Design of tree-type support structure of free-form shell generated using fractal geometry

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1 6 30 September, 06, Too, Japan K. Kawaguhi, M. Ohsai, T. Taeuhi (eds.) Design of tree-tpe support struture of free-form shell generated using fratal geometr Jinglan CUI*, Guanghun ZHOU a, Maoto OHSAKI b *Department of Arhiteture and Arhitetural Engineering, Koto Universit Koto-Daigau Katsura, Nishio, Koto , Japan ui.jinglan.63w@st.oto-u.a.jp a Harbin Institute of Tehnolog b Koto Universit Abstrat Owing to the development of arhitetural design method and tehnologies for fabriation and onstrution, we have more hane to design a omple shape for long-span strutures []. For realizing a rational form of free-surfae, the load path to the support struture is a signifiant fator. Hene, optimization of support struture is equivalentl important as optimization of the shell surfae. Tree-tpe support struture, first proposed b German arhitet/engineer Frei Otto in 960, was applied to Stuttgart airport. Sine then, more and more tree-tpe support struture were built, for instane the London Stansted airport, the Lisbon Oriente station and the Mumbai Tote restaurant. However, the eisting tree-tpe support strutures were designed based mainl on aestheti fator rather than on phsial rationalit. The idea of biomimetis [], whih designs artifats b mimiing the harateristis of organisms lie the artifiial life [3], is not new; however, reent development of omputer tehnolog enabled us to appl it to various fields of engineering inluding arhitetural design, onstrution, planning and environment [4,5]. In order to ahieve a tree-tpe support struture onsidering the mehanial rationalit, various methods have been proposed. Rational tree-tpe strutures were obtained b Kolodziejz [6] through wet sil model method for reverse-hanging eperiment, and b von Buelow [7] through dr sil model method. Moreover, von Buelow [8] investigated the method that generates rational tree-tpe struture b utilizing geneti algorithm. Hunt [9] proposed a tree-tpe struture that transmits loads with aial fore onl b transforming rigid onnetions into pins. Cui and Jiang [0] minimized the strain energ of struture using the sensitivit oeffiients, and presented a method for simultaneousl optimizing the shape, topolog and setional area of seletal strutures inluding tree-tpe strutures. Among the methods of biomimetis and artifiial life, the fratal geometr proposed b Mandelbrot [] an epress the tree shape in nature. The L-sstem is a tpial appliation of fratal geometr to generate tree-tpe [] strutures. In this stud, we propose a method of optimizing a tree-tpe support struture, onsidering the aestheti effet, to evolve the struture to a rationalized form. The initial shape is generated using the fratal geometr [3], and the B-spline urve is used to generate the boundaries and ave of the free-form shell. It is shown in the numerial eamples that the strain energ of overall struture an be redued b optimizing the shape of the tree-tpe struture.. Method for generating the shape of a tree-tpe support struture We use the Iterative Funtion Sstem (IFS) of fratal geometr to generate the branhes of the tree-tpe Copright 06 b <J.L. Cui, G.C. Zhou and M. Ohsai> Published b the International Assoiation for Shell and Spatial Strutures (IASS) with permission.

2 struture. IFS is based on affine transformation in the 3-dimensional spae. The global oordinate sstem and the loal oordinate sstem of a member are denoted b (,, z ) and ( ', ', ), respetivel. As indiated in Fig., we denote the projetion of branh ab to XY plane as ab. ' ' The angle between ab and -ais and the angle between ab ' ' and z-ais are denoted b θ and θ z, respetivel. The global oordinate (,, z ) is translated so that its origin oinides with the end point b of branh ab, followed b rotation of around z-ais, and rotation of z around -ais, to generate a new loal oordinate sstem ( ', ', ), as shown in Fig (b). In this loal oordinate sstem, -ais is loated in the member diretion from point b, and '-ais eists in the plane formed b branh ab and its projetion ab. ' ' z z z a z b o b z z ( ) o a b a a b a branh loation of all oordinates b loal oordinate of branh end Fig. Definition of loal oordinate sstem b affine L transformation m6 ' L m L m L m3 L m7 O 4 3 L m5 L m4 Fig. Possible loations of branhes New branhes are generated using the loal oordinate sstem ( ', ', ). The branhes ma eist in positions Lm,, Lm 7, shown in Fig.. and are the angles between z'-ais and the projetion of ( Lm, Lm 3) and ( Lm 4, Lm 6 ), respetivel, to ' ' plane. The angles between z'-ais and Lm and Lm 5 are also and, respetivel.,, 3, 4, respetivel, are the angles between '-ais and the projetion of Lm, Lm3, Lm4, Lm 6 to ' plane. ' The projeted lines of Lm and Lm 5 to ' ' plane are loated on '-ais. The new branh angles,,,, 3, 4 are related to the width of the upper part of the tree. To prevent too large angle between branhes, we assign 0 in advane, and ompute from as 0 () In addition, the length L and ross-setional area s of the branh are dereased with the progress of branhing b the ratios γ and, respetivel as ' L m L m6 L L m7 m L O 4 3 L m5 m3 L m4 ' ( ) ( ) L L s ( ) ( ), (0 ) s, (0 ) ()

3 where, is the step number of branhing. Proeedings of the IASS Annual Smposium 06 Lm6 Lm Lm6 Lm Lm Lm5 Lm5 Lm Lm Lm4 Lm4 Lm3 Lm3 ' o' ' o' ' o' ' o ' ' o' Tpe Tpe Tpe3 Tpe4 Tpe5 Lm6 Lm6 Lm Lm Lm7 Lm5 Lm7 Lm7 Lm7 Lm7 Lm5 Lm4 Lm Lm Lm Lm3 ' ' o' o' o' ' ' ' o' o ' Tpe6 Tpe7 Tpe8 Tpe9 Tpe0 The new branh is determined b a ombination of lm to lm7; however, the ombinations are lassified into 0 different tpes. In five stage branh, for eample, if tpes 3, 4, 4, 5, 3 are applied at the stages,, 3, 4, 5, respetivel, then the tree generation grammar is defined as A (3,4,4,5,3). Figure 4 shows a result of eight stage branh with A (3,4,4,3,4,4,4,4). We an see a realistiall imitating natural tree shape. However, simpl using the above mentioned methods will lead to a shape with sagging branhes. This is benefiial in nature, but is not aepted for a support struture. Therefore, when alulating the angle z between z-ais in the global oordinate and -ais in the loal oordinate, is bounded b (90 ) as Z Fig.3 Branh tpes (90 Z ) ( 0<.0) (3) Figure 5 shows the result using the same grammar as Fig. 4 with the ondition (3) so that there is no sagging branh. Fig.4 An eample of tree-tpe struture Fig.5 Modified tree-tpe struture. Definition of initial shape of tree-tpe support struture and free-form shell We suppose the shape of free-form shell is defined to meet the operating and onstrution requirements, and onl optimize the shape of tree-tpe support struture. In this setion, generation method of initial shape before optimization of the tree-tpe support struture is desribed... Determination of plane boundar In order to model the irregular boundar shape, B-spline urves with odd order (m ) is etended and used for defining the urve with several intervals. Suppose there are N piees of losed urves along the inside and outside the boundaries. The i th losed urve has M i intervals, and eah interval is defined as 3

4 T N Mi ( t) B,m ( t) i j S N M T i ( t) B,m ( t) i j S (4) where, S and T are the numbers of first and last basis funtions, and (m ) is the order of basis funtion of the i th losed urve. B, m () and are the linear ombination oeffiients of B, () t, and t is the B-spline basis funtion that an be obtained b the reursive formula [4] of de Boor and Co. For eah eternal boundaries and internal boundaries, we speif the sets ( t, ) and ( t, ), and solve Eq. (4) for the oeffiients and. Finall, equall divide ( t, t ) to determine using Eq. (4) the oordinates of the points, where the nodes of finite element analsis are loated... Generation of free-form surfae shape The shape of the free-form surfae is modeled using the tensor produt B-spline surfaes. The order of basis funtions in - and -diretions are denoted b m and n, respetivel. The number of basis funtions Bi, m ( ) and Bj, n ( ) in - and -diretions are denoted b M and N, respetivel. Using the linear ombination oeffiients i, j, the surfae z(,) is defined as i j i, j i, m j,n m M N z(, ) B ( ) B ( ) (5) The eternal boundaries of shell ontained in the retanglular domain is divided into grids on the plane. The oeffiients i, jare determined b assigning the height z at the point ( i, j) and solving Eq. (5). a Setting of plane grid Fig.6 B-spline surfae shape b Surfae shape.3. Generation of initial shape For the given initial shape of tree-tpe support, planar boundar shape, and free-form surfae shape, we optimize the shape the tree-tpe support. Therefore, we obtain the intersetion between the branhes of tree-tpe struture and free-form surfae. The intersetion point is the anhored point to free-form surfae. Then we arr out Delauna triangulation b speifing the support points of tree struture and nodes on inside and outside boundaries on the -plane. The points and lines of triangulation are projeted onto the shell surfae to obtain the triangular mesh for finite element analsis. 4

5 3. Optimization problem of strutural shape We optimize the nodal oordinates of the tree-tpe struture. The vetor of design variables is denoted X, whih onsists of all variable oordinates of nodes. The objetive funtion is the total strain energ of tree-tpe struture and surfae under vertial live load and self-weight. Let KX, ( ) UX ( ), and FX ( ) denote the stiffness matri of whore struture with respet to global oordinates, the displaement vetor, and the load vetor, respetivel. The stiffness equation is written as Strain energ C an be alulated from The optimization problem is formulated as where, 0 is the feasible region of X. K( X) U( X) F( X ) (6) T C( X) U( X) K( X) U( X ) (7) find X min. C( X) subjet to X 3.. Sensitivit oeffiients of strain energ The stiffness matri of the urved shell struture and frame struture are omposed of plane triangular elements and beam elements represented, respetivel, b K and K ; i.e., s h 0 s h K K K (9) The oordinates of support points of the tree-tpe struture are fied, and the support points on the surfae are onstrained to move onl on the surfae. Therefore, variable vetor is divided into free nodes F C X and onstrained nodes X. Onl the beam elements are related to the free nodes; however, both plane triangular elements and beam elements are related to the onstrained nodes. F F F F F F The sensitivit oeffiients d i, d i, di z of strain energ with respet to the oordinates i, i, z i of the i th free node, respetivel, are epressed as d d d U F T h i F i U F T h i F i U K K K F T h iz F zi However, the differentiation of stiffness matri is arried out onl for equations related to the i th free node. z -oordinate of a onstrained node is a funtion of - and -oordinates. The sensitivit oeffiients d i i, d of strain energ with respet to the oordinates are epressed as C i, U U U (8) (0) C i of the i th onstrained node, respetivel, 5

6 T K K zi di U ( ) U i zi i T K K zi di U ( ) U i zi i () Substituting Eq. (5) into Eq. (), we obtain d, d as i i M N T K K B,m( ) di U [, j B j,n ( )] U i zi j M N, ( ) T K K Bj n di U [, jb,m ( ) ] U i zi j () 3.. Optimization method of tree-tpe support struture for free-form surfae Using the sensitivit oeffiients of strain energ with respet to the nodal oordinate vetor X, the gradient C( X ) an be obtained, and the optimal solution is found using the steepest desent method. From the solution X of step, the solution is updated using step length as follows: X X C( X ) (3) The step size is determined b the golden setion. The termination ondition is given as or ma C C (4) 8 where is a suffiientl small value, herein set Eamples Figure 7(a) shows the 40.0m 40.0m square region of designing a free-form shell. The shell surfae is pin-supported to stiff olumns at A, B, and C, and support b a tree-tpe struture with pipe setion at C(-0.0, 0.0,0.0) B(0.0, 0.0,0.0) D(-3.5,-.5,0.0) A(-0.0,- 0.0,0.0) a Triangulation of grid Fig.7 Formation of initial shape b Initial shape D. We use the grammar Am (5,4,4,4,4,4) to generate the shape of a tree struture. The length and diameter of the main pole (the lowest vertial pole) are L 3.5m and D 5.0m, respetivel. The standard length and diameter are L0 6.0m and D0 0.0m. 6

7 Moreover, the redution rates of branh length and area are γ=0.7 and ρ= 0.8, respetivel, and the parameter of the branh is =0.7. The thiness of the pipe is 0.0m, and the material is steel with elasti modulus 0GPa and Poisson s ratio 0.3. The thiness of upper free-form surfae is 0.0m, and the material is reinfored onrete with elasti modulus 30GPa and Poisson s ratio 0.. Fig.8 Tree-tpe struture evolution stereogram Figure 8 shows the optimal shape obtained from Fig. 7. The evolution proess is shown in the plan and elevation in Fig. 8, where the solid line is the position of initial shape of the tree-tpe struture, and dotted line is the shape in the evolution proess. As we an be seen from Fig. 8, the loations of free and onstrained nodes of the optimal solution are far from the intial loations; the lowest branh onneting to the base of the optimal shape is muh shorter than that of the initial shape. Figure 9(a) shows the redution ratio of the strain energ of whole struture and tree struture in evolution proess. The strain energ of the tree struture rapidl dereases to 40% of the initial struture at step 0, and further dereases to 30%, while the strain energ of whole struture dereases onl to. Figure 9(b) shows the redution ratios of the strain energ related to aial fore and bending moment, whih are hereafter simpl alled aial fore and bending moment. As a result of optimization, the aial fore redues to 8% of the initial value, while the bending moment redues to 4%. Bending moment hanges muh greater than the aial fore, espeial in the first 5 steps of optimization. Figure 9() shows the redution ratios of strain energ orresponding to membrane stress and bending stress, whih are hereafter simpl alled membrane stress and bending stress, of upper free-form surfae. B optimizing of nodal loations of tree-tpe struture, the membrane stress of upper surfae redues to 75% of the initial value, while bending stress dereases to 5%. This wa, the deformation of the upper surfae is improved b optimizing the shape of lower struture. 7

8 a Strain energ b Tree-tpe struture Upper free-form surfae Fig.9 Change of mehanial properties The appliation of fratal geometr to generate tree-tpe support struture of the free-form surfae shell is ver effetive to design a mehaniall rational struture. Furthermore, the tree-tpe support struture redues the bending deformation to generate a rational load path dominated b aial fore to the base. In the earl stage of optimization, minor adjustments of nodes an signifiantl improve the mehanial properties. 5. Conlusion The method of shape generation of supporting struture of free-form surfae shape has been presented b using fratal geometr. The omple shape of boundar and interior region of upper free-form shell is modelled using B-spline urves and surfaes. A method has also been presented for modelling a realisti tree shape using IFS based on affine transformation in the 3-dimensional spae. The branhing tpes are lassified into 0 tpes, whih are ombined to define the shape generation grammar. The formulas have been derived for sensitivit analsis of total strain energ with respet to the oordinates of nodes of the tree-tpe struture, whih are onstrained on the B-spline surfae. It has been demonstrated in the numerial eample that optimization of the tree-tpe support struture redues the bending deformation of whole struture generate a rational load path dominated b aial fore to the base. In the earl stage of optimization, minor adjustments of nodes an signifiantl improve the mehanial properties. The method presented in this paper is onstrution intentions, moreover, full aount of the optimization methods under various boundar onditions, not just the mehanis and rational of a struture, but also taing the emotional elements into aount, we loo forward to help in the atual building design. 8

9 Referenes [] S. Adriaessens, P. Blo, D. Veenendaal and C. Williams (Eds.), Shell Struture for Arhiteture, Routledge, 04. [] A. Bejan: Shape and Struture from Engineering to Nature, Cambridge Universit Press, 000. [3] E. Thro, Artifiial Life Eplorer s Kit, Sams Publishing, 993. [4] A. Bejan, Shape and Struture, From Engineering to Nature, Cambridge Universit Press, 000. [5] P. Gruber, Biomimetis in Arhiteture, Springer, 00. [6] M. Kolodziejz: Verzweigungen mit Fäden, Einige Aspete der Formbildung mittels Fadenmodellen. Verzweigungen, Natürlihe Konstrutionen - Leihtbau in Arhitetur und Natur., pp. 0-6, 99.4 [7] P. von Buelow: Following a thread: a tree olumn for a tree house, Pro. IASS Smposium, Being, 006. [8] P. von Buelow: A geometri omparison of branhing strutures in tension and ompression versus minimal paths, Pro. IASS Smposium, Venie, 007. [9] J. Hunt,W. Haase and W. Sobe: A design tool for spatial tree strutures, J. Int. Asso. Shell and Spatial Strut.,Vol. 50(), pp. 3-0, 009. [0] C.-Y. Cui and B.-S. Jiang: A morphogenesis method for shape optimization of framed strutures subjet to spatial onstraints, Eng. Strut., Vol. 77, pp. 09 8, 04. [] B. B. Mandelbrot, Fratals: Form, Change, and Dimension, W. H. Freeman & Compan, 977. [] P. Prusiniewiz and A. Lindenmaer, The Algorithmi Beaut of Plants, Springer-Verlag, 990. [3] C. Bovill, Fratal Geometr in Arhiteture and Design, Birhäuser, 996. [4] G. Farin, Curves and Surfaes for CAGD, Morgan Kaufmann Publishers,