Structural Analysis (Aero 103)
Thin-walled Pressure Vessels

a Tank or pipe carrying a fluid or gas under a pressure is subjected to tensile forces, which resist bursting, developed across longitudinal and transverse sections.

TANGENTIAL STRESS, σt (Hoop Stress)
Consider the tank shown being subjected to an internal pressure . The length of the tank is and the wall thickness is . Isolating the right half of the tank:
The forces acting are the total pressures caused by the internal pressure  and the total tension in the walls .
F=PA=P*D*L
T=σttL

σh=PD/2t
If there exist an external pressure  and an internal pressure , the formula may be expressed as:
σh=(Pi-Po)D/2t

## LONGITUDINAL STRESS

Consider the free body diagram in the transverse section of the tank:

The total force acting at the rear of the tank  must equal to the total longitudinal stress on the wall . Since  is so small compared to , the area of the wall is close to PiDt

And Therefore                           σt=PD/4t
If there exist an external pressure  and an internal pressure , the formula may be  expressed as:
σt=(Po-Pi)D/4t

From This it can be easily observed that the tangential stress (  Hoop ) is twice the longitudinal stress .
σh = 2 σt

## Spherical Shell :

If a spherical tank of diameter  and thickness  contains gas under a pressure of , the stress at the wall can be expressed as:

σt=(D/Po-Pi)4t