Question

A thin spherical pressure vessel of 200 mm diameter and 1 mm thickness is subjected to an internal pressure varying from 4 to 8 MPa. Assume that the yield, ultimate, and endurance strength of material are 600, 800 and 400 MPa respectively. The factor of safety as per Goodman's relation is

• A. 1.2
• B. 1.6
• C. 1.4
• D. 2.0

Answer 1

The correct option is B 1.6

Stress induced
${\sigma }_1={\sigma }_2=\frac{pr}{2t}$

${\sigma }_{1max}=\frac{8\times 100}{2\times 1}=400MPa$

${\sigma }_{1min}=\frac{4\times 100}{2\times 1}=200MPa$

${\sigma }_{2max}=400MPa$
${\sigma }_{2min}=200MPa$
${\sigma }_{1m}=300MPa$
${\sigma }_{1a}=100MPa$
${\sigma }_{2m}=300MPa$
${\sigma }_{2a}=100MPa$

Equivalent Stresses
${\sigma }_{me}=\sqrt{{\sigma }_{1m}^2+{\sigma }_{2m}^2-{\sigma }_{1m}{\sigma }_{2m}}$

$=\sqrt{{300}^2+{300}^2-300\times 300}$

$=300MPa$
Similarly,
${\sigma }_{ae}=100MPa$
Goodman equation,
$\frac{{\sigma }_{ae}}{S_e}+\frac{{\sigma }_{me}}{S_{ut}}=\frac{1}{n}$

$\Rightarrow \frac{100}{400}+\frac{300}{800}=\frac{1}{n}$

n = 1.6