Shaft: axial force, bending moment and torque from three gauge bridges

What is calculated?

In sec. 10.4 Keil describes how three differently wired full bridges on a rod-shaped body separate the load components: the axial bridge (Fig. 10.38a) responds only to tension/compression, the bending bridge (b) only to bending, the torsion bridge (c) only to twist – in each case because the other components cancel in adjacent bridge arms. Hooke's law turns the three strains into stresses, and area, section modulus and polar section modulus turn them into the force and the two moments (Keil: N_z = σ·A, M = σ·W, M_d = τ·W_p). To judge the material effort, the von Mises equivalent stress (Keil eq. 13.1) combines normal and shear stress.

Equations

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

σ_N = E·ε_N, N = σ_N·A (Keil Abschn. 10.2.1, 10.4)

σ_b = E·ε_b, M_b = σ_b·W_b, W_b = π/(32·d_a)·(d_a⁴ − d_i⁴) (Keil Gl. 10.6, Abschn. 10.4; W_b = W_p/2)

τ = E·ε₄₅/(1+ν), M_t = τ·W_p, W_p = π/(16·d_a)·(d_a⁴ − d_i⁴) (Keil Gl. 10.35–10.38)

σ_v = √((σ_N+σ_b)² + 3τ²) (aus Keil Gl. 13.1 mit σ₃ = 0)

Assumptions and typical mistake

Circular or annular section without notches at the gauge location; bending about one axis (the bending bridge must lie in the bending plane, otherwise it measures only one component). Ideal separation of the components presupposes exact geometry and gauge alignment – Keil points to crosstalk that only a calibration captures (see the crosstalk-correction calculator). Linear-elastic.

Entering the sum of the bridge strains instead of the strain of one gauge (error factor 2 to 4); placing the bending bridge 90° away from the bending plane (zero reading despite bending); using a gauge factor in signal mode different from the one set in the amplifier.

What you enter – and where the values come from

What is your measured value?

Choose which numbers you have. If the amplifier already shows a strain in µm/m for each bridge (micrometres per metre; 1000 µm/m = 0.1 % change of length), choose strain – then enter the strain of one single gauge per bridge. If it shows the raw bridge signals in mV/V (millivolts of output per volt of excitation), choose signal; gauge factor and bridge layout are then needed for the conversion.

Strain, axial bridge ε_N [µm/m]

The longitudinal strain of one single gauge of the axial bridge (Keil Fig. 10.38a: longitudinally bonded gauges wired so that bending cancels). Positive = tension, negative = compression. Source: the amplifier reading for this bridge after zero balancing with the part unloaded.

Strain, bending bridge ε_b [µm/m]

The outer-fibre strain of one single gauge of the bending bridge (Keil Fig. 10.38b: longitudinally bonded gauges top and bottom in the bending plane, wired so that axial force cancels). Positive on the tension side. Source: the amplifier reading for this bridge after zero balancing.

Strain, torsion bridge ε₄₅ [µm/m]

The strain of one single gauge of the torsion bridge bonded at 45° to the shaft axis (Keil Fig. 10.38c: two ±45° pairs opposite each other, axial force and bending cancel). Sign gives the direction of rotation. Source: the amplifier reading for this bridge after zero balancing.

Signal, axial bridge [mV/V]

The raw signal of the axial bridge in mV/V (output divided by excitation), signal mode only. The calculator converts it with gauge factor and the chosen bridge layout into the strain of one gauge. Source: amplifier in mV/V mode after zero balancing.

Signal, bending bridge [mV/V]

The raw signal of the bending bridge in mV/V, signal mode only. The bending bridge is a full bridge with two gauges top and two bottom, bridge factor 4 (Keil eq. 10.14: U_M/U_B = k·ε). Source: amplifier in mV/V mode after zero balancing.

Signal, torsion bridge [mV/V]

The raw signal of the torsion bridge in mV/V, signal mode only. The torsion bridge is a full bridge of two ±45° pairs, bridge factor 4 (Keil eq. 10.41: U_M/U_B = k·ε₄₅). Source: amplifier 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.

Layout of the axial bridge

How the axial bridge is built – relevant only in signal mode. Two longitudinal and two transverse gauges (Keil's support rod, sec. 10.2.1) have bridge factor 2(1+ν) because the transverse gauges also measure the lateral contraction. Two opposite longitudinal gauges with two unloaded compensation gauges have bridge factor 2. 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 area squared and the section moduli to the third power – small measuring errors affect the moments noticeably. 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. The bore reduces the area more than the section moduli because the material at the centre carries little in bending and torsion anyway. Source: drawing or 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

Service-load measurement on drive and gearbox shafts, robot links and grippers (Keil's six-bridge example), drill strings, axles and journals where tension/compression, bending and torsion occur simultaneously.

What comes next

1. Install three full bridges per Fig. 10.38: axial and bending bridge with longitudinally bonded gauges, torsion bridge with ±45° gauges; balance all with the part unloaded.
2. Under load read the three strains or signals and enter them, enter the shaft diameters at the gauge location and the material data.
3. Compare N, M_b, M_t with the expected loads; check σ_v against the yield strength (with a safety factor).

How to read the results

• The first three results are the loads (force, bending moment, torque), the next three the corresponding stresses, the last two combine the stresses for the strength check.
• Each bridge delivers exactly one component – the results are only as cleanly separated as the installation. If the torsion bridge shows a deflection under pure bending, crosstalk is present (Keil sec. 10.4); then use the crosstalk-correction calculator.
• On rotating shafts the bending stress is alternating, axial and torsional stresses are usually static or pulsating – consider them separately for a fatigue check (e.g. DIN 743).

What each value means

Axial force N [kN]

The axial force the shaft transmits at the gauge location (axial stress times area). Positive = tension, negative = compression. Compare with the expected axial load (e.g. from helical gearing or preload).

Bending moment M_b [N·m]

The bending moment at the gauge location in the plane of the bending bridge (bending stress times section modulus). It depends on the location – closer to a bearing or the load introduction it differs. Compare with the expected moment from bearing distance and transverse force.

Torque M_t [N·m]

The torque the shaft transmits (shear stress times polar section modulus) – usually the most important quantity. Compare with the drive's rated torque; power = 2π·n·M_t/60.

Axial stress σ_N [N/mm²]

The tension or compression stress from the axial force, uniform over the section. Usually small compared with the bending stress.

Bending stress σ_b (outer fibre) [N/mm²]

The bending stress at the outer fibre – tension on one side, compression on the opposite side. On a rotating shaft it changes sign every revolution: this is the alternating stress that governs fatigue.

Shear stress τ [N/mm²]

The shear stress from torsion at the surface, computed from the 45° strain with the factor E/(1+ν). Compare with the allowable torsional stress.

Largest normal stress |σ_N|+|σ_b| [N/mm²]

The largest normal stress on the circumference: where axial and bending stress act in the same sense their magnitudes add. This is the worst point of the section.

Equivalent stress σ_v (von Mises) [N/mm²]

The von Mises equivalent stress combines the largest normal stress and the shear stress into one number that can be compared directly with the yield strength from the tensile test (Keil ch. 13). Safety = yield strength / σ_v; below about 1.5 it gets critical.

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.4 simultaneous measurement of several load components, Figs. 10.38–10.40; secs. 10.2.1, 10.2.2 eq. (10.6), 10.2.3 eqs. (10.35)–(10.38); ch. 13, eqs. (13.1)–(13.3); sec. 4.2, eq. (4.8).

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

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