Intro
Preface
Contents
1 Mathematical Background
1.1 Introduction
1.2 Matrices and Vectors
1.3 Matrix Algebra
1.3.1 Addition and Subtraction
1.3.2 Multiplication
1.3.3 Inverse of a Matrix
1.4 Linear Systems of Equations
1.4.1 Inhomogeneous and Homogeneous Systems of Equations
1.4.2 Existence of Solutions
1.4.3 Numerical Solution Methods
1.4.4 Norms and Condition Number
1.5 Eigenvalue Problems
1.5.1 General Eigenvalue Problem
1.5.2 Numerical Solution of Eigenvalue Problems
Exercises
References

2 Basic Equations of the Theory of Elasticity
2.1 Types of Structures
2.2 One-Dimensional and Two-Dimensional States of Stress
2.3 Beams and Plates in Bending
2.4 Spatial Structures
2.4.1 Generals
2.4.2 Beams
2.4.3 Solids
References
3 Truss and Beam Structures
3.1 Introduction
3.1.1 The Finite Element Method in Structural Analysis
3.1.2 Nodal Points, Degrees of Freedom, and Finite Elements
3.1.3 Computational Method
3.2 Introductory Example: Plane Truss System
3.2.1 Structural System
3.2.2 Element Stiffness Matrix of a Truss Element

3.2.3 Coordinate Transformation
3.2.4 Global Stiffness Matrix
3.2.5 Support Conditions
3.2.6 Solution of the System of Equations
3.2.7 Support Forces and Element Stresses
3.2.8 Flexibility Matrix
3.3 Elastic Springs
3.3.1 Elastic Support of Nodal Points
3.3.2 Spring Elements
3.4 Beams in Bending
3.4.1 Stiffness Matrix of the Beam Element
3.4.2 Element Loads
3.4.3 Extension of the Stiffness Matrix of the Beam Element
3.4.4 Coordinate Transformation
3.4.5 Hinges
3.5 Combined Beam and Truss Systems
3.6 Spatial Trusses and Beam Structures

3.6.1 3D Truss Element
3.6.2 3D Beam Element
3.6.3 Beam Element with Warping Torsion
3.7 Modeling of Beam Structures
3.7.1 Supports
3.7.2 Springs
3.7.3 Beam Structures
3.7.4 Symmetrical Structures
3.8 Quality Assurance and Documentation
3.8.1 Sources of Error
3.8.2 Checking of Beam Structures Computations
3.8.3 Documentation of Finite Element Analyses
Exercises
References
4 Plate, Shell, and Solid Structures
4.1 Historical Background
4.2 Basic Concepts
4.3 Approximation Character of the Finite Element Method

4.3.1 One-Dimensional Introductory Example
4.3.2 Analytical Solution
4.3.3 FEM Approximate Solution with Linear Shape Functions
4.3.4 FEM Approximate Solution with Quadratic Shape Functions
4.3.5 Properties of the Finite Element Approximate Solution
4.4 Rectangular Elements for Plates in Plane Stress
4.4.1 Shape Functions
4.4.2 Strains and Stresses
4.4.3 Stiffness Matrix
4.4.4 Element Loads
4.4.5 Examples
4.5 Finite Elements for Plates in Plane Stress
4.5.1 Properties of Finite Elements
4.5.2 Displacement-Based Elements with Compatible Shape Functions