A 12-in-long strip of steel is 1/8in thick and 1 in wide, as shown in Fig. 3–28. If the allowable shear stress is 11 500 psi and the shear modulus is 11.5(106) psi, find the torque corresponding to the allowable shear stress and the angle of twist, in degrees,(a) using Eq. (3–47) and (b) using Eqs. (3–40) and (3–41).

Question

A 12-in-long strip of steel is [latex]\frac{1}{8} [/latex]in thick and 1 in wide, as shown in Fig. 3–28.

If the allowable shear stress is 11 500 psi and the shear modulus is [latex]11.5\left(10^6\right) [/latex]psi, find the torque corresponding to the allowable shear stress and the angle of twist, in degrees,

(a) using Eq. (3–47) and (b) using Eqs. (3–40) and (3–41).

Accepted Answer

(a) The length of the median line is 1 in. From Eq. (3–47)

[latex] au =G heta _1c = \frac{3T}{Lc^2} [/latex] ,

[latex]T=\frac{L_c^2 au}{3} =\frac{\left(1 ight)\left({1}/{8} ight)^211500 }{3} =59.90\ lbf.in[/latex]

[latex] heta = heta _1l=\frac{ au l}{Gc} =\frac{11500\left(12 ight) }{11.5\left(10^6 ight)\left({1}/{8} ight) } =0.0960 \ rad=5.5°[/latex]

A torsional spring rate [latex]k_t[/latex] can be expressed as[latex]{T}/{ heta }:\k_t={59.90}/{0.0960}=624 \ lbf.in/rad[/latex]

(b) From Eq. (3–40)

[latex] au _{max} =\frac{T}{abc^2} =\frac{T}{bc^2}\left(3+\frac{1.8}{{b}/{c}} ight) [/latex]

[latex]T= \frac{ au_{max} bc^2 } {3+{1.8}/{ \left({1}/{0.125} ight) } } =55.72 \ lbf.in [/latex]

From Eq. (3–41)

[latex] heta =\frac{Tl}{\beta bc^3G} \ , \ with \ {b}/{c}={1}/{0.125}=8[/latex] ,

[latex] heta =\frac{Tl}{\beta bc^3G} =\frac{55.72\left(12 ight) }{0.307\left(1 ight)0.125^3\left(11.5 ight)10^6 } =0.0970 \ rad=5.6° [/latex]

[latex]k_t={55.72}/{0.0970}=574 \ lbf.in/rad[/latex]

The cross section is not thin, where b should be greater than c by at least a factor
of 10. In estimating the torque, Eq. (3–47) provides a value of 7.5 percent higher than
Eq. (3–40), and is 8.5 percent higher than when the table on page 102 is used.

Question 1.3.12: A 12-in-long strip of steel is 1/8in thick and 1 in wide, as...

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