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

Practical Programming of Finite Element Procedures for Solids and Structures with MATLAB®: From Elasticity to Plasticity provides readers with step-by-step programming processes and applications of the finite element method (FEM) in MATLAB®, as well as the underlying theory. The hands-on approach covers a number of structural problems such as linear analysis of solids and structural elements, as well as nonlinear subjects, including elastoplasticity and hyperelasticity. Each chapter begins with foundational topics to provide a solid understanding of the subject and then progresses to more complicated problems with supporting examples for constructing the appropriate program.

The book focuses on topics commonly encountered in civil, mechanical, and aerospace engineering, with special situations in structural analysis, 2D and 3D solids with various mesh elements, surface and body loading, incremental solution process, elastoplasticity, and finite deformation hyperelastic analysis each covered. Code that can be implemented and further extended is also provided.

Cover image
Title page
Table of Contents
Copyright
Dedication
Preface
Aims and scope
Subjects and contents
Chapter 1. A Brief Overview of MATLAB® Programming Language
Abstract
Table of Contents
1.1 Introduction
1.2 Variables
1.3 Vectors
1.4 Matrices
1.5 Export and import data
1.6 Loops
1.7 Conditional statements
1.8 The switch function
1.9 2D and 3D Plotting
1.10 Programming a function
1.11 Chapter overview
Exercises
References
Chapter 2. Matrix Analysis of Framed Structures
Abstract
Table of Contents
2.1 Introduction
2.2 EXAMPLE 2.1: Determining the general form of the stiffness matrix
2.3 Plane trusses
2.4 EXAMPLE 2.2: Matrix analysis of a plane truss
2.5 EXAMPLE 2.3: Matrix analysis of a plane truss
2.6 Space trusses
2.7 EXAMPLE 2.4: Matrix analysis of a space truss
2.8 EXAMPLE 2.5: Matrix analysis of a space truss
2.9 Plane frames
2.10 EXAMPLE 2.6: Matrix analysis of a plane frame
2.11 EXAMPLE 2.7: Matrix analysis of a plane frame
2.12 Space frames
2.13 EXAMPLE 2.8: Matrix analysis of a space frame
2.14 EXAMPLE 2.9: Matrix analysis of a space frame
2.15 Grids
2.16 EXAMPLE 2.10: Matrix analysis of a grid structure
2.17 EXAMPLE 2.11: Matrix analysis of a grid structure
2.18 Special cases
2.19 EXAMPLE 2.12: Matrix analysis of a plane frame subjected to a distributed member loading
2.20 EXAMPLE 2.13: Matrix analysis of a plane frame subjected to support settlements
2.21 EXAMPLE 2.14: Matrix analysis of a plane frame subjected to temperature variations
2.22 EXAMPLE 2.15: Matrix analysis of a plane truss frame affected by member mismatch
2.23 EXAMPLE 2.16: Matrix analysis of a plane frame with released members
2.24 EXAMPLE 2.17: Matrix analysis of a plane frame having an elastic support
2.25 Chapter overview
Exercises
References
Chapter 3. Elastic analysis of structures using finite element procedure
Abstract
Table of Contents
3.1 Introduction
3.2 FEM programming for continuum elements
3.3 EXAMPLE 3.1: Obtaining the stiffness matrix of a two-node truss member
3.4 EXAMPLE 3.2: Obtaining the stiffness matrix of a three-node truss member
3.5 EXAMPLE 3.3: Linear-elastic analysis of a one-dimensional structure with varying cross-sections containing a spring
3.6 EXAMPLE 3.4: Linear-elastic analysis of a one-dimensional structure with varying cross-sections
3.7 EXAMPLE 3.5: Linear-elastic analysis of a plane truss using FEM
3.8 EXAMPLE 3.6: Linear-elastic analysis of plane stress structure subjected to tensile surface load using quadrilateral elements
3.9 EXAMPLE 3.7: Linear-elastic analysis of plane stress structure subjected to uniform bending and surface tractional loads using quadrilateral elements
3.10 EXAMPLE 3.8: Linear-elastic analysis of a plane strain structure using quadrilateral elements
3.11 EXAMPLE 3.9: Linear-elastic analysis of an axisymmetric structure using quadrilateral elements
3.12 EXAMPLE 3.10: Linear-elastic analysis of a plane stress structure using triangular elements
3.13 EXAMPLE 3.11: Linear-elastic analysis of a plane stress structure subjected to body loads using triangular elements
3.14 EXAMPLE 3.12: Linear-elastic analysis of a three-dimensional structure subjected to a surface load
3.15 EXAMPLE 3.13: Linear-elastic analysis of a three-dimensional structure subjected to a body load
3.16 FEM programming for structural elements
3.17 EXAMPLE 3.14: Linear-elastic analysis of a clamped Timoshenko beam subjected to a uniform distributed load
3.18 EXAMPLE 3.15: Linear-elastic analysis of a Timoshenko beam having simple and shear supports subjected to a uniform distributed load
3.19 EXAMPLE 3.16: Linear-elastic analysis of a simply supported Mindlin–Reissner plate
3.20 EXAMPLE 3.17: Linear-elastic analysis of a clamped supported Mindlin–Reissner plate
3.21 Chapter overview
Exercises
References
Chapter 4. Elastoplastic Analysis of Structures Using Finite Element Procedure
Abstract
Table of Contents
4.1 Introduction
4.2 Basics
4.3 Elastoplasticity
4.4 Yield criteria
4.5 Hardening laws
4.6 Stress integration using the von Mises yield criterion
4.7 Solving nonlinear problems
4.8 Finite element programming considering the nonlinear behavior of materials
4.9 EXAMPLE 4.1: Elastoplastic analysis of structure with plane strain condition and kinematic hardening using four-node rectangular quadrilaterals subjected to a uniform surface displacement
EXAMPLE 4.2: Elastoplastic analysis of structure with plane strain condition and isotropic hardening using four-node rectangular quadrilaterals subjected to a uniform surface displacement
EXAMPLE 4.3: Elastoplastic analysis of structure with plane strain condition and isotropic hardening using four-node rectangular quadrilaterals subjected to a uniform surface load
EXAMPLE 4.4: Elastoplastic analysis of structure with plane strain condition and combined hardening using four-node rectangular quadrilaterals subjected to a uniform surface load
EXAMPLE 4.5: Elastoplastic analysis of structure with plane strain condition and power-law isotropic hardening using four-node rectangular quadrilaterals under uniform surface displacement
4.10 Extension to three-dimensional elements
4.11 Elastoplastic tangent operator
4.12 Convergence criteria
4.13 Chapter overview
Exercises
References
Further reading
Chapter 5. Finite deformation and hyperelasticity
Abstract
Table of Contents
5.1 Introduction
5.2 Strain and stress measures in finite deformation
5.3 EXAMPLE 5.1: Programming for obtaining the deformation gradient, the Lagrangian strain, and the second Piola–Kirchhoff stress
5.4 Analysis procedures
5.5 EXAMPLE 5.2: Finite strain analysis of a uniaxial truss element subjected to a tension load using the TL formulation
5.6 EXAMPLE 5.3: Finite strain analysis of a uniaxial truss element subjected to a tension load using the UL formulation
5.7 Hyperelastic materials
5.8 EXAMPLE 5.4: Fitting test data to the neo-Hookean model
5.9 EXAMPLE 5.5: Fitting test data to the Mooney–Rivlin model
5.10 EXAMPLE 5.6: Fitting test data to the Yeoh model
5.11 EXAMPLE 5.7: Fitting test data to the Ogden model
5.12 EXAMPLE 5.8: Finite deformation analysis of a structure with plane strain condition and hyperelastic material using four-node rectangular quadrilaterals subjected to a uniform tension load
5.13 EXAMPLE 5.9: Step-by-step finite deformation analysis of a structure with plane strain condition and hyperelastic material using four-node rectangular quadrilaterals subjected to a uniform tension load
5.14 EXAMPLE 5.10: Step-by-step finite deformation analysis of a structure with plane strain condition and nearly incompressible hyperelastic material using four-node rectangular quadrilaterals subjected to uniform flexural and traction loads
5.15 Chapter overview
Exercises
References
Further reading
Chapter 6. Finite strain
Abstract
Table of Contents
6.1 Introduction
6.2 Finite rotation and objective rates
6.3 EXAMPLE 6.1: Step-by-step finite strain analysis of a structure with plane strain condition using four-node rectangular quadrilaterals subjected to uniform flexural and traction loads via the TL formulation and Jaumann stress rate
6.4 EXAMPLE 6.2: Step-by-step finite strain analysis of a structure with plane strain condition using four-node rectangular quadrilaterals subjected to uniform flexural and traction loads via the UL formulation and Jaumann stress rate
6.5 Finite deformation elastoplasticity
6.6 Chapter overview
Exercises
References
Further reading
Chapter 7. Solution to systems of linear equations
Abstract
Table of Contents
7.1 Introduction
7.2 Solution to systems of linear equations
7.3 EXAMPLE 7.1: Programming for the LDLT decomposition
7.4 EXAMPLE 7.2: Programming of the Gauss–Seidel iterative algorithm
7.5 EXAMPLE 7.3: Programming of the successive overrelaxation algorithm
7.6 EXAMPLE 7.4: Programming of the modified Richardson iterative algorithm
7.7 EXAMPLE 7.5: Programming of the conjugate gradient iterative algorithm
7.8 EXAMPLE 7.6: Programming of the conjugate residual iterative algorithm
7.9 EXAMPLE 7.7: Programming of the preconditioned conjugate gradient iterative algorithm
7.10 EXAMPLE 7.8: Programming of the MINRES iterative algorithm
7.11 EXAMPLE 7.9: Programming of the BiCG iterative algorithm
7.12 EXAMPLE 7.10: Programming of the BiCG stabilized iterative algorithm
7.13 EXAMPLE 7.11: Programming of the CGS iterative algorithm
7.14 EXAMPLE 7.12: Programming of the QMR iterative algorithm
7.15 EXAMPLE 7.13: Programming of the TFQMR iterative algorithm
7.16 EXAMPLE 7.14: Programming of the GMRES iterative algorithm
7.17 Solution to eigenproblems
7.18 EXAMPLE 7.15: Programming for transforming a general eigenproblem into the standard form
7.19 EXAMPLE 7.16: Programming of the Jacobi method
7.20 EXAMPLE 7.17: Programming of the generalized Jacobi method
7.21 EXAMPLE 7.18: Programming of the HQRI method
7.22 EXAMPLE 7.19: Programming of the vector inverse method
7.23 EXAMPLE 7.20: Programming of the forward iteration method
7.24 EXAMPLE 7.21: Programming of the shifted vector iteration method
7.25 EXAMPLE 7.22: Programming of the Rayleigh Quotient iteration method
7.26 EXAMPLE 7.23: Programming of the Rayleigh Quotient iteration algorithm with the Gram–Schmidt method
7.27 EXAMPLE 7.24: Programming of the implicit polynomial iteration method
7.28 EXAMPLE 7.25: Programming for solving a generalized eigenproblem using the Lanczos transformation method
7.29 EXAMPLE 7.26: Programming of the convergence check for the Lanczos transformation method
7.30 EXAMPLE 7.27: Programming of the subspace method
7.31 Chapter overview
References
Further reading
Index

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

From Elasticity to Plasticity

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Salar Farahmand-Tabar

Kian Aghani