Strain gauges · Measurement

0°/45°/90° Strain Rosette Calculator

Calculate principal strains ε₁ and ε₂, maximum shear strain and the principal direction from the three strains of a rectangular 0°/45°/90° rosette.

DMSKeil 2017

Inputs

Strain ε_a (0°, reference) [µm/m]Strain ε_b (45°) [µm/m]Strain ε_c (90°) [µm/m]

Homogeneous strain field under the rosette, plane stress at the free surface; prefer stacked rosettes under steep gradients.

Results

Enter values and run the calculation.

Method

What is calculated?

Three strains in known directions define the strain circle uniquely. For the 0°/45°/90° rosette a and c lie on one diameter, so their mean gives the circle centre; b fixes the angle. Because tan 2α is ambiguous, Keil's Tab. 9.1 assigns the angle uniquely within 0° ≤ α < 180° from the signs of numerator and denominator. The angle is measured counter-clockwise from reference direction a to principal direction 1 (ε₁).

Equations

m = (ε_a+ε_c)/2 (Keil Gl. 9.51)

tan 2α = (2ε_b − ε_a − ε_c)/(ε_a − ε_c) (Gl. 9.55), Quadrant nach Tab. 9.1

r = ½·√2·√[(ε_a−ε_b)² + (ε_c−ε_b)²] (Gl. 9.63)

ε₁,₂ = m ± r (Gl. 9.65), γ_max = 2r (Gl. 9.64)

Limits

Assumptions and typical mistake

Homogeneous strain field under the rosette, plane stress at the free surface; prefer stacked rosettes under steep gradients.

Swapping grids b and c or measuring the angle clockwise – both give a principal direction off by 90°.

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

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 rosette – the grid whose direction serves as reference (0°). The angle to the principal direction is counted from it. Source: channel a of the measurement, mind the sign.
Strain ε_b (45°) [µm/m]: Displayed strain of grid b, lying 45° counter-clockwise from a. It carries the information about the shear deformation that a and c alone cannot capture. Source: channel b; check grid labelling on the rosette or datasheet.
Strain ε_c (90°) [µm/m]: Displayed strain of grid c, lying exactly 90° from a. Grids a and c together form a perpendicular pair; their mean is the mean strain at the point. Source: channel c of the measurement, mind the sign.

Context

What this calculator is for

Experimental stress analysis on components with unknown principal directions: castings, welded structures, pressure vessels, chassis components.

What comes next

Check grid labelling a, b, c per Keil fig. 9.23 (b is 45° counter-clockwise from a).
Enter strains with sign; correct transverse sensitivity beforehand if needed.
Transfer the principal strains to the principal-stress calculator.

Results

How to read the results

ε₁ and ε₂ are the extreme strains at this point; all other directions lie in between. They feed the principal-stress calculator.
α shows where the largest strain points: rotate counter-clockwise by α from the measuring direction of grid a. The crack-prone direction is perpendicular to σ₁.
γ_max and the radius r describe the shear content; for r ≈ 0 the state is equi-biaxial (e.g. diaphragm centre) and the angle is meaningless.

Result quantities

What each value means

Centre m = (ε_a+ε_c)/2 [µm/m]: Centre of the strain circle = mean strain; it is the same in every direction and equals (ε₁+ε₂)/2.
Radius r [µm/m]: Radius of the strain circle: half the difference of the principal strains, a measure of the shear content of the state.
Principal strain ε₁ [µm/m]: Largest strain at this point, in principal direction 1 (angle α from grid a). The largest normal stress acts in this direction.
Principal strain ε₂ [µm/m]: Smallest strain, perpendicular to principal direction 1 (α + 90°). Can be negative (compression).
Maximum shear strain γ_max = 2r [µm/m]: Maximum shear strain, the greatest angular distortion of the material at this point. It occurs at 45 degrees to the principal directions. For stress assessment, continue with Young's modulus and Poisson's ratio in the principal-stress calculator.
Orientation angle α (from a to direction 1, counter-clockwise) [°]: Angle from the measuring direction of grid a (0°) to principal direction 1, counter-clockwise, unique between 0° and 180°. Principal direction 2 lies at α + 90°.
Shear strain γ_ac = 2ε_b − ε_a − ε_c [µm/m]: Shear strain referred to the directions of grids a and c. Zero means those perpendicular grids already lie in the principal directions; a large magnitude indicates that the strain state is strongly rotated relative to the rosette.

Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 9.8 evaluation of 0°/45°/90° rosettes, eqs. (9.48)–(9.66), Tab. 9.1, figs. 9.23, 9.25.

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

FAQ

Frequently asked questions

Why does Z = 0, N > 0 give α = 0°?

Then ε_b = m and ε_a > ε_c: direction a already is principal direction 1.

What is the third principal strain?

Normal to the surface: ε₃ = −ν/(1−ν)·(ε₁+ε₂) per eq. 9.66; the principal-stress calculator reports it.

Does the evaluation apply dynamically?

Yes, if all three grids are sampled simultaneously; otherwise the time offset distorts the angle (Keil 10.3.3).

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