vti structural strain 0309

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949 • 955 • 1894

Technical

Practical Strain Gage

Introduction

With today’s emphasis on product liability and energy eiciency, designs must not only be lighterand stronger, but also more thoroughly tested than ever beore. This places new importance on the

subject o experimental stress analysis and the techniques or measuring strain. The main theme othis application note is aimed at strain measurements using bonded resistance strain gages. We wilintroduce considerations that aect the accuracy o this measurement and suggest procedures orimproving it. We will also emphasize the practical considerations o the strain gage measurement,with an emphasis on computer controlled instrumentation.

Appendix B contains schematics o many o theways strain gages are used in bridge circuits and theequations which apply to them.

Stress and Strain

The relationship between stress and strain is one othe most undamental concepts rom the study omechanics o materials and is o paramount importanceto the stress analyst. In experimental stress analysiswe apply a given load and then measure the strain onindividual members o a structure or machine. Thenwe use the stress strain relationships to compute thestresses in those members to veriy that these stressesremain within the allowable limits or the particularmaterials used.

Strain

When a orce is applied to a body, the body deorms.In the general case this deormation is called strain. Inthis application note we will be more speciic and deinethe term STRAIN to mean deormation per unit length

or ractional change in length and give it the symbol, ε.See Figure 1.

This is the strain that we typically measure with abonded resistance strain gage. Strain may be eithertensile (positive) or compressive (negative). See

Figure 2. When written in equation orm ε = ∆L/L , wesee that strain is a ratio and, thereore, dimensionless.To maintain the physical signiicance o strain, it is otenwritten with units o inch/inch. For most metals thestrains measured in experimental work are typically lessthan 0.005000 inch/inch.

Since practical strain values are so small, they are oten

expressed in microstrain which is ε x 10-6 (note this isequivalent to parts per million or ppm) and is expressed

by the symbol µε. Still another way to express strain is

as percent strain, which is ε x 100. For example: 0.005

inch/inch = 5000 µε = 0.5%.

As described to this point, strain is ractional change inlength and is directly measurable. Strain o this type isalso oten reerred to as normal strain.

Shearing Strain

Another type o strain, called SHEARING STRAIN is ameasure o angular distortion. Shearing strain is alsodirectly measurable, but not as easily as normal strain.I we had a thick book sitting on a table top and weapplied a orce parallel to the covers, we could see theshear strain by observing the edges o the pages. SeeFigure 3. Shearing strain, g , is deined as the angularchange in radians between two line segments thatwere orthogonal in the undeormed state. Since thisangle is very small or most metals, shearing strain isapproximated by the tangent o the angle

Figure 1Uni-axial Force Applied

Figure 2Cantilever in bending

Figure 3Visualizing Shearing Strain



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Technical


Poisson Strain

In Figure 4, a bar is shown with a uniaxial tensile orceapplied, like the bar in Figure 1. The dashed lines show

the shape o the bar ater deormation, pointing outanother phenomenon, that o Poisson strain . The dashedlines indicate that the bar not only elongates, but thatits girth contracts. This contraction is a strain in thetransverse direction due to a property o the materialknown as Poisson’s ratio. Poisson’s ratio, u, is deinedas the negative ratio o the strain in the transversedirection to the strain in the longitudinal direction.It is interesting to note that no stress is associatedwith the Poisson strain. Reerring to Figure 4, the

equation or Poisson’s Ratio is u = - εt / ε1. Note that u isdimensionless

Normal Stress

While orces and strains are measurable quantities used

by the designer and stress analyst, stress is the termused to compare the loading applied to a material withits ability to carry the load. Since it is usually desirableto keep machines and structures as small and light aspossible, the parts should be stressed, in service, tothe highest permissible level. STRESS reers to orceper unit area on a given plane within a body. The barin Figure 5 has a uniaxial tensile orce, F, applied alongthe x-axis. I we assume the orce to be uniormlydistributed over the cross-sectional area, A, the average

stress on the plane o the section is F/A. This stressis perpendicular to the plane and is called NORMAL

STRESS , s expressed in equation orm, s = F/A. and hasunits o orce per unit area. Since the normal stress is inthe x-direction and there is no component o orce in they-direction, there is no normal Stress in that direction.The normal stress is in the positive x-direction and istensile.

Shear Stress

Just as there are two types o strains, there is also asecond type o stress called SHEAR STRESS . Wherenormal stress is normal to the designated plane, shearstress is parallel to the plane and is expressed by thesymbol, t. In the example shown in Figure 5, there isno y-component o orce, thereore, no orce parallel tothe plane o the section, so there is no shear stress on

that plane. Since the orientation o the plane is arbitrarywhat happens i the plane is oriented other than normalto the line o action o the applied orce? Figure 6demonstrates this concept with a section taken on then-t coordinate system at some arbitrary angle, f, to theline o action o the orce.

We see that the orce vector, F, can be broken into twocomponents, Fn and Ft, that are normal and parallelto the plane o the section. This plane has a cross-sectional area o A’ and has both normal and shearstresses applied. The average normal stress, s, is inthe n-direction and the average shear stress, t is in thet-direction. Their equations are: s = Fn /A’ and t = Ft /A’.Note that it was the orce vector that was broken intocomponents, not the stresses, and that the resulting

stresses are a unction o the orientation o the section.This means that stresses (and strains), while havingboth magnitude and direction, are not vectors and donot ollow the laws o vector addition, except in certainspecial cases, and they should not be treated as such.We should also note that stresses are derived quantitiescomputed rom other measurable quantities, and arenot directly measurable.

Figure 4Poisson Strain

Figure 5Normal Stress

Figure 6Shear Stress





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Principal Axes

In the preceding examples the x-y axes are also thePRINCIPAL AXES or the uniaxially loaded bar. Bydeinition, the principal axes are the axes o maximumand minimum normal stress. They have the additionalcharacteristic o zero shear stress on the planes that liealong these axes. In Figure 5 the stress in the x-directionis the maximum normal stress, and we noted thatthere was no orce component in the y-direction andthereore zero shear stress on the plane. Since there isno orce in the y-direction, there is zero normal stressin the y-direction, and in this case zero is the minimumnormal stress. So the requirements or the principalaxes are met by the x-y axes. In Figure 6 the x-y axesare the principal axes since that bar is also loadeduniaxially. The n-t axes in Figure 6 do not meet thezero shear stress requirement o the principal axes. The

correspondingSTRAINS

on the principal axes are alsomaximum and minimum and the shear strain is zero.

The principal axes are very important in stress analysisbecause the magnitudes o the maximum and minimumnormal stresses are usually the quantity o interest.Once the principal stresses are known, the normal andshear stresses in any orientation may be computed. Ithe orientation o the principal axes is known, throughknowledge o the loading conditions or experimentaltechniques, the task o measuring the strains andcomputing the stresses is greatly simpliied.

In some cases we are interested in the average valueo stress or load on a member, but oten we want todetermine the magnitude o the stresses at a point. The

material will ail at the point where the stress exceedsthe load-carrying capacity o the material. This ailuremay occur because o excessive tensile or compressivenormal stress or excessive shearing stress. In actualstructures, the area o this excessive stress level maybe quite small. The usual method o diagramming thestress at a point is to use an ininitesimal element thatsurrounds the point o interest. The stresses are then a

unction o the orientation o this element and, in oneparticular orientation, the element will have its sides

parallel to the principal axes. This is the orientation thatgives the maximum and minimum normal stresses onthe point o interest.

Stress-Strain Relationships

Now that we have deined stress and strain, we needto explore the stress-strain relationship. It is thisrelationship that allows us to calculate stresses rom themeasured strains. I we have a bar made o mild steeland incrementally load it in uniaxial tension and plotthe strain versus the normal stress in the direction othe applied load, the plot will look like the stress-straindiagram in Figure 7.

From Figure 7 we can see that, up to a point called

the proportional limit, there is a linear relationshipbetween stress and strain. Hooke’s Law describes thisrelationship. The slope o this straight line portion o thestress-strain diagram is the MODULUS OF ELASTICITY or YOUNG’S MODULUS or the material. The moduluso elasticity, E, has the same units as stress (orceper unit area) and is determined experimentally ormaterials. Written in equation orm this stress-strain

relationship is s = E * ε. Some materials, or example,cast iron and concrete, do not have a linear portionto their stress-strain diagrams. To do accurate stressanalysis studies or these materials it is necessaryto determine the stress-strain properties, includingPoisson’s ratio, or the particular material on a testingmachine. Also, the modulus o elasticity may varywith temperature. This variation may need to be

experimentally determined and considered whenperorming stress analysis at temperature extremes.There are two other points o interest on the stress-strain diagram in Figure 7, the yield point and theultimate strength value o stress. The yield point is thestress level at which strain will begin to increase rapidlywith little or no increase in stress. I the material isstressed beyond the yield point, and then the stress isremoved, the material will not return to its original sizebut will retain a residual oset or strain.

Figure 7Stress - Strain diagramor mild steel





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The ultimate strength is the maximum stress developedin the material beore rupture.

The examples we have examined to this point have

been examples o uniaxial orces and stresses. Inexperimental stress analysis the biaxial stress state isthe most common. Figure 8 shows an example o ashat with both tension and torsion applied. The point ointerest is surrounded by an ininitesimal element,Figure 8 with its sides oriented parallel to the x-yaxes. The point has a biaxial stress state and a triaxialstrain state (remember Poisson’s ratio). The element,rotated to be aligned with the principal (p-q) axes, isalso shown in Figure 8. Figure 9 shows the elementremoved with arrows added to depict the stressesat the point or both orientations o the element. Wesee that the element oriented along the x-y axes hasa normal stress in the x-direction, zero normal stressin the y-direction, and shear stresses on its suraces.The element rotated to the p-q axes orientation has

normal stress in both directions but zero shear stressas it should, by deinition, i the p-q axes are theprincipal axes. The normal stresses, sp and sq, are themaximum and minimum normal stresses or the point.The strains in the p-q direction are also the maximumand minimum, and there is zero shear strain alongthese axes. Appendix C gives the equations relatingstress to strain or the biaxial stress state. I we knowthe orientation o the principal axes, we can thenmeasure the strain in those directions and computethe maximum and minimum normal stresses and themaximum shear stress or a given loading condition. We

don’t always know the orientation o the principal axes,but i we measure the strain in three separate directionswe compute the strain in any direction including theprincipal axes directions. Three and our element rosette

strain gages are used to measure the strain when theprincipal axes orientation is unknown. The equationsor computing the orientation and magnitudes o theprincipal strains rom 3-element rosette strain data areound in Appendix C.

Measuring Strain

Stress in a material can’t be measured directly. It mustbe computed rom other measurable parameters.Thereore, the stress analyst uses measured strainsin conjunction with other properties o the material tocalculate the stresses or a given loading condition.There are methods o measuring strain or deormationbased on various mechanical, optical, acoustical,pneumatic, and electrical phenomena. This section

briely describes several o the more common methodsand their relative merits.

Gage Length

The measurement o strain is the measurement o thedisplacement between two points some distance apart.This distance is the GAGE LENGTH and is an importantcomparison between various strain measurementtechniques. Gage length could also be described as thedistance over which the strain is averaged. For examplewe could (on some simple structure such as the part inFigure 10) measure the part length with a micrometerboth beore and during loading. Then we would subtracthe two readings to get the total deormation o the partDividing this total deormation by the original length

would yield an average value o strain or the entirepart.

The gage length would be the original length o thepart. I we used this technique on the part in Figure 10,the strain in the reduced width region o the part wouldbe locally higher than the measured value because othe reduced cross-sectional area carrying the load. The

Figure 8Shat in Torsion and Tension

Figure 9Element on X-Y Axes and Principal Axes




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stresses will also be highest in the narrow region andthe part will yield there beore the measured averagestrain value indicates a magnitude o stress greater thanthe yield point o the material.

Ideally, we want the strain measuring device to have anininitesimal gage length so we can measure strain ata point. I we had this ideal strain gage we would placeit in the narrow portion o the specimen in Figure 10to measure the high local strain in that region. Otherdesirable characteristics or this ideal strain measuringdevice are small size and mass, easy attachment, highsensitivity to strain, low cost, and low sensitivity totemperature and other ambient conditions.

Mechanical Devices

The earliest strain measurement devices weremechanical in nature. We have already considered anexample using a micrometer to measure strain and

observed a problem with that approach. Extensometersare a class o mechanical devices used or measuringstrain that employ a system o levers to ampliy theminute strains to a level that can be read. A minimum

gage length o 1/2 inch and a resolution o about 10 µε is the best that can be achieved with purely mechanicaldevices. The addition o light beam and mirrorarrangements to extensometers improves resolution

and shortens gage length, allowing 2 µε resolution andgage lengths down to 1/4 inch.

Still another type o device, the photoelectric gage, usesa combination o mechanical, optical, and electricalampliication to measure strain. This is done by using alight beam, two ine gratings, and a photocell detectorto generate an electrical current that is proportional

to strain. This device comes in gage lengths as shortas 1/16 inch but it is costly and delicate. All o thesemechanical devices tend to be bulky and cumbersometo use, and most are only suitable or static strainmeasurements.

Optical Methods

Several optical methods are used or strainmeasurement. One o these techniques uses theintererence ringes produced by optical lats to

measure strain. This device is sensitive and accurate butthe technique is so delicate that laboratory conditionsare required or its use.

Brittle Coatings

Brittle coating techniques are another way to indicatestatic strain. They are oten used in conjunctionwith strain gages. The test object is coated with abrittle lacquer and the load is applied in increments,when possible. The lacquer will crack irst in theregion o highest surace strain and the cracks willbe perpendicular to the highest tensile strain. Arepresentation o the ull strain ield is obtainedindicating where to locate the strain gages and in whatorientation. Under avorable conditions a reasonableestimate o the magnitude o the strain can be obtained.Getting goad data rom brittle lacquer coatings is asmuch art as science. For high-temperature applications,a brittle ceramic coating may be used instead o the

lacquer.

Electrical Devices

Another class o strain measuring devices depends onelectrical characteristics which vary in proportion tothe strain in the body to which the device is attached.Capacitance and inductance strain gages have beenconstructed but sensitivity to vibration, mountingdiiculties, and complex circuit requirements keep themrom being very practical or stress analysis work. Thesedevices are, however, oten employed in transducers.The piezoelectric eect o certain crystals has also beenused to measure strain. When a crystal strain gage isdeormed or strained, a voltage dierence is developedacross the ace o the crystal. This voltage dierence

is proportional to the strain and is o a relatively highmagnitude. Crystal strain gages are airly bulky, veryragile, and not suitable or measuring static strains.

Probably the most important electrical characteristic,which varies in proportion to strain, is that o electricalresistance. Devices whose output depend on thischaracteristic are the piezoresistive or semiconductorgage, the carbon-resistor gage, and the bonded metallicwire and oil resistance gages. The carbon resistor gageis the orerunner o the bonded resistance wire straingage. It is low in cost, can have a short gage length,

Figure 10Area o High Local Strain and Stress




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and is very sensitive to strain. A high sensitivity totemperature and humidity are the disadvantages o thecarbon-resistor strain gage.

The semiconductor strain gage is based on thepiezoresistive eect in certain semiconductor materialssuch as silicon and germanium. Semiconductor gageshave elastic behavior and can be produced to haveeither positive or negative resistance changes whenstrained. They can be made physically small whilestill maintaining a high nominal resistance. The strain

limit or these gages is in the 1000 to 10000 µε range

with most tested to 3000 µε in tension. Semiconductorgages exhibit a high sensitivity to strain but thechange in resistance with strain is nonlinear. Theirresistance and output are temperature sensitive andthe high output, resulting rom changes in resistanceas large as 10-20%, can cause measurement problems

when using the devices in a bridge circuit. However,mathematical corrections or the temperaturesensitivity, the nonlinearity o output, and the nonlinearcharacteristics o the bridge circuit (i used), can bemade automatically when using computer controlledinstrumentation to measure strain with semiconductorgages. They can be used to measure both static anddynamic strains. When measuring dynamic strains,temperature eects are usually less important than orstatic strain measurements, and the high output o thesemiconductor gage is an asset.

The bonded resistance strain gage is by ar the mostwidely used strain measurement tool or today’sexperimental stress analyst. It consists o a grid o veryine wire, or more recently, o thin metallic oil bonded

to a thin insulating backing called a carrier matrix. Theelectrical resistance o this grid material varies linearlywith strain. In use, the carrier matrix is attached to thetest specimen with an adhesive. When the specimenis loaded, the strain on its surace is transmitted tothe grid material by the adhesive and carrier system.The strain in the specimen is ound by measuring thechange in the electrical resistance o the grid material.The bonded resistance strain gage is low in cost, canbe made with a short gage length, is only moderatelyaected by temperature changes, has small physical sizeand low mass, and has airly high sensitivity to strain.It is suitable or measuring both static and dynamicstrains. The remainder o this application note deals withthe instrumentation considerations or making accurate,practical strain measurements using the bonded

resistance strain gage.

The Bonded Resistance Strain Gage

The term “bonded resistance strain gage” can apply tothe nonmetallic (semiconductor) gage or to the metallic(wire or oil) gage. Wire and oil gages operate on

the same basic principles and both can be treated inthe same ashion rom the measurement standpoint.The semiconductor gage, having a much highersensitivity to strain than metallic gages, can have otherconsiderations introduced into its measurement. Wewill use the term STRAIN GAGE or GAGE to reer to theBONDED METALLIC FOIL GRID RESISTANCE STRAINGAGE throughout the rest o this application note.These oil gages are sometimes reerred to as metal ilmgages.

Strain gages are made with a printed circuit processusing conductive alloys rolled to a thin oil. The alloysare processed, including controlled atmosphere heattreating, to optimize their mechanical propertiesand temperature coeicient o resistance. A gridconiguration or the strain sensitive element isused to allow higher values o gage resistance whilemaintaining short gage lengths. Gage resistancevalues range rom (30 to 3000) Ω, with 120 Ω and350 Ω being the most commonly used values or stressanalysis. Gage lengths rom 0.008 inch to 4 inches arecommercially available. The conductor in a oil gridgage has a large surace area or a given cross-sectionaarea. This keeps the shear stress low in the adhesiveand carrier matrix as the strain is transmitted by them.This larger surace area also allows good heat transerbetween grid and specimen. Strain gages are small andlight, will operate over a wide temperature range, andcan respond to both static and dynamic strains. Theyhave wide application and acceptance in transducers aswell as stress analysis.

In a strain gage application, the carrier matrix and theadhesive must work together to aithully transmit

the strains rom the specimen to the grid. They alsoserve as an electrical insulator between the grid andthe specimen and must transer heat away rom thegrid. Three primary actors inluencing gage selectionare operating temperature, state o strain (includinggradients, magnitude and time dependence), andstability requirements or the gage installation. Theimportance o selecting the proper combination ocarrier material, grid alloy, adhesive, and protectivecoating or the given application cannot be over-emphasized. Strain gage manuacturers are the bestsource o inormation on this topic and have manyexcellent publications to assist the customer in selectingthe proper strain gages, adhesives, and protectivecoatings.

Gage Factor When a metallic conductor is strained, it undergoes achange in electrical resistance, and it is this change thatmakes the strain gage a useul device. The measure othis resistance change with strain is GAGE FACTOR,





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GF. Gage actor is deined as the ratio o the ractionalchange in resistance to the ractional change in length(strain) along the axis o the gage. Gage actor is adimensionless quantity and the larger the value, themore sensitive the strain gage. Gage actor is expressedin equation orm as:

Equation 1

It should be noted that the change in resistance withstrain is not just due to the dimensional changes inthe conductor, but that the resistivity o the conductormaterial also changes with strain. The term gage actorapplies to the strain gage as a whole, complete withcarrier matrix, not just to the strain sensitive conductor.The gage actor or constantan and nickel-chromiumalloy strain gages is nominally 2 and various gage andinstrumentation speciications are usually based on thisnominal value.

Transverse Sensitivity

I the strain gage were a single straight lengtho conductor, and o small diameter with respectto its length, it would respond to strain along itslongitudinal axis and be essentially insensitive tostrain perpendicular or transverse to this axis. For anyreasonable value o gage resistance, it would alsohave a very long gage length. When the conductor is inthe orm o a grid to reduce the eective gage length,there are small amounts o strain sensitive materialin the end loops or turnarounds that lie transverse tothe gage axis. This end loop material gives the gage anon-zero sensitivity to strain in the transverse direction.TRANSVERSE SENSITIVITY FACTOR, Kt is deined by

and is usually expressed in percent. Values o Kt rangerom 0 to 10%. To minimize this eect, extra material isadded to the conductor in the end loops and the gridlines are kept close together. This serves to minimizethe resistance in the transverse direction. Correction ortransverse sensitivity may be necessary or short, widegrid gages, or where there is considerable misalignment

between the gage axis and the principal axis, or inrosette analysis where high transverse strain ieldsmay exist. Data supplied by the manuacturer withthe gage can be entered into the computer controllingthe instrumentation and corrections or transversesensitivity made to the strain data as it is collected.

Temperature Effects

Ideally we would preer the strain gage to changeresistance only in response to the stress induced strainin the test specimen, but the resistivity and the strainsensitivity o all known strain sensitive materials varywith temperature. O course that means the gageresistance and the gage actor will change when thetemperature changes. This change in resistance withtemperature or a mounted strain gage is a unctiono the dierence in the thermal expansion coeicientsbetween the gage and the specimen and o thethermal coeicient o resistance o the gage alloy. Sel-temperature compensating gages may be producedor speciic materials by processing the strain sensitivealloy such that it has thermal resistance characteristicsthat compensate or the eects o the mismatch inthermal expansion coeicients between the gage andthe speciic material. A temperature compensated gageproduced in this manner is accurately compensatedonly when mounted on a material that has a speciiccoeicient o thermal expansion. Table 2 is a listo common materials or which sel temperaturecompensated gages are available.

______________________________________________

Material PPM/˚C______________________________________________

Quartz 0.5Titanium 9Mild Steel 11Stainless Steel 16Aluminum 23

Magnesium 26______________________________________________

Table 2Thermal expansion coeicients o some commonmaterials or which temperature compensated straingages are available

The compensation is only eective over a limitedtemperature range because o the nonlinear charactero both the thermal coeicient o expansion and thethermal coeicient o resistance. The gage manuacturersupplies inormation speciying the accuracy o thetemperature compensation in the orm o an APPARENTSTRAIN curve. This is a plot o temperature-inducedapparent strain versus temperature, or the gage,mounted on a speciic material with a speciied

coeicient o thermal expansion. The equation or thiscurve can be obtained rom the gage manuacturer byapplying curve itting techniques to the graph suppliedwith the gage, or by generating an apparent strain curvewith the actual gage ater installation.

I we monitor the temperature at the gage during thestrain measurement we can solve this equation tocompensate or temperature-induced apparent strain.A later section o this application note discusses





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temperature compensation using this technique andalso a correction method or using a temperaturecompensated gage on a material with a thermal

coeicient o expansion dierent rom that usedby the manuacturer or the apparent strain curve.The manuacturer also supplies data, usually in theorm o a graph, that shows how gage actor varieswith temperature, so that the strain data can also becorrected or this temperature eect.

The Measurement

From the gage actor equation we see that it is theFRACTIONAL CHANGE in resistance that is theimportant quantity rather than the absolute resistancevalue o the gage. Let’s see just how large this

resistance change will be or a strain o 1 µε. I we usea 120 Ω strain gage with a gage actor o +2, the gage

actor equation tells us that 1 µε applied to a 120 Ω gage

produces a change in resistance o

∆R = 120 x 0.000001 x 2 = 0.000240 Ω,

or 240 µΩ. That means we need to have microohmsensitivity in the measuring instrumentation. Since it isthe ractional change in resistance that is o interest andsince this change will likely be only tens o milliohms,some reerence point is needed rom which to begin themeasurement. The nominal value o gage resistance hasa tolerance equivalent to several hundred microstrainand will usually change when the gage is bonded tothe specimen, so this nominal value can’t be used as areerence.

An initial, unstrained, gage resistance is used as the

reerence rom which strain is measured. Typically, thegage is mounted on the test specimen and wired to theinstrumentation while the specimen is maintained in anunstrained state. A reading taken under these conditionsis the unstrained reerence value and applying a strainto the specimen will result in a resistance change romthis value. I we had an ohmmeter that was accurateand sensitive enough to make the measurement, wewould measure the unstrained gage resistance andthen subtract this unstrained value rom the subsequentstrained values. Dividing the result by the unstrainedvalue would give us the ractional resistance change

caused by strain in the specimen. In some cases it ispractical to use just this method, and these cases willbe discussed in a later section o this application note.

A more sensitive way o measuring small changes inresistance is with the use o the Wheatstone bridgecircuit, and in act, most instrumentation or measuringstatic strain uses this circuit.

Measurement Methods

Wheatstone Bridge Circuit

Because o its outstanding sensitivity, the Wheatstonebridge circuit (depicted in Figure 11) is the mostrequently used circuit or static strain measurements.This section examines this circuit and its application tostrain gage measurements. By using a computer inconjunction with the measurement instrumentation,

we can simpliy using the bridge circuit, increasemeasurement accuracy, and compile large quantitieso data rom multi-channel systems. The computeralso removes the requirement or balancing the bridge,compensates or nonlinearities in output, and handlesthe switching and data storage in multi-channelapplications.

Balanced Bridge Strain Gage Measurement

In Figure 11, VIN is the input voltage to the bridge, Rg isthe resistance o the strain gage, R1, R2 and R3 are theresistances o the bridge completion resistors, and VOUTis the bridge output voltage. A 1/4-bridge conigurationexists when one arm o the bridge is an active gage andthe other arms are ixed value resistors or unstrained

gages, as is the case in this circuit. Ideally, the straingage, Rg, is the only resistor in the circuit that varies,and then only due to a change in strain on the suraceo the specimen to which it is attached. VOUT is aunction o VIN, R1, R2. R3 and Rg.This relationship is:

Equation 2

Figure 11Wheatstone Bridge Circuit




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When (R1 / R2) = (Rg / R3), VOUT becomes zero andthe bridge is balanced. I we could adjust one o the

resistor values, R2 or example, then we could balancethe bridge or various values o the other resistors.Figure 12 shows a schematic o this concept.

Reerring to the gage actor equation,

Equation 1

we see that the quantity we need to measure is theractional change in gage resistance rom the unstrainedvalue to the strained value. I, when the gage isunstrained, we adjust R2 until the bridge is balancedand then apply strain to the gage, the change in Rg dueto the strain will unbalance the bridge and V

OUTwill

become non-zero. I we adjust the value o R2 to onceagain balance the bridge, the amount o the changerequired in resistance R2 will equal the change in Rg due to the strain. Some strain indicators work on thisprincipal by incorporating provisions or inputtingthe gage actor o the gage being used and indicatingthe change in the variable resistance, R2, directly inmicrostrain.

In the previous example, the bridge becomesunbalanced when the strain is applied. VOUT is ameasure o this imbalance and is directly related tothe change in Rg, the quantity o interest. Instead orebalancing the bridge we could install an indicator,calibrated in microstrain, that responds to VOUT. Reerto Figure 12. I the resistance o this indicator is muchgreater than that o the strain gage, its loading eecton the bridge circuit will be negligible. i.e., negligiblecurrent will low through the indicator. This methodoten assumes: 1) a linear relationship betweenVOUT and strain, 2) a bridge that was balanced in theinitial, unstrained, state, 3) a known value o VIN. Ina bridge circuit the relationship between VOUT andstrain is nonlinear, but or strains up to a ew thousandmicrostrain, the error is usually small enough to beignored. At large values o strain, corrections must beapplied to the indicated reading to compensate or thisnonlinearity.

The majority o commercial strain indicators use someorm o balanced bridge or measuring resistance straingages. In multi-channel systems the number o manualadjustments required or balanced bridge methodsbecomes cumbersome to the user. Multi-channelsystems, under computer control, eliminate theseadjustments by using an unbalanced bridge technique.

Unbalanced Bridge Strain Gage Measurement

The equation or VOUT can be rewritten in the orm othe ratio o VOUT to VIN

Equation 3

This equation holds or both the unstrained and thestrained condition. Deining the unstrained value ogage resistance as Rg and the change due to strain as∆Rg, the strained value o gage resistance is Rg + ∆Rg.The actual eective value o resistance in each bridgearm is the sum o all the resistances in that arm andmay include such things as lead wires, printed circuitboard traces, switch contact resistance, interconnects,etc. As long as these resistances remain unchangedbetween the strained and unstrained readings, themeasurement will be valid. Let’s deine a new term, Vr,as the dierence o the ratios o VOUT to VIN rom theunstrained to the strained state:

Equation 4

By substituting the resistor values that correspond tothe two (VOUT /VIN) terms into this equation, we canderive an equation or ∆Rg /Rg.

This new equation is:

Equation 5

Figure 12Bridge circuit with provisionor balancing the bridge




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Note that it was assumed in this derivation that ∆Rg wasthe only change in resistance rom the unstrained tothe strained condition. Recalling the equation or gageactor:

Equation 1

and combining these two equations we get an equationor strain in terms o Vr and GF

Equation 6

The schematic in Figure 13 shows how we caninstrument the unbalanced bridge.

A constant voltage power supply urnishes VIN anda digital voltmeter (DVM) is used to measure VOUT.The DVM or this application should have a high(greater than 109 Ω) input resistance and 1 µV or betterresolution. The gage actor is supplied by the gagemanuacturer. In practice we would use a computer tohave the DVM read and store VOUT under unstrainedconditions, then take another reading o VOUT ater thespecimen was strained. Since the values or gage actorand the excitation voltage, VIN, are known, the computercan calculate the strain value indicated by the changein bridge output voltage. I the value o VIN is unknownor subject to variations with time, we can have the DVMmeasure it at the time VOUT is measured to get a moreprecise value or Vr. This timely measurement o VIN

greatly reduces the stability requirements o the powersupply, allowing a lower cost unit to be used. Note thatin the preceding 1/4-bridge example, the bridge was notassumed to be balanced nor its output approximated astruly linear. Instead we derived the equation or strain interms o quantities that are known, or can be measured,and let the computer solve the equation to obtain theexact strain value.

In the preceding example we made some assumptionsthat impact the accuracy o the strain measurement:

• Resistance in the three inactive bridge arms remainedconstant rom unstrained to strained readings.

• DVM accuracy, resolution, and stability were adequateor the required measurement.

• Resistance change in the active bridge arm was dueonly to change in strain.

• VIN and gage actor were both known quantities.

Appendix B shows schematics o several conigurationso bridge circuits, using strain gages, and gives theequation or strain as a unction o Vr or each.

Practical Strain Measurement

Shielding and Guarding Interference Rejection

The low output level o a strain gage makes strain

measurements susceptible to intererence rom othersources o electrical energy. Capacitive and magneticcoupling to long cable runs, electrical leakage rom thespecimen through the gage backing, and dierencesin grounding potential are but a ew o the possiblesources o diiculty. The results o this type o electricalintererence can range rom a negligible reduction inaccuracy to rendering the data invalid.

The Noise Model

In Figure 14, the shaded portion includes theWheatstone bridge strain gage measuring circuit seenpreviously in Figures 12 and 13. The single activegage, Rg, shown mounted on a test specimen - e.g.,an airplane tail section. The bridge excitation sourceVIN, bridge completion resistors, R1, R2 and R3. andthe DVM represent the measurement equipmentlocated a signiicant distance (e.g., 100 eet) romthe test specimen. The strain gage is connected tothe measuring equipment via three wires havingresistance R l in each wire. The electrical intererencewhich degrades the strain measurement is coupled intothe bridge through a number o parasitic resistanceand capacitance elements. In this context, the term“parasitic” implies that the elements are unnecessaryto the measurement, are basically unwanted, and are tosome extent unavoidable. The parasitic elements result

Figure 13Instrumentation or unbalancedbridge strain gage measurement




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rom the act that lead wires have capacitance to othercables, gages have capacitance to the test specimen,and gage adhesives and wire insulation are not perectinsulators - giving rise to leakage resistance. Examiningthe parasitic elements in more detail, the active gage,

Rg, is shown made up o two equal resistors with Ciso connected at the center. Ciso represents the capacitancebetween the airplane tail section and the gage oil. Sincethe capacitance is distributed uniormly along the gagegrid length, we approximate the eect as a “lumped”capacitance connected to the gage’s midpoint. Riso andCiso determine the degree o electrical isolation rom thetest specimen which is oten electrically grounded ormaintained at some “loating” potential dierent thanthe gage. Typical values o Riso, and Ciso, are 1000 MΩ and 100 pF, respectively. Elements Cc and Rc representthe wire-to-wire capacitance and insulation resistancebetween adjacent power or signal cables in a cable vaultor cable bundle. Typical values or Cc and Rc are 30pF and 1012 Ω per oot or dry insulated conductors inclose proximity.

The power supply exciting the bridge is characterizedby parasitic elements Cps and Rps. A line-powered,“loating output” power supply usually has nodeliberate electrical connection between the negativeoutput terminal and Earth via the third wire o its powercord. However, relatively large amounts o capacitanceusually exist between the negative output terminalcircuits and the chassis, and between the primary andsecondary windings o the power transormer. Theresistive element, Rps, is due to imperect insulators andmay be reduced several decades by ionic contaminationor moisture due to condensation or high ambienthumidity. I the power supply does not eature loatingoutput, Rps may be a raction o an ohm. It will beshown that use o a non-loating or grounded output

power supply drastically increases the mechanismscausing electrical intererence in a practical, industrialenvironment. Typical values or Cps and Rps or loatingoutput, laboratory grade power supplies are 0.01 µFand 100 MΩ respectively. It is important to realizethat neither the measuring equipment nor the gageshave been grounded at any point. The entire system isloating to the extent allowed by the parasitic elements.

To analyze the sources o electrical intererence wemust irst establish a reerence potential. Saety

considerations require that the power supply, DVM,bridge completion, cabinets, etc., all be connectedto earth ground through the third wire o theirpower cords. In Figure 14, this reerence potential isdesignated as the measurement earth connection. Thetest specimen is oten grounded (or saety reasons) tothe power system at a point some distance away romthe measurement equipment. This physical separationoten gives rise to dierent grounding potentials asrepresented by the voltage source, Vcm. In some cases,unctional requirements dictate that the test specimenbe “loated” or maintained many volts away rompower system ground by electronic power supplies orsignal sources. In either case, Vcm may contain constantand/or time varying components - most oten at powerline related requencies.

Typical values o Vcm, the common mode voltage,range rom millivolts due to IR drops in “clean”power systems to 250 V or specimens loating atpower line potentials, or example, parts o an electric

motor. The disturbing source, Vs, is shown connectedto measurement earth and represents the electricalpotential o some cable in close proximity, but unrelatedunctionally, to the gage wires. In many applications,these adjacent cables may not exist or may be so arremoved as to not aect the measurement. They will beincluded here to make the analysis general and morecomplete.

Shielding of Measurement Leads

The need or using shielded measurement leads can beseen by examining the case shown in Figure 15. Here aninsulation ailure (perhaps due to moisture) has reducedparasitic Rc to a ew thousand ohms and dc currentis lowing through the gage measurement leads as aresult o the source Vs. Negligible current lows throughthe DVM because o its high impedance. The currentsthrough Rg and R l , develop error-producing IR dropsinside the measurement loops.

In Figure 16, a shield surrounds the three measurementleads, and the current has been intercepted by theshield and routed to the point where the shield isconnected to the bridge. The DVM reading error hasbeen eliminated. Capacitive coupling rom the signal

Figure 14Remote quarter-bridge measurementillustrating parasitic elements andintererence sources





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cable to unshielded measurement leads will producesimilar voltage errors, even i the coupling occursequally to all three leads. In the case where Vs is a highvoltage sine wave power cable, the DVM error will besubstantially reduced i the voltmeter integrates theinput or a time equal to an integer number o periods

(e.g., 1, 10, or 100) o the power line wave orm. Theexact amount o the error reduction depends upon theDVM’s normal mode rejection, which can be as largeas 60 dB - 140 dB or 103:1 - 107:1. I the DVM is o atype having a very short sampling period, i.e., less than100 µs, it will measure the instantaneous value o dcsignal (due to strain) plus intererence. Averaging theproper number o readings can reduce the error due topower line or other periodic intererence.

In the situation where the measurement leads runthrough areas o high magnetic ields, near high currentpower cables, etc., twisted measurement leads minimizethe loop areas ormed by the bridge arms and the DVMthereby reducing measurement degradation as a resulto magnetic induction. The lat, three conductor, side-

by-side, molded cable commonly used or strain gagework approaches the eectiveness o a twisted pair

by minimizing the loop area between wires. The useo shielded, twisted leads and a DVM which integratesover one or more cycles o the power line waveormshould be considered whenever leads are long, traversea noisy electromagnetic environment, or when highestaccuracy is required.

Guarding the Measuring Equipment

Figure 17 shows the error producing current pathsdue to the common mode source, Vcm, entering themeasurement loop via the gage parasitic elements,Ciso and Riso. In the general case, both ac and dccomponents must be considered. Again, current lowthrough gage and lead resistances results in errorvoltages inside the bridge arms. Tracing either looprom the DVM’s negative terminal to the positiveterminal will reveal unwanted voltages o the samepolarity in each loop. The symmetry o the bridgestructure in no way provides cancellation o the eectsdue to current entering at the gage.

Whereas, shielding kept error-producing currents outo the measurement loop by intercepting the current,guarding controls current low by exploiting the act thatno current will low through an electrical componenthaving both o its terminals at the same potential.

In Figure 18, a guard lead has been connected betweenthe test specimen (in close proximity to the gage)and the negative terminal o the power supply. Thisconnection orces the loating power supply and allthe measuring equipment - including the gage - to thesame electrical potential as the test specimen. Since thegage and the specimen are at the same potential, noerror-producing current lows through Ciso and Riso intothe measuring loops. Another way o interpreting theresult is to say that the guard lead provides an alternate

current path around the measuring circuit.

It should be observed that i the power supply and therest o the measuring circuits could not loat aboveearth or chassis potential, the guarding technique wouldreduce the intererence by actors o only 2:1 or 4:1.Proper guarding with a loating supply should yieldimprovements on the order o 105:1 or 100 dB.

In situations where it is possible to ground the testspecimen at measurement earth potential, the commonmode source, Vcm, will be essentially eliminated.

Figure 15Current leakage rom adjacent cablelows through gage wires causingmeasurement error

Figure 16Addition o a metal shield around thegage wires keeps current due to Vs out omeasurement leads





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Bridge Excitation Level

The bridge excitation voltage level aects both theoutput sensitivity and the gage sel heating. From the

measurement standpoint, a high excitation level isdesirable but a lower level reduces gage sel heating.The electrical power in the gage is dissipated asheat which must be transerred rom the gage to thesurroundings. In order or this heat transer to occur, thegage temperature must rise above that o the specimenand the air. The gage temperature is thereore a unctiono the ambient temperature and the temperature risedue to power dissipation.

An excessive gage temperature can cause variousproblems. The carrier and adhesive materials are nolonger able to aithully transmit the strain rom thespecimen to the grid i the temperature becomestoo high. This adversely aects hysteresis and creepand may show up as instability under load. Zero or

unstrained stability is also aected by high gagetemperatures. Temperature compensated gages suer aloss o compensation when the temperature dierencebetween the gage grid and the specimen becomes toolarge, when the gage is mounted on plastics. Excessivepower dissipation can elevate the temperature o thespecimen under the gage to the point that the materialproperties o the specimen change.

The power that must be dissipated as heat by the gagein a bridge circuit with equal resistance arms is given bythe ollowing equation:

Equation 7

Where P is the power in watts, Rg is the gageresistance, I is the current through the gage and V isthe bridge excitation voltage. From Equation 7 we seethat lowering the excitation voltage (or age current) orincreasing the gage resistance will decrease the powerdissipation. When sel-heating may be a problem,higher values o gage resistance should be used.

The temperature rise o the grid is diicult to calculatebecause many actors inluence the heat balance.Unless the gage is submerged in a liquid, most

o the heat transer will occur by conduction to thespecimen. Generally, cooling o the gage by convectionis undesirable because o the possibility o creating timevariant thermal gradients on the gage. These gradientscan generate voltages due to the thermocouple eect atthe lead wire junctions causing errors in the bridge outputvoltage. Heat transer rom the gage grid to the specimen

is via conduction. The grid surace area and the materialsand thicknesses o the carrier and adhesive inluencegage temperature. The heat sink characteristics o thespecimen are also important.

POWER DENSITY is a parameter used to evaluate aparticular gage size and excitation voltage level ora particular application. Power density is the powerdissipated by the gage divided by the gage grid areaand has units o watts/square inch . Recommendedvalues o power density vary, depending upon accuracyrequirements, rom 2-10 or good heat sinks such as heavyaluminum or copper sections, to 0.01-0.05 or poor heatsinks such as unilled plastics. Stacked rosettes create aspecial problem in that the temperature rise o the bottomgage adds to the two gages above it and that o the center

gage adds to the top gage. These may require a very lowvoltage or dierent voltages or each o the three gages tohave the same temperature at each gage.

Figure 17Error producing common modecurrent path

Figure 18Guard wire diverts common modecurrent away rom gage wires




949 • 955 • 1894

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One way we can determine the maximum excitationvoltage that can be tolerated is by increasing the voltageuntil a noticeable zero instability occurs. We thenreduce the voltage until the zero is once more stableand without a signiicant oset relative to the zero point

at a low voltage. The bridge completion resistors alsodissipate power and in practice may be more susceptibleto drit rom sel-heating eects than the strain gage.The stability o the bridge completion resistors is relatedto load-lie, and maintaining only a raction o ratedpower in them will give better long-term stability. I theabove method o inding the maximum voltage level isused, care should be exercised to insure that the powerrating o the completion resistors is not exceeded as thevoltage is increased.

Reducing the bridge excitation voltage dramaticallyreduces gage power, since power is proportional to thesquare o voltage. However, bridge output voltage isproportional to excitation voltage, so reducing it lowerssensitivity. I the DVM used to read the output voltage

has 1 µV resolution, 1 µε resolution can be maintained,with a 1/4-bridge coniguration, using a 2 V bridgeexcitation level. I the DVM has 0.1 µV resolution,the excitation voltage can be lowered to 0.2 V whilemaintaining the same strain resolution.

Lead Wire Effects In the preceding chapter, reerence was made tothe eects o lead wire resistance on the strainmeasurement or the various conigurations. In a bridge

circuit, the lead wire resistance can cause two types oerrors. One error is due to resistance changes in thelead wires that are indistinguishable rom resistancechanges in the gage. The other error is known as LEADWIRE DESENSITIZATION and becomes signiicant whenthe magnitude o the lead wire resistance exceeds 0.1%o the nominal gage resistance. The signiicance o thissource o error is shown in Table 4.

I the resistance o the lead wires is known, thecomputed values o strain can be corrected orLEADWIRE DESENSITIZATION. In a prior section, wedeveloped equations or strain as a unction o themeasured voltages or a 1/4-bridge coniguration:

Equation 5

Equation 6

These equations are based on the assumptions that Vr,is due solely to the change in gage resistance, ∆Rg, andthat the total resistance o the arm o the bridge thatcontained the gage was Rg. Reerring to Figure 20 wesee that one o the lead wire resistances, R l ,, is in serieswith the gage so the total resistance o that bridge armis actually Rg + R l ,. I we substitute this into Equation 5

it becomes:

Equation 9

Figure 19Two-wire 1/4-bridge Connection

Lead Wire Desensitization (Refer to Figure 20)1/4 and 1/2-bridge, 3-wire connections

AWG Rg= 120 Ω R

g= 350Ω

18 0.54% 0.19%

20 0.87% 0.30%

22 1.38% 0.47%

24 2.18% 0.75%

26 3.47% 1.19%

28 5.52% 1.89%

30 8.77% 3.01%

Magnitudes o computed strain values are low byabove percent per 100 eet o hard drawn solid cop-per lead wire at 25 ˚C (77 ˚C)

Table 4





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Technical


Rewriting the equation or strain, we see that theprevious strain equation is in error by a actor o theratio o the lead wire resistance to the nominal gageresistance.

Equation 10

This actor is lead wire desensitization and we seerom Equation 10 and rom Table 4 that the eect isreduced i the lead wire resistance is small and/or thenominal gage resistance is large. I ignoring this term(1 +R l ,/Rg) will cause an unacceptable error, then itshould be added to the computer program such thatthe strains computed with Equation 6 are multiplied bythis actor. Appendix B gives the equations or variousbridge conigurations and the lead wire resistancecompensation terms that apply to them. Appendix A has

a table containing the resistance, at room temperature,o some commonly used sizes o copper wire.

The most common cause o changes in lead wireresistance is temperature change. The copper used orlead wires has a nominal temperature coeicient oresistance, at 25 ̊ C, o 0.00385 Ω / Ω /˚C. For the 2-wirecircuit in Figure 19, this eect will cause an error i thetemperature during the unstrained reading is dierentthan the temperature during the strained reading. Anerror occurs because any change in resistance in thegage arm o the bridge during this time is assumedto be due to strain. Also, both lead wire resistancesare in series with the gage in the bridge arm, urthercontributing to the lead wire desensitization error.

The THREE-WIRE method o connecting the gage, shownin Figure 20, is the preerred method o wiring straingages to a bridge circuit. This method compensates orthe eect o temperature on the lead wires. For eectivecompensation, the lead wires must have approximatelythe same nominal resistance, the same temperaturecoeicient o resistance, and be maintained at the sametemperature. In practice, this is eected by using thesame size and length wires and keeping them physicallyclose together.

Temperature compensation is accomplished because theresistance changes occur equally in adjacent arms o thebridge and thereore the net eect on the output voltageo the bridge is negligible. This technique works equallywell or 1/4 and 1/2-bridge conigurations. The lead

wire desensitization eect is reduced over the two-wireconnection because only one lead wire resistance is inseries with the gage. The resistance o the signal wireto the DVM doesn’t aect the measurement becausethe current low in this lead is negligible due to the highinput impedance o the DVM. Mathematical correctionor lead wire desensitization requires the resistanceso the lead wires to be known. The values given inwire tables can be used, but or temperature extremes,measurement o the wires ater installation is requiredor utmost accuracy. Two methods or arriving at theresistance o the lead wires rom the instrumentationside o the circuit in Figure 20 ollow:

(1) I the three wires are the same size and length, theresistance measured between points A and B, beore the

wires are connected to the instrumentation, is 2R l ,.

(2) Measure the voltage rom A-B (which is equivalentto B-C) and the voltage rom B-D. Since R3 is typicallya precision resistor whose value is well known. Thecurrent in the C-D leg can be computed using Ohm’s law.This is the current that lows through the lead resistanceso the value o R l , can be computed, since the voltagerom B-C is known. The equation or computing R l , is:

Equation 11

Figure 20Three-wire 1/4-bridge Connection




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______________________________________________________________________________________________________________________________________________________________

Appendix B: Bridge Circuits______________________________________________________________________________________________________________________________________________________________

Strain Gage Bridge Circuits and Equations

Quarter-bridge Configurations

Wire Resistance(Solid Copper wire)

AWG Ω /t (25˚C) Diameter (in.)

18 0.0065 0.040

20 0.0104 0.032

22 0.0165 0.0253

24 0.0262 0.0201

26 0.0416 0.0159

28 0.0662 0.0126

30 0.105 0.010

32 0.167 0.008

Average Properties of Selected Engineering Materials(exact values may vary widely)

Material Poisson’s Ratio,u Modulus o Elasticity, E psi x 106

Elastic strength (a)Tension (psi)

ABS (unflled) - 0.2 - 0.4 4500-7500

Aluminium (2024 - T4) 0.32 10.6 48000

Aluminium (7065 - T6) 0.32 10.4 72000

Red Brass, Sot 0.33 15 15000

Iron-Gray Cast - 13 - 14 -

Polycarbonate 0.285 0.3 - 0.38 8000-9500

Steel - 1018 0.285 30 32000

Steel - 4130/4340 0.28 - 0.29 30 45000

Steel - 304 SS 0.25 28 35000

Steel - 410 SS 0.27 - 0.29 29 40000

Titanium Alloy 0.34 14 135000

(a) Elastic strength may be represented by proportional limit, yield point,or yield strength at 0.2 percent oset.

______________________________________________________________________________________________________________________________________________________________

Appendix A: Tables______________________________________________________________________________________________________________________________________________________________

Appendices



Technical

Practical Strain Gage______________________________________________________________________________________________________________________________________________________________

Appendix B: Bridge Circuits (Continued)______________________________________________________________________________________________________________________________________________________________

Half-bridge Configurations

Full-bridge Configurations

______________________________________________________________________________________________________________________________________________________________

Appendix C: Equations______________________________________________________________________________________________________________________________________________________________

Biaxial Stress State

Rosette equations

Rectangular Rosette

Delta Rosette


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Measurement error (in µε) due to the instrumentation is oten diicult to determine rompublished speciications. However, accuracy can be computed using the ollowing simpliied errorexpressions. For the 1/4-bridge, add equations 1-6 (N=1). For the 1/2-bridge with two active arms,add equations 2-6 (N=2). For the ull bridge with our active arms, add equations 3-6 (N=4).

The total or a measurement must also include gage, lead wire, and i applicable, bridgenonlinearity errors. These are discussed in the body o this application note. Additionally, otherequipment imperections which vary rom instrument to instrument must occasionally beconsidered (e.g., osets caused by leakage currents due to humidity or ionic contamination onprinted circuit boards and connectors).

(1) R3 change rom unstrained to strained reading (due to temperature, load lie, etc.)

(2) R1 /R2 change rom unstrained to strained reading (due to temperature, load lie, etc.)

Digital voltmeters and A/D converters are speciied in terms o a ± gain error (% o reading) and a ± oset error (number o counts, in volts).Since strain calculations require two measurements, a repeatable oset error, e.g., due to relay thermal EMF, etc., will cancel, but oset dueto noise and drit will not. Assuming that noise and drit dominate, the oset on the two readings will be the root sum o squares o the twoosets. This is incorporated into the ormulas.

(3) DVM oset error on bridge measurement

Error terms 4-6 can usually be ignored when using high accuracy DVMs (e.g., 5 1/2 digit). These error terms are essentially the product osmall bridge imbalance voltages with small gain or oset terms. For equations 4-6, VOUT (the bridge imbalance voltage) is the measuredquantity which varies rom channel to channel. To calculate worst case perormance, the equations use resistor tolerances and measurestrain, eliminating the need or an exact knowledge o VOUT.

(4) DVM gain error on bridge measurement

The bridge excitation supply can be monitored with a DVM or preset using a DVM, and allowed to drit. In the irst case, supply related errorsare due only to DVM gain and oset terms, assuming a quiet supply. In the second case, since power supply accuracy is usually speciiedin terms o a ± gain and a ± oset rom the initial setting, identical equations can be used. Also or the second case, note that the strainedreading gain error is the sum o the DVM and excitation supply gain errors, while the strained reading oset error is the root sum o squareso the DVM and excitation supply oset errors.

Technical


______________________________________________________________________________________________________________________________________________________________

Appendix D: Instrument Accuracy______________________________________________________________________________________________________________________________________________________________

1/4-bridge Circuit




tii t t /St t lSt i T t

Technical


(5) Oset error on supply measurement (or on supply drit)

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

Equation 1/4-bridge 1/2-bridge Full Bridge

(1) R3

15.5 - -

(2) R1 /R

231.0 15.5 -

(3) Vout

oset 2.3 1.1 0.6

(4) Vout

gain 0.4 0.4 0.3

(5) Vin

oset 0.3 0.2 0.2

(6) Vin

gain 1.3 1.1 1.0

Sum ±50.8 µε ±18.3 µε ±2.1 µε

Example

Evaluate the error or a 24 hour strain measurement with a ± 5 ˚Cinstrumentation temperature variation. This includes the DVM and the bridgecompletion resistors but not the gages. The hermetically sealed resistors havea maximum TCR o ±3.1 ppm/˚C, and have a ±0.1% tolerance. The DVM/scannercombination, over this time and temperature span. has a 0.004% gain error anda 4 µV oset error on the 0.1 V range where the bridge output voltage, VOUT,will be measured. The excitation supply is to be set at 5 V using the DVM. TheDVM has a 0.002% gain error and a 100 µV o-set error on the 10 V range. Overthe given time and temperature span, the supply has a 0.015% gain error anda 150 µV oset error and will not be remeasured. The mounted gage resistance

tolerance is assumed to be ±0.5% or better. The strain to be measured is 3000 µε and the gage actor is assumed to be +2.

Notice that the temperature, as given, can change by as much as ± I0 ˚C betweenthe unstrained and strained measurements. This is the temperature change thatmust be used to evaluate the resistor changes due to TCR. The R1/R2 ratio has

the tolerance and TCR o two resistors included in its speciication so the ratio tolerance is ±0.2% and the ratio TCR is±6.2 ppm/˚C. The gain error change on the bridge output measurement, and on the excitation measurement, can be asmuch as twice the gain error speciication. The table above shows the total error and the contribution o the individualerror equations 1-6.

Conclusions

Based upon this example, several important conclusions can be drawn: • Surprisingly large errors can result even when using state-o-the-art bridge completion resistors and measuring

equipment.• Although typical measurements will have a smaller error, the numbers computed relect the guaranteed

instrumentation perormance.

• Measuring the excitation supply or both the unstrained and strained readings not only results in smaller errors butallows the use o an inexpensive supply.

• Bridge completion resistor drit limits quarter and hal-bridge perormance.• Changes due to temperature, moisture absorption, and load lie, require the use o ultra-stable hermetically sealed

resistors.

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