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DYNAMICS OF FIXED MARINE
STRUCTURES
Third edition
N. D. P. BarltropA. J. Adams
Atkins Oil & Gas Engineering Limited, Epsom, UK
UTTERWORTH
E I N E M A N NTHE MARINE TECHNOLOGY DIRECTORATE LIMITED
Contents
Foreword xvi
Preface xviii
1 Introduction 1
1.1 Outline of the contents 1
1.2 Layout 2
1.3 Sections which help with the selection of analysis strategy 2
1.4 Use of the book as a technical reference 2
1.5 Use of the book as an introductory text
2 Dynamics with deterministic loading 7
2.1 Linear single degree of freedom systems: SDOF 7
2.1.1 Units 8
2.2 Oscillation of an SDOF with neither forcing nor damping 10
2.3 Steady state oscillation of an SDOF with forcing and viscous damping 12
2.3.1 Steady state solution using real algebra 13
2.3.2 Dynamic amplification factor 14
2.3.3 Significance of forcing and natural frequencies 15
2.3.4 Steady state solution using complex algebra 18
2.3.5 Complex number representation of response 21
2.3.6 Steady state response of a non-linear SDOF 21
2.4 Damped decay and build-up of oscillation 22
2.4.1 Viscous, damped decay of oscillation 23
2.4.2 Damping ratio and logarithmic decrement 25
2.4.3 Response to an impulse 25
2.4.4 Viscous damped build-up of natural frequency oscillation 26
2.5 Damping28
2.5.1 Hysteretic damping 28
2.5.2 Friction damping 29
2.5.3 Typical structural damping 31
2.6 Modelling multidegree of freedom structures: MDOFs 31
2.6.1 Natural frequencies of a 2 degree of freedom system 32
2.6.2 Modelling frame structures 34
2.6.3 Beam element stiffness 34
2.6.4 Global axes 35
2.6.5 Axis transformation 36
2.6.6 Assembly of global stiffness matrix 37
2.6.7 Damping 37
2.6.8 Mass 38
2.6.9 Supports38
2.6.10 Forces applied at nodes 39
2.6.11 Forces applied to members 39
2.6.12 Constraints 41
2.6.13 Joints 42
2.6.14 Geometric stiffness 44
2.6.15 Hydrostatic stiffness and effective tension 45
2.6.16 Modelling continuous structures using plate, shell
and brick elements 46
2.6.17 Substructures47
2.7 Static analysis of MDOF structures 47
2.7.1 Quasi-static analysis48
2.8 Steady state solution using complex matrix algebra 49
vi Dynamics of fixed marine structures
2.9 Natural frequencies of MDOFs 49
2.9.1 Eigenvalue form 50
2.9.2 Jacobi method 51
2.9.3 Householder QR/QL method 52
2.9.4 Polynomial solution 53
2.9.5 Vector iteration methods 53
2.9.6 More complicated methods 54
2.9.7 Selection of frequency/mode shape calculation method 54
2.9.8 Some frequencies of commonly used structural elements 55
2.10 Normal mode (or principal or generalised) coordinates 57
2.10.1 Forced vibration of MDOF systems 57
2.10.2 Other uses of principal/generalised coordinates 59
2.11 Time history solution methods 59
2.11.1 Convolution integral 61
2.11.2 Time stepping methods 63
2.11.3 Central difference (explicit) method 64
2.11.4 Runge-Kutta (explicit) method 66
2.11.5 Newmark ft (implicit) method 68
2.12 Economic solution of large dynamic problems 68
2.12.1 Separate, simpler model 69
2.12.2 Guyan reduction or static condensation 70
2.12.3 Static improvement 71
Notation 72
Bibliography 74
References 74
3 Statistical and spectral description of random loading and response 77
3.1 Short term, frequency and sequence independent properties of y(t) 82
3.1.1 Measures of location 82
3.1.2 Measures of spread 83
3.1.3 Probability density function (PDF) 84
3.1.4 Cummulative distribution function (CDF) 85
3.1.5 Moments of a PDF 86
3.1.6 Gaussian (normal distribution) 87
3.1.7 Non-Gaussian distributions 88
3.2 Sequence dependent properties of a time history y(t) 88
3.2.1 Autocovariance 89
3.2.2 Autocorrelation function Ryy(x) 89
3.2.3 Autocorrelation coefficient or normalised autocovariance 90
3.2.4 Time scale 91
3.3 Fourier analysis and spectra of y(t) 92
3.3.1 Fourier series 92
3.3.2 Fourier transform representation of a random time history 94
3.3.3 Alternative forms of the Fourier transform 96
3.3.4 The discrete Fourier transform 97
3.3.5 The Fourier transform pair 97
3.3.6 Integral form of the Fourier transform pair 98
3.3.7 Spectral density 98
3.3.8 Spectral analysis of a dynamic system subject to loadingdefined by one variable 101
3.4 Relationship between autocovariance and the energy spectrum 103
3.5 Short term frequency and sequence independent statistics of
simultaneous samples from several time histories: y^t), y2(t) ... 105
3.5.1 Covariance of yt(t) and y2(t) 105
3.5.2 Correlation coefficient or normalised covariance 105
3.5.3 Statistical properties of a + byt(t) + cy2(t) 106
3.5.4 Statistical properties of yt(t) x y2(t) 107
3.5.5 Joint probability of n random variables 108
Contents vll
3.5.6 Gaussian multivariate distribution 109
3.6 Sequence dependent properties of samples from several time histories 109
3.6.1 Cross-covariance 109
3.6.2 Cross-correlation coefficient or normalised cross-covariance 111
3.6.3 Cross-correlation function 111
3.6.4 Nomenclature 111
3.7 Cross spectral density and coherence 111
3.7.1 Cross spectral density 111
3.7.2 Single-sided cross spectral density 112
3.7.3 Co- and quad-spectral density 113
3.7.4 Coherence 114
3.7.5 Spectral analysis of the response to a summation of
random signals 114
3.8 Relationship between the cross-covariance and the cross-spectrum 114
3.9 Some further derivations based on spectral properties 116
3.9.1 Velocity and acceleration spectra 117
3.9.2 Spectral moments 117
3.9.3 Bandwidth 118
3.9.4 Crossing periods and peak distributions 118
3.9.5 Level crossing periods and the zero crossing period Tz 119
3.9.6 The crest frequency fc and period Tc 122
3.9.7 Distribution of amplitudes in a narrow banded spectrum 122
3.9.8 Rayleigh distribution 124
3.9.9 Predicting the amplitude exceeded with a given probabilityor in a given number of cycles 124
3.9.10 Distribution of the extreme values of a Rayleigh distribution 127
3.10 Extreme value distributions for environmental data 131
3.10.1 Types of extreme value distribution 131
3.10.2 Selection of extreme value distribution 134
3.10.3 Return period 134
Notation 136
Commonly used symbols 136
Summary 137
Bibliography 143
References 144
4 Structural response to random loading 147
4.1 Wave, wind and earthquake - differences leading to different
analysis methods 148
4.2 Structural response in waves, wind and earthquake 150
4.2.1 Structural response to a unidirectional sea 150
4.2.2 Structural response to wind turbulence 152
4.2.3 Structural response to earthquakes 155
4.2.4 Structural response to waves, wind and earthquake: summary 156
4.3 Examples of non-linearities 156
4.3.1 The effect of non-linear drag loading 157
4.3.2 The effect of intermittent loading in the splash zone 157
4.3.3 The effect of non-linear drag for a structure in the wind 161
4.3.4 The effect of non-linear guy wire behaviour on a structure
in the wind 162
4.3.5 The effect of yielding on a structure in an earthquake 162
4.4 Time history analysis methods 163
4.4.1 Time history analysis of a structure in a unidirectional sea 164
4.4.2 Time history analysis of a structure in a spread sea 166
4.4.3 Time history analysis of a structure in a turbulent wind 166
4.5 Conclusion 167
Notation 167
References 168
viii Dynamics of fixed marine structures
Foundations 171
5.1 Introduction 171
5.1.1 Safety factors for foundations 173
5.2 Introduction to soil behaviour 174
5.2.1 Permeability 174
5.2.2 Effective stress 174
5.2.3 Failure of soils 175
5.2.4 Mohr's circle 176
5.2.5 Application of Mohr's circle in conjunction with the
soil failure criterion 178
5.2.6 Drained and undrained loading and liquefaction of sands 181
5.2.7 Consolidation of clays 181
5.2.8 Soil structure, relative density and clay remoulding 182
5.2.9 Stiffness of soils 184
5.2.10 Soil damping 186
5.2.11 Indicative soil properties 187
5.3 Site investigation and testing 188
5.3.1 In-situ measurements 188
5.3.2 Laboratory tests for soil strength 190
5.3.3 Consolidated-drained (CD) triaxial test 191
5.3.4 Consolidated-undrained (CU) triaxial test 192
5.3.5 Unconsolidated-undrained (UU) triaxial test 192
5.3.6 Unconfined compression test 193
5.3.7 Differences between soil properties estimated from drained
and undrained tests 193
5.4 Stability of the seabed surface 194
5.4.1 Scour 194
5.4.2 Mudslides 195
5.4.3 Sand waves, dunes, banks, etc. 196
5.4.4 Subsidence 196
5.5 Gravity structures 196
5.5.1 Finite element (FE) methods 200
5.5.2 Half-space theory 201
5.5.3 Ultimate capacity of gravity foundations 203
5.5.4 Piping 205
5.5.5 Effect of consolidation on bearing capacity 206
5.5.6 Bearing capacity from published factors 206
5.5.7 Bearing capacity calculated by the method of slices 208
5.5.8 More advanced analysis of foundation capacity 209
5.5.9 Jack-up platforms 209
5.6 Single piles 209
5.6.1 Development of lateral force-deflection (p-y) curves 211
5.6.2 Calculation of Pu 211
5.6.3 p-y curve for clay 213
5.6.4 p-y behaviour in clay under cyclic conditions 215
5.6.5 p-y curve for sand 216
5.6.6 Compression capacity of piled foundations 217
5.6.7 Tension capacity 218
5.6.8 Scour and cavities 219
5.6.9 Shaft resistance in sand 219
5.6.10 Shaft resistance in clay 219
5.6.11 Shaft resistance - displacement (t-z) curves 220
5.6.12 End bearing capacity of piles 220
5.6.13 Axial end bearing - displacement (q-z) curves 221
5.6.14 Torsional moment-rotation curves 222
5.6.15 Piles in calcareous soils 222
Contents lx
5.7 Including foundation behaviour in global structural analysis 222
5.7.1 The use of substructuring for the quasi-static analysisof structures on piled foundations 223
5.7.2 Linearised foundation tangent stiffness for quasi-static
analysis of structures on piled foundations 229
5.7.3 Linearised foundation secant stiffness for dynamic analysisof structures on piled foundations 236
5.8 Pile groups 240
5.8.1 Pile group axial capacity 240
5.8.2 Pile group lateral capacity 241
5.8.3 Force-deflection analysis of piles in groups 241
Notation 242
References 244
6 Vaves and wave loading 249
6.1 Introduction 249
6.2 Waves and currents 249
6.2.1 Regular waves 249
6.2.2 Particle motions 252
6.2.3 Mass transport 253
6.2.4 Group velocity Q 253
6.2.5 Ocean waves 255
6.2.6 Sea 255
6.2.7 Swell 255
6.2.8 Significant wave height and mean zero crossing period 256
6.2.9 Spectrum 257
6.2.10 Scatter diagrams 257
6.2.11 Persistence diagrams 258
6.2.12 Sea-state cycles 259
6.2.13 Effect of the seabed on wave characteristics 260
6.2.14 Shoaling 261
6.2.15 Diffraction 261
6.2.16 Refraction 261
6.2.17 Reflection 262
6.2.18 Absorption 262
6.2.19 Wave breaking 262
6.2.20 Currents 263
6.3 Measurement 265
6.3.1 Water surface elevation 265
6.3.2 Water particle velocities 266
6.4 Forecasting 267
6.4.1 General 267
6.4.2 Extrapolation to extreme values from measurements 267
6.4.3 Obtaining a long term description of the sea from measurements 269
6.4.4 Forecasting wave height and period from wind and fetch 269
6.4.5 Forecasting long term statistics of wave height and period 270
6.4.6 Forecasting currents 271
6.4.7 Computer modelling 271
6.4.8 Joint probability 271
6.5 Water surface elevation spectra 272
6.5.1 Introduction 272
6.5.2 Bretschneider and Pierson-Moscowitz spectra 275
6.5.3 JONSWAP spectra 276
6.5.4 Effect of alternative frequency units 278
6.5.5 Directional spectra 279
6.5.6 Selection of spectral shape 282
6.6 Individual wave scatter diagrams 283
6.6.1 Introduction 283
x Dynamics of fixed marine structures
6.6.2 The wave height exceedence method 283
6.6.3 Individual wave height - period joint probability diagrams 284
6.7 Wave modelling 285
6.7.1 Introduction 285
6.7.2 Basic physics 286
6.7.3 Mathematical manipulations 288
6.7.4 Wave theories 293
6.7.5 Regular wave theories 293
6.7.6 Linear wave theory 293
6.7.7 Stokes' wave theories 296
6.7.8 Cnoidal regular theory 298
6.7.9 Stream function wave theories 299
6.7.10 Other regular wave theories 299
6.7.11 Selection of suitable regular wave theory 300
6.7.12 Irregular (but specified profile) wave theories 304
6.7.13 Random wave theories 304
6.7.14 Breaking waves 305
6.7.15 Wave current interaction 306
6.8 Hydrodynamic loading 307
6.8.1 Introduction 307
6.8.2 Morison's equation 308
6.8.3 Selection of Cd and C,,, 308
6.8.4 Diffraction 321
6.8.5 Interference 321
6.8.6 Wave slam and slap 323
6.8.7 Structural motion, hydrodynamic added mass and damping 327
6.9 Analysis of structures subject to extreme and fatigue hydrodynamic
loading 328
6.9.1 Discussion of wave loading on offshore structures 328
6.9.2 Sine wave fitting and complex number methods 329
6.9.3 Analysis of wave frequency loading and structural response 330
6.9.4 Deterministic analysis 330
6.9.5 Frequency domain spectral analysis 334
6.9.6 Time domain spectral analysis with linear random wave theory 336
6.9.7 Time domain spectral analysis - non-linear random wave theory 337
Notation 337
References 339
7 Vortex-induced forces 345
7.1 The forces on stationary circular cylinders 347
7.2 Flow speeds for response of cylinders in steady flow 353
7.2.1 Critical velocities for cross-flow motion 353
7.2.2 Critical velocities for in-line motion 354
7.3 Structural response in steady flow 356
7.3.1 Harmonic model 356
7.3.2 Effective mass per unit length: me 360
7.3.3 Criteria for vortex-induced response 361
7.3.4 Predictions of amplitude of response of risers 363
7.4 Vortex shedding in waves 364
7.4.1 Introduction 364
7.4.2 A stationary cylinder in waves 3657.4.3 Effects of irregular waves, cylinder orientation, wave
directionality, currents, roughness and interference 369
7.4.4 A compliant cylinder in waves 3707.5 Devices for preventing vortex-induced oscillations 372
7.5.1 Strakes 373
7.5.2 Shrouds 3757.5.3 Fairings 375
Contents xl
7.5.4 Air bubbles 378
7.5.5 Structural damping devices 378
7.6 The effect of other flow and structural properties 378
7.7 Flow calculations 382
7.7.1 Hydrodynamic damping 382
7.7.2 Computational flow techniques 384
7.8 Analysis sequence 385
Notation 388
References 389
8 Wind turbulence 395
8.1 Introduction 395
8.2 The structure of strong winds 398
8.2.1 Origin of the wind 398
8.2.2 Weather systems 398
8.2.3 The atmospheric boundary layer 399
8.2.4 Atmospheric stability 400
8.2.5 Equilibrium 402
8.2.6 Summary 403
8.3 Statistical description of turbulence 403
8.3.1 Turbulence statistics 403
8.3.2 Turbulence - single point statistics 406
8.3.3 Turbulence - two point statistics 413
8.4 Wind data 418
8.4.1 The mean wind 419
8.4.2 The turbulent gusts 427
8.4.3 Non-neutral wind conditions 443
8.5 Turbulence loads 444
8.5.1 Aerodynamic loading 444
8.5.2 Aerodynamic damping 448
8.6 Calculation of response 451
8.6.1 Theory 451
8.6.2 Calculation of response- lattice structures 452
8.6.3 Calculation of response- single members 454
8.6.4 Extreme value analysis 461
8.6.5 Fatigue life analysis 462
8.7 Choice of method 463
8.7.1 Comparison of methods 463
8.7.2 Analysis hints 465
Notation 466
Bibliography 468
References 469
Annex 8A ESDU data items 471
Annex 8B Derivation of theory 471
8.B. 1 Turbulence loads (direct method, ESDU method) 471
8.B.2 Single-member methods 479
8.B.3 General methods 491
9 Installation 503
9.1 Introduction 503
9.2 Transportation 503
9.2.1 Barge motions 503
9.2.2 Cargo loading and response 505
9.2.3 Barge flexibility 507
9.2.4 Slam 508
9.2.5 Self-floating substructures 510
9.3 Launch and up-ending 511
9.3.1 Jacket launch analysis 512
xli Dynamics of fixed marine structures
9.3.2 Analysis method 513
9.4 Lift 514
9.4.1 Single degree of freedom lift analysis 516
9.4.2 Computer analysis of crane dynamic response 517
9.4.3 Selection of load conditions for analysis 518
9.5 Docking over a template 518
9.6 On-bottom stability 518
9.7 Pile driving 520
9.7.1 Mathematical analysis 523
9.8 Installation of gravity structures 527
Notation 528
References 529
10 Earthquakes 531
10.1 Introduction 531
10.2 Design philosophy for earthquake loads 531
10.3 Theory 534
10.3.1 The response spectrum method - overview 534
10.3.2 SDOF lumped-mass system 536
10.3.3 Derivation of response spectra 537
10.3.4 Use of response spectra - SDOF structure 538
10.3.5 MDOF lumped-mass system 539
10.4 Design data 544
10.4.1 Accelerograms 544
10.4.2 Response spectra 546
10.4.3 Directionality of loading 547
10.5 Specification of design earthquakes 548
10.5.1 Earthquake magnitude and intensity 548
10.5.2 Source evaluation 550
10.5.3 Source-to-site attenuation 552
10.5.4 Construction of the response spectrum 554
10.5.5 Site response analysis 556
10.5.6 Design data for North Sea sites 559
10.6 Calculation of structural response 561
10.6.1 Foundation model 562
10.6.2 Structure model 566
10.6.3 Analysis methods 569
10.6.4 Choice of analysis 578
10.6.5 Analysis of secondary systems 580
10.7 Structural configuration for seismic resistance 580
10.7.1 Global configuration (jacket structures) 583
10.7.2 Joint detailing (jacket structures) 583
10.7.3 Gravity structures 584
Annex 10A Sources of accelerogram data 585
Notation 586
Bibliography 587
References 587
11 Strength and fatigue 593
11.1 Introduction 593
11.1.1 Limit states 593
11.1.2 Safety factors 594
11.1.3 Unity checks 596
11.1.4 Non-linear complications with dynamic analysis 597
11.2 Strength assessment 598
11.2.1 Local modes of failure (yield, fracture, buckling) 598
11.2.2 Yield 598
11.2.3 Buckling 599
Contents xill
11.2.4 Beam columns 601
11.2.5 Joint strength 602
11.2.6 Deterministic quasi-static strength analysis 602
11.2.7 Frequency domain 'spectral' analysis 603
11.2.8 Response spectra analysis 606
11.2.9 Avoiding non-linearities in frequency domain analysis 606
11.2.10 Possible methods of linearisation 607
11.2.11 Time history analysis 608
11.3 Fatigue assessment 609
11.3.1 S-N curves 609
11.3.2 Miner's rule 610
11.3.3 Deterministic fatigue analysis 611
11.3.4 Spectral fatigue analysis 6i3
11.3.5 Narrow band spectra 613
11.3.6 Broad band spectra 616
11.3.7 Stress concentration factors 622
11.3.8 Non-linearities which affect spectral fatigue analysis 623
11.4 Fracture assessment 625
11.4.1 Brittle fracture 626
11.4.2 Application of fracture mechanics to fast fracture 628
11.4.3 Crack propagation 634
11.5 Overall analysis methods 636
11.5.1 Dynamic characteristics of environmental loading 636
11.5.2 Methods of handling the frequency content 637
11.5.3 Methods of structural analysis 638
11.5.4 Wave frequency loading 638
11.5.5 Wave slam and slap 640
11.5.6 Vortex shedding loading 640
11.5.7 Wind loading 640
11.5.8 Earthquake loading 641
Notation 642
References 644
12 Examples 649
12.1 Analyses of a single pile platform 650
12.1.1 Modelling method 651
12.1.2 Preliminary estimate of natural period 652
12.1.3 Foundation model: p-y curves 652
12.1.4 Time history dynamic analysis 653
12.1.5 Secant stiffness, linearised foundation, for frequency domain
dynamic analysis 657
12.1.6 Linear frequency domain analysis 660
12.1.7 Comparison of time and frequency domain analysis 661
12.1.8 Fatigue analysis 662
12.1.9 Semi-probabilistic fatigue analysis 662
12.1.10 Spectral fatigue analysis 666
12.1.11 The 2.5 second rule 667
12.1.12 Comparison of fatigue analysis methods 669
12.2 Dynamic response of a jack-up platform 670
12.2.1 Problem definition 670
12.2.2 Outline methodology 670
12.2.3 Estimation of natural period 672
12.2.4 Selection of extreme regular wave 675
12.2.5 Wave theory 676
12.2.6 Regular wave loading 676
12.2.7 Structural analysis of static response to regular wave
plus current 677
12.2.8 Results of regular wave analysis 678
xlv Dynamics of fixed marine structures
12.2.9 Spectrum for random wave, frequency domain, spectral analysis 678
12.2.10 Selection of linear wave theory 679
12.2.11 Calculation of wave particle kinematics at a range of depthsand wave periods 679
12.2.12 Combination of particle velocities with spectrum to determine
the rms velocity and linearised drag force equation at any
location 681
12.2.13 Mode shape and the consistent natural period 683
12.2.14 Hydrodynamic and structural damping 685
12.2.15 Spectral calculation of additional dynamic response to loadingin the vicinity of the structural natural period 686
12.2.16 Frequency multiplying effects 690
12.2.17 Wind force on the structure 690
12.2.18 Summation of the separately calculated deflections 690
12.3 Vortex shedding example 691
12.3.1 Basic data 692
12.3.2 Calculation of mode 1 frequency and mode shape 693
12.3.3 Calculation of mode 1 reduced velocity, stability parameter and
response 695
12.3.4 Calculation of mode 2 frequency and mode shape 696
12.3.5 Calculation of mode 2 reduced velocity, stability parameter
and response 697
12.3.6 Calculation of mode 3 frequency and mode shape 698
12.3.7 Combination of in-line and cross-flow response 699
12.3.8 Vortex shedding in waves 699
12.3.9 Wave synchronised vortex shedding 701
References 701
12.4 Wind turbulence example 702
12.4.1 Extreme response analysis 702
Static design 702
Direct method 703
ESDU method 715
WINDSPEC method 721
Summary 721
12.4.2 Fatigue life analysis 722
Omnidirectional analysis (u-component only) 722
Directional analysis (u-component only) 723
Directional analysis (u and v-components) 724
Summary 725
12.5 Earthquake example 72712.5.1 Modelling 72712.5.2 Member stiffness matrix 729
12.5.3 Formation of global stiffness matrix 730
12.5.4 Deflection under a static horizontal force 730
12.5.5 Mass matrix 731
12.5.6 Polynomial method for the calculation of natural frequencies 731
12.5.7 Vector iteration method for the calculation of mode shapes 735
12.5.8 Generalised mass for each mode 736
12.5.9 Spectral displacement and acceleration for each natural
frequency 738
12.5.10 Response to horizontal ground acceleration 73912.5.11 Response to vertical ground motion 74112.5.12 Summation of directions 74312.5.13 Static coefficient method 743References 744
Appendix A Complex number representation of amplitude and phase 745
A. 1 Plotting on the complex phase - phasor diagrams 745
Contents xv
A. 2 Calculations using 0° and -90° loading and response as real and
imaginary parts 746
A. 3 e<x+iy> 747
A. 4 Negative frequencies 747
A.5 Complex number multiplication and division 748
A. 6 Complex number inversion: 1/Z 748
A.7 Phase lead and lag 749
Appendix B The Gamma Function 750
Appendix C Consistent units 751
Appendix D Stiffness matrix of a 3-d beam element 752
Appendix E Useful data and formulae 754
Index 757
