Mohr's Circle
Analyze stress states with Mohr's circle — compute σ₁, σ₂, max shear, von Mises, and orientation angle. Free online stress tool for engineers.
About This Calculator
Mohr's circle is a powerful graphical method used in solid mechanics and engineering to analyze the state of stress at a point in a material. Named after the German engineer Otto Mohr, it provides a visual representation of how normal and shear stresses transform as the coordinate system rotates. This calculator computes all key stress parameters from your input: normal stresses σₓₓ and σᵧᵧ, and shear stress τₓᵧ.
The Mohr's circle is constructed with center at σ_avg = (σₓₓ + σᵧᵧ)/2 on the normal stress axis and radius R = √[((σₓₓ − σᵧᵧ)/2)² + τₓᵧ²]. Points where the circle intersects the normal stress axis give the principal stresses σ₁ (maximum) and σ₂ (minimum). The topmost point of the circle corresponds to the maximum shear stress τ_max = R. The angle from the original x-axis to the principal plane is θ = 0.5 × arctan(2τₓᵧ / (σₓₓ − σᵧᵧ)).
This calculator computes six results: maximum principal stress (σ₁), minimum principal stress (σ₂), maximum shear stress (τ_max), angle of orientation (θ in degrees), von Mises stress (σ_v = √(σ₁² + σ₂² − σ₁σ₂)), and mean stress (σ_m = (σ₁ + σ₂)/2). The interactive Mohr's circle chart visualizes the stress state graphically, showing the circle with principal stress points highlighted in red.
This tool is designed for mechanical engineers, civil engineers, aerospace engineers, geotechnical engineers, materials scientists, and engineering students who need to perform stress transformation analysis. It is applicable worldwide for structural design, failure analysis, and material strength assessment. Engineers in India, the US, the UK, and globally use Mohr's circle for stress analysis in beams, shafts, pressure vessels, soil mechanics, and finite element verification.
Frequently Asked Questions
What is Mohr's circle?
Mohr's circle is a graphical representation of the stress state at a point in a material. It plots normal stress on the horizontal axis and shear stress on the vertical axis, forming a circle whose center is the mean stress and radius equals the maximum shear stress. It helps visualize how stresses transform under rotation, making it essential for stress analysis in mechanical, civil, and aerospace engineering.
How do you calculate principal stresses using Mohr's circle?
Principal stresses are found by identifying where Mohr's circle intersects the horizontal (normal stress) axis. Maximum principal stress σ₁ = σ_avg + R and minimum principal stress σ₂ = σ_avg - R, where σ_avg = (σxx + σyy)/2 is the center of the circle and R = √[((σxx − σyy)/2)² + τxy²] is the radius. These are the normal stresses on planes with zero shear stress.
What is maximum shear stress and how is it found?
The maximum shear stress τ_max equals the radius R of Mohr's circle: τ_max = √[((σxx − σyy)/2)² + τxy²]. It can also be calculated from principal stresses as τ_max = (σ₁ − σ₂)/2. The maximum shear stress occurs on planes oriented at 45° to the principal planes and is critical for predicting material yield failure under the Tresca criterion.
What is von Mises stress and why is it important?
Von Mises stress σ_v is a scalar value derived from principal stresses: σ_v = √(σ₁² + σ₂² − σ₁σ₂) for 2D stress states. It represents the equivalent tensile stress that causes the same distortion energy as the actual multiaxial stress state. Engineers use von Mises stress with the von Mises yield criterion — if σ_v exceeds the material's yield strength, yielding is predicted.
What does the angle of orientation θ represent?
The angle θ (theta) is the angle from the original x-axis to the principal plane where maximum principal stress acts. It is calculated as θ = 0.5 × arctan(2τxy / (σxx − σyy)). A positive angle means counterclockwise rotation. This angle is essential for orienting structural components and failure planes in geotechnical and mechanical engineering.
What is mean stress and how is it used?
Mean stress σ_m = (σ₁ + σ₂)/2 = (σxx + σyy)/2 is the hydrostatic component of the stress state. It represents the center of Mohr's circle. Mean stress is used in fatigue analysis (mean stress correction models like Goodman, Soderberg, and Gerber) and in geotechnical engineering where the effective mean stress governs soil behavior.
What is the difference between Mohr's circle and principal stresses?
Principal stresses are the maximum and minimum normal stresses at a point, occurring on planes with zero shear stress. Mohr's circle is the graphical tool that shows all possible stress states as the plane rotates. The circle's intersection with the horizontal axis gives the principal stresses, the topmost point gives maximum shear stress, and the angle on the circle (doubled) corresponds to the physical rotation angle.
What units should I use for stress inputs?
You can use any consistent stress unit (MPa, psi, Pa, kPa, etc.) for all three inputs — the calculator performs the same mathematical transformation regardless of unit. The results will be in the same unit you entered. MPa is most common for engineering applications. For accurate results, ensure all inputs use the same unit system.