**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## Concept of Stress and Strain

** Concept of Stress and Strain: **If any body is subjected to external forces, then the effect of these forces will be spread all over the body. The point, line or surface over which any exlernal force is acting will have a tendency to move along with this force. In other words, the body will have il tendency of deformation on account of these external forces. This deformation will however be resisted by the adjacent particles surrounding it. As a result of this, there will be internal forces caused by the reaction of the surrounding particles. The resistance set up in the body increases as the deformation increases and when full resistance to the external force is developed, the process of deformation stops. This deformation is called strain and the acconipanying resistance to deformation is called stress.

** INTERNAL FORCES AND STRESSES: **Internal Forces-Method of Sections The existence of internal forces in a body as discussed above will become clear if the

body shown in Figure 11.1 (a) is considered. At the point or section where he internal forces are to be studied, thc body is imagined to be cut into parts (1) & (2) by the plane a-a.

fig..(1)

Since the whole body is in equilibrium as supporled under the action of the external forces P, & P, and reactive forces R, & R, , i.t can be concluded that each of the parts (1) & (2) are also in equilibrium. The equilibriv,m of each part will be studied separately. For this purpose. both the parts (1) & (2) are shown separately in Figure 1 (b). The two parts (1) & (2) can remain in equiliibrium only when forccs are acting at the cut section a-a as shown irn figure. It is these forces which are termed as internal forces. These forces are produced by the mutual action and reaction of the particles in each part. The effect of part (2) on part (1) is such that part (1) is in equilibrium under the action of external forces acting on this part and internal forces. By Newton's third law, the effect of part (1)

on part (2) at section a-a can be repre.c;ented by these same internal forces but acting in the opposite directions on part (2'). Tnese forces will keep part (2) also in equilibrium. ' The above ideas are made clear .m Figure 1 (b). It will be seen that with the help of the above method known as "method of sections", the internal forces acting in the body can be revealed and calculated. The above considerations lead to the following fundamental conclusion: "The external forcta to one side of an arbitrary cut must be balanced by the internal forces developed at the cut". The above concept will be relied upon as a first step in solving all problems where the internal forces are being investigated.

In discussing the method of sections, it is significant to note that some bodies, although not in static equilibrium, may be in dynamic equilibrium. These problems can be reduced to problems of static equilib~iiumF. irst, the acceleration 'a' of the par1 in question is computed, then it is multiplied by the mass of the body, giving a force F = ma. If the force so computed is applied to the body at its mass centre in a direction opposite to the acceleration the dynamic problem is reduced to one of statics. This is the so-called D'Alembert prinlciple which you have already studied. With this point in view, all bodies can be thought of' as being instantaneously in a state of static equilibrium. Hence for any body, whether in static or dynamic equilibrium, a free-body diagram can be prepared on which the necessary forces to maintain the body as a whole in equilibrium can be shown. From there on, the problem is the same as discussed above.

**Stresses - Normal and Shear:**** **In general, the internal forces acting on infinitesimal areas of a cut section may be of varying magnitudes and directions. These internal forces are vectorial in nature and maintain in equilibrium the externally applied forces. In mechanics of materials, it is particularly significant to determine the intensity of these forces on the various portions of the cut, as resistance lo deformation and the capacity of the materials to resist forces depend on these intensities. With respect to the body shown in Figure 11.1, consider the part (1) acted upon by external forces P, & R,. Figure 11.2 (a) shows the section made by plane a-a. Let AF be the magnitude of the internal force acting on an elementary area A A. The symbol 'A' is to be understood as referring lo a small quantity. In general, the force AF acting

fig..(2)

on the elementary area Δ A varies from point to point and is inclined with respect to the plane of the cut, The ratio means the average force acting on area Δ A or the force per unit area acting on Δ A. The force 'P ' is also called the resultant stress acting on area Δ A.