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

a
d
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THE FIRST MOMENT OF AREA, Q
V
b e
y
V+ΔV
x
e
τ V
M
M+ΔM
τ H
c f
τ H = τ V = τ
Δ x
FIGURE 9.8 Shear stress at point e.
The direction of the shear stress acting on a transverse plane is the same as the direction
of the internal shear force. As illustrated in Figure 9.8, the internal shear force acts
downward on section d–e–f. The shear stress acts in the same direction on the vertical
plane. Once the direction of the shear stress on one face has been determined, the shear
stresses acting on the other planes are known.
Although the stress given by Equation (9.2) is associated with a particular point in a
beam, it is averaged across the thickness t and hence is accurate only if t is not too great.
For a rectangular section having a depth equal to twice its width, the maximum stress computed
by methods that are more rigorous is about 3 percent greater than that given by Equation
(9.2). If the cross section is square, the error is about 12 percent. If the width is four
times the depth, the error is almost 100 percent! Furthermore, if the shear stress formula is
applied to a cross section in which the sides of the beam are not parallel, such as a triangular
section, the average stress is subject to additional error because the transverse variation
of stress is greater when the sides are not parallel.
9.4 The First moment of Area, Q
Calculation of the first moment of area, Q, for a specific y location in a beam cross section
is initially one of the most confusing aspects associated with shear stress in flexural members.
The reason for the confusion is that there is no unique value of Q for a particular cross
section—there are many values of Q. For example, consider the box-shaped cross section
shown in Figure 9.9a. In order to calculate the shear stress associated with the internal
shear force V at points a, b, and c, three different values of Q must be determined.
What is Q? Q is a mathematical abstraction termed a first moment of area. Recall that
a first-moment-of-area term appears as the numerator in the definition of a centroid:
y
=
∫A
∫
A
ydA
dA
(a)
Q is the first moment of area of only portion A′ of the total cross-sectional area A. Equation
(a) can be rewritten in terms of A′ instead of the total area A and multiplied by the denominator
of the right side (in terms of A′) to give a useful formulation for Q:
∫ A ∫ A
Q = ydA′ = y′ dA′ = y′ A′
′
′
(9.3)
Here, y – ′ is the distance from the neutral axis of the cross section to the centroid of area A′.
To determine Q at point a in Figure 9.9a, the cross-sectional area is subdivided at a by
slicing parallel to the neutral axis (which is perpendicular to the direction of the internal shear