Dynamics of Orthotropic Plates Using Characteristic Orthogonal Polynomial- Galerkin’s Method

ABSTRACT The assumed deflection shapes used in the approximate method such as in the Galerkin method were normally formulated by inspection and sometimes by trial and error , until recently, when a systematic method of constructing such a function in the form of Characteristic Orthogonal Polynomials (COPs) was developed in 1985. However, vibrational analysis of plates with different support and boundary conditions are much more complicated. This project used Characteristic Orthogonal Polynomial to obtain satisfactory approximate shape functions of plate of different support and boundary conditions. The functions were applied to Galerkin indirect variational method. To this end, new sets of stress- strain relations for orthotropic plate were derived. The principle of force of inertia was introduced, yielding the corresponding dynamic governing equation of orthotropic plate. This results obtained show an improvement to previous work done in this regard. The result were reasonable when compared with somewhat obtained in the previous work. The study have contributed in addressing problem of dearth of literature orthotropic plate vibration problems.

TABLE OF CONTENTS

 Page

Title Page i

Certification Page ii

Dedication iii

Acknowledgement iv

Abstract v

Table of Contents vi-ix

List of Tables x-xi

List of Figures xii

List of Symbols xiii-xiv

CHAPTER ONE

1.1 Background of Study 1

1.1.1 Importance of Vibration Analysis 3

1.1.2 Dynamics of Plate 4

1.2 Statement of Problem 5

1.2.1 Structural Analysis 6

1.2.2 Material Properties 7

1.2.3 Directional Materials 8

1.3 Objectives of Study 9

1.4 Significance of Work 10

1.5 Scope of Work 10

1.6 Limitation of Work 11

CHAPTER TWO: LITERATURE REVIEW

2.1 History of Plate theory development 12

9

2.2 Types of Plates 21

2.2.1 Kirchhoff’s Hypothesis 22

2.2.2 Boundary condition 24

2.3 Governing equation for deflection of plates in Cartesian coordinates 28

2.4 Vibrations of Plates 31

2.4.1 D’Alambert Principle 32

2.4.2 Governing Differential Equations of Free Motion of Plates 33

2.5 Bending of Anisotropic Plates 35

2.5.1 Basic Relationships 37

2.5.2 Determination of Rigidities for a Two-Way Reinforced Concrete Slab 41

2.5.3 Governing Differential Equation of Free Motion of Orthotropic Plates 43

2.6 Galerkin’s Method 45

CHAPTER THREE: METHODOLGY

3.1 Characteristic Orthogonal Polynomials Shape Functions 50

3.1.1 All Round Simply Supported Thin Rectangular Plate (SSSS) 56

3.1.2 All Round Clamped Thin Rectangular Plate (CCCC) 59

3.1.3 Rectangular Plate with One Edge Clamped and all Other Edges

 Simply Supported (CSSS) 63

3.1.4 Thin Rectangular Plate with One Edge Clamped and all Other Edges

 Simply Supported (CCSC) 66

3.1.5 Thin Rectangular Plate Clamped on Two Opposite Two Edges and Simply

 Supported on Two Other Opposite Edges (CSCS) 70

3.1.6 Thin Rectangular Plate Clamped on Two Adjacent Near Edges and Simply

 Supported on Two Adjacent Far Edges (CCSS) 73

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3.2 Fundamental Natural Frequency Equations of Orthotropic Plates 77

3.2.1 Fundamental Natural Equation for all Edges Simply Supported Plate

(SSSS) 77

3.2.2 Fundamental Natural Equation for all Clamped Plate Edges (CCCC) 81

3.2.3 Fundamental Frequency Equation for One Edge Clamped and all

 Other Edges Simply Supported (CSSS) 85

3.2.4 Fundamental Frequency Equation for One Edge Simply Supported

and all Other Edges Clamped Rectangular Plate (CCSC) 89

3.2.5 Fundamental Frequency Equation for Rectangular Plate Clamped on

 Two Opposite Edges and Simply Supported on Two Other Opposite Edges

 (CSCS) 93

3.2.6 Fundamental Frequency Equation for Two Adjacent Near Edges and Simply

Supported on Two Adjacent Far Edges 97

CHAPTER FOUR : RESULTS AND DISCUSSIONS

4.1 Calculations for Doubly Reinforced Sections 101

4.2 Discussion of Results 122

CHAPTER FIVE

5.1 Conclusion 124

5.2 Recommendations 124

References 125