Mohr's Circle Calculator

Mohr's circle transforms two-dimensional stress components into geometric coordinates. Given an initial normal stress in the x-direction ($̃ͷ\sigma_x$), normal stress in the y-direction ($̃ͷ\sigma_y$), and shear stress ($̃ͷ\tau_{xy}$), the calculator applies fundamental continuum mechanics formulas. The mean normal stress represents the center of the circle and is calculated as $̃ͷ\sigma_{\text{mean}} = (\sigma_x + \sigma_y) / 2$. The radius of the circle, which corresponds to the maximum shear stress ($̃ͷ\tau_{\text{max}}$) and the variation range for principal stresses, is found using the Pythagorean relation $̃ͷ\tau_{\text{max}} = \sqrt{((\sigma_x - \sigma_y)/2)^2 + \tau_{xy}^2}$. Consequently, the principal stresses are determined by adding and subtracting this radius from the mean stress center.

Consider a structural element subjected to a horizontal normal stress $̃ͷ\sigma_x = 100\text{ MPa}$, a vertical normal stress $̃ͷ\sigma_y = 40\text{ MPa}$, and a shear stress $̃ͷ\tau_{xy} = 30\text{ MPa}$. First, calculate the mean stress: $̃ͷ\sigma_{\text{mean}} = (100 + 40) / 2 = 70\text{ MPa}$. Next, compute the maximum shear stress radius: $̃ͷ\tau_{\text{max}} = \sqrt{((100 - 40)/2)^2 + 30^2} = \sqrt{30^2 + 30^2} = \sqrt{1800} \approx 42.43\text{ MPa}$. From these values, the maximum principal stress is $̃ͷ\sigma_1 = 70 + 42.43 = 112.43\text{ MPa}$, and the minimum principal stress is $̃ͷ\sigma_2 = 70 - 42.43 = 27.57\text{ MPa}$. Finally, the principal angle is derived using half the arctangent of the shear-to-normal stress ratio, yielding the exact orientation of the primary load-bearing planes.

Always maintain consistent sign conventions when entering your data; positive normal stresses indicate tension while negative values indicate compression. Double-check your shear stress directions, as contrasting signs can shift your calculated principal orientation angles by 90 degrees. Use this calculator alongside von Mises yield criteria to comprehensively evaluate ductile material safety under complex multidirectional loading scenarios.