L. Della Croce

Refined shell elements for the analysis of functionally graded structures

Abstract The present paper considers the static analysis of plates and shells made of Functionally Graded Material (FGM), subjected to mechanical loads. Refined models based on the Carrera’s Unified Formulation (CUF) are employed to account for grading material variation in the thickness direction. The governing equations are derived from the Principle of Virtual Displacement (PVD) in order to apply the Finite Element Method (FEM). A nine-nodes shell element with exact cylindrical geometry is considered. The shell can degenerate in the plate element by imposing an infinite radius of curvature. The Mixed Interpolation of Tensorial Components (MITC) technique is extended to the CUF in order to contrast the membrane and shear locking phenomenon. Different thickness ratios and orders of expansion for the displacement field are analyzed. The FEM results are compared with both benchmark solutions from literature and the results obtained using the Navier method that provides the analytical solution for simply-supported structures subjected to sinusoidal pressure loads. The shell element based on refined theories of the CUF turns out to be very efficient and its use is mandatory with respect to the classical models in the study of FGM structures.

MITC9 Shell Elements Based on Refined Theories for the Analysis of Isotropic Cylindrical Structures

In this work a nine-nodes shell finite element, formulated in the framework of Carrera’s Unified Formulation (CUF), is presented. The exact geometry of cylindrical shells is considered. The Mixed Interpolation of Tensorial Components (MITC) technique is applied to the element in order to overcome shear and membrane locking phenomenon. High-order equivalent single layer theories contained in the CUF are used to perform the analysis of shell structures. Benchmark solutions from the open literature are taken to validate the obtained results. The mixed-interpolated shell finite element shows good properties of convergence and robustness by increasing the number of used elements and the order of expansion of displacements in the thickness direction.

Hierarchic finite elements for thin Naghdi shell model

Abstract Two approaches have traditionally been used when general shell structures have been analysed. The first approach has been devised by Kirchhoff and Love and later the model has been improved by Koiter. A second class of models is based on the notion of surface introduced by Cosserat. Naghdi has developed this model, where the Reissner-Mindlin-type assumptions are taken into account. In this paper we consider the shell model arising from the Naghdi formulation. It is known that finite element schemes for this model suffer from both shear and membrane locking. Several solutions to avoid the numerical locking have been proposed. Here a displacement finite element scheme is developed using C0 finite elements of hierarchic type with degrees ranging from one to four. Two severe test problems are solved. The results show that good performances are achieved by using high-order finite elements to solve the shell problem in its displacement formulation. The numerical results indicate that high-order elements perform very well in both test problems and match all the available benchmark results.

Hierarchic finite elements for moderately thick to very thin plates

We consider the numerical solution of Reissner-Mindlin plates. The model is widely used for thin to moderately thick plates. Generally low order finite elements (with degree one or two) are used for the approximation. Unfortunately the solution degenerates very rapidly for small thickness (locking phenomenon). Non standard formulations of the problem are usually combined with low order finite elements to weaken or possibly eliminate the locking of the numerical solution. We introduce a family of hierarchic finite elements and we present a set of numerical results for the plate problem in its plain formulation. We show that reliable solutions are achieved for all thicknesses of practical interest by using high order finite elements. Moreover, the hierarchic structure allows a low cost assessment of the quality of the solution.

Hierarchic finite elements with selective and uniform reduced integration for Reissner-Mindlin plates

It is well known that the numerical solution of Reissner-Mindlin plates degenerates very rapidly for small thickness (locking phenomenon) when standard finite elements are used for the approximation. We have introduced a family of hierarchic high order finite elements in order to assess reliability and robustness with respect to the locking behavior. In a previous note we have given numerical results obtained with exact numerical integration. In this paper we present the results obtained with selective and uniform reduced integration. The results show that, compared with exact integration, selective reduced techniques improve the quality of the numerical performance and are preferable since computational cost is made smaller.

Solving cylindrical shell problems with a non-standard finite element

The subject of this work is the numerical analysis of a special finite element for the numerical solution of Naghdi cylindrical shell problems. Its discretization with standard finite elements suffers from the locking phenomenon, i.e., when small thickness shells are solved, the numerical results are affected by a large error. Several solutions to avoid the numerical locking have been proposed: mixed formulations, reduced and/or selective integration. In this paper we present a finite element based on a non-standard formulation of the discrete approximation of the shell problem. The main feature of the element proposed is the introduction of a linear operator that reduces the influence on the numerical solution of both shear and membrane energy terms. The performance of the new element is tested solving a benchmark problem involving very thin shells and severely distorted decompositions. The results show both properties of convergence and robustness.