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CONVERSIONS BETWEEN U.S. CUSTOMARY UNITS AND SI UNITS Times conversion factor U.S. Customary unit Equals SI unit Accurate Practical Acceleration (linear) 2 2 foot per second squared ft/s 0.3048* 0.305 meter per second squared m/s inch per second squared in./s 0.0254* 0.0254 meter per second squared m/s2 Area square foot ft 0.09290304* 0.0929 square meter m 2 2 2 square inch in. 645.16* 645 square millimeter mm Density (mass) slug per cubic foot slug/ft 515.379 515 kilogram per cubic meter kg/m 3 Density (weight) pound per cubic foot lb/ft 157.087 157 newton per cubic meter N/m 3 3 pound per cubic inch lb/in. 271.447 271 kilonewton per cubic meter kN/m 3 Energy; work foot-pound ft-lb 1.35582 1.36 joule (N▯m) J inch-pound in.-lb 0.112985 0.113 joule J kilowatt-hour kWh 3.6* 3.6 megajoule MJ British thermal unit Btu 1055.06 1055 joule J Force 2 pound lb 4.44822 4.45 newton (kg▯m/s ) N kip (1000 pounds) k 4.44822 4.45 kilonewton kN Force per unit length pound per foot lb/ft 14.5939 14.6 newton per meter N/m pound per inch lb/in. 175.127 175 newton per meter N/m kip per foot k/ft 14.5939 14.6 kilonewton per meter kN/m kip per inch k/in. 175.127 175 kilonewton per meter kN/m Length foot ft 0.3048* 0.305 meter m inch in. 25.4* 25.4 millimeter mm mile mi 1.609344* 1.61 kilometer km Mass 2 slug lb-s /ft 14.5939 14.6 kilogram kg Moment of a force; torque pound-foot lb-ft 1.35582 1.36 newton meter N·m pound-inch lb-in. 0.112985 0.113 newton meter N·m kip-foot k-ft 1.35582 1.36 kilonewton meter kN·m kip-inch k-in. 0.112985 0.113 kilonewton meter kN·m CONVERSIONS BETWEEN U.S. CUSTOMARY UNITS AND SI UNITS (Continued) Times conversion factor U.S. Customary unit Equals SI unit Accurate Practical Moment of inertia (area) 4 inch to fourth power in. 416,231 416,000 millimeter to fourth power mm 4 4 ▯6 ▯6 4 inch to fourth power in. 0.416231 ▯ 10 0.416 ▯ 10 meter to fourth power m Moment of inertia (mass) 2 2 slug foot squared slug-ft 1.35582 1.36 kilogram meter squared kg·m Power foot-pound per second ft-lb/s 1.35582 1.36 watt (J/s or N·m/s) W foot-pound per minute ft-lb/min 0.0225970 0.0226 watt W horsepower (550 ft-lb/s) hp 745.701 746 watt W Pressure; stress pound per square foot psf 47.8803 47.9 pascal (N/m2) P pound per square inch psi 6894.76 6890 pascal Pa kip per square foot ksf 47.8803 47.9 kilopascal kPa kip per square inch ksi 6.89476 6.89 megapascal MPa Section modulus 3 3 inch to third power in. 16,387.1 16,400 millimeter to third power mm inch to third power in.3 16.3871 ▯ 10 ▯6 16.4 ▯ 10▯6 meter to third power m 3 Velocity (linear) foot per second ft/s 0.3048* 0.305 meter per second m/s inch per second in./s 0.0254* 0.0254 meter per second m/s mile per hour mph 0.44704* 0.447 meter per second m/s mile per hour mph 1.609344* 1.61 kilometer per hour km/h Volume cubic foot ft 0.0283168 0.0283 cubic meter m 3 3 ▯6 ▯6 3 cubic inch in3 16.3871 ▯ 10 16.4 ▯ 10 cubic meter m 3 cubic inch in. 16.3871 16.4 cubic centimeter (cc) cm gallon (231 in. ) gal. 3.78541 3.79 liter L 3 3 gallon (231 in. ) gal. 0.00378541 0.00379 cubic meter m *An asterisk denotes an exact conversion factor Note: To convert from SI units to USCS units, divide by the conversion factor Temperature Conversion Formulas T(°C) ▯ ▯[T(°F) ▯ 32] ▯ T(K) ▯ 273.15 9 5 T(K) ▯ ▯▯[T(°F) ▯ 32] ▯ 273.15 ▯ T(°C) ▯ 273.15 9 9 9 T(°F) ▯ 5T(°C) ▯ 32 ▯ 5T(K) ▯ 459.67 Mechanics of Materials SEVENTH EDITION James M. Gere Professor Emeritus, Stanford University Barry J. Goodno Georgia Institute of Technology Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Mechanics of Materials, Seventh Edition © 2009 Cengage Learning James M. Gere and Barry J. Goodno ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored or used in any form or by any means graphic, electronic, Director, Global Engineering Program: Chris Carson Senior Developmental Editor: Hilda Gowans or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval Permissions: Nicola Winstanley systems, except as permitted under Section 107 or 108 of the 1976 United States Production Service: RPK Editorial Services, Inc Copyright Act, without the prior written permission of the publisher. Copy Editor: Rose Kernan For product information and technology assistance, contact us at Proofreader: Martha McMaster Cengage Learning Customer & Sales Support, 1-800-354-9706 Indexer: Shelly Gerger-Knechtl For permission to use material from this text or product, submit all requests online at Creative Director: Angela Cluer cengage.com/permissions Text Designer: Peter Papayanakis Further permissions questions can be emailed to permissionrequest@cengage.com Cover Designer: Andrew Adams Cover Image: Guggenheim Bilbao Photo © graficart.net/Alamy; Library of Congress Control Number: 2008923451 Lee Chin Crystal Building Photo Courtesy of Alan ISBN-13: 978-0-534-55397-5 Marsh/First Light Compositor: Integra ISBN-10: 0-534-55397-4 Printer: Quebecor World Cengage Learning 1120 Birchmount Road Toronto ON M1K 5G4 Canada Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education Ltd. For your course and learning solutions, visit academic.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States ofAmerica 1 2 3 4 5 6 7 11 10 09 08 Contents James Monroe Gere ix Photo Credits x Preface xi Symbols xv Greek Alphabet xviii 1 Tension, Compression, and Shear 2 1.1 Introduction to Mechanics of Materials 5 1.2 Normal Stress and Strain 7 1.3 Mechanical Properties of Materials 15 1.4 Elasticity, Plasticity, and Cree24 1.5 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio27 1.6 Shear Stress and Strain 32 1.7 Allowable Stresses and Allowable Loads 43 1.8 Design for Axial Loads and Direct Shear 49 Chapter Summary & Review 55 Problems 57 2 Axially Loaded Members 88 2.1 Introduction 91 2.2 Changes in Lengths of Axially Loaded Members 91 2.3 Changes in Lengths Under Nonuniform Conditions 100 2.4 Statically Indeterminate Structures107 2.5 Thermal Effects, Misfits, and Prestrains116 2.6 Stresses on Inclined Sections 128 2.7 Strain Energy 140 ★ 2.8 Impact Loading 153 ★2.9 Repeated Loading and Fatigue 162 ★2.10 Stress Concentrations 164 ★2.11 Nonlinear Behavior 170 ★Specialized and/or advanced topics iii iv CONTENTS ★2.12 Elastoplastic Analysis 175 Chapter Summary & Review 181 Problems 182 3 Torsion 220 3.1 Introduction 222 3.2 Torsional Deformations of a Circular Bar 223 3.3 Circular Bars of Linearly Elastic Material226 3.4 Nonuniform Torsion 238 3.5 Stresses and Strains in Pure Shear245 3.6 Relationship Between Moduli of Elasticity E and G 252 3.7 Transmission of Power by Circular Shafts 254 3.8 Statically Indeterminate Torsional Members 259 3.9 Strain Energy in Torsion and Pure Shear 263 3.10 Thin-Walled Tubes 270 ★3.11 Stress Concentrations in Torsion 279 Chapter Summary & Review 282 Problems 283 4 Shear Forces and Bending Moments 304 4.1 Introduction 306 4.2 Types of Beams, Loads, and Reactions 306 4.3 Shear Forces and Bending Moments 313 4.4 Relationships Between Loads, Shear Forces, and Bending Moments 320 4.5 Shear-Force and Bending-Moment Diagrams 325 Chapter Summary & Review 337 Problems 338 5 Stresses in Beams (Basic Topics) 350 5.1 Introduction 353 5.2 Pure Bending and Nonuniform Bending 353 5.3 Curvature of a Beam 354 5.4 Longitudinal Strains in Beams 356 5.5 Normal Stresses in Beams (Linearly Elastic Materials) 361 5.6 Design of Beams for Bending Stresses 374 5.7 Nonprismatic Beams 383 5.8 Shear Stresses in Beams of Rectangular Cross Section 387 5.9 Shear Stresses in Beams of Circular Cross Section 397 5.10 Shear Stresses in the Webs of Beams with Flanges 400 CONTENTS v ★★5.11 Built-Up Beams and Shear Flow 408 ★★ 5.12 Beams with Axial Loads 412 ★★5.13 Stress Concentrations in Bending 418 Chapter Summary & Review 421 Problems 424 6 Stresses in Beams (Advanced Topics) 454 6.1 Introduction 457 6.2 Composite Beams 457 6.3 Transformed-Section Method 466 6.4 Doubly Symmetric Beams with Inclined Loads 472 6.5 Bending of Unsymmetric Beams 479 6.6 The Shear-Center Concept 487 6.7 Shear Stresses in Beams of Thin-Walled Open Cross Sections 489 6.8 Shear Stresses in Wide-Flange Beams 492 6.9 Shear Centers of Thin-Walled Open Sections 496 ★★6.10 Elastoplastic Bending 504 Chapter Summary & Review 514 Problems 516 7 Analysis of Stress and Strain 536 7.1 Introduction 539 7.2 Plane Stress 540 7.3 Principal Stresses and Maximum Shear Stresses 548 7.4 Mohr’s Circle for Plane Stress558 7.5 Hooke’s Law for Plane Stress 575 7.6 Triaxial Stress 580 7.7 Plane Strain 584 Chapter Summary & Review 600 Problems 602 8 Applications of Plane Stress (Pressure Vessels, Beams, and Combined Loadings) 618 8.1 Introduction 621 8.2 Spherical Pressure Vessels 621 8.3 Cylindrical Pressure Vessels627 8.4 Maximum Stresses in Beams 635 8.5 Combined Loadings 645 Chapter Summary & Review 661 Problems 663 ★★ Advanced topics vi CONTENTS 9 Deflections of Beams 676 9.1 Introduction 679 9.2 Differential Equations of the Deflection Curve679 9.3 Deflections by Integration of the Bending-Moment Equation 685 9.4 Deflections by Integration of the Shear-Force and Load Equations 696 9.5 Method of Superposition 702 9.6 Moment-Area Method 711 9.7 Nonprismatic Beams 720 9.8 Strain Energy of Bending 725 ★★ 9.9 Castigliano’s Theorem 731 ★★ 9.10 Deflections Produced by Impact 744 ★★ 9.11 Temperature Effects 746 Chapter Summary & Review 749 Problems 751 10 Statically Indeterminate Beams 770 10.1 Introduction 773 10.2 Types of Statically Indeterminate Beams 773 10.3 Analysis by the Differential Equations of the Deflection Curve 777 10.4 Method of Superposition 784 ★★ 10.5 Temperature Effects 797 ★★ 10.6 Longitudinal Displacements at the Ends of a Beam 801 Chapter Summary & Review 805 Problems 806 11 Columns 816 11.1 Introduction 819 11.2 Buckling and Stability 819 11.3 Columns with Pinned Ends 823 11.4 Columns with Other Support Conditions 834 11.5 Columns with Eccentric Axial Loads 845 11.6 The Secant Formula for Columns 850 11.7 Elastic and Inelastic Column Behavior 856 ★★Advanced topics CONTENTS vii 11.8 Inelastic Buckling 858 11.9 Design Formulas for Columns 863 Chapter Summary & Review 882 Problems 883 12 Review of Centroids and Moments of Inertia 900 12.1 Introduction 902 12.2 Centroids of Plane Areas 902 12.3 Centroids of Composite Areas 905 12.4 Moments of Inertia of Plane Areas 909 12.5 Parallel-Axis Theorem for Moments of Inertia 912 12.6 Polar Moments of Inertia 916 12.7 Products of Inertia918 12.8 Rotation of Axes 921 12.9 Principal Axes and Principal Moments of Inertia923 Problems 927 References and Historical Notes 935 Appendix A Systems of Units and Conversion Factors 943 A.1 Systems of Units 943 A.2 SI Units 944 A.3 U.S. Customary Units 951 A.4 Temperature Units 952 A.5 Conversions Between Units 953 Appendix B Problem Solving 956 B.1 Types of Problems 956 B.2 Steps in Solving Problems 957 B.3 Dimensional Homogeneity 958 B.4 Significant Digits 959 B.5 Rounding of Numbers 961 Appendix C Mathematical Formulas 962 Appendix D Properties of Plane Areas 966 Appendix E Properties of Structural-Steel Shapes 972 viii CONTENTS Appendix F Properties of Structural Lumber 983 Appendix G Deflections and Slopes of Beams 984 Appendix H Properties of Materials 990 Answers to Problems 995 Name Index 1016 Subject Index 1017 James Monroe Gere 1925–2008 James Monroe Gere, Professor Emeritus of Civil Engineering at Stanford Un ▯ iversity, died in Portola Valley, CA, on January 30, 2008. Jim Gere was born on June 14, 1925, in Syracus ▯ e, NY. He joined the U.S. Army Air Corps at age 17 in 1942, serving in England, France and Germany. After the war, he earned undergraduate and master’s degrees in Civil Engineering from the Rensselaer Polytechnic Institute in 1949 and 1951, respectively.▯ He worked as an instructor and later as a Research Associate for Rensselaer between 1949 and 1952. He was awarded one of the first NSF Fellowships, and chose to study at Stanford.▯ He received his Ph.D. in 1954 and was offered a faculty position in Civil Engineering, beginning a 34-year career of engaging his students in challenging topics in mechanics, and ▯ tructural and earth- quake engineering. He served as Department Chair andAssociate Dean of Engineering and in 1974 co-founded the JohnA. Blume Earthquake Engineering Center at Stanford. In 1980, Jim Gere also became the founding head of the Stanford Committee on Earthquak ▯ e Preparedness, which urged campus members to brace and strengthen office equipment, furniture, and other contents items that could pose a life safety hazard in the event of a▯ rthquake. That same year, he was invited as one of the first foreigners to study the earthquake-devastated city of Tangshan, China. Jim retired from Stanford in 1988 but con- tinued to be a most valuable member of the Stanford community as he gave▯ freely of his time to advise students and to guide them on various field trips to the California earthquake country. Jim Gere was known for his outgoing manner, his cheerful personality and wonderful smile, his athleticism, and his skill as an educator in Civil Enginee▯ . He authored nine text- books on various engineering subjects starting in 1972 with echanics of Materials, a text that was inspired by his teacher and mentor Stephan P. Timoshenko. His other well-known text- books, used in engineering courses around the world, include: Theory of Elastic Stability, co-authored with S. Timoshenko; Matrix Analysis of Framed Structures and Matrix Algebra for Engineers, both co-authored with W. Weaver; Moment Distribution; Earthquake Tables: Structural and Construction Design Manua,lco-authored with H. Krawinkler; andTerra Non Firma: Understanding and Preparing for Earthquake,sco-authored with H. Shah. Respected and admired by students, faculty, and staff at Stanford University, Professor Gere always felt that the opportunity to work with and be of service to▯ oung people both inside and outside the classroom was one of his great joys. He hiked fr▯ uently and regu- larly visited Yosemite and the Grand Canyon national parks. He made over 20 ascents of Half Dome in Yosemite as well as “John Muir hikes” of up to 50 miles in a day. In 1986 he Jim Gere in the Timoshenko hiked to the base camp of Mount Everest, saving the life of a companion o▯ n the trip. James Library at Stanford holding aas an active runner and completed the Boston Marathon at age 48, in a ti▯ me of 3:13. copy of the 2nd edition of this James Gere will be long remembered by all who knew him as a considerate a ▯ nd loving text (photo courtesy of Richard whose upbeat good humor made aspects of daily life or work easier to▯ bear. His last proj- Weingardt Consultants, Inc.)ect (in progress and now being continued by his daughter Susan of Palo A▯ lto) was a book based on the written memoirs of his great-grandfather, a Colonel (122d NY) in the Civil War. ix Photo Credits Chapter 1. 2: Photo by Bryan Tokarczyk, PE/KPFF Tower Engineers 15: Courtesy of MTS Systems Corporation 16: Courtesy of MTS Systems Corporation 18: Courtesy of MTS Systems Corporation 32: © Barry Goodno 60: © Barry Goodno 66: Vince Streano/Getty Images 67: © Barry Goodno 67: © Barry Goodno 67: © Barry Goodno 68: © Barry Goodno 72: © Barry Goodno 76: Courtesy ofAmeri- can Superconductor83: © Barry Goodno86: Courtesy Tilt-Up Concrete Association. Chapter 2. 88:Joe Raedle/Getty Images93: © Barsik/Dreamstime.com118:Barros & Barros/Getty Images163:Courtesy of MTS systems corporation188:© Barry Goodno Chapter 3. 220: Harald Sund/Getty Images 232: Louie Psihoyos/Getty Images 233: Peter Ginter/Getty Images 290: © Barry Goodno 290: Bontrager Race XXX Lite Flat Handlebar, used Courtesy of Bontrager Chapter 4. 304:© Jupiter Images, 2007307: Joe Gough/Shuttterstock309: Courtesy of the National Information Service for Earthquake Engineering EERC, Un▯ ersity of California, Berkeley.339: Thomasz Gulla/Shuttterstock Chapter 5. 350: Lester Lefkowitz/Getty Images 374: Courtesy of AISC 413: Lester Lefkowitz/Getty Images427:Gabriel M. Covian/Getty Images Chapter 6. 454: Chris Harvey/Shutterstock 479: Franz Pfluegl/Shutterstock 527: © Barry Goodno 527: © Barry Goodno Chapter 7. 536:Alfred Pasieka/PeterArnold, Inc.548: Courtesy Eann Patterson548: Frans Lemmens/Getty Images594: Courtesy Omega Engineering Inc. Chapter 8. 618: Courtesy of Christian Michel, www.modernairships.info 621: Harald Høilan Tjøstheim/Getty Images 627: Wayne Eastep/Getty Images Chapter 9. 676: Courtesy of the National Information Service for Earthquake Engi- neering EERC, University of California, Berkeley. 698: Courtesy of the National Information Service for Earthquake Engineering EERC, University of Cali▯ rnia, Berkeley. 700: Tom Brakefield/Getty Images 709: Courtesy of the National Informa- tion Service for Earthquake Engineering EERC, University of Californi▯ erkeley. 720: Malcolm Fife/Getty Images Chapter 10. 770: © david sanger photography/Alamy 776: Lopatinsky Vladislav/ Shutterstock812: Courtesy of the National Information Service for Earthquake Engi- neering EERC, University of California, Berkeley. Chapter 11. 816:LUSHPIX/UNLISTED IMAGES, INC.833:Lester Lefkowitz/Getty Images834:Digital Vision/Getty Images887:© Barry Goodno Chapter 12. 900:Bob Scott/Getty Images903: Photo courtesy of Louis Geschwinder. 906: Don Farrall/Getty ImagesPreface. Richard Weingardt Consultants Inc. x Preface Mechanics of Materialsis a basic engineering subject that must be under- stood by anyone concerned with the strength and physical performance of structures, whether those structures are man-made or natural. The subject matter includes such fundamental concepts as stresses and strains, defor- ▯ mations and displacements, elasticity and inelasticity, strain energy, and load-carrying capacity. These concepts underlie the design and analysis of a huge variety of mechanical and structural systems. At the college level, mechanics of materials is usually taught during the sophomore and junior years. The subject is required for most stu- dents majoring in mechanical, structural, civil, biomedical, aeronautica▯l, and aerospace engineering. Furthermore, many students from such diverse fields as materials science, industrial engineering, architectur▯e, and agricultural engineering also find it useful to study this subject. About this Book The main topics covered in this book are the analysis and design of structural members subjected to tension, compression, torsion, and bending, including the fundamental concepts mentioned in the first para-▯ graph. Other topics of general interest are the transformations of stres▯s and strain, combined loadings, stress concentrations, deflections of beams, and stability of columns. Specialized topics include the following: Thermal effects, dynamic loading, nonprismatic members, beams of two materials, shear centers, pressure vessels, and statically indeterminate beams. For completeness and occasional reference, elementary topics such as shear forces, bending ▯ moments, centroids, and moments of inertia also are presented. As an aid to the student reader, each chapter begins with a Chapter Overview and closes with a Chapter Summary & Review in which the key points pre- sented in the chapter are listed for quick review (in preparation for examinations on the material). Each chapter also opens with a photo- graph of a component or structure which illustrates the key concepts to be discussed in that chapter. Much more material than can be taught in a single course is included in this book, and therefore instructors have the opportunity to▯ select the topics they wish to cover. As a guide, some of the more specialized topics are identified in the table of contents by stars. xi xii PREFACE Considerable effort has been spent in checking and proofreading the text so as to eliminate errors, but if you happen to find one, no matter▯ how trivial, please notify me by e-mail (bgoodno@ce.gatech.edu). Then we can correct any errors in the next printing of the book. Examples Examples are presented throughout the book to illustrate the theoretical▯ concepts and show how those concepts may be used in practical situations.▯ In some cases, photographs have been added showing actual engineering structures or components to reinforce the tie between theory and applica- ▯ tion. The examples vary in length from one to four pages, depending upon the complexity of the material to be illustrated. When the emphasis is on concepts, the examples are worked out in symbolic terms so as to better illustrate the ideas, and when the emphasis is on problem-solving, the examples are numerical in character. In selected examples throughout the text, graphical display of results (e.g., stresses in beams) has been a ▯ dded to enhance the student’s understanding of the problem results. Problems In all mechanics courses, solving problems is an important part of the learning process. This textbook offers more than 1,000 problems for homework assignments and classroom discussions. Approximately 40% of the problems are new or significantly revised in the seventh edition.▯ The problems are placed at the end of each chapter so that they are easy▯ to find and don’t break up the presentation of the main subject matter. Also, an unusually difficult or lengthy problem is indicated by attaching one or more stars (depending upon the degree of difficulty) to the prob- lem number, thus alerting students to the time necessary for solution. In general, problems are arranged in order of increasing difficulty. Answers to all problems are listed near the back of the book. Units Both the International System of Units (SI) and the U.S. Customary System (USCS) are used in the examples and problems. Discussions of both systems and a table of conversion factors are given in Appendix A. For problems involving numerical solutions, odd-numbered problems are in USCS units and even-numbered problems are in SI units. This conven- tion makes it easy to know in advance which system of units is being used in any particular problem. In addition, tables containing properties ▯ of structural-steel shapes in both USCS and SI units have been added to Appendix E so that solution of beam analysis and design examples and end-of-chapter problems can be carried out in either USCS or SI units. References and Historical Notes References and historical notes appear immediately after the last chapter ▯ in the book. They consist of original sources for the subject matter plus brief biographical information about the pioneering scientists, engineers ▯ , PREFACE xiii and mathematicians who created the subject of mechanics of materials. A separate name index makes it easy to look up any of these historical figu▯ res. Appendixes Reference material appears in the appendixes at the back of the book. Muc ▯ h of the material is in the form of tables—properties of plane areas, p▯ operties of structural-steel shapes, properties of structural lumber, deflections and slopes of beams, and properties of materials (Appendixes D through H, respectively). In contrast, Appendixes A and B are descriptive—the former gives a detailed description of the SI and USCS systems of units, and the latter▯ presents the methodology for solving problems in mechanics. Included in the latter are topics such as dimensional consistency and significant▯ digits. Lastly, as a handy time–saver, Appendix C provides a listing of commonly used mathematical formulas. S.P. Timoshenko (1878–1972) and J.M. Gere (1925–2008) Many readers of this book will recognize the name of Stephen P. Timoshenko—probably the most famous name in the field of applied mechanics. Timoshenko is generally recognized as the world’s most out- standing pioneer in applied mechanics. He contributed many new ideas and concepts and became famous for both his scholarship and his teach- ing. Through his numerous textbooks he made a profound change in the teaching of mechanics not only in this country but wherever mechanics is▯ taught. Timoshenko was both teacher and mentor to James Gere and provided the motivation for the first edition of this text, authored by James M. Gere and published in 1972; the second and each subsequent edition of this book were written by James Gere over the course of his long and distinguished tenure as author, educator, and researcher at Stanford University. James Gere started as a doctoral student at Stanford in 1952 and retired from Stanford as a professor in 1988 having authored▯ this and eight other well known and respected text books on mechanics, and structural and earthquake engineering. He remained active at Stanford as Professor Emeritus until his death in January of 2008. A brief biography of Timoshenko appears in the first reference at the back of the book, and also in an August 2007 STRUCTURE maga- zine article entitled “ Stephen P. Timoshenko: Father of Engineering Mechanics in the U.S. ” by Richard G. Weingardt, P.E. This article pro- vides an excellent historical perspective on this and the many other engineering mechanics textbooks written by each of these authors. Acknowledgments To acknowledge everyone who contributed to this book in some manner is clearly impossible, but I owe a major debt to my former Stanford teachers ▯ , especially my mentor and friend, and lead author, James M. Gere. I am also indebted to the many teachers of mechanics and reviewers of the book who have helped to shape this textbook in its various editions over▯ xiv PREFACE the years. With each new edition, their advice has resulted in significant improvements in both content and pedagogy. I wish to also acknowledge my Structural Engineering and Mechanics colleagues at Georgia Institute of Technology: James Craig, Reggie DesRoches, Mulalo Doyoyo, Bruce Ellingwood, Leroy Emkin, Rami Haj-Ali, Larry Jacobs, Larry Kahn, Kim Kurtis, Roberto Leon, Yang Wang, Don White, Kenneth (Mac) Will, Arash Yavari, and Abdul Zureick. I am especially grateful to Jim Craig, Rami Haj-Ali, Larry Jacobs, Larry▯ Kahn, Roberto Leon, Don White, Mac Will and Abdul Zureick, all of whom provided valuable advice on various aspects of the revisions and additions leading to the seventh edition. It is a privilege to work with all of these educators and to learn from them in almost daily interactions ▯ d discussions about structural engineering and mechanics in the context of▯ research and higher education. Two of my graduate research assistants, Mr. Kanoknart Leelard- charoen and Ms. Jee-Eun Hur, provided invaluable assistance in evaluating and solving many of the new and revised problems. Their careful attention to detail was an important contribution to the current▯ edition. The editing and production aspects of the book were always in skill- ful and experienced hands, thanks to the talented and knowledgeable personnel of Cengage Learning (formerly Thomson Learning). Their goal was the same as mine—to produce the best possible seventh editio▯n of this text, never compromising on any aspect of the book. The people with whom I have had personal contact at Cengage Learning are Christopher Carson, Director, Global Engineering Program, Cengage Learning, who, along with Jim Gere, helped to involve me and then guide me through the project; Hilda Gowans, Senior Developmental Editor, Cengage Learning, Engineering, who was always available to provide information and encouragement; Nicola Winstanley who man- aged all aspects of new photo selection; Andrew Adams, who created the covers; Peter Papayanakis, who created the interior book design; and Lauren Betsos, Global Marketing Services Coordinator, who developed promotional material in support of the text. I would like to especially acknowledge the work of Rose Kernan of RPK Editorial Services, who edited the manuscript and laid out the pages. To each of these individuals I express my heartfelt thanks not only for a job well done but also for the friendly and considerate way in which it was handled. I am deeply appreciative of the patience and encouragement pro- vided by my family, especially my wife, Lana, throughout this project. Finally, I am honored and extremely pleased to be involved in this endeavor, at the invitation of my mentor and friend of thirty eight years, Jim Gere, which extends this textbook toward the forty year mark. I too▯ am committed to the continued excellence of this text and welcome all comments and suggestions. Please feel free to provide me with your critical input at bgoodno@ce.gatech.edu. B ARRY J. GOODNO Atlanta, Georgia Symbols A area A f Aw area of flange; area of web a, b, c dimensions, distances C centroid, compressive force, constant of integration c distance from neutral axis to outer surface of a beam D diameter d diameter, dimension, distance E modulus of elasticity Er, Et reduced modulus of elasticity; tangent modulus of elasticity e eccentricity, dimension, distance, unit volume change (dilatation) F force f shear flow, shape factor for plastic bending, flexibility, frequency (Hz) T torsional flexibility of a bar G modulus of elasticity in shear g acceleration of gravity H height, distance, horizontal force or reaction, horsepower h height, dimensions I moment of inertia (or second moment) of a plane area I , I , I moments of inertia with respect to x, y, and z axes x y z Ix1, y1 moments of inertia with respect to1x and1y axes (rotated axes) I product of inertia with respect to xy axes xy Ix1y1 product of inertia with respect 1 1x y axes (rotated axes) P polar moment of inertia I1, 2 principal moments of inertia J torsion constant K stress-concentration factor, bulk modulus of elasticity, effective length factor for a column k spring constant, stiffness, symbol for EI▯ xv xvi SYMBOLS k torsional stiffness of a bar T L length, distance LE effective length of a column ln, log natural logarithm (base e); common logarithm (base 10) M bending moment, couple, mass M , M plastic moment for a beam; yield moment for a beam P Y m moment per unit length, mass per unit length N axial force n factor of safety, integer, revolutions per minute (rpm) O origin of coordinates O▯ center of curvature P force, concentrated load, power P allowable load (or working load) allow P cr critical load for a column P plastic load for a structure P Pr, Pt reduced-modulus load for a column; tangent-modulus load for a column PY yield load for a structure p pressure (force per unit area) Q force, concentrated load, first moment of a plane area q intensity of distributed load (force per unit distance) R reaction, radius r radius, radius of gyration (r ▯/A ) S section modulus of the cross section of a beam, shear center s distance, distance along a curve T tensile force, twisting couple or torque, temperature T P, Y plastic torque; yield torque t thickness, time, intensity of torque (torque per unit distance) tf w thickness of flange; thickness of web U strain energy u strain-energy density (strain energy per unit volume) ur, ut modulus of resistance; modulus of toughness V shear force, volume, vertical force or reaction v deflection of a beam, velocity v▯, v▯, etc. dv/dx, d2v/dx , etc. W force, weight, work w load per unit of area (force per unit area) x, y, z rectangular axes (origin at point O) xc yc,cz rectangular axes (origin at centroid C) x, y, z coordinates of centroid ▯▯▯ SYMBOLS xvii Z plastic modulus of the cross section of a beam a angle, coefficient of thermal expansion, nondimensional ratio b angle, nondimensional ratio, spring constant, stiffness bR rotational stiffness of a spring g shear strain, weight density (weight per unit volume) g , g , g shear strains in xy, yz, and zx planes xy yz zx gx1 1 shear strain with respect to1 1y axes (rotated axes) g shear strain for inclined axes u d deflection of a beam, displacement, elongation of a bar or spring ▯T temperature differential d ,d plastic displacement; yield displacement P Y e normal strain ex, ey, z normal strains in x, y, and z directions e , e normal strains in x and y directions (rotated axes) 1 y1 1 1 e normal strain for inclined axes u e1, 2 , 3 principal normal strains e▯ lateral strain in uniaxial stress e thermal strain T eY yield strain u angle, angle of rotation of beam axis, rate of twist of a bar in torsion▯ (angle of twist per unit length) u angle to a principal plane or to a principal axis p u angle to a plane of maximum shear stress s k curvature (k ▯ 1/r) l distance, curvature shortening n Poisson’s ratio r radius, radius of curvature (r ▯ 1/k), radial distance in polar coordinates, mass density (mass per unit volume) s normal stress s , s , s normal stresses on planes perpendicular to x, y, and z axes x y z s , s normal stresses on planes perpendicular to x y axes (rotated axes) x1 y1 1 1 s u normal stress on an inclined plane s , s , s principal normal stresses 1 2 3 s allowable stress (or working stress) allow s cr critical stress for a column cr ▯ PcrA) s proportional-limit stress pl s residual stress r sT thermal stress s , s ultimate stress; yield stress U Y xviii SYMBOLS t shear stress txy,yz ,zx shear stresses on planes perpendicular to the x, y, and z axes and acting parallel to the y, z, and x axes tx1 1 shear stress on a plane perpendicula1 to the x axis and acting parallel to the 1axis (rotated axes) tu shear stress on an inclined plane t allowable stress (or working stress) in shear allow tU, Y ultimate stress in shear; yield stress in shear f angle, angle of twist of a bar in torsion c angle, angle of rotation v angular velocity, angular frequency (v ▯ 2pf ) ★A star attached to a section number indicates a specialized or advanced to▯pic. One or more stars attached to a problem number indicate an increasing le▯vel of difficulty in the solution. Greek Alphabet ▯ a Alpha ▯ n Nu ▯ b Beta ▯ j Xi ▯ g Gamma ▯ o Omicron ▯ d Delta ▯ p Pi ▯ e Epsilon ▯ r Rho ▯ z Zeta ▯ s Sigma ▯ h Eta ▯ t Tau ▯ u Theta ▯ y Upsilon ▯ i Iota ▯ f Phi ▯ k Kappa ▯ x Chi ▯ l Lambda ▯ c Psi ▯ m Mu ▯ v Omega Mechanics of Materials This telecommunications tower is an assemblage of many members that act ▯primarily in tension or compression. NPR. Used with permission. 1 Tension, Compression, and Shear CHAPTER OVERVIEW In Chapter 1, we are introduced to mechanics of materials, which exam- ines the stresses, strains, and displacements in bars of various materials acted on by axial loads applied at the centroids of their cross sections▯. We will learn about normal stress ( and normal strain (▯) in materials used for structural applications, then identify key properties of variou▯s materials, such as the modulus of elasticity ( E) and yiyld (d ulti- mate (▯u) stresses, from plots of stress (▯) versus strain (▯). We will also plot shear stress ( versus shear strain ( ▯) and identify the shearing modulus of elasticity ( G). If these materials perform only in the linear range, stress and strain are related by Hooke’s Law for normal stress and strain ( ▯ E ▯) and also for shear stress and strain ( ▯ ▯ G ▯). We will see that changes in lateral dimensions and volume depend upon Poisson’s ratio (v). Material properties E, G, and v, in fact, are directly related to one another and are not independent properties of the materia▯l. Assemblage of bars to form structures (such as trusses) leads to consideration of average shear (▯) and bearing ( ▯b) stresses in their connections as well as normal stresses acting on the net area of the cross section (if in tension) or on the full cross-sectional area (if▯ in compression). If we restrict maximum stresses at any point to allow- able values by use of factors of safety, we can identify allowable levels of axial loads for simple systems, such as cables and barFactors of safety relate actual to required strength of structural members and account for a variety of uncertainties, such as variations in material properties and probability of accidental overload. Lastly, we will con- sider design: the iterative process by which the appropriate size of structural members is determined to meet a variety of both strength and stiffness requirements for a particular structure subjected to a variety of different loadings. 3 4 CHAPTER 1 Tension, Compression, and Shear Chapter 1 is organized as follows: 1.1 Introduction to Mechanics of Materials 5 1.2 Normal Stress and Strain 7 1.3 Mechanical Properties of Materials 15 1.4 Elasticity, Plasticity, and Cree24 1.5 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio 27 1.6 Shear Stress and Strain 32 1.7 Allowable Stresses and Allowable Loads 43 1.8 Design for Axial Loads and Direct Shear 49 Chapter Summary & Review 55 Problems 57 SECTION 1.1 Introduction to Mechanics of Materials 5 1.1 INTRODUCTION TO MECHANICS OF MATERIALS Mechanics of materials is a branch of applied mechanics that deals with the behavior of solid bodies subjected to various types of loading.▯ Other names for this field of study are strength of materials and mechanics of deformable bodies . The solid bodies considered in this book include bars with axial loads, shafts in torsion, beams in bending,▯ and columns in compression. The principal objective of mechanics of materials is to determine the stresses, strains, and displacements in structures and their compo- nents due to the loads acting on them. If we can find these quantities fo ▯ r all values of the loads up to the loads that cause failure, we will have▯ a complete picture of the mechanical behavior of these structures. An understanding of mechanical behavior is essential for the safe design of all types of structures, whether airplanes and antennas, buildi▯ ngs and bridges, machines and motors, or ships and spacecraft. That is why mechanics of materials is a basic subject in so many engineering fields.▯ Stat- ics and dynamics are also essential, but those subjects deal primarily wi▯ th the forces and motions associated with particles and rigid bodies. In mechanics of materials we go one step further by examining the stresses a ▯ nd strains inside real bodies, that is, bodies of finite dimensions that def▯ orm under loads. To determine the stresses and strains, we use the physical p ▯ rop- erties of the materials as well as numerous theoretical laws and concepts ▯ . Theoretical analyses and experimental results have equally important

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