Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Presenting Author, Another Author(s)

Investigation of the elastic behavior of reciprocal systems using

homogenization techniquesLorenzo GRECO

University of Pisa

Arthur LEBÉELaboratorie Navier, ENPC

Cyril DOUTHEIFSTTARR

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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What’s a reciprocal system?

1. Sketch of a roofing, by Villard de Honnecourt 13th century

2. Drawing of a bridge, by Leonardo da Vinci 16th century

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Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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What’s a nexorade nowadays?

3. Workshop on Nexorade construction, NIB Salerno Italy, D. Picker and G. Salvatore

4. Graphic 3D representation

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Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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What’s a nexorade nowadays?

5. Bibracte, sheltering over an archeological site, RFR and TESS

6. 3D representation of the Bibracte structural system

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Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Homogenization Technique outcomes

• Better understanding of structural behaviour

• Influence of each parameter on the whole structure

• Optimization

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Homogenization Technique

7. Just a few bricks

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Homogenization Technique

7. Just a few bricks

8. A brick wall

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Homogenization Technique

7. Just a few bricks

8. A brick wall

9. Homogenized plate describing the average brick wall behaviour

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Homogenization Technique

10. Applying strain on average on the unit cell

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Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Homogenization Technique for discrete media

11. The Nexorade studied and the Unit Cell

Unit cell

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Homogenization Technique outcomes

• Better understanding of structural behaviour

• Influence of each parameter on the whole structure

• Optimization

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

12

Nexorade configuration

φ = 0

a/l = 0

φ = φ*

0 ≤ a/l ≤ 0.5

φ = 45°

a/l = 0.5

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Degenerated Kirchhoff-Love plate model

• Only biaxial bending stiffness, linked to mean curvature

• Torsion leads to free motion → no stress

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Out of Plane Shear and Bending moment

12. Out of plane bending moment

13. Out of plane shear

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Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Out of Plane Shear and Bending moment

Shear as function of a/l Bending moment as function of a/l

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Out of Plane Stiffness

Stiffness as function of engagment

ratio a/l.

Shear stiffness is never

negligible

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Out of Plane FEM Analysis

14. Shear

15. Bending moment

16. Displacement

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Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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FEM validation

17. Displacement convergence, error as function of the number of unit cells

18. Displacement values comparison, Analytical vs FEM (Abaqus)

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Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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Conclusion and Outlooks

• Other nexor pattern?

• In-plane behavior?

• Homogenization as a shell?

• Good understanding of nexorade stress and displacement fields

• Correlation between geometric variables and structural behavior

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

Lorenzo GRECO, Arthur LEBÉE, Cyril DOUTHE

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THANKS FOR YOUR KIND ATTENTION