Mohr's Circle Calculator — Principal Stresses, Maximum Shear, von Mises & Mohr's Circle Diagram | Metric & Imperial

Mohr's Circle Calculator

The Mohr's circle calculator lets you calculate the principal stresses from a 2D (plane) stress state. Using the values of the normal stresses and the shear stress acting on a body, the calculator returns the principal stresses of the system. Stresses are one of the most fundamental aspects of designing any body or system (metal, for example), so this tool incorporates Mohr's circle equations to give you a complete picture of the stress state.

With this calculator you can find the principal stresses (minimum and maximum), the maximum shear stress, the angle of orientation, along with the von Mises and mean stress. The maximum stress comes in handy when finding the stress concentration factor. Read on to understand what principal stress is and how to draw Mohr's circle.

What is a stress state and principal stress?

A stress state of a body is the combination of stresses at a point considering all three directions, i.e., X, Y, and Z (or 1, 2, and 3). There are three normal stresses (acting perpendicular to the face) — σ₁₁, σ₂₂, and σ₃₃ — and six shear stresses (acting along the plane) — τ₁₂, τ₂₃, τ₁₃, τ₂₁, τ₃₂, and τ₃₁.

Considering equilibrium acting on the body, the shear stresses reduce to three values: τ₁₂ = τ₂₁, τ₁₃ = τ₃₁, and τ₂₃ = τ₃₂. Therefore, a stress state can be defined by six stresses: three normal stresses and three shear stresses. If we consider only in-plane directions and set τ₁₃ = τ₃₁ = 0 and τ₂₃ = τ₃₂ = 0, the 2D stress state can be defined using three stresses: two normal stresses (σ_x, σ_y) and one shear stress (τ_xy).

What is principal stress and how to calculate principal stress?

Consider a state at which only normal stress acts on the plane. The stresses at that state are known as principal stresses. They are obtained by transforming the current stress state — that is, by reducing the shear stresses to zero. Mathematically, the principal stresses for a 2D stress state are:

σ_1,2 = (σ_x + σ_y) / 2 ± √[ ((σ_x − σ_y) / 2)² + τ_xy² ]

where σ₁ is the maximum principal stress and σ₂ is the minimum principal stress. The maximum in-plane shear stress is equal to the radius of Mohr's circle:

τ_max = √[ ((σ_x − σ_y) / 2)² + τ_xy² ]

What is Mohr's circle — using Mohr's circle to estimate principal stress

Mohr's circle is a graphical representation of the transformation of stresses. On a plot with normal stress σ on the horizontal axis and shear stress τ on the vertical axis, the stress state is represented by a circle. The center of the circle lies on the σ-axis at the average normal stress:

σ_avg = (σ_x + σ_y) / 2

and the radius equals R = τ_max. The points where the circle crosses the σ-axis are the principal stresses σ₁ and σ₂, and the top and bottom of the circle give the maximum shear stress. The orientation of the principal plane is given by:

θ_p = ½ · arctan[ 2τ_xy / (σ_x − σ_y) ]

The plane of maximum shear stress is oriented 45° from the principal plane, i.e., θ_s = θ_p − 45°.

How to use the Mohr's circle calculator

1. Choose your stress unit — metric (Pa, kPa, MPa, GPa) or American/imperial (psi, ksi).
2. Enter the normal stress along the x-axis, σ_x.
3. Enter the normal stress along the y-axis, σ_y.
4. Enter the shear stress τ_xy (use negative values where appropriate).
5. Read off the principal stresses σ₁ and σ₂, the maximum shear stress τ_max, the orientation angles, the von Mises stress, and the mean stress.

FAQs

What is the maximum principal stress for σx = 80, σy = 20, τxy = 30 MPa?

The center is σ_avg = (80 + 20) / 2 = 50 MPa and the radius is R = √[((80 − 20)/2)² + 30²] = √[30² + 30²] ≈ 42.43 MPa. Hence σ₁ ≈ 92.43 MPa and σ₂ ≈ 7.57 MPa.

Is the maximum shear stress the same as the radius of Mohr's circle?

Yes. For a 2D stress state, the maximum in-plane shear stress equals the radius of Mohr's circle, τ_max = R = (σ₁ − σ₂) / 2.

How is the von Mises stress computed?

For plane stress, the von Mises (equivalent) stress is σ_v = √(σ₁² − σ₁σ₂ + σ₂²). It is used together with a material's yield strength to predict the onset of yielding.

Can I enter stresses in psi or ksi?

Yes. Mohr's circle math is unit-agnostic as long as all inputs share the same unit. Pick metric (Pa, kPa, MPa, GPa) or American/imperial (psi, ksi) from the unit selector, and the results are returned in that same unit.