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

Longitudinal plane
of symmetry
y
B′ B
= 0
y
B′
τ B = 0
B
363
SHEAR STRESS ANd SHEAR
FLOw IN THIN-wALLEd
MEMbERS
z
x
z
V
x
τ B′
τ C′ = 0
C′
C
τ C = 0
C′
(a) Closed thin-walled section with a longitudinal
plane of symmetry
C
FIGURE 9.23 Shear stress in a thin-walled box cross section.
Shear flow and shear stress
must vanish on a longitudinal
plane of symmetry.
(b) Shear stresses at the plane of symmetry
ExAmpLE 9.7
A beam with the thin-walled inverted-tee-shaped cross section shown is
subjected to a vertical shear force V = 37 kN. The location of the neutral
axis is shown on the sketch, and the moment of inertia of the inverted-tee
shape about the neutral axis is I = 11,219,700 mm 4 . Determine the shear
stresses in the tee stem at points a, b, c, and d, and in the tee flange at points
e and f. Plot the distribution of shear stress in both the stem and flange.
Plan the Solution
The location of the neutral axis and the moment of inertia of the invertedtee
shape about the neutral axis are given. The value of Q associated with
each point will be determined from Q = y – ′A′ for the applicable portion A′
of the cross-sectional area. After Q is determined, the shear stress will be
calculated from Equation (9.16).
143.0 mm
43.0 mm
8 mm
50 mm
a
50 mm
b 186 mm
N.A.
c 8 mm
V
d
f e
55 mm
220 mm
SolutioN
Points a, b, and c are located in the stem of the inverted-tee shape. A horizontal cutting
plane that is perpendicular to the walls of the stem defines the boundary of area A′. For
these locations, area A′ begins at the cutting plane and reaches upward to the top of the
stem. Point d is located at the junction of the stem and the flange. For this location, the area
A′ is simply the area of the flange. Point e is also at the junction of the stem and the flange;
however, the shear stress in only the flange is to be determined at e. The area A′ corresponding
to point e extends from the left end of the flange to a vertical cutting plane located at the
centerline of the stem. (Note that the centerline location for the cutting plane is acceptable
because the shape is thin walled.) For point f in the flange, a vertical cutting plane defines
the boundary of area A′, which extends horizontally from the cutting plane to the outer edge
of the flange. For all points, the first moment Q is the moment of the area A′ about the neutral
axis of the inverted-tee shape. The shear stress at each point is calculated from
τ =
where V = 37 kN and I = 11,219,700 mm 4 . The thickness t is 8 mm for each location. The
results of these analyses are summarized in the following table:
VQ
It