Question 5.7: Calculate the tensile stresses (circumferential and longitud......
Calculate the tensile stresses (circumferential and longitudinal) developed in the walls of a cylindrical pressure vessel with inside diameter 18 in and wall thickness 1/4 in. The vessel is subjected to an internal gage pressure of 300 psi and a simultaneous external axial tensile load of 50,000 lb.
Given: Dimensions of and loading on cylindrical pressure vessel.
Find: Hoop and longitudinal normal stresses.
Assume: We will test whether thin-walled theory may be applied to this vessel.
Does thin-walled theory apply? Is the thickness t ≤ 0.1r_i ?
0.25 \text { in } \leq 0.1 \cdot(9 \text { in })=0.90 \text { in } \checkmarkWe can use thin-walled theory.
The circumferential, or hoop stress, is calculated:
\sigma_{\theta \theta}=\frac{p r_{\mathrm{i}}}{t}=\frac{(300 \mathrm{psi})(9 \mathrm{in})}{0.25 \mathrm{in}}=10.8 \mathrm{ksi} .
The longitudinal stress due to the internal pressure may be combined with the normal stress due to the axial load by straightforward superposition, as these stresses are due to forces in the same direction and act normal to areas with the same orientation:
\begin{aligned} \sigma_{x x} & =\frac{p r_i}{2 t}+\frac{P}{A}=\frac{(300 \mathrm{psi})(9 \mathrm{in})}{0.5 \mathrm{in}}+\frac{50,000 \mathrm{lb}}{\left(2 \pi r_i\right) t}, \\ \sigma_{x x} & =5.4 \mathrm{ksi}+3.5 \mathrm{ksi}=8.9 \mathrm{ksi}. \end{aligned}Note: the area on which P acts can also be calculated as \pi r_o^2-\pi r_i^2 ; this gives essentially the same results for thin circular sections.