Costin Pacoste

Co-rotational beam elements with warping effects in instability problems

The present paper investigates the formulation of 3D co-rotational beam elements for the buckling and post-buckling analysis of frame structures. Following Pacoste and Eriksson [Comput. Methods App ...

Beam elements in instability problems

This paper deals with the formulation of beam elements for the numerical analysis of instability phenomena in frame-type structures. Total versus co-rotational approaches are discussed comparatively, for both two-dimensional and three-dimensional problems, and the similarities between the two types are outlined. In the context of 3D beam elements, special attention is given to the parameterisation of the orthogonal transformation used to define the rotational field of the beam. The technique advocated in the paper is based on the so-called rotational vector. This leads to symmetric stiffness matrices and avoids the need for special updating procedures for the rotational variables. A set of test problems, for which the critical behaviour is governed by fold, cusp and butterfly catastrophes, is used to assess the performances of the considered element types. It is shown that analytically verified identities in element formulation, also hold in numerical application. The examples also show how complex instability behaviour can be reproduced by all elements, where sufficient accuracy is introduced into the kinematic expressions. The analytical derivation of element expressions, with symbolic manipulations from stated basic assumptions, is consistently used in the paper.

Co-rotational flat facet triangular elements for shell instability analyses

Abstract This paper investigates the formulation of co-rotational flat facet triangular elements for the numerical analysis of instability phenomena in shell structures. The elements have three nodes with six degrees of freedom at each node. The term ‘co-rotational’ relates here to the provision of a local system that continuously rotates and translates with the element. Following mainly Nour-Omid and Rankin [B. Nour-Omid and C.C. Rankin, Finite rotation analysis and consistent linearization using projectors, Comput. Methods Appl. Mech. Engrg. 93 (1991) 353–384], the definition of an element resorts to a change of variables from the local frame to the global one. This is done through the use of a projector matrix which relates the variations of the local displacements to the variations of the global ones, by extracting the rigid body modes from the latter. The main difference from the original formulation lies in the parameterization of 3D finite rotations. In contrast to the paper by Nour-Omid and Rankin, a parameterization based on the rotational vector is here adopted and thus, an additional change of variables has to be performed. As a result, the rotational variables become additive and the necessity of a special updating procedure is avoided. The main feature of the adopted formulation is its independence of the local assumptions used to derive the internal forces and tangent stiffness in local coordinates. For a certain class of elements (i.e. elements with the same number of nodes and degrees of freedom) the main co-rotational framework is the same. Using this property, three types of local formulations are considered. A set of carefully chosen test problems is used in order to assess the performances of the three element types.

Element formulation and numerical techniques for stability problems in shells

In the context of instability problems in shells or shell like structures, the objective of the present paper is twofold. Primarily, the paper describes how quasi-static, conservative instability problems can be considered in a multi-parametric context, where generalized path-following procedures for augmented equilibrium problems are used as computational tools. These allow systematic treatment of the higher-dimensional solution sets generated under the variations of certain parameters deemed relevant for the given problem. The efficient implementation of the above mentioned procedures requires however, as an essential ingredient, a non-linear finite element which is not only accurate but also inexpensive. To this end, a systematic view on a corotational Total Lagrangian formulation is described. The TRIC element of Argyris and coworkers is slightly modified, and introduced as core element formulation. Special emphasis is given to the alternative methods for treatment of finite three-dimensional rotations, with reference to both the element definition and solution algorithms. Numerical examples verify the element capabilities, and the possibility to completely describe instability phenomena of large, discretized models.

Plastic instability of beam structures using co-rotational elements

In a previous paper [Comput. Methods Appl. Mech. Engrg. 191 (2002) 1755], the authors have presented a 3D co-rotational elastic beam element including warping effects. This formulation is now further developed in order to incorporate elasto-plastic deformations. The element possesses seven degrees of freedom at each node and can be used to model beams with arbitrary cross-sections. Thus, within the present approach, the centroid and shear center of the cross-section are not necessarily coincident. The main purpose of this element is to model elasto-plastic instability problems. In this context, two methods of branch-switching are tested and discussed. In the first one, the bifurcation point is isolated by successive bisections and the branch-switching is operated by using the eigenvector associated to the negative eigenvalue. In the second one, introduced by Petryk, an energy approach is used to select automatically the stable post-bifurcation path. Six examples, including large displacement and stability problems, are used in order to assess the performances of the element.

Numerical analysis of complex instability behaviour using incremental-iterative strategies

The paper describes how quasi-static, conservative instability problems can be analysed in a multi-parametric space, using generalised path-following procedures for augmented equilibrium problems. ...

Element behavior in post-critical plane frame analysis

Depending on the type of uni-modal (cuspoid) catastrophe which governs the local behavior of a structure at a simple critical point, correct prediction of the post-critical path requires an accurat ...

Improved minimal augmentation procedure for the direct computation of critical points

This paper presents a new numerical procedure for the direct computation of critical points for elastic beam structures undergoing large displacements and rotations. Compared to the approach described by Wriggers et al. [Comput. Methods Appl. Mech. Engrg. 70 (1988) 329; Int. J. Numer. Methods Engrg. 30 (1990) 155], two main modifications are introduced. First, following Eriksson [Comput. Methods Appl. Mech. Engrg. 114 (1994) 77; Comput. Methods Appl. Mech. Engrg. 156 (1998) 45; Comput. Methods Appl. Mech. Engrg. 179 (1999) 265; Int. J. Struct. Stability Dynam. 1(1) (2001)], the condition of criticality is expressed by a scalar equation instead of a vectorial one. Next, the present procedure does not use exclusively the extended system obtained from the equilibrium equations and the criticality condition, but also introduces intermediate iterations based purely on equilibrium equations under load or displacement control. Eight numerical examples, presenting bifurcation and limit points, are used in order to compare the performances of this new method and the one presented in [Comput. Methods Appl. Mech. Engrg. 70 (1988) 329; Int. J. Numer. Methods Engrg. 30 (1990) 155].

Symbolic software tools in the development of finite elements

Symbolic software has been used in a number of projects concerned with the development of finite element procedures, primarily aiming at complex, i.e. interacting and higher order instabilities, where high accuracy in formulations is required.The symbolic tools improve the eAciency and documentation of the developed procedures, in order to facilitate comparisons between diAerent element assumptions. Beam formulations for plane and space models were developed, in total displacement and co-rotational contexts, respectively. Symbolic derivation allowed analytical verification of equivalence between certain formulations within these two contexts. Treatment of finite space rotations, based on the rotational vector makes the history-less treatment of rotations easier, which is needed in the evaluation of critical equilibrium subsets in higher-dimensional parameter space. A co-rotational viewpoint, where local element displacements can be obtained from global variables in a systematic manner, allowed diAerent element expressions in a common framework. DiAerent simple, linear elements have been tested with respect to computational eAciency. A field consistence approach was used to develop highly accurate beam and plane stress elements. The common element formulations, based on the matrix multiplication B T DB, is often ineAcient, due to the large number of operations needed in the matrix product. Other formulations, based on an analytical integration and diAerentiation of the strain energy, producing explicit expressions for the stiAness terms, were considerably more eAcient for certain elements. # 1999 Elsevier Science Ltd. All rights reserved.

SOLUTION SURFACES AND GENERALIZED PATHS IN NON-LINEAR STRUCTURAL MECHANICS

The paper describes how quasi-static, conservative instability problems can be seen in a multi-dimensional context, and how one- and two-dimensional solution manifolds can reveal further information on the structural response. The discussed viewpoint can be seen as the natural extension of the common one-dimensional path-following methods, when additional variables are introduced to describe the parameter dependence in structural response, instability analyses and optimization. The paper describes the general setting of the generalized equilibrium problems, and discusses their numerical treatment for the cases of resulting one- and two-dimensional solution sets. Numerical examples show some applications of these models, and describe the possibilities and properties of the obtained solution sets.