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

Abstract

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.

Hybrid-interface finite element for laminated composite and sandwich beams

“…It can be seen that the results from the present theory are in very good agreement with the Liou and Sun [42] Engblomand Ochoa [41] Carrera LM4 [21] Carrera EMZC3 [21] x (a / 2, Z) elasticity solution [14]. It should also be noted that the hybrid stress models of Liou and Sun [42] and Carrera and Damasi [20,21] have yielded good results (see Figs. 5(a-d)).…”

“…The elasticity solution for composite laminates under cylindrical bending by Pagano [14], FE solution using multilayer plate bending element by Engblom and Ochoa [41] and Liou and Sun [42], higher order FE solutions by Lo et al [13] and closed-form solution for bending of laminated plates by Ren [43] have been considered for comparison of results obtained from the present FE model. FE solution using HZZT (EMZC3) and higher order multilayer plate bending (LM4) based on Reissner's mixed variational theorem by Carrera and Demasi [20,21] have also been used for comparison. On the basis of extensive numerical experimentation with different lay-ups, a minimum of s 1.4 + 1 number of elements along X-direction and element aspect ratio L x /L z between 10 and 15 have been found to provide a converging solution.…”

“…[20,21,38]) that stress fields are independent of displacement fields. Therefore fundamental elasticity relations cannot be satisfied and the continuity of some relevant variables is only guaranteed in weak sense.…”

“…[14][15][16]) indicate that interlaminar continuity of the transverse normal and shear stresses as well as the layer-wise continuous displacement field through the thickness are essential requirements in analysis. Thus, a layer-wise analysis is often required for laminated composite structures even if it is computationally [20][21][22], and others, providing satisfactory results for both the global values (e.g. deflections and flexural stresses) as well as the local values (e.g.…”

Interlaminar stress recovery for three-dimensional finite elements

“…Although primarily motivated by avoidance of the locking phenomena in finite element solutions, mixed formulations based on the Reissner or Hu-Washizu functional [6,7], with displacements, stresses and/or strains as field variables, are so closely related to stress recovery that the formulations can also be viewed as stress recovery methods [8]. Mixed Layer-Wise theories are exemplified by the work of Carrera [9], where a Reissner's mixed variational equation [10,11] is used to derive the governing equations in terms 2 of displacements and transverse stress variables [12,13]. On the one hand assumed stress methods do not require any post-processing for the accurate recovery of stresses.…”