Principal and von Mises Stress Calculator

Transform a Cartesian point-stress state into principal stresses and directions, von Mises equivalent stress, maximum shear, and optional yield screening.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Stress state

Von Mises equivalent stress, σv

—MPa

Enter a valid stress state to calculate.

Maximum principal, σ1— MPa

Intermediate principal, σ2— MPa

Minimum principal, σ3— MPa

Absolute maximum shear, τmax— MPa

Principal angle, θp—

Maximum in-plane shear— MPa

Yield utilization—

Safety factor, n—

Principal directions

σ1: —

σ2: —

σ3: —

Mohr-circle stress map

Normal stress horizontal · shear stress vertical

How to use

Select Plane stress when σz, τyz, and τzx are negligible, or select Full stress tensor when all six independent Cartesian components are known.
Select one stress unit. Enter signed normal stresses σx, σy, and (in 3D) σz; tension is positive and compression negative.
Enter signed shear components using the positive-face/positive-axis convention. Use one coordinate system and consistent units.
Optionally enter a positive yield strength for the same material condition to calculate utilization and safety factor.
Select Calculate or edit a value to update the results and Mohr-circle plot. Reset restores the worked plane-stress example.
Interpret σ1≥σ2≥σ3 as principal normal stresses, θp as the counter-clockwise 2D principal direction, σv as the ductile-yield comparison stress, and τmax as half the extreme principal-stress range.
Correct any warning before using a result. Do not use this point-stress model for brittle fracture, fatigue, buckling, crack growth, anisotropic yielding, contact-field reconstruction, or code compliance.

Equations used

With normal tension positive and a symmetric Cauchy stress tensor:

σ = [[σx, τxy, τzx], [τxy, σy, τyz], [τzx, τyz, σz]]

Plane stress: σz = τyz = τzx = 0

σavg = (σx + σy)/2; R = √{[(σx − σy)/2]² + τxy²}

σ1,2(2D) = σavg ± R; θp = ½ atan2(2τxy, σx − σy)

3D principal stresses: det(σ − λI) = 0, ordered σ1 ≥ σ2 ≥ σ3

σv = √{½[(σx−σy)²+(σy−σz)²+(σz−σx)²] + 3(τxy²+τyz²+τzx²)}

τmax = (σ1 − σ3)/2; U = 100σv/Sy; n = Sy/σv

All stress terms use the selected unit. θp is reported in degrees; U is percent; n is dimensionless. atan2 preserves the stress quadrant.

Theory and method

Principal stresses are the eigenvalues of the real symmetric stress tensor. Their normalized eigenvectors give the principal directions—orthogonal directions in which the transformed shear components vanish. Plane-stress mode also evaluates the familiar Mohr-circle center σavg, radius R, and principal angle θp. Full 3D mode solves the symmetric eigenproblem numerically and plots the circles for the σ1–σ3, σ1–σ2, and σ2–σ3 pairs.

Von Mises equivalent stress follows the distortion-energy criterion. Hydrostatic stress does not contribute, so equal normal stresses with zero shear produce σv=0. For a simple pure-shear state, σv=√3|τ|. Comparison with Sy is appropriate as a first screen for isotropic ductile yielding, but not for brittle, anisotropic, fatigue, creep, instability, or fracture-controlled behavior.

Assumptions and limits

Components represent the same material point, coordinate frame, unit, sign convention, and loading condition.
Plane stress is valid only when out-of-plane components are negligible; otherwise use 3D.
Stress gradients, stress concentrations not already represented in the inputs, residual stress, temperature effects, uncertainty, and plastic redistribution require separate treatment.
Yield strength must match material condition, temperature, environment, strain rate, and governing design basis.

Worked example

1. Select Plane stress and MPa. Enter σx=100 MPa, σy=40 MPa, τxy=30 MPa, and Sy=250 MPa. 2. Calculate. Expected: σ1=112.426 MPa, in-plane σ2=27.574 MPa, θp=22.5°, σv=101.489 MPa, utilization=40.596%, and n=2.463. 3. The entered state is below Sy by this screening criterion; further checks may still govern.

References

Budynas, R. G., and Nisbett, J. K., Shigley’s Mechanical Engineering Design, 11th ed., sections on stress transformation and distortion-energy theory, McGraw-Hill, 2020.
Timoshenko, S. P., and Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, 1970.
Principal-stress functional reference and von Mises functional reference, accessed 15 August 2026; equations independently implemented and benchmarked.

Continue in TriboSolver: use TriboSolver to refine component loading, contact conditions, material behavior, and independent simulation checks when a point-stress screen is not sufficient.