10.1 Introduction

One of the major reasons for the failure of materials or components is stress. The understanding and analysis of stress is a very important step in any design. Our primary objective here is to identify the stress, and then discuss the scheme of analysis for simple and complex situations.

10.2 Definitions

Stress is defined as the intensity of internal reaction. The intensity is measured with respect to the area. Consider a body in equilibrium under the action of certain forces as shown in Figure 10.1a.

6.1 Introduction

Kinematics is the study of the motion of particles and rigid bodies disregarding the forces associated with these motions.

6.2 Kinematics of particle

Kinematics of particle involves the study of position, velocity and acceleration of the particle without any consideration of the forces working on it. Particle can move on a straight line, on a plane or in space. We will restrict our study to plane motion only. The plane motion of the particle is classified as:

(a) Straight line motion

(b) Motion on curved path

Following classification of motion on curved path is useful from application point of view

(a) Position, velocity and acceleration in terms of Cartesian components

(b) Position, velocity and acceleration in terms of path variables and

(c) Position, velocity and acceleration in terms of polar coordinates

11.1 Introduction

A mechanical member subjected to a force system perpendicular to its axis is called beam. The moment of force system at any point on the beam is perpendicular to the axis of the beam. Stress developed in the beam due to bending moment is called bending stress. The bending stress developed on any section of the beam is normal to the section. The stress developed is such that the resultant reaction on the section is zero.

In general, the shear force and bending moment both are present at a section of a beam. In particular, if the shear force is absent and only bending moment is present on a beam, the beam is said to be in pure bending. Due to pure bending, the beam bends in the form of a circular arc as shown in Figure 11.1. The bending moment, which develops curvature upwards, is also called as sagging moment (Figure 11.1b). Similarly, the moment, which produces curvature downwards is called hogging moment (Figure 11.1a). It is clear from Figure 11.1 that the fibres toward the center of curvature are compressed and the outer fibres are elongated. The stresses developed in both the regions are shown in the Figure 11.1.

3.1 Introduction

A mechanical member is, in general, subjected to different types of loads. The mechanical members, depending on the type of loading, have been given different names in technical literature. Some technical names and loadings are given as follows:

(a) Column: Member subjected to compressive force along its axis (Figure 3.1a). Visible effect: Shortening in axial length of the member.

(b) Shaft: Member subjected to moments along the axis. Such a moment is called torque or twisting moment (Figure 3.1b). Visible effect: The relative rotation of the sections about the axis.

(c) Beam: Member subjected to forces and/or moments perpendicular to the axis (Figure 3.1c). Visible effect: The axis becomes curved. This is called bending and the moment responsible for it is called bending moment.

3.2 Terminology

(a) On the basis of supports, the beams are classified as:

(i) Simply supported: Both the supports are at the end and are either pin or roller (Figure 3.2a).

(ii) Cantilever: Fixed at one end and free at the other (Figure 3.2b).

(iii) Overhanging: Beam is larger than the distance between the supports, and is projected beyond the supports (Figure 3.2c).

(iv) Fixed beam: Both the ends are fixed (Figure 3.2d).

(v) Continuous beam: Beams having more than two supports (Figure 3.2e).

9.1 Introduction

Vibration is a very common phenomenon in machines. In certain situations, the control of vibration is very essential. To control the vibration, proper understanding of the phenomenon is desired. The essence of vibration is discussed below.

A mechanical system is said to vibrate if it executes a simple harmonic motion or oscillation.