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

the distribution of shear stress magnitudes over the depth of the web, while still parabolic,
is relatively uniform (Figure 9.13a). The minimum horizontal shear stress occurs at the
junction between the web and the flange, and the maximum horizontal shear stress occurs
at the neutral axis. For wide-flange steel beams, the difference between the maximum and
minimum web shear stresses is typically in the range of 10–60 percent.
In deriving the shear stress for mula, it was assumed that the shear stress across the
width of the beam (i.e., in the z direction) could be considered constant. This assumption,
however, is not valid for the flanges of
beams; therefore, shear stresses computed
for the top and bottom flanges
τ min (1)
from Equation (9.2) and plotted in
y
Figure 9.13a are fictitious. Shear
stresses are developed in the flanges
x
τ max
z
(1) of a wide-flange beam, but they
V
t
act in the x and z directions, not the x
w
and y directions. Shear stresses in
τ min
thin-walled members, such as wideflange
shapes, will be discussed in
(a) Shear stress distribution
more detail in Section 9.9.
FIGURE 9.13 Shear stress distribution in a wide-flange shape.
339
SHEAR STRESSES IN wEbS OF
FLANgEd bEAMS
a
b f
(2)
y
(b) Cross section
t f
d—
2
d—
2
ExAmpLE 9.3
A concentrated load P = 36 kN is applied to the upper
end of a pipe as shown. The outside diameter of
the pipe is D = 220 mm, and the inside diameter is
d = 200 mm. Determine the vertical shear stress on
the y–z plane of the pipe wall.
36 kN
y
Plan the Solution
The shear stress in a pipe shape can be determined
from the shear stress formula [Equation (9.2)] with
the use of the first moment of area, Q, calculated
from Equation (9.10).
SolutioN
Section Properties
The centroid location for the tubular cross section can be determined from symmetry.
The moment of inertia of the cross section about the z centroidal axis is equal to
z
x
d
x
z
Pipe cross section.
y
36 kN
D
π
I D d
64 [ ] π
z = - = [(220 mm) - (200 mm) ] = 36,450,329 mm
64
4 4 4 4 4
Equation (9.10) is used to compute the first moment of area for a pipe shape:
1
Q D d
12 [ ] 1
= - = [(220 mm) - (200 mm) ] = 220,667 mm
12
3 3 3 3 3
z
x