Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 301054, 9 pages

http://dx.doi.org/10.1155/2015/301054

## Robustness of Hierarchical Laminated Shell Element Based on Equivalent Single-Layer Theory

^{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 15 September 2014; Accepted 26 February 2015

Academic Editor: Sellakkutti Rajendran

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

This paper deals with the hierarchical laminated shell elements with nonsensitivity to adverse conditions for linear static analysis of cylindrical problems. Displacement approximation of the elements is established by high-order shape functions using the integrals of Legendre polynomials to ensure continuity at the interface between adjacent elements. For exact linear mapping of cylindrical shell problems, cylindrical coordinate is adopted. To find global response of laminated composite shells, equivalent single-layer theory is also considered. Thus, the proposed elements are formulated by the dimensional reduction from three-dimensional solid to two-dimensional plane which allows the first-order shear deformation and considers anisotropy due to fiber orientation. The sensitivity tests are implemented to show robustness of the present elements with respect to severe element distortions, very high aspect ratios of elements, and very large radius-to-thickness ratios of shells. In addition, this element has investigated whether material conditions such as isotropic and orthotropic properties may affect the accuracy as the element distortion ratio is increased. The robustness of present element has been compared with that of several shell elements available in ANSYS program.

#### 1. Introduction

Finite element methods generally involve finding approximate solutions in a space of piecewise polynomials of degree, which is often designated by the letter , on a grid of mesh size . In general, conventional finite elements based on mesh refinement have given reliable solutions when discretizing mesh is regular, while poor performance would be obtained when the element geometry is distorted. Also, it is known well that the interpolation precision of quadrilateral finite elements deteriorates if the element geometry considerably differentiated itself with a square. In this sense loss of element performance is commonly associated with a gradual increase of the stiffness, leading to sort of locking the element’s response. It brings about the inability of the element to find a good approximation of the solution. Moreover things would turn for the worse when conventional low-order finite element formulation is used in thin plates or shells since shear or membrane locking phenomena arise. Various robust schemes have been suggested for the problems involving locking. One possibility is to use higher-order elements. It is known that the locking completely eliminates certain types of locking [1]. The high-order finite element implementations have also been advocated in recent years as a means of eliminating the locking phenomena completely. The relevant works have been implemented by some researchers [2–5]. The issue of locking using lower-order elements has been most prominently addressed through the use of low-order finite technology using some mixed variational principles [6–8]. However, these depend upon reformulating the problems in special ways which have not been required in higher-order elements. In this paper, we will address only the finite element formulation for laminated shell behavior using the -version approach. The first successful -version formulation related to shells was reported by Woo and Basu [9] who presented the hierarchical -shell element formulation in the cylindrical coordinates associated with a suitable transfinite mapping function to represent the curved geometry. This element showed not only a constant strain and a rigid body motion from patch and eigenvalue tests but also a strong robustness with regard to very large aspect ratio and severely distorted mesh. The proposed hierarchical -shell element can also be successfully extended to singularity problems including a crack and a cut-out. Surana and Sorem [10] developed a three-dimensional curved shell element with a higher-order hierarchical displacement approximation in the thickness direction of the shell. The extension of this idea to laminated plates and shells was accomplished by the same authors [11]. Laminated plates and shells have usually been analyzed by the use of ESL (equivalent single-layer) theories based on either the classical Kirchhoff-Love hypothesis or first-order shear deformation theories. A drawback of ESL theories in modeling composite laminated plates and shells is that the transversal strain components are continuous across interfaces between dissimilar materials for the perfectly bonded layers. Therefore, the transverse stress components are discontinuous at the layer interfaces. On the other hand, layerwise theories are considerably more accurate than the preceding theories. Thus, Ahn and Woo [12, 13] extended the -version approach to efficient “mixed model analysis,” often called “global-local approach” to obtain interlaminar stresses at free edges in the laminated composite plate under extension and flexure on the basis of layerwise theories. The aim of this study is to present a simple high-order hierarchical -element for laminated composite shells using ESL concepts prior to layerwise theories in the cylindrical coordinates. This approach is computationally less expensive as compared to those obtained by three-dimensional elasticity solutions and layerwise finite element models.

In general, the most important symptoms of accuracy failure in modern finite elements are spurious mechanism, locking, elementary defects like violation of rigid body property, and invariance to node numbering. Also, parameters which affect accuracy are loading, element geometry, problem geometry, material properties, material anisotropy, and so on. Because of these reasons, governmental concern in the USA for the accuracy of finite element analysis is evidenced by NRC (Nuclear Regulatory Commission) requirement for structural analysis computer program validation, and the formation of NAFEM (National Agency for Finite Element Methods and Standards) in the United Kingdom.

In this study, the sensitivity test has been carried out to verify the robustness of proposed element in relation to distortion effect of mesh, very high aspect ratio, and very large radius-to-thickness ratio. In addition to these, the orthotropic cylindrical shell stacking with different fiber orientations is tested to check whether the anisotropy of materials may affect the accuracy as the element distortion ratio is increased. The numerical solutions obtained by present element are compared with several shell elements available in ANSYS program [14].

#### 2. Formulation of Hierarchical Laminated Shell Element

##### 2.1. Hierarchical Shape Functions for Quadrilaterals

A standard quadrilateral element is shown in Figure 1. Two-dimensional shape functions in the element can be classified into three groups such as nodal, edge, and internal shape functions. The nodal shape functions can be defined as and denote the local coordinate of the th node in the Figure 1. Also, the edge shape functions with any order , which vanish in all other edges, are defined separately for each individual edge. ConsiderThe quantity refers to the Kronecker tensor. and denote the local coordinate of both nodes in the th edge shown in the Figure 1. The functions which have one independent variable of the two variables and are defined as follows:where the functions are the Legendre polynomials. These functions with integrals of Legendre polynomials are well suited for computer implementation and have very favorable properties from the point of view of numerical stability [15]. Lastly, there are internal shape functions in . These can be written as