FINITE ELEMENT ANALYSIS

AS COMPUTATION

FINITE ELEMENT ANALYSIS AS COMPUTATION

What the textbooks don't teach you about finite elementanalysis

Gangan Prathap

iii

Preface

These course notes are specially prepared for instruction into the principlesunderlying good finite element analysis practice. You might ask: Why anotherbook on FEA (Finite Element Analysis) and the FEM (Finite Element Method) whenthere are hundreds of good textbooks in the market and perhaps thousands ofmanuals, handbooks, course notes, documentation manuals and primers available onthe subject.

Some time earlier, I had written a very advanced monograph on the analytical andphilosophical bases for the design and development of the finite elements whichgo into structural analysis software packages. Perhaps a hundred persons work onelement development in the world and out of these, maybe a dozen would probedeep enough to want to explore the foundational bases for the method. This wastherefore addressed at a very small audience. In the world at large are severalhundred thousand engineers, technicians, teachers and students who routinely usethese packages in design, analysis, teaching or study environments. Their needsare well served by the basic textbooks and the documentation manuals thataccompany installations of FEM software.

However, my experience with students and teachers and technicians and engineersover two decades of interaction is that many if not most are oblivious to thebasic principles that drive the method. All of them can understand thesuperficial basis of the FEM - what goes into the packages; what comes out; howto interpret results and so on. But few could put a finger on why the methoddoes what it does; this deeper examination of the basis of the method isavailable to a very few.

The FEM is now formally over thirty-five years old (the terminology "finiteelement" having been coined in 1960). It is an approximate method of solvingproblems that arise in science and engineering. It in fact originated and grewas such, by intuition and inspired guess, by hard work and trial and error. Itsorigins can be traced to aeronautical and civil engineering practice, mainlyfrom the point of view of structural engineering. Today, it can be used, withclever variations, to solve a wide variety of problems in science andengineering. There are billions of dollars worth of installed software andhardware dedicated to finite element analysis all over the world and perhapsbillions of dollars are spent on analysis costs using this software every year.It is therefore a remarkable commercial success of human imagination, skill andcraft.

The science of structural mechanics is well known. However, FEA, being anapproximate method has a uniqueness of its own and these curious features aretaken for closer examination in this course. The initial chapters are short onesthat examine how FEM is the art of numerical and computational analysis appliedto problems arising from structural engineering science. The remaining chapterselucidate the first principles, which govern the discretisation process by which"exact” theory becomes numerical approximations.

We shall close these notes with a telescopic look into the present scenario andthe shape of things to come in the future.

iv

Acknowledgements: I am extremely grateful to Dr T S Prahlad, Director, N.A.L.for his encouragement and support. Many colleagues were associated at variousstages of writing this book. They include Dr B P Naganarayana, Dr S Rajendranand Mr Vinayak Ugru, Dr Sudhakar Marur and Dr Somenath Mukherjee. I owe a greatdeal to them. Mr Arjun provided secretarial help and also prepared the linedrawings and I am thankful to him. I am also thankful to Latha and Rahul fortheir patience and understanding during the preparation of this manuscript.

Gangan PrathapBangalore2001

v

Contents

Preface iii

1. Introduction: From Science to Computation 1

2. Paradigms and some approximate solutions 11

3. Completeness and continuity: How to choose shapefunctions? 28

4. Convergence and Errors 33

5. The locking phenomena 45

6. Shear locking 57

7. Membrane locking, parasitic shear and incompressiblelocking 79

8. Stress consistency 97

9. Conclusion 111

1

Chapter 1

Introduction: From Science to Computation

1.1 From Art through Science to Computation

It will do very nicely to start with a description of what an engineer does,before we proceed to define briefly how and where structural mechanics plays acrucial role in the life of the engineer. The structural engineer's appointedoffice in society is to enclose space for activity and living, and sometimesdoes so giving time as well - the ship builder, the railway engineer and theaerospace engineer enable travel in enclosed spaces that provide safety withspeed in travel. He did this first, by imitating the structural forms alreadypresent in Nature.

From Art imitating Life, one is led to codification of the accumulatedwisdom as science - the laws of mechanics, elasticity, theories of the variousstructural elements like beams, plates and shells etc. From Archimedes' use ofthe Principle of Virtual Work to derive the law of the lever, through Galileoand Hooke to Euler, Lagrange, Love, Kirchhoff, Rayleigh, etc. we see thetheoretical and mathematical foundations being laid. These where then copiouslyused by engineers to fabricate structural forms for civil and militaryfunctions. Solid and Structural Mechanics is therefore the scientific basis forthe design, testing, evaluation and certification of structural forms made frommaterial bodies to ensure proper function, safety, reliability and efficiency.

Today, analytical methods of solution, which are too restricted inapplication, have been replaced by computational schemes ideal forimplementation on the digital computer. By far the most popular method inComputational Structural Mechanics is that called the Finite Element Method.People who use computational devices (hardware and software) with hardly anyreference to experiment or theory now perform design and Analysis. Fromadvertising claims made by major fem and cad software vendors (5000 sites orinstallations of one package, 32,000 seats of another industry standard vendor,etc.); it is possible to estimate the number of fem computationalists oranalysts as lying between a hundred thousand and two hundred thousand. It is tothem that this book is addressed.

1.2 Structural Mechanics

A structure is "any assemblage of materials which is intended to sustain loads".Every plant and animal, every artifact made by man or beast has to sustaingreater or less forces without breaking. The examination of how the laws ofnature operate in structural form, the theory of structural form, is what weshall call the field of structural or solid mechanics. Thus, the body ofknowledge related to construction of structural form is collected, collated,refined, tested and verified to emerge as a science. We can therefore think ofsolid and structural mechanics as the scientific basis for the design, testing,evaluation and certification of structural forms made from material bodies toensure proper function, safety, reliability and efficiency.

To understand the success of structural mechanics, one must understand itsplace in the larger canvas of classical mechanics and mathematical analysis,

2

especially the triumph of the infinitesimal calculus of continuum behavior inphysics. The development and application of complex mathematical tools led tothe growth of the branch of mathematical physics. This therefore encouraged thestudy of the properties of elastic materials and of elasticity - the descriptionof deformation and internal forces in an elastic body under the action ofexternal forces using the same mathematical equipment that was been used inother classical branches of physics. Thus, from great mathematicians such asCauchy, Navier, Poisson, Lagrange, Euler, Sophie Germain, came the formulationof basic governing differential equations which originated from the applicationof the infinitesimal calculus to the behavior of structural bodies. The study isthus complete only if the solutions to such equations could be found. For nearlya century, these solutions were made by analytical techniques however, thesewere possible only for a very few situations where by clever conspiracy, theloads and geometries were so simplified that the problem became tractable.However, by ingenuity, the engineer could use this limited library of simplesolutions to construct meaningful pictures of more complicated situations.

It is the role of the structural designer to ensure that the artifacts hedesigns serve the desired functions with maximum efficiency, but putting only asmuch flesh as is needed on the bare skeleton. Often, this is measured in termsof economy of cost and/or weight. Thus for vehicles,low weight is of theessence, as it relates directly to speed of movement and cost of operation. Theefficiency of such a structure will depend on how every bit of material that isused in components and joints is put to maximum stress without failing. Strengthis therefore the primary driver of design. Other design drivers are stiffness(parts of the structures must not have excessive deflections), buckling (membersshould not deflect catastrophically under certain loading conditions), etc. Itis not possible to do this check for structural integrity without havingsophisticated tools for analysis. FEM packages are therefore of crucialrelevance here. In fact, the modern trend is to integrate fem analysis toolswith solid modeling and CAD/CAM software in a single seamless chain or cycle allthe way from concept and design to preparation of tooling instructions formanufacture using numerically controlled machines.

1.3 From analytical solutions to computational procedures

1.3.1 Introduction

We have seen earlier in this chapter how a body of knowledge that governs thebehavior of materials, solids and structures came to exist. Giftedmathematicians and engineers were able to formulate precise laws that governedthe behavior of such systems and could apply these laws through mathematicalmodels that described the behavior accurately. As these mathematical models hadto take into account the continuum nature of the structural bodies, thedescription often resulted in what are called differential equations. Dependingon the sophistication of the description, these differential equations could bevery complex. In a few cases, one could simplify the behavior to grossrelationships - for a bar, or a spring, it was possible to replace thedifferential equation description with a simple relation between forces anddisplacements; we shall in fact see that this is a very simple and direct actof the discretization process that is the very essence of the finite elementprocess. For a more general continuum structure, such discretizationsor simplifications were not obvious at first. Therefore, there was no recourse

Fig.

except to find bthe governing difof complex geomethowever with ingestructural formsviscoelastic behalast two centurie

It was recognizedifficult to implanswers. In factgeneral than scomputational schthis chapter, wesimple problem anwill reveal to usproblem.

1.3.2 The simple

Perhaps the earliin civil engineerone-dimensional scompression, andconfiguration -meaning that onestrains and strecompute the analyelementary engine

1.3.2.1 Analytic

The problem is cathis problem, theequation describi

where E is the Yfunction describidifferentiationequilibrium descr

3

1.1 A simple bar problem under axial load

y mathematical and analytical techniques, viable solutions toferential equations. This can become very formidable in casesry, loading and material behavior, and often were intractable;nuity, solutions could be worked out for a useful variety of. Thus a vast body of solutions to the elastic, plastic,vior of bars, beams, plates and shells has been built over thes.

d quite early that where analytical techniques fail or wereement, approximate techniques could be devised to compute the, much before algebra and analysis arrived, in contexts moreolid or structural mechanics, simple approximation andemes were used to solve engineering and scientific problems. Inshall review what we mean by an analytical solution for a veryd then proceed to examine some computational solutions. Thishow the discretization process is logically set up for such a

bar problem

est application of the discretization technique as it appeareding practice was to the bar problem. The bar is a prismatictructure which can resist only axial loads, in tension or incannot take any bending loads. Figure 1.1 shows its simplesta cantilever bar. This is a statically determinate problem,can obtain a complete solution, displacements as well as

sses from considerations of equilibrium alone. We shall nowtical solution to the problem depicted in Fig.1.1 using veryering mathematics.

al solution

tegorized as a boundary value problem. We presume here that forreader is able to understand that the governing differential

ng the situation is

EA u,xx = 0 (1.1)

oung’s modulus, A is the area of cross-section, u(x) is theng the variation of displacement of a point, and ,x denoteswith respect to coordinate x. This is the equation ofibing the rate at which axial force varies along the length of

4

the bar. In general, governing differential equations belong to a categorycalled partial differential equations but here we have a simpler form known asan ordinary differential equation. Equation (1.1) is further classified as afield equation or condition as one must find a solution to the variable u whichmust satisfy the equation over the entire field or domain, in this case, for xranging from 0 to L. A solution is obviously

u = ax + b (1.2)

where a and b are as yet undetermined constants. To complete the solution, i.e.to determine a and b in a unique sense for the boundary (support) and loadingconditions shown in Fig. 1.1, we must now introduce what are called the boundaryconditions. Two boundary conditions are needed here and we state them as

u = 0 at x = 0 (1.3a)EA u,x = P at x = L (1.3b)

The first relates to the fact that the bar is fixed at the left end and thesecond denotes that a force P is applied at the free end. Taking these twoconditions into account, we can show that the following description completelytakes stock of the situation:

u(x) = Px/EA (1.4a) ε(x) = P/EA (1.4b)

σ(x)= Eε(x)= P/A (1.4c)P(x) = Aσ(x) = P (1.4d)

where ε(x), σ(x) and P(x) are the strain, stress and axial force (stressresultants) along the length of the bar. These are the principal quantities thatan engineer is interested in while performing stress analysis to check theintegrity of any proposed structural design.

1.3.2.2 Approximate solutions

It is not always possible to determine exact analytical solutions to mostengineering problems. We saw in the example above that an analytical solutionprovides a unique mathematical expression describing in complete detail thevalue of the field variable at every location in the body. From this expression,other expressions can be derived which describe further quantities of practicalinterest, e.g. strains, stresses, stress resultants, etc.

In most problems where analytical solutions appear infeasible, it isnecessary to resort to some approximate or numerical methods to obtain thesevalues of practical interest. One obvious method at this stage is to start withthe equations we have already derived, namely Equations (1.1) and (1.3). Thiscan be achieved using a technique known as the finite difference method. Weshall briefly describe this next.

Other approxidescriptions or varproblem at a levelmore specialized deseen in Equations (quantities and disccan appreciate thanalytically fromvariational or virtuthat an approximativariation carried ouas well as the apprwill be the basis fsubsections.

1.3.2.3 Finite dif

For a problem whereknown as the finitenumerical solution.it was this methodproblems in structur

Figure 1.2 shpoints that can beand (1.3). The largespacing h), the greataken to be known (

at these nodal poindegrees of freedom.equations and boundfinite difference f(again, details areanalysis or fem) tha

5

Fig. 1.2 Grid for finite difference scheme

mate solutions are based on what are called functionaliational descriptions of the problem. These describe thewhich is more fundamental in the laws of physics than thescriptions in terms of governing differential equations as1.1) and (1.3) above. We deal directly with energy or workretization or approximation is applied at that level. If weat the governing differential equations were derivedsuch energy or work principles using what is called aal work approach, then it would become easier to understandon applied directly to the energy or work quantities and at subsequently on these terms will preserve both the physicsoximating or numerical solution in one single process. Thisor the finite element method as we shall see in subsequent

ference approximations

the differential equations are already known, the techniquedifference method can be used to obtain an approximate orIn fact, before the finite element method was established,which was used to obtain solutions for very complicated

al mechanics, fluid dynamics and heat transfer.

ows a very simple uniformly spaced mesh or grid of nodalused to discretize the problem described by Equations (1.1)r the number of nodal points used (i.e. the smaller the gridter will be the accuracy involved. The field variable is nowor very strictly, unknown, at this stage of the computation)

ts. Thus, u1,..., ui,..., un, are the unknown variables orThe next step is to rewrite the governing differential

ary conditions in finite difference form. We see that theorms for u,x and u,xx are required. It is easy to showomitted, as these can be found in most books on numericalt at a grid point i.

u,x = (ui+1 – ui)/h (1.5a)

u,xx = (ui+1 – 2ui + ui-1)/h2 (1.5b)

6

We now attempt a solution in which four points are used in the finite differencegrid. The governing differential equations and boundary conditions are replacedby the following discrete set of linear, algebraic equations:

u1 = 0 (1.6a)

u3 – 2u2 + u1 = 0 (1.6b)

u4 - 2u3 + u2 = 0 (1.6c)

u4 - u3 = Ph/EA (1.6d)

where h=L/3 and Equations (1.6a) and (1.6d) represent the boundary conditions atx = 0 and L respectively. The reader can easily work out that the solutionobtained from this set of simultaneous algebraic equations is, u2 = PL/3EA,

u3 = 2PL/3EA and u4 = PL/EA. Comparison with Equation (1.4a) will confirm thatwe have obtained the exact answers at the grid points. Other quantities ofinterest like strain, stress, etc. can be computed from the grid point valuesusing the finite difference forms of these quantities.

The above illustrates the solution to a very simple one-dimensionalproblem for which the finite difference procedure yielded the exact answer.Generalizations to two and three dimensions for more complex continuous problemscan be made, especially if the meshes are of regular form and the boundary linescoincide with lines in such regular grids. For more complex shapes, the finitedifference approach becomes difficult to use. It is in such applications thatthe finite element method proves to be more versatile than the finite differencescheme.

1.3.2.4 Variational formulation

We began this chapter with an analytical solution to the differential equationsgoverning the problem. We did not ask how these equations originated. There aretwo main ways in which it can be done. The first, and the one that is most oftenintroduced at the earliest stage of the study of structural mechanics, is toformulate the equations of equilibrium in terms of force or stress quantities.Then, depending on considerations of static determinacy, make the problemdeterminate by introducing stress-strain and strain-displacement relations untilthe final, solvable, set of governing equations is obtained.

The second method is one that originates from a more fundamental statementof the principles involved. It recognizes that one of the most basic and elegantprinciples known to man is the law of least action, or minimum potential energy,or virtual work. It is a statement that Nature always chooses the path ofminimum economy. Thus, the law of least action or minimum total potential is asaxiomatic as the basic laws of equilibrium are. One can start with one axiom andderive the other or vice-versa for all of mechanics or most of classicalphysics. In dynamics, the equivalent principle is known as Hamilton's principle.

In structural mechanics, we start by measuring the total energy stored ina structural system in the form of elastic strain energy and potential due toexternal loads. We then derive the position of equilibrium as that state wherethis energy is an extremum (usually a minimum if this is a stable state ofequilibrium). This statement in terms of minimum energy is strictly true onlyfor what are called conservative systems. For non-conservative systems, a more

general statement known as the principle of virtual work would apply. From thesebrief general philosophical reflections we shall now proceed to the special caseat hand to see how the energy principle applies.

The total energy is stated in the form of a functional. In fact, most ofthe problems dealt with in structural mechanics can be stated in a variationalform as the search for a minimal or stationary point of a functional. Thefunctional, Π, is an integral function of functions which are defined over thedomain of the structure. Thus, for our simple bar problem, we must define termssuch as the strain energy, U, and the potential due to the load, V, in terms ofthe field variable u(x). This can be written as

Then,

Using standard variational prshow that the extremum or stat

If this variation is carried oterms, we will obtain the samthat appeared in Equations (1show what can be done with thethe approximation or discretiwhere the total potential or f

1.3.2.5 Functional approxim

There are several ways in whicenergy, virtual work or weigh(R-R) procedure, the Galerkinfirst choose a set of indepeessential (also called kinemafunctions are called admissiblin Equation (1.3a) is the geEquation (1.3b) is called a noThe admissible functions needfact, it can be shown that ideterioration in performance).field variable, say ( )xu , is o

using unknown coefficients orgeneralized coordinates. Thesethe extremum or stationary st

Π

(1.7a)

V = P ux=L (1.7b)

�= dxuEAL

021U x

2,/

7

(1.8)

ocedures from the calculus of variations, we canionary value of the functional is obtained as

δΠ = 0 (1.9)

ut, after integrating by parts and by regroupinge equation of equilibrium and boundary conditions.1) and (1.3) above. Our interest is now not tose differential equations but to demonstrate thatzation operation can be implemented at the stageunctional is defined.

ation

h an approximation can be applied at the level ofted residual statements, e.g. the Rayleigh-Ritzprocedure, the least-squares procedure, etc. Wendent functions which satisfy the geometric ortic) boundary conditions of the problem. Thesee functions. In this case, the condition specifiedometric boundary condition. The other condition,nessential or natural or force boundary condition.not satisfy the natural boundary conditions (inf they are made to do so, there can even be aThe approximate admissible configuration for the

btained by a linear combination of these functions

parameters, also known as degrees of freedom orunknown constants are then determined to satisfy

atement. Methods like the Rayleigh-Ritz procedure

( )( ) LxxL

02 uP-dxuEA21x =�= ,/υ

(the one going to be used now), the Galerkin method, collocation methods, leastsquare methods, etc. all proceed with approximating fields chosen in this way.

For our problem, let us choose only one simple linear function, using apolynomial form for the purpose. The advantages of using polynomial functionswill become clear later in the book. Thus, only one constant is required, i.e.

(1.10)

Note that this configuration is an admissible one satisfying the geometriccondition ( ) 00u = . By substituting in Equation (1.8) and carrying out the

variation prescribed in Equation (1.9), we are trying to determine the value ofa1 which provides the best functional approximation to the variational problem.It is left to the reader to show that the approximate solution obtained is

Thus, for this problem, thesolution. Obviously, we cannotinvolved in approximation by theapproximate solutions will be bufunctions and convergence to theterms used and the type of functio

The R-R procedure, when appdomains that constitute the entireThus, an understanding of how tunderstanding of the finite elemen

1.3.2.6 Finite element approxim

We shall now describe an approximshall use a form known as the dstiffness formulation). The simpnoded line element such as shown

bar, truss or rod element. It has

u2 are prescribed. These are als

indicated in Fig. 1.3 are the forc

Thus, any one-dimensional stplaced from end to end. The onelement is that stored as nodal dnow see how this piecewise Ritz ap

(((( )))) xaxu 1====

(1.11a)

li

e

( ) Px/EAxu =

(1.11b)( ) P/EAx =ε

(1.11c)( ) P/A(x)Ex == εσ

8

(1.11d)

approximate solution coincides with the exactdraw any conclusions here as to the errorsR-R procedure. In more complicated problems,ilt up from several constants and admissibleexact solution will depend on the number ofns used.

lied in a piecewise manner, over the elementstructure, becomes the finite element method.

he R-R method works will be crucial to ourt method.

ation

ate solution by the finite element method. Weisplacement-type formulation (also called theest element known for our purpose is a twon Fig. 1.3. This element is also known as the

two nodes at which nodal displacements u1 and

o called the nodal degrees of freedom. Also

es P1 and P2 acting at these nodes.

ructure can be replaced by contiguous elementsly information communicated from element togrees of freedom and the nodal forces. We canproximation is performed.

( ) P(x)AxP == σ

We first derfield variable, u

nodal degrees of f

where N1 =(1-ξ)/2functions. Note thin our R-R approxiai being now replcompute the energy

and the potential

Thus, if a variatequilibrium as

This matrix represassembled and maniattempt to solve oone element. Nodewith the free end

1 is simply u1 = 0

to show from theseobtained.

What we havfinite element met

Fig. 1.3 A two-node bar element

ive an expression for the approximate representation of the( )x , within the region of the element by relating it to the

reedom. It can be shown that,

and N2at thismationaced bstored

due to t

ion is

entatiopulatedur bar1 is thwhere th

and P1that u

e seenhod tha

(1.12)

=(1+ξ)/2,where ξ=x/l. N1 and N2 are called the shapeform is exactly the same as the linear function used

earlier, Equation (1.10), with generalized coordinatesy nodal quantities ui. It is possible therefore toin a beam element as

( ) 2211 NuNuxu +=

(1.13)

he applied loads as

{ }��

��

�

��

��

�

��

���

2

121

u

u

11-

1-1

2l

EAuu

(1.14)

taken over u1 and u2, we can write the equation of

{ }���

���

2

121

P

Puu

9

(1.15)

n is typical of the way finite element equations areautomatically on the digital computer. We shall nowproblem using a finite element model comprising justerefore placed at the fixed end and node 2 coincidese load P2 = P is applied. The fixity condition at node

indicates the reaction at this end. It is very simple

2 = PL/EA. We can see that the exact solution has been

above is one of the simplest demonstrations of thet is possible. Generalizations of this approach to two

���

���

=���

���

��

�

�

2

1

2

1

P

P

u

u

11-

1-1

2l

EA

10

and three dimensional problems permit complex plate, shell and three dimensionalelasticity problems to be routinely handled by ready made software packages.

1.4 Concluding remarks

In this chapter, we have briefly investigated how classical analyticaltechniques, which are usually very limited in scope, can give way tocomputational methods of obtaining acceptable solutions. We have seen that thefinite element method is one such approximate procedure. By using a very largenumber of elements, it is possible to obtain solutions of greater accuracy. Overthe years, finite element theorists and analysts have produced a very large bodyof work that shows how such solutions are managed for a great variety ofproblems. This is in the first-order tradition of the method. The remaining partof this book hopes to address questions like: How good is the approximatesolution? What kinds of situation affect these approximations in a trivial andin not-so-trivial ways? Since solutions depend very much on the number ofelements used and the type of shape function patterns used within each element,we must ask, what patterns are the best to assume? All these questions arelinked to one very fundamental question: In what manner does the finite elementmethod actually compute the solution? Does it make its first approximation ofthe displacement field directly or on the stress field? The exercises andexposition in this book are aimed at throwing more light on these questions.

11

Chapter 2

Paradigms and some approximate solutions

2.1 Introduction

We saw in the preceding chapter that a continuum problem in structural or solidmechanics can either be described by a set of partial differential equations andboundary conditions or as a functional Π based on the energy principle whoseextremum describes the equilibrium state of the problem. Now, a continuum hasinfinitely many material points and therefore has infinitely many degrees offreedom. Thus, a solution is complete only if analytical functions can be foundfor the displacement and stress fields, which describe these states, exactlyeverywhere in the domain of the problem. It is not difficult to imagine thatsuch solutions can be found only for a few problems.

We also saw earlier that the Rayleigh-Ritz (RR) and finite element (fem)approaches offer ways in which approximate solutions can be achieved without theneed to solve the differential equations or boundary conditions. This is managedby performing the discretization operation directly on the functional. Thus, areal problem with an infinitely large number of degrees of freedom is replacedwith a computational model having a finite number of degrees of freedom. In theRR procedure, the solution is approximated by using a finite number ofadmissible functions ƒi and a finite number of unknown constants ia so that the

approximate displacement field is represented by a linear combination of thesefunctions using the unknown constants. In the fem process, this is done in apiecewise manner - over each sub-region (element) of the structure, the

displacement field is approximated by using shape functions Ni within each sub-

region and nodal degrees of freedom ui at nodes strategically located so thatthey connect the elements together without generating gaps or overlaps. The

functional now becomes a function of the degrees of freedom (ai or ui as thecase may be). The equilibrium configuration is obtained by applying thecriterion that Π must be stationary with respect to the degrees of freedom.

It is assumed that this solution process of seeking the stationary orextremum point of the discretized functional will determine the unknownconstants such that these will combine together with the admissible or shapefunctions to represent some aspect of the problem to some "best" advantage.Which aspect this actually is has been a matter of some intellectualspeculation. Three competing paradigms present themselves.

It is possible to believe that by "best" we mean that the functions tendto satisfy the differential equations of equilibrium and the stress boundaryconditions more and more closely as more terms are added to the RR series ormore elements are added to the structural mesh. The second school of thoughtbelieves that it is the displacement field, which is approximated to greateraccuracy with improved idealization. The "aliasing" paradigm, which will becritically discussed later, belongs to this school. It follows from this thatstresses, which are computed as derivatives of the approximate displacementfields, will be less accurate. In this book however, we will seek to establishthe currency of a third paradigm - that the RR or fem process actually seeks tofind to best advantage, the state of stress or strain in the structure. In this

scenario, the displacement fields are computed from these "best-fit" stresses asa consequence.

Before we enter into a detailed examination of the merits or faults ofeach of these paradigms, we shall briefly introduce a short statement on what ismeant by the use of the term "paradigm" in the present context. We shall followthis by examining a series of simple approximations to the cantilever barproblem but with more and more complex loading schemes to see how the overallpicture emerges.

2.2 What is a "paradigm"?

Before we proceed further it may be worthwhile to state what we mean by aparadigm here. This is a word that is uncommon to the vocabulary of a trainedengineer. The dictionary meaning of paradigm is pattern or model or example.This does not convey much in our context. Here, we use the word in the greatlyenlarged sense in which the philosopher T. S. Kuhn introduced it in his classicstudy on scientific progress [2.1]. In this sense, a paradigm is a "framework ofsuppositions as to what constitutes problems, theories and solutions. It can bea collection of metaphysical assumptions, heuristic models, commitments, values,hunches, which are all shared by a scientific community and which provide theconceptual framework within which they can recognize problems and solve them[2.2]. The aliasing and best-fit paradigms can be thought of as two competingscenarios, which attempt to explain how the finite element method computesstrains and stresses. Our task will therefore be to establish which paradigm hasgreater explanatory power and range of application.

Fig.

12

2.1 Bar under uniformly distributed axial load

2.3 Bar under uniformly distributed axial load

Consider a cantilever bar subjected to a uniformly distributed axial load ofintensity 0q per unit length (Fig. 2.1). Starting with the differential

equation of equilibrium, it is easy to show that the analytical solution to theproblem is

(2.1a)

(2.1b)

Consider a one-term RR solution based on xau 1r ==== , where the subscript r denotes

the use of the RR approach. It is left to the reader to show that theapproximate solution obtained is

We now compare this wiThe solution obtained w

We see that the RR anbecause the fem soluticurious coincidence whethe tip. However, from

solutions to u. It is a

of the beam. It is po

relationship to the tru

Consider next whnoded) bar elements areout that the solutionnote that the distributthe physical load sysdescribed in Fig. 2.1 o

stairstep distributionFig. 2.2) which is senset of algebraic equati

We see once agaicoincidence for this pinto this fact. More st

(((( )))) (((( )))) (((( ))))2xLxEAqxu 20 −−−−====

(((( )))) (((( )))) (((( ))))xLAqx 0 −−−−====σ

(2.2a)( ) ( ) ( )2LxEAqxu 0r =

(2.2b)

th an fem solution based on a two-noded linear element.ill be

dorF

l

s

e

a

detr

σseo

nar

(((( )))) (((( )))) (((( ))))2LAqx 0r ====σ

(2.3a)( ) ( ) ( )2LxEAqxu 0f =

13

(2.3b)

fem solutions are identical. This is to be expectedn is effectively an RR solution. We can note also thee all three solutions predict the same displacement atig. 2.1 we can see that the ru and fu are approximate

so clear from Fig. 2.1 that σσσ == fr at the mid-point

sible to speculate that rσ and fσ bear some unique

variation σ.

t will happen if two equal length linear (i.e., two-used to model the bar. It is left to the reader to workescribed in Fig. 2.2 will be obtained. First, we mustd axial load is consistently lumped at the nodes. Thusem that the fem equations are solving is not thatFig. 2.2 as σ. Instead, we must think of a substitute

f, produced by the consistent load lumping process (seed by the fem stiffness matrix. Now, a solution of thens will result in fσ and fu as the fem solution.

that the nodal predictions are exact. This is only articular type of problem and nothing more can be readiking is the observation that the stresses computed by

(((( )))) (((( )))) (((( ))))2LAqx 0f ====σ

the finite element system now approximates the original true stress in astairstep fashion.

It also seems reasonable to conclude that within each element, the truestate of stress is captured by the finite element stress in a "best-fit" sense.In other words, we can generalize from Figs. 2.1 and 2.2, that the finiteelement stress magnitudes are being computed according to some precise rule.

Also, there is the promise that by carefully understanding what this rule is, itwill be possible to derive some unequivocal guidelines as to where the stressesare accurate. In this example, where an element capable of yielding constantstresses is used to model a problem where the true stresses vary linearly, thecentroid of the element yields exact predictions. As we take up further exampleslater, this will become more firmly established.

A cursory comparison of Figs. 2.1 and 2.2 also indicates that in a generalsense, the approximate displacements are more accurate than the approximatestresses. It seems compelling now to argue that this is so because theapproximate displacements emerge as "discretized" integrals of the stresses orstrains and for that reason, are more accurate than the stresses.

2.4 Bar under linearly varying distributed axial load

We now take up a slightly more difficult problem. The cantilever bar has a loaddistributed in a linearly varying fashion (Fig. 2.3). It is easy to establishthat the exact stress distribution in this case will be quadratic in nature. Wecan write it as

(2.4)

Some interesting features about this equation can be noted. A dimensionlesscoordinate, ξ = 2x/L-1 has been chosen so that it will also serve as a naturalcoordinate system taking on values -1 and 1 at the ends of the single three-nodebar element shown as modeling the entire bar in Fig.2.3. We have also verycuriously expanded the quadratic variation using the terms 1, ξ, (1-3ξ2). Thesecan be identified with the Legendre polynomials and its relevance to thetreatment here will bec me more obvious as we proceed further.

Fig. 2.2 Two element

( ) ��

���

����

���−�

��

���= 33ξ-1-2ξ348ALqx 22

0σ

o

14

model of bar under uniformly distributed axial load.

Fig. 2.3 Bar under linearly varying axial load

We shall postpone the first obvious approximation, that of using a one-term series xau 1r ==== till later. For now, we shall consider a two-term series

221r xaxau += . This is chosen so that the essential boundary conditions at x=0

is satisfied. No attempt is made to satisfy the force boundary condition at x=L.The reader can verify by carrying out the necessary algebra associated with theRR process that the solution obtained will yield an approximate stress patterngiven by

This is plotted on Fig. 2.3 asand (2.5) reveals an interestinterms are retained. Taking intoare orthogonal, what this means

a manner that seems to satisfy t

That is, the RR procedure has d

state of stress σ in the sensEquation (2.6). This is a resuvarious exercises we have condufrom any general principle thathere up till now.

�

(2.5)

the dashed line. A comparison of Equations (2.4)g fact - only the first two Legendre polynomialaccount the fact that the Legendre polynomialsis that in this problem, we have obtained rσ in

he following integral condition:

( ) ( ) ( )ξσ 2348ALqx 20r −=

15

(2.6)etermined a rσ that is a "best-fit" of the true

e described by the orthogonality condition inlt anticipated from our emerging results to thected so far. We have not been able to derive itthis must be so for stronger reasons than shown

( ) 0dr11- r =− ξσσσδ

Let us now proceed to an fem solution. It is logical to st rt here withthe three-node element that uses the quadratic shape functions,

and

at the nodes du

following schem

andform of a streFig. 2.3. Thus,stress system b

this step-wise

finite elementfor the three-nreader can assu

This tur

process in Equunexpected. Botquadratic admisboth have thecomplicated strbest approximattaken care of

quadraticallythrough Equatio

in the fem provaries in themanner did fσparadigm?

Let us n

the stairstep f

We shall detershown below is

The reader can

3LqP 202 =

( ) ,21N1

−= ξξ

( )2

21N ξ−=

16

We first compute the consistent loads to be placed

e to the distributed load using

e of loads at the three nodes i

. The resulting load confiss system shown as σf, represenany fem discretization automaty a step-wise system as shown

system that the finite element

computations are actually perode element and the consistentre himself, that the computed f

ns out to be exactly the sam

ation (2.5). At first sight, th the RR process and the fem psible functions for the displcapability to represent lineaess fields by an approximate liion. On second thought however,. In the RR process, the co

varying system σ (see Fig.n (2.6) that rσ responded to σcess, the load system that is bstairstep fashion. The questiorespond to σf - is it also

ow assume an unknown field σ ====

ield given by 3L2q 20f =σ in 0

mine the constants c0 and c1satisfied:

work out that this leads to

( ) .21N3

+= ξξ

0P3 =

( ) (σ 2348ALqx 20f −�

���

��=

d3L2qσ0

1

20 +�

��

��� − ��− ξσδ

( ) ( ) ( ) ,ξσ 2348ALqx 20 −=

. This resu ts in the

dentified in Fig. 2.3:

guration can be represeted by the dashed-dottically replaces a smootby fσ in Figs. 2.2 and

solution fσ responds

formed using the stiffload vector, it turnsem stress, pattern will

e as the rσ computed

his does not seem torocess here have startacement fields. This ir stress fields exactnear field that is in sthere is some more submputed rσ was respo

2.3). We could easilyin a "best-fit" manne

eing used is the σf syn confronting us now

consistent with the

ξ10 cc ++++ which is a "b

<x <L/2 and σf = 0 in

so that the "best-fit

�= dxqNP ii

)ξ

0d0σ1

0=�

��

��� − ξσδ

l

,6LqP 201 =

nted in theed lines inhly varying2.3. It is

to. If the

ness matrixout, as thebe

(2.7)

by the RR

be entirelyed out withmplies thatly or moreome sense atlety to bending to a

establishr. However,

stem, whichis, in what"best-fit"

est-fit" of

L/2 <x <L.

" variation

(2.8)

a

(2.9)

which is identical to the result obtained in Equation (2.7) by carrying out thefinite element process. In other words, the fem process follows exactly the"best-fit" description of computing stress fields. Another important lesson tobe learnt from this exercise is that the consistent lumping process preservesthe "best-fit" nature of the stress representation and subsequent prediction.Thus, is a "best-fit" of both σ and σf! Later, we shall see how the best-fitnatur ts disturbed if the loads are not consistently derived.

direc

procestresto asstart

appro

compa

are t

at ξmethofirstexist

appro

theseLegeninterof thariseboth,polynalsopolyn

see isinglcapabquadran opexistinteg

the o

e gefσ

It again seems reasonable to argue that the nodal displacements computedtly from the stiffness equations from which the stress field fσ has been

ssed can actually be thought of as being "integrated" from the "best-fit"s approximation. It seems possible now to be optimistic about our abilityk and answer questions like: What kind of approximating fields are best towith? and, How good are the computed results?

Before we pass on, it is useful for future reference to note that theximate solutions rσ or fσ intersect the exact solution σ at two points. A

rison of Equation (2.4) with Equations (2.5) and (2.7) indicates that these

he points where the quadratic Legendre polynomial, ( )231 ξ− vanishes, i.e.,

31±±±±==== . Such points are well known in the literature of the finite element

d as optimal stress points or Barlow points, named after the person whodetected such points. Our presentation shows clearly why such points

, and why in this problem, where a quadratic stress state is sought to be

ximated by a linear stress state, these points are at 31±=ξ . Curiously,

are also the points used in what is called the two-point rule for Gauss-dre numerical integration. Very often, the two issues are confused. Ourpretation so far shows that the fact that such points are sampling pointse Gauss-Legendre rule has nothing to do with the fact that similar pointsas points of optimal stress recovery. The link between them is that inthe Legendre polynomials play a decisive role - the zeros of the Legendre

omials define the optimal points for numerical integration and these pointshelp determine where the "best-fit" approximation coincides with a

omial field which is exactly one order higher.

We shall now return to the linear Ritz admissible function, xau 1r ==== , to

f it operates in the best-fit sense. This would be identical to using ae two-node bar element to perform the same function. Such a field isle of representing only a constant stress. This must now approximate theatically varying stress field σ(x) given by Equation (2.4). This gives usportunity to observe what happens to the optimal stress point, whether ones, and whether it can be easily identified to coincide with a Gaussration point.

Again, the algebra is very simple and is omitted here. One can show thatne-term approximate solution would lead to the following computed stress:

17

(2.10)(((( )))) (((( )))) (((( ))))348ALqx 20r ====σ

What becomes obvious by comparing this with the true stress σ(x) in Equation(2.4) and the computed stress from the two-term solution, (((( ))))xrσ in Equation

(2.5) is that the one-term solution corresponds to only the constant part of theLegendre polynomial expansion! Thus, given the orthogonal nature of the Legendrepolynomials, we can conclude that we have obtained the "best-fit" state ofstress even here. Also, it is clear that the optimal stress point is not easy toidentify to coincide with any of the points corresponding to the various Gauss

integration rules. The optimal point here is given by 341 −=ξ .

2.5 The aliasing paradigm

We have made out a very strong case for the best-fit paradigm. Let us nowexamine the merits, if any, of the competing paradigms. The argument that femprocedures look to satisfy the differential equations and boundary conditionsdoes not seem compelling enough to warrant further discussion. However, thebelief that finite elements seek to determine nodal displacements accurately wasthe basis for Barlow's original derivation of optimal points [2.3] and is alsothe basis for what is called the "aliasing" paradigm [2.4].

The term aliasing is borrowed from sample data theory where it is used todescribe the misinterpretation of a time signal by a sampling device. Anoriginal sine wave is represented in the output of a sampling device by analtered sine wave of lower frequency - this is called the alias of the truesignal. This concept can be extended to finite element discretization - thesample data points are now the values of the displacements at the nodes and thealias is the function, which interpolates the displacements within the elementfrom the nodal displacements. We can recognize at once that Barlow [2.3]developed his theory of optimal points using an identical idea - the termsubstitute function is used instead of alias.

Let us now use the aliasing concept to derive the location of the optimalpoints, as Barlow did earlier [2.3], or as MacNeal did more recently [2.4]. Weassume here that the finite element method seeks discretized displacementfields, which are substitutes or aliases of the true displacement fields bysensing the nodal displacements directly. We can compare this with the"best-fit" interpretation where the fem is seen to seek discretizedstrain/stress fields, which are the substitutes/aliases of the truestrain/stress fields in a "best fit" or "best approximation" sense. It isinstructive now to see how the alternative paradigm, the "displacement aliasing"approach leads to subtle differences in interpreting the relationship betweenthe Barlow points and the Gauss points.

Table 2.1 Barlow and Gauss points for one-dimensional case

Barlow pointsp Nodesat u ε

Gausspoints ‘best-fit’ aliasing

1 ± 1 ξ2 ξ ξ 1 0 0 0

2 0, ± 1 ξ3 ξ2 ξ2 ξ ± 1/√3 ± 1/√3 ± 1/√3

3 ±1/3, ± 1 ξ4 ξ3 ξ3 ξ2

1, ξ..,ξ4 indicate polynomial order

u ε

18

s

530, ±

0, ± √5/3530, ±

2.5.1 A one dimensional problem

We again take up the simplest problem, a bar under axial loading. We shallassume that the bar is replaced by a single element of varying polynomial orderfor its basis function (i.e. having varying no. of equally spaced nodes). Thus,from Table 2.1, we see that p=1,2,3 correspond to basis functions of linear,quadratic and cubic order, implying that the corresponding elements have 2,3,4nodes respectively. These elements are therefore capable of representing aconstant, linear and quadratic state of strain/stress, where strain is taken tobe the first derivative of the displacement field. We shall adopt the followingnotation: The true displacement, strain and stress field will be designated by

σε andu, . The discretized displacement, strain and stress field will be

designated by σε and,u . The aliased displacement, strain and stress field

will be designated by ua, εa and σa. Nodal displacements will be represented byui.

We shall examine three scenarios. In the simplest, Scenario a, the truedisplacement field u is exactly one polynomial order higher than what the finiteelement is capable of representing - we shall see that the Barlow points can bedetermined exactly in terms of the Gauss points only for this case. In Scenariob, we consider the case where the true field u is two orders higher than thediscretized field u . The definition of an identifiable optimal point becomesdifficult. In both Scenarios a and b, we assume that the rigidity of the bar isa constant, i.e., σ = D ε. In Scenario c, we take up a case where the rigiditycan vary, i.e., σ = D(ξ) ε. We shall see that once again, it becomes difficultto identify the optimal points by any simple rule.

We shall now take for granted that the best-fit rule operates according tothe orthogonality condition expressed in Equation (2.6) and that it can be usedinterchangeably for stresses and strains. We shall designate the optimal pointsdetermined by the aliasing algorithm as ξa, the Barlow points (aliasing), and

the points determined by the best-fit algorithm as ξb, the Barlow points (best-

fit). Note that ξa are the points established by Barlow [2.3] and MacNeal [2.4],

while ξb will correspond to the points given in References 2.6 and 2.7. Note

that the natural coordinate system ξ is being used here for convenience.

Thus, for Scenarios a and b, this leads to

whereas for Scenario c, it

Note that we can considercondition in Equation (corresponds to one inpolynomials can be made.this case, one can determ

points, which are the zelist of unnormalised Lege

(2.11)

becomes

( )� =− 0dVT εεεδ

19

(2.12)

Equation (2.11) as a special case of the orthogonality2.12) with the weight function D(ξ)=1. This casewhich a straightforward application of Legendre

This point was observed very early by Moan [2.5]. Inine the points where εε ==== as those corresponding to

ros of the Legendre polynomials. See Table 2.2 for andre polynomials. We shall show below that in Equation

( ) ( )� =− 0dVDT εεξεδ

20

(2.11), the points of minimum error are the sampling points of the GaussLegendre integration rule only if ε is exactly one polynomial order lower than

ε. And in Equation (2.12), the optimal points no longer depend on the nature ofthe Legendre polynomials, making it difficult to identify the optimal points.

Scenario a

We shall consider fem solutions using a linear (two-node), a quadratic (three-node) and a cubic (four-node) element. The true displacement field is taken tobe one order higher than the discretized field in each case.

Linear element (p = 1)

u = quadratic = bo + b1 ξ + b2 ξ2 (2.13)

ε = linear = u,ξ = b1 + 2b2 ξ = (((( ))))����====

====

1p

0sss P ξε (2.14)

Note that we have written ε in terms of the Legendre polynomials for futureconvenience. Note also that we have simplified the algebra by assuming thatstrains can be written as derivatives in the natural co-ordinate system. It isnow necessary to work out how the algebra differs for the aliasing and best-fitapproaches.

Aliasing: At ;, iaii uu1 ====±±±±====ξ then points where εε ====a are given by ξa=0. (The

algebra is very elementary and is left to the reader to work out). Thus, theBarlow point (aliasing) is ξa = 0, for this case.

Best-fit: u =linear, is undetermined at first. Let 0c====ε , as the element is

capable of representing only a constant strain. Equation (2.11) will now give

10 bc ==ε . Thus, the optimal point is ξb = 0, the point where the Legendre

polynomial P1(ξ) = ξ vanishes. Therefore, the Barlow point (best-fit) for this

example is ξb = 0.

Table 2.2 The Legendre polynomials Pi

Order of polynomial i PolynomialPi

0

1

2

3

4

1

ξ

(1-3ξ2)

(3ξ-5ξ3)

(3-30ξ2+35ξ4)

21

Quadratic element (p = 2)

u = cubic = b0 + b1 ξ + b2 ξ2 + b3 ξ3

ε = quadratic = u,ξ (2.15)

(((( )))) (((( ))))����====

====������������

������������ −−−−−−−−++++++++====

0sss

23231 P31b2bbb ξεξξ (2.16)

Aliasing: At ;iaii uu1,0, =±=ξ ; then points where aε = ε are given by 31a ±=ξ .

(Again, the algebra is left to the reader to work out). Thus, the Barlow points

(aliasing) are 31a ±±±±====ξ , for this case.

Best-fit: u = quadratic. Let ξε 10 cc ++++==== as this element is capable of

representing a linear strain. Equation (2.11) will now give (((( )))) ξε 231 2bbb ++++++++==== .

Thus, the optimal points are 31b ±±±±====ξ , the points where the Legendre

polynomial (((( )))) (((( ))))22 31P ξξ −−−−==== vanishes. Therefore, the Barlow points (best-fit) for

this example are 31b ±±±±====ξ .

Note that in these two examples, i.e. for the linear and quadratic elements, theBarlow points from both schemes coincide with the Gauss points (the points wherethe corresponding Legendre polynomials vanish). In our next example we will findthat this will not be so.

Cubic element (p = 3)

u = quartic = 44

33

2210 bbbbb ξξξξ ++++ (2.17)

ε = cubic = ξ,u

( ) ( ) ( ) ( )34

234231 5354b31b512b2bbb ξξξξ −−−−+++=

(((( ))))����====

========

3p

0sssP ξε (2.18)

Aliasing: At iaii uu1,31 =±±= ,ξ ; then points where εε ====a are given by

350a ±= ,ξ . Thus, the Barlow points (aliasing) are 350a ±±±±==== ,ξ for this case.

Note that the points where the Legendre polynomial ( ) ( )33 53P ξξξ −= vanishes are

530,0 =ξ !

Best-fit: u = cubic. Let ( )2210 31ccc ξξε −++= , as this element is capable of

representing a quadratic strain. Equation (2.11) will now give

(((( )))) (((( )))) (((( ))))234231 3ξ1b512b2bbb −−−−−−−−++++++++++++==== ξε . Thus, the Barlow points (best-fit) for

22

this example are 530,b =ξ , the points where the Legendre polynomial

( ) ( )33 53P ξξξ −= vanishes.

Therefore, we have an example where the aliasing paradigm does not givethe correct picture about the way the finite element process computes strains.However, the best-fit paradigm shows that as long as the discretized strain isone order lower than the true strain, the corresponding Gauss points are theoptimal points. Table 2.1 summarizes the results obtained so far.

The experience of this writer and many of his colleagues is that the best-fit model, is the one that corresponds to reality. If one were to actuallysolve a problem where the true strain varies cubically using a 4-noded element,which offers a discretized strain which is of quadratic order, the points ofoptimal strain actually coincide with the Gauss points.

Scenario b:

So far, we have examined simple scenarios where the true strain is exactly onepolynomial order higher than the discretized strain with which it is replaced.If ( )ξpP , denoting the Legendre polynomial of order p, describes the order by

which the true strain exceeds the discretized strain, the simple rule is that heoptimal points are obtained as ( ) 0P bp =ξ . These are therefore the set of p Gauss

points at which the Legendre polynomial of order p vanishes. Consider now a casewhere the true strain is two orders higher than the discretized strain - e.g. aquadratic element (p = 2) modeling a region where the strain and stress fieldvary cubically. Thus, we have,

( ) ( ) ( ) ( )34

234231 5354b31b512b2bbb ξξξξε −−−−+++=

ξε 10 cc += (2.19)

Equation (2.11) allows us to determine the coefficient ci in terms of bi; itturns out that

( ) ( ) ,ξε 512b2bbb 4231 +++= (2.20)

a representation made very easy by the fact that the Legendre polynomials areaorthogonal and that therefore ε can be obtained from ε by simple inspection.

It is not however a simple matter to determine whether the optimal pointscoincide with other well-known points like the Gauss points. In this example, wehave to seek the zeros of

( ) ( )34

23 5354b31b ξξξ −+− (2.21)

Since b3 and b4 are arbitrary, depending on the problem, it is not possible toseek universally valid points where this would vanish, unlike in the case ofScenario a earlier. Therefore, in such cases, it is not worthwhile to seekpoints of optimal accuracy. It is sufficient to acknowledge that the finite

23

element procedure yields strains ε , which are the most reasonable one can

obtain in the circumstances.

Scenario c

So far, we have confined attention to problems where σ is related to ε through asimple constant rigidity term. Consider an exercise where (the one-dimensionalbar again) the rigidity varies because the cross-sectional area varies orbecause the modulus of elasticity varies or both, i.e. σ = D(ξ)ε. Theorthogonality condition that governs this case is given by Equation (2.12).Thus, it may not be possible to determine universally valid Barlow points apriori if D(ξ) varies considerably.

2.6 The `best-fit' rule from a variational theorem

Our investigation here will be complete in all respects if the basis for thebest-fit paradigm can be derived logically from a basic principle. In fact, somerecent work [2.6,2.7] shows how by taking an enlarged view of the variationalbasis for the displacement type fem approach we will be actually led to theconclusion that strains or stresses are always sought in the best-fit manner.

The `best-fit' manner in which finite elements compute strains can beshown to follow from an interpretation uses the Hu-Washizu theorem. To see howwe progress from the continuum domain to the discretized domain, we will find itmost convenient to develop the theory from the generalized Hu-Washizu theorem[2.8] rather than the minimum potential theorem. As we have seen earlier, thesetheorems are the most basic statements that can be made about the laws ofnature. The minimum potential theorem corresponds to the conventional energytheorem. However, for applications to problems in structural and solidmechanics, Hu proposed a generalized theorem, which had somewhat moreflexibility [2.8]. Its usefulness came to be recognized when one had to grapplewith some of the problems raised by finite element modeling. One such puzzle isthe rationale for the "best-fit" paradigm.

Let the continuum linear elastic problem have an exact solution describedby the displacement field u, strain field ε and stress field σ. (We project thatthe strain field ε is derived from the displacement field through the strain-displacement gradient operators of the theory of elasticity and that the stressfield is derived from the strain field through the constitutive laws.) Let usnow replace the continuum domain by a discretized domain and describe thecomputed state to be defined by the quantities ε,u and σ , where again, we

take that the strain fields and stress fields are computed from the strain-displacement and constitutive relationships. It is clear that ε is an

approximation of the true strain field ε. What the Hu-Washizu theorem does,following the interpretation given by Fraejis de Veubeke [2.9], is to introducea dislocation potential to augment the usual total potential. This dislocation

potential is based on a third independent stress field σ which can beconsidered to be the Lagrange multiplier removing the lack of compatibilityappearing between the true strain field ε and the discretized strain field ε .

24

Note that, σ is now an approximation of σ . The three-field Hu-Washizu theoremcan be stated as

0=δπ (2.22)

where

(((( ))))����������������

������������

++++−−−−++++==== dvP2T

T εεσεσπ (2.23)

and P is the potential energy of the prescribed loads.

In the simpler minimum total potential principle, which is the basis for thederivation of the displacement type finite element formulation in mosttextbooks, only one field (i.e., the displacement field u) is subject tovariation. However, in this more general three field approach, all the threefields are subject to variation and leads to three sets of equations which canbe grouped and classified as follows:

Variation on Nature Equation

u Equilibrium ∇ σ + terms from P = 0 (2.24a)

(((( ))))ityCompatibilityOrthogonalσ (((( )))) 0dv

T====−−−−���� εεσδ (2.24b)

( )mEquilibriuityOrthogonalε 0dvT ====����

��������

���������������� σ−σεδ (2.24c)

Equation (2.24a) shows that the variation on the displacement field u requires

that the independent stress field σ must satisfy the equilibrium equations (∇signifies the operators that describe the equilibrium condition. Equation(2.24c) is a variational condition to restore the equilibrium imbalance between

σ and σ . In the displacement type formulation, we choose σ=σ . This satisfies

the orthogonality condition seen in Equation (2.24c) identically. This leaves uswith the orthogonality condition in Equation (2.24b). We can now argue that thistries to restore the compatibility imbalance between the exact strain field εand the discretized strain field ε . In the displacement type formulation this

can be stated as,

(((( )))) 0dVT ====���� ε−εσδ (2.25)

Thus we see very clearly, that the strains computed by the finite elementprocedure are a variationally correct (in a sense, a least squares correct)`best approximation' of the true state of strain.

25

2.7 What does the finite element method do?

To understand the motivation for the aliasing paradigm, it will help toremember that it was widely believed that the finite element method soughtapproximations to the displacement fields and that the strains/stresses werecomputed by differentiating these fields. Thus, elements were believed to be"capable of representing the nodal displacements in the field to a good degreeof accuracy." Each finite element samples the displacements at the nodes, andinternally, within the element, the displacement field is interpolated using thebasis functions. The strain fields are computed from these using a process thatinvolves differentiation. It is argued further that as a result, displacementsare more accurately computed than the strain and stress field. This follows fromthe generally accepted axiom that derivatives of functions are less accuratethan the original functions. It is also argued that strains/stresses are usuallymost inaccurate at the nodes and that they are of greater accuracy near theelement centers - this, it is thought, is a consequence of the mean valuetheorem for derivatives.

However, we have demonstrated convincingly that in fact, the Ritzapproximation process, and the displacement type fem which can be interpreted asa piecewise Ritz procedure, do exactly the opposite - it is the strain fieldswhich are computed, almost independently as it were within each element. Thiscan be derived in a formal way. Many attempts have been made to give expressionto this idea, (e.g., Barlow [2.1] and Moan [2.7]), but it appears that the mostintellectually satisfying proof can be arrived at by starting with a mixedprinciple known as the Hu-Washizu theorem [2.8]. This proof has been taken up ingreater detail in Reference 2.3 and was reviewed only briefly in the precedingsection. Having said that the Ritz type procedures determine strains, it followsthat the displacement fields are then constructed from this in an integralsense. The stiffness equation actually reflecting this integration process andthe continuity of fields across element boundaries and suppression of the fieldvalues at domain edges being reflected by the imposition of boundary conditions.It must therefore be argued that displacements are more accurate than strainsbecause integrals of smooth functions are generally more accurate than theoriginal data. We have thus turned the whole argument on its head.

2.8 Conclusions

In this chapter, we postulated a few models to explain how the displacementstype fem works. We worked out a series of simple problems of increasingcomplexity to establish that our conjecture that strains and stresses appear ina "best-fit" sense could be verified (falsified, in the Popperian sense) bycarefully designed numerical experiments.

An important part of this exercise depended on our careful choice and useof various stress terms. Thus terms like σ and σf was actual physical states

that were sought to be modeled. The stress terms rσ and fσ were the quantities

that emerged in what we can call the "first-order tradition" analysis in thelanguage of Sir Karl Popper [2.10] - where the RR or fem operations aremechanically carried out using functional approximation and finite elementstiffness equations respectively. We noticed certain features that seemed torelate these computed stresses to the true system they were modeling in apredictable or repeatable manner. We then proposed a mechanism to explain how

26

this could have taken place. Our bold conjecture, after examining thesenumerical experiments, was to propose that it is effectively seeking a best-fitstate.

To confirm that this conjecture is scientifically coherent and complete,we had to enter into a "second-order tradition" exercise. We assumed that thisis indeed the mechanism that is operating behind the scenes and derived

predicted quantities rσ that will result from the best-fit paradigm when thiswas applied to the true state of stress. These predicted quantities turned outto be exactly the same as the quantities computed by the RR and fem procedures.In this manner, we could satisfy ourselves that the "best-fit" paradigm hadsuccessfully survived a falsification test.

Another important step we took was to prove that the "best-fit" paradigmwas neither gratuitous nor fortuitous. In fact, we could also establish thatthis could be derived from more basic principles - in this regard, thegeneralized theorem of Hu was found valuable to determine that the best-fitparadigm had a rational basis. In subsequent chapters, we shall use thisfoundation to examine other features of the finite element method.

One important conclusion we can derive from the best-fit paradigm is that,an interpolation field for the stresses σ (or stress resultants as the case may

be) which is of higher order than the strain fields ε. On which it must `dowork' in the energy or virtual work principle is actually self-defeating becausethe higher order terms cannot be `sensed'. This is precisely the basis for deVeubeke's famous limitation principle [2.9], that ‘it is useless to look for abetter solution by injecting additional degrees of freedom in the stresses.’ Wecan see that one cannot get stresses, which are of higher order than arereflected in the strain expressions.

2.9 References

2.1. T. S. Kuhn, The Structure of Scientific Revolution, University of ChicagoPress, 1962.

2.2. S. Dasgupta, Understanding design: Artificial intelligence as anexplanatory paradigm, SADHANA, 19, 5-21, 1994.

2.3. J. Barlow, Optimal stress locations in finite element models, Int. J. Num.Meth. Engng. 10, 243-251 (1976).

2.4. R. H. MacNeal, Finite Elements: Their Design and Performance, MarcelDekker, NY, 1994.

2.5 T. Moan, On the local distribution of errors by finite elementapproximations, Theory and Practice in Finite Element Structural Analysis.Proceedings of the 1973 Tokyo Seminar on Finite Element Analysis, Tokyo,43-60, 1973.

2.6. G. Prathap, The Finite Element Method in Structural Mechanics, KluwerAcademic Press, Dordrecht, 1993.

27

2.7. G. Prathap, A variational basis for the Barlow points, Comput. Struct. 49,381-383, 1993.

2.8. H. C. Hu, On some variational principles in the theory of elasticity andthe theory of plasticity, Scientia Sinica, 4, 33-54 (1955).

2.9. B. F. de Veubeke, Displacement and equilibrium models in the finite elementmethod, in Stress Analysis, Ellis Horwood, England, 1980.

2.10. B. Magee, Popper, Fontana Press, London, 1988.

28

Chapter 3

Completeness and continuity: How to choose shape functions?

3.1 Introduction

We can see from our study so far that the quality of approximation we canachieve by the RR or fem approach depends on the admissible assumed trial, fieldor shape functions that we use. These functions can be chosen in many ways. Anobvious choice, and the one most universally preferred is the use of simplepolynomials. It is possible to use other functions like trigonometric functionsbut we can see that the least squares or best-fit nature of stress predictionthat the finite element process seeks in an instinctive way motivates us toprefer the use of polynomials. When this is done, interpretations using Legendrepolynomials become very convenient. We shall see this happy state of affairsappearing frequently in our study here.

An important question that will immediately come to mind is - How does theaccuracy or efficiency of the approximate solution depends on our choice ofshape functions. Accuracy is represented by a quality called convergence. Byconvergence, we mean that as we add more terms to the RR series, or as we addmore nodes and elements into the mesh that replaces the original structure, thesequence of trial solutions must approach the exact solution. We want quantitiessuch as displacements, stresses and strain energies to be exactly recovered,surely, and if possible, quickly. This leads to the questions: What kind ofassumed pattern is best for our trial functions? What are the minimum qualities,or essences or requirements that the finite element shape functions must show ormeet so that convergence is assured. Two have been accepted for a long time:continuity (or compatibility) and completeness.

3.2 Continuity

This is very easy to visualize and therefore very easy to understand. Astructure is sub-divided into sub-regions or elements. The overall deformationof the structure is built-up from the values of the displacements at the nodesthat form the net or grid and the shape functions within elements. Within eachelement, compatibility of deformation is assured by the fact that the choice ofsimple polynomial functions for interpolation allows continuous representationof the displacement field. However, this does not ensure that the displacementsare compatible between element edges. So special care must be taken otherwise,in the process of representation, gaps or overlaps will develop.

The specification of continuity also depends on how strains are defined interms of derivatives of the displacement fields. We know that a physical problemcan be described by the stationary condition δΠ=0, where Π=Π(φ) is thefunctional. If Π contains derivatives of the field variable φ to the order m,

then we obviously require that within each element, φ , which is the approximate

field chosen as trial function, must contain a complete polynomial of degree m.

So that φ is continuous within elements and the completeness requirements (see

below) are also met. However, the more important requirement now is thatcontinuity of field variable φ must be maintained across element boundaries -

29

this requires that the trial function φ and its derivatives through order m-1

must be continuous across element edges. In most natural formulations in solidmechanics, strains are defined by first derivatives of the displacement fields.In such cases, a simple continuity of the displacement fields across elementedges suffices - this is called C0 continuity. Compatibility between adjacent

elements for problems which require only C0 -continuity can be easily assured ifthe displacements along the side of an element depend only on the displacementsspecified at all that nodes placed on that edge.

There are problems, as in the classical Kirchhoff-Love theories of platesand shells, where strains are based on second derivatives of displacementfields. In this case, continuity of first derivatives of displacement fieldsacross element edges is demanded; this is known as C1-continuity. Elements,which satisfy the continuity conditions, are called conforming or compatibleelements.

We shall however find a large class of problems (Timoshenko beams, Mindlinplates and shells, plane stress/strain flexure, incompressible elasticity) wherethis simplistic view of continuity does not assure reasonable (i.e. practical)rates of convergence. It has been noticed that formulations that take liberties,e.g. the non-conforming or incompatible approaches, significantly improveconvergence. This phenomenon will be taken up for close examination in a laterchapter.

3.3 Completeness

We have understood so far that in the finite element method, or for that matter,in any approximate method, we are trying to replace an unknown function φ(x),which is the exact solution to a boundary value problem over a domain enclosed

by a boundary by an approximate function (((( ))))xφ which is constituted from a set of

trial, shape or basis functions. We desire a trial function set that will ensurethat the approximation approaches the exact solution as the number of trialfunctions is increased. It can be argued that the convergence of the trial

function set to the exact solution will take place if (((( ))))xφ will be sufficient to

represent any well behaved function such as φ(x) as closely as possible as thenumber of functions used becomes indefinitely large. This is called thecompleteness requirement.

In the finite element context, where the total domain is sub-divided intosmaller sub-regions, completeness must be assured for the shape functions usedwithin each domain. The continuity requirements then provide the compatibilityof the functions across element edges.

We have also seen that what we seek is a best-fit arrangement in some

sense between (((( ))))xφ and φ(x). From Chapter 2, we also have the insight that this

best-fit can be gainfully interpreted as taking place between strain or stressquantities. This has important implications in further narrowing the focus ofthe completeness requirement for finite element applications in particular.

By bringing in the new perspective of a best-fit strain or stressparadigm, we are able to look at the completeness requirements entirely from

30

what we desire the strains or stresses to be like. Of paramount importance nowis the idea that the approximate strains derived from the set of trial functionschosen must be capable in the limit of approximation (i.e. as the number ofterms becomes very large) to describe the true strain or stress fields exactly.This becomes very clear from the orthogonality condition derived in Equation2.25.

From the foregoing discussion, we are now in a position to make a moremeaningful statement about what we mean by completeness. We would likedisplacement functions to be so chosen that no straining within an element takesplace when nodal displacements equivalent to a rigid body motion of the wholeelement are applied. This is called the strain free rigid body motion condition.In addition, it will be necessary that each element must be able to reproduce astate of constant strain; i.e. if nodal displacements applied to the element arecompatible with a constant strain state, this must be reflected in the strainscomputed within the element. There are simple rules that allow these conditionsto be met and these are called the completeness requirements. If polynomialtrial functions are used, then a simple assurance that the polynomial functions

contain the constant and linear terms, etc. (e.g. 1, x in a one-dimensional C0

problem;) 1, x, y in a two-dimensional C0 problem so that each element iscertain to be able to recover a constant state of strain, will meet thisrequirement.

3.3.1 Some properties of shape functions for C0 elements

The displacement field is approximated by using shape functions Ni within each

element and these are linked to nodal degrees of freedom ui. It is assumed thatthese shape functions allow elements to be connected together without generatinggaps or overlaps, i.e. the shape functions satisfy the continuity requirementdescribed above. Most finite element formulations in practical use in generalpurpose application require only what is called C0 continuity - i.e. a field

quantity φ must have interelement continuity but does not require the continuityof all of its derivatives across element boundaries.

C0 shape functions have some interesting properties that derive from thecompleteness requirements imposed by rigid body motion considerations. Considera field φ (usually a displacement) which is interpolated for an n-noded elementaccording to

����====

====n

1iiiN φφ (3.1)

where the Ni are assumed to be interpolated using natural co-ordinates (e.g. ξ,η for a 2-dimensional problem). It is simple to argue that one property the

shape functions must have is that Ni must define the distribution of φ within

the element domain when the degree of freedom φi at node i has unit value and

all other nodal φ’s are zero. Thus

i) Ni = 1 when x = xi (or ξ = ξi) and Ni = 0 when x = xj (or ξ = ξj) for i ≠ j.

31

Consider now, the case of the element being subjected to a rigid body motion so

that at each node a displacement ui=1 is applied. It is obvious from physicalconsiderations that every point in the element must have displacements u=1. Thusfrom Eq 3.1 we have

ii) Σ Ni = 1.

This offers a very simple check for shape functions. Another useful check comesagain from considerations of rigid body motion - that a condition of zero strainmust be produced when rigid body displacements are applied to the nodes. Thereader is encouraged to show that this entails conditions such as

iii) Σ Ni,ξ = 0, Σ Ni,η = 0

for a 2-dimensional problem.

Note that for C1 elements, where derivatives of φ are also used as nodal degrees

of freedom, these rules apply only to those Ni which are associated with thetranslational degrees of freedom.

3.4 A numerical experiment regarding completeness

Let us now return to the uniform bar carrying an axial load P as shown inFig. 2.1. The exact solution for this problem is given by a state of constantstrain. Thus, the polynomial series (((( )))) xaaxu 10 ++++==== (see Section 2.3) will contain

enough terms to ensure completeness. We could at the outset arrange for a0 to be

zero to satisfy the essential boundary condition. With (((( )))) xaxu 11 ==== we were able

to obtain the exact solution to the problem. We can now investigate what will

happen if we omit the a1x term and start out with the trial function set

(((( )))) 222 xaxu ==== . It is left to the reader to show that this results in a computed

stress variation 2EAL3Px2 ====σ ; i.e. a linear variation is seen instead of the

actual constant state of stress. The reader is now encouraged to experiment with

RR solutions based on other incomplete sets like (((( )))) 333 xaxu ==== and

(((( )))) 33

2232 xaxaxu ++++====++++ . Figure 3.1 shows how erroneous the computed stresses σ2, σ3

and σ2+3 are.

The reason for this is now apparent. The term that we have omitted is theone needed to cover the state of constant strain. In this case, this very termalso provides the exact answer. The omission of this term has led to the loss ofcompleteness. In fact, the reader can verify that as long as this term is leftout of any trial set containing polynomials of higher order, there will not beany sort of clear convergence to the true state of strain. In the presentexample, because of the absence of the linear term in u (the constant strainterm therefore vanishes), the computed strain vanishes at the origin!

Fig. 3.1 S

3.5 Concluding

We have now laidtrial functionscorrect results.from coarse mesacceptably rapidparadigm arounderrors and conve

Unfortunatcan behave in udifficulties arproblems encountproblems may havclassify them. Athat curved beamthe strain-freeerror inducing mcalled membranetake these stran

32

tresses from RR solutions based on incomplete polynomials

remarks

down some specific qualities (continuity and completeness) thatmust satisfy to ensure that approximate solutions converge toIt is also hoped that if these requirements are met, solutions

hes will be reasonably accurate and that convergence will bewith mesh refinement. This was the received wisdom or reigning1977. In the next chapter, we shall look at the twin issues ofrgence from this point of view.

ely, elements derived rigorously from only these basic paradigmsnreasonably erratic ways in many important situations. Thesee most pronounced in the lowest order finite elements. Theered were called locking, parasitic shear, etc. Some of thee gone unrecorded with no adequate framework or terminology tos a very good example, for a very long time, it was believedand shell elements performed poorly because they could not meetrigid body motion condition. However, more recently, the correctechanism has been discovered and these problems have come to belocking. An enlargement of the paradigm is now inevitable toge features into account. This shall be taken up from Chapter 5.

33

Chapter 4

Convergence and Errors

4.1 Introduction

Error analysis is that aspect of fem knowledge, which can be said to belong tothe "second-order tradition" of knowledge. To understand its proper place, letus briefly review what has gone before in this text. We have seen structuralengineering appear first as art and technology, starting as the imitation ofstructural form by human fabrication. The science of solid and structuralmechanics codified this knowledge in more precise terms. Mathematical modelswere then created from this to describe the structural behavior of simple andcomplex systems analytically and quantitatively. With the emergence of cheapcomputational power, techniques for computational simulation emerged. Thefinite element method is one such device. It too grew first as art, bysystematic practice and refinement. In epistemological terms, we can see thisas a first-order tradition or level or inner loop of fem art and practice.Here, we know what to do and how to do it. However, the need for an outer loopor second-order tradition of enquiry becomes obvious. How do we know why weshould do it this way? If the first stage is the stage of action, this secondstage is now of reflection. Error analysis therefore belongs to this traditionof epistemological enquiry.

Let us now translate what we mean by error analysis into simpler terms inthe present context. We must first understand that the fem starts with thebasic premises of its paradigms, its definitions and its operational proceduresto provide numerical results to physical problems which are already describedby mathematical models that make analytical quantification possible. The answermay be wrong because the mathematical model wrongly described the physicalmodel. We must take care to understand that this is not the issue here. If themathematical model does not explain the actual physical system as observedthrough carefully designed empirical investigations, then the models must berefined or revised so that closer agreement with the experiment is obtained.This is one stage of the learning process and it is now assumed that over thelast few centuries we have gone through this stage successfully enough toaccept our mathematical models without further doubt or uncertainty.

Since our computational models are now created and manipulated usingdigital computers, there are errors which occur due to the fact thatinformation in the form of numbers can be stored only to a finite precision(word length as it is called) at every stage of the computation. These arecalled round-off errors. We shall assume here that in most problems we dealwith, word length is sufficient so that round-off error is not a majorheadache.

The real issue that is left for us to grapple with is that thecomputational model prepared to simulate the mathematical model may be faultyand can lead to errors. In the process of replacing the continuum region byfinite elements, errors originate in many ways. From physical intuition, we canargue that this will depend on the type and shape of elements we use, thenumber of elements used and the grading or density of the mesh used, the waydistributed loads are assigned to nodes, the manner in which boundary

34

conditions are modeled by specification of nodal degrees of freedom, etc. Theseare the discretization errors that can occur.

Most of such errors are difficult to quantify analytically or determinein a logically coherent way. We can only rely on heuristic judgement tounderstand how best to minimize errors. However, we shall now look only at thatcategory of discretization error that appears because the computational ordiscretized model uses trial functions, which are an approximation of the truesolution to the mathematical model. It seems possible that to some extent,analytical quantification of these errors is possible.

We can recognize two kinds of discretization error belonging to thiscategory. The first kind is that which arises because a model replaces aproblem with an infinitely large number of degrees of freedom with a finitenumber of degrees of freedom. Therefore, except in very rare cases, thegoverning differential equations and boundary conditions are satisfied onlyapproximately. The second kind of error appears due to the fact that byoverlooking certain essential requirements beyond that specified by continuityand completeness, the mathematical model can alter the physics of the problem.In both cases, we must be able to satisfy ourselves that the discretizationprocess, which led to the computational model, has introduced a certainpredictable degree of error and that this converges at a predictable rate, i.e.the error is removed in a predictable manner as the discretization is improvedin terms of mesh refinement. Error analysis is the attempt to make suchpredictions a priori, or rationalize the errors in a logical way, a posteriori,after the errors are found.

To carry out error analysis, new procedures have to be invented. Thesemust be set apart from the first-order tradition procedures that carry out thediscretization (creating the computational model from the mathematical model)and solution (computing the approximate results). Thus, we must designauxiliary procedures that can trace errors in an a priori fashion from basicparadigms (conjectures or guesses). These error estimates or predictions can beseen as consequences computed from our guesses about how the fem works. Theseerrors must now be verified by constructing simple digital computationexercises. This is what we seek to do now. If this cycle can be completed, thenwe can assure ourselves that we have carved out a scientific basis for erroranalysis. This is a very crucial element of our study. The fem, or for thatmatter, any body of engineering knowledge, or engineering methodology, can besaid to have acquired a scientific basis only when it has incorporated withinitself, these auxiliary procedures that permit its own self-criticism.Therefore, error analysis, instead of being only a posteriori or post mortemstudies, as it is usually practised, must ideally be founded on a prioriprojections computed from intelligent paradigms which can be verified(falsified) by digital computation.

In this chapter, we shall first take stock of the conventional wisdomregarding convergence. This is based on the old paradigm that fem seeks toapproximate displacements accurately. We next take note of the newlyestablished paradigm that the Ritz-type and fem approximations seekstrains/stresses in a `best-fit' manner. From such an interpretation, weexamine if it is possible to argue that errors, whether in displacements,stresses or energies, due to finite element discretization must diminish

rapidly, at least in a (l/L)2 manner or better, where a large structure (domain)of dimension L is sub-divided into elements (sub-domains) of dimension l. Thus,

35

with ten elements in a one-dimensional structure, errors must not be more thana few percent. This is the usual range of problems where the continuity andcompleteness paradigms explain completely the performance of finite elements.In subsequent chapters, we shall discover however that a class of problemsexist where errors are much larger - the discretization fail in a dramaticfashion. Convergence and error analysis must now be founded on a more complexconceptual framework - new paradigms need to be introduced and falsified. Thiswill be postponed to subsequent chapters.

4.2 Traditional order of error analysis

The order of error analysis approach that is traditionally used is inheritedfrom finite difference approaches. This seems reasonable because quite oftensimple finite difference and finite element approximations result in identicalequations. It is also possible with suitable interpretations to cast finitedifference approximations as particular cases of weighted residualapproximations using finite element type trial functions. However, there aresome inherent limitations in this approach that will become clear later.

It is usual to describe the magnitude of error in terms of the mesh size.Thus, if a series of approximate solutions using grids whose mesh sizes areuniformly reduced is available, it may be possible to obtain more informationabout the exact answer by some form of extrapolation, provided there is somemeans to establish the rate at which errors are removed with mesh size.

The order of error analysis proceeds from the understanding that thefinite element method seeks approximations for the displacement fields. Theerrors are therefore interpreted using Taylor Series expansions for the truedisplacement fields and truncations of these to represent the discretizedfields. The simple example below will highlight the essential features of thisapproach.

4.2.1 Error analysis of the axially loaded bar problem

Let us now go back to the case of the cantilever bar subjected to a uniformlydistributed axial load of intensity q0 (Section 2.3). The equilibrium equationfor this problem is

0AEqu, 0xx ====++++ (4.1)

We shall use the two-node linear bar elements to model this problem. We haveseen that this leads to a solution, which gives exact displacements at thenodes. It was also clear to us that this did not mean that an exact solutionhad been obtained; in fact while the true solution required a quadraticvariation of the displacement field u(x), the finite element solution (x)u waspiecewise linear. Thus, within each element, there is some error at locationsbetween the nodes.

Fig. 4.1 shows the displacement and strain error in the region of anelement of length 2l placed with its centroid at xi in a cantilever bar of

length L. If for convenience we choose 1AEq0 ==== , then we can show that

2xLxu(x) 2−−−−==== (4.2a)

2)x2xx(lLx(x)u 2ii

2 −+−= (4.2b)

36

u

Fig. 4.1 Displacement and strain error in a uniform bar uload 0q

ε(x)=L–x

ixL(x) −−−−====ε

If we denote the errors in the displacement field and ste’(x) respectively, we can show that

( ) ( ) ( ) 2lxxlxxxe ii −−+−=

( ) ( )ixxxe −=′

From this, we can argue that in this case, the strainelement centroid, x=xi, and that it is a maximum at theclear from Fig. 4.1. It is also clear that for this proerror is a maximum at x=xi. What is more important to usfor these errors in terms of the element size l, or morethe parameter Llh = , which is the dimensionless quant

mesh division relative to the size of the structure. Ithave these errors normalized with respect to typical disoccurring in the problem. From Equations (4.2) and (4.3maximum normalized errors to be of the following ordersstands for "order",

( )2max

hOue =

∈

( )x∈

nder

rain

erroelemblemisuse

ity

willplac), wof ma

∈

distributed axial

(4.2c)

(4.2d)

field by e(x) and

(4.3a)

(4.3b)

r vanishes at theent nodes. This is, the displacementto derive measuresfully, in terms ofthat indicates the

also be useful toements and strainse can estimate thegnitude, where “O”

(4.4a)

( )x

37

( )hOemax

=′ ε (4.4b)

Note that the order of error we have estimated for the displacement field isthat for a location inside the element as in this problem, the nodaldeflections are exact. The strain error is O(h) while the displacement error isO(h2). At element centroids, the error in strain vanishes. It is not clear to us

in this analysis that the discretized strain ( )xε is a "best-fit" of the true

strain ( )xε . If this is accepted, then it is possible to show that the error in

the strain energy stored in such a situation is O(h2) as well.

It is possible to generalize these findings in an approximate ortentative way for more complex problems discretized using elements of greaterprecision. Thus, if an element with q nodes is used, the trial functions are ofdegree q-1. If the exact solution requires a polynomial field of degree q at

least, then the computed displacement field will have an error O(hq). If strains

for the problem are obtained as the rth derivative of the displacement field,

then the error in the strain or stress is O(hq-r). Of course, these measures aretentative, for errors will also depend on how the loads are lumped at the nodesand so on.

4.3 Errors and convergence from "best-fit" paradigm

We have argued earlier that the finite element method actually proceeds tocompute strains/stresses in a `best-fit' manner within each element. We shallnow use this argument to show how convergence and errors in a typical finiteelement model of a simple problem can be interpreted. We shall see that a muchbetter insight into error and convergence analysis emerges if we base ourtreatment on the `best-fit' paradigm.

We must now choose a suitable example to demonstrate in a very simplefashion how the convergence of the solution can be related to the best-fitparadigm. The bar under axial load, an example we have used quite frequently sofar, is not suitable for this purpose. A case such as the uniform bar withlinearly varying axial load modeled with two-node elements gives nodaldeflections, which are exact, even though the true displacement field is cubicbut between nodes, in each element, only a linear approximate trial function ispossible. It can actually be proved that this arises from the specialmathematical nature of this problem. We should therefore look for an examplewhere nodal displacements are not exact.

The example that is ideally suited for this purpose is a uniform beamwith a tip load as shown in Fig. 4.2. We shall model it with linear (two-node)Timoshenko beam elements which represent the bending moment within each elementby a constant. Since the bending moment varies linearly over the beam for thisproblem, the finite element will replace this with a stairstep approximation.Thus, with increase in number of elements, the stress pattern will approach thetrue solution more closely and therefore the computed strain energy due tobending will also converge. Since the applied load is at the tip, it is veryeasy to associate this load with the deflection under the load usingCastigliano's theorem. It will then be possible to discern the convergence forthe tip deflection. Our challenge is therefore to see if the best-fit paradigmcan be used to predict the convergence rate for this example from firstprinciples.

38

Fig. 4.2 Cantilever beam under tip load

4.3.1 Cantilever beam idealized with linear Timoshenko beam elements

The dimensions of a cantilever under tip load (Fig. 4.2) are chosen such thatthe tip deflection under the load will be w=4.0. The example chosen representsa thin beam so that the influence of shear deformation and shear strain energyis negligible.

We shall now discretize the beam using equal length linear elements basedon Timoshenko theory. We use this element instead of the classical beam elementfor several reasons. This element serves to demonstrate the features of shearlocking which arise from an inconsistent definition of the shear strains (It istherefore useful to take up this element for study later in Chapter 6). Aftercorrecting for the inconsistent shear strain, this element permits constantbending and shears strain accuracy within each element - the simplestrepresentation possible under the circumstances and therefore an advantage inseeing how it works in this problem.

We shall restrict attention to the bending moment variation as we assumethat the potential energy stored is mainly due to bending strain and that wecan neglect the transverse shear strain energy for the dimensions chosen.

Figure 4.3 shows the bending moment diagrams for a 1, 2 and 4 elementidealizations of the present problem using the linear element. The true bendingmoment (shown by the solid line) varies linearly. The computed (i.e.discretized) bending moments are distributed in a piecewise constant manner asshown by the broken lines. In each case, the elements pick up the bendingmoment at the centroid correctly - i.e. it is doing so in a `best-fit' manner.What we shall now attempt to do for this problem is to relate this to theaccuracy of results. We shall now interpret accuracy in the conventionalsense, as that of the deflection at the tip under the load. Table 4.1 shows the

Table 4.1 Normalized tip deflections of a thin cantilever beam, L/t = 100

No. of elements 1 2 3Predicted rate .750 .938 .984Element without RBF .750 .938 .984Element with RBF 1.000 1.000 1.000

39

normalized tip deflection with increasing idealizations (neglecting a verysmall amount due to shear deformation). An interesting pattern emerges. Iferror is measured by the norm ( ){ } wfemww − , it turns out that this is given

exactly by the formula 24N1 where N is the number of elements. It can now be

seen that this relationship can be established by arguing that this featureemerges from the fact that strains are sought in the `best-fit' manner shown inFig. 4.3.

Consider a beam element of length 2l (Fig. 4.4). Let the moment and shear forceat the centroid be M and V. Thus the true bending moment over the elementregion for our problem can be taken to vary as M+Vx. The discretized bendingmoment sensed by our linear element would therefore be M. We shall now computethe actual bending energy in the element region (i.e. from a continuumanalysis) and that given by the finite element (discretized) model. We can showthat

Fig. 4.3 Bending moment diagrams for a one-, two- and four-element dealizationsof a cantilever beam under tip load.

Energy in continuum model = )3lV(M)EIl( 222 + (4.5)

Energy in discretized model = )(MEI)l( 2 (4.6)

Thus, as a result of the discretization process involved in replacing eachcontinuum segment of length 2l by a linear Timoshenko beam element which cangive only a constant value M for the bending moment, there is a reduction(error) in energy in each element equal to (l/EI) (V2l2/3). It is simple now toshow from this that for the cantilever beam of length L with a tip load P, the

total reduction in strain energy of the discretized model for the beam is U/4N2

where U=P2L3/6EI is the energy of the beam under tip load.

We are interested now to discover how this error in strain energytranslates into an error in the deflection under load P. This can be veryeasily deduced using Castigliano's second theorem. It is left to the reader toshow that the tip deflections of the continuum and discretized model will

differ as ( ){ } 24N1wfemww =− .

Table 4.1 shows this predicted rate of convergence. Our foregoinganalysis shows that this follows from the fact that if any linear variation isapproximated in a piecewise manner by constant values as seen in Fig. 4.3, thisis the manner in which the square of the error in the stresses/strains (or,equivalently, the difference in work or energy) will converge withidealization. Of course, in a problem where the bending moment is constant, therate of convergence will be better than this (in fact, exact) and in the casewhere the bending moment is varying quadratically or at a higher rate, the rateof convergence will be decidedly less.

Fig.

40

4.4 Bending moment variation in a linear beam element.

41

We also notice that convergence in this instance is from below. This canbe deduced from the fact that the discretized potential energy U is less thanthe actual potential energy U for this problem. It is frequently believed thatthe finite element displacement approach always underestimates the potentialenergy and a displacement solution is consequently described as a lower boundsolution. However, this is not a universally valid generalization. We can seebriefly below (the reader is in fact encouraged to work out the case in detail)where the cantilever beam has a uniformly distributed load acting on it usingthe same linear Timoshenko beam element for discretization that this is not the

case. It turns out that tip rotations converge from above (in a 22N1 rate)

while the tip deflections are fortuitously exact. The lower bound solutionnature has been disturbed because of the necessity of altering the load systemat the nodes of the finite element mesh under the `consistent load' lumpingprocedure.

As promised above, we now extend the concept of "best-fit" andvariationally correct rate of convergence to the case of uniformly distributedload of intensity q on the cantilever with a little more effort. Now, when afinite element model is made, two levels of discretization error areintroduced. Firstly, the uniformly distributed load is replaced by consistentloads, which are concentrated at element nodes. Thus, the first level ofdiscretization error is due to the replacement of the quadratically varyingbending moment in the actual beam with a linear bending moment within each beamelement. Over the entire beam model, this variation is piecewise linear. Thenext level of error is due to the approximation implied in developing thestiffness matrix which we had considered above this effectively senses a "best-approximated" constant value of the bending moment within each element of thelinear bending moment appearing to act after load discretization.

With these assumptions, it is a simple exercise using Castigliano'stheorem and fictitious tip force and moment P and M respectively to demonstratethat the finite element model of such a problem using two-noded beam elements

will yield a fortuitously correct tip deflection ( )8EIqLw 4= for all

idealizations (i.e. even with one element!) and a tip rotation that converges

at the rate 22N1 from above to the exact value ( )6EIqL3=θ . Thus, as far as

tip deflections are concerned, the two levels of discretization errors havenicely cancelled each other to give correct answers. This can deceive an unwaryanalyst into believing that an exact solution has been reached. Inspection ofthe tip rotation confirms that the solution is approximate and converging.

We see from the foregoing analysis that using linear Timoshenko beamelements for the tip loaded cantilever, the energy for this problem convergesas O(h2) where h=2l=L/N is the element size. We also see that this order ofconvergence carries over to the estimate of the tip deflections for thisproblem. Many text-books are confused over such relationships, especially thosethat proceed on the order of error analysis. These approaches arrive atconclusions such as strain error is proportional to element size, i.e. O(h) and

displacement error proportional to the square of the element size, i.e. O(h2)for this problem. We can see that for this problem (see Fig. 4.3) this estimateis meaningful if we consider the maximum error in strain to occur at elementnodes (at centroids the errors are zero as these are optimal strain points). Wealso see that with element discretization, these errors in strain vanish as

42

O(h). We can also see that the strain energies are now converging at the rate

of O(h2) and this emerges directly from the consideration that the discretizedstrains are `best-fits' of the actual strain. This conclusion is not so readilyarrived at in the order of error analysis methods, which often argue that the

strains are accurate to O(h), then strain energies are accurate to O(h2) becausestrain energy expressions contain squares of the strain. This conclusion isvalid only for cases where the discretized strains are `best-fit'approximations of the actual strains, as observed in the present example. Ifthe `best-fit' paradigm did not apply, the only valid conclusion that could bedrawn is that the strains that have O(h) error will produce errors in strainenergy that are O(2h).

4.4 The variationally correct rate of convergence

It is possible now to postulate that if finite elements are developed in aclean and strictly "legal" manner, without violating the basic energy orvariational principles, there is a certain rate at which solutions willconverge reflecting the fidelity that the approximate solution maintains withthe exact solution. We can call this the variationally correct rate ofconvergence. However, this prescription of strictly legal formulation is notalways followed. It is not uncommon to encounter extra-variational devices thatare brought in to enhance performance.

4.5 Residual bending flexibility correction

The residual bending flexibility (RBF) correction [4.1,4.2] is a device used toenhance the performance of 2-node beam and 4-node rectangular plate elements(these are linear elements) so that they achieve results equivalent to that

obtained by elements of quadratic order. In these C0 elements (Timoshenko theoryfor beam and Mindlin theory for plate) there is provision for transverse shearstrain energy to be computed in addition to the bending energy (which is the

only energy present in the C1 Euler-Bernoulli beam and Kirchhoff plateformulations). The RBF correction is a deceptively simple device that enhancesperformance of the linear elements by modifying the shear energy term tocompensate for the deficient bending energy term so that superior performanceis obtained. To understand that this is strictly an extra-variational trick (or"crime"), it is necessary to understand that a variationally correct procedurewill yield linear elements that have a predictable and well-defined rate ofconvergence. This has been carried out earlier in this chapter. It is possibleto improve performance beyond this limit only by resorting to some kind ofextra-variational manipulation. By a variationally correct procedure, we meanthat the stiffness matrix is derived strictly using a BTDB triple product, whereB and D are the strain-displacement and stress-strain matrices respectively.

4.5.1 The mechanics of the RBF correction.

MacNeal [4.2] describes the mathematics of the RBF correction using an originalcubic lateral displacement field (corresponding to a linearly varying bendingmoment field) and what is called a aliased linearly interpolated field (givinga constant bending moment variation for the discretized field). We might recallthat we examined the aliasing paradigm in Section 2.5 earlier. We find that aquicker and more physically insightful picture can be obtained by usingEquations (4.5) and (4.6) above. Consider a case where only one element of

43

length 2l is used to model a cantilever beam of length L with a tip load P.Applying Castigliano's theorem, we can show very easily that the continuum anddiscretized solutions will differ by,

( )kAG13EILPLw 2 += (4.7)

( )kAG14EILPLw 2 += (4.8)

Note the difference in bending flexibilities. This describes the inherentapproximation involved in the discretization process if all variational normsare strictly followed. The RBF correction proposes to manipulate the shearflexibility in the discretized case so that it compensates for the deficiencyin the bending flexibility for this problem. Thus if k* is the compensated shearcorrection factor, from Eq (4.7) and (4.8), we have

AG*k14EILkAG13EIL 22 +=+

3EIlkAG1AG*k1 2+= (4.9)

It is not accurate to say here that the correction term is derived from thebending flexibility. The bending flexibility of an element that can representonly a constant bending moment (e.g. the two-noded beam element used here) is

4EIL2 . The missing 12EIL2 (or 3EIl2 ) is now brought in as a compensation

through the shear flexibility, i.e. k is changed to k*. This "fudge factor"therefore enhances the performance of the element by giving it a rate ofconvergence that is not variationally permissible. Two wrongs make a righthere; or do they? Table 4.1 shows how the convergence in the problem shown inFig. 4.2 is changed by this procedure.

4.6 Concluding remarks.

We have shown that the best-fit paradigm is a useful starting point forderiving estimates about errors and convergence. Using a simple example andthis simple concept, we could establish that there is a variationally correctrate of convergence for each element. This can be improved only by takingliberties with the formulation, i.e. by introducing extra-variational steps.The RBF is one such extra-variational device.

It is also possible to argue that errors, whether in displacements, stresses orenergies, due to finite element discretization must converge rapidly, at leastin a O(h2) manner or better. If a large structure (domain) of dimension L issub-divided into elements (sub-domains) of dimension l, one expects errors of

the order of ( )2Ll . Thus, with ten elements in a one-dimensional structure,

errors must not be more than a few percent. We shall discover however that aclass of problems exist where errors are much larger - the discretizations failin a dramatic fashion, and this cannot be resolved by the classical (pre-1977)understanding of the finite element method. A preliminary study of the issuesinvolved will be taken up in the next chapter; the linear Timoshenko beamelement serves to expose the factors clearly. Subsequent chapters willundertake a comprehensive review of the various manifestations of such errors.It is felt that this detailed treatment is justified, as an understanding of

44

such errors has been one of the most challenging problems that the finiteelement procedure has faced in its history. Most of the early techniques toovercome these difficulties were ad hoc, more in the form of `art' or `blackmagic'. In the subsequent chapters, our task will be to identify the principlesthat establish the methodology underlying the finite element procedure usingcritical, rational scientific criteria.

4.7 References

4.1. R. H. MacNeal, A simple quadrilateral plate element, Comput. Struct., 8,175-183, 1978.

4.2. R. H. MacNeal, Finite Elements: Their Design and Performance, MarcelDekker, NY, 1994.

45

Chapter 5

The locking phenomena

5.1 Introduction

A pivotal area in finite element methodology is the design of robust andaccurate elements for applications in general purpose packages (GPPs) instructural analysis. The displacement type approach we have been examining sofar is the technique that is overwhelmingly favored in general purpose software.We have seen earlier that the finite element method can be interpreted as apiecewise variation of the Rayleigh-Ritz method and therefore that it seeksstrains/stresses in a `best-fit' manner. From such an interpretation, it ispossible to argue that errors, whether in displacements, stresses or energies,

due to finite element discretization must converge rapidly, at least in a ( )2Ll

manner or better, where a large structure (domain) of dimension L is sub-dividedinto elements (sub-domains) of dimension l. Thus, with ten elements in a one-dimensional structure, errors must not be more than a few percent.

By and large, the elements work without difficulty. However, there werespectacular failures as well. These are what are now called the ‘locking’problems in C0 finite elements - as the element libraries of GPPs stabilizedthese elements came to be favored for reasons we shall discuss shortly. Bylocking, we mean that finite element solutions vanish quickly to zero (errorssaturating quickly to nearly 100%!) as certain parameters (the penaltymultipliers) become very large. It was not clear why the displacement typemethod, as it was understood around 1977, should produce for such problems,answers that were only a fraction of a per cent of the correct answer with apractical level of discretization. Studies in recent years have established thatan aspect known as consistency must be taken into account.

The consistency paradigm requires that the interpolation functions chosento initiate the discretization process must also ensure that any specialconstraints that are anticipated must be allowed for in a consistent way.Failure to do so causes solutions to lock to erroneous answers. The paradigmshowed how elements can be designed to be free of these errors. It also enablederror-analysis procedures that allowed errors to be traced to theinconsistencies in the representation to be developed. We can now develop afamily of such error-free robust elements for applications in structuralmechanics.

In this chapter, we shall first introduce the basic concepts needed tounderstand why such discretized descriptions fail while others succeed. We

compare the equations of the Timoshenko beam theory (a C0 theory) to the

classical beam theory (a C1 theory) to show how constraints are generated insuch a model. This permits us to discuss the concept of consistency and thenature of errors that appear during a Ritz type approximation. These same errorsare responsible for the locking seen in the displacement type finite elementmodels of similar problems.

46

5.2 From C1 to C0 elements

As the second generation of GPPs started evolving around the late 70s and early80s, their element libraries replaced what were called the C1 elements with what

were known as the C0 elements. The former were based on well known classicaltheories of beams, plates and shells, reflecting the confidence structuralanalysts had in such theories for over two centuries - namely the Euler-Bernoulli beam theory, the Kirchhoff-Love plate theory and the equivalent shelltheories. These theories did not allow for transverse shear strain and permittedthe modeling of such structures by defining deformation in terms of a singlefield, w, the transverse deflection of a point on what is called the neutralaxis (in a beam) and neutral surface of a plate or shell. The strains could thenbe computed quite simply from the assumption that normal to the neutral surfaceremained normal after deformation. One single governing differential equationresulted, although of a higher order (in comparison to other theories we shalldiscuss shortly), and this was considered to be an advantage.

There were some consequences arising from such an assumption both for themathematical modeling aspect as well as for the finite element (discretization)aspect. In the former, it turned out that to capture the physics of deformationof thick or moderately thick structures, or the behavior of plates and shellsmade of newly emerging materials such as high performance laminated composites,it was necessary to turn to more general theories accounting or transverse sheardeformation as well - these required the definition of rotations of normalswhich were different from the slopes of the neutral surface. Some of thecontradictions that arose as a result of the old C1 theories - e.g. the use ofthe fiction of the Kirchhoff effective shear reactions, could now be removed,restoring the more physically meaningful set of three boundary conditions on theedge of a plate or shell (the Poisson boundary conditions as they are called) tobe used. The orders of the governing equations were correspondingly reduced. Asalutary effect that carried over to finite element modeling was that theelements could be designed to have nodal degrees of freedom which were the sixbasic engineering degrees of freedom - the three translations and the threerotations at a point. This was ideal from the point of view of the organizationof a general purpose package. Also, elements needed only simple basis functionsrequiring only the continuity of the fields across element boundaries - theseare called the C0 requirements. In the older C1 formulations, continuity ofslope was also required and to achieve this in arbitrarily oriented edges, aswould be found in triangular or quadrilateral planforms of a plate bendingelement, it was necessary to retain curvature degrees of freedom (w,xx, w,xy,

w,yy) at the nodes and rely on quintic polynomials for the element shape orbasis functions. So, as general purpose packages ideal for production runanalyses and design increasingly found favour in industry, the C0 beam, plate

and shell elements slowly began to replace the older C1 equivalents. It may beinstructive to note that the general two-dimensional (i.e. plane stress, planestrain and axisymmetric) elements and three-dimensional (solid or brick as they

are called) elements were in any case based on C0 shape functions - thus this

development was welcome in that universally valid C0 shape functions and theirderivatives could be used for a very wide range of structural applications.

47

However, life was not very simple - surprisingly dramatic failures came tobe noticed and the greater part of academic activity in the late seventies, mostof the eighties and even in the nineties was spent in understanding andeliminating what were called the locking problems.

5.3 Locking, rank and singularity of penalty-linked stiffness matrix,and consistency of strain-field

When locking was first encountered, efforts were made to associate it with therank or non-singularity of the stiffness matrix linked to the penalty term (e.g.the shear stiffness matrix in a Timoshenko beam element which becomes very largeas the beam becomes very thin, see the discussion below). However, onreflection, it is obvious that these are symptoms of the problem and not thecause. The high rank and non-singularity is the outcome of certain assumptionsmade (or not made, i.e. leaving certain unanticipated requirements unsatisfied)during the discretization process. It is therefore necessary to trace this tothe origin. The consistency approach argues that it is necessary in suchproblems to discretize the penalty-linked strain fields in a consistent way sothat only physically meaningful constraints appear.

In this section, we would not enter into a formal finite elementdiscretization (which would be taken up in the next section) but instead,illustrate the concepts involved using a simple Ritz-type variational method ofapproximation of the beam problem via both classical and Timoshenko beam theory[5.1]. It is possible to show how the Timoshenko beam theory can be reduced tothe classical thin beam theory by using a penalty function interpretation and indoing so, show how the Ritz approximate solution is very sensitive to the way inwhich its terms are chosen. An `inconsistent' choice of parameters in a loworder approximation leads to a full-rank (non-singular) penalty stiffness matrixthat causes the approximate solution to lock. By making it `consistent', lockingcan be eliminated. In higher order approximations, `inconsistency' does not leadto locked solutions but instead, produces poorer convergence than wouldotherwise be expected of the higher order of approximation involved. It is againdemonstrated that a Ritz approximation that ensures ab initio consistentdefinition will produce the expected rate of convergence - a simple example willillustrate this.

5.3.1 The classical beam theory

Consider the transverse deflection of a thin cantilever beam of length L underan uniformly distributed transverse load of intensity q per unit length of thebeam. This should produce a linear shear force distribution increasing from 0 atthe free end to qL at the fixed end and correspondingly, a bending moment that

varies from 0 to 2qL2 . Using what is called the classical or Euler-Bernoulli

theory, we can state this problem in a weak form with a quadratic functionalgiven by,

( )� −=L 2

xx0

dxqww,EI21Π (5.1)

This theory presupposes a constraint condition, assuming zero transverseshear strain, and this allows the deformation of a beam to be described entirely

48

in terms of a single dependent variable, the transverse deflection w of pointson the neutral axis of the beam. An application of the principle of minimumtotal potential allows the governing differential equations and boundaryconditions to be recovered, but this will not be entered into here. A simpleexercise will establish that the exact solution satisfying the governingdifferential equations and boundary conditions is,

( ) ( ) ( ) ( ) 4322 x24EIqx6EIqLx4EIqLxw −−= (5.2a)

( ) ( )2xx xL2qw,EI −= (5.2b)

( )xLqw,EI xxx −−= (5.2c)

5.3.1.1 A two-term Ritz approximation

Let us now consider an approximate Ritz solution based on two terms,3

32

2 xbxbw += . Note that the constant and linear terms are dropped,

anticipating the boundary conditions at the fixed end. One can easily work outthat the approximate solution will emerge as,

( ) ( ) ( ) 322 x12EIqLx24EI5qLxw −= (5.3a)

so that the approximate bending moment and shear force determined in this Ritzprocess are,

( ) ( )x2qL125qL,wEI 2xx −= (5.3b)

2qL,wEI xxx −= (5.3c)

If the expressions in Equations (5.2) and (5.3) are written in terms of thenatural co-ordinate system ξ, where ( ) 2L1x ξ+= so that the ends of the beam

are represented by ξ = -1 and +1, the exact and approximate solutions can beexpanded as,

( ) ( ))( 22xx 31312348qLw,EI ξξ −−−= (5.4)

( ) ( )ξ2348qL,wEI 2xx −= (5.5)

The approximate solution for the bending moment is seen to be a `best-fit' or

`least-squares fit' of the exact solution, with the points 31±=ξ , which are

the points where the second order Legendre polynomial vanishes, emerging aspoints where the approximate solution coincides with the exact solution. FromEquations (5.2c) and (5.3c), we see that the shear force predicted by the Ritzapproximation is a 'best-fit' of the actual shear force variation. Once again weconfirm that the Ritz method seeks the 'best-approximation' of the actual stateof stress in the region being studied.

49

5.3.2 The Timoshenko beam theory and its penalty function form

The Timoshenko beam theory [5.1] offers a physically more general formulation ofbeam flexure by taking into account the transverse shear deformation. Thedescription of beam behavior is improved by introducing two quantities at eachpoint on the neutral axis, the transverse deflection w and the face rotation θso that the shear strain at each point is given by xw,−= θγ , the difference

between the face rotation and the slope of the neutral axis.

The total strain energy functional is now constructed from the twoindependent functions for w(x) and θ(x), and it will now account for the bending(flexural) energy and an energy of shear deformation.

( )( )� −−+=L 2

x2x

0dxwqw,21EI21 θαθΠ , (5.6)

where the curvature κ=θ,x, the shear strain γ=θ-w,x and α=kGA is the shearrigidity. The factor k accounts for the averaged correction made for the shearstrain distribution through the thickness. For example, for a beam ofrectangular cross-section, this is usually taken as 5/6.

The Timoshenko beam theory will asymptotically recover the elementary beamtheory as the beam becomes very thin, or as the shear rigidity becomes very

large, i.e. α→∞. This requires that the Kirchhoff constraint θ-w,x→0 must

emerge in the limit. For a very large α, these equations lead directly to thesimple fourth order differential equation for w of elementary beam theory. Thus,this is secured very easily in the infinitesimal theory but it is this very samepoint that poses difficulties when a simple Ritz type approximation is made.

5.3.2.1 A two-term Ritz approximation

Consider now a two term Ritz approximation based on xa1=θ and xbw 1= . This

permits a constant bending strain (moment) approximation to be made. The shearstrain is now given by

11 bxa −=γ (5.7)

and it would seem that this can represent a linearly varying shear force. TheRitz variational procedure now leads to the following set of equations,

��

���

��

���

=���

���

��

���

��

���

��

��

�

−

−+�

��

2qL

0

b

a

L2L

2L3L

00

0LEI

21

1

2

23

α (5.8)

Solving, we get

2

3

12

2

1L12EI

1.5qL

2

qLb

L12EI

3qLa

ααα +−−=

+

−= ; (5.9)

50

As α→∞, both a1 and b1→0. This is a clear case of a solution `locking'. Thiscould have been anticipated from a careful examination of Equations (5.6) and(5.7). The penalty limit α→∞ in Equation (5.6) introduces a penalty conditionon the shear strain and this requires that the shear strain must vanish in theRitz approximation process - from Equation (5.7), the conditions emerging area1→0 and b1→0. Clearly, a1→0 imposes a zero bending strain θ,x→0 as well -this spurious constraint produces the locking action on the solution. Thus, ameaningful approximation can be made for a penalty function formulation only ifthe penalty linked approximation field is consistently modeled. We shall see howthis is done next.

5.3.2.2 A three-term consistent Ritz approximation

Consider now a three term approximation chosen to satisfy what we shall call the

consistency condition. Choose xa1=θ and 221 xbxbw += . Again, only a constant

bending strain is permitted. But we now have shear strain of linear order as,

( )x2bab 211 −+−=γ (5.10)

Note that as α→∞, the constraint α1-2b2→0 is consistently balanced and willnot result in a spurious constraint on the bending field. The Ritz variationalprocedure leads to the following equation structure:

���

���

�

���

���

�

=��

��

�

��

��

�

��

�

��

�

�

��

�

��

�

�

����

����

�

−

−

−−

+���

���

�

3qL

2qL

0

b

b

a

34LL32L

LL2L

32L2L3L

000

000

00L

EI

3

2

2

1

1

323

22

323

α (5.11)

It can be easily worked out from this that the approximate solution is given by,

x6EI

qL2−=θ (5.12a)

22

2 x12EI

qLx

2

qx

qLw +−=

αα(5.12b)

6qLEI 2x =− ,θ (5.12c)

( ) ( )xLq,w x −=−θα (5.12d)

There is no locking seen at all - the bending moment is now a least squarescorrect constant approximation of the exact quadratic variation (this can beseen by comparing Equation (5.12c) with Equations (5.4) and (5.5) earlier). Theshear force is now correctly captured as a linear variation - the consistentlyrepresented field in Equation (5.12) being able to recover this even as a α→∞!

51

A comparison of the penalty linked matrices in Equations (5.8) and (5.11)shows that while in the former, we have a non-singular matrix of rank 2, in thelatter we have a singular matrix of rank 2 for a matrix of order 3. It is clearalso that the singularity (or reduced rank) is a result of the consistentcondition represented by (α1-2b2) in the linear part of the shear straindefinition in Equation (5.10) - as a direct consequence, the third row of thepenalty linked matrix in Equation (5.11) is exactly twice the first row. It isthis aspect that led to a lot of speculation on the role the singularity or rankof the matrix plays in such problems. We can further show that non-singularityof penalty linked matrix arises in an inconsistent formulation only when theorder of approximation is low, as seen for the two-term Ritz approximation. Wecan go for a quadratic inconsistent approximation (with linear bending strainvariation) in the next section to show that there is no locking' of the solutionand that the penalty linked matrix is not non-singular - however the effect ofinconsistency is to reduce the performance of the approximation to a sub-optimallevel.

5.3.2.3 A four-term inconsistent Ritz approximation

We now take up a four term approximation which provides theoretically for a

linear variation in the approximation for bending strain, i.e. 221 xaxa +=θ and

221 xbxbw += so that x2aa 21x +== ,θκ and the shear strain is

( ) 22211 xax2bab +−+−=γ (5.13)

Note now that the condition α→∞ forces a2→0 this becomes a spurious constrainton the bending strain field. We shall now see what effect this spuriousconstraint has on the approximation process. The Ritz variational procedureleads to the following set of equations:

���

�

���

�

�

+

�������

�

�

�������

0000

0000

0034LL

00LL

EI32

2

(5.14)

��

�

��

�

�

��

�

��

�

�

=

��

�

��

�

�

��

�

��

�

�

���

�

���

�

�

������

������

�

−−

−−

−−

−−

3qL

2qL

0

0

b

b

a

a

34LL2L3L2

LL3L2L

2L3L5L4L

32L2L4L3L

3

2

2

1

2

1

3243

232

4354

3243

α

It can be seen that the penalty linked 4×4 matrix is singular as the fourth rowis exactly twice the second row - this arises from the consistent representation( )21 2ba − of the linear part of shear strain in Equation (5.13). The rank of the

52

matrix is therefore 3 and the solution should be free of "locking" - however theinconsistent constraint forces 0a2 → and this means that the computed bending

strain 1a→κ ; i.e. it will have only a constant bending moment prediction

capability. What it means is that this four term inconsistent approach willproduce answers only as efficiently as the three term consistent Ritzformulation. Indeed, one can work out from Equation (5.14) that,

2

22

1L60EI

15qL

6EI

qLa

α+−−= (5.15a)

2

2

2L60EI

15qLa

α+= (5.15b)

( )2

3

1L60EI2

5qLqLb

αα +−−= (5.15c)

α2q

6EI

qLb

2

2 +−= (5.15d)

As α→∞, the bending moment and shear force variation are given by,

6qLEI 2x =− ,θ (5.16a)

( ) ( ) ( )( ( ) )1Lx6Lx62.5qLxLq,w 2x +−+−=−θα (5.16b)

The solution can sense only a constant bending moment in the thin beam limit.There are now violent quadratic oscillations in the shear force and these

oscillations can be shown to vanish at the points ( )31121Lx += and

( )31121 − , or 31±=ξ , which are the points corresponding to the 2 point

Gauss-Legendre integration rule. The effect of the inconsistent representationhas been to reduce the effectiveness of the approximation. We shall next see howthe effectiveness can be improved by making the approximation consistent beforethe variational process is carried out.

5.3.2.4 A four-term consistent Ritz approximation

Let us now take up a Ritz approximation with a consistently represented function

for the shear strain defined as γ . This can be achieved by noting that in

Equation (5.13), the offending term is the quadratic term associated with a2. Wealso see from Equation (5.16b) that this leads to a spuriously excited quadratic

form ( )22 L6Lx6x +− . Our experience with consistent finite element formulations

[5.2] allows us to replace the x2 term in Equation (5.13) with 6LLx 2− so that,

53

( ) ( )6LLxax2bab 22211 −+−+−=γ

( ) ( )x2b-Laa6Lab 2212

21 ++−−= (5.17)

In fact, it can be proved using the generalized (mixed) variational theorem suchas the Hellinger-Reissner and the Hu-Washizu principles [5.2] that the

variationally correct way to determine γ from the inconsistent γ is,

� =���

��� − 0dx

Tγγγδ

and this will yield precisely the form represented in Equation (5.17). The Ritzvariational procedure now leads to,

���

�

���

�

�

+

�������

�

�

�������

0000

0000

0034LL

00LL

EI32

2

(5.18)

��

�

��

�

�

��

�

��

�

�

=

��

�

��

�

�

��

�

��

�

�

���

�

���

�

�

������

������

�

−−

−−

−−

−−

3qL

2qL

0

0

b

b

a

a

34LL2L3L2

LL3L2L

2L3L367L4L

32L2L4L3L

3

2

2

1

2

1

3243

232

4354

3243

α

It is important to recognize now that since the penalty linked matrix emerges

from the terms - ( )6Lab 221 + and ( )221 2b-Laa + there would only be two linearly

independent rows and therefore the rank of the matrix is now 2. The approximatesolution is then given by,

x2

qLqL

12

5EI 2 −=− κ (5.19a)

( )x-Lq=γα (5.19b)

Comparing Equation (5.19a) with Equation (5.3b) we see that the approximatesolution to the Timoshenko equation for this problem with a consistent shearstrain assumption gives exactly the same bending moment as the Ritz solution tothe classical beam equation could provide. We also see that Equation (5.19b) isidentical to Equation (5.12b) so that the shear force is now being exactly

computed. In other words, as α→∞, the terms a1, a2, b1 and b1 all yieldphysically meaningful conditions representing the state of equilibriumcorrectly.

54

5.4 Consistency and C0 displacement type finite elements

We shall see in the chapters to follow that locking, poor convergence andviolent stress oscillations seen in C0 displacement type finite elementformulations are due to a lack of consistent definition of the critical strainfields when the discretization is made - i.e. of the strain-fields that areconstrained in the penalty regime.

The foregoing analysis showed how the lack of consistency translates intoa non-singular matrix of full rank that causes locking in low-order Ritzapproximations of such problems. It is also seen that in higher orderapproximations, the situation is not as dramatic as to be described as locking,but is damaging as to produce poor convergence and stress oscillations. It iseasy to predict all this by examining the constrained strain-field terms fromthe consistency point of view rather than performing a post-mortem examinationof the penalty-linked stiffness matrix from rank or singularity considerationsas is mostly advocated in the literature.

We shall now attempt a very preliminary heuristic definition of theserequirements as seen from the point of view of the developer of a finite elementfor an application in a constrained media problem. We shall see later that if asimple finite element model of the Timoshenko beam is made), the results are invery great error and that these errors grow without limit as the beam becomesvery thin. This is so even when the shape functions for the w and θ have beenchosen to satisfy the completeness and continuity conditions. We saw in our Ritzapproximation of the Timoshenko beam theory in this section that as the beambecomes physically thin, the shear strains must vanish and it must begin toenforce the Kirchhoff constraint and that this is not possible unless theapproximate field can correctly anticipate this. In the finite element statementof this problem, the shape functions chosen for the displacement fields cannotdo this in a meaningful manner - spurious constraints are generated which causelocking. The consistency condition demands that the discretized strain fieldinterpolations must be so constituted that it will enforce only physically trueconstraints when the discretized functionals for the strain energy of a finiteelement are constrained.

We can elaborate on this definition in the following descriptive way. Inthe development of a finite element, the field variables are interpolated usinginterpolations of a certain order. The number of constants used will depend onthe number of nodal variables and any additional nodeless variables (thosecorresponding to bubble functions). From these definitions, one can compute thestrain fields using the strain-displacement relations. These are obtained asinterpolations associated with the constants that were introduced in the fieldvariable interpolations. Depending on the order of the derivatives of each fieldvariable appearing in the definition of that strain field (e.g. the shear strainin a Timoshenko theory will have θ and the first derivative of w), thecoefficients of the strain field interpolations may have constants from all thecontributing field variable interpolations or from only one or some of these. Insome limiting cases of physical behavior, these strain fields can be constrainedto be zero values, e.g. the vanishing shear strain in a thin Timoshenko beam.Where the strain-field is such that all the terms in it (i.e. constant, linear,quadratic, etc.) have, associated with it, coefficients from all the independentinterpolations of the field variables that appear in the definition of that

55

strain-field, the constraint that appears in the limit can be correctlyenforced. We shall call such a representation field-consistent. The constraintsthus enforced are true constraints. Where the strain-field has coefficients inwhich the contributions from some of the field variables are absent, theconstraints may incorrectly constrain some of these terms. This field-inconsistent formulation is said to enforce additional spurious constraints. Forsimple low order elements, these constraints are severe enough to producesolutions that rapidly vanish - causing what is often described as locking.

5.5 Concluding remarks

These exercises show us why it is important to maintain consistency of the basisfunctions chosen for terms in a functional, which are linked, to penaltymultipliers. The same conditions are true for the various finite elementformulations where locking, poor convergence and stress oscillations are knownto appear. It is also clear why the imposition of the consistency condition intothe formulation allows the correct rank or singularity of the penalty linkedstiffness matrix to be maintained so that the system is free of locking or sub-optimal convergence. Again, it is worthwhile to observe that non-singularity ofthe penalty linked matrix occurs only when the approximate fields are of verylow order as for the two-term Ritz approximation. In higher order inconsistentformulations, as for the four-term inconsistent Ritz approximation, solutionsare obtained which are sub-optimal to solutions that are possible if theformulation is carried out with the consistency condition imposed a priori. Weshall see later that the use of devices such as reduced integration permits theconsistency requirement to be introduced when the penalty linked matrix iscomputed so that the correct rank which ensures the imposition of the trueconstraints only is maintained.

In the next chapter, we shall go to the locking and other over-stiffeningphenomena found commonly in displacement finite elements. The phenomenon ofshear locking is the most well known - we shall investigate this closely. TheTimoshenko beam element allows the problem to be exposed and permits amathematically rigorous error model to be devised. The consistency paradigm isintroduced to show why it is essential to remove shear locking. The correctnessconcept is then brought in to ensure that the devices used to achieve aconsistent strain interpolation are also variationally correct. This exampleserves as the simplest case that could be employed to demonstrate all theprinciples involved in the consistency paradigm. It sets the scene for the needfor new paradigms (consistency and correctness) to complement the existing ones(completeness and continuity) so that the displacement model can bescientifically founded. The application of these concepts to Mindlin plateelements is also reviewed.

The membrane locking phenomenon is investigated in Chapter 7. The simplecurved beams are used to introduce this topic. This is followed up with studiesof parasitic shear and incompressible locking.

In Chapters 5 to 7 we investigate the effect of constraints on strains dueto the physical regimes considered, e.g. vanishing shear strains in thinTimoshenko beams or Mindlin plates, vanishing membrane strains in curved beamsand shells etc. In Chapter 8 we proceed to a few problems where consistencybetween strain and stress functions are needed even where no constraints on the

56

strains appear. These examples show the universal need of the consistency aspectand also the power of the general Hu-Washizu theorem to establish the complete,correct and consistent variational basis for the displacement type finiteelement method.

5.6 References

5.1 S. P. Timoshenko, On the transverse vibrations of bars of uniform cross-section, Philosophical Magazine 443, 125-131, 1922.

5.2 G. Prathap, The Finite Element Method in Structural Mechanics, KluwerAcademic Press, Dordrecht, 1993.

57

Chapter 6

Shear locking

In the preceding chapter, we saw a Ritz approximation of the Timoshenko beamproblem and noted that it was necessary to ensure a certain consistentrelationship between the trial functions to obtain accurate results. We shallnow take up the finite element representation of this problem, which isessentially a piecewise Ritz approximation. Our conclusions from the precedingchapter would therefore apply to this as well.

6.1 The linear Timoshenko beam element

An element based on elementary theory needs two nodes with 2 degrees of freedomat each node, the transverse deflection w and slope dw/dx and uses cubicinterpolation functions to meet the C1 continuity requirements of this theory(Fig. 6.1). A similar two-noded beam element based on the shear flexibleTimoshenko beam theory will need only C0 continuity and can be based on simplelinear interpolations. It was therefore very attractive for general purposeapplications. However, the element was beset with problems, as we shallpresently see.

6.1.1 The conventional formulation of the linear beam element

The strain energy of a Timoshenko beam element of length 2l can be written asthe sum of its bending and shear components as,

( )dxkGA21EI21 TT� + γγχχ (6.1)

where

x,θχ = (6.2a)

xw,−= θγ (6.2b)

In Equations (6.2a) and (6.2b), w is the transverse displacement and θ thesection rotation. E and G are the Young's and shear moduli and the shearcorrection factor used in Timoshenko's theory. I and A are the moment of inertiaand the area of cross-section, respectively.

(a) 1 2w ww,x w,x

(b) 1 2w wθ θ

Fig. 6.1 (a) Classical thin beam and (b) Timoshenko beam elements.

58

In the conventional procedure, linear interpolations are chosen for thedisplacement field variables as,

( ) 21N1 ξ−= (6.3a)

( ) 21N2 ξ+= (6.3b)

where the dimensionless coordinate ξ=x/l varies from -1 to +1 for an element oflength 2l. This ensures that the element is capable of strain free rigid bodymotion and can recover a constant state of strain (completeness requirement) andthat the displacements are continuous within the element and across the elementboundaries (continuity requirement). We can compute the bending and shearstrains directly from these interpolations using the strain gradient operatorsgiven in Equations (6.2a) and (6.2b). These are then introduced into the strainenergy computation in Equation (6.1), and the element stiffness matrix iscalculated in an analytically or numerically exact (a 2 point Gauss Legendreintegration rule) way.

For the beam element shown in Fig. 6.1, for a length h the stiffnessmatrix can be split into two parts, a bending related part and a shear relatedpart, as,

�������

�

�

�������

�

� −

=

������

�

�

������

�

�

−

−=

3h2h-6h2h

2h-12h-1-

6h2h-3h2h

2h12h1

hkGtk

1010

0000

1010

0000

hEIk

22

22

sb

We shall now model a cantilever beam under a tip load using this element,considering the case of a "thin" beam with E=1000, G=37500000, t=1, L=4, using afictitiously large value of G to simulate the "thin" beam condition. Table 6.1shows that the normalized tip displacements are dramatically in error. In factwith a classical beam element model, exact answers would have been obtained withone element for this case. We can carefully examine Table 6.1 to see the trendas the number of elements is increased. The tip deflections obtained, which areseveral orders of magnitude lower than the correct answer, are directly relatedto the square of the number of elements used for the idealization. In otherwords, the discretization process has introduced an error so large that the

resulting answer has a stiffness related to the inverse of N2. This is clearlyunrelated to the physics of the Timoshenko beam and also not the usual sort ofdiscretization errors encountered in the finite element method. It is this veryphenomenon that is known as shear locking.

Table 6.1 - Normalized tip deflections

No. of elements “Thin” beam1

2

4

816

0.200 × 10-5

0.800 × 10-5

0.320 × 10-4

0.128 × 10-3

0.512 × 10-3

59

The error in each element must be related to the element length, andtherefore when a beam of overall length L is divided into N elements of equallength h, the additional stiffening introduced in each element due to shearlocking is seen to be proportional to h2. In fact, numerical experiments showedthat the locking stiffness progresses without limit as the element depth tdecreases. Thus, we now have to look for a mechanism that can explain how this

spurious stiffness of (h/t)2 can be accounted for by considering the mathematicsof the discretization process.

The magic formula proposed to overcome this locking is the reducedintegration method. The bending component of the strain energy of a Timoshenkobeam element of length 2l shown in Equation (6.1) is integrated with a one-pointGaussian rule as this is the minimum order of integration required for exactevaluation of this strain energy. However, a mathematically exact evaluation ofthe shear strain energy will demand a two-point Gaussian integration rule. It isthis rule that resulted in the shear stiffness matrix of rank two that locked.An experiment with a one-point integration of the shear strain energy componentcauses the shear related stiffness matrix to change as shown below. Theperformance of this element was extremely good, showing no signs of locking atall (see Table 4.1 for a typical convergence trend with this element).

�������

�

�

�������

�

�

−

−−−

−

−

=

������

�

�

������

�

�

−

−=

4h2h4h2h

2h12h1

4h2h4h2h

2h12h1

hkGtk

1010

0000

1010

0000

hEIk

22

22

sb

6.1.2 The field-consistency paradigm

It is clear from the formulation of the linear Timoshenko beam element usingexact integration (we shall call it the field-inconsistent element) thatensuring the completeness and continuity conditions are not enough in someproblems. We shall propose a requirement for a consistent interpolation of theconstrained strain fields as the necessary paradigm to make our understanding ofthe phenomena complete.

If we start with linear trial functions for w and θ, as we had done inEquation 6.3 above, we can associate two generalized displacement constants witheach of the interpolations in the following manner

( )lxaaw 10 += (6.4a)

( )lxbb 10 +=θ (6.4b)

We can relate such constants to the field-variables obtaining in thiselement in a discretized sense; thus, a1/l=w,x at x=0, b0=θ and b1/l=θ,x at x=0.This denotation would become useful when we try to explain how thediscretization process can alter the infinitesimal description of the problem ifthe strain fields are not consistently defined.

60

If the strain-fields are now derived from the displacement fields given inEquation (6.4), we get

( )lb1=χ (6.5a)

( ) ( )lxblab 110 +−=γ (6.5b)

An exact evaluation of the strain energies for an element of length h=2l willnow yield the bending and shear strain energy as

( ) ( ) ( ){ } 21B lb2lEI21U = (6.6a)

( ) ( ) ( ){ }21

210s b31lab2lkGA21U +−= (6.6b)

It is possible to see from this that in the constraining physical limit of avery thin beam modeled by elements of length 2l and depth t, the shear strainenergy in Equation (6.6b) must vanish. An examination of the conditions producedby these requirements shows that the following constraints would emerge in sucha limit

0lab 10 →− (6.7a)

0b1 → (6.7b)

In the new terminology that we had cursorily introduced in Section 5.4,constraint (6.7a) is field-consistent as it contains constants from both thecontributing displacement interpolations relevant to the description of theshear strain field. These constraints can then accommodate the true Kirchhoffconstraints in a physically meaningful way, i.e. in an infinitesimal sense, thisis equal to the condition (θ-w,x)→0 at the element centroid. In direct

contrast, constraint (6.7b) contains only a term from the section rotation θ. Aconstraint imposed on this will lead to an undesired restriction on θ. In aninfinitesimal sense, this is equal to the condition θ,x→0 at the elementcentroid (i.e. no bending is allowed to develop in the element region). This isthe `spurious constraint' that leads to shear locking and violent disturbancesin the shear force prediction over the element, as we shall see presently.

6.1.3 An error model for the field-consistency paradigm

We must now determine that this field-consistency paradigm leads us to anaccurate error prediction. We know that the discretized finite element modelwill contain an error which can be recognized when digital computations madewith these elements are compared with analytical solutions where available. Theconsistency requirement has been offered as the missing paradigm for the error-free formulation of the constrained media problems. We must now devise anoperational procedure that will trace the errors due to an inconsistentrepresentation of the constrained strain field and obtain precise a priorimeasures for these. We must then show by actual numerical experiments with theoriginal elements that the errors are as projected by these a priori errormodels. Only such an exercise will complete the scientific validation of theconsistency paradigm. Fortunately, a procedure we shall call the functional re-constitution technique makes it possible to do this verification.

61

6.1.4 Functional re-constitution

We have postulated that the error of shear locking originates from the spuriousshear constraint in Equation (6.7b). We must now devise an error model for thecase where the inconsistent element is used to model a beam of length L anddepth t. The strain energy for such a beam can be set up as,

( ){ }� −+=L 2

x2x

0dxw,kGA21EI21 θθΠ , (6.8)

If an element of length 2l is isolated, the discretization process producesenergy for the element of the form given in Equation (6.6). In this equation,the constants, which were introduced due to the discretization process, can bereplaced by the continuum (i.e. the infinitesimal) description. Thus, we notethat in each element, the constants in Equations (6.6a) and (6.6b) can be tracedto the constants in Equations (6.4a) and (6.4b) and can be replaced by thevalues of the field variations θ, θ,x and w,x at the centroid of the element.Thus, the strain energy of deformation in an element is,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2x22

x2

xe ,kGAl61w,2lkGA21,2lEI21 θθθπ +−+= (6.9)

Thus the constants in the discretized strain energy functional have been re-constituted into an equivalent continuum or infinitesimal form. From this re-constituted functional, we can argue that an idealization of a beam region oflength 2l into a linear displacement type finite element would produce amodified strain energy density within that region of,

( ) ( ) ( ) ( )2x2

x2

e w,kGA21,3kGAlEI21 −++= θθπ (6.10)

This strain energy density indicates that the original physical system has beenaltered due to the presence of the inconsistent term in the shear strain field.Thus, we can postulate that a beam of length L modeled by equal elements oflength 2l will have a re-constituted functional

( ) ( ) ( ) ( ){ } dxw,kGA21,3kGAlEI21L 2

x2

x2

0� −++= θθΠ (6.11)

We now understand that the discretized beam is stiffer in bending (i.e. its

flexural rigidity) by the factor 3EIkGAl2 . For a thin beam, this can be very

large, and produces the additional stiffening effect described as shear locking.

6.1.5 Numerical experiments to verify error prediction

Our functional re-constitution procedure (note that this is an auxiliaryprocedure, distinct from the direct finite element procedure that yields thestiffness matrix) allows us to critically examine the consistency paradigm. Itindicates that an exactly-integrated or field-inconsistent finite element modeltends to behave as a shear flexible beam with a much stiffened flexural rigidityI’. This can be related to the original rigidity I of the system by comparingthe expressions in Equations (6.8) and (6.11) as,

3EIkGAL1II 2+=′ (6.12)

62

We must now show through a numerical experiment that this estimate for theerror, which has been established entirely a priori, starting from theconsistency paradigm and introducing the functional re-constitution technique,anticipates very accurately, the behavior of a field-inconsistent linearlyinterpolated shear flexible element in an actual digital computation. Exactsolutions are available for the static deflection W of a Timoshenko cantileverbeam of length L and depth t under a vertical tip load. If femW is the result

from a numerical experiment involving a finite element digital computation usingelements of length 2l, the additional stiffening can be described by a parameteras,

1WWe femfem −= (6.13)

From Equation (6.12), we already have an a priori prediction for this factor as,

3EIkGAl1IIe 2=−′= (6.14)

We can now re-interpret the results shown in Table 6.1 for the thin beamcase. Using Equations (6.13) and (6.14), we can argue a priori that theinconsistent element will produce normalized tip deflections ( ) ( )e11WWfem += .

Since e>>1, we have

( ) 52fem 105NWW −×= (6.15)

for the thin beam. Table 6.2 shows how the predictions made thus compare withthe results obtained from an actual finite element computation using the field-inconsistent element.

This has shown us that the consistency paradigm can be scientifically verified.Traditional procedures such as counting constraint indices, or computing therank or condition number of the stiffness matrices could offer only a heuristicpicture of how and why locking sets in.

It will be instructive to note here that conventional error analysis norms inthe finite element method are based on the percentage error or equivalent insome computed value as compared to the theoretically predicted value. We haveseen now that the error of shear locking can be exaggerated without limit, asthe structural parameter that acts as a penalty multiplier becomes indefinitely

Table 6.2 - Normalized tip deflections for the thin beam (Case 2) computed fromfem model and predicted from error model (Equation (6.15)).

N Computed (fem) Predicted

1

24

8

16

0.200 × 10-4

0.800 × 10-4

0.320 × 10-3

0.128 × 10-3

0.512 × 10-3

0.200 × 10-4

0.800 × 10-4

0.320 × 10-3

0.128 × 10-3

0.512 × 10-3

large. The percentage error norms therefore saturate quickly to a valueapproaching 100% and do not sensibly reflect the relationship between error andthe structural parameter even on a logarithmic plot. A new error norm called theadditional stiffening parameter, e can be introduced to recognize the manner inwhich the errors of locking kind can be blown out of proportion by a largevariation in the structural parameter. Essentially, this takes into account, thefact that the spurious constraints give rise to a spurious energy term andconsequently alters the rigidity of the system being modeled. In many otherexamples (e.g. Mindlin plates, curved beams etc.) it was seen that the rigidity,I, of the field consistent system and the rigidity, I’, of the inconsistent

system, were related to the structural parameters in the form, I’/I = α(l/t)2where l is an element dimension and t is the element thickness. Thus, if w isthe deflection of a reference point as predicted by an analytical solution tothe theoretical description of the problem and wfem is the fem deflectionpredicted by a field inconsistent finite element model, we would expect therelationship described by Equation 6.14. A logarithmic plot of the new errornorm against the parameter (l/t) will show a quadratic relationship that willcontinue indefinitely as (l/t) is increased. This was found to be true of themany constrained media problems. By way of illustration of the distinction made

Fig. 6.2 Variatcantil

63

ion of error norms e, E with structural parameter kGL2/Et2 for aever beam under tip shear force.

64

by this definition, we shall anticipate again, the results above. If werepresent the conventional error norm in the form ( ) WWWE fem−= , and plot both

E and the new error norm e from the results for the same problem using 4 FIelements against the penalty multiplier (l/t)2 on a logarithmic scale, thedependence is as shown in Fig. 6.2. It can be seen that E saturates quickly to avalue approaching 100% and cannot show meaningfully how the error propagates asthe penalty multiplier increases indefinitely. On the other hand, e capturesthis relationship, very accurately.

6.1.6 Shear Force Oscillations

A feature of inconsistently modeled constrained media problems is the presenceof spurious violently oscillating strains and stresses. It was not understoodfor a very long time that in many cases, stress oscillations originated from theinconsistent constraints. For a cantilever beam under constant bending momentmodeled using linear Timoshenko beam elements, the shear force (stresses)displays a saw-tooth pattern (we shall see later that a plane stress model using4-node elements will also give an identical pattern on the neutral bendingsurface). We can arrive at a prediction for these oscillations by applying thefunctional re-constitution technique.

If V is the shear force predicted by a field-consistent shear strainfield (we shall see soon how the field-consistent element can be designed) and Vthe shear force obtained from the original shear strain field, we can write fromEquation (6.5b),

( )labkGAV 10 −= (6.16a)

( )lxbkGAVV 1+= (6.16b)

We see that V has a linear term that relates directly to the constant thatappeared in the spurious constraint, Equation (6.7b). We shall see below fromEquation (6.17) that b1 will not be zero, in fact it is a measure of the bendingmoment at the centroid of the element. Thus, in a field-inconsistentformulation, this constant will activate a violent linear shear force variationwhen the shear forces are evaluated directly from the shear strain field givenin Equation (6.5b). The oscillation is self-equilibrating and does notcontribute to the force equilibrium over the element. However, it contributes afinite energy in Equation (6.9) and in the modeling of very slender beams, thisspurious energy is so large as to completely dominate the behavior of the beamand cause a locking effect.

Figure 6.3 shows the shear force oscillations in a typical problem - astraight cantilever beam with a concentrated moment at the tip. One to ten equallength field-inconsistent elements were used and shear forces were computed atthe nodes of each element. In each case, only the variation within the elementat the fixed end is shown, as the pattern repeats itself in a saw-tooth mannerover all other elements. At element mid-nodes, the correct shear force i.e. V=0is reproduced. Over the length of the element, the oscillations are seen to belinear functions corresponding to the kGA b1 (x/l) term. Also indicated by thesolid lines, is the prediction made by the functional re-constitution exercise.We shall explore this now.

Fig. 6.3 Shear forcemodels of a

Consider a straend. This should proshear force Q in thewill now respond in tthe average of the lelement. If the elem

after accounting forat the element centro

In a field-incoconsider the modified

to 1b ′ , that is,

where 3EIkGAle 2= .

Thus, in a fie

by the factor e; the

65

oscillations in element nearest the root, for N elementcantilever of length L = 60.

ight cantilever beam with a tip shear force Q at the freeduce a linearly varying bending moment M and a constantbeam. An element of length 2l at any station on the beamhe following manner. Since, a linear element is used, onlyinearly varying bending moment is expected in each finiteent is field-consistent, the constant b1 can be associated

discretization, to relate to the constant bending moment M0id as,

lbEIM 10 = or

EIlMb 01 = (6.17)

nsistent problem, due to shear locking, it is necessary toflexural rigidity I’ (see Equation 6.17) that modifies b1

IElMb 01 ′=′( ){ }e1EIlM0 +=

( )e1b1 += (6.18)

ld-inconsistent formulation, the constant b1 gets stiffened

constant bending moment M0 is also underestimated by the

66

same factor. Also, for a very thin beam where e>>1, the centroidal moment M0predicted by a field-consistent element diminishes in a t2 rate for a beam ofrectangular cross-section. These observations have been confirmed throughdigital computation.

The field-consistent element will respond with QVV 0 == over the entire

element length 2l. The field-inconsistent shear force V from Equations (6.16)and (6.18) can be written for a very thin beam (e>>1) as,

( ) ( )lxl3MQV 0+= (6.19)

These are the violent shear force linear oscillations within each element, whichoriginate directly from the field-inconsistency in the shear strain definition.

These oscillations are also seen if field-consistency had been achieved inthe element by using reduced integration for the shear strain energy. Unless theshear force is sampled at the element centroid (i.e. Gaussian point, x/l=0),these disturbances will be much more violent than in the exactly integratedversion.

6.1.7 The consistent formulation of the linear element

We can see that reduced integration ensures that the inconsistent constraintdoes not appear and so is effective in producing a consistent element, at leastin this instance. We must now satisfy ourselves that such a modification did notviolate any variational theorem.

The field-consistent element, as we now shall call an element version freeof spurious (i.e. inconsistent) constraints, can and has been formulated invarious other ways as well. The `trick' is to evaluate the shear strain energy,in this instance, in such a way that only the consistent term will contribute tothe shear strain energy. Techniques like addition of bubble modes, hybridmethods etc. can produce the same results, but in all cases, the need forconsistency of the constrained strain field must be absolutely met.

We explain now why the use of a trick like the reduced integrationtechnique, or the use of assumed strain methods allows the locking problem to beovercome. It is obvious that it is not possible to reconcile this within theambit of the minimum total potential principle only, which had been the startingpoint of the conventional formulation.

We saw in Chapter 2, an excellent example of a situation where it wasnecessary to proceed to a more general theorem (one of the so-called mixedtheorems) to explain why the finite element method computed strain and stressfields in a `best-fit' sense. We can now see that in the case of constrainedmedia problems, the mixed theorem such as the Hu-Washizu or Hellinger-Reissnertheorem can play a crucial role in proving that by modifying the minimum totalpotential based finite element formulation by using an assumed strain field toreplace the kinematically derived constrained field, no energy, or workprinciple or variational norms have been violated.

67

To eliminate problems such as locking, we look for a consistentconstrained strain field to replace the inconsistent kinematically derivedstrain field in the minimum total potential principle. By closely examining thestrain gradient operators, it is possible to identify the order up to which theconsistent strain field must be interpolated. In this case, for the lineardisplacement interpolations, Equations (6.5b), (6.7a) and (6.7b) tell us thatthe consistent interpolation should be a constant. At this point we shall stillnot presume what this constant should be, although past experience suggests itis the same constant term seen in Equation (6.7a). Instead, we bring in theHellinger-Reissner theorem in the following form to see the identity of theconsistent strain field clearly. For now, it is sufficient to note that theHellinger-Reissner theorem is a restricted case of the Hu-Washizu theorem. Inthis theorem, the functional is stated in the following form,

( )dxkGAkGA21EIEI21 TTTT� +−+− γγγγχχχχ (6.20)

where χ and γ are the new strain variables introduced into this multi-field

principle. Since we have difficulty only with the kinematically derived γ we canhave χχ = and recommend the use of a γ which is of consistent order to replace

γ. A variation of the functional in Equation (6.20) with respect to the as yetundetermined coefficients in the interpolation for γ yields

( )� =− 0dxT γγγδ (6.21)

This orthogonality condition now offers a means to constitute the coefficientsof the consistent strain field from the already known coefficients of thekinematically derived strain field. Thus, for γ given by Equation (6.5b), it ispossible to show that ( )lb 10 αγ −= . In this simple instance, the same result is

obtained by sampling the shear strain at the centroid, or by the use of one-point Gaussian integration. What is important is that, deriving the consistentstrain-field using this orthogonality relation and then using this to computethe corresponding strain energy will yield a field-consistent element which doesnot violate any of the variational norms, i.e. an exact equivalence to the mixedelement exists without having to go through the additional operations in a mixedor hybrid finite element formulation, at least in this simple instance. We saythat the variational correctness of the procedure is assured. The substitutestrain interpolations derived thus can therefore be easily coded in the form ofstrain function subroutines and used directly in the displacement type elementstiffness derivations.

6.1.8 Some concluding remarks on the linear beam element

So far we have seen the linear beam element as an example to demonstrate theprinciples involved in the finite element modeling of a constrained mediaproblem. We have been able to demonstrate that a conceptual framework thatincludes a condition that specifies that the strain fields which are to beconstrained must satisfy a consistency criterion is able to provide a completescientific basis for the locking problems encountered in conventionaldisplacement type modeling. We have also shown that a correctness criterion(which links the assumed strain variation of the displacement type formulation

68

to the mixed variational theorems) allows us to determine the consistent strainfield interpolation in a unique and mathematically satisfying manner.

It will be useful now to see how these concepts work if a quadratic beamelement is to be designed. This is a valuable exercise as later, the quadraticbeam element shall be used to examine problems such as encountered in curvedbeam and shell elements and in quadrilateral plate elements due to non-uniformmapping.

6.2 The quadratic Timoshenko beam element

We shall now very quickly see how the field-consistency rules explain thebehavior of a higher order element. We saw in Chapter 5 that the conventionalformulation with lowest order interpolation functions led to spuriousconstraints and a non-singular assembled stiffness matrix, which result inlocking. In a higher order formulation, the matrix was singular but the spuriousconstraints resulted in a system that had a higher rank than was felt to bedesirable. This resulted in sub-optimal performance of the approximation. We cannow use the quadratic beam element to demonstrate that this is true in finiteelement approximations as well.

6.2.1 The conventional formulation

Consider a quadratic beam element designed according to conventional principles,i.e. exact integration of all energy terms arising from a minimum totalpotential principle. As the beam becomes very thin, the element does not lock;in fact it produces reasonably meaningful results. Fig. 6.4 shows a typicalcomparison between the linear and quadratic beam elements in its application toa simple problem. A uniform cantilever beam of length 1.0 m, width 0.01 m anddepth 0.01 m has a vertical tip load of 100 N applied at the tip. For E=1010

N/m2 and µ=0.3, the engineering theory of beams predicts a tip deflection ofw=4.0 m. We shall consider three finite element idealizations of this problem -with the linear 2-node field-consistent element considered earlier in thissection (2C, on the Figure), the quadratic 3-node field-inconsistent elementbeing discussed now (3I, on the Figure) and the quadratic 3-node field-consistent element which we shall derive later (3C). It is seen that for thissimple problem, the 3C element produces exact results, as it is able to simulatethe constant shear and linear bending moment variation along the beam length.The 3I and 2C elements show identical convergence trends and behave as if theyare exactly alike. The curious aspects that call for further investigation are:the quadratic element (3I) behaves in exactly the same way as the field-consistent linear element (2C), giving exactly the same accuracy for the samenumber of elements although the system assembled from the former had nearlytwice as many nodes. It also produced moment predictions, which were identical,i.e., the quadratic beam element, instead of being able to produce linear-accurate bending moments could now yield only a constant bending moment withineach element, as in the field-consistent linear element. Further, there were nowquadratic oscillations in the shear force predictions for such an element. Notenow that these curious features cannot be explained from the old arguments,which linked locking to the non-singularity or the large rank or the spectralcondition number of the stiffness matrix. We shall now proceed to explain thesefeatures using the field-consistency paradigm.

Fig.conve

If quadraticand θ in the followi

the shear strain int

γ

Again, we emphasizeLegendre polynomialorthogonal nature obecomes the sum ofpolynomials. Indeed,

21Us =

Therefore, whbeam the shear straof the strain fieldseparately. In this

69

6.4 A uniform cantilever beam with tip shear force -rgence trends of linear and quadratic elements.

isoparametric functions are used for the field-variables wng manner

( ) ( )2210 lxalxaaw ++=

( ) ( )2210 lxblxbb ++=θ

erpolation will be,

( ) ( ) ( )2221120 313bl2abla3bb ξξ −−−+−+= (6.22)

the usefulness of expanding the strain field in terms of thes. When the strain energies are integrated, because of thef the Legendre polynomials the discretized energy expressionthe squares of the coefficients multiplying the Legendrethe strain energy due to transverse shear strain is,

( ) ( ) ( ) ( ){ }454bl2ab31la3bb2lkGA 22

221

2120 +−+−+ (6.23)

en we introduce the penalty limit condition that for a thinin energies must vanish, we can argue that the coefficientsexpanded in terms of the Legendre polynomials must vanishcase, three constraints emerge:

( ) 0la3bb 120 →−+ (6.24a)

70

( ) 0l2ab 21 →− (6.24b)

0b2 → (6.24c)

Equations (6.24a) and (6.24b) represent constraints having contributionsfrom the field interpolations for both w and θ. They can therefore reproduce, ina consistent manner, true constraints that reflect a physically meaningfulimposition of the thin beam Kirchhoff constraint. This is therefore the field-consistent part of the shear strain interpolation.

Equation (6.24c) however contains a constant only from the interpolationfor θ. This constraint, when enforced, is an unnecessary restriction on thefreedom of the interpolation for θ, constraining it in fact to behave only as a

linear interpolation as the constraint implies that θ,xx→0 in a discretizedsense over each beam element region. The spurious energy implied by such aconstraint does not contribute directly to the discretized bending energy,unlike the linear beam element seen earlier. Therefore, field-inconsistency inthis element would not cause the element to lock. However, it will diminish therate of convergence of the element and would induce disturbances in the form ofviolent quadratic oscillations in the shear force predictions, as we shall seein the next section.

6.2.2 Functional reconstitution

We can use the functional re-constitution technique to see how theinconsistent terms in the shear strain interpolation alter the description ofthe physics of the original problem (we shall skip most of the details, as thematerial is available in greater detail in Ref. 6.1).

The b2 term that appears in the bending energy also makes an appearance inthe shear strain energy, reflecting its origin through the spurious constraint.We can argue that this accounts for the poor behavior of the field-inconsistentquadratic beam element (the 3I of Fig. 6.4). Ref. 6.1 derives the effect moreprecisely, demonstrating that the following features can be fully accounted for:

i) the displacement predictions of the 3I element are identical to that made bythe 2C element on an element by element basis although it has an additional mid-node and has been provided with the more accurate quadratic interpolationfunctions.ii) the 3I element can predict only a constant moment within each element,exactly as the 2C element does.iii) there are quadratic oscillations in the shear force field within eachelement.

We have already discussed earlier that the 3I element (the field-inconsistent 3-noded quadratic) converges in exactly the same manner as the 2Celement (the field-consistent linear). This has been explained by showing using

the functional re-constitution technique, that the b2 term, which describes thelinear variation in the bending strain and bending moment interpolation, is"locked" to a vanishingly small value. The 3I element then effectively behavesas a 2C element in being able to simulate only a constant bending-moment in eachregion of a beam, which it replaces.

71

6.2.3 The consistent formulation of the quadratic element

As in the linear element earlier, the field-consistent element (3C) can beformulated in various ways. Reduced integration of the shear strain energy usinga 2-point Gauss-Legendre formula was the most popular method of deriving theelement so far. Let us now derive this element using the `assumed' strainapproach. We use the inverted commas to denote that the strain is not assumed inan arbitrary fashion but is actually uniquely determined by the consistency andthe variational correctness requirements. The re-constitution of the field is tobe done in a variationally correct way, i.e. we are required to replace γ inEquation (6.22) which had been derived from the kinematically admissibledisplacement field interpolations using the strain-displacement operators withan `assumed' strain field γ which contains terms only upto and including the

linear Legendre polynomial in keeping with the consistency requirement. Let uswrite this in the form

ξγ 10 cc += (6.25)

The orthogonality condition in Equation (6.21) dictates how γ should replace γover the length of the element. This determines how c0 and c1 should be

constituted from b0, b1 and b2. Fortunately, the orthogonal nature of theLegendre polynomials allows this to be done for this example in a very trivialfashion. The quadratic Legendre polynomial and its coefficient are simply

truncated and c0=b0 and c1=b1 represent the variationally correct field-consistent `assumed' strain field. The use of such an interpolation subsequentlyin the integration of the shear strain energy is identical to the use of reducedintegration or the use of a hybrid assumed strain approach. In a hybrid assumedstrain approach, such a consistent re-constitution is automatically implied inthe choice of assumed strain functions and the operations leading to thederivation of the flexibility matrix and its inversion leading to the finalstiffness matrix.

6.3 The Mindlin plate elements

A very large part of structural analysis deals with the estimation of stressesand displacements in thin flexible structures under lateral loads using what iscalled plate theory. Thus, plate elements are the most commonly used elements ingeneral purpose structural analysis. At first, most General Purpose Packages(GPPs) for structural analysis used plate elements based on what are called theC1 theories. Such theories had difficulties and limitations and a1so attention

turned to what are called the C0 theories.

The Mindlin plate theory [6.2] is now the most commonly used basis for thedevelopment of plate elements, especially as they can cover applications tomoderately thick and laminated plate and shell constructions. It has beenestimated that in large scale production runs using finite element packages, thesimple four-node quadrilateral plate element (the QUAD4 element) may account foras much as 80% of all usage. It is therefore important to understand that theevolution of the current generation of QUAD4 elements from those of yester-year,over a span of nearly three decades was made difficult by the presence of shearlocking. We shall now see how this takes place.

72

The history behind the discovery of shear locking in plate elements isquite interesting. It was first recognized when an attempt was made to representthe behavior of shells using what is called the degenerate shell approach [6.3].In this the shell behavior is modeled directly after a slight modification ofthe 3D equations and shell geometry and domain are represented by a 3D brickelement but its degrees of freedom are condensed to three displacements and twosection rotations at each node. Unlike classical plate or shell theory, thetransverse shear strain and its energy is therefore accounted for in thisformulation. Such an approach was therefore equivalent to a Mindlin theoryformulation. These elements behaved very poorly in representing even the trivialexample of a plate in bending and the errors progressed without limit, as theplates became thinner. The difficulty was attributed to shear locking. This isin fact the two-dimensional manifestation of the same problem that weencountered for the Timoshenko beam element; ironically it was noticed first inthe degenerate shell element and was only later related to the problems indesigning Timoshenko beam and Mindlin plate elements [6.4]. The remedy proposedat once was the reduced integration of the shear strain energy [6.5,6.6]. Thiswas only partially successful and many issues remained unresolved. Some of thesewere,

i) the 2×2 rule failed to remove shear locking in the 8-node serendipity plateelement,ii) the 2×2 rule in the 9-node Lagrangian element removed locking but introducedzero energy modes,iii) the selective 2×3 and 3×2 rule for the transverse shear strain energies

from γxz and γyz recommended for a 8-node element also failed to remove shearlocking,iv) the same selective 2×3 and 3×2 rule when applied to a 9-noded element isoptimal for a rectangular form of the element but not when the element wasdistorted into a general quadrilateral form,v) even after reduced integration of the shear energy terms, the degenerateshell elements performed poorly when trying to represent the bending of curvedshells, due to an additional factor, identified as membrane locking [6.7],originating now from the need for consistency of the membrane straininterpolations. We shall consider the membrane-locking phenomenon in anothersection.

We shall confine our study now to plate elements without going into thecomplexities of the curved shell elements.

In Kirchhoff-Love thin plate theory, the deformation is completelydescribed by the transverse displacement w of the mid-surface. In such adescription, the transverse shear deformation is ignored. To account fortransverse shear effects, it is necessary to introduce additional degrees offreedom. We shall now consider Mindlin's approximations, which have permittedsuch an improved description of plate behavior. The degenerate shell elementsthat we discussed briefly at the beginning of this section can be considered tocorrespond to a Mindlin type representation of the transverse shear effects.

In Mindlin's theory [6.2], deformation is described by three quantities,the section rotations θx and θy (i.e. rotations of lines normal to themidsurface of the undeformed plate) and the mid-surface deflection w. Thebending strains are now derived from the section rotations and do not cause any

73

difficulty when a finite element model is made. The shear strains are nowcomputed as the difference between the section rotations and the slopes of theneutral surfaces, thus,

xxxz w,−= θγ

yyyz w,−= θγ (6.26)

The stiffness matrix of a Mindlin plate element will now have terms from thebending strain energy and the shear strain energy. It is the inconsistentrepresentation of the latter that causes shear locking.

6.3.1 The 4-node plate element

The 4-node bi-linear element is the simplest element based on Mindlin theorythat could be devised. We shall first investigate the rectangular form of theelement [6.4] as it is in this configuration that the consistency requirementscan be easily understood and enforced. In fact, an optimum integration rule canbe found which ensures consistency if the element is rectangular. It wasestablished in Ref. 6.4 that an exactly integrated Mindlin plate element wouldlock even in its rectangular form. Locking was seen to vanish for therectangular element if the bending energy was computed with a 2×2 Gaussianintegration rule while a reduced 1-point rule was used for the shear strainenergy. This rectangular element behaved very well if the plate was thin but theresults deteriorated as the plate became thicker. Also, after distortion to aquadrilateral form, locking re-appeared. A spectral analysis of the elementstiffness matrix revealed a rank deficiency - there were two zero energymechanisms in addition to the usual three rigid body modes required for such anelement. It was the formation of these mechanisms that led to the deteriorationof element performance if the plate was too thick or if it was very looselyconstrained. It was not clear why the quadrilateral form locked even afterreduced integration. We can now demonstrate from our consistency view-point whythe 1-point integration of the shear strain energy is inadequate to retain allthe true Kirchhoff constraints in a rectangular thin plate element. However, weshall postpone the discussion on why such a strategy cannot preserve consistencyif the element was distorted to a later section.

Following Ref. [6.4], the strain energy for an isotropic, linear elasticplate element according to Mindlin theory can be constituted from its bendingand shear energies as,

SB UUU +=

( ) [{� � ++−

= yyxxy2yx

2x2

2

2,,124

Et ,, θθνθθν

( ) ( ) ] dydx21 2xy,yx, θθν +−+ (6.27)

( ) ( ) ( )[ ] }� � −+−−+ dydxw,w,t

16k 2yy

2xx2

θθν

74

Fig. 6.5 Cartesian and natural coordinate system for a four-node rectangularplate element.

where x, y are Cartesian co-ordinates (see Fig. 6.5), w is the transversedisplacement, θx and θy are the section rotations, E is the Young's modulus, νis the Poisson's ratio, k is the shear correction factor and t is the platethickness. The factor k is introduced to compensate for the error inapproximating the shear strain as a constant over the thickness direction of aMindlin plate.

Let us now examine the field-consistency requirements for one of the shearstrains, γxz, in the Cartesian system. The admissible displacement fieldinterpolations required for a 4-node element can be written in terms of theCartesian co-ordinates itself as,

xyayaxaaw 3210 +++= (6.28a)

xybybxbb 3210 +++=θ (6.28b)

The shear strain field derived from these kinematically admissible shapefunctions is,

( ) ( ) xybxbyabab 313210xz ++−+−=γ (6.29)

As the plate thickness is reduced to zero, the shear strains must vanish. Thediscretized constraints that are seen, to be enforced as 0xz →γ in Equation

(6.29) are,0ab 10 →− (6.30a)

0ab 32 →− (6.30b)

0b1 → (6.30c)

0b3 → (6.30d)

75

The constraints shown in Equations (6.30a) and (6.30b) are physically meaningfuland represent the Kirchhoff condition in a discretized form. Constraints (6.30c)and (6.30d) are the cause for concern here - these are the spurious or`inconsistent' constraints which lead to shear locking. Thus, in a rectangularelement, the requirement for consistency of the interpolations for the shearstrains in the Cartesian co-ordinate system is easily recognized as thepolynomials use only Cartesian co-ordinates. Let us now try to derive theoptimal element and also understand why the simple 1-point strategy of Ref. 6.4led to zero energy mechanisms.

It is clear from Equations (6.29) and (6.30) that the terms b1x and b3xyare the inconsistent terms which will contribute to locking in the form ofspurious constraints. Let us now look for optimal integration strategies forremoving shear locking without introducing any zero energy mechanisms. We shallconsider first, the part of the shear strain energy contributed by γxz. We must

integrate exactly, terms such as (b0-a1), (b2-a3)y, b1x, and b3xy. We now

identify terms such as (b0-a1), (b2-a3), b1, and b3 as being equivalent to the

quantities (θx-w,x)0, (θx-w,x),y0, (θx,x)0, and (θx,xy)0 where the subscript ‘0’denotes the values at the centroid of the element (for simplicity, we let thecentroid of the element lie at the origin of the Cartesian co-ordinate system).

An exact integration, that is a 2×2 Gaussian integration of the shearstrain energy leads to

( ) ( ) ( ) ( )[ ]� � ++−+−= 20xyx

2220xx

2y0

2xx

220xx

2xz ,9lh,3lw,3hw,4lhdydx θθθθγ , (6.31)

In the penalty limit of a thin plate, these four quantities act as constraints.The first two reproduce the true Kirchhoff constraints and the remaining two actas spurious constraints that cause shear locking by enforcing θx,x→0 and θx,xy→0in the element.

If a 1×2 Gaussian integration is used, we have,

( ) ( )[ ]� � −+−= y02

xx22

0xx2xz w,3hw,4lhdydx ,θθγ (6.32)

Thus, only the true constraints are retained and all spurious constraints areremoved. This strategy can also be seen to be variationally correct in thiscase; we shall see later that in a quadrilateral case, it is not possible toensure variational correctness exactly. By a very similar argument, we can showthat the part of the shear strain energy from γyz will require a 2×1 Gaussianintegration rule. This element would be the optimal rectangular bi-linearMindlin plate element.

Let us now look at the 1-point integration strategy used in Ref. 6.4. Thiswill give shear energy terms such as,

( )[ ]� � −= 20xx

2xz w,4lhdydx θγ (6.33)

76

We have now only one true constraint each for the shear energy from γxzand γyz respectively while the other Kirchhoff constraints ( ) 0w, y0xx →− ,θ and

( ) 0w, x0yy →− ,θ are lost. This introduces two zero energy modes and accounts for

the consequent deterioration in performance of the element when the plates arethick or are very loosely constrained, as shown in Ref. 6.4.

We have seen now that it is a very simple procedure to re-constitutefield-consistent assumed strain fields from the kinematically derived fieldssuch as shown in Equation (6.29) so that they are also variationally correct.This is not so simple in a general quadrilateral where the complication arisingfrom the isoparametric mapping from a natural co-ordinate system to a Cartesiansystem makes it very difficult to see the consistent form clearly. We shall seethe difficulties associated with this form in a later section.

6.3.2 The quadratic 8-node and 9-node plate elements

The 4-node plate element described above is based on bi-linear functions. Itwould seem that an higher order element based on quadratic functions would befar more accurate. There are now two possibilities, an 8-node element based onwhat are called the serendipity functions and a 9-node element based on theLagrangian bi-quadratic functions. There has been a protracted debate on whichversion is more useful, both versions having fiercely committed protagonists.By now, it is well known that the 9-node element in its rectangular form is freeof shear locking even with exact integration of shear energy terms and that itsperformance is vastly improved when its shear strain energies are integrated ina selective sense (2×3 and 3×2 rules for xzγ and yzγ terms respectively). It is

in fact analogous to the quadratic Timoshenko beam element, the field-inconsistencies not being severe enough to cause locking. This is however nottrue for the 8-node element which was derived from the Ahmad shell element [6.3]and which actually pre-dates the 4-node Mindlin element. An exact integration ofbending and shear strain energies resulted in an element that locked for mostpractical boundary suppressions even in its rectangular form. Many ad-hoctechniques e.g. the reduced and selective integration techniques, hybrid andmixed methods, etc. failed or succeeded only partially. It was thereforeregarded for some time as an unreliable element as no quadrature rule seemed tobe able to eliminate locking entirely without introducing other deficiencies. Itseems possible to attribute this noticeable difference in the performance of the8- and 9-node elements to the missing central node in the former. This makes itmore difficult to restore consistency in a simple manner.

6.3.3 Stress recovery from Mindlin plate elements

The most important information a structural analyst looks for in a typicalfinite element static analysis is the state of stress in the structure. It istherefore very important for one to know points of optimal stresses in theMindlin plate elements. It is known that the stress recovery at nodes fromdisplacement elements is unreliable, as the nodes are usually the points wherethe strains and stresses are least accurate. It is possible however to determinepoints of optimal stress recovery using an interpretation of the displacementmethod as a procedure that obtains strains over the finite element domain in aleast-squares accurate sense. In Chapter 2, we saw a basis for thisinterpretation. We can apply this rule to determine points of accurate stress

77

recovery in the Mindlin plate elements. For a field-consistent rectangular 4-node element, the points are very easy to determine [6.8] (note that in a field-inconsistent 4-node element, there will be violent linear oscillations in theshear stress resultants corresponding to the inconsistent terms). Thus, Ref. 6.8shows that bending moments and shear stress resultants Qxz and Qyz are accurate

at the centroid and at the 1×2 and 2×1 Gauss points in a rectangular element forisotropic or orthotropic material. It is coincidental, and therefore fortuitous,that the shear stress resultants are most accurate at the same points at whichthey must be sampled in a selective integration strategy to remove the field-inconsistencies! For anisotropic cases, it is safest to sample all stressresultants (bending and shear) at the centroid.

Such rules can be extended directly to the 9-node rectangular element. Thebending moments are now accurate at the 2×2 Gauss points and the shear stressresultants in an isotropic or orthotropic problem are optimal at the same 2×3and 3×2 Gauss points which were used to remove the inconsistencies from thestrain definitions. However, accurate recovery of stresses from the 8-nodeelement is still a very challenging task because of the difficulty informulating a robust element. The most efficient elements known today arevariationally incorrect even after being made field-consistent and need specialfiltering techniques before the shear stress resultants can be reliably sampled.

So far, discussion on stress sampling has been confined to rectangularelements. When the elements are distorted, it is no simple matter to determinethe optimal points for stress recovery - the stress analyst must then exercisecare in applying these rules to seek reliable points for recovering stresses.

6.4 Concluding remarks

We can conclude this section on shear locking by noting that the availableunderstanding was unable to resolve the phenomena convincingly. The proposedimprovement, which was the consistency paradigm, together with the functionalre-constitution procedure, allowed us to derive an error estimate for a caseunder locking and we could show through numerical (digital) experiments thatthese estimates were accurate. In this way we are convinced that a theory withthe consistency paradigm is more successful from the falsifiability point ofview than one without.

6.5 References

6.1 G. Prathap and C. R. Babu, Field-consistent strain interpolations for thequadratic shear flexible beam element, Int. J. Num. Meth. Engng. 23, 1973-1984, 1986.

6.2 R. D. Mindlin, Influence of rotary inertia and shear on flexural motion ofelastic plates, J. Appl. Mech. 18, 31-38, 1951.

6.3 S. Ahmad, B. M. Irons and O. C. Zienkiewicz, Analysis of thick and thinshell structures by curved finite elements, Int. J. Num. Meth. Engng. 2,419-451, 1970.

6.4 T. J. R. Hughes, R. L. Taylor and W. Kanoknukulchal, A simple and efficientfinite element for plate bending, Int. J. Num. Meth. Engng. 411, 1529-1543, 1977.

6.5 S. F. Pawsey and R. W. Clough, Improved numerical integration of thick shellfinite elements, Int. J. Num. Meth. Engng. 43, 575-586, 1971.

78

6.6 O. C. Zienkiewicz, R. L. Taylor and J. M. Too, Reduced integration techniquein general analysis of plates and shells, Int. J. Num. Meth. Engng. 43, 275-290, 1971.

6.7 H. Stolarski and T. Belytschko, Membrane locking and reduced ingression forcurved elements, J. Appl. Mech. 49, 172-178, 1982.

6.8 G. Prathap and C. R. Babu, Accurate force evaluation with a simple bi-linearplate bending element, Comp. Struct. 25, 259-270, 1987.

79

Chapter 7

Membrane locking, parasitic shear and incompressible locking

7.1 Introduction

The shear locking phenomenon was the first of the over-stiffening behavior thatwas classified as a locking problem. Other such pathologies were noticed in thebehavior of curved beams and shells, in plane stress modeling and in modeling ofthree dimensional elastic behavior at the incompressible limit (ν→0.5). Thefield-consistency paradigm now allows all these phenomena to be traced toinconsistent representations of the constrained strain fields. A unifyingpattern is therefore introduced to the understanding of the locking problems inconstrained media elasticity - whether it is shear locking in a straight beam orflat plate element, membrane locking in a curved beam or shell element,parasitic shear in 2D plane stress and 3D elasticity or incompressible lockingin 2D plane strain and 3D elasticity as the Poisson's ratio ν→0.5.

This chapter will now briefly summarize this interpretation. Unlike thechapter on shear locking, a detailed analysis is omitted as it would be beyondthe scope of the present book and readers should refer to the primary publishedliterature.

7.2 Membrane locking

Earlier, we argued that the most part of structural analysis deals with thebehavior of thin flexible structures. One popular and efficient form ofconstruction of such thin flexible structures is the shell - enclosing spaceusing one or more curved surfaces. A shell is therefore the curved form of aplate and its structural action is a combination of stretching and bending. Itis possible to perform a finite element analysis of a shell by using what iscalled a facet representation - i.e. the shell surface is replaced with flattriangular and/or quadrilateral plate elements in which a membrane stiffness(membrane element) is superposed on a bending stiffness (plate bending element).Such a model is understandably inaccurate in that with very coarse meshes, theydo not capture the bending-stretching coupling of thin shell behavior. Hence,the motivation for designing elements with mid-surface curvature taken intoaccount and which can capture the membrane-bending coupling correctly. There aretwo ways in which this could be done. One is to use elements based on specificshell theories (e.g. the Donnell, Flugge, Sanders, Vlasov theories, etc.). Thereare considerable controversies regarding the relative merits of these theoriesas each has been obtained by carrying out approximations to different degreeswhen the 3-dimensional field equations are reduced to the particular class ofshell equations. The second approach is called the degenerate shell approach -three dimensional solid elements can be reduced (degenerated) into shellelements having only mid-surface nodal variables - these are no longer dependenton the various forms of shell theories proposed and should be simple to use.They are in fact equivalent to a Mindlin type curved shell element.

However, accurate and robust curved shell elements have been extremelydifficult to design - a problem challenging researchers for nearly threedecades. One key issue behind this difficulty is the phenomenon of membranelocking - the very poor behavior of curved elements, as they become thin. This

F

was neither recognifirst that this wasrigid body motion csection, we shallrecognize that a cavoid the phenomenconsistency ratheris the key factor.dimensional curved b

In Chapter 6,deformation was presdeformable bending aplate regimes wherewith the bending strhow the field-consiscientifically satis

We shall nowin which both flexperform very poorlywas predominant, icompared to the bemembrane locking.

In this sectioelement and the lincurvilinear co-ordin

7.2.1 The classica

Figure 7.1 describescurvature R based ofreedom requireddisplacement w. TheThe membrane straindisplacement relatio

A C0 description fadmissible displacem

80

ig. 7.1 The classical thin curved beam element.

zed nor understood for a very long time. It was believed atbecause curved elements could not satisfy the strain-free

ondition because of their inherent curved geometry. In thisapply the field-consistency paradigm to the problem to

onsistent representation of membrane strains is desired toon called membrane locking and that it is the need forthan the requirement for strain-free rigid body motion whichThis is most easily identified by working with the one-eam (or arch) elements.

we studied structural problems where both bending and shearent. We saw that simple finite element models for the shearction of a beam or plate behaved very poorly in thin beam orthe shear strains become vanishingly small when compared

ains. This phenomenon is known as shear locking. We also sawstency paradigm was needed to explain this behavior in afying way.

see that a similar problem appears in curved finite elementsural and membrane deformation take place. Such elementsin cases where inextensional bending of the curved structure.e. the membrane strains become vanishingly small whennding strains. This phenomenon has now come to be called

n, we shall examine in turn, the classical thin curved beamear and quadratic shear flexible curved beam elements in aate system.

l thin curved beam element

the simplest curved beam element of length 2l and radius ofn classical thin beam theory. The displacement degrees ofare the circumferential displacement u and the radialco-ordinate s follows the middle line of the curved beam.ε and the bending strain χ are described by the strain-

ns

Rwu,s +=ε (7.1a)

sss w,Ru, −=χ (7.1b)

or u and a C1 description w is required. Kinematicallyent interpolations for u and w are

81

Fig. 7.2 Geometry of a circular arch

ξ10 aau += (7.2a)

33

2210 bbbbw ξξξ +++= (7.2b)

where ls=ξ and a0 to b3 are the generalized degrees of freedom which can be

related to the nodal degrees of freedom u, w and w,s at the two nodes.

The strain field interpolations can be derived as

( ) ( )ξχ 23

221 l6bl2bRla −−= (7.3a)

( ) ( ) ( ) ( )33

2231201 535Rb313Rb5R3bRb3RbRbla ξξξξε −−−−++++= (7.3b)

If a thin and deep arch (i.e. having a large span to rise ratio so that L/t>>1and R/H is small, see Fig. 7.2) is modeled by the curved beam element, thephysical response is one known as inextensional bending such that the membranestrain tends to vanish. From Equation (7.3b) we see that the inextensibilitycondition leads to the following constraints:

03RbRbla 201 →++ (7.4a)

053bb 31 →+ (7.4b)

0b2 → (7.4c)

0b3 → (7.4d)

Following the classification system we introduced earlier, we can observe thatconstraint (7.4a) has terms participating from both the u and w fields. It cantherefore represent the condition 0Rwu,s →+=ε in a physically meaningful way,

82

i.e. it is a true constraint. However, the three remaining constraint (7.4b) to(7.4d) have no participation from the u field. Let us now examine in turn whatthese three constraints imply for the physical problem. From the threeconstraints we have the conditions 0b0,b 21 →→ and 0.b3 → Each in turn

implies the conditions 0w,0,w, sss →→ and 0w,sss → . These are the spurious

constraints. This element has been studied extensively and the numericalevidence given there confirms that membrane locking is mainly determined by thefirst of these three constraints. Next, we shall briefly describe the functionalre-constitution analysis that will yield semi-quantitative error estimates formembrane locking in this element.

We can now use the functional re-constitution technique, (which in Chapter6 revealed how the discretization process associated with the linear Timoshenkobeam element led to large errors known as shear locking) to see how theinconsistent constraints in the membrane strain field interpolation (Equations(7.4b) to (7.4d)) modify the physics of the curved beam problem in thediscretization process.

The analysis is a little more difficult than for the shear locking caseand would not be described in full here. For a curved beam of length 2l, momentof inertia I and area of cross-section A, the strain energy can be written as,

( )� += dsEA21EI21 TTe εεχχπ (7.5)

where ε and χ are as defined in Equation (7.1). For simplicity assume that thecross-section of the beam is rectangular so that if the depth of the beam is t,

then 12AtI 2= . After discretisation, and reconstitution of the functional, we

can show that the inconsistent membrane energy terms disturb the bending energyterms - the principal stiffening term is physically equivalent to a spurious in-

plane or membrane stiffening action due to an axial load ( )22 3REAlN = .

This action is quite complex and it is not as simple to derive usefuluniversally valid error estimates as was possible with the linear Timoshenkobeam element. We therefore choose a problem of simple configuration which canreveal the manner in which the membrane locking action takes place. We considera shallow circular arch of length L and radius of curvature R, simply supportedat the ends. The loading system acting on it is assumed to induce nearlyinextensional bending. We also assume that the transverse deflection w can beapproximated in the form w=csin(πs/L). If the entire arch is discretized bycurved beam elements of length 2l each, the strain energy of the discretizedarch (again assuming that χ=-w,ss and neglecting the consistently modeled partof the membrane energy) one can show that due to membrane locking, the bendingrigidity of the arch has now been altered in the following form:

( ) ( ) ( ) ( ){ }22 RtLl41EIEI π+=′ (7.6)

We can therefore expect membrane locking to depend on the structural parameterof the type (Ll/Rt)2.

Numericalevaluate the vainconsistent eleThe consistent e

This clasdiscredited forthe element shouits very poor bemembrane strainfree rigid bodyerrors cannot bethe element intomodels) by desirequirements.

7.2.2 The Mindl

The exercise abtraced to the ibeam element useand used a curvunderstand howcurved shell eleinto the librariis called the detheory and usecapture the curv

To placecorrectly, it wibeam elements.displacement u,to the midline θthe rotation of

shear strain atis specified.

The strai

respective contr

83

Fig. 7.3 The Mindlin curved beam element.

experiments with various order integration rules applied torious components of the stiffness matrix indicated that thements locked exactly as predicted by the pattern describe above.lement was entirely free of locking.

sical curved beam element (the cubic-linear element) wasa very long time because it was not possible to understand whyld perform so poorly. We have now found that the explanation forhavior could be attributed entirely to the inconsistency in thefield interpolations. The belief that the lack of the strainmotion in the conventional formulation was the cause of thesesubstantiated at all. We have also seen how we could restorea very accurate one (having the accuracy of the straight beam

gning a new membrane strain field which met the consistency

in curved beam elements

ove showed that the locking behavior of curved beams could benconsistent definition of the membrane strain field. The curvedd to demonstrate this was based on classical thin beam theoryilinear co-ordinate description. Of interest to us now is tothe membrane locking phenomenon can take place in a generalment. Most general shell elements which have now been acceptedes of general purpose finite element packages are based on whatgenerate shell theory which is equivalent to a shear deformablequadratic interpolations (i.e. three nodes on each edge toed geometry in a Cartesian system).

the context of membrane locking in general shell elementsll be useful to investigate the behavior of the Mindlin curvedThese are now described by the circumferential (tangential)the radial (normal) displacement w and the rotation of a normal. As in the Timoshenko beam element, this rotation differs fromthe mid-line w,s by an amount sw,−= θγ which describes the

the section. Fig. 7.3 shows how the Mindlin curved beam element

n energy in a Mindlin curved beam is U=UM+UB+US where the

ibutions to U arise from the membrane strain ε, the bending

84

strain χ and the transverse shear strain γ. For an element of length 2l andradius R,

( ) ( ) � +== Rwu,wheredsEA21U s2

M εε (7.7a)

( ) ( ) � −== ss2

B Ru,wheredsEI21U ,θχχ (7.7b)

( ) ( ) � −== s2

S wwheredskGA21U ,θγγ (7.7c)

where EA, EI and kGA represent the respective rigidities.

There are no difficulties with the bending strain. However, depending onthe regime, both shear strains and membrane strains can be constrained to vanishleading to shear and membrane locking. The arguments concerning shear lockingare identical to those expressed in Chapter 6, i.e. while the inconsistency inthe shear strain field is severe enough to cause locking in the former (asignificant stiffening which caused errors that propagated indefinitely as the

structural parameter 22 EtkGl became very large), it is much milder in the

latter and is responsible only for a poorer rate of convergence (the performanceof the element having reduced to that of the linear field-consistent element).However with membrane locking, the inconsistencies at the quadratic level cancause locking - therefore both the linear and quadratic elements must beexamined in turn to get the complete picture.

For the linear Mindlin curved beam element, as a C0 displacement typeformulation is sufficient, we can choose linear interpolations of the followingform,

ξ10 aau += (7.8a)

ξ10 bbw += (7.8b)

ξθ 10 cc += (7.8c)

where ls=ξ is the natural co-ordinate along the arch mid-line. These

interpolations lead to the following membrane strain field interpolation:

( ) ( ) ξε RbRbla 101 ++= (7.9)

There is now one single spurious constraint in the limit of inextensionalbending of an arch, i.e. b1→0, which implies a constraint of the form w,s→0 atthe centroid of the element. The membrane locking action is similar to butsimpler than that seen for the classical thin curved beam element above. In athin beam, in the Kirchhoff limit, θ→w,s, which implies that membrane locking

induces the section rotations θ to lock. It can therefore be seen that aconstraint of this form leads to an in-plane stiffening action of the formcorresponding to the introduction of a spurious energy

( )( ) dsw3REAl212

s22� , (7.10)

85

The membrane locking action is therefore identical to that predicted for theclassical thin curved beam. A variationally correct field-consistent element caneasily be derived using the Hu-Washizu theorem. This is a very accurate element.

The quadratic Mindlin curved beam element can also be interpreted in thelight of our studies of the classical curved beam earlier. Spurious constraintsat both the linear (i.e. w,s→0) and quadratic (i.e w,ss→0) levels lead to

errors that propagate at a ( )2RtLl rate and ( )22 Rtl rate respectively. We

expect now that in a quadratic Mindlin curved beam element, the latterconstraint is the one that is present and that these errors are large enough tomake the membrane locking effect noticeable. This also explains why thedegenerate shell elements, which are based on quadratic interpolations, sufferfrom membrane locking.

A quadratic (three-node) element having nodes at s=-l, 0 and l (see Fig.7.3) would have a displacement field specified as,

2210 aaau ξξ ++= (7.11a)

2210 bbbw ξξ ++= (7.11b)

2210 ccc ξξθ ++= (7.11c)

The membrane strain field can be expressed as:

( ) ( ) ( )2212201 313RbRbl2a3RbRbla ξξε −−++++= (7.12)

The spurious constraint is now b2→0 a discretized equivalent of imposing the

constraint w,ss→0 at the centroid of each element. To understand that this can

cause locking, we note that in a thin curved beam, sss w,→,θ and therefore a

spurious constraint of this form acts to induce a spurious bending energy. Afunctional re-constitution approach will show that the bending energy for a thincurved beam modifies to,

( ) ( ) ( ) ( ) dsw,Rtl1541EI21U 2ss

22B � �

�

���

� += (7.13)

This can be interpreted as spurious additional stiffening of the actual bending

rigidity of the structure by the ( ) ( )22 Rtl154 term. It is a factor of this form

that caused the very poor performance of the exactly integrated quadratic shellelements.

A variationally correct field consistent element is obtained by using theHu-Washizu theorem or very simply by using a two-point integration rule for theshear and membrane strain energy energies. This is an extremely accurateelement.

It is interesting to compare the error norm for locking for this element,

i.e. ( )22 Rtl with the corresponding error norm for locking in the inconsistent

86

linear Mindlin curved beam element, which is ( )2RtlL . It is clear that since

L>>l when an arch is discretized with several elements, the locking in thequadratic element is much less than in the linear element. One way to look atthis is to examine the rate at which locking is relieved as more elements areadded to the mesh, i.e. as l is reduced. In the case of the inconsistent linearelement, locking is relieved only at an l2 rate whereas in the case of the

quadratic element, locking is relieved at the much faster l4 rate. However,locking in the quadratic element is significant enough to degrade theperformance of the quadratic curved beam and shell elements to a level where itis impractical to use in its inconsistent form. This is clearly seen from thefact that to reduce membrane-locking effects to acceptable limits, one must use

element lengths Rtl << . Thus for an arch with R=100 and t=1, one must useabout a 100 elements to achieve an accuracy that can be obtained with two tofour field-consistent elements.

7.2.3 Axial force oscillations

Our experience with the linear and quadratic straight beam elements showed thatinconsistency in the constrained strain-field is accompanied by spurious stressoscillations corresponding to the inconsistent constraints. We can nowunderstand that the inconsistent constraint b1→0 will induce linear axial forceoscillations in the linear curved beam element. Similarly, the inconsistentquadratic constraint will trigger off quadratic oscillations in the quadraticelement.

7.2.4 Concluding remarks

So far, we have examined the phenomenon of membrane locking in what were calledthe simple curved beam elements. These were based on curvilinear geometry andthe membrane locking effect was easy to identify and quantify. However, curvedbeam and shell elements which are routinely used in general purpose finiteelement packages are based on quadratic formulations in a Cartesian system. Weshall call these the general curved beam and shell elements. Membrane locking isnot so easy to anticipate in such a formulation although the predictions made inthis section have been found to be valid for such elements.

7.3 Parasitic shear

Shear locking and membrane locking are phenomena seen in finite element modelingusing structural elements based on specific theories, e.g. beam, plate and shelltheories. These are theories obtained by simplifying a more general continuumdescription, e.g. the three dimensional theory of elasticity or the twodimensional theories of plane stress, plane strain or axisymmetric elasticity.The simplifications are made by introducing restrictive assumptions on strainand/or stress conditions like vanishing of shear strains or vanishing oftransverse normal strains, etc. In the beam, plate and shell theories, thisallows structural behavior to be described by mid-surface quantities. However,these restrictive assumptions impose constraints which make finite elementmodeling quite difficult unless the consistency aspects are taken care of. Weshall now examine the behavior of continuum elements designed for modelingcontinuum elasticity directly, i.e. 2D elements for plane strain or stress oraxisymmetric elasticity and solid elements for 3D elasticity. It appears that

87

Fig. 7.4 (a) Constant strain triangle, (b) Rectangular bilinear element,(c) Eight-node rectangle and (d) Nine-node rectangle.

under constraining physical regimes, locking behavior is possible. One suchproblem is parasitic shear. We shall see the genesis of this phenomenon and showthat it can be explained using the consistency concepts. The problems of shearlocking, membrane locking and parasitic shear obviously are very similar innature and the unifying factor is the concept of consistency.

The finite element modeling of plane stress, plane strain, axisymmetricand 3D elasticity problems can be made with two-dimensional elements and three-dimensional elements (sometimes called continuum elements to distinguish themfrom structural elements), of which the linear 3-noded triangle and bi-linear 4-noded rectangle (Fig. 7.4) are the simplest known in the 2D case. In mostapplications, these elements are reliable and accurate for determining generaltwo-dimensional stress distributions and improvements can be obtained by usingthe higher order elements, the quadratic 6-noded triangle and 8- or 9-nodedquadrilateral elements (Fig. 7.4).

However, in some applications, e.g. the plane stress modeling of beam flexure orin plane strain and axisymmetric applications where the Poisson's ratioapproaches 0.5 (for nearly incompressible materials or for materials undergoingplastic deformation), considerable difficulties are seen. Parasitic shear occursin the plane stress and 3D finite element modeling of beam flexure. Under purebending loads, for which the shear stress should be zero, the elements yieldlarge values of shear stress except at certain points (e.g. centroids in linearelements). In the linear elements, the errors progress in the same fashion asfor shear locking in the linear Timoshenko beam element, i.e. the errorsprogress indefinitely as the element aspect ratio increases. Reduced integrationof the shear strain energy or addition of bubble modes for the 4-noded

88

Fig. 7.5 Plane stress model of beam flexure.

rectangular element completely eliminates this problem. The 8- and 9-nodedelements do not lock but improve in performance with reduced integration. Theseelements, in their original (i.e. field-inconsistent) form show severe linearand quadratic shear stress oscillations.

Here, we shall extend the consistency paradigm to examine this phenomenonin detail, confining attention to the rectangular elements.

7.3.1 Plane stress model of beam flexure

Figure 7.5 shows a two-dimensional plane stress description of the problem ofbeam flexure. The beam is of length L, depth T and thickness (in the normal tothe plane direction) b. Two independent field variables are required to describethe problem, the displacements u and v (Fig. 7.5). The strain energy functionalfor this problem consists of energies from the normal strains, UE and shear

strains, UG. For an isotropic case, where E is the Young's modulus, G is the

shear modulus and ν is Poisson's ratio,

GE UUU +=

( ) ( ) ( ) dydxvu2Gbvu,2,v,u-12

Eb 2xyyxy

2x

22

T0

L0

���

�

���

�++++= �� ,,,ν

ν(7.14)

A one-dimensional beam theory can be obtained by simplifying this problem. For avery slender beam, the shear strains must vanish and this would then yield theclassical Euler-Bernouilli beam theory. Note that although shear strains vanish,shear stresses and shear forces would remain finite. We shall see now that it isthis constraint condition that leads to difficulties in plane stress finiteelement modeling of such problems.

7.3.2 The 4-node rectangular element

An element of length 2l and depth 2t is chosen. The rectangular form is chosenso that the issues involved can be identified clearly. There are now 8 degrees

of freedom, u1-u4 and v1-v4 at the four nodes. The field-variables are nowinterpolated in the following fashion:

89

xyayaxaau 3210 +++= (7.15a)

xybybxbbv 3210 +++= (7.15b)

We need to examine only the shear strain field that will becomeconstrained (i.e. become vanishingly small) while modeling the flexural actionof a thin beam. This will be interpolated as,

( ) ybxabav,u, 3312xy +++=+=γ (7.16)

and the shear strain energy within the element will be,

( ) ( )�� +++=t

t-

23312

ll-e dydxybxaba2GbGU

( ) ( ) ( ) ( ) ( )[ ]3tb3laba4tl2Gb 23

23

212 +++= (7.17)

Equation (7.17) suggests that in the thin beam limit, the following constraintswill emerge:

0ba 12 →+ (7.18a)

0a3 → (7.18b)

0b3 → (7.18c)

We can see that Equation (7.18a) is a constraint that comprises constants fromboth u and v functions. This can enforce the true constraint of vanishing shearstrains in a realistic manner. In contrast, Equations (7.18b) and (7.18c) imposeconstraints on terms from one field function alone in each case. These areundesirable constraints. We shall now see how these lead to the stiffeningeffect that is called parasitic shear.

In the plane stress modeling of a slender beam, we would be using elementswhich are very long in the x direction, i.e. elements with l>t would be used.One can then expect from Equation (7.17) that as l2>>t2, the constraint a3→0

will be enforced more rapidly than the constraint b3→0 as the beam becomesthinner. The spurious energies generated from these two terms will also be in asimilar proportion. In Equation (7.17), one can interpret the shear strain fieldto comprise a constant value, which is `field-consistent' and two linearvariations which are `field-inconsistent'. We see now that the variation alongthe beam length is the critical term. Therefore, let us consider the use of theshear strain along y=0 as a measure of the averaged shear strain across avertical section. This gives

( ) xaba 312 ++=γ (7.19)

Thus, along the length of the beam element, γ have a constant term that reflectsthe averaged shear strain at the centroid, and a linear oscillating term alongthe length, which is related to the spurious constraint in Equation (7.18b). Asin the linear beam element in Section 4.1.6, this oscillation is self-equilibrating and does not contribute to the force equilibrium over the element.However, it contributes a finite energy in Equation (7.17) and in the modeling

90

of very slender beams, this spurious energy is so large as to completelydominate the model of the beam behavior and cause a locking effect. We shall nowuse the functional re-constitution technique to determine the magnitude oflocking and the extent of the shear stress oscillations triggered off by theinconsistency in the shear strain field.

We shall simplify the present example and derive the error estimates fromthe functional re-constitution exercise. We shall denote by 3a , the coefficient

determined in a solution of the beam problem by using a field-consistentrepresentation of the total energy in the beam, and by 3a , the value from a

solution using energies based on inconsistent terms. The field-consistent andfield-inconsistent terms will differ by,

( )2233 EtGl1aa += (7.20)

This factor, ( )22L EtGl1e += is the additional stiffening parameter that

determines the extent of parasitic shear.

We can relate a3 to the physical parameters relevant to the problem of a

beam in flexure to show that if 1eL >> , as in a very slender beam, there are

shear force oscillations,

( ) ( )lxl3MVV 00 += (7.21)

The oscillations are proportional to the centroidal moment, and for 1eL >> , the

oscillations are inversely proportional to the element length, i.e. they becomemore severe for smaller lengths of the plane stress element. It is important tonote that a field-inconsistent two node beam element based on linearinterpolations for w and θ (see Equation (7.19)) produces an identical stressoscillation.

7.3.3 The field-consistent elements

It is well known that the plane stress elements behave reliably in general 2Dapplications where no constraints are imposed on the normal or shear strainfields. The need for field-consistency becomes important only when a strainfield is constrained. Earlier, we saw that in the plane stress modeling of thinbeam flexure, the shear strain field is constrained and that these constraints

are enforced by a penalty multiplier 22 EtGl . The larger this term is, the more

severely constrained the shear strain becomes. Of the many strategies that areavailable for removing the spurious constraints on the shear strain field, theuse of a variationally correct re-distribution is recommended. This requires thefield-consistent γ to be determined from γ using the orthogonality condition

( )� =− 0dydxT γγγδ (7.22)

Thus instead of Equation (7.16), we can show that the shear strain-fieldfor the 4-node rectangular plane stress element will be,

91

( )12 ba +=γ (7.23)

It is seen that the same result is achieved if reduced integration using a 1Point Gaussian integration is applied to shear strain energy evaluation. In avery similar fashion, the consistent shear strain field can be derived for thequadratic elements.

7.3.4 Concluding remarks on the rectangular plane stress elements

The field-consistency principle and the functional re-constitution technique canbe applied to make accurate error analyses of conventional 4-node and 8-nodeplane stress elements. These a priori estimates have been confirmed throughnumerical experiments.

A field-consistent re-distribution strategy allows these elements to befree of locking and spurious stress-oscillations in modeling flexure. Theseoptions can be built into a shape function sub-routine package in a simplemanner so that where field-consistency requirements are paramount, theappropriate smoothed shape functions will be used.

7.3.5 Solid element model of beam/plate bending

So far, we have dealt with the finite element modeling of problems which weresimplified to one-dimensional or two-dimensional descriptions. There remains alarge class of problems which need to be addressed directly as three dimensionalstates of stress. Solid or three dimensional elements are needed to carry outthe finite element modeling of such cases. A variety of 3D solid elements exist,e.g. tetrahedral, triangular prism or hexahedral elements. In general cases of3D stress analysis, no problems are encountered with the use of any of theseelements as long as a sufficiently large number of elements are used and noconstrained media limits are approached. However, under such limits thetetrahedral and triangular prism elements cannot be easily modified to avoidthese difficulties. It is possible however to improve the hexahedral 8-node and27-node elements so that they are free of locking and of other ill-effects likestress oscillations. One problem encountered is that of parasitic shear or shearlocking when solid elements are used to model regions where bending action ispredominant. In fact, in such models, parasitic shear and shear locking mergeindistinguishably into each other. We briefly examine this below.

The 8-noded brick element is based on the standard tri-linearinterpolation functions and is the three dimensional equivalent of the bi-linearplane stress element. It would appear now that the stiffness matrix for theelement can be derived very simply by carrying out the usual finite elementoperations, i.e. with strain-displacement matrix, stress-strain matrix andnumerically exact integration (i.e. 2×2×2 Gaussian integration). Although thiselement performs very well in general 3D stress analysis, it failed to representcases of pure bending (parasitic shear) and cases of near incompressibility(incompressible locking).

The element is unable to represent the shear strains in a physicallymeaningful form when the shear strains are to be constrained, as in a case ofpure bending. The phenomenon is the 3D analogue of the problem we noticed for

92

the bi-linear plane stress element above. The problem of designing a useful 8-node brick element has therefore received much attention and typical attempts toalleviate this have included techniques such as reduced integration, addition ofbubble functions and assumed strain hybrid formulations. We shall now look atthis from the point of view of consistency of shear strain interpolation.

We can observe here that the shear strain energy is computed fromCartesian shear strains xzxy γγ , and yzγ . In the isoparametric formulation,

these are to be computed from terms involving the derivatives of u, v, w withrespect to the natural co-ordinates ξ, η, ζ and the terms from the inverse ofthe three dimensional Jacobian matrix. It is obvious that it will be a verydifficult, if not impossible task, to assure the consistency of theinterpolations for the Cartesian shear strains in a distorted hexahedral.Therefore attention is confined to a regular hexahedron so that consistencyrequirements can be easily examined.

To facilitate understanding, we shall restrict our analysis to a rectangularprismatic element so that the (x, y, z) and (ξ, η, ζ) systems can be usedinterchangeably. We can consider the tri-linear interpolations to be expanded inthe following form, i.e.

xyzaxzayzaxyazayaxaau 87654321 +++++++= (7.24a)

xyzbxzbyzbxybzbybxbbv 87654321 +++++++= (7.24b)

xyzcxzcyzcxyczcycxccw 87654321 +++++++= (7.24c)

where a1 to a8 and b1 to b8 are related to the nodal degrees of freedom u1 to u8and v1 to v8 respectively. We shall now consider a case of a brick elementundergoing a pure bending response requiring a constrained strain field

0vu, xyxy →+= ,γ . From Equations (7.24a) and (7.24b) we have,

( ) ( ) yzbxzaybxazbaba 88557623xy +++++++=γ (7.25)

When Equation (7.25) is constrained, the following spurious constraintconditions appear: a5→0; b5→0; a8→0 and b8→0.

Our arguments projecting these as the cause of parasitic shear would beproved beyond doubt if we can derive a suitable error model. These error modelsare now much more difficult to construct, but have been achieved using ajudicious mix of a priori and a posteriori knowledge of zero strain and stressconditions to simplify the problem. These error models have been convincinglyverified by numerical computations [7.1].

A field-consistent representation will be one that ensures that theinterpolations for the shear fields will contain only the consistent terms. Asimple way to achieve this is to retain only the consistent terms from theoriginal interpolations. Thus, for γxy we have,

( ) ( )zbaba 7623xy +++=γ (7.26)

93

Similarly, the other shear strain fields can be modified. These strain fieldsshould now be able to respond to bending modes in all three planes withoutlocking.

7.4 Incompressible locking

Conventional displacement formulations of 2D plane strain and 3D elasticity failwhen Poisson's ratio ν→0.5, i.e. as the material becomes incompressible. Suchsituations can arise in modeling of material such as solid rocket propellants,saturated cohesive soils, plastics, elastomers and rubber like materials and inmaterials that flow, e.g. incompressible fluids or in plasticity. Displacementfields lock and highly oscillatory stresses are seen - drawing an analogy withshear and membrane locking, we can describe the phenomenon as incompressiblelocking. In fact, a penalty term (bulk modulus→∞ as ν→0.5) is easilyidentifiable as enforcing incompressibility. Since this multiplies thevolumetric strain, it is easy to argue that a lack of consistency in thedefinition of the volumetric strain will cause locking and will appear asoscillatory mean stresses or pressures.

7.4.1 A simple one-dimensional case - an incompressible hollow sphere

It is instructive to demonstrate how field-consistency operates in the finiteelement approximation of a problem in incompressible (or nearly-incompressible)elasticity by taking the simplest one-dimensional case possible. A simpleproblem of a pressurized elastic hollow sphere serves to illustrate this.

Consider an elastic hollow sphere of outer radius b and inner radius awith an internal pressure P. If the shear modulus G is 1.0, the radialdisplacement field for this case is given by,

( )( ) �

�

���

�

+−+= r12

21

4r

1u

2 ννρ (7.27)

where ( )[ ]333 abGPba −=ρ and r=R/b non-dimensional radius and ν is the Poisson's

ratio.

A finite element approximation approaches the problem from the minimumtotal potential principle. Thus, a radial displacement u leads to the followingstrains:

ruruu,rr === φθ εεε

and the volumetric strain is

r2uu,rr +=++= φθ εεεε

The elastic energy stored in the sphere (when G=1) is

( )� +++=a

22222r

1

drr21 εαεεε φθU (7.28)

94

and ( )ννα 212 −= when G=1 and baa = is the dimensionless inner radius. The

displacement type finite element representation of this very simple problem runsinto a lot of difficulties when ν→0.5, i.e., when α→∞ in the functionalrepresented by Equation (7.28).

These arguments now point to the volumetric strain field in Equation(7.28) as the troublesome term. When α→∞, this strain must vanish. In aphysical situation, and in an infinitesimal interpretation of Equation (7.28),this means that ε→0. However, when a finite element discretization isintroduced, this does not happen so simply. To see this, let us examine whatwill happen when a linear element is used to model the problem.

7.4.2 Formulation of a linear element

The displacement field u and the radial distance r are interpolated bylinear isoparametric functions as

( ) ( )ξ1221 rr0.5rr0.5r −++= (7.29a)

( ) ( )ξ1221 uu0.5uu0.5u −++= (7.29b)

where the nodes of the element are at r1 and r2.

The troublesome term in the potential energy functional is the volumetricstrain and this contribution can be written as,

( )� � += dr2uru,drr 2r

22ε (7.30)

It is the discretized representation of this term ( )2uru,r + that will

determine whether there will be locking when α→∞. Thus, unlike in previoussections, the constraint is not on the volumetric strain ε but on rε because ofthe use of spherical coordinates to integrate the total strain energy. It istherefore meaningful to define this as a pseudo-strain quantity γ=rε. To see howthis is inconsistently represented, let us now write the interpolation for thisderived kinematically from the displacement fields as,

( ) ( )( ) ( ) ( ) ( )[ ] ξ122112

12

21r uu23uuuu

rr2

rr2uru, −+�

�

���

�++−

−+

=+ (7.31)

Consider now the constraints that are enforced when this discretized fieldis used to develop the strain energy arising from the volumetric strain as shownin Equation (7.31) when α→∞.

( )( ) ( ) ( ) 0uuuu

rr2

rr2112

12

21 →++−−

+(7.32a)

( ) 0uu 12 →− (7.32b)

95

Condition (7.32a) represents a meaningful discretized version of thecondition of vanishing volumetric strain and is called a true constraint in ourfield-consistency terminology. However, condition (7.32b) leads to an undesiredrestriction on the u field as it implies 0u,r → . This is therefore the spurious

constraint that leads to locking and spurious pressure oscillations.

7.4.3 Incompressible 3D elasticity

We now go directly to modeling with 3D elasticity. We re-write the strain energyfor 3D elasticity in the following form:

�� += dV21dVDG21U nTnd

Td εελεε (7.33)

In Equation (7.33), εd is the distortional strain and εn is the volumetricstrain. In this form, it is convenient to describe the elasticity matrix interms of the shear modulus, G and the bulk modulus K, where ( )[ ]ν+= 12EG ,

( )[ ]ν2-13EK = and ( ) ( ) ( )[ ]νννλ 2-11E32GK +=−= .

Consider the fields given in Equations (7.24) for the 8-noded brickelement. An elastic response for incompressible or nearly-incompressiblematerials will require a constrained strain field,

0w,vu zyxn →++= ,,ε (7.34)

where εn is the volumetric strain. From Equations (7.24) and (7.34) we have,

( ) ( ) ( ) ( ) ( )xycxzbyzazbaycaxcbcba 8z8676575432n +++++++++++=ε (7.35)

Equation (7.35) can be constrained to zero only when the coefficients of each ofits terms vanish, giving rise to the constraint conditions,

0cba 432 →++ (7.36a)

0cb 75 →+ (7.36b)

0ca 65 →+ (7.36c)

0ba 67 →+ (7.36d)

0a8 → (7.36e)

0b8 → (7.36f)

0c8 → (7.36g)

Equation (7.36b) to (7.36) are the inconsistent terms that cause locking. Toensure a field-consistent representation, only the consistent terms should beretained, giving

432n cba ++=ε (7.37)

This field will now be able to respond to the incompressible or nearlyincompressible strain states and the element should be free of locking.

96

7.4.4 Concluding remarks

It is clear from this section that incompressible locking is analogous to shearlocking, etc. A robust formulation must therefore ensure consistentrepresentations of volumetric strain in cases where incompressibility or nearincompressibility is expected.

7.5 References

7.1 G. Prathap, The Finite Element Method in Structural Mechanics, KluwerAcademic Press, Dordrecht, 1993.

97

Chapter 8

Stress consistency

8.1 Introduction

In Chapters 6 to 7 we examined the difficulties experienced by the displacementapproach to the finite element formulation of problems in which some strainfields are constrained. To obtain accurate solutions at reasonable levels ofdiscretization, it was necessary to modify these strain fields and use these incomputing the stiffness matrix and also in stress recovery. The criteriongoverning the relationship between the various terms in the modified strainfield interpolation was described as consistency - i.e. the strain fieldinterpolations must maintain a consistent balance internally of its contributingterms. This allows the constraints that emerge after discretization to remainphysically faithful to the continuum problem. This was the rule that guided thelocking-free design of all elements discussed so far.

In this chapter, we take a look at a class of problems where noconstraints are imposed on the strains but there is a need to relax thesatisfaction of the constitutive relationship linking discretised stress todiscretised strain so that again a certain degree of consistency is maintained.Such situations develop where there are structural regions in which the rigidityvaries spatially due to varying elastic moduli or cross-sectional area and alsoin initial strain problems, of which the thermal strain problem is the mostcommonly encountered.

In structural regions with varying rigidity the spatial variation ofstrain-fields and stress or stress-resultant fields will not match. In thediscretization of such cases, it is necessary to consider a form of externalconsistency requirement between the discretized strain fields and thediscretized stress or stress-resultant fields. This is necessary so that acorrect interpretation of computed stresses and stress resultants is possible;otherwise, oscillations will be seen.

In initial strain problems, the initial strain variation and the totalstrain variations may be of different order. This is a familiar problem infinite element thermal stress analysis. Here, it is necessary to obtainkinematically equivalent thermal nodal forces from temperature fields and alsoensure that the discretised thermal strains corresponding to this are consistentwith the discretised description of total strains. Again, one must carefullyidentify the conflicting requirements on the order of discretised functions tobe used.

8.2 Variable moduli problems

8.2.1 A tapered bar element

We shall now conduct a simple numerical experiment with a tapered bar element toshow that the force field computed directly from strains in a structural elementof varying sectional rigidities has extraneous oscillations. A linear elementwould have sufficed to demonstrate the basic principles involved. However, weuse a quadratic tapered bar element so that the extraneous oscillations, which

98

for a general case can be of cubic form, are not only vividly seen but also needspecial care to be reduced to its consistent form. In a linear element, thisexercise becomes very trivial, as sampling at the centroid of the element givesthe correct stress resultant.

We shall consider an isoparametric formulation for a quadratic bar elementof length 2l with mid-node exactly at the mid-point of the element. Then theinterpolations for the axial displacement u and the cross sectional area A interms of their respective nodal values are,

( ) ( ) 2321132 2u2uu2uuuu ξξ +−+−+=

( ) ( ) 2321132 2A2AA2AAAA ξξ +−+−+=

We shall first examine how the element stiffness is formulated when the minimumtotal potential principle is used. We start with a functional written as,

� −= WdxN21 T επwhere,

dxdu=ε the axial strain,

( ) εxAEN = the kinematically constituted axial force,

�= dxpuW the potential of external forces,

p distributed axial load,E Young's modulus of elasticity

After discretization, ε will be a linear function of ξ but N will be acubic function of ξ. The strain energy of deformation is then expressed as,

�= dxN21U T ε

From this product the terms of the stiffness matrix emerge. Due to theorthogonal nature of the Legendre polynomials, terms from N which are linkedwith the quadratic and cubic Legendre polynomials, N3 and N4 respectively, willnot contribute to the energy and therefore will not provide terms to thestiffness matrix! It is clear that, the displacements recovered from such aformulation cannot recognize the presence of the quadratic and cubic terms N3and N4 in the stress field N as these have not been accounted for when thestiffness matrix was computed. Hence, in a displacement type finite elementformulation, the stresses recovered from the displacement vector will haveextraneous oscillations if N3 and N4 are not eliminated from the stress fieldduring stress recovery. We shall designate by BAR3.0, the conventional elementusing N for stiffness matrix evaluation and recovery of force resultant.

Next, we must see how the consistent representation of the force fielddenoted by N must be made. N should comprise only the terms that willcontribute to the stiffness and strain energy and the simplest way to do this isto expand the kinematically determined N in terms of Legendre polynomials, and

retain only terms that will meaningfully contribute to the energy in ( )εTN .

99

Thus, N must be consistent with ε, i.e. in this case, retain only up to linearterms:

ξ21 NNN += (8.1)

Such an element is denoted by BAR3.1. It uses N for stiffness matrix evaluationand recovery of forces resultant. To see the variational basis for the procedureadopted so far, the problem is re-formulated according to the Hu-Washizuprinciple which allows independent fields for assumed strain and assumed stressfunctions.

8.2.2 Numerical experiments

We shall perform the computational exercises with the two versions of theelement; note that in both cases, the stiffness matrices and computeddisplacements are identical. Figure 8.1 shows a tapered bar clamped at node-1and subjected to an axial force P at node-3. The taper is defined by theparameters,

( ) 213 2AAA −=α and ( ) 2321 2AA2AA +−=β

Finite element results from a computational exercise using the two versionsdescribed above for a bar with cross section tapering linearly from the root tothe tip for which β=0 are obtained. Thus α is given by,

( ) ( )3113 AAAA +−=α

Thus α can vary from 0 to –1.0. Fig. 8.2 shows the axial force patternsobtained from the finite element digital computation for a case with α=-0.9802(with A3=0.01 A1) It can be noted here that the results are accurate at

31±=ξ .

Fig. 8.1 A cantilever bar modeled with a single element.

100

Fig. 8.2 Axial force pattern for linearly tapered bar (α=0.9802 and β=0.0) with

A3=0.01 A1.

A general case of taper is examined next where A1=1.0, A2=0.36 and

A3=0.04, and the area ratios are α=-4/3 and β=4/9. Fig. 8.3 shows the resultsfrom the finite element computations. Due to the presence of both quadratic andcubic oscillations, there are no points which can be easily identified foraccurate force recovery! Therefore it is necessary to perform a re-constitutionof the force resultant fields on a consistency basis using the orthogonalityprinciple as done here before reliable force recovery can be made.

8.2.3 Reconstitution of the stress-resultant field using the Hu-Washizuprinciple

In forming the Hu-Washizu functional for the total potential, an as yetundetermined assumed force function N is introduced but the assumed strainfield ε can be safely retained as ε (note that in a constrained media problemit will be required to introduce a field-consistent ε that will be different

from the kinematically derived and therefore usually field-inconsistent ε) - thefunctional now becomes

( ) ( ){ }� −−+= WdxNEA21 TT εεεεπ

101

Fig. 8.3 Axial force pattern for bar with combined linear and quadratic taper(α=-4/3 and β=4/9).

A variation of the Hu-Washizu energy functional with respect to thekinematically admissible degree of freedom u, gives the equilibrium equation,

{ }� =− 0dxpdxNduTδ

Variation with respect to the assumed strain field ε gives rise to aconstitutive relation

{ }� =+ 0dxEAN-T εεδ (8.2)

and variation with respect to the assumed force field N gives rise to thecondition

{ }� =0dx-NT εεδ (8.3)

Now Equations (8.2) and (8.3) are the orthogonality conditions requiredfor reconstituting the assumed fields for the stress resultant and the strain.The consistency paradigm suggests that the assumed stress resultant field N

102

should be of the same order as the assumed strain field ε . Then Equation (8.3)gives the orthogonality condition for strain field-redistribution.

On the other hand, orthogonality condition (8.2) can be used toreconstitute the assumed stress-resultant field N from the kinematicallyderived field N. Now, Equation (8.2) can be written as,

{ }� =0dxN-NTεδ (8.4)

Thus, if N is expanded in terms of Legendre polynomials, it can be provedthat N which is consistent and orthogonally satisfies Equation (8.4) isobtained very simply by retaining all the Legendre polynomial terms that areconsistent with ε , i.e. as shown in Equation (8.1). Thus the procedure adoptedin the previous section has variational legitimacy.

8.3 Initial strain/stress problems

Finite element thermal stress analysis requires the formulation of what iscalled an initial strain problem. The temperature fields which are imposed mustbe converted to discretised thermal (initial) strains and from thiskinematically equivalent thermal nodal forces must be computed. The usualpractice in general purpose codes is to use the same shape functions tointerpolate the temperature fields and the displacement fields. Thermal stressescomputed directly from stress-strain and strain-displacement matrices after thefinite element analysis is performed thus show large oscillating errors. Thiscan be traced to the fact that the total strains (which are derived fromdisplacement fields) are one order higher than the thermal strains (derived fromtemperature fields). Some useful rules that are adopted to overcome thisdifficulty are that the temperature field used for thermal stress analysisshould have the same consistency as the element strain fields and that ifelement stresses are based on Gauss points, the thermal stresses should also bebased on these Gauss point values. This strategy emerged from the understandingthat the unreliable stress predictions originate from the mismatch between theelement strain ε and the initial strain due to temperature ε0. We shall now showthat this is due to the lack of consistency of their respective interpolationswithin the element.

Earlier in this chapter, we saw that stress resultant fields computed fromstrain fields in a displacement type finite element description of a domain withvarying sectional rigidities showed extraneous oscillations. This was traced tothe fact that these stress resultant fields were of higher interpolation orderthan the strain fields and that the higher degree stress resultant terms did notparticipate in the stiffness matrix computations. In this section, we show thatmuch the same behavior carries over to the problem of thermal stresscomputations.

8.3.1 Description of the problem

With the introduction of initial strains ε0 due to thermal loading, the stressto strain relationship has to be written as

103

( ) m0 D-D εεεσ == (8.5)

The strain terms now need to be carefully identified. {ε} is the total strain

and {εm} is the mechanical or elastic strain. The free expansion of materialproduces initial strains

T0 αε = (8.6)

where T is the temperature relative to a reference value at which the body isfree of stress and α is the coefficient of thermal expansion. The total strains(i.e. the kinematically derived strains) are defined by the strain-displacementmatrix,

{ε}=[B]{d} (8.7)

where {d} is the vector of nodal displacements. In a finite element description,the displacements and temperatures are interpolated within the domain of theelement using the same interpolation functions. The calculation of the totalstrains {ε} involves the differentiation of the displacement fields and thestrain field functions will therefore be of lower order than the shapefunctions. The initial strain fields (see Equation (8.6)) involve thetemperature fields directly and this is seen to result in an interpolation fieldbased on the full shape functions. The initial strain matrix is of higher degreeof approximation than the kinematically derived strain fields if the temperaturefields can vary significantly over the domain and are interpolated by the sameisoparametric functions as the displacement fields. It is this lack ofconsistency that leads to the difficulties seen in thermal stress prediction.This originates from the fact that the thermal load vector is derived from apart of the functional of the form,

{ } { }� dVD 0T εεδ (8.8)

Again, the problem is that the `higher order' components of the thermal (orinitial) stress vector are not sensed by the total strain interpolations in theintegral shown above. In other words, the total strain terms "do work" only onthe consistent part of the thermal stress terms in the energy or virtual workintegral. Thus, a thermal load vector is created which corresponds to a initialstrain (and stress) vector that is `consistent' with the total strain vector.The finite element displacement and total strain fields which are obtained inthe finite element computation then reflect only this consistent part of thethermal loading. Therefore, only the consistent part of the thermal stressshould be computed when stress recovery is made from the nodal displacements;the inclusion of the inconsistent part, as was done earlier, results in thermalstress oscillations.

We demonstrate these concepts using a simple bar element.

8.3.2 The linear bar element - Derivation of stiffness matrix andthermal load vector

Consider a linear bar element of length 2l. The axial displacement u, the totalstrain ε, the initial strain ε0 and stress σ are interpolated as follows:

( ) ( ) 2uu2uuu 1221 −++= ξ (8.9a)

104

Fig. 8.4 Linear bar element.

( ) 2luuu, 12x −==ε (8.9b)

( ) ( ){ }2TT2TT 12210 −++= ξαε (8.9c)

( ) ( ) ( ){ }2TT2TTE2luuE 122112 −++−−= ξασ (8.9d)

where lx=ξ (see Fig.8.4), E is the modulus of elasticity, A the area of

cross-section of the bar and α the coefficient of expansion.

To form the element stiffness and thermal load vector, an integral of theform

� dxTσδε (8.10)

is to be evaluated. This leads to a matrix equation for each element of theform,

( )���

���

=���

���−+

−���

���

��

�

�

−−

2

121

2

1

F

F

1

1

2

TTEA

u

u

11

11

2l

EA α (8.11)

where F1 and F2 are the consistently distributed nodal loads arising from thedistributed external loading. By observing the components of the interpolationfields in Equations (8.9b) to (8.9d) carefully (i.e. constant and linear terms)and tracing the way they participate in the `work' integral in Equation (8.10),it is clear that the ( ) 2TT 12 − term associated with the linear (i.e. ξ) term in

σ (originating from ε0) cannot do work on the constant term in δεT and thereforevanishes from the thermal load vector; see Equation (8.11). Thus the equilibriumequations that result from the assembly of the element equilibrium equationsrepresented by Equation (8.11) will only respond to the consistent part of theinitial strain and will give displacements corresponding to this part only.

If these computed displacements are to be used to recover the initialstrains or thermal stresses, only the consistent part of these fields should beused. The use of the original initial strain or stress fields will result inoscillations corresponding to the inconsistent part. We shall work these out by

105

Fig. 8.5 Clamped bar subject to varying temperature

hand using a simple example below and compare it with the analytical solution.

8.3.3 Example problem

Figure 8.5 shows a bar of length L=4l clamped at both ends and subjected to avarying temperature field. We shall consider a case where two conventionallyderived linear elements are used, so that we require the nodal temperatures T1,

T2 and T3 as input. The nodal reactions are F1 and F3 (corresponding to the

clamped conditions u1=u3=0). We have the assembled equations as,

( )( )( )��

��

�

��

��

�

+−+−

+��

��

�

��

��

�

=��

��

�

��

��

�

���

���

�

−−−

−

32

31

21

3

1

2

TT

TT

TT

2

EA

F

0

F

0

u

0

11

121

11

2l

EA α(8.12)

From this, we can compute the displacements and nodal reactions as,

( ) 2TTlu 312 −= α and ( ) 4T2TTEAFF 32131 ++=−= α (8.13)

and these are the correct answers one can expect with such an idealization. Ifthe stress in the bar is computed from the nodal reactions, one gets a constantstress ( ) 4T2TTE 321 ++−= ασ , in both elements, which is again the correct answer

one can expect for this idealization. It is a very trivial exercise to showanalytically that a problem in which the temperature varies linearly from 0 to Tat both ends will give a constant stress field 2TEασ −= , which the above model

recovers exactly.

Problems however appear when the nodal displacement computed fromEquations (8.12) is used in Equations (8.9b) to (8.9d) to compute the initialstrains and stresses in each element. We would obtain now, the stresses as(subscripts 12 and 23 denote the two elements)

( ) ( ) 2TTE4T2TTE 1232112 −−++−= αξασ (8.14a)

( ) ( ) 2TTE4T2TTE 2332123 −−++−= αξασ (8.14b)

It is now very clear that a linear oscillation is introduced into each elementand this corresponds to that inconsistent part of the initial strain or stressinterpolation which was not sensed by the total strain term. This offers us astraightforward definition of what consistency is in this problem - retain onlythat part of the stress field that will do work on the strain term in the

106

functional. To see how this part can be derived in a variationally correctmanner, we must proceed to the Hu-Washizu theorem.

8.3.4 A note on the quadratic bar element

We may note now that if a quadratic bar element had been the basis for thefinite element idealization, the total strains would have been interpolated to alinear order; the initial strain and thermal stress field will now have aquadratic representation (provided the temperature field has a quadratic

variation) and the inconsistency will now be of the ( )231 ξ− type; thus thermal

stresses derived using a formal theoretical basis will show these quadraticoscillations which will vanish to give correct answers at the points

corresponding to 31±=ξ ; i.e. the points corresponding to the 2-point Gauss

integration rule.

8.3.5 Re-constitution of the thermal strain/stress field using the Hu-Washizu principle

We now seek to find a variational basis for the use of the re-constitutedconsistent initial strain and stress interpolations in the Hu-Washizu principle.

The minimum total potential principle states the present problem as, findthe minimum of the functional,

{ }� += dVP2mT

MTP εσΠ (8.15)

where σ and εm are as defined in (8.5) to (8.7) and the displacement and strainfields are interpolated from the nodal displacements using the element shapefunctions and their derivatives and P is the potential energy of the prescribedloads.

We now know that the discretized stress field thus derived, σ, is

inconsistent to the extent that the initial strain field ε0 is not of the same

order as the total strain field ε. It is therefore necessary to reconstitute thediscretized stress field into a consistent stress field σ without violating anyvariational norm. In the examples above, we had seen a simple way in which thiswas effected.

To see how we progress from the inconsistent discretized domain (i.e.involving ε0 and σ) to the consistent discretized domain (i.e. introducing 0εand σ , it is again convenient to develop the theory from the generalized Hu-Washizu mixed theorem. We shall present the Hu-Washizu theorem from the point ofview of the need to re-constitute the inconsistent ε0 to a consistent 0ε without

violating any variational norms. We proceed thus:

Let the continuum linear elastic problem have a discretized solution basedon the minimum total potential principle described by the displacement field u,strain field ε and stress field σ (we project that the strain field ε is derivedfrom the displacement field through the strain-displacement gradient operators

107

of the theory of elasticity and that the stress field σ is derived from thestrain field ε through the constitutive laws as shown in (8.2). Let us nowreplace the discretized domain by another discretized domain corresponding tothe application of the Hu-Washizu principle and describe the computed state tobe defined by the quantities ε, 0ε and σ , where again, we take that the stress

fields σ are computed using the constitutive relationships, i.e. ( )0D εεσ −= .

It is clear that 0ε is an approximation of the strain field ε0. Note that we

also argue that we can use εε = as there is no need to introduce such adistinction here (in a constrained media elasticity problem it is paramount thatε be derived as the consistent substitute for ε.)

What the Hu-Washizu theorem does, following the interpretation given by deVeubeke, is to introduce a "dislocation potential" to augment the usual totalpotential. This dislocation potential is based on a third independent stress

field σ which can be considered to be the Lagrange multiplier removing the lack

of compatibility appearing between the kinematically derived strain field ε0 and

the independent strain field 0ε . The three-field Hu-Washizu theorem can be

stated as,

0HW =Πδ (8.16)

where

( )� ���

��� +−+= dVP2 mm

Tm

THW εεσεσΠ (8.17)

where ( )0m εεε −= . At this stage we do not know what σ or 0ε are except that

they are to be of consistent order with ε.

In the simpler minimum total potential principle, which is the basis forthe derivation of the displacement type finite element formulation in mosttextbooks, only one field (i.e. the displacement field u), is subject tovariation. However, in this more general three field approach, all three fieldsare subjected to variation and leads to three sets of equations which can begrouped and classified as follows:

Variation on Nature Equation,

u Equilibrium +∇ σ terms from P=0 (8.18a)

σ Orthogonality ( )� =− 0dV00T εεσδ (8.18b)

(Compatibility)

0ε Orthogonality � =���

��� − 0dV

T

0 σσεδ (8.18c)

(Equilibrium)

108

Fig. 8.6 (a) Clamped bar under linear temperature variation and its (b) Barelement model, (c) Plane stress model.

Let us first examine the orthogonality condition in (8.18c). We caninterpret this as a variational condition to restore the equilibrium imbalance

between σ and σ . In this instance this condition reduces to σσ = . Note thatin a problem where the rigidity modulus D can vary significantly over the

element volume, this condition allows σ to be reconstituted from σ in aconsistent way.

The orthogonality condition in (8.18b) is now very easily interpreted.

Since we have σσ = consistent with ε, this condition shows us how to smooth 0εto 0ε to maintain the same level of consistency as ε. This equation therefore

provides the variationally correct rule or procedure to determine consistentthermal strains and stresses from the inconsistent definitions.

Therefore, we now see how the consistent initial strain field 0ε can be

derived from the inconsistent initial strain field 0ε without violating any

variational norm. We thus have a variationally correct procedure for re-constituting the consistent initial strain field - for the bar element above,this can be very trivially done by using an expansion of the strain fields interms of the orthogonal Legendre polynomials, as done in the previous sectionfor the consistent stress-resultants.

109

8.3.6 Numerical examples

In this section we take up the same example of a bar with fixed ends subjectedto linearly varying temperature along its axis and model it with the bar elementand a plane stress element for demonstrating the extraneous stress oscillationsresulting from the lack of consistency in initial strain definition.

A bar of length 8 units, depth 2 units subjected to a linear temperaturedistribution as shown in Fig 8.6a is modeled first with two bar elements (seeFig 4. 18.6b). Let BAR.0 and BAR.1 represent the bar element versions withinconsistent and consistent thermal stress fields. As already predicted, bothgive accurate displacements (-0.002 units at the mid point), see Equation(9.13)); BAR.1 gives exact stress throughout while BAR.0 shows linearoscillations (see Equations (8.14a) and (8.14b)) as shown in Fig 8.7.

Fig.8.7 Consistent and inconsistent thermal stress recovery for a clamped barproblem.

110

8.3.7 Concluding remarks

Before we close this section, it will be worthwhile to discuss the "correctness"of the various approaches from the variational point of view. Note that Equation(9.2) and (9.3) would have been fulfilled if an assumed stress-resultant fieldof the same order as N had been used in place of the N of consistent order.This would result in the Hu-Washizu formulation yielding the same stresspredictions as the minimum potential principle. Thus, both N and N are equally"correct" in the Hu-Washizu sense, but only N is "correct" with the minimumpotential energy formulation. Where they differ is in the consistency aspect -i.e. N is consistent with ε whereas N isn't. It follows that from thetheoretical point of view of the variational or virtual work methods, thepotential energy formulation and the Hu-Washizu formulation describe thecontinuum physics in exactly the same way. Both the potential energy and Hu-Washizu formulations are equally valid as far as variational correctness isconcerned. However, when approximations are introduced in the finite elementmethod, the Hu-Washizu formulation gives decidedly better results than thepotential energy formulation because of the flexibility it provides to modifystress fields and strain fields to satisfy the consistency requirements insituations where they play a prominent role. Note that the consistency paradigm,in this case that used to justify the truncation of N (obtained directly fromthe displacement approach) to a consistent N on the argument that the truncatedterms are not sensed in the discretized strain energy computations, lies clearlyoutside the variational correctness paradigm. Similarly, in the earlier chaptersof this book, we saw another variation of the consistency paradigm - the need toensure a proper balance in the discretized representation of constrained strainfields, which again lies outside the scope of the variational correctnessparadigm. It is very important to understand therefore that both consistency andcorrectness paradigms are mutually exclusive but are needed together to ensurethat the finite element formulations are correctly and consistently done.

The stress field-consistency paradigm introduced here also applies tofinite element applications where certain terms in either the stress or strainfields do not participate in the displacement recovery due to their inconsistentrepresentation e.g. initial stress/strain problems, problems with varyingmaterial properties within an element etc. In the next section, we shall extendthis paradigm to extract consistent thermal stress and/or strains from adisplacement type formulation.

In this section, we have demonstrated another variation of the consistencyparadigm - in applications to evaluation of stresses and stress resultants infinite element thermal stress computations. It is shown that such stresses mustbe computed in a consistent way, recognizing that the displacement type finiteelement procedure can sense only stresses which have the same consistency as thetotal strain field interpolations used within each element. Thus, in an elementwhere the temperature field interpolations do not have the same consistency asthe strain fields, it is necessary to derive a consistent initial strain-fieldfor purposes of recovery of stresses from the computed displacement field. Asimple and variationally correct way to do this has been established from theHu-Washizu theorem.

111

Chapter 9

Conclusion

9.1 Introduction

Technology and Science go hand in hand now, each fertilizing the other andproviding an explosive synergy. This was not so in the beginning; as there wasTechnology much before there was Science. In fact, there was technology beforethere were humans and it was technology which enabled the human race to evolve;science came later. The FEM is also an excellent example of a body of knowledgethat began as Art and Technology; Science was identified more slowly. Thischapter will therefore sum up the ideas that constitute the science of thefinite element method and also point to the future course of the technology.

9.2 The C-concepts: An epistemological summing-up

One would ideally like to find a set of cardinal or first principles that governthe entire finite element discretisation procedure, ensuring the quality ofaccuracy and efficiency of the method. At one time, it was believed that tworules, namely continuity and completeness, provided a necessary and sufficientbasis for the choice of trial functions to initiate the discretisation process.Chapter 3 of this book reviewed the understanding on these two aspects. However,finite element practitioners were to be confronted with a class of problemswhere the strict implementation of the continuity condition led to conflictswhere multiple strain fields were required. This was the locking problem whichwe examined carefully in chapters 5 to 7. Let us recapitulate our fresh insightin an epistemological framework below.

9.2.1 Continuity conflict and the consistency paradigm

We saw in chapters 5 to 7 that in a class of problems where some strain fieldsneed to be constrained, the implementation of the continuity rules on thedisplacement trial functions blindly led to a conflict whereby excessiveconstraining of some strain terms caused a pathological condition calledlocking.

Let us again take up the simple problem of a Timoshenko beam element. Twostrain energy components are important; the bending energy which is constitutedfrom the bending strain as x,θκ = and the shear strain energy which is based on

the shear strain xw,−= θγ . The simplistic understanding that prevailed for a

long time was that the existence of the x,θ and xw, terms in the energy

expression demanded that the trial functions selected for the θ and w fields

must be C0 continuous; i.e. continuity of θ and w at nodes (or across edges in a2D problem, etc.) was necessary and sufficient.

However, this led to locking situations. To resolve this conflict, itbecame necessary to treat the two energy components on separate terms. The shearstrain energy was constrained in the regime of thin beam bending to becomevanishingly small. This required that the shear strains must vanish without

112

producing spurious constraints. The idea of field consistent formulation was toanticipate these conditions on the constrained strain field. Thus for xw,−θ to

vanish without producing gratuitous constraints, the trial functions used for θand w in the shear strain field alone must be complete to exactly the same order- this is the consistency paradigm. We can also interpret this as a multiplecontinuity requirement. Note that although the bending strain requires θ to be

C0 continuous, in the shear strain component, we require θ not to have C0

continuity, i.e. say a C-1 continuity, is. Thus, such a conflict is resolved byoperating with consistently represented constrained strain fields.

9.2.2 The correctness requirement

The consistency requirement now demands that when trial functions are chosen fordisplacement fields, these fields (such as θ in xw,−= θγ ) which need to be

constrained must have a reconstituted consistent definition. The minimum totalpotential principle, being a single field variational principle, does notprovide a clue as to how this reconstitution of the strain fields can beperformed without violating the variational rules.

In the course of this book, we have come to understand that a multi-fieldvariational principle like the generalized Hu-Washizu theorem provided exactlythat flexibility to resolve the conflicting demands between consistency andcontinuity. This, we called the correctness requirement - that the re-constitution of the strain field which needed a reduced-continuity satisfyingdisplacement field from an inconsistent strain field kinematically derived fromthe original continuous displacement fields must be done strictly according tothe orthogonality condition arising from the HW theorem.

9.2.3 The correspondence principle

The study so far with consistency and correctness showed the primacy of the Hu-Washizu theorem in explaining how the finite element method worked. It alsobecame very apparent that if the internal working of the discretisationprocedure is examined more carefully, it turns out that it is strain and stressfields which are being approximated in a "best-fit" sense and not thedisplacement fields as was thought earlier. This brings us to the stresscorrespondence paradigm, and the discovery that this paradigm can be axiomatisedfrom the Hu-Washizu theorem, as shown in chapter 2.

9.3 How do we learn? How must we teach? In a chronological sense or in alogical sense?

In an epistemological sense, often, it is phenomena which first confronts us andis systematically recognized and classified. The underlying first principlesthat explain the phenomena are identified very much later, if not at all. Theclear understanding of first principles can also lead to the unraveling of newphenomena that was often overlooked.

Thus, if and when the first principles are available, then the logicalsequence in which one teaches a subject will be to start with first principlesand derive the various facets of the phenomena from these principles. Inpedagogical practice, we can therefore have a choice from two courses of action:

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teach in a chronological sense, explaining how the ideas unfolded themselves tous, or teach in the logical sense, from first principles to observable facts. Itis not clear that one is superior to the other.

In my first book [9.1], I choose the chronological order. The conflictscreated by the locking phenomena were resolved by inventing the consistencyparadigm. This led to the correctness principle and the recognition that the Hu-Washizu principle formed a coherent basis for understanding the fem procedure.From these, one could axiomatise the correspondence rule - that the fem approachmanipulates stresses and strains directly in a best-approximation sense. Thus,chapter 12 of Reference 9.1 sums up the first principles.

In the present book, I have preferred a logical sequence - the HW andcorrespondence rules are taken up first and then the various phenomena likelocking, etc.

9.4 Finite element technology: Motivations and future needs

Advances in computer science and technology have had a profound influence onstructural engineering, leading to the emergence of this new discipline we callcomputational structural mechanics (CSM). Along with it a huge software industryhas grown. CSM has brought together ideas and practices from several disciplines- solid and structural mechanics, functional analysis, numerical analysis,computer science, and approximation theory.

CSM has virtually grown out of the finite element method (FEM). Algorithmsfor the use of the finite element method in a wide variety of structural andthermomechanical applications are now incorporated in powerful general purposesoftware packages. The use of these packages in the Computer-Aided-Design/Computer-Aided-Manufacturing cycle forms a key element in newmanufacturing technologies such as the Flexible Manufacturing Systems. Theseallow for unprecedented opportunities for increase in productivity and qualityof engineering by automating the use of structural analysis techniques to checkdesigns quickly for safety, integrity, reliability and economy. Very largestructural calculations can be performed to account for complex geometry,loading history and material behavior. Such calculations are now routinelyperformed in aerospace, automotive, civil engineering, mechanical engineering,oil and nuclear industries.

Modern software packages, called general purpose programmes, couple FEMsoftware with powerful graphics software and the complete cycle of operationsinvolving pre-processing (automatic description of geometry and subsequent sub-division of the structure) and post-processing (projecting derived informationfrom FEM analysis on to the geometry for color coded displays to simplifyinterpretation and make decision making that much easier). Already artificialintelligence in the form of knowledge based expert systems and expert advisersand optimization procedures are being coupled to FEM packages to reduce humanintervention in structural design to a bare minimum.

It is not difficult to delineate many compelling reasons for the vigorousdevelopment of CSM [9.2]. These are:

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1. There are a large number of unsolved practical problems of current interestwhich still await experimental and/or numerical solutions. Some of thesedemand large computational power. Some of the examples described in reference9.2 are: simulation of response of transportation vehicles tomultidirectional crash impact forces, dynamics of large flexible structurestaking into account joint nonlinearities and nonproportional damping, studyof thermoviscoelatic response of structural components used in advancedpropulsion systems etc. In many structural problems, the fundamentalmechanics concepts are still being studied (e.g. in metal forming, adequatecharacterization of finite strain inelasticity is still needed).

2. Computer simulation is often required to reduce the dependence on extensiveand expensive testing; in certain mission critical areas in space, computermodeling may have to replace tests. Thus, for large space structures (e.g.large antennas, large solar arrays, the space station), it may not bepossible for ground-test technology in 1-g environment to permit confidenttesting in view of the large size of the structures, their low naturalfrequencies, light weight and the presence of many joints.

3. Emerging and future computer systems are expected to provide enormous powerand potential to solve very large scale structural problems. To realize thispotential fully, it is necessary to develop new formulations, computationalstrategies, and numerical algorithms that exploit the capabilities of thesenew machines (e.g. parallelism, vectorization, artificial intelligence).

Noor and Atluri [9.2] also expect that high-performance structures willdemand the following technical breakthroughs:

1. Expansion of the scope of engineering problems modeled, such as:

a) examination of more complex phenomena (e.g. damage tolerance of structuralcomponents made of new material systems);

b) study of the mechanics of high-performance modern materials, such as metal-matrix composites and high-temperature ceramic composites;

c) study of structure/media interaction phenomena (e.g. hydrodynamic/structuralcoupling in deep sea mining, thermal/control/structural coupling in spaceexploration, material/aerodynamic/structural coupling in composite wingdesign, electromagnetic/thermal/structural coupling in microelectronicdevices);

d) use of stochastic models to account for associated with loads, environment,and material variability

e) development of efficient high-frequency nonlinear dynamic modelingcapabilities (with applications to impulsive loading, high-energy impact,structural penetration, and vehicle crash-worthiness);

f) improved representation of structural details such as damping and flexiblehysteritic joints:

g) development of reliable life-prediction methodology for structures made ofnew materials, such as stochastic mechanisms of fatigue, etc.

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h) analysis and design of intelligent structures with active and/or passiveadaptive control of dynamic deformations, e.g. in flight vehicles, largespace structures, earthquake-resistant structures

i) Computer simulation of manufacturing processes such as solidification,interface mechanics, superplastic forming.

2. Development of practical measures for assessing the reliability of thecomputational models and estimating the errors in the predictions of themajor response quantities.

3. Continued reduction of cost and/or time for obtaining solutions toengineering design/analysis problems.

Special hardware and software requirements must become available to meet theneeds described above. These include:

1. Distributed computing environment having high-performance computers for largescale calculations, a wide range of intelligent engineering workstations forinteractive user interface/control and moderate scale calculations.

2. User-friendly engineering workstations with high-resolution and high speedgraphics, high speed long distance communication etc.

3. Artificial intelligence-based expert systems, incorporating the experienceand expertise of practitioners, to aid in the modeling of the structure, theadaptive refinement of the model and the selection of the appropriatealgorithm and procedure used in the solution.

4. Computerized symbolic manipulation capability to automate analyticcalculations and increase their reliability.

5. Special and general purpose application software systems that have advancedmodeling and analysis capabilities and are easy to learn and use.

9.5 Future directions

It is expected that CSM will continue to grow in importance. Three areas whichare expected to receive increasing attention are: 1) modeling of complexstructures; 2) predata and postdata processing and 3) integration of analysisprograms into CAD/CAM systems.

The accurate analysis of a complex structure requires the proper selectionof mathematical and computational models. There is therefore a need to developautomatic model generation facilities. Complex structures will require anenormous amount of data to be prepared. These can be easily performed by usingpredata and postdata processing packages using high resolution, high throughputgraphic devices. The generation of three dimensional color movies can help tovisualize the dynamic behavior of complex structural systems.

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9.6 Concluding remarks

In this concluding chapter, we have summarized the ideas that should inform therational development of robust finite elements and we have also described thecurrent trends in the use of structural modeling and analysis software inengineering design. It has been seen that CSM has greatly improved ourcapabilities for accurate structural modeling of complex systems. It has thepotential not only to be a tool for research into the basic behavior ofmaterials and structural systems but also as a means for design of engineeringstructures.

9.7 References

9.1 G. Prathap, The Finite Element Method in Structural Mechanics, KluwerAcademic Press, Dordrecht, 1993.

9.2 A. K. Noor and S. N. Atluri, Advances and trends in computational structuralmechanics, AIAA J, 25, 977-995 (1987).