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Strain gauges · Measurement

Torque Transducer Calculator for Shafts

Calculate shear stress, ±45° principal strains and the full-bridge signal of a solid or hollow shaft torque transducer from torque, diameters and material data.

DMSKeil 2017
01

Inputs

Circular or annular section without keyways, Saint-Venant torsion; misalignment of the 45° gauges reduces the signal and couples in bending.

02

Results

Enter values and run the calculation.

Method

What is calculated?

Pure torsion produces principal normal stresses at ±45° to the axis equal in magnitude to the shear stress, with opposite signs. Two perpendicular gauge pairs at 45°, wired alternately into a full bridge, add all four strain magnitudes and are insensitive to axial force, bending and temperature. Keil's example: hollow shaft 24/14 mm gives W_p = 2400 mm³; at 400 N·m τ = 167 N/mm² and ε = 1054 µm/m.

Equations

τ = M_t/W_p (Keil Gl. 10.35)

W_p = π/(16d_a)·(d_a⁴ − d_i⁴) (Gl. 10.36)

ε₁ = −ε₂ = τ·(1+ν)/E (Gl. 10.38, 10.39)

U_M/U_B = k·ε₁ (Vollbrücke, Gl. 10.41)

Limits

Assumptions and typical mistake

Circular or annular section without keyways, Saint-Venant torsion; misalignment of the 45° gauges reduces the signal and couples in bending.

Using longitudinal strain instead of 45° strain or computing τ with the bending section modulus.

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 torsion: 1 gauge pair at ±45° B = 2

Layout: One perpendicular gauge pair at ±45° to the shaft axis at one spot of the circumference, in adjacent bridge arms; two fixed resistors complete the bridge.

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

Use: When only one spot of the circumference is accessible or a single twin-grid gauge suffices.

  • One twin-grid gauge suffices; temperature, axial force and bending cancel within the pair.
  • Half the signal of the full bridge.
  • Asymmetric layout: bending is compensated only if both grids see the same bending strain.

Keil Abschn. 10.2.3

Full bridge torsion: 2 gauge pairs at ±45° B = 4

Layout: Two perpendicular gauge pairs at 45° to the shaft axis, 180° apart around the circumference; the two gauges of a pair sit in adjacent arms.

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

Use: The ideal arrangement for measuring torque.

  • All four magnitudes of the 45° principal strains add.
  • Insensitive to axial force, bending and temperature: both gauges of a pair are loaded alike and cancel.
  • Exact 45° alignment required; suppliers offer twin-grid (“herringbone”) gauges for it.
  • On rotating shafts signal transmission (slip ring, telemetry) is required.

Keil Abb. 4.16, Gl. 4.18; Abschn. 10.2.3

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

Torque M_t [N·m]
The torque twisting the shaft – rated torque of the transducer or expected service torque. Source: drive design (P = 2π·n·M), nameplate or comparison measurement. 1 N·m = 1 N at 1 m distance.
Outer diameter d_a [mm]
Outer diameter of the shaft where the gauges are bonded. It enters to the third power: a slightly thinner measuring section multiplies the signal. Source: drawing or calliper.
Inner diameter d_i (0 = solid) [mm]
Inner bore diameter for a hollow shaft; enter 0 for a solid shaft. A larger bore increases surface strain for the same torque but reduces strength and stiffness. Source: manufacturing drawing or measurement at an accessible end face, always for the measuring section.
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
One gauge pair at ±45° (half bridge) or two pairs on opposite sides of the shaft (full bridge, cancels bending and axial force – the standard circuit for torque shafts). Source: accuracy demand; with overhung loads or belt pull always use a full bridge.
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

Torque measuring shafts, drivetrain measurements and retrofit instrumentation of shafts, including slip-ring or telemetry transmission.

What comes next

  1. Enter torque, diameters and material data.
  2. Choose the full bridge with two 45° pairs 180° apart to compensate bending.
  3. Match the signal with the amplifier range and rated strain (≈ 1000 µm/m).
Results

How to read the results

  • ε₄₅ is the strain under one 45° gauge; the full-bridge signal is k·ε₄₅ because four equal magnitudes add.
  • Compare τ with the permissible shear stress of the shaft material; for transducers ε₄₅ ≈ 500–1000 µm/m at rated torque is usual.
Result quantities

What each value means

Polar section modulus W_p [mm³]
Polar section modulus of the solid or hollow shaft, the geometric resistance to torsional stress. It can convert another torque directly into surface shear stress; it contains no material properties.
Shear stress τ [N/mm²]
Shear stress at the shaft surface. It is also the magnitude of the principal normal stresses at ±45°.
Strain at 45° ε₁ = −ε₂ [µm/m]
Strain under a gauge bonded at 45° to the axis: +ε in one, −ε in the other 45° direction. A longitudinal gauge would read zero under pure torsion.
Output signal U_M/U_B [mV/V]
Output per volt of excitation; at rated torque the torque-transducer sensitivity.
Output voltage U_M [mV]
Raw signal in millivolts at the entered excitation.
Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.3 torsionally loaded objects, eqs. (10.35)–(10.41); sec. 4.5, eq. (4.18).

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

FAQ

Frequently asked questions

Why ε = τ(1+ν)/E?

From Hooke's law with σ₁ = −σ₂ = τ, ε₁ = (σ₁ − νσ₂)/E = τ(1+ν)/E, equivalent to τ/(2G).

Is one gauge pair enough?

As a half bridge yes, but bending is then not compensated; Keil recommends two opposite pairs.

How do long rotor leads matter?

Keil's fig. 10.20 shows leads should be placed in adjacent bridge arms so their resistance changes cancel.

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