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Introduction to Strain Gages

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Strain Gages

Have you ever seen the Birdman Contest, an annual event held at Lake Biwanear Kyoto? Many people in Japan know the event since it is broadcast every yearon TV. Cleverly designed airplanes and gliders fly several hundred meters on humanpower, teaching us a great deal about well-balanced airframes.

However, some airframes have their wings regrettably broken upon flying andcrash into the lake. Such crashes provoke laughter and cause no problem sinceairplane failures are common in the Birdman Contest.

Today, every time a new model of an airplane, automobile or railroad vehicle isintroduced, the structure is designed to be lighter to attain faster running speed andless fuel consumption. It is possible to design a lighter and more efficient product byselecting lighter materials and making them thinner for use. But the safety of theproduct is compromised unless the required strength is maintained. By the sametoken, if only the strength is taken into consideration, the weight of the productincreases and the economic feasibility is impaired.

Thus, harmony between safety and economics is an extremely important factorin designing a structure. To design a structure which ensures the necessary strengthwhile keeping such harmony, it is significant to know the stress borne by eachmaterial part. However, at the present scientific level, there is no technology whichenables direct measurement and judgment of stress. So, the strain on the surface ismeasured in order to know the internal stress. Strain gages are the most commonsensing element to measure surface strain.

Let’s briefly learn about stress and strain and strain gages.

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1

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Stress and StrainStress is the force an object generates inside byresponding to an applied external force, P. See Fig.1. If an object receives an external force from thetop, it internally generates a repelling force to main-tain the original shape. The repelling force is calledinternal force and the internal force divided by thecross-sectional area of the object (a column in thisexample) is called stress, which is expressed as aunit of Pa (Pascal) or N/m2. Suppose that the cross-sectional area of the column is A (m2) and the ex-ternal force is P (N, Newton). Since external force= internal force, stress, σ (sigma), is:

σ = P (Pa or N/m2) A

Since the direction of the external force is verticalto the cross-sectional area, A, the stress is calledvertical stress.

When a bar is pulled, it elongates by ∆L, and thusit lengthens to L (original length) + ∆L (change inlength). The ratio of this elongation (or contrac-tion), ∆L, to the original length, L, is called strain,which is expressed in ε (epsilon):

ε1 = ∆L (change in length) L (original length)

Strain in the same tensile (or compressive) direc-tion as the external force is called longitudinalstrain. Since strain is an elongation (or contrac-tion) ratio, it is an absolute number having no unit.Usually, the ratio is an extremely small value, andthus a strain value is expressed by suffixing “x10–6

(parts per million) strain,” “µm/m” or “µε.”

Fig. 1

Ext

erna

l for

ce, P

Inte

rnal

for

ce

Cross-sectional area, A

Fig. 1d

0

d0

– ∆d

L ∆L

Hooke’s law (law of elasticity)In most materials, a proportional relation is found between stress and strain borne, aslong as the elastic limit is not exceeded. This relation was experimentally revealed byHooke in 1678, and thus it is called “Hooke’s law” or the “law of elasticity.” The stresslimit to which a material maintains this proportional relation between stress and strainis called the “proportional limit” (each material has a different proportional limit andelastic limit). Most of today’s theoretical calculations of material strength are based onthis law and are applied to designing machinery and structures.

Robert Hooke (1635-1703)English scientist.Graduate of Cambridge University.Having an excellent talent especiallyfor mathematics, he served as a professor of geometry at Gresham College.Heexperimentally verified that the center of gravity of the earth traces an ellipse aroundthe sun, discovered a star of the first magnitude in Orion, and revealed the renowned“Hooke’s law” in 1678.

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3 With various materials, the relation between strainand stress has already been obtained experimen-tally. Fig. 3 graphs a typical relation between stressand strain on common steel (mild steel). The re-gion where stress and strain have a linear relationis called the proportional limit, which satisfies theHooke’s law.

σ = E.ε or σ = E ε

The proportional constant, E, between stress andstrain in the equation above is called the modulusof longitudinal elasticity or Young’s modulus, thevalue of which depends on the materials.

As described above, stress can be known throughmeasurement of the strain initiated by externalforce, even though it cannot be measured directly.

The pulled bar becomes thinner while lengthen-ing. Suppose that the original diameter, d0, is madethinner by ∆d. Then, the strain in the diametricaldirection is:

ε2 = –∆d d0

Strain in the orthogonal direction to the externalforce is called lateral strain. Each material has acertain ratio of lateral strain to longitudinal strain,with most materials showing a value around 0.3.This ratio is called Poisson’s ratio, which isexpressed in ν (nu):

ν = ε2 = 0.3 ε1

Str

ess,

σ

Elastic region Plastic region

Proportionallimit

Strain, ε

Fig. 3

Simeon Denis Poisson (1781-1840)French mathematician/mathematical physicist. Born in Pithiviers, Loiret, Franceand brought up in Fontainebleau. He entered l’Ecole Polytechnique in 1798 andbecame a professor following Fourier in 1806. His work titled “Treate du mechanique(Treatise on Mechanics)” long played the role of a standard textbook.Especially renowned is Poisson’s equation in potential theory in mass. In themathematic field, he achieved a series of studies on the definite integral and theFourier series. Besides the abovementioned mechanics field, he is also known inthe field of mathematical physics, where he developed the electromagnetic theory,and in astronomy, where he published many papers.Late in life, he was raised to the peerage in France. He died in Paris.

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StrainMagnitude of StrainHow minute is the magnitude of strain? To under-stand this, let’s calculate the strain initiated in aniron bar of 1 square cm (1 x 10–4m2) which verti-cally receives an external force of 10kN (approx.1020kgf) from the top.First, the stress produced by the strain is:

σ = P = 10kN (1020kgf) = 10 x 103N

A 1 x 10–4m2 (1cm2) 1 x 10–4m2

= 100MPa (10.2kgf/mm2)

Substitute this value for σ in the stress-strain rela-tional expression (page 5) to calculate the strain:

ε = σ = 100MPa = 100 x 106 = 4.85 x 10–4

E 206GPa 206 x 109

Since strain is usually expressed in parts per mil-lion,

ε = 485 = 485 x 10–6

1000000

The strain quantity is expressed as 485µm/m,485µε or 485 x10–6 strain.

Polarity of StrainThere exist tensile strain (elongation) and compres-sive strain (contraction). To distinguish betweenthem, a sign is prefixed as follows:

Plus (+) to tensile strain (elongation)Minus (–) to compressive strain (contraction)

Young’s modulusAlso called modulus of elasticity in tension or modulus of longitudinal elasticity. With materials obeying Hooke’s law,Young’s modulus stands for a ratio of simple vertical stress to vertical strain occurring in the stress direction within theproportional limit. Since this modulus was determined first among various coefficients of elasticity, it is generally expressedin E, the first letter of elasticity. Since the 18th century, it has been known that vertical stress is proportional to verticalstrain, as long as the proportional limit is not exceeded. But the proportional constant, i.e. the value of the modulus oflongitudinal elasticity, had been unknown. Young was first to determine the constant, and thus it was named Young’smodulus in his honor.

Thomas Young (1773-1829)English physician, physicist and archaeologist. His genius early asserted itself and he has been known as a pioneer inreviving the light wave theory. From advocating the theory for several years, he succeeded in discovering interference oflight and in explaining Newton’s ring and diffraction phenomenon in the wave theory. He is especially renowned forpresenting Young’s modulus and giving energy the same scientific connotation as used at the present.

Fig. 4 10kN(approx. 1020kgf)

Iron bar (E = 206GPa)of 1 x10–4m2 (1 sq.cm)

Prefixes meaning powers of 10

SymbolGMkhdadcmµ

NameGiga-Mega-Kilo-Hecto-Daka-Deci-Centi-Milli-Micro-

Multiple109

106

103

102

101

10–1

10–2

10–3

10–6

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Strain GagesStructure of Strain GagesThere are many types of strain gages. Among them,a universal strain gage has a structure such that agrid-shaped sensing element of thin metallicresistive foil (3 to 6µm thick) is put on a base ofthin plastic film (15 to 16µm thick) and is laminatedwith a thin film.

Laminate film

Metallic resistive foil(sensing element)

Plastic film (base)

Principle of Strain GagesThe strain gage is tightly bonded to a measuringobject so that the sensing element (metallic resistivefoil) may elongate or contract according to the strainborne by the measuring object. When bearingmechanical elongation or contraction, most metalsundergo a change in electric resistance. The straingage applies this principle to strain measurementthrough the resistance change. Generally, thesensing element of the strain gage is made of acopper-nickel alloy foil. The alloy foil has a rate ofresist-ance change proportional to strain with acertain constant.

Types of strain measuring methodsThere are various types of strain measuring methods, which may roughly beclassified into mechanical, optical, and electrical methods. Since strain on asubstance may geometrically be regarded as a distance change between twopoints on the substance, all methods are but a way of measuring such a distancechange. If the elastic modulus of the object material is known, strainmeasurement enables calculation of stress. Thus, strain measurement is oftenperformed to determine the stress initiated in the substance by an externalforce, rather than to know the strain quantity.

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Let’s express the principle as follows:

∆R = K.ε R

where, R: Original resistance of strain gage, Ω (ohm)∆R: Elongation- or contraction-initiated resistance change, Ω (ohm)K: Proportional constant (called gage factor)ε: Strain

The gage factor, K, differs depending on the metallicmaterials. The copper-nickel alloy (Advance)provides a gage factor around 2. Thus, a straingage using this alloy for the sensing element enablesconversion of mechanical strain to a correspondingelectrical resistance change. However, since strainis an invisible infinitesimal phenomenon, theresistance change caused by strain is extremelysmall.For example, let’s calculate the resistance changeon a strain gage caused by 1000 x10–6 strain.Generally, the resistance of a strain gage is120Ω,and thus the following equation is established:

∆R = 2 x 1000 x10–6

120 (Ω)

∆R = 120 x 2 x 1000 x10–6 = 0.24Ω

The rate of resistance change is:

∆R = 0.24 = 0.002 = 0.2% R 120

In fact, it is extremely difficult to accurately meas-ure such a minute resistance change, which can-not be measured with a conventional ohmmeter.Accordingly, minute resistance changes are meas-ured with a dedicated strain amplifier using an elec-tric circuit called a Wheatstone bridge.

Strain measurement with strain gagesSince the handling method is comparatively easy, a strain gage has widely been used, enabling strain measurement toimply measurement with a strain gage in most cases. When a fine metallic wire is pulled, it has its electric resistancechanged. It is experimentally demonstrated that most metals have their electrical resistance changed in proportion toelongation or contraction in the elastic region. By bonding such a fine metallic wire to the surface of an object, strain onthe object can be determined through measurement of the resistance change. The resistance wire should be 1/50 to1/200mm in diameter and provide high specific resistance. Generally, a copper-nickel alloy (Advance) wire is used.Usually, an instrument equipped with a bridge circuit and amplifier is used to measure the resistance change. Since astrain gage can follow elongation/contraction occurring at several hundred kHz, its combination with a proper measuringinstrument enables measurement of impactive phenomena. Measurement of fluctuating stress on parts of running vehiclesor flying aircraft was made possible using a strain gage and a proper mating instrument.

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Fig. 5

Fig. 6

Fig. 7

Wheatstone BridgeWhat’s the Wheatstone Bridge?The Wheatstone bridge is an electric circuit suit-able for detection of minute resistance changes. Itis therefore used to measure resistance changes ofa strain gage. The bridge is configured by combin-ing four resistors as shown in Fig. 5.Suppose:

R1 = R2 = R3 = R4, orR1 x R3 = R2 x R4

Then, whatever voltage is applied to the input, theoutput, e, is zero. Such a bridge status is called“balanced.” When the bridge loses the balance, itoutputs a voltage corresponding to the resistancechange.As shown in Fig. 6, a strain gage is connected inplace of R1 in the circuit. When the gage bearsstrain and initiates a resistance change, ∆R, thebridge outputs a corresponding voltage, e.

e = 1 . ∆R . E 4 R

That is,

e = 1 . K . ε . E 4

Since values other than ε are known values, strain,ε, can be determined by measuring the bridge out-put voltage.

Bridge StructuresThe structure described above is called a 1-gagesystem since only one gage is connected to thebridge. Besides the 1-gage system, there are 2-gage and 4-gage systems.• 2-gage systemWith the 2-gage system, gages are connected tothe bridge in either of two ways, shown in Fig. 7.

R1 R2

R3R4

input, E

Out

put

, e

Gage

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Fig. 8

Fig. 9

Fig. 10 (a)

(b)

•Output voltage of 4-gage systemThe 4-gage system has four gages connected oneeach to all four sides of the bridge. While this sys-tem is rarely used for strain measurement, it is fre-quently applied to strain-gage transducers.When the gages at the four sides have their resist-ance changed to R1 + ∆R1, R2 + ∆R2, R3 + ∆R3

and R4 + ∆R4, respectively, the bridge output volt-age, e, is:

e = 1 ∆R1 – ∆R2 + ∆R3 – ∆R4 E 4 R1 R2 R3 R4

If the gages at the four sides are equal in specifica-tions including the gage factor, K, and receivestrains, ε1, ε2, ε3 and ε4, respectively, the equa-tion above will be:

e = 1 . K (ε1 – ε2 + ε3 – ε4) E 4

( )

•Output voltage of 1-gage systemIn the cited equation for the 4-gage system, the 1-gage system undergoes resistance change, R1, atone side only. Thus, the output voltage is:

e = 1 . ∆R1 . E 4 R1

or, e = 1 . K . ε1 . E 4

In almost all cases, general strain measurement isperformed using the 1-gage system.

•Output voltage of 2-gage systemTwo sides among the four initiate resistance change.Thus, the 2-gage system in the case of Fig. 10 (1),provides the following output voltage:

e = 1 ∆R1 – ∆R2 E 4 R1 R2

or, e = 1 K (ε1 – ε2) E 4

In the case of Fig. 10 (b),

e = 1 ∆R1 + ∆R3 E 4 R1 R3

or, e = 1 K (ε1 + ε3) E 4

( )

( )

R1 R2

R3R4

e

E

R1

e

E

R1

e

E

R1

e

E

R2

R2

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Fig. 11

That is to say, the strain borne by the second gageis subtracted from, or added to, the strain borneby the first gage, depending on the sides to whichthe two gages are inserted, adjacent or opposite.

•Applications of 2-gage systemThe 2-gage system is mostly used for the followingcase. To separately know either the bending ortensile strain an external force applies to acantilever, one gage is bonded to the same positionat both the top and bottom, as shown in Fig. 11.These two gages are connected to adjacent or op-posite sides of the bridge, and the bending or ten-sile strain can be measured separately. That is, gage1 senses the tensile (plus) strain and gage 2 sensesthe compressive (minus) strain. The absolute strainvalue is the same irrespective of polarities, pro-vided that the two gages are at the same distancefrom the end of the cantilever.To measure the bending strain only by offsettingthe tensile strain, gage 2 is connected to the ad-jacent side of the bridge. Then, the output, e, ofthe bridge is:

e = 1 K (ε1 – ε2) E 4

Since tensile strains on gages 1 and 2 are plusand the same in magnitude, (ε1 – ε2) in the equa-tion is 0, thereby making the output, e, zero.On the other hand, the bending strain on gage 1is plus and that on gage 2 is minus. Thus, ε2 isadded to ε1, thereby doubling the output. That is,the bridge configuration shown in Fig. 12 enablesmeasurement of the bending strain only.

If gage 2 is connected to the opposite side, theoutput, e, of the bridge is:

e = 1 K (ε1 + ε2) E 4

Thus, contrary to the above, the bridge output iszero for the bending strain while doubled for thetensile strain. That is, the bridge configurationshown in Fig. 13 cancels the bending strain andenables measurement of the tensile strain only.

Gage 1

Gage 2

Fig. 12

Fig. 13

E

ee

E

Gage 1 Gage 2

Gage 1

Gage 2

Fig. 11

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Temperature CompensationOne of the problems of strain measurement isthermal effect. Besides external force, changingtemperatures elongate or contract the measuringobject with a certain linear expansion coefficient.Accordingly, a strain gage bonded to the objectbears thermally-induced apparent strain. Tempera-ture compensation solves this problem.

Active-Dummy MethodThe active-dummy method uses the 2-gage systemwhere an active gage, A, is bonded to the measur-ing object and a dummy gage, D, is bonded to adummy block which is free from the stress of themeasuring object but under the same temperaturecondition as that affecting the measuring object.The dummy block should be made of the samematerial as the measuring object.As shown in Fig. 14, the two gages are connectedto adjacent sides of the bridge. Since the measur-ing object and the dummy block are under the sametemperature condition, thermally-induced elonga-tion or contraction is the same on both of them.Thus, gages A and B bear the same thermally-in-duced strain, which is compensated to let the out-put, e, be zero because these gages are connectedto adjacent sides.

Self-Temperature-Compensation MethodTheoretically, the active-dummy method describedabove is an ideal temperature compensationmethod. But the method involves problems in theform of an extra task to bond two gages and installthe dummy block. To solve these problems, theself-temperature-compensation gage (SELCOM®

gage) was developed as the method of compensat-ing temperature with a single gage.With the self-temperature-compensation gage, thetemperature coefficient of resistance of the sens-ing element is controlled based on the linear ex-pansion coefficient of the measuring object. Thus,the gage enables strain measurement without re-ceiving any thermal effect if it is matched with themeasuring object. Except for some special models,all recent KYOWA strain gages apply the self-tem-perature-compensation method.

Fig. 14

A

D

Input, E

Out

put

, e

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1

Principle of Self-Temperature-Compensation Gages

Applicable materials

Composite materials, diamond, etc.

Composite materials, silicon, sulfur, etc.

Composite materials, lumber, tungsten, etc.

Composite materials, tantalum, etc.

Composite materials, titanium, platinum, etc.

Composite materials, SUS 631, etc.

Linear expansioncoefficient

1 x10–6/°C 3 x10–6/°C 5 x10–6/°C 6 x10–6/°C 9 x10–6/°C11 x10–6/°C

Applicable materials

Corrosion/heat-resistant alloys, nickel, etc.

Stainless steel, SUS 304, copper, etc.

2014-T4 aluminum, brass, tin, etc.

Magnesium alloy, composite materials, etc.

Acrylic resin, polycarbonate

Linear expansioncoefficient

13 x10–6/°C16 x10–6/°C23 x10–6/°C27 x10–6/°C65 x10–6/°C

SELCOM® gage applicable materials

Resistive element of strain gage(linear expansion coefficient, βg)

Measuring object(linear expansion coefficient, βs)

Fig. 15

As described in the previous section, except forsome special models, all recent KYOWA straingages are self-temperature-compensation gages(SELCOM® gages). This section briefly describesthe principle by which they work.

Principle of SELCOM® GagesSuppose that the linear expansion coefficient ofthe measuring object is βs and that of the resistiveelement of the strain gage is βg. When the straingage is bonded to the measuring object as shownin Fig. 15, the strain gage bears thermally-inducedapparent strain/°C, εT, as follows:

εT = α + (βs – βg) Ks

where, α: Temperature coefficient of resistance ofresistive element

Ks: Gage factor of strain gageThe gage factor, Ks, is determined by the materialof the resistive element, and the linear expansioncoefficients, βs and βg, are determined by the materialsof the measuring object and the resistive element,respectively. Thus, controlling the temperaturecoefficient of resistance, α, of the resistive elementsuffices to make the thermally-induced apparentstrain, εT, zero in the above equation.

α = –Ks (βs – βg) = Ks (βs – βg)

The temperature coefficient of resistance, α, of theresistive element can be controlled through heattreatment in the foil production process. Since it isadjusted to the linear expansion coefficient of theintended measuring object, application of the gageto other than the intended materials not only voidstemperature compensation but also causes largemeas-urement errors.

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Gage

Rg

Leadwiresr1

r2

r3

R2

R4 R3

Input, E

Out

put

, e

Fig. 17

Leadwire Temperature CompensationThe use of the self-temperature-compensation gage(SELCOM® gage) eliminates the thermal effectfrom the gage output. But leadwires between thegage and the strain-gage bridge are also affectedby ambient temperature. This problem too shouldbe solved.With the 1-gage 2-wire system shown in Fig. 16,the resistance of each leadwire is inserted in seriesto the gage, and thus leadwires do not generateany thermal problem if they are short. But if theyare long, leadwires adversely affect measurement.The copper used for leadwires has a temperaturecoefficient of resistance of 3.93 x10–3/°C. For ex-ample, if leadwires 0.3mm2 and 0.062Ω/m eachare laid to 10m length (reciprocating distance:20m), a temperature increase by 1°C produces anoutput of 20 x10–6 strain when referred to a strainquantity.

Fig. 16

Gage Leadwires

Out

put

, e

Input, EThe 3-wire system was developed to eliminate thisthermal effect of leadwires. As shown in Fig. 17,the 3-wire system has two leadwires connected toone of the gage leads and one leadwire connectedto the other.

Unlike the 2-wire system, the 3-wire systemdistributes the leadwire resistance to the gage sideof the bridge and to the adjacent side. In Fig. 17,the leadwire resistance r1 enters in series to Rgand the leadwire resistance r2 enters in series toR2. That is, the leadwire resistance is distributed toadjacent sides of the bridge. The leadwire resistancer3 is connected to the outside (output side) of thebridge, and thus it produces virtually no effect onmeasurement.

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Strain Gage Bonding ProcedureThe strain-gage bonding method differs depending on the type of the strain gage, the applied adhesiveand operating environment. Here, for the purpose of strain measurement at normal temperatures in aroom, we show how to bond a typical leadwire-equipped KFG gage to a mild steel specimen using CC-33A quick-curing cyanoacrylate adhesive.

Select the strain gagemodel and gage lengthwhich meet thereqirements of themeasuring object andpurpose. For the linearexpansion coefficient ofthe gage applicable tothe measuring object,refer to page 13. Selectthe most suitable onefrom the 11 choices.

Using a sand cloth(#200 to 300), polishthe strain-gage bondingsite over a wider areathan the strain-gagesize.Wipe off paint, rust andplating, if any, with agrinder or sand blastbefore polishing.

Using a #2 pencil or amarking-off pin, markthe measuring site inthe strain direction.When using a marking-off pin, take care not todeeply scratch thestrain-gage bondingsurface.

Using an industrial tissuepaper (SILBON paper)dipped in acetone, cleanthe strain-gage bondingsite. Strongly wipe thesurface in a singledirection to collect dustand then remove bywiping in the samedirection. Reciprocalwiping causes dust tomove back and forth anddoes not ensure cleaning.

Ascertain the back andfront of the strain gage.Apply a drop of CC-33A adhesive to theback of the strain gage.Do not spread theadhesive. If spreadingoccurs, curing isadversely accelerated,thereby lowering theadhesive strength.

After applying a drop ofthe adhesive, put thestrain gage on themeasuring site whilelining up the centermarks with the marking-off lines.

Cover the strain gagewith the accessorypolyethylene sheet andpress it over the sheetwith a thumb.Quickly perform steps (5)to (7) as a series ofactions. Once the straingage is placed on thebonding site, do not lift itto adjust the position. Theadhesive strength will beextremely lowered.

After pressing the straingage with a thumb forone minute or so,remove the polyethylenesheet and make sure thestrain gage is securelybonded. The abovesteps complete thebonding work. However,good measurementresults are available after60 minutes of completecuring of the adhesive.

(1) Select strain gage. (2) Remove dust and paint.

(3) Decide bonding position. (4) Remove grease from bonding surface and clean.

(5) Apply adhesive. (6) Bond strain gage to measuring site.

(7) Press strain gage. (8) Complete bonding work.

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Reliability through integration

KYOWA ELECTRONIC INSTRUMENTS CO., LTD.Overseas Department:1-22-14, Toranomon, Minato-ku, Tokyo 105-0001, JapanTel: (03) 3502-3553 Fax: (03) 3502-3678http://www.kyowa-ei.com e-mail: overseas@kyowa-ei.co.jp

Specifications are subject to change without notice for improvement.

Manufacturer,s Distributor

Safety precautions Be sure to observe the safety pre-cautions given in the instruction manual, in order to ensure correct and safe operation.

Printed in Japan on Recycled Paper 03/05 ocsCat. No. 107B-U53

For more useful information, contactyour local Kyowa sales/service distributoror Kyowa (overseas@kyowa-ei.co.jp).Thank you.