WORLD FOR CIVIL

Topic 5.3: Beams -Horizontal Shear Stress

In addition to the bending (axial) stress which develops in a loaded beam, there is also a shear stress which develops, including both a Vertical Shear Stress, and a Horizontal (longitudinal) Shear Stress. It can be shown that at any given point in the beam, the values of vertical shear stress and the horizontal shear stress must be equal, at that point, for static equilibrium. As a result it is usual to discuss and calculate the horizontal shear stress in a beam (and simply remember that the vertical shearing stress is equal in value to the horizontal shear stress at any given point).We will take a moment to derive the formula for the Horizontal Shear Stress. In Diagram 1, we have shown a simply supported loaded beam.

In Diagram 2a, we have cut a section dx long out of the left end of the beam, and have shown the internal horizontal forces acting on the section.

In Diagram 2b, we have shown a side view of section dx. Notice that the bending moment is larger on the right hand face of the section by an amount dM. (This is clear if we make the bending moment diagram for the beam, in which we see the bending moment increases from a value of zero at the left end to a maximum at the center of the beam.)

In Diagram 2c, we have shown a top slice of section dx. Since the forces are different between the top of the section and the bottom of the section (less at the bottom) there is a differential (shearing) force which tries to shear the section, shown in Diagram 2c, horizontally. This means there is a shear stress on the section, and in terms of the shear stress, the differential shearing force, F, can be written as F = times the longitudinal area of the section (b dx). A second way of expressing the shear force is by expressing the forces in terms of the bending stress, that is F₁ = (My/I) dA, and F2 = (M+dM)y/I dA, then the differential force is (dM y/I)dA. If we now combine the two F = expressions, we have:

F = * b dx = (dM y/I)dA, and then rewriting to solve for the shear stress:
= [(dM/dx)/Ib] y dA, however dM/dx is equal to the shear force V (as discussed in the previous topic), and y dA is the first moment of the area of the section, and may be written as A y', where A is the area of the section and y' is the distance from the centroid of the area A to the neutral axis of the beam cross section. Rewriting in a final form we have:

Horizontal Shear Stress: = VAy'/Ib, where

V = Shear force at location along the beam where we wish to find from the horizontal shear stress
A = cross sectional area, from point where we wish to find the shear stress at, to an outer edge of the beam cross section (top or bottom)
y' = distance from neutral axis to the centroid of the area A.
I = moment of inertia for the beam cross section.
b = width of the beam at the point we wish to determine the shear stress.
(In some texts, the product Ay' is given the symbol Q and used in the shear stress equation)

If we consider our shear relationship a little, we observe that the Horizontal Shear Stress is zero at the outer edge of the beam - since the area A is zero there. The Horizontal Shear Stress is (normally) a maximum at the neutral axis of the beam. This is the opposite of the behavior of the Bending Stress which is maximum at the other edge of the beam, and zero at the neutral axis.

To help clarify the Horizontal Shear Stress equation we will now look at at several example of calculating the Horizontal Shear Stress.

Please select:
Topic 5.3a: Horizontal Shear Stress - Example 1
Topic 5.3b: Horizontal Shear Stress - Example 2