Giovanni Garcea

KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES BY A FINITE ELEMENT APPROACH

SUMMARY This paper refers to the analysis of the postbuckling behaviour of thin-walled structures by means of an asymptotic approach based on a finite element implementation of Koiter’s non-linear theory of instability. The analysis has been accomplished by using the following assumptions: (i) the structure is described as an assemblage of flat slender rectangular panels; (ii) a non-linear Kirchhoff-type plate theory is used to model each panel; (iii) HC finite elements discretization is used; (iv) linear and quadratic extrapolations are assumed for the fundamental and the postbuckling paths, respectively; (v) multimodal buckling is considered; and (vi) imperfection sensitivity analysis is performed in both multimodal and monomodal form based on the steepest-descent path criterion. Several numerical results are presented and discussed. Comparisons with numerical solution obtained by standard incremental codes are given, which show the accuracy and reliability of the proposed approach.

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.

PERTURBATION APPROACH TO ELASTIC POST-BUCKLING ANALYSIS

Abstract This paper aims to show that effective and reliable computer codes can be obtained by a suitable finite element implementation of the Koiters perturbation method. However, careful attention has to be paid to all the implementation details in order to avoid kinematical inconsitencies that can strongly affect the results.

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.

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 node-based smoothed finite element method (MNS-FEM) for elasticity

In this paper, an alternative formulation of the NS-FEM based on an assumed stress field is presented to include drilling rotations. Within each triangular element the displacement field is described by a revised Allman triangle interpolation, while the stress field is assumed as linear or linear reduced on the conflict domain of the background grid. The elastic solution is constructed through the stationarity condition of a constrained mixed Hellinger–Reissner principle. The numerical experiments show that the proposed model performs well in elastic problems, in particular in the case of incompressibility, and takes advantage of the enrichment of the interpolation functions from quadratic contributions to the displacement field. The paper also shows a way to improve the description of the stress field.

An Efficient Algorithm for Shakedown Analysis Based on Equality Constrained Sequential Quadratic Programming

A new iterative algorithm to evaluate the elastic shakedown multiplier is proposed. On the basis of a three field mixed finite element, a series of mathematical programming problems or steps, obtained from the application of the proximal point algorithm to the static shakedown theorem, are obtained. Each step is solved by an Equality Constrained Sequential Quadratic Programming (EC-SQP) technique that retain all the equations and variables of the problem at the same level so allowing a consistent linearization that improves the computational efficiency. The numerical tests performed for 2D problems show the good performance and the great robustness of the proposed algorithm.

Decomposition Methods and Strain Driven Algorithms for Limit and Shakedown Analysis

A mathematical programming formulation of strain-driven path-following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in a FEM context, is presented. From the optimization point of view, standard arc–length strain driven elastoplastic analysis, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other nonlinear programming methods, simpler convergence proofs and duality to be exploited. Due to the unified approach in terms of total stresses, the strain driven algorithms become more effective and less nonlinear with respect to a self equilibrated stress formulation and easier to implement in existing codes performing elastoplastic analysis.

MIXED SOLID MODELS IN NUMERICAL ANALYSIS OF SLENDER STRUCTURES

The reasons of the better performances of mixed, stress–displacements, 3D solid finite elements in the analysis of slender elastic structures are explained. It will be shown that mixed or compatible description, also when derived from the same finite element and then completely equivalent from the discretization point of view, behave very differently when implemented in both asymptotic and path–following solution strategies due to the occurrence of a pathological locking phenomenon in the compatible formulation. The notable advantages of the used of a 3D mixed solid finite element in Koiter asymptotic analysis are also highlighted.

COMPOSITE FEM MODELS FOR LIMIT AND SHAKEDOWN ANALYSIS

The paper improved S-FEMs formulations with an enriched displacement field, making use of modified Allman’s shape functions. This mixed interpolation is the natural context in performing lower bound strategy for shakedown, limit analysis and elastoplastic analysis. The model takes advantages from the simplicity and few addressed requirements for good performances in nonlinear analysis. The simple assumption made for the stress field regards the convenience of using self-equilibrated stress interpolations in Cartesian coordinates. In the proposed composite elements the stress is discontinuous on the element and across their sides and the mesh of the elements is coincident with the discretization of the geometry. This stress interpolation is able to address the discontinuities in the plastic strain and, in such a way, to define in their description a finer mesh with respect the basic grid.