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STRAIN GAUGES

An investigation into the measurement of strain by electricalmeans

JONATHAN USBORNE06-MAR-2012

SUMMARYAn understanding and knowledge of material behavior is essential in engineeringdesign. The ability to accurately measure mechanical properties allows for thesuitability of a material for a specific application to be concluded. Varioustechniques can be employed to measure strain on a specimen. This report outlinesthe use of a strain gauge to determine deformation on a cantilevered steel beam,the objective to identify and quantitatively examine improvements to the technique.Load is applied to the free end of the beam causing deformation along thespecimen and the change in resistance within a strain gauge at the pinned node istranslated into a voltage differential. The circuitry is manipulated to improve theaccuracy of the readings. The initial strain resolution of a simple circuit iscalculated at 215µε. Measuring the voltage difference through a Wheatsonebridge, and amplifying the output, the resolution is improved to 6.1µε.ME10285: Strain Gauge Jonathan Usborne

TABLE OF CONTENTS1. Introduction................................................................................................... 32. Theory ........................................................................................................... 33. Method .......................................................................................................... 54. Results .......................................................................................................... 85. Discussion ................................................................................................... 106. Conclusion.................................................................................................. 117. Bibliography................................................................................................ 11

Appendix A: Raw Data .................................................................................. 12Appendix B: TI INA126 Data Sheet ............................................................... 13

2ME10285: Strain Gauge Jonathan Usborne

1. INTRODUCTION

Strain is the measure of deformation of a body in terms of relative displacementof particles within it. Strain can be calculated in numerous ways, mostcommonly through the use of a strain gauge. An electrical system, a straingauge circuit allows for more convenient and efficient data collection andmanipulation. First used by Simmons and Ruge in 1938[1], a strain gauge issimply a foil component mounted on a flexible, insulated base which, whenattached to specimen, translates mechanical strain into an electrical signal.

𝜀 = 3𝛿𝑑

𝑅 = 𝜌 𝐿

Using the application of a cantilevered steel beam, this report outlines theprocess for effectively collecting strain data electronically. Suggestedimprovements in the electrical circuitry are trialed and quantitatively evaluated.

2. THEORY

The electrical resistance of a conductor is related to its geometricalproperties such that

𝐴 (1)

where 𝜌 is the resistivity[2]. Hence the proportional and inversely proportionalrelationships between resistance and the length (𝐿), and resistance and thearea (𝐴) respectively, implies that stretching a conductor will increase itsresistance. There is additionally a small change to the material’s structure understrain which similarly affects the resistivity of the specimen.

To measure strain, a gauge is fitted to the sample such that when the sampleitself deforms, so does the gauge. As this deformation causes the foil elementof the gauge to stretch or compress, the electrical resistance of the foil thereforealters as well. It is this change in resistance that provides a measurableelectrical variable directly related to the mechanical change.

The 𝒅𝑹 of the gauge is almost proportional to the applied strain henceΔ 𝑅 𝑅 =

𝐾𝜀where 𝐾 is the gauge factor and the relative strain 𝜀 = ∆𝐿 𝐿. The gauge factoris assumed to be 2 for this report.

The strain on the beam is determined by the distanced along its length. Thestrain gauges in this experiment are mounted close to the pinned end (asillustrated in fig 3) and as such the strain is defined by the relationship

2𝐿!

3

𝑅 = 𝐾

3𝛿𝑑

Stra

in

(2)

(3)

ME10285: Strain Gauge Jonathan Usborne

given the deflection 𝛿, beam thickness 𝑑, and length 𝐿[3]. Plotting equation [3]

produces a calibration graph as in Fig 1.

Figure 1: Calibration Graph (equation 3)

0.0045

0.004

0.0035

0.003

0.0025

0.002

0.0015

0.001

0.0005

0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

0.04

Displacement (mm)

Combining equations 2 and 3 gives

∆ 𝑅 2𝐿!This implies a relatively small change in resistance given a change in str

ain.This difference can be calculated through a potential divider by measuring

theo

utput voltage over the gauge whilst in series

𝑉! = 𝑉!" − 𝑉!" 𝐺

with a fixed resistance. Given thesmall amplitude of the change however, a configuration known as a

Wheatstonebridge can further be employed to measure the change in voltage itself ra

therthan the total voltage over the gauge alone.

A Wheatstone bridge is effectively two potential dividing circuits connected in

parallel. Assuming the bridge is balanced (the voltage out at the center of both

rails is equal) then the ratio of resistances between the two sets of resistors is

equal, and the out of balance voltage is given by[2]

∆𝑉 = 𝑉 4 ∙ ∆𝑅

𝑅 = 𝑉 4 ∙ 2 ∆𝐿

𝐿 = 𝑉 2 ∙ 𝜀

Using four nominally identical strain gauges as resistors (two perpotential

divider) maximizes the bridge sensitivity, and also provides compensation for

temperature errors by essentially comparing the active gauge on the beam to

inactive, unstrained, gauges. Whilst this approach increases the sensitivity of

the measurement, the change in voltage is still small and can finally be

processed through an amplifier. (4)

(5)


A differential amplifier amplifies the difference between two input voltages –

int

his c

ase the voltages out between the two gauges of each set. The outputvoltage of the INA126 in-amp used in this experiment is defined in its

datasheet(Appendix B) as

! !where the gain (𝐺) is adjustable and calculated in terms of the resistance 𝑅!placed between pins 1 and 8 of the chip such that

𝐺 = 5 + 80𝐾Ω 𝑅!

3. METHOD

The apparatus consists of a cantilevered steel ruler of thickness (𝑑) 0.8mm with

a pair of 120Ω strain gauges adhered to the upper and lower surfaces at the

fixed end of the beam. Strain is applied via a threaded bolt housed in a frame

around the free end, 250mm from the pinned node.

Figure 2: Apparatus diagram

𝐿 = 250mm

(6)

(7)

Upper strain gauge

Lower strain gauge

Inactive gauges

Figures 3(a), 3(b): Top and end view of apparatus

Load

𝑉!" − 1𝑅!" = 1500

gauge

strain ~ 15V

1.5k

Ω


Initially the vertical offset was calibrated with respect to angular displacement of

the bolt head in order to provide a more accurate estimation of 𝛿.Simple Circuit :

Using a breadboard the upper strain gauge was configured according to Fig 4.

The voltages over each component were recorded, and a strain resolution

determined by applying strain until a measurable difference in voltage was

observed.

The 1.5kΩ resister in Fig 4 is connected in series with the strain gauge to

produce a potential dividing circuit. This allows the resistance of the gauge to be

calculated using a measurement of voltage over it given

𝑉!Figure 4: Simple Circuit diagram

𝑉!

𝑉!"

0

W

h

e

at

st

o

n

e

B

ri

d

g

e :

To improve results the upper gau

~ 15V

~ 15V

gauge

active

gauge

active

gauge

inactive

gauge

inactive INA126

gauge

inactive

gauge

inactive

gauge

inactive

gauge

inactive

ge and three inactive gauges were configuredusing a Wheatstone bridge (discussed in section 2) as per figure 5.

Theresolution of this configuration was similarly determined. Strain was applied

andthe voltage imbalance across the bridge was recorded at various stage

s ofbeam displacement.

(8)


Figure 5: Wheatstone Bridge circuit diagram

680 Ω

𝑉!"#

Differential Amplifier :

While the Wheatstone bridge provides more accurate readings, the output islow and can be amplified through a differential amplifier. The Wheatstone bridgecircuit was therefore connected to the differential amp as per figure 6. As beforethe resolution of the amp output was determined, and voltage recorded atvarious levels of strain. A datasheet for the differential amplifier (INA126) isattached in Appendix B.

Figure 6: Differential amplifier circuit diagram

680 Ω

82 Ω

𝑉!"#


4. RESULTSThe initial displacement calibration equated a 360° ration of the bolt head to a1.6mm vertical displacement on the beam.

The input voltage (𝑉!) was more accurately determined to be 15.38V

Simple Circuit :

The voltage measured over the active gauge (𝑉!") in an unloaded state was1.15V. Given the potential dividing nature of the circuit this implied an unloadedgauge resistance of 121.2Ω calculated using eqn [8].

𝑉 = 15 ∙ 120

The estimated strain resolution of this circuit, determined from the smallestmeasurable displacement of 11.2mm, was 215µε.

Wheatstone Bridge :

The out of balance voltages and strains were recorded and derived respectivelyat varying levels of displacement after the circuit was reconfigured into theWheatstone bridge arrangement. This data is reproduced in Appendix A.

The estimated strain resolution using the Wheatstone bridge, determined fromthe smallest measurable displacement of 1.8mm, was 34.6µε.

The derived strain was plotted as a function of the recorded voltage in Figure 7alongside the function of theoretical voltage given by eqn [5] where the voltage𝑉 over the gauge is

120 + 680 = 2.25

Figure 7: Wheatstone bridge out of balance voltage against strain


Differential Amplifier :

The gain of the differential amplifier (𝑉!"# 𝑉!"), as defined by eqn [7], wascalculated to give 980.6dB using 𝑅! = 82Ω as set out in section 3.

Again, voltage and strain were recorded at varying levels of displacement, and

values are reproduced in Appendix A.

The estimated strain resolution using the Wheatstone bridge and the differentialamplifier, determined from the smallest measurable displacement of 0.32mm,was 6.1µε.

The derived strain was plotted as a function of the recorded voltage in Figure 8.

Figure 8: In-amp output voltage against strain


5. DISCUSSION

The initial simple circuit uses a 1.5kΩ resistor. This allows for the creation of apotential divider when placed in series with the gauge. In the Wheatstone bridgeconfiguration however this is replaced with a 680Ω resistor. At approximatelyhalf the resistance, this reduces the current over the two rails of the bridge to asimilar value of that over the gauge in the simple circuit.

Whilst there was discrepancy between the gradients of strain as a function ofthe out-of-balance voltage from the Wheatstone bridge compared to thetheoretical voltage, both displayed a linear relationship. This conformed to thetheory outlined initially and systematic error accounts for the slight divergenceof experimental data from the theoretical values.

During the experiment however the Wheatstone bridge was not balanced beforea load was applied, adding an unnecessary degree of complexity to the dataanalysis. If the procedure was to be repeated, balancing the bridge through thelower strain gauge or a variable resistor would be an improvement to themethod discussed above.

The strain resolution of the Wheatstone bridge circuit was better than the simplecircuit, and the differential amplifier improved it further. This was expected asthe strain resolution was linked to the resolution of the voltage read by thedigital voltmeter, which itself was constant. As such, the increase in outputvoltage from each circuit subsequently increased the resolution of thecalculated strain.

In addition to the voltage resolution, another factor constraining the strainresolution was the ratio of resistances in the circuit. Using the in-amp forexample, the resolution could have been altered by controlling the gain of theamp via the resistance 𝑅!.Whilst temperature error was compensated for using inactive gauges in theWheatstone bridge, further errors will have accumulated due to lead resistancewithin the cables linking the gauges together. To accommodate this the circuitcould further be improved by using a three lead connection to the gauges wherethe voltage difference is taken from the end terminal of a third lead of equallength, acting to balance the bridge from the resistance in the wiring from theother two leads.

10


6. CONCLUSIONThe objective of this experiment was to evaluate methods of improving thetechnique for collecting strain data via electrical means. By making tworelatively simple alterations to the circuitry this was very much achieved.Quantitatively speaking the resolution was improved by over 3500% suggestingoverall success. The most important conclusion is the realisation that this wasachieved through electrical manipulation and conversion alone, with no changesmade to the physical or mechanical components of the experiment or thesensor itself.

Whilst results generally conformed to hypothesised linear realationships, somediscrepancy was observed between experimental and theoretical data sets.Retrospectively however, further improvements were identified which wouldcompensate for some of the errors that contributed to that discrepancy.

The predominant teaching point conveyed during this laboratory exercise wasthe need to consider all stages of a measurement system. Here manipulation ofthe variable conversion and signal processing components of the system had amonumental impact on the accuracy of measurement, thus identifying theimportance of the electrical design for the entire process.

7. BIBLIOGRAPHY

[1] Wikipedia, Strain Gauge [online]. St Petersburg, Florida: WikipediaFoundation. Available from http://bit.ly/6EWwmB

[2] Ngwompo, R. Sensor & Electronics (Lecture Notes) – Instrumentation,Electronics & Electrical Drives. 2011. University of Bath.

[3] Ngwompo, R. Strain Gauges (ME10285 lab sheet). University of Bath.

[4] Davison, M. Strain Gauges and the Wheatstone bridge. University ofPaisley. 1997. Available from http://bit.ly/ADmMy2

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APPENDIX A: RAW DATA

Displacement(mm)

Strain(µε)

WheatstoneBridge In-Amp

Voltage(mV)

0 0 2.2 17.43.2 61.4 2.3 17.56.4 122.9 2.5 17.689.6 184.3 2.7 17.8512.8 245.8 2.9 18.0116 307.2 3 18.1819.2 368.6 3.2 18.3522.4 430.1 3.4 18.5225.6 491.5 3.5 18.6928.8 553.0 3.7 18.8532 614.4 3.9 19.0235.2 675.8 4.1 19.1938.4 737.3 4.2 19.36


APPENDIX B: INA126 DATASHEET

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