Circular Representation of State of Stress:

From a state of stress defined by the components s_x, s_y and t_xy, we express the stress components on an arbitrary plane as

                       t= - (s_x-s_y )/2)sin 2q + t_xy cos 2q

which we might rewrite in general as

s =  a + b cos 2q + c sin 2q       ---------(i)

                t = - b sin 2q + c cos 2q        -------------(ii)

Now Let us try to establish direct relationship between s and t by eliminating q between

Eqs. (i) & (ii).

To make simplify the effort, let us take the origin of coordinate at (a, 0), so that the new variable   = (s - a) is let for building the relationship.

 --------------(iii)

t = - b sin 2q + c cos 2q   -------------- (iv)

By Squaring & adding, we obtain

+ b²  sin ²  2q + c2  cos2  2q - 2bc cos 2q sin 2q

= b²  (cos²  2q + sin ²  2q) + c²  (sin ²  2q+ cos²  2q)

i.e.

Since b and c are constants let b²  + c²  = r ²

∴       

Eq. shows that if s and t, the normal and shear stress components on any arbitrary plane are plotted as coordinates, the locus of the point will be a circle whose centre will

(s_x  + s _y)/2     , 0  and radius will be

In other terms, the state of stress at particular point might be represented by a circle & a point on the circle represents the normal & shear stress components, on some plane as horizontal & vertical coordinates.