A stress state is not a number. The same loaded point in a shaft, a bracket or a weld carries a different normal stress and a different shear stress on every plane you can cut through it, and the numbers you happen to measure depend entirely on how you oriented your axes. Mohr's circle is the graphical device that makes that whole family of values visible at once: plot normal stress horizontally and shear stress vertically, and the stresses on every possible plane trace out a single circle. Rotating the element by an angle moves you twice that angle round the circle.
This Mohr's circle calculator from Arb Digital does the arithmetic that the drawing encodes. Give it the three numbers that define a plane-stress element and it returns the principal stresses, the orientation of the planes they act on, the largest shear stress the element sees, and the stresses on any rotated plane you nominate. It is a stress-transformation and teaching tool, not a structural design: it tells you what the stress state is, never whether a part is adequate. That judgement belongs to a licensed engineer working to the governing code.
What This Mohr's Circle Calculator Does
The input is a plane-stress element: two normal stresses acting on perpendicular faces and one shear stress acting on both. Plane stress means the third principal stress is zero, which is the correct assumption for a thin plate loaded in its own plane, a free surface anywhere on a body, or a thin-walled shell away from its supports. Since the highest stresses in most components occur at a free surface, plane stress covers a large fraction of real analysis.
From those three numbers the tool computes the circle's centre, which is the average of the two normal stresses, and its radius, which is the maximum in-plane shear stress. Adding and subtracting the radius from the centre gives the two principal stresses — the largest and smallest normal stresses at that point, acting on the two planes where the shear stress vanishes entirely. The principal angle says how far you would have to rotate the element to land on them.
Three further outputs earn their place. The von Mises equivalent stress collapses the state into the single number that ductile yield criteria are written against. The absolute maximum shear stress accounts for the out-of-plane principal stress of zero, which frequently governs even though it never appears on the in-plane circle. And the query angle lets you read the stresses on a specific plane — a weld line, a bond line, a bedding plane — rather than only the extremes.
How to Use It
- Fix your axes before you type anything. The three inputs are meaningless without an orientation. Sketch the element, label which face is x, and take every number from that sketch.
- Enter normal stresses with signs. Tension is positive and compression is negative. A compressive stress entered as a positive number produces a completely different circle in a completely different part of the plane.
- Enter the shear stress once. Complementary shear means the value on the y face equals the one on the x face, so the tool needs only one figure. The sign convention used here treats shear as positive when it acts in the positive y direction on the positive x face.
- Set a query angle if a particular plane matters. Angles are measured anticlockwise from the x axis in the physical element, and the circle rotates at double that rate.
- Read the note, not only the grid. It states which principal stress is which, whether the state is uniaxial, biaxial or pure shear, and which shear value actually governs.
The Formulas and a Worked Example
The centre of the circle sits at σₐᵛₕ = (σₓ + σₖ)/2 and the radius is R = √[((σₓ − σₖ)/2)² + τₓₖ²]. The principal stresses are σ₁ = σₐᵛₕ + R and σ₂ = σₐᵛₕ − R, and the maximum in-plane shear stress is R itself. The principal angle follows from tan(2θₚ) = 2τₓₖ / (σₓ − σₖ), which the calculator evaluates with a two-argument arctangent so that the quadrant is never ambiguous. Stresses on a plane rotated by θ come from σ′ = σₐᵛₕ + ((σₓ − σₖ)/2)cos2θ + τₓₖsin2θ and τ′ = −((σₓ − σₖ)/2)sin2θ + τₓₖcos2θ.
Work the default. With σₓ = 80, σₖ = 20 and τₓₖ = 30 MPa, the centre is at 50 MPa and the radius is √(30² + 30²) = 42.43 MPa. So σ₁ = 92.43 MPa, σ₂ = 7.57 MPa, and the maximum in-plane shear is 42.43 MPa. The principal angle is half of arctan(60/60), which is 22.5 degrees. Note what happened to the peak normal stress: it rose from the 80 MPa you could see on the x face to 92.43 MPa on a plane rotated 22.5 degrees, an increase of more than fifteen per cent that no single face of the original element revealed. That hidden increase is the entire reason the transformation is worth doing. The Engineering LibreTexts chapter on stress transformations derives the same relations and constructs the circle step by step.
The Shear Stress That Does Not Appear on the Circle
This is the single most common serious error in plane-stress work. The circle drawn from σₓ, σₖ and τₓₖ is the in-plane circle, and its radius is the largest shear stress on any plane perpendicular to the sheet. But plane stress has a third principal stress, equal to zero, acting out of plane. Sorting all three principal stresses and taking half the difference between the largest and the smallest gives the absolute maximum shear stress, and it is not always the in-plane radius.
When the two in-plane principal stresses have opposite signs, zero lies between them and the in-plane circle is the largest of the three; the in-plane value governs. When both are positive, or both negative, zero lies outside the pair, and one of the out-of-plane circles is larger. Take equal biaxial tension of 60 MPa: the in-plane circle collapses to a point of zero radius, so there is no in-plane shear at all, yet the absolute maximum shear stress is 30 MPa on a plane at 45 degrees to the sheet. A ductile material yields on shear, so treating the in-plane figure as the answer would predict a component that never yields. The calculator reports both values and says which one is larger.
Why the Circle Turns at Twice the Element's Angle
The doubling catches people out constantly, and it is not an arbitrary convention. Stress is a second-rank tensor, and its transformation involves the product of two direction cosines rather than one. Squaring a cosine produces a double-angle term, so a physical rotation of θ degrees appears on the circle as 2θ, as the tensor transformation derivation in Roylance's Mechanics of Materials sets out. The practical consequence is that the two principal planes, which are 180 degrees apart on the circle, are only 90 degrees apart in the metal. Perpendicular planes always sit at opposite ends of a diameter.
The same doubling explains why the planes of maximum shear lie exactly 45 degrees from the principal planes rather than 90. On the circle, the top and bottom of the circle are a quarter-turn from the horizontal diameter, so 2θ = 90 degrees and θ = 45 degrees. This is why a mild-steel tensile specimen tears along a cone inclined at roughly 45 degrees to the pull, and why Lüders bands appear at that angle: the material is failing on the plane of greatest shear, not the plane of greatest tension. A brittle material, which fails on tension instead, breaks square across.
What Plane Stress Assumes, and When It Stops Holding
Everything here rests on three assumptions. The third principal stress is zero; the material is continuous and behaves as one piece; and the stress state is homogeneous over the small element being analysed. Each fails in an identifiable way.
The zero out-of-plane stress fails inside thick sections. A thick plate constrained through its thickness develops a plane-strain condition instead, where the out-of-plane strain rather than the out-of-plane stress goes to zero, and the third principal stress becomes Poisson's ratio times the sum of the other two. That changes the von Mises figure and the absolute shear, and it is exactly why fracture toughness measured on thin sheet is higher than on a thick block. The Poisson's ratio calculator covers the coupling that drives it.
Homogeneity fails at stress raisers. A hole, a fillet, a keyway or a weld toe produces a stress that varies sharply over a very short distance, and a nominal stress computed from load over area is not the stress at the notch root. The transformation on this page is exact for whatever state you feed it, but if you feed it a nominal stress where a concentration factor should have been applied first, the principal stresses will be right for a state that does not exist. Anisotropy fails the material assumption: composites, rolled sheet and timber have direction-dependent strength, so a plane can be critical because of grain rather than because of stress magnitude.
Reading von Mises Against Tresca
Once you have principal stresses, the usual next step is a yield criterion, and the two in common use disagree. Von Mises, appropriate for ductile metals, gives an equivalent stress of √(σ₁² − σ₁σ₂ + σ₂²) in plane stress. Tresca, more conservative and simpler, says yield occurs when the absolute maximum shear stress reaches half the tensile yield strength, so its equivalent stress is the largest principal minus the smallest.
For the default state, von Mises gives 88.88 MPa and Tresca gives 92.43 MPa, a gap of about four per cent. In pure shear the gap is at its widest: Tresca predicts yield at a shear stress of half the tensile yield, von Mises at 0.577 of it, so Tresca is roughly fifteen per cent conservative there. Neither is a licence to declare a part safe. This page publishes no allowable-stress table and no material strength data of any kind, for the same reason the live breaker size calculator publishes no ampacity table: those figures are code-controlled, edition-specific and jurisdiction-specific, and a number typed onto a web page is exactly the wrong place to get them. Take strengths from the governing standard and the material certificate, and let a qualified engineer apply them.
How This Differs From the Adjacent Arb Digital Tools
This page transforms a two-dimensional stress state between orientations and finds its extremes; nothing else on the site does that. The stress and strain calculator handles a single axial stress with no shear and no transformation. The Young's modulus calculator works on the stiffness relation rather than the orientation question. The hoop stress calculator generates the biaxial state a pressure vessel wall carries, which you can then bring here to find its principal values. The section modulus calculator gives the bending stress on a cross-section before any transformation, and the factor of safety calculator compares a stress you already have against a strength you already have. The column buckling calculator deals with a stability limit that stress transformation cannot see at all.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering compression as a positive number — the sign is load-bearing information. A compressive 40 typed as +40 puts the circle on the wrong side of the origin and inverts which principal stress is critical.
- Taking the in-plane shear as the maximum — when both principal stresses share a sign, the out-of-plane circle is larger and it is the one that governs ductile yield.
- Halving or doubling the angle by accident — the circle turns through 2θ while the element turns through θ. Read principal angles from the physical element, not from the circle's own scale.
- Transforming a nominal stress at a notch — apply the stress concentration factor before the transformation, not after. The circle is exact for the state you give it and cannot know the state was wrong.
- Treating a von Mises number as a verdict — it is one input to a design check that also involves the governing standard, load factors, fatigue and the actual material. It is not a pass mark.
Related Free Tools From Arb Digital
Build the stress state first with the stress and strain calculator, the hoop stress calculator or the section modulus calculator, then bring it here. Material behaviour is covered by the Young's modulus calculator and the Poisson's ratio calculator, and margin arithmetic by the factor of safety calculator and the margin of safety calculator. For torsion rather than plane stress, use the polar moment of inertia calculator. The full free online tools hub lists everything Arb Digital has published.
Frequently Asked Questions
A graphical representation of a two-dimensional stress state in which normal stress is plotted horizontally and shear stress vertically. Every plane through the point appears as one point on the circle, so the extremes of normal and shear stress can be read off directly instead of being searched for algebraically.
The largest and smallest normal stresses at a point, acting on the two perpendicular planes where the shear stress is exactly zero. They are the horizontal extremes of the circle, and they are the values most yield and fracture criteria are written in terms of.
Because stress is a second-rank tensor and its transformation involves a product of two direction cosines, which generates double-angle terms. A physical rotation of thirty degrees moves you sixty degrees around the circle, and perpendicular planes end up at opposite ends of a diameter.
Only for the in-plane planes. Plane stress has a third principal stress of zero, and when both in-plane principal stresses share a sign, an out-of-plane circle is larger. The absolute maximum shear is half the difference between the largest and smallest of all three principal stresses.
The faces you entered are already the principal planes, so the principal stresses equal the normal stresses you typed and the principal angle is zero. If both normal stresses are also equal, the circle shrinks to a point, the state is isotropic in plane, and no orientation is special.
It changes the sign of the principal angle but not the magnitudes of the principal stresses or the maximum shear. Flipping the shear sign reflects the circle about the horizontal axis, which moves where the principal planes are without moving how large the stresses are.
No. The tool assumes the out-of-plane stress is zero, which is correct at a free surface or in a thin sheet loaded in its plane. Thick sections approach plane strain, where the out-of-plane stress is Poisson's ratio times the sum of the other two, and a general three-dimensional state needs the full characteristic equation.
This tool is provided for educational and preliminary analysis use. It performs a stress transformation only and is not a structural design, is not stamped, and does not replace a licensed engineer working to the governing code. It publishes no allowable stress or material strength values.
