Strain Gages, Accessoriesand Instrumentation

Click on a title or page number below to jump to that section.ETechnical Articles and Specifications E-3

General Purpose Foil Gages E-8

Pre-Wired Gages E-10

Perpendicular Grids for Axial Strain E-11

45º Grids for Shear Strain E-12

Rosette Strain Gages E-13

Crack Propagation and Diaphragm Gages E-17

Transducer Grade Strain Gages E-18

Span Compensation Resistors E-21

Resistance Wire and Hot Curing Adhesive E-22

Article: Transducer Bridge Completion E-23

Terminal Pads, Completion Resistors, and Cold Cure Adhesives E-25

Fast Response Thermocouple E-26

Strain Gage Application Kit E-27

Fine Gage Hook-Up Wire and Multi-Wire Cable E-28

Strain Meters E-29

Strain Gage Instrumentation E-31

Technical Article: Practical Strain Gage Measurements E-45

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PRACTICAL STRAIN GAGE MEASUREMENTS

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INTRODUCTION

W ith today’s emphasis onproduct liability and energy

efficiency, designs must not onlybe lighter and stronger, but alsomore thoroughly tested than everbefore. This places newimportance on the subject ofexperimental stress analysis andthe techniques for measuringstrain. The main theme of thisapplication note is aimed at strainmeasurements using bondedresistance strain gages. We willintroduce considerations that affectthe accuracy of this measurementand suggest procedures forimproving it.

We will also emphasize the practicalconsiderations of strain gagemeasurement, with an emphasis oncomputer controlled instrumentation.

Appendix B contains schematics ofmany of the ways strain gages areused in bridge circuits and theequations which apply to them. Readers wishing a more thoroughdiscussion of bridge circuit theory are invitedto read Item 7 referencedin the Bibliography.

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STRESS & STRAIN

The relationship between stressand strain is one of the most

fundamental concepts from thestudy of the mechanics of materialsand is of paramount importance tothe stress analyst. In experimentalstress analysis, we apply a givenload and then measure the strain onindividual members of a structure ormachine. Then we use the stress-strain relationships to compute thestresses in those members to verifythat these stresses remain withinthe allowable limits for the particularmaterials used.

STRAINWhen a force is applied to a body,the body deforms. In the generalcase, this deformation is calledstrain. In this application note, wewill be more specific and define theterm STRAIN to mean deformationper unit length or fractional changein length and give it the symbol, «.See Figure 1. This is the strain thatwe typically measure with a bondedresistance strain gage. Strain maybe either tensile (positive) orcompressive (negative). See Figure2. When this is written in equation

form, « = DL/L, we can see thatstrain is a ratio and, therefore,dimensionless.

To maintain the physicalsignificance of strain, it is oftenwritten in units of inches/inch. Formost metals, the strains measuredin experimental work are typicallyless than 0.005000 inch/inch. Sincepractical strain values are so small.they are often expressed as micro-strain, which is « x 106 (note this isequivalent to parts per million orppm) with the symbol m«. Stillanother way to express strain is aspercent strain, which is « x 100. Forexample: 0.005 inch/inch = 5000m«=0.5%.

As described to this point, strain isfractional change in length and isdirectly measurable. Strain of thistype is also often referred to asnormal strain.

SHEARING STRAINAnother type of strain, calledSHEARING STRAIN, is a measureof angular distortion. Shearing strainis also directly measurable, but notas easily as normal strain. If we hada thick book sitting on a table topand we applied a force parallel to

the covers, we could see the shearstrain by observing the edges of thepages.

See Figure 3. Shearing strain, g, isdefined as the angular change inradians between two line segmentsthat were orthogonal in theundeformed state. Since this angleis very small for most metals,shearing strain is approximated bythe tangent of the angle.

POISSON STRAINIn Figure 4 is a bar with a uniaxialtensile force applied, like the bar inFigure 1. The dashed lines show theshape of the bar after deformation,pointing out another phenomenon,that of Poisson strain. The dashedlines indicate that the bar not onlyelongates but that its girth contracts.This contraction is a strain in thetransverse direction due to aproperty of the material known asPoisson’s Ratio. Poisson’s ratio, n,is defined as the negative ratio of the

SYMBOLSs normal stress

t shear stress

« strain (normal)

m« micro-strain ( « x 106)

g shear strain

E modulus of elasticityor Young’s modulus

n Poisson Ratio

GF gage factor

Rg gage resistance in ohms

Kt transverse sensitivity ratio

L length

DL change in length

DRg change in gage resistance(due to strain)

%DGF % change in gage factor(due to temperature)

Rl lead wire resistance

T temperature in °C

VIN bridge excitation voltage

VOUT bridge output voltage

Vr [(VOUT/VIN)[strained] — (VOUT/VIN)[unstrained] ]

Figure 1: Uniaxial Force Applied

Figure 2: Cantilever in Bending

Figure 3: Visualizing Shearing Strain

Figure 4: Poisson Strain

PRACTICAL STRAIN GAGE MEASUREMENTS

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STRESS AND STRAIN (continued)

strain in the transverse direction to thestrain in the longitudinal direction. It isinteresting to note that no stress isassociated with the Poisson strain.Referring to Figure 4, the equation forPoisson’s ratio is n = – «t /«1. Notethat n is dimensionless.

NORMAL STRESSWhile forces and strains are measurablequantities used by the designer andstress analyst, stress is the term used tocompare the loading applied to amaterial with its ability to carry the load.Since it is usually desirable to keepmachines and structures as small andlight as possible, component partsshould be stressed, in service, to thehighest permissible level. STRESSrefers to force per unit area on a givenplane within a body.

The bar in Figure 5 has a uniaxialtensile force, F, applied along the x-axis. If we assume the force to be

uniformly distributed over the cross-sectional area, A, the “average” stresson the plane of the section is F/A. Thisstress is perpendicular to the planeand is called NORMAL STRESS, s.Expressed in equation form, s = F/A,and is denoted in units of force perunit area. Since the normal stress is inthe x direction and there is nocomponent of force in the y direction,there is no normal stress in thatdirection. The normal stress is in thepositive x direction and is tensile.

SHEAR STRESSJust as there are two types of strain,there is also a second type of stress

called SHEAR STRESS. Wherenormal stress is normal to thedesignated plane, shear stress isparallel to the plane and has thesymbol t. In the example shown inFigure 5, there is no y component offorce, therefore no force parallel to theplane of the section, so there is noshear stress on that plane. Since theorientation of the plane is arbitrary,what happens if the plane is orientedother than normal to the line of actionof the applied force? Figure 6demonstrates this concept with asection taken on the n-t coordinatesystem at some arbitrary angle, f, tothe direction of action of the force.

We see that the force vector, F, can bebroken into two components, Fn andFt , that are normal and parallel to theplane of the section. This plane has across-sectional area of A' and has bothnormal and shear stresses applied.The average normal stress, s, is inthe n direction and the average shearstress, t, is in the t direction. Theirequations are: s = Fn /A' and t = Ft /A'. Note that it was the forcevector that was broken intocomponents, not the stresses, and thatthe resulting stresses are a function ofthe orientation of the section. Thismeans that stresses (and strains),while having both magnitude anddirection, are not vectors and do notfollow the laws of vector addition,except in certain special cases, andthey should not be treated as such.We should also note that stresses arederived quantities computed fromother measurable quantities, and arenot directly measurable. [3]

PRINCIPAL AXESIn the preceding examples, the x-yaxes are also the PRINCIPAL AXES

for the uniaxially loaded bar. Bydefinition, the principal axes are theaxes of maximum and minimumnormal stress. They have theadditional characteristic of zero shearstress on the planes that lie alongthese axes. In Figure 5, the stress inthe x direction is the maximum normalstress, and we noted that there wasno force component in the y directionand therefore zero shear stress onthe plane. Since there is no force inthe y direction, there is zero normalstress in the y direction and in thiscase zero is the minimum normalstress. So the requirements for theprincipal axes are met by the x-yaxes. In Figure 6, the x-y axes are theprincipal axes, since that bar is alsoloaded uniaxially. The n-t axes inFigure 6 do not meet the zero shearstress requirement of the principalaxes. The corresponding STRAINSon the principal axes is alsomaximum and minimum and theshear strain is zero.

The principal axes are very importantin stress analysis because themagnitudes of the maximum andminimum normal stresses are usuallythe quantities of interest. Once theprincipal stresses are known, then thenormal and shear stresses in anyorientation can be computed. If theorientation of the principal axes isknown, through knowledge of theloading conditions or experimentaltechniques, the task of measuring thestrains and computing the stresses isgreatly simplified.

In some cases, we are interested inthe average value of stress or load ona member, but often we want todetermine the magnitude of thestresses at a specific point. Thematerial will fail at the point where thestress exceeds the load-carryingcapacity of the material. This failuremay occur because of excessivetensile or compressive normal stressor excessive shearing stress. Inactual structures, the area of thisexcessive stress level may be quitesmall. The usual method ofdiagramming the stress at a point isto use an infinitesimal element that

Figure 5: Normal Stress

Figure 6: Shear Stress

surrounds the point of interest. Thestresses are then a function of theorientation of this element, and, inone particular orientation, the elementwill have its sides parallel to theprincipal axes. This is the orientationthat gives the maximum andminimum normal stresses on thepoint of interest.

STRESS-STRAINRELATIONSHIPSNow that we have defined stress andstrain, we need to explore the stress-strain relationship, for it is thisrelationship that allows us to calculatestresses from measured strains. If wehave a bar made of mild steel andincrementally load it in uniaxial tensionand plot the strain versus the normalstress in the direction of the appliedload, the plot will look like the stress-strain diagram in Figure 7.

From Figure 7, we can see that, up toa point called the proportional limit,there is a linear relationship betweenstress and strain. Hooke’s Lawdescribes this relationship. The slopeof this straight-line portion of thestress-strain diagram is theMODULUS OF ELASTICITY orYOUNG’S MODULUS for thematerial. The modulus of elasticity, E,has the same units as stress (forceper unit area) and is determinedexperimentally for materials. Writtenin equation form, this stress-strainrelationship is s = E •«. Somematerials do not have a linear portion(for example, cast iron and concrete)to their stress-strain diagrams. To doaccurate stress analysis studies forthese materials, it is necessary todetermine the stress-strain properties,

including Poisson’s ratio, for theparticular material on a testingmachine. Also, the modulus ofelasticity may vary with temperature.This variation may need to beexperimentally determined andconsidered when performing stressanalysis at temperature extremes.There are two other points of intereston the stress-strain diagram in Figure7: the yield point and the ultimatestrength value of stress.

The yield point is the stress level atwhich strain will begin to increaserapidly with little or no increase instress. If the material is stressedbeyond the yield point, and then thestress is removed, the material will notreturn to its original dimensions, butwill retain a residual offset or strain.The ultimate strength is the maximumstress developed in the materialbefore rupture.

The examples we have examined tothis point have been examples ofuniaxial forces and stresses. Inexperimental stress analysis, thebiaxial stress state is the mostcommon. Figure 8 shows an exampleof a shaft with both tension andtorsion applied. The point of interest issurrounded by an infinitesimalelement with its sides oriented parallelto the x-y axes. The point has a biaxialstress state and a triaxial strain state(remember Poisson’sratio). The element,rotated to be alignedwith the principal (p-q)axes, is also shown inFigure 8. Figure 9shows the elementremoved with arrowsadded to depict the stresses at the

point for both orientations of theelement.

We see that the element orientedalong the x-y axes has a normalstress in the x direction, zero normalstress in the y direction and shearstresses on its surfaces. The elementrotated to the p-q axes orientation hasnormal stress in both directions butzero shear stress as it should, bydefinition, if the p-q axes are theprincipal axes. The normal stresses,sp and sq , are the maximum andminimum normal stresses for thepoint. The strains in the p-q directionare also the maximum and minimum,and there is zero shear strain alongthese axes. Appendix C gives theequations relating stress to strain forthe biaxial stress state.

If we know the orientation of theprincipal axes, we can then measurethe strain in those directions andcompute the maximum and minimumnormal stresses and the maximumshear stress for a given loadingcondition. We don’t always know theorientation of the principal axes, but ifwe measure the strain in threeseparate directions, we can computethe strain in any direction including theprincipal axes’ directions. Three- andfour-element rosette strain gages areused to measure the strain when theprincipal axes’ orientation is unknown.The equations for computing theorientation and magnitude of theprincipal strains from 3-elementrosette strain data are found inAppendix C.

For further study of the mechanics ofmaterials, refer to Items 1, 4, and 6referenced in the Bibliography.Properties of several commonengineering materials are listed inAppendix A.

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Figure 7: Stress-Strain Diagram for Mild SteelFigure 8: Shaft in Torsion and Tension

Figure 9: Element on X-Y Axes and Principal Axes

PRACTICAL STRAIN GAGE MEASUREMENTS

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MEASURING STRAIN

S tress in a material can’t bemeasured directly. It must be

computed from other measurableparameters. Therefore, the stressanalyst uses measured strains inconjunction with other properties ofthe material to calculate the stressesfor a given loading condition. Thereare methods of measuring strain ordeformation based on variousmechanical, optical, acoustical,pneumatic, and electrical phenomena.This section briefly describes severalof the more common methods andtheir relative merits.

GAGE LENGTHThe measurement of strain is themeasurement of the displacementbetween two points some distanceapart. This distance is the GAGELENGTH and is an importantcomparison between various strainmeasurement techniques. Gagelength could also be described as thedistance over which the strain isaveraged. For example, we could, onsome simple structure such as thepart in Figure 10, measure the partlength with a micrometer both beforeand during loading. Then we wouldsubtract the two readings to get thetotal deformation of the part. Dividingthis total deformation by the originallength would yield an average value ofstrain for the entire part. The gagelength would be the original length ofthe part.

If we used this technique on the partin Figure 10, the strain in the reducedwidth region of the part would belocally higher than the measuredvalue because of the reduced cross-

sectional area carrying the load. Thestresses will also be highest in thenarrow region; the part will rupturethere before the measured averagestrain value indicates a magnitude ofstress greater than the yield point ofthe material as a whole.

Ideally, we want the strain measuringdevice to have an infinitesimal gagelength so we can measure strain at apoint. If we had this ideal strain gage,we would place it in the narrow portionof the specimen in Figure 10 tomeasure the high local strain in thatregion. Other desirable characteristicsfor this ideal strain measuring devicewould be small size and mass, easyattachment, high sensitivity to strain,low cost and low sensitivity totemperature and other ambientconditions. [2,6]

MECHANICAL DEVICESThe earliest strain measurementdevices were mechanical in nature. Wehave already considered an example(using a micrometer to measure strain)and observed a problem with thatapproach. Extensometers are a classof mechanical devices used formeasuring strain that employ a systemof levers to amplify minute strains to alevel that can be read. A minimumgage length of 1⁄2 inch and a resolutionof about 10 me is the best that canbe achieved with purely mechanicaldevices. The addition of a light beamand mirror arrangements toextensometers improves resolutionand shortens gage length, allowing2 me resolution and gage lengthsdown to 1⁄4 inch.

Still another type of device, thephotoelectric gage, uses acombination of mechanical, optical,and electrical amplifications tomeasure strain. This is done by usinga light beam, two fine gratings and aphotocell detector to generate anelectrical current that is proportional tostrain. This device comes in gagelengths as short as 1⁄16 inch, but it iscostly and delicate. All of thesemechanical devices tend to be bulky

and cumbersome to use, and mostare suitable only for static strainmeasurements.

OPTICAL METHODSSeveral optical methods are used forstrain measurement. One of thesetechniques uses the interferencefringes produced by optical flats tomeasure strain. This device issensitive and accurate, but thetechnique is so delicate that laboratoryconditions are required for its use.Item 5 referenced in the Bibliographygives excellent introductions to theoptical methods of photoelasticity,holography, and the moiré method ofstrain analysis. [2,5]

ELECTRICAL DEVICESAnother class of strain measuringdevices depends on electricalcharacteristics which vary inproportion to the strain in the body towhich the device is attached.Capacitance and inductance straingages have been constructed, butsensitivity to vibration, mountingdifficulties, and complex circuitrequirements keep them from beingvery practical for stress analysis work.These devices are, however, oftenemployed in transducers. Thepiezoelectric effect of certain crystalshas also been used to measure strain.When a crystal strain gage isdeformed or strained, a voltagedifference is developed across theface of the crystal. This voltagedifference is proportional to the strainand is of a relatively high magnitude.Crystal strain gages are, however,fairly bulky, very fragile, and notsuitable for measuring static strains.

Probably the most important electricalcharacteristic which varies inproportion to strain is electricalresistance. Devices whose outputdepends on this characteristic are thepiezoresistive or semiconductor gage,the carbon-resistor gage, and thebonded metallic wire and foil

Figure 10

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Figure 11: Large Area Strain Gages

temperature effects are usually lessimportant than for static strainmeasurements and the high output ofthe semiconductor gage is an asset.

The bonded resistance strain gage isby far the most widely used strain measurement tool for today’s experimental stress analyst. It consists of a grid of very fine wire (or, more recently, of thin metallic foil) bonded to a thin insulating backing called a carrier matrix. The

electrical resistance of this grid material varies linearly with

strain. In use, the carrier matrix is attached to the

test specimen with an adhesive.

When the specimen is loaded, the strain on its surface is transmitted to the grid material by the adhesive and carriersystem. The strain in the specimen isfound by measuring the change in theelectrical resistance of the grid

material. Figure 12 is a picture of abonded resistance strain gage with aConstantan foil grid and polyimidecarrier material. The bondedresistance strain gage is low in cost,can be made with a short gage length,is only moderately affected bytemperature changes, has smallphysical size and low mass, and hasfairly high sensitivity to strain. It issuitable for measuring both static anddynamic strains. The remainder of thisapplication note deals with theinstrumentation considerations formaking accurate, practical strainmeasurements using the bondedresistance strain gage. [2, 5, 6]

resistance gage. The carbon-resistorgage is the forerunner of the bondedresistance wire strain gage. It is low incost, can have a short gage length,and is very sensitive to strain. A highsensitivity to temperature andhumidity are the disadvantages of the carbon-resistor strain gage.

The semiconductor strain gage isbased on the piezoresistive effect incertain semiconductor materials suchas silicon and germanium.

Semiconductor gages have elasticbehavior and can be produced tohave either positive or negativeresistance changes when strained.They can be made physically smallwhile still maintaining a high nominal resistance. The strain limit for these gages is in the 1000 to 10000 me range, with most tested to 3000 me in tension.Semiconductor gages exhibit a highsensitivity to strain, but the change inresistance with strain is nonlinear.Their resistance and output aretemperature sensitive, and the highoutput, resulting from changes inresistance as large as 10-20%, can cause measurement problems when using the devices in a bridge circuit. However, mathematical correctionsfor temperature sensitivity, thenonlinearity of output, and thenonlinear characteristics of the bridgecircuit (if used) can be madeautomatically when using computer-controlled instrumentation to measurestrain with semiconductor gages.They can be used to measure bothstatic and dynamic strains. Whenmeasuring dynamic strains, Figure 12: Foil Bonded Resistance Strain Gages

applies to the strain gage as a whole,complete with carrier matrix, not justto the strain-sensitive conductor. Thegage factor for Constantan andnickel-chromium alloy strain gages isnominally 2, and various gage andinstrumentation specifications areusually based on this nominal value.

TRANSVERSE SENSITIVITYif the strain gage were a singlestraight length of conductor of smalldiameter with respect to its length, itwould respond to strain along itslongitudinal axis and be essentiallyinsensitive to strain appliedperpendicularly or transversely to thisaxis. For any reasonable value ofgage resistance, it would also have avery long gage length. When theconductor is in the form of a grid toreduce the effective gage length,there are small amounts of strain-sensitive material in the end loops orturn-arounds that lie transverse to thegage axis. See Figure 13. This endloop material gives the gage a non-zero sensitivity to strain in thetransverse direction. TRANSVERSESENSITIVITY FACTOR, Kt, isdefined as:

and is usually expressed in percent.Values of K range from 0 to 10%.

To minimize this effect, extra materialis added to the conductor in the endloops, and the grid lines are keptclose together. This serves tominimize resistance in the transversedirection. Correction for transversesensitivity may be necessary forshort, wide-grid gages, or wherethere is considerable misalignmentbetween the gage axis and theprincipal axis, or in rosette analysiswhere high transverse strain fieldsmay exist. Data supplied by themanufacturer with the gage can beentered into the computer thatcontrols the instrumentation, andcorrections for transverse sensitivity

PRACTICAL STRAIN GAGE MEASUREMENTS

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THE BONDEDRESISTANCE STRAINGAGE

T he term “bonded resistancestrain gage” can apply to the

nonmetallic (semiconductor) gage orto the metallic (wire or foil) gage.Wire and foil gages operate on thesame basic principle, and both canbe treated in the same fashion fromthe measurement standpoint. Thesemiconductor gage, having a muchhigher sensitivity to strain thanmetallic gages, can have otherconsiderations introduced into itsmeasurement. We will use the termSTRAIN GAGE or GAGE to refer tothe BONDED METALLIC FOIL GRIDRESISTANCE STRAIN GAGEthroughout the rest of this applicationnote. These foil gages are sometimesreferred to as metal-film gages.

Strain gages are made with a printedcircuit process using conductivealloys rolled to a thin foil. The alloysare processed, including controlled-atmosphere heat treating, to optimizetheir mechanical properties andtemperature coefficients of resistance.A grid configuration for the strainsensitive element is used to allowhigher values of gage resistancewhile maintaining short gage lengths.Gage resistance values range from30 to 3000 Ω, with 120 Ω and 350 Ωbeing the most commonly usedvalues for stress analysis. Gagelengths from 0.008 inch to 4 inchesare commercially available. Theconductor in a foil grid gage has alarge surface area for a given cross-sectional area. This keeps the shearstress low in the adhesive and carriermatrix as the strain is transmitted bythem. This larger surface area alsoallows good heat transfer betweengrid and specimen. Strain gages aresmall and light, operate over a widetemperature range, and can respondto both static and dynamic strains.They have wide application andacceptance in transducers as well as

in stress analysis.

In a strain gage application, the carriermatrix and the adhesive must worktogether to faithfully transmit strainfrom the specimen to the grid. Theyalso act as an electrical insulatorbetween the grid and the specimenand must transfer heat away from thegrid. Three primary factors influencinggage selection are 1) operatingtemperature; 2) state of strain(including gradients, magnitude andtime dependence); and 3) stabilityrequirements for the gage installation.The importance of selecting the propercombination of carrier material, gridalloy, adhesive, and protective coatingfor the given application cannot beover-emphasized. Strain gagemanufacturers are the best source ofinformation on this topic and havemany excellent publications to assistthe customer in selecting the properstrain gages, adhesives and protective coatings.

GAGE FACTORWhen a metallic conductor isstrained, it undergoes a change inelectrical resistance, and it is thischange that makes the strain gage auseful device. The measure of thisresistance change with strain isGAGE FACTOR, GF. Gage factor isdefined as the ratio of the fractionalchange in resistance to the fractionalchange in length (strain) along theaxis of the gage. Gage factor is adimensionless quantity, and the largerthe value, the more sensitive thestrain gage. Gage factor is expressedin equation form as:

It should be noted that the change inresistance with strain is not due solelyto the dimensional changes in theconductor, but that the resistivity ofthe conductor material also changeswith strain: The term gage factor

Equation No. 10

GF =DR/R

=DR/R

DL/L e

Kt =GF (transverse)

GF (longitudinal)

can thus be made to the strain dataas it is collected.

TEMPERATURE EFFECTSIdeally, we would prefer the straingage to change resistance only inresponse to stress-induced strain inthe test specimen, but the resistivityand strain sensitivity of all knownstrain-sensitive materials vary withtemperature. Of course this meansthat the gage resistance and thegage factor will change when thetemperature changes. This change inresistance with temperature for amounted strain gage is a function ofthe difference in the thermal

expansion coefficients between thegage and the specimen and of thethermal coefficient of resistance of thegage alloy. Self-temperature-compensating gages can beproduced for specific materials byprocessing the strain-sensitive alloy insuch a way that it has thermalresistance characteristics thatcompensate for the effects of themismatch in thermal expansioncoefficients between the gage and thespecific material. A temperaturecompensated gage produced in thismanner is accurately compensatedonly when mounted on a material thathas a specific coefficient of thermalexpansion. Table 2 is a list of commonmaterials for which self-temperature-compensated gages are available.

which to begin the measurement.The nominal value of gageresistance has a tolerance equivalentto several hundred microstrain, andwill usually change when the gage isbonded to the specimen, so thisnominal value can’t be used as areference.

An initial, unstrained gage resistanceis used as the reference againstwhich strain is measured. Typically,the gage is mounted on the testspecimen and wired to theinstrumentation while the specimenis maintained in an unstrained state.A reading taken under theseconditions is the unstrainedreference value, and applying astrain to the specimen will result in aresistance change from this value. Ifwe had an ohmmeter that wasaccurate and sensitive enough tomake the measurement, we wouldmeasure the unstrained gageresistance and then subtract thisunstrained value from thesubsequent strained values. Dividingthe result by the unstrained valuewould give us the fractionalresistance change caused by strainin the specimen. In some cases, it ispractical to use this very method, andthese cases will be discussed in alater section of this application note.A more sensitive way of measuringsmall changes in resistance is withthe use of a Wheatstone bridgecircuit, and, in fact, mostinstrumentation for measuring staticstrain uses this circuit. [2,5,6,7,8]

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Figure 14: Typical Temperature-Induced Apparent Strain

Figure 13: Strain Gage Nomenclature

APPROXIMATE THERMAL EXPANSION COEFFICIENT

MATERIAL PPM/°CQuartz 0.5

Titanium 9

Mild Steel 11

Stainless Steel 16

Aluminum 23

Magnesium 26

Table 2: Thermal Expansion Coefficientsof Some Common Materials for WhichTemperature Compensated Strain GagesAre Available

The compensation is only effectiveover a limited temperature rangebecause of the nonlinear character ofboth the thermal coefficient ofexpansion and the thermal coefficientof resistance.

THE MEASUREMENTFrom the gage factor equation, wesee that it is the FRACTIONALCHANGE in resistance that is theimportant quantity, rather than theabsolute resistance value of thegage. Let’s see just how large thisresistance change will be for a strainof 1me. If we use a 120 Ω straingage with a gage factor of +2, thegage factor equation tells us that 1me applied to a 120 Ω gageproduces a change in resistance of

or 240 micro-ohms. That means weneed to have micro-ohm sensitivityin the measuring instrumentation.Since it is the fractional change inresistance that is of interest, andsince this change will likely be onlyin the tens of milliohms, somereference point is needed from

DR = 120 x 0.000001 x 2 = 0.000240 Ω

Overall Pattern Length

Gage LengthSolder TabLength

Grid Width

Solder Tab Width

Grid Center Alignment Marks Inner Grid Lines

Outer Grid Lines

Tra

nsiti

on

Tab Spacing

EndLoops

PRACTICAL STRAIN GAGE MEASUREMENTS

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MEASUREMENTMETHODS

WHEATSTONE BRIDGECIRCUITBecause of its outstandingsensitivity, the Wheatstone bridgecircuit (depicted in Figure 15) is themost frequently used circuit for staticstrain measurement. This sectionexamines this circuit and its

application to strain gagemeasurement. By using a computerin conjunction with the measurementinstrumentation, we can simplify useof the bridge circuit, increasemeasurement accuracy, andcompile large quantities of data frommultichannel systems. The computeralso removes the necessity ofbalancing the bridge, compensatesfor nonlinearities in output, andhandles switching and data storagein multichannel applications.

BALANCED BRIDGE STRAINGAGE MEASUREMENTIn Figure 15, VIN is the input voltageto the bridge, Rg is the resistance ofthe strain gage, R1, R2 and R3 arethe resistances of the bridgecompletion resistors, and VOUT is thebridge output voltage. A 1⁄4 bridgeconfiguration exists when one arm ofthe bridge is an active gage and theother arms are fixed value resistorsor unstrained gages, as is the casein this circuit. Ideally, the straingage, Rg, is the only resistor in thecircuit that varies, and then only due

to a change in strain on the surfaceof the specimen to which it isattached. VOUT is a function of VIN,R1, R2, R3 and Rg. This relationship is:

When (R1/R2) = (Rg/R3), VOUTbecomes zero and the bridge isbalanced. If we could adjust one ofthe resistor values (R2, for example),then we could balance the bridge forvarying values of the other resistors.Figure 16 shows a schematic of thisconcept.

Referring to the gage factor equation,

we see that the quantity we need tomeasure is the fractional change ingage resistance from the unstrainedvalue to the strained value. If, whenthe gage is unstrained, we adjust R2until the bridge is balanced and thenapply strain to the gage, the changein Rg due to the strain will unbalancethe bridge and VOUT will becomenonzero. If we adjust the value of R2to once again balance the bridge,the amount of the change required inresistance R2 will equal the changein Rg due to the strain. Some strainindicators work on this principle byincorporating provisions for inputtingthe gage factor of the gage beingused and indicating the change inthe variable resistance, R2, directlyin micro-strain.

In the previous example, the bridge

becomes unbalanced when strain isapplied. VOUT is a measure of thisimbalance and is directly related tothe change in Rg, the quantity ofinterest. Instead of rebalancing thebridge, we could install an indicator,calibrated in micro-strain, thatresponds to VOUT. Refer to Figure16. If the resistance of this indicatoris much greater than that of thestrain gage, its loading effect on thebridge circuit will be negligible, i.e.,negligible current will flow throughthe indicator. This method oftenassumes: 1) a linear relationshipbetween VOUT and strain; 2) a bridgethat was balanced in the initial,unstrained, state; and 3) a knownvalue of VIN. In a bridge circuit, therelationship between VOUT and strainis nonlinear, but for strains up to afew thousand micro-strain, the erroris usually small enough to beignored. At large values of strain,corrections must be applied to theindicated reading to compensate forthis nonlinearity.

The majority of commercial strainindicators use some form ofbalanced bridge for measuringresistance strain gages. Inmultichannel systems, the number ofmanual adjustments required forbalanced bridge methods becomescumbersome to the user.Multichannel systems, undercomputer control, eliminate theseadjustments by using an unbalancedbridge technique.

UNBALANCED BRIDGE STRAINGAGE MEASUREMENTThe equation for VOUT can be rewrittenin the form of the ratio of VOUT to VIN:

This equation holds for both theunstrained and strained conditions.Defining the unstrained value ofgage resistance as Rg and thechange due to strain as DRg, thestrained value of gage resistance is

Figure 15: Wheatstone Bridge Circuit

Figure 16: Bridge Circuit with Provisionfor Balancing the Bridge

Equation No. 11

VOUT = VIN

R3 – R2[ R3 + Rg R1 + R2 ]

Equation No. 12

VOUT =R3 –

R2

VIN[ R3 + Rg R1 + R2

]

Equation No. 10

GF = DRg/Rg

e

Rg + DRg. The actual effective valueof resistance in each bridge arm isthe sum of all the resistances in thatarm, and may include such things aslead wires, printed circuit boardtraces, switch contact resistance,interconnections, etc. As long asthese resistances remain unchangedbetween the strained and unstrainedreadings, the measurement will bevalid. Let’s define a new term, Vr, asthe difference of the ratios of VOUT toVIN from the unstrained to thestrained state:

By substituting the resistor valuesthat correspond to the two (VOUT/VIN) terms into this equation, we canderive an equation for DRg/Rg. Thisnew equation is:

Note that it was assumed in thisderivation that DRg was the onlychange in resistance from theunstrained to the strained condition.Recalling the equation for gage factor:

and combining these two equations,we get an equation for strain interms of Vr and GF:

The schematic in Figure 17 showshow we can instrument theunbalanced bridge.

A constant voltage power supplyfurnishes VIN, and a digital voltmeter(DVM) is used to measure VOUT. TheDVM for this application should havea high (greater than 109 Ω) inputresistance, and 1 microvolt or betterresolution. The gage factor issupplied by the gage manufacturer.In practice, we would use a computer

from unstrained to strainedreadings,

• DVM accuracy, resolution, andstability were adequate for therequired measurement,

• resistance change in the activebridge arm was due only tochange in strain, and

• VIN and the gage factor were bothknown quantities.

Appendix B shows the schematicsof several configurations of bridgecircuits using strain gages, andgives the equation for strain as afunction of Vr for each.

MULTICHANNELWHEATSTONE BRIDGEMEASUREMENTSIn the preceding example,measurement accuracy wasdependent upon all four bridge arms’resistances remaining constant fromthe time of the unstrained reading tothe time of the strained reading,except for the change in the gageresistance due to strain. If any of thebridge arm resistances changedduring that time span, there would bea corresponding change in bridgeoutput voltage which would beinterpreted as strain-induced, so wewould see an error. The same wouldbe true of any other variation thatchanged the bridge output voltage.Any switching done in the bridgearms can cause a change inresistance due to variations in theswitch or relay contact resistance

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Figure 18: Switching Inside Bridge Arms

to have the DVM read and storeVOUT under unstrained conditions,then take another reading of VOUTafter the specimen is strained. Sincethe values for gage factor andexcitation voltage, VIN, are known,the computer can calculate thestrain value indicated by the changein bridge output voltage. If the valueof VIN is unknown or subject tovariation over time, we can have theDVM measure it at the time VOUT ismeasured to get a more precisevalue for Vr. This “timely”measurement of VIN greatly reducesthe stability requirements of thepower supply, allowing a lower-costunit to be used. Note that, in thepreceding 1⁄4 bridge example, thebridge was not assumed to bebalanced nor its outputapproximated as truly linear.Instead, we just derived the equationfor strain in terms of quantities thatare known or can be measured, andlet the computer solve the equationto obtain the exact strain value.

In the preceding example, we madesome assumptions that affect theaccuracy of the strain measurement:

• resistance in the three inactive bridge arms remained constant

Equation No. 10

GF = DRg/Rg

e

Figure 17:Instrumentation for Unbalanced Bridge

Strain Gage Measurement

Equation No. 15

e =-4Vr

GF(1 + 2Vr )

Equation No. 14

DRg =

-4Vr

Rg 1 + 2Vr

Equation No. 13

Vr = VOUT

–VOUT[(VIN

)strained (VIN )unstrained ]

PRACTICAL STRAIN GAGE MEASUREMENTS

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MEASUREMENTMETHODS (CONTINUED)

and can affect the bridge outputvoltage. For that reason, it is notdesirable to do switching inside thebridge arms for multichannelsystems, but, rather, to allow thoseinterconnections to be permanentlywired and switch the DVM frombridge to bridge. Since a DVM hasextremely high input impedancecompared to the bridge arms, itdoesn’t load the bridge, and switchingthe DVM has no effect on the bridgeoutput voltage level. Figures 18 and19 show the schematics of these twomethods of switching. We can seethat switching inside the bridge armsallows the same bridge completionresistors to be used for multiplegages, but that the power to the gageis removed when it is not being read.Also, any variations in switch contactresistance will appear in series withthe gage resistance and will beindistinguishable from resistancechanges due to strain.

Figure 19 shows a multiple-channelarrangement that switches the DVMand also shares the power supplyand internal half-bridge. This circuit isknown as a “Chevron Bridge” and isoften used for strain measurement onrotating machine elements to minimizethe number of slip rings. One channelis shown as a 1⁄4 bridge and the otheras a 1⁄2 bridge (two active gages). Themidpoint of the internal half-bridge foreither of these configurations servesas a voltage reference point for theDVM and isn’t affected by strain.Since the bridge completion resistors

must have excellent stabilityspecifications, they are relativelyexpensive, and there is a costadvantage to sharing the internalhalf-bridge in multichannel systems.

For this method to function properly,the circuit must be designed andconstructed such that a change incurrent due to strain in one arm doesnot change the current in any of theother arms. Also, the excitationvoltage, VIN, must be measuredacross points A-B, and it may bedesirable to measure this voltageeach time a new set of readings istaken from this group of channels.The DVM is switched between pointsC-D, C-E, etc., to read the outputvoltages of the various channels inthe group. This method keeps all ofthe gages energized at all times,which minimizes dynamic heatingand cooling effects in the gages andeliminates the need for switchinginside the bridge arms. If the DVMhas good low-level measurementcapability, the power supply voltagecan be maintained at a low level,thereby keeping the gage's self-heating effects to a minimum. Forexample, using a 2 volt power supplyfor the bridge yields a powerdissipation, in a 350 Ω gage, of only3 milliwatts. Yet even with this lowpower, 1 microstrain sensitivity is stillmaintained with a 1⁄4 bridgeconfiguration (assuming GF=2),when using a DVM with 1 microvoltresolution. Since several channelsare dependent upon one powersupply and one resistor pair, a failureof one of these components willcause several channels to becomeinoperative. However, themeasurement of the excitation

voltage permits the power supply todrift, be adjusted, or even bereplaced with no loss inmeasurement accuracy.

FOUR-WIRE OHM STRAINGAGE MEASUREMENTAs we mentioned before, we canmeasure the change in absolutevalue of gage resistance to computestrain. This can be done quiteaccurately using a four-wire Ωmeasurement technique with a highresolution (e.g., 1 milliohm per leastsignificant digit) digital multimeter(DMM). Figure 20 depicts the four-wire Ω method of resistance

measurement. The current source isconnected internally in the DMM tothe Ω source terminals, while thevoltmeter is connected to the Ωsense terminals of the DMM. Whena measurement is taken, the currentsource supplies a known fixed valueof direct current through the circuitfrom the Ω source terminals, whilethe voltmeter measures the dcvoltage drop across the gageresistance. The absolute resistancevalue is computed from the values ofcurrent and voltage by the DMM anddisplayed or output to a computer.The lead resistances, Rl , from the Ωsource terminals to the gage, are inseries with the gage resistance, butdo not affect the accuracy of themeasurement, since the voltage isread directly across the gage. Theinput impedance to the senseterminals is extremely high, so thecurrent flow in that loop is negligible.The source current value is typicallyvery low, which means the powerdissipated in the strain gage is alsovery low, and self-heating effects arevirtually eliminated. For example,

Figure 19: Schematic of Bridge Circuit with Shared Internal Half-Bridge and Power Supply

Figure 20: Schematic of Four-Wire Ohm Circuit

1 milliamp is a typical value for thesource current, and this correspondsto a power dissipation of120 microwatts in a 120 Ω gage or350 microwatts in a 350 Ω gage.

A technique for voltage offsetcompensation can be used withfour-wire Ω measurements tocorrect for these effects. This isaccomplished by first measuring thevoltage across the gage withoutcurrent flow from the sourceterminals, and then subtracting thisvalue from the voltage read withsource current flow. The resultingnet voltage is then used to computethe gage resistance. Offsetcompensated four-wire Ωmeasurements can be madeautomatically by the DMM if it hasthat capability, or the offsetcompensation can be accomplishedby the computer controlling theinstrumentation.

To use four-wire Ω for measuringstrain, we first make a resistancemeasurement of the gage in theunstrained condition and store thisreading. Then we apply strain to thespecimen and make anothermeasurement of gage resistance.The difference between these tworeadings divided by the unstrainedreading is the fractional change inresistance that we use in the gagefactor equation to compute strain. Ofcourse the DMM can input thesereadings directly to a computer,which calculates strain using thegage factor for the particular gage.This technique also lends itself tomultichannel systems, sincevariations in switch resistance in thecircuit have the same effect as leadresistances and do not affect theaccuracy of the measurement.

CONSTANT CURRENTTECHNIQUESIn the discussion of bridge circuits,we assumed that the bridgeexcitation was furnished by aconstant voltage source. We couldhave assumed constant currentexcitation for those discussions andderived the corresponding equations

strain can be observed, while slowly-changing effects, such astemperature, are rejected.

The use of very sensitive DMM’s tomeasure the bridge imbalancevoltage or the gage resistancedirectly with four-wire Ω limits thespeed at which the measurement canbe taken, and only low frequencydynamic strains can be measuredwith these methods. Higher speedanalog-to-digital converters typicallyhave lower sensitivities, so highersignal levels are needed whenmeasuring higher frequency dynamicor transient strains. One way toachieve this is to amplify the bridgeoutput voltage to an acceptable level.Another method is to use asemiconductor strain gage andexploit its large gage factor. Asemiconductor gage can be used in a bridge circuit (such as Figure 19)with a DVM having lower resolutionand higher speed than that requiredwith metal gages. A semiconductorgage can also be used in a circuitsimilar to that for four-wire Ω (seeFigure 22). In this case, the current

source and the DVM should beseparate instruments, to allow thecurrent level to be adjusted to obtainthe best output voltage for theexpected maximum strain level.

The lead wires do not affect themeasurement, since the voltage, asin four-wire Ω, is measured directlyacross the gage. This arrangementalso allows the use of a lesssensitive, higher speed DVM whilemaintaining reasonable strainresolution. For example, a DVM with100 microvolt sensitivity gives astrain resolution of 6 me with a0.44 milliamp current source (350 Ωsemiconductor gage with GF = 100).

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for strain as a function of voltage outand current supplied. In the exampleof Figure 19, the constant voltagesupply which is shared by multiplebridges cannot be directly replacedby a constant current source, sincewe wouldn’t know how the currentwas divided among the variousbridge circuits. In some cases, thebridge output is more nearly linearwhen using constant current ratherthan constant voltage excitation, butthat is of little consequence if wesolve an equation for strain versusoutput voltage with a computer. Theuse of a constant current source fora full-bridge configuration doeseliminate the need to sense thevoltage at the bridge, whicheliminates the need to run two wiresto the bridge. In general, there is noreal measurement advantage tousing constant current rather thanconstant voltage excitation for bridgecircuits as applied to strain gagemeasurements.

The four-wire Ω measurementdiscussed in the preceding sectionused a constant current source forexcitation, and we noted that thelead wires had no effect on themeasurement. That method requiredfour wires to be connected to thegage. Constant current excitation issometimes used with a two-wiregage connection for dynamic strainmeasurements where temperaturedrift effects are negligible or can befiltered out from the strain data. Inthe circuit of Figure 21, changes ingage resistance result inproportional changes in VOUT. Notethat VOUT is also affected by

changes in the lead resistances, Rl .By measuring only the time-varyingcomponent of VOUT, the dynamic

Figure 21: Constant Current CircuitDynamic Strain Measurement

Figure 22: Circuit for SemiconductorGage and High Speed Digital Voltmeter

PRACTICAL STRAIN GAGE MEASUREMENTS

E-57

PRACTICAL STRAINMEASUREMENT

SHIELDING AND GUARDINGINTERFERENCE REJECTIONThe low output level of a strain gagemakes strain measurementssusceptible to interference from othersources of electrical energy.Capacitive and magnetic coupling tolong cable runs, electrical leakagefrom the specimen through the gagebacking, and differences in groundingpotential are but a few of the possiblesources of difficulty. The results ofthis type of electrical interference canrange from a negligible reduction inaccuracy to deviances that render thedata invalid.

THE NOISE MODELIn Figure 23, the shaded portionincludes a Wheatstone bridge straingage measuring circuit seenpreviously in Figures 15 and 17. Thesingle active gage, Rg, is shownmounted on a test specimen — e.g.,an airplane tail section. The bridgeexcitation source, VIN, bridgecompletion resistors, R1 , R2 and R3,and the DVM represent themeasurement equipment located asignificant distance (say, 100 feet)from the test specimen. The straingage is connected to the measuringequipment via three wires havingresistance Rl in each wire. Theelectrical interference whichdegrades the strain measurement iscoupled into the bridge through anumber of parasitic resistance andcapacitance elements. In thiscontext, the term “parasitic” impliesthat the elements are unnecessaryto the measurement, are basicallyunwanted, and are to some extentunavoidable. The parasitic elementsresult from the fact that lead wireshave capacitance to other cables,gages have capacitance to the testspecimen, and gage adhesives andwire insulation are not perfect

insulators — giving rise to leakageresistance.

Examining the parasitic elements inmore detail, the active gage Rg isshown to be made up of two equalresistors with Ciso connected at thecenter. Ciso represents thecapacitance between the airplanetail section and the gage foil. Sincethe capacitance is distributeduniformly along the gage grid length,we approximate the effect as a“lumped” capacitance connected tothe gage’s midpoint. Riso and Cisodetermine the degree of electricalisolation from the test specimen,which is often electrically groundedor maintained at some “floating”potential different than the gage.Typical values of Riso and Ciso are1000 megohms and 100 pF,respectively. Elements Cc and Rcrepresent the wire-to-wirecapacitance and insulationresistance between adjacent poweror signal cables in a cable vault orcable bundle. Typical values for Ccand Rc are 30 pF and 1012 Ω per footfor dry insulated conductors in closeproximity.

The power supply exciting the bridgeis characterized by parasiticelements Cps and Rps. A line-powered, “floating output” powersupply usually has no deliberate

electrical connection between thenegative output terminal and earthvia the third wire of its power cord.However, relatively large amounts ofcapacitance usually exist betweenthe negative output terminal circuitsand the chassis and between theprimary and secondary windings ofthe power transformer. The resistiveelement Rps is caused by imperfectinsulators, and can be reducedseveral decades by ioniccontamination or moisture due tocondensation or high ambienthumidity. If the power supply doesnot feature floating output, Rps maybe a fraction of an Ω. It will be shownthat use of a non-floating orgrounded output power supplydrastically increases themechanisms causing electricalinterference in a practical, industrialenvironment. Typical values for Cpsand Rps for floating output, laboratorygrade power supplies are 0.01 mfand 100 megohms, respectively. It isimportant to realize that neither themeasuring equipment nor the gageshave been “ grounded” at any point.The entire system is “floating” to theextent allowed by the parasiticelements.

To analyze the sources of electricalinterference, we must first establisha reference potential. Safetyconsiderations require that the

Figure 23: Remote Quarter-Bridge MeasurementIllustrating Parasitic Elements and Interference Sources

Rl

Rl

RlRg

Riso

CisoR3

R1

R2

Vcm

Cc

100 FT

DVM

RpsCps

VINVs

RcMeasurement Unit

Measurement EarthConnection

Adjacent Power or Signal Cable

+

–

+

–

+ –+ –

– +– +

power supply, DVM, bridgecompletion, etc., cabinets all beconnected to earth ground throughthe third wire of their power cords. InFigure 23, this reference potential isdesignated as the measurementearth connection. The test specimenis often grounded (for safetyreasons) to the power system at apoint some distance away from themeasurement equipment. Thisphysical separation often gives riseto different grounding potentials asrepresented by the voltage sourceVcm. In some cases, functionalrequirements dictate that the testspecimen be “floated” or maintainedmany volts away from the powersystem ground by electronic powersupplies or signal sources. In eithercase, Vcm may contain dc and time-varying components — most oftenat power-line related frequencies.

Typical values of Vcm, the commonmode voltage, range from millivoltsdue to IR drops in “clean” powersystems to 250 volts for specimensfloating at power-line potentials (forexample, parts of an electric motor).The disturbing source, Vs, is shownconnected to measurement earthand represents the electricalpotential of some cable in closeproximity (but unrelated functionally)to the gage wires. In manyapplications, these adjacent cables

Figure 24. Here, an insulation failure(perhaps due to moisture) hasreduced parasitic dc to a fewthousand Ω, and dc current is flowingthrough the gage measurementleads as a result of the source Vs.Negligible current flows through theDVM because of its high impedance.The currents through Rg and Rldevelop error-producing IR dropsinside the measurement loops.

In Figure 25, a shield surrounds thethree measurement leads, and thecurrent has been intercepted by theshield and routed to the point wherethe shield is connected to the bridge.The DVM reading error has beeneliminated. Capacitive coupling fromthe signal cable to unshieldedmeasurement leads will producesimilar voltage errors, even if thecoupling occurs equally to all threeleads. In the case where Vs is a highvoltage sine wave power cable, theDVM error will be substantiallyreduced if the voltmeter integratesthe input for a time equal to anintegral number of periods (e.g., 1,10, or 100) of the power line waveform. The exact amount of the errorreduction depends upon the DVM’snormal mode rejection, which can beas large as 60-140 dB or 103:1 -107:1. If the DVM is of a type having avery short sampling period, i.e., lessthan 100 msec, it will measure the

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may not exist or may be so farremoved as to not affect themeasurement. They will be includedhere to make the analysis generaland more complete.

SHIELDING OFMEASUREMENT LEADSThe need for using shieldedmeasurement leads can be seen byexamining the case shown in

Figure 25: Addition of a Metal Shield Around the Gage WiresKeeps Current Due to Vs out of Measurement Leads

Figure 24: Current Leakage from Adjacent Cable Flows Through Gage Wires Causing Measurement Error

Rl

Rl

RlRg

Riso

Ciso

R3

R1

R2

Vcm

Cc

DVM

RpsCps

VINVs

RcMeasurement Unit

+

–

+

–

+ –+ –

– +– +

Rl

Rl

RlRg

Riso

Ciso

R3

R1

R2

Vcm

Cc

DVM

RpsCps

VINVs

Rc

+

–

+

–

+ –+ –

– +– +

PRACTICAL STRAIN GAGE MEASUREMENTS

E-59

PRACTICAL STRAINMEASUREMENT(CONTINUED)

instantaneous value of the dc signal(due to strain) plus interference.Averaging the proper number ofreadings can reduce the error due topower line or other periodicinterference.

Where the measurement leads runthrough areas of high magnetic field,near high-current power cables, etc.,using twisted measurement leads willminimize the loop areas formed bythe bridge arms and the DVM,thereby reducing measurementdegradation as a result of magneticinduction. The flat, three-conductorside-by-side, molded cablecommonly used for strain gage workapproaches the effectiveness of atwisted pair by minimizing the looparea between the wires. The use ofshielded, twisted leads and a DVMwhich integrates over one or morecycles of the power line wave formshould be considered wheneverleads are long, traverse a noisyelectromagnetic environment, orwhen the highest accuracy isrequired.

GUARDING THE MEASURINGEQUIPMENTFigure 26 shows the error-producingcurrent paths due to the commonmode source, Vcm, entering themeasurement loop via the gageparasitic elements, Ciso and Riso. Inthe general case, both ac and dccomponents must be considered.Again, current flow through gage

and lead resistances result in errorvoltages inside the bridge arms.Tracing either loop from the DVM’snegative terminal to the positiveterminal will reveal unwantedvoltages of the same polarity in eachloop. The symmetry of the bridgestructure in no way providescancellation of the effects due tocurrent entering at the gage.

Whereas shielding kept error-producing currents out of themeasurement loop by interceptingthe current, guarding controls currentflow by exploiting the fact that nocurrent will flow through an electricalcomponent that has both of itsterminals at the same potential.

In Figure 27, a “guard” lead hasbeen connected between the testspecimen (in close proximity to thegage) and the negative terminal ofthe power supply. This connectionforces the floating power supply andall the measuring equipment —including the gage — to the sameelectrical potential as the testspecimen. Since the gage and thespecimen are at the same potential,no error-producing current flowsthrough Riso and Ciso into themeasuring loops. Another way ofinterpreting the result is to say thatFigure 27: Guard Wire Diverts Common Mode Current away from Gage Wires

Rl

Rl

RlRg

Riso

CisoR3

R1

R2

Vcm

Cc

DVM

RpsCps

VINVs

Rc

+

–

+

–

+ –+ –

– +– +

Figure 26: Error-Producing Common Mode Current Path

Rl

Rl

RlRg

Riso

CisoR3

R1

R2

Vcm

Cc

DVM

RpsCps

VINVs

Rc

+

–

+

–

+ –+ –

– +– +

Guard Lead

the guard lead provides an alternatecurrent path around the measuringcircuit. It should be observed that, ifthe power supply and the rest of themeasuring circuits could not floatabove earth or chassis potential, theguarding technique would reduce theinterference by factors of only 2:1 or4:1. Proper guarding with a floatingsupply should yield improvements onthe order of 105:1 or 100 dB.

In situations where it is possible toground the test specimen atmeasurement earth potential, thecommon mode source, Vcm, will beessentially eliminated.

EXTENSION TOMULTICHANNELMEASUREMENTSFigure 28 shows the extension of theguarding technique to a multi-channel strain gage measurementusing a shared power supply andinternal half-bridge completionresistors. For simplicity, only thecapacitive parasitic elements areshown. In ordinary practice,capacitive coupling is usually moresignificant and more difficult to avoidthan resistive coupling. Forgenerality, we’ve used two test

to the gage) keeps the common modecurrent out of the gage leads selectedfor the measurement. Common modecurrent flows harmlessly through thegage leads of the unselected channel.It should be noted that each lead wireshield is “grounded” at only a singlepoint. The common mode currentthrough each combined guard andshield is limited by the relatively highimpedance of the parasitic elementCps, and should not be confused withthe “heavy” shield current which mightoccur if a shield were grounded atboth ends, creating a “ground loop”.

CMR LIMITATIONSThe schematics and discussion ofguarding presented thus far mightconvey the impression that infiniterejection of common modeinterference is possible. It seemsreasonable to ask what, if anything,limits common mode rejection? Figure30 includes a new parasitic element,Cug, the unguarded capacitance tochassis associated with the DVM andmultiplexer. In practice, the DVM andmultiplexer are usually realized asguarded instruments [Reference 13]featuring three-wire switching andmeasurements, but the guardisolation is not perfect. Capacitance

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specimens at different potentials withrespect to measurement earth. Theswitching shown in the figure allowssimultaneous selection of the DVMand the associated guard connection.

Figure 29 illustrates the currentsflowing due to the specimenpotentials Vcm1 and Vcm2. Note that,regardless of which channel isselected, the guard line (alsofunctioning as the shield for the wires

Figure 29: Multichannel Guard Switch Keeps Common Mode Currentout of Selected Gage Leads

Figure 28: Multichannel Strain Measurement Including Two Separate Test Specimens

Shield and Guard Combined

Rg1 Ciso1

R32

R1

R2

DVM

R31

Cps

VIN

Measurement Unit

+

–

+

–

– +– +

Vcm2

+

–

Vcm1

Ciso2

Rg2

DVMSwitch

GuardSwitch

Shield and Guard Combined

Rg1 Ciso1

R32

R1

R2

DVM

R31

Cps

VIN

+

–

+

–

– +– +

Vcm2

+

–

Vcm1

Ciso2

Rg2

DVM Switch

GuardSwitch

PRACTICAL STRAINMEASUREMENT(CONTINUED)

ranging from 15 pF to 20 mf can befound between the instrument lowconnection and chassis. In Figure 30,this capacitance causes a portion ofthe common mode current in theselected channel to flow through theinternal half-bridge resistors R1 andR2, giving rise to a measurementerror. In a multichannel system, all ofthe unselected channels (gages)sharing the same power supply alsocontribute current, but this currentexits the bridge via the power supplyand returns through the guard wire,causing no additional error.

In Figure 30, the ac interferencevoltage presented to the terminals ofthe DVM causes an error becausethe dc measuring voltmeter does nottotally reject the ac. A DVM’s abilityto measure dc voltage in thepresence of ac interference is calledthe normal mode rejection ratio(NMRR) and is usually stated for 50and 60 Hz interference.

A dc voltmeter’s NMRR is a functionof input filtering and the analog-to-digital conversion techniqueemployed.

Additionally, the DVM andmultiplexer system reject acinterference via guarding and designcontrol of parasitics. The quantitativemeasure of a system’s ability toreject common mode ac voltage isthe common mode rejection ratio,(CMRR), defined as:

where Vcm and VDVM are bothsinusoids at the power-linefrequency of interest - 50, 60, or400 Hz. Note that VDVM is an ac wave

the parasitics are controlled in thesystem and the samplingcharacteristics of the DVM, i.e.,integrating or instantaneoussampling.

Reference 10 provides additionalinformation on the subjects offloating, guarded measurements andrejection ratios. Appendix D containsmeasurement sensitivity data whichcan be used to computemeasurement error (in me) as afunction of DVM, power supply, andbridge completion resistorspecifications.

BRIDGE EXCITATION LEVELThe bridge excitation voltage levelaffects both the output sensitivity andthe gage self-heating. From ameasurement standpoint, a highexcitation level is desirable, but alower level reduces gage self-

PRACTICAL STRAIN GAGE MEASUREMENTS

E-61

Figure 30: Unguarded Capacitance of Multiplexer and DVM Result in Measurement Error Due to Vcm2 of Selected Channel

Rg1 Ciso1

R1

R2

DVM

R31

Cps

VIN

+

–

+

–

– +– +

Vcm2

+

–

Vcm1

Ciso2

Rg2

DVM Switch

GuardSwitch

Cps

R32

Equation No. 16

NMRR = 20 log VNM (ac)

VDVM (dc)

Equation No. 17

CMRR = 20 log Vcm (ac)

VDVM (ac)

form presented to the terminals of adc voltmeter. Thus, CMRR is an acvoltage transfer ratio from thecommon mode source to the DVMterminals. Caution must beexercised in comparing CMRRspecifications to insure that identicalprocedures are employed in arrivingat the numerical result.

The overall figure of merit for ameasurement system is the effectivecommon mode rejection ratio(ECMRR), which reflects thesystem’s ability to measure dcvoltage (strain) in the presence of ac

common mode interference. If allmeasurements are made at thesame frequency, and if the rejectionratios are expressed in dB,

Thus, ECMRR describes how well

Equation No. 18

EMCRR = CMRR x NMRR

ac interference = ac interference x ac @DVM

dc response error ac @DVM dc response error

transmission dc voltmetervia parasitics response toand guarding ac input

Equation No. 19

ECMRR(dB) = CMRR(dB) + NMRR(dB)

heating. The electrical power in thegage is dissipated as heat whichmust be transferred from the gage tothe surroundings. In order for thisheat transfer to occur, the gagetemperature must rise above that ofthe specimen and the air. The gagetemperature is therefore a functionof the ambient temperature and thetemperature rise due to powerdissipation.

An excessive gage temperature cancause various problems. The carrierand adhesive materials will nolonger be able to transmit strainfaithfully from the specimen to thegrid if the temperature becomes toohigh. This adversely affectshysteresis and creep and may showup as instability under load. Zero orunstrained stability is also affectedby high gage temperature.Temperature-compensated gagessuffer a loss of compensation whenthe temperature difference betweenthe gage grid and the specimenbecomes too large. When the gageis mounted on plastics, excessivepower dissipation can elevate thetemperature of the specimen underthe gage to the point that theproperties of the specimen change.

The power that must be dissipatedas heat by the gage in a bridgecircuit with equal resistance arms isgiven by the following equation:

a particular application. Powerdensity is the power dissipated bythe gage divided by the gage gridarea, and is given in units ofwatts/in2. Recommended values ofpower density vary, depending uponaccuracy requirements, from 2-10 forgood heat sinks (such as heavyaluminum or copper sections), to0.01-0.05 for poor heat sinks (suchas unfilled plastics). Stacked rosettescreate a special problem, in that thetemperature rise of the bottom gageadds to that produced by the twogages above it, and that of the centergage adds to the top gage’s. It mayrequire a very low voltage or differentvoltages for each of the three gagesto maintain the same temperature ateach gage. [6 ]

One way we can determine themaximum excitation voltage that canbe tolerated is by increasing thevoltage until a noticeable zeroinstability occurs. We then reducethe voltage until the zero is oncemore stable and without a significantoffset relative to the zero point at alow voltage. Bridge completionresistors also dissipate power and inpractice may be more susceptible todrift from self-heating effects than thestrain gage. The stability of thebridge completion resistors is relatedto load-life, and maintaining only afraction of rated power in them willgive better long term stability. If theabove method of finding themaximum voltage level is used, careshould be exercised to insure thatthe power rating of the completionresistors is not exceeded as thevoltage is increased.

Reducing the bridge excitationvoltage dramatically reduces gagepower, since power is proportional tothe square of voltage. However,bridge output voltage is proportionalto excitation voltage, so reducing itlowers sensitivity. If the DVM used toread the output voltage has1 microvolt resolution, 1 micro-strainresolution can be maintained with a1⁄4 bridge configuration, using a 2 voltbridge excitation level. If the DVMhas 0.1 microvolt resolution, theexcitation voltage can be lowered to0.2 volts while maintaining the samestrain resolution. From Table 3 we

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where P is the power in watts, Rg isthe gage resistance, I is the currentthrough the gage, and V is thebridge excitation voltage. FromEquation 20, we see that loweringthe excitation voltage (or gagecurrent) or increasing the gageresistance will decrease powerdissipation. Where self-heating maybe a problem, higher values of gageresistance should be used. Table 3illustrates the relationship betweenvoltage, gage resistance and powerdissipation.

The temperature rise of the grid isdifficult to calculate because manyfactors influence heat balance.Unless the gage is submerged in aliquid, most of the heat transfer willoccur by conduction to thespecimen. Generally, cooling of thegage by convection is undesirablebecause of the possibility of creatingtime-variant thermal gradients on thegage. These gradients can generatevoltages due to the thermocoupleeffect at the lead wire junctions,causing errors in the bridge outputvoltage. Heat transfer from the gagegrid to the specimen is viaconduction. Therefore, the gridsurface area and the materials andthicknesses of the carrier andadhesive influence gagetemperature. The heat sinkcharacteristics of the specimen arealso important.

POWER DENSITY is a parameterused to evaluate a particular gagesize and excitation voltage level for

STRAIN GAGE POWER DISSIPATIONBRIDGE GAGE POWER IN MILLIWATTS

EXCITATIONVOLTAGE 1000 Ω 500 Ω 350 Ω 120 Ω

0.1 0.0025 0.005 0.007 0.0210.2 0.010 0.020 0.029 0.0830.5 0.0625 0.125 0.179 0.5211.0 0.250 0.500 0.714 2.0832.0 1.000 2.000 2.857 8.3333.0 2.250 4.500 6.429 18.7504.0 4.000 8.000 11.429 33.3335.0 6.250 12.500 17.857 52.08310.0 25.000 50.000 71.400 208.300

Equation No. 20

P = V2/4Rg = (I2)Rg

Table 3

gage was Rg. Referring to Figure 32,we see that one of the lead wireresistances, Rl, is in series with thegage, so the total resistance of thatbridge arm is actually Rg + Rl. If wesubstitute this into Equation 14, itbecomes:

Rewriting the equation to solve forstrain, we see that the previous strainequation is in error by a factor of theratio of the lead wire resistance to thenominal gage resistance.

This factor is lead wire desensitization,and we see from Equation 22 andfrom Table 4 that the effect is reducedif the lead wire resistance is smalland/or the nominal gage resistance islarge. If ignoring this term (1 + Rl /Rg)will cause an unacceptable error,then it should be added to thecomputer program such that the

strains computed in Equation 15 aremultiplied by this factor. Appendix Bgives the equations for various bridgeconfigurations and the lead wireresistance compensation terms thatapply to them. Appendix A has atable containing the resistance, atroom temperature, of somecommonly used sizes of copper wire.

The most common cause of changesin lead wire resistance is temperaturechange. The copper used for leadwires has a nominal temperaturecoefficient of resistance, at25°C/77°C, of 0.00385 Ω/Ω °C. Forthe 2-wire circuit in Figure 31, thiseffect will cause an error if thetemperature during the unstrainedreading is different than the

PRACTICAL STRAIN GAGE MEASUREMENTS

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PRACTICAL STRAINMEASUREMENT(CONTINUED)

see that, at these excitation levels.the power dissipated by a 350 ohmgage goes from 2.857 to 0.029milliwatts. Thus, using a sensitiveDVM for measuring the bridge outputpermits the use of low excitationvoltages and low gage self-heatingwhile maintaining good measurementresolution.

The four-wire Ω technique is also agood way to keep the power in thegage extremely low. This is due tothe low value of constant currentsupplied to the gage by the DMM,typically 1 milliamp. This current (1 milliamp) corresponds to a powerdissipation of 0.12 milliwatts in a 120 Ω gage and 0.35 milliwatts in a350 Ω gage. With four-wire Ω, a gageis energized only when it is selectedand is actually being measured bythe DMM. As mentioned previously,resolution will be lower using four-wire Ω than with a bridge, but will beadequate for many applications.

LEAD WIRE EFFECTSIn the preceding chapter, referencewas made to the effects of lead wireresistance on strain measurement forvarious configurations. In a bridgecircuit, the lead wire resistance cancause two types of error. One is dueto resistance changes in the leadwires that are indistinguishable from

resistance changes in the gage. Theother error is known as LEAD WIREDESENSITIZATION and becomessignificant when the magnitude of thelead wire resistance exceeds 0.1% ofthe nominal gage resistance. Thesignificance of this source of error isshown in Table 4.

If the resistance of the lead wires isknown, the computed values of straincan be corrected for LEAD WIREDESENSITIZATION. In a priorsection, we developed equations forstrain as a function of the measuredvoltages for a 1⁄4 bridge configuration:

These equations are based on theassumptions that Vr is due solely tothe change in gage resistance, DRg,and that the total resistance of thearm of the bridge that contained the

Figure 31: Two-Wire 1⁄4 Bridge Connection

LEAD WIRE DESENSITIZATION(REFER TO FIGURE 32)

1⁄4 AND 1⁄2 BRIDGE, 3-WIRE CONNECTIONS

AWG Rg = 120 Ω Rg = 350 Ω18 .54% .19%20 .87 .3022 1.38 .4724 2.18 .7526 3.47 1.1928 5.52 1.8930 8.77 3.01

Magnitudes of computed strain values will below by the above percent per 100 feet of hard drawn solid copper lead wire at 25°C (77°F)

Table 4

Equation No. 14

DRg =

-4Vr

Rg (1 + 2Vr)

Equation No. 15

e =-4Vr

GF(1 + 2Vr )

Equation No. 21

DRg =

-4Vr Rg + Rl Rg

(1 + 2Vr )( Rg

)

Equation No. 22

e =-4Vr Rl

GF(1 + 2Vr ) • (1 +

Rg)

Error Term

temperature during the strainedreading. Error occurs because anychange in resistance in the gage armof the bridge during this time isassumed to be due to strain. Also,both lead wire resistances are inseries with the gage in the bridgearm, further contributing to the leadwire desensitization error.

The THREE-WIRE method ofconnecting the gage, shown in Figure32, is the preferred method of wiringstrain gages to a bridge circuit. Thismethod compensates for the effect oftemperature on the lead wires. Foreffective compensation, the leadwires must have approximately thesame nominal resistance, the sametemperature coefficient of resistanceand be maintained at the sametemperature. In practice, this iseffected by using the same size andlength wires and keeping themphysically close together.

Temperature compensation ispossible because resistance changesoccur equally in adjacent arms of thebridge and, therefore, the net effecton the output voltage of the bridge isnegligible. This technique worksequally well for 1⁄4 and 1⁄2 bridgeconfigurations. The lead wiredesensitization effect is reduced overthe two-wire connection becauseonly one lead wire resistance is inseries with the gage. The resistanceof the signal wire to the DVM doesn’taffect the measurement, because thecurrent flow in this lead is negligibledue to the high input impedance ofthe DVM.

Mathematical correction for lead wiredesensitization requires that theresistances of the lead wires beknown. The values given in wiretables can be used, but, for extremetemperatures, measurement of thewires after installation is required forutmost accuracy. Two methods forarriving at the resistance of the leadwires from the instrumentation side ofthe circuit in Figure 32 follow:

(1) If the three wires are the samesize and length, the resistancemeasured between points A andB, before the wires areconnected to the instrumentation,

connected to the instrumentation. Thistest will help identify gages that mayhave been damaged duringinstallation. Under laboratoryconditions with room-temperaturecuring adhesives, the mountedresistance value of metal foil gageswill usually fall within the packagetolerance range for the gage. Underfield conditions, the shift in gageresistance will usually be less than2%. Greater shifts may indicatedamage to the gage. The farther thegage resistance value deviates fromthe nominal value, the larger theunstrained bridge output voltage. Thislimits the strain range at maximumresolution when using the unbalancedbridge technique. The easiest, mostaccurate way to measure thisresistance is with the four-wire Ωfunction of a DMM.

GAGE ISOLATIONThe isolation resistance from thegage grid to the specimen, if thespecimen is conductive, should alsobe measured before connecting thelead wires to the instrumentation. Thischeck should not be made with ahigh-voltage insulation tester,because of possible damage to thegage, but rather with an ohmmeter. Avalue of isolation resistance of lessthan 500 MΩ usually indicates thepresence of some type of surfacecontamination. Contamination oftenshows up as a time-varying highresistance shunt across the gage,which causes an error in the strainmeasurement. For this reason, anisolation resistance value of at least150 MΩ should be maintained.

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is 2Rl.

(2) Measure the voltage from A-B(which is equivalent to B-C) andthe voltage from B-D. Since R3 istypically a precision resistorwhose value is well known, thecurrent in the C-D leg can becomputed using Ohm’s Law. Thisis the current that flows throughthe lead resistance, so the valueof Rl can be computed, since thevoltage from B-C is known. Theequation for computing Rl is:

These measured values for leadresistance should be retained for latercalculations.

DIAGNOSTICSTo insure strain data that is as errorfree as possible, various diagnosticchecks can be performed on thegage installation andinstrumentation. In a stress analysisapplication, the entire gageinstallation can’t be calibrated ascan be done with certaintransducers. Therefore, potentialerror sources should be examinedprior to taking data.

MOUNTED GAGE RESISTANCEThe unstrained resistance of the gageshould be measured after the gage ismounted but before the wiring is

Figure 32: Three-Wire 1⁄4 Bridge Connection

Equation No. 23

Rl =VAB • R3VBD

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PRACTICAL STRAINMEASUREMENT(CONTINUED)

A properly mounted gage with fullycured adhesive will usually have1000 MΩ or higher isolationresistance, so any gages with lowvalues should be suspect.[2]

DIAGNOSTIC BRIDGEMEASUREMENTSAdditional errors occur when voltagesare induced in the measurementcircuit by sources other than strain.These voltages may be in the form ofstatic offsets (such as a thermallyinduced voltage) or time-varyingdisturbances (such as a magneticallyinduced voltage). Other sources ofinterference are: capacitive couplingof signals to the gage or wiring;resistive leakage paths to the gage orfrom the wiring to adjacent signalcarriers; a leakage path in theexcitation supply; a poor connectionto a guard; or a damaged shield.Since error-producing interferencecan arise from so many unexpectedsources, what can be done to detectthe presence of unwanted voltages?

The first step is to disconnect theexcitation supply from the bridge andpower up all equipment that is to beoperating during the test. This insuresthat all possible interference sourcesare activated. Next, take severalconsecutive bridge output voltage

readings for each strain gagechannel. The voltages should be verynearly zero. If there is an offsetvoltage, it could be thermally inducedor due to a resistive leakage path. Atime-varying cyclic voltage could becaused by resistive, magnetic orcapacitive coupling to the interferingsource. Erratic voltage readings couldbe due to an open input to the DVM.An integrating voltmeter whichsamples over a whole number ofpower line cycles greatly increasesrejection of magnetic induction andother interference sources at power-line frequency. When we use non-integrating voltmeters, severalreadings can be averaged tominimize the effect on static strainreadings.

Thermally induced voltages arecaused by thermocouple effects atthe junctions of dissimilar metalswithin the measurement circuits in thepresence of temperature gradients.These can occur at connectors,where the lead wire meets the gagemetal, in switches or in the DVM.Magnetically induced voltages occurwhen the wiring is located in a time-varying magnetic field. Magneticinduction can be controlled by usingtwisted lead wires and formingminimum but equal loop areas ineach side of the bridge. These loopsshould be arranged as shown inFigure 33 to have minimum effect onbridge output. In severe magneticfields, magnetic shielding for thewiring may be required.

The next step is to connect theexcitation supply to the bridge. Aseries of readings taken by the DVMof the excitation voltage is a good

verification that the excitation supply isset to the correct voltage level and isstable enough to allow the accuracyexpected. Some thermally inducedvoltages may be due to heatingeffects from power dissipated in thebridge circuit, so a check should bemade with the power applied. This isdone by taking a sequence ofreadings of the bridge output, thenreversing the polarity of the excitationsupply and repeating the sequence.One-half the difference in the absolutevalues of the bridge output voltages isthe thermally induced voltage. If thetemperature and power levels willremain at this level during the test,then subsequent voltage readingscould be corrected by this offsetvoltage amount. To monitor thethermally induced voltages, the bridgepower can be connected with switchesso that the voltage readings can betaken with both power supply polarities.If the measured thermally inducedvoltages are more than a few micro-volts, the source should be found andeliminated rather than trying to correctthe voltage readings. If, after areasonable time for the gage andbridge resistors to reach steady statetemperatures, the voltage is stilldrifting, the excitation level may betoo high.

Another test on the gage, particularlyon the gage bond, can be performedat this time. While monitoring thebridge output with the DVM, presslightly on the strain gage with a pencil

Figure 34: Typical Installation

Figure 33: Gage Wiring to Minimize Magnetic Induction

eraser. The output voltage shouldchange slightly but then return to theoriginal value when the pressure isremoved. If the output voltagedoesn’t return to the original value orbecomes erratic, the gage isprobably imperfectly bonded or isdamaged and should be replaced.

The unstrained bridge output voltagelevel also has diagnostic value. Ashorted or open gage will give anoutput of approximately one-half theexcitation voltage. In many cases,the unstrained bridge output shouldbe 2 millivolts or less per excitationvolt. For example, if each of the fourbridge arms had a tolerance of ±1%,the unstrained output would at mostbe 10 millivolts per excitation volt. So,if the unstrained output is more thana few millivolts per volt of excitation,the installation should be inspected. Ifthe test entails some type oftemperature cycle and a temperaturecompensated gage is utilized,recording the unstrained output overthe temperature cycle is a method ofverifying the adequacy of thecompensation.

SHUNT CALIBRATION(VERIFICATION)When using the unbalanced bridgemethod of strain measurement withinstrumentation under computercontrol, there are no adjustments forbridge balance or span. Since shunt

TEMPERATURE EFFECTSWe have examined ways tocompensate for the effects oftemperature on the lead wires to thegage. Now let’s look at some methodsto compensate for the temperatureeffects on the gage resistance andthe gage factor. Some of thesemethods require the temperature tobe measured at the gage. This can beaccomplished by several differenttemperature sensors such asthermocouples, thermistors andresistance temperature detectors(RTD’s). Since we want to sense thetemperature of the strain gage itself,problems can arise when largethermal gradients exist or when thetemperature is rapidly changing. Weneed a sensor that has adequatethermal response, and we need tolocate it such that it senses the sametemperature that exists at the gage.

GAGE FACTOR VERSUSTEMPERATUREThe gage manufacturer supplies anominal gage factor and tolerancewith each gage. If this gage factor isper NAS 942, Reference [9], it is thenominal gage factor and tolerance asmeasured at room temperature, in auniaxial stress field, on a material witha Poisson’s ratio of 0.285, for thatparticular lot of gages. The tolerance

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calibration was originally used toadjust the span of balanced bridgeinstruments, what is the role of shuntcalibration with an unbalancedbridge? Shunt calibration with thistechnique might more correctly betermed “shunt verification,” since theinstrumentation won’t actually becalibrated by shunt calibration. Shuntverification is the placing in parallelwith one of the bridge arm resistors,or gages, of a resistor of knownvalue. This will change the bridgeoutput voltage by a predictableamount and, if we measure thisoutput change just as if it werecaused by strain, we can computethe equivalent strain value. Since wealready know the changein resistance from the parallelcombination of resistors, we cancompute the equivalent strain valuefor a given gage factor, i.e.,e = (1/GF)(DR/R). By using thesame program subroutines andinstrumentation which will be used inthe actual test, we verify most of thesystem and gain confidence in thetest setup.

The value of the shunt resistor isoften in the 10-500 kΩ range, so thecurrent through it is low, less than 1milliamp. This resistor is also outsidethe bridge arms, so the effects ofswitching and lead wires are not asimportant as for the gage. Any of thebridge arms for any bridgeconfiguration can be shunted and acorresponding value of equivalentstrain computed.

PRACTICAL STRAIN GAGE MEASUREMENTS

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PRACTICAL STRAINMEASUREMENT(CONTINUED)

on the gage factor directly affects theaccuracy of the strain computation. Inother words, the computed strainvalue will have a tolerance at least asgreat as the gage factor tolerance. Aplot showing how gage factor varieswith temperature is also furnishedwith the gage. This plot is in the formof % gage factor variation (%DGF)versus temperature (T). Thetemperature at which these variationsbecome significant depends upon thegage alloy and the accuracy required.

In practice, the temperature must bemeasured at the gage during thestrained measurement and the gagefactor variation computed or “lookedup.” The actual gage factor is thencomputed using this variation and thenominal gage factor.

This actual gage factor, GFA, is thenused in the equation for computingstrain, e.g., Equation 15, instead ofGF. The value of strain thuscomputed is compensated for theeffect of temperature on the gagefactor.

For most metallic gage alloyscommonly used for static strainmeasurement, the gage factorvariation with temperature is nearlylinear over a broad temperature rangeand is less than ±1% for temperatureexcursions of ±100°C/180°F. Forexample, the equation for gage factorvariation versus temperature in °C fora typical temperature-compensatedConstantan alloy gage, as taken fromthe plot enclosed by the manufacturer,was found to be:%DGF=0.007T–0.1.For gage alloys with nonlinearcharacteristics, we need to use apoint-by-point correction or some typeof curve-fitting routine to approximate

the temperature dependence. Ingeneral, gage factor temperaturecompensation is required only forlarge temperature extremes or fortests requiring the utmost accuracy.

TEMPERATURE-INDUCEDAPPARENT STRAINFor temperature compensated straingages, the manufacturer supplies aplot of temperature-inducedAPPARENT STRAIN versustemperature. This plot is obtained byinstalling a sample of gages from thelot on a piece of unstrained materialhaving a thermal coefficient ofexpansion matching that for whichthe compensated gage was intended,and then varying the temperature.The apparent strain value can thusbe computed and plotted versustemperature. The apparent straincurve may have been plotted byusing a gage factor of +2. This shouldbe considered when using this plot,since the actual gage value may bedifferent and temperature-dependent.A fourth- or fifth-order polynomial canbe used to describe the apparentstrain curve and can be obtainedfrom the manufacturer or derivedfrom the plot. Thermally inducedapparent strain occurs becauseperfect temperature compensationover a broad range can’t beachieved. It results from theinteraction of the thermal coefficientof resistance of the gage and thedifferential thermal expansionbetween the gage and the specimen.Also, the specimen will seldom be theexact alloy used by the gagemanufacturer in determining theapparent strain curve. Apparent strainis, of course, zero for the temperatureat which the gage is mounted. If thattemperature were maintained for theduration of the test, no correctionwould be required, but if thetemperature varies during the courseof the test, compensation for theapparent strain may be requireddepending upon the temperaturechanges, the gage alloy and theaccuracy required.

If the temperature changes betweenthe time of the unstrained and

strained readings, errors may beincurred, as can be seen from theapparent strain plot. These errors arein the form of a strain offset. If thegage temperature and the apparentstrain characteristics are known, thisoffset can be calculated and thestrain value compensatedaccordingly. Another way of achievingcompensation is to use an unstrained“dummy” gage mounted on the samematerial and subjected to the sametemperature as the active gage. Thisdummy gage and the active gagesthat are to be compensated should allbe from the same manufacturer’s lotso they all have the same apparentstrain characteristics. The dummycan be used in a bridge arm adjacentto the active gage, thereby effectingelectrical cancellation of the apparentstrain. For multichannel systemswhere many gages are mounted inan area of uniform temperature, it ismore efficient to read the dummygage directly. The value of strain readfrom the dummy gage will be thevalue of the apparent strain. Thestrain readings from the active gagesthat are mounted on the samematerial at the same temperature canthen be corrected by subtracting thisamount from them.

There are some cases where it isdesirable to generate a thermallyinduced apparent strain curve for theparticular gage mounted on the testspecimen. Such would be the case ifa compensated gage weren’tavailable to match the thermalcoefficient of expansion of thespecimen material, or if thecompensation weren’t adequate forthe desired accuracy. Any time thetemperature varies during the test,the accuracy of the apparent straincompensation can be improved byusing the actual characteristics of themounted gage. To accomplish this,the mounted but mechanicallyunstrained gage must be subjected totemperature variation, and theapparent strain computed atappropriate values of measuredtemperature. With computercontrolled instrumentation, the datacan be taken automatically while thetemperature is varied. If thetemperatures of the actual test areknown, the apparent strain values

Equation No. 24

GFA = GF (1 + % DGF)

100

with the media available today, andfrequent storage of data is goodinsurance against power interruptionsand equipment failures.

Previously we discussed using onepower supply for several differentchannels of strain gages andmeasuring it with an accurate DVM.This enables us to use aninexpensive supply and also allowsits replacement should it fail, with noloss in measurement accuracy. Wealso discussed a circuit that used acommon internal half-bridge forseveral channels of strain gages. Forvery expensive and/or long term testswhen using this technique, it may bedesirable to have some type of“backup” for this resistor pair, since

several data channels will be lostshould a resistor fail. This can beaccomplished by reading the voltageacross each of the two resistors andthe power supply voltage, and storing

can be recorded at only thosetemperatures and used as a “look up”table for correction of the test data.The temperature compensated gagefactor of the mounted gage should beused for computing these apparentstrain values. If the test temperaturesat which data will be taken are notknown, then it will be necessary togenerate the equation for theapparent strain curve over thetemperature range of interest. Curve-fitting computer programs areavailable to generate an equationthat approximates the measuredcharacteristics. [5,6]

DATA: INPUT, OUTPUT,STORAGEWhen using unbalanced bridgetechniques with computer control,data storage becomes an importantconsideration. Storage of theunstrained bridge imbalance voltageratio is especially critical, since forsome tests it may be impossible toreturn to the unstrained condition.This unstrained data should bestored in nonvolatile media such asmagnetic tape or disc, with aredundant copy if the test is critical orof long duration. Storage of thesubsequent strained readings can bedone during or after the test asrequired for data reduction or archivalpurposes. Large amounts of data canbe stored quickly and inexpensively

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Unstrained Data Should Be Stored in Nonvolatile Media

UNSTRAINED RAW DATACHAN Vout Vin RATIO

0 -0.000589 1.980 -0.0002981 -0.000528 1.980 -0.0002672 -0.000065 1.980 -0.0000333 -0.000101 1.980 -0.0000514 -0.000128 1.980 -0.0000655 -0.000418 1.980 -0.0002116 -0.000275 1.980 -0.0001397 -0.001345 1.980 -0.0006798 -0.000276 1.980 -0.0001399 -0.000244 1.980 -0.000123

Strain Readings from 10 Channels

STRAIN READINGSCHANNEL MICROSTRAIN

0 –2861 4102 11653 4174 2915 7766 2577 1428 3519 117

Plot of Stress vs. Time Computed fromStrain Gage Mounted on a Cantilever Beam

DATE AND TIME:

these voltages in a nonvolatilemedium. Should the resistor pair fail,they can be replaced with a new pairand a new set of voltage readingstaken. These two sets of readingswould then be used to compute anoffset voltage to compensate for thedifference in the ratio of the two pairsof resistors. This offset voltage wouldbe added to all strained imbalancevoltage readings taken with the newpair of resistors. This technique canresult in as little as ±10 microstrainloss in measurement accuracy.

Use of a computer to controlinstrumentation, data manipulation,and storage gives us almost unlimiteddata output capability. With the widevariety of printers, displays andplotters available, the test data can bereduced and output by the computerin almost any conceivable format,often while the test is still in progress.With computational power and “smart”instrumentation, we can greatlyincrease the speed and accuracy ofthe measurement while eliminatingthe tedious manual-adjustmentprocess. Now we have more time toconcentrate on the test results.

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APPENDICES ANDBIBLIOGRAPHY

APPENDIX A: TABLES

Quarter-Bridge Configurations

e =–4Vr Rl

GF(1 + 2Vr ) • (1 +

Rg)

AVERAGE PROPERTIES OF SELECTED ENGINEERING MATERIALSEXACT VALUES MAY VARY WIDELY

MODULUS OF ELASTICPOISSON'S ELASTICITY, E STRENGTH (*)

MATERIAL RATIO, n psi X 10 6 TENSION (psi)ABS (unfilled) — 0.2-0.4 4500-7500Aluminum (2024-T4) 0.32 10.6 48000Aluminum (7075-T6) 0.32 10.4 72000Red Brass, soft 0.33 15 15000Iron-Gray Cast — 13-14 —Polycarbonate 0. 285 0.3-0.38 8000-9500Steel-1018 0.285 30 32000Steel-4130/4340 0.28-0.29 30 45000Steel-304 SS 0.25 28 35000Steel-410 SS 0.27-0.29 29 40000Titanium alloy 0.34 14 135000

(*) Elastic strength can be represented by proportional limit, yield point, or yield strength at 0.2 percent offset.

APPENDIX B: BRIDGE CIRCUITS

WIRE RESISTANCESOLID COPPER WIRE

AWG Ω/FOOT (25°C) DIAMETER (IN)18 0.0065 0.04020 0.0104 0.03222 0.0165 0.025324 0.0262 0.020126 0.0416 0.015928 0.0662 0.012630 0.105 0.01032 0.167 0.008

Vr = [(VOUT/VIN)strained – (VOUT/VIN)unstrained]:

e = Strain: Multiply by 106 for microstrain:tensile is (+) and compressive is (–)

e =–4Vr Rl

GF[(1 + n ) -2Vr(n – 1)] • (1 +Rg)

Equations compute strain from unbalanced bridge voltages:

e =–2Vr Rl GF • (1 +

Rg)

OR

Half-Bridge Configurations (AXIAL)

sign is correct for VIN and VOUT as shown

GF = Gage Factor n = Poisson’s ratio:

(BENDING)

WHERE: ep,q = Principal strains

sp,q = Principal stresses

Up,q = the acute angle from the axis of gage 1 to the nearest principal axis. When positive,the direction is the same as that of the gage numbering and, when negative, opposite.

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e =–2Vr

GF[(n + 1) – Vr (n – 1)] e =

–2Vr

GF(n + 1)e =

–Vr

GF

Full-Bridge Configurations (BENDING)

APPENDIX C: EQUATIONSBIAXIAL STRESS STATE EQUATIONS

ROSETTE EQUATIONSRectangular Rosette:

Delta Rosette:

ex =sX – n

sy

E E

ey =sy – n

sx

E E

ez = – n sX – n

sy

E E

sx =E

(ex + n ey )1 - n2

sy =E

(ex + n ex)1 - n2

sz = 0

ep,q =1

2 [ ]sp,q =

E

2 [ ]Up,q =

1 TAN -1

2e2 – e1 – e3

2 e1 – e3

ep,q =1

3 [ ]sp,q =

E

3 [ ]Up,q =

1 TAN -1

3 (e2 e3 )

2 2e1 – e2 – e3

=

e1 + e2 + e3 ±1

1 – n 1 + n

e1 + e3 ±1

1 – n 1 + n

e1 + e3 ± (e1 – e3)2 + (2e2 – e1 – e3)2

= (e1 – e3)2 + (2e2 – e1 – e3)2

=2[(e1 – e2)2 + (e2 – e3)2 + (e3 – e1)2]

=2[(e1 – e2)2 + (e2 – e3)2 + (e3 – e1)2]e1 + e2 + e3 ±

2

3

1

23

1

45°

45°

60°

60°

=

(AXIAL)

1⁄4 Bridge Circuit

PRACTICAL STRAIN GAGE MEASUREMENTS

E-71

APPENDICES ANDBIBLIOGRAPHY(CONTINUED)

APPENDIX D: INSTRUMENTATION ACCURACYMeasurement error (in me) due to instrumentation is often difficult todetermine from published specifications. However, accuracy can becomputed using the following simplified error expressions. For the 1⁄4 bridge,add equations 1-6 (N = 1). For the 1⁄2 bridge with two active arms, addequations 2-6 (N=2). For the full bridge with four active arms, add equations3-6 (N = 4).

The total error for a measurement must also include gage, lead wire, and, ifapplicable, bridge nonlinearity errors. These are discussed in the body ofthis application note. Additionally, other equipment imperfections which varyfrom instrument to instrument must occasionally be considered (e.g., offsetscaused by leakage currents due to humidity or ionic contamination on PCboards and connectors).

(1) R3 change from unstrained to strained reading (due to temperature, load life, etc.) eerror ' –

DR3 /R3

GF

(2) R1

R2eerror ' D

R1 R1

R2 R2

GF•N

Digital voltmeters and A/D converters are specified in terms of a ± gain error (% of reading) and a ± offseterror (number of counts, in volts). Since strain calculations require two measurements, a repeatable offseterror, e.g., due to relay thermal EMF, etc., will cancel, but offset due to noise and drift will not. Assumingthat noise and drift dominate, the offset on two readings will be the root sum of squares of the two offsets.This is incorporated into the formulas.

,eerror ' –4(Offset error

strained)2 + (Offset error

unstrained)2

VIN•GF•N •

Error terms 4-6 can usually be ignored when using high accuracy DVM’s (e.g., 51⁄2 digit). These error termsare essentially the product of small bridge imbalance voltages with small gain or offset terms. For equations4-6, VOUT, the bridge imbalance voltage, is a measured quantity which varies from channel to channel. Tocalculate worst case performance, the equations use resistor tolerances and measured strain, eliminatingthe need for an exact knowledge of VOUT.

(3) DVM offset error on bridge measurement

The bridge excitation supply can be monitored with a DVM or preset using a DVM and allowed to drift. Inthe first case, supply related errors are due only to DVM gain and offset terms, assuming a quiet supply. Inthe second case, since power supply accuracy is usually specified in terms of a ± gain and a ± offset fromthe initial setting, identical equations can be used. Also for the second case, note that the strained readinggain error is the sum of the DVM and excitation supply gain errors, while the strained reading offset error isthe root sum of squares of the DVM and excitation supply offset errors.

(4) DVM gain error on bridge measurement

=

eerror ' –4 [(VOUT)•(Gain Error)strained reading

–(VOUT)• (Gain Error)unstrained reading

]GF•VIN•N •

,' – emeasured • (Gain Error)strained reading

– S tolerances on R 1/R2,R3,Rg • (Gain Error change )GF•N strained-unstrained

change from unstrained tostrained reading (due totemperature, load life, etc.)

E-72

E

ST

RA

IN G

AG

ES

CONCLUSIONSBased upon this example, severalimportant conclusions can be drawn:• Surprisingly large errors can result

(5) Offset error on supply measurement (or on supply drift)

(6) Gain error on supply measurement (or on supply drift)

BIBLIOGRAPHY AND CREDITS

EXAMPLEEvaluate the error for a 24-hour strainmeasurement with a ±5°C/9°Finstrumentation temperature variation. Thisincludes the DVM and the bridgecompletion resistors, but not the gages.The hermetically sealed resistors have amaximum TCR of ±3.1 ppm/°C, and havea ±0.1% tolerance. The DVM/Scannercombination, over this time andtemperature span, has a 0.004% gainerror and a 4 mV offset error on the0.1 volt range where the bridge outputvoltage, VOUT, will be measured. Theexcitation supply is to be set at 5 V usingthe DVM. The DVM has a 0.002% gainerror and a 100 mV offset error on the 10volt range. Over the given time andtemperature span, the supply has a0.015% gain error and a 150 mV offseterror and will not be remeasured. Themounted gage resistance tolerance isassumed to be ±0.5% or better. The strainto be measured is 3000 me and the gagefactor is assumed to be +2.

Notice that the temperature, as given, canchange by as much as ±10°C/18°Fbetween the unstrained and strainedmeasurements. This is the temperaturechange that must be used to evaluate theresistor changes due to TCR. The R1/R2ratio has the tolerance and TCR of tworesistors included in its specification, so theratio tolerance is ±0.2% and the ratio TCRis ±6.2 ppm/°C. The gain error change onthe bridge output measurement and onthe excitation measurement can be asmuch as twice the gain error specification.The following table shows the total errorand the contribution of individual errorequations 1-6.

OMEGA ENGINEERING, INC. gratefully acknowledges THE HEWLETT-PACKARD COMPANY for permission to reproduce their article

Application Note 290-1—Practical Strain Gage Measurements

1. Higdon, Ohlsen, Stiles, Weese and Riley: MECHANICS OF MATERIALS, 3rd Edition,John Wiley & Sons, Inc., New York, 1976.

2. C.C. Perry and H.R. Lissner: THE STRAIN GAGE PRIMER. McGraw-Hill, Inc., New York, 1962.

3. I.S. Sokolnikoff: TENSOR ANALYSIS (Theory and Applications to Geometry andMechanics of Continua), Second Edition, John Wiley & Sons, New York, 1964.

4. Hornsey, McFarland, Muhlbauer, and Smith: MECHANICS OF MATERIALS (An Individualized Approach), Houghton Mifflin Company, Boston, 1977.

5. MANUAL ON EXPERIMENTAL STRESS ANALYSIS, Third Edition, Society forExperimental Stress Analysis, Westport, CT, 1978.

6. James W. Daily and William F. Riley: EXPERIMENTAL STRESS ANALYSIS, Second Edition, McGraw-Hill, Inc., New York, 1978.

7. William M. Murray and Peter K. Stein: STRAIN GAGE TECHNIQUES, 1958.

8. Peter K. Stein: ADVANCED STRAIN GAGE TECHNIQUES, Stein EngineeringServices, Inc., Phoenix, AZ, 1962.

9. National Aerospace Standard 942, Revision 2. 1 July 1964: STRAIN GAGES, BONDEDRESISTANCE, Aerospace Industries Association of America, Inc., Washington DC

10. Hewlett-Packard Co., Application Note 123: FLOATING MEASUREMENTSAND GUARDING, Hewlett-Packard Co., Palo Alto, CA, 1970.

even when using state-of-the-artbridge completion resistors andmeasuring equipment.

• Although typical measurements willhave a smaller error, the numberscomputed reflect the guaranteedinstrumentation performance.

• Measuring the excitation supply forboth the unstrained and strainedreadings not only results in smallererrors, but allows the use of aninexpensive supply.

• Bridge completion resistor drift limitsquarter- and half-bridge performance.Changes due to temperature,moisture absorption and load liferequire the use of ultra-stablehermetically sealed resistors.

EQUATION 1⁄4 BRIDGE 1⁄2 BRIDGE FULL BRIDGE

(1) R3 15.5 — —

(2) R1/R2 31.0 15.5 —

(3) VOUT offset 2.3 1.1 0.6

(4) VOUT gain 0.4 0.4 0.3

(5) VIN offset 2 0.2 0.2

(6) VIN gain 1.3 1.1 1.0

Sum ±50.08 me ±18.3 me ±2.1 me

eerror ' 4 [(VOUT)•(Offset Error)strained reading

–(VOUT)• (Offset Error)unstrained reading

]GF•VIN2•N•

eerror ' 4 [(VOUT)•(Gain Error)strained reading

–(VOUT)• (Gain Error)unstrained reading

]GF•VIN•N •

,'emeasured • (Gain Error)strained reading

+ S tolerances on R 1/R2,R3,Rg • (Gain Error change )GF•N strained-unstrained

,'emeasured • (Offset Error)strained reading

+ S tolerances on R 1/R2,R3,Rg • (Offset errorstrained)

2 + (Offset errorunstrained)

2

VIN•GF•N=

VIN

Reproduced Courtesy of Hewlett Packard Co.