Computer aided design and pattering of tensioned fabric structures

Transcription

1 Compuer aided design and paering of ensioned fabric srucures Bharah Gowda Designer/Engineer, Advanced Srucures Inc Glencoe Ave., Marina del Rey, CA 90292, USA. Telphone: Fax: gowda@asidesign.com. Hamsa G. Krishnaraj Sofware Consulan, No. 756, 10h Main, 3 Block, 3 Sage, Basaveshwara Nagar, Bangalore , Karnaaka, India. hamsa_gayahri@yahoo.com Absrac This paper presens an overview for a Compuer-Aided Design and Paerning of Tensioned Fabric Srucures. Afer a brief inroducion o ensioned srucures and heir applicaions in building, a descripion of ension fabric srucure design is presened. A compuer program called Mahform, which is used for analysis design and paerning of ensioned fabric srucures uilizing he Dynamic Relaxaion mehod, is also discussed. Finally wo design examples creaed wih he Mahform program are presened. Inroducion [ASI 00] Tension fabric srucures are one of he mos exciing and rapidly developing echnologies in he building indusry oday. Maerial advances in coaed woven exiles combined wih design and numerical echniques for developing membrane srucures have yielded a new building form for permanen archiecural applicaions (see figure 1). Among hem are complee archiecural fabric enclosures for buildings, airpor erminals, resaurans, and oher public spaces: large span srucures such as sadiums, arena enclosures and reracable membrane sysems for covering indoor and oudoor spaces. Figure 1- Examples of Membrane Forms Perhaps he mos exciing aspecs of fabric srucures are he remarkable variey of aniclasic forms ha can be realized. These include hyperbolic shapes, saddles, cones, domes, vauls, and waved and plae ypes. The choices are endless. The range of forms is augmened hrough he use of suppor and resrain elemens such as cables, mass, russes, and rigid nodes. Cable-membrane srucures are referred o as form acive sysems since he form being derived from he direc relaionship beween force and cable srucures. This concep may be referred o as "form follows force".

2 Design of Tension Fabric Srucures The design of ension fabric srucures begins wih a form conceived by he designer. A drawing or a physical model usually represens his form. The desiger s form provides basic concep and suppor condiions which allow an engineer o find he rue shape of he srucure. (I is ineresing o noe he concep of form finding and no form giving). The engineer usually employees he following approaches for form finding: Qualiaively - hrough physical modeling hisorically used by designers like Frei Oo o analyze and design membrane srucures. Physical modeling involves creaing a scaled model made of maerials, which depic he acual srucure (e.g. exile cloh for he fabric, wire for edge cables ec.). If he srucural properies of he maerial of he model are known his model can also be a srucural esing model for load analysis. The grea advanage of he physical modeling mehod is he explanaion of he physical behavior of he acual srucure. Quaniaively[CAP97] - using mahemaical ools. Mahemaics does no explain physical behavior; i only describes i [SAL75]. However, in recen years, wih he help of powerful compuers, engineers can easily solve nonlinear equaions and rack ou complex rajecories ha canno be drawn. Mahemaical descripions are now so efficien ha powerful compuers can easily and fully conceive and explain membrane srucure behavior. Compuer simulaion of he srucure has become a valuable ool o help he designer find realisic shapes. The design of membrane srucures regardless of he mehods used, has hree seps: 1) Form-finding or Iniial geomery formulaion 2) Engineering analysis and membrane design 3) Paerning Compuaional mehods Tension fabric srucures can be designed eiher by using physical modeling or compuer mehods. Due o he variey of alernae design soluions o a fabric problem ha can be quickly achieved uilizing compuers, compuer mehods would be he favored ool of engineers for he design of fabric srucures. A number of compuer mehods have been developed for analysis of geomerically non-linear fabric srucures, which include shape (form-finding). The following is a brief descripion of a compuer program, Mahform. The program code is being wrien in Visual Basic and embedded ino AuoCAD. I uses Microsof Access as he backend sorage. The minimum hardware requiremen for running Mahform is a Penium processor PC (or higher) wih a Windows operaing sysem. Noe ha he program can be uilized for analysis of space russ srucures, however, he emphasis here is on he applicaion of he program for he analysis and design of fabric srucures. Mahform Program Descripion The analysis porion of he Mahform program is based on he published works of Dr. Michael Barnes [BAR77] [BAR86] [BAR84] and he work of Dr. Tajav Deganayar [DEG], who was one of he firs o implemen he dynamic relaxaion mehod in he Unied Saes o develop a sofware program called SOFTSPACE (wrien in FORTRAN). This program has been adoped by designers and engineers a Advanced Srucures Inc. o design fabric membrane srucures. Mahform is a nonlinear analysis and paerning program. The analysis secion of he program is based on he dynamic relaxaion mehod. The basis of he mehod is a sep-by-sep algorihm racing he moion of a srucure unil he srucure reaches equilibrium due o damping. The dynamic relaxaion mehods solve he geomeric nonlinear problem of form finding by equaing i o a dynamic problem. The dynamic problem is hen solved using he principle of dynamics. I is suiable for compuer simulaions of ensile srucures. The mehod can easily ake ino accoun nonlinear behavior resuling from large deformaions.

3 The analysis of fabric srucure by Mahform uilizing Dynamic Relaxaaion wih kineic damping involves he following seps: 1- Esablish a coordinae sysem wih nodal poin coordinaes and consider a line wih wo connecing nodes I and K in an arbirary locaion in space, as a hree dimensional linear elasic russ elemen. Le XYZ form he global coordinae sysem as shown in figure Esablish elemen ypes o be used for modeling various elemens. a) Consan ension fabric elemen (usually used during form finding). b) Consan force densiy fabric elemen (usually used during form finding). c) Tension only elemen (capable of aking only ension). d) Three dimensional russ elemen (capable of aking boh ension and compression). e) Slack elemen (zero siffness elemen). 3- Generaion of elemen properies, conneciviy. a) Preension (T). b) Modulous of elasiciy (E). c) Elemen lengh from node conneciviy (RL). 4- Define suppor boundary condiion. 5- Consider a simple nework of four cables as shown in figure-3.

4 R' n P F' R' := F' P ( 1) n Residual force a node i in he direcion being considered. Sum of he inernal member forces in he direcion being considered. Applied load a node in he direcion being considered. Any residual force will be aribued o he dynamic behaviour a he node. R T From Newon's second law R T := M ( i ) a ( 2) V T+ From Newon's equaion of moion V T + := V T + a. ( 3) M( i) Mass a node i a Acceleraion of he node in he ime inerval. V T Iniial velociy of he node a he sar of ime inerval. V T+ Final velociy of he node a he end of ime inerval. 6- Generae mass marix for each elemen. M := ( E A + T( R) ) 7- Compue mass marix for each node by summing of he mass of individual member coming o he node. RL ( 4) 8- Compue he velociy a ime T+. M( i) := M ( 5) n V T + := V T + RT ( 6) M ( i ) 9- Compue displacemen D a he end of ime inerval. := V T+ ( 7) 10- Updae X T+, Y T+ & Z T+ coordinaes of node i a he end of ime inerval. X ( T + ) := X T + ( 8) 11- Compue change in coordinaed Dx, Dy & Dz Dx T + := X T + X T ( 9) 12- Compue new ensions a he end of ime inerval. T T+ T E A RL T+ RL T := + RL T ( 10)

5 13- Updae residuals a node i a he end of ime inerval. 14- Check for local peak in kineic energy. R ( T+ ) ( T+ ) Dx := P xi + RL T T+ ( 11) Compue old & new kineic energy using velociies V T & V T+ M ( ) 2 1 KE old V T 1 := M ( 12) KE new := 2 2 M ( V T+ ) If KE new is found o be less han KE old han he peak has been passed and velociies are se o zero (Figure 4 depics a ypical kineic energy peaks & rese). 16- The firs velociies on resaring he process (by assuming he peak o occurs a mid poin of he firs ime sep) are given by: ( 13) 2 V := 2 M ( i ) RT ( 14) * K.E * K.E PEAK ** RESET * * * ** ** ** ** Time Sep Figure 4 Kineic Energy Plo 17-Repea sep 8 o updae velociies unil he nex energy peak. Ierae unil he soluion is arrived a when moion of a node comes o res due o damping (i.e. he residual forces are sufficienly small) and updae he final geomery. 18- Using he new geomery, apply loads and desired load combinaions. 19- Solve sysem of equaions by ieraion and find nodal displacemen and member forces. 20- Review he resuls and, if necessary, reanalyze he modifying inpu parameers. Applicaion of Mahform for form finding: In he Mahform form-finding process he form could be found eiher by specifying a consan ension or by specifying a force densiy raio (given by ension/lengh) during sep-12 while updaing ensions. Specifying consan ension would give minimal surface and using force densiy would form a curvaure in form finding.

6 The following is a flowchar for he program. START READ IN THE GEOMETRY FROM ACAD, ESTABLISH COORDINATE SYSTEM AND GENERATE NODAL POINT COORDINATES 1) NUMBER OF MEMBERS(M) 2) NUMBER OF JOINTS (N) 3) NUMBER OF SUPPORT RESTRAINS(NR) READ THE MATERIAL PROPERTIES AND ELEMENT PRETENSION READ NODE LOADS, SET TO 0 FOR FORM FINDING GET COORDINATES FROM AUTOCAD MODEL CALCULATE ORIGINAL LENGTH OF ELEMENTS SET MEMBER INFORMATION 1) DESIGNATION OF J END OF MEMBER (JJ) 2) DESIGNATION OF K END OF MEMBER (JK) 3) CROSS SECTIONAL AREA (A) 4) ELEMENT LENGTH (EL) 5) ELEMENT PRETENSION (P) 6) MODULUS OF ELASTICITY (E) SET THE JOINT RESTRAINTS SET THE JOINT RESIDULES COMPUTE THE MASS ARRAY GENERATE THE VELOCITY AND K.E. VECTORS ITIITILAZE THE LOAD, DISPLACEMENT, VELOCITY, AND THE MASS VECTORS COMPUTE THE VELOCITY AT A NEW TIME STEP COMPUTE THE DISPLACEMENT AT A NEW TIME STEP COMPUTE THE NEW ELEMENT TENSIONS & RESIDUALS COMPUTE KINETIC ENERGY USING CURRENT AND PREVIOUS VELOITIES. KE OLD > KE NEW FALSE TRUE RESET VELOCITY AND RESTART WITH REVISED VELOCITY CONVERGED, COMPUTE FINAL REACTIONS & STATIC CHECK UPDATE COORDINATES & RESIDUALS TRUE UPDATE AUTOCAD MODEL STOP Paerning Fabric srucures usually cover a hree-dimensional space even hough he membrane cover by iself is a wo dimensional surface. Paerning is a process of mapping a curvilinear surface o a fla surface or, mahemaically speaking, i is he process of ransforming a wo-dimensional surface in a hree-dimensional coordinae sysem ino a wo-dimensional coordinae sysem wih geomerical conformiy. The paerning of fabric srucures using Mahform involves he following seps: 1) Triangulae he model using Delauney s [HAN86] riangulaion and using he coordinaes of he riangles o draw 3dface in he AuoCAD model. 2) Flaen of seleced srips from he AuoCAD 3Dface model by ransforming 3Dfaces in X, Y, Z coordinaes o 3Dfaces in X, Y plane coordinae. 3) Calculae flaened srip (free of preension) by applying appropriae compensaion facor in he warp and fill direcion of he fabric srip 4) Creae compensaed srip drawings for fabricaion.

7 Design of Membrane Srucures using Mahform Mahform is used for hree purposes: a) As a form finding ool b) As an engineering ool and c) As a paerning ool. The following is a brief explanaion of how o execue Mahform. In order o run Mahform, he fabric model is generaed in Auo CAD. The process of generaing a hree dimensional ension membrane model involves creaing a fla finie elemen fabric ne model, wih properies in warp and fill direcion ha correspond o he behavior of he real fabric. The fabric ne mesh is srucured and uniform. Edge cables, mass, webbing or fabric reinforcemen, having differen properies, are modeled as a sequence of differen line elemens. The suppor boundary condiions are applied by resraining he seleced nodes (figures 5 & 7). The aribues for elemens like he maerial propery and preension are specified. The form finding process, which is eiher minimal surface or force densiy mehod selecion, complees he model generaion process for form-finding. Form-finding Form-finding, or iniial geomery formulaion, provides a deailed geomeric descripion of he srucure. In his phase, he shape of he srucure is deermined by assigning he proper presress forces o he fabric nework and specifying he suppor boundary condiion such as: mass, arches, perimeer beams, ec. The iniial shape of he fabric is approximaed (figures 5 & 7) and hen he preension analysis esablishes he final shape of he srucure (figures 6 & 8). Engineering analysis and membrane design Afer he final shape of he srucure is compued hrough form finding, he srucure is loaded for differen load cases. Due o he lighweigh of he srucure, he dead and seismic loads are negleced and wind or snow loads usually govern he design. Due o he nonlinear naure of he problem, he principle of superposiion does no hold. Hence he load analysis should be carried ou independenly for each load case. The compued forces in he fabric can be used o compare agains he allowable values. Criical reacions for he load case in he compuer runs are used for he design of he supporing srucure. The maximum displacemen of he srucure is used o compare agains he allowable deflecion per he appropriae code.

8 Paerning The form finding model is riangulaed and 3Dfaces are drawn on op of form-found model as shown in he figure 9. This process is called 3D facing. Afer 3Dfaceing is complee, srips are seleced for flaening. The following crieria are considered during srip selecion [HAN86]: Visual and archiecural effecs usually deermine he orienaion of he srips (seam alignmen). Parallel orienaion of hese srips is suiable for saddle-shaped (Hyper like) surface. Radial orienaion of he srips mus be applied for radial geomery srucures. Orienaion of each srip has o be fixed in such a direcion ha he warp and fill follow he direcion used in he form finding model. The widh of he membrane srips depends on he manufacured fabric widh and on minimizing he wase of maerial. Afer flaening he sress-free lenghs (slack lengh) of all cables, membrane pieces are deermined by applying he required compensaion refer o figures 10, 11 & 12. Conclusion Tension fabric srucures are a developing echnology, which gives archiecs and engineers he abiliy o experimen wih forms and creae exciing soluions o convenional design problems. Tension fabric srucures can be designed eiher by using physical modeling or compuer mehods. The possibiliy of realizing a diverse range of forms in a shor period of ime makes compuer simulaion he preferred mehod of designing ension fabric srucures in he fuure. Mahform is one of he few fabric srucure analysis sofware programs in he Unied Saes oday. Is fas solver, friendly inerface and inegraion wihin AuoCAD allow engineers o come up wih soluions o numerous srucural fabric applicaions, from he simple o he mos complex. Reference: [ASI 00] Advanced Srucures Inc. Fabric Lieraure (2000), 4094 Glencoe Ave., Marina del Rey, CA 90292, USA [BAR77] M.R. Barnes (1977), Form Finding and Analysis of Tension Srucures by Dynamic Relaxaion, PhD Thesis, The Ciy Universiy, Londan,. [BAR86] M.R. Barnes (1986), Compuer-aided design of Cable and membrane srucures, wih Applicaion o Expo88 and Riyadh, Lighweigh Srucures in Archiecure, Sydney, Ausralia. [BAR84] Michael R, Barnes (1984), David Wakefield, Dynamic Relaxaion Applied o Ineracive Form Finding and Analysis of Tension Srucres, Air-Suppored Srucures, The Insiue of Srucural Engineers, Brisol,. [DEG] Tajav Deganayar is Vice Presiden of Technology a Advanced Srucures Inc Glencoe Ave., Marina del Rey, CA 90292, USA. [HAN86] Ulrich Hangleier and Lohar Grundig (1986), Cuing Paern for Srucural Membranes, Lighweigh Srucures in Archiecure, Sydney, Ausralia. [KRI78] Prem Krishna (1978), Cable Suspended Roofs. McGraw-Hill Book Co. [SHA96] R.E. Shaeffer (1996), Tensioned Fabric Srucures: A Pracical Inroducion. American Sociey of Civil Engineers. [SAL75] Mario Salvadori (1975), Srucures in Archiecure. 2nd Ediion. Prenice Hall [CAP97] Frijof Capra (1997), The Web of Life. Anchor Books.