Mathematical Problems in Engineering

Volume 2013, Article ID 486453, 9 pages

http://dx.doi.org/10.1155/2013/486453

## Study on the Explicit Formula of the Triangular Flat Shell Element Based on the Analytical Trial Functions for Anisotropy Material

^{1}Department of Civil Engineering, China Agricultural University, Beijing 100083, China^{2}Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China

Received 17 July 2013; Accepted 13 August 2013

Academic Editor: Song Cen

Copyright © 2013 Xiang-Rong Fu 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 presents a novel way to formulate the triangular flat shell element. The basic analytical solutions of membrane and bending plate problem for anisotropy material are studied separately. Combining with the conforming displacement along the sides and hybrid element strategy, the triangular flat shell elements based on the analytical trial functions (ATF) for anisotropy material are formulated. By using the explicit integral formulae of the triangular element, the matrices used in proposed shell element are calculated efficiently. The benchmark examples showed the high accuracy and high efficiency.

#### 1. Introduction

The triangular flat shell element is included widely in finite element software packages. In order to improve the performances of the triangular flat shell element, most of studies focused on two challenges [1–15].

One challenge is the drilling degree of freedom in the membrane element. Introducing the in-plane drilling degree of freedom, the singularity of global stiffness can be avoided in the analysis of shell structures when the neighboring elements around a common node are close to coplanar. Olson introduced the drilling degree of freedom to describe the displacement of the flat shell element [4]. Mohr constructed the hybrid membrane element with the drilling degree of freedom [5]. Allman presented a rational interpolation function to construct the displacement fields associated with the drilling degree of freedom in the triangular membrane elements [6]. Many researchers made different possibilities to improve accuracy of the membrane element by introducing the drilling degree of freedom to the corner nodes of triangular or quadrilateral elements [7–9].

Another challenge is the shear locking of some bending plate elements based on the Mindlin/Reissner thick plate theory [10–15]. The pioneering work of Adini, Clough, and Melosh studied the famous thin plate element ACM, which showed the difficulties in constructing the conforming displacement along the boundaries of the thin plate elements [10]. The thick plate element based on the Mindlin/Reissner theory can avoid the strict conforming condition [11–15], but the shear locking problem in these elements became the main challenge, which raised the interest of many well-known researchers, including Zienkiewicz et al. [11], Hughes et al. [12], Hinton and Huang [13], Belytschko et al. [14], Bathe and Dvorkin [15].

Long gave a general strategy to deal with the nonconforming displacement along the boundary between the neighboring elements, which was named as the generalized conforming element [16]. Taking advantage of the conforming boundary displacement, the nonconforming displacement model can be used freely to describe the inner field of the element. The generalized conforming equations which are utilized to determine the parameters of the proposed displacement model were studied systematically in the past decades [17–20]. Cen and Long gave a rational way to define the displacement field in the thick plate elements [19]. Fu et al. proved that the hybrid element using the balanced stress fields and the generalized conforming element introducing the associated generalized conforming equations can derive the same transform matrix between the inner parameters and the nodal displacements in the element [20].

Fu et al. and Cen studied the hybrid element based on the variational principle containing the stress function [21]. A series of works showed that elements based on the analytical trial functions (ATF) would be insensitive to the distortion of the element shape [22–25]. The hybrid element strategy was improved through turning to the stress trial functions which are derived from the stress functions (e.g., the Airy stress functions for plane problem). The idea of ATF can be traced back to the first element proposed by Clough which is named as CST (constant stress triangular) element [26]. Some works showed that the element CST can be derived from three rigid body displacements and three displacements of constant stress [1]. On the other hand, the first hybrid element also employed five analytical trial functions of stress [27].

This paper studies the triangular flat shell element based on the analytical trial functions of anisotropy. There are mainly four parts as follows.(a)The first part extracts the general analytical solutions of plane and bending plate problem in anisotropy elasticity (in Section 2).(b)The second part gives the hybrid element strategy based on the analytical trial functions of stress derived in the first part (in Section 3).(c)The third part studies the explicit formulae of the hybrid element based on explicit integral formulae of the triangular element (in Section 4).(d)The fourth part shows some numerical examples as benchmark to study the accuracy and the efficiency of the proposed element model (in Section 5).

The proposed shell element of anisotropy has high accuracy and computational efficiency.

#### 2. The Analytical Trial Functions (ATF)

##### 2.1. The Analytical Trial Functions of the Plane Problem of Anisotropy

The general analytical solutions play very important roles in many methods of computational mechanics [21–27]. Article [28] provided an efficient approach named characteristic differential equation method (CDEM) to derive the general analytical solutions from the governing differential equations directly.

Following the strategy of CDEM to study the plane problem of anisotropy, stress-displacement relationship can be described as And the equilibrium equations in terms of displacement can be written as Utilizing the determinant expansion equation of the differential operator matrix in (2), the characteristic differential equation can be given as

Then the general analytical solutions of the displacement can be derived from the characteristic solution of (4): Substituting (5) into (1), the general stress solutions can be derived from the characteristic solutions . Table 1 gives seven fundamental analytical stress solutions of the anisotropy plane problem, which can be selected as trial functions of the following numerical method.

##### 2.2. The Analytical Trial Functions of the Bending Plate Problem of Anisotropy

The characteristic differential equation of the displacement solutions can be written as [29]

Deriving from the general analytical displacement solutions of (6), Table 2 presents seven general analytical stress solutions of the anisotropy plate problem, which will be selected as trial functions of the following hybrid element in Section 3.

#### 3. The Hybrid Element Strategy Based on the Analytical Trial Functions

##### 3.1. The Hybrid Membrane Element Based on the Analytical Trial Functions of Anisotropy

In the sense of novel hybrid element strategy [21], the basic stress solutions of the anisotropic plane problem in Table 1 can be employed as the trial function of the inner stress field in the element, which is denoted as where is matrix of the analytical trial functions, whose dimension is . will be determined by the number of trial functions involved in the element. is the vector of the unknowns of the inner parameters. The modified complementary energy can be presented as [21] where the complementary energy is given by The constitutive matrix is Substituting (7) into (9) yields where The additional complementary energy is given by in which and are the boundary forces along the side around the element, determined by the inner stress field in (7): where in which

As showed in Figure 1, is the angle between the side and the axis and is the length of the side .

The conforming displacements and along the side can be obtained from the displacements of and in Figure 1, which can be assumed as where is the natural coordinate along the side in the triangular element; its value is defined as and .

are the Hermite functions, which have the expressions as follows: is the linear shape function Involving the relationship between displacement and displacement , the conforming displacement along the side can be defined as It can also be denoted as where is the matrix of the trial functions of the displacements defined in side as

And is part of the unknown nodal displacement vector, Substituting (14) and (22) into (13), we have where

Thus the modified complementary energy can be presented as

With the principle of the modified complementary energy, ,

From the first part of (28), the relationship between the inner parameters and the nodal displacement can be determined as

The stiffness matrix can be calculated by

Two triangular membrane elements based on the analytical trial functions of anisotropy are studied in this paper. The one containing five items of basic stress solutions in Table 1 is named as ATF-TR5; the other containing seven items is named as ATF-TR7.

##### 3.2. The Hybrid Plate Element Based on the Analytical Trial Functions of Anisotropy

The basic stress solutions of the anisotropic plate problem in Table 2 can be employed as the trial function of the inner stress field in the plate element, which is denoted as where is matrix of the analytical trial functions, whose dimension is . will be determined by the number of trial functions involved in the element. is the vector of the unknowns of the inner parameters.

The modified complementary energy of the plate problem can be presented as where is the constitutive matrix of the anisotropy plate problem and is the vector of the boundary forces along the side of the element; it can be denoted as is the conforming displacement; it can be denoted as

As showed in Figure 2, the conforming displacement along the side can be defined as where And it is defined as Equation (36) can also be denoted as where is the matrix of the trial functions of the displacements defined in side as And is part of the unknown nodal displacement vector:

Following the hybrid element strategy proposed in Section 3.1, the modified complementary energy can be presented as

Similarly, the stiffness matrix can be calculated by

Two triangular plate elements based on the analytical trial functions of anisotropy are studied in this paper. The one containing five items of basic stress solutions in Table 2 is named as ATF-TP5; the other containing seven items is named as ATF-TP7.

#### 4. The Explicit Formulae of the Hybrid Element Based on the Analytical Trial Functions

The triangular flat shell elements based on the analytical trial functions of anisotropy can be presented in the natural coordinates of the triangular element. For example, the matrix and, of the stress trial functions contain the variables and , which can be written as while the matrix and matrix of the conforming displacements contain the variables , which can be denoted as the natural coordinate too, that is in the side

According to (16), the matrix and can be expressed in the coordinates of the nodes in the element. For example, in the side ,

In utilized equations (44)~(47), all variables in the matrices , , , , , and can be expressed in the natural coordinates , , and . To the integral formula of the natural coordinates in the triangular element, we have where is the area of the triangular element. Utilizing the explicit integral formulae of (48) in the triangular element, the explicit formula of the proposed elements in this paper can be obtained.

For example, in the plane elements ATF-TR5 and ATF-TR7, we have

In the plate elements ATF-TP5 and ATF-TP7, we have

#### 5. Numerical Examples

##### 5.1. Cantilever Shell

As shown in Figure 3, a slender cantilever shell has the length of 32 m, the height of 2 m, and the thickness of 0.01 m. It is made of the anisotropy T700, which has the material parameters GPa, GPa, GPa, and GPa, GPa, GPa. There is an angle between the principle axis 1 and the axis , designated as 0°, 45°, and 90°. Figure 4 shows four schemes of the mesh to the proposed cantilever shell.

Two load conditions are studied in the example. One loading is the unit in-plane moment N*·*m on the end side . Table 3 gives the results of the displacement obtained from the proposed elements ATF-TR5 and ATF-TR7, respectively. Comparing with the element S3, which is employed in the software ABAQUS, ATF-TR5 and ATF-TR7 are more accurate.

The other loading is the unit out-plane load N on the end point ; Table 4 gives the results of the displacement obtained from the proposed element, ATF-TP5 and ATF-TP7. Comparing with the element S3, ATF-TP5 and ATF-TP7 are more accurate.

##### 5.2. Clamped Square Plate

As shown in Figure 5, a clamped square plate has the length of 1 m in each side and the thicknesses of 0.001 m, 0.01 m, and 0.1 m. It is also made of the anisotropy T700 and is subjected to the uniformly distributed loading Pa. Table 5 gives the deflection of the central point in different mesh schemes.

##### 5.3. Cook’s Skew Beam

As showed in Figure 6, Cook’s skew flat shell [30] is studied. The thickness of the shell is 1 m. It is also made of the anisotropy T700 and is subjected to the uniformly distributed load which has the summation of N. Table 6 gives the deflection of the central point in different mesh schemes.

#### 6. Conclusions

This paper presents a novel way to formulate the flat shell element of anisotropy. The basic analytical solutions and the proposed conforming displacement are employed to formulate the triangular elements ATF-TR5, ATF-TR7, ATF-TP5, and ATF-TP7. The high accuracy and computational efficiency of these elements are proved by the benchmark.

#### Acknowledgments

This project is supported by the National Natural Science Foundation of China (no. 11272340) and the National Basic Research Programs of China (no. 2010CB731503).

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