Raffaele Casciaro

Mixed formulation and locking in path-following nonlinear analysis

Abstract The arc-length Riks strategy has rapidly become a standard tool for path-following analysis of nonlinear structures due to its theoretical ability to surpass limit points. The aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle ‘locking’ effect intrinsic to the nonlinear nature of the problem. As a consequence, its sanitization has to be pursued within a reformulation of the structural model. The use of a mixed (stress-displacement) variant of the algorithm, in particular, appears very promising in this respect. The topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in [39]. It is shown that, with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation, it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm.

An iterative method for shakedown analysis

Abstract Shakedown analysis for elastic–perfect plastic structures is discussed and a fast incremental-iterative solution method is proposed, suitable for the FEM analyses of large structures. The theoretical motivations of the proposed method are discussed in detail and an example of its implementation is described with reference to plane frame analysis. Some numerical results are presented showing the numerical performances of the method.

Extrapolation locking and its sanitization in Koiter's asymptotic analysis

Abstract This paper shows that the FEM implementation of Koiters asymptotic method [W.T. Koiter, On the stability of elastic equilibrium, 1970, Ph.D. Thesis, Delft, 1945. English transl. NASA TT-F10, 883, 1967, AFFDL-TR70-25] outlined by Casciaro et al. [Finite element asymptotic analysis of slender elastic structures: a simple approach, Int. J. Num. Meth. Eng. 35 (1992) 1397–1426] provides accurate and reliable results in the critical and post-critical analysis of non-linear elastic structures. Care, however, does have to be taken in implementing (apparently) minor details to avoid locking effects which adversely affect accuracy and which can destroy the method reliability. As the effects related to the finite element interpolation have been discussed before this paper focuses on the non-linear locking due to the use, implicit in the method, of finite distance extrapolations. Within this scope, it is shown that perturbation algorithms based on compatible formulations can imply a strong critical and post-critical locking when analysing structures characterized by high stiffness ratios in the presence of moderate pre-critical rotations. On the contrary, perturbation algorithms based on independent extrapolations of displacements and stresses furnish reliable results in excellent agreement with those provided by step-by-step analysis, at a small fraction of its computational cost.

MODE JUMPING AND ATTRACTIVE PATHS IN MULTIMODE ELASTIC BUCKLING

This paper summarizes a part of the first authors Ph.D. Thesis completely devoted to multimode elastic buckling within an FEM strategy. The theoretical arguments unfold among critical points on radial paths (the unique post-critical paths variationally defined), algebraic characterizations, proposition demonstrations and so on, by aiming to prove that the complexity of the phenomenon of multimode buckling (secondary bifurcations, post-critical attractive paths) can be theoretically explained.

A mixed solution strategy for the nonlinear analysis of brick masonry walls

The paper presents a discrete mechanical model for masonry walls based on a Lagrangean description where each brick is described as a rigid body and each mortar joint as an interface element. Constitutive assumptions, characterized by elasticity, damage and friction, are associated to the joints only. A numerical solution strategy, based on a mixed path-following approach in terms of stresses, strains, displacements, damage and load parameters, is proposed for avoiding convergence problems related to the joint softening behaviour. Some numerical results are also presented showing the robustness and effectiveness of this proposal.

On nonlinear beam models from the point of view of computational post-buckling analysis

Abstract The buckling and post-buckling analysis of elastic planar frames is considered and the use of geometrically exact beam models is thereby advocated. It is shown that usual technical beam models fail to predict correctly the curvature of the post-buckling curve at bifurcation even for standard problems of elastic stability theory. It is also argued that versatile and efficient computational procedures for bifurcation analysis of general planar frames are to be based on unconstrained beam models. Some remarks on finite element representation of nonlinear beam models are passed in conclusion.

Computational asymptotic post-buckling analysis of slender elastic structures

The lectures provide an introduction to the computational treatment of Koiter’s asymptotic strategy for post-buckling analysis of thin elastic structures.

Direct Evaluation of the Post-Buckling Behavior of Slender Structures Through a Numerical Asymptotic Formulation

The analysis of slender structures, characterized by complex buckling and postbuckling phenomena and by a strong imperfection sensitivity, is heavily penalized by the lack of adequate computational tools. Standard incremental iterative approaches are computationally expensive and unaffordable, while FEM implementation of the Koiter method is a convenient alternative. The analysis is very fast, its computational burden is of the same order as a linearized buckling load evaluation and the simulation of different imperfections costs only a fraction of that needed to characterize the perfect structure. In this respect it can be considered as a direct method for the evaluation of the critical and post-critical behaviour of geometrically nonlinear elastic structures. The main objective of the present work is to show that finite element implementations of the Koiter method can be both accurate and reliable and to highlight the aspects that require further investigation.

A Mixed 4-Node 3D Plate Element Based on Self-Equilibrated Isostatic Stresses

A new mixed stress 4-node flat shell finite element, designed for the linear and nonlinear analysis of folded plate structures, is presented. The kinematics of the element is defined by 24 dofs with in- and out-of-plane displacements assumed to be quadratic, controlled by displacement and rotation parameters through an Allman like interpolation, and flexural rotations assumed to be bilinear. The assumed stress approximation, described within a local Cartesian frame aligned with the element orientation, is self-equilibrated and ruled by the minimum number of parameters. The element does not suffer from kinematical locking and rank defectiveness. Many numerical tests show the very good performance of the element.

AN ADAPTIVE MULTIGRID SOLVER FOR PLATE VIBRATION AND BUCKLING PROBLEMS

Abstract The paper describes a finite element multigrid strategy for the solution of plate vibration and buckling problems. The solution procedure combines the residual iteration scheme used in Casciaro R, Aristodemo M. International Conference on Finite Elements Nonlinear Solid and Structural Mechanics, Gelio, Norway, 1977, (main loop) with the adaptive multigrid scheme proposed in Lopez S, Casciaro R. International Journal of Numerical methods in Engineering 1997;40:919–96. A subspace iteration technique allows several principal eigensolutions to be obtained simultaneously. The sequence of meshes used by the multigrid process is generated through an adaptive local refinement based on the Zienkiewicz–Zhu estimate of the discretization error Zienkiewicz OC, Zhu JZ. International Journal of Numerical Methods in Engineering 1987;24:337–57. A pointer-based data structure and an HC Aristoderno M. Computers and Structures 1985;21(5):987–93, finite element discretization make the solution process highly efficient. Some numerical tests show the effectiveness of the algorithm.