Overview

Mohr’s Circle is a graphical method used to determine the normal and shear stresses on an arbitrary oblique plane within a material element under plane stress. It provides an intuitive visualization of stress transformation and helps identify principal stresses and maximum shear stress.

2.4.1 Mohr’s Circle for Plane Stress

To construct Mohr’s Circle:

  1. Mark origin \( O \) as reference.
  2. Plot points \( A(\sigma_x, \tau_{xy}) \) and \( B(\sigma_y, -\tau_{xy}) \).
  3. Draw diameter \( AB \); its midpoint \( O' \) is the center of the circle.
  4. Draw the full circle with diameter \( AB \).
  5. To find stresses on a plane inclined at angle \( \theta \), rotate the diameter by \( 2\theta \) counterclockwise to locate the transformed stress point.

Circle Properties

Radius:

$$ R_\sigma = \sqrt{ \left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2 } $$

This is also the maximum shear stress.

Center Coordinates:

$$ OO' = \frac{\sigma_x + \sigma_y}{2} $$

The center lies on the x-axis.

Principal Planes and Stresses

Principal planes are those where shear stress is zero. They correspond to the extreme points on the x-axis of Mohr’s Circle.

Orientation of principal planes:

$$ \tan 2\phi = \frac{2\tau_{xy}}{\sigma_x - \sigma_y} $$

Principal stresses:

$$ \sigma_1 = OO' + R_\sigma, \quad \sigma_2 = OO' - R_\sigma $$

Maximum Shear Stress

Maximum shear stress occurs on planes oriented at \( 45^\circ \) to the principal planes. Its magnitude is: $$ \tau_{\text{max}} = R_\sigma $$