Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations

Abstract

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.

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.…”

“…[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.…”

“…[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.…”

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.…”