Strain gauges · Measurement

Principal Stress from Principal Strain Calculator

Calculate principal stresses, third principal strain, maximum shear stress and von Mises stress from two principal strains at a free surface.

DMSKeil 2017

Inputs

Principal strain ε₁ [µm/m]Principal strain ε₂ [µm/m]Young's modulus E [N/mm²]Poisson's ratio ν

Linear-elastic, isotropic, plane stress; for plastic deformation use the methods of Keil ch. 13.

Results

Enter values and run the calculation.

Method

What 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)

Limits

Assumptions and typical mistake

Linear-elastic, isotropic, plane stress; for plastic deformation use the methods of Keil ch. 13.

Computing σ = E·ε separately per direction and thereby ignoring lateral contraction.

Gauge layout

Where the gauges sit and what the bridge makes of them

Where the grids sit and how they are wired, plus which load components remain in the signal and which the circuit compensates.

What does the bridge factor B mean?

The general bridge equation is U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄): strains in opposite arms count positive, in adjacent arms negative. Expressing every active strain as a multiple of one reference strain ε, B is exactly that signed sum of multiples – e.g. B = 2 for two adding contributions, B = 4 for four. U_M/U_B = k·B·ε/4 summarises this.

To assess it: B is a pure gain of the circuit, not a statement about temperature compensation or which load type is in the signal – that is covered by “Use” and the response chips below. A larger B gives a larger, more easily measured signal U_M for the same mechanical strain; this calculator inserts B into the calculation automatically once you pick a circuit, so you never enter it yourself.

0°/45°/90° rosette (rectangular rosette) 3 × B = 1

Layout: Three grids a, b, c with measuring directions rotated 45° against each other; a is the reference direction and the order a → b → c runs counter-clockwise. Each grid is measured in its own bridge (usually circuit II).

ε_a, ε_b, ε_c in the signal
ΔT compensated

Use: Plane stress with unknown principal directions.

Yields the principal strains ε₁, ε₂ and the orientation angle α from three single measurements.
Stacked and side-by-side layouts are evaluated identically.

Three bridge circuits per rosette.
The angle α is correct only with the evaluation matching the grid labelling (Keil Tab. 9.1) – a wrong sense of rotation is the most common mistake.

Keil Abschn. 9.7, Abb. 9.23

0°/60°/120° rosette (delta rosette) 3 × B = 1

Layout: Three grids a, b, c with measuring directions rotated 60° against each other, reference direction a, counter-clockwise order. Each grid in its own bridge.

ε_a, ε_b, ε_c in the signal
ΔT compensated

Use: Plane stress with unknown principal directions; also called 0°/120°/240° or equiangular rosette.

Equal angular spacing of all three directions.
Triangular and star layouts are evaluated identically.

Three bridge circuits per rosette.
Evaluate only with the formula matching the labelling (Keil Tab. 9.2).

Keil Abschn. 9.7, Abb. 9.24

Bridge wiring diagrams after Keil, Dehnungsmessstreifen, 2nd ed. 2017 (secs. 4.5, 9.7, 10.2, 10.5) · bridge arms numbered R1–R4 with U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄).

Inputs

What you enter – and where the values come from

Principal strain ε₁ [µm/m]: The largest strain at this surface point (principal strain 1). Source: result of a rosette calculator, or the reading of a gauge already bonded along the known principal direction (e.g. along a tension rod).
Principal strain ε₂ [µm/m]: The strain perpendicular to it (principal strain 2), often negative. It is essential because lateral contraction co-determines the stress in direction 1 – σ₁ = E·ε₁ would be wrong. Source: rosette calculator or a second, transversely bonded gauge.
Young's modulus E [N/mm²]: Young's modulus describes how stiff the part's material is – how much stress it takes to produce a given strain (σ = E·ε). It is needed to turn measured or computed strains into stresses and vice versa. Source: material tables or datasheet; steel ≈ 210 000 N/mm², aluminium ≈ 70 000 N/mm², titanium ≈ 110 000 N/mm². E drops at elevated temperature.
Poisson's ratio ν: Poisson's ratio states how much a material contracts transversely when stretched longitudinally: ε_transverse = −ν·ε_longitudinal. It matters because transversely bonded gauges measure exactly this contraction and because both directions interact in biaxial stress states. Source: material tables; steel 0.28–0.30, aluminium 0.33, plastics 0.35–0.45.

Context

What this calculator is for

Evaluation of every strain-gauge stress analysis, utilisation checks at notches and transitions, validation of FE results with measurements.

What comes next

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).

Results

How to read the results

σ₁ and σ₂ act in the principal directions of the strains; τ_max acts at 45° to them.
For the strength check of ductile materials compare σ_v with the yield strength, for brittle materials σ₁ with the tensile strength.

Result quantities

What each value means

Principal stress σ₁ [N/mm²]: Largest normal stress in the surface, acting in principal direction 1 (direction of ε₁). Governs brittle materials and cracks perpendicular to it.
Principal stress σ₂ [N/mm²]: Smallest normal stress in the surface, perpendicular to σ₁. Not necessarily negative for negative ε₂ – lateral contraction can make it positive.
Strain normal to the surface ε₃ [µm/m]: Calculated strain normal to the free surface, representing local thickness change. It is supplementary only: a bonded strain gauge cannot measure this direction directly. For a thin component surface, zero stress normal to the surface is assumed.
Max. in-plane shear stress (σ₁−σ₂)/2 [N/mm²]: Largest shear stress in the surface plane, acting at 45° to the principal directions. Basis of the maximum-shear-stress hypothesis.
Equivalent stress σ_v (von Mises) [N/mm²]: Von Mises equivalent stress: one value combining both principal stresses for ductile metals. For a strength check, divide allowable stress or material yield strength by this value; the result is the safety factor.

Source

Technical 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.

FAQ

Frequently 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).

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