Based on known theoretical developments in linear dynamics of homogeneous beams, two homogenization
approaches of composite beams are developed further to an anterior work using two equivalent properties: the physical and the mechanic–geometrical properties. Further to the assumption of Euler–Bernoulli beams, dynamic parameters are needed. Equations of a given beam structure subjected to free un-damped and/or damped vibration are established. The natural frequency responses of the first five modes are obtained from both approaches, and then compared with those obtained from a finite element model approach, taking into account different slenderness and boundary conditions. The result shows good agreement. An extension to the equivalent physical parameter homogenization method using the behaviour law in nonlinear state is presented here. Thus, the homogenization is extended to an elastically equivalent model elaboration for a system having an elastic–plastic and bilinear behaviour. The aim is to use analytical expressions from an elastically equivalent model for a nonlinear system of multilayer beam type to obtain the new corrected dynamic
parameter. Based on the ductility factor method combined with the secant method that uses a substitute structure and secant stiffness to account for nonlinear behaviour, we developed a formula using global 

 

This paper intends to provide an equivalent method for the evaluation of natural frequency of isotropic and orthotropic thin rectangular plates with different restraint conditions. Starting from a simple and a general approximate formula for the frequency, which is the extension of Hearmon’s expressions presented for the fundamental mode; it is shown how to calculate the fundamental mode of isotropic and orthotropic rectangular plates using the proper coefficient values already available in the scientific literature. For the higher modal frequencies, a particular form of Rayleigh’s method is proposed, leading to a simple procedure to calculate the fundamental frequency. In fact the frequency calculation is reduced to the evaluation of the fundamental frequency of a special plate associated with the real one. An extensive finite element investigation was carried out to test the accuracy of this analytical short cut method. In addition a comparative study has shown good agreement between the frequencies responses obtained from both analytical (equivalent method) and finite element solution using ANSYS program. This method allows us to both avoid huge calculations, and produce a fast and simple approximate calculus of free vibration frequencies. This in turn is a necessary part of the preliminary design phase and the general and immediate verification of a construction project to its completion.