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Strain gauges · Measurement

Axial Force Transducer Calculator

Calculate strain, stress and bridge output of an axially loaded strain-gauge force transducer, including sensitivity in mV/V at rated force.

DMSKeil 2017
01

Inputs

Homogeneous uniaxial stress without notches; superimposed bending is compensated only in the full bridges, torsion does not affect longitudinal gauges.

02

Results

Enter values and run the calculation.

Method

What is calculated?

For a slender, axially loaded body the axial direction is principal direction 1. Two longitudinal and two transverse gauges wired alternately into a full bridge give the largest signal (B = 2(1+ν)) and cancel temperature and bending. Keil's force-transducer example: ν = 0.29 gives B = 2.58; with k = 2.1 and ε_l = 956 µm/m the sensitivity is 1.295 mV/V at 10 kN.

Equations

σ = F/A, ε_l = σ/E, ε_q = −ν·ε_l

U_M/U_B = k/4·(ε_l + νε_l + ε_l + νε_l) = k/4·ε_l·(2+2ν) (Keil Gl. 10.1, 10.2)

allgemein U_M/U_B = k·B·ε_l/4

Limits

Assumptions and typical mistake

Homogeneous uniaxial stress without notches; superimposed bending is compensated only in the full bridges, torsion does not affect longitudinal gauges.

Adding the transverse strain as a magnitude instead of through B, or entering A in cm².

Gauge layout

Where the gauges sit and what the bridge makes of them

One card per selectable circuit, in the order of the selector above: first, where the grids sit on the part and in which direction they are strained, then the Wheatstone bridge with Keil's arm occupancy – strains in opposite arms add, strains in adjacent arms cancel.

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.

Quarter bridge: 1 longitudinal gauge B = 1

Layout: One longitudinal gauge on the part; the other three bridge arms are fixed resistors in the instrument.

1RRRU_MU_MU_BU_BR = fixed resistor
  • F in the signal
  • M_b in the signal
  • M_t insensitive
  • ΔT not compensated

Use: Only at constant temperature and for a known uniaxial stress state along the principal direction.

  • Simplest installation, one gauge only.
  • Smallest measuring spot.
  • Temperature response enters the result additively – even with a four-wire connection.
  • Axial force and bending appear together in the signal.

Keil Abb. 4.8, Gl. 4.10

Half bridge: 1 longitudinal + 1 transverse gauge B = 1 + ν

Layout: Two gauges at the same spot, one along and one across the bar axis, in adjacent bridge arms. The transverse gauge measures −ν·ε.

12RRU_MU_MU_BU_BR = fixed resistor
  • F in the signal
  • M_b in the signal
  • M_t insensitive
  • ΔT compensated

Use: When a separate compensation gauge is inconvenient or impossible; uniaxial stress only.

  • Temperature compensation without a separate compensation block.
  • Signal larger than a single gauge by (1+ν).
  • The signal is the sum of both magnitudes – longitudinal and transverse strain can be separated only for uniaxial stress in the principal direction.
  • Bending remains in the signal.

Keil Abb. 4.10, Gl. 4.12

Full bridge: 2 longitudinal gauges opposite + 2 compensation gauges B = 2

Layout: Active longitudinal gauges on top and bottom in opposite arms; the two adjacent arms carry compensation gauges on unloaded blocks.

1K2KU_MU_MU_BU_B
  • F in the signal
  • M_b compensated
  • M_t insensitive
  • ΔT compensated

Use: Measuring axial force when bending is superimposed and transverse gauges cannot be used.

  • Bending cancels in the opposite arms, axial force counts twice.
  • Temperature compensated through the two compensation gauges.
  • Two extra compensation gauges on blocks of the same material, at the same temperature as the measuring point.
  • Without the compensation gauges (circuit V) the temperature response stays in the signal twice.

Keil Abb. 4.13, Gl. 4.15

Full bridge axial: 2 longitudinal + 2 transverse gauges B = 2(1 + ν)

Layout: One longitudinal and one transverse gauge each on top and bottom; the longitudinal gauges sit in opposite bridge arms.

1234U_MU_MU_BU_B
  • F in the signal
  • M_b compensated
  • M_t insensitive
  • ΔT compensated

Use: The typical circuit of axially loaded force-measuring elements and simple force transducers.

  • Fully self-compensated for temperature; bending cancels.
  • Transverse gauges add ν-fold signal without separate compensation gauges.
  • Only in uniaxial stress is the transverse strain −ν·ε_l; notches or load introduction near the gauges distort it.
  • Four gauges on both faces.

Keil Abb. 4.15, Gl. 4.17

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

Force F [N]
The force loading the measuring body along its axis – for transducer design the rated force, for a component measurement the expected service force. Tension positive, compression negative. Source: specification, design calculation or comparison measurement. 1 kN ≈ weight of 100 kg.
Cross-sectional area A [mm²]
Load-bearing cross-sectional area exactly where the strain gauges are bonded. Derive it from diameter for a round bar or outer and inner diameters for a tube. Area sets stress and strain and is the main design variable for a transducer. Source: technical drawing or measured dimensions; subtract holes and cut-outs.
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.
Gauge factor k
The gauge factor is the sensitivity of the strain gauge: it states how much the electrical resistance changes when the gauge is stretched (ΔR/R = k·ε). Without it no strain can be computed from an electrical signal. Source: printed on every gauge package or in the manufacturer datasheet, typically 2.0 to 2.1 for constantan foil gauges; valid at room temperature. The same value must be set in the amplifier.
Gauge circuit
Wiring of the gauges on the tension rod: one longitudinal gauge (simple, but bending and temperature interfere), half bridge longitudinal+transverse (temperature compensated), full bridge with two opposite longitudinal gauges plus two compensation gauges (compensates bending and temperature) or the classic force-transducer circuit with two longitudinal and two transverse gauges (largest signal, everything compensated). Source: decide by available channels and disturbances.
Excitation U_B [V]
The bridge excitation is the DC or AC voltage with which the amplifier supplies the bridge circuit. The signal is proportional to it – which is why signals are usually stated as a ratio in mV/V and the excitation is only needed for the absolute signal voltage. Source: amplifier setting, usually 2.5 V, 5 V or 10 V; the upper limit is set by gauge self-heating (see the excitation calculator).
Context

What this calculator is for

Designing self-built force transducers and tie rods, assessing signal yield in rod or anchor force measurements and converting amplifier signals into forces.

What comes next

  1. Enter rated force and measuring section; strain at rated load should be about 500–1500 µm/m.
  2. Choose material data and circuit.
  3. Match the sensitivity to the amplifier input range (typically 2 mV/V) and adjust the section if needed.
Results

How to read the results

  • ε is the axial strain under the longitudinal gauges, ε_q the one under the transverse gauges; both belong to the same location.
  • The signal at rated force is the transducer sensitivity: if well below 1 mV/V, reduce the section; above 2–3 mV/V the strain is usually too high for fatigue life.
Result quantities

What each value means

Axial stress σ [N/mm²]
Mean tensile or compressive stress in the measuring section. For mechanical design compare it with allowable stress or material yield strength, retaining adequate margin for overload, fatigue and temperature effects.
Longitudinal strain ε_l [µm/m]
Strain along the force under each longitudinally bonded gauge. This is the quantity the bridge factor refers to.
Transverse strain ε_q = −ν·ε_l [µm/m]
Strain transverse to the force (lateral contraction), i.e. what a transversely bonded gauge measures: same spot, opposite sign, ν times smaller.
Bridge factor B
Bridge factor of the chosen circuit; 2(1+ν) for 2 longitudinal + 2 transverse.
Output signal U_M/U_B [mV/V]
Output at the entered force per volt of excitation. Entered at rated force, this is the transducer sensitivity (usually 1–2 mV/V).
Output voltage U_M [mV]
Raw signal in millivolts at the amplifier input at the entered excitation.
Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.1 axially loaded objects, eqs. (10.1)–(10.5); sec. 4.5, eq. (4.17).

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

FAQ

Frequently asked questions

Why use transverse gauges?

They raise the signal by (1+ν) and provide temperature compensation within the bridge.

How do I reject bending?

Pairs of opposite longitudinal gauges in opposite bridge arms cancel bending components.

What rated strain is usual?

Commercial force transducers work at about 1000 µm/m at rated load to ensure fatigue life and linearity.

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