12.1 Introduction

Mechanical member subjected to external moment along the axis is called shaft. The moment is called torque. The state of shaft under torque is called torsion. Due to torsion, the sections of the shaft rotate relative to each other. This relative rotation is called angle of twist. On this basis, the torque is also called twisting moment.

Pure torsion: In general, a mechanical member may be subjected to axial force, shear force, bending moment and twisting moment, simultaneously. If a member is subjected to twisting moment alone, it is said to be in pure torsion as shown in Figure 12.1.

Torsion has wide application in engineering. Electric motors, generators, fans, a few drilling and boring tools are examples of the cases of torsion. The concept is extensively used in mechanical power transmission. A shear stress develops over the sections of the shaft subjected to torque. Estimation of shear stress in the shaft is very necessary for the design of shaft. The stress analysis is presented in section 12.2.

12.2 Analysis of shear stress in a shaft subjected to pure torque

As mentioned, the stress analysis is a statically indeterminate problem. The approach of strength of materials is to make assumptions regarding deformation, analyze strain, apply Hook's law and satisfy boundary conditions.

7.1 Introduction

Kinetics deals with the cause of motion. It is concerned about why the motion varies or how it may be produced. In other words, it looks into the following two aspects:

• Asserting force system required for producing a desired motion of the particle. Suppose a particle is required to move along a circular track, the forces required to perform this task may be estimated.

• Finding the motion of the particle subjected to a given force system. For example, the motion produced due to a sinusoidal variation of force on the particle.

The kinetic analysis of the particle is done with the help of Newton's laws of motion.

Truss is a structure made of straight mechanical members to support or transfer the loads to other bodies, e.g., structures used for supporting the power transmission lines, TV towers and the bridges. Structures which support the roof of sheds at railway stations, food grain stores the common examples of the application of truss. The truss is a very common structure in the engineering applications. Figure 4.1 shows a simple truss. AB, BC, AC, etc., are called members of the truss. The members are joined to form a rigid structure. A, B, C, D, etc., are called joints. These members are joined together at the joints by bolts, rivets or welds.

The truss supports the loads at joints and transfers them to the support. In the Figure 4.1, the supports are at A and E. These supports are either pin connected or roller.

The trusses are classified in different ways and in different groups in order to understand the procedure and simplification of analysis.

4.2 Construction of truss

Truss is made of straight members joined at the ends. Members may be joined by rivets or bolts or welded, but they are assumed to be joined by smooth pins. This is the idealization of the joints. It simplifies the analysis of truss.

4.3 Dimensionality of truss

The trusses are classified in two groups on the basis of dimensionality. These are:

(a) Two-dimensional or plane truss

(b) Three-dimensional or space truss

4.3.1 Plane Truss: If the center lines of all the members of the truss lie in a plane, the truss is called plane truss. It is also called two-dimensional truss.

4.3.2 Space Truss: If the center lines of all the members of the truss do not lie in a plane, the truss is called space truss or three-dimensional truss.

4.4 Rigidity of truss

Consider a triangular structure of three members ABC as shown in Figure 4.2a.

Mechanics is the science which describes and predicts the motion of bodies under the application of forces. Engineering mechanics is the branch of engineering which applies the laws of mechanics in design, and is at the core of every machine designed. It is almost impossible to over-emphasize the importance of this subject in engineering. It is apparent that having a sound knowledge of the fundamental principles of engineering mechanics and the techniques of their application are very essential for one to be considered a good engineer.

This book covers course material for the Engineering Mechanics syllabus of most universities in India. This textbook has been designed for undergraduate students pursuing an engineering course. It is expected that the book will also be useful to students preparing for India Engineering Services (IES), GATE and other PSU examinations.

The special features of the book are

• • Discussion on the limitations of the assumptions/idealizations

• • The elastic spring model for finding deflection in the elastic systems

• • Simple integration instead of double and triple integrals for finding moment of inertia

• • Numerous examples to cover all the fundamental principles: the reader is expected to study them carefully

• • Step-by-step problem-solving approach throughout the book

• • Summary in each chapter for quick reference

• • Concept Review Questions and Exercises aimed at strengthening the learnt methods for solving the problems

• • Objective Questions, True/False and Fill in the Blanks questions modeled from the main text to reinforce the understanding of fundamentals.

2.1 Introduction

Friction is a kind of natural force. It develops between two surfaces in contact when either there is a tendency of relative motion or a relative motion works between them. Friction is not a reaction force. It has got numerous applications in day to day life. It is the friction, which makes walking possible on the earth.

There are situations where friction is undesirable. For example, bearings, fluid transmission pipes, power screws, etc. There are many engineering applications where friction is very much desirable like brakes, clutches, power transmission devices, etc.

Frictional forces can be classified in following categories:

(i) Dry friction

(ii) Wet friction

Dry Friction: The friction between two solid surfaces is called dry friction.

Wet friction: The friction in thin film of liquid layer sandwiched between two solid surfaces is called wet friction.

Dry friction is generally large compared to wet friction. Lubricants are introduced to reduce the friction between two solid surfaces.

In this chapter, we shall confine the discussion to dry friction only. Dry friction can be further classified as:

Sliding friction: The friction between two plane surfaces, which slip or slide over each other, is called sliding friction.

Rolling friction: The friction between rolling surfaces is called rolling friction. Sliding friction occurs when both the surfaces in contact are plane, whereas rolling friction occurs when at least one of the surfaces is curved.

2.2 Nature of sliding friction

(i) Frictional force always tries to retard the relative motions between the surfaces in contact.

Suppose bodies A and B are in contact and are moving relative to each other (Figure 2.1a). Body B is moving to the right of A. The relative motion between two surfaces can be retarded by applying a force on A so that it moves towards left as shown in Figure 2.1b.

The equation 2.2 reveals that the force F is zero, when Ft is zero, and appears when Ft ≠ 0. Force Ft produces a tendency of relative motion. The force F is called friction force.

(ii) Friction force is a self-adjusting force up to a limit.

Experiments show that the body B remains stationary up to a certain value of Ft. If Ft is increased further, the body B starts motion with acceleration. It means that the friction force F can increase up to a certain limit.

5.1 Introduction

Centroid of area, center of mass, center of volume and center of gravity are very useful concepts in mechanics. These are called central points. The introduction of the concepts of centroid and center of mass greatly simplify the mathematical expressions of parallel axis theorems for areas and bodies, respectively. This will be shown in the latter part of this chapter.

It is proper to mention at this stage that the concept of center of mass offers wonderful simplifications in the calculations associated with any inertia related quantity. The mass moment of inertia, the area moment of inertia and the resultant of inertia force system are wonderfully simplified with the introduction of this concept. Kinetic energy expression for general motion of rigid bodies is greatly simplified by introducing the velocity of the center of mass. The resultant of inertia force passes through center of mass, and is directly related to the acceleration of this point. It is impossible to over praise the importance of this concept in the field of mechanics. This concept has been emphasized at its appearances in this book.

The force of buoyancy on body submerged in fluids always passes through center of volume. The center of gravity is the point at which the resultant of gravitational force on the body acts.

1.1 Introduction

Mechanics is the branch of science, which deals with forces. The entire classical mechanics is based on the Newton's laws of motion. Engineering mechanics is the application part of Newtonian mechanics. On one hand, it proposes suitable and simplified mathematical models for complicated situations, and, on the other hand, it develops the art and skill for efficient solutions.

Mechanics lies at the core of engineering analysis. Basic principles of mechanics are used in the study of subjects such as vibrations, strength of materials, stability of structures, robotics, rockets, space-craft, engine performance, fluid flow, electrical machines, etc.

1.2 Idealizations in mechanics

Some important idealizations in mechanics are defined and discussed as follows:

(a) Particle: A particle is considered as an object that has no size but mass. This is one of the widely used definitions in mechanics. In the true sense, nothing falls in this category. In practical sense, a body whose dimensions can be neglected in studying its motion or equilibrium is treated as a particle. For example, while studying the motion of moon around the earth, both the earth and the moon are treated as particles with mass concentrated at their center. Whenever the size of the body is unimportant in the mathematical formulation of the problem, the body is treated as a particle.

(b) Rigid body: A rigid body is the most commonly used concept in engineering mechanics. It is defined as the body, which does not deform under the application of forces. In the real sense, everybody undergoes deformation and the size of the body changes. This change, in most of the cases, is insignificant compared to the size of the body. Omission of this deformation does not affect the solution. In general, solid bodies are assumed to be rigid bodies, if the deformation is negligible compared to the size of the body.

(c) Space: The geometric region occupied by the body is termed as space. The position of the body is specified with reference to a co-ordinate system. The concepts of point, direction and length are required for measurement, and the location is space. A point is the indication of a location in space. A point has no size. Length is the concept for describing the size of the body by comparison with a second body of known size. For two-dimensional problems, two independent coordinates are needed.

8.1 Introduction

Virtual work method has many applications. It is more powerful than the traditional method in solving problems of frames and machines. It is especially suitable for problems involving degrees of freedom. The principle of virtual work is applicable to both types of problems: in static equilibrium and in dynamic condition. D’Alembert's principle is used to convert dynamic problem into static problem.

8.2 Degrees of freedom

To describe the physical motion of a system, a set of variables or coordinates are required. These are called generalized coordinates. The minimum number of independent coordinates needed to describe the motion of the system is called degrees of freedom.

A free particle is shown in Figure 8.1a. It requires three coordinates to completely specify its location. So, the degree of freedom of a particle in space is three.