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Consider a spherical pressure vessel with radius r and wall thickness t subjected to an internal gage pressure p.

For reasons of symmetry, all four normal stresses on a small stress element in the wall must be identical. Furthermore, there can be no shear stress.

The normal stresses s can be related to the pressure p by inspecting a free body diagram of the pressure vessel. To simplify the analysis, we cut the vessel in half as illustrated.

Since the vessel is under static equilibrium, it must satisfy Newton's first law of motion. In other words, the stress around the wall must have a net resultant to balance the internal pressure across the cross-section.

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 s_h and the longitudinal stress s_l are the principal stresses.

To determine the longitudinal stress s_l, 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 s_h, 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,