# Understanding the Mohr's Circle Solution (Conceptual, Not Analytical) Let's solve the principal stresses **using only the concept of Mohr's Circle**, not by formulas or calculation. --- ## Given State of Stress - \(\sigma_x = 50\) MPa - \(\sigma_y = 75\) MPa - \(\tau_{xy} = 25\) MPa --- ## Mohr's Circle Concept Steps ### 1. **Plotting the Points** - On the Mohr's circle, plot two points: - Point A: (\(\sigma_x\), \(\tau_{xy}\)) = (50, 25) - Point B: (\(\sigma_y\), \(-\tau_{xy}\)) = (75, -25) - These points represent the normal and shear stresses on two perpendicular planes. ### 2. **Drawing the Circle** - The diameter of the circle connects points A and B. - The center of the circle lies at the midpoint of the line joining A and B. - The radius of the circle is the distance from the center to either endpoint. ### 3. **Principal Stresses (Maximum and Minimum)** - On Mohr's circle, the intersection points with the horizontal (normal stress) axis are the **principal stresses**. - These points are the **farthest left** (minimum) and **farthest right** (maximum) on the circle. ### 4. **Interpretation** - The center of the circle is at the average of \(\sigma_x\) and \(\sigma_y\). - The principal stresses are always **further away from the center** by the radius, one on each side. --- ## **Result from the Image** Based on the Mohr’s circle concept (without formulas): - The **minimum principal stress** is the leftmost point on the circle. - The **maximum principal stress** is the rightmost point on the circle. From the image: - Minimum principal stress ≈ **37.55 MPa** - Maximum principal stress ≈ **90.45 MPa** --- ## **Summary Table** | | Value (MPa) | |------------------------|-------------| | Minimum Principal | 37.55 | | Maximum Principal | 90.45 | --- ### **Key Points** - **Mohr’s Circle** is a graphical way to find principal stresses. - It visually shows how normal and shear stresses transform, and where the extremum (principal) stresses occur. - **No formulas needed**: Just plot, draw the circle, and read off the principal points! --- **If you need a hand-drawn sketch of the Mohr's circle, let me know!**