AIAA JOURNAL Vol. 36, No. 5, May 1998 Evaluation of Layerwise Mixed Theories for Laminated Plates Analysis Erasmo Carrera* Turin Polytechnic Institute, Turin 10129, Italy The evaluation of mixed layerwise theories to calculate the in-plane and out-of-plane responses of thick plates in two-dimensional modeling of multilayered structures is made. The employedmodels, which were proposed by the author in earlier works, a priori fulfill the continuity of transverse shear and normal stress components at the interface between two adjacent layers. A Reissner's mixed variational equation is used to derive the governing equations, in terms of introduced stress and displacement variables. The interface continuity conditions are im- posed by writing the governing equations at a multilayered level. The related standard displacement formulations are also discussed for comparison purpose. Closed-form solutions are presented for plates made of orthotropic lamina and bent by harmonic distribution of transverse pressure. Symmetrically and unsymmetrically laminated, as well as sandwich, plates have been investigated. A comparison with a three-dimensional-elasticity analysis shows that present mixed layerwise models furnish a better description of the in-plane and out-of-plane response of thick plates with respect to existing layerwise and equivalent single-layer theories. In particular, the proposed models describe, with excellent accuracy, the transverse shear and normal stress fields. Unlike available current models, these fields are herein determined a priori without requiring implementation of any postprocessing procedures. The distribution of the transverse displacement and transverse normal stress in the plate thickness direction are also shown for most of the problems. I. Introduction T HE evaluations of transverse shear and normal stress and the related effects have played an important, constant role in thick- multilayered-plate analyses. Recent interest in such evaluations is due to the use of composite materials in primary thick components. In fact, as laminated materials undergo transition from secondary to primary structural components, the goals of analysis must be broadened to include the highly accurate assessment of localized regions where damage is likely to take place. As experienced by early three-dimensional elasticity analyses, 1 " 3 the variation of the mechanical properties in the thickness direction of laminated struc- tures, for compatibility and equilibrium reasons, leads to displace- ment and transverse stress fields, which reveal discontinuous deriva- tives in correspondence to each interface. The so-called zigzag form of displacement fields and interlaminar continuity for the trans- verse stresses were summarized in Ref. 4 using the term C" re- quirements, that is, displacement and transverse stress fields must be C°-continuous functions in the plate thickness direction z. Fur- thermore, the mentioned three-dimensional elasticity analyses un- derlined the fundamental role played by transverse normal stress. Nevertheless, three-dimensional-elasticity solutions are available in only a few cases, which are mainly related to simple geometries, specific stacking sequence of the lamina, and linear problems. In the most general cases and to minimize the computational effort, two-dimensional models are preferred in practice. In this respect, an enormous number of models and approximated techniques has been proposed over the last three decades. Exhaustive overviews on these topics can be found in many published review papers. Among these are Refs. 4-8. A short review, which is useful for the purpose of this work, is here given. Classical models, i.e., classical lamination theories (CLTs) and first-order shear deformation theory (FSDT), have been extended to laminated structures. 9 Higher-order expansions were applied to laminated plates by Lo et al. 1() These theories are based on the Received April 15, 1997; revision received Dec. 19, 1997; accepted for publication Dec. 23, 1997. Copyright © 1998 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. * Research Professor, Department of Aeronautics and Aerospace Engi- neering, Corso Duca degli Abruzzi, 24. E-mail: carrera@polito.it. Member AIAA. displacement formulation and belong to the group of equivalent single-layer models (ESLMs). These models preserve the indepen- dence of the number of the independent variables from the numbers of the layers. Unfortunately, CLT and FSDT, as well the model in Ref. 10, do not account for the C" requirements. Nevertheless, trans- verse shear stress can be evaluated at a postprocessing level through integration of the three-dimensional indefinite equilibrium equa- tions. The obtained results have, however, shown high inaccuracy in thick plate cases. 1 On the other hand, this postprocessing opera- tion cannot be implemented in the general case of a nonsymmetric in-plane stress response with respect to the plate middle surface, 11 as is in the case for nonlinear analysis. 12 Improvements, with respect to classical approaches, were achi- eved through partial fulfillment of the C" requirements using higher- order shear deformation theories (HSDTs). A pioneering analysis was presented by Yu, 13 where in-plane zigzag effects and transverse shear continuity were both fulfilled in correspondence to the two in- terfaces of a sandwich plate. Zero top-bottom plate conditions on the transverse shear stress were implemented to laminated structures by Reddy. 14 Cho and Parmerter 15 included in-plane zigzag effects and transverse shear continuity for arbitrarily laminated plates in a five- degree-of-freedom model. This work could be considered the best version of displacement-based theories that have originated from Refs. 16-18. Among HSDT models, particular mention should be made of the excellent work by Ren. 19 Ren extended very early work by Lekhnitstkii 20 ' 21 to anisotropic plates that had originally been presented for multilayered beams. In contrast to Ref. 15, zigzag kinematics were not assumed by Ren but were derived by means of a stress function formulation. The resulting seven-degree-of-freedom model (two more than the one by Cho and Parmerter 15 ) a priori ful- filled the three-dimensional indefinite equilibrium equations, show- ing excellent agreement with respect to three-dimensional elasticity solutions. Unfortunately, because of the intrinsic material couplings be- tween the transverse normal and in-plane stress components, all of the mentioned ESLMs experienced difficulties in extending the zigzag forms to the transverse displacement component or in ac- counting for the interlaminar continuity of the transverse normal stress. As a consequence, all of the related results have shown deficiencies in analyzing problems in which transverse normal stress plays a determinant role. 11 ' 22 The present paper does not overview results and theories based on the method of asymptotic 830 Downloaded by UNIVERSITY OF MICHIGAN on March 7, 2014 | http://arc.aiaa.org | DOI: 10.2514/2.444