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STRENGTH of MATERIALS B. Raghu Kumar M.Tech; Ph.D, LMISTE, LMCMSI Professor and Head Department of Mechanical Engineering Prasad V. Potluri Siddhartha Institute of Technology Vijayawada. BSP BS Publications A unit of BSP Books Pvt., Ltd. 4-4-309/316, Giriraj Lane, Sultan Bazar, Hyderabad -500 095. First published 2022 by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 © 2022 BS Publications CRC Press is an imprint of Informa UK Limited The right of B. Raghu Kumar to be identiﬁed as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750 8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identiﬁcation and explanation without intent to infringe. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record has been requested ISBN: 978-1-032-28838-3 (hbk) ISBN: 978-1-003-29874-8 (ebk) DOI: 10.1201/9781003298748 Dedicated to my Father Sri. B. Nageswara Rao Preface With nearly Twenty years of my teaching experience, I have strived to write this book on "Strength of materials" to provide solid foundation to students in this subject. A Simple English and user friendly language is used in this book, so that all the students can understand the basic concepts of strength of materials. The subject strength of materials is one of the core and very important subject for all the engineering students. During my long experience of interacting with students, I can realize the common difficulties faced by the students. An attempt has been made in this book to overcome these difficulties. While designing any structural or machine member the designer should see that the members should safe and reliable. To achieve this, the designer should know how to evaluate the maximum load carrying capacity of members subjected to various types of loading. This book deals with various structural members like prismatic bars, beams, columns, pressure vessels etc; Detailed analytical procedures are discussed to analyze various structural members subjected to different types of loading. Now-a-days people are using different softwares to solve these structural members. A separate chapter is included, which deals with a step-by-step procedure to solve the structural members using one of the mostly used software (ANSYS R). This book aims to be useful to undergraduate students of Mechanical and Civil engineering students as. Even Post Graduate students can also find the book useful to some extent in advanced topics like curved beams, shear center, unit load method etc; This book has been divided into sixteen chapters. Chapter 1 and Chapter 2 deals with different types of stresses and calculations of simple stresses for axially loaded members. Chapter 1 also deals with temperature stresses and statically indeterminate axially loaded members whereas chapter 2 deals with structural members subjected to torsion loading. Chapter 3 deals with principal stresses. Analytical and detailed graphical methods (Mohr's circle) are discussed. Chapters 4 to 7 deals with complete analysis of beams. These chapters' deals with starting from definition of beams to types, shear force and bending moment diagrams, shear stresses, bending stresses of different cross sections subjected to different types of loading, deflections of beams using various methods are discussed. Chapter 8 deals with statically indeterminate beams and various techniques to analyze these beams. Chapter 9 deals with thin and thick pressure vessels. Chapter 10 deals with columns and struts. Advanced topics are dealt from Chapters 11 to 15. These chapters include curved beams, centrifugal stresses, shear center, unit load method and strain energy methods. An exclusive Chapter 16 is included in this book, which deals with solving strength of materials problems using ANSYS. A Detailed step - by - step procedure is discussed to solve various types of problems. To easily understand the procedure the problems (v) (vi) I Contents solved as worked out examples in various chapters in this book are solved in ANSYS and presented in this chapter. The images used in this chapter is "courtesy of ANSYS, Inc" I sincerely express my sincere thanks to the management of Siddhartha Academy, Vijayawada for providing all facilities to write this book. I am very much thankful to Sri. K. Satyanarayana Garu, president, Mr. K. Lakshmana Ravish Garu, Vice-president, Mr. K. Raja Hareen Garu, Vice-president Koneru Lakshmaiah Education Foundation, K L University, Vijayawada for their constant encouragement from the beginning of my career. I would like to express my thanks to gurus Dr. K. V. Ramana, Professor, K L University and Dr. K. Mallikarjuna Rao, Emeritus Professor, JNTU Kakinada for their constant support. I express my sincere appreciation to my brother Mr. B. Madhusudhana Rao for standing behind me in all my successes and encouraging me continuously. Finally, I am deeply indebted to my family: My wife B. Uma Devi, my daughters B. Tanmayi and B. Aishwaraya Sai for motivating and encouraging me constantly to undertake and complete this work. Though every care has been taken to make the book error-free, still it is very difficult to claim absolute perfection. Suggestions for improving the quality of the book shall be gratefully acknowledged. Dr. B. Raghu Kumar Contents Preface (v) CHAPTERl Simple Stresses and Strains 1.1 Introduction 1 1.2 Normal Stress and Strain 2 1.3 Shear Stress and Shear Strain 4 1.4 Poisson's Ratio 6 1.5 Stress Strain Diagrams 7 1.6 Hooke's Law 10 1.7 Volumetric Strain (Dilation) 11 1.8 Statically Indeterminate Structures 12 1.9 Thermal Stresses 13 CHAPTER2 Torsion 2.1 Introduction 14 2.2 Torsion of Circular Bars 15 2.3 Power Transmitted by Shaft 18 2.4 Concept of Pure Shear and Complementary Shear 19 2.5 Relation between Elastic Constants E and G 20 2.6 Statically indeterminate Torsional Members 22 CHAPTER3 Principal Stresses 3.1 Introduction 23 3.2 Stresses on an Inclined Plane 25 3.3 Principal Stresses 29 3.4 Maximum Shear Stress 32 3.5 Mohr's Circle 35 CHAPTER4 Shear Force and Bending Moment 4.1 Introduction 38 4.2 Definitions 38 4.3 Types of Beams 39 (viii) I Contents 4.4 Shear Force and Bending Moment 41 4.5 Sign conventions 41 4.6 Shear Force and Bending Moment Diagrams 42 4.7 Relations between Load, Shear Force and Bending Moment 51 CHAPTERS Bending Stresses in Beams 5.1 Introduction 53 5.2 Bending Stress 54 5.3 Flexure Formula 56 5.4 Composite Beams 58 CHAPTER6 Shearing Stresses in Beams 6.1 Introduction 61 6.2 The Shear Stress Distribution 61 6.3 Shear Stress Distribution in a Rectangular Cross Section 64 6.4 Shear Stress Distribution in a Circular Cross Section 65 6.5 Shear Stress Distribution in a Triangular Cross Section 67 6.6 Shear Stress Distribution in a Symmetrical I Cross Section 69 CHAPTER 7 Deflection in Beams 7.1 Introduction 74 7.2 Differential Equation for Deflection of Beams 74 7.3 Moment Area Method 77 7.4 Macaulay's Method 79 7.5 Cantilever Beam subjected to different Loading Conditions 79 7.6 Simply supported Beam subjected to different Loads 103 CHAPTERS Statically Indeterminate Beams 8.1 Introduction 120 8.2 Analysis by the Differential Equations of Deflection Curve 122 8.3 Analysis by Moment Area Method 122 8.4 Propped cantilever beam subjected to different loading conditions 123 8.5 Fixed Beam Subjected to Different Loading Conditions 133 8.6 A Fixed Beam with Sinking/Yielding Support 148 8.7 Continuous Beam 149 Contents I( ix) CHAPTER9 Pressure Vessels 9.1 Introduction 153 9.2 Thin spherical Pressure Vessel 153 9.3 Thin cylindrical Pressure Vessel 155 9.4 Thick Cylinders 157 9.5 Lame's equation for Hoop & Circumferential Stresses 158 9.6 Longitudinal stresses and Maximum Shear Stress 161 9.7 Radial Strain 162 9.8 Compound Cylinder 162 9.9 Necessary difference of Radii for Shrinkage (Shrinkage Allowance) 165 CHAPTERlO Columns and Struts 10.1 Introduction 166 10.2 Buckling 166 10.3 Stability 167 10.4 Slenderness Ratio 168 10.5 Euler's Crippling of Columns for Different End Conditions 168 10.5.1 Euler's Crippling Load for Pinned - Pinned End Condition 169 10.5.2 Euler's Crippling Load for Fixed - Fixed End Condition 170 10.5.3 Euler's Crippling Load for Fixed - Free End Condition 172 10.5.4 Euler's Crippling Load for Fixed - Pinned End Condition 174 10.7 Rankine's Formula 177 10.8 Columns with Eccentric Loading (Secant Formula) 178 CHAPTERll Curved Beams 11.1 Introduction 181 11.2 Winkler - Bach Formula for Curved Beams 181 11.3 Position of Neutral Axis 186 11.4 Value of H2 187 11.5 Values of H2 for Different Cross Sections 187 11.5.1 Rectangular Cross Section 187 11.5.2 Circular Cross Section 188 11.5.3 Triangular Cross Section 190 11.5.4 Trapezium Cross Section 192