Mohr's Circle is a helpful tool used in engineering to understand how stress affects a material at a certain point. It gives a simple picture of the normal (straight) and shear (sliding) stresses that happen on different angles at that point. Let's break down how to make Mohr's Circle for 2D stress in a few easy steps.

Step 1: Identify the Stress Components

In a 2D stress situation, we look at:

• The normal stress on the x-axis, called $\sigma_x$.
• The normal stress on the y-axis, called $\sigma_y$.
• The shear stress on the x-y plane, called $\tau_{xy}$.

For example:

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

Step 2: Calculate the Center and Radius of Mohr's Circle

1. Center of the Circle: To find the center of Mohr's Circle, we average the normal stresses:

   $$C = \frac{\sigma_x + \sigma_y}{2} = \frac{50 + 30}{2} = 40 \, \text{MPa}$$

2. Radius of the Circle: We calculate the radius using this formula:

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

   Plugging in our values:

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

Step 3: Draw the Circle

• The center of the circle is at $(C, 0)$ in our stress graph, which is $(40, 0)$ for our example.
• The radius is about $22.36 \, \text{MPa}$, telling us how far the circle stretches from the center.

Step 4: Find the Principal Stresses

To find the main stresses, we look for points on Mohr's Circle where the shear stress is zero. This happens along the horizontal line:

• The main stresses can be calculated as:
  • $\sigma_1 = C + R$
  • $\sigma_2 = C - R$

Doing the math:

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

Step 5: Determine the Principal Directions

To find the angle of the main planes, we use this formula:

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

For our example:

$$\tan(2\theta_p) = \frac{2 \times 20}{50 - 30} = \frac{40}{20} = 2$$

This gives us an angle of about $\theta_p = 26.57^\circ$.

Conclusion

By following these steps, you can easily create Mohr's Circle for a 2D stress situation. This helps you find key information like the main stresses and how they are oriented. Mohr's Circle is an important method in engineering and materials science, helping us understand and predict how materials behave under force.