Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 676181, 11 pages

http://dx.doi.org/10.1155/2015/676181

## ESL Based Cylindrical Shell Elements with Hierarchical Shape Functions for Laminated Composite Shells

^{1}School of General Education, Yeungnam University, 280 Daehak-Ro, Gyeongsan, Gyeongbuk 712-749, Republic of Korea^{2}Department of Civil Engineering, Yeungnam University, 280 Daehak-Ro, Gyeongsan, Gyeongbuk 712-749, Republic of Korea

Received 7 October 2014; Accepted 7 February 2015

Academic Editor: Dane Quinn

Copyright © 2015 Jae S. Ahn et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce higher-order cylindrical shell element based on ESL (equivalent single-layer) theory for the analysis of laminated composite shells. The proposed elements are formulated by the dimensional reduction technique from three-dimensional solid to two-dimensional cylindrical surface with plane stress assumption. It allows the first-order shear deformation and considers anisotropic materials due to fiber orientation. The element displacement approximation is established by the integrals of Legendre polynomials with hierarchical concept to ensure the -continuity at the interface between adjacent elements as well as -continuity at the interface between adjacent layers. For geometry mapping, cylindrical coordinate is adopted to implement the exact mapping of curved shell configuration with a constant curvature with respect to any direction in the plane. The verification and characteristics of the proposed element are investigated through the analyses of three cylindrical shell problems with different shapes, loadings, and boundary conditions.

#### 1. Introduction

Shell structures are three-dimensional structures with any curvature, thin in one direction and long in the other two directions. In engineering design, they are among the most significant and ubiquitous structural components. Applications of them include pressure vessels, the bodies of automobiles and airplanes, bridges, buildings, roofs, the hulls of ships and submarines, and many other structures. Particularly, increasing application of laminated composite shell is evident in a variety of engineering structures and manufactured components, because of the well-recognized characteristics of superior strength-to-weight, stiffness-to-weight, and cost-to-weight ratios, compared to conventional materials. While the laminated composite materials provide the design flexibility to achieve desirable stiffness and strength through the choice of lamination scheme, the anisotropic constitution of laminated composite structures often results in stress concentrations near material and geometric discontinuities that can lead to damage in the form of delamination, adhesive bond separation, and matrix cracking. Recently, these problems have been mitigated by replacing conventionally used laminated composites with functionally graded materials where the materials properties are gradually varied at microscopic scale in the thickness direction [1].

Finite element methods are versatile numerical tools to solve differential equations related to physical phenomena. In the finite element applications for shell analysis, some types of shell elements are currently available. They are flat facet element, shell theory-based element, degenerated shell element, and solid-shell element and so forth. For the flat facet element, it does not have any curvature. Thus, the curved surface is approximately explained by the combination of several elements. It is very simple in formulations and has still been used for engineering applications. However, it cannot explain bending-stretching coupling behavior in an element level. On the other hand, the shell theory-based element with curvatures can handle the bending-stretching coupling properly. Also, the degenerated shell element can be used for arbitrary shapes of shell surface. Based on the flat facet element or shell theory-based element [2–6], two-dimensional shell elements have been introduced. Acknowledging the need of three-dimensional shell elements, several formulations have been presented based on a degenerated shell concept [7, 8]. Since 1990s, some solid-shell approaches, which have some benefits as compared to the degenerated shell element because of the simplicity of their kinematics, have been introduced by some researchers [9–11].

Meanwhile, a number of innovative approaches have been put forward for the analysis of laminated structural systems, to extend the capabilities of laminated anisotropic composites. As far as two-dimensional modeling is concerned, it is assumed that displacement components are continuously differentiable through the thickness regardless of the layer boundaries. Representatives of the theories are known as classical lamination theory (CLT) and first-order shear deformation theory (FSDT). Both of these models [12–14] are known as equivalent single-layer theories (ESLT) based on certain assumptions concerning the kinematics of deformation or stress across the total thickness. Although FSDT provides a sufficiently accurate description of the global laminate response for thin to moderately thick plates, it cannot allow direct calculation of transverse stresses with acceptable accuracy. So a number of higher-order theories [12, 15–17] have been put forward using successively third- to higher-degree polynomials and other functions with continuous derivatives to yield more accurate interlaminar stress distributions. The deficiency of the theories has led to layerwise models in which the variation of displacement functions across the thickness is assumed for each layer separately. The layerwise models [18–21] require displacement continuity at layer interfaces. Such characterization of laminated systems can generally exhibit a rapid change of slopes of displacement fields at layer interfaces, often termed as the zig-zag effect. In order to satisfy the interlaminar continuity of transverse stresses at each layer interface, appropriate functional continuities are required for transverse displacements and stresses [22]. The number of modal degrees of freedom in normal layerwise models depends on the number of layers in the laminated system. In conventional finite element analysis based mostly on Lagrangian two-dimensional shape functions, the layerwise models can satisfy displacement continuity but not stress. It is thus true that normal layerwise models would be too expensive when it is intended to comply with transverse normal stress continuity. Thus, multiple model approaches [23–27] have also been attempted to reduce the overall number of modal degrees of freedom by optimizing computation process for maximum solution accuracy within a particular subregion of interest only and in the process reducing the computational effort.

It is well-known that low-order finite element implementation for shells suffers from various forms of locking whenever purely displacement-based formulations are employed. In recent years, the issue of locking has been most prominently addressed through the use of low-order finite technology using mixed variational principles. The assumed strain and enhanced strain formulations are among the successful low-order implementations. High-order finite element implementations have also been advocated in recent years as a means of eliminating the locking phenomena completely. Most notably, whenever a sufficient degree of polynomial-refinement is adopted, highly reliable locking free numerical solutions may be obtained in a purely displacement-based setting [28]. The first -version formulation related to shells, one of high-order approaches, was reported by Woo and Basu [29] who presented a cylindrical shell element formulation in the cylindrical coordinates associated with a suitable transfinite mapping function to represent the curved geometry. In this paper, we address the finite element formulation for the laminated cylindrical shell behavior using the -version approach. The approach assumes that a heterogeneous laminated shell stacked with several laminae is treated as a shell element using hierarchic interpolation functions. Thus, characteristics of the proposed approach are presented in detail. Since higher-order Lagrange shape functions cannot be used due to excessive round-off errors, all approximate functions for displacement fields are derived in terms of integrals of Legendre polynomials which are orthogonal in the energy norm.

#### 2. Formulation of Cylindrical Shell Element with Hierarchical Shape Function

##### 2.1. Geometry and Displacement Fields

In cylindrical coordinate shown in Figure 1, based on function theory with continuity at the interface between adjacent layers, dimensional reduction is carried out by incorporating the first-order shear deformation for bending behavior and the plane stress condition for membrane action. For geometry and displacement fields, the curvilinear coordinate system is considered in reference to the middle surface of laminated shells keeping a constant curvature with respect to any direction in the two-dimensional plane. Geometry fields on a surface defined by two axes are expressed by linear interpolation between and variables over the four vertex nodes only, as shown in Figure 1.