Mohr’s Circle Calculator
Stress Analysis Results
What Is a Mohr’s Circle Calculator?
A Mohr’s Circle Calculator is a stress analysis tool that converts three plane stress inputs into principal stress and maximum shear stress results. It uses standard solid mechanics equations to find the circle center, circle radius, principal stresses, principal angle, and max shear angle.
This Mohr’s Circle calculator takes σx, σy, and τxy as inputs and calculates σ1, σ2, τmax, σavg, θp, θs, C, and R. It helps users understand how stress changes when an element is rotated, including the angle where shear stress becomes zero.
The result is a numerical estimate for a two-dimensional stress condition. It does not draw the circle, choose material limits, check failure criteria, or decide whether a design is safe. Users should enter all stress values in the same unit, such as psi, ksi, Pa, or MPa, because the calculator does not convert units.
How the Mohr’s Circle Calculator Formula Works
The calculator first finds the average normal stress. This value is also the center of Mohr’s Circle. Then it finds the radius, which equals the maximum in-plane shear stress. The principal stresses are found by adding and subtracting the radius from the center.
In these formulas, σx is the normal stress in the x-direction, σy is the normal stress in the y-direction, and τxy is the shear stress on the xy plane. C is the circle center. R is the circle radius. σ1 is the maximum principal stress. σ2 is the minimum principal stress. θp is the principal angle in degrees, and θs is the max shear angle in degrees.
For example, use σx = 50, σy = -20, and τxy = 40. The center is (50 + -20) ÷ 2 = 15. The radius is √(((50 – -20) ÷ 2)² + 40²), which equals √(35² + 40²), or 53.15 after rounding to two decimals.
The principal stresses are 15 + 53.15 = 68.15 and 15 – 53.15 = -38.15. The maximum in-plane shear stress is also 53.15. The principal angle is one-half of atan2(80, 70), converted to degrees, which gives 24.40°. The max shear angle is 69.40°.
The calculator rounds stress results and angle results to two decimal places. If the radius is less than 0.00001, the tool treats the condition as a pure hydrostatic or isotropic stress state. In that case, the principal angle and max shear angle are shown as indeterminate.
How to Use the Mohr’s Circle Calculator: Step by Step
- Enter the normal stress in the x-direction in the Normal Stress X (σx) field.
- Enter the normal stress in the y-direction in the Normal Stress Y (σy) field.
- Enter the shear stress on the xy plane in the Shear Stress XY (τxy) field.
- Use the same unit for all three stress inputs, such as psi, ksi, Pa, or MPa.
- Select Calculate to display the stress analysis results.
- Select Reset if you want to clear the inputs and hide the current results.
The output shows principal stresses as σ1 / σ2. It also shows a plain-English interpretation, maximum in-plane shear stress, average normal stress, principal angle, max shear angle, circle center, and circle radius. Stress outputs use the same unit as your inputs. Angle outputs are shown in degrees.
How to Read Your Mohr’s Circle Calculator Results
The most important result is usually the pair of principal stresses. These are the normal stresses acting on planes where shear stress is zero. The calculator labels them as σ1 and σ2. The first value is the larger principal stress, and the second value is the smaller principal stress.
Principal stresses
Principal stresses show the largest and smallest normal stress values for the entered 2D stress state. A positive value is commonly read as tensile stress. A negative value is commonly read as compressive stress. The calculator follows this wording in its interpretation, but it does not judge whether the result is acceptable for a material.
Maximum in-plane shear stress
The maximum in-plane shear stress equals the circle radius. This means τmax and R display the same numeric value. This result helps show the greatest shear stress found by rotating the stress element within the same plane.
Angles and rotation direction
The principal angle tells how far the element must be rotated from the original x-axis to reach zero shear stress. The calculator uses the atan2 function, which helps place the angle in the correct quadrant. In the written interpretation, a positive principal angle is described as counterclockwise, and a negative angle is described as clockwise.
| Output | What It Means |
|---|---|
| σ1 / σ2 | Maximum and minimum principal stresses |
| τmax | Maximum in-plane shear stress |
| σavg | Average normal stress |
| θp | Principal angle where shear stress is zero |
| θs | Angle for maximum in-plane shear stress |
| C | Mohr’s Circle center |
| R | Mohr’s Circle radius |
These results are estimates based on the values you enter. They do not include material strength, safety factors, failure theories, stress concentration, fatigue, buckling, thermal stress, or 3D stress effects. For design decisions, compare the results with the correct engineering standard and professional judgment.
Frequently Asked Questions
What is a Mohr’s Circle calculator used for?
A Mohr’s Circle calculator is used to analyze a two-dimensional stress state. This calculator finds principal stresses, maximum in-plane shear stress, average normal stress, the circle center, the circle radius, and stress rotation angles from σx, σy, and τxy.
How do I calculate principal stress with Mohr’s Circle?
To calculate principal stress, enter σx, σy, and τxy into the calculator. The tool finds the average normal stress, computes the circle radius, then calculates σ1 as C + R and σ2 as C – R. The displayed result is shown as σ1 / σ2.
What inputs does this Mohr’s Circle calculator need?
This calculator needs three inputs: Normal Stress X (σx), Normal Stress Y (σy), and Shear Stress XY (τxy). All three fields must contain valid numbers. The calculator does not include a unit selector, so each stress value should use the same unit.
Why is the principal angle shown as indeterminate?
The principal angle is shown as indeterminate when the circle radius is less than 0.00001. The calculator treats this as a pure hydrostatic or isotropic stress state. In that case, the circle degenerates into a single point, so there are no unique principal planes.
Is maximum shear stress the same as the circle radius?
Yes, in this calculator the maximum in-plane shear stress is the same as the circle radius. The code sets τmax equal to R. Both values are calculated from the normal stress difference and the xy shear stress, then rounded to two decimal places.
How accurate is this Mohr’s Circle calculator?
This calculator follows standard plane stress equations and rounds displayed results to two decimal places. Its accuracy depends on the numbers you enter and whether a 2D stress model fits your problem. It does not verify material safety, units, loads, or engineering assumptions.
Does this calculator draw Mohr’s Circle?
No, this calculator does not draw a visual Mohr’s Circle. It calculates and displays the numerical results: principal stresses, max in-plane shear stress, average normal stress, principal angle, max shear angle, circle center, and circle radius.