# 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 ΔV_{t}. 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 G_{f} 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.

In the basic Wheatstone Bridge circuit the unknown resistance R_{x} 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 V_{o} = 0 is obtained, at which point the unknown resistance can be calculated using the bridge balance relationship:

Any variations in R_{x} subsequently produce a non-zero output voltage V_{o} proportional to the resistance change ΔR_{x}; the value of V_{o} therefore provides a measure of the force acting on the structure under test.

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 R_{x}(unstressed) = R_{1} = R_{2} = R_{3}, the output voltage variation ΔV_{0} is given by: This shows that the ΔV_{o}/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:

The level of the required mechanical stress on the structure under test can thus be measured by the out-of-balance output voltage ΔV_{o}, given that the bridge supply voltage V_{s}, Young’s modulus for the structure material, E and the force transducer gauge factor, G_{f} are known. To reduce the Wheatstone bridge temperature dependence a second strain gauge R_{x(comp)} can be inserted in the circuit in place of R_{2} 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 R_{y} can be mounted on the side opposite to R_{x} 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 R_{y} against temperature variations a fourth gauge R_{y(comp)} is mounted perpendicular to R_{y} in such a way as to minimize the effect of the strain on its resistance value (Figure 4).

The overall sensitivity of such bridge is slightly more than double because the compensating gauges also contribute a resistance change ΔR_{x(comp)} and ΔR_{y(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 R_{x} and R_{y} but add to the overall bridge performance because they are connected electrically on the opposite side of the Wheatstone bridge to R_{x} and R_{y} . The overall bridge sensitivity is given by: Using eqn (4) we can therefore re-write eqn (6) as:

where μ is Poisson’s ratio and ΔL/L is the strain. Equation (7) shows that the output voltage change ΔV_{o} 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 ΔV_{o} 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

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.