Page 134 - Shigley's Mechanical Engineering Design

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Load and Stress Analysis 109

EXAMPLE 3–11 Compare the shear stress on a circular cylindrical tube with an outside diameter of 1 in
and an inside diameter of 0.9 in, predicted by Eq. (3–37), to that estimated by
Eq. (3–45).

Solution From Eq. (3–37),
Tr Tr T(0.5)
τ max = = 4 = = 14.809T
4
J (π/32) d − d (π/32)(1 − 0.9 )
4
4
o i
From Eq. (3–45),
T T
τ = = 2 = 14.108T
2A m t 2(π0.95 /4)0.05
Taking Eq. (3–37) as correct, the error in the thin-wall estimate is −4.7 percent.

Open Thin-Walled Sections
When the median wall line is not closed, the section is said to be an open section. Fig-
ure 3–27 presents some examples. Open sections in torsion, where the wall is thin, have
8
relations derived from the membrane analogy theory resulting in:
3T
τ = Gθ 1 c = 2 (3–47)
Lc
where τ is the shear stress, G is the shear modulus, θ 1 is the angle of twist per unit
length, T is torque, and L is the length of the median line. The wall thickness is
designated c (rather than t) to remind you that you are in open sections. By study-
ing the table that follows Eq. (3–41) you will discover that membrane theory pre-
sumes b/c →∞. Note that open thin-walled sections in torsion should be avoided
in design. As indicated in Eq. (3–47), the shear stress and the angle of twist are
2
3
inversely proportional to c and c , respectively. Thus, for small wall thickness,
stress and twist can become quite large. For example, consider the thin round tube
with a slit in Fig. 3–27. For a ratio of wall thickness of outside diameter of
c/d o = 0.1, the open section has greater magnitudes of stress and angle of twist by
factors of 12.3 and 61.5, respectively, compared to a closed section of the same
dimensions.

Figure 3–27 c

Some open thin-wall sections.
L

8 See S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill, New York, 1970, Sec. 109.