Allowable Pressure in a Thick-Walled Cylinder: 22-Mec-A7 Advanced Strength of Materials, May 2013, 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. A thick cylinder with $p_i=3.5\,p_e$, yielding when the elastic limit is reached.

Given data
Inner radius / outer radius	a = 50 mm, b = 75 mm (a² = 2500, b² = 5625 mm²)
Elastic limit	σ_Y = 340 MPa
Poisson’s ratio	v = 0.32
Pressure ratio	p_i = 3.5 p_e

Find. The allowable internal pressure p_i under (a) Tresca (max shear) and (b) von Mises yield criteria.

Approach. Use the Lamé equations to express the bore stresses in terms of p_i (the critical location), then set each yield criterion equal to the elastic limit and solve for p_i. Closed-end axial stress $\sigma_z=(\sigma_r+\sigma_\theta)/2$ is used.

Lamé stresses at the bore (r = a). With $p_e=p_i/3.5$, the constants $A=0.2857p_i$ and $B/a^2=1.2857p_i$ give $$\sigma_r=-p_i,\qquad \sigma_\theta=1.5714\,p_i,\qquad \sigma_z=\tfrac12(\sigma_r+\sigma_\theta)=0.2857\,p_i.$$ These are the three principal stresses ($\sigma_\theta$ largest, $\sigma_r$ smallest).
(a) Maximum shear stress (Tresca). Yield when $\sigma_1-\sigma_3=\sigma_Y$: $$\sigma_\theta-\sigma_r=(1.5714+1)\,p_i=2.5714\,p_i=340.$$ $$\boxed{p_i=132.2\ \text{MPa}\ \ (\text{Tresca})}$$
(b) Von Mises. Yield when $$\tfrac12\big[(\sigma_\theta-\sigma_r)^2+(\sigma_r-\sigma_z)^2+(\sigma_z-\sigma_\theta)^2\big]=\sigma_Y^{2}.$$ Substituting the bore stresses gives an equivalent stress $2.227\,p_i=340$: $$\boxed{p_i=152.7\ \text{MPa}\ \ (\text{von Mises})}$$
Comparison. Tresca is the more conservative criterion, permitting about 13% less pressure than von Mises — the expected ordering, since Tresca ignores the intermediate principal stress.

Question 4 — results
Criterion	Allowable p_i
Maximum shear stress (Tresca)	132.2 MPa
Von Mises (distortion energy)	152.7 MPa

Check: closed-end axial stress is assumed. Open ends (σ_z = 0) change the von Mises result only slightly, to 151.4 MPa; the Tresca value is unaffected because σ_z is the intermediate principal stress. The corresponding external pressure is p_e = p_i/3.5.

Topic: Yielding of thick-walled pressure vessels; Tresca vs. von Mises.

Key relations: Lamé $\sigma_{r,\theta}=A\mp B/r^2$, $A=(p_ia^2-p_eb^2)/(b^2-a^2)$, $B=(p_i-p_e)a^2b^2/(b^2-a^2)$; Tresca $\sigma_1-\sigma_3=\sigma_Y$; von Mises equivalent stress.

Why this works: In a thick cylinder the hoop stress is largest at the bore and decays outward, so first yield always initiates at the inner surface; evaluating the criteria there governs the whole wall. The radial stress equals $-p_i$ at the bore, and for closed ends the axial stress is exactly the mean of radial and hoop, so it is the intermediate principal stress. Tresca depends only on the extreme principal stresses and so is blind to σ_z, whereas von Mises blends all three, which is why it permits a higher pressure.

Common pitfall: evaluating the stresses at the outer radius, or forgetting that the external pressure reduces both A and B; also, treating the state as purely biaxial when the axial stress matters for von Mises.

Source: Ugural & Fenster, Advanced Strength and Applied Elasticity, Ch. 8 (thick cylinders) and Ch. 4 (yield criteria); 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)