MATRIX ANALYSIS OF ANCHORED STRUCTURES
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1 SES It Cof o DMIL SSEMS ad COOL ece Ita oveber - pp-8 M LSIS OF CHOED SES IOS MSOIS Head of the Departet of Coputer Scece Mtar Ist of verst Educato / Heec ava cade era Hatraou 8 Praeus GEECE OLG M Departet of Matheatcs verst Potehca of Bucharest Spau Idepedete Bucharest OM bstract: - he purpose of ths paper s to aae a structure cosstg of bea bars ad achored cabe whch hod the trassso es whe s subected to a statc oadg For ths stud we use the atr dspaceet ethod Because the eastc cabes are eeets capabe to resst o aa teso forces the uerca agorth w have two steps I the frst step we use the dspaceet ethod for the beas bars ad a cabes I the secod step we repeat the cacuus after we reove the copressed cabes e-ords: atr dspaceet ethod statc oadg achored cabes Itroducto he prar fucto of a structure s to support ad to trasfer etera apped oads to the reacto pots whe at the sae te beg subected to soe specfed costrats sg the cocept of stffess atr whch has bee deat wth b JS Preeec [ OC eewc[ Brebba C[ the proposed uerca procedure ca be used to detere the equbru cofgurato of a -desoa asseb wth cabe copoets aass of the suspeso cabes ad the truss-sstes was preseted [ [ he geera assuptos used ths paper are: - dspaceets of a structura eeet are ot ver arge ad the geoetr of the sste s we defed before the aass s attept; - dspaceets ad stras of the oaded structure are sa ad hece ear eastct theor appes he agorth s preseted for the asseb show Fg he basc structure odes - s caped the pots ad 8 ad s hged the odes - ebers that e betwee odes -8 are cosdered beas he two stffeers: - ad - ad the cabes: are p-oted bars Stffess propertes of eeets I the oca coordate sste Fg the stffess atr for a bea eeet bouded b the odes s s I where: the cross-sectoa area I the aa oet of erta of the secto about aes ad the egth of the bea he for of ths atr s preseted [ S S S S S S Fg S 8 S S S S
2 SES It Cof o DMIL SSEMS ad COOL ece Ita oveber - pp-8 For a p-oted eeet whch s show the Fg the stffess atr s of the for S Fg E f where: the cross-sectoa area oet of ad the egth of the bar rasforato of coordate aes I order to detere the stffess propert of the copete structure a coo datu ust be estabshed for a uassebed structura eeets so that a the dspaceets ad ther correspodg forces w be referred to a coo coordate sste Sce the stffess atrces s ad f are ta cacuated oca coordates t s ecessar to troduce trasforato atrces chagg the frae of referece fro a oca to a datu coordate sste hs reatoshp s epressed b the atr equato u & u where s rotator atr betwee the eeet dspaceets u the oca sste ad the eeet dspaceets u the datu sste he eeets of are the coses of ages betwee the oca ad datu coordate sste For a bea eeet we have wth C C C C o S o o cos O O ; cos O O; cos O O ; o cos O O ; o ad o cos O O C o O o o o o o o o o o cos O O ; o cos O O ; cos O O; cos O O ; he atr for a p-oted bar s o o o o o o wth the above otatos he as epaed [ we obta the reatoshp betwee the oca stffess atr s or f ad these stffess atr wrtte a datu coordate sste s ad f respectve: s s f f I the et step we but the stffess atr for the copete structure sug a the overappg ters of the atr s or f whch correspod to the adacet eeets hus we ust uber the odes whch ae the coecto betwee the cabe ad the foudato to be after the odes of bea eeets Fa order to obta the oda dspaceets of the cosdered structure we ust fd out the souto of the foowg atr reato F where the etera oadg atr F correspod to the dspaceets s the above reato has bee estabshed for a free structure the atr F w aso cota the reactos ad w be a sguar atr I order to cacuate the uow dspaceets the actve forces drecto F as we as the reactos F ad the forces due to the posed dspaceets F I the equato w be wrtte as foows F FI F II I C 8 Here C s the uow dspaceets of odes the drectos wthout the posed dspaceets ad the actve forces Fro the above sste 8 C F F I ca be edate obtaed wth the et reatos:
3 SES It Cof o DMIL SSEMS ad COOL ece Ita oveber - pp-8 F C F F uerca resuts F I I Fgure shows a structure subected to the forces ad the oets the odes ad Because ths oadg we egect the dspaceets: u u u u for the bea eeets whch have ver sa vaues Hece four degrees of freedo per ode are ecessar For a ode we ote: u u u v u w ad u Fg he eeets: - - ad the cabes have two degrees of freedo per ode 8 he odes coordates a datu coordate sste the cross-sectoa area ad I erta oet of the eeets are preseted the tabes: abe ode abe Eeets [ I [ beas cabes - stffeers - he oduus of eastct s E da/ For each bea eeet we have s I / E O 8 I I I I I I I I I I I I I I I I I I I I ad s of the et for eceptg the eeet - for whch equas wth the ut atr Fg
4 ad the we use to obta s a datu coordate sste hece s ow cosder the bars ad for ths we have ad ad wth we cacuate f For our eape we fd the stffess atr for the copete structure wth et reatos If 8 8 for the bea eeets ad for the stffeers: If 8 for the cabe: [ 8 8 [ 8 [ 8 [ where we cosder for each eeet that the s or f atr s of the for rtg a progra MathD where we used the reatos ad the dvso atr ethod we fd the dspaceets step a cabes: 8 ad the reactos: - the odes 8: - the odes : SES It Cof o DMIL SSEMS ad COOL ece Ita oveber - pp-8
5 ag the sg of the cabe reactos resuts that these are copressed f > ad teso f < ow we repeat the prevous cacuus for the structure defed b the odes ad the sae atr F he ew vaues of the dspaceets ad of the reactos are Cocuso copete procedure for the statc aass of structure whch cude cabe eeets has bee preseted he agorth has the great advatage that a be used hepess a fte eeet progra ad t a be eteded to aae these coposte structures for a etera oadg I ths case a sge odfcato appears: the atr s w have the correspodg for to degrees of freedo per ode [ efereces [ C Brebba C ees IC robe Boudar Eeet echques Sprger- erag ew or 8 [ O Bre Geera Souto of suspeded Cabe Probes Joura of the Structura Dvso SCE Proceedgs febr [ Perot H Gouos M ass of cabe structures Coputers & Structures vo pp 8-8 Pergao Press Ltd [ Pssareo G aovev de eore de résstace des atérau Edtos de Moscou 8 [ JS Preeec heor of atr structura aass ew or McGraw H Boo Co 8 [ O C eewc Fte Eeet Procedures the Souto of Pate ad She Joh e & Sos Ic ew or SES It Cof o DMIL SSEMS ad COOL ece Ita oveber - pp-8
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