Allowable Pressure in a Thick-Walled Cylinder: 22-Mec-A7 Advanced Strength of Materials, May 2014, Q4

Question 4: Allowable Pressure in a Thick-Walled Cylinder (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. $r_i=50$ mm, $r_o=80$ mm; elastic limit $\sigma_Y=280$ MPa; $v=0.28$; $p_i=5\,p_e$.

Find. The allowable internal pressure $p_i$ by (a) von Mises and (b) maximum-shear (Tresca).

Approach. The worst point is the inner wall. Use Lamé’s equations to express the three principal stresses ($\sigma_r,\sigma_\theta,\sigma_z$) there in terms of $p_e$, then equate the von Mises and Tresca equivalent stresses to $\sigma_Y$ and back out $p_i=5p_e$.

Lamé stresses at the inner wall. With $p_i=5p_e$, $p_e=p_e$, evaluating $\sigma=A\pm B/r^2$ at $r=r_i$ gives (per unit $p_e$) $$\sigma_r=-5\,p_e,\qquad \sigma_\theta=\frac{p_i(r_i^2+r_o^2)-2p_e r_o^2}{r_o^2-r_i^2}=8.128\,p_e,\qquad \sigma_z=\tfrac12(\sigma_r+\sigma_\theta)=1.564\,p_e,$$ where $\sigma_z$ is the closed-end axial stress (the intermediate principal stress).
von Mises criterion. $\sigma_{\text{vM}}=\sqrt{\tfrac12\big[(\sigma_\theta-\sigma_z)^2+(\sigma_z-\sigma_r)^2+(\sigma_r-\sigma_\theta)^2\big]}=11.37\,p_e$. Setting this to $\sigma_Y$: $$p_e=\frac{280}{11.37}=24.6\ \text{MPa}\;\Rightarrow\;\boxed{p_i=5p_e=123.1\ \text{MPa}}.$$
Maximum-shear (Tresca) criterion. The extreme principals are $\sigma_\theta$ and $\sigma_r$, so $\sigma_\theta-\sigma_r=13.128\,p_e=\sigma_Y$: $$p_e=\frac{280}{13.128}=21.3\ \text{MPa}\;\Rightarrow\;\boxed{p_i=5p_e=106.6\ \text{MPa}}.$$
Compare. Von Mises permits $15\%$ more pressure than Tresca, because Tresca ignores the intermediate stress $\sigma_z$ and is therefore the more conservative (safer) criterion.

Check / assumption. The ends are taken as closed, so an axial stress $\sigma_z=\tfrac12(\sigma_r+\sigma_\theta)$ acts. For open ends ($\sigma_z=0$) the von Mises result would differ slightly; the closed-end reading is standard for a pressurised cylinder and is stated here explicitly.

Criterion	Allowable $p_e$	Allowable $p_i=5p_e$
von Mises	$24.6$ MPa	$123.1$ MPa
Maximum shear (Tresca)	$21.3$ MPa	$106.6$ MPa

Topic: Yield of a thick-walled pressure cylinder under combined internal and external pressure.

Key relations: Lamé $\sigma_{r,\theta}=A\mp B/r^2$; closed-end $\sigma_z=\tfrac12(\sigma_r+\sigma_\theta)$; von Mises $\sigma_{\text{vM}}=\sqrt{\tfrac12\sum(\sigma_i-\sigma_j)^2}$; Tresca $\sigma_{\max}-\sigma_{\min}=\sigma_Y$.

Why this works: Lamé’s solution gives the exact axisymmetric elastic stress field, whose hoop stress peaks at the bore—so first yield always begins there. The three principal stresses at that point feed directly into either yield criterion. Because $\sigma_z$ lies between $\sigma_\theta$ and $\sigma_r$, Tresca (which uses only the largest difference $\sigma_\theta-\sigma_r$) predicts yield sooner than von Mises, which also credits the intermediate stress.

Common pitfall: Evaluating the stresses at the outer wall (they are smaller there), dropping the external pressure term, or forgetting the axial stress and thereby mis-scoring the von Mises value. Keep $p_i=5p_e$ throughout—solve for $p_e$ first, then multiply.

Source: Ugural & Fenster, Advanced Strength and Applied Elasticity 4th ed., Ch. 8 (axisymmetric thick cylinders); Boresi & Schmidt Ch. 11.

Question 4 of 8: Allowable Pressure in a Thick-Walled Cylinder

Question 4: Allowable Pressure in a Thick-Walled Cylinder (20 marks)