José L.F. Freire Pontifical Catholic University of Rio de Janeiro – PUC-Rio Key Words: strain, measurement, circuits, adhesives, Wheatstone bridge, potenciometer circuit, sensors, transducers, load cells, applications. Contents 1. Introduction 2. Sensitivity of a thin metallic conductor to strain 3. Strain gage conductor materials 4. Characteristics of modern strain gages 5. Other topics concerning modern strain gages 5.1 Conductor and strain gage sensitivities 5.2. Temperature response of strain gage installations 5.3. Gage response in non-uniform strain fields 5.4. Gage response in dynamic strain fields 6. Strain gage circuits 6.1 Potenciometer circuit 6.2. Constant voltage Wheatstone bridge circuits 6.3. Constant current Wheatstone bridge circuits 6.4. Constant voltage Wheatstone bridge circuits – specific application cases 6.5. Shunt Calibration 6.6 Quarter bridge: three-wire connection of a strain gage 6.7 Null balance bridges 7. Strain gage measurement systems 8. Examples of strain gage applications 9. Conclusions Glossary Bibliography Summary Electrical resistance strain gages are sensors fabricated from thin foil or wire-type conductors that respond to variations in their length with variations in their electrical resistance. Strain gages are used to measure linear strains that occur at surface points of an object when it responds to an actuating load, Strain gages are either bonded with adhesives to the surfaces of structures or are welded on. Strain gages are used to determine strains in localized areas of structural components in the laboratory, in the field or as sensors in transducers such as resistive accelerometers or load cells. They are accurate “point” measuring elements and can be used in cases of static or dynamic loading. The measurement systems employ conventional and in-shelf circuitry elements, and have the important advantage of leading to electrical and treatable output. Circuitries commonly used are the potentiometer and the Wheatstone bridge circuit.

ELECTRICAL RESISTANCE STRAIN GAGES

1. Introduction Electrical resistance strain gages are sensors made of thin foil or wire-type conductors that respond to variations in length with variations in electrical resistance. Strain gages are used to measure linear strains that occur at surface points of an object when it responds to some actuating load, as shown in Figure 1. This figure shows a surface point on the object before and after a load was applied. The strain gage is bonded to the surface with an adhesive. Deformation of the surface element forces the strain gage to change its length. For the special conductor gage materials, the variation in length of the parallel segments of the wire-type conductor will be directly proportional to the variation in electrical resistance of the conductor. Equation (1) shows the relationship between the variation in resistance of gage ∆R and strain ε to be determined, where K is the gage factor. The static or dynamic variation in resistance is measured and registered by an auxiliary circuitry. Notation of variables used in equation (1) is presented in Figure 1.

Figure 1: Sketch of a strain gage bonded to the surface point of an object before and after loading was applied

ε.. KRR

LLL

KR

RR

i

if

i

if =∆

∴−

=−

(1)

Figure 1 and equation (1) show that strain gages measure the relative displacement between two points located at the extremities of their grids with changes in resistance. These displacement changes can be so small that other mechanical devices would not be able to measure them with the necessary resolution and accuracy. An example of the necessary resolution is given in Figure 2. It shows a prismatic object loaded with an axial force P, as occurs, for example, in a tensile test used to measure the elastic and plastic mechanical properties of a structural material. Inside the elastic range, the uniaxial stress-strain relationships given in equations (2) hold, where E and µ are the Young modulus and Poisson coefficient, respectively. For low carbon steel – for example, ASTM A 36 – the yield strength is 250MPa or greater. Using E=200GPa, µ=0.3, load P=12.5 kN and assuming that the cross section area of the prismatic object is square with sides equal to w=10mm, it can be calculated that the tensile stress is σ=125MPa and the

linear strain measured by the strain gage is ε=612.5 x 10-6. The relative displacement occurring between two points, A and B, located at the surface of the prismatic object and far apart at L=10mm will be ∆L=0.0006125mm. These numbers show that working stresses as high as half the yield strength of the material of a structural member will cause very small displacements in a relatively large gage length of 10mm. Any device or method employed to measure this displacement will need a resolution of 1µm or at least 10µm to indicate strain steps of 1 to 10x10-6. Electrical resistance strain gages can be used with this resolution and with an accuracy varying from numbers as low as 1% to as high as 0.1%, depending on the set ups and circuitry used in each measurement.

Figure 2: Elastic stress-strain uniaxial mechanical behavior relations

⎪⎪⎩

⎪⎪⎨

⎧

=−=−=

=

zx

xy

xx

E

E

εσ

µµεε

σε

(2)

2. Sensitivity of a thin metallic conductor to strain Figure 3 shows a thin wire-type conductor subjected to a normal positive or negative change in length.

Figure 3: Thin wire-type conductor being strained

Resistance R depends on resistivity ρ, length L and area A, as shown in equation (3). The relative change in resistance is given by equation (4) in terms of relative changes of variables ρ, L and A.

ALR .ρ= (3)

AdA

LdLd

RdR

−+=ρρ (4)

Term L

dL is the instantaneous strain.

LdLd =ε (5)

If the conductor has a circular cross section with diameter D, the infinitesimal variation of the area is:

DdD

AdADA .2

4. 2

=→=π

The relative variation of the area can be written in terms of the infinitesimal strain and the Poisson coefficient:

εµεµε dA

dAddD

dDxy ..2. −=∴−==

The relative change in resistivity is assumed to be proportional to the relative change in volume, the proportionality coefficient designated as c, the Bridgeman constant.

VdVcd .=

ρρ (6)

The change in volume of the conductor is written as:

εµε ddA

dAL

dLVdV

ALV

..2

.

−=+=

=

Therefore, it is possible to determine the dependence of the relative change in resistance R in terms of ε as:

( )[ ]

ε

εµν

dKR

dR

dcR

dR

.

.1.2.21

=

++−= (7)

Integration of (7) gives (8) if K is considered constant:

i

f

i

f

LL

KRR

ln.ln = (8)

Equation (9) will be valid if the total strain value is low and if coefficients c and µ remain constant.

LfL

LiL

LL

LdLd

i

f ∆≅

∆≅=== ∫∫ lnεε

ε.KRR=

∆ (9)

In the case of c being equal to unity, which happens for a conductor made of the alloy Constantan (55%Cu + 45%Ni), proportionality constant K will be equal to 2 and will not depend on the change in value of µ in the elastic-plastic transition. 3. Strain gage conductor materials Electrical conductor materials for strain gages must possess the capability of being formed as thin wires or very thin metal foils, and must have constant K values over a wide range of elastic (and plastic if possible) strains and temperature. The ones that are used the most are listed in Table 1. Constantan and Karma alloys are the ones used the most in general applications within temperature ranges of [-30

oC,200oC] and [-50 oC,250oC], respectively.

Table 1: Stain gage conductor materials Material Alloys K Application

Advance or Constantan

45% Ni – 55% Cu 2.1 General use up to 8% strain. Widely used

Karma 74% Ni - 20% Cr – 3% Al – 3% Fe 2.0 Wide range for temperature

compensation, high fatigue strength

Isoelastic 36% Ni – 8% Cr – 0.5%Mo – 55.5% Fe 3.6 General use. Temperature sensitive.

Nichrome V 80% Ni – 20% Cr 2.1 Platinum-Tungsten

92% Pt – 8% W 4.0

Armour D 70% Fe – 20% Cr – 10%Al 2.0

High temperature applications (may be over 250oC)

4. Characteristics of modern strain gages The first strain gages were made from thin wire conductors that were bent to form several rows of parallel sensing legs to produce the sensing grids. These grids were cemented to thin special china paper backings to allow for electrical isolation and easy handling and mounting. Modern strain gages are made from thin conductor foils (e.g. Constantan foils) backed by flexible and non-conducting thin foils of polyamide, phenol or epoxy resins. The strain sensitive grids are produced by a photoresistive etching process. The backings give some stiffness to the foil grids to make their

handling process relatively easy and to electrically isolate the grids from the prototype metallic materials. The adhesives used to bond the strain gages to the prototype play an important role in the measurement system. They must be easy to apply, be compatible with both the prototype and backing materials, have a linear response to strain along the entire range of the measurement, and be time independent. Figure 4 and Table 2 summarize most of the technological firsthand information needed to give sound and important information to a beginner. Circuitry for electrical measurements of resistance variation ∆R are commented on in section 3 of this topic.

Figure 4: Sketch of a strain gage layout The process of bonding a gage is described in Table 3. Of course, there are other processes and adhesives for more specific applications such as strain gage bonding for transducers, high temperature applications, or large strain measurements.

Table 2: Characteristics of modern strain gages Topic Description

History Lord Kelvin worked on the basics in 1856. First wire grid gages were presented in mid-1930, by Ruge and Simmons independently of each other. Foil gages were first presented by Sanders and Roe after 1952.

Grid materials Advance or Constantan, Karma alloy, Isoelastic, Nichrome V, Armour D and Pt-W (see Table 1) Backing or carrier materials Thin paper, polyimide film, epoxy film, glass fiber epoxy-phenolic reinforced polymer film

Adhesive Materials Epoxy (more stable, used in transducer applications), cyanoacrylate (very easy to use, general application up to 65oC), polyester, ceramic (high temperature applications)

Strain gage configurations

Uniaxial

Biaxial rosette

Triaxial stacked rosette

Triaxial 45 degree rosette

Weldable

Linear array

High temperature (weldable)

Residual stress rosette

Special types of strain gages

Embedded (concrete)

Concrete

Stability Combined effect of time of exposure to strain, temperature and humidity on the gage installation and measurement system can affect stability. Zero drift can be as low as a few micro-strains to several hundred, depending on foil and carrier material, adhesive, installation cleanliness, and stability of instrumentation.

Gage Factor Constantan gages have gage calibration factor K around 2.0.

Transverse sensitivity Depends on gage geometry and type. Kt = St/Sa = 0 to 5%, usually < 0.5%

Temperature compensation Available for prototype materials with expansion coefficients varying from zero to 26x10-6/oC

Grid dimensions

Grid dimensions vary in such a way that the most suitable geometry for a specific application will be found most of the time. There are different W/L ratios. Minimum and maximum L range from 1.0mm (stress concentration) to 150mm (concrete) applications.

Initial resistance Common: 120 and 350Ω Possible: 500, 1000, 2000, 5000Ω

Accuracy Gage resistance = +0.3%, Gage factor= +1%. Transducers that employ strain gages have a much better accuracy due to calibration procedures and temperature corrections

Resolution 1µε. Can be lower or higher, depending on the circuitry and measurement range and devices Range + 1%, special gages with annealed foils may work up to 20% Circuitry Potenciometer and Wheatstone circuits. Four wire resistance direct reading circuit

Acquisition rate and dynamic range

Static to dynamic range. Dynamic measurements may require several MHz acquisition rate. Circuitry, amplifiers and reading-recording devices limit strain gage installation capabilities to measuring 100 ns pulses. Temperature -40oC<θ<60oC general use- polyimide carriers and cyanoacrylate adhesive. -195oC<θ<230oC Karma alloy fully encapsulated and epoxy adhesive. Special high temperature gages - PtW alloy - weldable or bonded with ceramic adhesive (~600oC)

Humidity Must be protected to avoid degradation of backing and adhesive and undesirable loss of isolation

Radiation Large apparent strains are induced with time (zero drift) Influence of other

parameters

Fatigue and strain cycling Fatigue strength varies from +1500x105 to +2200x108, depending on the gage type. Fatigue strength depends on the quality of the installation

Magnetic fields High field gradients influence the response of the gage and associated cabling

Hydrostatic pressure Minimal influence (20ηε) up to about 20MPa pressure

Application Widespread: experimental stress analysis, transducers (accelerometers with DC response and load cells), application to integrity monitoring, control of mixtures in chemical processes, weight control and measurement

Table 3: Steps for bonding, cabling installation, quality control and protection of strain gage installations

Step Description

Measurement system design

Selection of points where to measure strains, characteristics of measurement: static or dynamic, low or high temperature, low or high strain gradient, general stress analysis or transducer application, etc. Selection of gage, selection of adhesive and protection. Locate point at surface of prototype. Degreasing entire bonding and adjacent area. Check compatibility of prototype exposed surface with degreaser to avoid undesirable chemical attack. Thorough cleaning of surface, taking off all paint and oxidation. Expose surface to bare metal. Use hand sand paper up to number 220. Clean bonding area completely with clean cotton and Freon, alcohol or acetone. With hard lead pencil, mark position where to set gage. Place gage with adhesive tape and check for position. Pull back adhesive tape just enough to expose back side of gage that will receive adhesive. Put small drop of adhesive on surface of gage and reposition adhesive tape to previous well-positioned place. Press gage against prototype surface using tip of finger on adhesive tape for about two minutes. A small silicone rubber pad may be used to spread pressure over gage and protect finger skin from excess adhesive. Peel off adhesive tape. Clean surrounding gage area of excess adhesive. Cement solder auxiliary tabs with cyanoacrylate.

Example: bonding strain gage with cyanoacrylate

Solder lead wires. Use 20 to 30W soldering iron. Measure resistance of gage and circuit using ohmmeter. Check electrical

contacts Measure resistance between gage and surface of metallic prototype. Resistance should be above 500 MΩ.

Provide protection to installation

Using one or more coats, protect gage, electrical contacts and portion of lead wire that reaches installation. These should be selected to protect against electrical contact with environment, humidity and touch.

The high resistance of the gages is achieved with the use of relatively long and thin etched conductors. To make them relatively small in size, the gages are constructed with several parallel thin conductors that form a grid of dimensions L and W as showed in Figure 5. The initial resistance of normally marketed strain gages is 120 or 350Ω. Strain gages may also be found with resistances as high as 1,000 or 2,000Ω to be applied in special measurements; for example, when direct measurement of ∆R is required.

Figure 5: Sketch of strain gage showing grid dimensions, axial (measurement direction) and transverse direction. The terminal tabs used for soldering the cables or for connecting the parallel conductors that form the grid are made much thicker and wider. This procedure helps to decrease their resistance and, therefore, their variation under strain. As a consequence, almost 100% of the resistance variation is caused by the straining of the thin parallel conductors, and the spurious response of the gages to transverse strains is usually very low.

Figure 6: Constantan foil strain gage with polyimide backing. Original size is L = 75 mm to be applied in concrete structures. Note the large, thick tabs for soldering the connecting cables and the 180o turning corner connections of the wire-type parallel foil conductors. Manufacturers of commercial strain gages state the strain gage calibration factor K and the transverse sensitivity Kt (see item 5.1). Approximate values of K are given in Table 1. The calibration of gage factor K follows standardized procedures. The value of K is determined with the help of an instrumented cantilever steel beam with a 0.285 Poisson coefficient. 5. Other topics concerning modern strain gages This section discusses topics that deserve attention concerning measurements where modern strain gages are used. 5.1 Conductor and strain gage sensitivities The relative variation in the gage resistance is given in equation (10), where Sa, St and Sat are the sensitivities of the gage to the strain state (see Figure 5 for directions a and t).

SaStSatSatStSaRR

atta <<<<++=∆ γεε ... (10)

In light of the fact that Sat is very small when compared with Sa and St, and defining the transverse sensitivity as Kt = St/Sa, equation (11) is obtained.

⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

∆

a

tta

a

ta KSa

SaStSa

RR

εε

εεε

ε .1...1.. (11)

In the beam calibration test, the apparent axial strain indicated by the gage a

aε will be given by equation (12) as

aa

a

tta KKSa

RR ε

εε

ε ..1.. =⎟⎟⎠

⎞⎜⎜⎝

⎛+=

∆ (12)

Noting that the uniaxial stress state in the beam gives at εε .285.0−= , the gage factor K will be given by (13) as

( )tKSaK 285.01. −= (13) In actual experiments, the transverse sensitivity is neglected most of the time because the Kt value of modern gages is very low (<0.5%). In cases where the errors associated with the existence of the transverse sensitivity are considered relevant, two strain gages mounted along orthogonal directions must be used to determine the apparent strains in the axial and transverse directions. The actual strains will be determined by using equations (14) and the Kt value of the strain gages.

( )

( )aat

tt

t

tt

ttt

aa

t

ta

KK

K

KK

K

εεε

εεε

.1

285.01

.1

285.01

−+

−=

−+

−=

(14)

The error associated with the measurement of strain along the prescribed gage axial direction is given by equation (15).

t

a

tt

a

aaa

K

KError

285.01

285.0

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ − εε

εεε

(15)

As an example of the error measurement given, assume a thin walled cylindrical pressure vessel (external diameter D and wall thickness t) loaded with internal pressure p. Stresses and strains in the axial (longitudinal l) and transversal (circunferencial c) directions and their ratios are given by equations (16), where a Poisson coefficient µ=0.3 is used.

( ) ( ) 23.05.01

5.05.02

,5.012

5.04

,2

=−−

=∴−=−=

=∴==

µµ

εε

µεµε

σσ

σσ

c

llc

c

llc

tEpD

tEpD

tpD

tpD

(16)

Considering a transverse sensitivity equal to 5% (a value that may be considered very high for modern strain gages), the errors generated in considering the apparent strains as the actual ones in the circumferential and longitudinal directions will be 2.6% and 23%, respectively. The errors generated in calculating the circumferential and longitudinal stresses considering the apparent strains will be 3.6% and 12%, respectively. 5.2. Temperature response of strain gage installations Strain gages installations are temperature dependent and their response depends on the variation of temperature ∆θ that occurs during the test time period. Equation (17) shows that an apparent strain arises due to ∆θ, where α and β are expansion coefficients of the gage and prototype materials, respectively, and γ reflects the dependence of the material’s resistivity to temperature.

AdA

ldld

RdR

AAllRR

−+=

====

ρρ

θεθεθερρθε ),(),,(),,(),,(

( ) ( ) θαγθαβεθ

∆−+∆−=⎟⎠⎞

⎜⎝⎛ ∆=∴ ...K

RR

aparent (17)

Modern strain gages are fabricated to be temperature compensated for the most-used prototype materials such as carbon steel, stainless steel, aluminum alloy, glass and polymer, to name a few. Even if constructed in this way, they can still show some apparent strain when large variations in temperature occur. Even small changes in temperature can cause large apparent strains if the initial temperature is relatively far from room temperature (temperature compensation setup). Strain gage dealers furnish charts or appropriate equations to correct for the apparent strains generated by temperature variations. Sometimes, the user can calibrate the measurement system beforehand by varying the temperature conditions of the test setup. On other occasions it is possible to take advantage of the circuitry used to allow for electrical temperature compensation (see section 6). Strain gages are resistors that generate heat due to the flow of an electrical current. The circuitry used for resistance measurement increases the local temperature. Therefore, the heat dissipation of the local gage installation (including prototype material and gage protection) must be checked and taken into consideration in the measurement design. Suggestions for maximum allowable power density of heat generated in the strain gage grids are given in Table 4 for some prototype materials. The values depend on the ability that each material has to dissipate the heat generated in the installation, helping to keep temperature as uniform as possible. These power densities are calculated as the ratio between the heat generated by the strain gage and its grid area. The maximum allowable circuit voltage measurement can be selected using the power densities given in Table 4. The last column of this table gives allowable source voltages from

calculations that took into consideration equation (18), developed from the theory of the Wheatstone bridge circuit presented in section 6. Voltages in Table 4 were calculated using the mid range value of the power densities Pd, 120 Ω strain gages and grid areas equal to 10 mm2.

Table 4: Allowable maximum power densities Pd

Power density Pd = P/A (watts/in2) (kwatts/m2

)

Voltage from equation (18) in volts, A = 10 mm2, R=120 Ω

Thick sections of aluminum and cooper alloy 5 - 10 8 - 16 8

Thick sections of steel parts 2 - 5 3 - 8 5 Thin sections of steel parts 1 - 2 1.5 - 3.0 3 Ceramic, glass, composite materials 0.2 - 0.5 0.3 - 0.8 1.6

Polymers 0.02 - 0.05 0.03 - 0.08 0.5

RAPV d ..2= (18) 5.3. Gage response in non-uniform strain fields The resistance variation response given by the gage encompasses all the parallel sensing elements of its grid. Therefore, the response is related to the average strain occurring in the prototype material underneath the grid. This means that the effect of the strain gradient is averaged. Strain gages with small grids should be used to give accurate strain information in sharp strain gradients, as illustrated in Figure 7. Strips of small gages may be used to determine the strain gradient and the strain at the hot spot in a welded structure.

Figure 7: Example of an inappropriately large strain gage placed in the stress concentration area to measure the peak strain at the root of a U-notch. The average strain results from the total response of the grid which includes elements positioned in the x and y directions. 5.4. Gage response in dynamic strain fields

Strain gage response to dynamic loading can unfold in three parts. The first is related to the influence of the gage installation on the inertia and stiffness of the component. In general, relatively heavyweight and large-size prototypes are not influenced by gage installations, but some care has to be taken with the associated gage cabling. The second part has to do with the time required by the conducting grid and associated backing carrier and adhesive to respond to the prototype strain wave. This time is on the order of 100ns. The third part relates the grid length with the size of the wave pulse. Figure 8 shows the distorted response given by a gage to a rectangular pulse that has a travel speed equal to c.

Figure 8: Distorted response of a strain gage to a rectangular wave

The bibliography cited at the end of the paper also shows the influence of different environments (such as humidity, extremely low or high temperature, hydrostatic pressure, radiation) or type of loading (creep, cyclic deformation or fatigue) on the strain gages and associated installations. 6. Strain gage circuits Two types of signal conditioning circuits are commonly used to measure the variations in resistance of strain gages: the potenciometer circuit (Figures 9 and 10) and the Wheatstone circuit (Figure 11), both with configurations that use constant voltage or constant current energy supply sources. These circuits measure the drop in voltage between specific points (Figures 9 and 11). The constant current potenciometer circuit is commonly used with a direct resistance measurement circuit, as shown in Figure 10. 6.1 Potenciometer circuit The potentiometer circuit (Figure 9) can be activated by constant voltage or constant current sources. It is recommended that the potenciometer circuit be coupled with highly accurate and very fast response voltmeters since it does not have the hardware capability of zeroing the initial voltage readings between the strain gage terminals. The voltage variation response of both potenciometer circuits is given by equations (20) and (21), respectively, for the constant voltage and constant current circuits. The constant voltage circuit presents a nonlinear term, given in equation (20).

Figure 9: The potenciometer circuit: constant voltage on left and constant current on right Their response and connection with the strain gage response are respectively given by equations (20) and (21).

( )( )

1

2

2

2

1

12

2

1

12

111

1111

1RRr

RRr

RR

rRR

RR

rVR =

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆+

∆+

+−=−⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆−

∆+

=∆ ηη (20)

1

11..

RR

RIVR∆

=∆ (21)

Two advantages of the potenciometer constant current circuit are: the absence of the nonlinear term and the absence of influence on the voltage output of the resistance of the circuit’s auxiliary cables.

Figure 10: The four-wire direct resistance measurement circuit The so-called four-wire direct resistance measurement circuit is a constant current potenciometer circuit that uses a highly accurate ohmmeter to measure the strain gage resistance. It is shown in Figure 10. This circuit is essentially the same as the one in the right of Figure 9. The ohmmeter needs a resolution of at least 10-3 Ω in order to measure 1 to 10 µε accurately if low resistance (120 Ω) strain gages are used. Application of this circuit is optimized if strain gages with higher initial resistance (e.g. 1,000 or 2,000 Ω) are used. In any case, the accuracy and resolution required for the ohmmeters imply static measurements due to the need for several measurements (around a thousand) to be automatically taken by the instrument

and later repeated by the measurement dedicated system (around 30 measurements). The mean response of these measurements is used to avoid reading errors. Constant current circuits are able to measure with an acquisition rate of one reading per second per channel of measurement. This type of circuit can be very useful for measuring static phenomena that span small variations during long duration measurements. 6.2. Constant voltage Wheatstone bridge circuit The Wheatstone bridge is the most-used strain gage signal conditioning circuit. Due to its capability of initial circuit balancing (initial output voltage zeroing), it has the great advantage of allowing the measurement of small variations of resistance. The circuit is shown in Figure 11.

Figure 11: The Wheatstone bridge Basically, the Wheatstone bridge consists of four resistances placed in the four arms of the circuit, a constant voltage or current source, and a voltage reading and recording system. Equations (22) and (23) present the development and the final response of output voltage E of the constant voltage bridge.

43

4

21

1

2411

432432

211211

21

....

).(

).(

RRRV

RRRV

E

iRiREVRR

ViRRiVV

RRViRRiVV

iii

BD

AC

AC

+−

+=

−==+

=→+==

+=→+==

+=

(22)

( ) ( )4321

4231 ...

RRRRRRRR

VE+×+

−= (23)

The numerator of (23) will be zero when the bridge is balanced. Using this capability, small variations in resistance occurring in any of the arms will be included in the total variation of the output voltage reading by the voltage measurement device. Equation (24) is arrived at under the assumption of initial bridge balance, E = 0. Development of equation (24) leads to the basic bridge relation (25) and to the nonlinear term η depicted in equation (26). This term is usually neglected. Generally, it is seen as being very small when compared to the unity if the variations in resistances ∆R1, ∆R2, ∆R3 and ∆R4 are small (as they are expected to be if elastic strains are being measured).

( )( ) ( )( )( ) ( )44332211

44223311 ...

0

RRRRRRRRRRRRRRRR

VE

E

∆++∆+×∆++∆+∆+∆+−∆+∆+

=∆

= (24)

( )( )

1

2

4

4

3

3

2

2

1

12 1..

1.

RR

rRR

RR

RR

RR

rrVE =−⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆−

∆+

∆−

∆+

=∆ η (25)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆+

∆+

∆+

∆+

+=

3

3

2

2

4

4

1

1 .

11

1

RR

RRr

RR

RR

rη (26)

For example, using r = 1 and variations of ∆R of around 1% (usually very large in strain gage measurements) the nonlinear term η of equation (26) becomes approximately 0.02 <<1.00. Therefore, equation (25) is written without the nonlinear term as

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−

∆+

∆−

∆+

=∆4

4

3

3

2

2

1

12 .

1.

RR

RR

RR

RR

rrVE (27)

6.3. Constant current Wheatstone bridge circuit When compared with the constant voltage power supply circuit, the Wheatstone bridge circuit can also employ a constant current power supply with the advantage of reducing the nonlinear effect. The output voltage from an initially balanced constant current bridge is given by equation (28). It shows that nonlinearity is caused by the sum of the resistance variations, located in the denominator of the expression, and by the products of the ∆R terms, located inside the second bracket in the numerator.

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆∆−

∆∆+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆−

∆+

∆−

∆∆+

=∆∑∑ 4

4

2

2

3

3

1

1

4

4

3

3

2

2

1

131 ..

.RR

RR

RR

RR

RR

RR

RR

RR

RRRR

IEii

(28)

6.4. Constant voltage Wheatstone bridge circuits – specific cases The relationship between the output voltage and the sensed strain is given for three installation cases, illustrated in Figures 12 to 14. The first case (Figure 12) represents the majority of installations used in field measurements where only one arm of the bridge is used. It is known as the “quarter bridge” configuration. This case is used here to show that an amplification in output is required to couple the bridge with ordinary accuracy and resolution voltmeters. For example, a relatively average to high strain value of around 1,000 µε will produce an output of only 1mv if the excitation voltage is equal to 2v, gage factor K is 2 and ratio r is 1.

Figure 12: The Wheatstone bridge - quarter bridge configuration The second case uses two arms of the bridge and is commonly known as the “half bridge” configuration (Figure 13). The gages are mounted in arms R1 and R2 or R4. Generally, it is used when one wants to increase the output signal and minimize the temperature output. The employment of such configurations requires knowledge of the strain distribution in order to place the gages in the most appropriate positions. Two examples may be given for this case. One is the use of two gages on opposite surfaces of the cross section of a beam under flexure loading. The output of the bridge will be proportional to twice the strain on those surfaces if the gages are placed in arms R1 and R2 or R4. The second example is the installation of two gages in the longitudinal and transverse directions of a prismatic bar under axial loading, where the output will be given by the longitudinal strain increased by factor (1+µ), where µ is the Poisson coefficient. In both cases, it can be seen, using equation (27), that any undesirable spurious

temperature output from the gages will be cancelled out due to the opposite signal in the adjacent arms of the bridge.

Figure 13: The Wheatstone bridge - half bridge configuration The third case employs four gages placed in the arms of the bridge (Figure 14). This configuration is called the “full bridge”. It is used to cancel out spurious temperature outputs from the gages, while at the same time amplifying the output signal by multiplying the existing strains by four or by 2(1+ µ), respectively, in instances such as a beam under bending or a bar under axial loading. This case is encountered in most load cell circuits. It is also used in field measurements when certain types of loadings are to be measured in a structure but other loads are not.

Figure 14: The Wheatstone bridge - full bridge configuration

6.5. Shunt Calibration The shunt calibration is used to calibrate bridge output. It employs the placement of a parallel or shunt resistance along one of the arms of the bridge. A switch is also set in place to connect or disconnect the parallel resistance. When the switch is off, the arm resistance is R. When the switch is on, the arm resistance will change to a lower resistance. The parallel resistance can be calculated to simulate the output signal equivalent to a given strain; for example, -1000µε. If the gage factor is assumed to be K=2.0, the shunt calibration required for an output of -1000µε will be 59880Ω or 174650Ω, respectively, for strain gage resistances of 120 Ω and 350Ω. The development for the cases of 120Ω and 350Ω strain gages that have resistances Rs placed in parallel with the arm is illustrated in Figure 15. Equation (29) gives the values of Rs as a function of gage factor K and of desired calibration strain ε, which must be entered as a negative value since ∆R caused by the parallel resistance is negative.

Figure 15: Example of application of parallel resistance Rs to strain gage instrumented arm of Wheatstone bridge to give electrical output equivalent to strain equal to -1000µε

⎟⎠⎞

⎜⎝⎛ +−=

ε.11.

KRRs g (29)

6.6 Quarter bridge: three-wire connection of a strain gage A common practice used in quarter bridge installations is the use of the three-wire connection instead of the two-wire connection. Two-wire connections are sensitive to temperature output from the cabling and they also decrease the sensitivity output if the cables are very long. Figure 16 compares both types of connections with long cables. It can be seen that the resistance of the R1 arm of the bridge is composed of the strain gage resistance and the resistance of both connecting wires. The resistance

variation caused by the temperature variation occurring in both cables will aid the strain output sensed by the gage and will increase the denominator of ∆R/R. The three-wire arrangement relocates the resistances of the cables in the adjacent arms of the bridge. One cable resistance is placed in arm R1, the other in arm R4, and the third cable resistance is placed in the voltage source arm of the bridge, causing a minimum interference in the bridge output. The temperature output generated in the wires placed in arms R1 and R4 will be canceled out in the measurement of ∆E, as can be seen by application of equation (27).

Figure 16: Quarter bridges with two and three-wire connections 6.7 Null balance bridges The term “balancing” means the adjustment of the resistances of the arms of the bridge so that voltage output ∆E becomes zero. In most cases, a precision resistor with total resistance R5 + R6 is placed in parallel with arms R2 and R3, as depicted in Figure 17. Resistances R5 and R6 are varied (∆R5 = - ∆R6) to achieve identity of products R1 and R3eq, and R4 and R2eq. In the null balancing bridge, the variation of the strain gage resistance in arm 1 is determined by the measurement of the variation of the precision resistor placed in arm 5-6. The measurement is accomplished by counting the number of turns given in the resistor, which nulls the electrical current transiting galvanometer G, placed between points B and D of the bridge. Presently, in many applications balancing is achieved by software. The measuring circuits use highly accurate voltage meters that measure the initial imbalance of the circuit and subtract it from the voltage readings whenever a strain measurement is required.

Figure 17: Balancing the Wheatstone bridge with a precision variable resistor 7. Strain gage measurement systems There is a reasonable number of strain gage conditioner makers world wide. It is recommended that a thorough search be made in order to select the most adequate conditioner for a given application. Figure 18 and Table 5 illustrate and list the most important features and functions a measurement system should have.

Table 5: Features and functions of a strain gage measurement system Basic features Static measurement capability

Use of at least one strain gage in a quarter bridge configuration Constant voltage source, generally equal to 2V. Other maximum voltages are possible; for example, 2.5V, 5V, 10V, or continuous variable voltage source from 0 to 10 or 15V Battery or electrical DC or AC energy feed Built-in amplifier with gain equal to 100, 200 or 500 Output voltage terminal to connect to output voltage reading device, or built-in voltage reading device and recorder Bridge completion resistors - possibility of selecting circuit to work with quarter, half or full bridge Adjustment of gage factor K circuit Calibration circuit (shunt calibration) Initial balancing circuit

Advanced features

Channel multiplexing - manual or automatic devices. Static or dynamic measurements with acquisition rates of 2kHz or more per channel Data recording for 1, 2, 8, …, 1028 channels or more Noise filters Software-based recorder and data analyzers with functions ranging from rosette stress analysis to spectral analysis and FFT

Figure 18: Main features of a strain gage signal conditioner with multiplexing capability 7. Examples of strain gage applications Basically, strain gages are used as sensor elements in a large variety of transducers in field or laboratory applications for direct strain measurements. Strain gages are applied as sensor elements in transducers to measure force, such as in load cells used in weight scales and other measurement devices such as elongation clip gages, inclinometers and accelerometers to measure, respectively, displacement, inclination angles and acceleration. In all these applications, one or more full Wheatstone bridges are used. The gages are located and bonded in appropriate geometric configurations of spring elements in order to amplify as much as possible the signal to be measured and to avoid other possible spurious signals generated by unpredictable or undesirable loads. Examples of spring elements for load cells are given in Figures 19a to f.

Figure 20: Illustrations of spring elements for load cells. Geometric arrangements depend on type of load effort related to load to be measured: a) normal load, b) basic geometry for bending load; c)

advanced geometry for bending load, d) advanced binocular geometry for bending load; e) cantilever geometry for shear load; f) short pin geometry for shear load

In the case of precision transducers, there are certain practices for avoiding small uncertainties caused by the temperature response of the spring material and strain gages. Specifically, calculated or experimentally dimensioned Constantan, Cupper and Balco (a nickel-iron alloy) resistors are placed to adjust the sensitivity of the installation, to keep response at zero with the variation of temperature, and to give a null response under zero load. Figure 20 presents the general arrangement of the complete full bridge for load cells.

(a)

(b)

(c) (d)

(e)

(f)

Figure 20: General arrangement of complete full bridge for load cells, including resistors to adjust for

sensitivity, bridge balance and influence of temperature. Laboratory and field use for direct measurement of strains with the objective of understanding the mechanical behavior of large or small structures represents a very important application of strain gages. Figure 21 shows an example of the instrumentation of a steel pipe with a machined defect simulating metal thickness loss caused by corrosion. The steel pipe had closing caps and was hydrostatically tested to rupture. The strain gage was used to measure the circumferential strain at the instant of rupture. In this case, a special uniaxial strain gage capable of reaching and measuring large strains was used.

Figure 21: Bonding a strain gage rosette at the center region of a pipe with external machined defect to

determine the maximum circumferential strain in a laboratory burst pressure test. Figure 22 presents pictures of two sections of a stacker machine that was tested to confirm assumptions made at the design stage in order to make an increase in its load capability possible.

Figure 22: Instrumentation of a large iron-ore pile stacker machine

8. Conclusions This article has presented the principal concepts regarding strain gages. These gages are used to determine strains in localized areas of structural components in the laboratory or field, or as sensors in transducers such as resistive accelerometers or load cells. They are accurate “point” measuring elements and can be used in cases of static or dynamic loading. The measurement systems employ

Location of structural members that were monitored

(a)

(b) uniaxial gage at a member under nominal loading conditions

(c) rosette at stress concentration

(d) signal conditioner and data acquisition system

conventional and on-shelf circuitry elements, and they present the important advantage of leading to electrical and treatable output. Glossary Measurement system: Consists of a device or set of devices associated with a protocol, and are used to make a measurement. Sensor: The basic sensing element of a transducer. Transforms an input into an output. Constantan: 55% Cu, 45% Ni alloy used in the form of a thin wire or thin foils as the sensor element in strain gages. Advance: Same as Constantan Karma: 74% Ni, 20% Cr, 3% Al, 3% Fe alloy used as the sensor element in strain gages. Cyanoacrylate: An adhesive compound that polymerizes quickly when spread as a very thin film. Transducer: The measurement system element that is affected by the quantity to be measured. Helps the sensor to receive input and to transmit output. Potenciometer circuit: Electrical circuit that results from connecting the strain gage to a voltage meter and a power source. Used to determine the variation in the resistance of the strain gage. Can be powered by a constant voltage or constant current source. Wheatstone circuit: Electrical circuit composed of four resistors, voltage meter and power source. Used to determine variation in the resistance of the one, two or four strain gages that are placed in its four arms. Can be powered by a constant voltage or constant current source. Shunt calibration: Placement of a known resistor in parallel with a strain gage or arm of the measuring system circuit in order to cause a known resistance variation which will be used for calibration purposes. Null balance bridge: Wheatstone bridge circuit with adjustable resistors placed in parallel with two of its four arms in order to make its voltage output equal to zero. Bibliography 1. “Experimental Stress Analysis”, J.W. Dally and W.F. Riley, 4th ed., College House Enterprises,

LLC, 5713 Glen Cove Drive, Knoxville Tennessee, USA, 2006, jdally@collegehousebooks.com [a comprehensive approach to Experimental Mechanics. The topic strain gages is discussed with depth]

2. “Instrumentation for Engineering Measurements”, J.W. Dally, W.F. Riley and K.G. MacConnell, John Wiley & Sons, 2nd ed., 1993 [A comprehensive approach to measurement in mechanics. The topic circuits for strain gages is discussed in depth].

3. Strain Gage User’s Handbook, edited by R.L. Hannah and S.E. Reed, Society for Experimental Mechanics, Elsevier Applied Science [The whole book thoroughly discusses all topics regarding strain gages].

4. Handbook of Experimental Mechanics, edited by A.S. Kobayashi, Society for Experimental Mechanics, Prentice-Hall, 1987 [A comprehensive approach to Experimental Mechanics. The topic strain gages is discussed in depth in one of its chapters].

5. Springer Handbook of Experimental Solid Mechanics, 1st Edition, edited by William N. Sharpe, Society for Experimental Mechanics, Springer, 2008 [A comprehensive approach to Experimental Mechanics. The topic strain gages is discussed in depth].

6. Manual on Experimental Stress Analysis, edited by, J.F. Doyle and J.W. Phillips, Society for Experimental Mechanics, 7 School Street, Bethel, CT, USA, 1989 [An introductory approach to Experimental Mechanics. The topic strain gages is discussed in one of its chapters].

7. “The Strain Gage Primer”, C.C. Perry and H.R. Lissner, 2nd edition, McGraw-Hill, 1962 [A comprehensive approach to strain gages is discussed in depth in the book’s several chapters].

Author’s biographical sketch José L.F. Freire: B.S. (1972) and M.Sc. (1975) degrees in mechanical engineering from the Catholic University of Rio de Janeiro (PUC-Rio); Ph.D. (1979) in Engineering Mechanics from Iowa State University of Science and Technology (ISU); associate professor of Mechanical Engineering at PUC-Rio and chairman of the Structural Integrity Laboratory; member of the Society for Experimental Mechanics and is its past president; major areas of research: Experimental Stress Analysis, Pipeline Engineering and Structural Integrity of Equipment and Structures.