ABSTRACT
The assumed deflection shapes used in the approximate methods such asin the Rayleigh–Ritz method were normally formulated by inspection andsometimes by trial and error, until recently, when a systematic method ofconstructing such a function in the form of characteristic orthogonal polynomials(COPs) was developed in 1985. However, vibrational analyses of plates with all edges free or clamped are much more complicated. This project aims at establishing particular expressions for the fundamental natural frequency of rectangular plates using Rayleigh Ritz Method by obtaining approximate shape functions of plate of different support conditions through characteristics orthogonal polynomials. To this end, new sets of stress – strain relations for orthotropic plates were derived. The principle of force of inertia was introduced, yielding the corresponding dynamic governing equation of orthotropic plate and hence the strain energy equation of orthotropic plate.Rayleigh – Ritz quotient was obtained by equating the strain energy equation of the plate to the kinetic energy equation of the plate. From the rayleigh’s quotient, an expression for the natural frequency of the plate was obtained in terms of the shape functions of the plate. The shape function of plate of different support conditions was obtained through characteristics orthogonal polynomials. A spreadsheet programme was developed to aid the solutions of the equations and the results show that values of fundamental natural frequencies obtained using the present studies (Rayleigh – Ritz Method) for all round clamped plates with aspect ratio, r between 0.5 to 1.2 are four times greater than that obtained from that of previous studies (Galerkin’s method). Moreso, for plates with mixed support conditions, values of fundamental natural frequencies obtained using the present study are higher than that of previous studies (Galerkin’s Method) their difference ranging from 33% for r = 0.5 to 95% for r = 1.2 while results for plate all round simply supported for both present and previous studies are the same.
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND OF STUDY
Study of vibration of plates is an extremely important area owing to its wide variety of engineering applications such as in aeronautical, civil, and mechanical engineering. Since the members, viz., beams, plates, and shells, form integral parts of structures, it is essential for a design engineer to have a prior knowledge of the first few modes of vibration characteristics before finalizing the design of a given structure. In particular, plates with different shapes, boundary conditions at the edges, and various complicating effects have often found applications in different structures such as aerospace, machinedesign, telephone industry, nuclear reactor technology, naval structures, and earthquake-resistant structures.
A plate may be defined as a solid body bounded by two parallel, flat surfaces and having two dimensions far greater than the third. The vibration of plates is an old topic in which a lot of work has already been done in the past decades. In earlier periods, results were computed for simple cases only where the analytical solution could be found. The lack of good computational facilities made it almost impossible to get reasonably accurate results even in these simple cases. With the invention of fast computers, there was a tremendous increase in the research work using approximate and numerical methods for simple as well as complex plate vibration problems.
Now, we have some very fast and efficient algorithms that can solve these problems in a very short time and give comparatively accurate results. It is also worth mentioning that methods like finite element method, boundary integral equation method, finite difference method, and the methods of weighted residuals have made handling any shape and any type of boundary conditions possible. Different theories have been introduced to handle the vibration of plate
problems. Correspondingly, many powerful new methods have also been developed to
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analyze these problems. In thisstudy, an overviewof basic equations, theories, stress–strain relations, etc., were addressed related to the vibration basics for plates.These basics wereextended to the orthotropic plate, which is a counterpart of isotropic plate. Though there are efforts in the past towards analyzing such a plate, this study intends to improve on what have previously been done in the recent time.
Most loads acting on structures are dynamic in origin [1]. In his work, Mitchell [17]concurred that all loadings, from whatever source are variable with time, due either to variation of field strength, variations in the structure or its contents or inherent nature of the loadings.Thereon, as loads acting on a structure have dynamic implications, loadings, stresses, deflections, etc. might be best determined from the perspective of their dynamic behaviors. This would permit provision of adequate and approximate internal reactions to various loads on structures.
As noted earlier, the continuing tendency to reduce the weight and cost of structure and to increase the height to weight ratio of structure enhanced the need to predict the vibration response levels. In addition to the recent trend in construction industries, where for example building and bridges are made lighter, more flexible; and are made of materials that provide much lower energy dissipation. All of which made the structure more susceptible to critical vibration.
Dynamic analysis of structure is therefore even more important for modern structures, and this trend is likely to continue. The analysis of structural response is of considerable importance in the design of structure as a result of the following reasons:
a) Under certain situations, vibration may cause large deformation and severe stress in the structure. This may happen particularly when the frequency of existing force coincides with the natural frequency of the structure.
b) Fluctuating stresses, even of moderate intensity, may cause material failure through
fatigue, if the number of repetitions is large enough.
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c) Oscillating motion may at times cause wearing and malfunctioning.
d) When structure is designed for human use, vibrating motion may result in severe discomfort to the occupants[23].
Rectangular plates have wide applications in Civil and Mechanical engineering. Their dynamic characteristics are important in engineering designs [10]. The dynamics of plates, which are continuous elastic systems, can be modeled mathematically by partial differential equations based on the considerations of virtual work [32]. In practical applications, only the lateral vibration is of interest; and the effects of the extensional vibrations in the middle plane may be neglected. Therefore, the inertia forces, associated with the lateral translation of the plate, are considered.
Damping effects are caused either by internal friction or by the surrounding media. Although structural damping is theoretically present in all plate vibrations, as a consequence, the amplitude of free vibrations remains constant with time; but experience has shown that the amplitude diminishes with time and that the vibrations are gradually damped out. In the case of forced vibrations, the theory indicates that the amplitude can grow without limit at resonance. However, we know that because of damping, there is always some finite amplitude of steady-state response, even at resonance [28].
Structural damping is theoretically present in all plate vibrations [31]; it has usually little or no effect on:
a. The natural frequency.
b. The steady-state amplitudes.
Consequently, it can be safely ignored in the initial treatment of the plate problem.
1.2 STATEMENT OF PROBLEM
Thomas et al [27] noted that all systems possessing mass and elasticity are capable of free
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vibration, or vibration that takes place in the absence of external excitation. Previous researchers have observed the dependence of the natural frequencies on theamplitude of vibration [34]. Reinforced concrete slabs, exhibit force due to mass of inertia; thus, they are susceptible to free vibration.
Structural engineers have long been trying to develop solutions using the full potential of composing materials. This has been the structural solution progress directly towards increase in materials science knowledge. Thus, the constituent materials of reinforced concrete slab would not be exception in predicting the dynamic regime of structural elements, hencethis work. It is therefore, believed that investigating thiswouldgive a safe, economical and aesthetic reinforced concrete slab design [7].
Today, most structural designs carried out in our design offices are done without having recourse to the dynamic effects of the static loads and self-weight of the structure. In the words of Harr[9] ” When a plate is loaded statically, the elastic reaction of the plate is everywhere, equal and opposite to the applied loading, q. If there is no external applied loading but the plate is vibrating, the elastic reaction action on each element of the plate (measured in the direction of negative deflection, w produces an acceleration of each element of the plate in the same direction. The magnitude of the elastic reaction is expressed
as:
| మ |
− m(x, y)డ ௪ డ௧మ
(1.0)
Where, m(x, y) is the mass per unit area of the plate.However, Ventsel and Krauthammer [31] reminded that dynamic loads may be created by moving vehicles, wind gusts, unbalance machine vibrations, flight loads and sound.
1.3 OBJECTIVES OF STUDY
The main objective of this dissertation is toanalyze orthotropic plates in free vibration regime using Rayleigh – ritz method. It is however realized that the fulfillment of the following sub-
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objectives would in-turn fulfill the main objective
a) To obtain approximate shape functions of plates of different support conditions through characteristic orthogonal polynomials.
b) To establish particular expressions for the natural frequencies of rectangular plate using Rayleigh – Ritz’s approximate method.
c) To compare the results of the natural frequencies of plates obtained from the previous study (Galerkin’s Method) and the results of the natural frequencies of plates obtained from present study (Rayleigh – Ritz method).
d) To develop and run a spreadsheet programme for the computation of the natural frequencies of orthotropic plate in free vibration regime.
1.4 GENERAL METHODOLOGY
The approximate shape functionsfor rectangular plate with various boundary conditions were sought through the Characteristics Orthogonal Polynomial method. Consequently, the strain energy and the total potential energy of the plate were sought. Further, D’Alambert principle [9] for anybody undergoing displacement was introduced giving the potential energy of the plate. The solutions of the equations were sought in the form of Rayleigh Ritz approximate methods. More so, particular supports conditions of the platewere considered and their results compared with previous results where the plate is said to be orthotropic. Expressions relating to natural frequency of such an orthotropic plate were then established.
1.5 SIGNIFICANCE OF STUDY
The significance of this study lies on the following points:
a) The reinforced concrete slab design.
b) To the structural engineering students and other practicing engineers. c) The structural design regulatory bodies.
d) The government and their parastatals; also to the private sectors.
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e) Structural design consultants; and contractors and their clients.
It is also perceived that this study has the potential of generating further research in the following areas:
i. Dynamics of plates of various shapes and edge support conditions
ii. Dynamics effects of rectangular plates due to in plane loading (dynamic effect on plate due to buckling).
iii. Forcing Dynamic regime analysis of plates.
iv. In addition, dynamics of other anisotropic members.
1.6 SCOPE OF WORK
This workanalyzed orthotropic plates in free vibration regimeusing Rayleigh – ritz method by obtaining the approximate shape function of rectangular plates of different support conditions through characteristic orthogonal polynomials. It also derived new sets of stress-strain relations for orthotropic plates as well as the dynamic governing equation of orthotropic plate by the introduction of the force of inertia. A spreadsheet programme was developed to aid the solution of the equations.
1.7 LIMITATION OF STUDY This work is limited to the fundamental natural frequency of only rectangular orthotropic plates in free vibration regime. The support conditions of the plates considered were all edges clamped, all edges simply supported and mixed support conditions. Only avertical distributed load was considered on the rectangular orthotropic plate. The bending of the plate is only considered; but here, its dynamic response.No further attempts were made in experimenting the properties of these materials.
This material content is developed to serve as a GUIDE for students to conduct academic research
ANALYSIS OF ORTHOTROPIC PLATES IN FREE VIBRATION REGIME USING RAYLEIGH – RITZ METHOD
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