Thick-walled hydraulic cylinder — pressure at yield: 22-Mec-A7 Advanced Strength of Materials, December 2019, Q3

Question 3: Thick-walled hydraulic cylinder — pressure at yield

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 under internal pressure only, with negligible axial stress (open ends).

Given data
Quantity	Symbol	Value
Inner radius	$r_i$	50 mm
Outer radius	$r_o$	90 mm
Yield strength	$\sigma_Y$	200 MPa
Factor of safety	$N$	3
Axial stress	$\sigma_z$	$\approx 0$

Find. Maximum internal pressure $p_i$ by (a) the distortion-energy (von Mises) and (b) the maximum-shear (Tresca) theories, each with $N=3$.

Approach. The most-stressed point is the inner wall; write the three Lamé principal stresses there in terms of $p_i$, set the effective stress of each theory equal to $\sigma_Y/N$, and solve for $p_i$. The elastic constants are not needed for either strength check.

Lamé stresses at the inner wall ($p_o=0$). With the geometry factor $\dfrac{r_o^2+r_i^2}{r_o^2-r_i^2}=\dfrac{8100+2500}{8100-2500}=1.893$, $$\sigma_\theta=p_i\frac{r_o^2+r_i^2}{r_o^2-r_i^2}=1.893\,p_i,\qquad \sigma_r=-p_i,\qquad \sigma_z=0$$ so the ordered principals are $\sigma_1=1.893p_i\gt\sigma_2=0\gt\sigma_3=-p_i$.
Allowable effective stress. With the factor of safety, $$\sigma_{\text{all}}=\frac{\sigma_Y}{N}=\frac{200}{3}=66.7\ \text{MPa}$$
(a) Distortion-energy (von Mises). $$\sigma_{vM}=\sqrt{\tfrac{1}{2}\!\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]}=2.545\,p_i$$ Setting $\sigma_{vM}=66.7$ MPa, $$\boxed{p_i=\frac{66.7}{2.545}=26.2\ \text{MPa}}$$
(b) Maximum-shear (Tresca). $$\sigma_1-\sigma_3=\left(1.893+1\right)p_i=2.893\,p_i=\frac{\sigma_Y}{N}$$ $$p_i=\frac{66.7}{2.893}=23.0\ \text{MPa}$$

Results — Question 3
Failure theory	Effective stress	Max internal pressure $p_i$
(a) Distortion energy (von Mises)	$2.545\,p_i$	26.2 MPa
(b) Maximum shear (Tresca)	$2.893\,p_i$	23.0 MPa

Topic. Thick-walled pressure vessels; yield criteria.

Key relations. Lamé at inner wall: $\sigma_\theta=p_i\frac{r_o^2+r_i^2}{r_o^2-r_i^2}$, $\sigma_r=-p_i$; Tresca $\sigma_1-\sigma_3=\sigma_Y/N$; von Mises $\sqrt{\tfrac12\sum(\sigma_i-\sigma_j)^2}=\sigma_Y/N$.

Why this works. Because $\sigma_z=0$ is the intermediate principal stress, Tresca (which ignores it) is the more conservative criterion and predicts the lower pressure, while von Mises accounts for $\sigma_2$ and allows slightly more.

Common pitfall. Taking $\sigma_z$ as the axial-vessel value $\tfrac12(\sigma_r+\sigma_\theta)$; here the ends are open, so $\sigma_z=0$ and the von Mises/Tresca ratio is not the closed-end $2/\sqrt3$.

Source. Boresi & Schmidt, Advanced Mechanics of Materials, Ch. 11 (thick-wall cylinders) and Ch. 4 (yield criteria).

Question 3 of 7: Thick-walled hydraulic cylinder — pressure at yield

Question 3: Thick-walled hydraulic cylinder — pressure at yield