Analysis of Structures E-Book

Joe Eisley

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 576

Veröffentlichungsjahr: 2011

Beschreibung

Analysis of Structures offers an original way of introducing engineering students to the subject of stress and deformation analysis of solid objects, and helps them become more familiar with how numerical methods such as the finite element method are used in industry. Eisley and Waas secure for the reader a thorough understanding of the basic numerical skills and insight into interpreting the results these methods can generate. Throughout the text, they include analytical development alongside the computational equivalent, providing the student with the understanding that is necessary to interpret and use the solutions that are obtained using software based on the finite element method. They then extend these methods to the analysis of solid and structural components that are used in modern aerospace, mechanical and civil engineering applications. Analysis of Structures is accompanied by a book companion website href="http://www.wiley.com/go/waas">www.wiley.com/go/waas housing exercises and examples that use modern software which generates color contour plots of deformation and internal stress.It offers invaluable guidance and understanding to senior level and graduate students studying courses in stress and deformation analysis as part of aerospace, mechanical and civil engineering degrees as well as to practicing engineers who want to re-train or re-engineer their set of analysis tools for contemporary stress and deformation analysis of solids and structures. * Provides a fresh, practical perspective to the teaching of structural analysis using numerical methods for obtaining answers to real engineering applications * Proposes a new way of introducing students to the subject of stress and deformation analysis of solid objects that are used in a wide variety of contemporary engineering applications * Casts axial, torsional and bending deformations of thin walled objects in a framework that is closely amenable to the methods by which modern stress analysis software operates.

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

About the Authors

Preface

Chapter 1: Forces and Moments

1.1 Introduction

1.2 Units

1.3 Forces in Mechanics of Materials

1.4 Concentrated Forces

1.5 Moment of a Concentrated Force

1.6 Distributed Forces—Force and Moment Resultants

1.7 Internal Forces and Stresses---Stress Resultants

1.8 Restraint Forces and Restraint Force Resultants

1.9 Summary and Conclusions

Chapter 2: Static Equilibrium

2.1 Introduction

2.2 Free Body Diagrams

2.3 Equilibrium—Concentrated Forces

2.4 Equilibrium—Distributed Forces

2.5 Equilibrium in Three Dimensions

2.6 Equilibrium—Internal Forces and Stresses

2.7 Summary and Conclusions

Chapter 3: Displacement, Strain, and Material Properties

3.1 Introduction

3.2 Displacement and Strain

3.3 Compatibility

3.4 Linear Material Properties

3.5 Some Simple Solutions for Stress, Strain, and Displacement

3.6 Thermal Strain

3.7 Engineering Materials

3.8 Fiber Reinforced Composite Laminates

3.9 Plan for the Following Chapters

3.10 Summary and Conclusions

Chapter 4: Classical Analysis of the Axially Loaded Slender Bar

4.1 Introduction

4.2 Solutions from the Theory of Elasticity

4.3 Derivation and Solution of the Governing Equations

4.4 The Statically Determinate Case

4.5 The Statically Indeterminate Case

4.6 Variable Cross Sections

4.7 Thermal Stress and Strain in an Axially Loaded Bar

4.8 Shearing Stress in an Axially Loaded Bar

4.9 Design of Axially Loaded Bars

4.10 Analysis and Design of Pin Jointed Trusses

4.11 Work and Energy---Castigliano's Second Theorem

4.12 Summary and Conclusions

Chapter 5: A General Method for the Axially Loaded Slender Bar

5.1 Introduction

5.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix

5.3 The Assembled Global Equations and Their Solution

5.4 A General Method---Distributed Applied Loads

5.5 Variable Cross Sections

5.6 Analysis and Design of Pin-jointed Trusses

5.7 Summary and Conclusions

Chapter 6: Torsion

6.1 Introduction

6.2 Torsional Displacement, Strain, and Stress

6.3 Derivation and Solution of the Governing Equations

6.4 Solutions from the Theory of Elasticity

6.5 Torsional Stress in Thin Walled Cross Sections

6.6 Work and Energy—Torsional Stiffness in a Thin Walled Tube

6.7 Torsional Stress and Stiffness in Multicell Sections

6.8 Torsional Stress and Displacement in Thin Walled Open Sections

6.9 A General (Finite Element) Method

6.10 Continuously Variable Cross Sections

6.11 Summary and Conclusions

Chapter 7: Classical Analysis of the Bending of Beams

7.1 Introduction

7.2 Area Properties—Sign Conventions

7.3 Derivation and Solution of the Governing Equations

7.4 The Statically Determinate Case

7.5 Work and Energy—Castigliano's Second Theorem

7.6 The Statically Indeterminate Case

7.7 Solutions from the Theory of Elasticity

7.8 Variable Cross Sections

7.9 Shear Stress in Non Rectangular Cross Sections—Thin Walled Cross Sections

7.10 Design of Beams

7.11 Large Displacements

7.12 Summary and Conclusions

Chapter 8: A General Method (FEM) for the Bending of Beams

8.1 Introduction

8.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix

8.3 The Global Equations and their Solution

8.4 Distributed Loads in FEM

8.5 Variable Cross Sections

8.6 Summary and Conclusions

Chapter 9: More about Stress and Strain, and Material Properties

9.1 Introduction

9.2 Transformation of Stress in Two Dimensions

9.3 Principal Axes and Principal Stresses in Two Dimensions

9.4 Transformation of Strain in Two Dimensions

9.5 Strain Rosettes

9.6 Stress Transformation and Principal Stresses in Three Dimensions

9.7 Allowable and Ultimate Stress, and Factors of Safety

9.8 Fatigue

9.9 Creep

9.10 Orthotropic Materials—Composites

9.11 Summary and Conclusions

Chapter 10: Combined Loadings on Slender Bars—Thin Walled Cross Sections

10.1 Introduction

10.2 Review and Summary of Slender Bar Equations

10.3 Axial and Torsional Loads

10.4 Axial and Bending Loads—2D Frames

10.5 Bending in Two Planes

10.6 Bending and Torsion in Thin Walled Open Sections—Shear Center

10.7 Bending and Torsion in Thin Walled Closed Sections—Shear Center

10.8 Stiffened Thin Walled Beams

10.9 Summary and Conclusions

Chapter 11: Work and Energy Methods—Virtual Work

11.1 Introduction

11.2 Introduction to the Principle of Virtual Work

11.3 Static Analysis of Slender Bars by Virtual Work

11.4 Static Analysis of 3D and 2D Solids by Virtual Work

11.5 The Element Stiffness Matrix for Plane Stress

11.6 The Element Stiffness Matrix for 3D Solids

11.7 Summary and Conclusions

Chapter 12: Structural Analysis in Two and Three Dimensions

12.1 Introduction

12.2 The Governing Equations in Two Dimensions—Plane Stress

12.3 Finite Elements and the Stiffness Matrix for Plane Stress

12.4 Thin Flat Plates—Classical Analysis

12.5 Thin Flat Plates—FEM Analysis

12.6 Shell Structures

12.7 Stiffened Shell Structures

12.8 Three Dimensional Structures—Classical and FEM Analysis

12.9 Summary and Conclusions

Chapter 13: Analysis of Thin Laminated Composite Material Structures

13.1 Introduction to Classical Lamination Theory

13.2 Strain Displacement Equations for Laminates

13.3 Stress-Strain Relations for a Single Lamina

13.4 Stress Resultants for Laminates

13.5 CLT Constitutive Description

13.6 Determining Laminae Stress/Strains

13.7 Laminated Plates Subject to Transverse Loads

13.8 Summary and Conclusion

Chapter 14 : Buckling

14.1 Introduction

14.2 The Equations for a Beam with Combined Lateral and Axial Loading

14.3 Buckling of a Column

14.4 The Beam Column

14.5 The Finite Element Method for Bending and Buckling

14.6 Buckling of Frames

14.7 Buckling of Thin Plates and Other Structures

14.8 Summary and Conclusions

Chapter 15: Structural Dynamics

15.1 Introduction

15.2 Dynamics of Mass/Spring Systems

15.3 Axial Vibration of a Slender Bar

15.4 Torsional Vibration

15.5 Vibration of Beams in Bending

15.6 The Finite Element Method for all Elastic Structures

15.7 Addition of Damping

15.8 Summary and Conclusions

Chapter 16: Evolution in the (Intelligent) Design and Analysis of Structural Members

16.1 Introduction

16.2 Evolution of a Truss Member

16.3 Evolution of a Plate with a Hole—Plane Stress

16.4 Materials in Design

16.5 Summary and Conclusions

Appendix A: Matrix Definitions and Operations

A.1 Introduction

A.2 Matrix Definitions

A.3 Matrix Algebra

A.4 Partitioned Matrices

A.5 Differentiating and Integrating a Matrix

A.6 Summary of Useful Matrix Relations

Appendix B: Area Properties of Cross Sections

B.1 Introduction

B.2 Centroids of Cross Sections

B.3 Area Moments and Product of Inertia

B.4 Properties of Common Cross Sections

Appendix C: Solving Sets of Linear Algebraic Equations with Mathematica

C.1 Introduction

C.2 Systems of Linear Algebraic Equations

C.3 Solving Numerical Equations in Mathematica

C.4 Solving Symbolic Equations in Mathematica

C.5 Matrix Multiplication

Appendix D: Orthogonality of Normal Modes

D.1 Introduction

D.2 Proof of Orthogonality for Discrete Systems

D.3 Proof of Orthogonality for Continuous Systems

References

Index

This edition first published 2011 © 2011 John Wiley & Sons, Ltd

Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Eisley, Joe G. Analysis of structures : an introduction including numerical methods / Joe G. Eisley, Anthony M. Waas. p. cm. Includes bibliographical references and index. ISBN 978-0-470-97762-0 (cloth) 1. Structural analysis (Engineering)--Mathematics. 2. Numerical analysis. I. Waas, Anthony M. II. Title. TA646.W33 2011 624.1′71--dc22 2011009723

A catalogue record for this book is available from the British Library.

Print ISBN: 9780470977620

E-PDF ISBN: 9781119993285

O-book ISBN: 9781119993278

E-Pub ISBN: 9781119993544

Mobi ISBN: 9781119993551

We would like to dedicate this book to our families.

To Marilyn, Paul and Susan—JoeTo Dayamal, Dayani, Shehara and Michael—Tony

About the Authors

Joe G. Eisley received degrees from St. Louis University, BS (1951), and the California Institute of Technology, MS (1952), PhD (1956), all in the field of aeronautical engineering. He served on the faculty of the Department of Aerospace Engineering from 1956 to 1998 and retired as Emeritus Professor of Aerospace Engineering in 1998. His primary field of teaching and research has been in structural analysis with an emphasis on the dynamics of structures. He also taught courses in space systems design and computer aided design. After retirement he has continued some part time work in teaching and consulting.

Anthony M. Waas is the Felix Pawlowski Collegiate Professor of Aerospace Engineering and Professor of Mechanical Engineering, and Director, Composite Structures Laboratory at the University of Michigan. He received his degrees from Imperial College, Univ. of London, U.K., B.Sc. (first class honors, 1982), and the California Institute of Technology, MS (1983), PhD (1988) all in Aeronautics. He joined the University of Michigan in January 1988 as an Assistant Professor, and is currently the Felix Pawlowski Collegiate Professor. His current teaching and research interests are related to lightweight composite aerostructures, with a focus on manufacturability and damage tolerance, ceramic matrix composites for “hot” structures, nano-composites, and multi-material structures. Several of his projects have been funded by numerous US government agencies and industry. In addition, he has been a consultant to several industries in various capacities. At Michigan, he has served as the Aerospace Engineering Department Graduate Program Chair (1998–2002) and the Associate Chairperson of the Department (2003–2005). He is currently a member of the Executive Committee of the College of Engineering. He is author or co-author of more than 175 refereed journal papers, and numerous conference papers and presentations.

Preface

This textbook is intended to be an introductory text on the mechanics of solids. The authors have targeted an audience that usually would go on to obtain undergraduate degrees in aerospace and mechanical engineering. As such, some specialized topics that are of importance to aerospace engineers are given more coverage. The material presented assumes only a background in introductory physics and calculus. The presentation departs from standard practice in a fundamental way. Most introductory texts on this subject take an approach not unlike that adopted by Timoshenko, in his 1930 Strength of Materials books, that is, by primarily formulating problems in terms of forces. This places an emphasis on statically determinate solid bodies, that is, those bodies for which the restraint forces and moments, and internal forces and moments, can be determined completely by the equations of static equilibrium. Displacements are then introduced in a specialized way, often only at a point, when necessary to solve the few statically indeterminate problems that are included. Only late in these texts are distributed displacements even mentioned. Here, we introduce and formulate the equations in terms of distributed displacements from the beginning. The question of whether the problems are statically determinate or indeterminate becomes less important. It will appear to some that more time is spent on the slender bar with axial loads than that particular structure deserves. The reason is that classical methods of solving the differential equations and the connection to the rational development of the finite element method can be easily shown with a minimum of explanation using the axially loaded slender bar. Subsequently, the development and solution of the equations for more advanced structures is facilitated in later chapters.

Modern advanced analysis of the integrity of solid bodies under external loads is largely displacement based. Once displacements are known the strains, stresses, strain energies, and restraint reactions are easily found. Modern analysis solutions methods also are largely carried out using a computer. The direction of this presentation is first to provide an understanding of the behavior of solid bodies under load and second to prepare the student for modern advanced courses in which computer based methods are the norm.

Analysis of Structures: An Introduction Including Numerical Methods is accompanied by a website (www.wiley.com/go/waas) housing exercises and examples that use modern software which generates color contour plots of deformation and internal stress. It offers invaluable guidance and understanding to senior level and graduate students studying courses in stress and deformation analysis as part of aerospace, mechanical and civil engineering degrees as well as to practicing engineers who want to re-train or re-engineer their set of analysis tools for contemporary stress and deformation analysis of solids and structures.

We are grateful to Dianyun Zhang, Ph.D candidate in Aerospace Engineering, for her careful reading of the examples presented.

Corrections, comments, and criticisms are welcomed.

Joe G. Eisley Anthony M. WaasJune 2011Ann Arbor, Michigan

2

Static Equilibrium

2.1 Introduction

While all solid bodies deform to some extent under loads these deformations are often so small that the bodies may be considered to be rigid for certain purposes. A rigid body is an idealization that considers a body to have no deformation when subject to loads. When all forces, both known applied forces and unknown restraint forces, are identified and we attempt to sum the forces and moments we can have two possible situations:

1. The number of unknown forces is less than the number of independent equations of motion. Rigid body motion may result. This is the subject of rigid body dynamics. We shall not study this case here.

2. The number of unknown forces is equal to or greater than the number of independent equations of motion. There is no rigid body motion and we have static equilibrium. There are two subcases:

a. The body is statically determinate when the number of unknown forces is equal to the number of independent equations of static equilibrium. All the unknown forces can be determined by the summation of forces and moments without regard to what the body is made of or to its deformation.

b. The body is statically indeterminate when the number of unknown forces is greater than the number of independent equations of static equilibrium. The summation of forces and moments is not sufficient to find the unknown forces. We need to consider the deformation of the body and the physical properties that contribute to resisting deformation.

In the case of a statically indeterminate body we must introduce additional equations to obtain a solution. In this chapter we shall be interested primarily in statically determinate bodies. The subsequent chapters will contend with both statically determinate and statically indeterminate problems.

2.2 Free Body Diagrams

We shall illustrate each subcase with a simple example in two dimensions using concentrated forces only. In these examples it is assumed that the applied forces initially are known and the restraint forces are unknown.

Consider the rigid body shown in Figure 2.2.1. We introduce some common symbols to illustrate forces and restraints. For now we consider only concentrated loads and, in this case, assume they act entirely in the xy plane. We add concentrated restraints. The restraint on the lower left prevents movement in both the x and y directions but does allow rotation about the z axis. The restraint on the lower right prevents movement only in the y direction, since it is assumed to roll freely (without friction) in the x direction. Since it is assumed that there are no forces in the z direction there is no need to consider restraints in the z direction.

Figure 2.2.1

By replacing the restraint symbols in Figure 2.2.1 with visual representations of the characteristics to which we have assigned them, we create what is called a free body diagram. We draw the free body in Figure 2.2.2 showing all the applied forces and restraint forces. Unless otherwise stated we shall use the symbol F for concentrated applied forces and R for concentrated restraint forces. Appropriate subscripts will be added to identify individual forces and components. It is common to represent both the applied forces and the restraint forces in terms of their components in the appropriate coordinate directions. For the applied forces these components are

(2.2.1)

Figure 2.2.2

According to standard sign convention, restraint force components generally are drawn directed along the positive x and y directions, as defined by the coordinate system. If a force component acts in the direction it is drawn, it has a positive value. If it acts in the opposite direction, it has a negative value.

We have static equilibrium between the known applied forces and the unknown restraint forces. There are four known applied force components (F1x, F1y, F2x, and F2y), three unknown restraint force components (RAx, RAy, and RBy) and three equations of static equilibrium (summation of forces in the x and y directions and summation of moments about the z axis). The summation of moments about the z axis may be taken about any point in the xy plane. In the following equations the summation of moments is taken about point A.

(2.2.2)

In this case, the unknown restraint forces can be found by solving the equations of static equilibrium. Bodies that satisfy this condition are called statically determinate. These bodies will play a prominent role in subsequent chapters.

The statically determinate case is considered in some detail in this text. Many structures are naturally, or are purposely made to be, statically determinate; however, that will not be the case for many of our problems in later chapters. Indeterminate, also called redundant, structures have important positive characteristics and are widely used.

Consider the body in Figure 2.2.3. Suppose we remove the rollers from the bottom right restraint and add a third restraint at the top of the body as shown.

Figure 2.2.3

The free body diagram is shown in Figure 2.2.4.

Figure 2.2.4

Just by removing the rollers on the bottom right support we add a fourth restraint force component RAx. This alone would make the number of unknowns greater than the number of equations of static equilibrium. Add the third support and we then have six unknown restraint force components. We can write down the three equations of static equilibrium and solve them for three of the unknowns in terms of the remaining three unknowns, but, we shall need additional information in order to find all the restraint forces. You shall soon see that, for a deformable body, equations based on distributed displacements and material properties will supply the additional necessary information. Distributed displacements of the deformed body will play a prominent role in subsequent chapters.

The focus of most of this text will be the subject of statics. The bodies that we study will be in static equilibrium. Accordingly, all forces and all moments acting on those bodies will always sum to zero. In dynamics, on the other hand, bodies often experience rigid body motion and/or nonzero net forces. Some dynamic considerations are introduced in Chapter 14.

So far our examples have been two dimensional. In three dimensions, when presented in rectangular Cartesian coordinates, the equations of statics are

(2.2.3)