Concerning membrane and membrane + bending stresses see a FAQ in this forum.
The definition of a primary stress appears simple in wording, but is quite complex in the consequences: it is that part of the stress due to the external loads that is necessary to satisfy the equations of equilibrium (hence they are not required to be compliant).
The consequences of this definition might seem unrelated to strength calculations, but they are, as a primary stress is not self limiting: if the load increases, the stress will increase till rupture.
The commonest source of primary stress in a pressure vessel is pressure: however not all the stresses due to pressure are primary (e.g. continuity stresses at the junction between a cylinder and a head).
The way to understand primary stress better is to draw a comparison to secondary stress. Primary stress is not self-limiting and can cause failure or gross distorsion. Examples are membrane stress in shell and bending stress in flat head. Primary stress can be further classified as general or local which have different stress limit. Secondary stress is self-limiting (by stress redistribution) and will not cause failure. Examples are thermal stress and local bending stress. Membrane (or bending stress) can be primary or secondary.
At the joint between head and shell, membrane stress is primary (local), bending stress is secondary.
