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