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Modeling of frame structures undergoing largedeformations and large rotationsHui LiuPurdue University

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PURDUE UNIVERSITYGRADUATE SCHOOL

Thesis/Dissertation Acceptance

This is to certify that the thesis/dissertation prepared

By

Entitled

For the degree of

Is approved by the final examining committee:

To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material.

Approved by Major Professor(s):

Approved by:Head of the Departmental Graduate Program Date

Hui Liu

Modeling of frame structures undergoing large deformations and large rotations

Doctor of Philosophy

Arun PrakashChair

Shirley J. Dyke

Chin-Teh Sun

Michael E. Kreger

Arun Prakash

Rao Govindaraju 6/10/2016

MODELING OF FRAME STRUCTURES UNDERGOING LARGE

DEFORMATIONS AND LARGE ROTATIONS

A Dissertation

Submitted to the Faculty

of

Purdue University

by

Hui Liu

In Partial Fulfillment of the

Requirements for the Degree

of

Doctor of Philosophy

August 2016

Purdue University

West Lafayette, Indiana

ii

To my family and friends

for their love, support and encouragement

iii

ACKNOWLEDGMENTS

First of all, I would like to thank my advisor Professor Arun Prakash who has

been patiently guiding and enlightening me in studying computational mechanics.

He has been teaching me not only the knowledge and skills, but also how to become

a professional researcher.

I would also like to thank my examination committee members, Professor Shirley

Dyke, Professor Michael E. Kreger and Professor Chin-Teh Sun for their valuable

comments and suggestions on my research.

Special thanks to my colleagues from the CSSML research group with whom I have

studied computational mechanics and conducted research. I thank Hansen Pitandi,

Xiaowo Wang, Gregory Bunting, Payton E. Lindsey, Boyuan Liu and Joselito Wong

Yau for the helpful discussions. It has been a pleasant experience working with them.

Last but not least, I would like to thank my family for all the care and support

without which I could not have come this far.

iv

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Review of Existing Methods . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Simplified models for non-linear problems . . . . . . . . . . . . . . . 4

2.2 Panel zone models . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Geometrically exact beams for large deformations . . . . . . . . . . 7

2.4 Coupled models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Coupling using transition elements . . . . . . . . . . . . . . 8

2.4.2 Multi-point coupling . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Multi-time-step Methods . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Reduced-order Models for frame structures . . . . . . . . . . . . . . . . . 12

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Existing Approaches for Numerical Simulation . . . . . . . . 12

3.1.2 Beam-frame Elements for Large Deformations and Rotations 13

3.1.3 Simplified Models Based on Enriched Macro-Elements . . . . 14

3.2 Formulation of Reduced-order Spring-based Frame element . . . . . 16

3.2.1 Kinematics of Deformation and Internal Forces and Moments 17

3.2.2 Global Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.3 Solution by Consistent Linearization . . . . . . . . . . . . . 24

3.2.3.1 Linearization of Kinematic Quantities . . . . . . . 26

3.2.3.2 Linearization of Internal Forces . . . . . . . . . . . 29

3.2.4 Identification of Spring Stiffnesses and Material Behavior . . 30

v

Page

3.2.5 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Static Simulation Examples . . . . . . . . . . . . . . . . . . 33

3.3.1.1 Buckling of a Right-angle Frame in 2D . . . . . . . 34

3.3.1.2 Buckling of a 3D Frame . . . . . . . . . . . . . . . 36

3.3.2 Two-story Frame Structure Subject to Earthquake Loading . 37

3.3.3 Error Measures . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Geometrically Consistent Coupling of Beam and Continuum Elements . . 46

4.1 Large deformation formulation for continua . . . . . . . . . . . . . . 47

4.1.1 Verification of 3D continuum model . . . . . . . . . . . . . . 50

4.2 3D non-linear beam theory . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.2 Balance laws & equations of motion . . . . . . . . . . . . . . 53

4.2.3 Admissible variations . . . . . . . . . . . . . . . . . . . . . . 54

4.2.4 Weak form and linearization . . . . . . . . . . . . . . . . . . 55

4.2.5 Discretization and constitutive model . . . . . . . . . . . . . 57

4.2.6 Verification of 3D beam models . . . . . . . . . . . . . . . . 58

4.3 Coupling of Beam and Continuum Models-Static . . . . . . . . . . . 58

4.3.1 Lagrange-multiplier-based coupling . . . . . . . . . . . . . . 60

4.3.2 Virtual work functional . . . . . . . . . . . . . . . . . . . . . 60

4.3.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Special case: Geometrically consistent coupling for planar problems 67

4.4.1 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 69

4.4.1.1 Elastic buckling of a right-angle frame . . . . . . . 69

4.4.2 Pushover analysis of a portal frame with plasticity . . . . . . 71

vi

Page

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Dynamics of Coupled Beam and Continuum Models . . . . . . . . . . . . 76

5.1 Dynamics of continuum models . . . . . . . . . . . . . . . . . . . . 76

5.1.1 Midpoint time integration . . . . . . . . . . . . . . . . . . . 77

5.2 Dynamics of 3D geometrically exact beams . . . . . . . . . . . . . . 78

5.2.1 Exact energy and momentum conserving algorithms . . . . . 78

5.2.2 Verification of dynamics of 3D geometrically exact beam . . 81

5.2.2.1 Free flying beam . . . . . . . . . . . . . . . . . . . 81

5.2.2.2 Circular beam . . . . . . . . . . . . . . . . . . . . 82

5.3 Dynamic coupling of beam and continuum models . . . . . . . . . . 84

5.3.1 Coupling based on continuity of displacements . . . . . . . . 84

5.3.2 Coupling based on continuity of velocity . . . . . . . . . . . 86

5.3.3 Special case: Dynamic coupling for 2D problems . . . . . . . 89

5.3.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . 91

5.4 Multi-time-step method for dynamics of coupled models . . . . . . . 91

5.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 96

5.4.2.1 Split-SDOF coupling . . . . . . . . . . . . . . . . . 96

5.4.2.2 3D Cantilever beam with Mid-point method . . . . 97

5.4.2.3 2D Cantilever beam with Newmark method . . . . 99

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 105

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

vii

LIST OF TABLES

Table Page

3.1 Average global errors ε× 100 (%) for the two-story frame model . . . . 43

viii

LIST OF FIGURES

Figure Page

1.1 A severely damaged school building that is impending collapse after earth-quake in Haiti, 2010 (cf. [1]) . . . . . . . . . . . . . . . . . . . . . . . . 1

2.1 Plastic hinges at beam-column joint . . . . . . . . . . . . . . . . . . . . 6

2.2 Five deformation modes at joints . . . . . . . . . . . . . . . . . . . . . 6

2.3 Coupling of different time steps with a ratio of m between them. . . . . 11

3.1 Modeling a frame structure with reduced-order spring-based elements . 17

3.2 Kinematics of spring-based frame elements . . . . . . . . . . . . . . . 18

3.3 Internal forces within a typical extensional spring element m . . . . . 19

3.4 Original and deformed positions, angles and unit vectors associated withelement m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Internal moment in the rotational spring represented as a force coupleacting at the end nodes of element m . . . . . . . . . . . . . . . . . . . 22

3.6 Cantilever beam subjected to (a) Axial load, (b) Transverse load . . . . 32

3.7 Load vs. displacement of the free-end of the cantilever beam subject toaxial load and transverse loads . . . . . . . . . . . . . . . . . . . . . . 33

3.8 Geometry, loading and collapse sequence of a 2D right angle frame . . . 34

3.9 Load vs. displacement for point C of the 2D right-angle frame . . . . . 35

3.10 Geometry, loading and collapse sequence of a 3D frame . . . . . . . . . 36

3.11 Load vs. displacement for point D of the 3D frame . . . . . . . . . . . 37

3.12 Two-story frame subjected to earthquake ground motion . . . . . . . . 38

3.13 Comparison of natural frequencies . . . . . . . . . . . . . . . . . . . . . 38

3.14 Ground acceleration and displacement used for numerical study . . . . 39

3.15 Displacement, velocity and acceleration of point A in the x-direction forthe two-story frame subjected to earthquake load . . . . . . . . . . . . 39

3.16 Distribution of average local errors εi in x-displacement . . . . . . . . . 41

ix

Figure Page

3.17 Time history of instantaneous global error εj in x-displacement . . . . . 42

4.1 An example of the decomposition of a structural model into beam andcontinuum regions (c.f. Pitandi [74]). . . . . . . . . . . . . . . . . . . . 46

4.2 The configurations of a body before and after deformation . . . . . . . 48

4.3 Patch test on a 3D cube . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Plots of stresses from patch test . . . . . . . . . . . . . . . . . . . . . . 51

4.5 Kinematic description of 3D beam, orthogonal moving frame . . . . . . 52

4.6 Stress resultants on a cross section of beam . . . . . . . . . . . . . . . 53

4.7 Illustration of exponential map . . . . . . . . . . . . . . . . . . . . . . 56

4.8 Cantilever beam in 3D subjected to pure bending . . . . . . . . . . . . 58

4.9 Beam-continuum coupled model in 3D . . . . . . . . . . . . . . . . . . 59

4.10 Geometric properties of curve shaped cantilever beam . . . . . . . . . . 67

4.11 Deformation sequence of bathe-bolourchi beam modeled with (a)Pure beammodel, (b)Pure continuum model and (c)Beam-continuum coupled model 67

4.12 Load vs displacement plot of bathebolourchi beam at free end. . . . . . 68

4.13 Hinged frame collapse sequences showing (a) S11, (b) S22 and (c) S12stresses normalized to the maximum value. . . . . . . . . . . . . . . . 70

4.14 Laod vs. Displacements of a point under applied load. . . . . . . . . . 71

4.15 Portal frame collapse sequences using coupled model with stresses S11,S22 and S12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.16 Load vs. displacements of the point of applied load. . . . . . . . . . . . 73

5.1 Wave propagation inside a cantilever beam . . . . . . . . . . . . . . . . 78

5.2 Geometry of initially straight beam undergo large overall motion . . . . 82

5.3 Deformation sequence of initially straight beam and time history of totalenergy plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Geometry of initially circular beam subjected to two point loads . . . . 83

5.5 Deformation sequence of initially circular beam and time history of totalenergy plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.6 Cantilever beam simulated using (a) 20-element beam model, (b) 80-element continuum model and (c) coupled model . . . . . . . . . . . . 92

x

Figure Page

5.7 Time history plot of (a) displacements, (b) velocities and (c) accelerationsat the free end of the cantilever beam . . . . . . . . . . . . . . . . . . 93

5.8 Coupling of different time steps with a ratio of m between them. . . . . 94

5.9 Split SDOF problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.10 Time history plot of (a) displacements, (b) velocities ,(c) accelerations and(d) Lambda for STS and MTS method . . . . . . . . . . . . . . . . . . 98

5.11 3D cantilever beam subjected to a point load simulated using beam-continuum coupled model with MTS method. . . . . . . . . . . . . . . 99

5.12 Time history plot of (a) displacements, (b) velocities for 3D cantileverbeam using STS and MTS methods . . . . . . . . . . . . . . . . . . . 100

5.13 2D cantilever beam subjected to a point load. . . . . . . . . . . . . . . 101

5.14 Time history plot of displacements, velocity and acceleration of cantileverbeam at the interface simulated using STS and MTS with time-step ratio2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.15 Time history plot of displacements, velocity and acceleration of cantileverbeam at the free end simulated using STS and MTS with time-step ratio2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xi

ABSTRACT

Liu, Hui Ph.D., Purdue University, August 2016. Modeling of Frame Structures Un-dergoing Large Deformations and Large Rotations. Major Professor: Arun Prakash.

Numerical simulation of large-scale problems in structural dynamics, such as struc-

tures subject to extreme loads, can provide useful insights into structural behavior

while minimizing the need for expensive experimental testing for the same. These

types of problems are highly non-linear and usually involve material damage, large

deformations and sometimes even collapse of structures. Conventionally, frame struc-

tures have been modeled using beam-frame finite elements in almost all structural

analysis software currently being used by researchers and the industry. However,

there are certain limitations associated with this modeling approach. This research

focuses on two issues, in particular, of modeling frame structures undergoing large

deformations and rotations when subject to extreme loads such as high intensity

earthquakes.

One of the issues with using beam-frame models is that the theoretical formula-

tion and numerical implementation of such models are rather complicated and are

not well understood by the average engineer using such computer programs. The

complications arise primarily due the non-additive nature of three dimensional rota-

tional degrees of freedom, especially under large rotations. Further, ensuring that the

time integration schemes used for such models provide stable and accurate solutions

is still an active and challenging area of research. To address this issue, a reduced-

order model that idealizes a frame structure as a network of rotational and extensional

springs is developed. This formulation eliminates all the rotational degrees of freedom

in the system by expressing the force-displacement and moment-rotation relationships

only in terms of nodal coordinates. This not only simplifies the formulation, making

xii

it similar in complexity to a network of truss elements, but also avoids the numerous

implementational hurdles associated with large three dimensional rotations. Several

numerical examples are presented to verify and validate the performance of this ap-

proach against conventional beam-frame elements.

Existing models that attempt to capture the non-linear behavior of structures

undergoing large deformations and damage, which often occurs across multiple scales

of space and time, are either limited in the level of fidelity they offer or have an

extremely high computational cost associated with them. A computationally advan-

tageous way of approaching such problems is to decompose the structural domain into

two regions, one comprising most structural elements where beam-frame elements can

be used, and the other consisting of joint and connection regions where more detailed

continuum elements can be used as needed. This allows one to model the critical

structural components with great fidelity, while still using beam elements for the rest

of the model to keep the total computational cost in check. An essential ingredient

for this approach is the formulation of a geometrically consistent coupling of beams

and continuum elements, especially in the presence of large deformations and large

rotations. In addition to spatial coupling of beam and continuum elements, a multi-

time-step method is also formulated to allow the beam and continuum elements to be

simulated at different time scales. This further adds to the computational efficiency

of this approach. Numerical characteristics of such coupled models are verified with

a variety of static and dynamic benchmark problems.

1

1. INTRODUCTION

Structures are usually designed for different combinations of static loading scenar-

ios and are rather vulnerable to high-intensity dynamic loads such as strong earth-

quakes, impact or blast. Under strong dynamic loads, structures suffer severe damage

resulting in the failure of key structural components and in extreme cases that may

lead to collapse. One of such extreme cases is shown in Figure 1.1. It is clear from

such cases that structures subject to extreme events undergo significant local damage

and deformation especially at the locations of joints between structural members.

Fig. 1.1.: A severely damaged school building that is impending collapse after earth-quake in Haiti, 2010 (cf. [1])

In situations of impending collapse, conventional tools of structural analysis that

are usually based on assumptions of linear response and small deformations are in-

adequate to model the structural behavior. For instance these tools cannot be used

to determine whether a structure would sustain total, partial or no collapse under

heavy damage and/or large deformations. In the past, structural engineers have used

a variety of approximations such as enriching conventional models consisting of beam

2

elements with plastic hinges at the joints to model damage and have had some suc-

cess with these models in modeling and predicting structural behavior beyond the

linear regime. Another widely used approach to model joint behavior is the use of

panel zone models which usually are usually comprise an assembly of springs-type

elements. However, most of these models also have limited applicability and are not

able to capture the behavior of the joints in sufficient detail especially under large

deformations and damage. Another approach that has been adopted in the existing

literature is to use simplified equivalent models that are representative of the struc-

ture under consideration (see Sopanen and Mikkola [3]). This idea has been widely

used in modeling flexible multi-body systems and there are many existing simplified

models that use different types of elements such as rigid links sliders and springs to

model such systems. However, a majority of these models are limited to two dimen-

sions and some of the approaches are computationally complex to implement. Further

discussion on existing methods for modeling of structures is presented in Chapter 2.

Numerical simulation of structures using advanced computational tools that are

capable of modeling the behavior of structural components including local damage

and deformation more realistically can provide insight into the mechanisms leading

to collapse. For instance, it is possible to construct very detailed finite element mod-

els of an entire structure where the response of every component can be captured in

great detail ranging from evolution of local material damage to fracture and failure of

major structural components. Advances in constitutive models, element technology,

solution methods and modeling capabilities have made the construction of such de-

tailed models possible, at least in theory. However, the computational cost associated

with such models is usually so large that it renders them infeasible for practical ap-

plications. Further, the development of such detailed models has not reached a level

of maturity that they can be used reliably by practicing engineers.

Computationally efficient models and methods and needed to reduce the cost of

using detailed realistic models. One popular approach is to use spatial domain de-

composition to model only the critical regions of a structure using a refined model

3

and to couple these refined models with a coarser model for the rest of the structure.

In addition to spatial domain decomposition, resolution of different temporal scales

can also lead to gains in computational efficiency for large-scale dynamic problems.

With efficient multi-time-step coupling methods (see Prakash and Hjelmstad [4]), a

large-scale structure can be decomposed into smaller sub-domains which are solved

independently with different time steps that are appropriate for modeling the indi-

vidual dynamics of these sub-domains and then the solutions from these sub-domains

are coupled back together to obtain the global solution for the entire structure. A

multi-scale model that is capable of modeling across multiple scales in both space

and time can put the modeling of collapse within reach for practicing engineers. Key

issues of concern with such models are having consistency in the coupling of multiple

scales and stability and accuracy of the resulting solution.

In this dissertation, formulations, implementation and results from a set of com-

putational tools that enable efficient numerical modeling of problems in solid and

structural mechanics involving large deformations, large rotations, and problems that

may contain multiple spatial and temporal scales, is presented. First a review of

relevant current literature is presented in Chapter 2. Theoretical formulations, im-

plementation of its numerical solution, and results from a reduced-order model that

utilizes rotational and extensional springs to model frame structures is presented in

Chapter 3. Chapter 4 presents a geometrically consistent spatial coupling of beam

and continuum elements for static problems along with several numerical examples

to verify the performance of such coupled models in comparison to conventional ap-

proaches. In addition to spatial coupling, a multi-time-step method is presented in

Chapter 5 that allows the beam-frame elements and continuum element subdomains

to be integrated with different time-steps to enable a further resolution in time of the

critical regions within a structure. Finally, a summary and conclusions drawn from

the study are noted in Chapter 6.

4

2. REVIEW OF EXISTING METHODS

This chapter summarizes some of the approaches that have been commonly adopted

in the literature for modeling non-linear behavior of frame structures. Simplified

models, such as those based on spring elements and panel-zone models, are briefly

discussed first. An introduction to some research on the geometrically exact theory

of beams that is able to capture the behavior of structural components under large

deformations and large rotations encountered in situations of impending collapse is

given next. Finally, a review of existing beam-continuum coupled models along with

an overview of multi-time-step methods that allow the use of different time-steps in

different regions of a model is presented.

2.1 Simplified models for non-linear problems

In the simulation of large-scale highly non-linear problems, such as building col-

lapse, computational cost could be very high. It is therefore impractical to use a

detailed model for the entire building and different types of simplified models have

emerged (see Bao et al. [5]) as approximations. Mattern et al. [6] used a simplified

multi-rigid-body model to simulate building collapse and compared the results with

a finite element model. The simplified model treats columns as rigid-bodies that are

connected by hinges or springs at the top and bottom.The comparison shows that

the simulated collapse behavior is very similar and the simplified model is much more

computationally efficient. However, the comparison is not quantified and is simply

based on observations from a building collapse simulation.

Numerical models in structural engineering frequently employ beam elements in-

stead of continuum elements for modeling of structural members. However it is worth-

while to point out that, amongst the structural components, joints are most likely

5

to be the weakest link in the building due to the lack of detailing (see Pampanin et

al. [7]). Therefore it is reasonable to expect that large deformations and rotations take

place at the joint regions in extreme loading scenarios (see Popov [8]). Due to this

fact, beam elements may not be able to capture the flexibility of joints and thus tend

to make the structure stiffer than it actually is. Modification to element strength and

stiffness is one approach that is commonly adopted to make beam element models

more realistic. Kaewkulchai et al. [9], for instance, developed a beam-column element

and a solution procedure to take into account the flexibility and damage of joints. The

stiffness degradation is controlled through a damage-dependent constitutive relation-

ship. In cases of extreme loading, a more detailed joint model is required to obtain a

better simulation since localized large deformations and damage are involved. Panel

zone or coupled models are useful in modeling the joint response in such situations.

2.2 Panel zone models

In structural engineering, spring elements are often used for modeling structures

subjected to cyclic loading such as earthquakes (see Linde and Bachmann [10], De la

Llera et al. [11], Jiang and Saiidi [12]). Rotational springs are usually used as plastic

hinges at joints of beams and columns to model the non-linear behavior of joints.

In reinforced concrete structures, this non-linearity can result from yielding of the

reinforcement or from concrete damage, and in steel structures this could be due to

local yielding and buckling at the joint. Marante and Florez-Lopez [13] developed

a model based on lumped damage mechanics and the concept of plastic hinge. The

effects of concrete cracks and reinforcement yielding at joints are represented by an

elastic beam-column and two-plastic-hinge mechanism as shown in Figure 2.1. The

flexibility of the joint is then determined using damage variables and yield functions.

Their model simplifies the analysis by lumping and quantifying damage at the joints.

Only damage related to flexural effects are captured in the model while torsional and

axial influence is neglected.

6

Fig. 2.1.: Plastic hinges at beam-column joint

Kim and Engelhardt [14] developed a panel zone model where a rotational spring is

used to transfer the moment based on the change of angle between beam and column.

The panel zone element is dimensionless and it is used to study the steel moment

frames in monotonic and cyclic loadings. The effects of stiffness contribution from

beams, columns and slabs are presented, however, quantifying these effects precisely

over a range of deformations remains challenging in this model.

Fig. 2.2.: Five deformation modes at joints

Mulas [15] presented a more detailed panel zone model that is able account for

the finite dimension of joints. The joint deformation is determined by adding five

possible deformation modes as illustrated in Figure 2.2. This model is effective in

capturing the primary modes of deformation of the panel zone but more detailed

models are needed to simulate the behavior of panel zones under large deformations

and the resulting damage.

7

2.3 Geometrically exact beams for large deformations

For most structures the onset of non-linear behavior results from material yield-

ing and damage. However, in situations of impending collapse, structures are usually

subject to large deformations and large rotations. Capturing these non-linear geo-

metric effects accurately is essential for determining whether a structure will collapse

or not. This is a crucial performance criterion used in performance-based design - a

design philosophy that is increasingly gaining traction in the structural engineering

community.

In the past few decades, there have been a number of publications on the statics

and dynamics of geometrically exact beams that are able to capture the non-linear

geometric effects mentioned above very well. Simo [16] laid out the formulation and

numerical implementation for finite strain 3D beam theory which is considered as a

generalization of Reissner’s planar beam theory for static problems (see Reissner [17])

to 3D dynamical case as well as a 3D extension of Antman’s Kirchhoff-Love rod

model [18]. Later Simo and Vu-quoc [19, 20] presented the variational formulation

and numerical implementation of the statics and dynamics of 3D finite-strain rods

using the finite element method. Jelenic and Crisfield [21] commented that the update

of 3D rotations based on stored configurations from the last converged step is more

suitable rather than interpolating rotational variables along the length of beam. This

was also shown in the work of Cardona and Geradin [22]. Several subsequent works

have improved the formulation and numerical implementation of geometrically exact

beam elements and also proposed different alternative formulations. In this research,

the formulation geometrically exact beams will be considered.

As mentioned previously, the reduction in dimensionality gained from beam the-

ory leads to a great reduction of the computational cost associated with these models.

However, capturing the material damage and its evolution at the joints and connec-

tions becomes challenging. In order to meet this challenge, researchers have also

8

developed some coupled models that allow one to combine beam elements with con-

tinuum elements to model joint damage better.

2.4 Coupled models

In contrast to simplified models, the recent growth in computing power has also

led some researchers to explore detailed high fidelity models of structures that utilize

continuum finite elements for simulating structural behavior. However, the compu-

tational cost of such detailed models can become infeasible for large structures very

quickly. In order to keep the computational cost down and still avail the benefits of

detailed high-fidelity models, researchers have also developed domain decomposition

and coupling techniques (see Blanco et al. [23], Song [24]). These methods allow

one to divide a large structure into two or more regions, where in large parts of the

structure, reduced-dimension beam-frame elements are used and continuum elements

are employed only in those regions where they are needed for better accuracy. When

coupling these different types of elements, the main difficulty is the incompatibility

across the interface between them. A discussion of the approaches that have been pro-

posed by researchers to overcome this incompatibility is summarized in the following

subsections.

2.4.1 Coupling using transition elements

Transition elements can be used to couple models that have incompatible degrees

of freedom. Many of the transition elements are formulated to couple solid and shell

elements. Surana [25] formulated a 3D solid-shell transition element for small de-

formation linear analysis and later extended the formulation for non-linear analysis.

The proposed transition element is valid for large deformation and large rotation (see

Surana [26]). Gmur and Kauten [27] presented a solid-beam transition element for

dynamics using reduced integration. Admissible shape functions for the transition

element are constructed in the way that satisfy continuity, completeness and smooth-

9

ness conditions within the element. Results from their numerical study have shown

that the transition element is reliable in modeling the region between 3D solid and

beam elements of the structure.

Unfortunately, there are drawbacks with transition elements that limit their ap-

plication. Transition elements need to be specially designed to couple a specific pair

of element-types. In order to couple different types of beam and continuum elements,

a new transition element needs to be derived. Further, the transition element allows

transition from a beam element to one continuum element. Additional refinement

from one element to a refined continuum element region needs additional constraints

and approximations.

2.4.2 Multi-point coupling

Another approach that can be used to couple different types of elements is the

multi-point coupling approach. Monaghan et al. [28] developed a coupling of 1D

beam elements to a 3D body by balancing the work done by the 3D and 1D nodes

at the interface. A similar approach can be found in the work of McCune et al. [29]

where different types of incompatible element coupling are discussed. Consider a 2D

continuum to 1D beam coupling, for instance. The work done at the 2D continuum

and 1D beam are written as:

Πc =

∫A

(σxU + τxyV )dA (2D)

Πb = (Pu+Qv −Mθ) (1D) (2.1)

where σx and τxy denote stress components from the continuum element, U and

V denote the corresponding displacements, P , Q and M denote the axial, shear and

moment resultants for the beam and u, v and θ represent the corresponding kinematic

variables. By equating the two expressions, the relationships between beam and

continuum displacements can be established. These relationships are then used to

derive constraint equations.

10

Song [24], on the other hand, developed a multi-point coupling for beam and

continuum elements by enforcing the displacement and force continuities. The conti-

nuities are enforced by transformation matrices. The advantage of this model is that

it doesn’t introduce additional nodes. However, the assumptions on stress distribu-

tions which is required to obtain the transformation matrix may not always be valid

for non-linear problems.

Other methods to enforce constraints include the penalty method and the La-

grange multiplier method. The penalty method introduces a large penalty parameter

to enforce the constraints. Selection of this parameter needs to be done through trial

and error while for the Lagrange multiplier method, the multiplier is solved for as an

additional unknown.

Blanco et al. [23] proposed a variational framework to couple kinematically in-

compatible models, including 3D solid and 2D shell or Bernoulli 1D beam. A real

parameter γ ∈ [0, 1] is used to manipulate the terms in redefined governing equa-

tion to modify the continuity. Ho et al. [30] developed a multi-point constraint for

beam-shell, beam-solid and shell-solid coupling in a dynamic explicit solver.

All the methods available in the literature to coupe beam and continuum elements

make approximations that affect their performance in terms of stability and accuracy

of the coupling. In this research, a geometrically consistent coupling method will

be described to enable coupling of geometrically exact beams with continuum finite

elements.

2.5 Multi-time-step Methods

In addition to spatial coupling of beam and continuum elements, one must also

consider the different time scales that are associated with these types of elements.

For the continuum elements in critical regions of the structure where likelihood of

damage is high, one needs to use small time steps in conjunction with a refined mesh

of continuum elements to capture the progression of damage accurately. For beam

11

elements in the rest of the structure, relatively large time steps can be used to keep the

computational cost down while still capturing the overall dynamics of the structure.

Fig. 2.3.: Coupling of different time steps with a ratio of m between them.

Belytschko and Mullen [31] developed a basic multi-time-step integration method

where different time steps are used for different sub-domains of a model. Another

approach, known as the mixed method in time, allows three types of coupling algo-

rithms, ‘explicit-explicit’, ‘explicit-implicit’ and ‘implicit-implicit’ algorithms to be

used (see Miranda et al. [32], Hughes and Liu [33]). Gravouil and Combescure [34]

presented a multi-time-step method with domain decomposition which enables arbi-

trary numeric schemes of the Newmark family to be coupled with different time steps.

Prakash and Hjelmstad [4] improved upon this multi-time-step method by eliminat-

ing artificial numerical dissipation introduced by the coupling and also improved the

computational efficiency of the method.

12

3. REDUCED-ORDER MODELS FOR FRAME

STRUCTURES

3.1 Introduction

Structures subject to intense dynamic loading during strong earthquakes and other

extreme events usually exhibit significant deformations and severe damage that can

lead to disastrous consequences (c.f. Fu [35]). Due to concerns about seismic safety,

the study of building structures subjected to earthquakes has become an important

topic and has attracted extensive investigations (Lu et al. [36], Takewaki et al. [37]).

In the past few decades, various experimental studies such as shake table tests (Ji et

al. [38]) on full-scale structures (Van de Lindt et al. [39]) or down-scaled structures

(Li et al. [40]) have been conducted to study seismic response of different types of

structures. While experimental investigations can provide a lot of data and infor-

mation about the behavior of structural components and systems, they are usually

labor intensive and not cost-effective. Numerical simulations, on the other hand, are

relatively inexpensive and yet they can provide detailed insight into the behavior ob-

served in an experimental test or practice. However, numerical simulations are useful

only when the computational models being used have been verified and validated

against benchmark problems and real experimental data.

3.1.1 Existing Approaches for Numerical Simulation

One of the conventional approaches for structural dynamic analysis and design is to

use lumped mass-spring model (Chopra and Goel [41]). This approach is widely used

due to its simplicity and ability to capture most of the primary deformation modes of

the structure. However in some cases, for example when localized damage develops,

13

the vibration modes of structure may change (Pandey et al. [42]). Mass-spring models

are usually unable to capture such changes, especially in three-dimensional response

of a structure that includes torsional modes. Using a 3-element reference model,

Correnza et al. [43] demonstrated that the response of a structure changes signifi-

cantly when torsional effects are considered. A 3D model that is capable of capturing

such effects is necessary. Furthermore, such models do not provide the necessary

information about localized damage which can occur during extreme events.

In order to overcome the shortcomings of lumped models, a more realistic 3D mod-

eling approach was developed by Ohsaki et al. [44] who studied the collapse behavior

of buildings by carrying out modal analysis and simulation of local buckling. They

showed that highly refined FE models of steel structures, especially for connections,

are able to capture the damage evolution and eventual collapse of the structure in

great detail. Collins et al. [45] also showed that detailed continuum finite element

(FE) models are able to capture more realistic global and local behavior of light

wood-frame structures. However it is not practical to use detailed continuum FE

models when simulating large-scale structures and therefore alternative models that

incur lower computational cost are desirable. Beam-frame elements are associated

with significantly lesser computational cost compared to continuum FE models and

are discussed next.

3.1.2 Beam-frame Elements for Large Deformations and Rotations

The computational advantage of using beam elements over 3D continuum ele-

ments for modeling frame structures is evident from the fact that the total number of

degrees of freedom (DOF) in a beam-element model is far lesser compared to a 3D FE

model. However, since linear beam theory assumes small deformations and rotations,

it is unable to capture the response of structures that exhibit geometrically non-linear

behavior. In the past few decades, there have been a number of publications on the

development of beam element formulations, for both statics and dynamics, which ac-

14

count for large deformations and large rotations in 3D. One may refer to papers by

Bathe and Bolourchi [46], Simo [16] and Jelenic and Crisfield [47] for details on such

formulations. However, all these formulations have to navigate the numerical com-

plexities that arise due to the non-additive nature of large 3D rotations (Romero [48]).

Special treatment of large 3D rotations is required for the interpolating the rotational

DOFs. Romero [49] summarized the different types of interpolation methods com-

monly used and discussed their advantages and drawbacks. The treatment gets even

more complicated for dynamic simulations when using 3D geometrically exact beams

because at every time step the rotational DOFs are required to satisfy the constraints

of the SO(3) configurational space. Time integration schemes for finite rotations in

dynamic problems are discussed in papers by Simo et al. [50], Jelenic and Crisfield [51]

and Romero and Armero [52]. Bottasso and Borri [53] also formulated a generalized

Runge-Kutta scheme that satisfies this constraint by design.

3.1.3 Simplified Models Based on Enriched Macro-Elements

In structural dynamics, macro-element models are usually used to study nonlinear

behavior of structures. In the past, researchers have developed macro-element models

using spring-type elements for several different applications. One of the simplest

approaches is to use rotational springs to model formation for plastic hinges at the

joints and connections between structural elements. For example, Mattern et al. [6]

presented a multi-rigid-body model that treats columns as rigid bodies connected

with rotational springs. This model was used to study the collapse of a building and

it was compared to continuum models to demonstrate its computational efficiency.

Linde and Bachmann [10] developed a macro-element model consisting of four non-

linear springs to simulate earthquake-resistant walls. Similar macro-element models

with spring elements can be found in several other papers (see De la Llera et al. [11],

Jiang and Saiidi [12]). These models are used to simulate nonlinear response of the

structural components such as walls and columns. Another common application of

15

spring elements is to construct joint models to simulate beam-column connections

under seismic loads. Kim and Engelhardt [14] developed a panel zone model using

non-linear rotational springs to simulate beam-column joints under cyclic loading.

The panel zone model is dimensionless and the rotational spring is used to transfer

moments based on the change of angles between beams and columns. A more realistic

model was developed by Mulas [54] which takes into account the dimensions of the

panel zone as well. It was demonstrated that the model was able to capture the

primary deformation modes.

Spring elements are also widely used in other types of modeling such as flexible

multibody systems (see Shabana [55], Yoo et al. [56] and Mayo et al. [57] for examples).

In these types of models, spring elements are usually used as connectors to structural

components such as beams, rigid links, sliders etc. Such multibody systems are

developed to capture the 2D structural behavior in large deformations and large

rotations using a simplified equivalent mechanism. When deriving the multibody

system models, absolute nodal coordinates are usually used for large deformation

problems (for example Shabana et al. [58], Escalona et al. [59] and Dombrowski [60]).

Using this approach, the rotational DOFs are eliminated so that the computation is

simpler and faster. Wasfy [61] also presented a 2D torsional spring-like beam element

for dynamic analysis. This beam element consists of a spring element that captures

the bending response and two truss elements that model the axial response.

In the present study, an approach for developing computationally efficient reduced-

order models capable of simulating structures undergoing large deformations and large

rotations is developed. The proposed model consists of extensional and rotational

springs in 3D. However, the formulation is based only on the global coordinates of

nodes which are the only DOFs associated with the model. This not only reduces

the total number of DOFs in the model, but also circumvents all the complexities

associated with large 3D rotations, making it computationally very efficient. Another

motivation for developing this approach for reduced order modeling of frame struc-

tures is related to applications such as real-time structural health monitoring (Ko and

16

Ni [62]) and real-time hybrid simulation (Shao et al. [63], Carrion et al. [64], Chen et

al. [65], Dyke et al. [66]). In such applications, one usually has tight constraints on

the amount of time allowed to solve the numerical model at every time step. Due to

its computational efficiency, the proposed model may be appropriate for use in such

applications as well.

It is pointed out that despite the seemingly simple approach adopted in this work

(based on extensional and rotational springs), the formulation is by no means trivial,

and is not available in the existing literature or in existing software programs, as of

this writing. Even though there are several similar models described in the literature

(as discussed in the next section), the work presented herein is, to the best of the

author’s knowledge, new and different from any existing method or any capability

in existing software programs. In the following sections, detailed formulations are

presented followed by numerical examples to demonstrate the performance of this

modeling approach.

3.2 Formulation of Reduced-order Spring-based Frame element

In this study, behavior of different structural components in a frame structure is

modeled with extensional and rotational spring elements. As shown in Figure 3.1,

every beam and column is represented by one or more extensional springs to capture

the axial deformations. In addition, every pair of adjoining structural elements is also

connected with a rotational spring to model the bending response. In the following

subsections, the formulation of the proposed model is presented in detail. First, the

kinematics of deformation is described based on the coordinates of the end nodes of the

extensional springs. Equations of equilibrium are then derived for these nodes, while

also accounting for the internal moments generated within the rotational springs due

to bending. Finally, a consistent linearization of these nonlinear governing equations

using the Newton’s method is presented and the formulation for dynamics is discussed.

17

Fig. 3.1.: Modeling a frame structure with reduced-order spring-based elements

3.2.1 Kinematics of Deformation and Internal Forces and Moments

A typical spring-based frame element m, shown in Figure 3.2, consists of an exten-

sional spring and multiple rotational springs at its end nodes I and J . The rotational

springs connect all pairs of adjacent elements at any given node. Nodes I and J

may be connected to other elements ai (1 ≤ i ≤ K) and bi (1 ≤ i ≤ L) respectively.

In Figure 3.2, as an example, element ai adjoining node I and element bi adjoining

node J are highlighted to present the kinematics of element m. Elements ai and bi

are assumed to connect to nodes Ai and Bi at the opposite ends. In general, it is

also possible for either end node not to have any other adjoining element present,

as shown in Figure 3.4. In cases where node I or node J do not have an adjoining

element, the node may have a specified displacement boundary condition or it may

be a free end. If the node has a specified displacement boundary condition, then an

artificial rotational spring, connecting the deformed and undeformed configurations

of element m, is assumed to be present at node I. On the other hand, if the node

is free, then no rotational spring is assumed to be present at this node and this case

does not require any special attention. Thus, the following formulation for element

m requires consideration of two cases:

Case (a) node I connects to an adjoining element a, and

18

Case (b) node I has a specified displacement boundary condition.

Similar arguments hold for node J and the formulation for those cases is identical to

the treatment for node I presented below.

Fig. 3.2.: Kinematics of spring-based frame elements

Remark 3.2.1 Note that the current rotation-free formulation permits only specified

displacement boundary conditions and not specified rotations. Further, the loads act-

ing at the nodes of the structure are also permitted to be only in the form of applied

forces and not applied moments.

In what follows, the subscript i on the adjoining elements ai and bi of element m is

omitted to avoid a profusion of sub- and super-scripts. The formulation is presented

in terms of a particular pair of adjoining elements m and a at node I, and elements m

and b at node J , highlighted in Figure 3.2. The relationships that will be established

for these specific pairs of elements also hold for all other pairs of elements connected

to nodes I and J .

Figure 3.3 shows the original and deformed configurations of element m. As the

structure deforms, the original and deformed positions of a node K are denoted as

XK and xK respectively. The axial forces developed in element m are related to the

19

Fig. 3.3.: Internal forces within a typical extensional spring element m

original and deformed lengths, Lm and lm respectively. These lengths for the element

and are computed as:

Lm =√

(XJ −XI) · (XJ −XI) , (3.1)

lm =√

(xJ − xI) · (xJ − xI) . (3.2)

As depicted in Figure 3.3, the axial force developed in the extensional spring can then

be computed as:

fE = k (lm − Lm)nI , (3.3)

where k is the stiffness of the extensional spring and nI is the unit vector pointing

from node I to node J in the deformed configuration.

Similar to the internal forces in the extensional springs, the internal moment de-

veloped in a rotational spring at node I depends upon the change in angles either

between adjoining elements (case (a)) or between the original and undeformed con-

figurations of the element (case (b)). Case (a) is depicted in Figure 3.4(a) where the

plane formed by the deformed configurations of elements m and a is highlighted. In

general, this plane can be different from the plane formed by these elements in their

undeformed configurations. However, the rotational spring connecting elements m

20

Fig. 3.4.: Original and deformed positions, angles and unit vectors associated withelement m

and a is assumed to always be oriented in the plane of the adjoining elements. The

original and deformed angles between these elements are thus given by:

Θ = cos−1(NI ·NA) , (3.4)

θ = cos−1(nI · nA) , (3.5)

where the unit vectors NI , and NA, for the original configuration are defined as:

NI =XJ −XI

Lm, NA =

XA −XI

La, (3.6)

and the unit vectors nI , and nA, for the deformed configuration are defined as:

nI =xJ − xIlm

, nA =xA − xI

la. (3.7)

Note that the expressions for Θ and θ (Equations (3.4) and (3.5)) are also valid for

cases when the elements are aligned directly opposite to each other (or are coincident

21

upon each other). For case (b) when there are no adjoining elements at node I and a

displacement boundary condition is specified, depicted in Figure 3.4(b), the original

and deformed angles for the artificial rotational spring are defined as:

Θ = 0 , and (3.8)

θ = cos−1(nI ·NI) , (3.9)

respectively.

In terms of the angle change, θ − Θ, the internal moment generated within the

rotational spring and node I is given by:

m = κ(θ −Θ)hI (3.10)

where the unit vector hI is defined as:

hI =nI × nA‖nI × nA‖

Case (a) (3.11)

hI =nI ×NI

‖nI ×NI‖Case (b) (3.12)

and points perpendicular to the plane of elements m and a. Note that hI defines the

direction of internal moment m developed in the rotational spring as shown in Figure

3.5.

Remark 3.2.2 There may be situations where the adjacent elements m and a (and

consequently the vectors nI and nA), may be coincident (θ = 0) or diametrically op-

posite (θ = π). Such situations occur commonly in case (a) when a flexural structural

element, such as a beam or column, is discretized with multiple spring-based elements.

Additionally, such situations may arise in case (b) as well when nI and NI are co-

incident or diametrically opposite. In such cases, the expressions for hI in equations

(3.11)-(3.12) are not well defined because there are infinitely many orientations of the

rotational spring possible. However, such configurations are usually unstable (unless

22

θ = Θ and both are 0 or π). Thus, in such cases, one may assume that the inter-

nal moment m = 0 even though the vector hI remains undefined. This assumption

is akin to assuming that in such situations the rotational spring occupies all of its

infinitely many possible orientations simultaneously.

Fig. 3.5.: Internal moment in the rotational spring represented as a force couple actingat the end nodes of element m

In order to make the formulation free of rotational degrees of freedom, the moment

developed in the rotational spring m which also acts on element m (assumed to be

rigid against bending), needs to be modeled as an equivalent force couple (fR, and

−fR) acting at the end nodes I and J of element m. These couple forces can be

computed as:

fR =1

lm(m · hI)tI =

1

lmκ(θ −Θ)tI (3.13)

where the vector tI can be defined as:

tI = hI × nI (3.14)

23

and represents a unit vector perpendicular to element m in the plane of the rotational

spring. For every rotational spring at node I, connecting elements m and ai, there

are a pair of transverse couple forces fR acting at the end nodes I and J .

Remark 3.2.3 Note that, as with Remark 3.2.2, if hI is not well defined, then tI

is not well defined. Consistent with the assumption of m = 0, one needs to assume

fR = 0 as well in such situations.

Having computed the deformations and the element internal forces and moments

(in the form of transverse force-couples), the equations for global equilibrium of the

structure can be formulated.

3.2.2 Global Equilibrium

The total internal forces for element m at nodes I and J consist of the force fE

in the extensional spring, and the couple forces fR(m,ai) and fR(m,bj) created due to

the presence of the rotational springs at nodes I and J :

f intI = fE +K∑i=1

fR(m,ai) +L∑j=1

fR(m,bj) , (3.15)

f intJ = −f intI , (3.16)

where the superscript R(m, ∗) denotes the contribution of the rotational spring con-

necting a particular pair of elements m and ∗ at nodes I and J as shown in Figure 3.2.

Therefore for the spring element m that has K neighboring elements at node I and

L neighboring elements at node J , there will be a total of K + L internal transverse

forces applied at both end nodes. Equilibrium can be enforced for each element m by

defining the residual forces at nodes I and J as the difference between the internal

and external forces at these nodes.

gI = f intI − f extI , (3.17)

gJ = f intJ − f extJ , (3.18)

24

The global equilibrium equations for a structure with M elements can be written by

assembling the equilibrium equations from each element m:

g(x) =M∑m=1

Am

gI

gJ

(3.19)

where g denotes the global residual vector and x represents the vector of deformed

positions of all the nodes in the structure. For a 3-dimensional structure with N

nodes, the sizes of vectors g and x are 3N × 1 each. The matrix Am is the global

assembly operator (of size 3N × 6) that aggregates the contributions of element m

into the correct locations within the global residual vector g. Note that equilibrium

of element moments is automatically satisfied since the moment has already been

accounted for in the transverse couple forces.

Remark 3.2.4 While successfully avoiding the complications associated with large

rotations in 3D, the reduced-order model does suffer from a deficiency that it does not

account for torsional moments within the members. Nevertheless, as shown in the

later sections with numerical examples, this deficiency does not affect the performance

of this model for most real-life problems.

3.2.3 Solution by Consistent Linearization

The global equilibrium equation presented in Equation (3.19) is a set of nonlin-

ear equations that can be solved by consistent linearization using Newton Raphson

method. The iterative update equation is:

g(xi+1) ≈ g(xi) +

[∂g(xi)

∂x

]∆xi = 0 ⇒ ∆xi =

[Ki

T

]−1 −g(xi) (3.20)

25

where the superscript i indicates the iteration number and KiT represents the tangent

stiffness matrix of the structure given by:

KiT =

[∂g(xi)

∂x

](3.21)

Omitting the superscript i for the iteration number, the tangent stiffness matrix of

partial derivatives can be computed as:

∂g

∂x=

∂

∂x

M∑m=1

Am

gI

gJ

=M∑m=1

Am∂

∂x

gI

gJ

(3.22)

For each element m, the element stiffness matrix will be of the form:

∂

∂x

gI

gJ

=

∂gI∂xI

∂gI∂xJ

∂gI∂xA1

· · · ∂gI∂xAK

0 · · · 0

∂gJ∂xI

∂gJ∂xJ

0 · · · 0∂gJ∂xB1

· · · ∂gJ∂xBL

NTm (3.23)

where the left hand side of the equation is a 6×3N matrix of tangent stiffness terms.

The matrix on the right hand side is divided into three parts. The first part contains

terms that are obtained by taking partial derivatives of the residual g with respect

to the coordinates of nodes I and J . The other two parts of the matrix are obtained

by taking partial derivatives of the residual g with respect to the coordinates of the

nodes of the neighboring elements ai and bi. Therefore the size of this matrix is

6× (6 + 3K + 3L). The matrix Nm denotes a neighborhood assembly matrix (of size

3N×(6+3K+3L)) that acts on a vector xI ,xJ |xA1 ,xA2 · · ·xAK|xB1 ,xB2 · · ·xBL

T

belonging to the nodes associated with element m and its immediate neighbors, and

returns a global vector x of the same quantities assembled into the correct locations

associated with all the nodes in the structure. The global stiffness matrix KiT is

obtained by assembling all the M element stiffness matrices and is of the size 3N×3N

where N is the total number of nodes in the structure. Detailed calculations of the

terms in Equation (3.23) are presented next.

26

3.2.3.1 Linearization of Kinematic Quantities

In order to compute the derivatives for linearization of the residuals, defined in

the preceding section, it is useful to first compute the derivatives of the common

kinematic variables introduced earlier. First, the derivatives of the deformed length

of element m, as defined in equation (3.2), can be computed as:

∂lm∂xI

=2(xJ − xI)(−1)

2lm= −nI , (3.24)

∂lm∂xJ

=2(xJ − xI)

2lm= nI . (3.25)

Derivatives of the unit vectors nI and tI are computed next. Taking derivatives

of the vector nI with respect to xI and xJ , one obtains:

∂nI∂xI

=1

lm(−I + nI ⊗ nI) , (3.26)

∂nI∂xJ

=1

lm(I + nI ⊗ (−nI)) . (3.27)

where I denotes the 3×3 identity tensor and ⊗ denotes the dyadic tensor product of

two vectors. The derivative of the transverse unit vector tI with respect to coordinates

of the adjacent nodes needs to be computed differently for the two cases (a) and (b)

mentioned previously.

For case (a) derivative of the transverse unit vector tI with respect to the coordi-

nates xK of an adjacent node K can be written as:

∂tI∂xK

=∂(hI × nI)

∂xK=

∂

∂xK

((nI × nA)× nI‖nI × nA‖

)=

∂

∂xK

(n⊥A

‖nI × nA‖

)=

1

‖nI × nA‖

(∂n⊥A∂xK

)− n⊥A‖nI × nA‖3

⊗

([∂(nI × nA)

∂xK

]T(nI × nA)

). (3.28)

where the identity for vector triple products (a× b)× c = (a · c)b− (b · c)a has been

used as (nI × nA) × nI = nA − (nA · nI)nI . Further, note that nA − (nA · nI)nIis simply the component of nA that is perpendicular to nI i.e. if nA = n

‖A + n⊥A

27

where n‖A = (nA · nI)nI , then (nI × nA)× nI = nA − (nA · nI)nI = n⊥A. Thus, its

derivatives in the first term of the right hand side of equation (3.28) with respect to

the position vectors xI , xJ and xA can be computed as:

∂n⊥A∂xI

=∂nA∂xI

− nI ⊗

([∂nI∂xI

]TnA +

[∂nA∂xI

]TnI

), (3.29)

∂n⊥A∂xJ

=∂nA∂xJ

− nI ⊗

([∂nI∂xJ

]TnA +

[∂nA∂xJ

]TnI

), (3.30)

∂n⊥A∂xA

=∂nA∂xA

− nI ⊗

([∂nA∂xA

]TnI

), (3.31)

respectively. The derivative expression in the second term on the right hand side of

equation (3.28) can be computed using the standard indicial notation (with respect

to the basis e1,e2,e3) as:

[∂(nI × nA)

∂xK

]= εijk

(∂(nI)i∂(xK)m

(nA)j + (nI)i∂(nA)j∂(xK)m

)[ek ⊗ em] . (3.32)

For case (b) when there is no adjoining element at node I, (nI ×NI) × nI =

NI − (NI ·nI)nI = N⊥I . The derivative of the transverse unit vector tI with respect

to nodal positions xK of a node K is:

∂tI∂xK

=∂(hI × nI)

∂xK=

∂

∂xK

((nI ×NI)× nI‖nI ×NI‖

)=

∂

∂xK

(N⊥I

‖nI ×NI‖

)=

1

‖nI ×NI‖

(∂N⊥I∂xK

)− N⊥I‖nI ×NI‖3

⊗

([∂nI ×NI

∂xK

]T(nI ×NI)

)(3.33)

where the derivatives of the vector triple product with respect to xI and xJ can be

computed as:

∂N⊥I∂xI

= nI ⊗

([∂nI∂xI

]TNI

)− (nI ·NI)

∂nI∂xI

(3.34)

∂N⊥I∂xJ

= nI ⊗

([∂nI∂xJ

]TNI

)− (nI ·NI)

∂nI∂xJ

(3.35)

28

In addition to the derivatives of the lengths and unit vectors defined above, one

also needs the derivatives of the change in angle θ−Θ, defined in equations (3.4)-(3.5)

for case (a), and equations (3.8)-(3.9) for case (b). The derivatives of θ − Θ can be

computed with respect to xI , xJ and xA for case (a) as:

∂(θ −Θ)

∂xI=

−1√1− (nA · nI)2

([∂nA∂xI

]TnI +

[∂nI∂xI

]TnA

), (3.36)

∂(θ −Θ)

∂xJ=

−1√1− (nA · nI)2

([∂nI∂xJ

]TnA

), (3.37)

∂(θ −Θ)

∂xA=

−1√1− (nA · nI)2

([∂nA∂xI

]TnI

), (3.38)

and with respect to xI and xJ for case (b) as:

∂(θ −Θ)

∂xI=

−1√1− (nI ·NI)2

([∂nI∂xI

]TNI

), (3.39)

∂(θ −Θ)

∂xJ=

−1√1− (nI ·NI)2

([∂nI∂xJ

]TNI

). (3.40)

Remark 3.2.5 Note that when the angle θ is 0 or π, the derivatives in Equations

(3.36)-(3.40) become infinite and lead to a singularity. Consistent with the assump-

tions in Remarks 3.2.2 and 3.2.3, we further assume that the derivatives of θ are also

zero (only in the particular configuration when θ is 0 or π). When solving the prob-

lem with a numerical method that makes non-linear iterations, this assumption may

sometimes lead to a singular tangent stiffness matrix. However, this does not pose

a problem as one can employ any of the standard arc-length-type (see Crisfield [67])

continuation methods to solve for the equilibrium configuration in such situations.

The following subsection presents computations of the derivatives of the internal

forces and moments which are required to linearize the global equilibrium equations

(3.19).

29

3.2.3.2 Linearization of Internal Forces

The internal forces and moments for element m include the axial forces and trans-

verse forces given in Equation (3.15). As shown in Equations (3.3) and (3.13), the

variables involved in the calculations of internal forces are the direction unit vectors

nI , tI , the change in length lm−Lm and change in angles θ−Θ. Utilizing the deriva-

tives already computed in Section 3.2.3.1, the derivatives of the internal axial and

transverse forces can be computed as follows.

The derivatives of the internal axial force fE with respect to positions of nodes I

and J are computed as :

∂fE

∂xI= k

(−I(

1− Lmlm

)− Lm

lm(nI ⊗ nI)

), (3.41)

∂fE

∂xJ= k

(I

(1− Lm

lm

)+Lmlm

(nI ⊗ nI)). (3.42)

The derivatives of transverse force fR with respect to xI , xJ and xA are obtained

as:

∂fR

∂xI= κ

(−(θ −Θ)

l2mtI ⊗

(∂lm∂xI

)+tIlm⊗(∂(θ −Θ)

∂xI

)+

(θ −Θ)

lm

∂tI∂xI

), (3.43)

∂fR

∂xJ= κ

(−(θ −Θ)

l2mtJ ⊗

(∂lm∂xJ

)+tIlm⊗(∂(θ −Θ)

∂xJ

)+

(θ −Θ)

lm

∂tI∂xJ

), (3.44)

∂fR

∂xA= κ

(tIlm⊗(∂(θ −Θ)

∂xA

)+

(θ −Θ)

lm

∂tI∂xA

)(3.45)

where the derivatives of lm, θ − Θ and tI are defined in Equations (3.24) - (3.25),

(3.36) - (3.40), (3.28)and (3.33).

Finally the terms in element stiffness matrix as shown in Equation 3.23 can be

calculated using the formulations shown in subsection 3.2.3.1.

30

3.2.4 Identification of Spring Stiffnesses and Material Behavior

The material properties required for the spring-based frame element proposed

herein include the spring constants, k and κ, of the extensional and rotational springs.

The relationship between the spring constants and the geometric and elastic material

properties of the frame element have been studied by researchers such as Howell et

al. [68] and Wasfy [61]. It can be shown that the extensional and rotational spring

constants are given by:

k =EA

L(3.46)

κ =EI

L(3.47)

where E is the Young’s modulus, A is the area of cross-section, I is the second moment

of area of the cross-section, and L is the length of the element.

In addition to elastic properties of the springs, one may also employ plasticity for

problems that involve large deformations. In this work, a basic J2-plasticity model

with isotropic and kinematic hardening (Simo & Hughes [69]) is employed to describe

post-yielding behavior of the springs. The yielding axial force of extensional spring

fEy is calculated using the uni-axial yield-stress of material σy as:

fEy = σyA (3.48)

and the yield moment my of the rotational spring is calculated as:

my = σyZP (3.49)

where ZP denotes the plastic section modulus of the cross-section.

31

3.2.5 Dynamics

For dynamic problems, one can employ a time-stepping scheme to enforce the

equation of motion at discrete instants of time and advance the solution from a

known state at time tn to time tn+1 = tn + ∆t progressively. The nonlinear dynamic

governing equation of motion for the spring-based frame element model is given by

g(an+1,vn+1,dn+1) = 0, where an+1, vn+1, dn+1 denote the acceleration, velocity and

displacement vectors for all the nodes in the model and the residual g is defined as:

g(an+1,vn+1,dn+1) = M an+1 + f intn+1 − f extn+1 . (3.50)

In Equation (3.50) above, M denotes the mass matrix, f intn+1 and f extn+1 denote the

internal and external force vectors, respectively. Note that the internal force vector

can be modified to account for damping as f intn+1 = f intn+1 + Cvn+1 where the matrix

C denotes the damping matrix.

To compute the mass matrix M for the model, the element mass for an element

m is lumped at its two end nodes I and J such that:

M =M∑m=1

Am(1

2ρALmI6×6) (3.51)

where ρ is the density of material and I6×6 denotes a 6 × 6 identity matrix for 3-

dimensional problems.

In this work, the widely used Newmark time integration scheme ( [70]) is used for

numerical time integration. At time step n + 1, the velocity and displacement are

expressed as:

vn+1 = vn + ∆t[(1− γ)an + γan+1] (3.52)

dn+1 = dn + ∆tvn + ∆t2[(12− β)an + βan+1] (3.53)

32

where γ and β are algorithmic parameters and usually take the values of γ = 12

&

β = 0 for an explicit method or γ = 12

& β = 14

for an implicit method ( [70]).

3.3 Numerical Examples

In this section, static and dynamic numerical simulations using the proposed

spring-based frame element model are presented. In each of these numerical ex-

amples a reference model was also built using the commercial finite element software,

ABAQUS (Dassault Systemes Simulia Corp. [71]), where 3-dimensional beam-frame

elements were employed (with the non-linear geometry option turned on). Results

from this reference model are used for verification of the proposed reduced-order

model. A set of error measures are defined for dynamic simulations to quantify the

differences between the results of the proposed model and the reference ABAQUS

model.

(a) Load case 1 (b) Load case 2

Fig. 3.6.: Cantilever beam subjected to (a) Axial load, (b) Transverse load

First, a couple of simple verification problems are solved using the proposed model.

A cantilever beam subject to axial and transverse end loads as shown in Figure 3.6

is used to verify the calibration of the spring constants and plastic behavior of the

spring-based frame element. The beam is assumed to be 5 m long with circular

cross section and radius 0.025 m. Material properties of steel are adopted in this

example with Young’s modulus E = 205 GPa, density ρ = 7860 kg/m3 and yield

stress σy = 330 MPa. Two load cases are considered including a nominal axial load

of F1 = 100 KN and a nominal transverse load of F2 = 1 KN.

33

0

1

2

3

4

5

6

7

8

9

-0.25 -0.2 -0.15 -0.1 -0.05 0

Lo

ad

Pro

po

rtio

na

lity F

acto

r

x-displacements (m)

(a)

Load case 1

Beam 10 elemsSpr 10 elems

-4 -3 -2 -1 0

x-displacements (m)

(b)

Load case 2

-5 -4 -3 -2 -1 0

y-displacements (m)

(c)

Load case 2

Fig. 3.7.: Load vs. displacement of the free-end of the cantilever beam subject toaxial load and transverse loads

Figure 3.7 compares the results for the displacement at the free-end of the can-

tilever beam obtained from the proposed model using 10 spring-based frame elements

(denoted Spr) and the reference ABAQUS model also using 10 shear-flexible beam-

frame elements (denoted Beam). The load proportionality factor on the y-axis repre-

sents a multiple of the nominal load. The results show a good correspondence between

the two models.

3.3.1 Static Simulation Examples

Two static examples of buckling of frames in 2D and 3D are considered. The

material of the frames in both examples is assumed to be elastic with Young’s modulus

E = 205 GPa. Arc-length control ( [67]) is implemented in order to capture the

buckling and snap through behavior of these frames.

34

3.3.1.1 Buckling of a Right-angle Frame in 2D

A right-angle frame shown in Figure 3.8 is used as 2D static example (see [19], [72]).

Each leg of the frame is 1 m long and is assumed to have a circular cross-section with

radius 0.025 m. The two ends A and B of the frame are pinned and a downward point

load is applied to the frame at point C as shown. The frame is simulated using the

proposed reduced-order model with 5 and 10 spring-based frame elements for each

leg. A reference ABAQUS model using beam element is used to verify the proposed

model. The color-bar depicts the values of the forces in the extensional springs and

the moments in the rotational springs normalized between -1 and 1.

Fig. 3.8.: Geometry, loading and collapse sequence of a 2D right angle frame

Figure 3.8 also shows the collapse sequence for this right-angle frame and Figure

3.9 shows the load-displacement plots for the x and y directions at node C. The

collapse sequence is marked with numbers in these two figures to show the deformed

configurations along with its position on the load-displacement path. Upon loading

the frame reaches a limit point near configuration 3, and exhibits a snap-through

behavior near configuration 6. It is clear from these results that the proposed reduced-

order model captures the large deformation behavior of the frame very well.

35

-1 -0.8 -0.6 -0.4 -0.2 0

y-displacements (m)

(a)

1

2

3

4

5

6

7

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1

Lo

ad

Pro

po

rtio

na

lity F

acto

r

x-displacements (m)

(b)

Beam 10 elemsSpr 5elems

Spr 10 elems

1

2

3

4

5

6

7

Fig. 3.9.: Load vs. displacement for point C of the 2D right-angle frame

36

3.3.1.2 Buckling of a 3D Frame

In order to verify the behavior of the proposed reduced order model for large

deformation problems in 3D, a right-angle frame with three legs is considered as

shown in Figure 3.10. The three legs, each 1 m long, are oriented along the x-, y- and

z-directions and the end nodes A, B and C are pinned. Two concentrated loads of

F = 1000 KN are applied at point D along the directions 0, 0,−1 and 0, 1, 0 as shown

in the figure.

Fig. 3.10.: Geometry, loading and collapse sequence of a 3D frame

Figure 3.10 shows the collapse sequence of this 3D frame. Similar to the 2D

frame, the response includes large deformations, a limit point near configuration 1,

and snap-through near configuration 3. The load vs. displacement in the x-, y-

and z-directions at point D are shown in Figure 3.11. Two sets of results obtained

from spring-based frame element model with 5 and 10 elements per leg are compared

against the reference results from ABAQUS. Results obtained from the proposed

reduced-order model match well with the reference ABAQUS model.

Having verified the behavior of the proposed model for static problems over a wide

range of response regimes including large deformations, large rotations, limit points

37

0

0.5

1

1.5

0 0.2 0.4 0.6

Lo

ad

Pro

po

rtio

na

lity F

acto

r

x-displacements (m)

(a)

Beam 10 elemsSpr 5 elems

Spr 10 elems

1

2

3

4

0 0.2 0.4 0.6

y-displacements (m)

(b)

1

2

3

4

-0.5 -0.4 -0.3 -0.2 -0.1 0

z-displacements (m)

(c)

1

2

3

4

Fig. 3.11.: Load vs. displacement for point D of the 3D frame

and snap-through, the performance of this model for dynamic problems is evaluated

next.

3.3.2 Two-story Frame Structure Subject to Earthquake Loading

In this subsection, dynamic simulation of a two-story steel frame subjected to

earthquake ground motions is presented. Dimensions of this two-story structure are

shown in Figure 3.12. All the beams are 7 m long with area of cross-section A = 0.002

m2 and second moment of area I = 3.068 × 10−7 m4. The first story columns are 5

m long while second story columns are 4 m long, all with the same cross sectional

properties as the beams. Distributed line loads of 0.5 KN/m are applied to the beams

as shown in Figure 3.12. The material properties of steel as described in Section 3.2.4

are adopted in this example. Additionally, a Rayleigh damping is assumed to compute

the damping matrix C = αDM + βDK where the damping coefficients are chosen

38

as αD = 0.2 and βD = 0.01. This two-story frame is simulated using the proposed

reduced-order using 2, 5 and 10 spring-based frame elements per beam or column.

Fig. 3.12.: Two-story frame subjected to earthquake ground motion

5

6

7

8

9

Nat

ural

Fre

quen

cy (

Hz)

0

1

2

3

4

0 5 10 15 20 25 30

Nat

ural

Fre

quen

cy (

Hz)

Spr 10 elems

Beam 10 elems

0 5 10 15 20 25 30Mode number

Fig. 3.13.: Comparison of natural frequencies

The results are compared to a reference model constructed using shear-flexible

beam elements in ABAQUS where all the element lengths are 0.5 m. As a first

comparison, Figure 3.13 compares the first 30 natural frequencies and differences in

the mode shapes obtained from the two models. The natural frequencies for the two

39

models are obtained from the linearized system corresponding to the structure in its

undeformed configuration and are very similar to each other.

Fig. 3.14.: Ground acceleration and displacement used for numerical study

-0.4

-0.2

0

0.2

0.4

Dis

pla

cem

ents

(m

)

(a)

Beam 10 elemsSpr 2 elemsSpr 5 elems

Spr 10 elems

-0.9

-0.6

-0.3

0

0.3

0.6

0.9

Velo

citie

s (

m/s

)

(b)

-3

-2

-1

0

1

2

3

0 5 10 15 20 25 30 35 40 45

Accele

rations (

m/s

2)

time (s)

(c)

Fig. 3.15.: Displacement, velocity and acceleration of point A in the x-direction forthe two-story frame subjected to earthquake load

For time history analysis, an earthquake loading based on the 1940 El-Centro

earthquake is used in the current study (see Appendix 6 in [73]). Ground accelera-

tion and ground displacement for this loading are shown in Figure 3.14. This ground

40

displacement is specified as a boundary condition in the x-direction for all the nodes

at the ground level at each time step. Figure 3.15 shows the time history of displace-

ments, velocities and accelerations at node A as indicated in Figure 3.12. This node

is one of the nodes with the largest drift and is expected to have the largest difference

in response when compared to the reference model from ABAQUS. In Figure 3.15,

results obtained from the proposed spring-based model with 2, 5 and 10 elements

per beam or column are compared against the results obtained from the reference

model. It can be seen from the time history plots that as more spring-based frame

elements are used in the reduced-order model, the results match better with the ref-

erence model. For the case when 10 elements are used, the time history curves are

almost the same as the reference results.

In order to quantify the differences between the time histories obtained from

the proposed reduced-order and the reference model and to get a global picture of

these differences for all the nodes in the model, one must define objective measures of

differences or “errors.” The subsequent section defines a few local and global measures

of error used in this study to compare the results of the dynamic simulations obtained

using the proposed reduced-order model to the reference model.

3.3.3 Error Measures

In order to quantify the differences/errors between the results from the reduced-

order model and the reference model, four measures of errors are defined. Let the

time history for displacement, velocity or acceleration of the ith degree of freedom be

represented with a set of numbers, xij for the proposed reduced-order model, and X ij

for the reference model, where the subscript denotes the instant of time tj within the

duration of the simulation t0 ≤ tj ≤ tF .

First, the instantaneous local error at a node at a particular instant of time tj

is simply defined as the difference between the values from the two models at that

41

instant, normalized against the difference between maximum and minimum values of

the response during the entire time history:

εij ≡|xij −X i

j|maxFk=1X

ik −minFk=1X

ik

(3.54)

The average local error at a degree of freedom i over the duration of the entire

simulation from t0 to tF is defined as:

εi ≡

F∑j=1

|xij −X ij|

F (maxFk=1Xik −minFk=1X

ik)

(3.55)

The measure of average local error enables one to see the distribution of errors in the

entire model so that they can decide if mesh refinement is necessary around a certain

location. Figure 3.16 shows an example of the distribution of average local error εi

in x-displacements over the entire structure. It shows that errors are larger at the

nodes farther away from the base and that node A is indeed one of the nodes with

the largest nodal errors as mentioned earlier.

Fig. 3.16.: Distribution of average local errors εi in x-displacement

42

An instantaneous global error is defined as:

εj ≡1

N

N∑i=1

εij (3.56)

This instantaneous global error can be plotted as a time history and enables one

to see how global errors vary over time in order to identify the duration of time

with high transients in the response and then one may choose to use a smaller time

step during those times. Figure 3.17 shows the time history of instantaneous global

error in x-displacements. Errors are larger during the initial stages of the loading

when the largest structural response is observed and then dissipate over time as

shown. An interesting behavior that the instantaneous global error exhibits is the

periodic “bouncing” it displays between almost zero to its peak values during the

simulation. This phenomenon occurs because, despite the small differences in the

natural frequencies of the two models (the proposed reduced-order model and the

reference model), their deformed shapes becomes almost identical at regular intervals

as they oscillate back and forth around each other. This fact is also evident from

Figure 3.15 where one can observe that that the response from the reduced-order

model crosses the time history response from the reference model at multiple instants

of time during the simulation.

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 35 40 45

Err

ors

time (s)

(a)

Spr 2 elemsSpr 5 elems

Spr 10 elems

Fig. 3.17.: Time history of instantaneous global error εj in x-displacement

43

Finally an average global error measure is also defined to compare entire models

as a whole. Assuming a total of N nodes in the structure, the cumulative global error

of the entire model can be defined as the average of all the nodal errors:

ε =1

N

N∑i=1

εi (3.57)

Global errors are calculated for accelerations, velocities and displacements in the

x-, y- and z-directions of the frame with different mesh refinements, using 2, 5 or

10 elements per beam or column. Table 3.1 gives a summary of the global errors

calculated for acceleration, velocity and displacement in all directions with different

mesh refinements. The global error measure enables one to judge how much error

is associated with a particular model and how it changes when different numbers of

elements are used. It clearly shows that as the mesh is refined, average global error

decreases, suggesting that the results may convergence with mesh refinement.

Table 3.1.: Average global errors ε× 100 (%) for the two-story frame model

Elements / Acceleration (%) Velocity (%) Displacement (%)component x y z x y z x y z

2 8.20 0.26 6.01 11.10 0.07 6.43 9.48 1.15 42.905 6.64 0.20 7.76 9.55 0.06 6.46 7.92 0.19 10.7110 3.52 0.17 4.75 4.55 0.04 3.60 3.66 0.09 5.77

The different measures of errors help in evaluating the performance of proposed

reduced-order model and it clearly shows a good correspondence with the more so-

phisticated reference ABAQUS model without the complications associated with large

rotations in 3D. It is also worth mentioning that despite not being able to capture

the torsional response of individual structural members, the overall response of the

structure is captured very well by the reduced-order model for a variety of responses

including large deformations, buckling, limit points, snap-through, and dynamics.

44

3.4 Summary

A reduced-order model is presented that utilizes a network of extensional and ro-

tational springs to represent 3D frame structures and is formulated only in terms of

the coordinate positions of nodes. Compared to 3D geometrically exact beam mod-

els, the proposed model avoids the complex computations associated with 3D finite

rotational degrees of freedom. In addition, the proposed reduced-order model also

has fewer degrees of freedom in comparison to 3D beam-frame elements as the only

unknowns to be solved for comprise the translation degrees of freedom at each node.

This also helps to reduce the computational cost of this approach. The total com-

puational cost is reduced to half comparing to a 3D beam-frame element model with

same mesh size. A limitation of the proposed approach is that torsional deformation

of individual elements cannot be captured. However, through several numerical exam-

ples, it is shown that this limitation does not affect the performance of the proposed

reduced-order model.

The performance of the proposed reduced-order model is evaluated using both

2D and 3D large deformation problems under both, static and dynamic loadings. For

static problems, it has been shown that the proposed model is able to simulate a wide

variety of non-linear response regimes including large deformations, large rotations,

buckling, limit points and snap-through and post-buckling response. Results from the

proposed model are compared with results from a reference model constructed using

fully non-linear 3D shear-flexible beam-frame elements in the commercial software

program ABAQUS (Dassault Systemes Simulia Corp. [71]). It is shown that proposed

model matches the results of the reference model very well. For dynamic verification,

a two-story frame structure is subjected to the El-Centro earthquake ground motion.

Time history of displacement, velocity and acceleration show very good agreement

with results from a reference model. Different error measures used to quantify the

differences between the proposed reduced-order model and the reference model show

45

that errors reduce when progressively more spring-based frame elements are used,

suggesting possible convergence with model refinement.

46

4. GEOMETRICALLY CONSISTENT COUPLING OF

BEAM AND CONTINUUM ELEMENTS

As discussed in Chapter 2, it is difficult to model the effect of local damage at

structural joints and connections on the global behavior of a large structure using only

beam elements in combination with simplified spring-based damage models (e.g. plas-

tic hinges), especially under large deformations such as those encountered in situations

of impending collapse. In order to facilitate the use of more realistic continuum-based

damage models for specific regions of interest in a structure, a domain decomposition

approach is adopted here. In this approach, a large structural model is decomposed

into two or more different types of regions based on the level of damage expected in

those regions.

Fig. 4.1.: An example of the decomposition of a structural model into beam andcontinuum regions (c.f. Pitandi [74]).

For instance, as shown in Figure 4.1, all the joints, connections and other critical

sections of the beams and columns, where more damage is expected, are grouped

47

into one type of region, called the continuum regions henceforth. The rest of the

structure consisting of all the interior spans of structural members, are grouped into

a different type of region, called beam regions henceforth. Such a decomposition

facilitates the separation of disparate spatial scales in a problem. It permits the use

of computationally efficient beam elements in large parts of structural model to keep

the overall computational cost down, while still allowing a more detailed realistic

modeling of damage in the critical regions of the model.

The domain decomposition approach described above creates the need for devising

effective coupling of beam and continuum elements at the interface between the beam

and continuum regions. In the present research, a geometrically consistent method for

coupling beam and continuum finite elements is formulated that accounts for large

deformations and large rotations in two and three dimensions. The key idea is to

impose the geometric constraint of rigid sections, that is the basis of beam theory,

onto the interface between the beam and continuum regions in a mathematically

consistent manner.

In the next few sections, formulations for beam and continuum models in large

deformations are summarized first. Detailed derivations of the consistent coupling of

beam and continuum models based on the Lagrange multiplier method are presented

thereafter.

4.1 Large deformation formulation for continua

This section presents a summary on formulation for continuum elements in large

deformation. Detailed formulations can be found in standard texts on the topic such

as Bonet and Wood [75].

48

Consider the configurations of a body before and after deformation as illustrated

in Figure 4.2. The positions of a point p in the body before and after deformation

are denoted by X and x respectively. The deformation gradient tensor is defined as:

F =∂x

∂X(4.1)

Fig. 4.2.: The configurations of a body before and after deformation

The governing equation of equilibrium for any point p in the body is expressed as:

divσ + b = 0 (4.2)

where σ is the Cauchy stress tensor and b is the body force. The virtual work

functional is given as:

G(u, u) =

∫Ω

ε : σdΩ−∫

Ω

u · bdΩ−∫

ΓN

u · tdΓN (4.3)

where ε, the virtual strain, is defined as:

ε =1

2(∇u+∇uT ) (4.4)

49

The linearized weak form can then be written as:

G(u, u) = G(u, u) +DG(u, u) · [∆u] (4.5)

where the second term on the right is expressed as:

DG(u, u) ·∆u = DW I · [∆u]−DWE · [∆u] (4.6)

It is assumed here that the external virtual work doesn’t depend upon displacement

and therefore the second term on the right can be neglected. The final linearized

weak form then becomes:

DG(u, u) ·∆u = DW I · [∆u] (4.7)

=

∫Ω

ε : c∆ε+

∫Ω

σ :1

2((∇x(∆u))T∇xu+ (∇xu)T∇x(∆u))dΩ

where:

∇x(∆u) =∂∆u

∂x

∆ε =1

2(∇x(∆u) +∇x(∆u)T ) (4.8)

In the above expressions, c denotes the spatial elasticity tensor. An approximate

solution to the weak form can be obtained with the finite element method using

shape functions defined in reference (or parent) coordinates ξ. The transformation

between the parent and actual coordinates is through isoparametric mapping:

X =n∑

α=1

XαNα(ξ); x =n∑

α=1

xαNα(ξ) (4.9)

50

Upon discretization and approximation with finite elements, one obtains the following

system of equations:

gfe ≡ fint(d)− fext ; gfe = 0 (4.10)

where fint and fext denote the vectors of internal and external forces respectively.

Note that for linear problems internal forces can be simply expressed as fint = Kd

where K represents the stiffness matrix and d the vector of displacements. For non-

linear problems, this results in a non-linear system of algebraic equations which can

be linearized as:

Kfe ∆ufe = −gfe (4.11)

where Kfe is the tangent matrix obtained from the spatial discretization of equa-

tion 4.7. Further details on the finite element discretization and resulting numerical

solutions can be found in most standard texts on the topic.

4.1.1 Verification of 3D continuum model

In this subsection, a static patch test is conducted on a small cube to verify

the computer implementation of 3D continuum elements in the present study. As

illustrated in Figure 4.3, tensile traction tx = 100Pa is applied on a face of the cube

along the x-axis and the boundary conditions are on the opposite face such that,

arrows denote constraints in x & z or x & y displacements. The cube is discretized

using two different number of tetrahedral (T4) elements, one with 6 elements and

another with 184 elements. The material properties for the cube are taken as E = 500

Pa, ν = 0.1 and ρ = 10 kg/m3.

Figure 4.4 shows the σxx stress plots of the cubes and it is apparent that the

normal stress in the x-direction is uniform. This example serves as a verification of

the computer implementation of 3D continuum elements being used in the current

study.

51

Fig. 4.3.: Patch test on a 3D cube

−10

12

−1

0

1

2−1

0

1

2

x

Stresses Sxx − Averaged

y

z

100

100

100

100

100

100

100

100

100

100

100

(a)

−5

0

5

−5

0

5

−6

−4

−2

0

2

4

6

x

Stresses Sxx − Averaged

y

z

100

100

100

100

100

100

100

100

100

100

100

(b)

Fig. 4.4.: Plots of stresses from patch test

4.2 3D non-linear beam theory

In this section, the formulation for finite strain 3D beam elements is summarized

based on Simo and Vu-Quoc’s work [16], [19].

4.2.1 Kinematics

The configuration of the 3D beam is described by a vector field that gives the

current line of centroids and an orthogonal frame attached to the cross-section as

52

shown in Figure 4.5. The reference coordinate system, deformed configuration and

the orthonormal basis vector tI(S, t)I=1,2,3 of the moving frame at each point of the

curve S → φ0(S) are also defined in Figure 4.5. The parameter S ∈ [0, L] ⊂ R

denotes the coordinate along the line of centroids of undeformed beam and t ∈ R+

is a time parameter. Note that t3 is not tangent to the line of centroids but normal

to the cross section at all time. The notation t3(S) ≡ n(S) = t1(S)× t2(S) is often

used. EI(S)I=1,2,3 is the fixed reference (material) basis and ei(S)i=1,2,3 denotes the

fixed spatial basis.

Fig. 4.5.: Kinematic description of 3D beam, orthogonal moving frame

The position x0 of the centroid of the cross-section, i.e. the origin of the moving

frame, is defined by:

x0 = φ0(S, t) = φ0i(S, t)ei (4.12)

The basic kinematic assumption is that admissible configurations of the beam take

the explicit form:

x = ϕ(ξ1, ξ2, S) ≡ ϕ0(S) +2∑

Γ=1

ξΓtΓ(S) (4.13)

53

The relationships between the two base vectors are defined through an element Λ ∈

SO(3), where SO(3) represents the special orthogonal (Lie) group. The properties of

orthogonality detΛ = 1 and Λ−1 = Λt ensure that:

tI(S, t) = Λ(S, t)EI or tI(S) = ΛiIei, I = 1, 2, 3 (4.14)

such that the body frame tI can be obtained by rigid rotations of the material frame

EI .

4.2.2 Balance laws & equations of motion

Consider a cross section of the beam as shown in Figure 4.6. Expressions for the

stress resultant n and spatial stress couple m are given by:

n =

∫Ω

t3(x)dA, m =

∫Ω

p(x1, x2)× t3(x)dA (4.15)

Fig. 4.6.: Stress resultants on a cross section of beam

According to Newton’s second law, the spatial form of equations of motion are:

∂

∂Sn+ n = ρAφ0

∂

∂Sm+

∂φ0

∂S× n+ m = ρIw +w × [ρIw] (4.16)

54

where n and m are specified force and moment terms.

The translational and rotational strain measures are defined as:

Spatial Material

Axial-Shear γ =∂Φ0(S, t)

∂S− t3 Γ = Λt∂Φ0(S, t)

∂S−E3

Curvatures ω κ = Λtω

Derivatives of the moving frame are summarized by Simo and Vu-Quoc [19] as

following:

Spatial Material

∂Λ(S, t)

∂S= Ω(S, t)Λ(S, t)

∂Λ(S, t)

∂S= Ω(S, t)K(S, t)

Ω =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

K =

0 −κ3 κ2

κ3 0 −κ1

−κ2 κ1 0

∂Λ(S, t)

∂t= W (S, t)Λ(S, t)

∂Λ(S, t)

∂t= Λ(S, t)W (S, t)

where W (S, t)) is spatial skew-symmetric tensor which defines the spin of the

moving frame.

4.2.3 Admissible variations

The configuration space C that is defined as:

C , φ ≡ (φ0,Λ)|φ0 : (0, L)→ R3,Λ : (0, L)→ SO(3) (4.17)

This configuration space for 3D beam is a non-linear differentiable manifold due to

the presence of SO(3), we need to make use of admissible variations. The ‘perturbed’

55

configuration relative to φ0(S) is constructed as φε(S) ≡ (φ0ε(S),Λε(S)) ∈ C such

that:

φ0ε(S) = φ0(S) + εη0(S), Λε(S) = exp[εΘ(S)]Λ(S) (4.18)

where η0(S) is a vector field interpreted as an infinitesimal displacement of line of

centroids. Θ(S) is a skew-symmetric tensor field interpreted as superposed infinitesi-

mal rotation onto the moving frame Λ(S), with an axial vector ϑ(S). Finite rotations

are defined by orthogonal transformations and infinitesimal rotations are defined by

skew-symmetric transformations. One can obtain an orthogonal matrix by exponen-

tiation of a skew-symmetric matrix. It is often convenient to work with the associated

axial vector ϑ that:

Θh = ϑ× h for any h ∈ R3 (4.19)

Figure 4.7 shows the geometric interpretation of the exponential map. Geometrically

Θ defines an tangent field onto the current configuration defined by Λn. A subsequent

configuration can be obtained by the exponential map such that Λn+1 = exp[Θ]Λn.

The linearized strain measures are summarized as:

Spatial Material

δγ · η = η′0 − ϑ× φ′0 DΓ · η = Λt[η′0 − ϑ× φ′0]

δω · ϑ = ϑ′ Dκ · ϑ = Λtϑ′

4.2.4 Weak form and linearization

The weak form of the balance equations are:

G(φ,η) =

∫[0,L]

[(dn

dS+ n) · η0 + (

dm

dS+dφ0

dS× n+ m) · ϑ]dS = 0 (4.20)

56

Fig. 4.7.: Illustration of exponential map

Integration by parts:

G(φ,η) =

∫[0,L]

n · [dη0

dS− ϑ× dφ0

dS] +m · dϑ

dSdS −

∫[0,L]

(n · η0 + m · ϑ)dS (4.21)

The material expression of weak form is:

G(φ,η) =

∫[0,L]

N ·Λt[dη0

dS− ϑ× dφ0

dS] +M ·Λtdϑ

dSdS

−∫

[0,L]

(n · η0 + m · ϑ)dS (4.22)

The linear part of the functional G(φ,η) is obtained through directional derivative:

L[G(φ,η)] = G(φ, η) +DG(φ, η) ·∆φ (4.23)

57

4.2.5 Discretization and constitutive model

The incremental displacement and rotation are solved using finite element method

using shape functions as:

uh0e(S) =nel∑I=1

NI(S)uhI , ϑhe (S) =nel∑I=1

NI(S)ϑhI (4.24)

where nel denotes the number of nodes of beam element Ihe .

The material elasticity tensor is defined as:

C(S,Γ ,κ) ≡

∂Ψ

∂Γ ∂Γ

∂Ψ

∂Γ ∂κ∂Ψ

∂Γ ∂κ

∂Ψ

∂κ∂κ

(4.25)

The constitutive relations for elastic material can be often assumed as linear be-

tween the stress and stress-couple and material strain measures. Therefore the elas-

ticity tensor is usually assumed constant and diagonal as:

C = diag[GA1, GA2, EA,EI1, EI2, GJ ] (4.26)

The non-linear system of algebraic equations resulting from equations 4.21 and

4.22 can be expressed as:

gb ≡ fint(φ)− fext ; gb = 0 (4.27)

This system of non-linear algebraic equations can be linearized as:

Kb ∆φ = −gb (4.28)

where Kb is the tangent matrix obtained from the spatial discretization of equation

4.23.

58

4.2.6 Verification of 3D beam models

In order to verify the computer implementation of 3D beam-frame elements in

this study, a static problem of pure bending of a cantilever beam oriented in different

3D directions is studied. As illustrated by Figure 4.8(a), the beam has a length of

L = 1 oriented in the direction (1, 1, 1) and bending stiffness EI = 2. It is fixed at

one end and is subjected to a bending moment of M = 8π at the free end.

(a)

0.20.4

0.60.8

−0.2

0

0.2

0.4

0.6

0

0.2

0.4

0.6

(b)

Fig. 4.8.: Cantilever beam in 3D subjected to pure bending

Figure 4.8(b) shows the deformed configurations of the 3D beam while the loading

is increasing. The beam eventually bend into a full closed circle at load M =2πEI

L=

4π and then starts to wind around itself.

Having verified the implementation of 3D beam and continuum elements, the

coupling of these elements is addressed next.

4.3 Coupling of Beam and Continuum Models-Static

Consider an interface of continuum elements and beam elements as illustrated

in Figure 4.9. Consistent coupling of the beam and continuum elements is achieved

by enforcing the continuities of displacements at all points on the common interface

between the beam and continuum elements.

59

Fig. 4.9.: Beam-continuum coupled model in 3D

The position of an arbitrary point P on the interface can be expressed in terms

of the displacements and rotations of the beam element as:

xp = φ0 +2∑I=1

tI = X0 + ub +2∑I=1

ξIΛEI , (4.29)

where ub is the translational displacements of beam elements. On the other hand,

the position of point P can also be expressed in terms of the displacements of the

continuum element:

xp = Xfe + ufe = X0 +2∑I=1

ξIEI + ufe , (4.30)

where ufe denotes the displacements of continuum elements.

The geometric/kinematic constraint equation to ensure continuity of displace-

ments between the beam and continuum elements can be written as:

gc = ub − ufe + (Λ− I)p , (4.31)

where ufe denotes the displacement of points on the interface that belong to the

continuum regions, ub represents the displacement of the beam node on the interface,

Λ denotes the rotation of the interface as introduced in Section 4.2, I is the identity

tensor, and p =∑2

I=1 ξIEI represents the position vector of a general point on the

interface measured from the beam node.

60

4.3.1 Lagrange-multiplier-based coupling

In this research, the augmented Lagrangian approach is used to enforce the cou-

pling constraint 4.31. This is done by adding a term Ec to the total energy associated

with the coupled model. The total energy functional of the beam and continuum

elements is thus given as:

ET = Eb + Efe + Ec , (4.32)

where Eb and Efe are energy functional of the beam and continuum finite elements

respectively. Detailed expression for deriving the energy functional can be found

in [76] and [75].

Here is the focus will be on the constraint term Ec defined as:

Ec =

∫ΓI

λTgcdΓI , (4.33)

where ΓI denotes the interface for coupling and λ is the Lagrange multiplier to enforce

the constraint equation 4.31.

4.3.2 Virtual work functional

The total virtual work functional can be obtained by minimizing the energy func-

tional by taking directional derivatives with respect to the virtual displacements as

follows:

G(ufe,φ,λ,ufe,η,λ) = Gfe(ufe,ufe) + Gb(φ,η) + Gc(ufe,φ,λ,ufe,η,λ) , (4.34)

where Gfe and Gb are the virtual work functionals for continuum and beam elements

and are given in Equations 4.3 and 4.21-4.22 respectively. The third term of the total

virtual work functional is the additional term due to constraint and can be obtained

61

by taking directional derivatives of energy functional defined in Equation 4.33 with

respect to ufe, η and λ:

Gc(ufe,φ,λ,ufe,η,λ) = DEc · [ufe] +DEc · [η] +DEc · [λ] (4.35)

where:

DEc · [ufe] = −∫

ΓI

λTufedΓI (4.36)

DEc · [η] =

∫ΓI

λT [η0 + (ΘΛ)p]dΓI (4.37)

DEc · [λ] =

∫ΓI

λT

[ub − ufe + (Λ− I)p]dΓI . (4.38)

Combining 4.36-4.38 one obtains the final expression of the virtual work functional

for the constraint term:

Gc(ufe,φ,λ,ufe,η,λ) =

∫ΓI

λT [η0 + (ΘΛ)p− ufe]dΓI (4.39)

+

∫ΓI

λT

[ub − ufe + (Λ− I)p]dΓI (4.40)

4.3.3 Linearization

In order to solve the non-linear equations iteratively using Newton’s method, the

virtual work functional defined in Equation 4.34 needs to be linearized:

G = Gb + Gfe + Gc , (4.41)

where the first two terms are linearization of virtual work functionals for the beam

and continuum elements. Detailed expressions for these two terms can be found in

Simo and Vu-Quoc’s work ( [16], [19]) and Bonet and Wood’s book [75] and are also

62

given in the previous sections. The third term is obtained by linearization of the

additional virtual work functional due to constraint:

Gc = DGc · [∆ufe] +DGc · [∆φ] +DGc · [∆λ] , (4.42)

where:

DGc · [∆ufe] = −∫

ΓI

λT

∆ufedΓI , (4.43)

DGc · [∆φ] =

∫ΓI

∆λT (ΘΨΛp)dΓI +

∫ΓI

λT

(ub + ΨΛp)dΓI , (4.44)

DGc · [∆λ] =

∫ΓI

∆λT (η0 + ΘΛp− ufe)dΓI . (4.45)

Recall that Ψ and Θ are the superposed infinitesimal rotations associated with axial

vectors such that Ψ = [ψ×] and Θ = [ϑ×]. Utilizing the identity of vector triple

product defined in Section 3.2.3.1, Equations 4.44 and 4.45 can be further expanded

in terms of the incremental displacement and axial vector for rotation (ub,ψ) as:

DGc · [∆φ] =

∫ΓI

λT (ϑ× (ψ × (Λp)))dΓI +

∫ΓI

λT

(ub +ψ × (Λp))dΓI

=

∫ΓI

λT [ψ(ϑT

(Λp))−Λp(ϑTψ)]dΓI

+

∫ΓI

λT

[ub − (Λp)×ψ]dΓI , (4.46)

DGc · [∆λ] =

∫ΓI

∆λT [η0 − ϑ× (Λp)− ufe]dΓI . (4.47)

Note that the linearized virtual work functional now does not involve the rotation

matrix but their corresponding axial vectors which are easier to work with.

4.3.4 Discretization

At the coupling interface between beam and continuum element, there is one beam

node and n continuum elements involved for the coupling. Let the following quantities

63

denote the real and the virtual incremental displacements and velocities at the beam

node I:

∆φI = (uIb ,ψI) , ηI = (ηI0,ϑ

I) . (4.48)

Let the real and virtual displacements at all the n coupled continuum element nodes

be:

∆umfe = (∆u1fe,∆u

2fe, · · · ,∆uNfe, · · · ,∆unfe) , (4.49)

umfe = (u1fe,u

2fe, · · · ,uNfe, · · · ,unfe) , (4.50)

umfe = (u1fe,u

2fe, · · · ,uNfe, · · · ,unfe) . (4.51)

Lastly, define the vectors that contains all the Lagrange multipliers for the constraint

as:

∆λm = (∆λ1,∆λ2, · · · ,∆λN , · · · ,∆λn) , (4.52)

λm = (λ1,λ2, · · · ,λN , · · · ,λn) , (4.53)

λm

= (λ1,λ

2, · · · ,λN , · · · ,λn) . (4.54)

The displacement and rotation of the coupled beam node on the interface can be

described using the beam element shape functions as:

uIb = Nb∆φI , ηI0 = Nbη

I , ψI = Nθ∆φI , ϑ

I= Nθη

I (4.55)

where at the node location, the shape functions are defined simply to pick the real

and virtual displacements and rotations as follows:

Nb =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

, Nθ =

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

. (4.56)

64

Similarly, the displacements and the Lagrange multipliers at any coupled continuum

element node on the interface can be picked up as:

∆uNfe = NNfe∆b

mfe , uNfe = NN

feumfe , uNfe = NN

feumfe ,

∆λN = NNλ ∆λm , λN = NN

λ λm , λ

N= NN

λ λm. (4.57)

where the shape functions at that continuum are defined again to simply pick the

DOFs associated with N th node from the n coupled continuum nodes as:

NNfe =

0 · · · 0 1 0 0 0 · · · 0

0 · · · 0 0 1 0 0 · · · 0

0 · · · 0 0 0 1 0 · · · 0

, NNλ = NN

fe . (4.58)

From the virtual work functional defined in Equations 4.36-4.38, the contribution

of the constraint to the residual is:

gcfe = −n∑

N=1

(NNfe)

TNNλ λ

m , (4.59)

gcb =n∑

N=1

(NTb −NT

θ [(Λp)×]T )NNλ λ

m , (4.60)

gc =n∑

N=1

(NNλ )T [ub − uNfe + (Λ− I)p] . (4.61)

65

The contribution of the constraint terms to the tangent stiffness matrix is obtained

from Equations 4.43, 4.46 and 4.47:

Gcfe = −(λm

)T

[n∑

N=1

(NNλ )TNN

fe

]∆umfe , (4.62)

Gcb = (ηI)T

[n∑

N=1

(NT

θ Λp(λN)TNθ −NTθ ((λN)TΛp)Nθ

)]∆φI

+(λm

)T

[n∑

N=1

(NNλ )T (Nb − [(Λp)×]Nθ)

]∆φI , (4.63)

Gcλ = (ηI)T

[n∑

N=1

(NT

b Nλ −NTθ [(Λp)×]TNλ

)]∆λm

−(umfe)T

[n∑

N=1

(NNfe)

TNλ

]∆λm . (4.64)

In the above expressions, the terms inside the square brackets are the terms to be

assembled into the global tangent stiffness matrix. For sake of simplicity, the following

variables are defined to represent these terms:

Cfe = −n∑

N=1

(NNλ )TNN

fe , (4.65)

Cb =n∑

N=1

(NNλ )T (Nb − [(Λp)×]Nθ) , (4.66)

Kbc =n∑

N=1

(NT

θ Λp(λN)TNθ −NTθ ((λN)TΛp)Nθ

). (4.67)

The sizes of Cfe , Cb and Kbc are 3n × 3n, 3n × 6 and 6 × 6 respectively. The

expressions for Equations 4.62-4.64 are simplified as:

Gcfe = (λm

)TCfe∆umfe , (4.68)

Gcb = (ηI)TKbc∆φI + (λ

m)TCb∆φ

I , (4.69)

Gcλ = (ηI)TCTb ∆λ∆λm + (umfe)

TCTfe∆λ

m . (4.70)

66

Finally, the constraint residual and tangent stiffness matrix derived in Equations

4.59-4.61 and 4.65-4.67 are assembled into the global Newton’s update equation:

Kb + Kbc 0 CT

b

0 Kfe CTfe

Cb Cfe 0

∆φ

∆ufe

∆λ

= −

gb + gcb

gfe + gcfe

gc

, (4.71)

where:

gcb = Agbgcb , gcfe = Agfegcfe , gc = Acgc , (4.72)

Kbc = AbcKbcATbc , Cb = Ab1CbAb2 , Cfe = Afe1CfeAfe2 . (4.73)

The matrices A∗ denote the corresponding assembly operators to place Kbc, Cb and

Cfe into the correct locations within the global residual vector and stiffness matrix.

4.3.5 Numerical Examples

The Bathe-Bolourchi beam [46] has been used as benchmark problem by many

researchers to verify geometrically exact beam under large deformations (see [21]).

It is a curve shaped cantilever beam subjected to a tip load at the free end and the

geometry properties are shown in Figure 4.10. The cross section of the beam is square

with the area A = 1m2. Material properties of steel are adopted in this example with

Young’s modulus E = 205GPa, density ρ = 7800kg/m3 and Poisson’s ratio µ = 0.3.

A nominal load of 4MN is applied along the z-axis at the free end.

Figure 4.11 shows the bathe-bolourchi beam which is simulated using 79 con-

tinuum elements, 79 beam elements and the beam-continuum coupled model which

consists of 39 continuum elements and 39 beam elements. Mises stress distribution

normalized with respect to largest absolute stress is plotted in the continuum element

region. The figure shows that the three types of models follow the same deformation

sequence.

67

Fig. 4.10.: Geometric properties of curve shaped cantilever beam

Fig. 4.11.: Deformation sequence of bathe-bolourchi beam modeled with (a)Purebeam model, (b)Pure continuum model and (c)Beam-continuum coupled model

Figure 4.12 shows the comparison of load-displacement path along the x, y and

z-directions at the free end of bathe-bolourchi beam simulated using the three types

of models. From the figure, one may note that the results obtained from beam-

continuum coupled model is almost the same as the ones obtained from pure contin-

uum model while the pure beam model gives a slightly softer response.

4.4 Special case: Geometrically consistent coupling for planar problems

As a special case of 3D beam-continuum coupling, the formulation and implemen-

tation of planar beam and continuum elements is also investigated. This coupling for

planar problems has been studied by several authors including Pitandi [74] and the

68

0

1

2

3

4

5

6

7

8

9

-30 -20 -10 0

Lo

ad

Pro

po

rtio

nalit

y F

acto

r

x-displacements (m)

(a)

-50 -40 -30 -20 -10 0

y-displacements (m)

(b)

0 10 20 30 40 50 60 70

y-displacements (m)

(c)

FEBM

CoupledCoupled-Refine

Fig. 4.12.: Load vs displacement plot of bathebolourchi beam at free end.

references within. Note that the rotation matrix in the constraint equation 4.31 for

planar beam takes the form:

Λ =

cos θ −sin θ

sin θ cos θ

. (4.74)

The virtual work functional is expressed as:

DEc · [u] =

∫ΓI

λT[ub − ucz + θΛ′p

]dΓI , (4.75)

DEc ·[λ]

=

∫ΓI

λT

[ub − ucz + (Λ− I)p] dΓI . (4.76)

Note that Λ′ is the derivative of rotation matrix with respect to θ. To wit:

Λ′ =

−sin θ −cos θ

cos θ −sin θ

. (4.77)

69

Linearizing the virtual work functional with respect to ∆u and ∆λ, one obtains:

DGc · [∆u] =

∫ΓI

−λT θ Λ ∆θ p dΓI +

∫ΓI

λT

[∆ub −∆ucz + ∆θ Λ′ p] dΓI , (4.78)

DGc · [∆λ] =

∫ΓI

∆λ[ub − ucz + θ Λ′ p

]dΓI . (4.79)

This planar coupling formulation is tested with numerical examples in the following

section.

4.4.1 Numerical examples

This section presents two verification examples for the planar case of geometri-

cally consistent coupling of beam and continuum models. The benchmark right-angle

frame buckling problem is used to verify large deformation of the 2D coupled model.

Verification of plasticity in the continuum elements of coupled model is then carried

out through static pushover analysis of a portal frame.

4.4.1.1 Elastic buckling of a right-angle frame

The right angle frame buckling example similar to the example used in Section

3.3.1.1 is adopted here to verify the large deformation and post buckling behavior of

the beam-continuum coupled model in 2D. Each leg of this right angle frame is 120

m long with cross sectional area A = 3m2 and second moment of area I = 2.25m4.

The material properties adopted in this example are such that Young’s modulus

E = 7.2MPa and Poisson’s ratio ν = 0.3. The frame is modeled with 10-element

pure beam model, 1280-element pure continuum model and beam-continuum coupled

model in which the continuum element region is located at the connection zone of the

2 legs.

Collapse sequences of the right angle frame modeled with beam and continuum

elements coupled model is depicted in figure 4.13. Modeling of this frame using

70

Fig. 4.13.: Hinged frame collapse sequences showing (a) S11, (b) S22 and (c) S12stresses normalized to the maximum value.

only beam or continuum elements show very similar responses to figure 4.13. The

displacement history under applied load is plotted in figure 4.14 for all three cases

and it is clear that the coupled models are able to represent the behavior of the frame

well.

71

0 20 40 60 80 100

x-displacements (m)

(a)

-1

-0.5

0

0.5

1

1.5

2

2.5

-100 -80 -60 -40 -20 0

Load P

roport

ionalit

y F

acto

r

y-displacements (m)

(b)

Beam-MatBeam-Abq

Continuum-MatContinuum-Abq

Coupled-MatCoupled-Abq

Fig. 4.14.: Laod vs. Displacements of a point under applied load.

4.4.2 Pushover analysis of a portal frame with plasticity

This section present a pushover analysis of a portal frame, which has 5m long

columns and a 6m long beam. The cross sections for both columns and beam are of

the size 1m× 0.5m. This model incorporates large deformation plasticity within the

continuum elements, to verify the plasticity response of the beam-continuum coupled

model. The frame has fixed boundary conditions and is subjected to a horizontal point

load. The distribution of beam and continuum regions for the coupled model is shown

in Figure 4.15 and its behavior is compared with pure continuum models in MATLAB

and ABAQUS. The continuum region for the coupled model is discretized into 72, 288

and 1152 elements while the beam region is discretized into 36 elements. The material

properties adopted for this example are such that Young’s modulus E = 205GPa,

Poisson’s ratio ν = 0.3. Plasticity properties for the continuum element regions are

such that the yield stress σy = 330MPa with isotropic hardening κ = 2.05MPa.

Figure 4.16 shows the load-displacement plots in the x- and y-directions at the

load point for the three models. It’s shown that the results obtained from the beam-

72

Fig. 4.15.: Portal frame collapse sequences using coupled model with stresses S11,S22 and S12

73

continuum coupled model converges to the pure continuum model with mesh refine-

ment in the continuum region. The coupled model with 1152 continuum elements

exhibits very similar behavior to the pure continuum models. The beam-continuum

coupled model is able to capture plastic behavior of structures with proper domain

decomposition.

0 1 2 3 4 5

x-displacements (m)

(b)

0

1

2

3

4

5

6

7

8

9

-3 -2 -1 0

Load P

roport

iona

lity F

acto

r

y-displacements (m)

(a)

FE-AbqFE-Mat

Coupled-Mat1Coupled-Mat2Coupled-Mat3

Fig. 4.16.: Load vs. displacements of the point of applied load.

74

4.5 Summary

In this chapter, a geometrically consistent coupling of beam and continuum models

is presented. This coupled model is developed by applying a constraint equation that

enforces continuity of displacements on the coupling interface using the Lagrange

multiplier method. An initially curved beam was simulated in three different ways,

using only beam elements, using only continuum elements, and using a coupled model

for verification of the coupling method under large 3D deformations. Comparing the

results obtained from the three types of models, the pure beam model shows a softer

behavior while the result obtained from coupled model matches very well with the

pure continuum model. This demonstrates that the coupled model is able to capture

the structural response as well as the pure continuum model while reducing the total

computational cost.

The case of beam-continuum coupling for planar problems is also presented. A

right angle frame buckling problem is used to verify the large deformation and post

buckling behavior of the beam-continuum coupled model. The results show that the

coupled model displays a very similar behavior compared to pure continuum and pure

beam models. A pushover analysis of a portal frame is also carried out to verify the

elasto-plastic response of the beam-continuum coupled models. It is shown that the

results converge to the pure continuum model with refinement of the mesh in the

continuum regions where material plasticity is defined.

Comparing to a high fidelity pure continuum model, the computational cost of

coupled model is reduced depending on percentage of critical and non-critical regions.

For instance, if half of the structure is non-critical and can be simulated using simpler

beam elements, the total DOFs to be solved are reduced to approximately half of

the original. Assuming a solver complexity of n3, one can expect eight times lesser

computational cost associated with the beam-continuum coupled model.

This chapter presents the formulation, numerical implementation and verification

examples of geometrically consistent coupling of beam and continuum elements for

75

static problems. The numerical examples show that the coupled model are capable

of simulating static response of frame structures under large deformations. The next

chapter extends the study of beam-continuum coupled models to dynamic problems.

76

5. DYNAMICS OF COUPLED BEAM AND CONTINUUM

MODELS

This chapter presents the formulation for dynamics of the geometrically consistent

coupling of beam and continuum models. In the following sections, a summary of

the formulations for the dynamics of beam and continuum models is presented first.

The dynamic coupling of beam and continuum models is presented next exploring

different coupling constraints such as continuity of displacements and continuity of

velocities across the coupling interface. Lastly, a multi-time-step (MTS) method for

the beam-continuum coupled model is formulated. Numerical examples are presented

to characterize the performance of the beam-continuum coupled models.

5.1 Dynamics of continuum models

This section presents a brief summary of formulations on dynamics of continuum

models. The governing equation for continuum dynamics is written as:

divσ + b− ρu = 0 . (5.1)

Premultiplying with a virtual displacement u and integration by parts, the weak form

is obtained as:

G(u, u) =

∫Ω

u · ρudΩ +

∫Ω

∇u : σdΩ−∫

Ω

u · bdΩ−∫

ΓN

utNdΓN . (5.2)

Discretization in space leads to the following semi-discrete system of ordinary

differential equations:

g(d, d) ≡Md+ fint(d, d)− fext ; g(d, d) = 0 (5.3)

77

where M represents the mass matrix and a superimposed dot represents a time-

derivative.

The semi-discrete equations of motion can be discretized in time using time-

stepping schemes that compute the time history response of the body or the structure

by advancing the solution one time-step at a time in a sequential manner.

5.1.1 Midpoint time integration

In the current study, the midpoint time integration scheme is used for the 3D

coupled model. It has been shown that this integration scheme is energy conserving

for the 3D geometrically exact beams.

Fully discrete equations of motion using midpoint rule at tn+1/2 are given by:

M fean+1/2 + f int(vn+1/2,dn+1/2)− f extn+1/2 = −gfe(an+1/2,vn+1/2,dn+1/2) (5.4)

an+1/2 =1

2(an + an+1) =

1

∆t(vn+1 − vn) (5.5)

vn+1/2 =1

2(vn + vn+1) =

1

∆t(dn+1 − dn) (5.6)

Substituting equations 5.5 and 5.6 into 5.4, one obtains a non-linear system of

algebraic equations. This system of non-linear equations can be linearized in a similar

way as that for the static problems in the previous chapter to give:

(Kfen+1/2 +

2

∆t2M fe

n+1/2)∆dfen+1 = −gfe(dfen+1) (5.7)

This system can be solved for dn+1 and vn+1 in a time stepping manner.

An example of wave propagation inside a cantilever beam is used to verify the

dynamics of continuum in large deformation. As shown in Figure 5.1(a), a beam is

fixed at its lower end and subjected to a traction of τ = (0, 0, 2) × 104KPa on the

other end.Material properties adopted in this example are such that Young’s modulus E =

205GPa, density ρ = 7800kg/m3 and poison’s ratio ν = 0.3. Figure 5.1(a) shows

78

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014D

ispl

acem

ent (

m)

Time (s)

Fig. 5.1.: Wave propagation inside a cantilever beam

stress Szz distribution inside the beam and Figure 5.1(b) shows the time history of

displacement at the free end. They show that wave transfers smoothly inside the

beam under large deformation.

5.2 Dynamics of 3D geometrically exact beams

In the current study, energy-momentum conserving algorithm developed by Simo

et al. [50] is used for dynamics of 3D geometrically exact beam part of the coupled

model. The following sections present a brief summary of the formulation and some

verification examples.

5.2.1 Exact energy and momentum conserving algorithms

The local governing equations based on balance of momentum are expressed as:

p = n′ + n , (5.8)

π = m′ +ϕ′ × n+ m , (5.9)

79

where n and m denote the applied body forces and moments and the generalized

momenta are defined as:

p = Aρϕ , (5.10)

π = iρω , (5.11)

where ϕ denotes the position of the beam’s centroidal axis, ω denotes the angular

velocity, Aρ is mass per unit length, and iρ denotes the spatial moment of inertia of

the cross section. The dynamic virtual work functional is obtained as:

G(ϕ,ω,η,µ) =

∫ L

0

[π · µ+ p · η] dS +

∫ L

0

[n · (η′ − µ×ϕ′) +m · µ′] dS

−Gext(R;η,µ) (5.12)

A midpoint approximation that has been proven to conserve momentum is used.

Consider a typical time interval [tn, tn+1]. The incremental displacements and rota-

tions are related to their velocities with the following relations:

∆u = ∆t1

2(vn + vn+1) , (5.13)

ϑ = ∆t1

2(ωn+1 + cay[ϑ]ωn) , (5.14)

where ϑ is the rotation vector and its Cayley transform is used to update the rotation

field:

Λn+1 = cay[ϑ]Λn . (5.15)

80

The time discretization of the weak form 5.12 within the time interval [tn, tn+1] is

expressed as:

1

∆t

∫ L

0

[η · (pn+1 − pn) + µ · (πn+1 − πn)] dS

+

∫ L

0

[n · (η′ − µ×ϕ′

n+1/2) +m · µ′] = Gext(R;η,µ) . (5.16)

In equation 5.16, R = (n,m) denotes the stress resultants and the configuration at

the mid-time-step tn+1/2 is defined as:

ϕn+1/2 =1

2(ϕn +ϕn+1) , (5.17)

Λn+1/2 =1

2(Λn + Λn+1) . (5.18)

In order the ensure conservation of energy, algorithmic constitutive equations are

constructed as:

N = CN1

2[Γn+1 + Γn] , (5.19)

M = CM1

2[Ωn+1 + Ωn] . (5.20)

where N and M denote the material counterparts of the stress resultants and Γ and

Ω represent the strains and curvatures.

Following the same procedure for spatial discretization and linearization as that

for the static formulation in the previous chapter, one obtains the update equation

to solve for the incremental displacements:

(Kbn+1/2 +

2

∆t2M b

n+1/2)∆dbn+1 = −g(dbn+1) (5.21)

Updates to translational and angular velocities are obtained by linearizing equations

5.13 and 5.14. Within each time-step, these non-linear equations are solved iteratively

81

and, upon convergence, the process is repeated to advance the solution from one time

step to the next.

5.2.2 Verification of dynamics of 3D geometrically exact beam

This section presents two free motion numerical examples to verify the 3D geo-

metrically exact beam model under large overall motion. The time history of energy

and angular momentum are recorded to demonstrate the energy conserving property

of the energy-momentum method.

5.2.2.1 Free flying beam

A free flying beam that is initially straight is subjected to a combination of force

and moments at node 11 as shown in Figure 5.2(a). Material properties adopted

in this example are such that Young’s modulus E = 10000, cross sectional area

A = 1, density ρ = 10, moment of inertia I = 2. The beam is subjected to a force

F (t) = 0.1p(t) along the x-direction, and two moments such that M1(t) = p(t) in the

y-direction and M2(t) = 0.5p(t) in the z-direction, where p(t) is defined as:

p(t) =

600t for 0 < t < 2.5

1500− 600(t− 2.5) for 2.5 < t < 5

0 for 5 < t

(5.22)

Figure 5.2(b) shows the deformation sequence of the free flying beam from the

view of x-z plane. It shows that the beam undergoes large deformation and overall

motion under the combination of force and moments. Figure 5.3(a) shows the time

history of energy over a time period 0 s to 25 s and it is apparent that the energy is

conserved through out this period.

82

Fig. 5.2.: Geometry of initially straight beam undergo large overall motion

0.0E+0

2.0E+3

4.0E+3

6.0E+3

8.0E+3

1.0E+4

1.2E+4

1.4E+4

1.6E+4

1.8E+4

0 5 10 15 20 25 30Time (s)

Total energy

Fig. 5.3.: Deformation sequence of initially straight beam and time history of totalenergy plot

5.2.2.2 Circular beam

A closed circular beam as shown in Figure 5.4 is used by Simo et al. [50] to

verify the non-linear dynamic response of beams with initial non-zero curvature. The

material properties adopted for this example are such that Young’s modulus E =

10000, cross sectional area A = 1, density ρ = 10, moment of inertia I = 0.05. Two

83

Fig. 5.4.: Geometry of initially circular beam subjected to two point loads

0.0E+0

1.0E+2

2.0E+2

3.0E+2

4.0E+2

5.0E+2

6.0E+2

0 5 10 15 20 25 30Time (s)

Total energy

Fig. 5.5.: Deformation sequence of initially circular beam and time history of totalenergy plot

out-of-plane forces that are of equal magnitude and opposite directions are applied at

nodes 1 and 5 which are located 90°apart. Magnitude of the forces are specified as:

84

p(t) =

40t for 0 < t < 2.5

100− 40(t− 2.5) for 2.5 < t < 5

0 for 5 < t

(5.23)

Figure 5.5(a) illustrates the large deformation and overall motion of the circular

beam under the two forces. The time history of energy shown in Figure 5.5(b) again

demonstrates the energy conserving property of the energy-momentum method.

5.3 Dynamic coupling of beam and continuum models

In this section, the formulation for dynamic coupling of beam and continuum

models is presented in detail. Two coupling methods based on different constraints,

continuity of displacements and continuity of velocities at the interface, are investi-

gated. For each method, the constraint equation is first presented and then enforced

using Lagrange multipliers into the energy functional. The weak form is then de-

veloped and a consistent linearization is carried out to obtain the Newton update

equations. Finally, the contribution of the constraint into the global stiffness matrix

and residual are obtained.

5.3.1 Coupling based on continuity of displacements

The constraint equation based on continuity of displacements is expressed at time

step tn+1 as:

gcn+1 = [ub − ufe + (Λ− I)p]n+1 (5.24)

85

The contribution of this constraint to the energy functional (Lagrangian) can be

expressed using Lagrange multipliers as:

Ec =

∫ΓI

λTn+1[ub − ufe + (Λ− I)p]n+1dΓI (5.25)

The contribution of this term to the weak form is:

Gc = DEc · ub +DEc · ufe +DEc · ϑ+DEc · λ

=

∫ΓI

λTn+1[ub − ufe − [(Λn+1p)×]ϑ]dΓI

+

∫ΓI

λT [ubn+1 − ufen+1 + (Λn+1 − I)p]dΓI . (5.26)

Linearizing the weak form with respect to incremental displacement quantities and

Lagrange multipliers at time step tn+1 the following expression can be obtained:

Gc = DGc ·∆ubn+1 +DGc ·∆ufen+1 +DGc ·∆ϑ+DGc ·∆λ

=

∫ΓI

∆λTn+1[ub − ufe − [(Λn+1p)×]ϑ]dΓI

+

∫ΓI

λT[∆ubn+1 −∆ufen+1 + [(Λn+1p)×] ∆ϑ

]dΓI . (5.27)

Discretizing using the same shape functions shown in Equation 4.56 and 4.58, the

contribution of constraint term to the global residual is:

gcfen+1 = −n∑

N=1

(NNfe)

TλNn+1 ,

gcbn+1 =n∑

N=1

(NTb −NT

θ [(Λn+1p)×]T)λNn+1 ,

gcn+1 =n∑

N=1

(NNλ )T

(ub,In+1 − u

fe,Nn+1 + (Λn+1 − I)p

). (5.28)

86

Contribution to the stiffness matrix:

Cfen+1 = −

n∑N=1

2

∆t(Nλ,N)TN fe,N , (5.29)

Cbn+1 =

n∑N=1

(Nλ,N)T[N b −N θ([Λn+1p]×)T

], (5.30)

Kbcn+1 =

n∑N=1

(N θ)T[(Λn+1pλ

T )− (Λn+1p)′λ]N θ . (5.31)

The equations to be solved for beam and continuum parts are:

(Kbn+1/2 +

2

∆t2M b

n+1/2 +Kbcn+1)∆φbn+1 +Cb

n+1∆λ = −gb (5.32)

(Kfen+1/2 +

2

∆t2M fe

n+1/2)∆dfen+1 +Cfen+1∆λ = −gfe (5.33)

∆ubn+1 −∆ufen+1 + [(Λn+1p)×] ∆ϑ = −gc (5.34)

In the matrix form, these equations can be expressed as:

Kb + Kbc 0 CT

b

0 Kfe CTfe

Cb Cfe 0

∆φ

∆dfe

∆λ

= −

gb + gcb

gfe + gcfe

gc

, (5.35)

After ∆db and ∆dfe are obtained, the displacements, velocities and accelerations

at time step n+ 1 can be updated using Equations 5.5, 5.6, 5.13 and 5.14.

5.3.2 Coupling based on continuity of velocity

Differentiating the constraint equation based on displacements with respect to

time t, the constraint equation based on continuity of velocity can be expressed as:

gc =d

dt[ub − ufe + (Λ− I)p]n+1 (5.36)

= [vb − vfe + Λp]n+1 (5.37)

87

Since Λ = ωΛ, one can write the constraint equation as:

gc = [vb − vfe + ωΛp]n+1 (5.38)

Using the relations from the mid-point time integration:

vn+1 = −vn +2

∆t(un+1 − un) , (5.39)

ωn+1 = −cay[ϑ]ωn +2

∆tϑ , (5.40)

contribution of the constraint term into the weak form is:

Gc = DEc · [ub] +DEc · [ufe] +DEc · [ϑ] +DEc · [λ] (5.41)

For simplicity, define the following vectors and their associated skew symmetric ten-

sors:

a = cay[ϑ]ωn , A = a , (5.42)

b = Λn+1p , B = b . (5.43)

The weak form can be written as:

Gc =

∫ΓI

λTn+1

[2

∆t(ub − ufe)

]dΓI +

∫ΓI

λT[vbn+1 − v

fen+1 + ωΛn+1p

]dΓI

+

∫ΓI

ϑT (−ATBT − 2

∆tBT +BTAT − 2

∆tBT ϑT )λn+1dΓI . (5.44)

88

Linearized weak form is obtained as:

Gc = DGc · [∆ubn+1] +DGc · [∆ufen+1] +DGc · [∆ϑ] +DGc · [∆λn+1] (5.45)

=

∫ΓI

[2

∆t(ub − ufe)

]T∆λn+1dΓI

+

∫ΓI

λT[

2

∆t(∆ubn+1 −∆ufen+1)− (BA+

2

∆tB −AB +

2

∆tϑB)∆ϑ

]dΓI

+

∫ΓI

ϑT (−ATBT − 2

∆tBT +BTAT − 2

∆tBT ϑT )∆λn+1dΓI

+

∫ΓI

ϑT[bλn+1A+ (aTλn+1)B +

2

∆t(bλT − bTλn+1)− (bTλn+1)A

−aλTn+1B +2

∆t(λn+1b

T − bTλn+1) +2

∆t(ϑλn+1B − λn+1ϑ

TB)

]∆ϑdΓI

Discretizing using the same shape functions shown in Equation 4.56 and 4.58, the

contribution of constraint term to the global residual is:

gcfen+1 = −n∑

N=1

2

∆t(N fe,N)TNλ,Nλmn+1 , (5.46)

gcbn+1 =n∑

N=1

[2

∆t(N b)T + (N θ)T (−ATBT − 2

∆tBT +BTAT

− 2

∆tBT ϑT )

]Nλ,Nλmn+1 , (5.47)

gcn+1 =n∑

N=1

(Nλ,N)T(vbn+1 − v

fe,Nn+1 + ωn+1Λn+1p

). (5.48)

The contribution to the stiffness matrix from the above equation can be obtained

as: Kb + Kbc 0 CT

b

0 Kfe CTfe

Cb Cfe 0

∆φ

∆ufe

∆λ

= −

gb + gcb

gfe + gcfe

gc

, (5.49)

89

where Cfe, Cb and Kbc are obtained by assembling the contributions of the following

matrices respectively:

Cfen+1 = −

n∑N=1

2

∆t(Nλ,N)TN fe,N , (5.50)

Cbn+1 =

n∑N=1

(Nλ,N)T[

2

∆tN b − (BA+

2

∆tB −AB +

2

∆tϑB)N θ

], (5.51)

Kbcn+1 =

n∑N=1

(N θ)T[b(λN)TA− bTλNA+ aTλNB − a(λN)TB +

2

∆t

(b(λN)T

−2bTλN + λNbT + ϑ(λN)TB − λN ϑTB)]N θ . (5.52)

The update equation 5.49 can be solved iteratively the beam and continuum ele-

ment displacements and the Lagrange multipliers.

5.3.3 Special case: Dynamic coupling for 2D problems

The constraint equation at tn+1 based on continuity of velocities for 2D problems

is written as:

gcn+1 = vbn+1 − vfen+1 + θn+1Λ

′n+1p (5.53)

The contribution of the constraint to the energy functional is:

Ecn+1 =

∫ΓI

λTn+1

(vbn+1 − v

fen+1 + θn+1Λ

′n+1p

)dΓI (5.54)

The contribution to the weak form is expressed as:

Gcn+1 = DEc · [vb] +DEc · [vfe] +DEc · [θ] +DEc · [λ]

=

∫ΓI

λT(vbn+1 − v

fen+1 + θn+1Λ

′n+1p

)dΓI

+

∫ΓI

λTn+1

(vb − vfe + ¯θΛ′n+1p−

β∆t

γ¯θθn+1Λn+1p

)dΓI (5.55)

90

Linearizing the above expression one obtains:

Gcn+1 = DGcn+1 · [∆vbn+1] +DGcn+1 · [∆vfen+1] +DGcn+1 · [∆θn+1] +DGcn+1 · [∆λn+1]

=

∫ΓI

λT(

∆vbn+1 −∆vfen+1 + ∆θn+1Λ′n+1p

−β∆t

γθn+1Λn+1p∆θn+1

)dΓI (5.56)

After discretization, the contributions to the residuals are:

gcfen+1 = −n∑

N=1

(N fe,N)Tλ , (5.57)

gbn+1 =n∑

N=1

N bTλ+N θT (Λ′n+1p)Tλ− β∆t

γN θT θn+1(Λn+1p)Tλ , (5.58)

gcn+1 =n∑

N=1

(Nλ,N)T (vbn+1 − vfen+1Λ

′n+1p) . (5.59)

Similarly, contributions of the constraint terms to the tangent matrices are:

Cfen+1 = −

n∑N=1

(Nλ,N)TN fe,N , (5.60)

Cbn+1 =

n∑N=1

N b + Λ′n+1pNθ − θn+1(Λn+1p)N θ , (5.61)

Kbcn+1 = −

n∑N=1

(N θ)T (θn+1(pTΛ′Tn+1)λ)N θ (5.62)

Assembling this contributions to the global update equations, once again leads to a

similar system of equations as that for the 3D coupling method:

Kb + Kbc 0 CT

b

0 Kfe CTfe

Cb Cfe 0

∆φ

∆ufe

∆λ

= −

gb + gcb

gfe + gcfe

gc

, (5.63)

where the expressions for the variables are given in equations 5.60 - 5.62.

91

5.3.4 Numerical example

This section presents numerical example that demonstrates the performance of

consistent coupling of beam and continuum models in dynamic large deformation

problem.

A 5m long cantilever beam with 0.5 × 0.5m2 cross-section is used to study the

non-linear dynamic response of the beam-continuum coupled model. Elastic material

properties of steel are adopted for this example again. The structure is simulated using

a 20-element pure beam model, an 80-element pure continuum model and coupled

model consists of 10 beam elements and 40 continuum elements as shown in Figure

5.6. A point load F = 20MN is applied at the free end of the cantilever beam.

Figure 5.7 shows the time histories of displacements, velocities and accelerations

at the free end of the cantilever beam obtained using the three models. It can be

observed that the pure beam model shows slightly softer response in comparison to

the other two models. On the other hand, the pure continuum model and beam-

continuum coupled model show very similar responses. This example shows that the

coupled model is capable of simulating the dynamic behavior of a frame structure with

a similar fidelity as the pure continuum model at a fraction of the computational cost.

5.4 Multi-time-step method for dynamics of coupled models

This section presents a multi-time-step (MTS) method developed for the beam-

continuum coupled model based on the study by Prakash et al. 2014 [77]. As midpoint

integration is used for the 3D coupling in the current study while Prakash et al.

used Newmark integration, the following section only summarizes the key differences

resulting from adoption of the midpoint time integration.

92

Fig. 5.6.: Cantilever beam simulated using (a) 20-element beam model, (b) 80-elementcontinuum model and (c) coupled model

93

-1.6

-1.2

-0.8

-0.4

00 0.1 0.2 0.3 0.4 0.5

Dis

plac

emen

t (m

)

TIme (s)

BM20FE 80FE40BM10

-100

-75

-50

-25

0

25

50

75

100

0 0.1 0.2 0.3 0.4 0.5

Velo

city

(m/s

)

TIme (s)

BM20FE 80FE40BM10

-3.0E+05

-2.0E+05

-1.0E+05

0.0E+00

1.0E+05

2.0E+05

3.0E+05

0 0.1 0.2 0.3 0.4 0.5

Acc

eler

atio

n (m

/s²)

TIme (s)

BM20FE 80FE40BM10

Fig. 5.7.: Time history plot of (a) displacements, (b) velocities and (c) accelerationsat the free end of the cantilever beam

94

5.4.1 Formulation

Formulations for the MTS method for beam-continuum coupled model are sum-

marized in this section. Let the beam subdomain (Subdomain A) be integrated with

a large time step ∆T and the continuum subdomain (Subdomain B) be integrated

with a small time step ∆t as depicted in Figure 5.8

Fig. 5.8.: Coupling of different time steps with a ratio of m between them.

Let z = a,v,dT represent the state of the beam or finite element subdomains.

The residual equations to advance the beam domain from time t0 to tm are given by:

gb(zbm, zb0,λm) =

M babm/2 + pb(dbm/2,v

bm/2) +CbTλm/2 − f bm/2

vbm + vb0 − 2∆T

(dbm − db0)

abm + ab0 − 4∆T 2 (dbm − db0 −∆Tvb0)

(5.64)

Residual equations to advance the continuum domain from time step j − 1 to j:

gfe(zfej , zfej−1,λj) =

M feafej−1/2 + pfe(dfej−1/2,v

fej−1/2) +CfeTλj−1/2 − f fej−1/2

vfej + vfej−1 − 2∆t

(dfej − dfej−1)

afej + afej−1 − 4∆t2

(dfej − dfej−1 −∆tvfej−1)

(5.65)

Enforcing interface constraint at time step tm:

gIm(zbm, zfem ) = Cbvbm +Cfevfem (5.66)

95

Using an approximation for the Lagrange multipliers, λj can be calculated by linear

interpolation between time steps t0 and tm:

λj =

(1− j

m

)λ0 +

(j

m

)λm . (5.67)

After linearization, the Newton’s update equations for the beam domain is ob-

tained as:

Mbm∆zbm + CbT∆λm = −gb(zbm, zb0,λm) (5.68)

where:

Mbm =

M b Db

m Kbm

∆T2Ib −Ib 0

∆T 2

4Ib 0 −Ib

; Cb = Cbm,0,0 . (5.69)

Similarly for continuum domain, the update equations are:

Mfej ∆zfej + Nfe∆zfej−1 + CfeT∆λj = −gfe(zfej , z

fej−1,λj) (5.70)

where:

Mfej =

M fe Dfe

j Kfej

∆t2Ife −Ife 0

∆t2

4Ife 0 −Ife

; Cfe =Cfej ,0,0

. (5.71)

Matrix Nfe for mid-point method is:

Nfe =

M fe Dfe

j Kfej

Ife 2∆tIfe 0

∆t2Ife 2Ife 2

∆tIfe

(5.72)

96

The global system of equations to be solved to obtain the Newton updates for beam

and continuum domains is:

Mbm CbT

Mfe1

1mCfeT

Nfe Mfe2

2mCfeT

. . . . . ....

Nfe Mfem

mmCfeT

Bb Bfe

∆zbm

∆zfe1

∆zfe2

...

∆zfem

∆λm

=

−gb(zbm, zb0,λm)

−gfe(zfe1 , zfe0 ,λ1)

−gfe(zfe2 , zfe1 ,λ2)

...

−gfe(zfem , zfem−1,λm)

−gI(zbm, zfem )

(5.73)

where B is obtained from the constraint equation 5.66.

5.4.2 Numerical examples

This section presents numerical examples to verify the MTS method for beam-

continuum coupling with the midpoint time integration method.

5.4.2.1 Split-SDOF coupling

A split SDOF problem as illustrated in Figure 5.9 is used to study the MTS

method with mid-point integration scheme. This problem was used as a verification

example by Prakash et al. 2014 [77] for MTS method. The single spring-mass system

is separated into two domains A and B as shown in the figure.

Fig. 5.9.: Split SDOF problem

97

The non-linear spring in domain A has the following properties:

FAint = kA0 + 3kA1 d

2A (5.74)

where k0 = 1 and k1 = 1.

The properties for non-linear spring in domain B are such that:

FBint =

kB0 sin(π3(dB/d0)) for 0 < dB ≤ d0

c2(dB/d0)0.5 + c3 for d0 < dB(5.75)

where:

c1 = 3 (5.76)

c2 = 2c1pi

3cos(

pi

3) (5.77)

c3 = c1 sin(pi

3)− 2c1

pi

3cos(

pi

3) (5.78)

Mass for the two domains are mA = 1 and mB = 5. Two external forces applied

on the two domains are fA = 1 and fB = 5. The time step used for domains A

and B are 0.1 and 0.05 for MTS method. Figure 5.10 show the time history plot of

displacements, velocities, accelerations as well as the Lagrange multipliers at the two

domains with STS and MTS methods used. From the figure, the results from MTS

and STS are very similar.

5.4.2.2 3D Cantilever beam with Mid-point method

A 20m long cantilever beam with 1 × 1m2 cross-section is used to study the

performance of MTS method for beam-continuum coupled model. Elastic material

properties of steel are adopted for this example again. The structure is simulated

using coupled model consists of 5 beam elements and 5 continuum elements as shown

98

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Dis

plac

emen

t (m

)

Time (s)

STS -ASTS -BMTS -AMTS -B

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

0 5 10 15 20

Velo

city

(m/s

)

Time (s)

STS -ASTS -BMTS -AMTS -B

-2.2-1.8-1.4

-1-0.6-0.20.20.6

11.41.82.2

0 5 10 15 20

Acc

eler

atio

n (m

²/s)

Time (s)

STS -ASTS -BMTS -AMTS -B

-1.2

-0.8

-0.4

0

0.4

0.8

0 5 10 15 20Lam

bda

Time (s)

MTSSTS

Fig. 5.10.: Time history plot of (a) displacements, (b) velocities ,(c) accelerations and(d) Lambda for STS and MTS method

99

in Figure 5.11. A point load F = 100KN is applied at the free end of the cantilever

beam.

Fig. 5.11.: 3D cantilever beam subjected to a point load simulated using beam-continuum coupled model with MTS method.

Figure 5.12 shows the time histories of displacements and velocities at the free end

of the 3D cantilever beam simulated using beam-continuum coupled model with STS

and MTS methods. It can be observed that the results obtained from MTS methods

matches very well with the STS method.

5.4.2.3 2D Cantilever beam with Newmark method

The cantilever beam simulated using beam-continuum coupled model shown in

Figure 5.13 is used to examine the performance of multi-time-step method for beam

and continuum coupling. The beam is 3m long and has a cross section of 0.5m×0.5m.

Material properties of steel are adopted in this example and the beam is subjected to a

100MN downward load at the free end. For this example, Newmark time integration

is used for both beam and continuum regions.

The time history plot of displacements, velocities and accelerations obtained using

∆T = ∆t, called single-time-step (STS) method, and multi-time-step (MTS) method

with time step ratio m = ∆T/∆t = 2 are compared at the interface node and at the

100

-3

-2.5

-2

-1.5

-1

-0.5

00 0.05 0.1 0.15 0.2

z-D

ispla

cem

ents

(m)

Time (s)

STSMTS

-150

-100

-50

0

50

100

150

0 0.05 0.1 0.15 0.2

z-Ve

loci

ties

(m)

Time (s)

STSMTS

Fig. 5.12.: Time history plot of (a) displacements, (b) velocities for 3D cantileverbeam using STS and MTS methods

101

Fig. 5.13.: 2D cantilever beam subjected to a point load.

free end. Figure 5.14 shows the time history plots at the centroid on the interface

where the beam element node and a continuum element node coincide and Figure

5.15 shows the same for the free end of cantilever beam where load is applied. It can

be observed that the results from the MTS coupling method match well with the STS

coupling method.

5.5 Summary

This chapter presents dynamic coupling of beam and continuum models based on

continuity of displacements and continuity of velocities across the coupling interface.

Examples are first presented to verify the large deformation dynamics and overall

motion of 3D geometrically exact beam models and the energy conserving property

of the mid-point time integration method. An example of the cantilever beam is

then used to verify the coupling of beam and continuum elements for dynamic prob-

lems. The results obtained from the coupled model show a very close match to pure

continuum model while the pure beam model shows a bit softer response, similar to

the observation made for static problems. The results demonstrate that the coupled

model is able to capture the structural response similar to that of the pure continuum

model at a much lesser computational cost since the coupled model has about half

the number of degrees of freedom compared to the full continuum model.

The MTS method for the beam-continuum coupled model is also developed by

extending the formulation of of Prakash et. al 2014 [77] to couple the mid-point time

integration scheme. As an illustration of the MTS coupling method, first an example

102

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.02 0.04 0.06

y-D

ispl

acem

ent (

m)

Time (s)

STSMTS

-60

-40

-20

0

20

40

60

0 0.02 0.04 0.06y-Ve

loci

ty (m

/s)

Time (s)

STSMTS

-1.0E+5-8.0E+4-6.0E+4-4.0E+4-2.0E+40.0E+02.0E+44.0E+46.0E+48.0E+41.0E+5

0 0.02 0.04 0.06

y-A

ccel

erat

ion

(m/s

²)

Time (s)

STSMTS

Fig. 5.14.: Time history plot of displacements, velocity and acceleration of cantileverbeam at the interface simulated using STS and MTS with time-step ratio 2

103

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

00 0.02 0.04 0.06

y-D

ispl

acem

ent (

m)

Time (s)

STSMTS

-150

-100

-50

0

50

100

150

0 0.02 0.04 0.06y-Ve

loci

ty (m

/s)

Time (s)

STSMTS

-2.5E+5

-2.0E+5

-1.5E+5

-1.0E+5

-5.0E+4

0.0E+0

5.0E+4

1.0E+5

1.5E+5

0 0.02 0.04 0.06

y-A

ccel

erat

ion

(m/s

²)

Time (s)

STSMTS

Fig. 5.15.: Time history plot of displacements, velocity and acceleration of cantileverbeam at the free end simulated using STS and MTS with time-step ratio 2

104

of the split single degree of freedom problem is presented. The results obtained

using MTS method with mid-point time integration scheme are very similar to the

ones obtained using STS method. MTS method for beam-continuum coupled model is

verified using a 3D cantilever beam problem. The results obtained using MTS method

with midpoint integration scheme show very good match with the results obtained

using STS method. A 2D cantilever beam problem is solved using a beam-continuum

coupled model with continuity of velocities at the interface and with the MTS method

for Newmark schemes. The time history plots at the beam-continuum interface and

at the free end of the cantilever beam are compared for the MTS and STS methods

showing very similar behavior. This demonstrates that the beam-continuum coupled

model together with the MTS method can be used to simulate structural responses

with the same fidelity as that of a detailed pure continuum model while reducing the

total computational cost.

105

6. SUMMARY AND CONCLUSIONS

In this research, two issues with the current state of the art in modeling of frame

structures under large deformations and large rotations are studied. One issue is

that the numerical treatment of rotational degrees of freedom for problems with large

three dimensional rotations is quite complicated and is not easily accessible to most

practicing structural engineers who use these models. Another issue is that, under

large deformations, frame structures usually sustain significant damage within the

joint and connection regions of the structure and most existing approaches that use

non-linear spring elements or macro-elements are unable to model the full range of

structural behavior starting from the initiation and evolution of damage and leading

to potential collapse. On the other hand, more detailed high-fidelity models, that

usually employ continuum finite elements, are able to model the damage processes

much better, but are computationally very intensive.

To overcome the numerical difficulties associated with large three dimensional

rotations, a reduced-order model that eliminates the rotational degrees of freedom

and is still able to capture response of frame structures under large deformations

and large rotations is presented. The model utilizes a network of extensional and

rotational springs for modeling three dimensional frames. It eliminates the rotational

degrees of freedom at the nodes by expressing the change of angles between adjacent

elements only in terms of the nodal coordinates of these elements. This not only

reduces the complexity of formulation to a level similar to that of solving a truss

structure, but also reduces the total number of degrees of freedom in the model. One

limitation of the reduced-order model is that the torsional deformation of individual

elements is not captured, but of a structure as a whole is still captured. For problems

dominated by flexure, this limitation does not affect the performance of the reduced-

order model. Numerical examples of 2D and 3D right-angle frames are used to verify

106

the reduced-order model for large deformation buckling and post buckling behavior.

It is demonstrated that the results of the reduced order model match well with those

obtained from a reference model that uses shear-flexible 3D beam elements. Dynamics

of this model is also studied with a two-story frame structure subjected to earthquake

ground motion. Time history of displacements, velocities and accelerations show

very good agreement with reference model. Four different error measures are used to

quantify the differences between the results obtained from reduced-order model and

the reference model and these error measures show convergence of the results with

refinement of the reduced-order model.

A geometrically consistent coupling of beam and continuum models for problems

involving large deformations and large rotations is also presented. This approach

enables one to divide the structure into critical and non-critical regions and model

them using continuum and beam elements accordingly. The key idea of this approach

is to enforce geometric consistency at the interface between beam and continuum

elements by constraining the continuum nodes with the kinematic hypothesis of the

adjacent beam regions. The performance of this coupled formulation is studied with a

numerical example of an initially curved cantilever beam modeled with coupled beam-

continuum elements under 3D large deformations and large rotations. Results from

the coupled model are found to be very similar to the response of a pure continuum

model while the pure beam model exhibits a slightly softer response. Coupling of

beam and continuum elements for planar problem is also presented. The elastic

buckling of a right-angle frame is used to verify the performance of the coupled

model under large deformations and the results show that the coupled approach is

able to capture the buckling and post-buckling behavior of the frame well. Finally, an

example considering the pushover analysis of a portal frame is carried out to verify the

elasto-plastic behavior of the coupled model. With mesh refinement in the continuum

regions of the beam-continuum coupled model, the results are shown to converge to a

reference model constructed with only continuum elements. This fact demonstrates

107

that the coupled model is able to capture the geometric and material non-linearities

as well as the pure continuum models at a fraction of the computational cost.

Formulation for the dynamics of beam and continuum coupled models is also de-

veloped for the mid-point rule of time integration. Two different coupling methods

based on the continuity of displacements and continuity of velocities across the in-

terface are investigated. A cantilever beam subjected to a point load at the free

end is used as an example for verifying the dynamics of the coupled model. Time

histories of displacement, velocity and acceleration obtained from the coupled model

show a very close match with the reference model constructed using continuum el-

ements only while the pure beam model, once again, shows slightly softer response.

A multi-time-step method for coupling beam and continuum elements is also devel-

oped to allow beam and continuum regions to be integrated with different time steps.

The multi-time-step formulation for the mid-point time integration method is veri-

fied using a split-single degree of freedom problem. A 3D cantilever beam example

is used to demonstrate the performance of the multi-time-step method with the case

of using the same time-step for the beam and continuum regions as a reference. An

example of a 2D cantilever beam is also presented to compare the performance of the

multi-time-step with Newmark time integration.

The advances made in this research provide the capabilities necessary for simulat-

ing the behavior of frame structures more realistically over a wide range of large de-

formations and large rotations. Future studies may be directed towards incorporating

the effect of torsion within the reduced-order models for frame structures. Coupling

of these reduced-order models with continuum elements to completely eliminate the

numerical issues with the rotational degrees of freedom for large three dimensional

rotations may be another interesting research topic. For the coupled beam-continuum

models, future research may be directed towards investigating the effect of different

coupling constraints and different parametric choices on the stability and accuracy

of the coupling for both static and dynamic problems. A detailed study comparing

the performance of the coupled models to existing approaches for problems involv-

108

ing large deformations and possible collapse of frame structures may help establish

the coupled approach as a viable alternative to simplified empirical models and to

computationally intensive detailed high-fidelity models.

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109

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VITA

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VITA

Hui Liu obtained her bachelor’s degree and master’s in Civil Engineering from

Nanyang Technological University, Singapore in 2004 and 2009 respectively. She

has been pursuing a doctorate in Civil Engineering in Purdue University since 2010.

During her study in Purdue University, she has been employed as research assistant

under the guidance of Professor Arun Prakash. She has been doing research in the

field of computational mechanics to develop computationally efficient models.