ENGINEERINGMECHANICS

P.N. Chandramouli

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New Delhi-1100012011

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ENGINEERING MECHANICSP.N. Chandramouli

© 2011 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this bookmay be reproduced in any form, by mimeograph or any other means, without permission inwriting from the publisher.

ISBN-978-81-203-4422-8

The export rights of this book are vested solely with the publisher.

Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus,New Delhi-110001 and Printed by Mohan Makhijani at Rekha Printers Private Limited,New Delhi-110020.

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Preface xv

1. Mechanics: An Overview 1–19

1.1 What Is Engineering Mechanics? 11.2 Historical Background 11.3 Classification of Engineering Mechanics 21.4 Continuum 41.5 Introduction to Engineering Mechanics 41.6 Basic Concepts of Mechanics 51.7 Basic Principles of Mechanics 7

1.7.1 Parallelogram Law for Addition of Forces 71.7.2 Principle of Transmissibility 71.7.3 Newton’s Laws 81.7.4 Newton’s Law of Gravitation 10

1.8 Units 101.9 Dimensional Homogeneity 121.10 Vector and Scalar Quantities: Elementary Operations 121.11 Procedures for the Solution of Mechanics Problems 17

Summary 17Review Questions 18Objective Type Questions 19

2. Forces and Basic Principles of Statics 20–95

2.1 Introduction 202.2 Force 20

viii Contents

2.3 Characteristics of Force 212.3.1 Specifying the Forces/Naming the Forces—Bow’s Notation 22

2.4 Types of Forces 232.4.1 External Forces (Contacting or Applied Forces) 232.4.2 Body Forces (Non-contacting or Non-Applied Forces) 24

2.5 Pressure and Stress 252.6 Free Body Diagram 252.7 Types of Force Systems 272.8 Resolution of a Force 302.9 Vector Analysis 33

2.9.1 Orthogonal Projection of a Vector on an Axis 342.10 Multiplication of Vectors 522.11 Resultant of Two Concurrent Forces by Parallelogram Construction 58

2.11.1 Composition by Graphical Method 582.11.2 Composition by Analytical Method 59

2.12 Principles of Force Equilibrium 762.12.1 Equilibrium of Two-force Members 762.12.2 Equilibrium of Three-force Members 772.12.3 Equilibrium of Four-force Members 79

2.13 Law of Superposition of Forces 79

Summary 89Review Questions 90Numerical Problems 91Objective Type Questions 94

3. Coplanar, Concurrent Force System 96–150

3.1 Introduction 963.2 Resultant of Coplanar, Concurrent Force Systems 96

3.2.1 Graphical Method: Polygon Law of Forces 973.2.2 Analytical (Scalar) Method: Principle/Theorem of Resolved Parts 97

3.3 Equilibrium of Coplanar, Concurrent Force System 117

Summary 144Review Questions 145Numerical Problems 145Objective Type Questions 150

4. Coplanar, Non-concurrent Force System 151–243

4.1 Introduction 1514.2 Moment of a Force 151

4.2.1 Vector Formulation 1524.2.2 Unit of Moment 153

4.3 Types of Moments 1534.3.1 Scalar Formulation 153

Contents ix

4.4 Derivation of Varignon’s Theorem 1544.4.1 Different Expressions for the Moment of a Force

(Vector Formulation) 1554.5 Geometrical Meaning of Moment 1704.6 The Principle of Moments: Varignon’s Theorem 1714.7 Parallel Forces 171

4.7.1 Resultant of Two Parallel Forces 1724.7.2 Resultant of Two Unequal Parallel Forces Acting

in Opposite Directions 1734.7.3 General Case of Parallel Forces in a Plane 174

4.8 Couple (C) 1794.8.1 Vector Form 1804.8.2 Scalar Form 1804.8.3 Sense of a Couple 1814.8.4 Resolution of a Force into a Couple 184

4.9 Resultant of a Coplanar Non-concurrent Force System 1884.10 Equilibrium of Coplanar Force Systems 206

4.10.1 Types of Equilibrium 2104.10.2 Beams 2114.10.3 Types of Loading 2114.10.4 Support and Support Reactions 213

Summary 237Review Questions 237Numerical Problems 238Objective Type Questions 241

5. Analysis of Framed Structures 244–313

5.1 Introduction 2445.2 Elements of a Truss 2445.3 Determinacy and Stability 2465.4 Analysis of Simple Truss 2495.5 Method of Joints 2505.6 Method of Sections (Ritter’s Method) 300

Summary 310Review Questions 311Numerical Problems 311Objective Type Questions 313

6. Virtual Work 314–338

6.1 Introduction 3146.2 Work of a Force (U) 3146.3 Principle of Virtual Work 3156.4 Sign Conventions 317

x Contents

6.5 Applications of the Principle of Virtual Work 3186.5.1 Application of the Principle of Virtual Work for Problems on Beam 3186.5.2 Application of the Principle of Virtual Work for

Problems on Framed Structures 3306.5.3 Application of the Principle of Virtual Work for Problems on Ladder 332

Summary 335Review Questions 336Numerical Problems 336Objective Type Questions 337

7. Centroids and Centre of Gravity 339–422

7.1 Introduction 3397.2 Centre of Gravity 3397.3 Concept of First Moment 3407.4 Sign Conventions 3417.5 Centroid of a Length 3427.6 Centroid of a Sector of Circle 3447.7 Centroid of Composite Lines 3467.8 Centroid of an Area 3497.9 Methods of Finding the Centroid of a Plane Figure 351

7.9.1 By Geometrical Consideration 3517.9.2 By the Method of Moments 352

7.10 Centroids of Simple Geometric Areas 3537.10.1 Centroid of a Triangular Section 3537.10.2 Centroid of a Trapezoidal Section 354

7.11 Centroid of the Plane Area in Polar Coordinates 3557.11.1 Centroid of a Semicircular Lamina 355

7.12 Problems on Centroid of Area with a Parabolic Boundary 3577.13 Procedures to Find the Centre of Gravity (C.G.) of a Composite Figure 3647.14 Fluid Pressure Acting on Submerged Surfaces 385

7.14.1 Horizontal Plane Surface 3867.14.2 Vertical Plane Surface 3867.14.3 Centre of Pressure 387

7.15 Vertical Plane Surface of Arbitrary Shape Submerged in Liquid 3877.15.1 Centre of Pressure (h*) 388

7.16 Inclined Plane Surface Submerged in Liquid 4047.16.1 Centre of Pressure 405

7.17 Curved Surface Submerged in Liquid 4097.18 Centre of Gravity of a Body 413

Summary 415Review Questions 416Numerical Problems 416Objective Type Questions 422

Contents xi

8. Area Moments of Inertia 423–471

8.1 Introduction 4238.1.1 Basic Concept of Inertia 423

8.2 Definition of Moment of Inertia 4248.3 Polar Moment of Inertia 425

8.3.1 Units of Moment of Inertia 4258.4 Radius of Gyration 4258.5 Theorem’s of Moment of Inertia 426

8.5.1 Parallel Axis Theorem (Transfer Axis Theorem) 4268.5.2 Perpendicular Axis Theorem 427

8.6 Moments of Inertia of Simple Areas 4288.6.1 Rectangular Section 4298.6.2 Hollow Rectangular Section 4318.6.3 I-Section 4328.6.4 Circular Section 4338.6.5 Hollow Circular Section 4348.6.6 Triangular Section 4348.6.7 Quarter Circular Section 436

8.7 Composite Section 4438.8 Product of Inertia for an Area 4588.9 Parallel Axis Theorem 4598.10 Moment of Inertia of an Area about Inclined Axes 4648.11 Principal Moments of Inertia 465

Summary 469Review Questions 469Numerical Problems 470Objective Type Questions 470

9. Friction 472–523

9.1 Introduction 4729.2 Coefficient of Friction 4739.3 Angle of Friction 4759.4 Angle of Repose 4769.5 Cone of Friction 4779.6 Laws of Friction 477

9.6.1 Laws of Static Friction 4779.6.2 Laws of Dynamic or Kinetic Friction 478

9.7 Classification of Problems of Dry Friction 4789.8 Ladder Friction 5049.9 Wedge Friction 5109.10 Belt Friction 5139.11 Rolling Resistance 517

continue

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