Geometrically non-linear transverse vibrations of C–S–S–S and C–S–C–S1 rectangular plates

This paper is concerned with a new improved formulation of the theoretical model previously developed by Benamar et al. based on Hamilton's principle and spectral analysis, for the geometrically non-linear vibrations of thin structures. The problem is reduced to a non-linear algebraic system, the solution of which leads to determination of the amplitude-dependent fundamental non-linear mode shapes, the frequency parameters, and the non-linear stress distributions. The cases of C–S–C–S and C–S–S–S rectangular plates are examined, and the results obtained are in a good qualitative and quantitative agreement with the previous available works, based on various methods. In order to obtain explicit analytical solutions for the first non-linear mode shapes of C–S–C–S RP2 and C–S–S–S RP, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. For beams and fully clamped rectangular plates, has been slightly modified, and adapted to the above cases, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values up to 0.75 and 0.6 for the first non-linear mode shapes of C–S–C–S RP and C–S–S–S RP, respectively.