Pressure Vessels
Thin-walled pressure vessels
- Inner radius: r
- Wall thickness: t
- Thin walls: $t/r << 1$
- Assume that stress distribution through the thin wall is uniform
- Pressure vessel contains fluid under pressure p (understood to be gauge pressure, i.e., $p = \Delta p = p_i - p_o$)
Cylindrical Vessels
Cylindrical pressure vessels are an axisymmetric problem, there is no shear stress on an element parallel to the axis of the cylinder. Therefore, the only normal stresses are in the directions of axis and circumference.
Hoop Stress (circumferential stress): \[\sigma_1 = \sigma_h = \frac{pr}{t}\]
\[\sum F_z: \sigma_h (2t \Delta x) - p(2r \Delta x)=0\]
\[\therefore \sigma_h = \frac{pr}{t}\]
Axial Stress (longitudinal stress): \[\sigma_2 = \sigma_a = \frac{pr}{2t}\]
The force of the fluid in the longitudinal direction will be the internal pressure, $p$, times the area of the fluid within the vessel, $\pi r^2$. Similarly, the force within the vessel wall in the longitudinal direction will be the stress, $\sigma_a$, times the area of the wall. We assume we can unravel the thin wall of the pressure vessel, simplifying the area of the thin wall as the circumference, $2\pi r$, multiplied by the thickness, $t$. \[\sum F: \sigma_a (2\pi r t) - p(\pi r^2)=0\]
\[\therefore \sigma_a = \frac{pr}{2t}\]
Note: A more accurate derivation of the longitudinal stress may be calculated with the true area of the pressure vessel $(A_{wall} = \pi(r+t)^2 - \pi r^2 = \pi 2rt + \pi t^2)$, leading to an axial stress of: $\sigma_a = \frac{pr^2}{2rt + t^2}$. The approximation used in this course is only sufficient when $\frac{r}{t} \ge 10$.
Radial Stress:
Because $t/r << 1$: \[\sigma_h,\sigma_a >> \sigma_r\] so in general we can neglect $\sigma_r$
Spherical Vessels
Due to symmetry: \[\sigma_1 = \sigma_2 = \frac{pr}{2t}\]