MethodWhat is calculated?
At a free surface the stress state is plane, but the strains are triaxial. Hooke's law for two axes accounts for lateral contraction: a measured 1000 µm/m in steel does not mean 210 N/mm² but also depends on the second principal strain – with 30 % weight at ν = 0.3. The von Mises equivalent stress governs for ductile metals.
Equations
σ₁ = E/(1−ν²)·(ε₁ + νε₂) (Keil Gl. 9.27)
σ₂ = E/(1−ν²)·(ε₂ + νε₁) (Gl. 9.28)
ε₃ = −ν/(1−ν)·(ε₁+ε₂) (Gl. 9.66)
σ_v = √(σ₁² − σ₁σ₂ + σ₂²) (Gestaltänderungsenergiehypothese, Keil Abschn. 14.9)
ProcedureStep by step
- Take the principal strains from the rosette evaluation or two gauges in known principal directions.
- Enter E and ν of the object material at the measuring temperature.
- Compare σ_v with the yield strength; use σ₁ for brittle materials (maximum normal stress hypothesis).
Typical mistake
Computing σ = E·ε separately per direction and thereby ignoring lateral contraction.
PracticeApplication and limits
Evaluation of every strain-gauge stress analysis, utilisation checks at notches and transitions, validation of FE results with measurements.
Linear-elastic, isotropic, plane stress; for plastic deformation use the methods of Keil ch. 13.
SourceTechnical basis
Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 9.4 biaxial stress state, eqs. (9.24)–(9.29); sec. 9.8, eq. (9.66); sec. 14.9 distortion-energy hypothesis.
The source supports the equation structure and worked examples; this calculator does not replace calibration of the measuring chain.
FAQFrequently asked questions
How large is the error without ν?
Zero for ε₂ = −νε₁ (uniaxial); for ε₁ = ε₂ however σ = Eε underestimates the stress by 1/(1−ν) ≈ 1.43.
What is ε₃?
The thickness strain normal to the surface; it follows from σ₃ = 0 and is not needed for the surface analysis.
Which hypothesis for grey cast iron?
The maximum normal stress hypothesis: σ₁ governs (Keil sec. 14.4).
In the same cluster
All strain-gauge calculators