ADVANCED MECHANICS OF MATERIALS

Transcription

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2 SIXTH EDITION ADVANCED MECHANICS OF MATERIALS ARTHUR P. BORES1 Professor Emeritus Civil and Architectural Engineering The University of Wyoming at Laramie and Professor Emeritus Theoretical and Applied Mechanics University of Illinois at Urbana-Champaign RICHARD J. SCHMIDT Professor Civil and Architectural Engineering The University of Wyoming at Laramie JOHN WILEY & SONS, INC.

3 Acquisitions Editor Joseph Hayton Marketing Manager Katherine Hepburn Senior Production Editor Valerie A. Vargas Senior Designer Karin Kincheloe Production Management Services Argosy This book was set in 10/12 Times Roman by Argosy and printed and bound by Hamilton Printing. The cover was printed by Lehigh Press. This book is printed on acid-free paper. 00 Copyright John Wiley & Sons, Inc. 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, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , PERMREQ@WILEY.COM. To order books please call 1 (800) Library of Congress Cataloging in Publication Data: Boresi, Arthur P. (Arthur Peter), Advanced mechanics of materials /Arthur P. Boresi, Richard J. Schmidt. 4th ed. p. cm. Includes bibliographical references and index. ISBN (cloth : alk. paper) 1. Strength of materials. I. Schmidt, Richard J. (Richard Joseph), Title. TA405.B '12--d~ ISBN Printed in the United States of America

4 PREFACE This book is written as a text for advanced undergraduates and graduate students in aerospace, civil, and mechanical engineering and applied mechanics. It is also intended as a reference for practitioners. The book contains topics sufficient for two academic semesters or three quarters. Thus, there is enough variety that instructors of a one-semester course or one- or two-quarter courses can choose topics of interest to students. New to this Edition In this sixth edition, we have attempted to thoroughly review the fifth edition with the intention of clarifying and condensing the presentation, updating many of the examples and homework problems, and adding selected new topics. In the face of these additions, we have also attempted to control the growth in size of the book. Such is a difficult task. Based on a survey of the market, the final chapter, finite element methods, has been removed from the book and is available at no cost from the book Web site: Those topics that we have retained are presented with detail and precision, as in previous editions of the book. Throughout the revision pro-cess, the philosophy of previous editions has been maintained. That is, we have attempted to develop the topics from basic principles so that the applicability and limitations of the methods are clear. Without an understanding of the underlying principles and assumptions upon which analysis methods are based, users of those methods may be limited to applica-tion of the methods to known problems. Furthermore, they may not have the necessary understanding to extend or adapt the theories and developments to their own applications. Hence, we regard concepts and fundamentals as no less important than application of solu-tion methods in problem solving. Organization In Chapter 1 basic concepts of one-dimensional load-stress, load-deflec-tion, and stress-strain diagrams are introduced. A discussion of the tension test and associ-ated material properties is presented, followed by a brief introduction to failure theories. Theories of stress and strain follow in Chapter 2. Definitions of the stress tensor and vari-ous stress quantities are developed from detailed examination of equilibrium conditions for a body. Likewise, definitions of strain are developed from a consideration of deforma-tion of a body. Here, the independence and similarity of the theories of stress and strain become evident. Chapter 3 joins the theories of stress and strain by the theory of linear stress-strain-temperature relations, based upon the requirements of the first law of ther-modynamics. Stress-strain relations and material constants for anisotropic, orthotropic, and isotropic materials are discussed. Yield theory is developed in Chapter 4. Starting with one-dimensional stress-strain behavior, the concepts of yield criteria, yield functions, and yield surfaces are developed to describe nonlinear material response for multiaxial stress states. The von Mises and Tresca criteria are discussed and compared in detail. The appli-cation of energy methods, Chapter 5, includes a discussion of the dummy load method and its relation to the Castigliano method. vii

5 PREFACE Chapters 6-12 treat classical topics in mechanics of materials. That is, these chapters use fundamental concepts of equilibrium, compatibility conditions, constitutive relations, and material response to study the behavior of selected mechanical and structural members. Specifically, the following topics are considered: torsion, nonsymmetrical bending, shear center, curved beams, beams on elastic foundations, thick-wall cylinders, and column stability. Key kinematic and material response assumptions are emphasized in order to highlight the applicability and limitations of the analysis methods. Chapters contain selected topics that are not generally treated by the mechanics of materials method, but are nevertheless areas of interest and advanced study for practicing engineers. This part of the book contains a mix of topics involving both behavior of structural and mechanical systems (flat plates, stress concentrations, and contact stresses) as well as detailed study of material behavior (fracture mechanics and fatigue). Acknowledgments We thank Joe Hayton, Engineering Editor at Wiley, for his help and advice during the development of this edition. In addition, we also appreciate the help of Valerie Vargas, Production Editor, Adriana Lavergne at Argosy for work on composition, and David Wood at Wellington Studios for work on the illustration program. We acknowl-edge the help of the following reviewers, who offered useful insights into developing this edition: Abhijit Bhattacharyya, University of Alberta Bill Y. J. Chao, University of South Carolina Ali Fatemi, University of Toledo Stephen Folkman, Utah State University Stephen M. Heinrich, Marquette University Thomas Lacy, Wichita State University Craig C. Menzemer, University of Akron James A. Nemes, McGill University Steven O Hara, Oklahoma State University Pizhong Qiao, University of Akron Robert Yuan, University of Texas-Arlington Supplements Instructors who adopt the book for their courses may access solutions to the homework problems from the John Wiley & Sons Web site: collegelboresi.contact your local sales representative for additional details. Finally, we welcome comments, suggestions, questions, and corrections that you might wish to offer. Send your remarks to Dr. Arthur P. Boresi, Department of Civil and Archi-tectural Engineering, University of Wyoming, Laramie, WY

6 ~~~~~~ CONTENTS CHAPTER 1 INTRODUCTION Review of Elementary Mechanics of Materials Axially Loaded Members Torsionally Loaded Members Bending of Beams Methods of Analysis Method of Mechanics of Materials Method of Continuum Mechanics and the Theory of Elasticity Deflections by Energy Methods Stress-Strain Relations Elastic and Inelastic Response of a Solid Material Properties Failure and Limits on Design Modes of Failure 19 Problems 22 References 24 CHAPTER 2 THEORIES OF STRESS AND STRAIN Definition of Stress at a Point Stress Notation Symmetry of the S t r e s s h a y and Stress on an Arbitrarily Oriented Plane Symmetry of Stress Components Stresses Acting on Arbitrary Planes Normal Stress and Shear Stress on an Oblique Plane Transformation of Stress, Principal Stresses, and Other Properties Transformation of Stress Principal Stresses Principal Values and Directions Octahedral Stress Mean and Deviator Stresses Plane Stress Mohr s Circle in Two Dimensions Mohr s Circles in Three Dimensions Differential Equations of Motion of a Deformable Body Specialization of Equations Deformation of a Deformable Body Strain Theory, Transformation of Strain, and Principal Strains Strain of a Line Element Final Direction of a Line Element Rotation Between Two Line Elements (Definition of Shear Strain) Principal Strains Small-DisplacementTheory Strain Compatibility Relations Strain-Displacement Relations for Orthogonal Curvilinear Coordinates Strain Measurement and Strain Rosettes 70 Problems 72 References 78 CHAPTER 3 LINEAR STRESS-STRAIN- TEMPERATURE RELATIONS First Law of Thermodynamics, Internal-Energy Density, and Complementary Internal-Energy Density Elasticity and Internal-Energy Density Elasticity and Complementary Internal-Energy Density Hooke s Law: Anisotropic Elasticity Hooke s Law: Isotropic Elasticity Isotropic and Homogeneous Materials Strain-Energy Density of Isotropic Elastic Materials Equations of Thennoelasticity for Isotropic Materials Hooke s Law: Orthotropic Materials 93 Problems 101 References 103 CHAPTER 4 INELASTIC MATERIAL. BEHAVIOR Limitations on the Use of Uniaxial Stress-Strain Data Rate of Loading Temperature Lower Than Room Temperature Temperature Higher Than Room Temperature 105 ix

7 CONTENTS Unloading and Load Reversal Multiaxial States of Stress Nonlinear Material Response Models of Uniaxial StressStrain Curves Yield Criteria: General Concepts Maximum Principal Stress Criterion Maximum Principal Strain Criterion Strain-Energy Density Criterion Yielding of Ductile Metals Maximum Shear-Stress (Tresca) Criterion Distortional Energy Density (von Mises) Criterion Effect of Hydrostatic Stress and the z-plane Alternative Yield Criteria Mohr-Coulomb Yield Criterion Drucker-Prager Yield Criterion Hill s Criterion for Orthotropic Materials General Yielding Elastic-Plastic Bending Fully Plastic Moment Shear Effect on Inelastic Bending Modulus of Rupture Comparison of Failure Criteria Interpretation of Failure Criteria for General Yielding 137 Problems 142 References 146 CHAPTER 5 APPLICATIONS OF ENERGY METHODS Principle of Stationary Potential Energy Castigliano s Theorem on Deflections Castigliano stheorem on Deflections for Linear Load-Deflection Relations Strain Energy U, for Axial Loading Strain Energies U, and Us for Beams Strain Energy U, for Torsion Deflections of Statically Determinate Structures Curved Beams Treated as Straight Beams Dummy Load Method and Dummy Unit Load Method Statically Indeterminate Structures Deflections of Statically Indeterminate Structures 180 Problems 187 References 199 CHAPTER 6 TORSION Torsion of a Prismatic Bar of Circular Cross Section Design of Transmission Shafts Saint-Venant s Semiinverse Method Geometry of Deformation Stresses at a Point and Equations of Equilibrium Boundary Conditions Linear Elastic Solution Elliptical Cross Section Equilateral Triangle Cross Section Other Cross Sections 216 The Prandtl Elastic-Membrane (Soap-Film)Analogy Remark on Reentrant Corners Narrow Rectangular Cross Section Cross Sections Made Up of Long Narrow Rectangles Torsion of Rectangular Cross Section Members Hollow Thin-Wall Torsion Members and Multiply Connected Cross Sections Hollow Thin-Wall Torsion Member Having Several Compartments Thin-Wall Torsion Members with Restrained Ends I-Section Torsion Member Having One End Restrained from Warping 235 Various Loads and Supports for Beams in Torsion Numerical Solution of the Torsion Problem Inelastic Torsion: Circular Cross Sections Modulus of Rupture in Torsion Elastic-Plastic and Fully Plastic Torsion Residual Shear Stress Fully Plastic Torsion: General Cross Sections 250 Problems 254 References 262 CHAPTER 7 BENDING OF STRAIGHT BEAMS Fundamentals of Beam Bending Centroidal Coordinate Axes Shear Loading of a Beam and Shear Center Defined Symmetrical Bending Nonsymmetrical Bending Plane of Loads: Symmetrical and Nonsymmetrical Loading Bending Stresses in Beams Subjected to Nonsymmetrical Bending Equations of Equilibrium Geometry of Deformation StressStrain Relations Load-Stress Relation for Nonsymmetrical Bending Neutral Axis More Convenient Form for the Flexure Stress O,, Deflections of Straight Beams Subjected to Nonsymmetrical Bending 280

8 CONTENTS xi 7.4 Effect of Inclined Loads Fully Plastic Load for Nonsymmetrical Bending 285 Problems 287 References 294 CHAPTER 8SHEAR CENTER FOR THIN-WALL BEAM CROSS SECTIONS Approximations for Shear in Thin-Wall Beam Cross Sections Shear Flow in Thin-Wall Beam Cross Sections Shear Center for a Channel Section Shear Center of Composite Beams Formed from Stringers and Thin Webs Shear Center of Box Beams 306 Problems 312 References 318 CHAPTER 9 CURVED BEAMS Introduction Circumferential Stresses in a Curved Beam Location of Neutral Axis of Cross Section Radial Stresses in Curved Beams Curved Beams Made from Anisotropic Materials Correction of Circumferential Stresses in Curved Beams Having I, T, or Similar Cross Sections Bleich's Correction Factors Deflections of Curved Beams Cross Sections in the Form of an I, T, etc Statically Indeterminate Curved Beams: Closed Ring Subjected to a Concentrated Load Fully Plastic Loads for Curved Beams Fully Plastic Versus Maximum Elastic Loads for Curved Beams 351 Problems 352 References 356 CHAPTER 10 BEAMS ON ELASTIC FOUNDATIONS General Theory Infinite Beam Subiected to a Concentrated Load: Boundary Conditions Method of Superposition Beam Supported on Equally Spaced Discrete Elastic Supports Infinite Beam Subjected to a Distributed Load Segment Uniformly Distributed Load P L ' I z PL'+ m Intermediate Values of PL' Triangular Load Semiinfinite Beam Subjected to Loads at Its End Semiinfinite Beam with Concentrated Load Near Its End ShortBeams Thin-Wall Circular Cylinders 378 Problems 384 References 388 CHAPTER 1'1 THE THICK-WALL CYLINDER Basic Relations Equation of Equilibrium Strain-Displacement Relations and Compatibility Condition Stress-Strain-Temperature Relations Material Response Data Stress Components at Sections Far from Ends for a Cylinder with Closed Ends Open Cylinder Stress Components and Radial Displacement for Constant Temperature Stress Components Radial Displacement for a Closed Cylinder Radial Displacement for an Open Cylinder Criteria of Failure Failure of Brittle Materials Failure of Ductile Materials Material Response Data for Design Ideal Residual Stress Distributions for Composite Open Cylinders Fully Plastic Pressure and Autofrettage Cylinder Solution for Temperature Change Only Steady-State Temperature Change (Distribution) Stress Components Rotating Disks of Constant Thickness 411 Problems 419 References 422 CHAPTER 12 ELASTIC AND INELASTIC STABILITY OF COLUMNS Introduction to the Concept of Column Buckling Deflection Response of Columns to Compressive Loads Elastic Buckling of an Ideal Slender Column Imperfect Slender Columns The Euler Formula for Columns with Pinned Ends The Equilibrium Method Higher Buckling Loads; n > The Imperfection Method The Energy Method 433

9 CONTENTS 12.4 Euler Buckling of Columns with Linearly Elastic End Constraints Local Buckling of Columns Inelastic Buckling of Columns Inelastic Buckling Two Formulas for Inelastic Buckling of an Ideal Column Tangent-Modulus Formula for an Inelastic Buckling Load Direct Tangent-Modulus Method 446 Problems 450 References 455 CHAPTER 13 FLAT PLATES Introduction Stress Resultants in a Flat Plate Kinematics: Strain-Displacement Relations for Plates Rotation of a Plate Surface Element Equilibrium Equations for Small-Displacement Theory of Flat Plates Stress-Strain-Temperature Relations for Isotropic Elastic Plates Stress Components in Terms of Tractions and Moments Pure Bending of Plates Strain Energy of a Plate Boundary Conditions for Plates Solution of Rectangular Plate Problems Solution of vzvzw= g for a Rectangular Plate 477 D Westergaard Approximate Solution for Rectangular Plates: Uniform Load Deflection of a Rectangular Plate: Uniformly Distributed Load Solution of Circular Plate Problems Solution of v 2 v 2 W = g for a Circular Plate 486 D Circular Plates with Simply Supported Edges Circular Plates with Fixed Edges Circular Plate with a Circular Hole at the Center Summary for Circular Plates with Simply Supported Edges Summary for Circular Plates with Fixed Edges Summary for Stresses and Deflections in Flat Circular Plates with Central Holes Summary for Large Elastic Deflections of Circular Plates: Clamped Edge and Uniformly Distributed Load Significant Stress When Edges Are Clamped Load on a Plate When Edges Are Clamped Summary for Large Elastic Deflections of Circular Plates: Simply Supported Edge and Uniformly Distributed Load Rectangular or Other Shaped Plates with Large Deflections 498 Problems 500 References 501 CHAPTER 14 STRESS CONCENTRATIONS Nature of a Stress Concentration Problem and the Stress Concentration Factor Stress ConcentrationFactors: Theory of Elasticity Circular Hole in an Infinite Plate Under Uniaxial Tension Elliptic Hole in an Infinite Plate Stressed in a Direction Perpendicular to the Major Axis of the Hole Elliptical Hole in an Infinite Plate Stressed in the Direction Perpendicular to the Minor Axis of the Hole Crack in a Plate Ellipsoidal Cavity Grooves and Holes Stress Concentration Factors: Combined Loads Infinite Plate with a Circular Hole Elliptical Hole in an Infinite Plate Uniformly Stressed in Directions of Major and Minor Axes of the Hole Pure Shear Parallel to Major and Minor Axes of the Elliptical Hole Elliptical Hole in an Infinite Plate with Different Loads in Two Perpendicular Directions 517 Stress Concentration at a Groove in a Circular Shaft Stress Concentration Factors: Experimental Techniques Photoelastic Method Strain-Gage Method Elastic Torsional Stress Concentration at a Fillet in a Shaft 525 Elastic Membrane Method: Torsional Stress concentration Beams with Rectangular Cross Sections Effective Stress Concentration Factors Definition of Effective Stress Concentration Factor Static Loads Repeated Loads 532

10 CONTENTS xiii Residual Stresses Very Abrupt Changes in Section: Stress Gradient Significance of Stress Gradient Impact or Energy Loading Effective Stress Concentration Factors: Inelastic Strains Neuber s Theorem 537 Problems 539 References 541 CHAPTER 15 FRACTURE MECHANICS Failure Criteria and Fracture Brittle Fracture of Members Free of Cracks andflaws Brittle Fracture of Cracked or Flawed Members The Stationary Crack Blunt Crack Sharpcrack Crack Propagation and the Stress Intensity Factor Elastic Stress at the Tip of a Sharp Crack Stress Intensity Factor: Definition and Derivation Derivation of Crack Extension Force G Critical Value of Crack Extension Force Fracture: Other Factors Elastic-Plastic Fracture Mechanics Crack-Growth Analysis Load Spectra and Stress History Testing and Experimental Data Interpretation 563 Problems 564 References 565 CHAPTER 16 FATIGUE: PROGRESSNE FRACTURE Fracture Resulting from Cyclic Loading Stress Concentrations Effective Stress Concentration Factors: Repeated Loads Effective Stress Concentration Factors: Other Influences Corrosion Fatigue Effect of Range of Stress Methods of Reducing Harmful Effects of Stress Concentrations Low Cycle Fatigue and the E-N Relation Hysteresis Loop 580 XiV CONTENTS Fatigue-Life Curve and the E-N Relation 581 Problems 585 References 588 CHAPTER 17 CONTACT STRESSES Introduction The Problem of Determining Contact Stresses Geometry of the Contact Surface Fundamental Assumptions Contact Surface Shape After Loading Justification of Eq Brief Discussion of the Solution Notation and Meaning of Terms Expressions for Principal Stresses Method of Computing Contact Stresses Principal Stresses Maximum Shear Stress Maximum Octahedral Shear Stress Maximum Orthogonal Shear Stress Curves for Computing Stresses for Any Value of BIA Deflection of Bodies in Point Contact Significance of Stresses Stress for Two Bodies in Line Contact: Loads Normal to Contact Area Maximum Principal Stresses: k = Maximum Shear Stress: k = Maximum Octahedral Shear Stress: k = O Stresses for Two Bodies in Line Contact: Loads Normal and Tangent to Contact Area Roller on Plane Principal Stresses Maximum Shear Stress Maximum Octahedral Shear Stress Effect of Magnitude of Friction Coefficient Range of Shear Stress for One Load Cycle 619 Problems 622 References 623 CHAPTER 18 CREEP: TIME-DEPENDENT DEFORMATION Definition of Creep and the Creep Curve The Tension Creep Test for Metals One-DimensionalCreep Formulas for Metals Subjected to Constant Stress and Elevated Temperature 626

11 18.4 One-Dimensional Creep of Metals Subjected to Variable Stress and Temperature Preliminary Concepts Similarity of Creep Curves Temperature Dependency Variable Stress and Temperature Creep Under Multiaxial States of Stress General Discussion Flow Rule for Creep of Metals Subjected to Multiaxial States of stress Steady-State Creep Nonsteady Creep An Application of Creep of Metals Summary Creep of Nonmetals Asphalt Concrete Wood References This page has been reformatted by Knovel to provide easier navigation.