THRUST NETWORK ANALYSIS: A NEW METHODOLOGY FOR THREE-DIMENSIONAL EQUILIBRIUM

Transcription

1 TRUST NETWORK ANALYSIS: A NEW METODOLOGY FOR TREE-DIMENSIONAL EQUILIBRIUM Phppe BLOCK Research Assstant Budng Technoogy, MIT Cambrdge, MA, USA John OCSENDORF Assocate Professor Budng Technoogy, MIT Cambrdge, MA, USA Summary Ths paper presents a new methodoogy for generatng compresson-ony vauted surfaces and networs. The method fnds possbe funcuar soutons under gravtatona oadng wthn a defned enveope. Usng proectve geometry, duaty theory and near optmsaton, t provdes a graphca and ntutve method, adoptng the same advantages of technques e graphc statcs, but offerng a vabe extenson to fuy three-dmensona probems. The proposed method s appcabe for the anayss of vauted hstorca structures, specfcay n unrenforced masonry, as we as the desgn of new vauted structures. Ths paper ntroduces the method and shows exampes of appcatons n both feds. Keywords: Compresson-ony structures; Unrenforced masonry vauts; Funcuar anayss; Thrust Networ Anayss; Recproca dagrams; Form-fndng; Lower-bound anayss; Structura optmzaton. 1. Introducton Medeva vaut buders created compex forms carefuy baanced n compresson. The structura propertes of these sophstcated forms are st poory understood because of a ac of approprate anayss methods,.e. methods reatng stabty and form. Understandng the mechancs of these vauted structures eads to new nsghts for both anayss and desgn. Thrust Lne Anayss s a powerfu graphca method for cacuatng the range of ower-bound equbrum soutons of compresson-ony systems, such as unrenforced masonry structures (Fg. 1). It vsuases the stabty of these structures and suggests possbe coapse mechansms [1]. Unfortunatey, thrust ne anayss s prmary sutabe for -D cases and ths mtaton has prevented t from beng used for the assessment of compex 3-D structures. Whe numerca methods based on eastc soutons gve one possbe answer, they no onger suggest better form as was nherent to the more hostc graphca methods. (a) (b) Fg. 1 (a) Two possbe compresson-ony equbrum shapes for a random set of oads, and (b) an nteractve thrust-ne appcaton deveoped n [1]: the user can adapt the geometry by draggng contro ponts and the structura feedbac, n the form of a thrust-ne, s updated n rea-tme. The magntudes of the forces n the system are vsuazed n the accompanyng funcuar poygon (rght).

2 There s a rea need for toos to better understand and vsuase the stabty of compresson-ony structures, such as hstorc unrenforced masonry structures, as we as desgn toos that suggest better form. Both probems are reated to fndng axa force structures n equbrum actng ony n compresson or tenson. Currenty, graphc statcs provdes a hostc anayss and desgn too for two-dmensona structures. Wth today s avaabty of powerfu vrtua 3-D and parametrc envronments, the foowng queston arses: can a fuy three-dmensona verson of thrust-ne anayss provde the same freedom to expore the nfnte equbrum soutons for a certan oadng condton?. Methodoogy The Thrust-Networ Anayss method presented n ths paper s nspred by O Dwyer s wor on funcuar anayss of vauted masonry structures []. It s extended by addng the concept of duaty between geometry and the n-pane nterna forces of networs [3]..1 Recproca fgures The duaty between the geometry of a networ and ts nterna forces s an od concept, frst expaned by Maxwe [4]. e caed ths reatonshp recproca and defned t as foows: Two pane fgures are recproca when they consst of an equa number of nes, so that correspondng nes n the two fgures are parae, and correspondng nes whch converge to a pont n one fgure form a cosed poygon n the other. Ths means that the equbrum of a node n the frst dagram s represented by a cosed poygon n the second dagram and vce versa (Fg. ). Graphc statcs s based on ths prncpe [5]. (a) (b) Fg. (a) The prma grd Γ and dua grd Γ are reated by a recproca reatonshp. The equbrum of a node n one of them s guaranteed by a cosed poygon n the other and vce versa. The abeng uses Bow s notaton [6]. Ths ustraton demonstrates the dea on a pece of the networs n Fg. 3.. Assumptons The proposed method produces funcuar (compresson-ony) soutons for oadng condtons where a oads are apped n the same drecton, as s the case for gravtatona oadng. Snce the soutons are compresson-ony ths aso means that the vauts can never cur bac onto themseves, whch woud demand some eements to go nto tenson. The resutng three-dmensona networs can represent oad paths throughout a structure. There s no constrant on the ength of the branches or the panarty of the facets of the souton..3 Thrust Networ Anayss Key eements n the proposed process are (1) force networs, representng possbe forces n equbrum n the structure; () nteractve recproca dagrams, vsuasng the proportona reatonshp of the horzonta forces n the networ and provdng a hgh eve of contro for the user to manpuate the force dstrbutons n the system; (3) the use of enveopes defnng the souton space; and (4) near optmsaton, resutng n fast computaton of resuts.

3 .3.1 Overvew of man steps Thrust Networ Anayss has been mpemented usng Matab [7] and RhnoScrptng n Rhnoceros [8]. The set-up of the program s expaned n more deta beow: Fg. Reatonshp between compresson she (G), ts panar proecton (prma grd Γ) and the recproca dagram (dua grd Γ ) to determne equbrum. (a) Defnng a souton enveope: The soutons must e wthn gven boundares defned by an ntrados and an extrados (Fg. 6b). These put heght constrants on the nodes of the souton. These mts can be the desgn enveope or the actua vaut geometry for the anayss of exstng masonry vauts. (b) Constructng the prma grd Γ: In pan, a possbe force pattern topoogy s constructed. Ths s the prma grd Γ n Fg. 3. The branches represent possbe oad paths throughout the structure. The oaded nodes represent the horzonta proectons of centrods (cf. step c). These force patterns can be drawn by the user or generated automatcay. The prma grd Γ s the horzonta proecton of the fna souton G. (c) Attrbutng weghts: The weghts attrbuted to the oaded nodes come from umpng the dead oad of the 3-D trbutary area around those nodes. In addton to sef weght, oads such as asymmetrc ve oads can be apped. (d) Generatng the dua grd Γ: The dua grd Γ s produced from the prma grd Γ accordng to Maxwe s defnton of recproca fgures: correspondng branches stay parae and noda equbrum n the prma grd s guaranteed by cosed poygons n the dua grd. Ths s beng soved automatcay by two consecutve near optmsaton probems wth the constrants comng from Maxwe s defnton. The detas of ths optmsaton set-up w be eaborated on n future pubcatons [9]. The apped oads do not appear n the dua grd because they dsappear n the horzonta proecton. Therefore, the dua grd has an unnown scae ζ snce the reaton between the prma and dua grd s true regardess of ther reatve scaes (Fg. 4).

4 Fg. 4 For a reguar square pow shape wth contnuous supports on a edges and equa thrust n both man drecton, decreasng the scae factor ζ of the dua grd means gobay ower horzonta forces n the system and hence a deeper souton whch s n equbrum for the same set of apped oads. (e) Updatng the dua grd: In the case of an ndetermnate prma grd,.e. a grd wth nodes wth a hgher vaency than 3, the user can manuay change the force dstrbuton by manpuatng the dua grd (Fg. 5). (a) (b) Fg. 5 For a determnate (.e. 3-vaent) prma grd, there s a unque reatonshp between prma and dua (a). For an ndetermnate prma grd, mutpe dua grds, whch a satsfy Maxwe s defnton, are possbe (b). (f) Sovng for the resut G: Usng the geometry of both prma (Γ) and dua (Γ) grd, the weghts apped at the nodes and the boundary condtons, ths probem can be soved usng a one-step near optmsaton. We sove smutaneousy for the noda heghts of G and the scae of the dua grd ζ. The horzonta components of the forces n the souton G can easy be found by measurng the engths of the branches n the dua grd and mutpyng them by the actua scae ζ..3. Lnear optmsaton formuaton The frst set of constrants comes from enforcng statc equbrum at a nodes (Fg. 6a). The vertca equbrum of a typca nterna node gves V V V F F + F = P We descrbe (1) as a functon of the horzonta components of the forces ( z z ) ( x x ) + ( y y ) + (1) ( z z ) ( x x ) + ( y y ) ( z z ) ( x x ) + ( y y ) F + F + F = P The engths of branch n the prma and dua grds are defned respectvey as, and,. The horzonta components of the forces n the branches, F, can be expressed as a functon of the dua branch engths,, measured from the dua grd Γ and mutped by the as-yet unnown scae factor ζ. () F = ζ, F = ζ, F = ζ (3),,,

5 (a) (b) Fg. 6 The constrants come from (a) statc equbrum n every node under the apped oadng and (b) the gven boundares, resutng n noda heght constrants. Rearrangng equaton () and wrtng t as a functon of the branch engths n both grds usng equatons (3) gves,,,,,, + + z z z z P r = 0 (4),,,,,, where r s the nverse of the unnown scae of the dua grd, ζ. We can wrte equaton (4) as C z + C z + C z + C z P r = 0 (5) The constants C of the near functon (5) are a functon of the prma and dua branch engths. The equbrum constrants of the nodes can be wrtten as a near combnaton of z, the unnown noda heghts, and r. Ths emphasses the mportance of usng the nformaton provded by the dua grd (3). Thans to ths nsght, the nonnear constrants () can be made near by treatng r as a varabe. Note that, because engths (absoute vaues) are used, ths formuaton guarantees that a soutons G w be compresson-ony. A second set of constrants comes from the mts put on the noda heghts (Fg. 6b). We want the soutons to e wthn the gven boundares defned by an ntrados and an extrados. z I z z (6) Snce we are nterested n the range of possbe soutons that ft wthn the gven enveope, we want to mnmse or maxmse r (= 1/ ζ), resutng n respectvey the shaowest or deepest souton st contaned wthn the mts, for a chosen combnaton of prma and dua grd. Ths then becomes the obectve functon of the near optmsaton probem. E 3. Appcatons for the anayss of vauted masonry structures Usng the proposed methodoogy for the assessment of unrenforced masonry structures fts wthn the ream of owerbound anayss. Put smpy, f a compresson-ony networ can be found that fts wthn the boundares of a vaut, then the vaut w stand n compresson. Ths s a powerfu concept for understandng the stabty and proxmty to coapse of such structures. Addtona readng on ths topc can be found n eyman [10], O Dwyer [], Boothby [11] and Boc et a. [1]. The method s partcuary approprate for hstorc masonry structures because ther sef-weght s the domnant oad. The range of possbe equbrum states, bounded by a mnmum and maxmum thrust, can be produced (Fg. 7a). We are nterested n ths range of thrust vaues because t provdes the most usefu characterzaton of the structura behavour of the vaut. The mnmum (or passve) thrust state represents the east amount ths vaut w push sdeways onto ts neghbourng eements, ust because of ts sef-weght and shape. The maxmum (or actve) state of thrust on the other hand represents the argest horzonta force ths vaut can provde. So, ths vaue demonstrates how much horzonta pushng ths vaut can safey tae from ts neghbours.

6 (a) (b) (c) Fg. 7 (a) Possbe thrust vaues for ths gron vaut range from 45% to 70% of ts tota weght. (b) 3-D web and rb acton wth the forces many spannng between the rbs as represented n the dua grd (c). Fgure 7b shows a souton wth three-dmensona web acton. The dstrbuton of the horzonta components n the networ s represented n ts dua grd (Fg. 7c). Such a 3-D equbrum anayss s vauabe because t nforms us very ceary about the stabty of a vaut. A forthcomng pubcaton w show more detaed appcatons of ths methodoogy to masonry vauts [1]. 4. Appcatons for the desgn of compresson-ony structures Fgure 8 gves a seres of compresson-ony soutons for a unformy apped oadng, startng from a reguar rectanguar grd. It shows the reatonshp between the dua grd and the correspondng souton. From the dua grds, the nterna dstrbuton of a horzonta forces n the networs can be understood n a gmpse, and snce a dua grds are drawn at the same scae, the overa magntude of the forces n the dfferent soutons can be compared drecty. Some speca features are that force nes do not have to go through oaded nodes (8e-) or that the edges can be freed, archng n space (8h-). The exampes n Fgure 8 show the potenta of the method for desgn through a seres of exampes, whch were possbe because of the fexbty and ntutveness of the method: forces can be redstrbuted nternay wthn the networ by tweang the dua grds (e.g. 8a versus 8b); more force can be attracted n prmary force nes (e.g. 8e or 8f); and dfferent boundary condtons can be expored (e.g. 8h). (a) (b) (c) (d) (e) (f) (g) (h) () Fg. 8 A seres of exampes, startng from a reguar rectanguar grd, showng the reatonshp between the prma and dua grd and the correspondng souton.

7 The foowng exampe n Fgure 9 shows a possbe force pattern for a spherca dome nspred by structures by Per Lug Nerv. The generated dua grd (Fg. 9c), whch represents the equbrum of the forces n the prma grd (Fg. 9b), has a partcuary beautfu shape and coud become more nterestng as force pattern than the orgna prma grd. Snce the prma and dua grd have a dua reatonshp, ths means that the prma grd coud aso be seen as the representaton of the forces n the dua grd. So, notce that the force pattern wth an ocuus n Fg. 9c s a souton wth approxmatey equa horzonta forces n a eements because a branch engths n Fg. 9b are approxmatey the same. (a) (b) (c) Fg. 9 For a spherca dome wth an ange of embrace of 90, a Nerv-nspred force pattern s chosen (b). The dua grd (c) represents the forces n the prma grd. (a) shows the deepest souton under ts sef-weght for ths choce of prma and dua grd that fts wthn the dome s secton, representng the mnmum state of thrust of ths dome. Ths exampe shows that the generaton of a dua grd not ony aows to represent the forces n the system n a very vsua manner, but that t aso can surprse and start to nspre new desgns. It becomes more nterestng f both grds can freey be exchanged and changes can be made n ether grd, nfuencng and updatng the other. 5. Dscusson and concusons Ths paper has proposed the Thrust Networ Anayss method. It provdes - a vabe three-dmensona extenson for thrust-ne anayss; - a fexbe, ntutve and nteractve desgn too for fndng three-dmensona equbrum of compresson-ony surfaces and systems; and - an mproved ower-bound method for the assessment of the stabty of masonry vauts wth compex geometres. Key features are - cear graphca representaton of forces n the system (through the use of force dagrams,.e. the dua grds); - a hgh eve of contro, aowng the exporaton of dfferent possbe equbrum soutons; and - fast sovng tmes because of the formuaton as a smpe near optmsaton probem. Currenty, the number of eements n the networ s mted by the mpementaton n Matab and the user has to swtch between programs. Future wor ncudes gong towards true nteractvty and b-drectonaty between both grds, mpemented n a fuy parametrc envronment, and the automatc generaton of possbe networ topooges accordng to e.g. curvature, openngs, support condtons or archtectura preferences. We beeve that ths methodoogy has great potenta for the use n both desgn and anayss of compresson-ony vauted structures. 6. Acnowedgments The authors woud e to than Professor Chrs Wams of Bath Unversty, for hs comments on Bow s notaton for 3-D networs that spared the deas underyng ths wor. The frst author s aso gratefu to Xuan Vnh Doan at MIT for hs hep n settng up the near optmsaton probems and for the fnanca support of the arod orowtz (1951) Student Research Fund.

8 7. References [1] BLOCK P., CIBLAC T., OCSENDORF J., Rea-Tme Lmt Anayss of Vauted Masonry Budngs, Computers and Structures, Vo. 84, No. 9-30, 006, pp [] O'DWYER D., Funcuar anayss of masonry vauts, Computers and Structures, Vo. 73, 1999, pp [3] WILLIAMS C.J.K. "Defnng and desgnng curved fexbe tense surface structures," n The mathematcs of surfaces, J.A. Gregory (Ed.), Oxford, Carendon Press, 1986, pp [4] MAXWELL J.C., On recproca fgures and dagrams of forces, Ph. Mag. Seres, Vo. 4, 7, 19864, pp [5] CREMONA L. Le Fgure Recproche nea Statca Grafca. Man, Urco oep, [6] BOW R.. Economcs of constructon n reaton to frames structures. London, Spon, [7] Matab: [8] Rhnoceros: [9] BLOCK P. Three-dmensona Equbrum Approaches for Structura Anayss and Desgn, PhD thess, Department of Archtecture, MIT, Cambrdge, MA, USA, Expected 009. [10] EYMAN J. The Stone Seeton: Structura engneerng of masonry archtecture. Cambrdge, Cambrdge Unversty Press, [11] BOOTBY T.E., Anayss of masonry arches and vauts, Progress n Structura Engneerng and Materas, Vo. 3, 001, pp [1] BLOCK P. and OCSENDORF J. Lower-bound anayss of unrenforced masonry vauts, Submtted for the 6th Internatona Conference on Structura Anayss of storca Constructon, Bath, UK, 008.