Resistance Measurement Of Strain Gauges
Resistance Measurement Of Strain Gauges
Strain gauge fixed to a structure under traction will exhibit an increase in length and, a change in the output parameter ΔVt. For strain gauges this latter represents a change in resistance ΔR; can therefore be re-written as eqn (1) in which the only unknown is ΔR, since Gf and R (gauge factor and unstressed gauge resistance) can be obtained from the transducer manufacturer’s data sheet, while E and A (Young’s modulus and the cross-sectional area) are provided by the support structure design.
A measurement of the strain gauge resistance change ΔR is therefore sufficient to obtain the value of the force acting on the structure. Although several different methods of resistance measurement could be used for this purpose, the most popular one, in view of its sensitivity and flexibility of operation, is the Wheatstone Bridge which is the only method described in this book.
Figure 1 Basic Wheatstone bridge circuit
In the basic Wheatstone Bridge circuit the unknown resistance Rx is measured by comparison with three known values, as shown in Figure 1; this method of measurement is based on varying the known resistance values until the bridge is balanced, that is the condition Vo = 0 is obtained, at which point the unknown resistance can be calculated using the bridge balance relationship:
Any variations in Rx subsequently produce a non-zero output voltage Vo proportional to the resistance change ΔRx; the value of Vo therefore provides a measure of the force acting on the structure under test.
Figure 2 Three different methods of zeroing a Wheatstone bridge
A practical bridge circuit is composed of four equal resistances, that is the three fixed resistances are chosen to be of the same value as the strain gauge unstressed resistance, plus a bridge balancing network. This latter can be of three basic types, as shown in Figure 2. For these types of circuit where Rx(unstressed) = R1 = R2 = R3, the output voltage variation ΔV0 is given by: 
This shows that the ΔVo/ΔR relationship is only linear for small resistance changes, that is small ΔR/R. This is in fact the case in practice and we can therefore approximate eqn (2) to :
where L is the unstressed length of the strain gauge and ΔL its increment under stress. The ratio ΔL/L is a measurement of the mechanical strain and can therefore be inserted in Young’s modulus formula E = stress/strain: 
Figure 3 Temperature-compensated Wheatstone bridge
The level of the required mechanical stress on the structure under test can thus be measured by the out-of-balance output voltage ΔVo, given that the bridge supply voltage Vs, Young’s modulus for the structure material, E and the force transducer gauge factor, Gf are known. To reduce the Wheatstone bridge temperature dependence a second strain gauge Rx(comp) can be inserted in the circuit in place of R2 and mounted on the structure perpendicular to the first gauge so as to minimize any longitudinal elongation, as shown in Figure 3.
To double the bridge sensitivity, another ‘active’ gauge Ry can be mounted on the side opposite to Rx on the structure under test, in such a way that both gauges are subject to the longitudinal elongation produced by the strain, that is their axes are both parallel to the structure longitudinal axis. To also compensate Ry against temperature variations a fourth gauge Ry(comp) is mounted perpendicular to Ry in such a way as to minimize the effect of the strain on its resistance value (Figure 4).
Figure 4 Example of traction force measured using four strain gauge in a dual-active temperature-compensated configuration
The overall sensitivity of such bridge is slightly more than double because the compensating gauges also contribute a resistance change ΔRx(comp) and ΔRy(comp) due to variations in gauge width. However, Poisson’s ratio shows that an increase in width must be followed by a reduction in length, which means that the compensating gauges’ resistance changes are negative with respect to Rx and Ry but add to the overall bridge performance because they are connected electrically on the opposite side of the Wheatstone bridge to Rx and Ry . The overall bridge sensitivity is given by: 
Using eqn (4) we can therefore re-write eqn (6) as: 
Figure 5 Example of bending force measured using four strain gauges in a dual-acting temperature-compensated configuration
where μ is Poisson’s ratio and ΔL/L is the strain. Equation (7) shows that the output voltage change ΔVo is still proportional to the mechanical stress acting on the structure, as required, but that, for the same stress (that is, the same variations in gauge resistances ΔR), the bridge output voltage ΔVo is greater by more than a factor of two when compared with the basic circuit.
Similar derivations can be obtained for bending and twisting force applications to increase the strain measurement sensitivity and reduce temperature dependence. The physical locations of the gauges in these latter cases are shown in Figures 5 and 6. Note that for the twisting force
Figure 6 Example of twisting force measured using four strain gauges in a temperature-compensated configuration
case all the gauges are ‘active’, that is they all receive dimensional variations along their length and there the overall bridge sensitivity is four times that of the single-gauge uncompensated circuit. Also it should be pointed out that any bending moments acting on the structure will introduce an error since the mounting configuration shown does not compensate for them.
