Modeling of Swivel Joint in two Dimensional Beams Framework

Transcription

1 Volme No., Janary ISSN -577 Internatonal Jornal of Scence and Technology IJST. All rghts reserved Modelng of Swvel Jont n two Dmensonal Beams ramework G. E. Ntamack, M. Dawoa aotong, T. Beda, S. Charf D Oazzane Grope de Mécanqe et des Matérax, GMM, Département de Physqe, aclté des Scences, Unversté de Ngaonderé B.P. 5 Ngaondéré, Cameron Laboratore de Mécanqe, Thermqe et Matérax, LMTM, Ecole Natonale de l Indstre Mnérale, ENIM, B.P. 75 Rabat, Maroc ABSTRACT Generally swvel jonts often frameworks, notably beams one, and allow them to swvel n a pont arond the three space dmensons. In the modelng of those frameworks, swvel jont s spposed to be n a node. Between two elements at a node where there s a swvel jont, dsplacements are contnos and rotatons are not. Ths paper s a contrbton to the calclaton of the rotatons n a swvel pont. Or calclaton method s based on the fnte elements method and the dstrbance of the elementary stffness matrx throgh the statc condensaton. At the end, we present analytcal reslts obtaned on the connected elements by movng the swvel jont from one element to another and the proposton to pt smltaneosly two swvel jont n two related elements. eywords: Beams, Swvel Jont, nte Elements Method, Statc Condensaton.. INTRODUCTION In the fnte elements method, the calclaton of the degree of freedom of a framework passes throgh the assembly of all elementary stffness matrces of the framework []. Takng nto accont of the swvel jont on a beam n the elementary stffness matrx reqres the dstrbance of ths matrx accordng to the place of the swvel jont []. In the frst part of ths work we sggest a modelng of the swvel jont n a beam element by dstrbance of the elementary stffness matrx. Afterwards, we propose a calclaton method of the rotaton on a swvel jont whch allows obtanng the same vale of the rotaton ndscrmnately of ts poston n the two related elements. We also present the techncal dstrbance when we have swvel-swvel jont n two related elements. The fnal part of ths work s booked to the dscsson of or reslts.. STINESS MATRIX O THE TWO DIMENSIONAL BEAMS RAMEWOR Wth the fnte elements method, by takng two nodes elements, we can dvde the beams framework n two elements of fgre. The frst element have the nodes (,), and the second elements have the node (,): gre : Plan of a beams framework made wth two elements If we sppose that each element has two degrees of freedom, the dsplacement followng the (oy) axs:, and the rotaton arond the (oz) axs:. As shown n fgre, there are for degrees of freedom for one element. gre : Degrees of freedom on a beam element If we consder that at each node, there s elementary load wth two components: followng the (oy) axs, f m, f : the component load m : the component load arond the (oz) axs. The elementary system of the eqlbrm eqatons of the element () can be formlated as follows: f m f m () The elementary system of the eqlbrm eqatons of element () s of the same knd. By assmng that node () s set n, the assembly of the elementary stffness matrces and the elementary load vector wth the lmt condtons, the global fnal eqatons system obtaned s:

2 Volme No., Janary ISSN -577 Internatonal Jornal of Scence and Technology IJST. All rghts reserved f f m m In ths system, where components and () element, we can determne the prght dsplacement () f are related to and the rotaton arond of the set node (). When there s swvel at the node (), the calclatons are dfferent.. TAING INTO ACCOUNT O SWIVEL JOINTS Assmng that we have a swvel jont n the node (), as represented n the fgre (), there are two ways to modelng ths jont. f m f m () rom statc condensaton [], on can determne the local rotaton of the swvel rotaton of element () throgh the l followng relatonshp: (5) l The vale l s njected n the elementary eqatons system whch becomes dstrbed and can be wrtten as: gre : Beams framework wth a swvel jont n node () The swvel jont can ether be on the second node of element () or on the frst node of element ().. Case where the swvel jont s spposed to be n the second node of element () In ths case, we have the followng fgre : gre : Swvel jont n the second node of element () The rotaton vale of node () s dscontned becase of the presence of the swvel jont. At ths pont, there s a swvel node and a settng n one. The rotaton of the swvel jont node s calclated wth elementary eqatons and the rotaton of the set n node s calclated by the global eqatons. Consderng the swvel jont on node () of the element () conssts n cancellng the nternal force at ths node. At ths node the nternal force whch s related to the bendng momentm mst also be cancelled: m []. The elementary system eqatons of ths element s dstrbed and becomes: = f m f m Ths on can realze that the statc condensaton don t concern elementary external forces, t only affects the elementary rgdty matrx. In order to calclate dsplacement and rotaton of set node whch s global, we mst proceed to the g assembly of the elementary external loads and the assembly of the elementary stffness matrces. And we obtan the eqatons system (7): (6)

3 Volme No., Janary ISSN -577 Internatonal Jornal of Scence and Technology IJST. All rghts reserved f f (7) g m. Case where the swvel jont s on the element () Ths case s llstrated by fgre 5. We have the plan: f f g m () Ths system evalates the expresson of g whch s the rotaton of the set node at ths pont. gre 5: Case where the swvel jont s spposed to be n the frst node on the second element Pttng swvel jont on the local node () of element () amonts to cancel the nternal force that s, the bendng momentm at ths node. In the smlar way as we dd before by sng ths tme the local node () of element (), bt whch s the global node () of or framework. We obtan eqatons system (8): f m f (8) m Throgh statc condensaton, we calclate l whch s the rotaton of the swvel jont node. And we obtan the vale: l When we pt the algebrac expresson of l n the system (8), and we make the assembly of the elementary external forces and stffness matrx, we obtan the followng global eqatons system: (9). Case where the swvel jont s jontly on the two elements In that case, we have the followng plan: gre 6: Case where the swvel jont s smltaneosly on two related elements In the case, we sggest to dstrb smltaneosly the elementary rgdty matrces of the two elements. In addtons, as n the case where the global node () was spposed to set n the two related elements, we sggest ts rotaton to be contned between the two elements. In order to solve ths problem, we sggest to calclate ths nqe rotaton vale after dstrbance of both elementary matrces rgdty and ts assembly. Or sggeston drng dstrbance concerns conservaton of vales of the lne and to have zero vales n the node where there s a swvel jont n order to evalate the algebrac expresson on the mtal rotaton. Hence we obtan the followng matrx by elmnatng the matrx elements of the forth colmn except the element of the swvel node () of the element (). We se the same method as prevosly, bt ths tme for the second colmn and the element stffness matrx of the node () of element (). After assembly, we obtan the followng system eqatons: f f ll ()

4 Volme No., Janary ISSN -577 Internatonal Jornal of Scence and Technology IJST. All rghts reserved The vale ll of the global node (), where we have two swvel jonts, s obtan after resolton of the precedng system: ll () Accordng to or modelng, three beams are related by a swvel jont as t s shown on fgre 7: gre 7: Beams framework wth three elements related by a swvel jont Ths framework cold only be modellng ths by the followng plans (fgre 8: a, b, c): s charged, wth nodes () and () set n as border condtons: =. Hence we have: f m f m or, we have: f m f m. (6) We obtan the followng reslts: (5). n the case where the swvel jont s on the node () of element (): The rotaton of the swvel node s: gre 8.a gre 8.b l (7) and the rotaton of settng node s: g (8). In the case where the swvel jont s on the node () of element (), we have : gre 8.c gre 8: Dfferent ways of modelng a swvel jont between three beams. ANALYTICAL RESULTS AND DISCUSSION We take agan the framework shown n fgre. By sggestng that, beams are dentcal and that, only element () The rotaton of the swvel node s: l (9) And the rotaton of settng node s: g (). or the case of doble swvel jont, we have the followng vale of the rotaton at ths node:

5 Volme No., Janary ISSN -577 Internatonal Jornal of Scence and Technology IJST. All rghts reserved ll () If the beams of the framework are dentcal, t comes that: = l and l g = g. More over, we obtan n the case of doble swvel jont case: ll = l l. Lke n the case where node () s settng n both related beams, we have the rotatons n contne. Also we have establshed that, when the node s swvel jonted n both related beams, rotaton vale s also contne. 5. SUMMARY In ths contrbton, throgh techncal matrces based on the fnte element method, we have calclated rotatons vales arond the swvel jont. We have shown that, no matter the swvel jont s spposed to be n two related beams n a two dmensonal framework, we obtan the same vale. Moreover, we have sggested a modelng of pttng a swvel jont n two related beams whch permts to have a nqe rotaton vale n node, as we have a nqe rotaton vale n a jont settng node. REERENCES []. Dhatt G, Tozot, (98), «Une présentaton de la méthode des éléments fns». Malone. []. Imbert J.. (99), «Analyse des strctres par éléments fns». Ed. Cepadès-Edtons. []. Brocato M., (7), «Cors de constrcton. Statqes des strctres composées de potres-efforts ntérers». Unversté de Lège. []. NTAMAC G.E. (995), «Thèse de ème cycle». aclté des Scences, Unversté Mohamed V. Rabat- Maroc. [5]. Boabd H., Charf D Oazzane S., ass-ehr O., Znedne (), Rotaton as a means of constrant desgn for optmzaton of swvel-jonted strctres. LMCM-ENIM, Rabat. 5