Strain gauges · Measurement

Axial and Bending Strain Separation Calculator

Calculate axial strain, bending strains in two planes, bending-plane orientation and stresses from four gauges at 0°, 90°, 180° and 270° around a bar.

DMSKeil 2017

Inputs

Strain gauge 1 (0°, e.g. front) [µm/m]Strain gauge 2 (90°, e.g. port) [µm/m]Strain gauge 3 (180°, e.g. aft) [µm/m]Strain gauge 4 (270°, e.g. starboard) [µm/m]Young's modulus E [N/mm²]

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.

Results

Enter values and run the calculation.

Method

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)

Limits

Assumptions and typical mistake

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.

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

Gauge layout

Where the gauges sit and what the bridge makes of them

Where the grids sit and how they are wired, plus which load components remain in the signal and which the circuit compensates.

What does the bridge factor B mean?

The general bridge equation is U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄): strains in opposite arms count positive, in adjacent arms negative. Expressing every active strain as a multiple of one reference strain ε, B is exactly that signed sum of multiples – e.g. B = 2 for two adding contributions, B = 4 for four. U_M/U_B = k·B·ε/4 summarises this.

To assess it: B is a pure gain of the circuit, not a statement about temperature compensation or which load type is in the signal – that is covered by “Use” and the response chips below. A larger B gives a larger, more easily measured signal U_M for the same mechanical strain; this calculator inserts B into the calculation automatically once you pick a circuit, so you never enter it yourself.

4 longitudinal gauges around the circumference (0°, 90°, 180°, 270°) 4 × B = 1

Layout: Four longitudinal gauges 90° apart at equal radius, numbered counter-clockwise; each gauge measured individually (circuit II).

F in the signal
M_b (2 Ebenen) in the signal
M_t insensitive
ΔT compensated

Use: Masts, columns, piles, tie rods: separate axial force and bending in two planes numerically.

Mean = axial strain, half differences of opposite gauges = bending strains; the bending plane follows from both.
Alternatively electrical: 1 and 3 in opposite arms (VI) → 2ε_N, in adjacent arms (IV) → 2ε_b.

Gauges exactly 90° apart and labelled in circumferential order, otherwise the bending plane is wrong.
Four channels to be recorded simultaneously.

Keil Abschn. 10.2.1, Abb. 10.3–10.5, Tab. 10.1

Bridge wiring diagrams after Keil, Dehnungsmessstreifen, 2nd ed. 2017 (secs. 4.5, 9.7, 10.2, 10.5) · bridge arms numbered R1–R4 with U_M/U_B = k/4·(ε₁ − ε₂ + ε₃ − ε₄).

Inputs

What you enter – and where the values come from

Strain gauge 1 (0°, e.g. front) [µm/m]: Reading of the first longitudinal strain gauge, defining the zero-degree reference position around the circumference. All four gauges must lie in one section and parallel to the bar axis. Source: strain reading from channel 1 after common zeroing; enter compressive strain with a negative sign.
Strain gauge 2 (90°, e.g. port) [µm/m]: Reading of the second longitudinal gauge, sitting 90° counter-clockwise from the first around the circumference (looking along the bar). Together with gauge 4 it yields the bending in the second plane. Source: channel 2 of the measurement.
Strain gauge 3 (180°, e.g. aft) [µm/m]: Reading of the third longitudinal gauge, exactly opposite the first (180°). Half its difference from gauge 1 is the bending strain in this plane. Source: channel 3 of the measurement.
Strain gauge 4 (270°, e.g. starboard) [µm/m]: Reading of the fourth longitudinal gauge, exactly opposite the second (270°). Its difference from gauge 2 carries the bending in this plane; the mean of all four carries the axial force. Source: channel 4 of the measurement.
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.

Context

What this calculator is for

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

What comes next

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

Results

How to read the results

ε_N is the axial component (mean); the bending components are the half differences of opposite gauges.
The maximum stress does not necessarily occur under one of the four gauges but at angle φ – hence σ_max can exceed the largest single reading times E.

Result quantities

What each value means

Axial strain (mean) ε_N [µm/m]: Strain component due to axial force – identical in all four gauges. Negative = compression.
Bending strain plane 1–3 ε_b,x = (ε₁−ε₃)/2 [µm/m]: Bending strain in the plane of gauges 1 and 3: positive if gauge 1 is on the tension side.
Bending strain plane 2–4 ε_b,y = (ε₂−ε₄)/2 [µm/m]: Bending strain in the plane of gauges 2 and 4: positive if gauge 2 is on the tension side.
Resultant bending strain ε_b [µm/m]: Largest bending strain around the circumference (outer fibre of the resultant bending plane) – where bending is actually maximal, usually between two gauges.
Direction of max. bending strain (from gauge 1, counter-clockwise) [°]: Angle from gauge 1 counter-clockwise to the circumferential position where bending tension is maximal. The compression side lies opposite (φ + 180°).
Axial stress σ_N [N/mm²]: Normal stress from axial force acting uniformly over the section. It is the common base component at all four measuring positions and is superposed with bending stress.
Bending stress σ_b (outer fibre) [N/mm²]: Largest magnitude of bending stress at the outer fibre in the resultant bending plane. Do not assess the component from this alone; combine it with axial stress and inspect the maximum and minimum total stresses.
Maximum axial stress σ_N + σ_b [N/mm²]: Largest axial stress around the circumference (axial + bending) – the governing value for the strength check on the tension side.
Minimum axial stress σ_N − σ_b [N/mm²]: Smallest axial stress around the circumference (axial − bending) – governing on the compression side and for buckling considerations.

Source

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.

FAQ

Frequently asked questions

Why four gauges instead of two?

Two opposite gauges separate bending in one plane only; the bending plane is usually unknown.

Can I separate the components electrically?

Yes: gauges 1 and 3 in opposite arms (circuit VI) give 2ε_N, in adjacent arms (IV) 2ε_b.

What about temperature?

A common compensation gauge on unloaded bar material, as in Keil's example, cancels the temperature response for all four points.

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