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

SHEAR STRESS IN bEAMS
y
z
z
a
c
b
V
(a) Box shape
(b) Area A′ for calculating Q at
point a
z
A′
a
A′
y
y
(c) Area A′ for calculating Q at
point b
V
V
b
y′
y′
force V). The area A′ begins at this cut line and extends away from the neutral axis to the free
surface of the beam. (Recall the free-body diagram in Figure 9.7 used to evaluate the horizontal
equilibrium of area A′ in the derivation of Equation 9.1.) The area A′ to be used in calculating
Q at point a is highlighted in Figure 9.9b. The centroid of the highlighted area relative to
the neutral axis (the z axis in this instance) is determined, and Q is calculated from the product
of this centroidal distance and the area of the shaded portion of the cross section.
A similar process is used to calculate Q at point b. The box shape is sectioned at b parallel
to the neutral axis. (Note: V is always perpendicular to its corresponding neutral axis.) The
area A′ begins at this cut line and extends away from the neutral axis to the free surface, as
(d) Area A′ for calculating Q at
point c
FIGURE 9.9 Calculating Q at different locations in a box-shaped
cross section.
Generally, if the point of interest
is above the neutral axis, it is
convenient to consider an area A′
that begins at the point and
extends upward. If the point of
interest is below the neutral axis,
consider the area A′ that begins
at the point and extends
downward.
330
z
A′
c
(3)
y
V
y 1
(1)
2
Vy
(2)
b
z
y
(e) Calculation process
shown in Figure 9.9c. The centroidal location y – ′ of the highlighted
area relative to the neutral axis is determined, and Q
is calculated from Q = y – ′A′.
Point c is located on the neutral axis for the box shape;
thus, area A′ begins at the neutral axis (Figure 9.9d). For
points a and b, it was clear which direction was meant by
the phrase “away from the neutral axis.” However, in this
instance c is actually on the neutral axis—a situation that
raises the question, Should area A′ extend above the neutral
axis or below the neutral axis? The answer is, Either direction
will give the same Q at point c. Although the area above
the neutral axis is highlighted in Figure 9.9d, the area below
the neutral axis would give the same result. The centroidal
location y – ′ of the highlighted area relative to the neutral axis
is determined, and Q is calculated from Q = y – ′A′.
The first moment of the total cross-sectional area A
taken about the neutral axis must be zero (by definition of
the neutral axis). While the illustrations given here have
shown how Q can be calculated from an area A′ above
points a, b, and c, the first moment of the area below
points a, b, and c is simply the negative. In other words,
the value of Q calculated with the use of an area A′ below
points a, b, and c must have the same magnitude as Q
calculated from an area A′ above points a, b, and c. It is usually easier to calculate Q with
the use of an area A′ that extends away from the neutral axis, but there are exceptions.
Let us consider the calculation of Q at point b (Figure 9.9c) in more detail. The area
A′ can be divided into three rectangular areas (Figure 9.9e) so that A′ = A 1 + A 2 + A 3 . The
centroid location of the highlighted area with respect to the neutral axis is
The value of Q associated with point b is
yA + yA + yA
Q = y′ A′ =
A + A + A
y′
y 3
yA + yA + yA
y′ =
A + A + A
1 1 2 2 3 3
1 2 3
1 1 2 2 3 3
1 2 3
( A + A + A ) = y A + y A + y A
1 2 3 1 1 2 2 3 3
This result suggests a more direct calculation procedure that is often expedient. For cross
sections that consist of i shapes,
∑
Q = y A
(9.4)
where y i is the distance between the neutral axis and the centroid of shape i and A i is the
area of shape i.
i
i
i