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DYNAMIC ANALYSIS OF RECTANGULAR THIN ISOTROPIC PLATES USING BEAM ANALOGY METHOD
ABSTRACT
This work aims at obtaining an exact solution to the general plate equation for dynamic loading (free and forced vibration) by using a simple, economic and straight-forward technique termed ‘beam analogy’. This technique analyzes a plate as an array of beams arranged in perpendicular directions x and y. The end conditions (simple, free and clamped) are combined to give 5 beams strips, in each direction. So, for a two-dimensional array like a plate, we have 25 plates, but due to symmetry, we have only 15 plates to analyze. Polynomial shape functions were used in this work because it is known that a direct and continuous integration of the plate equation would yield a polynomial solution. Both sides of the plate equation for dynamic loading were multiplied by a deflection term,
, integrated and simplified, to obtain the energy equation. For free vibration analysis (loading q =
0), the assumed shape function and aspect ratio (β = ) of each plate was substituted in the energy equation to obtain the natural frequency ω of the plate in terms of a non-dimensional frequency parameter H. In the forced vibration analysis (q ≠ 0) the forcing frequency Ω was substituted into the energy equation to obtain the deflection constant, A, for any particular value of ‘frequency factor’ N (Ω = Nω, N = 0, 0.2, 0.4, 0.6, 0.8, 1.0). The numerical coefficient in the expression for A was recorded as G for each plate. It is from the values of G, A and N that the following quantities: Dynamic Load Factor DLF, the full shape function, the bending moments and shears were determined. The values of H obtained increased as the aspect ratio β decreased, and also as the number of clamped edges in the plate increased. These results agreed well with those of past researchers that used polynomial shape functions. From the values of G obtained, it was observed that, for the same plate and aspect ratio β, the quantities (Dynamic Load Factor DLF, deflection, slope, bending moments and shears) increased as the value of N increased. For the same plate and frequency factor N, the quantities increased as the aspect ratio β increased. These quantities were checked and found to satisfy the natural and geometric boundary conditions of the plates.
Key words: Dynamic analysis, rectangular plate, beam analogy method, natural frequency, forcing frequency, dynamic load factor, resonance.
TABLE OF CONTENTS
Title Page i
Certification ii
Dedication iii
Acknowledgements iv
Abstract vi
Table of Contents vii
List of Tables xii
List of Figures xvii
Notations xviii
CHAPTER ONE: INTRODUCTION
1.1 Background of the Study 1
1.2 Statement of the Problem 3
1.3 Objectives of the Study 3
1.4 Justification of the Study 4
1.5 Scope of the Study 4
CHAPTER TWO: LITERATURE REVIEW
2.1 Plate Dynamics 6
2.2 The Governing Differential Equation of Motion 6
2.2.1 Historical Development of the Plate Equation 7
2.2.2 Plate Equation for Forced Vibration 8
2.2.3 Plate Equation for Free Vibration 9
2.3 Solutions to the Plate Equation 11
2.4 The Boundary Conditions 20
2.4.1 Physical Conditions at the Boundaries 20
2.4.2 Number of Boundary Conditions 21
2.4.3 Mathematical Expressions for Various Boundary Conditions 21
2.5 Classical Methods for Analysis of Free Vibration 23
2.5.1 The Equilibrium Methods 23
2.5.1.1 Navier’s Solution 23
2.5.1.2 Levy’s Solution 24
2.5.1.3 Chakraverty’s Analytical Solution 26
2.5.2 Energy Methods 28
2.5.2.1 The Principle of Conservation of Energy 28
2.5.2.2 Rayleigh’s Method 30
2.5.2.3 Ritz’s Method 32
2.5.2.4 Galerkin’s Method 33
2.5.2.5 Natural Frequencies Obtained from Static Deflections 34
2.5.3 Numerical Methods 36
2.5.3.1 Finite Difference Method 36
2.5.3.2 Finite Element Method 37
2.6 Results from Previous Researchers 40
2.6.1 Natural Frequencies of SSSS Plate 40
2.6.2 Natural Frequencies of CCCC Plate 42
2.6.3 Natural Frequencies of CCSC Plate 44
2.6.4 Natural Frequencies of CSSS Plate 44
2.6.5 Natural Frequencies of CCSS Plate 45
2.6.6 Natural Frequencies of CSCS Plate 45
2.6.7 Natural Frequencies of Plates with One Free End 46
2.6.7.1 Natural Frequencies of CCFC Plate 46
2.6.7.2 Natural Frequencies of CSFS Plate 46
2.6.7.3 Natural Frequencies of SCFC Plate 47
2.6.7.4 Natural Frequencies of CCFS Plate 47
2.6.7.5 Natural Frequencies of SSFS Plate 48
2.6.7.6 Natural Frequencies of SCFS Plate 48
2.6.8 Natural Frequencies of Plates with Two Free Edges 49
2.6.8.1 Natural Frequencies of SCFF Plate 49
2.6.8.2 Natural Frequencies of SSFF Plate 49
2.6.8.3 Natural Frequencies of CCFF Plate 49
2.6.9 Results for Haφ from Ibearugbulem et al. (2014) 50
2.7 Classical Methods of Analyzing Forced Vibration 52
2.7.1 Navier’s Method 52
2.7.2 Levy’s Method 53
2.7.3 Finite Difference Method 53
2.7.4 Finite Element Method 54
2.7.4.1 Estimating Dynamic Response using DLF 54
2.7.4.2 Harmonic Analysis 54
2.8 Characteristics of the Exact Solution 55
2.9 Summary of Past Works 56
2.10 Evaluation of Past Works 72
CHAPTER THREE: METHODOLOGY
3.1 Formulation of an Exact Solution to the Governing Differential Equation 74
3.2 Development of Shape Functions of Plates Subjected to Free Vibration 75
3.2.1 Beam Analogy 75
3.2.2 Structural Descriptions 76
3.2.3 Boundary Conditions and Nomenclature of Beams 77
3.2.4 Plate Dimensions and Boundary Conditions 79
3.2.5 Nomenclature and Types of Plates 80
3.2.6 Shape Functions for Various Beam Strips 83
3.2.7 Shape Functions for the Plates 85
3.3 Development of Fundamental Natural Frequencies of the Plates
for Free Vibration 89
3.4 Derivation of Equation for Dynamic Load Factor (DLF) for the Plates
Using Beam Analogy 124
3.5 Deflections, Slopes, Moments and Shears Induced on the Plates
by Forced Vibration 126
3.5.1 Deflection and Slope 126
3.5.2 Moment and Shear Equations for a Rectangular Plate 145
3.5.3 Equations for Bending and Twisting Moments 146
3.5.3.1 Bending and Twisting Moment Equations for Plates without Free Edges 146
3.5.3.2 Bending and Twisting Moment Equations for Plates with One Free Edge 147
3.5.3.3 Bending and Twisting Moment Equations for Plates with Two Free Edges 147
3.5.3.4 Values of Bending and Twisting Moments for the Plates 147
3.5.4 Equations for Shear Forces 166
3.5.4.1 Shear Force Equations for Plates without Free Edges 166
3.5.4.2 Shear Force Equations for Plates with One Free Edge 166
3.5.4.3 Shear Force Equations for Plates with Two Free Edges 167
3.5.4.4 Values of Shear Forces at the Edges of the Plates 167
3.5.5 Checks on the Satisfaction of Boundary Conditions 179
3.6 Comparison of the Frequencies Obtained in this Work with the Ones
Obtained by Past Researchers 192
CHAPTER FOUR: RESULTS AND DISCUSSION
4.1 Results 194
4.1.1 Exact Solution to the Governing Differential Equation of Motion of
the Plates 194
4.1.2 Shape Functions of Plate Subjected to Free Vibration 194
4.1.3 Fundamental Natural Frequencies of the Plates 195
4.1.4 Dynamic Load Factors DLF for the Plates Using Beam Analogy Method 200
4.1.5 Deflections, Slopes, Moments and Shears Induced on Plates by
Forced Vibration 200
4.1.5.1 Deflection and Slope 200
4.1.5.2 Bending and Twisting Moments 212
4.1.5.3 Shear Forces 220
4.1.6 Comparison of the Frequencies Obtained in this Work with the Ones
Obtained by Past Researchers 225
4.2 Discussions 241
4.2.1 Exact Solution to the Governing Differential Equation of Motion of the
Plates under Free Vibration 241
4.2.2 Shape Functions of Plate Subjected to Free Vibration 241
4.2.3 Fundamental Natural Frequencies of the Plates 241
4.2.4 Dynamic Load Factors DLF for the Plates Using Beam Analogy Method 244
4.2.5 Deflections, Slopes, Moments and Shears Induced on Plates by
Forced Vibration 245
4.2.5.1 Deflection and Slope 245
4.2.5.2 Moments and Shears 245
4.2.6 Comparison of the Frequencies Obtained in this Work with the Ones
Obtained by Past Researchers 246 CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions 247
5.2 Recommendations 249
5.3 Contributions to Knowledge 250
REFERENCES 252
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
A plate can be defined as a structural component having significant dimensions in two directions. It is seen as a continuum, in which one dimension is much smaller than the other two dimensions. A plate is flat (i.e. it has level surface) before loading as contrasted from a shell which is curved before loading. It is described as thin if the ratio of its thickness to the smaller span length is less than 1/20, otherwise it is thick. When the stiffness and strength of a plate in all directions is identical, the plate is said to be isotropic, otherwise it is orthotropic. Most plates are either rectangular, circular, or any other geometry in shape (Chakraverty, 2009). Plates are integral parts of any structure. They occur as shear walls, floor panels and shelves. In steel structures, plates occur as components of I-, H-, T-, or Channel sections. In structural hollow sections, sheets used to enclose lift shafts, or walls or cladding in framed structures, are plates.
Every structural component is designed to carry loads. The loads that act on plates are either static or dynamic. These loads can either be in-plane or out-of-plane. In-plane loads cause stretching of the plate when tensile, and buckling when compressive. Out-of-plane loads cause bending of the plate. When the load is not suddenly applied to the plate, and it does not change with time and position, it is said to be a static load. A load whose magnitude, direction and/or position varies with time is a dynamic load. Forces suddenly applied to structures, are termed shock or impact loads, and result in dynamic loading. Shock loading is produced by a sudden application of force or motion to a structural member, whereas impact loading results from the collision of bodies. When the time of application of a load, is equal to or smaller than the largest natural period of vibration of the structural component, shock or impact loading is produced.
Owens, Knowles, and Dowling (2000) stated that the sources of dynamic loads on structures include the following:
- Forces generated inside the structure, from vibrating machinery/equipment, impacts, human activity, like walking, dancing, etc.
- External forces, from wind buffeting and other aerodynamic effects, waves on off-shore structures, impacts from moving vehicles, etc. on bridges, etc.
- Ground motions, due to earthquakes, seismic disturbances, ground-borne vibration caused by railways, roads, pile driving, etc.
The effects of dynamic loads on structures include reduction in strength, initiation and propagation of fatigue cracks, destruction and breakage of certain sensitive equipment, feeling of danger and insecurity in the minds of occupants of the structure, and in extreme cases, collapse leading to loss of lives and property.
Structural dynamics deals with time-dependent motions of structures and the analyses of the internal forces associated with them, with the objective of determining the effects of vibrations on the performance of the structure. Usually, a dynamic force acts to modify the static stress and strain fields as well as the resistance properties of a structural material. The problem is that dynamic behaviour is influenced by a larger number of parameters than static behaviour. And so, assessing the structural response to dynamic loads is not easy.
Researchers in the past, like Euler, Bernoulli and Timoshenko, have been able to formulate a governing differential equation describing the response of a rectangular thin isotropic plate subjected to a dynamic load (Szilard, 2004; Ventsel & Krauthammer, 2001). This resulted into a fourth order partial differential equation. Obtaining the exact solution to this equation has been the subject of research for engineers and mathematicians for several years. The existing conventional methods of obtaining solutions are the equilibrium approach developed by Euler, the energy approach and the numerical approach. Navier’s and Levy’s methods are examples of equilibrium approach. The energy approach includes the Ritz, Raleigh-Ritz, Garlekin, minimum potential energy methods, etc. Examples of the numerical approach are finite difference, finite strip, Runge-Kutta and finite element methods, among others. Each of these methods has limitations, problems, and most of them only give approximate solutions.
1.2 Statement of the Problem
The effects of dynamic loads on a structure are more severe and dangerous than those of static loads. As stated earlier, a dynamic force acts to modify the static stress and strain fields as well as the resistance properties of a structural material.
The dynamic behaviour of a structure is influenced by a larger number of parameters than the static behaviour. Therefore, the assessment of the structural responses to dynamic loads is rigorous and complicated. Most existing solution methods for the governing differential equation of motion of the structural plate subjected to dynamic loads only give approximate solutions and not the exact solution. Numerical methods seem to produce near satisfactory results, but the amount of work involved in the formulation, and the expertise required in the use of computer, is much. This is so because the accuracy of the numerical methods depends on the number of finite units the plate is divided into in the analysis; and, the analysis of plates divided into many finite units involves a great deal of computation.
Deflections and natural frequencies, if not well calculated and taken care of in design, can lead to errors in safety and economy, and total collapse of the structure, especially when the vibration reaches resonance. Consequently, there is need to develop a simpler and less time consuming method, which predicts accurately the dynamic response of plates subjected to dynamic loads or motions. This research work is concerned with the dynamic analysis of rectangular, thin, isotropic plates using beam analogy method, which offers the advantages of simplicity, accuracy, and saves time.
1.3 Objectives of the Study
The main objective of this research is the dynamic analysis of thin, isotropic, rectangular plates using beam analogy method. The specific objectives are:
- To formulate an alternative to the governing differential equation of a thin isotropic rectangular plate subjected to dynamic loading using integral calculus.
- To develop shape functions for plates subjected to free vibration, for various boundary conditions, using beam analogy method.
- To develop fundamental natural frequencies for plates subjected to free vibration, for various boundary conditions and different aspect ratios, using beam analogy method. To derive the equation for dynamic load factor for plates using beam analogy method.
- To obtain the deflections, slopes, shears and moments induced on plates by forced vibration under different boundary conditions.
- To compare the frequencies obtained herein for undamped free vibration analysis with the corresponding solutions obtained by previous researchers.
1.4 Justification of the Study
When the objectives above are met, the following benefits shall accrue:
- The exact values of dynamic responses (i.e. deflections, slopes, frequencies, shears, moments and dynamic load factors) of rectangular, thin, isotropic plates to loads can be more easily and economically obtained.
- Design of structures involving plates, walls and slabs, will be more accurate, and hence result into safer structures.
- It will provide a guide for the prevention of resonance in structures that comprise plates.
- It will provide a database containing the dimensionless parameters of natural frequencies for professional engineers and designers.
- It will provide a basis for future research on design and dynamicanalysis of structural plates.
1.5 Scope of the Study
This study was limited to the analysis of a thin, rectangular, isotropic, plate subjected to dynamic loads. The governing differential equation of motion of the plate was solved mathematically by using integral calculus of characteristic orthogonal polynomial functions.
Twenty-five types of plates were studied in this work. The various types of plates, with their different boundary conditions, covered include the following: SSSS, CSCS, CSSS, SSFS, CSFS, SCSC, CCCC, CCSC, SCFC, CCFC, SCSS, CCCS, CCSS, SCFS, CCFS, SSSF, CSCF, CSSF, SSFF, CSFF, SCSF, CCCF, CCSF, SCFF and CCFF (where C, F and S
designate clamped, free and simple supported edges of the loaded plate). The aspect ratios, φ = , considered in the analyses are 1 to 2, at increaments of 0.1, (b > a). These are the limits of aspect ratio considered in the structural design of two-way slabs. Thus, the limits of the other aspect ratio β = are 0.5 and 1. The equations and values of the natural frequencies and their dimensionless parameters were obtained and compared with those obtained from previous researches in this field. Also, equations and values of the deflections, shears, moments and dynamic load factors were obtained for designers
DYNAMIC ANALYSIS OF RECTANGULAR THIN ISOTROPIC PLATES USING BEAM ANALOGY METHOD
