Nonlinear dynamic analysis of offshore towers

NONLINEAR DYNAMIC ANALYSIS OF

OFFSHORE TOWERS

by

ARVIND KUMAR JAIN

THESIS SUBMITTED TO THE

INDIAN INSTITUTE OF TECHNOLOGY, DELHI FOR THE AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY

Department of Civil Engineering

INDIAN INSTITUTE OF TECHNOLOGY, DELHI INDIA

JUNE, 1984

TO MY PARENTS

CERTIFICATE

This is to certify that the thesis entitled

"NON LINEAR DYNAMIC ANALYSIS OF OFFSHORE TOWERS" being submitted by Mr. Arvind Kumar Jain to the Indian Insti- tute of Technology, Delhi for the award of the degree of DOCTOR OF PHILOSOPHY is a record of the bonafide research work carried out by him.

Mr. Arvind Kumar Jain has fulfilled the require- ments for the submission of this thesis which to my knowledge has reached the requisite standard.

The thesis, or any part thereof, has not been submitted to any other University or Institute for the award of any Degree or Diploma.

( Dr. T.K. Datta ) AssiStant Professor

Department of Civil Engineering I.I.T. Delhi

Hauz Khas, New Delhi-110 016.

ACKNOWLEDGEMENT

I am grateful to Dr. A.K. Basu for the initiation of the problem and guidance in the initial stages of the work.

I am really indebted to Dr. T.K. Datta for his guidance and encouragement throughout the course of the work.

Without his sincere help the work would not have been completed.

I am thankful to the competent authority of "Indian Institute of Technology, Delhi" for giving me the permi-

ssion to carry out this study as a part-time research scholar. Co-operation extended by Civil Engineering

Department and Computer Services Centre,, Delhi is also duly acknowledged.

I must acknowledge my sincere appreciation for the kind cooperation and help extended by Dr. R.P. Singh.

My sincere thanks are due to Dr. A. Dutta, Mr. Arun Agarwal, Mr. A. Gupta, Mr. J. Verma for their friendship and all the help extended to me.

V

I wish to thank Mr. R.V. Aggarwal for tracing the figures and Mrs. Rama Sharma for typing this thesis.

Finally, I must acknowledge the understanding, coope- ration and assistance extended by my wife during the final

stages of this thesis.

( Arvind Kumar Jain )

ABSTRACT

This study is concerned with the treatment of the nonlinearities due to the drag component of the wave force and the variable submergence of the structure near the

splash zone, in the frequency domain analysis of fixed base offshore structures. It also aims at evaluating the

severity of these effects, on the response of the structure., under both regular and random waves in the presence of

steady current.

For treating the nonlinearity due to the drag component of the wave force, in the frequency domain analysis, six different methods have been developed for solving a SDOF system. They are : (i) Exact Newton-Raphson Method (Method A), (ii) Approximate Newton-Raphson Method (Method B),

(iii) Hydrodynamic Damping Method (Method C), (iv) simple Iteration Method (Method D), (v) Extrapolation Method

(Method E) and (vi) Approximate Linearized Method (Method F).

In Method A, the drag related nonlinearity is handled by using the Newton-Raphson method for simultaneous equations, whereas the other effects ate incorporated by 'correcting' the loading term in every iteration. The Newton-Raphson iteration method requires, the derivatives of the harmonics of the drag 1dading with respect to the harmonics of thee displacement in its solution procedure. Methods B and C

VII

are also based on Newton-Raphson method. The main difference between these methods is in the use of the iteration equa- tion employed in their solution procedure. In Method B,

the off-diagonal submatrices of the Jacobian matrix are taken as null, whereas in Method C, the fourier transforms of the relative velocity, used in the iteration equation,

are taken as zero. In Method D, the hydrodynamic damping (as found in Method C) is taken as Zero in the iteration

equation. The Method D has linear convergence, whereas Method Abased on the exact version of the Newton-Raphson method converges quadratically. Method E accelerates the convergence of Method D by using the delta - square extra- polation technique, Method F uses changed system parameters (frequency and damping), which take-care of the full non-

linearity of the drag force, thus obviating the need for any iteration. Out of the above methods, the one which is based on the Exact Newton-Raphson Method is found to have the fastest convergence.

Three methods namely, (i) Approximate Newton-Raphson Method, (ii) Simple Iterative Method and (iii) Extrapola-

tion Method are then extended for solving MDOF system (an idealized fixed base offshore tower), subjected to hydro- dynamic forces produced by regular waves and steady current, The methods also take into consideration the effect of

viii

variable submergence. This effect is included in the compu- tation of hydrodynamic loading by two approaches namely, Chakrabarti's and Hogben's. Variable submergence is found

to have considerable influence on the response, specially upon the harmonic contents of the hydrodynamic loading.

Inclusion of current in the computation of hydrodynamic loading also changes its harmonic contents.

The two iterative frequency domain methods namely, (i) Approximate Newton-Raphson Method and (ii) Simple Iter-

ative Method are then applied for calculating the dynamic stochastic response of MDOF system. The effect of variable submergence is included in the computation of hydrodynamic loading in an approximate manner, by using Chakrabarti's

and Hogben's approaches. Iterative frequency domain methods use the fourier transforms of the time histories of the

random wave forces, at each iterative cycle. These time histories are generated with the help of simulation proce- dure (Wave superposition technique). For the proper simul- ation, of the time histories, optimum values of some para- meters like the number of wave components and the record length etc. are obtained through a separate numerical study.

The efficiency of the two iterative methods in obtain- ing the response of a fixed base offshore tower, under

IX

random waves is compared with the help of a parametric study.

The effect of the variable submergence and nonlinearity of the drag component of wave loading, on the statistical characteristics of the response, are also examined. Suit- able statistical models to describe the extreme values of response, for the case of drag nonlinearity and variable submergence, are then finally proposed.

CONTENTS

Pae

TITLE PAGE

DEDICATION II

CERTIFICATE III

ACKNOWLEDGEMENT IV

ABSTRACT VI

CONTENTS

LIST OF TABLES XIX

LIST OF FIGURES XXII

CHAPTER I : INTRODUCTION AND LITERATURE REVIEW

1.1 General 1

1.2 Nonlinerrities Involved in the Analysis of Fixed Base Offshore

Structures 2

1.3 Dynamic Analysis of Fixed Base Structures under Random Wave

Loading 7

1.4 Chapterwise Detail 10

1.5 References 17

CHAPTER II : SINGLE DEGREE--OF-FREEDOM TOWER

XI

Page

MODEL UNDERREGULARWAVES

2.1 Introduction 22

2.2 ObjLietiVes 28

2.3 Theory 29

2.3.1 Idealization of Single Degree-of-

Freedom Tower Model 29

2.3.2 Derivation of Equations 31 2.3.3 Computational Schemes 37 2.3.3.1 Scheme A : Exact Newton--Raphson

Method 38

2.3.3.2 Scheme B e Approximate Newton-

Raphson Method 39

2.3.3.3 Scheme C s Hydrodynamic Damping

Method 40

2.3.3.4 Scheme D : Simple Iteration method 43 2.3.3.5 Scheme E a Extrapolation Method 43

2.4 Numerical Study and Discussion of

Results 44

2.5 Conclusions 51

2.6 Nomenclature 52

2.7 References 53

XII

CHAPTER III :

3.1 3.1.1

A LINEARIZED SOLUTION METHOD FOR

Pale

55 55 TREATING DRAG NONLINEARITY

Introduction

Linearization techniques

3.2 Objectives 60

3.3 Theory 60

3.3.1 Reduction to Simple Structure (SDOF

System) 60

3.3.2 Method A 64

3.3.3 Approximate Linearized Method ⁶⁴ 3.3.4 Determination of Changed Parameters 65

3.4 Numerical Study 67

3.4.1 Convergence Criteria for Method A 68 3.5 Discussion of Results 68

3.6 Conclusions 81

3.7 Derivation of

I

V

3.7.1 Lumped Area, A 83

3.7.2 Lumped Volume, V 84

XIII

3.8 3.9

CHAPTER IV

Nomenclature References

: FIXED-BASE TOWER UNDER REGULAR

Page 86 88

WAVES

4.1 Introduction 89

4.1.1

Cylinders 91

4.1.3 Morison Equation for a Flexible Cylinder & Uncertainties Regarding

the Values of CD and Cm 93 4,1.4 Wave - Current Loading ₉₆ 4.1.5 Different Approaches to Incorpo.

rate Variable Submergence

97

4.1.6 Methods to Deal with the Nonlinear- ities in the Hydrodynamic _{Forces 99}

4.2 Objectives 101

4.3 Theory 102

4.3.1 Theoretical Model 102

4.3.2 Structural Model 105

XIV

Page 4.3.3 Computation of Natural Frequencies

and Mode Shapos 107

4.3.4 Evaluation of Wave Length 111 4.3.5 Evaluation of Water Particle Velo-

city and Acceleration 112 4.3.6 Generation of the Wave Loading

Vector 113

4.3.7 Solution of Equations of Motion 116 4.3.7.1 Mode Summation Method 116

4.4 Frequency Domain Iterative Method

Applied to MDOF 118

4.4.1 Approximate Newton-Raphson Iterative

Method 118

4.4.2 Simple Iterative Method 131 4.4.3 Simple Iterative Method Using

6 2

Extrapolation 131

4.5 Numerical Study 132

4.5.1 Model Tower 133

4.5.2 Convergence Criteria 138 4.6 Discussion of Results 138

XV

Page 4.6.1 Effect of Variable Submergence 140 4.6.2 Comparison of Iteration Methods 144 4.6.3 Effect of Fluid-Structure Interaction 148 4.6.4 Comparison of the Efficiencies of

Method A (Proposed Iterative Method in this Study) and Method B (Penzien's

Linearized Method)

153

4.6.S

4,7 Conclusions 162

4,8 Nomenclature 165

4.9 References 167

CHAPTER V: STUDIES ON SPECTRA

5.1 Introduction 170

5.1.1 Ocean Environment 171

5.1.2 Description of Sea State 171 5.1.3 Short-Term Description 172 5.1.4 Representation of Wave Loading 176

5.1.5 Simulation Procedures 180

5.2 Objectives of the Study 184

XVI

119.22

5.3 Theory 185

5.3.1 Simulation of Irregular Waves 185 5.3.2 Response Evaluation of Inertia Domi.

nated System 191

5.4 Numerical Study 193

5.4.1 Sea Surface Elevation Spectrum 193

5.4.2 Response Spectrum 196

5.5 Discussion of Results 198

5.5.1 Number of Wave Components 198 5.5.2 Duration of Simulated Wave Record 206

5.5.3 Effective Band _Width 206

5.5.4 Simulation with Optimum Parameters 212

5.5.5 Response Spectra 216

5.6 Conclusions 216

5.7 Nomenclature 218

5.8 References 221

CHAPTER VI : FIXED BASE TOWER UNDER RANDOM WAVES

XVII

Lase

6.1 Introduction 229

6.1.1 Effect of Nonlinear Drag Forces 231 6.1.2 Effect of Variable Submergence 232

6.2 Objectives 233

6.3 Theory 233

6.3.1 Equation of Motion 233

6.3.2 Evaluation of the Time History of

Wave Force 235

6.3.3 Simulation of Wave Forces 237 6.3.4 Evaluation of Response 238 6.3.5 Solution-of Equation of Motion 240

6.4 Statistics of Response 242

6.4.1 Analysis of the Response Time History 243

6.5 Numerical Study 246

XVII

I

Page

6.6 Discussion of Results 249

6.6.1 Modelling of the Statistics of the

Response 262

6.7 Conclusions 269

6,8 Nomenclature 272

6.9 References 275

CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

7,1 Concluding Remarks 278

7.2 Recommend tions for Future Work 280

APPENDIX : ANALYSIS OF TIME SERIES 283