## Pressure Cylinder

Consider an air cylinder is constructed from a 12 mm rolled steel plate with an internal diameter of 700 mm. If the tangential stress inside the tank is 50 MPa, what is the pressure gauge reading inside the cylinder in kPa?

Hint
First determine if the cylinder is thick or thin walled. A cylinder can be considered thin-walled if:
$$t\leq \frac{1}{10}r_i$$$where $$t$$ is the cylinder wall’s thickness, and $$r_i$$ is the internal radius. Hint 2 Tangential (hoop) stress of a thin-walled cylinder: $$\sigma_t=\frac{P_ir}{t}$$$
where $$P_i$$ is the internal pressure, $$r$$ is the average radius, and $$t$$ is the wall thickness.
First, let’s determine if the cylinder is thick or thin walled. A cylinder can be considered thin-walled if:
$$t\leq \frac{1}{10}r_i$$$where $$t$$ is the cylinder wall’s thickness, and $$r_i$$ is the internal radius. $$0.012m\leq \frac{1}{10}(0.7m)=0.07$$$
In this situation, the pressure vessel is thin-walled. The tangential (hoop) stress of a thin-walled cylinder:
$$\sigma_t=\frac{P_ir}{t}$$$where $$P_i$$ is the internal pressure, $$r$$ is the average radius, and $$t$$ is the wall thickness. To find $$r$$ : $$r=\frac{r_i+r_o}{2}$$$
where $$r_i$$ is the inner radius, and $$r_o$$ is the outer radius. Thus,
$$r=\frac{(0.7m/2)+((0.7m/2)+0.012m)}{2}=\frac{0.712m}{2}=0.356\:m$$$Therefore, $$P_i=\frac{\sigma_t(t)}{r}=\frac{50\cdot 10^6Pa(0.012m)}{0.356m}=1,685\:kPa$$$
1,685 kPa