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Strain gauges · Measurement

Shear Force Transducer Calculator (45° Gauges)

Calculate mean shear stress in the neutral fibre, 45° strains and the half- or full-bridge signal from shear force and web area.

DMSKeil 2017
01

Inputs

Mean shear stress of thin webs; for solid sections τ_max = 1.5·F/A (rectangle) or 4/3·F/A (circle). Gauges must sit in the neutral fibre.

02

Results

Enter values and run the calculation.

Method

What is calculated?

At the mid-web of a beam the bending stress is zero and the shear force produces pure shear; principal strains lie at ±45° with equal magnitude and opposite sign. Gauge pairs at 45° on both sides of the web, wired into a full bridge, measure shear force largely independent of bending moment – the shear-beam load-cell principle. Keil uses τ ≈ F/A_web for thin-walled sections.

Equations

τ_max ≈ F/A_Steg (Keil Gl. 10.42)

ε₁ = −ε₂ = τ(1+ν)/E

U_M/U_B = k/4·(4ε) = k·ε (Vollbrücke, Gl. 10.43)

Limits

Assumptions and typical mistake

Mean shear stress of thin webs; for solid sections τ_max = 1.5·F/A (rectangle) or 4/3·F/A (circle). Gauges must sit in the neutral fibre.

Using the whole section instead of the web area, or installing gauges off the neutral fibre where bending strain superimposes.

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.

Half bridge: 1 pair +45°/−45° on one web face B = 2

Layout: One perpendicular gauge pair at ±45° at mid-web on one face, in adjacent arms; two fixed resistors.

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

Use: Shear force on beams whose web is accessible from one side only.

  • Both 45° principal strains add; temperature compensated.
  • At mid-web the bending stress is zero – the signal is largely independent of the bending moment.
  • Half the signal of the full bridge.

Keil Abschn. 10.2.4, Gl. 10.43

Full bridge: 1 pair at ±45° on each web face B = 4

Layout: One gauge pair at ±45° at mid-web on each face of the web, wired alternately into a full bridge.

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

Use: The principle of the shear-beam load cell and of shear-type force transducers.

  • Largest signal, fully temperature compensated.
  • Shear beams are robust, stiffer than bending beams and insensitive to side loads.
  • Pure shear causes no volume change – an advantage for precision transducers.
  • Both web faces must be accessible; the approximation τ ≈ F/A_web holds for thin-walled sections.

Keil Abschn. 10.2.4, Abb. 10.25

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

Shear force F [N]
The shear force transmitted in the beam section under consideration (e.g. the support reaction of a beam or the load of a shear-beam load cell). Source: statics of the system or rated load of the transducer.
Web height h [mm]
Height of the web, i.e. the thin vertical plate carrying most of the shear force in I or box sections (the flanges mainly carry bending). Source: section table or drawing; for rolled sections the clear height between the flanges.
Web thickness t [mm]
Web thickness, i.e. the wall thickness of the vertical plate. Force divided by the web area h·t gives the mean shear stress the gauges sense. Source: section table or 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.
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
Strain-gauge arrangement: one pair at plus/minus 45 degrees on one web face as a half bridge, or one pair on each face as a full bridge. The full bridge rejects temperature and bending across the web more effectively. Source: bonding and wiring plan, or the datasheet/connection diagram of a finished shear beam.
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

Shear-beam load cells, shear-force measurement on beams, bearing-load determination on axles and crane runways, support reactions in civil engineering.

What comes next

  1. Enter shear force and web dimensions at the measuring section.
  2. For precise values use τ = F·S/(I·t) with the first moment of area and compare.
  3. Choose the circuit; the full bridge compensates bending and temperature.
Results

How to read the results

  • ε₄₅ applies to gauges exactly in the neutral fibre; further out bending strain superimposes, which the full bridge compensates only for a symmetric layout.
  • The signal is proportional to shear force and independent of the load position along the beam – which makes shear-beam load cells insensitive to load placement.
Result quantities

What each value means

Mean shear stress τ = F/A_web [N/mm²]
Mean shear stress in the web at the neutral fibre (approximation F/A_web for thin-walled sections).
Strain at 45° ε₁ = −ε₂ [µm/m]
Strain under a gauge bonded at 45° to the beam axis at mid-web (+ε in the tension direction, −ε perpendicular).
Shear strain γ = 2ε₁ [µm/m]
Shear strain: angular distortion of a small web element under shear force. It equals the difference between the plus/minus 45-degree strains and can be used for plausibility checks or comparison with a simulation.
Output signal U_M/U_B [mV/V]
Output of the circuit per volt of excitation at the entered shear force.
Output voltage U_M [mV]
Raw signal in millivolts at the entered excitation.
Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.4 shear-loaded objects, eqs. (10.42), (10.43).

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 shear measurement independent of bending?

In the neutral fibre bending strain is zero and the ±45° layout responds only to shear strain.

Can I design a load cell with it?

Yes, roughly: choose the web area so that ε at rated load is about 500–1000 µm/m.

What is the shear strain γ?

The angular distortion of the element; it equals the diameter of the strain circle, γ = ε₁ − ε₂ = 2ε₁.

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