04-25-2023, 08:51 AM

Dynamic Analysis of Structures

Author(s)/Editor(s): john Katsikadelis | Size: 26 MB| Format: PDF| Publisher: Academic Press| Year: 2020| pages: 755 | ISBN: Paperback ISBN: 9780128186435 eBook ISBN: 9780128186442

Dynamic Analysis of Structures reflects the latest application of structural dynamics theory to produce more optimal and economical structural designs. Written by an author with over 37 years of researching, teaching and writing experience, this reference introduces complex structural dynamics concepts in a user-friendly manner. The author includes carefully worked-out examples which are solved utilizing more recent numerical methods. These examples pave the way to more accurately simulate the behavior of various types of structures. The essential topics covered include principles of structural dynamics applied to particles, rigid and deformable bodies, thus enabling the formulation of equations for the motion of any structure.

Preface

General concepts and principles of structural dynamics

Chapter outline

Introduction

Types of dynamic loads

Dynamic degrees of freedom

Dynamic model and formulation of the equation of motion of SDOF systems

Derivation of the equations of motion using dAlemberts principle

Principle of virtual displacements

Hamiltons principle

Lagranges equations

Derivation of Lagranges equations

Lagrange multipliers

Small displacements

Potential energy and stiffness matrix

Kinetic energy and mass matrix

Raleighs dissipation function

Influence of the gravity loads

Problems

References and further reading

Single-degree-of-freedom systems: Free vibrations

Chapter outline

Introduction

Free undamped vibrations

Free damped vibrations

Critically damped system

Underdamped system

Overdamped system

Conservation of energy in an undamped system

Problems

References and further reading

Chapter 3: Single-degree-of-freedom systems: Forced vibrations

3.1. Introduction

3.2. Response to harmonic loading

3.2.1. Response of undamped systems to harmonic loading

3.2.2. Response of damped systems to harmonic loading

3.3. Response to arbitrary dynamic loading-Duhamels integral

3.3.1. Undamped vibrations

3.3.2. Damped vibrations

3.4. Analytical evaluation of the Duhamel integral-applications

3.4.1. Response to step function load

3.4.2. Response to ramp function load

3.4.3. Response to step function load with finite rise time. Static load

3.5. Response to impulsive loads

3.5.1. Rectangular pulse load

3.5.2. Triangular pulse load

3.5.3. Asymmetrical triangular pulse load

3.5.4. Response to piecewise linear loading

3.6. Response to a periodic loading

3.6.1. Periodic loads

3.6.2. Fourier series

3.6.3. Response of the SDOF system to periodic excitation

3.7. Response to unit impulse

3.7.1. The delta function or Diracs delta function

3.7.2. Response to unit impulse

3.7.3. Response to arbitrary loading

3.7.4. The reciprocal theorem in dynamics

3.8. Problems

References and further reading

Numerical integration of the equation of motion

Chapter outline

Introduction

The central difference method

The average acceleration method

The analog equation method

Stability of the numerical integration methods

Errors in the numerical integration

Difference equations

Difference equations and stability of the numerical integration methods

Stability of the central difference method

Stability of the average acceleration method

Stability of the analog equation method

Accuracy of the numerical integration

Problems

References and further reading

Chapter 5: Nonlinear response: Single-degree-of-freedom systems

Chapter outline

5.1. Introduction

5.2. The central difference method

5.3. The average acceleration method

5.4. The analog equation method

5.5. Problems

References and further reading

Response to ground motion and vibration isolation

Chapter outline

Introduction

Equation of motion: Relative displacement

Response spectra

Equation of motion in terms of the total displacement

Vibration isolation

Transmission of force

Transmission of motion

Problems

References and further reading

Damping in structures

Chapter outline

Introduction

Loss of energy due to damping

Equivalent viscous damping

Hysteretic damping

Coulomb damping

Free vibrations with Coulomb damping

Forced vibrations with Coulomb damping

Damping modeling via fractional derivatives

Introduction

The fractional derivative

Measurement of damping

Free vibration decay method

Resonance amplitude method

Width of response curve method

Problems

References and further reading

Generalized single-degree-of-freedom systems-Continuous systems

Chapter outline

Introduction

Generalized single-degree-of-freedom systems

Continuous systems

Introduction

Solution of the beam equation of motion

Free vibrations of beams

The simply supported beam

The cantilever beam

Orthogonality of the free-vibration modes

Forced vibrations of beams

Problems

References and further reading

Analysis in the frequency domain

Chapter outline

Introduction

Complex form of the Fourier series

Complex dynamic response to periodic load

Fourier integral representation of a nonperiodic load

Response to a nonperiodic load

Discrete Fourier transform

Application of the discrete Fourier transform to dynamic analysis

Fast Fourier transform

The Sande-Tukey algorithm

Problems

References and further reading

Multi-degree-of-freedom systems: Models and equations of motion

Introduction

Systems with localized mass and localized stiffness

Systems with distributed mass and localized stiffness

Systems with localized mass and distributed stiffness

The method of influence coefficients

Elastic forces

Damping forces

Inertial forces

Systems with distributed mass and distributed stiffness

The method of global shape functions

Mixed systems

Transformations of the equations of motion

Problems

References and further reading

The finite element method

Introduction

The finite element method for the plane truss

Properties of the plane truss element

The method of the Lagrange equations

The method of virtual work

Transformation of the nodal coordinates of the truss element

Equation of motion of the plane truss

Steps to formulate the equations of motion for a plane truss by the finite element method

Modification of the equations of motion due to the supports of the structure

The finite element method for the plane frame

Properties of the plane frame element

The method of the Lagrange equations

The method of virtual work

Transformation of the nodal coordinates of the plane frame element

Static condensation: Guyans reduction

Flexural vibrations of a plane frame

Reduction of the degrees of freedom due to constraints

Axial constraints in the plane frame

The finite element method for the plane grid

Properties of the plane grid element

Transformation of the nodal coordinates of the plane grid element

The finite element method for the space frame

Properties of the space frame element

Transformation of the nodal coordinates of the space frame element

The finite element method for the space truss

Properties of the space truss element

Transformation of the nodal coordinates of the space truss element

Rigid bodies within flexible skeletal structures

Rigid bodies in spaces frames

Rigid bodies in spaces trusses, plane grids, plane frames, and plane trusses

Problems

References and further reading

Multi-degree-of-freedom systems: Free vibrations

Chapter outline

Introduction

Free vibrations without damping

Orthogonality of eigenmodes

Eigenmodes of systems with multiple eigenfrequencies

The linear eigenvalue problem

The standard eigenvalue problem of linear algebra

Properties of the eigenvalues and eigenvectors

The generalized eigenvalue problem

The Rayleigh quotient

Properties of eigenfrequencies and modes of MDOF systems without damping: A summary

Solution of the vibration problem without damping

The method of mode superposition

Solution of the vibration problem with damping

Direct solution of the differential equation

Linearization of the quadratic eigenvalue problem

The use of a proportional viscous damping matrix

Construction of a proportional damping matrix

Rayleigh damping

Additional orthogonality conditions: Caughey damping matrix

Construction of the proportional damping matrix using the modal matrix

Problems

References and further reading

Numerical evaluation of the eigenfrequencies and eigenmodes

Chapter outline

Introduction

The vector iteration method

The inverse vector iteration method

Convergence of the inverse vector iteration method

Computation of higher-order eigenpairs

The vector purification method

The inverse vector iteration method with shifts

Free or partially supported structure

Problems

References and further reading

Multi-degree-of-freedom systems: Forced vibrations

Introduction

The mode superposition method

Modal contribution in the mode superposition method

Modal participation

Static correction method

Error in mode superposition method due to truncation of higher modes

Reduction of the dynamic degrees of freedom

Static condensation

Kinematic constraints

Rayleigh-Ritz method

Ritz transformation

Approximation using Ritz vectors

Selection of Ritz vectors

Method of natural mode shapes

The method of derived Ritz vectors

Support excitation

Multiple support excitation

Uniform support excitation

The response spectrum method

Comparison of mode superposition method and Rayleigh-Ritz method

Numerical integration of the equations of motions-Linear MDOF systems

The central difference method (CDM)-Linear equations

The average acceleration method (AAM)-Linear equations

The analog equation method (AEM)-Linear equations

Numerical integration of the equations of motions-Nonlinear MDOF systems

The average acceleration method (AAM)-Nonlinear equations

The analog equation method (AEM)-Nonlinear equations

Problems

References and further reading

Dynamic analysis of multistory buildings

Chapter outline

Introduction

The multistory building

The concept of the multistory element

Nodal displacement matrix, nodal force matrix, transformation matrix, and stiffness matrix of the MSE

Mass matrix of the MSE and multistory building

Equation of motion of the multistory building

Dynamic response of multistory buildings due to ground motion

Problems

References and further reading

Base isolation

Chapter outline

Introduction

Analysis of the one-story building with base isolation

Linear response of the isolation systems

Modeling of nonlinear response of isolation systems

Linear springs or laminated rubber bearings with flat sliders

Linear springs or rubber bearings and nonlinear dampers

Friction pendulum bearing

High damping rubber bearing or lead rubber bearing-Bilinear model

Hysteretic isolators-Bouc-Wen model

The multistory building with base isolation

The equation of motion of the multistory building with base isolation

Reduction of the DOF of the superstructure using mode shapes

Reduction of the superstructure DOF using Ritz vectors

Linear response of the isolation system

Nonlinear response of the isolation system

Problems

References and further reading

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