Principal Stress Calculator (2D & 3D)
Compute principal stresses, principal angles, maximum shear stress, and von Mises equivalent stress — with a live Mohr's circle diagram. Free, 100% client-side, no sign-up.
2D Principal Stress
Out-of-plane condition
Mohr's circle
Convention: plotted points X = (σx, −τxy) and Y = (σy, +τxy); vertical axis is shear stress. Angles on the circle are doubled (2θ).
3D Principal Stresses (full stress tensor)
The tensor is symmetric, so τxy=τyx etc. The three principal stresses are the roots of the characteristic cubic λ³ − I₁λ² + I₂λ − I₃ = 0, solved with the closed-form trigonometric method (all three roots real).
Method & worked examples
2D formulas
3D formulas
Worked example 1 — uniaxial tension
σx=100, σy=0, τxy=0 MPa → avg=100, R=100 → σ1=100.0000, σ2=0.0000, τmax=50.0000, von Mises=100.0000 MPa.
Worked example 2 — biaxial with shear
σx=80, σy=40, τxy=30 MPa → avg=60, R=√(400+900)=36.0555 → σ1=96.0555, σ2=23.9445, θp=28.155°, τmax=36.0555, von Mises=86.6025 MPa.
Frequently asked questions
What is principal stress?
Principal stresses are the maximum and minimum normal stresses acting on a material element, found on planes where the shear stress is zero. In 2D they are called sigma1 and sigma2 (ordered sigma1 >= sigma2); in 3D there are three: sigma1 >= sigma2 >= sigma3. They completely describe the extreme normal-stress state at a point.
How do you calculate principal stresses in 2D?
For a 2D state (sigmax, sigmay, tauxy): sigma1,2 = (sigmax + sigmay)/2 +/- sqrt(((sigmax - sigmay)/2)^2 + tauxy^2). The average normal stress (sigmax + sigmay)/2 is the center of Mohr's circle, and the square-root term is its radius R. This calculator evaluates the formula live and also draws the circle.
What is the principal angle theta_p and how is it found?
theta_p is the angle (from the original x-axis) at which the element must be rotated to reach a principal plane, where shear stress vanishes: theta_p = (1/2) * atan2(2*tauxy, sigmax - sigmay). There are always two principal planes, 90 degrees apart: theta_p and theta_p + 90 degrees. One carries sigma1, the other sigma2.
What is maximum shear stress?
In-plane maximum shear stress is tau_max = (sigma1 - sigma2)/2 and it acts on planes at 45 degrees to the principal planes. The absolute (3D) maximum shear stress is (sigma_max - sigma_min)/2 taken over all three principal stresses, including the out-of-plane direction. The absolute value is never smaller than the in-plane value.
What is von Mises equivalent stress and when is it used?
The von Mises equivalent stress reduces a multiaxial stress state to a single scalar: in 2D, sigma_vm = sqrt(sigma1^2 - sigma1*sigma2 + sigma2^2); in 3D, sigma_vm = sqrt(((sigma1-sigma2)^2 + (sigma2-sigma3)^2 + (sigma3-sigma1)^2)/2). It is the standard yield criterion for ductile metals: yielding is predicted when sigma_vm reaches the material's tensile yield strength.
What is the difference between plane stress and plane strain in this calculator?
Plane stress (thin plates, free surfaces) assumes the out-of-plane normal stress is zero (sigmaz = 0). Plane strain (thick sections, constrained deformation) assumes the out-of-plane normal strain is zero, which produces a non-zero sigmaz = nu*(sigmax + sigmay), where nu is Poisson's ratio. In plane-strain mode this tool sorts sigma1, sigma2 and sigmaz into three ordered principal stresses and reports the true absolute maximum shear stress.
How does the 3D principal stress calculator work?
The 3D calculator takes the full symmetric stress tensor (sigmaxx, sigmayy, sigmazz, tauxy, tauxz, tauyz) and solves the characteristic cubic equation lambda^3 - I1*lambda^2 + I2*lambda - I3 = 0, where I1, I2, I3 are the stress invariants. Because the tensor is symmetric, all three roots are real, so the equation is solved with the closed-form trigonometric (Casus irreducibilis) method - no iteration, no library.
Why are there two principal angles, and which one gives sigma1?
The formula theta_p = (1/2)*atan2(2*tauxy, sigmax - sigmay) returns one principal direction; adding 90 degrees gives the perpendicular principal plane. Which one carries sigma1 (the larger) depends on the sign of the stresses: this tool evaluates both angles numerically and labels each with its resulting normal stress, so there is no guessing.
What units does the calculator use?
Inputs are entered in the selected unit - MPa, psi, or ksi - and every result is shown in the same unit. Conversions use 1 MPa = 145.038 psi = 0.145038 ksi. Switching units re-renders all inputs, results, and the Mohr's circle axis labels consistently.
What sign convention is used for shear stress?
This calculator uses the standard mechanics-of-materials convention: tauxy is positive when it acts on the positive x-face in the positive y-direction. On Mohr's circle the plotted points are X = (sigmax, -tauxy) and Y = (sigmay, +tauxy), with the vertical axis being shear stress. The circle geometry and all angles are consistent with this convention.
Can principal stresses be negative?
Yes. A negative principal stress is a compressive stress. The tool always orders results so sigma1 >= sigma2 (>= sigma3 in 3D), so a state like pure shear (sigmax = sigmay = 0, tauxy = 50 MPa) correctly gives sigma1 = +50 MPa and sigma2 = -50 MPa.
How is absolute maximum shear stress different from in-plane maximum shear?
In-plane maximum shear, (sigma1 - sigma2)/2, only considers the two in-plane principal stresses. The absolute maximum shear stress considers all three principal directions: tau_abs_max = (sigma_max - sigma_min)/2. For example, with sigma1 = 96 MPa, sigma2 = 24 MPa and an out-of-plane stress of 0, the in-plane value is 36 MPa but the absolute value is 48 MPa - the out-of-plane shear governs. This tool reports both.
What does the Mohr's circle diagram show?
The diagram plots the stress state as a circle centered at the average normal stress (sigmax + sigmay)/2 with radius R = sqrt(((sigmax - sigmay)/2)^2 + tauxy^2). It marks the input points X and Y, both principal stresses where the circle crosses the normal-stress axis, the principal angle 2*theta_p, and the top and bottom points where shear stress is maximal. It is drawn live as SVG with no external libraries.
Is this calculator suitable for pressure vessel or structural design?
This tool is an educational and pre-design calculator: its math (principal stresses, Mohr's circle, von Mises) is exact, but real design must also follow the applicable code (ASME VIII, EN 13445, AISC 360, etc.), include safety factors, stress concentrations, fatigue, and a qualified engineer's review. Use the results to check hand calculations and build intuition, not as a final design sign-off.
Scope boundary
This tool computes principal stresses (2D and 3D), principal angles, in-plane and absolute maximum shear stress, von Mises equivalent stress, and Mohr's circle. It is an educational stress-transformation calculator, not a full finite-element or code-based design tool.
Anti-cannibalization check: no existing tool in the nrtechstudio catalog covers principal stress, Mohr's circle, or von Mises equivalent stress (grep of tools-catalog.txt and BUILD-QUEUE.md: CLEAN), so there are no overlapping cross-links to add. This tool is the catalog's first entry for stress transformation.
Free tool by NRTechStudio