The shearing stress on a cross section of beam of rectangular cross section may be found out by the relation
where is the first moment of the portion of the cross-sectional area above the point where shear stress is required about neutral axis, Vis the transverse shear force,b is the width of the rectangular cross-section and Izz is the moment of inertia of the cross-sectional area about the neutral axis. Due to shear stress, the angle between the lines which are originally at right angle will change. The shear stress varies across the height in a parabolic manner in the case of a rectangular cross-section. Also, the shear stress distribution is different for different shape of the cross section. However, to simplify the computation shear stress is assumed to be uniform (which is strictly not correct) across the cross section. Consider a segment of length ds subjected to shear stress τ. The shear stress across the cross section may be taken as
in which A is area of the cross-section and k is the form factor which is dependent on the shape of the cross section. One could write, the deformation du as
where Δy is the shear strain and is given by
Hence, the total deformation of the beam due to the action of shear force is
Now the strain energy stored in the beam due to the action of transverse shear force is given by,
The strain energy due to transverse shear stress is very low compared to strain energy due to bending and hence is usually neglected. Thus the error induced in assuming a uniform shear stress across the cross section is very small.
