Micro-Mesuremens Srin Gges nd Insrumens Errors Due o Trnsverse Sensiivi in Srin Gges Tech Noe TN-509 Trnsverse Sensiivi Trnsverse sensiivi in srin gge refers o he behvior of he gge in responding o srins which re perpendiculr o he primr sensing xis of he gge. Idell, i would be preferble if srin gges were compleel insensiive o rnsverse srins. In prcice, mos gges exhibi some degree of rnsverse sensiivi; bu he effec is ordinril quie smll, nd of he order of severl percen of he xil sensiivi. In plne wire srin gges, rnsmission of srin ino he wire from direcion perpendiculr o he wire xis is nerl negligible. As resul, he rnsverse sensiivi of hese gges is due lmos enirel o he fc h porion of he wire in he end loop lies in he rnsverse direcion. Becuse of his, he sign of he rnsverse sensiivi for plne wire gge will lws be posiive, nd he mgniude of he effec cn be clculed quie closel from he geomer of he grid. This semen does no ppl o he smll wrpround gges hving he wire wound on flened core. Such gges ofen exhibi negive rnsverse sensiiviies. In foil srin gges, on he oher hnd, he rnsverse sensiivi rises from much more complex phenomen, nd i is ffeced b lmos ever spec of grid design nd gge consrucion. In ddiion o end loop effecs, he foil gridlines, hving lrge rio of widh o hickness, re srined significnl b rnsverse srins. The mgniude of rnsverse srin rnsmission ino he gridlines is deermined b he relive hicknesses nd elsic moduli of he bcking nd foil, b he widh-o-hickness rio of he foil gridlines, nd, o lesser degree, b severl oher prmeers, including he presence or lck of n encpsuling ler over he grid. Depending upon he foil meril nd is mellurgicl condiion, he conribuion o rnsverse sensiivi from he rnsmission of rnsverse srin ino he gridlines cn be eiher posiive or negive. Becuse of his, he overll rnsverse sensiivi of foil srin gge cn lso be eiher posiive or negive. While he rnsverse sensiivi of foil gge is hus subjec o greer degree of conrol in he design of he gge, he compromises necessr o opimize ll specs of gge performnce generll limi he inble reducion in rnsverse sensiivi. Errors Due o Trnsverse Sensiivi Errors in srin indicion due o rnsverse sensiivi re generll quie smll since he rnsverse sensiivi iself is smll. However, in bixil srin fields chrcerized b exreme rios beween principl srins, he percenge error in he smller srin cn be ver gre if no correced for rnsverse sensiivi. On he oher hnd, in he priculr cse of unixil sress in meril wih Poisson s rio of 0.85, he error is zero becuse he gge fcor given b he mnufcurer ws mesured in such unixil sress field nd lred includes he effec of he Poisson srin. I is imporn o noe h when srin gge is used under n condiions oher hn hose emploed in he gge-fcor clibrion, here is lws some degree of error due o rnsverse sensiivi. In oher words, n gge which is: () inslled on meril wih differen Poisson s rio; or (b) inslled on seel, bu subjeced o oher hn unixil sress se; or (c) even inslled on seel wih unixil sress se, bu ligned wih oher hn he mximum principl sress, exhibis rnsverse-sensiivi error which m require correcion. The hisoricl prcice of quoing gge fcors which, in effec, msk he presence of rnsverse sensiivi, nd which re correc in hemselves for onl specific sress field in specific meril, is n unforune one. This pproch hs generll compliced he use of srin gges, while leding o errors nd confusion. Alhough he unixil sress field is ver common, i is no highl significn o he generl field of experimenl sress nlsis. There is no priculr meri, herefore, in combining he xil nd rnsverse sensiiviies for his cse. In generl, hen, srin gge cull hs wo gge fcors, nd, which refer o he gge fcors s deermined in unixil srin field (no unixil sress) wih, respecivel, he gge xes ligned prllel o nd perpendiculr o he srin field. or n srin field, he oupu of he srin gge cn be expressed s: R R + (), srins prllel o nd perpendiculr o he gge xis, or he gridlines in he gge. xil gge fcor. rnsverse gge fcor. Tech Noe Documen Number: 059 Revision: 0-Nov-00 or echnicl suppor, conc micro-mesuremens@vishpg.com www.micro-mesuremens.com 9
TN-509 Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges Or, R R + () n + ν 0 ν 0 00 (5) rnsverse sensiivi coefficien, referred o from here on s he rnsverse sensiivi. When he gge is clibred for gge fcor in unixil sress field on meril wih Poisson s rio, ν 0, Therefore, or, ν 0 R R R R ν 0 ν (3) 0 The srin gge mnufcurers commonl wrie his s: n he error s percenge of he cul srin long he gge xis. ν 0 he Poisson s rio of he meril on which he mnufcurer s gge fcor,, ws mesured (usull 0.85)., respecivel, he cul srins prllel nd perpendiculr o he primr sensing xis of he gge.* rom he bove equion, i is eviden h he percenge error due o rnsverse sensiivi increses wih he bsolue vlues of nd /, wheher hese prmeers re posiive or negive. Equion (5) hs been ploed in igure for convenience in judging wheher he mgniude of he error m be significn for priculr srin field. igure lso ields n pproxime rule-of-humb for quickl esiming he error due o rnsverse sensiivi h is, n x 00 (percen) Tech Noe R R (3) mnufcurer s gge fcor, which is decepivel simple in ppernce, since, in reli: ν (4) 0 urhermore, is cull, he srin long he gge xis (nd onl one of wo srins sensed b he gge during clibrion) when he gge is ligned wih he mximum principl sress xis in unixil sress (no unixil srin) field, on meril wih ν 0 0.85. Errors nd confusion occur hrough filure o full comprehend nd lws ccoun for he rel menings of nd s used b he mnufcurers. I is imperive o relize h for n srin field excep h corresponding o unixil sress field (nd even in he ler cse, wih he gge mouned long n direcion excep he mximum principl sress xis, or on n meril wih Poisson s rio oher hn 0.85), here is lws n error in srin indicion if he rnsverse sensiivi of he srin gge is oher hn zero. In some insnces, his error is smll enough o be negleced. In ohers, i is no. The error due o rnsverse sensiivi for srin gge oriened n ngle, in n srin field, on n meril, cn be expressed s: As Equion (5) shows, his pproximion holds quie well s long s he bsolue vlue / is no close o ν 0. or n exmple, ssume he sk of mesuring Poisson (rnsverse) srin in unixil sress field. In his cse, he Poisson srin is represened b, he srin long he gge xis, nd he longiudinl srin in he es member b, since he ler is rnsverse o he gge xis (see skech nd foonoe below). P ν ν * Subscrips () nd () lws refer o he xil nd rnsverse direcions wih respec o he gge (wihou regrd o direcions on he es surfce), while subscrips (x) nd () refer o n rbirr se of orhogonl xes on he es surfce, nd subscrips ( p) nd (q) o he principl xes. P www.micro-mesuremens.com 9 or echnicl quesions, conc micro-mesuremens@vishpg.com Documen Number: 059 Revision: 0-Nov-00
Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges TN-509 5 / / 5 Consider firs he wo-gge 90-degree rosee, wih he gge xes ligned wih wo orhogonl xes, x nd, on he es surfce. When using his pe of rosee, he x nd xes would ordinril be he principl xes, bu his need no necessril be so. The correc srins long n wo perpendiculr xes cn lws be clculed from he following equions in erms of he indiced srins long hose xes: x ν ˆ ˆ 0 x ν0 ˆ ˆ x (6) (7) igure If he es specimen is n luminum llo, wih ν 0.3, hen / /ν 3.. Assuming h he rnsverse sensiivi of he srin gge is 3% (i.e., 0.03*), he rule of humb gives n pproxime error of +9.3%. The cul error, clculed from Equion (5), is +8.5%. ˆ ˆ x ˆ he indiced (uncorreced) srin from gge no.. ˆ he indiced (uncorreced) srin from gge no.. x, correced srins long he x nd xes, respecivel. The ( ) erm in he denominors of Equions (6) nd (7) is generll in excess of 0.995, nd cn be ken s uni: ν ˆ 0 ˆ (6) x x Correcing for Trnsverse Sensiivi The effecs of rnsverse sensiivi should lws be considered in he experimenl sress nlsis of bixil sress field wih srin gges. Eiher i should be demonsred h he effec of rnsverse sensiivi is negligible nd cn be ignored, or, if no negligible, he proper correcion should be mde. Since wo- or hree-gge rosee will ordinril be used in such cses, simple correcion mehods re given here for he wo-gge 90-degree rosee, he hree-gge recngulr rosee, nd he del rosee. Unless oherwise noed, hese correcions ppl o rosees in which he rnsverse sensiiviies of he individul gge elemens in he rosees re equl o one noher, or pproximel so. Generlized correcion equions for n combinion of rnsverse sensiiviies re given in he Appendix. * or subsiuion ino n equion in his Tech Noe, mus lws be expressed decimll. Thus, he vlue of (in percen) from he gge pckge d shee mus be divided b 00 for conversion o is deciml equivlen. ν0 ˆ ˆ (7) x D reducion cn be furher simplified b seing he gge fcor conrol on he srin-indicing insrumenion insed of, he mnufcurer s gge fcor. Since, ν 0 Equions (6) nd (7) cn be rewrien: ˆ ˆ x x ˆ ˆ x ˆ, ˆ x srins s indiced b insrumenion wih gge fcor conrol se (6b) (7b) Tech Noe Documen Number: 059 Revision: 0-Nov-00 or echnicl quesions, conc micro-mesuremens@vishpg.com www.micro-mesuremens.com 93
TN-509 Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges ν0 As n lernive o he preceding mehods, quick grphicl correcion for he rnsverse sensiivi cn be mde hrough he use of igure. To use he grph, he firs sep is o clcule: ˆ ˆ ˆ ˆ ˆ ˆ x ˆ ˆ ˆ x ˆ ˆ ˆ (Gge No.) (Gge No.) Hving done his, i is onl necessr o ener he grph he pproxime vlue of, move upwrd o he line (or inerpoled line) represening he observed (indiced) srin rio, ( ˆ / ˆ )for h priculr rosee elemen, nd horizonll o he vericl scle on he lef o red he correcion fcor. Then, x C ˆ 5 ˆ / ˆ ˆ / ˆ 5 Similrl, C ˆ ollowing is numericl exmple uilizing firs Equions (6) nd (7), nd hen igure. Assume h he indiced srins for rosee elemens () nd () long he x nd xes re, respecivel: igure IN% ˆ + 530µ ˆ + 90µ Assume lso h 0.06. Subsiuing ino Equions (6) nd (7), wih ν 0 0.85, ollowing he line for 0.06 upwrd, inerpoling he locion of ( ˆ / ˆ ) 0.6, nd ( ˆ / ˆ ).65, nd reding he respecive vlues of he correcion fcor, C.06; C. rom which, Tech Noe x ( + 0.85 x 0.06) (530 + 0.06 x 90) 6μ ( + 0.85 x 0.06) (90 + 0.06 x 530) 09μ or use wih he correcion grph, igure, ˆ.. ˆ 90 0 60 530 06 ˆ ˆ 530. 663 65. 90 C ˆ 06. x 530 60µ x x C ˆ. x 90 030 µ Correcion or Sher Srin A wo-gge, 90-degree rosee, or T -rosee, is someimes used for he direc indicion of sher srin. I cn be shown h he sher srin long he bisecor of he gge xes, is, in his cse, numericll equl o he difference in norml srins on hese xes. Thus, when he wo gge elemens of he rosee re conneced in djcen rms of Whesone bridge, he indiced srin is equl o he indiced sher srin long he bisecor, requiring mos correcion for www.micro-mesuremens.com 94 or echnicl quesions, conc micro-mesuremens@vishpg.com Documen Number: 059 Revision: 0-Nov-00
Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges TN-509 he error due o rnsverse sensiivi. The ler error cn be correced for ver esil if boh gges hve he sme rnsverse sensiivi, since he error is independen of he se of srin. The correcion fcor for his cse is: C γ ν0 (8) The cul sher srin is obined b mulipling he indiced sher srin b he correcion fcor. Thus, Correcion cor Cγ ν γ Cγ ˆ γ Cγ ( ˆ ) x ˆ 0 x ˆ ˆ ν Cγ 0 ν 0 85 0. γ ν 0 Wih his chnge, he srin indicor will indice he cul sher srin long he bisecor of he gge xis, lred correced for rnsverse sensiivi in he srin gges. Three-Gge Recngulr (45 ) Rosee When he direcions of he principl xes re unknown, hree independen srin mesuremens re required o compleel deermine he se of srin. or his purpose, hree-gge rosee should be used, nd he recngulr rosee is generll he mos convenien form. If he rnsverse sensiivi of he gge elemens in he rosee is oher hn zero, he individul srin redings will be in error, nd he principl srins nd sresses clculed from hese d will lso be incorrec. Correcion for he effecs of rnsverse sensiivi cn be mde eiher on he individul srin redings or on he principl srins or principl sresses clculed from hese. Numbering he gge elemens consecuivel, elemens () nd (3) correspond direcl o he wo-gge, 90-degree rosee, nd correcion cn be mde wih Equions (6) nd (7), or (6) nd (7), or (b properl seing he gge fcor conrol on he srin indicor) wih Equions (6b) nd (7b). The cener gge of he rosee requires specil correcion relionship since here is no direc mesuremen of he srin perpendiculr o he grid. The correcion equions for ll hree gges re lised here for convenience: ν0 ν0 ( ˆ ˆ ) 3 ˆ ( ˆ + ˆ 3 ˆ ) (9) (0) () ν0 3 [ ˆ ˆ ] 3 () igure 3 IN% or convenience, he sher srin correcion fcor is ploed in igure 3 gins, wih ν 0 0.85. Since his correcion fcor is independen of he se of srin, i cn gin be incorpored in he gge fcor seing on he srinindicing insrumenion if desired. This cn be done b seing he gge fcor conrol : ˆ, ˆ, ˆ 3 indiced srins from he respecive gge elemens.,, 3 correced srins long he gge xes. I should be noed h Equions (0), (), nd () re bsed upon he ssumpion h he rnsverse sensiivi is he sme, or effecivel so in ll gge elemens, s i is in scked rosees. This m no be rue for plnr foil rosees, Tech Noe Documen Number: 059 Revision: 0-Nov-00 or echnicl quesions, conc micro-mesuremens@vishpg.com www.micro-mesuremens.com 95
TN-509 Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges since he individul gge elemens do no ll hve he sme orienion wih respec o he direcion in which he foil ws rolled. I is common prcice, however, o ech he rosee in posiion of smmer bou he foil rolling direcion, nd herefore he rnsverse sensiiviies of gge elemens () nd (3) will be nominll he sme, while h of elemen () m differ. Correcion equions for rosees wih nonuniform rnsverse sensiiviies mong he gge elemens re given in he Appendix. Del Rosees A del srin gge rosee consiss of hree gge elemens in he form of n equilerl ringle or Y wih equll spced brnches. The del rosee offers ver sligh poenil dvnge over he hree-gge recngulr rosee in h he lowes possible sum of he srin redings obinble in priculr srin field is somewh higher hn for hree-gge recngulr rosee. This is becuse he hree gge elemens in he del rosee re he grees possible ngle from one noher. However, he d reducion for obining he principl srins or correcing for rnsverse sensiivi is lso more involved nd lengh hn for recngulr rosees. As in he cse of recngulr rosees, plne foil del rosees re mnufcured smmericll wih respec o he rolling direcion of he foil. Thus, wo of he gge elemens will ordinril hve he sme nominl rnsverse sensiivi, nd hird m differ. Correcion equions for his condiion re given in he Appendix. In he scked del rosee, ll hree gges hve he sme nominl sensiivi. The individul srin redings from del rosee cn be correced for rnsverse sensiivi wih he following relionships when single vlue of cn be used for he rnsverse sensiivi: ν 0 Correcion of Principl Srins Wih n rosee, recngulr, del, or oherwise, i is lws possible (nd ofen mos convenien) o clcule he indiced principl srins direcl from he compleel uncorreced gge redings, nd hen ppl correcions o he principl srins. This is rue becuse of he fc h he errors in principl srins due o rnsverse sensiivi re independen of he kind of rosee emploed, s long s ll gge elemens in he rosee hve he sme nominl rnsverse sensiivi. Since Equions (6) nd (7) ppl o n wo indiced orhogonl srins, he mus lso ppl o he indiced principl srins. Thus, if he indiced principl srins hve been clculed from srin redings uncorreced for rnsverse sensiivi, he cul principl srins cn redil be clculed from he following: p q ν 0 ν 0 ( ˆ p ˆ q ) ( ˆ q ˆ p ) (6) (7) urhermore, Equions (6) nd (7) cn be rewrien o express he cul principl srin in erms of he indiced principl srin nd correcion fcor. Thus, ν ˆ q p ˆ 0 p ˆ p (8) Tech Noe ν 0 + 3 ˆ ( ˆ + ˆ ) 3 3 ν0 + ˆ ( ˆ + ˆ ) 3 3 3 ν0 + ˆ 3 3 ( ˆ + ˆ ) 3 3 (3) (4) (5) As before, simplificion cn be chieved b reing ( ) s uni, nd b incorporing he quni ( ν 0 ) ino he gge fcor seing for he srin insrumenion. When doing his, he gge-fcor conrol is se : ν q ˆ 0 q ˆ p ˆ q (9) Since Equions (8) nd (9) re he sme relionship used o plo he correcion grph of igure, his grph cn be used direcl o correc indiced principl srins b he procedure described erlier, merel noing h: ˆ ˆ q when correcing ˆ ˆ ˆ p p www.micro-mesuremens.com 96 or echnicl quesions, conc micro-mesuremens@vishpg.com Documen Number: 059 Revision: 0-Nov-00
Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges TN-509 nd ˆ ˆ ˆ p when correcing ˆ ˆ q q In fc, he indiced srins from hree gges wih n relive ngulr orienion define n indiced Mohr s circle of srin. When emploing d-reducion scheme h produces he disnce o he cener of Mohr s circle of srin, nd he rdius of he circle, sill noher simple correcion mehod is pplicble. To correc he indiced Mohr s circle o he cul Mohr s circle, he disnce o he cener of he indiced circle should be muliplied b ( ν 0 )/( + ), nd he rdius of he circle b ( ν 0 )/( - ). The mximum nd minimum principl srins re he sum nd difference, respecivel, of he disnce o he cener nd he rdius of Mohr s circle of srin. Bibliogrph ASTM Sndrd E5, Pr III. Sndrd Tes Mehod for Performnce Chrcerisics of Bonded Resisnce Srin Gges. Avril, J. L Effe Lérl des Juges Élecriques. GAMAC Conference. April 5, 967. Bumberger, R. nd. Hines. Prcicl Reducion ormuls for Use on Bonded Wire Srin Gges in Two- Dimensionl Sress ields. Proceedings of he Socie for Experimenl Sress Anlsis II: No., 3-7, 944. Bossr,. J. nd G. A. Brewer. A Grphicl Mehod of Rosee Anlsis. Proceedings of he Socie for Experimenl Sress Anlsis IV: No., -8, 946. Cmpbell, W, R, Performnce Tess of Wire Srin Gges: IV Axil nd Trnsverse Sensiiviies. NACA TN04, 946. Gu, W. M. A Simplified Mehod for Elmining Error of Trnsverse Sensiivi of Srin Gge. Experimenl Mechnics : No. 6-8, Jnur 98. Meier, J.H. The Effec of Trnsverse Sensiivi of SR-4 Gges Used s Rosees. Hndbook of Experimenl Sress Anlsis, ed. b M. Heénri, John Wile & Sons, pp. 407-4, 950. Meier, J. H. On he Trnsverse-srin Sensiivi of oil Gges. Experimenl Mechnics : 39-40, Jul 96. Meer, M.L. A Unified Rionl Anlsis for Guge cor nd Cross-Sensiivi of Elecric-Resisnce Srin Guges. Journl of Srin Anlsis : No. 4, 34-33, 967. Meer, M. L. A Simple Esime for he Effec of Cross Sensiivi on Evlued Srin-gge Mesuremen. Experimenl Mechnics 7: 476-480, November 967. Murr, W.M. nd P.. Sein. Srin Gge Techniques. Msschusees Insiue of Technolog, Cmbridge, Msschuses, pp. 56-8, 959. Nsudevn, M. Noe on he Effec of Cross-Sensiivi in he Deerminion of Sress. STRAIN 7: No., 74-75, April 97. Srr, J.E. Some Unold Chpers in he Sor of he Mel ilm Srin Gges. Srin Gge Redings 3: No. 5, 3, December 960 Jnur 96. Wu, Chrles T. Trnsverse Sensiivi of Bonded Srin Gges. Experimenl Mechnics : 338-344, November 96. Tech Noe Documen Number: 059 Revision: 0-Nov-00 or echnicl quesions, conc micro-mesuremens@vishpg.com www.micro-mesuremens.com 97
TN-509 Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges APPENDIX The following relionships cn be used o correc for rnsverse sensiivi when he gge elemens in rosee do no ll hve he sme vlue of. In ech cse, ν 0 is he Poisson s rio of he meril on which he mnufcurer s gge fcor ws mesured (usull 0.85). Two-Gge, 90-Degree rosee ˆ ( ν0 ) ˆ ν 0 ˆ ( ν0 ) ˆ ν 0 (0) () ˆ, ˆ indiced srins from gges () nd (), uncorreced for rnsverse sensiivi., rnsverse sensiiviies of gges () nd ()., cul srins long gge xes () nd (). Three-Gge Recngulr (45-Degree) Rosee ˆ ( ν0 ) ˆ ν 0 3 3 3 ˆ ( ν ν 0 ) ( 0 ) ( 3)+ ˆ ( ν0 ) ˆ ν 3 3 0 3 3 3 ( ) ( ) ˆ ˆ3 ν 0 3 3 () (3) (4) Tech Noe When he rnsverse sensiiviies of he orhogonl gges () nd (3) re nominll he sme, le 3 3 www.micro-mesuremens.com 98 or echnicl quesions, conc micro-mesuremens@vishpg.com Documen Number: 059 Revision: 0-Nov-00
Micro-Mesuremens Errors Due o Trnsverse Sensiivi in Srin Gges TN-509 Then: ν0 ( ˆ ˆ ) 3 3 3 3 ( ) + ( ) ( ν0 ) + ˆ 3 ν 0 ˆ 3 ˆ 3 + 3 (5) (6) 0 3 ν 3 ˆ 3 ˆ 3 3 ˆ, ˆ, ˆ 3 indiced srins from gges (), (), nd (3), uncorreced for rnsverse sensiivi.,, 3 rnsverse sensiiviies of gges (), (), nd (3). (7) 3 rnsverse sensiivi of orhogonl gges () nd (3).,, 3 cul srins long gge xes (), (), nd (3). Del Rosee ˆ ( ν0 ) ( 3 3 3) ˆ ( ν0 )( )+ ˆ ( ν 0 ) 3 +3 3 3 3 3 3 3 3 3 ˆ ( ν0 3 )( 3 3 ) ˆ 3( ν0 )( )+ ˆ ( ν 0 ) 3 +3 3 3 3 3 3 3 ˆ 3 ν0 3 3 ( ) ˆ 3 ( ν0 )( )+ ˆ ( ν 0 ) 3 +3 3 3 3 3 (8) (9) (30) When wo of he gges, for exmple, () nd (3), hve he sme nominl rnsverse sensiivi, ˆ ( ν0 3 )( 3 3 3 ) ˆ 3 ( ν0 )( )+ ˆ ( ν0 ) 3 + 3 3 3 3 3 3 3 3 (3) ˆ ( ν0 ν ) ( 3+ 3) ( ˆ + ˆ 3) ( 03 ) 3 + 3 3 3 ˆ 3( ν0 ) 3 3 ( 3 3 ) ˆ 3 ( ν 0 3 )( )+ ˆ ν 3 0 3 + 3 3 3 3 3 ( ) 3 The subscrips in Equions (8) hrough (33) hve he sme significnce s in Equions () hrough (7), excep h he wo gges wih common rnsverse sensiivi, 3, re no orhogonl. (3) (33) Tech Noe Documen Number: 059 Revision: 0-Nov-00 or echnicl quesions, conc micro-mesuremens@vishpg.com www.micro-mesuremens.com 99