Comprehensive Theory

with Solved Examples and Practice Questions

Strength of Materials

Civil Engineering

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MADE EASY Publications

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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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