The analysis of piping under pressure, weight and thennal expansion is complex. This complexity can be understood by knovledge of Principal Axis System.
Stress is considered as the ratio of Force to Area. To find the stress in the small element, say cube of a piece of pipe, construct a three-dimensional, mutually perpendicular principal axis system with each axis perpendicular to the face of the cube it intersects.
Each force, acting on the cube can be resolved into force components, acting along each of the axis. Each force, acting on the face of the cube divided by area of the cube face is called the principal stress.
The principal stress acting along the centerline of the pipe is called Longitudinal principal stress. This stress is caused by longitudinal bending, axial force loading or pressure.
Radial principal stress acts on a line from a radial line from center of pipe through the pipe wall. This stress is compressive stress acting on pipe inside diameter caused by internal pressure or a tensile stress caused by vacuum pressure.