Discrete elastica model for shape design of grid shells

Transcription

1 Abstracts for IASS Annua Symposium th September, 017, Hamburg, Germany Annette Böge, Manfred Grohmann (eds.) Discrete eastica mode for shape design of grid shes Yusuke SAKAI* and Makoto OHSAKI a * Department of Architecture and Architectura Engineering, Kyoto University Kyoto-Daigaku Katsura, ishikyo, Kyoto , Japan sakai.yusuke.5v@st.kyoto-u.ac.jp a Kyoto University,Japan Abstract A grid she is designed by bending beams connected by hinge joints. This paper presents an approach to designing grid she structures made of stee or wood, considering them as an assemby of discretized piecewise inear curves, which are caed discrete eastica. Eastica is defined as the shape of bucked beam-coumn with arge defection. Therefore, a grid she consisting of eastica curves can reduce the interaction forces between the curved beams at joints. Shape parameters such as span, height at the support, and height at an interna joint are assigned for the piecewise inear curve. The inear segments are connected with springs at nodes. Externa moments are aso appied to generate various shapes of discrete eastica. We address the probem of designing discrete eastica by minimizing the strain energy function defined with equivaence to the strain energy of a continuous beam. It is shown that the Lagrange mutipiers correspond to the reaction forces at supports. In the numerica exampes, a gridshe surface with discrete eastica is generated using the proposed method. Large deformation anaysis is carried out for verification of the shape generated using discrete eastica. Keywords grid she, discrete eastica, strain energy, arge-deformation anaysis, active bending 1. Introduction Grid she structure (Ohsaki et a. [1], Matsuo et a. [], Adriaenssens et a. [3]) is one of the most efficient roof structures in view of construction period of time and cost. Shapes of gridshes are generated by straight beams mutuay connected by hinge joints. Then forced deformation and externa moments are given at the boundary to obtain a curved surface. A curved surface with uniform grid size is generated by using various existing methods such as compass method and partice-spring method (Tayeb et a. [4], Douthe et a. [5], Bouhaya et a. [6]). However, it is very difficut to generate sef-equiibrium shape of a curved surface from pane grid. Furthermore, the interaction forces at joints are too arge, if the shape is not appropriate. Hence, the method for reducing the interaction forces between beams have been presented (Ohsaki et a. [1], Matsuo et a. []). They defined the target shape of a curved beam as eastic, which is the shape of a bucked beam-coumn with arge defection. However, to obtain the target shape, we need to cacuate a differentia equation with respect to the arc-ength parameter. In this study, we present a method for designing shapes of grid she structures using discrete eastica mode (Bruckstein et a. [7], Charame et a. [8]), which has been studied in the fied of computer science. This mode enabes us to generate easiy the target shape of curved beams that are in sef-equiibrium Copyright 017 by Y. Sakai and M. Ohsaki Pubished by the Internationa Association for She and Spatia Structures (IASS) with permission.

2 Proceedings of the IASS Annua Symposium 017 state. In addition, we show that a curved surface in three-dimensiona space can be generated by connecting severa discrete easticas... Shape design of discrete eastica mode Eastica is defined as a shape of bucked beam-coumn under point oads at both ends. The defected equiibrium shape of eastica is obtained by soving a differentia equation with respect to the arc-ength parameter (Watson and Wang [9]). Since an expicit soution is not possibe, the shape of eastica can be obtained using a forward difference approach. The shape can aternativey be obtained by minimizing an energy function. The bending stiffness of continuous beam, curvatures, and arc-ength parameter are denoted by EI, κ, and s, respectivey. A parameter β is given to penaize axia deformation. Then, the penaized strain energy function is defined as (EIκ / + β) ds, which is to be minimized under appropriate boundary and oading conditions. However, it is very difficut to obtain a compex shape composed of mutipe curves easticay supported at the ends and connections of curves. Meanwhie, discrete eastica mode is defined as discretized piecewise inear curve with segments that have the same ength as shown in Figure 1. The mode has + 1 nodes denoted by P i (i = 0,, + 1). The defection ange of a segment connecting nodes i and i + 1 from x-axis, and the ange between the segments i 1 and i are denoted by Ψ i and θ i (= Ψ i 1 Ψ i ), respectivey. By soving an energy minimization probem, we obtain a sef-equiibrium shape of discrete eastica mode. Figure 1: Piecewise inear panar curve with equa-ength segments. The tota number of segments is +1 in accordance with Ref. [7]. We assign rotationa springs at a nodes and both ends. The springs represent bending stiffness of members and supporting coumns. The externa moments are given at both ends to generate various shapes. The objective function is the tota potentia energy consisting of the discretized form of penaized strain energy (EIκ / + β) ds and the externa work corresponding to the externa moments M 0 and M +1. The stiffness of each rotationa spring is derived from the equivaence of the strain energy between the discrete eastica and the continuous beam. Deformation of the continuous beam, which has stiffness EI and a constant curvature κ, is equivaent to the deformation of discrete eastica mode under the condition that the reation between θ and κ is given as κ = θ assuming uniform curvature. Hence, the strain energy S is defined as Thus, the stiffness of rotationa springs is EI. S = 1 EIκ = 1 EI (θ ) = EI θ.

3 Proceedings of the IASS Annua Symposium 017 We assign constraints on the span L and the height difference H between the eft support P 0 and the right support P +1. The design variabes are Ψ (= Ψ 0,, Ψ ) and. The number of nodes and segments are + and + 1, respectivey. The optimization probem, which is an energy minimization probem, is formuated as min. Ψ, subject to Π(Ψ, ) = [ ( EI (Ψ i Ψ i 1 i=1 ) + 1) ] M 0 Ψ 0 M +1 Ψ (1) cos Ψ i = L () sin Ψ i = H. (3) Let λ 1 and λ denote the Lagrange mutipiers for constraints () and (3), respectivey. The Lagrangian is formuated as L(Ψ,, λ 1, λ ) = Π + λ 1 ( cos Ψ i L) + λ ( sin Ψ i H). Taking derivatives with respect to + 4 variabes, we have the foowing stationary conditions. EI( Ψ i 1 + Ψ i Ψ i+1 ) λ 1 i sin Ψ i + λ i cos Ψ i = 0, (i = 1,, 1) 1 [EI(Ψ 1 Ψ 0 ) M 0 ] λ 0 1 sin Ψ 0 + λ 0 cos Ψ 0 = 0 (5) 1 [EI(Ψ Ψ 1 ) M +1 ] λ 1 sin Ψ + λ cos Ψ = 0 (6) EI ( Ψ i Ψ i 1 ) (4) + + λ 1 cos Ψ i + λ sin Ψ i = 0. (7) Athough the detais are omitted, the Lagrange mutipiers λ 1 and λ represent the support reaction forces in the horizonta and vertica directions, respectivey, and Eqs. (4), (5), and (6) are the equiibrium equations at nodes. The arger β becomes, the shorter the ength of the segments becomes. Thus, in the foowing exampes, we set β = 1 for simpicity. 3. Comparison between discrete and continuous eastica modes. In this section, we compare the shapes obtained by discrete and continuous eastica modes. We use sequentia quadratic programming avaiabe in the ibrary SOPT Ver.7 (Gi et a. [10]) for energy minimization of the discrete eastica, and we approximate the sensitivity coefficients by finite difference approach. Abaqus Ver (Dassaut Systèmes [11]) is used for arge deformation anaysis of continuous beams. Forced dispacements and externa moments are given at both ends of a straight beam on a pane. Length of the beam is equa to the tota ength of the discrete eastica obtained by optimization. The beam has pin support at the eft end, and a forced dispacement is given at the right end assuming the transationa dispacements are constrained by a rigid coumn. In addition, the externa moments, whose vaues are equa to ones that are appied to the boundaries of discrete eastica mode, are given at the both supports. 3

4 Proceedings of the IASS Annua Symposium 017 The materia of beam is eastic with Young s moduus 10.0 GPa and Poisson s ratio 0.3. The beam is composed of a pate with width 0.10 m and thickness 0.0 m. The oading (path) parameter t is increased from 0.0 to.0. Upward virtua oad equivaent to sefweight is appied to a members from t = 0.0 to 1.0. In the period 1.0 t.0, the virtua oad is ineary removed, whie the forced dispacements and externa moments are ineary increased at both ends of the beam. As shown in Fig., continuous eastica mode deforms. Figure : Continuous eastica mode Figures 3(a) and (b) show the resuts of arge deformation anaysis with the parameter vaues in Tabe 1. The beam is discretized into 0 segments. Shapes of continuous beam and discrete eastica mode are expressed, respectivey, in soid and dashed ines. The tota engths of modes 1 and are 1.81 m and m, respectivey. It is seen from Figs. 3(a) and (b) that the shapes of beam and discrete eastica mode are very cose. Tabe 1: Parameters of Modes 1 and Mode 1 Mode M 0 [m] M +1 [m] L [m] H [m] 0 4 EI [m ] (a) Mode 1 (b) Mode Figure 3: Shapes of Modes 1 and ; dashed ine: discrete eastica with 0 segments. 4

5 Proceedings of the IASS Annua Symposium Shape design by connecting mutipe curves of discrete eastica We generate shapes of mutipe curves of discrete eastica, which are sequentiay connected in the same pane. For simpicity, we connect two curves, which has + 3 nodes denoted by P i (= P 0,, P (+1) ). The first curve consists of nodes from P 0 to P +1, and the second curve from P +1 to P (+1). We assign rotationa spring at the interna boundary, node P +1, between the two curves, and define both ends as pin-joint. The rotationa spring has very sma stiffness α (= 0.001). We specify span ength L for both curves, and heights H 1 and H at the interna boundary and the right support. The optimization probem to find the shape of consisting of two discrete easticas is formuated as min. Ψ, +1 Π(Ψ, ) = [ ( EI (Ψ i Ψ i 1 ) + 1) i=1 ( EI α ) ( Ψ +1 Ψ ) ] M 0 Ψ 0 M (+1) Ψ +1 (8) subject to cos Ψ i = L (9) sin Ψ i = H 1 (10) +1 i=+1 +1 cos Ψ i = L (11) sin Ψ i = H 1 + H. (1) i=+1 5. Comparison between connected discrete and continuous easticas We sove the optimization probem with parameter vaues in Tabe in order to design shapes of connected modes, and compare the obtained shapes with continuous beams. The tota engths of modes 3 and 4 are m and.074 m, respectivey. The eft end of the connected discrete eastica is pin supported, the right end is roer supported, and the interna joint is fixed in the horizonta direction. For arge deformation anaysis, the stiffness of the interna joint of two beams is equa to that of the discrete eastica modes. The deformation of connected continuous eastica mode is shown in Fig. 4. Figures 5(a) and (b) show the resuts of arge deformation anaysis. It is seen from these figures that the shapes of discrete and continuous eastica modes are very cose. Figure 4: Connected continuous eastica mode 5

6 Proceedings of the IASS Annua Symposium 017 Tabe : Parameters of modes 3 and 4 Mode 3 Mode 4 M 0 [m] M (+1) [m] L [m] 0 0 H 1 [m] H [m] EI[m ] (a) Mode 3 (b) Mode 4 Figure 5: Shapes of Modes 3 and 4; dashed ine: discrete eastica with 40 segments. 6. Generating surfaces with discrete eastica mode In this section, we generate a target surface of gridshe using discrete eastica. Furthermore, we compare the target shape with the shape of grid she, which is generated by forced dispacements and externa moments to the continuous beams connected with atera beams. Figure 6(a) shows the target surface generated by connecting six easticas. The pan of surface is a 10 m 10 m square. The parameter vaues are isted in Tabe 3. Mode (a) represents the curves aong the exterior boundary, whose height difference is m. Modes (b) and (c) represent the curves aong the diagona ines, which intersect at the center. The both ends of (b) are on the ground and the both ends of (c) are connected to coumns with the height m. Figure 6(b) shows the target shape of gridshe, which is generated by assigning boundary conditions and externa moments to the primary beams in Fig. 6(a). The primary beams have the width 0.10 m and thickness 0.00 m. Meanwhie, we assign sender secondary beams, which have the width m and thickness m, to reduce the interaction forces at nodes. Figures 6(c)-(f) show that the shapes of beams and discrete easticas are very cose. As the resut of arge deformation anaysis, a member stresses are confirmed to be smaer than the yied stress 35 MPa. Tabe 4 shows reaction forces at supports. 6

7 Proceedings of the IASS Annua Symposium 017 Tabe 3: Parameters of mode (a), (b) and (c) Mode (a) Mode (b) Mode (c) M 0 [m] M +1 [m] L [m] H [m] 0 0 EI [m ] z y x (a) The target surface (b) Grid she composed of continuous beams (c) Mode (a) (d) Mode (b) (e) Mode (c) (f) Pan Figure 6: The comparison between the target shape and grid shes composed of continuous beams. These beams have 0 segments. (f) shows the pans of discrete and continuous easticas. Continuous eastica is three dimensiona shape. Thus, it is defected in x-y direction. Tabe 4: Reaction forces Mode (a) Mode (b) Mode (c) λ 1 [] λ []

8 7. Concusions Proceedings of the IASS Annua Symposium Curved beams of gridshe can be modeed as discrete eastica, which is composed by rigid bars and rotationa springs. The stiffness of each rotationa spring is derived from the equivaence of the strain energy between the discrete and continuous eastica modes.. The shape of discrete eastica can be found by soving an optimization probem. The objective function is defined as the tota potentia energy consisting of discretized form of penaized strain energy and the externa work corresponding to the externa moments. Soving the probem under the constraints with the span ength and the height of supports, we can generate various shapes of discrete eastica. Furthermore, the reaction forces needed for deformation can be found from the Lagrange mutipiers. 3. Target shape of a curved surface of gridshe can be obtained by connecting severa discrete easticas with thin secondary beams. It has been confirmed by arge deformation anaysis that the surface shape obtained by assigning forced dispacements and externa moments is cose to the target shape defined using the discrete easticas. Therefore, we can expect that the interaction forces at the connections are very sma. Acknowedgement This study is partiay supported by JSPS KAKEHI o. 16K References [1] M. Ohsaki, K. Seki and Y. Miyazu, Optimization of ocations of sot connections of gridshes modeed using eastica, Proc. IASS Symposium 016, Tokyo, Int. Assoc. She and Spatia Struct., Paper o. CS5A-101, 016. [] A. Matsuo, M. Ohsaki, Y. Miyazu, Optimization of easticay deformed gridshe with partiay reeased joints, Proceedings of the IASS WORKIG GROUPS Internationa Cooquium 015, 15T-03, 015. [3] S. Adriaenssens, P. Bock, D. Veenendaa and C. Wiiams (Eds.), She Structures in Architecture, Routedge, 014. [4] F. Tayeb, B. Lefebre, O. Bavere, JF. Caron and L. Du Peoux, Design and reaization of composite gridshe structures, J. IASS., Vo. 56, pp.49-59, 015. [5] C. Douthe, J.F. Caron and O. Bavere, Form-finding of a grid she in composite materias, J. IASS., Vo. 47, pp. 53-6, 006. [6] L. Bouhaya, O. Bavere and J.-F. Caron, Optimization of gridshe bar orientation using a simpified genetic approach, Structura Mutidiscipinary Optimization, Vo. 50, pp , 014. [7] A. M. Bruckstein, R. J. Hot, and A.. etravai, Discrete eastica, Apicabe Anaysis, Vo. 78, pp , 010. [8]. Chaame, A. Kocsis and C. M. Wang, Discrete and non-oca eastica, Internationa Journa of on-linear Mechanics Vo. 77, pp , 015. [9] L. T. Watson and C. Y. Wang, A homotopy method appied to eastica probems, Int. J. Soids Structures,. Vo. 17, pp. 9-37, [10] P. E. Gi, W. Murray and M. A. Saunders, SOPT: An SQP agorithm for arge-scae constrained optimization. SIAM J. Opt., Vo. 1, , 00. [11] Dassaut Systèmes, ABAQUS User s Manua Ver. 6.13,