Strain gauges · Measurement

Bending moment and bending force from measured strain

Work back from the longitudinal strain measured on the tension or compression side of a beam – or from the bridge signal – to the bending stress, the bending moment at the gauge location and, via the lever arm, the force that caused it.

DMSKeil 2017
01

Inputs

Straight beam of constant section at the gauge location, force perpendicular to the beam, bending about a principal axis of the section. Gauges on the outer fibre (surface) and away from the load introduction. Superimposed axial force is cancelled only by the top/bottom half and full bridges; a single gauge measures it too. Linear-elastic.

02

Results

Enter values and run the calculation.

Method

What is calculated?

The largest longitudinal strain occurs at the surface of a bent beam: tension on one side, compression on the other, of almost equal magnitude. By Hooke's law the bending stress is E·ε_b, and Keil eq. (10.6) links it via the section modulus to the bending moment at the gauge: M_b = σ_b·W_b. If the force acts at distance x, F = M_b/x. Keil's bending spring (1 × 10 mm of 50CrV4, E = 210 000 N/mm², gauges 18 mm from the load point) gives exactly the limit force ±32.6 N at ±1676 µm/m; the full bridge of two gauges top and bottom produces k·ε = 3.44 mV/V.

Equations

ε_b = 4·(U_M/U_B)/(k·B) (Keil Gl. 4.8 bzw. 10.12–10.14, nur Modus Signal)

σ_b = E·ε_b (Hookesches Gesetz, Keil Gl. 9.24)

W_b = b·h²/6 (Keil Gl. 10.7) bzw. π·d³/32 (Kreis, Dubbel C 2.4)

M_b = σ_b·W_b (Umkehrung von Keil Gl. 10.6)

F = M_b/x (Keil Gl. 10.6: M_b = F·x)

Limits

Assumptions and typical mistake

Straight beam of constant section at the gauge location, force perpendicular to the beam, bending about a principal axis of the section. Gauges on the outer fibre (surface) and away from the load introduction. Superimposed axial force is cancelled only by the top/bottom half and full bridges; a single gauge measures it too. Linear-elastic.

Swapping width and height – the height in the bending direction enters squared; measuring the lever arm to the fixed end instead of to the load point; a gauge bonded at the load point or inside the clamp gives wrong values due to local stress disturbances.

Inputs

What you enter – and where the values come from

What is your measured value?
Choose which number you have. If the amplifier already shows a strain in µm/m (micrometres per metre; 1000 µm/m = 0.1 % change of length), choose strain. If it only shows the raw Wheatstone-bridge signal in mV/V (millivolts of output per volt of excitation), choose signal; gauge factor and bridge circuit are then needed for the conversion.
Measured outer-fibre strain ε_b [µm/m]
The longitudinal strain shown by a gauge bonded on the beam surface along the beam. On the stretched side it is positive, on the compressed side negative – in bending both of equal magnitude. For half or full bridges enter the strain of one single gauge, not the sum. Source: the amplifier reading after zero balancing with the beam unloaded.
Bridge signal U_M/U_B [mV/V]
The raw bridge signal: Wheatstone-bridge output divided by the excitation, in mV/V. Keil's bending spring gives 3.44 mV/V at limit load. 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
How the gauges are wired – this sets the bridge factor B, i.e. how many times the outer-fibre strain is contained in the signal. One gauge: B = 1. One gauge top and one bottom in adjacent arms: the opposite strains add, B = 2. Two top and two bottom (Keil's bending spring, Fig. 10.10): B = 4, plus temperature and axial load compensated. Source: your own wiring or datasheet.
Cross-section
How the section modulus W_b is determined – the geometry figure linking bending stress and bending moment. Rectangle from width and height, circle from the diameter, or directly from a section table (e.g. for I-beams or tubes). Only the section exactly at the gauge position counts.
Width b [mm]
The width of the rectangular section across the bending direction – the dimension of the face the gauge is bonded on (Keil's spring: 10 mm). Enters linearly. Source: calliper or drawing.
Height h [mm]
The height of the rectangular section in the bending direction, i.e. the distance from the tension to the compression side (Keil's spring: 1 mm). It enters squared – a 1 % error in h is a 2 % error in the moment. Source: calliper or drawing.
Diameter d [mm]
The diameter of a round beam (axle, shaft journal, round bar) at the gauge location. It enters to the third power – a 1 % error in diameter is a 3 % error in the moment. Source: calliper or drawing.
Section modulus W_b [mm³]
The section modulus in bending in mm³, entered directly – only when ‘enter directly’ is selected. For rolled sections it is listed in section tables (often called W_y or W_el), for tubes π/32·(d_o⁴−d_i⁴)/d_o. It must belong to the axis about which bending actually occurs.
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.
Lever arm x (0 = moment only) [mm]
The distance from the centre of the gauge grid to the point where the force acts perpendicular to the beam – 18 mm for Keil's spring. The bending moment at the gauge is force times this distance, so x turns the moment into the force. Enter 0 if you only need the moment or the load point is unknown. Source: measurement on the part or drawing.
Context

What this calculator is for

Force measurement with bending springs and beams (Keil's measuring wheel for camshaft torque, rowing oarlock), determination of the actual bending moment on girders, crane jibs, axles and shaft journals from a strain measurement.

What comes next

  1. Install gauges longitudinally on the tension and/or compression side, as close to the fixed end as possible (largest moment there), balance the bridge unloaded.
  2. Under load read the strain or the signal and enter it; enter the section at the gauge location and Young's modulus.
  3. Measure the lever arm x from the gauge centre to the load point (or leave 0); compare M_b or F with expectation and allowable values.
Results

How to read the results

  • The calculator works backwards: strain → bending stress → bending moment at the gauge → (with lever arm) force. The stress is the value for the strength check, moment and force are the load quantities.
  • The sign follows the side that was measured: positive strain = tension side. With a single gauge part of it may belong to a superimposed axial force – measure once on the opposite side to check.
  • "Bending moment per 1 µm/m" is the conversion number of the measuring point for all further readings as long as section and material stay the same.
Result quantities

What each value means

Outer-fibre strain used [µm/m]
The outer-fibre strain of one single gauge used in the calculation – in strain mode your entered value, in signal mode the strain converted back from mV/V, gauge factor and bridge factor. Lets you check the bridge choice.
Bending stress σ_b [N/mm²]
The bending stress at the surface, tension on one side and compression on the other. Compare with the allowable bending stress; Keil uses 50 % of the yield strength as the upper limit for springs.
Section modulus W_b [mm³]
The section modulus in bending used – a geometry figure of the section. Check: rectangle 10 × 1 mm → 1.667 mm³; round 10 mm → 98.2 mm³.
Bending moment M_b at the gauge [N·m]
The bending moment exactly where the gauge is bonded – the real answer when the moment is wanted. Note: closer to the fixed end the moment is larger for a single force (F·l instead of F·x).
Force F at lever arm x [N]
The force acting perpendicular to the beam at lever arm x that produces the moment – 0 if no lever arm was entered. Keil's spring: 32.6 N. For simply supported beams this is the support reaction.
Bending moment per 1 µm/m [N·mm]
The sensitivity of the measuring point: this much bending moment (in N·mm) corresponds to 1 µm/m of strain (E·W_b). For converting further readings of the same point in your head or as a calibration value.
Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.2 bending-loaded objects, eqs. (10.6)–(10.14), (10.24)–(10.33); sec. 9.4, eq. (9.24); sec. 4.2, eq. (4.8). Circular section: Dubbel, sec. C 2.4 (W_b = π·d³/32).

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

FAQ

Frequently asked questions

What is the difference between the moment at the gauge and the moment at the fixed end?

The gauge only measures the moment exactly at its position. With a single force at the beam end, the moment rises linearly with the distance from the load point; at the fixed end it is F·l, larger than at the gauge (F·x).

Why is the top/bottom full bridge insensitive to axial load?

Axial load stretches both sides equally; in adjacent bridge arms equal strains cancel. Bending stretches the top and compresses the bottom – this difference is amplified (Keil sec. 10.2.2.1).

Can I use the calculator for a simply supported beam?

Yes, with x as the distance from the support to the gauge; F is then the support reaction. For a central load the total force is twice that (Keil oarlock, eqs. 10.30–10.33).