SUMMARYThis paper deals with the formulation of ÿnite plate elements for an accurate description of stress and strain ÿelds in multilayered, thick plates subjected to static loadings in the linear, elastic cases. The so-called zig-zag form and interlaminar continuity are addressed in the considered formulations. Two variational statements, the principle of virtual displacements (PVD) and the Reissner mixed variational theorem (RMVT ) are employed to derive ÿnite element matrices. Transverse stress assumptions are made in the framework of RMVT and the resulting ÿnite elements describe a priori interlaminar continuous transverse shear and normal stresses. Both modellings which preserve the number of variables independent of the number of layers (equivalent single-layer models, ESLM) and layer-wise models (LWM) in which the same variables are independent in each layer, have been treated. The order N of the expansions assumed for both displacement and transverse stress ÿelds in the plate thickness direction z as well as the number of element nodes Nn have been taken as free parameters of the considered formulations. By varying N , Nn, variable treatment (LW or ESL) as well as variational statements (PVD and RMVT), a large number of newly ÿnite elements have been presented. Finite elements that are based on PVD and RMVT have been called classical and advanced, respectively.In order to write the matrices related to the considered plate elements in a concise form and to implement them in a computer code (see Part 2), extensive indicial notations have been set out. As a result, all the ÿnite element matrices have been built from only ÿve arrays that were called fundamental nuclei (four are related to RMVT applications and one to PVD cases). These arrays have 3 × 3 dimensions and are therefore constituted of only nine terms each. The di erent formulations are then obtained by expanding the indices that were introduced for the N -order expansion, for the number of nodes Nn and for the constitutive layers N l . Compliances and=or sti ness are accumulated from layer to multilayered level according to the corresponding variable treatment (ESLM or LWM). The numerical evaluations and assessment for the presented plate elements have been provided in the companion paper (Part 2), where it has been concluded that it is convenient to refer to RMVT as a variational tool to formulate multilayered plate elements that are able to give a quasi-three-dimensional description of stress=strain ÿelds in multilayered thick structures.
SUMMARYThis paper presents numerical evaluations related to the multilayered plate elements which were proposed in the companion paper (Part 1). Two-dimensional modellings with linear and higher-order (up to fourth order) expansion in the z-plate=layer thickness direction have been implemented for both displacements and transverse stresses. Layer-wise as well as equivalent single-layer modellings are considered on both frameworks of the principle of virtual displacements and Reissner mixed variational theorem. Such a variety has led to the implementation of 22 plate theories. As far as ÿnite element approximation is concerned, three quadrilaters have been considered (four-, eight-and nine-noded plate elements). As a result, 22×3 di erent ÿnite plate elements have been compared in the present analysis. The automatic procedure described in Part 1, which made extensive use of indicial notations, has herein been referred to in the considered computer implementations. An assessment has been made as far as convergence rates, numerical integrations and comparison to correspondent closed-form solutions are concerned. Extensive comparison to early and recently available results has been made for sample problems related to laminated and sandwich structures. Classical formulations, full mixed, hybrid, as well as three-dimensional solutions have been considered in such a comparison. Numerical substantiation of the importance of the fulÿlment of zig-zag e ects and interlaminar equilibria is given. The superiority of RMVT formulated ÿnite elements over those related to PVD has been concluded.Two test cases are proposed as 'desk-beds' to establish the accuracy of the several theories. Results related to all the developed theories are presented for the ÿrst test case. The second test case, which is related to sandwich plates, restricts the comparison to the most signiÿcant implemented ÿnite elements. It is proposed to refer to these test cases to establish the accuracy of existing or new higher-order, reÿned or improved ÿnite elements for multilayered plate analyses.
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