A Conforming Triangular Plane Element with Rotational Degrees of Freedom

This paper presents a novel way to formulate the triangular plane element with rotational degrees of freedom (RDOF). The linear distribution of rotational displacement is assumed. The conforming displacement along the sides based on the rotational displacement assumption is derived, and the triangular plane element TR3 for isotropic material is formulated. By using the explicit integral formulae of the triangular element, the matrices used in the proposed plane element TR3 are calculated efficiently. The benchmark examples showed thier high accuracy and high efficiency.


Introduction
The triangular plane element is widely used [1][2][3].In order to improve the performance of the triangular plane element, rotational degrees of freedom (RDOF) has been introduced.The other advantage of plane elements with RDOF is that the singularity of global stiffness can be avoided in the analysis of shell structures.Mervyn and Terrence introduced the RDOF to describe the displacement of the flat shell element [4].Mohr constructed the hybrid membrane element with RDOF [5].Allman et al. presented some rational displacement-type plane elements with RDOF [6][7][8][9][10].Many researchers improve accuracy of the membrane element by this way [11][12][13][14][15], and the plane elements with RDOF have been widely used in analysis of shell structure.
Long et al. studied the generalized conforming plane elements [16][17][18][19][20] with RDOF.Taking advantage of the conforming boundary displacement, the nonconforming displacement model can be involved freely by the generalized conforming equations [17][18][19][20].Chen et al. proved that the hybrid element using the balanced stress fields can play the same role in the element formulation as the generalized conforming element introducing the associated generalized conforming equations [20].On the other hand, the hybrid element employed analytical trial functions of stress needs conforming boundary displacement [21][22][23][24][25][26].
This paper studies the triangular plane element based on the rational assumption of the rotation displacement field.There are mainly four steps as follows in Section 2.
(a) The first step assumes the rational distribution of the rotational displacement (in Section 2.1).
(b) The second step gives the displacement field based on the assumed rotational displacement (in Section 2.2).
(c) The third step formulates the stiffness matrix of the element TR3 based on the associated strain fields (in Section 2.3).
(d) The fourth step calculates the stiffness matrix using the explicit integral formulae of the triangular element (in Section 2.4).
In Section 3, some numerical examples are shown as benchmark to study the accuracy and the efficiency of the proposed element model TR3.
In Section 4, some conclusions are given.The rotational displacement field in the triangular element with three nodes can be assumed as follows: where where in which  1 ,  2 , and  3 and  1 ,  2 , and  3 are coordinates of three nodes in the triangular element. 1 ,  2 , and  3 are rotational displacement of three nodes.

The Distribution of the Displacement.
According to the definition of the rotation displacement in the elasticity, the relationship between displacement symbols , V and the rotational displacement   can be denoted as From (1), From ( 4) and ( 5), we can assume where we can give two assumptions, On the other hand, we can define the displacement fields in the triangular element as According to ( 6)-( 8), the nodal conforming requirement can be given as Then, we can find the relationship between nodal displacement symbols  1 ,  2 ,  3 , V 1 , V 2 , and V 3 and inner parameters  0 ,   ,   , V 0 , V  , and V  ; it is a linear translation matrix.By solving the associated equations ( 9), we can get the distribution of the displacement as follows: It can be proved that the assumed displacement is conforming.

The Stiffness Matrix of the Plane Element TR3 with RDOF.
From (10), taking into account the definition of the strain field, we can get the strain matrix about the nodal displacement.It can be denoted as in which Then, the stiffness matrix can be obtained: where

The Stiffness Matrix Using the Explicit Integral Formulae.
The strain matrix of the triangular plane element TR3 with RDOF can be presented in the natural coordinates of the triangular element.The stress trial functions contain the variables  and , which can be written as Then, the matrix [] can be expressed in the natural coordinates  1 ,  2 , and  3 .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 ( 16) in the triangular element, the explicit formula of the proposed elements in this paper can be obtained.

Benchmark of the Proposed Element TR3
3.1.Cantilever Beam.As shown in Figure 1, a slender cantilever beam has the length  of 32 m, the height ℎ of 2 m, and the thickness  of 1 m.It is made of the isotropy material, which has the material parameters  = 768 Pa,  = 0.25.
Figure 2 shows four schemes of the mesh to the proposed cantilever beam.
The load is an in-plane moment   = 1 N⋅m on the end side .Table 1 gives the results of the displacement   obtained by the proposed element TR3.Comparing with other elements, TR3 can give higher precise results.

Cook's Skew Beam.
As showed in Figure 3, the Cook's skew beam [10] is studied.The thickness  of the shell is 1 m.It is also made of the isotropy, whose material parameters  = 1 Pa,  = 0.3333, and is subjected to the uniformly distributed load which has the sum of  = 1 N. Table 2 gives the deflection of the central point  in different mesh scheme.

Conclusion
This paper present a new type of plane element with RDOF: the assumptions of the rotational displacement and the displacement are proved to be useful.The accuracy and the efficiency of the proposed element model TR3 are shown in the benchmark problems.

Table 1 :
Deflection   /m of the cantilever beam subjected to   .

Table 2 :
Deflections (m) of the central point  in different mesh scheme.