Problem 2.6. Design pressure vessel

Required thickness according to Tresca criterion (rounded to the nearest mm), t [mm]=

Required thickness according to von Mises criterion (rounded to the nearest mm), t [mm]=

Step 1. Write the principal stresses of the pressure vessel as a function of the unknown thickness, t.

Check principal stressesHide principal stresses

Stresses can be determined from the pressure vessel formula (Eqs.11-8 and 11-10).

$$\begin{gathered} {\sigma }_1=\frac{pR}{t}=\frac{1.6\times 790}{t}=\frac{1264}{t} \\ {\sigma }_2=\frac{pR}{2t}=\frac{1.6\times 790}{2t}=\frac{632}{t} \end{gathered}$$

N and m are used (1.6 MPa = 1.6 N/mm²).

Step 2. Determine the required thickness from Tresca failure criterion.

Check Tresca's limitHide Tresca's limit

Tresca failure criterion gives upper limits to the principal stresses and to the maximum shear stress, from which lower limits of the thickness can be determined:

Eq.(2-23).

$$\begin{gathered} \left|{\sigma }_1\right|\le f \to \frac{1264}{t}\le 200 \to t\ge \frac{1264}{200}=6. 32 \mathrm{mm} \\ \left|{\sigma }_2\right|\le f \to \frac{632}{t}\le 200 \to t\ge \frac{632}{200}=3.16 \mathrm{mm} \\ \left|{\tau }^{\max }\right|\le \frac{f}{2} \to \left|{\sigma }_1–{\sigma }_2\right|\le \frac{f}{2} \to \left|\frac{1264}{t}–\frac{632}{t}\right|\le \frac{f}{2} \to t\ge \frac{632\times 2}{200}=6.32 \mathrm{mm} \end{gathered}$$

The applied thickness of the wall must satisfied all the above conditions

${\mathit{t}}_{\mathbf{a}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{e}\mathbf{d}}\mathbf{=}\mathbf{7}\mathbf{mm}\ge \max \left(6.32, 3.16, 6.32\right)$

Step 3. Determine the required thickness from von Mises failure criterion.

Check von Mises criterionHide von Mises criterion

Von Mises failure criterion results in the following limit

Eq.(2-24).

$$\begin{gathered} {\sigma }_1^2+{\sigma }_2^2–{\sigma }_1{\sigma }_2\le f^2 \to \frac{{1264}^2}{t^2}+\frac{{632}^2}{t^2}–\frac{1264\times 632}{t^2}\le f^2 \\ \to t^2\ge \frac{({1264}^2+{632}^2–1264\times 632)}{{200}^2}=29.96 \to t\ge 5.473\mathrm{mm} \\ \to {\mathit{t}}_{\mathbf{applied}}\mathit{=}\mathbf{6}\mathbf{ }\mathbf{mm} \end{gathered}$$

Step 4. Compare results from the two criteria.

Check comparisonHide comparison

First the principal stresses of the pressure vessel are written as a function of the unknown thickness, t. N and m are used below(1.6 MPa = 1.6 N/mm²).

See pressure vessel formula (Eqs.11-8 and 11-10).

$$\begin{gathered} {\sigma }_1=\frac{pR}{t}=\frac{1.6\times 790}{t}=\frac{1264}{t} \\ {\sigma }_2=\frac{pR}{2t}=\frac{1.6\times 790}{2t}=\frac{632}{t} \end{gathered}$$

Tresca failure criterion gives upper limits to the principal stresses and to the maximum shear stress, from which lower limits of the thickness can be determined:

Eq.(2-23).

$$\begin{gathered} \left|{\sigma }_1\right|\le f \to \frac{1264}{t}\le 200 \to t\ge \frac{1264}{200}=6. 32 \mathrm{mm} \\ \left|{\sigma }_2\right|\le f \to \frac{632}{t}\le 200 \to t\ge \frac{632}{200}=3.16 \mathrm{mm} \\ \left|{\tau }^{\max }\right|\le \frac{f}{2} \to \left|{\sigma }_1–{\sigma }_2\right|\le \frac{f}{2} \to \left|\frac{1264}{t}–\frac{632}{t}\right|\le \frac{f}{2} \to t\ge \frac{632\times 2}{200}=6.32 \mathrm{mm} \end{gathered}$$

The applied thickness of the wall must satisfied all the above conditions

${\mathit{t}}_{\mathbf{a}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{e}\mathbf{d}}\mathbf{=}\mathbf{7}\mathbf{mm}\ge \max \left(6.32, 3.16, 6.32\right)$

Von Mises failure criterion results in the following limit

Eq.(2-24).

$$\begin{gathered} {\sigma }_1^2+{\sigma }_2^2–{\sigma }_1{\sigma }_2\le f^{ 2} \to \frac{{1264}^2}{t^2}+\frac{{632}^2}{t^2}–\frac{1264\times 632}{t^2}\le f^{ 2} \\ \to t^2\ge \frac{({1264}^2+{632}^2–1264\times 632)}{{200}^2}=29.96 \to t\ge 5.473\mathrm{mm} \\ \to {\mathit{t}}_{\mathbf{applied}}\mathit{=}\mathbf{6}\mathbf{ }\mathbf{mm} \end{gathered}$$

Comparing the results from the two criteria shows that Tresca criterion results in larger thickness.