Improvement of the Semi-Analytical Method, for Determining the Geometrically Non-Linear Response of Thin Straight Structures. Part I: Application to Clamped–clamped and Simply Supported–clamped Beams

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

“…In order to obtain explicit analytical solutions for the first nonlinear axisymmetric mode shape of the FG circular plates, 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, Benamar, and White 28 and El Kadiri and Benamar 29 for beams and rectangular plates has been developed and adapted for the FG circular plate, which are shown to be in a good agreement with the iterative method.…”

“…(37), necessary to obtain the clamped FG circular plate nonlinear axisymmetric mode shape and associated nonlinear resonant frequencies at large vibration amplitudes by an explicit solution, which may be appropriate for engineering purposes or for further analytical investigations. This explicit solution is obtained by applying and adapting the so-called first formulation developed for many beams and plates cases studied by El Kadiri, Benamar, and White 28 and El Kadiri and Benamar. 29 A comparison is then made between the two solutions-numerical iterative and analytical-in order to determine exactly the range of validity of the last approximate approach.…”

Improvement of the Semi-Analytical Method, for Determining the Geometrically Non-Linear Response of Thin Straight Structures: Part Ii—first and Second Non-Linear Mode Shapes of Fully Clamped Rectangular Plates

“…A comparison between Table 2 and Table 1 shows notable di!erences between contributions. These di!erences can be explained as follows: in the beam cases presented in reference [20], the bases used to expand the displacement function series were those of the linear normal modes of vibration of the beam considered in each case, in which the mass and rigidity matrices are diagonal. In the FCRP case considered here, the basic functions used in the de#ection function expansions have been calculated as products of x-and y-beam functions, as explained in section 3.1.…”

“…This may be attributed, as stated in reference [20], among other reasons, to the fact that the various attempts to describe mathematically the non-linear structural dynamic behaviour which have been developed and presented in the literature still appear somewhat esoteric and di$cult to deal with in practical situations. Also, no equivalence has been found, in the non-linear case, to the powerful and nice description of the linear dynamic behaviour of a structure, in terms of its natural frequencies and natural modes of vibration.…”

“…It was also hoped that such an attempt could provide users with simple formulae, ready to use for analytical and engineering purposes, which would be much more practical than the published tables of data, obtained from the numerical iterative solution of a non-linear algebraic system, which necessitates the use of appropriate software in each case. Part I of this series of papers [20] was concerned with beams, with the objective of developing a practical simple &&multi-mode theory'', based on the linearization of the non-linear algebraic equations, written on the modal basis, in the neighbourhood of each resonance.…”