$(20$ points$)$ A pressurized cylinder has an inner diameter of $4$ inches and an outer diameter of $8$ inches. The maximum allowable

strength of the material is $100,000$ psi. Assuming an external pressure of zero and using a factor of safety of $2$ :

a$)$ what is the maximum internal pressure that the cylinder can withstand?

Special case of zero outside pressure, $p_0=0.$

${}_t=\frac{r_i^2{p}_i}{r_o^2-r_i^2}(1+\frac{r_o^2}{r^2})$

${}_r=\frac{r_i^2{p}_i}{r_o^2-r_i^2}(1-\frac{r_o^2}{r^2})$

In this situation, the max stress is at $r_i.$

$({}_i{)}_{\text{m}\text{a}\text{x}}=p_i(\frac{r_0^2+r_i^2}{r_0^2-r_j^2})$

$({}_r{)}_{\text{m}\text{a}\text{x}}=-p_i$