Geometrical nonlinear analysis by plane quadrilateral element

Document Type : Article


1 Civil Engineering Department, Ferdowsi University of Mashhad , Mashhad, Khorasan Razavi, Iran .

2 Civil Engineering Department, Ferdowsi University of Mashhad , Mashhad, Khorasan Razavi, Iran


Various corotational schemes for solid, shell, bending plate and beam elements have been proposed so far. Nevertheless, this approach has rarely been utilized for membrane problems. In this paper, a new quadrilateral element will be suggested for solving nonlinear membranes. The simplicity, rapid convergence and high accuracy of the formulation are the three main characteristics of the presented element. It is worth emphasizing; the recommended element can solve structures with irregular geometry and distorted mesh. This element is insensitive to the aspect ratio. In addition, using this element leads to high-accuracy results. Several numerical examples will be tested to prove the high precision of authors' element in coarse distorted meshes with a large aspect ratio.


Main Subjects

1. Felippa, C.A. and Militello, C. \Developments in variational
methods for high performance plate and shell
elements", In Analytical and Computational Models for
Shells, CED 3, Ed. by A.K. Noor, T. Belytschko and
J.C. Simo, pp. 191-216, ASME, New York (1989).
2. Felippa, C.A. \A survey of parametrized variational
principles and applications to computational mechanics",
Invited Chapter in Science and Perspectives
in Mechanics, Ed. by B. Nayroles, J. Etay and D.
Renouard, pp. 1-42, ENS Grenoble, Grenoble, France
(1994), Expanded version in Computer Methods in
Applied Mechanics and Engineering, 113, pp. 109-139
3. Bergan, P.G. and Hanssen, L. \A new approach for
deriving 'good' nite elements", In the Mathematics
of Finite Elements and Applications, Ed. By J.R.
Whiteman, 2, pp. 483-497, Academic Press, London
4. Felippa, C.A. \The extended free formulation of nite
elements in linear elasticity", Journal of Applied Mechanics,
56(3), pp. 609-616 (1989).
5. Bergan, P.G. and Nygard, M.K. \Finite elements
with increased freedom in choosing shape functions",
International Journal for Numerical Methods in Engineering,
20, pp. 643-664 (1984).
6. Park, K.C. and Stanley, G.M. \A curved C0 shell
element based on assumed natural-coordinate strains",
Journal of Applied Mechanics, 53, pp. 278-290 (1986).
7. Militello, C. and Felippa, C.A. \A variational justi-
cation of the assumed natural strain formulation of
nite elements: I. Variational principles", Computers
and Structures, 34, pp. 431-438 (1990).
8. Militello, C. and Felippa, C.A. \The rst ANDES
elements: 9-DOF plate bending triangles", Computer
Methods in Applied Mechanics and Engineering, 93,
pp. 217-246 (1991).
9. Felippa, C.A. and Militello, C. \Membrane triangles
with corner drilling freedoms II. The ANDES element",
Finite Elements Analysis and Design, 12, pp.
189-201 (1992).
10. Felippa, C.A. and Militello, C. \The variational
formulation of high performance nite elements:
parametrized variational principles", Computers and
Structures, 36, pp. 1-11 (1990).
11. Felippa, C.A. \Recent advances in nite element templates",
Chapter 4, In Computational Mechanics for
the Twenty-First Century, Ed. by B.H.V. Topping, pp.
71-98, Saxe-Coburn Publications, Edinburgh (2000).
12. Rezaiee-Pajand, M. and Yaghoobi, M. \Formulating
an e ective generalized four-sided element", European
Journal of Mechanics A/Solids, 36, pp. 141-155
13. Rezaiee-Pajand, M. and Yaghoobi, M. \A free of
parasitic shear strain formulation for plane element",
Research in Civil and Environmental Engineering, 1,
pp. 1-27 (2013).
14. Rezaiee-Pajand, M. and Yaghoobi, M. \A robust triangular
membrane element", Latin American Journal
of Solid and Structures, 11, pp. 2648-2671 (2014).
15. Rezaiee-Pajand, M. and Yaghoobi, M. \Two new
quadrilateral elements based on strain states", Civil
Engineering Infrastructures Journal, 48, pp. 133-156
16. Felippa, C.A. \Supernatural QUAD4: A template
formulation", Comput. Methods Appl. Mech. Engrg.,
195, pp. 5316-5342 (2006).
17. Grover, N., Singh, B.N., and Maiti, D.K. \Analytical
and nite element modeling of laminated composite
and sandwich plates: An assessment of a new shear
deformation theory for free vibration response", International
Journal of Mechanical Sciences, 67, pp. 89-99
18. Mehar, K. and Panda, S.K. \Geometrical nonlinear
free vibration analysis of FG-CNT reinforced composite

at panel under uniform thermal eld", Composite
Structures, 143, pp. 336-346 (2016).
2500 M. Rezaiee-Pajand and M. Yaghoobi/Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 2488{2500
19. Hari Kishore, M.D.V., Singh, B.N., and Pandit, M.K.
\Nonlinear static analysis of smart laminated composite
plate", Aerospace Science and Technology, 15, pp.
224-235 (2011).
20. Mehar, K. and Panda, S.K. \Numerical investigation of
nonlinear thermomechanical de
ection of functionally
graded CNT reinforced doubly curved composite shell
panel under di erent mechanical loads", Composite
Structures, 161, pp. 287-298 (2017).
21. Mehar, K. and Panda, S.K. \Thermoelastic analysis
of FG-CNT reinforced shear deformable composite
plate under various loadings", International Journal of
Computational Methods, 14(1), p. 1750019 (22 pages)
22. Dow, J.O., A Uni ed Approach to the Finite Element
Method and Error Analysis Procedures, Academic
Press (1999).
23. Felippa, C.A. \A template tutorial", K.M. Mathisen,
T. Kvamsdal and K.M. Okstad, Eds., Computational
Mechanics: Theory and Practice, pp. 29-68, CIMNE,
Barcelona (2004).
24. Felippa, C.A. \A study of optimal membrane triangles
with drilling freedoms", Comput. Methods Appl. Mech.
Engrg., 192, pp. 2125-2168 (2003).
25. Battini, J.M. \A non-linear corotational 4-node plane
element", Mechanics Research Communications, 35,
pp. 408-413 (2008).
26. Du, Y. and Cen, S. \Geometrically nonlinear analysis
with a 4-node membrane element formulated by the
quadrilateral area coordinate method", Finite Elements
in Analysis and Design, 44, pp. 427-438 (2008).
27. Wisniewski, K. and Turska, E. \Improved 4-node
Hu-Washizu elements based on skew coordinates",
Computers and Structures, 87, pp. 407-424 (2009).
28. Choi, N., Choo, Y.S., and Lee, B.C. \A hybrid
Tre tz plane elasticity element with drilling degrees of
freedom", Comput. Methods Appl. Mech. Engrg., 195,
pp. 4095-4105 (2006).
29. Kien, N.D. \A Timoshenko beam element for large
displacement analysis of planar beam and frames",
International Journal of Structural Stability and Dynamics,
12(6), p. 1250048 (9 pages) (2012).
30. Yau, J.D. and Yang, Y.B. \Geometrically nonlinear
analysis of planar circular arches based on rigid element
concept-A structural approach", Engineering
Structures, 30, pp. 955-964 (2008).
31. Yang, Y.B. and Kuo, S.R., Theory and Analysis of
Nonlinear Framed Structures, Singapore, Prentice Hall