Mohr Circle Interactive Calculator

Calculating principal stresses and maximum shear stress from a combined stress state is a routine job in structural analysis, but it's also easy to slip up on signs or conventions. If you need to find principal stresses, transformed stresses at an angle, or maximum shear from a set of normal and shear stresses, this Mohr Circle Interactive Calculator will do the basic number crunching for you. These calculations come up in most engineering fields wherever loads don't act along obvious axes. If you want to check or understand the math by hand, you'll find the formulas, a real shaft example, and a detailed breakdown and FAQ below.

What is a Mohr Circle?

Mohr's circle is a diagram that helps you see how the stress at a point in a material changes as you consider different planes through that point. If you know the normal and shear stresses, you can use the circle to quickly read off the maximum (and minimum) possible normal stresses and figure out the angles where they occur.

Simple Explanation

The stress at a point isn't just one value—it's different depending on what direction or plane you look at. Mohr’s circle is a fast way to capture all those possible stresses: just plot the input values, and you’ll see the worst-case scenarios right away. The far right of the circle is the highest normal stress; the very top is the peak shear. No guesswork, and you skip the trig.

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Stress Element Diagram

Interactive Mohr Circle Calculator

How to Use This Calculator

1. Pick what you want to calculate from the dropdown (principal stresses, stresses at an angle, max shear, circle info, or the reverse case).
2. Input the normal stresses σ_x and σ_y (in MPa).
3. Add the shear stress τ_xy (and rotation angle if needed).
4. Hit Calculate. The results will output below.

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Governing Equations

The formulas here let you work out principal stresses from any 2D stress state.

The following equation is how you get the principal angle.

The formula for maximum in-plane shear stress is shown below.

To get the transformed (rotated) stresses, use the equations below.

Mohr circle center and radius are found as follows.

Simple Example

Given: σ_x = 100 MPa, σ_y = 40 MPa, τ_xy = 30 MPa

Circle center: σ_avg = (100 + 40)/2 = 70 MPa

Circle radius: R = √[(30)² + (30)²] = √(900 + 900) = √1800 ≈ 42.43 MPa

Principal stresses: σ₁ = 70 + 42.43 = 112.43 MPa, σ₂ = 70 − 42.43 = 27.57 MPa

Maximum shear stress: τ_max = 42.43 MPa

Theory & Practical Applications

Fundamentals of Stress Transformation

The Mohr circle, introduced by C.O. Mohr in 1882, is a practical solution for finding what stress exists on any arbitrary plane through a material point. Practically, you’re working with the fact that certain combinations of stresses don’t change no matter how you rotate your coordinate axes. The sum σ_x + σ_y is always the same; it’s called a stress invariant. On the circle, this sum gives you the center of the circle along the σ axis. The radius measures how much those stress values can shift as you pick different angles.

Mohr’s circle saves you from cranking through the trig every time you want to look at another stress orientation. Every location on the circle matches a possible stress combination for some orientation. But here’s the catch: if you rotate the actual element in space by θ, the corresponding location on Mohr’s circle rotates by 2θ, because the transformation equations have cos(2θ) and sin(2θ) terms in them. So, things move around the circle twice as fast as you’d expect compared to plain geometry.

Construction Methodology and Sign Conventions

To keep sign errors from tripping you up, follow standard conventions: normal stresses go on the horizontal (right is positive tension, left is negative compression); shear stresses go on the vertical. For shear, use the rule that positive shear (producing a clockwise moment on the face) is plotted downwards. This might seem reversed compared to most graphs, but it’s set up so your angle rotations on the object match movements around the circle without confusion.

Start by plotting two points: A at (σ_x, τ_xy), and B at (σ_y, -τ_xy). These always represent stresses on perpendicular faces of your element. Draw a line between these—this is your diameter. The center is halfway between, at (σ_x + σ_y)/2 and zero shear. The length from center to A or B is your radius. Where the circle meets the horizontal axis, that’s your maximum (right) and minimum (left) principal stresses. Far up or down, the radius gives you the maximum possible shear—this happens where the normal stress difference vanishes.

Critical Engineering Limitations

Mohr’s circle as shown here only applies to plane stress—the kind of cases you see in thin plates or surface layers, where the out-of-plane stress is pretty much zero. If your problem has nonzero σ_z (like thick pipes or triaxial soil test samples), you’ll need three Mohr circles (one for each unique stress pair). The circle connecting the highest and lowest principal stresses gives you the true maximum shear in the material, and this is often higher than just the in-plane shear you get from a 2D analysis. Don’t use the plane stress Mohr circle blindly in cases where out-of-plane effects might be large; it may miss the critical shear in thick components or confined environments.

What you get out of Mohr’s circle depends on what failure theory you plan to use. For glass or ceramics (brittle materials), the maximum normal stress (which you see as far right on the circle) matters most. For metals and ductile stuff, shear is usually more important. Tresca's criterion says yielding kicks off when the maximum shear stress equals half of the yield strength, so the Mohr circle’s radius (for the biggest circle if using 3D) gives you a number to check against. Von Mises is a bit more involved, but again, the circle gives you the principal stresses you need for the calculation. Always use the criterion that matches the material and mode you’re concerned about.

Industrial Applications Across Disciplines

In geotech work, Mohr's circle is used together with the Mohr-Coulomb line to check slope or excavation stability. The stress circle must not touch the failure envelope τ_f = c + σ tan(φ). Take a vertical cut: if you lose horizontal support, σ_x may drop to zero but σ_y (vertical) remains high. Draw the new Mohr circle and see if it stays inside the failure line. If not, expect slip or collapse.

In pressure vessel calculations, it's common to see Mohr’s circle built from hoop and longitudinal stresses. For thin-wall cylinders with negligible radial stress, the circle radius is half the difference between hoop and longitudinal. Watch out: the true maximum shear occurs through the wall, not just in the cylinder’s surface plane. Check the full 3D stress if wall thickness is comparable to diameter, or if you need extreme accuracy (for example, in deep submarine hulls or high-pressure reactors).

Worked Example: Shaft Under Combined Torsion and Bending

Suppose you have a solid shaft, 50 mm in diameter, with a 1.8 kN·m bend and 1.2 kN·m torque. To find critical principal stresses and orientation:

Step 1: Calculate stresses
Section modulus: S = πd³/32 = π(0.05)³/32 = 1.534 × 10⁻⁵ m³
Polar moment: J = πd⁴/32 = π(0.05)⁴/32 = 6.136 × 10⁻⁷ m⁴
Bending: σ_x = 1800 / 1.534e-5 ≈ 117.35 MPa
Torsion: τ_xy = 1200 × 0.025 / 6.136e-7 ≈ 48.90 MPa
Normal stress perpendicular to bending (σ_y) is zero

Step 2: Mohr circle
Center: (117.35 + 0)/2 = 58.675 MPa
Radius: sqrt(58.675² + 48.90²) ≈ 76.38 MPa

Step 3: Principal stresses
σ₁ = 58.675 + 76.38 = 135.06 MPa
σ₂ = 58.675 - 76.38 = -17.70 MPa
Principal angle: tan(2θ) = (2 × 48.90) / 117.35 = 0.8333
2θ = arctan(0.8333) = 39.81° ⇒ θ = 19.91° from shaft axis

Step 4: Max shear
Max shear = radius = 76.38 MPa, at 64.91° from shaft axis
Normal stress at this orientation is the center value: 58.675 MPa

Why it matters: Combined loading can push your peak stress well beyond what you’d calculate from bending alone; here, you get a 15% increase. The most loaded plane is not perpendicular to the shaft—the critical stresses act at an angle. Use the actual principal values (not just original bending or torsion numbers) if you want to avoid under-design in shafts or axles.

Advanced Considerations in Pole Method

The pole method offers a shortcut for finding the stress on any plane graphically. Once you’ve drawn the Mohr circle, find the pole by drawing a line from (σ_x, τ_xy) parallel to one face of your physical element. Wherever this hits the circle, you’ve located the pole. Then, any line through the pole and the circle gives you the stress for planes parallel to that orientation—no recalculating or angle-doubling needed, which speeds up work when you need many orientations evaluated quickly.

The pole unlocks a direct mapping between physical and graphical orientation—no need to halve or double angles. That’s useful for hand sketches, rapid checks, or when working without a calculator. It works best for plane stress, and is less suited if you need numeric accuracy (then, just do the math).

Additional Engineering Insights

Always check that σ_x + σ_y matches σ₁ + σ₂; if not, there's an error. On the circle, points directly opposite each other (180° apart) represent perpendicular planes and should show shear stresses equal in size but opposite in sign. If τ_xy (input) is zero, your coordinate system is already aligned with the principal directions and the circle shrinks to a line—good for a sense check. For more calculator options—including von Mises and Tresca—you can find other tools at FIRGELLI's engineering calculator hub.

Frequently Asked Questions

📹 Video Walkthrough — How to Use This Calculator

📹 Video Walkthrough — How to Use This Calculator