strain gage measurements and application

8/4/2019 Strain Gage Measurements and Application

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Strain Gage Rosettes:Selection, Application and Data Reduction

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Strain Gages and Instruments

F thnl ppt, [email protected]

www.m-mmnt.m151

rvn 12-sp-2010

1.0 Introduction

A strain gage rosette is, by denition, an arrangemento two or more closely positioned gage grids, separatelyoriented to measure the normal strains along dierentdirections in the underlying surace o the test part.Rosettes are designed to perorm a very practical and

important unction in experimental stress analysis. It canbe shown that or the not-uncommon case o the generalbiaxial stress state, with the principal directions unknown,three independent strain measurements (in dierentdirections) are required to determine the principal strainsand stresses. And even when the principal directions areknown in advance, two independent strain measurementsare needed to obtain the principal strains and stresses.

To meet the oregoing requirements, the Micro-Measurements manuactures three basic types o straingage rosettes (each in a variety o orms):

Tee: two mutually perpendicular grids.

45-Rectangular: three grids, with the second andthird grids angularly displaced rom the rst grid by45 and 90, respectively.

60-Delta: three grids, with the second and third grids60 and 120 away, respectively, rom the rst grid.

Representative gage patterns or the three rosette types arereproduced in Figure 1.

In common with single-element strain gages, rosettesare manuactured rom dierent combinations o gridalloy and backing material to meet varying application

requirements. They are also oered in a number o gagelengths, noting that the gage length specied or a rosettereers to the active length o each individual grid withinthe rosette. As illustrated in Figure 2, rectangular anddelta rosettes may appear in any o several geometricallydierent, but unctionally equivalent, orms. Guidancein choosing the most suitable rosette or a particular

application is provided in Section 2.0, where selectionconsiderations are reviewed.

Fg 1 B tt typ, lfd by gd nttn: () t; (b) 45-tngl; () 60 dlt.

5X

()

5X

(b)

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()

rtngl

Dlt

Fg 2 Gmtlly dnt, bt ntnlly

qvlnt nfgtn tngl nd dlt tt.
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Strain Gage Rosettes: Selection, Application and Data Reduction

Since biaxial stress states occur very commonly in machine

parts and structural members, it might be presumed thathal or so o the strain gages used in experimental stressanalysis would be rosettes. This does not seem to be thecase, however, and ten percent (or less) rosette usage may bemore nearly representative. To what degree this pattern ousage refects an inclination or on-site makeup o rosettesrom single-element gages, or simply an undue tendency toassume uniaxiality o the stress state, is an open question.At any rate, neither practice can generally be recommendedor the accurate determination o principal strains.

It must be appreciated that while the use o a straingage rosette is, in many cases, a necessary conditionor obtaining the principal strains, it is not a sufcientcondition or doing so accurately. Knowledgeability inthe selection and application o rosettes is critical to theirsuccessul use in experimental stress analysis; and theinormation contained in this Tech Note is intended to helpthe user obtain reliably accurate principal strain data.

2.0 Rosette Selection Considerations

A comprehensive guide or use in selecting Micro-Measurements strain gages is provided in Reerence 1. Thispublication should rst be consulted or recommendationson the strain-sensitive alloy, backing material, sel-temperature-compensation number, gage length, andother strain gage characteristics suitable to the expected

application. In addition to basic parameters such as theoregoing, which must be considered in the selection o anystrain gage, two other parameters are important in rosetteselection. These are: (1) the rosette type tee, rectangular,or delta; and (2) the rosette construction planar (single-plane) or stacked (layered).

The tee rosette should be used only when the principal straindirections are known in advance rom other considerations.Cylindrical pressure vessels and shats in torsion are twoclassical examples o the latter condition. However, caremust be exercised in all such cases that extraneous stresses(bending, axial stress, etc.) are not present, since thesewill aect the directions o the principal axes. Attentionmust also be given to nearby geometric irregularities,such as holes, ribs, or shoulders, which can locally alterthe principal directions. The error magnitudes due tomisalignment o a tee rosette rom the principal axes aregiven in Reerence 2. As a rule, i there is uncertainty aboutthe principal directions, a three-element rectangular ordelta rosette is preerable. When necessary (and, using theproper data-reduction relationships), the tee rosette can beinstalled with its axes at anyprecisely known angle rom theprincipal axes; but greatest accuracy will be achieved byalignment along the principal directions. In the latter case,except or the readily corrected error due to transversesensitivity, the two gage elements in the rosette indicate thecorresponding principal strains directly.

Where the directions o the principal strains are unknown,

a three-element rectangular or delta rosette is alwaysrequired; and the rosette can be installed without regardto orientation. The data-reduction relationships given inSection 4.0 yield not only the principal strains, but also thedirections or the principal axes relative to the reerencegrid (Grid 1) o the rosette. Functionally, there is littlechoice between the rectangular and delta rosettes. Becausethe gage axes in the delta rosette have the maximumpossible uniorm angular separation (eectively 120), thisrosette is presumed to produce the optimum sampling othe underlying strain distribution. Rectangular rosetteshave historically been the more popular o the two,primarily because the data-reduction relationships aresomewhat simpler. Currently, however, with the widespread

access to computers and programmable calculators, thecomputational advantage o the rectangular rosette iso little consequence. As a result o the oregoing, thechoice between rectangular and delta rosettes is moreapt to be based on practical application considerationssuch as availability rom stock, compatibility with thespace available or installation, convenience o solder tabarrangement, etc.

All three types o rosettes (tee, rectangular, and delta)are manuactured in both planar and stacked versions.As indicated (or the rectangular rosette) in Figure 3, the

Fg 3 rtngl tt ( th m gg

lngth) n pln nd tkd nttn.

stkd

Pln
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planar rosette is etched rom the strain-sensitive oil as

an entity, with all gage elements lying in a single plane.The stacked rosette is manuactured by assembling andlaminating two or three properly oriented single-elementgages. When strain gradients in the plane o the test partsurace are not too severe, the normal selection is theplanar rosette. This orm o rosette oers the ollowingadvantages in such cases:

Thinandexible,withgreaterconformabilitytocurved suraces;

Minimalreinforcingeffect;

Superiorheatdissipationtothetestpart;

Availableinallstandardformsofgageconstruction,

and generally accepts all standard optional eatures; Optimalstability;

Maximumfreedominleadwireroutingandattachment.

The principal disadvantages o the planar rosette arise romthe larger surace area covered by the sensitive portion othe gage. When the space available or gage installationis small, a stacked rosette may t, although a planar onewill not. More importantly, where a steep strain gradientexists in the surace plane o the test part, the individualgage elements in a planar rosette may sense dierent strainelds and magnitudes. For a given active gage length, thestacked rosette occupies the least possible area, and has

the centroids (geometric centers) o all grids lying overthe same point on the test part surace. Thus, the stackedrosette more nearly approaches measurement o the strainsat a point. Although normally a trivial consideration, it canalso be noted that all gages in a stacked rosette have thesame gage actor and transverse sensitivity, while the gridsin a planar rosette will dier slightly in these properties,due to their dierent orientations relative to the rollingdirection o the strain-sensitive oil. The technical datasheet accompanying the rosettes ully documents theseparate properties o the individual grids.

It should be realized, however, that the stacked rosetteis noticeably stier and less conormable than its planar

counterpart. Also, because the heat conduction paths orthe upper grids in a stacked rosette are much longer, the heatdissipation problem may be more critical when the rosetteis installed on a material with low thermal conductivity.Taking into account their poorer heat dissipation and theirgreater reinorcement eects, stacked rosettes may not bethe best choice or use on plastics and other nonmetallicmaterials. A stacked rosette can also give erroneous strainindications when applied to thin specimen in bending, sincethe grid plane o the uppermost gage in a three-gage stackmay be as much as 0.0045 in [0.11 mm] above the specimensurace. In short, the stacked rosette should ordinarilybe reserved or applications in which the requirement or

minimum surace area dictates its selection.

3.0 Gage Element Numbering

Numbering, as used here, reers to the numeric (oralphabetic) sequence in which the gage elements in arosette are identiied during strain measurement, andor substitution o measured strains into data-reductionrelationships such as those given in Section 4.0. Contrary to awidely held impression, the subject o gage numbering is notnecessarily a trivial matter. It is, in act, basic to the proper,and complete, interpretation o rosette measurement.3With any three-element rosette, misinterpretation o therotational sequence (CW or CCW), or instance, can leadto incorrect principal strain directions. In the case o therectangular rosette, an improper numbering order willproduce completely erroneous principal strain magnitudes,

as well as directions. These errors occur when the gageusers numbering sequence diers rom that employed inthe derivation o the data-reduction relationships.

To obtain correct results using the data-reductionrelationships provided in Section 4.0 (and in the Appendix),the grids in three-element rosettes must be numbered ina particular way. It is always necessary in a rectangularrosette, or instance, that grid numbers 1 and 3 be assigned totwo mutually perpendicular grids. Any other arrangementwill produce incorrect principal strains. Following arethe general rules or proper rosette numbering. With arectangular rosette, the axis o Grid 2 must be 45 awayrom that o Grid 1; and Grid 3 must be 90 deg away, in the

same rotational direction. Similarly, with a delta rosette,the axes o Grids 2 and 3 must be 60 and 120 away,respectively, in the same direction rom Grid 1.

In principle, the preceding rules could be implementedby numbering the grids in either the clockwise orcounterclockwise direction, as long as the sequence iscorrect. Counterclockwise numbering is preerable,however, because it is consistent with the usual engineeringpractice o denoting counterclockwise angular measure-ment as positive in sign. The gage grids in all Micro-Measurements general-purpose, three-element planarrosettes (rectangular and delta) are numerically identied,and numbered in the counterclockwise direction.*

Examples o the grid numbering or several representativerosette types are illustrated in Figure 4. At rst glimpse, itmight appear that gage patterns (b) and (c) are numberedclockwise instead o counterclockwise. But when thesepatterns are examined more closely, and when the axiso Grid 2 is transposed across the grid-circle diameter tosatisy the oregoing numbering rules, it can be seen thatthe rosette numbering is counterclockwise in every case.

* Micro-Measurements also supplies special-purpose planarrectangular rosettes designed exclusively or use with the hole-drilling method o residual stress analysis. Since these rosettesrequire dierent data-reduction relationships, procedures, andinterpretation, they are numbered clockwise to distinguish

them rom general-purpose rosettes.


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Micro-Measurements stacked rosettes are not numbered,as a matter o manuacturing economics. The user shouldnumber the gages in stacked rosettes according to the rulesgiven here and illustrated in Figures 2 and 4.

4.0 Principal Strains and Directions

from Rosette MeasurementsThe equations or calculating principal strains rom threerosette strain measurements are derived rom what isknown as a strain-transormation relationship. Asemployed here in its simplest orm, such a relationshipexpresses the normal strain in any direction on a testsurace in terms o the two principal strains and the anglerom the principal axis to the direction o the speciedstrain. This situation can be envisioned most readily withthe aid o the well-known Mohrs circle or strain, as shownin Figure 5**. It can be seen rom Figure 5a (noting that theangles in Mohrs circle are double the physical angles onthe test surace) that the normal strain at any angle rom

the major principal axis is simply expressed by:

=

++P Q P Q

2 22

cos

(1)

Fg 4 cntlkw nmbng gd

n M-Mmnt gnl-pp

tn gg tt ( txt).

** The Mohrs circle in Figure 5 is constructed with positive shearstrain plotted downward. This is done so that the positiverotational direction in Mohrs circle is the same (CCW) asor the rosette, while maintaining the usual sign conventionor shear (i.e., positive shear corresonds to a reduction in theinitial right angle at the origin o the X-Y axes as labeled inFigure 5b).

Fg 5 stn tnmtn m th pnpl tn t

th tn n ny dtn: () n tm pnpl tn P,

nd Q, hwn by Mh l tn; (b) tngltt ntlld n tt , wth Gd 1 t th

bty ngl m th mj pnpl x; () x thtngl tt pmpd n Mh l tn.

()

(b)

()


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Figure 5b represents a small area o the test surace, with

a rectangular rosette instal led, and with the reerence grid(#1) oriented at degrees rom p. Mohrs circle, with theaxes o the rosette superimposed, is shown in Figure 5c.

By successively substituting into Equation (1) the angles orthe three grid directions, the strain sensed by each grid canbe expressed as ollows:

1

2

2 22

2 22

=+

+

=+

+ +

P Q P Q

P Q P Q

cos

cos 445

2 22 903

o

o

( )

=+

+ +( )

P Q P Q

cos

(2a)

(2b)

(2c)

When the rosette is installed on a test part subjected toan arbitrary strain state, the variables on the right-handside o Equations (2) are unknown. But the strains 1, 2and 3 can be measured. Thus, by solving Equations (2)simultaneously or the unknown quantities P, Q, and ,the principal strains and angle can be expressed in terms othe three measured strains. Following is the result o thisprocedure:

P Q,

=+

( ) + ( )

=

1 31 2

22 3

2

1

2

1

2

1

2tan1

22 2 3

1 3

+

(3)

(4)

I the rosette is properly numbered, the principal strainscan be calculated rom Equation (3) by substituting themeasured strains or 1, 2 and 3. The plus and minus

alternatives in Equation (3) yield the algebraicallymaximum and minimum principal strains, respectively.Unambiguous determination o the principal angle romEquation (4) requires, however, some interpretation, asdescribed in the ollowing. To begin with, the angle represents the acute angle rom the principal axis to thereerence grid o the rosette, as indicated in Figure 5. In thepractice o experimental stress analysis, it is somewhat moreconvenient, and easier to visualize, i this is reexpressedas the angle rom Grid 1 to the principal axis. To changethe sense o the angle requires only reversing the sign oEquation (4). Thus:

P Q,

= =

1

2

2 2 1 3

1 3

tan1

(5)

The physical direction o the acute angle given by eitherEquation (4) or Equation (5) is always counterclockwisei positive, and clockwise i negative. The only dierenceis that is measured rom the principal axis to Grid 1,while is measured rom Grid 1 to the principal axis.Unortunately, since tan 2tan 2(+ 90), the calculatedangle can reer to either principal axis; and hence theidentiication in Equation (5) as P,Q. This ambiguitycan readily be resolved (or the rectangular rosette) byapplication o the ollowing simple rules:

(a) i1

> 3

, then P,Q

= P

(b) i1 < 3, then P,Q = Q

(c) i1 = 3 and 2 < 1, then P,Q = P = 45

(d) i1 = 3 and 2 > 1, then P,Q = P = +45

(e) i1 = 2 = 3, then P,Q is indeterminate (equalbiaxial strain).

The reasoning which underlies the preceding rules becomesobvious when a sketch is made o the corresponding Mohrscircle or strain, and the rosette axes are superimposed asin Figure 5c. A similar technique or assuring correctinterpretation o the angle is given in the orm o a step-by-step algorithm in Reerence 3.

The preceding development has been applied to therectangular rosette, but the same procedure can be usedto derive corresponding data-reduction equations or thedelta rosette shown in Figure 6. When grid angles , +60, and + 120 are successively substituted into Equation(1), the resulting three equations can again be solvedsimultaneously or the principal strains and angle. Thus,or the delta rosette:

P Q, =

+ + ( ) + ( ) + ( )1 2 3 1 2

22 3

23 1

2

3

2

3 (6)

= ( )

1

2

3

2

3 2

1 2 3

tan1

(7)

As in the case o Equation (4), the angle calculated romEquation (7) reers to the angular displacement o Grid 1rom the principal axis. The sense o the angle can again bechanged by reversing its sign to give the angle rom Grid 1to a principal axis:

P Q,

= =

( )

1

2

3

2

2 3

1 2 3

tan1 (8)


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In every case [Equations (4), (5), (7), and (8)], the anglesare to be interpreted as counterclockwise i positive, andclockwise i negative.

Equation (8) embodies the same ambiguity with respect tothe tan 2and tan 2(+ 90) as Equation (5). As beore, theambiguity can easily be resolved (or the delta rosette) byconsidering the relative magnitudes (algebraically) amongthe individual strain measurements; namely:

(a) i then

(b) i

1

2 3

1

2>

+=

= = +P Q P

then is indeterminate 1 2 3= = ,P Q((equal biaxial strain)

When the principal angle is calculated automatically bycomputer rom Equation (5) or Equation (8), it is alwaysnecessary o course, to avoid the condition o division byzero i1 = 3 with a rectangular rosette, or 1 = (2 + 3)/2with a delta rosette. For this reason, the computer shouldbe programmed to perorm the oregoing (c) and (d) tests,in each case, prior to calculating the arc-tangent.

OncetheprincipalstrainshavebeendeterminedfromEquation (3) or Equation (6), the strain state in the suraceo the test part is completely deined. I desired, the

maximum shear strain can be obtained directly rom:

MAX = P Q (9)

Intuitive understanding o the strain state can also beenhanced by sketching the corresponding Mohrs circle,approximately to scale. In Equations (3) and (6), the rstterm represents, in each case, the distance rom the originto the center o the circle, and the second term representsthe radius, or MAX/2. With the angle calculated, urtherinsight can be gained by superimposing the rosette gridaxes on the Mohrs circle, as in Figures 5c and 6b.

5.0 Principal Stresses from Principal Strains

As previously noted, a three-element strain gage rosettemust be employed to determine the principal strains ina general biaxial stress state when the directions o theprincipal axes are unknown. The usual goal o experimentalstress analysis, however, is to arrive at the principal stresses,or comparison with some criterion o ailure. With thestrain state ully established as described in Section 4.0,the complete state o stress (in the surace o the test part)can also be obtained quite easily when the test materialmeets certain requirements on its mechanical properties.Since some types o strain gage instrumentation, such as

Fg 6 Dlt tt: () ntlld n tt ,

wth Gd 1 t th ngl m th mj pnpl tn

dtn; (b) tt gd x pmpd n Mhl tn. Nt tht Gd 2 t b vwd +60

(ccW) m Gd 1 n th tt, nd +120 n Mh l.

()

(b)


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our System 6000, calculate both the principal strains and

the principal stresses, the ollowing material is provided asbackground inormation.

I the test material is homogeneous in composition, and isisotropic in its mechanical properties (i.e., the propertiesare the same in every direction), and i the stress/strainrelationship is linear, with stress proportional to strain,then the biaxial orm o Hookes law can be used to convertthe principal strains into principal stresses. This procedurerequires, o course, that the elastic modulus (E) andPoissons ratio () o the material be known. Hookes lawor the biaxial stress state can be expressed as ollows:

P P QE

= +

( )1 2 (10a)

Q Q P

E= +( )

1 2 (10b)

The numerical values o the principal strains calculatedorm Equation (3) or Equation (6) can be substituted intoequations (10), along with the elastic properties, to obtainthe principal stresses. As an alternative, Equation (3)or Equation (6) depending on the rosette type) can besubstituted algebraically into Equations (10) to express theprincipal stresses directly in terms o the three measured

strains and the material properties. The results are asollows:

Rectangular:

P Q

E,

=

+

+( ) + ( )

2 1

2

11 3

1 22

2 32

(11)

Delta:

P Q

E,

=

+ +

+( ) + ( ) +

3

1

2

1

1 2 3

1 22

2 32

312

( )

(12)

When the test material is isotropic and linear-elastic inits mechanical properties (as required or the validity othe preceding strain-to-stress conversion), the principalstress axes coincide in direction with the principal strains.Because o this, the angle rom Grid 1 o the rosette to

the principal stress direction is given by Equation (5)

or rectangular rosettes, and by Equation (8) or deltarosettes.

6.0 Errors, Corrections, and Limitations

The obvious aim o experimental stress analysis is todetermine the signicant stresses in a test object as accuratelyas necessary to assure product reliability under expectedservice conditions. As demonstrated in the precedingsections o this Tech Note, the process o obtaining theprincipal stresses involves three basic, and sequential, steps:(1) measurement o surace strains with a strain gage rosette;(2) transormation o measured strains to principal strains;and (3) conversion o principal strains to principal stresses.

Each step in this procedure has its own characteristic errorsources and limits o applicability; and the stress analystmust careully consider these to avoid potentially seriouserrors in the resulting principal stresses.

Ofrstimportanceisthatthemeasuredstrainsbeasfreeas possible o error. Strain measurements with rosettesare subject, o course, to the same errors (thermal output,transverse sensitivity, leadwire resistance eects, etc.) asthose with single-element strain gages. Thus, the samecontrolling and/or corrective measures are required toobtain accurate data. For instance, signal attenuationdue to leadwire resistance should be eliminated by shuntcalibration4, or by numerically correcting the strain data or

the calculated attenuation, based on the known resistanceso the leadwires and strain gages.

Because at least one o the gage grids in any rosette will inevery case be subjected to a transverse strain which is equalto or greater than the strain along the grid axis, considerationshould always be given to the transverse-sensitivity errorwhen perorming rosette data reduction. The magnitude othe error in any particular case depends on the transverse-sensitivity coeicient (Kt) o the gage grid, and on theratio o the principal strains (P /Q). In general, whenKt 1%, the transverse-sensitivity error is small enough to beignored. However, at larger values oKt, depending on therequired measurement accuracy, correction or transversesensitivity may be necessary. Detailed procedures, as wellas correction equations or all cases and all rosette types,are given in Reerence 5.

When strain measurements must be made in a variablethermal environment, the thermal output o the strain gagecan produce rather large errors, unless the instrumentationcan be zero-balanced at the testing temperature, understrain-ree conditions. In addition, the gage actor o thestrain gage changes slightly with temperature. Reerence6 provides a thorough treatment o errors due to thermaleects in strain gages, including specic compensation andcorrection techniques or minimizing these errors.

Ater making certain that strain measurement errors such


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as the oregoing have been eliminated or controlled to the

degree easible, attention can next be given to possibleerrors in the strain-transormation procedure or obtainingthe principal strains. A potentially serious source o errorcan be created when the user attempts to make up a rosetteon the specimen rom three conventional single-elementgages. The error is caused by misalignment o the individualgages within the rosette. I, or example, the second andthird gages in a rectangular rosette conguration are notaccurately oriented at 45 and 90, respectively, rom therst gage, the calculated principal strains will be in error.

The magnitude o the error depends, o course, on themagnitude (and direction) o the misalignment; but italso depends on the principal strain ratio, P /Q, and onthe overall orientation o the rosette with respect to theprincipal axes. For certain combinations o principal strainratio and rosette orientation, 5-degree alignment errors ingages 2 and 3 relative to Gage 1 can produce an error o 20percent or more in one o the principal strains.

Since it is very dicult or most persons to install a smallstrain gage with the required precision in alignment, the useris well-advised to employ commercially available rosettes.The manuacturing procedures or Micro-Measurementsstrain gage rosettes are such that errors due to gridalignment within the rosette need never be considered. Forthose cases in which it is necessary, or whatever reason, toassemble a rosette rom single-element gages, extreme careshould be exercised to obtain accurate gage alignment.

And when the principal strain directions can be predictedin advance, even approximately, alignment o Gage 1 or3 in a rectangular rosette, or alignment o any gage in adelta rosette, with a principal axis, will minimize the errorin that principal strain caused by inter-gage misalignment.

The strain-transormation relationships and data-reductionequations given in Section 4.0 assume a uniorm state ostrain at the site o the rosette instal lation. Since the rosettenecessarily covers a nite area o the test surace, severevariations in the strain eld over this area can producesignicant errors in the principal strains particularlywith planar rosettes.7 For this type o application, thestacked rosette is distinctly superior; both because it

covers a much smaller area (or the same gage length), andbecause the centroids o all three grids lie over the samepoint on the test surace.

The requirements or a homogeneous material anduniorm strain state can be (and are) relaxed under certaincircumstances. A case in point is the use o strain gagerosettes on ber-reinorced composite materials. I thedistance between inhomogeneities in the material (i.e.,ber-to-ber spacing) is small compared to the gage lengtho the rosette, each grid will indicate the macroscopic oraverage strain in the direction o its axis. These measuredstrains (ater the usual error corrections) can be substitutedinto Equation (3) or Equation (6) to obtain the macroscopic

principal strains or use in the stress analysis o test objects

made rom composite materials.

8

As noted later in thissection, however, Equations (10)-(12) cannot be used orthis purpose.

There is an additional limitation to the strain-transormat-ion relationship in Equation (1) which, although notrequently encountered in routine experimental stressanalysis, should be noted. The subject o the straindistribution about a point, as universally treated inhandbooks and in mechanics o materials textbooks, isdeveloped rom what is known as innitesimal-straintheory. That is, in the process o deriving relatively simplerelationships such as Equation (1), the strain magnitudesare assumed to be small enough so that normal- andshear-strain approximations o the ollowing types can beemployed without introducing excessive error:

+ 2 (13)

sin tan (14)

Although oten unrecognized, these approximationsare embodied in the equations used throughout thecontemporary practice o theoretical and experimentalstress analysis (where strain transormation is involved).This includes the concept o Mohrs circle or strain,and thus all o the equations in Section 4.0, which areconsistent with the strain circle. Innitesimal-strain theoryhas proven highly satisactory or most stress analysisapplications with conventional structural materials, sincethe strains, i not truly ininitesimal, are normallyvery small compared to unity. Thus, or a not-untypicalworking strain level o 0.002 (2000), the error in ignoring2 compared to is only about 0.2 percent.

However, strain gage rosettes are sometimes used in themeasurement o much larger strains, as in applicationson plastics and elastomers, and in post-yield studies ometal behavior. Strain magnitudes greater than about 0.01(10 000) are commonly reerred to as large or nite,and, or these, the strain-transormation relationship inEquation (1) may not adequately represent the actual

variation in strain about a point. Depending on the strainmagnitudes involved in a particular application, and onthe required accuracy or the principal strains, it may benecessary to employ large-strain analysis methods orrosette data reduction.9

The nal step in obtaining the principal stresses is theintroduction o Hookes law [Equations (10)] or the biaxialstress state. To convert principal strains to principalstresses with Hookes law requires, o course, that theelastic modulus and Poissons ratio o the test materialbe known. Since the calculated stress is proportion to E,any error in the elastic modulus (or which a 3 to 5 percentuncertainty is common) is carried directly through to the


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principal stress. An error in Poissons ratio has a much

smaller eect because o the subordinate role o in therelationship.

It is also necessary or the proper application o Hookeslaw that the test material exhibit a linear relationshipbetween stress and strain (constant E) over the rangeo working stresses. There is normally no problem insatisying this requirement when dealing with commonstructural materials such as the conventional steel andaluminumalloys.Othermaterials(e.g.,someplastics,cast iron and magnesium alloys, etc.) may, however, bedistinctly nonlinear in their stress/strain characteristics.Since the process o transormation rom measured strainsto principal strains is independent o material properties,the correct principal strains in such materials can bedetermined rom rosette measurements as described inthis Tech Note. However, the principal strains cannot beconverted accurately to principal stresses with the biaxialHookes law i the stress/strain relationship is perceptiblynonlinear.

A urther requirement or the valid application o Hookeslaw is that the test material be isotropic in its mechanicalproperties (i.e., that the elastic modulus and Poissonsratio be the same in every direction). Although severelycold-worked metals may not be perectly isotropic, thisdeviation rom the ideal is commonly ignored in routineexperimental stress analysis. In contrast, high-perormancecomposite materials are usually abricated with directional

iber reinorcement, and are thus strongly directional(orthotropic or otherwise anisotropic) in their mechanicalproperties. Hookes law as expressed in Equations (10) isnot applicable to these materials; and special constitutiverelationships are required to determine principal stressesrom rosette strain measurements.8

References

1. Micro-Measurements Tech Note TN-505, Strain GageSelection Criteria, Procedures, Recommendations.

2. Micro-Measurements Tech Note TN-511, Errors Dueto Misalignment o Strain Gages.

3. Perry, C.C., Data-Reduction Algorithms orStrain Gage Rosette Measurements, ExperimentalTechniques. May, 1989, pp. 13-18.

4. Micro-Measurements Tech Note TN-514, ShuntCalibration o Strain Gage Instrumentation.

5. Micro-Measurements Tech Note TN-509, Errors Dueto Transverse Sensitivity in Strain Gages.

6. Micro-Measurements Tech Note TN-504, Strain Gage

ThermalOutputandGageFactorVariationwithTemperature.

7. Troke, .W., Flat versus Stacked Rosettes,Experimental Mechanics, May, 1967, pp. 24A-28A.

8. Tsai, S.W. and H.T. Hahn, Introduction to CompositeMaterials, Technomic Publishing Company, 1980.

9. Meyer, M.L., Interpretation o Surace-StrainMeasurements in terms o Finite HomogeneousStrains. Experimental Mechanics, December, 1963,pp. 294-301.


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Appendix

Derivation of Strain-Transformation Relationship

[Equation (1) in text] from Deformation Geometry

Consider a small area o a test surace, as sketched inFigureA-1.ThelineO-P,oflengthLO, and at the angle rom the X axis, is scribed on the surace in the unstrainedstate. When uniorm principal strains P and Q are appliedin the directions o the X and Y axes, respectively, the pointP moves to P as a result o the displacements X and Y(greatly exaggerated in the sketch).

It is evident rom the gure that:

X = P (LO cos ) (A-1)

Y =Q (LO sin ) (A-2)

It can also be seen (Figure A-2), by enlarging the detail inthe vicinity o points P and P , that or small strains:

L X cos + Y sin (A-3)

Substituting rom Equations (A-1) and (A-2),

L LO (P cos2+ Q sin

2) (A-3)

Or,

=

= +

L

LoP Qcos sin

2 2

(A-5)

But,

cos cos

sin cos

2

2

12

1 2

1

21 2

= +( )

= ( )

Ater substituting the above identities,

=

++P Q P Q

2 22

cos

(A-6)

Alternate Data Reduction Equations

In the extensive technical literature dealing with straingage rosettes, the user will oten encounter data-reductionrelationships which are noticeably dierent rom oneanother, and rom those in the body o this Tech Note. Asa rule, these published equations yield the same results,and dier only in algebraic ormat although provingso in any given case may be rather time consuming.Since certain orms o the equations may be preerred ormnemonic reasons, or or computational convenience,several alternative expressions are given here. All o theollowing are equally correct when the gage elements in the

rosette are numbered as described in this Tech Note.

Fg a-1 Fg a-2


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Rectangular Rosette:

where:

P Q

P Q

,

,

=+

( ) + +( ) 1 3

1 32

2 1 3

2

2

1

22

==+

( ) + ( )

=

1 31 2

22 3

2

1

22 /

,P Q C C(( ) + ( )

=+

22

2

1 3

2

C

C

(A-7)

(A-8)

(A-9)

(A-9)

Delta Rosette:

(A-10)

(A-11)

(A-12)

where:

P Q,

=+ +

+( )

+ (1 2 3 1 2 3

2

2 33

2

3

1

3))

=+ +

( ) + ( ) + (

2

1 2 31 2

22 3

23 1

32

P Q, ))

= ( ) + ( ) + ( )

2

12

22

32

9

2

/

, P Q C C C C

=+ +

/ 3

3

1 2 3C

Cartesian Strain Components from Rosette Measurements

It is sometimes desired to obtain the Cartesian components o strain (X, Y, and XY) relative to a specied set o X-Ycoordinate axes. This need can arise, or example, when making strain measurements on orthotropic composite materials.The Cartesian strain components are also useul when calculating principal strains rom rosette data using matrixtransormation methods.*

When the X axis o the coordinate system coincides with the axis o the reerence grid (Grid 1) o the rosette, the Cartesiancomponents o strain are as ollows:

Rectangular Rosette:

X= 1

Y= 3

XY= 22 (1 + 3)

Y

3

2

1

X

* Milner, R.R., A Modern Approach to Principal Stresses and Strains, Strain, November, 1989, pp.135-138.


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Delta Rosette:

/

/

X

Y

XY

=

= +( )

= ( )

1

2 3 1

2 3

2 3

2 33

The oregoing assumes in each case that the gage elements in the rosette are numbered counterclockwise as indicated. Whenthe calculated XY is positive in sign, the initial right angle at the origin o the X-Y coordinate system is decreased by the

amount o the shear strain.

Y

3 2

1

X

strain gage measurements and application

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