Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

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SHEAR STRESS IN bEAMS
Equation (f) defines the horizontal shear stress in a beam. Note that shear stress will exist
in a beam at those locations where the bending moment is not constant (i.e., where dM/dx ≠ 0).
As discussed previously, the first moment of area, Q, varies in value for every possible y in
the beam cross section. Depending upon the shape of the cross section, the width t may also
vary with y. Consequently, the horizontal shear stress varies over the depth of the cross section
at any location x along the beam span.
The simple investigation presented in Section 9.2 and the equations derived in this section
have illustrated the concept that is essential to understanding shear stresses in beams:
Horizontal shear forces and, consequently, horizontal shear stresses are caused in a flexural
member at those locations where the internal bending moment is changing along
the beam span. The imbalance in the resultant bending stress forces acting on a portion
of the cross section requires an internal horizontal shear force for equilibrium.
Equation (f) gives an expression for the horizontal shear stress developed in a beam. Still,
although the term dM/dx helps to clarify the source of shear stress in beams, it is awkward
for calculation purposes. Then, is there an equivalent expression for dM/dx? Recall the
relationships developed in Section 7.3 between the internal shear force and the internal
bending moment. Equation (7.2) defined the following relationship:
dM
dx
= V
in other words, wherever the bending moment is changing, there is an internal shear
force V. The term dM/dx in Equation (f) can be replaced by the internal shear force V to
give an expression for τ H that is easier to use:
VQ
τ H = (g)
It
z
The terms horizontal shear
stress and transverse shear
stress are both used in reference
to beam shear stress. Since shear
stresses on perpendicular planes
must be equal in magnitude,
these two terms are effectively
synonyms in that both refer to
the same numerical shear
stress value.
The moment of inertia I z in
Equation (9.2) is the moment
of inertia of the entire cross
section about the z axis.
Section 1.6 demonstrated that a shear stress never acts on just one surface. If there is a shear
stress τ H on a horizontal plane in the beam, then there is also a shear stress τ V of the same
magnitude on a vertical plane (Figure 9.8). Since the horizontal and vertical shear stresses
are equal, we will let τ H = τ V = τ; thus, Equation (g) can be simplified into a form commonly
known as the shear stress formula:
VQ
τ = (9.2)
It z
Because Q varies with area A′, the value of τ varies over the depth of the cross section. At the
upper and lower boundaries of the cross section (e.g., points a, c, d, and f in Figure 9.8), the
value of Q is zero, since area A′ is zero. The maximum value of Q occurs at the neutral axis
of the cross section. Accordingly, the largest shear stress τ is usually located at the neutral
axis; however, this is not necessarily so. In Equation (9.2), the internal shear force V and the
moment of inertia I z are constant at any particular location x along the span. The value of Q
is clearly dependent upon the particular y coordinate being considered. The term t (which is
the width of the cross section in the z direction at any specific y location) in the denominator
of Equation (9.2) can also vary over the depth of the cross section. Therefore, the maximum
horizontal shear stress τ occurs at the y coordinate that has the largest value of Q/t—most
often, but not necessarily, at the neutral axis.