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A number of devices are used to measure strain. Among them, the electrical resistance strain gauges are, by far, the most commonly used. The electrical resistance of a wire increases when the wire is stretched. The ratio between the change in resistance and the corresponding percent increase in length is called the gauge factor, k,

(1)

For the metals in common use, k varies between 0-0.5 and 5. Semiconductor gauges of silicon or germanium have gauge factors as high as 150.

A resistance strain gauge consists of a conductor, bonded to a carrier which is, in turn, fixed on to the structure or machine (base). The carrier may be in the form of a foil, a sheath or a frame and it may be cemented or welded to the base (Figure 1). The sheath and frame types are usually referred to as unbonded gauges.

When the temperature of gauge and base change by ΔT, the gauge grid stretches by αΔT while the carrier and base stretch by βΔT, α and β being respectively the thermal expansion coefficients of the grid and of the base metal. At the same time, the change in temperature results in a change in resistivity of γΔT (ohm/ohm). Assuming that the base is far more rigid than the gauge itself, the strain to which the gauge grid is subjected is (B − α) ΔT and the total change in resistance is,

(2)

In some commercially available gauges, the parameters α and γ are chosen in such a way that the term in brackets in the previous expression is zero, for a given base material. Such gauges are called self-compensated.

The change in resistance is measured by either a Wheatstone bridge or a potentiometer circuit. The former is used for static measurements (null balance method in general) and dynamic measurements (deflection method). The latter is only used for dynamic measurements (with temperature compensated gauges usually).

In the bridge shown in Figure 2, the terminals are connected to input source (V), BD to output (E). Arms' resistance AB = R1, BC = R2, CD = R3, DA = R4. Output is given by

(3)

When the bridge is initially balanced, R1R3 = R2R4. In the null balance method the bridge is continuously balanced in such a way that

(4)

For small changes,

(5)

If R4 is the measuring resistor, such as a precision decade box, potentiometer or slide wire, for a quarter bridge configuration in which R1 is the strain gauge, the strain is

(6)

In a half bridge with two strain gauges R1, R2 normally used to measured bending, the tensile strain measured by R1 is equal in magnitude to the compressive strain measured by R2,

(7)

In the deflection method the output E is measured by means of a millivoltmeter or resistance RL. In general, R1, R2, R3, and R4 are approximately equal and much smaller than RL. It can then be shown that for a quarter bridge,

(8)

and for the half bridge

(9)

In the potentiometer circuit (Figure 3) the gauge is in series with a ballast resistor. Input terminals A, C, output B, C. Resistance AB = R2, BC = R1 (gauge). Open circuit output is

(10)

The circuit cannot be balanced, so that the initial reading of the measuring instrument (large impedance) is E. Upon application of strain,

(11)

hence

(12)

Since V is of the order of 10 V, ΔR of the order of a few mV, the measuring instrument must have high resolution plus extended full-scale range. For this reason, only dynamic measurements are possible with DC current and filter to block the steady state output, letting only ΔE through (ΔE cyclic).

The nominal resistance of most strain gauges is 120 Ω. For a strain sensitivity of 10−5 (10 microstrain) and a maximum strain of 10−2 (1%) with a voltage input of 10 V, the output of a Wheatstone bridge will only be about 100 mV full scale deflection with 0.1 mV sensitivity.

The material used for the strain gauges is Cu/Ni alloy up to temperatures of 300°C. For higher temperatures, Pt may be used. The base may be epoxy, polyester or, in wire strain gauges, resin impregnated paper. A variety of cements, ranging from epoxies to cyanocrylates are available. There is a large variety of gauge materials, base materials and cements and for the selection of a commercially available system it is necessary to consult suppliers' catalogues.

Protection of the strain gauge against the environment is also an important issue.

#### REFERENCES

Dally, J. W. and Riley, W. F. (1965) Experimental Stress Analysis, McGraw-Hill.

Kobayashi, A. S. (1991) Handbook of Experimental Mechanics, Soc.Exp. Mech.

#### References

1. Dally, J. W. and Riley, W. F. (1965) Experimental Stress Analysis, McGraw-Hill.
2. Kobayashi, A. S. (1991) Handbook of Experimental Mechanics, Soc.Exp. Mech.