Question 12.5: Compute the magnitude of the maximum tangential and radial s...

Objective   Compute the maximum stresses and specify a material.

Given         Pressure = p = 69 MPa; D_{o} = 200 mm; D_{i} = 160 mm

Analysis    Use Procedure C from this section. These data are the same as those used in Example Problem 12–4. Some values will be carried forward.

Results       Steps 1, 2, and 3. Sphere is thick walled.

Step 4. Use equations from Table 12–1. a = 80 mm; b = 100 mm

The tangential stress is equal in all directions:

\sigma_{1} =  \sigma_{2} = \frac{p(b^{3} + 2a^{3})}{2 (b^{3}-a^{3})} = \frac{(69  MPa)[100^{3}+2(80)^{3}]mm^{3}}{2(100^{3}-80^{3})mm^{3}} 

\sigma_{1} =  \sigma_{2} = 143.1 MPa tangential

The radial stress is compressive and equal to the applied internal pressure.

\sigma_{3} = =-p = -69 MPa

Each of these stresses is a maximum at the inner surface.

Steps 5 and 6. For a maximum stress of 143 MPa, the required yield strength for the mate-rial is

s_{y} =  N( \sigma_{2})  = 4(143 MPa) = 572 MPa

Step 7. From Appendix A–10, we can specify SAE 4140 OQT 1300 steel that has a yield strength of 696 MPa. Others could be used.

Comment The maximum stress in the sphere is less than half that in the cylinder of the same size, allowing a material with a much lower strength to be used. Alternatively, it would be pos-sible to design the sphere with the same material but with a smaller wall thickness.