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Strain gauges · Measurement

Bending Beam Strain Gauge Calculator

Calculate bending stress, strain at the gauge, deflection at the load point and bridge signal of a cantilevered rectangular beam with strain gauges.

DMSKeil 2017
01

Inputs

Bernoulli beam with rectangular section, gauge grid centred at x, no shear deformation, no notch effect at the clamp.

02

Results

Enter values and run the calculation.

Method

What is calculated?

On a cantilever the bending moment grows linearly towards the clamp; gauges are installed close to the clamp on top and bottom where tensile and compressive strains are equal in magnitude. Four alternately wired gauges add all individual signals (U_M/U_B = k·ε) and reject axial force and temperature. Keil's spring-tongue example: 32.6 N at a 30 mm lever arm gives ±1676 µm/m and 1.68 mm deflection.

Equations

σ_b = ±M_b/W_b = ±F·x/W_b (Keil Gl. 10.6)

W_b = b·h²/6 (Gl. 10.7)

ε = ±F·x/(E·W_b) (Gl. 10.11)

f = F·l³/(3EI), I = b·h³/12 (Gl. 10.10)

U_M/U_B = k·ε (Vollbrücke, Gl. 10.14) bzw. k·B·ε/4

Limits

Assumptions and typical mistake

Bernoulli beam with rectangular section, gauge grid centred at x, no shear deformation, no notch effect at the clamp.

Using the gauge distance instead of the free length for deflection, or swapping h and b (ε depends on h²).

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 gauge top + 1 gauge bottom B = 2

Layout: One longitudinal gauge each on the top and bottom face of the beam, in adjacent bridge arms (top +ε, bottom −ε).

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

Use: The ideal half bridge for bending measurements on bars.

  • Pure bending: axial force and temperature act equally on both gauges and cancel.
  • Twice the signal of a single gauge.
  • Both faces of the part must be accessible.
  • Install the gauges exactly opposite each other at equal distance from the neutral axis.

Keil Abb. 4.11, Gl. 4.13

Full bridge bending: 2 gauges top + 2 gauges bottom B = 4

Layout: Two longitudinal gauges each on top and bottom, wired alternately: gauges strained alike sit in opposite arms, oppositely strained ones in adjacent arms.

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

Use: Ideal for bending-loaded measuring elements and transducers (bending-beam load cells).

  • Largest bending signal: all four magnitudes add.
  • Fully self-compensated for temperature; axial force cancels.
  • Four gauges, both faces accessible, symmetric installation required.

Keil Abb. 4.14, Gl. 4.16

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 at the beam tip [N]
The force acting transversely at the free end of the beam and bending it. Source: rated load of the transducer or expected service force; sign is free (it only decides which side is in tension).
Distance gauge ↔ load point x [mm]
Distance from the load point to the centre of the strain-gauge grid, measured along the beam. This lever arm sets the bending moment at the measuring point, so a gauge nearer the clamp sees more strain. Source: technical drawing or direct measurement from load point to grid centre.
Free length l (for deflection) [mm]
Free beam length from the clamp to the load point. It is needed only for the tip deflection, not for the strain at the gauge; usually it is somewhat larger than the gauge distance x. Source: drawing or measurement.
Beam width b [mm]
Width of the rectangular section – the dimension across the bending direction, i.e. the face the gauge is bonded to. It enters linearly: double width = half strain. Source: calliper or drawing.
Beam thickness h [mm]
Beam thickness in the bending direction. It enters the strain squared: half the thickness = four times the strain. Source: calliper/drawing.
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.
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
One gauge on one side (quarter bridge), one each top and bottom (half bridge, cancels axial force and temperature) or two each top and bottom (full bridge, largest signal). Source: number of gauges available and accuracy demand.
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

Bending-beam load cells, spring tongues as force or displacement transducers, stress measurement on cantilevers and booms.

What comes next

  1. Enter beam geometry, force and the gauge distance from the load point.
  2. Check that σ_b stays below roughly 50 % of the yield strength (Keil).
  3. Choose the circuit and compare the signal with the amplifier range.
Results

How to read the results

  • ε is the surface strain at the gauge location, not at the clamp; the closer to the clamp, the larger.
  • Signal and deflection both scale linearly with force – the deflection checks whether the beam as a spring stays elastic.
Result quantities

What each value means

Section modulus W_b [mm³]
Bending section modulus of the rectangular section. It describes geometry alone and links bending moment to surface stress. It can be reused in other beam calculations for a strength check.
Bending stress σ_b at the gauge [N/mm²]
Bending stress in the outer fibre at the gauge location: tension on one side, compression on the other, equal magnitude.
Outer-fibre strain ε [µm/m]
Strain under the gauge on the top (+) or bottom (−) side. The bridge factor refers to this.
Deflection at the load point [mm]
Deflection at the load point – for plausibility and for use as a displacement transducer.
Output signal U_M/U_B [mV/V]
Output of the chosen circuit per volt of excitation at the entered force.
Output voltage U_M [mV]
Raw signal in millivolts at the entered excitation.
Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.2.2 cantilever beam, eqs. (10.6)–(10.14); sec. 4.5, eqs. (4.13), (4.16).

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

FAQ

Frequently asked questions

Why not measure right at the clamp?

Notch effects and clamp reactions disturb the field there; Keil recommends a small offset, accounted for in the lever arm x.

What changes with a half bridge?

The signal halves relative to the full bridge; axial force and temperature remain compensated.

Does it apply to simply supported beams?

Only for the cantilever moment distribution; for other supports determine M_b at the gauge separately (Keil 10.2.2.3/10.2.2.4).

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