International Journal of Civil and Structural Engineering Research ISSN 2348-7607 (Online) Vol. 6, Issue 2, pp: (207-226), Month: October 2018 - March 2019, Available at: www.researchpublish.com
Dynamic Response of Rectangular Orthotropic Plates Based on Characteristic Orthogonal Polynomials and Rayleigh-Ritz Model M. E. Ephraim1, T. Ode2, C. A. Ohia3 1, 2 3
Lecturer, Civil Engineering Department, Rivers State University, Port Harcourt, Nigeria PG Student, Civil Engineering Department, Rivers State University, Port Harcourt, Nigeria Corresponding author:3chibuike.ohia@ust.edu.ng
Abstract: In this study, the frequencies and mode shapes of thin rectangular plates have been obtained using the Rayleigh Ritz method and characteristic orthogonal polynomials for plates of various boundary conditions. The maximum and minimum frequencies of 23.95 and 9.43 rad/sec were obtained for reinforced concrete plates with all round rigid fixity for aspect ratios of 0.5 and 1.0 respectively. These results clearly demonstrate the crucial role of aspect ratios and boundary restraints. Plates clamped on two adjacent edges and simply supported on the other two (CCSS) and plates clamped on two opposite edges and simply supported on the other two opposite edges (CSCS) yielded intermediate values of fundamental frequencies with the plate with similar restraints on contiguous sides showing the lesser values. It is noteworthy that plate clamped on opposite side and simply supported on the opposite side gave a maximum frequency almost equal to that for all round clamped plate. Rectangular plates with simply supported edges showed the least values of maximum and minimum of 12.08 and 5.17 rad/sec for aspect ratio of 0.5 and 1.0 respectively. The mode shapes followed a sequence of transverse vibrations along the longitudinal, transverse and diagonal axes of the plates. However, there were variations dependent on aspect ratio and boundary conditions. The output of the proposed model is in close agreement with those of other researchers and standard computer software. Keywords: rayleigh-ritz, characteristic orthogonal polynomials, reinforced concrete, rectangular plates, vibration, ANSYS
1. INTRODUCTION Plates are extensively used in several engineering fields such as civil, military, marine, and aerospace. In Civil Engineering, they are used in bridge decks, offshore platforms and are predominant in multi-storey buildings. Such wide application of plates across various engineering disciplines makes their vibrational behaviour an important consideration in their structural analysis and design. The transverse vibration of plates has been a subject of many excellent books including Timoshenko and Krieger [1] among others. A lot of research papers have also been published especially on the transverse vibration of rectangular isotropic plates much more than for orthotropic plates. However, the governing equations for a plate, represented by higher order differential equations, are rather difficult to solve analytically except for simple loadings and boundary conditions. This informed the need to evolve approximate methods which are easy to use and can be applied to plates with various configurations. Approximate general solutions for the natural frequency of orthotropic plates have been achieved with Rayleigh-Ritz method in combination with characteristic beam functions by Hearmon [2], decomposition techniques by Bazely et al. [3] and Marangoni et al.[4] and Gorman superstition by Yu [5]. Further developments include the use of pseudo-Fourier expansion by Rossi et al. [6] and the finite difference method by Chen [7] and Huang et al. [8]. The finite element method has also been used to study dynamic response of complicated cases of laminated skewed sandwich plates and tapered beams in the works of Chalak et al. [9] and Bayat et al. [10] respectively.
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