This paper gives a historical review of the theories that have been developed for the analysis of multilayered structures. Attention has been restricted to the so-called Zig-Zag theories, which describe a piecewise continuous displacement field in the plate thickness direction and fulfill interlaminar continuity of transverse stresses at each layer interface. Basically, plate and shell geometries are addressed, even though beams are also considered in some cases. Models in which the number of displacement variables is kept independent of the number of constitutive layers are discussed to the greatest extent. Attention has been restricted to those plate and shell theories which are based on the so-called method of hypotheses or axiomatic approach in which assumptions are introduced for displacements and/or transverse stresses. Mostly, the work published in the English language is reviewed. However, an account of a few articles originally written in Russian is also given. The historical review conducted has led to the following main conclusions. 1! Lekhnitskii ~1935! was the first to propose a Zig-Zag theory, which was obtained by solving an elasticity problem involving a layered beam. 2! Two other different and independent Zig-Zag theories have been singled out. One was developed by Ambartsumian ~1958!, who extended the well-known Reissner-Mindlin theory to layered, anisotropic plates and shells; the other approach was introduced by Reissner ~1984!, who proposed a variational theorem that permits both displacements and transverse stress assumptions. 3 ! On the basis of historical considerations, which are detailed in the paper, it is proposed to refer to these three theories by using the following three names: Lekhnitskii Multilayered Theory, ~LMT!, Ambartsumian Multilayered Theory ~AMT!, and Reissner Multilayered Theory ~RMT!. As far as subsequent contributions to these three theories are concerned, it can be remarked that: 4! LMT although very promising, has almost been ignored in the open literature. 5! Dozens of papers have instead been presented which consist of direct applications or particular cases of the original AMT. The contents of the original works have very often been ignored, not recognized, or not mentioned in the large number of articles that were published in journals written in the English language. Such historical unfairness is detailed in Section 3.2. 6! RMT seems to be the most natural and powerful method to analyze multilayered structures. Compared to other theories, the RMT approach has allowed from the beginning development of models which retain the fundamental effect related to transverse normal stresses and strains. This review article cites 138 references. @DOI: 10.1115/1.1557614#

Historical review of Zig-Zag theories for multilayered plates and shells

This work is an overview of available theories and finite elements that have been developed for multilayered, anisotropic, composite plate and shell structures. Although a comprehensive description of several techniques and approaches is given, most of this paper has been devoted to the so called axiomatic theories and related finite element implementations. Most of the theories and finite elements that have been proposed over the last thirty years are in fact based on these types of approaches. The paper has been divided into three parts. Part I, has been devoted to the description of possible approaches to plate and shell structures: 3D approaches, continuum based methods, axiomatic and asymptotic two-dimensional theories, classical and mixed formulations, equivalent single layer and layer wise variable descriptions are considered (the number of the unknown variables is considered to be independent of the number of the constitutive layers in the equivalent single layer case). Complicating effects that have been introduced by anisotropic behavior and layered constructions, such as high transverse deformability, zig-zag effects and interlaminar continuity, have been discussed and summarized by the acronimC
 -Requirements. Two-dimensional theories have been dealt with in Part II. Contributions based on axiomatic, asymtotic and continuum based approaches have been overviewed. Classical theories and their refinements are first considered. Both case of equivalent single-layer and layer-wise variables descriptions are discussed. The so-called zig-zag theories are then discussed. A complete and detailed overview has been conducted for this type of theory which relies on an approach that is entirely originated and devoted to layered constructions. Formulas and contributions related to the three possible zig-zag approaches, i.e. Lekhnitskii-Ren, Ambartsumian-Whitney-Rath-Das, Reissner-Murakami-Carrera ones have been presented and overviewed, taking into account the findings of a recent historical note provided by the author. Finite Element FE implementations are examined in Part III. The possible developments of finite elements for layered plates and shells are first outlined. FEs based on the theories considered in Part II are discussed along with those approaches which consist of a specific application of finite element techniques, such as hybrid methods and so-called global/local techniques. The extension of finite elements that were originally developed for isotropic one layered structures to multilayerd plates and shells are first discussed. Works based on classical and refined theories as well as on equivalent single layer and layer-wise descriptions have been overviewed. Development of available zig-zag finite elements has been considered for the three cases of zig-zag theories. Finite elements based on other approches are also discussed. Among these, FEs based on asymtotic theories, degenerate continuum approaches, stress resultant methods, asymtotic methods, hierarchy-p,_-s global/local techniques as well as mixed and hybrid formulations have been overviewed.

Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shells

The first journal article on neural network application in civil/structural engineering was published by in this journal in 1989. This article reviews neural network articles published in archival research journals since then. The emphasis of the review is on the two fields of structural engineering and construction engineering and management. Neural networks articles published in other civil engineering areas are also reviewed, including environmental and water resources engineering, traffic engineering, highway engineering, and geotechnical engineering. The great majority of civil engineering applications of neural networks are based on the simple backpropagation algorithm. Applications of other recent, more powerful and efficient neural networks models are also reviewed. Recent works on integration of neural networks with other computing paradigms such as genetic algorithm, fuzzy logic, and wavelet to enhance the performance of neural network models are presented.

Neural Networks in Civil Engineering: 1989–2000

This review article is devoted to the use of the Reissner Mixed Variational Theorem ~RMVT! forward two-dimensional modeling of flat and curved, multilayer structures. A thorough review of the literature involving the use in the modeling of multilayered plates and shells using RMVT is also presented. In the first part, the paper overviews relevant key points that should be taken into account for an accurate description of strain and stress fields in multilayered plate and shell analysis. It is then shown that RMVT has been originated in view of the fulfillment of such key points, herein referred to as C-Requirements ~zig-zag form of the displacement fields in the thickness direction and continuity of transverse normal and shear stresses at each layer interface!. Classical variational statements are used to introduce Reissner’s Theorem. In the second part, the paper presents various ways in which RMVT can be used to develop plate and shell theories in a systematic manner. The so called layer-wise and equivalent single layer variable description are considered. Both strong and weak ~finite element! forms of governing equations have been derived. A Weak Form of Hooke’s Law ~WFHL!, is also discussed as an idea to eliminate transverse stress variables leading to standard classical models with only displacement unknowns. Two appendices display details of governing equations related to multilayered doubly curved shells and to finite element matrices of multilayered plates. A third part reviews the works that have appeared in literature which make use of RMVT. Mainly papers on multilayered plate and shell modelings have been addressed. The final part of the paper is devoted to giving an overview with selected results of numerical performance that can be acquired by RMVT applications; extensive comparison to elasticity solutions and to other significant analyses, based on classical and refined approaches, are given. It is concluded that Reissner’s Mixed Theorem should be considered as a natural tool for multilayered structure analyses; it plays a similar role to that of the Principle of Virtual Displacement in the analysis of isotropic single-layer structures. This review article includes 119 references. @DOI: 10.1115/1.1385512#

Developments, ideas, and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells

The review outlines modern trends in theoretical developments, novel designs and modern applications of sandwich structures. The most recent work published at the time of writing of this review is considered, older sources are listed only on as-needed basis. The review begins with the discussion on the analytical models and methods of analysis of sandwich structures as well as representative problems utilizing or comparing these models. Novel designs of sandwich structures is further elucidated concentrating on miscellaneous cores, introduction of nanotubes and smart materials in the elements of a sandwich structure as well as using functionally graded designs. Examples of problems experienced by developers and designers of sandwich structures, including typical damage, response under miscellaneous loads, environmental effects and fire are considered. Sample applications of sandwich structures included in the review concentrate on aerospace, civil and marine engineering, electronics and biomedical areas. Finally, the authors suggest a list of areas where they envision a pressing need in further research.

Review of current trends in research and applications of sandwich structures

The focus of this review is on the hierarchy of computational models for sandwich plates and shells, predictor-corrector procedures, and the sensitivity of the sandwich response to variations in the different geometric and material parameters. The literature reviewed is devoted to the following application areas: heat transfer problems; thermal and mechanical stresses (including boundary layer and edge stresses); free vibrations and damping; transient dynamic response; bifurcation buckling, local buckling, face-sheet wrinkling and core crimping; large deflection and postbuckling problems; effects of discontinuities (eg, cutouts and stiffeners), and geometric changes (eg, tapered thickness); damage and failure of sandwich structures; experimental studies; optimization and design studies. Over 800 relevant references are cited in this review, and another 559 references are included in a supplemental bibliography for completeness. Extensive numerical results are presented for thermally stressed sandwich panels with composite face sheets showing the effects of variation in their geometric and material parameters on the accuracy of the free vibration response, and the sensitivity coefficients predicted by eight different modeling approaches (based on two-dimensional theories). The standard of comparison is taken to be the analytic three-dimensional thermoelasticity solutions. Some future directions for research on the modeling of sandwich plates and shells are outlined.

Ahmed K. Noor

Papers

Computational Models for Sandwich Panels and Shells

Computational strategies for tire modeling and analysis

A hybrid neurocomputing/numerical strategy for nonlinear structural analysis

Buckling and postbuckling of composite panels with cutouts subjected to combined edge shear and temperature change

Sensitivity analysis for large-deflection and postbuckling responses on distributed-memory computers