Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 1: Derivation of finite element matrices

This paper deals with the formulation of nite plate elements for an accurate description of stress and strain elds 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 nite element matrices. Transverse stress assumptions are made in the framework of RMVT and the resulting nite 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 elds 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 nite 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 nite element matrices have been built from only ve 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 Nl. 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 elds in multilayered thick structures