Question 8 of 8: Thick cylinder: Tresca vs Von Mises

Question 8: Thick cylinder: Tresca vs Von Mises (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. A thick-walled cylinder under internal pressure eight times the external pressure.

Find. the allowable internal pressure by (a) the maximum-shear (Tresca) and (b) the Von Mises yield criteria.

Approach. Evaluate the Lamé stresses at the inner wall (the critical radius), take the closed-end axial stress as the mean of radial and hoop, then apply each yield criterion at first yield.

1. Lamé stresses at the bore. With $r_i=55$, $r_o=87.5$ mm and $p_e=p_i/8$, $$\sigma_\theta=1.893\,p_i,\qquad \sigma_r=-p_i,\qquad \sigma_z=\tfrac12(\sigma_\theta+\sigma_r)=0.447\,p_i\ \text{(closed end)}.$$
2. Maximum-shear (Tresca). The extreme principals are $\sigma_\theta$ and $\sigma_r$, so $\sigma_\theta-\sigma_r=\sigma_Y$ gives $$2.893\,p_i=289\ \Rightarrow\ \boxed{p_i=99.9\ \text{MPa}}.$$
3. Von Mises. For $\sigma_z$ equal to the mean, $\sigma_{vM}=\tfrac{\sqrt3}{2}(\sigma_\theta-\sigma_r)=\sigma_Y$, hence $$p_i=\frac{289}{2.505}=\boxed{115.4\ \text{MPa}}.$$

Von Mises permits about 15% more pressure than Tresca — the fixed factor 2/√3 that always separates the two criteria when the axial stress is the intermediate principal. Tresca is the conservative choice for a code-compliant design.