Question 12.6: A cylindrical vessel has an outside diameter of 400 mm and a...
A cylindrical vessel has an outside diameter of 400 mm and an inside diameter of 300 mm. For an internal pressure of 20.1 MPa, compute the hoop stress σ1 at the inner and outer surfaces and at points within the wall at intervals of 10 mm. Plot a graph of σ_{1} radial position in the wall.
Objective Compute the hoop stress at specified positions in the wall of the cylinder.
Given Pressure = p = 20.1 MPa; Do = 400 mm; Di = 300 mm
Use 10 mm increments for radius within the wall from the outside surface to the inside surface.
Analysis Use Steps 1 through 4 from Procedure A from this section.
Results Step 1. D_{m} = (D_{o} + D_{i}) /2 = (400 + 300)/2 = 350 mm
Step 2. t = (D_{o} + D_{i}) /2 = (400 – 300)/2 = 50 mm
D_{m} /2 = 350/50 = 7.00 <20; thick-walled cylinder
Step 3. This step does not apply.
Step 4. Use the equation for tangential stress from Table 12–1.
| TABLE 12–1 Stresses in thick-walled cylinders and spheres .^{a} | ||
| Stress at position r | Maximum stress | |
| Thick-walled cylinder | ||
| Hoop (tangential) | \sigma_{1} = \frac{pa^{2}(b^{2}+r^{2})}{r^{2}(b^{2}-a^{2})} | \sigma_{1} = \frac{p(b^{2}+a^{2})}{b^{2}-a^{2}} (at inner surface) |
| Longitudinal | \sigma_{2} = \frac{pa^{2}}{b^{2}-a^{2}} | \sigma_{2} = \frac{pa^{2}}{b^{2}-a^{2}} (uniform throughout wall) |
| Radial | \sigma_{3} = \frac{-pa^{2}(b^{2}-r^{2})}{r^{2}(b^{2}-a^{2})} | \sigma_{3} = -p (at inner surface) |
| Thick-walled sphere | ||
| Tangential | \sigma_{1} = \sigma_{2} = \frac{pa^{3}(b^{3} + 2r^{3})}{2r^{3}(b^{3}-a^{3})} | \sigma_{1} = \sigma_{2} = \frac{p(b^{3} + 2a^{3})}{2(b^{3}-a^{3})} (at inner surface) |
| Radial | \sigma_{3} = \frac{-pa^{3}(b^{3}-r^{3})}{r^{3}(b^{3}-a^{2})} | \sigma_{3} = -p (at inner surface) |
\sigma_{3} = \frac{pa^{2}(b^{2}-r^{2})}{r^{2}(b^{2}-a^{2})}
a = D_{i}/2 = 300/2 = 150 mm
b = D_{o}/2 = 300/2 = 150 mm
The results are shown in tabular form in the following.
| r(mm) | \sigma_{2} (MPa) | |
| 200 | 51.7 | ( Maximum at outer surface ) |
| 190 | 54.5 | |
| 180 | 57.7 | |
| 170 | 61.6 | |
| 160 | 66.2 | |
| 150 | 71.8 | ( Maximum at inner surface ) |
Comment Figure 12–8 shows the graph of tangential stress versus position in the wall. The graph illustrates clearly that the assumption of uniform stress in the wall of a thick-walled cylin-der would not be valid.
