THE SHEAR FORMULA

Because the strain distribution for shear is not easily defined, as in the
case of axial load, torsion, and bending, we will obtain the shear-stress
distribution in an indirect manner. To do this we will consider the
horizontal force equilibrium of a portion of an element taken from the
beam in Fig. 7–4a. A free-body diagram of the entire element is shown in
Fig. 7–4b. The normal-stress distribution acting on it is caused by the
bending moments M and M + dM. Here we have excluded the effects of
V, V + dV, and w(x), since these loadings are vertical and will therefore
not be involved in a horizontal force summation. Notice that Fx = 0 is
satisfied since the stress distribution on each side of the element forms
only a couple moment, and therefore a zero force resultant
Now let’s consider the shaded top portion of the element that has been
sectioned at y from the neutral axis, Fig. 7–4c. It is on this sectioned
plane that we want to find the shear stress. This top segment has a width t
at the section, and the two cross-sectional sides each have an area A . The
segment’s free-body diagram is shown in Fig.7–4d. The resultant moments
on each side of the element differ by dM, so that Fx = 0 will not be
satisfied unless a longitudinal shear stress t acts over the bottom
sectioned plane. To simplify the analysis, we will assume that this shear
stress is constant across the width t of the bottom face. To find the
horizontal force created by the bending moments, we will assume that
the effect of warping due to shear is small, so that it can generally be
neglected. This assumption is particularly true for the most common case
of a slender beam, that is, one that has a small depth compared to its
length. Therefore, using the flexure formula,
t = the shear stress in the member at the point located a distance y
from the neutral axis. This stress is assumed to be constant and
therefore averaged across the width t of the member
V = the shear force, determined from the method of sections and
the equations of equilibrium
I = the moment of inertia of the entire cross-sectional area
calculated about the neutral axis
t = the width of the member’s cross section, measured at the point
where t is to be determined
Q = yA, where A is the area of the top (or bottom) portion of the
member’s cross section, above (or below) the section plane
where t is measured, and y is the distance from the neutral axis
to the centroid of A
Although for the derivation we considered only the shear stress acting
on the beam’s longitudinal plane, the formula applies as well for finding
the transverse shear stress on the beam’s cross section, because these
stresses are complementary and numerically equal.