Elastic transducers and strain gauge load cells

Principle: an applied force deforms an elastic member (a spring, ring, beam or column) within its elastic limit. The deflection or strain is proportional to the force ($F = kx$ or $\varepsilon = F/AE$) and is measured by a displacement transducer or by strain gauges.

Proving ring

A steel ring is loaded along a diameter. The change in diameter is proportional to the force.

        F
        v
      .---.
     /     \    deflection measured by
    |       |   micrometer / LVDT / strain gauges
     \     /
      '---'
        ^
        F

Advantages: high accuracy and good linearity; used to calibrate testing machines. Limitations: bulky, and each ring covers only a limited range.

Strain-gauge column load cell

A solid or hollow steel column carries the force. Four gauges are bonded: two axial (aligned with the load, in compression) and two transverse (at $90^\circ$, in tension because of the Poisson effect).

        F
        v
     +-----+   Axial gauges: R1 (front face), R3 (back face)
     | R1  |   Transverse gauges: R2 (front face), R4 (back face)
     | R2  |
     +-----+
        ^
        F (reaction)

Derivation of the bridge output

Axial strain (compressive): $\varepsilon_a = -\dfrac{F}{AE}$. Transverse strain: $\varepsilon_t = +\nu\dfrac{F}{AE}$.

Connect the axial gauges $R_1$, $R_3$ in opposite arms and the transverse gauges $R_2$, $R_4$ in the other two arms. Then

$$V_o = \frac{V_{ex}G}{4}\left(\varepsilon_1 - \varepsilon_2 + \varepsilon_3 - \varepsilon_4\right) = \frac{V_{ex}G}{4}\left[2\varepsilon_a - 2\varepsilon_t\right]$$ $$V_o = \frac{V_{ex}G}{2}\left(-\frac{F}{AE} - \nu\frac{F}{AE}\right) = -\frac{V_{ex}\,G\,(1+\nu)}{2AE}\,F$$

The output magnitude is proportional to $F$ and is $(1+\nu)/2$ times that of a full bridge with all four gauges at the axial strain. The arrangement also compensates for temperature (all four gauges change alike) and cancels bending (gauges of each pair are on opposite faces).

Example sensitivity: with $G = 2$, $\nu = 0.3$, $\varepsilon_a = 1000\ \mu\varepsilon$ and $V_{ex}=10$ V, $V_o = 13$ mV (1.3 mV/V).

Other elastic members: cantilever beam (bending), S-type, shear-beam and ring-type cells, all with strain gauges wired as a bridge.