Question

Which of the plot(s) shown is/are valid Mohr's circle representations of a plane stress state in a material? (The center of each circle is indicated by O.) 

 

• A. M1
• B. M2
• C. M3
• D. M4

Hint:

The fastest way to check the validity of a Mohr's circle diagram is to check the location of its center. If the center is NOT on the horizontal (\(\sigma\)) axis, it is not a valid Mohr's circle.

Answer 1

The Correct Option is A, C

Step 1: Understanding the Concept:
Mohr's circle is a graphical representation of the state of stress at a point. It plots the normal stress (\(\sigma\)) on the horizontal axis and the shear stress (\(\tau\)) on the vertical axis. A fundamental property of Mohr's circle for a 2D or 3D stress state is that its center must always lie on the horizontal (\(\sigma\)) axis.
Step 2: Key Formula or Approach:
For a plane stress state with stresses \((\sigma_x, \sigma_y, \tau_{xy})\), the coordinates of the center (\(C\)) and the radius (\(R\)) of Mohr's circle are given by: - Center: \(C = \sigma_{\text{avg}} = \frac{\sigma_x + \sigma_y}{2}\) - Radius: \(R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}\) The key point is that the center of the circle is at the coordinate \((\sigma_{\text{avg}}, 0)\). This means the center must always be on the \(\sigma\)-axis.
Step 3: Detailed Explanation:
Let's analyze each plot:
- M1: The center O is at the origin (0, 0). This corresponds to a state where \(\sigma_{\text{avg}} = 0\), which means \(\sigma_x + \sigma_y = 0\) or \(\sigma_x = -\sigma_y\). A state of pure shear is an example (\(\sigma_x = \sigma_y = 0\)). Since the center is on the \(\sigma\)-axis, this is a valid Mohr's circle.
- M2: The center O is located on the \(\tau\)-axis, not the \(\sigma\)-axis. Its coordinate is of the form \((0, \tau_c)\) where \(\tau_c \neq 0\). This violates the fundamental rule that the center must lie on the normal stress axis. This is an invalid representation.
- M3: The center O is located on the \(\sigma\)-axis at a point where \(\sigma \neq 0\). This corresponds to a general plane stress state where \(\sigma_{\text{avg}} = (\sigma_x + \sigma_y)/2 \neq 0\). Since the center is on the \(\sigma\)-axis, this is a valid Mohr's circle.
- M4: The center O is located in the first quadrant, with both a non-zero \(\sigma\) coordinate and a non-zero \(\tau\) coordinate. Like M2, this violates the rule that the center must lie on the \(\sigma\)-axis. This is an invalid representation.
Step 4: Final Answer:
The plots M1 and M3 are the only valid representations of Mohr's circle.
Step 5: Why This is Correct:
The construction of Mohr's circle is based on stress transformation equations, which dictate that the center of the circle, representing the average normal stress, must lie on the normal stress axis (\(\tau=0\)). Only M1 and M3 satisfy this condition.