Mechanics of Materials by Andrew Paytel PDF

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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 cont...

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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.

Mechanics of Materials Second Edition

Mechanics of Materials Second Edition

Andrew Pytel The Pennsylvania State University

Jaan Kiusalaas The Pennsylvania State University

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Mechanics of Materials, Second Edition

ª 2012, 2003 Cengage Learning

Andrew Pytel & Jaan Kiusalaas

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.

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Library of Congress Control Number: 2010938461 ISBN-13: 978-0-495-66775-9 ISBN-10: 0-495-66775-7

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Printed in the United States of America 1 2 3 4 5 6 7 13 12 11 10

To Jean, Leslie, Lori, John, Nicholas and To Judy, Nicholas, Jennifer, Timothy

Preface This textbook is intended for use in a first course in mechanics of materials. Programs of instruction relating to the mechanical sciences, such as mechanical, civil, and aerospace engineering, often require that students take this course in the second or third year of studies. Because of the fundamental nature of the subject matter, mechanics of materials is often a required course, or an acceptable technical elective in many other curricula. Students must have completed courses in statics of rigid bodies and mathematics through integral calculus as prerequisites to the study of mechanics of materials. This edition maintains the organization of the previous edition. The first eight chapters are dedicated exclusively to elastic analysis, including stress, strain, torsion, bending and combined loading. An instructor can easily teach these topics within the time constraints of a two-or three-credit course. The remaining five chapters of the text cover materials that can be omitted from an introductory course. Because these more advanced topics are not interwoven in the early chapters on the basic theory, the core material can e‰ciently be taught without skipping over topics within chapters. Once the instructor has covered the material on elastic analysis, he or she can freely choose topics from the more advanced later chapters, as time permits. Organizing the material in this manner has created a significant savings in the number of pages without sacrificing topics that are usually found in an introductory text. The most notable features of the organization of this text include the following:

.

. .

Chapter 1 introduces the concept of stress (including stresses acting on inclined planes). However, the general stress transformation equations and Mohr’s circle are deferred until Chapter 8. Engineering instructors often hold o¤ teaching the concept of state of stress at a point due to combined loading until students have gained su‰cient experience analyzing axial, torsional, and bending loads. However, if instructors wish to teach the general transformation equations and Mohr’s circle at the beginning of the course, they may go to the freestanding discussion in Chapter 8 and use it whenever they see fit. Advanced beam topics, such as composite and curved beams, unsymmetrical bending, and shear center, appear in chapters that are distinct from the basic beam theory. This makes it convenient for instructors to choose only those topics that they wish to present in their course. Chapter 12, entitled ‘‘Special Topics,’’ consolidates topics that are important but not essential to an introductory course, including energy methods, theories of failure, stress concentrations, and fatigue. Some, but not all, of this material is commonly covered in a three-credit course at the discretion of the instructor. vii

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Preface

. .

Chapter 13, the final chapter of the text, discusses the fundamentals of inelastic analysis. Positioning this topic at the end of the book enables the instructor to present an e‰cient and coordinated treatment of elastoplastic deformation, residual stress, and limit analysis after students have learned the basics of elastic analysis. Following reviewers’ suggestions, we have included a discussion of the torsion of rectangular bars. In addition, we have updated our discussions of the design of columns and reinforced concrete beams.

The text contains an equal number of problems using SI and U.S. Customary units. Homework problems strive to present a balance between directly relevant engineering-type problems and ‘‘teaching’’ problems that illustrate the principles in a straightforward manner. An outline of the applicable problemsolving procedure is included in the text to help students make the sometimes di‰cult transition from theory to problem analysis. Throughout the text and the sample problems, free-body diagrams are used to identify the unknown quantities and to recognize the number of independent equations. The three basic concepts of mechanics—equilibrium, compatibility, and constitutive equations—are continually reinforced in statically indeterminate problems. The problems are arranged in the following manner:

. . .

Virtually every section in the text is followed by sample problems and homework problems that illustrate the principles and the problemsolving procedure introduced in the article. Every chapter contains review problems, with the exception of optional topics. In this way, the review problems test the students’ comprehension of the material presented in the entire chapter, since it is not always obvious which of the principles presented in the chapter apply to the problem at hand. Most chapters conclude with computer problems, the majority of which are design oriented. Students should solve these problems using a high-level language, such as MATHCAD= or MATLAB=, which minimizes the programming e¤ort and permits them to concentrate on the organization and presentation of the solution.

Ancillaries 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, where these resources can be found, for instructors and students. The following ancillaries are available at www.cengagebrain.com.

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Study Guide to Accompany Pytel and Kiusalaas Mechanics of Materials, Second Edition, J. L Pytel and A. Pytel, 2012. The goals of the Study Guide are twofold. First, self-tests are included to help the student focus on the salient features of the assigned reading. Second, the study guide uses ‘‘guided’’ problems which give the student an opportunity to work through representative problems before attempting to solve the problems in the text. The Study Guide is provided free of charge. The Instructor’s Solution Manual and PowerPoint slides of all figures and tables in the text are available to instructors through http://login.cengage.com.

Preface

Acknowledgments We would like to thank the following reviewers for their valuable suggestions and comments: Roxann M. Hayes, Colorado School of Mines Daniel C. Jansen, California Polytechnic State University, San Luis Obispo Ghyslaine McClure, McGill University J.P. Mohsen, University of Louisville Hassan Rejali, California Polytechnic State University, Pomona In addition, we are indebted to Professor Thomas Gavigan, Berks Campus, The Pennsylvania State University, for his diligent proofreading. Andrew Pytel Jaan Kiusalaas

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1st Pass Pages

Contents CHAPTER 1

Stress

1

1.1 Introduction 1 1.2 Analysis of Internal Forces; Stress 1.3 Axially Loaded Bars 4

CHAPTER 4

2

Shear and Moment in Beams

a. Centroidal (axial) loading 4 b. Saint Venant’s principle 5 c. Stresses on inclined planes 6 d. Procedure for stress analysis 7 1.4 Shear Stress 18 1.5 Bearing Stress 19

Strain

31

2.1 Introduction 31 2.2 Axial Deformation; Stress-Strain

2.5 2.6

Diagram 32 a. Normal (axial) strain 32 b. Tension test 33 c. Working stress and factor of safety 36 Axially Loaded Bars 36 Generalized Hooke’s Law 47 a. Uniaxial loading; Poisson’s ratio 47 b. Multiaxial loading 47 c. Shear loading 48 Statically Indeterminate Problems 54 Thermal Stresses 63

CHAPTER 3

Torsion

75

3.1 Introduction 75 3.2 Torsion of Circular Shafts

a. b. c. d. e. f.

76 Simplifying assumptions 76 Compatibility 77 Equilibrium 77 Torsion formulas 78 Power transmission 79 Statically indeterminate problems

107

4.1 Introduction 107 4.2 Supports and Loads 108 4.3 Shear-Moment Equations and

CHAPTER 2

2.3 2.4

3.3 Torsion of Thin-Walled Tubes 91 *3.4 Torsion of Rectangular Bars 99

80

Shear-Moment Diagrams 109 a. Sign conventions 109 b. Procedure for determining shear force and bending moment diagrams 110 4.4 Area Method for Drawing Shear-Moment Diagrams 122 a. Distributed loading 122 b. Concentrated forces and couples 124 c. Summary 126 CHAPTER 5

Stresses in Beams

139

5.1 Introduction 139 5.2 Bending Stress 140

a. b. c. d. e.

Simplifying assumptions 140 Compatibility 141 Equilibrium 142 Flexure formula; section modulus 143 Procedures for determining bending stresses 144 5.3 Economic Sections 158 a. Standard structural shapes 159 b. Procedure for selecting standard shapes 160 5.4 Shear Stress in Beams 164 a. Analysis of flexure action 164 b. Horizontal shear stress 165 c. Vertical shear stress 167 * Indicates optional sections.

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Contents

d. Discussion and limitations of the shear stress formula 167 e. Rectangular and wide-flange sections 168 f. Procedure for analysis of shear stress 169 5.5 Design for Flexure and Shear 177 5.6 Design of Fasteners in Built-Up Beams 184 CHAPTER 6

Deflection of Beams

195

6.1 Introduction 195 6.2 Double-Integration Method

196 a. Di¤erential equation of the elastic curve 196 b. Double integration of the di¤erential equation 198 c. Procedure for double integration 199 6.3 Double Integration Using Bracket Functions 209 *6.4 Moment-Area Method 219 a. Moment-area theorems 220 b. Bending moment diagrams by parts 222 c. Application of the moment-area method 225 6.5 Method of Superposition 235

CHAPTER 7

Statically Indeterminate Beams 249 7.1 Introduction 249 7.2 Double-Integration Method 250 7.3 Double Integration Using Bracket

Functions

Combined Axial and Lateral Loads 284 8.4 State of Stress at a Point (Plane Stress) 293 a. Reference planes 293 b. State of stress at a point 294 c. Sign convention and subscript notation 294 8.5 Transformation of Plane Stress 295 a. Transformation equations 295 b. Principal stresses and principal planes 296 c. Maximum in-plane shear stress 298 d. Summary of stress transformation procedures 298 8.6 Mohr’s Circle for Plane Stress 305 a. Construction of Mohr’s circle 306 b. Properties of Mohr’s circle 307 c. Verification of Mohr’s circle 308 8.7 Absolute Maximum Shear Stress 314 a. Plane state of stress 315 b. General state of stress 316 8.8 Applications of Stress Transformation to Combined Loads 319 8.9 Transformation of Strain; Mohr’s Circle for Strain 331 a. Review of strain 331 b. Transformation equations for plane strain 332 c. Mohr’s circle for strain 333 8.10 The Strain Rosette 338 a. Strain gages 338 b. Strain rosette 339 c. The 45 strain rosette 340 d. The 60 strain rosette 340 8.11 Relationship between Shear Modulus and Modulus of Elasticity 342 8.3

256

*7.4 Moment-Area Method 260 7.5 Method of Superposition 266

CHAPTER 9

Composite Beams CHAPTER 8

Stresses Due to Combined Loads 277 8.1 8.2

Introduction 277 Thin-Walled Pressure Vessels a. Cylindrical vessels 278 b. Spherical vessels 280

* Indicates optional sections.

349

9.1 Introduction 349 9.2 Flexure Formula for Composite

Beams 350 278

9.3 Shear Stress and Deflection in Composite

Beams 355 a. Shear stress 355 b. Deflection 356 9.4 Reinforced Concrete Beams 359 a. Elastic Analysis 360 b. Ultimate moment analysis 361

Contents

CHAPTER 10

Columns

371

10.1 Introduction 371 10.2 Critical Load 372

CHAPTER 11

397

Introduction 397 Shear Flow in Thin-Walled Beams 398 Shear Center 400 Unsymmetrical Bending 407 a. Review of symmetrical bending 407 b. Symmetrical sections 408 c. Inclination of the neutral axis 409 d. Unsymmetrical sections 410 11.5 Curved Beams 415 a. Background 415 b. Compatibility 416 c. Equilibrium 417 d. Curved beam formula 418 11.1 11.2 11.3 11.4

CHAPTER 12

Special Topics

458

CHAPTER 13

a. Definition of critical load 372 b. Euler’s formula 373 10.3 Discussion of Critical Loads 375 10.4 Design Formulas for Intermediate Columns 380 a. Tangent modulus theory 380 b. AISC specifications for steel columns 381 10.5 Eccentric Loading: Secant Formula 387 a. Derivation of the secant formula 388 b. Application of the secant formula 389

Additional Beam Topics

12.5 Stress Concentration 452 12.6 Fatigue Under Repeated Loading

425

12.1 Introduction 425 12.2 Energy Methods 426

a. Work and strain energy 426 b. Strain energy of bars and beams 426 c. Deflections by Castigliano’s theorem 428 12.3 Dynamic Loading 437 a. Assumptions 437 b. Mass-spring model 438 c. Elastic bodies 439 d. Modulus of resilience; modulus of toughness 439 12.4 Theories of Failure 444 a. Brittle materials 445 b. Ductile materials 446

Inelastic Action

463

Introduction 463 Limit Torque 464 Limit Moment 466 Residual Stresses 471 a. Loading-unloading cycle 471 b. Torsion 471 c. Bending 472 d. Elastic spring-back 473 13.5 Limit Analysis 477 a. Axial loading 477 b. Torsion 478 c. Bending 479 13.1 13.2 13.3 13.4

APPENDIX A

Review of Properties of Plane Areas A.1 First Moments of Area; Centroid A.2 Second Moments of Area 488

487 487

a. Moments and product of inertia 488 b. Parallel-axis theorems 489 c. Radii of gyration 491 d. Method of composite areas 491 A.3 Transformation of Second Moments of Area 500 a. Transformation equations for moments and products of inertia 500 b. Comparison with stress transformation equations 501 c. Principal moments of inertia and principal axes 501 d. Mohr’s circle for second moments of area 502 APPENDIX B

Tables B.1 B.2 B.3

509 Average Physical Properties of Common Metals 510 Properties of Wide-Flange Sections (W-Shapes): SI Units 512 Properties of I-Beam Sections (S-Shapes): SI Units 518

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Contents

xiv B.4 B.5 B.6

B.7

Properties of Channel Sections: SI Units 519 Properties of Equal and Unequal Angle Sections: SI Units 520 Properties of Wide-Flange Sections (W-Shapes): U.S. Customary Units 524 Properties of I-Beam Sections (S-Shapes): U.S. Customary Units 532

B.8 B.9

Properties of Channel Sections: U.S. Customary Units 534 Properties of Equal and Unequal Angle Sections: U.S. Customary Units 535

Answers to Even-Numbered Problems Index

539 547

List of Symbols A A0 b c C Cc D; d d E e f F G g H h I I I1 ; I 2 J J k L Le M ML M nom M ult Myp m N n P Pcr Pdes P p Q q R r

area partial area of beam cross section width; distance from origin to center of Mohr’s circle distance from neutral axis to extreme fiber centroid of area; couple critical slenderness ratio of column diameter distance modulus of elasticity eccentricity of load; spacing of connectors frequency force shear modulus gravitational acceleration horizontal force height; depth of beam moment of inertia of area centroidal moment of inertia of area principal moments of inertia of area polar moment of inertia of area centroidal polar moment of inertia of area stress concentration factor; radius of gyration of area; spring sti¤ness length e¤ective length of column bending moment limit moment ultimate nominal bending moment ultimate bending moment yield moment mass factor of safety; normal force; number of load cycles impact factor; ratio of moduli of elasticity force; axial force in bar critical (buckling) load of column design strength of column power pressure first moment of area; dummy load shear flow radius; reactive force; resultant force radius; least radius of gyration of cross-sectional area of column xv

xvi

List of Symbols

S s T TL Typ t t U u; v v V W w x; y; z x; y; z

section modulus; length of median line distance kinetic energy; temperature; tensile force;...