Bending of Super-Elliptical Mindlin Plates by Finite Element Method

Year 2018, Volume: 29 Issue: 4, 8469 - 8496, 01.07.2018

Murat Altekin

https://doi.org/10.18400/tekderg.332384

Cited By: 6

Abstract

Bending of shear deformable super-elliptical
plates under transverse load was investigated using the Mindlin plate theory by
means of the finite element method. Four-noded isoparametric quadrilateral
plate bending element with three degrees of freedom per node was used.
Parametric results for the maximum deflections were presented via sensitivity
analysis for several geometric characteristics such as thickness, aspect ratio,
and super-elliptical power. Good agreement with the solutions of elliptical and
rectangular plates was obtained using fine mesh. The results revealed that the
deflections of clamped and point supported super-elliptical plates lie in the
range bounded by elliptical and rectangular plates. However, the bending
response of simply supported plates was observed to be entirely different. It
was shown that high rate of convergence is required to obtain such a relation
and using insufficient number of degrees of freedom results in finding a
totally different trend for the clamped case.

Keywords

Plate, bending, deflection, finite element

Bending of Super-Elliptical Mindlin Plates by Finite Element Method

Abstract

Kayma şekil değiştirmesi yapabilen süper-eliptik plakların transvers yük altında eğilmesi sonlu eleman yöntemiyle incelendi. Her düğüm noktasında üç serbestlik derecesine sahip dört düğüm noktalı izoparametrik dörtgen plak eğilme elemanı kullanıldı.Duyarlık analizi yapılarak kalınlık, en-boy oranı, süper-eliptik üs gibi geometrik özellikler için en büyük çökme değerleri parametrik olarak belirlendi. Ankastre ve nokta mesnetli süper-eliptik plakların çökmesinin eliptik ve dikdörtgen plakların arasında olduğu ortaya kondu. Bununla birlikte basit mesnetli plaklarda durumun tamamen farklı olduğu gözlemlendi. Bu ilişkilerin belirlenmesinde yüksek yakınsama elde edilmesinin gerektiği gösterildi ve yetersiz sayıda serbestlik tanıtılması durumunda ankastre mesnetli plaklar için aynı davranışın bulunamadığı gösterildi. 

Keywords

plak, çökme, eğilme, sonlu eleman

References

• [1] Reddy, J.N., Chao, W.C., Large-deflection and large-amplitude free vibrations of laminated composite-material plates, Computers and Structures, 13 (1-3), 341-347, 1981. [2] Mbakogu, F.C., Pavlovic, M.N., Bending of clamped orthotropic rectangular plates: a variational symbolic solution, Computers and Structures, 77 (2), 117-128, 2000. [3] Bayer, I., Guven, U., Altay, G., A parametric study on vibrating clamped elliptical plates with variable thickness, Journal of Sound and Vibration, 254 (1), 179-188, 2002. [4] Ozkul, T.A., Ture, U., The transition from thin plates to moderately thick plates by using finite element analysis and the shear locking problem, Thin-Walled Structures, 42 (10), 1405-1430, 2004. [5] Setoodeh, A.R., Karami, G., Static, free vibration and buckling analysis of anisotropic thick laminated composite plates on distributed and point elastic supports using a 3-D layer-wise FEM, Engineering Structures, 26 (2), 211-220, 2004. [6] Algazin, S.D., Vibrations of a free-edge variable-thickness plate of arbitrary shape in plan, Journal of Applied Mechanics and Technical Physics, 52 (1), 126-131, 2011. [7] Cai, Y.C., Tian, L.G., Atluri, S.N., A simple locking-free discrete shear triangular plate element, CMES, 77 (4), 221-238, 2011. [8] Kutlu, A., Omurtag, M.H., Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method, International Journal of Mechanical Sciences, 65 (1), 64-74, 2012. [9] Sapountzakis, E.J., Dikaros, I.C., Large deflection analysis of plates stiffened by parallel beams, Engineering Structures, 35, 254-271, 2012. [10] Thai, H.-T., Choi, D.-H., Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates, Applied Mathematical Modelling, 37 (18-19), 8310-8323, 2013. [11] Rao, L.B., Rao, C.K., Buckling of annular plates with elastically restrained external and internal edges, Mechanics Based Design of Structures and Machines, 41 (2), 222- 235, 2013. [12] Sanusei, S., Mazhari, E., Shahidi, A., Analysis of buckling behavior of elliptical plate with non-concentric elliptic hole, International Journal of Materials Engineering and Technology, 13 (1), 81-108, 2015. [13] Szilard, R., Theories and Applications of Plate Analysis, USA, John Wiley & Sons Inc.,2004. [14] Altay, G., Dokmeci, M.C., A polar theory for vibrations of thin elastic shells, International Journal of Solids and Structures, 43 (9), 2578–2601, 2006. [15] Lee, S.L., Ballesteros, P., Uniformly loaded rectangular plate supported at the corners, International Journal of Mechanical Sciences, 2 (3), 206-211, 1960. [16] Szilard, R., Theory and Analysis of Plates, Englewood Cliffs, USA, Prentice Hall, 1974. [17] Rajaiah, K., Rao, A.K., Collocation solution for point-supported square plates, Journal of Applied Mechanics, 45 (2), 424-425, 1978. [18] Shanmugam, N.E., Huang, R., Yu, C.H., Lee, S.L., Uniformly loaded rhombic orthotropic plates supported at corners, Computers & Structures, 30 (5), 1037-1045, 1988. [19] Nong, L., Bao-lian, F., The symmetrical bending of an elastic circular plate supported at K internal points, Applied Mathematics and Mechanics, 12 (11), 1091-1096, 1991. [20] Liew, K.M., Han, J.B., Bending analysis of simply supported shear deformable skew plates, Journal of Engineering Mechanics, 123 (3), 214-221, 1997. [21] Han, J.B., Liew, K.M., An eight-node curvilinear differential quadrature formulation for Reissner/Mindlin plates, Computer Methods in Applied Mechanics and Engineering, 141 (3-4), 265-280, 1997a. [22] Han, J.B., Liew, K.M., Analysis of moderately thick circular plates using differential quadrature method, Computer Methods in Applied Mechanics and Engineering, 123 (12), 1247-1252, 1997b. [23] Wang, C.M., Lim, G.T., Bending solutions of sectorial Mindlin plates from Kirchhoff plates, Journal of Engineering Mechanics, 126 (4), 367-372, 2000. [24] Wang, C.M., Lim, G.T., Reddy, J.N., Lee, K.H., Relationships between bending solutions of Reissner and Mindlin plate theories, Engineering Structures, 23 (7), 838- 849, 2001. [25] Wang, C.M., Wang, Y.C., Reddy, J.N., Problems and remedy for the Ritz method in determining stress resultants of corner supported rectangular plates, Computers and Structures, 80 (2), 145-154, 2002. [26] Lim, G.T., Reddy, J.N., On canonical bending relationships for plates, International Journal of Solids and Structures, 40 (12), 3039-3067, 2003. [27] Reddy, J.N., Theory and Analysis of Elastic Plates and Shells, Second Edition, Boca Raton, CRC Press, 2007. [28] Lim, C.W., Yao, W.A., Cui, S., Benchmark symplectic solutions for bending of corner- supported rectangular thin plates, The IES Journal Part A: Civil & Structural Engineering, 1 (2), 106-115, 2008. [29] Civalek, O., Ersoy, H., Free vibration and bending analysis of circular Mindlin plates using singular convolution method, Communications in Numerical Methods in Engineering, 25 (8), 907-922, 2009. [30] Batista, M., An elementary derivation of basic equations of the Reissner and Mindlin plate theories, Engineering Structures, 32 (3), 906-909, 2010a. [31] Batista, M., New analytical solution for bending problem of uniformly loaded rectangular plate supported on corners, The IES Journal Part A: Civil & Structural Engineering, 3 (2), 75-84, 2010b. [32] Nguyen-Xuan, H., Tran, L.V., Thai, C.H., Nguyen-Thoi, T., Analysis of functionally graded plates by an efficient finite element method with node-based strain smoothing, Thin-Walled Structures, 54, 1-18, 2012. [33] Asemi, K., Ashrafi, H., Salehi, M., Shariyat, M., Three-dimensional static and dynamic analysis of functionally graded elliptical plates, employing graded finite elements, Acta Mechanica, 224 (8), 1849-1864, 2013. [34] Wang, C. Y., Vibrations of completely free rounded rectangular plates, Journal of Vibration and Acoustics, 137 (2), doi:10.1115/1.4029159, 2015a. [35] Wang, C. Y., Vibrations of completely free rounded regular polygonal plates, International Journal of Acoustics and Vibration, 20 (2), 107-112, 2015b. [36] Li, R., Wang, P., Tian, Y., Wang, B., Li, G., A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates, Scientific Reports DOI: 10.1038/srep17054, 2016. [37] Falsone, G., Settineri, D., A Kirchhoff-like solution for the Mindlin plate model: A new finite element Approach, Mechanics Research Communications, 40, 1– 10, 2012. [38] Endo, M., Study on an alternative deformation concept for the Timoshenko beam and Mindlin plate models, International Journal of Engineering Science, 87, 32–46, 2015. [39] Li, R., Wang, B., Li G., Benchmark bending solutions of rectangular thin plates point- supported at two adjacent corners, Applied Mathematics Letters, 40, 53-58, 2015. [40] Liew, K.M., Kitipornchai, S., Lim, C.W., Free vibration analysis of thick superelliptical plates, Journal of Engineering Mechanics, 124 (2), 137-145, 1998. [41] DeCapua, N.J., Sun, B.C., Transverse vibration of a class of orthotropic plates, Journal of Applied Mechanics, 39 (2), 613-615, 1972. [42] Irie, T., Yamada, G., Sonoda, M., Natural frequencies of square membrane and square plate with rounded corners, Journal of Sound and Vibration, 86 (3), 442-448, 1983. [43] Wang, C.M., Wang, L., Vibration and buckling of super elliptical plates. Journal of Sound and Vibration, 171 (3), 301-314, 1994. [44] Lim, C.W., Liew, K.M., Vibrations of perforated plates with rounded corners, Journal of Engineering Mechanics, 121 (2), 203-213, 1995. [45] Lim, C.W., Kitipornchai, S., Liew, K.M., A free-vibration analysis of doubly connected super-elliptical laminated composite plates, Composites Science and Technology, 58 (3-4), 435-445, 1998. [46] Chen, C.C., Lim, C.W., Kitipornchai, S., Liew, K.M., Vibration of symmetrically laminated thick super elliptical plates, Journal of Sound and Vibration, 220 (4), 659- 682, 1999. [47] Chen, C.C., Kitipornchai, S.,Free vibration of symmetrically laminated thick perforated plates, Journal of Sound and Vibration, 230 (1), 111-132, 2000. [48] Liew, K.M., Feng, Z.C., Three-dimensional free vibration analysis of perforated superelliptical plates via the p-Ritz method, International Journal of Mechanical Sciences, 43 (11), 2613-2630, 2001. [49] Zhou, D., Lo, S.H., Cheung, Y.K., Au, F.T.K., 3-D vibration analysis of generalized super elliptical plates using Chebyshev-Ritz method, International Journal of Solids and Structures, 41 (16-17), 4697-4712, 2004. [50] Wu, L., Liu, J., Free vibration analysis of arbitrary shaped thick plates by differential cubature method. International Journal of Mechanical Sciences, 47 (1), 63-81, 2005. [51] Altekin, M., Free linear vibration and buckling of super-elliptical plates resting on symmetrically distributed point-supports on the diagonals, Thin-Walled Structures, 46 (10), 1066-1086, 2008. [52] Altekin, M., Free vibration of orthotropic super-elliptical plates on intermediate supports, Nuclear Engineering and Design, 239 (6), 981-999, 2009. [53] Altekin, M., Free in-plane vibration of super-elliptical plates, Shock and Vibration, 18 (3), 471-484, 2010b. [54] Altekin, M., Free transverse vibration of shear deformable super-elliptical plates, Wind and Structures, 24 (4), 307-331, 2017. [55] Ceribasi, S., Altay, G., Free vibration of super elliptical plates with constant and variable thickness by Ritz method, Journal of Sound and Vibration, 319 (1-2), 668- 680, 2009. [56] Jazi, S.R., Farhatnia, F., Buckling analysis of functionally graded super elliptical plate using pb-2 Ritz method, Advanced Materials Research, 383-390, 5387-5391, 2012. [57] Zhang, D.G., Zhou, H.M., Nonlinear symmetric free vibration analysis of super elliptical isotropic thin plates, CMC: Computers, Materials & Continua, 40 (1), 21-34, 2014. (the author does not have access to this paper) [58] Hasheminejad, S.M., Keshvari, M.M., Ashory, M.R., Dynamic stability of super elliptical plates resting on elastic foundations under periodic in-plane loads, Journal of Engineering Mechanics, 140 (1), 172-181, 2014. [59] Ghaheri, A., Nosier, A., Keshmiri, A., Parametric stability of symmetrically laminated composite super-elliptical plates, Journal of Composite Materials DOI: 10.1177/0021998316629481, 2016. [60] Altekin, M., Altay, G., Static analysis of point-supported super-elliptical plates, Archive of Applied Mechanics, 78 (4), 259-266, 2008. [61] Ceribasi, S., Altay, G., Dökmeci, M.C., Static analysis of super elliptical clamped plates by Galerkin’s method, Thin-Walled Structures, 46 (2), 122-127, 2008. [62] Tang, H.W., Yang, Y.T., Chen, C.K., Application of new double side approach method to the solution of super-elliptical plate problems, Acta Mechanica, 223 (4), 745-753, 2012. [63] Ceribasi, S., Static and dynamic analysis of thin uniformly loaded super elliptical FGM plates, Mechanics of Advanced Materials and Structures, 19 (5), 325-335, 2012. [64] Zhang, D.G., Non-linear bending analysis of super-elliptical thin plates, International Journal of Non-Linear Mechanics, 55, 180-185, 2013. [65] Altekin, M., Bending of orthotropic super-elliptical plates on intermediate point supports, Ocean Engineering, 37 (11-12), 1048-1060, 2010a. [66] Altekin, M., Large deflection analysis of point supported super-elliptical plates, Structural Engineering and Mechanics, 51 (2), 333-347, 2014. [67] Tang, H.W., Lo, C.Y., Application of double side approach method on super elliptical Winkler plate, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 8 (8), 1472-1476, 2014. [68] Altunsaray, E., Static deflections of symmetrically laminated quasi-isotropic super- elliptical thin plates, Ocean Engineering, 141, 337-350, 2017. [69] Reddy, J.N., An Introduction to the Finite Element Method, Second Edition, Singapore, McGraw-Hill International editions, 1993. [70] Hughes, T.J.R., Taylor, R.L., Kanoknukulchai, W., A simple and efficient finite element for plate bending, International Journal for Numerical Methods in Engineering, 11 (10), 1529-1543, 1977. [71] Krishnamoorthy, C.S., Finite Element Analysis: Theory and Programming, (Second Edition), New Delhi, Tata McGraw-Hill Publishing Company Limited, 1994. [72] Timoshenko, S. P., Woinowsky-Krieger, S., Theory of Plates and Shells, Singapore, McGraw-Hill International Editions, 1959.