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EXTREME RESPONSE OF NONLINEAR OCEAN STRUCTURES: IDENTIiICATION OF MINIMAL STOCHASTIC WAVE INPUT FOR TIME-DOMAIN SIMULATION
TORHAUG, RUNE
DEGREE DATE: 1996
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EXTREME RESPONSE OF NONLINEAR OCEAN
STRUCTURES: IDENTIFICATION OF MINIMAL
STOCHASTIC WAVE INPUT FOR TIME-DOMAIN
SIMULATION
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF CIVIL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
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DOCTOR OF PHILOSOPHY
By Rune Torhaug
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© Copyright 1996 by Rune Torhaug
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as a dissertation for the deree of Doctor of Philosophy.
Steven R. Winterstein (Principal Adviser)
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
/
.f 'L
C. AlUn Cornell
Sverre Haver
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Ill
Anne ST Kidjan
Approved for the University Committee on Graduate
Studies:
ï
Abstract
The wave loads on complex ocean structures typically vary nonlinearly with the wave elevation. For these nonlinear ocean structures, time-domain simulation remains one of the few general techniques for estimating response statistics under random wave loads. The major drawback of time-domain simulation lies in its computational ex-pense. We here critically evaluate strategies to identify minimal portions of stochastic wave input to form reliable extreme response estimates of nonlinear ocean structures.
Recently a "Guideline for site-specific assessment of mobilejackup units" (SNAME.
1993) has been developed. This guideline represents state-of-the-art industry practice for jackup analyses. Motivated by the recommended use of time-domain analysis for flexible jackups in this guideline, we consider first the nonlinear dynamic response of
a jackup structure under random wave loads. For a simplified jackup model, average behavior and variability in extreme forces and responses are found from simulation over many 6-hour seastates. Weibull and Hermite analytical models of response ex-tremes are also presented and evaluated These models use shorter, less expensive simulations to estimate a limited number of response statistics, such as moments or
parameters of the response peak distribution, and fit analytical models to estimate
global extremes. We therefore refer to them as hybrid simulation-analytical methods.
Necessary simulation lengths are established both for direct simulation of extremes. and for analytical extreme models.
Next, we consider how the use of "design seastate histories" reduces the cost of time-domain response analysis. This is motivated by the fact that the inherent
vari-ability in the response process leads to rather long simulations aimed at achieving
reliable estimates of the extreme response However, the computationally expensive part of time-domain simulation lies not in the simulation of multiple wave histories,
but rather in the propagation of each through the hydrodytiamical and structural
model to obtain forces and responses. If we could exclude wave histories when their wave characteristics differ too much from their theoretical ensemble average values and only simulate response in a few most representative "design seastate histories", this would be a promising means of reducing the variability in the response estimates without need for an increase in the number of simulations. Such a procedure has been suggested for jackups (SNAME, 1993). We undertake here a careful study of which wave characteristic are most informative in describing extreme jackup response. In quasi-static cases, the maximum wave crest height, is shown to best explain ex-treme deck sway. For more flexible structures, we introduce a new wave characteristic,
SD,based on response spectral concepts from earthquake engineering (Newmark and
Rosenblueth, 1971). We conclude that for stiff jackup structures, design wave
meth-ods based on hourly maximum crest heights accurately predict mean response.
The need for simulation and analysis of an entire sea-state history is then avoided. Motivated by the success of these single wave cycle analyses for the fixed jackup Structure we study the nonlinear extreme response of a flared container ship through short time-domain analyses. The study is also motivated by the recent development
of computer intensive tools for fully nonlinear analysis of 3-D (fluid-structure) ship problems. Examples include the multi-phase projects behind the codes LA1 from
SAIC Inc. (Lin et al, 1994, Lin and Yue, 1990), SWAN from MIT (Kring, 1994, Nakos et al, 1994) and USAERO from the University of Michigan (Beck and Magee, 1991). We focus on the use of "critical wave episodes", which are short wave segments which are likely candidates to produce extreme response in the hourlong history. We hence
leave the lengthy hourly simulations and concentrate on propagating through the
structure only a few critical wave episodes (each critical wave episode is only a few wave cycles long) per hourly simulation. We discuss how we can identify the location of critical wave histories within a longer wave history. Once a location is identified
we discuss how long (how many cycles) a critical wave episode must be to ensure
accurate results.
Acknowledgments
This work would have been much poorer in both content and quality without the help from my advisor. Dr. Steven R. Winterstein. Thanks Steve for being extremely help-ful and supportive throughout the course of this work. Thanks are also extended to Professor C. Alun Cornell for valuable contributions and for the perspective provided. Dr. Sverre Haver has improved this work as a member of the reading committee.
I am most thankful for his insightful comments and for the fruitful discussions we
have had over the last three years.
I thank the third member of my reading committee, Professor Anne S. Kiremid-jian, for many years of confidence and encouragement at Stanford.
I would like to thank roy friends and colleagues at Det Norske Ventas. Pâl Bergan and Peter Bjerager deserve special thanks for believing in me and, by doing so, making a three year stav at Stanford University financially possible. Thanks are also extended
to Arne Braathen, Lars Christian Christensen, Espen Cramer, øistein Hagen, Kim
Mørk and Gudfinnur Sigurdsson for being very helpful during different phases of this work.
The assistance and friendship from my office mates over the years at Stanford, Paolo Bazzurro, Abk Jha, Karthik Knishnamurthv, Satyendra Kumar, Clifford Lange,
Doug Schmucker, Nelish Shome and Todd Ude have been of crucial importance for the succsful completion of this work. I will always remember your kind friendship
and support.
During my stay at Stanford Paula and Wendell Gilland have shared their kindness and generosity. Thanks for making everything so much more fun.
My wife Cecilie Schjander has been a tremendous helper and support. Without
her I had not been able to start nor to complete this work. Cecilie, your sacrifices have been many, I sincerely hope I can make up to you some day. Special thanks
are also extended to our kids, Eirik and Ingvild. lt was you that made everything
worthwhile.
Finally, the financial support from The Royai Norwegian Council for Scientific and Industrial Research arid Det Norske Ventas is gratefully acknowledged.
Contents
Abstract
y Acknowledgments viiii
Introduction
i
1.1 Objectives of Work 1 L2 Background 21.3 Scope and Organization 7
2
Hybrid Simulation-Analytical Methods
io
2.1 Overview io
2.2 Introduction 11
2.3 Jackup Force and Response Model 12
2.4 Simulated Results 15
2.4.1 Required Simulation Length 18
2.4.2 Dynamic Amplification Factors 22
2.5 Analytical Extreme Estimates 24
2.5.1 Weibull Model of Peaks 24
xi
2.5.2 Hermite Transformation Model 25
2.5.3 Numerical Results 26
2.6 Conclusions 29
2.7 Appendix 1: Weibull Model Statisti 32
3 Design Sectate Histories
343.1 Overview 34
3.2 Introduction 36
3.3 Design Seastate Histories 37
3.4 Jackup Model 39
3.5 Numerical Results 40
3.5.1 Wave Statistics 40
3.6 Design Seastate Simulation 48
3.7 Design Wave Models 51
3.8 Conclusions 53
4
Critical Wave Episodes
574.1 Overview 57
4.2 Problem Statement and Solution Approach 58
4.3 Model of Ship and Wave Environment 61
4.4 Identification of Critical Wave Episodes 66
4.5 Implementation of Critical Wave Episodes 72
4.5.1 Initial Conditions from Complete Irregular Linear Analysis 73
4.5.2 Initial Conditions from Linear Analysis with a Single Harmonic
4.6 Experimental Design 79
4.6.1 General 79
4.6.2 How do We Assess Statistical Uncertainty Introduced Through
Limited Information9 81
4.7 Conclusions and Recommendations for Further Work 88
5 Summary and Conclusions
91List of Tables
2.1 .Jackup force and response statistics. 23
3.1 Hourly Mean Max Deck Displacements for Relative and Absolute Fluid
Velocities 40
4.1 Non-Linear Bending Moments [kNm] from Selected Identification Schemes 69
4.2 Non-Linear Bending Moments [kNm] PREDICTED by critical wave
episodes of length N wave cycles, Identified by the 3 largest linear
bending moments, Initial Conditions from "Exact" Linear Analysis 79
4.3 Non-Linear Bending Moments [kNm] PREDICTED by critical wave
episodes of length N wave cycles, Identified by the 3 largest linear
bending moments, Initial Conditions from "Average" Linear Analysis 80
4.4 Bootstrapped Statistical Uncertainty (CoV), BIAS and Normalized
MSE, H5 = 15.5, T = 16.5, n= 3 lbms, n =3 cycles 83
4.5 nHfornencng =36
844.6 MSE,ZØ,.T,. 50% fractile, Tie identified with Linear Bending Moments,
41 MSEIW,fl 50% fractile: i identified with Wave Heights, H = 10.Om,
= 10.Os 86
4.8 MSE,WrTTh for n = 3 87
4.9 MSE,.,T. 50% fractile 87
List of Figures
2.1 Jackup Structure 14
2.2 Variation in hourly maxima of waves and resuLting forces. 15
2.3 Relation between times, T1 and T2 when wave and for is largest . 17
2.4 Six-hour maxima of deck sway; different natural periods. 22
2.5 Dynamic amplification factor (DAF) for 6-hour maxima. 24
2.6 Average 6-hour maximum deck sway versus period for 3% damping. 27
2.7 Average 6-hour maximum deck sway versus damping for T=6s. . . . 27
2.8 Distribution of local maxima, on Weibull scale. 29
3.1 Sway response versus max crest, wave heights 42
3.2 Sway response versus wave skewness. kurtosis 42
3.3 Standard deviation reduction SR: various wave statistics 43
3.4 Standard deviation reduction SR: 17 and other wave statistics. 44
3.5 Standard deviation reduction SR, based on spectral displacement SD(T, Ç). 49
3.6 Response CoV, Vy,,, Ordinary and Design Seastates. 51
3.7 Predicted mean response; various methods 53
4.1 Body and Strip Plan of the Vessel 63
4.2 Non-Linear Bending Moment History 64
4.3 100 Year Environmental Contour 64
4.4 Normalized Wave Spectra and Transfer Function from Wave to Linear
Bending Moment 65
45 The Largest Wave Height as Identifier of Critical Wave Episodes(H. =
155m, = 16.5s) 69
4.6 The Three Largest Wave Heights as Identifiers of Critical Wave Episodes(H, =
155m, T = 16.5s)
4.7 The Largest Linear Bending Moments as identifiers of Critical Wave
Episodes(H, = 15.5rn,T = 16.5s)
4.8 The Three Largest Linear Bending Moments as Identifiers of Critical
Wave Episodes (H, = 15.5m,T = 16.5s) 72
4.9 1 and 3 Cycles Predictions Starting from "Exact" Linear Initial
Con-ditions (H, = 15.5m,T = 16.5s) 74 4.10 1 and 3 Cycles Predictions Starting From Exact" Linear Initial
Con-ditions (H, = 10.0m.T = 10.Os) 75
4.11 1, 2, 3 and 4 Cycles Predictions Starting From "Average" Linear Initial
Conditions (H, = 15.5m.2 = 16.5s)
4.12 1,2,3 and 4 Cycles Predictions Starting From "Average" Linear Initial
Conditions (H, = 10.0m,T = 10.Os) 78
Chapter 1
Introduction
1.1
Objectives of Work
In this work we critically evaluate strategies to identify minimal portions of stochastic
wave mput to form reliable estimates of extreme response statistics for nonlinear ocean structures under random wave loads. Specifically, we consider the extreme structural response from nonlinear wave loads (the wave to load relation is nonlinear) of selected ocean structures, the fixed offshore jackup structure and the moving ship under random wave actions. We present methodologies and strategiesthat are general and the strategies can be used to estimate any statistics of the extreme response. We have chosen to concentrate on the mean extreme response of the fixed jackup structure
in Chapters 2 and 3. Elficient strategies for estimating the mean extreme response within a severe storm are useful for design calculations where the main goal is to
estimate accurately the mean extreme value corresponding to a given return period
(typically loo years). An accurate estimate of the mean extreme response within a
CHAPTER 1 INTRODUCTION 2
seastate is also useful as input to the long-term analysis given that, from experience, we have knowledge about the conditional short-term response variability.
In Chapter 4 we focus on forming reliable estimates of not only the mean but also the variability of the extreme response of the moving ship. For these various applica-tions and strategies we also study the uncertainty in extreme response estimates due to limited-duration wave input. Adequate durations are suggested.
1.2
Background
Ocean structures are exposed to wave, wind and current processes that define the en-vironment that the structures have to withstand over their entire design life, typically 20-30 years. Waves, winds and currents are all meteorological phenomena combining the effects of gravity and those of temperature and pressure in the air and in the sea. Because of the random nature of the ocean environment, we are not able to estimate with certainty the most severe weather conditions an ocean structure will experience
during its design life We can only estimate the probability that a certain weather
condition will occur within a given period of time. To predict the future of a structure exposed to forces from the ocean environment we are left, therefore, with the tools of probabilistic modeling.
We find it beneficial for reasons of analysis to separate the random nature of the
en-vironmental processes into variability over long-term and short-term time scales. We model the continuously-evolving environment as a series of statistically-independent and stationary "seastates". Seastate durations are commonly taken to be on the order of one to six hours. Over these durations, we can consider environmental effects such
CHAPTER L ENTRODUCiION 3
as the wave elevation, wind speed, etc., as realizationsof stationary random proCesses.
The analysis of structural responses within a seastate is what we coairnonly refer to as short-term analysis.
The waves are the primary loading source for many ocean structures. We typically describe the wave elevation within a seastate by its power spectrum, representing the energy in the waves due to different frequencies. The wave spectrum models are commonly pararneterized in terms of signi&ant wave height, H, and peak spectral period, T. H, is a measure of the energy in the waves and is equal to four times the standard deviation of the wave elevation 7; measures the location of the spectral peak, indicating the primary frequency content in the waves.
The long-term evolution of the stationary seastates is what we refer to as
long-term uncertainty. By integrating the structural response results from the short-long-term
analysis which are conditional on the short-term environmentoverthe long-term
environmental variability, we achieve the long-term distribution of structural
re-sponses. From this long-term distribution of structural responses wecan, for example. derive the response with a loo year return period.
Linear analysis of ocean structures exposed to Gaussian waves efficiently takes
place in the frequency domain where the response to any time trace of the ocean
surface will be the superimposed response from each of the Fourier components of the same time trace. Once a transfer-function from wave to response is established, we can efficiently estimate the power spectrum for the response within each seastate. From the response power spectrum we can achieve both short-term distributions of the structural peak response and estimates of response mean up-crossing rates. The
CHAPTER 1. INTRODUCTION 4
fairly general assumptions for linear systems exposed to Gaussian waves (Sarpakaya and lsaacson, 1981). The long-term peak response distribution can further be reached as a weighted integration (over the long-term environment) of the short-term response distributions. The weights in the integration over the long term environment are the
probabilities of occurrence of the independent and stationary short-term seastates,
adjusted by a factor accounting for the variation in response frequency between the\ seastates (Sarpakava and Tsaacson, 1981).
The computational time involved in the estimation of long-term extreme
re-sponse of structures responding linearly to Gaussian waves is hence marginal once the transfer-functions from wave to response are established.
If any nonlinearity in the wave-to-response relation is introduced, the principle of superposition is no longer valid. The wave loads on complex ocean structures typically
vary nonlinearly with the wave elevation; for these nonlinear ocean structures, time-domain simulation remains one of the few general techniques for estimating response
statistics under random wave loads. For slender ocean structures (e.g., the jackup and the jacket structures with their slender legs as compared to the wave length of
the extreme waves), the main nonlinearities in the wave-to-force relation arise from
the quadratic drag term in the force model and from wave inundation, in which
wave kinematios are "stretched" to the exact, time-varying wave surface. For
mono-hull ships with flared mono-hull geometries (the outward leaning walls of the mono-hull), the buoyancy forces on the ship will vary nonlinearly with displacement from the still
water equilibrium. Typically, these nonlinear effects are strongest for the large wave conditions that govern design. In addition to the nonlinearities in the wave-to-force relation the wave process often is nonlinear by itself. In this study, however, we focus
CHAPTER 1. IIVTRODUCTION 5
exclusively on response from linear Gaussian waves.
For both slender offshore structures and ships, complex mechanical models are
necessary to assess the loads and actions of the global structure and the structurai
details. For free moving ships, research efforts are increasingly aimed towards a
more complete nonlinear, time-domain analysis of 3-D (fluid-structure) ship problems. Examples include the multi-phase projects behind the codes LAMP from SAIC Inc.
(Lin et al, 1994, Lin and Yue, 1990), SWAN from MiT (Kring, 1994, Nakos et al, 1994)
and USAEp.a from the University of Michigan (Beck and Magee, 1991). The jackup is analyzed through a complicated structural model, typically a finite element model with hundreds of degrees of freedom (SNAME. 1993).
These increased analysis capabilities do not come without cost, however. A com-plex hydrodynarnical or mechanical model leads to involved computations for each
time step in the simulation. Despite growing computationalefficiencies and effective vectorization, the promise of "real time" simulation appears a challenge we have yet to overcome. For example, simulation of the fully nonlinear response of a 3-D ship to 1 wave-minute may require not only i min of CPU time, but something closer to 102
CPU minutes or more.
The stochastic nature of the loading from the waves mandates simulationof irregular waves and corresponding structural responseover multiple storm hours,
(ten to hundreds) if the statistical uncertainty in the structural response estimates is to be reduced. lt hence remains difficult to simulate an ideal load and structural
modeltypically a finite element model with hundreds of degrees of freedom for the jackup and a fully nonlinear three dimensional model of the interaction between the
CHAPTER 1. rNTRODUCTION 6
These growing computational demands imply increased challenges for practical, reliability-based ocean structure analyses and designs. As we discussed earlier, relia-bility studies require recognition of uncertainties, both in the "long-term" variation
of wave spectral parameters (e.g., significant wave height, H, and peak spectral period, 7,) and in the "short-term" variation of wave elevation given H, T, and
other spectral parameters. As response calculations grow in expense, an exhaustive simulation of all likely wave conditionseither short-term or long-termbecomes
in-feasible. It has thus become increasingly important to develop efficient stochastic
analysis methodseither analytical or clever/selective simulationto permit these
more realistic mechanical models to reflect realistic uncertainty levels also.
This work focuses on the short-term problem (variation of the response given the input wave spectral parameters) and the objective of this work is to evaluate, quan-tify and reduce the extent of nonlinear simulations needed to form reliable extreme response estimates under random wave loads. The methodologies and strategies that we present are general and can be used to directly estimate any statistics of the ex-treme response. For example, possible uses of the strategies in full reliability analysis
could be:
Estimating two statistics of the extreme response, i.e, the mean and the standard deviation or the 50% and the 75% fractile, then fitting a distribution based on these statistics.
Estimating only the mean or the median of the extreme responseacknowledging the short-term variability in the response process by inflating the seastate dis-tribution contour (Winterstein et al, 1996).
CHAPTER 1. INTRODUCTION 7 3. Estimating directly only one fractile (larger than the median) of the extreme re-sponse. Again, acknowledging the short-term variability in the response process, by achieving the optimal fractile from FORM (first order reliability methods) omission factors (Madsen et al, 1986, 'Winterstein et al, 1996).
In Chapters 2 and 3 we focus on the mean extreme response of a jackup offshore
structure. An accurate estimate of the mean extreme response within a seastate is useful for design considerations if we anal ze the 'correct" seastate. The "correct"
seastate in this context is the seastate that provides the response with 100 year return period (given that this is the goal). The seastate that provides the 100 year response
will have a longer return period than 100 years By estimating the mean extreme
response from a seastate with a longer return period than 100 years, weacknowledge
the short-term variability in the response process (Winterstein et al, 1996). An accu-rate estimate of the mean extreme response within a seastate is also useful as input to the long-term analysis given that, from experience, we bave knowledge about the conditional short-term response variability.
In Chapter 4 we focus on forming reliable estimates of not only the mean but,
also the variability of the extreme response. We use the moving ship as an example application.
1.3
Scope and Organization
In practice, the statistical model is often simplified and the extreme response in a
seastate is sometimes estimated by "hybrid" simulation-analyticalmethods. These methods employ one or several full time-domain simulations over a limited duration
CHAPTER 1. LVTRODUCTÏON 8
with typical duration ranging from 20-60 minutes. An idealized analytical model,
such as a Weibull model of response peak statistics, is then fit to extrapolate limited-duration results to predict average behavior over many 6hour seastates.
In Chapter 2 we investigate such "hybrid" simulation-analytical methods and
apply them to extreme response of a jackup structure. We have two main objectives in doing this:
To understand the level of uncertainty (as a function of simulation time) in actual 6-hour response estimates, and hence the simulation time required to
forni reliable estimates.
To use accurate extreme estimates, based on long simulations, to evaluate and calibrate conventional "hybrid" extreme estimates.
In Chapter 3 we introduce the use of "design seastates", hourly-long wave histories selected because certain statistics of the histories correspond to theoretical ensemble average values (SNAME, 1993), as an unoertainty-reducing technique. We apply these "design seastates" to estimate extreme response of the same jackup structure that we
focus on in Chapter 2. Multiple wave histories are simulated and screened. Only a
few wave hours are finally propagated through the structural model. The motivation for this screening process and selective simulation is:
To reduce uncertainty in response estimates without increasing simulation length (of structural response).
To take advantage of the fact that the computational expense lies not in sim-ulation of multiple wave histories, but rather in propagating each through the structural model to obtain forces and responses.
CHAPTER 1. ThITRODUCTION 9
One of the main conclusions in this chapter is that in quasi-static cases, a good design seastate (the goal is to estimate the mean extreme response within the seastate) has
ecL to .,.e C
maximum wave crest near its average value. The maximum response in these
reses will occur together in time with The need for simulation and analysis of an entire seastate is then avoided.
Motivated by the success of single wave cycle analyses for the fixed offshore jackup
structure in Chapter 3, we study the nonlinear extreme response of a flared container
ship through short time-domain simulations. Chapter 4 focuses on use of "critical
wave episodes", which are short wave segments which are likely candidates to produce
the actual extreme response in the hourlong history. We hence leave the lengthy
hourly simulations and concentrate ori propagating through the structure only a few
critical wave episodes (each critical wave episode is only a few wave cycles long)
per hourly simulation. We discuss how we can identify the location of critical wave
episodes within a longer wave history. Once a location is identified we discuss how long
(how many wave cycles) a critical wave episode must be to assure accurate results.
We apply the ucritical wave episodes" to estimate extreme response of a container
ship. The motivation for this is,
Significant reduction of computational resources can be achieved if the location of critical wave episodes can be identified in advance and short simulations of the structural response can be performed only with these critical wave episodes.
Chapter 2
Estimating Extreme Response of
Jackup Structures From Limited
Time Domain Simulation
2.1
Overview
The chapter examines the nonlinear dynamic response of a jackup structure under
random wave loads. For a simplified jackup model, average behavior and variability in extreme forces and responses are found from simulation over many 6-hour sea_states. Weibull and Hermite analytical models of response extremes are also presented and evaluated. These models use shorter, less expensive simulations to estimate a limited
number of response statisti, such as moments or parameters of the response peak
distribution, and fit analytical models to estimate global extremes. We therefore refer to them as hybrid simulation-analytical methods. Necessary simulation lengths are
CHAPTER 2. HYBRID SIMULATIOIANALYrICAL 'vfETHODS 11
established both for direct simulation of extremes and for hybrid simulation-analytical extreme models.
2.2
Introduction
As we have discussed in Chapter 1, the major drawback of simulation is its
computa-tional expense. It remains difficult to simulate a realistic structural modeltypically a finite element model (FEM) with hundreds of degrees of freedomover a statisti-cally sufficient duration. In practice it is often the statistical model, rather than the
structure, that is simplified. Specifically, the 6-hour extreme response is sometimes estimated by "hybrid" simulation-analytical methods, in which
One or several FEM simulations are done over a limited duration; typical du-rations are 20-60 minutes.
An idealized analytical model, such as a Weibull model of response peak
statis-tics, is fit to extrapolate limited-duration results to predict average extreme
behavior over many 6-hour seastates (e.g., Karunakaran. 1991).
One drawback to such procedures is that, lacking more extensive simulations, their accuracy cannot be verified.
In this study we instead simplify the structural model, adopting a simple SDOF (single degree of freedom) jackup model. This.permits exhaustive time-domain sim-ulation of jackup forces and responses. We have two main objectives for these
CHAPTER 2. HYBRID SIMULATION-ANALYTICAL METHODS 12
To understand the level of uncertainty in actual 6-hour response extremes and. hence, the simulation time required to form reliable extreme estimates; and To use accurate extreme estimates, based on these long simulations, to evaluate and calibrate conventional "hybrid" extreme response estimates.
With the first item we hope to provide a broader perspective regarding the ac-curacy of simulation, which is most often studied under linear/Gaussian behavior.
Note that a quadratic nonlinearity may be expected to roughly double the response
variabilityan observation our jackup results confirm. This, in turn, implies a
low-fold increase in the simulation length needed to keep a constant confidence interval (which depends on the ratio of standard deviation to in terms of the simulation length n). The second point is motivated by the observation that advanced research practice may continue to use hybrid methods based on only one or several short
sim-ulations. We therefore seek to compare various hybrid methods and quantify their
biases and uncertainties.
2.3
Jackup Force and Response Model
The gross effects of waves and current are reflected by the total base shear and overturning moment they cause. We focus here on the total wave force F(t) on the jackupLe., its quasi-static base shearand the corresponding deck sway Y(t).
The total wave force is calculated in a rather severe storm typical of 100-year North
CHAPTER Z RYBRID SIM1TLATION-ANALYTICAL METHODS 13
A Gaussian wave elevation i7(t) is first simulated as
i7(t)= E t2kcos(Wkt
+
9k) (2.1)The magnitudes of theakare related to the one-sided JONSWAP power spectrum of
the waves S(wk)by
= .J2S,,(wk).A (2.2)
where Wk is the circular frequency of a particular harmonic component and 6k is a random phase.
The applied force f(z, z) per unit length, at elevation z and leg location, z, is
calculated from the Morison equation:
f(z, z) = PRDCD(U(Z-. z) + to)Iu(x, z) + t
±p7rR1CMù(z, z) (2.3)
We use the absolute (and not relative) fluid velocity, u(x, z). The total forces are then found by integrating Morison's equation to the exact free surface. When this
is above the mean water level, kinematics from linear wave theory are extended by constant stretching (Sarpa1ya and I.saacson, 1981).
Any relative velocity effects are modeled here through the response damping,
which includes hydrodynamic as well as structural and foundation effects.
We consider an idealized jackup at water depth d=100m from the mean surface, and three legs at x=O, O, and 65m in the along-wave direction; see Figure 2.1. For
CHAPTER 2. HYBRID SIMULATION-ANALI'TICAL METHODS 14 MWL Maa Deck DD=On DC=2rien dalied Lap 65
o
Wave DtecimnIicp (evt,oe) Toç View
Figure 2.1: Jackup Structure
assign drag radius R=R,=1.25n., and assume that this implies an effective drag
coefficient Cd=2.5. This is typical of values found forallorth Sea jackup (Løseth et al, 1990), assuming local Cd values between 0.65-()80 for the lattice legs and applying standard methods (Det Norske \Teritas Classification, 1984). Finally, assuming a
linear mode shape, we estimate the deck sway Y(t) by applying two-thirds of the total base shear to a SDOF oscillator. The structural propertieshere, the natural period and damping of the equivalent SDOF modelare varied from 2s to 8s and from 1% to 5%. This range of total dampinghydrodynamic and foundation as
well as structuralis intended to reflect observed jackup behavior (e.g., Brecke et al, 1990; Hambly et al, 1990). The modal mass m'=22.5 (kg.106] is assumed, based on representative deck and leg masses- Note that because the natural frequency and
damping ratio (are fixed, ail responses will scale directly if this m is changed (i.e.,
CHAPTER 2. HYBRID SIMULATION-ANALYTICAL METHODS 15 3e+C7 .a5e+O7 ç, o IL a, > a, 2es-07 l.5e+07 i e-s-07 Conirrent rnaxs O Nm-concurrent maxs +
y=Cx2
Figure 2.2: Variation in hourly maximaof waves and resulting forces. This simple model is intended to reflect relative responsevarIabilitynot necessarily the absolute jack-up deck sway.
2.4
Simulated Results
This sectìon describes simulated results for 6-hour extreme forces and responses. To reflect current practice, we simulate series of shorter, 1-hour response histories. Each is formed from N=16384 simulated time points at constant dtO.238s, after discarding the first 1260 points (5 minutes) as 'transient" effects due to zero initial conditions. For each case we perform 180 simulations (hours), which are grouped into sets of 6 to yield 30 estimates of 6-hour maxima. Thefollowing section compares these 6-hour extremes with analytical estimates, based on the shorter 1-hour histories.
12 13 14 15 16
X = 1-Hr Max Wave Crest [m}
17 18
11
CHAPTER 2. 1-IYBRID SiMULATION-ANALYTICAL METHODS 16
Force Statistics
Figure 2.2 shows how the 180 hourly maxima of wave force, F, vary with the
corresponding wave elevation maxima, in these hours. Figure 2.2 reveals that, as might be expected from nonlinear drag and stretching, there is a roughly quadratic
relation between
and F,,, that is,
Ci7 (2.4)
for an appropriately chosen constant C. Note also from Figure 2.2 that there is still significant scatter in F,,. given Thus, the extreme force is not that well
explained by knowledge of only the extreme wave level. This might occur, for example,
if the maximum wave and force levels occur at different times, say T1 and T2, during the hour. To investigate this, it is easy to identify hours in which the maxima occur concurrently (T1 within seconds of T2). Figure 2.3 reveals that this occurs roughly 90% of the time. However, Figure 2.2 shows that these rare "non-concurrent" maxima do not produce appreciable scatter (and, in fact, they generally produce lower, less interesting force levels). We conclude that the scatter in Figure 2.2 should be reflected
by considering "concurrent" cases, but requires a more detailed description of the wave
profile (and its uncertainty) around its largest crest. We will return to this topic in
Chapter 3.
To the degree that Equation 2.4 holds, it suggests that should show greater
scatter than If the CoV, the ratio between the standard deviation and the
mean value of m, is small and the mean of Fm, is well estimated
CHAPTER 2. HYBRID SIMULATION-ANALYTICAL METHODS 17 3500 3000 2500 2000 1500 1000 500
-
! I Concurrent maxs o Nonconcurrent maxs + + + + + 4 (4.j
500 1000 1500 2000 2500 3000 3500Ti =Time duringhourwhen wave crest ismax[secj
Figure 2.3: Relation between times, T1 and T2 when wave and force is largest twice the CoV
2%Ç (2.5)
A virtue of this estimate is that because i(t) is a Gaussian process, analytical
estimates of are available (e.g., Davenport, 1964):
[.450± 1.56 lnN]: N
=TIT: (2.6)Where T is the total duration of the wave histor3
In our case, with T=16s and 73.3 we find T:=12.43s. The expected number of
cycles in 6 hours is then N
=
T/T;=6 .3600/12.43=1738. Hence, Equations 2.5 - 2.6predict =0.083 and =0.166. These values agree fairly well with the
±
+ +
CHAPTER 2. HYBRID SIMULATION-ANALYTICAL METHODS 18
2. For each simulated wave history, a calculation of the corresponding response
observed values:
V
=07; V1
.14 (2.7)2.4.1
Required Simulation Length
These estimates of wave and force variability directly suggest simulation lengths. To illustrate how let us say we want our mean-maxestimate, , to be within 10% on either side of the true mean value, my, and we want to achieve this error tolerance
with 95% confidence. In our case is the maximum deck sway, however Y, may
in other cases be the crest height, or the force, F,,,,,. In time-domain simulation it is common to discretize r7(t) into equally spaced observations i(t1)
-. r7(t). The
response Y may then be viewed as simply a function of a (large) number of randomvariables:
Y = g(X); X = [i(t1)
.. -ii(t)]
(28)We further assume that only the mean maximum response, my,,r, is sought.
For jackup structures, interest often focuses on the mode of Y,, ("most probable maximum"). The mean my, however, is most directly estimated from time-domain simulations, which consists of the following steps:
1. Generation of discretized wave histories, X1, over multiple seastate realizations
CHAPTER 2. fyBRD SIMULATION-ANALYTICAL METHODS 19
Extraction of the maximum response, Y,,., from each simulated history.
Estimation of my,, by the average response over all simulations: Y,Y/N3S.
The uncertainty associated with estimating the true mean value my,, by the average value of a limited number of outcomes, N, is reflected through its unit-less coefficient of variation (CoV):
where Vy,, is the CoV of the maximum response in a single sea-state r and reflects
the variability (randomness) in the process itself.
We can then create a 95% confidence band for the mean-max-estimate being within
10% on either side of the true mean value:
P[0.9 . <1.1 . my,,..j = 0.95 (2.13)
(29) =
The numerator and denominator in Equation 2.9 can be written as:
o_Y (2.10)
=c1:):
(2.11)
n=E[=E{H=
Hence the CoV of Y,, can be written as:
(2.12) 14
Finally, solving for to satisfy Equation 2.18 we need
(2.17)
= (20. (2.19)
CHAPTER 2. HYBRiD SIMULATION-ANALYTICAL METHODS 20
In practice } is typically the average maximum response of a small number of
simulations (N33 4 10) and the normalized variable, U,
-
my,(2.14)
hence approximately follows a Gaussian distribution. We can express the following relation.
P[k
<U
k) = 0.95 (2.15) Inserting Equation 2.14 in Equation 2.15 we can write:P[my_,,
(2.16)To satisfy these equations, k will be approximately 2.0. By using Equations 2.13 and 2.16 we can then write
2.0 .cry
my,,,3r+ 1.1.Tflyfl_
Equation 2.17 can be rewritten as
CHAPTER 2. HYBRID SIMULATION-ANALYTICAL METHODS 21
This result for Nsq can apply to estimates of the maximum wave crest, force or
response, provided reflects the relevant CoV value. We see from Equation 219
that a doubling of the CoV of Y,, results in a fourfold increase in required simulation
length, N53q In particular, we can note for the example here that Equations 2.7 and 2.19 suggest N=8 samples (48 hours) for extreme forces. We show in Section
2.5, Analytical Extreme Estimates, how this may be reduced with analytical models
of extremes.
Note also that the required simulation length will change as a function of the
degree of accuracy we demand in the mean-max-estimate. For example, if we demand 95% confidence that the mean-max-estimate lies within 5% of either side of the true
mean value (instead of 10%), the required simulation length, (Nsq = (40
Vy,.j2),
will increase fourfold to 32 samples (l92hours) for extreme forces.By relaxing the demand for accuracy to 90% confidence (instead of 95%) in the mean-max-estimate being within 10% of the true mean value, k in Equation 2.16 will be 1.65 (instead of 2.0). This will lead to a reduction in required simulation length,
(N55 = (16.5 . Vyr)2) to 5-6 samples (30-36 hours) for extreme forces.
Response Statistics
We consider now the corresponding deck sway, with jackup natural periods from
T=2s to T,=Ss.
As T,, increases, one might expect the increased dynamic ef-fects (1) to "Gaussianize" the non-Gaussian wave loads (Winterstein and Løseth, 1990, Manuel, 1992) and, hence, (2) to reduce the relative response tails andex-treme variability. Table 2.1 confirms this first effect, showing that as T,, grows the
CHAPTER2. RYBPJDSIMULATION-ANALYTICAL METHODS 22 E >. cl,
o
0 20 40 60 80 loo 120 140 160 180 Simulation length [hrsjFigure 2.4: Six-hour maxima of deck sway; different natural periods.
(ci=E[(Ymy)/c,-y). Nonetheless, response variability is not significantly altered:
the CoV of extreme response remains close to that of the wave force, see Table 2.1.
This is also shown by Figure 2.4, in which 6-hour response maxima for various T, have
similar ranges of relative variability. Hence, the foregoing results can also be used to predict response CoV values (Equations 2.5-2.6) and required simulation length (Equation 2.19).
2.4.2
Dynamic Amplification Factors
Following common practice, we are also interested in ìnvestigating the potential of focusing first on the static maximum response, E[Y3.], and then applying a dynamic amplification factor (DAF):
I I
Tn = 2s
-w Tn = 4s Tn = Ss Tn =SS - .. o-1Table 2.1: Jackup force and response statistics.
DF
EX]
E[Y5] (2.20)We choose the ratio of mean maidma, rather than the mean ratio, inview of its
common use as a correction factor to E{Y3J. Figure 2.5 shows t)AF estimates, found
from simulation for various periods T and damping Ç Also shown is the standard steady-state result for sinusoidal loads with period T::
i
= /(1
- r2)2T,.
(2çr)2'
r=-
(2.21) This result is found to predict the average trend with T fairly well, although it underestimates its variation with damping Ç. Similar conclusions have been found from full FEM simulations (Grenda, 1986). The alternative proposed DAF,1/(1 - r) (McDonald and Bea, 1990), appears somewhat conservative in this case.Wave Elevation Wave
Force T=2s
Deck Sway.T=4s
Ç=.03T=6s T,=Ss
Average skewness, -.004 1.81 1.69 1.03 .569 .188 Average kurtosis, 2.99 8.98 8.96 7.53 5.24 5.16 C0V, observed y in 6-hr periods .074 J41 .145 .163 .155 .147 CoV, predicted from Hermite model in 1-hr periods .081 .139 .144 .150J48
.174CHAPTER 2. HYBRID SIMJJLATTON.ANALYTICAL METHODS 24
2.2
Anarylicaf: Damping=5%
-Analylical;Damping=3%
2 -
AnaíylcatDamping=W---SAnulated: Damping =5%ie-4
Simulated: Damping =3%
l-+--1.8 Simulated; Damping =1% lBI
0.8 I I
0 1 2 3 4 5 6 7 8
Natural period. Tn [si
Figure 2.5: Dynamic amplification factor (DAF) for 6-hour maxima.
2.5
Analytical Extreme Estimates
Finally, we compare these simulated response extremes with analytical estimates formed from shorter, 1-hour histories Two analytical models are pursued: the Weibull model of peaks and the Hermite transformation model.
2.5.1
Weibull Model of Peaks
Given short response histories, one may seek to model the distribution of cadi local maximum, M. A three-parameter Weibull model is commonly used for this purpose:
G1(z)=P[M>Y]=exp[
(Y_YO)Th;YY0
(2.22)t6
1.4
CHAPTER 2. HYBRID SrMUL4TION-ANALYTTCAL METHODS 25
A nonlinear least-squares algorithm is used here to fit Equation 2.22 to the
ob-served distribution of maxima (Det Norske Ventas Sesam, 1989). If these maxima are assumed independent, statistics of the global maximum Y, in N cycles-e.g.,
its mode, median, and meanreadily follow (Section 21 Appendix 1: Weibufl Model
Statistics). We focus here on the mean maximum in Equation 2.23, a
com-mon statistical choice and the one most directly estimated from data:
.577
±a(lnN)'(1+
ßlnN
Because all maxima have been fit, N is taken here as the expected number of
maxima in the period of ìnterest (6 hours).
2.5.2
Hermite Transformation Model
The Herrnite model assumes a functional relation between the non-Gaussian process
of interest, Y(t), and a standard normal process u(t):
Y=g(u)=my±cy{u±c3(tz-1)+c4(u3-3u)]; cr(1±2c+6c*2
(2.24)
The coefficients c3 and c4 are based on the observed higher moments a3 and a4
of Y(t) (Equation 2.24 considers here a. > 3, i.e.
a fatter tail than the normal
distributionthe common case of interest for jackup forces and responses). In its simplest "2nd-order" form, analytical estimates of C and c4 are used to roughly
match a3 and a4. More recent implementations choose c3 and c4 to minimize errors
CL4PTER 2. HYBRiD SIMtTLATION-A.NALYTICAL METHODS 26
in matching and a, while requiring that g in Equation 2.24 remain monotone. This has not been found computationally burdensome, and source code and documentation are available (Winterstein et al, 1994). Empiricalfits of the resulting c3 and c4 values
give cx3y
1-= 6
1 ± 0.2(4y - 3)
1.43c [1 + 1.25(a4y 3)}1/3- I
C40 = c4 - c40[1 - (co4y - 3) IOOnce e3 and c4 have been estimated, the mean of Y, can be estimated from the g-function in Equation 2.24:
(2.27)
Here is the corresponding mean extreme of ZL(t), found from Equation
2.23 with xo=0, 3=2 and c=v' Similarly, g can also be used to estimate modes
and medians e.g.,
Ym=g(U,1d,,j.
2.5.3
Numerical Results
Figure 2.6 compares 6-hour simulated extreme deck sway with predictions from both the Weibull and Hermite models. Simulated results, based on the 30 observed ex-tremes in 180 hours, are shown with ±1-sigma error bars. The Weibull and Hermite
models are then applied to each 1-hour segment to estimate the mean 6-hour
(2.25)
CHAPTER 2. 1-JYBRID SI ULATION-ANALYTICAL METHODS 27 1.8 1.6 1.4 1.2 0.8 0.6 0.4 0.2 o 0.95 0.9 0.85 0.8 0.75 0.7 0.65
- Simulated; 3% damping I4-4
Optimal HerTnite 2ndorder Hermite -3-parameter Weibull
----'z..
L
Simulated; 1n6 sec i-e---t Optimal Hermite Weibull; all data .
-Weibull; upper 80% Weibull; upper 50%
-4 1
Figure 2.7: Average 6-hour macimum deck sway versus damping for T=6s.
2 3 4 5 6 7 8
Natural peñod, In [si
Figure 2.6: Average 6-hour maximum deck sway versus period for3% damping.
0.01 0.03 0.05
CHAPTER 2. HYBRID SrMULATION-ANALYTICAL METHODS 28
extreme. Each model thus gives 180 values, one per hour, whose average is
shown in Figure 2.6.
In particular, the "optimal" Hermite prediction appears to have Little bias.
More-over, its single-hour estimate shows similar variabilityCoV values between
14-17% (Table 2.1)as the actual observed maximum in a szx-hour period. This implies a roughly sixfold reduction in simulation length: with the Hermite mode'
Equation 219 now implies about 8-12 single-hour simulations, rather than 8-12 sim-ulations of 6-hour seastates.
Figure 2.6 also shows that the 2nd-order Hermite model tends to overestimate
extremes, because of its conservative choice of c3 and c4, given a3 and cx. In contrast, it appears somewhat non conservative here to fit the Weibull model to all response
peaks. For fairly dynamic systems (T,. near 6-8s) the maximum deck sway tends to be under-predicted by around 15%. This may arise, for example, if the actual
distribution of local maxima is somewhat broader in its tails than a body-fit model would suggest. This is supported by Figure 2.7, which shows alternative tail-fit Weibull models from the upper 80% and 50% of all maxima. Weibull estimates
of grow as we focus increasingly on the upper tail: an optimal Weibull fit appears here to use between 50% and 80% of the largest maxima. This is not
universal, however; for stiffer systems the best Weibull fit often uses between 80% and 100% of all maxima. We therefore hesitate to recommend ari optimal Weibull fitting procedure for all jackups (Winterstein and Torhaug, 1996).
The Weibull assumption can also be studied by viewing the observed distribution of response peaks on Weibull scale. Figure 2.8 shows the result for a typical 1-hour
CHAPTER 2. HYBRID SIM LATION-ANALYTICAL METHODS 29
*
u-lo
t
Weibull 3moment fit
-Weibufl least-sq fit all data Weibull fit upper 8001 Weibull fIt upper 50% i-hr simulation; Tri=8 sec O
6hr median max leve!
-I I I
I 1111
0.1 0.2 0.5 i 2
Lol maxima of deck sway, x [m]
Figure 2.8: Distribution of local maxima, on Weibull scale.
begins to flatten. While apparently a small effect, whenextrapolated to the 6-hour
median levelthe level for which FN=O.5__it leads to the systematic differences between body- and tai1-t Weibull models. Similar effects, and non conservative
extreme estimates from body-fit Weibull models. have been found for some flexible riser systems (Sødahl. 1991).
2.6
Conclusions
We have studied the dynamic response of a jackup structure under random wave
loads. For a simplified jackup model we have found average behavior and variability in extreme forces and responses over many 6-hour seastates. We have also evalu-ated Weibull and Hermite models of extreme response. We have discussed necessary
CFL4PTER 2. RYBPIJD SIMULATlON-ANALTICAL METHODS 30
simulation lengths for both direct simulations of extremes and for hybrid simulation-analytical extreme models.
We have focused on one particular jackup geometry in a specific extreme seastate. The chosen jackup geometry has, however, representative leg diameters and spacings between platform legs for North-Sea applications.
The conclusions drawn should generally be applicable to comparable jackup
con-figurations also in other extreme seastates given that the nature of the structural
loading and response does not change significantly (relative importance of drag to inertia forces etc.).
We have investigated a simple SDOF model of the jackup structure however, the important nonlinearities due to drag-forces and stretching of wave kinematics to the free surface are included in the model. These wave-to-load nonlinearities are shown to be important contributers to the total CoV in the extreme jackup responses (Manuel,
1992). Analyses of the CoV of the extreme deck sway of a comparable jackup geometry
modeled with a refined finite element modelincluding geometrical nonlinearities and nonlinearities from soil-structure interaction show comparable response variability in 6-hour response maxima (CoVs 13%) (Kariinakaran et al, 1993) to what we have found in our analyses of the simplified structure. We hence expect the CoVs and the required simulation lengths we have found for 6-hour maximum forces and responses to be fairly representative also for more refined structural models.
Based on the preceding results and discussions, we have come to the following
conclusions:
Because of nonlinear drag and wave stretching, extreme wave forces vary roughly
CIL4PTER 2. RYBPJD SIMULATION-ARAL'fTICAL METHODS 31
simple analytical estimates can be constructed for the coefficient of variation of extreme wave and force (Equations 2.5-2.6) and, hence, the required number of
simulations (eg., Equation 2.19) to estimate the mean extreme responsewith
a specified confidence
Typical CoVs of 6-hour maximum forces and responses are found to lie be-tween 14-17% (e.g, Table 2.1). Comparable CoVs from the 6-hour maximum forces and responses are found from complete FEM-analyses (Karunakaran et
al, 1993). With Equation 2.19, this suests theneed to simulate 8-12 maxima,
each over a 6-hour seastate. This is roughly a fourfold increase over the simula-tion length needed for mean extreme estimates of Gaussian processes (e.g., the
wave elevation).
The (optimal) Herinite model appears to have little systematic bias in predicting mean 6-hour maximum deck sway (Figures 2.6-2.7) from 1-hour long response histories. Moreover, its variability in a single hour is comparable to that in
observed maxima over 6-hour seastates (Table2.1) The needed simulation length is thus reduced roughly sixfold. For example, 95% confidence bands
within ±10% (Equation 2.19) of mean 6-hour extreme response now require
about 8-12 single-hour simulations with the Hermite model, rather than 8-12 simulations of 6-hour seastates.
For some flexible jackups (T=6s or larger), local response maxima appear to have somewhat broader distribution tails than a body-fit Weibull model would suggest. Such Weibull models, fit by nonlinear least squares, are then found to underestimate the 6-hour maximum response by 10-15% ( Figures 2.6-2.8).
CFL4PTER 2. HYBRID SIMuLATI0N-ANALy'rIcAL METHODS 32
2.7
Appendix 1: Weibull Model Statistics
We derive here the extremeresponse statistics for the Weibull model. We first assume that local maxima M, are independent, with a common comp'ementary distribution
function G1(Y)=P[M > Y]. The distribution of Y,,, the global maximum in N
cycles, is then
F(Y) = P[YWWW <Y] = ]1 - G1(Y)]' exp[-NG1(y)] (228) To estimate maxima statistics we frequently define a new variable z such that NG1(Y) = exp(z), so that Equation 2.28 becomes a standard Gumbel distribution. For the 3-parameter Weibtill model of G1(Y) from Equation 2.22, this z variable
becomes
z = - ln[Gj(Y)] - in N ('
- Yoß
in Na
tnverting this relation for Y, one finds
Y-Yo-
- (in N + z)'1(in N)"[1 ±
N'
a
Finally, one can estimate the mode, median, and mean of Y by substituting
into Equation 2.30 the corresponding values for the standard Gumbel variable z (z=0,
.367, and .577):
+ a(1nN)1 (2.31)
(2.29)
CHAPTER 2. HYBRJD SIMULATION-ANALYTICAL METHODS 3,3
367
= Yo + c(lnN + .367)' Yo + a(1nN)1(l +
3lN
(2.32).577
Yo+a(lnN)1(1+
8lN
Either or Y, may be taken as a representative extreme
value. Note that S in common practice, with rel-ative differences decreasing as N increases. Note also the slightly different form of Equation 2.32. Because medians are preserved under nonlinearoperations, the last approximation in Equation 2.30 (linearization in z) is not necessary. Finally, because
r2=x2/6, Equation 2.30 also leads to the following standard deviation of Y,:
7ra
= (lnN)
(2.33)
Chapter 3
Design Seastate Histories For
Extreme Response of Jackup
Structures
3.1
Overview
In this chapter we continue to study the extreme response of the jackup structure that
we studied in Chapter 2 (Figure 2.1). We consider how the use of "design seastate histories" can reduce the cost of time-domain response analysis. This is motivated by what we concluded in Chapter 2: the inherent variability in the response pro-cess leads to a rather long simulation time if one is to achieve reliable estimates of
the mean extreme response. However, the computationally expensive part of time-domain simulation lies not in simulation of multiple wave histories, but rather in the propagation of each through the hydrodynamical and structural model that obtains
CHAPTER 3. DESIGN SEASTATE HISTORIES 35
forces and responses. If we could exclude wave histories if their observed wave char-acteristics differ too much from their theoretical ensemble average values and only
simulate response in the few remaining "design seastate histories", this would be a
promising way of reducing the uncertainty in the mean maximum response estimates
without need for an increase in the number of response simulations. Note that we
have studied jackup response in Gaussian Waves. We do expect, however, that our findings will be applicable also for analyses of structural response in Non-Gaussian waves.
We first identify critical wave characteristics for extreme response prediction. In
quasi-static cases, the maximum wave crest height, is shown to best explain
extreme deck sway. For more flexible structures, we introduce a new wave
character-istic,SD,based on response spectral concepts from earthquake engineering (Newmark
and Rosenbluetb, 1971). Finally, we show how accurate mean maximum response estimates can require fewer time-domain analyses, provided design seastates are pre-selected to ensure that or SD is near its average value. With respect to standard Monte-Carlo simulation, these design seastates are found to achieve at least a 50% reduction in response CoV, and hence at least a fourfold savings in needed simulation cost. These critical characteristics lend insight not only into time-domain simulation but also into more fundamental questions of jackup behavior. They also suggest that at least in quasi-static cases, still simpler design wave methods based on '7m may
suffice. We illustrate and evaluate one design wave method, the "New Wave" model (Tromans et al, 1991).
CHAPTER 3. DESiGN SEASTATE HISTORIES 36
3.2
Introduction
We continue to study the maximum deck sway, Y, of the chosen jackup structure
over severe 1-hour seastates. In Section 2.4.1 we demonstrated how the uncertainty associated with estimating the true mean-max-response, my,,, by the sample aver-age, can be reflected through its unitless coefficient of variation (CoV), Vr-=
I
V ir
mar'
/(.577+3lnN)
This result for the CoV of maximum response in a single seastate, considers Y,, the maximum of Nr independent response peaks, each peak following Weibull
distribution with shape parameter ß. Vy,,. can be derived as the ratio between
Equations 2.34 and 2.33. For example, if we seek to use Equation 3.1 to describe the
maximum wave crest 77,,, in a 1-hour seastate, ß=2 (Rayleigh peaks) and N=T/T, or 3600/124=290 cycles. Equation 3.1 then yields V,=0.108, which agrees quite
well with the result Vy,,,=0.104 from 300 1-hour simulations.
The drawback in the time-domain simulation is the cost of estimating the re-sponse Y=g(X) in each seastate. Estimating the rere-sponse involves nonlinear
dy-namic analysis, often of a large finite element structural model. For multiple
simu-lations (Ng5 > 1), this is especially burdensome. In practi, N55 may be chosen to achieve a desirably small Vi in Equation 3.1. For example, if we seek
Vi-0.05 in estimating the maximum wave
17,
with Vy, = 0.108 we need N55=4-5simulations. Still more simulations are typically required for the response, because of nonlinear effects, as we discussed in Chapter 2. In particular, we concluded in (3.1)
CHAPTER 3 DESIGN SEASTATE HISTORIES 37
Chapter 2 that a quadratic nonlinearity leads to a halving of 3 (e.g., from Rayleigh to exponential peaks) and, hence, to a roughly fourfold increase in the number of
required simulations, N55.
Note that we avoid here cases where the simulation period T is less than the storm
duratione.g., 3-6 hoursover which the extreme response is sought. Vv'hile rather
common, this situation requires extrapolation to estimate extremes over a
longer-than-simulated history. This leads to more complicated estimates, i.e., by fitting either Weibull or Hermite models as we discussed in Chapter 2. Each of these assigns a general probabilistic model of all extremes, and hence may have its own bias and uncertainty.
3.3
Design Seastate Histories
Basic Monte-Carlo simulation seeks, to the degree feasible, to reflect the actual
stochastic nature of the wave and response processes. It is thus precisely as
vari-able as the natural phenomenon itself; e.g., theCoV, of maximum response is
fixed by the stochastic dynamics of the problem. Our only option, then, to reduce
the variability in the estimate of the mean-max-response,
-, in Equation 2.12 is
to increase the number of simulations, N55.To reduce this cost, we seek a strate' to create (one or a few) "design" seastates.
As with other variance-reduction techniques, the goal is to modify basic Monte-Carlo simulation to reduce the response variability from each simulation, while
retain-ing the same mean response my, (Le., avoiding bias). This modification promises
CHAPTER 3. DESIGN SEASTATE HISTORIES 38
The general underlying premise is that if input wave characteristics are held near
their mean levels, the output should also lie near its mean, Tfly,,, with com-paratively little scatter. We will confirm the premise later in the chapter.
We thus seek a reduced set of wave statistics, 0, with two properties. First, we wish 0 to "explain" most of the variability of Y,,,, suggesting formally that the
response Y is of the form
Ymo.z = gi(8) ± (3.2)
in which , the error in replacing the full vector X from Equation 2.8 by the limited
statistics G. has much less scatter (standard deviation) than ':
SR =«1
(3.3)Beyond seeking a low value of SR (standard deviation reduction ratio), we also
re-quire that the mean response, is well estimated by fixing the statistics
8 near their mean values:
E[e] O; E[YTW.X] E[g1(8)] g.(E[8J) (3.4)
Various choices of critical parameters have been proposed. For example, the
Newwave model (Tromans et al, 1991) uses a deterministic response model based
on only the maximum wave crest;
Y,_-gi(i,); ¿=O
(3.5)cHAPTER 3. DESIGN SEASTATE HISTOPJES 39
is described further in Section 3.7.
Less drastically, simulated histories may simply be excluded if their observed
statistics differ too greatly from theoretical ensemble average values. This has been
suggested for jackupsscreening with respect both to and to moments a3 and a4and performing the relatively expensive response analysis with the remaining
subset of "qualified" wave histories (SNAME, 1993).
Note also the "Designer Wave" proposal (Morrison and Leonard, 1993), which analyzes many hours of wave data to find (1) the true average global response (dy-narnic base shear, overturning moment, etc.) and (2) the 20-second wave segments during which the response takes on this average value. The latter is used in the more
expensive, detailed response.analyses. We seek here to avoid such lengthy screening analyses and instead identify representative wave historiesusually full 1-hour
seast-ates but also shorter design wave profilesfrom wave characteristics that are more
easily calculated. We thus undertake here a detailed study of which wave character-istics are most informative in describing extreme jackup response.
3.4
Jackup Model
We consider the same structural model as we studied in Chapter 2. The structure is exposed to the same climate and force model as in the previous chapter. Note,
however, that the wave kinematics are estimated, and integrated, to the exact surface by Wheeler stretching, while a constant stretching was used in Chapter 2. We continue to use here the absolute (and not relative) fluid velocity, u(x, z). A damping ratio of
CHAPTER 3. DESIGN SEASTATE HISTOPJES 40
Table 3.1: Hourly Mean Max Deck Displacements for Relative and Absolute Fluid
Velocities
we find quite similar resultsand similar benefits of design
seastatesif we instead
use Morison models based on relative velocity formulations with (=0.01 structural damping. Table 3.1 shows representative comparisons.
3.5
Numerical Results
3.5.1
Wave Statistics.
We have simulated 300 individual hours of waves and deck sway for the jackup. From
each hour we select the maximum deck sway response and a set of wave characteristics
9.
Consider first a relatively stiff jackup, with period T=2s to reflect shallow-water conditions. Figures 3.1 and 3.2 show how well various wave characteristics, 9, (one by one) are correlated with the response Y in this quasi-static case. The
charac-teristics, 9, span out the x-axis and the responses, Y,, span out the y-axis. We
search for characteristics whose pairs, (O, Y,,), fall as close as possible to the perfect Table 3.1: Hourly Mean Max Deck Displacements {m]
for Relative and Absolute Fluid Velocities in Morisons Equation for Structural Periods 2-8s. H, = 15m, TD = 16s
Natural Period T AbsouteMori son
