Torque on a shaft from measured strain

What is calculated?

In pure torsion the shaft surface carries principal normal stresses at ±45° to the axis equal in magnitude to the shear stress; a gauge at 45° therefore measures ε₄₅ = τ(1+ν)/E. This relation can be inverted: the measured strain gives the shear stress, and the polar section modulus gives the torque. Keil's example run backwards: 1054 µm/m on a 24/14 mm hollow shaft with E = 206 000 N/mm² gives τ ≈ 167 N/mm² and M_t ≈ 400 N·m.

Equations

ε₄₅ = 4·(U_M/U_B)/(k·B) (Keil Gl. 4.8, nur Modus Signal)

τ = ε₄₅·E/(1+ν) (Umkehrung von Keil Gl. 10.38)

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

M_t = τ·W_p (Umkehrung von Keil Gl. 10.35)

Assumptions and typical mistake

Pure torsion of a circular or annular section without notches at the gauge location; superimposed bending or axial force is rejected only by the full bridge with two opposite pairs. Linear-elastic; temperature response and lead resistance not included.

Entering a strain measured along the shaft – it is zero in pure torsion and yields no torque; or choosing the wrong bridge factor in signal mode (a factor-of-2 or -4 error).

What you enter – and where the values come from

What is your measured value?

Choose which number you have. Modern amplifiers in strain-gauge mode display a strain directly in µm/m (micrometres per metre, 1000 µm/m = 0.1 % change of length) – then choose strain. If the instrument only shows the raw Wheatstone-bridge signal in mV/V (millivolts of output per volt of excitation), choose signal; the calculator then also needs the gauge factor and bridge circuit.

Measured strain at 45° ε₄₅ [µm/m]

The strain shown by a strain gauge bonded at 45° to the shaft axis. For a full bridge of two pairs this is the strain of one single gauge, not the sum. Positive when the gauge lies in the tensile direction; the sign gives the direction of the torque. Source: the amplifier reading after zero balancing with the shaft unloaded.

Bridge signal U_M/U_B [mV/V]

The raw bridge signal: Wheatstone-bridge output voltage divided by the excitation, in mV/V. A torque transducer typically gives 1–2 mV/V at rated torque. Source: the amplifier reading in mV/V mode after zero balancing.

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.

Bridge circuit (signal mode only)

How the 45° gauges are wired – this sets the bridge factor B, i.e. how many times the strain of one gauge is contained in the signal: 1 gauge = B 1, one pair (+45°/−45°) in a half bridge = B 2, two opposite pairs in a full bridge = B 4. Relevant only in signal mode. Source: your own wiring or the measuring shaft's datasheet.

Outer diameter d_a [mm]

The shaft's outer diameter exactly where the gauges are bonded. It enters the torque to the third power – a 1 % error in diameter is a 3 % error in torque. Source: calliper or drawing.

Inner diameter d_i (0 = solid) [mm]

The bore diameter if the shaft is hollow; enter 0 for a solid shaft. A bore removes material that carries little in torsion anyway (the centre is almost stress-free), so it changes the torque less than the outer diameter does. Source: drawing or a measurement at the end face.

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.

What this calculator is for

Retrofit torque measurement on drive and gearbox shafts, test-rig evaluation, calibration checks of torque-measuring shafts and determination of the actual service load when only a strain signal is available.

What comes next

1. Install gauges at 45° to the shaft axis and balance the bridge with the shaft unloaded.
2. Under load, read the strain (or the mV/V signal) and enter it here; choose the matching mode.
3. Enter the diameters at the gauge location and the material data; compare the resulting M_t with the allowable or expected torque.

How to read the results

• The calculator works backwards: from what you measure (strain or signal) to what you want to know (torque). The intermediate shear stress is at the same time the value for the strength comparison.
• The sign of the torque follows the sign of the strain – which direction of rotation is positive depends on which gauge of the full bridge sits in arm 1; check once with a known rotation direction if needed.
• "Torque per 1 µm/m" is the conversion number for all further readings from the same measuring point as long as diameter and material stay the same.

What each value means

Strain at 45° used [µm/m]

The strain of one single 45° gauge used in the calculation – in strain mode your entered value, in signal mode the strain converted from mV/V, gauge factor and bridge factor. Lets you check the bridge factor was chosen correctly.

Surface shear stress τ [N/mm²]

The shear stress at the shaft surface that produced this strain. Compare with the material's allowable torsional stress (e.g. τ_allow from Roloff/Matek, or a safety factor against yield).

Polar section modulus W_p [mm³]

The polar section modulus of the section – a pure geometry figure linking shear stress and torque. The larger it is, the more torque the same stress represents.

Torque M_t [N·m]

The torque the shaft actually transmits at the gauge location – the calculator's real answer. Sign as the strain. Compare with the drive's rated torque or the allowable torque; multiply by speed for power (P = 2π·n·M_t/60).

Torque per 1 µm/m [N·m]

The sensitivity of the measuring point: this much torque corresponds to 1 µm/m of strain. Handy as a calibration factor for the amplifier or for quick mental conversion of further readings.

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.3 torsionally loaded objects, eqs. (10.35)–(10.41); sec. 4.2, eq. (4.8); sec. 9.4, eq. (9.25).

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

Frequently asked questions

In the same cluster

• Strain-gauge basic equation: resistance change from strain and gauge factor
• Transverse sensitivity: correction for two perpendicular gauges
• Temperature response: apparent strain of a gauge under temperature change
• Bridge excitation voltage and gauge self-heating
• Wheatstone bridge: output voltage of the unbalanced bridge
• Basic strain-gauge bridge equation: signal from four strains

All strain-gauge calculators