STATIC ANALYSIS OF TENSEGRITY STRUCTURES

Transcription

1 SI NYSIS O ENSEGIY SUUES JUIO ES OE HESIS PESENED O HE GDUE SHOO O HE UNIVESIY O OID IN PI UIEN O HE EQUIEENS O HE DEGEE O SE O SIENE UNIVESIY O OID

2 o m mother for her fte geerost.

3 KNOWEDGENS I wat to tha to Dr. arl rae ad Dr. l Serg, memers of m commttee for ther overseeg of the thess ad for ther valuale suggestos. specal thas go to Dr. Joseph Duff, m commttee charma, for showg cofdece m wor, ad for hs support ad dedcato. ore tha academc owledge I have leared from hm that wsdom ad smplct go together.

4 E O ONENS page KNOWEDGENS... S... v HPES INODUION... SI ONEPS...4. he Prcple of Vrtual Wor...4. Plücer oordates rasformato atrces....4 eacto orces ad eacto omets Numercal Eample Verfcato of the Numercal esults GENE EQUIONS O HE SIS O ENSEGIY SUUES Geeraled oordates he Prcple of Vrtual Wor for esegrt Structures oordates of the Eds of the Struts Ital odtos he Vrtual Wor Due to the Eteral orces he Vrtual Wor Due to the Eteral omets he Potetal Eerg he Geeral Equatos NUEI ESUS alss of esegrt Structures ther Uloaded Postos alss of oaded esegrt Structures Eample : alss of a esegrt Structure wth 3 Struts alss for the Uloaded Posto alss for the oaded Posto...66 v

5 4.4 Eample : alss of a esegrt Structure wth 4 Struts alss for the Uloaded Posto alss for the oaded Posto Eample 3: alss of a esegrt Structure wth 6 Struts alss for the Uloaded Posto alss for the oaded Posto ONUSIONS... PPENDIES IS EQUIIIU EQUION O HE SIS O ENSEGIY SUUE WIH 3 SUS...4 SOWE O HE SI NYSIS O ENSEGIY SUUE...5 EEENES...7 IOGPHI SKEH...9 v

6 stract of hess Preseted to the Graduate School of the Uverst of lorda Partal ulfllmet of the equremets for the Degree of aster of Scece SI NYSIS O ENSEGIY SUUES Julo ésar orrea ugust harma: Dr. Joseph Duff aor Departmet: echacal Egeerg esegrt structures are three dmesoal assemlages formed of rgd ad elastc elemets. he hold the promse of ovel applcatos. However ther ehavor s ot completel uderstood at ths tme. hs research addresses the statc aalss prolem ad determes the posto assumed the structure whe eteral loads are appled. he dervato of the mathematcal model for the equlrum postos of the structure s ased o the vrtual wor prcple together wth cocepts related to geometr of les. he soluto for the resultat equatos s performed usg umercal methods. Several eamples are preseted to demostrate ths approach ad all the results are carefull verfed. software that s ale to geerate ad to solve the equlrum equatos s developed. he software also permts oe to vsuale dfferet equlrum postos for the aaled structure ad ths wa to ga sght the phscs ad the geometr of tesegrt sstems. v

7 HPE INODUION esegrt structures are spatal structures formed a comato of rgd elemets (the struts) ad elastc elemets (the tes). No par of struts touch ad the ed of each strut s coected to three o-coplaar tes []. he struts are alwas compresso ad the tes teso. he etre cofgurato stads tself ad matas ts form solel ecause of the teral arragemet of the tes ad the struts []. esegrt s a arevato of teso ad tegrt. gure. shows a umer of at-prsm tesegrt structures formed wth 3, 4 ad 5 struts respectvel. gure.. esegrt structures coformed 3, 4 ad 5 struts. he developmet of tesegrt structures s relatvel ew ad the wors related have ol ested for the 5 ears. Keer [3] estalshes the relato etwee the rotato of the top ad ottom tes. oe [] presets procedures for the geerato of tesle structures phscal ad graphcal meas. Y []

8 otas Keer s results usg eerg derato ad fds the equlrum posto for the uloaded tesegrt prsms. Ster [4] develops geerc desg equatos to fd the legths of the struts ad elastc tes eeded to create a desred geometr. Sce o eteral forces are dered hs results are referred to the uload posto of the structure. Kght [5] addresses the prolem of stalt of tesegrt structures for the desg of a deploale atea. he prolem of the determato of the equlrum posto of a tesegrt structure whe eteral forces ad eteral momets act o the structure has ot ee studed prevousl. hs s the focus of ths research. It s ow that whe the sstems ca store potetal eerg, as the case of the elastc tes of a tesegrt structure, the eerg methods are applcale. or ths reaso the vrtual wor formulato was selected from several possle approaches to solve the curret prolem. Despte ther complet, at-prsm tesegrt structures eht a patter ther cofgurato. hs fact s used to develop a stet omeclature vald for a structure ad wth ths ass to develop the equlrum equatos. o smplf the dervato of a mathematcal model some assumptos are cluded. hose smplfcatos are related ascall wth the asece of teral dsspatve forces ad wth the umer ad fasho that eteral loads are appled to the struts of the structure. Eve for the smplest case the resultat equatos are legth ad hghl coupled. Numercal methods offer a alteratve to solve the equatos. Parallel to the research preseted here a software atla was developed. ascall

9 3 the software s ale to develop the equlrum equatos for a gve tesegrt structure ad to solve them whe the eteral loads are gve. he software uses the well ow Newto aphso method whch s mplemeted the fucto fsolve of atla. o avod the lmtatos of umercal methods to coverge to a aswer, the proper selecto of the tal codtos was dered carefull together wth the gudace of the soluto through small cremets of the eteral loads. Oce the equatos are solved, the output data sts of ascall lstg of the varous coordates of the eds of the structure epressed a gloal coordate sstem for the equlrum posto. Whe dealg wth three dmesoal sstems, the umercal results themselves are ot suffcet to uderstad the ehavor of the sstems. o assst to the compreheso of the results the software developed provdes graphc outputs. I ths wa the comple equlrum equatos are coected a eas wa to the phscal stuato. Oe mportat questo that arose was the valdt of the umercal results. hs pot s specall mportat whe oe ders the complet of the equatos. depedet valdato of the results was realed usg Newto s hrd aw. hs thess s ascall as follows: hapter troduces the asc cocepts related to the tools requred to develop the mathematcal model for a tesegrt structure, hapter 3 develops a sstematc omeclature for the elemets of a tesegrt structure ad presets the mathematcal model. hapter 4 provdes eamples to llustrate the applcato of the model.

10 HPE SI ONEPS he ma oectve of ths research s to fd the fal equlrum posto of a geeral at-prsm tesegrt structure after a artrar load ad or momet has ee appled. I ths chapter the ma cocepts volved the dervatos of the equatos that gover the statcs of the structure are preseted.. he Prcple of Vrtual Wor he prcple of vrtual wor for a sstem of rgd odes for whch there s o eerg asorpto at the pots of tercoecto estalshes that the sstem wll e equlrum f [6] N δ W δ r (.) where δw : vrtual wor. : force appled to the sstem at pot. δ r : vrtual dsplacemet of the vector r. N : umer of appled forces. he vrtual dsplacemet represets magar ftesmal chages δ r of the posto vector r that are stet wth the trats of the sstem ut otherwse artrar [7]. he smol δ s used to emphase the vrtual character 4

11 5 of the stataeous varatos. he vrtual dsplacemets oe the rules of dfferetal calculus. If the sstem has p degrees of freedom there are p geeraled coordates q,,,..., q, the the varato of r must e evaluated p wth respect to each geeraled coordate. ( q q... q ) r r,,, p δ r r r δq δq... q q r q p δq p p r δ r δq q (.) he prcple of vrtual wor ca e modfed to allow for the cluso of teral ervatve forces terms of potetal fuctos [6]. I geeral the vrtual wor cludes the cotruto of oth ervatve ad o-ervatve forces δ W δw c δw c (.3) where the suscrpts c ad c deote ervatve ad o-ervatve vrtual wor respectvel. he vrtual wor performed the o-ervatve forces ca e epressed as c δ W. δ r (.4) c

12 6 where c s the o-ervatve force ad s the umer of oervatve forces. Susttutg (.) to (.4) elds, c p c q q r δw δ. (.5) he vrtual wor performed the ervatve force ca e epressed the form [7] c V W δ δ (.6) where ),...,, ( p q q q V V s the potetal eerg assocated wth the ervatve force. herefore p p c q q V q q V q q V W δ δ δ δ... (.7) d the total vrtual wor performed the ervatve forces s gve m p c q q V δw δ (.8) where m s the umer of ervatve forces. Wth the ad of (.5) ad (.8), equato (.3) ca e rewrtte the form m p c p q q V q r δw δ m c p q q V q r δ (.9)

13 7 he prcple of vrtual wor requres that the precedg epresso vashes for the equlrum. ecause the geeraled vrtual dsplacemets δ q are all depedet ad hece etrel artrar, (.9) ca e satsfed [7], f ad ol f c r q V q m m V Q,,,, q where p (.) Q c r q (.) he term Q s ow as the geeraled forces ad despte ts ame ma clude oth the vrtual wor due to eteral o-ervatve forces ad the vrtual wor due to eteral o-ervatve momets. If the lower eds of the struts of a tesegrt sstem are traed to move o the horotal plae ad also the rotato aout the logtudal as of the strut s traed, the each strut has 4 degrees of freedom ad the whole sstem has p 4* _ struts (.) degrees of freedom where _ struts s the umer of struts of the structure.

14 8 (. Plücer oordates he coordates of a le og two fte pots wth coordates,, ) ad,, ) ca e wrtte as ( S $ (.3) S o where S s the drecto alog the le ad S s the momet of the le aout the org O. S ad S ca e evaluated from the coordates of the pots as follows [8] S (.4) where N (.5) (.6) N (.7) ad P S o Q (.8) where P (.9) Q (.) (.)

15 9 he umers,, N, P, Q ad are called the Plücer le coordates ad the caot e smultaeousl equal to ero. he Plücer le coordates ca e epressed uted form dvdg the vectors S ad S N provded, ad N are ot all equal to ero. S s $ (.) N S o s o force ca e epressed as a scalar multple of the ut vector s oud to the le. he momet of the force aout a referece pot O ca e epressed as a scalar multple of the momet vector s [9], therefore s $ f $ f (.3) s o where f stads for the magtude of the force. If, ad N are all equal to ero the uted Plücer le coordates have the form $ P Q S o s o (.4) d the Plücer le coordates of a pure momet are $ m $ m (.5) so where m stads for the magtude of the momet.

16 osder two coordates sstems show gure.. he org of sstem X ' Y ' Z' s traslated (,, ) ad rotated artrarl wth respect to sstem XYZ. he Plücer coordates of the le $ epressed the sstem X ' Y ' Z' ca e trasformed to the sstem XYZ usg the followg relato [9], $ e $' (.6) where $ : Plücer coordates of the le epressed the sstem XYZ $' : Plücer coordates of the le epressed the sstem X ' Y ' Z' ad e 3 O 3 (.7) where : rotato matr of the sstem X ' Y ' Z' wth respect to the sstem XYZ O : eroes 33 matr 3 3 (.8) oversel f the Plücer coordates of the le are gve the sstem X ' Y ' Z' ad t s desred to epress them the sstem XYZ, from (.6) $' e $ (.9)

17 $ ' ' ' gure.. Geeral chage of a coordate sstem. where e 3 O 3 (.3).3 rasformato atrces gure. shows a artrar pot P located o a strut of legth s. I a referece sstem D whose as s alog the as of the strut ad wth ts org D s located at the lower ed of the strut, the coordates of P are smpl (,, l ). However frequetl s more coveet for purposes of aalss to epress the locato of P the gloal referece sstem. hs ca e accomplshed a trasformato matr. If the lower ed of the strut s traed to move o the horotal plae ( ), ad also the rotato aout ts logtudal as s traed, the strut

18 ca e modeled a uversal ot. I ths wa the ot provdes the 4 degrees of freedom assocated wth the strut. D P s r l Po P gure.. Strut a artrar posto. he algmet of the as o the fed sstem wth the as of the rod ca e accomplshed usg the followg three ecutve trasformatos [] : raslato, t ( a,,), gure.3. Note that the coordate s ero ecause of the restrcto mposed to the movemet of the lower ed of the strut. otato, aout the curret as ( ), gure.4. otato, aout the curret as ( ), gure.5.

19 3 a t gure.3. raslato ( a,,) the sstem., gure.4. otato aout as the sstem.

20 4 gure.5. otato aout curret as the sstem. he coordates of P measured the gloal referece sstem are,, P P D D a (.3) where:,, a a (.3) s s (.33) s s D (.34) D, D D

21 5 D P l (.35) Susttutg the aove prevous epressos to (.3) elds P l s a l s l (.36) Whe the values of (,, ) are ow, the agles ad ca e calculated from (.36) ad ta (.37) ta a s (.38) or ta a (.39) s the sgs are ow for each umerator ad deomator, equatos (.37) through (.39) gve uque values for ad. he geeraled coordates assocated wth the degrees of freedom of the strut are a,,, ad ; therefore the vrtual dsplacemet δ r of r P gve (.36) ca e evaluated usg (.) as follows

22 6 δ r δ δ δ r a δa r δ r δ r δ or, δ δ δ δa δ s δ s s s δ ad therefore, δ δ a l δ (.4) δ δ l δ l s s δ (.4) δ l s δ l s δ (.4).4 eacto orces ad eacto omets he vrtual wor approach does ot eld the reacto forces ad reacto momets. he are otaed usg Newto s hrd aw. Several eteral forces have ee appled at artrar pots o the strut show gure.6a together wth a eteral momet whch s the resultat of the eteral momets appled alog the as of the uversal ot. oth eteral forces ad eteral momet are epressed the gloal referece sstem. gure.6 shows the reacto force ad the reacto momet eerted the support. he equlrum equato usg Plücer coordates epressed the gloal referece sstem s $ $ $ $ (.43)

23 7 r, r r (a) t () gure.6. Statc aalss of a strut. a) Eteral loads; ) eactos where: $ : Plücer coordates of the eteral force. $ : Plücer coordates of the eteral momet. $ : Plücer coordates of the reacto force.

24 8 $ : Plücer coordates of the reacto momet. : umer of eteral forces Sce $ ad $ are pure momets (.43) ca e rewrtte the form r t (.44) Usuall the frst ad secod terms together wth the posto vector t the thrd term of (.44) are ow ecause the correspod to ow data or as a result of the vrtual wor aalss. Hece the reacto force ad the reacto momet ca e solved easl from (.44)..5 Numercal Eample he followg eample helps to clarf the cocepts dscussed so far ad also troduces to the umercal techques emploed to solve the resultat equatos. gure.7 shows a massless strut of legth s oed to the horotal plae a uversal ot wthout frcto ts movg parts. he support of oe of the as of the uversal ot s frml attached to the groud therefore the ot caot perform a logtudal dsplacemet. he strut s tall equlrum ad the coordates of the upper ed, P,, the sstem are ow for the tal posto. he a tat force

25 9 P, s,,,, S.5 m N P, m.5.3 N m N m gure.7. Data for the statc aalss of a strut. ad two tat momets alog the as of the uversal ot are appled as t s show gure.7. he force s epressed a gloal referece sstem whose org s located at the tersecto of the aes of the uversal ot. Sce the coordates sstems ad are cocdet, the vector t whch represets the locato of the org of the sstem wth respect to the sstem s. It s requred to determe the fal equlrum posto of the strut ad the reacto force ad the reacto momet the support of the strut. he umercal values for P,,, ad the magtudes of the momets ad are llustrated gure.7.

26 our coordates sstems are defed followg the gudeles preseted o gures. 3 through.5. Sstem : gloal referece coordate sstem. Sstem : otaed after a traslato (,,) of sstem. Sstem : otaed after a rotato aout. Sstem D : otaed after a rotato aout Sstems, ad are show gure. 7. Wth ths otato epressed the sstem are. ad.5 N m (.45).3 N m (.46) he strut has degrees of freedom gve the rotatos of the uversal ot. he soluto of the prolem sts o fdg the value of that rotatos, e ad. he fal posto of the upper ed of the strut ca e foud wth the ad of P (.36) otg that r, f, l s, a ad. s s r P, f s s (.47) s

27 where r has ee epressed rectagular coordates stead of homogeeous coordates. he vrtual dsplacemet otg that r r(, ). δ r s otaed from (.) δ r r δ r δ rom (.47) s δ r δ s s s s δ (.48) s s s s Notg that the eteral force has o compoet, the vrtual wor δ W performed the eteral force s gve δw s. δ r s δ s s s δ s s s s d after smplfg δw δ s δ s δ (.49) S S S he vrtual wor due to the eteral momets δ W s gve δw δ δ (.5) s the scalar or dot product s varat uder coordate trasformato the last epresso ca e evaluated easl f the terms o the rght sde are epressed the sstem. Sce

28 ad the δ δ (.5) ad δ δ (.5) Susttutg (.45), (.46), (.5) ad (.5) to (.5) the vrtual wor due to the eteral momets s smpl δw δ δ (.53) he total vrtual wor s gve the sum of (.49) ad (.53) ad the equlrum must e ero, the S δ s δ s δ δ δ d re-groupg S S ( s ) δ ( s ) δ S S S (.54) Sce equato (.54) s vald for all values of δ ad δ whch are ot geeral equal to ero the

29 3 S s (.55) ad S S s (.56) or ths eample the resultat equatos (.55) ad (.56) are ot strogl coupled ad t s possle to ota a soluto closed form, however the most geeral prolems ths s ot the case ad t wll e show that umercal solutos are easer to mplemet. ver well ow umercal techque s the Newto-aphso method. he fucto fsolve of atla s used to mplemet the Newto-aphso algorthm. I order to use t s ecessar to specf the set of equatos to e solved, for stace (.55) ad (.56) the curret eample, together wth the tal values of ad. he tal values of ad, ( ad ) ca e calculated from (.37) ad (.38) otg that a..5 ta o o ta o o s s 48. Wth these tal codtos the results gve the software are o 7.5 ad o 7.7 (.57) Susttutg these values ad the value of s to (.47) elds

30 4 P.37 f,.8 m (.58).4 he result s llustrated gure.8. gure.8. al equlrum posto of the strut. soluto umercal methods s hghl sestve to a correct selecto of the tal values. or ths eample the locato of P, was gve eplctl ad ths fact permtted to evaluate ad, ut the aalss of tesegrt structures t s ecessar to fd them usg aother approach. hs topc wll e dscussed detal Secto 3.4. ale. shows the results otaed whe artrarl aother set of agles ad are chose as tal guesses. lthough the Newto-aphso algorthm stll elds umercal results ad that results are equlrum postos, the solutos lsted ale. are ot compatle wth the tal codtos of ths eercse. I geeral f the tal values are ot correct the algorthm wll ot coverge to a soluto or to fd aswers that caot e realed practcall.

31 5 ale.. Numercal solutos for dfferet tal codtos. 35º -º 8.7º.7º 5º 3º 7.5º 7.7º 35º 5º 6.3º -.7º other mportat derato to assure the qualt of the umercal solutos s to avod large cremets the put values. It s alwas possle to crease graduall the value of the eteral momets ad forces, for the statc case. I ths wa the umercal soluto s guded wthout dffcult. Oce the equlrum posto s solved the et step s to evaluate the reacto force ad the reacto momet. or ths eample there s ol a sgle eteral force ad t s appled at the upper ed of the strut, ad due to the fact sstems ad are cocdet, the vector t s ero, as show gure.7. or the fal equlrum posto of the strut, (.44) ecomes, f P (.59) f P, (.59) s gve (.47). Usg ths result the frst term of (.59) ca e epaded as

32 6 P, f s ( s ) s( s ) ( ) s s s (.6) he eteral momet s geerated the eteral momets ad. However the were epressed the sstem, (see gure.7). However (.59) requres them to e epressed sstem. It s ot dffcult to estalsh the geometrc relatoshps etwee sstems ad. Here the use of the geeral relatos (.6) to (.8) s preferred ecause the are more useful more comple stuatos. s $ s the resultat of ad oth epressed the sstem $ e e ( $ $ ) (.6) where e : matr that trasforms a le epressed the sstem to the sstem. $ : Plücer coordates of the sstem. $ : Plücer coordates of the sstem. atr e s otaed usg (.7) ad for ths case e 3 3 (.6) s gve Sce the orgs of sstems ad are cocdet the 3 (see (.8))

33 7 3 (.63) he rotato matr s otaed from the followg trasformato, (.64) rom gure.7 s apparet that sstems ad are parallel, the (.65) rom gures.4 ad.7 s clear that sstem s otaed after a rotato aout, the s s (.66) rom (.65) ad (.66) s apparet that s s (.67) Susttutg (.67) together wth (.63) to (.6) elds

34 8 s s s s e (.68) he Plücer coordates of gve (.45) are $ (.69) he Plücer coordates of gve (.46) are $ (.7) Susttutg (.68), (.69) ad (.7) to (.6) elds s $ (.7) Susttutg (.6) ad (.7) to (.59) ad solvg for uows

35 9 ( ) S s (.7) ( ) s S ( ) s s s S ecallg the data provded gure.7 ad the results otaed (.57) N S 5m. m N m N he reacto force ad reacto momet ca e otaed from (.7). her umercal values are m N m N m N N N N.36.43

36 3.6 Verfcato of the Numercal esults s t wll e show the et chapter the aalss of tesegrt structures volves ver comple ad legth equatos. If there s a error the dervato of the equato the umercal methods stll gve a aswer. However the aswer does ot of course correspod to the real stuato. It s desrale to verf the valdt of the aswers otaed usg the vrtual wor approach. Newto s hrd aw asssts the verfcato. ascall the dea s to state the equlrum equato such a wa that some of the reactos vash. he resultat equato depeds ol o the put data ad o the geeraled coordates. If the umercal values of the geeraled coordates otaed usg the vrtual wor approach are correct, the must satsf the equlrum equatos otaed usg the Newtoa approach. hese cocepts are demostrated usg the last eample. he equlrum equato (.43) the sstem for the strut of Secto.5 s $ $ $ $ (.74) $ s otaed epressg $ the sstem usg (.9) ad (.3) ad otg that the term correspodg to the traslato dsplacemet s ero $ e $ (.75) where e O 3 O 3 (.76)

37 3 was otaed (.67). Susttutg the traspose of (.67) to (.76) elds s s s s e (.77) $ s gve (.6). Susttutg (.77) ad (.6) to (.75) elds ( ) ( ) ( ) s s s s s s s s $ S S S (.78) $ s gve the Plücer coordates of ad, equatos (.69) ad (.7) $ (.79) $ s gve the Plücer coordates of a force passg through the org of the sstem, therefore t alwas has the form

38 3 $ (.8) all the sstem the uversal ot caot provde momet reactos alog ts movg aes, the $ has the form $ (.8) Susttutg (.78), (.79), (.8) ad (.8) to (.74) elds ( ) ( ) ( ) s s s s s s s s S S S (.8) rom the forth ad ffth rows (.8) s possle to defe g ad g as ( ) g S s (.83) ( ) g S s s s (.84)

39 33 Equatos (.83) ad (.84) volve ol the put data ad the geeraled coordates ad whose values are ow from the vrtual wor approach. fter susttutg ad ad the put data to (.83) ad (.84), g ad g must e ero f the values of ad correspod to a equlrum posto. Susttutg ac the values for S,,,, ad gve gure. 7 ad (.57) to the last epressos elds g.5( 7.7 )( )s(7.5 ).5 g.5( ( 7.7 ) ( )(7.5 )s( 7.7 )).3 s oth g ad g vash, the results otaed usg the vrtual wor for calculatg ad correspod to a equlrum posto.

40 HPE 3 GENE EQUIONS O HE SIS O ENSEGIY SUUES Whe a eteral wrech s appled to a tesegrt structure the tes are deformed ad the struts go to a ew equlrum posto. hs ew posto would e perfectl defed usg the coordates of the lower ad upper eds of the struts a gloal referece sstem. However the are uow. Equatos are developed ths secto usg the prcple of vrtual wor to solve ths prolem. esegrt structures eht a patter ther cofgurato ad t s possle to tae advatage of that stuato to geerate geeral equatos for the statc aalss. efore startg to mplemet the method t s ecessar to estalsh the omeclature for the sstem ad some assumptos to smplf the prolem. gure 3.a shows a tesegrt structure coformed struts each oe of legth S. gure 3. shows the same structure ut wth ol some of ts struts. he selecto of the frst strut s artrar ut oce t s chose t should ot e chaged. he ottom eds of the strut are laeled ecutvel as E, E,, E,, E where detfes the frst strut ad stads for the last strut. Smlarl the top eds of the struts are laeled as,,,,,, as show gure

41 35 Strut op te Strut oectg te s ottom te (a) Strut E E E E E () gure 3.. Nomeclature for tesegrt structures. a) Geerc ames; ) Specfc omeclature.

42 36 I ever structure t s possle to detf the top tes, the ottom tes ad the lateral or coectg tes, as show gure 3.a. he curret legth of the top, ottom ad lateral tes are called, ad respectvel. he top te eteds etwee the top eds ad f < ad etwee ad f. he ottom te eteds etwee the ottom eds E ad E f < ad etwee E ad E f. he lateral te eteds etwee the top ed ad the ottom ed E f < ad etwee ad E f. I Secto.3 t was estalshed that the moto of a artrar strut ca e descred modelg ts lower ed wth a uversal ot traed to move the horotal plae. he same model s used ow for the dervato of the equlrum equatos for a geeral tesegrt structure. I addto the followg assumptos are made wthout loss of geeralt: he eteral momets are appled alog the aes of the uversal ots. he struts are massless. ll the struts have the same legth. Ol oe eteral force s appled per strut. here are o dsspatve forces actg o the sstem. ll the tes are teso at the equlrum posto;.e., the curret legths of the tes are loger tha ther respectve free legths. he free legths of the top tes are equal. he free legths of the ottom tes are equal.

43 37 he free legths of the coectg tes are equal. here are o terfereces etwee struts. he stffess of all the top tes s the same. he stffess of all the ottom tes s the same. he stffess of all the coectg tes s the same. he ottom eds of the strut rema the horotal plae for all the postos of the structure. 3. Geeraled oordates Due to the fact the lower ed of each strut s traed to move the horotal plae ad there s o moto alog the logtudal as sce t s traed a uversal ot, each strut has four degrees of freedom ad the total sstem has 4 degrees of freedom whch meas there are 4 geeraled coordates. or each strut the geeraled coordates are the horotal dsplacemets a,, as llustrated gure 3., of the lower ed of the strut together wth two rotatos aout the aes of the uversal ot. he agular coordates assocated wth the strut are ad where correspods to the rotato of the strut aout the curret as ad correspods to the rotato aout as, as t was show gures.4 ad.5. ale 3. shows the geeraled coordates assocated wth each strut.

44 38 E a O gure 3.. oordates of the eds of a strut the gloal referece sstem wth referece pot O. ale 3.. Geeraled coordates assocated wth each strut. Strut Geeraled coordates a a a a 3. he Prcple of Vrtual Wor for esegrt Structures Equatos (.) ad (.) of Secto. estalshed the codtos for the equlrum of a sstem of rgd odes. he otato used there assumes that the geeraled coordates are grouped a vector q such that ( q q... ) q,,, q p where p s the umer of geeraled coordates. However, sce the otato used for the tesegrt structures dffers from Secto., there s ol oe eteral force per strut ad the momets act ol

45 39 alog the aes of the uversal ot t s more coveet to state the equlrum equatos usg the curret otato ad tag accout the smplfcatos troduced here. rom (.3) δ W δw c δw c (3.) where δ W s the total vrtual wor, δ Wc s the vrtual wor performed for oervatve forces ad momets ad δ Wc s the vrtual wor performed ervatve forces. δ Wc ca e represeted as δ W δw δw (3.) c where δ W s the total vrtual wor performed o-ervatve forces ad δ W s the total vrtual wor performed o-ervatve momets. I (.6) was estalshed that the vrtual wor performed the ervatve force, δ Wc s δwc δv where δ V s the potetal eerg assocated wth the ervatve force, therefore the total cotruto of the ervatves forces δ Wc s δw c δv (3.3) where δ V s the summato over all the δ V preset the structure. Susttutg (3.) ad (3.3) to (3.) elds δw δw δw δv (3.4)

46 4 I equlrum the vrtual wor descred (3.4) must e ero, the the equlrum codtos ca e deduced from δ W δw δv (3.5) I what follows each term the epresso (3.5) wll e determed. 3.3 oordates of the Eds of the Struts he coordates of the lower eds ca e epressed drectl the gloal referece sstem. he lear dsplacemets assocated wth the strut are a ad, the correspod to the coordates, measured the sstem. herefore the coordates of the lower ed E epressed the gloal referece sstem, (see gure 3.), are smpl a E (3.6) E a O gure 3.. oordates of the eds of a strut the gloal referece sstem wth referece pot O. he coordates of the upper ed of the strut are evaluated wth the ad of equato (.36),

47 4 s s l l a l P (.36) Whe the agles ad for the -th strut are replaced ad respectvel ad l s replaced S, (.36) elds s s s a s s (3.7) Now t s possle to ota epressos for the legths of the top, ottom ad lateral tes. he legths of the top tes are gve ( ) ( ) ( ) ( ) / ( ) ( ) ( ) ( ) / ( ) ( ) ( ) ( ) /,,,,,, (3.8) f the he legths of the ottom tes are gve ( ) ( ) ( ) ( ) / E E E E E E ( ) ( ) ( ) ( ) / E E E E E E ( ) ( ) ( ) ( ) /,,,,,, E E E E E E (3.9)

48 4 f the he legths of the lateral tes are gve ( E ) ( E ) ( E ) ) / ( E ) ( E ) ( E ) ) / ( E ) ( E ) ( E ) ) /,,,,,, (3.) f the 3.4 Ital odtos I the eample of Secto.5 t was estalshed that the umercal methods are hghl sestve to the selecto of the tal values. he prolem of the tal posto of a tesegrt structure, ths s the posto of the structure ts uloaded posto were addressed Y []. I ths secto hs results are preseted wthout proof ad are adapted to the curret omeclature. he free legths of the top ad ottom tes ad the curret legths of the top ad ottom tes satsf the relatos llustrated gure 3.3, therefore o (3.) γ s o (3.) γ s γ s (3.3) γ s (3.4)

49 43 where ad are the free legths of the top ad ottom tes respectvel ad ad are the curret legths of the top ad ottom tes for the uloaded posto. he agle γ depeds o the umer of struts ad s gve π γ (3.5) where s the umer of struts o γ o γ o o o γ γ (a) o () gure 3.3. elatos for the top ad ottom tes of a tesegrt structure. a) es wth ther free legths; ) es after elogato. I the uloaded posto the quattes, ad the curret legth of the lateral tes satsf the followg equatos o γ ( o ) s (3.6) o γ ( o ) s (3.7)

50 44 [ ( α γ ) α] (3.8) where s : stffess of the top tes. : stffess of the ottom tes. : stffess of the lateral tes. S : legth of the struts. : free legth of the lateral tes. α : agle related to the rotato of the polgo coformed the top ed wth respect to the polgo coformed the ottom eds of the struts ad s gve α π π (3.9) he soluto of (3.6), (3.7) ad (3.8) ca e carred out umercall. Oce, (ad ) have ee evaluated the values of ad are calculated from (3.3) ad (3.4). Summarg, whe the free legths of the top, ottom ad lateral tes of a tesegrt structure are gve, together wth ther stffess, strut legths ad umer of struts, equatos (3.6), (3.7) ad (3.8) eld the curret values of the top, ottom ad lateral tes ts uloaded posto. lthough the wor of Y [], the followg relatos are ot estalshed eplctl, t ca e show that f the gloal referece sstem s oreted such a wa that ts as passes through the ottom of oe of the struts whe the structure s ts uloaded posto, the the coordates of the top ad lower

51 45 eds of the strut for ts tal posto a gloal referece sstem, (see gure 3.4), are ( ( ) γ ) ( ( ) ),,,..., a, E o, s γ ( ( ) γ α ) ( ( ) γ ),,,..., o s α H (3.) (3.) where f the. urther, H s s (3.) γ H represets the heght etwee the platform defed the lower eds of the struts ad the platform defed the upper eds of the struts.,,, H E, E, gure 3.4. Ital posto of a tesegrt structure. E,

52 46 Oce the coordates of E, ad, are otaed, the tal agles, ad, α correspodg to the rotato of each strut are gve (.37), (.38) or (.39) ad, ta (.37) ta a s (.38) or ta a (.39) where a, are replaced a,,, gve (3.) ad (,, ) s replaced, gve (3.), the ta,, s( ( ) γ α) (3.3) H ( ( ) γ α) a, ta, (3.4), s( ( ) γ α) s, or ( ( ) γ α) a, ta, (3.5) H,

53 he Vrtual Wor Due to the Eteral orces s t s assumed that there s ol oe eteral force actg o each strut, the vrtual wor δ W performed all the eteral forces s gve δw δ r (3.6) where s the eteral force actg the strut, r s the vector to the pot of applcato of the eteral force. I (3.6) oth ad r must e epressed the same coordate sstem. If the sstem chose s the gloal referece sstem the the terms satsfg (3.6) have the form δ r δr δr δr (3.7) If the dstace etwee the pot of applcato of the force ad the lower ed of the strut s f, see gure 3.5, the a epresso for r the gloal sstem ca e otaed from (.36) where the agles ad ad the dstaces a,, l ad are susttuted,, a, ad respectvel. r r r r s a s (3.8)

54 s 48 r a gure 3.5. ocato of the eteral force actg o the strut. he vrtual dsplacemets ca e deduced from equato (3.8) where the geeraled coordates for the strut are,, a ad δ r δr δr δr δ δa δ δ s s δ s δ s δ (3.9) Susttutg (3.9) to (3.7) regroupg terms, ad susttutg to (3.6), the geeral epresso for the vrtual wor performed eteral forces s gve δw ( [ s ] [ s s s ] δ δ δa δ ) (3.3)

55 he Vrtual Wor Due to the Eteral omets Provded that ths model of the tesegrt structure the eteral momets ca e eerted ol alog the as of the uversal ot, the vrtual wor performed the eteral momets s gve W δ δ δ (3.3) s efore all the elemets of equato (3.3) must e epressed the same coordate sstem. However as the scalar product s varat uder trasformato of coordates a coveet coordate sstem ma e selected. It was estalshed at Secto.5 that whe (3.3) s epressed a referece sstem otaed traslatg the geeral referece sstem to the ase of strut ad the rotatg aout the curret as, (see gure 3.6) the terms (3.3) have the form (3.3) the δ δ (3.33) ad (3.34)

56 5 Strut a, gure 3.6. Eteral momets ad coordate sstems at the ase of strut. the δ δ (3.35) Susttutg (3.3), (3.33), (3.34) ad (3.35) to (3.3) elds δw δ δ (3.36) 3.7 he Potetal Eerg Provded that the struts are dered massless the term related to the potetal eerg the prcple of vrtual wor s the resultat of the elastc potetal eerg cotrutos gve the tes. he potetal elastc eerg for a geeral te s gve [6] V ( ) w w (3.37) where

57 5 : V elastc potetal eerg for te : te stffess : w curret legth of the te : w free legth of the te herefore the dfferetal potetal eerg for te s w w w V δ δ ) ( (3.38) he dfferetal of the potetal eerg for all the tesegrt structure, V δ, s the resultat of the cotrutos of the top tes, the ottom tes ad the lateral tes ad ca e epressed as ( ) ( ) ( ) o o o V δ δ δ δ (3.39) where,, are the stffess of the top, ottom ad lateral tes respectvel. he curret legths of the tes are fuctos of some sets of the geeraled coordates for the structure, show (3.4) ( ) a a a,,,,...,,,,,,,, ( ) a a a,,,,...,,,,,,,, ( ) a a a,,,,...,,,,,,,, (3.4) herefore (3.39) ca e epaded the form

58 5 ( ) ( ) ( ) a a a a a a a a a a a a V δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ (3.4) 3.8 he Geeral Equatos Now that each oe of the terms cotrutg to the vrtual wor has ee evaluated, the equlrum codto for the geeral tesegrt structure ca e estalshed. Susttutg (3.3), (3.36) ad (3.4) to (3.5) ad re-groupg elds f f f f f f a f a f a f δ δ δ δ δ δ δ δ δ f f f δ δ δ (3.4) where

59 53 ( ) ( ) o o a a f ( ) o a (3.43),...,, ( ) ( ) o o f ( ) o (3.44),...,,

60 54 [ ] ( ) ( ) o o f s ( ) o (3.45),...,, [ ] ( ) ( ) o o f 3 s s s ( ) o (3.46),...,, Equato (3.4) must e satsfed for all the values of the geeraled coordates whch geeral are dfferet from ero, the

61 55 f f f 4 (3.47) where f s gve equatos (3.43) to (3.46). Equatos (3.47) represet a strogl coupled sstem of 4 equatos depedg ol o the 4 geeraled coordates. he soluto s otaed umercall. he tal codtos for a,, ad are gve (3.), (3.3), (3.4) or (3.5). he equlrum posto for a geeral tesegrt structure s otaed solvg the set (3.47) for a,,,,..., a,,,. Equatos (3.6) ad (3.7) are eplct epressos for the coordates of the eds of the struts the gloal coordate sstem.

62 HPE 4 NUEI ESUS hs chapter presets the methodolog to fd the equlrum posto for tesegrt structures. hree umercal eamples are provded to llustrate the cocepts dscussed the prevous sectos. esegrt structures wth dfferet umer of struts ad dfferet eteral loads are aaled. Each eample s developed detal utl to ota the umercal solutos. I addto to the umercal results, the graphcs of the structures ther equlrum postos are also provded. he statc aalss s performed two steps: tall the equlrum posto of the structure ts uloaded posto s evaluated, the the eteral loads are dered ad the ew equlrum posto s foud. he umercal results are otaed here evaluatg some of the equatos what where derved detal hapter 3. he author has repeated some of these equatos the preset chapter for coveece order to mme repeated referece to the pages of hapter alss of esegrt Structures ther Uloaded Postos. Whe there are o eteral loads appled, the equlrum posto ca e determed usg Y s results. Numercal values are gve Secto 4.3. I order to determe the uloaded equlrum posto the legths of the struts are specfed, S, whch are assumed to e all the same, together wth the stffess 56

63 57 of the top tes (assumed equal), ottom tes (assumed equal), coectg tes (assumed equal) ad the free legths of the top tes (assumed equal), ottom tes (assumed equal) ad coectg tes (assumed equal). I order to fd the equlrum posto of the structure ts uloaded posto, the coordates of the eds of all the struts measured a gloal referece sstem are determed. hs s accomplshed frst computg the three uows, ad the legth of the coectg tes the followg equatos gve hapter 3 (see also gures 3.3 (a) ad ()). o γ ( o ) s (3.6) o γ ( o ) s (3.7) [ ( α γ ) α] (3.8) s where o (3.) γ s o (3.) γ s d the agles γ ad α are gve γ π (3.5) α π π (3.9) where s the umer of struts.

64 58 he values of ad are the susttuted to the equatos (3.), (3.) ad (3.) whch eld the coordates E, o ad, o, ths s the coordates of the lower ad the upper eds of the struts the gloal referece sstem respectvel. Note that the su-de dcates the uloaded posto. ( ( ) γ ) ( ( ) ),,,..., a, E,, s γ ( ( ) γ α ) ( ( ) γ ),,,...,, s α H (3.) (3.) where f the, ad H s s (3.) γ 4. alss of oaded esegrt Structures he eteral loads actg o a tesegrt structure ma e eteral forces ad eteral momets. ccordg to the restrctos of ths stud, ol oe eteral force ad two eteral momets ma e appled per strut. I addto the drectos of the eteral momets are alog the as of the uversal ot used to model the strut. o e ale to perform the statc aalss the compoets,, ) ( ad the pot of applcato measured alog the strut for each force must e

65 s 59 ow, together wth the drectos of the eteral momets ad, see gures 4. ad 3.5. E, gure 4.. Eteral loads appled to oe of the struts of a tesegrt structure. r a gure 3.5. ocato of the eteral force actg o the strut.

66 6 strut of a tesegrt structure traed to rema o the horotal plae has four degrees of freedom, two assocated wth ts logtudal dsplacemets a ad, ad two assocated wth ts rotatos ad, see gure 4.. herefore the whole structure posses 4 * degrees of freedom where s the umer of struts. However f some of the freedoms of the sstem are traed the degrees of freedom decrease. Hece addto to the owledge of the eteral loads t s ecessar to ow the umer of freedoms of the structure. a gure 4.. Degrees of freedom assocated wth oe of the struts of a tesegrt structure. he equlrum posto of the structure s determed for a sstem of p equatos where p s the umer of freedoms of the sstem. hese equatos are otaed epadg equatos (3.43) through (3.46) for each oe of the geeraled coordates of the sstem.

67 6 ( ) ( ) o o a a f ( ) o a (3.43),...,, ( ) ( ) o o f ( ) o (3.44),...,,

68 6 f [ s ] ( ) o ( ) o ( ) o (3.45),,..., [ s s ] f3 s ( ) o ( ) o ( ) o (3.46),,..., s the resultat sstem must e solved umercall the the tal values of the geeraled coordates must e evaluated pror to the mplemetato of the umercal method. he geeraled coordates a,,,,, ad,

69 63 correspodg to the tal values for the strut are otaed from equatos (3.), (3.3) ad (3.4) a,, s ( ( ) γ ) ( ( ) γ ),,,..., (3.) ta,, s( ( ) γ α) (3.3) H ( ( ) γ α) a, ta, (3.4), s( ( ) γ α) s, d all the terms (3.), (3.3) ad (3.4) have ee defed prevousl. Now the equatos ca e solved ad umercal values for the geeraled coordates a,, ad are otaed, therefore the equlrum posto for the tesegrt structure has ee foud. I order to ehace the performace of the umercal method t s advsale to crease the eteral loads graduall a step step procedure. I ths wa the geeraled coordates evaluated at each step are the tal values for the et step. Equatos (3.6) ad (3.7) determe the coordates of the lower ad upper eds of the struts, sstem. E ad respectvel, the gloal referece

70 64 a E (3.6) s s a s s s (3.7) 4.3 Eample : alss of a esegrt Structure wth 3 Struts 4.3. alss for the Uloaded Posto. tesegrt structure wth 3 struts has the stffess ad free legths show ale 4.. Each of ts struts has a legth evaluate ts uloaded equlrum posto. s mm. It s requred to ale 4.. Stffess ad free legths for the structure of eample. Stffess (N/mm) ree legths (mm) op tes.5 35 ottom tes.3 5 oectg tes 8 he soluto of the sstem o γ ( o ) s (3.6) o γ ( o ) s (3.7) [ ( α γ ) α] (3.8) where s

71 65 π 36 γ (3.5) 3 π π 9 9 α 5 (3.9) 3 o 35 mm. 7 γ s 6 s o 5 mm 3. γ s 6 s elds mm mm (3.) (3.) mm.84 mm he coordates of the eds of the struts for the uloaded posto are otaed from ( ( ) γ ) ( ( ) ),,,..., a, E,, s γ ( ( ) γ α ) ( ( ) γ ),,,...,, s α H (3.) (3.) where f the, ad H γ s s mm he results are summared ale 4.. gure 4.3 shows the structure ts uloaded posto.

72 66 ale 4.. ower ad upper coordates for the uloaded posto for the structure of eample (mm). Strut Strut Strut 3 E E E E E E 3 gure 4.3. Uloaded posto for the structure of eample alss for the oaded Posto. If oe eteral force s appled at the upper ed of each strut whch compoets ad pot of applcato are preseted ale 4.3 ad there s o trats actg o the struts of the structure, t s requred to evaluate the fal equlrum posto of the structure.

73 67 ale 4.3. Eteral forces actg o the structure of eample. Strut Strut Strut 3 (N) (N) (N) (mm) Sce the sstem has 3 struts ad there s o trats the there are degrees of freedom ad therefore equatos are requred, oe per each geeraled coordate. he equatos are geerated followg the procedure descred secto 3.8: Equato (3.43) elds f, f ad f 3 Equato (3.44) elds f 4, f5 ad f 6 Equato (3.45) elds f 7, f8 ad f 9 Equato (3.46) elds f, f ad f Each f s equated to ero ad the the sstem s solved umercall. s a eample the frst equato, (3.43), s show the apped. It s clear that the complete set s etremel large ad coupled. efore attemptg to ota a soluto t s ecessar to evaluate the tal codtos,.e. the values of the geeraled coordates the uloaded posto. hs s accomplshed usg (3.), (3.3) ad (3.4) a,, s ( ( ) γ ) ( ( ) γ ),,,..., (3.)

74 68 ta,, s( ( ) γ α) (3.3) H ( ( ) γ α) a, ta, (3.4), s( ( ) γ α) s, d all the terms (3.), (3.3) ad (3.4) have ee defed prevousl. he results are summared ale 4.4. ale 4.4. Ital values of the geeraled coordates for the structure of eample. Strut Strut Strut 3 a (mm) (mm) (rad) (rad) It s ow possle to mplemet the umercal method. he magtude of the eteral force s creased steps of N ad the equlrum posto s evaluated for each step. he fal values for the geeraled coordates of the structure for a eteral force of N are show ale 4.5. ale 4.5. Geeraled coordates for the fal posto for the structure of eample. Strut Strut Strut 3 a (mm) (mm) (rad) (rad)

75 69 Usg these values, equatos (3.6) ad (3.7) ca ow e used to ota the coordates of the eds of the struts for the fal posto. a E (3.6) s s a s s s (3.7) he results are summared ale 4.6 ad gure 4.4 shows the structure ts fal equlrum posto. ale 4.6. ower ad upper coordates for the fal posto of the structure of eample (mm). Strut Strut Strut 3 E E E

76 7 3 E E E 3 gure 4.4. al equlrum posto for the structure of eample. gure 4.5 llustrates the secod strut modeled wth a uversal ot ts frst ad fal posto. It ca e apprecated ts logtudal ad agular dsplacemets. gure 4.6 shows a top vew of the structure ts tal ad fal postos. It should e oted that the ase E, E, E3 creases se ut matas ts oretato whlst the top,, 3 creases se ad also udergoes a rotato.

77 7 3 E E E 3 (a) 3 E E E 3 () gure 4.5. Secod strut of the structure of eample. a)uloaded posto; )ast posto.

78 7 E 3 E E 3 (a) E 3 E E 3 () gure 4.6. Pla vew structure eample. a)uloaded posto; )ast posto.

79 73 ecause of the smmetr of the eteral loads the heght of the structure decreases uforml,.e. the coordate for the pots E, E ad E 3 remas the same for each posto. hese results are llustrated gure 4.7. gure 4.7. Heght of the 3 struts Vs magtude of the eteral force for the structure of eample. gure 4.8 llustrates the varato of the tes legths for each cremet the eterall appled load. It should e oted that for the last posto the legth of the coectg tes s 8 mm whch s approachg the free legth. hs meas that f a larger force s appled to the structure, t caot loger rema as a tesegrt structure. lthough there could e other equlrum postos the model developed ths research s ot vald amore.

80 74 oectg tes ottom tes op tes gure 4.8. Strg legths of the 3 struts structure Vs magtude of the eteral force. ecause of the complet of the equatos that defe equlrum postos t s essetal to verf the aswers otaed depedetl. he software developed for ths purpose performs ths terall for each strut ad for each posto of the strut. o clarf ths pot, the verfcato of the aswer s demostrated here for the strut the last posto. gure 4.9 shows a free od dagram for the secod strut ad the locato of all the ed pots of the structure for the last posto. It also cludes the reacto force. Note that the drecto for the force the tes s dered as postve whe the force goes from a pot wth sude to aother pot wth

81 75 sude. or eample gure 4.9 the force goes from to 3 therefore s postve whle the force goes from to hece s egatve. If the sstem s a equlrum posto, the the summato of momets wth respect to E must e ero ad r r r r (4.) 3 r E E E 3 gure 4.9. ree od dagram for the secod strut of the structure of eample the last posto. he vector r from E to s gve r E mm (4.)

82 76 d from ale 4.3 the eteral force at the strut s N (4.3) he force actg the top te s gve s 3 (4.4) where ( ) (4.5) o s the curret legth of the top te mm ad are gve ale 4..5 N mm o 35 mm Susttutg the values of, ad to (4.5) N (4.6) s 3 are the uted Plücer coordates of the le passg through 3 ad form equatos (.5), (.6), (.7) ad (.) s (4.7) Susttutg (4.6) ad (4.7) to (4.4) elds

83 N (4.8) Smlarl, the force actg the top te s gve s (4.9) where ( ) (4.) o he curret legth of the top te,, s mm herefore N (4.) he uted Plücer coordates of the le passg through are s (4.) Susttutg (4.) ad (4.) to (4.9) (4.3) all the force actg o the lateral te s gve s E3 (4.4) where ( ) (4.5) o

84 78 d E mm ad are gve ale 4. N mm o 8 mm Susttutg the values of, ad to (4.5). 74 N (4.6) he uted Plücer coordates of the le passg through E 3 are s E (4.7) Susttutg (4.6) ad (4.7) to (4.4) N (4.8).975 Now the summato of momets ca e evaluated susttutg (4.), (4.3), (4.6), (4.3) ad (4.8) to (4.) N mm N mm N mm N mm N mm

85 79 Sce the summato of momets wth respect to E s, the umercal result cofrms that the curret posto for the strut of the structure correspods to a equlrum posto. 4.4 Eample : alss of a esegrt Structure wth 4 Struts 4.4. alss for the Uloaded Posto. It s requred to evaluate the uloaded equlrum posto of a tesegrt structure wth 4 struts wth the stffess ad free legths show ale 4.7. Each of ts struts has a legth s mm. ale 4.7. Stffess ad free legths for the structure of eample. Stffess (N/mm) ree legths (mm) op tes.5 4 ottom tes.5 4 oectg tes.5 4 he soluto of the sstem o γ ( o ) s (3.6) o γ ( o ) s (3.7) [ ( α γ ) α] (3.8) s where π 36 γ 9 (3.5) 4

86 8 π π 9 9 α. 5 (3.9) 4 o 4 mm γ s 45 s o 4 mm γ s 45 s elds mm mm (3.) (3.) 4.58 mm 4.58 mm he coordates of the eds of the struts for the uloaded posto are otaed from ( ( ) γ ) ( ( ) ),,,..., a, E,, s γ ( ( ) γ α ) ( ( ) γ ),,,...,, s α H (3.) (3.) where f the, ad H γ s s mm he results are summared ale 4.8 ad gure 4. shows the structure ts uloaded posto.

87 8 ale 4.8. ower ad upper coordates for the uloaded posto for the structure of eample (mm). Strut Strut Strut 3 Strut 4 E E E E 4 E E 3 gure 4.. Uloaded posto for the structure of eample. E 4.4. alss for the oaded Posto. It s requred to evaluate the fal equlrum posto of the structure whe the eteral momets lsted ale 4.9 are appled alog the as of the uversal ots that model the structure, see gure 4., ad the lower eds of the struts are traed such a wa that the caot move the horotal plae.

88 8 ale 4.9. Eteral momets actg o the structure of eample. Strut Strut Strut 3 Strut 4 (N.mm) (N.mm) gure 4.. Drectos of the eteral momets for the structure of eample. ecause there are trats per strut there are 8 degrees of freedom for ths sstem, ad the are assocated wth the rotatos of the struts. he geeraled coordates are,,,, 3, 3 ad 4, 4, where the suscrpt dcates the umer of the strut. he equlrum equatos are otaed as follows Equato (3.45) elds f 9, f, f ad f

89 83 Equato (3.46) elds f 3, f4, f5 ad f 6 he tal codtos, ths s the values of the geeraled coordates the uloaded posto, are otaed usg (3.), (3.3) ad (3.4) a,, s ( ( ) γ ) ( ( ) γ ),,,..., (3.) ta,, s( ( ) γ α) (3.3) H ( ( ) γ α) a, ta, (3.4), s( ( ) γ α) s, d all the terms (3.), (3.3) ad (3.4) have ee defed prevousl. he results are preseted ale 4.. ale 4.. Ital values of the geeraled coordates for the structure of eample. Strut Strut Strut 3 Strut 4 (rad) (rad) I order to avod evaluatg postos that do ot correspod to the real prolem s essetal to crease the load smoothl small cremets rather tha trg to ota the fal values of the momets a sgle step. he umer of steps was chose artrarl as. ale 4. shows the values of the eteral momets for each step. he results for the fal posto are lsted ale 4.

90 84 ale 4.. Eteral momets at each step for the structure of eample N.mm. Step Strut Strut Strut 3 Strut ale 4.. Geeraled coordates for the fal posto for the structure of eample. Strut Strut Strut 3 Strut 4 (rad) (rad) Wth values lsted ale 4., equatos (3.6) ad (3.7) permt to ota the coordates of the eds of the struts for the fal posto. a E (3.6) s s a s s s (3.7) he results are summared ale 4.3 ad gure 4. shows the structure ts fal equlrum posto.