Mohr Circle

Definition

A graphical representation of normal stresses and shear stresses was developed by Professor Otto Mohr (1835–1918) who recognized that the equations could be rearranged into the form describing a circle:

$$ {\left[{\sigma}_{\beta }-\left(\frac{\sigma_x+{\sigma}_y}{2}\right)\right]}^2+{\left[{\tau}_{\beta }-0\right]}^2={\left[\sqrt{{\left(\frac{\sigma_x-{\sigma}_y}{2}\right)}^2+{\tau}_x^2}\ \right]}^2 $$

(1)

where the second part of the right-hand term is the x coordinate of the circle center, the second part of the middle term is the y coordinate of the circle center, and the term inside the right-hand bracket is the circle radius. The Mohr circle graph contains three circles in a plane stress example: σ₂ = 0. A similar graph showing three-dimensional information would have two additional circles inset into each of the three circles shown that intersect the normal stress x axis at the value of σ₂ with diameters equal to (σ₁−σ₃) (shown), (σ₁−σ₂), and (σ₂−σ₃). The smaller...