*ABSTRACT*

*This research work presents the analysis of rectangular plate on Winkler’s foundation using second degree characteristic orthogonal polynomials. The plates were of three different edge conditions: Thin Rectangular Plate pinned on all edges (SSSS), Thin Rectangular Plate clamped on all edges (CCCC) and Thin Rectangular Plate pinned on two opposite edges and clamped on the other two edges (CSCS). The exact displacement and bending moment functions of these plates under uniformly distributed in–plane loading on their longitudinal edges on Winkler foundation were obtained by applying work principle approach and total minimum energy method. These methods were applied on the governing differential equation of plates so as to obtain the deflection coefficient Wuv equations. These coefficients were obtained by finding the second degree of freedom functional equation and substituting them on work done expression for plates putting into consideration the aspect ratio of the rectangular plate. The expression gotten were later rearranged in matrix form to give the stiffness matrix of the plates which were later substituted back to the defection coefficient equation, Wuv The bending moments were then obtained by substituting the shape functions in the moment equation expressed in non-dimensional parameters and aspect ratio. The values of midspan deflection gotten for SSSS plate as aspect ratio varies from 1.0 to 2.0 are (0.000062590, 0.000062748, 0.000062859,* *0.000062939, 0.000063000, 0.000063082, 0.000063111, 0.000063135,and 0.000063154)m. The difference from Timoshenko and Woinowsky values reflects that second degree characteristic orthogonal polynomials produces better results when used to analyse higher degree polynomials. This also featured on results for CCCC plate-(0.000065330, 0.000065680, 0.000065923, 0.000066098,* *0.000066227, 0.000066325, 0.000066401, 0.000066460, 0.000066508, and 0.000066546) m and* *CSCS plate-(0.0011881, 0.0011898, 0.0011910, 0.0011917, 0.0011923, 0.0011927, 0.0011930,* *0.0011932, 0.0011934, and 0.0011935) m. In conclusion higher degrees of freedom Characteristic orthogonal polynomials shape functions for rectangular plates are satisfactory in approximating the deformed shape of thin rectangular plates of various boundary conditions. The results from the study* *compares very well with those of the literature. Though they are upper bound, they are adequate to be used in design as they are safer and straight forward.*

**CHAPTER ONE**

**INTRODUCTION**

**1.1 Background of Study**

There are manyhypotheses on models of elastic foundationsproven by many researchers. The simplest of them has been suggested by Winkler (Ventseland Krauthammer, 2001). It is based on the assumption that the foundation’s reaction q(x,y) can be described by the following relationship:

� = 𝑘� 1.1 where *k *is a constant termed the foundation modulus, which has the dimensions of force per unit surface area of the plate per unit deflection, N/m2 (1/m) in SI units. Values of k for various soils are given in numerous works, which is the resisting pressure of the foundation and w is the deflection of the plate.

Thin plates are initially flat structural members bounded by two parallel planes called faces, and either a plane or cylindrical surface, called an edge or boundary. The distance between the plane faces is called the thickness (h) of the plate. The mechanical properties of plates can be isotropic or anisotropic. The classical theory of elasticity assumes the material is homogeneous and isotropic, i.e., its mechanical (material) properties are the same in all directions and at all points. Many construction materials such as steel, aluminum, etc., fall into this category. However, certain materials display direction-dependent properties. Consequently, they are referred to as anisotropic for examples wood, plywood, delta wood, fiber-reinforced, plastics, etc. A number of manufactured plates made of isotropic materials also may fall into the category of anisotropic plates: examples include corrugated and stiffened plates, etc. These examples of anisotropy is referred to as structural anisotropy plate. Practical applications of orthotropic plates in civil, marine, and aerospace engineering are numerous and include decks of contemporary steel bridges, composite-beam grid works, and platesreinforced with closely spaced flexible ribs, and reinforced concrete plates.

The plate can also be rectangular, circular or any other polygonal shape. Rectangular plates are those plates that have four plane surfaces (edges). They have wide applications in Civil and Mechanical engineering. They have three dimensions a, b and h. Where b and a are respectively secondary and

primary in-plane dimensions and h is the plate thicknesswhich could be uniform or varying. The ratio, a/h is used to classify a plate as thick, stiff, thin or membrane. If the ratio, a/h is less than ten (10) then the plate is thick. If the range, 10 ≤ a/h ≤ 100 holds then the plate is thin. If the ratio, a/h is greater than hundred (100) then the plate is a membrane. In this research, emphasis was on isotropic thin rectangular plates of constant thickness.

Thin rectangular isotropic plate has four edges and the numbering of the edges is shown in Figure 1.1. Some of the boundary conditions of the edges of a thin rectangular isotropic plate are: S – designates simply support, C – designates clamped support and F – designates free support. A rectangular plate is unique from the other by the conditions of its four edges. These conditions can be same or mixed for a particular plate type. Such as SSSS, CCCC, CSCS, CSSS, CCSS, CCCS, etc. Any plate is named

according to conditions at the edges in line with the order of their arrangement shown in Figure 1.1.

Figure 1.1: Rectangular plate with edge numbering

Three approaches are used in the solution of thin rectangular plate analysis. There are;

I. The equilibrium (Euler) approach, II. The numerical approach.

III. The energy (approximate) approach

The Euler approach tends to find solution of the governing differential equation by direct integration and satisfying the boundary conditions of the four edges of the plate. Numerical approach is a good alternative to the Euler approach. Some examples of this approach include truncated double Fourier series, finite difference, finite strip, Runge-Kutta and finite element methods among others.

Energy approach is another method that can be used and is quite different from Euler and numerical approaches. The solution from it agrees approximately with the exact solution. Typical examples of energy approaches are Ritz, Raleigh-Ritz; Garlekin, minimum potential energy etc. These methods are called variational methods. They seek to minimize the total potential energy functional in order to get the solution matrix and accuracy of the solution is dependent on the accuracy of the approximate deflection function (shape function). Approximate shape function is substituted in the total potential energy functional, and the resulting equation is partially differentiated. The total potential energy is said to be minimized when its partial derivative is equated to zero. This implies that the difference between the approximate and exact solutions is zero (Iyengar, 1988).The energy approach can be of direct variational approach or by indirect variational approach. In the direct variational approach, the energy functional is minimized to arrive at equilibrium of forces equations. That is, differentiating the energy functional with respect to displacement gives force function and equating that function to zero. The energy functional is said to have been minimized. Indirect variational approach doesn’t convert the energy functional to force function, rather uses the principle of conservation of energy. That is, total energy (input and output) in a continuum that is in static equilibrium is always equal to zero. Typical examples of direct energy variational approach are Rayleigh, Ritz and Rayleigh-Ritz method. Typical examples of indirect energy variational approach are the finite difference method, the methods of boundary collocations, the boundary element method, and the Galerkin method.

Methods of numerical approaches have the capacity of handling plates of various boundary conditions. It has been shown from past works that in most cases, the solution from numerical approach approximate closely to those of the exact approach (Ventsel and Krauthammer, 2001). The problem with these numerical solutions is that the accuracy of the solution is dependent on the amount of work to be done. For instance, if one is using finite element method, the more the number of elements used in the analysis, the closer the approximate solution to the exact solution. Hence, when a plate has to be divided into several elemental plates for an accurate solution to be reached, then the extensive analysis is involved, requiring enormous time to be invested. A sound knowledge in mathematics and skilful

experience in computer programming are inevitable in this case. At this point one will see vividly that the problem one is trying to avoid in equilibrium approach is still found in numerical approach.

**1.2. STATEMENT OF PROBLEM**

The exact and explicit elastic analysis of isotropic rectangular plate have been a subject of continuous study from the conceptual time to the recent time. Just recently, attentions were moving away from finding the solution of plates’ problem through assumption that its solutions existed in the single degree of freedom domain. Engineering members are more of multi – degrees of freedom systems by their weight and positions. Researches on engineering members were focused presently on the behaviour and such solutions when derived would give improved expressions and enhance convergence of their mechanical behaviour.Galerkin energy method is not an adequate tool for a continuum whose deflection function has up to two degrees of freedom. Ritz and Rayleigh-Ritz which could be adequate for single and multi-degrees of freedoms make use of total potential energy approach that involves squares of derivatives. This approach usually culminate to large number of terms in the resulting expression, which could lead to difficulty in the final computation and susceptible to high computational errors (Ibearugbulem, et. al; 2014). The situation is worse with trigonometric function solutions for plates’ support other than simple supports.No researcher to the best ofmy knowledgein the course of this research have bothered to apply the recently developed work principle technique by Ibearugbulem et. al (2014) on elastic analysis of isotropic rectangular plates. In the light of the above problems, the work undertook a closed – form analysis of elastic isotropic rectangular plate by using characteristic orthogonal polynomial approach to solve for deflection functions of the plates for three different edge conditions. The new theories for work, termed: “work principle and minimum work error theory” were used to evaluate the elastic behaviour of thin rectangular plates under uniformly distributed lateral loading and expressions of their critical mechanical characteristics such as maximum deflections and moments

**1.3 OBJECTIVE OF STUDY**

The main objective of this study was to carry out analysis of rectangular plate on Winkler’s

Foundation using Characteristic Orthogonal Polynomials method, while the specific objectives are:

i. To explore the potentials and functionalities of work principle technique on the elastic analysis of isotropic rectangular plates’ problem of different edge conditions and subjected to uniformly distributed transverse load using multi – degree of freedom deflection functions.

ii. To use orthogonal polynomials to obtain the shape functions for thin rectangular isotropic plate analysis

iii. To use orthogonal polynomials to obtain the defection functions for thin rectangular isotropic plate analysis.

iv. To use orthogonal polynomials to obtain the bending moment functions for thin rectangular isotropic plate analysis

v. To compare results of the study with those of literature where Euler method was used in solving rectangular plate problems; and subsequently making justifiable inferences.

**1.4 SCOPE OF STUDY**

This work centers on classical thin rectangular plate analysis using second degrees of freedom characteristic orthogonal polynomials on three cases of plates ‘namely*: *Thin Rectangular Plate pinned on all boundaries ( SSSS), Thin Rectangular Plate clamped on all edges (CCCC) and Thin Rectangular Plate pinned on two opposite edges and clamped on the other two edges (CSCS).

**1.5 JUSTIFICATION OF THE STUDY**

This research is set to introduce the use of multi – degrees of freedom characteristic orthogonal polynomials displacement equations in work principle technique of thin rectangular plate problems and to open new frontier in classical structural mechanics for plane continua analysis. It will also give clearly distinctions among the energy methods in terms of the suitability in elastic analysis of isotropic rectangular plates’ problems.

This material content is developed to serve as a **GUIDE** for students to conduct academic research

ANALYSIS OF RECTANGULAR PLATE ON WINKLERS FOUNDATION USING SECOND DEGREE CHARACTERISTIC ORTHOGONAL POLYNOMIALS

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