الفهرس 
Preamble.Only 14 pages are availabe for public view
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Abstract
Preamble
Nowadays, Studying the static and dynamic characteristics of
elasticity problems grabbed scientists’ attention as they can be employed in
extensive applications such as structural components, tanks, aerospace
industry, space crafts and deck plates in launch vehicles. Since circular
plate is considered one of the most meaningful examples of elasticity
problems through which we can review different elasticity theories and
implementation of energy and variational principles, in this work, analysis
of circular plates is considered.
from manufacturing and economic point of view, producing structural
components that have high strength with low weight and can perform well
under extraordinary conditions is one of the most important research fields.
Consequently, the idea of composite materials arised to achieve that
purpose. Recently, a new class of composite materials called functionally
graded material (FGM), an inhomogeneous composite usually made from a
mixture of ceramics and metal by gradually varying material properties
through thickness direction. This type of composite materials is chosen to
describe the inhomogeneity of the material of the plate. Furthermore,
different parameters can influence the behavior of the plate, for instance,
thickness profile, existence of elastic foundation and boundary conditions.
Dealing with such complicated problems that include different
parameters reflects on formulation of the mathematical model. Employing
conventional analytical methods in this case is not practical as it can be
extremely hard to present an analytical solution. Instead, numerical
approaches are the best choice to treat such problems. One of the most
Chapter 1 Introduction
2
popular approaches is differential quadrature method (DQM), where the
differential equation of the plate is discretized at grid points which result in
a set of linear algebraic equations that can be solved together to present the
solution. The merit of this method is the high level of accuracy with low
computational effort in comparison with other numerical methods.
1.2 Problem Depiction
Most of research work considering the bending and free vibration
analysis of circular plates only focus on simple types of plates, for
example, isotropic plates with uniform thickness, circular plates resting on
uniform one-parameter elastic foundation and circular plates fully resting
on elastic foundation. But only few studied the effects of different
parameters such as material gradient, variable thickness, two-parameters
elastic foundation and the variation of soil subgrade modulus. However, no
research work has been conducted to study all these parameters together
deeply, besides investigating the real behavior of the soil underneath elastic
foundation to present a formula describes the variation in the soil subgrade
modulus.
1.3 Thesis Objectives
In this thesis, a numerical solution to the bending and free vibration
analysis of FGM circular plate resting of Winkler-Pasternak foundation
will be presented using differential quadrature method. The study will
include investigation of different parameters on transverse displacements,
radial stress, natural frequencies and modal shapes. Moreover, the behavior
of soil will be demonstrated to develop an equation describes the variation
of the soil subgrade modulus. The parametric study will provide a more-
general overview about the behavior of circular plates resting on elastic
foundation.
Chapter 1 Introduction
3
1.4 Procedure of Analyzing Circular Plates
1.4.1 Derivation of the Governing Equation
The governing equation is driven based on the classical plate theory as
follows:
1. Determination of displacements field (,,)uvw .
2. Find strain-displacement and stress relations.
3. Applying Hamilton’s principle.
1.4.2 Implementation of DQM
The numerical solution of the governing equation of circular plate will be
introduced using differential quadrature method as follow:
1. Computation of weighting coefficients.
2. Discretization of governing equation and boundary conditions.
3. Solve linear system of equations to obtain the solution.
1.4.3 Parametric Study
The analysis of deflection, radial stress, natural frequency and mode shapes
will be discussed in terms of different parameters, such as
1. Material gradient index.
2. Non-uniform thickness.
3. Two-parameter elastic foundation with variable subgrade
modulus.
4. Fully and partially supported area on elastic foundation.
5. Clamped and simply supported boundary conditions.
1.5 Thesis Organization
Chapter 1 Introduction
4
Organization of thesis chapters is listed as follow
Chapter (1): Introduction
This chapter consists of introduction, problem depiction, thesis
objectives, procedure of analyzing circular plates.
Chapter (2): Literature Review
This chapter reviews past research work and efforts to analyze
elasticity problems using various analytical and numerical methods.
Chapter (3): Classical Plate Theory
This chapter presents the derivation of the governing equation of
circular plate based on the classical plate theory using Hamilton’s
principle.
Chapter (4): Differential Quadrature Method
This chapter reviews the differential quadrature method, computation
of weighting coefficients, selection of grid points and the techniques
used in the implementation of boundary conditions
Chapter (5): Axisymmetric Bending of FGM Circular Plate
This chapter investigates the static analysis of FGM circular plate
with variable thickness fully or partially resting on non-uniform
Winkler-Pasternak foundation using DQM.
Chapter (6): Axisymmetric Free Vibration of FGM Circular Plate
This chapter investigates the dynamic analysis of FGM circular plate
with variable thickness resting on non-uniform Winkler-Pasternak
Foundation using DQM.
Chapter 1 Introduction
5
Chapter (7): Distribution of Soil Subgrade Modulus
This chapter presents a formula describes the variation in the subgrade
modulus of the soil underneath the circular plates based on regression
analysis of different elastic foundation models.
Chapter (8): Conclusion and Recommendations for Future Work
This chapter concludes the main outcomes of the present work and
focuses on the future research work.
References: The thesis contains 117 references
Appendix: Provides a MATLAB code to solve bending and free
vibration of axisymmetric circular plates with variable thickness and
resting on non-uniform two-parameter elastic foundation.
Arabic Summary: Provides a summary of the work in Arabic