info MECHANICS OF SOLIDS This page intentionally left blank www.org MECHANICS OF SOLIDS S. Bhavikatti Emeritus Fellow (AICTE) BVB College of Engineering and Technology, Hubli (Formerly Principal, RYMEC, Bellary S. Bhavikatti Professor & Dean SDMCET, Dharwad and NITK, Surathkal) Copyright © 2010, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd.info All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to rights@newagepublishers.com ISBN (13) : 978-81-224-2858-2 PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.com Preface Mechanics of Solids is an important course for all engineering students by which they develop analytical skill. In this course, laws of mechanics are applied to parts of bodies and skill is developed to get solution to engineering problems maintaining continuity of the parts. The author has clearly explained theories involved and illustrated them by solving a number of engineering problems. Neat diagrams are drawn and solutions are given without skipping any step. SI units and standard notations as suggested by Indian Standard Code are used throughout. The author has made this book to suit the latest syllabus of Gujarat Technical University. Author hopes, the students and teachers of Gujarat Technical University will receive this book whole-heartedly as most of the earlier books of the author have been received by the students and teachers all over India. The suggestions and corrections, if any, are most welcome. The author acknowledges the efforts of M/s. New Age International Publish- ers in bringing out this book in nice form. He also acknowledges the opportu- nity given by AICTE for associating him with B. Engineering College, Hubli.org This page intentionally left blank www.info Contents Preface v 1 INTRODUCTION TO MECHANICS OF SOLIDS 1–14 1.1 Basic Terminologies in Mechanics .3 Scalar and Vector Quantities .4 Composition and Resolution of Vectors . 14 Problems for Exercise . 14 2 FUNDAMENTALS OF STATICS 15–64 2.1 Principles of Statics .2 System of Forces .3 Moment of a Force .6 Transfer of a Force to Parallel Position .7 Composition of Concurrent Coplanar Forces .8 Equilibriant of a Force System .9 Composition of Coplanar Non-concurrent Force System .10 X and Y Intercepts of Resultant .11 Types of Forces on a Body .12 Free Body Diagram .13 Equilibrium of Bodies .14 Equilibrium of Concurrent Force Systems .15 Equilibrium of Connected Bodies .16 Equilibrium of Non-concurrent Force Systems . 58 Problems for Exercise .1 Perfect, Deficient and Redundant Trusses .3 Nature of Forces in Members .4 Methods of Analysis .5 Method of Joints .6 Method of Section . 87 Problems for Exercise . 88 4 DISTRIBUTED FORCES, CENTRE OF GRAVITY AND MOMENT 94–160 OF INERTIA 4.1 Determination of Areas and Volumes .2 Centre of Gravity and Centroids .3 Centroid of a Line .4 First Moment of Area and Centroid .5 Second Moments of Plane Area .6 Moment of Inertia from First Principles .7 Moment of Inertia of Composite Sections .8 Theorems of Pappus-Guldinus .9 Centre of Gravity of Solids .info Important formulae . 152 Problems for Exercise .1 Coefficient of Friction .2 Laws of Friction .3 Angle of Friction, Angle of Repose and Cone of Friction .4 Problems on Blocks Resting on Horizontal and Inclined Planes .5 Application to Wedge Problems .6 Application to Ladder Problems . 187 Problems for Exercise .3 Law of Machine .4 Variation of Mechanical Advantage .5 Variation of Efficiency .6 Reversibility of a Machine .9 Wheel and Axle .10 Wheel and Differential Axle .11 Weston Differential Pulley Block .14 Differential Screw Jack . 224 Problems for Exercise . 225 7 PHYSICAL AND MECHANICAL PROPERTIES OF 228–233 STRUCTURAL MATERIALS 7. 233 8 SIMPLE STRESSES AND STRAINS 234–282 8.1 Meaning of Stress .2 Unit of Stress .5 Stress-Strain Relation .6 Nominal Stress and True Stress .7 Factor of Safety .9 Extension/Shortening of a Bar .10 Bars with Cross-sections Varying in Steps .11 Bars with Continuously Varying Cross-sections .17 Relationship between Modulus of Elasticity and Modulus of Rigidity .18 Relationship between Modulus of Elasticity and Bulk Modulus .19 Composite/Compound Bars .21 Thermal Stresses in Compound Bars . 277 CONTENTS Important Formulae . 279 Problems for Exercise .2 Types of Supports .3 Types of Beams .4 Types of Loading .5 Reactions from Supports of Beams .6 Shear Force and Bending Moment .8 Relationship between Load Intensity, Shear Force and Bending Moment .9 Shear Force and Bending Moment Diagrams .10 SFD and BMD for a few Standard Cases .11 Short-cut Procedure . 310 Problems for Exercise . 310 10 STRESSES IN BEAMS 313–345 10.info Bending Equation .3 Locating Neutral Axis .4 Moment Carrying Capacity of a Section .5 Section Moduli of Standard Sections .7 Shear Stress Distribution .8 Shear Stresses in Built-up Sections . 343 Problems for Exercise . 343 11 PRINCIPAL STRESSES AND STRAINS 346–373 11.1 Stresses on Inclined Planes .2 Principal Stresses and Planes .3 Principal Stresses in Beams .5 Measurement of Strain . 372 Problems for Exercise . 372 1 Introduction to Mechanics of Solids The state of rest and the state of motion of the bodies under the action of different forces has engaged the attention of mathematicians and scientists for many centuries. The branch of physical science that deal with the state of rest or the state of motion of bodies is termed as mechanics. Starting from the analysis of rigid bodies under gravitational force and application of simple forces the mechanics has grown into the analysis of complex structures like multistorey buildings, aircrafts, space crafts and robotics under complex system of forces like dynamic forces, atmospheric forces and temperature forces. Archemedes (287–212 BC), Galileo (1564–1642), Sir Issac Newton (1642–1727) and Einstein (1878–1955) have contributed a lot to the development of mechanics. Contributions by Varignon, Euler, and D. Alemberts are also substantial. The mechanics developed by these researchers may be grouped as (i) Classical mechanics/Newtonian mechanics (ii) Relativistic mechanics (iii) Quantum mechanics/Wave mechanics. Sir Issac Newton, the principal architect of mechanics, consolidated the philosophy and experimental findings developed around the state of rest and state of motion of the bodies and putforth them in the form of three laws of motion as well as the law of gravitation. The mechanics based on these laws is called Classical mechanics or Newtonian mechanics. Albert Einstein proved that Newtonian mechanics fails to explain the behaviour of high speed (speed of light) bodies. He putfourth the theory of Relativistic mechanics. Schrödinger (1887–1961) and Broglie (1892–1965) showed that Newtonian mechanics fails to explain the behaviour of particles when atomic distances are concerned. They putforth the theory of Quantum mechanics. Engineers are keen to use the laws of mechanics to actual field problems. Application of laws of mechanics to field problems is termed as Engineering mechanics. For all the problems between atomic distances to high speed distances there are various engineering problems for which Newtonian mechanics has stood the test of time and hence is the mechanics used by engineers. The various bodies on which engineers are interested to apply laws of mechanics may be classified as (i) Solids and (ii) Fluids.org 2 MECHANICS OF SOLIDS The bodies which do not change their shape or size appreciably when the forces are applied are termed as Solids while the bodies which change their shape or size appreciably even when small forces are applied are termed as Fluids. Stone, steel, concrete etc. are the example of solids while water, gases are the examples of fluids. In this book application of Newtonian mechanics to solids is dealt with.1 BASIC TERMINOLOGIES IN MECHANICS The following are the terms basic to the study of mechanics, which should be understood clearly. Mass The quantity of the matter possessed by a body is called mass. The mass of a body will not change unless the body is damaged and part of it is physically separated. If the body is taken out in a space craft, the mass will not change but its weight may change due to the change in gravitational force. The body may even become weightless when gravitational force vanishes but the mass remain the same. Time The time is the measure of succession of events. The successive event selected is the rotation of earth about its own axis and this is called a day. To have convenient units for various activities, a day is divided into 24 hours, an hour into 60 minutes and a minute into 60 seconds. Clocks are the instruments developed to measure time. To overcome difficulties due to irregularities in the earths rotation, the unit of time is taken as second which is defined as the duration of 9192631770 period of radiation of the cesium-133 atom.info The geometric region in which study of body is involved is called space. A point in the space may be referred with respect to a predetermined point by a set of linear and angular measurements. The reference point is called the origin and the set of measurements as coordinates. If the coordinates involved are only in mutually perpendicular directions, they are known as cartesian coordination. If the coordinates involve angles as well as the distances, it is termed as Polar Coordinate System. Length It is a concept to measure linear distances. The diameter of a cylinder may be 300 mm, the height of a building may be 15 m, the distance between two cities may be 400 km. Actually metre is the unit of length. However depending upon the sizes involved micro, milli or kilo metre units are used for measurements. A metre is defined as length of the standard bar of platinum-iradium kept at the International Bureau of weights and measures. To overcome the difficulties of accessibility and reproduction now metre is defined as 1690763.73 wavelength of krypton-86 atom. Continuum A body consists of several matters. It is a well known fact that each particle can be subdivided into molecules, atoms and electrons. It is not possible to solve any engineering problem by treating a body as conglomeration of such discrete particles. The body is assumed to be a continuous distribution of matter. In other words the body is treated as continuum. INTRODUCTION TO MECHANICS OF SOLIDS 3 Rigid Body A body is said to be rigid, if the relative positions of any two particles do not change under the action of the forces acting on it.1 (a), point A and B are the original positions in a body. After the application of forces F1, F2, F3, the body takes the position as shown in Fig. A′ and B′ are the new positions of A and B. If the body is treated as rigid, the relative position of A′B′ and AB are the same i. A′B′ = AB Many engineering problems can be solved by assuming bodies rigid F2 B B′ A′ A F1 F3 (a) (b) Fig.1 Particle A particle may be defined as an object which has only mass and no size. Theoretically speaking such a body cannot exist. However in dealing with problems involving distances considerably larger compared to the size of the body, the body may be treated as a particle, without sacrificing accuracy. For example: — A bomber aeroplane is a particle for a gunner operating from the ground. — A ship in mid sea is a particle in the study of its relative motion from a control tower. — In the study of movement of the earth in celestial sphere, earth is treated as a particle. Force Force is an important term used in solid mechanics. Newton’s first law states that everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled by an external agency acting on it.
