Hydroelastic analysis of circular very large floating structures

HYDROELASTIC ANALYSIS OF CIRCULAR VERY LARGE FLOATING STRUCTURES LE THI THU HANG NATIONAL UNIVERSITY OF SINGAPORE 2005 HYDROELASTIC ANALYSIS OF CIRCULAR VERY LARGE FLOATING STRUCTURES BY LE THI THU HANG B.E. (Hanoi University of Civil Engineering) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 ACKNOWLEDGEMENTS I wish to convey my sincere gratitude to my supervisor Professor Wang Chien Ming for his encouragements, critical comments and suggestions throughout the research work. His invaluable guidance and advice have greatly shaped my thinking over the past two years and what I have learnt and experience will undoubtedly be useful for my future studies. I am indeed grateful to Professor Tomoaki Utsunomiya from Kyoto University for his advice and useful discussions on this research study. I would like to express my thanks to the National University of Singapore for providing the financial support in the form of the NUS scholarship and facilities to carry out the research. Thanks are also extended to my colleagues in Civil Engineering Department for their kind assistance. Finally, special thanks to my family and my friends for their encouragements and love in many respects. Le Thi Thu Hang i TABLE OF CONTENTS ACKNOWLEGEMENTS ..................................................................................................... i TABLE OF CONTENTS…………………………………………………………………...ii SUMMARY……………………………………………………………………………...........v NOTATIONS………………………………………………………………………………..vii LIST OF FIGURES………………………………………………………………………....x LIST OF TABLES……………………………………………………………………..….xiv CHAPTER 1 INTRODUCTION ............................................................................... 1 1.1 BACKGROUND INFORMATION ON VLFS ........................................................1 1.2 LITERATURE REVIEW ........................................................................................5 1.3 OBJECTIVE OF RESEARCH .................................................................................8 1.4 LAYOUT OF THESIS..............................................................................................9 CHAPTER 2 VIBRATION ANALYSIS OF UNIFORM CIRCULAR PLATES....................................................................... 11 2.1 PROBLEM DEFINITION ......................................................................................11 2.2 GOVERNING EQUATIONS AND METHOD OF SOLUTION ..........................12 2.3 RESULTS AND DISCUSSIONS ...........................................................................16 2.4 CONCLUDING REMARKS..................................................................................31 CHAPTER 3 VIBRATION ANALYSIS OF STEPPED CIRCULAR PLATES....................................................................... 32 3.1 PROBLEM DEFINITON .......................................................................................32 3.2 METHOD OF SOLUTION AND MATHEMATICAL MODELLING.................32 ii 3.3 RESULTS AND DISCUSSIONS ...........................................................................38 3.4 CONCLUDING REMARKS..................................................................................61 CHAPTER 4 HYDROELASTIC ANALYSIS OF UNIFORM CIRCULAR VLFS ............................................................................ 62 4.1 BASIC ASSUMPTIONS AND PROBLEM DEFINITION ...................................62 4.2 BOUNDARY VALUE PROBLEMS AND GOVERNING EQUATIONS ...........63 4.3 MODAL EXPANSION OF MOTION ...................................................................66 4.4 SOLUTIONS FOR RADIATION POTENTIALS .................................................67 4.5 SOLUTIONS FOR DIFFRACTION POTENTIALS .............................................69 4.6 EQUATION OF MOTION IN MODAL COORDINATES ...................................70 4.7 NUMERICAL RESULTS.......................................................................................72 4.8 CONCLUDING REMARKS..................................................................................76 CHAPTER 5 HYDROELASTIC ANALYSIS OF STEPPED CIRCULAR VLFS ............................................................................ 77 5.1 PROBLEM DEFINITION ......................................................................................77 5.2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS ......................77 5.3 EQUATIONS OF MOTION IN MODAL COORDINATES.................................80 5.4 RESULTS AND DISCUSSIONS ...........................................................................83 5.5 CONCLUDING REMARKS..................................................................................89 CHAPTER 6 CONCLUSIONS............................................................................. 90 6.1 CONCLUSIONS.....................................................................................................90 6.2 RECOMMENDATIONS ........................................................................................91 iii REFERENCES ..................................................................................................................... 92 APPENDICES ...................................................................................................................... 97 APPENDIX 1 ELEMENTS OF MATRIX [K]9x9 FOR NON-AXISYMETRIC VIBRATION OF STEPPED CIRCULAR PLATE ..............................97 APPENDIX 2 ELEMENTS OF MATRIX [K]6x6 FOR AXISYMETRIC VIBRATION OF STEPPED CIRCULAR PLATE ............................102 iv SUMMARY This thesis presents a hydroelastic analysis of pontoon-type, circular, very large floating structure (VLFS) under action of waves. The coupled fluid-structure interaction problem may be solved by firstly decomposing the unknown deflection of the plate into modal functions associated with a freely vibrating plate in air. The second step involves substituting the modal functions into the hydrodynamic equations and solving the boundary value problem using the boundary element method. The modal amplitudes of the set of equations of motion obtained are then back substituted into the modal functions and the stress-resultants functions for the deflections and stress-resultants of the VLFS under the action of waves. Although one may use any form of admissible functions for the vibration modes, it is essential that the final stress-resultants satisfy the natural boundary conditions along the free edges of the plate. Recently, Wang et al. (2001) and Xiang et al. (2001) showed that the use of the classical thin plate theory for modeling the pontoon-type VLFS and well-known energy methods (such as the Ritz method and the finite element method) for vibration analysis yield modal stress resultants that (a) do not satisfy the natural boundary conditions and (b) contain oscillations/ripples in their distributions, affecting the accuracy of the peak values and their locations. When these modal solutions are used in the hydrodynamic analysis, the final stress-resultants will also contain these aforementioned inaccuracies. The use of the more refined plate theory of Mindlin (1951) that incorporates the effects of transverse shear deformation and rotary inertia, the accuracy of the stress-resultants, especially the transverse shear forces and twisting moments maybe improved. In order to develop accurate numerical v solution for detecting the hydroelastic response of VLFS, there is a need to obtain benchmark solutions, especially the vibration modes and modal stress-resultants of freely vibrating plates. As circular plate with free edge is the one can be obtained the exact vibration results, this study focuses on VLFS with a circular planform. By obtaining exact mode shapes and modal stress-resultants of circular Mindlin plate, the hydroelastic results are expected to be accurate. More specifically, we consider circular VLFS with constant thickness as well as thickness variation. A comparative study on deflection and stress-resultants between two kinds of circular plates (by keeping constant volume of material) is conducted. Numerical results show that the stepped circular VLFS gives much better performance than uniform circular plate because the final deflection and modal stress-resultants maybe reduced. Therefore, it would be beneficial to design stepped circular VLFS instead of uniform thickness one. The formulations for vibration analysis and hydroelastic analysis for uniform and stepped circular VLFS are given in explicit forms and the solutions obtained maybe regarded as almost exact. Such exact solutions should be extremely useful for the preliminary design of a circular VLFS. vi NOTATIONS A amplitude of incident wave (m) b step location for step thickness junction D plate rigidity of uniform circular plate (kNm) D0 plate rigidity of reference uniform circular plate (kNm) D1 plate rigidity of annular sub-plate (kNm) D2 plate rigidity of core circular sub-plate (kNm) E Young modulus (kN/m2) G shear modulus (kN/m2) g gravitational acceleration (m/sec2) h thickness of uniform circular plate (m) h0 thickness of reference uniform circular plate (m) h1 thickness of annular sub-plate (m) h2 thickness of core circular sub-plate (m) k wave number M rr bending moment per unit length the radial direction (kNm) M rr = M rr R / D0 = non-dimensional bending moment M rθ twisting moment (kNm) M rθ = M rθ R / D0 = non-dimensional twisting moment n number of nodal diameter Qr shear forces (kN) Qr =Q r R 2 / D0 = non-dimensional shear force vii R radius of circular plate (m) r radial coordinate (m) S non-dimensional plate rigidity of uniform circular plate s number of sequence for each mode S0 non-dimensional plate rigidity of reference uniform circular plate w vertical displacement (m) w = w / R = non-dimensional vertical displacement wmax maximum deflection (m) α step ratio of thicknesses = h2 h1 χ non-dimensional radial coordinate = r / R φ velocity potential φ Dn diffraction potential φ ns radiation potential γ density of plate material (kg/m3) κ2 correction factor λ = ϖR 2 γh D non-dimensionalized frequency parameter of circular plate λ0 = ϖ 0 R 2 γh0 D 0 non-dimensionalized frequency parameter of reference circular plate λs = ϖ s R 2 γh0 D 0 non-dimensionalized frequency parameter of stepped circular plate ν Poisson’s ratio Θi potential functions θ circumferential coordinate (radiants) viii ρ density of the fluid (kg/m3) τ thickness ratio of uniform circular plate = τ0 thickness ratio of reference uniform circular plate = τ1 thickness ratios of stepped circular plate = h1 R τ2 thickness ratios of stepped circular plate = h2 R ω natural frequency of uniform circular plate ω0 natural frequency of reference uniform circular plate ωs natural frequency of stepped circular plate ψr rotary displacement along radial axis of circular plate ψθ rotary displacement along circumferential of circular plate ζ ns modal apmplitude ∇(•) Laplacian operator ix h0 R h0 R LIST OF FIGURES Figure 1.1 Mega Float in Tokyo Bay, Japan............................................................ 2 Figure 1.2 Floating Oil Storage at Kamigoto, Japan................................................ 2 Figure 1.3 Yumeshima-Maishima Floating Bridge in Osaka, Japan ....................... 3 Figure 1.4 Floating Rescue Emergency Base at Tokyo Bay, Japan ........................ 3 Figure 1.5 Floating island at Onomichi Hiroshima, Japan....................................... 3 Figure 1.6 Floating pier at Ujina Port Hiroshima, Japan ......................................... 3 Figure 1.7 Floating Restaurant in Yokohoma, Japan............................................... 3 Figure 1.8 Floating heliport in Vancouver, Canada................................................. 3 Figure 1.9 Nordhordland Brigde Floating Bridge, Norway.................................... 3 Figure 1.10 Hood Canal Floating Bridge, USA......................................................... 3 Figure 1.11 Types of Floating Structures ................................................................... 4 Figure 2.1 Geometry of a Circular Mindlin Plate ................................................... 12 Figure 2.2a SAP2000 modal stress resultants associated with the fundamental frequency of a uniform circular plate with free edges .......................... 21 Figure 2.2b Exact modal stress resultants associated with the fundamental frequency of a uniform circular plate with free edges ........................................... 21 Figure 2.2c Mode shapes and modal stress resultants for free circular plates based on classical thin plate theory and Mindlin plate theory ............................ 21 Figure 2.3 3D-mode shape plots of uniform circular Mindlin plate.............. 22 to 24 Figure 2.4a Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.01. The number of nodal diameters n = 0 (axisymmetric modes) ........................................................................... 22 x Figure 2.4b Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.01. The number of nodal diameters is n = 1, 2, 3 and 4, respectively.......................................................................... 23 Figure 2.4c Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.01. The number of nodal diameters is n = 5, 6, 7 and 8, respectively.......................................................................... 24 Figure 2.5a Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.10. The number of nodal diameters is n = 0 (axisymmetric modes) ........................................................................... 25 Figure 2.5b Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.1. The number of nodal diameters is n = 1, 2, 3 and 4, respectively......................................................................... 26 Figure 2.5c Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.1. The number of nodal diameters is n = 5, 6, 7 and 8, respectively.......................................................................... 27 Figure 3.1 Geometry of a Stepped Circular Plate ................................................... 29 Figure 3.2 Frequency parameter λ s versus step location b for Mindlin plates with reference thickness ratio τ 0 = 0.1, α = 0.5 to 2 .................................... 43 Figure 3.3 Frequency parameter λ s versus step location b for plates with τ 0 = 0.1, α = 2 and n = 2 .................................................................................... 45 Figure 3.4a Frequency parameter λ s versus reference thickness ratios τ 0 for plates with step location b = 0.5 and stepped thickness ratio α = 0.5 ............ 46 Figure 3.4b Frequency parameter λ s versus reference stepped thickness ratio α for plates with step location b=0.5 and reference thickness ratio τ 0 =0.1 .. 46 xi Figure 3.5a Mode shapes (with n = 2, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 47 Figure 3.5b Mode shapes (with n = 0, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 48 Figure 3.5c Mode shapes (with n = 3, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 49 Figure 3.5d Mode shapes (with n = 1, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 50 Figure 3.5e Mode shapes (with n = 4, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 51 Figure 3.5f Mode shapes (with n = 5, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 52 Figure 3.5g Mode shapes (with n = 2, s = 2) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 53 Figure 3.5h Mode shapes (with n = 0, s = 2) and modal stress resultants for stepped plates and their reference constant thickness plates ............................ 54 Figure 3.6 Three Dimensional Stress-resultant Plots of Uniform and Stepped Circular Plates ............................................................................. 58 to 60 Figure 4.1 Geometry of an Uniform Circular VLFS ............................................. 57 Figure 4.2 Deflection for Problem 1, Real part and Imaginary part ...................... 68 Figure 4.3 Deflection Amplitude for Problem 1 .................................................... 68 Figure 4.4 Deflection for Problem 2, Real part and Imaginary part ...................... 68 Figure 4.5 Deflection Amplitude for Problem 2 .................................................... 68 Figure 4.6 Bending moment amplitude for Problem 2 .......................................... 69 Figure 4.7 Twisting moment amplitude for Problem 2.......................................... 69 xii Figure 4.8 Shear force amplitude for Problem 2.................................................... 69 Figure 5.1 Geometry of a Stepped Circular VLFS ................................................ 72 Figure 5.2 Displacements and Bending Moments for stepped VLFSs and the reference constant thickness VLFS ...................................................... 81 Figure 5.3 Twisting Moments and Shear Forces for stepped VLFSs and the reference constant thickness VLFS ...................................................... 82 Figure 5.4 Stresses M rri R /(τ i2 D0 A) , M rθi R /(τ i2 D0 A) , Qri R 2 /(τ i D0 A) for stepped VLFSs and the reference constant thickness VLFS ............................. 83 xiii LIST OF TABLES Table 2.1 Frequency parameters λ = ωR 2 γh / D for free circular Mindlin plates (ν = 0.3 , κ 2 = 5 / 6 )................................................................................. 20 Table 3.1a Frequency parameter λs for stepped plates with step location at b = 1/2, reference constant thicknesses τ o = 0.01 and 0.1 ................................. 39 Table 3.1b Frequency parameter λs for stepped plates with step location at b = 1/2 , reference constant thicknesses τ o = 0.125 and 0.15 .............................. 40 Table 3.2a Frequency parameter λs for stepped plates with step location at b = 1/3, reference constant thicknesses τ o = 0.01 and 0.1 .................................. 41 Table 3.2b Frequency parameter λ s for stepped plates with step location at b = 1/3, reference constant thicknesses τ o = 0.125 and 0.15 ............................. 42 Table 4.1 Parameters for Analyzed Circular VLFSs................................................ 66 Table 5.1 Parameters for Analyzed Stepped Circular VLFSs.................................. 79 xiv Chapter 1 INTRODUCTION This chapter introduces the very large floating structures (VLFSs) and their applications. A literature review on hydroelastic analysis of pontoon-type VLFS, the objective of research work and layout of the thesis are presented. 1.1 BACKGROUND INFORMATION ON VLFS With a growing of population and a corresponding expansion of urban development in land-scarce island countries and countries with long coastlines, the governments of these countries have resorted to land reclamation from the sea in order to ease the pressure on existing heavily-used land space. There are, however, constraints on land reclamation works, such as the negative environmental impact on the country’s and neigbouring countries’ coastlines and marine eco-system as well as the huge economic costs in reclaiming land from deep coastal waters, especially when the cost of sand for reclamation is very high. In response to both the aforementioned needs and problems, engineers have proposed the construction of very large floating structures (VLFS) for industrial space, airports, storage, facilities and even habitation. Japan, for instance, has constructed the Mega-Float in the Tokyo Bay (Fig. 1.1), the floating oil storage base Shirashima and Kamigoto (Fig. 1.2), the Yumeshima-Maishima floating bridge in Osaka (Fig. 1.3), the floating emergency rescue bases in Tokyo Bay, Osaka Bay (Fig. 1.4), the floating island at Onomichi Hiroshima (Fig. 1.5), the floating pier at Ujina Port Hiroshima (Fig. 1.6), and the floating restaurant in Yokohoma (Fig.1.7). Canada has constructed a floating heliport in Vancouver (Fig. 1.8) and the Kelowna floating bridge on Lake On in British Columbia. Norway has the Bergsoysund floating bridge 1 Introduction and Nordhordland Brigde (Fig. 1.9), while the United States has the Hood Canal floating bridge (Fig. 1.10) and Korea has a floating hotel. These VLFSs have advantages over the traditional land reclamation solution in the following aspects: • They are cost effective when the water depth is large and sea bed is soft • Environmentally friendly as they do not damage the marine eco-system or silt-up deep harbours or disrupt the ocean currents • They are easy and fast to construct and therefore sea-space can be speedily exploited • They can be easily removed or expanded • The structures on VLFSs are protected from seismic shocks since the floating structure is inherently base isolated • They do not suffer from differential settlement due to reclaimed soil consolidation • Their positions with respect to the water surface are constant and thus facilitate small boats and ship to come alongside when used as piers and berths • Their location in coastal water provide scenic body of water all around, making them suitable for developments associated with leisure and water sport activities. Figure 1.1 Mega Float in Tokyo Bay, Figure Japan 2 1.2 Floating Oil Kamigoto, Japan Storage at Introduction Figure 1.3 Yumeshima-Maishima Floating Bridge in Osaka, Japan Figure 1.4 Floating Rescue Emergency Base at Tokyo Bay, Japan Figure 1.5 Floating island at Onomichi Hiroshima, Japan Figure 1.6 Floating pier at Ujina Port Hiroshima, Japan Figure 1.7 Floating Restaurant in Yokohoma, Japan Figure 1.8 Floating heliport in Vancouver, Canada Figure 1.9 Nordhordland Brigde Floating Figure 1.10 Hood Canal Floating Bridge, Bridge, Norway USA _________________________ Figures courtesy of Prof E Watanabe, Kyoto University 3 Introduction VLFS may be classified under two categories, namely the semi-submersible type and the pontoon-type (see Fig. 1.11). The semi-submersibles type is designed to minimize the effects of waves while maintaining a constant buoyant force. Thus it can reduce the wave-induced movement of the structure, and therefore it is suitable for areas where the water is very deep and very high waves. The semi-submersibles are kept in position by either tethers or thrusters. In contrast, pontoon-type floating structures lie on the sea level like a giant plate floating on water (see Fig. 1.1). Pontoon-type floating structures are suitable for use in only calm waters, often near the shoreline. The pontoon-type VLFS is very flexible when compared to other kinds of offshore structures and so the elastic deformations are more important than their rigid body motions. Semi-submersible-type Pontoon-type Figure 1.11 Types of Floating Structures This thesis deals with the hydroelastic analysis of pontoon-type circular VLFSs under action of waves. Both uniform circular VLFS and stepped circular VLFS’s solutions are considered. This study develops analytical approach for hydroelastic analysis of these VLFS structures. Exact deflections and stress resultants are given and should be useful as they served as benchmark solutions for verification of numerical programs such as BEM or FEM for VLFS analysis. 4 Introduction 1.2 LITERATURE REVIEW The hydroelastic analysis of very large floating structures has attracted the attention of many researchers, especially with the construction of the Mega-Float in Tokyo Bay in 1995. Many researchers analyzed pontoon-type VLFS of a rectangular planform (Utsunomiya et al. 1998, Mamidipudi and Webster 1994, Endo 2000, Ohkusu and Namba 1998, Namba and Ohkusu 1999), mainly because of practical reasons for this shape and also it lends itself for the construction of semi-analytical methods for solution. There are very few studies on non-rectangular VLFS. Hamamota and Fujita (2002) treated L-shaped, T-shaped, C-shaped and X-shaped VLFSs. It was suggested that hexagonal shaped VLFSs be constructed as Japanese Society of Steel Construction (1994). Circular pontoon-type VLFSs are considered by Hamamoto (1994), Zilman and Miloh (2000), Tsubogo (2001), Peter et al. (2003) and Watanabe et al. (2003). So it so appears that more studies on VLFSs of circular shape should be carried out. The hydroelastic analysis of VLFS may be conducted in the frequency domain or in the time domain. Most hydroelastic analyses are carried out in the frequency-domain, being the simpler of the two. The commonly-used approaches for the analysis of VLFS in the frequency domain are the modal expansion method and the direct method. The principal difference between the modal superposition method and the direct method lies in the treatment of the radiation motion for determining the radiation pressure. In the direct method, the deflection of the VLFS is determined by directly solving the motion of equation without any help of eigenmodes. Mamidipudi and Webster (1994) pioneered this direct method for a VLFS. In their solution procedure, the potential of diffractions and radiation problems were established first, and the deflection of VLFS was determined by solving the combined hydroelastic equation via the finite difference scheme. Their method was modified by Yago and Endo (1996) 5 Introduction who applied the pressure distribution method and the equation of motion was solved using the finite element method. Ohkusu and Namba (1996) proposed a different type of direct method which does away with the commonly-used two-step modal expansion approach. Their approach is based on the ideal that the thin plate is part of the water surface but with different physical characteristics than those of the free surface of the water. The problem is considered as a boundary value problem in hydrodynamics rather than the determination of action. In Kashiwagi’s direct method (1998), the pressure distribution method was applied and the deflection was solved from the vibration equation of the structure. In order to archive a high level of accuracy in a very short wave length regime as well as short computational times and fewer unknowns, he uses bi-cubic B-spline functions to present the unknown pressure and a Galerkin’s method to satisfy the boundary conditions. His method for obtaining accurate results in the short wave length regime is a significant improvement over the numerical techniques proposed by other researchers (Maeda et al. 1995, Takaki and Gu 1996, Yago 1995, Wang et al. 1997), who have also employed the pressure distribution method. The modal expansion method consists of separating the hydrodynamic analysis and the dynamic response analysis of the plate. The deflection of the plate with free edges is decomposed into vibration modes that can be arbitrarily chosen. In this regard, researchers have adopted different modal functions such as products of free-free beam modes (Maeda et al. 1995, Wu et al. 1995, 1996 and 1997, Kashiwagi 1998, Nagata et al. 1998, Utsunomiya et. al. 1998, Ohmatsu 1998), B-spline function (Lin and Takaki 1998), Green function (Eatock Taylor and Ohkusu 2000), two-dimensional polynomial functions (Wang et al. 2001) and finite element solutions of freely vibrating plates (Takaki and Gu 1996). Also, it should be remarked that the modes may be that of the 6 Introduction dry type or the wet type. While the most analyses used the dry-mode approach (Wu et al. 1997) because of its simplicity and numerical efficiency, Hamamoto et al. (1995, 1996, 1997, 2002) have conducted studies using the wet-mode approach. Although the dry-modes superposition and wet-modes superposition can lead to the same solution, the wet-mode superposition approach is considered to be rather complex (for example, an iterative procedure is needed to obtain a wet-mode). In order to validate the numerical methods and to check the accuracy and convergence of solutions, analytical solutions are important needed for hydrodynamic response of VLFSs. Moreover it was shown that numerical techniques such as the finite element method (FEM) and the Rayleigh-Ritz method actually do not furnish accurate distributions of modal stress-resultants (Wang et al. 2001, Xiang et al. 2001). In fact, the distributions of the numerically obtained modal stress-resultants contain oscillations and they do not satisfy completely the natural boundary conditions at the free edges. The reason for this shortcoming is that the FEM and the Rayleigh-Ritz method do not directly enforce the natural boundary conditions as is done for the essential boundary conditions. Therefore exact vibration solutions, especially exact modal stress resultants, for free plates are important to have as benchmark solutions for assessing the accuracy of numerical results. A plate shape that admits the derivation of exact solutions for plates with free edges is the circular shape. Probably, the first paper on hydroelastic analysis of circular VLFS is the one written by Hamamoto and Tanaka (1992). They developed an analytical approach to predict the dynamic response of a flexible circular floating island subjected to stochastic wind-waves and seaquakes (see also Hamamoto, 1994). Their approach was based on the superposition of wet-modes (free vibration modes in still-water). 7 Introduction Researchers have also been seeking analytical solutions. Zilman and Miloh (2000) obtained a closed form solutions of the hydroelastic response of a circular floating plate in shallow waters. Tsubogo (2000, 2001) solved the same floating circular plate problem independently. However, the assumption of shallow water limits the applicability range, and the extension of their method to finite-water depth has not yet been made. 1.3 OBJECTIVE OF RESEARCH Complementing the above studies, this study will develop analytical approach for hydroelastic analysis of a circular VLFS. The analysis is carried out in the frequency domain using the modal expansion method (dry-mode superposition). The aims of the present study are • to determine the exact mode shape and modal stress-resultants of freely vibrating circular plate with uniform thickness as well as stepped thickness variation. • to solve the hydroelastic problem of pontoon-type circular VLFS under action of waves. In the open literature, many analysts used the classical thin plate theory for modeling the pontoon-type VLFS. For more accurate evaluation of modal stress resultants, the more refined plate theory proposed by Mindlin (1951) should be adopted instead. The Mindlin plate theory allows for the effect of transverse shear deformation and rotary inertia which become significant in high vibration modes. Moreover, the stress-resultants are evaluated from first order derivatives of deflection and rotations. In contrast, the stress-resultants in the classical thin plate theory are expressed in terms of second order and third order derivatives of deflection. Therefore, more accurate stress-resultants can be obtained by using the Mindlin plate theory. 8 Introduction VLFSs are usually designed as optimally as possible with properties sometimes varying abruptly over their cross-sections for economic distribution of materials. Owing to the variations in structural properties, the deflection pattern may have a very different spatial character from a similar structure with uniform structural property characteristics. Therefore, vibration problem of stepped circular plate is tackled with and the hydroelastic analysis solution for stepped circular VLFS is presented in this thesis. The influence of the stepped thickness design on the vibration frequencies, mode shapes and modal stress resultants is explored by comparing with the corresponding results of a reference circular plate of constant thickness and equal volume. Comparison study of the deflections and stress-resultants of stepped circular VLFS and its reference uniform thickness circular VLFS are also given. These exact solutions and research findings should be useful in the hydroelastic analysis and costeffective design of circular VLFSs with a stepped thickness variation. 1.4 LAYOUT OF THESIS This thesis comprises of two parts. Part 1, consisting of Chapters 2 and 3, deals with the free vibration analysis of a uniform and non-uniform circular plates vibrating in air, normally reformed to dry mode solution. Part 2, consisting of Chapters 4 and 5, is concerned with the hydroelastic analysis of these circular VLFSs under actions of waves. More specifically, Chapter 2 deals with the free vibration analysis of circular plates with uniform thickness. Adopting the Mindlin plate theory, the governing equations and the boundary condition are presented. They are solved analytically and the natural frequencies, mode shapes and modal stress-resultants are given. 9 Introduction Chapter 3 is concerned with the free vibration solution of stepped circular plates. In solving such a stepped plate problem, the stepped plate is decomposed into two subplates, a core circular plate and an outer annular plate. The Mindlin plate theory is also adopted. The boundary conditions are those of free edges the continuity conditions at the interface between two sub-plates. By keeping the volume of stepped plates a constant, the frequency values, mode shape and modal stress-resultants are investigated with respect to those of a corresponding circular plate with constant thickness. The influence of the stepped thickness design on the vibration frequencies, mode shapes and modal stress resultants is also explored. In the hydrodynamic analysis of a VLFS structure, the mode shapes and modal stress resultants from the free vibration analysis of the structure are utilized to predict the dynamic responses of the structure. Following studies on the free vibration analysis, Chapter 4 and 5 deal with hydroelastic analysis of uniform circular VLFS and stepped circular VLFS, respectively. The analysis of VLFS is carried out in the frequency domain using modal expansion matching method. Firstly, decomposing the deflection of circular Mindlin plates given in Chapter 2 and 3 into vibration modes and then the hydrodynamic diffraction and radiation forces are evaluated by using eigenfunction expansion matching method. The modal deflection and stress resultants of both uniform and nonuniform circular VLFS are served as benchmark solution for checking the validity, convergence and accuracy of numerical solutions and methods for analysis of pontoontype VLFSs. In Chapter 6, the conclusions and some suggestions for future research work on circular VLFS are presented. 10 Chapter 2 VIBRATION ANALYSIS OF UNIFORM CIRCULAR PLATES Presented herein are exact vibration solutions of freely vibrating, circular Mindlin plates with free edges. The natural frequencies, mode shapes and modal stress-resultant are given for various plate thickness to radius ratios. As the vibration analysis is carried out analytically the stress-resultants obtained completely satisfy the natural boundary conditions. 2.1 PROBLEM DEFINITION Considered as an isotropic, flat circular plate of radius R, thickness h, mass density γ , Poisson’s ratio ν , Young’s modulus E and shear modulus G ( = E /[2(1 + ν )] ). The plate is free from any attachment/support as shown in Fig. 2.1 R r, χ θ o Free Edge h Figure 2.1 Geometry of a Circular Mindlin Plate 11 Vibration Analysis of Uniform Circular Plate The problem at hand is to determine the modal displacement fields and stress resultants for the freely vibrating circular plate. To allow for the effects of transverse shear deformation and rotary inertia the Mindlin plate theory is adopted instead of the commonly used classical thin plate theory. 2.2 GOVERNING EQUATIONS AND METHOD OF SOLUTION Following the work by Mindlin and Deresiewicz (1951), the rotations (ψ r ,ψ θ ) and transverse displacements w may be expressed as functions of three potentials Θ1 , Θ 2 and Θ 3 in the following manner: ψ r = (σ 1 − 1) ∂Θ ∂Θ 1 1 ∂Θ 3 + (σ 2 − 1) 2 + ∂χ χ ∂θ ∂χ ψ θ = (σ 1 − 1) 1 ∂Θ 1 χ ∂θ + (σ 2 − 1) 1 ∂Θ 2 χ ∂θ − (2.1) ∂Θ 3 ∂χ (2.2) w = Θ1 + Θ 2 (2.3) where σ1,σ 2 = (δ 22 , δ 12 ) ⎡τ 2 λ2 6(1 − ν )κ 2 ⎤ − ⎢ ⎥ τ2 ⎣ 12 ⎦ = 2(δ 22 , δ 12 ) δ 32 (1 − ν ) ⎡ (2.4) 2 (2.5) 2 ⎡τ 2 λ2 6(1 − ν )κ 2 ⎤ − δ = ⎥ (1 − ν ) ⎢⎣ 12 τ2 ⎦ (2.6) 2 2 λ 2 ⎢τ 2 ⎤ ⎛τ 2 ⎞ τ2 τ2 4 ⎜ ⎟ + 2⎥ + ± − δ ,δ = 2 2 ⎟ ⎜ 2 ⎢ 12 6(1 − ν )κ λ ⎥ ⎝ 12 6(1 − ν )κ ⎠ 2 1 ⎣ ⎦ 2 3 χ= γh r h , τ = , λ = ωR 2 D R R (2.7 a, b, c) 12 Vibration Analysis of Uniform Circular Plate in which r and θ are the radial and circumferential coordinates of the polar coordinate system, w, ψ r and ψ θ the transverse displacement and rotations in the Mindlin plate theory, w is the transverse displacement of the plate, χ the non-dimensionalised radial coordinate (see Figure 2.1), κ 2 the shear correction factor, and λ the nondimensionalised frequency parameter. In view of the three potential functions Θ1 , Θ 2 and Θ 3 the governing differential equations of the vibrating circular plate, in polar coordinates, may be compactly expressed as (Mindlin 1951) (∇ 2 + δ 12 ) Θ1 = 0 (2.8) (∇ 2 + δ 22 ) Θ 2 = 0 (2.9) (∇ 2 + δ 32 Θ 3 = 0 ) (2.10) where the Laplacian operator ∇ 2 (•) = ∂ 2 (•) 1 ∂ (•) 1 ∂ 2 (•) + + 2 χ ∂χ ∂χ 2 χ ∂θ 2 (2.11) The general solutions to equations (2.8) to (2.10) may be expressed as Θ1 = A1 Rn (∆ 1 χ ) cos nθ + B1 S n (∆ 1 χ ) cos nθ (2.12) Θ 2 = A2 Rn (∆ 2 χ ) cos nθ + B2 S n (∆ 2 χ ) cos nθ (2.13) Θ 3 = A3 Rn (∆ 3 χ )sin nθ + B3 S n (∆ 3 χ )sin nθ (2.14) ⎧⎪ δ ∆j =⎨ j ⎪⎩Im δ j , j = 1, 2, 3 (2.15) if δ j2 ≥ 0 , j = 1, 2, 3 if δ j2 < 0 (2.16) where ( ) if δ j2 ≥ 0 if δ j2 < 0 ( ( ⎧⎪ J ∆ χ Rn ∆ j χ = ⎨ n j ⎪⎩ I n ∆ j χ ( ) ) ) 13 Vibration Analysis of Uniform Circular Plate ( ( ⎧⎪ Y ∆ χ Sn ∆ j χ = ⎨ n j ⎪⎩ K n ∆ j χ ( ) ) ) if δ j2 ≥ 0 if δ j2 < 0 , j = 1, 2, 3 (2.17) in which A j and B j ( j = 1, 2 and 3) are the arbitrary constants that will be determined by the free boundary conditions of the plate, n is the number of nodal diameters of a vibration mode, J n (•) and I n (•) are the first kind and the modified first kind Bessel functions of order n, and Yn (•) and K n (•) the second kind and the modified second kind Bessel functions of order n. For a circular plate, the arbitrary constants B j must be set to zero in order to avoid singularity for the displacement fields w , ψ r and ψ θ at the plate centre ( χ = r / R = 0 ). The displacement fields and the stress resultants of the circular plate are therefore expressed in terms of the arbitrary constants A j . The boundary conditions of circular Mindlin plate with free edge given by Qr = 0, M rr = 0, M rθ = 0 (2.18 a-c) where the transverse shear force Qr , the radial bending moment M rr and the twisting moment M rθ are given by ⎛ ∂w Qr = κ 2 Gh⎜⎜ +ψ r ⎝ ∂χ ⎞ ⎟⎟ ⎠ (2.19) M rr = ∂ψ θ D ⎡ ∂ψ r ν ⎛ + ⎜⎜ψ r + ⎢ R ⎢⎣ ∂χ χ⎝ ∂θ ⎞⎤ ⎟⎟⎥ ⎠⎥⎦ (2.20) M rθ = D ⎛ 1 - ν ⎞ ⎡ 1 ⎛ ∂ψ r −ψ θ ⎜ ⎟⎢ ⎜ R ⎝ 2 ⎠ ⎣⎢ χ ⎜⎝ ∂θ ⎞ ∂ψ θ ⎤ ⎟⎟ + ⎥ ⎠ ∂χ ⎥⎦ (2.21) By substituting Eqs. (2.12) to (2.14) into Eqs. (2.1) to (2.3) and then into Eqs. (2.18 a-c), one obtain a homogeneous system of equations which may be expressed as 14 Vibration Analysis of Uniform Circular Plate ⎧ A1 ⎫ [K ]⎪⎨ A2 ⎪⎬ = {0} ⎪A ⎪ ⎩ 3⎭ (2.22) and [K ] is a 3 x 3 matrix where the elements are given by k11 = (σ 1 − 1)[ J n'' (δ 1 ) + νJ n' (δ 1 ) − νn 2 J n (δ 1 )] (2.23) k12 = (σ 2 − 1)[ J n'' (δ 2 ) + νJ n' (δ 2 ) − νn 2 J n (δ 2 )] (2.24) k13 = n(1 − ν )[ J n' (δ 3 ) − J n (δ 3 )] (2.25) k 21 = −2n(σ 1 − 1)[ J n' (δ 1 ) − J n (δ 1 )] (2.26) k 22 = −2n(σ 2 − 1)[ J n' (δ 2 ) − J n (δ 2 )] (2.27) k 23 = − J n'' (δ 3 ) + J n' (δ 3 ) − n 2 J n (δ 3 )] (2.28) k 31 = σ 1 J n' (δ 1 ) (2.29) k 32 = σ 2 J n' (δ 2 ) (2.30) k 33 = nJ n (δ 3 ) (2.31) By setting the determinant of [K ] in equation (2.22) to be zero and solving the characteristic equation for the root, we obtain the natural frequency of vibration. The modal displacement fields w , ψ r and ψ θ and modal stress resultants Qr , M rr and M rθ are calculated from the angular frequency ω and the corresponding eigenvector [A1 A2 A3 ] . In presenting the vibration modes and modal stressT resultants, we normalize the maximum transverse displacement w= w ; wmax = 1 R (2.32 a,b) 15 Vibration Analysis of Uniform Circular Plate The corresponding bending moment, twisting moment and shear force are presented in their non-dimensional forms as follows: M rr = R M rr D (2.33) M rθ = R M rθ D (2.34) R2 Qr = Qr D (2.35) 2.3 RESULTS AND DISCUSSIONS Before presenting vibration solutions for circular plates, we demonstrate the shortcomings of the finite element method in obtaining accurate modal stress resultants. Take for instance, the problem of a circular plate with free edges and its thickness to radius ratio being equal to 0.01. We compute the fundamental vibration frequency of this plate using well-known finite element software packages such as SAP2000 and ABAQUS. Fig. 2.2a shows clearly that SAP2000 modal stress resultants do not satisfy the natural boundary conditions at the free edge, especially the twisting moment and the transverse shear force. On the other hand, Fig. 2.2b shows the corresponding exact solutions that satisfy the boundary conditions. Moreover, the peak value of modal stress resultants of SAP2000 have not converged to the exact values even though a very fine mesh design was used (see Fig. 2.2a for the mesh design). For example, the peak value of modal transverse shear force Qr = 1.315 was obtained by SAP2000 while the corresponding exact peak value is Qr = 0.926 , a difference of 42%. Moreover, Figure 2.2c compares the exact modal displacement w and modal stress resultants Qr , M rr and M rθ for a free circular plate obtained on the basis of the classical thin plate theory (Leissa, 1969) and of the Mindlin plate theory. The 16 Vibration Analysis of Uniform Circular Plate normalised effective shear force Vr is calculated based on its definition in the classical thin plate theory. The plate thickness ratio h/R is taken to be 0.01 and the number of nodal diameters n and the mode sequence s are set to be 4 and 1, respectively. It shows that the mode shape w and modal stress resultants from the thin and thick plate theories are almost the same except for Qr and M rθ near the vicinity of the plate edge. Unfortunately, the discrepancies found at the vicinity of the free edge also contain the peak values of the stress-resultants. And the boundary conditions Qr = 0 and M rθ = 0, are not satisfied when using the classical thin plate theory due to the free edge conditions based on the thin plate theory are Vr = 0 and M rr = 0. Clearly, these shows the importance of exact free vibration solutions that we shall be presenting below for benchmark purposes as well as for use in the hydroelastic analysis of circular VLFS. The Poisson ratio ν = 0.3 and the shear correction factor κ 2 = 5 / 6 are adopted for all calculations. Exact vibration frequency parameters λ = ωR 2 γh / D for free circular Mindlin plates with thickness to radius ratios of 0.005, 0.01, 0.1, 0.125 and 0.15 are presented in Table 2.1. The number of nodal diameters n varies from 1 to 8 and the mode sequence number s (for a given n value) for 1 to 4, respectively. For a better view of how a circular plate deflect regarding to number of n and s, one may refer to the 3D-plots of mode shapes as given in Figure 2.3 In the hydrodynamic analysis of a VLFS structure, the mode shapes and modal stress resultants from the free vibration analysis of the structure are utilized to predict the dynamic responses of the structure. The exact mode shapes and modal stress resultants for free circular Mindlin plates are presented herein thus serve as important benchmark values for researchers to verify their numerical models for circular Mindlin plate analysis. The mode shapes and modal stress-resultants with frequency values that 17 Vibration Analysis of Uniform Circular Plate are boldfaced (in Table 2.1) are depicted in Figures 2.4a-c and 2.5a-c, respectively. Note that the modal displacement fields and modal stress resultants in Figures 2.4 and 2.5 are plotted along radial direction where their peak values are found. The modal displacements w and ψ r , and modal stress resultants Qr and M rr in the circumferential direction vary with cos nθ , while the modal displacement ψ θ and modal stress resultant M rθ vary with sin nθ . Figures 2.4a and 2.5a present the normalized modal displacement fields and modal stress resultants along the radial direction for a thinner circular Mindlin plate (h/R = 0.01) and a thicker plate (h/R = 0.10), respectively. The plates vibrate in axisymmetric modes (n = 0). The modal displacement fields and modal stress resultants for the thinner and thicker plates show very similar trends. The values of the modal rotation ψ r and the modal stress resultants Qr and M rr for the thicker plate are smaller than the ones for the thinner plate. As expected, the rotation ψ θ and moment M rθ on the whole plate and the rotation ψ r and shear force Qr at the centre of the plates ( χ = r/R = 0) are zero due to the plates vibrating in axisymmetric modes. The modal stress resultants Qr , M rr and M rθ vanish at the plate free edge ( χ = r/R = 1). The number of nodal circumferential lines of the modal displacements w and ψ r and modal stress resultant M rr increases from 1 to 4 as the mode sequence number s varies from 1 to 4. However, the number of nodal circumferential lines of the modal stress resultant Qr changes from 2 to 5 while the mode sequence number s increases from 1 to 4. Figures 2.4b, 2.4c, 2.5b and 2.5c show the normalized modal displacement fields and modal stress resultants along the radial direction for a thinner circular Mindlin plate (h/R = 0.01) and a thicker plate (h/R = 0.10), respectively. The vibration of the plates is non-axisymmetric. Similar trends are observed for the modal displacement 18 Vibration Analysis of Uniform Circular Plate fields and modal stress resultants of the thinner and thicker plates. The values of the modal displacements ψ r and ψ θ and the modal stress resultants Qr and M rr for the thicker plate are smaller than the ones for the thinner plate. The mode sequence number s is fixed at 1 and the number of nodal diameters n varies from 1 to 8. It is interesting to observe that there are two nodal circumferential lines for the modal displacement w if the plates vibrate with one nodal diameter (n = 1). The modes with two nodal diameters (n = 2) are the fundamental modes as shown by the frequency values in Table 2.1. The modal displacement fields and stress resultants for the modes with 3 and more nodal diameters (i.e. n ≥ 3 ) show similar trends in general. The modal displacement fields w , ψ r and ψ θ and stress resultants Qr , M rr and M rθ are zero at the centre of the plates ( χ = r/R = 0). The values of the displacement fields w , ψ r and ψ θ increase monotonically with increasing radial coordinate ( χ = r/R) except for the rotation ψ θ of the thicker plate near the free edge where a slight decrease of ψ θ is observed. It is observed that as the number of nodal diameters n increases from 3 to 8, the vibration of the plates is more concentrated on the portion of the plates near the free edge. It can be seen that the stress resultants Qr , M rr and M rθ satisfying the natural boundary condition at free edge for all cases shown in Figures 2.4a to c and 2.5a to c. It is noted that for the thinner plate (h/R = 0.01), there are sharp variations in stress resultants Qr and M rθ near the vicinity of the free edge when the number of nodal diameters n varies from 2 to 8 as shown in Figures 2.4b and 2.4c. For the thicker plate (h/R = 0.10), however, the variation of the stress resultants Qr , M rr and M rθ near the vicinity of the free edge becomes quite smooth (see Figures 2.5b and 2.5c) and the peak values of the shear force Qr near the free edge for the thicker plate is much smaller than the ones for the thinner plate. 19 Vibration Analysis of Uniform Circular Plate Table 2.1 Frequency parameters λ = ωR 2 γh / D for free circular Mindlin plates ( ν = 0 .3 , κ 2 = 5 / 6 ) n s 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 0.005 9.00279 38.4365 87.7151 156.706 20.4698 59.7918 118.889 197.689 5.35655 35.2426 84.3196 153.183 12.4320 52.9684 111.856 190.492 21.8188 73.4713 142.283 230.727 33.4641 96.6411 175.508 273.816 47.3269 122.400 211.462 319.699 63.3767 150.689 250.085 368.322 81.5905 181.462 291.329 419.639 Thickness Ratio, h/R 0.01 0.10 0.125 8.79655 9.00175 8.86877 34.9310 38.4164 36.0592 72.3414 87.6099 76.7577 116.194 156.370 126.483 20.4613 19.7165 19.3460 59.7396 54.2993 51.8997 118.692 100.071 93.1010 197.151 153.044 139.024 5.24446 5.35453 5.27822 35.2140 33.0500 32.0525 84.2088 73.9519 69.7839 152.848 123.973 113.996 11.9070 12.4237 12.0667 52.9040 48.2623 46.2614 111.655 94.6541 88.2630 189.964 148.269 134.907 20.3694 21.7983 20.8089 73.3521 64.9534 61.5587 141.953 116.137 107.120 229.944 172.818 155.757 30.3583 33.4237 31.2861 96.4439 82.8204 77.6511 175.006 138.198 126.204 272.708 197.516 176.495 41.6200 47.2564 43.2853 122.097 101.624 94.3233 210.736 160.684 145.411 318.185 222.285 197.089 53.9310 63.2633 56.6081 150.248 121.173 111.415 249.078 183.476 164.667 366.313 247.068 217.516 67.0998 81.4185 71.0734 180.845 141.313 128.807 289.975 206.485 183.920 417.037 271.824 237.763 0.15 8.71132 33.7076 67.9521 106.673 18.9273 49.4103 86.4264 126.415 5.20584 30.9716 65.6276 104.738 11.7271 44.1745 82.1058 122.837 19.8850 58.1397 98.6884 140.694 29.3621 72.6068 115.277 158.297 39.8807 87.3984 131.809 175.638 51.2114 102.392 148.245 192.713 63.1697 117.501 164.557 209.519 Note: n is the number of nodal diameters of the mode and s is the mode sequence for a given n value. The cases with the boldfaced values have their modes and modal stress resultants presented. 20 Vibration Analysis of Uniform Circular Plate Nondimensionalized Stress Resultants 2 h/R = 0.01, λ = 5.39672 M rr Qr χ =1 = 1 . 315 M rθ Qr 1 M rr 0.0 0.2 0.4 0.6 0.8 χ =1 = 0.074 χc 0 1.0 Mrθ χ=1 = −0.44 -1 Mesh Design -2 Figure 2.2a SAP2000 modal stress resultants associated with the fundamental frequency of a uniform circular plate with free edges Nondimensionalized Stress Resultants 2 h/R = 0.01, λ = 5.35453 R M rr M rθ 1 r, χ Qr χc 0 0.0 0.2 0.4 0.6 0.8 θ o 1.0 -1 -2 Figure 2.2b Exact modal stress resultants associated with the fundamental frequency of a uniform circular plate with free edges 60 M odal Stress Resultants M odal D isplacem ent w 1 0.75 Thick Plate Theory Thin Plate Theory 0.5 0.25 χ 0 0.0 0.2 0.4 0.6 0.8 1.0 Qr M rr M rθ 40 Thick Plate Theory Thin Plate Theory Vr Effective Shear ForceVr 20 0 χ -20 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2.2c Mode shapes and modal stress resultants for free circular plates based on classical thin plate theory and Mindlin plate theory ( h R = 0.01, n = 4, s = 1 ) 21 n s n s 0 1 1 1 0 2 1 2 0 3 1 3 0 4 1 4 Mode shape Figure 2.3 3D-mode shape plots of uniform circular Mindlin plate 22 Mode shape n s n s 2 1 3 1 2 2 3 2 2 3 3 3 2 4 3 4 Mode shape Figure 2.3 (Contd.) 3D-mode shape plots of uniform circular Mindlin plate 23 Mode shape n s n s 4 1 5 1 4 2 5 2 4 3 5 3 4 4 5 4 Mode shape Figure 2.3 (Contd.) 3D-mode shape plots of uniform circular Mindlin plate 24 Mode shape Vibration Analysis of Uniform Circular Plate Modal Displacement Fields Modal Stress Resultants n = 0, s = 1 1 n = 0, s = 1 20 Qr 15 M rr 0 0.0 0.2 0.4 0.6 0.8 1.0 -1 M rθ 10 5 0 w -2 0.0 ψr 0.2 0.4 0.6 0.8 1.0 0.8 1.0 0.8 1.0 0.8 1.0 -5 ψθ -3 -10 n = 0, s = 2 4 n = 0, s = 2 100 Qr M rr 50 2 M rθ 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 -50 w -2 ψr -100 ψθ -150 -4 n = 0, s = 3 n = 0, s = 3 4 600 w Qr ψr 2 M rr 400 ψθ M rθ 0 200 0.0 0.2 0.4 0.6 0.8 1.0 -2 0 0.0 -4 -200 -6 -400 0.2 n = 0, s = 4 900 6 600 4 300 2 0 0 -300 0.0 0.0 -4 0.2 0.4 0.6 0.6 n = 0, s = 4 8 -2 0.4 0.8 0.2 0.4 0.6 1.0 -600 w ψr -900 ψθ -6 Qr M rr M rθ -1200 Figure 2.4a Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.01. The number of nodal diameters n = 0 (axisymmetric modes) 25 Vibration Analysis of Uniform Circular Plate Modal Displacement Fields Modal Stress Resultants n = 1, s = 1 n = 1, s = 1 6 75 w 4 50 ψr 25 ψθ 2 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 -25 -2 -50 -4 -75 -6 -100 Qr M rr M rθ n = 2, s = 1 n = 2, s = 1 3 2 w ψr 2 ψθ 1 1 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.0 0.2 0.4 0.6 -1 0.8 1.0 Qr -1 M rr M rθ -2 -2 n = 3, s = 1 n = 3, s = 1 12 3 w ψr 2 Qr 9 M rr ψθ M rθ 6 1 3 0 0.0 0.2 0.4 0.6 0.8 1.0 -1 0 -2 -3 -3 -6 0.0 0.2 n = 4, s = 1 0.4 0.6 0.8 1.0 n = 4, s = 1 25 4 w 3 ψr ψθ 2 20 Qr 15 M rr M rθ 10 1 5 0 0.0 0.2 0.4 0.6 0.8 0 1.0 0.0 -1 0.2 0.4 0.6 0.8 1.0 -5 -2 -10 -3 -15 Figure 2.4b Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.01. The number of nodal diameters is n = 1, 2, 3 and 4, respectively 26 Vibration Analysis of Uniform Circular Plate Modal Displacement Fields Modal Stress Resultants n = 5, s = 1 w 4 ψr ψθ 3 n = 5, s = 1 50 5 40 Qr 30 M rr M rθ 2 20 1 10 0 0 0.0 0.2 0.4 0.6 0.8 0.0 1.0 -1 -10 -2 -20 0.2 0.4 0.6 0.8 1.0 0.8 1.0 -30 -3 n = 6, s = 1 n = 6, s = 1 75 6 w Qr ψr 4 50 ψθ M rr M rθ 25 2 0 0 0.0 0.2 0.4 0.6 0.8 0.0 1.0 -2 -25 -4 -50 0.2 n = 7, s = 1 0.4 0.6 n = 7, s = 1 8 100 w ψr 6 Qr 75 ψθ M rr 50 M rθ 4 25 2 0 0.0 0 0.0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.0 -25 1.0 -2 -50 -4 -75 n = 8, s = 1 n = 8, s = 1 8 150 w 6 ψr Qr 100 ψθ 4 M rr M rθ 50 2 0 0.0 0.2 0.4 0.6 0.8 0 1.0 0.0 -2 0.2 0.4 0.6 0.8 1.0 -50 -4 -6 -100 Figure 2.4c Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.01. The number of nodal diameters is n = 5, 6, 7 and 8, respectively 27 Vibration Analysis of Uniform Circular Plate Modal Displacement Fields Modal Stress Resultants n = 0, s = 1 n = 0, s = 1 1 20 Qr 15 M rr 0 0.0 0.2 0.4 0.6 0.8 1.0 -1 M rθ 10 5 0 w -2 0.0 ψr 0.2 0.4 0.6 0.8 1.0 -5 ψθ -3 -10 n = 0, s = 2 n = 0, s = 2 4 100 Qr M rr 50 2 M rθ 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 -50 w -2 ψr -100 ψθ -4 -150 n = 0, s = 3 n = 0, s = 3 600 4 w Qr ψr 2 M rr 400 ψθ M rθ 200 0 0.0 0.2 0.4 0.6 0.8 1.0 0 -2 0.0 -4 -200 -6 -400 0.2 n = 0, s = 4 900 6 600 4 300 2 0 0 -300 0.0 0.0 -4 0.2 0.4 0.6 0.6 0.8 1.0 0.8 1.0 n = 0, s = 4 8 -2 0.4 0.8 0.2 0.4 0.6 1.0 -600 w ψr -900 ψθ Qr M rr M rθ -1200 -6 Figure 2.5a Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.10. The number of nodal diameters is n = 0 (axisymmetric modes) 28 Vibration Analysis of Uniform Circular Plate Modal Displacement Fields Modal Stress Resultants n = 1, s = 1 n = 1, s = 1 75 6 50 w 4 ψr 25 ψθ 2 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 -2 0.2 0.4 0.6 0.8 1.0 0.8 1.0 -25 -50 -4 -75 -6 -100 Qr M rr M rθ n = 2, s = 1 n = 2, s = 1 3 2 w ψr 2 ψθ 1 1 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.0 0.2 0.4 0.6 Qr -1 -1 M rr M rθ -2 -2 n = 3, s = 1 n = 3, s = 1 12 3 w ψr 2 Qr 9 M rr ψθ M rθ 6 1 3 0 0.0 0.2 0.4 0.6 0.8 1.0 -1 0 -2 -3 -3 -6 0.0 0.2 n = 4, s = 1 0.4 0.6 0.8 1.0 n = 4, s = 1 25 4 w 3 ψr ψθ 2 20 Qr 15 M rr M rθ 10 1 5 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.0 -1 0.2 0.4 0.6 0.8 1.0 -5 -2 -10 -3 -15 Figure 2.5b Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.1. The number of nodal diameters is n = 1, 2, 3 and 4, respectively 29 Vibration Analysis of Uniform Circular Plate Modal Displacement Fields Modal Stress Resultants n = 5, s = 1 n = 5, s = 1 5 50 w 4 ψr ψθ 3 40 Qr 30 M rr M rθ 2 20 1 10 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 -1 -10 -2 -20 -3 -30 0.2 n = 6, s = 1 0.4 0.6 0.8 1.0 0.8 1.0 n = 6, s = 1 6 75 w Qr ψr 4 50 ψθ M rr M rθ 2 25 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 -2 -25 -4 -50 0.2 n = 7, s = 1 0.4 0.6 n = 7, s = 1 100 8 w Qr 75 ψr 6 M rr ψθ 50 M rθ 4 25 2 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.8 1.0 -25 -2 -50 -4 -75 n = 8, s = 1 n = 8, s = 1 8 150 w 6 ψr Qr 100 ψθ 4 M rr M rθ 50 2 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.0 -2 0.2 0.4 0.6 -50 -4 -6 -100 Figure 2.5c Mode shapes and modal stress resultants for free circular Mindlin plates with thickness ratio h/R = 0.1. The number of nodal diameters is n = 5, 6, 7 and 8, respectively 30 Vibration Analysis of Uniform Circular Plate 2.4 CONCLUDING REMARKS Presented in this chapter are the exact vibration frequencies and their associated mode shapes and modal stress-resultants of circular Mindlin plates with free edges. The exact free vibration solutions, when employed in the hydrodynamic analysis, will yield highly accurate deflections and stress-resultants of circular VLFS under the action of waves. This point will be discussed in chapter 4. 31 Chapter 3 VIBRATION ANALYSIS OF STEPPED CIRCULAR PLATES This chapter deals with the exact vibration solutions of stepped circular Mindlin plates with free edges. By keeping a constant material volume of stepped plates with the corresponding uniform circular plates, a comparative study on frequency values, mode shape and modal stress-resultants is conducted. The influence of the stepped thickness design on the vibration frequencies, mode shapes and modal stress resultants is also explored. 3.1 PROBLEM DEFINITION Consider an isotropic, stepped, circular plate of radius R, Young’s modulus E, shear modulus G = E/[2(1 + ν)], mass density γ and Poisson’s ratio ν . The plate edge is completely free. The outer annular sub-plate 1 has a constant thickness h1 while the inner circular sub-plate 2 has a constant thickness h2 . The separation between two sub-plates occurs at stepped location r = bR . Two Design Types of stepped circular plates are investigated. Design Type I plate is regarded as sub-plate 1 thinner than subplate 2 while Design Type II plate is regarded as sub-plate 2 thinner than sub-plate 1 (see Fig. 3.1). The natural frequencies, the mode shapes and the modal stress-resultants of the freely vibrating stepped circular plates need to be determined. 3.2 METHOD OF SOLUTION AND MATHEMATICAL MODELLING To solve such a circular plate problem, the commonly used method is decomposing stepped plate into two sub-plates where the separation boundary at r = bR. Similar to 32 Vibration Analysis of Stepped Circular Plate the uniform circular plate solved in Chapter 2, Mindlin plate theory be adopted where the rotations (ψ r ,ψ θ ) and transverse displacements w may be expressed as functions of three potential functions Θ1 , Θ 2 and Θ 3 in the stepped circular plate as the followings Sub-plate 1 R χ, r O Free edge θ bR Sub-plate2 Step location h1 h2 O Design Type I ( α = h2 >1) h1 bR R h2 h1 O Design Type II ( α = h2 1 (designated as Design Type I), the circular sub-plate is thicker than the annular sub-plate and when α 1 ) may have frequency parameters reaching maximum or minimum values at certain step locations for each mode (n, s). The frequency parameters have an interesting relationship with the mode sequence s for a given number n. It can be seen from Figs. 3.3a to 3.3d that the number of “peak” (or maximum) values of frequency parameters λs coincides with the number of sequence s for a given n value (n = 2 in this case). This implies that for a mode (n, s), one should be able to find s locations of peaks where its frequency values are local maxima. 39 Vibration Analysis of Stepped Circular Plate Fig. 3.4a shows the variation of the first 4 frequency parameters λs of modes (n, s) = (2, 1) , (0, 1), (3, 1), (1, 1) with respect to reference thickness ratio τ o varying from 0.05 to 0.25. A step location b = 0.5 and a stepped thickness ratio α = 0.5 are chosen. For this range of reference thickness ratios, the frequency parameters λs of stepped plates decrease with respect to increasing thickness ratio of their corresponding reference plates. Fig. 3.4b shows the variation of the first four frequency parameters λs of modes (n, s) = (2, 1), (0, 1), (3, 1), (1, 1) with respect to stepped thickness ratio α varying from 0.5 to 2.5 for a step location b = 0.5 and a reference thickness ratio τ o = 0.1 . While the frequency parameter λs of the mode (2, 1) increases, the frequency parameter λs of mode (3, 1) decreases as the stepped thickness ratio α varies from 0.5 to 2.5. The frequency parameters λs of modes (0, 1) and (1, 1) increase initially and then decrease in the considered range of stepped thickness ratios. These results show that the introduction of stepped thickness will only improve certain frequency values over its uniform thickness counterpart. Therefore, it is necessary to know the dominant modes in the hydroelastic analysis if an optimal stepped thickness is to be selected for the circular VLFS. The cases that are highlighted by boldfacing values have their mode shapes and modal stress resultants depicted in Figs. 3.5a to 3.5h. The mode shapes and modal stress resultants in Figs. 3.5a to 3.5h are plotted along the radial direction where their peak values are found. The modal displacement fields w and ψ r , and modal stress resultants Q r and M rr in the circumferential direction vary with cos(nθ), while the modal displacement ψ θ and modal stress resultant M rθ vary with sin(nθ). Figs. 3.5a to 3.5h also show the differences between mode shapes and modal stress resultants of Design Type II plates ( α = 2 ), Design Type I plates ( α = 0.5 ) and those of reference 40 Vibration Analysis of Stepped Circular Plate constant thickness plates. The first 8 modes of the reference plate with thickness ratio ho / R = 0.125 are chosen. Both stepped plates have a step location at b = 0.5. Stepped plates have similar mode shapes and modal stress resultant distributions as those of their corresponding reference constant thickness plates. Although the step variation exists in the plates, the transverse displacement w for all cases are smooth at the step location b = 0.5. However, a kink in the slope variation at the step location is observed for modal displacement ψ r and ψ θ for all cases except for cases with n = 0 (axissymmetric modes) where ψ θ = 0. The kink is caused by the terms in Eqs. 3.1 to 3.3 through hi and λi which are related to the stepped thickness ratio α . Therefore, the stresses are affected by the stepped plate design as well. In almost cases of modes (n, s), the mode shapes and modal stress-resultants of Design Type II stepped plate are much smaller than those of reference plate while those of Design Type I should be higher or lower depending each mode (n, s). This finding shows that for the same material volume, one should choose stepped circular plates other than uniform circular plates for designing circular VLFS because their displacements and final stressresultants under action of waves may be reduced. The modal stress-resultants in three dimension plots are given in Fig. 3.6 for the first 3 modes (2, 1), (0, 1) and (3, 1). 41 Table 3.1a Frequency parameter λs for stepped plates with step location at b = 1/2, reference constant thicknesses τ o = 0.01 and 0.1 Mode Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 α = 0. 5 τ o = 0.01 α =1 α =2 5.04362(2,1) 7.78131(0,1) 13.1987(3,1) 17.4539(1,1) 24.1569(4,1) 28.7181(0,2) 32.9294(2,2) 37.6866(5,1) 49.2354(1,2) 51.8818(3,2) 53.6764(6,1) 72.0814(7,1) 72.2963(2,3) 74.0596(0,3) 74.3952(4,2) 5.35452(2,1) 9.00175(0,1) 12.4237(3,1) 20.4613(1,1) 21.7983(4,1) 33.4237(5,1) 35.2140(2,2) 38.4164(0,2) 47.2564(6,1) 52.9040(3,2) 59.7398(1,2) 63.2633(7,1) 73.3521(4,2) 81.4185(8,1) 84.2088(2,3) 7.14972(2,1) 10.7670(0,1) 12.3215(3,1) 19.0274(4,1) 19.2338(1,1) 27.7242(5,1) 33.7390(2,2) 38.3989(6,1) 41.3685(0,2) 50.9719(7,1) 52.8634(3,2) 65.3713(8,1) 66.5661(1,2) 71.6324(4,2) 81.5423(9,1) α = 0. 5 τ o = 0.1 α =1 α =2 4.92495(2,1) 7.66442(0,1) 12.7058(3,1) 16.5598(1,1) 22.7912(4,1) 27.5878(0.2) 30.4884(2,2) 34.7115(5,1) 46.0646(1,2) 46.8779(3,2) 48.1356(6,1) 62.8187(7,1) 65.1960(4,2) 65.3939(2,3) 67.1498(0,3) 5.27822(2,1) 8.86877(0,1) 12.0667(3,1) 19.7165(1,1) 20.8089(4,1) 31.2861(5,1) 33.0500(2,2) 36.0592(0,2) 43.2853(6,1) 48.2623(3,2) 54.2993(1,2) 56.6081(7,1) 64.9534(4,2) 71.0734(8,1) 73.9519(2.3) 6.93210(2,1) 10.5758(0,1) 11.8362(3,1) 18.1896(4,1) 18.5547(1,1) 28.5986(4,1) 31.6860(2,2) 36.0023(6,1) 38.2474(0,2) 78.8671(0,3) 47.1030(7,1) 47.8779(3,2) 58.8407(1,2) 59.4138(8,1) 63.1753(4,2) Note: The values in brackets (n,s) denote the number of nodal diameters (n) and the mode sequence (s) 42 Table 3.1b Frequency parameter λs for stepped plates with step location at b = 1/2 , reference constant thicknesses τ o = 0.125 and 0.15 Mode Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 α = 0 .5 τ o = 0 . 125 α =1 4.87769(2,1) 7.60084(0,1) 12.4918(3,1) 16.2018(1,1) 22.1974(4,1) 27.01 (0,2) 29.5012(2,2) 33.4578(5,1) 44.5591(1,2) 5.24446(2,1) 8.79655(0,1) 11.907 (3,1) 19.346 (1,1) 20.3694(4,1) 30.3583(5,1) 32.0525(2,2) 34.931(0,2) 41.62 (6,1) 44.8505(3,2) 45.9053(6,1) 59.2822(7,1) 61.6042(4,2) 62.4519(2,3) 73.3952(8,1) 46.2614(3,2) 51.8997(1,2) 53.931(7,1) 61.5587(4,2) 69.7839(2,3) 72.3414(0,3) α =2 α = 0 .5 τ o = 0.15 α =1 6.85254(2,1) 10.4726(0,1) 11.6553(3,1) 17.8559(4,1) 18.2229(1,1) 25.6978(5,1) 30.7451(2,2) 34.9795(6,1) 36.8216(0,2) 45.4638(7,1) 4.82476(2,1) 7.52571(0,1) 12.2530(3,1) 15.8119(1,1) 21.5521(4,1) 26.3565(0,2) 28.4516(2,2) 32.1385(5,1) 42.7558(3,2) 42.9279(1,2) 43.6335(6,1) 55.7903(7,1) 58.0210(4,2) 59.4173(2,3)) 61.0051(0,3) 5.20584(2,1) 8.71132(0,1) 11.7271(3,1) 18.9273(1,1) 19.8850(4,1) 29.3621(5,1) 30.9716(2,2) 33.7076(0,2) 39.8807(6,1) 44.1745(3,2) 49.4103(1,2) 51.2114(7,1) 58.1397(4,2) 63.1697(8,1) 65.6276(2,3) 45.8453(3,2) 55.6397(1,2) 59.9548(4,2) 70.1209(2,3) 73.8172(0,3) α =2 6.76752(2,1) 10.3515(0,1) 11.4623(3,1) 17.4951(4,1) 17.8489(1,1) 25.0487(5,1) 29.7298(2,2) 33.8812(6,1) 35.3173(0,2) 43.7331(7,1) 43.7558(3,2) 52.4138(1,2) 54.3910(8,1) 56.7349(4,2) 65.5697(2,3) Note: The values in brackets (n,s) denote the number of nodal diameters (n) and the mode sequence (s). The cases with the boldfaced values have their modes and modal stress resultants presented in Figs. 3.5a-h. 43 Table 3.2a Frequency parameter λs for stepped plates with step location at b = 1/3, reference constant thicknesses τ o = 0.01 and 0.1 Mode Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 α = 0. 5 τ o = 0.01 α =1 α =2 5.17083(2,1) 8.08945(0,1) 12.9218(3,1) 19.0753(1,1) 22.9933(4,1) 34.7076(2,2) 35.3569(5,1) 36.6343(0,2) 50.0179(6,1) 53.5734(3,2) 54.4449(1,2) 66.9659(7,1) 69.0213(0,3) 75.9501(4,2) 78.1126(2,3) 5.35452(2,1) 9.00175(0,1) 12.4237(3,1) 20.4613(1,1) 21.7983(4,1) 33.4237(5,1) 35.2140(2,2) 38.4164(0,2) 47.2564(6,1) 52.9040(3,2) 59.7398(1,2) 63.2633(7,1) 73.3521(4,2) 81.4185(8,1) 84.2088(2,3) 6.07063(2,1) 9.28275(0,1) 11.4005(3,1) 18.2020(1,1) 19.0224(4,1) 28.5082(5,1) 34.5394(2,2) 38.9908(0,2) 39.5321(6,1) 48.0724(3,2) 51.8752(7,1) 53.6635(1,2) 61.9126(4,2) 65.3670(8,1) 73.9618(2,3) α = 0. 5 τ o = 0.1 α =1 α =2 5.06704(2,1) 7.97508(0,1) 12.4993(3,1) 18.1813(1,1) 21.8422(4,1) 32.4754(2,2) 32.8769(5,1) 34.7624(0,2) 45.4202(6,1) 48.6546(3,2) 49.8784(1,2) 59.2803(7,1) 63.0600(0,3) 66.6417(4,2) 68.7871(2,3) 5.27822(2,1) 8.86877(0,1) 12.0667(3,1) 19.7165(1,1) 20.8089(4,1) 31.2861(5,1) 33.0500(2,2) 36.0599(0,2) 43.2853(6,1) 48.2623(3,2) 54.2993(1,2) 56.6081(7,1) 64.9534(4,2) 71.0734(8,1) 73.9519(2.3) 6.07063(2,1) 9.28275(0,1) 11.4005(3,1) 18.2020(1,1) 19.0224(4,1) 28.5082(5,1) 34.5394(2,2) 38.9908(0,2) 39.5321(6,1) 48.0724(3,2) 51.8752(7,1) 53.6635(1,2) 61.9126(4,2) 65.3670(8,1) 73.9618(2,3) Note: The values in brackets (n,s) denote the number of nodal diameters (n) and the mode sequence (s) 44 Table 3.2b Frequency parameter λ s for stepped plates with step location at b = 1/3, reference constant thicknesses τ o = 0.125 and 0.15 Mode Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 α = 0 .5 τ o = 0 . 125 α =1 α =2 5.0256 (2,1) 7.91279(0,1) 12.3142(3,1) 17.8023(1,1) 21.3358(4,1) 31.4687(2,2) 31.8133(5,1) 33.8405(0,2) 43.5239(6,1) 46.5556(3,2) 47.9454(1,2) 56.2537(7,1) 60.3979(0,3) 62.9541(4,2) 65.1928(2,3) 5.24446(2,1) 8.79655(0,1) 11.907 (3,1) 19.346 (1,1) 20.3694(4,1) 30.3583(5,1) 32.0525(2,2) 34.931 (0,2) 41.62 (6,1) 46.2614(3,2) 51.8997(1,2) 53.931 (7,1) 61.5587(4,2) 67.0998(8,1) 69.7839(2,3) 6.00116(2,1) 9.20176(0,1) 11.2479(3,1) 17.8642(1,1) 18.6723(4,1) 27.7867(5,1) 33.3653(2,2) 37.438 (0,2) 38.2293(6,1) 46.0281(3,2) 49.7568(7,1) 51.0313(1,2) 58.8788(4,2) 62.1838(8,1) 69.5939(2,3) α = 0 .5 τ o = 0.15 α =1 α =2 4.97921(2,1) 7.83914(0,1) 12.1070(3,1) 17.3858(1,1) 20.7814(4,1) 30.3827(2,2) 30.6813(5,1) 32.8233(0,2) 41.5642(6,1) 44.3796(3,2) 45.9293(1,2) 53.2157(7,1) 57.6210(0,3) 59.2779(4,2) 61.6002(2,3) 5.20584(2,1) 8.71132(0,1) 11.7271(3,1) 18.9273(1,1) 19.8850(4,1) 29.3621(5,1) 30.9716(2,2) 33.7076(0,2) 39.8807(6,1) 44.1745(3,2) 49.4103(1,2) 51.2114(7,1) 58.1397(4,2) 63.1697(8,1) 65.6276(2,3) 5.92774(2,1) 9.10629(0,1) 11.0809(3,1) 17.4835(1,1) 18.2857(4,1) 27.0019(5,1) 32.1238(2,2) 35.7900(0,2) 36.8443(6,1) 43.9255(3,2) 47.5595(7,1) 48.3152(1,2) 55.8110(4,2) 58.9626(8,1) 65.2900(2,3) Note: The values in brackets (n,s) denote the number of nodal diameters (n) and the mode sequence (s) 45 λs λs Mode n = 2, s = 1 α =2 α = 1.5 α =1 α = 0.5 8 7 14 10 5 8 b 4 0.3 α =2 α = 1.5 α =1 α = 0.5 12 6 0.1 Mode n = 0, s = 1 0.5 0.7 b 6 0 0.9 0.2 0.4 0.8 1 (b) (a) λs 0.6 λs Mode n = 3, s = 1 15 Mode n = 1, s = 1 26 α =2 α = 1.5 α =1 α = 0.5 14 13 α =2 α = 1.5 α =1 α = 0.5 23 20 17 12 b 11 0 0.2 0.4 0.6 0.8 b 14 0 1 0.2 0.4 0.6 0.8 1 (d) (c) Figure 3.2 Frequency parameter λ s versus step location b for Mindlin plates with reference thickness ratio τ 0 = 0.1, α = 0.5 to 2 46 λs λs Mode n = 4, s = 1 α =2 α = 1.5 α =1 α = 0.5 25 23 37 34 21 31 19 28 17 0 0.2 Mode n = 5, s = 1 α =2 α = 1.5 α =1 α = 0.5 0.4 0.6 0.8 1 b 25 0 0.2 0.4 0.6 (e) λs λs α =2 α = 1.5 α =1 α = 0.5 37 29 29 b 25 0.5 Mode n = 0, s = 2 39 34 0.3 b 44 33 0.1 1 (f) Mode n = 2, s = 2 41 0.8 0.7 α =2 α = 1.5 α =1 α = 0.5 b 24 0.9 0 (g) 0.2 0.4 0.6 0.8 1 (h) Figure 3.2 (Contd.) Frequency parameter λ s versus step location b for plates with reference thickness ratio τ 0 = 0.1, α = 0.5 to 2 47 s=1 λ lss s=2 8 λlss 40 7 37 6 34 5 31 b b 4 0 0.2 0.4 0.6 0.8 b b 28 1 0 0.2 (a) λlss 134 75 128 70 122 65 0.4 1 0.6 0.8 1 140 80 0.2 0.8 s=4 85 0 0.6 (b) s=3 λlss 0.4 0.6 0.8 1 bb b b 116 0 (c) 0.2 0.4 (d) Figure 3.3 Frequency parameter λ s versus step location b for plates with τ 0 = 0.1, α = 2 and n = 2 48 α ≥1 α ≤1 λls λls α =1 23 18 18 13 (1,1) (1,1) (3,1) 8 (3,1) 13 (0,1) (0,1) (2,1) (2,1) 8 3 0.05 ttoo 0.1 0.15 0.2 αa 3 0.5 0.25 1 1.5 2 2.5 Figure 3.4b Frequency parameter λ s versus reference stepped thickness ratio α for plates with step location b = 0.5 and reference thickness ratio τ 0 = 0.1 Figure 3.4a Frequency parameter λ s versus reference thickness ratios τ 0 for plates with step location b = 0.5 and stepped thickness ratio α = 0.5 49 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 6.85254 λo = 5.24446 λs = 4.87769 Modal tress resultants Modal tress resultants Modal tress resultants 4 16 M rr M rr M rθ Qr 8 M rθ Qr 2 2 M rr 1 M rθ Qr 0 0.0 0.2 0.4 0.6 0.8 1.0 0.8 1.0 0 0 0.0 0.2 0.4 0.6 0.8 0.0 1.0 0.2 0.8 1.0 2.0 Mode shape Mode shape ψθ ψθ ψθ 0.0 0 0 0.2 0.4 0.6 0.8 1.0 wr 1 wr 1 wr 0.0 ψr 2 ψr 2 ψr 1.0 -1 -2 Mode shape -2.0 0.6 -2 -8 -1.0 0.4 0.0 0.2 0.4 0.6 0.8 0.0 1.0 -1 -1 -2 -2 0.2 0.4 0.6 Figure 3.5a Mode shapes (with n = 2, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates 50 30 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 10.4726 λo = 8.79655 λs =7.60084 Modal tress resultants Modal tress resultants Modal tress resultants 20 M rr M rθ Qr 15 M rθ Qr 10 0 0.2 0.4 0.6 0.8 1.0 -15 0.2 0.6 0.8 0 0.8 1.0 0.6 0.8 1.0 0.8 1.0 1 0 0.0 0.2 0.4 0.6 0.8 0.0 1.0 0.2 0.4 0.6 ψr ψr ψθ 0.4 Mode shape 0 0.6 0.2 -4 2 0.4 0.0 1.0 Mode shape 2 -2 ψr -1 -4 ψθ wr ψθ wr -4 0.4 -10 0.2 M rθ Qr 0 0.0 Mode shape 0.0 M rr 4 0 0.0 -2 8 M rr wr -2 Figure 3.5b Mode shapes (with n = 0, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates 51 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 11.6553 λo = 11.907 λs =12.4918 Modal tress resultants Modal tress resultants 10 Modal tress resultants 4 3 M rr 0 0.2 0.4 -10 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0 0.0 M rr M rθ Qr -20 0.0 -3 M rr -6 -30 Mode shape 3.0 ψθ 1.5 0.2 0.4 0.6 0.8 0.0 1.0 -1.5 -1.5 -3.0 -3.0 0.8 1.0 wr 0 0.0 0.0 1.0 ψθ 2 wr wr 0.8 ψr ψr ψθ 0.6 Mode shape 4 ψr 0.4 -4 Mode shape 3.0 0.0 0.2 -2 M rθ Qr -9 1.5 M rθ Qr 2 0 0.0 0.2 0.4 0.6 0.8 0.0 1.0 0.2 0.4 0.6 -2 -4 Figure 3.5c Mode shapes (with n = 3, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates 52 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 18.2229 λo = 19.346 λs = 16.2018 Modal tress resultants Modal tress resultants 40 50 0 0 Modal tress resultants 30 M rr M rθ Qr 15 0.0 0.2 0.4 0.6 0.8 M rr -40 0.0 1.0 0.2 0.4 0.6 0.8 1.0 -50 M rθ Qr 0.0 M rθ Qr 0.2 6 0.4 0.6 -2 0.8 -4 wr 0 ψθ 1.0 ψθ 3 wr 1.0 ψr 0.8 ψr ψθ 3 0.2 1.0 6 ψr 0.0 0.8 Mode shape Mode shape 0 0.6 -30 Mode shape 2 0.4 -15 -100 -80 0 M rr 0 0.0 0.2 0.4 0.6 0.8 0.0 1.0 -3 -3 -6 -6 0.2 0.4 0.6 wr -6 Figure 3.5d Mode shapes (with n = 1, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates 53 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 17.8559 λo = 20.3694 λs = 22.1974 Modal tress resultants Modal tress resultants 10 Modal tress resultants 6 5 0 0 0.0 0.2 0.4 0.6 0.8 0.0 1.0 0.2 0.6 0.8 0 1.0 0.0 -5 -10 M rr -20 M rθ Qr -15 -40 1.0 0.8 1.0 Mode shape 4 ψr ψr ψr ψθ 2 ψθ 2 wr wr wr 0 0 0 0.2 0.8 -18 4 ψθ 0.6 M rr Mode shape 4 0.4 M rθ Qr -12 -20 Mode shape 0.0 0.2 -6 M rr -10 M rθ Qr -30 2 0.4 0.4 0.6 0.8 0.0 0.0 1.0 -2 -2 -4 -4 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 -2 -4 Figure 3.5e Mode shapes (with n = 4, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates 54 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 25.6978 λo = 30.3583 λs = 33.4578 Modal tress resultants Modal tress resultants 10 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 -10 0.0 0.2 M rr -30 -40 0.8 0.0 1.0 M rθ Qr 4 ψr ψθ 2 ψθ 2 wr wr 0 0.6 0.8 1.0 ψθ 1.0 Mode shape ψr 0 0 ψr 0.8 -40 Mode shape 2 0.4 0.6 M rθ Qr -30 4 0.2 0.4 M rr -20 -40 4 0.0 0.2 -10 Mode shape -6 0.6 M rr -20 M rθ Qr -30 -4 0.4 -10 -20 -2 Modal tress resultants 10 10 0.0 0.2 0.4 0.6 0.8 0.0 1.0 -2 -2 -4 -4 0.2 0.4 0.6 0.8 wr Figure 3.5f Mode shapes (with n = 5, s = 1) and modal stress resultants for stepped plates and their reference constant thickness plates 55 1.0 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 30.7451 λo = 32.0525 λs =29.5012 Modal tress resultants Modal tress resultants Modal tress resultants 120 120 40 M rr M rr 80 40 M rθ Qr 60 M rθ Qr 0 0.0 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 Mode shape 0.8 1.0 6 ψθ ψr ψθ 4 Mode shape ψr 8 1.0 -80 Mode shape 8 0.8 M rθ Qr -60 -120 0.6 M rr -40 -40 -80 0.4 wr 4 wr 3 0 0 0.0 -4 0.0 0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 -4 0.6 0.8 1.0 -3 -6 0.2 0.4 0.6 ψr ψθ wr Figure 3.5g Mode shapes (with n = 2, s = 2) and modal stress resultants for stepped plates and their reference constant thickness plates 56 Design Type I ( α = 2, b = 0.5) Reference plate with constant thickness( α = 1, τ o = 0.125 ) Design Type II ( α = 0.5, b = 0.5) λs = 36.8216 λo = 34.931 λs = 27.01 Modal tress resultants Modal tress resultants Modal tress resultants 120 240 160 80 40 M rr M rr M rθ Qr M rθ Qr 60 M rr 0 0 0.0 0 0.0 0.2 0.4 0.6 0.8 0.2 0.6 0.8 1.0 -60 8 0.4 0.6 0.8 1.0 0.8 1.0 Mode shape 2 ψr ψr ψθ 2 ψθ 0 wr wr 0.0 0 0.0 0 0.2 0.2 Mode shape 4 0.0 0.0 -20 Mode shape -4 0.4 1.0 -80 4 M rθ Qr 20 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 -2 -2 0.2 0.4 0.6 ψr ψθ wr -4 -4 Figure 3.5h Mode shapes (with n = 0, s = 2) and modal stress resultants for stepped plates and their reference constant thickness plates 57 n = 2, s = 1 n = 2, s = 1 n = 2, s = 1 Bending Moment Bending Moment Bending Moment Twisting Moment Twisting Moment Twisting Moment Shear Force Shear Force Shear Force Figure 3.6 Three Dimensional Stress-resultant Plots of Uniform and Stepped Circular Plates 58 n =0, s = 1 n =0, s = 1 n =0, s = 1 Bending Moment Bending Moment Bending Moment Twisting Moment Twisting Moment Twisting Moment Figure 3.6 (Contd.) Three Dimensional Stress-resultant Plots of Uniform and Stepped Circular Plates 59 n =3, s = 1 Bending Moment n =3, s = 1 Bending Moment n =3, s = 1 Bending Moment Twisting Moment Twisting Moment Twisting Moment Shear Force Shear Force Shear Force Figure 3.6 (Contd.) Three Dimensional Stress-resultant Plots of Uniform and Stepped Circular Plates 60 Vibration Analysis of Stepped Circular Plate 3.4 CONCLUDING REMARKS Presented in this chapter are exact vibration solutions of circular stepped plates with free edges. The solutions include the frequency values, the mode shapes and the modal stress resultants. The natural boundary conditions of stepped circular plate are also completely satisfied. A comparison study on the vibration solutions of a stepped plate with constant thickness plates is made with a view to ascertain the optimal usage of materials for designing stepped circular VLFSs. Thickness ratio and step location is rather sensitive to mode sequence. However, these results show that the stepped circular plate always yields smaller stress-resultants than its reference uniform circular plate. Therefore, the exact vibration solutions of stepped plates, when employed in the hydroelastic analysis, will yield smaller value of stress-resultants than the uniform circular VLFS. 61 Chapter 4 HYDROELASTIC ANALYSIS OF UNIFORM CIRCULAR VLFS This chapter presents the solution for hydroelastic problem of uniform circular VLFS. The analysis is carried out in frequency domain using modal expansion matching method. The diffraction and radiation forces are evaluated by using eigen function expansion matching method. The modal deflection and stress-resultants given here will serve as accurate solutions for engineers when developing their numerical methods for analyzing VLFSs. 4.1 BASIC ASSUMPTIONS AND PROBLEM DEFINITION In a basic hydroelastic analysis of pontoon-type VLFSs, the following assumptions are made: • The VLFS is modeled as an isotropic flat plate with free edges. • The fluid is incompressible, inviscid and its motion is irrotational so that the velocity potential exists. • The amplitude of the incident wave and the motions of the VLFS are both small and only the vertical motion of structure is considered. • There are no gap between the VLFS and the free surface. The fluid-structure system and the cylindrical coordinate system are shown in Fig. 4.1. The origin of the coordinate system is on the flat sea-bed and the z- axis is pointing upwards. The undisturbed free surface is on the plane z = d , and the sea-bed is assumed to be flat at z = 0 . The floating, flat, circular plate has a radius of R and a thickness of h . The zero draft is assumed for simplifying the fluid-domain analysis. 62 Hydroelastic Analysis of Uniform Circular Plate The incident wave impacts the plate at θ = 0 . The problem at hand is to determine the deflections and stress-resultants of the uniform circular plate under action of wave forces. Below, the governing equations and boundary conditions for the hydroelastic analysis are presented. The hydroelastic analysis is performed in the frequency domain. y r, χ R incident wave q x z h z=d Sea bed, z = 0 O Figure 4.1 Geometry of an Uniform Circular VLFS 4.2 BOUNDARY VALUE PROBLEMS AND GOVERNING EQUATIONS Considering time-harmonic motion with the complex time dependence e iϖt being applied to all first-order oscillatory quantities, where i represents the imaginary unit, ϖ the angular frequency and t the time, the complex velocity potential φ (r ,θ , z ) is governed by the Laplace’s equation ∇ 2φ ( r , θ , z ) = 0 (4.1) 63 Hydroelastic Analysis of Uniform Circular Plate in the fluid domain, and it must satisfy the boundary conditions on the free surface, on the sea-bed, and on the wetted bottom surface of the floating body ∂φ (r , θ , z ) ϖ 2 = φ (r ,θ , z ) ∂z g on z = d , r > R (4.2) ∂φ (r ,θ , z ) =0 ∂z on z = 0 (4.3) ∂φ (r ,θ , z ) = iϖw(r , θ ) ∂z on z = d , r < R (4.4) where w(r ,θ ) is the vertical complex displacement of the plate, and g is the gravitational acceleration. The radiation condition for the scattering and radiation potential is also applied at infinity ⎡ ∂ (φ − φ I ) ⎤ lim r ⎢ + ik (φ − φ I )⎥ = 0 as r → ∞ r →∞ ⎣ ∂r ⎦ (4.5) where r is the radial coordinate measured from the centre of the VLFS, k is the wave number, and φ I the potential representing the undisturbed incident wave φI = ∞ igAM 01 / 2 igA cosh kz ikx e = f 0 ( z )∑ ε n i n J n (kr ) cos nθ ϖ cosh kd ϖ cosh kd n =0 (4.6) where ε 0 = 1, ε n = 2 (n ≥ 2) ; A the amplitude of the incident wave; J n the Bessel function of the first kind of order n ; and k tanh kd = ϖ 2 / g (4.7) f ( z ) = M −1 / 2 cosh kz (4.8) 1 ⎛ sinh 2kd ⎞ ⎜1 + ⎟ 2⎝ 2kd ⎠ (4.9) M = 64 Hydroelastic Analysis of Uniform Circular Plate By assuming the circular VLFS to be an elastic, isotropic, shear deformable plate, the motion of the floating body is governed by the following Mindlin plate equations ∂M r 1 ∂M rθ M r − M θ γh 3 + + − Qr + ϖ 2 ψr = 0 ∂r r ∂θ r 12 (4.10) 3 ∂M rθ 1 ∂M rθ 2M rθ 2 γh + + − Qθ + ϖ ψθ = 0 ∂r r ∂θ r 12 (4.11) ∂Qr 1 ∂Qθ Qr + + + ϖ 2 γhw = p (r , θ ) ∂r r ∂θ r (4.12) where the bending moments M r , M θ , twisting moment M rθ , and the shear forces Qr , Qθ can be calculated from the following relations ⎧ ∂ψ ∂ψ θ ⎞⎫ ν⎛ M r = D ⎨ r + ⎜ψ r + ⎟⎬ r⎝ ∂θ ⎠⎭ ⎩ ∂r (4.13) ⎧ ∂ψ r 1 ⎛ ∂ψ θ ⎞⎫ M θ = D ⎨ν + ⎜ψ r + ⎟⎬ r⎝ ∂θ ⎠⎭ ⎩ ∂r (4.14) M rθ = ⎧ ∂ψ ⎫ D (1 − ν )⎨ θ + 1 ⎛⎜ ∂ψ r − ψ θ ⎞⎟⎬ 2 r ⎝ ∂θ ⎠⎭ ⎩ ∂r (4.15) ∂w ⎞ ⎛ Qr = κ 2 Gh⎜ψ r + ⎟ ∂r ⎠ ⎝ (4.16) 1 ∂w ⎞ ⎛ Qθ = κ 2 Gh⎜ψ θ + ⎟ r ∂θ ⎠ ⎝ (4.17) in which D is the plate rigidity, γ the mass per unit area of the plate, ν the Poisson’s ratio, κ 2 the Mindlin shear correction factor, ρg the hydrostatic restoring force factor, where ρ is the density of the fluid, and p (r ,θ ) the pressure on the bottom surface of the plate. The pressure p (r ,θ ) is related to the velocity potential φ (r ,θ , z ) by Newman (1994) 65 Hydroelastic Analysis of Uniform Circular Plate p (r ,θ ) = −iρϖφ (r , θ , d ) − ρgw(r ,θ ) (4.18) The floating body subjected to no constraint in the vertical direction along its edges must satisfy the zero bending moment, zero twisting moment and zero shear force conditions for a free edge M r = 0, M rθ = 0, Qr = 0 (4.19) 4.3 MODAL EXPANSION OF MOTION In order to decouple the fluid-structure interaction into the hydrodynamic problem in terms of the velocity potential and the mechanical problem for the vibration of the circular plate, the motion of the plate is expanded by the modal functions which consist of the product of the natural dry modes of circular Mindlin plates with free edges. The exact vibration solutions for uniform circular Mindlin plates with free edge are already presented in Chapter 2. Using the superposition of the natural dry modes and the two rigid-body motions (heave and pitch), the final solution of the plate deflection is given by N M w(r ,θ ) = ζ 00 w00 + ζ 10 w10 cos θ + ∑∑ ζ ns wns (r ) cos nθ (4.20) n = 0 s =1 N M ψ r (r ,θ ) = ∑∑ ζ nsψ r ,ns (r ) cos nθ (4.21) n = 0 s =1 N M ψ θ (r ,θ ) = ∑∑ ζ nsψ θ ,ns (r ) sin nθ (4.22) n = 0 s =1 where wns ,ψ r ,ns ,ψ θ ,ns represent the natural dry modes (mode shape of free vibration) which are mostly given in chapter 2; n is the number of nodal diameters of the mode ( n = 0,1,..., N ) ; s the sequence for a given n value ( s = 1,2,..., M ); and 66 Hydroelastic Analysis of Uniform Circular Plate w00 = 2 , w10 = 2r / R (4.23) The complex modal amplitudes ζ ns are the unknowns which are to be determined. 4.4 SOLUTIONS FOR RADIATION POTENTIALS The velocity potential φ is then decomposed into diffraction and radiation potentials by using the same modal amplitudes as (Newman, 1994) N N φ (r ,θ , z ) = ∑ φ Dn (r , z ) cos nθ + iϖ ∑ ∑ζ ns n = 0 s = 0 ( n = 0 ,1) s =1( n ≥ 2 ) n =0 φ ns (r , z ) cos(nθ ) (4.24) Then the boundary condition on the free surface and wetted bottom surface of the floating body, Eqs. (4.2) and (4.4) are modified to ∂φ Dn (r , z ) =0 ∂z on z = d , r > R (4.25) ∂φ ns (r , z ) = wns (r ) on z = d , r < R ∂z (4.26) The general solutions for radiation potentials may be given by the following equations (Hamamoto and Tanaka, 1992) φ ns( e ) (r , z ) = C ns ,0 ∞ K n (k j r ) H n( 2 ) (kr ) f ( z ) + C ns , j f j ( z) ∑ 0 ( 2) K n (k j R) H n (kR) j =1 n ∞ I n (l j r ) ⎛r⎞ φ (r , z ) = Dns ,0 ⎜ ⎟ g 0 ( z ) + ∑ {Dns , j g j ( z) + I n (l j R) ⎝R⎠ j =1 (4.27) (i ) ns J n (λ nj r / R) cosh(λ nj z / R) R ⋅ ⋅ wns (r ) J n (λ nj r / R)rdr} Rλ nj J n2+1 (λ nj ) sinh(λ nj d / R ) ∫0 (4.28) 2 Here, the supercripts e and i represent the external domain ( r > a) and the internal domain ( r < a ) , respectively; H n( 2 ) , I n and K n represent Hankel function of the second-kind, modified Bessel function of the first and second-kind of order n , 67 Hydroelastic Analysis of Uniform Circular Plate respectively; λ nj is the j -th positive root of the equation J n (λ nj ) = 0 ; C ns , j and Dns , j are the unknown coefficients to be determined; and f j ( z ) = M −j 1 / 2 cos(k j z ) Mj = ( j ≥ 1) (4.29) ( j ≥ 1) (4.30) 1 ⎛ sin( 2k j d ) ⎞ ⎜1 + ⎟ 2 ⎜⎝ 2kd ⎟⎠ k j tan(k j d ) = ϖ2 g (k j > 0; j ≥ 1) g j ( z ) = ε 1j / 2 cos(l j z ) , l j = jπ d (4.31) (4.32 a,b) Note that when j = 0, we have k 0 = ik and Eq. (4.31) reduces to the dispersion relation given in Eq. (4.7). The above situation where j ≥ 1 implies that we are referring to evanescent wave. The following orthogonal relationships are satisfied 1 d 1 d f i ( z ) f j ( z )dz = ∫ g i ( z ) g j ( z )dz = δ ij ∫ 0 d d 0 ⎧1 (i = j ) where δ ij = ⎨ ⎩0 (i ≠ j ) (4.33) (4.34) Based on the continuity of the potentials on r = R and the application of d 1 ...g l ( z )dz to the continuity equation, one obtains: d ∫0 ∞ Dns.l = ∑ C ns , j E jl (4.35) j =0 d E jl = 1 f j ( z ) g l ( z )dz d ∫0 (4.36) From the continuity of the horizontal velocity on r = R and the application of d 1 ... f p ( z )dz to the continuity equation, one gets d ∫0 68 Hydroelastic Analysis of Uniform Circular Plate C ns , 0 C ns , p ⎫ (For p = 0) ⎪ K n (k p R) n ⎪ ⎬ = Dns , 0 E p 0 ( 2 )' R k p H n (k p R) ⎪ p (For 1 ) ≥ ⎪ H n( 2 ) (k p R) ⎭ ' ' ⎧ ⎫ l j I n (l j R) 2 J n (λ nj ) Dns , j E pj + 2 2 ⎪ ⎪ ∞ I n (l j R) a J n +1 (λ nj ) ⎪ ⎪ + ∑⎨ d ⎬ a cosh(λ nj z / R) j =1 ⎪ 1 ⋅ f ( z )dz.∫ wns (r ) J n (λ nj r / R)rdr ⎪ ⎪ d ∫ sinh(λ d / R) p ⎪ nj 0 ⎩ 0 ⎭ kK n' (k p R) (4.37) By substituting Eqs. (4.35) into Eqs. (4.37) and rearranging the equation, we have ~ ∑ C {H ∞ q =0 ns , q ~ np δ pq − Gnpq } ⎧ 2 J n' (λ nj ) 1 d cosh(λ nj z / R) ⎫ ⋅ f ( z ) dz ⎪ ⎪ p 2 2 ∫ ∞ ⎪ R J n +1 (λ nj ) d 0 sinh(λ nj d / R) ⎪ = ∑⎨ ⎬ (4.38) R j =1 ⎪ ⎪ ⎪⋅ ∫ wns (r ) J n (λ nj r / R)rdr ⎪ ⎭ ⎩ 0 where ⎧ kH n( 2 ) ' (kR) (for p = 0) ⎪ ( 2) H kR ( ) ⎪ n =⎨ ⎪ k p K n ' (k p R) (for p ≥ 1) ⎪ K (k R) n p ⎩ (4.39) l j I n' (l j R) n ∞ ~ Gnpq = E p 0 E q 0 + ∑ E pj E qj R j =1 I n (l j R ) (4.40) ~ H np By solving Eq. (4.38) with respect to C ns ,q (where the infinite-sum should be truncated at some number), and then substituting them in to Eq. (4.35), the unknown coefficients for the radiation potentials φ ns (r , z ) are obtained. 4.5 SOLUTIONS FOR DIFFRACTION POTENTIALS The general solution for diffraction potential may be given by the following equations 69 Hydroelastic Analysis of Uniform Circular Plate (e) (r , z ) = C n,0 φ Dn ∞ K n (k j r ) H n( 2 ) (kr ) f ( z ) + Cn, j f j ( z ) + φ In (r ) f 0 ( z ) (4.41) ∑ 0 ( 2) K n (k j R) H n (kR) j =1 ⎛r⎞ ⎝R⎠ n ∞ I n (l j r ) j =1 I n (l j R) (i ) ( r , z ) = Dn , 0 ⎜ ⎟ g 0 ( z ) + ∑ Dn , j φ Dn g j ( z) (4.42) where φ In = igAM 01 / 2 ε n i n J n (kr ) ϖ cosh kd (4.43) By applying a similar procedure as for the radiation potentials, the following sets of equations are finally obtained ∞ Dn ,l = ∑ C n , j E jl + φ In ' ( R) E 0l (4.44) j =0 ~ ∑ C {H ∞ q =0 n ,q ~ np } ~ δ pq − Gnpq = −φ In ' ( R )δ 0 p +φ In ( R)Gnp 0 (4.45) By solving Eq. (4.45) with respect to C n ,q , and then substituting them into Eq. (4.44), all of the unknown coefficients for the diffraction potential φ Dn (r , z ) are determined. 4.6 EQUATION OF MOTION IN MODAL COORDINATES In order to derive the equations of motion in modal coordinates, we consider the kinetic energy T , the strain energy U and the energy associated with the pressure V T= h2 2 1 2π R 2⎧ 2 γ h ϖ w ψ r + ψ θ2 + ⎨ ∫ ∫ 0 0 2 12 ⎩ 1 U= 2 ( 2π R 0 0 ∫ ∫ )⎫⎬rdrdθ (4.46) ⎭ ν ∂ψ r ⎛ ∂ψ θ ⎞ ⎛ ∂ψ r ⎞ +ψ r ⎟ {D[⎜ ⎜ ⎟ +2 ⋅ r ∂r ⎝ ∂θ ⎝ ∂r ⎠ ⎠ 2 ∂ψ θ ∂ψ r ⎞ 1 ⎛ ∂ψ ⎞ 1 −ν ⎛ + 2 ⎜ θ + ψ r ⎟ + 2 ⋅ ⎜ψ θ − r − ⎟ ] ∂r ∂θ ⎠ 2r ⎝ r ⎝ ∂θ ⎠ 2 2 1 ⎛ ∂w ⎞ + κ Gh[⎜ +ψ r ⎟ + 2 r ⎝ ∂r ⎠ 2 2 70 ⎛ ∂w ⎞ + rψ θ ⎟ ]}rdrdθ ⎜ ⎝ ∂θ ⎠ 2 (4.47) Hydroelastic Analysis of Uniform Circular Plate V = −∫ 2π 0 R ∫ 0 p (r , θ , d ) wrdrdθ (4.48) The Hamilton’s principle can be given as − δT + δU + δV = 0 (4.49) By substituting w, ψ r and ψ θ which are represented by Eq. (4.20)-(4.22), and applying the Galerkin’s method, we obtain M ∑ R ζ ns [−γhϖ 2 ∫ {wns wnp + s = 0 ( n = 0 ,1) s =1( n ≥ 2 R 0 + ∫ {D[ 0 + + ∂ψ r ,ns ∂ψ r ,np ν ∂ψ r ,ns (nψ θ ,np + ψ r ,np ) ⋅ + ⋅ ∂r ∂r ∂r r ν ∂ψ r ,np ∂r r 1 −ν 2r 2 h2 (ψ r ,nsψ r ,np + ψ θ ,nsψ θ ,np )}rdr 12 (nψ θ , ns + ψ r ,ns ) + 1 (nψ θ ,ns + ψ r ,ns ) ⋅ (nψ θ ,np + ψ r ,np ) r2 (4.50) ∂ψ θ ,np ∂ψ θ ,ns ⎞ ⎛ ⎞ ⎛ ⎜⎜ψ θ ,ns − r + nψ r ,ns ⎟⎟ ⋅ ⎜⎜ψ θ ,np − r + nψ r ,np ⎟⎟] ∂r ∂r ⎝ ⎠ ⎝ ⎠ ⎞ ⎛ ∂w ⎞ ⎛ ∂wnp + κ 2 Gh[⎜ ns + ψ r ,ns ⎟ ⋅ ⎜⎜ + ψ r ,np ⎟⎟ ⎝ ∂r ⎠ ⎝ ∂r ⎠ + 1 (− nwns + rψ θ ,ns )(−nwnp + rψ θ ,np )]}dr ] r2 In view of the fact that the normal modes satisfy Eqs. (4.10)-(4.17) and Eq. (4.19) , Eq. (4.50) can be simplified as M ∑ R ζ ns [−ϖ γhR δ ps + ϖ γhR δ ps − ϖ ρ ∫ φ ns (r , d ) wnp rdr 2 2 2 ns s = 0 ( n = 0 ,1) s =1( n ≥ 2 ) 2 2 0 (4.51) R R 0 0 + ρg ∫ wns wnp rdr ] = − ρiϖ ∫ φ Dn (r , d ) wnp rdr where the normalization of the modal vectors is made such that ⎧ ⎫ τ2 2 + w w ∫0 ⎨⎩ ns np 12 (ψ r ,nsψ r ,np + ψ θ ,nsψ θ ,np )⎬⎭rdr = R δ R 71 ps (4.52) Hydroelastic Analysis of Uniform Circular Plate and ϖ ns represents the natural frequency, Eq. (4.51) may be represented in a nondimensional form by M ∑ ζ ns [− s = 0 ( n = 0 ,1) s =1( n ≥ 2 ) ⎞ ϖ 2R ⎛ γ ⎞ ϖ 2 R ⎛⎜ 1 1 ⎟ τ ⎜⎜ ⎟⎟δ ps + λ2ns Sδ ps − φ w χ d χ ns np ∫ ⎟ g ⎝ρ⎠ g ⎜⎝ R 0 ⎠ (4.53) 1 + ∫ wns wnp χdχ ] = − 0 iϖ φ Dn wnp χdχ g ∫0 1 where χ = r / R , τ = h / R and S = D /( ρgR 4 ) . It is to be noted that Eq. (4.68) can be solved separately for each n (number of nodal diameters or Fourier modes). 4.7 NUMERICAL RESULTS We analyze two uniform circular plates with difference of plate rigidity S as shown in Table 4.1. The first problem is a circular plate with thickness ratio h / R = 0.1 , radius R = 50m while the other problem involves a rather thin plate with thickness ratio h / R = 0.01 , R = 200m . The incident wave length λ = 50m is taken for both problems. In calculating the plate rigidity D , we assume that the plated structure is made of steel with Young’s modulus E = 206 GPa and top and bottom plate thicknesses t = 15 mm (this thickness is not to be confused with the total thickness of the floating structure denoted by h). Therefore one can find that the plate rigidity ( ) D = 2 Et (h / 2) 2 / 1 − ν 2 . In order to check the convergence of the solutions for truncation of the infinite sums in the formulations, several truncation numbers are examined at T = 10,20,40 . Other necessary parameters for analyzing are shown in Table 4.1. Table 4.1 Parameters for Analyzed Circular VLFSs 72 Hydroelastic Analysis of Uniform Circular Plate Item Problem 1 Problem 2 Radius R 50 200 Thickness h 5 2 Water Depth d 20 20 Density Ratio γ / ρ 0.1 0.25 Non-dimensional Plate Rigidity S 0.7 0.000433 Poinsson’s Ratio ν 0.3 0.3 Shear Correction Factor κ 2 5/6 5/6 Incident Wave Length λ 50 50 Number of Nodal Diameters of Mode N 14 14 Number of Sequence for Each Mode M 5 5 Truncation Number for Infinite Sums T 10, 20, 40 10, 20 Note that the natural dry modes and stress-resultants of uniform circular plates in Problem 1 and Problem 2 are extracted from examples given in Section 2.3 (corresponding to thickness ratio h / R equals to 0.1 and 0.01, respectively). FORTRAN Code is employed to develop the program to analyze the hydroelastic analysis. The deflection, the bending moment, the twisting moment and the shearing force are plotted along the center-line of the circular plate (along x -axis) in Figs. 4.2 to 4.8. Figure 4.2 shows that Problem 1 model behaves almost like a rigid plate, although slight elastic deformation is observed. The convergence of the results in terms of the truncation of infinite sums is presented in Fig. 4.3 at T = 10, 20 and 40 . It can be seen that the truncation numbers examined here are good enough for convergence. In Fig. 4.4, it seems clear that the Problem-2-circular plate has a considerable elastic deformation. As pointed out by Zilman and Miloh (2000), the hydroelastic effect becomes important when S ≤ 0.001 . The results presented herein are also consistent with their arguments. The convergence check for the truncation number T is also 73 Hydroelastic Analysis of Uniform Circular Plate observed in Fig. 4.5. The truncation number T = 20 presented here gives a reasonable result for engineering purposes. Figures 4.6, 4.7 and 4.8 show the bending moment, the twisting moment and the shearing force of circular VLFS for Problem 2. As expected, the free-edge boundary conditions are exactly satisfied because of the ultilization of the Mindlin plate theory with exact solutions. 0.400 w/A 0.200 Real part 0.000 -1.0 -0.5 0.0 0.5 Imaginary part 1.0 -0.200 -0.400 x/R Figure 4.2 Deflection for Problem 1, Real part & Imaginary part 0.50 w/A T=10 0.25 T=20 T=40 0.00 -1.0 -0.5 0.0 0.5 1.0 x/R Figure 4.3 Deflection Amplitude for Problem 1 Wave direction 0.2 w/A 0.1 -1.0 0.0 -0.5 0.0 0.5 1.0 Real part Imaginary part -0.1 -0.2 x/R Figure 4.4 Deflection for Problem 2, Real part & Imaginary part 74 Hydroelastic Analysis of Uniform Circular Plate Wave direction 0.3 0.2 w/A T=10 T=20 0.1 0.0 -1.0 -0.5 0.0 0.5 1.0 x/R Figure 4.5 Deflection Amplitude for Problem 2 Wave direction 15 MrrR/(DA) 12 9 6 3 0 -1.0 -0.5 0.0 0.5 1.0 x/R Figure 4.6 Bending moment amplitude for Problem 2 Wave direction 5 MrθR/(DA) 4 3 2 1 0 -1.0 -0.5 0.0 0.5 1.0 x/R Figure 4.7 Twisting moment amplitude for Problem 2 Wave direction Qr R2/(DA) 125 100 75 50 25 0 -1.0 -0.5 0.0 0.5 x/R Figure 4.8 Shear force amplitude for Problem 2 75 1.0 Hydroelastic Analysis of Uniform Circular Plate 4.8 CONCLUDING REMARKS In this Chapter, the hydroelastic problem for a circular VLFS subjected to wave is analyzed in an exact manner for both plate and fluid parts. The implementations if the method presented herein is not so complicated for engineers to obtain accurate solutions for their hydroelastic analysis. Numerical results themselves presented herein serve as benchmark solutions. Although the formulations are given in explicit formula, infinite sums are included. Thus, the convergence check is observed. Most importantly, the theory used here is based on the more refined Mindlin plate theory, instead of the commonly used classical thin plate theory. With this advanced feature, we can obtain exact stress resultants that satisfy free-edge boundary conditions. Note that in an earlier study, Wang et al. (2000) showed that finite element and Ritz analyses of such plates could not produce stress-resultants that satisfy the natural boundary conditions. Therefore the presented results should be useful as they serve as benchmark solutions for verification of numerical programs such as BEM or FEM for VLFS analysis. 76 Chapter 5 HYDROELASTIC ANALYSIS OF STEPPED CIRCULAR VLFS In this chapter, the hydroelastic analysis of stepped circular VLFS is carried out in an exact manner for both plate and fluid parts. The exact modal deflection and stress-resultants are given. A numerical result is examined and a comparative study between stepped VLFS and uniform VLFS in terms of modal deflection and stresses is also investigated. 5.1 PROBLEM DEFINITION The stepped circular VLFS system is shown in Fig. 5.1. The cylindrical coordinate system (r , θ , z ) is introduced, where the origin is on the flat sea-bed and the z- axis is pointing upwards. The undisturbed free surface is on the plane z = d , and the sea-bed is assumed to be flat at z = 0 . The floating flat stepped circular plate has a radius of R and step thicknesses h1 for ( R ≥ r ≥ bR ) and h2 for ( bR ≥ r ≥ 0 ) (see Fig. 5.1). The bottom surface of the stepped plate is assumed to be flat and a zero draft is assumed for simplifying the fluid-domain analysis. We wish to determine the deflections and stressresultants of the stepped circular VLFS under action of wave forces. 5.2 GOVERNING EQUATIONS AND BOUNADRY CONDITIONS Following the same procedure for hydroelastic analysis of uniform circular VLFS in Chapter 4, the time-harmonic motion with the complex time dependence e iϖt being applied to all first-order oscillatory quantities, the complex velocity potential φ (r ,θ , z ) 77 Hydroelastic Analysis of Stepped Circular Plate is governed by the Laplace’s equation ∇ 2φ = 0 in the fluid domain, and it must also satisfy the boundary conditions on the free surface, on the sea bed, and on the wetted bottom surface of the floating body of equations which are the same as Eqs. (4.2), (4.3) and (4.4), respectively. incident wave r,χ R q x bR z=d z h2 h1 Sea bed, z = 0 O Figure 5.1 Geometry of a Stepped Circular VLFS ∂φ (r , θ , z ) ϖ 2 = φ (r ,θ , z ) ∂z g on z = d , r > R (5.1) ∂φ (r ,θ , z ) =0 ∂z on z = 0 (5.2) ∂φ (r ,θ , z ) = iϖw(r , θ ) ∂z on z = d , r < R (5.3) The radiation condition ⎡ ∂ (φ − φ I ) ⎤ lim r ⎢ + ik (φ − φ I )⎥ = 0 as r → ∞ r →∞ ⎣ ∂r ⎦ 78 (5.4) Hydroelastic Analysis of Stepped Circular Plate and φ I the potential representing the undisturbed incident wave ∞ igAM 01 / 2 igA cosh kz ikx φI = e = f 0 ( z )∑ ε n i n J n (kr ) cos nθ ϖ cosh kd ϖ cosh kd n =0 (5.5) where ε 0 = 1, ε n = 2 (n ≥ 2) ; A the amplitude of the incident wave; J n the Bessel function of the first kind of order n ; and k tanh kd = ϖ 2 / g (5.6) f ( z ) = M −1 / 2 cosh kz (5.7) 1 ⎛ sinh 2kd ⎞ ⎜1 + ⎟ 2kd ⎠ 2⎝ (5.8) M = The governing equations of the floating stepped circular plate is followed the Mindlin plate equations as γh ∂M rri 1 ∂M rθi M rri − M θθi + + − Qri + ϖ 2 i ψ ri = 0 r ∂θ r ∂r 12 3 (5.9) ∂M rθi 1 ∂M rθi 2M rθi γh + + − Qθi + ϖ 2 i ψ θi = 0 ∂r r ∂θ r 12 (5.10) ∂Qri 1 ∂Qθi Qri + + + ϖ 2 γhi wi = p (r , θ ) ∂r r ∂θ r (5.11) 3 where the bending moments M rri , M θθi , twisting moment M rθi , the shear forces Qri , Qθi of sub-plate i (e.g i = 1 representing annular sub-plate and i = 2 representing core circular sub-plate) can be calculated from the constitutive equations (Mindlin 1951) ⎧ ∂ψ ∂ψ θi ν⎛ M rri = Di ⎨ ri + ⎜ψ ri + r⎝ ∂θ ⎩ ∂r ⎞⎫ ⎟⎬ ⎠⎭ ⎧ ∂ψ ri 1 ⎛ ∂ψ θi M θθi = Di ⎨ν + ⎜ψ ri + r⎝ ∂θ ⎩ ∂r ⎞⎫ ⎟⎬ ⎠⎭ 79 (5.12) (5.13) Hydroelastic Analysis of Stepped Circular Plate M rθi = ⎫ ⎧ Di (1 − ν )⎨ ∂ψ θi + 1 ⎛⎜ ∂ψ ri − ψ θi ⎞⎟⎬ 2 r ⎝ ∂θ ⎠⎭ ⎩ ∂r (5.14) ∂w ⎞ ⎛ Qr = κ 2 Gh⎜ψ r + ⎟ ∂r ⎠ ⎝ (5.15) 1 ∂w ⎞ ⎛ Qθ = κ 2 Gh⎜ψ θ + ⎟ r ∂θ ⎠ ⎝ (5.16) By assuming the bottom surface of the plate to be flat (see Fig. 5.1), the pressure p (r ,θ ) underneath the stepped plate is related to the velocity potential by φ (r ,θ , z ) by p (r ,θ ) = −iρϖφ (r , θ , d ) − ρgw(r ,θ ) . (5.17) The floating body subjected to no constraint in the vertical direction along its edges must satisfy the zero bending moment, zero twisting moment and zero shear force conditions for a free edge of equations Qr1 = 0, M rr1 = 0, M rθ 1 = 0 . (5.18, 19, 20) and the matching conditions at the interface of stepped location ( χ = r / R = b ) w1 = w2 , ψ r1 = ψ r 2 , ψ θ 1 = ψ θ 2 (5.21, 22, 23) M rr1 = M rr 2 , M rθ 1 = M rθ 2 , Qr1 = Qr 2 (5.24, 25, 26) 5.3 EQUATIONS OF MOTION IN MODAL COORDINATES The modal expansion of the stepped circular VLFS’s motion which consist of the product of the natural dry modes of stepped circular Mindlin plate with free edge N M w(r ,θ ) = ζ 00 w00 + ζ 10 w01 cos θ + ∑∑ ζ ns wns (r ) cos nθ (5.27) n = 0 s =1 N M ψ r (r ,θ ) = ∑∑ ζ nsψ r ,ns (r ) cos nθ n = 0 s =1 80 (5.28) Hydroelastic Analysis of Stepped Circular Plate N M ψ θ (r ,θ ) = ∑∑ ζ nsψ θ ,ns (r ) sin nθ (5.29) n = 0 s =1 where wns ,ψ r ,ns ,ψ θ ,ns represent the natural dry modes of stepped circular plate; n is the number of nodal diameters of the mode ( n = 0,1,..., N ) ; s the sequence for a given n value ( s = 1,2,..., M ); and mode shapes of rigid body modes (heave and pitch) are given w00 = 2 , w10 = 2r / a (5.30) The velocity potential φ is then decomposed into a diffraction potential φ Dn and a radiation potential φ ns whose solutions are given in Chapter 4, section 4.4 and 4.3, respectively. The kinetic energy T , the strain energy U for stepped circular VLFS are decomposed into two sub-plates as ⎧ 2 h1 2 2 ⎫ 1 2π R 2 T = ∫ ∫ γh1ϖ ⎨w1 + ψ r1 + ψ θ21 ⎬rdrdθ 2 0 bR 12 ⎩ ⎭ 2 ⎫ ⎧ 2 π bR h 1 2 + ∫ ∫ γh2ϖ 2 ⎨w2 + 2 ψ r22 + ψ θ22 ⎬rdrdθ 2 0 0 12 ⎭ ⎩ 2π R ν ∂ψ r1 ∂ψ θ 1 1 ∂ψ ( U = ∫ ∫ {D1[( r1 ) 2 + 2 ⋅ + ψ r1 ) 2 0 bR ∂r r ∂r ∂θ ( ) ( ) 1 ∂ψ θ 1 1 −ν ∂ψ θ 1 ∂ψ r1 2 ( ) ] +ψ r1 ) 2 + 2 (ψ θ 1 − r − 2 2r r ∂θ ∂r ∂θ 1 ∂w ∂w + κ 2Gh1[( 1 +ψ r1 ) 2 + 2 ( 1 + rψ θ 1 ) 2 ]}rdrdθ ∂r r ∂θ 2π bR 1 ν ∂ψ r 2 ∂ψ θ 2 ∂ψ ( + ∫ ∫ {D2 [( r 2 ) 2 + 2 ⋅ +ψ r 2 ) 2 0 0 ∂r r ∂r ∂θ (5.31) + (5.32) 1 ∂ψ θ 2 1 −ν ∂ψ θ 2 ∂ψ r 2 2 ( ) ] +ψ r 2 ) 2 + 2 (ψ θ 2 − r − 2 2r r ∂θ ∂r ∂θ 1 ∂w ∂w + κ 2Gh2 [( 2 +ψ r 2 ) 2 + 2 ( 2 + rψ θ 2 ) 2 ]}rdrdθ ∂r r ∂θ and the energy associated with the pressure V + V = −∫ 2π 0 ∫ R 0 p (r , θ , d ) wrdrdθ (5.33) 81 Hydroelastic Analysis of Stepped Circular Plate The Hamilton’s principle can be given as − δT + δU + δV = 0 (5.34) By substituting wi , ψ ri and ψ θi which are given by Eq. (5.27) to (5.29) and applying Galerkin’s method, we obtain M ∑ R ζ ns [−γh1ϖ 2 ∫ {w1ns w1np + s = 0 ( n = 0 ,1) s =1( n ≥ 2 bR bR 2 h1 (ψ r1,nsψ r1,np + ψ θ 1,nsψ θ 1,np )}rdr 12 2 h2 (ψ r 2,nsψ r 2,np + ψ θ 2,nsψ θ 2,np )}rdr 12 0 R ∂ψ r1,ns ∂ψ r1,np ν ∂ψ r1,ns + ∫ {D1 [ ⋅ + ⋅ (nψ θ 1,np + ψ r1,np ) ∂r ∂r ∂r r bR 1 ν ∂ψ r1,np + (nψ θ 1,ns + ψ r1,ns ) + 2 (nψ θ 1,ns + ψ r1,ns ) ⋅ (nψ θ 1,np + ψ r1,np ) r ∂r r ∂ψ θ 1,np ∂ψ θ 1,ns 1 −ν + 2 (ψ θ 1,ns − r + nψ r1,ns ) ⋅ (ψ θ 1,np − r + nψ r1,np )] ∂r ∂r 2r ∂w1np ∂w + κ 2 Gh1 [( 1ns + ψ r1,ns ) ⋅ ( + ψ r1,np ) (5.35) ∂r ∂r 1 + 2 (− nw1ns + rψ θ 1,ns ).(− nw1np + rψ 1θ ,np )]}dr ] r bR ∂ψ r 2,ns ∂ψ r 2,np ν ∂ψ r 2,ns + ∫ {D 2 [ ⋅ + ⋅ (nψ θ 2,np + ψ r 2,np ) ∂r ∂r ∂r r 0 1 ν ∂ψ r 2,np + (nψ θ 2,ns + ψ r 2,ns ) + 2 (nψ θ 2,ns + ψ r 2,ns ) ⋅ (nψ θ 2,np + ψ r 2,np ) r ∂r r ∂ψ θ 2,np ∂ψ θ 2,ns 1 −ν + 2 (ψ θ 2,ns − r + nψ r 2,ns ) ⋅ (ψ θ 2,np − r + nψ r 2,np )] ∂r ∂r 2r ∂w2 np ∂w + κ 2 Gh2 [( 2 ns + ψ r 2,ns )( + ψ r 2,np ) ∂r ∂r 1 + 2 (− nw2 ns + rψ θ 2,ns ) ⋅ (− nw2 np + rψ 1θ ,np )]}dr ] r − γh2ϖ 2 ∫ {w2 ns w2 np + In view of the fact that the normal modes satisfy Eqs. (5.9) to (5.16) and Eqs. (5.18) to (5.26), Eq.(5.35) can be simplified as M ∑ ζ ns [−ϖ 2 γR 2 (h1 2 Λ 1 + h2 2 Λ 2 ) + ϖ ns2 γR 2 (h1 2 Λ 1 + h2 2 Λ 2 ) s = 0 ( n = 0 ,1) s =1( n ≥ 2 ) R R R 0 0 0 (5.36) − ϖ 2 ρ ∫ φ ns (r , d ) wnp rdr + ρg ∫ wns wnp rdr ] = − ρiϖ ∫ φ Dn (r , d ) wnp rdr R where Λ 1 = ∫ {w1ns w1np + bR τ 12 12 (ψ r1,nsψ r1,np + ψ θ 1,nsψ θ 1,np )}rdr 82 (5.37) Hydroelastic Analysis of Stepped Circular Plate R Λ 2 = ∫ {w2 ns w2 np + bR τ 22 12 (ψ r 2,nsψ r 2,np + ψ θ 2,nsψ θ 2,np )}rdr (5.38) and ϖ ns represents the natural frequency. Equation (5.36) may be represented in a nondimensional form by M ∑ ζ ns [− s = 0 ( n = 0 ,1) s =1( n ≥ 2 ) − (5.39) ϖ R 1 2 g 1 ϖ 2R γ ( )(τ 1 Λ 1 + τ 2 Λ 2 ) + λ2ns S 0 (τ 1 Λ 1 + τ 2 Λ 2 ) g ρ τ0 ( 1 φ R∫ 0 1 ns wnp χdχ ) + ∫ wns wnp χdχ ] = − 0 iϖ φ Dn wnp χdχ g ∫0 1 where χ = r / R , τ 1 = h1 / R , τ 2 = h2 / R , τ 0 = h0 / R , S 0 = D0 /( ρgR 4 ) . Also, the frequency parameter λ ns and the coressponding mode shapes wns of stepped cicular plate are extensively given in Chapter 3. The homogeneous sytem of equations (5.39) can be solved separately with respect to modal amplitudes ζ ns of each mode n. Then they are back substituted into the Eqs. (5.27) to (5.29) to obtain the total responses of w(r , θ ) and stress-resultants. 5.4 RESULTS AND DISCUSSIONS The analysis has been made for two Design Types of stepped circular VLFS and their reference uniform circular VLFS as shown in Table 5.1. The stepped thickness ratios are set as α = 2 and α = 0.5 for Design Type 1 plate and Design Type 2 plate, respectively. The stepped location of both types is at χ = 0.5. Their reference plate which has the same volume of material with the stepped plates (see Chapter 3 for defining a reference plate of a stepped plate) is the uniform circular plate with reference thickness ratio t 0 = h0 / R = 0.125, radius R = 500m . Be noted that the plate ( rigidity of reference circular VLFS D = 2 Et (h / 2) 2 / 1 − ν 2 83 ) in which the top and Hydroelastic Analysis of Stepped Circular Plate bottom plate thickness t = 20 mm and Young’s modulus E = 206 GPa. And the mode shapes and modal stress-resultants of these stepped circular plates and their reference uniform plate are given in Chapter 3. These dry mode solutions should be developed into hydroelastic analysis by using modal expansion matching method to final results for these particular cases of stepped VLFS. Other parameters for the hydroelastic solutions are listed in the followings Table 5.1 Parameters for Analyzed Stepped Circular VLFSs Design Reference Design Type 1 Plate Type 2 500 500 500 2 1 0.5 Thickness ratio of annular sub-plate h1 0.1 0.125 0.143 Thickness ratio of core circular sub-plate h2 0.2 0.125 0.071 Reference thickness Ratio h0 / R N/A 0.125 N/A Step Location b 0.5 N/A 0.5 Water Depth d 25 25 25 Density Ratio γ / ρ 0.33 0.33 0.33 Non-dimensional Plate Rigidity S 0 N/A 0.014 N/A Poinsson’s Ratio ν 0.3 0.3 0.3 Shear Correction Factor κ 2 5/6 5/6 5/6 Incident Wave Length λ 50 50 50 Number of Nodal Diameters of Mode N 14 14 14 Number of Sequence for Each Mode M 5 5 5 Item Radius R Stepped Thickness Ratio α The deflection, the bending moment, the twisting moment and the shearing force are presented in non-dimensional forms M rri R /( D0 A) , M rθi R /( D0 A ) and Qri R 2 /( D0 A) , respectively. The displacement amplitudes and bending moment amplitudes are shown in Fig. 5.2. The displacement results reveal that the maximum deflections of the both two 84 Hydroelastic Analysis of Stepped Circular Plate Design Type stepped circular VLFS (0.024333 for Design Type I and 0.020004 for Design Type II) less than that of the reference constant thickness VLFS (0.055656 for uniform plate). However, the bending moment of Design Type II might be higher and the bending moment of Design Type I might be lower than that of reference circular VLFS (see Fig. 5.2). The twisting moment amplitudes of both Design Type I and Design Type II plates could be lower than that of corresponding reference plate (see Fig. 5.3). The shear force amplitude of Design Type I plate could be much lower than that of reference plate while the shear force of Design Type II plate is not much difference to that of reference plate (see Fig. 5.3). In order to observe more clearly the distribution of stresses on each sub-plate of stepped VLFS, we depict them in terms of non-dimensional stresses as can be seen in Fig. 5.4. The non-dimensional stresses corresponding to bending moment, twisting moment and shear force are M rri R / τ i2 D0 A , M rθi R / τ i2 D0 A , Qri R 2 / τ i D0 A , respectively. The stresses in both core circular and outer annular sub-plates of two type stepped plates become smaller than those of the reference uniform plate except for the stresses such as stress M rri R / τ i2 D0 A near the center of core sub-plate of Design Type II plate and stress M rθi R / τ i2 D0 A near the thinner sub-plate of both two types of stepped plates or stress Qri R 2 / τ i D0 A near the step location of Design Type II plate. This finding reveals that when designing stepped position details of VLFS, one may pay more attention for the stress-concentration at these kinds of location. The stresses of M rri R / τ i2 D0 A , Qri R 2 / τ i D0 A of Design Type I plate in this certain case decreases over the whole platform when comparing with that of the reference uniform circular VLFS. Over all, these results show that stepped circular VLFS designs take more advantage than their uniform circular VLFS in terms of lowering stresses. 85 Reference plate with constant thickness ( α = 1, τ o = 0.125 ) 0.03 0.02 0.04 0.02 w/A 0.06 0.01 0.02 0.00 0.00 0.0 0.5 -1.0 1.0 -0.5 1.0 -0.5 0.16 0.08 0.5 0.0 0.5 1.0 -0.5 x/R 0.5 1.0 -1.0 -0.5 0.0 x/R Figure 5.2 Displacements and Bending Moments Amplitudes for stepped VLFSs and the reference constant thickness VLFS 86 0.5 1.0 0.5 0.0 0.0 x/R 1.0 1.0 0.0 -1.0 0.5 1.5 1.0 0.00 0.0 x/R 1.5 M A) rriR/(D0 A) MrriR/(D MrriR/(D0A) A) MrriR/(D Bending Moments 0.5 -1.0 x/R 0.24 -0.5 0.00 0.0 x/R -1.0 0.01 M A) rriR/(D0 A) MrriR/(D -0.5 h2 0.03 w/A -1.0 Design Type II ( α = 0.5, b = 0.5) h1 ho w/A Displacements h1 Design Type I ( α = 2, b = 0.5) h2 h1 ho 0.2 0.1 0.0 0.5 -1.0 1.0 -0.5 0.1 0.0 0.0 x/R 3 6 QQr i R2/(D /(D0A) A) x/R 2 1 0.5 -1.0 1.0 -0.5 2 0.0 0.5 1.0 -0.5 0.0 x/R x/R 1.0 0.5 1.0 4 2 0.5 1.0 0 -1.0 -0.5 0.0 x/R Figure 5.3 Twisting Moments and Shear Forces Amplitudes for stepped VLFSs and the reference constant thickness VLFS 87 0.5 6 4 -1.0 0.0 x/R 0 0 -0.5 0.2 0.0 0.0 -1.0 Mrθq R/(D i R/(DA) 0 A) M M R/(D0A) A) MrqθiR/(D M Mrθq R/(D A) i R/(D0A) 0.1 -0.5 0.3 0.3 0.2 Q R2/(D /(D0A) A) Qr i R Shear Forces Twisting Moments 0.3 -1.0 Design Type II ( α = 0.5, b = 0.5) h2 Reference plate with constant thickness ( α = 1, τ o = 0.125 ) Qr i R 2/(D Q /(D0A) A) h1 Design Type I ( α = 2, b = 0.5) h2 24 60 16 8 20 0.5 -1.0 1.0 -0.5 -1.0 -0.5 0.5 1.0 8 -1.0 -0.5 x/R 0.0 0.5 QriQR2R/(τ/(iτDD 0 A) A) 10 x/R 0.5 1.0 20 -1.0 -0.5 0.0 0.5 1.0 60 40 20 0 0.0 1.0 0 -0.5 x/R 40 0 0.5 20 -1.0 1.0 60 20 1.0 40 x/R 30 0.5 60 0 0.0 0.0 x/R 16 0 2 D A) Qri RQ /(τRi D/(τ 0 A) 1.0 2 MrMrθiR/( iD q i R/(τ τ i D0 A) A) 2 MMrθiR/( D0A) A) rq i R/(τ τii D M MrθiR/( τi2iDD0A)A) rq i R/(τ 8 -0.5 0.5 24 16 -1.0 0 0.0 x/R 24 -0.5 80 0 0.0 x/R -1.0 160 2 Qri R A) A) Q /(Rτi/(Dτ0 D -0.5 240 40 0 -1.0 Design Type II ( α = 0.5, b = 0.5) 2 MrrMrriR/( iD i R/(τ τ 0 A) i DA) 2 MMrriR/( DA) i iD rr i R/(τ τ 0 A) Reference plate ( α = 1, τ o = 0.125 ) 2 DA) MMrriR/( iiD rr i R/(ττ 0 A) Design Type I ( α = 2, b = 0.5) 0 0.0 x/R 0.5 1.0 -1.0 -0.5 0.0 x/R Figure 5.4 Stresses M rri R /(τ i2 D0 A) , M rθi R /(τ i2 D0 A) , Qri R 2 /(τ i D0 A) for stepped VLFSs and the reference constant thickness VLFS 88 Hydroelastic Analysis of Stepped Circular Plate 5.5 CONCLUDING REMARKS In this chapter, the hydroelastic problem for a stepped circular VLFS subjected to wave is analyzed in an exact manner for both plate and fluid parts. This new exact hydroelastic solution of stepped circular VLFS is compared with the results of the reference uniform circular VLFS (which has the same material volume) in order to assess the advantages of the stepped circular VLFS over uniform circular VLFS. A numerical example showed that the deflection and stresses of circular VLFS could be reduced by having an approximately design stepped plate. The presented exact deflections and stress-resultants of stepped circular VLFS should be very useful for engineers who may wish to check the accuracy of FEM or BEM results of stepped circular VLFS. 89 Chapter 6 CONCLUSIONS Conclusions drawn from the studies on the vibration and hydroelastic analysis of circular VLFS are presented in this chapter. Future studies in this research area are also suggested. 6.1 CONCLUSIONS In this thesis, the hydroelastic problem of pontoon-type circular VLFS subjected to wave is analyzed in an exact method not only for plate but also for the fluid part. The hydroelastic analysis consists of separating the hydrodynamic analysis from the dynamic response analysis or free vibration analysis of the VLFS. The deflection of the plate with free edges is decomposed into vibration modes which can be obtained in an exact manner. Then the hydrodynamic radiation forces are evaluated for unit amplitude motions of each mode together with the diffraction forces. The hydrodynamic forces have been evaluated by the eigenfunction expansion matching method, by which analytical solutions can be obtained in an exact manner. The Galerkin’s method, by which the governing equation of the plate is approximately satisfied, is used to calculate the modal amplitudes, and then the modal responses are summed up to obtain the total response. The Mindlin plate theory is employed instead of the commonly used classical thin plate theory to produce the accurate stress-resultants which are difficult to obtain using numerical methods (see Fig. 2.2). These accurate vibration solutions when employed in the hydrodynamic analysis yield highly accurate deflection and stress-resultants of circular VLFS under action of waves (see Figs. 4.6 to 4.7 and Figs. 5.2 to 5.4). 90 Conclusions Moreover, the uniform and stepped circular plates are presented in this research in order to assess the advantages of the stepped circular VLFS over uniform circular VLFS. The research findings showed that the deflections and stresses of circular VLFS could be reduced by using stepped plates (see Fig. 5.2 and Fig. 5.4). Hence, the stepped circular plate is recommended when designing circular VLFS under action of waves for more economic use of materials. The formulations for vibration analysis and hydroelastic analysis are given in explicit forms. Hence, the implementation of the hydroelastic analysis is more tractable for engineers to obtain accurate solutions. These accurate results should be useful as benchmark solutions for engineers and researchers who are developing numerical techniques for the hydroelastic analysis of circular VLFS. 6.2 RECOMMENDATIONS Despite much research being done on the hydroelastic analysis of pontoon-type circular VLFS, there is still much work to be done. Future studies could investigate the followings • With the advantage features of stepped plate, multiple stepped circular VLFS and the optimal design of step locations should be considered. • The same study may be repeated for hydroelastic analysis of circular VLFS with a central circular cutout. • Another possible research work on this area is to develop a simplify methods for the analysis of circular VLFS. 91 REFERENCES Eatock, T.R. and Okushu M. (2000), “Green functions for hydroelastic analysis of vibrating free-free beams and plates”, Applied Ocean Research, 22, pp. 295-314. Endo, H. 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(2000), "Hydroelastic buoyant circular plate in shallow water: a closed form solution", Applied Ocean Research, 22, pp. 191-198. 96 [...]...ρ density of the fluid (kg/m3) τ thickness ratio of uniform circular plate = τ0 thickness ratio of reference uniform circular plate = τ1 thickness ratios of stepped circular plate = h1 R τ2 thickness ratios of stepped circular plate = h2 R ω natural frequency of uniform circular plate ω0 natural frequency of reference uniform circular plate ωs natural frequency of stepped circular plate ψr... floating structures are suitable for use in only calm waters, often near the shoreline The pontoon-type VLFS is very flexible when compared to other kinds of offshore structures and so the elastic deformations are more important than their rigid body motions Semi-submersible-type Pontoon-type Figure 1.11 Types of Floating Structures This thesis deals with the hydroelastic analysis of pontoon-type circular. .. comprises of two parts Part 1, consisting of Chapters 2 and 3, deals with the free vibration analysis of a uniform and non-uniform circular plates vibrating in air, normally reformed to dry mode solution Part 2, consisting of Chapters 4 and 5, is concerned with the hydroelastic analysis of these circular VLFSs under actions of waves More specifically, Chapter 2 deals with the free vibration analysis of circular. .. results of a reference circular plate of constant thickness and equal volume Comparison study of the deflections and stress-resultants of stepped circular VLFS and its reference uniform thickness circular VLFS are also given These exact solutions and research findings should be useful in the hydroelastic analysis and costeffective design of circular VLFSs with a stepped thickness variation 1.4 LAYOUT OF. .. Analyzed Circular VLFSs 66 Table 5.1 Parameters for Analyzed Stepped Circular VLFSs 79 xiv Chapter 1 INTRODUCTION This chapter introduces the very large floating structures (VLFSs) and their applications A literature review on hydroelastic analysis of pontoon-type VLFS, the objective of research work and layout of the thesis are presented 1.1 BACKGROUND INFORMATION ON VLFS With a growing of population... LITERATURE REVIEW The hydroelastic analysis of very large floating structures has attracted the attention of many researchers, especially with the construction of the Mega-Float in Tokyo Bay in 1995 Many researchers analyzed pontoon-type VLFS of a rectangular planform (Utsunomiya et al 1998, Mamidipudi and Webster 1994, Endo 2000, Ohkusu and Namba 1998, Namba and Ohkusu 1999), mainly because of practical reasons... solutions for assessing the accuracy of numerical results A plate shape that admits the derivation of exact solutions for plates with free edges is the circular shape Probably, the first paper on hydroelastic analysis of circular VLFS is the one written by Hamamoto and Tanaka (1992) They developed an analytical approach to predict the dynamic response of a flexible circular floating island subjected to stochastic... dynamic responses of the structure Following studies on the free vibration analysis, Chapter 4 and 5 deal with hydroelastic analysis of uniform circular VLFS and stepped circular VLFS, respectively The analysis of VLFS is carried out in the frequency domain using modal expansion matching method Firstly, decomposing the deflection of circular Mindlin plates given in Chapter 2 and 3 into vibration modes and... view of how a circular plate deflect regarding to number of n and s, one may refer to the 3D-plots of mode shapes as given in Figure 2.3 In the hydrodynamic analysis of a VLFS structure, the mode shapes and modal stress resultants from the free vibration analysis of the structure are utilized to predict the dynamic responses of the structure The exact mode shapes and modal stress resultants for free circular. .. 3 Figure 1.5 Floating island at Onomichi Hiroshima, Japan 3 Figure 1.6 Floating pier at Ujina Port Hiroshima, Japan 3 Figure 1.7 Floating Restaurant in Yokohoma, Japan 3 Figure 1.8 Floating heliport in Vancouver, Canada 3 Figure 1.9 Nordhordland Brigde Floating Bridge, Norway 3 Figure 1.10 Hood Canal Floating Bridge, USA 3 Figure 1.11 Types of Floating Structures .. .HYDROELASTIC ANALYSIS OF CIRCULAR VERY LARGE FLOATING STRUCTURES BY LE THI THU HANG B.E (Hanoi University of Civil Engineering) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING... Figure 1.11 Types of Floating Structures This thesis deals with the hydroelastic analysis of pontoon-type circular VLFSs under action of waves Both uniform circular VLFS and stepped circular VLFS’s... VLFS analysis Introduction 1.2 LITERATURE REVIEW The hydroelastic analysis of very large floating structures has attracted the attention of many researchers, especially with the construction of