Static Analysis of Prestressed Tensegrity Structures
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1 Sttc lss Prestressed esegrt Structures Jul Crre Jseph u Crl. Cre III eserch ssstt Grdute eserch Pressr Pressr Ceter r Itellget Mches d tcs, eprtmet Mechcl geerg, Uverst lrd, Gesvlle, 6 cmr@cmr.me.ul.edu strct I ths pper the mthemtcl mdel t perrm the sttc lss prestressed tprsm tesegrt structure suected t rtrr reduct ts cectg tes s ddressed. vrtul wr pprch s used t deduce the equlrum equts d the umercl results re vered usg Newt pprch. Oe emple s prvded t llustrte the mthemtcl mdel.. Itrduct esegrt structures re sptl structures rmed cmt rgd elemets (the struts d elstc elemets (the tes. N pr struts tuch d the ed ech strut s cected t three cplr tes []. he struts re lws cmpress d the tes tes. he etre cgurt stds tsel d mts ts rm slel ecuse the terl rrgemet the tes d the struts []. esegrt s revt tes d tegrt. he develpmet tesegrt structures s reltvel ew d the wrs relted hve l ested r the 5 ers. Keer [] estlshed the relt etwee the rtt the tp d ttm tes. e [] preseted prcedures r the geert tesle structures phscl d grphcl mes. Y [] ted Keer s results usg eerg csderts d ud the equlrum pst r the ulded tesegrt prsms. Ster [4] develped geerc desg equts t d the legths the struts d elstc tes eeded t crete desred gemetr. Sce eterl rces re csdered hs results re reerred t the ulded pst the structure. Kght [5] ddressed the prlem stlt tesegrt structures r the desg deplle te. Mst recetl Crre [6] ted the equlrum pst r geerl tprsm tesegrt structure suected t wde vret eterl lds usg vrtul wr. hs methd s used here. It s sd tht structure s prestressed whe the ree legth e r severl the cectg tes s decresed. I ths pper the prlem the determt the equlrum pst prestressed tprsm tesegrt structure s ddressed. ls stwre le t geerte d slve the equts ecessr t mdel the structure ws develped Mtl.. Nmeclture gure shws tesegrt structure rmed struts ech e legth S. I ever structure t s pssle t det the tp tes, the ttm tes d the lterl r cectg tes whch re deted s, d respectvel. gure shws the sme structure. he ttm eds ech strut s leled csecutvel s,,,,, where detes the rst strut d stds r the lst strut. Smlrl the tp eds the struts re leled s,,,,, rtrr ut ce t s chse t shuld t e chged.. he select the rst strut s
2 p te Cectg te Strut ttm te ( gure. Nmeclture r tesegrt structures. Cmpets; Strut eds. (. Geerled Crdtes d rsrmts Mtrces gure shws rtrr pt P lcted strut legth s. I reerece sstem whse s s lg the s the strut d wth ts rg lcted t the lwer ed the strut, the crdtes P re smpl (,,l. Hwever requetl s mre cveet r purpses lss t epress the lct P the gll reerece sstem. I the lwer ed e strut s cstred t mve the hrtl ple d ls the rtt ut ts lgtudl s s cstred, the strut c e mdeled uversl t. I ths w the t prvdes the 4 degrees reedm sscted wth the strut. he ttl sstem hs 4* degrees reedm whch mes there re 4* geerled crdtes. r ech strut the geerled crdtes re the hrtl dsplcemets, the lwer ed the strut tgether wth tw rtts ut the es the uversl t, ε d. ε crrespds t the rtt the strut ut s d crrespds t the rtt ut C s, see gure. C P l S C, t ε, C ( ( gure. egrees reedm sscted wth e the struts tesegrt structure. rtrr pt the strut; Strut mdeled s uversl t.
3 he lgmet the s the ed sstem wth the s the rd c e ccmplshed usg the llwg three csecutve trsrmts, [7]: trslt, t(,,, rtt ε ut the curret s ( d rtt ut the curret s ( C. he crdtes P epressed the gll reerece sstem re C P P (,, C ε where,, ( cs ε s ε C ε ( s ε cs ε cs s (4 s cs C P ] [ l (5 Susttutg the ve three epresss t ( elds P l s l s ε cs l csε cs (6 I ddt t the cstrt mpsed tht the lwer eds re t rem the hrtl ple d r ech strut t vd the rtt ut ts lgtudl s the llwg ssumpts re mde wthut lss geerlt: he struts re mssless. ll the struts hve the sme legth. here re dssptve rces ctg the sstem. he ree legths the tp tes re equl. he ree legths the ttm tes re equl. he ree legths the cectg tes re the sme t the ustressed pst. he stess ll the tp tes s the sme. he stess ll the ttm tes s the sme. he stess ll the cectg tes s the sme. 4. Crdtes he ds the Struts he Crtes crdtes the lwer eds, epressed the gll reerece sstem, re ted terms the geerled crdtes susttutg l (6, the (7
4 4 Smlrl the crdtes the upper ed the struts re evluted susttutg l the legth the struts S (6 s s s s ε cs (8 s csε cs quts (7 d (8 permt e t t epresss r the legths the tp, ttm d lterl tes terms the geerled crdtes s llws / (,, (,, (,, / (,, (,, (,, / ( ( ( ( (,,,,,, where (9 ( ( the 5. he Prcple Vrtul Wr r esegrt Structures δw he vrtul wr r sstems le t stre ptetl eerg c e stted rm [8] δw δw δv ( M where δ W s the ttl vrtul wr, δ W s the ttl vrtul wr perrmed -cservtve rces, δ WM s the ttl vrtul wr perrmed -cservtve mmets d δv s the deretl the ptetl eerg. he prcple vrtul wr sttes tht the equlrum ( vshes. I ddt sce there re eterl lds ( c e rewrtte s δ V ( 8. he Ptetl erg Sce the struts re csdered mssless the term relted t the ptetl eerg the prcple vrtul wr s the resultt the elstc ptetl eerg ctruts gve the tes. he ptetl elstc eerg r geerl te s gve, [8] ( V w w (4 where V s the elstc ptetl eerg r te, the te stess, w the curret legth the te d w the ree legth the te. herere the deretl the ptetl eerg r te s δ V δw (5 ( w w he deretl the ptetl eerg r ll the tesegrt structure, δ V, s the resultt the ctruts the tp tes, the ttm tes d the lterl tes d c e epressed s δv ( δ ( δ ( δ,, where, re the stess the tp, ttm d cectg tes respectvel, d re gve (9, ( d ( d re ucts sme the geerled crdtes d, d re the ree legths the tp, ttm d cectg tes respectvel. It shuld e ted sce the prestressg s ted reducg the ree legths the cectg tes, the vlues m chge r ech cectg te. (6
5 5 9. he Geerl quts Nw tht ech e the terms ctrutg t the vrtul wr hs ee evluted, the equlrum cdt r the geerl tesegrt structure c e estlshed. Susttutg (6 t ( d re-grupg elds δ δ δ δ δ δ δ δ δ δε δε δε (7 where ( ( ( (8 ( ( ( (9 ( ( ( ε ε ε ( ( ( ( (,...,, qut (7 must e stsed r ll the vlues the vrtul dsplcemets whch geerl re deret rm er, the 4 M ( where s gve equts (8 t (. quts ( represet strgl cupled sstem 4* equts depedg l the 4* geerled crdtes. he equlrum pst r geerl tesegrt structure s ted slvg umercll the set ( r,,, ε,..., ε,,,. ter tht equts (7 d (8 eld eplctl epresss r the crdtes the eds the struts the gll crdte sstem.. Itl Cdts e le t slve ( t s ecessr t d prper set vlues r the geerled crdtes the ulded pst. hs s ccmplshed usg Y s methd [], whch s preseted here wthut demstrt. I ths methd the three uws, d the legth the cectg tes equts (, (4 d (5 re slved ( s ( ( s (4 ( [ ] cs cs α α s (5 where s d s (6 d the gles d α re gve
6 6 π π π d α (7 where s the umer struts. he vlues d re the susttuted t the equts (8 thrugh ( whch eld the geerled crdtes r the ulded pst. ( (,,,... ( (,,,..., cs, (8, s, (9 tε,, s( ( α ( H cs( ( α, t, (, s( ( α s ε, where H s s ( d the. Verct the Numercl esults ecuse the cmplet the equlrum equts t s essetl t ver the swers s ted. depedet vldt the results c e ccmplshed sltg strut d perrmg the summt mmets wth respect t the lwer ed the strut, see gure r r r ( where r (4 ( (5 ( (6 ( (7 I ter susttutg the umercl vlues d the evlut ( s the er vectr the the curret pst s equlrum pst.. mple: lss esegrt Structure wth 5 Struts. NYSIS O H UNPSSS POSIION It s requred t evlute the uprestressed equlrum pst tesegrt structure wth 5 struts d wth the stess d ree legths shw le. ch the struts hs legth s 65mm. le. Stess d ree legths r the structure. Stess (N/mm ree legths (mm p tes.5 5 ttm tes. 45 Cectg tes.4,tl 65
7 7 r r ths emple gure. ree d dgrm rtrr strut. 5 the (6 d (7 eld 7, α 54, mm d 8. 79mm. he slut (, (4 d (5 elds 4. 5mm,. 98mm d rm (, H mm. he tl vlues the geerled crdtes re ted rm (7 thrugh ( d re lsted le. le. Itl vlues the geerled crdtes r the structure. Strut Strut Strut Strut 4 Strut 5 (mm (mm ε (rd (rd Usg the vlues le, equts (7 d (8 eld the crdte the eds the structure r the ulded pst. he results re llustrted le 4 d gure 4.. NYSIS O H PSSS POSIION It s requred t evlute the l equlrum pst the structure whe the ree legths the cectg tes ttched t the upper eds d 5 re decresed t 5 mm d the hrtl dsplcemets the lwer eds d re cstred, see gure 4. Sce the sstem hs 5 struts d there re 4 cstrts (,, d the there re 6 degrees reedm d therere 6 equts re requred, e per ech geerled crdte. qut (8 elds, 4 d 5, equt (9 elds 8, 9 d, equt ( elds,,, 4 d 5 d equt ( elds 6, 7, 8, 9 d. It shuld e ted tht ( 5 5mm d 4 65mm ch s equted t er d the the sstem s slved umercll usg the stwre develped d the tl vlues lsted le. he vlues the prmeters tht determe the l pst re lsted le. Nte tht the vlues,, d les d d t chge ecuse the re cstred
8 8 le. Prmeters r the l pst r the structure. Strut Strut Strut Strut 4 Strut 5 (mm (mm ε (rd (rd Usg the vlues les, equts (7 d (8 eld the crdtes the eds the struts r the tl d l pst. he results re preseted le 4 d gure 4. le 4. wer d upper crdtes r the ulded d l pst r the structure (mm. Strut Strut Strut Strut 4 Strut 5 Itl l Itl l Itl l Itl l Itl l ( ( gure 4. Uprestressed d l psts r the structure. gure 5 shws the ree d dgrm r the secd strut ts l pst. qut ( r the preset strut hs the rm r r r (8 Wth the d the tl vlues gve le, the crdtes r the l pst lsted le 4 d recllg tht the ree legth the cectg te rems uchged 65 mm, equts (4 thrugh (7 eld
9 9 r ( mm ( N N.88 ( N r 5 4 gure 5. ree d dgrm r the secd strut the l pst. ter susttutg the prevus vlues (8 the result s whch mes the curret pst s equlrum pst. he stwre des terll the sme pert r ech strut.. Ccluss he mdel develped here llws e t le geerl prestressed tprsm tesegrt structure suected t prestressg ct. he mdel s develped usg the vrtul wr pprch d ll the results re checed usg the Newt s hrd w. hs verct ssures e tht the swers prduced the umercl methd ccurtel crrespd t equlrum psts. he presett the results grphc rm ds t the uderstdg the cmple cgurt tht structure c ssume ter rtrr prestressg ts cectg tes.
10 cwledgemets he uthrs wuld le t grteull cwledge the supprt the r rce Oce Scetc eserch (Grt Numer d the U.S. eprtmet erg (Grt Numer - G4-86N7967. eereces. Y, J.P., lss r the esg Sel-eplle esegrt d erced esegrt Prsms wth lstc es, eprt r the Ceter r Itellget Mches d tcs, Uverst lrd, Gesvlle,.. e,.s., eprt Iqur t he stece, rmt d epresett esle Structures, Mster Idustrl esg hess, Prtt Isttute, New Yr, Keer, H., Gedesc Mth d Hw t Use It, Uverst Clr Press, erele d s geles, Clr, Ster, I.P. evelpmet esg quts r Sel-eplle N-Strut esegrt Sstems, Mster Scece hess, Uverst lrd, Gesvlle, Kght,.. eplle te Kemtcs usg esegrt Structure esg, ctr Phlsph ssertt, Uverst lrd, Gesvlle,. 6. Crre, J.C. Sttc lss esegrt Structures, Mster Scece hess, Uverst lrd, Gesvlle,. 7. Cre, C., u, J., Kemtc lss t Mpultrs, Cmrdge Uverst Press, New Yr, ught, S., Mechcs Mches, Jh Wle & Ss, Ic., New Yr, 988.
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