Shear force on a shear beam from measured 45° strain

What is calculated?

A shear force produces shear stresses in the web of a beam. Shear only distorts the material (angle change) and cannot be measured with gauges directly – but at 45° to the shear direction two equal principal normal stresses of opposite sign act (Keil Fig. 10.23), and a 45° gauge measures their strains. Hooke's law for plane stress gives τ = E·ε₄₅/(1+ν); with Keil's estimate τ ≈ F/A_web for mid-web the shear force follows. Keil's shear beam (web 46 × 4.6 mm, E = 209 000 N/mm², ν = 0.3) gives τ = 140 N/mm² and F ≈ 30 kN at 870 µm/m – exactly the rated force it was designed for.

Equations

ε₄₅ = 4·(U_M/U_B)/(k·B) (Keil Gl. 4.8 bzw. 10.43, nur Modus Signal)

σ₁ = −σ₂ = E/(1−ν²)·(ε₁ + ν·ε₂) mit ε₂ = −ε₁ ⇒ σ₁ = E·ε₄₅/(1+ν) (Keil Gl. 10.44)

τ_max = σ₁ (reiner Schub: Hauptspannungen unter 45°, Keil Abb. 10.23)

F = τ_max·A_Steg (Umkehrung von Keil Gl. 10.42)

Assumptions and typical mistake

Pure shear state at mid-web; τ ≈ F/A_web is an approximation for thin webs (Keil: "accuracy sufficient for the estimate") – the exact shear distribution per Dubbel is somewhat higher at mid-web. Gauges exactly on the neutral axis, otherwise bending enters. Linear-elastic; excitation temperature compensation not included.

Using the area of the whole section instead of the web only (flanges carry no appreciable shear); bonding gauges along the beam instead of at 45° – there the strain from shear is zero; choosing the wrong bridge factor in signal mode.

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 strain at 45° ε₄₅ [µm/m]

The strain shown by a gauge bonded at mid-web at 45° to the force direction. At +45° it is positive (tension), at −45° equally large and negative (compression) – the hallmark of pure shear. For half or full bridges enter the strain of one single gauge. 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 shear beam is designed for 1.74 mV/V at 30 kN (before excitation temperature compensation). 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 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. One gauge: B = 1. One pair +45°/−45° in adjacent arms: the opposite strains add, B = 2. One pair on each web face (Keil's shear beam, eq. 10.43): B = 4, plus temperature and bending compensation. Source: your own wiring or datasheet.

Web area

How the web area is determined: from web height times web thickness, or entered directly (e.g. from the drawing if the web is not rectangular). Meant is only the thin part of the section standing along the force that carries the shear – not the flanges.

Web height h [mm]

The web height in the direction of the shear force, i.e. the distance between the two flanges or the height of the shear diaphragm (Keil: 46 mm). Source: drawing or measurement at the milled pocket.

Web thickness b [mm]

The web thickness, i.e. the remaining wall between the two milled pockets on whose floor the gauges are bonded (Keil: 4.6 mm). The thinner, the larger the strain for the same force. Source: drawing or measurement.

Web area A_web [mm²]

The web area directly in mm², only when ‘enter directly’ is selected. Keil's example: 215 mm² (46 × 4.66 mm rounded). For rolled sections: web height between the flanges times web thickness from the section table.

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

Shear-beam force transducers and load cells, force measurement on short cantilevers, pins and axles with an I-shaped measuring section, furnace and conveyor equipment (Keil's carrier bars), crane-hook load pins.

What comes next

1. Install gauges at mid-web (neutral axis, no bending stress there) at ±45° to the force direction, balance the bridge unloaded.
2. Under load read the strain or the signal and enter it; enter the web dimensions or area and the material data.
3. Compare F with the expected or rated force; use the sensitivity as calibration value and confirm it by a testing-machine calibration.

How to read the results

• The calculator works backwards: 45° strain → shear stress (via Hooke's law with the factor 1/(1+ν)) → shear force (shear stress times web area). The shear stress is the value for the strength check, the shear force the load quantity.
• The shear force is an estimate using Keil's approximation τ ≈ F/A_web; for shear-beam transducers the sensitivity is confirmed by calibration with a known force.
• If the strains at +45° and −45° are not equal with opposite sign, the state is not pure shear – then bending or axial force is in the signal and the gauges are not exactly on the neutral axis.

What each value means

Strain at 45° used [µm/m]

The 45° 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.

Shear stress τ at mid-web [N/mm²]

The shear stress at mid-web that produced this strain – at the same time the principal normal stress at 45°. Compare with the material's allowable shear stress (about 0.58·yield strength as the shear yield limit).

Web area A_web [mm²]

The web area used – the part of the section carrying the shear force. Check: 46 × 4.6 mm = 211.6 mm².

Shear force F [kN]

The shear force actually loading the web at the gauge location – the calculator's real answer. Sign as the strain (direction of the force). Compare with the rated or allowable force.

Shear force per 1 µm/m [N]

The stiffness of the measuring point: this much shear force corresponds to 1 µm/m of strain (E·A_web/(1+ν)). For converting further readings of the same point in your head.

Sensitivity of the measuring point [mV/V je kN]

The bridge signal per kilonewton – the value you set in the amplifier so it reads directly in kN. Keil's shear beam: 0.058 mV/V per kN, slightly less after excitation temperature compensation with the factor 0.92.

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.4 shear-loaded objects, eqs. (10.42)–(10.45), Figs. 10.23–10.24; 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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