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

This paper presents a novel way to formulate the triangular flat shell element. The basic analytical solutions of membrane and bending plate problem for anisotropymaterial 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.

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][8][9].
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][18][19][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][23][24][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.The proposed shell element of anisotropy has high accuracy and computational efficiency.

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][22][23][24][25][26][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 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.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.

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 [ S] is matrix of the analytical trial functions, whose dimension is 3 × . will be determined by the number of trial functions involved in the element.{ β} is the vector of the unknowns of the inner parameters.The modified complementary energy can be presented as [21] Π = Π − HPC , where the complementary energy Π is given by The constitutive matrix [ D] is Substituting ( 7) into (9) yields where The additional complementary energy HPC is given by in which   and   are the boundary forces along the side Γ around the element, determined by the inner stress field in (7): Mathematical Problems in Engineering 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 ũ and ṽ along the side  − can be obtained from the displacements of ũ and ũ 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   = 0 and   = 1.
() are the Hermite functions, which have the expressions as follows: () is the linear shape function Involving the relationship between displacement ũ /ũ  and displacement ũ/Ṽ, the conforming displacement along the side  − can be defined as It can also be denoted as where [ Ñ] is the matrix of the trial functions of the displacements defined in side  − as And {q  } is part of the unknown nodal displacement vector, Substituting ( 14) and ( 22) into (13), we have where Thus the modified complementary energy Π can be presented as With the principle of the modified complementary energy,  Π = 0, From the first part of (28), the relationship between the inner parameters { β} and the nodal displacement {q} can be determined as Substituting ( 29) into (27), 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.

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 [S] is matrix of the analytical trial functions, whose dimension is 5 × . 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 [A] is the constitutive matrix of the anisotropy plate problem and {F} is the vector of the boundary forces along the side of the element; it can be denoted as {d} 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 (37)

Mathematical Problems in Engineering
And it is defined as Equation ( 36) can also be denoted as where [N] is the matrix of the trial functions of the displacements defined in side  − as And {q  } is part of the unknown nodal displacement vector: Following the hybrid element strategy proposed in Section 3.1, the modified complementary energy Π 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.

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 [ S] and, [S] of the stress trial functions contain the variables  and , which can be written as while the matrix [ Ñ] and matrix [N] 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 [ L] and [L] 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 [K] 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 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   = 1 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   = 1 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.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  = 1 Pa.Table 5 gives the deflection   of the central point  in different mesh schemes.

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  = 1 N. Table 6 gives the deflection of the central point  in different mesh schemes.

Conclusions
This paper presents a novel way to formulate the flat shell element of anisotropy.The basic analytical solutions and the and ATF-TP7.The high accuracy and computational efficiency of these elements are proved by the benchmark.
(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).

Figure 2 :
Figure 2: The conforming displacement along the side of the element.

Table 3 :
Deflection   /10 −6 m of the cantilever shell subjected to   .

Table 4 :
Deflection   /m of the cantilever shell subjected to   .
C Figure 5: Schemes of the mesh.

Table 5 :
Deflections   of the central point .

Table 6 :
Deflections (10 −9 m) of the central point  in different mesh scheme.