "W denotes the gradient (a column vector) of W . If (6.7) holds, the differential of workdW as defined in (6.1) is

dW = ("W )T du = !W!ux

dux + !W!uy

duy + !W!uz

duz . (6.8)

This is now an exact differential, and consequently the path integral (6.4) evaluates to (6.5) inde-pendently of the xA # xB path. If a work potential exists, the negated function

V = $W, (6.9)

is called the potential energy function or simply potential energy, associated with the force f =$"V . The name arises from the fact that energy is the capacity to do work. If WBA = W (xB) $W (xA) is negative, VBA = V (xB) $ V (xA) = $WBA is positive, so energy has been stored whenmoving the force from A to B. If B and A coalesce so that the path returns to A, as pictured inFigure 6.1(b), the net work spent, or energy stored, is zero.Any force that cannot be expressed as (6.7) is called nonconservative.What happens if the gradient (6.7) vanishes? Then f = 0. If the system consists of just the particleat point P , it is in static equilibrium. The generalization to an arbitrary conservative system linksthe total force residual r to the gradient of a total potential energy, as covered in §6.3.If all forces acting on a mechanical system are conservative, the system is called conservative,and nonconservative otherwise. Often the acting forces can be decomposed into conservative andnonconservative, as discussed in the last Chapters.The foregoing derivations, which consider a point P moving along a path, and acted upon by apoint force, can be generalized in several ways as discussed next.

§6.2.4. Work Under Multiple Forces

Suppose that point P in Figure 6.1(a) is acted upon by several concurrent forces, say f1, f2 and f3.Then the work differential can be expanded as in (6.2), which is reproduced for convenience:

dW = f T1 du+ f T2 du+ f3 T du = (f T1 + f T2 + f3 T ) du = f T du. (6.10)

Here f is the vectorial resultant f = f1 + f2 + f3. It follows that multiple forces acting at one pointmay be replaced by their resultant without modifying the work function. This is a consequence ofthe additivity of energy, a property discussed in §6.5 in more detail.

6–5


§6.2.5. Work in a Force FieldA force-per-unit-volume (force density) that depends on position is called a field. Well knownexamples are gravity, centrifugal and electrostatic fields. In this case the force f acting on a body isobtained by integrating the force field over the volume of the body. For example, consider a gravityfield of constant magnitude g directed along the!z axis. Suppose that a body in this field occupiesthe domain ! and has constant mass density ". Then the acting force in terms of components is

f =! 0

0"

!" (!g) d!

#

=! 0

0!" g

"

!d!

#

=! 0

0!" g V

#

, (6.11)

in which V ="

!d! denotes the volume measure. A similar scheme can be used for position-

dependent fields, although integrals can become more complicated. The force field is called con-servative if the integrated forces over an arbitrary body are.

§6.2.6. Work Associated With Multiple PointsThis scenario arises when a continuous system is discretized into a model with arbitrary numberof degrees of fredom (DOF) assigned to multiple locations. If the discretization is done by theFinite Element method (FEM) those locations are called nodes. The assignment of forces tonodes is carried out by techniques collectively known as force lumping. These are covered in theintroductory FEM course [120]. The total work differential is then computed by adding up nodalcontributions.

§6.2.7. Separation Into Internal and ExternalThis generalization is the most important one for conservative systems that involve flexible bodies.Work is separated into two components, which together form the total potential energy functiondefined as

# = U ! W. (6.12)HereU is the internal energy or stored energy whereas W is the work potential function of appliedor external forces, introduced in §6.2.3. The internal energy is that stored in the body as a result ofits deformation under stresses, which ultimately (at the molecular scale) resolves into interatomicforces.3 This separation is convenient for various reasons, both physical and computational, whichare explained later.Frequently asked question: why doU andW appear with different signs in (6.12)? Physical reason:U is an energy, that is, the capacity to do work, whereas W is work spent. So (6.12) expressesenergy balance: internal energy balances external work. If W is replaced by !V as per (6.9), #takes the form

# = U + V . (6.13)This states that the total energy of the system is the sum of the internal and external energies.Although (6.13) is perhaps easier to comprehend, it is not popular in FEM formulations becausethe idea of force balance takes precedence there. It is used more frequently in Lagrange andHamiltonian dynamics to express total-energy conservation laws.

3 Other components of internal energy, for example those associated with thermomechanical, electromagnetic or chemicalreaction effects, are not considered here.

6–6

6–7 §6.3 FORCE RESIDUAL OF CONSERVATIVE SYSTEMS

§6.3. Force Residual of Conservative Systems

The concepts and generalizations so far discussed in §6.2 justify the linkage of force residuals toenergy functions. This connection was introduced in previous Chapters as recipes, but can nowbe restated in a more systematic form. The total residual force vector r may be expressed as thegradient of the total potential energy ! with respect to the state vector. For the case of mulitplecontrol parameters covered in Chapter 3:

r(u,!) = "!(u,!)

"u, (6.14)

whereas for the case of a single control parameter (the staging parameter), introduced in Chapter 4:

r(u, #) = "!(u, #)

"u, (6.15)

Furthermore, the energy decomposition ! = U ! W and the force decomposition r = p ! f arerelated in the sense that

p = "U"u

, f = "W"u

. (6.16)

in which p and f are the internal and external forces, respectively. As discussed in ? U is theinternal energy— which reduces to the strain energy in the problems considered in this course—and W is the external work function.The force equilibrium equations r = 0 or f = p express the fact that the total potential energy isstationary with respect to variations of the state vector when the structure is in static equilibrium.Mathematically:

$! = rT $u =!

"!

"u

"T

$u = 0. (6.17)

where $u denotes a virtual displacement, $ being the variation symbol. Since $u is arbitrary, (6.17)implies that r = 0.

§6.3.1. Advantages of Energy DerivationIf the structural system is conservative there are several advantages in taking advantage of thatproperty:(1) If discrete force equilibrium equations are worked out by hand (either for complete structures

or finite elements) derivation from a potential is usually simpler than direct use of equilibrium,because differentiation is a straightforward and less error prone operation, especially as regardssigns. Exercise 6.3 gives an example of this.

(2) The transformation of residual equations to different coordinate systems is simplified becauseof the invariance properties of energy functions.

(3) The conventional finite element discretization method relies on the availability of an internalenergy functional.

(4) The tangent stiffness matrix is guaranteed to be symmetric. Consequently equation solvers(and eigensolvers) can take advantage of this property.

6–7


Deformed

Undeformed

Fu

No change in loadmagnitude or direction

Figure 6.2. Structure under concentrated (point) dead load F

(5) Loss of stability can be assessed by the singular stiffness criterion introduced in Chapter 5,which is static in nature. If the system is nonconservative, loss of stability may have to betested by a dynamic criterion, which is more error-prone and computationally expensive.

§6.4. Construction of Work Functions

The concept of load potential is the easiest to understand. This function, called W , is the potentialof the work done by the applied or prescribed forces working on the displacements of the pointson which those forces act. The negative of this function, V = !W is called the external potentialfunction, but in the present course we shall primarily use W .Next we illustrate how to buildW for systemswith finite degrees of freedom. The presentation is notgeneral in nature but relies on a few simple examples complemented with exercises. The material isintended to serve as a “bridge” to the formulation of geometrically nonlinear finite elements, whichstarts in Chapter 8.

§6.4.1. Point Loads

For a concrete example, consider a structure loaded by a single point force F that does not changein magnitude or direction as the structure displaces. See Figure 6.2. A force with these propertiesis called a dead load.If u is the deflection of the point of application of F in the direction of the force, then the workperformed is obviously Fu. Consequently,

W = Fu. (6.18)

If the structure is subjected to n loads Fk (k = 1, . . . n) and the corresponding deflections in thedirection of the forces are called uk , then

W =n

!

i=1Fkuk . (6.19)

In general these forces will be defined by their three components along the axes x, y, z and are moreproperly represented by vectors fk . For example, if at location k = 3 we have a force F3 acting in

6–8

6–9 §6.4 CONSTRUCTION OF WORK FUNCTIONS

Undeformed

Deformed A By

y

x

y

f (x)

u (x)

Figure 6.3. Structure under under line load fy(x) (directed upward) over segment AB.

the y-direction,

f3 =! 0

F30

"

. (6.20)

Likewise, the displacement of points of application of fk is denoted by vector uk . The vectorgeneralization of (6.19) is the sum of n inner products:

W =n

#

k=1fTk uk . (6.21)

Finally, if all applied force components are collected in the external force vector f (augmented withzero entries as necessary to be in one-to-one correspondence with the state vector u) then we havethe compact inner-product expression

W = f Tu. (6.22)

§6.4.2. Distributed Dead Loads

For distributed forces invariant in magnitude and direction, a spatial integration process is necessaryto obtain P . These forces may include line loads, surface loads or volume loads (body forces).For example, consider the structure of Figure ?, on which a dead line load fy(x) acts in the ydirection along segment AB of the x axis. Then

P =$ xB

xA

fy(x) uy(x) dx, (6.23)

where uy(x) is the y-displacement component of points on segment (A,B). A similar techniquecan be used for volume (body) forces as illustrated in Exercise 6.1.

6–9


δk x

21Figure 6.4. Linear spring of stiffness k deforming along its axis.

Remark 6.2. Substantial mathematical complications arise if some forces are functions of the displacements.For example, in slender structures under aerodynamic pressure loads the change of direction of the forces asthe structure deflects may have to be considered in the stability analysis. These so-called “follower” forces,which introduce force B.C. nonlinearities, are considered later in the course. Suffices to say here that no loadspotential W generally exist in such cases and the system is nonconservative.

§6.4.3. The Internal Energy: A Linear Spring

The internal energy, called U , is the recoverable mechanical work “stored” in the material of thestructure by virtue of its elastic deformation. When this work is expressed in terms of strains andstresses, as in following Chapters, it is called the strain energy. Note that only flexible bodies canstore strain energy; a rigid body cannot.We shall illustrate the internal energy concept here by considering the simplest of all structuralelements already encountered in linear finite element analysis: a linear spring of stiffness k,illustrated in Figure 6.4. Generalization to more complicated structures and structural componentswill be made in subsequent Chapters.

If the spring is undeformed, its internal energy U can be conventionally taken as zero (because anenergy function can be adjusted by an arbitrary constant without changing its gradients). Now letthe spring deform slowly (to avoid inertial effects) such that its two ends separate by a distance! called the elongation. The internal spring force f for an intermediate elongation 0 ! ! ! ! isf = k!. An elementary result of mechanics is that the strain energy taken up by the spring in itsdeformed state is

U =! !

0spring-force" d(elongation) =

! !

0(k!) d ! = 1

2k!2. (6.24)

Suppose that the spring is fixed at end 1 and that end 2 can move only along the x axis, as inFigure 6.4. Call u the x displacement of end 2. Then ! = u # 0 = u and the strain energy isU = 1

2ku2. According to (6.2) the internal force, which in this case is just the spring axial force p,

is the derivative of U with respect to u:

p = "U"u

= ku. (6.25)

This is linear in the displacement u so nothing has changed so far with respect to linear finiteelement analysis.

§6.4.4. The Internal Energy: How Geometric Nonlinearities Arise

Now suppose that the spring can move arbitrarily on the plane x , y, as depicted in Figure 6.5. Theposition of the deformed spring is completely defined by the four displacement components ux1,

6–10

6–11 §6.4 CONSTRUCTION OF WORK FUNCTIONS

x

y k

Deformed

Undeformed

1 11(x ,y )

2 22(x ,y )

u u

u

u

x1 y1

y2

x2

Figure 6.5. Linear spring of stiffness k displacing on the x, y plane.

uy1, ux2 and uy2, which we collect in the state vector

u =

!

"

#

ux1uy1ux2uy2

$

%

&. (6.26)

Let ! and !d denote the spring lengths in the undeformed and deformed configurations, respectively.The elongation " is given by

" = !d ! ! ='

(!x + #x )2 + (!y + #y)2 !'

!2x + !2y, (6.27)

in which#x = ux2 ! ux1,#y = uy2 ! uy1, !x = x2 ! x1, !y = y2 ! y1, in which x1, y1, x2 and y2denote the x , y coordinates of the end nodes of the undeformed spring. Consequently

U = 12k"2 = 1

2k(!2 + !2d ! 2!!d)

= 12k (2!2 + 2!x#x + #2

x + 2!y#y + #2y ! 2!

'

(!x + #x )2 + (!y + #y)2.(6.28)

The components of the internal forces are

p = $U$u

=

!

"

"

"

"

"

"

#

$U$ux1$U$uy1$U$ux2$U$uy2

$

%

%

%

%

%

%

&

. (6.29)

The actual expressions of the components in (6.29) which are nonlinear functions of the displace-ments, are worked out in Exercise 6.2.

6–11


The important points that emerge from this example are:

1. The internal forces are nonlinear functions of the displacements, al-though the spring itself remains constitutively linear. This nonlinearitycomes in as a result of geometric effects, and is thus properly calledgeometric nonlinearity.

2. The effect of geometric nonlinearities can be traced to the change indirection of the spring. Because if the spring stretches along its originalaxis the internal force remains linear in the displacements. This changeof direction is measured by rotations.

Even for this simple case the exact nonlinear equations are quite nasty, involving irrational functionsof the displacements. The second property, however, shows that approximations to the exactnonlinear equations may be made when the change in direction is “small” in some sense. Thisfeature is illustrated in Exercise 6.3.

§6.5. Internal Energy: Additivity Property

If the structure consists of m linear springs, each of which absorbs an internal energy Uk , the totalinternal energy is the sum of the individual spring energies:

U = U1 +U2 + . . . +Um . (6.30)

This additivity property is of course general because energies are scalar quantities. It applies toarbitrary structures decomposed into structural components such as finite elements. Furthermore,(6.30) is not affected by whether the structure is linear or nonlinear.The last property explains why finite element equations should be derived from energy functions ifsuch functions exist. That is not, however, always possible.

§6.6. *Derivatives of Energy FunctionsIf the residual r(u, !) is derivable from a total potential energy function "(u, !) as in (3.2), then the stiff-ness matrix and incremental load vector appear naturally as components of the following matrix of secondderivatives:

!

"

#2"#u#u

#2"#u #!

#

#2"#!#u

$T#2"#!#!

%

& =' K !q

!qT a

(

(6.31)

where a = #2"/#!2 has not been introduced previously. Obviously the tangent stiffnessmatrixK (theHessianof ") is now symmetric. Note also that

#q#u

= #3"

#u #!#u= #

#!

#2"

#u#u= #K

#!= K!, (6.32)

is a symmetric matrix.

6–12

6–13 §6.7 *ENERGY INCREMENTS

The complementary energy function !! may be defined from the dual Legendre transformation (see e.g.,Chapter 2.5 of Sewell’s book [598]) as

! + !! = ui"!

"ui= uT r = rTu. (6.33)

This gives!!(r, #) = rTu" ! with u eliminated from r(u, #) = 0, so now the residual forces are the activevariables. Obviously

u = "!!

"r, or ui = "!!

"ri. (6.34)

The matrix of second derivatives of !! is!

"

"2!!

"r"r"2!!

"r"##

"2!!

"#"r$T

"2!!

"#"#

%

& =' F vvT b

(

. (6.35)

These are linked to the quantities that appear in (6.31) by the matrix relations

F = K"1, v = K"1q = Fq, b = qTK"1q" a. (6.36)

The converse relations areK = F"1, q = Kv, a = vTKv" b. (6.37)

The tangent flexibility matrix F = K"1 (the Hessian of K ) is now symmetric. Note also that

"u"r

= "3!!

"r "#"r= "

"#

"2!!

"r"r= "F

"#= F#, (6.38)

is a symmetric matrix.

Remark 6.3. The following matrix appears (as amplification matrix) in the study of the stability of incremental methods:

A = "v"u

= "(Fq)"u

= "F"uq+ F "q

"u= "F

"rKq+ F "K

"#. (6.39)

AlthoughA is unsymmetric, under some general conditions it has real eigenvalues. To show that we expressA as the productof two symmetric matrices:

A = "v"u

= "v"r

"r"u

= "F"#K = F#K, (6.40)

where the relation (6.38) has been used. If F# is nonsingular, the eigensystem Axi = µixi can be transformed to thegeneralized symmetric eigenproblem

Kxi = µiF"1# xi . (6.41)

If K is positive definite this system has nonzero real roots µi . If F# is singular but K positive definite, consideration of thealternative eigensystem

F#yi = µiK"1yi = µiFyi , (6.42)shows that such a singularity contributes only zero roots.

Remark 6.4. Another quantity that appears in the analysis of incremental methods is the vector

v# = "v"#

= "v"u

"u"#

= Av = F#Kv = F#q. (6.43)

Remark 6.5. Two other Legendre transforms may be constructed: X ($,u) and Y ($, r), in which $ = "!/"# (a generalizeddisplacement if # is a load multiplier) is the active variable and either u or r take the role of passive variables. X and Ytogether with ! and K form a closed chain of Legendre transformations. The functions X and Y are, however, of limitedinterest in the present context.

6–13


§6.7. *Energy IncrementsIn this section we continue to assume that r is derivable from the potential ! = U ! W . For questions suchas positive path traversal it is interesting to obtain an expression of the energy increment on passing from anequilibrium position (u, ") to a neighboring configuration (u+ #u, " + #") on the equilibrium path:

#! = !(u+ #u, " + #") ! !(u, "). (6.44)

First we note that adding an arbitrary function of " to !

! + F("), (6.45)

does not change the equilibrium equations or rate forms. To second order in the increments we get

#! = rT#u+ A#" + 12#u

TK#u! qT#u#" + 12a(#")2, (6.46)

withA = $!

$", a = $2!

$"2, (6.47)

evaluated at (u, "). But we can always adjust F(") in (6.44) so that A = a = 0. Furthermore at an equilibriumposition r = 0, and along the equilibrium path #u = K!1q#" = uT#". Substituting we find for the energyincrement

#! = #U ! #W = 12q

Tu(#")2 ! qTu(#")2 = ! 12q

Tu(#")2. (6.48)

This formula displays the important function of the product qTu in the energy increment. By extension wemay call

#W = qTu(#")2 (6.49)

the external work increment even if r does not derive from a potential.To fix the ideas assume that r derives from a quadratic potential

! = 12u

TKu! qTu" + C" + D, (6.50)

where C and D are arbitrary constants. Then the increment #! from an equilibrium position (u, ") thatsatisfies the linear relation Ku = qT", to an arbitrary configuration (u+ #u, " + #") is

#! = #uT (Ku! qT") + #"(qTu! C) = !#"(qTu! C) = !(qT v" ! C)#". (6.51)

Since C is arbitrary, chose it so that $!/$" = !qTu+ C = 0. Then

#! = !qT v#( 12"2). (6.52)

6–14

6–15 Exercises

Homework Exercises for Chapter 6Conservative Systems

Note: the use of a symbolic algebra package, such asMathematica orMathCad, is recommended for Exercises6.3 and 6.4 to avoid tedious algebra and generate plots quickly. (There could be a gain from hours to minutes).

EXERCISE 6.1 [A:15] A body of volume V and density ! is in an uniform gravity field g acting along the!z axis. The body displaces to another position defined by the small-displacement field u(x, y, z). Find theexpression of the load potential P as an integral over the body if the change in shape of the body is negligible.

EXERCISE 6.2 [A:20] Work out the expression of the internal forces for (6.51). Then extend this relationto the three-dimensional case in which the ends of the spring move by ux1, uy1, uz1, ux2, uy2, uz2 in the x, y, zspace.

EXERCISE 6.3 [A+N/C:30] Consider the shallow arch model shown in Figure E6.1. This consists of twoidentical linear springs of axial stiffness k pinned to each other and to unmoving pinned supports as shown.The springs are assumed able to resist both tensile and compressive forces. The distance between the supportsis 2L . The undeformed springs form an angle " with the horizontal axis.

The central pin in loaded by a dead vertical force of magnitude f , positive downwards, which is parametrizedas f = #kL . Only symmetrical deformations of the arch are to be considered for this Exercise. Consequentlythe system has just one degree of freedom which we take to be the displacement u under the load, also positivedownwards. The response of this system exhibits the snap-through behavior sketched in Figure E6.2.

u

f

" $

k k

L L

Figure E6.1. Structure under under line load fy(x) (directed upward) over segment AB.

L1

L2

#

u or µ

Figure E6.2. Snap-through response of shallow arch (sketch).

6–15


(a) Show that the internal energy U and load potential P of the two-spring system are given by

U = kL2!

1cos!

! 1cos "

"2

, P = f u, (E6.1)

where " is the angle shown in Figure E6.1, which is linked to u by the relation tan " + u/L = tan!.

(b) Derive the exact equilibrium equation

r(u, #) = $%

$u= 0, (E6.2)

in which% = U ! P is the total potential energy, and # = f/(kL) is the dimensionless state parameter.For convenience rewrite this as

r(µ, #) = 0, (E6.3)

in terms of the dimensionless state parameter

µ = uL tan!

. (E6.4)

(c) Derive the exact equation for the limit load parameters

$#(µ)

$µ

#

#

#

#

µ=µL ,#=#L

= 0. (E6.5)

(Hint: the exact equation in terms of the angular coordinate " is cos3 "L = cos!). Solve this trigonometricequation4 for the limit-load parameters #L1 and #L2 and the dimensionless displacements µL1 and µL2at those points assuming that ! = 30".

4 Equation (E6.5) is equivalent to det K = 0 because for a one-DOF system det K = K = $#/$µ.

6–16

6–17 Exercises

(d) If the arch initially is and remains sufficiently “shallow” throughout its snap-through behavior, we maymake the small-angle approximations,

cos! ! 1" 12!

2, cos " ! 1" 12 "

2, sin! ! tan! ! !, sin " ! tan " ! " . (E6.6)

Recast the energy, equilibrium equations, and limit load equations in terms of these approximations,obtaining U as a quartic polynomial in " , r as a cubic polynomial in " , etc, then replace in terms of µ.As a check, the residual equation in terms of # and µ should be given by (4.16). Calculate the limit loadparameters #L1 and #L2, and the dimensionless displacements µL1 and µL2 at those loads. Verify thatthese displacements correspond to the angles "L = ±!/

#3.

(e) Draw the control-state response curves r(µ, #) = 0, derived using the exact nonlinear equations andthose from the small-angle approximations on the #, µ plane (as in the sketch of Figure E6.2, going upto µ ! 2.5) for ! = 30$.

EXERCISE 6.4 [A+N:15] Derive the current stiffness parameter $ defined in Equations (5.16) and (5.17) forthe approximate (small-angle) model of the two-spring arch of Exercise 6.3. Plot the variation of $(µ) as µ

varies from 0 to µL1 at the first limit point, with µ along the horizontal axis. Does $ vanish at the limit point?

6–17

.

7Review of

ContinuumMechanics

7–1

Chapter 7: REVIEW OF CONTINUUM MECHANICS 7–2

TABLE OF CONTENTS

Page§7.1. Introduction 7–3§7.2. Notational Systems 7–3§7.3. The Continuum Model 7–4

§7.3.1. Particle Motion . . . . . . . . . . . . . . . . . 7–4§7.3.2. Configurations . . . . . . . . . . . . . . . . 7–5§7.3.3. Distinguished Configurations . . . . . . . . . . . . 7–5§7.3.4. Kinematic Descriptions . . . . . . . . . . . . . 7–7§7.3.5. Coordinate Systems . . . . . . . . . . . . . . . 7–7§7.3.6. Configurations and Staged Analysis . . . . . . . . . 7–10

§7.4. Nonlinear Kinematics 7–11§7.4.1. Deformation and Displacement Gradients . . . . . . . . 7–11§7.4.2. Stretch and Rotation Tensors . . . . . . . . . . . . 7–12§7.4.3. Green-Lagrange Strain Measure . . . . . . . . . . . 7–13§7.4.4. Strain-Gradient Matrix Expressions . . . . . . . . . 7–14§7.4.5. Pull Forward and Pull Back . . . . . . . . . . . . . 7–15

§7.5. Stress Measure 7–15§7.6. Constitutive Equations 7–16§7.7. Strain Energy Density 7–17§7. Exercises . . . . . . . . . . . . . . . . . . . . . . 7–18

7–2

7–3 §7.2 NOTATIONAL SYSTEMS

§7.1. Introduction

Chapters 3–6 covered general properties of the governing force residual equations of geometricallynonlinear structural systems with finite number of degrees of freedom (DOF). The DOFs are col-lected in the state vector, and driven by control parameters. The residual equations, being algebraic,are well suited for numerical computations.Continuum models of actual structures, however, are expressed as ordinary or partial differentialequations in space or space-time. As such, thosemodels possess an infinite number ofDOFs. Exceptfor simple (typically linear) models, they cannot be directly solved analytically. The reduction toa finite number is accomplished by discretization methods. It was observed in Chapter 1 that fornonlinear problems in solid and structural mechanics the finite element method (FEM) is the mostwidely used discretization method.This Chapter provides background material for the derivation of geometrically nonlinear finite ele-ments from continuum models. It is essentially an overview of kinematic, kinetic and constitutiverelations of 3D continuum mechanics of an elastic deformable body, as needed in following Chap-ters. Readers already familiar with continuum mechanics should just peruse it to absorb notation.

§7.2. Notational Systems

Continuummechanics dealswith vector and tensorfields such as displacements, strains and stresses.Four notational systems are in common use.Indicial Notation. Also called component notation. The key concept is that of an index. Indicesidentify components of vectors and tensors. It has convenient abbreviation rules, such as commasfor partial derivatives and Einstein’s summation convention. The notation is general and powerful,and as such is preferred in analytic developments as well as publication in theoretical journals andmonographs. It readily handles arbitrary tensors of any order, curvilinear coordinate systems andnonlinear expressions. When used in non-Cartesian coordinates, it sharply distinguishes betweencovariant and contravariant quantities. The main disadvantages are: (i) physics is concealed behindthe index jungle, and (ii) highly inefficient for expressing numeric computations. Because of (i), itis not suitable for elementary instruction.Direct Notation. Sometimes called algebraic notation. Vectors and tensors are represented bysingle symbols, usually bold letters. These are linked by the well known operators of mathematicalphysics, such as . for dot product, ! for cross products, and " for gradient (or divergence). Hasthe advantage of compactness and quick visualization of intrinsic properties. Some operations,however, become undefined beyond a certain range. Some overlap with matrix algebra while othersdo not. This fuzzyness can lead to confusion in computational work.Matrix Notation. This is similar to the previous one, but entities are appropriately recast so thatonly matrix operations are used. The form can be directly mapped to discrete equations as well asmatrix-oriented programming languages such as Matlab. It has the disadvantage of losing contactwith the original physical entities along the way. For example, stress is a symmetric second-ordertensor, but is recast as a 6-vector for FEMdevelopments. This change, however, may forgo essentialproperties. For instance, it makes sense to say that principal stresses are eigenvalues of the stresstensor. But those get lost (or moved to the background) when recast as a vector.

7–3


Full Notation. In the full-form notation every term is spelled out. No ambiguities of interpretationcan arise, Consequently this works well as a notation of last resort, and also as a “comparisontemplate” against which one can check out the meaning of more compact expressions. It is alsouseful for programming in low-order languages.In this and following Chapters the direct, matrix and full notation are preferred, whereas the indicialnotation is used if either complicated tensor forms are needed, or nonlinear expressions not amenableto other notations appear. Often the expression isfirst given in direct form and confirmed by full formif feasible. Then it is transformed to matrix notation for use in FEM developments. The decisionleads to possible ambiguities against reuse of vector symbols in two contexts: continuummechanicsand FEM discretizations. Such ambiguities are resolved in favor of keeping FEM notation simple.

Example 7.1. Consider the well known dot product between two physical vectors in 3D space, a = (a1, a2, a3)and b = (b1, b2, b3) written in the four different notations:

ai bi!"#$

indicial

= a.b!"#$

direct= aT b

!"#$

matrix= a1b1 + a2b2 + a3b3

! "# $

full

. (7.1)

Example 7.2. Take the internal static equilibrium equations of a continuum body, expressed in terms of Cauchystresses and body forces per unit volume:

!i j, j + bi = 0! "# $

indicial

, !! + b = 0! "# $

direct

, DT!v + b = 0! "# $

matrix

,"!11

"x1+ "!12

"x2+ "!13

"x3+ b1 = 0, plus 2 more

! "# $

full

. (7.2)

In the third (matrix notation) form, !v denotes the stress tensor reformatted as a 6-vector.

Example 7.3. If a discrete mechanical system is conservative, it was shown in previous Chapters that the totalforce residual is the gradient of a total potential energy function with respect to the state:

ri = "#

"uidef= #,i

! "# $

indicial

, r = !#! "# $

direct, r = "#

"u! "# $

matrix

, r1 = "#

"u1, r2 = "#

"u2, . . .

! "# $

full

. (7.3)

The indicial form requires defining#,i as abbreviation of partial derivative with respect to u j . The direct formdepends on the gradient operator symbol chosen. Some authors may write r = grad#.

§7.3. The Continuum Model

In the present section a structure is mathematically treated as a continuum body B. In this model,the body is considered as being formed by a set of points P called particles, which are endowedwith certain mechanical properties. For FEM analysis the body is divided into elements, whichinherit the properties of the continuum model.

§7.3.1. Particle Motion

Particles displace or move in response to external actions characterized by the control parameters$i introduced in Chapter 3. Following the reduction process discussed in that Chapter, in eachstage the body responds to the single stage parameter %.

7–4

7–5 §7.3 THE CONTINUUM MODEL

A one-parameter series of positions occupied by the particles as they move in space is called amotion. The motion may be described by the displacement u(P) ! u(x) of the particles1 withrespect to a base or reference state in which particle P is labelled P0. The displacements of allparticles u(x) such that x ! {x, y, z} " B, constitutes the displacement field.The motion is said to be kinematically admissible if:1. Continuity of particle positions is preserved so that no gaps or interpenetration occurs.2

2. Kinematic constraints on the motion (for example, support conditions) are preserved.A kinematically admissible motion along a stage will be called a stage motion. For one such motionthe displacements u(x) characterize the state and the stage control parameter ! characterizes thecontrol. Both will be generally parametrized by the pseudo-time t introduced in Chapter 3, and soa stage motion can be generally represented by

! = !(t), u = u(x, t), x " B. (7.4)

§7.3.2. Configurations

If in (7.4) we freeze t , we have a configuration of the structure. Thus a configuration is formally theunion of state and control. It may be informally viewed as a “snapshot” taken of the structure and itsenvironment when the pseudotime is frozen. If the configuration satisfies the equilibrium equations,it is called an equilibrium configuration. In general, however, a randomly given configuration isnot in equilibrium unless artificial body and surface forces are applied to it.A staged response, or simply response, can be nowmathematically defined as a series of equilibriumconfigurations obtained as ! is continuously varied, starting from 0.

§7.3.3. Distinguished Configurations

A particular feature of geometrically nonlinear analysis is the need to carefully distinguish amongdifferent configurations of the structure. As defined above, set of kinematically admissible dis-placements u(x) plus a staged control parameter ! at a frozen t defines a configuration. This isnot necessarily an equilibrium configuration. In fact it will not usually be one. It is also importantto realize that an equilibrium configuration is not necessarily a physical configuration assumed bythe actual structure.3 Configurations that are important in geometrically nonlinear analysis receivespecial qualifiers:

admissible, perturbed, deformed, base, reference, iterated, target, corotated, aligned

This terminology is collected in Figure 7.1 in a tabular format. Of the nine listed there, the first fourare used extensively in theoretical and applied mechanics, the last four exclusively in computationalmechanics, and one (reference) used in both, although usually with different meanings.

1 The underlining in u is used to distinguish the physical displacement vector from the finite element node displacementarray, which is a computational vector.

2 The mathematical statement of this compatibility condition is quite complicated for finite displacements, and will not begiven here. The finite element formulations worked out later will automatically satisfy the requirement.

3 Recall the suspension bridge under zero gravity of Chapter 3.

7–5


Name Alias Definition Equilibrium Identification Required?

Admissible A kinematically admissible configuration No

Perturbed Kinematically admissible variation of No of an admissible configuration

Current Deformed Any admissible configuration taken during the No or Spatial analysis process. Contains all others as special cases

Base Initial The configuration defined as the origin of Yes , or Undeformed displacements. Strain free but not necessarily Material stress free

Reference Configuration to which stepping computations TL,UL: yes TL: UL: in an incremental solution process are referred CR: no, yes CR: and

Iterated Configuration taken at the kth iteration No of the nth increment step

Target Equilibrium configuration accepted Yes after completing the nth increment step

Corotated Shadow Body- or element-attached configuration obtained No Ghost from through a RBM (CR description only)

Aligned Preferred A fictitious body ot element configuration aligned No Directed with a particular set of axes (usually global axes)

Definitions with blue background are used only in theoretical and applied mechanics for analytical formulations. Definitions with yellow background are only used in computational mechanics. Definitions with green background are used in both, hence the color choice. The meaning of reference configuration, however, may differ. The one stated above is for computational mechanics.

The base configuration is often the same as the natural state in which body (or element) is undeformed and stress free.

In dynamic analysis using the CR kinematic description, and are called the inertial and dynamic reference configurations, respectively, when applied to an entire structure such as an airplane (e.g., autopilot simulations)

Figure 7.1. Distinguished configurations in geometrically nonlinear analysis.

The three most important configurations insofar as a FEM implementation is concerned, are: base,reference and current. These are pictured in Figure 7.2. An examination of this figure indicates thatthe choice of the reference configuration depends primarily on the kinematic description chosen.Such a choice is examined in the next subsection.

Remark 7.1. Many names can be found for the configurations listed in Figure 7.1 in the literature dealign withfinite elements and continuum mechanics. Here are some of those alternative names.

Perturbed configuration: adjacent, deviated, disturbed, incremented, neighboring, varied, virtual.Current configuration: arbitrary, deformed, distorted, pull-forward, moving, present, spatial, varying.Base configuration: baseline, initial, global, material, natural, original, overall, undeformed, undistorted.Reference configuration: fixed, frozen, known, pull-back.Iterated configuration: corrected, intermediate, stepped, transient, transitory.Target configuration: converged, equilibrated spatial, unknown.Corotated configuration: attached, convected, ghost, phantom, shadow.Aligned configuration: directed, body-matched, preferred.

7–6


Current Configuration

Reference Configuration(identifier depends on

kinematic description chosen)

Base Configuration , or

or

Figure 7.2. The three most important configurations for geometrically nonlinear analysis. Specializa-tion to the kinematic descriptions tabulated in Figure 7.3 are pictured in Figures 7.4, 7.5, and 7.6.

§7.3.4. Kinematic Descriptions

Three kinematic descriptions of geometrically nonlinear finite element analysis are in current use inprograms that solve nonlinear structural problems: Total Lagrangian or TL, Updated Lagrangianor UL, and Corotational or CR. They are described in Figure 7.3 They can be distinguished by thechoice of reference configuration. The important configurations for these three descriptions arepictured in Figures 7.4, 7.5, and 7.6, respectively.The TL formulation remains the most widely used in continuum-based FEM codes.4 The CRformulation is gaining in popularity for structural elements such as beams, plates and shells, espe-cially in Aerospace.5 The UL formulation is useful in treatments of vary large strains and flow-likebehavior, as well as in processes involving topology and/or phase changes; e.g., metal forming.

§7.3.5. Coordinate Systems

Configurations taken by a body or element during the response analysis are linked by a Cartesianglobal frame, to which all computations are ultimately referred.6 There are actually two suchframes:(i) The material global frame with axes {Xi } or {X, Y, Z}.(ii) The spatial global frame with axes {xi } or {x, y, z}.7

4 A key reason is historic: three of the original nonlinear FEM codes: MARC, ABAQUS and ANSYS, originally imple-mented elements based on that description.

5 Geometrically nonlinear problems in Aerospace Engineering tend to involve large motions, in particular large rotations,but small strains. Reason: structures are comparatively thin to save weight.

6 In dynamic analysis the global frame may be moving in time as a Galilean or inertial frame. This is convenient to trackthe trajectory motion of objects such as aircraft or satellites.

7 The choice between {X1, X2, X3} versus {X, Y, Z} and likewise {x1, x2, x3} versus {x, y, z} is a matter of notationalconvenience. For example, when developing specific finite elements it is preferable to use {X, Y, Z} or {x, y, z} so asto reserve coordinate subscripts for node numbers. On the other hand, in derivations that make heavy use of indicialnotation, the Xi and xi notation is more appropriate.

7–7


Name Acronym Definition Primary applications

Total Lagrangian TL Base and reference configurations Solid and structural mechanics with finite coalesce and remain fixed throughout but moderate displacements and strains. the solution process Primarily used for elastic material. Unreliable for flow-like behavior or topology changes Updated Lagrangian UL Base configuration remains fixed but Solid and structural mechanics with finite reference configuration is periodically. displacements and possibly large strains. updated. Most common update strategy Handles material flow-like behavior well, is to set reference configuration to last (e.g., forming processes) as well as topology converged solution changes (fracture) Corotational CR Reference configuration is split into base Solid and structural mechanics with arbitrarily and corotated. Strains and stresses are large finite motions, but small strains and measured from corotated to current, while elastic material behavior. Extendible to nonlinear base configuration is maintained as materials if inelasticity is localized so most of reference to measure rigid body motions structure stays elastic.

All three descriptions are Lagrangian: computations are always referred to a previous configuration (base and/or reference).Eulerian formulations, which are common in fluid mechanics, are not popular in solid and structural mechanics

Figure 7.3. Kinematic descriptions used in FEM programs that handle geometricallynonlinear problems in solid and structural mechanics.

The material frame tracks the base configuration whereas the spatial frame tracks all others. Thisdistinction agrees with the usual conventions of classical continuum mechanics. In this book bothframes are taken to be identical, as nothing is gained by separating them. Thus only one set ofglobal axes, with dual labels, is drawn in Figure 7.7In stark contrast to global frame uniqueness, the presence of elements means there are many localframes to keep track of. More precisely, each element is endowed with two local Cartesian frames:(iii) The element base frame with axes {Xi } or {X , Y , Z}.(iv) The element reference frame with axes {xi } or {x, y, z}.The base frame is attached to the base configuration. It remains fixed if the base is fixed. It ischosen according to usual FEM practices. For example, in a 2-node spatial beam element, X1 isdefined by the two end nodes whereas X2 and X3 lie along principal inertia directions. The originis typically placed at the element centroid.The meaning of the reference frame depends on the description chosen:

Total Lagrangian (TL). The reference and base frames coalesce.Updated Lagrangian (UL). The reference frame is attached to the reference configuration, andrecomputed when the reference configuration (often taken as the previous converged solution) isupdated. It remains fixed during an iterative (corrective) process.Corotational description (CR). The reference frame is renamed corotated frame or CR frame. Itremains attached to the element and continuously moves with it.

7–8



Base and Reference Configuration =

TOTAL LAGRANGIAN (TL)Kinematic Description

0

Figure 7.4. Important configurations in Total Lagrangian (TL) kinematic description.


Base Configuration

UPDATED LAGRANGIAN (UL)Kinematic Description

B

nReference Configuration usually updated after each incremental step

Figure 7.5. Important configurations in Updated Lagrangian (UL) kinematic description.


Base Configuration

COROTATIONAL (CR)Kinematic Description

0

RCorotated Configurationa rigid motion of the base configuration

Figure 7.6. Important configurations in Corotational (CR) kinematic description. The Note:the corotated and current configuration are shown highly offset for visualization convenience.

In practical use they highly overlap; for example, the centroids coincide.

7–9


V

P0(X)

P(x)

u = x ! X

X, x Y, y

Z , z

X " x0

x = x(t)

Base configuration (for drawing simplicity, assumed tocoalesce with reference, as in TL)

Current configuration

Figure 7.7. The geometrically nonlinear problem in a Lagrangian kinematics: coordinate systems, referenceand current configurations, and displacements. To keep the figure simple it has been assumed that base and

reference configurations coalesce, as happens in the Total Lagrangian (TL) description.

The transformationx = X + u, (7.5)

maps the position of base particle P(X, Y, Z) to P(x, y, z). See Figure 7.7. Consequently theparticle displacement vector is defined as

u =! uXuYuZ

"

=! x ! Xy ! Yz ! Z

"

= x ! X. (7.6)

in which (X, Y, Z) and (x, y, z) pertain to the same particle.

Remark 7.2. Variations of this notation scheme are employed as appropriate to the subject under consideration.For example, the coordinates of P in a target configuration Cn may be called (xn, yn, zn).

Remark 7.3. In continuum mechanics, (X, Y, Z) and (x, y, z) are called material and spatial coordinates,respectively. In general treatments both systems are curvilinear and need not coalesce. The foregoing relationsare restrictive in two ways: the base coordinate systems for the reference and current configurations coincide,and that system is Cartesian. This assumption is sufficient, however, for the problems addressed here.

Remark 7.4. The dual notation (X, Y, Z) " (x0, y0, z0) is introduced on two accounts: (1) the use of(x0, y0, z0) sometimes introduces a profusion of additional subscripts, and (2) the notation agrees with thattraditionally adopted in continuum mechanics for the material coordinates, as noted in the previous remark.The identification X " x0, Y " y0, Z " z0 will be employed when it is convenient to consider the referenceconfiguration as the initial target configuration; cf. Remark 7.1.

§7.3.6. Configurations and Staged Analysis

The meaning of some special configurations can be made more precise if the nonlinear analysisprocess is viewed as a sequence of analysis stages, as discussed in Chapter 3. We restrict attentionto the Total Lagrangian (TL) and Corotational (CR) kinematic descriptions, which are the onlyones covered in this book. In a staged TL nonlinear analysis, two common choices for the referenceconfiguration are:

7–10

7–11 §7.4 NONLINEAR KINEMATICS

(1) Reference ! base. The base configuration is maintained as reference configuration for allstages.

(2) Reference ! stage start. The configuration at the start of an analysis stage, i.e. at ! = 0, ischosen as reference configuration.

A combination of these two strategies can be of course adopted. In a staged CR analysis thereference is split between base and corotated. The same update choices are available for the base.This may be necessary when rotations exceed 2" ; for example in aircraft maneuvers.Theadmissible configuration is a “catch all” concept that embodies all others as particular cases. Theperturbed configuration is an admissible variation from a admissible configuration. An ensembleof perturbed configurations is used to establish incremental or rate equations.The iterated and target configurations are introduced in the context of incremental-iterative solutionprocedures for numerically tracing equilibriumpaths. The target configuration is the “next solution”.More precisely, an equilibrium solution (assumed to exist)which satisfies the total residual equationsfor a given value of the stage control parameter !. While working to reach the target, a typicalsolution process goes through a sequence of iterated configurations that are not in equilibrium.The corotated configuration is a rigid-body rotation of the reference configuration that “follows”the current configuration like a “shadow”. It is used in the corotational (CR) kinematic descriptionof nonlinear finite elements. Strains measured with respect to the corotated configuration maybe considered “small” in many applications, a circumstance that allows linearization of severalrelations and efficient treatment of stability conditions.

§7.4. Nonlinear KinematicsThis section presents the essential kinematics necessary for geometrically nonlinear analysis.

§7.4.1. Deformation and Displacement GradientsThe derivatives of (x, y, z) with respect to (X, Y, Z), arranged in Jacobian format, constitute theso-called deformation gradient matrix:

F = #(x, y, z)#(X, Y, Z)

=

!

""

#

#x#X

#x#Y

#x#Z

#y#X

#y#Y

#y#Z

#z#X

#z#Y

#z#Z

$

%%

&. (7.7)

The inverse relation gives the derivatives of (X, Y, Z) with respect to (x, y, z) as

F"1 = #(X, Y, Z)

#(x, y, z)=

!

"""

#

#X#x

#X#y

#X#z

#Y#x

#Y#y

#Y#z

#Z#x

#Z#y

#Z#z

$

%%%

&

. (7.8)

These matrices can be used to relate the coordinate differentials

dx =' dxdydz

(

= F' dXdYdZ

(

= F dX, dX = F"1 dx. (7.9)

7–11


The indicial version of the foregoing definitions is compact: Fi, j = !xi/!X j , F!1i, j = !Xi/!x j .

The displacement gradientswith respect to the reference configuration can be presented as the 3"3matrix

G = F! I =

!

""

#

!x!X ! 1 !x

!Y!x!Z

!y!X

!y!Y ! 1 !y

!Z!z!X

!z!Y

!z!Z ! 1

$

%%

&=

!

""

#

!uX!X

!uX!Y

!uX!Z

!uY!X

!uY!Y

!uY!Z

!uZ!X

!uZ!Y

!uZ!Z

$

%%

&= #u. (7.10)

Displacement gradients with respect to the current configuration are given by

J = I! F!1 =

!

"""

#

1! !X!x

!X!y

!Z!x

!Y!x 1! !Y

!y!Y!z

!Z!x

!Z!y 1! !Z

!z

$

%%%

&

=

!

"""

#

!uX!x

!uX!y

!uX!z

!uY!x

!uY!y

!uY!z

!uZ!x

!uZ!y

!uZ!z

$

%%%

&

. (7.11)

For the treatment of the Total Lagrangian description it will found to be convenient to arrange thedisplacement gradients of (7.10) as a 9-component vector (written as row vector to save space):

gT = [ g1 g2 g3 g4 g5 g6 g7 g8 g9 ]

=' !uX

!X!uY!X

!uZ!X

!uX!Y

!uY!Y

!uZ!Y

!uX!Z

!uY!Z

!uZ!Z

(

.(7.12)

For arbitrary rigid-body motions (motions without deformations) FTF = FFT = I, that is, F is anorthogonal matrix, and G becomes a rotation matrix.In nonlinear continuum mechanics, displacement gradients play an important role that is absent inthe infinitesimal theory. This is especially true in the Total Lagrangian description.

Remark 7.5. Displacement gradient matrices are connected by the relations

G = (I! J)!1 ! I, J = I! (I+G)!1. (7.13)

For small deformations G $ J!1 and J $ G!1.

Remark 7.6. The ratio between infinitesimal volume elements dV = dx dy dz and dV0 = dX dY dZ in thecurrent and reference configuration appears in several continuum mechanics relations. Because of (7.9) thisratio may be expressed as

dVdV0

= "0

"= detF, (7.14)

where " and "0 denote themass densities in the current and reference configuration, respectively. This equationexpresses the law of conservation of mass.

§7.4.2. Stretch and Rotation Tensors

Tensors F andG are the building blocks of various deformation measures used in nonlinear contin-uum mechanics. The whole subject is dominated by the polar decomposition theorem: any particle

7–12

7–13 §7.4 NONLINEAR KINEMATICS

deformation can be expressed as a pure deformation followed by a rotation, or by a rotation followedby a pure deformation. Mathematically this is written as multiplicative decompositions:

F = RU = VR. (7.15)Here R is an orthogonal rotation tensor, whereas U and V are symmetric positive definite matricescalled the right and left stretch tensors, respectively. If the deformation is a pure rotation, U =V = I. Premultiplying (7.15) by FT = URT gives U2 = FTF and consequently U =

!FTF.

Postmultiplying (7.15) by FT = RTV givesV2 = FFT and consequentlyV =!FFT . Upon taking

the square roots, the rotation is then computed as either R = FU"1 or R = V"1F. ObviouslyU = RVRT and V = RTUR.The combinations CR = FTF and CL = FFT are symmetric positive definite matrices that arecalled the right and left Cauchy-Green stretch tensors, respectively. To get U and V as square rootsit is necessary to solve the eigensystem of CR and CL , respectively.To convert a stretch tensor to a strain tensor one substracts I from it or takes its log, so as to have ameasure that vanishes for rigid motions. EitherU"I orV"I represent appropriate strain measuresfor geometrically nonlinear analysis. These are difficult, however, to express analytically in termsof the displacement gradients because of the intermediate eigenproblem. A more convenient strainmeasure is described next.

§7.4.3. Green-Lagrange Strain MeasureA convenient finite strain measure is the Green-Lagrange (GL) strain tensor.8 Its three-dimensionalexpression in Cartesian coordinates is

e = 12!

FTF" I"

= 12 (G+GT ) + 1

2GTG =

# eXX eXY eX ZeY X eYY eY ZeZ X eZY eZ Z

$

, (7.16)

Identifying the components of FTF" I or 12 (G+GT ) + 12G

TG with the tensor conponents we get

eXX = !uX!X

+ 12

#%

!uX!X

&2+

%

!uY!X

&2+

%

!uZ!X

&2$

eYY = !uY!Y

+ 12

#%

!uX!Y

&2+

%

!uY!Y

&2+

%

!uZ!Y

&2$

eZ Z = !uZ!Z

+ 12

#%

!uX!Z

&2+

%

!uY!Z

&2+

%

!uZ!Z

&2$

eY Z = 12

%

!uY!Z

+ !uZ!Y

&

+ 12

'

!uX!Y

!uX!Z

+ !uY!Y

!uY!Z

+ !uZ!Y

!uZ!Z

(

= eZY ,

eZ X = 12

%

!uZ!X

+ !uX!Z

&

+ 12

'

!uX!Z

!uX!X

+ !uY!Z

!uY!X

+ !uZ!Z

!uZ!X

(

= eXZ ,

eXY = 12

%

!uX!Y

+ !uY!X

&

+ 12

'

!uX!X

!uX!Y

+ !uY!X

!uY!Y

+ !uZ!X

!uZ!Y

(

= eY X .

(7.17)

8 A more appropriate name would be Green-St.Venant strain tensor. Actually Lagrange never used it but his name appearsbecause of the connection to the Lagrangian kinematic description. Many authors call it the Green strain tensor.

7–13


If the nonlinear portion (that enclosed in square brackets) of these expressions is neglected, oneobtains the infinitesimal strains !xx , !yy, . . . !zx = 1

2"zx , !xy = 12"xy encountered in linear finite

element analysis. For future use in finite element work we shall arrange the components (7.17) asa 6-component strain vector e constructed as follows:

e =

!

"""""

#

e1e2e3e4e5e6

$

%%%%%

&

=

!

"""""

#

eXXeYYeZ Z

eY Z + eZYeZ X + eXZeXY + eY X

$

%%%%%

&

=

!

"""""

#

eXXeYYeZ Z2eY Z2eZ X2eXY

$

%%%%%

&

. (7.18)

Remark 7.7. Other finite strain measures are used in nonlinear continuum mechanics. Their common charac-teristic is that theymust predict zero strains for arbitrary rigid-bodymotions, andmust reduce to the infinitesimalstrains if the nonlinear terms are neglected. This topic is further explored in Exercise 7.5.

§7.4.4. Strain-Gradient Matrix Expressions

For the development of the TL core-congruential formulation presented in Chapters 10-11 (omittedin this course offering), it is useful to have a compact matrix expression for the Green-Lagrangestrain components of (7.18) in terms of the displacement gradient vector (7.12). To this end, notethat (7.17) may be rewritten as

e1 = g1 + 12 (g

21 + g22 + g23),

e2 = g5 + 12 (g

24 + g25 + g26),

e3 = g9 + 12 (g

27 + g28 + g29),

e4 = g6 + g8 + g4g7 + g5g8 + g6g9,e5 = g3 + g7 + g1g7 + g2g8 + g3g9,e6 = g2 + g4 + g1g4 + g2g5 + g3g6.

(7.19)

These relations may be collectively embodied in the quadratic form

ei = hTi g+ 12g

THig, (7.20)

where hi are sparse 9! 1 vectors:

h1 =

!

"""""""""""

#

100000000

$

%%%%%%%%%%%

&

, h2 =

!

"""""""""""

#

000010000

$

%%%%%%%%%%%

&

, h3 =

!

"""""""""""

#

000000001

$

%%%%%%%%%%%

&

, h4 =

!

"""""""""""

#

000001010

$

%%%%%%%%%%%

&

, h5 =

!

"""""""""""

#

001000100

$

%%%%%%%%%%%

&

, h6 =

!

"""""""""""

#

010100000

$

%%%%%%%%%%%

&

, (7.21)

7–14

7–15 §7.5 STRESS MEASURE

and Hi are very sparse 9! 9 symmetric matrices:

H1 =

!

"""""""""""

#

1 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

$

%%%%%%%%%%%

&

, etc. (7.22)

Remark 7.8. For strain measures other than Green-Lagrange’s, expressions similar to (7.20) may be con-structed. But although the hi remain the same, the Hi become complicated functions of the displacementgradients.

§7.4.5. Pull Forward and Pull Back

Most of the foregoing material is classical continuum mechanics as covered in dozens of scholarlybooks. Next is a kinematic derivation scheme that is quintaessential FEM. Consider the motion ofan elastic bar element in the 2D plane as depicted in Figure 7.8.(To be expanded, subsection unnished but not necessary for rest of Chapter)

§7.5. Stress Measure

Associated with each finite strain measure is a corresponding stress measure that is conjugate toit in the sense of virtual work. That corresponding to the Green-Lagrange strain is the secondPiola-Kirchhoff symmetric stress tensor, often abbreviated to “PK2 stress.” The three-dimensionalcomponent expression of this tensor in Cartesian coordinates is

s =' sXX sXY sX ZsY X sYY sY ZsZ X sZY sZ Z

(

, (7.23)

in which sXY = sY X , etc. As in the case of strains, for future use in finite element work it isconvenient to arrange the components (7.23) as a 6-component stress vector s:

sT = [ s1 s2 s3 s4 s5 s6 ] = [ sXX sYY sZ Z sY Z sZ X sXY ] . (7.24)

Remark 7.9. The physical meaning of the PK2 stresses is as follows: si j are stresses “pulled back” to thereference configuration C0 and referred to area elements there.

7–15


θ

θ

B

X

X

x

Y

Y

y

~ ~

_

_

Globallyaligned

Base

Deformed

Corotated

Figure 7.8. Rigid motion of bar in 2D illustrating concept of globally aligned configuration.

.

Remark 7.10. The PK2 stresses are related to the Cauchy (true) stresses !i j through the transformation

!

""""

#

sXXsYYsZ ZsY ZsZ XsXY

$

%%%%

&

= "0

"

!

"""""""""""

#

#X#x

#X#x

#X#y

#X#y

#X#z

#X#z

#X#y

#X#z

#X#z

#X#x

#X#x

#X#y

#Y#x

#Y#x

#Y#y

#Y#y

#Y#z

#Y#z

#Y#y

#Y#z

#Y#z

#Y#x

#Y#x

#Y#y

#Z#x

#Z#x

#Z#y

#Z#y

#Z#z

#Z#z

#Z#y

#Z#z

#Z#z

#Z#x

#Z#x

#Z#y

#Y#x

#Z#x

#Y#y

#Z#y

#Y#z

#Z#z

#Y#y

#Z#z

#Y#z

#Z#x

#Y#x

#Z#y

#Z#x

#X#x

#Z#y

#X#y

#Z#z

#X#z

#Z#y

#X#z

#Z#z

#X#x

#Z#x

#X#y

#X#x

#Y#x

#X#y

#Y#y

#X#z

#Y#z

#X#y

#Y#z

#X#z

#Y#x

#X#x

#Y#y

$

%%%%%%%%%%%

&

!

""""

#

!XX!YY!Z Z!Y Z!Z X!XY

$

%%%%

&

, (7.25)

!

""""

#

!XX!YY!Z Z!Y Z!Z X!XY

$

%%%%

&

= "

"0

!

""""""""""

#

#x#X

#x#X

#x#Y

#x#Y

#x#Z

#x#Z

#x#Y

#x#Z

#x#Z

#x#X

#x#X

#x#Y

#y#X

#y#X

#y#Y

#y#Y

#y#Z

#y#Z

#y#Y

#y#Z

#y#Z

#y#X

#y#X

#y#Y

#z#X

#z#X

#z#Y

#z#Y

#z#Z

#z#Z

#z#Y

#z#Z

#z#Z

#z#X

#z#X

#z#Y

#y#X

#z#X

#y#Y

#z#Y

#y#Z

#z#Z

#y#Y

#z#Z

#y#Z

#z#X

#y#X

#z#Y

#z#X

#x#X

#z#Y

#x#Y

#z#Z

#x#Z

#z#Y

#x#Z

#z#Z

#x#X

#z#X

#x#Y

#x#X

#y#X

#x#Y

#y#Y

#x#Z

#y#Z

#x#Y

#y#Z

#x#Z

#y#X

#x#X

#y#Y

$

%%%%%%%%%%

&

!

""""

#

sXXsYYsZ ZsY ZsZ XsXY

$

%%%%

&

. (7.26)

The density ratios that appears in these equations may be obtained from (7.14). If all displacement gradientsare small, both transformations reduce to the identity, and the PK2 and Cauchy stresses coalesce.

§7.6. Constitutive EquationsThroughout this course we restrict our attention to constitutive behavior in which conjugate strainsand stresses are linearly related. For the Green-Lagrange and PK2 measures used here, the stress-strain relations will be written, with the summation convention implied,

si = s0i + Ei j e j , (7.27)

7–16

7–17 §7.7 STRAIN ENERGY DENSITY

in which ei and si denote components of the GL strain and PK2 stress vectors defined by (7.18) and(7.24), respectively, s0i are PK2 stresses in the reference configuration (also called initial stressesor prestresses), and Ei j are constant elastic moduli with Ei j = E ji . In full matrix notation,

!

"""""

#

s1s2s3s4s5s6

$

%%%%%

&

=

!

"""""

#

s01s02s03s04s05s06

$

%%%%%

&

+

!

"""""

#

E11 E12 E13 E14 E15 E16E12 E22 E23 E24 E25 E26E13 E23 E33 E34 E35 E36E14 E24 E34 E44 E45 E46E15 E25 E35 E45 E55 E56E16 E26 E36 E46 E56 E66

$

%%%%%

&

!

"""""

#

e1e2e3e4e5e6

$

%%%%%

&

, (7.28)

or in compact form,s = s0 + Ee. (7.29)

Remark 7.11. For an invariant reference configuration, PK2 and Cauchy (true) prestresses obviously coincide(see previous Remark). Thus !0 ! s0 in such a case. However if the reference configuration is allowed tovary often, as in the UL description, things get more complicated.

§7.7. Strain Energy Density

We conclude this review by giving the expression of the strain energy density U in the currentconfiguration reckoned per unit volume of the reference configuration:

U = s0i ei + 12 (si " s0i )ei = s0i ei + 1

2ei Ei j e j , (7.30)

or, in matrix formU = eT s0 + 1

2eTEe. (7.31)

If the current configuration coincides with the reference configuration, e = 0 and U = 0. It can beobserved that the strain energy density is quadratic in the Green-Lagrange strains. To obtain thisdensity in terms of displacement gradients, substitute their expressions into the above form to get

U = s0i (hTi g+ gTHig) + 12'

(gThi + 12g

THig)Ei j (hTj g+ 12g

TH jg)(

. (7.32)

Because hi and Hi are constant, this relation shows that the strain energy density is quartic in thedisplacement gradients collected in g.The strain energy in the current configuration is obtained by integrating this energy density overthe reference configuration:

U =)

V0U dX dY dZ . (7.33)

This expression forms the basis for deriving finite elements based on the Total Lagrangian (TL)description.

7–17


Homework Exercises for Chapter 7Review of Continuum Mechanics

EXERCISE 7.1 [A:15] Obtain the expressions of H3 and H5.

EXERCISE 7.2 [A:15] Derive (7.30) by integrating si de!i from C0 (e!

i = 0) to C (e!i = ei ) and using (7.27).

EXERCISE 7.3 [A:20] A bar of length L0 originally along the X " x axis (the reference configuration C0) isrigidly rotated 90# to lie along the Y " y axis while retaining the same length (the current configuration C).Node 1 at the origin X = Y = 0 stays at the same location.(a) Verify that the motion from C0 to C is given by

x = $Y, y = X, z = Z . (E7.1)

(b) Obtain the displacement field u, the deformation gradient matrix F, the displacement gradient matrix Gand the Green-Lagrange axial strain e = eXX . Show that the Green-Lagrange measure correctly predictszero axial strain whereas the infinitesimal strain measure ! = !XX = "uX/"X predicts the absurd valueof $100% strain.

EXERCISE 7.4 [A:20] Let L0 and L denote the length of a bar element in the reference and current con-figurations, respectively. The Green-Lagrange finite strain e = eXX , if constant over the bar, can be definedas

e =L2 $ L202L20

. (E7.2)

Show that the definitions (E7.2) and of e = eXX in the GL strain definition (7.17) are equivalent. (Hint: expressL0 and L in terms of the coordinates and displacements in the bar system.)

EXERCISE 7.5 [A:25] The Green-Lagrange strain measure is not the only finite strain measure used instructural and solid mechanics. For the uniaxial case of a stretched bar that moves from a length L0 in C0 to alength L in C, some of the other measures are defined as follows:

(a) Uniaxial Almansi strain:

eA =L2 $ L202L2

. (E7.3)

(b) Uniaxial Hencky strain, also called logarithmic or “true” strain:eH = log(L/L0), (E7.4)

where log denotes the natural logarithm.(c) Uniaxial midpoint strain9

eM =L2 $ L20

2[(L + L0)/2]2. (E7.5)

(d) Uniaxial engineering strain:eE = ! = L $ L0

L0. (E7.6)

If L = (1+!)L0 , show by expanding eA, eH and eM in Taylor series in ! (about ! = 0) that these measures, aswell as the Green-Lagrange axial strain (E7.2) agree with each other to first order [i.e., they differ by O(!2)]as ! % 0.

9 The midpoint strain tensor, which is a good approximation of the Hencky strain tensor but more easily computable, isfrequently used in finite element plasticity or viscoplasticity calculations that involve large deformations, for example inmetal forming processes.

7–18

7–19 Exercises

EXERCISE 7.6 [A:35] (Advanced). Extend the definition of the Almansi, Hencky, midpoint and engineeringstrain to a three dimensional strain state. Hint: use the spectral decomposition of FTF and the concept offunction of a symmetric matrix.

EXERCISE 7.7 [A:35] (Advanced). Extend the definition of engineering strain to a three-dimensional strainstate. The resulting measures (there are actually two) are called the stretch tensors. Hint: use either thespectral decomposition of FTF, or the polar decomposition theorem of tensor calculus.

EXERCISE 7.8 [A:40] (Advanced). Define the stress measures conjugate to the Almansi, Hencky, midpointand engineering strains.

7–19

.

8The TL Bar

Element:Formulation

8–1

Chapter 8: THE TL BAR ELEMENT: FORMULATION 8–2

TABLE OF CONTENTS

Page§8.1. Introduction 8–3§8.2. The Two-Dimensional Bar Element 8–3

§8.2.1. Element Kinematics . . . . . . . . . . . . . . . 8–3§8.2.2. GL Axial Strain . . . . . . . . . . . . . . . . 8–5§8.2.3. Strain Derivatives . . . . . . . . . . . . . . . . 8–7§8.2.4. PK2 Axial Stress . . . . . . . . . . . . . . . . 8–7§8.2.5. Total Potential Energy . . . . . . . . . . . . . . 8–7§8.2.6. Internal Force Vector . . . . . . . . . . . . . . 8–8§8.2.7. The Tangent Stiffness Matrix . . . . . . . . . . . . 8–8

§8. Exercises . . . . . . . . . . . . . . . . . . . . . . 8–11

8–2

8–3 §8.2 THE TWO-DIMENSIONAL BAR ELEMENT

§8.1. Introduction

The basic concepts of nonlinear continuum mechanics reviewed in Chapter 7 are applied to thedevelopment of finite element equations of a two-dimensional (plane) bar element based on theTotal Lagrangian (TL) kinematic description. This will be referred to as a TL bar element forbrevity. There are two ways to construct TL finite elements:1. The Standard Formulation (SF)2. The Core Congruential Formulation (CCF).The first method is easier to describe and is that followed in this Chapter. The second one is moreflexible and powerful but it is far more difficult to teach because it proceeds in stages. The CCFwill not be covered in this course, but older Chapters explaining it are posted on the web site.

§8.2. The Two-Dimensional Bar Element

The element developed in this Chapter is a prismatic bar element that can be used to model pin-jointed plane truss structures of the type sketched in Figure 8.1. These structures may undergolarge displacements and rotations but their strains are assumed to remain small so that the materialbehavior stays in the linear elastic range. These assumptions allows us to consider only geometricnonlinear effects.1

A two-node bar element appropriate to model members of such truss structures is shown in Fig-ure 8.2. The element moves in the (X, Y ) plane. In the reference (base) configuration the elementhas cross section area A0 (constant along the element) and length L0. In the current configurationthe cross section area and length become A and L , respectively. Thematerial has an elastic modulusE that links the axial-stress and axial-strain measures defined below.Because thisChapter deals primarilywith the formulation of an individual element, the identificationsuperscript (e) will be omitted to reduce clutter until assemblies are considered.The element has four node displacements and associated node forces. These quantities are collectedin the vectors

u =

!

"

#

uX1uY1uX2uY2

$

%

&, f =

!

"

#

fX1fY1fX2fY2

$

%

&, (8.1)

The loads acting on the nodes will be assumed to be conservative.

X(!) ='

X (!)

Y (!)

(

=' 12 (1! !)X1 + 1

2 (1+ !)X212 (1! !)Y1 + 1

2 (1+ !)Y2

(

='

N1X1 + N2X2N1Y1 + N2Y2

(

x(!) ='

x(!)y(!)

(

=' 12 (1! !)x1 + 1

2 (1+ !)x212 (1! !)y1 + 1

2 (1+ !)y2

(

='

N1x1 + N2y2N1y1 + N2y2

(

1 An important application inAerospace is the deployment of space trusses. InCivil Engineering itwould be the deploymentof geodesic domes. For those applications a three-dimensionaal version of this bar element would be required.

8–3


Motion

Reference configuration (same as base in TL description)

Current configuration

Figure 8.1. A plane truss structure undergoing large displacements while its material stays in thelinear elastic range.

u

u

u

u

X2

Y2

Y1

X2

Y2X1X1

Y1

f

f

f

f

2 2

Area A , length L0 0

Area A, length L

Current configurationPK2 stress s, GL strain e

C

Reference configurationPK2 stress s , GL strain e = 0

C 0

0 0

2 (X ,Y )

2 22 (x ,y )

1 11 (X ,Y )

1 11 (x ,y )

Y, y

X, x

X,Y : material framex,y : spatial frame

Figure 8.2. The geometrically nonlinear, two-node, plane bar element in the Total Lagrangian (TL)kinematic description. Can be used to model members of a plane truss such as that shown in Figure 8.1.

§8.2.1. Element KinematicsIn accordance with bar theory, to describe the element motion it is sufficient to consider a genericpoint P0 of coordinates X located on the longitudinal axis of the reference configuration C0. Thatpoint maps to point P at x in the current configuration C. The bar element remains straight in anyconfiguration. These coordinates can be parametrically interpolated from the end nodes as

X(!) =!

X (!)

Y (!)

"

=! 1

2 (1 ! !)X1 + 12 (1 + !)X2

12 (1 ! !)Y1 + 1

2 (1 + !)Y2

"

=!

N1(!)X1 + N2(!)X2N1(!)Y1 + N2(!)Y2

"

, (8.2)

x(!) =!

x(!)

y(!)

"

=! 1

2 (1 ! !)x1 + 12 (1 + !)x2

12 (1 ! !)y1 + 1

2 (1 + !)y2

"

=!

N1(!)x1 + N2(!)x2N1(!)y1 + N2(!)y2

"

. (8.3)

8–4


X(!) =X (!)

Y (!)=

12 (1 ! !)X1 + 1

2 (1 + !)X21 (1 ! !)Y + 1 (1 + !)Y2 1 2 2

x(!) =x(!)

y(!)=

12 (1 ! !)x1 + 1

2 (1 + !)x212 (1 ! !)y1 + 1

2 (1 + !)y2

u(!) = x(!)!X(!) =u X (!)

uY (!)=

! 12 (1 ! !)vX1 + 1

2 (1 + !)vX212 (1 ! !)vY 1 + 1

2 (1 + !)vY 2

Bar longitudinal axis

Bar longitudinal axis

2 2

0

0

2 (X ,Y )

2 22 (x ,y )

1 11 (X ,Y )

1 11 (x ,y )

C

C

X, x

Y, y

P (X)

P(x)

Figure 8.3. The definition of displacement field for the plane TL bar element.

Here ! is the usual isoparametric coordinate that varies from !1 at node 1 to +1 at node 2, whereasN1(!) = 1

2 (1 ! !) and N2(!) = 12 (1 + !) are the well known linear shape functions in terms of ! .

The displacement field is obtained by subtracting the foregoing position vectors:

u(!) = x(!) ! X(!) ="

u X (!)

uY (!)

#

="

N1(!) u X1 + N2(!) u X2N1(!) uY 1 + N2(!) uY 2

#

, (8.4)

Equation (8.4) may be expressed in matrix form as

u(!) ="

u X (!)

uY (!)

#

="

N1(!) 0 N2(!) 00 N1(!) 0 N2(!)

#

$

%

&

u X1uY 1u X2uY 2

'

(

)= N(!)u. (8.5)

Here N(!) is the 2 " 4 shape function matrix. The element kinematic defined by these equations ispictured in Figure 8.3

§8.2.2. GL Axial Strain

As discussed in Chapter 7, in the Total Lagrangian (TL) description the Green-Lagrange (GL)strains and the second Piola-Kirchhoff (PK2) stresses are frequently used as conjugate measures inthe formulation of the internal energy. The only GL strain that appears in the energy expression isthe GL axial strain e1 # e, which is most expediciously defined using the length change as

e = L2 ! L20

2L20

, (8.6)

8–5


1(X1, Y1)

2(X2, Y2)

C0

C

X21

Y21

L0

L

!

!0

aX = X21L0

= cos!0

aY = Y21L

= sin!00

ay = y21L0

= LL0sin!1(x1 = X1+ u uX1, y1 = Y1+ Y1)

u u2(x2 = X2+ X2, y2 = Y2+ Y2)

ux21 = X21 + X21

uy21 = Y21 + Y21

ax = x21L0

= LL0cos!

Y, y

X, x

Figure 8.4. Geometric interpretation of quantities used in the study of element kinematics.

rather than through displacement gradients. Because of the linear displacement assumptions (8.5)the strain e is constant over the element.2

This expression can be maneuvered into a matrix function of the node displacements. To expeditethe procedure it is convenient to introduce the following auxiliary variables:

X21 = X2 ! X1, Y21 = Y2 ! Y1, u X21 = u X2 ! u X1, uY21 = uY2 ! uY1,

umX = u X2 + u X1

2, um

Y = uY2 + uY12

, aX = X21L0

= cos!0, aY = Y21L0

= sin!0,

ax = aX + u X21L0

= x21L0

= LL0cos!, ay = aY + uY21

L0= y21

L0= L

L0sin!.

(8.7)

Some of these quantities can be geometrically interpreted as illustrated in Figure 8.4. In particular,um

X and umY are the {X, Y } displacements of the bar midpoints; aX and aY are the {X, Y } direction

cosines of the bar longitudinal axis in the reference configuration. On the other hand, ax and ayare scaled direction cosines, but of the current configuration. The squared bar expressions in e interms of nodal displacements are

L2 = (X21 + u X21)2 + (Y21 + uY21)

2, L2 ! L20 = 2X21 u X21 + 2Y21 uY21 + X221 + Y 2

21, (8.8)whence the GL axial strain (8.6) becomes

e = L2 ! L202L20

= 1L0

(aX u X21 + aY uY21) + 12L20

!

u2X21 + u2Y21"

= 1L0[!aX !aY aX aY ]u+ 1

L20[!um

X !umY um

X umY ]u.

(8.9)

2 This is in fact the only use of the displacement interpolation (8.5) in the ensuing derivations.

8–6


This may be compactly written as

e = B0 u+ 12u

T Hu, (8.10)

in which

B0 = 1L0[!aX !aY aX aY ] , H = 1

L20

!

"

#

1 0 !1 00 1 0 !1

!1 0 1 00 !1 0 1

$

%

&. (8.11)

For further use defineB = B0 +Hu = 1

L0[!ax !ax ax ax ] , (8.12)

which is similar to B0 but with entries evaluated in the current configuration.As can be seen the GL strain splits naturally into two parts: e = eL + eN , in whicheL = B0 u, where B0 is a constant-over-element, 1" 4 rectangular matrix given by (8.11), dependslinearly on the node displacements u. This is the linear part of the GL strain.eN = 1

2uT Hu, whereH is a constant-over-element, 4"4 symmetric square matrix given by (8.11),

depends quadratically on the node displacements. This is the nonlinear part of the GL strain.

§8.2.3. Strain DerivativesFor further use in the computation of internal forces and stiffness matrix, the first and secondderivatives of e with respect to the nodal displacements will be needed. The derivative of e withrespect to u is

!e!u

=!(B0 u+ 1

2uT Hu)

!u= BT0 + uTH = BT . (8.13)

The second derivative of e with respect to u is

!2e!u !u

= !

!u!e!u

= !(BT0 + uTH)

!u= !BT

!u= H. (8.14)

§8.2.4. PK2 Axial StressThe stress measure conjugate to GL strains is the second Piola-Kirchhoff (PK2) stress tensor. Theonly component that appears in the internal energy is the axial stress s, which is related to e throughthe constitutive equation

s = s0 + E e, (8.15)where s0 is the axial stress in the reference configuration (assumed constant over the element), andE is the elastic modulus.The axial force based on this stress is

F = A0 s. (8.16)Note that this is not the true axial force in the current configuration C , which would be

Ftrue = A ", (8.17)

in which " denotes the true or Cauchy stress in C and A is the actual cross-section area there.

8–7


§8.2.5. Total Potential Energy

In what follows it is assumed that the element is subjected only to node forces f that are conservativeand proportional, so that f = !q, where q is the incremental load vector.The Total Potential Energy (TPE) of the element in the current configuration, expressed in terms ofGL strains and PK2 stresses, is

" = U ! W =!

V0(s0 e + 1

2 E e2) dV0 ! fTu =

!

L0A0(s0 e + 1

2 Ee2) d X ! !qTu. (8.18)

where X is directed along the bar longitudinal axis in C0, as shown in Figure 8.4. All integrals arecarried out over the reference (=base) configuration C0. Since the integrands are constant, we get

" = U ! W, U = V0 (s0 e + 12 Ee

2), W = !qT u. (8.19)

Here V0 = A0L0 is the bar volume in C0. This energy expression is separable because the internalenergy U depends only on u through e, and not on !.

§8.2.6. Internal Force Vector

The finite element residual equations are obtained by taking the gradient of (8.19) with respect to u.Since " is separable, r = p! f, where f = #W/#u = !q, and the internal force can be expandedas follows

p = #U#u

= V0(s0#e#u

+ Ee#e#u

) = V0 s#e#u

. (8.20)

Using (8.13) we arrive at the compact expression

p = F L0 BT = F

"

#

$

!ax!ayaxay

%

&

'. (8.21)

Here F = A0 s is the axial force in the current configuration measured per unit area of the referenceconfiguration.3 As regards the geometric interpretation of ax and ay , see Figure 8.5. The relationbetween F = A0s and the true axial force Ftrue = A$ can be worked out from inspection of thisdiagram.

§8.2.7. The Tangent Stiffness Matrix

Because the residual equations are separable the tangent stiffness matrix is obtained simply bydifferentiating the internal force with respect to the node displacements u:

K = #p#u

= #(F L0 BT )

#u= KM +KG . (8.22)

3 This is the PK2 axial force; cf. (8.16) .

8–8


1

2

= !F0 ax

= !F0 ay

= F0 ax

= F0 ay

X2

Y2

X1

Y1

f

f

f

f

F0

Figure 8.5. Geometrical interpretation of the internal force vector. The axial force F0 = A0swould be positive as shown.

The above expression expresses thatK splits naturally into two parts: KM andKG , which are calledthe material stiffness matrix and geometric stiffness matrix, respectively, in the FEM literature. Toget KM note that

!s!u

= !(s0 + Ee)!u

= E!e!u

= E BT . (8.23)

ConsequentlyKM = E A0L0BTB. (8.24)

Inserting the expression (8.12) for B yields

KM = E A0L0

!

"

#

a2x axay !a2x !axayaxay a2y !axay !a2y!a2x !axay a2x axay

!axay !a2y axay a2y

$

%

&(8.25)

This component of K looks formally similar to the stiffness matrix of a linear bar element,4 exceptthat B now depends on u. The dependence of KM on the material properties (here the elasticmodulus E) explains the name “material stiffness” given in the FEM literature.The other component can be obtained using (8.14) and the result is

KG = F L0H = FL0

!

"

#

1 0 !1 00 1 0 !1

!1 0 1 00 !1 0 1

$

%

&. (8.26)

This component ofK depends only on the stress state in the current configuration, because F = A0s.No material properties appear. Thus the name “geometric stiffness” applied to KG .5

4 To which it reduces if u = 0. In that case ax and ay become the sine and cosine of the angle "0 shown in Figure 8.4.5 In the pre-1970 FEM literature, the name “initial stress stiffness” was used for KG by some authors.

8–9


Remark 8.1. Assuming that E , A0 and L0 are nonzero, the rank of KM is obviosly one because B is a 1! 4matrix. On the other hand the rank of the numerical matrix in H is 2 (because its eigenvalues are 2, 2, 0, 0).ConsequentlyKG has rank 2 if s is nonzero and 0 otherwise. Combining these results it can be shown that therank ofK = KM +KG is 1 if the current configuration is unstressed and 2 otherwise. In other words, the rankdeficiency is 3 and 2, respectively. The implications of this property in the analysis of stability are consideredin later Chapters.

Remark 8.2. The addition of KG increases the bar stiffness if the current configuration is in tension (s > 0),but it reduces it if the current configuration is in compression (s < 0). This is in accord with physical intuition.The main effect of this stiffness is on the rotational rigid-body motions of the bar about the Z axis.

8–10

8–11 Exercises


The TL Bar Element

!

SS

H

X, xY, y

(2)(1)

(3)

1

2

4

3

uX

uY

fY = "

E, A0

E, A0

E, A0

Figure E8.1. A 3-bar FEM model for Exercise 8.1.

EXERCISE 8.1 [A/C:25] You go to work as a nonlinear-FEM engineer for a car company. Your supervisorassigns you the job of designing a component of a wheel suspension system that can be modeled by the 3-barstructure depicted in Figure E8.1. The model has the dimensions and properties shown and is only subjectedto vertical loads at node 2. The length S, bar section areas A0 and elastic modulus E are known, but the riseangle ! > 0 is a design variable. Find the largest ! for which bifurcation, which is bad for the wheel, cannotoccur. (For the 2-bar arch example structure that maximum ! was shown to be defined by tan! !

"2/2.)

To study bifurcation, it is enough to set up the tangent stiffness matrix assuming that the X (horizontal)displacement uX is zero. The following tangent stiffness matrix is obtained for S = 2, E = 1, A0 = 1 anduX = 0:

K =! KXX KXYKXY KYY

"

=

#

$

%

uY4 + u2Y

16 + 2+ 2HuY + u2Y&

(1+ H 2)30

0 12 + 3uY

4 + 6u2Y16 + 2H 2 + 6HuY + 3u2Y

&

(1+ H 2)3

'

(

)

(E8.1)Bifurcation points occur if KXX , which is quadratic in uY : a+ b uY + c u2Y , vanishes. Real bifurcation pointsoccur if the discriminant b2 # 4ac $ 0. Study when this happens as a function of H , and deduce the largest! for which bifurcation cannot occur.

8–11


L

X, x

Y, y

(1)

1 uX

uY

2

m

g

Bar element modelsweightless string inreference configuration

E, A0

Point mass

Gravity field

Figure E8.2. Model of a classic pendulum for Exercise 8.2.

EXERCISE 8.2 [A:C:20] Although this course focuses on statics, this exercise deals with the effect of thegeometric stiffness on vibrations. Consider the pendulum configuration idealized in Figure E8.2. A lumpedmass m is suspended by a weightless elastic string. The string is modeled as a 2-node bar element. Thiselement is under a tensile prestress s0 = mg/A0, where g is the accelaration of gravity. The tangent stiffnessmatrix for the cable element in the reference configuration is K = KM +KG , which is 2 ! 2 upon removingthe degrees of freedom at the fixed node 2. Because of the prestress the geometric stiffness does not vanish.The order-2 vibration eigenproblem is

Kzi = !2iMzi , i = 1, 2 (E8.2)

where i is the mode index, !i is the i th circular frequency in radians per second, zi the associated eigenvectorthat include the horizontal and vertical displacements of node 1, and the mass matrix is

M =!m 0

0 m

"

(E8.3)

Compute the two frequencies !1 and !2. One of them, say !1, describes pendulum motions while the otherone pertains to a “bar mode” associated with axial motions. Discuss what happens to !1 and !2 if E " #,which characterizes the “inextensional string” limit, and whether the classical pendulum small-oscillationsfrequency !P =

$g/L is correct.

EXERCISE 8.3 (Requires knowledge of continuum mechanics.) [A:15] Suppose that the bar-element materialis linear isotropic, with elastic modulus E and " is Poisson’s ratio ". Find the relation between the true (Cauchy)axial stress # = #xx in the bar and the PK2 axial stress s = sXX . Hint: study the change in cross section areaas function of ".

8–12

.

9The TL

TimoshenkoPlane Beam

Element

9–1

Chapter 9: THE TL TIMOSHENKO PLANE BEAM ELEMENT 9–2

TABLE OF CONTENTS

Page§9.1. Introduction 9–3§9.2. Beam Models 9–3

§9.2.1. Basic Concepts and Terminology . . . . . . . . . . . 9–3§9.2.2. Mathematical Models: Classical and Timoshenko . . . . . 9–4§9.2.3. Finite Element Models . . . . . . . . . . . . . . 9–5§9.2.4. Bernoulli-Euler versus Timoshenko Beam Elements . . . . 9–7

§9.3. X -Aligned Reference Configuration 9–9§9.3.1. Element Description . . . . . . . . . . . . . . . 9–9§9.3.2. Motion . . . . . . . . . . . . . . . . . . . 9–9§9.3.3. Displacement Interpolation . . . . . . . . . . . . . 9–12§9.3.4. Strain-Displacement Relations . . . . . . . . . . . 9–12§9.3.5. *Consistent Linearization . . . . . . . . . . . . . 9–13

§9.4. Arbitrary Reference Configuration 9–14§9.4.1. Strain-Displacement Matrix . . . . . . . . . . . . 9–14§9.4.2. Constitutive Equations . . . . . . . . . . . . . . 9–15§9.4.3. Strain Energy . . . . . . . . . . . . . . . . . 9–16

§9.5. The Internal Force 9–16§9.6. The Stiffness Matrix 9–17

§9.6.1. The Material Stiffness Matrix . . . . . . . . . . . . 9–17§9.6.2. Eliminating Shear Locking by RBF . . . . . . . . . 9–18§9.6.3. The Geometric Stiffness Matrix . . . . . . . . . . . 9–19

§9.7. A Commentary on the Element Performance 9–21§9.8. Summary 9–22§9. Exercises . . . . . . . . . . . . . . . . . . . . . . 9–23

9–2

9–3 §9.2 BEAM MODELS

§9.1. Introduction

In the present Chapter the Standard Formulation of Total Lagrangian (TL) kinematics is used toderive the finite element equations of a two-node Timoshenko plane beam element. This derivationis more typical of the general case. It is still short, however, of the enormous complexity involved,for instance, in the FEM analysis of nonlinear three-dimensional beams or shells. In fact the latterare still doctoral thesis topics.In the formulation of the bar element in Chapter 8, advantage was taken of the direct expression ofthe axial strain in terms of reference and current element lengths. That shortcut bypasses the useof displacement gradients and coordinate transformations. The simplification works equally wellfor bars in three-dimensional space.A more systematic but lengthier procedure is unavoidable with more complicated elements. Theprocedure requires going through the displacement gradients to construct a strain measure. Some-times this measure is too complex and must be simplified while retaining physical correctness.Then the stresses are introduced and paired with strains to form the strain energy function of theelement. Repeated differentiations with respect to node displacements yield the expressions of theinternal force vector and tangent stiffness matrix. Finally, a transformation to the global coordinatesystem may be required.In addition to giving a better picture of the general procedure, the beam element offers an illustrationof the treatment of rotational degrees of freedom.

§9.2. Beam Models

§9.2.1. Basic Concepts and Terminology

Beams represent the most common structural component found in civil and mechanical structures.Because of their ubiquity they are extensively studied, from an analytical viewpoint, in MechanicsofMaterials courses. Such a basic knowledge is assumed here. The followingmaterial recapitulatesdefinitions and concepts that are needed in the finite element formulation.A beam is a rod-like structural member that can resist transverse loading applied between itssupports. By “rod-like” it is meant that one of the dimensions is considerably larger than the othertwo. This dimension is called the longitudinal dimension and defines the longitudinal directionor axial direction. Directions normal to the longitudinal directions are called transverse. Theintersection of planes normal to the longitudinal direction with the beam are called cross sections,just as for bar elements. The beam longitudinal axis is directed along the longitudinal direction andpasses through the centroid of the cross sections.1.Beams may be used as isolated structures. But they can also be combined to form frameworkstructures. This is actually the most common form of high-rise building construction. Individualbeam components of a framework are called members, which are connected at joints. Frameworks

1 If the beam is built of several materials, as in the case of reinforced concrete, the longitudinal axis passes through thecentroid of a modified cross section. The modified-area technique is explained in elementary courses of Mechanics ofMaterials

9–3


motion

current configuration

reference configuration

Figure 9.1. A geometrically nonlinear plane framework structure.

can be distinguished from trusses by the fact that their joints are sufficiently rigid to transmit bendingmoments between members.In practical structures beam members can take up a great variety of loads, including biaxial bending,transverse shears, axial forces and even torsion. Such complicated actions are typical of spatialbeams, which are used in three-dimensional frameworks and are subject to forces applied alongarbitrary directions.A plane beam resists primarily loading applied in one plane and has a cross section that is symmetricwith respect to that plane. Plane frameworks, such as the one illustrated in Figure 9.1, are assembliesof plane beams that share that symmetry. Those structures can be analyzed with two-dimensionalidealizations.A beam is straight if the longitudinal direction is a straight line. A beam is prismatic if the crosssection is uniform. Only straight, prismatic, plane beams will be considered in this Chapter.

§9.2.2. Mathematical Models: Classical and Timoshenko

Beams are actually three-dimensional solids. One-dimensional mathematical models of planebeams are constructed on the basis of beam theories. All such theories involve some form ofapproximation that describes the behavior of the cross sections in terms of quantities evaluated atthe longitudinal axis. More precisely, the element kinematics of a plane beam is completely definedif the following functions are given: the axial displacement uX (X), the transverse displacementuY (X) and the cross section rotation !Z (X) ! !(X), where X denotes the longitudinal coordinatein the reference configuration. See Figure 9.2.Two beam models are in common use in structural mechanics:Euler-Bernoulli (EB) Model. This is also called classical beam theory or the engineering beamtheory and is the one covered in elementary treatments of Mechanics of Materials. This modelaccounts for bending moment effects on stresses and deformations. Transverse shear forces arerecovered from equilibrium but their effect on beam deformations is neglected. Its fundamentalassumption is that cross sections remain plane and normal to the deformed longitudinal axis. This

9–4


motion



Current cross section

Referencecross section

X

X

, x

Y, y

uX (X)

uY (X)

!Z (X) ! !(X)

Figure 9.2. Definition of beam kinematics in terms of the three displacement functionsuX (X), uY (X) and !(X). The figure actually depicts the EB model kinematics.In the Timoshenko model, !(X) is not constrained by normality (see next figure).

rotation occurs about a neutral axis that passes through the centroid of the cross section.

TimoshenkoModel. This model corrects the classical beam theory withfirst-order shear deformationeffects. In this theory cross sections remain plane and rotate about the same neutral axis as the EBmodel, but do not remain normal to the deformed longitudinal axis. The deviation from normalityis produced by a transverse shear that is assumed to be constant over the cross section.Both the EB and Timoshenko models rest on the assumptions of small deformations and linear-elastic isotropic material behavior. In addition both models neglect changes in dimensions of thecross sections as the beam deforms. Either theory can account for geometrically nonlinear behaviordue to large displacements and rotations as long as the other assumptions hold.

§9.2.3. Finite Element Models

To carry out the geometrically nonlinear finite element analysis of a framework structure, beammembers are idealized as the assembly of one or more finite elements, as illustrated in Figure 9.3.The most common elements used in practice have two end nodes. The i th node has three degrees offreedom: two node displacements uXi and uYi , and one nodal rotation !i , positive counterclockwisein radians, about the Z axis. See Figure 9.4.The cross section rotation from the reference to the current configuration is called ! in both models.In the BE model this is the same as the rotation " of the longitudinal axis. In the Timoshenko

9–5


motion


reference configuration finite element idealizationof reference configuration

finite element idealizationof current configuration

Figure 9.3. Idealization of a geometrically nonlinear beam member (as taken,for example, from a plane framework structure likethe one in Figure 9.1) as an assembly of finite elements.

uX1

uY1

uX2

uY2

!1

!2

uX1

uY1

uX2

uY2

!1

!2

1 2 1 2

(b) C (Timoshenko) model

X, x

Y, y

(a) C (BE) model1 0

Figure 9.4. Two-node beam elements have six DOFs, regardless of the model used.

model, the difference " = # ! ! is used as measure of mean shear distortion.2 These angles areillustrated in Figure 9.5.Either the EB or the Timoshenko model may be used as the basis for the element formulation.Superficially it appears that one should select the latter only when shear effects are to be considered,as in “deep beams” whereas the EB model is used for ordinary beams. But here a “twist” appearbecause of finite element considerations. This twist is one that has caused significant confusionamong FEM users over the past 25 years.

2 It is # ! ! instead of ! ! # because of sign convention, to make eXY positive.

9–6


normal to deformed beam axis

90!

normal to referencebeam axis X

// X (X = X)

ds

θ

ψ

ψdirection of deformedcross section

_γ =

ψ−θ

_

Note: in practice γ << θ; typically 0.1% or less. The magnitudeof γ is grossly exaggerated in the figure for visualization convenience.

__

Figure 9.5. Definition of total section rotation ! and BE section rotation " in theTimoshenko beam model. The mean shear deformation is # = ! " " , whichis constant over the cross section. For small deformations of typical engineeringmaterials # << 1; for example, typical values for |# | are O(10"4) radianswhereas rotations " and ! may be of the order of several radians.

Although the Timoshenko beam model appears to be more complex because of the inclusion ofshear deformation, finite elements based on this model are in fact simpler to construct! Here arethe two main reasons:(i) Separate kinematic assumptions on the variation of cross-section rotations are possible, as

made evident by Figure 9.5. Mathematically: !(X) may be assumed independenltly of u X (X)

and uY (X). As a consequence, two-node Timoshenko elements may use linear variations inboth displacement and rotations. On the other hand a two-node EB model requires a cubicpolynomial for uY (X) because the rotation !(X) is not independent.

(ii) The linear transverse displacement variation matches that commonly assumed for the axialdeformation (bar-like behavior). The transverse and axial displacements are then said to beconsistent.

The simplicity is even more important in geometrically nonlinear analysis, as strikingly illustratedin the two-node elements depicted in Figure 9.6. Although as shown in that figure both of theseelements have six degrees of freedom, the internal kinematics of the Timoshenko model is farsimpler.

§9.2.4. Bernoulli-Euler versus Timoshenko Beam Elements

In the FEM literature, a BE-based model such as the one shown in Figure 9.4(a) is called a C1

beam because this is the kind of mathematical continuity achieved in the longitudinal directionwhen a beam member is divided into several elements (cf. Figure 9.3). On the other hand theTimoshenko-based element shown in Figure 9.4(b) is called a C0 beam because both transversedisplacements, as well as the rotation, preserve only C0 continuity.

9–7


2-node (cubic) elementfor Euler-Bernoulli beam model:plane sections remain plane andnormal to deformed longitudinal axis

2-node linear-displacement-and-rotations element for Timoshenko beam model:plane sections remain plane but notnormal to deformed longitudinal axis

(a) (b)

C1

element with same DOFsC1

C0

Figure 9.6. Sketch of the kinematics of two-node beam finite element models based on(a) Euler-Bernoulli beam theory, and (b) Timoshenko beam theory. Thesemodels are called C1 and C0 beams, respectively, in the FEM literature.

What would be the first reaction of an experienced but old-fashioned (i.e, “never heard aboutFEM”) structural engineer on looking at Figure 9.6? The engineer would pronounce the C0 elementunsuitable for practical use. And indeed the kinematics looks strange. The shear distortion impliedby the drawing appears to grossly violate the basic assumptions of beam behavior. Furthermore, ahuge amount of shear energy would be require to keep the element straight as depicted.The engineer would be both right and wrong. If the two-node element of Figure 9.6(b) wereconstructed with actual shear properties and exact integration, an overstiff model results. Thisphenomenom is well known in the FEM literature and receives the name of shear locking. To avoidlocking while retaining the element simplicity it is necessary to use certain computational devices.The most common are:1. Selective integration for the shear energy.2. Residual energy balancing.These devices will be used without explanation in some of the derivations of this Chapter. Fordetailed justification the curious reader may consult advanced FEM books such as Hughes’.3

In this Chapter the C0 model will be used to illustrated the TL formulation of a two-node, geomet-rically nonlinear beam element.

Remark 9.1. As a result of the application of the aforementioned devices the beam element behaves like a BEbeam although the underlying model is Timoshenko’s. This represents a curious paradox: shear deformation

3 T. J. R. Hughes, The Finite Element Method, Prentice-Hall, 1987.

9–8

9–9 §9.3 X -ALIGNED REFERENCE CONFIGURATION

is used to simplify the kinematics, but then most of the shear is removed to restore the correct stiffness.4 As aresult, the name “C0 element” is more appropriate than “Timoshenko element” because capturing the actualshear deformation is not the main objective.

Remark 9.2. The two-node C1 beam element is used primarily in linear structural mechanics. (It is in factthe beam model used in the “Introduction to FEM” course.) This is because some of the easier-constructionadvantages cited for the C0 element are less noticeable, while no artificial devices to eliminate locking areneeded. The C1 element is also called the Hermitian beam element because the shape functions are cubicpolynomials specified by Hermite interpolation formulas.

§9.3. X-Aligned Reference Configuration

§9.3.1. Element Description

We consider a two-node, straight, prismatic C0 plane beam element moving in the (X, Y ) plane, asdepicted in Figure 9.7(a). For simplicity in the following derivation the X axis system is initiallyaligned with the longitudinal direction in the reference configuration, with origin at node 1. Thisassumption is relaxed in the following section, once invariant strain measured are obtained.The reference element length is L0. The cross section area A0 and second moment of inertia I0with respect to the neutral axis5 are defined by the area integrals

A0 =!

A0d A,

!

A0Y d A = 0, I0 =

!

A0Y 2 d A, (9.1)

In the current configuration those quantities become A, I and L , respectively, but only L is frequentlyused in the TL formulation. The material remains linearly elastic with elastic modulus E relatingthe stress and strain measures defined below.As in the previous Chapter the identification superscript (e) will be omitted to reduce clutter untilit is necessary to distinguish elements within structural assemblies.The element has the six degrees of freedom depicted in Figure 9.4. These degrees of freedom andthe associated node forces are collected in the node displacement and node force vectors

u =

"

#

#

#

#

#

$

uX1uY1!1uX2uY2!2

%

&

&

&

&

&

'

, f =

"

#

#

#

#

#

$

fX1fY1f!1fX2fY2f!2

%

&

&

&

&

&

'

. (9.2)

The loads acting on the nodes will be assumed to be conservative.

4 The FEM analysis of plates and shells is also rife with such paradoxes.5 For a plane prismatic beam, the neutral axis at a particular section is the intersection of the cross section plane X =constant with the plane Y = 0.

9–9


1 2

θ = ψ − γ

P(x,y)

C

X

XP

YC

YC

P (X,Y)o

C(x ,y )

C (X,0)C (X)

o

oYu

YPu

XCuXC

y

y

x

Cx

u

CC

1

2

1 2

θ(X)

X

Y

C(X+u ,u )

X

X Y

u (X) = uu (X) = u

1

2

C0

C

(a) (b)

X, x

Y, y

oL

ψ

_L

Figure 9.7. Lagrangian kinematics of the C0 beam element with X -aligned referenceconfiguration: (a) plane beam moving as a two-dimensional body; (b) reductionof motion description to one dimension measured by coordinate X .

§9.3.2. Motion

The kinematic assumptions of the Timoshenko model element have been outlined in §9.2.2. Ba-sically they state that cross sections remain plane upon deformation but not necessarily normal tothe deformed longitudinal axis. In addition, changes in cross section geometry are neglected.To analyze the Lagrangian kinematics of the element shown in Figure 9.6(a), we study the motion ofa particle originally located at P0(X, Y ). The particle moves to P(x, y) in the current configuration.The projections of P0 and P along the cross sections at C0 and C upon the neutral axis are calledC0(X, 0) and C(xC , yC), respectively. We shall assume that the beam cross section dimensions donot change, and that the shear distortion ! << 1 so that cos ! can be replaced by 1. Then

x = xC ! Y (sin" + sin ! cos") = xC ! Y [sin(" + ! ) + (1 ! cos ! ) sin"] .= xC ! Y sin #,y = yC + Y (cos" ! sin ! sin") = yC + Y [cos(" + ! ) + (1 ! cos ! ) cos"] .= yC + Y cos # .

(9.3)

But xC = X + u XC and yC = u XC . Consequently x = X + u XC ! Y sin # and y = uY C + Y cos # .From now we shall call u XC and uY C simply u X and uY , respectively, so that the Lagrangianrepresentation of the motion is

!

xy

"

=!

X + u X ! Y sin #uY + Y cos #

"

(9.4)

9–10

9–11 §9.3 X -ALIGNED REFERENCE CONFIGURATION

in which uX , uY and ! are functions of X only. This concludes the reduction to a one-dimensionalmodel, as sketched in Figure 9.7(b).For future use below it is convenient to define an “extended” internal displacement vector w, andits gradient or X derivative:

w =! uX (X)

uY (X)

!(X)

"

, w! = dwdX

=! duX/dXduY /dXd!/dX

"

=! u!

Xu!Y! !

"

, (9.5)

in which primes denote derivatives with respect to X . The derivative ! ! is also as " , which has themeaning of beam curvature in the current configuration. Also useful are the following differentialrelations

1+ u!X = s ! cos#, u!

Y = s ! sin#, s ! = dsdX

=#

(1+ u!X )2 + (u!

Y )2, (9.6)

in which ds is the differential arclength in the current configuration; see Figure 9.5. If u!X and u!

Yare constant over the element,

s ! = L/L0, 1+ u!X = L cos#/L0, u!

Y = L sin#/L0. (9.7)

Remark 9.3. The replacement of 1" cos $ by zero in (9.3) is equivalent to saying that Y (1" cos $ ) can beneglected in comparison to other cross section dimensions. This is consistent with the uncertainty in the crosssection changes, which would depend on the normal stress and Poisson’s ratio effects. The motion expression(9.4) has the virtue of being purely kinematic and of leading to (exactly) eYY = 0.

9–11


§9.3.3. Displacement Interpolation

For a 2-node C0 element it is natural to express the displacements and rotation functions as linearin the node displacements:

w =! uX (X)

uY (X)

!(X)

"

= 12

! 1! " 0 0 1+ " 0 00 1! " 0 0 1+ " 00 0 1! " 0 0 1+ "

"

#

$

$

$

$

$

%

uX1uY1!1uX2uY2!2

&

'

'

'

'

'

(

= Nu,

(9.8)in which " = (2X/L0) ! 1 is the isoparametric coordinate that varies from " = !1 at node 1 to" = 1 at node 2. Differentiating this expression with respect to X yields the gradient interpolation:

w" =! u"

Xu"Y! "

"

= 1L0

! !1 0 0 1 0 00 !1 0 0 1 00 0 !1 0 0 1

"

#

$

$

$

$

$

%

uX1uY1!1uX2uY2!2

&

'

'

'

'

'

(

= Gu. (9.9)

§9.3.4. Strain-Displacement Relations

The deformation matrix of the motion (9.4) is

F =! #x#X

#x#Y

#y#X

#y#Y

"

=)

1+ u"X ! Y$ cos ! ! sin !

u"Y ! Y$ sin ! cos !

*

(9.10)

where primes denote derivatives with respect to X , and $ = ! ". The displacement gradient matrixis6

GF = F! I =)

u"X ! Y$ cos ! ! sin !u"Y ! Y$ sin ! cos ! ! 1

*

, (9.11)

from which the Green-Lagrange (GL) strain tensor follows:

e =)

eXX eXYeY X eYY

*

= 12 (F

TF! I) = 12 (GF +GT

F ) + 12G

TFGF

= 12

)

2(u"X ! Y$ cos !) + (u"

X ! Y$ cos !)2 + (u"Y ! Y$ sin !)2 !(1+ u"

X ) sin ! + u"Y cos !

!(1+ u"X ) sin ! + u"

Y cos ! 0

*

(9.12)It is seen that the only nonzero strains are the axial strain eXX and the shear strain eXY +eY X = 2eXY ,whereas eYY vanishes. Through the consistent-linearization techniques described in the subsection

6 This is denoted by GF to avoid clash with the shape function gradient matrix G = N" introduced in (9.9).

9–12

9–13 §9.3 X-ALIGNED REFERENCE CONFIGURATION

below, it can be shown that under the small-strain assumptions made precise therein, the axial straineXX can be replaced by the simpler form

eXX = (1+ u!X ) cos ! + u!

Y sin ! " Y" " 1, (9.13)

in which all quantities appear linearly except ! . The nonzero axial and shear strains will be arrangedin the strain vector

e =!

e1e2

"

=!

eXX2eXY

"

=!

(1+ u!X ) cos ! + u!

Y sin ! " Y ! ! " 1"(1+ u!

X ) sin ! + u!Y cos !

"

=!

e " Y"#

"

. (9.14)

The three strain quantities introduced in (9.14):

e = (1+ u!X ) cos ! + u!

Y sin ! " 1, # = "(1+ u!X ) sin ! + u!

Y cos !, " = ! !, (9.15)

characterize axial strains, shear strains and curvatures, respectively. These are collected in thefollowing generalized strain vector:

hT = [ e # " ] (9.16)

Because of the assumed linear variation in X of uX (X), uY (X) and !(X), e and # only depend on! whereas " is constant over the element. Making use of the relations (9.7) one can express e and# in the geometrically invariant form

1+ e = s ! cos(! " $) = L cos #L0

, # = "s ! sin(! " $) = L sin #L0

(9.17)

In theory one could further reduce e to L/L0 and # to L # /L0, but these “simplifications” actuallycomplicate the strain variations taken in the following Section.

§9.3.5. *Consistent Linearization

The derivation of the consistent linearization (9.14) is based on the following study, known in continuummechanics as a polar decomposition analysis of the deformation gradient. Introduce the matrix

!(%) =# cos% " sin%sin% cos%

$

(9.18)

which represents a two-dimensional rotation (about Z ) through an angle %. Since! is an orthogonal matrix,!T = !"1. The deformation gradient (9.9) can be written

F = !($)

# s ! 00 0

$

+ !(!)

#"Y ! ! 00 1

$

. (9.19)

where s ! is defined in (9.6). Premultiplying both sides of (9.19) by !("!) gives the modified deformationgradient

F = !("!)F =# s ! cos(! " $) " Y ! ! 0

"s ! sin(! " $) 1

$

(9.20)

Now the GL strain tensor 2e = FTF " I does not change if F is premultiplied by an orthogonal matrix !

because FT!T!F = FTF. Consequently 2e = FT F " I. But if the strains remain small, as it is assumed inthe Timoshenko model, the following are small quantities:

9–13


(i) s ! " 1 = (L/L0) " 1 because the axial strains are small;(ii) Y ! ! = Y" because the curvature " is of order 1/R, R being the radius of curvature, and |R| >> |Y |

according to beam theory because Y can vary only up to the cross section in-plane dimension;(iii) # = $ " ! , which is the mean angular shear deformation.Then F = I+L+ higher order terms, where L is a first-order linearization in the small quantities s ! " 1, Y ! !

and # = $ " ! . It follows that

e = 12 (L+ LT ) + higher order terms (9.21)

Carrying out this linearization one finds that eXY and eYY do not change, but that eXX simplifies to (9.13). Itcan also be shown that 2eXY = #

.= # within the order of approximation of (9.21).

§9.4. Arbitrary Reference Configuration

In the general case the reference configuration C0 of the element is not aligned with X . Thelongitudinal axis X forms an angle % with X , as illustrated in Figure 9.8. The six degrees offreedom of the element are indicated in that Figure. Note that the section rotation angle ! ismeasured from the direction Y , normal to X , and no longer from Y as in Figure 9.6.Given the node coordinates (X1, Y1) and (X2, Y2), the reference angle % is determined by

cos% = X21/L0, sin% = Y21/L0, X21 = X2 " X1, Y21 = Y2 " Y1, L20 = X221 + Y 221. (9.22)

The angle & = $+% formed by the current longitudinal axis with X (see Figure 9.12) is determinedby

cos& = cos($ + %) = x21/L , sin& = sin($ + %) = y21/L , withx21 = x2 " x1 = X21 + uX2 " uX1, y21 = y2 " y1 = Y21 + uY2 " uY1, L2 = x221 + y221.

(9.23)Solving the trigonometric relations (9.22)-(9.23) for $ gives

cos$ = X21x21 + Y21y21LL0

= X21(X21 + uX2 " uX1) + Y21(Y21 + uY2 " uY1)LL0

,

sin$ = X21y21 " Y21x21LL0

= X21(Y21 + uY2 " uY1) " Y21(X21 + uX2 " uX1)LL0

.

(9.24)

It follows that L sin$ and L cos$ are exactly linear in the translational node displacements. Thisproperty simplifies considerably the calculations that follow.

§9.4.1. Strain-Displacement Matrix

For the generalized strains it is convenient to use the invariant form (9.17), which does not dependon %. The variations 'e, '# and '" with respect to nodal displacement variations are required in theformation of the strain displacement relation 'h = B 'u. To form Bwe take partial derivatives of e,# and " with respect to node displacements. Here is a sample of the kind of calculations involved:

(e(uX1

= ([L cos(! " $)/L0 " 1](uX1

= ([L(cos ! cos$ + sin ! sin$)/L0 " 1](uX1

= "X21 cos ! + Y21 sin !L20

= " cos% cos ! + sin% sin !L0

= "cos)L0

(9.25)

9–14

9–15 §9.4 ARBITRARY REFERENCE CONFIGURATION

C0

C

ϕ

φ = ψ+ϕ

ϕ

ψ

uX

1(X , Y )1 1

1(x ,y )1 1

2(x ,y )2 2

2(X , Y )2 2uY 1 uX2

uY 21θ

2θ

X, x

Y, y

!// X!// X_

!// Y_

!// Y_

!X_

!Y_

Figure 9.8. Plane beam element with arbitrarily oriented reference configuration.

in which " = # +$, and where use is made of (9.24) in a key step. These derivatives were checkedwith Mathematica. Collecting all of them into matrix B:

B = 1L0

! ! cos" ! sin" L0N1! cos" sin" L0N2!

sin" ! cos" !L0N1 (1 + e) ! sin" cos" !L0N2 (1 + e)0 0 !1 0 0 1

"

. (9.26)

Here N1 = (1 ! %)/2 and N2 = (1 + %)/2 are abbreviations for the element shape functions(caligraphic symbols are used to lessen the chance of clash against axial force symbols).

§9.4.2. Constitutive Equations

Because the beam material is assumed to be homogeneous and isotropic, the only nonzero PK2stresses are the axial stress sX X and the shear stress sXY . These are collected in a stress vector srelated to the GL strains by the linear elastic relations

s =#

sX XsXY

$

=#

s1s2

$

=#

s01 + Ee1

s02 + Ge2

$

=#

s01

s02

$

+#

E 00 G

$ #

e1e2

$

= s0 + Ee, (9.27)

where E is the modulus of elasticity and G is the shear modulus. We introduce the prestressresultants

N 0 =%

A0

s01 d A, V 0 =

%

A0

s02 d A, M0 =

%

A0

!Y s01 d A, (9.28)

9–15


C0

C

M0N0

V0

MN

V

Figure 9.9. Beam stress resultants (internal forces)in the reference and current configurations.

which define the axial forces, transverse shear forces and bending moments, respectively, in thereference configuration. We also define the stress resultants

N = N 0 + E A0 e, V = V 0 + GA0 ! , M = M0 + E I0 ". (9.29)

These represent axial forces, tranverse shear forces and bending moments in the current configura-tion, respectively, defined in terms of PK2 stresses. See Figure 9.9 for signs. These are collectedin the stress-resultant vector

zT = [ N V M ] . (9.30)

§9.4.3. Strain Energy

As in the case of the bar element, the total potential energy # = U ! P is separable becauseP = $qTu. The strain (internal) energy is given by

U =!

V0

"

(s0)T e+ 12e

TEe#

dV =!

A0

!

L0

"

(s01e1 + s02e2) + 12 (Ee

21 + Ge22)

#

d A d X . (9.31)

Carrying out the area integrals while making use of (9.27) through (9.30), U can be written as thesum of three length integrals:

U =!

L0(N 0e + 1

2 E A0 e2) d X +

!

L0(V 0! + 1

2GA0!2) d X +

!

L0(M0" + 1

2 E I0 "2) d X ,

(9.32)The three terms in (9.32) define the energy stored through bar-like axial deformations, shear dis-tortion and pure bending, respectively.

9–16

9–17 §9.6 THE STIFFNESS MATRIX

§9.5. The Internal Force

The internal force vector can be obtained by taking the first variation of the internal energy withrespect to the node displacements. This can be compactly expressed as

!U =!

L0

"

N !e + V !" + M !##

d X =!

L0zT !h d X =

!

L0zTB d X !u. (9.33)

Here h and B are defined in equations (9.16) and (9.26) whereas z collects the stress resultants inC as defined in (9.28) through (9.30). Because !U = p T !u, we get

p =!

L0BT z d X . (9.34)

This expression may be evaluated by a one point Gauss integration rule with the sample point at$ = 0 (beam midpoint). Let %m = (%1 + %2)/2, &m = %m + ', cm = cos&m , sm = sin&m ,em = L cos(%m ! ()/L0 ! 1, "m = L sin(( ! %m)/L0, and

Bm = B|$=0 = 1L0

$

%

!cm !sm ! 12 L0"m cm sm ! 1

2 L0"msm !cm 1

2 L0(1+ em) sm !cm 12 L0(1+ em)

0 0 !1 0 0 1

&

' (9.35)

where subscript m stands for “beam midpoint.” Then

p = L0BTmz =

$

%

!cm !sm 12 L0"m cm sm 1

2 L0"msm !cm ! 1

2 L0(1+ em) !sm cm ! 12 L0(1+ em)

0 0 !1 0 0 1

&

'

T( NVM

)

(9.36)

§9.6. The Stiffness Matrix

The first variation of the internal force vector (9.34) defines the tangent stiffness matrix

!p =!

L0

"

BT !z+ !BT z#

d X = (KM +KG) !u = K !u. (9.37)

This is again the sum of the material stifness KM and the geometric stiffness KG .

§9.6.1. The Material Stiffness Matrix

The material stiffness comes from the variation !z of the stress resultants while keeping B fixed.This is easily obtained by noting that

!z =(

!N!V!M

)

=( E A0 0 0

0 GA0 00 0 E I0

) (

!e!"

!#

)

= S !h, (9.38)

where S is the diagonal constitutive matrix with entries E A0, GA0 and E I0. Because !h = B !u,the term BT !z becomes BTSB !u = KM !u whence the material matrix is

KM =!

L0BTSB d X . (9.39)

9–17


This integral is evaluated by the one-point Gauss rule at ! = 0. Denoting again by Bm the matrix(9.35), we find

KM =!

L0BTmSBm d X = Ka

M +KbM +Ks

M (9.40)

where KaM , Kb

M and KsM are due to axial (bar), bending, and shear stiffness, respectively:

KaM = E A0

L0

"

#

#

#

#

#

$

c2m cmsm !cm"mL0/2 !c2m !cmsm !cm"mL0/2cmsm s2m !"mL0sm/2 !cmsm !s2m !"mL0sm/2

!cm"mL0/2 !"mL0sm/2 " 2mL20/4 cm"mL0/2 "mL0sm/2 " 2mL20/4!c2m !cmsm cm"mL0/2 c2m cmsm cm"mL0/2

!cmsm !s2m "mL0sm/2 cmsm s2m "mL0sm/2!cm"mL0/2 !"mL0sm/2 " 2mL20/4 cm"mL0/2 "mL0sm/2 " 2mL20/4

%

&

&

&

&

&

'

(9.41)

KbM = E I0

L0

"

#

#

#

#

#

$

0 0 0 0 0 00 0 0 0 0 00 0 1 0 0 !10 0 0 0 0 00 0 0 0 0 00 0 !1 0 0 1

%

&

&

&

&

&

'

(9.42)

KsM = GA0

L0

"

#

#

#

#

#

$

s2m !cmsm !a1L0sm/2 !s2m cmsm !a1L0sm/2!cmsm c2m cma1L0/2 cmsm !c2m cma1L0/2

!a1L0sm/2 cma1L0/2 a21L20/4 a1L0sm/2 !cma1L0/2 a21L20/4!s2m cmsm a1L0sm/2 s2m !cmsm a1L0sm/2cmsm !c2m !cma1L0/2 !cmsm c2m !cma1L0/2

!a1L0sm/2 cma1L0/2 a21L20/4 a1L0sm/2 !cma1L0/2 a21L20/4

%

&

&

&

&

&

'

(9.43)in which a1 = 1+ em .

§9.6.2. Eliminating Shear Locking by RBF

How good is the nonlinear material stiffness (9.42)-(9.43)? If evaluated at the reference configura-tion aligned with the X axis, cm = 1, sm = em = "m = 0, and we get

KM =

"

#

#

#

#

#

#

#

#

#

#

#

$

E A0L0 0 0 ! E A0

L0 0 0

0 GA0L0

12GA0 0 !GA0

L012GA0

0 12GA0

E I0L0 + 1

4GA0L0 0 ! 12GA0 ! E I0

L0 + 14GA0L0

! E A0L0 0 0 E A0

L0 0 0

0 !GA0L0 ! 1

2GA0 0 GA0L0 ! 1

2GA0

0 12GA0 ! E I0

L0 + 14GA0L0 0 ! 1

2GA0E I0L0 + 1

4GA0L0

%

&

&

&

&

&

&

&

&

&

&

&

'

(9.44)This is the well known linear stiffness of the C0 beam. As noted in the discussion of Section9.2.4, this element does not perform as well as the C1 beam when the beam is thin because too

9–18

9–19 §9.6 THE STIFFNESS MATRIX

much strain energy is taken by shear. The following substitution device, introduced by MacNeal,7removes that deficiency in a simple way. The shear rigidityGA0 is formally replaced by 12E I0/L20,and magically (9.44) becomes

KM =

!

"

"

"

"

"

"

"

"

"

"

"

"

"

"

#

E A0L0 0 0 ! E A0

L0 0 0

0 12E I0L30

6E I0L20

0 !12E I0L30

6E I0L20

0 6E I0L20

4E I0L0 0 !6E I0

L202E I0L0

! E A0L0 0 0 E A0

L0 0 0

0 !12E I0L30

!6E I0L20

0 12E I0L30

!6E I0L20

0 6E I0L20

2E I0L0 0 !6E I0

L204E I0L0

$

%

%

%

%

%

%

%

%

%

%

%

%

%

%

&

. (9.45)

This is the well known linear stiffness matrix of the C1 (Hermitian) beam based on the Euler-Bernoulli model. That substitution device is called the residual bending flexibility (RBF) correc-tion.8 Its effect is to get rid of the spurious shear energy due to the linear kinematic assumptions. Ifthe RBF is formally applied to the nonlinear material stiffness one gets KM = Ka

M + KMb, whereKaM is the same as in (9.43) (because the axial stiffness if not affected by the substitution), whereas

KbM and Ks

M merge into

KbM = E I

L30

!

"

"

"

"

"

#

12s2m !12cmsm 6a1L0sm !12s2m 12cmsm 6a1L0sm!12cmsm 12c2m !6cma1L0 12cmsm !12c2m !6cma1L06a1L0sm !6cma1L0 a2L20 !6a1L0sm 6cma1L0 a3L20!12s2m 12cmsm !6a1L0sm 12s2m !12cmsm !6a1L0sm12cmsm !12c2m 6cma1L0 !12cmsm 12c2m 6cma1L06a1L0sm !6cma1L0 a3L20 !6a1L0sm 6cma1L0 a2L20

$

%

%

%

%

%

&

(9.46)in which a1 = 1+ em , a2 = 4+ 6em + 3e2m and a3 = 2+ 6em + 3e2m .

Remark 9.4. MacNeal actually proposed the more refined substitution

replace1

GA0by

1GAs

+L20

12E I0(9.47)

where GAs is the actual shear rigidity; that is, As is the shear-reduced cross section studied in Mechanicsof Materials. The result of (9.47) is the C1 Hermitian beam corrected by shear deformations computedfrom equilibrium considerations.9 If the shear deformation is negligible, the right hand side of (9.47) isapproximately L20/(12E I0), which leads to the substitution used above.

7 R. H. MacNeal, A simple quadrilateral shell element, Computers and Structures, 8, 1978, pp. 175-183.8 RBF can be rigurously justified through the use of a mixed variational principle, or through a flexibility calculation.9 See, e.g., J. Przemieniecki, Theory of Matrix Structural Analysis, Dover, New York, 1968.

9–19


§9.6.3. The Geometric Stiffness Matrix

The geometric stiffness KG comes from the variation of B while the stress resultants in z are keptfixed. To get a closed form expression it is convenient to pass to indicial notation, reverting tomatrix notation later upon “index contraction.” Let the entries of KG , B, u and z be denoted asKGi j , Bki , u j and zk , where indices i , j and k range over 1–6, 1–6, and 1–3, respectively. CallA j = !B/!u j , j = 1, . . . 6. Then using the summation convention,

KGi j"u j =!

L0"BT z dX =

!

L0

!Bki!u j

"u j zk dX =!

L0A jki zk dX "u j , (9.48)

whence

KGi j =!

L0zk A j

ki d X , (9.49)

Note that in carrying out the derivatives in (9.49) by hand one must use the chain rule because B isa function of e, # and $ , which in turn are functions of the node displacements u j . To implementthis scheme we differentiate B with respect to each node displacement in turn, to obtain:

A1 = !B!uX1

= 1L0

" 0 0 N1 sin% 0 0 N2 sin%0 0 N1 cos% 0 0 N2 cos%0 0 0 0 0 0

#

,

A2 = !B!uY1

= 1L0

" 0 0 !N1 cos% 0 0 !N2 cos%0 0 N1 sin% 0 0 N2 sin%0 0 0 0 0 0

#

,

A3 = !B!$1

= N1

L0

" sin% ! cos% !N1L0(1+ e) ! sin% cos% !N2L0(1+ e)cos% sin% !N1L0# ! cos% ! sin% !N2L0#0 0 0 0 0 0

#

,

A4 = !B!uX2

= 1L0

" 0 0 !N1 sin% 0 0 !N2 sin%0 0 !N1 cos% 0 0 !N2 cos%0 0 0 0 0 0

#

,

A5 = !B!uY2

= 1L0

" 0 0 N1 cos% 0 0 N2 cos%0 0 !N1 sin% 0 0 !N2 sin%0 0 0 0 0 0

#

,

A6 = !B!$2

= N2

L0

" sin% ! cos% !N1L0(1+ e) ! sin% cos% !N2L0(1+ e)cos% sin% !N1L0# ! cos% ! sin% !N2L0#0 0 0 0 0 0

#

.

(9.50)

To restore matrix notation it is convenient to define

WNi j = A j1i , WVi j = A j

2i , WMi j = A j3i , (9.51)

as the entries of three 6 " 6 “weighting matrices”WN ,WV andWM that isolate the effect of thestress resultants z1 = N , z2 = V and z3 = M . The first, second and third row of each A j becomes

9–20

9–21 §9.7 A COMMENTARY ON THE ELEMENT PERFORMANCE

the j th column ofWN ,WV andWM , respectively. The end result is

WN = 1L0

!

"

"

"

"

"

#

0 0 N1 sin! 0 0 N2 sin!0 0 !N1 cos! 0 0 !N2 cos!

N1 sin! !N1 cos! !N 21 L0(1+ e) !N1 sin! N1 cos! !N1N2L0(1+ e)

0 0 !N1 sin! 0 0 !N2 sin!0 0 N1 cos! 0 0 N2 cos!

N2 sin! !N2 cos! !N1N2L0(1+ e) !N2 sin! N2 cos! !N 22 L0(1+ e)

$

%

%

%

%

%

&

(9.52)

WV = 1L0

!

"

"

"

"

"

#

0 0 N1 cos! 0 0 N2 cos!0 0 N1 sin! 0 0 N2 sin!

N1 cos! N1 sin! !N 21 L0" !N1 cos! !N1 sin! !N1N2L0"

0 0 !N1 cos! 0 0 !N2 cos!0 0 !N1 sin! 0 0 !N2 sin!

N2 cos! N2 sin! !N1N2L0" !N2 cos! !N2 sin! !N 22 L0"

$

%

%

%

%

%

&

(9.53)andWM = 0. Notice that the matrices must be symmetric, sinceKG derives from a potential. Then

KG ='

L0(WN N +WV V ) d X = KGN +KGV . (9.54)

Again the length integral should be done with the one-point Gauss rule at # = 0. Denoting againquantities evaluated at # = 0 by an m subscript, one obtains the closed form

KG = Nm2

!

"

"

"

"

"

#

0 0 sm 0 0 sm0 0 !cm 0 0 !cmsm !cm ! 1

2 L0(1+ em) !sm cm ! 12 L0(1+ em)

0 0 !sm 0 0 !sm0 0 cm 0 0 cmsm !cm ! 1

2 L0(1+ em) !sm cm ! 12 L0(1+ em)

$

%

%

%

%

%

&

+ Vm2

!

"

"

"

"

"

#

0 0 cm 0 0 cm0 0 sm 0 0 smcm sm ! 1

2 L0"m !cm !sm ! 12 L0"m

0 0 !cm 0 0 !cm0 0 !sm 0 0 !smcm sm ! 1

2 L0"m !cm !sm ! 12 L0"m

$

%

%

%

%

%

&

.

(9.55)

in which Nm and Vm are N and V evaluated at the midpoint.

§9.7. A Commentary on the Element Performance

The material stiffness of the present element works fairly well once MacNeal’s RBF device is done.On the other hand, simple buckling test problems, as in Exercise 9.3, show that the geometricstiffness is not so good as that of the C1 Hermitian beam element.10 Unfortunately a simple

10 In the sense that one must use more elements to get equivalent accuracy.

9–21


X

X

Y

Y

Eqs (9.5), (9.8)

Eq. (9.9)

Node Displacements u

Element displacement field w = [u , u , θ]

Displacement gradients w' = [u' , u' , θ' ]

Generalized strains h = [ e , γ, κ ]

Stress resultants z = [ N ,V, M ]

Strain energy U

Eq. (9.15)

Eq. (9.29)

T

T

T

T

Tangent stiffness matrix K = K + K M G

Internal forces p

Eq. (9.34)

Eq. (9.40)

TT TδU = ! z B dX δu = p δuL 0vary U:

_

T Tδp = ! (B δz + δB z) dX δu = (K + K )L0

M Gvary p:_

Figure 9.10. Main steps in the derivation of the C0 plane beam element.

substitution device such as RBF cannot be used to improve KG , and the problem should be viewedas open.An intrinsic limitation of the present element is the restriction to small axial strains. This wasdone to facilitate close form derivation. The restriction is adequate for many structural problems,particularly in Aerospace (example: deployment). However, it means that the element cannot modelcorrectly problems like the snap-through and bifurcation of the arch example used in Chapter 8, inwhich large axial strains prior to collapse necessarily occur.

§9.8. Summary

Figure 9.10 is a roadmap that summarizes the key steps in the derivation of the internal force andtangent stiffness matrix for the C0 plane beam element.

9–22

9–23 Exercises

Homework Exercises for Chapter 9The Plane Beam Element

EXERCISE 9.1 [A+N:20] Consider a plane prismatic beam of length L0, cross section area A0 and secondmoment of inertia I0. In the reference configuration the beam extends from node 1 at (X = Y = 0) to node2 at (X = L0, Y = 0). The beam is under axial prestress force N 0 != 0 in the reference configuration, whileboth V 0 and M0 vanish.(a) Obtain the internal force p, the material stiffness matrix KM and the geometric stiffness matrix KG in

the reference configuration, for which u X1 = uY 1 = u X2 = uY 2 = !1 = !2 = 0.(b) The beam rotates 90" rigidly to a current configuration for which u X1 = uY 1 = 0, u X2 = #L0, uY 2 = L0,

!1 = !2 = 90". Check that the stress resultants do not change (that is, N = N0 and V = M = 0 becausee = " = # = 0), and obtain KM and KG in that configuration.

EXERCISE 9.2

[N:20] Analyze the pure bending of a cantilever discretized into an arbitrary number of elements.


FEM discretization intoN equal length elements

fixed end P

P

e

C0

L

Buckling sketch

P = -λ EI/Lcr cr2

Figure E9.1. Structure for Exercises 9.3 and 9.4.

EXERCISE 9.3

[N:25] This exercise and the next one pertain to the simple structure shown in Figure E9.1. It is a cantilever,plane beam-column of length L , modulus E , area A and inertia I , loaded by an axial force P as shown. Thestructure is discretized into Ne plane beam elements of equal length. The structures moves in the plane of thefigure. The objective is to compute the classical buckling load Pcr and compare with the known analyticalvalue.In the classical buckling analysis, deformations prior to buckling are neglected. The structure stiffness K =KM + KG is evaluated on the reference configuration, with KG evaluated from the internal force state. Thisstability model is called linearized prebuckling or LPB, and is studied in detail in later Chapters. Under theLPB assumptionsKM is constant whileKG linear in the applied forces. Critical point analysis leads to a lineareigensystem called the buckling eigenproblem. The smallest eigenvalue characterizes the critical load, whichfor the LPB model can be shown to produce bifurcation.The classical buckling load (also called Euler load) for the configuration of Figure E9.1 is Pcr = $cr E I/L2

with $cr = #%2/4 (negative because P has to compress the beam-column to achieve buckling). FEM resultsfor Ne = 1, 2, . . . can be obtained by running the code of Figure A9.7. This script assemblesK taking in 0 withE = I = A = L = 1 to simplify computations while leaving P = $ symbolic. The only nonzero internalforce is N0 = P . The determinant det(K) is formed explicitly as a polynomial in P , called the characteristicpolynomial. All of its roots are computed via NSolve. The root closest to zero defines the critical load.

9–23


As answers to the exercise run the script until the largest Ne that can be reasonably handled by yourMathematicaversion (this will depend on memory and CPU speed). Termination can be controlled by adjusting the loopon k, which doubles Ne on each pass.Report on the following items:(a) Your computed !cr ’s for the Ne you were able to run. Comment on whether they converge toward!"2/4.Is the convergence monotonic?(b) Why does NSolve give 2Ne roots?11

(c) Why are the roots real?12

(d) Why do the computations get rapidly very slow as Ne increases?13

EXERCISE 9.4 [N:20] The brute-force technique used to find !cr in Exercise 9.3 is easy to implement but itis extremely inefficient as the number of elements increase.14

Amore effective tecnique is implemented in the script shown inFigureA9.8 of theAddendum. OnMathematicathis works fine on myMac up to 128 elements, beyond which point memory and CPU time requirements growtoo large. As answer to this exercise:(a) Describe what is going on in the script.(b) Discuss why the solution is more efficient and robust than the previous script.15

11 Only one of which is of practical interest.12 Assuming that exact integer arithmetic is used to form the determinant. If you form the determinant in floating-pointcomplex roots will likely emerge because of numerical imprecision.

13 If you can’t guess, try printing the expanded determinant.14 Forming the characteristic polynomial by expanding det(K) is frowned upon by numerical analysts for several reasons,one of which is that the polynomial coefficients tend to get enormously large. This requires either extended precision ifdone in integer arithmetic (as in the script) or rapidly lead to overflow if done in floating point.

15 It is still far, however, from the optimal way to compute !cr . The practical technique used in production FEM codes is avariation on that script, using an eigensolution method called inverse power iteration.

9–24

9–25 Exercises

Addendum for Homework Exercises 9.3 and 9.4

The Mathematica Notebook PlaneBeam.nb contains several modules and scripts that support Exercises 9.3and 9.4. The Notebook file is posted on the courseWeb site. The important pieces of code are briefly describedin this Addendum.Figure A9.1 shows module FormIntForceC0TwoNodePlaneBeam. This forms the internal force vector p ofa two-node, C0, plane beam formulated with the TL description. Although this module is not directly calledin Exercises 9.3 and 9.4, it is used later in the course. It is also used in the verifivation of the stiffness matrixmodules by finite differences.

FormIntForceC0TwoNodePlaneBeam[XY1_,XY2_,u1_,u2_,S0_,z0_]:=Module[{X1,Y1,X21,Y21,L0,uX1,uY1,theta1,uX2,uY2,theta2,x21,y21,thetam,ctheta,stheta,Lcpsi,Lspsi,N0,V0,M0,EA0,GA0,EI0,cphi,sphi,cm,sm,Nm,Vm,Mm,kappa,p},{X1,Y1}=XY1; {X2,Y2}=XY2; X21=X2-X1; Y21=Y2-Y1;{uX1,uY1,theta1}=u1; {uX2,uY2,theta2}=u2;x21=X21+uX2-uX1; y21=Y21+uY2-uY1;L0=PowerExpand[Sqrt[X21^2+Y21^2]]; thetam=(theta1+theta2)/2;ctheta=Cos[thetam]; stheta=Sin[thetam];Lcpsi=Simplify[(X21*x21+Y21*y21)/L0];Lspsi=Simplify[(X21*y21-Y21*x21)/L0];em= (ctheta*Lcpsi+stheta*Lspsi)/L0-1;gm=-(stheta*Lcpsi-ctheta*Lspsi)/L0;kappa=(theta2-theta1)/L0;{N0,V0,M0}=z0; {EA0,GA0,EI0}=S0;Nm=Simplify[N0+EA0*em]; Vm=Simplify[V0+GA0*gm];Mm=Simplify[M0+EI0*kappa];cphi=X21/L0; sphi=Y21/L0;cm=ctheta*cphi-stheta*sphi; sm=stheta*cphi+ctheta*sphi;Bm= (1/L0)*{{-cm,-sm, L0*gm/2, cm,sm, L0*gm/2 },

{ sm,-cm,-L0*(1+em)/2,-sm,cm,-L0*(1+em)/2},{ 0, 0, -1, 0, 0, 1 }};

p=L0*Transpose[Bm].{{Nm},{Vm},{Mm}};Return[Simplify[p]] ];

p=FormIntForceC0TwoNodePlaneBeam[{0,0},{L/Sqrt[2],L/Sqrt[2]},{0,0,Pi/2},{-2*L/Sqrt[2],0,Pi/2},{EA,GA0,EI},{0,V0,0}];

Print["p=",p];

Figure A9.1. Evaluation of internal force vector p for plane beam element.

The 6 arguments of this module are XY1, XY2, u1, u2, S0 and z0. XY1 lists the coordinates {X1, Y2} of node1 whereas XY2 lists the coordinates {X1, Y2} of node 2, in the reference configuration. u1 passes the threedisplacements: {uX1, uY1, !1} of node 1, and u2 does the same for node 2. S0 collects the section integratedconstitutive properties {E A0,GA0, E I0}. To apply MacNeal’s RBF, GA0 should be replaced by 12E I0/L20as discussed in Section 9.5. Finally z0 passes the internal forces {N0, V0,M0} in the reference configuration;

9–25


FormMatStiffC0TwoNodePlaneBeam[XY1_,XY2_,u1_,u2_,S0_,z0_]:=Module[{X1,Y1,X21,Y21,L0,uX1,uY1,theta1,uX2,uY2,theta2,x21,y21,thetam,ctheta,stheta,Lcpsi,Lspsi,N0,V0,M0,EA0,GA0,EI0,cphi,sphi,cm,sm,a1,KM},{X1,Y1}=XY1; {X2,Y2}=XY2; X21=X2-X1; Y21=Y2-Y1;{uX1,uY1,theta1}=u1; {uX2,uY2,theta2}=u2;x21=X21+uX2-uX1; y21=Y21+uY2-uY1;L0=PowerExpand[Sqrt[X21^2+Y21^2]]; thetam=(theta1+theta2)/2;ctheta=Cos[thetam]; stheta=Sin[thetam];Lcpsi=Simplify[(X21*x21+Y21*y21)/L0];Lspsi=Simplify[(X21*y21-Y21*x21)/L0];em= (ctheta*Lcpsi+stheta*Lspsi)/L0-1;gm=-(stheta*Lcpsi-ctheta*Lspsi)/L0;cphi=X21/L0; sphi=Y21/L0;cm=ctheta*cphi-stheta*sphi; sm=stheta*cphi+ctheta*sphi;{N0,V0,M0}=z0; {EA0,GA0,EI0}=S0; a1=1+em;KM = (EA0/L0)*{{ cm^2,cm*sm,-cm*gm*L0/2,-cm^2,-cm*sm,-cm*gm*L0/2},{ cm*sm,sm^2,-gm*L0*sm/2,-cm*sm,-sm^2,-gm*L0*sm/2},{-cm*gm*L0/2,-gm*L0*sm/2,gm^2*L0^2/4,cm*gm*L0/2,gm*L0*sm/2,gm^2*L0^2/4},

{-cm^2,-cm*sm,cm*gm*L0/2,cm^2,cm*sm,cm*gm*L0/2},{-cm*sm,-sm^2,gm*L0*sm/2,cm*sm,sm^2,gm*L0*sm/2},{-cm*gm*L0/2,-gm*L0*sm/2,gm^2*L0^2/4,cm*gm*L0/2,gm*L0*sm/2,gm^2*L0^2/4}}+

(EI0/L0)*{{0,0,0,0,0,0}, {0,0,0,0,0,0},{0,0,1,0,0,-1}, {0,0,0,0,0,0},{0,0,0,0,0,0}, {0,0,-1,0,0,1}}+

(GA0/L0)*{{sm^2,-cm*sm,-a1*L0*sm/2,-sm^2,cm*sm,-a1*L0*sm/2},{-cm*sm,cm^2,cm*a1*L0/2,cm*sm,-cm^2,cm*a1*L0/2},{-a1*L0*sm/2,cm*a1*L0/2, a1^2*L0^2/4,a1*L0*sm/2,-cm*a1*L0/2, a1^2 L0^2/4},

{-sm^2,cm*sm, a1*L0*sm/2,sm^2,-cm*sm,a1*L0*sm/2},{ cm*sm,-cm^2,-cm*a1*L0/2,-cm*sm,cm^2,-cm*a1*L0/2},{-a1*L0*sm/2,cm*a1*L0/2,a1^2*L0^2/4,a1*L0*sm/2,-cm*a1*L0/2, a1^2*L0^2/4}};

Return[KM] ];

Figure A9.2. Evaluation of material stiffness matrix KM for plane beam element.

these forces are assumed to be constant along the element. Thmodule forms the internal force vector p with aone point integration rule as discussed earlier, and returns p as function value.Figures A9.2 and A9.3 show modules FormMatStiffC0TwoNodePlaneBeam andFormGeoStiffC0TwoNodePlaneBeam. As their name suggest, these form thematerial and geometric stiffnesscomponents, respectively, of the plane beam element. They have been separated for convenience although formost applications they are combined to form the tangent stiffness matrix. The 6 arguments are exactly the

9–26

9–27 Exercises

FormGeoStiffC0TwoNodePlaneBeam[XY1_,XY2_,u1_,u2_,S0_,z0_]:=Module[{X1,Y1,X21,Y21,L0,uX1,uY1,theta1,uX2,uY2,theta2,x21,y21,thetam,ctheta,stheta,Lcpsi,Lspsi,N0,V0,M0,EA0,GA0,EI0,cphi,sphi,cm,sm,Nm,Vm,KG},{X1,Y1}=XY1; {X2,Y2}=XY2; X21=X2-X1; Y21=Y2-Y1;{uX1,uY1,theta1}=u1; {uX2,uY2,theta2}=u2;x21=X21+uX2-uX1; y21=Y21+uY2-uY1;L0=PowerExpand[Sqrt[X21^2+Y21^2]]; thetam=(theta1+theta2)/2;ctheta=Cos[thetam]; stheta=Sin[thetam];Lcpsi=Simplify[(X21*x21+Y21*y21)/L0];Lspsi=Simplify[(X21*y21-Y21*x21)/L0];em= (ctheta*Lcpsi+stheta*Lspsi)/L0-1;gm=-(stheta*Lcpsi-ctheta*Lspsi)/L0;kappa=(theta2-theta1)/L0;{N0,V0,M0}=z0; {EA0,GA0,EI0}=S0;Nm=Simplify[N0+EA0*em]; Vm=Simplify[V0+GA0*gm];cphi=X21/L0; sphi=Y21/L0;cm=ctheta*cphi-stheta*sphi; sm=stheta*cphi+ctheta*sphi;KG = Nm/2*{{ 0, 0, sm, 0, 0, sm}, {0, 0, -cm, 0, 0,-cm},

{sm, -cm, -(L0/2)*(1+em), -sm, cm,-(L0/2)*(1+em)},{0, 0, -sm, 0, 0, -sm}, {0, 0, cm, 0, 0, cm},{sm, -cm, -(L0/2)*(1+em), -sm, cm,-(L0/2)*(1+em)}}+

Vm/2*{{0, 0, cm, 0, 0, cm}, {0, 0, sm, 0, 0, sm},{cm, sm, -(L0/2)*gm, -cm, -sm,-(L0/2)*gm},{0, 0, -cm, 0, 0, -cm}, {0, 0, -sm, 0, 0, -sm},{cm, sm, -(L0/2)*gm, -cm, -sm,-(L0/2)*gm}};

Return[KG] ];

Figure A9.3. Evaluation of geometric stiffness matrix KG for plane beam element.

FormTanStiffC0TwoNodePlaneBeam[XY1_,XY2_,u1_,u2_,S0_,z0_]:=Module[{},KM=FormMatStiffC0TwoNodePlaneBeam[XY1,XY2,u1,u2,S0,z0];KG=FormGeoStiffC0TwoNodePlaneBeam[XY1,XY2,u1,u2,S0,z0];Return[KM+KG]];

ClearAll[EA,GA,EI];KM=FormMatStiffC0TwoNodePlaneBeam[{0,0},{10,0},{0,0,0},{0,0,0},

{EA,GA,EI},{0,0,0}]; Print[KM];KG=FormGeoStiffC0TwoNodePlaneBeam[{0,0},{4,3},{0,0,0},{0,0,0},

{1,1,1},{10,30,20}]; Print[KG];Print[Chop[Eigenvalues[N[KG]]]];

Figure A9.4. Evaluation of tangent stiffness matrix K for plane beam element,along with test statements.

9–27


MergeElemIntoMasterIntForce[pe_,eftab_,pm_]:=Module[{i,ii,nf=Length[eftab],p}, p=pm;

For[i=1, i<=nf, i++, ii=eftab[[i]];If [ii>0,p[[ii,1]]+=pe[[i,1]]]

]; Return[p]];

MergeElemIntoMasterStiff[Ke_,eftab_,Km_]:=Module[{i,j,ii,jj,nf=Length[eftab],K}, K=Km;

For[i=1, i<=nf, i++, ii=eftab[[i]]; If[ii==0,Continue[]];For[j=i, j<=nf, j++, jj=eftab[[j]];If [ii>0 && jj>0,K[[jj,ii]]=K[[ii,jj]]+=Ke[[i,j]]]]

]; Return[K]];

Figure A9.5. Merge modules for master internal forceand master stiffness of a plane beam structure.

same as for the internal force module, Both matrices are formed with the one-point integration rule.and arereturn as function values. The correctness of the implementation is checked with a finite difference techniqueimplemented in a script not listed here.

Figure A9.4 lists module FormTanStiffC0TwoNodePlaneBeam, which returns the tangent stiffness matrixK = KM +KG . It simple calls the previous two modules and returns the matrix sum as function value.

Figure A9.5 lists two modules that merge the internal force vector and stiffness matrix, respectively, of oneindividual element into the corresponding master quantities for the entire structure.

FigureA9.6 lists twomodules: AssembleMasterStiffOfCantBeam andAssembleMasterIntForceOfCantBeamthat assemble p and K, respectively, for one specific structure. This structure is the cantilever beam-columnunder axial force P shown in Figure E9.1, discretized into Ne ! 1 plane beam elements.

Figures A9.7 and A9.8 lists two scripts for use in Exercises 9.3 and 9.4, respectively.

9–28

9–29 Exercises

AssembleMasterIntForceOfCantBeam[{L_,P_,Ne_},{Em_,A0_,I0_},u_]:=Module[{e,numnod,numdof,u1,u2,u3,Le,X1,X2,S0,z0,eftab,

pe,p},numnod=Ne+1; numdof=3*numnod; Le=L/Ne;p=Table[0,{numdof},{1}];z0={P,0,0}; S0={Em*A0,12*Em*I0/Le^2,Em*I0}; (* RBF *)For [e=1, e<=Ne, e++,X1=(e-1)*Le; X2=e*Le;If [e==1,u1={0,0,0}; u2=Take[u,{3*e-2,3*e}];

eftab={0,0,0,3*e-2,3*e-1,3*e}];If [e>1, u1=Take[u,{3*e-5,3*e-3}]; u2=Take[u,{3*e-2,3*e}];

eftab={3*e-5,3*e-4,3*e-3,3*e-2,3*e-1,3*e}];pe=FormIntForceC0TwoNodePlaneBeam[{X1,0},{X2,0},u1,u2,S0,z0];(*Print["pe=",pe];*)p=MergeElemIntoMasterIntForce[pe,eftab,p];

];Return[Simplify[p]]

];

AssembleMasterStiffOfCantBeam[{L_,P_,Ne_},{Em_,A0_,I0_},u_]:=Module[{e,numnod,numdof,u1,u2,u3,Le,X1,X2,S0,z0,eftab,

pe,p},numnod=Ne+1; numdof=3*numnod-3; Le=L/Ne;K=Table[0,{numdof},{numdof}];z0={P,0,0}; S0={Em*A0,12*Em*I0/Le^2,Em*I0}; (* RBF *)For [e=1, e<=Ne, e++,X1=(e-1)*Le; X2=e*Le;If [e==1,u1={0,0,0}; u2=Take[u,{3*e-2,3*e}];

eftab={0,0,0,3*e-2,3*e-1,3*e}];If [e>1, u1=Take[u,{3*e-5,3*e-3}]; u2=Take[u,{3*e-2,3*e}];

eftab={3*e-5,3*e-4,3*e-3,3*e-2,3*e-1,3*e}];Ke=FormTanStiffC0TwoNodePlaneBeam[{X1,0},{X2,0},u1,u2,S0,z0];(*Print["Ke=",Ke];*)K=MergeElemIntoMasterStiff[Ke,eftab,K];

];Return[Simplify[K]]

];

Figure A9.6. Assembly modules for cantilevered beam-column under axial load.

9–29


ClearAll[L,P,Em,A0,I0]; Em=A0=I0=L=1; Ne=1;For [k=1, k<=6, k++,

Print["Number of elements=",Ne];LPNe={L,P,Ne}; EAI={Em,A0,I0}; u=Table[0,{3*Ne+3}];K=AssembleMasterStiffOfCantBeam[LPNe,EAI,u];(*Print["K=",K//MatrixForm];*)detK=Det[K];(*Print["det(K)=",detK//InputForm];*)roots=NSolve[detK==0,P];Print["roots of stability det=",roots]; Ne=2*Ne;

];Print["exact buckling load coeff is ",-N[Pi^2/4]];

Figure A9.7. Script for Exercise 9.3. It uses assembly,merge and element formation modules.

ClearAll[L,P,Em,A0,I0]; Em=A0=I0=L=1; Ne=1;For [k=1, k<=8, k++,

Print["Number of elements=",Ne];LPNe={L,P,Ne}; EAI={Em,A0,I0}; u=Table[0,{3*Ne+3}];K=AssembleMasterStiffOfCantBeam[LPNe,EAI,u];KM=Coefficient[K,P,0]; KG=Coefficient[K,P,1];SG=LinearSolve[N[KM],N[KG]];(*Print["KM=",KM//MatrixForm]; Print["KG=",KG//MatrixForm];*)(*Print["SG=",SG//MatrixForm];*)emax=-Max[Eigenvalues[SG]]; Print["FEM lambda cr=",1/emax];Ne=2*Ne;

];Print["exact buckling lambda coeff is ",-N[Pi^2/4]//InputForm];

Figure A9.8. Script for Exercise 9.4. It uses assembly,merge and element formation modules.

9–30

.

10The Core-Congruential

Formulation:Core Equations

10–1

Chapter 10: THE CORE-CONGRUENTIAL FORMULATION: CORE EQUATIONS 10–2

TABLE OF CONTENTS

Page§10.1. Introduction 10–3§10.2. Overview 10–3

§10.2.1. Basic Concepts . . . . . . . . . . . . . . . . . 10–4§10.2.2. Direct and Generalized CCF . . . . . . . . . . . . 10–4§10.2.3. The CCF Philosophy: Divide and Conquer . . . . . . . 10–5

§10.3. Historical Background 10–6§10.4. Core Stiffness Equations 10–8

§10.4.1. TL Description of Particle Motion . . . . . . . . . . 10–8§10.4.2. Energy Variations . . . . . . . . . . . . . . . . 10–9§10.4.3. Parametrized Forms . . . . . . . . . . . . . . . 10–9§10.4.4. Spectral Forms . . . . . . . . . . . . . . . . . 10–11§10.4.5. Generalization to H(g) . . . . . . . . . . . . . . 10–11

§10.5. Core Stiffness Derivation Examples 10–12§10.5.1. Bar in 3D Space . . . . . . . . . . . . . . . . 10–12§10.5.2. Plate in Plane Stress . . . . . . . . . . . . . . . 10–14§10.5.3. Plate Bending . . . . . . . . . . . . . . . . . 10–16§10.5.4. 2D Timoshenko Beam . . . . . . . . . . . . . . 10–18§10.5.5. 3D Timoshenko Beam: Kinematics . . . . . . . . . . 10–20§10.5.6. 3D Timoshenko Beam: Core equations . . . . . . . . 10–23

§10.6. References 10–24§10. Exercises . . . . . . . . . . . . . . . . . . . . . . 10–25

10–2

10–3 §10.2 OVERVIEW

In Chapter 8 it was noted that two methods for developing geometrically nonlinear elements basedon the Total Lagrangian description exist: the Standard Formulation (SF) and theCore-CongruentialFormulation or CCF. Chapters 10 and 11 cover the second method. The present exposition is takenlargely from a recent survey article [1].

§10.1. Introduction

There is an elegant Total Lagrangian (TL) formulation of geometrically nonlinear mechanical finiteelements that has received little attention in the literature. This will be referred to as the Core-Congruential Formulation, or CCF, in the sequel. The key concepts, presented by Rajasekaran andMurray [2] in 1973, evolved from the analysis and reinterpretation of the pioneer work of MalletandMarcal [3] as well as Murray’s previous work in geometrically nonlinear finite element analysis[4]. The discussion of Reference [2] by Felippa [5] provided parametric expressions for the stiffnessmatrices that appear at various levels of the discrete governing equations. This work originatedwhat is called here the Direct Core Congruential Formulation, or DCCF.In 1987 this course presented the derivation of several elements using the DCCF. Preparationof homework assignments and feedback from students in this and follow-up offerings helped tostreamline the material. Subsequently Crivelli’s doctoral thesis [6] used the CCF in the systematicdevelopment of a three-dimensional nonlinear Timoshenko beam element capable of undergoingarbitrarily large rotations. Challenges posed by this application pushed this formulation beyondfrontiers hitherto deemed impassable by a TL element with rotational degrees of freedom. Thisdevelopment was summarily reported in a survey article by Felippa and Crivelli [7] and explainedin more detail in a subsequent paper by Crivelli and Felippa [8].A lesson gained from this research is that, when dealing with 3D finite rotations, the CCF shouldbe applied in a staged fashion that allows the systematic examination of additional terms arising inthe transformations to physical degrees of freedom. That transformation methodology gave rise towhat is here called the Generalized CCF, or GCCF.Both DCCF and GCCF share the same “divide and conquer” philosophy. However, the coreequations as well as subsequent steps that transform those equations to physical freedoms varyin complexity. To simplify the exposition while focusing on the essential aspects, Sections 10.3through 10.7 focus on the DCCF. Examples of application to elements amenable to the directtreatment are presented. The GCCF is discussed in Chapter 11, and illustrated with applications to2D and 3D beam elements.

Remark 10.1. Several authors have expressed the belief that the approximation performance of TL-basedelements degrades beyond moderate rotations, and an updated Lagrangian or corotational description is nec-essary for handling truly large motions. For example, in 1986 Mathiasson, Bengtsson and Samuelsson [9]concluded that “The TL formulation can only be used in problems with small or moderate displacements.”More recently Bergan and Mathisen [10] voice a similar opinion: “it is commonly known that in a step by stepTL formulation artificial strains easily arise in beam elements due to nonhomogeneities in the displacementexpansions in transverse and longitudinal directions.” Our experience shows that such limitations are not in-herent in the TL description but instead emerge when a priori kinematic approximations are made to simplifyelement derivations. The 3D beam element just cited exhibits computational and approximation performancefor very large rotations comparable to those based on the co-rotational and Updated Lagrangian descriptionswhile retaining certain advantages listed in the Conclusions.

10–3


§10.2. Overview

§10.2.1. Basic Concepts

The original development of the CCF was concerned with the construction of TL stiffness matricesfor geometrically nonlinear analysis through the congruential-transformation pattern

Klevel =!

V0GTSlevelG dV, (10.1)

where S is the core stiffness matrix, K the physical stiffness in terms of the nodal degrees offreedom v, G a core-to-physical-freedom transformation matrix assumed to be independent ofv, V0 the appropriate reference integration volume, and in which “level” identifies the governingequation level at which the stiffness matrix is used.The three variational levels of interest in practice are: energy (level 0), force equilibrium (level 1),and first-order incremental equilibrium (level 2). Qualifiers “residual-force” and “secant-stiffness”are also used for level 1, and “tangent-stiffness” used for level 2.The core stiffness matrix is expressed in terms of the displacement gradients at each material point.Displacement gradients g make a better choice of core variables than finite strains because forelements with translational degrees of freedom (DOFs) they can be expressed linearly in terms ofnode displacements v as g = Gv, a property that validates (10.1) for all levels. As discussed below,such elements fall under the purview of the Direct CCF.The qualifier “core” emphasizes the goal of independence of Slevel with respect to discretizationdecisions such as element geometry, shape functions, and choice of nodal degrees of freedom.Such a dependence is introduced by the congruential transformation indicated in (10.1) and theintegration over the element volume.

§10.2.2. Direct and Generalized CCF

The basic schematics of the CCF, mathematically expressed through (10.1), may be diagrammedas

CoreSti!nessEquations

!"Congruential

TransformationEquations

!"Physical-DOF

Sti!nessEquations

But this panoramic view needs to be rendered more precise. If the relation between core DOFs(the displacement gradients g) and the physical DOFs (the node displacements v of a finite elementmodel) is linear, these transformations do not depend on level:

(0) Core Energy Sti!ness(1) Core Secant Sti!ness(1) Core Internal Force(2) Core Tangent Sti!ness

!"Congruential

TransformationEquations

!"Physical Energy Sti!nessPhysical Secant Sti!nessPhysical Internal Force

Physical Tangent Sti!ness

10–4

10–5 §10.2 OVERVIEW

In this diagram, numbers annotated within the “core box” denote the variational level of the gov-erning equation in use. Internal force and secant stiffness are two alternative governing-equationexpressions at level 1. The energy level (level 0) may also be expressed in several ways, but this isnot shown in the diagrams to reduce clutter. Under the aforementioned assumption we obtain theDirect Core Congruential Formulation, or DCCF.If the relation between displacement gradients g and node displacements v is nonlinear, the trans-formations sketched above are not only more complex but depend on variational level and possiblythe expression form used within a level. This complication arises when elements with rotationaldegrees of freedom such as beams, plates and shells are considered. It gives rise to the GeneralizedCore Congruential Formulation, or GCCF.Two variants of the GCCF may be distinguished. If the relation between g and v is nonlinearbut algebraic, the transformation equations do vary with level but in principle are still possible asillustrated in the following diagram.

(0) Core Energy Sti!ness(1) Core Secant Sti!ness

(1) Core Internal Force(2) Core Tangent Sti!ness

!"

!"

S-CongruentialTransformation

Equations

T-CongruentialTransformation

Equations

!"

!"

Physical Energy Sti!nessPhysical Secant Sti!ness

Physical Internal ForcePhysical Tangent Sti!ness

Here “T-Congruential” and “S-Congruential” are abbreviations for “Tangent Congruential” and“Secant-Congruential,” respectively. Such a distinction is elaborated upon in Chapter 11.If the relation between g and v is nonlinear and can be expressed only in non-integrable differentialform, the “Secant Transformation Equations” of the preceding diagram do not generally exist, andthe diagram must be truncated:

(0) Core Energy Sti!ness(1) Core Secant Sti!ness

(1) Core Internal Force(2) Core Tangent Sti!ness

!"T-CongruentialTransformation

Equations

!" Physical Internal ForcePhysical Tangent Sti!ness

These two variants of the GCCF are called Algebraic GCCF and Differential GCCF and denoted byacronyms AGCCF and DGCCF, respectively, in the sequel. The main distinction between AGCCFand DGCCF is that it makes no sense to talk about missing quantities, such as the physical secantstiffness, with the latter.The original development of the CCF outlined in the Introduction focused on elements withtranslational-degree-of-freedom configurations. For such elements the Direct form of the CCF,or DCCF, is sufficient. Sections §10.3 through §10.7 focus on that form, leaving the developmentand application of the GCCF to the next Chapter.

10–5


§10.2.3. The CCF Philosophy: Divide and Conquer

The CCF derivation of the finite element equations naturally reflects the outlined framework. Itproceeds through two phases: a core phase followed by a transformation phase. In the initial phasecore energy, secant and tangent stiffness matrices as well as internal force vectors are obtained.These matrices and vectors pertain to individual particles. For the stiffness matrices they arecollectively represented by the term Slevel in (10.1).The key goal is to try to make such core equations as independent as possible with respect to finite-element discretization decisions such as element geometry, shape functions, selection of nodaldegrees of freedom and (in the case of rotational DOFs) rotational parametrizations. To emphasizethis independence, the term core was coined. Complete independence is in fact achievable if therelation between displacement gradients g and v is linear, which characterizes the DCCF. The goalhas to be tempered if the relation is nonlinear because dependenciesmay arise at the tangent stiffnesslevel. Such dependencies create the so-called complementary geometric stiffness terms, which arecharacteristic of elements that fall under purview of the GCCF.In the transformation phase, these core forms are transformed to physical DOFs, i.e. element nodedisplacements. The transformation may be done directly for simple elements and in multistagefashion for complex ones. In particular, multistage transformations are recommended for elementsthat require the Differential GCCF such as 3D beam and shell elements. In this case the trans-formation phase is decomposed into transformation stages that progressively “bind” particles intolines, areas or volumes through kinematic constraints, and eventually link the element domain tothe nodal degrees of freedom. Decisions such as the choice of specific parametrizations for finiterotations may be deferred to final stages.What are the differences between the CCF and the more conventional Total Lagrangian formulationof nonlinear finite elements? If kinematic exactness is maintained throughout, the final discreteequations are identical. This is shown in Appendix 1 for the DCCF applied to continuum elements.But in geometrically nonlinear analysis approximations of various kinds are common, especiallyin structural elements with rotational degrees of freedom such as beams, plates and shells. In theconventional formulation it is quite difficult to assess a priori the effect of seemingly innocuousapproximations “thrown into the pot,” and a posteriori exhaustive testing of complex situationsbecomes virtually impossible. Sample: how does the neglect of higher order terms in the axialdeformation of a spinning 3D beam affects torsional buckling?The staged approach recommended for the GCCF permits a better control over such assumptions.The core equations are physically transparent, clearly displaying the effect of material behavior,displacement gradients and prestresses. In the ensuing transformation sequence the origin of eachterm can be accurately traced, and on that basis informed decisions on retention or dropping made.This process can be aided by computer by testing subproblems that isolate the physics modeled byspecific terms.From this discussion it follows that, from the standpoint of element development, evaluation andtesting, the most significant advantage that can be claimed for the CCF is the clean separation ofphysical effects. The importance of this factor should not be underestimated, because physicaltransparency is the key to success in nonlinear analysis.

§10.3. Historical Background

10–6


In 1968 Mallet and Marcal [3] attempted to establish a standard nomenclature for geometricallynonlinear finite element structural analysis based on the Total Lagrangian (TL) kinematic descrip-tion. Consider a discrete, finite element model of a static structural system under dead loadingwith nodal displacement degrees of freedom collected in array v. Displacements are measuredfrom a fixed reference configuration C0 to a current configuration C. The virtual-work conjugateforces, independent of v, are collected in array p. The system has a total potential energy function! = U ! P that is the difference between the strain energy U and the loads potential P = pT v.The residual node forces are r = "!/"v, and the symbol # denotes increment associated withthe variation of the current configuration. (In keeping up with the spirit of Reference [3] actualvariations are used below rather than virtual ones; the latter are identified by the usual $ prefix.)Mallet and Marcal expressed the total potential energy, the residual (force-balance) equilibriumequations, and the incremental equilibrium equations as follows:

! = U ! P = 12v

T !K0 + 1

3N1 + 16N2

"v! pT v, (10.2)

r = "!

"v=

!K0 + 1

2N1 + 13N2

"v! p = 0, (10.3)

#r = [K0 + N1 + N2] #v!#p = 0. (10.4)

Here K0 is the linear stiffness matrix evaluated at the reference configuration, whereas N1 and N2are nonlinear stiffness matrices, also evaluated at the reference configuration, that depend linearlyand quadratically, respectively, on the node displacements v. The N matrices were said “to repeat”in the foregoing expressions. (This old notation has not survived; presently symbol N is mostcommonly used to identify matrices of element shape functions.)Five years later Rajasekaran and Murray [2] examined more critically the structure of the matricesthat appear in the above equations. In that investigation they chose to start from the “core” stiffnessmatrices corresponding to K, N1 and N2 expressed in terms of displacement gradients, and indoing so laid down the main idea of the CCF. Working with specific elements they showed thatthe nonlinear stiffness matrices N1 and N2 are not uniquely determined. Indeed (10.2)-(10.4) aswritten are unique only for a single degree of freedom. They did not present, however, a generalexpression valid for arbitrary elements. This was partly done by Felippa [5], who in the discussionof Reference [2] considered again those equations, rewritten here in a more general and compactform:

! = 12v

TKUv+#p0 ! p

$T v, (10.5)r = Kr v+ p0 ! p = f! p = 0, (10.6)

#r = K#v!#p = 0, (10.7)

in which the notation of this paper— rather than that of Reference [5]— is used. Here KU , Kr

and K denote the energy, secant and tangent stiffness matrices, respectively. (Energy and secantstiffnesses are not denoted by Ke and Ks because such symbols are used for other purposes inthe finite element course noted in the Introduction.) In addition, p0 is the prestress force vector,which vanishes if the reference configuration is stress free and was omitted in that discussion, [5]and f = Krv + p0 is the internal force vector. The tangent stiffness is of course fundamental inincremental-iterative solution methods and stability analysis, while the secant stiffness (by itself or

10–7


in the internal-force form Krv + p0 ) is important in pseudo-force methods. The energy stiffnessenjoys limited application per se but has theoretical importance as source for the other two.In linear problemsKU = Kr = K = K0 and the three stiffness matrices coalesce. But in nonlinearproblems not only do the matrices differ but, as shown in the next section, KU andKr may involvearbitrary scalar coefficients. Such parametrized expressions were given by Felippa [5] under thefollowing restrictions:

(R1) Kr is symmetric.(R2) The reference configuration is stress free.(R3) The finite strain measure is quadratic in the displacement gradients.(R4) The transformation between core and physical freedoms is linear.

The following treatment eliminates restrictions (R1) and (R2) altogether, and the other two se-lectively. It should be noted that restriction (R4) is the condition that, with present terminology,characterizes the DCCF.

§10.4. Core Stiffness Equations

§10.4.1. TL Description of Particle Motion

A conservative, geometrically nonlinear structure under dead loading is viewed as a continuumundergoing finite displacements u. These displacements are measured from a fixed referenceconfiguration C0 to a variable current configuration C. No discretization into finite elements isimplied at this stage. We confine our attention to the case inwhich thematerial behavior stayswithinthe linear elastic range, thus implying small deformational strains but arbitrarily large rotations.Corresponding points or particles in the reference and current configuration are referred to a fixedCartesian coordinate system and have the coordinates Xi and xi (i = 1, . . . nd ), respectively, wherend is the number of space dimensions. The displacement field components are ui = xi ! Xi .Let the state of strain at a particle in the current configuration be characterized by ns strainsei (i = 1, 2, . . . ns) collected in an array e, and let the corresponding conjugate stresses be si(i = 1, 2, . . . ns), collected in an array s. Using the summation convention the elastic stress-strainrelations are written

si = s0i + Ei j e j , with Ei j = E ji , or s = s0 + Ee, (10.8)

where s0i are stresses in the reference configuration (stresses that remain if ei = 0, also calledprestresses) and Ei j are elastic moduli arranged as a ns " ns square array in the usual manner.Let ! , U , P ,!," and# denote the analogues of", U , P , p, f and r, respectively, at the particlelevel. (The first three acquire the meaning of energy densities, whereas! is a dead-loading bodyforce density independent of u.) The strain energy density can be expressed as

U = ei s0i + 12ei Ei j e j = eT s0 + 1

2eTEe. (10.9)

The total strain energyU is obtained by integrating (10.9) over the structure volume: U =!V0 U dV ;

the integration taking place— as can be expected in a TL description— over the reference config-uration geometry.

10–8

10–9 §10.4 CORE STIFFNESS EQUATIONS

Next, introduce the ng displacement gradients gmn = !um/!Xn . These are subsequently identifiedas gi (i = 1, 2, . . . ng) so they can be conveniently arranged in a one-dimensional array g. FollowingRajasekaran andMurray [2] and Felippa [5] assume that the strains ei are linked to the displacementgradients through matrix relations of the form

ei = hTi g+ 12g

THi g , i = 1, 2, . . . ns (10.10)

where hi and Hi are arrays of dimension ng ! 1 and ng ! ng , respectively, with Hi symmetric.In the original References [2,5] it was assumed that Hi is independent of g, which is the case forthe Green-Lagrange strain measure. This restriction, labeled (R3) in §10.3, will be enforced belowexcept in §10.4.5.

§10.4.2. Energy Variations

As noted previously, for deriving core equations we regard the displacement gradients g as degreesof freedom. On substituting (10.8) and (10.10) into (10.9) we obtain the “core counterparts” of(10.5)–(10.7), in which v has become g:

" = U " P = 12g

TSUg+ (! 0 " !)T g, (10.11)

" = !H!g

= Srg+ ! 0 " ! = # " ! = 0, (10.12)

#" = S#g"#! = 0. (10.13)

Here SU , Sr and S denote the energy, secant and tangent core stiffness matrices, and!0, which isindependent of g, is the core counterpart of p0.With this notation the first and second variations of the strain energy density can be expressed as

$U = $gT (SUg+ !0) + 12g

T $SUg = $gT!Srg+ !0" = $gT#, (10.14)

$2U = $gTSr $g+ $gT $Srg+ ($2g)T# = $gTS $g+ ($2g)T#. (10.15)

These variational equations implicitly determine Sr , # and S from SU and !0. If the linearityrestriction (R4) holds, the term in $2g drops out as explained in the Remark below, and

$2U = $gTS $g. (10.16)

Remark 10.2. If g = Gv with G independent of v, $2 g = G $2v = 0 because v are independent variables.On the other hand, if displacement gradients are nonlinear functions of node displacements expressable asgi = gi (v j ), then

$gi = !gi!v j

$v j = Gi j $v j , $2gi = !2gi!v j!vk

$v j$vk + !gi!v j

$2v j = Fi jk $v j $vk!0. (10.17)

Thus $g is still G $v but $2g = (F $v) $v, where F is a cubic array. The presence of the term $2g is taken intoaccount in the GCCF discussed in Chapter 11.

10–9


§10.4.3. Parametrized Forms

For convenience introduce the following ng ! ng matrices (with summation convention on i, j =1, . . . ng implied):

S0 = Ei j hih j , S1 = Ei j higTH j , S"1 = Ei j (hTi g)H j ,

S2 = Ei j Hi ggTH j , S"2 = Ei j (gTHig)H j ,

(10.18)

in which parentheses are used to emphasize the grouping of scalar quantities such as gTHig. Itmay be then verified that, if assumptions (R3)-(R4) of §10.3 hold, the core stiffnesses and prestressvector in (10.13)–(10.15) possess the general form:

SU (!,") = S0 + 12!(S1 + ST1 ) + (1# !)S"

1 + 14"S2 + 1

4 (1# ")S"2 + s0i Hi

= S0 + 12!(S1 + ST1 ) + ( 12 # !)S"

1 + 14"(S2 # S"

2) + 12 (s

0i + si )Hi ,

= S0 + 12!(S1 + ST1 ) # !S"

1 + 14"S2 # 1

4 (1+ ")S"2 + siHi ,

Sr (#,$) = S0 + 12S1 + #ST1 + (1# #)S"

1 + 14 (2# $)S2 + 1

4$S"2 + s0i Hi

= S0 + 12S1 + #ST1 + ( 12 # #)S"

1 + 14 (2# $)S2 + 1

4 ($ # 1)S"2 + 1

2 (s0i + si )Hi ,

= S0 + 12S1 + #ST1 # #S"

1 + 14 (2# $)(S2 # S") + siHi ,

S = S0 + S1 + ST1 + S"1 + S2 + 1

2S"2 + s0i Hi = S0 + S1 + ST1 + S2 + siHi ,

! 0 = s0i hi .(10.19)

Here !, ", # and$ are arbitrary scalar coefficients in the sense that gTSUg and Srg are independentof them. In fact,

" = Srg+ !0 = sibi , (10.20)where bi is defined in (10.25) below. The expressions (10.19) are more general than those originallygiven by Felippa [5] because restrictions (R1)-(R2) noted in §10.3 are no longer enforced. Notethat the secant core stiffness Sr becomes symmetric if # = 1/2.The “repeatable forms” (10.2)–(10.4) of Mallet and Marcal are obtained if ! = " = $ = 2/3 and# = 1/2, in which case the combinations S1+ST1 +S"

1 and S2+ 12S

"2 become the core counterparts

of N1 and N2, respectively. But this observation has largely historical interest. More physicallyrelevant are the following combinations:

SD = S1 + ST1 + S2, SM = S0 + SD,

SG = S"1 + 1

2S"2 + s0i Hi = siHi .

(10.21)

These are the core versions of the initial-displacement, material and geometric stiffness, respec-tively. The core tangent stiffness is S = S0 + SD + SG = SM + SG .If the Generalized CCF is required for downstream element development as explained in Chapter11, SG = siHi is called the principal core geometric stiffness and is denoted by SGP . In this casethe combination

S = SM + SGP , (10.22)receives the name principal core tangent stiffness.

10–10

10–11 §10.4 CORE STIFFNESS EQUATIONS

Remark 10.3. Finite element practicioners may be surprised at the nonuniqueness of SU and Sr . It appears tocontradict the fact that, given two square matricesA1 andA2 and an arbitrary nonzero test vector x,A1x = A2xfor all x implies A1 = A2. But this is not necessarily true if A1 and A2 are functions of x. More precisely, theenergy core stiffness is not unique because

gT (S1 ! S"1)g = 0, gT (ST1 ! S"

1)g = 0, gT (S2 ! S"2)g = 0, (10.23)

and the secant core stiffness is not unique because

(ST1 ! S"1)g = 0, (S2 ! S"

2)g = 0. (10.24)

Adding “gage terms” such as those of (10.24) multiplied by arbitrary coefficients does not change !U andconsequently the secant stiffness acquires two free parameters. Uniqueness holds for the tangent stiffnessbecause the test vectors are the virtual displacement gradient variations, and S is not a function of !g.

Remark 10.4. Because of (10.23), an additional free parameter appears in SU if unsymmetry is allowed. Ifsymmetry is enforced the first two gage expressions must be combined to read gT (S1 + ST1 ! 2S"

1)g = 0.

§10.4.4. Spectral Forms

There is a more compact alternative expression of the core stiffnesses that offers theoretical as wellas implementational advantages at the cost of some generality. Define vectors bi and ci as

ei = cTi g, ci = hi + 12Hig, bi = "ei

"g= hi +Hig. (10.25)

Then the spectral forms (so called because of the formal similarity of equations (10.26)–(10.28)with the spectral decomposition of a matrix as the sum of rank-one matrices) are

SU (1, 1) = SU!!#=$=1 = Ei j cicTj + s0i Hi , (10.26)

Sr (0, 0) = Sr!!%=&=0 = Ei j bicTj + s0i Hi , (10.27)

Sr ( 12 , 1) = Sr!!%= 12 ,&=1

= Ei j cicTj + 12 (si + s0i )Hi , (10.28)

S = Ei j bibTj + siHi = SM + SG . (10.29)

Note that Sr ( 12 , 1) is symmetric but Sr (0, 0) is not. It is seen that for energy and secant stiffnesses,

compactness is paid in terms of settling for specific coefficients.

Remark 10.5. The foregoing relations may be easily verified by noting that

Ei j cicTj = S0 + 12 (S1 + ST1 ) + 1

4S2,Ei j bicTj = S0 + 1

2S1 + ST1 + 12S2,

Ei j bibTj = S0 + S1 + ST1 + S2,

Ei j"(cicTj )"g

= Ei j

"ci

#"c j"g

$T

+#"ci"g

$cTj

%= S"

1 + 12S

"2 = Ei j e jHi = (si ! s0i )Hi ,

Ei j"2(cicTj )"g2

= 2Ei j"ci"g

#"c j"g

$T

= 12S

"2,

(10.30)

and seeking these patterns in the general parametrized expressions (10.20).

10–11


§10.4.5. Generalization to H(g)

If the Hi depend on g, as it generally happens if strain measures other than Green-Lagrange’sare used, the secant and tangent stiffness core equations become more complex because of thepresence of first and second g-derivatives of Hi . The changes in the core variational equations(10.14)–(10.15) can be succintly expressed as

!U = !gT!(Sr +"Sr )g+ !0 + "!0# = !gT (" + ""), (10.31)

!2U = !gT (S+"S) !g+ (!2g)T (" + ""). (10.32)

where"Sr ,"S and "" are additional core terms that arise on account of the dependence of theHi on g.The parametrization and efficient characterization of such terms for several strain measures ofinterest in practice, notably logarithmic and midpoint strains, are presently open problems. Suchtopics would in fact be good candidates for term projects in advanced nonlinear finite elementcourses.

§10.5. Core Stiffness Derivation Examples

Because the core equations reflect the motion of an individual particle, their form is primarilydetermined by the choice of components of s, e and g that are retained in the strain energy density.This choice is in turn a byproduct of themathematical idealization of the actual structure or structuralcomponent.

Several cases are worked out below to illustrate the basic steps. The core expressions developed inthese examples do not force commitment to specific elements, only to a mathematical model. Forexample the bar core equations may be subsequently used to develop 2-node straight elements or3-node curved ones. Some specific elements based on these equations are derived in Chapter 11.

§10.5.1. Bar in 3D Space

The particle belongs to a bar moving in 3D space. The only energy contribution is due to the axial(longitudinal) stress. We have nd = 3, ns = 1 and ng = 3. To simplify node subscripting, Cartesiansystems and displacement components will be denoted by {X, Y, Z}, {x, y, z} and {uX , uY , uZ }rather than {X1, X2, X3}, {x1, x2, x3} and {u1, u2, u3}, respectively. In the reference configurationC0 the bar is referred to a local Cartesian system {X , Y , Z}, with X located along the bar axis. SeeFigure 10.1.

With reference to this local system, the motion of a particle initially at X is defined by the displace-ment components u X = u X (X), uY = uY (X) and u Z = u Z (X). The three displacement gradientsthat intervene in the definition of nonlinear strains are

g =$ g1g2g3

%

=$" u X/" X" uY /" X" u Z/" X

%

. (10.33)

10–12

10–13 §10.5 CORE STIFFNESS DERIVATION EXAMPLES

vX1vX2

vY1

vY2

vZ1

vZ2

X, xY, y

Z , z

X

YZ

E, A0

L0

1

2

1'

2'

C0

C

Figure 10.1. A 2-node bar element in 3D space.

As uniaxial strain measure we adopt the Green-Lagrange (GL) axial strain, defined as

e ! e1 = ! u X! X

+ 12

!"! u X! X

#2+

"! uY! X

#2+

"! u Z! X

#2$

= g1 + 12 (g

21 + g22 + g23)

=! 100

$T ! g1g2g3

$

+ 12

! g1g2g3

$T ! 1 0 00 1 00 0 1

$ ! g1g2g3

$

= hT g+ 12g

THg.

(10.34)

Thus for this choice of strain, hT1 ! hT = [ 1 0 0 ] and H1 ! H is the 3 " 3 identity matrix.The conjugate stress measure s1 ! s is the second Piola-Kirchhoff (PK2) axial stress. The stress-strain relation is s = s0 + Ee, where s0 and s are PK2 axial stresses in the reference and currentconfigurations, respectively, and E is Young’s modulus.BecauseH is independent of g, to form the core stiffnesses in local coordinates we can directly usethe spectral expressions (10.26)–(10.29). First construct the vectors

c ! c1 =

%

&1+ 1

2g112g212g3

'

( , b ! b1 =! 1+ g1

g2g3

$

, (10.35)

which inserted into the spectral forms yield

10–13


X, x

Y, y 1 2

34

1'

2'

3'

4'

vX1vY1

C0

C

Figure 10.2. A 4-node plane stress element in 2D space.

SU (1, 1) = E ccT + s0H = E

!

"(1+ 1

2 g1)2 1

2 g2(1+ 12 g1)

12 g3(1+ 1

2 g1)14 g

22

14 g2g3

symm 14 g

23

#

$ + s0% 1 0 00 1 00 0 1

&, (10.36)

Sr ( 12 , 1) = E ccT + smH = E

!

"(1+ 1

2 g1)2 1

2 g2(1+ 12 g1)

12 g3(1+ 1

2 g1)14 g

22

14 g2g3

symm 14 g

23

#

$+ sm% 1 0 00 1 00 0 1

&, (10.37)

S = E bbT + sH = E

!

"(1+ g1)2 g2(1+ g1) g3(1+ g1)

g22 g2g3symm g23

#

$ + s

% 1 0 00 1 00 0 1

&. (10.38)

In equation (10.37), sm = 12 (s

0 + s) = s0 + 12 Ee is the average or “half-way” stress. The clean

separation into material and geometric (initial-stress) stiffnesses should be noted.

§10.5.2. Plate in Plane Stress

As second example we consider a particle that pertains to a plate in plane stress (membrane),constrained to move in its plane. See Figure 10.2. As usual we consider only the motion ofthe midplane. The Cartesian reference system and displacement components will be denoted by{X, Y }, {x, y} and {uX , uY } rather than {X1, X2}, {x1, x2} and {u1, u2}, respectively. The elementdisplacement field of a generic particle originally at (X, Y ) is defined by the two componentsuX = uX (X, Y ) and uY = uY (X, Y ). Three in-plane PK2 stresses contribute to the strain energy

10–14


and four displacement gradients appear in the corresponding GL strain. Consequently nd = 2,ns = 3 and ng = 4. The four displacement gradients are arranged as

g =

!

"#

g1g2g3g4

$

%& =

!

"#

!uX/!X!uY /!X!uX/!Y!uY /!Y

$

%& (10.39)

The strain measures chosen are the three components ei (i = 1, 2, 3) of the GL strains defined inthe usual manner:

e1 = eXX = g1 + 12 (g

21 + g22) =

!

"#

1000

$

%&

T

g+ 12g

T

!

"#

1 0 0 00 1 0 00 0 0 00 0 0 0

$

%& g, (10.40)

e2 = eYY = g4 + 12 (g

23 + g24) =

!

"#

0001

$

%&

T

g+ 12g

T

!

"#

0 0 0 00 0 0 00 0 1 00 0 0 1

$

%& g, (10.41)

e3 = eXY + eY X = g2 + g3 + g1g3 + g2g4 =

!

"#

0110

$

%&

T

g+ 12g

T

!

"#

0 0 1 00 0 0 11 0 0 00 1 0 0

$

%& g, (10.42)

from which expressions for hi and Hi (i = 1, 2, 3) follow. For brevity, only the derivation of thetangent stiffness matrix will be described. Begin by forming the vectors

b1 =

!

"#

1+ g1g200

$

%& , b2 =

!

"#

00g3

1+ g4

$

%& , b3 =

!

"#

g31+ g41+ g1g2

$

%& . (10.43)

Then from (10.29) we get the core stiffness

S = Ei jbibTi + siHi = SM + SG, (10.44)

where si = s0i + Ei j e j , (i, j = 1, 2, 3), are the PK2 stresses in the current configuration.In full and using the abbreviations a1 = 1+ g1, a4 = 1+ g4 we get

SM =

!

"#

E11a21 + 2E13a1g3 + E33g23 E11a1g2 + E13(a1a4 + g2g3) + E33a4g3E11g22 + 2E13a4g2 + E33a24

symmE12a1g3 + E13a21 + E23g23 + E33a1g3 E12a1a4 + E13a1g2 + E23a4g3 + E33g2g3

E12g2g3 + E13a1g2 + E23a4g3 + E33a1a4 E12a4g2 + E13g22 + E23a24 + E33a4g2E22g23 + 2E23a1g3 + E33a21 E22a4g3 + E23(a1a4 + g2g3) + E33a1g2

E22a24 + 2E23a4g2 + E33g22

$

%&

(10.45)

10–15


SG =

!

"#

s1 0 s3 0s1 0 s3

s2 0symm s2

$

%& . (10.46)

§10.5.3. Plate Bending

This is similar to the previous example in that the structure is a flat thin plate but now motionin 3D space {X, Y, Z} is allowed. With this increased freedom the plate is capable of membranestretching and bending. For the latter a Kirchhoff mathematical model is assumed. The threeenergy-contributing GL strains are now functions of six gradients. Consequently nd = 3, ns = 3and ng = 6. The contributing gradients are arranged as

g =

!

"""""#

g1g2g3g4g5g6

$

%%%%%&=

!

"""""#

!uX/!X!uY /!X!uZ/!Z!uX/!Y!uY /!Y!uZ/!Y

$

%%%%%&(10.47)

The three GL strains are defined as

e1 = eXX = g1 + 12 (g

21 + g22 + g23) =

!

"""""#

100000

$

%%%%%&

T

g+ 12g

T

!

"""""#

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

$

%%%%%&g, (10.48)

e2 = eYY = g5 + 12 (g

24 + g25 + g26) =

!

"""""#

000010

$

%%%%%&

T

g+ 12g

T

!

"""""#

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

$

%%%%%&g, (10.49)

e3 = eXY + eY X = g2 + g4 + g1g4 + g2g5 + g3g6 =

!

"""""#

010100

$

%%%%%&

T

g+ 12g

T

!

"""""#

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

$

%%%%%&g,

(10.50)which definehi andHi , i = 1, 2, 3. When one reaches this level of bookkeeping it ismore expedientand less error-prone to obtain the core matrices through symbolic manipulation. For example, thefollowingMacsyma program forms SM and SG in matrices SM and SG, respectively:

10–16


h1: matrix([1],[0],[0],[0],[0],[0])$h2: matrix([0],[0],[0],[0],[1],[0])$h3: matrix([0],[1],[0],[1],[0],[0])$g: matrix([g1],[g2],[g3],[g4],[g5],[g6])$HH1:matrix([1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],

[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0])$HH2:matrix([0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0],

[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1])$HH3:matrix([0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],

[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0])$b1:h1+HH1.g$ b2:h2+HH2.g$ b3:h3+HH3.g$SM:E11* b1.transpose(b1)+E22*b2.transpose(b2)+E33*b3.transpose(b3)+ E12*(b1.transpose(b2)+b2.transpose(b1))+ E13*(b1.transpose(b3)+b3.transpose(b1))+ E23*(b2.transpose(b3)+b3.transpose(b2))$ratvars(g6,g5,g4,g3,g2,g1,a5,a1,E11,E12,E13,E22,E23,E33)$SM:ratsimp(SM)$SG:ratsimp(s1*HH1+s2*HH2+s3*HH3)$

These matrices may be automatically converted to TEX by appropriate Macsyma statements (notshown above). That output was reformatted by hand for inclusion here. For the core tangentstiffness this semi-automated process yields

10–17


SM (1, 1) = E33g24 + 2E13(1+ g1)g4 + E11(1+ g1)2

SM (1, 2) = E13((1+ g1)(1+ g5) + g2g4) + E33g4(1+ g5) + E11(1+ g1)g2SM (1, 3) = E13((1+ g1)g6 + g3g4) + E33g4g6 + E11(1+ g1)g3SM (1, 4) = E23g24 + E33(1+ g1)g4 + E12(1+ g1)g4 + E13(1+ g1)2

SM (1, 5) = E12((1+ g1)(1+ g5)) + E23g4(1+ g5) + E33g2g4 + E13(1+ g1)g2SM (1, 6) = E12(1+ g1)g6 + E23g4g6 + E33g3g4 + E13(1+ g1)g3SM (2, 2) = E33(1+ g5)2 + 2E13g2(1+ g5) + E11g22SM (2, 3) = E33(1+ g5)g6 + E13(g2g6 + g3(1+ g5)) + E11g2g3SM (2, 4) = E33((1+ g1)(1+ g5)) + E23g4(1+ g5) + E12g2g4 + E13(1+ g1)g2SM (2, 5) = E23(1+ g5)2 + E33g2(1+ g5) + E12g2(1+ g5) + E13g22SM (2, 6) = E23(1+ g5)g6 + E12g2g6 + E33g3(1+ g5) + E13g2g3SM (3, 3) = E33g26 + 2E13g3g6 + E11g23SM (3, 4) = E33(1+ g1)g6 + E23g4g6 + E12g3g4 + E13(1+ g1)g3SM (3, 5) = E23(1+ g5)g6 + E33g2g6 + E12g3(1+ g5) + E13g2g3SM (3, 6) = E23g26 + E33g3g6 + E12g3g6 + E13g23SM (4, 4) = E22g24 + 2E23(1+ g1)g4 + E33(1+ g1)2

SM (4, 5) = E23((1+ g1)(1+ g5) + g2g4) + E22g4(1+ g5) + E33(1+ g1)g2SM (4, 6) = E23((1+ g1)g6 + g3g4) + E22g4g6 + E33(1+ g1)g3SM (5, 5) = E22(1+ g5)2 + 2E23g2(1+ g5) + E33g22SM (5, 6) = E22(1+ g5)g6 + E23(g2g6 + g3(1+ g5)) + E33g2g3SM (6, 6) = E22g26 + 2E23g3g6 + E33g23

(10.51)

(which can be further compacted by introducing the auxiliary symbols a1 = 1+ g1 and a5 = 1+g5as done in §10.5.3) and

SG = SGP =

!

"""""#

s1 0 0 s3 0 00 s1 0 0 s3 00 0 s1 0 0 s3s3 0 0 s2 0 00 s3 0 0 s2 00 0 s3 0 0 s2

$

%%%%%&. (10.52)

Remark 10.6. If the plate element to which the particle belong has (as usual) rotational freedoms, an additionalgeometric stiffness (the complementary geometric stiffness) appears in the transformation phase. Because ofthis, the core geometric stiffness (10.52) has been relabeled as SGP , where subscript P means “principal.”

Remark 10.7. The core stiffness matrices may also be used for part of the formulation of thin-shell facetelements, with the proviso that global reference axes {X, Y, Z} are to be replaced by a local coordinate system{X , Y , Z} with Z normal to the element midplane.

10–18


Y

X

x

X

u

θ

1

u0

θ2

θ1

X0

RTζ

ζ 2

Figure 10.3. Kinematics of 2D Timoshenko beam element

§10.5.4. 2D Timoshenko Beam

Consider next an isotropic Timoshenko plane beam that moves in the (X ,Y ) plane. For notationalsimplicity it is assumed that the longitudinal axis of the beam is aligned with X . The only PK2stresses that contribute to the strain energy are the axial stress s1 ! sXX and the mean shear stresss2 ! sXY . The corresponding GL strains are the axial strain e1 ! eXX and the section-averagedshear strain e2 ! !XY = eXY+eY X . The constitutive equations are s1 = s01+Ee1 and s2 = s02+G e2,where E and G are the Young’s modulus and shear modulus, respectively, of the material. Thetreatment outlined below is slightly modified from that of a course term project by Alexander, dela Fuente and Haugen. [11]The finite displacements are described in a local coordinate system that is attached to the initialposition of the beam, as illustrated in Figure 1. Under the usual kinematic assumptions of theTimoshenko beam model (plane sections remain plane but not necessarily normal to the deformedcentroidal axis) the coordinates of a particle in the underformed and deformed configurations maybe written

X = X0 + !, X0 =!X0

", ! =

!0Y

", (10.53)

x = x0 + RT !, x0 =!X + u0Xu0Y

", RT =

!cos " " sin "sin " cos "

", (10.54)

where u0X and u0Y are the components of the centroidal displacement vectoru0. Subtracting (10.53)

10–19


from (10.54) gives the element displacement field

u = x! X =!uXuY

"=

!u0X ! Y sin !

u0Y + Y (cos ! ! 1)

". (10.55)

Four displacement gradients contribute to the GL strains. Thus for this case we have nd = 2, ns = 2and ng = 4. The four contributing displacement gradients are arranged in the usual pattern:

g =

#

$%

g1g2g3g4

&

'( =

#

$%

"uX/"X"uY /"X"uX/"Y"uY /"Y

&

'( . (10.56)

For future use in Chapter 11 we note that the gradients can be written in terms of generalized sectionfreedoms as

g =

#

$%

g1g2g3g4

&

'( =

#

$%

# ! Y$ cos !% ! Y$ sin !

! sin !cos ! ! 1

&

'( , (10.57)

in which # = "u0X/"X is a generalized axial strain, % = "u0Y /"X a generalized shear strain, and$ = "!/"X is the beam curvature.The matrix form of the GL strains is

e1 = g1 + 12 (g

21 + g2)2 =

#

$%

1000

&

'(

T

g+ 12g

T

#

$%

1 0 0 00 1 0 00 0 0 00 0 0 0

&

'( g, (10.58)

e2 = g2 + g3 + g1g3 + g2g4 =

#

$%

0110

&

'(

T

g+ 12g

T

#

$%

0 0 1 00 0 0 11 0 0 00 1 0 0

&

'( g, (10.59)

which define h1, h2, H1 and H2. On introducing the auxiliary vectors

b1 =

#

$%

1+ g1g200

&

'( , b2 =

#

$%

g31+ g41+ g1g2

&

'( , c1 =

#

$%

1+ 12g1

12g200

&

'( , c2 =

#

$$%

12g3

1+ 12g4

1+ 12g1

12g2

&

''( , (10.60)

the spectral core stiffness matrices and internal force vector can be written

SU = Ec1cT1 + G c2cT2 + s01H1 + s02H2 , (10.61)

Sr = Ec1cT1 + G c2cT2 + 12 (s

01 + s1)H1 + 1

2 (s02 + s2)H2 , (10.62)

S = SM + SGP , SM = Eb1bT1 + Gb2bT2 , SGP = s1H1 + s2H2 , (10.63)! = s1b1 + s2b2. (10.64)

Because beam elements have rotational freedoms, a complementary geometric stiffness matrixappears when carrying out the transformation phase. This term is considered in the subsequentGCCF treatment of this element in Chapter 11.

10–20


0X

Y

Z

u0

x0

a2

a3

a1

1

n2

n3

X, n

Figure 10.4. Kinematics of 3D Timoshenko beam element.

§10.5.5. 3D Timoshenko Beam: Kinematics

The last example of derivation of core equations involve a TL 3D Timoshenko beam capable ofarbitrarily large rotations. The followingmaterial is largely extracted from a recent paper by Crivelliand Felippa [8] as well as Crivelli’s thesis [6] and is continued with the DGCCF transformationphase in Chapter 11. The notation used in those references has been slightly edited to fit that of thepresent article.As in the 2D case, the beam is isotropically elastic with Young’s modulus E and shear modulus G.The reference configuration of the beam is straight and prismatic although not necessarily stressfree. A local reference frame ni is attached to it, with n1 directed along the longitudinal axis (thelocus of cross section centroids). Axes n2 and n3 are in the plane of the left-end cross section; thesewill be eventually aligned with the principal inertia axes to simplify some algebraic expressions.Along these axes we attach the coordinate system {X, Y, Z}. This description is schematicallyshown in Figure 2. We further define a set of moving frames, denoted by {a1, a2, a3}, parametrizedby the longitudinal coordinate X . Initially these frames coincide with {n1,n2,n3}, and displacerigidly attached to the cross-sections of the moving current configuration.A beam particle originally at (X, Y, Z) displaces to

x(X) = x0(X) + RT (X)!(Y, Z), !T = [ 0 Y Z ] , (10.65)

where x0 describes the position of the centroid of the given cross-section, R is a 3-by-3 orthogonalmatrix function that orients the displaced cross section, and ! is a cross-section position vector. Thedisplacement field is

u = x! X = u0 + (RT ! I)!. (10.66)

where u0(X) = x0(X) ! X0(X) is the centroidal displacement (see Figure 2).

10–21


In the sequel 3! 3 skew-symmetric matrices are consistently denoted by placing a tilde over theiraxial 3-vector symbol; for example

a = spin (a) =! 0 a3 "a2

"a3 0 a1a2 "a1 0

"

, a =! a1a2a3

"

= axial (a). (10.67)

The skew-symmetric curvaturematrix ! is defined by ! = R(dRT /dX), which is the rate of changeof the orthogonal rotationmatrixRwith respect to the longitudinal coordinate. The curvature vectoris ! = axial (!). We shall also require later the variation of angular orientation !!, defined asthe axial vector of the skew matrix R !RT :

#!! = R !RT = "!RRT , !! = axial (#!!), (10.68)

All displacement gradients gi j appear in the GL strain measures. To maintain compactness the ninegradients are partitioned into three 3-vectors:

g1 =!"uX/"X"uY /"X"uZ/"X

"

, g2 =!"uX/"Y"uY /"Y"uZ/"Y

"

, g3 =!"uX/"Z"uY /"Z"uZ/"Z

"

, (10.69)

The 9-component gradient vector is gT = [gT1 gT2 gT3 ], but this symbol is not used directly here.Also introduce the 3-vectors

h1 =! 100

"

, h2 =! 010

"

, h3 =! 001

"

. (10.70)

With the help of these quantities, explicit expressions for the displacement gradient vectors g canbe given as

g1 = du0dX

+ RT !" = du0dX

+ RT "T!,

g2 = (RT " I)h2, g3 = (RT " I)h3.(10.71)

The only nonzero components of the GL strain tensor can be written

e1 # e11 = hT1 g1 + 12g

T1Hg1,

e2 # #12 = 2e12 = hT2 g1 + hT1 g2 + 12 (g

T1Hg2 + gT2Hg1),

e3 # #13 = 2e13 = hT3 g1 + hT1 g3 + 12 (g

T1Hg3 + gT3Hg1),

(10.72)

where H is here the 3! 3 identity matrix. Note that from the orthogonality of the rotation matrixR we find

e22 = hT2 g2 + 12g

T2 g2

= R22 " 1+ 12$R221 + (R22 " 1)2 + R223

%= R22 " 1+ 1

2 (2" 2R22) = 0,2e23 = hT2 g3 + hT3 g2 + gT2 g3

= R32 + R23 + R21R31 + R22R32 " R32 + R23R33 " R23 = 0,

(10.73)

10–22


and similarly e33 = 0. This confirms that the only nonzero strains are (10.72).The strains (10.72) may be rewritten in a more physically suggestive form:

e1 = e11 = eb + e f ,

eb =!du0dX

"T !h1 + 1

2du0dX

",

e f = !TKe + 12"

T !!T" ! !TKe,

! = !12 + !13,

e2 = !12 = !2 + ! 2 = hT2 # + hT2 !T",

e3 = !13 = !3 + ! 3 = hT3 # + hT3 !T".

(10.74)

Here eb, e f are stretching and flexural normal strains, !2 and !3 represent bending-induced shearstrains, and ! 2, ! 3 are torsion-induced shear strains. The last term in e f represents a squared-curvature contribution to flexure, which can usually be neglected (cf. Remark 9.2). The strainenergy stored in the current configuration is

U =#

L0

#

A0U d A dX, with U = 1

2 Ee21 + 1

2G$e22 + e23

%+ s01e1 + s02e12 + s03e3. (10.75)

§10.5.6. 3D Timoshenko Beam: Core equations

The PK2 stresses associated with the GL strains (10.72) are s1 " s11 = sXX , s2 " s12 = sXY ands3 " s13 = sX Z . The constitutive equations are s1 = s01 + Ee1, s2 = s02 + Ge2 and s3 = s03 + Ge3.The spectral core stiffnesses can be compactly expressed in terms of the vectors ci = hi + 1

2Hgiand bi = hi + Hgi for i = 1, 2, 3, where no subscript is needed in H " I. Applying the spectralformulas of §10.4.4 we obtain for the 9# 9 core energy stiffness

SU =

&

'ESU1 + G(SU2 + SU3 ) GSU4 GSU5

GSU4T GSU1 0

GSU5T 0 GSU1

(

) +* s01H s02H s03Hs02H 0 0s03H 0 0

+

, (10.76)

where SU1 = c1cT1 , SU2 = c2cT2 , S

U3 = c3cT3 , S

U4 = c2cT1 and S

U5 = c3cT1 . At the residual level

we obtain for Sr a form similar to (10.76) except that the prestresses s0i , i = 1, 2, 3 have tobe replaced by the midpoint stresses 1

2 (s0i + si ). The internal force vector conjugate to "g is

! = Srg+ !0 = !# + !$ , in which

!# =* s1b100

+

!$ =* s2b2 + s3b3

s2b1s3b1

+

, (10.77)

represent the contribution of the normal and shear stresses, respectively.The principal core tangent stiffness matrix S = SM + SGP is obtained from (10.29). The materialstiffness is

SM =* ES1 + G(S2 + S3) GS4 GS5

GST4 GS1 0GST5 0 GS1

+

, (10.78)

10–23


where S1 = b1bT1 , S2 = b2bT2 , S3 = b3bT3 , S4 = b2bT1 and S5 = b3bT1 . The principal geometricstiffness is

SGP =! s1H s2H s3Hs2H 0 0s3H 0 0

"

=!

(s01 + Ee1)H (s02 + Ge2)H (s02 + Ge3)H(s02 + Ge2)H 0 0(s02 + Ge3)H 0 0

"

. (10.79)

The contribution of#!2g

$T ! to the complementary geometric stiffness depends on the targetvariables in the ensuing transformation phase. Because this transformation requires the DGCCF, itis taken up in Chapter 11.

§10.6. References

[1] C. A. Felippa, L. A. Crivelli and B. Haugen, A Survey of the Core-Congruential Formulation for Geo-metrically Nonlinear TL Finite Elements, Archives of Computational Methods in Engineering, 1, 1994,pp. 1–48

[2] S. Rajasekaran and D. W. Murray, Incremental finite element matrices, J. Str. Div. ASCE, 99, pp. 2423–2438, 1973.

[3] R. H. Mallet and P. V. Marcal, Finite element analysis of nonlinear structures, J. Str. Div. ASCE, 94, pp.2081–2105, 1968.

[4] D.W.Murray, Finite element nonlinear analysis of plates, Ph. D. Dissertation, Dept. of Civil Engineering,University of California, Berkeley, California, 1967.

[5] C. A. Felippa, Discussion of Reference 1, J. Str. Div. ASCE, 100, pp. 2519–2521, 1974.

[6] L. A. Crivelli, A Total-Lagrangian beam element for analysis of nonlinear space structures, Ph. D.Dissertation, Dept. of Aerospace Engineering Sciences, University of Colorado, Boulder, CO, 1990.

[7] C. A. Felippa and L. A. Crivelli, A congruential formulation of nonlinear finite elements, in NonlinearComputational Mechanics - The State of the Art, ed. by P. Wriggers and W. Wagner, Springer-Verlag,Berlin, pp. 283–302, 1991.

[8] L. A. Crivelli and C. A. Felippa, A three-dimensional non-linear Timoshenko beam element based onthe core-congruential formulation, Int. J. Numer. Meth. Engrg., 36, pp. 3647–3673, 1993.

[9] K.Mathiasson, A. Bengtsson andA. Samuelsson, On the accuracy and efficiency of numerical algorithmsfor geometrically nonlinear structural analysis, in Finite Element Methods for Nonlinear Problems, ed.by P. G. Bergan, K. J. Bathe and W. Wunderlich, Springer-Verlag, Berlin, pp. 3–24, 1986.

[10] P. G. Bergan and K. M. Mathisen, Large displacement analysis of highly flexible offshore structures,in Nonlinear Computational Mechanics - The State of the Art, ed. by P. Wriggers and W. Wagner,Springer-Verlag, Berlin, pp. 303–331.

[11] S.Alexander, H.M. de la Fuente andB.Haugen, Correspondence betweenCC-TL andC-TL formulationsA 2D Timoshenko beam element using the Total-Lagrangian Core Congruential Formulation, in TermProjects in Nonlinear Finite Element Methods, ed. by C. A. Felippa, Report CU-CSSC-91-12, Centerfor Space Structures and Controls, University of Colorado, Boulder, CO, May 1991.

10–24

10–25 Exercises


Not assigned

EXERCISE 10.1

Show that the “core” forms gTSUg andSr g, withSU andSr given by (10.19), are independent of the coefficients!, ", # and $ . It is sufficient to work this exercise for for i, j = 1, that is, only one stress, strain and modulusis considered. Keep, however, g, h and H generic.Note: This is a good exercise in matrix gymnastics. If you are “rusty” in matrix magic, reading the Addendumbelow is strongly recommended.

Addendum: Matrix Product Properties

In carrying out the manipulations required by Exercise 10.1, the following properties of matrix products oughtto be kept in mind.

(1) Let x and y be two conforming vectors. Since their inner product is a scalar, obviously

xT y = yT x.

(2) If x and y are two conforming vectors and A a conforming matrix, the quadratic form xTAy is also ascalar; consequently

xTAy = yTAT x.

Furthermore, if A is square and symmetric, xTAy = yTAx.

(3) If x and y are conforming vectors, xyT and yxT are rank-one square matrices (the transpose of eachother). Furthermore, xxT is a symmetric matrix.

(4) Scalars can be moved to any position within a matrix product. If the scalar is in itself the result of avector or matrix product, the components may be transposed as per rules (1) and (2). For example, if A,B and C are conforming matrices, and x and y are conforming vectors,

xT yAB = AxT yB = ABxT y = AByT x, etc.

xTAyBC = yTAT xBC = BxTAyC = ByTAT xC, etc.

10–25

.

11The Core Congruential

Formulation:Sti!ness Equations

11–1

Chapter 11: THE CORE CONGRUENTIAL FORMULATION: STIFFNESS EQUATIONS11–2

TABLE OF CONTENTS

Page§11.1. DCCF Transformation to Physical Freedoms 11–3§11.2. DCCF Transformation Examples 11–4

§11.2.1. The Bar Element . . . . . . . . . . . . . . . . 11–4§11.2.2. Iso-P Plane Stress Element . . . . . . . . . . . . 11–4

§11.3. The Generalized CCF 11–5§11.3.1. Generalized Coordinates as Generic Target . . . . . . . 11–6§11.3.2. Algebraic Transformation . . . . . . . . . . . . . 11–6§11.3.3. Differential Transformation . . . . . . . . . . . . . 11–7§11.3.4. Multistage Transformation . . . . . . . . . . . . 11–7

§11.4. A 2-Node 2D Timoshenko Beam Element 11–8§11.4.1. Generalized Coordinates and Stress Resultants . . . . . . 11–8§11.4.2. Transformation Matrices . . . . . . . . . . . . . 11–9§11.4.3. Internal Force Vector . . . . . . . . . . . . . . . 11–10§11.4.4. Tangent Stiffness Matrix . . . . . . . . . . . . . 11–10§11.4.5. Can a Secant Stiffness be Constructed? . . . . . . . . . 11–13

§11.5. A 2-Node 3D Timoshenko Beam Element 11–13§11.5.1. Transformation to Generalized Gradients . . . . . . . . 11–13§11.5.2. Transformation to the Rotational Vector . . . . . . . . 11–18§11.5.3. Transformation to Finite Element Freedoms . . . . . . . 11–19

§11.6. Equivalence of DCCF and Standard TL Formulation 11–20§11.7. References 11–22§11. Exercises . . . . . . . . . . . . . . . . . . . . . . 11–24

11–2

11–3 §11.1 DCCF TRANSFORMATION TO PHYSICAL FREEDOMS

The present Chapter completes the development of the Core Congruential Formulation (CCF) ofTotal Lagrangian (TL) elements. It present procedures to transform core equations to finite elementstiffness equations.

§11.1. DCCF Transformation to Physical FreedomsCore expressions for the internal-force vector and stiffness matrices of an individual TL element aregiven in (10.20)and (10.26)-(10.29), respectively. These expressions pertain to material particlesof the structure. The behavior of each particle is expressed in terms of its displacement gradientscollected in vector g. To create a discrete model the structure is subdivided into finite elements.Finite elements equations in terms of the physical DOFs collected in vector v are constructedthrough a combination of core-to-physical transformations and integration over element domains.In this section we stay within the scope of the Direct CCF by assuming that the transformationsbetween g and v are linear. Because all subsequent developments pertain to an individual element,no element identifiers are used to reduce indexing clutter.Over an individual element the displacement field uT = (u1, u2, u3) is interpolated as

u = Nv, (11.1)

where v now collects the element node-displacement degrees of freedom (DOFs) and N =N(X1, X2, X3) is a matrix of shape functions independent of v. Differentiating (11.1) with re-spect to the Xi and taking the first two v variations yields

g = Gv, !g = G !v, !2g = 0, (11.2)

(for the last one see Remark 10.2). Invariance of the strain energy variations !U and !2U obtainedby integrating (10.14)-(10.15) over the element reference volume yields

KU =!

V0GTSUG dV, Kr =

!

V0GTSrG dV, K =

!

V0GTSG dV, (11.3)

f =!

V0GT! dV, p =

!

V0GT" dV, p0 =

!

V0GT"0 dV, (11.4)

Although the dependency of Slevel and " on g is not made implicit in these equations, it must beremembered that the transformation g = Gv also appears there. Because of the ensuing algebraiccomplexity, numerical integration is generally required unless the gradients are constant over theelement.Often G is expressed as a chain of transformations, some of which are position dependent andremain inside the element integral whereas others are not and may be taken outside. For example,in the bar element treated below, G = T G, where G transforms g to local node displacementswhile T transforms local to global node displacements.

11–3


§11.2. DCCF Transformation Examples

§11.2.1. The Bar Element

The core equations for a geometrically nonlinear TL bar were derived in Section 5.1. These equa-tions are now applied to the formulation of a two-node, linear-displacement, prismatic TL barelement. The element has constant reference area A0 and initial length L0. The two end nodes arelocated at (X1, Y1, Z1) and (X2, Y2, Z2), respectively. The node displacements are (vX1, vY1, vZ1)and (vX2, vY2, vZ2). The element displacement field in local coordinates {X , Y , Z} may be inter-polated as

u =! u XuYu Z

"

=! N1 0 0 N2 0 00 N1 0 0 N2 00 0 N1 0 0 N2

"#

$$$%

vX1vY1vZ1vX2vY2vZ2

&

'''(= N v, (11.5)

where N1 = 1! X/L0 and N2 = X/L are linear shape functions. Differentiating with respect tothe reference coordinate we get

G = 1L0

!!1 0 0 1 0 00 !1 0 0 1 00 0 !1 0 0 1

"

v = 1L0Gv, (11.6)

This transformation may be applied to the core matrices and vectors derived in Chapter 10. Forexample, application to the core tangent stiffness (10.38) yields

K = 1L20

)

V0GTSG dV = A0

L0GT

(EbbT + sH) G, (11.7)

Finally, transformation to node displacements (vXi , vY i , vZi ), i = 1, 2 is handled in the usualmanner by writing the local-to-global transformation equation

u =! u XuYu Z

"

=! TXX TXY TXZTY X TYY TY ZTZX TZY TZ Z

"! uXuYuZ

"

= Tu, (11.8)

which is valid for both end nodes giving vi = Tv, i = 1, 2. Consequently the element tangentstiffness matrix in local coordinates is given by

K = A0L0

*TT 00 TT

+GT

(EbbT + sH) G*T 00 T

+. (11.9)

For this simple element all entries may be obtained in closed form and no numerical integration isnecessary.

11–4

11–5 §11.3 THE GENERALIZED CCF

§11.2.2. Iso-P Plane Stress Element

For the case of plane stress considered in §10.5.2, we shall assume that the associated finite elementsare isoparametric displacement models with n nodes, and that (as usual for such models) the nodalfreedoms are of translational type. The transformation to physical DOFs can then be handled withinthe purview of the DCCF.As in §10.5.2 the reference system, current system and in-plane displacement components aredenoted by {X, Y }, {x, y} and {uX , uY }, respectively. The element nodes are located at {Xi , Yi },(i = 1, . . . n) in the reference configuration C0 and move to {xi = Xi + uXi , yi = Yi + uYi },(i = 1, . . . n) in the current configuration C. The element displacement field may be expressed as

!uXuY

"=!N1 0 N2 . . . 00 N1 0 . . . Nn

"

#

$$$$%

vX1vY1vX2...

vYn

&

''''(= Nv, (11.10)

in which Ni are appropriate isoparametric shape functions written in terms of natural coordinatessuch as ! and " for quadrilaterals. The G matrix follows upon differentiation with respect to Xand Y , and all core equations transformed as per (11.3)–(11.4). For example, the physical tangentstiffness is

K =)

V0GT (SM + SG)G dV, (11.11)

where SM and SG are given by (10.45) and (10.46), respectively. As in the case of linear elements,(11.11) is most conveniently evaluated by numerical integration. Because several of the integrandmatrices are sparse, in the interest of efficiency in the computer implementation the integrandmay be symbolically evaluated through a computer algebra system such as Macsyma, Mapleor Mathematica, and automatically converted to Fortran or C program statements before beingencapsulated in the Gauss quadrature loop.

§11.3. The Generalized CCFAs discussed in §10.2, the Generalized Core Congruential Formulation or GCCF is required whenthe relation between displacement gradients g and finite element degrees of freedom v is nonlinear.This complication occurs in elements with rotational freedoms, such as beams, plates and shells, iffinite rotations are exactly treated.Recall the expression (10.15) of the second variation #2U of the internal energy density. Thisexpression has the core tangent stiffness S as kernel of the quadratic form in #g. The core internalforce ! also appears in the inner product

*#2g+T !. This second term may either survive or drop

out depending on the relation of g with the target physical or generalized coordinates (the latterterm is explained below) chosen in the CCF transformation phase. In the case of the DCCF, thisterm drops out and

S = SM + SG (11.12)is the tangent core stiffness, which forward transforms as per (11.3). This is the situation consideredso far. But if that term survives two things happen. First, (11.12) is relabeled as

S = SM + SGP , (11.13)

11–5


in which S and SGP are called the principal core tangent stiffness and principal geometric stiffness,respectively. Second, transforming the term

!!2g"T ! to freedoms v produces a extra term in

accordance with the schematics

K = KM +KGP +KGC , SM ! KM , SGP ! KGP ,!!2g"T ! "! !vTKGC !v, (11.14)

where! and "! symbolize DCCF-transformation and GCCF-transformation-styles, respectively.As can be seen the transformation phase produces a new term KGC called the complementarygeometric stiffness. That term cannot be expressed in terms of the variation !g of the displacementgradients. Consequently there is no “core complementary core geometric stiffness” SGC that can beadded to (11.13). Instead it appears as a “carry forward term” that materializes as a quadratic-formkernel upon transforming.

§11.3.1. Generalized Coordinates as Generic Target

For elements that require the GCCF treatment a one-shot transformation between g and v is oftenreplaced by a multistage transformation. The degree of freedom sets used as intermediate targetsof this process will be collectively referred to as “generalized coordinates” and identified as q.Of course the final target: element node displacements v, is a particular instance of such array ofchoices.In §10.2 it was noted that two variants of the GGCF, qualified as algebraic and differential, should bedistinguished in terms of consequences on the existence of physical stiffness equations at variousvariational levels. These variants are examined below. The ensuing development examines thetransformation from displacement gradients g to a “generic target” set of generalized coordinatesqi collected in vector q. These coordinates are assumed to be independent, a restriction removedlater. Symbols K and f are used to denote tangent stiffness matrices and internal force vectors,respectively, in terms of q.

§11.3.2. Algebraic Transformation

TheAlgebraicGCCF, orAGCCF, applies if the relation betweeng (source) andq (target) is nonlinearbut algebraic. We have g = g(q) or in index notation, gi = gi (q j ). Differentiating with respect tothe qi variables yields

!gi = "gi"q j

!q j = Gi j!q j , or !g = G !q,

!2gi = "2gi"q j"qk

!q j!qk + "gi"q j

!2q j = Fi jk!q j!qk, or !2g = (F !q) !q!0,

(11.15)

Here (F!q) is the matrix Fi jk !qk = Fki j !qk ; F being a cubic array. The arrayG receives the nametangent transformation matrix. The second term in the expansion of !2gi vanishes because the qiare assumed to be independent target variables.Enforcing invariance of !2U yields the tangent stiffness transformation

K =#

V0

$GT (SM + SGP)G+Q

%dV = KM +KGP +KGC = KM +KG, (11.16)

11–6

11–7 §11.3 THE GENERALIZED CCF

where the entries ofQ are (cf. Remark 4.1) Qi j = Q ji = Fki j!k with summation on k = 1, . . . ng .Note that Q is symmetric because Fki j = Fkji . Integration of Q over V0 yields the complementaryportion KGC of the geometric stiffness KG .The internal, applied and prestress force vectors transform according to the formulas in (11.4) withthe G defined in (11.15):

f =!

V0GT! dV, p =

!

V0GT" dV, p0 =

!

V0GT"0 dV . (11.17)

What happens to KU and Kr? They can be obtained, somewhat artificially, by constructing thematrix equation

g =Wq, (11.18)

whereW is called a secant transformation matrix. Generally this matrix is far from unique becauseits ng ! nq entries must satisfy only ng conditions. (Care has often to be given to the q j " 0 if0/0 limits appear inW.) Using (11.18) we can proceed to form

KU =!

V0WTSUW dV, Kr =

!

V0GTSrW dV . (11.19)

Because in generalW #= G, symmetry in the secant stiffness Kr cannot be expected even if Sr issymmetric.

Remark 11.1. The AGGCF is applicable to finite elements with degrees of freedoms that include fixed-axisrotations, because such rotations are integrable. Examples are provided by two-dimensional beams as well asplane stress (membrane) elements with drilling freedoms if only in-plane motions are allowed.

Remark 11.2. Why is KGC called a geometric stiffness? Because it vanishes if the current configuration isstress free, in which case the core internal force! vanishes and so does Q.

§11.3.3. Differential Transformation

The Differential GCCF, or DGCCF, is required if the relation between g (source) and q (target) isonly available as a non-integrable differential form between their variations:

"gi = Gi j"q j , or "g = G "q,

"2gi = #Gi j

#qk"q j "qk = Fi jk "q j "qk, or "2g = (F"q) "q.

(11.20)

The transformation equation (11.16) still applies forK whereas (11.17) holds for the force vectors.But no integral g = g(q) as in the AGGCF exists. Consequently KU and Kr , which require asecant matrix relation of the form (11.18), cannot be constructed. FurthermoreQ is not necessarilysymmetric; a condition for that being Fki j = Fkji or equivalently #Gki/#q j = #Gkj/#qi .

Remark 11.3. For mechanical finite elements the DGCCF naturally arises when three-dimensional finiterotations are present as nodal degrees of freedom, because such rotations are non-integrable.

11–7


Remark 11.4. The relations (11.20) have points of resemblance with the case of non-holonomic constraintsin analytical dynamics.

§11.3.4. Multistage Transformation

Up to this point the q have been assumed to be independent variables. But as previously noted,for complicated elements the GCCF transformations are more conveniently applied in stages. Thetarget variables in one stage become the source variables for the next one.What happens if the q are intermediate variables in a transformation chain? If the q are linear in thefinal independent degrees of freedom v, all previous formulas hold because the DCCF applies forthe remaining transformations, which are strictly congruential. But if the q are nonlinear in v, oronly a non-integrable differential relation exists, term (!gi/!q j ) "2q j = Gi j "

2q j in the second of(11.15) survives. The net effect is that the geometric stiffness acquires a higher order component,implicitly defined as the kernel of !

V0#iGi j "

2q j dV, (11.21)

This term cannot be resolved (“resolution” meaning explicit extraction of its stiffness kernel in theform of a complementary geometric stiffness) until the transformation chain reaches downstreamvariables that either are the final degrees of freedom (and thus independent), or depend linearlyon such. It is difficult to state detailed rules that encompass all possible situations. Instead thetreatment of the 2D and 3D beam element transformations in §11.4 and §11.11 illustrates the basictechniques for “carrying forward” terms such as (11.21).

§11.4. A 2-Node 2D Timoshenko Beam ElementWecontinue herewith the derivation of a 2D, isotropic Timoshenko beam element started in §10.5.4.This example serves to illustrate the Algebraic GCCF. The specific element constructed here hastwo end nodes, six degrees of freedom, and reference length L0. The cross section area A ! A0 andmoment of inertia I =

"A Y

2 d A are constant along the element. Axis X is made to pass through thecentroid so that

"A Y d A = 0. Furthermore it is assumed that the cross section is doubly symmetric

so that"A Y

3 d A = 0.The element displacement field, defined by u0X (X), u0Y (X) and $(X), is interpolated with linearshape functions:

# u0Xu0Y$

$

=# N1 0 0 N2 0 00 N1 0 0 N2 00 0 N1 0 0 N2

$

%

&&&&&'

vx1vy1$1vx2vy2$2

(

)))))*= Nv, (11.22)

where N1 = 1" (X/L0) and N2 = 1" N1 = X/L0. Consequently

% = !u0X!X

= vX2 " vX1

L0, & = !u0Y

!X= vX2 " vX1

L0, ' = !$

!X= $2 " $1

L0, (11.23)

are constant over the element.

11–8

11–9 §11.4 A 2-NODE 2D TIMOSHENKO BEAM ELEMENT

§11.4.1. Generalized Coordinates and Stress Resultants

As intermediate set of generalized coordinates we take qT = [ ! " # $ ]. These four quantitiesare constant over each cross section and may be viewed as cross-section orientation coordinates.Consequently when obtaining stiffness matrices and internal forces in terms of q it is convenient tointegrate over the beam cross section. The resulting quantities appear naturally in terms of crosssection stress-resultants as shown below. In terms of these generalized coordinates the auxiliaryvectors bi listed in (10.61) become

b1 =

!

"#

1+ g1g200

$

%& =

!

"#

1+ ! ! Y# cos $" ! Y# sin $

00

$

%& , b2 =

!

"#

g31+ g41+ g1g2

$

%& =

!

"#

! sin $

cos $1+ ! ! Y# cos $

" ! Y# sin $

$

%& ,

(11.24)The well known stress resultants of beam theory are the axial force N , transverse shear force Vand bending moment M . They are obtained by integrating the PK2 stresses over the beam crosssection:

N ='

A0s1 d A = E A

(! + 1

2 (!2 + " 2)

)+ 1

2 E I#2 + N 0,

V ='

A0s2 d A = GAs%" + V 0,

M ='

A0s1Y d A = !E I#%! + M0,

(11.25)

where %! = (1+!) cos $ +" sin $ and %" = " cos $ ! (1+!) sin $ can be viewed as generalizedskew strains. In (11.25) N 0, V 0 and M0 denote initial-stress resultants (stress resultants in C0, alsocalled prestress forces), A " A0, I =

*A0 Y

2 d A, and As = µA, in which µ is the usual shearcorrection factor of Timoshenko beam theory. Because of the doubly-symmetric cross-sectionassumption, a term containing the third-section-moment

*A0 Y

3 d A has been omitted from theexpression for M .In addition to N , V and M , the following higher order moment, which is absent from the lineartheory, appears in the residual force and tangent stiffness:

C ='

A0s1Y 2 d A = E I

((! + 1

2 (!2 + " 2)

)+ 1

2 EH#2)+ C0, (11.26)

in which H =*A0 Y

4 d A. If terms in #2 are neglected,

C ! C0 = (N ! N 0)(I/A) = (N ! N 0)r2, (11.27)

where r =#I/A is the radius of gyration of the cross section. If such terms are retained this

relation is only exact if r2 = H/I and approximate otherwise.

Remark 11.5. One may verify that*A s2Y d A vanishes identically. This serves as a check of the strain

distribution equations.

11–9


§11.4.2. Transformation Matrices

The differential relations required to establish the tangent transformation are obtained from (10.57)as

!g = "g"q

!q =

!

"#

1 0 !Y cos # Y$ sin #

0 1 !Y sin # !Y$ cos #0 0 0 ! cos #0 0 0 ! sin #

$

%&

!

"#

!%

!&

!$

!#

$

%& = G1 !q, (11.28)

!q = "q"v

!v = 1L0

!

"#

!1 0 0 1 0 00 !1 0 0 1 00 0 !1 0 0 10 0 L0 ! X 0 0 X

$

%&

!

""#

!vX1!vY1

...

!#2

$

%%& = G2 !v, (11.29)

The transformation relating !g = G !v may be obtained as the product

G = G1G2 = 1L0

!

"#

!1 0 Y (cos # + (L0 ! X)$ sin #) 1 0 Y (! cos # + X$ sin #)

0 !1 Y (sin # ! (L0 ! X)$ cos #) 0 1 Y (! sin # ! X$ cos #)

0 0 !(L0 ! X) cos # 0 0 !X cos #0 0 !(L0 ! X) sin # 0 0 !X sin #

$

%&

(11.30)

but it is more instructive (as well as conducive to higher efficiency in the computer implementation)to perform the transformation phase in two stages.Observe that the first transformation (from g to q) is nonlinear and algebraic whereas the secondone (from q to v) is linear. Consequently we have to use the AGCCF for the first transformationbut the second one can be done simply through the DCCF.


The internal force vector in terms of q, denoted by fq , is obtained from the core expression (10.64)for ! and the matrix G1 given in (11.28):

fq ='

A0GT1 ! d A0 =

!

"#

f q%f q&f q$f q#

$

%& =

!

"#

N (1+ %) ! M$ cos # ! V sin #

N& ! M$ sin # + V cos #!M'% + C$

!M$'& ! V'%

$

%& . (11.31)

Finally, application of (11.29) and integration over the element length yields

f =' L0

0GT2 fq dX =

!

"""""""#

! f q%! f q&

! f q$ + 12 L0 f

q#

f q%f q&

f q$ + 12 L0 f

q#

$

%%%%%%%&

. (11.32)

This vector satisfies translational equilibrium.

11–10


§11.4.4. Tangent Stiffness Matrix

Transforming to generalized coordinates q produces three components of the tangent stiffnessmatrix:

Kq =!

A0

"GT1 (SM + SG)G1 +Q

#d A = Kq

M +KqGP +Kq

GC . (11.33)

The entries of KqM , obtained through symbolic manipulation, are

KqM (1, 1) = E A(1+ !)2 + GAs sin2 " + E I#2 cos2 ",

KqM (1, 2) = E A(1+ !)$ ! GAs sin " cos " + E I#2 sin " cos ",

KqM (1, 3) = E I#

"(1+ !)(1+ cos2 ") + $ sin " cos "

#,

KqM (1, 4) = E I#2%$ cos " + GAs%! sin ",

KqM (2, 2) = E A$ 2 + GAs cos2 " + E I#2 sin2 ",

KqM (2, 3) = E I#

"(1+ !) sin " cos " + $ (1+ sin2 ")

#,

KqM (2, 4) = E I#2%$ sin " ! GAs%! cos ",

KqM (3, 3) = E I%!

2 + EH#2,

KqM (3, 4) = E I#

"(1+ !)$ (cos2 " ! sin2 "

#+"$ 2 ! (1+ !)2

#sin " cos "

#,

KqM (4, 4) = E I#2&2g + GAs%!

2.

(11.34)

The principal geometric stiffness, which is readily worked out by hand, is

KqGP =

$

%&

N 0 !M cos " M# sin " ! V cos "N !M sin " !M# cos " ! V sin "

C 0symm C#2

'

() (11.35)

The new term contributed by the AGCCF toKq is the complementary geometric stiffnessKqGC . Its

source is the matrix Q introduced in Section 8.2. The entries of Q are Qi j = ('2gk/'qi'q j )(k ,where the components of g and ! = s1b1 + s2b2 may be obtained from (10.57) and (11.24),respectively.The entries of Q were symbolically generated by the followingMathematica module:

QmatrixOf2DTimoBeamElement[eps_,gamma_,kappa_,theta_,Em_,Gm_,Y_]:=Module[{g,h1,h2,H1,H2,e1,e2,s1,s2,b1,b2,phi,i,j,k},q={eps,gamma,kappa,theta}; phi={1,1,1,1};g={eps-Y*kappa*Cos[theta],gamma-Y*kappa*Sin[theta],

-Sin[theta],Cos[theta]-1};gg={{g[[1]]},{g[[2]]},{g[[3]]},{g[[4]]}};h1={{1},{0},{0},{0}}; h2={{0},{1},{1},{0}};H1={{1,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}};H2={{0,0,1,0},{0,0,0,1},{1,0,0,0},{0,1,0,0}};e1=(Transpose[h1].gg+(1/2)*Transpose[gg].H1.gg)[[1,1]];e2=(Transpose[h2].gg+(1/2)*Transpose[gg].H2.gg)[[1,1]];s1=Simplify[Em*e1]; s2=Simplify[Gm*e2];

11–11


b1={1+eps-Y*kappa*Cos[theta],gamma-Y*kappa*Sin[theta],0,0};b2={-Sin[theta],Cos[theta],1+eps-Y*kappa*Cos[theta],

gamma-Y*kappa*Sin[theta]};phi=Simplify[s1*b1+s2*b2];Q=Table[0,{4},{4}];For[i=1,i<=4,i++, For[j=1,j<=4,j++, For[k=1,k<=4,k++,Q[[i,j]]=Q[[i,j]]+(D[D[g[[k]],q[[i]]],q[[j]]])*phi[[k]] ]]];

Return[Q]];

The output of this module was integrated over the cross section and pattern matched with theexpression of the stress resultants (11.25)-(11.26) to produce

KqGC =

!

"#

0 0 0 00 0 0

0 !M!"

symm !V!" + M#!$ ! C#2

$

%& , (11.36)

which added to (11.35) yields the geometric stiffness

KqG =

!

"#

N 0 !M cos % M# sin % ! V cos %N !M sin % !M# cos % ! V sin %

C !M!"

symm !V!" + M#!$

$

%& . (11.37)

Finally, the tangent stiffness in terms of q is Kq = KM +KG . Denoting the entries of Kq by Kqi j ,

i, j = 1, . . . 4 the tangent stiffness matrix K in terms of node displacements v is formed throughthe DCCF transformation

K =' L0

0GT2KqG2 dX =

!

"""""""#

Kq11 Kq

12 Kq13 ! 1

2 L0Kq14

Kq22 Kq

23 ! 12 L0K

q24

Kq33 ! L0Kq

34 + 13 L

20K

q44

symm!Kq

11 !Kq12 !Kq

13 ! 12 L0K

q14

!Kq12 !Kq

22 !Kq23 ! 1

2 L0Kq24

!Kq13 + 1

2 L0Kq14 !Kq

23 + 12 L0K

q24 !Kq

33 + 16 L

20K

q44

Kq11 Kq

12 Kq13 + 1

2 L0Kq14

Kq22 Kq

23 + 12 L0K

q24

Kq33 + L0Kq

34 + 13 L

20K

q44

$

%%%%%%%%&

.

(11.38)

The above rule can be applied to KM and KG should separate formation be desirable, as whensetting up a stability eigenproblem.If the reference configuration is not aligned with X , the preceding expressions apply to the localsystem {X , Y }. A final local-to-global transformation step, similar to that discussed for the 3D bar

11–12


in §11.3, is then necessary. This step can be handled by a simple DCCF transformation, becausethe finite rotation ! remains the same in global coordinates.

Remark 11.6. The foregoing exact expressions contain curvature-squared terms typically in the combinationI"2. This can be shown to be of order (r/R)2 compared to other terms, where r is the radius of gyrationof the cross section and R = 1/" the radius of curvature of the current configuration. For typical beams(r/R)2 is 10!6 or less; consequently all such tiny terms may be dropped without visible loss of accuracy. Forhighly-bent extremely-thin beams, however, that ratio may go up to 0.01 in which case the "2 terms mighthave a noticeable though small effect if retained.

§11.4.5. Can a Secant Stiffness be Constructed?

To attempt the construction of a secant stiffness Krq in terms of generalized coordinates q oneshould obtain a secant matrix form of the relationship g = g(q). As noted previously such form isfar from unique. One possible choice is

g =

!

"#

g1g2g2g4

$

%& =

!

"#

1 0 !Y cos ! 00 1 !Y sin ! 00 0 0 ! sin !/!

0 0 0 (cos ! ! 1)/!

$

%&

!

"#

#

$

"

!

$

%& =W1q, (11.39)

which has themerit of not being too dissimilar fromG1. Note that some caremust be taken as regardssome 0/0 limits. ThenKrq =

'AG

T1 SrW1 d A, which may be easily worked out in closed form but

is unsymmetric. Because q is linear in v, the next transformation is simplyKr =' L00 GT

2KrqG2 dXwhich can be handled through a scheme similar to (11.38) but with an unsymmetric kernel matrix.

§11.5. A 2-Node 3D Timoshenko Beam ElementWe continue here the development of a two-node 3D Timoshenko beam element started in §10.5.5.As can be surmised, the development is more complex and demanding than for its 2D counterpart.Only a summary taken from Crivelli’s thesis [1] and Crivelli and Felippa [3] is presented here. Thetransformation phase to pass from the core equations to the element nodal degrees of freedom iscarried out in three stages:1. From particle displacement gradients g to generalized gradients w at each cross section. An

integration over the cross section area is involved.2. From generalized gradients w to cross-section orientation coordinates q. The rotational

parametrization is introduced at this stage.3. From cross-section orientation to finite-element nodal degrees of freedom v. An integration

over the element length, as defined by the shape functions, is involved.These transformation stages are summarized in Tables 11.1 and 11.2, which together also serve todefine notation

§11.5.1. Transformation to Generalized Gradients

The first set of target variables are the generalized gradients w(X) at each reference cross sectiondefined by the longitudinal coordinate X . The components of w are indirectly given through theirfirst variation:

%w =( d %u0dX

d%!dX %!

)T, (11.40)

11–13


Table 11.1 Internal energy and its variations for 3D Timoshenko beam element

Core Section Gradients Section Orientation Physical DOF

Particle Cross-Section Cross-Section Whole Elementg w z v

U = 12g

TSUg+ gT!0 — — —

!U = !gT (Srg+ !0) !UG = !wT R !Uz = !zT fz !U = !vT f

!2U = !gT S !g+ !2gT" !2UG = !wTS !w+ F !2Uz = !zTKz !z !2U = !vTK !v

Table 11.2. Core-to-physical-DOFs transformations for 3D beam element

Core Level Section Gradients Section Orientation Physical DOF

Particle Cross-Section Cross-Section Whole Elementg w z v

" R =!

A0

WT" d A fz = ZTR f =! L0

0GTz fz d X

S S =!

A0

WTSW d A Kz = ZTSZ + SGCz K =! L0

0GTz Kz Gz d X

!g = W !w !w = Z !z !z = Gz !v

where !!, defined in (10.68), measures the variation of angular orientation. Because this quantityis not generally integrable for three-dimensional motions, it is not possible to express! as a uniquefunction of the displacements. The variation of g1 is

!g1 = d!u0dX

+ RT !T d!!dX

+ RT !T"!! + RT "!!!

T", (11.41)

where we used the relation [1] !" = d!!/dX + "!!. On using the commutative law ab = bT aand Jacobi’s identity #ab = ab! ba we may rewrite (11.41) as

!g1 = " !u0"X

+ RT !T " !!

"X+ RT "T ! !! (11.42)

For the other gradient vectors we have !g2 = !RTh2 = RT "!!h2 = RT hT2 !! and !g3 = RT hT3 !!,

11–14


which can be collected in matrix form as

!g =

!

"!g1!g2!g3

#

$ =

!

%"

I RT !T RT "T !

0 0 RT hT20 0 RT hT3

#

&$

!

%%"

d !u0dXd !!dX!!

#

&&$ =

!

%"W1

W2

W3

#

&$ !w =W !w, (11.43)

where I is the 3-by-3 identity matrix andWi are 3-by-9 matrices. The second variation of g, whichis required for the complementary geometric stiffness, is

!2g1 = RT '!!!T d !!dX

+ RT '!! !T" !! + RT '!! !

T d !!dX

+ RT '!!!T"!! + !2RT !

T" + RT !

T!2",

!2g2 = !2RT i2, !2g3 = !2RT i3

(11.44)

At this point it is appropriate to introduce the following section resultants:

P = A"b + P0, sb = Eeb,Q = µs A + Q0, # = #2 + #3, #2 = G#2h2, #3 = G#3h3,

M" = E ISKe + M0" , IS =

(

A0!!T d A, Ke = $ ",

M$ = µtG IP" + M0$ , IP =

(

A0! !

Td A.

(11.45)

Here P,Q,M" andM$ are axial forces, shear forces, bending moments and torsional moments,respectively, at the current configuration C; P0, Q0, M0

" and M0$ are similar quantities at the

reference configuration C0; µs and µt are transverse-shear and torsion coefficients that account forthe actual shear stress distributions, respectively; and IS and IP are the cartesian and polar inertiatensors, respectively, of the cross section. Should the axes Y and Z be aligned with the principalinertia axes the latter simplified to

IS =) 0 0 00 I22 00 0 I33

*

, IP =) I22 + I33 0 0

0 I33 00 0 I22

*

. (11.46)

Because the relation between g and w is of differential type the applicable transformation rules arethose the DGCCF, and no energy or secant stiffness survives. Thus only the internal force vectorR and tangent stiffness S associated with w are derived below.

Internal Force Vector. The generalized internal force vector is

R =(

A0WT" d A =

+

i

(

A0siWTbi d A = R" + R$ , (11.47)

11–15


where R! and R" are the contributions of the normal and shear stresses respectively. Detailedcalculations result [1] in the following exact expressions:

R! =

!

"RT (P! + "M! )

!TM!

#KTe M!

$

% , R" =

!

"RTQM"

!TQ + "TM"

$

% . (11.48)

For small deformations in which the squared curvature may be neglected, R ! I, ! !#h1, Ke ! "and "M! ! 0. If these approximations are made,

R! =

!

&"Ph1#hT1 M!

0

$

'% , R" =

!

"Q

M"

#hT1 Q

$

% . (11.49)

These resemble the classic linearized theory equations. Furthermore observe that the term PRT!corresponds to the internal force of the TL 3D bar.

Tangent Stiffness. For the tangent stiffness we have the decomposition

S = SM + SGP + SGC . (11.50)

Furthermore, since w is nonlinear in downstream variables, the complementary geometric stiffnesssplits into two components:

SGC = SGCw + SGCq , (11.51)

where SGCw and SGCq contains terms that depend on the first and second variations, respectively,of R and ". The notation is suggested by the fact that SGCw can be merged into SGP to yield thegeometric stiffness SGw = SGP +SGCw, which is associated with the generalized gradients w andindependent of the rotational parametrization selected in the next set of target variables q. On theother hand, the kernel SGCq cannot be extracted at the w level and must be carried forward to theq level because it is parametrization dependent. Each of the components in (11.50)-(11.51) maybe expressed as the sum of two contributions, one from the normal stresses and one from the shearstresses:

SM = SM! + SM" , SGP = SGP! + SGP" , SGCx = SGCx! + SGCx" , x = w, q. (11.52)

Material Stiffness. The generalized core material stiffness is given by the congruential transforma-tion

SM =(

A0WTSMW d A =

)

i

(

A0EiWTbibTi W d A = SM! + SM" . (11.53)

Carrying out the algebraic manipulations one obtains

SM! = E

!

"RT (!!T + "T IS")R RT "IS! RT "IS#Ke

!T IS! !

T IS#Ke

symm #KTe IS#Ke

$

% , (11.54)

11–16


SM! = µG

!

"ART I!R 0 ART I!!

IP IP "

symm A!T I!! + "T IP "

#

$ , in which I! =% 0 0 00 1 00 0 1

&

. (11.55)

The contribution RT!!TR is the core material stiffness of a TL 3D bar.

Geometric Stiffness due to Normal Stresses. It is convenient to work out together all geometricstiffness terms produced by the normal stresses, i.e.

SG" = SGP" + SGCw" + SGCq" = SGw" + SGCq" . (11.56)

The appropriate definitions are

SGP" ='

A0s11WT

1HW1 d A,

SGC" ='

A0s11b1 #2g d A = #wTSGCw" #w+ F(#2R, #2"),

(11.57)

where F contains SGCq as q level kernel. Carrying out the algebraic manipulations one obtains

SGw" = SGP" + SGCw" =

!

"P I RT (MT

" RT "T (M"

0 (M" !

symm !(M" " + "T (M" !

#

$ (11.58)

The term P I corresponds to the core geometric stiffness of the 3D Tl bar.The higher order term in (11.57) may be expressed as

F" (#2R, #2") = MT" !#2" + !TR#2RT "M" #qT

)V(!

TM" ) + U("M" ; !)*

#q, (11.59)

ConsequentlySGCq" = V(!

TM" ) + U("M" ; !). (11.60)

Because the next-level target variables q include the finite rotation parametrization, matrices V andU depend on that choice. They are the source of unsymmetries in the stiffness matrices when certainrotational parametrizations are adopted, such as the incremental rotation vector. If the rotationalvector is chosen these matrices are symmetric.

Geometric Stiffness due to Shear Stresses. The contribution of the shear stresses to the geometricstiffness is

SG! = SGP! + SGCw! + SGCq! = SGw! + SGCq! . (11.61)

The appropriate definitions are

SGP! ='

A0s12(WT

1HW2 +W2HW1) + s13(WT1HW3 +W3HW1) d A

SGC! ='

A0(s12b2 + s13b3) #2g d A = #wTSGCw! #w+ F! (#

2R, #2").

(11.62)

11–17


Carrying out manipulations one obtains the surprisingly simple form for SGw!

SGw! = SGP! + SGCw! =

!

"0 0 RT #QT

0 0symm 0

$

% . (11.63)

The terms due to the second variation of g become

F! = QT "2R! + MT! "2!. (11.64)

The kernel carried forward to the q level is

SGCq! = V(M! ) + U(Q ;!). (11.65)

§11.5.2. Transformation to the Rotational Vector

The second transformation stage passes fromw to z, which is a vector of generalized displacements,also associated with a beam section, which embodies the parametrization of the cross sectionrotation:

z =&du0dX

d"

dX"

'T, "z =

&d "u0dX

d ""

dX""

'T. (11.66)

Here " denotes the rotational vector parametrization defined by the standard formulas

" = axial ("), R = exp("T ), (11.67)

and which may be extracted from R by

" = logR = arcsin(! )

2!axial (RT ! R), ! = 1

2 || axial (RT ! R)||. (11.68)

Because only the variations ofw are known the relation betweenw and z is also of differential type:

"w = Z "z, or "w =

!

"I 0 00 Y(z) dY(z)

dX0 0 Y(z)

$

%

!

("

d "u0dXd ""dX""

$

)% , (11.69)

in whichY(#) = sin |"|

|"|I+

*1! sin |"|

|"|

+""T

|"|2! 1! cos |"|

|"|2". (11.70)

On applying the transformations (11.69)we find for the internal force and thematerial and principal-geometric components of the tangent stiffness matrix:

fq = ZT (R$ + R! ), KMq = ZT (SM)Z, KGPq = ZT (SGwZ. (11.71)

The materialization of the geometric stiffness terms SGCq$ and SGCq! for the rotational vectorneeds additional work. We state here only the final result:

U(# ;!) =

!

"0 0 0

0 0symm U!

$

% , T(M! ) =

!

"0 0 0

0 T!1

symm T!2

$

% . (11.72)

11–18


where

U! = c1!T!I+ c2!!!T + !T!

"+ c3

#!T!""T + !T "!I+ !T!"T + "!T !

$

+ c5!"T!

!!"T + "!T

"+ "T!

!"!T + !"T "+ !T""T!I

"

+ c4 !T "!""T + c6!T""T!""T ,

V!1 = c2%M

T! + c3"MT

! + c5""T %M! + c7!M!"

T + "TM! I"+ c8"TM!""T ,

V!2 = !c3

d"

dX

TM! I! c4

d"

dX

TM!""T + c5

&'d"

dX

T

M!"T + "

'd"

dXMT

! + "T'd"

dXM! I

(

+ c6"T'd"

dXM!""T + c7

)d"

dXMT

! + M!

d"

dX

T*+

+ c8d"

dX

T"!"MT

! + M!"T + "TM! I

"+ c9

d"

dX

T""TM!""T ,

(11.73)in which

c1 = ! sin"

",

c4 = !c1 + 3c3"2

,

c7 = 1+ c1"2

,

c2 = 1! cos""2

,

c5 = !c1 + 2c2"2

,

c8 = 3c3 ! 2c2"2

,

c3 = sin" ! " cos""3

,

c6 = !c3 + 4c5"2

,

c9 = c5 ! 5c8"2

.

(11.74)

A similar approach can be taken with (11.69), which defines F# . The tangent stiffness matrix canbe obtained by superposing all contributions.

§11.5.3. Transformation to Finite Element Freedoms

The final stage introduces a finite element representation for the degrees of freedom. The beamor beam assembly is divided into a set of two-node finite elements. Each of these nodes has threedisplacement degrees of freedom and three rotational degrees of freedom corresponding to the three{"X , "Y , "Z } components of the rotational vector ". Each element in turn has twelve freedomswhich are collected in the array vT = {un "n}T where dn collects the six translational freedomswhile "n collects the six rotations. The cross-section state vector z is approximated inside eachelement by

z =

+

,N 00 d N

d X0 N

-

./ dn

"n

0= Gz

/ dn"n

0= Gz v. (11.75)

where N is a matrix of linear shape functions. Since $q = Gz$v the final internal force vector f andtangent stiffness matrix K of each element are obtained through the DCCF transformations

f =1 L0

0GTz fz d X, K =

1 L0

0GTz KzGz d X. (11.76)

11–19


The choice of shape functions for the rotational vector poses some subtle questions. In small-deflection analysis it is common practice to select all Timoshenko beam shape functions to be linearin X . This choice obviously enforces nodal compatibilitywhile preserving constant curvature states.But for finite deflections a linear interpolation for the rotational vector components cannot exactlyrepresent a constant curvature state unless the rotations are about a single axis (plane rotations).The same is true if the rotation matrix R(X) is interpolated linearly. On the other hand, linearinterpolation of Euler parameters does preserve the constant curvature state. This motivated thedevelopment of an interpolation scheme that starts from the 4 Euler parameters !i (X), i = 0, 1, 2, 3,!

i !2i = 1 that orient the normal of a cross section at X . These are collected in the 4-vector

! = [ !0 !1 !2 !3 ]T . Given the eight end values !(0) and !(L) the interpolation that can copya constant curvature vector " is found to be [1]

!(" ) = cos(" )

"1! tan(" )

tan("L)

#!(0) + sin(" )

sin("L)!(L), (11.77)

where " = 12#X , "L = 1

2#L , # ="

"T". The constant curvature vector can be extracted from theend values through the formula

" = 1$2L

$%&!(L) ! 2!0(L)I'!(0) !

%&!(0) ! 2!0(0)I'!(0)

(, (11.78)

This interpolation is then transformed to the variations in terms of the rotational vector. Details areprovided in Reference [1].

§11.6. Equivalence of DCCF and Standard TL FormulationThe correspondence between theDirect Core Congruential Formulation (DCCF) and the Standard Formulation(SF) of the Total Lagrangian (TL) kinematic description is established below for 3D continuum finite elements.This connection was worked out in a course term project [4]. Such elements fit within the DCCF frameworkbecause their physical DOFs (node displacements) are of translational type.The Standard Formulation is based on the same scheme used for linear finite elements: first interpolate, thenvary. As in the linear case, the departure point is extremization of the Total Potential Energy functional (TPE)over the element domain:

% = U ! W =)

V0

eT s0 dV + 12

)

V0

eTEe dV !)

V0

uTb dV !)

St0

uT t dS, (11.79)

where as usual conservative dead loading is assumed. In (11.79), b is the prescribed body force field, t aresurface tractions prescribed over portion St0 of the boundary in C0, and other quantities are as defined in Section4. The weak equilibrium equations are obtained on making (11.79) stationary:

&% = &U ! &W =)

V0

&eT s0 dV +)

V0

&eTEe dV !)

V0

&uTb dV !)

St0

&uT t dS = 0. (11.80)

The displacement and strain fields are interpolated in terms of the element degrees of freedom v:

u = Nv, &u = N &v, &e = B &v, (11.81)

where B = B(v) depends in v but N does not. Substituting these interpolations into (11.80) yields the residualequilibrium equations

&% = &vT r = &vT (f! p) = 0. (11.82)

11–20

11–21 §11.6 EQUIVALENCE OF DCCF AND STANDARD TL FORMULATION

wheref =

!

V0

BT (s0 + Ee) dV =!

V0

BT s dV, p =!

V0

NTb dV +!

St0

Ns dS, (11.83)

where f and p are the internal and external force vectors, respectively, and s = s0 +Ee are the PK2 stresses inC. Because the variations !v are arbitrary, the residual-force nonlinear equilibrium equation is r = f! p = 0or f = p. The tangent stiffness matrix is given by

K = "r"v

= "f"v

, (11.84)

because p (for conservative dead loading) does not depend on v. Splitting B = Bc + Bv(v), where Bc isconstant but Bv depends on v, gives the well known decomposition

K = K0 +KD +KG, (11.85)

where K0, KD and KG denote the linear, initial-displacement and geometric stiffness matrices, respectively.These are given by

K0 =!

V0

BTc EBc dV,

KD =!

V0

(BTc EBv + BTv EBc + BTv EBv) dV,

KG !v =!

V0

!BT s dV .

(11.86)

To correlate these standard forms with those produced by the DCCF, we note that the GL strains can be alsosplit as e = ec + ev , where ec and ev are linear and nonlinear in v, respectively. The latter may be expressedin terms of the displacement gradients as

ev = 12Ag, (11.87)

where A is the 6" 9 matrix

A =

"

####$

gT1 0 00 gT2 00 0 gT30 gT3 gT2gT3 0 gT1gT2 gT1 0

%

&&&&'=

"

####$

g1 g2 g3 0 0 0 0 0 00 0 0 g4 g5 g6 0 0 00 0 0 0 0 0 g7 g8 g90 0 0 g7 g8 g9 g4 g5 g6g7 g8 g9 0 0 0 g1 g2 g3g4 g5 g6 g1 g2 g3 0 0 0

%

&&&&', (11.88)

in which the displacement gradients are vector-arranged as

gT = [ g1 g2 · · · g8 g9 ] =( "u1

"X1"u2"X1 · · · "u2

"X3"u3"X3

). (11.89)

Comparing!ev = 1

2 !Ag+ 12A !g = A !g, (11.90)

to the DCCF transformation relation !g = G !v, in which G is independent of v, we see that

Bv = AG. (11.91)

The other expression we require is !AT s, which appears in the geometric stiffness matrix contracted with !v:

KG !v =!

V0

!BT s dV =!

V0

GT !AT s dV . (11.92)

11–21


It is well known— see for instance Chapter 19 of Zienkiewicz [5]— that

!AT s = M !g = MG !v, with M =

!s1I s4I s5Is4I s2I s6Is5I s6I s3I

", (11.93)

where I is the 3 ! 3 identity matrix and si , i = 1, . . . 6 are components of the PK2 stress tensor ordereds1 = s11, s2 = s22, . . . s6 = s23. Using this relation, KG can be placed in the standard form

KG =#

V0

GTMG dV, (11.94)

which by inspection is seen to be the DCCF-transformation of the core geometric stiffnessM " SG = siHi ,with the Hi matrices defined in (10.10).To correlate other terms, write the linear part of the GL strains in terms of gradients as

ec = Dg = DG !v, with D =

$

%%%%&

1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 10 1 0 1 0 0 0 0 00 0 1 0 0 0 1 0 00 0 0 0 0 1 0 1 0

'

((((). (11.95)

The numerical D matrix can be easily related to the hi vectors introduced in (10.10). Because both Dand G are independent of v it follows that !ec = DG !v and consequently Bc = DG. Partitioning A as[ aT1 aT2 . . . aT6 ] one easily finds that ai = Hig. Now the following identities can be verified throughsimple algebra:

DTED = Ei jhihTi = S0,DTEA = Ei jhiaTj = Ei jhigTH j = S1, ATED = ST1 ,

ATEA = Ei jaia j = Ei jHigigTj H j = S2 = ST2 ,

M = s0i Hi + Ei jhigH j + 12 (g

THig)H j = s0i Hi + S#1 + 1

2S#2 = siHi .

(11.96)

Comparing these to the expressions of §10.4.3 we conclude that

K0 =#

V0

GTDTEDG dV =#

V0

GTS0G dV,

KD =#

V0

GT (DTEA+ ATED+ ATEA)G dV =#

V0

GTSDG dV,

KG =#

V0

GTMG dV =#

V0

GTSGG dV,

(11.97)

which displays the equivalence of both formulations when no approximations are made. This proof may beextended without difficulty to the AGCCF in which caseG is a function of v, although as noted in the text thatsituation is sometimes mishandled in the Standard Formulation through the introduction of a priori kinematicapproximations. The equivalence between DGCCF and SF is more difficult to prove because there is no TPEfunctional from which the latter can be derived, and such connection should be regarded as an open problem.

11–22

11–23 §11.7 REFERENCES

§11.7. References

[1] L. A. Crivelli, A Total-Lagrangian beam element for analysis of nonlinear space structures, Ph. D.Dissertation, Dept. of Aerospace Engineering Sciences, University of Colorado, Boulder, CO, 1990.

[2] C. A. Felippa and L. A. Crivelli, A congruential formulation of nonlinear finite elements, in NonlinearComputational Mechanics - The State of the Art, ed. by P. Wriggers and W. Wagner, Springer-Verlag,Berlin, pp. 283–302, 1991.

[3] L. A. Crivelli and C. A. Felippa, A three-dimensional non-linear Timoshenko beam element based onthe core-congruential formulation, Int. J. Numer. Meth. Engrg., 36, pp. 3647–3673, 1993.

[4] F. Abedzadeh Anaraki, A. Barzegar Mehrabi and H. R. Lofti, Correspondence between CC-TL andC-TL formulations, in Term Projects in Nonlinear Finite Element Methods, ed. by C. A. Felippa, ReportCU-CSSC-91-12, Center for Space Structures and Controls, University of Colorado, Boulder, CO, May1991.

[5] O. C. Zienkiewicz, The Finite Element Method, 3rd ed., McGraw-Hill, London, 1976.

11–23



Not Assigned

EXERCISE 11.1

Consider a two-node geometrically TL nonlinear bar in three dimensions built with the CCF.Write subroutinesto compute the tangent stiffness matrixK and the PK2 stress s in the current configuration given the followinginput data:

(1) The coordinates of both element nodes in the reference configuration global Cartesian system X, Y, Z .(2) The X, Y, Z displacements vXi , vY i , vZi of the two element end nodes i = 1, 2.(3) The elastic modulus E and the reference bar cross section A0.(4) The bar axial stress s0 in the reference configuration.

The subroutines you have to write (in Fortran, C or C++) have the following names and calling sequenceinterface:

BAR3F (X0, v, E, A0, s0, F, status)BAR3K (X0, v, E, A0, s0, K, status)BAR3S (X0, v, E, s0, s, status)

BAR3F and BAR3K compute the internal force vector f and tangent stiffness matrixK, respectively, in the currentconfiguration. Subroutine BAR3S computes the PK2 stress s in the current configuration.

Input variables (Fortran assumed in description):

Variable Declaration DescriptionX0 double precision x0(3,2) Global coordinates of end nodes.

The coordinates of node i go in thei th column of X0

v double precision v(3,2) vX , vY , vZ displacements of end nodes.The displacements of node i go in thei th column of v

E double precision e Elastic modulusA0 double precision a0 Reference cross section areas0 double precision s0 PK2 stress in reference configuration

The output variables are:

Variable Declaration DescriptionF double precision f(6) internal force vectorK double precision k(6,6) tangent stiffness matrixS double precision s PK2 axial stress s in current configurationSTAT character*(*) stat Blank if no error; else error message

11–24

11–25 Exercises

For intermediate manipulations, construct the local bar axis X joining nodes 1 to 2 in C0. Then construct Y ,and Z normal to X forming a right-handed system. As noted §11.2.1, there is some arbitrariness because Y , Zmay be “gyrated” around X without changing the final answer; select whatever orientation rule seem to bemore computationally “robust” in the sense that it should not fail for arbitrary bar orientations.

The answer for this exercise should be a listing of BAR3F, BAR3K and BAR3S.

EXERCISE 11.2

Explain the rule you chose to orient Y and Z . (Appropriate comments in the BARK source code should besufficient to answer this one.)

EXERCISE 11.3

Test the subroutines on the following input data:

Argument Input value(s)X0 [1.23, 2.34, 3.45, 5.43, 4.32, 3.21 ]v [0.76, !2.12, 1.67, !2.45, 3.01, !3.28]E 1.82A0 0.765S0 3.21

To feed these values write a short test driver that calls the two subroutines in turn; a suggested driver is listedunder “Programming Recommendations.”Compare the output of BAR3K with the following tangent stiffness matrix:

K =

!

""""#

0.935471 0.097502 !0.071172 !0.935471 !0.097502 0.0711720.097502 1.622137 !0.511148 !0.097502 !1.622137 0.511148

!0.071172 !0.511148 1.295011 0.071172 0.511148 !1.295011!0.935471 !0.097502 0.071172 0.935471 0.097502 !0.071172!0.097502 !1.622137 0.511148 0.097502 1.622137 !0.5111480.071172 0.511148 !1.295011 !0.071172 !0.511148 1.295011

$

%%%%&(E11.1)

Compare the output of BAR3F to the following internal force vector:

f =

!

""""#

!0.912675!6.5546684.7846310.9126756.554668

!4.784631

$

%%%%&(E11.2)

The computed PK2 stress s in the current configuration returned by BAR3S should be 5.603088.

11–25

.

12CR Formulation

Overview I

12–1

Chapter 12: CR FORMULATION OVERVIEW I 12–2

TABLE OF CONTENTS

Page§12.1. Introduction 12–3§12.2. The Emergence of CR 12–3

§12.2.1. Continuum Mechanics Sources . . . . . . . . . . . . 12–3§12.2.2. FEM Sources . . . . . . . . . . . . . . . . . . 12–4§12.2.3. Shadows of the Past . . . . . . . . . . . . . . . . 12–5§12.2.4. Linking FEM and CR . . . . . . . . . . . . . . . 12–6§12.2.5. Element Independent CR . . . . . . . . . . . . . . 12–7

§12.3. Corotational Kinematics 12–9§12.3.1. Configurations . . . . . . . . . . . . . . . . . 12–9§12.3.2. Coordinate Systems . . . . . . . . . . . . . . . . 12–9§12.3.3. Coordinate Transformations . . . . . . . . . . . . . 12–11§12.3.4. Rigid Displacements . . . . . . . . . . . . . . . . 12–11§12.3.5. Rotator Formulas . . . . . . . . . . . . . . . . 12–12§12.3.6. Degrees of Freedom . . . . . . . . . . . . . . . . 12–12§12.3.7. EICR Matrices . . . . . . . . . . . . . . . . . 12–13§12.3.8. Deformational Translations . . . . . . . . . . . . . . 12–16§12.3.9. Deformational Rotations . . . . . . . . . . . . . . 12–17

12–2

12–3 §12.2 THE EMERGENCE OF CR

Note: This and the following Chapters on the CR formulation of geometrically nonlinear FEM are new.References and Appendices for this material have been placed in Appendix R.


Three Lagrangian kinematic descriptions are in present use for finite element analysis of geometricallynonlinear structures: (1) Total Lagrangian (TL), (2) Updated Lagrangian (UL), and (3) Corotational(CR). The CR description is the most recent of the three and the least developed one. Unlike the others,its domain of application is limited by a priori kinematic assumptions:

Displacements and rotations may be arbitrarily large, but deformations must be small. (12.1)

Because of this restriction, CR has not penetrated the major general-purpose FEM codes that cater tononlinear analysis. A historical sketch of its development is provided in Section 2.As typical of Lagrangian kinematics, all descriptions: TL, UL and CR, follow the body (or element) asit moves. The deformed configuration is any one taken during the analysis process and need not be inequilibrium during a solution process. It is also known as the current, strained or spatial configurationin the literature, and is denoted here by CD . The new ingredient in the CR description is the “splitting”or decomposition of the motion tracking into two components, as illustrated in Figure 12.1.1. The base configuration C0 serves as the origin

of displacements. If this happens to be oneactually taken by the body at the start of theanalysis, it is also called initial or undeformed.The name material configuration is used pri-marily in the continuum mechanics literature.

2. The corotated configuration CR varies fromelement to element (and also from node tonode in some CR variants). For each individ-ual element, its CR configuration is obtainedthrough a rigid body motion of the elementbase configuration. The associated coordinatesystem is Cartesian and follows the elementlike a “shadow” or “ghost,” prompting namessuch as shadow and phantom in the Scandi-navian literature. Element deformations aremeasured with respect to the corotated con-figuration.

Rigid body motion

Deformational motion

Globalframe

Deformed (current,spatial) CD

Base (initial, undeformed,material) configuration C0

Motion splits intodeformational and rigid

Corotated CR

Figure 12.1. The CR kinematic description.Deformation from corotated to deformed (current)

configuration grossly exaggerated for visibility.

In static problems the base configuration usually remains fixed throughout the analysis. In dynamicanalysis the base and corotated configurations are sometimes called the inertial and dynamic referenceconfigurations, respectively. In this case the base configuration may move at uniform velocity (a Galileaninertial system) following the mean trajectory of an airplane or satellite.From a mathematical standpoint the explicit presence of a corotated configuration as intermediary be-tween base and current is unnecessary. The motion split may be exhibited in principle as a multiplicativedecomposition of the displacement field. The device is nonetheless useful to teach not only the physicalmeaning but to visualize the strengths and limitations of the CR description.

12–3


§12.2. The Emergence of CR

The CR formulation represents a confluence of developments in continuum mechanics, treatment offinite rotations, nonlinear finite element analysis and body-shadowing methods.

§12.2.1. Continuum Mechanics Sources

In continuum mechanics the term “corotational” (often spelled “co-rotational”) appears to be first men-tioned in Truesdell and Toupin’s influential exposition of field theories [R.81, Sec. 148]. It is usedthere to identify Jaumann’s stress flux rate, introduced in 1903 by Zaremba. By 1955 this rate had beenincorporated in hypoelasticity [R.82] along with other invariant flux measures. Analogous differentialforms have been used to model endochronic plasticity [R.85]. Models labeled “co-rotational” have beenused in rheology of non-Newtonian fluids; cf. [R.17,R.78]. These continuum models place no majorrestrictions on strain magnitude. Constraints of that form, however, have been essential to make the ideapractical in nonlinear structural FEA, as discussed below.The problem of handling three-dimensional finite rotations in continuum mechanics is important in allLagrangian kinematic descriptions. The challenge has spawned numerous publications, for example[R.1,R.4,R.5,R.34,R.42,R.43,R.62,R.71,R.72]. For use of finite rotations in mathematical models, par-ticularly shells, see [R.60,R.70,R.77]. There has been an Euromech Colloquium devoted entirely to thattopic [R.61].The term “corotational” in a FEM paper title was apparently first used by Belytschko and Glaum [R.8].The survey article bt Belystchko [R.9] discusses the concept from the standpoint of continuummechanics.

§12.2.2. FEM Sources

In the Introduction of a key contribution, Nour-Omid and Rankin [R.54] attribute the original conceptof corotational procedures in FEM to Wempner [R.86] and Belytschko and Hsieh [R.7].The idea of a CR frame attached to individual elements was introduced by Horrigmoe and Bergan[R.39,R.40]. This activity continued briskly under Bergan at NTH-Trondheim with contributions byKrakeland [R.46], Nygard [R.15,R.56,R.57], Mathisen [R.48,R.49], Levold [R.47] and Bjærum [R.18].It was summarized in a 1989 review article [R.56]. Throughout this work the CR configuration is labeledas either “shadow element” or “ghost-reference.” As previously noted the device is not mathematicallynecessary but provides a convenient visualization tool to explain CR. The shadow element functionsas intermediary that separates rigid and deformational motions, the latter being used to determine theelement energy and internal force. However the variation of the forces in a rotating framewas not directlyused in the formation of the tangent stiffness, leading to a loss of consistency. Crisfield [R.22–R.24]developed the concept of “consistent CR formulation” where the stiffness matrix appears as the truevariation of the internal force. An approach blending the TL, UL and CR descriptions was investigatedin the mid-1980s at Chalmers [R.50–R.52].In 1986 Rankin and Brogan at Lockheed introduced [R.63] the concept of “element independent CRformulation” or EICR, which is further discussed below. The formulation relies heavily on the use ofprojection operators, without any explicit use of “shadow” configurations. It was further refined byRankin, Nour-Omid and coworkers [R.54,R.64–R.67], and became essential part of the nonlinear shellanalysis program STAGS [R.68].The thesis ofHaugenonnonlinear thin shell analysis [R.37] resulted in the development of the formulationdiscussed in this article. This framework is able to generate a set of hierarchical CR formulations. Thework combines tools from the EICR (projectors and spins) with the shadow element concept and assumed

12–4


P0

Base (initial) configuration

Corotated (shadow) configuration

PR

P

Corotated (a.k.a.dynamic reference) frame

Base (a.k.a. inertial) frame

CR

C0

C

C0

Deformed configuration(Shown separate from for visualization convenience)

CD

CR

CR

Figure 12.2. The concept of separation of base (a.k.a. inertial) and CR (a.k.a. dynamic)configurations in aircraft dynamics. Deformed configuration (with deformations grossly exaggerated)

shown separate from CR configuration for visibility. In reality points C and CR coincide.

strain element formulations. Spins (instead of rotations) are used as incremental nodal freedoms. Thissimplifies the EICR “front end” and facilitates attaining consistency.Battini and Pacoste at KTH-Stockholm [R.2,R.3,R.58] have recently used the CR approach, focusingon stability applications. The work by Teigen [R.79] should be cited for the careful use of offset nodeslinked to element nodes by eccentricity vectors in the CR modeling of prestressed reinforced-concretemembers.

§12.2.3. Shadows of the Past

The CR approach has also roots on an old idea that preceeds FEM by over a century: the separation of rigidbody and purely deformational motions in continuum mechanics. The topic arose in theories of smallstrains superposed on large rigid motions. Truesdell [R.80, Sec. 55] traces the subject back to Cauchy in1827. In the late 1930s Biot advocated the use of incremental deformations on an initially stressed bodyby using a truncated polar decomposition. However this work, collected in a 1965 monograph [R.16],was largely ignored as it was written in an episodic manner, using full notation by then out of fashion.A rigurous outline of the subject is given in [R.83, Sec. 68] but without application examples.Technological applications of this idea surged after WWII from a totally different quarter: the aerospaceindustry. The rigid-plus-deformational decomposition idea for an entire structurewas originally used byaerospace designers in the 1950s and 1960s in the context of dynamics and control of orbiting spacecraftas well as aircraft structures. The primary motivation was to trace the mean motion.The approach was systematized by Fraeijs de Veubeke [R.25], in a paper that essentially closed the subjectas regards handling of a complete structure. The motivation was clearly stated in the Introduction of thatarticle, which appeared shortly before the author’s untimely death:“The formulation of the motion of a flexible body as a continuum through inertial space is unsatisfactory fromseveral viewpoints. One is usually not interested in the details of this motion but in its main characteristicssuch as the motion of the center of mass and, under the assumptions that the deformations remain small, thehistory of the average orientation of the body. The last information is of course essential to pilots, real andartificial, in order to implement guidance corrections. We therefore try to define a set of Cartesian meanaxes accompanying the body, or dynamic reference frame, with respect to which the relative displacements,

12–5


BaseBase

Deformed (membrane)Deformed (bar)

Deformed (shell)Deformed (beam)

(a) (b)

Figure 12.3. Geometric tracking of CR frame: (a) Bar or beam elementin 2D; (b) Membrane or shell element in 3D.

velocities or accelerations of material points due to the deformations are minimum in some global sense.If the body does not deform, any set of axes fixed into the body is of course a natural dynamic referenceframe.”

Clearly the focus of this article was on a whole structure, as illustrated in Figure 12.2 for an airplane.This will be called the shadowing problem. A body moves to another position in space: find its meanrigid body motion and use this information to locate and orient a corotated Cartesian frame.Posing the shadowing problem in three dimensions requires fairly advanced mathematics. Using two“best fit” criteria Fraeijs de Veubeke showed that the origin of the dynamic frame must remain at thecenter of mass of the displaced structure: CR in Figure 12.2. However, the orientation of this frameleads to an eigenvalue problem that may exhibit multiple solutions due to symmetries, leading to nonuniqueness. (This is obvious by thinking of the polar and singular-value decompositions, which were notused in that article.) That this is not a rare occurrence is demonstrated by considering rockets, satellitesor antennas, which often have axisymmetric shape.

Remark 12.1. Only CD (shown in darker shade in Figure 12.2) is an actual configuration taken by the picturedaircraft structure. Both reference configurations C0 and CR are virtual in the sense that they are not generallyoccupied by the body at any instance. This is in contrast to the FEM version of this idea.

§12.2.4. Linking FEM and CR

The practical extension of Fraeijs de Veubeke’s idea to geometrically nonlinear structural analysis byFEM relies on two modifications:1. Multiple Frames. Instead of one CR frame for the whole structure, there is one per element. This

is renamed the CR element frame.2. Geometric-Based RBM Separation. The rigid body motion is separated directly from the total

element motion using elementary geometric methods. For example in a 2-node bar or beam oneaxis is defined by the displaced nodes, while for a 3-node triangle two axes are defined by the planepassing through the points. See Figure 12.3.

The first modification is essential to success. It helps to fulfill assumption (12.1): the element defor-mational displacements and rotations remain small with respect to the CR frame. If this assumptionis violated for a coarse discretization, break it into more elements. Small deformations are the key toelement reuse in the EICR discussed below. If intrinsically large strains occur, however, the breakdownprescription fails. In that case CR offers no advantages over TL or UL.

12–6


CR "Filters"Finite Element

Library

Assembler

Solver

Incorporaterigid bodymotions

Form elementmass, stiffness

& forces

Extractdeformational

motions

Evaluateelement stressesTotal

displacements

Global element equations

System equationsof motion

Deformational element

equations

Deformational displacements

Figure 12.4. The EICR as a modular interface to a linear FEM library. Theflowchartis mainly conceptual. For computational efficiency the interface logic

may be embedded with each element through inlining techniques.

The second modification is inessential. Its purpose is to speed up the implementation of geometricallysimple elements. The CR frame determination may be refined later, using more advanced tools such aspolar decomposition and best-fit criteria, if warranted.

Remark 12.2. CR is ocassionally confused with the convected-coordinate description of motion, which is used inbranches of fluid mechanics and rheology. Both may be subsumed within the class of moving coordinate kinematicdescriptions. The CR description, however, maintains orthogonality of the moving frame(s) thus achieving anexact decomposition of rigid-body and deformational motions. This property enhances computational efficiency astransformation inverses become transposes. On the other hand, convected coordinates form a curvilinear systemthat “fits” the change of metric as the body deforms. The difference tends to disappear as the discretization becomesprogressively finer, but the fact remains that the convected metric must encompass deformations. Such deformationsare more important in solid than in fluid mechanics (because classical fluid models “forget” displacements). Theidea finds more use in UL descriptions, in which the individual element metric is updated as the motion progresses.

§12.2.5. Element Independent CR

As previously noted, one of the sources of the present work is the element-independent corotational(EICR) description developed by Rankin and coworkers [R.54,R.63–R.67]. Here is a summary descrip-tion taken from the Introduction to [R.54]:

“In the co-rotation approach, the deformational part of the displacement is extracted by purging the rigidbody components before any element computation is performed. This pre-processing of the displacementsmay be performed outside the standard element routines and thus is independent of element type (exceptfor slight distinctions between beams, triangular and quadrilateral elements).”

Why is the EICR worth study? The question fits in a wider topic: why CR? That is, what can CR dothat TL or UL cannot? The topic is elaborated in the Conclusions section, but we advance a practicalreason: reuse of small–strain elements, including possibly materially nonlinear elements.The qualifier element independent does not imply that the CR equations are independent of the FEMdiscretization. Rather it emphasizes that the key operations of adding and removing rigid body motionscan be visualized as a front end filter that lies between the assembler/solver and the element library, assketched in Figure 12.4. The filter is purely geometric. For example, suppose that a program has fourdifferent triangular shell elements with the same node and degree-of-freedom configuration. Then the

12–7


Table 12.1 Configurations in Nonlinear Static Analysisby Incremental-Iterative Methods

Name Alias Explanation Equilibrium IdentificationRequired?

Generic Admissible A kinematically admissible configuration No C

Perturbed Kinematically admissible variation No C + !Cof a generic configuration.

Deformed Current Actual configuration taken during No CD

Spatial the analysis process. Containsothers as special cases.

Base! Initial The configuration defined as the Yes C0

Undeformed origin of displacements.Material

Reference Configuration to which TL,UL: Yes. TL: C0, UL: Cn"1,computations are referred CR: CR no, C0 yes CR: CR and C0

Iterated† Configuration taken at the kth No Cnk

iteration of the nth increment step

Target† Equilibrium configuration accepted Yes Cn

at the nth increment step

Corotated‡ Shadow Body or element-attached configuration No CR

Ghost obtained from C0 through a rigidbody motion (CR description only)

Globally- Connector Corotated configuration forced to align No CG

aligned with the global axes. Used as “connector”in explaining the CR description.

! C0 is often the same as the natural state in which body (or element) is undeformed and stress-free.† Used only in Part II [R.38] in the description of solution procedures.‡ In dynamic analysis C0 and CR are called the inertial and dynamic-reference configurations,respectively, when they apply to the entire structure.

front end operations are identical for all four. Adding a fifth small-strain element of this type incursrelatively little extra work to “make it geometrically nonlinear.”This modular organization is of interest because it implies that the element library of an existing FEMprogram being converted to the CR description need not be drastically modified, as long as the analysisis confined to small deformations. Since that library is typically the most voluminous and expensive partof a production FEM code, element reuse is a key advantage because it protects a significant investment.For a large-scale commercial code, the investment may be thousands of man-years.Of course modularity and computational efficiency can be conflicting attributes. Thus in practice thefront end logic may be embedded with each element through techniques such as code inlining. If so theflowchart of Figure 12.4 should be interpreted as conceptual.

12–8

12–9 §12.3 COROTATIONAL KINEMATICS

§12.3. Corotational Kinematics

This section outlines CR kinematics of finite elements, collecting the most important relations. Mathe-matical derivations pertaining to finite rotations are consigned to Appendix A. The presentation assumesstatic analysis, with deviations for dynamics briefly noted where appropriate.

§12.3.1. Configurations

To describe Lagrangian kinematics it is convenient to introduce a rich nomenclature for configurations.For the reader’s convenience those used in geometrically nonlinear static analysis using the TL, ULor CR descriptions are collected in Table 12.1. Three: base, corotated and deformed, have alreadybeen introduced. Two more: iterated and target, are connected to the incremental-iterative solutionprocess covered in Part II [R.38]. The generic configuration is used as placeholder for any kinematicallyadmissible one. The perturbed configuration is used in variational derivations of FEM equations.Two remain: reference and globally-aligned. The reference configuration is that to which elementcomputations are referred. This depends on the description chosen. For Total Lagrangian (TL) thereference is base configuration. For Updated Lagrangian (UL) it is the converged or accepted solutionof the previous increment. For corotational (CR) the reference splits into CR and base configurations.The globally-aligned configuration is a special corotated configuration: a rigid motion of the base thatmakes the body or element align with the global axes introduced below. This is used as a “connector”device to teach the CR description, and does not imply the body ever occupies that configuration.The separation of rigid and deformational components of motion is done at the element level. As notedpreviously, techniques for doing this have varied according to the taste and background of the investigatorsthat developed those formulations. The approach covered here uses shadowing and projectors.


A typical finite element, undergoing 2D motion to help visualization, is shown in Figure 12.5. Thisdiagram as well as that of Figure 12.6 introduces kinematic quantities. For the most part the notationfollows that used by Haugen [R.37], with subscripting changes.Configurations taken by the element during the response analysis are linked by a Cartesian global frame,to which all computations are ultimately referred. There are actually two such frames: the materialglobal frame with axes {Xi } and position vector X, and the spatial global frame with axes {xi } andposition vector x. The material frame tracks the base configuration whereas the spatial frame tracksthe CR and deformed (current) configurations. The distinction agrees with the usual conventions ofdual-tensor continuum mechanics [R.81, Sec. 13]. Here both frames are taken to be identical, since forsmall strains nothing is gained by separating them (as is the case, for example, in the TL description).Thus only one set of global axes, with dual labels, is drawn in Figures 12.5 and 12.6.Lower case coordinate symbols such as x are used throughout most of the paper. Occasionally it isconvenient for clarity to use upper case coordinates for the base configuration, as in Appendix C.The global frame is the same for all elements. By contrast, each element e is assigned two local Cartesianframes, one fixed and one moving:{xi } The element base frame (blue in Figure 12.5). It is oriented by three unit base vectors i0i , which arerows of a 3! 3 orthogonal rotation matrix (rotator) T0, or equivalently columns of TT0 .{xi } The element corotated or CR frame (red in Figure 12.5). It is oriented by three unit base vectors iRi ,which are rows of a 3! 3 orthogonal rotation matrix (rotator) TR , or equivalently columns of TT0 .

12–9


Base (initial,undeformed)

Deformed (current)

Corotated

xPC0

_x1

x1~

x2~

//x1~

_x2

~//x2

Pu

udP

P

C0

P0

rigid bodyrotation

global frame(with material &spatial coalesced)

elementbase frame element

CR frame

RC C

X ,x1 1

X ,x2 2

C is element centroid in statics,but center of mass in dynamics.

O

PR

xP

a b

uPR

xPCR

xPR

xP0

c

Figure 12.5. CR element kinematics, focusing on the motion of generic point P .Two-dimensional kinematics pictured for visualization convenience.

deformationalrotation R(a "drilling rotation" in 2D)

d

_x1

_x2

P

P0

globalframe

elementbase frame

elementCR frame

TT0

c

ab

O

C0

R0

R (depends on P)

RC C

R

TT

x1~

x2~ X ,x1 1

X ,x2 2

Figure 12.6. CR element kinematics, focusing on rotational transformation betweenframes.

Note that the element index e has been suppressed to reduced clutter. That convention will be followedthroughout unless identification with elements is important. In that case e is placed as supercript.

The base frame {xi } is chosen according to usual FEM practices. For example, in a 2-node spatial beamelement, x1 is defined by the two end nodes whereas x2 and x3 lie along principal inertia directions. Animportant convention, however, is that the origin is always placed at the element centroid C0. For eachdeformed (current) element configuration, a fitting of the base element defines its CR configuration, alsoknown as the element “shadow.” Centroids C R and C ! C D coincide. The CR frame {xi } originatesat C R . Its orientation results from matching a rigid motion of the base frame, as discussed later. Whenthe current element configuration reduces to the base at the start of the analysis, the base and CR framescoalesce: {xi ! xi }. At that moment there are only two different frames: global and local, which agreeswith linear FEM analysis.

Notational conventions: use of G, 0, R and D as superscripts or subscripts indicate pertinence to theglobally-aligned, base, corotated and deformed configurations, respectively. Symbols with a overtildeor overbar are measured to the base frame {xi } or the CR frame {xi }, respectively. Vectors without asuperposed symbol are referred to global coordinates {xi ! xi }. Examples: xR denote global coordinatesof a point in CR whereas xG denote base coordinates of a point in CG . Symbols a, b and c = b " a areabbreviations for the centroidal translations depicted in Figure 12.5, and more clearly in Figure 12.7(b).A generic, coordinate-free vector is denoted by a superposed arrow, for example #u, but such entitiesrarely appear in this work.

The rotators T0 and TR are the well known local-to-global displacement transformations of FEM analysis.

12–10


CD

C(fixed)0

C(moving)

R

C(fixed)

G

CR

(a) (c)(b)

_x1

x 2

x 1~

~_x2

X , x1 1

X , x2 2a

c

b

ϕ

ϕ

0

R

CGCG

C0

C0CR

Figure 12.7. Further distillation to essentials of Figure 12.5. A bar moving in 2D is shown:(a) Rigid motion from globally-aligned to base and corotated configurations; (b) key geometricquantities that define rigid motions in 2D; (c) as in (a) but followed by a stretch from corotatedto deformed. The globally-aligned configuration is fictitious: only a convenient link up device.

Given a global displacement u, u = T0u and u = TRu.

§12.3.3. Coordinate Transformations

Figures 12.5 and 12.6, although purposedly restricted to 2D, are still too busy. Figure 12.7, whichpictures the 2D motion of a bar in 3 frames, displays essentials better. The (fictitious) globally-alignedconfiguration CG is explicitly shown. This helps to follow the ensuing sequence of geometric relations.Begin with a generic point xG in CG . This point is mapped to global coordinates x0 and xR in the baseand corotated configurations C0 and CR , respectively, through

x0 = TT0 xG + a, xR = TT

R xG + b, (12.2)

in which rotators T0 and TR were introduced in the previous subsection. To facilitate code checking, forthe 2D motion pictured in Figure 12.7(b) the global rotators are

T0 =! c0 s0 0

!s0 c0 00 0 1

"

, TR =! cR sR 0

!sR cR 00 0 1

"

, c0 = cos!0, s0 = sin!0, etc. (12.3)

When (12.2) are transformed to the base and corotated frames, the position vector xG must repeat:x0 = xG and xR = xG , because the motion pictured in Figure 12.7(a) is rigid. This condition requires

x0 = T0 (x0 ! a), xR = TR (xR ! b). (12.4)

These may be checked by inserting x0 and xR from (12.2) and noting that xG repeats.

§12.3.4. Rigid Displacements

The rigid displacement is a vector joining corresponding points in C0 and CR . This may be referred to theglobal, base or corotated frames. For convenience call the C0"CR rotator R0 = TT

R T0. Also introducec = TT

0 c and c = TTR c. Some useful expressions are

ur = xR ! x0 = (TTR ! TT

0 )xG + c = (R0 ! I)TT0 xG + c = (R0 ! I)TT

0 x0 + c

= (R0 ! I)TT0 xR + c = (R0 ! I)(x0 ! a) + c = (I! RT

0 )(xR ! b) + c,ur = T0 ur = T0(R0 ! I)TT

0 x0 + c = (R0 ! I)x0 + c,

ur = TR ur = TR (I! RT0 )TT

R xR + c = (I! RT0 )xR + c.

(12.5)

12–11


Here I is the 3 ! 3 identity matrix, whereas R0 = T0R0TT0 and R0 = TRR0TT

R denote the C0"CR

rotator referred to the base and corotated frames, respectively.

§12.3.5. Rotator Formulas

Traversing the links pictured in Figure 12.8 shows that any rotatorcan be expressed in terms of the other two:

T0 = TR R0, TR = T0 RT0 , R0 = TT

R T0, RT0 = TT

0 TR . (12.6)

In the CR frame: R0 = TR R0 TTR , whence

R0 = T0 TTR, RT

0 = TR TT0 . (12.7)

Notice thatT0 is fixed sinceCG andC0 are fixed throughout the anal-ysis, whereas TR and R0 change. Their variations of these rotatorsare subjected to the following constraints:

globalframeX, x

elementbase frame

X~

elementCR frame

x_

TT

T

R0

0T0

R0T

TT R

R

Figure 12.8. Rotator frame links.

!T0 = !TT0 = 0, !TR = T0 !RT

0 , !TTR = !R0 TT

0 , !R0 = !TTR T0,

!RT0 = TT

0 !TR, TTR !TR + !TT

R TR = 0, RT0 !R0 + !RT

0 R0 = 0.(12.8)

The last two come com the orthogonality conditionsTTR TR = I andRT

0 R0 = I, respectively, and provide!R0 = #R0 !RT

0 R0, !RT0 = #RT

0 !R0 RT0 , etc.

We denote by ! and ! the axial vectors of R0 and R0, respectively, using the exponential map form ofthe rotator described in Section A.10. The variations !! and !! are used to form the skew-symmetricspin matrices Spin(!!) = !R0 RT

0 = #Spin(!!)T and Spin(!!) = !R0 RT0 = #Spin(!!)T . These

matrices are connected by congruential transformations:

Spin(!!) = TT0 Spin(!!) T0, Spin(!!) = T0 Spin(!!) TT

0 . (12.9)

Using these relations the following catalog of rotator variation formulas can be assembled:

!TR = T0 !RT0 = #TR !R0 RT

0 = #TR Spin(!!) = #RT0 Spin(!!) T0,

!TTR = !R0 TT

0 = #R0 !RT0 TT

R = Spin(!!) TTR = TT

0 Spin(!!) R0.

!R0 = !TTR T0 = #R0 !RT

0 R0 = Spin(!!) R0 = TT0 Spin(!!) R0 T0,

!RT0 = TT

0 !TR = #RT0 !R0 RT

0 = #RT0 Spin(!!) = #TT

0 RT0 Spin(!!) T0,

!R0 = T0 !R0 TT0 = #T0 R0 !RT

0 TTR = T0 Spin(!!) TT

R = Spin(!!) R0,

!RT0 = T0 !RT

0 TT0 = #TR !R0 RT

0 TT0 = #TR Spin(!!) TT

0 = #RT0 Spin(!!).

(12.10)

12–12


Table 12.2. Degree of Freedom and Conjugate Force Notation

Notation Frame Level Description

v = [ v1 . . . vN ]T

with va =! uaRa

"

Global Structure Total displacements and rotations at structure nodes.Translations: ua , rotations: Ra , for a = 1, . . . N .

!v = [ !v1 . . . !vN ]T

with !va =!

!ua!!a

"

Global Structure Incremental displacements and spins at structurenodes used in incremental-iterative solution procedure.Translations: !ua , spins: !!a ; conjugate forces: na andma , respectively, for a = 1, . . . N .

!ve = [ !ve1 . . . !veNe ]T

with !vea =!

!uea!!e

a

"

Local CR Element Localization of above to element e in CR frame. Trans-lations: !uea , spins: !!e

a ; conjugate forces: nea and mea ,

respectively, for a = 1, . . . Ne.

ved = [ ved1 . . . vedNe ]T

with veda =! ueda

"eda

"

Local CR Element Deformational displacements and rotations at elementnodes. Translations: ueda , rotations: "

eda ; conjugate

forces: na and ma , respectively, for a = 1, . . . Ne.

N = number of nodes in structure; Ne = number of nodes in element e; a, b : node indices.

§12.3.6. Degrees of Freedom

For simplicity it will be assumed that an Ne-node CR element has six degrees of freedom (DOF) per node:three translations and three rotations. This assumption covers the shell and beam elements evaluated inPart II [R.38]. The geometry of the element is defined by the Ne coordinates x0a , a = 1, . . . Ne in thebase (initial) configuration, where a is a node index.The notation used for DOFs at the structure and element level is collected in Table 12.2. If the structurehas N nodes, the set {ua,Ra} for a = 1, . . . N collectively defines the structure node displacement vectorv. Note, however, that v is not a vector in the usual sense because the rotators Ra do not transform asvectors when finite rotations are considered. The interpretation as an array of numbers that defines thedeformed configuration of elements is more appropriate.The element total node displacements ve are taken from v in the usual manner. Given ve, the key CRoperation is to extract the deformational components of the translations and rotations for each node. Thatsequence of operations is collected in Table 12.3. Note that the computation of the centroid is done bysimply averaging the coordinates of the element nodes. For 2-node beams and 3-node triangles this isappropriate. For 4-node quadrilaterals this average does not generally coincides with the centroid, butthis has made little difference in actual computations.

§12.3.7. EICR Matrices

Before studying element deformations, it is convenient to introduce several auxiliary matrices: P =Pu ! P", S, G, H and L that appear in expressions of the EICR front-end. As noted, elements treatedhere possess Ne nodes and six degrees of freedom (DOF) per node. The notation and arrangement usedfor DOFs at different levels is defined in Table 12.2. Subscripts a and b denote node indices that run from1 to Ne. All EICR matrices are built node-by-node from node-level blocks. Figure 12.9(a) illustratesthe concept of perturbed configuration CD + !C, whereas Figure 12.9(b) is used for examples. The CRand deformed configuration are “frozen”; the latter being varied in the sense of variational calculus.

12–13


Table 12.3. Forming the Deformational Displacement Vector.

Step Operation for each element e and node a = 1, . . . a

1. From the initial global nodal coordinates xea compute centroid position ae =xeC0 = (1/Ne)

!Ne

a=1 xea . Form rotator Te0 as per element type convention.

Compute node coordinates in the element base frame: xea = Te0 (xea ! ae).

2. Compute node coordinates in deformed (current) configuration: xea = xea + ueaand the centroid position vector be = xeC = (1/Ne)

!Ne

a=1 xea . Establish the

deformed local CR systemTe by a best-fit procedure, andRe0 = Te(Te0)T . Form

local-CR node coordinates of CR configuration: xeRa = Te (xea ! be).

3. Compute the deformational translations uda = xea ! xeRa . Rd = TnRaTT0 Com-pute the deformational rotator Re

da = TeRea(T

e0)T . Extract the deformational

angles ! eda from the axial vector of Reda .

The translational projector matrix Pu or simply T-projector is dimensioned 6Ne " 6Ne. It is built from3" 3 numerical submatrices Uab = ("ab ! 1/Ne) I, in which I is the 3" 3 identity matrix and "ab theKronecker delta. Collecting blocks for all Ne nodes and completing with 3" 3 zero and identity blocksas placeholders for the spins and rotations gives a 6Ne " 6Ne matrix Pu . Its configuration is illustratedbelow for Ne = 2 (e.g., bar, beam, spar and shaft elements) and Ne = 3 (e.g., triangular shell elements):

Ne = 2: Pu =

"

#

#

$

12 I 0 ! 1

2 I 00 I 0 0

! 12 I 0 1

2 I 00 0 0 I

%

&

&

'

, Ne = 3: Pu =

"

#

#

#

#

#

#

#

$

23 I 0 ! 1

3 I 0 ! 13 I 0

0 I 0 0 0 0! 13 I 0 2

3 I 0 ! 13 I 0

0 0 0 I 0 0! 13 I 0 ! 1

3 I 0 23 I 0

0 0 0 0 0 I

%

&

&

&

&

&

&

&

'

. (12.11)

For any Ne # 1 it is easy to verify thatP2u = Pu , with 5Ne unit eigenvalues and Ne zero eigenvalues. ThusPu is an orthogonal projector. Physically, it extracts the deformational part from the total translationaldisplacements.Matrix S is called the spin-lever or moment-arm or matrix. It is dimensioned 3Ne " 3 and has theconfiguration (written in transposed form to save space):

S = [!ST1 I !ST2 I . . . !STNe I ]T , (12.12)

in which I is the 3 " 3 identity matrix and Sa are node spin-lever 3 " 3 submatrices. Let xa =[ x1a x2a x3a ]T generically denote the 3-vector of coordinates of node a referred to the element centroid.Then Sa = Spin(xa). The coordinates, however, may be those of three different configurations: C0, CR

and CD , referred to two frame types: global or local. Accordingly superscripts and overbars (or tildes)are used to identify one of six combinations. For example

S0a =

"

$

0 !x03a!a3 x02a!a2x03a!a3 0 !x01a!a1

!x02a!a2 x01a!a1 0

%

' , SRa =

"

$

0 !x R3a x R2ax R3a 0 !x R1a

!x R2a x R1a 0

%

' , SDa =

"

$

0 !x D3a x D2ax D3a 0 !x D1a

!x D2a x D1a 0

%

' ,

(12.13)

12–14


Perturbed C + δCD

DDeformed C

D

RCorotated C

C + δC

C + δC

Instantaneousrotation axis

x–

δω–

δv (includesdisplacements and spins)

–

δc

–

––

–

δω x

aperturbed a

δθ

a

Dx– a

Dx–aa

1iδu_

3iδu (+up)_

3jδu (+up)_ 1jδu

_2jδu_

iD

iR

jRjD

DDeformed C

RCorotated C

Perturbed C + δCD

2iδu_

_x1_x1

_x2

_x (+up)3

_x2

_x3

RC C

RC C

(b)(a)

_δω3

L

Figure 12.9. Concept of perturbed configuration to illustrate derivation of EICR matrices:(a) facet triangular shell element moving in 3D space; (b) 2-node bar element also in 3Dbut depicted in the {x1, x2} plane of its CR frame. Deformations grossly exaggerated for

visualization convenience; strains and local rotations are in fact infinitesimal.

are node spin-lever matrices for base-in-global-frame, CR-in-local-frame and deformed-in-local-frame,respectively. The element matrix (12.12) inherits the notation; in this case S0, SR and SD , respectively.For instance, S matrices for the 2-node space i ! j bar element pictured in Figure 12.9(b) are 12 " 3. Ifthe length of the bar in CD is L , the deformed bar spin-lever matrix referred to the local CR frame is

SD = 12 L

! 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 !1 0 0 00 !1 0 0 0 0 0 1 0 0 0 0

"T

. (12.14)

The first row is identically zero because the torque about the bar axis x1 vanishes in straight bar models.Matrix G, introduced by Haugen [R.37], is dimensioned 3 " 6N e, and will be called the spin-fittermatrix. It links variations in the element spin (instantaneous rotations) at the centroid of the deformedconfiguration in response to variations in the nodal DOFs. See Figure 12.9(a). G comes in two flavors,global and local:

!!def= G !ve =

#

aGa !ve

a, !!def= G !ve =

#

aGa !ve

a, with#

a#

#N e

a=1. (12.15)

Here the spin axial vector variation !!e denotes the instantaneous rotation at the centroid, measuredin the global frame, when the deformed configuration is varied by the 6N e components of !ve. Whenreferred to the local CR frame, these become !!e and !ve, respectively. For construction, both G andG may be split into node-by-node contributions using the 3 " 6 submatrices Ga and Ga shown above.As an example, G matrices for the space bar element shown in Figure 12.9(b) is 3 " 12. The spin-levermatrix in CD referred to the local CR frame is

GD = 1L

! 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 !1 0 0 00 !1 0 0 0 0 0 1 0 0 0 0

"

. (12.16)

The first row is conventionally set to zero as the spin about the bar axis x1 is not defined by the nodalfreedoms. This “torsion spin” is defined, however, in 3D beam models by the end torsional rotations.

12–15


Unlike S, the entries ofG depend not only on the element geometry, but on a developer’s decision: howthe CR configuration CR is fitted to CD . For the triangular shell element this matrix is given in AppendixB. For quadrilateral shells and space beam elements it is given in Part II [R.38].

Matrices SD and GD satisfy the biorthogonality property

GS = D. (12.17)

where D is a 3! 3 diagonal matrix of zeros and ones. A diagonal entry of D is zero if a spin componentis undefined by the element freedoms. For instance in the case of the space bar, the product GDSD of(12.14) and (12.16) is diag(0, 1, 1). Aside from these special elements (e.g., bar, spars, shaft elements),D = I. This property results from the fact that the three columns of S are simply the displacementvectors associated with the rigid body rotations !"i = 1. When premultiplied byG one merely recoversthe amplitudes of those three modes.The rotational projector or simplyR-projector is generically defined asP" = SG. Unlike the T-projectorPu such as those in (12.11), the R-projector depends on configuration and frame of reference. Those areidentified in the usual manner; e.g., PR" = SR" G

R" . This 6Ne ! 6Ne matrix is an orthogonal projector of

rank equal to that of D = GS. If GS = I, Pr has rank 3. The complete projector matrix of the elementis defined as

P = Pu " P". (12.18)

This is shown to be a projector, that is P2 = P, in Section 4.2.Two additional 6Ne ! 6Ne matrices, denoted by H and L, appear in the EICR. H is a block diagonalmatrix built of 2Ne 3! 3 blocks:

H = diag [ I H1 I H2 . . . I HNe ] , Ha = H(!a), H(!) = #!/#". (12.19)

HereHa denotes the Jacobian derivative of the rotational axial vector with respect to the spin axial vectorevaluated at node a. An explicit expression ofH(!) is given in (R.48) of Appendix A. The local versionin the CR frame is

H = diag [ I Hd1 I Hd2 . . . I HdNe ] , Hda = H(!da), H(!d) = #!d/#"d . (12.20)

L is a block diagonal matrix built of 2Ne 3! 3 blocks:

L = diag [ 0 L1 0 L2 . . . 0 LNe ] , La = L(!a,ma). (12.21)

where ma is the 3-vector of moments (conjugate to !"a) at node a. The expression of L(!,m) isprovided in (R.49) of Appendix A. The local form L has the same block organization with La replacedby La = L(!da, ma).

§12.3.8. Deformational Translations

Consider a generic point P0 of the base element of Figure 12.5, with global position vector x0P . P0 rigidlymoves to PR in CR with position vector xRP = x0P + uRP = x0P + c+ xRPC . Next the element deforms tooccupy CD . PR displaces to P , with global position vector xP = x0P + uP = x0P + c+ xRPC + udP .The global vector from C0 to P0 is x0P " a, which in the base frame becomes x0P = T0(x0P " a). Theglobal vector fromCR # C to PR is xRP "b, which in the element CR frame becomes xRP = TR(xRPC "b).

12–16


But x0P = xRP since the C0!CR motion is rigid. The global vector from PR to P is udP = xP " xRP ,which represents a deformational displacement. In the CR frame this becomes udP = TR(xP " xRP).The total displacement vector is the sum of rigid and deformational parts: uP = ur P + udP . The rigiddisplacement is given by expressions collected in (12.5), of which ur P = (R0 " I) (x0P " a) + c is themost useful. The deformational part is extracted as udP = uP " ur P = uP " c + (I " R0) (x0P " a).Dropping P to reduce clutter this becomes

ud = u" c+ (I" R0) (x0 " a). (12.22)

The element centroid position is calculated by averaging its node coordinates. Consequently

c = (1/Ne)!

bub, ua " c =

!

bUab ub, with

!

b#

!Ne

b=1 (12.23)

in which Uab = (!ab " 1/Ne) I is a building block of the T-projector introduced in the foregoingsubsection. Evaluate (12.22) at node a, insert (12.23), take variations using (12.10) to handle !R0, use(12.2) to map R0(x0 " a) = xR " b, and employ the cross-product skew-symmetric property (R.3) toextract !!:

!uda = !(ua"c) " !R0 (x0a"a) =!

bUab !ub " Spin(!!)R0 (x0a"a)

=!

bUab !ub " Spin(!!) (xRa " b) =

!

bUab !ub + Spin(xRa " b) !!

=!

bUab !ub +

!

bSRa Gb !vb.

(12.24)

Here matrices S andG have been introduced in (12.12)–(12.15). The deformational displacement in theelement CR frame is ud = TR ud . From the last of (12.5) we get ud = u " c " (I " RT

0 )xR , whereR0 = TR R0TTR . Proceeding as above one gets

!uda =!

bUab !ub +

!

bSDa Gb !vb. (12.25)

The node lever matrix SRa of (12.24) changes in (12.25) to SDa , which uses the node coordinates of the

deformed element configuration.

§12.3.9. Deformational Rotations

Denote by RP the rotator associated with the motion of the material particle originally at P0; seeFigure 12.6. Proceeding as in the translational analysis this is decomposed into the rigid rotation R0and a deformational rotation: RP = RdP R0. The sequence matters because RdP R0 $= R0 RdP . Theorder RdP R0: rigid rotation follows by deformation, is consistent with those used by Bergan, Rankinand coworkers; e.g. [R.54,R.56]. (From the standpoint of continuum mechanics based on the polardecomposition theorem [R.81, Sec. 37] the left stretch measure is used.) Thus RdP = RP RT

0 , whichcan be mapped to the local CR system as Rd = TR Rd TTR . Dropping the label P for brevity we get

Rd = RRT0 = RTT0 TR, Rd = TR Rd TTR = TR RTT0 . (12.26)

The deformational rotation (12.26) is taken to be small but finite. Thus a procedure to extract a rotationaxial vector "d from a given rotator is needed. Formally this is "d = axial

"

Loge(Rd)], but this can beprone to numerical instabilities. A robust procedure is presented in Section A.11. The axial vector isevaluated at the nodes and identified with the rotational DOF.

12–17


Evaluating (12.26) at a node a, taking variations and going through an analysis similar to that carriedout in the foregoing section yields

!!da = "!da""da

!

b

""da

""b!"b = Ha

!

b

"

!ab [ 0 I ]!Gb#

!vb.

!!da = "!da""da

!

b

""da

""b!"b = Ha

!

b

"

!ab [ 0 I ]! Gb#

!vb.(12.27)

where Gb is defined in (12.15) and Ha in (12.19).

Overview continues in next Chapter.

12–18

.

13CR Formulation

Overview II

13–1

Chapter 13: CR FORMULATION OVERVIEW II 13–2

TABLE OF CONTENTS

Page§13.1. Internal Forces 13–3

§13.1.1. Force Transformations . . . . . . . . . . . . . . . 13–3§13.1.2. Projector Properties . . . . . . . . . . . . . . . . 13–4

§13.2. Tangent Stiffness 13–5§13.2.1. Definition . . . . . . . . . . . . . . . . . . . 13–5§13.2.2. Material Stiffness . . . . . . . . . . . . . . . . 13–5§13.2.3. Geometric Stiffness . . . . . . . . . . . . . . . . 13–6§13.2.4. Consistency Verification . . . . . . . . . . . . . . 13–7

§13.3. Three Consistent CR Formulations 13–7§13.3.1. Consistent CR formulation (C) . . . . . . . . . . . . 13–7§13.3.2. Consistent Equilibrated CR Formulation (CE) . . . . . . . 13–8§13.3.3. Consistent Symmetrizable Equilibrated CR Formulation (CSE) . . 13–8§13.3.4. Formulation Requirements . . . . . . . . . . . . . 13–8§13.3.5. Limitations of the EICR Formulation . . . . . . . . . . 13–9

§13.4. Conclusions 13–10

13–2

13–3 §13.1 INTERNAL FORCES

§13.1. Internal Forces

The element internal force vector pe and tangent stiffnessmatrix Ke are computed in the CR configurationbased on small deformational displacements and rotations. Variations of the element DOF, collectedin ved as indicated in Table 13.2, must be linked to variations in the global frame to flesh out the EICRinterface of Figure 12.4. This section develops the necessary relations.

§13.1.1. Force Transformations

Consider an individual element e with Ne nodes with six DOF (three translations and three rotations) ateach. Assume the element to be linearly elastic, undergoing only small deformations. Its internal energyis assumed to be a function of the deformational displacements: Ue = Ue(ved), with array ved organizedas shown in Table 13.2. Ue is a frame independent scalar. The element internal force vector pe in theCR frame is given by pe = !Ue/! ved . For each node a = 1, . . . Ne:

pea = !Ue

! veda, or

!

peuape"a

"

=

#

$

!Ue

!uda!Ue

!!da

%

& (13.1)

where the second form separates the translational and rotational (moment) forces. To refer these to theglobal frame we need to relate local-to-global kinematic variations:

!

#ueda#!

eda

"

=Ne'

b=1Jab

!

#uea#"e

a

"

, Jab =

#

(

$

!uedb!uea

!uedb!"e

a

!!edb

!uea!!

edb

!"ea

%

)

&. (13.2)

From virtual work invariance, (peu)T #ued + (pe" )T #!ed = (peu)T #ue + (pe" )T #"e, whence

!

peuape"a

"

=Ne'

b=1JTab

!

peuape"a

"

, a = 1, . . . Ne. (13.3)

It is convenient to split the Jacobian in (13.2) as Jab = HbPabTa and JTab = TTa PTabH

Tb . These matrices

are provided from three transformation stages, flowcharted in Figure 13.1:!

#uedb#!

edb

"

=!

I 00 Hdb

" !

#ueb#"e

b

"

, with Hdb =*

!!edb

!"edb

+

,

!

#ueb#"e

b

"

= Pab!

#uea#"e

a

"

, with Pab =

#

(

$

!uedb!uea

!uedb!"e

a!"e

db!uea

!"edb

!"ea

%

)

&,

!

#uea#"e

a

"

= Ta!

#uea#"e

a

"

=!

TR 00 TR

" !

#uea#"e

a

"

,

(13.4)

The 3! 3 matrix L is the Jacobian derivative already encountered in Chapter 12. An explicit expressionin terms of ! is given in Appendix R. To express compactly the transformations for the entire element itis convenient to assemble the 6Ne ! 6Ne matrices

13–3


P =

!

"

#

P11 P12 . . . P1Ne

P21 P22 . . . P2Ne

. . . . . . . . . . . .

PNe1 PNe2 . . . PNeNe

$

%

&,

T = diag [TR TR . . . TR ] .

(13.5)

and H is defined in Appendix R. Then the elementtransformations can be written

!ved = H P T !ve, pe = TT PT HT pe. (13.6)

CR deformationaldisplac & rotations

δu , δθ

CR deformationaldisplac & spins

δu , δω

CR totaldisplac & spins

δu, δω

global totaldisplac & spins

δu, δω_

_ _ __

_

H

P = P + P

T

ωu

H P T

rotation-to-spin Jacobian

Projector

global-to-CR frame rotator

d

d d

d

_

_ _ _

_ _

Figure 13.1. Staged transformation sequencefrom deformed to global DOFs.

The 6 ! 6 matrix Pab in (13.4) extracts the deformational part of the displacement at node b in termsof the total displacement at node a, both referred to the CR frame. At the element level, !ved = P !veextracts the deformational part by “projecting out” the rigid body modes. For this reason P is called aprojector. As noted in Section 3.5, Pmay be decomposed into a translational projector or T-projector Puand a rotational projector or R-projector P", so that P = Pu + P". Each has a rank of 3. The T-projectoris a purely numeric matrix exemplified in Chapter 12. The R-projector can be expressed as P" = SG,where S and G are defined in Chapter 12. Additional properties are studied below.Remark 3. Rankin and coworkers [R.54,R.63–R.67] use an internal force transformation in which the incrementalnodal rotations are used instead of the spins. This results in an extra matrix, H"1 appearing in the sequence (13.6).The projector derived in those papers differs from the one constructed here in two ways: (1) only the R-projector isconsidered, and (2) the origin of the CR frame is not placed at the element centroid but at an element node definedby local node numbering. Omitting the T-projection is inconsequential if the element is “clean” with respect totranslational rigid body motions [R.30, Sec. 5].

§13.1.2. Projector Properties

In this section the bar over P, etc is omitted for brevity, since the properties described below are frameindependent. In Section 12.5 it was stated without proof that the orthogonal projector property P2 = Pwas verified. Since P2 = (Pu " P")2 = P2u " 2PuP" + P2", satisfaction requires P2u = Pu , P2" = P", andPuP" = 0. Verification of P2u = Pu is trivial. Recalling that P" = SG we get

P2" = S (G S) G = S I G = S G = P". (13.7)

This assumesD = I; verification for non-identityD is immediate upon removal of zero rows and columns.The orthogonality property Pu P" = Pu S G = 0 follows by observing that Spin(xC) = 0, where xC arethe coordinates of the element centroid in any frame with origin at C .In the derivation of the consistent tangent stiffness, the variation of PT contracted with a force vector f,where f is not varied, is required. The variation of the projector can be expressed as

!P = !Pu " !P" = "!P" = "!S G " S !G. (13.8)

For the tangent stiffness one needs !PT f. This vector can be decomposed into a balanced (self-equilibrated) force fb = Pf and an unbalanced (out of equilibrium) force fu = (I " P)f. Then

!PT f = "(GT !ST + !GT ST ) (fb + fu) = "GT !ST PT f " (GT !ST + !GT ST ) fu= "GT !ST PT f + !PT fu .

(13.9)

13–4

13–5 §13.2 TANGENT STIFFNESS

where ST fb = 0 was used. This comes from the fact that the columns of S are the three rotational rigidbody motions, which do not produce work on an self-equilibrated force vector.The term !PT fu will be small if element configurations CR and CD are close because in this case fu willapproach zero. If G has the factorizable form shown below, however, we can show that !PT fu = 0identically, regardless of how close CR and CD are, as long as f is in translational equilibrium. Assumethat G can be factored as

G = !", with !G = !!". (13.10)

where ! is a coordinate dependent invertible 3 ! 3 matrix, and " is a constant 3 ! 6Ne matrix. SinceGS = I, !"1 = "S, and !GS+G !S = !!"S+G !S = 0, whence !! = "G !S!. Then

!PT fu = "(GT !ST + "T !!T ST ) fu = "(GT !ST " "T !T !ST ,GT ST ) fu= "(GT !ST "GT !ST PT") fu = "(GT !ST (I" PT") fu = "GT !ST (I" PT") fu= "GT !ST (I" PT")(I" PT ) f = 0,

(13.11)

if f is in translational equilibrium: f = PT" f. This is always satisfied for any element that represents rigidbody translations correctly [R.30, Sec. 5].

§13.2. Tangent Stiffness

We consider here only the stiffness derived from the internal energy. The load stiffness due to noncon-servative forces, such as aerodynamic pressures, has to be treated separately.

§13.2.1. Definition

The consistent tangent stiffness matrix Ke of element e is defined as the variation of the internal forceswith respect to element global freedoms:

!pe def= Ke !ve, whence Ke = #pe

#ve. (13.12)

Taking the variation of pe in (13.6) gives rise to four terms:

!pe = !TT PT HT pe + TT !PT HT pe + TT PT !HT pe + TT PT HT!pe

= (KeGR +Ke

GP +KeGM +Ke

M) !ve.(13.13)

The four terms identified in (13.13) receive the following names. KM is the material stiffness, KGM themoment-correction geometric stiffness,KGP the equilibrium projection geometric stiffness, andKGR therotational geometric stiffness. If nodal eccentricities treated by rigid links are considered, one more termappears, called the eccentricity geometric stiffness. This term is studied in great detail in [R.37].

§13.2.2. Material Stiffness

The material stiffness is generated by the variation of the element internal forces pe:

KeM !ve = TTR P

T HT!pe. (13.14)

The linear stiffness matrix in terms of the deformational freedoms in ved is defined as the Hessian of theinternal energy:

Ke = #2U e

# ved# ved= #pe

# ved(13.15)

13–5


Using the transformation of !ve in (13.6) gives

KeM = TT PT HT Ke H P T. (13.16)

Thus the material stiffness is given by a congruential transformation of the local stiffness Ke to the globalframe. This is formally the same as in linear analysis but here the transformation terms depend on thestate. The expression (13.16) is valid only if Ke is independent of the deformational ved freedoms.

§13.2.3. Geometric Stiffness

To express compactly the geometric stiffness components it is convenient to introduce the arrays

peP = PT HT pe =

!

"

"

"

"

#

ne1me1

...

neNe

meNe

$

%

%

%

%

&

, Fn =

!

"

"

"

"

#

Spin(ne1)0...

Spin(neNe)

0

$

%

%

%

%

&

, Fmn =

!

"

"

"

"

#

Spin(ne1)Spin(me

1)...

Spin(neNe)

Spin(meNe)

$

%

%

%

%

&

. (13.17)

These are filled with the projection node forces peP . Only the final form of the geometric stiffnesscomponents is given below, omitting the detail derivations of [R.37]. The rotational geometric stiffnessis generated by the variation of T: Ke

RG !ve = !TT PT HT pe and can be expressed as

KGR = !TT Fnm G T, (13.18)

KeRG is the gradient of the internal force vector with respect to the rigid rotation of the element. This

interpretation is physically intuitive because a rigid rotation of a stressed element necessarily reorientsthe stress vectors by that amount. Consequently the internal element forcesmust rigidly rotate to preserveequilibrium.The moment-correction geometric stiffness is generated by the variation of the jacobian H: Ke

RG!ve =TTR P

T!HT pe. It is given by

KeGM = TT PT L PT. (13.19)

where L is defined in Section 3.5.The equilibrium projection geometric stiffness arises from the variation of the projector P with respectto the deformed element geometry: Ke

GP!ve = TTR !PT HT pe. As in Section 4.2, decompose pe into abalanced (self-equilibrated) force peb = PT pe and an unbalanced force peu = pe ! peb. If !P

T peu is eitheridentically zero or may be neglected as discussed in Section 4.2, Ke

GP is given by

KGP = !TT GT Fn P T, (13.20)

in which the balanced force peb is used in (13.17) to get Fn .

If TT !PT peu cannot be neglected, as may happen in highly warped shell elements in a coarse mesh, thefollowing correction term may be added to Ke

RG :

"KGP = !TT'

GT Fnu P+ #G#vS peu

(

T, (13.21)

where Fnu is Fn of (13.17) when peu is inserted instead of pe. In the computations reported in Part II[R.38] this term was not included.

13–6

13–7 §13.3 THREE CONSISTENT CR FORMULATIONS

KeRG expresses the variation of the projection of the internal force vector pe as the element geometry

changes. This can be interpreted mathematically as follows: In the vector space of element force vectorsthe subspace of self-equilibrium force vectors changes as the element geometry changes. The projectedforce vector thus has a gradient with respect to the changing self-equilibrium subspace, even though theelement force fe does not change.The complete form of the element tangent stiffness, excluding correction terms (13.21) for highly warpedelements, is

Ke = TT!

PT HT Ke H P+ PT L P! Fnm G! GT FTn P"

T = TT KePT, (13.22)

in which KeR , which is the local tangent stiffness matrix (the tangent stiffness matrix in the local CR

frame of the element) is given by the parenthesized expression.

§13.2.4. Consistency Verification

The local tangent stiffness matrix KeR given in (13.22) has some properties that may be exploited to verify

the computer implementation [R.54,R.37]:

KeR S = !Fnm, ST Ke

R = !FTn , KM S = 0, KGR S = !Fnm,

KGP S = 0, ST KM = 0, ST (KGR + KGP) = !FTn .(13.23)

In addition, rigid-body-mode tests on the linear stiffness matrix Ke using linearized projectors arediscussed in [R.30]. The set (13.23) tests the programming of the nonlinear projector P since it checksthe null space of P. It also indicates whether the projector matrix is used correctly in the stiffnessformulation. However, satisfaction does not fully guarantee consistency between the internal forceand the tangent stiffness because H and L are left unchecked. Full verification of consistency can benumerically done through finite difference techniques.

§13.3. Three Consistent CR Formulations

From the foregoing unified forms of the internal force and tangent stiffness, three CR consistent for-mulations can be obtained by making simplifying assumptions at the internal force level. These satisfyself-equilibrium and symmetry to varying degree. The following subsections describe the three versionsin order of increasing complexity. For all formulations one can take in account DOFs at eccentric nodesas described in [R.37].

§13.3.1. Consistent CR formulation (C)

This variant is that developed by Bergan and coworkers in the 1980s at Trondheim and summarized inthe review article [R.56]. The internal force (13.6) is simplified by taking H = I and P = I, whileretaining !ved = H P !ve for recovery of deformational DOFs. Since !P = !H = 0, the expression forthe tangent stiffness of (13.13) simplifies to the material and rotational geometric stiffness terms:

pe = TT Ke ved , Ke = TT (Ke H P! Fnm G)T. (13.24)

Here Fnm is computed according to (13.17) with pe = Kevd . The internal force is in equilibrium withrespect to the CR configuration CR . For a shell structure, the material stiffness approaches symmetry asthe element mesh is refined if the membrane strains are small. As the mesh is refined, the deformationalrotation axial vectors !da become smaller and approach vector properties that in turn make H(!da

13–7


approach the identity matrix. With small membrane strains Ke is indifferent with respect to post-multiplication with P because the CR and C configurations will be close and KeP ! Ke. The consistentgeometric stiffness is always unsymmetric, even at equilibrium. Because of this fact one cannot expectquadratic convergence for this formulation if a symmetric solver is used.This formulation may be unsatisfactory for warped quadrilateral shell elements since the CR and CD

reference configurations may be far apart. Only in the limit of a highly refined element mesh will theCR and CD references in general be close, and satisfactory equilibrium ensured.

§13.3.2. Consistent Equilibrated CR Formulation (CE)

The internal force (13.6) is simplified by taking H = I so !H = 0, but the projector P is retained. Thisgives

pe = TT PT Ke ved , Ke = TT (PT Ke H P" Fmn G" GT FTn P)T. (13.25)

where Fnm and Fn are computed according to (13.17) with pe = PT Keved .Due to the presence of P on both sides, the material stiffness of the CE formulation approaches symmetryas the mesh is refined regardless of strain magnitude. The geometric stiffness at the element level is non-symmetric, but the assembled global geometric stiffness will become symmetric as global equilibrium isapproached, provided that there are no applied nodalmoments and that displacement boundary conditionsare conserving. A symmetrized global tangent stiffness maintains quadratic convergence for refinedelement meshes with this formulation.

§13.3.3. Consistent Symmetrizable Equilibrated CR Formulation (CSE)

All terms in (13.6) are retained, giving

pe = TT PT HT Ke ved , Ke = TT (PT HT KeH P+ PT L P" Fnm G" GT FTn P)T. (13.26)

where Fnm and Fn are computed according to (13.17) with f = PT H Ke ved . The assembled globalgeometric stiffness for this formulation becomes symmetric as global equilibrium is approached, asin the CE case, as long as there are no applied nodal moments and the loads as well as boundaryconditions are conserving. Since the material stiffness is always symmetric, quadratic convergence witha symmetrized tangent stiffness can be exppected without the refined-mesh-limit assumption of the CEformulation.

Remark 4. The relative importance of including theHmatrix, which is neglected by most authors, and the physicalsignificance of this Jacobian term are discussed in Part II [R.38].

§13.3.4. Formulation Requirements

It is convenient to set forward a set of requirements for geometrically nonlinear analysis with respectto which different CR formulations can be evaluated. They are listed below in order of decreasingimportance.Equilibrium. By this requirement is meant: to what extent the finite element internal force vector p is inself-equilibrium with respect to the deformed configuration CD? This is a fundamental requirement fortracing the correct equilibrium path in an incremental-iterative solution procedure.Consistency. A formulation is called consistent if the tangent stiffness is the gradient of the internal forceswith respect to the global DOF. This requirement determines the convergence rate of an incremental-iterative solution algorithm. An inconsistent tangent stiffness may give poor convergence, but does not

13–8

13–9 §13.3 THREE CONSISTENT CR FORMULATIONS

Table 13.4. Attributes of Corotated Formulations C, CE and CSE.

Formulation Self-equil.(1) Consistent(2) Invariant(3) Symmetriz.(4) Elem.Indep.(5)

C! ! !

CE! ! ! !

CSE! ! ! ! !

(1) checked if element is in self-equilibrium in deformed configuration CD .(2) checked if tangent stiffness is the v gradient of the element internal force.(3) checked if the formulation is insensitive to choice of node numbering.(4) checked if formulation maintains quadratic convergence of a true Newton solution

algorithm with a symmetrized tangent stiffness matrix.(5) checked if the matrix and vector operations that account for geometrically nonlinear

effects are the same for all elements with the same node and DOF configuration.

alter the equilibrium path since this is entirely prescribed by the foregoing equilibrium requirement.However, lack of consistency may affect the location of bifurcation (buckling) points and the branchswitching mechanism for post-buckling analysis. In other words, an inconsistent tangent stiffness matrixmay detect (“see”) a bifurcationwhere equilibrium is not satisfied from the residual equation. Subsequenttraversal by branch-switching will then be difficult because the corrector iterations need to jump to thesecondary path as seen by the residual equation.Invariance. This requirement refers to whether the solution is insensitive to internal choices that maydepend on node numbering. For example, does a local element-node reordering give an altered equi-librium path or change the convergence characteristics for the analysis for an otherwise identical mesh?The main contributor to lack of invariance is the way the deformational displacement vector is extractedfrom the total displacements, if the extraction is affected by the choice of the local CR frame. If lack ofinvariance is observed, it may be usually traced to the matrix G, which links the variation of the rigidbody rotation to that of the nodal DOF degrees of freedom.Symmetrizability. This means that a symmetrized K can be used without loss of quadratic convergencerate in a true Newton solver even when the consistent tangent stiffness away from equilibrium is notsymmetric. In the examples studied in Part II [R.38] this requirement was met when the material stiffnessof the formulation was rendered symmetric.Element Independence. This is used in the sense of the EICR discussed in Section 2.5. It means thatthe matrix and vector operations that account for geometrically nonlinear effects are the same for allelements that possess the same node and DOF configuration.Attributes of the C, CE and CSE formulations in light of the foregoing requirements are summarized inTable 13.4.

§13.3.5. Limitations of the EICR Formulation

The present CR framework, whether used in the C, CE or CSE formulation variants, is element inde-pendent in the EICR sense discussed in Section 2.5 since it does not contain gradients of intrinsicallyelement dependent quantities such as the strain-displacement relationship. This treatment is appropriatefor elements where the restriction to small strains automatically implies that the CR and deformed ele-ment configurations are close. This holds automatically for low order models such as two-node straightbars and beams, and three-node facet shell elements.The main reason for limiting element independence to low-order elements is the softening effect of thenonlinear projector P. The use of P to restore the correct rigid body motions, and hence equilibrium with

13–9


respect to the deformed element geometry, effectively reduces the eigenvalues of the material stiffnessrelative to the CR material stiffness Ke before projection. This softening effect becomes significant ifthe CR and CD geometries are far apart.Such softening effects are noticeable in four-node initially-warped shell elements. Assume that theelement is initially warped with “positive” warping, and consider only the effect of P. The elementmaterial stiffness of this initial positive warping is then K+ = PT+KeP+ = Ke. Apply displacementsthat switch this warping to the opposite of the initial one; that is, a “negative” warping. The newelement material stiffness then becomesK! = PT!KeP! "= Ke One will intuitively want the two elementconfigurations to have the same rigidity in the sense of the dominant nonzero eigenvalues of the tangentstiffness matrix. But it can be shown that the eigenvalues of the projected material stiffness matrix K!can be significantly lower than those of the initial stiffness matrix K+. If the element stiffness Ke isreferred to the flat element projection, one can restore symmetry ofK+ andK! with respect to dominantnonzero eigenvalues, but it is not possible to remove the softening effect.This argument also carries over to higher order bar, arch and shell elements that are curved in the initialreference configuration. It follows that the EICR is primarily useful for low-order elements of simplegeometry.

§13.4. Conclusions

This article presents a unified formulation for geometrically nonlinear analysis using the CR kinematicdescription, assuming small deformations. Although linear elastic material behavior has been assumedfor brevity, extension to materially nonlinear behavior such as elastoplasticity and fracture within theconfines of small deformations, is feasible as further discussed below. All terms in the internal force andtangent stiffness expressions are accounted for. It is shown how dropping selected terms in the formerproduces simpler CR versions used by previous investigators.These versions have been tested on thin shell and flexible-mechanism structures, as reported in Part II[R.38]. Shells are modeled by triangle and quadrilateral elements. The linear stiffness of these elementsis obtained with the ANDES (Assumed Natural DEviatoric Strains) formulation of high performanceelements [R.27–R.32,R.53]. Test problems include benchmarks in buckling, nonlinear bifurcation andcollapse.Does the unified formulation close the book on CR? Hardly. Several topics either deserve furtherdevelopment or have been barely addressed:(A) Relaxing the small-strain assumption to allow moderate deformations.(B) Robust handling of extremely large rotations involving multiple revolutions.(C) Integrating CR elements with rigid links and joint elements for flexible multibody dynamics.(D) Using substructuring concepts for CR modeling of structural members with continuum elements.(E) Achieving a unified form for CR dynamics, including nonconservative effects and multiphysics.Topic (A)means the use of CR for problemswhere strains may locally reachmoderate levels, say 1–10%,as in elastoplasticity and fracture, using appropriate strain and stress measures in the local frame. Thechallenge is that change of metric of the CR configuration should be accounted for, even if it meansdropping the EICR property. Can CR compete against the more established TL and UL descriptions?It seems unreasonable to expect that CR can be of use in overall large strain problems such as metalforming, in which UL reigns supreme. But it may be competitive in localized failure problems, wheremost of the structure remain elastic although undergoing finite rotations.

13–10

13–11 §13.4 CONCLUSIONS

Somedata points are available: previous large-deformationworkpresented in [R.46,R.47,R.50,R.51,R.57].More recently Skallerud et al. reported [R.75] that a submerged-pipeline failure shell code using the AN-DES CR quadrilateral of [R.37] plus elastoplasticity [R.74] and fracture mechanics [R.21] was able tobeat a well known commercial TL-based code by a factor of 600 in CPU time. This speedup is of obviousinterest in influencing both design cycle and deployment planning.Topic (B) is important in applications where a floating (free-free) structure undergoes several revolutions,as in combat airplane maneuvers, payload separation or orbital structure deployment. The technicaldifficulty is that expressions presented in the Appendix cannot handle finite rotations beyond ±2! , andthus require occasional resetting of the base configuration. While this can be handled via restarts forstructures such as full airplanes, it can be more difficult when the relative rotation between componentsexceeds ±2! , as in separation, fragmentation or deployment problems.Topics (C) and (D) have been addressed in the FEDEM program developed by SINTEF at Trondheim,Norway. This program combines CR shell and beam elements of [R.37], grouped into substructures,with kinematic objects typical of rigid-body dynamics: eccentric links and joints. Basic tools used inFEDEM for combining joint models with flexible continuum elements are covered in a recent book[R.73].Finally, topic (E) is fertile ground for research. The handling of model components such as mass,damping and nonconservative effects in fluid-structure interaction and aeroelasticity is an active ongoingresearch topic. For example a recent paper [R.26] describes flight maneuver simulations of a completeF-16 fighter using CR elements to model the aircraft. As in statics, a key motivation for CR in dynamicsis reuse of linear FEM force-stiffness libraries. Can that reuse extend to mass and damping libraries?And how do standard time integration methods perform when confronted with unsymmetric matrices?These topics have barely been addressed.

13–11

.

14The CR Formulation:

Space Bar

14–1

Chapter 14: THE CR FORMULATION: SPACE BAR 14–2

TABLE OF CONTENTS

Page§14.1. Introduction 14–3§14.2. Line Segment Moving in 3D 14–3

§14.2.1. Line Segment Derivatives . . . . . . . . . . . . . 14–4§14.2.2. Derivatives of Length Functions . . . . . . . . . . . 14–5§14.2.3. Mathematica Implementation and FD Verification . . . . . 14–6

§14.3. The CR Bar Element 14–9§14.3.1. Internal Energy, Force and Stiffness . . . . . . . . . 14–9§14.3.2. Matrices for Specific Strain Measures . . . . . . . . . 14–10§14.3.3. Mathematica Implementation . . . . . . . . . . . 14–11

§14. Exercises . . . . . . . . . . . . . . . . . . . . . . 14–14

14–2

14–3 §14.2 LINE SEGMENT MOVING IN 3D


The expressions provided in the foregoing two Chapters and in Appendix R address the formation ofincremental equations for an arbitrary CR element. The main restriction is the assumption of smalllocal deformations, which permit the use of the linearized equations for the deformational energy.Closed form of the incremental expressions become intractable for elements of arbitrary geometry,such as curved shells or beams, because c and R are complicated functions of the element displacementfield, which is turn determined by u. Fortunately the CR approach is often used with elements ofsimple geometry in which rotational freedoms, if any, may be ignored in defining the corotatedconfiguration. Under those conditions one can work out the base-to-deformed transformation arraysdirectly from geometric arguments. The transformations may be then systematically applied toexisting linear elements through a modular interface, as illustrated in Figure 12.7.In particular, for a simplex element (a constant strain element without rotational freedoms) it ispossible to work out all transformation from the intrinsic geometry of a line segment, triangle ortetrahedron moving in 3D space. (By “intrinsic” is meant changes in edge lengths, face areas andvolume dimensions.) For the line segment modeling a space bar the formulas are worked out andcollected in the next sections. These apply to several types of finite elements, such as bars and cables.The results may be specialized to two dimensions, if desired, by setting the third coordinate to zero.It is important that the kinematic analysis be exact so that arbitrary rigid body motions can beaccomodated. Restrictions on local deformations in the motion from CR to CD can then be madewhen considering specific elements, particularly those endowed with rotational DOFs.Several of the following results are new. Their closed form derivation was made possible because ofthe use of Mathematica to synthesize abtruse algebraic expressions containing symbolic terms.

§14.2. Line Segment Moving in 3D

X, xZ, z

Y, y uY1

X1u

uX2 uZ2

uZ1uY2

0base length a

deformed length a

0

0Base configuration C0

Deformedconfiguration CD

1 1 11 (X ,Y ,Z )

2 22 2 (X ,Y ,Z )

2 2 22(x ,y ,z )

Global system

1 1 1 1(x ,y ,z )

Figure 14.1. A line segment moving in 3D space.

Consider the line segment shown in Figure 14.1, defined by the end nodes 1-2. The segment movesin three-dimensional space. The global axes will be denoted by {X, Y, Z} instead of {X1, X2, X3} so

14–3


X, xZ, z

Y, y

0

a = a + a + a

02

0 1

Global system

1

2x y z2 2 2 2

a = a + a + aX0 Y0 Z02 2 2 2

Y0a = Y21

Z0a = Z21

a = XX0 21

ya = y21

za = z21

xa = x21

Figure 14.2. Line segment components.

that {Xn, Yn, Zn} can be used for the coordinates of node n. Global axes {x, y, z}, which are usedfor the deformed configuration, coalesce with {X, Y, Z},The base line configuration C0 is specified by the coordinates (X1, Y1, Z1) and (X2, Y2, Z2) of theline end nodes. The line moves to the deformed (current) configuration CD of length a defined bycoordinates {x1 = X1+u X1, y1 = Y1+uY 1, z1 = Z1+uZ1} and {x2 = X2+u X2, y2 = Y2+uY 2, z2 =Z2 + uZ2}, where u X1 through uZ2 are the node displacements. Node coordinate and displacementdifferences are abbreviated by X21 = X2 ! X1, x21 = x2 ! x1, u X21 = u X2 ! u X1, etc. As illustratedin Figure 14.2, the line lengths are given by

a20 = a2

X0 + a2Y 0 + a2

Z0 = X221 + Y 2

21 + Z221, a2 = a2

x + a2y + a2

z = x221 + y2

21 + z221. (14.1)

For further use define the following vectors

a0 =! aX0

aY 0aZ0

"

, a0 = 1a0

! aX0aY 0aZ0

"

, a =! ax

ayax

"

, a = 1a

! axayaz

"

, u21 = a ! a0 =! u X21

uY 21uZ21

"

, (14.2)

Here a0 = a0/a0 and a = a/a denote the direction cosine vectors of the base and current linesegment, respectively. This “hat convention” will be used to identify direction vectors normalized tounit length. It is important not to confuse the 3-vector u21 with the 6-vector of node displacements

u = [ u X1 uY 1 uZ1 u X2 uY 2 uZ2 ]T (14.3)

In fact, !u21/!u is the 3 " 6 matrix [ !I I ], where I is the 3 " 3 identity matrix.

§14.2.1. Line Segment Derivatives

Suppose that the node displacements are functions of two variables, generically denoted by " and#: u X1 = u X1(", #), etc. The partial derivative of a with respect to " is

!a!"

= ax

a!u X21!"

+ ay

a!uY 21!"

+ az

a!uZ21!"

= aT u21,", in which u21," =

#

$

$

$

%

!u X21!"

!uY 21!"

!uZ21!"

&

'

'

'

(

. (14.4)

14–4


Note that the base configuration is only “remembered” through the displacements since the initiallength a0 does not appear explicitly.To obtain the second derivative we take the partial of (14.4) with respect to the generic variable !,which yields aT "2u21/("#"!)+(" aT /"!) ("u21/"#). While the first term is easy, the second onerequires the derivatives of a. Since this involves the variation of a fixed-length (unit) vector, it canbe expected to involve the orthogonal projector associated with a. The final result can be presentedin the compact matrix form

"2a"#"!

= uT21,#Hu21,! + aT u21,#!, (14.5)

in which

u21,#! =

!

"

"

"

"

#

"2uX21"#"!"2uY21"#"!"2uZ21"#"!

$

%

%

%

%

&

, H = 1a3

!

#

a2y + a2z !axay !axaz!ayax a2z + a2x !ayaz!azax !azay a2x + a2y

$

& = 1a

(I! aaT ) = 1aPa . (14.6)

Here I denotes the identity matrix of order 3 whereas Pa = I ! aaT is the orthogonal projectorassociated with the direction a. (To show this, square Pa and verify that P2a = Pa .)The results (14.4) and (14.5) can be specialized to various choices. For example, if the displacementsare viewed as functions of real time (or a time-like parameter) t , we have # " ! " t , and

a = aT u21, a = aT u21 + uT21H u21 (14.7)

where a superposed dot denotes derivative with respect to t . Of more interest for element derivationis to set # and ! in turn to the six entries of the node displacement vector u arranged as (14.3). Inthis case the term u21,#! in (14.5) vanishes, and we obtain

"a"u

= 1a[!ax !ay !az ax ay az ]T = [!aT aT ]T def= h,

"2a"u "u

='

H !H!H H

(

def= G.

(14.8)Vector h and matrix G will appear in the derivation of the space bar element in §14..3. In additionthe following definition and relations are useful there:

J def='

I !I!I I

(

= aG+ hhT , G = 1a

)

J! hhT*

. (14.9)

§14.2.2. Derivatives of Length Functions

Often the derivatives of a function F(a) are needed, for example for bar strain measures other thanengineering strains. The function F(a) is assumed twice differentiable respect to a. The a-derivativeabbreviations are

F,a = "F(a)"a

, F,aa = "2F(a)"a2

, (14.10)

14–5


(Partials are used since F could be a function of other variables in addition to a.) Using the previousresults and the chain rule we obtain

!F(a)!"

= !F(a)!a

!a!"

= F,a aT u21,",

!2F(a)!"!#

= !F,a

!a!a!#

aT u21," + F,a!2a

!"!#= uT21,"

!

F,aa aaT + F,a H"

u21,# + F,a aT u21,"#

(14.11)For F(a) = a the previous results are recovered. The case of powers of a: F(a) = an has immediateapplications. For n = 2, with" and# specializedfirst to t and then tou, we obtain the squared-lengthderivatives:

a2 = 2a a u21, a2 = 2uT u21 + 2au21 (14.12)!a2

!u= 2a [!aT aT ]T = 2a h,

!2a2

!u !u= 2

#

I !I!I I

$

= 2J. (14.13)

The simplicity of the Hessian of a2 will be a cogent argument for the use of the Green measureof strain in bars. For n = !1, with " and # specialized as above, we obtain the inverse-lengthderivatives:

˙1/a = !(1/a2) a u, ¨1/a = (1/a3) uT21 (3a aT ! I) u21 ! (1/a2) u21 (14.14)

!(1/a)!u

= !(1/a2) [!a a ] ,!2(1/a)!u !u

= (1/a3)#

3a aT ! I !3a aT + I!3a aT + I 3a aT ! I

$

(14.15)

The foregoing equations may be specialized to two-dimensional motions of a segment moving in the{X, Y } plane by setting the Z component to zero and then removing that component from vectorsand matrices.All of the results given so far are geometrically exact and pose no limit on how much the segmentstretches or contracts. Restrictions in the form of small deformations will appear when the formulasare applied to a bar element in §14.3.

§14.2.3. Mathematica Implementation and FD Verification

The foregoing formulas for the first and second derivatives of an arbitrary function F(a), given in(14.11), have been implemented inMathematica in the form of twomodules. Results are numericallyverified by finite differences.Scripts for the computation of first partial derivatives of F(a) with respect to " are shown in Fig-ure 14.3. This is done by module LengthFunctionFirstDerivatives. The results when " isidentified with the node displacement vector are verified with central finite differences with moduleLengthFunctionFirstDerivativesByFD. The driver code is shown at the bottom of the figure.The computations are exercised for F(a) set to a, a2, 1/a, 1/a2,

"a and log(a) in the loop shown at

the bottom of Figure 14.3. The coordinate and displacement values used for numerical verificationare

(aX , aY , aZ ) = (11, 10, 2), (uX1, uY1, uZ1) = (3, !5, !5), (uX2, uY2, uZ2) = (8, !4, 1),(ax , ay, az) = (aX , aY , aZ )+(uX2, uY2, uZ2) ! (uX1, uY1, uZ1) = (16, 11, 8),

a0 =%

a2X+a2Y+a2Z =%

112+102+22 = 15, a =&

a2x+a2y+a2z =%

162+112+82 = 21.(14.16)

14–6


LengthFunctionFirstDerivative[{fa_,a_},aXYZ_,u21_,Φ_]:=Module[ {nu=Length[u21],nΦ=Length[Φ],axyz,u21Φ}, axyz=aXYZ+u21; u21Φ=Table[D[u21[[i]],Φ[[j]]],{i,1,nu},{j,1,nΦ}]; Return[(D[fa,a]*axyz/a).u21Φ] ]; LengthFunctionFirstDerivativeByFD[{fa_,a_},aXYZ_,u21_,δ_]:= Module[{ax,ay,az,axyz,axyzp,axyzm,axp,ayp,azp,axm,aym,azm, i,inc,dfada,da,d=Table[0,{6}]}, {ax,ay,az}=axyz=aXYZ+u21; dfada=D[fa,a]/.a->Sqrt[ax^2+ay^2+az^2]; inc= *{{-1,0,0},{0,-1,0},{0,0,-1},{1,0,0},{0,1,0},{0,0,1}}; For [i=1,i<=6,i++, axyz=aXYZ+u21; axyzp=axyz+inc[[i]]; axyzm=axyz-inc[[i]]; {axp,ayp,azp}=axyzp; {axm,aym,azm}=axyzm; da=Sqrt[axp^2+ayp^2+azp^2]-Sqrt[axm^2+aym^2+azm^2]; d[[i]]=dfada*da/(2*δ); ]; Return[d]];

ClearAll[a,axyz,a,aX,aY,aZ,uX1,uY1,uZ1,uX2,uY2,uZ2,Φ,δ

δ

δ

δ]; =1/100;rep={aX->11,aY->10,aZ->2,uX1->3,uY1->-5,uZ1->-5,uX2->8,uY2->-4, uZ2->1,a0->15,a->21};aXYZ={aX,aY,aZ}; u21={uX2-uX1,uY2-uY1,uZ2-uZ1}; axyz=aXYZ+u21; {ax,ay,az}=axyz;Print["aXYZ=",aXYZ/.rep," u21=",u21/.rep," axyz=",axyz/.rep, " a0=",Sqrt[aX^2+aY^2+aZ^2]/.rep," a=",Sqrt[ax^2+ay^2+az^2]/.rep];u={uX1,uY1,uZ1,uX2,uY2,uZ2};For [if=1,if<=6,if++, fa={a,a^2,1/a,1/a^2,Sqrt[a],Log[a]}[[if]]; d=Simplify[LengthFunctionFirstDerivative[{fa,a},aXYZ,u21,u]]; Print["fa= ",fa,", d=",d,"\n = ",N[d/.rep]]; dFD=LengthFunctionFirstDerivativeByFD[{fa,a},aXYZ/.rep,u21/.rep, ]; Print["d by FD= ", N[dFD/.rep]]; ];

Figure 14.3. Mathematica implementation of first-derivative computations in (14.11) for F(a) andnumerical check by central finite differences.

fa= a , d=

−aX − uX1 + uX2

2 a3/2, −

aY − uY1 + uY22 a3/2

, −aZ − uZ1 + uZ2

2 a3/2,

aX − uX1 + uX22 a3/2

,aY − uY1 + uY2

2 a3/2,

aZ − uZ1 + uZ22 a3/2

= −0.0831306, −0.0571523, −0.0415653, 0.0831306, 0.0571523, 0.0415653

d by FD= −0.0831306, −0.0571523, −0.0415653, 0.0831306, 0.0571523, 0.0415653

Figure 14.4. Partial results from script of Figure 14.3 for F(a) =!a and numerical values (14.16).

Displacement increment ! = 0.01 used for verification by central finite differences.

Figure 14.4 shows results for the case F(a) =!a. Numerical results correspond to the data (14.16).

It can be seen that the finite difference values, obtained with a displacement increment of ! = 0.01,agree to all places shown with the analytical results.Scripts for the computation of second partial derivative of F(a) with respect to " and # are shownin Figure 14.5. This is done by module LengthFunctionSecondDerivatives. The results whenboth" and# are identified with the node displacement vector are verified through finite differenceswith module LengthFunctionSecondDerivativesByFD. The driver code is shown at the bottomof the figure. The computations are exercised for F(a) set to a, a2, 1/a, 1/a2,

!a and log(a) in the

loop shown at the bottom of Figure 14.5.Figure 14.6 shows numerical results for F(a) =

!a and the data (14.16). Symbolic results are

omitted to reduce clutter. It can be seen that the finite difference values, obtained with a displacementincrement of ! = 0.01, agree to at least 5 places with the analytical results.

14–7


LengthFunctionSecondDerivative[{fa_,a_},aXYZ_,u21_,{Φ_,Ψ_}]:=Module[ {nc=Length[aXYZ],nu=Length[u21],nΦ=Length[Φ],nΨ=Length[Ψ], u21Φ,u21Ψ,J,H,P,S,Q,dfda,d2fdada}, axyz=aXYZ+u21; S=Q=Table[0,{6},{6}]; u21Φ=Table[D[u21[[i]],Φ[[j]]],{i,1,nu},{j,1,nΦ}]; u21Ψ=Table[D[u21[[i]],Ψ[[j]]],{i,1,nu},{j,1,nΨ}]; J=Table[axyz[[i]]*axyz[[j]]/a^2,{i,1,nc},{j,1,nc}]; H=(IdentityMatrix[nc]-J)/a; Q=Table[0,{nΦ},{nΨ}]; For [i=1,i<=nc,i++, Q=Q+(axyz[[i]]/a)* Table[D[u21[[i]],Φ[[j]],Ψ[[k]]],{j,1,nΦ},{k,1,nΨ}] ]; dfda=D[fa,a]; d2fdada=D[dfda,a]; S=Transpose[u21Φ].(d2fdada*J+dfda*H).u21Ψ; Return[{S,Q}]; ]; LengthFunctionSecondDerivativeByFD[{fa_,a_},aXYZ_,u21_, _]:= Module[{aX,aY,aZ,uX21,uY21,uZ21,ax,ay,az, i,inc,du,S=Table[0,{6},{6}]}, inc= *{{-1,0,0},{0,-1,0},{0,0,-1},{1,0,0},{0,1,0},{0,0,1}}; For [i=1,i<=6,i++, du=inc[[i]]; dp=LengthFunctionFirstDerivativeByFD[{fa,a},aXYZ,u21+du, ]; dm=LengthFunctionFirstDerivativeByFD[{fa,a},aXYZ,u21-du, ]; S[[i]]=(dp-dm)/(2* ); ]; Return[S]]; ClearAll[a,n,axyz,a,aX,aY,aZ,uX1,uY1,uZ1,uX2,uY2,uZ2,Φ, ]; =1/100;rep={aX->11,aY->10,aZ->2,uX1->3,uY1->-5,uZ1->-5,uX2->8,uY2->-4,uZ2->1,a0->15,a->21};aXYZ={aX,aY,aZ}; u21={uX2-uX1,uY2-uY1,uZ2-uZ1}; axyz=aXYZ+u21; {ax,ay,az}=axyz;Print["aXYZ=",aXYZ/.rep," u21=",u21/.rep," axyz=",axyz/.rep, " a0=",Sqrt[aX^2+aY^2+aZ^2]/.rep," a=",Sqrt[ax^2+ay^2+az^2]/.rep];u={uX1,uY1,uZ1,uX2,uY2,uZ2};For [if=1,if<=6,if++, fa={a,a^2,1/a,1/a^2,Sqrt[a],Log[a]}[[if]]; {S,Q}=Simplify[LengthFunctionSecondDerivative[{fa,a},aXYZ,u21,{u,u}]]; Print["fa= ",fa,", S=",S//MatrixForm]; Print["fa= ",fa,", S=",N[Simplify[S/.rep]]//MatrixForm]; Print["eigs of S=", Chop[Eigenvalues[N[S/.rep]]]]; SFD=LengthFunctionSecondDerivativeByFD[{fa,a},aXYZ/.rep,u21/.rep, ]; Print["S by FD: ",Chop[N[SFD],.000001]//MatrixForm]; Print["eigs of SFD=", Chop[Eigenvalues[N[SFD]],.00001]]; ];

δ

δ

δ

δδ

δ δ

δ

Figure 14.5. Mathematica implementation of second-derivative computations in (14.11) for F(a)and check by central finite differences.

fa= a , S=

0.000671548 −0.00311033 −0.00226206 −0.000671548 0.00311033 0.00226206−0.00311033 0.00305731 −0.00155516 0.00311033 −0.00305731 0.00155516−0.00226206 −0.00155516 0.00406464 0.00226206 0.00155516 −0.00406464−0.000671548 0.00311033 0.00226206 0.000671548 −0.00311033 −0.002262060.00311033 −0.00305731 0.00155516 −0.00311033 0.00305731 −0.001555160.00226206 0.00155516 −0.00406464 −0.00226206 −0.00155516 0.00406464

eigs of S= 0.0103913, 0.0103913, −0.00519566, 0, 0, 0

S by FD:

0.00067155 −0.00311033 −0.00226206 −0.00067155 0.00311033 0.00226206−0.00311033 0.00305731 −0.00155516 0.00311033 −0.00305731 0.00155516−0.00226206 −0.00155516 0.00406463 0.00226206 0.00155516 −0.00406463−0.00067155 0.00311033 0.00226206 0.00067155 −0.00311033 −0.002262060.00311033 −0.00305731 0.00155516 −0.00311033 0.00305731 −0.001555160.00226206 0.00155516 −0.00406463 −0.00226206 −0.00155516 0.00406463

eigs of SFD= 0.0103913, 0.0103913, −0.00519566, 0, 0, 0

Figure 14.6. Partial results from script of Figure 14.5 for F(a) =!a and numerical values (14.16).

Symbolic results not shown to reduce clutter. Displacement increment ! = 0.01 used for verificationby central finite differences.

14–8

14–9 §14.3 THE CR BAR ELEMENT

Deformed (current)configuration C

Length L

0

0Base configuration C0

yz

x∼

x−

∼

y−

∼

z−1 1 1 1(x ,y ,z )

1 1 1 1 (X ,Y ,Z )2 2 22 (X ,Y ,Z )

2 2 22(x ,y ,z )

0Length L

E, A constant

X, xZ, z

Y, y

Global system

D

Corotated configurationaligned with C , withidentical element midpoint

D

Figure 14.7. The 2-node bar element moving in 3D space.

§14.3. The CR Bar Element

The space bar element shown in Figure 14.7 will be presented here as an application of the foregoingresults on line segment moving through 3D space. The static EICR formulation is illustrated with alinearly elastic, prismatic, 2-node bar element moving in 3D space, as depicted in that figure. Theelement has six degrees of freedom collected in the 6-vector (14.3).The area and length in the base configuration C0 are A0 and L0, respectively. These become A andL in the deformed configuration CD . The element elongation is called d = L ! L0. The elasticmodulus is E . In accordance with the usual assumptions of the CR description, bar deformations areassumed to be small. This allows to carry all integrals over the initial volume A0L0.Application of the “best corotational fit criterion” shows that, as may be expected, the corotatedconfiguration CR is aligned with, and lies halfway from, the current end nodes. But this result willnot be required to develop the element equations.

§14.3.1. Internal Energy, Force and Stiffness

The axial strain and stress measures are denoted by e and s, respectively, with s being the energyconjugate of e. Both are assumed to be constant over the element volume. For the moment the choiceof e and s will be left open. The strain and stress in C0 are 0 and s0, respectively. In CD they becomee and s = s0 + Ee. The axial forces in C0 and CD are N0 = A0s0 and N = A0 s = N0 + E A0 e,respectively.The energy density in C0 is taken as zero. It becomes U = s0 e + 1

2 E e2 in CD , which is constant overthe element. The total internal energy in CD is

U =!

V0

U dV = U V0 = A0L0(s0 e + 12 E e2) = L0(N0 e + 1

2 E A0 e2). (14.17)

The strain measure is taken to be a function e = e(d, L0) = u(L , L0) to be chosen later. Sinced = L ! L0 and L0 is fixed, e,d = !e/!d = !e/!L = e,L and likewise e,dd = !2e/(!d!d) =

14–9


Table 14.1. Strain Measures for Bar Element and their Derivatives

Strain Measure Symbol Definition for bar !e!L

!2e!L2

Engineering eE (L ! L0)/L0 1/L0 0

Green-Lagrange eG (L2 ! L20)/(2L20) L/L20 1/L20

Hencky eH log(L/L0) 1/L !1/L2

Midpoint eM 2(L ! L0)/(L + L0) 4L0/(L + L0)2 !8L0/(L + L0)3

!2e/(!L!L) = e,LL . The derivatives of e with respect to the nodal displacements DOF (14.3) areobtained by the chain rule: !e/!u = e,L (!L/!u), etc. The necessary partials are already workedout for the line segment in §14.2, with the substitutions a " L , a " L, a" L, etc. Recall that

!L!u

= h =!

!LL

"

,!2L!u!u

= G =!

H !H!H H

"

= 1L

(J! hhT ), (14.18)

where L is the a given in in (14.2), H is defined in (14.8) and J in (14.9), with the replacementsindicated above.The internal force is the gradient of the internal energy with respect to the node displacements:

p = !U!u

= !U!e

!e!u

= L0 (N0 + E A0e)!e!u

= L0 N!e!L

!L!u

= L0 N!e!Lh. (14.19)

The tangent stiffness is the Hessian of U :

K = !2U!u!u

= !p!u

= L0!N!u

!e!L

!L!u

+ L0 N!2e!L2

!L!u

#

!L!u

$T

+ L0 N!e!L

!2L!u!u

= L0 E A0!e!L

!e!L

!L!u

#

!L!u

$T

+ L0 N

%

!2e!L2

!L!u

#

!L!u

$T

+ !e!L

!2L!u!u

&

'

= E A0 L0( !e!L

&2hhT + N L0

( !2e!L2

hhT + !e!LG

&

= E A0 L0( !e!L

&2hhT + N L0

!

1L

!e!LJ+

#

!2e!L2

! 1L

!e!L

$

hhT"

= KM +KG .

(14.20)

Here KM and KG denote the material and geometric stiffness, a decomposition already encounteredin the TL description of Chapters 8–9.

§14.3.2. Matrices for Specific Strain Measures

Some specific strain measures and the values of their partials with respect to L are collected in Table14.1. The appropriate choice should be replaced in (14.19) and (14.20) to get the final form of theinternal force vector and tangent stiffness matrix.

14–10


Example 14.1. If the engineering strain e = eE = (L ! L0)/L0 is used,

p = N h, KM = E A0L0

hhT , KG = NLG, K = KM +KG . (14.21)

Example 14.2. If the Green-Lagrange strain e = eG = (L2 ! L20)/(2L20) is used,

p = NLL0h, KM = E A0L2

L30hhT , KG = N

L0J = N

L0

!

"

"

"

"

#

1 0 0 !1 0 00 1 0 0 !1 00 0 1 0 0 !1

!1 0 0 1 0 00 !1 0 0 1 00 0 !1 0 0 1

$

%

%

%

%

&

, K = KM +KG .

(14.22)For this choice the geometric stiffnessKG has the same form as that of the TL bar element derived in Chapter 8.The material matrix, however, is different unless one sets L = L0.

Example 14.3. If the Hencky strain e = eH = log(L/L0) is used,

p = NL0Lh, KM = E A0L0

L2hhT , KG = NL0

L2(J! 2hhT ), K = KM +KG . (14.23)

CRSpaceBar2InternalForce[ncoor0_,Em_,A0_,ue_,N0_,sm_,numer_]:= Module[{X1,Y1,Z1,X2,Y2,Z2,LX,LY,LZ,LL0,L0,x1,y1,z1,x2,y2,z2, Lx,Ly,Lz,LL,L,Lavg,d,EA,e,dedL,d2edL2,ND,h,p}, {{X1,Y1,Z1},{X2,Y2,Z2}}=ncoor0; {{uX1,uY1,uZ1},{uX2,uY2,uZ2}}=ue; {LX,LY,LZ}={X2-X1,Y2-Y1,Z2-Z1}; LL0=LX^2+LY^2+LZ^2; L0=Sqrt[LL0]; {{x1,y1,z1},{x2,y2,z2}}=ncoor0+ue; {Lx,Ly,Lz}={x2-x1,y2-y1,z2-z1}; LL=Lx^2+Ly^2+Lz^2; L=Sqrt[LL]; d=L-L0; Lavg=(L+L0)/2; e=Null; If [sm=="eE", e=d/L0; dedL=1/L0; d2edL2=0]; If [sm=="eG", e=d*Lavg/L0^2; dedL=L/L0^2; d2edL2=1/L0^2]; If [sm=="eH", e=Log[L/L0]; dedL=1/L; d2edL2=-1/L^2]; If [sm=="eM", e=d/Lavg; dedL=L0/Lavg^2; d2edL2=-L0/Lavg^3]; If [e==Null,Print["CRSpaceBar2Force: Illegal sm arg"]; Return[Null]]; ND=N0+Em*A0*e; h={-Lx,-Ly,-Lz,Lx,Ly,Lz}/L; p=ND*L0*dedL*h; If [numer,p=N[p]]; Return[p]];

ClearAll[ ]; ncoor0=N[{{0,0,0},{11,10,2}}]; Em=5000; A0=3; N0=10; numer=True;For [iε=1, iε<=4, iε++, ={0,0.0001,0.01,0.1}[[iε]]; ue=N[{{2,3,-4},{-4,-5,8}}*(1+ε)^2]; For [ism=1,ism<=4,ism++, sm={"eE","eG","eH","eM"}[[ism]]; p=CRSpaceBar2InternalForce[ncoor0,Em,A0,ue,N0,sm,numer]; Print["sm: ",sm,", ε=", ε,", p=",p]; ]];

Figure 14.8. Mathematica implementation of internal force vector calculation for 2-node space bar.

§14.3.3. Mathematica Implementation

The calculation of the internal force vector for the CR space bar is implemented via moduleCRSpaceBar2InternalForce. This is listed in Figure 14.8, along with test statements describedbelow.

14–11


sm: eE, ε=0, p={−3.33333, −1.33333, −9.33333, 3.33333, 1.33333, 9.33333}sm: eG, ε=0, p={−3.33333, −1.33333, −9.33333, 3.33333, 1.33333, 9.33333}sm: eH, ε=0, p={−3.33333, −1.33333, −9.33333, 3.33333, 1.33333, 9.33333}sm: eM, ε=0, p={−3.33333, −1.33333, −9.33333, 3.33333, 1.33333, 9.33333}

sm: eE, ε=0.0001, p={−3.87431, −1.54886, −10.8525, 3.87431, 1.54886, 10.8525}sm: eG, ε=0.0001, p={−3.87476, −1.54904, −10.8538, 3.87476, 1.54904, 10.8538}

sm: eH, ε=0.0001, p= {−3.87386, −1.54868, −10.8513, 3.87386, 1.54868, 10.8513}sm: eM, ε=0.0001, p={−3.87386, −1.54868, −10.8513, 3.87386, 1.54868, 10.8513}

sm: eE, ε=0.01, p={−56.577, −21.3257, −165.128, 56.577, 21.3257, 165.128}sm: eG, ε=0.01, p={−57.5008, −21.6739, −167.824, 57.5008, 21.6739, 167.824}sm: eH, ε=0.01, p={−55.6686, −20.9833, −162.477, 55.6686, 20.9833, 162.477}sm: eM, ε=0.01, p={−55.6664, −20.9824, −162.47, 55.6664, 20.9824, 162.47}

sm: eE, ε=0.1, p={−430.732, −36.854, −1902.59, 430.732, 36.854, 1902.59}sm: eG, ε=0.1, p={−517.786, −44.3025, −2287.12, 517.786, 44.3025, 2287.12}

sm: eH, ε=0.1, p={−358.761, −30.6961, −1584.69, 358.761, 30.6961, 1584.69}

sm: eM, ε=0.1, p={−356.998, −30.5453, −1576.9, 356.998, 30.5453, 1576.9}

Figure 14.9. Results obtained by running the script of Figure 14.8.

The module is invoked as

p = CRSpaceBar2InternalForce[ncoor0, Em, A0, ue, N0, sm, numer ] (14.24)

The arguments are

ncoor0 Node coordinates of element base configuration, arranged as{ { X1,Y1,Z1 },{ X2,Y2,Z2 } }.

Em Elastic modulus.

A0 Cross section area in base configuration.

ue Node displacements from base to deformed configuration, arranged as{ { uX1,uY1,uZ1 },{ uX2,uY2,uZ2 } }.

N0 Axial force in base configuration.

sm A two-character string that specifieswhich strainmeasure to use for the bar constitutiveequations: "eE", "eG", "eH" and "eM" for the engineering, Green-Lagrange, Henckyand midpoint measures, respectively. If sm is not one of these, an error message isprinted and the module returns Null.

numer A logical logical flag with the value True or False. If True the computations arecarried out in floating-point arithmetic. If False symbolic processing is assumed.

The module returns as function valuep The internal forcevector arranged as aone-dimensional list: { pX1,pY1,pZ1,pX2,pY2,pZ2 }.

The module is numerically exercised by the test statements shown at the bottom of Figure 14.8.The test case corresponds to the following data: E = 5000, A0 = 3, N0 = 10, base configurationnode coordinates (X1, Y1, Z1) = (0, 0, 0) and (X2, Y2, Z2) = (11, 12, 2) and node displacements(uX1, uY1, uZ1 = (2, 3, !4) " (1 + !)2 and (uX2, uY2, uZ2) = (!4, !5, 8) " (1 + epsilon)2, in

14–12


which ! is an adjustable parameter. If ! = 0 the bar moves in space but does not change length:L0 = L = 15. If ! > 0 the bar stretches by approximately !L0.The four strain measures eE , eG , eH and eM are exercised by cycling over sm = "eE", "eG", "eH",and "eM", respectively, in the inner test loop. For values of !: 0, 0.0001, 0.01 and 0.1 are tested inthe outer test loop.As can be observed the internal force vectors are identical if ! = 0, since for zero strain it does notmatter which measure is used. The forces are very close for the different measures if ! = 0.0001,which is very small strain (roughly 100 micros) but they differ substantially as ! gets large.The implementation and verification of the tangent stiffness matrix is deferred to Exercises 14.3 and14.4.

The formulation of the mass and damping matrices for dynamic analysis is not provided in thisChapter.

14–13



The CR Formulation: Space Bar

EXERCISE 14.1 [A:15] Verify by hand the formulas given in §14..3.2.

EXERCISE 14.2 [A:15] Verify the expressions given in Examples 14.1–3, and append to these the p, KM andKG matrices for the choice e = eM .

EXERCISE 14.3 [C:20] Implement the calculation of the tangent stiffness matrix for the space bar element(any language is OK) to return KM and KG . If done in Mathematica, the code of Figure 14.8, which is postedon the web site linked to the Chapter 14 Index, may be used as template. It is convenient to compute and returnKM and KG as two separate matrices as function value in Mathematica: Return[{ KM,KG }].As numerical test, run the bar used in the test statements of that figure using ! = 0 and any strain measure. Printout KM , KG , K = KM +KG and the eigenvalues of the three matrices. Validation check: the rank of KM , KGandK should be 1, 3 and 3, respectively (except in the case of engineering strain, in which case the rank shouldbe 1, 2 and 3, respectively.)

EXERCISE 14.4 [C:20] Using the module developed in the previous Exercise, compute and showKM ,KG andK for the bar data used in the example of Figure 14.8. For the inner loop cycle over the 4 strain measure choicesused in the internal force test. For the outer test loop, cycle over ! equal to 0, 0.01 and !0.01. Compute andshow the eigenvalues of the three matrices. Validation check: the rank of KM , KG and K should be 1, 3 and 3,respectively. For ! = !0.01 you may see some negative eigenvalues in KG and K; do not be alarmed.

14–14

.

15The CR Description:

C1 Plane Beam

15–1

Chapter 15: THE CR DESCRIPTION: C1 PLANE BEAM 15–2

TABLE OF CONTENTS

Page§15.1. Introduction 15–3§15.2. CR Beam Kinematics 15–3

§15.2.1. Coordinate Systems . . . . . . . . . . . . . . . 15–3§15.2.2. Degrees of Freedom . . . . . . . . . . . . . . . 15–3§15.2.3. Partial Derivatives . . . . . . . . . . . . . . . . 15–5§15.2.4. Arbitrary Initial Configuration . . . . . . . . . . . 15–7§15.2.5. Stress Resultants . . . . . . . . . . . . . . . . 15–8

§15.3. The Deformational Strain Energy 15–8§15.4. Internal Force Vector and Tangent Stiffness Matrix 15–9

§15.4.1. Internal Force Vector . . . . . . . . . . . . . . 15–9§15.4.2. Material Stiffness Matrix . . . . . . . . . . . . . 15–9§15.4.3. Geometric Stiffness Matrix . . . . . . . . . . . . 15–10

§15. Exercises . . . . . . . . . . . . . . . . . . . . . . 15–11

15–2

15–3 §15.2 CR BEAM KINEMATICS


In this Chapter we use the CR description to construct a geometrically nonlinear, 2-node Bernoulli-Euler plane beam. Unlike Chapter 9 we will do a C1 (Hermitian) beam from the start, since with CRit is as easy to do C1 or C0, and the former has a much better geometric stiffness matrix.

§15.2. CR Beam Kinematics

The CR formulation of the beam motion is quite similar to that of the bar element in many respects,and much of the development can be reused. Only the major differences will be noted here.


As in Chapter 9, we consider a plane, straight, prismatic beam element with two nodes. The elementis initially aligned with the global X axis in the initial configuration C0, with the origin O0 locatedat the element midpoint. This configuration is assumed to be straight and undeformed although itmay be under initial uniform axial stress with resultant N 0. The bar properties include the elasticmodules, E , the cross section area A0 and the moment of inertia I0 about the neutral axis. The lengthin C0 is L0.The motion on the {X, Y } plane carries it to the current configuration C. The corotated configurationCR is selected as depicted in Figure 15.1:1. The longitudinal axis passes through the current position of the end nodes. This defines the

local axis xe. The origin of {xe, ye} is placed halfway between the nodes. This forms an angle! with X .

2. The CR nodes are placed at an equal distance from the C nodes. Hence the corotated axes{xeR, yeR}, including origin, coincide with {xe, ye}.

The new ingredient is the rotation angle " about Z or z. With CR chosen as indicated, the deformationpart of these rotations is easily extracted: " = " ! ! .Other possibilities for selecting CR are possible. The foregoing choice has the advantage of beingcompatible with that of the bar element discussed in the previous Chapter.

§15.2.2. Degrees of Freedom

The beam element has six degrees of freedom, which are placed in the vectors

u =

!

"

"

"

"

"

"

"

#

uX1uY1"1

uX2uY2"2

$

%

%

%

%

%

%

%

&

, ue =

!

"

"

"

"

"

"

"

#

ueX1ueY1" e1ueX2ueY2" e2

$

%

%

%

%

%

%

%

&

=

!

"

"

"

"

"

"

"

#

! 12d0"112d0"2

$

%

%

%

%

%

%

%

&

. (15.1)

See Figure 15.2 for a picture of the global displacements and Figure 15.3 for the deformation dis-placements.

15–3


Base or initial configuration

CurrentC

C0

ϕ

φ = ψ+ϕ

ϕ

ψ

uX11(X , Y )1 1

1(x ,y )1 1

2(x ,y )2 2

2(X , Y )2 2uY1 uX2

uY21θ

2θ

X, x

Y, y

// X// X

_// Y

_

// Y_

X_

!Y_

CRCorotated

Figure 15.1. Kinematics of corotational C1 plane beam element.

0 0

uX1

uX2

uY1

uY2

//X, x

//X, x

C

C0

1θ

2θ

1

1

2

2

Figure 15.2. Global element displacements upon aligning X and X .

xe

ye

C

CR1

2θ2

_1θ_

−d/2

L

d/2

L0

Figure 15.3. Deformational displacements in element system.

15–4


Proceeding as in the general formulation specialized to the 2D case, we can obtain the followingrelation:

ue =

!

"

"

"

"

"

"

"

#

uex1uey1! e1uex2uey2! e2

$

%

%

%

%

%

%

%

&

=

!

"

"

"

"

"

"

"

#

c" s" 0 0 0 0!s" c" 0 0 0 00 0 1 0 0 00 0 0 c" s" 00 0 0 !s" c" 00 0 0 0 0 1

$

%

%

%

%

%

%

%

&

!

"

"

"

"

"

"

"

#

uX1 ! uX0uY1 ! uY0

!1

uX2 ! uX0uY2 ! uY0

!2

$

%

%

%

%

%

%

%

&

+

!

"

"

"

"

"

"

"

"

#

12 L0(1! c" )

12 L0s"!"

12 L0(c" ! 1)

! 12 L0s"!"

$

%

%

%

%

%

%

%

%

&

(15.2)

Here c" and s" and the angle" are implicitly defined by the displacements through the trigonometricrelations

s" = sin" = LyL

, c" = cos" = LxL

, " = arctanLyLx

(15.3)

where Lx = L0 + uX2 ! uX1, Ly = uY2 ! uY1, and

L ='

L2x + L2y (15.4)

is the bar length in the current configuration, ignoring the bending deformation.We note the following relations

#L#uX2

= ! #L#uX1

= c" ,#L#uY2

= ! #L#uY1

= s" ,#L#!1

= #L#!2

= 0,

#c"#uX2

= ! #c"#uX1

=s2"L

,#c"#uY2

= ! #c"#uY1

= ! s"c"L

,#c"#!1

= #c"#!2

= 0,

#s"#uX2

= ! #s"#uX1

= ! s"c"L

,#s"#uY2

= ! #s"#uY1

=c2"L

,#s"#!1

= #s"#!2

= 0,

#"

#uX2= ! #"

#uX1= s"

L,

#"

#uY2= ! #"

#uY1= !c"

L,

#"

#!1= #"

#!2= 0.

(15.5)

which are useful in the calculations that follow.

§15.2.3. Partial Derivatives

The first and second partial derivatives of the deformations d, !1 and !2 with respect to the nodedisplacements are necessary for the computations of internal forces and stiffness matrices.Using (15.5) and Mathematica, one obtains for the first derivatives:

!

"

"

"

"

"

"

"

#

$uex1$uey1$! e1$uex2$uey2$! e2

$

%

%

%

%

%

%

%

&

=

!

"

"

"

"

"

"

"

#

12c"

12 s" 0 ! 1

2c" ! 12 s" 0

!s"c" L0/L c2" L0/L 0 s"c" L0/L !c2" L0/L 0!s"/L c"/L 1 s"/L !c"/L 0! 12c" ! 1

2 s" 0 12c"

12 s" 0

s"c" L0/L !c2" L0/L 0 !s"c" L0/L c2" L0/L 0!s"/L c"/L 0 s"/L !c"/L 1

$

%

%

%

%

%

%

%

&

!

"

"

"

"

"

"

"

#

$uX1$uY1$!1

$uX2$uY2$!2

$

%

%

%

%

%

%

%

&

(15.6)

15–5


C0

C

ϕ

φ = ψ+ϕ

ϕ

ψ

uX1

1(X , Y )1 1

1(x ,y )1 1

2(x ,y )2 2

2(X , Y )2 2uY 1 uX2

uY 2

1θ

2θ

X, x

Y, y

// Y_

// Y_

// X

_

// X_

X

_Y

Figure 15.4. Beam element with arbitrarily oriented initial configuration C0,forming an angle ! with X . Corotated configuration not shownto reduce clutter.

Since !uex1 = ue

x2 = 12 d, " e

1 = "1, and " e2 = "2 we get

!

"

#d#"1

#"2

#

$ =

!

"

!c$ !s$ 0 c$ s$ 0!s$/L c$/L 1 s$/L !c$/L 0!s$/L c$/L 0 s$/L !c$/L 1

#

$

!

%

%

%

%

%

%

%

"

#u X1

#uY 1

#"1

#u X2

#uY 2

#"2

#

&

&

&

&

&

&

&

$

(15.7)

The second derivatives of deformation variables are

%2d%u %u

= 1L

!

%

%

%

%

%

%

"

s2$ !s$c$ 0 !s2

$ s$c$ 0!s$c$ c2

$ 0 s$c$ !c2$ 0

0 0 0 0 0 0!s2

$ s$c$ 0 s2$ !s$c$ 0

s$c$ !c2$ 0 !s$c$ c2

$ 00 0 0 0 0 0

#

&

&

&

&

&

&

$

(15.8)

%2"1

%u %u= 1

L2

!

%

%

%

%

%

%

"

!2s$c$ c2$ ! s2

$ 0 2s$c$ s2$ ! c2

$ 0c2$ ! s2

$ 2s$c$ 0 s2$ ! c2

$ !2s$c$ 00 0 0 0 0 0

2s$c$ s2$ ! c2

$ 0 !2s$c$ c2$ ! s2

$ 0s2$ ! c2

$ !2s$c$ 0 c2$ ! s2

$ 2s$c$ 00 0 0 0 0 0

#

&

&

&

&

&

&

$

(15.9)

15–6


!2"2

!u !u= 1

L2

!

"

"

"

"

"

"

#

!2s#c# c2# ! s2# 0 2s#c# s2# ! c2# 0c2# ! s2# 2s#c# 0 s2# ! c2# !2s#c# 00 0 0 0 0 0

2s#c# s2# ! c2# 0 !2s#c# c2# ! s2# 0s2# ! c2# !2s#c# 0 c2# ! s2# 2s#c# 00 0 0 0 0 0

$

%

%

%

%

%

%

&

(15.10)

§15.2.4. Arbitrary Initial Configuration

The foregoing relations can be generalized to the case of a initial configuration C0 not alignedwith the X axis as shown in Figure 15.4. Given the node coordinates and displacements shownin the figure, it is easily shown (Section §9.4) that cos$ = X21/L0, sin$ = Y21/L0, cos% =cos(# + $) = x21/L , sin% = sin(# + $) = y21/L , cos# = (X21x21 + Y21y21)/(LL0) andsin# = (X21y21 ! Y21x21)/(LL0).The preceding transformation rules remain correct if # is replaced by % = $ + # , except for thedeformation angle computation, which remain "1 = "1 ! # and "2 = "2 ! # because the "s aremeasured from X .The relation between deformational and global displacements become

d = L ! L0 = uX21 c% + uY21 s% + L0(1! c%)"1 = "1 ! #

"2 = "2 ! #

(15.11)

The first derivatives of d=eformation variables are

!

#

&d&"1

&"2

$

& =

!

#

!c% !s% 0 c% s% 0!s%/L c%/L 1 s%/L !c%/L 0!s%/L c%/L 0 s%/L !c%/L 1

$

&

!

"

"

"

"

"

"

"

#

&uX1&uY1&"1

&uX2&uY2&"2

$

%

%

%

%

%

%

%

&

(15.12)

The second derivatives of deformation variables are

!2d!u !u

= 1L

!

"

"

"

"

"

"

#

s2% !s%c% 0 !s2% s%c% 0!s%c% c2% 0 s%c% !c2% 00 0 0 0 0 0

!s2% s%c% 0 s2% !s%c% 0s%c% !c2% 0 !s%c% c2% 00 0 0 0 0 0

$

%

%

%

%

%

%

&

(15.13)

!2"1

!u !u= 1

L2

!

"

"

"

"

"

"

#

!2s%c% c2% ! s2% 0 2s%c% s2% ! c2% 0c2% ! s2% 2s%c% 0 s2% ! c2% !2s%c% 00 0 0 0 0 0

2s%c% s2% ! c2% 0 !2s%c% c2% ! s2% 0s2% ! c2% !2s%c% 0 c2% ! s2% 2s%c% 00 0 0 0 0 0

$

%

%

%

%

%

%

&

(15.14)

15–7


C0

C

N0

N0

V0

V0

N

N

V

VM02

M2

M01

M1

Figure 15.5. Beam stress resultants depicting positive sign conventions. Axial forces Nand transverse shear forces V are constant along the length, but the bendingmoments M vary linearly. Hence two nodal values of M are required.

!2"2

!u !u= 1

L2

!

"

"

"

"

"

"

#

!2s#c# c2# ! s2# 0 2s#c# s2# ! c2# 0c2# ! s2# 2s#c# 0 s2# ! c2# !2s#c# 00 0 0 0 0 0

2s#c# s2# ! c2# 0 !2s#c# c2# ! s2# 0s2# ! c2# !2s#c# 0 c2# ! s2# 2s#c# 00 0 0 0 0 0

$

%

%

%

%

%

%

&

(15.15)

§15.2.5. Stress Resultants

The stress resultants in the reference configuration (either C0 or CR) are N 0, M01 and M0

2 . The initialshear force is V 0 = (M0

1 ! M02 )/L0. See Figure 15.5 for sign conventions.

Denote by N , V and M the stress resultants in the current configuration. Whereas N and V areconstant along the element, M = M(xe) varies linearly along the length because this is a Hermitianor model, which relies on cubic transverse displacements. Consequently we will define its variationby the two node valuesM1 andM2. The shear V is recovered from equilibrium as V = (M1!M2)/L ,which is also constant. The stress resultants can be obtained from the deformations as

N = N 0 + E A0L0

d, M1 = M01 ! 2E I0

L0(2"1 + "2),

M2 = M02 + 2E I0

L0("1 + 2"2), V = M1 ! M2

L= V 0

L0L

+ 2E ILL0

("1 ! "2).

(15.16)

§15.3. The Deformational Strain Energy

The next step in the CR formulation is to work out the deformational strain energy of the beam. Thebasic choices are:

15–8

15–9 §15.4 INTERNAL FORCE VECTOR AND TANGENT STIFFNESS MATRIX

1. A linear beam2. A nonlinear TL beamThe strain energy of the beam for small strains can be written

U = Ua +Ub +Ug (15.17)

where Ua , Ub and Ug are the energy taken by axial (bar) deformation, bending deformation, andinitial-stress geometric effects, respectively. We adopt the following energy expressions:

Ua = N 0 d + 12 (N ! N 0)d2 = N 0 L0 e + 1

2 E A0L0 e2,

Ub = M02 !2 ! M0

1 !1 + 12

!

!1!2

"T E I0L0

!

4 22 4

" !

!1!2

"

,

Ug = 12

!

!1!2

"T N 0L030

!

4 !1!1 4

" !

!1!2

"

.

(15.18)

The 2" 2 matrices appearing in Ub and Ug may be derived from those given in Chapters 5 and 15,respectively, of Przemieniecki’s book.1 This book, howevr, omits the initial stress terms.

§15.4. Internal Force Vector and Tangent Stiffness Matrix

The internal force vector and tangent stiffness matrix of the corrotational element are then obtainedby the usual formulas:

p = "U"u

, K = "p"u

= KM +KG (15.19)

To develop these quantities it is necessary to find the first and second partial derivatives of d, !1 and!2 in terms of the node displacements.


Using the partial derivatives compiled above andMathematica, one obtains the following expressionfor the internal forces.

p = pa + pb + pg (15.20)

where

pa = "Ua

"u= N [!c# !s# 0 c# s# 0 ]T

pb = "Ub

"u= [ V s# !Vc# !M1 !V s# Vc# M2 ]T

pg = "Ug

"u= N 0L0

30[!3s#(!1 + !2)/L 3c#(!1 + !2)/L 4!1 ! !2

3s#(!1 + !2)/L !3c#(!1 + !2)/L 4!2 ! !1 ]T

(15.21)

1 J. S. Przemieniecki, Theory of Matrix Structural Analysis, Dover, New York, 1985.

15–9


§15.4.2. Material Stiffness Matrix

Carrying out the computations one obtains the following compact expression for thematerial stiffness:KM = TTKM0T (15.22)

where

KM0 =

!

"

"

"

"

"

"

"

"

"

"

"

"

#

E AL 0 0 ! E A

L 0 0

0 12E IL3

6E IL2

0 !12E IL3

6E IL2

0 6E IL2

4E IL 0 !6E I

L22E IL

! E AL 0 0 E A

L 0 0

0 !12E IL3

!6E IL2

0 12E IL3

!6E IL2

0 6E IL2

2E IL 0 !6E I

L24E IL

$

%

%

%

%

%

%

%

%

%

%

%

%

&

(15.23)

is the stiffness matrix of the linear beam element, and T is the transformation matrix

T =

!

"

"

"

"

"

#

c! s! 0 0 0 0!s! c! 0 0 0 00 0 1 0 0 00 0 0 c! s! 00 0 0 !s! c! 00 0 0 0 0 1

$

%

%

%

%

%

&

(15.24)

which introduces the effect of finite rigid body motions.

§15.4.3. Geometric Stiffness Matrix

The expression for the geometric stiffness is a bit more complicated. It can be presented in a compactform as follows:

KG = TTKNGT+KV

G (15.25)whereT is the transformationmatrix (15.24),KN

G is thewell knowngeometric stiffness for aHermitianbeam element under axial force:

KNG = N

30L

!

"

"

"

"

"

#

0 0 0 0 0 00 36 3L 0 !36 3L0 3L 4L2 0 !3L !L20 0 0 0 0 00 !36 !3L 0 36 !3L0 3L !L2 0 !3L 4L2

$

%

%

%

%

%

&

(15.26)

and the remaining term introduces the effect of varying moments through the transverse shear forcein C:

KVG = V

L

!

"

"

"

"

"

#

sin 2! ! cos 2! 0 ! sin 2! cos 2! 0! cos 2! ! sin 2! 0 cos 2! sin 2! 0

0 0 0 0 0 0! sin 2! cos 2! 0 sin 2! ! cos 2! 0cos 2! sin 2! 0 ! cos 2! ! sin 2! 00 0 0 0 0 0

$

%

%

%

%

%

&

(15.27)

in which sin 2! = 2s!c! and cos 2! = c2! ! s2! .

15–10

15–11 Exercises

Homework Exercises for Chapter 15The Corotational Description: 2D C1 Beam

EXERCISE 15.1 Complete the derivation of p for the 2-node C1 beam element and implement inMathematica,using the same inputs as in Chapter 9 Addendum. (Implemented and posted on Web)

EXERCISE 15.2 Complete the derivation ofK for the 2-nodeC1 beam element and implement inMathematica,using the same inputs as in Chapter 9 Addendum. (Implemented and posted on Web)

EXERCISE 15.3 Aplane 2-nodeC1 beam element has properties L0 = 6, E = 3000, A0 = 2, I0 = 12, N 0 = 5in the initial state C0 along X , with node 1 at (0,0) and node 2 at (L0, 0). The beam rotates by 45! about the originso that at the current configuration C node 1 stays at {0, 0}while node 2 moves to {(L0 +d)/

"2, (L0 +d)/

"2},

where d = L0/1000. The rotational freedoms at C are !1 = !2 = 45! = "/4 radians. Compute p, KM andKG at the current configuration, and compare those quantities with those of the C0 beam element presented inChapter 9, using RBF for the latter.Note: A Mathematica implementation of this C1 element has been posted on the Web as a Mathematica 4.1Notebook PlaneBeamC1.nb. The element checks out when moving about the reference configuration C0. Itgives excellent buckling values for the problem of Exercise 9.3. More tests are needed, however, for an arbitraryconfiguration to make sure the internal force vector and the tangent stiffness are consistent.

EXERCISE 15.4 Confirm the previous statement by repeating the buckling calculations of Exercise 9.3 using theCR beam element provided in the Mathematica Notebook mentioned above (extract the material and stiffnessmatrices, ignore the rest). Compare the speed of convergence of the CR and TL element for the cantileverbuckling problem.

15–11

.

16Overview of

Solution Methods

16–1

Chapter 16: OVERVIEW OF SOLUTION METHODS 16–2

TABLE OF CONTENTS

Page§16.1. Introduction 16–3

§16.1.1. Stages, Increments and Iterations . . . . . . . . . . . 16–3§16.1.2. Why Incrementation? . . . . . . . . . . . . . . 16–4

§16.2. Advancing the Solution: Increment Control 16–5§16.3. Advancing the Solution: Prediction 16–5§16.4. Advancing the Solution: Correction 16–7§16.5. Traversing Equilibrium Path in Positive Sense 16–7

§16.5.1. Positive External Work . . . . . . . . . . . . . . 16–8§16.5.2. Angle Criterion . . . . . . . . . . . . . . . . 16–8

§16.6. Constraint Strategy 16–9§16.6.1. ! Control . . . . . . . . . . . . . . . . . . . 16–9§16.6.2. State Control . . . . . . . . . . . . . . . . . 16–9§16.6.3. Arclength Control . . . . . . . . . . . . . . . . 16–11§16.6.4. (Global) Hyperelliptic Control . . . . . . . . . . . 16–11§16.6.5. Local Hyperelliptic Control . . . . . . . . . . . . . 16–12

§16.7. Practical Solution Requirements 16–12§16.7.1. Tracing the Response . . . . . . . . . . . . . . 16–12§16.7.2. Finding a Nonlinear Solution . . . . . . . . . . . . 16–13§16.7.3. Stability Assessment . . . . . . . . . . . . . . 16–13§16.7.4. Post-buckling and Snap-through . . . . . . . . . . . 16–13§16.7.5. Multiple Load Parameters . . . . . . . . . . . . . 16–14

§16. Exercises . . . . . . . . . . . . . . . . . . . . . . 16–15

16–2

16–3 §16.1 INTRODUCTION

In previous Chapters we have covered the governing equations of geometrically nonlinear structuralanalysis and the discretization of those equations by finite element methods. The result is a set ofparametrized nonlinear algebraic equations called residual force equations.The solution of these equations as the control parameters are varied varied provides the equilibriumresponse of the structure. In this Chapter we begin the coverage of solution methods suitable fordigital computation.


It was noted in Chapter 1 that all solution procedures of practical importance are strongly rooted inthe idea of “advancing the solution” by continuation. The basic idea is to follow the equilibriumresponse of the structure as the control and state parameters vary by small amounts. The motivationin terms of circumventing the “solution morass” is described in that Chapter.This overarching framework gives rise tomany variants called solution schemes. A common featureis that continuation is a multilevel process, as illustrated in Figure 16.1. The process involvesa hierarchical breakdown into stages, incremental steps, and iterative steps. The middle level:incrementation, is always present. Staging may be missing if there is only one control parameter.Iteration may be missing if there if no correction process.In the present Chapter multilevel continuation is described in general terms, with the goal ofmaintaining independence from specific solution schemes. The final subsections describe how thegeneral procedure is adapted to the analysis of problems encountered in engineering practice.

§16.1.1. Stages, Increments and Iterations

As discussed in Chapter 4, processing a complex nonlinear problem generally involves performinga series of analysis stages. Multiple control parameters are not varied independently in eachstage and may therefore be characterized by a single stage control parameter !. Stages are onlyweakly coupled in the sense that the end solution of one may provide the starting point for another.Throughout this and following Chapters attention is focused on a generic stage and there is no needto use an identifying index for it.To advance the solution, the stage is broken down into incremental steps, or increments for short. Ifnecessary incremental steps will be identified by the subscript n; for example, the state vector afterthe nth increment is un and the state vector before any increment (at stage start) is u0. Over eachincremental step the state vector u and stage control parameter ! undergo finite changes denotedby "u and "!, respectively.Incremental solution methods can be divided into two broad classes:1. Purely incremental methods, also called predictor-only methods.2. Corrective methods, also called predictor-corrector or incremental-iterative methods.In purely incremental methods the iteration level is missing. In corrective methods a predictor stepis followed by one or more iteration steps. The set of iterations is called the corrective phase. Itspurpose is to eliminate or reduce the so called drifting error, discussed in §16.4, which plaguespurely incremental methods.Iteration steps will be usually identified by the superscript k; for example {ukn, !kn} may denote thesolution after the kth iteration of thenth step, whereas {u0n, !0n} is the predicted solutionbefore starting

16–3


StagesIncrements

Iterations

Figure 16.1. Nested hierarchy in nonlinear solution methods:stages, increments and iterations.

the corrective process. This superscript is enclosed in parentheses if there is potential confusionwith exponents. Iterative changes in !u and !" are often shortened to d and #, respectively.

Remark 16.1. Solutions accepted after each increment following completion of the corrective process, areoften of interest to users because they represent approximations to equilibrium states. They are therefore savedas they are computed. On the other hand, intermediate results of iterative processes are rarely of interest unlessone is studying the “insides” of solution processes. Hence most production programs discard them.

Remark 16.2. The terminology of nonlinear static analysis is far from standardized. Despite their practicalimportance, few authors recognize the existence of stages. Many use the term step to mean incremental stepwhereas the terms substep, subincrement and cycle are used for the iteration level. There is more uniformityin dynamic analysis, possibly because there is only one advancing level: step is universally used to denote thechange over a time increment.

§16.1.2. Why Incrementation?

The use of increments may seem at first sight unnecessary if one is interested primarily in the finalsolution. But breaking up a stage into increments may serve other purposes:

Helping convergence. Success in a corrective process done by a Newton-like method may hinge onhaving a good initial guess supplied by the predictor, since such methods are notoriously finicky.The quality of this guess can be improved by reducing the increment.

Sidestepping extraneous roots. Incrementation helps the solution procedure from falling into the“root morass” discussed in Chapter 1.

Gaining insight into structural behavior. As noted in Remark 16.1, programs often save convergedsolutions after each increment and for a good reason: a response plot can teach the engineer moreabout the structural behavior than simply knowing the final solution.

Avoiding surprises. Critical points may occur before the stage ends. There are problems in whichsuch points, notably bifurcation, may be masked if coarse increments are taken.

16–4

16–5 §16.3 ADVANCING THE SOLUTION: PREDICTION

Alleviating path dependence. Although the focus of this course is on path-independent problems,it should be noted that the presence of path-dependent effects severely restricts increment sizesbecause of history-tracing constraints. For example, in plasticity analysis stress states must not beallowed to stray too far outside the yield surface.

§16.2. Advancing the Solution: Increment Control

A nonlinear analysis program is “marching” along a stage. Assume that n incremental steps havebeen completed. The last accepted solution is un , which corresponds to !n . Performing the (n+1)thstep entails the calculation of the increments

"un = un+1 ! un, "!n = !n+1 ! !n, (16.1)

that satisfy the residual equilibrium equations r(u, !) = 0 to requested accuracy. As stated the taskis not fully defined because there are less equations than unknowns, which makes the incrementsizes indeterminate. The problem is closed by adopting an increment control strategy. The strategymay be expressed in general form as a constraint condition:

c("un,"!n) = 0, (16.2)

which equalizes the number of equations to the number of unknowns.A rate form of the constraint equation (16.2) is obtained by differentiating with respect to the pseudotime t :

aT u+ g! = 0, (16.3)

whereaT = #c

#u, g = #c

#!. (16.4)

Remark 16.3. The addition of the constraint equation serves two purposes: it makes the algebraic problemdeterminate, and it can be used to control the increment size directly or indirectly to enhance robustness andconvergence.

Remark 16.4. Note that the constraint c = 0 is expressed in terms of the increments {"un,"!n} from the lastsolution and not in terms of the total values. This localization condition is essential to maintaining invariancewith respect to the origin chosen for u and !. Once a step is finished, the constraint is reset for the next oneby moving its origin.

Remark 16.5. Specific choices for (16.4) are discussed in §164 below but for some developments it is possibleto keep c arbitrary. Furthermore, it is also possible to specify the constraint directly in the rate form (16.3)without an explicit integral. An example is Fried’s orthogonal trajectory accession method.1

1 I. Fried, Orthogonal Trajectory Accession to the Nonlinear Equilibrium Curve, Comp. Meth. Appl. Mech. Engrg., 47,283–297, (1984).

16–5


§16.3. Advancing the Solution: Prediction

Having decided upon an increment control strategy, to start up the (n+1)th incremental step, aninitial approximation

!u0n, !"0n, (16.5)

to the increments (16.1) is calculated by a prediction step. These values are called the predictedincrements and the formula used is called a predictor or extrapolator.Most predictors are based on the first-order path equation derived in Chapter 4 and repeated herefor convenience:

r = 0, or Ku = q ", (16.6)

Assuming K to be nonsingular, the forward Euler method furnishes the simplest predictor:

!u0n = K!1n qn!"0n = vn !"0n, (16.7)

in which v is the incremental velocity vector defined in Chapter 4. The process is completed byselecting an increment control strategy through the constraint (16.2). Two examples follow.

Example 16.1. For the prescribed-load-value strategy in which!"n is specified to be #n (positive or negative),the constraint is

c(!un,!"n) = !"n ! #n = 0. (16.8)

Then the increments are directly given by (16.6), i.e.

!u0n = vn#n, !"n = #n . (16.9)

This formula obviously fails whenKn is singular, i.e. at critical points, because there vn becomes either infinite(at limit points) or nonunique (at bifurcation points). This suggests that the solution process will break downat those points.

Example 16.2. For the arclength strategy in which the absolute value of the distance (4.23) is specified to be#n > 0, the constraint is

c(!un,!"n) = |!sn| ! #n = 1fn

!!!vTn!un +!"n

!!! ! #n = 0, (16.10)

where fn = +"1+ vTn vn . Substitution into (16.6) yields

!"0n = #n fn±(vTn vn + 1)

= #n

±"vTn vn + 1

= ± #nfn

, !u0n = ±vn#nfn

. (16.11)

In this case two signs for the increment are obtained. The proper one is obtained by applying one of the “pathadvancing” criteria discussed below.Note also that (16.10) does not fail at isolated limit points if one properly passes to the limit v/|v| " z, as perRemark 4.2. This limit process yields

!"0n = 0, !u0n = ±#nz (16.12)

The normalized v near the limit point serves as a good approximation for z. It should be noted, however, thatthe formula fails at multiple limit points and at bifurcation points; thus the arclength strategy is no panacea.

16–6

16–7 §16.5 TRAVERSING EQUILIBRIUM PATH IN POSITIVE SENSE

Actual Equilibrium Path

Computed solutions

Drift error

λ

v

Figure 16.2. Drift error in purely incremental solution procedure.

Both of the foregoing examples above contain a specified length !n . For the first step, !0 is normallychosen by the user. If the predictor is followed by a corrective process, in subsequent steps !n maybe roughly adjusted according to the “last iteration count” rule of Crisfield,2 which works well inpractice. If no corrective phase follows, the proper selection of !n is discussed later in the sectiondealing with purely incremental methods.

§16.4. Advancing the Solution: CorrectionIf the predicted increments (16.5) are inserted in the residual equation r(u, ") = 0, there willgenerally be a departure from equilibrium:

r0n = r!un +#u0n, "n +#"0n

"!= 0. (16.13)

This departure is called drift error. A corrective process is an iterative scheme that eliminates, orat least reduces, the drift error by producing a sequence of values

#ukn, #"kn, (16.14)

that as k " # hopefully tend to the increments (16.2) that satisfy equilibrium and meet incrementcontrol specifications. Popular corrective methods are studied in subsequent Chapters.As previously noted, there are purely incrementalmethods that omit the corrective phase. They arecovered in following Chapters. See Figure 16.2 for an illustration of the drift error phenomenonthat occurs when a corrector is not applied.

Remark 16.6. An even simpler predictor consists of setting#u0n = 0, #"0n = 0. The corrective process thenstarts from the previous solution. This overcautious approach is rarely used in practice.

2 M. A. Crisfield, An Incremental-Iterative Algorithm that Handles Snap-Through, Computer & Structures, 16, 55–62(1981)M. A. Crisfield, An Arc-Length Method Including Line Searches and Accelerations, Int. J. Num. Meth. Engrg., 19,1269–1289 (1983).

16–7


§16.5. Traversing Equilibrium Path in Positive Sense

In Example 16.2 two signs were obtained for the predicted !"0n and !u0n . This is typical ofconstraints that are reversible about the last solution point; that is, reversing the signs of both !uand!" satisfies c = 0.3 In that case the resulting algebraic system usually provides two solutions:

±!"0n, ±!u0n. (16.15)

Even in Example 16.1 there is an ambiguity because the specified #n may be positive or negative.The sign ambiguity arises because, as explained in Chapter 4, the tangent at regular points of anequilibrium path has two possible directions, which generally intersect the constraint hypersurfacein at least two points. Thus it becomes necessary to chose the direction corresponding to a positivepath traversal. Two rules for chosing the proper sign are described below.

§16.5.1. Positive External Work

The simplest rule requires that the external work expenditure over the predictor step be positive:

!W = qT!u0n = qT vn!"n > 0. (16.16)

That is, !" should have the sign of qT v = qTK!1q.This condition works well when “the structure follows the load” and is particularly effective at limitpoints. It fails if q and v are orthogonal:

qT v = 0, (16.17)

because then the condition (16.16) is vacuous. This happens in the following cases.

Bifurcation points. As a bifurcation point B is approached, v/|v| " z, achieving equality at B.Since qT z = 0, it follows that (16.16) fails at B.

Incremental velocity reversal. If the structure becomes “infinitely stiff” at a point in the equilibriumpath v vanishes. This case is rarer than the previous one, but may arise in the vicinity of turningpoints.

Bifurcation points demand special treatment and cannot be easily passed through simple predictormethods. One way out is to insert artificial purtuebations that transformperturbations are inserted.However, the case v " 0 can be overcome by a modification of the previous rule.

§16.5.2. Angle Criterion

There are problems in which the structure gains suddenly stiffness, as for example in the vicinity ofa turning point T . If the positive work criterion is used eventually the solution process “turns back”and begins retracing the equilibrium path. When it reaches the high stiffness point again it doesanother U-turn and so on. The net result of this “ping pong” effect is that the solution process getsstuck. Physically a positive work rule is incorrect because the structure needs to release externalwork to continue along the equilibrium path.

3 Some authors call such constraints symmetric.

16–8

16–9 §16.6 CONSTRAINT STRATEGY

To get over this difficulty a condition on the angle of the prediction vector is more effective. Lettn!1 be the tangent at the previous solution. Then chose the positive sense so that

tTn tn!1 > 0. (16.18)

Once the “ping-pong” region is crossed, the work criterion should be reversed so the external workis negative.

Remark 16.7. Other geometric criteria are given by Crisfield (loc. cit. in footnote 2) and Skeie and Felippa4

§16.6. Constraint Strategy

So far the form of the constraint equation, (16.2) or (16.3), has been left arbitrary. In the sequel welist, roughly in order of ascending complexity, instances that are either important in the applicationsor have historical interest. In what follows ! is always a dimensionless scalar that characterizes thesize of the increment. Six constraints are pictured in Figure 16.3. In this figure, c is the constraintcurve, S is the last solution point, P the predicted point and C the converged solution.

§16.6.1. " Control

At each step #"n = !n , where ! is a dimensionless scalar. The constraint equation is (16.8) listedin Example 16.1. This is generally called "-control. Often the parameter " is associated with aloading amplitude, in which case this is called load control. The physical analogy would be a testmachine in which the operator increases the load to specific values.The differential form (16.3) has

a = 0, g = 1. (16.19)

As noted in Example 16.1, this constraint form fails as critical points are approached.

§16.6.2. State Control

This consists of specifying a norm of #un , for example the Euclidean norm:

c(#un) " (#uTn#un)2 ! !n u2 = 0, (16.20)

where u is a reference value with dimensions of displacement, which is introduce for scalingpurposes. An alternative way of doing that consists of using the scaled increment of §4.6:

#!uTn#!un ! !2n = 0. (16.21)

(See also Remark below.) The differential form (16.3) has

aT = 2#un, g = 0. (16.22)

4 G. Skeie and C. A. Felippa, A Local Hyperelliptic Constraint for Nonlinear Analysis, Proceedings of NUMETA’90Conference, Swansea, Wales, Elsevier Sci. Pubs, 1990.

16–9


(a) (b) (c)

(d) (e) (f)

C C

CC

C

C

P

P

P

P

P

P

S

S

S

S

S

S

c=0

c=0

c=0

c=0

c=0c=0

Figure 16.3. Geometric representation of constraint equations for a one-dof problem, withstate u and control parameter ! plotted horizontally and vertically, respectively.(a) load control, (b) state control, (c) arclength control, (d) hyperspherical control,(e) global hyperelliptical control, and (f) local hyperelliptic control.

Remark 16.8. In the finite element literature the term displacement control has been traditionally associatedwith the case in which the magnitude of only one of the components of u, say ui , is specified, which istantamount to choosing a special infinity norm of u. This old technique was used in the mid-1960s by Argyrisand Felippa.5 There is a generalization of single displacement control in which several reference displacementsare used. This multiple dimensional hyperplane control has been investigated by Powell, Bergan and others.6

5 J. H. Argyris, Continua and Discontinua, in Proceedings Conference on Matrix Methods in Structural Engineering,AFFDL-TR-66-80, Wright-Patterson AFB, Dayton, Ohio, 11–189 (1966).C. A. Felippa, Refined Finite Element Analysis of Linear and Nonlinear Two-dimensional Structures, Ph.D. thesis, Dept.of Civil Engrg, University of California, Berkeley (1966).

6 G. H. Powell and J. Simons, Improved Iteration Strategy for Nonlinear Structures, Int. J. Num. Meth. Engrg., 17, 1655–1667 (1981)P. G. Bergan, G. Horrigmoe, B. Krakeland and T. H. Søreide, Solution Techniques for Nonlinear Finite Element Problems,Int. J. Num. Meth. Engrg., 12, 1677–1696 (1978)P. G. Bergan, Solution Algorithms for Nonlinear Structural Problems, Computers & Structures, 12 497–509 (1980)P. G. Bergan and J. Simons, Hyperplane Displacement Control Methods in Nonlinear Analysis, in Innovative Methodsfor Nonlinear Problems, ed. by W. K. Liu, T. Belytschko and K. C. Park, Pineridge Press, Swansea, U.K., 345–364(1984)

16–10

16–11 §16.6 CONSTRAINT STRATEGY

§16.6.3. Arclength Control

Arclength control consists of specifying a distance |!s| = " along the path tangent. The constraintequation is (16.10) in Example 16.2. This form has scaling problems since it intermixes u and #.It is generally preferable to work with the scaled quantities of §4.6 in which case the constraintbecomes

!!sn ! "n = 1!fn

""!vTn !!un +!#n"" ! "n = 0, (16.23)

The differential form (16.3) for the unscaled form (16.9) is

aT = vn/ fn, g = 1/ fn. (16.24)

and for (16.24)aT = vnS2/ !fn, g = 1/ !fn. (16.25)

Without the scaling this becomes the constraint of Riks andWempner,7 also called arclength control.Geometrically the unscaled equation represents a hyperplane normal to t, located a distance "nfrom the last solution point S(un, #n) in the state-control space. The scaled form admits a similarinterpretation in the scaled state-control space space (Su, #).

Remark 16.9. The “orthogonal trajectory” constraint discussed by Fried (see footnote 1) may be regarded asa generalization of the arclength constraint in which a traversal orthogonality condition is applied throughoutthe corrective phase. This differential constraint is interesting in that it does not fit the form (16.2) and mayin fact be followed independently of the the predictor and past solution. But following the trajectory dependson v = K!1q being frequently updated and is practical only with a true Newton corrector.

§16.6.4. (Global) Hyperelliptic Control

There is a wide family of constraints that combine the magnitude of !#n and a norm of !un . Afrequently used combination is the hyperelliptic constraint

a2n!uTn!un + b2n (!#n)2 = "2n, (16.26)

where scalar coefficients a and b may not be simultaneously zero.More effective in practice is the scaled form of the above, namely

a2n!!uTn!!un + b2n (!#n)2 = "2n, (16.27)

where all quantities are now dimensionless.Geometrically these constraints corresponds to an hyperellipse that has the last solution as center,and includes other constraints as degenerate cases. The scaling parameters a and b were introduced

7 E. Riks, The Application of Newton’s Method to the Problem of Elastic Stability, Trans. ASME, J. Appl. Mech., 39,1060–1065 (1972)G.A.Wempner, DiscreteApproximations Related toNonlinear Theories of Solids, Int. J. Solids Structures, 7, 1581–1599(1971).

16–11


by Padovan and Park.8 The expression was rendered dimensionless by Felippa9 who introducedscaling parameters and and discussed appropriate choices. If a = b = 1 in the unscaled form(16.27) we recover the hyperspherical constraint proposed (but not used) by Crisfield (loc. cit. infootnote 4).The constraint gradients are

a = 2a2!u, g = 2b2!". (16.28)

§16.6.5. Local Hyperelliptic Control

This is a variation of the previous one in which we take a combination of !" and a norm of !u,where !" and !u are to be determined according to a local coordinate system at S(un, "n):

c(u, ") = a2(u! un)TS(u! un) + b2("! "n)2 ! #2n = 0, (16.29)

where a and b are scalar coefficients and #n is prescribed. Geometrically this is a hyperellipse withprincipal axes in a coordinate system defined by !" and !u. An attractive choice for the localsystem is provided by the path tangent vector tn and the normal hyperplane at point S(un, "n).These are given by by (4.16) and (4.20) respectively, with v " vn .Near critical points, v # $. In such a case we would like to recover the global system to avoidnumerical difficulties. This is achieved by defining the new variables !" and !u according to

!" = vT (!u! v!"), !u = v(vT!u+!"). (16.30)

Scaling of this constraint to achieve consistency is discussed by Skeie and Felippa (work cited infootnote 4), where additional computational details may be found. It turns out that this constraintcan include all ones previously discussed as special regular or limit cases.

Remark 16.10. Another interesting strategy: the work constraint of Bathe and Dvorkin 10 limits the totalexternal work spent during the corrective phase.

Remark 16.11. In path-independent problems that involve only geometric or conservative boundary-conditionnonlinearities, it is generally best to maximize step lengths subject to stability and equilibrium accuracyconstraints. Stability depends on the curvature of the response path, presence of critical points, and solutionmethod used. Equilibrium accuracy depends chiefly on whether a corrective process is applied.

8 J. Padovan andS. Tovichakchaikul, Self-Adaptive Predictor-CorrectorAlgorithm for StaticNonlinear StructuralAnalysis,Computers & Structures, 15, 365–377 (1982).K. C. Park, A Family of Solution Algorithms for Nonlinear Structural Analysis Based on the Relaxation Equations, Int.J. Num. Meth. Engrg., 18, 1637–1647 (1982).

9 C. A. Felippa, Dynamic Relaxation under General Increment Control, in Innovative Methods for Nonlinear Problems,ed. by W. K. Liu, T. Belytschko and K. C. Park, Pineridge Press, Swansea, U.K., 103–163 (1984).

10 K. J. Bathe and E. Dvorkin, On the Automatic Solution of Nonlinear Finite Element Equations, Computers & Structures,17, 871–879 (1983).

16–12

16–13 §16.7 PRACTICAL SOLUTION REQUIREMENTS

§16.7. Practical Solution Requirements

The remaining subsections describe various types of nonlinear structural analyses encountered inengineering practice, and the requirements they pose on solution procedures.

§16.7.1. Tracing the Response

“Tracing the response” is of interest for many nonlinear problems. For a typical stage, perform asequence of incremental steps to find equilibrium states

un, !n, n = 1, 2, . . .

in sufficient number to ascertain the response u = u(!) of the structure within engineering require-ments.If the control parameter is associated with a fundamental load system, the response path is knownas the fundamental equilibrium path, as it pertains to the service range in which the structure issupposed to operate.One class of problems that fit this requirement is that in which structural deflections, rather thanstrength, are of primary importance in the design. For example, some large flexible space structuresmust meet rigorous “dimensional stability” tolerances while in service.

§16.7.2. Finding a Nonlinear Solution

A variant of the foregoing occurs if the primary objective of the analysis is to find a solution ucorresponding to a given ! (for example, ! = 1), whereas tracing of the response path is in itselfof little interest.Very flexible structures that must operate in the nonlinear regime during service fit this problemclass. The example of the suspension bridge under its own weight, discussed in §3.4, providesa good illustration. The undeflected “base” configuration u = 0 is of little interest as it has nophysical reality and the bridge never assumes it. It is merely a reference point for measuringdeflections.Under such circumstances, the chief consideration is that the accuracy with which the response pathis traced is of little concern. Getting the final answer is the important thing. Once this referenceconfiguration is obtained, “excursions” due to live loads, temperature variations, wind effects andthe like may be the subject of further analysis staging.

§16.7.3. Stability Assessment

This is perhaps the most important application of nonlinear static analysis. The analyst is concernedwith the value (or values) of ! closest to 0 at which the structure behavior is not uniquely determinedby !. These are the critical points discussed in Chapter 5. In physical terms, the system becomesuncontrollable and may “take off” dynamically.Problem of this nature arise in stability design. The determination of limit points is called collapseor snapping analysis. The determination of bifurcation points is called buckling analysis.

16–13


§16.7.4. Post-buckling and Snap-through

Occassionally it is of interest to continue the nonlinear analysis beyond a limit or bifurcationpoint. Continuation past a limit point is post-collapse or snap-through analysis; continuation pasta bifurcation point is post-buckling analysis.Post-critical analyses are less commonly encountered in practice than the previous two types. Theyare of interest to ascertain imperfection sensitivity of primary structural components, or to assessstrength reserve in fail-safe analysis under abnormal conditions such as construction, deploymentor accidents.Conventional load control is not generally sufficient to trace snap-through. This may be achieved,however, with the aid of the more general increment control strategies discussed above. Traversingbifurcation points is notoriously more difficult; a technique applicable to well isolated bifurcationpoints is discussed later in the context of augmented equations and auxiliary systems.

§16.7.5. Multiple Load Parameters

As discussed in Chapter 3, the case of multiple control parameters is reduced to a sequence ofone-parameter analyses. The previous classification apply to individual stages, and not all stagesnecessarily fit the same type of analysis requirements.The systematic determination of a complete equilibrium surface as the envelope of all response pathsis rarely pursued in practice aside from academic examples. For practical structures, an investigationof this type would put enormous demands on human and computer time and is doubtful whetherthe additional insight would justify such expenditures.There is, however, a special case of multiparameter investigation that is gaining popularity fordesigning lightweight structures: stability interaction curves as envelopes of critical points.

16–14

16–15 Exercises

Homework Exercises for Chapter 16Overview of Solution Methods

EXERCISE 16.1

[C:20] Consider the following residual equilibrium equation:

r(!, ") = sec(#!!)

!"sec# (2+" sin!) ! 2 sec(#!!)

#tan(#!!) ! " cos! sec#

$= 0, (E16.1)

in which # is a problem parameter, " the control parameter, and ! the only degree of freedom. This r comesfrom the 2-bar arch problem already studied in Exercise 6.2. Here # is the initial arch rise angle whereas! = # ! $ is the angle change from the reference state, at which !0 = 0 and "0 = 0. A plot of the exact"(!) for ! = [0, 60"] = [0,%/3] is shown in Figure E16.1(a); the fundamental path ends at limit point L .

0.2 0.4 0.6 0.8 1

−0.06

−0.04

−0.02

0.02

0.04

0.06

0.08

0.2 0.4 0.6 0.8 1

−0.06

−0.04

−0.02

0.02

0.04

0.06 Exact response forExercise 16.1

Exact response forExercise 16.2

λλ

ψ (rad) ψ (rad)

LB(a) (b)

Figure E16.1. Exact responses for Exercises E16.1 and E16.2.

(a) Derive the first-order rate form K ! = q " by taking t # ", and convert to u = du/d" = v. (Recall thatK = &r/&! , q = !&r/&", and v = K!1q.)

(b) Integrate numerically the rate equation u = v found in (a) by the purely incremental, forward-Eulermethod with load control over " = [0, 0.1]. Start from "0 = 0 and !0 = 0. (All angles should be inradians.) Use # = 30" = %/6 as arch rise angle and take 10 load increments of 'n = 0.01 (same for allsteps). Are you able to detect and traverse the limit point L?

(c) Repeat the run twice, each time cutting 'n by 1/4 and quadrupling the number of steps. Is limit pointdetection and traversal improved?

Hints. If using Mathematica the 10-step forward Euler script followed by the response plot could be imple-mented as

v=q/K; Eulersol={{0,0}}; psin=0; lambdan=0; ns=10; ln=0.01;For [i=1,i<=ns,i++,

vn=N[v/.{lambda->lambdan,psi->psin}];lambdanp1=lambdan+ln; psinp1=psin+vn*ln;Eulersol=AppendTo[Eulersol,N[{psinp1,lambdanp1}]];lambdan=lambdanp1; psin=psinp1];

ListPlot[Eulersol,PlotJoined->True];

EXERCISE 16.2

[C:20] Repeat (a)–(c) of the foregoing Exercise for the residual

r = r"1/4! 16"+ cos(4!)

#. (E16.2)

in which r is the residual (E16.1). The fundamental path now ends at a bifurcation point B, as pictured inFigure E16.1(b). Use # = 30" and same solution method. Are you able to detect the bifurcation point?

16–15

.

17Purely Incremental

Methods:Load Control

17–1

Chapter 17: PURELY INCREMENTAL METHODS: LOAD CONTROL 17–2

TABLE OF CONTENTS

Page§17.1. Governing Differential Equation 17–3§17.2. Forward Euler Integration 17–3§17.3. More Accurate Integration 17–4§17.4. Numerical Stability of Forward Euler 17–5§17.5. Accuracy Monitoring 17–7

17–2

17–3 §17.2 FORWARD EULER INTEGRATION

Incremental methods calculate the nonlinear response through the numerical integration of a rateform of the equilibrium equations as the stage control parameter ! is varied. In the nomenclatureintroduced in Chapter 16, we can characterize these as predictor-only methods: no correctiveiterations to recover equilibrium are performed. They are also known as step-by-step, initial-value or marching methods in the engineering literature. The qualifier “purely” distinguishes theseincremental methods from those that make use of the pseudo-force concept, and which are coveredin Chapter 19.

The present Chapter emphasizes purely incremental methods in which the first-order rate equationsare integrated by a forward Euler scheme. Furthermore, for simplicity we focus on the simplestincrement control strategy: load control, in which ! is treated as an independent variable. Thisrestriction allows subjects such as stability and accuracy to be discussed in a straightforwardmanner.An arclength-parametrized version, which allows the introduction of more robust increment controltechniques and the automatic traversal of limit points, is presented in the following Chapter.

§17.1. Governing Differential Equation

Recall the first-order rate equation r = Ku! q! = 0 specialized to t " !:

r# = Ku# ! q = 0, (17.1)

where primes denote differentiation with respect to !. If the stiffness matrix is nonsingular, thisequation uniquely relates the differential of u to that of !:

u# = dud!

= K!1q = v, (17.2)

where as usual v denotes the incremental velocity vector. Purely incremental methods with ! asindependent variable are based on the numerical integration of (17.2) to generate an approximateresponse u = u(!) given the initial condition

u = u0 at ! = 0. (17.3)

Remark 17.1. The exact integral of (17.2) with the initial conditions (17.3) is

r(u, !) = r0 (17.4)

where r0 = r(u0, 0). Thus an initial equilibrium error does not decay even if the integration were carried outexactly. This is the source of the drifting error that afflicts purely incremental methods. The error committedat each step moves the equilibrium point to a neighboring curve in the incremental flow (see Figure 16.2).Consequently the solution may “drift away” quickly when the incremental flow paths “flare out” from theequilibrium path.

17–3


§17.2. Forward Euler Integration

In the remaining subsections of thisChapterwe consider that the incrementationprocess is controlleddirectly by varying the stage parameter !, which thus assumes the role of independent variable.This is tantamount to using the !-control increment discussed in Chapter 16. This restriction isremoved in the next Chapter.The simplest incrementation scheme is obtained by using the forward Euler integrator

un+1 = un + "!u!n, (17.5)

where n is the incremental step index, undef= u(!n) and

"!n = !n+1 " !n, (17.6)

is the stage parameter stepsize. Treating (17.2) with this integrator yields the scheme

"un = K"1n qn "!n = vn "!n,

un+1 = un + "un.(17.7)

In the actual computer implementation of (17.7) the linear system Knvn = qn is preprocessed byassembling and factoring Kn . The right hand side qn is solved for to get vn . This is multiplied by"!n , which is either prescribed or (better) adjusted by the stepsize-control techniques discussedbelow.

Remark 17.2. As discussed inChapter 16, (17.7) is also the usual predictor for incremental-correctivemethods.

§17.3. More Accurate Integration

To increase accuracy, more refined integration formulas have been proposed. An attractive second-order choice is the explicit midpoint rule (also called Heun’s rule by some authors):

un+1/2 = un + 12K

"1n qn "!n,

Kn+1/2def= K(un+1/2), qn+1/2

def= q(un+1/2),un+1 = un +K"1

n+1/2 qn+1/2 "!n.

(17.8)

This scheme was used in the author’s thesis1 to treat problems with combined geometric andmaterial nonlinearities. The midpoint rule has attractive features for flow-plasticity studies, sincelocal elastic unloading can be detected during the first “trial” step andKn+1/2 adjusted accordingly.The same feature can be used to advantage in bifurcation analysis if a stiffness-determinant changeis detected betweenKn andKn+1/2. But note that the stiffness matrix has to be formed and factoredtwice per incremental step.

1 C. A. Felippa, Refined Finite Element Analysis of Linear and Nonlinear Two-dimensional Structures, Ph.D. thesis, Dept.of Civil Engineering, University of California, Berkeley (1966)

17–4

17–5 §17.4 NUMERICAL STABILITY OF FORWARD EULER

Natural extensions of (17.8) are third and fourth-order Runge-Kutta (RK) formulas, which requirethree and four stiffness evaluations and factorizations per step, respectively. These more refinedmethods, however, are rarely used in structural mechanics for the amount of work per step isconsiderable. Remark 17.8, however, indicates a possible niche for the classical fourth-order RKin nonconservative problems.

§17.4. Numerical Stability of Forward Euler

Can the integration process (17.7) become numerically unstable? The subject is rarely mentionedin the finite element literature. For simplicity we begin with the one-degree-of-freedom counterpartu! = v of u! = v. The right-hand side v = K"1q is Taylor-series expanded in!u = u " un aboutun as

u! = un + µ !u + O(!u2) with µ = "v

"u. (17.9)

For the linearized stability analysis only the homogeneous part of (17.9) is retained, which yieldsthe model equation

u! = µu. (17.10)

Consider the case in which µ is negative real and h = !# > 0. Then the solution u = u(#) ofthe model equation is exponentially decreasing as # increases. The forward Euler integration isabsolutely stable2 if

|1+ hµ| # 1, or h # "2/µ. (17.11)

If h exceeds this value, the computed solution exhibits oscillatory instability. Ifµ is positive real thesolution of the model equation grows exponentially as # increases and the forward Euler integrationis “relatively stable” for all h > 0.Now if # decreases so that h = !# < 0 the roles are reversed (cf. Remark 17.3). The stabilitycondition is h $ "2/µ if µ > 0. If # is a load parameter, loading and unloading sequences maybe viewed as equally likely; consequently a safe stability constraint is

|!#| # 2|µ|

. (17.12)

For the general system (17.2), let µi (i = 1, 2 . . . N , N being the number of degrees of freedom)be the eigenvalues of the so-called amplification matrix

A = "v"u

= "(K"1q)"u

. (17.13)

It is shown in Remark 5.8 that this matrix, although generally unsymmetric, has real eigenvalues ifthe problem is conservative, i.e. K is the Hessian of a potential$(u, #) for fixed #, andK is positive

2 C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J.(1971)L. Lapidus and J. H. Seinfield, Numerical Solution of Ordinary Differential Equations, Academic Press, New York(1971).

17–5


definite. The eigenvalues are given by the eigensystems (5.22) or (5.23). Under such conditions asafe increment is given by (17.12), where now

µ = µmax = maxi

|µi |, i = 1, . . . N . (17.14)

If K does not depend on ! (as in the linear case) all µi vanish and the increment is unrestricted. IfK depends midly upon !, eigensystems (5.21)–(5.22) show that the largest |µi | are associated withthe smallest eigenvalues of K, i.e. the fundamental stiffness modes (see also Remark 17.4).Of course the actual calculation of all µi at each step would be a formidable computational task.But the following finite-difference “path” estimate is easily obtained:

µ ! "vn+1 # vn""un+1 # un"

= "vn+1 # vn""vn"

1"!n

= an"!n

, (17.15)

where an = "vn+1 # vn"/"vn" and " denotes the 2-norm or Euclidean norm of a vector: "x" =$xT x. (an may be viewed as a kind of “incremental acceleration”.) Unfortunately this quantity

is not available until the nth step is completed, and to get a practical estimate we replace it by theprevious step estimate:

µ ! an#1"!n#1

, an#1 = "vn # vn#1""vn#1"

. (17.16)

Insertion into (17.12) yields the stability condition

|"!n| % 2 |"!n#1|an#1

. (17.17)

If A can have complex eigenvalues, however, this simple rule does not apply (see Remark 17.8).

Remark 17.3. The fact that h = "! can be either positive or negative is a distinguishing feature of incrementalstatic analysis. In the numerical integration of actual dynamical systems, the time increment h = "# is nevernegative; therefore stability results from conventional ODE theory should be used with caution.

Remark 17.4. Having the largest µi associated with the smallest eigenvalues of K represents another note-worthy difference with dynamic problems. In the latter, the stability limit of explicit integrators such as (17.7)is determined by the highest frequencies of the system. As discussed in Chapters dealing with dynamic relax-ation methods, the difference is due to the fact that the matrix multiplying the highest derivative is the mass(stiffness) in a dynamic (static) problem.

Remark 17.5. If the midpoint rule (17.8) is used, the stability limit remains the same for real µi .

Remark 17.6. Replacing "vn#1" by "vn" in (17.16) is inconsequential, as it is only a gross estimate. In fact,choosing the smallest of the two norms would be the more conservative policy. A more serious objection isthe choice of the 2-norm unless the problem is well scaled and all degrees of freedoms have common physicaldimension. Otherwise a diagonal scaling matrix may have to be introduced as discussed in Chapter 4; forexample

a2n#1 = |(vn # vn+1)TS2(vn # vn+1)|min(|vTn S2vn|, |vTn#1S

2vn#1|). (17.18)

17–6

17–7 §17.5 ACCURACY MONITORING

Remark 17.7. As K approaches singularity, !v! " # and the stable !" approaches zero. This is anindication of the problems encountered by this type of incremental method at critical points. “Flat” limitpoints can be traversed using the step-controlled parametric formulation discussed in the next Chapter. Atthose points !" changes sign. But as the limit point becomes progressively sharper, numerical difficultiesincrease. Bifurcation points, which in some sense may be viewed as infinitely sharp limit points, are difficultto traverse without resort to either perturbation or special techniques that necessarily involve buckling modeestimation, as discussed later.

Remark 17.8. If the problem is not derivable from a potential orK is not positive definite,Amay have complexeigenvalues such as µi = #i + j$i , j =

$%1. Let h = !". If h#i < 0, the appropriate stability condition

for forward Euler reads(1+ h#i )2 + h2$2i & 1. (17.19)

If the imaginary component $i dominates, the stable stepsize may be sharply reduced over that of the potentialcase, and if #i = 0 there is no stable h. It is not difficult to construct “load follower” problems that yield almostimaginary µi . The morale is that purely incremental methods should be used with caution in nonconservativeproblems. For this class of problems, third- and fourth-order Runge-Kutta methods do enjoy a substantialstability edge; see, for example, the stability charts on page 120 of Lapidus and Seinfeld (referenced cited infootnote 2).

§17.5. Accuracy MonitoringIf the response is twice differentiable, the local truncation error of the Euler integrator (17.5) at aregular point is easily obtained from the truncated Taylor expansion about (un, "n) as3

e = 12 (!")2 u''

% = 12 (!")2 v'

% , (17.20)

where the symbol v'% = u''

% denotes the second derivative &2u/&"2 = &v/&" evaluated at points%i ( ["n, "n+1], which generally differ from component to component. To assess the magnitudee = !e! of this error we need an estimate of the norm of v'. An obvious finite-difference estimatefor this quantity is (vn+1 % vn)/!"n but — as in the stability analysis – !"n and vn+1 are notavailable until the step is completed. For a practical estimation of e we are forced to use previousstep values:

e = ||e|| ) 12 (!"n)

2 !vn % vn%1!!"n%1

. (17.21)

For accuracy monitoring a convenient dimensionless measure is the ratio ' of e to the incrementlength !!un! = !!"nvn!:

' = 12

!

!

!

!

!

!"n

!"n%1

!

!

!

!

!

an%1 with an%1 =!vn % vn%1!

!vn!. (17.22)

Here we again denote by a a ratio similar to the one in (17.16) but with !vn! in the denominator,which is an inconsequential change. To strive for uniform local accuracy the basic idea is to specify' (say ' ) 0.1) and adjust the stepsize so that

|!"n| &2' |!"n%1|

an%1. (17.23)

3 P. Henrici, Error Propagation for Difference Methods, Wiley, New York (1963).

17–7


This increment sizemust also be subjected to other bounds provided by rules discussed later. Similaraccuracy monitoring techniques can be devised for more complicated integration schemes such asthe midpoint rule.

Remark 17.9. Comparing (17.23) with (17.17) furnishes a simple rule: choosing ! < 1 takes care of stabilityif the roots of A are real. In any case the similarity between the stability and accuracy control rules is striking.

Remark 17.10. For poorly scaled problems the use of a weighted norm, as in (17.18), is advisable.

Remark 17.11. Given bounds on e at each step, bounds on the accumulated drift error can be obtained butthey are usually so pessimistic as to be useless, unless some detailed problem information is available; seee.g. Gear (loc.cit. in footnote 2). The only reliable way to assess global accuracy is to rerun the problem withseveral values of !, for example ! = 0.2, 0.1 and 0.05.

Remark 17.12. This truncation error monitoring technique works in static nonlinear analysis because eis controlled by physically relevant low-frequency modes. It fails on direct time integration of dynamicalproblems— see e.g. Park4— because e is then controlled by physically irrelevant high-frequency modes. Innumerical analysis parlance, problems in structural dynamics are said to be stiff.

4 K. C. Park and C. A. Felippa, Direct Time Integration Methods in Nonlinear Structural Dynamics, Comp. Meth. Appl.Mech. Engrg., 17/18, pp. 277–313 (1979)K. C. Park, Time Integration of Structural Dynamics: A Survey, Ch. 4.2 inPressure Vessels and PipingDesign Technology— A Decade of Progress ASME, New York (1982).

17–8

.

18Purely Incremental

Methods:General Control

18–1

Chapter 18: PURELY INCREMENTAL METHODS: GENERAL CONTROL 18–2

In this Chapter we continue the development of the purely incremental methods under generalincremental control conditions. A general nonlinear transformation of state and control parametersto pseudo-time is introduced. Computable forms of this constraint are specialized to arclengthand hyperelliptic control. The computational implications of these decisions are discussed, and apractical implementation of hyperplane-distance arclength control outlined.

§18.1. Parametric Form

In previous Chapters it was noted that continuation solution methods under ! (load) control havedifficulty traversing limit points. This shortcoming can be circumvented through the use of moregeneral increment control schemes. This generalization can be practically effected by adjoiningalgebraic constraint equations such as those listed in Chapter 16.These more general forms of increment control can be described on a uniform basis as follows.Because the main effect of enforcing the constraint c("u, "!) = 0, is to link the increments of uand !, we express the response in the pseudo time parametric form

u = u(t), ! = !(t). (18.1)

If t ! ! we would of course regress to the !-control parametrization

u = u(!), (18.2)

of the equilibrium path, and nothing would be gained. But the pseudotime t in (18.1) is now at ourdisposal and we can try to do better. Differentiating (18.1) with respect to t we get

du = u dt, d! = ! dt = (1/ f ) dt, (18.3)

where we have called for convenience

f = 1!

= dtd!

. (18.4)

§18.1.1. Requirements

The incremental path equation becomes

r = #r(u, !)

#t= #r

#uu" #r

#!! = Ku" q/ f = 0. (18.5)

At limit points d! = 0. By “smooth traversal” of a limit point is meant that t varies regularly as a“vehicle odometer” as that limit point is crossed. Consequently f = dt/d! must go to infinity ata limit point. This reasoning shows why t ! ! does not work, because if so f = 1 everywhere. Italso follows that the relation between ! and a useful t must be necessarily nonlinear, for if we taket = c1! + c2 with c1 #= 0, f is always 1/c1.It is computationally desirable, however, that in “almost linear” portions of the response f ap-proaches a finite value, because in the limit we would like to recover the conventional methods oflinear structural analysis.

18–2

18–3 §18.1 PARAMETRIC FORM

§18.1.2. The Arclength Choice

Nowwhich quantity becomes infinite at limit points? An obvious choice is the incremental velocityv = K!1q or, more precisely, some norm of it. A particularly attractive choice in light of itsgeometric significance is

f =!

1+ vT v, (18.6)

This corresponds to takingdt " ds = f d! = v du+ d!, (18.7)

As shown in Chapter 4, ds is the differential arclength of the response curve in state-control space.Consequently s is the arclength traversed along this curve measured from some arbitrary point,such as the last solution. Note the following attractive features of this choice:

(a) At limit points f #$, as required.

(b) At turning points where v # 0, f # 1, so (unlike state control) traversal controlled byincreasing arclength is not affected.

(c) If the response is linear f maintains a constant value.

The arclength choice is no panacea, however, because three computational problems remain, listedbelow in order of increasing difficulty.

(d) The exact arclength sn from the last computed solution at Pn(un, !n) to another point P(u, !)

on the equilibrium path is given by the path integral

sn ="

Pn#Pds =

"

Pn#P

!

1+ vT v ds. (18.8)

But as written this expression is not directly computable because it requires knowledge ofv = K!1q along the equilibrium path that emanates from the last solution, which is preciselywhat we want to compute. Consequently an increment control constraint such as

sn = "n, (18.9)

makes no computational sense. This minor difficulty is eliminated by using an approximation#sn to the arclength. The approximation is directly or indirectly effected by introducing anappropriate constraint equation, as discussed below.

(e) The choice (18.6) intermixes state parameters, which have generally physical dimension (ofdisplacement), with the dimensionless scalar 1. This mixture can introduce difficulties inthat the computed solutions are not invariant with respect to the choice of state parameterdimensions. It can be corrected by employing appropriate state-scaling techniques as discussedin Chapter 4.

(f) Traversal of bifurcation points remain difficult or impossible without additional “tricks”. Thekey roadblock is that the continuous parametrization (18.1) breaks down at bifurcation points,

18–3


because it cannot simultaneously represent the two or more branches that intersect there.1Circumventing this difficulty within the context of purely incremental methods is not easy,and consideration is deferred to later Chapters.

We now study two computable approximations to the arclength parametrization.

§18.2. Hyperplane Distance Control

The simplest computable approximation to the exact arclength constraint sn = !n is "sn = !n ,where"sn is the hyperplane distance (4.23) to the last solution un, #n . Thus the increment controlconstraint is

"sn = (vTn "un + "#n)/ fn = !n, where fn =!

1+ vTn vn. (18.10)

in which !n , which controls the magnitude of the nth incremental step, is either specified or auto-matically adjusted through some accuracy control rule, as discussed below.Substituting "un = vn"#n into the above we get the Forward Euler incremental scheme

vn = K!1n qn, fn =!

1+ vTn vn, "#n = ± !n

fn,

un+1 = un + vn"#n, #n+1 = #n + "#n.

(18.11)

It remains to select the sign of "#n and the magnitude of !n . The proper sign for "#n can bechosen according to the criteria discussed in §16.7. If the positive external work criterion (16.16)is used, "# is chosen to have the sign of qT v. As noted there, this simple rule fails at bifurcationand turning points, wherein the angle criterion discussed there should be used.As for the magnitude of !n , two possibilities exist. Either !n is kept constant and equal to the given!0, or it may be automatically adjusted in an adaptive scheme. If the latter strategy is adopted,parameter $ is used to bound the local error as explained in §16.5. Without going through thederivation, which is given in §16.4 below, the result is

!n = 2$ !n!1

an!1(18.12)

where an!1 is estimated by (16.23) in which v is replaced by v/ f . This must be complemented bya minimum arclength travel distance condition:

!n " !min, (18.13)

which avoids “getting stuck” at limit points under certain conditions, aswell as amaximumarclengthtravel distance condition

!n # !max , (18.14)

to avoid surprises in hitting rapidly changing portions of the response.

1 A second order rate form is in fact required in the vicinity of a simple bifurcation point.

18–4

18–5 §18.3 GLOBAL HYPERELLIPTIC CONSTRAINT

One simple way to define these minmax values is to set

!min = !0

! f ac, !max = !0 ! f ac, (18.15)

where the factor ! f ac, which is typically 1 to 20, is part of the input data. If ! f ac = 1 this ruleeffectively forces fixed a fixed !n = !0.

Remark 18.1. The essential difference between the arclength and load control schemes is the arclengthform does not have trouble crossing limit points. As the limit point is approached, "# is driven to zero andautomatically changes sign upon crossing it. The only numerical danger is that of hitting the singularity exactlyso that the factorization of K fails; however, handling of this emergency in the computer implementation isnot difficult.

Remark 18.2. In a poorly scaled problem the previous scheme should be modified by replacing f by its scaledequivalent:

fn =!

1+ vTn S2vn, (18.16)

where S2 is a diagonal scaling matrix. Some appropriate choices for S2 are discussed in the next Chapter.

§18.3. Global Hyperelliptic Constraint

As a second example, if we adopt the unscaled global hyperelliptical constraint (16.27) reproducedhere for convenience

a2n"uTn "un + b2n = !2n, (18.17)where an , bn and !n are scalars given at each step, we get the formula

"#n = !n

±!

a2nvTn vn + b2n, (18.18)

which is followed by solving for "un .For poorly scaled problems, or problems in which u collects quantities of different physical mag-nitudes, we should used the scaled form

a2n"uTn S2 "un + b2n = !2n,

where the diagonal matrix S2 is chosen to take care of scaling. Then

"#n = !n

±!

a2nvTn S2vn + b2n(18.19)

One interesting possibility is to choose S2 = K. This may be viewed as an energy constraintbecause vTKv = qT v is external incremental work. With that choice, and adopting the positive-external-work criterion to choose the sign of "#, we have

"#n = !

+"

a2n |qTn vn| + b2nsign(qTn vn). (18.20)

18–5


Remark 18.3. Equation (18.19) represents a “closed” constraint surface (a hyperellipse in state-control space)and thus it may be thought of as “safer” than the hyperplane distance constraint, which is “open”. Althoughthis consideration has some merit if a corrective phase that moves on that surface follow, it has no weightin a purely incremental method. One drawback of the hyperelliptic form is the need to choose an and bn inaddition to !n at each step. Although this adds flexibility, it can complicate the implementation and requiremore run-time decisions.

Remark 18.4. The scalar "uTK "u is not necessarily positive if K is indefinite; hence the need for takingthe absolute value of qT v in (18.20).

Remark 18.5. Diagonal scaling on displacement increments provides an intermediate choice between unscaledand K-scaled forms. That is, choose S2 = diag(K).

§18.4. Accuracy Control

Assuming that t = s so that arclength increment control is used, the Taylor expansion about a solution pointreads

!

"u"#

"

=! v f

(1/ f )

"

"s + 12

#

$

%

dv fds

d(1/ f )ds

&

'

(

("s)2 + O("s3), (18.21)

where v f = v/ f = v/!1+ vT v. The truncation error is the quadratic term in "s. Proceeding as in the load

control case we obtain! ev

e#

"

= 12

! |dv f /ds||d(1/ f )/ds|

"

("s)2. (18.22)

where ev and e# are the norms of the local truncation errors associatedwith u and #, respectively. We shall focuson controlling accuracy by monitoring ev , since sufficient accuracy on # generally follows. Again defining asin §14.5 the desired local accuracy level by the scalar $, we arrive at the rule: adjust !n by

|"sn| "2$|"sn1 |an#1

(18.23)

in which an#1 is estimated byan#1 = vn/ fn # vn#1/ fn#1

vn/ fn, (18.24)

Observe that this is the same as the adjustment rule (14.23)–(14.24) but with v replaced by v f = v/ f .

§18.5. Numerical Stability

Following the same procedure as §14.4 it can be shown that the linearized problem that governs numericalstability becomes

#

$

%

d"udsd"#

ds

&

'

(

=

#

$

%

%v f%u

%v f%#

%(1/ f )%u

%(1/ f )%#

&

'

(

!

"u"#

"

, (18.25)

in which v f = v/ f = v/!1+ vT v. The amplification matrix A is that shown in brackets above. So far this

problem appears to be analytically intractable and no general conclusions may be drawn so far. However,numerical evidence show that arclength-control incremental methods are as stable as those using load control.Thus the empirical rule of Chapter 14 may be used; namely if ! is adjusted for accuracy, setting $ < 1 takescare of stability.

18–6

18–7 §18.8 ASSESSMENT OF PURELY INCREMENTAL METHODS

§18.6. What Happens at a Limit Point

Suppose that the computed solution process approaches an isolated limit point at which K has thenull eigenvector z normalized to unit length, that is

Kz = 0, zT z = 1. (18.26)

As the critical point is approached, it can be shown (using the theory of inverse iteration) that

d!! 0, v = K"1q! "z, "! ±#, (18.27)

That is, v tends to become parallel to z although its norm goes to (plus or minus) infinity. But thef -normalized v does approach the normalized eigenvector:

v f = 1/ f v = v$1+ vT v

! ±z. (18.28)

and the incremental equations approach!

#u#!

"

=!

±z0

"

$ (18.29)

Therefore the f -normalization of v automtically take care of aligning the incremental directionnormally to the ! axis.

Remark 18.6. Some minor safeguards remain: if the tangent stiffness is evaluated at or very near the limitpoint, K(u) may be numerically singular, in which case a simple remedy is to change u by a tiny amount andtry again (a more theoretically sound approach based on penalty spring stabilization is discussed later). Thissimple technique is used in the computer program for Exercises 18.1–18.5. Also if $ is automatically adjustedby accuracy requirements it must not be allowed to fall under a minimum value.

§18.7. An Automated Incremental Algorithm

We are now in a position to give the outline of a arclength-controlled incremental algorithm essen-tially based on the scheme described previously. This is done in Table 18.1. The steps listed thereinare for a single stage. (In programs that accept multistage analysis, “stop” is replaced by a save,exit, check error conditions and recover if possible, reset and restart process with the next stage.Although programming that sequence can be elaborate, it does not involve conceptual difficulties.)

§18.8. Assessment of Purely Incremental Methods

The simplest load-controlled incremental scheme described in Chapter 17 has been extensivelyused as a stand-alone method (that is, not combined with equilibrium corrections) in early imple-mentations (1955-1965) of finite-element-based nonlinear structural analysis. It survives in thisprimitive form in a surprisingly large proportion of finite element computer programs; particularlythose aimed at treating highly nonlinear material behavior. From current perspective it suffers fromtwo serious disadvantages.

Drift Error. The residual force vector r never enters the calculations. Consequently, the deviationfrom the equilibrium path due to the propagation and accumulation of local integration errors cannot

18–7


Table 18.1 - Arclength-Controlled Incremental SolutionForward Euler Procedure for a Single Stage

At ! = 0 we know u = u0. We want to advance ! in a external-work-increasing processuntil either the norm of the state vector u exceeds umax , the magnitude of ! exceeds!max , the number of incremental steps exceeds nmax , or an impassable bifurcation pointis reached. Parameters " < 1, and #0 are specified. Set n = 0 and perform the followingsteps.

Step 1. Form and factor stiffness matrix Kn . If the factorization fails on account ofsingularity, perturb u by a tiny amount and repeat. If this failure repeats aftera certain number of tries, stop with appropriate error message.

Step 2. Form right-hand side qn and solve Knvn = qn for the incremental velocity vn .Form fn =

!

1+ vTn vn .Step 3. If an adaptive stepsize scheme is used, adjust #n as per §18.2, else keep #n = #0.

Set $!n = #n/ fn , and give it the sign of qTn vn .Step 4. Compute $un = vn$!n . Advance un+1 = un + $un and !n+1 = !n + $!n .

If adaptive # control is used, save v f to be used in the estimation of an!1.Step 5. If |un+1| exceeds umax , or |!n+1| exceeds !max , or n exceeds nmax , stop. Else

set n" n + 1 and return to Step 1.

If scaling of v is introduced because of the physical dimensionality discrepancy between! and u, the simplest implementation is to define a reference length Lref as part of theinputs. Then fn =

"

1+ vTn vn , in which vn = vn/Lref .

be eliminated by corrective iteration. In practical terms, this means that realistic estimates of theresponse tracing accuracy can be obtained only by rerunning the problem with several incrementsizes.

Computational Expense. To keep the drifting error down, many small steps may be required,particularly in “difficult” regions of the response. But at each step the stiffness matrix must beformed and factored. This can be an expensive proposition in two- and three-dimensional problems.To reduce the stiffness recalculation cost, the “pseudo-force”methods discussed in Chapter 19 havebeen used extensively in plasticity and viscoelasticity calculations. These methods are affected,however, by serious numerical stability difficulties.What are the advantages? First, given the early applications of finite element methods (see Remarkbelow), incremental methods are quite easy to program as extensions of linear analysis codes. Theabsence of the residual vector is in fact helpful, since linear analysis does not require it whereasthe stiffness matrix is always available. And the underlying concept (follow the physics) is readilyunderstood by practicing engineers. The virtues of simplicity and physical transparency should

18–8

18–9 §18.8 ASSESSMENT OF PURELY INCREMENTAL METHODS

never be underestimated!Second, in problems that exhibit strong path-dependency, as typified by flow plasticity models,the many-small-steps requirement is not necessarily a hindrance, since the increments have to bekept small anyway to avoid unacceptable element-level errors (for example, yield surface drift) inmaterial-law calculations.Finally, the high frequency of stiffness matrix assembly and refactorization has a silver lining: it isdifficult to miss critical points. The fact that purely incremental methods have trouble tracing post-buckling branches beyond bifurcation points is irrelevant to situations in which the determinationof such points, rather than traversing them, is the main objective of the analysis.

Remark 18.7. The preference of incremental over corrective methods in the early implementation of nonlinearfinite element analysis has historical roots. Finite element methods were invented in the aircraft/aerospaceindustry where linear analysis dominates. (This also helps to explain the initial popularity of the force method,which is ill-suited for nonlinear analysis.) Furthermore, early excursions in the nonlinear world involved fairlymild nonlinearities. Thus, developers of displacement-based finite element codes passed naturally from thelinear stiffness equations

Ku = q (18.30)

to the incremental formK!u = !q,

unew = uold + !u(18.31)

and gave scant thought to the nonlinear equilibrium equations.

18–9


Homework Exercise for Chapters 16-18

Incremental Solution Methods

Exercises 18.1 and 18.2 pertain to the response analysis by purely incremental methods, of the nonlinearresponse of the two-bar arch example structure of Chapter 8, which was treated by the TL description.The structure properties are: span S = 2, height H = 1 (hence the rise angle is ! = 45!), elastic modulusE = 1 and cross section area A0 = 1. The applied loads are fX = 0 and fY = "F where F = "1 is constant.This gives a downward vertical applied load if " > 0. For this rise angle, the first limit point of the TotalLagrangian model analytically occurs at load level "L # 0.136 and a vertical deflection of uY L # "0.425.Traversal of limit points is done through a positive-work advancing criterion: qT v > 0.Exercises 18.1 and 18.2 should be done with the Mathematica Notebook called IncSolTwoBarArch.nb,which has been posted on the course Web site. This is a very rough conversion from the ancient Fortran codeused in previous offerings. [The conversion took much longer than anticipated despite the simplicity of thecode.]Despite the roughness of this implementation, it can be used to illustrate the behavior of two integrators:Forward Euler (FE) andMidpoint Rule (MR), combined with three increment control strategies: Load Control(LC), Displacement Control (DC) and Arclength Control (AC). [The Classical Runge Kutta (RK4) integratorwill be added later as an Exercise.] One significant advantage ofMathematica over Fortran is the availabilityof built-in graphics. Thus response plots, for example, can be immediately generated as part of the output,helping quick visualization of method performance.

EXERCISE 18.1 [D:15] Prepare a hierarchical diagram of Cells 1-12 of the IncSolTwoBarArch.nb Note-book, beginning with the main program given in Cell 12. Note which module calls which and write down thepurpose of each module in one or two lines along the module name. Return this diagram as answer to thehomework.*

EXERCISE 18.2 [C:20]Run the four scripts (mainprograms) inCells 12 through15ofIncSolTwoBarArch.nb.Briefly explain what they do, and how the four method combinations driven by them stack up in terms of (i)robustness in traversing limit points, (ii) accuracy in locating the first limit point (analytical values are givenabove), and (iii) accuracy in crossing " = 0 at uY = "1. Please attach the four response plots of uY versus "

to your returned homework (do not bother with printouts). (I forgot to include method labels in the ListPlotcommands to identify which method is which; these may be written by hand).Note: before you can run those scripts, Cells 1-11 should be initialized. A quick way to accomplish that inversion 3.0 is to click Kernel$ Evaluate$ Evaluate Initialization. Do this twice to get rid of error messageboxes.

* If you are not sure of what a hierachical diagram is, go to the IFEM Web page:http://caswww.colorado.edu/courses.d/IFEM.d/Home.htmland look up Exercise 15.7 in Chapter 15.

18–10

18–11 Exercises

EXERCISE 18.3 [C:25] Consider the one-DOF nonlinear problem (not a structure) governed by the forceresidual equation

(!! 1)2 + (u + 1)2 ! 2 = 0 (E18.1)

This represents a circle of radius"2 and center (1,!1) in the (!, u) plane. Suppose that one starts at the

reference state u = ! = 0 and tries to trace the response, with a constant stepsize, by going around the circlecounterclockwise and eventually returning to the origin. In doing so you need to cross two turning points andtwo limit points. While traversing limit points is easy using Arclength Control and a postive work criterion,turning points are trouble.Program from scratch an incremental solution to this problem using the Midpoint Rule, Arclength Control,constant ", and the angle criterion2 to keep a positive traversal direction, and try it.3 If your implementationworks all the way you are ahead of all commercial nonlinear FEM codes in this regard.

EXERCISE 18.4 [C:25] The fixed-step classical 4th order Runge Kutta integrator (RK4) is implemented inCell 10 of the IncSolTwoBarArch.nb Notebook.4 It is identified as integ="RK4". Check visually if theimplementation is correct by comparing to the description, for example, in Numerical Recipes.Run the same arch problem as in Exercise 8.2 using RK4 combined with Load Control and Arclength Control;experiment with ell and nmax and report on whether you can obtain high accuracy (see 8.2 for how to assessit) with a fairly large stepsize, say ell=0.1.

EXERCISE 18.5 [C:25] It was shown in Chapter 8 that if the rise angle # of the two-bar arch example structureexceeds 60#, i.e. tan# >

"3, it will fail by bifurcation first. For example S = 2 and H = 2 would do it

because tan# = 2. However if you set those inputs and run the programs in IncSolTwoBarArch.nb thebifurcation point will be completely masked; you will see only the symmetric solution passing two limit points.One simple technique to make “dumb incremental solvers,” like those provided in the Notebook, pay attentionto bifurcation points is to inject artificial imperfections. This can be done, for example, by putting a fictitiousbut tiny load system that disturbs the symmetric response. For example, define the reference crown load inforce as {-0.001,-1} instead of {0,-1}. Set S = H = 2 and play with the tiny X -force, the incrementlength ell and number of steps nmax until you see a decent tracing of bifurcation post-buckling: at a certainload level uX will increase rapidly, signaling that the arch is buckling horizontally. To see that better, do theListPlot of uX versus ! collected in list uXvslambda found near the bottom of the driver cell. Comment onwhat combination of method and solution parameters let you succeed.Note (1). Bifurcation experiments can be found in the Notebook IncBifSolTwoBarArch.nb also posted onthe web site.Note (2). If you take the response long enough you may be able to have the structure return to the primarysymmetric path upon passing through the second bifurcation point, but that may take lots and lots of stepssince the implementation uses a constant stepsize ".

2 The computation of the angle between vn and vn!1 is illustrated in the posted Notebook. It is saved as part of the solutiontable, although it is not used in the solution procedure therein. The variable is called a or an and is actually the cosineof that angle, which is simply the dot product of those velocity vectors normalized to unit length.

3 The positive-work criterion qT v > 0 fails because one needs to release work along some parts of the response trajectory.In fact the net work on doing a complete circle is zero.

4 For a description of the RK4 algorithm see any book on numerical methods for ODEs. For example the widely usedNumerical Recipes in Fortran; it is presented in Section 16.1 of the second edition.

18–11

.

19Pseudo-Force

IncrementalMethods

19–1

Chapter 19: PSEUDO-FORCE INCREMENTAL METHODS 19–2

TABLE OF CONTENTS

Page§19.1. Pseudo Force Formulation 19–3§19.2. Computing the Reference Stiffness and Internal Force 19–4§19.3. Integration of Pseudo-force Rate Equation 19–5

§19.3.1. Forward Euler Integration under Load Control . . . . . . 19–5§19.3.2. Pseudo-Force Extrapolation . . . . . . . . . . . . 19–6§19.3.3. Iterative Improvement . . . . . . . . . . . . . . 19–6

§19.4. Numerical Stability 19–7§19.5. Accuracy Control 19–8§19.6. Secant Estimation of n ! 19–8§19.7. General Increment Control 19–8

19–2

19–3 §19.1 PSEUDO FORCE FORMULATION

Thedisadvantages of purely incrementalmethods in termsof solution “drift” andhigh computationalexpense were recognized as applications of nonlinear finite element analysis expanded to coverwider classes of problems. Many techniques aimed at avoiding these difficulties were proposedand tested during the late 1960s and early 1970s. From the mass of experience accumulated duringthis period, two principal strategies emerged.For path-independent, smooth nonlinearities typified by finite deflection and nonlinear-elastic be-havior, the power of Newton-like corrective methods was eventually recognized. In such problems,incremental methods were relegated to the secondary role of predictors for starting a correctiveprocess.For path-dependent material nonlinearities, however, purely incremental methods have remainedimportant because of the reasons noted in Chapter 18. These methods underwent modificationsaimed primarily at reducing the computational expense while retaining the advantages of numericalstability, implementation simplicity and physical transparency. Unfortunately these goals, beinglargely contradictory, can only be met half-way. The most successful attempt in this direction hasbeen the development of pseudo-force incremental methods, which are covered below.

§19.1. Pseudo Force Formulation

In the pseudo-force reformulation of incrementalmethods the pervasive role that the tangent stiffnessmatrix plays in the methods discussed in Chapters 17-18 is relaxed. Instead the deviation from a“reference linear response” is collected in a pseudo-force vector. This approach allows a referencestiffness matrix to be reused over many incremental steps. Since this avoids having to repeatedlyassemble and factor the tangent stiffness matrix, the gain in speed per step over the conventionalincremental methods may be very substantial in two- and three-dimensional problems. There is nofree lunch, however, for the speed gain is counterbalanced by two disadvantages:

1. Pronounced accuracy loss as nonlinearities become severe, which may force extremely smallincrements to be taken.

2. Increasing danger of numerical instability, especially in “hardening” portions of the response.

The pseudo-force method can be explained more conveniently by starting from the “force balance”form (4.9) of the residual:

r = f! p = 0, (19.1)

where f and p are the internal and external (applied) force vectors, respectively. Furthermore forsimplicity we shall assume the separable form (4.10), that is f(u) = p(!).Decompose the internal residual force f as follows:

f = Kre f (u! ure f ) + n, (19.2)

whereKre f is a nonsingular reference stiffness matrix that is kept fixed as long as possible and ure fis the state at ! = !re f (usually 0). The deviation

n = f!Kre f (u! ure f ), (19.3)

19–3


is called the pseudo-force vector. Inserting (19.2) in the residual expression (19.1) we get

r = Kre f (u! ure f ) + n! p = 0. (19.4)

Differentiating (19.4) with respect to ! yields the rate form

r" = Kre f u" + n " ! q = 0, (19.5)

where q = "p/"! is the incremental load vector. Since Kre f is assumed nonsingular, solving foru" yields

u" = K!1re f (q! n") = v f , (19.6)

Vector v f is called the pseudo incremental velocity. It plays a similar role to that of v = K!1q inthe conventional incremental methods discussed in Chapters 14 and 15.

Remark 19.1. The pseudo-force vector n may be viewed as a “force deviation from the linear referenceresponse” Kre f (u! ure f ). Now if the structural response is linear so that

K0u = p, (K0 constant), (19.7)

choosing Kre f equal to the linear stiffness K0 and ure f = 0 gives n = 0. For this reason many authors call nthe nonlinear force vector.

Remark 19.2. Differentiating (19.3) with respect to u gives, for a separable residual,

"r"u

= K = Kre f +Knon, Knon = "n"u

. (19.8)

If Kre f is kept equal to the linear stiffness throughout, Knon is called the nonlinear stiffness matrix.

§19.2. Computing the Reference Stiffness and Internal Force

A very common choice forKre f , though far from the only one, is the stiffnessK0 at the start ! = 0of the response-calculation stage. Choosing ure f = u0 accordingly, (19.3) becomes

r = K0(u! u0) + n! p = 0. (19.9)

Response calculation procedures based on the choice (19.9) are sometimes called the initial stiffnessmethod.Another common strategy is to start with the initial stiffness and continue with as many incrementalsteps as possible, resetting Kre f = Kn if numerical stability or accuracy problems are detected atthe (n + 1)th step. Still a third approach is to keep Kre f for a preassigned number of incrementalsteps, saym, and updateKre f at steps 0,m, 2m, . . ., unless numerical stability or accuracy problemsare encountered.The effectiveness of pseudo-force methods in finite element programming depends largely on theability to compute the internal force vector f directly on an element-by-element basis. Recall, forexample, that in the core-congruential formulation of the Total Lagrangian (TL) description,

f =!

V0GT si bi dV0, (19.10)

19–4

19–5 §19.3 INTEGRATION OF PSEUDO-FORCE RATE EQUATION

whereV0 is the reference volume,G the transformation betweendisplacement gradients andphysicaldegrees of freedom, si are PK stresses in the current configuration, and bi are the vectors definedin §8.5. Note that expression does not explicitly involve material properties, and is consequentlyapplicable to problems with material nonlinearities. More precisely: the only requirement for using(19.9) is the availabilibity of the stresses si in the current configuration whereas the procedure bywhich such stresses are obtained is irrelevant.

Remark 19.3. In finite element work the synonyms initial force method, initial stress method and initial strainmethod have been associated with restricted versions of what we call here pseudo-force methods. These namesfocus attention on various physical interpretations of the calculation of the f term. A heated controversy as towhich version was the best took place in the late sixties; from current perspective such arguments have onlyhistorical interest.

§19.3. Integration of Pseudo-force Rate Equation

In this subsection we assume that the calculation of f, given the necessary ingredients to apply(19.5), is more practical than that of f !. An estimation of this rate is done through finite differenceapproximations. This leads to very simple and fast implementations at the cost of numericalreliability and accuracy. A more expensive but reliable alternative technique for evaluating f! isdiscussed later.

§19.3.1. Forward Euler Integration under Load Control

The simplest incremental algorithm results on treating (19.5) by the forward Euler method with abackward-difference estimation of f ! and assuming that the increments of ! are prescribed:

nn = fn "Kre f (un " ure f ),

n!n = nn " nn"1

"!n"1,

"un = K"1re f (qn " n!

n) "!n = v fn "!n,

un+1 = un + "un, !n+1 = !n + "!n.

(19.11)

where as usual we denote

fn # f(un), nn = n(un), etc.

The pseudo incremental velocity v fn = K"1re f (qn " n!

n) plays the role of the incremental velocityvector, as can be seen by comparing the advancing equations (17.7) for purely incremental methods.The scheme (19.11) applies if n $ 1 and as long as Kre f and ure f are kept fixed. At the startn = 0, nn"1 = n"1 is not known. But if Kre f = K0 and ure f = u0, n! = 0. The same conditionis applicable when Kre f and ure f are reset, if one chooses the tangent stiffness at ure f as referencestiffness.This advancing scheme has poor accuracy characteristics unless nonlinearities are mild (say within±20% of the reference response). There are three ways of improving accuracy: pseudo-force ex-trapolation, iteration and resetting the reference stiffness. The first two are described in subsectionsbelow.

19–5


Example 19.1. Solve the residual equation

r = 5u ! u3 ! ! = 0, for ! = 4, (19.12)

using the pseudo-forcemethod (19.11)with two"! increments of 2.0, Kref = K0 = 5 and ure f = u0 = 0. Theexact solution on the fundamental path is v(!) = v(4) = 1. Since q = !#r/#! = 1, and n = 5u!u3!5u =!u3, the rate form is r " = 5u" ! (u3)" ! 1 = 0.The first increment, with !0 = 2 specified, is

n"0 = 0, "u0 = K!1

re f (q0 ! n"0)"!0 = 0.4, u1 = u0 + "u0 = 0.4, !1 = !0 + "!0 = 2.

The second increment, with "!1 = 2 again, is

n"1 = !(u31 ! u30)/"!0 = !0.032, "u1 = K!1

re f (q1 ! n"1)"!0 = 0.4128,

u2 = u1 + "u1 = 0.8128, !2 = !1 + "!1 = 4.

Repeating these computations with 4, 8 and 17 equal increments of ! gives 0.8461, 0.8841 and 0.9190,respectively. As can be observed the accuracy attained is low. Table 19.1 compares these values with thoseobtained with other methods.

§19.3.2. Pseudo-Force Extrapolation

Accuracy improves if the first of (19.10) is replaced by a central difference estimator:

n"n = nn+1 ! nn!1

"!n + "!n!1(19.13)

Since nn+1 is not known, it has to be predicted by extrapolation. The simplest extrapolator is

nn+1 # nPn+1 = n(uPn+1), with uPn+1 = un + (un!un!1)"!n

"!n!1. (19.14)

For constant"!, the predicted uPn+1 is simply 2un ! un!1. The advancing algorithm is identical to(19.11) with the second equation replaced by (19.13) and (19.14).The result of applying this technique to the example equation (19.12) with fixed ! increments ispresented in Table 17.1. The accuracy obtained now is similar to that of the conventional purelyincremental method with Forward Euler. Also given there are the results of using the midpoint rule(17.8), which as can be seen delivers higher accuracy.

§19.3.3. Iterative Improvement

Another way to improve accuracy while avoiding the reset of Kre f is to iterate on un+1 while keeping "!nfixed. To derive an iteration scheme, write the residual form (19.4) at n+1 and n, subtract, and solve for"un :

"un = K!1re f (nn+1 ! nn) ! qn "!n (19.15)

Let k be an iteration step index and u1n+1 be the value obtained from the increment equation. The resultingiterative scheme is

"ukn = K!1re f (nkn+1 ! nn) ! qn "!n,

uk+1n+1 = un + "ukn .

!

k = 1, . . . (19.16)

19–6

19–7 §19.4 NUMERICAL STABILITY

Table 19.1 Computed Incremental Solutions for (19.12)Load Control, Equal ! Increments

Steps PFI-FE PFI-FE-X PI-FE PI-MR

1 0.8000 0.8000 0.8000 0.88502 0.8128 0.8512 0.8425 0.93604 0.8461 0.8835 0.8884 0.97608 0.8841 0.9139 0.9276 0.988817 0.9190 0.9407 0.9565 0.996432 0.9470 0.9617 0.9755 0.999064 0.9675 0.9768 0.9868 0.9997128 0.9812 0.9867 0.9931 0.9999

PFI-FE: Pseudo-force incremental with Forward Euler (19.11)PFI-FE-X: PFI-FE with extrapolation (19.13)-(19.14) for n!

nPI-FE: Purely incremental with Forward Euler (17.7)PI-FE: Purely incremental with Midpoint Rule (17.8)

Using the fact that q is independent of u, this can be rewritten in the “subincremental” form

uk+1n+1 = ukn+1 "K"1re f rkn+1. (19.17)

If Kre f # Kn , this is precisely the modified Newton-Raphson (MNR) method with unit steplength. Thisshows that the the iterated pseudo-force incremental method is a MNR method with an arbitrary selection ofreference stiffness. The properties of thesemethods are investigated inChapters that dealwithNewton-Raphsoncorrective methods.

§19.4. Numerical StabilityThe purely incremental tangent-stiffness methods studied in Chapters 17-18 are highly stable if some mildprecautions are heeded. On the other hand, pseudo-force methods are much less robust. For a single degreeof freedom, the homogeneous model equation corresponding to (17.10) is

v! = K"1re f n

! = K"1re f

"n"v

v! = Av!, (19.18)

where A = Knon/Kref . Unlike (17.10) this is no longer a differential equation in v, v! but a difference equationin v!, with A as amplification number. The iteration process to solve this equation is stable if

|A| < 1. (19.19)

This condition is independent of the stepsize #!. It is seen that the key for numerical stability is that thereference stiffness “dominates” the nonlinear stiffness in the sense (19.20). That is, from that standpoint it isbetter to overestimate Kref .The generalization of (19.19) to N degrees of freedom is

u! = K"1re f u! = K"1

re f"n"uu! = K"1

re fKnon = Au!. (19.20)

19–7


Stability is controlled by the N ! N amplification matrix

A = K"1re f

!n #

!u= K"1

re f Knon . (19.21)

Assume that this amplificationmatrixA has eigenvaluesµi , i = 1, . . . N , and letµ = maxi |µi |. The conditionfor numerical stability is

µ < 1. (19.22)

This condition holds regardless of the stepsize "#. Likewise, the iteration (19.17) converges only if thecondition (19.22) holds.A practical estimator for this eigenvalue is

µ $|K"1

re f (nn " nn"1)||un " un"1|

. (19.23)

Remark 19.4. This result is another aspect of the close relationship between pseudo-force incrementalmethodsand modified-Newton corrective methods. If nonlinearities are substantial, the method diverges regardless ofthe increment length used.

Remark 19.5. It is beneficial from a stability standpoint to have Kre f “dominate” Knon . This happens insoftening structures when the elastic stiffness is selected as Kre f , and explains the success of the method inplasticity analysis. On the other hand, if the structure hardens as # increases (examples: cable and pneumaticstructures), the stability condition is easily violated.

Remark 19.6. The simplest cure to numerical instability is to recompute the reference stiffness. Another (asyet unexplored) possibility is to correct Kre f with a rank-one matrix.

§19.5. Accuracy Control

Accuracy control may be effected as in the case of conventional incremental methods if one sub-stitutes v f for v.

§19.6. Secant Estimation of n #

The finite difference estimators for n# described in §17.3 are easy to implement but decidedly sufferfrom a lack of robustness unless the problem is only mildly nonlinear. An alternative estimateof n # can be obtained through the following “secant approximation” technique. This estimate iscomputationally slower but more reliable. Recall that

n = p"Kre f (u" ure f ). (19.24)

n# = q"Kre f u# = q"Kre f (K"1q) = (I"Kre fK"1)q. (19.25)We now replace the exact inverse of the tangent matrix by a secant approximation:

K"1 $ Fs . (19.26)

Fs is a low-rank correction (typically rank one or two) of Fre f = K"1re f that is constructed on the

basis of the following increments:

"us = "un"1 = un " un"1, "rs = r(un, #n) " r(un"1, #n), (19.27)

19–8

19–9 §19.7 GENERAL INCREMENT CONTROL

Note that the residual!rs is computed by holding " constant and equal to "n and is not rn ! rn!1.Two choices for the inverse stiffness secant approximant are the Davidon-Fletcher-Powell (DFP)rank-two update formula

Fs = Fre f + !us!uTs!uTs !rs

! Fre f !rs!rTs Fre f!rTs Fre f !rs

, (19.28)

and Davidon’s rank-one update formula

Fs = Fre f + (!us ! Fre f !rs)(!us ! Fre f !rs)T

(!us ! Fre f !rs)T!rs. (19.29)

These formulas are collectively called Quasi-Newton updates in the numerical analysis literature,although to be more precise what we have shown above is just the first member of such updates.

§19.7. General Increment Control

The preceding developments assume that!"n is prescribed. But we can readily extend the pseudo-force technique to general increment control by following the procedures discussed in Chapter 18.For that we must replace vn by

v fn = K!1re f (qn ! nn). (19.30)

where the superposed dot denotes derivative respect to the pseudotime parameter t pertinent to theincrement strategy chosen. Of particular importance is arclength control, in which t becomes thearclength s. Criteria for stability and accuracy are readily converted to this case.

19–9

.

20Conventional

NewtonMethods

20–1

Chapter 20: CONVENTIONAL NEWTON METHODS 20–2

TABLE OF CONTENTS

Page§20.1. Introduction 20–3§20.2. Stage Analysis Review 20–3§20.3. Problem Statement 20–3§20.4. The Corrective Phase 20–4§20.5. Solving the Newton Systems 20–5§20.6. Termination Tests 20–5§20.7. The Ordinary Newton Method 20–6§20. Exercises . . . . . . . . . . . . . . . . . . . . . . 20–10

20–2

20–3 §20.3 PROBLEM STATEMENT


In the overview of solutionmethods given in Chapter 16 it was noted that solution methods based oncontinuation generally included two phases: incremental and corrective. In the purely incrementalmethods covered in Chapter 17–19 the corrective phase is absent. If the corrective phase is present,the incremental formula simply functions as a predictor that provides a starting point for thecorrective iteration. The purpose of this iteration is to eliminate (or at least reduce) the driftingerror by moving towards the equilibrium path along the constraint hypersurface.

Solution methods that include a corrective phase will be collectively called corrective methods,although perhaps a more appropriate name would be predictor-corrector methods. There are purelycorrective methods that lack a predictor phase entirely (for example, the orthogonal trajectoryaccession method) but they have not proven important in practical applications.

The most important class of corrective methods pertains to the Newton-Raphson method and itsnumerous variants: modified, modified-delayed, damped, quasi, and so forth. These are collectivelycalled Newton-like methods, and only require access to the past solution. In the present section westudy the conventional Newton method under general increment control.

§20.2. Stage Analysis Review

Let us recall that our purpose is to solve the residual equations

r(u, !) = 0 (20.1)

over a loading stage as the control parameter is incremented from 0. As previously discussed theadditional equation that makes (20.1) determinate is the increment constraint equation

c(u, !) = 0. (20.2)

Starting from ! = 0, we want to calculate a series of solutions

u0, !0, u1, !1, . . . un, !n . . . (20.3)

that characterizes numerically the response u = u(!) while satisfying the residual equations (20.1)within prescribed accuracy.

The purely incremental methods covered in the three previous Chapters compute a sequence ofvalues such as (20.3) by direct integration of the first-order rate equations Ku! = q or Kre f u! =q " f !. The methods considered here implement a corrective phase in which one iterates forequilibrium while satisfying the increment constraint. The starting point for the corrective phaseis the solution predicted by the incremental method. Consequently, these methods are often calledincremental-iterative methods.

20–3


§20.3. Problem Statement

Assume that n incremental steps of the stage analysis have been performed. The last acceptedsolution is

un, !n (20.4)

We want to compute the solutionun+1, !n+1 (20.5)

that satisfies the nonlinear algebraic system

r(un+1, !n+1) = 0,c("un, "!n) = 0,

(20.6)

where"un = un+1 ! un, "!n = !n+1 ! !n (20.7)

Although the above increment constraint is a special case of (20.2), it befits those most commonlyused in practical calculations.The predicted solution

u0n, !0n, (20.8)

is typically obtained by performing an incremental step as described in Chapters 17 and 18.

§20.4. The Corrective Phase

All that computations that follow pertain to the nth incremental step. Hence for simplicity we shallomit the subscript n from the formulas.Starting from the predicted approximation (20.8),

u0 " u0n, !0 " !0n (20.9)

the conventional Newton method applied to (20.6) generates a sequence of iterates

uk, !k, (20.10)

where k = 1, 2 . . . is an iteration step index.The conventional Newton method is based on the truncated Taylor expansion of the system r = 0,c = 0 about (uk, !k):

rk+1 = rk + #r#u

(uk+1 ! uk) + #r#!

(!k+1 ! !k) + H.O. = 0,

ck+1 = ck + #c#u

(uk+1 ! uk) + #c#!

(!k+1 ! !k) + H.O. = 0.(20.11)

where ‘H.O.’ denote higher order terms that are quadratic or higher in the changes uk+1 ! uk and!k+1 ! !k , and all derivatives are evaluated at (uk, !k). Discarding such terms and recalling thatK = #r/#u, q = !#r/#!, aT = #c/#u, g = #c/!, we obtain for the corrections

d = uk+1 ! uk, $ = !k+1 ! !k (20.12)

20–4

20–5 §20.6 TERMINATION TESTS

the linear algebraic system!

K !qaT g

" !

d!

"

= !!

rc

"

, (20.13)

where

K = "r"u

, q = ! "r"#

, aT = "c"u

, g = "c"#

, (20.14)

and all known quantities are evaluated at uk , #k . Note that for notational simplicity this superscripthas been kept out of d, !, r, etc, unless it is desirable to make the dependency on the iteration indexk explicit. If the tangent stiffness matrixK is of order N , the coefficient matrix of the linear system(20.13) has order N + 1. This matrix is called the augmented stiffness matrix.

Note that although generally K is symmetric and sparse, the augmented stiffness is generally un-symmetric (but see Exercise 20.2), and its sparseness is detrimentally affected by the augmentation.It is therefore of interest to treat the linear system (20.13) with techniques that preserve those at-tributes. The solution procedures described below make use of auxiliary systems of equations toachieve that goal. The number of auxiliary systems depends on whether the tangent stiffness Kis nonsingular (regular points) or singular (critical points). For the latter we have to distinguishbetween limit points and bifurcation points. In the present section we shall concentrate on thetreatment of regular points.

§20.5. Solving the Newton Systems

Recall from Chapter 4 that regular points of the system (20.1) are equilibrium solutions (u, #) atwhich the tangent stiffness matrixK is nonsingular. If this property holds, we can perform forwardGauss elimination on (20.13) to get rid of d and produce the following scalar equation for !:

(g + aTK!1q) ! = !c + aTK!1r. (20.15)

Let dr and dq denote the solution of the symmetric linear systems

Kdr = !r, Kdq = q. (20.16)

Then

! = ! c + aTdrg + aTdq

, d = dr + !dq . (20.17)

It is seen that two right hand sides, r and q, have to be generally solved for at each Newton step. Thenumber reduces to one for k > 1, however, if modified Newton is used so that K is held fixed forseveral steps and q does not vary. The last assumption holds in structural mechanics applicationsif the loading is conservative and proportional. (The modified Newton method is described in thenext Chapter).

20–5


§20.6. Termination Tests

At which point should we stop the Newton iteration? There are several convergence criteria thatcan be applied.

1. Displacement convergence test. The change in the last correction d of the state vector u, asmeasured in an appropriate norm, should not exceed a given tolerance !d . For example, usingthe 2-norm (Euclidean norm)

!d! =!

dTd " !d . (20.18)

2. Residual convergence test. Since the residual r measures the departure from equilibrium,another appropriate convergence test is

!r! " !r . (20.19)

Some comments are now in order.1. The two tests may be applied in an “and” or “or” matter as iteration stopping criterion. It is

also possible to combine both tests in the form of an “work change” criterion, for example

|rTd| " !d!r (20.20)

2. Since d and r have usually physical dimensions, so do necessarily !d and !r . For a generalpurpose implementation of Newton iteration this dependency on physical units is undesirableand it is more convenient to work out with ratios that render the !r and !d dimensionless. Forexample:

!r!!r0!

" !r (20.21)

where r0 is the residual after the predictor step; now !r can be dimensionless. A similar ratiocan be used for the displacement convergence test, but here the reference value should be atotal or accumulated displacement; for example:

!d!!u0!

" !d (20.22)

3. Divergence Safeguards. The Newton iteration is not guaranteed to converge. There shouldtherefore be divergence detection tests that will cause the iteration to be interrupted. For ex-ample, turning the above ratios around, divergence may be diagnosed if either of the followinginequalities occur:

!r!!r0!

# gr ,!d!!u0!

# gd (20.23)

where gr and gd are “dangerous growth” factors, for example gr = gd = 1000.Occasionally the Newton iteration will neither diverge not converge but just “bounce around”(oscillatory behavior). To avoid excessive wheel spinning in such cases it is always a goodpractice to put a maximum number of iterations per step in the program. Typical limits mightbe 20 to 50.

20–6

20–7 §20.7 THE ORDINARY NEWTON METHOD

§20.7. The Ordinary Newton Method

The Newton iteration discussed in the mathematical literature on solving nonlinear systems assumethat ! is held constant. This corresponds to ! control or load control in our terminology. Aspreviosuly explained, fixing ! makes critical points impassable. However, the resultant methodprovides good examples to watch the typical behavior of the Newton iteration process.If ! is kept constant the incremental step constraint is "!n = #n , which has derivatives 0 and 1with respect to u and !, respectively. System (20.13) simplifies to

!

K !q0 1

" !

d$

"

= !!

r0

"

, (20.24)

where c = 0 because the constraint is satisfied exactly. Since $ = 0, the “bordering” disappearsand the Newton iteration reduces to

uk+1 = uk ! (Kk)!1rk, !k+1 = !k = !n + #n (kept fixed) (20.25)

Note that the incremental load vector q disappeared entirely. This is the method found in standardnumerical analysis texts. This version is used in the examples that follow.

Example 20.1. The computation of the square root +"a of a scalar number a > 0 by Newton iteration is

set up as follows. The square root satisfies the equation r(x) = f (x) = x2 ! a = 0. Starting from an initialvalue x0 > 0, the Newton iteration computes

xk+1 = xk ! f (xk)/ f #(xk) = xk ! ((xk)2 ! a)/(2xk). (20.26)

where prime denotes derivative with respect to x . The results for a = 3 and x0 = 1 are illustrated by theMathematica program below.

f[x_,a_]:=x^2-a; Df[x_]:=2*x;a=3.; xk=1.; Print["x0=",xk];For[k=0, k<6, k++, xkp1=xk-f[xk,a]/Df[xk];

Print["x",k+1,"=",xkp1//InputForm]; xk=xkp1];

x0=1.x1=2.x2=1.75x3=1.732142857142857143x4=1.732050810014727541x5=1.732050807568877295x6=1.732050807568877294

After six cycles the iteration yields 16 places of accuracy for"3. Note that the number of exact digits roughly

doubles from k = 2 onwards. This is typical of the Newton iteration once it “locks in” a root because theprocess has asymptotically quadratic convergence.The numerical process shown above is actually that used by mathematical software libraries of languages likeFortran or C for the computation of the square root function. However the initial value is determined by ascaled rational interpolant that gives 2-4 digits of accuracy for x0; as a result only 2 or 3 cycles are needed toachieve double-precision accuracy for most inputs.

20–7


Example 20.2. This example is more typical of a structural application. Consider the residual equation for a2-DOF system

r(u, !) =! r1r2

"

=! u1 + 2u31 ! u22 ! 2!3u2 ! 2u1u2 ! !

"

(20.27)

The tangent stiffness matrix is

K = "r"u

=! 1+ 6u21 !2u2

!2u2 3! 2u1

"

(20.28)

For ! = 1 the residual equations have three real roots which to six digits of accuracy are

u1 =! 11

"

, u2 =! 1.184001.58227

"

, u3 =! 1.66726

!2.98938

"

. (20.29)

The following Mathematica programs starts from the initial values u01 = u02 = 0.8 and quickly finds thenearest root u1 = u2 = 1, delivering 16 digits of accuracy after 5 cycles:

ClearAll[r,u1,u2,lambda];lambda=1;r[u1_,u2_]:={{u1+2*u1^3-u2^2-2*lambda},{3*u2-2*u1*u2-lambda}};Kt[u1_,u2_]:={{1+6*u1^2,-2*u2}, {-2*u2,3-2*u1}}uk={{0.8},{0.8}};Print["Starting v0=",uk//InputForm];For [k=0, k<5, k++,

{u1,u2}={uk[[1,1]],uk[[2,1]]};ukp1 = uk - Inverse[Kt[u1,u2]].r[u1,u2];Print["Cycle k= ",k," u",k+1,"=",ukp1//InputForm];uk=ukp1 ];

{u1,u2}={uk[[1,1]],uk[[2,1]]};Print["Final residual=",r[u1,u2]];

Starting v0={{0.8}, {0.8}}Cycle k= 0 u1={{1.025426944971537002}, {0.9719165085388994309}}Cycle k= 1 u2={{1.001827210881738689}, {1.005246766090385063}}Cycle k= 2 u3={{0.999984431106495672}, {0.9999493398104184104}}Cycle k= 3 u4={{0.9999999985779087286}, {0.9999999955786530219}}Cycle k= 4 u5={{0.9999999999999999893}, {0.9999999999999999659}}

-18 -17Final residual={{-7.48099 10 }, {-1.25767 10 }}

But changing the initial values to u0 = 0.8 and u1 = 1.1, which is even closer to the u1 = u2 = 1 root, theprocess converges to the second root in (20.29), reaching 16 digits of accuracy after 8 cycles:

ClearAll[r,u1,u2,lambda];lambda=1;r[u1_,u2_]:={{u1+2*u1^3-u2^2-2*lambda},{3*u2-2*u1*u2-lambda}};Kt[u1_,u2_]:={{1+6*u1^2,-2*u2}, {-2*u2,3-2*u1}}uk={{0.8},{1.1}};Print["Starting u0=",uk//InputForm];For [k=0, k<5, k++,

20–8

20–9 §20.7 THE ORDINARY NEWTON METHOD

{u1,u2}={uk[[1,1]],uk[[2,1]]};ukp1 = uk - Inverse[Kt[u1,u2]].r[u1,u2];Print["Cycle k= ",k," u",k+1,"=",ukp1//InputForm];uk=ukp1 ];

{u1,u2}={uk[[1,1]],uk[[2,1]]};Print["Final residual=",r[u1,u2]];

Starting u0={{0.8}, {1.1}}Cycle k= 0 u1={{1.188636363636363636}, {1.325}}Cycle k= 1 u2={{1.215044904251438747}, {1.718220285975100653}}Cycle k= 2 u3={{1.189744133901914791}, {1.602104015495415557}}Cycle k= 3 u4={{1.184187697535010493}, {1.582880662857412937}}Cycle k= 4 u5={{1.183998614947570215}, {1.582271181104649434}}Cycle k= 5 u6={{1.183998417328768284}, {1.582270556284133866}}Cycle k= 6 u7={{1.183998417328558547}, {1.582270556283474425}}Cycle k= 7 u8={{1.183998417328558548}, {1.582270556283474426}}

-19Final residual={{0.}, {-8.67362 10 }}

This illustrates the “finicky” nature of Newton iteration. It can do (and often does) the unexpected, such asdiverging or converging to the “wrong” root. In fact the a whole subset of fractal or chaotic mathematicsis devoted to the understanding of “domains of attraction” of roots. Because of this capricious behavior, inpractical use of the Newton corrector numerous safeguards are implemented to avoid surprises. But the wholesubject is too lenghty for coverage in an introductory treatment.

20–9



Conventional Newton Methods

EXERCISE 20.1 (A:15) If the case of load control, the incremental stepsize constraint reduces to

c = !"n ! #n = 0 (E20.1)

where #n is prescribed. After the predictor step, c = 0. Show that the Newton method reduces to a standardform that requires only the solution of one of the auxiliary systems (20.16).

EXERCISE 20.2 (A:15) What algebraic constraint c = 0 would make the augmented tangent stiffness sym-metric?

EXERCISE 20.3 (C:25) Enlarge the program that solves Exercise 18.2 by using conventional Newton ascorrector, the incremental stepsize constraint

c = |!sn| ! #n = 0, (E20.2)

(where # = #n is keptfixed throughout) and the termination tolerance |rk/r0| " 10!3 or 20 iterations, whicheveroccurs first. Suggested values for #n for all n is 0.01, but the results should be insensitive to that choice.

EXERCISE 20.4 (A:30) (Advanced, paper level) Obtain the expression of the augmented stiffness if theorthogonal trajectory accession method is used and write out the Newton algorithm for this case.

20–10

.

21Newton-Like

Methods

21–1

Chapter 21: NEWTON-LIKE METHODS 21–2

TABLE OF CONTENTS

Page§21.1. Introduction 21–3§21.2. Newton Iteration as a Dynamical System 21–3

§21.2.1. Corrective Process for Fixed ! . . . . . . . . . . . . 21–4§21.2.2. Corrective Process for Varying ! . . . . . . . . . . 21–6

§21.3. Relaxed Newton Methods 21–6§21.4. Damped Newton Methods 21–6§21.5. Chord and Modified Newton Methods 21–7§21.6. Quasi-Newton Methods 21–8§21.7. *Convergence of Modified Newton 21–8§21. Exercises . . . . . . . . . . . . . . . . . . . . . . 21–10

21–2

21–3 §21.2 NEWTON ITERATION AS A DYNAMICAL SYSTEM


The conventional Newton method (CNM) described in Chapter 20 is hindered by two major short-comings:High cost. The tangent stiffness matrix Kk = K(uk, !k) has to be formed and factored at eachiteration step.Low Reliability. Convergence to the desired solution is not guaranteed unless the initial estimateis sufficiently close. The method may diverge, or converge to an unwanted solution. This is quitelikely in the vicinity of bifurcation points.Because of these shortcomings many variations of CNM, collectively called Newton-like methods,have been proposed and implemented over the past four decades, with varying degree of success.Among the most important are1. Relaxed Newton Methods (RNM): for reliability2. Damped Newton Methods (DNM): for reliability3. Modified Newton Methods (MNM): for efficiency4. Quasi-Newton Methods (QNM): for efficiencyAs can be seen by the large number of variations, nomodification can be said to be uniformly superiorto others. The above list covers the most important so-called Newton-like methods. Question arise,however, as to how far the offsprings can deviate from the parent and still be called Newton-like.Authors have different opinions in this matter. To further complicate things, problem-adaptivecombinations of these techniques are often used in advanced nonlinear solvers.Some of these variants, notably the Relaxed Newton methods (RNM), are more easily derivedby interpreting the Newton method in the context of a dynamical system. This interpretation isdiscussed next.

§21.2. Newton Iteration as a Dynamical System

Figure 21.1 sketches what the chief goal of the Newton method as corrector is: to allow largeincremental steps by eliminating the drift error.As discussed in previous Chapters, the incremental phase is driven by the first order rate form r = 0,in which the pseudo-time t is measured by an “increment clock.” Penalize the drift error by addinga term proportional to the residual r:

r+Wr = 0, (21.1)

in whichW is a positive-definite residual weighting matrix, which for the moment is left arbitrary.This is called a first order corrective form, and also a first order relaxation form. It obviouslyreduces to r = 0 on an equilibrium path r = 0. The job of the penalty term Wr is to force thesolution trajectories of (21.1) to approach r = 0 as the pseudotime t runs along a “corrective clock”See Figure 21.2.Figures 21.1 and 21.2 are a bit deceptive in that they depict corrective processes for a one-DOFproblem. A more realistic state of affairs can be observed in Figure 21.3, which depicts trajectoriesof a corrective process on a constraint surface for the case of two DOFs.

21–3


Correct

Equilibrium Path

Pred

ict

0

1

2 3

u

λ

Δu0

Δu1

Δλ0

Δλ1

Increment Control Constraint

R

Figure 21.1. Sketch of how an incremental-iterative solution method works.

To bring explicitly the stiffness matrix and incremental load vector into play, insert r = Ku ! q!into the above and transfer q! to the right hand side:

Ku+Wr = q!. (21.2)

Remark 21.1. A second order corrective form generalizes (21.1) by taking the second order differentialequation

r+W1r+W2 r = 0, (21.3)

in whichW1 andW2 are weighting matrices. These have different function: W1 provides damping whileW2is a “conditioner.” This more general form is not analyzed here.

§21.2.1. Corrective Process for Fixed !

Suppose that we are at {uk, !k} at which the tangent stiffness Kk is nonsingular. We want to moveto a new state {uk+1, !k} closer to r = 0 while keeping ! = !k fixed. This can be done by treating(21.2) with the Forward Euler integrator

uk+1 = uk + huk . (21.4)

where h is the integration steplength. The integrated corrector equation is

uk+1 = uk ! h FkWk rk, (21.5)

where F = K!1 = ("r/"u)!1 is a flexibility matrix. Calling d = uk+1 ! uk the correction indisplacements and passing K!1 to the right hand side,

Kk d = !hWk rk (21.6)

If now we takeW = I, h = 1 for any k, (21.7)

21–4

21–5 §21.2 NEWTON ITERATION AS A DYNAMICAL SYSTEM

0 u

λ

R

t=tt=t

corrective clock

incr

emen

tal c

lock

0

K

r = 0

Figure 21.2. Figure 21.2. Pseudo time t running along an “incremental clock”interspersed by a“corrective clock.” K is the total number of corrective iterations.

1u

2u

λ

r = 0

Incremental contraintsurface c = 0

Figure 21.3. The corrective process for two degrees of freedom. The challenge is toend at a solution no matter where one starts on the constraint surface.

we obtain the conventional Newton method (CNM) for fixed !, as can be easily verified. A variantcalled Relaxed Newton, discussed below, results by letting h be adjustable.

21–5


§21.2.2. Corrective Process for Varying !

Suppose next that ! is to be let vary while satisfying the scalar constraint c("u, "!) = 0 thatcontrols the increment size. The corrective equation can be generalized as

!

r+Wrc

"

=!

00

"

(21.8)

Inserting r = Ku! q! and c = aT u + g!, in which a = #c/#u and g = #c/#!, one gets!

Ku+WraT u+ g!

"

=!

q!0

"

(21.9)

Treat (21.9) with the Forward Euler integrator on both u and !:

uk+1 = uk + huk, !k+1 = !k + h!k (21.10)

Integrating (21.9) with (21.10), followed by setting W = I and h = 1, yields the conventionalNewton method for general increment control treated in the previous Chapter. The verification isthe matter of an exercise.

§21.3. Relaxed Newton Methods

One commonly used variant of CNM aims to increase the reliability but not necessarily lower thecost per iteration. This is done by deriving CNM from the dynamical process described in theprevious section, and letting the steplength h be a variable. The

!

K !qaT g

" !

d$

"

= !!

rc

"

, (21.11)

where supercript k is suppressed fromK, q, etc., to reduce clutter. Solving for d and $ as explainedin the previous Chapter, one then corrects

uk+1 = uk + hd, !k+1 = !k + h$. (21.12)

It is understood that h may change from integration to iteration, that is, h = hk .The method (21.11)-(21.12) is called the relaxed Newton-Raphson method, or RNR.1 There arethree possibilities as regards hk :1. If hk < 1, the iteration step is said to be underrelaxed and hk is an underrelaxation parameter.2. If hk > 1, the iteration step is said to be overrelaxed and hk is an overrelaxation parameter.3. If hk = 1 for all k, RNM reduces to CNM.How is the steplength h chosen? Rules to this effect are discussed in Chapter 22.

1 Some authors called this the damped Newton-Raphson method but that name is reserved here for the variant discussedin the next section.

21–6

21–7 §21.5 CHORD AND MODIFIED NEWTON METHODS

§21.4. Damped Newton Methods

The Relaxed Newton Methods provide gains in reliability as long as the stiffness matrix is notsingular or ill-conditioned. But it does not help in the vicinity of critical points. For example ifK is exactly singular, system (21.11) is not solvable by the double RHS method discussed in theprevious Chapter, and the variable steplength device does not help.Critical points may come in many flavors. In order of increasing traversal difficulty: isolated limitpoints, isolated bifurcation points, initially singular structures, and clustered limit and/or bifurcationpoints.For the less difficult cases, moving away slightly from the singularity often works. Much tougheris the case when stiffness matrix at the start of the analysis, or of an analysis stage, may be highlysingular. This happens, for instance, in some cable, pneumatic and biological structures that aremechanisms in the reference configuration and acquire stiffness as they deform. For such cases avariants collectively known as the Damped Newton Method or DNM, can be effective at the coistof programming complexity. [The name of Regularized Newton is also used.]DNM overcomes the singularity problem by adding a diagonal correction to the stiffness matrix.Instead of (21.10) one solves

!

K+ !D !qa g

" !

d"

"

=!

rc

"

, (21.13)

where D is a nonnegative diagonal matrix ! " 0 is a “numeric damping” coefficient, and ksupercripts have been omitted. The correction can then be applied with a steplength h:

uk+1 = uk + hd, #k+1 = #k + h". (21.14)

If the damping coefficient ! is zero and h is unity, the CNM results. As ! is increased the methodapproaches steepest descent ifD = I and scaled steepest descent for generalD. This has been usefulin conjunction with cable net structures that must traverse highly singular regions. Two choices forD tried in that case are

D = $I, $ = rTKrrT r

D = $DK , $ = rTKKrKrTK r

(21.15)

where DK = diagK and rK = D!1K r. Practical values for the damping coefficient ! may be

characterized as follows:

! " 1 very heavy damping1 " ! " 0.1 heavy damping0.1! " 0.01 moderate damping0.01! " 0.001 light damping

The best results for cable net structures were obtained with light damping. Once the structureacquires sufficient stiffness by deforming, the correction terms may be removed by setting ! = 0.

21–7


§21.5. Chord and Modified Newton Methods

The problem of high computational cost of CNR per step can be alleviated if the same stiffnessmatrix is maintained for several iteration steps. This general class of methods, collectively knownas chord methods is based on the iteration scheme

!

K !qaT g

" !

d!

"

= !!

rc

"

, (21.16)

uk+1 = uk + d, "k+1 = "k + ! (21.17)Here K and q denote an approximation to K and q in some sense, which is maintained fixed forseveral or all iteration steps. On the other hand r and c are changed at each iteration. Severalvariants result of this general scheme result according to two criteria: (a) How K and q are chosenand updated. (b) How a and g are chosen and updated.Two specializations of the chord method have proven effective in practice. If K = Kn , which isthe stiffness matrix at the start of the nth increment, which is kept fixed thereafter, the modifiedNewton method (MNM) method results. There is a variant called delayed modified Newton method(DMNM) for which K = K0, which is the stiffness matrix evaluated after the predictor step, andwhich again is kept fixed for all k.Updating versions of MNM and DMNM, identified by acronyms UMNM and UDMNM, respec-tively, emerge if K is allowed to vary during the iterative process. Several strategies to that effectcan be devised. Only three, ranging from the simplest to the most sophisticated, are mentionedhere:

1. Periodic update: RecomputeK everym " 1 iterations. The “period”m is chosen on the basisof prior experience, relative computational cost of factorization versus solving, etc. Obviouslym = 1 gives back the conventional Newton method.

2. Residual monitoring: If the residual norm ||rk || does not steadily decreases over a certain“subperiod” m#, K is recomputed. Typically m# = 3 or 4, which allows for “residual spikes”common asK is reset. This strategy is best combined with the previous one by choosing m asa multiple of m#.

3. Progressive update: This merges chord methods with a nonunitary steplength h.

§21.6. Quasi-Newton Methods

Quasi-Newton (QN) methods represent a refinement of the MNM methods. The stiffness K isupdated at each iteration step with rank-one or rank-two matrices built up from information fromthe previous iteration. In this way a better approximation of the actual stiffness matrix is obtainedwhile still avoiding revaluation and factorization.The idea comes from the field of optimization, where QN methods (also called variable metricmethods) have enjoyed great success. They were proposed for solving nonlinear structural prob-lems in the late 1970with high hopes. Evidence shows, however, that themoderate improvements inreducing the number of iterations to convergence does not compensate for the increase in program-ming complexity and storage. The idea has some uses, however, in the derivation of acceleratorand secant formulas presented in Chapter 22.

21–8

21–9 §21.7 *CONVERGENCE OF MODIFIED NEWTON

§21.7. *Convergence of Modified Newton

(ASEN 5107 students pls ignore this advanced material. It is placed here for eventual development)Assuming for simplicity that ! is kept fixed, then the limit relaxation equation becomes

Ku = !K(x) (21.18)

where x = u! u(") is the distance to the equilibrium solution at t = ". This can be modally decomposedas

y = !µy (21.19)

in which µ are the roots of the symmetric eigenproblem K0z = Kz and y are modal amplitudes. Theappropriate eigenvalue for the Newton direction u can be estimated by the Rayleigh quotient

µ = uTKuuTKu

(21.20)

The structural behavior can be characterized as follows.1. If µ < 1, the structure is softening in the mode y;2. If µ > 1, the structure is hardening in the mode y.For the MNM to converge,

|"| = |1! hµ| < 1 (21.21)

If the structure softens, MNM converges but the converges rate deteriorates unless h is increased (overrelax-ation). If the structure hardens, MNM diverges unless h is cut (underelaxation). But the structure hardens insome modes while softening in others, MNM cannot be continued, and a refactoring of the stiffness matrix iscalled for.The preceding observations are well know to experienced investigators. They have observed that MNMworksquite well in problems when the structure experiences overall softening.

Remark 21.2. To reduce the variation of µ, one may reduce the incremental step, or proceed to reform thestiffness matrix. In some programs the strategy control attempts to cut the incremental steplengths; after twoor three unsuccesful attempts the stiffness is reformed.

21–9



Newton-Like Methods

EXERCISE 21.1 [A:10] (Very easy, just to get acquanted with the relaxation equation). Verify that (21.4) and(21.5) followed byW = I and h = 1 lead to the Conventional Newton Method (20.25) for load control.

EXERCISE 21.2 [A:20] Starting from (21.9) and (21.10) derive the general form of the Relaxed Newtonmethod. Verify that if W = I and h = 1 this reduces to the Conventional Newton system for generalincrement control defined by (20.12)-(20.14).

EXERCISE 21.3 [A:20] Find out whichW leads to the Damped Newton system (21.13).

21–10

.

22Accelerators

and Line Search

22–1

Section 22: ACCELERATORS AND LINE SEARCH 22–2

TABLE OF CONTENTS

Page§22.1. Introduction 22–3§22.2. General Comments 22–3§22.3. Accelerator Models 22–3

§22.3.1. Tangent Accelerators for Fixed ! . . . . . . . . . . . 22–3§22.4. References 22–10

22–2

22–3 §22.3 ACCELERATOR MODELS


Newton-like methods were initially used in geometrically nonlinear analysis with a unit h. This isoptimal for conventional Newton (CNM) near a solution, but not necessarily so otherwise. Sincea key reason for letting h vary is to accelerate convergence to equilibrium, schemes that utilizenonunitary stepsizes are called accelerators. Another application is diagnosing and overcomingdivergence in “difficult” regions of the response. The latter application is tied upwith the line searchtechnique that comes from the field of optimization. In the present Chapter we study accelerationformulas and line search schemes based on the relaxation equations.

§22.2. General Comments

The device of letting h be a free parameter has proven useful in the following situations.

1. Oscillatory Response. Successive iterates “bounce” around so that the residuals at successivesteps are almost equal and opposite. This behavior is quite common when the structure“stiffens” globally. The cure is to go half-way, which means h ! 0.5. Taking h < 1 to killoscillations is sometimes called underrelaxation.

2. SlowMonotonicConvergence. Successive residuals “point” the sameway, but hardly decrease.This behavior is quite common when the structure “softens” globally, as in extensive creep orplasticity. The remedy is to “overrelax” with h > 1.

3. Erratic Behavior. The response of many nonlinear structures exhibits “difficult regions”where sucessive residuals do not show significant correlation, and divergence is likely. If thenonlinearities are smooth, this behavior is observed as one approaches bifurcation points orsharp limit points. It is especially pronounced when several such points are clustered, as oftenhappens in optimized structures. In hard nonlinear problems, such as contact and cracking,this behavior emanates from local effects. The recommended cure is cautious line search. Ifthe search delivers a very small h, say h < 0.1, the program should recompute the stiffnessmatrix.

The determination of h at each iteration step can be used for monitoring these situations. Forexample, if h ! 0.5 for two or three consecutive steps, chances are that the iterates are oscillating.Note that it does not necessarily follow that such h is in fact applied. For example, the author hassucessfully used the rule that if the predicted h is in the range 0.7 to 1.7, a value of h = 1 is used.The reason for this strategy is that the application of a nonunitary h generally demands additionalresidual calculations.From the preceding considerations it follows that a “best” value of h need not be know precisely;e.g., there is little practical difference between h = 1.8 and h = 2.0. This low accuracy requirementallows wide latitude in simplifying optimal-stepsize formulas so they are practical to use.

§22.3. Accelerator Models

Two models may be use to derive optimal stepsize formulas. At an equilibrium solution, r = 0 and(if not a critical point) u = 0. Accelerators based on these two models are studied in the followingsubsections.

22–3


§22.3.1. Tangent Accelerators for Fixed !

We begin by studying accelerators based on the zero-velocity model. If the nonlinearities aresmooth, and an estimate of K or of Ku can be easily procured, a technique due to Park [3] may beused for estimating h. Take the fixed-! second-order relaxation equation in which the term Wr isneglected:

Ku+ (K+WK)u = 0. (22.1)Treat this equation with the forward Euler integrator on u:

u(k+1) ! u(k) = hu(k), (22.2)and require that h be such that

u(k+1) " 0, (22.3)whence

(K(k) +W(k)K(k)) u(k)h " K(k)u(k). (22.4)For fixed ! we have r = Ku, which can be rewritten as

(K(k)u(k) +W(k)r(k)) h " r(k). (22.5)This is a highly undetermined system for h. Three main avenues of research open up here, leadingto a scalar stepsize, a full-matrix stepsize and a diagonal-matrix stepsize, respectively. Only thefirst case is treated in some detail; the others are relegated to Remarks below.

Scalar Stepsize. The straightforward way to solve (22.5) is to premultiply both sides by a “weight-ing” vector w. Park [3] proposes to use w = u, a choice that makes h a Rayleigh quotient withrespect to the direction u; the underlying idea being to anihilate the dominant error term. The resultis

h = u(k) T r(k)

u(k) T K(k)u(k) + u(k) T W(k) r(k)(22.6)

This formula can now be specialized to various Newton-like methods by inserting the appropriateW.

Remark 22.1. Other weighting vectors may be chosen. For example, "u, r or r. The latter choice yieldsaccelerators of “minimal residual form”, which are studied later as part of the r " 0 model.

Conventional Newton. IfW = I, (22.6) specializes to

hCNM = 11+ #

, where # = u(k) T K(k)u(k)

u(k) T r(k). (22.7)

If K is constant (linear system), then # = 0 and h = 1, which of course gives the correct answer.If the structure is stiffening (softening) in the u direction, then h is less (greater) than 1, which isthe right trend. Numerical experiments with this accelerator suggest that the “averaged” stepsize

hHalley = 12 (1+ 1

1+ #). (22.8)

gives marginally better performance; for a scalar equation the use of in fact yields Halley’s thirdorder iteration function; see e.g. Traub [18, p. 232].

22–4


Remark 22.2. For the quadratic scalar equation r = u2 = 0, u(0) != 0, we have r = 2uu, K = 2u, ! = "0.5and hCNM = 2, which yields the exact solution u = 0 in one step. Similarly, for the equation r = um = 0 weget hCNM = m and again we arrive at the solution in one step.

Remark 22.3. At noncritical points the performance of hCNM is not so dramatic as above; in fact it tends to“overshoot” the solution by the same amount as h = 1 tends to undershoot it. This is the rationale behind theHalley accelerator.

Modified Newton . Insertion ofW = KF into yields

hMNM = u(k) T r(k)

u(k) T K(k)u(k) + r(k) T F r(k)(22.9)

If the term involving K is neglected,

h#MNM = u(k) T r(k)

r(k) T F r(k)(22.10)

which is more easily calculable given the information available in a MNM iteration. (A similarsimplification in (80) would be too drastic, for it would reduce it to hCNM = 1; but Ku is moreeasily calculable in CNM.) Accelerators for the damped Newton method (73) can be constructedin a similar manner.Formulas such as (79) and (82) are called tangent accelerators as they depend only on the state atpseudotime tk and t-derivatives there. But the presence of rates such as K and r causes computationaldifficulties because the rates can be readily calculated only in simple problems. The problem canbe overcome by using secant information as discussed later.

Remark 22.4. Stepsize Matrix. A second approach to solving (78) is to transmute h into a “stepsize matrix”H given by

H = (K+WK)"1K. (22.11)

This may be inserted to produce the iteration family

u(k+1) = u(k) " (K(k) +W(k)K(k))"1W(k) r(k), (22.12)

into which we may replace K by the rank-two estimate derived from Quasi-Newton formulas in Felippa [19].This approach is presently unexplored.

22–5


Remark 22.5. Diagonalised Stepsize Matrix. Yet a third approach that merits attention is to transmute h intoa diagonal matrix, HD . This effectively assigns stepsizes component by component, which is in line with theempirical “alpha method” of Nayak and Basu [20]. To get the stepsize hi associated with the i th componentof u, premultiply both sides by eTi , which is a row vector with unit i th component and zero otherwise. Theresult is

hi =e(k)i

T r(k)

e(k)iT K(k)u(k) + e(k)i

T W(k) r(k)(22.13)

AppropiateW may now be inserted and rates replaced by secant relations.

Tangent Accelerators for Variable !

Let now ! vary during the corrective process. Take the following second-order relaxation equationwhich results from neglecting the terms Wr and q!:

Ku+ (K+WK)u = q! +Wq!. (22.14)

Treat this equation by the integrators

u(k+1) ! u(k) = hu(k), !(k+1) ! !(k) = h!(k), (22.15)

and again determine h fromu(k+1) " 0, !(k+1) " 0. (22.16)

Upon replacing Ku ! q! by r in the resulting equation, we obtain again the previous result.Therefore, all previous formulas remain valid for the variable-! case provided that the assumptionsmade above hold. Note that the result is independent of the constraint condition assumed forincrement control.

Secant Accelerators

The presence of rates such as K and r in the tangent accelerators gives rise to computationaldifficulties in complex nonlinear problems. More practical is to use secant-type (finite difference)information to obtain secant accelerators. The idea is to formally replace

K # "K(k) = K(k+1) !K(k),

u # d(k) = u(k+1) ! u(k) = "u(k+1) ! "u(k),

r # g(k) = r(k+1) ! r(k),(22.17)

in the tangent accelerators. This formal procedure, however, has the disadvantage that"K is not onlyunwieldy to calculate in CNM but unavailable in MNM. Fortunately in the tangent accelerators Kappears always in the combination Ku; consequently"K is always postmultiplied by d in its secantcounterparts. A convenient expression for"Kd can be obtained from the Quasi-Newton formulas,which are multidimensional generalizations of the scalar secant method. Pertinent expressions forthe Broyden family [21] are worked out in Fletcher [19], from where one gets

Ku# "K(k) d(k) = g(k) !K(k)d(k). (22.18)

22–6


Substituting the secant expressions yields

h = g(k) T d(k)

g(k) T W(k) d(k) + d(k) T (g(k) !K(k)d(k))(22.19)

On replacing the appropriateW, secant counterparts of hCNM, hMNM and h"MNM are obtained as

hCNM = 11+ !

, ! = d(k) T K(k)d(k)

g(k) T d(k) , (22.20)

hMNM = g(k) T d(k)

g(k) T Fg(k) + d(k) T (g(k) !K(k)d(k)). (22.21)

h"MNM = g(k) T d(k)

g(k) T Fg(k), (22.22)

Remark 22.6. The important equation (22.18) is not sensitive to the choice of Quasi-Newton formula. In factone always has

K(k+1)d(k) = (K(k) + "K(k))d(k) = g(k), (22.23)

which is called the Quasi-Newton condition and serves as a point of departure for defining such formulas.

A Secant Accelerator Algorithm

For the computer implementation it is important to note that secant accelerators involve u(k+1) andr(k+1). Since these are not known until the stepsize h has been selected, the process is inherentlyiterative. The basic structure of a model algorithm follows.

1. Select an initial h equal to the final stepsize used in the previous iteration step; if the first step,set h = 1.

2. Compute u(k+1) and r(k+1) based on this h.

3. Calculate h. If this value differs substantially from the assumed one (say by more than±50%)repeat steps 2 and 3 but no more than a fixed number of times (typically 1).

4. If a h has been accepted, use to calculate u(k+1). Otherwise branch to a line search procedure.

Many stylistic variations are possible. For example, one might decide never to repeat steps 2–3.This is the same as extrapolating h from the previous-step information, which is economical butpotentially dangerous.

Minimum Residual Accelerator

Accelerators may also be derived from the r # 0 model, in which the residual is viewed as afunction of the stepsize:

r(h) = r(u(k) + c(k+1) + hs(k)) (22.24)

22–7


where s(k) is defined by (65). Note that for variable ! vector c(k+1) = v"!(k+1) depends indirectlyon h. The stepsize h is to be chosen so that a norm of r is minimized. Most commonly used is thescaled Euclidean norm:

r = 12r

T Dr, (22.25)

where D is a positive-definite scaling matrix (usually diagonal) A non-identity diagonal D is usefulwhen the residual components have different physical units. The choice of D ! F = K"1 (ifpositive definite) is also of some interest because the residual norm becomes an energy norm.The general approach to minimizing r(h) is through a line-search method as described in the nextsubsection. But if the dependence of r on h is mildly nonlinear, an approximate minimizer can beobtained by assuming a linear dependence of r on h:

r(h) # h r(0) + (1" h)r(1), (22.26)

where r(0) and r(1) denote the residual vectors evaluated at h = 0 and h = 1, respectively; if k > 0then r(0) should be available from the previous iteration. The minimizing condition #r/#h = 0yields

h = r(0)TDr(0) " r(1)TDr(0)r(0)TDr(0) " 2r(1)TDr(0) + r(1)TDr(1)

(22.27)

Inasmuch as the assumption (22.26) is only strictly valid for linear response behavior, it is recom-mended to check whether

r(h) < r(1) and r(h) < r(0) (22.28)

holds before accepting h. If this condition is not verified, a line search may be called for.

Remark 22.7. Unlike the accelerators based on the zero-velocity model, the formula (22.27) depends on thecorrective method only indirectly through r(1). This gives it an implementation edge over secant acceleratorsin the sense that the stepsize calculations are less method dependent. On the other hand, its performance isnot usually as good as that of secant accelerators.

Remark 22.8. If D is the identity matrix, $ is the residual magnitude ratio $ = $r(1)$/$r(0)$ =!

r(1)T r(1)/r(0)T r(0) and % is the angle subtended by r(1) and r(0), the above formula can be presented as

h = 1" $ cos%1" 2$ cos% + $ 2

(22.29)

which admits of a geometric interpretation studied in Felippa [22].

Remark 22.9. Is it worthwhile accepting a non-unitary h? If so, an extra residual evaluation at h wouldbe required. A detailed analysis by Felippa [22] indicates that h should be accepted only if it is near 0.5(underrelaxation) or near 2.0 (overrelaxation).

A Minimum Residual Accelerator Algorithm

The following model algorithm has some points in common with that described for a secant ac-celerator, but is more cautious. It is offered as an illustration of the kind of strategies that may beimplemented in a general-purpose nonlinear analysis program.

22–8


1. Select initial stepsize h = 1. If the first iteration, calculate r(0).

2. Evaluate the residual r(1) and calculate h from (100). If the first iteration, accept h = 1 butrecord h; else drop through.

3. Branch according to h: (a) if h is outside a “reasonable range” (for example, 0.4 to 2.5) fortwo consecutive steps, branch to a line search procedure. (b) if outside the range for this step,accept h = 1 if h > 2.5, else set h = 0.5, and proceed to next iteration. (c) if 0.75 ! h ! 1.7,accept h = 1 and proceed to next iteration. Else drop through.

4. Set h = h and evaluate r(h). If condition (101) is verified, accept this h and go to the nextiteration. Else branch to a line search procedure.

Line Search

A line search is a systematic procedure to find a stepsize h that approximately minimizes a residual-magnitude measure, which for definiteness shall be assumed to be (98). Vector s(k) is called thesearch direction.Line search procedures are highly developed in optimization work, where they are essential com-ponents of general-purpose function minimization programs: a line search is carried out at eachiteration. On the other hand, their role in nonlinear equation solving is secondary. Because linesearch is fairly expensive in terms of residual evaluations, it is invoked only when a solutionprocedure runs into severe difficulties.The acceptance tests stated below involve the directional derivative g = !r(h)/!h along s. If D isconstant,

g(h) = !r!h

= rTD!r!h

= rTDK(s(k) + v!"

!h) (22.30)

The last expression is computationally cumbersome because it involves the tangent stiffness matrix.More practical is to use finite differences to estimate the factor !r/!h.Line search algorithms “backtrack” h in a systematic fashion until an acceptance condition isverified. A widely used acceptance condition in optimization programs of the the early 1970s isthe Goldstein-Armijo (GA) rule which may be stated as

0 < "#1hg(h) ! r(0) " r(h) ! "#2hg(h), (22.31)

where #1 and #2 are scalars that satisfy 0 < #1 ! #2 < 1; typical values being #1 # 0.1 and#2 # 0.9. The upper and lower bounds in (104) insure that h is neither “too large” nor “too small”.In more modern optimization work [23], the GA rule is replaced by slope conditions of the form

|g(h)| #!

!

!

g(h) " g(h1)h " h1

!

!

!! "#g(0), (22.32)

r(0) " r(h) $ "$hg(0). (22.33)

where 0 ! # < 1 and 0 < $ ! 0.5, $ ! #, and 0 ! h1 < h. A value of # # 0.8, which gives a notvery restrictive line search, can be recommended.

A Line Search Algorithm

22–9


There are many variations on this theme and readers interested in additional details are referred tothe abundant literature on practical optimization methods. The following algorithm is meant onlyto illustrate the basic procedural steps.

1. Estimate g(0) by finite differences, for example

g(0) ! r(0)TD(r(0.1) " r(0))

0.1(22.34)

If g(0) # 0 exit with h = 0, which should trigger a refactoring. Else set h1 = 0, h2 = 10(say), h = 1.

2. Evaluate r(h).

3. Test for (22.32). If not satisfied, compute h by restricted interpolation from data at h1 and h;set h2 = h, h = h, and return to 2.

4. Test for (22.31). If satisfied, exit. If not, compute h by restricted interpolation from data at h1and h; reset h1 = h, h = h and go to 2.

If the exit h < 0.1 (say), the iteration step should be abandoned and the stiffness matrix refactoredat h = 0.As stressed, a line search procedure should be initiated only when erratic or divergent behavior issuspected. This being the case, the initial steps of the search procedure should take advantage ofpreviously computed information such as r(0) and r(1). Similarly (22.27), suitably safeguarded,can be utilized for the interpolation process in steps 3 and 4; its key advantage being that it doesnot require g derivatives.

§22.4. References

[22.1] OTTER, J. R. H., “Computations for Prestressed Concrete Reactor Pressure Vessels Using DynamicRelaxation,” Nucl. Struct. Engng., 1, 61–75, 1965.

[22.2] UNDERWOOD, P. G., “Dynamic Relaxation— A Review,” Ch. 5 in Computational Methodsfor Transient Dynamic Analysis, (eds. T. Belytschko and T. J. R. Hughes), North-Holland,Amsterdam, 1983.

[22.3] PARK, K. C., “A Family of Solution Algorithms for Nonlinear Structural Analysis Based on theRelaxation Equations,” Int. J. Num. Meth. Engng., 18, 1337–1347, 1982.

[22.4] RIKS, E., “The Application of Newton’s Method to the Problem of Elastic Stability,”, J. Appl.Mech., 39, 1060–1065, 1972.

[22.5] CRISFIELD, M. A., “Incremental/Iterative Solution Procedures for Nonlinear Structural Analy-sis,” in Numerical Methods for Nonlinear Problems, Vol. 1, (eds. C. Taylor, E. Hinton andD. R. J. Owen), Pineridge Press, Swansea, U. K., 261–290, 1980.

[22.6] CRISFIELD, M. A., “An Incremental-Iterative Algorithm that Handles Snap-Through,”Computer& Structures, 13, 55–62, 1981.

[22.7] KUBICEK,M. andHLAVACEK,V.,Numerical Solution of Nonlinear Boundary Value Prob-lems with Applications, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.

22–10

22–11 §22.4 REFERENCES

[22.8] BAKHVALOV, N. S., Numerical Methods, Mir Publishers, Moscow, 1975.

[22.9] ORTEGA, J. M. and RHEINBOLDT, W. C., Iterative Solution of Nonlinear Equations inSeveral Variables, Academic Press, New York, 1970.

[22.10] BOGGS, P. T., “The Solution of Nonlinear Systems of Equations by A-Stable Integration Tech-niques,” SIAM J. Num. Anal., 8, 767–785, 1971.

[22.11] BRANIN, F. H. and HOO, S. K., “A Method for Finding Multiple Extrema of a Function of nVariables,” inNumerical Methods for Nonlinear Optimization, (ed. F. A. Lootsma), AcademicPress, London, 1972.

[22.12] KELLER, H. B., “Global Homotopies and Newton Methods,” inRecent Advances in NumericalAnalysis, (eds. C. de Boor and G. H. Golub), Academic Press, New York, 1978.

[22.13] WACKER, Hj. (ed.), Continuation Methods, Academic Press, NY, 1978.

[22.14] ALLGOWER, E. L., “A Survey of Homotopy Methods for Smooth Mappings,” in NumericalSolution of Nonlinear Equations, (eds. E. L. Allgower et al.), Lecture Notes in Mathematics878, Springer-Verlag, Berlin, 1981.

[22.15] PADOVAN, J., “Self-Adaptive Predictor-Corrector Algorithm for Static Nonlinear Structural Anal-ysis,” Report NASA CR-165410 to Lewis Research Center, The University of Akron, Akron, Ohio,1981.

[22.16] BATHE, K. J. and DVORKIN, E., “On the Automatic Solution of Nonlinear Finite Element Equa-tions,” Computer & Structures, 17, 871–879, 1983.

[22.17] FELIPPA, C. A., “Finite Element Analysis of Three-Dimensional Cable Structures,” in Compu-tational Methods in Nonlinear Mechanics, (ed. J. T. Oden et al.), The Texas Institute forComputational Mechanics, University of Texas, Austin, Texas, 311–324, 1974.

[22.18] TRAUB, J. F., Iterative Methods for the Solution of Equations, Prentice-Hall, EnglewoodCliffs, New Jersey, 1964.

[22.19] FELIPPA, C. A., “Dynamic Relaxation and Quasi-Newton Methods,” in Numerical Methods forNonlinear Problems 2, (eds C. Taylor, E. Hinton, D. R. J. Owen & E. Onate), Pineridge Press,Swansea, U. K., 27–38, 1984.

[22.20] BASU, A. K., “New Light on the Nayak Alpha Technique,” Int. J. Num. Meth. Engng., 6,152–153, 1973.

[22.21] BROYDEN, C. G., “Quasi-Newton Methods and their Application to Function Minimization,”Maths. Comput., 21, 368–381, 1967.

[22.22] FELIPPA, C. A., “Procedures for Computer Analysis of Large Nonlinear Structural Systems,” inLarge Engineering Systems, (ed. A. Wexler), Pergamon Press, Oxford, 60–101, 1977.

[22.23] GILL, P. E., MURRAY, W. and WRIGHT, M. H., Practical Optimization, Academic Press,London, 1981.

22–11

.

23Detecting

and TraversingCritical Points

23–1

Chapter 23: DETECTING AND TRAVERSING CRITICAL POINTS 23–2

TABLE OF CONTENTS

Page§23.1. Introduction 23–3§23.2. Direct Methods and Test Functions 23–3

§23.2.1. Determinant as CPTF . . . . . . . . . . . . . . . 23–4§23.2.2. Lowest Eigenvalue as CPTF . . . . . . . . . . . . 23–4§23.2.3. Solving the Minimum-Eigenvalue CPTF System . . . . . . 23–5

§23.3. Indirect Methods 23–5§23.3.1. Refined Determination of a Critical Point . . . . . . . . 23–6

§23.4. Traversing Critical Points: Difficulties 23–7§23.5. Newton-Raphson near Critical Points 23–9

§23.5.1. Injecting Perturbations . . . . . . . . . . . . . . 23–9§23.5.2. Penalty Spring Stabilization . . . . . . . . . . . . 23–9§23.5.3. Thurston’s Equivalence Transformation . . . . . . . . . 23–10

§23.6. Crivelli’s Procedure 23–11§23.7. Predictor Stabilization 23–13§23.8. Critical-Point Test Functions 23–13§23.9. Branch switching 23–15

§23.9.1. Tangent Predictors . . . . . . . . . . . . . . . 23–15§23.9.2. Buckling Mode Injection . . . . . . . . . . . . . 23–17

§23.10.Correctors 23–18§23.11.Treating Bifurcation by Perturbation 23–20§23. Exercises . . . . . . . . . . . . . . . . . . . . . . 23–21

23–2

23–3 §23.2 DIRECT METHODS AND TEST FUNCTIONS


Critical points were defined and mathematically characterized in Chapter 5. Chapter 18 explainshow solution procedures of purely incremental type can be adjusted to traverse limit points, anddifficulties in traversing bifurcation points are noted. Chapters 24 and follwing discuss criticalpoints in the context of stability analysis.The present Chapter dealswith the determination or detection of critical points in statically nonlinearanalysis and with procedures for traversal of such points with incremental/corrective techniques.Emphasis is given to handling isolated bifurcation points, which offer moderate degree of difficulty:tougher than limit points, easier than multiple bifurcation. Much of the material that follows istaken from Chapter 5 of Crivelli’s thesis.1

There are two basic approaches to the problem of detecting critical points. We may attempt to findcritical points by solving an algebraic equation which has those point as a root, or we can try todetect critical points as we are marching along an equilibrium path.The first approach embodies what are collectively called direct methods. As the name implies,these methods look for critical points without being concerned, at least theoretically, with tracingequilibrium paths up to those points.The second approach embodies the so-called indirect methods. In these methods the detection of acritical point is intimately related to the continuation procedure.

Remark 23.1. The simplest example of a directmethod is the Linearized PreBuckling (LPB) analysis discussedin Chapters 24 and 25. This procedure, when applicable, sets up directly an eigenproblem in the referenceconfiguration and thus avoids tracing the nonlinear response.

§23.2. Direct Methods and Test Functions

A direct method for calculating bifurcation points consists of formulating a suitable set of equationswhich has the critical points as solutions. In the context of general nonlinear analysis (that is, whenno a priori simplifying assumptions are made as in the LPB analysis) this set of equations shouldinclude the equilibrium equation because only critical points on the equilibrium path are of interest.It follows that direct formulations are achieved by augmenting the residual force equilibrium equa-tion with a set of constraints that characterize the critical point. The main characteristic of thisprocedure is that it does not require, at least theoretically, equilibrium path tracing. Thus, ideally itshould be possible to obtain all the critical points first and then join them with equilibrium paths.Except for certain specialized situations this is not possible, however, because the solution of thenonlinear equations resulting from this approach requires a good initial guess. Since these equationsare nonlinear, they cannot be solved directly, and linearization coupled with an iterative procedureis required. This linearization accounts for most of the cost of direct methods.As mentioned above, the procedure consists of augmenting the original residual force equation bya set of constraints that define a critical point. This characterization of critical points is intimately

1 L. A. Crivelli, A Total-Lagrangian Beam Element for the Analysis of Nonlinear Space Structures, Ph. D. Dissertation,Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO, April 1991.

23–3


related to the idea of constructing a critical point test function or CPTF. By definition, a CPTF is afunction whose zeros are the critical points of the response.2

§23.2.1. Determinant as CPTF

Since the tangent stiffness matrix is singular at critical points, the most obvious test function is thedeterminant of K. To be able to use this function, however, in the context of a Newton-Raphsoniterative procedure we need expressions for the partial derivatives of the determinant with respectto the state parameters. This approach is generally impractical for several reasons:

1. Analytical expressions for the determinant are hopelessly complicated for any but trivial prob-lems.

2. The estimation of these derivatives by one-sided finite differences would be enormously ex-pensive, requiring the assembly and factorization of N + 1 tangent stiffness factorizations forN degrees of freedom3

3. The detrminant is a notoriously ill-behaved function.

§23.2.2. Lowest Eigenvalue as CPTF

A critical point can also be characterized by checking for a null eigenvalue of K. Thus a prototypeof direct methods, as described in Seydel4 may be expressed generally as

! r(u, !)K(u, !)z"(z)

"

= 0. (23.1)

The last equation, "(z) = 0, has been added as a constraint on z to rule out the trivial solutionz = 0 and to make the system determinate. There are several possible choices for the normalizationequation, the simplest being to require z to have unit length: zT z = 1.To solve this equation iteratively, define the augmented incremental vector

y =!d#

h

"

(23.2)

where d and # are the corrective changes in u and !, respectively, defined in Chapter 17, and andh = zk+1 ! zk . is the corrective change in z. Assuming that K does not depend on ! and that thederivative of K with respect to u, denoted by Kv , is commutative with respect to its two outermostindices, the resulting Newton system is

! K q 0Kvz 0 K0 0 aT

"

y = !! rr!"

"

(23.3)

2 The name “branching test function” is sometimes restricted to test functions that catch only bifurcation points. Here weshall use the generic term “critical test function” to embody all critical points.

3 That of the tangent stiffnessK, plus N tangent stiffnesses corresponding to the perturbation of each state vector component.4 R. Seydel, From Equilibrium to Chaos, Elsevier, New York, 1988.

23–4

23–5 §23.3 INDIRECT METHODS

where r is the residual from the equilibrium equations, r! is the residual from the eigenvalue problemand " measures the violation of the normalization condition.This approach has been explored by Wriggers and Simo5 in the context of nonlinear structuralanalysis. As can be seen from the previous equation the Jacobian of this augmented system isneither sparse nor symmetric, and has large storage requirements. In addition, the coefficientmatrix becomes singular at bifurcation points, and hence ill-conditioned in their neighborhood.

§23.2.3. Solving the Minimum-Eigenvalue CPTF System

It is desirable to solve (21.3) while taking advantage of the symmetry of K. This can be done byusing the concept of auxiliary systems. Assuming K is nonsingular, consider the five auxiliarysystems:

Kdr = !r, Kdq = q, Kdh = !r!, (23.4)Kdhr = Kvzdr , Kdhq = Kvzdq . (23.5)

Then the solution to the system can be written as:

# = !" ! aTdhraTdhq

, d = dr + #dq , h = dh ! dhr + #dhq . (23.6)

The solution of these five systems at each corrective iteration is expensive. However, the maindisadvantage of this approach is having to compute the u-derivative of the stiffness matrix. Since anexact derivation ofKv is only practically feasible for simple nonlinear problems, some investigators(e.g. Seydel and Wriggers-Simo, loc. cit.) propose using numerical differentiation to compute Kv .In this context it should be observed that only the product Kvz (a square matrix) is required.Numerical experiments reported in Seydel’s book show that the convergence rate of this approachdepends on whether the critical point is a limit or a bifurcation point. At limit points, the Jacobian ofthe augmented equation is nonsingular, which guarantees fast local convergence. At a bifurcationpoint, the vector [ zT 0T 0 ]T is a left null eigenvector, which shows that the Jacobian becomessingular. Thus, slower convergence rates may be expected at bifurcation points. However, experi-ence shows that for some problems the convergence rate is good for the ! component when a notvery stringent accuracy is required, as can be seen from the examples presented in Wriggers andSimo.

Remark 23.2. If the critical point is known to be a limit point, the penalty spring augmentation methoddescribed below is much simpler and inexpensive. And even the simplistic random perturbation schemedescribed there often works. Thus it appears that the direct method is advantageous in special instances, forexamplewhen direct determination of the CP locations as stability envelope, is desirable, and a good estimationof that location is available. This occurs in some applications such as optimization with stability constraints.

5 P.Wriggers and J.C. Simo, AGeneral Procedure for the Direct Computation of Turning and Bifurcation Points, Int. J. Nu-mer. Meth. Engrg., 30, 155–176, 1990.

23–5


§23.3. Indirect Methods

The underlying idea behind indirect methods is to recognize the occurrence of a critical point whilefollowing an equilibrium path by continuation. The key to success is having a good CPTF ! . Wedefer the discussion on how to construct such a CPTF to §21.4, and for the moment assume thatsuch a function is available.These methods benefit from data collected from the continuation procedure. A critical point ischaracterized by being a zero of the CPTF. Because hitting an exact zero is highly unlikely, wecheck for a critical point by invoking the condition

! (uk+1, "k+1) ! (uk, "k) < 0 (23.7)

This is called a bracketing or straddling condition. It means that a critical point occurs in theinterval or bracket ["k ! " ! "k+1]. The next task is to “zero in” this point by reducing the bracketsize.

§23.3.1. Refined Determination of a Critical Point

Assume a critical point has been located by verifying bracketing condition (21.7) on the CPTF. Further, assumethat the test function is a continuous function of " and has only one isolated zero in the interval. This conditionrequires using a sensible incremental step—which is not known a priori. However, the problemof determiningan appropriate step may be solved by a combination of intuition and an adaptive incremental control strategy,which monitors the convergence behavior of the solution.Once the algorithm finds that a critical point is bracketed or straddled by two equilibrium solutions, we mayobtain a first approximation to such point by using linear interpolation:

"c " "k + "k+1 # "k

! k # ! k+1! k (23.8)

We present below an adaptive method developed in Crivelli’s thesis (loc. cit. to improve the initial approxi-mation (21.8). It makes use of iterative inverse interpolation and divided differences.6 We have computed aset of m values of the CPTF at m unequally spaced values of the control parameter ". We seek a polynomialinterpolation for ! of the form

! (") = c0 + c1("# "0) + · · · + cm("# "0) . . . ("# "m#1) (23.9)

On defining the sequence of divided differences

! ["0; "] = ! (") # ! ("0)

"# "0

......

! ["0, . . . , "k#1; "] = ! ["0, . . . , "k#2; "]# ! ["0, . . . , "k#2; "k#1]"# "k#1

(23.10)

6 For background in such techniques, see for example G. Dahlquist and A. Bjorck, Numerical Methods, Prentice-Hall,Englewood Cliffs, 1974.

23–6

23–7 §23.4 TRAVERSING CRITICAL POINTS: DIFFICULTIES

the coefficients in equation (8) can be computed as

c0 = ! ("0)

c1 = ! ["0; "1]...

...

ck = ! ["0, . . . , "k!1; "k]

(23.11)

Observe that this formula is recursive and a new point can be added without great difficulty. Now the problemof locating a critical point is reduced to finding a zero of (21.9).

Remark 23.3. Observe that we may as well use equation to interpolate " as a function of ! and obtain "c justby setting ! = 0. However, both procedures are fundamentally different and may give different results. Forinstance, for very sharp limit points and for bifurcation points the test function could stay almost constant formost of the continuation and change to a steep slope while very close to the limit point. This type of behaviorcan be regarded as an exponential relation between ! and ". Such an exponential can be well approximatedby a polynomial, whereas its inverse—a logarithmic function— may pose severe difficulties. Thus we mayexpect faster convergence from the former procedure.

Based on the foregoing remark we choose inverse interpolation for finding a zero of (21.9). Forthis we recast that equation into the form

"c = #("c) (23.12)

or, more explicitly

"c = "0 ! 1c1[c0 + · · · + cm ("c ! "0) . . . ("c ! "m!1)] (23.13)

The equation is then solved by iteration, with "(0)c equal to the approximated "c obtained form the

last interpolation, starting with the linear interpolation given by (21.8). Convergence is usuallyfast because the error is inversely proportional to the first CPTF derivative, ! " = $!/d", which isexpected to be large near isolated critical points. Furthermore, should ! " be small, we can expect agood prediction from the linear interpolation formula.The main advantages of the foregoing procedure is that it is self adaptive and easily programmable.We can increment the number of points used in the interpolation equation with little extra effort.We can keep the number of points under a given maximum by discarding the outermost points whenwe add a new one. Thus the procedure is flexible and opens up several implementation alternatives.Furthermore, the storage requirements are kept modest and we can expect good convergence toreasonable accuracy.After a new approximation "c has been established, we have to compute the corresponding solutionuc. However, some care has to be exercised since the Jacobian matrix can be very ill-conditioned inthe critical point neighborhood. There are several alternatives to circumventing this ill-conditioningproblem, which merge with the general topic of traversing a critical point. This subject is taken upin the following subsections.

23–7


§23.4. Traversing Critical Points: Difficulties

In the remainder of this sectionwe consider the problemof traversing critical points using incremen-tal/corrective methods. Corrective solution methods, of which Newton-Raphson is the prototype,may run into difficulties at or near critical points. Essentially those are caused by the fact that thetangent stiffness matrix becomes singular at such points and consequently ill-conditioned in theirneighborhood. This ill-conditioning can introduce noise which may render the solution procedureunstable.Sometimes the problems associated with noise are easily circumvented, whereas in some casesthe problems may become computationally overwhelming and may require either specialized tech-niques or intensive human intervention to proceed. Generally speaking, the degree of difficulty isinfluenced by the following factors:1. Bifurcation points are more difficult to handle than limit points.2. Isolated critical points (at which the rank deficiency is one) are easier to handle than multiple

or compound critical points.3. Dynamic critical points of flutter type are more difficult to handle than static ones. Ultimate

in nastiness (sort of like the triple jump in ice skating) are problems of fracture or localiza-tion where an equililibrium path suddenly terminates, and dynamic methods are required forproceeding further.

In what follows we focus on the characteristics of Newton’s method near an isolated critical point ofstatic type. Even under those relatively benign conditions the topic is still an area of active research.

23–8

23–9 §23.5 NEWTON-RAPHSON NEAR CRITICAL POINTS

§23.5. Newton-Raphson near Critical Points

The behavior of Newton-Raphson methods at critical points have received much attention from thenumerical analysis community.7 Convergence rates have been established and accelerators havebeen proposed in a general framework. Under certain conditions, it can be established that theconvergence of the solution component in the null space is linear while that in the range spaceconverges superlinearly. This property can be exploited in the limit point case.The purpose of a Newton-Raphson critical point accelerator, or NRCPA, is to recover quadraticrate of convergence. To achieve this goal on a general setting NRCPAs need information providedby second order (and ocassionally higher) rate equations. This information is very expensive toobtain because the rate K of the tangent stiffness matrix, which is a “cubic array” discussed furtherin Chapter 24, makes its appearance. Because of such expense, NRCPAs based on higher orderrate equations are rarely used in nonlinear structural mechanics.The procedures described below represent more practical approaches to the traversal problem. Theyattempt to live with the first-order rate information, such as provided byK and q, which is normallyavailable in finite element programs.

§23.5.1. Injecting Perturbations

By far the simplest technique to attempt critical point traversal is the equivalent of the “Hail Mary”football play. If at a certain step of an incremental/iterative process the equation solver returns a“singular stiffness” diagnostic, perturb the state u by a minute random amount and try again. Thismay be repeated up to a certain number of “downs” (usually 2 or 3).This simplistic method sometimes works surprisingly well for crossing “nice” limit points in con-junction with the arclength increment control method. But at sharp limit points, bifurcation pointsor regions where the tangent stiffness has high rank deficiency this technique often fails.

§23.5.2. Penalty Spring Stabilization

As emphasized several times before, the essential difficulty lies in the ill-conditioning of K nearcritical points. To stabilize this matrix, Felippa8 has proposed to combine the solution of threelinear systems. The coefficient matrix is rendered nonsingular by adding a fictitious penalty springstiffness s to the i th equation:

(K+ sEi )dsr = !r, (K+ sEi )dsq = q, (K+ sEi )dse = sei (23.14)

7 See, for example, the publications:D. W. Decker, H. B. Keller and C. T. Kelly, Convergence Rates for Newton’s Method at Singular Points, Siam J. Nu-mer. Anal., 20, 296–314, 1983.C. T. Kelly and R. Suresh, ANewAccelerationMethod for Newton’sMethod at Singular Points, Siam J. Numer. Anal.,20, 1001–1009, 1983.G. Reddien, On Newton’s Method for Singular Problems, Siam J. Numer. Anal., 15, 993–996, 1978.W. Rheinboldt, Numerical Analysis of Continuation Methods for Nonlinear Structural Problems, Computer & Struc-

tures, 15, 1–11, 1978.8 C. A. Felippa, Traversing critical points by penalty springs, Contrib. C2/1, Proc. NUMETA’87 Conf., M. Nijhoff Pubs.,Dordrecht, 1987.

23–9


Here ei is the elementary vector of order N all of whose entries are zero except the i th entry whichis one, and Ei = eieTi . Some possible choices for the index i are discussed in that article (handedout in class). One advantage to this approach is that the symmetry and sparseness of K is notaffected by this diagonal correction. The solution of the original Newton-Raphson system (17.13),reproduced here for convenience:

!

K !qaT g

" !

d!

"

= !!

rc

"

, (23.15)

can be expressed as a linear combination of the solutions of the three auxiliary problems defined in(21.14):

d = dsr + "qdsq + "edse,! = !

#

c + aTd$

/g,. (23.16)

Here the coefficients "q and "e are obtained by requiring that d and ! solve the original problem.The following 2" 2 unsymmetric system results:

!

g + aTdsq aTdse!eTi dsq 1! eTi dse

" !

"q"e

"

=!

!c ! aTdsreTi d

sr

"

(23.17)

The computation of ! from this equation breaks down if g = 0. In such a case, it is recommendedthat ! be recovered from the equivalent equation:

! = "q + s"edi ! d2idTq

(23.18)

The value of s in equation is irrelevant as long as it is sufficiently large to stabilize K.An advantage of this procedure is that it avoids the need for partitioning the stiffnessmatrix proposedby Rheinboldt (loc. cit.), which requires special treatment of the elements in the i th row and columnof K. Furthermore, none of the right-hand sides of the auxiliary systems in requires access toelements of K.Disadvantages of this procedure are as follows:1. It breaks down at bifurcation points, where the coefficient matrix of the 2" 2 system becomes

singular.2. Some overhead is necessary to trace the rate of change of the components of the state vectors,

which is used as a criterion in the selection of the index i .3. The equation solver should be able to undo part of the reduction, from the current value of the

row index to the selected value of i .

§23.5.3. Thurston’s Equivalence Transformation

A different approach has been proposed by Thurston, Brogan and Stehlin. 9 Here an equivalencetransformation is effected on the stiffness matrix to obtain a partitioning such that the singularity

9 G. A. Thurston, F. A. Brogan and P. Stehlin, Postbuckling Analysis Using a General Purpose Code, AIAA Journal, 24,1013–1020, 1986.

23–10

23–11 §23.6 CRIVELLI’S PROCEDURE

is confined to a small block-submatrix. To achieve this transformation it is necessary to computethe eigenvectors corresponding to the smallest eigenvalues of K. These eigenvectors enter the firstcolumns of the transformation matrix, which is then completed with a permutation matrix in sucha way that renders the full matrix non-singular. The solution vector is then split accordingly andthe components corresponding to the permutation part of the transformation matrix are condensedand solved in terms of the modal variables. Although this procedure works adequately for almostsingular matrices, it fails for truly singular matrices where the modal variables remain undefined.

§23.6. Crivelli’s Procedure

We present here an alternative procedure developed in Crivelli’s thesis (cited in footnote 1) whichdoes not involve partitioning but requires access to elements ofK. Furthermore, it avoids the need toestimate penalty coefficients and the choice of the index i is straightforward. It also requires minormodifications to the solver routine. It can be implemented without even modifying the solver, byforcing the solver to return when it finds a diagonal term smaller than a given threshold, performingthe required transformations outside the solver and then having the solver continue from the point ofinterruption. In addition, the proposed procedure furnishes an approximation to the null eigenvectorand provides a measure of the distance to the critical point.Define the permutation matrix Pi :

Pi =

!

"

"

"

"

"

"

"

"

"

"

#

i!

1 . . . 0 0 . . . 0 0...

. . ....

.... . .

......

0 . . . 1 0 . . . 0 0i " 0 . . . 0 0 . . . 0 0

0 . . . 0 1 . . . 0 0...

. . ....

.... . .

......

0 . . . 0 0 . . . 1 0

$

%

%

%

%

%

%

%

%

%

%

&

(23.19)

Observe that

PiPTi =' Ii#1

0In#i

(

= In # eieTi , (23.20)

where N is the dimension of Pi and Ik is the k $ k identity matrix. Define

K = PTi KP+ eneTn , h = PTi Kei = PTi ki , (23.21)

and denote z to be the solution of

Kz = h, z = Pz# ei . (23.22)

The vector z so defined satisfies

Kz = kTi zei , ki = Pi KPTi z+ Kiiei . (23.23)

We show below that ifK is singular with rank deficiency of one, and the index i is chosen properly,then z is the null eigenvector of K. If K is singular, there is a subset of the columns of K that are

23–11


linearly dependent. Denote by S = {i1, . . . , ik} a set of indices such that S ! {1, . . . , n}; this set Sis a subset of the column indices of K. This set is chosen so that there is a linear combination ofcolumns of K satisfying

!

ik!S!ikkik = 0 (23.24)

where ki is the i th column of K. Such a set exists because K is singular. Choose an index i fromsuch that !i "= 0. Column ki can be then expressed as a linear combination of the other columnsof K. Apply now the procedure with the index i chosen as described. It is immediately seen thatkTi z = 0; therefore z is the null eigenvector of K.IfK is almost singular, then z is an approximation to the null eigenvector. It can be shown that ifKis continuously differentiable and K is nonsingular, then z approaches z as (u, ") approach (uc, "c).To solve the original problem, define the following auxiliary systems:

Kdr = PTi r, dr = Pi dr , Kdq = PTi q, dq = Pi dq (23.25)

Observe that dTr ei = dTq ei = 0. Define

d = dr + #qdq + #zz (23.26)

To obtain the coefficients #q and #e we use a similar procedure as before. We have

Kdr = PiPTi KPi dr + kTi drei = Pi Kdr + kTi drei= #(I# eieTi )r+ kTi drei = #r+ (kTi dr + ri )ei

(23.27)

andKdq = q+ (kTi dq # qi )ei (23.28)

Also observe thatkTi dr = dTr (Pi Kz+ Kiiei ) = d

Tr Kz

= #rTPi z = #rT (z+ ei ) = #rT z# ri(23.29)

ThuskTi dr + ri = #rT z (23.30)

SimilarlykTi dq # qi = #qT z (23.31)

From previous equations it is immediately seen that

Kd# q$ = #r+"

#q + c + aTdg

#

q+$

kTi z#z + qT z#q # rT z%

ei (23.32)

To recover the original system, we require the terms inside squared brackets to vanish, which leadsto the auxiliary system

"

g + aTdq aT zqT z kTi z

# "

#q#z

#

="

#c # aTdrrT z

#

(23.33)

23–12

23–13 §23.8 CRITICAL-POINT TEST FUNCTIONS

Finally, we have to recover !. Instead of using We can obtain ! directly by requiring

! = "q (23.34)

This approach has some advantages over the one pertaining to the penalty spring method. First, if"z = 0 it reduces to the one used by classical continuation. Second, it enforces prior equations.Closer inspection of tells us that "e = di if exact arithmetic is used. Thus, the second term vanishes.When finite precision arithmetic is used, rounding errors may prevent this term from vanishing.If in addition, a sufficiently large value for the penalty spring s is chosen, could produce someartificially large value of ! with the consequence that the corrector step may move the solutionaway from convergence.Several remarks may be made in regard to the preceding derivation. At a limit point, neither qT znor aT z vanish and the equations are well behaved even though kTi z is zero. Thus this procedure cansatisfactory traverse limit points. At a bifurcation point the last row of equation is identically null,thus the value of "z is not defined. This is consistent with the definition of bifurcation point, since atthese points more than one solution is possible and we cannot continue tracing any branch withoutintroducing additional information. Thuswhen a bifurcation point is detected, this procedure breaksdown and we have to resort to a branch switching algorithm as described in the following sections.

§23.7. Predictor Stabilization

A similar procedure can be employed to stabilize the predictor, a subject that has not received muchattention in the literature. If K is almost singular, the predictor may be poor. Using dq defined inequation (11) we define the predictor step #u(0) as

#u(0) = dq#$(0) + "zz (23.35)

ThusK#u(0) ! q#$(0) =

!

#$(0)(kTi q! qi ) + "zkTi z"

ei (23.36)

Requiring the term in brackets to vanish and using equations (17) the load increment is obtained as:

#$(0) = kTi zqT z

"z (23.37)

Assuming we use the global hyperelliptic constraint we can obtain "q as

"z = %

±

#

$

$

$

%

a2

v2

&

'

(

kTi zqT z

)2

dTq Sdq + zTSz

*

++(

kTi zqT z

)2

b2

(23.38)

since by construction dTq z = 0.

Observe that at a limit point, kTi z = 0, qT z "= 0, equation (38) gives "z = %, #$(0) = 0, allowingthe solution to move away from the limit point. At a bifurcation point qT z = 0, and this equationdoes not give any solution for "z , as expected, since any solution with a nonzero value of "z will lieon the emanating branch. In this case, we simply set "z = 0 and use prior relations compute#$(0).

23–13


§23.8. Critical-Point Test Functions

In this section we investigate several candidates for CPTF. A CPTF is a function that allowsmonitoring the continuation procedure for occurrence of critical points. By definition, criticalpoints should be the only zeros of a CPTF. As mentioned before, critical points are characterizedby the tangent stiffness K being singular. Thus, test functions essentially monitor K to obtain anestimate of its proximity to a critical point.A highly accurate estimator of how singular is K is its smallest eigenvalue. Since K is real andsymmetric, its eigenvalues are all real; in addition, if K varies continuously, its eigenvalues alsovary continuously and singularity is simply detected by checking for a null eigenvalue. Thus, thesmallest eigenvalue ofKmakes a very attractive choice for a CPTF.However, an eigenvalue analysisat each increment is prohibitively expensive.Another possible test function is the determinant of K. As noticed by Abbott10 the order of mag-nitude of the determinant may be computationally inconvenient, especially for large stiff systems,when the solution is computed far away from bifurcation. ScalingK by 10 changes the determinantby 10N ! This difficulty can be overcome by computing the determinant as a pair of numbers, acharacteristic normalized between !1 and 1 and a mantissa. This decomposition is available inseveral direct solvers. A drawback of this approach occurs, however, if the null eigenvalue has aneven multiplicity, since in this case the determinant will not change sign and the tester will fail todetect a critical point.Two additional choices are considered in the following. Using previous equations the followingrelation results:

zTKzzT z

= kTi zzT z

(23.39)

If we regard the left-hand side of this equation as a Rayleigh quotient, and taking into account thatz is a close approximation to the smallest eigenvector ofKwhen the solution is sufficiently close tothe limit point, we may use the right-hand side of equation to monitor the smallest eigenvalue ofK.Thus, kiz is a candidate for a CPTF. A similar test function is proposed by Seydel (loc.cit.). Thisapproach is expensive if it is to be carried out at every increment, because the decomposition is onlycomputed near critical points. However, if by some other means we have detected the proximityof a critical point, this computation may be obtained as a by-product of the stabilization proceduredescribed in the previous Chapter. In such case, it becomes an attractive CPTF. We therefore lookfor a cheaper alternative that can provide a good estimate of the singularity of K.Since we use full Newton we generally have a decomposition of K of the form:

K = LTDL (23.40)

where L is an upper triangular matrix and D is diagonal. It is well known that

! = miniD(i) (23.41)

10 J. P. Abbott, An Efficient Algorithm for the Determination of Certain Bifurcation Points, J. Comput. Appl. Math., 4,19–27, 1978.

23–14

23–15 §23.9 BRANCH SWITCHING

also monitors the smallest eigenvalue of K. Moreover, ! = 0 when K is singular and ! < 0 ifK is indefinite. Thus, ! defined is readily available without any overhead and is the test functionadopted here. Once it has been determined that there is a critical point lying between two solutionswe may switch to the test function.

§23.9. Branch switching

In this Chapter we consider the computational treatment of bifurcation points. Such points arecharacterized by the fact that the augmented stiffness becomes singular there. The vector

!

zT 0"

is a left singular vector of the augmented stiffness, as can be seen by premultiplying the augmentedstiffness matrix by the vector

!

zT 0"

and taking into account that q is orthogonal to z. However,this relation shows that the problem is consistent and as such, a solution exits; more precisely, aninfinite set of solutions exits with any two of them differing by a vector lying in the null space ofK.The coefficient "z cannot be obtained because the second row of this equation vanishes identically.Thus, we cannot use here the same procedure used to traverse limit points. As previouslymentioned,at a bifurcation point more than one branch coexist. If "z is kept equal to zero, we may continuealong the current branch, while if another value of "z is chosen appropriately, we may switch to adifferent branch. It is seen that the problem is indeterminate and we need an additional conditionto specify on which branch of the solution the continuation will proceed. While continuing on theknown branch can be achieved fairly simply, switching to a crossing branch is not so obvious andrequires a more elaborate procedure.We will assume that K has a rank deficiency of one and the bifurcation point is isolated or simple.Under these conditions, there are only two intersecting branches at the bifurcation point. Theprocess of branch switching requires two steps. First we have to find some point sufficiently closeto the crossing branch. Second, we have to stay on that branch; this requires to avoid any iterationsending us back to the known branch. This Chapter will be concerned with the first step, calculatingone solution point on the emanating branch.Suppose we have detected a critical point, and that we have classified it as a bifurcation point. Thescenario is as follows: we have some solutions u(#) such that r(u, #) = 0. If the bifurcation pointhas been located by using the indirect method, wemay assume that we have one solution, say (u, #),that approximates the bifurcation point (uc, #c), and in addition we have a good estimate of thebuckling mode vector z. Denote a solution on the crossing branch by (w, #), i.e. r(w, #) = 0. Ourmain goal in this section is to find one such solution. This first solution is then used as a startingpoint to trace the entire branch.In general, any method for switching branches consists of two parts, first an approximation to apoint on the crossing branch is guessed by means of a predictor; then an iteration converging tothat branch must be established. In the following subsections we will describe different types ofpredictors.

§23.9.1. Tangent Predictors

Possibly themost accurate predictor is the one based on the tangent to the crossing branch. However,this procedure requires a rather accurate calculationof the bifurcationpoint (uc, #c) andof the secondderivatives of the residual vector, which is given in terms of the rate matrices K and q. Since we

23–15


Unknown branch

Known branch

Acceptableiteration

Unacceptableiteration

λPredictedsolution

known solutionC B

P

v

Figure 23.1. Schematic representation of a branch switching procedure.

have assumed that the bifurcation point is simple, there can be at most two intersecting branches.Since q is in the range of Kc, the system Kw = q is consistent at a bifurcation point. Hence, thereis a unique solution w lying in the range ofK(uc, !c) that is, satisfying the relation wT z = 0. Nowthe state variation rate u at a bifurcation point can be decomposed into a homogeneous solutioncomponent "z in the buckling mode direction, and a particular solution component, orthogonal toz, which is furnished by w:

u = (w+ "z) ! (23.42)

The next step is tofind" . It was pointed out that thefirst-order rate equations do not provided enoughinformation for computing " . To obtain the required information we must resort to the second-order system. Premultiplying equation by zT and taking into account the bifurcation condition 3 inDefinition 2 we get the scalar equation

zT Ku! zT q! = 0 (23.43)

The rate matrices K and q may be written as linear combinations of u and ! as:

K = Lu+ N!, q = !!

Nu+ j!"

(23.44)

ThenzT!

Lu+ N!"

u+ zT!

Nu+ j!"

! = 0 (23.45)

The replacement of u by its decomposition gives

zT!

L (w+ "z) !+ N!"

(w+ "z) !+ zT!

N (w+ "z) !+ j!"

! = 0 (23.46)

Insertion of the relations

a = zTLzz, b = zT (Lzw+ Nz) , c = zT (Lww+ 2Nw+ j) (23.47)

23–16

23–17 §23.9 BRANCH SWITCHING

yields the quadratic equationa! 2 + 2b! + c = 0 (23.48)

If at least one coefficient is nonzero, this quadratic equation provides two roots: !1 and !2 —root! = ! is acceptable. Since by hypothesis a branch reaches the bifurcation point, one of theroots must be real. Consequently, the other root must also be real. The condition that at least onecoefficient be nonzero serves as a criterion for a simple bifurcation point because it guarantees atransversal intersection of two branches. It is worth noting that a = 0 characterizes a pitchfork orsymmetric bifurcation while a "= 0 characterizes a transcritical or asymmetric bifurcation.Although this method is systematic and reliable, it suffers from the disadvantage of requiring secondderivatives of the residual in addition to requiring an accurate approximation of the bifurcationpoint. Both procedures are rather expensive, particularly the former, since it is not always possibleto obtain an analytical expression for the rate of the stiffnessmatrix in a reasonably simpleway. Thisis obviously the case when the formulation used corresponds to elaborate mathematical models.These problems motivated the search for other alternatives. In general, these alternatives are basedon buckling mode injection.

§23.9.2. Buckling Mode Injection

The purpose of these predictors is to obtain a sufficiently close approximation to the emanatingbrancheswithout resorting to the computation of the exact tangents to suchbranches. Theunderlyingidea is to obtain an increment vectord representing the difference between one solution in the knownbranch and a solution on the unknown branch for a given value of the load parameter ",

d = w# u, " = " (23.49)

Observe that the two tangents at the bifurcation point obtained by replacing the two roots span aplane. In a sufficiently small neighborhood of the bifurcation point (uc, "c) both branches lie onthat plane. If " is sufficiently close to "c it would be possible to approximate the vector d by a vectoralmost parallel to this tangent plane. If d could be obtained exactly it would be possible to switchbranches in a straightforward manner. However, in most practical problems only an approximationto d is possible. Such approximation is generally split into two steps

1. Find a direction pointing to the other branch2. Find a step along this direction so that the distance between the two solutions is minimized in a

given norm.The problem is then reduced to finding d and # such that

|u+ #d# w| = 0, r(u, ") = r(w, ") = 0 (23.50)

The first condition can be viewed in the more general context of a constraint relation between thesolutions on both branches, thus generalizing constraints.The first subproblem can be solved by taking d proportional to z since z lies on the plane definedby the two tangents. This can easily be seen just by taking the difference between the two tangentsand observing that this difference is a multiple of z. This procedure can be viewed as an injectionof the buckling mode into the known solution.

23–17


The solution to the second subproblem is more difficult. It does not seem to be possible to find a !that satisfies the previous equation for a given value of " without resorting to information providedby second or higher-order rate equations. We cope with this problem by shifting it to the corrector,which will adjust " instead of ! in such a way that a constraint condition between the solutions onboth branches is satisfied. The problem is so restated because the corrector is better equipped todeal with a problem posed in terms of changes in the state vector and control parameters, becausean incremental equation for ! is not available. Thus, at the predictor level, we will be content withan estimation of !. Some values dictated by experience are given in Seydel (loc.cit.).Note that two predictors are in fact available:

w1,2 = u± !z (23.51)

For perfect structures undergoing symmetric—pitchfork— bifurcation, the sign choice is irrelevant,but for asymmetric—transcritical—bifurcation, one branchwill be energy preferred. This difficultycan be overcome by calculating solutions on both half-branchs emanating from the bifurcation pointand choosing the one with least energy.The success of thewhole procedure strongly depends on an appropriate corrector. Correctors shouldbe carefully designed in order to display selective properties. This is meant as the capability of thecorrector to avoid converging towards the known branch. The design of such correctors is dealtwith in the next section.

§23.10. Correctors

One alternative is to enrich the displacement increment with the buckling mode. Felippa (loc.cit.)modifies equations so that

d = dsr + #qdsq + #edse ± #zz (23.52)

To provide sufficient equations a length constraint in injected, zTd = !. Again, the unknowncoefficients are obtained by requiring that d verifies the incremental equation which leads to thefollowing 3! 3 unsymmetric auxiliary system:

! g + aq ae az"dqi 1" dei lsq se 1

"!

#q#e

±#z

"

=!"c " ar

dri! " sr

"

(23.53)

in whichaz = zT a, dz = zTd, sr = zTdsr , sq = zTdsq , se = zdse (23.54)

The selective properties of this corrector are not clearly established.To obtain a corrector with enhanced selective properties, Skeie and Felippa11 propose using aconstraint that has greater affinity for the emanating branch than for the known branch. For thispurpose, they look at the limit behavior of the local hyperelliptic constraint. They show that thisconstraint reduces to a narrow cylinder around the known branch when a/$ # b/$. To obtain arobust corrector it is necessary to inject information from both branches into the constraint equation.

11 G. Skeie and C.A. Felippa, Detecting and Traversing Bifurcation Points in Nonlinear Structural Analysis, Report CU-CSSC-89-23, Center for Space Structures and Controls, University of Colorado, Boulder, 1989.

23–18

23–19 §23.10 CORRECTORS

This can easily be achieved if the constraint is expressed in terms of the difference between thesolutions on two different branches, as shown below.In the correction step, can be generalized to

c(w! u, !! !) = 0 (23.55)

Increments to both w and u and to ! are computed so that this constraint is enforced. What weare trying to avoid is an iteration converging towards the known branch. Experience shows thatthis frequently happens when ! = !c is kept fixed while trying to obtain a state vector u satisfyingthe residual equation. For example, consider the case of symmetric—pitchfork— bifurcation. Inthis case, the crossing branch exists for values of ! either greater or smaller than !c. Since wedo not know beforehand on which side of the bifurcation point the crossing branch lies, there isa 50% chance that no emanating branch exist for the chosen value of !. Besides, it is possiblethat the emanating branch could not be parametrized by ! close to the bifurcation point and a localparametrization may be required.Once again we can expand the solution space. A point in this expanded space is defined by (u,w, !)

and a parametrization for the solution in this new space is sought. In the case of prior equations theparameter was the arc-length "; here # plays the same role. The only difference is that we replace$u, which represents an increment of the solution on the known branch, by w! u which measuresthe difference between two solutions lying on different branches. For example, displacement controlresults in

wk ! uk = # (23.56)

In summary, we look for two state corrections dw and dv and a load correction % that simultaneouslysatisfy

r(uk + dkv, !k + %k) = r(wk + dkw, !k + %k) = 0 (23.57)and the constraint condition

c($uk + dkv,$wk + dkw;$!k + %k) = 0 (23.58)

These corrections are obtained from!Kv 0 !q0 Kw !qaT !aT g

"! dv

dw

%

"

= !! rvrwc

"

(23.59)

where Kv = K(uk, !k) and Kw = K(wk, !k).Again it is possible to solve the previous equation while preserving the symmetry and sparsenessof K. If K is nonsingular we may write

dru = !K!1v rv, drw = !K!1

w rw, dqu = K!1v q, dqw = K!1

w q (23.60)

Thus, the two incremental displacements and the load increment may obtained as:

dv = dru + %dqudw = drw + %dqw

% = ! c + aT (dru ! drw)

g + aT (dqu ! dqw)

(23.61)

23–19


By using the procedure described in previous equations we require much less storage than whenusing the original system, because Kv and Kw can overwrite each other. The extra storage re-quirement is for four N -dimensional auxiliary vectors. However, it is still demanding in terms ofcomputer time, because two linear systems of equations with different coefficient matrices must besolved at each iteration.As an alternative, we may keep the solution on the known branch fixed and search for anotherequilibrium solution that also verifies the constraint equation. That is, solve

!

r(w, !)

c(u;dw, !)

"

= 0 (23.62)

This equation resembles the original continuation equation; the main difference being a slightmodification of the constraint condition. However, there is no guarantee that a solution of thisequation will lie on the emanating branch; observe that u and w correspond to two different valuesof the load parameter ! and it is possible that two solutions on the same branch satisfy the constraintequation.In the examples presented in Crivelli’s thesis the first approach is used, since we are willing topay the additional price to guarantee the success of the branch switching. Furthermore this cost ismodest compared to that of the entire continuation process.A remark is in order here. The stiffness matrices Kv and Kw may become ill-conditioned and aprocedure similar to that leading to equations may be developed. However, note that we are notlooking for a highly accurate solution on the known branch, since we are using it just to repel aniteration from the crossing branch back to the known branch. Such a process is not required toprovide an highly accurate solution on the crossing branch either, because that solution is just usedto obtain a starting point on the new branch. We can thus apply the transformation to K to avoidnumerical instability, but regarding dru and drw as sufficiently good approximations to dru and drw,respectively. Once a point on the new branch is obtained, the solution can be further improved byswitching back to the continuation procedure used for regular points.

§23.11. Treating Bifurcation by Perturbation

The last approach to bementioned here is treating bifurcation by perturbation. The idea is to perturbthe residual equation r(u, !) in such a way that the underlying regularity intrinsic to a bifurcationpoint is destroyed. This is accomplished by introducing physical or numerical imperfections. In themathematical literature, this approach is referred to as unfolding. The perturbed system displayslimit point behavior rather than bifurcation point behavior. The key difficulty with this approach isto find a perturbation parameter " and an imperfection distribution that is critical to the underlyingregularity. For simple structures, this parameter " may be guessed from physical considerations.Once an appropriate value is found, the branch can be traced by the standard continuation procedureand the problem is reduced to limit point traversal.

23–20

23–21 Exercises


Detecting and Traversing Critical Points

EXERCISE 23.1

[C:25] For the steep 2-bar arch with S = 2 and arbitrary H and subject to a vertical load !!q, find the locusof all critical points directly by setting up and solving the r = 0 system augmented by det(K) = 0 as CPTF.Hint: you may try the effect of

ClearAll[Em,A0,S,vX,vY,H,lambda];Em=1; A0=1; S=2;Do [rX = (4*A0*Em*vX*(S^2 + 2*vX^2 + 4*H*vY + 2*vY^2))/(4*H^2 + S^2)^(3/2);rY = (8*A0*Em*(H + vY)*(vX^2 + 2*H*vY + vY^2))/(4*H^2 + S^2)^(3/2)-lambda;K={{(4*A0*Em*(S^2 + 6*vX^2 + 4*H*vY + 2*vY^2))/(4*H^2 + S^2)^(3/2),

(16*A0*Em*vX*(H + vY))/(4*H^2 + S^2)^(3/2)},{(16*A0*Em*vX*(H + vY))/(4*H^2 + S^2)^(3/2),(8*A0*Em*(2*H^2 + vX^2 + 6*H*vY + 3*vY^2))/(4*H^2 + S^2)^(3/2)}};

sol=NSolve[{rX==0,rY==0,Det[K]==0},{lambda,vX,vY}];Print["solution for H=",H," is ",({lambda,vX,vY}/.sol)//InputForm],{H,1/2,3,1/2} ];

EXERCISE 23.2 [A+C:25] The residual equation

r(", !) = " ! ! sin " = 0, (E23.1)

where v " " is the only degree of freedom, provides what is perhaps the simplest example of bifurcation.This has two solution paths, " = 0 (the fundamental path) and ! = "/ sin " (the post-buckling path), whichintersect at B (! = 1, " = 0.Solve this equation using the purely incremental arclength control method programmed in Chapter 18 anddevise a procedure bywhich the program, starting from! = " = 0 andmoving up the fundamental path, detectsB and (without cheating) follows one of the post-buckling paths. Probably the simplest one to implement isperturbation.

EXERCISE 23.3 [C:25] Solve the steep arch H = 2S under a vertical load using the program of Exercise20.3 and observe whether the program senses the bifurcation point or just marches along the fundamental path.Experiment with perturbing the problem by inserting a small horizontal load to see whether you can coax itinto the secondary path.

23–21

.

24Bifurcation:

LinearizedPrebuckling I

24–1

Chapter 24: BIFURCATION: LINEARIZED PREBUCKLING I 24–2

TABLE OF CONTENTS

Page§24.1. Introduction 24–3§24.2. Loss of Stability Criteria 24–4

§24.2.1. Static criterion . . . . . . . . . . . . . . . . . 24–4§24.2.2. Dynamic criterion . . . . . . . . . . . . . . . 24–4

§24.3. The Tangent Stiffness Test 24–5§24.4. Linearized Prebuckling 24–6§24.5. The LPB Eigensystem 24–7§24.6. Solving the Stability Eigenproblem 24–7§24.7. LPB Analysis Example 24–8§24.8. Summary of LPB Steps 24–11§24. Exercises . . . . . . . . . . . . . . . . . . . . . . 24–12

24–2

24–3 §24.1 INTRODUCTION


This Chapter starts a systematic study of the stability of elastic structures. We shall postpone themore rigorously mathematical definition of stability (or lack thereof) until later because the conceptis essentially dynamic in nature. For the moment the following physically intuitive concept shouldsuffice:

“A structure is stable at an equilibrium position if it returns to that positionupon being disturbed by an extraneous action”

Note that this informal definition is dynamic in nature, because the words “returns” and “upon”convey a sense of history. But it does not imply that the inertial and damping effects of true dynamicsare involved. So real time is not involved in the static case.A structure that is initially stable may lose stability as it moves to another equilibrium positionwhen the control parameter(s) change. Under certain conditions, that transition is associated withthe occurrence of a critical point. These have been classified into limit points and bifurcation pointsin Chapter 5.For the slender structures that occur in aerospace, civil and mechanical engineering, bifurcationpoints are more practically important than limit points. Consequently, attention will be initiallydirected to the phenomena of bifurcation or branching of equilibrium states, a set of phenomenaalso informally known as buckling. The analysis of what happens to the structure after it crosses abifurcation point is called post-buckling analysis.The study of bifurcation and post-buckling while carrying out a full nonlinear analysis is a math-ematically demanding subject. But in important cases the loss of stability of a geometricallynonlinear structure by bifurcation can be assessed by solving linear algebraic eigenvalue problemsor “eigenproblems” for short. This eigenanalysis provides the magnitude of the loads (or, moregenerally, of the control parameters) at which buckling is expected to occur. The analysis yields noinformation on post-buckling behavior. Information on the buckling load levels is often sufficient,however, for design purposes.The present Chapter covers the source of such eigenproblems for conservatively loaded elasticstructures. Chapters 26 through 28 discuss stability in the context of full nonlinear analysis. Thetwo final Chapters (29–30) extend these concepts to structures under nonconservative loading.Following a brief review of the stability assessment criteria the singular-stiffness test is described.Attention is then focused on the particular form of this test that is most used in engineering practice:the linearized prebuckling (LPB) analysis. The associated buckling eigenproblem is formulated.The application of LPB on a simple problem is worked out using the bar element developed inthe previous three sections. The assumptions underlying LPB and its range of applicability arediscussed in the next Chapter.

24–3


§24.2. Loss of Stability Criteria

For elastic, geometrically-nonlinear structures under static loadingwe can distinguish the followingtechniques for stability assessment.

Loading

!

"

"

"

"

#

"

"

"

"

$

Conservative%Static criterion (Euler method): singular stiffness

Dynamic criterion: zero frequency

Nonconservative%

Dynamic criterion% zero frequency (divergence)

frequency coalescence (flutter)

§24.2.1. Static criterion

The static criterion is also known as Euler’s method, since Euler introduced it in his famousinvestigations of the elastica published in 1744. Other names for it are energy method and methodof adjacent states. To apply this criterionwe look atadmissible static perturbations of an equilibriumposition1. These perturbations generate adjacent states or configurations, which are not generallyin equilibrium.Stability is assessed by comparing the potential energy of these adjacent configurations with thatof the equilibrium position. If all adjacent states have a higher potential energy, the equilibrium isstable. If at least one state has a lower (equal) potential energy the equilibrium is unstable (neutrallystable). This comparison can be expressed in terms of the second variation of the potential energyand hence can be reduced to the assessment of the positive definite character of the tangent stiffnessmatrix.Although stability is a dynamic phenomenon, no true-dynamics concepts such as mass or dampingare involved in the application of the static criterion, which is a key reason for its popularity. Butthe reasoning behind it makes it strictly applicable only to conservatively loaded systems, becausea load potential function is assumed to exist.

§24.2.2. Dynamic criterion

The dynamic criterion looks at dynamic perturbations of the static equilibrium position. In informalterms, “give the structure a (little) kick and see how it moves.” More precisely, we consider smalloscillations about the equilibrium position, and pose an eigenproblem that determines characteristicexponents and associated eigenmodes. The characteristic exponents are generally complex num-bers. If all characteristic exponents have no positive real components the equilibrium is dynamicallystable, and unstable otherwise.These exponents change as the control parameter ! is varied. For sufficiently small values thestructure is stable. Loss of stability occurs when a characteristic exponent enters the right-handcomplex plane. If that happens, the associated mode viewed as a displacement pattern will amplifyexponentially in the course of time. A deeper study of the stable-to-unstable transition mechanism

1 “Admissible” in the sense of the Principle of Virtual Work: variations of the state parameters that are consistent with theessential boundary conditions (kinematic constraints)

24–4

24–5 §24.3 THE TANGENT STIFFNESS TEST

reveals two types of instability phenomena, which are associated with the physically-oriented termsterms divergence and flutter.Divergence occurs when the characteristic exponent enters the right-hand plane through the origin,and it can therefore be correlated with the zero frequency test and the singular stiffness test.The dynamic criterion is applicable to both conservative and nonconservative systems. This widerrange of application is counterbalanced by the need of incorporating additional information (massand possibly damping) into the problem. Furthermore, unsymmetric eigenproblems arise in thenonconservative case, and these are the source of many computational difficulties.

§24.3. The Tangent Stiffness TestThe stability of conservative systems can be assessed by looking at the spectrum2 of the tangentstiffness matrix K. Let µi denote the i th eigenvalue of K. The set of µi ’s are the solution of thealgebraic eigenproblem

Kzi = µizi . (24.1)Since K is real symmetric3 all of its eigenvalues are real. Thus we can administer the followingtest:

(I) If all µi > 0 the equilibrium position is strongly stable(II) If all µi ! 0 the equilibrium position is neutrally stable(III) If some µi < 0 the equilibrium position is unstable

In engineering applications one is especially interested in the behavior of the structure as the stagecontrol parameter ! is varied, and so

K = K(!). (24.2)Given this dependence, a key information is the transition from stability to instability at the valueof ! closest to stage start, which is usually ! = 0. This is called the critical value of !, which weshall denote as !cr .If the entries of K depend continuously on ! the eigenvalues of K also depend continuously4 on!, although the dependence is not necessarily continuously differentiable. It follows that transitionfrom strong stability— case (I)— to instability— case (III)— has to go through case (II), i.e. azero eigenvalue. Thus a necessary condition is that K be singular, that is

detK(!cr ) = 0, (24.3)

or, equivalently,K(ucr , !cr )z = 0, (24.4)

where z "= 0 is the bucklingmode introduced in Chapters 4–5, where it was called a null eigenvector.Equation (24.3) or (24.4) is the expression of the static test for finding a stability boundary.

2 The spectrum of a matrix is the set of its eigenvalues.3 Because K = "2#/"u"u is the Hessian of the total potential energy #.4 Continuous dependence of eigenvalues on the entries is guaranteed by the perturbation theory for symmetric andHermitianmatrices. This continuous dependence does not hold, however, for eigenvectors.

24–5


Remark 24.1. Equation (24.3) is a nonlinear eigenvalue problem because: (a) K has to be evaluated at anequilibrium position, and (b)K is a nonlinear function of u, which in turn is a nonlinear function of ! as definedby the equilibrium path. It follows that in general a complete response analysis has to be conducted to solve(24.3). Such techniques were called “indirect methods” in the context of critical point location methods inChapter 23. This involves evaluatingK at each computed equilibrium position, and then finding the spectrumofK. An analysis of this nature is obviously computationally expensive. One way of reducing part of the costis noted in the following remark.

Remark 24.2. If K is known at a given !, an explicit solution of the eigenproblem (24.1) is not necessary forassessing stability. It is sufficient to factor K as

K = LDLT (24.5)

where L is unit lower triangular and D is diagonal. The number of negative eigenvalues of K is equal tothe number of negative diagonal elements (“pivots”) of D. Matrix factorization is considerably cheaper thancarrying out a complete eigenanalysis because sparseness can be exploited more effectively.

Remark 24.3. The condition (24.3) is not sufficient for concluding that a system that is stable for! < !cr will gounstable as ! exceeds !cr . A counterexample is provided by the stable-symmetric bifurcation point discussedin later Chapters. The Euler column furnishes a classical example. At such points (24.3) holds implyingneutral stability but the system does not lose stability as the bifurcation state is traversed. Nonetheless thedisplacements may become so large that the structure is practically rendered useless.

§24.4. Linearized Prebuckling

We investigate now the first critical state of an elastic system if the change in geometry prior to itcan be neglected. We shall see that in this case the nonlinear equilibrium equations can be partlylinearized, a process that leads to the classical stability eigenproblem or buckling eigenproblem.The eigenstability analysis procedure that neglects prebuckling displacements is known as lin-earized prebuckling (LPB). The modeling assumptions that are tacitly or explicitly made in LPBare discussed in some detail in the next Chapter, as well as the practical limitations that emanatefrom these assumptions. In the present Chapterwe discuss the formulation of the LPB eigenproblemand illustrate these techniques on a simple problem using the bar elements developed in previousChapters.

24–6

24–7 §24.6 SOLVING THE STABILITY EIGENPROBLEM

§24.5. The LPB Eigensystem

The two key results from the LPB assumptions (which are studied in the next Chapter) can be sum-marized as follows. Recall from Chapters 8–10 that the tangent stiffness matrix can be decomposedas the sum of material and geometric stiffness matrices:

K = KM +KG . (24.6)

Then the LPB leads to the following simplifications:

(1) The material stiffness is the stiffness evaluated at the reference configuration:

KM = K0. (24.7)

(2) The geometric stiffness is linearly dependent on the control parameter !:

KG = !K1. (24.8)

where K1 is constant and also evaluated at the reference configuration.

Now the stability test (24.3) requires that K be singular, which leads to the stability eigenproblem

Kz = (K0 + !K1) z = 0. (24.9)

In the following Chapter we shall prove that under certain restrictions the critical states deter-mined from this eigenproblem are bifurcation points and not limit points. That is, they satisfy theorthogonality test

zTq = 0. (24.10)

The eigenproblem (24.9) befits the generalized symmetric algebraic eigenproblem

Ax = !Bx, (24.11)

where both matrices A ! K0 and B ! "K1 are real symmetric, and x ! z are the buckling modeeigenvectors. If (as usual) the material stiffness K0 is positive definite, eigensystem theory saysthat all eigenvalues of (24.11) are real. We cannot in general make statements, however, about thesign of these eigenvalues. That will depend on the physics of the problem as well as on the signconventions chosen for the control parameter(s).

24–7


q λ

32

X, x

Y, y

1

(1)

k(1)

k(1)

)L(10

(2) k(2)

k(2)

L(2)0

<<

p =

Figure 24.1. LPB example involving two bar elements displacing on the X ! x, Y ! y plane.

§24.6. Solving the Stability Eigenproblem

In production FEMcodes, the stability eigenproblem (24.9) is generally treatedwith special solutiontechniques that take full advantage of the sparsity of both K0 and K1, such as subspace iteration orLanczos methods.For small systems an expedient solution method consists of reducing it to canonical form bypremultiplying both sides by the inverse of K0. This is possible if K0 is nonsingular, which meansthat the ! = 0 configuration is not a critical one. Calling A = K"1

0 K1 and µ = "1/! one gets

Azi = µizi (24.12)

This is a standard algebraic eiegnproblem, which can be solved by standard library routines for theeigenvalues µi and eigenvectors zi . For example, EigenSystem inMathematica or Eig inMatlab.The µi farthest away from zero gives the !i = "1/µi closest to zero.One disadvantage of this reduction is that A is unsymmetric even if K0 and K1 are. There aremore complicated reduction methods that preserve symmetry. These may be studied in standardnumerical analysis textbooks covering linear algebra; for example Golub and Van Loan.

§24.7. LPB Analysis ExampleTo illustrate the application ofLPB to a very simple example, the 2-bar assembly shown inFigure 24.1 is chosen.The bars can only displace on the x, y plane, thus the problem is two dimensional. The equivalent-springstiffness of the bars is denoted by

k(1) =E A(1)

0

L (1)0

, k(2) =E A(2)

0

L (2)0

, (24.13)

in which A(e)0 and L (e)

0 denote the cross sectional areas and lengths, respectively, of the eth bar in the referenceconfiguration, and E is the elastic modulus common to both bars.

24–8

24–9 §24.7 LPB ANALYSIS EXAMPLE

The figure shows the reference configuration C0 for the two bars. That configuration is taken when the appliedload is zero, that is, ! = 0. We shall assume that the stiffness of bar 1 is much greater than that of bar 2, i.e.,k(1) >> k(2) and is such that the vertical displacement uY2 of node 2 under the load is very small compared tothe dimensions of the structure.Now let the load p = !q be gradually applied by increasing !. The structure assumes a deformed currentconfiguration in equilibrium, that is, a target configuration C. According to the LPB basic assumption, thedisplacements prior to the buckling load level characterized by !cr are negligible. Therefore C ! C0 as longas |!| < |!cr |.The linear finite element equations for the example problem are as follows. For element (1):

k(1)

!

"

#

0 0 0 00 1 0 "10 0 0 00 "1 0 1

$

%

&

!

"

#

uX1uY1uX2uY2

$

%

&

=

!

"

#

000

"!q

$

%

&

. (24.14)

For element (2):

k(2)

!

"

#

1 0 "1 00 0 0 0

"1 0 1 00 0 0 0

$

%

&

!

"

#

uX2uY2uX3uY3

$

%

&

=

!

"

#

0000

$

%

&

. (24.15)

Assembling and applying the boundary conditions uX1 = uY1 = uX3 = uY3 = 0 we get' k(2) 00 k(1)

( ' uX2uY2

(

=' 0

"!q

(

. (24.16)

The linear solution is

uX2 = 0, uY2 = " !qk(1) = "

!qL (1)0

E A(1)0

. (24.17)

The axial linear strain and Cauchy (true) stress developed in element (1) are

"(1) = uY2L (1)0

= !q

E A(1)0

, # (1) = E"(1) = "!qA(1)0

. (24.18)

According to the assumptions stated above the change in geometry prior to buckling is neglected. Consequently

e(1) # "(1) s(1) # # (1)

The axial strain and stress of element (2) are zero.

The simplified nonlinear finite element equations are, for element (1))

*

+

*

,

k(1)

!

"

#

0 0 0 00 1 0 "10 0 0 00 "1 0 1

$

%

&

+ N (1)

L (1)0

!

"

#

1 0 "1 00 1 0 "1

"1 0 1 00 "1 0 1

$

%

&

-

*

.

*

/

!

"

#

uX1uY1uX2uY2

$

%

&

=

!

"

#

000

"!q

$

%

&

. (24.19)

where N (1) = A(1)0 s(1) = "p = "!q denotes the axial force in bar element (1).

24–9


For element (2) we have the same linear matrix equations as before because its geometric stiffness vanishes.Assembling and applying displacement boundary conditions we get the equations

!

"

k(2) ! !qL (2)0

0

0 k(1) ! !qL (1)0

#

$

% uX2uY2

&

=% 0

!!q

&

. (24.20)

One now regards K in (24.20) as unaffected by the displacements uX2 and uY2, which is consistent with theassumption that the change of geometry prior to buckling is neglected. This having being done, setting thedeterminant of K to zero yields the buckling eigenproblem:

det

!

"

k(2) ! !qL (1)0

0

0 k(1) ! !qL (1)0

#

$ = 0. (24.21)

This matrix is singular if either diagonal element vanishes, which yields the two eigenvalues

!cr1 = k(1)L (1)0 /q, !cr2 = k(2)L (1)

0 /q, (24.22)

as critical values of the load parameter. Since k(2) << k(1) the lowest critical load will be

pcr = !cr2q = k(2)L (1)0 . (24.23)

This is the buckling load obtained under the LPB assumptions.

24–10

24–11 §24.8 SUMMARY OF LPB STEPS

§24.8. Summary of LPB Steps

The foregoing example illustrates the key steps of LPB analysis. These are summarized below forcompleteness.

1. Assemble the linear stiffness K0 and solve the linear static problem

K0u = q0!, (24.24)

for ! = 1 and obtain the internal force (stress) distribution. Note: In staticallydeterminate structures, such as Exercise 24.4, the internal forces and stresses maybe obtained directly from equilibrium. However K0 is still necessary for step 3.

2. Form the reference geometric stiffness K1 for that internal force distribution. Thegeometric stiffness is KG = !K1.

3. Solve the stability eigenproblem

(K0 + !K1) zi = 0, or K0zi = !!iK1 zi , (24.25)

The eigenvalue !i closest to zero is the critical load multiplier, and the associatedeigenvector zi gives the corresponding buckling mode.

24–11



Bifurcation: Linearized Prebuckling I

EXERCISE 24.1 [A:15] Find the buckling mode (null eigenvector) z (normalized to unit length) for theexample problem of §24.6, and verify the orthogonality condition zTq = 0.

EXERCISE 24.2 [A:15] Find the buckling load for the two-bar problem if bar (2) forms an angle 0 ! ! < 90"

with the x axis. Assume still that the system displaces only on the x # y plane and that k(2) << k(1) so thatthe stress in bar (2) can be neglected in forming the geometric stiffness. Determine z and verify orthogonality.

EXERCISE 24.3 [A:20] Suppose the load of the two-bar example problem of §24.6 depends on the verticaldisplacement of point 2 as p = #"cu2Y2, where c is a constant with dimensions of stress. Show that even ifprebuckling deformations are neglected, the singular stiffness test leads to a nonlinear eigenvalue problem.

EXERCISE 24.4 [A:25]The “Euler column” shown inFigureE24.1 ismodelled byone2-nodeEuler-Bernoullibeam column element along its length:

λ2X, x

Y, y

1P

E, I constant

L

z

Figure E24.1. One-element model of Euler column

The state parameters are the nodal displacements degrees of freedom arranged as

u =

!

"

"

"

"

#

u X1uY1#z1u X2uY2#z2

$

%

%

%

%

&

(E24.1)

where #z1 and #z2 are (to first order) the end rotations, positive counterclockwise about z.

The linear material matrix in the reference (undeformed) configuration is

24–12

24–13 Exercises

K0 =

!

"

"

"

"

"

"

"

"

"

#

E AL 0 0 ! E A

L 0 012E IL3

6E IL2 0 !12E I

L36E IL2

4E IL 0 !6E I

L22E IL

E AL 0 0

12E IL3 !6E I

L2

symm 4E IL

$

%

%

%

%

%

%

%

%

%

&

(E24.2)

in which E is the elastic modulus, L the element length, A the cross section area, and I the moment of inertiaof the cross section about the z neutral axis.

The exactly-integrated geometric stiffness at the reference configuration is1

KG = !K1, K1 = P30L

!

"

"

"

"

#

0 0 0 0 0 036 3L 0 !36 3L

4L2 0 !3L !L2

0 0 036 !3L

symm 4L2

$

%

%

%

%

&

(E24.3)

where N is the axial force in the element (here obviously equal to the applied force !P because the structureis statically determinate.)For this problem:

(a) Check thatK0 andK1 satisfy translational infinitesimal rigid body motion conditions uX " 1 and uy " 1if the six degrees of freedom are left unconstrained. (Convert those modes to node displacements, thenpremultiply by the stiffness matrices.)

21

buckling mode

"z1

"z2 = !"z1

Figure E24.2. Symmetric buckling of one-FE model of Euler column.

(b) Set up the linearized prebuckling eigenproblem

(K0 + !K1)z = 0 (E24.4)

Apply the support end conditions to remove uX1, uY1 and uY2 as degrees of freedom.

1 See e.g. Przemieniecki’s Theory of Matrix Structural Analysis, loc. cit.

24–13


(c) Justify that freedom uX2 can be isolated from the eigenproblem, and proceed to drop it to reduce theeigenproblem to 2 ! 2.

(d) Reduce the 2 ! 2 eigenproblem to a scalar one for the symmetric buckling mode sketched in FigureE24.2. by setting !z1 = "!z2 (see Figure), and get the first critical load parameter "1.

(e) Repeat (e) for the antisymmetric buckling mode sketched in Figure E24.3 by setting !z1 = !z2 (seeFigure) and obtain the second critical load parameter "2.

21

buckling mode

!z1 !z2 = !z1

Figure E24.3. Antisymmetric buckling of one-FE model of Euler column.

(f) Compare the results of (d)–(e) to the exact critical load values

PE1 = "#2 E I

L2 , PE2 = "4#2 E I

L2 , (E24.5)

The first one was determined by Euler in 1744 and therefore is called the Euler critical load. For PE1 the

FEM result should be within 25%, which is good for one element.

(g) Repeat the calculations of "1 and "2 with the following reduced-integration geometric stiffness matrices6

K#1 = P

8L

!

"

"

"

"

#

0 0 0 0 0 09 3L 0 "9 3L

L2 0 "3L "L2

0 0 09 "3L

symm L2

$

%

%

%

%

&

(E24.6)

K##1 = P

24L

!

"

"

"

"

#

0 0 0 0 0 024 0 0 "24 0

2L2 0 0 "2L2

0 0 024 0

symm 2L2

$

%

%

%

%

&

(E24.7)

and comment on the relative accuracy obtained against the exact values.

(h) Repeat the calculations of steps (b) through (e) for a two equal-element discretization. Verify thatthe symmetric-mode buckling load is now "10E I/L2, which is (surprisingly) close to Euler’s valuePE

1 = "#2E I/L2.

6 These matrices are obtained by one-point and two-point Gauss integration, respectively, whereas the K1 of (E24.3) isobtained by either 3-point Gauss or analytical integration.

24–14

24–15 Exercises

EXERCISE 24.5 [A:25] This is identical in all respects to Exercise 24.4, except that the 2-node Timoshenkobeam model is used, with Mac Neals’s RBF correction for the material stiffness matrix. The net result isthat K0 is identical to (E24.2) but K1 is different. In fact !K1 is given by Equation (9.45) where V = 0and N = !P . Note: dont be surprised if the one-element results are poor when compared to the analyticalbuckling load.

EXERCISE 24.6 [A/C: 20] The column shown in Figure E24.4 consists of 3 rigid bars of equal length Lconnected by hinges and stabilized by two lateral springs of linear stiffness k. The applied axial load is!P , where ! > 0 means compression. Compute the two buckling loads !1P and !2P in terms of k and L ,assuming the LPB model of infinitesimal displacements from the initial state. Show that one load correspondsto a symmetric buckling mode and the other to an antisymmetric buckling mode, and find which one is critical.

λ P

L

L

L

k

A

B

1

2k

All 3 bars areconsidered rigid

Figure E24.4. Buckling of a segmented-hinged column propped by two springs.

Hint. The two degrees of freedom are the small lateral displacements u1 and u2 of hinges 1 and 2, whereu1 << L , u2 << L . Write the total potential energy as

" = U ! W, U = 12 ku

21 + 1

2 ku22, W = !P

2L!

u21 + (u1 ! u2)

2 + u22

"

. (E24.8)

Explain where the expression of W comes from. Once " is in hand, it is smooth sailing.

24–15

.

25Bifurcation:

LinearizedPrebuckling II

25–1

Chapter 25: BIFURCATION: LINEARIZED PREBUCKLING II 25–2

TABLE OF CONTENTS

Page§25.1. Introduction 25–3§25.2. State Decomposition at Bifurcation Point 25–3§25.3. LPB Assumptions 25–3§25.4. Limitations of LPB 25–5

§25.4.1. When LPB Works . . . . . . . . . . . . . . . . 25–6§25.4.2. And When It Doesn’t . . . . . . . . . . . . . . 25–7§25.4.3. How to Extend the Applicability of LPB . . . . . . . . 25–8

§25. Exercises . . . . . . . . . . . . . . . . . . . . . . 25–9

25–2

25–3 §25.3 LPB ASSUMPTIONS


This Chapter continues with the subject of linearized prebuckling (LPB) bifurcation analysis. Itgoes deeper than Chapter 24 in that it probes the assumptions (so far stated without proof) behindLPB, and the practical modeling implications that emanate from these assumptions.To present some of the derivations in mathematical terms it is necessary to introduce the conceptof state decomposition at the bifurcation point and to define homogeneous and particular solutions.This is done briefly in §24.2 primarily as a means of introducing notation for S24.3 and following.The detailed mathematical analysis of this decomposition is relegated to Chapter 26.

§25.2. State Decomposition at Bifurcation Point

Recall from previous Chapters that an isolated bifurcation point at !cr is characterized by a singulartangent stiffness at the equilibrium configuration,

K(ucr , !cr ) z = 0, (25.1)

and by the normalized null eigenvector (buckling mode) z != 0, "z" = 1, being orthogonal to theincremental load vector:

qT z = zTq = 0. (25.2)

Assume that we have located a bifurcation point B and computed the buckling mode z. Our nexttask is to examine the structural behavior in the neighborhood of B.We shall be content with lookingat the so-called branching direction information. This information characterizes the tangents to theequilibrium branches that cross at B.To carry out this task we borrow from algebraic ODE theory. Consider the variation in the statevector u measured from its value uB at buckling:

"u = u# uB (25.3)

Divide this increment by "t , t being the timelike parameter introduced in Chapter 3, and pass tothe limit:

u = limt$0

"u"t

. (25.4)

This variation rate u from the bifurcation point can be decomposed into a homogeneous solutioncomponent #z in the buckling mode direction, and a particular solution component y, which isorthogonal to z:

u = (y+ #z)!, yT z = zT y = 0, (25.5)

The particular solution solves the system

Ky = q, yT z = 0, (25.6)

which is simply the first-order incremental equation Ku = q! augmented by a normality con-straint. Imposing this constraint removes the singularity (rank deficiency) of K. The geometricinterpretation of this decomposition on the y, z plane is shown in Figure 24.1.

25–3


y

z

!z"

y"

zT y = 0||z||2 = 1

B

Figure 25.1. State decomposition at bifurcation point B.

§25.3. LPB Assumptions

With the notation introduced in §25.2 we may now state the key assumptions invoked in linearizedprebuckling (LPB). (Those collected in item (II) have already been formally stated and used inChapter 21.)

(I) The loading is conservative and proportional:

p = q0 + "q. (25.7)

and the structure is linearly elastic. Inother words, the residual equations are derivable froma potential energy function.

(II) The displacements and displacement gradients prior to the critical state are negligible in thesense that (a) the material stiffness matrix can be evaluated at the reference configuration,and (b) the geometric stiffness is proportional to the control parameter ":

KM ! K0, KG ! "K1, (25.8)

in which K0 is the material matrix evaluated at the reference configuration, also called thelinear stiffness, and K1 is the reference geometric stiffness. As discussed in the previousChapter the singular stiffness criterion detK = 0 leads to the eigenproblem

(K0 + "K1) z = 0. (25.9)

(III) The particular solution y defined in §25.2 is obtained by solving

(K0 + "crK1) y = q (25.10)

25–4

25–5 §25.4 LIMITATIONS OF LPB

under the constraint yT z = 0. Observe that from assumption (I) q is constant.

We now prove that if these assumptions hold, all critical points determined from the LPB eigenprob-lem are bifurcation points, that is, zTq vanishes. To show that, premultiply both sides of (25.10)by zT :

zTq = zT (K0 + !K1)y = yT (K0 + !K1)z = yT (Kz) = 0 (25.11)Note that the transformation zTKy = yTKz holds becauseK0 andK1 are symmetric on account ofthe conservativeness assumption (I).

Remark 25.1. Bifurcation points are classified in later sections into various types: unsymmetric, stable-symmetric, stable-unsymmetric, and so on. It will be shown later that, under most common assumptions, LPBbifurcation points are generally of symmetric type. The LPB model does not provide, however, informationas to the post-bifurcation stability, so we cannot say whether the bifurcation point is stable-symmetric orunstable-symmetric.

§25.4. Limitations of LPB

Linearized prebuckling (LPB) is used extensively in engineering design. Standard books in struc-tural stability1 concentrate upon it. In its finite element version LPB is a feature available in manyfinite element programs. Exercising this feature has the advantages of avoiding a full nonlinearanalysis, which can be expensive and time-consuming. Given its practical importance, structure de-signers (and most especially aerospace designers) should be familiar with the range of applicabilityof LPB. The limitations are discussed next.

1. Conservative loading. LPB is a restricted form of the static criterion also known as Euler’stest (see §24.2). If the loads are not conservative, the dynamic criterion should be used, atleast to check out whether a flutter condition may occur. If the dynamic criterion shows thatstability is lost by divergence, one may regress to the singular-stiffness test criterion.

2. Loss of stability must be by symmetric bifurcation. If the first critical point is a limit point orasymmetric bifurcation,2 LPB is not strictly applicable although in some cases it may providea sufficiently good approximation. Lacking experimental confirmation or a priori knowledge,the only practical way to check whether the first critical point is symmetric bifurcation is togo through a full nonlinear analysis.

3. Prebuckling deformations must be small. This assumption fits well many engineering struc-tures because of the nature of construction materials. The structures that best fit these assump-tions are straight columns, frameworks and flat plates, as illustrated in Figure 25.2. Care mustbe exercised for arches, shells, very thin members, and for imperfection-sensitive structuresin general.

4. Elasticmaterial behavior. If thematerial is inelastic the structure is not internally conservative.Then the tangent stiffness depends on the prior deformation history, and the LPB eigenproblem

1 For example, Timoshenko and Gere’s Theory of Elastic Stability.2 Symmetric bifurcation occurs when bucking in the z and !z directions is equally likely. Asymmetric bifurcation occurswhen one of the directions is physically more likely; for example axially compressed cylinders buckle inwards. Thisclassification of critical points is covered in more detail in Chapter 11 and following.

25–5


Figure 25.2. Structures that are adequately modeled by LPB assumptions.

negligible deformationprior to buckling

λ

v

B

R

Figure 25.3. Type of response expected under LPB assumptions.(Branch intersection at B not shown for clarity)

loses meaning. The topic of inelastic buckling (in particular creep and plastic buckling) is anenormous subject that falls outside the scope of this course.

5. Applied loads should not depend nonlinearly on the displacements. Such a dependence usuallyintroduces nonconservative effects, thus voiding the conservative-loading assumptions. Evenis the loads remain conservative, the reference geometric stiffness would depend on the loadlevel, thus leading to a nonlinear eigenproblem.

6. The effect of imperfections is negligible. Some structures are highly imperfection sensitive inthat the first critical load is strongly affected by the presence of imperfections. In such casesobviously LPB is of limited value or outright irrelevant.

§25.4.1. When LPB Works

25–6

25–7 §25.4 LIMITATIONS OF LPB

λ

v

L

R

BLPB Actual

(a)

λ

v

L

R

B

LPB

Actual

(b)

Figure 25.4. Two structures that fit the LPB assumptions poorly.

The systems that bestfit the LPBmodel are symmetrically loaded structures such as straight columnsand in-plane-loaded plates (laminas) which are not excessively thin. See Figure 25.2. The lateralbuckling of such structures occurs following very small deformations, as typified by the responsesketch in Figure 25.3.

§25.4.2. And When It Doesn’t

Two examples of structures that are not properly treated by the LPB model are shown in Figure25.4. The LPB predictions are way off in both cases, but for different reasons.Case (a) is an axially compressed cylindrical shell made up of almost flat panels joined by curvedpanels, forming like a “curved triangle” cross section seen in some combat helicopters and theSpace Shuttle fuselage. There is a substantial redistribution of stresses due to changes on geometry.The structure eventually collapses at a limit point substantially over the predicted LPB load. Thelatter is therefore overly safe.

25–7


On the other hand, the axially compressed circular cylinder of case (b) is highly imperfection-sensitive structure that fails at a substantially lower load than that predicted by LPB. Consequentlythe LPB prediction is highly unsafe.

§25.4.3. How to Extend the Applicability of LPB

One way to broaden the application of the LPB model is to update the reference configuration3 sothat the prebuckling deformations are reduced. If this is done the control parameter ! is of coursemeasured from the latest reference configuration and consequently becomes a true stage controlparameter. Limitations on the conservativeness of applied loads and types of critical point, however,cannot be readily circumvented by this “staging” technique.

3 As naturally done in the CR description, in which the deformational displacements are measured from a continuouslyvarying configuration, and also in the Updated Lagrangian description.

25–8

25–9 Exercises


Bifurcation: Linearized Prebuckling II

EXERCISE 25.1 [A:15] Find the particular solution y at the lowest bifurcation load of the two-bar exampleof Chapter 24.

EXERCISE 25.2 [A:15] Find the particular solution y at the symmetric and antisymmetric bifurcation loadsof the one-element Euler column example of Exercise 24.4.

EXERCISE 25.3 [A:25] The first order residual rate equations is r = 0, where r is given by

r = Ku! q! = 0, (E25.1)

(E25.1) holds at a bifurcation point whereK and q are the tangent stiffness matrix and incremental load vector,respectively, at bifurcation. Decompose u = (y + "z)!, where y is the particular solution and z "= 0 thebuckling mode normalized to length one. Show that the first-order differential equation system (E25.1) cannotgive information on the “buckling mode amplitude” " because one gets " = 0/0. (Hint: premultiply thatequation by an appropriate vector.)

25–9

.

26QualitativeAnalysis of

Critical Points

26–1

Chapter 26: QUALITATIVE ANALYSIS OF CRITICAL POINTS 26–2

TABLE OF CONTENTS

Page§26.1. General Notion of Stability 26–3§26.2. Stability of a Discrete Conservative System 26–3§26.3. Stability Transformation at a Limit Point 26–4§26.4. Stability Exchange at Bifurcation Points 26–7

§26.4.1. Asymmetric bifurcation . . . . . . . . . . . . . . 26–7§26.4.2. Stable-symmetric bifurcation . . . . . . . . . . . . 26–8§26.4.3. Unstable-symmetric bifurcation . . . . . . . . . . . 26–8

26–2

26–3 §26.2 STABILITY OF A DISCRETE CONSERVATIVE SYSTEM

§26.1. General Notion of Stability

In Chapter 24 stability was informally defined as the ability of a physical system to return toequilibrium when disturbed. If the equilibrium is static in nature, we speak of static stability. Fora more precise definition concerning a mechanical system, let’s hear Dirichlet:1

“The equilibrium [of a mechanical system] is stable if, in displacing the points of the system fromtheir equilibrium positions by an infinitesimal amount and giving each one a small initial velocity,the displacements of different points of the system remain, throughout the course of the motion,contained within small prescribed limits”

Some essential ingredients of this definition are:

(1) Stability is a quality of one solution— an equilibrium solution of the system.

(2) The problem of ascertaining the stability of a solution concerns the “neighborhood”of the particular solution and is therefore a local one.

(3) The concept of stability is inherently dynamic in nature. But for a conservativesystem dynamics can be “factored out” of the problem, and we are left with a staticcriterion.

§26.2. Stability of a Discrete Conservative System

As discussed in Chapter 24, the static stability of a conservative mechanical system can be testedcompletely using a static criterion. Such criterion, often referred to as the Euler stability test, theenergy test, or the method of adjacent states, relates to the positive definiteness character of thesecond variation of the potential energy.We know that a stationary value of the total potential energy with respect to the state variables isnecessary and sufficient for the equilibrium of the system. Proceeding one step further, a completerelative minimum of the total potential energy is necessary and sufficient for the stability of anequilibrium state.For a discrete system with a finite number of degrees of freedom the criterion can be enunciated interms of the definiteness of the tangent stiffness matrix K if all state variables are of displacementtype, which we assume in the sequel (see Remark below). For a conservative system we know thatK is a symmetric matrix.

1 As it appears in his Appendix to the German translation of Lagrange’sMecanique Analytique (1853).

26–3


!(

u

u

u, "2), "2 > "cr

"cr

!( , "cr )

!( , "1), "1 < "cr"

u

L

Primary (fundamental) path

Figure 26.1. Transformation of potential energy at a limit point.

Remark 26.1. The restriction to systems with displacement state variables aims to exclude those in whichLagrange multipliers are carried along as degrees of freedom. For such systems the criterion applies uponeliminating the multipliers, but such elimination is often messy and would complicate the exposition.

The stability criterion for a conservative system is summarized in the following table.

If K evaluated at an The potential energy ! Then the equilibriumequilibrium position is at that position has a position is

positive definite strict minimum stablepositive semidefinite cylindrical or inflexion point neutrally stableindefinite saddle point unstable

If the eigenvalues of K are easily available a test for stability is immediate.2 If all eigenvaluesare greater than zero, the matrix is positive definite and the equilibrium is stable. If one or moreeigenvalues are zero and the rest positive, the equilibrium is neutrally stable. If one or moreeigenvalues are negative, the equilibrium is unstable.In practice an eigenvalue test can be recommended only for small matrices, say of order less than 20or so. For larger matrices the same information can be obtained more economically by decomposingK using triangular factorization or Gauss elimination, as discussed in Remark 24.2. If all pivotsare positive, the equilibrium is stable. If at least one pivot is negative, the equilibrium is unstable.The border case of neutral stability is more difficult to detect in the presence of rounding errors.

2 See §24.2 for computational details.

26–4

26–5 §26.3 STABILITY TRANSFORMATION AT A LIMIT POINT

!

Primary (fundamental) equilibrium path

L1

L2

1

1

2

"( 1, 2, ! )

u

u

u u

Figure 26.2. Typical potential energy surface in snap-through response.

§26.3. Stability Transformation at a Limit Point

As discussed in previous sections, there is a close relationship between equilibrium configurations,occurrence of critical points, and the stability of the system. We now examine qualitatively, follow-ing the classical treatise of Thompson and Hunt,3 four types of critical points from the standpointof the variation of the total potential energy in the neighborhood of equilibrium states. This isdone with the typical response plots in which the control parameter ! is the vertical axis while arepresentative displacement u or deformation mode amplitude is shown along the horizontal axisas state parameter.Drawing conventions are as follows: heavy lines represent equilibrium paths, continuous linesdenoting stable paths while broken lines denote unstable paths. Plots of total potential energy"(u, !) at various fixed values of ! are shown as “shaded profile” energy surfaces. These surfacesdeform as the parameter ! changes. Equilibrium configurations correspond to stationary points of" with respect to u. Strong minima (maxima) of this surface are associated with stable (unstable)equilibrium configurations.We first consider the case of a limit point, which is shown in Figure 26.1. The fundamentalequilibrium path that starts from the origin (the reference configuration u = 0, ! = 0) is initiallystable. Stability is lost when the local maximum at ! = !cr is reached. ! = !cr . A “snap-through”response of this form is characteristic of shallow arches and domes. At a fixed value ! = !1 lessthan !cr the total potential energy "(u, !1) has a minimum with respect to the state parameteru on the stable rising region of the path and a maximum on the unstable falling region. As the

3 See References in Appendix Z

26–5


!( , "2), "2 > "cr

"cr

!( , "cr )

!( , "1), "1 < "cr"

B


Secondary path

u

u

u

u

Figure 26.3. Asymmetric bifurcation point.

prescribed value of " is increased the maximum and minimum approach each other and coalescewhen " = "cr . At this critical point the total potential energy !(u, ") has an horizontal point ofinflexion. At a higher value of ", say "2 > "cr , there are no local equilibrium states and the totalpotential energy !(u, "2) has no stationary point. The critical equilibrium state is seen to be itselfunstable, and the absence of local equilibrium states at values of " greater than "cr implies that aphysical system under slowly increasing " will eventually snap-through dynamically. Limit pointsare generally insensitive to imperfections.A more general schematic diagram is shown in three dimensions in Figure 26.2 on a plot of "

against two state parameters, v1 and v2. This plot includes a remote rising region of the equilibriumpath since this is often encountered with this type of behavior. A total potential energy surface!(v1, v2, ") is drawn for a fixed value of " < "cr . As " is slowly increased through its criticalvalue the system will “snap through” dynamically, eventually stop, and initiate a large amplitude,nonlinear vibration about the remote stable equilibrium path. In the presence of some damping thesystem will eventually rest on that path.These figures illustrate the physics well but if we are dealing with a system with many degrees offreedom care must be taken in drawing conclusions from these schematic figures. On an actual plotof " against one of the vi ’s, the limit point is normally seen as a smooth maximum, but it must berealized that for a certain choice of the state parameter vi the point might appear as a sharp cusp.The smooth maximum of a path in three-dimensional space can for example be seen as a cusp ifthe eye is directed along the horizontal tangent to the path.

26–6

26–7 §26.4 STABILITY EXCHANGE AT BIFURCATION POINTS

!( , "2), "2 > "cr

"cr

!( , "cr )

!( , "1), "1 < "cr

"

B

Secondary path


u

u

u

u

Figure 26.4. Stable-symmetric bifurcation point.

§26.4. Stability Exchange at Bifurcation Points

After the limit point we consider bifurcation or branching points. We cover the three most commontypes of bifurcation: asymmetric, stable-symmetric, and unstable symmetric.

§26.4.1. Asymmetric bifurcation

Figure 26.3 shows the case of an asymmetric point of bifurcation. The initially stable fundamentalequilibrium path that emanates from the origin loses its stability on intersecting a distinct andcontinuous secondary (post-buckling) equilibrium path. The intersection point B is a critical pointof bifurcation type. An asymmetric bifurcation point is characterized by the fact that both pathshave a nonzero slope with respect to " at B.With varying " the paths exhibit a phenomenon called exchange of stability. For "1 < "cr the totalpotential energy !(u, "1) has a minimum with respect to u on the stable region of the fundamentalpath and a maximum with respect to u on the unstable region of the post-buckling path. As " isincreased the maximum and minimum finally coalesce so that at " = "cr the total potential energy!(u, "cr ) has a horizontal point of inflexion at the critical equilibrium state. At " values over thecritical one the maximum and minimum exchange places. Since an unstable branch emanates fromB, the critical equilibrium state is unstable. In the presence of small disturbances a physical systemunder slowly increasing " would snap dynamically from this critical equilibrium state despite theexistence of stable equilibrium states at higher values of ".Critical points of this type are moderately to highly sensitive to the presence of loading or fabricationimperfections.

26–7


!( , "2), "2 > "cr

"cr

!( , "cr )

!( , "1), "1 < "cr

"

v

B


Secondary path

u

u

u

Figure 26.5. Unstable-symmetric bifurcation point.

§26.4.2. Stable-symmetric bifurcation

Symmetric bifurcation points are characterized by the fact that the intersecting path has zero slopewith respect to the control parameter at B. These points may be categorized into stable and unstable,depending on whether the intersecting post-buckling) path is “rising” or “falling”.Figure 26.4 depicts the case of an stable-symmetric point of bifurcation. Here a fundamentalequilibrium path rising monotonically from the reference state is seen to intersect a stable risingsecondary (post-buckling) path that passes smoothly through the critical equilibrium state with zeroslope. The continuation of the fundamental path beyond B is unstable. The total potential energy!(u, "1), where "1 < "cr , has a single stationary value with respect to u, namely the minimumon the stable region of the fundamental path, and as the value of " is increased this minimum istransformed into two minima and one maximum. The critical equilibrium state is neutrally stableand the secondary path is stable, so a physical system under slowly increasing " would exhibit nodynamic snap but would follow the stable rising post-buckling path, the direction taken dependingon the small disturbances or imperfections which are inevitably present.Critical points of this type are insensitive to the presence of imperfections.

§26.4.3. Unstable-symmetric bifurcation

The last configuration examined here is the unstable-symmetric point of bifurcation, shown inFigure 26.5. Here the fundamental path intersects an unstable falling path which as in the previouscase has a zero slope at the critical equilibrium state. At a prescribed value of " = "1 < "crthe total potential energy !(u, "1) has now three stationary values with respect to u, namely two

26–8

26–9 §26.4 STABILITY EXCHANGE AT BIFURCATION POINTS

maxima on the unstable post-buckling or secondary path, and a minimum on the stable region ofthe fundamental path. As the figure shows, these three stationary points transform into a singlemaximum with increasing !. The critical equilibrium state is seen to be unstable, so a physicalsystem would snap dynamically from the critical equilibrium state, the direction taken dependingon the postulated small disturbances or imperfections.Critical points of this type are highly sensitive to the presence of structural or loading imperfections.Sometimes the sensitivity is extreme, as in the classical case of the axially compressed cylindricalshell discussed in §25.4.

26–9

.

27Nonlinear

BifurcationAnalysis

27–1

Chapter 27: NONLINEAR BIFURCATION ANALYSIS 27–2

TABLE OF CONTENTS

Page§27.1. Introduction 27–3§27.2. Levels of Bifurcation Analysis 27–3§27.3. Recapitulation of Governing Equations 27–3

§27.3.1. Residual and Rate Equations . . . . . . . . . . . . 27–3§27.3.2. Stiffness and Load Rates . . . . . . . . . . . . . 27–5§27.3.3. Limitations of !-Parametrized Forms . . . . . . . . . 27–5

§27.4. A Deeper Look at Bifurcation 27–5§27.4.1. State Decomposition . . . . . . . . . . . . . . 27–6§27.4.2. Failure of first-order rate equations at bifurcation . . . . . 27–6

§27.5. Branch Analysis of Simple Bifurcation 27–6§27.5.1. State Decomposition . . . . . . . . . . . . . . 27–6§27.5.2. Finding " . . . . . . . . . . . . . . . . . . 27–8

§27.6. The Hinged Cantilever 27–10§27.6.1. Finding the Critical Point . . . . . . . . . . . . . 27–11§27.6.2. Branching Analysis . . . . . . . . . . . . . . . 27–11

§27. Exercises . . . . . . . . . . . . . . . . . . . . . . 27–14

27–2

27–3 §27.3 RECAPITULATION OF GOVERNING EQUATIONS


We initiated our study of stability of conservative systems in Chapters 24-25 by using the simplifiedmodel of linearized prebuckling (LPB). This was followed in Chapter 26 by a qualitative study ofthe more general stability model that led to classifying isolated critical points into four types: limitpoint, asymmetric bifurcation, stable-symmetric bifurcation, and unstable-symmetric bifurcation.This classification is especially helpful in understanding the effect of imperfections on stability.This Chapter presents a more detailed mathematical analysis of the phenomenon of bifurcation bystudying equilibrium branches in the vicinity of an isolated bifurcation point. The topic is coveredunder the name nonlinear bifurcation to emphasize that we are dealing with the general case asopposed to the LPB model. A simple example involving a one-degree of freedom system is thenworked out in some detail. The next Chapter takes up the subject of how physical or numericalimperfections affect structural behavior as regards both limit and bifurcation points.

§27.2. Levels of Bifurcation Analysis

Nonlinear bifurcation analysis can be carried out at different levels of detail, as demanded byapplication needs.1 Four levels of increasing detail are schematized in Figure 27.1, which assumethe occurrence of an isolated bifurcation point B.1. Locate: find where B occurs while tracing a response. Can be done by monitoring changes

of sign of the determinant of K or equivalently tracing the sign of factorization pivots (SeeChapter 21).

2. Determine subspace: having located B, determine vectors y (particular solution) and z (nulleigenvector for isolated bifurcation) that together with ! form an intrinsic subspace “wherethe action is.” Requires a partial eigensolution; more precisely getting the null eigenvector(s).

3. Branching analysis: having located B, and computed y and z, find the directions u1 and u2 oftangents to the equilibrium paths (branches) that pass through B. Requires an analysis of thesecond order rate equations r = 0.

4. Branch curvature analysis: having located B and determined y, z, u1 and u2, find the curvaturesof the equilibrium paths (branches) passing through B. Requires an analysis of the third orderequation ...r = 0.

The information necessary for level 3 is quite difficult to obtainfrom a general purpose finite elementprograms, while that needed for level 4 is truly inaccesible. For this reason most FE programs canprovide only levels 1 and 2 on a routine basis. In the present Chapter we study up to level 3(branching analysis), but the practical difficulties of implementing that level should be kept inmind.

§27.3. Recapitulation of Governing Equations

Below we recapitulate discrete governing equations derived in Chapters 3 and 4, and introduceadditional nomenclature required for the branching analysis carried out in §27.4.

1 For example in preliminary design only the location of the first bifurcation point would be of interest.

27–3


B

B

B

B

Rate Equation Order

1 (Factorization)

1 (Eigensolution)

2 if isolated

3 if isolated

Locate

Determineactive subspace

Branching analysis

Branchcurvature analysis

y

z

˙1

u

u

2

90!

Analysis Type

Figure 27.1. The four levels of information for nonlinear bifurcation analysis.

§27.3.1. Residual and Rate Equations

The one-parameter residual equilibrium equations are

r(u, !) = 0, (27.1)

where ! is the stage control parameter and u is the state vector. Solutions of this equation may beconveniently represented in parametric form

u = u(t), ! = !(t), (27.2)

where t is a dimensionless path parameter. Two important special choices for pseudotime are

t = !, t = s, (27.3)

which leads to the !-parametrized and arclength forms, respectively.

27–4

27–5 §27.4 A DEEPER LOOK AT BIFURCATION

Rate equations are systems of ordinary differential equations obtained by successive differentiationof (27.1) with respect to t . Recall that K and q denote the tangent stiffness matrix and incrementalload vector, respectively, whose entries are given by

Ki j = !ri!u j

, qi = !ri!"

. (27.4)

Using superposed dots to denote t-differentiation we obtain

r = Ku! q" = 0, (27.5)

r = Ku+ Ku! q" ! q" = 0, (27.6)...r = K...u + Ku+ Ku! q

..." ! q" ! q" = 0. (27.7)

Eq. (27.5) is a system of first-order rate equations, also called the incremental stiffness equationsor simply the stiffness equations. Eq. (27.6) is a system of second-order rate equations, also calledthe stiffness-rate equations. Eq. (27.7) is a system of third-order rate equations. And so on. Forthe branching analysis undertaken here we will go up to the second-rate equations (27.6).

§27.3.2. Stiffness and Load Rates

In the second-order system (27.6), the stiffness matrix rate K and incremental load vector rate qmay be expressed as linear combinations of u and ":

K = Lu+ N", q = !(Nu+ a") (27.8)

The entries of these new matrices and vectors are given by

Li jk = !2ri!u j!uk

, Ni j = !2ri!u j!"

= !Ki j

!", ai = !2ri

!"!"= !qi

!". (27.9)

Remark 27.1. Note that L is a three-dimensional array which may be called a cubic matrix to distinguish itfrom an ordinary square matrix. (Also referred to as a third order tensor.) Postmultiplying a cubic matrix bya vector yields an ordinary matrix. For example Lu is a matrix.

§27.3.3. Limitations of "-Parametrized Forms

If one chooses t = ", simplifications take place in systems (27.5)-(27.7) because " = 1 and" =

..." = 0. Using primes to denote differentiation with respect to ", the first two rate forms reduce

toKu" ! q = 0, (27.10)

Ku"" +K"u" ! q" = 0, (27.11)

in whichK" = Lu" + N, q" = !(Nu" + a). (27.12)

These forms are unsuitable, however, near a bifurcation point, because there the relationship betweenu and " ceases to be unique, and the more general parametrized forms such as (27.6) must be used.

27–5


§27.4. A Deeper Look at Bifurcation

At critical points K becomes singular and therefore possesses at least one null eigenvector, whichas usual is called z. This eigenvector will be normalized to unit length. As discussed in Chapter 5,if the eigenvector is orthogonal (non-orthogonal) to the incremental load vector, the critical pointis a bifurcation (limit) point.

§27.4.1. State Decomposition

The conditions for bifurcation may be summarily stated as

Kz = 0, !z! = zT z = 1, qT z = 0. (27.13)

Because the structural system is assumed to be conservative, K is symmetric. Consequently z isalso a left eigenvector of K = KT . In structural mechanics, the eigenvector z is called a bucklingmode or buckling shape. This term conveys the idea that the structure jumps from a prebucklingstate into the new shape. Although the name is appropriate in the LPB model, we shall see that itis not necessarily appropriate in the general case.

At the bifurcation point B the state vector u and the control parameter ! assume values uB and!B , respectively. As in Chapter 25 we study small deviations of u and ! in the neighborhood ofB. These deviations are denoted by "u = u " uB and "! = ! " !B , respectively. For smalldeviations from the bifurcation point the relation between "u and "! may be linearized as

"u = (#z+ y)"!, (27.14)

where y is the particular solution introduced in §25.2, and # is the buckling mode amplitude.

Dividing by "t and passing to the limit "t # 0 we obtain the rate form of the above equation:

u = (#z+ y)!. (27.15)

This decomposition of u in the y, z plane is depicted in Figure 27.2, a duplicate of Figure 25.1.

§27.4.2. Failure of first-order rate equations at bifurcation

At bifurcation points the first-order rate equations (27.5) yield no information on the bucklingmode amplitude. This is worked out in Exercise 25.3, which shows that # = 0/0 and is thereforeindeterminate. To get deterministic information in the vicinity of a bifurcation point it is necessaryto use information from higher-order rate equations. This is covered in the following subsectionfor a isolated (simple, distinct) bifurcation point. For this case the second-order rate equations(27.6) are usually sufficient.

27–6

27–7 §27.5 BRANCH ANALYSIS OF SIMPLE BIFURCATION

y

z

!z"

y"

zT y = 0||z||2 = 1

B

u

Figure 27.2. State decomposition at isolated bifurcation point B, depicted in the (y, z) plane.

§27.5. Branch Analysis of Simple Bifurcation

The subsequent analysis assumes that the rank deficiency of K at bifurcation is only one, and so zis the only null eigenvector. This is called an isolated, simple or distinct bifurcation point. We shallsee that at such points there can be at most two equilibrium paths that intersect at B. Such pathsare called branches.

§27.5.1. State Decomposition

Assume that we have located a bifurcation point B and computed the buckling mode z. Our nexttask is to examine the structural behavior in the neighborhood of B. This analysis is important toanswer questions pertaining to the safety of the structure and its sensitivity to imperfections.We have seen that the state variation rate u from the bifurcation point can be decomposed intoa homogeneous solution component !z in the buckling mode direction, and a particular solutioncomponent y, which is orthogonal to z:

u = (y+ !z)", yT z = 0 (27.16)

See the geometric interpretation in Figure 27.2.The particular solution vector y solves the system

Ky = q, zT y = 0, (27.17)

which is simply the first-order incremental flow equation augmented by a normality constraint.Imposing this constraint removes the singularity (rank deficiency) of K at B.

Remark 27.2. The homogeneous solution z lies in the null space ofK whereas the particular solution y lies inthe range space of K. In more physical terms we may say that y “responds” to the load (the incremental loadvector q) whereas z, like any homogeneous solution, is dictated by the boundary conditions.

27–7


z

B

Bran

ch 2

Bran

ch 1

Tangent 1

Tang

ent 2

y

˙

u

1

˙

u

2

y!

!

!

!

Figure 27.3. Intersection of two equilibrium paths at an isolated bifurcationpoint B, depicted in the y, z, !) subspace.

Remark 27.3. The decomposition (27.16) is analogous in many respects to the decomposition of elementmotions into purely-deformational and rigid-body, studied in Chapter 10. Here z take the role of rigid bodymode. The decomposition (27.16) is, however, expressed in terms of rates because it is local: it is restrictedto the vicinity of the bifurcation point.

§27.5.2. Finding "

The first-level information on the equilibrium branches at B is given by their tangents at B. Becausewe can obtain y and z from the first-order rate equations, these tangents are fully determined if inaddition we know " , u and ! at B.But as noted previously the first-order rate equations (27.5) do not provide information on thebuckling mode amplitude " . To get that information it is necessary to go to the second-ordersystem (27.6), which is repeated here for convenience:

Ku+ Ku! q! ! q! = 0 (27.18)

Premultiplying both sides by zT and taking account of the bifurcation conditions (27.13) we get atB the scalar equation

zT Ku! zT q! = 0 (27.19)

Replacement of K and q by the expressions (27.8) gives

zT (Lu+ N!)u+ zT (Nu+ a!)! = 0. (27.20)

Finally, substitution of u by its homogeneous-plus-particular decomposition (27.16) yields

zT!

L(y+ "z)! + N!"

(y+ "z)! + zT!

N(y+ "z)! + a!"

! = 0. (27.21)

27–8

27–9 §27.5 BRANCH ANALYSIS OF SIMPLE BIFURCATION

Removing the common differential factor (!)2 and collecting terms in " we arrive at the quadraticequation

a" 2 + 2b" + c = 0, (27.22)

in which

a = zTLzz, b = zT [Lzy+ Lyz+ 2Nz] , c = zT [Lyy+ 2Ny+ a] . (27.23)

This quadratic equation generally provides two roots: "1 and "2. In what follows we shall assumethat these two roots are real (see Remark 27.5 below).Substitution of "1 and "2 into (27.16) furnishes the branching directions at the bifurcation point:

u1 = (y+ "1z)!, u2 = (y+ "2z)!. (27.24)

These are sketched in Figure 27.3 in the three-dimensional space (y, z, !) with origin at B. Figure27.4 projects this picture onto the (y, z) plane for additional clarity.

B z

Branch 2

Branch 1Ta

ngent

1Tangent 2

y

˙1u u2y!

"2z! "1z!

Figure 27.4. Same as Figure 27.3 but looking down the ! axis onto the (y, z) plane.

The key result of this subsection is that there are at most two branches emanating from a simplebifurcation point. The classification of such points into asymmetric and symmetric bifurcationpoints according to the values of "1 and "2 appears in the Exercises. In the following section anillustrative example is worked out by hand.

27–9


A

rigid

L

θ

k

P = λ q

Figure 27.5. The hinged cantilever.

Remark 27.4. If a = 0 one root, say !2, becomes infinite while the other is !1 = !c/(2b), assuming b "= 0.Then u2 becomes aligned with z. Only in this case it is justified to call z the “buckling mode.”

Remark 27.5. Intuitively it appears that the two roots of (27.22) must be real. The argument goes as follows:one of the two branches is supposed to exist since B has been located by hypothesis on the equilibrium path.Its tangent at B must therefore correspond to one of the roots of (27.22). Since one of the roots is by hypothesisreal, the other must also be real because a, b and c are real coefficients.This indirect proof is not intellectually satisfying, especially to a mathematician. It would be preferable toprove the root reality by direct reasoning. However the writer has not been able to find such a proof in theliterature, and personal efforts (one hour trying) have been so far unrewarding.

Remark 27.6. If a = b = c = 0 the second-order rate form (27.6) does not provide any local informationas regards branches at B. Then one must continue to the third order rate form (27.7). This will give a cubicequation in ! with four real coefficients. Since such an equation can have one or three real roots, things get farmore complicated. If all four coefficients vanish, one must go to the fourth-order rate form, and so on. (For amathematician specialized in this kind of analysis, hell is a place where the first one million rate forms yieldno information.)

§27.6. The Hinged Cantilever

The branch analysis technique is illustrated on the hinged-cantilever problem depicted in Figure 27.5. A rigidrod of length L supported by a torsional spring of stiffness k is axially loaded by a dead force P = "q,q = k/L . Note that k has the physical dimension of force # length, i.e. of a moment. Hence the definitionP = "k/L renders " dimensionless, which is convenient for hand analysis.The dimensionless stage control parameter is " = P L/k. As state parameter we chose the tilt angle # as mostappropriate for hand analysis. The total potential energy is

$ = U ! V = 12 k#2 ! Pu = 1

2 k#2 ! P L(1! cos #) = k!

12 #

2 ! "(1! cos #"

. (27.25)

27–10

27–11 §27.6 THE HINGED CANTILEVER

The equilibrium equation in terms of ! is

r = "#

"!= k(! ! $ sin !) = 0, (27.26)

This has the two solutions! = 0, $ = !

sin !, (27.27)

which pertain to the primary (vertical or untilted) and secondary (tilted) equilibrium paths, respectively. Thetwo paths intersect at $ = 1, which is therefore a bifurcation point.

§27.6.1. Finding the Critical Point

The incremental equation in terms of ! isK ! ! q$ = 0, (27.28)

withK = "r

"!= k(1! $ cos !), q = ! "r

"$= k sin ! . (27.29)

On the primary path, ! = 0, the stiffness vanishes at

$ = 1, or P = k/L . (27.30)

On this path the stiffness is positive (negative) if $ < 1 ($ > 1), respectively. On the secondary path,$ = !/ sin ! , the stiffness is given by

K = k!

1! ! cos !sin !

"

, (27.31)

which vanishes at ! = 0 because !/ sin ! " 1 as ! " 0. If ! #= 0, K > 0. The various cases as regards thesign of K are summarized in Figure 27.6. Because K is a scalar, positive and negative values corresponds tostable and unstable equilibrium, respectively, with neutral stability at B. Stable (unstable) paths are shownswith full (dashed) lines.It is seen that ! = 0$ and $ = 1 is the only point at which K vanishes, and consequently is the only criticalpoint. Let us verify now that the critical point is a bifurcation point. Since the system has only one degree offreedom, the normalized null eigenvector is simply the scalar z = 1, and the inner product zTq reduces to

zq = q = k sin ! (27.32)

which vanishes at ! = 0$. Consequently ($ = 1, ! = 0$) is a bifurcation point.

§27.6.2. Branching Analysis

In this problem the particular solution y vanishes because there is only one degree of freedom. We maytherefore take

! = %z$ = % $ (27.33)

The second-order rate equation is$ sin ! ! ! ! 2 cos ! ! $ = 0,

which upon substituting ! = % $ yields the quadratic equation (27.22) with a = $ sin ! , b = !2, c = 0. Atthe bifurcation point ($ = 1, ! = 0) we get

0.% 2 ! 2% = !2% = 0 (27.34)

27–11


B

R

!

K > 0

K = 0

K > 0 K > 0

K < 0

"


Secondary path

Figure 27.6. The sign of the stiffness coefficient K for the hinged cantilever response.

The two roots of (27.34) as a quadratic equation are

#1 = 0, #2 = !, (27.35)

leading to the solutions

" = 0, ! = 0. (27.36)

These branches are the tangents to the primary (vertical bar) and secondary (tilted bar), respectively, at thebifurcation point. See Figure 27.7.

This Figure also sketches the post-buckling response, which for this problem is easily obtained from theexact equilibrium solutions (27.27). According to the qualitative classification of Chapter 26, the bifurcationpoint is of stable-symmetric type. This subclassification of a symmetric bifurcation point into stable andunstable cannot be discerned, however, from the branch-tangent analysis, because it requires information onthe curvature of the z-directed branch.

27–12

27–13 §27.6 THE HINGED CANTILEVER

B

R

!

"

Secondary path

Tangent 2: ! = 0

Tangent 1: " = 0

Figure 27.7. The two branch directions at bifurcation point of the hinged cantilever.

27–13


Homework Assignments for Chapter 27Nonlinear Bifurcation Analysis

EXERCISE 27.1 [A:15] Consider

L = !K!u

, N = !K!"

, a = !!q!"

. (E27.1)

Are these relations true?

EXERCISE 27.2 [A:20] If a " 0 in the quadratic equation (27.22) while b #= 0, one of the roots, say #1, goesto$ whereas the other one becomes #2 = !c/2b. This is called a symmetric bifurcation. Show that in sucha case the branch direction corresponding to #1 coincides with the buckling mode z, and draw a bifurcationdiagram similar to Figure 27.1.

EXERCISE 27.3 [A:40] Algebraically prove that the roots of the quadratic equation (27.5) are real2

EXERCISE 27.4 [A:25] The LPB first order rate equations are r = Ku! q" = 0, in which K = K0 + "K1and where K0, K1 and q are constant. Using Exercise 25.3 (posted solution), show that LPB can only predictsymmetric bifurcation. What wonderful thing happens if K1y = 0?

EXERCISE 27.5 [A:20] The propped cantilever shown in Figure 28.3 consists of a rigid bar of length Lpinned at A and supported by a linear extensional spring of stiffness k. The spring is assumed to be capableof resisting both tension and compression and retains its horizontal orientation as the system deflects. The barmay rotate all the way around the pin. The rigid bar is subjected to a vertical dead load P that remains vertical.Define dimensionless control and state parameters as

" = PkL

, µ = sin $ . (E27.2)

Analyze the stability of the propped cantilever in a manner similar to §27.5. Show that the secondary equi-librium path is the circle "2 + µ2 = 1 and sketch the response paths showing the complete circle. From thisdiagram, can you tell whether the bifurcation point at " = 1 is stable-symmetric or unstable-symmetric? Howabout the one at " = !1?

2 A very difficult assignment worth of a paper. I am not aware of anybody that has done for the general case.

27–14

.

28Imperfections

28–1

Chapter 28: IMPERFECTIONS 28–2

TABLE OF CONTENTS

Page§28.1. No Body is Perfect 28–3§28.2. The Imperfect Hinged Cantilever 28–3

§28.2.1. Equilibrium Analysis . . . . . . . . . . . . . . . 28–3§28.2.2. Critical Point Analysis . . . . . . . . . . . . . . 28–3§28.2.3. Discussion . . . . . . . . . . . . . . . . . . 28–4

§28.3. The Imperfect Propped Cantilever 28–5§28.4. Parametrizing Imperfections 28–7§28.5. Imperfection Sensitivity at Critical Points 28–8

§28.5.1. Limit Point . . . . . . . . . . . . . . . . . 28–9§28.5.2. Asymmetric Bifurcation . . . . . . . . . . . . . . 28–9§28.5.3. Stable Symmetric Bifurcation . . . . . . . . . . . 28–10§28.5.4. Unstable Symmetric Bifurcation . . . . . . . . . . . 28–10

§28.6. Extensions: Multiple Bifurcation, Continuous Systems 28–10§28. Exercises . . . . . . . . . . . . . . . . . . . . . . 28–12

28–2

28–3 §28.2 THE IMPERFECT HINGED CANTILEVER

§28.1. No Body is Perfect

In the previous four Chapters we have been concerned with the behavior of geometrically perfectstructures. For the geometrically nonlinear analysis of slender structures, such as those used inaerospace products, we must often take into account the presence of imperfections. It is useful todistinguish two type of imperfections, one associated with the physical structure, the other with thecomputational model.

Physical imperfections. Physical imperfections may be categorized into fabrication and load im-perfections. Real structures inevitably carry geometric imperfections inherent in their manufacture.In addition, loads on structural members that carry primarily compressive loads, such as columnsand cylindrical shells, are not necessarily centered. The load-carrying capacity of certain classesof structures, notably thin shells, may be significantly affected by the presence of physical imper-fections. We shall see that high sensitivity to the presence of small imperfections is a phenomenonassociated with certain types of critical points. Structures that exhibit high sensitivity are calledimperfection sensitive.

Numerical imperfections. Imperfections may be incorporated in the computational model for vari-ous reasons. Numerical imperfections may be used to either simulate actual physical imperfectionsor to “trigger” the occurrence of certain types of response. One common application of numericalimperfections is in fact to “nudge” the structure along a post-bifurcation path, as in Exercises 21.2and 21.3.We begin the study of the effect of imperfections through a simple yet instructive one-degree-of-freedom example: the imperfect hinged cantilever.

§28.2. The Imperfect Hinged CantileverWe take up again the critical-point analysis of the hinged cantilever already studied in §25.5. But we assumethat this system is geometrically imperfect in the sense that the rotational spring is unstrained when the rigidbar “tilts” by a small angle ! with the vertical. By varying ! we effectively generate a family of imperfectsystems that degenerate to the perfect system when ! ! 0.Denoting again the total rotation from the vertical by " as shown in Figure 28.1, the strain energy of theimperfect system can be written

U (", !) = 12 k(" " !)2. (28.1)

The potential energy of the imperfect system is

#(", $, !) = U " V = 12 k(" " !)2 " f L(1" cos ") = k

!

12 (" " !)2 " $(1" cos ")

"

, (28.2)

in which as before we take $ = f L/k as dimensionless control parameter.

§28.2.1. Equilibrium AnalysisThe equilibrium equation in terms of the angle " as degree of freedom is

r = %#

%"= k(" " ! " $ sin ") = 0. (28.3)

Therefore, the equilibrium path equation of an imperfect system is

$ = " " !

sin ". (28.4)

28–3


rigidL

θ

k

ε p

Figure 28.1. The imperfect hinged cantilever. The imperfectionparameter is the initial tilt angle !.

§28.2.2. Critical Point Analysis

The first-order incremental equation in terms of " is the same as in Chapter 25:

K " ! q# = 0, (28.5)

where

K = $r$"

= k(1! # cos "), q = $r$"

= k sin " . (28.6)

We have stability if K > 0, that is1! # cos " > 0, (28.7)

and instability if K < 0, that is1! # cos " < 0. (28.8)

Critical points are characterized by K (#cr ) = 1! #cr cos " = 0, or

#cr = 1cos "

. (28.9)

On equating this value of # with that given by the equilibrium solution (28.4) we obtain

" ! ! = tan " . (28.10)

This relation characterizes the locus of critical points as ! is varied. It is not difficult to show that these criticalpoints are limit points if ! "= 0 (imperfect systems) and a bifurcation point if and only if ! = 0 (perfect system).

28–4

28–5 §28.3 THE IMPERFECT PROPPED CANTILEVER

-2 -1 0 1 2

0.25

0.5

0.75

1

1.25

1.5

1.75

2

(rad)

Unstable

Stable

!

" > 0

" > 0

" < 0

" < 0

#

Figure 28.2. Equilibrium paths of the imperfect (" != 0)and perfect (" = 0) hinged cantilever.

§28.2.3. Discussion

The response of this family of imperfect systems is displayed in Figure 28.2.In this Figure heavy lines represent the response of the peerfect system whereas light lines represent theresponses of imperfect systems for fixed values of ". Furthermore continuous lines identify stable equilibriumpath portions whereas broken lines identify unstable portions. We see that systems with a positive " giveequilibrium paths in two opposite quadrants while systems with a negative " give equilibrium paths in theremaining two quadrants. The equilibrium paths of the imperfect systems collapse onto the equilibrium pathsof the perfect system as " goes to zero. The locus of critical-point equilibrium states given by (28.10) separatesthe stable and unstable domains and is shown in Figure 28.2 as curve ss.We see that a given imperfect system loaded from its unstrained state will give rise to a constantly rising pathso that no instability is encountered; the deflections merely growing more rapidly as the critical load of theperfect system is passed. In addition to this natural equilibrium path an imperfect system will also have acomplementary path which lies in the opposing quadrant. However, this path (partly stable and partly unstable)will not be encountered in a natural loading process that starts from ! = 0.The response shown in Figure 28.2 is well knwon to structural engineers and is exhibited by the familiar Eulercolumnwhich is taught in elementary courses of mechanics of materials. In §28.5 it is shown that this behavioris characteristic of systems that possess a stable-symmetric bifurcation point.

§28.3. The Imperfect Propped Cantilever

The perfect propped cantilever is shown in Figure 28.3. It differs from the hinged cantilever in that it issupported by an ordinary (rectilinear) spring of stiffness k attached to the top. An imperfect version is shownin Figure 28.4, where the initial horizontal displacement "L defines the imperfection parameter ".

28–5


rigid column

k

L!

u = L sin !

p

L(1 ! cos !)

Figure 28.3. The perfect propped cantilever.

k

L

L

!

p

" u = L sin !

L(1 ! cos !)

Figure 28.4. The imperfect propped cantilever. The imperfectionparameter is ", where "L is the displacement from thevertical at which the rectilinear spring is unstrained.

The potential energy of the imperfect structure is

#(u, f ) = U ! V = 12 k(u ! "L)2 ! f L (1 ! cos !) = 1

2 k(u ! "L)2 ! f L!

1 ! (u/L)2 (28.11)

where u = L sin ! is the total horizontal displacement from the vertical, and a constant term has been droppedfrom V . It is convenient to take the ratio $ = f L/k as dimensionless control parameter and µ = u/L as the

28–6

28–7 §28.4 PARAMETRIZING IMPERFECTIONS

! > 0

! > 0

! < 0

! < 0

"

Stable

-1 -0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

1.2

1.4

Unstable

µ

Figure 28.5. Equilibrium paths of the imperfect (! != 0)and perfect (! = 0) propped cantilever.

dimensionless state variable. Then the potential energy, upon dividing by kL2, becomes

#(µ, ") = 12 (µ " !)2 " "

!

1" µ2 (28.12)

The residual equation in terms of µ, " and the imperfection parameter is

r(µ, ", !) = µ " ! " "µ

!

1" µ2= 0 (28.13)

Carrying out the analysis as in the previous section, one finds the response paths depicted in Figure 28.5.The important difference is that the bifurcation point is now of unstable-symmetric type. The equilibriumpaths that emerge from the unloaded state of the imperfect systems are no longer rising but exhibit limit-pointmaxima that may be viewed as failure loads. These limit points occur at lower loads than the bifurcation loadof the perfect structure.Therefore, the load-carrying capacity of the propped cantilever is adverseley affected by the imperfection, andthe structure is said to be imperfection-sensitive.

§28.4. Parametrizing Imperfections

The treatment of imperfections in the foregoing two examples illustrates many features that recurin more complex cases. Imperfect systems are derived as perturbations of the perfect system.Imperfections in real structures are seldom known precisely. They are usually random quantitiesthat can be rigorously treated only by stochastic techniques. Such a treatment, however, would behopelessly expensive in nonlinear systems. A more practical deterministic approach consists oflooking at aparametrized familyof imperfect systems characterized by a dimensionless imperfectionparameter !.

28–7


!

!

> 0

! < 0

"

! = 0

"cr

L

µ

Figure 28.6. Effect of initial imperfectons at a limit point.

The system corresponding to ! = 0 is called the perfect system. Systems corresponding to ! != 0are described as imperfect systems. Parameter ! is inserted in the potential energy #(u, ", !). Foreach fixed ! the analysis proceeds along the usual lines: residual equilibrium equations, first orderincremental equations, finding critical points, and so on.The physical interpretation of the parameter depends on the type of structure. For example, in theanalysis of thin shells whose thickness is accurately controlled a natural choice would be the ratio ofthe expected imperfection amplitude to the thickness. If the thickness itself may vary locally aboutits nominal value— as it would happen, for example, in reinforced concrete shells— the thicknessvariation may be taken as the imperfection parameter. In general we may say that the choice ofparameter is tied up to the fabrication method whereas the value of the parameter is determined byfabrication or construction quality control. In the case of mass-produced systems imperfection datais sometimes available from actual field measurements.

§28.5. Imperfection Sensitivity at Critical Points

The effect of imperfections on the load-carrying capacity of a structure that fails at a critical pointmay drastically vary according to the type of critical point. Structures whose failure loads aresubstantially reduced by imperfections are called imperfection sensitive. In this Chapter we reviewthe question of sensitivity for the four types of critical points introduced in Chapter 23, using typicalresponse plots.In the following two-dimensional response plots, the dimensionless control parameter " is plottedagainst a representative state parameter. This " is assumed to be a load multiplier or load factor thatcharacterizes the strength of the structure. Following the same conventions as in the example of§25.2 heavy lines represent the equilibrium path of theperfect system while light lines represent theequilibrium paths of imperfect systems. Furthermore continuous lines represent stable equilibriumpath segments whereas broken lines represent unstable equilibrium segments.

28–8

28–9 §28.5 IMPERFECTION SENSITIVITY AT CRITICAL POINTS

!

"

B

! > 0

! > 0

! < 0

! < 0

"cr

µ

Figure 28.7. Effect of initial imperfectons at a asymmetric bifurcation point.

§28.5.1. Limit Point

Figure 28.6 pertains to a limit point. We see that the response of the imperfect system is notdissimilar from that of the corresponding perfect system. The peak or failure load measured by"cr varies quasi-linearly with the imperfection parameter ! and this variation "cr (!) is shown in theright-hand diagram of Figure 28.6. As can be observed the function "cr (!) has normally a finiteand nonzero slope and exhibits no singular behavior as ! ! 0. We may characterize a system thatfails at a limit point typified by the response of Figure 28.6 as being mildly imperfection sensitive.

§28.5.2. Asymmetric Bifurcation

Typical pictures for an asymmetric point of bifurcation are shown in Figure 28.7. Now we see thatimperfections play a far more significant role in changing the critical-point response of the systemthan in the previous case.For a small positive1 value of !, the system loses stability at a limit point that corresponds toa drastically reduced value of ". On the other hand a system with a small negative value of !

apparently exhibits no instability in the vicinity of the bifurcation point and follows a stable risingpath. We might note, however, that this continuously rising path travels a region of metastabilityand consequently it may not be reliable in the presence of small dynamic disturbances.The variation of the failure load factor"cr with the imperfection parameter !, shown in the right-handdisgram, is now of considerable interest. For positive imperfections the function "cr (!) is locallyparabolic, having no singularities as ! ! 0 but an infinite slope as shown. Thus there is an extremesensitivity to initial positive imperfections. Since systems with negative imperfections display nolocal failure loads their “buckling” is characterized by a more rapid growth of the deflections as the

1 Positive in the sense that it reinforces the bifurcation phenomenon.

28–9


!

!

> 0 ! < 0

"

! = 0

B

"cr

µ

Figure 28.8. Effect of initial imperfectons at a stable-symmetric bifurcation point.

critical load level of the perfect system is reached. It follows that there is no local branch of the"cr (!) curve for ! < 0.

§28.5.3. Stable Symmetric Bifurcation

Typical pictures for an stable symmetric point of bifurcation are shown in Figure 28.8. The generalbehavior is similar to that encountered in the hinged-cantilever example of §25.2. We see thatimperfections play here a relatively minor role in changing the response of the system. Smallpositive and small negative imperfections have similar effects, each yielding a continuously stableand rising equilibrium paths as shown. Therefore, imperfect systems of this type display no sharpfailure load, “buckling” being simply characterized by a more rapid growth of the deflections as thecritical load of the perfect system is approached.

§28.5.4. Unstable Symmetric Bifurcation

Finally, typical pictures for an unstable symmetric point of bifurcation are shown in Figure 28.9. Wesee that imperfections play here a significant role in modifying the behavior of the system althoughthe effect is not so drastic as in the asymmetric bifurcation case. Small positive and small negativeimperfections have symmetrical effects, each now inducing failure at a limit point that correspondsto a considerably reduced value of "cr (!). The variation of "cr with ! is shown in the right-handdiagram. For both positive and negative imperfections the function follows locally (that is, near! = 0) the so-called “two-thirds law:”

" ! !2/3, (28.14)

discovered originally by Koiter in the 1940s. This law yields a sharp “cusp“ at ! = 0 as shown. Wecan summarize this case as being one of high imperfection sensitivity to both positive and negativeimperfections.

28–10

28–11 §28.6 EXTENSIONS: MULTIPLE BIFURCATION, CONTINUOUS SYSTEMS

!

B

" > 0

"

"

> 0

" < 0

" < 0

!cr

µ

Figure 28.9. Effect of initial imperfectons at an unstable-symmetric bifurcation point.

§28.6. Extensions: Multiple Bifurcation, Continuous Systems

The preceding discussion, as well as the treatment of Chapter 24, pertains to limit points and isolatedbifurcation points of discrete structural systems. This rises questions as to what happens at multipleor compound bifurcation points, or critical points of continuous systems.At a multiple bifurcation point of order k (order being the rank deficiency of the tangent stiffnessmatrix there) a second-order analysis similar to that carried out in §27.4 show that one can expectin general 2k crossing branches. For an isolated bifurcation point k = 1 and 2k = 2 as previouslyfound. But if k ! 2much greater complexity of behavior can be expected. At a multiple bifurcationpoint the effect of initial imperfections is typically far more severe than at single bifurcation pointsif one or more of the emanating branches are unstable, as is usually the case. The systematicinvestigation of these effects remains a frontier research subject, which is nonetheless gaining inimportance because of its application to stability-optimized aerospace structures that are designedto simultaneously fail in several buckling modes.As for continuous systems, the aforementioned two-thirds power law for the imperfection sensitivityat an unstable-symmetric bifurcation point carries over to continuous systems if the geometricimperfection is assumed to have the shape of the buckling mode. If this common assumption is notmade new power laws may emerge from the continuous analysis. As for asymmetric buckling, itremains poorly understood in the continuous case.

28–11



Imperfections

EXERCISE 28.1 Work out the details of the analysis of the imperfect propped cantilever described in §28.3.In particular, verify that the diagram shown in Figure 28.5 is correct. Obtain the equation of the imperfectionsensitivity diagram !cr (") where !cr are load limits (limit points) obtained when " != 0. Plot this diagram forvalues of " = 0 to 1; note vertical-tangent “cusp” at " = 0! [Qualitatively, the diagram should look like theone in Figure 28.9.]

EXERCISE 28.2 A highly simplified, one-DOF “beer-can-like” structure has the total potential function

#(u, f, ") = U " V, U = kL2(!

1+ (u/L) " 1" $ ")2, V = f L"

1+ (u/L)" "!

1" (u/L)2#

(E28.1)

where u is the state variable, f is the applied load, k, L are structure property constants with dimensions ofspring constant and length, respectively, $ is a dimensionless constant, and " is a dimensionless geometricimperfection parameter. Reduce (E28.1) to a dimensionless form

#(µ, !, ") (E28.2)

by defining the state parameter µ = u/L and the control parameter ! = f/(kL), and dividing the wholething through by kL2. From then on take $ = 0. Form the equilibrium equations and generate a responsediagram for the imperfect structure similar to those shown in Figures 28.2 and 28.5, varying " in the range(-1,1), µ in the range (-0.99,0.99), and ! in (0,1.5). From visual inspection conclude whether the structureexperiences asymmetric bifurcation (that is, it has a preferred buckling direction) or a symmetric one. Drawthe imperfection sensitivity diagram of !cr versus ". What is the load capacity drop for imperfections ofmagnitude 0.01, 0.1 and 1?

Using Graphic Tools to Expedite HW

Use of built-in graphic tools such as those provided in Matlab, Mathematica or Maple can speed up signifi-cantly the generation of response diagrams for Exercises 28.1 and 28.2. For example, an initial version of thediagram shown in Figure 28.2 was produced by the following Mathematica script:

lam[theta_,eps_]:=(theta-eps)/Sin[theta];p1=Plot[{lam[theta,0.01],lam[theta,-0.01],lam[theta,0.1],lam[theta,-0.1],lam[theta,0.2],lam[theta,-0.2],lam[theta,0.5],lam[theta,-0.5]},

{theta,-Pi/1.2,Pi/1.2},PlotRange->{0,2},DisplayFunction->Identity];p2=Plot[lam[theta,0],{theta,-Pi/1.2,Pi/1.2},PlotRange->{0,2},DisplayFunction->Identity];p3=Plot[1/Cos[theta],{theta,-1.5,1.5},PlotRange->{0,2},DisplayFunction->Identity];Show[Graphics[Thickness[0.002]],p1, Graphics[Thickness[0.004]],p2,

Graphics[Thickness[0.004]],Graphics[AbsoluteDashing[{5,5}]],p3,PlotRange->{0,2},Axes->True,AxesLabel->{"theta","lambda"},DisplayFunction->$DisplayFunction];

The plot cell was then converted and saved as an Adobe Illustrator 88 file, picked up by Adobe Illustrator 6.0and “massaged” for bells and wistles such as Greek labels, dashed lines, shading of unstable region, etc.

28–12

.

29Nonconservative

Loading

29–1

Chapter 29: NONCONSERVATIVE LOADING 29–2

TABLE OF CONTENTS

Page§29.1. Introduction 29–3§29.2. Potential Force Example: Gravity 29–3§29.3. Follower Load and Associated Load Stiffness 29–5§29.4. General Characterization of the Load Stiffness 29–6§29.5. Forces Produced by Fluid Motion 29–7§29.6. Load Stiffness For 2D Fluid Motion 29–9§29. Exercises . . . . . . . . . . . . . . . . . . . . . . 29–11

29–2

29–3 §29.2 POTENTIAL FORCE EXAMPLE: GRAVITY

§29.1. IntroductionIn Chapter 5 a mechanical system was defined to be conservative when both external and internalforces are derivable from a potential. In this course we consider only elastic systems; conse-quently the internal forces are derivable from an strain (internal) energy potential U . Thus theconservative/nonconservative character depends on whether the external loads are conservative ornonconservative.Conservative applied forces fmay be derived from the external loads potential V by differentiatingwith respect to the state variables:

f = !V!u

. (29.1)

Nonconservative forces, on the other hand, are not expressable as (29.1). They have to be workedout directly at the force level.In the present Chapter we will give examples of both force types in conjunction with the TL-formulated two-node bar element The main result is that consideration of nonconservative loadscontributes an unsymmetric component, called load stiffness, to the tangent stiffness matrix. Treat-ing this effect in stability analysis requires a dynamic criterion, which is covered in Chapter 30.

Remark 29.1. The chief sources of nonconservative forces in various branches of engineering are:1. Aerodynamic forces (aerospace, civil); hydrodynamic forces (mechanical, marine, chemical); aircraft

and rocket propulsion forces (aerospace); frictional forces (mechanical, civil).2. Gyroscopic forces (aerospace, electrical).3. Active control systems (aerospace, electrical, mechanical).In this Chapter we consider only hydrodynamic (fluid motion) forces as prototype example.

§29.2. Potential Force Example: GravityConsider the two-node, three-dimensional bar element immersed in a gravity field of constantstrength g acting along the global !Z axis, as illustrated in Figure 29.1. The bar has referencelength L0, reference area A0 and mass density ". The element coordinate systems are labeled asfollows:

x0, y0, z0 in the reference configuration C0x, y, z in the current configuration C

This distinction between local coordinate systems is introduced here as it becomes necessary inlater Sections. Take a differential element of bar of length dx0 in C0. This moves to a corresponsingposition in C, with a vertical displacement of uz with respect to C0. See Figure 29.2. The workpotential gained by this displacement is

dV = !"gA0uz(x0) dx0 (29.2)

The external potential of the bar element is obtained by linearly interpolating uz = (1!# )uz1+#uz2,# = x/L0 and integrating over the bar length:

V = !! L

0"gA0uzd x0 = !

! 1

0A0g [1! # # ]

"

uz1uz2

#

L0d#

= !"gA0L0 12 (uz1 + uz2).(29.3)

29–3


1

2C

C0

x

x0

X, x

Y, y

Z , z

g = directed along ! z

E, A0, L0

Figure 29.1. TL bar element displacing in a gravity field g.

1

2C

C0

x

x0

X, x

Y, y

Z , zg

!A0 dx0

vz

Figure 29.2. Calculation of external potential.

(As usual in the TL kinematic description, all quantities are referred to C0.) It follows that theexternal force vector for the element is

fg = "V"u

=

!

"

"

"

"

"

#

"V/"ux1"V/"uy1"V/"uz1"V/"ux2"V/"uy2"V/"uz2

$

%

%

%

%

%

&

= ! 12!A0L0

!

"

"

"

"

"

#

001001

$

%

%

%

%

%

&

. (29.4)

This can also be derived through basic principles of statics. Note that this vector is independent of

29–4

29–5 §29.3 FOLLOWER LOAD AND ASSOCIATED LOAD STIFFNESS

C

!

X, x

1 2

L0

L

pd

pd

C0

Y, y

yx

x0 ! X

uY1

uX1

uX2

uY2

Figure 29.3. 2D bar under constant “follower” pressure pd .

the current configuration. This is a distinguishing feature of external work potentials that dependlinearly on the displacements, such as (29.3).

§29.3. Follower Load and Associated Load StiffnessTo illustrate the concept of load stiffness with a minimum of mathematics, let us consider a two-dimensional specialization. The bar element originally lies along the x axis in the reference con-figuration C0 and moves in the (x, y) plane to C, which forms an angle ! with x . The bar is under aa constant pressure pd that is always normal to the element as it displaces, as shown in Figure 29.3.This kind of applied force is called a follower load in the literature.1

From statics the external force vector is obviously

f = 12 pd L

!

"

"

"

"

"

#

" sin !

cos !0

" sin !

cos !0

$

%

%

%

%

%

&

(29.5)

From geometry

cos ! = L0 + uX21L

, sin ! = uY21L

, with uX21 = uX2 " uX1, uY21 = uY2 " uY1, (29.6)

1 Such loads are often applied by fluids at rest or in motion. The latter case is studied in Sections 29.4-5.

29–5


Consequently

f = 12 pd

!

"

"

"

"

"

#

!uY21L0 + uX21

0!uY21

L0 + uX210

$

%

%

%

%

%

&

. (29.7)

Take now the partial of the negative of this external load vector with respect to u. The result is amatrix with dimensions of stiffness, denoted by KL :

KL = ! !f!u

= 12 pd

!

"

"

"

"

"

#

0 !1 0 0 1 01 0 0 !1 0 00 0 0 0 0 00 !1 0 0 1 01 0 0 !1 0 00 0 0 0 0 0

$

%

%

%

%

%

&

. (29.8)

KL is called a load stiffness matrix. It arises from displacement-dependent loads.2 We can seefrom this example that KL is unsymmetric. A consequence of this fact is that (29.2) does not havea potential V that is a function of the node displacements.3

§29.4. General Characterization of the Load Stiffness

Suppose that we have a one-parameter conservative system with displacement dependent forces.Then

" = U (u) ! V (u, #), (29.9)where the external potential V = V (u, #) depends on the displacements u in a general fashion.Then

r = !"

!u= !U

!u! !V

!u= p! f, (29.10)

K = !r!u

= !p!u

! !f!u

. (29.11)

The partial !p/!u gives KM +KG , the material plus geometric stiffness, as discussed in previousChapters. The last term gives KL , the conservative load stiffness

KL = ! !f!u

= !!2V!u2

(29.12)

which is called the conservative load stiffness. This matrix is obviously symmetric because it is thenegated Hessian of V (u, #) with respect to u. Consequently

K = KM +KG +KL . (29.13)

2 This source of nonlinearity was called force B.C. nonlinearity in Chapter 2.3 If KL were symmetric we could work backwards and integrate (29.5), expressed in terms of the node displacements, tofind the potential function V .

29–6

29–7 §29.5 FORCES PRODUCED BY FLUID MOTION

These three components of K are symmetric, and so is K.Now consider a more general structural system subject to both conservative and non-conservativeloads:

r = p! fc ! fn, (29.14)

Here fc = !V/!u whereas fn collects external forces not derivable from a potential. Then

K = !r!u

= KM +KG +KLc +KLn. (29.15)

The nonconservative load stiffness matrix, KLn , is unsymmetric.

Remark 29.2. In practice one derives the total force f from statics, as in the example of §29.3, and obtainsKL by taking the partials with respect to the displacements in u. If the resulting stiffness is unsymmetric theload is nonconservative. The splitting of KL into a symmetric matrix KLc and unsymmetric part KLn can bedone in a variety of ways. (If the unsymmetric part is required to be antisymmetric, however, the splitting isunique.)

§29.5. Forces Produced by Fluid Motion

To study in more detail a frequent source of non-conservative follower loads, suppose that the barelement is submerged in a moving fluid whose flow is independent of time— i.e., a steady flow.See Figure 29.4. We neglect “feedback” effects on the flow due to the presence and motion of thebar. The steady notion can be described by the fluid-particle velocity field4

u f (X, Y, Z) =! u f X (X, Y, Z)

u f Y (X, Y, Z)

u f Z (X, Y, Z)

"

, (29.16)

For simplicity in the formulation below, we further assume that the velocity field is uniform, i.e.,does not depend upon (X, Y, Z), and that it is directed along the x axis:

u f =! u f X00

"

, (29.17)

where u f X is independent of position.By virtue of drag effects the fluid motion exerts a normal drag force pd (force per unit length) uponthe bar in the current configuration C. The drag force is normal to the bar longitudinal axis x andit is a function of the magnitude of the velocity component normal to that axis. Furthermore if thebar cross section is circular or annular, the force is coaxial with the normal velocity vector. Foradditional simplicity we shall assume that the cross section satisfies such a geometric constraint5

4 The symbol u and its vector counterparts u and "u are commonly used in fluid mechanics to denote velocities ratherthan displacements as in structural and solid mechanics. In fact displacements are rarely used in fluids. Subscript f isintroduced here to lessen the risk of confusion with structural displacements.

5 For arbitrary cross sections, the fluid motion exerts drag and lift forces, the latter being normal to the bar axis and to thenormal velocity vector. Lift forces are what makes airplanes fly. This more general situation is dealt with in treatises onaerodynamics, wind forces and hydraulics.

29–7


X, x

Y, y

Z , z

C

v f X in (x, X) plane

y

z

x

fluid velocity vector

C0

x0

Figure 29.4 Bar element in steady fluid flow.

For slow (laminar) flow the drag force is proportional to the magnitude of the normal velocitycomponent whereas if the motion is fully turbulent it is proportional to the square of that velocity.We assume here the latter case. Other drag-velocity dependencies can be similarly treated.

Consider the bar in the (x, X) plane as illustrated in Figure 29.5, and let y be defined as the normalto the element axis x that is located in this plane and forms an acute angle ! with x . The drag forceon the element per unit length is directed along y and has the value

pd = 12Cd " f d u2f n (29.18)

where Cd is the drag coefficient,6 " f the fluid mass density, d the “exposed width” (for a barof circular cross-section, its external diameter), and u f n the fluid-normal velocity u f X cos ! (seeFigure). The total force on the element is pd L , where L is the current length, and this force “lumps”into 1

2 pd L at each node.

In order to refer these forces to the global X, Y, Z axes, we need to know the direction cosines t21,t22 and t23 of y with respect to x, y, z. Then the hydrodynamic node force vector in the (X, Y, Z)

6 CD is a dimensionless number tabulated in fluid dynamic handbooks

29–8

29–9 §29.6 LOAD STIFFNESS FOR 2D FLUID MOTION

x

C

y, npaper is plane (x, X) ! plane (x, y)

v f X

v f n

!

X, x

1

2

Figure 29.5. Normal fluid velocity component in the current bar configuration

system is

f = 12 pd L

!

"

"

"

"

"

#

t21t22t23t21t22t23

$

%

%

%

%

%

&

(29.19)

To compute these direction cosines, one proceeds as follows:

(1) Compute the direction z by taking the cross product of x and X .(2) Compute the direction y by taking the cross product of z and x .

If x and X are parallel, step (1) does not define z but then the fluid flow occurs along the elementaxis and the pressure pd vanishes.

Remark 29.3. If the fluid flow is uniform with speed u f j along a general direction j ! "j , the precedingderivation must be modified by taking "z = "x # "j , "y = "z # "x , ! = angle("y, "j). Observe that it would beincorrect to decompose u f j onto its components in the X , Y and Z directions and superpose associated forces,because the drag force is nonlinear in the velocity.

Remark 29.4. If the flow is steady but nonuniform, numerical integration over elements is generally required.For this simple elkement integration with the flow velocity evaluated at the element center is often sufficient.

§29.6. Load Stiffness For 2D Fluid Motion

To show what kind of load stiffness is produced by fluid drag forces, consider again the case ofFigure 29.4 but now make pd depend on the “tilt” ! as explained in §29.3; see Figure 29.6. Sincea turbulent-motion-induced drag force is proportional to the square of u f n = u f X cos ! , it may beexpressed as

pd = pd0 cos2 ! (29.20)

where pd0 is pd for ! = 0 (bar normal to fluid motion).

29–9


C

!

X, x

1 2

u

L0

L

C0

Y, y

yx

x0 ! X

pd0

p d =pd0cos2 !

Y1

uX1

uX2

uY2

Figure 29.6. Follower pressure pd on a 2D bar that depends on the “tilt angle” ! .

The external load vector is

f = 12 pd0L

!

"

"

"

"

"

#

" sin ! cos2 !

cos3 !

0" sin ! cos2 !

cos3 !

0

$

%

%

%

%

%

&

(29.21)

To differentiate this expression under the assumption that pd0 does not depend on the node dis-placements, and that L is constant, we need partial derivative expressions such as

"(" sin ! cos2 !)

"uX21= "2 sin ! cos !

" cos !"uX21

" cos2 !" sin !

"uX21

= 1Lsc(c2 " 2s2) = 1

Lsc(1" 3s2),

(29.22)

etc. The resulting load stiffness KL = ""f/"u is more complicated than (29.8), but still can beobtained in closed form.If L is let to vary, then one can substitute cos ! = (L0 + uX21)/L and sin ! = uY21/L to put f interms of uX21 and uY21, and the differentiation to get KL becomes straightforward. Thus the exactexpression is in fact easier to work out than the approximate one. The details of the derivation areworked out in Exercise 29.5.

29–10

29–11 Exercises

Homework Exercise for Chapter 29

Nonconservative Loading

EXERCISE 29.1 (A:20) Work out fd for the case of a uniform flow of speed u f j in a general direction !j asdescribed in Remark 29.2.

EXERCISE 29.2 (A:15) Specialize the result of Exercise 29.1 to the two dimensional case (bar and flow inthe x, y plane). Differentiate to obtain KL , comparing with (29.20).

EXERCISE 29.3 (A:20) In the previous exercise take into account the effect of friction forces exerted on thebar by the flow. Use the linear model: the tangential friction force pt per unit length of the bar is directedalong x and has the value C f au f t , where C f is a friction coefficient, a is the “exposed perimeter” of the bar(for a circular cross section, a = 2!d), and u f t = u f j sin " is the tangential velocity (fluid velocity projectedon the current bar direction, with proper sign).

EXERCISE 29.4 (A:20) Prove the formulas (29.10).

EXERCISE 29.5 (A:20) Complete the derivation of KL in §29.6.

EXERCISE 29.6 (A:30) A simple example of a gyroscopic force is a torsional moment Mx directed along thelongitudinal axis x of a beam-column element, which keeps pointing in that direction as the element movesand rotates. Obtain the gyroscopic force vector fn and associated load stiffness KLn for a three-dimensionalbeam column of length L currently directed along the global x axis. The element degrees of freedom are

uT = [ ux1 uy1 uz1 "x1 "y1 "z1 ux2 uy2 uz2 "x2 "y2 "z2 ] . (E29.1)

For this “moment tilting” analysis it is sufficient to assume that: (a) node 1 stays fixed, (b) the element remainstraight, and (c) any deviations from the current x direction are infinitesimal.

29–11

.

30DynamicStabilityAnalysis

30–1

Chapter 30: DYNAMIC STABILITY ANALYSIS 30–2

TABLE OF CONTENTS

Page§30.1. Introduction 30–3§30.2. The Linearized Equations of Motion 30–3§30.3. The Characteristic Problem 30–4

§30.3.1. Connection with the Free-Vibration Eigenproblem . . . . . 30–5§30.4. Characteristic Exponents and Stability 30–5

§30.4.1. Negative Real Case: Harmonic Oscillations . . . . . . . 30–5§30.4.2. Positive Real Case: Divergence . . . . . . . . . . . 30–6§30.4.3. Complex Case: Flutter . . . . . . . . . . . . . . 30–7§30.4.4. Stable and Unstable Regions in the Complex Plane . . . . . 30–8

§30.5. Graphical Representations 30–8§30.5.1. Root locus plots . . . . . . . . . . . . . . . . 30–8§30.5.2. Amplitude Plots . . . . . . . . . . . . . . . . 30–10

§30.6. Regression to Zero Frequency and Static Tests 30–11§30. Exercises . . . . . . . . . . . . . . . . . . . . . . 30–12§30. Solutions to Exercises. . . . . . . . . . . . . . . . . . . . . . 30–16

30–2

30–3 §30.2 THE LINEARIZED EQUATIONS OF MOTION


If the loading is nonconservative the loss of stability may not show up by the system going intoanother equilibrium state but by going into unbounded motion. To encompass this possibility wemust consider the dynamic behavior of the system because stability is essentially a dynamic concept(recall the definition in §25.1).The essential steps are as follows. We investigate the motion that occurs after some initial per-turbation is applied to the equilibrium state being tested, and from the properties of the motionwe can infer or deny stability. It if turns out that the perturbed motion consists of oscillations ofincreasing amplitude, or is a rapidly increasing departure from the equilibrium state, the equilibriumis unstable; otherwise it is stable.The practicality of this approach depends crucially on the linearization of the equations of motion ofthe perturbation. Thus we avoid having to trace the ensemble of time histories for every conceivabledynamic departure from equilibrium— which for a system with many degrees of freedom wouldclearly be a computationally forbidding task.By linearizing we can express the perturbation motion as the superposition of complex exponentialelementary solutions. The characteristic exponents of these solutions can be determined througha characteristic value problem or eigenproblem. This problem includes the free-vibration naturalfrequency eigenproblem as particular case when the system is conservative and the tangent stiffnessmatrix is symmetric. Through the stability criterion discussed in §30.3, the set of characteristic ex-ponents gives complete information on the linearized stability of the system at the given equilibriumconfiguration.In practical studies the characteristic exponents are functions of the control parameter !. Assumingthat the system is stable for sufficiently small ! values, say ! = 0, we are primarily concerned withfinding the first occurrence of ! at which the system loses stability. The transition to instability mayoccur in two different ways, which receive the names divergence and flutter, respectively.1

The distinction between divergence and flutter instability is important in that the singular-stiffnesstest discussed in Chapter 26 remains valid if the stability loss occurs by divergence, although ofcourse the tangent stiffness is not necessarily symmetric. Therefore it follows that in that casewe may fall back upon the static criterion, which is simpler to apply because it does not involvesinformation about mass and damping. Such a regression is not possible, however, if the loss ofstability occurs by flutter.

§30.2. The Linearized Equations of Motion

The structure is in static equilibrium under a given value of the control parameter !. The equilibriumstate is defined by the state vector u. At time2 " = 0 apply a dynamic input (e.g., an impulse) to thisconfiguration and examine the subsequent motion of the system. Roughly speaking if the motionis unbounded (remains bounded) as " tends to infinity the system is dynamically unstable (stable).

1 These names originated in aeronautical engineering applications, more specifically the investigation of sudden airplane“blow ups” during the period 1910-1930. In the mathematical literature flutter goes by the name ‘Hopf bifurcation.’

2 The symbol " denotes real time because t is used throughout the course to denote a pseudo-time parameter. Only realtime is considered in this Chapter.

30–3


As noted in the Introduction, to simplify the mathematical treatment we consider only the localstability condition, in which the imparted excitation is so tiny that the subsequent motion can beviewed as a linearizable perturbation. We are effectively dealing with small perturbations aboutthe equilibrium position.Let M be the symmetric mass matrix, which is assumed positive definite, and K the tangentstiffness matrix, which is real but generally unsymmetric because of load nonconservativeness. Theperturbation motion is denoted as

d(! ) = u(! ) ! u(0), ! " 0+ (30.1)

The discrete, unforced, undamped governing equations of motion are

Md+Kd = 0, (30.2)

in which a superposed dot— unlike previous Chapters— denotes differentiation with respect toreal time. The ordinary differential equations (30.2) express the linearized dynamic equilibriumbetween stiffness and inertial forces. The stiffness forces generally include nonconservative loadingeffects.

Remark 30.1. In structuralwith rotationalDOFs,Mmight be only nonnegative definite because of the presenceof zero rotational masses. If so it is assumed that those DOFs have been eliminated by a static condensationprocess.The assumption of positive definiteness also excludes the presence of Lagrange multipliers in the state vectoru, because the associated masses of such degrees of freedom are zero. Again the stability criteria can beextended by eliminating the multipliers in the linearized equation of motion.

Remark 30.2. We shall ignore damping effects because of two reasons:

(1) The effect of diagonalizable, light viscous structural damping does not generally affect stability results(it certainly does not when stability loss is by divergence). See also Remark 30.4.

(2) The effect of more complicated nonlinear dampingmechanisms such as dry frictionmay not be amenableto linearization.

Thus cases when damping effects are significant lead to mathematics beyond the scope of this course. Readersinterested in pursuing this topic are referred to the vast literature on the subject of dynamic stability.

§30.3. The Characteristic Problem

The linear ODE system (30.2) can be treated by assuming the eigenmodal expansion

d(! ) =!

idi (! ) =

!

izi e pi ! , (30.3)

where i ranges over the number of degrees of freedom (number of state parameters). The pi aregenerally complex numbers called the characteristic exponents whereas the corresponding columnvectors zi are the characteristic modes or characteristic vectors.3

3 In his classical treatise Nonconservative Problems of the Theory of Elastic Stability, (Pergamon, 1963), Bolotin employss for what we call here p, and so do many other authors. This notation connects well to the common use of the Laplacetransform to do more complicated systems. However, we have already reserved s for Piola-Kirchhoff stresses as well asarclength.

30–4

30–5 §30.4 CHARACTERISTIC EXPONENTS AND STABILITY

Replacing di = p2i d into (30.2) yields

(K+ p2iM) zi = 0, (30.4)

which is the characteristic problem or eigenproblem that governs dynamic stability. This equationbefits the generalized unsymmetric eigenproblem of linear algebra

Axi = µiBxi (30.5)

in which matrix A ! K is real and generally unsymmetric whereas B ! M is real symmetricpositive definite. The eigenvaluesµi ! "p2i of this eigenproblemmay be either real or complex; ifthe latter, they occur in conjugate pairs. The square roots of these eigenvalues yield the characteristicexponents pi of the eigenmodal expansion (30.3).

§30.3.1. Connection with the Free-Vibration Eigenproblem

If the system is conservative and stable, K is symmetric and positive definite. If so all roots p2i of(30.4) are negative real and their square roots are purely imaginary numbers:

pi = ± j!i , (30.6)

where j =#

"1, and the nonnegative real numbers !i are the natural frequencies of free vibration.Because p2 = "!2i , (30.4) reduces to the usual vibration eigenproblem

(K" !2iM) zi = 0. (30.7)

Thus for the conservative case we regress to a well studied problem. In such a case the system willsimply vibrate, that is, perform harmonic oscillations about the equilibrium position because eachroot is associated with the solution

e j!i " = cos!i" + j sin!i". (30.8)

The presence of positive damping will of course damp out these oscillations and the system even-tually returns to the static equilibrium position.

§30.4. Characteristic Exponents and Stability

The characteristic exponents are generally complex numbers:

pi = #i + j!i , (30.9)

where #i and !i are real numbers, and j =#

"1. The component representation of the square ofpi is

p2i = (#2i " !2i ) + 2 j#i!i , (30.10)

The exponential of a complex number has the component representation

epi " = e(#i+ j!i )" = e#i " (cos!i" + j sin!i" ), (30.11)

On the basis of this representation we can classify the growth behavior of the subsequent motionand consequently the stability of the system as examined in the next 3 subsections.

30–5


!

i

2"/#i

d

Figure 30.1. Harmonic oscillatory motion for the case where root p2i of (30.4) is negativereal. Equivalently, pi = ± j#i where #i is the circular frequency.

§30.4.1. Negative Real Case: Harmonic Oscillations

Ifpi = ± j#i , d(! ) =

!

di (! ), di (! ) = Ai cos#i! + Bi sin#i!. (30.12)

where Ai and Bi are determined by initial conditions. The motion di associated with ± j#i isharmonic and bounded, as illustrated in Figure 30.1. The system is dynamically stable for thisindividual eigenvalue.If all eigenvalues are negative real and distinct, the system is dynamically stable because anysuperposition of harmonic motions of different periods is also a harmonic motion. If two or moreeigenvalues coalesce the analysis becomes more complicated because of the appearance of secularterms that grow linearly in time. These effects can be studied inmore detail in treatises inmechanicalvibrations.

!

e$i !

e!$i !

id

Figure 30.2. Aperiodic, exponentially growing motion for the real root case p2i = $2i ,pi = ±$i . Transition to this kind of instability is called divergence.

30–6

30–7 §30.4 CHARACTERISTIC EXPONENTS AND STABILITY

!

e"i !

2#/$i

!e"i !

id

Figure 30.3. Periodic, exponentially growing motion for case pi = +"i ± j$i withnonzero "i . Transition to this kind of instability is called flutter.

§30.4.2. Positive Real Case: Divergence

If p2i is positive real,pi = ±"i . (30.13)

The +"i square root will give rise to an aperiodic, exponentially growing motion. The other rootwill give rise to an exponentially decaying motion. When the two solutions are combined theexponentially growing one will dominate for sufficiently large ! as sketched in Figure 30.2, and thesystem is then exponentially unstable.As noted above p2i is generally a function of %. The transition from stability (in which all roots arenegative real) to this type of instability necessarily occurs when a eigenvalue p2i (%), moving fromleft to right as % varies, passes through the origin p2 = 0 of the p2 complex plane. This type ofinstability is called divergence.

§30.4.3. Complex Case: Flutter

If p2i is complex, solutions of the eigenproblem (30.4) occur in conjugate pairs because bothmatricesM and K are real. Consequently, if p2i = ("2i ! $2i ) + j (2"i$i ) is a complex eigenvalueso is its conjugate (p2i ) = ("2i !$2i )! j (2"i$i ). On taking the square root of this pair we find fourcharacteristic exponents

±"i ± j$i . (30.14)

Two of these square roots will have positive real parts (+") and for sufficiently large ! they willeventually dominate the other pair, yielding exponentially growing oscillations; see Figure 30.3.This is called periodic exponential instability or flutter instability.If the system is initially stable (i.e., all roots are negative real) then transition to this type of instabilityoccurs when at a certain value of % two real roots coalesce on the real axis and “branch out” intothe complex p2 plane. This loss of stability is called flutter.

30–7


Remark 30.3. Frequency coalescence is necessary but not sufficient forflutter. It is possible for two frequenciesto pass by other “like ships crossing in the night” without merging. This happens if there is no mechanism bywhich the two associated eigenmodes can exchange energy.

Remark 30.4. The fact that all characteristic motions are either harmonic or exponentially growing is aconsequence of the neglect of damping in setting up the stability problem. As noted in Remark 30.2, thepresence of damping or, in general, dissipative forces, introduces additional mathematical complications thatwill not be elaborated upon here. Suffices to say that the addition of damping to a conservative system hasalways a stabilizing effect (Rayleigh’s theorem). For non-conservative systems, the preceding statement is nolonger true, and indeed several counterexamples involving destabilizing damping have been constructed overthe past 40 years. In spite of this the effect is not often observed in practice.

Remark 30.5. The occurrence of flutter requires the coalescence of two natural frequencies. Consequently,flutter cannot occur in systems with one degree of freedom (“it takes two toflutter”). The physical interpretationof the flutter phenomenon is that one vibration mode absorbs energy and feeds it into another; this transferenceor “energy resonance” becomes possible when the two modes have the same frequency.

UnstableStable

UnstableStable

(a) (b)2!" "

!

p planep2 plane

"2 ! !2

Figure 30.4. Stable and unstable regions in (a) the complex p2 plane, (b) thecomplex p plane. For the latter the stable region is the left-halfplane ! = "(p) # 0. For (a) it is the negative real axis.

§30.4.4. Stable and Unstable Regions in the Complex Plane

From the preceding study it follows that the only stable region in the complex p2-plane is thenegative real axis:

"(p2) < 0, $(p2) = 0. (30.15)

The rest of the p2 complex plane is unstable; see Figure 30.4(a).On the complex p-plane, the stable region is the left-hand plane

! = "(p) # 0. (30.16)

which includes the imaginary axis ! = 0 as stability boundary. The right-hand p-plane ! > 0 isunstable. See Figure 30.4(b).

30–8

30–9 §30.5 GRAPHICAL REPRESENTATIONS

p planep2 plane

DivergenceDivergence

p2i (!)

j pi (!)

! j pi (!)root trajectory

root trajectory

Figure 30.5. Root locus plots on the complex p2 and p planes for divergence instability.

p planep2 plane

Flutter

Flutterp21(!)

j p1(!)

! j p1(!)

p22(!) j p2(!)

! j p2(!)trajectory of interacting roots

trajectory of interacting roots

Figure 30.6. Root locus plots on the complex p2 and p planes for flutter instability.

§30.5. Graphical Representations

§30.5.1. Root locus plots

Graphical representations of the “trajectories” of the eigenvalues pi (!) as ! is varied on the complexp2 or p planes are valuable insofar as enhancing the understanding of the differences betweendivergence and flutter. These are called root locus plots4 and are illustrated in Figures 30.5 and30.6.Figure 30.5 illustrates loss of stability by divergence. As ! is varied, eigenvalue p2

i passes from theleft-hand plane to the right-hand plane through the origin p2 = 0. Stability loss occurs at the ! forwhich p2

i vanishes. The right-hand diagram depicts the same phenomenon on the p plane, for theroot pair ±pi .

4 The word root in root-locus is used as abbreviation for characteristic root or eigenvalue

30–9


|p|

!cr

!

Divergence

|p1| = |"1|

Figure 30.7. Root amplitude plot illustrating loss of stability by divergence at !cr .

|p|

!cr!

|p1| = |"1|

|p1,2|

Flutter

|p2| = |"2|

Figure 30.8. Root amplitude plot illustrating loss of stability by flutter at !cr .

Figure 30.6 illustrates loss of stability by flutter. As ! is varied, two interacting eigenvalues, labeledas p21 and p22, coalesce on the negative real axis of the p2 plane and branch out into the unstableregion. The right-hand diagram depicts the same phenomenon on the p plane for the interactingroots, which appears in complex-conjugate pairs.

§30.5.2. Amplitude Plots

Another commonly used visualization technique is the characteristic root amplitude or simply rootamplitude plots. These plots show the magnitude of pi (!), that is |pi (!)| on the vertical axis against! on the horizontal axis. If the eigenvalue is real, |pi | is simply its absolute value whereas if it iscomplex |pi | is its modulus.This graphical representation enjoys the following advantages: (a) the critical value of ! is displayed

30–10

30–11 §30.6 REGRESSION TO ZERO FREQUENCY AND STATIC TESTS

more precisely than with a locus or trajectory plot, (b) all related square roots such as ±!i ± "i“collapse” into a single value, and (c) the variation of several important roots (for several values ofi) may be shown without cluttering the picture.Figures 30.7 and 30.8 illustrate typical root-amplitude plots in loss of stability by divergence andflutter, respectively.

§30.6. Regression to Zero Frequency and Static Tests

The stability loss by divergence occurs when an eigenvalue pi vanishes. Because "i = 0 if pi = 0,this is equivalent to a zero-frequency test on the eigenproblem

(!"2iM+K) zi = 0. (30.17)

But if "i = 0 andM is positive definite, which we assume, then K must be singular. Therefore wecan regress to the static criterion or singular tangent stiffness test

detK(#) = 0, (30.18)

which allows us to discard the mass matrix. This regression may be useful if one is solving a seriesof closely related problems, for example during the design of a structure which is known a priorito become unstable by divergence.It should be cautioned, however, that the tangent stiffness matrix K for nonconservative systemsis generally unsymmetric (Chapter 29), and that the test for singularity must take account of thatproperty.

30–11



Dynamic Stability Analysis

EXERCISE 30.1 (A+C:25) This Exercise studies the stability of the “follower load” nonconservative systemshown in Figure E30.1.

1 1

3

2

C

C0

u

u

x

y

L

L

!P90!

90!k1

k2

Bar (1)

Bar (2)

2

3

x

y

Figure E30.1. Structure for Exercise 30.1.

Two elastic bars, (1) and (2), are supported at 1 and 3 and hinged at 2. The bars have length L , axial stiffnessesk1 and k2, respectively, and can only move in the x, y plane. Bar (1) is loaded at node 2 by a force !P1,directed upwards, that stays normal to bar (1) as it displaces. Bar (2) is loaded at node 2 by a force !P2,directed leftwards, that stays normal to bar (2) as it displaces.For the present exercise set P1 = P2 = P . Furthermore the following simplifying assumptions are to be made:

(A1) The displacements from the reference configuration are so small that C " C0 insofar as setting up thestability eigensystem5

(A2) The contribution of the geometric stiffness is neglected.

(a) Show that under the simplifying assumptions (A1)–(A2), the tangent stiffness at C " C0 in terms of thetwo degrees of freedom ux = ux2 and uy = uy2, is

K =! k1 00 k2

"

+ !PL

! 0 1#1 0

"

. (E30.1)

5 This is similar to LPB (Chapters 24-25), but here a dynamic analysis is involved.

30–12

30–13 Exercises

The first component of K is the material stiffness whereas the second component is the load stiffness.Hint for the latter: use the results of Remark 30.4

(b) The linearized dynamic eigenproblem (30.4) is

(p2iM+K)zi = 0, i = 1, 2. (E30.2)

The exponents pi (the square roots of p2i ) are generally complex numbers:

pi = !i + j"i , (E30.3)

where ! and " are the real and imaginary part of pi , respectively, zi are associated eigenmodes, andMis the diagonal mass matrix

M =!M 00 M

"

, (E30.4)

where M is the lumped mass at node 2 (half of the sum of the bar masses). By appropriate normalizationshow that the eigenproblem can be reduced to the dimensionless form

#

p2! 1 00 1

"

+!

# 00 1

"

+ $

! 0 1!1 0

"$

zi = 0, (E30.5)

where # = k1/k2, p and $ are dimensionless.

(c) Show that the critical positive $cr at which the eigenvalues p2i coalesce is given by the relation

$cr = |1! #|2

. (E30.6)

Further show that if $ > $cr the roots pi become complex and hence explain whether loss of stabilityoccurs. Is it divergence or flutter?

(d) For # = 0.01, 1.0, 4.0 and 100 plot the dependence of | pi | (i = 1, 2) (where |.| denotes the modulus ofa complex number) on $ using

| p|/"

#, $/"

#, (E30.7)

as vertical and horizontal axes, respectively. Go from $ = 0 up to 2$cr or 1.0, whichever is greater, anduse sufficient steps to get reasonable graphical accuracy.

EXERCISE 30.2 (A+C:25) Do the previous exercise removing assumption (A2), that is, considering now theeffect of the geometric stiffness KG but still assuming C # C0. Is there any difference with the critical loadresult (E30.6)?

EXERCISE 30.3 (A+C:30) Beck’s column6 is the simplest follower-load problem involving a cantileveredbeam-column.7 This problem is shown in Figure E30.2.The beam-column has length L , elastic modulus E and smallest moment of inertia I . It is loaded by acompressive force$P which after deformation rotateswith the end section of the column and remains tangentialto its deformed axis (see Figure above). The mass M (half of the column mass) is lumped at its free end.

6 M. Beck, Die Knicklast des eiseiting eigenspannen, tangential gedruckten Stabes, Z. angew. Math. Phys., 3, No. 3, 1952.7 It is sometimes used as a very simple model to illustrate stability analysis of rockets against the “pogo” effect.

30–13


θ

λP1

2

x

y

M

L

E, A, I constant

CC0

Figure E30.2. Beck’s column: structure for Exercise 30.3.

If this problem is treated by the static criterion (Euler’s method) one erroneously concludes that the beamcolumn cannot lose stability for any value of the load !P8 A dynamic stability analysis, first carried out byBeck (loc.cit.), shows that stability is lost by flutter at the critical load

!Pcr = 20.05093E IL2

. (E30.8)

(a) Find the critical dynamic load given by the finite element method if one Euler-Bernoulli beam-columnelement is used along the length of Beck’s column. Lateral displacements may be considered infinites-imal; hence sin " ! " , cos " ! 1, and the axial force is simply !P . The degrees of freedom are ux1,uy1 and "z1. Use the material and geometric stiffness matrices given in equations (E24.2) and (E24.3),respectively, to which an unsymmetric load stiffness matrix KL , which couples the "z1 and uy1 degreesof freedom, should be added.

(b) Repeat the analysis for two and four elements of equal length along the column. For two elements thethree nodes are 1 (top), 2 (middle of column) and 3 (root). Use lumped masses with MX2 = MY2 equalto one half of the total column mass and MX1 = MY1 = MX2/2 = MY2/2. For four elements there arefive nodes, etc. Use of Mathematica or a similar program is recommended.

8 See for example, pp. 7–8 of Bolotin’s book cited in footnote 3.

30–14

30–15 Exercises

Solution of Exercise 30.3(a) for one-element discretization:The dynamic matrix perturbation equation taking C ! C0 is

!M 0 00 M 00 0 0

"! uxu y!z

"

+

#

$

%

&

E AL 0 00 12E I

L3"6E I

L20 "6E I

L24E IL

'

( " P30L

! 0 0 00 36 "3L0 "3L 4L2

"

+P

! 0 0 00 0 10 0 0

")! uxuy!z

"

=! 000

"

(E30.9)

where for simplicity ux = ux1, uy = uy1, !z = !z1. The first dynamic equation in ux uncouples and has no effect in theanalysis. The last equation is static in nature because the rotational mass is zero. Thus, we can solve for !z in terms of uy :

!z ="6E I

L2+ P10

4E IL

" 4PL30

uy = NDuy (E30.10)

where N and D denote the numerator and denominator, respectively, of the relation that links !z to uy . The eigenvalueequation becomes

*

p2M + 12E IL3

" 6E IL2

ND

" 36P30L

+ 3LND

P30L

+ PND

+

uy = 0. (E30.11)

One of the bending eigenvalues p2 of (E30.9) is always# because the rotational mass is zero. Flutter occurs when the twobeding eigenvalues coalesce at infinity. The finite p2 becomes infinite if D = 0 while N $= 0. Thus the critical load for“flutter at infinity” is

Pcr = 30E IL2

(E30.12)

which is about 50% in error with respect to the analytical value 20.05093E I/L2 quoted in the exercise statement.

EXERCISE 30.4 (A:25) Do the previous exercise for a one-element discretization if the line of action ofthe applied end load is forced to pass through the cantilever root (point 2). Does the structure loses stabilitydynamically or statically?

30–15

.

HThe Small Strain

TL C1 Plane Beam

H–1

Appendix H: THE SMALL STRAIN TL C1 PLANE BEAM H–2

§H.1 SUMMARY

This Appendix derives the discrete equations of a geometrically nonlinear, C1 (Hermitian), pris-matic, plane beam-column in the framework of the Total Lagrangian (TL) description. The formu-lation is restricted to the three deformational degrees of freedom: d, !1 and !2 shown in Figure H.1.The element rigid body motions have been removed by forcing the transverse deflections at the endnodes to vanish. The strains are assumed to be small while the cross section rotations ! are smallbut finite.Given the foregoing kinematic limitations, this element is evidently of no use per se in geometricallynonlinear analysis. Its value is in providing the local equations for a TL/CR formulation

§H.2 FORMULATION OF GOVERING EQUATIONS

§H.2.1 Kinematics

We consider a geometrically nonlinear, prismatic, homogenous, isotropic elastic, plane beam ele-ment that deforms in the x, y plane as shown in Figure H.1. The element has cross section area A0and moment of inertia I0 in the reference configuration, and elastic modulus E .

x

y

C

CR1 2

θ21θ-d/2

L

d/2

L0

Figure H.1 Kinematics of TL Hermitian beam element

The plane motion of the beam is described by the two dimensional displacement field{ux (x, y), uy(x, y)} where ux and uy are the axial and transverse displacement components, re-spectively, of arbitrary points within the element. The rotation of the cross section is !(x), whichis assumed small. The following kinematic assumptions of thin beam theory are used

!

ux (x, y)uy(x, y)

"

=#

uax (x) ! y"uay(x)

"xuay(x)

$

=!

uax (x) ! y!(x)uay(x)

"

(H.1)

where uax and uay denote the displacements of the neutral axes, and !(x) = "uay/"x is the rotationof the cross section. The three degrees of freedom of the beam element are

ue =# d

!1!2

$

(H.2)

H–2

H–3 §H.2 FORMULATION OF GOVERING EQUATIONS

§H.2.2 Strains

We introduce the notation

! = "ux"x

, # = "$

"x=

"2uay" x2

. (H.3)

for engineering axial strain and beam curvature, respectively. The exact Green-Lagrange measureof axial strain is

e = "ux"x

+ 12

!

"ux"x

"2+ 1

2

!

"uy"x

"2= ! ! y# + 1

2 (! ! y#)2 + 12$2 (H.4)

This can be expressed in terms of the displacement gradients as follows:

e = hT g+ 12g

THg = cT g (H.5)

where

g =

#

$

"uax/"x"uay /"x

"2uay /"x2

%

& ='

!

$

#

(

, h =' 10

!y

(

, H =' 1 0 !y0 1 0

!y 0 y2

(

(H.6)

We simplify this expression by dropping all y dependent terms form the H matrix:

H =' 1 0 00 1 00 0 0

(

(H.7)

The simplified axial strain is

e = hT g+ 12g

T Hg = ! ! y# + 12!2 + 1

2$2 (H.8)

The rational for this selective simplification is that ea = ! + 12!2 is the GL mean axial strain. If the

12!2 term is retained, a simpler geometric stiffness is obtained. The term 1

2$2 is the main nonlinear

effect contributed by the section rotations.

The vectors that appear in the CCF formulation of TL finite elements discussed in Chapters 10-11are

b = h+Hg =

#

$

1+ !

$

!y

%

& , c = h+ 12Hg =

#

$

1+ 12!

12$

!y

%

& , (H.9)

H–3


C0

C

N0N0V0

V00 0

NN

V

V

M02

M22

2

M01

M1

1

1

Figure H.2. Stress resultants in reference and current configurations.Configurations shown offset for clarity.

§H.2.3 Stresses and Stress Resultants

The stress resultants in the reference configuration are N 0, M01 and M0

2 . The initial shear force isV 0 = (M0

1 ! M02 )/L0. The axial force N 0 and transvese shear force V 0 are constant along the

element, whereas the bending moment M0(x) is linearly interpolated from M0 = M01 (1! x/L0)+

M02 x/L0. See Figure H.2 for sign conventions. The initial PK2 axial stress is computed using beam

theory:

s0 = N 0

A0! M0y

I0(H.10)

Denote by N , V and M the stress resultants in the current configuration. Whereas N and V areconstant along the element, M = M(x) varies linearly along the length because this is a Hermitianmodel, which relies on cubic transverse displacements. Consequently wewill define its variation bythe two node values M1 and M2. The shear V is recovered from equilibrium as V = (M1!M2)/L ,which is also constant. The PK2 axial stress in the current state is s = s0 + Ee = s0 + EcT g, orinserting (H.9):

s = s0 + E!

! + 12!2 + 1

2"2 ! y#

"

(H.11)

§H.2.4 Constitutive Equations

Integrating (H.11) over the cross section one gets the constitutive equations in terms of resultants:

N =#

A0s d A = s0A0 + E A0(! + 1

2!2 + 1

2"2) = N 0 + E A0(ea + 1

2"2),

M = !#

A0ys d A = M0 + E I0#

(H.12)

§H.2.5 Strain Energy Density

H–4

H–5 §H.2 FORMULATION OF GOVERING EQUATIONS

We shall use the CCF formulation presented in Chapter 10 to derive the stiffness equations. Using! = " = 1 (not a spectral form) one obtains the core energy of a beam particle as

U =

= 12g

T

!

"E

#

$

(1+ 12#)

2 + 14$2 ! 1

3 y%13$ !y(1+ 1

3#)13$

14 (#

2 + $2) + 13 (# ! y%) ! 1

3 y$!y(1+ 1

3#) ! 13 y$ y2

%

&+ s0' 1 0 00 1 00 0 0

(

)

* g

(H.13)Integration over this cross section yields the strain energy per unit of beam length:

UA = 12g

T+

A0(Ec cT + s0H) d A g

= 12g

T

!

"E

#

$

(1+ 12#)

2A0 12 (1+ 1

2#)$ A0 012 (1+ 1

2#)$ A014$2A0 0

0 0 I0

%

&+ N 0' 1 0 00 1 00 0 0

(

)

* g(H.14)

To obtain the element energy it is necessary to specify the variation of #, $ and % along the beam.At this point shape functions have to be introduced.

§H.2.6 Shape Functions

Define the isoparametric coordinate & = 2x/L0. The displacement interpolation is taken to be thesame used for the linear beam element:

,

uaxuay

-

=, 12& 0 00 1

8 L0(1! &)2(1+ &) 18 L0(1+ &)2(1! &)

-

' d$1$2

(

. (H.15)

From this the displacement gradients are

g ='

#

$

%

(

= 1L0

' 1 0 00 1

4 L0(& ! 1)(3& + 1) 14 L0(1+ &)(3& ! 1)

0 3& ! 1 3& + 1

(' d$1$2

(

= Gue. (H.16)

The rotation $ varies quadritically and the curvature $ linearly. The node values are obtained onsetting & = ±1:

g1 ='

#1$1%1

(

= 1L0

' 1 0 00 L0 00 !4 !2

(

ue, g2 ='

#2$2%2

(

= 1L0

' 1 0 00 0 L00 2 4

(

ue (H.17)

§H.2.7 Element Energy

The strain energy of the element can be now obtained by expressing the gradients g = Gue andintegrating over the length. the result can be expressed as

Ue =+

12 L0

! 12 L0UA d A =

+ 1

!1UA

12 L0 d& = 1

2 (ue)TKUue (H.18)

H–5


where the energy stiffness is the sum of three contributions: KU = KUa +KU

b +KUN . These come

from the axial deformations, bending deformations and initial stress, respectively:

KUa = E A0

L0

!

"

"

"

"

#

(1+ 12!)

2 (1+ 12!)(4"1 ! "2)L0

60(1+ 1

2!)(!"1 + 4"2)L060

(1+ 12!)(4"1 ! "2)L0

60(12"21 ! 3"1"2 + "22 )L

20

840(!3"21 + 4"1"2 ! 3"22 )L

20

1680(1+ 1

2!)(!"1 + 4"2)L060

(!3"21 + 4"1"2 ! 3"22 )L20

1680("21 ! 3"1"2 + 12"22 )L

20

840

$

%

%

%

%

&

,

KUb = E I0

L0

' 0 0 00 4 20 2 4

(

, KUN = N0

L0

' 1 0 00 2L20/15 L20/300 L20/30 2L20/15

(

.

(H.19)

§H.3 INTERNAL FORCE

The internal force p is obtained as the derivative

p = #Ue

#ue=)

KU + 12 (u

e)T#KU

#ue

*

ue = Kpue (H.20)

The internal force stiffness is again the sum of three contributions: Kp = Kpa +Kp

b +KpN . These

come from the axial deformations, bending deformations and initial stress, respectively:

Kpa = E A0

L0

!

"

"

"

#

1+ 32! + 1

2!2 (3+ 2!)(4"1 ! "2)L0

120(3+ 2!)(!"1 + 4"2)L0

120(3+ 2!)(4"1 ! "2)L0

120(12"21 ! 3"1"2 + "22 )L

20

420(!3"21 + 4"1"2 ! 3"22 )L

20

840(3+ 2!)(!"1 + 4"2)L0

120(!3"21 + 4"1"2 ! 3"22 )L

20

840("21 ! 3"1"2 + 12"22 )L

20

420

$

%

%

%

&

,

Kpb = E I0

L0

' 0 0 00 4 20 2 4

(

, KpN = N0

L0

' 1 0 00 2L20/15 L20/300 L20/30 2L20/15

(

.

(H.21)

§H.4 TANGENT STIFFNESS

The tangent stiffness K is obtained as the derivative

K = #p#ue

=+

Kr + (ue)T#Kr

#ue

,

ue (H.22)

This is again the sum of three contributions: K = Ka + Kb + KN , which come from the axial

H–6

H–7 §H.4 TANGENT STIFFNESS

deformations, bending deformations and current stress, respectively:

Ka = E A0L0

!

"

"

"

#

(1+ !)2(1+ !)(4"1 ! "2)L0

30(1+ !)(!"1 + 4"2)L0

30(1+ !)(4"1 ! "2)L0

30(12"21 ! 3"1"2 + "22 )L

20

210(!3"21 + 4"1"2 ! 3"22 )L

20

420(1+ !)(!"1 + 4"2)L0

30(!3"21 + 4"1"2 ! 3"22 )L

20

420("21 ! 3"1"2 + 12"22 )L

20

210

$

%

%

%

&

,

Kb = E I0L0

' 0 0 00 4 20 2 4

(

, KN = N30L0

' 30 0 00 4L20 !L200 !L20 4L20

(

.

(H.23)The material stiffness is KM = Ka +Kb and the geometric stiffness is KG = KN .

H–7

.

RReferences

(in progress)

R–1

Appendix R: REFERENCES (IN PROGRESS) R–2

TABLE OF CONTENTS

Page§R.1. Foreword R–3§R.2. Reference Database R–3

R–2

R–3 §R.2 REFERENCE DATABASE

§R.1. ForewordCollected references for most Chapters (except those in progress) for books

Introduction to Finite Element Methods, abbrv. IFEMAdvanced Finite Element Methods; abbrv. AFEMNonlinear Finite Element Methods; abbrv. NFEMMatrix Finite Element Methods in Statics; abbrv. MFEMSMatrix Finite Element Methods in Dynamics; abbrev. MFEMD

Margin letters are to facilitate sort; will be removed on completion.Note: Many of the books listed below are out of print. The advent of the Internet has meant that it is easier tosurf for used books across the world without moving from your desk. There is a fast search “metaengine” forcomparing prices at URL http://www.addall.com: clink on the “search for used books” link. Amazon.comhas also a search engine, which is poorly organized, confusing and full of unnecessary hype, but links to onlinereviews. [Since about 2008, old scanned books posted online on Google are an additional potential source;free of charge if the useful pages happen to be displayed.]

§R.2. Reference Database

A

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