Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 486453, 9 pages
http://dx.doi.org/10.1155/2013/486453
Research Article

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

1Department of Civil Engineering, China Agricultural University, Beijing 100083, China
2Department 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 [115].

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 [79].

Another challenge is the shear locking of some bending plate elements based on the Mindlin/Reissner thick plate theory [1015]. 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 [1115], 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 [1720]. 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 [2225]. 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 [2127]. 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.

tab1
Table 1: The fundamental stress solutions of the anisotropic plane problem.
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.

tab2
Table 2: The fundamental stress solutions of anisotropic material used in bending plate problem.

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 .

486453.fig.001
Figure 1: Normal direction and tangential direction.

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

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.

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:

486453.fig.002
Figure 2: The conforming displacement along the side of the element.

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.

486453.fig.003
Figure 3: Cantilever beam subjected to force/moment.
fig4
Figure 4: Schemes of the mesh.

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.

tab3
Table 3: Deflection /10−6 m of the cantilever shell subjected to .

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.

tab4
Table 4: Deflection /m of the cantilever shell subjected to .
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.

tab5
Table 5: Deflections of the central point C.
486453.fig.005
Figure 5: Schemes of the mesh.
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.

tab6
Table 6: Deflections (10−9 m) of the central point C in different mesh scheme.
486453.fig.006
Figure 6: Cook’s skew shell.

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).

References

  1. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method for Solid and Structural Mechanics, Elsevier Butterworth-Heinnemann, Oxford, UK, 6th edition, 2005. View at Zentralblatt MATH
  2. Y. Q. Long, S. Cen, and Z. F. Long, Advanced Finite Element Method in Structural Engineering, Springer, Berlin, Germany, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G. P. Bazeley, Y. K. Cheung, B. M. Irons, and O. C. Zienkiewicz, “Triangular element in plate bending-conforming and nonconforming solutions,” in Proceedings of the 1st Conference on Matrix Methods in Structural Mechanics, pp. 547–576, Wright Patterson Air Force Base, Dayton, Ohio, USA, 1965.
  4. M. D. Olson and T. W. Bearden, “A simple flat shell element revisited,” International Journal for Numerical. Methods in Engineering, vol. 11, pp. 1041–1043, 1979. View at Google Scholar
  5. G. A. Mohr, “Finite element formulation by nested interpolations: application to drilling freedom problem,” Computers & Structures, vol. 19, pp. 1–8, 1984. View at Google Scholar
  6. D. J. Allman, “A compatible triangular element including vertex rotations for plane elasticity analysis,” Computers and Structures, vol. 19, no. 1-2, pp. 1–8, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. R. D. Cook, “Modified formulations for nine-D.O.F. plane triangles that include vertex rotations,” International Journal for Numerical Methods in Engineering, vol. 31, no. 5, pp. 825–835, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. R. H. MacNeal and R. L. Harder, “A refined four-noded membrane element with rotational degrees of freedom,” Computers and Structures, vol. 28, no. 1, pp. 75–84, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. A. Ibrahimbegovic and E. L. Wilson, “A unified formulation for triangular and quadrilateral flat shell finite elements with six nodal degrees of freedom,” Communications in Applied Numerical Methods, vol. 7, no. 1, pp. 1–9, 1991. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. A. Adini and R. W. Clough, “Anslysis of plate bending by the finite element method,” Tech. Rep. Grant G7337, The National Science Foundation, Washington, DC, USA, 1960. View at Google Scholar
  11. Z. O. Zienkiewicz, T. R. Taylor, and T. J. Too, “Reduced integration technique in general analysis of plates and shells,” International Journal for Numerical Methods in Engineering, vol. 3, pp. 275–290, 1971. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. T. J. Hughes, R. L. Taylor, and W. Kanoknukulchai, “Simple and efficient finite element for plate bending,” International Journal for Numerical Methods in Engineering, vol. 11, no. 10, pp. 1529–1543, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. E. Hinton and H. C. Huang, “A family of quadrilateral Mindlin plate elements with substitute shear strain fields,” Computers and Structures, vol. 23, no. 3, pp. 409–431, 1986. View at Publisher · View at Google Scholar · View at Scopus
  14. T. Belytschko, C. S. Tsay, and W. K. Liu, “A stabilization matrix for the bilinear mindlin plate element,” Computer Methods in Applied Mechanics and Engineering, vol. 29, no. 3, pp. 313–327, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. K. J. Bathe and E. N. Dvorkin, “Short communication: a four-node plate bending element based on Mindlin/reissner plate theory and a mixed interpolation,” International Journal for Numerical Methods in Engineering, vol. 21, no. 2, pp. 367–383, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. Y. Q. Long and J. Q. Zhao, “A new generalized conforming triangular element for thin plates,” Communications in Applied Numerical Methods, vol. 4, no. 6, pp. 781–792, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. Y. Q. Long and K. G. Xin, “A universal method for including shear deformation in thin plates,” International Journal For Numerical Methods in Engineering, vol. 34, pp. 171–177, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. S. Cen, X. R. Fu, Y. Q. Long, H. G. Li, and Z. H. Yao, “Application of the quadrilateral area coordinate method: a new element for laminated composite plate bending problems,” Acta Mechanica Sinica/Lixue Xuebao, vol. 23, no. 5, pp. 561–575, 2007. View at Publisher · View at Google Scholar · View at Scopus
  19. S. Cen, Y.-Q. Long, Z.-H. Yao, and S.-P. Chiew, “Application of the quadrilateral area co-ordinate method: a new element for Mindlin-Reissner plate,” International Journal for Numerical Methods in Engineering, vol. 66, no. 1, pp. 1–45, 2006. View at Publisher · View at Google Scholar · View at Scopus
  20. X. R. Fu, Y. Q. Long, M. W. Yuan, and S. Cen, “A generalized conforming plane element with internal parameters based on analytical trial functions,” Gongcheng Lixue/Engineering Mechanics, vol. 23, no. 1, pp. 1–5, 2006 (Chinese). View at Google Scholar · View at Scopus
  21. X.-R. Fu, S. Cen, C. F. Li, and X.-M. Chen, “Analytical trial function method for development of new 8-node plane element based on the variational principle containing airy stress function,” Engineering Computations, vol. 27, no. 4, pp. 442–463, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. S. Cen, X.-R. Fu, and M.-J. Zhou, “8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 29–32, pp. 2321–2336, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. S. Cen, M.-J. Zhou, and X.-R. Fu, “A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions,” Computers and Structures, vol. 89, no. 5-6, pp. 517–528, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. S. R. Cen, X. H. Fu, G. Zhou, M. J. Zhou, and C. F. Li, “Shape-free finite element method: the plane hybrid stress-function (HS-F) element method for anisotropic materials,” Science China Physics, Mechanics and Astronomy, vol. 54, no. 4, pp. 653–665, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. S. Cen, G. H. Zhou, and X. R. Fu, “A shape-free 8-node plane element unsymmetric analytical trial function method,” International Journal for Numerical Methods in Engineering, vol. 91, pp. 158–185, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. R. W. Clough, “The finite element method in plane stress analysis,” in Proceedings of 2nd ASCE Conference on Electronic Computation, vol. 9, pp. 8–9, Pittsburgh, Pa, USA, 1960.
  27. T. H. H. Pian, “Derivation of element stiffness matrices by assumed stress distributions,” AIAA Journal, vol. 2, no. 7, pp. 1333–1336, 1964. View at Publisher · View at Google Scholar
  28. X. R. Fu, M. R. Yuan, S. Cen, and G. Tian, “Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity,” Applied Mathematics and Mechanics, vol. 33, no. 10, pp. 1253–1264, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. W. J. Wang, G. Tian, X. R. Fu, and Y. Zhao, “A strategy for formulating the fundamental analytical solutions by solving the characteristic equations of plane problems with anisotropic materials,” Journal of Engineering Mechanics, vol. 29, no. 9, pp. 11–16, 2012 (Chinese). View at Google Scholar
  30. R. D. Cook, D. S. Malkus, and M. E. Plesha, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York, NY, USA, 1989.