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

Three-dimensional Problems of the Theory of Elasticity. By A. I. Lur’e.1964. (Interscience Publishers)

A large variety of plate theories are described and assessed in the present work to evaluate the bending and vibration of sandwich structures. A brief survey of available works is first given. Such a survey includes significant review papers and latest developments on sandwich structure modelings. The kinematics of classical, higher order, zigzag, layerwise, and mixed theories is described. An exhaustive numerical assessment of the whole theories is provided in the case of closed form solutions of simply supported panels made of orthotropic layers. Reference is made to the unified formulation that has recently been introduced by the first author for a plate/shell analysis. Attention has been given to displacements, stresses (both in-plane and out-of-plane components), and the free vibration response. Only simply supported orthotropic panels loaded by a transverse distribution of bisinusoidal pressure have been analyzed. Five benchmark problems are treated. The accuracy of the plate theories is established with respect to the length-to-thickness-ratio (LTR) geometrical parameters and to the face-to-core-stiffness-ratio (FCSR) mechanical parameters. Two main sources of error are outlined, which are related to LTR and FCSR, respectively. It has been concluded that higher order theories (HOTs) can be conveniently used to reduce the error due to LTR in thick plate cases. But HOTs are not effective in increasing the accuracy of the classical theory analysis whenever the error is caused by increasing FCSR values; layerwise analysis becomes mandatory in this case.

A Survey With Numerical Assessment of Classical and Refined Theories for the Analysis of Sandwich Plates

Recent research advances in the dynamic behavior of shells: 1989–2000, Part 2: Homogeneous shells

A unified formulation to assess theories of multilayered plates for various bending problems

An approach is proposed to solve three-dimensional stress problems for noncircular hollow cylinders. The end conditions are such that the problem can be reduced to a two-dimensional problem. This problem is reduced to a one-dimensional problem by introducing additional functions into the resolvable system of equations. These functions are determined using discrete Fourier series. The one-dimensional problem is solved by a stable numerical method. As an example, the stress state of cylinders with an elliptic cross section is analyzed depending on their thickness and degree of ellipticity.

On One Approach to the Solution of Stress Problems for Noncircular Hollow Cylinders

The approach developed to solve two-dimensional static problems for nonthin conical shells of varying thickness is used to examine the effect of the geometrical parameters on the stress-strain state of shells. The approach is based on spline-approximation and a stable numerical method of solving one-dimensional problems

Stress-strain analysis of closed nonthin orthotropic conical shells of varying thickness

An approach developed earlier to solve boundary-value problems is used to analyze the behavior of the stress-strain state of orthotropic elliptic cylindrical shells with variation in the geometric parameters of their cross section at constant volume (weight)

Refined analysis of the stress state of orthotropic elliptic cylindrical shells with variable geometrical parameters

Stress problems for noncircular cylindrical shells in classical, refined, and spatial statements are solved using nonconventional approaches based on discrete Fourier series and spline functions. Solutions for isotropic and orthotropic shells are presented as plots and tables

Ya. M. Grigorenko

Papers

On One Approach to the Solution of Stress Problems for Noncircular Hollow Cylinders

Stress-strain analysis of closed nonthin orthotropic conical shells of varying thickness

Refined analysis of the stress state of orthotropic elliptic cylindrical shells with variable geometrical parameters

Nonconventional approaches to static problems for noncircular cylindrical shells in different formulations

Approaches to the numerical solution of linear and nonlinear problems in shell theory in classical and refined formulations