Form-finding of grid shells with continuous elastic rods

Transcription

1 Page of 0 Form-fndng of grd shells wth contnuous elastc rods Jan-Mn L PhD student Insttute of Buldng Structures and Structural Desgn (tke), Unversty Stuttgart Stuttgar, Germany quantumamn@gmal.com Jan Knppers Prof. Dr.-Ing. Insttute of Buldng Structure and Structural Desgn (tke), Unversty Stuttgart Stuttgart, Germany j.knppers@tke.un-stuttgart.de Keywords: grd shells; form-fndng; elastcty; free-form; dynamc relaxaton; NURBS; least stran energy; constrants; mappng; resdual forces. Abstract Grd shells wth contnuous elastc rods have the advantages to generate curved spaces wth unform members and jonts. The jonts connectng the crossng rods are capable of n-plane rotaton. Ths makes grds lack the n plane rgdty and allows grds to take large deformatons durng erecton/formaton process. Besdes, members n these elastc structures are n bended states, thus when they are all combned together they can easly generate sngle or double curved space. Ths character makes elastc grd shell have the potental to be the structures of free-form archtectures. However, fndng the boundary condtons, ncludng the grd pattern and bearng postons, whch lead to a specfc geometry, s not an easy task. Desgners have to keep equal grd lengths, mnmse the resdual forces, ensure the smoothness and attan the desred geometry smultaneously. In short of an approprate numercal method, desgners of the projects lke Mannhem Multhall and Downland Museum returned to physcal models to fnd the boundary condtons. The complcated and mult-steps form-fndng process makes ths knd of structure less popular to archtects/engneers and only a few buldng projects were realzed. Ths paper descrbes a new numercal method whch can derve the grd pattern and bearng postons accordng to a desred geometry. Ths s done by fndng the least stran energy state of the elastc grd n the soluton doman defned by constrants. Ths method can provde archtects a grd pattern that satsfes all the geometrcal demands. At the same tme, a result wth least stran energy s favoured by engneers. Ths s especally mportant for an elastc grd shell, whose structural stablty s largely affected by the resdual forces. The rest of the paper s organzed as follows. In secton, we dscuss related works and gve our assessment to each method. Secton gves a detaled descrpton of the theorem that our method s based on. A physcal model s used to help readers to buld a general dea of our method. Then, more detaled descrpton of our numercal method s presented. A grd shell wth the shape of Downland museum s generated by our method as llustraton (Fg.). In secton, we dscuss the deformatons of elastc grd shells, caused by resdual forces, by analyzng the llustratve example n secton. Fnally, a concluson s gven. tern = tern = tern = 0 Fg. : stages of teraton tern = 00

2 Page of 0 Summary Form-fndng of grd shells wth contnuous elastc rods Jan-Mn L PhD student Insttute of Buldng Structures and Structural Desgn (tke), Unversty Stuttgart Stuttgar, Germany quantumamn@gmal.com Jan Knppers Prof. Dr.-Ing. Insttute of Buldng Structure and Structural Desgn (tke), Unversty Stuttgart Stuttgart, Germany j.knppers@tke.un-stuttgart.de Grd shells wth contnuous elastc rods have the advantages to generate curved spaces wth unform members and jonts. However, fndng the boundary condtons, ncludng the grd pattern and bearng postons, whch lead to a specfc geometry, s not an easy task. Desgners have to keep equal grd lengths, mnmse the resdual forces and ensure the smoothness of geometres smultaneously. In ths paper, we present a new numercal method whch can derve the grd pattern and bearng postons n accordance wth a desred geometry. Ths s done by fndng the least stran energy state of the elastc grd n the soluton doman defned by constrants. Ths method can provde archtects a grd pattern that satsfes all the geometrcal demands. At the same tme, a structure wth less stran energy s favoured by engneers. Ths s especally mportant for elastc grd shells, whose structural stablty s largely affected by the resdual forces. Keywords: grd shells; form-fndng; elastcty; free-form; dynamc relaxaton; NURBS; least stran energy; constrants; mappng; resdual forces.. Introducton Grd shells wth contnuous elastc rods have the advantages to generate curved spaces wth unform members and jonts. The jonts connectng the crossng rods are capable of n-plane rotaton. Ths makes grds lack the n plane rgdty and allows grds to take large deformatons durng erecton/formaton process. Besdes, members n these elastc structures are n bended states, thus when they are all combned together they can easly generate sngle or double curved space. Ths character makes elastc grd shell have the potental to be the structures of free-form archtectures. However, fndng the boundary condtons, ncludng the grd pattern and bearng postons, whch lead to a specfc geometry, s not an easy task. Desgners have to keep equal grd lengths, mnmse the resdual forces, ensure the smoothness and attan the desred geometry smultaneously. In short of an approprate numercal method, desgners of the projects lke Mannhem Multhall and Downland Museum returned to physcal models to fnd the boundary condtons. The complcated and mult-steps form-fndng process makes ths knd of structure less popular to archtects/engneers and only a few buldng projects were realzed. Ths paper descrbes a new numercal method whch can derve the grd pattern and bearng postons accordng to a desred geometry. Ths s done by fndng the least stran energy state of the elastc grd n the soluton doman defned by constrants. Ths method can provde archtects a grd pattern that satsfes all the geometrcal demands. At the same tme, a result wth least stran energy s favoured by engneers. Ths s especally mportant for an elastc grd shell, whose structural stablty s largely affected by the resdual forces. The rest of the paper s organzed as follows. In secton, we dscuss related works and gve our assessment to each method. Secton gves a detaled descrpton of the theorem that our method s based on. A physcal model s used to help readers to buld a general dea of our method. Then, more detaled descrpton of our numercal method s presented. A grd shell wth the shape of

3 Page of 0 Downland museum s generated by our method as llustraton. In secton, we dscuss the deformatons of elastc grd shells, caused by resdual forces, by analyzng the llustratve example n secton. Fnally, a concluson s gven.. Related works and our assessment Hangng chans models can generate funcular geometres that have only tenson forces whle takng gravty load. Invertng the geometres of funcular, desgners can get deal shell geometres that have only compresson forces when takng gravty load. The dea was frst ntroduced by the physcan Hook n the th century, further developed by the archtect Gaud n the th [], and appled n the form-fndngs of grd shells by Fre Otto n 0s[]. The geometres derved n ths way have good performance whle takng gravty load, however when lateral or uneven loadng s domnant, ths method loses ts advantages. Besdes, gravty loads takes the man role n the formatons of hangng chans models. Ths wll lmt possble geometres of grd shells. The method of the Chebyshev net, also known as the compass method, s a geometrcal method. The drawng of the Chebyshev net starts from two arbtrary ntersectng curves on a surface. Each curve s composed of segments wth the same mesh wdth. The rest nodal ponts are only determned by fndng ntersecton from two adjacent nodes wth the same mesh wdth (Fg.). The method was frst Fg. : the Chebyshev net / the compass method, IL() seen n the work of P.L. Chebyshev n [] and further researched by Fre Otto n 0s. It has the advantage to adapt to free form surfaces. But t does not take account of bendng behavours of elastc materals. Elastc grd shells wth the geometres derved n ths way are not n statc equlbrum and may transform to other shapes. Ths wll brng addtonal stresses n members and extra dffcultes for erecton. Fg. : Downland museum, desgned by Edward Cullnan, engneered by Buro Happold, Sngleton England, 00, ( Chrs Wllam and M.R. Barnes developed a way usng dynamc relaxaton method to compute the mechancs of elastc rods []. Chrs Wllam appled the method to buld the numercal model of Downland Museum (Fg.) []. In that case, physcal models were stll used to decde the boundary condtons []. The team of Insttut Naver also appled ths method to desgn a grd shell n composte materal []. The form-fndng n that case started from a specfc cuttng pattern and stopped whle an aesthetc shape s formed durng the upwards pushng process. The relatonshp between the boundary condtons and the correspondngly generated form remans unclear. The team of Insttut Naver also proposed a way of mappng contnuous elastc grds on gven surfaces []. The dynamc explct fnte element analyss was used to smulate plane elastc grds lad on mposed surfaces under the tracton of gravty. Ths method s able to create a grd shell structure complyng wth specfc surfaces whle takng bendng behavours nto consderaton. However, the tracton by gravty wll further twst grd geometres and lead geometres to follow the drecton of gravty. Ths phenomenon wll lmt possble geometres and brng addtonal stresses n elastc rods. The tracton by gravty could be seen as an addtonal constrant n our method by takng t as extra nodal loads durng the form-fndng process.

4 Page of 0. Mappng grds by fndng the mnmal stran energy state Our method s based on the phenomenon that f a grd composed of contnuous elastc rods s constraned on a surface, t shall transform nto a specfc geometry wth the mnmal stran energy. Ths s the key pont of our mappng method. In the followng text, a physcal model s used to help readers to buld a general dea of our method. Then, more detaled descrpton of our numercal method s presented. A grd shell wth the shape of Downland museum s generated by our method as llustraton.. Physcal model The mechansm of least stran energy can be more easly understood by the physcal model as n Fg.. The grd s composed of 0 contnuous elastc rods. The jonts between the crossng rods are desgned to allow n-plane rotaton. The char back surface and the ponts fxed by the hand are taken as constrants. It shows that wth only a few nodes constraned on the surface the rest nodes Fg. : physcal model wll ft nto the surface by the mechansm of elastcty. If we fx the nodes n other postons, the grd wll automatcally adjust ts whole shape and form another smooth geometry mmedately. After fxng the boundares and removng the mould (the char n Fg. ) the geometry of the grd wll change slghtly untl an equlbrum state s reached.. Numercal model We dvde the process of buldng numercal model nto two stages. Frst, we use dynamc relaxaton method to buld a model whch can smulate the mechansm of elastc rods. Second, we derve the geometrcal nformaton from NURBS and apply t as surface and curve constrants to the model... Smulatng mechansm by dynamc relaxaton method Usng the method of dynamc relaxaton (DR), we can transform the elastc rod system nto a partcle system []. The axal forces and bendng momentums n rods can be calculated from the dstances between partcles and the ncluded angles defned by lnes whch connect partcles []. We can derve the veloctes and postons of partcles n the next tme step by values already known n the prevous tme step. Therefore, we can get the dynamc of the system n tme hstory. Furthermore, we apply knetc dampng [] nto the dynamc system such that the system wll gradually run out of ts knetc energy and fnally reach a statc equlbrum state... Applyng constrants We apply constrants by confnng the partcles always movng on the constrant surfaces and curves. It s done by projectng each partcle to the nearest pont on the constrant surface/curve n each teraton crcle as well as modfyng each partcle s velocty as n () that only the tangental part s actve. The unt vector nˆ s normal to the assgned surface/curve at the poston of node. v V tan v = V v ( V nˆ ) nˆ We can detect f the equlbrum state s reached by checkng the tangental part of the net force s smaller than a gven value for every partcle. ()

5 Page of 0. Example ntal square grd surface constrant curve constrant nodes on longer sdes Fg. : elastc grd and constrants (surface wth span.m, heght.m; mesh length m) tern = tern = tern = 0 Fg. : stages of teraton tern = 00. Geometres of bearng structures We further llustrate our numercal method by generatng a grd shell wth the geometry of Downland Museum. One mportant characterstc of ths trple-bulb double hourglass grd shell s that t s generated from a square grd []. After the formaton, the nodes on the longer sdes of the ntal square grd are restuated on the curved buldng foundaton. The shell s boundary condton can not be found by the exstng numercal form-fndng methods as descrbed n secton. Physcal models were used by the desgn team to fnd the postons and the tltng angles of bearng ponts []. Below we show how we solve ths dffculty wth our new method. The ntal square grd s composed of crossng elastc rods wth profle xcm n wood wth E modulus N/mm (Fg.). The nodes on the longer sdes are constraned to move along the edge curves whle the rest nodes are constraned to move on the surface. Fg. shows that the grd goes through dfferent stages and reaches the fnal equlbrum state wth a smooth grd confguraton wth equal grd lengths. Intal length of grd members s m. The lengths after mappng are changed due to the axal forces n members. The longest member s 0.mm longer than ntal length and the shortest member s 0.mm shorter than the ntal length. Mantanng shape s the basc ssue for elastc grd shells. One crucal queston arses here. Once the constrants are removed, wll the grd structure stay n place and keep ts geometry? In order to answer ths, we need to transform our mappng results nto real bearng structures. We can make t by replacng prevous constrants by real bearng condtons, addng optonal bracngs, and addng edge beams f there are open ends. After that, we need to relax the structure once agan to get a shell geometry whch s n balance wth resdual forces.. The deformaton by resdual forces In order to answer the above-mentoned queston, we transform the grd pattern derved n. to a bearng structure and observe ts deformatons by resdual forces. We remove the surface and curve constrants and assgn bearng ponts as hnge. Edge beams are added to the two open ends. Each edge beam s composed of coupled elastc rods wth profle xcm n wood (Fg.). edge beam Fg. : bearng structure bearng ponts

6 Page of 0 A B C D The deformatons after the surface s removed are shown n Table. The largest deformaton s on the top of the edge beam, wth magntude 0cm, whch s manly caused by the elastcty n edge beams. The second largest deformaton occurs on the shell s wast area, whch s n the secton C (Fg.), wth magntude cm. Then, we try another bearng condton by changng the bearng ponts as fxed jonts such that the postons and tltng angles are fxed. The largest deformaton occurs stll on the tops of the edge beams, wth Fg. : plane vew and secton axes magntude 0cm, / compared wth the heght of the structure. The second largest deformaton occurs n wast areas, wth magntude cm. These varatons from the constrant geometry are acceptable for us, snce the structure stll mantans ts characterstc geometry and smooth pattern. We made another test by addng bracngs as n Downland museum and usng hnges nstead of fxed jonts for bearng ponts. The result shows that the deformaton on the tops of the edge beams shrnks dramatcally to cm. The elastc shell matches the constrant surface very well. Table : deformatons under dfferent condtons deformaton Hnge bearngs Fxed bearngs Perspectve Sde vew Secton A Secton B Secton C Secton D Hnge bearngs wth bracngs

7 Page of 0. Concluson We propose a new numercal method whch can derve the grd pattern and bearng postons n accordance wth a desred geometry. Ths s done by fndng the least stran energy state of the elastc grd n the soluton doman defned by constrants. Ths method can provde archtects a grd pattern that satsfes all the geometrcal constrants. At the mean tme, a result wth least stran energy s favoured by engneers. Ths s especally mportant for an elastc grd shell, whose structural stablty s largely affected by the resdual forces. We further show the feasblty of ths method by generatng a grd shell wth the geometry of Downland museum. References [] Graefe R., On the form development of archs and vaults (n German), Journal of Hstory of Archtecture, Munchen-Berln,, pp. - [] IL, IL Grd Shells, Insttute for the Lghtweght Structures,, Stuttgart. [] Eugene Vladmrovch Popov, Geometr Approach to Chebyshev Net Generaton Along an Arbtrary Surface Represented By NURBS, Internatonal Conference Graphcon 00, Nzhny Novgorod, Russa [] S. Adraenssens, M.R. Barnes, and C.J.K. Wllams, A new Analytc and Numercal bass for the Form-fndng and Analyss of Splne and Grd-shell, Computng Developments n Cvl and Structural Engneerng,, pp. -0 [] Olle Kelly, Rchard Harrs, Mchael Dckson, and James Rowe, Constructon of the Downland grd shell, Structural Engneer, :, 00, pp. -. [] Rowe, J. and Harrs, R. The structural engneerng of the Downland Grdshell, IABSE Conference: Innovatve Wooden Structures and Brdges, 00, pp. - August, Laht, Fnland. [] C. Douthe, O. Baverel, J.-F. Caron, Form-fndng of a grd shell n composte materals, Journal of IASS, 00. [] Lna BOUHAYA, Olver BAVEREL, Jean-Franços CARON, Mappng two-way contnuous elastc grd on an mposed surface, Proceedngs of the IASS Symposum, 00, pp. -,Valenca. [] M.R. Barnes, Form fndng and analyss of tenson structures by dynamc relaxaton, Internatonal Journal of Space Structures, Vol. No.,, pp. -.