Formula used
For small strains, the strain tensor is symmetric. In 2D, principal strains are the eigenvalues of: [ εx γxy/2; γxy/2 εy ].
- ε̄ = (εx + εy)/2
- R = sqrt(((εx − εy)/2)^2 + (γxy/2)^2)
- ε1,2 = ε̄ ± R
- θp = 0.5·atan2(γxy, εx − εy) (direction of ε1)
- γmax (engineering) = ε1 − ε2 = 2R
In 3D, principal strains are eigenvalues of the 3×3 tensor with off-diagonals γ/2. This tool uses a stable Jacobi rotation for symmetric matrices.
How to use this calculator
- Select 2D if you only have εx, εy, and γxy.
- Select 3D if you measured εz, γyz, and γzx too.
- Pick a unit: microstrain (µε) or dimensionless strain.
- Enter strain components as signed numbers, then click Calculate.
- Download CSV or PDF after results appear under the header.
| Case | εx (µε) | εy (µε) | γxy (µε) | ε1 (µε) | ε2 (µε) | θp (deg) |
|---|---|---|---|---|---|---|
| 2D example | 350 | -120 | 220 | 400.598 | -170.598 | 13.283 |
| Pure shear | 0 | 0 | 400 | 200 | -200 | 22.5 |
| Equal biaxial | 250 | 250 | 0 | 250 | 250 | 0 |
Professional article
1) Why principal strains matter
Principal strains are the maximum and minimum normal strains at a point after rotating axes so shear vanishes. They simplify comparison with allowable limits, help interpret rosette measurements, and provide a coordinate‑independent deformation summary. The associated principal directions form an orthogonal basis for reporting.
2) Typical strain magnitudes in practice
In many service‑loaded metallic parts, strains commonly range from a few dozen to several thousand microstrain. Brittle materials usually tolerate much less tensile strain before cracking, so the maximum principal tensile strain is often the critical value. Temperature effects and gauge noise can add apparent strain. For structural steel, 2000 µε corresponds to about 0.2% strain, a common yield‑level indicator in simple tension tests practice.
3) Converting rosette readings to εx, εy, γxy
A three‑element rosette measures strain along different angles. After reducing those readings to in‑plane components εx, εy, and γxy, the 2D mode delivers ε1, ε2, and θp immediately. If you already have components from software, enter them directly and keep a consistent sign convention. Document gauge angles and wiring so the reduction can be audited later.
4) Understanding the principal angle output
The calculator reports θp counter‑clockwise from the x‑axis using an atan2 convention to handle all quadrants. Changing axis labels or sign conventions can shift θp by 90° or 180° without changing the physical principal directions. Use the same reference frame in testing, analysis, and drawings.
5) Engineering vs tensor shear strain
Most instrumentation outputs engineering shear strain γ, which equals 2εxy in tensor form. This tool assumes γ inputs and internally uses γ/2 in the symmetric tensor. In 2D, ε1 − ε2 equals γmax (engineering). The Mohr radius R provides a check because 2R must match that difference.
6) Extending to 3D principal strains
For multiaxial states, principal strains are the eigenvalues of the full 3D tensor built from εx, εy, εz and γxy, γyz, γzx. The results return ε1 ≥ ε2 ≥ ε3 plus direction cosines. This is valuable for thermal gradients, constrained assemblies, and complex load paths.
7) Quick quality checks you should run
First confirm units: microstrain for gauges, dimensionless strain for many simulations. Ensure tensor symmetry by processing shear consistently. In 2D, verify ε1 + ε2 = εx + εy and ε1 − ε2 = γmax. In 3D, check ordering and that eigenvectors are normalized. Large rounding can hide small shear; increase precision during verification steps.
8) Turning results into decisions
Maximum principal tensile strain often correlates with crack initiation, while high compressive principal strain can indicate crushing or buckling‑related deformation. Pair principal values with material limits, load cases, and uncertainty. Export CSV for spreadsheets and PDF for traceable reporting across reviews and audits. Store inputs with results to support future recalculation and comparison.
FAQs
What is the difference between strain and microstrain?
Strain is dimensionless deformation (ΔL/L). Microstrain expresses the same value scaled by 10⁻⁶, which matches typical gauge magnitudes. Selecting microstrain lets you enter familiar numbers like 350 instead of 0.000350.
Should I enter γxy or εxy for shear?
Enter engineering shear strain γxy as most instruments report it. The calculator converts internally to tensor shear εxy = γxy/2 when forming the symmetric strain tensor used for principal strains.
Why can the principal angle be negative or larger than 90°?
The angle is computed with atan2, so it spans all quadrants. Negative simply means clockwise from the x‑axis. Adding or subtracting 180° represents the same physical direction, and ε2 is always 90° from ε1.
When should I use 3D mode instead of 2D?
Use 3D when out‑of‑plane strain or shear is significant, such as thick parts, constrained assemblies, thermal gradients, or multiaxial loading. If εz, γyz, and γzx are negligible, 2D is usually sufficient.
Do eigenvector signs or order affect interpretation?
Eigenvector signs can flip without changing the axis direction, so a sign change is not an error. The calculator sorts principal strains so ε1 ≥ ε2 ≥ ε3; compare using that ordering consistently in reports.
What if ε1 equals ε2 in 2D?
When ε1 and ε2 are equal, the Mohr radius is zero and the state is isotropic in-plane. In that case, any orthogonal axes are principal, and the reported angle becomes less meaningful.
How should I handle noisy strain measurements?
Average repeated readings, compensate temperature where possible, and keep consistent wiring and calibration. Small noise in γ can noticeably shift θp when εx≈εy, so report uncertainty and avoid over‑interpreting minor angle changes.
Notes and practical interpretation
Principal strains represent the normal strains measured along directions where shear strain vanishes. In materials testing, they help interpret rosette readings, validate finite element results, and estimate maximum shear effects.
In 2D, the angle θp gives the orientation of the ε1 axis relative to the x-axis using a consistent atan2 convention. If your sign convention differs, keep the same convention for all components and interpret the direction accordingly.
Accurate principal strains help you design safer structures always.