On geometrically exact post-buckling of composite columns with interlayer slip - The partially composite elastica

International audienceThis paper is focused on the geometrically exact elastic stability analysis of two interacting kinematically constrained, flexible columns. Possible applications are to partially composite or sandwich columns. A partially composite column composed of two inextensible elastically connected sub-columns is considered. Each sub-column is modeled by the Euler-Bernoulli beam theory and connected to each other via a linear constitutive law for the interlayer slip. The paper discusses the validity of parallel and translational kinematics beam assumptions with respect to the interlayer constraint. Buckling and post-buckling behavior of this structural system are studied for cantilever columns (clamped-free boundary conditions). A variational formulation is presented in order to derive relevant boundary conditions in a geometrically exact framework The exact post-buckling behavior of this partially composite beam-column is investigated analytically and numerically. The Euler elastica problem is obtained in the case of non-composite action. The "partially composite elastica" is then treated analytically and numerically, for various values of the interaction connection parameter. An asymptotic expansion is performed to evaluate the symmetrical pitchfork bifurcation, and comparisons are made with some exact numerical results based on the numerical treatment of the non-linear boundary value problem. A boundary layer phenomenon, similar to that also observed for the linearized bending analysis of partially composite beams, is observed for large values of the connection parameter. This boundary layer phenomenon is investigated with a straightforward asymptotic expansion, that also is valid for large rotations. Finally, the paper analyses the effect of some additional imperfection eccentricities in the loading mode, that lead to some pre-bending phenomena

Quelques arguments énergétiques en faveur de certains modèles d'endommagements non-locaux

Une des sources de non-localité macroscopique est liée à l'hétérogénéité de la fissuration et aux interactions entre fissures. L'idée du modèle développé est de considérer un volume représentatif en présence de gradient de fissuration, ou gradient d'endommagement. Le milieu effectif endommagé se modélise ainsi comme un milieu non-homogène avec gradient de fissuration (ou gradient de propriétés). L'évolution des variables macroscopiques de propagation se déduit par formulation duale variationnelle. Un modèle similaire au modèle pionnier de Pijaudier-Cabot et Ba ant est ainsi obtenu

Boundary layer effect in composite beams with interlayer slip

International audienceAn apparent analytical peculiarity or paradox in the bending behavior of elastic-composite beams with interlayer slip, sandwich beams, or other similar problems subjected to boundary moments exists. For a fully composite beam subjected to such end moments, the partial composite model will render a nonvanishing uniform value for the normal force in the individual subelement. This is from a formal mathematical point of view in apparent contradiction with the boundary conditions, in which the normal force in the individual subelement usually is assumed to vanish at the extremity of the beam. This mathematical paradox can be explained with the concept of boundary layer. The bending of the partially composite beam expressed in dimensionless form depends only on one structural parameter related to the stiffness of the connection between the two subelements. An asymptotic method is used to characterize the normal force and the bending moment in the individual subelement to this dimensionless connection parameter. The outer expansion that is valid away from the boundary and the inner expansion valid within the layer adjacent to the boundary (beam extremity) are analytically given. The inner and outer expansions are matched by using Prandtl’s matching condition over a region located at the edge of the boundary layer. The thickness of the boundary layer is the inverse of the dimensionless connection parameter. Finite-element results confirm the analytical results and the sensitivity of the bending solution to the mesh density, especially in the edge zone with stress gradient. Finally, composite beams with interlayer slip can be treated in the same manner as nonlocal elastic beams. The fundamental differential equation appearing in the constitutive law associated with the partial-composite action in a nonlocal elasticity framework is discussed. Such an integral formulation of the constitutive equation encompassing the behavior of the whole of the beam allows the investigation of the mechanical problem with the boundary-element method

On the occurrence of flutter in the lateral torsional instabilities of circular arches under follower loads

International audienceThe lateral-torsional stability of circular arches subjected to radial and follower distributed loading is treated herein. Three loading cases are studied, including the radial load with constant direction, the radial load directed towards the arch centre, and the follower radial load (hydrostatic load), as treated by Nikolai in 1918. For the three cases, the buckling loads are first obtained from a static analysis. As the case of the follower radial load (hydrostatic load) is a non-conservative problem, the dynamic approach is also used to calculate the instability load. The governing equations for out-of-plane vibrations of circular arches under radial loading are then derived, both with and without Wagner's effect. Flutter instabilities may appear for sufficiently large values of opening angle, but flutter cannot occur before divergence for the parameters of interest (civil engineering applications). Therefore, it is concluded that the static approach necessarily leads to the same result as the dynamic approach, even in the non-conservative case

Discrete and nonlocal models of Engesser and Haringx elastica

International audienceIn this paper, a generalized discrete elastica including both bending and shear elastic interactions is developed and its possible link with nonlocal beam continua is revealed. This lattice system can be viewed as the generalization of the Hencky bar-chain model, which can be retrieved in the case of infinite shear stiffness. The shear contribution in the discrete elastica is introduced by following the approach of Engesser (normal and shear forces are aligned with and perpendicular to the link axis, respectively) and that of Haringx (shear force is parallel to end section of links), both supported by physical arguments. The nonlinear analysis of the shearable-bendable discrete elastica under axial load is accomplished. Buckling and post-buckling of the lattice systems are analyzed in a geometrically exact framework. The buckling loads of both the discrete Engesser and Haringx elastica are analytically calculated, and the post-buckling behavior is numerically studied for large displacement. Nonlocal Timoshenko-type beam models, including both bending and shear stiffness, are then built from the continualization of the discrete systems. Analytical solutions for the fundamental buckling loads of the nonlocal Engesser and Haringx elastica models are given, and their first post-budding paths are numerically computed and compared to those of the discrete Engesser and Haringx elastica. It is shown that the nonlocal Timoshenko-type beam models efficiently capture the scale effects associated with the shearable-bendable discrete elastica

Comportement du béton de chanvre en compression simple et cisaillement

International audience Dans la littérature, la plupart des mélanges chaux/chanvre étudiés montrent un comportement fragile et une très faible résistance mécanique. Les formulations sont généralement riches en liant et légèrement compactées. Jusqu'à présent, ce matériau n'est pas considéré comme un matériau porteur et est principalement utilisé comme isolant de remplissage, combiné avec des composants de structure en bois, en béton ou en maçonnerie. Le présent travail est une étude expérimentale du comportement du béton de chanvre à la compression et au cisaillement, afin d'évaluer la capacité porteuse et au contreventement de ce matériau bio-sourcé, tout en assurant de bonnes qualités d'isolation thermique. Deux séries de tests sont effectuées. Le premier est un test de compression uniaxiale dans chaque direction pour caractériser l'anisotropie mécanique du matériau. Le second permet de caractériser le comportement au cisaillement de différentes configurations de la composition. Les expériences réalisées montrent une ductilité élevée de ce matériau en cisaillement, ce qui confirme les résultats de la littérature sur les structures de paroi. Ces résultats sont très prometteurs, notamment pour des applications parasismiques.</p

On the stability of nonconservative elastic systems under mixed perturbations

International audienceThis paper shows that the loading mode strongly influences the stability of discrete non-conservative elastic systems. The stability of the constrained system is compared to the stability of the unconstrained system, through the incorporation of Lagrange multipliers. Initially, the approach is illustrated for a two-degrees-of-freedom generalized Ziegler's column. Then, it is applied to a two-degrees-of-freedom model representing a soil constrained with isochoric loading. The isochoric instability load is not necessarily greater than the instability load of the free problem. Excluding flutter instabilities, it is shown that the second-order work criterion is not only a lower bound of the stability boundary of the free system, but also the boundary of the stability domain, in presence of mixed perturbations based on proportional kinematic conditions.Cet article étudie l'influence du mode de chargement sur la stabilité de systèmes élastiques discrets non conservatifs. La stabilité du système contraint est comparée à celle du système libre, par l'introduction de multiplicateurs de Lagrange. L'approche est illustrée avec le pendule généralisé de Ziegler. Elle est ensuite appliquée à un modèle à deux degrés de liberté représentant un sol contraint par un chargement isochore. On montre que le chargement isochore affecte sensiblement la frontière de stabilité pour le problème conservatif et pour le problème non conservatif. En dehors des instabilités par flottement, le critère de travail du second-ordre constitue une borne inférieure de la frontière de stabilité du système libre ainsi que la frontière du domaine de stabilité du système sous chargements mixtes proportionnels en déplacement

Stability of non-conservative elastic structures under additional kinematics constraints

International audienceIn this paper, the specific effect of additional constraints on the stability of undamped non-conservative elastic systems is studied. The stability of constrained elastic system is compared to the stability of the unconstrained system, through the incorporation of Lagrange multipliers. It is theoretically shown that the second-order work criterion, dealing with the symmetric part of the stiffness matrix corresponds to an optimization criterion with respect to the kinematics constraints. More specifically, the vanishing of the second-order work criterion corresponds to the critical kinematics constraint, which can be interpreted as an instability direction when the material stability analysis is considered (typically in the field of soil mechanics). The approach is illustrated for a two-degrees-of-freedom generalised Ziegler's column subjected to different constraints. We show that a particular kinematics constraint can stabilize or destabilize a non-conservative system. However, for all kinematics constraints, there necessarily exists a constraint which destabilizes the non-conservative system. The constraint associated to the lowest critical load is associated with the second-order criterion. Excluding flutter instabilities, the second-order work criterion is not only a lower bound of the stability boundary of the free system, but also the boundary of the stability domain, for all mixed perturbations based on proportional kinematics conditions