The stress state at a point in a material under plane stress condition is equi-biaxial tension with a magnitude of 10 MPa. If one unit on the σ – π plane is 1 MPa, the Mohr's circle representation of the state-of-stress is given by

Correct Answer: a point on the σ axis at a distance of 10 units from the origin

Explanation:

To understand why the Mohr's circle for this state of stress is represented by a single point on the normal stress ($\sigma$) axis, let us analyze the given information and the provided diagram step-by-step.

1. Analysis of the Given State of Stress
From the problem statement and the left side of the attached image, we observe a square stress element subjected to:
- A horizontal normal tensile stress:

$${\sigma }_x=10\text{ MPa}$$

- A vertical normal tensile stress:

$${\sigma }_y=10\text{ MPa}$$

- No shear stress acting on these planes:

$${\tau }_{xy}=0\text{ MPa}$$

This condition where the two principal stresses are equal and positive is known as equi-biaxial tension.

2. Finding the Center of Mohr's Circle
The center of Mohr's circle on the $\sigma$ – $\tau$ plane (referred to as the σ – π plane in the question text) is given by the coordinate:

$$C=\left({\sigma }_{\text{avg}},0\right)$$

where the average normal stress ${\sigma }_{\text{avg}}$ is calculated as:

$${\sigma }_{\text{avg}}=\frac{{\sigma }_x+{\sigma }_y}{2}$$

Substituting the given values:

$${\sigma }_{\text{avg}}=\frac{10+10}{2}=10\text{ MPa}$$

Therefore, the center of the circle lies at $\left(10,0\right)$.

3. Finding the Radius of Mohr's Circle
The radius $R$ of Mohr's circle is defined by the formula:

$$R=\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$

Substituting our values:

$$R=\sqrt{{\left(\frac{10-10}{2}\right)}^2+0^2}=0$$

Because the radius is $0$, Mohr's circle degenerates into a single point.

4. Plotting the Point
Since one unit on the graph represents $1\text{ MPa}$, a stress of $10\text{ MPa}$ corresponds to $10$ units from the origin. As shown on the right side of the diagram:
- The point is plotted on the horizontal normal stress axis ($\sigma$).
- The distance from the vertical axis ($\tau$) to this point is labeled as $10$ units.
Thus, the state-of-stress is represented by a single point on the $\sigma$ axis at a distance of 10 units from the origin.