New Method for Large Deflection Analysis of an Elliptic Plate Weakened by an Eccentric Circular Hole

Document Type : Research Paper


1 Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran

2 Department of Mechanical Engineering, Damavand Branch, Islamic Azad University, Damavand, Iran


The bending analysis of moderately thick elliptic plates weakened by an eccentric circular hole has been investigated in this article. The nonlinear governing equations have been presented by considering the von-Karman assumptions and the first-order shear deformation theory in cylindrical coordinates system. Semi-analytical polynomial method (SAPM) which had been presented by the author before has been used. By applying SAPM method, the nonlinear partial differential equations have been transformed to the nonlinear algebraic equations system. Then, the nonlinear algebraic equations have been solved by using Newton–Raphson method. The obtained results of this study have been compared with the results of other references and the accuracy of the results has been shown. The effect of some important parameters on the results such as the location of the circular hole, the ratio of major to minor radiuses of elliptical plate, the size of circular hole and boundary conditions have been studied. It is concluded that applying the presented method is very convenient and efficient. So, it can be used for analyzing the mechanical behavior of elliptical plates, instead of relatively complicated formulations in elliptic coordinates system.


[1] Dastjerdi Sh., Jabbarzadeh M., 2016, Non-local thermo-elastic buckling analysis of multi-layer annular/circular nano-plates based on first and third order shear deformation theories using DQ method, Journal of Solid Mechanics 8(4): 859-874.
[2] Dastjerdi Sh., Jabbarzadeh M., 2016, Nonlocal bending analysis of bilayer annular/circular nano plates based on first order shear deformation theory, Journal of Solid Mechanics 8(3): 645-661.
[3] Sato K., 2006, Bending of an elliptical plate on elastic foundation and under the combined action of lateral load and in-plane force, III European Conference on Computational Mechanics.
[4] Datta S., 1976, Large deflections of elliptic plates exhibiting rectilinear orthotropy and placed on elastic-foundation, Journal of Applied Mechanics 43(4): 690-692.
[5] Kutlu A., Hakkı Omurtag M., 2012, 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.
[6] Zhong H., Li X., He Y., 2005, Static flexural analysis of elliptic Reissner–Mindlin plates on a Pasternak foundation by the triangular differential quadrature method, Archive of Applied Mechanics 74(10): 679-691.
[7] Parnes R., 1989, Bending of simply-supported elliptic plates: B.P.M. solutions with second-order derivative boundary conditions, Journal of Applied Mechanics 56(2): 356-363.
[8] Parnes R., 1988, A BPM solution for elliptical plates subjected to eccentric loads, International Journal of Solids and Structures 24(8): 761-776.
[9] Wang Z.Q., Jiang J., Tang B.T., Zheng W., 2014, Numerical solution of bending problem for elliptical plate using differentiation matrix method based on Barycentric Lagrange interpolation, Applied Mechanics and Materials 638: 1720-1724.
[10] Hsieh M.C., Hwu Ch., 2002, Bending of an anisotropic plate weakened by an elliptical hole, The 3rd Asian-Australasian Conference on Composite Materials, New Zealand.
[11] Leissa A.W., 1967, Vibration of a simply-supported elliptical plate, Journal of Sound and Vibration 6(1): 145-148.
[12] Leissa A.W., 1969, Vibration of Plates, Washington, Office of Technology Utilization, SP-160, NASA.
[13] Leissa A.W., 1978, Recent research in plate vibrations: Classical theory, The Shock and Vibration Digest 9: 13-24.
[14] Leissa A.W., 1978, Recent research in plate vibrations: Complicating effects, The Shock and Vibration Digest 9: 21-35.
[15] Leissa A.W., 1981, Plate vibration research, 1976-1980: Classical theory, The Shock and Vibration Digest 13: 11-22.
[16] Leissa A.W., 1981, Plate vibration research: Classical theory, The Shock and Vibration Digest 13: 19-36.
[17] Leissa A.W., 1987, Recent studies in plate vibrations: part I, Classical theory, The Shock and Vibration Digest 19: 11-18.
[18] Leissa A.W., 1987, Recent studies in plate vibrations: part II, Complicating effects, The Shock and Vibration Digest 19: 10-24.
[19] Wang C.M., Wang L., Liew K.M., 1994, Vibration and buckling of super elliptical plates, Journal of Sound and Vibration 171(3): 301-314.
[20] Altekin M., Altay G., 2008, Static analysis of point-supported super-elliptical plates, Archive of Applied Mechanics 78(4): 259-266.
[21] Altekin M., 2008, 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.
[22] Altekin M., 2009, Free vibration of orthotropic super-elliptical plates on intermediate supports, Nuclear Engineering and Design 239(6): 981-999.
[23] Altekin M., 2010, Bending of orthotropic super-elliptical plates on intermediate point supports, Ocean Engineering 37(11): 1048-1060.
[24] Laura P.A., Rossit C., 1998, Thermal bending of thin, anisotropic, clamped elliptic plates, Ocean Engineering 26(5): 485-488.
[25] Zhang D.G., 2013, Non-linear bending analysis of super elliptical thin plates, International Journal of Non-Linear Mechanics 55: 180-185.
[26] Mc Nitt R.P., 1963, Free vibration of a damped semi-elliptical plate and a quarter-elliptical plate, AIAA Journal 29(9): 1124-1125.
[27] Hasheminejad S.M., Vaezian S., 2014, Free vibration analysis of an elliptical plate with eccentric elliptical cut-outs, Meccanica 49: 37-50.
[28] Irie T., Yamada G., 1979, Free vibration of an orthotropic elliptical plate with a similar hole, Bulletin of JSME 22: 1456-1462.
[29] Ghaheri A., Keshmiri A., Taheri-Behrooz F., 2014, Buckling and vibration of symmetrically laminated composite elliptical plates on an elastic foundation subjected to uniform in-plane force, Journal of Engineering Mechanics 140 (7).
[30] Biswas P., 1975, Large deflection of a heated elliptical plate under stationary temperature, Defense Science Journal 26: 41-46.
[31] Dastjerdi Sh., Lotfi M., Jabbarzadeh M., 2016, The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory, Composites Part B 98: 78-87.
[32] Guminiak M., Szajek K., 2014, Static analysis of circular and elliptic plates resting on internal flexible supports by the boundary element method, Journal of Applied Mathematics and Computational Mechanics 13(2): 21-32.