Pressure Vessels

Thin-Walled Pressure Vessels

Inner radius: $ r $
Wall thickness: $ t $
Thin walls: $ \dfrac{t}{r} <<1 $
Assume that stress distribution through the thin wall is uniform
Pressure vessel contains fluid under pressure ($ p $), usually called gauge pressure (i.e., $ p = \Delta p = p_i - p_o $).

Cylindrical Vessels

Cylindrical pressure vessel behind Transportation Building.

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 (circumfrential) stress. #hp-str

$$ \sigma_1 = \sigma_h = \frac{pr}{t}\ $$

$$ \sum F_z: \sigma_h (2t \Delta x) - p(2r \Delta x)=0\ $$

$$ \therefore \sigma_h = \frac{pr}{t}\ $$

Axial (longitudinal) stress. #axl-str

$$ \sigma_2 = \sigma_a = \frac{pr}{2t}\ $$

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 $.

Spherical Vessels

Due to symmetry.

$$ \sigma_1 = \sigma_2 = \frac{pr}{2t}\ $$

Radial Stress

In addition to hoop and axial stresses, there is also a radial stress $ \sigma_r $ that acts normal to the pressure vessel walls.

Inner Surface: $ \sigma_r = -p $
Outer Surface: $ \sigma_r = 0 $

Recall the hoop stress $ \sigma_h =p\left(\dfrac{r}{t}\right) $ and axial stress $ \sigma_h =p\left(\dfrac{r}{2t}\right) $. Because $ \dfrac{t}{r} << 1 $, it is easy to see that $ \sigma_h,\sigma_a >> \sigma_r\ $. We therefore neglect $ \sigma_r $ in thin-walled pressure vessel calculations because it is small compared to the other stresses experienced in the vessel.