Question

The intensity of horizontal shear stress at the elemental part of a beam section

Answer 1

Analysis of Horizontal Shearing Stress

In this section, fv will be used for shearing

stress instead of the standard symbol τ.

For the upper shaded portion of the beam, the forces acting are the total normal forces FR and FL due to the bending stresses to the left and to the right of the beam. These forces will be resisted by the shearing force fvb dx acting at the boundary surface between the shaded and the unshaded portions.

For equilibrium of the upper shaded portion

FL+FV−FR=0

FV=FR−FL

Where

FV=fvbdx

FL=∫fb1dA;fb1=MyI

FR=∫fb2dA;fb2=(M+dM)yI

Fv=∫(M+dM)yIdA−∫MyIdA

fvbdx=∫MyIdA+∫dMIydA−∫MyI

fvbdx=dMI∫ydA

fv=1IbdMdx∫ydA

But dMdx=V, where V represents the shear at the section in N, and ∫ydA=Ay¯ represents the first moment of an area of the shaded section about N.A. in mm3 which we will denote as Q, then

fv=VQIb