Lower bound static approach for the yield design of thick plates

International audienceThe present work addresses the lower bound limit analysis (or yield design) of thick plates under shear-bending interaction. Equilibrium finite elements are used to discretize the bending moment and the shear force fields. Different strength criteria, formulated in the five-dimensional space of bending moment and shear force, are considered, one of them taking into account the interaction between bending and shear resistances. The criteria are chosen to be sufficiently simple so that the resulting optimization problem can be formulated as a second-order cone programming problem, which is solved by the dedicated solver MOSEK. The efficiency of the proposed finite element is illustrated by means of numerical examples on different plate geometries, for which the thin plate solutions as well as the pure shear solutions are accurately obtained as two different limit cases of the plate slenderness ratio. In particular, the proposed element exhibits a good behavior in the thin plate limit

Yield design computations on homogenized periodic plates

International audienceHomogenization approaches have frequently been proposed to evaluate the mechanical properties of highly heterogeneous structures. The determination of such homogenized or macroscopic properties is performed by solving a specific auxiliary problem formulated on an elementary representative volume or a unit cell in the case of periodically heterogeneous materials. Once such properties have been determined, the initial heterogeneous problem is substituted by an equivalent homogeneous one. If global elastic computations using a quite limited number of homogenized moduli are straightforward, this is not the case as regards strength properties. Homogenized yield design or limit analysis computations require, indeed, a semi-analytical description of the homogenized yield surface, simple enough to be efficiently used in an optimization solver. The following work presents a combined homogenization/approximation approach to perform global computations on periodically heterogeneous thin plates in bending. Homogenization theory in limit analysis or yield design is applied to a thin plate model and macroscopic yield surfaces are derived by solving the auxiliary problem, by means of thin plate finite elements and second-order cone programming. An original approximation procedure is used to express the so-obtained yield surface as a convex hull of ellipsoids. This simple description enables to formulate yield design problems on a homogenized structure very easily. In particular, a specific attention will be devoted to the formulation of the corresponding static and kinematic approaches as second-order cone programs as well. An important feature of the method is that upper bound and lower bound status are still preserved on the homogenized problems, so that arising approximation errors can be safely estimated and controlled. Homogenized limit loads can then be bracketed with a relatively good accuracy. Numerical illustrative applications will be presented on various types of structures like reinforced and perforated plates

A computational homogenization approach for the yield design of periodic thin plates. Part I: Construction of the macroscopic strength criterion

International audienceThe purpose of this paper is to propose numerical methods to determine the macroscopic bending strength criterion of periodically heterogeneous thin plates in the framework of yield design (or limit analysis) theory. The macroscopic strength criterion of the heterogeneous plate is obtained by solving an auxiliary yield design problem formulated on the unit cell, that is the elementary domain reproducing the plate strength properties by periodicity. In the present work, it is assumed that the plate thickness is small compared to the unit cell characteristic length, so that the unit cell can still be considered as a thin plate itself. Yield design static and kinematic approaches for solving the auxiliary problem are, therefore, formulated with a Love-Kirchhoff plate model. Finite elements consistent with this model are proposed to solve both approaches and it is shown that the corresponding optimization problems belong to the class of second-order cone programming (SOCP), for which very efficient solvers are available. Macroscopic strength criteria are computed for different type of heterogeneous plates (reinforced, perforated plates,...) by comparing the results of the static and the kinematic approaches. Information on the unit cell failure modes can also be obtained by representing the optimal failure mechanisms. In a companion paper, the so-obtained homogenized strength criteria will be used to compute ultimate loads of global plate structures

A greedy algorithm for yield surface approximation

International audienceThis Note presents an approximation method for convex yield surfaces in the framework of yield design theory. The proposed algorithm constructs an approximation using a convex hull of ellipsoids such that the approximate criterion can be formulated in terms of second-order conic constraints. The algorithm can treat bounded as well as unbounded yield surfaces. Its efficiency is illustrated on two yield surfaces obtained using up-scaling procedures

A computational homogenization approach for the yield design of periodic thin plates. Part II : Upper bound yield design calculation of the homogenized structure

International audienceIn the first part of this work (Bleyer and de Buhan, 2014), the determination of the macroscopic strength criterion of periodic thin plates has been addressed by means of the yield design homogenization theory and its associated numerical procedures. The present paper aims at using such numerically computed homogenized strength criteria in order to evaluate limit load estimates of global plate structures. The yield line method being a common kinematic approach for the yield design of plates, which enables to obtain upper bound estimates quite efficiently, it is first shown that its extension to the case of complex strength criteria as those calculated from the homogenization method, necessitates the computation of a function depending on one single parameter. A simple analytical example on a reinforced rectangular plate illustrates the simplicity of the method. The case of numerical yield line method being also rapidly mentioned, a more refined finite element-based upper bound approach is also proposed, taking dissipation through curvature as well as angular jumps into account. In this case, an approximation procedure is proposed to treat the curvature term, based upon an algorithm approximating the original macroscopic strength criterion by a convex hull of ellipsoids. Numerical examples are presented to assess the efficiency of the different methods

Yield surface approximation for lower and upper bound yield design of 3d composite frame structures

International audienceThe present contribution advocates an up-scaling procedure for computing the limit loads of spatial structures made of composite beams. The resolution of an auxiliary yield design problem leads to the determination of a yield surface in the space of axial force and bending moments. A general method for approximating the numerically computed yield surface by a sum of several ellipsoids is developed. The so-obtained analytical expression of the criterion is then incorporated in the yield design calculations of the whole structure, using second-order cone programming techniques. An illustrative application to a complex spatial frame structure is presented

Locking-free discontinuous finite elements for the upper bound yield design of thick plates

International audienceThis work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner-Mindlin thick plates in shear-bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner-Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second-order cone programming (SOCP) problem and is solved using the industrial software package Mosek. A comprehensive study of the shear-locking phenomenon is also conducted and it is shown that discontinuous elements avoid such a phenomenon quite naturally, whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so-obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates

A viscous active shell theory of the cell cortex

The cell cortex is a thin layer beneath the plasma membrane that gives animal cells mechanical resistance and drives most of their shape changes, from migration, division to multicellular morphogenesis. It is mainly composed of actin filaments, actin binding proteins, and myosin molecular motors. Constantly stirred by myosin motors and under fast renewal, this material may be well described by viscous and contractile active-gel theories. Here, we assume that the cortex is a thin viscous shell with non-negligible curvature and use asymptotic expansions to find the leading-order equations describing its shape dynamics, starting from constitutive equations for an incompressible viscous active gel. Reducing the three-dimensional equations leads to a Koiter-like shell theory, where both resistance to stretching and bending rates are present. Constitutive equations are completed by a kinematical equation describing the evolution of the cortex thickness with turnover. We show that tension and moment resultants depend not only on the shell deformation rate and motor activity but also on the active turnover of the material, which may also exert either contractile or extensile stress. Using the finite-element method, we implement our theory numerically to study two biological examples of drastic cell shape changes: osmotic shocks and cell division. Our work provides a numerical implementation of thin active viscous layers and a generic theoretical framework to develop shell theories for slender active biological structures.Comment: 37 pages, 13 figures, 1 appendi

An optimization method for approximating the macroscopic strength criterion of stone column reinforced soils

International audienceIn this contribution, the yield design homogenization method is applied to the evaluation of the ultimate bearing capacity of a purely cohesive soil reinforced by a periodic array of columnar inclusions, made of a purely frictional material (stone column technique). The method is implemented following a three-step procedure. a) First, the numerical determination of the macroscopic strength criterion is performed using the kinematic approach of yield design. b) Second, an easier to handle formulation of the criterion is obtained as the sum of a few ellipsoids in the stress space. c) Finally, the so-obtained approximation is incorporated into a numerical code, leading to the determination of an optimized upper bound for the ultimate load bearing capacity of a reinforced soil foundation, which is compared with previously obtained estimates for the same problem

Upper bound limit analysis of plates using a rotation-free isogeometric approach

International audienceThis paper presents a simple and effective formulation based on a rotation-free isogeometric approach for the assessment of collapse limit loads of plastic thin plates in bending. The formulation relies on the kinematic (or upper bound) theorem and namely B-splines or non-uniform rational B-splines (NURBS), resulting in both exactly geometric representation and high-order approximations. Only one deflection variable (without rotational degrees of freedom) is used for each control point. This allows us to design the resulting optimization problem with a minimum size that is very useful to solve large-scale plate problems. The optimization formulation of limit analysis is transformed into the form of a second-order cone programming problem so that it can be solved using highly efficient interior-point solvers. Several numerical examples are given to demonstrate reliability and effectiveness of the present method in comparison with other published methods