Fundamentals of Structural Dynamics: Theory and Computation

ISBN-13: 9783030899431 / Angielski / Twarda / 2022 / 572 str.

Keith D. Hjelmstad

Spis treści

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 Di_erential Equations

2.1 Why numerical methods?

2.2 Practical implementation

2.3 Analysis of a first order equation

2.4 Analysis of second order di_erential equations

2.4.1 The central di_erence 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

xiii

xiv Contents

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

Contents xv

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 _nite element analysis

9.5.1 Dynamic principle of virtual work

9.5.2 Finite element functions

9.5.3 A slightly di_erent 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 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

xvi Contents

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 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 _nite 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 Veri_cation of element performance

12.3.5 Wave propagation in the Timoshenko beam

Contents xvii

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 Veri_cation of static code

13.6 Dynamic analysis of nonlinear planar beams

13.6.1 Solution of the nonlinear di_erential 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 Speci_c 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

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

xviii Contents

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

 

O autorze

Dr. Keith D. Hjelmstad is President's Professor and Program Chair of the Department of Civil, Environmental, and Sustainable Engineering, Sustainable Engineering and the Built Environment, Arizona State University.

Blurb

This text closes the gap between traditional textbooks on structural dynamics and how structural dynamics is practiced in a world driven by commercial software, where performance-based design is increasingly important. The book emphasizes numerical methods, nonlinear response of structures, and the analysis of continuous systems (e.g., wave propagation). Fundamentals of Structural Dynamics: Theory and Computation builds the theory of structural dynamics from simple single-degree-of-freedom systems through complex nonlinear beams and frames in a consistent theoretical context supported by an extensive set of MATLAB codes that not only illustrate and support the principles, but provide powerful tools for exploration. The book is designed for students learning structural dynamics for the first time but also serves as a reference for professionals throughout their careers.