Static and dynamic behavior of axially functionally graded structural elements with different boundary conditions

Abstract

The present thesis investigates static and dynamic behaviour of axially functionally graded structural elements (beams and plates) on elastic foundation with different boundary conditions. For beams, Euler-Bernoulli and Timoshenko beam model are separately considered, whereas, the plate is taken as thin plate. The material of structural elements is considered to be functionally graded continuously along longitudinal direction. To incorporate the material gradation, different material models are selected depending on the gradation of the elastic modulus and density in the axial direction. Non-uniform structural geometry has also been taken into account in the present thesis considering variation in thickness along the axial direction. For that purpose, linear, parabolic and exponential taper patterns are chosen for thickness. The structural elements are considered to be resting on elastic foundation with different classical boundary conditions and subjected under externally applied uniformly distributed load. The foundation has been mathematically incorporated into the analysis as a set of linear springs attached uniformly at the bottom surface of the structure. A displacement based semi-analytical method associated with the whole physical domain of the system is utilized for formulation of the problems throughout the thesis. Geometric nonlinearity is also included in the present thesis considering nonlinear straindisplacement relations. The governing set of nonlinear equations of the system are derived adopting suitable energy methods and solved by numerical application of suitable iterative methods. For beam (thin & thick), study of free vibration and forced vibration characteristics are performed, whereas, in case of plate, static and free vibration analysis are taken up. The main concern of the static analysis is to represent the load versus deflection plot and deflected shape plot under the application of steady state loading considering the effect of various parameters viz. material model, taper pattern, system geometry and elastic foundation. The governing set of nonlinear equation in static analysis is derived utilizing principle of minimum total potential energy and unknown co-efficient of the governing equations are solved using an iterative method (direct substitution with relaxation). The main focus of the free vibration analysis is to represent backbone curves and corresponding mode shapes. In order to generate these, it is necessary to find out the natural xxviii frequencies of the system under undeformed and deformed conditions. The problem is divided in two distinct parts. Firstly, the static problem is carried out through an iterative scheme using a relaxation parameter and later on the subsequent dynamic problem is solved as a standard eigen value problem. The obtained results are validated from previously published results and are found to be in good agreement. The free vibrational frequencies are tabulated for different taper profile, taper parameter and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane. Investigation of mode switching for AFG plate on elastic foundation is also a vital consideration and leads to identification of particular conditions for which the above mentioned phenomenon is observed. Linear and nonlinear mode shape plots are also presented to compare the free vibration behaviour. Forced vibration analysis is conducted with an objective to find out the response of the system, in terms of displacement amplitude, under externally applied time varying excitations. The derivation of governing equations is accomplished following Hamilton’s principle. In the present work, only steady-state response is presented and frequency of response of the undamped system is assumed to be equal to that of the external excitation. An indirect approach is adopted for solving the problem, where it is reduced to a static scenario by assuming that under maximum amplitude of excitation, i.e., when the system suffers maximum deformation, the dynamic system fulfils force equilibrium conditions. Broyden method, which is a multidimensional secant method used for numerically solving a system of nonlinear equations. A convergence study is performed to determine the values for various parameters related to the numerical scheme. Established result from existing literature is used to provide validation for the adopted method and solution procedure. The geometric nonlinear forced vibration characteristic of the system is represented through frequency response curves in non-dimensional excitation frequency-maximum response amplitude diagrams. the effects of excitation frequency on the ODS (Operational Deflection Shape) is also investigated. New results, capable of acting as benchmark results, are provided for a ombination of different flexural boundary conditions, various material models and foundation stiffness values.