Mohr's Circle Calculator

Plane stress transformation — visualize principal stresses, maximum shear, and stress rotation

Stress State (Plane Stress)

Enter normal stresses σ_x, σ_y and shear stress τ_xy. Sign convention: τ positive CCW on x-face.

σ_x

σ_y

τ_xy

Unit

Stress Element

Mohr's Circle Formulas

For a plane stress state with normal stresses σx, σy and shear stress τxy:

C = \frac{σ _{x} + σ _{y}}{2} R = (\frac{σ _{x} - σ _{y}}{2})^{2} + τ_{x y}^{2}

σ_{1, 2} = C \pm R τ_{ma x} = R tan 2 θ_{p} = \frac{2 τ _{x y}}{σ _{x} - σ _{y}}

$C$ is the center of the circle (the average normal stress), $R$ is its radius, and $θ_{p}$ is the angle from the x axis to the principal plane.

How to Draw Mohr's Circle by Hand

Plot point X at (σx, τxy) and point Y at (σy, −τxy) on axes of σ (horizontal) and τ (vertical).
Join X and Y. The line crosses the σ axis at the center C.
Draw the circle through X and Y about C. Its radius is R.
Read σ1 and σ2 where the circle meets the σ axis, and τmax at the top and bottom of the circle.
The angle from CX to the σ axis is 2θp. Rotate the element by θp in the same direction.

Worked Example

σx = 100 MPa, σy = −50 MPa and τxy = 30 MPa (the calculator's default values, so you can compare directly):

Center: C = (100 + (−50)) / 2 = 25 MPa
Radius: R = √(75² + 30²) = √6525 ≈ 80.8 MPa
Principal stresses: σ1 = 25 + 80.8 ≈ 105.8 MPa and σ2 = 25 − 80.8 ≈ −55.8 MPa
Maximum in-plane shear: τmax = R ≈ 80.8 MPa
Principal angle: θp = ½ tan⁻¹(60 / 150) ≈ 10.9°

Mohr's Circle FAQ

What does Mohr's circle show?

Every possible combination of normal and shear stress on planes through a point under plane stress. Each point on the circle is one plane orientation; rotating the plane by θ moves 2θ around the circle.

How do I find the principal stresses?

They are where the circle crosses the σ axis: σ1 = C + R and σ2 = C − R, where C is the average normal stress and R is the radius. Shear stress is zero on the principal planes.

What is the maximum in-plane shear stress?

It equals the radius R of the circle and acts on planes 45° from the principal planes, where the normal stress equals the average stress C.

Is Mohr's circle in the FE exam handbook?

Yes. The FE Reference Handbook includes the plane stress transformation and Mohr's circle equations in the Mechanics of Materials chapter.