Static and modal analysis of in-plane functionally graded structures using extended Kantorovich method
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First time, the elasticity based accurate analytical solutions are presented for static and free vibration analysis of in-plane functionally graded composite and piezoelectric structures. Modified Hamiltons principle is applied to derive the weak form of coupled governing equations in which, stresses and displacements variables acting as primary variables. Interface continuity and boundary conditions are satisfied in exact pointwise manners, which ensures the same order of accuracy for all the variables (stresses, displacements, and electric variables). Further multi-term extended Kantorovich method, recently developed powerful iterative technique, is employed to reduce the governing equation into sets of ordinary differential equations (ODEs). These sets of ODEs are solved analytically, which is key in the accurate prediction of free edge stresses. A novel power series based algorithm is developed and proposed to extract the static response, flexural frequencies and mode shape from a set of non-homogeneous ODEs having variable coefficients. A detailed numerical study established excellent agreement of the present iterative series solution with existing literature results and 2D/3D FE results. Further, the influence of the in-plane graded material properties on the static and dynamic response of structures is studied extensively for different support conditions and thickness ratios. The present solution is valid for thick as well as thin FGM structures (beams, plates and panels). Benchmark results are presented for both homogeneous and in-plane graded rectangular plates for various configurations and different combinations of boundary conditions. The significant effect of in-plane gradation of material properties is observed on the static and free vibration response of the functionally graded structures. The present analytical solution for in-plane functionally graded structures serves as benchmarks in the assessment of the accuracy of the numerical models and 1D/2D theories. This development will also assist in taking suitable kinematic assumption which is very helpful in the development of refined beam and plate theories for in-plane functionally graded structures.
Supervisor: Poonam Kumari