5.1 INTRODUCTION

In the previous chapter, we have seen how to quantify stresses developed due to bending moment (BM). In developing the expression for bending stresses, it was assumed that BM is constant and shear force (SF) is zero. But this is a rare condition; generally, in all practical problems SF and BM occur together. Thus, the bending is not uniform bending but it is varying fl exure. The varying BM introduces SF there by shearing stresses in addition to the normal stress due to fl exure. In this chapter, we quantify the shearing stresses produced due to varying BM.

5.2 EXPRESSION FOR FLEXURAL SHEAR STRESS

Consider a beam subjected to varying moment. Take an elemental strip of length ‘δx’ in the longitudinal direction of the beam as shown in Figure 5.1.

At section (1)-(1), the BM is M, while the BM at section (2)-(2) is M + δM. At section (1)-(1), normal stress develops due to BM.

The normal stress at section (1)-(1) is M/I ξ , in which ξ is the location of elemental area dA with in the cross-section from neutral axis (NA).

The normal stress at the section (2)-(2) is

The normal stresses (bending) at sections (1)-(1) and (2)-(2) were shown in Figure 5.2.

Let us consider the equilibrium of force acting at sections (1)-(1) and (2)-(2) just above the fi ber considered, where shearing stress are to be found, that is, y distance from NA.

Stress at section (1)-(1) at ξ distance from

Force on the elemental area within the cross-section located ξ distance from

Total force acting on shaded area of cross-section shown in Figure 5.2

Similarly, the total force acting on the shaded area of the cross-section (2)-(2)

The free body diagram (FBD) of the shaded portion over length δx is shown in Figure 5.3.

For equilibrium, sum of forces in the horizontal direction/longitudinal direction of the element should be zero.

From the FBD, it is clear that there should be force equal to F2 – F1 to maintain equilibrium. Thus, force cannot act on the surface, and hence it has to act along the plane ABCD.