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Mechanics of Materials, SI Edition
Mechanics of Materials, SI Edition
James M. Gere, Barry J. Goodno
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The Eighth Edition of MECHANICS OF MATERIALS continues its tradition as one of the leading texts on the market. With its hallmark clarity and accuracy, this text develops student understanding along with analytical and problemsolving skills. The main topics include analysis and design of structural members subjected to tension, compression, torsion, bending, and more. The book includes more material than can be taught in a single course giving instructors the opportunity to select the topics they wish to cover while leaving any remaining material as a valuable student reference
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2012
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8
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Cengage Learning
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english
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1120 / 1124
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1111577749
ISBN 13:
9781111577742
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2 comments
BIGSIAW
Mechanics of Materials, 8th2013_(Barry J. Goodno and James M. Gere).pdf
pages: 1124
pages: 1124
01 August 2017 (14:00)
ahmad
where can i find the solution manual?
26 November 2018 (17:27)
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Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:40 PM Page i Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:40 PM Page ii Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:40 PM Page iii Mechanics of Materials Eighth Edition, SI 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 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be ; suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:40 PM Page iv Mechanics of Materials, Eighth Edition, SI © 2013, 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, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. Publisher, Global Engineering: Christopher M. Shortt Senior Acquisitions Editor: Randall Adams Senior Developmental Editor: Hilda Gowans Editorial Assistant: Tanya Altieri Team Assistant: Carly Rizzo Marketing Manager: Lauren Betsos Media Editor: Chris Valentine Manager, Content and Media Production: Patricia M. Boies Senior Content Project Manager: Jennifer Ziegler For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 18003549706. For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be emailed to permissionrequest@cengage.com. Production Service: RPK Editorial Services, Inc. Copyeditor: Shelly GergerKnechtl Library of Congress Control Number: 2012930214 Proofreader: Martha McMaster ISBN13: 9781111577742 Indexer: Paul Mailhot Compositor: Integra Senior Art Director: Michelle Kunkler Internal Designer: Juli Cook/PlanItPublishing Cover Designer: Andrew Adams/4065042 Canada Inc. Cover Image: © Arcaid/Corbis Rights Acquisitions Specialist: Deanna Ettinger Text and Image Permissions Researcher: Kristiina Paul Manufacturing Planner: Doug Wilke ISBN10: 1111577749 Cengage Learning 200 First Stamford Place, Suite 400 Stamford, CT 06902 USA 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 www.cengage.com/engineering. Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com. Printed in Canada 1 2 3 4 5 6 7 16 15 14 13 12 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:40 PM Page v CONTENTS James Monroe Gere ix Preface to the SI Edition xi Symbols xviii Greek Alphabet xx 1. TENSION, COMPRESSION, AND SHEAR 2 1.1 1.2 1.3 1.4 1.5 1.6 Introduction to Mechanics of Materials 4 Statics Review 6 Normal Stress and Strain 27 Mechanical Properties of Materials 37 Elasticity, Plasticity, and Creep 45 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio 52 1.7 Shear Stress and Strain 57 1.8 Allowable Stresses and Allowable Loads 68 1.9 Design for Axial Loads and Direct Shear 74 Chapter Summary & Review 80 Problems 83 2. AXIALLY LOADED MEMBERS 122 2.1 Introduction 124 2.2 Changes in Lengths of Axially Loaded Members 124 2.3 Changes in Lengths Under Nonuniform Conditions 134 2.4 Statically Indeterminate Structures 142 2.5 Thermal Effects, Misfits, and Prestrains 153 2.6 Stresses on Inclined Sections 168 2.7 Strain Energy 180 *2.8 Impact Loading 191 *2.9 Repeated Loading and Fatigue 199 *2.10 Stress Concentrations 201 *2.11 Nonlinear Behavior 209 *2.12 Elastoplastic Analysis 214 Chapter Summary & Review 220 Problems 222 3. TORSION 262 3.1 Introduction 264 3.2 Torsional Deformations of a Circular Bar 265 3.3 Circular Bars of Linearly Elastic Materials 268 3.4 Nonuniform Torsion 280 3.5 Stresses and Strains in Pure Shear 291 3.6 Relationship Between Moduli of Elasticity E and G 298 3.7 Transmission of Power by Circular Shafts 299 3.8 Statically Indeterminate Torsional Members 304 3.9 Strain Energy in Torsion and Pure Shear 308 3.10 Torsion of Noncircular Prismatic Shafts 315 3.11 ThinWalled Tubes 324 *3.12 Stress Concentrations in Torsion 332 Chapter Summary & Review 336 Problems 338 4. SHEAR FORCES AND BENDING MOMENTS 364 4.1 4.2 4.3 4.4 Introduction 366 Types of Beams, Loads, and Reactions 366 Shear Forces and Bending Moments 373 Relationships Between Loads, Shear Forces, and Bending Moments 383 4.5 ShearForce and BendingMoment Diagrams 387 Chapter Summary & Review 400 Problems 402 5. STRESSES IN BEAMS (BASIC TOPICS) 416 5.1 Introduction 418 5.2 Pure Bending and Nonuniform Bending 418 5.3 Curvature of a Beam 419 *Specialized and/or advanced topics v Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd vi 3/1/12 6:40 PM Page vi Contents 5.4 Longitudinal Strains in Beams 421 5.5 Normal Stresses in Beams (Linearly Elastic Materials) 426 5.6 Design of Beams for Bending Stresses 440 5.7 Nonprismatic Beams 449 5.8 Shear Stresses in Beams of Rectangular Cross Section 453 5.9 Shear Stresses in Beams of Circular Cross Section 462 5.10 Shear Stresses in the Webs of Beams with Flanges 465 *5.11 BuiltUp Beams and Shear Flow 472 *5.12 Beams with Axial Loads 476 *5.13 Stress Concentrations in Bending 482 Chapter Summary & Review 486 Problems 490 6. STRESSES IN BEAMS (ADVANCED TOPICS) 524 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 *6.10 Introduction 526 Composite Beams 526 TransformedSection Method 535 Doubly Symmetric Beams with Inclined Loads 544 Bending of Unsymmetric Beams 551 The ShearCenter Concept 559 Shear Stresses in Beams of ThinWalled Open Cross Sections 561 Shear Stresses in WideFlange Beams 564 Shear Centers of ThinWalled Open Sections 568 Elastoplastic Bending 576 Chapter Summary & Review 584 Problems 587 7. ANALYSIS OF STRESS AND STRAIN 608 7.1 Introduction 610 7.2 Plane Stress 610 7.3 Principal Stresses and Maximum Shear Stresses 618 7.4 Mohr’s Circle for Plane Stress 627 7.5 Hooke’s Law for Plane Stress 643 7.6 Triaxial Stress 649 7.7 Plane Strain 653 Chapter Summary & Review Problems 672 668 8. APPLICATIONS OF PLANE STRESS (PRESSURE VESSELS, BEAMS, AND COMBINED LOADINGS) 692 8.1 8.2 8.3 8.4 8.5 Introduction 694 Spherical Pressure Vessels 694 Cylindrical Pressure Vessels 700 Maximum Stresses in Beams 707 Combined Loadings 716 Chapter Summary & Review 734 Problems 736 9. DEFLECTIONS OF BEAMS 754 9.1 Introduction 756 9.2 Differential Equations of the Deflection Curve 756 9.3 Deflections by Integration of the BendingMoment Equation 761 9.4 Deflections by Integration of the ShearForce and Load Equations 772 9.5 Method of Superposition 778 9.6 MomentArea Method 786 9.7 Nonprismatic Beams 795 9.8 Strain Energy of Bending 800 *9.9 Castigliano’s Theorem 805 *9.10 Deflections Produced by Impact 817 *9.11 Temperature Effects 819 Chapter Summary & Review 824 Problems 826 10. STATICALLY INDETERMINATE BEAMS 848 10.1 Introduction 850 10.2 Types of Statically Indeterminate Beams 850 10.3 Analysis by the Differential Equations of the Deflection Curve 853 10.4 Method of Superposition 860 *10.5 Temperature Effects 873 *10.6 Longitudinal Displacements at the Ends of a Beam 881 Chapter Summary & Review 884 Problems 886 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:40 PM Page vii Contents 11. COLUMNS 900 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Introduction 902 Buckling and Stability 902 Columns with Pinned Ends 910 Columns with Other Support Conditions 921 Columns with Eccentric Axial Loads 931 The Secant Formula for Columns 936 Elastic and Inelastic Column Behavior 941 Inelastic Buckling 943 Chapter Summary & Review 950 Problems 952 12. REVIEW OF CENTROIDS AND MOMENTS OF INERTIA 968 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 Introduction 970 Centroids of Plane Areas 970 Centroids of Composite Areas 973 Moments of Inertia of Plane Areas 976 ParallelAxis Theorem for Moments of Inertia 979 Polar Moments of Inertia 983 Products of Inertia 985 Rotation of Axes 988 Principal Axes and Principal Moments of Inertia 990 Problems 994 vii REFERENCES AND HISTORICAL NOTES 1001 APPENDIX A: SYSTEMS OF UNITS AND CONVERSION FACTORS 1009 APPENDIX B: PROBLEM SOLVING 1019 APPENDIX C: MATHEMATICAL FORMULAS 1025 APPENDIX D: PROPERTIES OF PLANE AREAS 1031 APPENDIX E: PROPERTIES OF STRUCTURALSTEEL SHAPES 1037 APPENDIX F: PROPERTIES OF STRUCTURAL TIMBER 1043 APPENDIX G: DEFLECTIONS AND SLOPES OF BEAMS 1045 APPENDIX H: PROPERTIES OF MATERIALS 1051 ANSWERS TO PROBLEMS 1057 NAME INDEX 1091 SUBJECT INDEX 1092 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:40 PM Page viii Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page ix JAMES MONROE GERE 1925–2008 James Monroe Gere, Professor Emeritus of Civil Engineering at Stanford University, died in Portola Valley, CA, on January 30, 2008. Jim Gere was born on June 14, 1925, in Syracuse, 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 34year career of engaging his students in challenging topics in mechanics, and structural and earthquake engineering. He served as Department Chair and Associate Dean of Engineering and in 1974 cofounded the John A. Blume Earthquake Engineering Center at Stanford. In 1980, Jim (Ed Souza/Stanford News Service) Gere also became the founding head of the Stanford Committee on Earthquake Preparedness, which urged campus members to brace and strengthen office equipment, furniture, and other items that could pose a life safety hazard in the event of an earthquake. That same year, he was invited as one of the first foreigners to study the earthquakedevastated city of Tangshan, China. Jim retired from Stanford in 1988 but continued 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 Engineering. He authored nine textbooks on various engineering subjects starting in 1972 with Mechanics of Materials, a text that was inspired by his teacher and mentor Stephan P. Timoshenko. His other wellknown textbooks, used in engineering courses around the world, include: Theory of Elastic Stability, coauthored with S. Timoshenko; Matrix Analysis of Framed Structures and Matrix Algebra for Engineers, both coauthored with W. Weaver; Moment Distribution; Earthquake Tables: Structural and Jim Gere in the Timoshenko Construction Design Manual, coauthored with H. Krawinkler; and Terra Library at Stanford holding a Non Firma: Understanding and Preparing for Earthquakes, coauthored with copy of the 2nd edition of this text (photo courtesy of Richard H. Shah. Weingardt Consultants, Inc.) 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 young people both inside and outside the classroom was one of his great joys. He hiked frequently and regularly 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 hiked to the base ix Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd x 3/1/12 6:41 PM Page x James Monroe Gere camp of Mount Everest, saving the life of a companion on the trip. James was an active runner and completed the Boston Marathon at age 48, in a time of 3:13. James Gere will be long remembered by all who knew him as a considerate and loving man whose upbeat good humor made aspects of daily life or work easier to bear. His last project (in progress and now being continued by his daughter Susan of Palo Alto) was a book based on the written memoirs of his greatgrandfather, a Colonel (122d NY) in the Civil War. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page xi P R E FAC E TO T H E S I E D I T I O N Mechanics of materials is a basic engineering subject that, along with statics, must be understood by anyone concerned with the strength and physical performance of structures, whether those structures are manmade or natural. At the college level, statics is usually taught during the sophomore or junior year and is a prerequisite for the followon course in mechanics of materials. Both courses are required for most students majoring in mechanical, structural, civil, biomedical, petroleum, nuclear, aeronautical, and aerospace engineering. Furthermore, many students from such diverse fields as materials science, industrial engineering, architecture, and agricultural engineering also find it useful to study mechanics of materials. A FIRST COURSE IN MECHANICS OF MATERIALS In many university engineering programs today, both statics and mechanics of materials are now taught in large sections comprised of students from the variety of engineering disciplines listed above. Instructors for the various parallel sections must cover the same material, and all of the major topics must be presented so that students are well prepared for the more advanced courses required by their specific degree programs. An essential prerequisite for success in a first course in mechanics of materials is a strong foundation in statics, which includes not only understanding of fundamental concepts but also proficiency in applying the laws of statical equilibrium to solution of both two and three dimensional problems. This eighth edition begins with a new section on review of statics in which the laws of equilibrium and boundary (or support) conditions are reviewed, as well as types of applied forces and internal stress resultants, all based upon and derived from a properly drawn free body diagram. Numerous examples and end of chapter problems are included to help the student review the analysis of plane and space trusses, shafts in torsion, beams and plane and space frames and to reinforce basic concepts learned in the prerequisite course. Many instructors like to present the basic theory of say, beam bending, and then use real world examples to motivate student interest in the subject of beam flexure, beam design, etc. In many cases, structures on campus offer easy access to beams, frames, and bolted connections which can be dissected in lecture, or on homework problems, to find reactions at supports, forces and moments in members and stresses in connections. In addition, study of causes of failures in structures and components also offers the opportunity for students to begin the process of learning from actual designs and even past engineering mistakes. A number of the new example problems and also the new or revised endofchapter problems in this eighth edition are based upon actual components or structures and are accompanied by photographs so that the student can see the real world problem alongside the simplified mechanics model and free body diagrams to be used in its analysis. An increasing number of universities are using rich media lecture (and/or classroom) capture software in their large undergraduate courses in mathematics, physics, and engineering and the many new photos and enhanced graphics in the eighth edition are designed to support this enhanced lecture mode. xi Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd xii 3/1/12 6:41 PM Page xii Preface to the SI Edition NEW TO THE EIGHTH EDITION OF MECHANICS OF MATERIALS, SI EDITION 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 above. Other important topics are the transformations of stress and strain, combined loadings and combined stress, deflections of beams, and stability of columns. Some additional specialized topics include the following: stress concentrations, dynamic and impact loadings, nonprismatic members, shear centers, bending of beams of two materials (or composite beams), bending of unsymmetric beams, maximum stresses in beams, energy based approaches for computing deflections of beams, and statically indeterminate beams. Review material on centroids and moments of inertia is presented in Chapter 12. As an aid to the student reader, each chapter begins with a Chapter Overview which highlights the major topics to be covered in that chapter, and closes with a Chapter Summary & Review in which the key points as well as major mathematical formulas presented in the chapter are listed for quick review (in preparation for examinations on the material). Each chapter also opens with a photograph of a component or structure which illustrates the key concepts to be discussed in that chapter. Some of the notable features of this eighth edition, which have been added as new or updated material to meet the needs of a modern course in mechanics of materials, are as follows: • Statics review—A new section entitled Statics Review has been added to Chapter 1. New Section 1.2 includes four example problems which illustrate calculation of support reactions and internal stress resultants for truss, beam, circular shaft and plane frame structures. Twenty six endofchapter problems on statics provide the student with two and three dimensional structures to be used as practice, review or homework assignment problems of varying difficulty. • Expanded Chapter Overview and also Chapter Summary & Review sections– The Chapter Overview and Chapter Summary sections have been expanded and now include key equations presented in that chapter. These summary sections will serve as a convenient review for the student of key topics and equations presented in each chapter. • Increased emphasis on equilibrium, constitutive, and straindisplacement/ compatibility equations in problem solutions–Example problem and endofchapter problem solutions have been updated to emphasize an orderly process of explicitly writing out the equilibrium, constitutive and straindisplacement/ compatibility equations before attempting a solution. • New/expanded topic coverage—The following topics have been added or have received expanded coverage: stress concentrations in axially loads bars (Sec. 2.10); torsion of noncircular shafts (Sec. 3.10); stress concentrations in bending (Sec. 5.13); and transformed section analysis for composite beams (Sec. 6.3). • New example and endofchapter problems—Fortyeight new example problems have been added to the eighth edition. In addition, close to 800 of the endofchapter problems are new or revised, out of a total of almost 1200 problems. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page xiii Preface to the SI Edition • xiii Review problems—A total of one hundred and nineteen review problems have been added at the ends of chapters 1 to 11. The student must select from 4 available answers (A, B, C or D), only one of which is the correct answer. The correct answer choices are listed in the Answers section at the back of this text, and the detailed solution for each problem is available on the student website. Solution of these problems will provide the student with a quick check on his or her mastery of the subject matter presented in that chapter. 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, new photographs have been added showing actual engineering structures or components to reinforce the tie between theory and application. In both lecture and text examples, it is appropriate to begin with simplified analytical models of the structure or component and the associated freebody diagram(s) to aid the student in understanding and applying the relevant theory in engineering analysis of the system. The text 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 problemsolving, the examples are numerical in character. In selected examples throughout the text, graphical display of results (e.g., stresses in beams) has been added 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 1230 problems for homework assignments and classroom discussions. 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, problems are generally arranged in order of increasing difficulty thus alerting students to the time necessary for solution. Answers to all problems are listed near the back of the book. An Instructor Solution Manual (ISM) is available to registered instructors at the publisher’s web site. Considerable effort has been spent in checking and proofreading the text so as to eliminate errors. If you happen to find one, no matter how trivial, please notify me by email (bgoodno@ce.gatech.edu). We will correct any errors in the next printing of the book. UNITS The International System of Units (SI) is used in all examples and problems. Tables containing properties of selected structuralsteel shapes in SI units may be found in Appendix E; these tables will be useful in the solution of beam analysis and design examples and endofchapter problems in Chapter 5. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd xiv 3/1/12 6:41 PM Page xiv Preface to the SI Edition SUPPLEMENTS INSTRUCTOR RESOURCES An Instructor’s Solutions Manual (ISM) is available in both print and digital versions. The digital version is accessible to registered instructors at the publisher’s web site. This web site also includes both a full set of PowerPoint slides containing all graphical images in the text, and a set of LectureBuilder PowerPoint slides of all equations and Example Problems from the text, for use by instructors during lecture or review sessions. To access additional course materials, please visit www.cengagebrain.com. At the cengagebrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where these resources can be found. 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 outstanding pioneer in applied mechanics. He contributed many new ideas and concepts and became famous for both his scholarship and his teaching. 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 in the References and Historical Notes section, and also in an August 2007 STRUCTURE magazine article entitled “Stephen P. Timoshenko: Father of Engineering Mechanics in the U.S.” by Richard G. Weingardt, P.E. This article provides 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. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page xv Preface to the SI Edition xv I am grateful to my many colleagues teaching Mechanics of Materials at various institutions throughout the world who have provided feedback and constructive criticism about the text; for all those anonymous reviews, my thanks. With each new edition, their advice has resulted in significant improvements in both content and pedagogy. My appreciation and thanks also go to the reviewers who provided specific comments for this eighth edition: Jonathan Awerbuch, Drexel University Henry N. Christiansen, Brigham Young University Remi Dingreville, NYU—Poly Apostolos Fafitis, Arizona State University Paolo Gardoni, Texas A & M University Eric Kasper, California Polytechnic State University, San Luis Obispo Nadeem Khattak, University of Alberta Kevin M. Lawton, University of North Carolina, Charlotte Kenneth S. Manning, Adirondack Community College Abulkhair Masoom, University of Wisconsin—Platteville Craig C. Menzemer, University of Akron Rungun Nathan, The Pennsylvania State University, Berks Douglas P. Romilly, University of British Columbia Edward Tezak, Alfred State College George Tsiatis, University of Rhode Island Xiangwu (David) Zeng, Case Western Reserve University Mohammed Zikry, North Carolina State University I wish to also acknowledge my Structural Engineering and Mechanics colleagues at the Georgia Institute of Technology, many of whom provided valuable advice on various aspects of the revisions and additions leading to the current edition. It is a privilege to work with all of these educators and to learn from them in almost daily interactions and discussions about structural engineering and mechanics in the context of research and higher education. I wish to extend my thanks to my many Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd xvi 3/1/12 6:41 PM Page xvi Preface to the SI Edition current and former students who have helped to shape this text in its various editions. Finally, I would like to acknowledge the excellent work of German Rojas, PhD., PEng. who carefully checked the solutions of many of the new examples and end of chapter problems. The editing and production aspects of the book were always in skillful and experienced hands, thanks to the talented and knowledgeable personnel of Cengage Learning. Their goal was the same as mine—to produce the best possible new edition 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, Executive Director, Global Publishing Program, Christopher Shortt, Publisher, Global Engineering Program, Randall Adams and Swati Meherishi, Acquisitions Editors, who provided guidance throughout the project; Hilda Gowans, Senior Developmental Editor, Engineering, who was always available to provide information and encouragement; Kristiina Paul who managed all aspects of new photo selection and permissions research; Andrew Adams who created the cover design for the book; and Lauren Betsos, Global Marketing Manager, who developed promotional material in support of the text. I would like to especially acknowledge the work of Rose Kernan of RPK Editorial Services, and her staff, who edited the manuscript and managed it throughout the production process. To each of these individuals I express my heartfelt thanks for a job well done. It has been a pleasure working with you on an almost daily basis to produce this eighth edition of the text. I am deeply appreciative of the patience and encouragement provided by my family, especially my wife, Lana, throughout this project. Finally, I am very pleased to continue this endeavor, at the invitation of my mentor and friend, Jim Gere. This eighth edition text has now reached its 40th year of publication. I am committed to its continued excellence and welcome all comments and suggestions. Please feel free to provide me with your critical input at bgoodno@ce.gatech.edu. Barry J. Goodno Atlanta, Georgia Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page xvii Preface to the SI Edition xvii CengageNOW—Just What You Need To Know and Do NOW! CengageNOW Is An Online Teaching & Learning Resource CengageNOW offers all of your teaching and learning resources in one intuitive program organized around the essential activities you perform for class—lecturing, creating assignments, grading, quizzing, and tracking student progress and performance. CengageNOW’s intuitive “tabbed” design allows you to navigate to all key functions with a single click and a unique homepage tell you just what needs to be done and when. CengageNOW provides students access to an integrated eBook, interactive tutorials, and other multimedia tools that help them get the most out of your course. CengageNOW Provides More Control In Less Time CengageNOW’s flexible assignment and gradebook options provide you more control while saving you valuable time in planning and managing your course assignments. With CengageNOW you can automatically grade all assignments, weigh grades, choose points or percentages and set the number of attempts and due dates per problem to best suit your overall course plan. CengageNOW Delivers Better Student Outcomes CengageNOW Personalized Study is a diagnostic tool consisting of a chapterspecific PreTest, Study Plan and Posttest that utilizes valuable textspecific assets to empower students to master concepts, prepare for exams and be more involved in class. It’s easy to assign and if you want, results will automatically post to your gradebook. Results to Personalized Study provide immediate and ongoing feedback regarding what students are mastering and why they’re not—to both you and the student. In most cases, Personalized Study, in most cases links to an integrated eBook so students can easily review topics. Best of all, CengageNOW Personalized Study is designed to help your students get a better grade in your class. To purchase or access CengageNOWTM with eBook for Gere/Goodno, Mechanics of Materials, Eighth Edition go to www.cengagebrain.com. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page xviii SYMBOLS A Af, Aw a, b, c C c D d E Er, Et e F f fT G g H h I Ix, Iy, Iz Ix1, Iy1 Ixy Ix1y1 IP I1, I2 J K xviii k kT L LE ln, log M MP, MY m N n O O⬘ P Pallow Pcr area area of flange; area of web dimensions, distances centroid, compressive force, constant of integration distance from neutral axis to outer surface of a beam diameter diameter, dimension, distance modulus of elasticity reduced modulus of elasticity; tangent modulus of elasticity eccentricity, dimension, distance, unit volume change (dilatation) force shear flow, shape factor for plastic bending, flexibility, frequency (Hz) torsional flexibility of a bar modulus of elasticity in shear acceleration of gravity height, distance, horizontal force or reaction, horsepower height, dimensions moment of inertia (or second moment) of a plane area moments of inertia with respect to x, y, and z axes moments of inertia with respect to x1 and y1 axes (rotated axes) product of inertia with respect to xy axes product of inertia with respect to x1y1 axes (rotated axes) polar moment of inertia principal moments of inertia torsion constant stressconcentration factor, bulk modulus of elasticity, effective length factor for a column spring constant, stiffness, symbol for 2P/EI torsional stiffness of a bar length, distance effective length of a column natural logarithm (base e); common logarithm (base 10) bending moment, couple, mass plastic moment for a beam; yield moment for a beam moment per unit length, mass per unit length axial force factor of safety, integer, revolutions per minute (rpm) origin of coordinates center of curvature force, concentrated load, power allowable load (or working load) critical load for a column Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page xix Symbols PP Pr , Pt PY p Q q R r S s T TP , TY t tf , tw U u ur, ut V v v⬘, v⬘⬘, etc. W w x, y, z xc, yc, zc x, y, z Z α β βR γ γxy, γyz, γzx γx1y1 γθ δ ⌬T δP , δY ε εx, εy, εz εx1, εy1 εθ ε1, ε2, ε3 ε⬘ εT εY θ xix plastic load for a structure reducedmodulus load for a column; tangentmodulus load for a column yield load for a structure pressure (force per unit area) force, concentrated load, first moment of a plane area intensity of distributed load (force per unit distance) reaction, radius radius, radius of gyration (r ⫽ 2I/A) section modulus of the cross section of a beam, shear center distance, distance along a curve tensile force, twisting couple or torque, temperature plastic torque; yield torque thickness, time, intensity of torque (torque per unit distance) thickness of flange; thickness of web strain energy strainenergy density (strain energy per unit volume) modulus of resistance; modulus of toughness shear force, volume, vertical force or reaction deflection of a beam, velocity dv/dx, d 2v/dx 2, etc. force, weight, work load per unit of area (force per unit area) rectangular axes (origin at point O) rectangular axes (origin at centroid C) coordinates of centroid plastic modulus of the cross section of a beam angle, coefficient of thermal expansion, nondimensional ratio angle, nondimensional ratio, spring constant, stiffness rotational stiffness of a spring shear strain, weight density (weight per unit volume) shear strains in xy, yz, and zx planes shear strain with respect to x1y1 axes (rotated axes) shear strain for inclined axes deflection of a beam, displacement, elongation of a bar or spring temperature differential plastic displacement; yield displacement normal strain normal strains in x, y, and z directions normal strains in x1 and y1 directions (rotated axes) normal strain for inclined axes principal normal strains lateral strain in uniaxial stress thermal strain yield strain angle, angle of rotation of beam axis, rate of twist of a bar in torsion (angle of twist per unit length) Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd xx 3/1/12 6:41 PM Page xx Symbols θp θs κ ν ρ σ σx, σy, σz σx1, σy1 σθ σ1, σ2, σ3 σallow σcr σpl σr σT σU, σY τ τxy, τyz, τzx τx1y1 τθ τallow τU, τY φ ψ ω angle to a principal plane or to a principal axis angle to a plane of maximum shear stress curvature (κ ⫽ 1/ρ) distance, curvature shortening Poisson’s ratio radius, radius of curvature (ρ ⫽ 1/κ), radial distance in polar coordinates, mass density (mass per unit volume) normal stress normal stresses on planes perpendicular to x, y, and z axes normal stresses on planes perpendicular to x1y1 axes (rotated axes) normal stress on an inclined plane principal normal stresses allowable stress (or working stress) critical stress for a column (σcr ⫽ Pcr/A) proportionallimit stress residual stress thermal stress ultimate stress; yield stress shear stress shear stresses on planes perpendicular to the x, y, and z axes and acting parallel to the y, z, and x axes shear stress on a plane perpendicular to the x1 axis and acting parallel to the y1 axis (rotated axes) shear stress on an inclined plane allowable stress (or working stress) in shear ultimate stress in shear; yield stress in shear angle, angle of twist of a bar in torsion angle, angle of rotation angular velocity, angular frequency (ω ⫽ 2πf ) GREEK ALPHABET A B Γ ⌬ E Z H Θ I K Λ M α β γ δ ε ζ η θ ι κ λ μ Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu N Ξ O Π P Σ T ⌼ Φ X Ψ Ω ν ξ ο π ρ σ τ υ φ χ ψ ω Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega *A star attached to a section number indicates a specialized. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_00_fm_p00i001.qxd:77742_00_fm_p00i001.qxd 3/1/12 6:41 PM Page 1 Mechanics of Materials Eighth Edition, SI Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 1 3/2/12 2:16 PM Page 2 CHAPTER Tension, Compression, and Shear This telecommunications tower is an assemblage of many members that act primarily in tension or compression. (Péter budella/Shutterstock) Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:16 PM Page 3 I CHAPTER OVERVIEW In Chapter 1, we are introduced to mechanics of materials, which examines the stresses, strains, and displacements in bars of various materials acted on by axial loads applied at the centroids of their cross sections. After a brief review of basic concepts presented in statics, we will learn about normal stress (σ ) and normal strain (ε ) in materials used for structural applications, then identify key properties of various materials, such as the modulus of elasticity (E ) and yield (σy ) and ultimate (σu ) stresses, from plots of stress (σ ) versus strain (ε). We will also plot shear stress (τ ) versus shear strain (γ ) 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 (σ E • ε) and also for shear stress and strain (τ G • γ ). We will see that changes in lateral dimensions and volume depend upon Poisson’s ratio (ν). Material properties E, G, and ν, in fact, are directly related to one another and are not independent properties of the material. Assemblage of bars to form structures (such as trusses) leads to consideration of average shear (τ) and bearing (σb) 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 crosssectional area (if in compression). If we restrict maximum stresses at any point to allowable values by use of factors of safety, we can identify allowable levels of axial loads for simple systems, such as cables and bars. Factors 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 consider 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. Chapter 1 is organized as follows: 1.1 1.2 1.3 1.4 1.5 1.6 Introduction to Mechanics of Materials 4 Statics Review 6 Normal Stress and Strain 27 Mechanical Properties of Materials 37 Elasticity, Plasticity, and Creep 45 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio 52 1.7 Shear Stress and Strain 57 1.8 Allowable Stresses and Allowable Loads 68 1.9 Design for Axial Loads and Direct Shear 74 Chapter Summary & Review 80 Problems 83 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 4 3/2/12 2:16 PM Page 4 Chapter 1 Tension, Compression, and Shear 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 components due to the loads acting on them. If we can find these quantities for 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, buildings and bridges, machines and motors, or ships and spacecraft. That is why mechanics of materials is a basic subject in so many engineering fields. Statics and dynamics are also essential, but those subjects deal primarily with the forces and motions associated with particles and rigid bodies. However, most problems in mechanics of materials begin with an examination of the external and internal forces acting on a stable deformable body. We first define the loads acting on the body, along with its support conditions, then determine reaction forces at supports and internal forces in its members or elements using the basic laws of static equilibrium (provided that it is statically determinate). A wellconstructed freebody diagram is an essential part of the process of carrying out a proper static analysis of a structure. In mechanics of materials we go beyond the concepts presented in statics to study the stresses and strains inside real bodies, that is, bodies of finite dimensions that deform under loads. To determine the stresses and strains, we use the physical properties of the materials as well as numerous theoretical laws and concepts. Later, we will see that mechanics of materials provides additional essential information, based on the deformations of the body, to allow us to solve socalled statically indeterminate problems (not possible if using the laws of statics alone). Theoretical analyses and experimental results have equally important roles in mechanics of materials. We use theories to derive formulas and equations for predicting mechanical behavior, but these expressions cannot be used in practical design unless the physical properties of the materials are known. Such properties are available only after careful experiments have been carried out in the laboratory. Furthermore, not all practical problems are amenable to theoretical analysis alone, and in such cases physical testing is a necessity. The historical development of mechanics of materials is a fascinating blend of both theory and experiment—theory has pointed the way to useful results in some instances, and experiment has done so in others. Such famous persons as Leonardo da Vinci (1452–1519) and Galileo Galilei (1564–1642) performed experiments to determine the strength of wires, bars, and beams, although they did not develop adequate theories (by today’s standards) to explain their test results. By contrast, the famous Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:16 PM Page 5 1.1 Introduction to Mechanics of Materials 5 mathematician Leonhard Euler (1707–1783) developed the mathematical theory of columns and calculated the critical load of a column in 1744, long before any experimental evidence existed to show the significance of his results. Without appropriate tests to back up his theories, Euler’s results remained unused for over a hundred years, although today they are the basis for the design and analysis of most columns.* Problems When studying mechanics of materials, you will find that your efforts are divided naturally into two parts: first, understanding the logical development of the concepts, and second, applying those concepts to practical situations. The former is accomplished by studying the derivations, discussions, and examples that appear in each chapter, and the latter is accomplished by solving the problems at the ends of the chapters. Some of the problems are numerical in character, and others are symbolic (or algebraic). An advantage of numerical problems is that the magnitudes of all quantities are evident at every stage of the calculations, thus providing an opportunity to judge whether the values are reasonable or not. The principal advantage of symbolic problems is that they lead to generalpurpose formulas. A formula displays the variables that affect the final results; for instance, a quantity may actually cancel out of the solution, a fact that would not be evident from a numerical solution. Also, an algebraic solution shows the manner in which each variable affects the results, as when one variable appears in the numerator and another appears in the denominator. Furthermore, a symbolic solution provides the opportunity to check the dimensions at every stage of the work. Finally, the most important reason for solving algebraically is to obtain a general formula that can be used for many different problems. In contrast, a numerical solution applies to only one set of circumstances. Because engineers must be adept at both kinds of solutions, you will find a mixture of numeric and symbolic problems throughout this book. Numerical problems require that you work with specific units of measurement. This book utilizes the International System of Units (SI). A discussion of SI units appears in Appendix A, where you will also find many useful tables. All problems appear at the ends of the chapters, with the problem numbers and subheadings identifying the sections to which they belong. The techniques for solving problems are discussed in detail in Appendix B. In addition to a list of sound engineering procedures, Appendix B includes sections on dimensional homogeneity and significant digits. These topics are especially important, because every equation must be dimensionally homogeneous and every numerical result must be expressed with the proper number of significant digits. In this book, final numerical results are usually presented with three significant digits when a number begins with the digits 2 through 9, and with four significant digits when a number begins with the digit 1. Intermediate values are often recorded with additional digits to avoid losing numerical accuracy due to rounding of numbers. *The history of mechanics of materials, beginning with Leonardo and Galileo, is given in Refs. 11, 12, and 13. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 6 3/2/12 2:16 PM Page 6 Chapter 1 Tension, Compression, and Shear 1.2 STATICS REVIEW In your prerequisite course on statics, you studied the equilibrium of rigid bodies acted upon by a variety of different forces and supported or restrained in such a way that the body was stable and at rest. As a result, a properly restrained body could not undergo rigidbody motion due to the application of static forces. You drew freebody diagrams of the entire body, or of key parts of the body, and then applied the equations of equilibrium to find external reaction forces and moments or internal forces and moments at critical points. In this section, we will review the basic static equilibrium equations and apply them to the solution of example structures (both two and three dimensional) using both scalar and vector operations (both acceleration and velocity of the body will be assumed to be zero). Most problems in mechanics of materials require a static analysis as the first step, so all forces acting on the system and causing its deformation are known. Once all external and internal forces of interest have been found, we will be able to proceed with the evaluation of stresses, strains, and deformations of bars, shafts, beams, and columns in subsequent chapters. Equilibrium Equations The resultant force R and resultant moment M of all forces and moments acting on either a rigid or deformable body in equilibrium are both zero. The sum of the moments may be taken about any arbitrary point. The resulting equilibrium equations can be expressed in vector form as: R gF 0 (11) M gM g (r F) 0 (12) where F is one of a number of vectors of forces acting on the body and r is a position vector from the point at which moments are taken to a point along the line of application of any force F. It is often convenient to write the equilibrium equations in scalar form using a rectangular Cartesian coordinate system, either in two dimensions (x, y) or three dimensions (x, y, z) as gFx 0 gFy 0 gMz 0 (13) Eq. (13) can be used for twodimensional or planar problems, but in three dimensions, three force and three moment equations are required: gFx 0 gFy 0 gFz 0 (14) gMx 0 gMy 0 gMz 0 (15) If the number of unknown forces is equal to the number of independent equilibrium equations, these equations are sufficient to solve for all unknown reaction or internal forces in the body, and the problem is referred to as statically determinate (provided that the body is stable). If the body or structure is constrained by additional (or redundant) supports, it is statically indeterminate, and a solution is not possible using the laws of static equilibrium alone. For statically indeterminate structures, we must also examine the deformations of the structure, as will be discussed in the following chapters. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:16 PM Page 7 1.2 Statics Review 7 Fig. 11 Plane frame structure F q2 4 q0 3 e C E q1 D c b d 4 B 3 y a FB MA x A Applied Forces External loads applied to a body or structure may be either concentrated or distributed forces or moments. For example, force FB (with units of pounds, lb, or newtons, N) in Fig. 11 is a point or concentrated load and is assumed to act at point B on the body, while moment MA is a concentrated moment or couple (with units of lbft or N • m) acting at point A. Distributed forces may act alone or normal to a member and may have constant intensity, such as line load q1 normal to member BC (Fig. 11) or line load q2 acting in the y direction on inclined member DF; both q1 and q2 have units of force intensity (lb/ft or N/m). Distributed loads also may have a linear (or other) variation with some peak intensity q0 (as on member ED in Fig. 11). Surface pressures p (with units of lb/ft2 or Pa), such as wind acting on a sign (Fig. 12), act over a designated region of a body. Finally, a body force w (with units of force per unit volume, lb/ft3 or N/m3), Fig. 12 y Wind on sign p Ws P Wp H z x Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 8 3/2/12 2:16 PM Page 8 Chapter 1 Tension, Compression, and Shear such as the distributed selfweight of the sign or post in Fig. 12, acts throughout the volume of the body and can be replaced by the component weight W acting at the center of gravity (c.g.) of the sign (Ws ) or post (Wp ). In fact, any distributed loading (line, surface, or body force) can be replaced by a statically equivalent force at the center of gravity of the distributed loading when overall static equilibrium of the structure is evaluated using Eqs. (11) to (15). FreeBody Diagrams A freebody diagram (FBD) is an essential part of a static analysis of a rigid or deformable body. All forces acting on the body, or component part of the body, must be displayed on the FBD if a correct equilibrium solution is to be obtained. This includes applied forces and moments, reaction forces and moments, and any connection forces between individual components. For example, an overall FBD of the plane frame in Fig. 11 is shown in Fig. 13a; all applied and reaction forces are shown on this FBD and statically equivalent concentrated loads are displayed for all distributed loads. Statically equivalent forces Fq0, Fq1, and Fq2, each acting at the c.g. of the corresponding distributed loading, are used in the equilibrium equation solution to represent distributed loads q0, q1, and q2, respectively. Next, the plane frame has been disassembled in Fig. 13b, so that separate FBD’s can be drawn for each part of the frame, thereby exposing pinconnection forces at D (Dx, Dy). Both FBD’s must show all applied forces as well as reaction forces Ax and Ay at pinsupport joint A and Fx and Fy at pinsupport joint F. The forces transmitted between frame elements EDC and DF at pin connection D must be determined if the proper interaction of these two elements is to be accounted for in the static analysis. The plane frame structure in Fig. 11 will be analyzed in Example 12 to find reaction forces at joints A and F and also pinconnection forces at joint D using the equilibrium equations Eqs. (11) to (13). The FBD’s presented in Figs. 13a and 13b are essential parts of this solution process. A statics sign convention is usually employed in the solution for support reactions; forces acting in the positive directions of the coordinate axes are assumed positive, and the righthand rule is used for moment vectors. Reactive Forces and Support Conditions Proper restraint of the body or structure is essential if the equilibrium equations are to be satisfied. A sufficient number and arrangement of supports must be present to prevent rigid body motion under the action of static forces. A reaction force at a support is represented by a single arrow with a slash drawn through it (see Fig. 13) while a moment restraint at a support is shown as a doubleheaded or curved arrow with a slash. Reaction forces and moments usually result from the action of applied forces of the types described above (i.e., concentrated, distributed, surface, and body forces). A variety of different support conditions may be assumed depending on whether the problem is 2D or 3D. Supports A and F in the 2D plane frame structure shown in Fig. 11 and Fig. 13 are pin supports, Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:16 PM Page 9 1.2 Statics Review Fig. 13 F Fx Fq2 Fy Fq0 4 q0 9 3 (a) Overall FBD of plane frame structure from Fig. 11, and (b) Separate freebody diagrams of parts A through E and DF of the plane frame structure in Fig. 11 e C q1 D E Fq1 c d b 4 B 3 y a FB MA x A Ax Ay (a) Fq 2 Fx F q2 Fy Fq0 D q0 Dx Dy E Resultant D Dy Dx C D q1 Fq1 4 B 3 y FB MA x A Ax Ay Resultant A (b) while the base of the 3D sign structure in Fig. 12 may be considered to be a fixed or clamped support. Some of the most commonly used idealizations for 2D and 3D supports, as well as interconnections between members or elements of a structure, are illustrated in Table 11. The Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 10 3/2/12 2:16 PM Page 10 Chapter 1 Tension, Compression, and Shear Table 11 Reaction and Connection Forces in 2D or 3D Static Analysis Type of support or connection Simplified sketch of support or connection (1) Roller support— horizontal, vertical, or inclined Display of restraint forces and moments, or connection forces (a) Twodimensional roller support Horizontal roller support (constrains motion in both y and y directions) y x R y Bridge with roller support (The Earthquake Engineering Online Archive) Rx Vertical roller restraints x y x θ R Rotated or inclined roller support (b) Threedimensional roller support z y y x z (2) Pin support Ry x (a) Twodimensional pin support y x Rx Bridge with pin support (Courtesy of Joel Kerkhoff, P.Eng.) F Ry y Rx x Ry Pin support at F in Fig. 11 (b) Threedimensional pin support z z Pin support on old truss bridge (© Barry Goodno) Ry Rx Rz x y x y Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:16 PM Page 11 1.2 Statics Review (3) Sliding support 11 Table 11 (continued) y Mz Rx x Frictionless sleeve on vertical shaft (4) Clamped or fixed support A (a) Twodimensional fixed support A Weld y Mz x Rx Pole Ry Base plate y x Rx Concrete pier Fixed support at base of sign post (see Fig. 12) Mz Ry (b) Threedimensional fixed support y x Rz Mz Ry Rx Mx z My (5) Elastic or spring supports (a) Translational spring (k) y δy −kxδx kx ky δx x −kyδ y Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 12 3/2/12 2:16 PM Page 12 Chapter 1 Tension, Compression, and Shear Table 11 (continued) (b) Rotational spring (kr ) y θz kr x Rx Ry Mz = −kr θz (6) Pinned connection (from Figs. 11 and 13) D D Pin connection on old bridge (© Barry Goodno) Dx Dy Dy D x D Pinned connection at D between members EDC and DF in plane frame (Fig. 11) (7) Slotted connection (modified connection from that shown in Figs. 11 and 13) D Dy Dy D Alternate slotted connection at D on plane frame (Note that the plane frame in Fig. 11 is unstable if this slotted connection is used instead of a pin at D.) (8) Rigid connection (internal forces and moment in members joined at C of plane frame in Fig. 11) C Mc q1 Fq 1 Nc 4 B 3 Rigid connection at C on plane frame Vc Nc Vc Mc restraining or transmitted forces and moments associated with each type of support or connection are displayed in the third column of the table (these are not FBDs, however). The reactions forces and moments for the 3D sign structure in Fig. 12 are shown on the FBD in Fig. 14a; Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:16 PM Page 13 1.2 Statics Review Fig. 14 y y 13 (a) FBD of symmetric sign structure, and (b) FBD of eccentric sign structure Ws Ws P P Wp Wp H Mx Mx Rz z Ry x Rz z Mz (a) Ry x My (b) only reactions Ry, Rz, and Mx are nonzero because the sign structure and wind loading are symmetric with respect to the yz plane. If the sign is eccentric to the post (Fig. 14b), only reaction Rx is zero for the case of wind loading in the z direction. (See Prob. 1.716 at the end of Chapter 1 for a more detailed examination of the reaction forces due to wind pressure acting on the sign structure in Fig. 12; forces and stresses in the base plate bolts are also computed. Several eccentric sign structures are presented for analysis as end of chapter problems in Chapter 8.) Internal Forces (Stress Resultants) In our study of mechanics of materials, we will investigate the deformations of the members or elements which make up the overall deformable body. In order to compute the member deformations, we must first find the internal forces and moments (i.e., the internal stress resultants) at key points along the members of the overall structure. In fact, we will often create graphical displays of the internal axial force, torsional moment, transverse shear and bending moment along the axis of each member of the body so that we can readily identify critical points or regions within the structure. The first step is to make a section cut normal to the axis of each member so that a FBD can be drawn which displays the internal forces of interest. For example, if a cut is made at the top of member BC in the plane frame in Fig. 11, the internal axial force (Nc), transverse shear force (Vc) and bending moment (Mc) at joint C can be exposed as shown in the last row of Table 11. Fig. 15 shows two additional cuts made through members ED and DF Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 14 3/2/12 2:16 PM Page 14 Chapter 1 Tension, Compression, and Shear in the plane frame; the resulting FBD’s now can be used to find N, V, and M in members ED and DF of the plane frame. Stress resultants N, V, and M are usually taken along and normal to the member under consideration (i.e., local or member axes are used), and a deformation sign convention (e.g., tension is positive, compression is negative) is employed in their solution. In later chapters, we will see how these (and other) internal stress resultants are used to compute stresses in the member cross section. The following examples are presented as a review of application of the equations of static equilibrium in the solution for external reactions and internal forces in truss, beam, circular shaft, and frame structures. First, a truss structure is considered and both scalar and vector solutions for reaction forces are reviewed. Then member forces are computed using the method of joints. Properly drawn FBD’s are seen to be essential to the overall solution process. The second example involves static analysis of a beam structure to find reactions and internal forces at a particular section along the beam. In the third example, reactive and internal torsional moments in a stepped shaft are computed. And, finally, the fourth example presents the solution of the plane frame structure discussed here. Numerical values are assigned to applied forces and structure dimensions, and then reaction, pin connection, and selected internal forces in the frame are computed. Fig. 15 FBD’s for internal stress resultants in ED and DF Fq2 q2 M N Fq0 D q0 Dy M M N V N V Dy Fx F N Fy V FBDED E V M FBDDF Dx Dx C q1 D Fq2 B y FB x MA Ax A Ay Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:16 PM Page 15 15 1.2 Statics Review • • • Example 11 The plane truss shown in Fig. 16 is pin supported at A and has a roller support at B. Joint loads 2P and P are applied at joint C. Find support reactions at joints A and B, then solve for forces in members AB, AC and BC. Use numerical properties given below. Fig. 16 Example 11: Plane truss static analysis for joint loads Numerical data: y P 160 kN P C θc θA 60° b 2.2 m Solution L b A L 3m 2P θA = 60° θB B x c (1) Use the law of sines to find angles θB and θC , then find the length (c) of member AB. (2) Draw the FBD, then use equilibrium equations in scalar form [Eq. (13)] to find the support reactions. (3) Find member forces using the method of joints. (4) Repeat solution for support reactions using a vector solution. (5) Solve for support reactions and member forces for a 3D version of this plane (2D) truss. (1) Use the law of sines to find angles θB, θC then find the length (c) of member AB. See law of sines in Appendix C: Fig. 17 Example 11: FBD of plane truss b sin (θA) b 39.426° so θC 180° (θA θB ) 80.574° L and c L a sin (θA) sin (θC) b 3.417 m or c b cos (θA) L cos (θB) 3.417 m Note that the Law of Cosines also could be used: P C θB arcsina 2P θc c 3b2 L2 2bL cos (θC) 3.417 m b L θA = 60° Ax A θB B Note that the plane truss is statically determinate since there are (m r 6) unknowns (where m number of member forces and r number of reactions), but there are (2j 2 3 6) equations of statics from the method of joints (where j number of joints). c By Ay (2) Draw the FBD (Fig. 17), then use equilibrium equations in scalar form [Eq. (13)] to find the support reactions. Use equilibrium equations in scalar form to find support reactions. Sum moments about A to get reaction By : Fig. 18 Example 11: FBD of each joint of plane truss 2P FAC 230 kN Sum forces in y direction to get Ay : Ax 2 P 320 kN FBC FAB Ay c Sum forces in x direction to get Ax : FBC FAC Ax A [Pb cos (θA) (2 P)b sin (θA)] Ay P By 70 kN P C By B FAB By (3) Find member forces using the method of joints. Draw FBDs of each joint (Fig. 18) then sum forces in x and y directions to find member forces. Continues ➥ Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 16 ••• 3/2/12 2:16 PM Page 16 Chapter 1 Tension, Compression, and Shear Example 11  Continued Sum forces in y direction at joint A: FAC Ay 80.7 kN sin (θA) Sum forces in x direction at joint A: FAB Ax FAC cos (θA) 280 kN Sum forces in y direction at joint B: FBC By FBC 362 kN sin (θB ) Check equilibrium at joint C. (First at x direction, then y direction.) FAC cos (θA) FBC cos (θB) 2P 0 FAC sin (θA) FBC sin (θB) P 0 (4) Repeat solution for support reactions using a vector solution (show x, y, z components in vector format). Position vectors to B and C from A: c 3.4173 rAB £ 0 ≥ £ 0 ≥ m 0 0 b cos (θA) 1.1 rAC £ b sin (θA) ≥ £ 1.9053 ≥ m 0 0 Force vectors at A, B, and C: 0 2P Ax A £ A y ≥ B £ B y ≥ C £ P ≥ 0 0 0 Sum moments about point A, then equate each expression to zero: MA rAB B rAC C £ so By 0 0 ≥ 3.417 m By 785.7 m kN 785.7 230 kN 3.417 i j k i j k b 3 1 or Á 3 £ c 0 0≥ 3 4 § b 0¥ 4 785.68 k kN # m 3.4173 m By k 2 2 0 By 0 2P P 0 Now sum forces and equate each expression to zero: Ax 320 kN A B C £ Ay By 160 kN ≥ so Ax 320 kN 0 Ay 160 By 70 kN Reactions Ax , Ay , and By are the same as from the scalar solution approach. (5) Solve for support reactions and member forces for a 3D version of plane (2D) truss. To create a space truss from the plane truss, move joint A along the z axis a distance z while holding B on the x axis and constraining C to lie some Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:17 PM Page 17 1.2 Statics Review Fig. 19 Example 11: FBD of space truss (extended version of plane truss) y Cz b P C A Az z 2P θc y O z θA = 60° c Ax L x θB B Bz x 17 distance y along the y axis (see Fig. 19); hold member lengths (L, b, c) and angles (θA , θB , θC ) to the values used for the plane truss. Apply joint loads 2P and P at joint C. Add 3D pin support at A, two restraints at B (By , Bz ), and one restraint at C (Cz). Note that the space truss is statically determinate since there are (m r 9) unknowns (where m number of member forces and r number of reactions), but there are (3j 3 3 9) equations of statics from the method of joints (where j number of joints). First, find x, y, and z projections of members along coordinate axes. Then find angles OBC, OBA, and OAC in each plane. x B L2 b2 c2 2.81408 m 2 By z Ay y B L2 b2 c2 1.03968 m 2 L2 b2 c2 1.93883 m B 2 y OBC arctan a b 20.277° x z OBA arctan a b 34.566° x y OAC arctan a b 28.202° z Draw the overall FBD (see Fig. 19), then use a scalar solution to find reactions and member forces. (1) Sum moments about a line through A, which is parallel to the y axis (this will isolate reaction Bz, giving us one equation with one unknown): Bzx (2P )z 0 Bz 2P z 220 kN x This is based on a statics sign convention so the negative sign means that force Bz acts in the z direction. (2) Sum moments about the zaxis to find By , then sum forces in y direction to get Ay: By 2 P( y) x 118.2 kN so Ay P By 41.8 kN (3) Sum moments about the x axis to find Cz: Ay z Cz 77.9 kN y (4) Sum forces in the x and z directions to get Ax and Az: Ax 2P 320 kN Az Cz Bz 142.6 kN (5) Finally, use the method of joints to find member forces (a deformation sign convention is used here so positive () means tension and negative () means compression). Sum forces in x direction at joint A: x F Ax 0 c AB FAB c A x x FAB 389 kN Sum forces in y direction at joint A: y b FAC Ay 0 FAC b (Ay ) y FAC 88.4 kN Continues ➥ Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 18 ••• 3/2/12 2:17 PM Page 18 Chapter 1 Tension, Compression, and Shear Example 11  Continued Sum forces in y direction at joint B: y L F By 0 B FBC FBC 341 kN y y L BC Recompute reactions for the space truss using a vector solution. Find position (r) and unit (e) vectors from joint A to joints B and C: x rAB £ 0 ≥ z eAB 0.823 £ 0 ≥ rAB 0.567 0 rAC £ y ≥ z eAC 0 £ 0.473 ≥ rAC  0.881 rAB rAC Sum moments about point A, then equate each expression to zero: MA rAB B rAC C 0 2P MA rAB £ By ≥ rAC £ P ≥ Cz Bz £ i or 3 £ x 0 1.9388 m By 1.0397 m Cz 310.21 kN # m 2.8141 m Bz 620.43 kN # m ≥ 2.8141 m By 332.7 kN # m j k 0 z ≥ 3 1.9388 m By i 2.8141 m Bz j 2.8141 m By k By Bz i and 3 £ 0 2P j y P k z ≥ 3 1.0397 m Cz i 310.21 kN # m i Cz 620.43 kN # m j 332.7 kN # m k Collecting coefficients of j and solving: 620.43 220 kN 2.8141 Collecting coefficients of k and solving: Bz 332.7 118.2 kN 2.8141 Collecting coefficients of i and solving: By 310.21 1.9388 By 77.9 kN 1.0397 Complete the solution by summing forces and equating to zero: Cz 0 2P Ax 320.0 kN Ax £ Ay ≥ £ By ≥ £ P ≥ £ Ay41.8 kN ≥ Az Bz Cz Az 142.6 Ax 320 kN Ay 41.8 kN Az 142.6 kN Reactions Ax , Ay , Az and By , Bz are the same as from the scalar solution approach. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:17 PM Page 19 1.2 Statics Review 19 • • • Example 12 The simplysupported beam structure shown in Fig. 110 is subjected to moment MA at pinsupported joint A, inclined load FB applied at joint B, and uniform load with intensity q1 on member segment BC. Find support reactions at joints A and C, then solve for internal forces at the midpoint of BC. Use properly drawn freebody diagrams in your solution. Fig. 110 4 3 Example 12: Beam static analysis for support reactions FB MA q1 A C B x a b Numerical data (Newtons and meters): a 3m b 2m MA 380 N # m FB 200 N q1 160 N/m Solution (1) Draw the FBD of the overall beam. The solution for reaction forces at A and C must begin with a proper drawing of the FBD of the overall beam (Fig. 111). The FBD shows all applied and reactive forces. Fig. 111 b/2 Example 12: FBD of beam MA FB A (4/5)FB Fq1 (3/5)FB q1 C Ax B Ay a Cy b (2) Determine statically equivalent concentrated forces. Distributed forces are replaced by their statical equivalents (Fq1) and the components of the inclined concentrated force at B may also be computed: Fq1 q1 b 320 N FBx 4 F 160 N 5 B FBy 3 F 120 N 5 B ➥ (3) Sum the moments about A to find reaction force Cy . This structure is statically determinate because there are three available equations from statics (ΣFx 0, ΣFy 0, and ΣM 0) and three reaction unknowns (Ax, Ay , Cy ). It is convenient to start the static analysis using ΣMA 0, because we can isolate one equation with one unknown and then easily find reaction Cy . A statics sign convention is used (i.e., righthand rule or CCW is positive). Cy 1 b cMA FBya Fq1 aa b d 260 N 2 (a b) ➥ Continues ➥ Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 20 ••• 3/2/12 2:17 PM Page 20 Chapter 1 Tension, Compression, and Shear Example 12  Continued (4) Sum the forces in x and y directions to find reaction forces at A. Now that Cy is known, we can complete the equilibrium analysis to find Ax and Ay using ΣFx 0 and ΣFy 0. Then we can find the resultant reaction force at A using components Ax and Ay : Sum forces in x direction: Ax FBx 0 Ax FBx Ax 160 N Sum forces in y direction: Ay FBy Cy Fq1 0 Ay FBy Cy Fq1 Ay 60 N Resultant force at A: A 4A2x A2y A 171 N (5) Find the internal forces and moment at the midpoint of member segment BC. Now that reaction forces at A and C are known, we can cut a section through the beam midway between B and C, creating left and right FBDs (Fig. 112). Section forces Nc (axial) and Vc (shear) as well as section moment (Mc ) are exposed and may be computed using statics. Either FBD may be used to find Nc , Vc , and Mc ; the computed internal forces and moment Nc , Vc , and Mc will be the same. Calculations based on left FBD: ©Fx 0 N FBx Ax 0 N ©Fy 0 b V Ay FBy q1 a b 100 N 2 ©M 0 M MA Ay aa b b b b b FBy a b q1 a b a b 180 N # m 2 2 2 4 Calculations based on right FBD: ©Fx 0 N 0 b V q1 a b Cy 100 N 2 b b b ©M 0 M Cy a b q1 a b a b 180 N # m 2 2 4 ©Fy 0 The computed internal forces (N and V ) and internal moment (M) are the same and can be determined using either the left or right FBD. This applies for any section taken through the beam at any point along its length. Later, we will create plots or diagrams which show the variation of N, V, and M over the length of the beam. These diagrams will be very useful in the design of beams, because they readily show the critical regions of the beam where N, V, and M have maximum values. Fig. 112 b/2 Example 12: Left and right FBDs of beam MA A FB (3/5)FB b/2 q1 V (4/5)FB Ax M N B q1 M C N V Ay Cy a Left FBD Right FBD Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:17 PM Page 21 21 1.2 Statics Review • • • Example 13 A stepped circular shaft is fixed at A and has three gears that transmit the torques shown in Fig. 113. Find the reaction torque at A, then find the internal torsional moments in segments AB, BC, and CD. Use properly drawn freebody diagrams in your solution. Fig. 113 Example 13: Stepped circular shaft in torsion 1900 N·m A 1000 N·m 550 N·m B C x D Solution (1) Draw the FBD of the overall shaft structure. The cantilever shaft structure is statically determinate. The solution for the reaction moment at A must begin with a proper drawing of the FBD of the overall structure (Fig. 114). The FBD shows all applied and reactive torques. Fig. 114 Example 13: FBD of overall shaft MAx 1900 N·m 1000 N·m 550 N·m x A B C D (2) Sum the moments about the x axis to find the reaction moment MAx. This structure is statically determinate because there is one available equation from statics (ΣM x 0) and one reaction unknown (MAx). A statics sign convention is used (i.e., righthand rule or CCW is positive). MAx 1900 N # m 1000 N # m 550 N # m 0 MAx (1900 N # m 1000 N # m 550 N # m) 350 N # m The computed result for MAx is positive, so the reaction moment vector is in the positive x direction as assumed. (3) Find the internal torsional moments in each segment of the shaft. Now that reaction moment MAx is known, we can cut a section through the shaft in each segment creating left and right FBDs (Fig. 115). Internal Continues ➥ Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 22 ••• 3/2/12 2:17 PM Page 22 Chapter 1 Tension, Compression, and Shear Example 13  Continued torsional moments then may be computed using statics. Either FBD may be used; the computed internal torsional moment will be the same. Find the internal torque TAB (Fig. 115a). Fig. 115a Example 13: Left and right FBDs of shaft for each segment MAx A TAB 1900 N·m 1000 N·m 550 N·m x B C D Left FBD Right FBD (a) Left FBD: TAB MAx 350 N # m Right FBD: TAB 1900 N # m 1000 N # m 550 N # m 350 N # m Find the internal torque TBC (Fig. 115b). Fig. 115b 1900 N·m MAx A TBC B 1000 N·m 550 N·m x C D Left FBD Right FBD (b) Left FBD: TBC MAx 1900 N # m 1550 N # m Right FBD: TBC 1000 N # m 550 N # m 1550 N # m Find internal torque TCD (Fig. 115c). Fig. 115c 1900 N·m MAx A 1000 N·m B C TCD Left FBD 550 N·m x D Right FBD (c) Left FBD: TCD MAx 1900 N # m 1000 N # m 550 N # m Right FBD: TCD 550 N # m In each segment, the internal torsional moments computed using either the left or right FBDs are the same. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:18 PM Page 23 23 1.2 Statics Review • • • Example 14 The plane frame in Fig. 116 is a modified version of that shown in Fig. 11. Initially, member DF has been replaced with a roller support at D. Moment MA is applied at pinsupported joint A, and load FB is applied at joint B. A uniform load with intensity q1 acts on member BC, and a linearly distributed load with peak intensity q0 is applied downward on member ED. Find the support reactions at joints A and D, then solve for internal forces at the top of member BC. Use numerical properties given. As a final step, remove the roller at D, insert member DF (as shown in Fig. 11) and reanalyze the structure to find the reaction forces at A and F. Fig. 116 q0 Example 14: Plane frame static analysis for support reactions D C q1 E c b d B 4 3 y a FB MA x A Numerical data (Newtons and meters): a 3m MA 380 N # m b 2m FB 200 N c 6m d 2.5 m q0 80 N/m q1 160 N/m Solution (1) Draw the FBD of the overall frame. The solution for reaction forces at A and D must begin with a proper drawing of the FBD of the overall frame (Fig. 117). The FBD shows all applied and reactive forces. Fig. 117 q0 Example 14: FBD of plane frame Fq 0 D C q1 E Dy c Fq d b 4 B 3 y 1 a FB x MA A Ay Ax Continues ➥ Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 24 ••• 3/2/12 2:18 PM Page 24 Chapter 1 Tension, Compression, and Shear Example 14  Continued (2) Determine the statically equivalent concentrated forces. Distributed forces are replaced by their statical equivalents (Fq0 and Fq1). The components of the inclined concentrated force at B also may be computed: 1 q c 240 N 2 0 4 F 160 N 5 B Fq0 FBx Fq1 q1 b 320 N FBy 3 F 120 N 5 B (3) Sum the moments about A to find reaction force Dy . This structure is statically determinate because there are three available equations from statics (ΣFx 0, ΣFy 0, and ΣM 0) and three reaction unknowns ( Ax , Ay , Dy ). It is convenient to start the static analysis using ΣMA 0, because we can isolate one equation with one unknown and then easily find reaction Dy . Dy 1 b 2 c MA FBx a Fq1 aa b Fq0 ad cb d 152 N 2 d 3 (4) Sum the forces in the x and y directions to find the reaction forces at A. Now that Dy is known, we can find Ax and Ay using ΣFx 0 and ΣFy 0, and then find the resultant reaction force at A using components Ax and Ay. Sum forces in x direction: Ax FBx Fq1 0 Ax FBx Fq1 Ax 160 N Sum forces in y direction: Ay FBy Dy Fq0 0 Ay FBy Dy Fq0 Ay 208 N Resultant force at A: A 4A2x A2y A 262 N (5) Find the internal forces and moment at the top of member BC. Now that reaction forces at A and D are known, we can cut a section through the frame just below joint C, creating upper and lower FBDs (Fig. 118). Fig. 118 Example 14: Upper and lower FBDs of plane frame Fq0 D E c Dy C Mc d Vc Nc Nc Mc Vc b/2 Fq1 b/2 B y FB a 3 x MA 4 A Ax Ay Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 77742_01_ch01_p002121.qxd:77742_01_ch01_p002121.qxd 3/2/12 2:18 PM Page 25 25 1.2 Statics Review Section forces Nc (axial) and Vc (shear) as well as section moment (Mc) are exposed and may be computed using statics. Either FBD may be used to find Nc , Vc , and Mc ; the computed stress resultants Nc , Vc , and Mc will be the same. Calculations based on upper FBD: ©Fx 0 Vc 0 ©Fy 0 Nc Dy Fq0 88 N ©Mc 0 Mc Dyd Fq0 ad 2 cb 1180 N # m 3 Calculations based on lower FBD: ©Fx 0 Vc Fq1 FBx Ax 0 ©Fy 0 Nc FBy Ay 88 N ©Mc 0 Mc Fq1 b FBx b Ax(a b) MA 1180 N # m 2 (6) Remove the roller at D, insert member DF (as shown in Fig. 11) and reanalyze the structure to find reaction forces at A and F. Member DF is pinconnected to EDC at D, has a pin support at F, and carries uniform load q2 in the y direction. See Figs. 13a and 13b for the FBDs required in the solution. Note that there are now four unknown reaction forces (Ax, Ay, Fx and Fy ) but only three equilibrium equations available (Σ Fx 0, Σ Fy 0, Σ M 0) for use with the overall FBD in Fig. 13a. To find another equation, we will have to separate the structure at pin connection D to take advantage of the fact that the moment at D is known to be zero (friction effects are assumed to be negligible); we can then use Σ MD 0 for either the upper FBD or the lower FBD in Fig. 13b to develop one more independent equation of statics. Recall that a statics sign convention is used here for all equilibrium equations. Dimensions and loads for new member DF: e 5 m ex 3 4 e 3 m ey e 4 m 5 5 q2 180 N/m Fq2 q2e 900 N First, write equilibrium equations for the entire