A solid circular shaft, of diameter 50 mm and length 300 mm, is subjected to a gradually increasing torque T . The yield stress in shear for the shaft material is 120 MN/m² and, up to the yield point, the modulus of rigidity is 80 GN/m² (a) Determine the value of T and the associated angle of

Question

A solid circular shaft, of diameter 50 mm and length 300 mm, is subjected to a gradually increasing torque T . The yield stress in shear for the shaft material is [latex]120 MN/m^{2} [/latex]and, up to the yield point, the modulus of rigidity is [latex] 80 GN/m^{2}[/latex].
(a) Determine the value of T and the associated angle of twist when the shaft material first yields.
(b) If, after yielding, the stress is assumed to remain constant for any further increase in strain, determine the value of T when the angle of twist is increased to twice that at yield.

Accepted Answer

(a) For this part of the question the shaft is elastic and the simple torsion theory applies,

[latex]T=\frac{ au J}{R} =\frac{120 imes 10^{6} }{25 imes 10^{-3} } imes \frac{\pi \left(25 imes 10^{-3} ight)^{4} }{2} =2950[/latex]
[latex]=2.95 kNm[/latex]

[latex] heta =\frac{ au L}{GR} =\frac{120 imes 10^{6} imes 300 imes 10^{-3} }{80 imes 10^{9} imes 25 imes 10^{-3} } =0.018 radian[/latex]
[latex]=1.03°[/latex]

If the torque is now increased to double the angle of twist the shaft will yield to some radius [latex] R_{1}[/latex] . Applying the torsion theory to the elastic core only,

[latex] heta =\frac{ au L}{GR}[/latex]
[latex]2 imes 0.018=\frac{120 imes 10^{6} imes 300 imes 10^{-3} }{80 imes 10^{9} imes R_{1} }[/latex]

[latex]R_{1}= \frac{120 imes 10^{6} imes 300 imes 10^{-3} }{2 imes 0.018 imes 80 imes 10^{9} }=0.0125=12.5 mm[/latex]

Therefore partially plastic torque, from eqn. (3.12),

[latex]T_{pp}=\frac{\pi au _{y} }{6} \left[4R^{3}-R_{1} ^{3} ight][/latex] (3.12)

[latex]=\frac{\pi au _{y} }{6} \left[4R^{3}-R_{1} ^{3} ight][/latex]
[latex]=\frac{\pi imes 120 imes 10^{6} }{6} \left[4 imes 25^{3}-12.5^{3} ight] 10^{-9}[/latex]

Question 3.4: A solid circular shaft, of diameter 50 mm and length 300 mm,...

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