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Solid Mechanics

Strain Gauge Rosette Simulator

Solve the complete in-plane strain state from the readings of a three-gauge "rosette". Enter each gauge reading εa, εb, εc and the in-plane strains εx, εy, γxy, the principal strains, the principal direction and the Hooke's-law principal stresses are computed in real time, with a Mohr's circle of strain.

Parameters
Rosette type
Switches the layout angles of the three gauges
Gauge A reading εa
μ
Reading of the reference (0°) gauge, in microstrain
Gauge B reading εb
μ
Reading of the middle (45° or 60°) gauge
Gauge C reading εc
μ
Reading of the third (90° or 120°) gauge
Young's modulus E
GPa
Elastic modulus of the material. ~200 GPa for steel
Poisson's ratio ν
Degree of lateral contraction. ~0.3 for steel
Results
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Normal strain εx (μ)
—
Normal strain εy (μ)
—
Shear strain γxy (μ)
—
Principal strain ε₁ (μ)
—
Principal strain ε₂ (μ)
—
Principal stress σ₁ (MPa)
—
Rosette layout & principal-strain directions

The three gauges A/B/C are drawn at their layout angles on a material patch, with the solved principal strain axes ε₁ and ε₂ as arrows. The patch deforms (exaggerated) according to the principal strains.

Mohr's circle of strain
Strain components (microstrain)
Theory & Key Formulas

In-plane strains of a rectangular rosette (0°-45°-90°):

$$\varepsilon_x=\varepsilon_a,\quad \varepsilon_y=\varepsilon_c,\quad \gamma_{xy}=2\varepsilon_b-\varepsilon_a-\varepsilon_c$$

εa, εb, εc: the three gauge readings. εx, εy: normal strains, γxy: shear strain.

Principal strains:

$$\varepsilon_{1,2}=\frac{\varepsilon_x+\varepsilon_y}{2}\pm\sqrt{\left(\frac{\varepsilon_x-\varepsilon_y}{2}\right)^2+\left(\frac{\gamma_{xy}}{2}\right)^2}$$

The principal angle is θp = ½·atan2(γxy, εx−εy). A delta rosette uses a 0°/60°/120° layout, giving εy=(2εb+2εc−εa)/3 and γxy=2(εb−εc)/√3.

Hooke's-law principal stresses (plane stress):

$$\sigma_{1}=\frac{E}{1-\nu^{2}}(\varepsilon_1+\nu\varepsilon_2),\quad \sigma_{2}=\frac{E}{1-\nu^{2}}(\varepsilon_2+\nu\varepsilon_1)$$

E: Young's modulus, ν: Poisson's ratio. Strains are computed in microstrain (×10⁻⁶).

What is a Strain Gauge Rosette?

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When we bond strain gauges to a test part, why use a three-gauge "rosette" instead of just one? One gauge measures strain, doesn't it?
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Good question. A single gauge measures only "the strain in the one direction that gauge stretches or contracts". But the surface state of a part is really defined by three numbers: εx, εy and γxy. Three unknowns need three equations. So you bond three gauges at different angles at the same point and collect three readings. That is a rosette.
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I see. But if you already knew the principal direction — the direction that stretches the most — couldn't you just put one gauge there?
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Right — if the principal direction is known for certain, two gauges along and across it are enough. The problem is that "on a real part, the principal direction is almost never known". Around holes, welds or complex shapes, the direction the load flows in cannot be guessed by intuition. With a rosette you just bond three gauges without guessing the direction, and the calculation recovers even the principal direction. That is why "use a rosette by default" is the field rule.
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The "Rosette type" on the left has rectangular and delta. How do I choose between them?
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A rectangular rosette places the gauges at 0°, 45° and 90°, so you read εx=εa and εy=εc directly and the formulas are simple. It suits cases where the x and y directions can be roughly imagined. A delta rosette uses 0°, 60° and 120°, evenly over three directions; accuracy stays good whatever way it points, so use it when the principal direction is completely unreadable. Switch the type — you'll see the formulas for εy and γxy change for the same readings.
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It also outputs the "principal stress σ₁". It's a strain gauge — how can it give a stress?
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That is an important point. What a strain gauge directly measures is only "strain". The stress is "converted" through Hooke's law. With σ₁=E/(1−ν²)·(ε₁+ν·ε₂), Young's modulus E and Poisson's ratio ν turn the principal strains into principal stresses. So if you don't enter the material's E and ν correctly, the strain is right but the stress numbers are off. Move ν on the left and watch σ₁ change.
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What is the "Mohr's circle of strain" below for?
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It is the strain version of the stress Mohr's circle. The circle's centre is the average strain (εx+εy)/2, and the radius R represents the shear contribution. The points where the circle crosses the horizontal axis are the principal strains ε₁ and ε₂. From how far the circle is tilted from the point (εx, γxy/2), you can read at a glance how much the principal axes have rotated. A single circle conveys the strain state far more intuitively than a list of numbers — it is a staple tool in experimental stress analysis.

Frequently Asked Questions

A single strain gauge measures only the strain in the one direction it is aligned with. A general plane-strain state, however, is defined by three independent components: εx, εy and γxy. Three unknowns require three equations. So three gauges at different angles are bonded at the same point to obtain three readings — this is a rosette. The three readings are solved simultaneously for εx, εy and γxy, from which the principal strains, principal direction and principal stresses follow uniquely.
It is the gauge layout angles. A rectangular rosette uses three gauges at 0°, 45° and 90°, giving very simple relations: εx=εa, εy=εc, γxy=2εb−εa−εc. It suits parts whose x and y directions are roughly known. A delta rosette uses 0°, 60° and 120°; with the gauges evenly spread over three directions, it stays accurate even when the principal direction is completely unknown. Both fully determine the in-plane strain state, and the final principal strains and stresses agree.
A strain gauge only ever measures strain — stress cannot be measured directly. Assuming plane stress, Hooke's law converts the principal strains into principal stresses: σ1=E/(1−ν²)·(ε1+ν·ε2) and σ2=E/(1−ν²)·(ε2+ν·ε1). E is Young's modulus and ν is Poisson's ratio. The key point is that a correct stress requires accurate values of the material's E and ν. If the material is unknown, you can still measure strain but you cannot convert it to stress.
The principal direction is θp=½·atan2(γxy, εx−εy), the rotation from the x-axis to the ε1 principal axis. Using atan2 gives the correct angle including the quadrant that a plain atan cannot distinguish. A positive θp means the principal axes are rotated counter-clockwise, a negative θp clockwise. When the shear strain γxy is zero, θp=0 and the x and y axes are themselves the principal axes.

Real-World Applications

Field strain measurement on machine parts: Rosettes are bonded to the surface of in-service parts — pressure vessels, piping, rotating shafts, frames and brackets — to record strain during operation. At holes, shoulders and welds where stress concentrations occur, the principal direction cannot be read by intuition, so a rosette that resolves the direction with three gauges is indispensable. The resulting principal stresses are compared with the material's allowable stress and fatigue limit to assess safety.

FEM validation (correlation with experiment): Whether the stress predicted by a finite-element analysis is truly correct is ultimately confirmed by measurement. A rosette is bonded at a representative point on a specimen, the test is run, and the measured principal strains and stresses are compared with the FEM result at the same location. Agreement supports the boundary conditions and mesh; an order-of-magnitude difference points to a load or constraint setup error. A hand-calculation-level conversion like this tool is the basis of that correlation.

Residual stress measurement (hole-drilling method): Residual stresses left inside a part by welding or machining are estimated by drilling a small hole at the centre of a rosette and reading the strain change in three directions before and after drilling (per standards such as ASTM E837). The released residual strain is back-calculated from the differences in the three gauge readings — a flagship application of rosette analysis.

Structural health monitoring: Permanently bonded rosettes on bridges, cranes, wind turbines and aircraft structures record the history of principal strains over long periods. This allows early detection of unexpected loads or progressing fatigue damage, feeding inspection planning and life assessment. Handling multi-axial states is a strength that single-axis gauges lack.

Common Misconceptions and Pitfalls

The biggest misconception is assuming that "because the gauge reading εb is the normal strain in the 45° direction, the principal strain must coincide with one of the gauges". εa, εb and εc are simply "the normal strain in each gauge direction", which is different from the principal strains ε₁ and ε₂. The principal strains usually coincide with no gauge direction at all. As this tool shows, for the default values εa=600 gives ε₁≈612 and ε₂≈−212 — different from the gauge readings themselves. Confusing "measured value = principal strain" leads to a dangerous underestimate.

Next is the mistake of using the rectangular-rosette relation γxy=2εb−εa−εc for a delta rosette as well. The formulas that recover the in-plane strains from the readings depend entirely on the gauge layout angles. The coefficients for 0°/45°/90° and 0°/60°/120° are completely different; a delta rosette must use εy=(2εb+2εc−εa)/3 and γxy=2(εb−εc)/√3. Mixing up the type gives badly wrong principal strains and stresses for the same readings. In experiments, always confirm "which layout the bonded rosette is" first.

Finally, beware the loose phrase "a strain gauge measures stress". What the gauge physically senses is only strain (stretch and contraction); the stress is a conversion result using E and ν. Apparent strain from temperature change (thermal output), poor bonding, transverse sensitivity, and the material's non-linear or plastic range all break this conversion. In particular, after yielding Hooke's law itself fails, so outside the plane-stress, linear-elastic assumptions of this tool (small strain, before yield), the stress values it produces must not be taken at face value.

How to Use

  1. Input the three gauge readings (εₐ, εᵦ, εᶜ) in microstrain from your 45° rosette configuration. Standard rosettes use gauges at 0°, 45°, and 90° orientations.
  2. Enter the material's Young's modulus (E) in GPa. For steel use 200 GPa, aluminum 70 GPa, or titanium 105 GPa.
  3. The simulator calculates strain components εₓ, εᵧ, and γₓᵧ, then derives principal strains ε₁ and ε₂, and maximum principal stress σ₁ using Mohr's circle transformation.

Worked Example

A 304 stainless steel pressure vessel wall (E = 193 GPa, ν = 0.29) has rosette readings: εₐ = 450 μ, εᵦ = 280 μ, εᶜ = 120 μ. The simulator yields εₓ = 450 μ, εᵧ = 120 μ, γₓᵧ = 320 μ. Principal strain ε₁ = 520 μ, ε₂ = 50 μ. Using σ₁ = E·ε₁/(1-ν²), principal stress σ₁ ≈ 95 MPa, indicating biaxial tensile stress typical of hoop stress loading.

Practical Notes

  1. Zero your gauges before loading. Temperature compensation is critical—a 50°C rise causes ~2300 μ apparent strain in Inconel without correction.
  2. For shear-dominated loading (γₓᵧ > εₓ), verify gauge alignment to within ±0.5° or shear calculations will show 10% error.
  3. Negative principal strain indicates compression; use |σ| for buckling checks on thin composite panels.