During operation, pipes are subject to all of these types of stresses. Examining a small cube of metal from the most highly stressed point of the pipe wall, the stresses are distributed as so:
There are an infinite number of orientations in which this cube could have been selected, each with a different combination of normal and shear stresses on the faces. For example, there is one orientation of the orthogonal stress axes for which one normal stress is maximized, and another for which one normal stress is minimized — in both cases all shear stress components are zero. In orientations in which the shear stress is zero, the resulting normal components of the stress are termed the principal stresses. For 3-dimensional analyses, there are three of them, and they are designated as Si (the maximum), S2, and S3 (the minimum). Note that regardless of the orientation of the stress axes, the sum of the orthogonal stress components is always equal, i.e:
The converse of these orientations is that in which the shear stress component is maximized (there is also an orientation in which the shear stress is minimized, but this is ignored since the magnitudes of the minimum and maximum shear stresses are the same); this is appropriately called the orientation of maximum shear stress. The maximum shear stress in a three dimensional state of stress is equal to one-half of the difference between the largest and smallest of the principle stresses (Si and S3).
The values of the principal and maximum shear stress can be determined through the use of a Mohr's circle. The Mohr's circle analysis can be simplified by neglecting the radial stress component, therefore considering a less complex (i.e., 2-dimensional) state of stress. A Mohr's circle can be developed by plotting the normal vs. shear stresses for the two known orientations (i.e., the longitudinal stress vs. the shear and the hoop stress vs. the shear), and constructing a circle through the two points. The infinite combinations of normal and shear stresses around the circle represent the stress combinations present in the infinite number of possible orientations of the local stress axes.
A differential element at the outer radius of the pipe (where the bending and torsional stresses are maximized and the radial normal and force-induced shear stresses are usually zero) is subject to 2-dimensional plane stress, and thus the principal stress terms can be computed from the following Mohr's circle:
Introduction to Pipe Stress Analysis