Mohr's circle calculator

This Mohr circle calculator accepts finite sigma_x, sigma_y, and tau_xy, applies the closed-form identities that sigma_1 and sigma_2 equal the average normal stress plus or minus the Mohr radius and that tau_max equals that radius, and returns structured JSON you can pin in homework, lab notes, or a regression suite. When any stress component is missing, empty, infinite, or not a number, the task fails with invalid_input instead of emitting a silent NaN that would poison a downstream safety check.

What Mohr's circle reports for a 2D stress state

When the two normals are equal and shear vanishes, the radius collapses to zero, both principals equal the hydrostatic value, and there is no in-plane shear to rotate away — that hydrostatic case is a valid input, not an error. Search queries such as mohr circle calculator, principal stress calculator, and maximum shear from plane stress all want those three results as numbers, not a reminder that a construction exists. The two-dimensional plane-stress state of a material element is fully specified by three numbers: the normal stress on the x-face (sigma_x), the normal stress on the y-face (sigma_y), and the in-plane shear (tau_xy). Mohr's circle is the picture of that tensor: plot normal stress horizontally and shear vertically, mark the known point (sigma_x, tau_xy), and the circle centred on the average of the two normals with radius equal to the distance to that point contains every possible orientation of the element. The extreme horizontal intercepts are the principal stresses sigma_1 (algebraically larger) and sigma_2 (algebraically smaller); the extreme vertical extent is the in-plane maximum shear tau_max, equal to the radius. The same geometry is the eigenvalue problem of the two-by-two stress matrix, so this calculator does not draw a figure — it evaluates the closed-form identities sigma_1,2 equals (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, and tau_max equals that same square root. Delivering sigma_x, sigma_y, tau_xy, sigma_1, sigma_2, tau_max, center, radius, 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 Mohr centre as the average of sigma_x and sigma_y, the radius as the Euclidean combination of half the normal difference and tau_xy, then sigma_1 as centre plus radius, sigma_2 as centre minus radius, and tau_max as the radius. The success body echoes the three inputs and returns sigma_1, sigma_2, tau_max, center, radius, and a formula string restating the identities. Example: sigma_x equals eighty, sigma_y equals twenty, tau_xy equals forty yields centre fifty, radius fifty, principals one hundred and zero, and maximum shear fifty — the classic 3-4-5 triangle of Mohr's circle. Pair the response with your own plotting code if you need a sketch of the circle; this endpoint solves the invariants 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 Mohr-circle 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. Sign convention follows the usual continuum-mechanics two-dimensional tensor: the principal values and the in-plane maximum shear magnitude do not depend on which textbook draws shear positive-up or positive-down on Mohr's circle, because those values come from the invariants. 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. Failed validation does not consume balance. Classrooms and tutoring platforms use a Mohr circle calculator so learners can check principal stresses and maximum shear 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 2D 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. This calculator reports the in-plane tau_max equal to the Mohr radius; 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.