...

ABOUT MTD LTDThe Marine Technology Directorate Limited (MTD ltd) aims to promote, develop and advance, in the national i!lterest, research, training and information dissemination in marine technology, including all aspects of engineering, science and technology relating to the exploration and exploitation of the sea. MTD Ltd is an association of members having interests and capabilities in marine-related technology. They include industry, government, research establishments, academic and other learned S institutions, and the _ cience and Engineering Research Council (SERC). MTD Ltd advances marine research and development, primarily by means of its research activities in Higher Education Institutes and partly funded by SERC. MTD Ltd also provides an interface between such research and the requirements and expertise of its members. Since its absorption of UEG, the research and information group for the offshore and underwater engineering industries, MTD Ltd has expanded its interests to include multi-sponsor projects. For further details, contact: The Secretary The Marine Technology Directorate Limited 19 Buckingham Street, London WC2N 6EF Telephone 071-321 0674 Facsimile 071-930 4323

1\

DYNAMICS OF FIXED MARINE STRUCTURES Third editionN. D. P. Barltrop A. J. AdamsAtkins Oil & Gas Engineering L~mited, Epsom, UK

UTTERWORTHEINEMANN

THE MARINE TECHNOLOGY DIRECTORATE LIMITED

Butterworth-Heinemann Ltd Linacre House, Jordan Hill, Oxford OX2 8DP

t~

l'ART OF REED INIEltNATIONAL BOOKS

OXJIORD KUNial'100'0

LONDON BOSTON NEW DEUII SlNGAPORETORONTO WELUNGTON

SYDNEY

Fust publiShed 1m Second edition 1978 Third edition 1991

@The Marine Technology Directorate Ltd 1991AU rights reserved. No part of this publication may be reproduced in any material fonn (including photocopying or storing in any medium by electronic means an4 whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publishers.British Ubrary Cataloguiog in Publication Data A catalogue record for this book: is available from the British LibraryUbrary of Coagress Cataloguing in Publication Data A catalogue ttro&d for this book is available from the Library of Congress

ISBN 0 7506 1046 8

Printed in Great Britain by 1bomson Litho Ltd., East Kilbride, Scotland.

-

t\1

g:Contents

Foreword Preface

xvi xviii

1

--1.1 Outline of1.2 1.3 1.4 1.5

Introduction

the Layout Sections which help with the selection of analysis strategy Use of the book as a technieal reference Use of the book as an introductory text

contents

l1

2 2 27 7 8 10 12 13 14 15 18 21 2122

2 Dynamics with deterministic loading2.1l

2.2 2.3

2 :4

2.5

,2.6

'

\~

2.72.8

Linear single degree of freedom systems: SDOF 2.1.1 Units Oscillation of an SDOF with neither forcing nor damping Steady state oscillation of an SDOF with forcing and viscous damping 2.3.1 Steady state solution usingreal algebra 2.3.2 Dynamic amplification factor 2.3.3 Significance of forcing and natural frequencies 2.3.4 Steady state solution using complex algebra 2.3.5 Complex number representation of response 2.3.6 Steady state response of a non-linear SDOF Damped decay and build-up of oscillation 2.4. 1 Viscous. damped decay of . oscillation 2.4.2 Damping ratio and logarithmic decrement 2.4.3 Response to an impulse 2.4.4 Viscous damped build-up .o f natural frequency oscillation Damping 2.5.1 Hysteretic damping 2. 5. 2 Friction damping 2.5.3 Typical structural damping Modelling multidegree of freedom structures: MDOFs 2.6.1 Natural frequencies of a 2 degree of freedom system 2. 6. 2 Modelling frame structures 2.6.3 Beam element stiffness 2.6.4 Global axes 2.6.5 Axis transformation 2.6.6 Assembly of global stiffness matrix 2. 6. 7 Damping 2.6.8 Mass 2.6. 9 Supports 2.6.10 Forces applied at nodes 2.6.11 Forces applied to members 2.6.12 Constraints 2.6.13 Joints 2. 6. 14 Geometric stiffness 2.6.15 Hydrostatic stiffness and effective tension 2.6.16 Modelling continuous structures using plate. shell and brick elements 2. 6. 17 Substructures Static analysis of MDOF structures 2. 7.1 Quasi-static ari;ilysis Steady state solution using complex matrix algebra

23 25 25 26 28 28 29 31 31 32 34 34 35 36 37 37 38 38 39 39 41 42 44 4546

47 4748

49

vi

Dynamics .of fixed marine structures

freqpencies of MDOFs Eigenvalue form Jacobi method Householder QR/QL method Polynomial solution Vector iteration methods '. 2.9.6 More complicated methods 2. 9. 7 Selection of frequency/mode shape calculation method 2.~.8 _ Some frequencies of commonly used structural elements 2.10 Normal mode (or principal or generalised) coordinates 2.10.1 Forced vibration of MDOF systems 2.10.2 Other uses of principal/generalised coordinates 2.11 Time history solution methods 2.11.1 Convolution integral 2.11.2 Time stepping methods 2.11.3 Central difference (explicit) method 2.11.4 Runge-Kutta (explicit) method 2.11.5 Newmark {3 (implicit) method 2.1a Economic solution Or large dynamic problems 2.12.1 Separate. simpler model 2.12.2 Guyan reduction or static condensation 2.12.3 Static improvement Notation Bibliography References2.9

Natural 2. 9.1 2. 9.2 2. 9.2 2. 9.4 . 2.9.5

49 50 51

'

52 53 53 54 54

55

5157 59 59 61 63 64

66 68 68 6970

7172

74 7477

3 Stat istical and spectral description or random loading and _ responseShort term, f"requency and sequence independent properties of y(t) 3.1.1 Measures of location 3.1.2 Measures of spread 3.1.3 Probability density function (PDF) 3.1.4 Cummulative distribution function (CDF) 3.1.5 Moments of a PDF 3.1.6 Gaussian (normal distribution) 3. 1. 7 Non-Gaussian distributions 3.2 Sequence dependent properties of a time history y(t) 3.2.1 Autocovariance 3.2.2 Autocorrelation function Ryy(-r) 3.2.3 Autocorrelatioh coefficient or normalised autocovariance 3.2.4 Time scale 3.3 Fourier analysis and spectra of y(t) 3.3.1 Fourier series 3.3.2 Fourier transform representation of a random time history 3.3.3 Alternative forms of the Fourier transform 3.3.4 The discrete Fourier transform 3.3.5 The Fourier transform pair 3.3.6 Integral form of the Fourier transform pair 3.3.7 Spectral density _ 3.3.8 Spectral analysis of a dynamic system subject to loading defined by one variable Relationship between autocovariance and the energy spectrum ' 3.4 3.5 Short term frequency and sequence independent statistics of simultaneous samples from several time histories: y 1(t). y 2(t) --- = 3.5.1 Covariance of y 1 (t) and Y2(t) . 3.5.2 Correlation coefficient or normalised covariance 3.5.3 Statistical properties of a + by1(t) + cy2(t) 3.5.4 Statistical properties of y1 (t) x Y2(t) 3.5.5 Joint probability of n random variables3.1

82 82 8384

\

85

86 87 88 88 89 89 9091 92 92

94 9697 97

98 98101 103 105105 lOS

106 107 108

Contents

vii

'.

Gaussian multivariate distribution Sequence dependent properties of samples from several time histories 3.6.1 Cross- covariance 3.6.2 Cross- correlation coefficient or normalised cross- covariance 3.6.3 Cross-correlation function 3.6.4 Nomenclature 3.7 Cross spectral density and coherence 3.7.1 Cross spectral density 3. 7.2 Single- sided cross spectral density 3. 7.3 Co- and quad-spectral density 3. 7.4 Coherence 3. 7.5 Spectral analysis of the response to a summation of random signals 3 .8 Relationship be ween the cross-covariance and the cross-spectrum t 3 .9 Some further derivations based on _ spectral properties 3. 9.1 Velocity and acceleration spectra 3. 9.2 Spectral moments 3. 9.3 Bandwidth 3. 9.4 Crossing periods and peak distributions \ 3. 9.5 Level crossing periods and the zero crossing period T z 3.9.6 The crest frequency fc and period Tc 3. 9. 7 Distribution of amplitudes in a narrow banded spectrum 3. 9.8 Rayleigh distribution 3. 9. 9 Predicting the amplitude exceeded with a given probability or in a given number of cycles 3. 9.10 _.Distribution of the extreme valu~ of a Rayleigh distribution 3.10 Extreme value distributio~ for environmental data 3.10.1 Types of extreme value distribution 3.10.2 Selection of extreme value distribution 3.10.3 Return period Notation Commonly used symbols Summary Bibliography References3.5.6 3.6

109 109 109

111 111 111 111 111 112113 114

114114

116 117 117 118 118l19 122 122 124

124 127131 131 134 134 136 136 137 143 144

4 Structural response to random loadingWave, wind and eai:"thquake - differences leading to different analysis methods 4.2 Structural response in waves, wind and earthquake 4.2:-1 Structural response to a unidirectional sea 4.2.2 Structural response to wind turbulence 4.2.3 Structural response to earthquakes 4.2.4 Structural response to waves, wind and earthquake: summary 4.3 Examples of non-linearities 4.3.1 The effect of non- linear drag loading 4.3.2 The effect of intermittent loading in the splash zone 4.3.3 The effect of non- linear drag for a structure in the wind 4.3.4 The effect of non- linear guy wire behaviour on a structure in the wind The effect of yielding on a structure in an earthquake 4.3.5 4.4 T:me history analysis methods 4A.1 Time history analysis of a structure in a unidirectional sea Time history analysis of a structure in a spread sea 4.4.2 Time history analysis of a structure in a tUr-bulent wind 4.4.3 4.5 Conclusion Notation References4.1

147148 150 150 152 155 156 156 157

..

157161 162 162 163 164 166 166

161167

168

viii

Dynamics of fixed marine structures

5

Foundations 5.1 5.2 Introduction 5.1.1 Safety factors for foundations Introduction to soil behaviour 5.2.1 Permeability 5.2.2 Effective stress 5.2.3 Failure of soils 5.2.4 Mohr's circle 5.2.5 Application of Mohr's circle in conjunction with the soil failure criterion 5.2.6 Drained and undrained loading and liquefaction of sands 5.2. 7 Consolidation of clays 5.2.8 Soil structure, relative density and clay remoulding 5.2. 9 Stiffness of soils 5.2.10 Soil damping 5.2.11 Indicative soil properties Site investigation and testing 5.3.1 In-situ measurements 5.3.2 Laboratory tests for soil strength 5.3.3 Consolidated-drained (CD) triaxial test 5.3.4 Consolidated-undrained (CU) triaxial test 5.3.5 Unconsolidated-undrained (UU) triaxial test 5.3.6 Unconfined compression test 5.3.1 Differences between soil properties estimated from drained and undrained tests Stability of the seabed surface 5.4.1 Scour 5.4.2 Mudslides 5.4.3 Sand waves, dunes, banks, etc. 5.4.4 Subsidence Gravity structures 5.5.1 Finite element (FE) methods Half-space theory 5.5.2 5.5.3 Ultimate capacity of gravity foundations 5.5.4 Piping 5.5.5 Effect of consolidation on bearing capacity 5.5.6 Bearing capacity from published factors 5.5.7 Bearing capacity calculated by the method of slices 5.5.8 More advanced analysis of foundation capacity Jack-up platforms 5.5.9 Single piles 5.6.1 Development of lateral force-deflection (p-y) curves 5.6.2 Calculation of P u 5.6.3 p-y curve for clay 5.6.4 p-y behaviour in clay under cyclic conditions 5.6.5 p-y curve for sand 5.6.6 Compression capacity of piled foundations 5.6.7 Tension capacity 5.6.8 Scour and cavities 5.6.9 Shaft resistance in sand 5.6.10 Shaft resistance in clay 5.6.11 Shaft resistance - displacement (t-z) curves 5.6.12 End bearing capacity of piles 5.6.13 Axial end bearing - displacement (q-z) curves 5.6.14 Torsional moment-rotation curves 5.6.15 Piles in calcareous soils

171 171 173 174 174 174 175 176 178 181 181 182 184 186

..

187188

5.3l

188190

191192 192 193 193

5.4

194194 195 196 196 196 200 201 203 205 206 206 208 209 209 209 211 211 213 215 216 217

5.5

5.6

218219 219

219220 220 221 222 222

Contents

lx

"~

'

Including foundation behaviour in global structural analysis 5.7.1 The use of. substructuring for the quasi-static analysis of structures on piled foundations 5.7.2 Linearised foundation tangent stiffness for quasi-static analysis of structures on piled foundations 5.7.3 Linearised foundation secant stiffness for dynamic analysis of structures on piled foundations 5.8 Pile groups 5.8.1 Pile group axial capacity 5.8.2 Pile group lateral capacity 5.8.3 Force-deflection analysis of piles in groups Notation References

5.7

222 223

229 236 240 240 241 241 242 244 249 249 249 249 252 253 253 255 255 255 256 257 257 258 259 260 261 261 261 262 262 262 263 265 265 266 267 267 267 269 269 270 271271

6 Vaves and vave loading

..

l'_-

. ,\

'

.

J

'\

~

6.1 - Introduction 6.2 Waves and currents 6.2.1 Regular waves 6.2.2 Particle motions l 6.2.3 Mass transport 6.2.4 Group velocity Cc 6.2.5 Ocean waves 6.2.6 Sea 6.2.7 Swell 6.2.8 Significant wave height and mean zero crossing period 6.2.9 Spectrum ' 6.2.10 Scatter diagrams 6.2.11 Persistence diagrams 6.2.12 Sea-state cycles 6.2.13 Effect of the seabed on wave characteristics 6.2.14 Shoaling 6.2.15 Diffraction 6.2.16 Refraction 6.2.17 Reflection 6.2.18 Absorption 6.2.19 Wave breaking 6.2.20 Currents cr6.3 Measurement 6.3.1 Water surface' elevation 6.3.2 Water particle velocities 6.4 Forecasting General 6.4.1 6.4.2 Extrapolation to extreme values from measurements Obtaining a long term description of the sea from measurements 6.4.3 Forecasting wave height and period from wind and fetch 6.4.4 Forecasting long term statistics of wave height and period 6.4.5 6.4.6 Forecasting currents Computer modelling 6.4.7 6.4.8 Joint probability 6.5 Water surface elevation spectra Introduction 6.5.1 Bretschneider and Pierson-Moscowitz spectra 6.5.2 6.5.3 JONSWAP .. 6.5.4 Effect of spectra frequency units alternative 6.5.5 Directional spectra 6.5.6 Selection of spectral shape 6.6 Individual wave scatt~r diagrams 6.6.1 Introduction

-

271 272 272 275 276 278 279 282 283 283

X

Dynamics

of

fixed marine structures

\

6.6.2 The wave height exceedence method 6.6.3 Individual wave height - period joint probability diagrams Wave modelling 6.7 6. 7. 1 Introduction 6.7.2 Basic physics 6.7.3 Mathematical manipulations 6.7.4 Wave theories 6.7.5 Regular wave theories 6. 7. 6 Linear wave theory 6.7.7 Stokes' wave theories 6. 7. 8 Cnoidal regular theory 6.7.9 Stream function wave theories 6. 7.10 Other regular wave theories 6. 7.11 Selection of suitable regular wave theory 6. 7.12 Irregular (but specified profile) wave theories 6. 7.13 Random wave theories 6~7.14 Breaking waves 6.7. 15 Wave current interaction 6.&. Hydrodynamic loading 6.8.1 Introduction 6.8.2 Morison's equation 6.8.3 Selection of Cd and C. 6.8.4 Diffraction 6.8.5 Interference 6.8.6 Wave slam and slap 6.8.7 Structural motion, hydrodynamic added mass and damping 6. 9 Analysis of structures subject to extreme and fatigue hydrodynamic loading 6. 9.1 Discussion of wave loading on offshore structures 6. 9.2 Sine wave fitting and complex number methods 6. 9.3 Analysis of wave frequency loading and structural response 6. 9.4 Deterministic analysis 6. 9.5 Frequency domain spectral analysis 6. 9.6 Time domain spectral analysis with linear random wave theory 6. 9 . 7 Time domain spectral analysis - non-linear random wave theory Notation References

283

284 285 285 286 288 293 293 293 296 298 299 299 300 304 304 305 306 307 307 308 308 321 321 323 327 328 328 329 330 330 334 . 336 337 337 339 345 347 353 353 354 356 356 360 361 363 364 364 365 369 370 372373

7

Vorte x - induced for ces7.1 7.2

7.3

7.4

7.5

The forces on stationary circu lar cylinders Flow speeds for response of cylinders in steady flow 7.2.1 Critical velocities for cross-flow motion 7.2.2 Critical velocities for in-line motion Structural . response in steady flow 7.3.1 Harmonic model 7.3.2 Effective mass per unit length: me 7.3.3 Criteria for vortex-induced response 7 .3.4 Predictions of amplitude of response of risers Vortex shedding in waves 7 .4.1 Introduction 7.4.2 A stationary cylinder in waves 7.4.3 Effects of irregular waves, cylinder orientation, wave directionality, currents, roughness and interference 7.4.4 A compliant cylinder in waves Devices for preventing vortex-induced oscillations 7.5.1 Strakes 7.5.2 Shrouds 7.5.. 3 Fairings

375 375

Contents

xi

t,

\

7.5.4 Air bubbles 7.5.5 Stru

= ==

amplitude of the motion phase lag angle between the applied force and resulting motion. x

Equation 2.14 becomes or xX0(

=X

0

cos (wt - 4>)

(2.21}(2.22)

coswt cos 4> + sinwt sin4>)

by comparison with equation 2.14 a.. X0

=X

0

cos4>a2 +

b

=X

0

sin4>

(2.23)

=/

b2

and

4>

= tan-1 (

: )X0

(2.24) and 4> are

.. obtaining a and b from equation 2.20 and substituting into 2.24., given byXo

=\

fo/cK - Mw2)2 + .C2w2,

4>

= tan- 1

[ K

~ Mw2

]

(2.25)

2.3.2

Dynamic amplification factor

It is convenient to write these expressions in a dimensionless form. The following ratios are used: frequency ratio

w = n = -Wn

(2.26)

damping ratio

=

~

=

c2Mw 0

=

c

(2.27)

2~

The amplitude of the res ponse can also be represented in a dimens ionless form by defining a magnification factor, Q:

Q. Q

=

amplitude of di s placement equivalent static displacement

.

==

Xo

f

0

/K2~n1 - Q2

(2.28)

=

11(1 - Q2)2 +

4>

tan- 1 [

]

(2.29)

(2~Q)2

These equations are plotted . to give a dimens ionless measure of the amplitude (Figure 2.5) and phase angle of the response (Figure 2.6) as a function of the frequency ratio. The curves show that the presence of damping modifies the extreme frequency response curve in two ways:1.

2.

As the damping increases, the maximum amplitude reduces. As the damping increases, the peak of the curve appears at increasingly below the natural frequency.

a

frequency

The frequency of oscillation at which the peak displacement of a damped structure occurs, wp is given by:

Wp

= r~. The observed value of x at any time t is the real part of z For a phase lag positive sign convention: . z

phase lag 4> relative to x 0 and argument wt - .p. phase lag sign convention : Re(z}.

= z(wt) = Xiy

0

cos(wt - f>) + ix0 sin(wt - f/>)\o

=x

0

el{l4t-f)

(2.36)

\

X

Figure 2.9

Rotating phasor diagram

sh?w~ng

z lagging f 0

The equation of motion then becomes: Mz +

Cz

+ Kz = f 0 (cos(wt) + i sin(wt)) = f 0 el14 t

(2.37}

A solution is found to be:

z z .. z

= = =

Xoel(Ct>t-f) iwxoel(c.>t-f) i2w2x 0 eHc.>t-f)

= . -w2J;C

el(c.>t-f) 0

= iwz = -w2z

(2.38}

Substituting the solution back into the equation of motion: -Mw2z + iCwz + Kz .. (-Mw2 + iCw + K)z

=f =f0 0

0

elc.>t(2.39}

elc.>t

o"r

r-z = H(w}f el:J

(2.40}

where

I

H(w}

=

K - MW2 + iCw

1

-

-

1 - fi2 + i~Q

1/K

I

(2.41)

H(w} is the complex transfer function:Q

= w/wn

~

=

C/2~[MK]

Dynamics with deterministic loading

19

H(w)f 0 represents the response amplitude and the phase lag angle relative to the force f 0 ei"'t. H(w) is called the complex transfer function because it transfers input force to output deflection and maintains phase information, complex displacement

ie.

=

complex transfer function

'X

force amplitude

Although the derivation above assumes f work quite satisfactorily for both: a) b) f f

=

f 0 (cos(wt) + i sin (wt)) it will also

=f =f

0

(cos (wt - tfl) + i sin(wt - q,)) (cos(wt + tfl) - i sin(wt + q,))

0

H(w) can be presented in terms of amplitude and phase shift as in Figures 2.15 and 2.6. Alternatively it can be presented in terms of its real and imaginary parts: Figur12.10 shows KH(w) which is the complex equuivalent of the dynamic amplification factor.6

4 2Re (KH(c.l)]0

-2 -4

-60.10

10

Frequency ratio 0 = Real port64

GJ/GJ n

l

I

I

I

I

I I I

I

I

I

I

I

I I I

~ = 1 .0

~-

..

2~=0.5

lm[KH{c.l)] 0

-...-....

\

-2-4

~ y.;.--(=0.2 1--- ~=0.1I I II

'

/

-60.1

I

I

I

I

'

(=0.001I I I IIlLI

1----

0

10

Frequency ratio 0 = Imaginary port

"'/"'n

Figure 2.10

Real and imaginary parts of KH(w) - the complex equivalent of a dynamic amplification factor

20

Dynamics of fixed marine structures

K - Mw2 + iQ..) is plotted on complex a + ib plane in Figure 2.U. (The complex a,ib plane rotates at wt relative to the x,iy plane so that the constant with time phase relationships rather than the instantaneous absolute phases are shown).

the

K - Mw2 + iU.., has a phase lag angle of

e

= - tan- 1

[

K

~wMWZ

)

fo has, as shown in Figure 2.12, by the rules of complex K - Mw 2 + iCw number division (Appendix A.S) the following characteristics:

H(w)f o

=

phase lag angle 4> =

-e

= tan-1

(

K

~ MWZ

)

(2.42)

amplitude x 0

= 1z 1 =

fo

(2.43)+ (a,.,)2

\

~(K- Mw2)2

These agree with the results calculated using real algebra in Section 2.3.1.

ib

Figure 2.11

K - Mw2 + iCw .plotted on the complex plane (phase advance convention)

ib

iC GJ

V( K -

M GJ 2 ) 2

+ ( C GJ )

2

0

fo

;

..Jc

K -

M G) 2

>2 +

cc (.J)

2

Hf0

Figure 2.12

f 0 j(K - Mw2 +

iCw) plotted on the complex plane

Dynamics with deterministic loading

Zl

2.3.5 Complex number representation of responseThe complex form of the response equation may be used in place of the x 0 , form.z(c..~t)c..~t

- tfJ

= H(c..~)f0eil.)t

(2.44)

Because the phase lag relative to the applied force is constant for a given "' it is often even more convenient to work with the complex number representation of the amplitude and phase lag alone. We have already used this idea, in Figures 2.11 and 2.12, but now consider the practical details:cA(c..~)

= H(w)f = a(c..~)0

+ ib(w)

(2.45)

The modulus of H(w)f 0 is the response amplitude: X 0 = IcA I = ~ [a2 + b2]. The argument is the phase advance angle: -t/1. Because it is normal to use the phase lag sign convention it is convenient to define an alternative complex number, c(t/J) . as follows:

c(~) = cA(t/J) = H(c..~)f = a(w)0

- ib(w)

(2.46)

(The asterisk indicates the complex conjugate which simply means that the sign of the imaginary part is reversed). Note that z(c..~t) and f 0 ei(,)t rotate on th~ complex plane. In contrast c(t/J) and f 0 have fixed directions on the complex plane. When plotting, z(wt) on the complex plane the phase advance angle must have the same (usually anti clockwise) sense as c..~t. If a phase lag convention is used the positive sense of phase lag must be in the opposite direction to c..~t. When plotting relative phase angles the positive (anticlockwise) sense may represent phase advance (cA) or phase lag (c). If instead a phase advance sign convention was used the phase advance angles would usually be negative but the mathematics would be much neater. This relative (phase lag) complex number representation of a dynamic response will be used extensively in Chapter 3.

2.3.6 Steady state response of a non-linear SDOFNon-linearities arise in a number of areas of offshore structural analysis. Some of the most important non-lineartties are associated with loading. These are not too great . a problem if the loading is described deterministically but cause significant analysis difficulties if the loading is random. This aspect of non-linearity is discussed in Chapter 4. Some structures respond in a non- linear manner, even to static or regular sinusoidal loading. The non- linearity may either be caused by material or geometric effects. Yielding and strain hardening are effects which change the material stiffness according to the deflections and so prevent deflection being linearly proportional to the applied load. Large deflection effects may also result in non-linearities. Buckling is an example of this type of non-linearity. In general if significant non- linearities are present it will probably be necessary to perform a time history analysis as described in Section 2.11. It . is nevertheless useful to be able to visualise the likely effect of non- linearity on dynamic behaviour. The effect will depend on whether the structure stiffens or becomes more flexible as the. deflections increase. Figure 2.13 shows the characteristics of single degrees of freedom non-linear systems. The stiffness change with amplitude makes the apparent natural frequency increase

22

Dynamics of flxed marine structures

with amplitude for a stiffening system and decrease with amplitude for a softening system. This may cause unexpected resonances to occur if the natural frequency has only been calculated for small deflections. Also because the curves fold back on themselves there are often three solutions for the response amplitude. In pra~tice the response amplitude may suddenly and unexpectedly jump from say point c up to point b. Alternatively the systems . may not settle to any regular response at all. In the latter case the response is classified as chaotic. Fortunately the non-linear response of fixed structures is not usually high enough to result in jump or chaotic responses. However, these types of non-linearity have been observed in floating, moored structures.

0

.... a .. ,............

.//

't;\:E Q>Q) 0"0

''

c

0

4;

-

0..

E 0

--

~bI

''

''

'.

.... ,

---

c

)

I

r-\" ' ',

'

." " ',1.0

Each curve represents a different force amplitude

:;= ::>0~

0"0

c

4)

4; E

Q>Q.

____---

/.," /I -,I/

,"''

"

/I

...........

/' bI

/

/ . I \~

I

o

o

0.1c.J

10

0.1

1.0

10

I"' no

"'I"' noResponse of on SDOF system which stiffens with increasing deflection Farcing frequency Small deflection natural frequency

Response of on SDOF system which . softens with increasing deflectionc.J

=

"'no

=

Figure 2.13

SDOFs vith non-linear stiffness

2.4

Damped decay and builcl.;..up o oscillation

This section is relevant to (a) the convolution integral method which may be used for time history response analysis (Section 2.11.1); (b) vortex shedding in waves (Chapter 7), and (c) impulsh;e loading effects in waves (Sections 6.8.6 and 4.3.1) . . Apart from the background to the definitions of percentage critical damping and logarithmic decrement it is not ilJlportant for understanding the primary response of a structure to waves or the wind. If an SDOF system is subject to a change in applied loading its oscillation may be split into two parts:v -

1. 2.

A transient decaying oscillation at the damped natural period; A response at the forcing period.

In Section 2.4.1 we look at the transient oscillations caused by giving an SDOF an initial displacement or velocity. Section 2.4.3 develops the response to an applied impulse. This is used in the convolution integral technique. Section 2.4.4 considers the build-up of response when the forcing is at the natural period of the SDOF. This is the basis of a model f or the response of a member to vortex shedding in waves.

Dynamics with deterministic loading

23

2.4.1

Viscous, damped decay o oscillation

The equation of motion for the transient decay after an initial starting displacement and velocity is:MX+Cx+Kx

= oA.zt

(2.47)

The solution is of the form:X

=

a 1e

A.lt

+ a 2e

(2.48) is substituted into equation 2.47 (2.49)(2.50)

To find the solution, x ..

= eA-t=0

MA2 eA.t + CAeA.t + KeA.tMA2 + CA + K

=0-

\ A1,2

=

-C

JC 22M

4MK ,

(2.51)

Defining

~

=

c zJM~

(The reason for choosing this definition of below)

~

is explained

andA1;2

wn

= J~ = natural

(undamped) frequency (2.52)

= ( -~

~ ~2

-

1' ) Wn

For ~ < 1, A is complex and eA-t will result in a decaying sinusoidal response (see Figure 2.14). For ~ > 1, A is real and eA.t ._will result in a smooth overdamped decay of response without a sinusoidal component. (See Figure 2.15). ~ = 1 divides these regimes. It is the lowest value that gives no response oscillation and the amount of damping is termed critical.

X

Xo

xe- ~ "n t

t

.,..-/

.,....-,.,--

/

/

'

/ /

Figure 2.14

Underdamped response (~

50< 150

I

Subspace iteration

Householder

QL/R

I ..

Yes (

Allfrequencies required

No

Vector iteration

Figure 2.47

Selection of method for calculation of natural frequencies and mod.e shapes

Dynamics with deterministic loading

55

2.9.8 Some frequencies of commonly used. structural elementsThe following information is provided to assist with the estimation of the natural frequencies of individual members and complete structures. systemsm,KEnd moss M; spring moss m per unit length, spring stiffness K.

~IL

c.J_n

=

~ M+ :L/ 3

.,.1

~LS""'\

--_0

K,m

--

0.

Cantilever; end moss M; beam moss m per unit length. stiffness by Table 2.1."'" =

~ + 0~3mlM

~

Simply supported beam; central moss M; beam moss m per unit length, stiffness by Table 2.1 .

"'" =

~ M+O~ml

Longitudinal or torsional vibration of beams

!~.,.---

l _ _ _..~n = 0

longitudinal vibration of cantilever;A = cross section,

CJ.n

=

rr (n+-1)

2

\]~

(AE

E = modulus of elasticity.m n

~

"

---

,

.--- n=n=2

= moss per unit length, = 0, 1, 2. 3 = number of

nodes.

for steel, l

in meters

this becomes:

~

-