Comp Geo Mechanics Lecture Notes

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Lecture Notes onComputational Geomechanics:

Inelastic Finite Elements forPressure Sensitive Materials

Prof. Boris Jeremi;University of California, Davis, California, U.S.A.

with significant contributions, as noted in Chapters, by:

Prof. Zhaohui Yang;University of Alaska, Anchorage, AK, U.S.A.

Dr. Zhao Cheng;Itasca International Inc. Minneapolis, MN, U.S.A.

Dr. Guanzhou Jie;Wells Fargo Securities, New York, NY, U.S.A.

Prof. Kallol Sett;University of Akron, Akron, OH, U.S.A.

Prof. Mahdi Taiebat;

University of British Columbia, Vancouver, BC, Canada

Dr. Matthias PreisigEcole Polytechnique Fdrale de Lausanne, Lausanne, Suisse

Mr. Nima Tafazzoli;University of California, Davis, CA, U.S.A.

Ms. Panagiota Tasiopoulou;National Technical University of Athens, Greece

Version: March 21, 2012, 17:15

Copyright by Boris Jeremi

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Computational Geomechanics Group Lecture Notes 2

Copyright is held by Boris Jeremiunder

Attribution-NonCommercial-ShareAlike 3.0 Unported(CC BY-NC-SA 3.0) license:

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These notes used to be under a different license, given below:

The use of the modeling and simulation system FEI (these lecture notes and accompanying modeling, computational

and visualization tools) for teaching, research and professional practice is strictly encouraged. Copyright and

Copyleft are covered by GPL1 and Woodys (Guthrie) license (adapted by B.J.):

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caught using it without our permission, will be mighty good friends of ourn, cause we dont give a darn.

Read it.

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Purpose

The main purpose of the FEI system (these lecture notes and accompanying modeling tools, computational libraries

and visualization tools) is to help us at the Computational Geomechanics Group at the University of California,

Davis, research and teach numerical solution techniques for civil engineering mechanics problems. Focus is on the

development and use of methods that reduce Kolmogorov complexity and modeling uncertainty.

These lecture notes, in particular, are being developed to document some of the research, teaching and practical

problem solving work within the Computational Geomechanics Group at the University of California at Davis, as

well as to serve as the main reading material for a number of courses.

Work on these lecture notes was motivated by a number of books and lecture notes ( Bathe and Wilson (1976),

Felippa (1993), Lubliner (1990), Crisfield (1991), Chen and Han (1988), Zienkiewicz and Taylor (1991a,b),

Malvern (1969), Dunica and Kolundija (1986), Koji (1997), Hjelmstad (1997), Oberkampf et al. (2002)), that

I have enjoyed over many years.

Why OpenSource?

To allow interested readers from UCDs Computational Geomechanics research group and around the world to

access, use and contribute to a knowledge base (these notes and accompanying software system) that is managed,

organized and quality controlled.


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Comments

Comments are much appreciated! Special thanks to (in chronological order): Miroslav ivkovi (Miroslav

ivkovi), Dmitry J. Nicolsky, Andrzej Niemunis, Robbie Jaeger, (Yiorgos Perikleous),

Robert Roche,

The best way to send a comment on lecture notes is by email, however please read the following NOTE about

sending an email to me. It would be great if you can place the following in the subject line of your email: Lecture

Notes. This will be much appreciated as it will help me filter your email and place it in LectureNotes email-box

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Overview Table of Contents

I Theoretical and Computational Formulations 23

1 Introduction (1996-2003-) 25

2 Finite Element Formulation for Single Phase Material (1989-1994-) 29

3 Micromechanical Origins of Elasto-Plasticity (1994-2002-2010-) 55

4 Small Deformation Elasto-Plasticity (1991-1994-2002-2006-2010-) 57

5 Probabilistic Elasto-Plasticity and Spectral Stochastic Elastic-Plastic Finite Element Method

(2004-2006-2009-) 123

6 Large Deformation Elasto-Plasticity (1996-2004-) 125

7 Solution of Static Equilibrium Equations (1994-) 175

8 Solution of Dynamic Equations of Motion (1989-2006-) 189

9 Finite Element Formulation for Porous Solid-Pore Fluid Systems (1999-2005-) 193

10 Parallel Computing in Computational Geomechanics (1998-2000-2005-) 211

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II Software and Hardware Platform Design and Development 299

III Verification and Validation 301

IV Application to Practical Engineering Problems 303

11 Static Soil-Pile and Soil-Pile Group Interaction in Single Phase Soils (1999-2002-) 305

12 Earthquake-Soil-Structure Interaction, General Aspects (1989-2002-2009-2010-2011-) 367

13 Earthquake-Soil-Structure Interaction, Bridge Structures (2003-2007-2011-) 391

14 Earthquake-Soil-Structure Interaction, Nuclear Power Plants (2010-2011-2012) 479

15 Cyclic Mobility and Liquefaction (2002-2006-2009-) 515

16 Slope Stability in 2D and 3D (1999-2010-) 539

V References 541

VI Appendix 565

A Useful Formulae (1985-1989-1993-...) 567

B Body and Surface Wave Solutions (2005-2001-2010-2011-) 577

C The nDarray Programming Tool (1993-1995-1996-1999-) 597

D Closed Form Gradients to the Potential Function (1993-1994-) 613

E Hyperelasticity, Detailed Derivations (1995-1996-) 621


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Contents

I Theoretical and Computational Formulations 23

1 Introduction (1996-2003-) 25

1.1 Chapter Summary and Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2 Specialization to Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3 Tour of Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.1 Special Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.1.1 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.1.2 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.1.3 Failure Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.2 Generalized Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.3 Sources of Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Finite Element Formulation for Single Phase Material (1989-1994-) 29


2.2 Formulation of the Continuum Mechanics Incremental Equations of Motion . . . . . . . . . . . . 30

2.3 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Isoparametric Solid Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.1 8 Node Brick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.2 20 Node Brick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.3 27 Node Brick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4.4 Isoparametric 8 20 Node Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5 Isoparametric, 3D Beam-Column Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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2.6 Triangular Shell Finite Element with 6DOFs per Node . . . . . . . . . . . . . . . . . . . . . . . . 50

2.7 Two Node, 3D, Frictional Contact Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.7.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.8 Seismic Isolator Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.8.1 Two Node, 3D, Rubber Isolator Finite Element . . . . . . . . . . . . . . . . . . . . . . . 53

2.8.2 Two Node, 3D, Frictional Pendulum Finite Element . . . . . . . . . . . . . . . . . . . . . 53

3 Micromechanical Origins of Elasto-Plasticity (1994-2002-2010-) 55


3.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Particle Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Dilatancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Small Deformation Elasto-Plasticity (1991-1994-2002-2006-2010-) 57


4.2 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.1 Constitutive Relations for Infinitesimal Plasticity . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.2 On Integration Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.3 Midpoint Rule Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.3.1 Accuracy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.3.2 Numerical Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.4 Crossing the Yield Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.5 Singularities in the Yield Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.5.1 Corner Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.5.2 Apex Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.5.3 Influence Regions in Meridian Plane . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4 A Forward Euler (Explicit) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5 A Backward Euler (Implicit) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5.1 Single Vector Return Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5.2 Backward Euler Algorithms: Starting Points . . . . . . . . . . . . . . . . . . . . . . . . . 92


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4.5.2.1 Single Vector Return Algorithm Starting Point. . . . . . . . . . . . . . . . . . . 92

4.5.3 Consistent Tangent Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.5.3.1 Single Vector Return Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5.4 Gradients to the Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.5.4.1 Analytical Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.5.4.2 Finite Difference Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.6 ElasticPlastic Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6.1 Yield Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6.2 Plastic Flow Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.6.3 HardeningSoftening Evolution Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.6.4 Tresca Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.6.5 von Mises Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.6.5.1 Yield and Plastic Potential Functions: von Mises Model (form I) . . . . . . . . . 106

4.6.5.2 Yield and Plastic Potential Functions: von Mises Model (form II) . . . . . . . . 107

4.6.6 Drucker-Prager Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.6.6.1 Yield and Plastic Potential Functions: Drucker-Prager Model (form I) . . . . . . 109

4.6.6.2 Yield and Plastic Potential Functions: Drucker-Prager Model (form II) . . . . . 110

4.6.6.3 Hardening and Softening Functions . . . . . . . . . . . . . . . . . . . . . . . . 111

4.6.7 Modified Cam-Clay Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.6.7.1 Yield and Plastic Potential Functions: Cam-Clay Model . . . . . . . . . . . . . 115

4.6.8 Dafalias-Manzari Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5 Probabilistic Elasto-Plasticity and Spectral Stochastic Elastic-Plastic Finite Element Method

(2004-2006-2009-) 123

6 Large Deformation Elasto-Plasticity (1996-2004-) 125


6.2 Continuum Mechanics Preliminaries: Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.1 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2.2 Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.3 Strain Tensors, Deformation Tensors and Stretch . . . . . . . . . . . . . . . . . . . . . . 129


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6.2.4 Rate of Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3 Constitutive Relations: Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3.2 Isotropic Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.3.3 VolumetricIsochoric Decomposition of Deformation . . . . . . . . . . . . . . . . . . . . 139

6.3.4 SimoSerrins Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3.5 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.3.6 Tangent Stiffness Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.3.7 Isotropic Hyperelastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3.7.1 Ogden Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3.7.2 NeoHookean Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3.7.3 MooneyRivlin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3.7.4 Logarithmic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3.7.5 SimoPister Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4 Finite Deformation HyperelastoPlasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.3 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4.4 Implicit Integration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.4.5 Algorithmic Tangent Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.5 Material and Geometric NonLinear Finite Element Formulation . . . . . . . . . . . . . . . . . . 164

6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.5.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.5.3 Formulation of NonLinear Finite Element Equations . . . . . . . . . . . . . . . . . . . . 165

6.5.4 Computational Domain in Incremental Analysis . . . . . . . . . . . . . . . . . . . . . . . 167

6.5.4.1 Total Lagrangian Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.5.5 Finite Element Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.5.5.1 Strong Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.5.5.2 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172


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6.5.5.3 Linearized Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.5.5.4 Finite Element Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7 Solution of Static Equilibrium Equations (1994-) 175


7.2 The Residual Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.3 Constraining the Residual Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.4 Load Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.5 Displacement Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.6 Generalized, HyperSpherical Arc-Length Control . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.6.1 Traversing Equilibrium Path in Positive Sense . . . . . . . . . . . . . . . . . . . . . . . . 182

7.6.1.1 Positive External Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.6.1.2 Angle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.6.2 Predictor step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.6.3 Automatic Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.6.4 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.6.5 The Algorithm Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8 Solution of Dynamic Equations of Motion (1989-2006-) 189


8.2 The Principle of Virtual Displacements in Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.3 Direct Integration Methods for the Equations of Dynamic Equilibrium . . . . . . . . . . . . . . . 190

8.3.1 Newmark Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.3.2 HHT Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9 Finite Element Formulation for Porous Solid-Pore Fluid Systems (1999-2005-) 193


9.2 General form of upU Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.2 Governing Equations of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.2.1 The equilibrium equation of the mixture . . . . . . . . . . . . . . . . . . . . . . 195


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9.2.2.2 The equilibrium equation of the fluid . . . . . . . . . . . . . . . . . . . . . . . 195

9.2.2.3 Flow conservation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.2.3 Modified Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.2.3.1 Solid part equilibrium equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9.2.3.2 Fluid part equilibrium equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9.2.3.3 Mixture balance of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9.3 Numerical Solution of the upU Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 199

9.3.1 Numerical Solution of solid part equilibrium equation . . . . . . . . . . . . . . . . . . . . 200

9.3.2 Numerical Solution of fluid part equilibrium equation . . . . . . . . . . . . . . . . . . . . 201

9.3.3 Numerical Solution of flow conservation equation . . . . . . . . . . . . . . . . . . . . . . 203

9.3.4 Matrix form of the governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.3.5 Choice of shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.4 u-p Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.4.1 Governing Equations of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.4.2 Numerical Solutions of the Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 207

9.4.2.1 Numerical solution of the total momentum balance . . . . . . . . . . . . . . . . 207

9.4.2.2 Numerical solution of the fluid mass balance . . . . . . . . . . . . . . . . . . . 208

9.4.2.3 Matrix form of the governing equations . . . . . . . . . . . . . . . . . . . . . . 209

10 Parallel Computing in Computational Geomechanics (1998-2000-2005-) 211


10.2 Plastic Domain Decomposition Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

10.2.2 Inelastic Parallel Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

10.2.2.1 Adaptive Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

10.2.2.2 Multiphase Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.2.2.3 Multiconstraint Graph Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . 216

10.2.2.4 Adaptive PDD Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.2.3 Adaptive Multilevel Graph Partitioning Algorithm . . . . . . . . . . . . . . . . . . . . . . 218

10.2.3.1 Unified Repartitioning Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 222


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10.2.3.2 Study of ITR in ParMETIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.3 Performance Studies on PDD Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.3.2 Parallel Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

10.3.3 Soil-Foundation Interaction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

10.3.4 Numerical Study for ITR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

10.3.5 Parallel Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

10.3.5.1 Soil-Foundation Model with 4,035 DOFs . . . . . . . . . . . . . . . . . . . . . 236

10.3.5.2 Soil-Foundation Model with 4,938 Elements, 17,604 DOFs . . . . . . . . . . . . 240

10.3.5.3 Soil-Foundation Model with 9,297 Elements, 32,091 DOFs . . . . . . . . . . . . 245

10.3.6 Algorithm Fine-Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

10.3.7 Fine Tuning on Load Imbalance Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . 250

10.3.8 Globally Adaptive PDD Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

10.3.8.1 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

10.3.8.2 Performance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10.3.9 Scalability Study on Prototype Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

10.3.9.1 3 Bent SFSI Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . 261

10.3.9.2 Scalability Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

10.3.10Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

10.4 Application of Project-Based Iterative Methods in SFSI Problems . . . . . . . . . . . . . . . . . 271

10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

10.4.2 Projection-Based Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

10.4.2.1 Conjugate Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

10.4.2.2 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

10.4.2.3 BiCGStab and QMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

10.4.3 Preconditioning Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

10.4.4 Preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

10.4.4.1 Jacobi Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

10.4.4.2 Incomplete Cholesky Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . 276


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10.4.4.3 Robust Incomplete Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 277

10.4.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

10.4.6 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

10.5 Performance Study on Parallel Direct/Iterative Solving in SFSI . . . . . . . . . . . . . . . . . . . 290

10.5.1 Parallel Sparse Direct Equation Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

10.5.1.1 General Techniques SPOOLES . . . . . . . . . . . . . . . . . . . . . . . . . . 291

10.5.1.2 Frontal and Multifrontal Methods MUMPS . . . . . . . . . . . . . . . . . . . 292

10.5.1.3 Supernodal Algorithm SuperLU . . . . . . . . . . . . . . . . . . . . . . . . . 294

10.5.2 Performance Study on SFSI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

10.5.2.1 Equation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

10.5.2.2 Performance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

10.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

II Software and Hardware Platform Design and Development 299

III Verification and Validation 301

IV Application to Practical Engineering Problems 303

11 Static Soil-Pile and Soil-Pile Group Interaction in Single Phase Soils (1999-2002-) 305


11.2 Numerical Analysis of Pile Behavior under Lateral Loads in Layered ElasticPlastic Soils . . . . . 306

11.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

11.2.2 Constitutive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

11.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

11.2.3.1 Pile Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

11.2.3.2 Plastic Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

11.2.3.3 p y Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

11.2.3.4 Comparisons of Pile Behavior in Uniform and Layered Soils . . . . . . . . . . . . 315

11.2.3.5 Comparison to Centrifuge Tests and LPile Results . . . . . . . . . . . . . . . . 322


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11.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

11.3 Study of soil layering effects on lateral loading behavior of piles . . . . . . . . . . . . . . . . . . . 326

11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

11.3.2 Finite Element Pile Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

11.3.3 Constitutive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

11.3.4 Comparison of py Behavior in Uniform and Layered Soil Deposits . . . . . . . . . . . . . 330

11.3.4.1 Uniform Clay Deposit and Clay Deposit with an Interlayer of Sand. . . . . . . . 330

11.3.4.2 Uniform Sand Deposit and Sand Deposit with an Interlayer of Soft Clay. . . . . 332

11.3.5 Parametric Study for the Lateral Resistance Ratios in Terms of Stiffness and Strength

Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

11.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

11.4 Numerical Study of Group Effects for Pile Groups in Sands . . . . . . . . . . . . . . . . . . . . . 343

11.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

11.4.2 Pile Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11.4.3 Summary of Centrifuge Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11.4.4 Finite Element Pile Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11.4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

11.4.6 Plastic Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

11.4.7 Bending Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

11.4.8 Load Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

11.4.9 p y Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

11.4.10Comparison with the Centrifuge Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

11.4.11Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

12 Earthquake-Soil-Structure Interaction, General Aspects (1989-2002-2009-2010-2011-) 367


12.2 Seismic Energy Propagation and Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

12.2.1 Seismic energy input into SFS system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

12.2.2 Seismic energy dissipation in SFS system . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

12.3 Free Field Ground Motions in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370


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12.3.1 Seismic Motion Lack of Correlation (Incoherence) . . . . . . . . . . . . . . . . . . . . . . 372

12.3.2 Lack of Correlation Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . 373

12.4 Multi-Directional and Seismic Input Coming in at Inclined Angle . . . . . . . . . . . . . . . . . . 374

12.5 Dynamic Soil-Foundation-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

12.6 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

12.7 The Domain Reduction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

12.7.1 Method Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

12.7.2 Method Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

12.8 Numerical Accuracy and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

12.8.1 Grid Spacing h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

12.8.2 Time Step Length t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

12.8.3 Nonlinear Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

12.9 Domain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

12.9.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

12.9.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

13 Earthquake-Soil-Structure Interaction, Bridge Structures (2003-2007-2011-) 391


13.2 Case History: Earthquake-Soil-Structure Interaction for a Bridge System . . . . . . . . . . . . . . 393

13.2.1 Prototype Bridge Model Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

13.2.1.1 Soil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

13.2.1.2 Element Size Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

13.2.1.3 Time Step Length Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . 398

13.2.1.4 Domain Reduction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

13.2.1.5 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

13.2.1.6 Simulation Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

13.2.2 Earthquake Simulations - 1994 Northridge . . . . . . . . . . . . . . . . . . . . . . . . . . 405

13.2.2.1 Input Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

13.2.2.2 Displacement Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

13.2.2.3 Acceleration Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411


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13.2.2.4 Displacement Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 414

13.2.2.5 Acceleration Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

13.2.2.6 Structural Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

13.2.3 Earthquake Simulations - 1999 Turkey Kocaeli . . . . . . . . . . . . . . . . . . . . . . . . 424

13.2.3.1 Displacement Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

13.2.3.2 Acceleration Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

13.2.3.3 Displacement Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 433

13.2.3.4 Acceleration Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

13.2.3.5 Structural Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

13.2.4 Earthquake Soil Structure Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . 443

13.2.4.1 How Strength of Soil Foundations Affects ESSI . . . . . . . . . . . . . . . . . . 443

13.2.4.2 How Site Non-Uniformity Affects ESSI . . . . . . . . . . . . . . . . . . . . . . . 449

13.2.4.3 How Input Motion Affects ESSI . . . . . . . . . . . . . . . . . . . . . . . . . . 470

14 Earthquake-Soil-Structure Interaction, Nuclear Power Plants (2010-2011-2012) 479

14.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

14.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

14.3 Slipping behavior of SFSI models by considering 1D wave propagation . . . . . . . . . . . . . . . 484

14.3.1 Morgan Hill earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

14.3.2 Ricker wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

14.4 Slipping behavior of SFSI models by considering 3D wave propagation . . . . . . . . . . . . . . . 497

14.4.1 Ricker wave, with fault located at 45 towards the top middle point of the model . . . . . 499

14.4.2 Ricker wave, with fault located at 34 towards the top middle point of the model . . . . . 506

14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

15 Cyclic Mobility and Liquefaction (2002-2006-2009-) 515


15.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

15.3 Liquefaction of Level and Sloping Grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

15.3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

15.3.2 Behavior of Saturated Level Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520


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15.3.3 Behavior of Saturated Sloping Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

15.4 Pile in Liquefied Ground, Staged Simulation Model Development . . . . . . . . . . . . . . . . . . 524

15.4.1 First Loading Stage: Self Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

15.4.2 Second Loading Stage: PileColumn Installation . . . . . . . . . . . . . . . . . . . . . . . 527

15.4.3 Third Loading Stage: Seismic Shaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

15.4.4 Free Field, Lateral and Longitudinal Models . . . . . . . . . . . . . . . . . . . . . . . . . 529

15.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

15.5.1 Pore Fluid Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

15.5.2 Soil Skeleton Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

15.5.3 Pile Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

15.5.4 Pile Pinning Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

16 Slope Stability in 2D and 3D (1999-2010-) 539

V References 541

VI Appendix 565

A Useful Formulae (1985-1989-1993-...) 567

A.1 Chapter Summary and Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

A.2 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

A.2.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

A.2.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

A.3 Derivatives of Stress Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

B Body and Surface Wave Solutions (2005-2001-2010-2011-) 577

B.1 Matlab code body wave solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

B.2 Matlab code surface wave solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

B.3 Matlab code Ricker wavelet as an input motion . . . . . . . . . . . . . . . . . . . . . . . . . . 588

C The nDarray Programming Tool (1993-1995-1996-1999-) 597

C.1 Chapter Summary and Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598


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C.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

C.3 nDarray Programming Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

C.3.1 Introduction to the nDarray Programming Tool . . . . . . . . . . . . . . . . . . . . . . . 599

C.3.2 Abstraction Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

C.3.2.1 nDarray_rep class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

C.3.2.2 nDarray class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

C.3.2.3 Matrix and Vector Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

C.3.2.4 Tensor Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

C.4 Finite Element Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

C.4.1 Stress, Strain and Elastoplastic State Classes . . . . . . . . . . . . . . . . . . . . . . . . 604

C.4.2 Material Model Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

C.4.3 Stiffness Matrix Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

C.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

C.5.1 Tensor Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

C.5.2 Fourth Order Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

C.5.3 Elastic Isotropic Stiffness and Compliance Tensors . . . . . . . . . . . . . . . . . . . . . . 608

C.5.4 Second Derivative of Stress Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

C.5.5 Application to Computations in Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . 610

C.5.6 Stiffness Matrix Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

C.6 Performance Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

C.7 Summary and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

D Closed Form Gradients to the Potential Function (1993-1994-) 613

D.1 Chapter Summary and Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

E Hyperelasticity, Detailed Derivations (1995-1996-) 621

E.1 Chapter Summary and Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

E.2 SimoSerrins Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

E.3 Derivation of 2volW/(CIJ CKL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

E.4 Derivation of 2isoW/(CIJ CKL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

E.5 Derivation of wA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626


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E.6 Derivation of YAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627


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Part I

Theoretical and Computational

Formulations

23

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Chapter 1

Introduction (1996-2003-)

25

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1.1 Chapter Summary and Highlights

1.2 Specialization to Computational Mechanics

In this section we start from general mechanics and specialize our interest toward the field of computational

mechanics...

1.2.0.0.1 Mechanics

1.2.0.0.2 Computational Mechanics

1.2.0.0.3 Statics and Dynamics

1.2.0.0.4 Linear and Nonlinear Analysis

1.2.0.0.5 Elastic and Inelastic Analysis

1.2.0.0.6 Discretization Methods

1.2.0.0.7 The Solution Morass

1.2.0.0.8 Smooth and Rough nonlinearities

1.3 Tour of Computational Mechanics

In this section we describe various examples of equilibrium path and set up basic terminology.


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1.3.0.0.9 Equilibrium Path

1.3.1 Special Equilibrium Points

1.3.1.1 Critical Points

1.3.1.1.1 Limit Points

1.3.1.1.2 Bifurcation Points

1.3.1.2 Turning Points

1.3.1.3 Failure Points

1.3.2 Generalized Response

1.3.3 Sources of Nonlinearities

Tonti Diagrams


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Chapter 2

Finite Element Formulation for Single

Phase Material (1989-1994-)

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2.1 Chapter Summary and Highlights

2.2 Formulation of the Continuum Mechanics Incremental Equa-

tions of Motion

Consider1 the motion of a general body in a stationary Cartesian coordinate system, as shown in Figure (2.1),

and assume that the body can experience large displacements, large strains, and nonlinear constitutive response.

The aim is to evaluate the equilibrium positions of the complete body at discrete time points 0,t, 2t, . . . ,

where t is an increment in time. To develop the solution strategy, assume that the solutions for the static and

kinematic variables for all time steps from 0 to time t inclusive, have been obtained. Then the solution process

for the next required equilibrium position corresponding to time t+t is typical and would be applied repetitively

until a complete solution path has been found. Hence, in the analysis one follows all particles of the body in their

motion, from the original to the final configuration of the body. In so doing, we have adopted a Lagrangian ( or

material ) formulation of the problem.

t+ t

t+ t ui

tui

t+ tu ix i

0

ix0

ixt

0

n

n+1

1

22 20 t

2 3)P(

0

0V

t

t

AV

V

0

3330 tx x x

x x x

x x x,

, ,

t+ t

t+ t t+ t t+ t

t+ t

t+ t

t+ t

1tP( x , 2tx , )t 3x

01P( x )0 3x02x

0Configuration at time

t

Configuration at time

Configuration at time t+ t

, ,

,,

,

t+ t

i

i

ix

x

x

1x,1xt,1x

t

A

A

i=1,2,3

++

+=

=

=

Figure 2.1: Motion of body in stationary Cartesian coordinate system

In the Lagrangian incremental analysis approach we express the equilibrium of the body at time t+ t using the

1detailed derivations and explanations are given in Bathe (1982)


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principle of virtual displacements. Using tensorial notation2 this principle requires that:

t+tV

t+tij t+teijt+tdV = t+tR (2.1)

where the t+tij are Cartesian components of the Cauchy stress tensor, the t+teij are the Cartesian components

of an infinitesimal strain tensor, and the means "variation in" i.e.:

t+teij = 1

2

(ui

t+txj+

uj t+txi

)=

1

2

(ui

t+txj+

uj t+txi

)(2.2)

It should be noted that Cauchy stresses are "body forces per unit area" in the configuration at time t+ t, and

the infinitesimal strain components are also referred to this as yet unknown configuration. The right hand side of

equation (2.1), i.e. t+tR is the virtual work performed when the body is subjected to a virtual displacement attime t+ t:

t+tR =t+tV

(t+tfBi ut+ti

)ut+ti

t+tdV +

t+tS

t+tfSi ut+ti

t+tdS (2.3)

where t+tfBi andt+tfSi are the components of the externally applied body and surface force vectors, re-

spectively, and uit+t is the inertial body force that is present if accelerations are present3, ui is the ithcomponent of the virtual displacement vector.

A fundamental difficulty in the general application of equation (2.1) is that the configuration of the body at time

t+ t is unknown. The continuous change in the configuration of the body entails some important consequences

for the development of an incremental analysis procedure. For example, an important consideration must be that

the Cauchy stress at time t+ t cannot be obtained by simply adding to the Cauchy stresses at time t a stress

increment which is due only to the straining of the material. Namely, the calculation of the Cauchy stress at

time t+ t must also take into account the rigid body rotation of the material, because the components of the

Cauchy stress tensor are not invariant with respect to the rigid body rotations4.

The fact that the configuration of the body changes continuously in a large deformation analysis is dealt with in

an elegant manner by using appropriate stress and strain measures and constitutive relations. When solving the

general problem5 one possible approach6 is given in Simo (1988). Previous discussion was oriented toward small

deformation, small displacement analysis leading to the use of Cauchy stress tensor ij and small strain tensor

eij .

2Einsteins summation rule is implied unless stated differently, all lower case indices (i, j, p, q,m, n, o, r, s, t, . . . ) can have values

of 1, 2, 3, and values for capital letter indices will be specified where need be.3This is based on DAlemberts principle.4However, that problem will not be addressed in this work since this work deals with MaterialNonlinearOnly analysis of solids,

thus excluding large displacement and large strain effects.5That is, large displacements, large deformations and nonlinear constitutive relations.6This is still a "hot" research topic!


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In the following, we will briefly cover some other stress and strain measures particularly useful in large strain and

large displacement analysis.

The basic equation that we want to solve is relation (2.1), which expresses the equilibrium and compatibility

requirements of the general body considered in the configuration corresponding to time t+ t. The constitutive

equations also enter (2.1) through the calculation of stresses. Since in general the body can undergo large

displacements and large strains, and the constitutive relations are nonlinear, the relation in (2.1) cannot be solved

directly. However, an approximate solution can be obtained by referring all variables to a previously calculated

known equilibrium configuration, and linearizing the resulting equations. This solution can then be improved by

iterations.

To develop the governing equations for the approximate solution obtained by linearization we recall that the

solutions for time 0,t, 2t, . . . , t have already been calculated and that we can employ the fact that the 2nd

PiolaKirchhoff stress tensor is energy conjugate to the GreenLagrange strain tensor:

0V

t0Sij

t0ij

0dV =

0V

(0t

0txi,m

tmn0txj,n

) (t0xk,i

ttekl

t0 xl,j

)0dV =

0V

0t

tmn t0emn

0dV (2.4)

because:

t0xk,l

0txl,m = km

and since:

00dV = ttdV

we have:

0V

t0Sij

t0ij

0dV =

0V

tmn ttemn

tdV (2.5)

where 2nd PiolaKirchhoff stress tensor is defined as:

t0Sij =

0t

0txi,m

tmn0txj,n (2.6)


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where 0txj,n =0xitxm

and0t represents the ratio of the mass density at time 0 and time t, and the GreenLagrange

strain is defined as:

t0ij =

1

2

(t0ui,j +

t0uj,i +

t0uk,i

t0uk,j

)(2.7)

Then, by employing (2.5) we refer the stresses and strains to one of these known equilibrium configurations.

The choice lies between two formulations which have been termed total Lagrangian and updated Lagrangian

formulations.

In the total Lagrangian formulations, also termed Lagrangian formulation, all static and kinematic variables

are referred to the initial configuration at time 0, while in the updated Lagrangian formulation all static and

kinematic variables are referred to the configuration at time t. Both the total Lagrangian and updated Lagrangian

formulations include all kinematic nonlinear effects due to large displacement, large rotations and large strains,

but whether the large strain behavior is modeled appropriately depends on the constitutive relations specified.

The only advantage of using one formulation rather than the other lies in its greater numerical efficiency.

Using (2.5) in the total Lagrangian formulations we consider this basic equation:

0V

t+t0 Sij

t+t0 ij

0dV = t+tR (2.8)

while in the updated Lagrangian formulations we consider:

tV

t+tt Sij

t+tt ij

tdV = t+tR (2.9)

in which t+tR is the external virtual work as defined in (2.3). Approximate solution to the (2.8) and (2.9) can beobtained by linearizing these relations. By comparing the total Lagrangian and updated Lagrangian formulations

we can observe that they quite analogous and that, in fact, the only theoretical difference between the two

formulations lies in the choice of different reference configurations for the kinematic and static variables. If in the

numerical solution the appropriate constitutive tensors are employed, identical results are obtained.

Coupling of large deformation, large displacement and material nonlinear analysis is still the topic of research in the

research community. Possible direction may be the use of both Lagrangian and Eulerian formulation intermixed

in one scheme.


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2.3 Finite Element Discretization

Consider the equilibrium of a general threedimensional body such as in Figure (2.2) (Bathe, 1996). The external

forces acting on a body are surface tractions fSi and body forces fBi . Displacements are ui and strain tensor

7 is

eij and the stress tensor corresponding to strain tensor is ij .

x2

x1

x3

1

1

2

2

3

3f B

f Bf B

f Sf S

f S

1r

r3

r2

Figure 2.2: General three dimensional body

Assume that the externally applied forces are given and that we want to solve for the resulting displacements,

strains and stresses. One possible approach to express the equilibrium of the body is to use the principle of virtual

displacements. This principle states that the equilibrium of the body requires that for any compatible, small

virtual displacements8 imposed onto the body, the total internal virtual work is equal to the total external virtual

work. This statement can be mathematically expressed using equation (2.10) for the body at time t + t, and

since we are using the incremental approach let us drop the time dimension, so that all the equations are imposed

for the given increment9, at time t+ t. The equation is now, using tensorial notation10:

V

ij eij dV =

V

(fBi ui

)ui dV +

S

fSi ui dS (2.10)

The internal work given on the left side of (2.10) is equal to the actual stresses ij going through the virtual7 small strain tensor as defined in equation: eij = 12 (ui,j + uj,i).8which satisfy the essential boundary conditions.9t+ t will be dropped from now one in this chapter.

10Einsteins summation rule is implied unless stated differently, all lower case indices (i, j, p, q,m, n, o, r, s, t, . . . ) can have values

of 1, 2, 3, and values for capital letter indices will be specified where need be.


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strains eij that corresponds to the imposed virtual displacements. The external work is on the right side of (2.10)

and is equal to the actual (surface) forces fSi and (body) forces fBi ui going through the virtual displacements

ui.

It should be emphasized that the virtual strains used in (2.10) are those corresponding to the imposed body

and surface virtual displacements, and that these displacements can be any compatible set of displacements that

satisfy the geometric boundary conditions. The equation in (2.10) is an expression of equilibrium, and for different

virtual displacements, correspondingly different equations of equilibrium are obtained. However, equation (2.10)

also contains the compatibility and constitutive requirements if the principle is used in the appropriate manner.

Namely, the displacements considered should be continuous and compatible and should satisfy the displacement

boundary conditions, and the stresses should be evaluated from the strains using appropriate constitutive relations.

Thus, the principle of virtual displacements embodies all requirements that need be fulfilled in the analysis of a

problem in solid and structural mechanics. The principle of virtual displacements can be directly related to the

principle that total potential of the system must be stationary.

In the finite element analysis we approximate the body in Figure (2.2) as an assemblage of discrete finite elements

with the elements being interconnected at nodal points on the element boundaries. The displacements measured

in a local coordinate system r1, r2 and r3 within each element are assumed to be a function of the displacements

at the N finite element nodal points:

u ua = HI uIa (2.11)

where I = 1, 2, 3, . . . , n and n is number of nodes in a specific element, a = 1, 2, 3 represents a number of

dimensions (can be 1 or 2 or 3), HI represents displacement interpolation vector, uIa is the tensor of global

generalized displacement components at all element nodes. The use of the term generalized displacements means

that both translations, rotations, or any other nodal unknown are modeled independently. Here specifically only

translational degrees of freedom are considered. The strain tensor is defined as:

eab =1

2(ua,b + ub,a) (2.12)

and the by using (2.11) we can define the approximate strain tensor:

eab eab = 12

(ua,b + ub,a) =

=1

2

((HI uIa),b + (HI uIb),a

)=

=1

2((HI,b uIa) + (HI,a uIb)) (2.13)


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The most general stressstrain relationship11 for an isotropic material is:

ab = Eabcd(ecd e0cd

)+ 0ab (2.14)

where ab is the approximate Cauchy stress tensor, Eabcd is the constitutive tensor12, ecd is the infinitesimal

approximate strain tensor, e0cd is the infinitesimal initial strain tensor and 0ab is the initial Cauchy stress tensor.

Using the assumption of the displacements within each finite element, as expressed in (2.11), we can now derive

equilibrium equations that corresponds to the nodal point displacements of the assemblage of finite elements. We

can rewrite (2.10) as a sum13 of integrations over the volume and areas of all finite elements:

m

Vm

ab eab dVm =

m

Vm

(fBa ua

)ua dV

m +m

Sm

fSa uSa dS

m (2.15)

where m = 1, 2, 3, . . . , k and k is the number of elements. It is important to note that the integrations in (2.15)

are performed over the element volumes and surfaces, and that for convenience we may use different element

coordinate systems in the calculations. If we substitute equations (2.11), (2.12), (2.13) and (2.14) in (2.15) it

follows:

m

Vm

(Eabcd

(ecd e0cd

)+ 0ab

)

(1

2(HI,b uIa +HI,a uIb)

)dV m =

m

Vm

fBa (HI uIa) dVm

m

Vm

HJ uJa (HI uIa) dVm +

m

Sm

fSa (HI uIa) dSm (2.16)

or:

m

Vm

(Eabcd

((1

2(HJ,d uJc +HJ,c uJd)

) e0cd

)+ 0ab

)

(1


)dV m =

=m

Vm

fBa (HI uIa) dVm

m

Vm

HJ uJa (HI uIa) dVm +

m

Sm

fSa (HI uIa) dSm

(2.17)

We can observe that in the previous equations represents a virtual quantity but the rules for are quite similar to

regular differentiation so that can enter the brackets and "virtualize" the nodal displacement14. It thus follows:11in terms of exact stress and strain fields but it holds for approximate fields as well.12This tensor can be elastic or elastoplastic constitutive tensor.13Or, more correctly as a union

m since we are integrating over the union of elements.

14since they are driving variables that define overall displacement field through interpolation functions


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m

Vm

(Eabcd

((1


) e0cd

)+ 0ab

) (1


)dV m =

=m

Vm

fBa (HIuIa) dVm

m

Vm

HJ uJa (HIuIa) dVm +

m

Sm

fSa (HIuIa) dSm

(2.18)

Let us now work out some algebra in the left hand side of equation (2.18):

m

Vm

(Eabcd

((HJ,d uJc +HJ,c uJd)

2

) Eabcde0cd + 0ab

) ((HI,b uIa +HI,a uIb)

2

)dV m =

=m

Vm

fBa (HIuIa) dVm

m

Vm

HJ uJa HIuIa dVm +

m

Sm

fSa (HIuIa) dSm

(2.19)

and further:

m

Vm

((1


)Eabcd

(1


))dV m +

+m

Vm

(Eabcd e0cd

(1


))dV m +

+m

Vm

(0ab) (1


)dV m =

m

Vm

fBa (HIuIa) dVm

m

Vm

HJ uJa HIuIa dVm

+m

Sm

fSa (HIuIa) dSm (2.20)

Several things should be observed in the equation (2.20). Namely, the first three lines in the equation can be

simplified if one takes into account symmetries of Eijkl and ij . In the case of the elastic stiffness tensor Eijkl

major and both minor symmetries exist. In the case of the elastoplastic stiffness tensor, such symmetries exists if

a flow a rule is associated. If flow rule is nonassociated, only minor symmetries exist while major symmetry is

destroyed15. As a matter of fact, both minor symmetries in Eijkl are the only symmetries we need, and the first

line of (2.20) can be rewritten as:

15for more on stiffness tensor symmetries see sections (4.2, 4.4 and 4.5)


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m

Vm

((1

2(HJ,d uJc +HJ,c ujd)

)Eabcd

(1


))dV m =

=m

Vm

(HJ,d uJc) Eabcd (HI,b uIa) dVm =

=m

Vm

(HI,b uIa) Eabcd (HJ,d uJc) dVm (2.21)

Similar simplifications are possible in second and third line of equation (2.20). Namely, in the second line we can

use both minor symmetries of Eijkl so that:

m

Vm

(Eabcd e0cd

(1


))dV m =

=m

Vm

(Eabcd e0cd (HI,b uIa)) dV m (2.22)

and the third line can be simplified due to the symmetry in Cauchy stress tensor ij as:

m

Vm

(0ab) (1


)dV m =

=m

Vm

(0ab)

(HI,b uIa) dVm (2.23)

After these simplifications, equation (2.20) looks like this:

m

Vm

(HI,b uIa) Eabcd (HJ,d uJc) dVm +

+m

Vm

(Eabcd e0cd (HI,b uIa)) dV m +m

Vm

(0ab)

(HI,b uIa) dVm =

=m

Vm

fBa (HIuIa) dVm

m

Vm

HJ uJa HIuIa dVm +

m

Sm

fSa (HIuIa) dSm (2.24)

or if we leave the unknown nodal accelerations16 uJc and displacements uJc on the left hand side and move all

the known quantities on to the right hand side:16It is noted that uJc = ac uJa relationship was used here, where ac is the Kronecker delta.


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m

Vm

HJ ac uJc HIuIa dVm +

m

Vm

(HI,b uIa) Eabcd (HJ,d uJc) dVm =

=m

Vm

fBa (HIuIa) dVm +

m

Sm

fSa (HIuIa) dSm +

+m

Vm

(Eabcd e

0cd (HI,b uIa)

)dV m

m

Vm

(0ab)

(HI,b uIa) dVm (2.25)

To obtain the equation for the unknown nodal generalized displacements from (2.25), we invoke the virtual

displacement theorem which states that virtual displacements are any, non zero, kinematically admissible dis-

placements. In that case we can factor out nodal virtual displacements uIa so that equation (2.25) becomes:

[m

Vm

HJ ac uJc HI dVm +

m

Vm

(HI,b) Eabcd (HJ,d uJc) dVm

]uIa =

=m

[Vm

fBa HI dVm

]uIa +

m

[Sm

fSa HI dSm

]uIa +

+m

[Vm

(Eabcd e

0cd HI,b

)dV m

]uIa

m

[Vm

(0ab)HI,b dV

m

]uIa (2.26)

and now we can just cancel uIa on both sides:

m

Vm

HJ ac HI uJcdVm +

m

Vm

(HI,b) Eabcd (HJ,d uJc) dVm =

=m

Vm

fBa HI dVm +

m

Sm

fSa HI dSm +

+m

Vm

(Eabcd e

0cd HI,b

)dV m

m

Vm

(0ab)HI,b dV

m (2.27)

One should also observe that in the first line of equation (2.27) generalized nodal accelerations uJc and generalized

nodal displacements uJc are unknowns that are not subjected to integration so they can be factored out of the

integral:


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m

Vm

HJ ac HI dVm uJc

+m

Vm

HI,b Eabcd HJ,d dVm uJc

=m

Vm

fBa HI dVm +

m

Sm

fSa HI dSm +

+m

Vm

(Eabcd e

0cd HI,b

)dV m

m

Vm

(0ab)HI,b dV

m (2.28)

We can now define several tensors from equation (2.28):

(m)MIacJ =

Vm

HJ ac HI dVm (2.29)

(m)KIacJ =

Vm

HI,b Eabcd HJ,d dVm (2.30)

(m)FBIa =

Vm

fBa HI dVm (2.31)

(m)FSIa =

Sm

fSa HI dSm (2.32)

(m)Fe0mnIa =

Vm

Eabcd e0cd HI,b dV

m (2.33)

(m)F0mnIa =

Vm

0ab HI,b dVm (2.34)


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where (m)KIacJ is the element stiffness tensor, (m)FBIa is the tensor of element body forces,(m)FSIa is the tensor

of element surface forces, (m)F e0mn

Ia is the tensor of element initial strain effects,(m)F

0mnIa is the tensor of element

initial stress effects. Now equation (2.28) becomes:

(m)

(m)MIacJ uJc +(m)

(m)KIacJ uJc =m

(m)FBIa +m

(m)FSIa +m

(m)Fe0mnIa

m

(m)F0mnIa (2.35)

By summing17 all the relevant tensors, a well known equation is obtained:

MAacB uBc +KAacB uBc = FAa (2.36)

A,B = 1, 2, . . . ,# of nodes

a, c = 1, . . . ,# of dimensions (1, 2 or 3)

where:

MAacB =m

(m)MIacJ ; KAacB =m

(m)KIacJ (2.37)

are the system mass and stiffness tensors, respectively, uBc is the tensor of unknown nodal accelerations, and uBc

is the tensor of unknown generalized nodal displacements, while the load tensor is given as:

FAa =m

(m)FBIa +m

(m)FSIa +m

(m)Fe0mnIa

m

(m)F0mnIa (2.38)

After assembling the system of equations in (2.37) it is relatively easy to solve for the unknown displacements uLc

either for static or fully dynamic case. It is also very important to note that in all previous equations, omissions

of inertial force term (all terms with ) will yield static equilibrium equations. Description of solutions procedures

for static linear and nonlinear problems are described in some detail in chapter 7. In addition to that, solution

procedures for dynamic, linear and nonlinear problems are described in some detail in chapter 8.

A note on the final form of the tensors used is in order. In order to use readily available system of equation solvers

equation (2.37) will be rewritten in the following from:17Summation of the element volume integrals expresses the direct addition of the element tensors to obtain global, system tensors.

This method of direct addition is usually referred to as the direct stiffness method.


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MPQ uP +KPQ uP = FQ P,Q = 1, 2, . . . , (DOFperNode)N (2.39)

where MPQ is system mass matrix, KPQ is system stiffness matrix and FQ is the loading vector. Matrix form of

equation 2.37, presented as equation 2.39 is obtained flattening the system mass tensor MAacB , system stiffness

tensor KAacB , unknown acceleration tensor uBc, unknown displacement tensor uBc and the system loading tensor

FAa. Flattening from the fourth order mass/stiffness tensors to two dimensional mass/stiffness matrix is done by

simply performing appropriate (re) numbering of nodal DOFs in each dimension. Similar approach is used for

unknown accelerations/displacements and for loadings.

2.3.0.0.1 Static Analysis: Internal and External Loads.

Internal and external loading tensors can be defined as:

(fIa)int =(m)

(m)KIacJ uJc =m

Vm

ab HI,b dVm (2.40)

(fIa)ext =m

(m)FBIa +m

(m)FSIa +m

(m)Fe0mnIa

m

(m)F0mnIa (2.41)

where (fIa)int is the internal force tensor and (fIa)ext is the external force tensor. Equilibrium is obtained when

residual:

rIa(uJc, ) = (fIa (uJc))int (fIa)ext (2.42)

is equal to zero, r(u, ) = 0. The same equation in flattened form yields:

r(u, ) = fint(u) fext = 0 (2.43)


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Table 2.1: Values of , , and at each of the eight nodesNode

1 +1 +1 +1

2 -1 +1 +1

3 -1 -1 +1

4 +1 -1 +1

5 +1 +1 -1

6 -1 +1 -1

7 -1 -1 -1

8 +1 -1 -1

2.4 Isoparametric Solid Finite Elements

2.4.1 8 Node Brick

Figure 2.3: 8 node brick element

Shape function of the nodes which i indicates the node number:

N(e)i =

1

8(1 + i)(1 + i)(1 + i) (2.44)


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Table 2.2: Values of , , and at each of the 9th to 20th nodesNode

9 0 +1 +1

10 -1 0 +1

11 0 -1 +1

12 +1 0 +1

13 0 +1 -1

14 -1 0 -1

15 0 -1 -1

16 +1 0 -1

17 +1 +1 0

18 -1 +1 0

19 -1 -1 0

20 +1 -1 0

2.4.2 20 Node Brick


Shape function of the 8 corner nodes (1 to 8) which i indicates the node number:

N(e)i =

1

8(1 + i)(1 + i)(1 + i)(i + i + i 2) (2.45)


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Shape function of the node numbers 9, 11, 13, and 15 which i indicates the node number:

N(e)i =

1

4(1 2)(1 + i)(1 + i) (2.46)


N(e)i =

1

4(1 2)(1 + i)(1 + i) (2.47)


N(e)i =

1

4(1 2)(1 + i)(1 + i) (2.48)


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2.4.3 27 Node Brick


Table 2.3: Values of , , and at each of the 21th to 27th nodesNode

21 0 0 0

22 0 +1 0

23 -1 0 0

24 0 -1 0

25 +1 0 0

26 0 0 +1

27 0 0 -1

Shape function of the 8 corner nodes (1 to 8) which i indicates the node number:

N(e)i =

1

8(1 + i)(1 + i)(1 + i)(i)(i)(i) (2.49)


N(e)i =

1

4(1 2)(1 + i)(1 + i)(i)(i) (2.50)


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N(e)i =

1

4(1 + i)(1 2)(1 + i)(i)(i) (2.51)


N(e)i =

1

4(1 + i)(1 + i)(1 2)(i)(i) (2.52)

Shape function of the node number 21:

N(e)21 = (1 2)(1 2)(1 2) (2.53)

Shape function of the node numbers 22 and 24 which i indicates the node number:

N(e)i =

1

2(1 2)(1 + i)(1 2)(i) (2.54)


N(e)i =

1

2(1 + i)(1 2)(1 2)(i) (2.55)


N(e)i =

1

2(1 2)(1 2)(1 + i)(i) (2.56)

2.4.4 Isoparametric 8 20 Node Finite Element

The basic procedure in the isoparametric18 finite element formulation is to express the element coordinates and

element displacements in the form of interpolations using the local three dimensional19 coordinate system of the

element. Considering the general 3D element, the coordinate interpolations, using indicial notation20 are:

xi = HA (rk) xAi (2.57)18name isoparametric comes from the fact that both displacements and coordinates are defined in terms of nodal values. Super-

parametric and subparametric finite elements exists also.19in the case of element presented here, that is isoparametric 8 20 node finite element.20Einsteins summation rule is implied unless stated differently, all lower case indices (i, j, p, q,m, n, o, r, s, t, . . . ) can have values

of 1, 2, 3, and values for capital letter indices will be specified where need be.


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where A = 1, 2, . . . , n and n is the total number of nodes associated with that specific element, xAi is the i-th

coordinate of node A, i = 1, 2, 3, k = 1, 2, 3 and HA are the interpolation functions defined in local coordinate

system of the element, with variables r1, r2 and r3 varying from 1 to +1.

The interpolation functions HA for the isoparametric 820 node are the so called serendipity interpolation

functions mainly because they were derived by inspection. For the finite element with nodes numbered as in

Figure (2.6) they are given21 in the following set of formulae:

r1

r2

r33

49

11

14

15

x

x

x

17

12

1613

6

18

210

7

8

20

19

1

3

2

1 5

Figure 2.6: Isoparametric 820 node brick element in global and local coordinate systems

H20 =isp (20 ) (1 + r1) (1 r2)

(1 r23

)4

H19 =isp (19 ) (1 r1) (1 r2)

(1 r23

)4

H18 =isp (18 ) (1 r1) (1 + r2)

(1 r23

)4

H17 =isp (17 ) (1 + r1) (1 + r2)

(1 r23

)4

H16 =isp (16 ) (1 + r1)

(1 r22

)(1 r3)

4H15 =

isp (15 )(1 r21

)(1 r2) (1 r3)

4

H14 =isp (14 ) (1 r1)

(1 r22

)(1 r3)

4H13 =

isp (13 )(1 r21

)(1 + r2) (1 r3)

4

H12 =isp (12 ) (1 + r1)

(1 r22

)(1 + r3)

4H11 =

isp (11 )(1 r21

)(1 r2) (1 + r3)

4

H10 =isp (10 ) (1 r1)

(1 r22

)(1 + r3)

4H9 =

isp (9 )(1 r21

)(1 + r2) (1 + r3)

4

21for more details see Bathe (1982).


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H8 =(1 + r1) (1 r2) (1 r3)

8+h15 h16 h20

2

H7 =(1 r1) (1 r2) (1 r3)

8+h14 h15 h19

2

H6 =(1 r1) (1 + r2) (1 r3)

8+h13 h14 h18

2

H5 =(1 + r1) (1 + r2) (1 r3)

8+h13 h16 h17

2

H4 =(1 + r1) (1 r2) (1 + r3)

8+h11 h12 h20

2

H3 =(1 r1) (1 r2) (1 + r3)

8+h10 h11 h19

2

H2 =(1 r1) (1 + r2) (1 + r3)

8+h10 h18 h9

2

H1 =(1 + r1) (1 + r2) (1 + r3)

8+h12 h17 h9

2

where r1, r2 and r3 are the axes of natural, local, curvilinear coordinate system and isp (nod_num) is boolean

function that returns +1 if node number (nod_num) is present and 0 if node number (nod_num) is not present.

To be able to evaluate various important element tensors22, we need to calculate the straindisplacement transfor-

mation tensor23. The element strains are obtained in terms of derivatives of element displacements with respect

to the local coordinate system. Because the element displacements are defined in the local coordinate system,

we need to relate global x1, x2 and x3 derivatives to the r1, r2 and r3 derivatives. In order to obtain derivatives

with respect to global coordinate system, i.e. xa we need to use chain rule for differentiation in the following

form:

xk=raxk

ra= J1ak

ra(2.58)

while the inverse relation is:

rk=xark

xa= Jak

xa(2.59)

where Jak is the Jacobian operator relating local coordinate derivatives to the global coordinate derivatives:

22i.e. (m)KIacJ , (m)FBIa,(m)FSIa,

(m)F0mnIa ,

(m)F0mnIa , that are defined in chapter (2.3).

23from the equation ab = 12((HI,b uIa

)+(HI,a uIb

))Jeremi et al. University of California, Davis Version: March 21, 2012, 17:15

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Jak =xark

=

x1r1

x2r1

x3r1

x1r2

x2r2

x3r2

x1r3

x2r3

x3r3

(2.60)

The existence of equation (2.58) requires that the inverse of Jak exists and that inverse exists provided that there

is a onetoone24 correspondence between the local and the global coordinates of element.

It should be pointed out that except for the very simple cases, volume and surface element tensor25 integrals are

evaluated by means of numerical integration26 Numerical integration rules is quite a broad subject and will not

be covered here27.

2.5 Isoparametric, 3D Beam-Column Finite Element

Bathe and Wilson (1976); Bathe (1982)

2.6 Triangular Shell Finite Element with 6DOFs per Node

Bergan

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