text: Consider a point  in a continuum under a state of plane stress, or plane strain, with stress components  and all other stress components equal to zero  (Figure 8.1). From static equilibrium of an infinitesimal material element at  (Figure 8.2), the normal stress  and the shear stress  on any plane perpendicular to the - plane passing through  with a unit vector  making an angle of  with the horizontal, i.e.  is the direction cosine in the  direction, is given by:
  These equations indicate that in a plane stress or plane strain condition, one can determine the stress components at a point on all directions, i.e. as a function of , if one knows the stress components  on any two perpendicular directions at that point. It is important to remember that we are considering a unit area of the infinitesimal element in the direction parallel to the - plane.
  The principal directions (Figure 8.3), i.e., orientation of the planes where the shear stress components are zero, can be obtained by making the previous equation for the shear stress  equal to zero. Thus we have:
  and we obtain
  This equation defines two values  which are  apart (Figure 8.3). The same result can be obtained by finding the angle  which makes the normal stress  a maximum, i.e. 
  The principal stresses  and , or minimum and maximum normal stresses  and , respectively, can then be obtained by replacing both values of  into the previous equation for . This can be achieved by rearranging the equations for  and , first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have
  where
  which is the equation of a circle of radius  centered at a point with coordinates , called Mohr's circle. But knowing that for the principal stresses the shear stress , then we obtain from this equation:
  When  the infinitesimal element is oriented in the direction of the principal planes, thus the stresses acting on the rectangular element are principal stresses:  and . Then the normal stress  and shear stress  as a function of the principal stresses can be determined by making . Thus we have
  Then the maximum shear stress  occurs when  , i.e.  (Figure 8.3):
  Then the minimum shear stress  occurs when  , i.e.  (Figure 8.3):