Mohr's Circle

Mohr's Circle is a graphical representation of the state of stress at a point in a material. It is used in mechanical engineering, specifically in the field of machine design, to determine principal stresses, maximum shear stresses, and the orientations of these stresses.

Construction of Mohr's Circle

To construct Mohr's Circle, follow these steps:

1. Identify the Normal and Shear Stresses:

   • Normal stresses ($\sigma_x$ and $\sigma_y$)
   • Shear stress ($\tau_{xy}$)

2. Plot the Points:

   • Plot the points $(\sigma_x, \tau_{xy})$ and $(\sigma_y, -\tau_{xy})$ on the Cartesian plane.

3. Determine the Center and Radius:

   • The center $C$ of Mohr's Circle is located at: $$C = \left( \frac{\sigma_x + \sigma_y}{2}, 0 \right)$$
   • The radius $R$ of Mohr's Circle is: $$R = \sqrt{\left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2}$$

4. Draw the Circle:

   • With the center $C$ and radius $R$, draw Mohr's Circle.

Principal Stresses

The principal stresses ($\sigma_1$ and $\sigma_2$) are found at the intersections of Mohr's Circle with the horizontal axis:

$$\sigma_{1,2} = C \pm R$$

Maximum Shear Stress

The maximum shear stress ($\tau_{\text{max}}$) is the radius of Mohr's Circle:

$$\tau_{\text{max}} = R$$

Orientation of Principal Stresses

The orientation $\theta_p$ of the principal stresses is given by:

$$\tan(2\theta_p) = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$$

Example

Given:

• $\sigma_x = 50 \text{ MPa}$
• $\sigma_y = 30 \text{ MPa}$
• $\tau_{xy} = 20 \text{ MPa}$

1. Center of Mohr's Circle:

   $$C = \left( \frac{50 + 30}{2}, 0 \right) = (40, 0)$$

2. Radius of Mohr's Circle:

   $$R = \sqrt{\left( \frac{50 - 30}{2} \right)^2 + 20^2} = \sqrt{10^2 + 20^2} = \sqrt{100 + 400} = \sqrt{500} = 22.36 \text{ MPa}$$

3. Principal Stresses:

   $$\sigma_1 = 40 + 22.36 = 62.36 \text{ MPa}$$ $$\sigma_2 = 40 - 22.36 = 17.64 \text{ MPa}$$

4. Maximum Shear Stress:

   $$\tau_{\text{max}} = 22.36 \text{ MPa}$$

Mohr's Circle provides a clear visual tool to understand the stresses at a point and is an essential concept in the analysis and design of mechanical components.