Keywords: geometrically non-linear vibration, coupled transverse-in-plane displacements, rectangular plate.

In a previous series of papers [1,2], the geometrically non-linear free vibrations of isotropic thin rectangular plates with fully clamped edges, undergoing large amplitude flexural deflections, have been studied using Hamilton's principle and spectral analysis. In the previous formulation, called in what follows the "- formulation", the in-plane displacements intervening in the Von Kármán relations have been neglected, consequently, the vibration parameters have been over-estimated, and the associated membrane stresses could not be reasonably quantified.

The objective of this paper is to generalize and extend the model for non-linear free vibrations of a fully clamped thin isotropic rectangular plate developed previously in [1] in order to examine qualitatively and quantitatively the coupling between the transverse and the membrane displacements in the non-linear range. Three coupled sets of non-linear algebraic equations have been obtained, which reduces, once the in-plane inertia is neglected, to one set of non-linear algebraic equations in term of the contribution coefficient of the transverse displacement only. This set, which is similar to that obtained in [1,2,3], involves a new general term of the fourth non-linear rigidity tensor , expression of which depends henceforth on the two order tensors , and , and the three and due to the in-plane displacements, in addition to the n-linear fourth order tensor related to the transverse displacement.

Judicious choice of admissible and compatible basic functions for a fully clamped-immovable rectangular plate has been made and both iterative and explicit analytical method have been employed to solve the amplitude equations of motion in order to establish the validity of the present -- formulation through comparisons of the numerical analytical results obtained here with those found in the published literature [4].

Hence, accurate estimates of the fundamental non-linear natural frequency have been obtained for a wide range of vibration amplitudes. The fundamental amplitude dependent non-linear mode including both axial displacement and and the transverse displacements has been determined, and the associated non-linear membrane and bending stress distribution have also been presented. The latter results showed that the in-plane membrane stresses have a large contribution to the total maximum surface stresses when large vibration amplitudes occur, since its maximum exceed 30% of the maximum total stress at vibration amplitudes equal to about twice the plate thickness. Consequently, they cannot be neglected in the engineering design of large deflected structures

1

R. Benamar, M.M.K. Bennouna, R.G. White, "The effects of large vibration amplitudes on the fundamental mode shape of thin elastic structures, part I: Simply supported and clamped-clamped beams", Journal of Sound and Vibration 149, 179-195, 1991. doi:10.1016/0022-460X(91)90630-3

2

K. EL Bikri, R. Benamar, M.M. Bennouna , "Geometrically non-linear free vibrations of clamped simply supported rectangular plates. Part I: the effects of large vibration amplitudes on the fundamental mode shape", Computers & Structures, 81, 2029-2043, 2003. doi:10.1016/S0045-7949(03)00152-4

3

M. Haterbouch, R. Benamar, "The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part II: iterative and explicit analytical solution for non-linear coupled transverse and in-plane vibrations", (inpress), Journal of Sound and Vibration, 2003. doi:10.1016/j.jsv.2003.08.039

4

W. Han, M. Petyt, "Geometrically non-linear vibration analysis of thin, rectangular plates using the hierarchical finite element method-I: The fundamental mode of isotropic plates", Computers & Structures, 63, 295-308, 1997. doi:10.1016/S0045-7949(96)00345-8

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