Stepwise procedure of Mohr's circle

Mohr's circle is a graphical method that represents the state of plane stress at a point; every point on the circle gives the normal and shear stress on one plane.

Given: $\sigma_x$, $\sigma_y$ and $\tau_{xy}$ on an element, and a plane whose normal is at angle $\theta$ to the $x$-axis.

Procedure

1. Draw the $\sigma$-axis horizontally (tension positive, to the right of O) and the $\tau$-axis vertically through O. Select a suitable scale.
2. Sign rule for plotting shear: a shear stress that tends to turn the element clockwise is plotted above the $\sigma$-axis, and one that turns it anticlockwise is plotted below. With this rule, a rotation of $2\theta$ on the circle is in the same sense as the rotation $\theta$ of the plane. Plot point $X(\sigma_x, \tau_{xy})$ for the face on which $\sigma_x$ acts.
3. Plot point $Y(\sigma_y, -\tau_{xy})$ for the perpendicular face (its shear turns the element the opposite way).
4. Join X and Y. The line cuts the $\sigma$-axis at $C$, the centre, where $OC = \dfrac{\sigma_x+\sigma_y}{2}$.
5. With C as centre and $CX$ as radius, draw the circle. Radius $R = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}$.
6. The circle cuts the $\sigma$-axis at $P_1$ and $P_2$: $\sigma_1 = OC + R$, $\sigma_2 = OC - R$. The angle $XCP_1 = 2\theta_p$ gives the principal plane.
7. For the stresses on the given inclined plane, rotate the radius CX through $2\theta$ about C in the same sense as the plane is rotated, reaching point $Q$. The abscissa of Q is the normal stress $\sigma_n$ and the ordinate is the shear stress $\tau_\theta$. The diametrically opposite point $Q'$ gives the stresses on the perpendicular plane.
8. The resultant stress is $OQ$, and its obliquity is $\angle QOC$ measured from the $\sigma$-axis.
9. The top and bottom points of the circle give the maximum shear stress $\tau_{max} = R$.

Sketch

        tau
         |        T (tau max = R)
         |     .--*--.
         |   /    |    \
         |  *Q    |     * X(sx, txy)
 --------+-P2-----C------P1------- sigma
         O  \     |    /
         |   *Y   |   /
         |     '--*--'
         |        T'
 P2 = sigma_2, P1 = sigma_1, angle XCP1 = 2*theta_p