Discrete elastica model for shape design of gridshellsfor shape design of gridshells Yusuke Sakai...
Transcript of Discrete elastica model for shape design of gridshellsfor shape design of gridshells Yusuke Sakai...
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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.
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Problem of the generating process
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Desired shape
Initial shape
Generated shape
Large deformation analysis
Unknown:Forced disp., Load, External moment etcβ¦
Trial & Error
Interaction forces Too large
Inappropriate shape
Engineer
Designer
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[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. ππ = οΏ½12πΈπΈπΈπΈπ π 2 + π½π½ ππππ ,
π π ππ =ππΞ¨(ππ)ππππ
ππ: arc-length parameter, πΈπΈπΈπΈ: bending stiffness, π π ππ : curvature, π½π½: penalty parameter for the total length
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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.
Ξ¨ππ
ππ: Length of each segment,πΏπΏ = Ξ¨0, β¦ ,Ξ¨ππ ππ:Deflection angles between
each segment and π₯π₯ axis
ππ
π₯π₯
π₯π₯
Discrete elastica
Rotationalspring
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ππππ ππ = 0, β¦ ,ππ + 1 : Nodesππ: Length of each segmentπΏπΏ = Ξ¨0, β¦ ,Ξ¨ππ : Deflection angle of a segment from π₯π₯-axis
Equivalence of strain energyStiffness of rotational spring
Ξ¨ππ
Ξ¨ππβ1ππππ = Ξ¨ππβ1 β Ξ¨ππ
Curvature: π π = ππππππ
Strain energy: ππ = 12πΈπΈπΈπΈπ π 2ππ = 1
2πΈπΈπΈπΈ ππππ
ππ
2ππ = πΈπΈπΈπΈ
2ππππππ2
πΈπΈπΈπΈ/ππ
π₯π₯
ππ
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External workDiscretized form of ππ = β« 12 πΈπΈπΈπΈπ π
2 + π½π½ ππππ
π½π½ = 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
Ξ²β=
+
=
=
Ξ = Ξ¨ βΞ¨ + β Ξ¨ β Ξ¨
Ξ¨ =
Ξ¨ =
β
β
β
Ψ ΦΨ
: Span length
: Height of a support
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Stationary conditions = Equilibrium of segments Lagrange multipliers = Reaction forces
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Lagrangian (ππ1 and ππ2 are Lagrange multipliers)
β πΏπΏ, ππ, ππ1, ππ2 = Ξ + ππ1 οΏ½ππ=1
ππ
ππ cosΞ¨ππ β πΏπΏ + ππ2 οΏ½ππ=1
ππ
ππ sinΞ¨ππ β π»π»
Stationary conditions: Derivatives with respect to πΏπΏ = Ξ¨0, β¦ ,Ξ¨ππ ππ
πΈπΈπΈπΈππβΞ¨ππβ1 β Ξ¨ππ
ππ+Ξ¨ππ β Ξ¨ππ+1
ππβ ππ1 sinΞ¨ππ + ππ2 cosΞ¨ππ = 0
ππ = 1, β¦ ,ππ β 11πππΈπΈπΈπΈ Ξ¨1 β Ξ¨0
ππβ ππ0 β ππ1 sinΞ¨0 + ππ2 cosΞ¨0 = 0
1πππΈπΈπΈπΈ Ξ¨ππ β Ξ¨ππβ1
ππβ ππππ+1 β ππ1 sinΞ¨ππ + ππ2 cosΞ¨ππ = 0
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πΈπΈπΈπΈππ
βΞ¨ππβ1 β Ξ¨ππ
ππ+Ξ¨ππ β Ξ¨ππ+1
ππβ ππ1 sinΞ¨ππ + ππ2 cosΞ¨ππ = 0
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ππ1 and ππ2: Support reaction forces
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Curve 1 Curve 2
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Comparisons of shapes and reaction forces
Discrete elastica
Continuous beam
Curve 1 ππ1[kN] (x-dir.) 0.4342 0.4449ππ2[kN] (z-dir.) 0.0000 0.0002
Curve 2 ππ1[kN] (x-dir.) 0.8940 0.9140ππ2[kN] (z-dir.) 0.8212 0.8171
Reaction forces
ββββ
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β’ PlanοΌSquareοΌ10 mΓ10 mοΌ
β’ Curve A Boundary
(Height difference = 2m)β’ Curve B, C In the diagonal planeTarget surface composed of primary beams
(Discrete elastica)
Gridshell (Large deformation analysis)
Material: Glass fiber reinforced polymerοΌGFRPοΌ
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
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π‘π‘: Loading parameter 0.0 β€ π‘π‘ β€ 2.00.0 β€ π‘π‘ β€ 1.0: Upward virtual load (equivalent to self-weight)1.0 β€ π‘π‘ β€ 2.0: Forced displacement & external moments
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Forced disp.
External moments
Motion of large-deformation analysis
Virtual load
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Curve A Curve B
β’ All member stress < Yielding stress 200 MPaβ’ Discrete & continuous beams close
Target shape(Discrete elastica)
Continuous beam(Large deformationanalysis)
Curve C Plan
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Comparison
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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
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