Question 3 of 8: Thick-walled cylinder — allowable internal pressure

Question 3: Thick-walled cylinder — allowable internal pressure (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Closed thick cylinder, internal radius $r_i=0.05\text{ m}$, external radius $r_o=0.09\text{ m}$; loading $p_i=6p_e$.

Find. The allowable internal pressure $p_i$ by (a) von Mises and (b) maximum-shear (Tresca), for onset of yield at the bore (the most stressed point).

Approach. Evaluate the Lamé stresses at the inner wall with $p_e=p_i/6$, take the closed-end axial stress as the mean of radial and hoop, then apply each yield criterion at the bore where the stresses are largest.

1. Lamé stresses at the bore, in units of $p_i$. With $p_e=p_i/6$ and $r_o^2-r_i^2=5.6\times10^{-3}\text{ m}^2$, the inner-wall stresses become $$\sigma_r=-p_i,\quad \sigma_\theta=\frac{(r_o^2+r_i^2)-\tfrac{2}{6}r_o^2}{r_o^2-r_i^2}\,p_i=1.411\,p_i,\quad \sigma_z=\tfrac{1}{2}(\sigma_r+\sigma_\theta)=0.2054\,p_i.$$ (The closed-end axial stress equals the mean of $\sigma_r$ and $\sigma_\theta$, so it is the intermediate principal stress.)
2. (b) Maximum-shear (Tresca). The extreme principal stresses are $\sigma_\theta$ and $\sigma_r$: $$\sigma_\theta-\sigma_r=\sigma_Y\;\Rightarrow\;(1.411+1)\,p_i=285\;\Rightarrow\; p_i=\frac{285}{2.411}=\boxed{118.2\text{ MPa}}.$$
3. (a) Von Mises. Using all three principal stresses, $$\sigma_\text{vM}=\sqrt{\tfrac12\!\left[(\sigma_\theta-\sigma_r)^2+(\sigma_r-\sigma_z)^2+(\sigma_z-\sigma_\theta)^2\right]}=2.088\,p_i.$$ Setting $\sigma_\text{vM}=\sigma_Y$, $$p_i=\frac{285}{2.088}=\boxed{136.5\text{ MPa}}.$$
4. Compare. Von Mises permits the higher pressure (136.5 vs 118.2 MPa) because it credits the intermediate axial stress; Tresca ignores $\sigma_z$ and is the conservative design value.