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Comprehensive Theory
with Solved Examples and Practice Questions
Strength of Materials
Civil Engineering
Publications
MADE EASY Publications
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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: 2015Second 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.
Publications
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 Metals, 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, Shear Centre’ 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
Preface
Chapter 1Properties of Metals ................................ 1 1.1 Introduction ...................................................................... 1
1.2 Normal Stress ................................................................... 1
1.3 Strain ................................................................................... 2
1.4 Tension Test for Mild Steel ........................................... 3
1.4.1 Specifications of Specimen ........................... 4
1.4.2 Stress Strain Curve for Tension..................... 4
1.4.3 Actual Curve v/s Engg. Curve in Tension .. 5
1.4.4 Compression Curve for Mild Steel .............. 5
1.4.5 Stress-strain Curve for other ......................... 5
1.4.6 Stress-strain Curve for Various Materials .. 6
1.5 Properties of Metals ....................................................... 6
1.5.1 Ductility ................................................................ 6
1.5.2 Brittleness ............................................................ 6
1.5.3 Malleability ......................................................... 6
1.5.4 Hardness .............................................................. 6
1.6 Creep ................................................................................... 6
1.7 Stress Relaxation ............................................................. 7
1.8 Elasticity ............................................................................. 7
1.8.1 Proof Stress ......................................................... 8
1.8.2 Elasto-Plastic Behaviour of Metals .............. 8
1.8.3 Types of Material Behaviour ......................... 8
1.9 Toughness ......................................................................... 9
1.10 Fatigue ................................................................................ 9
1.11 Failure of Materials in Tension and Compression .............................. 10
1.11.1 Ductile Metals in Tension Test .................10
1.11.2 Brittle Metals in Tension Test ....................10
1.11.3 Ductile Metals in Compression Test ......10
1.11.4 Brittle Metals in Compression Test .........10
Objective Brain Teasers ................................................12
Chapter 2Simple Stress-strain and Elastic Constants ...............................................15 2.1 Stress .................................................................................15
2.1.1 Normal Stress ...................................................15
2.1.2 Shear Stresses or Tangential Stresses ......16
2.2 Matrix Representation of Stress and Strain .........17
2.2.1 Stress Tensor (3D Stress Element) .............17
2.2.2 Matrix Representation of Strains...............20
2.3 Differential Form of Strains .......................................20
2.4 Allowable Stresses ........................................................22
2.5 Saint Venant Principal .................................................23
2.6 Hooke’s Law ....................................................................24
2.6.1 Assumption in Hooke’s Law ........................24
2.7 Elastic Constants ...........................................................24
2.7.1 Relationship between Elastic Constants 25
2.8 Applications of Hooke’s Law .....................................27
2.9 Volumetric Strain (sV) ..................................................28
2.10 Deflection of Axially Loaded Members .................32
2.10.1 Principle of Superposition ........................34
2.10.2 Axial Deflection of Varying Cross-
Sectional Bar .................................................. 36
2.11 Statically Indeterminate Axial Loaded Structures . 46
2.12 Axial Deflection in Interconnected Members ....50
2.13 Temperature Stresses ..................................................53
2.13.1 Temperature Stresses in Composite Bar .. 58
2.14 Stresses in Bolts and Nuts ..........................................67
2.15 Strain Energy ..................................................................69
2.15.1 Strain Energy Due to Shear Force ...........70
2.15.2 Strain Energy in Terms of Principal
Stresses ........................................................ 70
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Strength of MaterialsContents
2.15.3 Strain Energy Stored Due to Bending
Moment ....................................................... 70
2.15.4 Strain Energy Stored Due to Torque ......70
Objective Brain Teasers .................................................78
Conventional Practice Questions ...............................85
Chapter 3Shear Force and Bending Moment .......87 3.1 Types of Loading ...........................................................87
3.2 Types of Supports .........................................................88
3.2.1 2D Supports ......................................................88
3.2.2 3D Supports ......................................................89
3.3 Types of Beam ................................................................90
3.4 Stability in 2-D Structures ..........................................91
3.5 Procedure of Analysis ..................................................91
3.6 Shear Force ......................................................................95
3.6.1 Sign Convention for Shear Force ..............96
3.7 Bending Moment ..........................................................96
3.7.1 Sign Convention for BM ...............................97
3.8 Important Points about SFD and BMD ..................99
3.9 Curve Tracing for SFD and BMD............................ 100
3.9.1 Maximum Bending Moment ................... 101
3.10 Shear Force and Bending Moment Diagrams . 102
3.10.1 SFD and BMD by Integration ................ 111
3.10.2 Effect of Concentrated Moment on SFD
and BMD ...................................................... 122
3.10.3 Shear Force and Bending Moment
Diagrams for Frames ............................... 130
3.11 Loading Diagram and BMD from SFD ................ 140
3.12 Loading Diagram from BMD .................................. 145
Objective Brain Teasers .............................................. 166
Conventional Practice Questions ............................ 178
Chapter 4Centroids and Moments of Inertia .....180 4.1 Centroid......................................................................... 180
4.2 Moment of Inertia ...................................................... 182
4.3 Product of Inertia ....................................................... 183
4.4 Parallel Axis Theorem ............................................... 184
4.5 Perpendicular Axis Theorem .................................. 185
4.6 Properties of Plane Areas ........................................ 185
4.7 Principal Axes and Principal Moments of Inertia .188
4.8 Rotation of Axes ......................................................... 189
Objective Brain Teasers .............................................. 197
Conventional Practice Questions ............................ 200
Chapter 5Bending Stress in Beams .....................202 5.1 Effect of Bending........................................................ 202
5.2 Simply Bending or Pure Bending ......................... 202
5.2.1 Assumptions in Theory of Pure Bending .203
5.2.2 Neutral Axis
5.2.3 Equation of Pure Bending ........................ 204
5.2.4 Limitations of Equation of Pure Bending 204
5.3 Nature of Bending Stress ........................................ 204
5.4 Sectional Modulus (Z) .............................................. 206
5.5 Moment of Resistance (MOR) ................................ 207
5.6 Bending Stresses in Axially Loaded Beams ...... 212
5.7 Force on a Partial Area of a Section ..................... 216
5.8 Bending Stress Distribution in Composite Beam 218
5.8.1 Equivalent Section ...................................... 219
5.9 Flitched Beam ............................................................. 224
5.9.1 Top and Bottom Flitched Beam
5.10 Beam of Uniform Strength ..................................... 228
5.11 Biaxial Bending ........................................................... 229
Objective Brain Teasers .............................................. 245
Conventional Practice Questions ............................ 251
Chapter 6Shear Stress in Beams .........................253 6.1 Shear Stress Distribution in Beams ...................... 253
6.1.1 Shear Stress Distribution in Rectangular Section ..............................................................255
6.1.2 Shear Stress Distribution in Triangular Section ..............................................................257
6.1.3 Shear Stress Distribution in Circular
Sections ............................................................259
6.1.4 Shear Stress Distribution in I-section ... 261
6.1.5 Shear Stress Distribution in Some Other Sections ............................................................262
6.2 Shear Stresses in Composite Sections ................ 264
.............................................. 270
Conventional Practice Questions ............................ 272
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Chapter 7Principal Stress-strain and Theories of Failure 273 7.1 Principal Plane ............................................................ 273
7.2 Principal Stresses ....................................................... 273
7.2.1 Analytical Method ....................................... 273
7.3 Principal Stress in Beams ......................................... 284
7.4 Graphical Method (Mohr’s Circle Method) ....... 290
7.4.1 Properties of Mohr’s Circle ....................... 291
7.4.2 Construction of Mohr’s Circle .................. 293
7.5 Analysis of Strain ........................................................ 299
7.5.1 Analytical Method ....................................... 299
7.5.2 Graphical Method (Mohr’s Circle Method) ..302
7.5.3 Properties of Strain Mohr’s Circle ........... 303
7.6 Total strain energy in terms of principal stress 304
7.7 Strain Rosette .............................................................. 305
7.8 Theories of Elastic Failure ........................................ 310
7.8.1 Maximum Principal Stress Theory
(Rankine’s Theory) ...................................... 310
7.8.2 Maximum Principal Strain Theory
(St. Venant’s Theory) .................................. 312
7.8.3 Maximum Shear Stress Theory (Guest &
Tresca’s) .......................................................... 314
7.8.4 Maximum Strain Energy Theory (Haigh
and Beltrami) ................................................ 315
7.8.5 Maximum Shear Strain Energy Theory or
Distortion Energy Theory (Mises-Henky
Theory) ........................................................... 318
7.8.6 Octahedral Shear Stress Theory ............. 319
Objective Brain Teasers .............................................. 327
Conventional Practice Questions ............................ 335
Chapter 8Torsion of Shafts ..................................336 8.1 Introduction ................................................................. 336
8.2 Difference between Bending Moment and
Twisting Moment ....................................................... 336
8.3 Assumptions Involved in the Theory of Pure Torsion 336
8.3.1 Sign Convention of Torque ...................... 337
8.3.2 Effects of Torsion .......................................... 337
8.3.3 Polar Section Modulus ............................... 339
8.4 Shear Stress Distribution in Circular Section ... 342
8.5 Design of Shaft ........................................................... 343
8.6 Power Transmitted by Shaft ................................... 345
8.7 Series Combination of Shaft .................................. 348
8.8 Parallel Combination of Shaft ................................ 348
8.9 Strain Energy in Torsion ........................................... 353
8.10 Torsion in Thin Walled Tubes.................................. 355
8.10.1 Angle of Twist in Thin Walled Tube ..... 356
8.11 Torsion of Non-circular Section ............................ 358
8.12 Indeterminate Shaft .................................................. 359
8.13 Shaft Subjected to Combined Bending Moment
and Twisting Moment .............................................. 362
8.13.1 Equivalent Bending Moment ................ 363
8.13.2 Equivalent Torque ..................................... 364
8.14 Shaft Subjected to Combined Axial Force and
Torsional Moment ...................................................... 366
8.15 Theories of Failure for Shaft Design .................... 369
Objective Brain Teasers .............................................. 387
Conventional Practice Questions ............................ 393
Chapter 9Deflection of Beams ............................395 9.1 Introduction ................................................................. 395
9.2 Methods for Determining Slope and Deflection .395
9.2.1 Double Integration Method .................... 395
9.2.2 Use of Discontinuity Function : Macaulay’s
Method .............................................................407
9.2.3 Area Moment Method: (Mohr’s Method) 418
9.2.4 Conjugate Beam Method ......................... 435
9.2.5 Strain Energy Method ................................ 444
9.2.6 Method of Superposition ......................... 453
9.2.7 Application of Maxwell’s Reciprocal
Theorem ...........................................................454
Objective Brain Teasers .............................................. 469
Conventional Practice Questions ............................ 477
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Chapter 10Pressure Vessels ...................................479 10.1 Thin Cylindrical Shell ............................................... 479
10.1.1 Stresses in the Thin Cylindrical Shell .. 479
10.2 Analysis of Thin Cylindrical Shell with Closed Flat Ends ................................................................................ 479
10.2.1 Hoop Stress or Circumferential Stress 480
10.2.2 Longitudinal Stress (sL) ........................... 480
10.2.3 Radial Stress (sR) ........................................ 481
10.3 Strains in Cylindrical Shell ....................................... 481
10.3.1 Hoop Strain (Major Principal Strain) ... 481
10.3.2 Longitudinal Strain ................................. 482
10.3.3 Volumetric Strain in Cylinder ................ 482
10.3.4 Maximum Shear Stress ............................ 485
10.4 Analysis of Thin Spheres .......................................... 486
10.5 Strains in Sphere ........................................................ 487
10.6 Stresses in Riveted Cylindrical Shell .................... 488
10.7 Thin Cylinders with Hemispherical Ends ........... 489
10.7.1 Thickness of Cylinder for Same Hoop
Stress ................................................................489
10.7.2 Thickness of Cylinder for No Distortion at
Junction ..........................................................489
10.8 Pressure Vessels Subjected to Axial Force ......... 490
10.9 Thick Cylinder .............................................................. 494
10.9.1 Analysis of Thick Cylinder ...................... 494
10.9.2 Stresses in Thick Cylinder ....................... 494
10.10 Analysis of Stresses.................................................... 495
10.10.1 Determination of A and B .................... 496
10.10.2 Variation of Radial and Hoop Stresses 497
10.11 Analysis of Thick Sphere .......................................... 497
10.11.1 Hoop and Longitudinal Stresses ...... 497
10.11.2 Radial Stress .............................................. 497
10.12 Design of Pressure Vessels ...................................... 500
10.13 Strengthening of Cylinder ...................................... 501
Objective Brain Teasers .............................................. 507
Conventional Practice Questions ............................ 509
Chapter 11Theory of Columns ..............................510 11.1 Compression Member ............................................. 510
11.2 Types of Equilibrium ................................................. 510
11.2.1 Elastic Instability and Critical Load ..... 511
11.3 Euler’s Theory for Buckling Failure ....................... 512
11.3.1 Assumptions of Euler’s Theory ............. 513
11.3.2 Effective Length of Column ................... 514
11.3.3 Critical Stress ............................................... 516
11.3.4 Limitations of Euler’s Theory ................. 517
11.3.5 Graph between s and l .......................... 517
11.4 Maximum Lateral Deflection of Column ........... 519
11.5 Rankine’s Gorden Theory ........................................ 520
11.6 Column with Eccentric Loading ........................... 522
11.6.1 Condition for No Tension ....................... 523
11.6.2 Middle Third Rule ...................................... 523
11.6.3 Middle Fourth Rule ................................... 524
11.7 Eccentric Loading about both x-axis and y-axis ...526
Objective Brain Teasers .............................................. 529
Conventional Practice Questions ............................ 532
Chapter 12Theoy of Springs ..................................534 12.1 Springs ........................................................................... 534
12.2 Types of Springs ......................................................... 534
12.2.1 Bending Springs ........................................ 534
12.2.2 Torsional Spring ......................................... 534
12.2.3 Helical Spring.............................................. 535
12.3 Springs in Series and Parallel ................................. 538
Objective Brain Teasers .............................................. 542
Conventional Practice Questions ............................ 544
Chapter 13Shear Centre .........................................545 13.1 Introduction ................................................................. 545
13.2 Location of Shear Centre ......................................... 545
13.3 Shear Flow .................................................................... 546
13.4 Shear Centres of Thin-walled Open Sections... 547
13.5 Thin Walled Semicircular Cross-section ............. 549
13.6 Shear Centres of Some Important Sections ..... 550
Objective Brain Teasers .............................................. 551
Conventional Practice Questions ............................ 552
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