Principal stress calculator

This principal stress calculator evaluates the closed-form identity that sigma_1 and sigma_2 equal the average of the two normals plus or minus the square root of half the normal difference squared plus the shear squared, and it returns both principals as structured JSON. Missing, empty, infinite, or non-numeric components fail with invalid_input instead of emitting a silent NaN.

What principal stresses mean for a plane-stress state

Principal stresses are the extreme normal stresses that appear when that element is rotated until shear on the faces is zero. They are the eigenvalues of the two-by-two stress matrix whose diagonals are the two normals and whose off-diagonal is the shear. When the two normals are equal and shear vanishes, both principals collapse to that hydrostatic value and the element is already in principal orientation — that is a valid input, not an error. Units are yours — megapascals, pounds per square inch, or any coherent stress unit — as long as all three components share that unit; the output uses the same unit. A plane-stress state on a material element is specified by two normal stresses and one in-plane shear: sigma_x on the x-face, sigma_y on the y-face, and tau_xy. The closed-form solution used by every mechanics-of-materials textbook is therefore an average plus or minus a radius: sigma_1 and sigma_2 equal (sigma_x plus sigma_y) divided by two, plus or minus the square root of ((sigma_x minus sigma_y) over two) squared plus tau_xy squared. The algebraically larger root is reported as sigma_1 and the algebraically smaller root as sigma_2, so clients never have to sort the pair. Search queries such as principal stress calculator, plane stress principals, and sigma 1 sigma 2 from shear want those two numbers as JSON, not a reminder that a Mohr circle exists as a drawing aid. This endpoint solves the invariants only: it does not plot a circle, does not introduce a third out-of-plane principal of zero, and does not compute von Mises or Tresca equivalent stress. Delivering sigma_x, sigma_y, tau_xy, sigma_1, sigma_2, and formula as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing the square-root arithmetic.

Input contract, response shape, and error behaviour

There is no extra domain restriction: compression (negative normals), tension, zero shear, and negative shear are all legal finite stress states. Results are rounded to twelve decimal places so goldens compare byte-for-byte across JavaScript engines; minus-zero is normalised to zero. POST a JSON body with three required finite numbers: sigma_x, the normal stress on the x-face; sigma_y, the normal stress on the y-face; and tau_xy, the in-plane shear that acts with them. Each field may arrive as a JSON number or as a numeric string so forms and CSV-style pipelines can pass values without an extra cast. Missing fields, empty strings, non-numeric text, NaN, and Infinity all fail immediately with invalid_input that names the offending field, and failed validation is not billed. After validation the engine computes the average of sigma_x and sigma_y, the Mohr-equivalent radius as the Euclidean combination of half the normal difference and tau_xy, then sigma_1 as that average plus the radius and sigma_2 as that average minus the radius. The success body echoes the three inputs and returns sigma_1, sigma_2, and a formula string restating the identity. Example: sigma_x equals eighty, sigma_y equals twenty, tau_xy equals forty yields principals one hundred and zero — the classic 3-4-5 right triangle of the two-dimensional stress transformation. Pair the response with your own plotting code if you need a sketch of the element or of Mohr's circle; this endpoint reports both principals only and does not emit images or step-by-step drafting prose beyond the formula field. The same pure function powers the free in-browser tool and the API path, so homework checks and production pipelines never disagree on a floating-point edge case.

Where a principal-stress API earns its keep, limits, and pricing

Exam-prep generators need a deterministic oracle that rejects a non-finite component instead of silently returning NaN that a buggy grader might treat as success. Units are yours as long as all three components share one coherent stress unit. Failed validation does not consume balance. Classrooms and tutoring platforms use a principal stress calculator so learners can check both principals for arbitrary sigma_x, sigma_y, and tau_xy without mixing the half-angle identities by hand. Strain-gauge reductions, weld and shaft checks, and finite-element snapshots often collapse a two-dimensional tensor to the same three components; calling a tiny parse API avoids pulling a full computer-algebra or continuum-mechanics library for an eigenvalue problem that has a closed form. Sign convention follows the usual continuum-mechanics two-dimensional tensor: the principal values do not depend on which textbook draws shear positive-up or positive-down on a Mohr diagram, because those values come from the invariants and tau_xy enters only as a square. This calculator reports the in-plane principals only; it does not introduce a third out-of-plane principal of zero, does not compute von Mises or Tresca equivalent stress, and does not rotate a three-dimensional tensor. Keep requests small, pin the rounded principals in tests, and you get reproducible sigma_1 and sigma_2 for homework, LMS plugins, and edge workers without standing up symbolic math infrastructure of your own.