Discussion
A closed thin cylindrical tank with a mean diameter d = 300 mm and thickness t = 2 mm, is subjected to a uniform internal gas pressure p. The allowable shear stress on the curved wall of the tank is 70 MPa.
Based on the Tresca criteria, which one of the following options for the maximum safe value of p (in MPa) is CORRECT?
Answer: C
The maximum safe pressure is 1.87 MPa.
For a thin closed cylinder the wall stresses are the hoop stress, the longitudinal stress and, on the inside, a radial stress that is small enough to take as zero:
\[ \begin{aligned} \sigma_{h} &= \frac{pd}{2t} = \frac{300p}{4} = 75p \\ \sigma_{l} &= \frac{pd}{4t} = 37.5p, \quad \sigma_{r} \approx 0 \end{aligned} \]
The Tresca criterion uses the largest shear stress in three dimensions, half the difference between the largest and smallest principal stresses, here \(\sigma_{h}\) and \(\sigma_{r} = 0\):
\[ \begin{aligned} \tau_{max} &= \frac{75p - 0}{2} = 37.5p = 70 \\ \Rightarrow p &= 1.87\ \text{MPa} \end{aligned} \]
3.73 MPa is what the in-plane shear stress \((\sigma_{h} - \sigma_{l})/2 = 18.75p\) alone would allow; it ignores the larger out-of-plane shear. 7.46 and 5.60 MPa would put the absolute maximum shear stress above 70 MPa.
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