Combined force and moment measurement

A bar loaded by an axial force $F$ and a bending moment $M$ has a surface strain that is the sum of an axial part $\varepsilon_a$ (the same on all faces) and a bending part $\pm\varepsilon_b$ (opposite on the top and bottom).

$$\varepsilon_{top} = \varepsilon_a + \varepsilon_b, \qquad \varepsilon_{bottom} = \varepsilon_a - \varepsilon_b$$

Solving:

$$\varepsilon_a = \frac{\varepsilon_{top} + \varepsilon_{bottom}}{2}, \qquad \varepsilon_b = \frac{\varepsilon_{top} - \varepsilon_{bottom}}{2}$$

Calculation

Given: $\varepsilon_{top} = +600\ \mu\varepsilon$, $\varepsilon_{bottom} = -200\ \mu\varepsilon$, $E = 200$ GPa, $b = 20$ mm, $h = 10$ mm.

$$\varepsilon_a = \frac{600 - 200}{2} = 200\ \mu\varepsilon, \qquad \varepsilon_b = \frac{600 + 200}{2} = 400\ \mu\varepsilon$$

Area: $A = 0.02 \times 0.01 = 2\times10^{-4}$ m$^2$.

$$F = EA\varepsilon_a = 200\times10^{9}\times 2\times10^{-4}\times 200\times10^{-6} = 8000\ \text{N} = 8\ \text{kN}$$

Section modulus about the bending axis:

$$Z = \frac{bh^2}{6} = \frac{0.02 \times (0.01)^2}{6} = 3.333\times10^{-7}\ \text{m}^3$$ $$M = EZ\varepsilon_b = 200\times10^{9}\times 3.333\times10^{-7}\times 400\times10^{-6} = 26.7\ \text{N m}$$

Answer: axial force $= 8$ kN (tension), bending moment $= 26.7$ N m.

Wiring to separate the effects

• For bending moment only: put a top gauge and a bottom gauge in adjacent arms of a half bridge (or two on each face in a full bridge). The axial strain is equal on both faces, so it cancels in adjacent arms, while the bending strain adds. Output $= V_{ex}G\varepsilon_b/2$.
• For axial force only: put a top gauge and a bottom gauge in opposite arms of the bridge, with dummy gauges or fixed resistors in the other two arms (or use a full bridge with Poisson gauges). The bending strain is opposite in sign and cancels, while the axial strain adds. Output $= V_{ex}G\varepsilon_a/2$.

 Bending bridge:  R_top (arm 1), R_bot (arm 2)  adjacent
 Axial bridge:    R_top (arm 1), R_bot (arm 3)  opposite
 (other arms: fixed resistors or dummy gauges)

In practice two gauges are bonded on each face: one top-bottom pair is wired into the bending bridge and the other pair into the axial bridge, each with its own excitation and readout channel (or a switch box connects the same pair to each circuit in turn).