Mechanics of Materials, Seventh Edition PDF

 Seventh Edition Mechanics of Materials Ferdinand P. Beer Late of Lehigh University E. Russell Johnston, Jr. Late of University of Connecticut John T. DeWolf University of Connecticut David F. Mazurek United...

 Seventh Edition Mechanics of Materials Ferdinand P. Beer Late of Lehigh University E. Russell Johnston, Jr. Late of University of Connecticut John T. DeWolf University of Connecticut David F. Mazurek United States Coast Guard Academy MECHANICS OF MATERIALS, SEVENTH EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2015 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2012, 2009, 2006, and 2002. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 QVR/QVR 1 0 9 8 7 6 5 4 3 2 1 0 ISBN 978-0-07-339823-5 MHID 0-07-339823-3 Senior Vice President, Products & Markets: Kurt L. Strand Vice President, General Manager: Marty Lange Vice President, Content Production & Technology Services: Kimberly Meriwether David Editorial Director: Thomas Timp Global Brand Manager: Raghothaman Srinivasan Brand Manager: Bill Stenquist Marketing Manager: Heather Wagner Product Developer: Robin Reed Director, Content Production: Terri Schiesl Content Project Manager: Jolynn Kilburg Buyer: Nichole Birkenholz Media Project Manager: Sandra Schnee Photo Research: Carrie K. Burger In-House Designer: Matthew Backhaus Cover Designer: Matt Backhaus Cover Image Credit: ©Walter Bibikow Compositor: RPK Editorial Services, Inc. Typeface: 9.5/12 Utopia Std Printer: Quad/Graphics All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. The photo on the cover shows the steel sculpture “Venture” by Alex Liberman (1912-1999) in front of the Bank of America Building in Dallas, Texas. The building is supported by a combination of structural steel and reinforced concrete. Library of Congress Cataloging-in-Publication Data on File The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. www.mhhe.com  About the Authors John T. DeWolf, Professor of Civil Engineering at the University of Con- necticut, joined the Beer and Johnston team as an author on the second edition of Mechanics of Materials. John holds a B.S. degree in civil engi- neering from the University of Hawaii and M.E. and Ph.D. degrees in structural engineering from Cornell University. He is a Fellow of the Amer- ican Society of Civil Engineers and a member of the Connecticut Academy of Science and Engineering. He is a registered Professional Engineer and a member of the Connecticut Board of Professional Engineers. He was selected as a University of Connecticut Teaching Fellow in 2006. Profes- sional interests include elastic stability, bridge monitoring, and structural analysis and design. David F. Mazurek, Professor of Civil Engineering at the United States Coast Guard Academy, joined the Beer and Johnston team as an author on the fifth edition. David holds a B.S. degree in ocean engineering and an M.S. degree in civil engineering from the Florida Institute of Technol- ogy, and a Ph.D. degree in civil engineering from the University of Con- necticut. He is a registered Professional Engineer. He has served on the American Railway Engineering & Maintenance of Way Association’s Com- mittee 15—Steel Structures since 1991. He is a Fellow of the American Society of Civil Engineers, and was elected into the Connecticut Academy of Science and Engineering in 2013. Professional interests include bridge engineering, structural forensics, and blast-resistant design. iii  Contents Preface ix Guided Tour xiii List of Symbols xv 1 Introduction—Concept of Stress 3 1.1 Review of The Methods of Statics 4 1.2 Stresses in the Members of a Structure 7 1.3 Stress on an Oblique Plane Under Axial Loading 27 1.4 Stress Under General Loading Conditions; Components of Stress 28 1.5 Design Considerations 31 Review and Summary 44 2 Stress and Strain—Axial Loading 55 2.1 An Introduction to Stress and Strain 57 2.2 Statically Indeterminate Problems 78 2.3 Problems Involving Temperature Changes 82 2.4 Poisson’s Ratio 94 2.5 Multiaxial Loading: Generalized Hooke’s Law 95 *2.6 Dilatation and Bulk Modulus 97 2.7 Shearing Strain 99 2.8 Deformations Under Axial Loading—Relation Between E, n, and G 102 *2.9 Stress-Strain Relationships For Fiber-Reinforced Composite Materials 104 2.10 Stress and Strain Distribution Under Axial Loading: Saint- Venant’s Principle 115 2.11 Stress Concentrations 117 2.12 Plastic Deformations 119 *2.13 Residual Stresses 123 Review and Summary 133 *Advanced or specialty topics iv  Contents v 3 Torsion 147 3.1 Circular Shafts in Torsion 150 3.2 Angle of Twist in the Elastic Range 167 3.3 Statically Indeterminate Shafts 170 3.4 Design of Transmission Shafts 185 3.5 Stress Concentrations in Circular Shafts 187 *3.6 Plastic Deformations in Circular Shafts 195 *3.7 Circular Shafts Made of an Elastoplastic Material 196 *3.8 Residual Stresses in Circular Shafts 199 *3.9 Torsion of Noncircular Members 209 *3.10 Thin-Walled Hollow Shafts 211 Review and Summary 223 4 Pure Bending 237 4.1 Symmetric Members in Pure Bending 240 4.2 Stresses and Deformations in the Elastic Range 244 4.3 Deformations in a Transverse Cross Section 248 4.4 Members Made of Composite Materials 259 4.5 Stress Concentrations 263 *4.6 Plastic Deformations 273 4.7 Eccentric Axial Loading in a Plane of Symmetry 291 4.8 Unsymmetric Bending Analysis 302 4.9 General Case of Eccentric Axial Loading Analysis 307 *4.10 Curved Members 319 Review and Summary 334 5 Analysis and Design of Beams for Bending 345 5.1 Shear and Bending-Moment Diagrams 348 5.2 Relationships Between Load, Shear, and Bending Moment 360 5.3 Design of Prismatic Beams for Bending 371 *5.4 Singularity Functions Used to Determine Shear and Bending Moment 383 *5.5 Nonprismatic Beams 396 Review and Summary 407 vi Contents 6 Shearing Stresses in Beams and Thin-Walled Members 417 6.1 Horizontal Shearing Stress in Beams 420 *6.2 Distribution of Stresses in a Narrow Rectangular Beam 426 6.3 Longitudinal Shear on a Beam Element of Arbitrary Shape 437 6.4 Shearing Stresses in Thin-Walled Members 439 *6.5 Plastic Deformations 441 *6.6 Unsymmetric Loading of Thin-Walled Members and Shear Center 454 Review and Summary 467 7 Transformations of Stress and Strain 477 7.1 Transformation of Plane Stress 480 7.2 Mohr’s Circle for Plane Stress 492 7.3 General State of Stress 503 7.4 Three-Dimensional Analysis of Stress 504 *7.5 Theories of Failure 507 7.6 Stresses in Thin-Walled Pressure Vessels 520 *7.7 Transformation of Plane Strain 529 *7.8 Three-Dimensional Analysis of Strain 534 *7.9 Measurements of Strain; Strain Rosette 538 Review and Summary 546 8 Principal Stresses Under a Given Loading 557 8.1 Principal Stresses in a Beam 559 8.2 Design of Transmission Shafts 562 8.3 Stresses Under Combined Loads 575 Review and Summary 591  Contents vii 9 Deflection of Beams 599 9.1 Deformation Under Transverse Loading 602 9.2 Statically Indeterminate Beams 611 *9.3 Singularity Functions to Determine Slope and Deflection 623 9.4 Method of Superposition 635 *9.5 Moment-Area Theorems 649 *9.6 Moment-Area Theorems Applied to Beams with Unsymmetric Loadings 664 Review and Summary 679 10 Columns 691 10.1 Stability of Structures 692 *10.2 Eccentric Loading and the Secant Formula 709 10.3 Centric Load Design 722 10.4 Eccentric Load Design 739 Review and Summary 750 759 11 Energy Methods 11.1 Strain Energy 760 11.2 Elastic Strain Energy 763 11.3 Strain Energy for a General State of Stress 770 11.4 Impact Loads 784 11.5 Single Loads 788 *11.6 Multiple Loads 802 *11.7 Castigliano’s Theorem 804 *11.8 Deflections by Castigliano’s Theorem 806 *11.9 Statically Indeterminate Structures 810 Review and Summary 823 viii Contents Appendices A1 A Moments of Areas A2 B Typical Properties of Selected Materials Used in Engineering A13 C Properties of Rolled-Steel Shapes A17 D Beam Deflections and Slopes A29 E Fundamentals of Engineering Examination A30 Answers to Problems AN1 Photo Credits C1 Index I1  Preface Objectives The main objective of a basic mechanics course should be to develop in the engineering stu- dent the ability to analyze a given problem in a simple and logical manner and to apply to its solution a few fundamental and well-understood principles. This text is designed for the first course in mechanics of materials—or strength of materials—offered to engineering students in the sophomore or junior year. The authors hope that it will help instructors achieve this goal in that particular course in the same way that their other texts may have helped them in statics and dynamics. To assist in this goal, the seventh edition has undergone a complete edit of the language to make the book easier to read. General Approach In this text the study of the mechanics of materials is based on the understanding of a few basic concepts and on the use of simplified models. This approach makes it possible to develop all the necessary formulas in a rational and logical manner, and to indicate clearly the conditions under which they can be safely applied to the analysis and design of actual engineering struc- tures and machine components. Free-body Diagrams Are Used Extensively. Throughout the text free-body diagrams are used to determine external or internal forces. The use of “picture equations” will also help the students understand the superposition of loadings and the resulting stresses and deformations. NEW The SMART Problem-Solving Methodology is Employed. New to this edition of the text, students are introduced to the SMART approach for solving engineering problems, whose acronym reflects the solution steps of Strategy, Modeling, Analysis, and Reflect & T hink. This methodology is used in all Sample Problems, and it is intended that students will apply this approach in the solution of all assigned problems. Design Concepts Are Discussed Throughout the Text Whenever Appropriate. A dis- cussion of the application of the factor of safety to design can be found in Chap. 1, where the concepts of both allowable stress design and load and resistance factor design are presented. A Careful Balance Between SI and U.S. Customary Units Is Consistently Main- tained. Because it is essential that students be able to handle effectively both SI metric units and U.S. customary units, half the concept applications, sample problems, and problems to be assigned have been stated in SI units and half in U.S. customary units. Since a large number of problems are available, instructors can assign problems using each system of units in what- ever proportion they find desirable for their class. Optional Sections Offer Advanced or Specialty Topics. Topics such as residual stresses, torsion of noncircular and thin-walled members, bending of curved beams, shearing stresses in non-symmetrical members, and failure criteria have been included in optional sections for use in courses of varying emphases. To preserve the integrity of the subject, these topics are presented in the proper sequence, wherever they logically belong. Thus, even when not ix x Preface covered in the course, these sections are highly visible and can be easily referred to by the students if needed in a later course or in engineering practice. For convenience all optional sections have been indicated by asterisks. Chapter Organization It is expected that students using this text will have completed a course in statics. However, Chap. 1 is designed to provide them with an opportunity to review the concepts learned in that course, while shear and bending-moment diagrams are covered in detail in Secs. 5.1 and 5.2. The properties of moments and centroids of areas are described in Appendix A; this material can be used to reinforce the discussion of the determination of normal and shearing stresses in beams (Chaps. 4, 5, and 6). The first four chapters of the text are devoted to the analysis of the stresses and of the corresponding deformations in various structural members, considering successively axial load- ing, torsion, and pure bending. Each analysis is based on a few basic concepts: namely, the conditions of equilibrium of the forces exerted on the member, the relations existing between stress and strain in the material, and the conditions imposed by the supports and loading of the member. The study of each type of loading is complemented by a large number of concept applications, sample problems, and problems to be assigned, all designed to strengthen the students’ understanding of the subject. The concept of stress at a point is introduced in Chap. 1, where it is shown that an axial load can produce shearing stresses as well as normal stresses, depending upon the section considered. The fact that stresses depend upon the orientation of the surface on which they are computed is emphasized again in Chaps. 3 and 4 in the cases of torsion and pure bending. However, the discussion of computational techniques—such as Mohr’s circle—used for the transformation of stress at a point is delayed until Chap. 7, after students have had the oppor- tunity to solve problems involving a combination of the basic loadings and have discovered for themselves the need for such techniques. The discussion in Chap. 2 of the relation between stress and strain in various materials includes fiber-reinforced composite materials. Also, the study of beams under transverse loads is covered in two separate chapters. Chapter 5 is devoted to the determination of the normal stresses in a beam and to the design of beams based on the allowable normal stress in the material used (Sec. 5.3). The chapter begins with a discussion of the shear and bending- moment diagrams (Secs. 5.1 and 5.2) and includes an optional section on the use of singularity functions for the determination of the shear and bending moment in a beam (Sec. 5.4). The chapter ends with an optional section on nonprismatic beams (Sec. 5.5). Chapter 6 is devoted to the determination of shearing stresses in beams and thin-walled members under transverse loadings. The formula for the shear flow, q 5 VQyI, is derived in the traditional way. More advanced aspects of the design of beams, such as the determination of the principal stresses at the junction of the flange and web of a W-beam, are considered in Chap. 8, an optional chapter that may be covered after the transformations of stresses have been discussed in Chap. 7. The design of transmission shafts is in that chapter for the same reason, as well as the determination of stresses under combined loadings that can now include the determination of the principal stresses, principal planes, and maximum shearing stress at a given point. Statically indeterminate problems are first discussed in Chap. 2 and considered through- out the text for the various loading conditions encountered. Thus, students are presented at an early stage with a method of solution that combines the analysis of deformations with the conventional analysis of forces used in statics. In this way, they will have become thoroughly familiar with this fundamental method by the end of the course. In addition, this approach helps the students realize that stresses themselves are statically indeterminate and can be com- puted only by considering the corresponding distribution of strains.  Preface xi The concept of plastic deformation is introduced in Chap. 2, where it is applied to the analysis of members under axial loading. Problems involving the plastic deformation of circu- lar shafts and of prismatic beams are also considered in optional sections of Chaps. 3, 4, and 6. While some of this material can be omitted at the choice of the instructor, its inclusion in the body of the text will help students realize the limitations of the assumption of a linear stress-strain relation and serve to caution them against the inappropriate use of the elastic torsion and flexure formulas. The determination of the deflection of beams is discussed in Chap. 9. The first part of the chapter is devoted to the integration method and to the method of superposition, with an optional section (Sec. 9.3) based on the use of singularity functions. (This section should be used only if Sec. 5.4 was covered earlier.) The second part of Chap. 9 is optional. It presents the moment-area method in two lessons. Chapter 10, which is devoted to columns, contains material on the design of steel, alumi- num, and wood columns. Chapter 11 covers energy methods, including Castigliano’s theorem. Supplemental Resources for Instructors Find the Companion Website for Mechanics of Materials at www.mhhe.com/beerjohnston. Included on the website are lecture PowerPoints, an image library, and animations. On the site you’ll also find the Instructor’s Solutions Manual (password-protected and available to instruc- tors only) that accompanies the seventh edition. The manual continues the tradition of excep- tional accuracy and normally keeps solutions contained to a single page for easier reference. The manual includes an in-depth review of the material in each chapter and houses tables designed to assist instructors in creating a schedule of assignments for their courses. The various topics covered in the text are listed in Table I, and a suggested number of periods to be spent on each topic is indicated. Table II provides a brief description of all groups of problems and a classification of the problems in each group according to the units used. A Course Organization Guide providing sample assignment schedules is also found on the website. Via the website, instructors can also request access to C.O.S.M.O.S., the Complete Online Solutions Manual Organization System that allows instructors to create custom homework, quizzes, and tests using end-of-chapter problems from the text. McGraw-Hill Connect Engineering provides online presentation, assignment, and assessment solutions. It connects your students with the tools and resources they’ll need to achieve success. With Connect Engineering you can deliver assignments, quizzes, and tests online. A robust set of questions and activities are presented and aligned with the textbook’s learning outcomes. As an instructor, you can edit existing questions and author entirely new problems. Integrate grade reports easily with Learning Management Systems (LMS), such as WebCT and Black- ® board—and much more. ConnectPlus Engineering provides students with all the advantages of Connect Engineering, plus 24/7 online access to a media-rich eBook, allowing seamless integration of text, media, and assessments. To learn more, visit www.mcgrawhillconnect.com. McGraw-Hill LearnSmart is available as a standalone product or an integrated feature of McGraw-Hill Connect Engineering. It is an adap- tive learning system designed to help students learn faster, study more efficiently, and retain more knowledge for greater success. LearnSmart assesses a student’s knowledge of course con- tent through a series of adaptive questions. It pinpoints concepts the student does not under- stand and maps out a personalized study plan for success. This innovative study tool also has features that allow instructors to see exactly what students have accomplished and a built-in assessment tool for graded assignments. Visit the following site for a demonstration. www. LearnSmartAdvantage.com xii Preface Powered by the intelligent and adaptive LearnSmart engine, SmartBook is the first and only continuously adaptive reading experience available today. Distinguishing what students know from what they don’t, and honing in on concepts they are most likely to forget, SmartBook personalizes content for each student. Reading is no longer a passive and linear experience but an engaging and dynamic one, where students are more likely to master and retain important concepts, coming to class better prepared. SmartBook includes powerful reports that identify specific topics and learning objectives students need to study. Craft your teaching resources to match the way you teach! With McGraw- Hill Create, www.mcgrawhillcreate.com, you can easily rearrange chapters, combine material from other content sources, and quickly upload your original content, such as a course syllabus or teaching notes. Arrange your book to fit your teaching style. Create even allows you to per- sonalize your book’s appearance by selecting the cover and adding your name, school, and course information. Order a Create book and you’ll receive a complimentary print review copy in 3–5 business days or a complimentary electronic review copy (eComp) via email in minutes. Go to www.mcgrawhillcreate.com today and register to experience how McGraw-Hill Create empowers you to teach your students your way. Acknowledgments The authors thank the many companies that provided photographs for this edition. We also wish to recognize the efforts of the staff of RPK Editorial Services, who diligently worked to edit, typeset, proofread, and generally scrutinize all of this edition’s content. Our special thanks go to Amy Mazurek (B.S. degree in civil engineering from the Florida Institute of Technology, and a M.S. degree in civil engineering from the University of Connecticut) for her work in the checking and preparation of the solutions and answers of all the problems in this edition. We also gratefully acknowledge the help, comments, and suggestions offered by the many reviewers and users of previous editions of Mechanics of Materials. John T. DeWolf David F. Mazurek  Guided Tour Chapter Introduction. Each chapter begins with an introductory section that sets up the purpose and goals of the chapter, describing in simple terms the material that will be covered and its application to the solution of engineering problems. Chapter Objectives provide students with a preview of chap- Introduction— 1 ter topics. Concept of Stress Stresses occur in all structures subject to loads. This chapter will examine simple states of stress in elements, such as in the two-force members, bolts and pins used in the structure shown. Chapter Lessons. The body of the text is divided Objectives Review of statics needed to determine forces in members of into units, each consisting of one or several theory simple structures. Introduce concept of stress. Define different stress types: axial normal stress, shearing stress and bearing stress. sections, Concept Applications, one or several Discuss engineer’s two principal tasks, namely, the analysis and design of structures and machines. Develop problem solving approach. Sample Problems, and a large number of homework Discuss the components of stress on different planes and under different loading conditions. Discuss the many design considerations that an engineer should review before preparing a design. problems. The Companion Website contains a Course Organization Guide with suggestions on each chapter lesson. bee98233_ch01_002-053.indd 2-3 11/8/13 1:45 PM Concept Application 1.1 Concept Applications. Concept Appli- Considering the structure of Fig. 1.1 on page 5, assume that rod BC is made of a steel with a maximum allowable stress sall 5 165 MPa. Can cations are used extensively within individ- rod BC safely support the load to which it will be subjected? The mag- nitude of the force FBC in the rod was 50 kN. Recalling that the diam- eter of the rod is 20 mm, use Eq. (1.5) to determine the stress created ual theory sections to focus on specific in the rod by the given loading. topics, and they are designed to illustrate P 5 FBC 5 150 kN 5 150 3 103 N 20 mm 2 A 5 pr2 5 pa b 5 p110 3 1023 m2 2 5 314 3 1026 m2 specific material being presented and facili- 2 P 150 3 103 N s5 5 5 1159 3 106 Pa 5 1159 MPa tate its understanding. A 314 3 1026 m2 Since s is smaller than sall of the allowable stress in the steel used, rod BC can safely support the load. Sample Problems. The Sample Prob- lems are intended to show more compre- hensive applications of the theory to the solution of engineering A B Sample Problem 1.2 problems, and they employ the SMART problem-solving methodology The steel tie bar shown is to be designed to carry a tension force of magnitude P 5 120 kN when bolted between double brackets at A that students are encouraged to use in the solution of their assigned and B. The bar will be fabricated from 20-mm-thick plate stock. For the grade of steel to be used, the maximum allowable stresses are s 5 175 MPa, t 5 100 MPa, and sb 5 350 MPa. Design the tie bar by problems. Since the sample problems have been set up in much the determining the required values of (a) the diameter d of the bolt, (b) the dimension b at each end of the bar, and (c) the dimension h of the bar. same form that students will use in solving the assigned problems, STRATEGY: Use free-body diagrams to determine the forces needed to obtain the stresses in terms of the design tension force. Setting these F1 they serve the double purpose of amplifying the text and demonstrat- F1 stresses equal to the allowable stresses provides for the determination of the required dimensions. MODELING and ANALYSIS: ing the type of neat and orderly work that students should cultivate in bee98233_ch01_002-053.indd 9 d F1  1 P P 11/7/13 3:27 PM a. Diameter of the Bolt. Since the bolt is in double shear (Fig. 1), 2 F1 5 12 P 5 60 kN. their own solutions. In addition, in-problem references and captions Fig. 1 Sectioned bolt. t 5 20 mm F1 60 kN 60 kN t5 5 1 100 MPa 5 d 5 27.6 mm have been added to the sample problem figures for contextual linkage 4p d 4p d 2 1 2 A h Use d 5 28 mm ◀ to the step-by-step solution. d At this point, check the bearing stress between the 20-mm-thick plate (Fig. 2) and the 28-mm-diameter bolt. b P 120 kN sb 5 5 5 214 MPa , 350 MPa OK td 10.020 m2 10.028 m2 Homework Problem Sets. Over 25% of the nearly 1500 home- Fig. 2 Tie bar geometry. t b. Dimension b at Each End of the Bar. We consider one of the end portions of the bar in Fig. 3. Recalling that the thickness of the work problems are new or updated. Most of the problems are of a prac- steel plate is t 5 20 mm and that the average tensile stress must not 1 exceed 175 MPa, write tical nature and should appeal to engineering students. They are a 2 P P' ⫽ 120 kN s5 1 2P 175 MPa 5 60 kN a 5 17.14 mm b d 1 ta 10.02 m2a primarily designed, however, to illustrate the material presented in the a 2 P b 5 d 1 2a 5 28 mm 1 2(17.14 mm) b 5 62.3 mm ◀ c. Dimension h of the Bar. We consider a section in the central text and to help students understand the principles used in mechanics Fig. 3 End section of tie bar. t 5 20 mm portion of the bar (Fig. 4). Recalling that the thickness of the steel plate is t 5 20 mm, we have of materials. The problems are grouped according to the portions of s5 P 175 MPa 5 120 kN h 5 34.3 mm th 10.020 m2h material they illustrate and are arranged in order of increasing diffi- P 5 120 kN Use h 5 35 mm ◀ culty. Answers to a majority of the problems are given at the end of the h REFLECT and THINK: We sized d based on bolt shear, and then checked bearing on the tie bar. Had the maximum allowable bearing book. Problems for which the answers are given are set in blue type in Fig. 4 Mid-body section of tie bar. stress been exceeded, we would have had to recalculate d based on the bearing criterion. the text, while problems for which no answer is given are set in red. xiii bee98233_ch01_002-053.indd 19 11/7/13 3:27 PM xiv Guided Tour Chapter Review and Summary. Each chapter ends with a review and summary of the material covered in that chapter. Subtitles are used to help students organize their Review and Summary review work, and cross-references have been included to help This chapter was devoted to the concept of stress and to an introduction them find the portions of material requiring their special to the methods used for the analysis and design of machines and load- bearing structures. Emphasis was placed on the use of a free-body diagram attention. to obtain equilibrium equations that were solved for unknown reactions. Free-body diagrams were also used to find the internal forces in the vari- ous members of a structure. Axial Loading: Normal Stress Review Problems. A set of review problems is included The concept of stress was first introduced by considering a two-force member under an axial loading. The normal stress in that member P (Fig. 1.41) was obtained by at the end of each chapter. These problems provide students s5 P (1.5) A further opportunity to apply the most important concepts The value of s obtained from Eq. (1.5) represents the average stress over the section rather than the stress at a specific point Q of the section. introduced in the chapter. A Considering a small area DA surrounding Q and the magnitude DF of the force exerted on DA, the stress at point Q is ¢F s 5 lim (1.6) ¢Ay0 ¢A In general, the stress s at point Q in Eq. (1.6) is different from the value of the average stress given by Eq. (1.5) and is found to vary across the section. However, this variation is small in any section away from the points of application of the loads. Therefore, the distribution of the normal P' stresses in an axially loaded member is assumed to be uniform, except in Review Problems Fig. 1.41 Axially loaded member with cross section normal to member used to the immediate vicinity of the points of application of the loads. For the distribution of stresses to be uniform in a given section, the line of action of the loads P and P9 must pass through the centroid C. Such define normal stress. a loading is called a centric axial loading. In the case of an eccentric axial 1.59 In the marine crane shown, link CD is known to have a uniform loading, the distribution of stresses is not uniform. cross section of 50 3 150 mm. For the loading shown, determine the normal stress in the central portion of that link. Transverse Forces and Shearing Stress 15 m 25 m 3m When equal and opposite transverse forces P and P9 of magnitude P are applied to a member AB (Fig. 1.42), shearing stresses t are created over B any section located between the points of application of the two forces. P 35 m A C B C 80 Mg 15 m P⬘ D A Fig. 1.42 Model of transverse resultant forces on either side of C resulting in shearing stress at section C. 44 Fig. P1.59 1.60 Two horizontal 5-kip forces are applied to pin B of the assembly bee98233_ch01_002-053.indd 44 11/7/13 3:27 PM shown. Knowing that a pin of 0.8-in. diameter is used at each connection, determine the maximum value of the average nor- mal stress (a) in link AB, (b) in link BC. 0.5 in. B 1.8 in. 5 kips 5 kips 0.5 in. A 60⬚ 45⬚ 1.8 in. C Computer Problems The following problems are designed to be solved with a computer. 1.C1 A solid steel rod consisting of n cylindrical elements welded together Element n Fig. P1.60 is subjected to the loading shown. The diameter of element i is denoted Pn 1.61 For the assembly and loading of Prob. 1.60, determine (a) the by di and the load applied to its lower end by Pi, with the magnitude Pi of average shearing stress in the pin at C, (b) the average bearing this load being assumed positive if Pi is directed downward as shown and stress at C in member BC, (c) the average bearing stress at B in negative otherwise. (a) Write a computer program that can be used with member BC. either SI or U.S. customary units to determine the average stress in each element of the rod. (b) Use this program to solve Probs. 1.1 and 1.3. 47 1.C2 A 20-kN load is applied as shown to the horizontal member ABC. Member ABC has a 10 3 50-mm uniform rectangular cross section and Element 1 is supported by four vertical links, each of 8 3 36-mm uniform rectan- P1 gular cross section. Each of the four pins at A, B, C, and D has the same diameter d and is in double shear. (a) Write a computer program to cal- Fig. P1.C1 bee98233_ch01_002-053.indd 47 11/7/13 3:27 PM culate for values of d from 10 to 30 mm, using 1-mm increments, (i) the maximum value of the average normal stress in the links connecting pins B and D, (ii) the average normal stress in the links connecting pins C Computer Problems. Computers make it possible for and E, (iii) the average shearing stress in pin B, (iv) the average shearing stress in pin C, (v) the average bearing stress at B in member ABC, and (vi) the average bearing stress at C in member ABC. (b) Check your pro- engineering students to solve a great number of challenging gram by comparing the values obtained for d 5 16 mm with the answers given for Probs. 1.7 and 1.27. (c) Use this program to find the permissible values of the diameter d of the pins, knowing that the allowable values problems. A group of six or more problems designed to be of the normal, shearing, and bearing stresses for the steel used are, respectively, 150 MPa, 90 MPa, and 230 MPa. (d) Solve part c, assuming solved with a computer can be found at the end of each chap- that the thickness of member ABC has been reduced from 10 to 8 mm. ter. These problems can be solved using any computer 0.4 m language that provides a basis for analytical calculations. 0.25 m C 0.2 m Developing the algorithm required to solve a given problem B will benefit the students in two different ways: (1) it will help 20 kN E D them gain a better understanding of the mechanics principles A involved; (2) it will provide them with an opportunity to apply the skills acquired in their computer programming course to Fig. P1.C2 the solution of a meaningful engineering problem. 51 bee98233_ch01_002-053.indd 51 11/7/13 3:27 PM  List of Symbols a Constant; distance PU Ultimate load (LRFD) A, B, C,... Forces; reactions q Shearing force per unit length; shear A, B, C,... Points flow A, A Area Q Force b Distance; width Q First moment of area c Constant; distance; radius r Radius; radius of gyration C Centroid R Force; reaction C1, C2,... Constants of integration R Radius; modulus of rupture CP Column stability factor s Length d Distance; diameter; depth S Elastic section modulus D Diameter t Thickness; distance; tangential e Distance; eccentricity; dilatation deviation E Modulus of elasticity T Torque f Frequency; function T Temperature F Force u, v Rectangular coordinates F.S. Factor of safety u Strain-energy density G Modulus of rigidity; shear modulus U Strain energy; work h Distance; height v Velocity H Force V Shearing force H, J, K Points V Volume; shear I, Ix,... Moment of inertia w Width; distance; load per unit length Ixy,... Product of inertia W, W Weight, load J Polar moment of inertia x, y, z Rectangular coordinates; distance; k Spring constant; shape factor; bulk displacements; deflections modulus; constant x, y, z Coordinates of centroid K Stress concentration factor; torsional Z Plastic section modulus spring constant a, b, g Angles l Length; span a Coefficient of thermal expansion; L Length; span influence coefficient Le Effective length g Shearing strain; specific weight m Mass gD Load factor, dead load (LRFD) M Couple gL Load factor, live load (LRFD) M, Mx,... Bending moment d Deformation; displacement MD Bending moment, dead load (LRFD) e Normal strain ML Bending moment, live load (LRFD) u Angle; slope MU Bending moment, ultimate load (LRFD) l Direction cosine n Number; ratio of moduli of elasticity; n Poisson’s ratio normal direction r Radius of curvature; distance; density p Pressure s Normal stress P Force; concentrated load t Shearing stress PD Dead load (LRFD) f Angle; angle of twist; resistance factor PL Live load (LRFD) v Angular velocity xv This page intentionally left blank  Seventh Edition Mechanics of Materials  Introduction— 1 Concept of Stress Stresses occur in all structures subject to loads. This chapter will examine simple states of stress in elements, such as in the two-force members, bolts and pins used in the structure shown. Objectives Review of statics needed to determine forces in members of simple structures. Introduce concept of stress. Define different stress types: axial normal stress, shearing stress and bearing stress. Discuss engineer’s two principal tasks, namely, the analysis and design of structures and machines. Develop problem solving approach. Discuss the components of stress on different planes and under different loading conditions. Discuss the many design considerations that an engineer should review before preparing a design. 4 Introduction—Concept of Stress Introduction Introduction The study of mechanics of materials provides future engineers with the 1.1 REVIEW OF THE means of analyzing and designing various machines and load-bearing METHODS OF STATICS structures involving the determination of stresses and deformations. This 1.2 STRESSES IN THE first chapter is devoted to the concept of stress. MEMBERS OF A Section 1.1 is a short review of the basic methods of statics and their STRUCTURE application to determine the forces in the members of a simple structure 1.2A Axial Stress consisting of pin-connected members. The concept of stress in a member 1.2B Shearing Stress of a structure and how that stress can be determined from the force in the 1.2C Bearing Stress in Connections member will be discussed in Sec. 1.2. You will consider the normal stresses 1.2D Application to the Analysis and in a member under axial loading, the shearing stresses caused by the appli- Design of Simple Structures cation of equal and opposite transverse forces, and the bearing stresses 1.2E Method of Problem Solution created by bolts and pins in the members they connect. 1.3 STRESS ON AN OBLIQUE Section 1.2 ends with a description of the method you should use PLANE UNDER AXIAL in the solution of an assigned problem and a discussion of the numerical LOADING accuracy. These concepts will be applied in the analysis of the members of 1.4 STRESS UNDER GENERAL the simple structure considered earlier. LOADING CONDITIONS; Again, a two-force member under axial loading is observed in COMPONENTS OF STRESS Sec. 1.3 where the stresses on an oblique plane include both normal and 1.5 DESIGN shearing stresses, while Sec. 1.4 discusses that six components are required CONSIDERATIONS to describe the state of stress at a point in a body under the most general 1.5A Determination of the Ultimate loading conditions. Strength of a Material Finally, Sec. 1.5 is devoted to the determination of the ultimate 1.5B Allowable Load and Allowable strength from test specimens and the use of a factor of safety to compute Stress: Factor of Safety the allowable load for a structural component made of that material. 1.5C Factor of Safety Selection 1.5D Load and Resistance Factor Design 1.1 REVIEW OF THE METHODS OF STATICS Consider the structure shown in Fig. 1.1, which was designed to support a 30-kN load. It consists of a boom AB with a 30 3 50-mm rectangular cross section and a rod BC with a 20-mm-diameter circular cross section. These are connected by a pin at B and are supported by pins and brackets at A and C, respectively. First draw a free-body diagram of the structure by detaching it from its supports at A and C and showing the reactions that these supports exert on the structure (Fig. 1.2). Note that the sketch of the structure has been simplified by omitting all unnecessary details. Many of you may have recognized at this point that AB and BC are two-force mem- bers. For those of you who have not, we will pursue our analysis, ignoring that fact and assuming that the directions of the reactions at A and C are unknown. Each of these reactions are represented by two components: Ax and Ay at A, and Cx and Cy at C. The equilibrium equations are. 1l o MC 5 0: Ax 10.6 m2 2 130 kN2 10.8 m2 5 0 Ax 5 140 kN (1.1) 1 o Fx 5 0: Ax 1 Cx 5 0 y Cx 5 2Ax Cx 5 240 kN (1.2) 1x o Fy 5 0: Ay 1 Cy 2 30 kN 5 0 Photo 1.1 Crane booms used to load and unload ships. Ay 1 Cy 5 130 kN (1.3)  1.1 Review of The Methods of Statics 5 C d 5 20 mm 600 mm A 50 mm B 800 mm 30 kN Fig. 1.1 Boom used to support a 30-kN load. We have found two of the four unknowns, but cannot determine the other two from these equations, and no additional independent equation can be obtained from the free-body diagram of the structure. We must now dismember the structure. Considering the free-body diagram of the boom AB (Fig. 1.3), we write the following equilibrium equation: 1l o MB 5 0: 2Ay 10.8 m2 5 0 Ay 5 0 (1.4) Substituting for Ay from Eq. (1.4) into Eq. (1.3), we obtain Cy 5 130 kN. Expressing the results obtained for the reactions at A and C in vector form, we have A 5 40 kN y Cx 5 40 kN z Cy 5 30 kNx Cy C Cx Ay Ay By 0.6 m B Ax A Ax A B Bz 0.8 m 0.8 m 30 kN 30 kN Fig. 1.2 Free-body diagram of boom showing Fig. 1.3 Free-body diagram of member AB freed applied load and reaction forces. from structure. 6 Introduction—Concept of Stress Note that the reaction at A is directed along the axis of the boom AB and causes compression in that member. Observe that the components Cx and Cy of the reaction at C are, respectively, proportional to the horizontal and vertical components of the distance from B to C and that the FBC FBC reaction at C is equal to 50 kN, is directed along the axis of the rod BC, 5 and causes tension in that member. 30 kN 3 These results could have been anticipated by recognizing that AB 4 B and BC are two-force members, i.e., members that are subjected to forces FAB FAB at only two points, these points being A and B for member AB, and B and C for member BC. Indeed, for a two-force member the lines of action of the resultants of the forces acting at each of the two points are equal and 30 kN opposite and pass through both points. Using this property, we could have (a) (b) obtained a simpler solution by considering the free-body diagram of pin B. Fig. 1.4 Free-body diagram of boom’s joint B and The forces on pin B, FAB and FBC, are exerted, respectively, by members associated force triangle. AB and BC and the 30-kN load (Fig. 1.4a). Pin B is shown to be in equi- librium by drawing the corresponding force triangle (Fig. 1.4b). Since force FBC is directed along member BC, its slope is the same as that of BC, namely, 3/4. We can, therefore, write the proportion FAB FBC 30 kN 5 5 4 5 3 from which FAB 5 40 kN FBC 5 50 kN Forces F9AB and F9BC exerted by pin B on boom AB and rod BC are equal and opposite to FAB and FBC (Fig. 1.5). FBC C FBC C D FBC F'BC D B F'BC B F'BC FAB A B F'AB Fig. 1.5 Free-body diagrams of two-force Fig. 1.6 Free-body diagrams of sections of rod BC. members AB and BC. Knowing the forces at the ends of each member, we can now deter- mine the internal forces in these members. Passing a section at some arbi- trary point D of rod BC, we obtain two portions BD and CD (Fig. 1.6). Since 50-kN forces must be applied at D to both portions of the rod to keep them in equilibrium, an internal force of 50 kN is produced in rod BC when a 30-kN load is applied at B. From the directions of the forces FBC and F9BC in Fig. 1.6 we see that the rod is in tension. A similar procedure enables us to determine that the internal force in boom AB is 40 kN and is in compression.  1.2 Stresses in the Members of a Structure 7 1.2 STRESSES IN THE MEMBERS OF A STRUCTURE 1.2A Axial Stress In the preceding section, we found forces in individual members. This is the first and necessary step in the analysis of a structure. However it does not tell us whether the given load can be safely supported. Rod BC of the example considered in the preceding section is a two-force member and, therefore, the forces FBC and F9BC acting on its ends B and C (Fig. 1.5) are directed along the axis of the rod. Whether rod BC will break or not under Photo 1.2 This bridge truss consists of two-force this loading depends upon the value found for the internal force FBC, the members that may be in tension or in compression. cross-sectional area of the rod, and the material of which the rod is made. Actually, the internal force FBC represents the resultant of elementary forces distributed over the entire area A of the cross section (Fig. 1.7). The average P FBC FBC ⫽ A A P ⫽ A A Fig. 1.7 Axial force represents the resultant of distributed elementary forces. intensity of these distributed forces is equal to the force per unit area, FBCyA, on the section. Whether or not the rod will break under the given loading depends upon the ability of the material to withstand the corre- sponding value FBCyA of the intensity of the distributed internal forces. Let us look at the uniformly distributed force using Fig. 1.8. The P' P' force per unit area, or intensity of the forces distributed over a given sec- (a) (b) tion, is called the stress and is denoted by the Greek letter s (sigma). The Fig. 1.8 (a) Member with an axial load. stress in a member of cross-sectional area A subjected to an axial load P (b) Idealized uniform stress distribution at an is obtained by dividing the magnitude P of the load by the area A: arbitrary section. P s5 (1.5) A ⌬F A positive sign indicates a tensile stress (member in tension), and a nega- ⌬A tive sign indicates a compressive stress (member in compression). Q As shown in Fig. 1.8, the section through the rod to determine the internal force in the rod and the corresponding stress is perpendicular to the axis of the rod. The corresponding stress is described as a normal stress. Thus, Eq. (1.5) gives the normal stress in a member under axial loading: Note that in Eq. (1.5), s represents the average value of the stress over the cross section, rather than the stress at a specific point of the cross section. To define the stress at a given point Q of the cross section, consider a small area DA (Fig. 1.9). Dividing the magnitude of DF by DA, you obtain the average value of the stress over DA. Letting DA approach zero, the stress at point Q is P' ¢F Fig. 1.9 Small area DA, at an arbitrary cross s 5 lim (1.6) section point carries/axial DF in this axial member. ¢Ay0 ¢A 8 Introduction—Concept of Stress P In general, the value for the stress s at a given point Q of the section  is different from that for the average stress given by Eq. (1.5), and s is found to vary across the section. In a slender rod subjected to equal and opposite concentrated loads P and P9 (Fig. 1.10a), this variation is small in a section away from the points of application of the concentrated loads  (Fig. 1.10c), but it is quite noticeable in the neighborhood of these points (Fig. 1.10b and d). It follows from Eq. (1.6) that the magnitude of the resultant of the distributed internal forces is  # dF 5 # s dA A But the conditions of equilibrium of each of the portions of rod shown in Fig. 1.10 require that this magnitude be equal to the magnitude P of the concentrated loads. Therefore, P' P' P' P' (a) (b) (c) (d) Fig. 1.10 Stress distributions at different sections P5 # dF 5 # s dA A (1.7) along axially loaded member. which means that the volume under each of the stress surfaces in Fig. 1.10 must be equal to the magnitude P of the loads. However, this is the only  P information derived from statics regarding the distribution of normal stresses in the various sections of the rod. The actual distribution of C stresses in any given section is statically indeterminate. To learn more about this distribution, it is necessary to consider the deformations result- ing from the particular mode of application of the loads at the ends of the rod. This will be discussed further in Chap. 2. Fig. 1.11 Idealized uniform stress distribution In practice, it is assumed that the distribution of normal stresses in implies the resultant force passes through the cross an axially loaded member is uniform, except in the immediate vicinity of section’s center. the points of application of the loads. The value s of the stress is then equal to save and can be obtained from Eq. (1.5). However, realize that when we assume a uniform distribution of stresses in the section, it follows from P elementary statics† that the resultant P of the internal forces must be applied at the centroid C of the section (Fig. 1.11). This means that a uni- form distribution of stress is possible only if the line of action of the concen- trated loads P and P9 passes through the centroid of the section considered (Fig. 1.12). This type of loading is called centric loading and will take place in all straight two-force members found in trusses and pin-connected structures, such as the one considered in Fig. 1.1. However, if a two-force member is loaded axially, but eccentrically, as shown in Fig. 1.13a, the con- C ditions of equilibrium of the portion of member in Fig. 1.13b show that the internal forces in a given section must be equivalent to a force P applied at the centroid of the section and a couple M of moment M 5 Pd. This distribution of forces—the corresponding distribution of stresses—cannot be uniform. Nor can the distribution of stresses be symmetric. This point will be discussed in detail in Chap. 4. P' † See Ferdinand P. Beer and E. Russell Johnston, Jr., Mechanics for Engineers, 5th ed., Fig. 1.12 Centric loading having resultant forces McGraw-Hill, New York, 2008, or Vector Mechanics for Engineers, 10th ed., McGraw-Hill, passing through the centroid of the section. New York, 2013, Secs. 5.2 and 5.3.  1.2 Stresses in the Members of a Structure 9 When SI metric units are used, P is expressed in newtons (N) and A P in square meters (m2), so the stress s will be expressed in N/m2. This unit is called a pascal (Pa). However, the pascal is an exceedingly small quantity and often multiples of this unit must be used: the kilopascal (kPa), the megapascal (MPa), and the gigapascal (GPa): P d 1 kPa 5 103 Pa 5 103 N/m2 d C M 6 6 2 1 MPa 5 10 Pa 5 10 N/m