Mohr's Circle
Introduced by Otto Mohr in 1882, Mohr's Circle illustrates principal stresses and stress transformations via a graphical format,
The two principal stresses are shown in red, and the maximum shear stress is shown in orange. Recall that the normal stesses equal the principal stresses when the stress element is aligned with the principal directions, and the shear stress equals the maximum shear stress when the stress element is rotated 45° away from the principal directions.
As the stress element is rotated away from the principal (or maximum shear) directions, the normal and shear stress components will always lie on Mohr's Circle.
Mohr's Circle was the leading tool used to visualize relationships between normal and shear stresses, and to estimate the maximum stresses, before hand-held calculators became popular. Even today, Mohr's Circle is still widely used by engineers all over the world.
Derivation of Mohr's Circle
To establish Mohr's Circle, we first recall the stress transformation formulas for plane stress at a given location,
Using a basic trigonometric relation (cos22q + sin22q = 1) to combine the two above equations we have,
This is the equation of a circle, plotted on a graph where the abscissa is the normal stress and the ordinate is the shear stress. This is easier to see if we interpret sx and sy as being the two principal stresses, and txy as being the maximum shear stress. Then we can define the average stress, savg, and a "radius" R (which is just equal to the maximum shear stress), 
The circle equation above now takes on a more familiar form,
The circle is centered at the average stress value, and has a radius R equal to the maximum shear stress, as shown in the figure below,
Cylindrical Pressure Vessel
Consider a cylindrical pressure vessel with radius r and wall thickness t subjected to an internal gage pressure p.
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 sh and the longitudinal stress sl are the principal stresses.
To determine the longitudinal stress sl, 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, 
and <this strees is a n axial stress&get;
· To determine the hoop stress sh, 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,
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