From Folds to Structures, a Review

International audienceStarting from simple notions of paper folding, a review of current challenges regarding folds and structures is presented. A special focus is dedicated to folded tessellations which are raising interest from the scientific community. Finally, the different mechanical modeling of folded structures are investigated. This reveals efficient applications of folding concepts in the design of structures

Raideur en cisaillement transverse du module à chevrons utilisé comme âme de panneaux sandwich = Transverse Shear Stiffness of a Chevron Folded Core Used in Sandwich Construction

National audienceEn se basant sur la méthode proposée par Kelsey et al. [1], les bornes supérieures et inférieures de la raideur en cisaillement transverse d'une âme pliée en module à chevrons sont déterminées analytiquement et comparées au calcul par éléments finis. On observe que ces bornes sont généralement assez larges et qu'il existe des configurations géométriques pour lesquelles le module à chevrons peut être jusqu'à 40% plus raide que les nids d'abeille

Homogenization of a space frame as a thick plate: Application of the Bending-Gradient theory to a beam lattice

International audienceThe Bending-Gradient theory for thick plates is the extension to heterogeneous plates of Reissner-Mindlin theory originally designed for homogeneous plates. In this paper the Bending-Gradient theory is extended to in-plane periodic structures made of connected beams (space frames) which can be considered macroscopically as a plate. Its application to a square beam lattice reveals that classical Reissner-Mindlin theory cannot properly model such microstructures. Comparisons with exact solutions show that only the Bending-Gradient theory captures second order effects in both deflection and local stress fields

A bending-gradient model for thick plates, I : theory

International audienceThis is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff-Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner-Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner-Mindlin model. In part two (Lebee and Sab, 2010a), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner-Mindlin theory and to full 3D Pagano's exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity

Quelques exemples d'application aux composites stratitfiés de la théorie Bending-Gradient pour les plaques

International audienceCe travail présente l'application aux composites fibrés d'une nouvelle théorie de plaque. Ce modèle destiné aux plaques épaisses et anisotropes utilise les six inconnues statiques de la theorie de Kirchhoff-Love auxquelles sont ajoutées six nouvelles inconnues représentant le gradient dumoment de flexion. Nommé théorie Bending-Gradient, ce nouveaumodèle peut être considéré comme une extension aux plaques hétérogènes dans l'épaisseurs du modèle de Reissner-Mindlin ; ce dernier étant un cas particulier lorsque la plaque est homogène. La théorie Bending-Gradient est appliquée aux plaques stratifiées et comparée à la solution exacte de Pagano [1] ainsi qu'à d'autres approches. Elle donne de bonnes prédictions pour la flèche, pour la distribution des contraintes de cisaillement transverse ainsi que pour les déplacements plans dans de nombreuses configurations matérielles

A Bending-Gradient theory for thick laminated plates homogenization

This work presents a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Love-Kirchhoff theory, to which six components are added representing the gradient of the bending moment. The Bending-Gradient theory is an extension to arbitrary multilayered plates of the Reissner-Mindlin theory which appears as a special case when the plate is homogeneous. The new theory is applied to multilayered plates and its predictions are compared to full 3D Pagano's exact solutions and other approaches. It gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity

Homogenization of thick periodic plates: Application of the Bending-Gradient plate theory to a folded core sandwich panel

International audienceIn a previous paper from the authors, the bounds from Kelsey et al. (1958) were applied to a sandwich panel including a folded core in order to estimate its shear forces stiffness (Lebée and Sab, 2010b). The main outcome was the large discrepancy of the bounds. Recently, Lebée and Sab (2011a) suggested a new plate theory for thick plates - the Bending-Gradient plate theory - which is the extension to heterogeneous plates of the well-known Reissner-Mindlin theory. In the present work, we provide the Bending-Gradient homogenization scheme and apply it to a sandwich panel including the chevron pattern. It turns out that the shear forces stiffness of the sandwich panel is strongly influenced by a skin distortion phenomenon which cannot be neglected in conventional design. Detailed analysis of this effect is provided

Stiffness and stress efficiency of periodic reinforcing pads for FPSO's hull

National audienceThe hull of FPSO units (Floating Production Storage and Offloading units) suffers from accelerated corrosion. It becomes now mandatory to restore the original thickness of the hull by means of reinforcing pads. For large areas, the pads follow a periodic pattern and are not connected together. Hence, in this work, we apply plate homogenization techniques coupled with finite elements simulations for assessing the behavior of the pads. It is found that, for most geometric configurations studied here, the pads do not provide sufficient strength reinforcement and requires their mechanical connection

The Bending-Gradient theory for laminates and in-plane periodic plates

DoctoralThe classical theory of plates, known also as Kirchhoff-Love plate theory is based on the assumption that the normal to the mid-plane of the plate remains normal after transformation. This theory is also the first order of the asymptotic expansion with respect to the thickness. Thus, it presents a good theoretical justification and was soundly extended to the case of periodic plates. It enables to have a first order estimate of the macroscopic deflection as well as local stress fields. In most applications the first order deflection is accurate enough. However, this theory does not capture the local effect of shear forces on the microstructure because shear forces are one higher-order derivative of the bending moment in equilibrium equations.Because shear forces are part of the macroscopic equilibrium of the plate, their effect is also of great interest for engineers when designing structures. However, modeling properly the action of shear forces is still a controversial issue.Revisiting the approach from Reissner directly with laminated plates, it appears that the transverse shear static variables which come out when the plate is heterogeneous are not shear forces but the full gradient of the bending moment. Using conventional variational tools, it is possible to derive a new plate theory, called Bending-Gradient theory. This new plate theory is considered as an extension of Reissner's theory to heterogeneous plates which preserves most of its simplicity. Originally designed for laminated plates, it is also extended to in-plane periodic plates using averaging considerations. The lecture will be illustrated with application to strongly heterogeneous plates such as cellular sandwich panels or periodic space frames

Investigation of the elastic behavior of reciprocal systems using homogenization techniques

International audienceIn this paper, the authors endeavour to develop design formulas for reciprocal systems using homogenization techniques. The theoretical background for homogenizing periodic beams systems as Kirchhoff-Love plates is first recalled. Then it is applied to a square reciprocal system. It is found that only biaxial bending (i.e. positive Gaussian curvature) generates stress inside the beams so that the equivalent plate model is a degenerated Kirchhoff-Love which is fully detailed. Then, some optimal configurations are investigated in terms of bending stiffness and strength. Finally, in order to validate the approach, full finite element simulations of simply supported reciprocal systems on square boundaries are compared with the homogenized solution previously derived. The convergence of the model and its accuracy for reasonable scale ratio is confirmed. Nexorade, Homogenization, Periodic plates, Space fram