Hydroelastic analysis of circular very large floating structures

21 Figure 2.2c Mode shapes and modal stress resultants for free circular plates based on classical thin plate theory and Mindlin plate theory .... 22 to 24 Figure 2.4a Mode shapes and m

HYDROELASTIC ANALYSIS OF CIRCULAR VERY

LARGE FLOATING STRUCTURES

LE THI THU HANG

NATIONAL UNIVERSITY OF SINGAPORE

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

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

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

3.3 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

APPENDICES 97

VIBRATION OF STEPPED CIRCULAR PLATE 97

VIBRATION OF STEPPED CIRCULAR PLATE 102

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

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

NOTATIONS

A amplitude of incident wave (m)

D plate rigidity of uniform circular plate (kNm)

g gravitational acceleration (m/sec 2)

h thickness of uniform circular plate (m)

M =M rθR/D0= non-dimensional twisting moment

n number of nodal diameter

r

Q shear forces (kN)

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

α step ratio of thicknesses =h2 h1

χ non-dimensional radial coordinate = r/R

ρ density of the fluid (kg/m 3)

τ thickness ratio of uniform circular plate =

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

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 λs versus step location b for Mindlin plates with

reference thickness ratio τ0 = 0.1, α =0.5 to 2 43

Figure 3.3 Frequency parameter λs versus step location b for plates with τ0= 0.1,

2

=

α and n = 2 45

Figure 3.4a Frequency parameter versus reference thickness ratios λs τ0 for plates

with step location b = 0.5 and stepped thickness ratio α =0.5 46

Figure 3.4b Frequency parameter versus reference stepped thickness ratio α for λs

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 4.8 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/(τi2D0A) M rθi R/(τi2D0A) Q ri R2/(τi D0A)for stepped

VLFSs and the reference constant thickness VLFS 83

LIST OF TABLES Table 2.1 Frequency parameters λ=ωR2 γh/D for free circular Mindlin plates

(ν =0.3, κ2 =5/6) 20

Table 3.1a Frequency parameter λs for stepped plates with step location at b = 1/2,

reference constant thicknesses τo =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 τo =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 τo =0.01 and 0.1 41

Table 3.2b Frequency parameter λs for stepped plates with step location at b = 1/3,

reference constant thicknesses τo =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

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

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 1.3 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

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

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

Semi-submersible-type Pontoon-type

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

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

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

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

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,

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 cost-effective 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

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 sub-plates, 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 non-uniform circular VLFS are served as benchmark solution for checking the validity, convergence and accuracy of numerical solutions and methods for analysis of pontoon-type VLFSs

In Chapter 6, the conclusions and some suggestions for future research work on circular VLFS are presented

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

Figure 2.1 Geometry of a Circular Mindlin 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 (ψr, ψθ) and

transverse displacements w may be expressed as functions of three potentials Θ1, Θ2

and Θ3 in the following manner:

∂θ

∂χ

∂χ

∂σ

∂χ

∂σ

2 1 1

1)

1()

σ

∂θ

∂χσ

2 1 1

1)1(

1)1

−

Θ

−+

κνλ

τ

δδσ

2)

1(612

)(

3

2 1 2 2 2

2 2

2

2 1 2 2 2

2 2

2 2

2

2 2

2 2 2 2

1

4)

1(612)

1(612

ττ

κν

ττ

λδ

2

τ

κνλ

τν

D

h R R

h R

in which r and θ are the radial and circumferential coordinates of the polar coordinate

system, w, ψr and ψθ 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), κ2 the shear correction factor, and λthe

non-dimensionalised frequency parameter

In view of the three potential functions Θ1,Θ and 2 Θ the governing differential 3

equations of the vibrating circular plate, in polar coordinates, may be compactly

2

2 ( ) 1 ( ) 1 ( )

)(

∂θ

∂χ

∂χ

∂χ

02

j j

δδ

(2.15)

0

02

if J

R

j j

n

j j

n j

δχ

( ) ( ) ( ) , 1,2,3

0

02

if Y

S

j j

n

j j

n j

δχ

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 Y n(•) 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 , ψr and ψθ

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 constantsA j

The boundary conditions of circular Mindlin plate with free edge given by

0,

where the transverse shear force Q , the radial bending moment r M and the twisting rr

moment M are given by rθ

=

∂θ

∂ψψχ

∂θ

∂ψχ

-(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

A A

A

(2.22) and [ ]K is a 3 x 3 matrix where the elements are given by

)]

()

()()[

()()[

1(

1(

()

31 σ J n δ

)( 2' 2

32 σ J n δ

)( 3

33 nJ n δ

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 and r ψθ and modal stress resultants Q , r M rr

and M are calculated from the angular frequency rθ ω and the corresponding

eigenvector [ ]T

A A

A1 2 3 In presenting the vibration modes and modal resultants, we normalize the maximum transverse displacement

stress-R w

The corresponding bending moment, twisting moment and shear force are presented in

their non-dimensional forms as follows:

2

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 Q r =1.315 was obtained by

SAP2000 while the corresponding exact peak value is Q r =0.926, a difference of

42% Moreover, Figure 2.2c compares the exact modal displacement w and modal

stress resultants Q , M and M for a free circular plate obtained on the basis of the

normalised effective shear force V is calculated based on its definition in the classical r 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 Q and r M near the vicinity of the plate edge rθ

Unfortunately, the discrepancies found at the vicinity of the free edge also contain the peak values of the stress-resultants And the boundary conditions Q = 0 and r M = 0, rθ

are not satisfied when using the classical thin plate theory due to the free edge conditions based on the thin plate theory are V = 0 and r M = 0 Clearly, these shows rr

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 κ2 =5/6 are adopted for all calculations Exact vibration frequency parameters λ=ωR2 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

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 , and modal stress resultants r Q and r M in the rr

circumferential direction vary with cosnθ , while the modal displacement ψθ and modal stress resultant M vary with rθ sinnθ

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

r

ψ and the modal stress resultants Q and r M for the thicker plate are smaller than rr

the ones for the thinner plate As expected, the rotation ψθ and moment M on the rθ

whole plate and the rotation ψ and shear force r Q at the centre of the plates ( = r χ r/R

= 0) are zero due to the plates vibrating in axisymmetric modes The modal stress resultants Q , r M rr and M vanish at the plate free edge ( = rθ χ r/R = 1) The number

of nodal circumferential lines of the modal displacements w and ψ and modal stress rresultant M increases from 1 to 4 as the mode sequence number s varies from 1 to 4 rr

However, the number of nodal circumferential lines of the modal stress resultant Q r 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

fields and modal stress resultants of the thinner and thicker plates The values of the modal displacements ψ and r ψθ and the modal stress resultants Q and r M for the rr

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 and r ψθ and stress resultants Q , r M rr and M are rθ

zero at the centre of the plates (χ = r/R = 0) The values of the displacement fields w ,

r

ψ and ψθ increase monotonically with increasing radial coordinate (χ = r/R) except

for the rotation ψθ of the thicker plate near the free edge where a slight decrease of ψθ

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 Q , r M rr and M satisfying the rθ

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 Q and r M near the vicinity of the free edge when the number of rθnodal 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 Q , r 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 Q near the free edge for the thicker plate is much r

Table 2.1 Frequency parameters λ=ωR2 γh/D for free circular Mindlin plates (ν =0.3, 6κ2 =5/ )

χ χ

Figure 2.2a SAP2000 modal stress resultants associated with the fundamental

frequency of a uniform circular plate with free edges

Figure 2.2b Exact modal stress resultants associated with the fundamental frequency

of a uniform circular plate with free edges

0.0 0.2 0.4 0.6 0.8 1.0

Thick Plate Theory Thin Plate Theory Effective Shear Force

Q r Mrr

M rθ

V r

Figure 2.2c Mode shapes and modal stress resultants for free circular plates based on

classical thin plate theory and Mindlin plate theory

1 =

= χ

rr

M

44 0

1 = −

=

χ θ

r

M

315 1

n s Mode shape n s Mode shape

n s Mode shape n s Mode shape

n s Mode shape n s Mode shape