A weld inspector points at a cracked bracket and asks a simple-sounding question: which way was the stress actually pulling when it broke? The crack runs at an angle that matches neither the bolt line nor the load direction. The answer is that the stress at that point was not aligned with the geometry at all. Every loaded point in a part carries normal and shear stress whose values depend on the direction you choose to look — and somewhere there is a direction where the shear vanishes and the normal stress reaches its peak.

Finding that direction, and the stresses that go with it, is what Mohr's circle does. This article explains the construction, works a full example by hand, and flags the sign conventions that trip people up.

Why this calculation matters

Real components rarely see pure tension or pure shear. A pressurized pipe wall, a loaded gear tooth, a shaft under combined torsion and bending — each carries a mix of normal and shear stress on whatever face you pick. Failure theories, though, are written in terms of principal stresses: the maximum and minimum normal stresses, acting on the planes where shear is zero. To apply a yield criterion or a fatigue model you first have to transform whatever stress components you measured or computed into those principal values.

You could memorize the transformation equations and grind through trigonometry every time. Mohr's circle gives you the same answer geometrically, and the geometry carries insight the algebra hides. You see at a glance how far the stress state sits from yield, where the maximum shear plane lies, and how the whole state rotates as you turn the element. For checking a finite-element result or sketching a quick estimate on a notepad, it is hard to beat.

The core method

Otto Mohr showed in 1882 that the stress transformation equations describe a circle when you plot normal stress on the horizontal axis and shear stress on the vertical axis. Given the plane-stress components sigma_x, sigma_y, and tau_xy, the circle is fully defined by two numbers.

The centre always sits on the normal-stress axis at the average of the two normal stresses:

c = (sigma_x + sigma_y) / 2

The radius is the distance from that centre to the plotted stress point:

R = sqrt( ((sigma_x - sigma_y)/2)^2 + tau_xy^2 )

Once you have the circle, every result you want is a feature of it. The principal stresses are where the circle crosses the normal-stress axis, because that is where shear is zero:

sigma_1 = c + R
sigma_2 = c - R

The maximum in-plane shear stress is simply the radius — the highest point on the circle:

tau_max = R

And the orientation of the principal planes comes from the angle of the line from the centre to the stress point:

tan(2 * theta_p) = 2 * tau_xy / (sigma_x - sigma_y)

The factor of two is the key idea: angles in the real component are doubled on the circle. A 90-degree rotation of the physical element is a full 180-degree trip around the circle, which is exactly why sigma_x and sigma_y always sit on opposite ends of a diameter.

A worked example

Consider a point in a loaded part with plane-stress components sigma_x = 80 MPa, sigma_y = 20 MPa, and shear tau_xy = 30 MPa.

Step 1 — centre of the circle.

c = (sigma_x + sigma_y) / 2 = (80 + 20) / 2 = 50 MPa

Step 2 — radius.

R = sqrt( ((80 - 20)/2)^2 + 30^2 )
R = sqrt( 30^2 + 30^2 )
R = sqrt(1800) = 42.4 MPa

Step 3 — principal stresses.

sigma_1 = c + R = 50 + 42.4 = 92.4 MPa
sigma_2 = c - R = 50 - 42.4 = 7.6 MPa

Step 4 — maximum in-plane shear stress.

tau_max = R = 42.4 MPa

Step 5 — principal-plane angle.

tan(2 * theta_p) = 2 * tau_xy / (sigma_x - sigma_y) = 60 / 60 = 1
2 * theta_p = 45 degrees
theta_p = 22.5 degrees

So the original 80/20/30 stress state is equivalent to a peak tension of 92.4 MPa and a minimum of 7.6 MPa, acting on a pair of planes rotated 22.5 degrees from the x-axis. The largest shear the point ever sees is 42.4 MPa, on planes 45 degrees from those principal directions. Same physical state, three different but equivalent descriptions — and Mohr's circle ties them together in one picture.

Common mistakes

Getting the shear sign wrong. The plotting convention for tau_xy is not universal. A common choice plots shear that tends to rotate the element clockwise as positive (or negative) depending on the textbook. Pick one convention and use it consistently, or the circle and the rotation direction will disagree.

Forgetting the doubled angle. Angles on Mohr's circle are twice the physical angle. A principal angle of 22.5 degrees in the part shows up as 45 degrees on the diagram. Reading the circle angle straight onto the component is a frequent error.

Treating it as a full 3D picture. The standard construction is for plane stress. The third principal stress is zero, and the true maximum shear in three dimensions may be larger than the in-plane R if both in-plane principal stresses share the same sign. For triaxial states you need all three Mohr circles.

Mixing up which root is sigma_1. By convention sigma_1 is the algebraically larger principal stress. With compressive values this matters: -10 MPa is larger than -50 MPa. Order them on the number line, not by magnitude.

Try the interactive NovaSolver calculator

Drawing the circle by hand is a good way to learn it, but for everyday work it helps to see it update live. The Mohr's Circle Stress Analyzer on NovaSolver takes the three stress components sigma_x, sigma_y, and tau_xy and draws the circle in real time, returning the principal stresses, the maximum shear stress, the principal angle, and the circle's centre and radius. You can drag points directly on the circle to set a stress state, toggle the principal-stress planes, and even animate the element as it rotates — a fast way to build a feel for how a stress state transforms.

Related calculators

• Stress concentration — find the local peak stress at holes, fillets, and notches before you transform it.
• Beam shear stress distribution — see how shear varies through a cross-section, a common source of the tau_xy you feed into Mohr's circle.
• Pressure vessel — for hoop and axial stress in cylindrical shells, a classic two-component state ready for principal-stress analysis.

The full set lives in the structural tools hub.

Closing note

Mohr's circle endures because it turns a piece of algebra into something you can see. Once the construction is in your hands, principal stresses, maximum shear, and the orientation of the critical planes all become measurements off a single diagram. Remember the two anchors: the centre is the average normal stress and the radius is the maximum shear. Keep your sign convention straight, respect the doubled angle, and the circle will give you a stress picture you can trust at a glance.