Strain gauge rosette calculator

How a strain gauge rosette is read

A single gauge measures strain along one line. Mounted at three different angles, three gauges give three equations, and these are enough to find the strain in any direction at that point: the normal strains ε_x and ε_y and the shear strain γ_xy. From those, Mohr’s circle of strain gives the principal strains and their directions.

1. Each gauge at angle θ reads ε_θ = ε_x cos²θ + ε_y sin²θ + γ_xy sin θ cos θ.
2. Solve the three equations for ε_x, ε_y, γ_xy.
3. Principal strains: ε_1,2 = ε_avg ± R, with R = √[((ε_x − ε_y) / 2)² + (γ_xy / 2)²].
4. With E and ν, use Hooke’s law for plane stress to get the stresses.

Worked example

A 45° rectangular rosette on a steel surface reads ε_a = 600 με at 0°, ε_b = 500 με at 45° and ε_c = −100 με at 90°. Take E = 200 GPa and ν = 0.3.

1. Strains from the rosette
   Each gauge reads ε_θ = ε_x cos²θ + ε_y sin²θ + γ_xy sin θ cos θ. Three readings give three equations, solved for ε_x, ε_y and γ_xy.
   For a 45° rectangular rosette this simplifies to ε_x = ε_a = 600 με, ε_y = ε_c = (−100) με, γ_xy = 2ε_b − ε_a − ε_c = 2 × 500 − 600 − (−100) = 500 με.
   Result: ε_x = 600 με, ε_y = −100 με, γ_xy = 500 με.
2. Centre and radius
   ε_avg = (ε_x + ε_y) / 2 = (600 + (−100)) / 2 = 250 με
   R = √[ ((ε_x − ε_y) / 2)² + (γ_xy / 2)² ] = 430 με
3. Principal strains
   ε₁ = ε_avg + R = 680 με
   ε₂ = ε_avg − R = −180 με
   ε₁ acts along the direction θ_p = 17.8° from the x axis (anticlockwise positive), and ε₂ is 90° from that.
4. Maximum in-plane shear strain
   γ_max = 2R = ε₁ − ε₂ = 860 με, on planes at −27.2° where the normal strain is ε_avg = 250 με.
5. Stresses from Hooke’s law (plane stress)
   Assume the surface is in plane stress (σ_z = 0) with E = 200 GPa and ν = 0.3.
   σ_x = E (ε_x + ν ε_y) / (1 − ν²) = 125 MPa
   σ_y = E (ε_y + ν ε_x) / (1 − ν²) = 17.6 MPa
   τ_xy = G γ_xy with G = E / (2(1 + ν)) = 76.9 GPa, so τ_xy = 38.5 MPa
   Principal stresses: σ₁ = 138 MPa, σ₂ = 5.26 MPa, maximum shear τ_max = 66.2 MPa. Von Mises stress: 135 MPa.
   Out-of-plane strain ε_z = −ν (ε_x + ε_y) / (1 − ν) = −214 με.

Related

Library formulas: Hooke’s law, Young’s modulus from a test, Poisson’s ratio from measured strains, Shear strain from shear stress, Shear modulus from Young’s modulus and Poisson’s ratio. If you already have stresses rather than strains, use the Mohr’s circle calculator.

Common questions

What is a strain gauge rosette?

A rosette is a group of three strain gauges at known angles on the same spot. One gauge measures strain in only one direction, so on a surface where the directions of the principal strains are not known, three readings at different angles are needed to find the full strain state.

How do I get εx, εy and γxy from the readings?

Each gauge at angle θ reads εθ = εx cos²θ + εy sin²θ + γxy sinθ cosθ. Three readings give three equations, solved for the three unknowns. For a 45° rectangular rosette (0°, 45°, 90°) this gives εx = εa, εy = εc and γxy = 2εb − εa − εc. For a 60° delta rosette (0°, 60°, 120°), εx = εa, εy = (2(εb + εc) − εa) / 3 and γxy = 2(εb − εc) / √3.

How are the principal strains found?

The principal strains are εavg ± R, where εavg = (εx + εy) / 2 and R = √[((εx − εy) / 2)² + (γxy / 2)²]. The direction of ε₁ comes from tan 2θp = γxy / (εx − εy), taken in the correct quadrant.

Why is the shear strain halved on Mohr’s circle?

The rotation formulas for strain match those for stress only when the tensor shear strain γ / 2 is used. The engineering shear strain γ is twice that, so the circle uses γ / 2 on its vertical axis and the maximum shear strain is twice the radius.

How do I get the stresses from the strains?

For a free surface in plane stress with a linear elastic, isotropic material, σx = E(εx + νεy) / (1 − ν²), σy = E(εy + νεx) / (1 − ν²) and τxy = Gγxy with G = E / (2(1 + ν)). Enter E in GPa and ν to get the stresses in MPa. The result is only as good as the material values and the plane-stress assumption.

Can I use my own gauge angles?

Yes. Choose custom angles and type three different angles in degrees from the x axis. The calculator solves the three equations directly and tells you if the angles cannot separate the strains.

Is it free, and does it keep my numbers?

It is free to use, with no sign-up, no adverts and no personal tracking. The numbers you type stay in your browser.