Discrete elasticafor shape design of gridshells
Yusuke Sakai (Kyoto University)Makoto Ohsaki (Kyoto University)
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Y. Sakai and M. Ohsaki, Discrete elastica for shape design of gridshells,Eng. Struct., Vol. 169, pp. 55-67, 2018.
Problem of the generating process
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Desired shape
Initial shape
Generated shape
Large deformation analysis
Unknown:Forced disp., Load, External moment etc…
Trial & Error
Interaction forces Too large
Inappropriate shape
Engineer
Designer
[1] Ohsaki M, Seki K, Miyazu Y. Optimization of locations of slot connections of gridshells modeled using elastica. Proc IASS Symposium 2016, Tokyo. Int Assoc Shell and Spatial Struct 2016;Paper No. CS5A-1012.
Solution & BackgroundTarget shape of a curved beam
Elastica: Shape of a buckled beam-column Reduce the interaction forces [1]
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Continuous elastica
min. 𝑈𝑈 = �12𝐸𝐸𝐸𝐸𝜅𝜅2 + 𝛽𝛽 𝑑𝑑𝑑𝑑 ,
𝜅𝜅 𝑑𝑑 =𝜕𝜕Ψ(𝑑𝑑)𝜕𝜕𝑑𝑑
𝑑𝑑: arc-length parameter, 𝐸𝐸𝐸𝐸: bending stiffness, 𝜅𝜅 𝑑𝑑 : curvature, 𝛽𝛽: penalty parameter for the total length
1. Initial shape (straight) Equilibrium shape2. Span & Height of a support Only a few parameters3. Uniaxial bending Planar model
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[2] A. M. Bruckstein, R. J. Holt and A. N. Netravali, Discrete elastica, Appl. Anal., Vol. 78, pp. 453-485, 2001.
[3] Y. SAKAI and M. OHSAKI, Discrete elastica for shape design of gridshell, Eng. Struct., Vol. 169, pp. 55-67, 2018.
Ψ𝑖𝑖
𝑙𝑙: Length of each segment,𝚿𝚿 = Ψ0, … ,Ψ𝑁𝑁 𝑇𝑇:Deflection angles between
each segment and 𝑥𝑥 axis
𝑙𝑙
𝑥𝑥
𝑥𝑥
Discrete elastica
Rotationalspring
𝑃𝑃𝑖𝑖 𝑖𝑖 = 0, … ,𝑁𝑁 + 1 : Nodes𝑙𝑙: Length of each segment𝚿𝚿 = Ψ0, … ,Ψ𝑁𝑁 : Deflection angle of a segment from 𝑥𝑥-axis
Equivalence of strain energyStiffness of rotational spring
Ψ𝑖𝑖
Ψ𝑖𝑖−1𝜃𝜃𝑖𝑖 = Ψ𝑖𝑖−1 − Ψ𝑖𝑖
Curvature: 𝜅𝜅 = 𝜃𝜃𝑖𝑖𝑙𝑙
Strain energy: 𝑆𝑆 = 12𝐸𝐸𝐸𝐸𝜅𝜅2𝑙𝑙 = 1
2𝐸𝐸𝐸𝐸 𝜃𝜃𝑖𝑖
𝑙𝑙
2𝑙𝑙 = 𝐸𝐸𝐸𝐸
2𝑙𝑙𝜃𝜃𝑖𝑖2
𝐸𝐸𝐸𝐸/𝑙𝑙
𝑥𝑥
𝑙𝑙
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External workDiscretized form of 𝑈𝑈 = ∫ 12 𝐸𝐸𝐸𝐸𝜅𝜅
2 + 𝛽𝛽 𝑑𝑑𝑑𝑑
𝛽𝛽 = 1000
Optimization problem (Minimize total potential energy)
( ) ( )21, , 10 0 1
0
0
min. ,2
subject to cos ,
sin
N
i il i
N NN
iiN
ii
EIl ll
M M
l L
l H
β−=
+
=
=
Π = Ψ −Ψ + − Ψ − Ψ
Ψ =
Ψ =
∑
∑
∑
Ψ ΦΨ
: Span length
: Height of a support
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Stationary conditions = Equilibrium of segments Lagrange multipliers = Reaction forces
Lagrangian (𝜆𝜆1 and 𝜆𝜆2 are Lagrange multipliers)
ℒ 𝚿𝚿, 𝑙𝑙, 𝜆𝜆1, 𝜆𝜆2 = Π + 𝜆𝜆1 �𝑖𝑖=1
𝑁𝑁
𝑙𝑙 cosΨ𝑖𝑖 − 𝐿𝐿 + 𝜆𝜆2 �𝑖𝑖=1
𝑁𝑁
𝑙𝑙 sinΨ𝑖𝑖 − 𝐻𝐻
Stationary conditions: Derivatives with respect to 𝚿𝚿 = Ψ0, … ,Ψ𝑁𝑁 𝑇𝑇
𝐸𝐸𝐸𝐸𝑙𝑙−Ψ𝑖𝑖−1 − Ψ𝑖𝑖
𝑙𝑙+Ψ𝑖𝑖 − Ψ𝑖𝑖+1
𝑙𝑙− 𝜆𝜆1 sinΨ𝑖𝑖 + 𝜆𝜆2 cosΨ𝑖𝑖 = 0
𝑖𝑖 = 1, … ,𝑁𝑁 − 11𝑙𝑙𝐸𝐸𝐸𝐸 Ψ1 − Ψ0
𝑙𝑙− 𝑀𝑀0 − 𝜆𝜆1 sinΨ0 + 𝜆𝜆2 cosΨ0 = 0
1𝑙𝑙𝐸𝐸𝐸𝐸 Ψ𝑁𝑁 − Ψ𝑁𝑁−1
𝑙𝑙− 𝑀𝑀𝑁𝑁+1 − 𝜆𝜆1 sinΨ𝑁𝑁 + 𝜆𝜆2 cosΨ𝑁𝑁 = 0
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𝐸𝐸𝐸𝐸𝑙𝑙
−Ψ𝑖𝑖−1 − Ψ𝑖𝑖
𝑙𝑙+Ψ𝑖𝑖 − Ψ𝑖𝑖+1
𝑙𝑙− 𝜆𝜆1 sinΨ𝑖𝑖 + 𝜆𝜆2 cosΨ𝑖𝑖 = 0
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𝜆𝜆1 and 𝜆𝜆2: Support reaction forces
Curve 1 Curve 2
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Comparisons of shapes and reaction forces
Discrete elastica
Continuous beam
Curve 1 𝜆𝜆1[kN] (x-dir.) 0.4342 0.4449𝜆𝜆2[kN] (z-dir.) 0.0000 0.0002
Curve 2 𝜆𝜆1[kN] (x-dir.) 0.8940 0.9140𝜆𝜆2[kN] (z-dir.) 0.8212 0.8171
Reaction forces
≃≃≃≃
• Plan:Square(10 m×10 m)
• Curve A Boundary
(Height difference = 2m)• Curve B, C In the diagonal planeTarget surface composed of primary beams
(Discrete elastica)
Gridshell (Large deformation analysis)
Material: Glass fiber reinforced polymer(GFRP)
Yield stress 200 MPaDiameter Thickness
Primary 0.08 m 0.008 m
Secondary 0.0475 m 0.003m
Sectional area: Hollow cyrinder
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2 m
𝑡𝑡: Loading parameter 0.0 ≤ 𝑡𝑡 ≤ 2.00.0 ≤ 𝑡𝑡 ≤ 1.0: Upward virtual load (equivalent to self-weight)1.0 ≤ 𝑡𝑡 ≤ 2.0: Forced displacement & external moments
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Forced disp.
External moments
Motion of large-deformation analysis
Virtual load
Curve A Curve B
• All member stress < Yielding stress 200 MPa• Discrete & continuous beams close
Target shape(Discrete elastica)
Continuous beam(Large deformationanalysis)
Curve C Plan
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Comparison
Conclusion1.Discrete elastica Equilibrium curved beams
derived from optimization Target shape of primary beams
of gridshell2.Span, height of a support, and external moments
Shape parameters3.Verification (Primary beams)
Target shape vs Large-deformation analysisVery close
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Other studies of gridshell Combinatorial optimization for arranging slot+hinge joints Spatial discrete elastica (extended from Discrete elastica) Development of 3D elastic beam model for large-deformation
analysis
Discrete elastica�for shape design of gridshellsスライド番号 2スライド番号 3スライド番号 4スライド番号 5スライド番号 6スライド番号 7スライド番号 8スライド番号 9スライド番号 10スライド番号 11スライド番号 12スライド番号 13