Question 7 of 8: Allowable internal pressure of a thick-walled cylinder

Question 7: Allowable internal pressure of a thick-walled cylinder

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. $r_i=150$ mm, $r_o=190$ mm, elastic limit $\sigma_Y=350$ MPa, $\nu=0.32$, $p_i=6p_e$; closed-end cylinder.

Find. the allowable internal pressure by (a) Tresca and (b) von Mises.

Approach. Evaluate the Lamé stresses at the inner wall (the critical radius), express them per unit external pressure, then apply each yield criterion.

1. Lamé stresses at the bore. With $p_i=6p_e$ and $r_o^2-r_i^2=13\,600\ \text{mm}^2$, at $r=r_i$ $$\sigma_\theta=\frac{p_i(r_i^2+r_o^2)-2p_er_o^2}{r_o^2-r_i^2}=20.54\,p_e,\quad \sigma_r=-p_i=-6\,p_e,\quad \sigma_z=\frac{\sigma_r+\sigma_\theta}{2}=7.27\,p_e.$$ The closed-end axial stress is the intermediate principal stress.
2. Maximum-shear-stress (Tresca) — part a. Yield when $\sigma_\theta-\sigma_r=\sigma_Y$: $$(20.54+6.00)\,p_e=350\;\Rightarrow\;p_e=13.19\ \text{MPa},\quad p_i=6p_e=\boxed{79.1\ \text{MPa}}$$
3. Von Mises — part b. With $\sigma_\theta-\sigma_z=\sigma_z-\sigma_r=13.27\,p_e$ and $\sigma_\theta-\sigma_r=26.54\,p_e$, $$\sigma_{vm}=\sqrt{\tfrac12\!\left[(\sigma_\theta-\sigma_z)^2+(\sigma_z-\sigma_r)^2+(\sigma_r-\sigma_\theta)^2\right]}=22.99\,p_e=350\;\Rightarrow\;p_e=15.22\ \text{MPa}$$ $$p_i=6p_e=\boxed{91.4\ \text{MPa}}$$ The ratio $p_{i,VM}/p_{i,Tresca}=2/\sqrt3=1.155$, confirming von Mises is the less conservative criterion.