Allowable pressure in a thick-walled cylinder: 22-Mec-A7 Advanced Strength of Materials, May 2016, Q3

Question 3: Allowable pressure in a thick-walled cylinder (equal value)

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. Inner radius r_i = 0.06 m, outer radius r_o = 0.11 m (radius ratio k = r_o/r_i = 1.833); yield σ_Y = 320 MPa; ν = 0.28; p_i = 6.5 p_e; closed ends assumed (axial stress carried by the wall).

Find. The allowable internal pressure p_i by (a) Tresca and (b) von Mises, using first yield at the bore.

Check: the source prints “0.11 m external diameter,” which is smaller than the 0.12 m internal diameter and therefore impossible; it is read as the external radius. Because Lamé stresses depend only on the ratio r_o/r_i = 1.833, the alternative reading (both values as diameters, r_i = 0.03 m, r_o = 0.055 m) yields the identical allowable pressures.

Thick cylinder cross-section; internal pressure 6.5× the external pressure.

Approach. Write the Lamé stresses at the bore (where they are largest), express them per unit p_i with p_e = p_i/6.5, then apply each yield criterion to the three principal stresses σ_θ > σ_z > σ_r.

Lamé stresses at the inner wall. With p_e = p_i/6.5 and r_o²−r_i² in the denominator: $$\sigma_\theta=\frac{p_i(r_o^2+r_i^2)-2p_e r_o^2}{r_o^2-r_i^2}=1.409\,p_i,\qquad \sigma_r=-p_i$$ $$\sigma_z=\frac{p_i r_i^2-p_e r_o^2}{r_o^2-r_i^2}=0.2045\,p_i=\tfrac12(\sigma_r+\sigma_\theta)\ \checkmark$$ The axial stress equals the average of radial and hoop, confirming the closed-end result and identifying σ_z as the intermediate principal stress.
Maximum-shear-stress (Tresca) criterion. Yield when σ_max − σ_min = σ_Y: $$\sigma_\theta-\sigma_r=(1.409+1)\,p_i=2.409\,p_i=\sigma_Y \;\Rightarrow\; p_i=\frac{320}{2.409}$$ $$\boxed{p_i^{\,\text{Tresca}}=132.8\ \text{MPa}}$$
Von-Mises (distortion-energy) criterion. Using all three principal stresses: $$\sqrt{\tfrac12\!\left[(\sigma_\theta-\sigma_r)^2+(\sigma_r-\sigma_z)^2+(\sigma_z-\sigma_\theta)^2\right]}=2.086\,p_i=\sigma_Y$$ $$\boxed{p_i^{\,\text{VM}}=\frac{320}{2.086}=153.4\ \text{MPa}}$$
Compare. Von Mises permits about 15.5% more pressure than Tresca, because Tresca ignores the intermediate stress σ_z and so is the more conservative (safer) design bound.

Question 3 — allowable internal pressure
Criterion	Allowable p_i
Maximum shear stress (Tresca)	132.8 MPa
Von Mises	153.4 MPa

Topic. First-yield of a thick pressurised cylinder (Lamé theory) under Tresca vs. von Mises.

Key relations. Lamé σ_θ,r = A ± B/r²; closed-end σ_z = (σ_r+σ_θ)/2; Tresca σ₁−σ₃ = σ_Y; von Mises as above.

Why this works. In a pressurised cylinder the largest stresses occur at the bore, where the hoop stress is maximum tensile and the radial stress equals −p_i. Yield therefore initiates there. Tresca depends only on the extreme principal stresses (hoop and radial) and ignores the axial value, giving the simpler and more conservative estimate. Von Mises weighs all three differences and, because the axial stress lies between the other two, predicts a higher onset pressure. Expressing every stress as a multiple of p_i reduces each criterion to a single scalar equation for the allowable pressure.

Common pitfall. Using open-end conditions (σ_z = 0) when the ends are closed, or forgetting the external pressure term (here p_e = p_i/6.5) in the hoop stress. Also, Poisson’s ratio does not enter the yield check for a closed thick cylinder — it only affects displacements.

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

Question 3 of 8: Allowable pressure in a thick-walled cylinder

Question 3: Allowable pressure in a thick-walled cylinder (equal value)