Shear Stresses in Beams

Shear Stress in Beams:
When a beam is subjected to nonuniform bending, both bending moments, M, and
shear forces, V, act on the cross section. The normal stresses, σx, associated with the
bending moments are obtained from the flexure formula. We will now consider the
distribution of shear stresses, τ, associated with the shear force, V.
Let us begin by examining a beam of rectangular cross section. We can reasonably
assume that the shear stresses τ act parallel to the shear force V. Let us also assume
that the distribution of shear stresses is uniform across the width of the beam.
Shear




Shear stresses on one side of an element are accompanied by shear stresses of equal
magnitude acting on perpendicular faces of an element. Thus, there will be horizontal
shear stresses between horizontal layers (fibers) of the beam, as well as, transverse
shear stresses on the vertical cross section. At any point within the beam these
complementary shear stresses are equal in magnitude.
The existence of horizontal shear stresses in a beam can be demonstrated as follows
 
A single bar of depth 2h is much stiffer that two separate bars each of depth h.

Shown below is a rectangular beam in pure bending.