| The first topic in this research work is about a new methodology for damage detection that allows one to separate the damage accumulated in structural members (in the form of microcracks) from that in the boundaries/joints by interpolating between known exact frequencies. This method is valid for simple continuous systems where exact frequency solutions are available, and its study presents an excellent start for this research in understanding the general behavior of structures with damaged boundaries.;Next, the natural frequencies of a tapered Euler beam with damaged boundaries were obtained, and the stability of uniform and non-uniform columns was investigated by applying the shifted Chebyshev polynomials directly to the governing equation and boundary conditions. The advantages of using these polynomials are in the ease of applying them in a symbolic way, and a relatively small number of polynomials are enough to attain convergence.;In a similar manner, calculating the natural frequencies continuous in systems with damaged and undamaged boundary conditions was carried out numerically through the application of the collocation method, which is basically a method that approximates the derivative of a function at a certain point by a linear weighted sum of the function values at all discrete points in the domain. This suggested method was applied in an efficient way by using the Kronecker product operator, which makes it easier to express the governing equations in matrix vector form, and thus simply obtaining the eigenvalues which represents the natural frequencies of the system. The proposed method is applied to complicated systems such as Mindlin plates with different geometries, tapered orthotropic Kirchhoff plates, and three dimensional thick plates, and for all undamaged cases very good results are achieved.;The second topic of this work is studying the stability analysis of conservative and non-conservative continuous systems by the collocation method, where the critical loads and the instability types are identified in a simple and efficient way. In this section, the proposed method is applied using three different approaches depending on the nature of the governing equation(s) and boundary conditions of the system. Good agreement is found between the results of this technique and other published results available in the literature. |
