For tube or cylinders where wall thickness is more than 1/20 of diameter.

Axial direction is one that goes along the tube longitudinally.

The stress in axial direction at a point in the tube or cylinder wall can be expressed as:

σ_a = (p_i r_i² - p_o r_o² )/(r_o² - r_i²)

where

σ_a = stress in axial direction (MPa, psi)
p_i = internal pressure in the tube or cylinder (MPa, psi)
p_o = external pressure in the tube or cylinder (MPa, psi)
r_i = internal radius of tube or cylinder (mm, in)
r_o = external radius of tube or cylinder (mm, in)

The stress in circumferential direction - hoop stress - at a point in the tube or cylinder wall can be expressed as:

σ_c = [(p_i r_i² - p_o r_o²) / (r_o² - r_i²)] - [r_i² r_o² (p_o - p_i) / (r² (r_o² - r_i²))]

where

σ_c = stress in circumferential direction (MPa, psi)
r = radius to point in tube or cylinder wall (mm, in) (r_i < r < r_o)

maximum stress when r = r_i (inside pipe or cylinder)

Radial direction is one going through the wall thickness, such as from outside surface to the inside surface.

The stress in radial direction at a point in the tube or cylinder wall can be expressed as:

σ_r = [(p_i r_i² - p_o r_o²) / (r_o² - r_i²)] + [r_i² r_o² (p_o - p_i) / (r² (r_o² - r_i²))]

maximum stress when r = ro (outside pipe or cylinder)

Combined stress in a single point in the cylinder wall cannot be described by a single vector using vector addition. Instead stress tensors (matrixes) describing the linear connection between two physical vectors quantities can be used.

Reference: Link

When a thin-walled tube or cylinder is subjected to internal pressure a hoop and longitudinal stress are produced in the wall.

For the thin walled equations below the wall thickness is less than 1/20 of tube or cylinder diameter.

The hoop stress is acting circumferential and perpendicular to the axis and the radius of the cylinder wall. The hoop stress can be calculated as

σ_h = p d / (2 t)

where

σ_h = hoop stress (MPa, psi)
p = internal pressure in the tube or cylinder (MPa, psi)
d = internal diameter of tube or cylinder (mm, in)
t = tube or cylinder wall thickness (mm, in)