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

(b) Maximum Horizontal Shear Stress in Cross Section
The maximum horizontal shear stress in the rectangular cross section occurs at the neutral
axis. To calculate Q, the area beginning at the z axis and extending upwards or extending
downwards may be used, as shown in the following two figures:
6 in.
12 in.
6 in.
z
3 in.
Neutral axis
12 in.
6 in.
z
Neutral axis
3 in.
6 in.
6 in.
For either area, the first moment of area is calculated as
Q = (6 in.)(6 in.)(3 in.) = 108 in.
max
The maximum value of Q always occurs at the neutral-axis location. Also, the maximum
horizontal shear stress usually occurs at the neutral axis. There are instances, however, in
which the width t of the stressed surface varies over the depth of the cross section. In such
instances, it is possible that the maximum horizontal shear stress will occur at a y location
other than the neutral axis.
The maximum horizontal shear stress in the rectangular cross section is computed as
3
3
VQmax
(4.5 kips)(108 in. )
τ max = = = 0.0938 ksi = 93.8 psi
4
It (864 in. )(6 in.)
z
Ans.
pRoBLEmS
For Problems P9.1 through P9.4, a beam segment subjected to internal
bending moments at sections A and B is shown along with a
sketch of the cross-sectional dimensions. For each problem,
(a) sketch a side view of the beam segment and plot the distribution
of bending stresses acting at sections A and B. Indicate
the magnitudes of key bending stresses in the sketch.
(b) determine the resultant forces acting in the x direction on the
specified area at sections A and B, and show these resultant
forces in the sketch.
(c) determine the horizontal force required to satisfy equilibrium
for the specified area and show the location and direction of
this force in the sketch.
Δx y
M A
M B
z
x
A
B
FIGURE p9.1a/2a Beam segment.
z
(1)
(2)
y
b
d 1
d 2
FIGURE p9.1b/2b
Cross section.
d
p9.1 The beam segment shown in Figure P9.1a/2a is subjected
to internal bending moments M A = 1,640 lb ⋅ ft and M B = 760 lb ⋅ ft.
The segment length is Dx = 7 in. Consider area (1) shown in Figure
P9.1b/2b, and use cross-sectional dimensions of b = 3 in., d = 9 in.,
and d 1 = 1.5 in.
p9.2 The beam segment shown in Figure P9.1a/2a is subjected to
internal bending moments M A = 1,430 N ⋅ m and M B = 3,140 N ⋅ m.
The segment length is Dx = 180 mm. Consider area (2) shown in
Figure P9.1b/2b, and use cross-sectional dimensions of b = 90 mm,
d = 240 mm, and d 2 = 70 mm.
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