Separating axial force and bending: four gauges around a bar

What is calculated?

For a slender bar under axial force and bending the axial direction is principal; each longitudinal gauge measures the sum of the normal strain and the bending component at its circumferential position. Two opposite gauges share the normal part but have opposite bending parts, so mean and half difference separate the components. Keil's mast example (Tab. 10.1) with −246/−340/−324/−241 µm/m yields −288 µm/m compression plus bending components aft and to port.

Equations

ε_N = (ε₁+ε₂+ε₃+ε₄)/4

ε_b,x = (ε₁−ε₃)/2, ε_b,y = (ε₂−ε₄)/2

ε_b = √(ε_b,x² + ε_b,y²), φ = atan2(ε_b,y, ε_b,x)

σ = E·ε (einachsig, Keil Tab. 10.1)

Step by step

1. Install four longitudinal gauges 90° apart and label them counter-clockwise.
2. Record and enter the strains of one load case simultaneously.
3. Compare stresses with permissible values; use the bending-plane direction to interpret the load.

Typical mistake

Not labelling the gauges in circumferential order, which corrupts the bending-plane direction.

Application and limits

Stress measurement on masts, columns, piles, tie rods and piston rods where axial force and bending superimpose.

Circular or doubly symmetric section, gauges exactly 90° apart at equal radius, no torsion effect on longitudinal gauges and no temperature difference between gauge locations.

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.1 yacht-mast example, Tab. 10.1, figs. 10.3–10.5; sec. 4.5, eqs. (4.14), (4.15).

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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