Analytical identification of damaged areas in porous circular plates resting on elastic foundation based on modal shapes derivatives

Structural health monitoring is essential to predict problems and maintain structure integrity which can be effective in prolonging the structure lifetime. The accumulation of failure in the structures causes a severe structural fracture; therefore, the development of damage detection methods for structural fracture is one of the most important points in maintaining the integrity and safety of the structures. In this paper, damage identification in functionally graded (FG) porous circular plates based on modal data and vibration analysis on elastic foundation is carried out. The vibration analysis process is performed based on first-order shear deformation theory and analytical method. Circular plates are widely used in the industry, for example in bower, balconies, screens, halls and swimming pool ceilings. Functionally graded materials are a new type of composite materials and may be characterized by the variation in composition and structure gradually over volume, resulting in desired changes in the characteristics of the material. The porous materials are lightweight, flexible and resistant to tiny cracks, these materials have two phases; their first phase is solid and the second phase is liquid or gas. The proposed method is achieved using modal analysis information extracted from a mathematical code in MAPLE. Based on this method, the plate can be examined with different boundary conditions including clamped, free and simply supported. Power series method has been used to solve the governing equations of circular plate. For the first part of the circular plate, the power series expands around the zero point and for other parts of the plate, the power series expands around the outer radius of its own part. In this study, first of all, the governing equations of circular plates were extracted based on first-order shear deformation theory and displacement functions. Two types of porosity have been used for applying porosity to the plate. Thereafter, the continuity conditions of displacement which generally include 6 items have been applied: continuity of in-plane displacement of the middle layer, continuity of rotation of the transverse surface, continuity of displacement of the plate, continuity of in-plane force, continuity of bending torque and continuity of transverse shear force per unit length. Furthermore, stress resultants in the points of connection have been applied which includes 3 items: in-plane force, bending torque and transverse shear force per unit length. The matrix determinant that is based on boundary and continuity conditions has been calculated to find the natural frequencies of the plate. After that, based on the Newton-Raphson method which is a numerical method for determining the root of a function, the initial natural frequencies of the circular plate were obtained and their modal shapes were plotted. At the end of the process, when the natural frequencies are determined, the unknowns were calculated. To prove the accuracy and efficiency of the proposed method, the natural frequencies obtained from this method were compared with the results presented in other papers. The comparisons have shown that the results obtained through this study had a good agreement with those of other ones and the proposed method could accurately show the damage location.