## F Experimental Measurement of Strain

90 Typically, the transducer to measure strains in a material is the strain gage. The most common type of strain gage used today for stress analysis is the bonded resistance strain gage shown in Figure 4.9.

GAGE LENGTH

Figure 4.9: Bonded Resistance Strain Gage

91 These gages use a grid of fine wire or a metal foil grid encapsulated in a thin resin backing. The gage is glued to the carefully prepared test specimen by a thin layer of epoxy. The epoxy acts as the carrier matrix to transfer the strain in the specimen to the strain gage. As the gage changes in length, the tiny wires either contract or elongate depending upon a tensile or compressive state of stress in the specimen. The cross sectional area will increase for compression and decrease in tension. Because the wire has an electrical resistance that is proportional to the inverse of the cross sectional area, R a A, a measure of the change in resistance can be converted to arrive at the strain in the material.

92 Bonded resistance strain gages are produced in a variety of sizes, patterns, and resistance. One type of gage that allows for the complete state of strain at a point in a plane to be determined is a strain gage rosette. It contains three gages aligned radially from a common point at different angles from each other, as shown in Figure 4.10. The strain transformation equations to convert from the three strains a t any angle to the strain at a point in a plane are:

ta = ex cos2 da + ty sin2 da + Yxy sin da cos da (4.184)

tb = tx cos2 db + ty sin2 db + Yxy sin db cos db (4.185)

tc = tx cos2 dc + ty sin2 dc + Yxy sin dc cos dc (4.186)

93 When the measured strains ta, tb, and ec, are measured at their corresponding angles from the reference axis and substituted into the above equations the state of strain at a point may be solved, namely, tx, ty, and Yxy. In addition the principal strains may then be computed by Mohr's circle or the principal strain equations.

94 Due to the wide variety of styles of gages, many factors must be considered in choosing the right gage for a particular application. Operating temperature, state of strain, and stability of installation all influence gage selection. Bonded resistance strain gages are well suited for making accurate and practical strain measurements because of their high sensitivity to strains, low cost, and simple operation.

9& The measure of the change in electrical resistance when the strain gage is strained is known as the gage factor. The gage factor is defined as the fractional change in resistance divided by the fractional

change in length along the axis of the gage. GF = -Art Common gage factors are in the range of 1.5-2

for most resistive strain gages.

96 Common strain gages utilize a grid pattern as opposed to a straight length of wire in order to reduce the gage length. This grid pattern causes the gage to be sensitive to deformations transverse to the gage length. Therefore, corrections for transverse strains should be computed and applied to the strain data. Some gages come with the tranverse correction calculated into the gage factor. The transverse sensitivity factor, Kt, is defined as the transverse gage factor divided by the longitudinal gage factor. Kt = These sensitivity values are expressed as a percentage and vary from zero to ten longitudinal percent.

97 A final consideration for maintaining accurate strain measurement is temperature compensation. The resistance of the gage and the gage factor will change due to the variation of resistivity and strain sensitivity with temperature. Strain gages are produced with different temperature expansion coefficients. In order to avoid this problem, the expansion coefficient of the strain gage should match that of the specimen. If no large temperature change is expected this may be neglected.

98 The change in resistance of bonded resistance strain gages for most strain measurements is very small. From a simple calculation, for a strain of 1 /e (/ = 10~6) with a 120 ^ gage and a gage factor of 2, the change in resistance produced by the gage is AR =1 x 10~6 x 120 x 2 = 240 x 10~6^. Furthermore, it is the fractional change in resistance that is important and the number to be measured will be in the order of a couple of / ohms. For large strains a simple multi-meter may suffice, but in order to acquire sensitive measurements in the range a Wheatstone bridge circuit is necessary to amplify this resistance. The Wheatstone bridge is described next.

### 4.10.1 Wheatstone Bridge Circuits

99 Due to their outstanding sensitivity, Wheatstone bridge circuits are very advantageous for the measurement of resistance, inductance, and capacitance. Wheatstone bridges are widely used for strain measurements. A Wheatstone bridge is shown in Figure 4.11. It consists of 4 resistors arranged in a diamond orientation. An input DC voltage, or excitation voltage, is applied between the top and bottom of the diamond and the output voltage is measured across the middle. When the output voltage is zero, the bridge is said to be balanced. One or more of the legs of the bridge may be a resistive transducer, such as a strain gage. The other legs of the bridge are simply completion resistors with resistance equal to that of the strain gage(s). As the resistance of one of the legs changes, by a change in strain from a resistive strain gage for example, the previously balanced bridge is now unbalanced. This unbalance causes a voltage to appear across the middle of the bridge. This induced voltage may be measured with a voltmeter or the resistor in the opposite leg may be adjusted to re-balance the bridge. In either case the change in resistance that caused the induced voltage may be measured and converted to obtain the

4.10.2 Quarter Bridge Circuits

100 If a strain gage is oriented in one leg of the circuit and the other legs contain fixed resistors as shown in Figure 4.11, the circuit is known as a quarter bridge circuit. The circuit is balanced when R1 = .

When the circuit is unbalanced Vout = Vin( rR+R _ R Raa+R3).

101 Wheatstone bridges may also be formed with two or four legs of the bridge being composed of resistive transducers and are called a half bridge and full bridge respectively. Depending upon the type of application and desired results, the equations for these circuits will vary as shown in Figure 4.12. Here E0 is the output voltage in mVolts, E is the excitation voltage in Volts, e is strain and v is Poisson's ratio.

102 In order to illustrate how to compute a calibration factor for a particular experiment, suppose a single active gage in uniaxial compression is used. This will correspond to the upper Wheatstone bridge configuration of Figure 4.12. The formula then is

Bridge

Description

Eo/E in mV/V ein microStrain

Single active gage in uniaxial compression or tension.

Two active gages in uniaxial E stress field. One "Poisson Gage" & one gage aligned with maximum principal strain.

Two active gages with equal E and opposite. Common for bending beam test.

Two active gages with equal strains of the same sign. Bending cancellation arrangement.

Four active gages in uniaxial stress field. Two "Poisson Gages" & two aligned with maximum principal strain. Column test.

Four active gages in uniaxial stress field. Two "Poisson Gages" & two aligned with maxium principal strain. Beam Test

Four active gages with pairs subjected to equal and opposite strains. Typical of a beam in bending or a shaft in torsion.

Figure 4.12: Wheatstone Bridge Configurations

103 The extra term in the denominator 2Fe(10-6) is a correction factor for non-linearity. Because this term is quite small compared to the other term in the denominator it will be ignored. For most measurements a gain is necessary to increase the output voltage from the Wheatstone bridge. The gain relation for the output voltage may be written as V = GE0(103), where V is now in Volts. so Equation 4.187 becomes

EG(103)

104 Here, Equation 4.188 is the calibration factor in units of strain per volt. For common values where F = 2.07, G = 1000, E = 5, the calibration factor is simply (2 07)(400 0)(5) or 386.47 microstrain per volt.

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