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

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SHEAR STRESS IN bEAMS
d
2
d
2
FIGURE 9.27 Shifting load
P away from the centroid.
x
A
A′
F f
e
y
Ff
P
B
Fw = V
z
The thickness of each flange is thin compared with the overall depth d of the channel
shape; therefore, the vertical shearing force transmitted by each flange is small and can be
neglected. (See Figure 9.16.) Consequently, the resultant shear force F w determined by integrating
the shear flow in the web must equal V. The resultant shear force F f produced in
each flange by the shear flow can be determined by integrating q over the width b of the
channel flange. The directions of the resultant shear forces in the flanges and in the web are
shown in Figure 9.26b. Since the forces F f are equal in magnitude, but act in opposite directions,
they form a couple that tends to twist the channel section about its longitudinal axis x.
This couple, which arises from the resultant shear forces in the flanges, causes the channel
to twist as it bends, as depicted in Figure 9.24b.
In Figure 9.27, the couple formed by the flange forces F f causes the channel to
twist in a counterclockwise direction. To counterbalance this twist, an equal clockwise
torsional moment is required. A torsional moment can be produced by moving the
external load P away from the centroid (i.e., to the right in Figure 9.27). Because there
is moment equilibrium about point B (located at the top of the channel web), the beam
will no longer have a tendency to twist when the clockwise moment Pe equals the
counterclockwise moment F f d. The distance e measured from the centerline of the
channel web defines the location of the shear center O. Furthermore, the location of
the shear center is solely a function of the cross-sectional geometry and dimensions,
and does not depend upon the magnitude of the applied loading, as will be demonstrated
in Example 9.11.
When the vertical external load P acts through the shear center O of the channel
(Figure 9.28a), the cantilever beam bends without twisting (Figure 9.28b).
The shear center of a cross section is always located on an axis of symmetry.
Thus, if the external load is applied in the horizontal direction through the centroid
of the channel, as shown in Figure 9.29a, there is no tendency for the channel to
twist as it bends (Figure 9.29b). The resultant shear forces in the flanges are equal
in magnitude, and both act to oppose the applied load P. In the channel web, there
e
P
e
P
O
O
(a) External load P acting through the shear
center O
FIGURE 9.28 Bending of the cantilever beam without twisting.
(b) Bending without twisting in response to
the applied load