The Bonded Electrical Resistance Strain Gage an Introduction 019507209X

Examination o f Eqs. (10.31 ) and (10.34 ) tells u s tha t whe n

the quantitie s i n th e squar e bracke t o f eac h equatio n wil l b e identical , s o that R/ R wil l b e directl y proportional t o a a. Thi s mean s tha t whe n on e computes th e correc t angl e betwee n th e tw o gages , o r grids , one wil l hav e a stress gage . Fro m Eq . (i) , the relativ e angl e o f inclination betwee n th e tw o

STRESS GAGE S 32

5

gages, o r grids , can b e established. Thus ,

This expressio n ca n als o b e writte n as

Since v will b e between 0 and 0.5 , and becaus e K ca n b e expected t o b e less tha n 0.04 , th e produc t v K wil l be ver y smal l wit h respec t t o unity , and so b y neglectin g th e product , vK , th e valu e of cos 2 become s

which is a very close approximation, especially when K tend s towards zero. The term , co s 2(/> , can als o b e writte n a s

Using th e valu e of co s 2 give n b y Eq . (10.36) , th e valu e of ta n becomes

If th e produc t v K i s ignored, the n

Taking th e transvers e sensitivit y factor, K, equa l t o zer o gives

Stress gage s ma y b e mad e usin g eithe r wir e or foil . A typical foi l stres s gage i s shown i n Fig . 10.6 .

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FIG. 10.6 . Foi l stress gage in V configuration. (Courtes y o f Measurement s Group, Inc.)

10.6. Application of a single strain gage to indicate principal stress (8) There ar e certai n situation s in whic h th e direction s o f the principa l stresse s are known . This may b e through the condition s o f symmetry, a preliminary study wit h a brittl e lacque r coat , o r b y som e othe r method . Unde r thes e conditions, i f tw o strai n gage s ar e installed , on e i n th e directio n o f eac h principal axis , th e tw o strai n reading s thu s obtaine d wil l provid e sufficien t information t o enable th e computation o f the two principa l stresses . At each point t o b e investigated, this saves the us e of at leas t one se t of instrumentation whe n al l reading s ar e require d t o b e mad e simultaneously . Also, th e time involve d wit h calculation s wil l b e muc h reduce d i n compariso n wit h that require d fo r standar d rosett e analysis , which might employ equipment for thre e or fou r observation s at eac h statio n fo r eac h loa d level . If on e i s onl y intereste d i n on e o f th e tw o principa l stress magnitudes, however, an d know s t o whic h axi s thi s corresponds , a singl e strai n gag e indication fo r eac h loa d a t eac h statio n is all that i s required t o provid e th e necessary information . Such a gag e i s shown in Fig . 10.7. The amoun t o f require d instrumentatio n is reduce d b y two-third s i n comparison wit h th e general rosett e method . Th e corresponding calculatio n time ca n als o b e reduce d b y a n eve n greate r proportion , especiall y i f th e equipment ca n b e calibrated t o giv e a direct readou t i n term s of the desire d data. Due t o th e fac t tha t th e strai n condition s ar e symmetrica l wit h respect to a principa l axis , i f a V-typ e stress gag e i s t o b e use d an d line d u p wit h the principa l axis , bot h halve s o f th e gri d wil l b e subjecte d t o exactl y th e same strai n condition s an d wil l sho w th e sam e uni t chang e i n resistance . This mean s tha t on e hal f of th e V-typ e gage wil l b e redundant . Becaus e o f this onl y on e hal f o f the gag e need s t o b e used , whic h means a singl e strain gage incline d a t th e correc t angl e wit h respec t t o th e principa l axi s wil

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327

FIG. 10.7. Singl e strain gage inclined a t a n angl e 0 wit h respec t to th e principa l axis.

yield al l the necessar y information to enabl e th e evaluatio n of the principa l stress magnitude. As shown in Section 10.5 , tan 0 = ^/ v — K. I f K = 0, then tan 10.7.

Determination of plane shearing stress

Wire and foi l gages have little response to shearing strains so, if this quantity is t o b e determined , i t i s necessar y t o mak e th e approac h throug h th e measurement o f linear strains , whic h can b e converte d int o th e equivalent of plane shea r strain , and the n into term s o f shearing stress by means of the shearing modulus of elasticity. In Chapter 8 on rosette analysis, it was shown that linea r strain s i n certai n give n direction s ca n b e converte d int o thei r equivalent value s in term s o f the hydrostati c component, S H, an d th e pur e shear component , s s. Thes e tw o values are writte n in terms o f the principa l strains a s follows : (position o f the cente r o f Mohr's circle) (radius o f Mohr's circle ) This is shown in Fig . 10.8 . As shearin g stresse s ar e currentl y bein g considered , th e hydrostati c component, which is equivalent to tw o principal strain s of equal magnitud e and lik e sign , wil l no t concer n us . Th e pur e shea r component , whic h i s equivalent t o tw o principa l strain s of equal magnitud e bu t o f opposite sign , will b e examined rather carefully . I t i s from th e pur e shear componen t tha t the shearin g conditio n i n an y particula r directio n ca n b e determined , a s

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FIG. 10.8 . Shea r strain i n term s of pure shear component.

shown i n Fig . 10.8 . Sinc e th e dat a produce d wit h any rosett e arrangemen t can b e converted int o it s hydrostatic and pur e shea r components , i t ca n b e seen that rosette s might, in general, be used to find the shearing strain in any particular direction . However, except in two special cases, the data reductio n involved i n thi s procedure i s too complicate d an d time-consuming . Let us now look into the simpler procedures which can be accomplishe d with just tw o strai n indications . Consider th e rectangula r rosett e wit h fou r strain observations , which , fo r thi s specia l case , ca n b e reduce d t o tw o measurements. Thi s rosett e consist s o f fou r strai n gage s mounte d i n th e directions OA, OB, OC, and OD , as indicated i n Fig. 10.9 . For thi s particula r rosette, th e value s o f % , e s, an d ta n 2 0 ar e give n b y Eqs . (8.41a) , (8.41b) , and (8.42) . These ar e

Furthermore, i f gages A an d C an d gage s B an d D ca n b e connecte d i n adjacent arm s o f th e Wheatston e bridge , th e differences , (c a — ec) an d

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329

FIG. 10.9. Arrangemen t of gage axe s fo r four-elemen t rectangula r rosette.

(sb — Ed), ca n b e read ou t directl y so that th e determinatio n o f the maximum shear strai n an d th e directions o f the principa l axe s may b e found fro m jus t two strain observations . Thi s save s both time and equipment. It shoul d als o be note d tha t wit h thi s for m o f rosette i t i s als o possibl e t o determin e th e magnitudes of the tw o principa l stresse s a s wel l a s th e direction s o f the tw o principal axes . When K ^ 0 , th e gage s respon d t o latera l strai n i n additio n t o axia l strain. Thus ,

There i s no chang e i n th e direction s o f the principa l axes , since the valu e of tan 2 9 i s independen t o f K . Figur e 10.1 0 show s a n availabl e stacke d four-element rectangula r rosett e o f the configuratio n show n i n Fig . 10.9 . If stres s gage s rathe r tha n strai n gage s wer e used , th e chang e i n thes e relations woul d b e du e essentiall y t o th e differenc e i n th e numerica l valu e of K , whic h woul d the n tak e o n th e particula r valu e o f K tha t woul d b e

FIG. 10.10. Four-elemen t rectangula r rosette. (Courtes y o f Measurements Group, Inc. )

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equal t o th e valu e o f Poisson's rati o o f the materia l upo n whic h th e gage s were installed . Hence ,

When K = v,

This ca n b e rewritten a s

or

where G , the shea r modulus , is

The shear gage In orde r t o determin e th e linea r strai n equivalen t o f shearin g strain , tw o gages ca n b e use d tha t ar e mounte d wit h thei r axe s havin g an y arbitrar y angle betwee n them. In orde r t o sho w this , the argumen t give n by Perr y (9) will be followed. The tw o gages are bonded t o the test material a s illustrated by Fig . 10.11 . The expression s fo r th e strains , e a and £ b, can b e writte n as

Solving Eqs . (10.46 ) an d (10.47 ) simultaneously for y xjl produce s

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331

FIG. 10.11. Arbitrar y gag e arrangement.

If gage s a an d b ar e bisecte d b y th e x axis , the n 6 cos 9A = cos 9B. In tha t cas e Eq . (10.48) i s rewritten as

A

= — 9B an d s o

Furthermore, si n 29A = —si n 26 B, whic h further reduce s Eq . (10.49) t o

Perry (9 ) generalizes these result s as follows : Th e difference i n normal strain sensed by any two arbitrarily oriented strain gages in a uniform field is proportional to the shear strain along an axis bisecting the strain gage axes, irrespective of the included angle between the gages. An examination o f Eq. (10.50) shows that if the tw o gages are 90 ° apart the denominato r become s unity , since 9 A = 45 ° (or 9 B = —45°) . Thus ,

Equation (10.51 ) tell s u s tha t th e shearin g strai n alon g th e bisecto r o f th e gages' axe s i s equal t o th e differenc e i n th e norma l strains . I t ca n b e seen , then, that a two-element rectangular rosette makes an ideal shear gag e when the two gages are arranged i n adjacent arms o f a Wheatstone bridge . Figure 10.12 show s the gag e an d bridg e arrangement . Unless th e tw o gage s happe n t o b e line d u p wit h th e principa l axe s (when th e individua l strai n indication s wil l correspon d t o eac h o f th e principal strains) , no informatio n abou t th e principal strai n magnitudes , o r the directions of the principal axes, is available from th e two gages. However,

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FIG. 10.12. Two-elemen t rectangula r rosette arrange d t o measure

if on e i s onl y intereste d i n shea r i n a give n direction , th e tw o gage s wil l provide th e necessar y data . Since tw o strai n gage s onl y occup y hal f o f the Wheatston e bridge, th e bridge outpu t ca n b e double d b y addin g tw o mor e gage s wit h thei r axe s parallel an d perpendicula r t o th e firs t two . I f th e bisecto r o f th e gag e axe s happens t o lin e u p wit h the principa l axes , the n twic e th e maximu m shea r is indicate d b y th e outpu t fro m th e complet e Wheatston e bridge . I n particular, i t shoul d b e note d that , sinc e th e fou r gage s i n th e Wheatston e bridge onl y measur e strain s i n tw o direction s (at righ t angles) , the y wil l d o nothing to determine th e directions of the principal strai n axes . Figure 10.1 3 shows severa l configuration s of commerciall y availabl e four-elemen t gage s for determinin g shear strain. Care mus t b e take n i n orde r t o avoi d confusio n betwee n th e four element shea r gag e an d th e four-elemen t rectangular rosette . Bot h emplo y four activ e strai n gages . Th e shea r gage , whic h measure s strai n i n tw o perpendicular directions , involve s a simpl e procedur e fo r establishin g th e shearing strain , and thu s the shearin g stress, but i t wil l onl y permi t on e t o find this i n a particula r direction . A rectangular rosette , o n th e othe r hand , is muc h mor e genera l i n nature , permittin g strai n observation s i n fou r different direction s space d successivel y at 45° . Wit h th e rosette , on e ca n determine th e tw o principa l strains , th e direction s o f th e tw o principa l axes, an d th e shearin g strai n i n an y direction , includin g th e maximu m value. However , th e correspondin g computation s ar e somewha t mor e elaborate. Equations (10.50 ) an d (10.51 ) wer e develope d b y considerin g th e transverse sensitivit y factor, K, t o b e zero. I f K i s to b e accounte d for , then sa an d E b in Eqs . (10.50 ) an d (10.51 ) will hav e t o b e modified . Thi s ca n b e accomplished b y returnin g to Eqs . (9.14 ) and (9.15) . Thes e ar e

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333

FIG. 10.13. Ful l bridge s fo r shea r measurement . (Courtes y o f Measurements Group, Inc.)

These equation s represen t strain s for a pai r o f orthogonal gages . Since the gage s use d t o indicat e shea r strai n ar e arbitraril y oriented , thes e tw o expressions wil l be rewritten . Thus ,

where £„' and ej, ' are strain s perpendicular t o e' a and e' b, respectively. The value s of e' ^ and ej, ' can b e determine d b y usin g th e firs t strai n invariant , s o tha t

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Substituting th e value s o f e ^ given b y Eq . (a ) int o Eq . (10.52) ,

Substituting th e valu e o f 4' give n b y Eq . (b ) int o Eq . (10.53) ,

Since th e shearin g strai n i s proportiona l t o (e a — eb ), we hav e fro m Eqs . (c ) and (d )

This reduce s t o

Note tha t £' a and e' b are indicate d strains . Equation (10.54 ) show s tha t Eqs . (10.50 ) an d (10.51 ) ca n b e correcte d for transvers e sensitivit y b y multiplyin g th e shearin g strai n b y ( 1 — v 0 K)/ (1 - X) . Therefore, Eq . (10.50) become s

Likewise, Eq . (10.51 ) fo r th e two-elemen t rectangula r rosett e become s

Problems 10.1. Tw o strai n gages with thei r axes perpendicular to each other are t o b e use d as a stres s gage . Th e followin g data ar e availabl e for th e gages : R a = 350 ohms, (Gr)a = 2.15 , K a = 0.007, R n = 12 0 ohms, (G f )„ = 2.05 , K n = 0.009. Wil l thi s arrangement b e suitabl e fo r a stres s gage? I f so, specif y th e materia l on which it ma y b e used.

STRESS GAGE S

335

10.2. Fo r th e V stress gag e show n i n Fig . 10.6 , determin e th e include d angle, using K = 0, when designed for use on materials having the following Poisson ratios:

(a) v = 0.25 ; (b ) v = 0.30 ; (c ) v = 0.34 . 10.3. A single strain gag e i s used t o measur e th e longitudina l stres s at a point o n a n aluminum cantilever beam. Using v = 0.3 3 and E = 10 x 10 6 psi, determine the following: (a) Th e angl e (f > between th e bea m axi s an d th e gag e axi s (b) Th e longitudina l stres s fo r a recorde d strai n o f 884 microstrain. 10.4. Thin-walle d pressur e vessel s are t o b e mad e fro m severa l differen t materials . They have an internal pressure p, a diameter d , a wall thickness t, Poisson rati o v, an d a modulu s of elasticity E. A singl e strain gag e i s to b e bonde d t o eac h vessel s o th e hoo p stress , a H, ma y b e monitored an d th e vesse l automatically shut dow n i f a specifie d stres s i s exceeded . Develo p a n expressio n fo r
FIG. 10.14 .

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REFERENCES 1. Williams , Sidney B. , "The Dyadi c Gage, " SESA Proceedings, Vol. I, No. 2 , 1944 , pp. 43-55. 2. "SR- 4 Stress-Strain-Gage, " Produc t Dat a 4323, BLH Electronics , Inc., 75 Shawmut Rd., Canton, M A 02021 , May , 1961 . (Now ou t o f print.) 3. Hines , Fran k F. , "Th e Stress-Strai n Gage, " Proc. 1s t International Congress o n Experimental Mechanics, 1963 , pp. 237-253 . 4. Kern , Rober t E. , "Th e Stres s Gage, " SESA Proceedings, Vol . IV , No . 1 , 1946 , pp. 124-129 . 5. Kern , Rober t E . an d Sidne y B . Williams , "Stres s Measuremen t b y Electrica l Means," Electrical Engineering, Transactions, Vol . 65, March 1946 , pp. 100-107 . 6. Williams , Sidne y B. , "Geometry i n th e Desig n o f Stres s Measuremen t Circuits ; Improved Method s Throug h Simple r Concepts, " SESA Proceedings, Vol . XVII, No. 2 , 1960 , pp . 161-178 . 7. Sevenhuijsen , Pieter J. , "Stres s Gages, " Experimental Techniques, Vol . 8 , No . 3 , March 1984 , pp. 26-27 . 8. Lissner , H . R . an d C . C . Perry , "Conventiona l Wir e Strai n Gag e Use d a s a Principal Stres s Gage, " SESA Proceedings, Vol. XIII, No . 1 , 1955, pp. 25-34 . 9. Perry , C. C., "Plane-shear Measuremen t wit h Strai n Gages, " Experimental Mechanics, Vol. 9, No. 1 , Jan. 1969 , pp. 19N-22N .

11 TEMPERATURE EFFECTS ON STRAIN GAGES 11.1. Introduction When usin g strai n gages , th e enginee r want s t o measur e strain s produce d only b y th e loadin g o n th e structur e and t o eliminat e strains produce d b y other variables, particularly temperature. Since metals change their resistance with temperature as well as with strain, the purpose of the electrical resistance strain gage is to measure the strain-induced resistance change independently of the temperature-induced resistance change. Therefore, we want to account , or t o compensat e automatically , for th e effect s o f temperature o n th e strai n observations. The physica l phenomen a occurrin g i n a strai n gag e bonde d t o a tes t specimen are complex when a change o f temperature takes place (1). Among them ar e th e following : 1. Th e bas e material expand s o r contracts . 2. Th e strain-sensitiv e filamen t o f the gag e expand s o r contracts . 3. Th e resistanc e o f the filamen t changes . 4. Th e gag e factor o f the gag e i s subject to variation . 5. Th e bon d betwee n the gag e an d th e bas e materia l ma y b e affected . 6. Du e t o the transverse sensitivity, dimensional changes whic h take place in th e latera l direction , eithe r i n th e gag e o r i n th e bas e material , will show a n indicate d chang e i n resistance . 7. Th e carrie r o n whic h the sensitiv e filament is mounted ma y chang e it s properties. 11.2.

Basic considerations of temperature-induced strain (2-4 )

In Chapte r 1 , in th e discussio n o f the strai n sensitivit y of a wire , Eq. (1.18 ) was developed . From this , th e expressio n fo r th e uni t chang e i n resistanc e can b e writte n as

where v

= Poisson' s rati o L = conducto r lengt h p = resistivity of the conducto r materia l

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Equation (11.1 ) show s that th e uni t chang e i n resistanc e i s dependen t on th e uni t change s i n lengt h an d resistivit y o f th e conductor . I f n o mechanical strai n take s place, a uni t resistance change ca n stil l occur when the conducto r i s subjected t o a temperatur e change. Consider a strai n gag e bonde d t o a bas e materia l an d connecte d t o a strain indicator . I f th e bas e materia l i s unrestraine d and the n undergoe s a temperature change , th e strai n indicato r wil l sho w a n indicate d strai n consisting o f the algebrai c su m o f three components. 1 . Th e base material to whic h the gage is bonded expands or contracts in the directio n o f the gag e axis . This uni t change i n length , or strain , is

where a

m = coefficien t o f thermal expansio n of the bas e material AT = temperature chang e fro m a referenc e temperature . 2. Th e strain gag e grid material expands or contracts du e to the tempera ture change . This uni t chang e i n length , or strain , is

where
where f t = resistance-temperature coefficien t o f th e strai n gag e gri d material. Equation (11.4 ) can be expressed in terms of strain by dividing both side s o f the equatio n b y G F, the gag e factor . Thus ,

The strain , calle d therma l outpu t (sometime s referre d t o a s apparen t strain), tha t wil l b e registere d o n th e strai n indicato r ma y b e expresse d a s the algebrai c su m o f the thre e strains . Hence,

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339

FIG. 11.1. Apparen t strain vs. temperature for strain gage alloys bonded to steel. (From ref. 2.)

or

We are not t o assum e that th e strain given b y Eq. (11.6) is a linear function of temperature. I t is not, since the coefficients a m, a g, and ft are also functions of temperature. We must know, therefore, the temperatur e characteristic s of each gag e use d a s wel l a s the temperatur e characteristic s o f the materia l o n which th e gag e is bonded . For illustratio n only , Fig . 11. 1 show s th e variatio n o f strai n wit h temperature fo r severa l strai n gag e material s bonde d t o steel . Th e figur e shows tha t larg e error s ca n occu r whe n th e strai n gag e an d th e material t o which it is bonded ar e subjected to temperatures differin g fro m th e reference , or bonding , temperature . Thi s illustrate s th e nee d fo r correctio n whe n th e strain gage system i s subjected t o temperature fluctuations. Corrections ma y

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be accomplishe d b y computatio n usin g a temperature-strai n calibratio n curve, o r b y usin g a compensating , o r dummy , gag e i n a n adjacen t ar m o f the bridg e that i s subject t o certai n restrictions . If the correctio n i s made b y computation, the n a temperatur e recor d mus t b e kep t durin g th e test . Gage factor variation with temperature The gag e factor , G F, als o varie s with temperature. I f the temperatur e rang e is smal l an d th e variatio n i n G F is slight, then a correctio n ma y b e ignored . If, however , the tes t temperatur e rang e i s large an d th e variatio n i n G F with temperature cannot b e disregarded, then, depending on the required accurac y of th e strai n measurement , a gag e facto r correction ma y b e necessary . Thi s is illustrate d i n Fig . 11.2 , which show s th e variatio n i n gag e facto r wit h temperature fo r severa l strai n gag e alloys . Severa l o f th e alloy s ar e linea r over a considerable temperatur e range an d sho w quit e a variatio n i n slopes . Constantan show s a n increas e o f les s tha n 1 percen t pe r 100° F wit h increasing temperature, while Nichrome V shows a decrease o f over 2 percent per 100° F wit h increasin g temperature . Isoelasti c ha s a ver y sligh t chang e in gag e facto r betwee n roo m temperatur e an d 200 CF, bu t change s quit e perceptibly outsid e o f thi s region. This latte r material , however , is use d fo r dynamic measurement s rathe r tha n stati c measurements . Unde r dynami c conditions, other error s may b e considerably greater than th e change in gage factor, an d s o correctin g the gag e facto r may b e inessential. In orde r t o correc t th e gag e facto r fro m it s valu e a t th e referenc e temperature t o it s valu e a t th e tes t temperature , a simpl e procedur e i s followed. I f Fig . 11. 2 is examined, i t i s see n tha t th e percen t chang e i n gag e factor i s plotted versu s temperature. The gag e factor , G F T, a t som e tempera ture differen t fro m th e referenc e temperature i s

where G

= gage facto r at th e reference temperature, generall y at roo m temperature wher e G fR = G f Gf(%) = percentag e chang e i n th e gag e facto r fro m th e referenc e temperature t o th e tes t temperatur e FR

The prope r sig n o f AG F(%) must, o f course, b e use d i n Eq . ( 1 1.7). Method of determining gage factor variation with temperature The method o f determining th e variation i n the gage factor with temperatur e for resistanc e strai n gage s i s give n b y ASTM(5) . Tw o methods , stati c an d dynamic, are discussed, bu t onl y the static method wil l be outlined here . The test apparatus, shown in Fig. 1 1.3, consists of a beam havin g a uniform stress

FIG. 11.2 . Gag e facto r variatio n wit h temperature fo r severa l strai n gag e alloys . (From ref . 3.)

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FIG. 11.3. Apparatu s for stati c determination o f gage-factor variatio n vs . temperature. (From ref. 5 with permission . © ASTM. )

area tha t i s directl y proportiona l t o th e deflectio n o f th e en d poin t containing th e rider , whic h is located a t th e ape x o f the angl e formed by th e beam sides . The fixture holding th e bea m mus t b e muc h mor e rigi d tha n th e bea m in orde r t o preven t error s du e t o it s ow n deformation . Th e slidin g steppe d block ha s fou r surface s that mus t b e o f nearly equal steps , with the surfaces parallel t o eac h othe r an d t o th e slidin g surfaces. The step s ar e suc h tha t a maximum surface strain of 1000 + 10 0 uin/in is produced o n the beam. Whe n the ride r i s resting o n th e lowes t surface of the slidin g block, th e en d o f th e beam shoul d b e deflecte d abou t 2 percen t o f it s tota l planne d deflectio n i n order t o insur e positiv e contact . The gages to be tested ar e symmetricall y mounted i n the constant-stres s area o f the beam an d aligne d wit h th e longitudinal axis. Thermocouples ar e mounted a s near the gages a s possible an d a t eac h en d o f the constant stres s area. The entire tes t uni t is then placed i n a temperatur e chamber , th e gage s are connected t o th e instrumentation , and th e fixture and bea m ar e allowe d to com e t o equilibriu m a t th e referenc e temperature, whic h is usually roo m temperature. Wit h th e ride r restin g o n th e lowes t ste p o f th e slidin g block , the instrumentatio n is balanced, the n gage outpu t i s recorded a s th e ride r i s displaced t o subsequen t step s o n th e slidin g block . Readings , take n thre e times, ar e recorde d fo r bot h increasin g and decreasin g deflections . The tes t chambe r i s brough t t o previousl y selected temperature s an d

TEMPERATURE EFFECT S O N STRAI N GAGE S 34

3

the process i s repeated afte r th e temperature ha s stabilized. Th e temperature difference ove r the constant stres s area shall not exceed 5°F (3°C) or 1 percent of th e temperatur e o f the gag e area , whicheve r is greater. Neithe r shal l th e temperature change more than 5°F (3°C) during a test at any temperature. The chang e i n gag e facto r i s compute d a s th e differenc e betwee n th e gage output due to the strain for a given temperature and that at the reference temperature. Thi s i s expresse d a s a percentag e change . Thus , th e percen t change i n gag e facto r i s

where E

t = gage outpu t a t tes t temperatur e Er = gage outpu t a t referenc e temperatur e

If mor e accurac y i s desired , correction s ca n b e mad e fo r th e therma l expansion o f the bea m an d th e steppe d block . Thi s give s

where a

b = coefficient o f thermal expansio n o f the bea m as = coefficien t of thermal expansion o f the slidin g bloc k AT = difference betwee n th e tes t an d referenc e temperatur e

11.3. Self-temperature-compensated strain gages (2 , 4) The manufacture r of strain gag e alloy s ca n contro l temperature-resistanc e coefficients withi n reasonabl e limits . Wit h carefu l selectio n o f particula r melts, followed by judicious process control, the alloy will exhibit a minimum temperature response ove r a given temperature rang e when bonded to a test specimen whos e coefficien t o f thermal expansio n matche s tha t o f the strai n gage alloy . B y choosin g a gag e tha t i s temperatur e compensate d fo r th e material bein g tested, a three-wire, quarter-bridge circui t may be used rathe r than usin g a half-bridg e circui t with a matchin g dummy , or compensating , gage. In th e case o f the quarter-bridge circuit , the Wheatstone bridg e can be completed b y usin g a stabl e precisio n resisto r i n th e adjacen t ar m a t th e instrument, o r b y usin g a n instrumen t tha t accommodate s th e three-wire , quarter-bridge circuit . This circui t (wit h lead-lin e resistance ) wa s discusse d in Chapte r 5 . Figur e 11. 4 show s th e three-wir e circui t (withou t lead-lin e resistance) wit h the bridge-completio n resistor . Self-temperature-compensated gage s ar e readil y availabl e fro m strai n gage manufacturers. In the gage designation code, a number usuall y appears that indicate s th e materia l fo r whic h th e gag e i s temperature-compensated . For instance , 6 or 06 indicates a gage compensated fo r mild steel , where the

344

THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

FIG. 11.4. Quarte r bridg e wit h thre e lea d wire s an d bridg e completio n resistor.

thermal expansion coefficient is 6 (uin/in)/°F. This is also designated as parts per million pe r degree Fahrenhei t an d abbreviate d t o ppm/°F . A graph ma y be include d i n th e gag e packe t showin g th e variatio n wit h temperatur e o f both th e thermal output , expressed i n microstrain, and the gage factor . Ther e may also be a polynomial expression givin g the thermal output as a functio n of temperature . Figur e 11. 5 shows a typica l grap h o f bot h therma l outpu t and gag e facto r variatio n vs . temperature. In developin g th e therma l outpu t vs . temperature curve, the data give n are for a foi l lo t rathe r tha n a gage designation. A test gag e made fro m a foi l lot i s bonde d t o a tes t specime n an d th e procedur e give n b y AST M i s followed (5) . The tes t specime n i s placed i n a temperatur e chambe r an d th e gage i s connecte d t o a strai n indicator , th e gag e facto r i s se t (usuall y 2.0 ) and th e instrumen t i s the n balance d a t th e referenc e temperatur e o f 75°F . The tes t specime n i s unrestrained an d allowe d t o expan d o r contrac t freel y as th e temperatur e i s varied . Sinc e n o mechanica l o r therma l stresse s ar e present a t th e equilibriu m temperatures , th e recorde d strai n a t thes e temperatures i s du e onl y t o th e therma l effects , thu s enablin g on e t o plo t the therma l outpu t vs . temperature . Th e therma l outpu t ma y als o b e expressed a s a polynomia l give n as

where e

= therma l outpu t i n microstrai n T = temperatur e

TO

The coefficients , A , B , C , D, an d E ma y b e give n fo r bot h th e Fahrenhei t and Celsiu s temperatur e scales . If greater accurac y i s required i n determining th e therma l outpu t when testing an actual structure , the gage, or gages, ma y be bonded t o the structure along wit h th e adjacentl y place d temperatur e senso r fo r eac h gage . Th e strain-measuring instrumen t i s balance d a t th e referenc e temperature , an d the structur e i n th e unrestraine d stat e (n o mechanica l o r therma l stresse s

TEMPERATURE EFFECT S O N STRAI N GAGES

345

Foil Lo t No . A38AD31 5

FIG. 11.5 . Therma l output and gag e factor variatio n vs. temperature.

present) is subjected t o th e tes t temperature , o r temperatures . The thermal output (microstrain ) a t eac h equilibriu m temperatur e i s recorded , whic h allows subsequent correctio n in order to obtain th e strains due to the loading. Table 11. 1 is a listin g of the coefficient s fo r Eq . (11.10 ) fo r severa l foi l lot numbers . The temperatur e coefficien t o f the gag e facto r fo r eac h foi l lo t is also included. Thermal output correction When th e structure carryin g th e bonded strai n gag e i s loaded and teste d a t some subsequen t temperature , th e strai n indicato r wil l sho w a n indicate d strain, e|, tha t i s made u p no t onl y of the therma l outpu t (apparen t strain ) but als o th e strai n produce d b y the mechanica l an d therma l stresse s due t o the loading . Settin g th e actua l valu e o f th e gag e factor , G F, give n o n th e strain gag e dat a sheet , on th e strai n indicator , a firs t approximatio n o f the strain a t th e test temperatur e is obtained b y subtracting the therma l outpu t from th e indicated strain . Whe n doin g this , car e mus t b e taken i n using th e proper sig n of the strain . This gives

Table 11. 1 Foil lo t no . Tes A11BP11 10

1 8 Steel

A12BJ01 101

8 Steel 2024-T4 A l

A12BJ03 101

8 Steel 2024-T4 A l

A38AD315 101

A

t materia l

8 Steel

B

2.31 x 103 1.37 x 103

C

D

E

-2.79 x 10' -4.59 x 10- 2 8.60 x 10-5 -5.51 x 10' 4.79 X 10-4 -1.23 x 10 -1 Temperature coefficient of gag e facto r = (0. 7 + 0.2)/100°C -5

-2.9 x 10 -8 -3.11 x 10 -7

°F °C

-8

=F 'C 3 F J C

7.03 x 4.21 x 2.98 x 1.88 x

102 102 102 102

-8.31 -1.78 -4.22 -1.67 x 101 -2.94 -1.76 -7.00 -3.89 Temperature coefficient

5.15 X 10 x 10 -2 -4 2.75 X 10 x 10 - 2 -2 -5 6.06 X 10 x 10 3.19 X 10-4 x 10 - 2 of gag e factor = (0. 8 + 0.2)/100 C

-3.37 x -3.53 x -4.59 x -4.82 x

10 10 10 10

7.47 x 4.47 x 2.40 x 1.54 x

102 102 102 102

-8.80 4.19 X 10-5 -1.86 x 10 -2 -1.78 x 10' -4.75 x 10-2 2.36 X 10-4 2.54 X 10-5 -2.26 -1.44 x 10 -2 -5.58 -3.84 x 10 - 2 1.58 X 10-4 Temperature coefficient of gag e facto r = (0. 8 + 0.2)/100°C

-1.09 x -1.14 x 1.26 x 1.32 x

10 10 10 10

101 10'

3.02 -2.89 x 10 -2 8.13 X 10-5 -4 -2 2.53 4.29 X 10 -6.96 x 10 Temperature coefficient of gag e factor = (0. 8 + 0.2)/100°C

-9.59 x -2.64 x

Source: Courtes y o f Measurement s Group, Inc .

-7 -8 -7

-8 -7

-8 -7

-5.99 x 10 - 8 -6.29 x 1 0 - 7

=F "C °F ! C

°F "C

TEMPERATURE EFFECT S O N STRAI N GAGES 34

where e,

7

' =strai n indicato r readin g unde r tes t condition s ETO = thermal outpu t fro m th e data shee t (microstrain )

The valu e o f e, - give n b y Eq . (11.11 ) ma y b e o f sufficien t accuracy , bu t e'i was obtaine d wit h the actua l valu e of G F set on th e strai n indicator , while the therma l output , S TO, wa s determine d wit h a differen t valu e o f G f (generally 2.0) . Therefore , if further accurac y i s desired, th e therma l outpu t should b e correcte d fo r th e differenc e i n gag e factors . Thi s i s a simpl e procedure, a s on e ma y recal l fro m Chapte r 5 . The correctio n i s

where G

F1 = GF2 = £i = £2 =

gag e facto r set on th e strai n indicato r gage factor of the strai n gag e indicated strai n o n th e strai n indicato r correcte d strai n

Since e TO wa s obtained b y setting a gage facto r o n th e strai n indicato r other tha n th e actua l gag e factor , a correcte d valu e o f th e therma l outpu t may b e calculate d b y usin g Eq . (11.12) . B y takin g G F1 = G$, G F2 = G F, e = £ i ro> an< 3 £2 =e ro> w e hav e th e corrected therma l outpu t as

where G f i s th e gag e facto r used i n determinin g the therma l outpu t curve . Using Eq . (11.13) , Eq . (11.11 ) ca n b e rewritten as

If desired , th e entir e therma l outpu t vs . temperatur e curv e coul d b e corrected beforehan d fo r th e actua l gag e facto r b y usin g Eq . (11.13 ) an d plotting a ne w curve . Note , however , tha t thi s correctio n woul d b e fo r reference temperatur e gag e factor s an d woul d no t accoun t fo r gag e facto r variation wit h temperature. Example 11.1. Tw o identica l gage s ar e attache d t o a structur e a t differen t locations. Th e gag e dat a an d therma l outpu t curv e ar e show n i n Fig . 11.5 . A gage facto r o f 2.05 i s se t o n th e strai n indicator , th e instrumen t i s balanced a t 75°F, an d th e structur e i s loaded an d brough t t o it s test temperatur e o f 300°F. The followin g readings ar e obtained : Gage 1 E Gage 2 s

I 2

= 1180|iin/i n = -2060uin/i n

348 TH

E BONDE D ELECTRICA L RESISTANC E STRAIN GAG E

Use Eq . (11.14 ) t o determin e th e corrected strai n fo r each gage. Solution. Fro m Fig . (11.5) , K ro = — 80 ^in/in (G f = 2. 0 fo r thi s curve) . Usin g Eq. (11.14) ,

Note tha t the gage factor correctio n amounte d to onl y 2 uin/in.

Correcting for thermal output and gage factor variation In th e precedin g example , th e therma l outpu t wa s correcte d fo r th e actua l gage factor at th e reference temperature, since that value was set on the strain indicator whe n makin g th e strai n measurements . I f th e gag e facto r i s als o to be corrected a s the temperature changes, then the indicated strai n as taken from th e strai n indicator , a s wel l a s th e therma l output , mus t b e correcte d for th e chang e i n gag e factor . I f th e therma l outpu t curv e wa s develope d using a gag e facto r o f G* = 2.0 , bu t a differen t valu e o f G F was se t o n th e strain indicato r whe n th e strain s wer e measured , the n eac h strai n valu e (indicated an d therma l output ) woul d hav e t o b e correcte d individuall y before makin g th e fina l correction . As lon g a s a fe w identical gage s wer e used i n a test , thi s process, while inconvenient, coul d b e used . Generally , however , ther e woul d likel y b e a mixture of gages with different gag e factors and therma l output curves. Under these conditions i t i s simpler t o us e a metho d correctin g bot h th e indicate d strain an d therma l outpu t i n on e operation . A simpl e metho d o f correctin g fo r th e therma l outpu t an d th e gag e factor variatio n i s available. Since th e therma l output curve s ar e develope d using a particular gag e factor o f Gf (usuall y 2.0) at the reference temperature, set the sam e gag e facto r on th e strai n indicato r whe n conducting tests . Th e strain readin g ca n the n b e correcte d fo r th e effec t o f the therma l outpu t b y using Eq . (11.11) . Th e nex t ste p i s t o correc t th e actua l gag e facto r t o it s proper valu e at th e tes t temperature . This ca n b e don e b y usin g Eq . (11.7). Combining thes e tw o equation s give s th e actua l strain , e , a t th e tes t temperature. Thus ,

TEMPERATURE EFFECT S O N STRAI N GAGE S 34

where E

9

= thermal outpu t a t th e tes t temperatur e £• = strai n indicato r readin g unde r tes t condition s Gj£ = gag e factor at whic h the therma l outpu t wa s recorde d GFR = actual gag e factor a t th e referenc e temperature AGF(%) = percen t variatio n i n gag e facto r a t tes t temperature , wit h the prope r sig n TO

Example 11.2. A strain gag e havin g a gag e facto r o f 2.1 5 i s bonde d t o a stee l structure an d a gage factor o f 2.0 is set on the strain indicator, which is balanced at roo m temperature . A t th e tes t temperatur e th e followin g dat a ar e recorded : Indicated strai n = 267 5 uin/i n Thermal outpu t = —85 0 |iin/in AGF(%) = 0.7 5 percen t Solution. Usin g Eq . (11.15) ,

= 325 5 uin/i n Note that the reference temperatur e gage factor. G FR, wil l be the manufacturer' s gage factor , G T, providing th e referenc e temperatur e fo r thi s tes t i s 75°F.

The precedin g metho d i s the easies t t o use , since th e correctio n ca n b e made i n on e step . Othe r way s may b e employed. 1. Th e manufacturer' s gag e facto r ca n b e se t o n th e strai n indicato r an d the instrumen t balanced . Thi s woul d requir e th e indicate d strai n an d the thermal output to be corrected separately for gage factor variation. 2. Se t an y arbitrar y gag e factor , or th e tes t temperatur e gag e factor , o n the instrumen t an d balanc e i t at th e referenc e temperature. 3. Se t th e tes t temperatur e gag e facto r o n th e strai n indicator . Brin g the tes t structur e t o th e tes t temperatur e an d balanc e th e indicato r before applyin g th e load . Th e drawbac k here , o f course , i s bein g sure th e structur e is stress fre e whe n balancing th e instrument . 11.4. Strain gage-test material mismatch (2) Self-temperature-compensated strai n gage s ar e manufacture d fo r material s that hav e coefficient s o f thermal expansio n rangin g fro m 0 t o 1 8 parts pe r million pe r degre e Fahrenhei t (ppm/°F) . Thes e value s cove r a rang e o f commonly used engineerin g materials. Gages use d o n plastics , however, are

350 TH

E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E Table T1.2. Thermal expansio n coefficient s o f commo n materials Expansion coefficient Material Aluminum, 2024-T4 , 7075-T 6 Beryllium coppe r 2 5 Brass, 30-7 0 Bronze, phospho r (10%) Copper Iron, gra y cas t Magnesium, AZ-31 B Molybdenum Monel Steel, 1008 , 101 8 Steel, 434 0 Steel, 30 4 stainles s Steel, 31 6 stainles s Tin, pur e Titanium, pur e

(per F )

(per C )

12.9 9.3 11.1 10.2 9.3 6.0 14.5 2.2 7.5 6.7 6.3 9.6 8.9 13.0 4.8

23.2 16.7 20.0 18.4 16.7 10.8 26.1 4.0 13.5 12.1 11.3 17.3 16.0 23.4 8.6

Source: referenc e 2 .

manufactured wit h coefficient s o f 30 , 40 , an d 5 0 ppm/°F. I f a strai n gag e compensated fo r steel , fo r instance , i s use d o n a secon d materia l wit h a different coefficien t o f therma l expansion , the n th e therma l outpu t curv e furnished wit h th e gage wil l no longe r b e directly applicable. Th e amoun t o f deviation wil l depend o n th e differenc e i n the therma l expansio n coefficient s of th e tw o materials . Tabl e 11. 2 i s a partia l listin g o f therma l expansio n coefficients fo r som e common engineerin g materials . If a strai n gag e i s use d o n a materia l fo r whic h i t i s no t compensated , and i f the differenc e i n therma l expansio n coefficient s i s not to o large , the n over a limite d temperatur e rang e nea r th e referenc e temperatur e th e erro r produced i n using the given thermal output curve may b e acceptable. A s the difference betwee n th e therma l expansio n coefficient s become s larger , som e steps shoul d b e taken t o determin e th e therma l outpu t fo r th e gag e whe n i t is used o n materia l fo r which it i s not compensated . Thi s i s particularly tru e for plastics , no t onl y becaus e o f th e wid e variety , but als o becaus e o f th e difference betwee n manufacturer s fo r supposedl y th e sam e typ e o f plastic . When suc h gage s ar e t o b e use d o n a specifi c application , i t migh t b e advisable t o determin e th e therma l outpu t curv e fo r tha t particula r strai n gage-plastic combination. Thi s can be done i n accordance wit h the procedur e discussed earlier . When a strai n gag e i s applied t o a material fo r whic h i t is mismatched , an approximat e correctio n ca n b e mad e b y usin g Eq . (11.6) , whic h i s th e expression fo r therma l output . I f th e strai n gag e i s applie d t o th e firs t

TEMPERATURE EFFECT S O N STRAI N GAGE S 35

1

material, then th e therma l outpu t is

where th e subscrip t 1 refers t o th e firs t materia l tested . I f a simila r gag e is applied t o a secon d material , then th e therma l outpu t is

where th e subscrip t 2 refer s t o th e secon d material . In Eqs . (a ) an d (b) , th e las t ter m o n th e right-han d sid e i s th e same , since i t refer s t o th e strai n gage . Thus ,

If the thermal output, £ T01, i s known, then the thermal output, er02 , can be approximated b y usin g Eq. (c) . This gives

Equation (11.16 ) give s a firs t approximatio n fo r th e therma l outpu t when the strain gage is applied to a second material. This amounts to rotating the give n therma l outpu t curv e abou t th e referenc e temperature . I f a m2 i s larger than
352 TH

E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

FIG. 11.6. Therma l outpu t an d gag e facto r variatio n vs . temperature.

where e, Tol i s the therma l output fo r 101 8 steel and e. T02 i s the calculate d valu e of th e therma l outpu t fo r 2024-T 4 aluminum . Determin e e rol a t variou s temperatures fro m th e curv e fo r 101 8 steel, o r comput e i t b y usin g th e polynomial coefficient s give n i n Tabl e 11.1 . Carrying ou t th e calculations , th e results ove r a temperatur e rang e fro m — 100°F t o SOO T are tabulated .

r, °F -100 -50 0

50 100 150 200 250 300 350 400 450 500

£TOl, fan/in

1398 1135 747

266 -278 -856 -1439 -2003 -2524 -2979 -3364 -3608 -3747

ETm, fan/in 313

360 282 111

-123 -391 -664 -918 -1129 -1274 -1349 -1283 -1112

TEMPERATURE EFFECT S O N STRAI N GAGES

353

FIG. 11.7. Therma l outpu t an d gag e facto r variatio n vs . temperature. The results are plotted i n Fig. 11.7 . Fo r th e values of the thermal expansio n coefficients used , a first approximation o f the therma l output , s T02, shows tha t the approximate d value s are positiv e an d slightl y higher tha n th e tes t value s at temperatures below 75°F , while the approximated value s are negative an d belo w the test values for temperatures abov e 75°F . The error i n the approximate value s ranges betwee n 1 5 and 2 0 percen t u p t o 300°F , the n increase s considerabl y above tha t temperature . I n lie u o f other information , however , thi s correctio n for th e therma l outpu t woul d no t b e unreasonable , particularl y i f the strain s imposed b y mechanica l an d therma l stresse s wer e large . I t doe s illustrate , though, the need for an actual test if more exact values of the strain ar e required .

11.5.

Compensating gage

It wa s pointed out i n Chapte r 5 that two identica l gages placed i n adjacent arms o f a half-bridg e circuit an d bonde d t o th e sam e materia l woul d giv e temperature compensation if both gages were subjected to th e same temperature. This also applies to a full-bridge circui t if all four gages , bonded t o th e same material, were always at the same temperature during the test. Furthermore, i n either of these circuits the lea d wire s mus t be route d together an d be at th e sam e temperature.

354 TH

E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E

FIG. 11.8 . Temperature-compensate d circui t usin g a dumm y gage.

A commo n arrangemen t fo r temperatur e compensation i s the us e o f a dummy gage, identical to the active gage, in a half-bridge circuit. The dummy gage must be bonded to a stress-free piece of material identical to the material on whic h the activ e gag e i s bonded an d place d a s clos e t o th e activ e gage as possibl e s o tha t i t experience s th e sam e temperature . Th e lea d wire s of both the active gage and dumm y gage should be routed together. This circuit is show n in Fig . 11.8. If th e bridg e i s initially balanced, then , fro m Eq . (5.1),

where R

! = activ e stain gag e R2 = dummy strain gag e

If ther e i s a chang e i n temperatur e only , AR 1 = AR 2, sinc e th e activ e an d dummy gage s ar e identical , are mounte d o n th e sam e typ e o f material, an d are maintaine d a t identica l temperatures . Therefore ,

This shows that th e bridge remains balanced, irrespectiv e of the temperature change, sinc e th e influenc e o f temperatur e ha s bee n eliminated , an d an y unbalance o f the bridg e wil l be due solely to mechanica l strai n o n th e active gage. An alternativ e method woul d employ th e dummy gage an d activ e gage in quarter-bridge circuits . The dummy gage would be placed adjacen t t o th e active gag e an d record s kep t o f bot h gag e outputs . Th e therma l outpu t recorded fro m th e dumm y gage woul d b e subtracte d fro m th e activ e gag e strain. This would , of course, double th e require d strai n gag e channels . If th e ful l bridg e i s considered, an d i t i s assumed fo r simplicit y tha t al l resistances ar e identica l strai n gage s o f resistanc e R an d mounte d o n th e

TEMPERATURE EFFECT S O N STRAI N GAGE S 35

5

same material , the n th e circui t outpu t give n by Eq . (5.6 ) is, taking n = 0,

If eac h of the resistanc e change s is compose d of load-induced chang e plu s temperature-induced change , and all gages have undergone the same temperature change, the n Eq . (11.19 ) can b e writte n as

where subscrip t L = load-induced resistanc e change subscript T = temperature-induced resistanc e chang e Since

This shows that we have temperature compensation fo r an initially balanced bridge a s lon g a s adjacan t arm s ar e mad e u p o f strai n gage s o f th e sam e type, bonded t o th e sam e material , an d kep t a t th e sam e temperature . Although the basic idea for temperature compensation is simple enough, nevertheless, like many other aspect s o f strain gage work , attention to detai l is essential if optimum results are to b e achieved. One must always remember that th e observation s fro m what , i n othe r respects , i s a practicall y perfec t test ca n b e made quite valueless by faulty temperatur e compensation. Som e points t o b e kept i n mind ar e th e following : 1. Th e magnitud e o f th e erro r include d i n th e indicate d observatio n depends upo n (a) Change s i n temperatur e betwee n active and dumm y gages. (b) Th e gage s an d materia l upo n whic h the y are mounted . (c) Th e operatin g temperatur e level . 2. Th e piec e o f materia l upo n whic h th e dumm y i s mounte d ma y b e unintentionally subjecte d t o mechanica l strain. 3. Th e therma l connectio n betwee n th e bloc k carryin g th e dumm y gag e and the material upon which the active gage is mounted may not be very good s o tha t a temperatur e differentia l i s set up. 4. Ther e wil l b e a differenc e betwee n gage s o f th e sam e lot , particularl y at th e highe r temperatures . There ar e condition s unde r whic h temperatur e compensatio n ca n b e attained b y havin g th e dumm y gag e pla y a n active , rathe r tha n a passive,

356

THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

FIG. 11.9. Cantileve r beam s wit h strai n gage s aligne d paralle l t o th e longitudina l axi s an d temperature-compensated.

role i n th e measuremen t o f stress-induce d strains , an d a t th e sam e tim e increase th e outpu t signal . On e suc h arrangemen t i s show n i n Fig . 11.9 , which consists o f a thin cantilever beam. Since top and botto m surfac e strains at a given section ar e equal i n magnitude bu t o f opposite sig n when the loa d F i s applied, either a half-bridg e circuit consisting of gages 1 and 2 (or 3 an d 4) i n adjacen t arms , o r a full-bridg e circuit, a s shown , ca n b e used . Thi s requires, o f course, tha t n o therma l gradien t exist s in th e bea m an d tha t al l gages ar e a t th e sam e temperature . Th e outpu t signa l i s either tw o o r fou r times tha t o f a singl e gage , dependin g o n th e circuit , and wil l giv e strain s due onl y t o th e bendin g cause d b y load F . Figure 11.1 0 show s a tensio n membe r wit h fou r gages . Unde r th e requirement tha t all gages ar e at the same temperature , the full-bridge circuit shown wil l b e temperature-compensated , an d th e outpu t signa l wil l b e 2(1 + v ) times the average longitudina l strain. This circui t wil l rea d onl y th e

FIG. 11.10. Tensio n membe r wit h strai n gages .

TEMPERATURE EFFECT S O N STRAI N GAGE S

357

FIG. 11.11. Strai n gage s arranged fo r measurin g torque.

effect o f the axia l load F and wil l cancel bending strains . On th e othe r hand , if only gages 1 and 2 were used in adjacen t arm s o f a half-bridg e circuit, the circuit woul d be temperature-compensated bu t th e relationship betwee n the strains, g j and e 2, would no t b e known unles s the load , F , was a pur e axial load. A thir d exampl e i s illustrate d i n Fig . 11.11 . I f gage s a , b, c, and d ar e arranged i n a full-bridg e circui t wit h gage s a an d b i n adjacen t arm s an d gages c an d d i n arm s opposit e a an d b , respectively , then th e bridg e (al l gages mus t b e a t th e sam e temperature) , wil l b e temperature-compensate d and th e outpu t wil l b e a functio n of the torqu e only. As a final example, an instrument was used in which a full-bridge circuit had t o b e wire d t o externa l bindin g post s tha t wer e arrange d i n a fairl y compact rectangl e o n th e instrument' s to p surfac e an d adjacen t t o th e balancing control . Tw o activ e gage s wer e arrange d i n opposit e arms , an d since testin g took plac e a t roo m temperature , tw o precisio n resistor s wer e placed i n th e othe r tw o arm s a t th e bindin g posts . Whe n balancin g th e instrument, driftin g was observed an d balanc e coul d no t b e obtained . Thi s continued fo r som e tim e unti l th e operato r realize d tha t hi s arm , whe n adjusting th e instrument , wa s directl y above on e o f the precisio n resistors , causing its temperature t o change. Replacing the two precision resistor s with two dumm y gages bonde d t o a piec e o f the tes t materia l an d movin g them away fro m th e instrumen t solve d th e problem .

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E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

Problems 11.1. A strain gage , whos e thermal outpu t curve is shown in Fig . 11.5 , is bonded t o a machin e element made o f mild steel. The strai n indicato r i s balanced a t 75° F using a gag e facto r o f 2.0 , the n th e machin e elemen t i s heate d t o th e tes t temperature of 200°F and a load i s applied. If G F = 2.05 from th e gage packag e data an d th e strai n indicato r shows , afte r loading , a n indicate d strai n o f K', = —246 5 u,in/in, determine the actua l strain, correcting fo r bot h th e therma l output an d th e gag e facto r variation. 11.2. Th e machin e element in Proble m 11. 1 has it s temperature raised t o 250° F an d a ne w loa d i s applied. Afte r a readin g i s taken th e temperatur e i s then raise d to 400° F and th e loadin g i s again changed . For th e followin g data , correc t th e strains a s i n Proble m 11. 1 and determin e th e differenc e i n strai n between th e two temperatures : r=250°F K T = 400°F f,

- = -187 5 uin/i n \ = - 362 8 uln/in

11.3. A strai n gag e o f th e sam e typ e a s use d i n Proble m 11. 1 is bonde d t o a tes t specimen an d th e indicato r i s balanced a t roo m temperature . A gage facto r of 3.0 was inadvertently set on th e strai n indicato r rathe r tha n 2.0 . When th e tes t specimen was brought to its test temperature of 300°F and loaded , the indicated strain wa s e.' t = 193 6 uin/in. Determine th e actua l strain. 11.4. Plo t th e therma l outpu t curve s for th e foil s give n in Tabl e 11.1 . 11.5. A strai n gag e wit h foi l lo t numbe r A12BJ0 1 is t o b e use d o n a magnesiu m member. Determin e th e approximat e therma l outpu t curve , £7-02 , usin g th e thermal outpu t curv e for steel fo r values of erol . 11.6. A strain gag e wit h foi l lo t numbe r A11BP1 1 is to b e use d o n a plasti c whose thermal expansion coefficien t i s 40 ppm/°F. Determine the approximate thermal output curve. 11.7. A rectangula r rosett e ha s a nomina l gag e facto r o f 2.1 2 for al l sections . Th e thermal outpu t curv e associate d wit h thi s rosett e wa s obtaine d o n 2024-T 4 aluminum. Th e rosett e i s bonde d t o a stee l tes t membe r an d a gag e facto r of 2.0 is set o n th e instrument , which is then balance d a t 75°C . The tes t membe r is loade d an d brough t t o a temperatur e o f 300°F . A t thi s temperatur e th e thermal outpu t (fo r aluminum ) i s t. TO = — 950 uin/in an d th e gag e facto r variation i s 1 percent. Gag e a i s aligne d alon g a chose n coordinat e axi s an d all angles ar e measure d fro m thi s axis . The followin g strains were recorded: < = 87 5 uin/in a t 0 = 0° 4 = - 196 0 uin/in at 0 = 45° s'c = - 157 5 uin/in at 9 = 90° (a) Determin e th e principa l strain s fo r the uncorrecte d readings . (b) Determin e th e principa l strain s for th e correcte d readings . 11.8. A rectangula r rosett e ha s a nomina l gag e facto r o f 2.14 5 for al l sections . Th e thermal outpu t curv e associated wit h this rosett e wa s obtaine d o n 101 8 steel . The rosett e i s applied t o a stee l test member , a gag e facto r o f 2.0 is set o n th e strain indicator , an d th e instrumen t i s balance d a t 75°F . Th e tes t membe r i s loaded an d brough t t o a temperatur e o f — 50°F, wher e the therma l outpu t i s — 500 uin/in and th e gag e variatio n facto r is —0. 5 percent . Gage a i s aligned

TEMPERATURE EFFECT S O N STRAI N GAGE S 35

9

along a chosen coordinat e axi s and al l angles are measured fro m thi s axis. The following strain s wer e recorded: s'a= -68 5 nin/i n at 9 = 0° 4= -182 5 uin/i n at 6 = 45° e'c= 133 5 uin/in a t 0 = 90° (a) Determin e th e principa l strain s fo r the uncorrecte d readings . (b) Determin e th e principa l strain s fo r the correcte d readings . 11.9. A delta rosette, bonde d t o aluminu m and havin g a nominal gage factor of 2.08 for al l sections, is loaded t o it s tes t temperatur e of 350°F. A t thi s temperature the therma l outpu t i s S TO = — 90 uin/in an d th e gag e variatio n facto r i s 1. 2 percent. The strain indicato r was initially balanced with G F = 2.0 at 75°F. Gag e a i s aligne d alon g a chose n referenc e axi s an d al l angle s ar e measure d fro m this axis . The followin g strain s were recorded : £'„= -53 5 uin/i n at 0 = 0° 4 = -84 5 uin/i n at 9 = 120° s'c= 18 0 uin/in at 0 = 240 ° (a) Determin e th e principa l strain s fo r the uncorrecte d readings . (b) Determin e the principa l strains for th e correcte d readings . REFERENCES 1. Murray , Willia m M . an d Pete r K . Stein , Strain Gage Techniques, Lectures an d laboratory exercises presented at MIT, Cambridge, MA: July 8-19,1963, pp. 95-96. 2. "Temperature-Induce d Apparen t Strai n an d Gag e Facto r Variatio n i n Strai n Gages," TN-504, Measurement s Group, Inc., P.O. 27777 , Raleigh, NC 27611,1983. 3. "Catalo g 500: Part B—Strai n Gage Technica l Data, " Measurements Group, Inc., P.O. Bo x 27777, Raleigh, NC 27611 , 1988 . 4. "SR- 4 Strain Gage Handbook," BLH Electronics, Inc., 75 Shawmut Road, Canton, MA 02021 , 1980 . 5. 798 6 Annual Book a t ASTM Standards, 191 6 Race St. , Philadelphia , P A 19103 , "Performance Characteristic s o f Bonde d Resistanc e Strai n Gages, " Vol . 03.01 , Designation: E251-86 , pp. 413-428. Copyright ASTM. Reprinted with permission.

12 TRANSDUCERS 12.1.

Introduction

When on e o r mor e strai n gage s ar e use d t o measur e som e quantit y whos e magnitude ca n b e determine d b y th e indicatio n o f strai n o n som e load bearing member, the whol e unit i s frequently describe d a s a transducer . Th e load-bearing membe r ma y hav e one , two , three , o r mor e strai n gage s mounted o n it , dependin g o n th e quantit y t o b e measured , th e precisio n desired, an d th e influenc e o f extraneou s effects , som e o f whic h ca n b e eliminated o r reduce d t o negligibl e proportions . In general , th e load-bearin g element s fo r transducer s ma y b e divide d into a fe w categories whic h depend upo n wha t i s to b e measured , a s wel l a s being dependent o n spac e requirements . These includ e direct stres s (tensio n or compression ) fo r th e measuremen t o f larg e forces , member s i n bendin g for determinin g mediu m o r smal l forces , th e indicatio n o f torsion , th e measurement o f fluid pressure, etc. Ther e ar e man y variation s o f apparatu s to accomplis h thes e end s an d considerabl e overlappin g o f th e differen t procedures, an d som e o f th e device s tha t hav e bee n develope d fo r specia l conditions ar e exceptionall y ingenious. A revie w o f some o f the mor e usua l types o f transducer i s presented i n thi s chapter . Let u s no w examin e a simpl e case involvin g fou r strai n gage s (on e fo r each ar m o f th e Wheatston e bridge ) a s indicate d i n Fig . 12.1 . Her e /?! = R 2 = R 3 = R 4 = R a and , i f idealized bridg e condition s ar e assume d and th e bridge is initially balanced , E = 0 and /^ ^= R 2R4- Fo r thi s case , then, th e bridg e rati o is

From Eq . (5.39) , th e bridg e output , A£ 0, ca n b e writte n as,

Since th e uni t change s i n resistanc e wil l b e smal l compare d t o unity , th e

TRANSDUCERS

361

FIG. 12.1. Wheatston e bridge .

nonlinearity ter m i s determined b y usin g Eq . (5.42) . Thus ,

Equation (12.1) , the general expressio n fo r the output fro m th e initially balanced Wheatston e bridge , tell s on e tha t th e bridg e outpu t i s directl y proportional t o th e applie d voltage , V , an d fo r smal l uni t change s i n resistance i s nearly proportional t o th e following : 1. Th e algebrai c differenc e betwee n th e uni t change s o f resistanc e i n adjacent arm s o f the bridge . 2. Th e algebrai c su m o f th e uni t resistanc e change s i n opposit e arm s of th e bridge . In additio n 3. I f tw o o r mor e gage s happe n t o b e connecte d i n serie s i n on e ar m o f the bridge , th e averag e valu e o f th e correspondin g strain s wil l b e reflected i n th e bridg e output . This mean s tha t i f the gage s ar e appropriatel y located , a bridg e outpu t wil l be produced representin g th e addition, subtraction, or th e average o f strains at certai n particula r locations . The full bridge Since Eq . (12.1 ) fo r A£ 0, th e outpu t o f a n initiall y balance d Wheatston e bridge, contains a n equa l numbe r o f terms wit h positiv e and negativ e signs, this suggests that if one were designing a transducer fo r full-bridge operation ,

362 TH

E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E

strain gage s wit h positiv e and negativ e gage factors might be considere d i n order t o achiev e th e maximu m output , o r indication , pe r uni t load . A s it may no t b e possible, or desirable , to us e gages wit h gag e factor s of opposite sign, it i s fortunate that th e sam e result can b e achieve d b y usin g lik e gage s and mountin g them alternately in regions o f tension an d compressio n o f the load-carrying elemen t o f th e transducer . This i s commo n practice , whic h works bes t whe n th e strain s i n tensio n an d compressio n ar e o f equa l magnitude. Equation (12.1 ) also tell s u s tha t i f all gage s ar e alik e an d th e gage s i n adjacent arm s o f th e bridg e ar e subjecte d t o strain s o f opposit e sign , th e indication A£ 0 wil l b e large r tha n tha t fro m a hal f bridge . I n th e sam e manner, i f the strain s o n th e gage s ar e o f the sam e sign , th e bridg e outpu t will b e les s tha n tha t fro m a hal f bridg e whos e gage s i n opposit e arm s ar e subjected t o the two largest strains. In the worst case, there may be no bridg e output al l (AE 0 = 0). The half bridge There are certain situations in which i t will be more convenient to us e a hal f bridge instea d o f a ful l bridge . I n thi s case tw o activ e gage s ar e employe d instead o f four. Th e tw o gages represente d b y R 3 an d R 4 i n Fig . 12. 1 can b e replaced b y any tw o equal fixed resistors (for initial bridge balance) , o r they may b e left out . In tha t even t one must b e sure that th e applie d voltag e doe s not sen d a curren t throug h th e gage s i n exces s o f th e norma l carryin g capacity, which is usuall y about 3 0 milliamperes. With tw o fixe d resistor s fo r R 3 an d R 4, AR 3 = A.R 4 = 0 . I n thi s cas e Eq. (12.1 ) reduces t o

Equation (12.3 ) is the same a s Eq . (4.26) for the potentiometri c circui t when o = l . Th e nonlinearit y factor i s given by

The hal f bridg e i s particularl y usefu l fo r bendin g member s wit h a symmetrical cros s sectio n i n whic h th e tensil e an d compressiv e strain s o n opposite surface s ar e o f equa l magnitude . Fo r thi s cas e n = 0 an d th e half-bridge outpu t become s

TRANSDUCERS 36

3

The quarter bridge When on e want s t o measur e strai n a t a singl e point , o r i n rar e cases , t o produce a transduce r wit h a singl e activ e gage , th e singl e activ e gage an d three fixe d resistor s ca n b e use d i n th e Wheatston e bridge . I n orde r t o accomplish bridg e balanc e a t zer o load , on e o f the thre e resistor s mus t b e equal to that o f the gage. The other two, then, can have any resistance values, but the y must b e equa l t o eac h othe r i f a four-ar m bridg e is being used . A convenien t way to provid e the thre e fixed resistors, although not th e only one , i s t o moun t thre e strai n gages , identica l t o th e activ e one, o n a piece o f materia l simila r t o tha t upo n whic h th e activ e gag e ha s bee n mounted. This arrangemen t give s an equal-ar m Wheatston e bridg e suitable for bot h static and dynami c measurements. If the material carrying the three inactive, o r dummy , gage s ha s th e sam e therma l characteristic s a s th e material carrying the active gage, the system will be temperature-compensated . If dynami c measurement s onl y ar e t o b e mad e an d temperatur e compensation i s of no concern , i t woul d b e preferabl e to us e th e potentio metric circuit and chang e the ballast rati o fro m 1 to about 10 . This increase s the circui t efficienc y fro m 5 0 percent t o abou t 9 0 percent. 12.2. Axial-force transducers Tension-compression load cell This typ e o f transducer, generally called a load cell , is one o f the earlies t t o be used . B y proper en d connection s eithe r tensil e o r compressiv e loads , o r both, ma y b e measured . Th e centra l sectio n wher e th e strai n gage s ar e bonded i s made long enough s o that the strains at th e gage location are no t affected b y th e en d conditions . Thi s sectio n i s designe d s o tha t maximu m possible strains are reached, yet the member remains within the elastic regio n and wel l belo w the yiel d point o f the materia l i n orde r t o reduc e hysteresis. The cross section of the load cell at the strain gag e location can have differen t geometries, with cylindrical, square, or tubular cross sections being common. Figure 12.2 shows a cylindrical load cel l for both tensile and compressiv e loads. Fou r gage s ar e shown ; gages 1 and 3 are 180 ° apar t an d aligne d i n the longitudina l direction , whil e gage s 2 and 4 ar e 180 ° apar t an d aligne d in th e transvers e direction . Th e gage s ar e arrange d int o a ful l bridg e a s illustrated, wit h E J = e 3 = E and e 2 = e 4 = — vs. Thi s bridg e arrangemen t cancels bendin g strain s an d i s temperature-compensate d a s lon g a s n o temperature gradient s exis t in th e member . Instead o f using Eqs. (12.1) and (12.2 ) in determining the bridge output, AE0, an d th e nonlinearit y factor , ( 1 — n), w e wil l retur n t o Eqs . (5.39 ) an d (5.40) for each case. The bridge output, A£0, for this bridge arrangement is

364

THE BONDE D ELECTRICA L RESISTANC E STRAIN GAG E

FIG. 12.2 . Loa d cel l fo r tensil e an d compressiv e loads . The nonlinearit y factor , ( 1 — n), i s

These result s correspon d t o th e result s obtaine d i n Exampl e 5.2 , where this bridge arrangemen t wa s examined . Equation (12.6 ) shows tha t th e bridg e give s an outpu t o f 2(1 + v ) times as grea t a s tha t o f a singl e longitudina l gage , considerin g th e nonlinearit y factor t o b e unity . Substituting the nonlinearit y factor, ( 1 — n), given b y Eq . (12.7) int o Eq . (12.6) , th e resultin g bridg e outpu t i s

The indicate d strain , £ t, in term s o f th e actua l strain , E , is

TRANSDUCERS 36

5

Solving Eq . (12.9 ) for e gives

The ratio , £/e ;, is

When usin g thes e equations , not e tha t th e strain s mus t b e entere d a s £ x 10~ 6 in/in. These equations sho w that the bridge output is nonlinear, since all gages do no t se e th e sam e strai n magnitudes . Th e transvers e strains , becaus e of the Poisson effect, ar e about 30 percent of the longitudinal strains. Equation (12.10) show s tha t th e actua l strai n wil l b e larger tha n th e indicate d strai n for a tensil e force , whil e th e convers e i s tru e fo r a compressiv e force . Th e bridge nonlinearit y a t a strai n leve l o f 100 0 uin/in , fo r eithe r tensio n o r compression strains , i s about 0.0 7 percent . Another nonlinearit y facto r i s present i n th e geometr y o f the loa d cel l in tha t th e area changes unde r load . Thi s can b e approximated fo r a roun d cross section by considering the change in diameter due to the Poisson effect . The diameter , d , at an y loa d withi n the elasti c region , is

where d 0 is the diameter a t no load. The bridge an d geometr y nonlinearities, however, ar e offsetting , wit h th e bridg e nonlinearit y being th e highe r o f the two. It may be desirable, a t times , to hav e only gages 1 and 3 active. In tha t case, precisio n resistor s coul d b e use d i n arm s 2 and 4 to complet e th e ful l bridge; however, a more convenien t wa y would be to bon d tw o strain gages to materia l simila r t o th e transduce r an d us e the m a s dumm y gages . Thi s would giv e temperatur e compensatio n provide d al l gage s remaine d a t th e same temperature. The bridg e output, A£ 0, woul d be double tha t o f a single gage an d woul d be nonlinear . Ring-type load cell The provin g ring ha s bee n i n use for years as a standar d fo r the calibratio n of tensile-testing machines . Th e diametra l deflectio n of the rin g is a measure of the applie d load, where the deflection is measured by means of a precisio n micrometer. Th e thicknes s o f th e cros s section , whic h i s th e differenc e between th e inne r an d oute r radii , i s small compare d t o th e mea n radius . Rather than measur e diametra l deflection , strain gages may b e bonde d

366 TH

E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E

FIG. 12.3. Ring-typ e loa d cell .

to th e ring , as shown in Fig. 12.3 , arranged int o a ful l bridge , and th e bridg e output use d a s a measur e o f the applie d load . A n axia l forc e an d bendin g moment ac t a t th e sectio n containin g th e strai n gages , a s illustrate d in th e free-body diagra m o f th e uppe r hal f o f th e loa d cell . Sinc e eac h gag e i s subjected t o the same axial strain du e to the axial force, F/2, thes e strains are canceled an d th e bridge responds onl y to the strains induced b y the bendin g moment, M 0. Fo r th e tensil e load, F , gage s 1 and 3 wil l b e i n compressio n due t o bending , an d gage s 2 an d 4 wil l b e i n tension . Th e convers e wil l b e true for a compressive load. Furthermore, thi s bridge arrangemen t give s ful l temperature compensation . The strai n at th e gages may b e estimated from th e bendin g stresses. The moment, M 0, is

Since th e cros s sectio n i s rectangular, the strai n du e t o M 0 i s

TRANSDUCERS

367

FIG. 12.4. Curve d beam cross section.

This reduce s t o

where h « R . If th e differenc e betwee n th e inne r an d oute r radi i increases , th e loa d cell wil l n o longe r b e considere d a thi n ring . I n thi s case , a bette r estimat e of the bending strain s du e to M 0 ma y be obtained fro m curve d beam theor y (1). In the curved beam, the centroidal axis and neutra l axis do not coincide , with th e neutra l axi s shifte d inwar d toward s th e cente r o f curvature . Th e geometry, show n in Fig . 12.4 , ha s th e followin g notation : Rj = radius o f inner fiber Rn = radius o f neutral axi s R = radiu s of centroidal axi s R0 = radius o f outer fiber e = distance betwee n the centroida l an d neutra l axe s h = section thickness , R 0 — Rt w = section widt h yt = distanc e fro m th e neutra l axi s to th e inner fiber y0 = distanc e fro m th e neutral axi s to th e oute r fiber The bending moment , M 0, now become s

368 TH

E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E

Since e = R — Rn, th e radius , /?„, must be computed . It i s

The stresse s a t th e inne r an d oute r fibers are

where A i s the cross-sectional area. For th e sens e of M 0 show n i n Fig . 12.3 , o-; will be a tensile stress and a 0 wil l be a compressive stress. The correspond ing strain s ar e

For a tensile force, F , acting o n th e loa d cell , we see that £ t = e 3 = — £0 and s2 = e 4 = E ; and s o th e bridg e outpu t i s nonlinear. Fo r th e developmen t of these equations , th e reade r i s referred to Referenc e 1 . The expressions for the strains at th e gag e locations ar e estimates, since the bosses where the load i s applied have a stiffening effect . The y can b e used for design , but calibratio n i s essential. 12.3. Simple cantilever beam A devic e ofte n use d a s a transduce r i s th e cantileve r beam . Amon g it s applications, i t ma y b e use d t o measur e force , t o serv e as a comparator , o r to determine deflections in areas not readil y accessible to othe r instruments. Single active gage The most basi c application uses a single active gage and thre e fixed resistors in th e Wheatston e bridge . Th e bea m ca n b e mad e fro m a piec e o f unifor m bar stoc k o f rectangula r cros s section , wit h th e strai n gag e mounte d nea r the fixed end o n th e longitudina l center lin e of the uppe r surface . Th e force , F, ca n the n b e measure d afte r a suitabl e calibration o f the bea m ha s bee n performed. Thi s devic e is subject t o th e followin g limitations:

TRANSDUCERS 36

9

1. Th e outpu t will be low because only one ar m o f the Wheatston e bridg e is active. 2. Th e lin e of action o f the applie d force , F , must alway s remain paralle l to itsel f (includin g calibration ) an d a t th e sam e distance , L , fro m th e center o f the gage . 3. Unles s a self-temperature-compensated strain gage is used to match the thermal properties o f the beam, the apparatus can only be used precisely at th e temperatur e o f calibration, otherwis e seriou s error s ma y occur . 4. N o compensatio n i s provide d fo r force s (i f any) , othe r tha n F , which ma y produc e latera l bending , torsion, o r direc t axia l thrust. The bridg e outpu t i s given as

The nonlinearit y facto r is

The indicate d strain , e t, in term s o f the actua l strain , £ , is

Solving Eq . (12.24 ) for e produce s

The rati o o f the actua l strai n t o th e indicate d strai n is

Longitudinal and transverse gages on the same side There may be some cases for which two gages mus t be mounted on the sam e side o f the beam . Her e on e ca n tak e advantag e o f the Poisso n effect , whic h produces a latera l strai n o f opposit e sig n fro m th e axia l strain . Thi s arrangement use s one gage mounted in a longitudinal direction an d a secon d

370 TH

E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

gage bonde d i n the transvers e direction. Thi s arrangemen t ca n b e used a s a half bridg e whe n th e gage s ar e connecte d i n adjacen t arms . Wit h th e tw o gages a t th e sam e temperature , th e bridg e outpu t wil l b e automaticall y temperature-compensated. However , al l othe r characteristic s o f th e singl e gage applicatio n als o appl y t o this . Depending upo n th e valu e o f Poisson's ratio for the ba r material , the output for this bridge wil l be about 3 0 percent greater tha n tha t o f a singl e gage. For thi s bridge, the longitudinal strain is £a = s and th e transverse strain is £„ = — ve. With the gage s placed i n bridge arms 1 and 2 , the uni t resistance changes ar e

From Eq . (7.21),

Using th e value s of R l/R1, R the bridg e output , E 0, i s

2/R2,

an d F a from Eqs . (a), (b), and (7.21) ,

The nonlinearit y factor, (1 — n), can b e writte n as

TRANSDUCERS 37

1

If K i s ignored ( K = 0), then A£ 0 an d ( 1 — ri) revert t o

Two longitudinal gages on opposite surfaces Provided tha t th e tw o side s o f th e bea m ar e fre e fro m an y obstruction , a considerable advantag e i n outpu t ca n b e obtaine d b y mountin g th e gage s back t o bac k o n opposit e surfaces . Becaus e the y are subjecte d t o strain s of equal magnitude but o f opposite sign , the y ca n b e placed i n adjacen t arm s of a hal f bridge . Sinc e s i = s an d & 2= — e, the bridg e output , A£ 0, wil l b e linear an d i s

This arrangemen t i s compensate d fo r temperatur e change s provide d both gage s ar e maintaine d a t lik e temperatures . I t i s also compensate d fo r direct axia l thrust , which wil l produc e th e sam e resistanc e changes i n bot h gages, althoug h axia l thrus t wil l produc e bridg e nonlinearity . For metalli c gages, the variatio n o f this nature in ( 1 — n) will usuall y be smal l enoug h t o be neglected . Full bridge, two gages back to back on opposite surfaces Four gage s mounted back t o bac k i n pairs o n opposit e surface s of the bea m and arrange d a s a ful l bridg e wil l giv e th e larges t bridg e output . Sinc e e = e i 3 =£ an d E 2 =£ 4 = ~ £> the bridge output , A£ 0, wil l b e linear an d i s

Provided al l gage s ar e maintaine d a t th e sam e temperature , thi s arrangement gives temperature compensation. A s in the two-arm bridge, the strains cause d b y axia l thrus t wil l b e nullified , althoug h thes e strain s wil l produce bridg e nonlinearity . There ar e severa l comments i n orde r concernin g th e cantileve r beam . 1. I f the line of action o f the force remains parallel t o itself , th e momen t a t the gage section decreases becaus e o f the shortening of the moment ar m due t o th e curvatur e of the beam .

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THE BONDE D ELECTRICA L RESISTANCE STRAI N GAG E

2. Th e strai n alon g th e lengt h o f the strai n gag e i s not constant . Thi s ca n be alleviated b y designin g a constant-stres s bea m o f uniform thickness and a triangularly shaped width , or unifor m widt h and a parabolicall y shaped thickness . Th e loa d i n eac h cas e i s applie d a t th e narrowes t point o f the beam . Fo r a tapere d widt h beam , se e Fig. 7.5 . 12.4.

Bending beam load cells

A variet y of load cell s ca n b e constructe d b y usin g different configuration s of beams. Whethe r or no t al l make satisfactory load-measuring devices must be determine d b y a combinatio n o f analysi s and testing . Severa l differen t types, amon g th e man y available , wil l b e discusse d here . Fixed-end beam One may consider a beam wit h fixed ends an d cente r loading , show n in Fig . 12.5, for use as a load cell . One placemen t o f the strai n gage s an d th e bridg e arrangement ar e shown , with gage s 1 and 3 being in compression an d gage s 2 an d 4 bein g i n tensio n fo r th e loadin g illustrated . Becaus e th e bea m i s symmetrical, th e reaction s a t eac h built-i n end ar e identical . Furthermore , the support s ar e ver y stif f compare d t o th e beam . Expression s fo r th e moments an d reactiv e forces a t th e fixed ends ma y b e developed o r foun d in a tex t o n mechanic s o f materials. Since th e end s o f the bea m ar e constraine d fro m movin g laterally, this influence wil l no t b e accounte d fo r i n th e expression s fo r th e en d reactions . Because of this constraint, a horizontal force is produced tha t affect s bendin g

FIG. 12.5. Bea m with fixe d ends.

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moments i n th e bea m an d therefor e the deflectio n curve . While th e strain s produced b y this force ar e canceled by the bridge arrangement, nevertheless, they wil l produc e nonlinearit y in th e bridg e output . Two fixed-end beams A beam-typ e loa d cel l tha t overcome s th e lac k o f latera l movemen t o f th e fixed end s i s show n i n Fig . 12.6 . Durin g loading , eithe r i n tensio n o r compression, the ends are free t o move laterally and thus eliminate horizontal forces o n th e beams . Th e centra l sectio n wher e the loa d i s applied an d th e two en d support s ar e ver y stif f compare d t o th e thinne d bea m sections , an d so practically all of the deflectio n is produced i n the thi n sections . Thi s loa d cell, however, has twice the deflection of the single beam shown in Fig. 12.5.

FIG. 12.6 . Dual-bea m load cell.

One arrangemen t o f four strai n gage s for a ful l bridg e i s shown. Gages 1 and 4 are subjected to strain s of like sign, and gage s 2 and 3 are subjecte d to strain s o f like sign. If the loa d i s compressive, for instance , gages 1 and 4 will hav e tensile strain s an d gage s 2 and 3 will hav e compressiv e strains. When designin g this load cell , one wants to estimat e th e strai n leve l at the gag e locations . I n orde r t o accomplis h this , a free-bod y diagram o f the upper bea m i s show n i n Fig . 12.7 . Section s A- B an d C- D hav e th e sam e moments o f inertia, whil e section B-C ha s a much larger momen t o f inertia in orde r t o reduc e th e deflectio n i n thi s section. Althoug h section B- C wil l deflect slightl y (dependen t o n th e valu e o f its momen t o f inertia compare d to section A-B), mos t o f the deflection wil l occur in sections A-B an d C-D. Since the beam i s symmetrical, the reactions a t both end s are equal; however, the beam i s statically indeterminate t o th e first degree, since M0 is unknown. Knowing th e slop e o f the deflectio n curv e i s horizontal a t poin t A an d a t the center under th e load, M 0 in terms of the beam dimension s and th e load, F, can b e computed. Th e moment-are a method , fo r instance, ma y easil y be used. If th e gage s ca n b e located s o th e strain s a t gage s 1 and 4 are equa l i n

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THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

FIG. 12.7 . Free-bod y diagram o f the uppe r beam.

magnitude but opposit e in sign to gages 2 and 3 , the bridge output, A£0, wil l be linear . Becaus e o f th e latera l movemen t o f th e ends , though , ther e ma y be som e nonlinearit y effec t becaus e o f th e sligh t chang e i n geometry . Also, if al l gage s ar e subjecte d t o th e sam e temperature , th e bridg e wil l b e temperature-compensated. S-shaped dual beam The S-shaped, dual beam load cell uses two beams attached t o sections whose stiffness i s muc h large r tha n tha t o f th e beams . I t i s use d fo r direc t tensil e or compressiv e loads , a s show n i n Fig . 12.8 . For bes t results , th e loa d cel l should b e machined fro m a solid bloc k o f material. Eccentri c loading error s are minimize d an d th e gage s ar e easil y protected . Figure 12. 9 shows the loadin g (fo r a tensil e force) acting o n on e o f th e beams, along wit h the deflection curve, for estimating the strains in the beam .

FIG. 12.8. S-shaped , dual-bea m load cell .

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FIG. 12.9. Fixed-en d beam wit h en d displacement.

From symmetry, the reactions a t each en d of the beam ar e identical. Durin g loading, th e right-han d sid e o f th e beam , relativ e t o th e left , move s u p through a distance , y , thereb y producin g th e reaction s shown . Th e shea r force, R 0, an d th e movement , M 0, ca n b e determine d i n term s o f th e deflection, y , an d th e bea m dimensions . Onc e again , th e moment-are a method lend s itsel f to th e determinatio n o f the reactions . Th e value s of R 0 and M 0 i n term s o f the bea m deflectio n and bea m dimension s ar e

For th e gage placements shown , gages 1 and 3 are in tension an d gages 2 and 4 are i n compression fo r a tensil e load, whil e the convers e applies fo r a compressiv e load . Furthermore , as lon g as all gages are subjecte d to the same temperature , th e bridg e wil l b e temperature-compensated . Th e non linearity of the bridge will depend on the values of the strains at each gage. There are a number of other beam-bending load cells in use or that could be constructe d fo r laborator y use . A n examinatio n o f a manufacturer' s catalog will show beam-bending loa d cells are used for applications involvin g loads fro m les s tha n 1-l b a t th e lo w end t o abou t 100 0 Ib at th e uppe r end . For load s i n exces s of 100 0 Ib, other design s ar e generall y utilized . Fo r a n excellent discussio n o f strain-gage-based transducers , se e Referenc e 2 . 12.5.

Shear beam load cell

The shea r bea m loa d cell , usually designed fo r hig h loads , i s in th e for m o f a cantilever beam wit h a cross sectio n larg e enough tha t th e beam deflection

376

THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

FIG. 12.10. Shea r bea m loa d cell .

is kep t t o a minimum . Sinc e th e bendin g stresse s o n th e oute r surfac e would b e quit e lo w unde r thes e conditions , a reces s i s machine d o n eac h side s o th e cros s sectio n forme d resemble s a n I-beam . Here , mos t o f th e bending moment i s resisted b y the flanges , whil e the vertica l shear i s carried by th e web . Th e shearin g stres s i s maximu m a t th e neutra l axis , an d s o the dimension s o f th e I-bea m sectio n ca n b e chose n s o tha t th e strain s will produc e a desire d bridg e output . Suc h a loa d cel l i s show n i n Fig . 12.10. Because ther e i s pur e shea r a t th e neutra l axis , th e principa l stresses , and therefor e th e principa l strains , are a t ±45 ° fro m th e neutra l axis . Two pairs o f strain gages, bonde d bac k t o bac k on opposit e surface s o f the recess, can b e centere d acros s th e neutra l axi s a t +45° . Althoug h th e gage s ar e subjected t o a slight amount o f bending strain becaus e the y extend o n either side o f th e neutra l axis , thi s effec t tend s t o b e self-canceling . A bette r arrangement, for instance, would use a two-elemen t 90° gage, generally used for torqu e measurements , o n eac h surface . Choosin g a torqu e gag e wit h electrically independen t element s whos e grid s ar e +45 ° t o th e gag e longi tudinal axi s allow s th e gag e t o b e bonde d s o tha t it s longitudina l axi s coincides wit h the neutra l axi s of the beam . I n thi s manner, th e element s of the gag e wil l experienc e bendin g strain s o f th e sam e magnitud e bu t o f opposite sign . Whe n th e gage s ar e arrange d int o a ful l bridge , th e bendin g strains wil l cancel . Furthermore , thi s arrangemen t wil l als o cance l an y bending du e t o sid e loading . As lon g a s th e loa d i s t o th e righ t sid e o f th e recess , a s show n i n Fig . 12.10, th e bridg e outpu t i s relativel y insensitiv e t o th e poin t o f loa d application. Althoug h i t i s desirable to kee p th e bea m a s shor t a s possible ,

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FIG. 12.11. Folde d shea r beam.

the load mus t be far enough fro m th e reces s tha t its localized effect s wil l no t influence th e bridg e output . The shearing stres s at th e neutral axis, and thereb y th e shearing strain , must b e determined i n order t o estimat e the bridge output , A£ 0, fo r a given load. Th e equatio n fo r shea r stres s in th e web , which can b e found in texts on mechanic s o f materials, is

where V

= vertical shea r forc e o n th e sectio n / = momen t o f inertia abou t th e neutral axis t = web thickness Q = first moment o f the are a abov e th e neutra l axis

The principa l stresses , a t 45 ° on eithe r sid e o f the neutra l axis , are equa l i n magnitude bu t opposit e i n sign , giving a 1 = — cr2 = T . The principa l strain s are

The shea r bea m loa d cel l may als o b e constructe d s o that it s profil e i s S-shaped, as shown in Fig. 12.11 . Thi s configuratio n is also referre d to a s a folded shear bea m b y some manufacturers . The lin e of action o f the applie d force goe s throug h th e cente r o f th e strai n gag e bridge , thu s eliminatin g bending a t tha t section .

378

THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

12.6. The torque meter Although many different type s of torque meters have been devised , probabl y the mos t commo n consist s o f a shaf t o f circular cross sectio n wit h fou r lik e strain gage s mounte d a t 45 ° to th e axi s of the shaft . Car e mus t b e take n i n assuring th e gages ar e mounte d a t precisel y 45°, and tha t companio n gage s subjected t o tensio n (o r compression ) ar e bonde d exactl y opposit e eac h other. A typica l torque meter i s shown i n Fig . 12.12 . In constructing a torque meter, one should b e aware of its characteristics , which ar e state d a s follows : 1. Th e uni t is automatically compensated fo r changes i n temperature. Thi s is due to th e fact tha t a uniform temperatur e change will produce equa l resistance change s i n all four arm s o f the bridge , thereb y producing n o change i n th e conditio n o f balance. 2. Theoretically , th e instrumen t wil l no t respon d t o th e effect s o f axial thrust, i f suc h shoul d exist . Thi s i s becaus e axia l thrus t wil l produc e equal resistanc e change s i n al l fou r arm s o f th e Wheatston e bridge ; therefore, ther e wil l b e n o chang e i n th e conditio n o f balance . 3. Ther e wil l be no respons e t o bending , if such shoul d occur , because th e resistance change in the two front gage s wil l be equal i n magnitude bu t opposite in sign to the resistance changes in the two gages at the back. 4. Th e outpu t o f th e bridg e wil l b e linea r wit h respec t t o th e torque , T , because th e nonlinearit y factor i s ( 1 — n) = 1 ; that is , n = 0 . Due t o th e location o f the gages , torque produce s resistance changes in

FIG. 12.12. Torqu e meter .

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9

each bridge arm. Since the gages are alike , th e bridge rati o i s a = 1 , and th e resistance changes ar e

Using these resistance changes, the bridge output, A£0, given by Eq. (12.1) is

This reduces t

Also, substitutin g thes e resistanc e change s int o Eq . (12.2 ) show s tha t th e nonlinearity term , n, is zero, givin g a linear bridg e output . The cas e ca n b e examined where there i s not onl y torque bu t a n axia l load actin g o n th e meter . Unde r thes e conditions , th e resistanc e chang e i n each gag e i s

The subscript s T an d A refe r t o torqu e an d axia l thrust , respectively . The bridge output no w becomes

Simplifying, thi s reduces t o

Equation (12.38) shows that the bridge output does not chang e because of th e axia l load , providin g th e nonlinearlit y factor, ( 1 — n), i s unity . Th e nonlinearity factor, however, must be examined to se e if it affect s th e bridg e

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E BONDE D ELECTRICA L RESISTANCE STRAIN GAGE

output. Rathe r tha n us e the approximat e expressio n for the nonlinea r ter m given b y Eq . (12.2) , Eq. (5.40 ) wil l b e used . I f th e resistanc e change s give n by Eqs . (a) , (b) , (c) , an d (d ) ar e substitute d into Eq . (5.40) , it wil l b e found the nonlinearit y factor reduce s t o

Equation (12.39 ) show s tha t th e nonlinearit y term wil l hav e a n effect , although minor , whe n a n axia l force is combined wit h torsion . Thi s mean s that th e nonlinearit y ter m wil l hav e a differen t valu e fo r eac h differen t combination o f axial thrust an d torque . If th e torqu e mete r i s use d i n a stationar y application , th e lea d wire s from th e strain gage bridge may b e readily connected t o a suitable indicator. For limite d angular motio n a t a low rate o f rotation, th e lea d wire s may b e of suc h a lengt h tha t windu p i s permitted . If , however , the torqu e mete r rotates, the n som e arrangemen t mus t b e mad e t o brin g th e signa l t o th e instrumentation, eithe r throug h sli p rings , radiotelemetry , o r som e othe r method. 12.7.

The strain gage torque wrench

Mechanical torqu e wrenche s have been in use for many years, with the mos t common havin g a pointe r attache d t o th e hea d en d an d extendin g ove r a scale, calibrate d t o rea d torque , attache d a t th e handl e end . Th e handl e i s pin-connected t o th e wrenc h bod y s o tha t th e forc e i s transmitte d t o th e body through the pin, thereby keeping the force at a fixed point. Rather than use a pointe r an d scale , strai n gage s coul d b e bonde d t o th e wrenc h bod y near th e hea d end , th e syste m calibrated, and th e torqu e rea d o n a suitabl e strain indicator . Th e force , however , woul d stil l hav e t o b e applie d a t a fixed point . Ca n one , then , arrang e strai n gage s s o tha t th e indicato r reading i s a measur e o f th e torqu e an d independen t o f th e poin t o f forc e application? Meier (3) investigated this problem and arrive d at a bridge arrangement so tha t th e bridg e outpu t wa s linearl y relate d t o th e torqu e a t th e wrench-head cente r line , yet wa s independen t o f the poin t o f force applica tion. Figur e 12.1 3 shows th e wrench , th e strai n gag e placement , an d th e bending momen t diagram . The bending moment is maximum at section 3 where the torque is being applied. Since it is impractical to measure bending at this section, the bending moment, M 3, can be related t o the bending moments, M 1 an d M 2, at section s 1 an d 2 , respectively. Any force s an d moment s applie d t o th e wrenc h must be t o th e lef t o f section 1 , with non e applie d betwee n sections 1 and 2 . Th e

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FIG. 12.13. Gag e arrangement on torque wrench for direct torque measurement. (From ref. 3.)

moments o f the thre e sections ar e

If Eqs . (b ) an d (c ) are solve d for th e force , F , then

From Eq . (d) , M 3 is

Taking L 2 = 2L 1(

Since th e bendin g moments , M i an d M 2, ca n b e expresse d i n term s of

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E BONDE D ELECTRICA L RESISTANC E STRAIN GAGE

strain,

where Z i s the section modulus for bending and E is the modulus of elasticity. Substituting th e value s of Mj an d M 2, given by Eqs . (e ) and (f) , respectively, into Eq . (12.41 ) produce s

The eight strain gages bonded t o th e wrench can b e arranged int o a ful l bridge t o produc e th e operation indicate d i n parentheses i n Eq. (12.42). For the bendin g momen t diagra m shown , gage s a , b , an d c wil l experienc e a compressive strai n o f — e2, whil e gage d wil l hav e a compressiv e strai n o f — Ej. Gage s e , f , an d g wil l hav e tensile strains o f e 2, whil e gage h wil l hav e a tensil e strain o f e l. Th e bridg e output, A£ 0, fo r th e give n strain s is

Comparing Eqs . (12.42 ) and (12.43) , it ca n b e see n tha t th e bridg e outpu t i s proportional t o th e torque , M 3. The torque wrenc h can easily be calibrated usin g known weights. Meier found th e calibratio n curv e o f indicator readin g agains t th e torque , M 3, t o be ver y consisten t an d straigh t ove r a wid e rang e o f level arms an d applie d weights. Whil e th e uni t ha s bee n describe d a s a torqu e wrench , i t ca n b e applied t o othe r situation s requirin g a torqu e arm . On e application , fo r instance, would be the determination of reaction torqu e for a cradle-mounted piece of equipment, using the device described a s the arm fo r the measurement of torqu e abou t th e cradl e axis . 12.8.

Pressure measurement

The measurement of pressure is often require d during the course of a project . There ar e man y device s available usin g pressur e forc e t o ac t o n a n elasti c mechanical element, thereby causing i t to deflect . Among thes e elements ar e the Bourdon tube with different configurations , diaphragms, bellows, straight tubes, and flattene d tubes . These elements are use d in conjunction with some sort o f measuring system , so thei r deflection is a n indicatio n o f pressure. I f the pressure-measurin g devic e i s t o b e constructe d rathe r tha n purchased , there ar e severa l options , dependin g o n projec t requirements .

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FIG. 12.14. Thin-walle d pressur e transducer.

Thin-walled cylindrical tube For stati c o r slowl y varying pressures, a simpl e an d effectiv e metho d i s t o construct a thin-walle d cylindrica l tube , wit h tw o gage s mounte d i n th e circumferential (hoop ) direction . A ful l bridg e ca n b e use d b y placin g these two activ e gage s i n opposit e bridg e arms , the n completin g th e bridg e b y bonding two dummy gages o n an unstrained piec e of similar material place d adjacent t o th e cylinder, or b y extending the soli d uppe r en d o f the cylinder and bondin g th e dumm y gage s to thi s unstraine d portion . If all gage s ar e maintained a t th e sam e temperature , th e bridg e wil l b e temperature compensated. A typical transducer o f this type is shown in Fig . 12.14 . The circumferentia l strain, E H, and th e longitudina l strain , e L, ar e

where p

= d= t= E= v=

internal pressur e inner diamete r wall thicknes s modulus of elasticity Poisso n rati o

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E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

With only the circumferential strai n gages activ e and i n opposite bridg e arms, the bridg e output , A£ 0, i s

The nonlinearit y factor ,

Although th e circumferentia l stres s i s twic e the longitudina l stress , th e same i s not tru e fo r th e strains . Usin g Eqs. (12.44 ) and (12.45) , the rati o of strains i s

For stee l with v = 0.3 , E H = 4.25s L. If all gages wer e bonded t o th e cylinder , two circumferentia l and tw o longitudinal , and arrange d int o a full y activ e bridge, th e bridg e outpu t woul d b e reduce d b y approximatel y 2 4 percent . This typ e o f pressure transduce r i s best use d a t relativel y high pressur e for a compac t design . A s Eq . (12.44 ) indicates , th e diameter , d , mus t b e increased and/o r the wall thickness, t, decreased i n order to obtain reasonabl e strain reading s fo r lowe r pressures . Onc e th e transduce r dimension s hav e been chosen , however , i t ca n b e constructe d an d calibrate d b y usin g a deadweight tester , fo r instance . Th e frequenc y respons e ca n b e improved b y reducing th e interna l volum e throug h th e insertio n o f a soli d plug , thu s reducing th e flow caused b y pressure variation . Diaphragm pressure transducer A second type o f pressure transducer uses a diaphragm. The diaphragm may be made fro m a thi n sheet o f flat material clampe d betwee n tw o element s of the transduce r body , o r i t ca n b e machine d a s a n integra l par t o f th e transducer body . Th e informatio n outlined her e ma y b e use d t o arriv e a t a preliminary design , bu t th e fina l outpu t o f th e instrumen t wil l hav e t o b e obtained b y calibration. I n determining the characteristics of the diaphragm , the followin g restrictions apply: 1. Th e diaphrag m i s rigidly clamped a t it s oute r edge . 2. Th e diaphrag m i s fla t an d o f unifor m thickness. 3. Th e deflectio n of th e cente r wil l no t excee d one-hal f o f th e diaphrag m thickness. 4. Th e natura l frequenc y o f th e diaphrag m mus t b e hig h enoug h t o respond adequatel y to fluctuating pressure .

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FIG. 12.15. Clampe d circula r plat e with strai n distribution .

In determinin g th e characteristic s o f the diaphragm , th e analysi s fo r a uniformly loade d thi n circula r plat e clampe d a t th e edg e ca n b e use d (4) . The plate an d it s loading ar e show n in Fig . 12.15 . The pressur e act s o n th e upper surfac e and th e strai n gage s ar e bonded t o th e unde r surface . The tangentia l bendin g moment , M t, an d th e radia l bendin g moment , Mr, a t an y radiu s ar e

The corresponding stresse s ar e

The strain s follo w a s

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E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

The strain s give n b y Eqs . (12.53 ) and (12.54 ) ar e als o plotte d i n Fig . 12.15 . At r = 0 the tangentia l and radia l strain s ar e identica l an d expresse d a s

At r = R th e tangentia l strain is zero and th e radial strain becomes

Equations (12.55 ) an d (12.56 ) show wher e the gage s shoul d b e placed . A pai r o f stacked orthogona l gage s coul d b e use d a t th e center , whil e tw o radial gage s coul d b e place d a s clos e t o th e boundar y a s possible , the n arranged int o a ful l bridge . Althoug h th e bridg e woul d b e temperatur e compensated, a n examinatio n o f Eq . (5.40) , usin g thes e strains , show s tha t the nonlinearit y factor i s not zero . Special gages , Fig . 12.16 , have bee n designe d fo r us e wit h diaphragm s (5). This gage takes advantage o f the strai n distribution show n i n Fig. 12.15 . Since the tangential strai n decreases mor e slowl y with increasing radiu s tha n does the radia l strain , the centra l elemen t is designed t o measur e tangential strain. Th e oute r element s ar e the n arrange d i n a radia l directio n t o tak e advantage o f the radia l strai n a t th e boundary , where it i s maximum. If th e strain i s average d ove r th e regio n covere d b y eac h element , an d usin g Gr = 2.0 , the bridg e outpu t i s approximately

The deflectio n a t an y radiu s is

The maximu m deflection , at th e cente r o f the plate , is

In order t o have the transducer respon d satisfactoril y to pressure pulses , the natura l frequenc y o f the diaphrag m mus t b e a t leas t thre e t o five times higher tha n th e forcin g frequenc y (5) . The undampe d natura l frequenc y o f

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FIG. 12.16. Diaphrag m strai n gag e fo r a pressur e transducer . (Courtes y o f Measurement s Group, Inc. )

the diaphrag m i s

where g

= gravitational constant, 386. 4 in/sec 2 y = specifi c weight o f diaphragm material , lb/in 3

Comments The transducer s describe d i n thi s chapter hav e th e inten t o f do-it-yourself , where suc h a n instrumen t wil l b e use d wit h existin g strain-measurin g instrumentation. They are , therefore, not designe d t o stan d alone . Wit h th e do-it-yourself transducer , desirable adjustments ca n be made a t th e instru ment (suc h a s gag e facto r adjustment) t o brin g i t withi n the desire d limits. For mor e precis e compensatio n procedures , th e reade r i s referre d t o th e paper b y Dorse y (6 ) or t o Referenc e 2.

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THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

Problems In all problems use steel with v = 0.3 and E = 30 x 106 psi. 12.1. Th e load cel l shown i n Fig. 12. 2 is used t o measur e loads between + 750001b . The loa d cel l ha s a diamete r o f 1.5 0 in, G F = 2.15 , and R g = 12 0 ohms. Wit h the loa d a t bot h extremes , determin e the following : (a) Th e bridg e nonlinearity. (b) Th e geometri c nonlinearity. 12.2. I n Proble m 12. 1 the bridg e is rearranged so that gages 1 and 3 are activ e gages , with R2 = R 4 = R g being dummy gages bonded t o a similar piece of unstrained material. Fo r th e sam e loadin g conditions , determine th e following : (a) Th e nonlinearit y factor. (b) Th e bridg e output , A£ 0, i f the suppl y voltage i s 1 0 volts. compressive force , F , acts on a ring-type load cell . By considering the strain s based o n a curved beam, <•: , = c 3 = E O and r, 2 = c 4 = — f,,., writ e th e expression for th e nonlinearit y factor , 1 — n. 12.4. A ring-typ e loa d cel l i s subjecte d t o a tensil e loa d o f F = lOOOOlb . It s dimensions are R0 = 3.0 in, K, = 1. 5 in, and w = 0.7 5 in. Determine the following: 12.3. A

(a) Th e percentag e differenc e i n M 0 betwee n Eqs . (12.13 ) an d (12.16) . (b) Th e strain s a t eac h gag e usin g thin-rin g equations. (c) Th e strain s at eac h gag e base d o n curved-bea m equations. 12.5. Th e cantileve r bea m i n Fig . 12.1 7 is t o hav e a constan t strai n o f I200uin/i n along its tapered lengt h when the load i s applied at th e verte x of the equilateral triangle forme d b y it s tw o sides . Determin e w.

FIG. 12.17 .

12.6. Desig n a loa d cel l havin g th e geometr y show n i n Fig . 12.6 . Th e widt h is limite d t o 0.62 5 in, th e rati o I BC/IAB i s 25 , th e tota l deflectio n i s no t to excee d 0.01 5 in , an d th e maximu m loa d i s 50 0 Ib. Fo r th e fina l gag e location, compute th e bridg e nonlinearit y at th e maximu m load .

TRANSDUCERS 38

9

12.7. Desig n a loa d cel l having the configuratio n shown i n Fig . 12.8 . Th e widt h is limited t o 1 in, and th e maximu m deflection must no t excee d 0.01 2 in a t th e rated loa d o f 700 Ib. Determine the gage locations and thei r strains . Calculat e the bridg e nonlinearit y at th e maximu m load . 12.8. Desig n a shea r bea m loa d cel l havin g th e configuratio n show n i n Fig . 12.10. Th e maximu m loa d o f 5000 0 Ib i s t o produc e A£ 0/K o f approxim ately 2 mV/V. 12.9. Usin g Eq . (5.40) , deriv e Eq . (12.39). 12.10. A torqu e mete r ha s a diamete r o f 1.2 5 in an d use s fou r 350-oh m gages , with G F = 2.10, t o for m a ful l bridge . I f A £ 0 / K = 2 m V / V a t maximu m torque, determin e the valu e of the torque . 12.11. I f the transvers e sensitivity of the gage s i n Proble m 12.1 0 is 0.9 percent, what will b e th e percentag e chang e i n torqu e i f th e transvers e sensitivit y i s considered? 12.12. Desig n a torqu e wrench, shown i n Fig . 12.13 , to th e followin g specifications: (a) Th e maximu m torque i s 200 ft-lb. (b) Th e overal l length of the wrenc h must no t excee d 1 8 in. (c) Th e maximu m strain a t sectio n 2 shall be 100 0 uin/i n at ful l torque . 12.13. A thin-walle d cylindrica l pressur e transduce r ha s a n interna l diamete r o f 1.25 i n an d a wal l thicknes s o f 0.0 5 in. Tw o circumferentia l gage s wit h Rg = 35 0 ohms an d G F = 2.10 ar e bonded t o th e cylinder . If the hoo p strain, EH, i s limite d t o 100 0 uin/in , determin e th e maximu m interna l pressur e an d the correspondin g valu e of A£0 /K 12.14. Mak e a preliminary design o f a diaphrag m pressur e transduce r t o measur e a peak pressur e o f 7 5 psi a t a frequenc y o f 10 0 cycles/sec. Th e desire d bridg e output, A£ 0 /K, is 1.5 mV/V at the peak pressure . Us e GF - 2.0 . REFERENCES 1. Cook , Rober t D . an d Warre n C . Young , Advanced Mechanics o f Materials, Ne w York, Macmillan , 1985 , Chap. 10. 2. Strain Gage Based Transducers, Their Design an d Construction. Measurement s Group, Inc. , P.O . Bo x 27777, Raleigh, NC 27611 , 1988. 3. Meier , J. H. , "Some Phases o f the Techniqu e of Recording Performanc e Data o n Large Machines, " SESA Proceedings, Vol. X, No. 1 , 1952, pp. 35-52 . 4. Timoshenko , S. , Strength o f Materials, 3 d edition , Par t II , Ne w York , Va n Nostrand Reinhold , 1958 , Chap. 4. 5. "Desig n Consideration s for Diaphragm Pressur e Transducers, " TN-510 , Measure ments Group , Inc. , P.O. Bo x 27777, Raleigh, NC 27611 , 1982 . 6. Dorsey , James, "Homegrow n Strain-gag e Transducers, " Experimental Mechanics, Vol. 17 , No. 7 , July 1977 , pp. 255-260 .

13 STRAIN GAGE SELECT10N AND APPLICATION 13.1.

General considerations

On first observation, the strain gage appears to be a rather simple instrument that ca n b e applie d wit h minimu m effort . Thi s ma y b e so , perhaps , i f th e gage i s to b e bonde d t o a fairl y larg e plan e are a wit h ample workin g room and usin g a quick-setting cement. The novice soon learns , however, that even the supposedl y simpl e tas k o f satisfactorily soldering lead wire s to th e gage's solder tab s i s not easy . Whe n h e moves o n t o bondin g gage s i n a confined space and the n attaching lead wires, his appreciation o f the talent of a skilled technician rise s rapidly . When selectin g a strai n gage , o r gages , fo r a project , th e condition s under whic h th e gag e wil l operat e mus t b e considered . Whe n al l o f th e variables that go into gage construction are examined (backing material, foil , gage length , configuration , etc.) , thousand s o f type s ar e available . A study of manufacturer's catalogs shows that gage s ar e divide d int o relate d groups , or series , o f th e sam e basi c construction . Sinc e gage s belongin g t o a serie s have simila r characteristic s an d capabilities , th e tas k o f choosing a gag e i s therefore reduced . The firs t ste p i n choosin g a strai n gag e installatio n i s t o lis t a s man y conditions a s possible affectin g th e system. Some of these are the following: 1. I s th e strai n t o b e measured i n elasti c or plasti c region? I f the strai n i s in the plastic region, for instance, then a post-yield gage will be chosen. 2. I f th e stres s fiel d i s uniaxial , a single-elemen t gag e aligne d alon g th e principal stres s directio n wil l suffice . I f th e fiel d i s biaxia l an d th e principal strain directions are known, a two-element rectangular rosett e can b e used . I f th e principa l strai n direction s ar e unknown , a three element rosett e wil l b e required . 3. Wha t is the duratio n o f the test ? Will it b e measured i n minutes, hours, or years ? The concer n her e i s the shiftin g o f the zer o referenc e point . 4. Ho w difficul t wil l th e installatio n o f gages be ? 5. Wil l th e test s b e stati c o r dynamic ? I f the y ar e purel y dynamic , the n consideration ha s t o b e give n t o a foi l tha t exhibit s goo d fatigu e properties. 6. Th e temperatur e rang e ove r which the gage wil l operate an d th e choice of it s self-temperature-compensatio n numbe r mus t b e considered .

STRAIN GAG E SELECTIO N AND APPLICATIO N 39

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7. Ar e strain gradients perpendicula r t o th e tes t surface or i n the plan e of the tes t surface ? 8. Th e choice of an adhesive is important an d canno t b e over emphasized. Depending o n the specia l requirements of a give n test, othe r condition s ca n be adde d t o thi s list . Th e cos t o f the strai n gages , however , may hav e low priority, sinc e th e gag e cos t i s generally smal l whe n compared t o th e tota l cost o f a test . After th e condition s tha t affec t th e syste m ar e listed , a manufacturer's catalog ca n b e consulte d i n orde r t o choos e a specifi c gage. Her e wil l b e found a designation code giving the features of the gage. They are as follows: 1. Gag e series an d typ e o f strain-sensing allo y 2. Backin g o r carrie r materia l o f the strain-sensin g alloy 3. Self-temperatur e compensation numbe r 4. Th e activ e gage length 5. Gri d an d ta b geometr y 6. Gag e resistanc e 7. Options , i f desired 13.2.

Strain gage alloys (1, 2)

Constantan One o f the most common strai n gage foils i s a copper-nickel alloy generally known as constantan. I t finds wide use in static strain measurements as well as bein g employe d i n transducers . I t als o show s goo d fatigu e lif e whe n applied t o alternatin g strains , providin g th e strai n level s ar e kep t belo w ±1500 uin/in. I t ha s a lo w an d controllabl e temperatur e coefficien t o f resistance as well as good strain sensitivity, which gives a nominal gage factor of 2.0 . Furthermore , th e gag e facto r i s relativel y insensitive to strai n leve l and temperature . Constantan can be processed fo r self-temperature compensation s o that it matche s th e therma l expansio n characteristic s o f a numbe r o f commo n engineering materials wit h thermal expansion coefficient s rangin g from zer o to 5 0 ppm/°F. W e have seen how mismatchin g of the coefficient s of thermal expansion o f th e gag e an d tes t materia l rotate s th e therma l outpu t curv e around th e referenc e temperatur e i n orde r t o obtai n th e mos t favorabl e results in a desired temperature range. An examination of strain gage catalogs shows that satisfactory gage resistance is obtainable even for very small gages made o f this material . If very large strains, on th e order o f 5 percent, are t o be measured, the n an anneale d constanta n foi l is used . If gage length s of |inc h or larger ar used, strains in excess of 20 percent ca n b e measured. Annealed constantan , however, i s not recommende d fo r cycli c strains, sinc e permanen t resistanc e change occur s a s a functio n o f number o f strain cycles.

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Constantan ha s severa l disadvantages . I f the tes t temperatur e i s abov e 150°F (66°C) , i t show s referenc e poin t drift , whic h i s undesirabl e fo r test s conducted ove r a lon g perio d o f time. The therma l output i s also ver y hig h at temperature s belo w -SO T (-45°C ) an d abov e 400 T (205°C) . Isoelastic Isoelastic, mos t generall y use d fo r dynamic strain measurement , i s a nickel chromium-iron allo y wit h molybdenu m added . It s hig h gag e facto r o f approximately 3.2 improves the signal-to-noise ratio in dynamic testing. This, coupled wit h superio r fatigu e life , make s i t particularl y usefu l fo r dynami c strain measurement . The therma l outpu t o f isoelastic is about 8 0 uin/in/T an d i t canno t b e self-temperature compensated , thereb y makin g i t generall y unsuitabl e fo r measuring static strains. This feature makes it undesirable for any long-rang e measurements i f a stabl e referenc e poin t mus t b e maintained. Furthermore , its respons e become s nonlinea r at strain s o n th e orde r o f 0.5 percent; hence , it i s confine d t o strai n measuremen t i n th e elasti c region . I n specia l cases , where a high-outpu t response i s desired, i t ma y b e use d wit h a full-bridg e circuit, thereb y obtainin g circui t temperatur e compensation . Karma Karma, a nickel-chromiu m allo y wit h smal l percentage s o f iro n an d aluminum, i s anothe r desirabl e material , sinc e gage s mad e o f thi s materia l show minima l referenc e point drif t wit h tim e an d temperature . Becaus e o f this stability , it is a fine choice fo r long-time stati c measurement s at o r nea r room temperature . I t i s recommende d fo r stati c strai n measurement s fro m -452T (-270°C ) t o SOO T (260°C) , bu t encapsulate d gage s ca n b e use d to 750 T (400°C ) fo r shor t tim e periods . Th e materia l als o exhibit s goo d fatigue lif e wit h minimum reference point drif t eve n after bein g cycle d a large number o f times . Becaus e o f it s hig h resistivity , smalle r gage s fo r a give n resistance can b e manufactured. Karma ca n b e self-temperature compensated ove r a broa d temperatur e range, bu t i t i s mor e limite d tha n constanta n i n th e numbe r o f therma l expansion coefficient s fo r whic h i t ma y b e compensated . A n advantage , however, i s a flatte r therma l outpu t curve . Another featur e is a gag e facto r that goe s negativ e wit h increasin g temperature , thu s compensatin g fo r the temperature-induce d chang e i n th e modulu s o f elasticit y o f th e tes t material. Karma ha s severa l disadvantages. I t i s difficul t t o solder , an d fo r thi s reason gage s wit h copper-clad tab s are available . Gage s o f this material ar e also mor e difficul t t o manufacture , making them mor e expensiv e than gage s using constantan .

STRAIN GAG E SELECTIO N AN D APPLICATIO N 39

3

Platinum-tungsten A platinum-tungste n allo y ha s bee n develope d fo r high-temperatur e use . I t has unusua l stability and fatigu e lif e a t temperature s abov e 750° F (400°C) , does no t underg o an y metallurgica l change s t o abou t 1650° F (900°C) , an d so it s resistanc e remain s essentiall y unchange d wit h time . I t ha s a high temperature coefficient o f resistance that i s not adjustable , although repeat able; thus , it cannot b e self-temperature compensated. I f temperature comp ensation i s desired, i t shoul d b e don e throug h circui t compensation . This materia l i s use d fo r dynami c strai n measurement s t o 1500° F (815°C) an d fo r stati c strai n measurement s t o 1200° F (650°C) . I t ha s a higher strai n sensitivit y tha n copper-nicke l o r nickel-chromiu m alloys , but i t i s nonlinear . Th e strai n rang e i s generall y limited t o approximatel y + 0.3 percent. 13.3.

Grid backing materials (1-4 )

The strain-sensin g elemen t (eithe r foi l o r wire ) o f a strai n gag e i s mounte d on a backing (carrier) material. The backing material serves several purposes. 1. I t protect s th e strain-sensin g gri d fro m damag e durin g handlin g an d installation. 2. I t provide s a bondin g surfac e t o th e tes t piece . 3. I t transmit s strai n fro m th e tes t piec e t o th e strain-sensin g alloy . It s stiffness mus t be low enough s o it can follow th e strains in the test piece without affectin g it . On th e othe r hand, it must be stif f whe n compare d to th e strain-sensin g allo y s o tha t th e conducto r materia l follow s th e strains withou t irregular distortion . 4. I t provide s electrical insulation between the strain-sensing element an d the tes t piece . Paper carriers One o f th e firs t backin g materials , an d on e stil l used , i s a nitrocellulos e paper. Strai n gage s usin g thi s readil y availabl e backin g materia l easil y conform t o th e surfac e o f a tes t specimen . Th e gage s ar e usuall y bonde d to a tes t piec e wit h a nitrocellulos e cemen t tha t impregnate s th e paper' s pores an d cure s b y evaporation. Gage s bonde d i n thi s manne r ca n operat e between - 100° F and 180° F (-7 3 t o 82°C), although the y can be used for short period s of time beyond th e upper temperatur e limit. At room tempera ture, this combination o f paper an d adhesive , when properly applied , ca n be subjected t o strain s in exces s of 1 0 percent befor e breaking down . Polyimide resins Polyimide resin s ca n b e provide d i n bot h cas t fil m an d glass-reinforce d laminated construction . I t i s a general-purpos e materia l use d fo r bot h

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static an d dynami c strains . Th e cas t fil m type s are tough , flexible , an d ca n be elongate d u p t o 2 0 percent . Becaus e o f thei r flexibility , the y ca n b e contoured t o fi t smal l radii . Thi s materia l ca n b e use d a t temperature s ranging from cryogeni c to 400°F (205°C). For highe r temperatures, however, the resi n ca n b e reinforce d with glass an d th e gag e encapsulate d fo r us e t o 700°F (370°C), although th e temperatur e can b e increased t o 750° F (400°C ) for short-duratio n tests. Epoxy resins Epoxy resin s reinforced with glass fibers were developed i n order t o improve temperature capabilities . Thi s material has a n operatin g temperatur e range , for bot h stati c an d dynami c strain measurements , fro m cryogeni c t o abou t 550°F (290°C), with an uppe r limit of 750°F (400°C) fo r short-duration tests. This backin g als o ha s improve d dimensiona l stabilit y fo r us e i n precisio n transducers. The glass reinforcement, however , reduce s the maximum strai n to abou t 1 percent bu t result s in a n extremel y thin carrier . Sinc e it i s mor e brittle tha n polyimide , it requires more car e i n handling in orde r t o preven t damage. Metallic carriers Metallic carrier s hav e bee n discusse d i n Chapte r 1 under weldabl e gages . Weldable wir e gage s ar e covere d i n Sectio n 1.5 , while weldable foi l gage s are discusse d i n Sectio n 1.6 . 13.4.

Gage length, geometry, and resistance (1, 2)

Gage length When referrin g t o gag e length , it i s th e activ e o r strain-measurin g portio n that is referred to, not th e overall or matrix length. A major purpose o f using strain gage s i s to determin e strains at critica l points o n a structure. Because these point s are ofte n wher e stress concentrations exist , thereby resultin g in strain gradient s whic h ma y b e quit e steep , consideratio n mus t b e give n t o the strai n gradien t alon g th e gag e length . Stee p strai n gradient s ma y als o occur i n dynami c measurements , suc h a s occu r whe n th e propagatio n o f stress wave s i n a materia l i s being studied . Sinc e a strai n gag e average s th e strain alon g it s activ e length , choosin g a gag e lengt h considerabl y longe r than th e pea k strai n regio n result s i n a strai n readin g o n th e lo w side ; therefore, a gag e lengt h consisten t wit h th e pea k strai n regio n shoul d b e chosen. For nonhomogeneou s materials , however, a gage length long enough to spa n th e representativ e structure of the materia l should b e used i n orde r to averag e the strai n ove r voids , etc. When possible , gage s wit h lengths from ^ in t o ^ in ar e preferable , since they ar e easie r to apply , offe r th e larges t numbe r o f geometries and options ,

STRAIN GAG E SELECTIO N AN D APPLICATIO N 39

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and ar e less expensive. For gage s o f identical resistance an d applie d voltage , the large r gag e wil l dissipat e hea t mor e easil y becaus e o f th e lowe r hea t generated pe r uni t area . Thi s i s particularl y importan t whe n th e gag e i s bonded t o a materia l wit h poo r heat-transfe r qualities. Gage geometry When choosin g a strain gage for a particular test, several elements ente r into the decision . Amon g thes e ar e th e shap e o f th e strain-sensin g grid , th e number o f grid s an d thei r orientatio n relativ e t o eac h other , solde r ta b arrangement, an d spac e availabl e fo r mounting . I f th e principa l stres s i s known to be uniaxial and it s direction is also known, then a single grid gage may b e used. This conditio n generall y does not exis t and singl e gages should be use d onl y whe n on e i s absolutely sur e one ha s a uniaxia l stress state . For th e biaxial stress state a three-element rosette is used if the principal stress direction s ar e unknown . Th e grid s o f th e rosett e ma y hav e an y orientation relativ e t o eac h other , bu t rosette s hav e bee n standardize d o n the delt a an d th e rectangula r configuration . Thi s make s dat a reductio n simpler, particularl y for the rectangula r rosette . Whe n bondin g a rosett e t o a tes t specimen , any orientatio n ca n b e used , but usuall y one rosett e le g is aligned alon g som e chose n axi s of the specimen . If mounting space is confined, stacked rosette s are preferred, particularly when ther e is a high strain gradien t i n the plan e o f the mountin g surface. I n this case they give a closer approximation of the strain at the point, but hea t dissipation ma y b e a problem . The y ar e stiffe r tha n th e plan e rosett e an d conform les s easily to curved surfaces. O n th e othe r hand , plan e rosettes ar e preferred whe n th e strai n gradien t i s normal t o th e surface , sinc e al l grid s are a s clos e t o th e specime n surfac e a s possible . When th e principa l stres s direction s ar e known , the n a two-elemen t rosette ma y b e used . Th e principa l stres s direction s ma y b e apparen t fro m the geometr y o f the tes t specimen , suc h a s a thin-walle d tub e wit h interna l pressure, fo r instance . Generally , th e principa l direction s ar e determine d through th e us e o f a brittl e lacque r coa t o r a photoelasti c coating . I t i s obvious tha t i f the principa l axe s ar e known , considerable saving s in tim e and labo r ca n b e attaine d i n wirin g a numbe r o f two-element gage s rathe r than three-elemen t gages . Special-purpose gages , suc h a s gage s fo r residua l stres s measurement , crack detection, or diaphragm gage s for pressure transducers, are available. When a high strain gradient transverse to the gage axis exists, a gage with a narro w grid widt h should b e chosen i n orde r t o giv e a bette r strai n average . The reduce d gag e area , though , will reduc e th e abilit y to dissipat e heat. Gage resistance An examinatio n o f manufacturer's catalogs show s that strai n gage s may b e obtained wit h resistances up to 100 0 ohms. The two most common resistanc e

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values, however , are 12 0 and 35 0 ohms. A s w e sa w i n Chapte r 5 , lead-line resistance desensitize s th e circuit . I f lead-lin e resistanc e o r othe r parasiti c resistances ar e present , the n choosin g a higher-resistanc e gag e wil l reduc e the circui t desensitization . Thi s i s illustrate d in Exampl e 5.4 . Fo r th e sam e applied voltage , a higher-resistanc e gag e reduce s th e hea t generated . A 120-ohm gage , fo r instance , generate s nearl y thre e time s th e wattag e o f a 350-ohm gage. Conversely, if the wattag e remains the same , a highe r voltage may b e use d o n th e highe r resistance gag e i n orde r t o increas e th e output. Self-temperature compensation Choosing a gag e fo r the prope r self-temperatur e compensation numbe r i s a matter o f examining a strain gage catalog i n order to determine the available thermal expansio n coefficients . Whe n th e desire d numbe r i s chosen , i t i s a matter o f addin g i t i n th e prope r plac e i n a manufacturer' s strai n gag e designation code . Self-temperatur e compensatio n an d it s us e hav e bee n covered i n Chapte r 11 . Options Both standar d option s an d specia l option s ar e available . Amon g thes e ar e attached lea d wires , gag e encapsulation , solde r dots , an d etche d integra l terminals, t o nam e a few . For a complet e descriptio n o f options , bot h standard an d special , consul t a manufacturer' s catalog . 13.5. Adhesive s (1, 2 , 4) Successful us e o f strai n gage s i s ver y muc h dependen t upo n satisfactoril y bonding th e gag e t o th e tes t specimen . Th e chose n adhesiv e mus t hav e sufficient shea r strengt h i n order t o transmi t strain s i n th e tes t specime n t o the strain-sensing grid, yet it must be compatible wit h both the gage backin g material an d th e tes t materia l s o tha t neithe r i s damaged . Further , th e adhesive shoul d hav e long-term stabilit y s o tha t i t doe s no t decompos e o r show appreciabl e cree p ove r th e test' s lifetime . The manufacturer' s instruction s i n th e us e o f th e adhesiv e mus t b e followed carefully , particularl y i f the adhesiv e call s fo r mixin g a resi n an d hardener. Th e adhesiv e mus t b e capabl e o f forming a thi n glu e line fre e o f voids, wit h minimum curing time being a desirabl e feature . I t als o help s t o electrically isolat e th e gri d fro m th e tes t material . Whe n checkin g th e resisance betwee n th e gri d an d tes t specimen , th e recommende d resistanc e is 1000 0 megohms minimum , but preferabl y higher . An adhesiv e should b e capable o f high elongation a s well as have the abilit y t o operat e ove r a wide temperature range . A large number of adhesives are available, each with detailed techniques for it s application . Manufacturer s wil l suppl y th e use r wit h instructions . One thin g that i s crucial to satisfactor y strai n gage performance , regardless

STRAIN GAG E SELECTIO N AN D APPLICATIO N 39

7

of th e adhesiv e used , i s cleanliness. Th e tes t surfac e mus t b e fre e o f grease, rust, or other contaminants s o that the bare base material i s exposed. Durin g the preparation proces s the hands must be kept clea n and car e taken no t t o touch th e surface . Afte r th e surfac e i s prepared th e gage s shoul d b e applie d without undu e delay. Although ther e ar e numerou s adhesive s available , onl y th e mor e commonly use d one s wil l b e discussed . Detaile d informatio n o n specifi c adhesives and their methods of application ca n be obtained fro m informatio n bulletins supplie d b y manufacturers. Nitrocellulose Nitrocellulose adhesive s (suc h a s Duco ) wer e onc e widel y use d whe n paper-backed gage s wer e prevalent . Thi s typ e o f adhesiv e set s b y solven t evaporation; thus , it s us e toda y i s limite d t o paper-backe d gage s o r gage s with a porou s backing . A minimu m pressur e ha s t o b e applie d durin g th e curing process, which is usually in excess of 24 hours, depending on humidity and temperarure . Applicatio n o f hea t wil l accelerat e th e curin g process , however. The curing process ma y be monitored b y periodically checking the gage resistance to ground, since the resistance increases as the adhesive sets. Gages bonde d wit h nitrocellulos e adhesive s may b e use d u p t o 180° F (82°C). The y ar e hygroscopi c (i.e. , the y absor b moistur e fro m th e air ) an d must be protected wit h a moisture-resistant coating once the adhesive is fully cured to ensure electrical and dimensiona l stability. Because adhesives of this type ar e vulnerabl e t o ketoni c solutions , the y ar e easil y remove d withou t surface damag e b y usin g a ketoni c solution. Cyanoacrylate Cyanoacrylate adhesive s ar e widel y use d a s general-purpos e cement s tha t are fas t curin g and simpl e to use , since no mixin g is required. While the lif e of an unopene d containe r o f Cyanoacrylate is approximately 9 months when stored a t roo m temperature , th e lif e ca n b e extende d b y refrigeratio n a t 40°F (4°C) . Whe n remove d fro m th e refrigerator , the adhesiv e shoul d b e allowed t o com e t o roo m temperatur e befor e openin g i n orde r t o preven t condensation an d possibl e damag e t o th e material . Onc e th e containe r ha s been opene d i t shoul d b e stored i n a cool , dar k are a rathe r tha n returning it t o th e refrigerator. In preparin g a gag e fo r bonding , th e gag e backin g materia l i s treate d with a catalyst , sparingl y used , an d allowe d t o dr y fo r approximatel y 1 minute. A thi n coa t o f adhesiv e i s pu t on , th e gag e i s place d o n th e tes t specimen, and thum b pressure i s applied t o th e gage . Polymerizatio n take s place i n th e adhesiv e fil m i n approximatel y 1 minute. The bonde d gag e i s ready fo r us e by th e tim e lead wire s are attached . The glue line is sensitive to moisture and must be protected b y a coating. A properl y protecte d gage , however , ca n b e use d i n we t atmosphere s fo r

398 TH

E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E

short-duration tests . Thes e adhesive s ar e excellen t fo r short-ter m test s bu t are seldo m use d fo r test s extendin g ove r lon g tim e periods , sinc e th e bon d is subjec t t o embrittlemen t wit h age. Gages bonde d wit h thes e cement s ca n be use d t o measur e strain s o f the orde r o f 1 5 percent an d ca n operat e ove r a temperatur e rang e o f -25 ° t o 150° F (-3 2 t o 65°C). Epoxies Epoxy adhesives, in use for many years, come in a wide selection o f two types; namely, on e wher e polymerizatio n take s plac e a t roo m temperatur e an d another tha t require s the application o f external heat fo r correct polymeriza tion. Wit h epoxies , ther e i s n o solven t evaporatio n involved , ver y littl e shrinkage, an d a goo d permanen t bon d i s forme d wit h a wid e variet y of materials. Epoxie s als o exhibi t excellent moistur e an d chemica l resistance , and can be used over a temperature range from cryogeni c to 600°F (315°C). One typ e o f epoxy, using an amin e catalyst , cures a t roo m temperatur e through th e exothermi c reactio n produce d whe n th e adhesiv e components , hardener an d resin , are mixe d together. Anothe r typ e o f epoxy, activated b y an acid anhydride catalyst, requires external heat fo r polymerization t o occu r properly. A temperatur e o f a t leas t 250° F (120°C ) mus t b e maintaine d fo r several hours . Bot h type s requir e a clampin g pressur e durin g th e curin g process. Furthermore , i f either typ e i s t o b e use d a t a temperatur e highe r than th e curin g temperature , the n a post-cur e temperatur e abov e th e expected maximu m test temperature should b e maintained for several hours . For th e room-temperature-curin g epoxy , th e post-cur e temperatur e shoul d be 70 degF to 8 5 degF (40 to 47 degC) abov e th e maximum test temperature. For th e hot-cur e epoxy , th e post-cur e temperatur e shoul d b e 8 5 deg F t o 115 degF (4 7 to 6 4 degC) abov e th e maximu m tes t temperature . Other adhesives Other availabl e adhesive s ar e generall y use d fo r mor e specialize d applica tions. Among these are phenolic, polyimide, and cerami c adhesives . Phenoli c adhesives ar e littl e used becaus e the y require complicated, lon g curin g cycles and hig h clampin g pressure . Polyimid e adhesive s ar e difficul t t o wor k wit h and th e solvents in them are not easily removed. Remaining solvents degrad e the adhesive properties. Cerami c adhesive s are applied t o free-filamen t gage s and thermocouple s fo r temperature s tha t excee d th e limit s o f organi c materials. Again, for special applications, consul t th e manufacturers and their application departments . 13.6.

Bonding a strain gage to a specimen

Bonding a strai n gag e appear s t o b e a simpl e process , bu t clos e attentio n must b e pai d t o eac h step . Thi s involve s surfac e preparatio n o f th e tes t specimen, cementing th e gage to that surface, soldering lead wires , and finall y

STRAIN GAG E SELECTIO N AN D APPLICATIO N 39

9

applying a protectiv e coa t t o th e installation . Cleanlines s canno t b e over emphasized; th e hand s shoul d b e washe d frequentl y durin g th e proces s o r cleaned wit h neutralizer , n o materia l shoul d b e reused , an d th e wor k are a must b e kept clean . Surface preparation

1. Usin g a degreasin g agent , suc h a s trichloroethylen e o r carbo n tetra chloride, clean the test surface, being sure to have adequate ventilation. 2. San d th e degrease d surfac e i n orde r t o remov e al l scale , dirt , o r dus t particles. 3. Clea n th e surfac e wit h a spong e o r tissu e saturate d wit h th e cleanin g solvent. 4. Usin g a metal conditioner, wet lap the area with silicon-carbide paper. 5. Usin g a clea n tissue , wipe the are a dr y wit h one stroke . D o no t reus e the tissue . Repeat severa l times. 6. Usin g a ballpoin t pe n o r 4- H pencil , locat e an d mar k referenc e line s for gag e alignment. Do not use a scribe: make certain you d o not scratch the surface. 1. Usin g a cotton-ti p swab , di p i t int o meta l conditione r an d scru b th e surface. Wip e dr y wit h on e strok e usin g a clea n tissue . Usin g a clea n swab an d tissue , repea t severa l time s unti l th e cotton-ti p show s n o foreign material . 8. Di p a cotto n swa b int o neutralize r an d scru b th e surface . Wip e clea n with one stroke usin g a clean tissue . Repeat severa l times to ensure the surface i s neutralized. 9. Instal l th e gag e a s soo n a s possible . Bonding the gage

Since bonding technique s wil l differ dependin g o n th e adhesive , th e metho d for a cyanoacrylat e adhesiv e wil l b e described , sinc e i t i s a widel y use d cement. 1. Remov e the gag e fro m it s packe t an d plac e it , bondin g sid e down , o n a clea n surface . Position a separat e termina l stri p relativ e t o th e gag e tabs. 2. Usin g a piec e o f cellophane tape , plac e i t ove r th e gag e an d termina l strip. Pul l th e tap e fro m th e surfac e at a shallo w angle , bein g certai n the gag e an d termina l stri p ar e firmly attached . 3. Plac e the gag e o n th e tes t specimen , alignin g the referenc e tabs o n th e gage wit h th e marke d referenc e syste m o n th e tes t surface . Th e tap e and gag e ar e no w i n th e desire d position . 4. Lif t on e end of the tape from th e test surface until the gage and termina l strip ar e just clear . Th e remainde r o f the tap e i s stil l attache d t o th e specimen.

400 TH

E BONDE D ELECTRICAL RESISTANC E STRAI N GAGE

5. Pul l the fre e en d o f the tap e bac k unti l the bondin g surface s o f the gag e and termina l stri p ar e exposed . Brus h catalys t sparingl y ont o th e bonding surfaces . Allow t o dr y fo r 1 minute. 6. Appl y one o r tw o drop s o f adhesiv e at th e boundar y lin e o f th e tap e and tes t surface . Pull th e fre e en d o f the tap e tau t and toward s th e test specimen, makin g a shallo w angle . A t th e sam e time , usin g a clea n tissue, wipe over the tap e fro m th e boundar y line towards the fre e en d so that th e cement spread s unde r the gage an d termina l strip , bonding them t o th e surface . Appl y thum b pressur e t o th e gag e an d termina l strip fo r approximatel y 1 minute. 7. Afte r severa l minutes, grasp one end of the tape and slowl y and carefull y pull back on itself until it is removed. The gag e is now read y for soldering. Completing the installation Now tha t th e gage is successfully bonde d t o th e test specimen, there remains the tas k o f attaching th e lea d wire s and the n applyin g a protectiv e coat t o the entir e installation . The procedur e i s outlined i n th e following: 1. I f the gage has an open grid, cover the grid area with a piece of masking tape, leavin g the solde r tab s exposed . 2. A 30-40-wat t solderin g iron wit h a smooth , tinne d ti p i s required. 3. Us e a fine rosin-core solde r whos e melting temperature i s compatibl e with th e tes t environment. 4. Wit h the soldering iron at the proper temperature, lay the solder across the gage tab an d appl y the iron firmly for a second. Lif t th e solder an d iron a t th e sam e time , leaving behind a shin y mound o f solder o n th e tab. 5. Lightl y tin th e termina l strips . 6. Separat e th e individua l leads o f the composit e lea d wir e and remov e about inc h t o f inc h insulation from each . O n eac h individua l lead, separate on e strand, twist the remaining strands togethe r and ti n for a short distanc e a t th e insulation . Snip of f the remainin g end, leavin g about inc h of the tinne d bundle. The singl e strand wil l b e use d a s a jumper wir e from th e termina l stri p to th e gag e tab . 7. Solde r th e tinne d lea d wire s to th e termina l strip. 8. Usin g the singl e strand o f each lead , solde r t o th e gag e solderin g tab , arranging it so there is some slack between the terminal strip and gag e tab. (Fin e insulate d wir e may b e use d i n plac e o f the singl e strand. ) 9. Clea n al l solde r joints wit h rosi n solvent , remove th e maskin g tape , and clea n th e gag e wit h rosin solvent . 10. Secur e the lea d wire s so the y cannot accidentl y be pulle d loose . 11. Chec k th e resistanc e betwee n the gag e an d th e specimen . It shoul d be at leas t 1000 0 megohms. 12. Appl y a protectiv e coatin g t o th e gage , termina l strips , an d a shor t distance ont o th e lea d wir e insulation.

STRAIN GAG E SELECTIO N AN D APPLICATIO N 40

1

REFERENCES 1. "Catalo g 500: Part B—Strai n Gage Technical Data, " Measurements Group, Inc. , P.O. Bo x 27777 , Raleigh, NC 27611 , 1988 . 2. "SR- 4 Strain Gage Handbook," BL H Electronics, Inc., 75 Shawmut Road, Canton, MA 02021 , 1980 . 3. "Weldabl e and Embedable Integral Lead Strain Gages," Applications and Installation Manual , Eato n Corp. , Ailtec h Strain Gag e Products , 172 8 Maplelaw n Rd., Troy, M I 48084 , 1985 . 4. Vaughn , John, Application o f B & K Equipment to Strain Measurements, Brue l & Kjaer, Naerum , Denmark , 1975 , Chaps. 3 and 4 .

ANSWERS TO SELECTED PROBLEMS 2.2. a , = 2 0 500 psi; a2 = 450 0 psi; 6 = 45°. 2.4. =
4.9. (a ) £ 9 = 185 5 nin/in; s b = -75 7 nin/in . (b) AR g = 0.4630 ohms; AR b = -0.188 9 ohms. (c) E = 0.033 96 volts.

ANSWERS T O SELECTE D PROBLEM S 40 5.2. A£ 0 /K = (G Fe)/(4 + 2G Fe); E/E, = 2/(2 - G F£;). 5.4. A£ 0 /K= (G F £)/2;£/£;= 1. 5.6. A£ 0 /F = G F(1 + v)e/2 ; e/g; = 1. 5.8. <7 t =3 2 490 psi; = 26.6° ; (b) 0 = 28.7° ; (c ) 0 = 30.2° . 10.4. CTH =-^= ——— [3(1 - v ) + (1 +v) cos 2 1 — v 8t( l — v)

3

404 TH

E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

10.6. y xy = 276 (iradians . 11.1. £ = -230 0 nin/in. 11.3. E =288 5 nin/in. 11.7. (a ) E ! = 167 3 |j.in/in ; £ 2 = —237 3 nin/in. (b) K ! =375 4 nin/in; e 2 = ~2 6 )j.in/in . 11.9. (a ) ei = 207 nin/in ; « 2 = - 100 7 jiin/in. (b) E I = 28 3 nin/in; e, 2 = —87 1 |j.in/in . 12.1. (a ) Tension : nonlinearit y factor = 0.9989 . Compression : nonlinearity factor 1.0011. (b) Tension : nonlinearit y facto r = 1.008 . Compression : nonlinearit y factor 0.9992. 12.3. 12.5. w= 1 . 6 i n . 12.10. r= 2106in-lb . 12.11. Percen t differenc e = 0.86. 12.13. A £ 0 / K = 1.05mV/V .

INDEX

Adhesives, 396- 8 cyanoacrylate, 397- 9 epoxies, 398 nitrocellulose, 39 7 Axial strai n sensitivity , 236 Backing material , 393- 4 Baker, M . A. , 9 Ballast circuit , se e Potentiometric circui t Ballast resistor , 100- 1 Biaxial stress , 45 Biermasz, A. J. , 1 6 Bonded wir e strain gage, 24- 7 Bridge inpu t resistance, 151-2, 173- 5 Bridge outpu t resistance , 151-2 , 176- 7 Bridge ratio , 155 , 159-61 Brittle lacque r coatings , 3 , 36- 8 Calibration potentiometric circuit , 141- 4 strain gag e us e under othe r conditions , 246-8 Wheatstone bridge , 193- 5 Circuits, elementary constant current , 94- 6 advantages, 96- 7 constant voltage , 91- 4 nonlinearity, 93- 4 Circuits, potentiometri c advantages an d limitations , 105- 6 applications, 104- 5 ballast resistor , 100- 1 calibration, 141- 4 characteristics, 102- 3 circuit analysis , 106- 9 circuit efficiency , 111-1 2 circuit equations , 101- 2 components, 10 0

dynamic strains , 118-1 9 gages i n series , 112-1 4 linearity consideration s fixed ballast resistance , 123- 4 variable ballast resistance, 121- 3 measurements, stati c vs. dynamic, 114-15 nonlinearity, 102 , 108- 9 signal measurement , 147- 9 static strain , 118 temperature effects , 129-4 1 ballast an d gag e leads, 133- 5 voltage limitation , 110 Coatings brittle lacquer , 3 , 36- 8 photoelastic, 3 8 Compensating strai n gage, 27-8, 353-7 , 363

Constantan, 14 , 339-41, 391- 2 Crack measurin g gage, 34- 5 Cyanoacrylate cement, 397-8 Data analysis , 253-6 Delta rosett e analysis, 267- 9 Mohr's circle, 269-73 principal stres s directions , 26 9 T-delta, 278-81 transverse sensitivity , 301- 6 Desensitization o f circuits full bridge , 227-31 half bridge , 218-2 5 kinds, 20 7 meter resistance , 175- 9 power suppl y resistance , 173- 5 reasons for varying, 205-6 single gage , 207-1 7 combination, serie s an d parallel, 211-1 6

406

Desensitization o f circuits (contd.) single gag e (contd.) resistance i n parallel , 209-1 1 resistance i n series, 207- 9 temperature effects, 21 6 Dorsey, J. , 8 , 387 Dummy gage, see Compensating strai n gage

Embedment gage , 3 6 Epoxy cement , 398 Equiangular rosette, see Delta rosett e Four-element rosett e rectangular, 275- 8 T-delta, 278-8 1 Friction gage , 35- 6 Gage factor determination, 242-3 manufacturers, 26 , 236 relation wit h axia l an d norma l strains , 240-2 variation wit h temperature , 340-3 Gages crack measuring , 34- 5 embedment, 3 6 friction, 35- 6 semiconductor, 32- 3 temperature, 33- 4 Mines, F . F. , 318 Hydrostatic strai n component , 6 9 Hydrostatic stres s component , 5 6 Indicated vs . actual strain, 165-8 , 206 Invariants strain, 81 stress, 8 1 Isoelastic, 240 , 392 Jones, E. , 1 0 Karma, 39 2 Kelvin, Lord , 5 Kern, R . E., 320 , 32 2

INDEX Lateral effect , 234-5 1 basic equations , 236-42 transverse sensitivit y factor, K, 238-4 0 Lead-line resistance , 180-9 1 full bridge , 180- 1 half bridge—fou r wire, 181- 4 half bridge—thre e wire , 184- 6 quarter bridge—thre e wire , 187- 8 quarter bridge—tw o wire, 188-90 Load cell axial force , 363- 5 bending beam , 372- 5 ring type , 365-8 shear beam , 375- 7 Maslen, K . R. , 10 Material, backing, 393- 4 epoxy resins , 394 metallic, 39 4 paper carrier , 393 polyimide resins , 393-4 Material, strai n gag e constantan, 14 , 339-41, 391- 2 isoelastic, 39 2 karma, 392 platinum tungsten , 39 3 properties desired , 10 McClintock, F . A. , 281 Measurements, fundamenta l laws, 97- 8 Meier, J. H. , 10 , 278, 380, 38 2 Meter resistance , 175- 9 Mohr's circl e delta rosette . 269-7 3 rectangular rosette , 261- 5 strain, 68-70 stress, 54- 7 Multiple circuits . 195- 7 Nitrocellulose cement , 397 Nonlinearity o f circuits elementary. 93- 4 potentiometric, 102 , 108- 9 Wheatstone bridge , 150-1 , 163- 4 Normal strai n sensitivity , 236 Perry, C . C., 330- 1 Photoelastic coating , 38 Plane shearing stress, determination, 327-30 Plane strain , 62- 5 Plane stress , se e Biaxial stres s Platinum tungsten , 393

INDEX Poisson's ratio , 17 , 73 Potentiometric circui t advantages an d limitations , 105- 6 applications, 104- 5 ballast resistor , 100- 1 calibration, 141- 4 characteristics, 102- 3 circuit analysis , 106- 9 circuit efficiency , 111-1 2 circuit equations , 101- 2 components, 10 0 dynamic strains, 118-1 9 gages i n series , 112-1 4 linearity consideration s fixed ballast resistance , 123- 4 variable ballast resistance , 121- 3 measurements, static vs. dynamic, 114-1 8 nonlinearity, 102 , 108-9 signal measurement , 147-9 static strains , 11 8 temperature effects , 129-4 1 ballast an d gag e leads , 133- 5 voltage limitation, 11 0 Pressure transduce r diaphragm, 384- 7 thin-walled cylinder , 383- 4 Principal strains , 64- 5 Principal stresses , 48-53 , 260-5, 269-73 Rectangular rosett e analysis, 258-61 four element , 275- 8 Mohr's circle, 261-5 principal stres s directions , 260- 1 transverse sensitivity , 296—300 Resistance, basi c equation s fo r uni t change , 236-8 Resistor, ballast , 100-1 , 119-26 Rosettes delta analysis, 267-9 Mohr's circle, 269-73 principal stres s directions , 26 9 transverse sensitivity , 301-6 geometry, 256-8 graphical solutions , 281- 7 rectangular analysis, 258-61 four element , 275-8 Mohr's circle, 261-5 principal stress directions, 261 transverse sensitivity , 296-300 stress equations , summary , 280-8 1

407

T-delta, 278-81 transverse sensitivity delta, 301- 6 rectangular, three-element , 296-300 two differen t orthogona l gages , 294- 6 two identica l orthogona l gages , 291- 4 Sanchez, J . C, 12 6 Semiconductor gages , 32-3 Semiconductor materials , 8- 9 Sensitivity variation full bridge , 227-3 1 half bridge , 218-25 reasons, 205- 6 single gage, 207-16 Shear gage , 330-4 Shear strain , Mohr' s circl e sign convention, 68-9 Shear stres s biaxial stres s state, 51- 2 determination o f plane, 327-30 Mohr's circl e sig n convention, 54- 6 Shoub, H. , 14 , 22 Stein, P . K. , 157 Strain apparent, se e Thermal outpu t basic concepts , 61- 2 correcting fo r thermal outpu t an d gag e factor variation , 348- 9 elastic, in metals, 7- 8 indicated vs . actual, 165-8 , 206 invariants, 81 Mohr's circle , 68-70 nonlinearity, 102 , 108-9, 163- 4 plastic, i n metals, 8 principal, 64- 5 shear, sign , 63- 4 small vs. large, 20- 4 temperature-induced, 337-4 0 thermal outpu t correction , 344- 7 transformation equations , 63- 5 Strain gag e alloys, 391- 3 basic principle , 5 bonding, 398-40 0 characteristics, 4- 5 compensating, 27-8, 353-7, 363 foil, 29-3 1 gage length, 394-5 general considerations , 390- 1 geometry, 39 5 lateral effect , 234-51 orthogonally crosse d pair , 248-51

408

Strain gag e (contd.) properties desired , 1 0 resistance, 395- 6 self-temperature compensated , 131-2 , 343-5 self-temperature compensation , 39 6 temperature effects , 33 7 test materia l mismatch , 349 5 1 use under conditions differin g fro m calibration, 246-8 weldable foil, 3 1 wire, 27- 9 wire, 2 4 9 Strain sensitivity analysis, 14-2 4 general case , 14-1 7 small vs . large strain , 17-2 4 uniform straigh t wire , 17-2 0 axial an d normal , 236- 8 gage facto r relation , 240- 2 definition, 5-8 , 23 6 material properties , desired , 1 0 numerical values , 11-1 3 reasons fo r varying , 205-6 Strain transformatio n equations, 63- 5 Stress basic concepts , 43- 4 biaxial, 4 5 circuit, indicatio n o f normal stress , 32 0 fields, 253-6 invariants, 81 Mohr's circle , 54- 7 principal, 48-53 , 260-5 , 269-73 using a singl e gage, 326- 7 sign convention , 45 transformation equations , 45-5 3 Stress gage normal, 310-1 2 single roun d wire , L configuration, 312-14 two orthogona l gages , 314-1 6 V-type, 321- 5 Stress-strain gage , 316-2 0 Stress-strain relations , 72- 7 Stress transformatio n equations , 45-5 3 Temperature gages , 33- 4 Temperature-induced strain , 337-4 0 Thermal expansio n coefficients , 350 Thermal output , 338 , 34 4 correction, 344- 7 Thevenin's theorem , 17 3

INDEX Torque meter , 378-8 0 Torque wrench , 380-2 Transducers axial force , 363- 5 bending beam , 372- 5 cantilever beam , 368-7 2 full bridge , 361- 2 half bridge , 36 2 pressure measurement diaphragm. 384-7 thin-walled cylinder , 383-4 quarter bridge . 36 3 ring-type, 365- 8 shear beam . 375- 7 torque meter , 378-8 0 torque wrench , 380-2 Transformation equation s strain, 63- 5 stress. 45-5 3 summary, strain , 65 summary, stress , 52- 3 Transverse sensitivity definition, 238-4 0 delta rosette , 301- 6 determination. 244-6 rectangular rosette , 296-30 0 two differen t orthogona l gages , 294- 6 two identica l orthogona l gages , 291- 4 typical values , 239-40 Unbonded wir e strai n gage , 2 4 Weibull, W. , 12 , 14 , 20, 2 2 Weldable strai n gag e foil, 3 1 wire, 27- 9 Weymouth, L . J. . 14 1 Wheatstone bridg e bridge inpu t resistance , 151-2 , 173- 5 bridge outpu t resistance , 151-2 , 176- 7 bridge ratio , 155 , 159-6 1 calibration, 193- 5 derivation o f elementary bridg e equations, 157-65 , 169-7 2 elementary bridg e equations , 149-5 2 general bridg e equations , 172- 9 lead-line resistance , 180-9 1 meter current , 152-3 , 179 meter resistance , 175- 9 nonlinearity. 150-1 , 163- 4 null balanc e referenc e bridge, 15 4 null balanc e svstem , 153

INDEX 40 reference system , 153- 4 unbalanc resistance i n serie s wit h bridge , 172- 5 Williams summary o f properties, 155- 7 Wnuk unbalance referenc e bridge, 15 4 Wright

9 e system , 153 , S. B., 320, 33 5 , S. P., Jr., 14 1 , W. V. , 126