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

A
B
y
O
z
P
E
D
1.038 in.
0.643 in.
(a) Load acting through centroid
A
B
y
T
z
O
P
E
D
1.038 in.
0.643 in.
(b) Equivalent loading at shear center
shape for a load applied at the shear center.
Therefore, it will be valuable to determine an
equivalent loading that acts at the shear center.
This equivalent loading will enable us to separate
the loading into components that cause (a)
bending and (b) torsion.
The actual load acts through the centroid,
as shown in Figure (a). The equivalent load at
the shear center consists of a force and a concentrated
moment, as shown in Figure (b). The
equivalent force at O is simply equal to the applied
load P. The concentrated moment will be
a torque of magnitude
T = (900 lb)(0.643 in. + 1.038 in.) = 1,513 lb⋅in.
A
B
a
A
B
Shear Stress due to Bending
The maximum shear stress due to bending
caused by the 900 lb load was determined in
Example 9.12. The flow of the shear stress is
shown in Figure (c). Recall that the maximum
shear stress due to this load occurred at the horizontal
axis of symmetry and had a value
C O
τ P
E
D
(c) Shear stress due to bending
C
b
τ c =
1, 038 psi
Shear Stress due to torsion
T
τ
The torque T causes the member to twist, and
the shear stress is greatest along the edges of the
cross section. Recall that torsion of noncircular
E
D
sections—particularly, narrow rectangular cross
sections—was discussed in Section 6.11. That
(d) Shear stress due to torsion discussion revealed that the maximum shear stress
and the shear stress distribution for a member of
uniform thickness and arbitrary shape is equivalent to that of a rectangular bar with a large
aspect ratio. (See Figure 6.20.) For the channel shape considered here, the shear stress can
be calculated from Equation (6.25):
a = 0.125 in.
b = 3.00 in. + 8.00 in. + 3.00 in. = 14.00 in.
3T
3(1,513 lb⋅in.)
= =
= 20,750 psi
ab (0.125 in.) (14.00 in.)
τ max 2 2
Maximum Combined Shear Stress
The maximum stress due to the combined bending and twisting occurs at the neutral axis
(i.e., point C) on the inside surface of the web. The value of this combined shear stress is
τmax = τbend + τ twist = 1, 038 psi + 20,750 psi = 21, 788 psi Ans.
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