a) Corresponding author. Email: batoz@utc.fr.

XXIV ICTAM, 21-26 August 2016, Montreal, Canada

ON THE MECHANICS OF BIO-INSPIRED STIFFENED SHELL STRUCTURES

Jean-Louis Batoz1a, Pascal Lardeur1, Eduard Antaluca2, Fabien Lamarque 2, Salim Bouabdallah3

1Sorbonne University, Université de Technologie de Compiègne, UMR 6253 Roberval, 60205 Compiègne, 2Sorbonne University, Université de Technologie de Compiègne, EA 7284, 60205 Compiègne, France

3Altair Engineering France, 5 Rue de la Renaissance, 92160 Antony, France

Summary The paper deals with the 3D finite element modelling of structures obtained by assemblages of thin shell parts. The whole structure represents a complex geometry inspired by vegetable nature and plants such as a maple leaf in the present study. This type of structure can become a bio-inspired architectural project at human scale. In the present study some challenging finite element analyses are discussed related to the complete integrated modelling: 3D scanning, geometrical model, appropriate finite element model for linear and nonlinear static and dynamic analysis of the stiffened 3D shell structures subjected to gravity and pressure loading. New finite element models are developed in the research project.

INTRODUCTION Nature has always been a source of inspiration for scientists and engineers in various domains, from ancient civilizations to nowadays. The geometry of plants, trees, branches and leaves has been the subject of interest and applications from the point of view of mathematicians and computer graphics specialists [1]. Wood and bones have been deeply studied from a material science point of view but also for their mechanical properties pertinent for the use in construction (for wood) or for biomechanics, or bioengineering applications (for bones) [2]. The structure of plants has been studied and has been an inspiration for design in architectural and mechanical engineering. One famous example is the Gaudi architecture in Barcelona as discussed in many documents such as [2]. In [3], chapter 8 and in [4], chapter 5, the authors (Andres, Ortega and Paloto) discussed the funicular polygon and the homeostatic model based on a biological principle that can be generated by the simultaneous action of thermal and mechanical loads on thermoplastic materials to design free form shapes. Natural structures such as wood and bones have also been the source of structural design and shape optimisation algorithms to reduce stress concentrations or to optimize the fibres orientations in composite man-made materials. Several books and papers have been published on the subject of optimization mechanics in nature such as [4] including a very interesting chapter on the structural efficiency of trees. The present paper is part of a general project related to the mechanics of plants in order to understand the mechanical behaviour under gravity and wind pressure loading in various situations of plant types and scale (trees, branches, leaves, flowers), in various space and life time conditions (behaviour of a deciduous leaf from spring to fall for example). The project is limited to the mechanical aspects, without coupling with the genetic bio-chemistry ones (which are essential for the explanation of the growth and attribution of material properties of any natural object). This means that the geometry and material properties relevant for the present mechanical analysis must be assumed for a given space and time situation. The project is also “biomimesis” oriented since we will analyse, design and possibly optimize engineering man-made structures inspired by nature. More precisely the present paper deals with the mechanics of a 3D structure with the shape of a maple leaf at a given stage of development (end of summer, say), Figure 1. The paper concentrates on the different models (geometry and physics) based on the Finite Element Method [5].

Figure 1. A typical maple leaf on the ground in the fall season.

THE FINITE ELEMENT MODELS We first collect a maple leaf of about 30 cm maximum length. The average thickness of the leaf is 0,18 mm. The mimetic structure to be analysed will have the same dimensions multiplied by a factor of 5, and the material properties will be those of an elastic isotropic steel with E=210 GPa, ν=0.3 and Yield stress σY = 400 MPa. The first step is to identify the shape of the contour, the main cantilevered beam connecting the leaf to the branch and all the ribs we can identify with a criteria of diameter more than 1,5 times the leaf thickness. That step can be performed by different means including laser scanning. The ribs of circular cross section are represented as stiffeners and are assumed to be non-eccentric with respect to the middle surface. There are around 20 secondary ribs per leaf with varying diameters. The main connecting beam has a diameter of 1,5 mm and a length of 5 cm. The leaf will be clamped at its root (cantilevered main beam in Figure 2).

Figure 2. A finite element model of a maple leaf The different finite element models considered in the present study are:

- the classical thin shell triangular and quadrilateral DKT18 and DKQ24 elements with 6 dof/node developed by the first author and colleagues [5],compatible with the classical 3D beam elements representing the ribs. This type of analysis can be performed with commercial finite element packages,

- new shell elements with one rotation per mid side [6], with a new representation for the beam stiffeners, - rotation free shell elements as discussed in different papers [7],[8].

The different analyses performed are: - linear static analyses considering gravity and pressure loadings, for different mesh densities and for different shell

formulations (as defined above), - dynamic analyses (frequencies and vibration modes) for a clamped leaf and a free leaf in space.

Results include displacements and stresses distributions, frequencies and natural modes (figure 2 as example).

CONCLUSIONS

The mechanics of a complex ribbed shell structure bio-inspired by a maple leaf has been studied in this paper using several finite element models. The objective is to understand the behaviour of the leaf structure under gravity and wind loads and to evaluate the stress distribution at any point. One difficult aspect of the study was the construction of relevant meshes for the different finite element models. The present study includes the development of new rotation free finite element shell models in the perspective of future works: dynamic and geometrical non-linear analysis. References [1] P. Prusinkiewicz, A. Lindenmayer: The algorithmic beauty of plants, Springer-Verlag, 1990. [2] J. Bassegoda I Nonell, P. Vivas, R. Pla: Gaudi, toutes les oeuvres, Triangle Postal, 2014 [3] M.W. Collins, M.A. Atherton, J.A. Bryant, Editors: Nature and design, WIT Press, 2005. [4] M.W. Collins, D.G. Hunt, J.A. Bryant, Editors: Optimisation mechanics in nature, WIT Press, 2004. [5] J.L. Batoz, G. Dhatt: Modélisation des structures par éléments finis. Vol. 2, Poutres et plaques, Vol. 3, Coques. Hermes Editeurs. Paris, 1992. [6] J.L. Batoz, C.L. Zheng, F. Hammadi: Formulation and evaluation of new triangular, quadrilateral, pentagonal and hexagonal Discrete Kirchhoff

plate/shell elements, International Journal for Numerical Methods in Engineering. Vol. 52, pp. 615-630, 2001. [7] E.M. Meghlat, M. Oudjene, H. Ait-Aider, J.L. Batoz: A new approach to model nailed and screwed timber joints using the finite element method,

Construction and Building Materials, Vol. 41, pp. 263-269, 2013. [8] Y.X. Zhou, K.Y. Sze: A geometric nonlinear rotation-free triangle and its application to drape simulation, International Journal for Numerical

Methods in Engineering, Vol. 89, pp. 509-536, 2012.

Shell elements

Secondary beam

Cantilevered main beam