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Strength Of Materials

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Strength Of Materials Fifth Edition William A. Nash, Ph.D. Former Professor of Civil Engineering University of Massachusetts Merle C. Potter, Ph.D. Professor Emeritus of Mechanical Engineering Michigan State University Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

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Copyright © 2011, 1998, 1994, 1972 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, 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 permission of the publisher. ISBN: 978-0-07-163507-3 MHID: 0-07-163507-6 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-163508-0, MHID: 0-07-163508-4. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the beneﬁ t of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected] This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that neither the author nor the publisher is engaged in rendering legal, accounting, securities trading, or other professional services. If legal advice or other expert assistance is required, the services of a competent professional person should be sought. —From a Declaration of Principles Jointly Adopted by a Committee of the American Bar Association and a Committee of Publishers and Associations Trademarks: McGraw-Hill, the McGraw-Hill Publishing logo, Schaum’s and related trade dress are trademarks or registered trademarks of The McGraw-Hill Companies and/or its afﬁ liates in the United States and other countries and may not be used without written permission. All other trademarks are the property of their respective owners. The McGraw-Hill Companies is not associated with any product or vendor mentioned in this book. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGrawHill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.

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Preface This fifth edition of Schaum’s Strength of Materials book has been substantially modified by the second author to better fit the outline of the introductory Strength of Materials (Solid Mechanics) course, and to better fit the presentation of material in most introductory textbooks on the subject. In addition, the fol- lowing changes have been made: 1. Problem solutions and Supplementary Problems are presented using the metric SI units only. 2. The computer programs have been omitted. The use of MATLAB or other programs are available to students if more complicated problems are of interest. 3. The more advanced materials and problems that are not found in an introductory course have been omitted for simplicity of presentation. This book is intended to be used in an introductory course only. 4. A short chapter on Fatigue, a subject included on the Fundamentals of Engineering Examination, has been added. It is a modified chapter, based on a section on Fatigue written by my friend and previous colleague, Charlie Muvdi, from “Engineering Mechanics of Materials,” by B. B. Muvdi and J. W. McNabb. 5. A section on Combined Loading has been added. 6. The chapter on Centroids and Moments of Inertia has been omitted; it is assumed to have been part of a Statics course that precedes Solid Mechanics. Strength of Materials, also called The Mechanics of Materials or Solid Mechanics, provides the basis for the design of the components that make up machines and load-bearing structures. In Statics, the forces and moments acting at various points in a structural component or at points of contact with other structures were determined. The forces, stresses, and strains existing within a component were not of interest. In Solid Mechanics, we will consider questions like, “What load will cause this structure to fail?”, “What maximum torque can this shaft transmit?”, “What material should be selected for this component?”, “At what load will this column buckle?” Such questions were not of interest in a Statics course. But, before any of these questions can be answered, we must calculate the forces and moments acting on the components that make up a structure or machine. So, Statics always precedes the study of Strength of Materials. Sometimes Statics is combined with Strength of Materials in one course since they are so closely related. I would like to thank the estate of the late William Nash for allowing me to create this fifth edition of a book that obviously required much diligent work by Professor Nash. Many thanks are also given to Dr. Charlie Muvdi who provided good advice on the content of this revision. It was a pleasure working with Kimberly Eaton of McGraw-Hill in making the many decisions required in such a venture. MERLE C. POTTER, E. LANSING, MI Michigan State University, 2010 v

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Contents CHAPTER 1 Tension and Compression 1 1.1 Internal Effects of Forces 1.2 Mechanical Properties of Materials 1.3 Statically Indeterminate Force Systems 1.4 Classification of Materials 1.5 Units CHAPTER 2 Shear Stresses 27 2.1 Shear Force and Shear Stress 2.2 Deformations Due to Shear Stresses 2.3 Shear Strain CHAPTER 3 Combined Stresses 35 3.1 Introduction 3.2 General Case of Plane Stress 3.3 Principal Stresses and Maximum Shearing Stress 3.4 Mohr’s Circle CHAPTER 4 Thin-Walled Pressure Vessels 49 4.1 Introduction 4.2 Cylindrical Pressure Vessels 4.3 Spherical Pressure Vessels CHAPTER 5 Torsion 59 5.1 Introduction 5.2 Torsional Shearing Stress 5.3 Shearing Strain 5.4 Combined Torsion and Axial Loading CHAPTER 6 Shearing Force and Bending Moment 73 6.1 Basics 6.2 Internal Forces and Moments in Beams 6.3 Shear and Moment Equations with Diagrams 6.4 Singularity Functions CHAPTER 7 Stresses in Beams 97 7.1 Basics 7.2 Normal Stresses in Beams 7.3 Shearing Stresses in Beams 7.4 Combined Loading CHAPTER 8 Deflection of Beams 123 8.1 Basics 8.2 Differential Equation of the Elastic Curve 8.3 Deflection by Integration 8.4 Deflections Using Singularity Functions 8.5 Deflections Using Superposition vii

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viii Contents CHAPTER 9 Statically Indeterminate Elastic Beams 143 9.1 Basics CHAPTER 10 Columns 159 10.1 Basics 10.2 Critical Load of a Long Slender Column 10.3 Eccentrically Loaded Columns 10.4 Design Formulas for Columns Having Intermediate Slenderness Ratios CHAPTER 11 Fatigue 175 Index 185

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CHAPTER 1 Tension and Compression 1.1 Internal Effects of Forces In this book we shall be concerned with what might be called the internal effects of forces acting on a body. The bodies themselves will no longer be considered to be perfectly rigid as was assumed in statics; instead, the calculation of the deformations of various bodies under a variety of loads will be one of our primary concerns in the study of strength of materials. Axially Loaded Bar The simplest case to consider at the start is that of an initially straight metal bar of constant cross section, loaded at its ends by a pair of oppositely directed collinear forces coinciding with the longitudinal axis of the bar and acting through the centroid of each cross section. For static equilibrium the magnitudes of the forces must be equal. If the forces are directed away from the bar, the bar is said to be in tension; if they are directed toward the bar, a state of compression exists. These two conditions are illustrated in Fig. 1-1. Under the action of this pair of applied forces, internal resisting forces are set up within the bar and their characteristics may be studied by imagining a plane to be passed through the bar anywhere along its length and oriented perpendicular to the longitudinal axis of the bar. Such a plane is designated as a-a in Fig. 1-2(a). If for purposes of analysis the portion of the bar to the right of this plane is considered to be removed, as in Fig. 1-2(b), then it must be replaced by whatever effect it exerts upon the left portion. By this technique of introducing a cutting plane, the originally internal forces now become external with respect to the remaining portion of the body. For equilibrium of the portion to the left this ‘‘effect’’ must be a horizontal force of mag- nitude P. However, this force P acting normal to the cross section a-a is actually the resultant of distributed forces acting over this cross section in a direction normal to it. At this point it is necessary to make some assumption regarding the manner of variation of these distrib- uted forces, and since the applied force P acts through the centroid it is commonly assumed that they are uniform across the cross section. a P P a (a) P P (b) Fig. 1-1 Axially loaded bars. Fig. 1-2 Internal force. Normal Stress Instead of speaking of the internal force acting on some small element of area, it is better for comparative purposes to treat the normal force acting over a unit area of the cross section. The intensity of normal force 1

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