Mohr’s Circle

Equations (3.46) for stresses at a point O can be represented conveniently by Mohr’s circle (Fig. 3.16). Normal stress ƒ is taken as the abscissa, and shear stress v is taken as the ordinate.
The center of the circle is located on the ƒ axis at (ƒ1 + ƒ2) /2, where ƒ1 and ƒ2 are the maximum and minimum principal stresses at the point, respectively. The circle has a radius of (ƒ1 - ƒ2) /2. For each plane passing through the point O there are two diametrically opposite points on Mohr’s circle that correspond to the normal and shear stresses on the plane. Thus Mohr’s circle can be used conveniently to find the normal and shear stresses on a plane when the magnitude and direction of the principal stresses at a point are known.
Use of Mohr’s circle requires the principal stresses ƒ1 and ƒ2 to be marked off on the
abscissa (points A and B in Fig. 3.16, respectively). Tensile stresses are plotted to the right of the v axis and compressive stresses to the left. (In Fig. 3.16, the principal stresses are indicated as tensile stresses.) A circle is then constructed that has radius (ƒ1 + ƒ2)/2 and passes through A and B. The normal and shear stresses ƒx, ƒy, and vxy on a plane at an angle  with the principal directions are the coordinates of points C and D on the intersection of