Practical Programming of Finite Element Procedures for Solids and Structures with MATLAB®

1. A brief overview of MATLAB® programming language 1.1 Introduction 1.2 Variables 1.3 Vectors 1.3.1 Vector operations 1.3.2 The linspace and logspace functions 1.4 Matrices 1.4.1 Singular, orthogonal, and positive definite matrices 1.4.2 Multi-dimensional matrices 1.4.3 Matrix operations 1.4.4 Matrix transpose and inverse 1.4.5 Concatenating matrices 1.4.6 Reshaping matrices 1.4.7 Solving the systems of linear equations in matrix form 1.5 Export and import data 1.6 Loops 1.6.1 The for loop 1.6.2 The while loop 1.7 Conditional statements 1.8 The switch function 1.9 2D and 3D plotting 1.10 Programming a function 1.10.1 Scripting 1.10.2 Functions 1.11 Chapter overview 2. Matrix analysis of framed structures 2.1 Introduction 2.1.1 Determining the Equation of the Stiffness Method 2.1.2 Forming the Stiffness Matrix of a Structural Member 2.1.3 Applying Boundary Conditions 2.2 Plane Trusses 2.2.1 Programming for the Matrix Analysis of Plane Trusses 2.2.2 Obtaining the Internal Forces of Plane Truss Members 2.3 Space Trusses 2.3.1 Programming for the Matrix Analysis of Space Trusses 2.3.2 Obtaining the Internal Forces of Space Truss Members 2.4 Plane Frames 2.4.1 Programming for the Matrix Analysis of Plane Frames 2.4.2 Obtaining the Internal Forces of Plane Frame Members 2.5 Space Frames 2.5.1 Programming for the Matrix Analysis of Space Frames 2.5.2 Obtaining the Internal Forces of Space Frame Members 2.6 Grids 2.6.1 Programming for the Matrix Analysis of Grids 2.6.2 Obtaining the Internal Forces of Grid Members 2.7 Special Cases 2.7.1 Member Loadings 2.7.2 Support Settlements 2.7.3 Temperature Variations 2.7.4 Non-Compliant Members 2.7.5 Released Members 2.7.6 Elastic Supports 2.8 Chapter Overview 3. Elastic analysis of structures using finite element procedure 3.1 Introduction to Finite Element Procedure 3.2 FEM Programming for Continuum Elements 3.2.1 One-Dimensional elements 3.2.2 Two-Dimensional elements 3.2.3 Three-Dimensional elements 3.3 FEM Programming for Structural Elements 3.3.1 Timoshenko Beam Theory 3.3.2 Mindlin-Reissner Plate Bending Theory 3.4 Chapter Overview 4. Elastoplastic analysis of structures using finite element procedure 4.1 Introduction 4.2 Basics 4.2.1 Stress invariants and deviatoric stresses 4.2.2 Strain invariants and deviatoric strains 4.2.3 Voigt notation 4.2.4 Haigh–Westergaard stress space 4.3 Elastoplasticity 4.3.1 Plastic flow rule 4.3.2 Yield function 4.4 Yield criteria 4.4.1 Von Mises yield criterion 4.4.2 Tresca yield criterion 4.4.3 Evolution of the yield surface 4.5 Hardening laws 4.5.1 Perfect-plasticity 4.5.2 Isotropic hardening 4.5.3 Kinematic hardening 4.5.4 Combined/Mixed kinematic and isotropic hardening 4.6 Stress integration using the von Mises yield criterion 4.6.1 Material with kinematic hardening 4.6.2 Material with isotropic hardening 4.6.3 Material with combined kinematic/isotropic hardening 4.6.4 Material with power-law Isotropic hardening 4.7 Solving non-linear problems 4.7.1 Incremental iterative solution method 4.7.2 General steps 4.7.3 Plasticity algorithm 4.8 Finite Element programming considering the non-linear behavior of materials 4.8.1 Materials with kinematic hardening 4.8.2 Materials with isotropic hardening 4.8.3 Materials with combined isotropic/kinematic hardening 4.8.4 Materials with power-law isotropic hardening 4.9 Extension to Three-Dimensional Elements 4.10 Elastoplastic tangent operator 4.10.1 Tangent operator for one-dimensional elastoplasticity 4.10.2 Tangent operator for multi-dimensional elastoplasticity 4.11 Convergence 4.11.1 rate of convergence 4.12 Chapter overview 5. Finite deformation and hyperelasticity 5.1 Introduction 5.2 Strain and stress measures in finite deformation 5.2.1 Deformation gradient 5.2.2 Lagrangian strain 5.2.3 Cauchy and Piola-Kirchhoff stresses 5.3 Analysis procedures 5.3.1 Introduction to linearization of the equilibrium equation 5.3.2 Total Lagrangian (TL) formulation 5.3.3 Updated Lagrangian (UL) formulation 5.4 Hyperelastic Materials 5.4.1 Strain energy density potential 5.4.2 The F-bar deformation gradient 5.4.3 Types of the strain energy density potential 5.4.4 Obtaining the Cauchy stress for the neo-Hookean, Mooney–Rivlin, and Yeoh models 5.4.5 Fitting hyperelastic material constants from test data 5.4.6 Programming for hyperelastic materials using the neo-Hookean model 5.5 Chapter overview 6. Finite strain 6.1 Introduction 6.2 Finite rotation and objective rates 6.2.1 Objective stress rates 6.2.2 Analysis procedures with the TL formulation and Jaumann stress rate 6.2.3 Analysis procedures with the UL formulation and Jaumann rate 6.2.4 Extension to elastoplasticity 6.3 Finite deformation elastoplasticity 6.3.1 Multiplicative decomposition 6.3.2 Formulation of the Finite deformation elastoplasticity 6.3.3 Integration procedure 6.3.4 Consistent tangent matrix 6.4 Chapter overview 7. Solution to systems of linear equations 7.1 Introduction 7.2 Solution to systems of linear equations 7.2.1 Direct methods 7.2.2 Iterative methods 7.3 Solution to eigen-problems 7.3.1 Transformation methods 7.3.2 Vector iteration methods 7.3.3 Implicit polynomial iteration method 7.3.4 Lanczos iteration method 7.3.5 Subspace iteration method 7.4 Chapter overvie