Question 3 of 8: Thick-Walled Cylinder — Lamé Stresses and Failure Criteria
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams, May 2016 — 98-Mar-A5 Advanced Strength of Materials, 3 hours, open book, non-communicating calculator permitted (any five of the eight problems constitute a complete paper, all equal value; all eight answered below for full study coverage).
Reference texts: Boresi & Schmidt, Advanced Mechanics of Materials, 6th ed.; Hibbeler, Mechanics of Materials, 10th ed.; Timoshenko & Goodier, Theory of Elasticity, 3rd ed.
It is solved as the strength-of-materials exam it actually is.
Given. Thick cylinder, closed ends, internal radius $a=0.06\text{ m}$; the source pairs this with an "external diameter of 0.11 m," which is geometrically impossible (it would make the outer radius 0.055 m, smaller than $a$) — see the callout below for how this is resolved. $\sigma_{\text{elastic limit}}=320\text{ MPa}$, $\nu=0.28$ (not needed for the stress/pressure calculation, only for strain problems), internal/external pressure ratio $P_i=6.5P_o$.
Given data
Quantity
Value
Internal radius $a$
0.06 m
External radius $b$
0.11 m (see callout)
Elastic limit $\sigma_y$
320 MPa
Pressure ratio $P_i/P_o$
6.5
Find. Allowable internal pressure $P_i$ by (a) Tresca and (b) von Mises.
Thick-cylinder cross-section: internal pressure $P_i$ acts outward at $r=a$, external pressure $P_o=P_i/6.5$ acts inward at $r=b$.
Approach. Use Lamé's equations for a thick cylinder under combined internal/external pressure to get $\sigma_r,\sigma_\theta$ at the critical inner surface, add the closed-end axial stress $\sigma_z$, then apply Tresca and von Mises with the three principal stresses.
Lamé stresses at $r=a$ (the critical surface). With $P_o=P_i/6.5$:
$$\sigma_r(a)=-P_i\qquad \sigma_\theta(a)=\frac{P_i(a^2+b^2)-2b^2P_o}{b^2-a^2}\qquad \sigma_z=\frac{P_ia^2-P_ob^2}{b^2-a^2}\ \ \text{(closed ends)}$$
Writing each as a coefficient times $P_i$ (since $P_o=P_i/6.5$ is linear in $P_i$): $\sigma_r=-1.0000\,P_i$, $\sigma_\theta=1.4090\,P_i$, $\sigma_z=0.2045\,P_i$.
Part (a) — Tresca (maximum shear stress). The extreme principal stresses are $\sigma_\theta$ (max) and $\sigma_r$ (min):
$$\sigma_\theta-\sigma_r=\sigma_y\ \Rightarrow\ (1.4090+1.0000)P_i=320\text{ MPa}$$
$$\boxed{P_{i,\text{allow}}=\frac{320}{2.4090}=132.83\text{ MPa}\ \ (P_{o}=20.44\text{ MPa})}$$
Part (b) — von Mises. With the three principal-stress coefficients from Step 1:
$$\sigma_{VM}=\sqrt{\tfrac12\!\left[(\sigma_r-\sigma_\theta)^2+(\sigma_\theta-\sigma_z)^2+(\sigma_z-\sigma_r)^2\right]}=2.0863\,P_i=\sigma_y$$
$$\boxed{P_{i,\text{allow}}=\frac{320}{2.0863}=153.38\text{ MPa}\ \ (P_o=23.60\text{ MPa})}$$
Question 3 — final results
Criterion
Allowable $P_i$
Corresponding $P_o$
Maximum shear stress (Tresca)
132.83 MPa
20.44 MPa
Von Mises
153.38 MPa
23.60 MPa
Check: as printed, "0.06 m internal radius and 0.11 m external diameter" is self-contradictory (external radius would be 0.055 m < internal radius 0.06 m). This solution reads "external diameter" as a typo for "external radius" (0.11 m), giving $b/a=1.833$, a normal thick-cylinder proportion. This choice does not actually affect the numeric answer: Lamé's stress coefficients depend only on the ratio $b/a$, and the alternative consistent reading (both figures as diameters, $a=0.03\text{ m}$, $b=0.055\text{ m}$) gives the identical ratio $b/a=1.833$ and hence identical allowable pressures.
