Consider a cylindrical pressure vessel with radius r and wall thickness t subjected to an internal gage pressure p.
The coordinates used to describe the cylindrical vessel can take advantage of its axial symmetry. It is natural to align one coordinate along the axis of the vessel (i.e. in the longitudinal direction). To analyze the stress state in the vessel wall, a second coordinate is then aligned along the hoop direction.
With this choice of axisymmetric coordinates, there is no shear stress. The hoop stress sh and the longitudinal stress sl are the principal stresses.
To determine the longitudinal stress sl, we make a cut across the cylinder similar to analyzing the spherical pressure vessel. The free body, illustrated on the left, is in static equilibrium. This implies that the stress around the wall must have a resultant to balance the internal pressure across the cross-section.
Applying Newton's first law of motion, we have,
To determine the hoop stress sh, we make a cut along the longitudinal axis and construct a small slice as illustrated on the right.
The free body is in static equilibrium. According to Newton's first law of motion, the hoop stress yields,