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# Theory of Matrix Structural Analysis

## Overview

This pioneering aerospace engineering text belongs on the shelf of every aerospace and structural engineering graduate student and professional engineer. Originally published in 1968, the treatment remains a valuable guide, tracing each procedure in a clear, step-by-step fashion and employing minimal mathematical rigor in its examples.

The text begins with an overview of matrix methods and their application to the structural design of modern aircraft and aerospace vehicles. Subsequent chapters cover the basic equations of elasticity, energy theorems, structural idealization, Castigliano's theorem, derivation of stiffness matrices from flexibility, and constant-shear-flow panels. Additional subjects include a comparison of force and displacement methods, analysis of substructures, structural synthesis, and nonlinear structural analysis. Abundant end-of-chapter supplements provide materials for classroom use.

## Product Details

ISBN-13: | 9780486649481 |
---|---|

Publisher: | Dover Publications |

Publication date: | 06/13/2012 |

Edition description: | 46th EDITION, DOVER EDITION |

Pages: | 480 |

Product dimensions: | 5.40(w) x 8.48(h) x 0.96(d) |

## About the Author

John Przemieniecki was a professor and Associate Dean for Research at the U.S. Air Force Institute of Technology and a member of the U.S. Senior Executive Service* . *President Reagan conferred upon him the rank of Distinguished Executive, and in 1999 he received an honorary doctorate from the Warsaw University of Technology.

## Read an Excerpt

John Przemieniecki was a professor and Associate Dean for Research at the U.S. Air Force Institute of Technology and a member of the U.S. Senior Executive Service* . *President Reagan conferred upon him the rank of Distinguished Executive, and in 1999 he received an honorary doctorate from the Warsaw University of Technology.

## First Chapter

John Przemieniecki was a professor and Associate Dean for Research at the U.S. Air Force Institute of Technology and a member of the U.S. Senior Executive Service* . *President Reagan conferred upon him the rank of Distinguished Executive, and in 1999 he received an honorary doctorate from the Warsaw University of Technology.

## Table of Contents

PREFACE

CHAPTER 1 MATRIX METHODS

1.1 Introduction

1.2 Design Iterations

1.3 Methods of Analysis

1.4 Areas of Structural Analysis

CHAPTER 2 BASIC EQUATIONS OF ELASTICITY

2.1 Strain-Displacement Equations

2.2 Stress-Strain Equations

2.3 Stress-Strain Equations for Initial Strains

2.4 Equations of Equilbrium

2.5 Compatibility Equations

CHAPTER 3 ENERGY THEOREMS

3.1 Introduction

3.2 Work and Complementary Work; Stain Energy and Complementary Strain

3.3 Green's Identity

3.4 Energy Theorems Based on the Principle of Virtual Work

3.5 Energy Theorems Based on the Principle of Complementary Virtual Work

3.6 Clapeyron's Theorem

3.7 Betti's Theorem

3.8 Maxwell's Reciprocal Theorem

3.9 Summary of Energy Theorems and Definitions

PROBLEMS

CHAPTER 4 STRUCTURAL IDEALIZATION

4.1 Structural Idealization

4.2 Energy Equivalence

4.3 Structural Elements

CHAPTER 5 STIFFNESS PROPERTIES OF STRUCTURAL ELEMENTS

5.1 Methods of Determining Element Force-Displacement Relationships

5.2 Determination of Element Stiffness Properties by the Unit-displacement Theorem

5.3 Application of Castigliano's Theorem (Part I) to Derive Stiffness Properties

5.4 Transformation of Coordinate Axes: ? Matrices

5.5 Pin-jointed Bar Elements

5.6 Beam Elements

5.7 Triangular Plate Elements (In-plane Forces)

5.8 Rectangular Plate Elements (In-plane Forces)

5.9 Quadrilateral Plate Elements (In-plane Forces)

5.10 Tetrahedron Elements

5.11 Triangular Plates in Bending

5.12 Rectangular Plates in Bending

5.13 Method for Improving Stiffness Matrices

PROBLEMS

CHAPTER 6 THE MATRIX DISPLACEMENT METHOD

6.1 Matrix Formulation of the Displacement Analysis

6.2 Elimination of the Rigid-body Degrees of Freedom: Choice of Reactions

6.3 Derivation of the Transformation Matrix V from Equilibrium Equations

6.4 Derivation of the Transformation Matrix T from Kinematics

6.5 Condensation of Stiffness Matrices

6.6 Derivation of Stiffness Matrices from Flexibility

6.7 Stiffness Matrix for Constant-shear-flow Panels

6.8 Stiffness Matrix for Linearly Varying Axial-force Members

6.9 Analysis of a Pin-jointed Truss by the Displacement Method

6.10 Analysis of a Cantilever Beam by the Displacement Method

6.11 Equivalent Concentrated Forces

PROBLEMS

CHAPTER 7 FLEXIBILITY PROPERTIES OF STRUCTURAL ELEMENTS

7.1 Methods of Determing Element Displacement-Force Relationships

7.2 Inversion of the Force-Displacement Equations: Flexibility Properties of Pin-jointed Bars and Beam Elements

7.3 Determination of Element Flexibility Properties by the Unit-load Theorem

7.4 Application of Castigliano's Theorem (Part II) to Derive Flexibility Properties

7.5 Solution of Differential Equations for Element Displacements to Derive Flexibility Properties

7.6 Pin-jointed Bar Elements

7.7 Beam Elements

7.8 Triangular Plate Elements (In-plane Forces)

7.9 Rectangular Plate Elements (In-plane Forces)

7.10 Tetrahedron Elements

7.11 Constant-shear-flow Panels

7.12 Linearly Varying Axial-force Members

7.13 Rectangular Plates in Bending

PROBLEMS

CHAPTER 8 THE MATRIX FORCE METHOD

8.1 Matrix Formulation of the Unit-load Theorem for External-force Systems

8.2 Matrix Formulation of the Unit-load Theorem for Internal-force Systems: Self-equilibrating Force Systems

8.3 Matrix Formulation of the Force Analysis: Jordanian Elimination Technique

8.4 Matrix Force Analysis of a Pin-jointed Truss

8.5 Matrix Force Analysis of a Cantilever Beam

8.6 Comparison of the Force and Displacement Methods

PROBLEMS

CHAPTER 9 ANALYSIS OF SUBSTRUCTURES

9.1 Substructure Analysis by the Matrix Displacement Method

9.2 Substructure Displacement Analysis of a Two-Bay Truss

9.3 Substructure Analysis by the Matrix Force Method

9.4 Substructure Force Analysis of a Two-bay Truss

PROBLEMS

CHAPTER 10 DYNAMICS OF ELASTIC SYSTEMS

10.1 Formulation of the Dynamical Problems

10.2 Principle of Virtual Work in Dynamics of Elastic Systems

10.3 Hamilton's Principle

10.4 Power-Balance Equation

10.5 Equations of Motion and Equilibrium

10.6 Static and Dynamic Displacements in a Uniform Bar

10.7 Equivalent Masses in Matrix Analysis

10.8 Frequency-dependent Mass and Stiffness Matrices for Bar Elements

10.9 Frequency-dependent Mass and Stiffness Matrices for Beam Elements

PROBLEMS

CHAPTER 11 INERTIA PROPERTIES OF STRUCTURAL ELEMENTS

11.1 Equivalent Mass Matrices in Datum Coordinate System

11.2 Equivalent Mass Matrix for an Assembled Structure

11.3 Condensed Mass Matrix

11.4 Pin-jointed Bar

11.5 Uniform Beam

11.6 Triangular Plate with Translational Displacements

11.7 Rectangular Plate with Translational Displacements

11.8 Solid Tetrahedron

11.9 Solid Parallelepiped

11.10 Triangular Plate with Bending Displacements

11.11 Rectangular Plate with Bending Displacements

11.12 Lumped-mass Representation

PROBLEMS

CHAPTER 12 VIBRATIONS OF ELASTIC SYSTEMS

12.1 Vibration Analysis Based on Stiffness

12.2 Properties of the Eigenmodes: Orthogonality Relations

12.3 Vibration Analysis Based on Flexibility

12.4 Vibration of Damped Structural Systems

12.5 Critical Damping

12.6 Longitudinal Vibrations of an Unconstrained Bar

12.7 Longitudinal Vibrations of a Constrained Bar

12.8 Transverse Vibrations of a Fuselage-Wing Combination

12.9 Determination of Vibration Frequencies from the Quadratic Matrix Equation

PROBLEMS

CHAPTER 13 DYNAMIC RESPONSE OF ELASTIC SYSTEMS

13.1 Response of a Single-degree-of-freedom System: Duhamel's Integrals

13.2 Dynamic Response of an Unconstrained (Free) Structure

13.3 Response Resulting from Impulsive Forces

13.4 Dynamic Response of a Constrained Structure

13.5 Steady-state Harmonic Motion

13.6 Duhamel's Integrals for Typical Forcing Functions

13.7 Dynamic Response to Forced Displacements: Response to Earthquakes

13.8 Determination of Frequencies and Modes of Unconstrained (Free) Structures Using Experimental Data for the Constrained Structures

13.9 Dynamic Response of Structural Systems with Damping

13.10 Damping Matrix Proportional to Mass

13.11 Damping Matrix Proportional to Stiffness

13.12 Matrix C Proportional to Critical Damping

13.13 Orthonormalization of the Modal Matrix p

13.14 Dynamic Response of an Elastic Rocket Subjected to Pulse Loading

13.15 Response Due to Forced Displacement at One End of a Uniform Bar

PROBLEMS

CHAPTER 14 STRUCTURAL SYNTHESIS

14.1 Mathematical Formulation of the Optimization Problem

14.2 Structural Optimization

CHAPTER 15 NONLINEAR STRUCTURAL ANALYSIS

15.1 Matrix Displacement Analysis for Large Deflections

15.2 Geometrical Stiffness for Bar Elements

15.3 Geometrical Stiffness for Beam Elements

15.4 Matrix Force Analysis for Large Deflections

15.5 Inelastic Analysis and Creep

15.6 Stability Analysis of a Simple Truss

15.7 Stability Analysis of a Column

15.8 Influence of a Constant Axial Force on Transverse Vibrations of Beams

PROBLEMS

APPENDIX A MATRIX ALGEBRA

APPENDIX B BIBLIOGRAPHY

INDEX

## Reading Group Guide

PREFACE

CHAPTER 1 MATRIX METHODS

1.1 Introduction

1.2 Design Iterations

1.3 Methods of Analysis

1.4 Areas of Structural Analysis

CHAPTER 2 BASIC EQUATIONS OF ELASTICITY

2.1 Strain-Displacement Equations

2.2 Stress-Strain Equations

2.3 Stress-Strain Equations for Initial Strains

2.4 Equations of Equilbrium

2.5 Compatibility Equations

CHAPTER 3 ENERGY THEOREMS

3.1 Introduction

3.2 Work and Complementary Work; Stain Energy and Complementary Strain

3.3 Green's Identity

3.4 Energy Theorems Based on the Principle of Virtual Work

3.5 Energy Theorems Based on the Principle of Complementary Virtual Work

3.6 Clapeyron's Theorem

3.7 Betti's Theorem

3.8 Maxwell's Reciprocal Theorem

3.9 Summary of Energy Theorems and Definitions

PROBLEMS

CHAPTER 4 STRUCTURAL IDEALIZATION

4.1 Structural Idealization

4.2 Energy Equivalence

4.3 Structural Elements

CHAPTER 5 STIFFNESS PROPERTIES OF STRUCTURAL ELEMENTS

5.1 Methods of Determining Element Force-Displacement Relationships

5.2 Determination of Element Stiffness Properties by the Unit-displacement Theorem

5.3 Application of Castigliano's Theorem (Part I) to Derive Stiffness Properties

5.4 Transformation of Coordinate Axes: ? Matrices

5.5 Pin-jointed Bar Elements

5.6 Beam Elements

5.7 Triangular Plate Elements (In-plane Forces)

5.8 Rectangular Plate Elements (In-plane Forces)

5.9 Quadrilateral Plate Elements (In-plane Forces)

5.10 Tetrahedron Elements

5.11 Triangular Plates in Bending

5.12 Rectangular Plates in Bending

5.13 Method for Improving Stiffness Matrices

PROBLEMS

CHAPTER 6 THE MATRIX DISPLACEMENT METHOD

6.1 Matrix Formulation of the Displacement Analysis

6.2 Elimination of the Rigid-body Degrees of Freedom: Choice of Reactions

6.3 Derivation of the Transformation Matrix V from Equilibrium Equations

6.4 Derivation of the Transformation Matrix T from Kinematics

6.5 Condensation of Stiffness Matrices

6.6 Derivation of Stiffness Matrices from Flexibility

6.7 Stiffness Matrix for Constant-shear-flow Panels

6.8 Stiffness Matrix for Linearly Varying Axial-force Members

6.9 Analysis of a Pin-jointed Truss by the Displacement Method

6.10 Analysis of a Cantilever Beam by the Displacement Method

6.11 Equivalent Concentrated Forces

PROBLEMS

CHAPTER 7 FLEXIBILITY PROPERTIES OF STRUCTURAL ELEMENTS

7.1 Methods of Determing Element Displacement-Force Relationships

7.2 Inversion of the Force-Displacement Equations: Flexibility Properties of Pin-jointed Bars and Beam Elements

7.3 Determination of Element Flexibility Properties by the Unit-load Theorem

7.4 Application of Castigliano's Theorem (Part II) to Derive Flexibility Properties

7.5 Solution of Differential Equations for Element Displacements to Derive Flexibility Properties

7.6 Pin-jointed Bar Elements

7.7 Beam Elements

7.8 Triangular Plate Elements (In-plane Forces)

7.9 Rectangular Plate Elements (In-plane Forces)

7.10 Tetrahedron Elements

7.11 Constant-shear-flow Panels

7.12 Linearly Varying Axial-force Members

7.13 Rectangular Plates in Bending

PROBLEMS

CHAPTER 8 THE MATRIX FORCE METHOD

8.1 Matrix Formulation of the Unit-load Theorem for External-force Systems

8.2 Matrix Formulation of the Unit-load Theorem for Internal-force Systems: Self-equilibrating Force Systems

8.3 Matrix Formulation of the Force Analysis: Jordanian Elimination Technique

8.4 Matrix Force Analysis of a Pin-jointed Truss

8.5 Matrix Force Analysis of a Cantilever Beam

8.6 Comparison of the Force and Displacement Methods

PROBLEMS

CHAPTER 9 ANALYSIS OF SUBSTRUCTURES

9.1 Substructure Analysis by the Matrix Displacement Method

9.2 Substructure Displacement Analysis of a Two-Bay Truss

9.3 Substructure Analysis by the Matrix Force Method

9.4 Substructure Force Analysis of a Two-bay Truss

PROBLEMS

CHAPTER 10 DYNAMICS OF ELASTIC SYSTEMS

10.1 Formulation of the Dynamical Problems

10.2 Principle of Virtual Work in Dynamics of Elastic Systems

10.3 Hamilton's Principle

10.4 Power-Balance Equation

10.5 Equations of Motion and Equilibrium

10.6 Static and Dynamic Displacements in a Uniform Bar

10.7 Equivalent Masses in Matrix Analysis

10.8 Frequency-dependent Mass and Stiffness Matrices for Bar Elements

10.9 Frequency-dependent Mass and Stiffness Matrices for Beam Elements

PROBLEMS

CHAPTER 11 INERTIA PROPERTIES OF STRUCTURAL ELEMENTS

11.1 Equivalent Mass Matrices in Datum Coordinate System

11.2 Equivalent Mass Matrix for an Assembled Structure

11.3 Condensed Mass Matrix

11.4 Pin-jointed Bar

11.5 Uniform Beam

11.6 Triangular Plate with Translational Displacements

11.7 Rectangular Plate with Translational Displacements

11.8 Solid Tetrahedron

11.9 Solid Parallelepiped

11.10 Triangular Plate with Bending Displacements

11.11 Rectangular Plate with Bending Displacements

11.12 Lumped-mass Representation

PROBLEMS

CHAPTER 12 VIBRATIONS OF ELASTIC SYSTEMS

12.1 Vibration Analysis Based on Stiffness

12.2 Properties of the Eigenmodes: Orthogonality Relations

12.3 Vibration Analysis Based on Flexibility

12.4 Vibration of Damped Structural Systems

12.5 Critical Damping

12.6 Longitudinal Vibrations of an Unconstrained Bar

12.7 Longitudinal Vibrations of a Constrained Bar

12.8 Transverse Vibrations of a Fuselage-Wing Combination

12.9 Determination of Vibration Frequencies from the Quadratic Matrix Equation

PROBLEMS

CHAPTER 13 DYNAMIC RESPONSE OF ELASTIC SYSTEMS

13.1 Response of a Single-degree-of-freedom System: Duhamel's Integrals

13.2 Dynamic Response of an Unconstrained (Free) Structure

13.3 Response Resulting from Impulsive Forces

13.4 Dynamic Response of a Constrained Structure

13.5 Steady-state Harmonic Motion

13.6 Duhamel's Integrals for Typical Forcing Functions

13.7 Dynamic Response to Forced Displacements: Response to Earthquakes

13.8 Determination of Frequencies and Modes of Unconstrained (Free) Structures Using Experimental Data for the Constrained Structures

13.9 Dynamic Response of Structural Systems with Damping

13.10 Damping Matrix Proportional to Mass

13.11 Damping Matrix Proportional to Stiffness

13.12 Matrix C Proportional to Critical Damping

13.13 Orthonormalization of the Modal Matrix p

13.14 Dynamic Response of an Elastic Rocket Subjected to Pulse Loading

13.15 Response Due to Forced Displacement at One End of a Uniform Bar

PROBLEMS

CHAPTER 14 STRUCTURAL SYNTHESIS

14.1 Mathematical Formulation of the Optimization Problem

14.2 Structural Optimization

CHAPTER 15 NONLINEAR STRUCTURAL ANALYSIS

15.1 Matrix Displacement Analysis for Large Deflections

15.2 Geometrical Stiffness for Bar Elements

15.3 Geometrical Stiffness for Beam Elements

15.4 Matrix Force Analysis for Large Deflections

15.5 Inelastic Analysis and Creep

15.6 Stability Analysis of a Simple Truss

15.7 Stability Analysis of a Column

15.8 Influence of a Constant Axial Force on Transverse Vibrations of Beams

PROBLEMS

APPENDIX A MATRIX ALGEBRA

APPENDIX B BIBLIOGRAPHY

INDEX

## Interviews

PREFACE

CHAPTER 1 MATRIX METHODS

1.1 Introduction

1.2 Design Iterations

1.3 Methods of Analysis

1.4 Areas of Structural Analysis

CHAPTER 2 BASIC EQUATIONS OF ELASTICITY

2.1 Strain-Displacement Equations

2.2 Stress-Strain Equations

2.3 Stress-Strain Equations for Initial Strains

2.4 Equations of Equilbrium

2.5 Compatibility Equations

CHAPTER 3 ENERGY THEOREMS

3.1 Introduction

3.2 Work and Complementary Work; Stain Energy and Complementary Strain

3.3 Green's Identity

3.4 Energy Theorems Based on the Principle of Virtual Work

3.5 Energy Theorems Based on the Principle of Complementary Virtual Work

3.6 Clapeyron's Theorem

3.7 Betti's Theorem

3.8 Maxwell's Reciprocal Theorem

3.9 Summary of Energy Theorems and Definitions

PROBLEMS

CHAPTER 4 STRUCTURAL IDEALIZATION

4.1 Structural Idealization

4.2 Energy Equivalence

4.3 Structural Elements

CHAPTER 5 STIFFNESS PROPERTIES OF STRUCTURAL ELEMENTS

5.1 Methods of Determining Element Force-Displacement Relationships

5.2 Determination of Element Stiffness Properties by the Unit-displacement Theorem

5.3 Application of Castigliano's Theorem (Part I) to Derive Stiffness Properties

5.4 Transformation of Coordinate Axes: ? Matrices

5.5 Pin-jointed Bar Elements

5.6 Beam Elements

5.7 Triangular Plate Elements (In-plane Forces)

5.8 Rectangular Plate Elements (In-plane Forces)

5.9 Quadrilateral Plate Elements (In-plane Forces)

5.10 Tetrahedron Elements

5.11 Triangular Plates in Bending

5.12 Rectangular Plates in Bending

5.13 Method for Improving Stiffness Matrices

PROBLEMS

CHAPTER 6 THE MATRIX DISPLACEMENT METHOD

6.1 Matrix Formulation of the Displacement Analysis

6.2 Elimination of the Rigid-body Degrees of Freedom: Choice of Reactions

6.3 Derivation of the Transformation Matrix V from Equilibrium Equations

6.4 Derivation of the Transformation Matrix T from Kinematics

6.5 Condensation of Stiffness Matrices

6.6 Derivation of Stiffness Matrices from Flexibility

6.7 Stiffness Matrix for Constant-shear-flow Panels

6.8 Stiffness Matrix for Linearly Varying Axial-force Members

6.9 Analysis of a Pin-jointed Truss by the Displacement Method

6.10 Analysis of a Cantilever Beam by the Displacement Method

6.11 Equivalent Concentrated Forces

PROBLEMS

CHAPTER 7 FLEXIBILITY PROPERTIES OF STRUCTURAL ELEMENTS

7.1 Methods of Determing Element Displacement-Force Relationships

7.2 Inversion of the Force-Displacement Equations: Flexibility Properties of Pin-jointed Bars and Beam Elements

7.3 Determination of Element Flexibility Properties by the Unit-load Theorem

7.4 Application of Castigliano's Theorem (Part II) to Derive Flexibility Properties

7.5 Solution of Differential Equations for Element Displacements to Derive Flexibility Properties

7.6 Pin-jointed Bar Elements

7.7 Beam Elements

7.8 Triangular Plate Elements (In-plane Forces)

7.9 Rectangular Plate Elements (In-plane Forces)

7.10 Tetrahedron Elements

7.11 Constant-shear-flow Panels

7.12 Linearly Varying Axial-force Members

7.13 Rectangular Plates in Bending

PROBLEMS

CHAPTER 8 THE MATRIX FORCE METHOD

8.1 Matrix Formulation of the Unit-load Theorem for External-force Systems

8.2 Matrix Formulation of the Unit-load Theorem for Internal-force Systems: Self-equilibrating Force Systems

8.3 Matrix Formulation of the Force Analysis: Jordanian Elimination Technique

8.4 Matrix Force Analysis of a Pin-jointed Truss

8.5 Matrix Force Analysis of a Cantilever Beam

8.6 Comparison of the Force and Displacement Methods

PROBLEMS

CHAPTER 9 ANALYSIS OF SUBSTRUCTURES

9.1 Substructure Analysis by the Matrix Displacement Method

9.2 Substructure Displacement Analysis of a Two-Bay Truss

9.3 Substructure Analysis by the Matrix Force Method

9.4 Substructure Force Analysis of a Two-bay Truss

PROBLEMS

CHAPTER 10 DYNAMICS OF ELASTIC SYSTEMS

10.1 Formulation of the Dynamical Problems

10.2 Principle of Virtual Work in Dynamics of Elastic Systems

10.3 Hamilton's Principle

10.4 Power-Balance Equation

10.5 Equations of Motion and Equilibrium

10.6 Static and Dynamic Displacements in a Uniform Bar

10.7 Equivalent Masses in Matrix Analysis

10.8 Frequency-dependent Mass and Stiffness Matrices for Bar Elements

10.9 Frequency-dependent Mass and Stiffness Matrices for Beam Elements

PROBLEMS

CHAPTER 11 INERTIA PROPERTIES OF STRUCTURAL ELEMENTS

11.1 Equivalent Mass Matrices in Datum Coordinate System

11.2 Equivalent Mass Matrix for an Assembled Structure

11.3 Condensed Mass Matrix

11.4 Pin-jointed Bar

11.5 Uniform Beam

11.6 Triangular Plate with Translational Displacements

11.7 Rectangular Plate with Translational Displacements

11.8 Solid Tetrahedron

11.9 Solid Parallelepiped

11.10 Triangular Plate with Bending Displacements

11.11 Rectangular Plate with Bending Displacements

11.12 Lumped-mass Representation

PROBLEMS

CHAPTER 12 VIBRATIONS OF ELASTIC SYSTEMS

12.1 Vibration Analysis Based on Stiffness

12.2 Properties of the Eigenmodes: Orthogonality Relations

12.3 Vibration Analysis Based on Flexibility

12.4 Vibration of Damped Structural Systems

12.5 Critical Damping

12.6 Longitudinal Vibrations of an Unconstrained Bar

12.7 Longitudinal Vibrations of a Constrained Bar

12.8 Transverse Vibrations of a Fuselage-Wing Combination

12.9 Determination of Vibration Frequencies from the Quadratic Matrix Equation

PROBLEMS

CHAPTER 13 DYNAMIC RESPONSE OF ELASTIC SYSTEMS

13.1 Response of a Single-degree-of-freedom System: Duhamel's Integrals

13.2 Dynamic Response of an Unconstrained (Free) Structure

13.3 Response Resulting from Impulsive Forces

13.4 Dynamic Response of a Constrained Structure

13.5 Steady-state Harmonic Motion

13.6 Duhamel's Integrals for Typical Forcing Functions

13.7 Dynamic Response to Forced Displacements: Response to Earthquakes

13.8 Determination of Frequencies and Modes of Unconstrained (Free) Structures Using Experimental Data for the Constrained Structures

13.9 Dynamic Response of Structural Systems with Damping

13.10 Damping Matrix Proportional to Mass

13.11 Damping Matrix Proportional to Stiffness

13.12 Matrix C Proportional to Critical Damping

13.13 Orthonormalization of the Modal Matrix p

13.14 Dynamic Response of an Elastic Rocket Subjected to Pulse Loading

13.15 Response Due to Forced Displacement at One End of a Uniform Bar

PROBLEMS

CHAPTER 14 STRUCTURAL SYNTHESIS

14.1 Mathematical Formulation of the Optimization Problem

14.2 Structural Optimization

CHAPTER 15 NONLINEAR STRUCTURAL ANALYSIS

15.1 Matrix Displacement Analysis for Large Deflections

15.2 Geometrical Stiffness for Bar Elements

15.3 Geometrical Stiffness for Beam Elements

15.4 Matrix Force Analysis for Large Deflections

15.5 Inelastic Analysis and Creep

15.6 Stability Analysis of a Simple Truss

15.7 Stability Analysis of a Column

15.8 Influence of a Constant Axial Force on Transverse Vibrations of Beams

PROBLEMS

APPENDIX A MATRIX ALGEBRA

APPENDIX B BIBLIOGRAPHY

INDEX

## Recipe

CHAPTER 1 MATRIX METHODS

1.1 Introduction

1.2 Design Iterations

1.3 Methods of Analysis

1.4 Areas of Structural Analysis

CHAPTER 2 BASIC EQUATIONS OF ELASTICITY

2.1 Strain-Displacement Equations

2.2 Stress-Strain Equations

2.3 Stress-Strain Equations for Initial Strains

2.4 Equations of Equilbrium

2.5 Compatibility Equations

CHAPTER 3 ENERGY THEOREMS

3.1 Introduction

3.2 Work and Complementary Work; Stain Energy and Complementary Strain

3.3 Green's Identity

3.4 Energy Theorems Based on the Principle of Virtual Work

3.5 Energy Theorems Based on the Principle of Complementary Virtual Work

3.6 Clapeyron's Theorem

3.7 Betti's Theorem

3.8 Maxwell's Reciprocal Theorem

3.9 Summary of Energy Theorems and Definitions

PROBLEMS

CHAPTER 4 STRUCTURAL IDEALIZATION

4.1 Structural Idealization

4.2 Energy Equivalence

4.3 Structural Elements

CHAPTER 5 STIFFNESS PROPERTIES OF STRUCTURAL ELEMENTS

5.1 Methods of Determining Element Force-Displacement Relationships

5.2 Determination of Element Stiffness Properties by the Unit-displacement Theorem

5.3 Application of Castigliano's Theorem (Part I) to Derive Stiffness Properties

5.4 Transformation of Coordinate Axes: ? Matrices

5.5 Pin-jointed Bar Elements

5.6 Beam Elements

5.7 Triangular Plate Elements (In-plane Forces)

5.8 Rectangular Plate Elements (In-plane Forces)

5.9 Quadrilateral Plate Elements (In-plane Forces)

5.10 Tetrahedron Elements

5.11 Triangular Plates in Bending

5.12 Rectangular Plates in Bending

5.13 Method for Improving Stiffness Matrices

PROBLEMS

CHAPTER 6 THE MATRIX DISPLACEMENT METHOD

6.1 Matrix Formulation of the Displacement Analysis

6.2 Elimination of the Rigid-body Degrees of Freedom: Choice of Reactions

6.3 Derivation of the Transformation Matrix V from Equilibrium Equations

6.4 Derivation of the Transformation Matrix T from Kinematics

6.5 Condensation of Stiffness Matrices

6.6 Derivation of Stiffness Matrices from Flexibility

6.7 Stiffness Matrix for Constant-shear-flow Panels

6.8 Stiffness Matrix for Linearly Varying Axial-force Members

6.9 Analysis of a Pin-jointed Truss by the Displacement Method

6.10 Analysis of a Cantilever Beam by the Displacement Method

6.11 Equivalent Concentrated Forces

PROBLEMS

CHAPTER 7 FLEXIBILITY PROPERTIES OF STRUCTURAL ELEMENTS

7.1 Methods of Determing Element Displacement-Force Relationships

7.2 Inversion of the Force-Displacement Equations: Flexibility Properties of Pin-jointed Bars and Beam Elements

7.3 Determination of Element Flexibility Properties by the Unit-load Theorem

7.4 Application of Castigliano's Theorem (Part II) to Derive Flexibility Properties

7.5 Solution of Differential Equations for Element Displacements to Derive Flexibility Properties

7.6 Pin-jointed Bar Elements

7.7 Beam Elements

7.8 Triangular Plate Elements (In-plane Forces)

7.9 Rectangular Plate Elements (In-plane Forces)

7.10 Tetrahedron Elements

7.11 Constant-shear-flow Panels

7.12 Linearly Varying Axial-force Members

7.13 Rectangular Plates in Bending

PROBLEMS

CHAPTER 8 THE MATRIX FORCE METHOD

8.1 Matrix Formulation of the Unit-load Theorem for External-force Systems

8.2 Matrix Formulation of the Unit-load Theorem for Internal-force Systems: Self-equilibrating Force Systems

8.3 Matrix Formulation of the Force Analysis: Jordanian Elimination Technique

8.4 Matrix Force Analysis of a Pin-jointed Truss

8.5 Matrix Force Analysis of a Cantilever Beam

8.6 Comparison of the Force and Displacement Methods

PROBLEMS

CHAPTER 9 ANALYSIS OF SUBSTRUCTURES

9.1 Substructure Analysis by the Matrix Displacement Method

9.2 Substructure Displacement Analysis of a Two-Bay Truss

9.3 Substructure Analysis by the Matrix Force Method

9.4 Substructure Force Analysis of a Two-bay Truss

PROBLEMS

CHAPTER 10 DYNAMICS OF ELASTIC SYSTEMS

10.1 Formulation of the Dynamical Problems

10.2 Principle of Virtual Work in Dynamics of Elastic Systems

10.3 Hamilton's Principle

10.4 Power-Balance Equation

10.5 Equations of Motion and Equilibrium

10.6 Static and Dynamic Displacements in a Uniform Bar

10.7 Equivalent Masses in Matrix Analysis

10.8 Frequency-dependent Mass and Stiffness Matrices for Bar Elements

10.9 Frequency-dependent Mass and Stiffness Matrices for Beam Elements

PROBLEMS

CHAPTER 11 INERTIA PROPERTIES OF STRUCTURAL ELEMENTS

11.1 Equivalent Mass Matrices in Datum Coordinate System

11.2 Equivalent Mass Matrix for an Assembled Structure

11.3 Condensed Mass Matrix

11.4 Pin-jointed Bar

11.5 Uniform Beam

11.6 Triangular Plate with Translational Displacements

11.7 Rectangular Plate with Translational Displacements

11.8 Solid Tetrahedron

11.9 Solid Parallelepiped

11.10 Triangular Plate with Bending Displacements

11.11 Rectangular Plate with Bending Displacements

11.12 Lumped-mass Representation

PROBLEMS

CHAPTER 12 VIBRATIONS OF ELASTIC SYSTEMS

12.1 Vibration Analysis Based on Stiffness

12.2 Properties of the Eigenmodes: Orthogonality Relations

12.3 Vibration Analysis Based on Flexibility

12.4 Vibration of Damped Structural Systems

12.5 Critical Damping

12.6 Longitudinal Vibrations of an Unconstrained Bar

12.7 Longitudinal Vibrations of a Constrained Bar

12.8 Transverse Vibrations of a Fuselage-Wing Combination

12.9 Determination of Vibration Frequencies from the Quadratic Matrix Equation

PROBLEMS

CHAPTER 13 DYNAMIC RESPONSE OF ELASTIC SYSTEMS

13.1 Response of a Single-degree-of-freedom System: Duhamel's Integrals

13.2 Dynamic Response of an Unconstrained (Free) Structure

13.3 Response Resulting from Impulsive Forces

13.4 Dynamic Response of a Constrained Structure

13.5 Steady-state Harmonic Motion

13.6 Duhamel's Integrals for Typical Forcing Functions

13.7 Dynamic Response to Forced Displacements: Response to Earthquakes

13.8 Determination of Frequencies and Modes of Unconstrained (Free) Structures Using Experimental Data for the Constrained Structures

13.9 Dynamic Response of Structural Systems with Damping

13.10 Damping Matrix Proportional to Mass

13.11 Damping Matrix Proportional to Stiffness

13.12 Matrix C Proportional to Critical Damping

13.13 Orthonormalization of the Modal Matrix p

13.14 Dynamic Response of an Elastic Rocket Subjected to Pulse Loading

13.15 Response Due to Forced Displacement at One End of a Uniform Bar

PROBLEMS

CHAPTER 14 STRUCTURAL SYNTHESIS

14.1 Mathematical Formulation of the Optimization Problem

14.2 Structural Optimization

CHAPTER 15 NONLINEAR STRUCTURAL ANALYSIS

15.1 Matrix Displacement Analysis for Large Deflections

15.2 Geometrical Stiffness for Bar Elements

15.3 Geometrical Stiffness for Beam Elements

15.4 Matrix Force Analysis for Large Deflections

15.5 Inelastic Analysis and Creep

15.6 Stability Analysis of a Simple Truss

15.7 Stability Analysis of a Column

15.8 Influence of a Constant Axial Force on Transverse Vibrations of Beams

PROBLEMS

APPENDIX A MATRIX ALGEBRA

APPENDIX B BIBLIOGRAPHY

INDEX