Principal stress 2D calculator

What Mohr's circle reports for a plane-stress element

The identities assume two-dimensional plane stress or the in-plane part of a plane-strain problem; they do not compute a third principal for a full three-dimensional tensor, do not apply von Mises or Tresca yield checks, and do not convert units. Flipping the shear sign changes the principal angle but not the two principal magnitudes, because shear enters the radius only as a square. A material point under plane stress is described by three in-plane components: the normal stress on the x-face sigma_x, the normal stress on the y-face sigma_y, and the complementary shear tau_xy. Those three numbers do not stay put when you rotate the inspection element. At some orientation the shear vanishes and the remaining normal stresses take their extreme values; those extremes are the principal stresses sigma_1 (algebraically larger) and sigma_2 (algebraically smaller). Halfway around Mohr's circle, at forty-five degrees from the principal axes in physical space, the in-plane shear reaches its peak magnitude tau_max. The same geometry is written without drawing a circle: the centre sits at the average of sigma_x and sigma_y, the radius is the square root of the squared half-difference of the normals plus the squared shear, the principals are centre plus and minus that radius, and tau_max equals that radius. Keep sigma_x, sigma_y, and tau_xy in one consistent stress unit—megapascals, pounds per square inch, or pascals—and the returned principals and shear share that unit. Sign convention follows the standard continuum statement: positive normal stress is tension, and positive tau_xy is the shear on the positive-x face acting in the positive-y direction.

How this calculator validates input and applies Mohr's identities

A hydrostatic or already-principal state, where the two normals are equal and the shear is zero, reports equal principals, a zero radius, and a principal angle of zero degrees. There is no unit conversion, no plot image, no three-dimensional eigenvalue solve, and no yield-criterion overlay; this is the pure Mohr-circle evaluation of a 2D stress element only. Send a JSON body with the three required fields sigma_x, sigma_y, and tau_xy. Every field must be a finite number or a numeric string that parses to one; missing fields, empty strings, non-numeric text, NaN, and Infinity all fail immediately with invalid_input before any result is returned, and failed validation is not charged. Optional aliases such as sx, sy, txy, stress_x, stress_y, shear_xy, sigmax, sigmay, tau, shear, sigma_xx, sigma_yy, sxx, syy, and tauxy are accepted when documented so forms and scripts can use textbook symbols. After validation the engine computes the Mohr-circle centre as the average of the two normals, the radius as the Euclidean norm of the half-difference and the shear, the principals as centre plus and minus that radius, and the angle theta_p in degrees as one-half of atan2 of twice the shear over the difference of the normals. That atan2 form selects the orientation of the algebraically larger principal without an extra branch: substituting the resulting double-angle into the stress-transformation equations recovers sigma_1 by construction. There is no division-by-zero path in these identities, so every finite triple yields a defined result; magnitude bounds exist only to keep intermediate products finite on a Worker isolate. Results are rounded to a stable twelve-significant-figure precision so the same input always yields the same JSON across JavaScript engines. The same pure function powers the free in-browser tool and the API path, with no network, no random seed, and no wall clock in the computation path.

Practical use, limits, pricing, and how this differs from F/A and My/I

Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: identical inputs always produce the same principals, the same tau_max, and the same formula and method labels. Wire the endpoint into mechanics drills, contest answer checkers, early-stage design worksheets, or preprocessing steps where you prefer explicit domain errors—such as rejecting a missing sigma_x—over opaque widgets that cannot explain why a bad input was refused.