Finite Element II, Solid Mechanics_Victor E.saouma_2001

DraftDRAFT

Lecture Notes in:

FINITE ELEMENT II

Solid Mechanics

CVEN 6525

cVICTOR E. SAOUMA,

SPRING 2001

Dept. of Civil Environmental and Architectural Engineering

University of Colorado, Boulder, CO 80309-0428

Draft02

Victor Saouma Finite Elements II; Solid Mechanics

Draft

Contents

1 PREREQUISITE 111.1 Variational Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Strain Displacement Relations . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1.1 Axial Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1.2 Flexural Members . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Virtual Displacement and Strains . . . . . . . . . . . . . . . . . . . . . . . 151.2.3 Element Stiness Matrix Formulation . . . . . . . . . . . . . . . . . . . . 15

1.2.3.1 Stress Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Direct Stiness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.1 Global Stiness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1.1 Structural Stiness Matrix . . . . . . . . . . . . . . . . . . . . . 171.3.1.2 Augmented Stiness Matrix . . . . . . . . . . . . . . . . . . . . 17

1.3.2 Logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2.1 Boundary Conditions, [ID] Matrix . . . . . . . . . . . . . . . . . 181.3.2.2 LM Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101.3.2.3 Assembly of Global Stiness Matrix . . . . . . . . . . . . . . . . 110

E 1-1 Assembly of the Global Stiness Matrix . . . . . . . . . . . . . . . . . . . 1101.3.2.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

E 1-2 Direct Stiness Analysis of a Truss . . . . . . . . . . . . . . . . . . . . . . 113E 1-3 Analysis of a Frame with MATLAB . . . . . . . . . . . . . . . . . . . . . 118E 1-4 Analysis of a simple Beam with Initial Displacements . . . . . . . . . . . 120

2 INTRODUCTION 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Elliptic, Parabolic and Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . 21

E 2-1 Seepage Problem;(Bathe 1996) . . . . . . . . . . . . . . . . . . . . . . . . 23E 2-2 Diusion Problem; (Bathe 1996) . . . . . . . . . . . . . . . . . . . . . . . 25E 2-3 Wave Equation, (Bathe 1996) . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Solution of Discrete-System Mathematical models . . . . . . . . . . . . . . . . . . 282.3.1 Steady State Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1.1 Elastic Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1.2 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2102.3.1.3 Hydraulic Network . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.3.1.4 DC Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

2.3.2 Equivalent Truss/Direct Stiness Models . . . . . . . . . . . . . . . . . 2132.3.2.1 Nonlinear Elastic Spring . . . . . . . . . . . . . . . . . . . . . . 215

Draft02 CONTENTS2.3.3 Propagation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

2.3.3.1 Dynamic Elastic System . . . . . . . . . . . . . . . . . . . . . . 2162.3.3.2 Transient Heat Flow . . . . . . . . . . . . . . . . . . . . . . . . . 216

2.3.4 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2172.3.4.1 Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2172.3.4.2 Column Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . 218

2.4 Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2202.4.1 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221E 2-4 Flexure of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

2.5 Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2232.6 Examples of applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

3 FUNDAMENTAL RELATIONS 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.1.1 Indicial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.1.2 Tensor Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.1.3 Voigt Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Vector Fields; Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.1.1 Force, Traction and Stress Vectors . . . . . . . . . . . . . . . . . 353.2.1.2 Traction on an Arbitrary Plane; Cauchys Stress Tensor . . . . . 36

E 3-1 Stress Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.2 Kinematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.3 Fundamental Laws of Continuum Mechanics . . . . . . . . . . . . . . . . 310

3.2.3.1 Conservation of Mass; Continuity Equation . . . . . . . . . . . . 3123.2.3.2 Linear Momentum Principle; Equation of Motion . . . . . . . . 3123.2.3.3 Conservation of Energy; First Principle of Thermodynamics . . 313

3.2.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3143.2.4.1 General 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3143.2.4.2 Transversly Isotropic Case . . . . . . . . . . . . . . . . . . . . . 3153.2.4.3 Special 2D Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

3.2.4.3.1 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . 3163.2.4.3.2 Axisymmetry . . . . . . . . . . . . . . . . . . . . . . . . 3163.2.4.3.3 Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . 316

3.2.4.4 Pore Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3173.2.5 Field Equations for Thermo- and Poro Elasticity . . . . . . . . . . . . . 317

3.3 Scalar Field: Diusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3193.3.1 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3203.3.2 Derivation of the Diusion Problem . . . . . . . . . . . . . . . . . . . . . 321

3.3.2.1 Simple 2D Derivation . . . . . . . . . . . . . . . . . . . . . . . . 3213.3.2.2 Generalized Derivation . . . . . . . . . . . . . . . . . . . . . . . 3223.3.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 324

3.4 Summary and Tonti Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324


DraftCONTENTS 034 MESH GENERATION 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Voronoi Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Delaunay Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.3 MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Finite Element Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.1 Boundary Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.2 Interior Node Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.3 Final Triangularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 VARIATIONAL and RAYLEIGH-RITZ METHODS 515.1 Multield Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Total Potential Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2.1 Static; Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2.2 Dynamic; Euler/Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . 54E 5-1 Hamiltons Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 General Hu-Washizu Variational Principle . . . . . . . . . . . . . . . . . . . . . . 595.4 Rayleigh Ritz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

5.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515E 5-2 Uniformly Loaded Simply Supported Beam; Polynomial Approximation . 515E 5-3 Heat Conduction; (Bathe 1996) . . . . . . . . . . . . . . . . . . . . . . . . 516

6 INTERPOLATION FUNCTIONS; NATURAL COORDINATE SYSTEMS616.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2.1 C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2.1.1 Truss element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2.1.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2.1.3 Constant Strain Triangle Element . . . . . . . . . . . . . . . . . 636.2.1.4 Further Generalization: Lagrangian Interpolation Functions . . . 656.2.1.5 Rectangular Bilinear Element . . . . . . . . . . . . . . . . . . . . 666.2.1.6 Solid Rectangular Trilinear Element . . . . . . . . . . . . . . . . 67

6.2.2 C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2.2.1 Flexural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2.2.2 C1: Hermitian Interpolation Functions . . . . . . . . . . . . . . 69

6.2.3 Characteristics of Shape Functions . . . . . . . . . . . . . . . . . . . . . . 6106.3 Natural Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

6.3.1 Straight Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.3.2 Triangular Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6126.3.3 Volume Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6146.3.4 Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

6.4 Pascals Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

7 FINITE ELEMENT DISCRETIZATION and REQUIREMENTS 717.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.1.1 Discretization of the Variational Statement for the General TPE Varia-tional Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


Draft04 CONTENTS7.1.2 Discretization of the Variational Statement for the HWVariational Principle73

7.2 General Element Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.3 Discretization Error and Convergence Rate . . . . . . . . . . . . . . . . . . . . . 787.4 Lower Bound Character of Minimum Potential Energy Based Solutions . . . . . 7107.5 Equilibrium and Compatibiliy in the Solution . . . . . . . . . . . . . . . . . . . . 710

8 STRAIGHT SIDED ELEMENTS; 1st GENERATION 818.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2 Rod Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.2.1 Truss Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.2.2 Beam Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3 Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.3.1 Cartesian Coordinate System (CST) . . . . . . . . . . . . . . . . . . . . . 838.3.2 Natural Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.3.2.1 Linear, T3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.3.2.2 Quadratic Element (T6) . . . . . . . . . . . . . . . . . . . . . . . 86

8.4 Bilinear Rectangular Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878.5 Element Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.5.1 CST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.5.2 BiLinear Rectangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

9 ISOPARAMETRIC ELEMENTS; 2nd GENERATION 919.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919.2 Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.2.1 Bar Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.2.2 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

9.2.2.1 Linear Element (Q4) . . . . . . . . . . . . . . . . . . . . . . . . . 959.2.2.1.1 Example: Jacobian Operators, (Bathe 1996) . . . . . . 910

9.2.2.2 Quadratic Element . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.2.2.2.1 Serendipity Element (Q8) . . . . . . . . . . . . . . . . . 9119.2.2.2.2 Lagrangian element (Q9) . . . . . . . . . . . . . . . . . 9139.2.2.2.3 Variable (Hierarchical) Element . . . . . . . . . . . . . 915

9.2.3 Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9169.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917

9.3.1 Newton-Cotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9189.3.2 Gauss-Legendre Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . 920

9.3.2.1 Legendre Polynomial . . . . . . . . . . . . . . . . . . . . . . . 9219.3.2.2 Gauss-Legendre Quadrature for n = 2 . . . . . . . . . . . . . . . 922

9.3.3 Rectangular and Prism Regions . . . . . . . . . . . . . . . . . . . . . . . . 9229.3.4 Triangular Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922

9.4 Stress Recovery; Nodal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9249.5 Nodal Equivalent Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925

9.5.1 Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9259.5.2 Traction Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9269.5.3 Initial Strains/Stresses; Thermal Load . . . . . . . . . . . . . . . . . . . . 928

9.6 Computer Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9299.6.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9299.6.2 MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930


DraftCONTENTS 059.6.2.1 stiff.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9309.6.2.2 dmat.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9329.6.2.3 sfr.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9339.6.2.4 jacob.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9359.6.2.5 bmatps.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935

9.6.3 Plott of Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 936

10 ELEMENT FORMULATION and STRAIN RECOVERY in HW FORMU-LATION 10110.1 Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.2 Strain Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

10.2.1 C-lumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.2.2 Strain smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.2.3 C-splitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

10.3 Uniqueness and Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . 105

11 WEIGHTED RESIDUAL METHODS 11111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.2 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

11.2.1 Dierential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.2.1.1 Application to 1D Axial Member . . . . . . . . . . . . . . . . . . 112

11.2.2 Residual Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.3 Weighted Residual Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

11.3.1 Point Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 11411.3.2 Subdomain Collocation Method . . . . . . . . . . . . . . . . . . . . . . . 11411.3.3 Least-Squares Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11511.3.4 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115E 11-1 String Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

11.4 Applications of the Galerkin Method to 3D Elasticity Problems . . . . . . . . . . 11711.4.1 Derivation of the Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . 11711.4.2 FE Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

12 FINITE ELEMENT DISCRETIZATION OF THE FIELD EQUATION 12112.1 Derivation of the Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12112.2 FE Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

12.2.1 No Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12312.2.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

12.3 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124E 12-1 Composite Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124E 12-2 Heat Transfer across a Fin . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

12.4 Comparison Between Vector and Scalar Formulations . . . . . . . . . . . . . . . . 1212

13 TOPICS in STRUCTURAL MECHANICS 13113.1 Condensation/Substructuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13113.2 Element Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

13.2.1 Patch Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13213.2.2 Eigenvalue Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13213.2.3 Order of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134


Draft06 CONTENTS13.2.3.1 Full Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 13413.2.3.2 Reduced Integration . . . . . . . . . . . . . . . . . . . . . . . . . 13513.2.3.3 Selective Reduced Integration . . . . . . . . . . . . . . . . . . . 136

13.3 Parasitic Shear/Incompatible Elements . . . . . . . . . . . . . . . . . . . . . . . . 13713.3.1 Q4, The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13713.3.2 Q6, The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13813.3.3 QM6, Further Enhancements . . . . . . . . . . . . . . . . . . . . . . . . . 138

13.4 Rotational D.O.F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13913.5 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310

14 GEOMETRIC NONLINEARITY 14114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

14.1.1 Strong Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14214.1.1.1 Lower Order Dierential Equation . . . . . . . . . . . . . . . . . 14214.1.1.2 Higher Order Dierential Equation . . . . . . . . . . . . . . . . 143

14.1.2 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14614.1.2.1 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14614.1.2.2 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

14.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14814.3 Elastic Instability; Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . 149

E 14-1 Column Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410E 14-2 Frame Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413

14.4 Second-Order Elastic Analysis; Geometric Non-Linearity . . . . . . . . . . . . . . 1415E 14-3 Eect of Axial Load on Flexural Deformation . . . . . . . . . . . . . . . . 1416E 14-4 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419

14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422

15 PLATES 15115.1 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

15.1.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15115.1.2 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15415.1.3 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

15.2 Plate Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15615.2.1 Reissner-Mindlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

15.2.1.1 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . 15615.2.1.2 Dierential Equation . . . . . . . . . . . . . . . . . . . . . . . . 15815.2.1.3 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . 158

15.2.2 Kirchho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15915.2.2.1 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . 15915.2.2.2 Dierential Equation . . . . . . . . . . . . . . . . . . . . . . . . 151015.2.2.3 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151015.2.2.4 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . 1511

15.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151115.3 Finite Element Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513

15.3.1 Rectangular Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151315.3.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151315.3.1.2 Shear Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516

15.3.2 Nonconforming Kirchho Triangular Element . . . . . . . . . . . . . . . . 1517


DraftCONTENTS 0715.3.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151715.3.2.2 Nonconformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519

15.3.3 Discrete Kirchho Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . 152015.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524

16 MATERIAL NONLINEARITIES 16116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

16.1.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16116.1.2 Solution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

16.2 Load Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16416.2.1 Newton-Raphson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

16.2.1.1 Newton-Raphson/Tangent Stiness Method . . . . . . . . . . . . 16416.2.1.2 Modied Newton-Raphson . . . . . . . . . . . . . . . . . . . . . 16616.2.1.3 Secant Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

16.2.2 Acceleration of Convergence, Line Search Method . . . . . . . . . . . . . 16716.2.3 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1610

16.3 Direct Displacement Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161016.4 Indirect Displacement Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613

16.4.1 Partitioning of the Displacement Corrections . . . . . . . . . . . . . . . . 161316.4.2 Arc-Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161516.4.3 Relative Displacement Criterion . . . . . . . . . . . . . . . . . . . . . . . 161716.4.4 IDC Methods with Approximate Line Searches . . . . . . . . . . . . . . . 1618

A VECTOR OPERATIONS A1A.1 Vector Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1

A.1.1 Derivative WRT to a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . A1E A-1 Tangent to a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A2A.1.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A3E A-2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5A.1.3 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5A.1.4 Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5E A-3 Gradient of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A6E A-4 Stress Vector normal to the Tangent of a Cylinder . . . . . . . . . . . . . A6

A.2 Vector Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A9A.2.1 Integral of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A9A.2.2 Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A9A.2.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A9A.2.4 Gauss; Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . A10A.2.5 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A10A.2.6 Green; Gradient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . A10E A-5 Physical Interpretation of the Divergence Theorem . . . . . . . . . . . . A10

B CASE-STUDY: FRACTURING of A DAM DUE TO THERMAL LOAD B1B.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B1

B.1.1 Elastic and Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . B1B.1.2 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B2

B.2 ANALYSIS II; Thermal Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . B2B.2.1 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B3


Draft08 CONTENTSB.2.2 Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B7B.2.3 Data Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B9

B.3 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B9

C MISC. C1C.1 Units & Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1C.2 Metric Prexes and Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . C2


Draft

List of Figures

1.1 Summary of Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Duality of Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Frame Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Example for [ID] Matrix Determination . . . . . . . . . . . . . . . . . . . . . . . 191.5 Simple Frame Analyzed with the MATLAB Code . . . . . . . . . . . . . . . . . . 1111.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131.7 Simple Frame Analyzed with the MATLAB Code . . . . . . . . . . . . . . . . . . 1181.8 Stiness Analysis of one Element Structure . . . . . . . . . . . . . . . . . . . . . 121

2.1 Finite Element Process, (Bathe 1996) . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Seepage Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 One Dimensional Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Rod subjected to Step Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 System of Rigid Carts Interconnected by Linear Springs, (Bathe 1996) . . . . . . 292.6 Slab Subjected to Temperature Boundary Conditions, (Bathe 1996) . . . . . . . 2102.7 Pipe Network, (Bathe 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.8 DC Network, (Bathe 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2122.9 Equivalent Trusses/Direct Stiness . . . . . . . . . . . . . . . . . . . . . . . . . . 2132.10 Heat Transfer Idealization in an Electron Tube, (Bathe 1996) . . . . . . . . . . . 2172.11 Stability of a Two Rigid Bars System . . . . . . . . . . . . . . . . . . . . . . . . 2182.12 Idealization, Discretization and Solution of a Numerical Simulation, (Felippa 2000)220

3.1 Stresses as Tensor Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Cauchys Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Flux Through Area dS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.4 Equilibrium of Stresses, Cartesian Coordinates . . . . . . . . . . . . . . . . . . . 3133.5 Flux vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3203.6 Flux Through Sides of Dierential Element . . . . . . . . . . . . . . . . . . . . . 3223.7 *Flow through a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3223.8 Components of Tontis Diagram, (Felippa 2000) . . . . . . . . . . . . . . . . . . . 3243.9 Fundamental Equations of Solid Mechanics and Heat Flow . . . . . . . . . . . . . 325

4.1 Voronoi and Delaunay Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Control Point for a 2D Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Control Point for a 3D Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 A Two Dimensional Triangularization AlgorithmControl Point for a 3D Mesh . . 45

5.1 Tonti Diagram for the Total Potential Energy, (Cervenka, J. 1994) . . . . . . . . 545.2 Tonti Diagram for Hu-Washizu, (Cervenka, J. 1994) . . . . . . . . . . . . . . . . 512

Draft02 LIST OF FIGURES5.3 Uniformly Loaded Simply Supported Beam Analysed by the Rayleigh-Ritz Method515

6.1 Axial Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2 Constant Strain Triangle Element . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3 Rectangular Bilinear Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4 Solid Trilinear Rectangular Element . . . . . . . . . . . . . . . . . . . . . . . . . 676.5 Flexural Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.6 Shape Functions for Flexure of Uniform Beam Element. . . . . . . . . . . . . . . 6106.7 Natural Coordinate System Along a Straight Line . . . . . . . . . . . . . . . . . 6116.8 Natural Coordinate System for a Triangle . . . . . . . . . . . . . . . . . . . . . . 6126.9 Integration over a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

7.1 Completness and Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 h, p and r Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.3 Interelement Continuity of Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 7117.4 Interelement Continuity of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

8.1 Rectangular Bilinear Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888.2 Linear Triangular Element Subjected to Pure Bending . . . . . . . . . . . . . . . 810

9.1 Two-Dimensional Mapping of Some Elements . . . . . . . . . . . . . . . . . . . . 919.2 Actual and Parent Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.3 Iso, Super, and Sub Parametric Elements . . . . . . . . . . . . . . . . . . . . . . 939.4 Three-Noded Quadratic Bar Element . . . . . . . . . . . . . . . . . . . . . . . . . 939.5 Four Noded Isoparametric Element . . . . . . . . . . . . . . . . . . . . . . . . . . 959.6 Dierential Element in Curvilinear Coordinate System . . . . . . . . . . . . . . . 989.7 Cross Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.8 *Elements with Possible Singular Jacobians . . . . . . . . . . . . . . . . . . . . . 999.9 Serendipity and Lagrangian Quadratic Quadrilaterals . . . . . . . . . . . . . . . 9119.10 Pascal Triangles for Quadrilateral and Triangle Elements . . . . . . . . . . . . . . 9129.11 Serendipity Isoparametric Quadratic Finite Element: Global and Parent Element 9139.12 Shape Functions for 8 Noded Quadrilateral Element . . . . . . . . . . . . . . . . 9149.13 Shape Functions for 9 Noded Quadrilateral Element . . . . . . . . . . . . . . . . 9159.14 Nodal Numbering for Isoparameteric Elements . . . . . . . . . . . . . . . . . . . 9169.15 Newton-Cotes Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . 9189.16 Gauss-Legendre Integration Over a Surface . . . . . . . . . . . . . . . . . . . . . 9239.17 Numerical Integration Over a Triangle . . . . . . . . . . . . . . . . . . . . . . . . 9239.18 Extrapolation from 4-Node Quad Gauss Points (Felippa 1999) . . . . . . . . . . . 9249.19 Gravity Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9269.20 Traction Load in Isoparametric Elements . . . . . . . . . . . . . . . . . . . . . . 9279.21 Traction Load in Contiguous Isoparametric Elements . . . . . . . . . . . . . . . . 928

10.1 Patch test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

12.1 Heat Flow in a Thin Rectangular Fin . . . . . . . . . . . . . . . . . . . . . . . . 129

13.1 Patch Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13213.2 Eigenvectors Corresponding to a) Non-Zero and b) Zero Eigenvalues for a Square

Bilinear element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134


DraftLIST OF FIGURES 0313.3 Independent Displacement Modes for a Bilinear Element . . . . . . . . . . . . . . 13613.4 Hourglas Modes in Under-Integrated Quadratic Element . . . . . . . . . . . . . . 13613.5 Rectangular Bilinear Element Subjected to Bending; Bilinear Element and Cor-

rect Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13713.6 Displacements Associated with Incompatible Modes for the Q6 Element . . . . . 13813.7 Side Displacements Induced by Drilling d.o.f. . . . . . . . . . . . . . . . . . . . . 139

14.1 Level of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14114.2 Euler Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14214.3 Simply Supported Beam Column; Dierential Segment; Eect of Axial Force P . 14414.4 Solution of the Tanscendental Equation for the Buckling Load of a Fixed-Hinged

Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14514.5 Summary of Stability Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423

15.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15115.2 Stresses in a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15215.3 Free Body Diagram of an Innitesimal Plate Element . . . . . . . . . . . . . . . 15315.4 Displacements in a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15415.5 Positive Moments and Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15615.6 Rectangular Plate Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151315.7 Triangular Plate Element in Natural Coordinate System . . . . . . . . . . . . . . 151715.8 Edges of Adjacent Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . 152015.9 Discrete Kirchho Triangular Element . . . . . . . . . . . . . . . . . . . . . . . . 1521

16.1 Test Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16316.2 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16516.3 Modied Newton-Raphson Method, Initial Tangent in Increment . . . . . . . . . 16616.4 Modied Newton-Raphson Method, Initial Problem Tangent . . . . . . . . . . . 16716.5 Incremental Secant, Quasi-Newton Method . . . . . . . . . . . . . . . . . . . . . 16816.6 Schematic of Line Search, (Reich 1993) . . . . . . . . . . . . . . . . . . . . . . . . 16916.7 Flowchart for Line Search Algorithm, (Reich 1993) . . . . . . . . . . . . . . . . . 16916.8 Divergence of Load-Controled Algorithms . . . . . . . . . . . . . . . . . . . . . . 161116.9 Hydrostatically Loaded Gravity Dam . . . . . . . . . . . . . . . . . . . . . . . . . 161316.10Load-Displacement Diagrams with Snapback . . . . . . . . . . . . . . . . . . . . 161316.11Flowchart for an incremental nonlinear nite element program with indirect dis-

placement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161616.12Two points on the load-displacement curve satisfying the arc-length constraint . 161716.13Flow chart for line search with IDC methods . . . . . . . . . . . . . . . . . . . . 1620

A.1 Dierentiation of position vector p . . . . . . . . . . . . . . . . . . . . . . . . . . A2A.2 Curvature of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A2A.3 Vector Field Crossing a Solid Region . . . . . . . . . . . . . . . . . . . . . . . . . A3A.4 Flux Through Area dA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A4A.5 Innitesimal Element for the Evaluation of the Divergence . . . . . . . . . . . . . A4A.6 Radial Stress vector in a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . A7A.7 Gradient of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A8A.8 Physical Interpretation of the Divergence Theorem . . . . . . . . . . . . . . . . . A11

B.1 Boundary Description of Dam for Transient Thermal Analysis . . . . . . . . . . . B4


Draft04 LIST OF FIGURESB.2 Heat of Hydration Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . B6B.3 Temperature Distribution in the Transient Thermal Analysis at Day 8 . . . . . . B7B.4 Maximum Principal Stresses and Deformed Mesh at Day 8 . . . . . . . . . . . . B8


Draft

List of Tables

1.1 Summary of Variational Terms Associated with One Dimensional Elements . . . 14

3.1 Selected Examples of Diusion Problems . . . . . . . . . . . . . . . . . . . . . . . 3193.2 Comparison of Scalar and Vector Field Problems . . . . . . . . . . . . . . . . . . 3263.3 Classication of various Physical Problems, (Kardestuncer 1987) . . . . . . . . . 327

5.1 Functionals in Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Comparison Between Total Potential Energy and Hu-Washizu Formulations . . . 512

6.1 Characteristics of Beam Element Shape Functions . . . . . . . . . . . . . . . . . 696.2 Interpretation of Shape Functions in Terms of Polynomial Series (1D & 2D) . . . 6166.3 Polynomial Terms in Various Element Formulations (1D & 2D) . . . . . . . . . . 616

8.1 Shape Functions and Derivatives for T6 Element . . . . . . . . . . . . . . . . . . 86

9.1 Shape Functions, and Natural Derivatives for Q8 Element . . . . . . . . . . . . . 9139.2 Shape Functions for Variable Node Elements . . . . . . . . . . . . . . . . . . . . 9169.3 Weights for Newton-Cotes Quadrature Formulas . . . . . . . . . . . . . . . . . . 9199.4 Integration Points and Weights for Gauss-Quadrature Formulaes Over the Inter-

val [1, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.5 Coordinates and Weights for Numerical Integration over a Triangle . . . . . . . . 9239.6 Natural Coordinates of Bilinear Quadrilateral Nodes . . . . . . . . . . . . . . . . 925

10.1 Polynomial orders of the shape functions. . . . . . . . . . . . . . . . . . . . . . . 10110.2 Table of coecients and spectral radii for CS technique. . . . . . . . . . . . . . 105

12.1 Comparison of Scalar and Vector Field Problems, Revisited . . . . . . . . . . . . 1212

13.1 Full and Reduced Numerical Integrations for Quadrilateral Elements . . . . . . . 13513.2 Bilinsear and Exact Displacements/Strains . . . . . . . . . . . . . . . . . . . . . 137

15.1 Comparison of Governing Equations in Elasticity and Plate Bending . . . . . . . 151215.2 Integration Rules for Mindlin Plate Elements . . . . . . . . . . . . . . . . . . . . 1516

A.1 Similarities Between Multiplication and Dierentiation Operators . . . . . . . . . A1

B.1 Concrete Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B2B.2 Thermal Properties of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . B2B.3 Interface Element Material Properties . . . . . . . . . . . . . . . . . . . . . . . . B2B.4 Loads applied on the Dam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B3B.5 Heat of Hydration From the Literature . . . . . . . . . . . . . . . . . . . . . . . . B5

Draft02 LIST OF TABLESB.6 Heat of Hydration Adopted in the Simulation; Days and J/Kg/Day . . . . . . . . B5B.7 Sresses Along the Interface Element; m] and [Pa] . . . . . . . . . . . . . . . . . . B8B.8 Crack Opening and Sliding Displacements; [m] . . . . . . . . . . . . . . . . . . . B9


DraftLIST OF TABLES 03NOTATION

SCALARSA Areac Specic heatE Elastic Modulush Film coecient for convection heat transferI Moment of inertiaJ St Venants torsional constantL LengthQ Rate of internal heat generation per unit volumet TimeT Temperatureu, v,w Translational displacements along the x, y, and z directionsU0 Strain energy densityU Strain energyU0 Complementary strain energy densityU Complementary strain energyW Work Potential energy Coecient of thermal expansion Shear modulus Poissons ratio mass density Rotational displacementM Virtual momentP Virtual force Virtual rotationu Virtual displacement Virtual curvatureU Virtual internal strain energyW Virtual external work

TENSORS order 1

a Vector of coecients in assumed displacement eldb Body forcec Nodal coordinatesF Unknown element forces and unknown support reactionsp Matrix of coecients of a polynomial seriesN Displacement shape functionsN Coordinate shape functionsp Element nodal forces = FP Structure nodal forcesq Flux per unit areaR Structure reactionsR Residualst Traction vector


Draft04 LIST OF TABLESt Specied tractions along tu Displacement vectoru(x) Specied displacements along uu Displacement vectorue Nodal element displacementsu Nodal displacements in a continuous systemu Structure nodal displacementsV Shear forces in a plate Vx, Vy Element nodal displacements Virtual strain vector Virtual stress vector Stress vector0 Initial stress vector

TENSORS order 2

d Element exibility matrixI Idendity matrixk Element stiness matrixkg Geometric element stiness matrixK Structure stiness matrixKg Structures geometric stiness matrixlij Direction cosine of rotated axis i with respect to original axis jM Moments in a plate Mxx,Mxy,Myx,MyyN Membrane forces Nxx, Nxy, Nyx, Nyy Shear deformations Transformation matrix Strain vector0 Initial strain vectork Conductivity Curvature

TENSORS order 4

D Constitutive matrix

CONTOURS, SURFACES, VOLUMES

Surfacet Boundary along which surface tractions, t are speciedu Boundary along which displacements, u are speciedT Boundary along which temperatures, T are speciedc Boundary along which convection ux, qc are speciedq Boundary along which ux, qn are specied Volume of body

FUNCTIONS, OPERATORS


DraftLIST OF TABLES 05u Neighbour function to u(x) Variational operatorB Discrete strain-displacement operatorL Linear dierential operator relating displacement to strains Divergence, (gradient operator) on scalar x y z T.u = div .u Divergence, (gradient operator) on vector uxx + uyy + uzz v 2 Euclidian norm. . . . Innity norm.

PROGRAM ARRAYS

ID Matrix relating nodal dof to structure dofLM structure dof of nodes connected to a given element


Draft06 LIST OF TABLES


DraftLIST OF TABLES 071 Jan. 16 Introduction; Course objective; Overview; Notation.

18 Mathematical Formulations.2 23 Elasticity

25 Direct Method; Field Eq.3 30 Variational Methods

Feb. 1 Mesh Generators; Laboratory4 6 Variational Methods, Mechanics; Laboratory

8 FE Discretization and Requirements; C0 Elements.5 13 Isoparametric Elements, Bar Element 6.1-6.14

15 Isoparamteric Element, Bilinear Element6 20 Isoparameteric Element; Quadratic, Hierarchical Elements;

Numerical integration22 Isoparameteric Element; Numerical integration

7 27 Laboratory29 Weighted Residuals

8 Mar. 5 Galerkin; 3D Elasticity; Field Equation7 Field Equation, Theory, application Ch. 16

9 12 Field Equation14 Exam I

10 19 Lab (Field Equations)21 Error Analysis

SPRING BREAK12 Apr. 2 Topics (Condensation, Transformation, Integration, Test)

4 Order of Integration, Eignevalu tests13 9 Plate Bending 9.1-9.6

11 Plate Bending 11.1-11.514 16 Plate Bending

18 Geometric Non Linearity Ch. 14.15 23 Geometric Nonlinearity

25 Material Nonlinearity Ch. 1716 30 Dynamics Ch. 13

May 2 Review


Draft08 LIST OF TABLES


Draft

Chapter 1

PREREQUISITE

1 In the rst course (CVEN4525/5525, Finite Element I; Framed Structures), the direct stinessmethod was rst introduced (element stiness matrix, transformation matrix, global stinessmatrix assembly, internal force recovery). As an interlude we then covered the exibility methodand stiness-exibility relationship. The second part of the course began with a thorough cov-erage of variational method (duality between extremization of a functional and a correspondingeuler dierential equation) followed by a rigorous introduction/derivation of the various energymethods.

1.1 Variational Formulations

2 A summary of the various methods introduced in Finite Element I; Framed Structures isshown in Fig. 1.1, Fig. 1.2, and Table 1.1.

1.2 Finite Element Formulation

1.2.1 Strain Displacement Relations

3 The displacement at any point inside an element can be written in terms of the shapefunctions N and the nodal displacements {}

= N{} (1.1)

The strain is then dened as: = [B]{} (1.2)

where [B] is the matrix which relates joint displacements to strain eld.

1.2.1.1 Axial Members

u = (1 xL) xL N

{u1u2

} {}

(1.3-a)

Draft12 PREREQUISITE

div + b = 0t t = 0 t

U0def= 0 d

Gauss

Du = 0u = 0 u

Principle of Virtual Work

Td uTbd t uT td = 0

Wi We = 0

Principle of StationaryPotential Energy

= 0 def= U We

= U0d (

uibid+

tuitid)

CastiglianosFirst Theorem

Wik

= Pk

Rayleigh-Ritz

uj ni=1

cjiji +

j0

cji= 0 i = 1, 2, , n; j = 1, 2, 3

ij 12 (ui,j + uj,i) = 0ui u = 0 u

U0def=0 d

Gauss

ij,j = 0ti = 0 t

Principle of Complementary

Virtual Work ijijd

uuitid = 0

W i W e = 0

Principle of ComplementaryStationary Potential Energy

= 0 def= W i +W

e

= U0d +

uuitid

CastiglianosSecond Theorem

W iPk

= k

Natural B.C.

Essential B.C.

Figure 1.1: Summary of Variational Methods


Draft1.2 Finite Element Formulation 13

Kinematically Admissible Displacements

Displacements satisfy the kinematic equationsand the the kinematic boundary conditions

Principle of StationaryComplementary Energy

Principle of ComplementaryVirtual Work

Principle of Virtual Work

Principle of StationaryPotential Energy

Statically Admissible Stresses

Stresses satisfy the equilibrium conditionsand the static boundary conditions

Figure 1.2: Duality of Variational Principles


Draft14 PREREQUISITEU Virtual Displacement U Virtual Force U

General Linear General Linear

Axial 12

L0

P 2

AEdx

L0dx

L0Edu

dx

d(u)dx

Adxd

L0dx

L0P

P

AE

dx

Shear ... L

0V xydx ...

L0V xydx ...

Flexure 12

L0

M2

EIzdx

L0Mdx

L0EIz

d2v

dx2

d2(v)dx2

dx

L0Mdx

L0M

M

EIz

dx

Torsion 12

L0

T 2

GJdx

L0Tdx

L0GJ

dxdx

d(x)dx

dx

L0Tdx

L0T

T

GJ

dx

W Virtual Displacement W Virtual Force W

P i 12Pii iPii iPiiM i 12Mii iMii iMii

w

L0w(x)v(x)dx

L0w(x)v(x)dx

L0w(x)v(x)dx

Table 1.1: Summary of Variational Terms Associated with One Dimensional Elements

= x =du

dx=

1LN1x

1LN2x

[B]

{u1u2

} {}

(1.3-b)

1.2.1.2 Flexural Members

Using the shape functions for exural elements previously derived in Eq. 6.41 we have:

=y

= y

d2v

dx2(1.4-a)

1

=M

EI(1.4-b)

= yd2v

dx2(1.4-c)

= y

6L2 (2 1) 2N1x2

2L(3 2) 2N2x2

6L2

(2 + 1) 2N3x2

2L(3 1) 2N4x2

[B]

v11v22

{}

(1.4-d)


Draft1.2 Finite Element Formulation 151.2.2 Virtual Displacement and Strains

= [N]{} (1.5-a) = [B]{} (1.5-b)

(1.5-c)

1.2.3 Element Stiness Matrix Formulation

Let us consider the most general case, or element with:

Initial strain: (temperature eect, support settlement, or other) such that:

x =xE

due to load

+ ixinitial strain

(1.6)

thus:x = Ex Eix (1.7)

or in matrix form:{} = [D]{} [D]{i} (1.8)

where [D] is the constitutive matrix which relates stress and strain.

Load: q(x) along it.

Let us apply the principle of virtual work.

U = W (1.9-a)

U =vol{}dvol (1.9-b)

{} = [D]{} [D]{i} (1.9-c){} = [B]{} (1.9-d){} = [B]{} (1.9-e) = [B]T (1.9-f)

(1.9-g)

Combining Eqns. 1.9-a, 1.9-b, 1.9-c, 1.9-f, and 1.9-e, the internal virtual strain energy isgiven by:

U =vol

[B]T [D][B]{} dvol vol[B]T [D]{i}dvol

= vol

[B]T [D][B] dvol{} vol

[B]T [D]{i}dvol (1.10-a)

the virtual external work in turn is given by:

W = Virt. Nodal Displ.

{F}Nodal Force

+

lq(x)dx (1.11)


Draft16 PREREQUISITEcombining this equation with:

{} = [N]{} (1.12)yields:

W = {F}+ l

0[N]T q(x) dx (1.13)

Equating the internal strain energy Eqn. 1.10-a with the external work Eqn. 1.13, we obtain:

vol

[B]T [D][B] dvol [K]

{} vol

[B]T [D]{i}dvol {Finit}

=

{F} + l

0[N]T q(x) dx

{Fe}

(1.14-a)

where:

The element stiness matrix:

[K] =vol

[B]T [D][B]dvol (1.15)

Element initial force vector:

{Fi} =vol

[B]T [D]{i}dvol (1.16)

Element equivalent load vector:

{Fe} = l

0[N] q(x) dx (1.17)

1.2.3.1 Stress Recovery

Recall that we have:{} = [D]{}{} = [B]{}

}{} = [D] [N]{} (1.18)

1.3 Direct Stiness Method

1.3.1 Global Stiness Matrix

4 The physical interpretation of the global stiness matrix K is analogous to the one of theelement, i.e. If all degrees of freedom are restrained, then Kij corresponds to the force alongglobal degree of freedom i due to a unit positive displacement (or rotation) along global degreeof freedom j.

5 For instance, with reference to Fig. 1.3, we have three global degrees of freedom, 1, 2, and


Draft1.3 Direct Stiness Method 17

PM

P/2

Hw

L/2 L/2

2 3

1

C C

BBAA

EI

P

w

C

AB

K

K1

K

12

3222

C

BAA B

C

K

K

K

13

23 33

1K

K11

31K21A B

C

1

Figure 1.3: Frame Example

3. and the global (restrained or structures) stiness matrix is

K =

K11 K12 K13K21 K22 K23K31 K32 K33

(1.19)and the rst column corresponds to all the internal forces in the unrestrained d.o.f. when aunit displacement along global d.o.f. 1 is applied.

1.3.1.1 Structural Stiness Matrix

6 The structural stiness matrix is assembled only for those active degrees of freedom whichare active (i.e unrestrained). It is the one which will be inverted (or rather decomposed) todetermine the nodal displacements.

1.3.1.2 Augmented Stiness Matrix

7 The augmented stiness matrix is expressed in terms of all the dof. However, it is parti-tioned into two groups with respective subscript u where the displacements are known (zerootherwise), and t where the loads are known.

{Pt

Ru?

}=[Ktt KtuKut Kuu

]{t?u

}

(1.20)

We note that Ktt corresponds to the structural stiness matrix.


Draft18 PREREQUISITE8 The rst equation enables the calculation of the unknown displacements.

t = K1tt (Pt Ktuu) (1.21)

9 The second equation enables the calculation of the reactions

Ru = Kutt +Kuuu (1.22)

10 For internal book-keeping purpose, since we are assembling the augmented stiness matrix,we proceed in two stages:

1. First number all the global unrestrained degrees of freedom

2. Then number separately all the global restrained degrees of freedom (i.e. those withknown displacements, zero or otherwise) starting with -1 this will enable us later on todistinguish the restrained from unrestrained dof.

11 The element internal forces (axial and shear forces, and moment at each end of the member)are determined from

p(e)int = k

(e)(e) (1.23)

at the element level where p(e)int is the six by six array of internal forces, k(e) the element

stiness matrix in local coordinate systems, and (e) is the vector of nodal displacements in localcoordinate system. Note that this last array is obtained by rst identifying the displacementsin global coordinate system, and then premultiplying it by the transformation matrix to obtainthe displacements in local coordinate system.

1.3.2 Logistics

1.3.2.1 Boundary Conditions, [ID] Matrix

12 Because of the boundary condition restraints, the total structure number of active degrees offreedom (i.e unconstrained) will be less than the number of nodes times the number of degreesof freedom per node.

13 To obtain the global degree of freedom for a given node, we need to dene an [ID] matrixsuch that:

ID has dimensions l k where l is the number of degree of freedom per node, and k is thenumber of nodes).

ID matrix is initialized to zero.

1. At input stage read ID(idof,inod) of each degree of freedom for every node such that:

ID(idof, inod) ={

0 if unrestrained d.o.f.1 if restrained d.o.f.

(1.24)


Draft1.3 Direct Stiness Method 192. After all the node boundary conditions have been read, assign incrementally equation

numbers

(a) First to all the active dof

(b) Then to the other (restrained) dof, starting with -1.

Note that the total number of dof will be equal to the number of nodes times the numberof dof/node NEQA.

3. The largest positive global degree of freedom number will be equal to NEQ (Number OfEquations), which is the size of the square matrix which will have to be decomposed.

14 For example, for the frame shown in Fig. 1.4:

Figure 1.4: Example for [ID] Matrix Determination

1. The input data le may contain:

Node No. [ID]T

1 0 0 02 1 1 03 0 0 04 1 0 0

2. At this stage, the [ID] matrix is equal to:

ID =

0 1 0 10 1 0 00 0 0 0

(1.25)3. After we determined the equation numbers, we would have:

ID =

1 1 5 32 2 6 83 4 7 9

(1.26)


Draft110 PREREQUISITE1.3.2.2 LM Vector

15 The LM vector of a given element gives the global degree of freedom of each one of the elementdegree of freedoms. For the structure shown in Fig. 1.4, we would have:

LM = 1 2 4 5 6 7 element 1 (2 3)LM = 5 6 7 1 2 3 element 2 (3 1)LM = 1 2 3 3 8 9 element 3 (1 4)

1.3.2.3 Assembly of Global Stiness Matrix

16 As for the element stiness matrix, the global stiness matrix [K] is such that Kij is theforce in degree of freedom i caused by a unit displacement at degree of freedom j.

17 Whereas this relationship was derived from basic analysis at the element level, at the struc-ture level, this term can be obtained from the contribution of the element stiness matrices[K(e)] (written in global coordinate system).

18 For each Kij term, we shall add the contribution of all the elements which can connect degreeof freedom i to degree of freedom j, assuming that those forces are readily available from theindividual element stiness matrices written in global coordinate system.

19 Kij is non-zero if degree of freedom i and degree of freedom j

1. Are connected by an element.

2. Share a node.

3. Are connected by an element and the corresponding value in the element stiness matrixin the global coordinate system is zero.

20 There are usually more than one element connected to a dof. Hence, individual elementstiness matrices terms must be added up.

21 Because each term of all the element stiness matrices must nd its position inside the globalstiness matrix [K], it is found computationally most eective to initialize the global stinessmatrix [KS ](NEQANEQA) to zero, and then loop through all the elements, and then througheach entry of the respective element stiness matrix K(e)ij .

22 The assignment of the element stiness matrix term K(e)ij (note that e, i, and j are all knownsince we are looping on e from 1 to the number of elements, and then looping on the rows andcolumns of the element stiness matrix i, j) into the global stiness matrix KSkl is made throughthe LM vector (note that it is k and l which must be determined).

23 Since the global stiness matrix is also symmetric, we would need to only assemble one sideof it, usually the upper one.

24 Contrarily to the previous method, we will assemble the full augmented stiness matrix.

Example 1-1: Assembly of the Global Stiness Matrix


Draft1.3 Direct Stiness Method 111

7.416 m 8 m

3 m

4 kN/m50kN

8 m

1

2 3

1

2

Figure 1.5: Simple Frame Analyzed with the MATLAB Code

As an example, let us consider the frame shown in Fig. 1.5.The ID matrix is initially set to:

[ID] =

1 0 11 0 11 0 1

(1.27)We then modify it to generate the global degrees of freedom of each node:

[ID] =

4 1 75 2 86 3 9

(1.28)Finally the LM vectors for the two elements (assuming that Element 1 is dened from node 1to node 2, and element 2 from node 2 to node 3):

[LM ] =[ 4 5 6 1 2 3

1 2 3 7 8 9]

(1.29)

Let us simplify the operation by designating the element stiness matrices in global coordinatesas follows:

K(1) =

4 5 6 1 2 34 A11 A12 A13 A14 A15 A165 A21 A22 A23 A24 A25 A266 A31 A32 A33 A34 A35 A361 A41 A42 A43 A44 A45 A462 A51 A52 A53 A54 A55 A563 A61 A62 A63 A64 A65 A66

(1.30-a)

K(2) =

1 2 3 7 8 91 B11 B12 B13 B14 B15 B162 B21 B22 B23 B24 B25 B263 B31 B32 B33 B34 B35 B367 B41 B42 B43 B44 B45 B468 B51 B52 B53 B54 B55 B569 B61 B62 B63 B64 B65 B66

(1.30-b)

We note that for each element we have shown the corresponding LM vector.


Draft112 PREREQUISITENow, we assemble the global stiness matrix

K =

A44 +B11 A45 +B12 A46 +B13 A41 A42 A43 B14 B15 B16A54 +B21 A55 +B22 A56 +B23 A51 A52 A53 B24 B25 B26A64 +B31 A65 +B32 A66 +B33 A61 A62 A63 B34 B35 B36

A14 A15 A16 A11 A12 A13 0 0 0A24 A25 A26 A21 A22 A23 0 0 0A34 A35 A36 A31 A32 A33 0 0 0B41 B42 B43 0 0 0 B44 B45 B46B51 B52 B53 0 0 0 B54 B55 B56B61 B62 B63 0 0 0 B64 B65 B66

(1.31)

We note that some terms are equal to zero because we do not have a connection betweenthe corresponding degrees of freedom (i.e. node 1 is not connected to node 3).

1.3.2.4 Algorithm

25 The direct stiness method can be summarized as follows:

Preliminaries: First we shall

1. Identify type of structure (beam, truss, grid or frame) and determine the

(a) Number of spatial coordinates (1D, 2D, or 3D)(b) Number of degree of freedom per node (local and global)(c) Number of cross-sectional and material properties

2. Determine the global unrestrained and restrained degree of freedom equation num-bers for each node, Update the [ID] matrix (which included only 0s and 1s in theinput data le).

Analysis :

1. For each element, determine

(a) Vector LM relating local to global degree of freedoms.(b) Element stiness matrix [k(e)](c) Angle between the local and global x axes.(d) Rotation matrix [(e)](e) Element stiness matrix in global coordinates [K(e)] = [(e)]T [k(e)][(e)]

2. Assemble the augmented stiness matrix [K(S)] of unconstrained and constraineddegree of freedoms.

3. Extract [Ktt] from [K(S)] and invert (or decompose into into [Ktt] = [L][L]T where[L] is a lower triangle matrix.

4. Assemble load vector {P} in terms of nodal load and xed end actions.5. Backsubstitute and obtain nodal displacements in global coordinate system.

6. Solve for the reactions.


Draft1.3 Direct Stiness Method 1137. For each element, transform its nodal displacement from global to local coordinates{} = [(e)]{}, and determine the internal forces [p] = [k]{}.

26 Some of the prescribed steps are further discussed in the next sections.

Example 1-2: Direct Stiness Analysis of a Truss

Using the direct stiness method, analyze the truss shown in Fig. 1.6.

8

7

65

4

3

2

1

54

321

100k

50k

12

1616

Figure 1.6:

Solution:

1. Determine the structure ID matrix

Node # Bound. Cond.X Y

1 0 12 0 03 1 14 0 05 0 0

ID =[0 0 1 0 01 0 1 0 0

](1.32-a)

=[Node 1 2 3 4 51 2 2 4 61 3 3 5 7

](1.32-b)

2. The LM vector of each element is evaluated next

LM1 = 1 1 4 5 (1.33-a)LM2 = 1 1 2 3 (1.33-b)LM3 = 2 3 4 5 (1.33-c)


Draft114 PREREQUISITELM4 = 4 5 6 7 (1.33-d)LM5 = 2 3 4 5 (1.33-e)LM6 = 2 3 6 7 (1.33-f)LM7 = 2 3 2 3 (1.33-g)LM8 = 2 3 6 7 (1.33-h)

3. Determine the element stiness matrix of each element in the global coordinate system notingthat for a 2D truss element we have

[K(e)] = [(e)]T [k(e)][(e)] (1.34-a)

=EA

L

c2 cs c2 cscs s2 cs s2c2 cs c2 cscs s2 cs s2

(1.34-b)where c = cos = x2x1L ; s = sin =

Y2Y1L

Element 1 L = 20, c = 16020 = 0.8, s =

12020 = 0.6,

EAL =

(30,000 ksi)(10 in2)20

= 15, 000 k/ft.

[K1] =

1 1 4 5

1 9, 600 7200 9, 600 7, 2001 7, 200 5, 400 7, 200 5, 4004 9, 600 7, 200 9, 600 7, 2005 7, 200 5, 400 7, 200 5, 400

(1.35)Element 2 L = 16

, c = 1 , s = 0 , EAL = 18, 750 k/ft.

[K2] =

1 1 2 3

1 18, 750 0 18, 750 01 0 0 0 02 18, 750 0 18, 750 03 0 0 0 0

(1.36)Element 3 L = 12

, c = 0 , s = 1 , EAL = 25, 000 k/ft.

[K3] =

2 3 4 5

2 0 0 0 03 0 25, 000 0 25, 0004 0 0 0 05 0 25, 000 0 25, 000

(1.37)Element 4 L = 16

, c = 1 , s = 0 , EAL = 18, 750 k/ft.

[K4] =

4 5 6 7

4 18, 750 0 18, 750 05 0 0 0 06 18, 750 0 18, 750 07 0 0 0 0

(1.38)


Draft1.3 Direct Stiness Method 115Element 5 L = 20

, c = 16020 = 0.8 , s = 0.6 , EAL = 15, 000 k/ft.

[K5] =

2 3 4 5

2 9, 600 7, 200 9, 600 7, 2003 7, 200 5, 400 7, 200 5, 4004 9, 600 7, 200 9, 600 7, 2005 7, 200 5, 400 7, 200 5, 400

(1.39)Element 6 L = 20

, c = 0.8 , s = 0.6 , EAL = 15, 000 k/ft.

[K6] =

2 3 6 7

2 9, 600 7, 200 9, 600 7, 2003 7, 200 5, 400 7, 200 5, 4006 9, 600 7, 200 9, 600 7, 2007 7, 200 5, 400 7, 200 5, 400

(1.40)Element 7 L = 16

, c = 1 , s = 0 , EAL = 18, 750 k/ft.

[K7] =

2 3 2 3

2 18, 750 0 18, 750 03 0 0 0 02 18, 750 0 18, 750 03 0 0 0 0

(1.41)Element 8 L = 12

, c = 0 , s = 1 , EAL = 25, 000 k/ft.

[K8] =

2 3 6 7

2 0 0 0 03 0 25, 000 0 25, 0006 0 0 0 07 0 25, 000 0 25, 000

(1.42)4. Assemble the global stiness matrix in k/ft Note that we are not assembling the augmentedstiness matrix, but rather its submatrix [Ktt].8

>

>

>

>

>

>

>

>

>

>

>

>

>

:

00

100k00

50k0

9

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

;

=

2

6

6

6

6

6

6

6

4

9, 600 + 18, 750 18, 750 0 9, 600 7, 200 0 09, 600 + (2) 18, 750 7, 200 0 0 9, 600 7, 200

5, 400 + 25, 000 0 25, 000 7, 200 5, 40018, 750 + (2)9, 600 7, 200 7, 200 18, 750 0

SYMMETRIC 25, 000 + 5, 400(2) 0 018, 750 + 9, 600 7, 200

25, 000 + 5, 400

3

7

7

7

7

7

7

7

5

8

>

>

>

>

>

>

>

>

>

>

>

>

>

:

u1u2v3u4v5u6v7

9

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

;

(1.43)

5. Convert to k/in and simplify

00

10000500

Pt

=

2, 362.5 1, 562.5 0 800 600 0 03, 925.0 600 0 0 800 600

2, 533.33 0 2, 083.33 600 4503, 162.5 0 1, 562.5 0

SYMMETRIC 2, 983.33 0 02, 362.5 600

2, 533.33

Ktt

U1U2V3U4V5U6V7

ut

(1.44)


Draft116 PREREQUISITE6. Invert stiness matrix and solve for displacements

U1U2V3U4V5U6V7

=

0.0223 in0.00433 in0.116 in0.0102 in0.0856 in0.00919 in0.0174 in

(1.45)

7. Solve for member internal forces (in this case axial forces) in local coordinate systems

{u1u2

}=[

c s c sc s c s

]U1V1U2V2

(1.46)Element 1

{p1p2

}1= (15, 000 k/ft)(

112

ftin)[

0.8 0.6 0.8 0.60.8 0.6 0.8 0.6

]0.0223

00.01020.0856

(1.47-a)=

{52.1 k52.1 k

}Compression (1.47-b)

Element 2

{p1p2

}2= 18, 750 k/ft(

112

ftin)[

1 0 1 01 0 1 0

]0.0233

00.004330.116

(1.48-a)=

{ 43.2 k43.2 k

}Tension (1.48-b)

Element 3

{p1p2

}3= 25, 000 k/ft(

112

ftin)[0 1 0 10 1 0 1

]0.004330.1160.01020.0856

(1.49-a)=

{ 63.3 k63.3 k

}Tension (1.49-b)


Draft1.3 Direct Stiness Method 117Element 4

{p1p2

}4= 18, 750 k/ft(

112

ftin)[

1 0 1 01 0 1 0

]0.01020.08560.009190.0174

(1.50-a)=

{ 1.58 k1.58 k

}Tension (1.50-b)

Element 5

{p1p2

}5= 15, 000 k/ft(

112

ftin)[ 0.8 0.6 0.8 0.6

0.8 0.6 0.8 0.6]

0.01020.0856

00

(1.51-a)=

{54.0 k54.0 k


Element 6

{p1p2

}6= 15, 000 k/ft(

112

ftin)[

0.8 0.6 0.8 0.60.8 0.6 0.8 0.6

]0.004330.1160.009190.0174

(1.52-a)=

{ 60.43 k60.43 k

}Tension (1.52-b)

Element 7

{p1p2

}7= 18, 750 k/ft(

112

ftin)[

1 0 1 01 0 1 0

]0.004330.116

00

(1.53-a)=

{6.72 k6.72 k


Element 8

{p1p2

}8= 25, 000 k/ft(

112

ftin)[0 1 0 10 1 0 1

]000.009190.0174

(1.54-a)=

{36.3 k36.3 k



Draft118 PREREQUISITE8. Determine the structures MAXA vector

[K] =

1 3 9 142 5 8 13 19 25

4 7 12 18 246 11 17 23

10 16 2215 21

20

MAXA =

1246101520

(1.55)

Thus, 25 terms would have to be stored.

Example 1-3: Analysis of a Frame with MATLAB

The simple frame shown in Fig. 1.7 is to be analyzed by the direct stiness method. Assume:E = 200, 000 MPa, A = 6, 000 mm2, and I = 200 106 mm4. The complete MATLAB solutionis shown below along with the results.

7.416 m 8 m

3 m

4 kN/m50kN

8 m

1

2 3

1

2

Figure 1.7: Simple Frame Analyzed with the MATLAB Code

% zero the matricesk=zeros(6,6,2); K=zeros(6,6,2); Gamma=zeros(6,6,2);% Structural properties units: mm^2, mm^4, and MPa(10^6 N/m)A=6000;II=200*10^6;EE=200000;% Convert units to meter and kNA=A/10^6;II=II/10^12;EE=EE*1000;% Element 1i=[0,0];j=[7.416,3]; [k(:,:,1),K(:,:,1),Gamma(:,:,1)]=stiff(EE,II,A,i,j);% Element 2i=j;j=[15.416,3]; [k(:,:,2),K(:,:,2),Gamma(:,:,2)]=stiff(EE,II,A,i,j);% Define ID matrixID=[

-4 1 -7;-5 2 -8;-6 3 -9];

% Determine the LM matrixLM=[


Draft1.3 Direct Stiness Method 119-4 -5 -6 1 2 3;1 2 3 -7 -8 -9];

% Assemble augmented stiffness matrixKaug=zeros(9); for elem=1:2

for r=1:6lr=abs(LM(elem,r));for c=1:6

lc=abs(LM(elem,c));Kaug(lr,lc)=Kaug(lr,lc)+K(r,c,elem);

endend

end% Extract the structures Stiffness MatrixKtt=Kaug(1:3,1:3);% Determine the fixed end actions in local coordinate systemfea(1:6,1)=0; fea(1:6,2)=[0,8*4/2,4*8^2/12,0,8*4/2,-4*8^2/12];% Determine the fixed end actions in global coordinate systemFEA(1:6,1)=Gamma(:,:,1)*fea(1:6,1); FEA(1:6,2)=Gamma(:,:,2)*fea(1:6,2);% FEA_Rest for all the restrained nodesFEA_Rest=[0,0,0,FEA(4:6,2)];% Assemble the load vector for the unrestrained nodeP(1)=50*3/8;P(2)=-50*7.416/8-FEA(2,2);P(3)=-FEA(3,2);% Solve for the Displacements in meters and radiansDisplacements=inv(Ktt)*P% Extract KutKut=Kaug(4:9,1:3);% Compute the Reactions and do not forget to add fixed end actionsReactions=Kut*Displacements+FEA_Rest% Solve for the internal forces and do not forget to include the fixed end actionsdis_global(:,:,1)=[0,0,0,Displacements(1:3)];dis_global(:,:,2)=[Displacements(1:3),0,0,0]; for elem=1:2

dis_local=Gamma(:,:,elem)*dis_global(:,:,elem);int_forces=k(:,:,elem)*dis_local+fea(1:6,elem)

end

function [k,K,Gamma]=stiff(EE,II,A,i,j)% Determine the lengthL=sqrt((j(2)-i(2))^2+(j(1)-i(1))^2);% Compute the angle theta (careful with vertical members!)if(j(1)-i(1))~=0

alpha=atan((j(2)-i(2))/(j(1)-i(1)));else

alpha=-pi/2;end% form rotation matrix GammaGamma=[ cos(alpha) sin(alpha) 0 0 0 0; -sin(alpha)cos(alpha) 0 0 0 0; 0 0 1 0 00; 0 0 0 cos(alpha) sin(alpha) 0; 0 0


Draft120 PREREQUISITE0 -sin(alpha) cos(alpha) 0; 0 0 0 0 01];% form element stiffness matrix in local coordinate systemEI=EE*II; EA=EE*A; k=[EA/L, 0, 0, -EA/L, 0, 0;

0, 12*EI/L^3, 6*EI/L^2, 0, -12*EI/L^3, 6*EI/L^2;0, 6*EI/L^2, 4*EI/L, 0, -6*EI/L^2, 2*EI/L;

-EA/L, 0, 0, EA/L, 0, 0;0, -12*EI/L^3, -6*EI/L^2, 0, 12*EI/L^3, -6*EI/L^2;0, 6*EI/L^2, 2*EI/L, 0, -6*EI/L^2, 4*EI/L];

% Element stiffness matrix in global coordinate systemK=Gamma*k*Gamma;

This simple proigram will produce the following results:

Displacements =

0.0010-0.0050-0.0005

Reactions =

130.497355.676613.3742

-149.247322.6734

-45.3557

int_forces = int_forces =

141.8530 149.24732.6758 9.3266

13.3742 -8.0315-141.8530 -149.2473-2.6758 22.67348.0315 -45.3557

We note that the internal forces are consistent with the reactions (specially for the second nodeof element 2), and amongst themselves, i.e. the moment at node 2 is the same for both elements(8.0315).

Example 1-4: Analysis of a simple Beam with Initial Displacements


Draft1.3 Direct Stiness Method 121The full stiness matrix of a beam element is given by

[ke] =

v1 1 v2 2

V1 12EI/L3 6EI/L2 12EI/L3 6EI/L2M1 6EI/L2 4EI/L 6EI/L2 2EI/LV2 12EI/L3 6EI/L2 12EI/L3 6EI/L2M2 6EI/L2 2EI/L 6EI/L2 4EI/L

(1.56)This matrix is singular, it has a rank 2 and order 4 (as it embodies also 2 rigid body motions).

27 We shall consider 3 dierent cases, Fig. 1.8

1 2-4-3

P2

-3 1 -4M

1-4

-2 -3

Figure 1.8: Stiness Analysis of one Element Structure

Cantilivered Beam/Point Load

1. The element stiness matrix is

k =

2

6

6

4

3 4 1 23 12EI/L3 6EI/L2 12EI/L3 6EI/L24 6EI/L2 4EI/L 6EI/L2 2EI/L1 12EI/L3 6EI/L2 12EI/L3 6EI/L22 6EI/L2 2EI/L 6EI/L2 4EI/L

3

7

7

5

2. The structure stiness matrix is assembled

K =

2

6

6

4


3

7

7

5

3. The global matrix can be rewritten as8

>

>

>

:

P0

R3?R4?

9

>

>

=

>

>

;

=

2

6

6

4

12EI/L2 6EI/L2 12EI/L3 6EI/L26EI/L2 4EI/L 6EI/L2 2EI/L12EI/L3 6EI/L2 12EI/L3 6EI/L26EI/L2 2EI/L 6EI/L2 4EI/L

3

7

7

5

8

>

>

>

:

1?2?

3

4

9

>

>

=

>

>

;

4. Ktt is inverted (or actually decomposed) and stored in the same global matrix2

6

6

6

6

4

L3/3EI L2/2EI 12EI/L3 6EI/L2

L2/2EI L/EI 6EI/L2 2EI/L

12EI/L3 6EI/L2 12EI/L3 6EI/L26EI/L2 2EI/L 6EI/L2 4EI/L

3

7

7

7

7

5


Draft122 PREREQUISITE5. Next we compute the equivalent load, Pt = Pt Ktuu, and overwrite Pt by Pt

Pt Ktuu =

8

>

>

>

>

>

:

P0

00

9

>

>

>

=

>

>

>

;

2

6

6

6

6

4




3

7

7

7

7

5

8

>

>

>

:

P0

0

0

9

>

>

=

>

>

;

=

8

>

>

>

>

>

:

P0

00

9

>

>

>

=

>

>

>

;

6. Now we solve for the displacement t = K1tt Pt, and overwrite Pt by t8

>

>

>

>

>

:

1

2

00

9

>

>

>

=

>

>

>

;

=

2

6

6

6

6

4




3

7

7

7

7

5

8

>

>

>

>

>

:

P0

00

9

>

>

>

=

>

>

>

;

=

8

>

>

>

>

>

>

>

:

PL3/3EIPL2/2EI

00

9

>

>

>

>

=

>

>

>

>

;

7. Finally, we solve for the reactions, Ru = Kuttt+Kuuu, and overwrite u by Ru8

>

>

>

>

>

:

PL3/3EIPL2/2EI

R3

R4

9

>

>

>

=

>

>

>

;

=

2

6

6

6

6

4

L3/3EI L2/2EI 12EI/L3 6EI/L2L2/2EI L/EI 6EI/L2 2EI/L

12EI/L3 6EI/L2 12EI/L3 6EI/L2

6EI/L2 2EI/L 6EI/L2 4EI/L

3

7

7

7

7

5

8

>

>

>

>

>

>

>

>

>

:

PL3/3EIPL2/2EI

0

0

9

>

>

>

>

>

=

>

>

>

>

>

;

=

8

>

>

>

>

>

:

PL3/3EIPL2/2EI

P

PL

9

>

>

>

=

>

>

>

;

Simply Supported Beam/End Moment


k =

2

6

6

4


3

7

7

5


K =

2

6

6

4

1 2 3 41 4EI/L 2EI/L 6EI/L2 6EI/L22 2EI/L 4EI/L 6EI/L2 6EI/L23 6EI/L2 6EI/L2 12EI/L3 12EI/L34 6EI/L2 6EI/L2 12EI/L3 12EI/L3

3

7

7

5

3. The global stiness matrix can be rewritten as8

>

>

>

>

>

>

>

:

0

M

R3?R4?

9

>

>

>

>

=

>

>

>

>

;

=

2

6

6

4

4EI/L 2EI/L 6EI/L2 6EI/L22EI/L 4EI/L 6EI/L2 6EI/L26EI/L2 6EI/L2 12EI/L3 12EI/L36EI/L2 6EI/L2 12EI/L3 12EI/L3

3

7

7

5

8

>

>

>

:

1?2?

3

4

9

>

>

=

>

>

;


Draft1.3 Direct Stiness Method 1234. Ktt is inverted

2

6

6

6

6

4

L3/3EI L/6EI 6EI/L2 6EI/L2

L/6EI L/3EI 6EI/L2 6EI/L26EI/L2 6EI/L2 12EI/L3 12EI/L36EI/L2 6EI/L2 12EI/L3 12EI/L3

3

7

7

7

7

5

5. We compute the equivalent load, Pt = Pt Ktuu, and overwrite Pt by Pt

Pt Ktuu =

8

>

>

>

:

0

M

00

9

>

>

=

>

>

;

2

6

6

6

6

4



3

7

7

7

7

5

8

>

>

>

:

0M

0

0

9

>

>

=

>

>

;

=

8

>

>

>

:

0

M

00

9

>

>

=

>

>

;

6. Solve for the displacements, t = K1tt Pt, and overwrite Pt by t8

>

>

>

>

>

:

1

2

00

9

>

>

>

=

>

>

>

;

=

2

6

6

6

6

4



3

7

7

7

7

5

8

>

>

>

:

0

M

00

9

>

>

=

>

>

;

=

8

>

>

>

>

>

>

>

:

ML/6EIML/3EI

00

9

>

>

>

>

=

>

>

>

>

;

7. Solve for the reactions, Rt = Kuttt +Kuuu, and overwrite u by Ru8

>

>

>

>

>

:

ML/6EIML/3EI

R1

R2

9

>

>

>

=

>

>

>

;

=

2

6

6

6

6

4

L3/3EI L/6EI 6EI/L2 6EI/L2L/6EI L/3EI 6EI/L2 6EI/L26EI/L2 6EI/L2 12EI/L3 12EI/L3


3

7

7

7

7

5

8

>

>

>

>

>

>

>

:

ML/6EIML/3EI

0

0

9

>

>

>

>

=

>

>

>

>

;

=

8

>

>

>

>

>

>

>

:

ML/6EIML/3EI

M/L

M/L

9

>

>

>

>

=

>

>

>

>

;

Cantilivered Beam/Initial Displacement and Concentrated Moment


k =

2

6

6

4


3

7

7

5


K =

2

6

6

4

1 2 3 41 4EI/L 6EI/L2 2EI/L 6EI/L22 6EI/L2 12EI/L3 6EI/L2 12EI/L33 2EI/L 6EI/L2 4EI/L 6EI/L24 6EI/L2 12EI/L3 6EI/L2 12EI/L3

3

7

7

5


Draft124 PREREQUISITE3. The global matrix can be rewritten as

8

>

>

>

:

M

R2?R3?R4?

9

>

>

=

>

>

;

=

2

6

6

4

4EI/L 6EI/L2 2EI/L 6EI/L26EI/L2 12EI/L3 6EI/L2 12EI/L32EI/L 6EI/L2 4EI/L 6EI/L2


3

7

7

5

8

>

>

>

:

1?

2

3

4

9

>

>

=

>

>

;

4. Ktt is inverted (or actually decomposed) and stored in the same global matrix2

6

6

6

4

L/4EI 6EI/L2 2EI/L 6EI/L26EI/L2 12EI/L3 6EI/L2 12EI/L32EI/L 6EI/L2 4EI/L 6EI/L2


3

7

7

7

5

5. Next we compute the equivalent load, Pt = Pt Ktuu, and overwrite Pt by Pt

Pt Ktuu =

8

>

>

>

:

M

000

9

>

>

=

>

>

;

2

6

6

6

4



3

7

7

7

5

8

>

>

>

>

>

:

1

0

0

0

9

>

>

>

=

>

>

>

;

=

8

>

>

>

>

>

:

M + 6EI0/L2

000

9

>

>

>

=

>

>

>

;

6. Now we solve for the displacements, t = K1tt Pt, and overwrite Pt by t8

>

>

>

:

1

000

9

>

>

=

>

>

;

=

2

6

6

6

4



3

7

7

7

5

8

>

>

>

>

>

:

M + 6EI0/L2

000

9

>

>

>

=

>

>

>

;

=

8

>

>

>

>

>

:

ML/4EI + 30/2L

000

9

>

>

>

=

>

>

>

;

7. Finally, we solve for the reactions, Rt = Kuttt+Kuuu, and overwrite u by Ru8

>

>

>

>

>

:

ML/4EI + 30/2L

R2

R3

R4

9

>

>

>

=

>

>

>

;

=

2

6

6

6

6

6

4

L/4EI 6EI/L2 2EI/L 6EI/L26EI/L2 12EI/L3 6EI/L2 12EI/L3

2EI/L 6EI/L2 4EI/L 6EI/L2


3

7

7

7

7

7

5

8

>

>

>

>

>

>

>

:

ML/4EI + 30/2L

0

0

0

9

>

>

>

>

=

>

>

>

>

;

=

8

>

>

>

>

>

>

>

>

>

:

ML/4EI + 30/2L

3M/2L 3EI0/L3

M/2 3EI0/L2

3M/2L + 3EI0/L3

9

>

>

>

>

>

=

>

>

>

>

>

;


Draft

Chapter 2

INTRODUCTION

2.1 Introduction

1 Whereas the rst course focused exclusively on one dimensional rod elements, this coursewill greatly expand our horizons by considering introducing a methodology to solve partialdierential equations, with special emphasis on solid mechanics.

2 The eld of mechanics, can itself be subdivided into four major disciplines:

Theoretical which deals with the fundamental laws and principles of mechanics. A ContinuumMechanics course is a must.

Applied mechanics seeks to apply the theoretical knowledge to engineering applications. Elas-ticity or Fracture Mechanics solutions are such an example of applied mechanics.

Computational mechanics combines mathematical models with numerical methods to solveproblems on a digital computer.

Experimental mechanics is conducted exclusively in a laboratory through physical measure-ments.

3 Any problem characterized by a PDE can be analyzed by the nite element method. Theprocess of nite element analysis is illustrated by Fig. 2.12.

2.2 Elliptic, Parabolic and Hyperbolic Equations

4 Since the nite element method is a numerical scheme to solve (partial) dierential equations,let us closely examine some of the major PDE which can be solved.

5 The general form of a partial dierential equation is (note that we adopt the tensor notationwhere u,x = dudx):

F (x, y, z, , u, u,x, u,y, u,z, , u,xx, u,yy, , u,xy, u,xz, ) = 0and the order of the PDE is dened by the order of the highest partial derivatives appearingin the equation. For instance

1u,xx + 2u,xy + 3u,yy + 4 = 0

Date post:	24-Nov-2015
Category:	Documents
Upload:	andrija-popovic
View:	147 times
Download:	4 times

Finite Element II, Solid Mechanics_Victor E.saouma_2001

Documents