How this instrument works
Stress at a point inside a loaded solid is not one number — it is a full tensor, and the normal and shear stress you would measure on any small plane through that point depends on which way the plane is cut. Mohr's circle is what results when you plot every possible cut: put normal stress on the horizontal axis and shear stress on the vertical axis, and as the plane's orientation sweeps through 360°, the point (σ, τ) traces a full circle once for every 180° of physical rotation. Otto Mohr worked this relationship out in 1882 as a drafting technique; this page reproduces it with algebra instead of a compass and protractor.
The center, C = (σx + σy) ⁄ 2, is simply the average of the two normal stresses you supplied — and, not coincidentally, it is the one quantity in the whole stress-transformation family that does not change as the cutting plane rotates. The radius, R = √(((σx−σy) ⁄ 2)²+τxy²), is a Pythagorean combination of the half-difference of the normal stresses and the shear stress, and it equals the single largest shear stress obtainable at that point, in any direction. Where the circle crosses the shear-free axis, at C + R and C − R, shear has vanished entirely and only a normal stress remains: the maximum and minimum principal stresses, σ1 and σ2.
This page solves the plane-stress, two-dimensional case, which is exact wherever the third direction genuinely carries no stress — a thin plate loaded in its own plane, or the free surface of almost any part. It is not automatically valid for a point buried deep inside a thick section, where a nonzero out-of-plane stress introduces a third principal stress this circle cannot show; combining all three correctly needs the full three-circle 3D construction or a tensor eigenvalue solve. A subtler trap is sign convention: swap σx and σy, or flip the sign on τxy, and the arithmetic still runs and the circle still closes — it simply describes a different physical rotation than the one on your free-body diagram.
- Enter the stress on the x-face into Normal stress, x-direction — tension positive, compression negative.
- Enter the stress on the y-face into Normal stress, y-direction, using that same sign convention.
- Enter the shear stress acting on that pair of faces into Shear stress, xy-plane.
- Read Mohr's circle center and Mohr's circle radius (= max shear stress) — the circle's midpoint and half-width.
- Read Maximum principal stress and Minimum principal stress, the shear-free extremes the circle predicts.
Worked example — 100 MPa, 40 MPa, and 30 MPa shear
Take a plate carrying 100 MPa tension in the x-direction, 40 MPa tension in the y-direction, and 30 MPa of shear on that face — a plausible reading near a bolted flange or a welded pressure-vessel seam. The center falls at C = (100 + 40) ⁄ 2 = 70 MPa. The radius is R = √(((100 − 40) ⁄ 2)² + 30²) = √(30² + 30²) = √1800 = 42.4264 MPa.
The principal stresses sit at the two ends of the circle: σ1 = 70 + 42.4264 = 112.4264 MPa, and σ2 = 70 − 42.4264 = 27.5736 MPa. Neither original input — 100 MPa nor 40 MPa — is the true peak stress; the shear term pushes the real maximum past 112 MPa, which is precisely the figure a yield check against, say, a 250 MPa allowable needs, not the 100 MPa someone might mistakenly read straight off the input field.
Questions
What do the center and radius of Mohr's circle actually represent?
The center is the average of the two normal stresses, σx and σy — the part of the stress state that does not depend on orientation. The radius is the maximum in-plane shear stress obtainable at that point, reached on planes at 45° to the principal directions. Together they fix the whole circle, and every possible cut through that spot in the material lands on some (normal, shear) pair sitting on it.
Why are the principal stresses always σ1 = C + R and σ2 = C − R?
Because the principal stresses are defined as the normal stresses on the one pair of perpendicular planes where shear is exactly zero, and those are precisely where the circle crosses the shear-free horizontal axis — one radius to the right of center, one radius to the left. No plane through that point can carry a normal stress higher than C + R or lower than C − R; that is the geometric meaning of the radius.
Does this handle compressive or negative stress values?
Yes — enter negative numbers into Normal stress, x-direction or y-direction to represent compression, and the same arithmetic applies unchanged. With σx = −50 MPa, σy = 20 MPa, and τxy = 15 MPa, the center simply comes out at −15 MPa; the formulas for radius and principal stress do not switch convention, only the sign of the input does.
Is this the same as a 3D Mohr's circle?
No. This solves the 2D, plane-stress case with two normal stresses and one shear stress, which is exact at a free surface or across a thin plate. A point carrying a nonzero third normal stress needs the full three-circle 3D construction, and its intermediate principal stress cannot be recovered from this page's three inputs alone.
Why does swapping σx and σy not change the principal stresses?
The formulas for C and R use (σx + σy) and (σx − σy)², and both are unaffected by swapping the two labels — squaring erases the sign of the difference. Swap σx and σy and σ1 and σ2 come out identical in magnitude; only which physical face each principal direction points toward changes, not the circle itself.
What is a real failure check that uses σ1 and σ2 directly?
The maximum-normal-stress (Rankine) criterion compares σ1 directly against a material's tensile yield or ultimate strength — a stress state that yields 112 MPa from inputs each safely under 100 MPa is exactly the kind of surprise this circle is built to catch before it becomes a fatigue crack. The Tresca and von Mises criteria used for ductile metals also build on σ1 and σ2 rather than the raw σx, σy, τxy triplet.