Computational Statics and Dynamics: An Introduction Based on the Finite Element Method

Preface to the Second Edition
Preface to the First Edition
Acknowledgements
Contents
Symbols and Abbreviations
1 Introduction to the Finite Element Method
	References
2 Rods and Trusses
	2.1 Introduction
	2.2 Derivation of the Governing Differential Equation
		2.2.1 Kinematics
		2.2.2 Constitutive Equation
		2.2.3 Equilibrium
		2.2.4 Differential Equation
	2.3 Finite Element Solution
		2.3.1 Derivation of the Principal Finite Element Equation
		2.3.2 Derivation of Interpolation Functions
		2.3.3 Assembly of Elements and Consideration of Boundary Conditions
		2.3.4 Post-Computation: Determination of Strain, Stress and Further Quantities
		2.3.5 Analogies to Other Field Problems
		2.3.6 Solved Rod Problems
	2.4 Assembly of Elements to Plane Truss Structures
		2.4.1 Rotational Transformation in a Plane
		2.4.2 Solved Truss Problems
	2.5 Supplementary Problems
	References
3 Euler–Bernoulli Beams and Frames
	3.1 Introduction
	3.2 Derivation of the Governing Differential Equation
		3.2.1 Kinematics
		3.2.2 Constitutive Equation
		3.2.3 Equilibrium
		3.2.4 Differential Equation
	3.3 Finite Element Solution
		3.3.1 Derivation of the Principal Finite Element Equation
		3.3.2 Derivation of Interpolation Functions
		3.3.3 Assembly of Elements and Consideration of Boundary Conditions
		3.3.4 Post-Computation: Determination of Strain, Stress  and Further Quantities
		3.3.5 Solved Beam Problems
	3.4 Assembly of Elements to Plane Frame Structures
		3.4.1 Rotation of a Beam Element
		3.4.2 Generalized Beam Element
		3.4.3 Solved Problems
	3.5 Supplementary Problems
	References
4 Timoshenko Beams
	4.1 Introduction
	4.2 Derivation of the Governing Differential Equation
		4.2.1 Kinematics
		4.2.2 Equilibrium
		4.2.3 Constitutive Equation
		4.2.4 Differential Equation
	4.3 Finite Element Solution
		4.3.1 Derivation of the Principal Finite Element Equation
		4.3.2 Linear Interpolation Functions for the Displacement and Rotational Field
		4.3.3 Higher-Order Interpolation Functions for the Beam with Shear Contribution
		4.3.4 Solved Problems
	4.4 Supplementary Problems
	References
5 Plane Elements
	5.1 Introduction
	5.2 Derivation of the Governing Differential Equation
		5.2.1 Kinematics
		5.2.2 Constitutive Equation
		5.2.3 Equilibrium
		5.2.4 Differential Equation
	5.3 Finite Element Solution
		5.3.1 Derivation of the Principal Finite Element Equation
		5.3.2 Four-Node Planar Element
		5.3.3 Solved Plane Elasticity Problems
	5.4 Supplementary Problems
	References
6 Classical Plate Elements
	6.1 Introduction
	6.2 Derivation of the Governing Differential Equation
		6.2.1 Kinematics
		6.2.2 Constitutive Equation
		6.2.3 Equilibrium
		6.2.4 Differential Equation
	6.3 Finite Element Solution
		6.3.1 Derivation of the Principal Finite Element Equation
		6.3.2 Rectangular Four-Node Plate Element
		6.3.3 Solved Classical Plate Element Problems
	6.4 Supplementary Problems
	References
7 Shear Deformable Plate Elements
	7.1 Introduction
	7.2 Derivation of the Governing Differential Equation
		7.2.1 Kinematics
		7.2.2 Constitutive Equation
		7.2.3 Equilibrium
		7.2.4 Differential Equation
	7.3 Finite Element Solution
		7.3.1 Derivation of the Principal Finite Element Equation
		7.3.2 Rectangular Four-Node Plate Element
		7.3.3 Solved Thick Plate Element Problems
	7.4 Supplementary Problems
	References
8 Three-Dimensional Elements
	8.1 Derivation of the Governing Differential Equation
		8.1.1 Kinematics
		8.1.2 Constitutive Equation
		8.1.3 Equilibrium
		8.1.4 Differential Equation
	8.2 Finite Element Solution
		8.2.1 Derivation of the Principal Finite Element Equation
		8.2.2 Hexahedron Solid Elements
		8.2.3 Solved Three-Dimensional Element Problems
	8.3 Supplementary Problems
	References
9 Principles of Linear Dynamics
	9.1 Newton's Laws of Motion
	9.2 Relationship Between Displacement, Velocity and Acceleration
	9.3 Solved Problems
	9.4 Supplementary Problems
	References
10 Integration Methods for Transient Problems
	10.1 Introduction
	10.2 Derivation of the Governing Differential Equation
		10.2.1 Kinematics
		10.2.2 Constitutive Equation
		10.2.3 Equilibrium
		10.2.4 Differential Equation
	10.3 Finite Element Solution
		10.3.1 Derivation of the Principal Finite Element Equation
		10.3.2 Consideration of Damping
		10.3.3 Transient Solution Schemes
		10.3.4 Solved Problems
	10.4 Supplementary Problems
	References
Appendix A Mathematics
A.1 Greek Alphabet
A.2 Frequently Used Constants
A.3 Special Products
A.4 Trigonometric Functions
A.5 Derivatives
A.6 Integrals
A.7 Integration by Parts
A.8 Integration and Coordinate Transformation
A.9 Numerical Integration
A.9.1 Simpson's Rule
A.9.2 Gauss–Legendre Quadrature
A.10 Taylor's Series Expansion
A.11 Matrix Operations
A.11.1 Matrix Multiplication
A.11.2 Scalar Product
A.11.3 Dyadic Product
A.11.4 Inverse of Matrices
A.12 Solution of Linear Systems of Equations
A.12.1 Elimination of Variables
A.12.2 Matrix Solution
A.13 Elementary Geometry
A.14 Analytical Geometry
A.14.1 Straight-Line Equations
A.14.2 Sign of Second Derivative of a Curve
A.14.3 Area of a Polygon
Appendix B Mechanics
B.1 Centroids
B.2 Second Moment of Area
B.3 Parallel-Axis Theorem
Appendix C Units and Conversion
C.1 SI Base Units
C.2 Coherent SI derived Units
C.3 Consistent Units
C.4 Conversion of Important English Units to the Metric System
Appendix D Triangular Elements
Appendix E Summary of Stiffness Matrices
E.1 One-Dimensional Elements
E.2 Two-Dimensional Elements
E.3 Three-Dimensional Elements
Appendix F Extrapolation from Integration Points to Nodes
Appendix G Answers to Supplementary Problems
G.1 Problems from Chap.2
G.2 Problems from Chap.3
G.3 Problems from Chap.4
G.4 Problems from Chap.5
G.5 Problems from Chap.6
G.6 Problems from Chap.7
G.7 Problems from Chap.8
G.8 Problems from Chap.9
G.9 Problems from Chap.10
Index