Fundamentals of Structural Dynamics: Theory and Computation

Preface
Contents
1 Foundations of Dynamics
	1.1 Kinematics of Particles
	1.2 Kinetics of Particles
	1.3 Power, Work, and Energy
	1.4 Conservation of Energy
	1.5 Dynamics of Rigid Bodies
	1.6 Example
	1.7 The Euler–Lagrange Equations
	1.8 Summary
2 Numerical Solution of Ordinary Differential Equations
	2.1 Why Numerical Methods?
	2.2 Practical Implementation
	2.3 Analysis of a First Order Equation
	2.4 Analysis of Second Order Differential Equations
		2.4.1 The Central Difference Method
		2.4.2 The Generalized Trapezoidal Rule
		2.4.3 Newmark's Method
	2.5 Performance of the Methods
	2.6 Summary
3 Single-Degree-of-Freedom Systems
	3.1 The SDOF Oscillator
	3.2 Undamped Free Vibration
	3.3 Damped Free Vibration
	3.4 Forced Vibration
		3.4.1 Suddenly Applied Constant Load
		3.4.2 Sinusoidal Load
		3.4.3 General Periodic Loading
	3.5 Earthquake Ground Motion
	3.6 Nonlinear Response
	3.7 Integrating the Equation of Motion
	3.8 Example
4 Systems with Multiple Degrees of Freedom
	4.1 The 2-DOF System as a Warm-up Problem
	4.2 The Shear Building
	4.3 Free Vibration of the NDOF System
		4.3.1 Orthogonality of the Eigenvectors
		4.3.2 Initial Conditions
	4.4 Structural Damping
		4.4.1 Modal Damping
		4.4.2 Rayleigh Damping
		4.4.3 Caughey Damping
		4.4.4 Non-classical Damping
	4.5 Damped Forced Vibration of the NDOF System
	4.6 Resonance in NDOF Systems
	4.7 Numerical Integration of the NDOF Equations
5 Nonlinear Response of NDOF Systems
	5.1 A Point of Departure
	5.2 The Shear Building, Revisited
	5.3 The Principle of Virtual Work
	5.4 Nonlinear Dynamic Computations
	5.5 Assembly of Equations
	5.6 Adding Damping to the Equations of Motion
	5.7 The Structure of the NDOF Code
	5.8 Implementation
6 Earthquake Response of NDOF Systems
	6.1 Special Case of the Elastic System
	6.2 Modal Recombination
	6.3 Response Spectrum Methods
	6.4 Implementation
	6.5 Example
7 Special Methods for Large Systems
	7.1 Ritz Projection Onto a Smaller Subspace
	7.2 Static Correction Method
	7.3 Summary
8 Dynamic Analysis of Truss Structures
	8.1 What Is a Truss?
	8.2 Element Kinematics
	8.3 Element and Nodal Static Equilibrium
	8.4 The Principle of Virtual Work
	8.5 Constitutive Models for Axial Force
	8.6 Solving the Static Equations of Equilibrium
	8.7 Dynamic Analysis of Truss Structures
	8.8 Distributed Element Mass
	8.9 Earthquake Response of Truss Structures
	8.10 Implementation
	8.11 Example
9 Axial Wave Propagation
	9.1 The Axial Bar Problem
	9.2 Motion Without Applied Loading
	9.3 Classical Solution by Separation of Variables
	9.4 Modal Analysis with Applied Loads
	9.5 The Ritz Method and Finite Element Analysis
		9.5.1 Dynamic Principle of Virtual Work
		9.5.2 Finite Element Functions
		9.5.3 A Slightly Different Formulation
		9.5.4 Boundary Conditions
		9.5.5 Higher Order Interpolation
		9.5.6 Initial Conditions
	9.6 Axial Bar Dynamics Code
10 Dynamics of Planar Beams: Theory
	10.1 Beam Kinematics
		10.1.1 Motion of a Beam Cross Section
		10.1.2 Strain–Displacement Relationships
		10.1.3 Normal and Shear Strain
	10.2 Beam Kinetics
	10.3 Constitutive Equations
	10.4 Equations of Motion
		10.4.1 Balance of Linear Momentum
		10.4.2 Balance of Angular Momentum
	10.5 Summary of Beam Equations
	10.6 Linear Beam Theory
		10.6.1 Linearized Kinematics
		10.6.2 Linearized Kinetics
		10.6.3 Linear Equations of Motion
		10.6.4 Boundary Conditions
		10.6.5 Initial Conditions
11 Wave Propagation in Linear Planar Beams
	11.1 Propagation of a Train of Sinusoidal Waves
		11.1.1 Bernoulli–Euler Beam
		11.1.2 Rayleigh Beam
		11.1.3 Timoshenko Beam
	11.2 Solution by Separation of Variables
	11.3 The Bernoulli–Euler Beam
		11.3.1 Implementing Boundary Conditions
		11.3.2 Natural Frequencies
		11.3.3 Orthogonality of the Eigenfunctions
		11.3.4 Implementing the Initial Conditions
		11.3.5 Modal Vibration
		11.3.6 Other Boundary Conditions
		11.3.7 Wave Propagation
		11.3.8 Example: Simple–Simple Beam
	11.4 The Rayleigh Beam
		11.4.1 Simple–Simple Rayleigh Beam
		11.4.2 Orthogonality Relationships
		11.4.3 Wave Propagation: Simple–Simple Beam
		11.4.4 Other Boundary Conditions
		11.4.5 Implementation
	11.5 The Timoshenko Beam
		11.5.1 Simple–Simple Beam
		11.5.2 Wave Propagation
		11.5.3 Numerical Example
	11.6 Summary
12 Finite Element Analysis of Linear Planar Beams
	12.1 The Dynamic Principle of Virtual Work
		12.1.1 The Ritz Approximation
		12.1.2 Initial Conditions
		12.1.3 Selection of Ritz Functions
		12.1.4 Beam Finite Element Functions
		12.1.5 Ritz Functions and Degrees of Freedom
		12.1.6 Local to Global Mapping
		12.1.7 Element Matrices and Assembly
	12.2 The Rayleigh Beam
		12.2.1 Virtual Work for the Rayleigh Beam
		12.2.2 Finite Element Discretization
		12.2.3 Initial Conditions for Wave Propagation
		12.2.4 The Rayleigh Beam Code
		12.2.5 Example
	12.3 The Timoshenko Beam
		12.3.1 Virtual Work for the Timoshenko Beam
		12.3.2 Finite Element Discretization
		12.3.3 The Timoshenko Beam Code
		12.3.4 Verification of Element Performance
		12.3.5 Wave Propagation in the Timoshenko Beam
13 Nonlinear Dynamic Analysis of Planar Beams
	13.1 Equations of Motion
	13.2 The Principle of Virtual Work
	13.3 Tangent Functional
	13.4 Finite Element Discretization
	13.5 Static Analysis of Nonlinear Planar Beams
		13.5.1 Solution by Newton's Method
		13.5.2 Static Implementation
		13.5.3 Verification of Static Code
	13.6 Dynamic Analysis of Nonlinear Planar Beams
		13.6.1 Solution of the Nonlinear Differential Equations
		13.6.2 Dynamic Implementation
		13.6.3 Example
	13.7 Summary
14 Dynamic Analysis of Planar Frames
	14.1 What Is a Frame?
	14.2 Equations of Motion
	14.3 Inelasticity
		14.3.1 Numerical Integration of the Rate Equations
		14.3.2 Material Tangent
		14.3.3 Internal Variables
		14.3.4 Specific Model for Implementation
	14.4 Element Matrices
		14.4.1 Finite Element Discretization
		14.4.2 Local to Global Transformation
	14.5 Static Verification
	14.6 Dynamics of Frames
		14.6.1 Earthquake Ground Motion
		14.6.2 Implementation
		14.6.3 Examples
		14.6.4 Sample Input Function
A Newton's Method for Solving Nonlinear Algebraic Equations
	A.1 Linearization
	A.2 Systems of Equations
B The Directional Derivative
	B.1 Ordinary Functions
	B.2 Functionals
C The Eigenvalue Problem
	C.1 The Algebraic Eigenvalue Problem
	C.2 The QR Algorithm
	C.3 Eigenvalue Problems for Large Systems
	C.4 Subspace Iteration
D Finite Element Interpolation
	D.1 Polynomial Interpolation
	D.2 Lagrangian Interpolation
	D.3 Ritz Functions with hp Interpolation
	D.4 Lagrangian Shape Functions
	D.5 C0 Bubble Functions
	D.6 C1 Bubble Functions
E Data Structures for Finite Element Codes
	E.1 Structure Geometry and Topology
	E.2 Structures with Only Nodal DOF
	E.3 Structures with Non-nodal DOF
F Numerical Quadrature
	F.1 Trapezoidal Rule
	F.2 Simpson's Rule
	F.3 Gaussian Quadrature
	F.4 Implementation
	F.5 Examples
Index