Mohr's Circle Calculator

Mohr's circle is a graphical representation of the state of stress at a point. Given the three components of a plane-stress state — the two normal stresses σx and σy and the shear stress τxy — every possible pair of normal and shear stresses (σx′, τx′y′) you could measure by rotating the element to a new angle θ lies on a single circle in the σ–τ plane. Christian Otto Mohr published the construction in 1882, and it is still the fastest way to see how stress transforms with orientation.

The circle has a centre on the σ-axis at the average normal stress σavg = (σx + σy)/2 and a radius R = √[((σx − σy)/2)² + τxy²]. Two facts follow immediately and for free: the principal stresses σ1 and σ2 are the points where the circle crosses the σ-axis (shear is zero there), and the maximum in-plane shear stress τmax equals the radius R, occurring on planes 45° from the principal directions. That is the whole point of the tool — reading σ1, σ2, τmax and the orientation θp straight off the geometry instead of memorising six transformation formulas.

This calculator is pre-solved the moment it loads with the classic textbook state σx = 80, σy = 20, τxy = 30 MPa (σ1 = 92.43, σ2 = 7.57, τmax = 42.43 MPa, θp = 22.5°), and — unlike the static images most free sites serve — the circle is live: drag the amber X′ point (or the θ slider) and watch the stresses on the physical element rotate with it.

The circle is just the geometric picture of the plane-stress transformation equations. For an element rotated counter-clockwise by an angle θ:

Quantity	Formula
Normal stress, x′-face	σx′ = (σx+σy)/2 + (σx−σy)/2·cos2θ + τxy·sin2θ
Normal stress, y′-face	σy′ = (σx+σy)/2 − (σx−σy)/2·cos2θ − τxy·sin2θ
Shear stress	τx′y′ = −(σx−σy)/2·sin2θ + τxy·cos2θ
Centre (average)	σavg = (σx + σy)/2
Radius	R = √[((σx−σy)/2)² + τxy²]
Major principal	σ1 = σavg + R
Minor principal	σ2 = σavg − R
Max in-plane shear	τmax = R (on the plane σ = σavg)
Principal angle	tan 2θp = 2τxy / (σx − σy) → θp to σ1
Max-shear angle	θs = θp − 45°

Two invariants make excellent sanity checks, and the calculator satisfies both exactly: the sum of the normal stresses is constant, σx′ + σy′ = σx + σy = σ1 + σ2 for any θ; and on the principal planes the shear vanishes (τx′y′ = 0), which is precisely why σ1 and σ2 sit on the σ-axis. Because the angle appears as 2θ in the formulas, a physical rotation of θ moves you 2θ around the circle — a 45° physical rotation is a 90° sweep on the circle, which is why the max-shear planes are 45° from the principal planes.

1. Enter your stress state — σx, σy and τxy in the boxes (MPa, or ksi with the SI ⇄ imperial toggle). Tension is positive; a positive τxy is the shear that tends to rotate the +x face counter-clockwise. Type a negative σ for compression.
2. Read the principal results instantly — the KPI strip shows σ1, σ2, the in-plane τmax, the principal angle θp, the absolute (out-of-plane) max shear and the von Mises equivalent stress. No "calculate" button: everything updates as you type.
3. Drag the circle. Grab the amber X′ point and sweep it around the circle, or use the θ slider. The physical stress element on the right rotates in lockstep and the live readout prints σx′, σy′ and τx′y′ for the current orientation — this is the intuition a static calculator can never give you.
4. Jump to the key orientations with one click — θp rotates the element to the principal planes (shear = 0), θs to the maximum-shear planes, 0° back to the reference element.
5. Try a preset — pure shear, uniaxial tension, biaxial tension–compression and equal-biaxial states are one click away and are great for building intuition about how the circle moves.
6. Run the code check — pick a real steel grade (fy is read from the same NBR 7007 / ASTM / EN / ZAR catalog that powers the CalcSteel editor) and a standard (NBR 8800 with γa1 = 1.10, or AISC 360 with φ = 0.90). The panel prints the utilization η = σvM / resistance with a PASS/FAIL verdict, and the σ1–σ2 plane draws the von Mises ellipse and Tresca hexagon with your state point on it — a design check, not just a number.
7. Go 3-D. Toggle triaxial (σz) and enter the out-of-plane principal stress: the tool draws the full three-circle construction, reports σI/σII/σIII, the absolute τmax and a true 3-D von Mises, and feeds them into the same code check — the genuine triaxial capability free 2-D tools defer.
8. Load a real solved section. After you analyse a model in the CalcSteel 3D editor, the "From your solved model" panel lists every solved member; pick a bar, an end and a fibre (extreme fibre or neutral axis) and the actual σ/τ the FEM solver produced there load straight into the circle — Mohr's circle becomes the inspection lens of the real solver.
9. Export or share for free — download SVG/PNG/CSV (no watermark), or hit Share link to copy a permalink that encodes the exact stress state, grade and standard — paste it in a thesis, a class or a message and it reopens pre-solved.

Tip: the pole P (origin of planes) is drawn on the circle. A line from the pole parallel to any physical plane meets the circle again at the stresses acting on that plane — a shortcut worth learning.

Principal stresses are the largest and smallest normal stresses at the point, and they act on planes that carry zero shear. Geometrically they are simply the two σ-axis crossings of the circle, σ1 = σavg + R and σ2 = σavg − R. Their orientation θp comes from tan 2θp = 2τxy/(σx − σy); the calculator reports it measured from the x-axis to the σ1 plane and reduces it to the (−90°, 90°] range engineers actually use. Principal stresses matter because most failure theories are written in terms of them — maximum-normal-stress (Rankine) for brittle materials, and the principal values feed straight into the distortion-energy and Tresca checks for ductile ones.

Maximum shear stress. The top and bottom of the circle give the maximum in-plane shear, τmax = R, acting on planes 45° from the principal directions and accompanied by the normal stress σavg on those same planes (a detail beginners often miss — the max-shear planes are not shear-only unless σavg = 0). For a plane-stress state the third principal stress is σ3 = 0, so there are actually three Mohr circles; the absolute maximum shear — the one that governs a Tresca yield check — is the radius of the largest of them, τabs = max(|σ1|, |σ2|, |σ1−σ2|)/2. When σ1 and σ2 have the same sign this out-of-plane value is larger than the in-plane τmax, so the calculator reports it separately.

The pole (origin of planes) is the elegant part. Once you locate the pole P on the circle, any line through P drawn parallel to a physical plane intersects the circle again exactly at the (σ, τ) acting on that plane — no angle-doubling arithmetic needed. It converts orientation questions into ruler-and-parallel-line constructions, which is why it is a staple of geotechnical and mechanics teaching. This tool plots and labels the pole so you can practise the method directly on your own numbers.

Most free Mohr tools stop at a von Mises number. This one closes the loop into an actual yield verification. Pick a real steel grade and the tool reads its yield strength fy from the same material catalog that drives the CalcSteel 3D editor and profile pages — NBR 7007 (MR-250, AR-350…), ASTM (A36, A572, A992…), EN 10025 (S235–S460), Australian and Indian grades, plus cold-formed ZAR / S___GD steels. The equivalent demand — von Mises σvM = √(σ1² − σ1σ2 + σ2²) in plane stress, or the full 3-D form in triaxial mode, with a Tresca option — is then compared against the code design resistance:

Standard	Design resistance	Utilization
NBR 8800:2008 (§5.4.2.2)	fy / γa1, γa1 = 1.10	η = σvM / (fy/γa1)
AISC 360-22	φ · fy, φ = 0.90	η = σvM / (φ·fy)

The result is a PASS/FAIL verdict and a utilization ratio η you can quote directly. Alongside it, the σ1–σ2 design plane draws the von Mises ellipse (distortion-energy locus at fy), the inscribed Tresca hexagon and the dashed design-resistance locus, with your state point plotted on it — if the point sits inside the design curve, the section is safe. Choosing a real profile and grade turns a commodity transformation calculator into a genuine design tool.

Load it from the real solver. The headline differential: after you analyse a structure in the CalcSteel 3D editor, this tool can read the solver's element-end forces (N, V, M, T) for any member, combine them with the section properties of the assigned profile, and place the resulting (σ, τ) at the fibre you choose — the extreme fibre (maximum bending stress, zero shear) or the neutral axis (zero bending, maximum shear) — directly onto the circle. No free competitor can do this because none of them own a solver; here, Mohr's circle becomes the inspection lens of your actual FEM model.

Sign conventions are the number-one source of confusion with Mohr's circle, so here is exactly what this tool does:

• Normal stress σ: tension is positive (plotted to the right on the σ-axis); compression is negative (to the left).
• Shear stress τxy: positive means the shear on the +x face points in the +y direction — equivalently, it tends to rotate the element counter-clockwise. This is the Gere/Hibbeler convention.
• Plotting: point X = (σx, −τxy) and point Y = (σy, +τxy) are plotted at opposite ends of a diameter. Plotting the x-face with −τxy is the standard trick that makes a counter-clockwise physical rotation θ correspond to a counter-clockwise sweep of 2θ on the drawn circle, so the circle and the element always turn the same way.
• Angle θ: measured counter-clockwise from the reference x-axis. The reported θp is the angle to the σ1 plane; θs = θp − 45° is the angle to the max-shear plane.
• Physical vs circle angle: remember the factor of two — θ on the element is 2θ on the circle.

If your textbook plots shear positive downward or measures angles clockwise, the numbers (σ1, σ2, τmax, |θp|) are identical; only the picture is mirrored. The transformation equations this calculator evaluates are convention-independent for the magnitudes.

The results are closed-form, not numerical — plane-stress transformation is exact algebra, so σ1, σ2, τmax and θp are computed to machine precision directly from σx, σy and τxy. There is no meshing, iteration or tolerance involved; the only rounding is in the display (the internal math keeps full double precision). Every value was cross-checked against the transformation equations: rotating the element to θp returns τx′y′ = 0 and σx′ = σ1, rotating to θs returns σx′ = σavg and |τx′y′| = τmax, and σx′ + σy′ stays equal to σx + σy at every angle — all satisfied to < 1 × 10⁻³.

Assumptions and scope: by default this is a plane-stress state — σz = τxz = τyz = 0, the right model for thin plates, free surfaces and the outer fibre of a beam or shaft — and the reported absolute max shear and von Mises already account for the σ3 = 0 circle. Switch on triaxial mode to supply an out-of-plane principal σz directly (still assuming z is a principal direction, τxz = τyz = 0): the tool then draws all three Mohr circles and computes the true 3-D von Mises √[((σI−σII)²+(σII−σIII)²+(σIII−σI)²)/2] and τabs = (σI−σIII)/2. The code check is real: the equivalent stress is compared against fy of a catalog steel grade with the standard's resistance factor (NBR 8800 γa1 = 1.10; AISC 360 φ = 0.90) to give a PASS/FAIL utilization. To build the governing stress from several load cases automatically, or to run the full member/connection verification, continue in the CalcSteel 3D editor — from which you can also load a solved section straight into this tool.