A TEXTBOOK OFENGINEERING MECHANICS

A TEXTBOOK OF

ENGINEERING MECHANICS(In SI Units)

For

B.E./B.Tech. Ist YEAR

Strictly as per the latest syllabus prescribed byGautam Buddh Technical University, Lucknow

By

Dr. R.K. BANSALB.Sc. Engg. (Mech.), M. Tech., Hons. (I.I.T., Delhi),

Ph.D., M.I.E. (India)Formerly

Professor of Mechanical EngineeringDepartment of Mechanical Engineering

Delhi College of EngineeringDelhi

AND

SANJAY BANSALB.E. (Computer)

LAXMI PUBLICATIONS (P) LTD(An ISO 9001:2008 Company)

BENGALURU • CHENNAI • GUWAHATI • HYDERABAD • JALANDHARKOCHI • KOLKATA • LUCKNOW • MUMBAI • RANCHI • NEW DELHI

BOSTON (USA) • NAIROBI (KENYA)

A TEXTBOOK OF ENGINEERING MECHANICS

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First Edition : 2011; Reprint : 2012, 2013, 2014, Eight Edition : 2014, 2016ISBN 978-81-318-0855-9

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ContentsChapters Pages

UNIT-I: FORCE SYSTEMS AND FRICTION

1. Basic Concepts of Engineering Mechanics 3–271.1. Definitions ... 3

1.2. Scalar and Vector Quantities ... 3

1.3. Free, Forced and Fixed Vectors ... 4

1.4. Law of Parallelogram of Forces ... 4

1.5. Lami’s Theorem ... 7

1.6. System of Units ... 7

1.7. Resolution of a Force ... 17

1.8. Moment of a Force ... 20

1.9. Laws of Mechanics ... 22

1.10. The Principle of Transmissibility of Forces ... 24

Highlights ... 25

Exercise 1 ... 25

2. Force System and Classification 28–482.1. Coplanar Collinear and Concurrent Forces ... 28

2.2. Classification of a Force System ... 28

2.3. Resultant of Several Forces ... 29

2.4. Resultant of Coplanar Forces ... 30

2.5. Resultant of Collinear Coplanar Forces ... 30

2.6. Resultant of Concurrent Coplanar Forces ... 31

2.7. Resultant of Non-concurrent Coplanar Forces ... 42

Highlights ... 46

Exercise 2 ... 46

3. Moment of a Force and Varignon’s Theorem 49–663.1. Parallel Forces ... 49

3.2. Moment of a Force ... 49

3.3. Varignon’s Theorem (or Principle of Moments) ... 50

3.4. Types of Parallel Forces ... 53

3.5. Resultant of Two Parallel Forces ... 53

3.6. Resolution of a Force into a Force and a Couple ... 58

3.7. General Case of Parallel Forces in a Plane ... 61

Highlights ... 63

Exercise 3 ... 63

( v )

4. Equilibrium of Coplanar Force System and Free Body Diagram 67–1024.1. Equilibrium of Coplanar Force System ... 67

4.2. Action and Reaction ... 78

4.3. Free Body Diagrams ... 79

4.4. Equilibrium of a Body Under Three Forces ... 97

4.5. Lami’s Theorem ... 98

Highlights ... 100

Exercise 4 ... 100

5. Determination of Support Reactions 103–1275.1. Introduction ... 103

5.2. Types of Supports ... 103

5.3. Types of Loading ... 104

5.4. Determination of Reactions of a Beam ... 105

Highlights ... 125

Exercise 5 ... 125

6. Friction 128–1736.1. Introduction ... 128

6.2. Co-efficient of Friction (µ) ... 128

6.3. Angle of Friction (φ) ... 129

6.4. Cone of Friction ... 130

6.5. Types of Friction ... 130

6.6. Coulomb’s Laws of Friction ... 130

6.7. Angle of Repose ... 135

6.8. Equilibrium of a Body Lying on a Rough Inclined Plane ... 136

6.9. Applications of Friction ... 148

6.10. Wedge Friction ... 149

6.11. Belt Friction ... 153

6.12. Brakes and Screws Friction ... 160

6.13. Friction in a Square Threaded Screw ... 164

Highlights ... 171

Exercise 6 ... 172

UNIT-II: BASIC STRUCTURAL ANALYSIS

7. Plane Truss 177–2147.1. Difference between Truss and Frame ... 177

7.2. Perfect and Imperfect Truss ... 178

7.3. Assumptions and Analysis of Plane Truss ... 179

7.4. Method of Joints ... 179

7.5. Method of Sections ... 200

Chapters Pages

( vi )

Chapters Pages

7.6. Zero Force Members ... 207

Highlights ... 208

Exercise 7 ... 208

8. Beams (Shear Force and Bending Moment) 215–2658.1. Types of Beams ... 215

8.2. Types of Load ... 216

8.3. Statically Determinate Beams ... 216

8.4. Shear Force and Bending Moment in Beams ... 217

8.5. Shear Force and Bending Moment Diagrams ... 217

8.6. Shear Force and Bending Moment Diagrams for a Cantilever ... 219

8.7. Shear Force and Bending Moment Diagrams for a Simply Supported Beam... 234

8.8. Shear Force and Bending Moment Diagrams for Overhanging Beams ... 247

8.9. Shear Force and Bending Moment Diagrams for BeamsSubjected to Couples ... 255

8.10. Relations between Load, Shear Force and Bending Moment ... 259

8.11. Support Reactions from given Shear Force Diagram or givenBending Moment Diagram ... 260

Highlights ... 262

Exercise 8 ... 263

UNIT-III: CENTROID AND MOMENT OF INERTIA

9. Centroid and Moment of Inertia 269–3259.1. Centre of Gravity ... 269

9.2. Centre of Mass ... 269

9.3. Centroid or Centre of Gravity of Simple Plane Figures ... 269

9.4. Centroid (or Centre of Gravity) of Areas of Plane Figures by theMethod of Moments ... 269

9.5. Centroid of Composite Bodies ... 271

9.6. Determination of Centroid by Integration ... 278

9.7. Centroid of Volume ... 290

9.8. Definition of Moment of Inertia of Area ... 293

9.9. Perpendicular Axis Theorem and Polar Moment of Inertia ... 295

9.10. Parallel Axis Theorem ... 295

9.11. Moment of Inertia of Simple Areas by Integration ... 296

9.12. Moment of Inertia of Composite Areas ... 304

9.13. Polar Moment of Inertia of Masses ... 311

9.14. Parallel Axis Theorem for Mass Moment of Inertia ... 312

9.15. Mass Moment of Inertia of Simple Bodies by Integration ... 312

9.16. Mass Moment of Inertia of Composite Bodies ... 318

Highlights ... 319

Exercise 9 ... 321

( vii )

UNIT-IV: KINEMATICS AND KINETICS OF RIGID BODIES

10. Kinematics of Rigid Bodies 329–39410.1. Introduction ... 329

10.2. Absolute Motion ... 329

10.3. Plane Rectilinear Motion of Rigid Body ... 329

10.4. Plane Curvilinear Motion of Rigid Body ... 347

10.5. Rotation of Rigid Bodies ... 350

10.6. x – y and n – t Components ... 358

10.7. Relative Motion ... 362

10.8. Plane Motion of Rigid Bodies ... 366

10.9. Instantaneous Centre of Zero Velocity ... 382

Highlights ... 390

Exercise 10 ... 391

11. Kinetics of Rigid Bodies 395–46811.1. Introduction ... 395

11.2. Force, Mass and Acceleration ... 395

11.3. Newton’s Laws of Motion ... 396

11.4. D’ Alembert’s Principle and Dynamic Equilibrium ... 401

11.5. Laws of Motion Applied to Planer Translation ... 407

11.6. Momentum and Angular Momentum (or Moment of Momentum) ... 428

11.7. Laws of Motion Applied to Rotary Motion ... 432

11.8. Laws of Motion Applied to Plane Motion ... 441

11.9. Work and Energy ... 442

11.10. Kinetic Energy ... 448

11.11. Principle of Work and Energy ... 449

11.12. Conservative Forces ... 451

11.13. Law of Conservation of Energy ... 451

11.14. Linear Impulse and Momentum ... 459

11.15. Conservation of Linear Momentum ... 462

Highlights ... 465

Exercise 11 ... 466

UNIT-V: MECHANICS OF DEFORMABLE SOLIDS

12. Simple Stresses and Strains 471–49712.1. Introduction ... 471

12.2. Simple Stress and Strain ... 471

12.3. Normal and Shear Stresses ... 472

12.4. Elasticity and Elastic Limit ... 475

12.5. One Dimensional Loading ... 476

( viii )Chapters Pages

12.6. Stress on an Inclined Plane ... 481

12.7. Members of Varying Cross-Sections ... 486

12.8. Bars in Series ... 491

12.9. Tensile Test Diagrams for Ductile and Brittle Materials ... 493

Highlights ... 494

Exercise 12 ... 495

13. Elastic Constants 498–51613.1. Introduction ... 498

13.2. Longitudinal Strain ... 498

13.3. Lateral Strain ... 498

13.4. Poisson’s Ratio ... 499

13.5. Volumetric Strain ... 501

13.6. Volumetric Strain of a Cylindrical Rod ... 507

13.7. Bulk Modulus ... 508

13.8. Expression for Young’s Modulus in Terms of Bulk Modulus ... 508

13.9. Relationship between Modulus of Elasticity and Modulus of Rigidity ... 510

Highlights ... 513

Exercise 13 ... 514

14. Strain Energy and Impact Loading 517–52614.1. Introduction ... 517

14.2. Some Definitions ... 517

14.3. Expression for Strain Energy Stored in a Body when the Load is

Applied Gradually ... 517

14.4. Expression for Strain Energy Stored in a Body when the Load is

Applied Suddenly ... 519

14.5. Expression for Strain Energy Stored in a Body when the Load is

Applied with Impact ... 521

Highlights ... 525

Exercise 14 ... 525

15. Bending of Beams 527–54315.1. Introduction ... 527

15.2. Theory of Pure Bending ... 527

15.3. Neutral Surface and Neutral Axis ... 530

15.4. Stresses in Beams of Different Cross-Sections ... 532

Highlights ... 541

Exercise 15 ... 542

16. Theory of Torsion 544–55916.1. Torque and Twist ... 544

16.2. Shear Stress Produced in a Circular Shaft Subjected to Torsion ... 544

Chapters Pages

( ix )

16.3. Maximum Torque Transmitted by a Circular Solid Shaft ... 546

16.4. Torque Transmitted by a Hollow Circular Shafts ... 548

16.5. Power Transmitted by Shafts ... 549

Highlights ... 557

Exercise 16 ... 558

Objective Type Questions 561–593

Examination Paper with Solutions 595–605

Chapters Pages

( x )

Preface to the Eighth EditionThe course contents of the eighth edition of the book entitled, ‘A Textbook of

Engineering Mechanics’ are planned in such a way that the book covers the complete courseof first year students of all disciplines of Gautam Buddh Technical University, Lucknow(Formerly known as U.P. Technical University, Lucknow) strictly as per the latest syllabusprescribed by G.B. Technical University, Lucknow. This edition has been thoroughly revisedand made up-to-date.

The book contains following five units:Unit I : Force Systems, Basic Concepts, Resultant of a Force System, Free Body

Diagrams, Equilibrium and Equation of Equilibrium and Friction.Unit II : Basic Structural Analysis, Plane Truss and Beam (Shear Force and Bending

Moment Diagrams).Unit III : Centroid and Moment of Inertia.Unit IV : Kinematics and Kinetics of Rigid Body.Unit V : Mechanics of Deformable Solids, Simple Stress and Strain, Bending of Beams

and Theory of Torsion.The following additions have been made in this edition:1. Free, Forced and Fixed Vectors. Application of Friction, Wedge, Belt, Brakes and

Screws.2. Zero Force Member and Statically determinate Beams.3. Mass Moment of Inertia of Composite Bodies.4. Relative Velocity, Plane Motion of Rigid Bodies and Conservative Forces.5. Tensile Test Diagrams for Ductile and Brittle Materials.

Also the latest solved papers of U.P. Technical University have been added atproper places. These additions makes the book more useful to the students.

The book is written in a simple and easy-to-follow languages, so that even an averagestudent can grasp the subject by self-study. At the end of each chapter, highlights, theoreticalquestions and many unsolved problems with answers are given for the students to solve them.

Though every care has been taken in checking the manuscripts and proofreading, yetclaiming perfection is very difficult. We shall be very grateful to the readers and users of thisbook for pointing any mistake that might have crept in. Suggestions for improvement aremost welcome and would be incorporated in the next edition with a view to make the bookmore useful.

—AUTHORS

( xi )

SYLLABUSB. Tech. Ist Year

NME-102 : ENGINEERING MECHANICS L T P [310]

UNIT I: FORCE SYSTEMS 9

• Basic concepts: Definitions, Basic assumptions, Scalar and Vector quantities, Free,Forced and fîxed vectors.

• Force System; Force, Classification and Representation, Force as a Vector, Compositionof forces, Parallelogram Law, Resolution, Principle of Transmissibility of forces.

• Moment of a force, Vector representation, Moment for coplanar force system, Varignon’stheorem.

• Couple, Vector representation, Resolution of a force into a force and a couple.• Force Systems: Coplanar Concurrent Force system and Coplanar Non-concurrent force

systems, Resultant of coplanar force system.• Equilibrium of coplanar force system, Free body diagrams, Determination of reactions,

Equilibrium of a body under three forces, Lami’s theorem.Friction

• Introduction, Wet and Dry friction, Theory of Dry friction, Angle of friction, Angle ofRepose, Cone of friction, Coulomb’s laws of friction, Applications of friction: Wedge,Belt, Brakes and Screws.

UNIT II: BASIC STRUCTURAL ANALYSIS 8

• Plane Truss: Difference between truss and frame, Perfect and imperfect truss,Assumptions and Analysis of Plane Truss, Method of joints, Method of section, Zeroforce members.

• Beams, Types of beams, Statically Determinate Beams, Shear force and bending momentin beams, Shear force and bending moment diagrams, Relationships between load, shearand bending moment.

UNIT III: CENTROID AND MOMENT OF INERTIA 8

• Center of Gravity, Center of Mass and Centroid of curves, areas, volumes, Determinationof centroid by integration, Centroid of composite bodies.

• Definition of Moment of inertia of area, Perpendicular axis theorem and Polar momentof Inertia, Parallel axis theorem, Moment of inertia of simple areas by integration,Moment of Inertia of Composite Areas.

• Moment of Inertia of masses, Parallel axis theorem for mass moment of inertia, Massmoment of inertia of simple bodies by intégration, Mass moment of inertia of compositebodies.

UNIT IV: KINEMATICS OF RIGID BODY 9

• Introduction, Absolute motion, Plane rectilinear motion of rigid body, Plane curvilinearMotion of rigid body, x–y and n–t components, Rotation of rigid bodies, Relative Motion,Plane Motion of rigid bodies, Instantaneous center of zero velocity

( xii )

Kinetics of Rigid Body• Introduction, Force, Mass and Acceleration, Newton’s law of motion, D’ Alembert’s Prin-

ciples and Dynamic Equilibrium, Laws of motion applied to planar translation, rotationand plane motion.

• Work and Energy, Kinetic energy, Principle of work and energy, Conservative forces,Law of conservation of energy.

• Linear Impulse and Momentum, Conservation of linear momentum.

UNIT V: MECHANICS OF DÉFORMABLE SOLIDS 8

• Simple Stress and Strain: Normal and shear stresses. One Dimensional Loading;Stress on an inclined plane, members of varying cross-section, bars in series. TensileTest diagram for ductile and brittle materials, Elastic constants, Strain energy, resilience.

• Bending of Beams: Theory of pure bending, neutral surface and neutral axis, stressesin beams of different cross-sections.

• Theory of Torsion: Torque and twist, Shear stress due to torsion circular sections.

( xiii )

A Textbook Of Engeneering Mechanics(Gautam Buddh Technical University)

Lucknow

Publisher : Laxmi Publications ISBN : 9788131808559Author : R. K. Bansal,Sanjay Bansal

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