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(Compression)

(Tension)

How strain gages work.

Strain, Stress, and Poisson's RatioWhen a material receives a tensile forceP, it has a stress thatcorresponds to the applied force. In proportion to the stress,the cross-section contracts and the length elongates by Lfrom the length L the material had before receiving the tensileforce (see upper illustration in Fig. 1).

Fig. 1

The ratio of the elongation to the original length is called atensile strain and is expressed as follows:

= LL : Strain

L: Original length

L: ElongationSee the lower illustration in Fig. 1. If the material receives acompressive force, it bears a compressive strain expressed asfollows:

= LL

For example, if a tensile force makes a 100mm long materialelongate by 0.01mm, the strain initiated in the material is asfollow:

= L = 0.01 = 0.0001 = 100 x106L 100

Thus, strain is an absolute number and is expressed with anumeric value plus x106 strain, or m/m.

The relation between stress and the strain initiated in a

material by an applied force is expressed as follows based onHooke's law:

= E : StressE: Elastic modulus: Strain

Stress is thus obtained by multiplying strain by the elasticmodulus. When a material receives a tensile force, it elongatesin the axial direction while contracting in the transversedirection. Elongation in the axial direction is called longitudinalstrain and contraction in the transverse direction, transversestrain. The absolute value of the ratio between the longitudinalstrain and transverse strain is called Poisson's ratio, which isexpressed as follows:

= 21 : Poisson's ratio

1: Longitudinal strainL orL (Fig. 1)L L

2: Transverse strain

Dor

D(Fig. 1)

D D

Poisson's ratio differs depending on the material. For refer-ence, major industrial materials have the following mechanicalproperties including Poisson's ratio.

Mechanical Properties of Industrial Materials

ShearingModulusG (GPa)

TensileStrength

(MPa)

Poisson'sRatio

Material

Carbon steel (C0.1 - 0.25%)

Carbon steel (C > 0.25%)Spring steel (quenched)

Nickel steel

Cast iron

Brass (casting)

Phosphor bronze

Aluminum

Concrete

363 - 441

417 - 569588 - 1667

549 - 657

118 - 235

147

431

186 - 500

0.28 - 0.3

0.28 - 0.30.28 - 0.3

0.28 - 0.3

0.2 - 0.29

0.34

0.38

0.34

0.1

78

7979 - 81

78

40

29

43

27

9 -13

205

206206 - 211

205

98

78

118

73

20 - 29

Young'sModulusE (GPa)

Principle of Strain GagesEach metal has its specific resistance. An external tensile force(compressive force) increases (decreases) the resistance byelongating (contracting) it. Suppose the original resistance is Rand a strain-initiated change in resistance is R. Then, the

following relation is concluded:R = Ks L = Ks.R L

where, Ks is a gage factor, the coefficient expressing straingage sensitivity. General-purpose strain gages use copper-nickel or nickel-chrome alloy for the resistive element, and thegage factor provided by these alloys is approximately 2.

Types of Strain GagesTypes of strain gages include foil strain gage, wire strain gageand semiconductor strain gage.

Structure of Foil Strain GageThe foil strain gage has metal foil photo-etched in a gridpattern on the electric insulator of the thin resin and gageleads attached, as shown in Fig. 2 below.

Fig. 2

Base length

Laminate

Example of KFG gage

Metal foil Laminate film

Bonded surface

Base

Center mark

Base

Gridwidth

Basewidth

Grid length (strain sensing part)

(Gage length)

Gage lead (silver-clad copper wire,0.12 to 0.16mm and 25mm long)

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Bridge voltage E

R1 R2

R3R4

Strain gage

Strain gageRg

Strain gageRg

Activegage

Activegage

Ac

tivegage

Dummygage

R R

R

R

R

R

R R

R

R

Rg2

Rg1

E

E

E

eo

eo

eo

eo

Outputvoltageeo

Suppose the resistance R1 is a strain gage and it changes byR due to strain. Then, the output voltage is,

Principle of Strain MeasurementStrain-initiated resistance change is extremely small. Thus, forstrain measurement a Wheatstone bridge is formed to convertthe resistance change to a voltage change. Suppose in Fig. 3resistances () are R1, R2, R3 and R4and the bridge voltage (V)is E. Then, the output voltage eo (V) is obtained with the

following equation:

eo = R1R3 R2R4 .E(R1 + R2) (R3 + R4)

The strain gage is bonded to the measuring object with adedicated adhesive. Strain occurring on the measuring site istransferred to the strain sensing element via the gage base.For accurate measurement, the strain gage and adhesiveshould match the measuring material and operating conditionsincluding temperature. For the method of bonding the straingage to metal, refer to Page 8.

eo = (R1 + R)R3 R2R4 .E

(R1 + R + R2) (R3 + R4)

If R1 = R2 = R3 = R4 = R,

eo =R2+ R R R

2 .E(2R + R) 2R

Since R may be regarded extremely larger than R,

eo1

. R

.E =

1.Ks

..E 4 R 4

Thus obtained is an output voltage that is proportional to achange in resistance, i.e. a change in strain. This microscopicoutput voltage is amplified for analog recording or digitialindication of the strain.

Fig. 3

Strain-gage Wiring SystemsA strain-gage Wheatstone bridge is configured with 1, 2 or 4gages according to the measuring purpose. The typical wiringsystems are shown in Figs. 4, 5 and 6. For varied strain-gagebridge formation systems, refer to Bridge.pdf.

1-gage systemWith the 1-gage system, a strain gage is connected to a sideof the bridge and a fixed resistor is inserted into each of theother 3 sides. This system can easily be configured, and thus itis widely used for general stress/strain measurement. The 1-gage 2-wire system shown in Fig. 4-1 receives much influenceof leads. Therefore, if large temperature changes are antici-pated or if the leadwire length is long, the 1-gage 3-wiresystem shown in Fig. 4-2 must be used. For the 1-gage 3-wiresystem, refer to "Method of Compensating Temperature Effectof Leadwire" (Page 5).

Fig. 4-1 Fig. 4-2

2-gage systemWith the 2-gage system, 2 strain gages are connected to thebridge, one each to the 2 sides or both to 1 side; a fixed resis-tor is inserted into each of the other 2 or 3 sides. See Figs. 5-1and 5-2 below. There exist the active-dummy method, whereone strain gage serves as a dummy gage for temperaturecompensation, and the active-active method, where both ga-

ges serve as active gages. The 2-gage system is used to elim-inate strain components other than the target strain; accordingto the measuring purpose, 2 gages are connected to thebridge in different ways. For details, refer to "How to FormStrain-gage Bridges" (Bridge.pdf).

Fig. 5-1

Fig. 5-2

4-gage systemSee Fig. 6. The 4-gage system has 4 strain gages connectedone each to all 4 sides of the bridge. This circuit ensures largeoutput of strain-gage transducers and improves temperature

compensation as well as eliminates strain components otherthan the target strain. For details, refer to "How to Form Strain-gage Bridges" (Bridge.pdf).

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L = 0L =2

M = W 8

2 L M =W L

3

2 4

0 L 2

M = W L

2 4

1

L =

2

M = WL

4

2 L

0 L 2

M = WL2

M = W( L)

2

0 L 1

1 L ( 1 + 2)

M = WL

M = W 1

1 bh26

1 . b(h23 h13)

6 h2

d3

32

.

d24 d1432 d2

Fig. 6

Rg4 Rg3

E

h

L

L

b

W

W

Strain gage

Strain gage

Strain gage

eo

Rg2Rg1

Typical Measurements with Strain Gages

Bending Stress Measurement(1) 1-gage SystemAs illustrated below, bond a strain gage on the top surface of acantilever with a rectangular section. If load W is applied to theunfixed end of the cantilever, the strain-gage bonding site hasthe following surface stress :

= 0. E

Strain 0 is obtained through the following equation:

0 = 6WLEbh2

where, b: Width of cantilverh: Thickness of cantileverL: Distance from the load point to the center of strain gage

Bending Stress Measurement with 1-gage System

Bending Stress Measurement with 2-gage System

(2) 2-gage SystemStrain gages bonded symmetrically on the front and rearsurfaces of the cantilever as illustrated below output plus andminus signals, respectively, with an equal absolute value. Ifthese 2 gages are connected to adjacent sides of the bridge,the output of the bridge corresponding to the bending strain isdoubled and the surface stress at the strain-gage bondingsite is obtained through the following equation:

= 0 .E

2

The 2-gage system discards strain-gage output correspondingto the force applied in the axial direction of the cantilever.

Equation to Obtain Strain on BeamsStrain 0 on beams is obtained through the following equation:

0 =MZE

where, M: Bending moment (refer to Table 1)Z: Section modulus (refer to Table 2)E: Young's modulus (refer to "Mechanical Properties of

Industrial Materials," page 6)

Typical shapes of beams and their bending moments M andsection moduli Z are shown in Tables 1 and 2.

Table 1. Typical Equations to Obtain Bending Moment

Table 2. Typical Equations to Obtain Section Modulus

Cross Section Section Modulus Z

Shape of Beam Bending Moment M

M = WL

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Cross Section Polar Modulus of Section Zp

Torsional and Shearing Stress Measurement of AxisWhen twisted, an axis has shearing stress , and in the 2directions inclined by 45 from the axial line it has tensile andcompres-sive stress in an equal magnitude to the shearingstress.In measuring strain on a twisted axis under simple shearingstress status, the strain gage does not directly measure theshearing strain but detects tensile or compressive straininitiated by tensile or compressive stress that is simultaneouslygenerated with the shearing stress. Stress conditions on amicroscopic part of the surface of the axis may be asillustrated below.

Tensile strain Compressive strain

Shearing stress

Tensile stress

Compressive stress

Shearing stress is defined as illustrated below, and themagnitude is calculated through the following equation:

= G

where, G: Shearing modulus (refer to "Mechanical Propertiesof Industrial Materials," page 1)

: Shearing stress

When the axis is twisted, point A moves to point B, therebyinitiating torsional angle .

(1) Stress Measurement with 1-gage SystemBond the strain gage on the twisted axis in the directioninclined by 45 from the axial line. The relation between strain0 and stress is expressed with the following equation toobtain tensile or compressive stress :

= = 2

d2

d

= 0. E1 +

where, 0: Indicated strainE: Young's modulus (refer to "Mechanical Properties of

Industrial Materials," page 1): Poisson's ratio

Stress and shearing stress are equal in magnitude, and

thus, =

(2) Stress Measurement with 2 or 4-gage System2 or 4 strain gages forming the strain-gage bridge are strainedin an equal magnitude to enable 2 or 4 times larger output. Accordingly, the stress is calculated by dividing the indicatedstrain by 2 or 4.For axial strain measurement, the 2 or 4-gage system is usedto eliminate strain caused by bending moment. Also, for meas-urement of tensile strain and compressive strain, strain gagesare symmetrically positioned from the center of the axis asshown below.

(3) Application to Torque MeasurementStrain on the surface of the axis is proportional to the torqueapplied to the axis. Thus, the torque is obtained by detectingthe strain on the surface.Shearing stress distributed on the lateral section is balancedwith the applied torque T, establishing the following equation:

T = . Zp

where, Zp: Polar modulus of section

This equation may be rewritten as follows by substituting theshearing stress with the relational expression of tensile strainand stress:

T = 0. E . Zp1+

The polar modulus of the section is specific to each shape ofthe cross-section as follows:

A strain-gage torque transducer can be designed using theaforementioned relational expression of 0 and T.Obtain 0 from the allowable stress for the material, anddetermine the width d of the axis which is matched with themagnitude of the applied torque. Then, amplify the strain out-put with a strain amplifier and read the output voltage with ameasuring instrument.

d3

16

d24 d14

16 d2

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*120 gage

L-5L-9L-6

0.50.110.08

0.73.24.4

11.350.669.0

Principle of Self-temperature-compensation Gages(SELCOM Gages)Suppose the measuring object and the resistive element of thestrain gage have linear expansion coefficients s and g,respectively. Then, the strain gage bonded on the surface ofthe object provides a thermally-induced apparent strain T per1C that is expressed with the following equation:

Resistive element (g)

Resistive element (s)

Temperature (C)

RgRg

R3 R3R2 R2

R1R1

Rg

Typical Temperature Characteristics ofSelf-temperature-compensation Foil Gage

40

20

020

40

60

80

Thermally-inducedapparentst

rainoutput(/C)

1.8/C

1.8

/C

T =

+ (s g)Ks

where,: Resistive temperature coefficient

of resistive elementKs: Gage factor of strain gage

The self-temperature-compensation gage is designed so that

T in the above equation is approximated to zero by controllingthe resistive temperature coefficient of the gage's resistiveelement according to the linear expansion coefficient of themeasuring object.When bonded to a suitable material, KYOWA's self-tempera-ture-compensation gage (SELCOM gage) minimizes apparentstrain in the compensated temperature range to 1.8/C(graph below shows apparent strain output of 3-wire straingage).

Linear Expansion Coefficients of Materials

MaterialQuartz glass

Amber

Brick

Tungsten

Lumber (grain dir.)

Molybdenum

Zirconium

Cobar

Concrete

Titanium alloy

Platinum

Soda-lime glass

SUS 631

SUS 630Cast iron

NiCrMo steel

MaterialBeryllium

Common steel

Inconel X

Nickel

Gold

SUS 304

Beryllium copper

Copper

Brass

2024-T4 aluminum

2014-T4 aluminum

Magnesium alloy

Lead

Acrylic resinPolycarbonate

Rubber

Linear Exp. Coef.11.5

11.7

12.1

13.3

14.0

16.2

16.7

16.7

21.0

23.2

23.4

27.0

29.0

Approx. 65 to 10066.6

Approx. 77

Linear Exp. Coef.0.4

1.1

3.0 to 5.0

4.5

5.0

5.2

5.4

5.9

6.8 to 12.7

8.5

8.9

9.2

10.3

10.610.8

11.3

(x106/C)

Temperature Effect of Leadwire with 2-wire SystemLeadwireModel

Cross-SectionalArea of Conductor

(mm2)

Reciprocating Resistanceof 10m long Leadwire

approx. ()

Apparent Strain*with 10m Extension

approx.(/C)

Thermally-induced apparent strain r (/C) is obtainedthrough the following equation.

r = r . Rg + r Ks

where, Rg: Resistance of strain gage ()r : Resistance of leadwire ()

Ks: Preset gage factor of strain amplifier, usually 2.00: Resistive temperature coefficient of copper wire

(R/R/C), 3.9 x103

1r

2

1r

2

Method of Compensating Temperature Effect ofLeadwire (3-wire System)

For effective self-temperature-compensation, SELCOM gagesadopt the 1-gage system. However, if the leadwire cable is a2-wire system, strain output from the bridge is affected bytemperature effect of the leadwire. To avoid such adverseeffect, the 3-wire system is adopted.If 3 leads are connected to the strain gage as shown below,one half the leadwire resistance is applied to the adjacent sideof the bridge to compensate the resistive components of the 2leads affected by a similar temperature change, and thus thebridge output is free from any temperature effect of theleadwire. The temperature effect of a third lead connecteddirectly to the amplifier can be ignored since the amplifierprovides a high input impedance.

As precautions in using the 3-wire system, the 3 leads shouldbe the same in type, length and cross-section to receive thesame temperature effect. If they are exposed to direct sunlight,the coating color too should be identical.

1

r2

1r

2

1r2

1r

21

r2

1r

2

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Resistive element

Gage base

Strain direction

2

1

0

tr

R

Output

R

Strain gageRg

Input

r1 r2

Insulation resistance

Bridge Circuit Designed with Insulation ResistanceTaken into Consideration

Influence of Insulation ResistanceThe insulation resistance of a strain gage including leads doesnot affect the measured value if it is higher than 100M. But ifthe insulation resistance changes drastically during measure-ment, it causes the measured value to include an error.

If the insulation resistance descends from r1 to r2 in the figureabove, error strain is:

Rg(r1 r2)

Ksr1r2

Suppose,Rg = 120 (resistance of strain gage)Ks = 2.00 (gage factor of strain gage)r1 = 1000M (original insulation resistance)r2 = 10M (changed insulation resistance)

Then, the error strain is approximately 6.In general strain measurement, such an error causes virtuallyno problem. In practice, however, the lowered insulation resist-ance, r2, is not kept constant but sharply changes due totemperature, humidity and other conditions. Thus, it is notpossible to specify to which part of the circuit the insulationresistance r is applied. Accordingly, precautions should betaken.

Resistance Change of Strain Gage Bonded toCurved Surface

The strain c occurring on the resistive element of a strain gagebonded to a curved surface may be expressed with the

following equation:

c =t

2r + t

where, t: Thickness of gage base plus adhesive layerr: Radius of gage bonding surface

For example, if a uniaxial KFG gage of which the gage baseincluding the adhesive layer is 0.015mm thick, is bonded to acurved surface of 1.5r, the strain gage receives strain ofapproximately 5000 under the mere bonding condition.If the gage factor Ks is 2.00,

R/R 10000

since R/R = .Ks.

If the gage resistance is 120, it increases by approximately1.2. If the gage is bonded inside the curve, the resistance

decreases.

t2

Strain Gage Bonded on Curved Surface

Method of Compensating Gage FactorsIf the gage factor of the strain gage is different from that (2.00)of the strain amplifier, the real strain can be obtained throughthe following equation:

=2.00 x i

Ks

where, i: Measured strainKs: Gage factor of strain gage

0 =1 {(1 + 2) + (12) cos 2 }2

If 2 = 1 (: Poisson's ratio) under the uniaxial stress condi-tion,

Misalignment EffectThe strain 0 measured by a strain gage that is misaligned byan angle from the direction of the principal strain is expressedwith the following equation:

0 =1

1{(1 ) + (1 +) cos 2 }2

Method of Compensating Leadwire Extension EffectIf the leadwire or cable is extended with the 1-gage or 2-gagesystem, additional resistance is initiated in series to the straingage, thereby decreasing the apparent gage factor. Forexample, if a 10m long leadwire with 0.3mm2 conductors isused, the gage factor decreases by 1%. In the case of the 4-gage system (transducer), the extension decreases the bridgevoltage too. In these cases, the real strain can be obtainedthrough the following equation (Supposing the gage factor Ksis 2.00):

=(1 + r )x iRg

where, i: Measured strainRg: Resistance of strain gage

r : Total resistance of leadwire (For reciprocatingresistance, see the table on the next page.)One-way resistance in the case of 3-wire system

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Method of Compensating Nonlinearity of 1-gage SystemNonlinearity beyond the specification in large strain measure-ment with the 1-gage system can be compensated throughthe following equation to obtain the real strain :

Reciprocating Resistance of Leadwire

Number/Diameterof Strands

Cross-Section(mm2)

ReciprocatingResistance per 10m

()Remarks

0.08

0.11

0.3

0.5

4.4

3.2

1.17

0.7

L-6, 7

L-9, 10

L-2

L-5

7/0.12

10/0.12

12/0.18

20/0.18

= 0 (x106)

1 0where, 0: Measured strain

Method of Obtaining Magnitude and Direction ofPrincipal Stress (Rosette Analysis)

Usually, if the direction of the principal stress is unknown instress measurement of structures, a triaxial rosette gage isused and multiple physical quantities are obtained by puttingmeasured strain values in the following equations. (Theseequations apply to right-angled triaxial rosette gages.)

Precautions in Analysis(1) Regard a b c as the forward direction.(2) Angle is:

Angle of the maximum strain to the a axis when a > c;Angle of the minimum strain to the a axis when a < c.Comparison between a and c in magnitude includes plusand minus signs.

Max. principal strain

Max. principal stress

Min. principal stress

Max. shearing strain

Max. shearing stress

Min. principal strain

Direction of principalstrain (from a axis)

Examples of Calibration Strain Value and Resistance(Rg = 120, Ks = 2.00)

Generating Calibration Value based on Tip ParallelResistance Method

When extending the leadwire by several hundred meters or toobtain an accurate calibration value, use the tip parallel resist-ance method. The parallel resistance r can be obtainedthrough the following equation:

r = RgKs .

where, Rg: Resistance of strain gageKs: Gage factor of strain gage: Calibration strain value

Resistance, r (approx.)Calibration Strain Value

100

200

500

1000

2000

600 k

300 k

120 k

60 k

30 k

max. =1 [

a + c + 2{(ab)2

+ (bc)2

} ]2

min. = 1 [a + c 2{(ab)2+ (bc)2} ]2= 1 tan1[ 2bac]2 ac

max. = 2{(ab)2+ (bc)2}

max. = E [ (1 +) (a + c) + (1 )2(1 2)x 2{(ab)2+ (bc)2} ]

min. = E [ (1 +) (a + c) (1 )2(1 2)x 2{(ab)2+ (bc)2} ]

max. = E x 2{(ab)2+ (bc)2}2(1 )

: Poisson's ratioE: Young's modulus(Refer to "Mechanical Properties of Industrial Materials" (page 6).

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Typical Strain Gage Bonding Method and Dampproofing TreatmentThe strain gage bonding method differs depending on the type of adhesive applied. The description below applies to a casewhere the leadwire-equipped KFG gage is bonded to a mild steel test piece with a representative cyanoacrylate adhesive,CC-33A. The dampproofing treatment is in the case of using an butyl rubber coating agent, AK-22.

Like drawing a circle withsandpaper (#300 or so),polish the strain gagebonding site in aconsiderably wider areathan the strain gage size.

(If the measuring object is a

practical structure, wipe offpaint, rust and plating witha grinder or sand blast.Then, polish withsandpaper.)

Make sure of the front(metal foil part) and theback of the strain gage.Apply a drop of adhesive tothe back and immediatelyput the strain gage on thebonding site.(Do not spread the adhesiveover the back. If so, curingis adversely accelerated.)

When the adhesive is cured,remove the polyethylenesheet and check thebonding condition. Ideally,the adhesive is slightlyforced out from around thestrain gage.

If the adhesive is widelyforced out from around thegage base, remove theprotruding adhesive with acutter or sandpaper.Place gage leads in a slightlyslackened condition.

Put up the leadwire frombefore the part where theadhesive is applied. Place ablock of the coating agentbelow the leadwire withgage leads slightlyslackened.

Completely cover the straingage, protruding adhesiveand part of the leadwire withanother block of the coatingagent. Do not tear the blockto pieces but slightly flattenit with a finger to closelycontact it with the straingage and part of theleadwire. Completely hideprotrusions including gage

leads behind the coatingagent.

Using an absorbent cotton,gauze or SILBON paperdipped in a highly volatilesolvent such as acetonewhich dissolves oils andfats, strongly wipe thebonding site in a singledirection to remove oils andfats. Reciprocated wiping

does not clean the surface.After cleaning, mark thestrain gage bondingposition.

Cover the strain gage withthe accessory polyethylenesheet and strongly press thestrain gage over the sheetwith a thumb for approxi-mately 1 minute (do notdetach midway). Quicklyperform steps 3 and 4.Otherwise, the adhesive iscured. Once the strain gageis put on the bonding site,

do not put it up to adjustthe position.

"Strain Gage Bonding Manual" is available from KYOWA at a price of 1,200 per copy. If required, contact your KYOWA sales representative.

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