Strain gauges · Measurement

0°/60°/120° Delta Rosette Calculator

Calculate principal strains and the principal direction from the three strains of a 0°/60°/120° delta rosette using the strain circle.

DMSKeil 2017

Inputs

Strain ε_a (0°, reference) [µm/m]Strain ε_b (60°) [µm/m]Strain ε_c (120°) [µm/m]

As for the rectangular rosette: homogeneous field, plane stress, equal gauge factors of all grids.

Results

Enter values and run the calculation.

Method

What is calculated?

In the delta rosette the three measuring directions are 60° apart, so the circle centre is the arithmetic mean of all three strains. The equations follow from the strain circle with angles 120° and 240° for grids b and c. Keil again resolves the tangent ambiguity via the signs of numerator Z* = √3(ε_b−ε_c) and denominator N* = 2ε_a−ε_b−ε_c in Tab. 9.2.

Equations

m = (ε_a+ε_b+ε_c)/3 (Keil Gl. 9.70)

tan 2α = √3(ε_b−ε_c)/(2ε_a−ε_b−ε_c) (Gl. 9.81), Quadrant nach Tab. 9.2

r = √[((2ε_a−ε_b−ε_c)/3)² + (ε_b−ε_c)²/3] (Gl. 9.90)

ε₁,₂ = m ± r (Gl. 9.91)

Limits

Assumptions and typical mistake

As for the rectangular rosette: homogeneous field, plane stress, equal gauge factors of all grids.

Using the 0°/45°/90° formulas or numbering grids clockwise.

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°/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

Strain ε_a (0°, reference) [µm/m]: Strain displayed by the amplifier for grid a of the delta rosette – the grid whose direction serves as reference (0°) and from which the angle to the principal direction is counted. Source: channel a of the measurement, mind the sign.
Strain ε_b (60°) [µm/m]: Displayed strain of grid b, lying 60° counter-clockwise from a. Check the grid labelling on the rosette or datasheet – swapping b and c rotates the computed principal direction. Source: channel b of the measurement.
Strain ε_c (120°) [µm/m]: Displayed strain of grid c, lying 120° counter-clockwise from a. All three grids of the delta rosette are equivalent; their mean is the mean strain at the point. Source: channel c of the measurement.

Context

What this calculator is for

Stress analysis with delta rosettes, which offer the least direction-dependent uncertainty for unknown principal directions thanks to their uniform angular spacing.

What comes next

Check grid labelling per Keil fig. 9.24 (a, b, c counter-clockwise).
Enter the strains with sign.
Convert the result to stresses with the principal-stress calculator.

Results

How to read the results

Results as for the rectangular rosette: ε₁/ε₂ are the extremes, α the direction of ε₁ from grid a counter-clockwise.
The centre m here is the mean of all three grids – a quick plausibility check of the readings.

Result quantities

What each value means

Centre m = (ε_a+ε_b+ε_c)/3 [µm/m]: Centre of the strain circle = mean strain = (ε₁+ε₂)/2, for the delta rosette the mean of all three grids.
Radius r [µm/m]: Radius of the strain circle, half the difference between largest and smallest strain. The larger it is relative to mean strain, the stronger the directional or shear-related component.
Principal strain ε₁ [µm/m]: Largest strain, in principal direction 1 at angle α from grid a.
Principal strain ε₂ [µm/m]: Smallest strain, perpendicular to it (α + 90°).
Maximum shear strain γ_max = 2r [µm/m]: Maximum shear strain, the greatest angular distortion of the material. It occurs at 45 degrees to the principal directions and is useful as a plausibility or simulation comparison value.
Orientation angle α (from a to direction 1) [°]: Counter-clockwise angle from measuring grid a to the direction of largest strain, between 0 and 180 degrees. Transfer it to the component sketch; the second principal direction lies exactly 90 degrees from it.

Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 9.9 evaluation of 60°/120°/180° rosettes, eqs. (9.67)–(9.91), Tab. 9.2, figs. 9.24, 9.28.

The source supports the equation structure and worked examples; this calculator does not replace calibration of the measuring chain.

FAQ

Frequently asked questions

Why is it also called a 60°/120°/180° rosette?

Keil counts directions until returning to the reference axis; geometrically it is the same rosette.

Is the delta rosette more accurate?

The principal-strain error depends less on the principal direction, but evaluation is more involved.

What if all three strains are equal?

Then r = 0 (equi-biaxial state) and the angle is undefined; the calculator returns 0°.

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