Mechanics of Materials

Mohr's Circle Calculator

Enter a plane stress state and get principal stresses, maximum shear, and the full Mohr's Circle diagram — with an interactive angle slider to inspect the stress on any rotated plane.

Plane Stress Principal Stresses Stress Transformation
Principal Stresses Real-time
s1 (max)
—
MPa
s2 (min)
—
MPa
t max
—
MPa
s avg
—
MPa
θp (plane 1)
—
deg
θp (plane 2)
—
deg
θs (max shear)
—
deg

Mohr's Circle

Stress at Angle θ
σ(θ)
—
MPa
τ(θ)
—
MPa

Rotated Stress Element

About This Simulator

Mohr's Circle is a graphical method for visualizing how normal and shear stress on a material element change as the plane of interest is rotated. Given a plane stress state (sx, sy, txy), every possible combination of normal stress s and shear stress t — on every plane through that point — falls on a single circle in the s–t plane.

The circle is centered at s_avg = (sx + sy)/2 on the s-axis, with radius R = v[((sx - sy)/2)² + txy²]. The points where the circle crosses the s-axis give the principal stresses s1 and s2 — the maximum and minimum normal stresses at that point, occurring on planes where shear is zero. The top and bottom of the circle give the maximum in-plane shear stress, tmax = R, which always occurs 45° from the principal planes.

This calculator supports both sign conventions found in textbooks: the Materials convention (Callister, Shackelford), where positive shear plots upward, and the Mechanics convention (Hibbeler, Beer & Johnston), where positive shear plots downward. The physics — s1, s2, tmax, and the angles to reach them — is identical either way; only the diagram's orientation changes.

How to Use

  1. Step 1. Enter sx, sy, and txy for your plane stress state, in any consistent unit.
  2. Step 2. Pick the sign convention that matches your textbook or course — it only changes how the diagram looks, not the results.
  3. Step 3. Read s1, s2, tmax, and the principal/max-shear angles directly from the result cards.
  4. Step 4. Drag the θ slider to inspect the normal and shear stress on a plane rotated by any angle — both the circle and the rotated stress element update live.

Understanding the Results

s1 and s2 (Principal Stresses)

The maximum and minimum normal stresses at the point, occurring on perpendicular planes where shear stress is exactly zero.

tmax (Maximum In-Plane Shear Stress)

Equal to the circle's radius R. It occurs on planes rotated 45° from the principal planes, where the normal stress equals s_avg.

θp and θs (Plane Angles)

θp is the physical rotation angle (from the x-face) to reach the first principal plane; the second principal plane is 90° away. θs is the angle to the maximum shear plane, 45° from θp.

σ(θ) and τ(θ)

The normal and shear stress on a plane rotated by the angle θ you set with the slider, computed with the general stress transformation equations — not just read off the circle.

Frequently Asked Questions

Why do the circle's orientation change with the sign convention, but not s1, s2, or tmax?
The convention only flips which direction on the page counts as "positive t." It's a drawing choice, not a physics choice — the stress state itself, and therefore every principal quantity, stays the same.
What does it mean when the circle degenerates to a point?
That's a hydrostatic stress state: sx = sy and txy = 0. Every plane is a principal plane, there is no shear on any plane, and s1 = s2 = s_avg.
Can I use any unit for sx, sy, and txy?
Yes — MPa, ksi, psi, or anything else — as long as all three inputs use the SAME unit. The unit selector only relabels the fields; it does not convert your numbers.
What are the limitations of this calculator?
This tool covers plane stress (2D) only — it does not compute the third principal stress or the full 3D Mohr's Circle. It assumes linear elastic behavior and does not account for material failure criteria.