Mechanical Engineering

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1 Mechanical Engineering Strength of Materials Comprehensive Theory with Solved Examples and Practice Questions Publications

2 Publications MADE EASY Publications Corporate Office: 44-A/4, Kalu Sarai (Near Hauz Khas Metro Station), New Delhi Contact: , Visit us at: Strength of Materials Copyright by MADE EASY Publications. All rights are reserved. No part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photo-copying, recording or otherwise), without the prior written permission of the above mentioned publisher of this book. First Edition: 2015 Second Edition (Revised and Updated): 2016 All rights reserved by MADE EASY PUBLICATIONS. No part of this book may be reproduced or utilized in any form without the written permission from the publisher.

Preface This book was motivated by the desire to further the evolution of a concise book on Strength of Materials.

accordingly. This edition has an expanded discussion of all relevant topics in the subject.

We identified their various problems like- lack of fundamentals of the subject, difficulty in solving simple solutions, shortage of a complete

The book addresses all the requirements of the students, i.e. comprehensive coverage of theory, fundamental concepts, objective type problems

The concise presentation will help the readers grasp the concepts with clarity and apply them with ease to solve problems quickly.

Topics like Properties of Materials, Simple Stress Strain Elastic Constant, Shear Force and Bending Moment, Centroids and Moments of Inertia,

Pressure Vessels, Theory of Column, Theory of Springs are given full coverage in line with our research on their importance in competitive

We have put in our sincere efforts to present elaborate solutions for various problems, different problem solving methodology, some useful

A summary of important points to remember is added at the end of each topic.

3 Preface This book was motivated by the desire to further the evolution of a concise book on Strength of Materials. Keeping in focus the importance of this subject in GATE and ESE, we have done a proper study and thereafter developed the content of the book accordingly. This edition has an expanded discussion of all relevant topics in the subject. Initially, we compiled the perceptions of our students on their problems in GATE and ESE while dealing with the questions from this subject. We identified their various problems like- lack of fundamentals of the subject, difficulty in solving simple solutions, shortage of a complete study package, etc. These strengthened our determination to present a complete edition of Strength of Materials textbook. The book addresses all the requirements of the students, i.e. comprehensive coverage of theory, fundamental concepts, objective type problems and conventional problems, articulated in a lucid language. The concise presentation will help the readers grasp the concepts with clarity and apply them with ease to solve problems quickly. The books not only covers the entire syllabus of GATE and ESE, but also addresses the need of many other competitive examinations. Topics like Properties of Materials, Simple Stress Strain Elastic Constant, Shear Force and Bending Moment, Centroids and Moments of Inertia, Bending Stresses in Beams, Shear Stress in Beams, Principal Stress-strain and Theories of Failure, Torsion of Shaft, Deflection of Beams, Pressure Vessels, Theory of Column, Theory of Springs are given full coverage in line with our research on their importance in competitive examinations. We have put in our sincere efforts to present elaborate solutions for various problems, different problem solving methodology, some useful quick techniques to save time while attempting MCQs without compromising the accuracy of answers. A summary of important points to remember is added at the end of each topic. For the convenience of readers, points to remember are specifically highlighted in the form of a note- both in theory as well as solved examples. At the end of each chapter, sets of practice question are given with their keys, that will allow the readers to evaluate their understanding of the topics and sharpen their problem solving skills. Our team has made their best efforts to remove all possible errors of any kind. Nonetheless, we would highly appreciate and acknowledge if you find and share with us any printing, calculation and conceptual error. It is impossible to thank all the individuals who helped us, but we would like to sincerely thank all the co-authors, editors and reviewers for putting in their efforts to publish this book. We also express our thanks to MADE EASY publications for completing and publishing the book on time. With Best Wishes B. Singh CMD, MADE EASY

4 Contents Strength of Materials Chapter 1 Properties of Materials Introduction Normal Stress Strain Tension Test for Mild Steel Specifications of Specimen Stress Strain Curve for Tension Actual Curve v/s Engg. Curve in Tension Compression Curve for Mild Steel Stress-strain Curve for other Stress-strain Curve for Various Materials Properties of Metals Ductility Brittleness Malleability Hardness Creep Stress Relaxation Elasticity Proof Stress Elasto-Plastic Behaviour of Metals Types of Material Behaviour Toughness Fatigue Failure of Materials in Tension and Compression Ductile Metals in Tension Test Brittle Metals in Tension Test Ductile Metals in Compression Test Brittle Metals in Compression Test...10 Objective Brain Teasers...12 Chapter 2 Simple Stress-strain and Elastic Constants Stress Normal Stress Shear Stresses or Tangential Stresses Matrix Representation of Stress and Strain Stress Tensor (3D Stress Element) Matrix Representation of Strains Differential Form of Strains Allowable Stresses Saint Venant Principal Hooke s Law Assumption in Hooke s Law Elastic Constants Relationship between Elastic Constants Applications of Hooke s Law Volumetric Strain ( V ) Deflection of Axially Loaded Members Principle of Superposition Axial Deflection of Varying Cross- Sectional Bar Statically Indeterminate Axial Loaded Structures Axial Deflection in Interconnected Members Temperature Stresses Temperature Stresses in Composite Bar Stresses in Bolts and Nuts Strain Energy Strain Energy Due to Shear Force Strain Energy in Terms of Principal Stresses (iv)

5 Strain Energy Stored Due to Bending Moment Strain Energy Stored Due to Torque...70 Objective Brain Teasers...78 Conventional Practice Questions...86 Chapter 3 Shear Force and Bending Moment Types of Loading Types of Supports D Supports D Supports Types of Beam Stability in 2-D Structures Procedure of Analysis Shear Force Sign Convention for Shear Force Bending Moment Sign Convention for BM Important Points about SFD and BMD Curve Tracing for SFD and BMD Maximum Bending Moment Shear Force and Bending Moment Diagrams SFD and BMD by Integration Effect of Concentrated Moment on SFD and BMD Shear Force and Bending Moment Diagrams for Frames Loading Diagram and BMD from SFD Loading Diagram from BMD Objective Brain Teasers Conventional Practice Questions Chapter 4 Centroids and Moments of Inertia Centroid Moment of Inertia Product of Inertia Parallel Axis Theorem Perpendicular Axis Theorem Properties of Plane Areas Principal Axes and Principal Moments of Inertia Rotation of Axes Chapter 5 Objective Brain Teasers Conventional Practice Questions Bending Stress in Beams Effect of Bending Simply Bending or Pure Bending Assumptions in Theory of Pure Bending Neutral Axis Equation of Pure Bending Limitations of Equation of Pure Bending Nature of Bending Stress Sectional Modulus (Z) Moment of Resistance (MOR) Bending Stresses in Axially Loaded Beams Force on a Partial Area of a Section Bending Stress Distribution in Composite Beam Equivalent Section Flitched Beam Top and Bottom Flitched Beam Beam of Uniform Strength Biaxial Bending Chapter 6 Objective Brain Teasers Conventional Practice Questions Shear Stress in Beams Shear Stress Distribution in Beams Shear Stress Distribution in Rectangular Section Shear Stress Distribution in Triangular Section Shear Stress Distribution in Circular Sections Shear Stress Distribution in I-section Shear Stress Distribution in Some Other Sections Shear Stresses in Composite Sections Objective Brain Teasers Conventional Practice Questions (v)

6 Chapter 7 Principal Stress-strain and Theories of Failure Principal Plane Principal Stresses Analytical Method Principal Stress in Beams Graphical Method (Mohr s Circle Method) Properties of Mohr s Circle Construction of Mohr s Circle Analysis of Strain Analytical Method Graphical Method (Mohr s Circle Method) Properties of Strain Mohr s Circle Total strain energy in terms of principal stress Strain Rosette Theories of Elastic Failure Chapter Maximum Principal Stress Theory (Rankine s Theory) Maximum Principal Strain Theory (St. Venant s Theory) Maximum Shear Stress Theory (Guest & Tresca s) Maximum Strain Energy Theory (Haigh and Beltrami) Maximum Shear Strain Energy Theory or Distortion Energy Theory (Mises-Henky Theory) Octahedral Shear Stress Theory Objective Brain Teasers Conventional Practice Questions Torsion of Shafts Introduction Difference between Bending Moment and Twisting Moment Assumptions Involved in the Theory of Pure Torsion Sign Convention of Torque Effects of Torsion Polar Section Modulus Shear Stress Distribution in Circular Section Design of Shaft Power Transmitted by Shaft Series Combination of Shaft Parallel Combination of Shaft Strain Energy in Torsion Torsion in Thin Walled Tubes Angle of Twist in Thin Walled Tube Torsion of Non-circular Section Indeterminate Shaft Shaft Subjected to Combined Bending Moment and Twisting Moment Equivalent Bending Moment Equivalent Torque Shaft Subjected to Combined Axial Force and Torsional Moment Theories of Failure for Shaft Design Objective Brain Teasers Conventional Practice Questions Chapter 9 Deflection of Beams Introduction Methods for Determining Slope and Deflection Double Integration Method Use of Discontinuity Function : Macaulay s Method Area Moment Method: (Mohr s Method) Conjugate Beam Method Strain Energy Method Method of Superposition Application of Maxwell s Reciprocal Theorem Objective Brain Teasers Conventional Practice Questions (vi)

7 Chapter 10 Pressure Vessels Thin Cylindrical Shell Stresses in the Thin Cylindrical Shell Analysis of Thin Cylindrical Shell with Closed Flat Ends Hoop Stress or Circumferential Stress Longitudinal Stress (s L ) Radial Stress (s R ) Strains in Cylindrical Shell Hoop Strain (Major Principal Strain) Longitudinal Strain Volumetric Strain in Cylinder Maximum Shear Stress Analysis of Thin Spheres Strains in Sphere Stresses in Riveted Cylindrical Shell Thin Cylinders with Hemispherical Ends Thickness of Cylinder for Same Hoop Stress Thickness of Cylinder for No Distortion at Junction Pressure Vessels Subjected to Axial Force Thick Cylinder Analysis of Thick Cylinder Stresses in Thick Cylinder Analysis of Stresses Determination of A and B Variation of Radial and Hoop Stresses Analysis of Thick Sphere Hoop and Longitudinal Stresses Radial Stress Design of Pressure Vessels Strengthening of Cylinder Objective Brain Teasers Conventional Practice Questions Chapter 11 Theory of Columns Compression Member Types of Equilibrium Elastic Instability and Critical Load Euler s Theory for Buckling Failure Assumptions of Euler s Theory Effective Length of Column Critical Stress Limitations of Euler s Theory Graph between s and l Maximum Lateral Deflection of Column Rankine s Gorden Theory Column with Eccentric Loading Condition for No Tension Middle Third Rule Middle Fourth Rule Eccentric Loading about both x-axis and y-axis Objective Brain Teasers Conventional Practice Questions Chapter 12 Theoy of Springs Springs Types of Springs Bending Springs Torsional Spring Helical Spring Springs in Series and Parallel Objective Brain Teasers Conventional Practice Questions (vii)

MECHANICS OF SOLIDS IM LECTURE HOURS PER WEEK STATICS IM0232 DIFERENTIAL EQUAQTIONS

MECHANICS OF SOLIDS IM LECTURE HOURS PER WEEK STATICS IM0232 DIFERENTIAL EQUAQTIONS COURSE CODE INTENSITY PRE-REQUISITE CO-REQUISITE CREDITS ACTUALIZATION DATE MECHANICS OF SOLIDS IM0233 3 LECTURE HOURS PER WEEK 48 HOURS CLASSROOM ON 16 WEEKS, 96 HOURS OF INDEPENDENT WORK STATICS IM0232