Strain gauges · Measurement

Diaphragm Pressure Transducer Calculator

Calculate stresses, strains, centre deflection with linearity limit and the full-bridge signal of an edge-clamped circular diaphragm under pressure.

DMSKeil 2017

Inputs

Pressure p [bar]Diaphragm radius R [mm]Diaphragm thickness s [mm]Young's modulus E [N/mm²]Poisson's ratio νGauge factor k

Thin plate (Kirchhoff), ideally rigid clamping, small deflection; the signal is the ideal value for grids exactly at centre and edge – real rosettes yield markedly less per Keil.

Results

Enter values and run the calculation.

Method

What is calculated?

In a pressure-loaded, edge-clamped circular diaphragm the radial and tangential directions are principal. At the centre both strains are equal and positive; at the edge the radial strain is negative and twice as large. Diaphragm rosettes exploit this: two tangential grids in the centre and two radial grids at the edge form a full bridge with opposite-sign strains in adjacent arms. The relation stays linear as long as the centre deflection is below a quarter of the thickness.

Equations

σ_z = 3p·R²(1+ν)/(8s²) (Keil Gl. 10.89)

ε_z = 3p·R²(1−ν²)/(8Es²) (Gl. 10.93)

ε_R = −3p·R²(1−ν²)/(4Es²) (Gl. 10.94)

f_z = 3R⁴p(1−ν²)/(16Es³) < s/4 (Gl. 10.95)

U_M/U_B = k/4·(2ε_z + 2|ε_R|) = 1,5·k·ε_z (ideal)

Limits

Assumptions and typical mistake

Thin plate (Kirchhoff), ideally rigid clamping, small deflection; the signal is the ideal value for grids exactly at centre and edge – real rosettes yield markedly less per Keil.

Using bar as N/mm² without conversion (1 bar = 0.1 N/mm²) or entering the diameter instead of the radius.

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.

Diaphragm rosette: 2 tangential inside + 2 radial outside full bridge, no B

Layout: On the clamped circular diaphragm two grids with tangential conductors in the centre (positive strain) and two radial grids near the edge (negative strain), wired into a full bridge.

p in the signal
ΔT compensated

Use: Pressure transducers, less often force elements.

Four grids on the smallest area, fully temperature compensated, opposite-sign strains in adjacent arms.
Solder terminals sit outside the maximum-strain regions.

Linear only while the diaphragm is loaded by bending alone – guideline: centre deflection below a quarter of the thickness.
Rosette diameter must match the diaphragm diameter.

Keil Abschn. 10.5, Abb. 10.44

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

Pressure p [bar]: The pressure acting uniformly on one side of the diaphragm – rated pressure of the transducer or service pressure. In bar (1 bar = 0.1 N/mm² ≈ atmospheric pressure). Source: specification or process data.
Diaphragm radius R [mm]: Radius of the free diaphragm area between the clamping edges (half the free diameter). Strains and deflection grow with R² and R⁴. Source: drawing of the diaphragm body.
Diaphragm thickness s [mm]: Diaphragm thickness. It enters the strain squared and the deflection cubed – the key design parameter. Source: drawing or 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.
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.
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.

Context

What this calculator is for

Designing pressure transducers with diaphragm rosettes, estimating strains on vessel ends and covers, choosing diaphragm thickness and rosette diameter.

What comes next

Enter rated pressure, radius and thickness.
Check f_z/s; at ≥ 0.25 stiffen the diaphragm.
Compare the signal with the actual rosette layout – real grids sense averages, not the extremes.

Results

How to read the results

ε_z is the strain the inner (tangential) grids see, ε_R that of the outer (radial) grids; the signs differ, which is why the magnitudes add in the full bridge.
f_z/s ≥ 0.25 means the diaphragm is too compliant for linear behaviour – increase thickness or reduce radius even though the signal drops.

Result quantities

What each value means

Stress at centre σ_z (radial = tangential) [N/mm²]: Stress at the diaphragm centre, equal in radial and tangential direction there (equi-biaxial).
Strain at centre ε_z [µm/m]: Strain at the diaphragm centre; the z index denotes the centre here. The two inner tangential grids should measure this positive value. Compare it with their reading to check placement and bonding.
Radial strain at the edge ε_R [µm/m]: Radial strain at the clamped edge; the R index denotes the rim radius. The two outer radial grids measure this negative value. In the full bridge its magnitude adds to the positive centre strain.
Centre deflection f_z [mm]: Deflection of the diaphragm centre under pressure. Compare it with available space and allowable stop travel; together with thickness it also determines whether linear plate theory remains valid.
Ratio f_z/s (linearity: < 0.25): Deflection relative to thickness. Below 0.25 the characteristic stays linear (< 0.3 % deviation); above, the diaphragm stretches and the signal turns non-linear.
Ideal full-bridge signal U_M/U_B [mV/V]: Ideal full-bridge signal per volt of excitation if the grids sensed exactly centre and edge; real rosettes yield less because their grids average over areas.

Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.5.1 diaphragm rosettes, eqs. (10.87)–(10.95), figs. 10.43–10.45.

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

FAQ

Frequently asked questions

Why is ε_R twice ε_z?

From eq. 10.92 with r = R, ε_r = −2·ε_z; the edge fibre is bent most.

What does the limit f_z < s/4 mean?

Up to there bending dominates; beyond it the diaphragm stretches and the characteristic turns non-linear (> 0.3 %).

Which rosette diameter is optimal?

Per Keil it should equal the diaphragm diameter so the edge radial grids capture the negative strains.

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