International audienceThis work addresses the numerical computation of the two-dimensional flow of yield stress fluids (with Bingham and Herschel-Bulkley models) based on a variational approach and a finite element discretization. The main goal of this paper is to propose an alternative op-timization method to existing procedures such as penalization and augmented Lagrangian techniques. It is shown that the minimum principle for Bingham and Herschel-Bulkley yield stress fluid steady flows can, indeed, be formulated as a second-order cone programming (SOCP) problem, for which very efficient primal-dual interior point solvers are available. In particular, the formulation does not require any regularization of the visco-plastic model as is usually the case for existing techniques, avoiding therefore the difficult choice of the regularization parameter. Besides, it is also unnecessary to adopt a mixed stress-velocity approach or discretize explicitly auxiliary variables as frequently proposed in existing meth-ods. Finally, the performance of dedicated SOCP solvers, like the Mosek software package, enables to solve large-scale problems on a personal computer within seconds only. The pro-posed method will be validated on classical benchmark examples and used to simulate the flow generated around a plate during its withdrawal from a bath of yield stress fluid
surface approximation for lower and upper bound yield design of 3d composite frame structures. Computers and Structures, Elsevier, 2013, 129, pp. 86-98. <10.1016/j.compstruc.2013
AbstractThe 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.
Summary
The objective of this contribution is to present some new recent developments regarding the evaluation of the ultimate bearing capacity of massive three‐dimensional reinforced concrete structures which cannot be modeled as 1D (beams) or 2D (plates) structural members. The approach is based upon the implementation of the lower bound static approach of yield design through a discretization of the three‐dimensional structure into tetrahedral finite elements, on the one hand, the formulation of the corresponding optimization problem in the context of semi‐definite programming techniques, on the other hand. Another key feature of the method lies in the treatment of the concrete‐embedded reinforcing bars not as individual elements, but by resorting to an extension of the yield design homogenization approach. The whole procedure is first validated on the rather simple illustrative problem of a uniformly loaded simply supported beam, then applied to the design of a bridge pier cap taken as an example of more complex and realistic structure.
SUMMARYThe ultimate bearing capacity problem of a strip foundation resting on a soil reinforced by a group of regularly spaced columns is investigated in the situation when both the native soil and reinforcing material are purely cohesive. Making use of the yield design homogenization approach, it is shown that such a problem may be dealt with as a plane strain yield design problem, provided that the reinforced soil macroscopic strength condition has been previously determined. Lower and upper bound estimates for such a macroscopic criterion are obtained, thus giving evidence of the reinforced soil strong anisotropy. Performing the upper bound kinematic approach on the homogenized bearing capacity problem, by using the classical Prandtl's failure mechanism, makes it then possible to derive analytical upper bound estimates for the reinforced foundation bearing capacity, as a function of the reinforced soil parameters (volume fraction and cohesion ratio), as well as of the relative extension of the reinforced area. It is shown in particular that such an estimate is closer to the exact value of the ultimate bearing capacity, than that derived from a direct analysis which implicitly assumes that the reinforced soil is an isotropic material.
This 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.
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