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

and the angle of twist for a 12 in. long bar is
TL
(2,000 lb⋅in.)(12in.)
φ = =
3 3
βabG
(0.249)(1.00 in.) (2.50 in.)(500,000 psi)
= 0.0771 rad Ans.
For bar (b), the long side of the bar is b = 1.875 in. and the short side is a = 1.25 in.; therefore,
b/a = 1.5. From Table 6.1, α = 0.231 and β = 0.196.
The maximum shear stress produced in bar (b) by a torque T = 2,000 lb · in. is
T
2,000 lb⋅in.
= =
= 2,960 psi
Ans.
αab
(0.231)(1.25 in.) (1.875 in.)
τ max 2 2
and the angle of twist for a 12 in. long bar is
TL
(2,000 lb⋅in.)(12in.)
φ = =
= 0.0669 rad Ans.
3 3
βabG
(0.196)(1.25 in.) (1.875 in.)(500,000 psi)
6.12 Torsion of Thin-Walled Tubes: Shear Flow
The elementary torsion theory presented in Sections 6.1, 6.2, and 6.3 is limited to circular
sections; however, one class of noncircular sections can be readily analyzed by elementary
methods. These shapes are thin-walled tubes such as the one illustrated in Figure
6.21a, which represents a noncircular section with a wall of variable thickness (i.e.,
t varies).
A useful concept associated with the analysis of thin-walled sections is the shear flow q,
defined as the internal shearing force per unit of length of the thin section. Typical units for
q are pounds per inch or newtons per meter. In terms of stress, q equals τ × t, where τ is the
average shear stress across the thickness t.
First, we will demonstrate that the shear flow on a cross section is constant even
though the wall thickness of the section may vary. Figure 6.21b shows a block cut from the
member of Figure 6.21a between A and B. Since the member is subjected to pure torsion,
dx
t B
T
y
V 3
B
B
V 4
V 2
dx
A
T
V 1
A
z
x
t A
(a)
t
(b)
FIGURE 6.21 Shear flow in thin-walled tubes.
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