Finite Elements in Structural Analysis: Theoretical Concepts and Modeling Procedures in Statics and Dynamics of Structures (Springer Tracts in Civil Engineering)

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
		4.5.3 Nonconforming Elements
		4.5.4 Hybrid Elements
		4.5.5 Other Element Types
		4.5.6 Element Types in Finite Element Software for Structural Analysis
	4.6 Rectangular Element for Plates in Bending
		4.6.1 Element Type
		4.6.2 Shape Functions
		4.6.3 Deformations and Section Forces
		4.6.4 Stiffness Matrix
		4.6.5 Element Loads
	4.7 Finite Elements for Plates in Bending
		4.7.1 Displacement-Based Shear Flexible Elements
		4.7.2 Displacement-Based Shear Rigid Plate Elements
		4.7.3 Hybrid Plate Elements
		4.7.4 Other Element Types
		4.7.5 Element Types in Finite Element Software for Structural Analysis
	4.8 Finite Elements for Shells
		4.8.1 Plane Shell Elements as Superimposed Membrane and Bending Elements
		4.8.2 Curved Shell Elements as “Degenerated” Solid Elements
		4.8.3 Axisymmetric Shell Elements
		4.8.4 Element Types in Finite Element Software for Structural Analysis
	4.9 Solid Elements
		4.9.1 Isoparametric Elements
		4.9.2 Other Element Types
		4.9.3 Axisymmetric Solid Elements
		4.9.4 Element Types in Finite-Element Software for Structural Analysis
	4.10 Transition between Beam, Plate and Solid Elements
		4.10.1 Generals
		4.10.2 Transformation of Element Matrices
		4.10.3 Connections with Displacement Assumptions (RDT)
		4.10.4 Connections with Stress Assumptions (EST)
		4.10.5 Engineering Models
		4.10.6 Other Element Transitions
	4.11 Modeling of Structural Elements and Buildings
		4.11.1 Structural Models
		4.11.2 Singularities of Stresses and Displacements
		4.11.3 Element Types and Meshing
		4.11.4 Mesh Generation
		4.11.5 Modeling of Plates in Plane Stress
		4.11.6 Modeling of Plates in Bending
		4.11.7 Foundation Slabs
		4.11.8 Modeling of Folded Plate and Shell Structures
		4.11.9 Three-Dimensional Building Models
		4.11.10 Interpretation of Results
	4.12 Quality Assurance and Documentation
		4.12.1 Types of Error
		4.12.2 Error Estimation and Adaptive Meshing
		4.12.3 Checking of Surface and Spatial Structure Computations
		4.12.4 Documentation of Finite Element Analyses of Surface and Spatial Structures
	Exercises
	References
5 Dynamic Analysis of Structures
	5.1 Introduction
	5.2 Basic Concepts of Dynamics
		5.2.1 Kinematics
		5.2.2 Inertial Forces
		5.2.3 Damping Forces
	5.3 Equations of Motion
	5.4 Free Vibrations
		5.4.1 Undamped Vibrations
		5.4.2 Damped Vibrations
	5.5 Forced Vibrations with Harmonic Excitation
	5.6 Forced Vibrations with General Dynamic Excitation
		5.6.1 Generals
		5.6.2 Direct Numerical Integration
		5.6.3 Modal Analysis
		5.6.4 Fourier Transformation
	5.7 Earthquake Excitation
		5.7.1 Generals
		5.7.2 Methods of Analysis
		5.7.3 Time History Analysis
		5.7.4 Response Spectrum Analysis
	5.8 Modeling for Dynamic Analysis
		5.8.1 Structural Model
		5.8.2 Finite Element Model
		5.8.3 Discretization in Time and Frequency Domain
		5.8.4 Building Models
		5.8.5 Soil–Structure Interaction
		5.8.6 Modeling and Validation
	Exercises
	References
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Index