Mohr's Circle Calculator

Mohr's circle is a graphical and analytical method for transforming a two-dimensional stress state. At a point in a loaded component, normal stress can act in the x and y directions while shear stress acts on the corresponding faces. Rotating the physical plane changes the components reported by those axes even though the underlying state of stress remains the same. Mohr's circle organizes every possible orientation on one circle and makes important extreme values easy to identify. The horizontal coordinate of the circle represents normal stress and the vertical coordinate represents shear stress. Its center is the average of the two input normal stresses. Its radius is the square root of the squared half-difference between the normal stresses plus the squared shear stress. Adding and subtracting the radius from the center gives the maximum and minimum principal stresses. At principal orientations, transformed shear stress is zero. The radius itself is the maximum in-plane shear stress. Angles on Mohr's circle are twice physical angles. The calculator obtains the principal plane orientation from one half of the two-argument arctangent of twice the input shear stress and the normal-stress difference. Using the two-argument form preserves the correct quadrant. For the user-entered transformation angle, the familiar double-angle stress equations calculate transformed normal stress in both perpendicular directions and transformed shear stress. The sign convention follows positive shear in the standard tensor transformation equations. The material buttons preserve the choices from the original calculator and help label the engineering context, but material type does not alter stress transformation. Mohr's circle depends on the current stress components, not elastic modulus or yield strength. Steel, aluminum, and concrete can have identical transformed stresses when subjected to the same plane-stress tensor, although their allowable stresses and failure criteria are very different. Positive normal values are treated as tension and negative values as compression. Keep all stress inputs in MPa so the outputs remain in MPa. The same equations work with another consistent stress unit, but the displayed unit would then need reinterpretation. The calculator handles pure tension, biaxial loading, pure shear, compression, and combined states. It does not calculate out-of-plane principal stress or three-dimensional maximum shear. Engineers use these results to orient strain gauges, inspect critical planes, compare stresses with yield criteria, and understand failure directions. For ductile metals, principal values often feed von Mises or Tresca checks. Brittle materials may require principal-stress or Mohr-Coulomb criteria. This calculator supplies the plane-stress transformation, but final design decisions must also include material properties, stress concentrations, fatigue, safety factors, and the applicable engineering code.