Calculate the principal stresses, maximum shear stress, and principal angle from a 2D plane-stress state (σx, σy, τxy) — the same transformation Mohr's circle visualizes graphically.
2D Stress State
Enter the normal stresses σx and σy and the shear stress τxy for the stress element. All three values can be positive (tension) or negative (compression / reversed shear).
The normal stress acting on the x-face of the stress element, in megapascals.
The normal stress acting on the y-face of the stress element, in megapascals.
The shear stress acting on the x-face in the y-direction, in megapascals.
Default example: σx = 100 MPa, σy = 20 MPa, τxy = 30 MPa — giving principal stresses of about 110 MPa and 10 MPa.
Result
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Enter σx, σy, and τxy to compute Mohr's circle.
σ1 (max)—
σ2 (min)—
τmax—
θp—
Stress on Any Plane
Use the slider to inspect the normal and shear stress on a plane rotated counterclockwise from the +x axis.
At θp, shear stress is zero and normal stress is σ1.
Type in the normal stresses σx and σy and the shear stress τxy for your stress element, in megapascals. All three can be positive or negative — the calculator updates instantly as you type.
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Read the principal stresses
The Principal Stresses tab headlines σ1 (the larger principal stress) and σ2 (the smaller). The stat grid below always shows all four derived values together: σ1, σ2, τmax, and θp.
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Check the maximum shear stress
Switch to the Max Shear tab to headline τmax, the largest shear stress the element experiences on any plane — equal to the radius of Mohr's circle.
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Find the principal angle
Switch to the Principal Angle tab to headline θp, the rotation (in degrees, from the x-axis) at which the element sees only normal stress and zero shear.
Mohr's circle plots every possible normal/shear stress pair (σ, τ) a stress element can see as it's rotated through 360°, as a circle in σ-τ space. The circle's center sits on the σ-axis at the average of the two normal stresses; its radius is the distance from that center out to the (σx, τxy) point, computed with the Pythagorean-style formula above.
Principal stresses, max shear, and principal angle
The principal stresses σ1 (maximum) and σ2 (minimum) are where the circle crosses the σ-axis — the center plus or minus the radius — and represent the orientation with zero shear stress. The radius itself is the maximum in-plane shear stress, τmax. The principal angle θp is half the angle (measured on the circle) from the σx point to the nearest principal-stress point, giving the physical rotation needed to align the element with its principal axes.
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Glossary
Key Terms Explained
7 terms▸
Mohr's circle ↗A graphical method for visualizing how normal and shear stress at a point change as the stress element is rotated — every possible (σ, τ) pair traces out a circle in stress space, centered on the σ-axis.
Principal stress ↗The normal stress (σ1 maximum, σ2 minimum) that acts on a plane where the shear stress is exactly zero. Every 2D stress state has exactly two principal stresses, 90° apart.
Maximum shear stress (τmax) ↗The largest shear stress the element experiences at any rotation angle, equal to Mohr's circle's radius and occurring 45° away from the principal planes.
Principal angle (θp) ↗The rotation angle, measured from the x-axis, at which the stress element aligns with its principal planes — the orientation where normal stress is maximized/minimized and shear stress is zero.
Stress transformation ↗The process of computing the normal and shear stress on a plane at an arbitrary angle from a known stress state (σx, σy, τxy) — Mohr's circle is a graphical shortcut for this transformation.
Normal stress (σ) ↗Stress acting perpendicular to a surface, either pulling it apart (tension, positive) or pushing it together (compression, negative). σx and σy are the normal stresses on the x- and y-faces of the stress element.
Shear stress (τ) ↗Stress acting parallel to a surface, sliding one layer of material past another. τxy is the shear stress on the x-face acting in the y-direction, equal by symmetry to the shear stress on the y-face acting in the x-direction.
Center = (100+20)/2 = 60 MPa, radius = √(40² + 30²) = 50 MPa. So σ1 = 60+50 = 110 MPa, σ2 = 60−50 = 10 MPa, τmax = 50 MPa, and θp ≈ 18.43°. The bracket's worst-case normal stress is 110 MPa — that's the number to compare against the material's yield strength.
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Structural engineer
Sizing a weld for maximum shear rather than normal stress
With σy negative (compression), center = 30 MPa and radius = √(50² + 40²) ≈ 64.03 MPa, giving τmax ≈ 64.03 MPa on the Max Shear tab. Because welds often fail in shear first, this value — not σ1 — is the governing design check here.
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Materials researcher orienting a strain gauge
Finding the rotation angle for a principal-stress reading
Since σx = σy and τxy = 0, the element is already in a principal state (radius = 0) — σ1 = σ2 = 50 MPa and θp reads 0°, meaning no rotation is needed to find a shear-free orientation. This is the degenerate case where Mohr's circle collapses to a single point.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
Mohr's circle is the classic graphical tool for stress transformation: instead of re-deriving the stress-transformation equations every time you need the normal and shear stress on a rotated plane, you plot one circle and read the answer off its geometry. This calculator does the same math numerically, returning the principal stresses, maximum shear stress, and principal angle directly from your σx, σy, and τxy inputs.
How the circle encodes every possible orientation
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Any 2D stress state can be described by three numbers: the normal stresses on two perpendicular faces (σx, σy) and the shear stress between them (τxy). As you imagine rotating the stress element, the normal and shear stress on its faces trace out a circle in σ-τ space — Mohr's circle. The circle is centered at (σx + σy)/2 on the σ-axis, with a radius of √(((σx − σy)/2)² + τxy²). Every point on the circle is a valid (σ, τ) pair for some rotation angle, which is why the circle contains all the information needed for stress transformation without repeating the trigonometry by hand.
Why the principal stresses matter
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The two points where the circle crosses the σ-axis — σ1 (maximum) and σ2 (minimum) — are the principal stresses: the normal stresses on the orientation where shear stress drops to zero. These are usually the numbers compared against a material's yield or ultimate strength in a failure check, since a material typically fails under the largest normal stress it experiences, regardless of which direction that stress happens to point in the original x-y coordinate system. See the companion Von Mises Stress Calculator for combining principal stresses into a single equivalent-stress failure criterion.
Maximum shear stress and the principal angle
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The circle's radius is also the maximum in-plane shear stress, τmax — it occurs 45° away from the principal planes, at the top and bottom of the circle. The principal angle, θp, is the physical rotation (measured from the x-axis) that aligns the stress element with its principal planes. Many failure modes — welds, adhesive joints, ductile-material yielding — are governed by shear rather than normal stress, so τmax can be the more important number depending on the material and the failure mode you're checking. Compare against the standalone Shear Stress Calculator for the simpler single-force case.
Limits of this 2D model
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This calculator handles plane stress (2D) — the common case for thin plates, shells, and surface-stress analysis where the out-of-plane stress is zero or negligible. A fully general 3D stress state has three principal stresses and needs three Mohr's circles (or a full eigenvalue analysis) to characterize completely. For a purely uniaxial or biaxial check without shear, see the Stress Calculator; for the full 3D failure criterion, use the Von Mises Stress Calculator's component-stress mode.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula behind Mohr's circle?+
The circle is centered at (σx + σy)/2 with radius √(((σx − σy)/2)² + τxy²). The principal stresses are the center plus or minus the radius (σ1,2 = center ± radius), the maximum shear stress equals the radius, and the principal angle is θp = ½·atan2(2τxy, σx − σy).
How do I find the maximum shear stress?+
Maximum shear stress equals the radius of Mohr's circle: τmax = √(((σx − σy)/2)² + τxy²). It's headlined on the Max Shear tab, alongside the principal stresses and angle in the stat grid for reference.
What does the principal angle actually mean?+
The principal angle θp is the rotation, in degrees from the x-axis, at which the stress element is oriented so that shear stress is zero and normal stress is at its maximum or minimum (principal) value. It's computed as half the angle to the τxy point on the circle: θp = ½·atan2(2τxy, σx − σy).
Does this calculator handle 3D stress states?+
No — this is a 2D (plane-stress) Mohr's circle, appropriate for thin plates, shells, and surface stress analysis where the out-of-plane stress is negligible. A full 3D stress state needs three principal stresses and a more general eigenvalue analysis.
What units does the calculator use?+
All three inputs (σx, σy, τxy) and all four outputs (σ1, σ2, τmax, θp) are treated in megapascals (MPa) for the stresses, with the principal angle in degrees. Enter values in any consistent stress unit and the outputs will be in that same unit.
What sign convention does this calculator use?+
Positive normal stress (σx, σy) means tension; negative means compression. Shear stress τxy follows the standard engineering convention where a positive value tends to rotate the element counterclockwise. The principal angle θp is measured counterclockwise from the positive x-axis.
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