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

Calculation of Shear Stress Distribution
The radius of the given aluminum tube is
r
=
D + d
4
=
8.00 in. + 7.50 in.
4
= 3.875 in.
Thus, the shear stress distribution is
V
rt cos 21,000 lb
τ = θ =
π (3.875 in.)(0.25 in.) cos θ
π
= (6,900 psi) cosθ
The direction of the shear stress is shown in the next figure, along with a graph of the
shear stress magnitude as a function of the angle θ:
z
N.A.
π−θ
y
V
θ
τ
7,000
6,000
5,000
τ 4,000
(psi)
3,000
2,000
1,000
6,900
6,800
6,480
5,980
5,290
4,440
3,450
2,360
1,200
0
0
10
20
30
40 50
θ (deg)
60
70
80
90
9.10 Shear centers of Thin-Walled open Sections
In Sections 8.1 through 8.3, the theory of bending was developed for prismatic beams. In
deriving this theory, beams were assumed to have a longitudinal plane of symmetry (Figure
8.2a) and loads acting on the beam, as well as the resulting curvatures and deflections, were
assumed to act only in the plane of bending. The only time that the requirement of symmetry
was removed was in Section 8.8, where it was shown that the bending moment could be resolved
into component moments about the principal axes of the cross section, provided that
the loading was pure bending (i.e., no shear forces were present). However, unsymmetrical
bending configurations in which shear forces were present were not considered.
If loads are applied in the plane of bending and the cross section is symmetric with
respect to the plane of bending, twisting of the beam cannot occur. However, suppose that
we consider bending of a beam that is (a) not symmetric with respect to the longitudinal
plane of bending and (b) subjected to transverse shear forces in addition to bending moments.
For beams such as this, the resultant of the shear stresses produced by the transverse
loads will act in a plane that is parallel to, but offset from, the plane of loading. Whenever
the resultant shear forces do not act in the plane of the applied loads, the beam will twist
about its longitudinal axis in addition to bending about its neutral axis. Bending without
twisting is possible, however, if the transverse loads pass through the shear center, which
can be simply defined as the location (to the side of the longitudinal axis of the beam)
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