# Quadrilateral Area Coordinate Systems, Part II — New Tools for Constructing Quadrilateral Elements

• Song Cen
• Zhi-Fei Long

## Abstract

This chapter focuses on the applications of the quadrilateral area coordinate systems discussed in the pre vious chapter. Here, a series of new quadrilateral elements formulated by the quadrilateral area coordinate methods are introduced in detail. Foll’ owing the Introduction in Sect. 17.1, the sensitivity analysis of the quadri lateral membrane elements to mesh distortion is discussed in Sect. 17.2. Then, a brief review of the construction of various quadrilateral elements formulated by the quadrilateral area coordinate methods is given in Sect. 17.3. In Sects. 17.4 to 17.7 of this chapter, various quadrilateral membrane elements (4-node element, 4-node element with drilling freedoms, 8-node element) for linear and nonlinear analyses, in which the are coordinate methods are adopted, are introduced in detail. And, in the last three sections (Sects. 17.8 to 17.10), the applications of the area coordinate methods for the quadrilateral thin plate, thick plate and composite laminated plate elements are descri bed, respectively. It is demonstrated that the area coordinate methods are efficient tools for developing simple, effective and reliabl e serendipity quadrilateral element models.

## References

1. [1]
Pian THH, Sumihara R (1984) Rational approach for assumed stress finite elements. International Journal for Numerical Methods in Engineering 20:1685–1695
2. [2]
Wilson EL, Ibrahimbegovic A (1990) Use of incompatible displacement modes for the calculation of element stiffness or stresses. Finite Elements in Analysis and Design 7: 229–241
3. [3]
Xu Y, Long ZF, Long YQ (1998) A generalized conforming quadrilateral membrane element insensitive to geometric distortion. Advances in Structural Engineering 1(3): 185–191Google Scholar
4. [4]
Lee NS, Bathe KJ (1993) Effects of element distortions on the performance of isoparametric elements. International Journal for Numerical Methods in Engineering 36: 3553–3576
5. [5]
Cen S, Long ZF, Zhang CS (1998) Two eight-node quadrilateral elements constructed by area coordinates. In: Proceedings of the Seventh National Conference on Structural Engineering (Vol. I). China, Shijia Zhuang, pp 237–241 (in Chinese)Google Scholar
6. [6]
Long YQ, Long ZF, Cen S (2001) Several problems and advances in finite element method. In: Proceedings of the Tenth National Conference on Structural Engineering (Vol. I). China, Nanjing, pp 34–51 (in Chinese)Google Scholar
7. [7]
Chen XM, Cen S, Long YQ, Yao ZH (2004) Membrane elements insensitive to distortion using the quadrilateral area coordinate method. Computers & Structures 82(1): 35–54
8. [8]
Cen S, Du Y, Chen XM, Fu XR (2007) The analytical element stiffness matrix of a recent 4-node membrane element formulated by the Quadrilateral Area Coordinate method. Communications in Numerical Methods in Engineering 23(12): 1095–1110
9. [9]
Du Y, Cen S (2008) Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method. Finite Elements in Analysis and Design 44(8): 427–438
10. [10]
Cen S, Chen XM, Fu XR (2007) Quadrilateral membrane element family formulated by the quadrilateral area coordinate method. Computer Methods in Applied Mechanics and Engineering 196(41–44): 4337–4353
11. [11]
Chen XM, Cen S, Fu XR, Long YQ (2008) A new quadrilateral area coordinate method (QACM-II) for developing quadrilateral finite element models. International Journal for Numerical Methods in Engineering 73(13): 1911–1941
12. [12]
Chen XM, Long YQ, Xu Y (2003) Construction of quadrilateral membrane elements with drilling DOF using area coordinate method. Gong Cheng Li Xue/Engineering Mechanics 20(6): 6–11 (in Chinese)Google Scholar
13. [13]
Soh AK, Long YQ, Cen S (2000) Development of eight-node quadrilateral membrane elements using the area coordinates method. Computational Mechanics 25(4): 376–384
14. [14]
Guan NX, Cen S, Chen XM (2007) Quadrilateral axisymmetric elements formulated by the area coordinate method. In: Yao ZH, Yuan MW (eds) Computational Mechanics (Proceedings of the ISCM 2007). Tsinghua University Press & Springer, China, Beijing, CD Rom pp 1055–1059Google Scholar
15. [15]
Long ZF, Li JX, Cen S, Long YQ (1997) A quadrilateral plate bending element by using area coordinate method. Gong Cheng Li Xue/Engineering Mechanics 14(4): 1–10 (in Chinese)Google Scholar
16. [16]
Soh AK, Long ZF, Cen S (2000) Development of a new quadrilateral thin plate element using area coordinates. Computer Methods in Applied Mechanics and Engineering 190(8–10): 979–987
17. [17]
Cen S, Long YQ, Yao ZH, Chiew SP (2006) Application of the quadrilateral area coordinate method: A new element for Mindlin-Reissner plate. International Journal for Numerical Methods in Engineering 66(1): 1–45
18. [18]
Cen S, Fu XR, Long YQ, Li HG, Yao ZH (2007) Application of the quadrilateral area coordinate method: a new element for laminated composite plate bending problems. Acta Mechanica Sinica 23(5): 561–575
19. [19]
Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. International Journal for Numerical Methods in Engineering 10: 1211–1219
20. [20]
Cen S, Long ZF (1998) A new triangular generalized conforming element for thin-thick plates. Gong Cheng Li Xue/Engineering Mechanics 15(1): 10–22 (in Chinese)Google Scholar
21. [21]
Wilson EL, Taylor RL, Doherty WP, Ghabussi T (1973) Incompatible displacement models. In: Fenven ST et al., eds. Numerical and Computer Methods in Structural Mechanics. Academic Press, New York, 43–57Google Scholar
22. [22]
Chen WJ, Tang LM (1981) Isoparametric quasi-conforming element. Journal of Dalian University of Technology 20(1): 63–74 (in Chinese)Google Scholar
23. [23]
Pian THH, Wu CC (1986) General formulation of incompatible shape function and an incompatible isoparametric element. In: Proceedings of the Invitational China-American Workshop on FEM. China, Chengde, pp 159–165Google Scholar
24. [24]
Piltner R, Taylor RL (1997) A systematic construction of B-bar functions for linear and nonlinear mixed-enhanced finite elements for plane elasticity problems. International Journal for Numerical Methods in Engineering 44: 615–639
25. [25]
Bassayya K, Bhattacharya K, Shrinivasa U (2000) Eight-node brick, PN340, representing constant stress fields exactly. Computers & Structures 74: 441–460
26. [26]
Long YQ, Xu Y (1994) Generalized conforming quadrilateral membrane element with vertex rigid rotational freedom. Computers & Structures 52(4): 749–755
27. [27]
MacMeal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elements in Analysis and Design 1(1): 3–20
28. [28]
Chen WJ, Cheung YK (1992) Three dimensional 8-node and 20-node refined hybrid isoparametric elements. International Journal for Numerical Methods in Engineering 35: 1871–1889
29. [29]
Andelfinger U, Ramm E (1993) EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. International Journal for Numerical Methods in Engineering 36: 1311–1337
30. [30]
MacNeal RH (1987) A theorem regarding the locking of tapered four-noded membrane elements. International Journal for Numerical Methods in Engineering 24: 1793–1799
31. [31]
Wu CC, Jiao ZP (1993) Geometrically nonlinear analyses for 2-D problems based on the incompatible finite elements with internal parameters. Acta Mechanica Sinica 25(4): 505–513 (in Chinese)Google Scholar
32. [32]
ABAQUS Inc. ABAQUS Documentation Version 6.5 (2004) ABAQUS Inc., Rawtucket, Rhode IslandGoogle Scholar
33. [33]
Bisshopp RE, Drucker DC (1945) Large deflection of cantilever beams. Quarterly of Applied Mathematics 3(3): 272–275
34. [34]
Weissmen SL, Taylor RL (1990) Resultant fields for mixed plate bending elements. Computer Methods in Applied Mechanics in Engineering 79: 321–355
35. [35]
Batoz JL, Tahar MB (1982) Evaluation of a new quadrilateral thin plate bending element. International Journal for Numerical Methods in Engineering 18: 1655–1677
36. [36]
Hu HC (1984) Variational principles of theory of elasticity with applications. Science Press, BeijingGoogle Scholar
37. [37]
Soh AK, Cen S, Long YQ, Long ZF (2001) A new twelve DOF quadrilateral element for analysis of thick and thin plates. European Journal of Mechanics A/Solids 20(2): 299–326
38. [38]
Kant T, Hinton E (1983) Mindlin plate analysis by the segmentation method. Journal of Engineering Mechanics, Div. ASCE, 109: 537–556
39. [39]
Lardeur P, Batoz JL (1989) Composite plate analysis using a new discrete shear triangular finite element. International Journal for Numerical Methods in Engineering 27: 343–359
40. [40]
Wu CC, Pian THH (1997) Incompatible numerical analysis and hybrid finite element method. Science Press, Beijing (in Chinese)Google Scholar
41. [41]
Pagano NJ, Hatfield SJ (1972) Elastic behavior of multilayered bidirectional composites. AIAA Journal 10: 931–933
42. [42]
Reddy JN (1997) Mechanics of laminated composite plates — theory and analysis. CRC Press, Boca Raton

© Tsinghua University Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg 2009

## Authors and Affiliations

• Song Cen
• 1
• Zhi-Fei Long
• 2
1. 1.Department of Engineering Mechanics, School of AerospaceTsinghua UniversityBeijingChina
2. 2.School of Mechanics & Civil EngineeringChina University of Mining & TechnologyBeijingChina