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On the estimation of design wave forces Ganapathy, Sumathy 1997

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ON THE ESTIMATION OF DESIGN WAVE FORCES by  SUMATHY GANAPATHY  B.E., South Gujarat University, India, 1988 M.Tech., Mangalore University, India, 1992  A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF T H E REQUIREMENTS FOR THE D E G R E E OF M A S T E R OF APPLIED SCIENCE  in  T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF CIVIL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A April, 1997 © Sumathy Ganapathy, 1997  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purpose may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Civil Engineering The University of British Columbia 2324 Main Mall Vancouver, B.C., V6T 1Z4 Canada  Abstract The determination of wave-induced loads on ocean structures is a key aspect in the design of such structures. In the present study, five alternative approaches to estimating the load levels associated with a specified return period are described. The alternative approaches are based on different treatments of available wave data, rather than on new force formulations.  The approaches include the estimation of a maximum wave height directly from the longterm distribution of significant wave heights (Method A); from the long-term distribution of individual wave heights (Method B); from the joint distribution of heights and periods (Method C); from a first order reliability method applied to estimating the maximum wave height (Method D); and from a first order reliability method applied to estimating the maximum wave force directly (Method E). These approaches to estimating the maximum wave force are carried out for small and large vertical circular cylinders. The analyses are also carried out to indicate the influence of wave period and the type of analytical distribution of significant wave height that is adopted.  The results show that the most conventional approach, Method A , predicts relatively low values of the maximum wave force for both the slender and large cylinder.  The  estimation by Method C is closer to Method A . The force values by other methods are significantly higher for the large cylinder than the slender cylinder. distribution for H  s  The Type HI  gives lower force values than Type I distribution. The force values  are higher when T = 4.43 -Jn  s  than T is constant. In general, the results indicate that For  both slender and large cylinder, the estimation of wave force by the Methods B , D and E might be on the conservative side, while by the Methods A and C may be less conservative.  ii  Table of Contents Abstract  '•  Table of Contents  ii iii  List of Tables  v  List of Figures  vi  List of Symbols  viii  Acknowledgments  x  Chapter 1 Introduction  1  1.1 General  1  1.2 Objectives  2  Chapter 2 Theoretical Background  3  2.1 Wave Force Formulation - Slender Cylinder  3  2.2 Wave Force Formulation - Large Cylinder  6  2.3 Exceedence Probability and Return Period  7  2.4 Wave Height Estimation  8  2.4.1 Description of Wave Data  8  2.4.2 Long-term Distribution of Significant Wave Height  10  2.4.3 Short-term Distribution of Individual Wave Height  12  2.4.4 Long-term Distribution of Individual Wave Height  13  2.4.4.1 Analytical Distributions of Significant Wave Height  13  2.4.4.2 Scatter Diagram  16  2.5 Reliability Method  18 iii  2.5.1 Description of the Method  18  2.5.2 Calculation of the Reliability Index using the Program R E L A N  20  2.6 Application of Reliability Method to Wave Height Estimation  20  2.7 Application of Reliability Method to Wave Force Estimation  21  2.8 Alternative Approaches to Design Force Estimation  23  Chapter 3 Results and Discussion  28  3.1 Problem Definition  28  3.2 Slender Cylinder  29  3.3 Large Cylinder  30  Chapter 4 Conclusions  32  References  34  i  iv  List of Tables Table 3.1.  Parameters a and b for Type I and Type LTJ H  distribution for the  S  slender cylinder Table 3.2.  36  Parameters a and b for Type I and Type m H  S  distributions for the  large cylinder Table 3.3.  36  Wave force on slender cylinder (Type I H  distribution and T = 8.5  s  sec) Table 3.4.  37  Wave force on slender cylinder (Type I H  s  distribution and  T = 4.43 VH7) Table 3.5.  38  Wave force on slender cylinder (Type HI H  s  distribution and T= 8.5  sec) Table 3.6.  39  Wave force on slender cylinder (Type HI H  s  distribution and  T = 4.43^/^7) Table 3.7.  40  Wave force on large cylinder (Type I H  distribution and T = 12  s  sec) Table 3.8.  Wave force on large cylinder (Type I T  Table 3.9.  41 H  s  distribution  and  = 4.43 7HT)  42  Wave force on large cylinder (Type HJ H  s  distribution and T = 12  sec)  43  Table 3.10. Wave force on large cylinder (Type III H T = 4.43  )  s  distribution and 44  v  List of Figures Fig. 2.1. Definition sketch  45  Fig. 2.2. Sketch of scatter diagram indicating calculation of numbers of waves from an entry wy and njj  46  Fig. 2.3. Geometric representation of F O R M / S O R M reliability calculation  47  Fig. 3.1. Wave scatter diagram (large cylinder)  48  Fig. 3.2. Wave scatter diagram (slender cylinder)  48  Fig. 3.3. Alternate predictions of wave force on the slender cylinder for a Type I H  s  distribution, (a) T = 8.5 sec. (b) T =4.43^/57  49  Fig. 3.4. Alternate predictions of wave force on the slender cylinder for a Type m H distribution, (a) T = 8.5 sec. (b) T =4.43 ^/rLJ s  50  Fig. 3.5. Effect of wave period assumption on force estimates on the slender cylinder (Type I H  distribution), (a) Method A . (b) Method B . (c)  s  Method C. (d) Method D. (e) Method E  51  Fig. 3.6. Effect of wave period assumption on force estimates on the slender cylinder (Type III H  distribution), (a) Method A . (b) Method B . (c)  s  Method C. (d) Method D. (e) Method E Fig. 3.7. Effect of Type of H  s  distribution on force estimates on the slender  cylinder (T = 8.5 sec), (a) Method A . (b) Method B . (c) Method C.  vi  52  (d) Method D. (e) Method E Fig. 3.8.  Effect of Type of H  53  distribution on force estimates on the slender  s  cylinder (T =4.43 7^7)•  (a) Method A. (b) Method B . (c) Method C.  (d) Method D. (e) Method E Fig. 3.9.  54  Alternate predictions of wave force on the large cylinder for a Type I H  S  distribution, (a) T = 12 sec. (b) T=4.43VH7  55  Fig. 3.10. Alternate predictions of wave force on the large cylinder for a Type m H  S  distribution, (a) T = 12 sec. (b) T =4.43 - /H7 N  56  Fig. 3.11. Effect of wave period assumption on force estimates on the large cylinder (Type I H  distribution), (a) Method A . (b) Method B . (c)  s  Method C. (d) Method D. (e) Method E  57  Fig. 3.12. Effect of wave period assumption on force estimates on the large cylinder (Type HI H distribution), (a) Method A . (b) Method B . (c) s  Method C. (d) Method D. (e) Method E Fig. 3.13. Effect of Type of H cylinder  distribution  s  58  on force estimates on the large  (T = 12 sec), (a) Method A. (b) Method B. (c) Method C.  (d) Method D. (e) Method E Fig. 3.14. Effect of Type of H  s  cylinder (T =4.43 JK^).  59  distribution on force estimates on the large (a) Method A . (b) Method B . (c) Method  C. (d) Method D. (e) Method E  vii  60  List of Symbols a  acceleration (m/sec ), 2  cylinder radius (m), constant o f Q ( H ) distribution S  b  constant of Q(H ) distribution  Cd  drag coefficient  C  inertia coefficient  S  m  d  water depth (m)  D  cylinder diameter (m)  f  average number of waves per unit time  F  m  maximum wave force (N)  F  D  drag force (N)  Fi  inertia force (N)  g  gravitational constant (m/sec )  G(x)  performance function  H  individual wave height (m)  2  H H  S  M  significant wave height (m) maximum wave height (m)  Ji  Bessel function of first kind and order 1  k  wave number (m" )  L  wave length (m)  1  viii  n  number of data points, number of random variables in performance function G(x)  N  total number of waves per year  P  cumulative probability  Pf  probability of failure  P  Rayleigh distribution cumulative probability  r  Q  exceedence probability  r  recording interval (hours)  r'  1/ f', recording interval for individual waves  t  time  T  wave period (sec)  TR  return period (years)  u  horizontal water particle velocity (m/sec)  u  horizontal water particle acceleration (m/sec )  Yi  Bessel function of second kind and order 1  X  discrete random variable  P  reliability index  (j)  standard normal probability distribution function  T|  water surface elevation  p  water density (kg/m )  x  duration (hours)  CO  angular frequency (rad/sec)  2  3  ix  Acknowledgments It is with great pleasure and gratitude that the author acknowledge those whose assistance contributed immensely in developing this thesis. The author would like to thank her supervisor Dr. Michael Isaacson for his guidance and encouragement throughout the preparation of this thesis. The author would like to thank her loving parents Mr. R. Ganapathy Iyer and Ms. G. Jaya Lakshmi and her brothers Mr. G. Venkat Raman and Mr. G. Kalyana Raman who were constant source of encouragement throughout her studies. The author would also like to thank her colleagues and friends for their help and support.  Financial support in the form of a Research Assistantship from the Department of Civil Engineering, University of British Columbia is gratefully acknowledged.  x  Chapter 1 Introduction  1.1 General The estimation of wave-induced loads on ocean structures is a key consideration in the design of such structures.  Due to the complex nature of the wave environment,  probabilistic methods are generally utilized in the estimation of such loads. Design load levels are associated with long return periods corresponding to acceptably low exceedence probabilities, and a number of alternative approaches have been adopted to estimating the load levels associated with any specified return period or exceedence probability.  The most commonly used structural member in ocean structures is a circular cylinder. Thus small diameter circular cylinders form a fundamental structural element found in many offshore structures in the form of support legs and braces, submarine risers, cables, and pipelines. Large diameter circular cylinders may approximate a monolith structure, corresponding, for example, to a storage tank, a berth or an oil production platform.  For design purposes, wave forces on circular cylinders are estimated using two conventional approaches. These correspond to the use of the Morison equation (1950) for the case of slender cylinders, and the MacCamy and Fuchs diffraction equation (1954) for 1  Chapter 1 Introduction the case of large cylinders (see, for example, Sarpkaya and Isaacson, 1981).  A risk analysis can be used to associate a specified return period with a load level. Alternative approaches to design so are available. Recently, a reliability analysis has proven to be an effective approach in estimating design load levels for offshore structures, especially when the events associated with the load are uncertain and when the influence of several loads acting simultaneously requires consideration.  1.2 Objectives The primary objective of this thesis is to compare alternative approaches to estimating maximum wave forces corresponding to a specified return period.  The alternative  approaches that have been considered are based on alternative treatments of available wave data, rather than in new force formulations, and thus the relatively fundamental cases of slender and large vertical cylinders have been adopted here. Thus, comparisons of the alternative treatments are carried out for a slender vertical cylinder on the basis of the Morison equation (1950), and for a large vertical cylinder on the basis of linear diffraction theory using the MacCamy and Fuchs (1954) closed-form solution. These comparisons are carried out on the basis of alternative assumptions relating to wave period and to the form of analytical distribution of the significant wave height.  2  Chapter 2 Theoretical Background  2.1 Wave Force Formulation - Slender Cylinder The wave force on a slender vertical cylinder is obtained from the Morison equation, first proposed by Morison et al. (1950).  This approach assumes that the structure is  sufficiently small so as not to disturb the incident wave field. Thus, it may be applied to conditions corresponding to D « L, where D is the cylinder diameter and L is the wave length.  A definition sketch is given in Fig. 2.1. The Morison equation expresses the horizontal force dF acting on a strip of height ds of a vertical circular cylinder as the sum of two force components, a drag force and an inertia force, and is given by:  1  . .  TZD  2  dF = - p D C u|u| ds + p — — C u d s d  m  (2.1)  Here, p is the mass density of water, u is the instantaneous horizontal velocity of the water, i i is the instantaneous horizontal acceleration of the water, D is the diameter of the section, Cd is the drag coefficient, and C  3  m  is the inertia or mass coefficient.  Chapter 2 Theoretical Background The use of the Morison equation requires assumed values of the two hydrodynamic coefficients Ca and C . These coefficients have generally been estimated on the basis of m  laboratory and field data. In the present analysis, they are assumed known and constant.  When the Morison equation is used in conjunction with linear wave theory, the horizontal particle velocity u and the acceleration ii based on linear wave theory may be applied to Eq. 2.1. The corresponding expressions for u, and u are given respectively as:  7i H cosh(ks) u= — . cos(kx-cot) T sinh(kd)  (2.2)  2 7 t H cosh(ks) 2  *  T  =  2  sinhtkd)  8 1  ^-^  (  2  3  )  where H is the wave height, T is the wave period, k = 27C./L is the wave number, L is the wave length, co = 2TC/T is the wave angular frequency, d is the water depth, t is time, x is the horizontal coordinate in the direction of wave propagation, z is the vertical coordinate measured upwards from the still water level, and s = z + d, is the vertical distance from seabed (see, Fig. 2.1). The wave frequency co and wave number k are related by the linear dispersion relation:  co =gktanh(kd)  (2.4)  2  4  Chapter 2 Theoretical Background In the case of a vertical circular cylinder extending from the seabed to the free surface, the total force on the cylinder may be obtained by a suitable integration of the sectional force over the length of the cylinder. Applying the linear wave theory expressions for u and u to Eq. 2.1, taking the cylinder axis to coincide with x = 0, and carrying out the corresponding integration, the total force F may eventually be expressed as:  F=Acos(cot) |cos(cot)| - Bsin(cot)  (2.5)  where A=—pgH DC 2  d  1+  2kd sinh(2kd)  B= |pgHD C tanh(kd)  (2.6)  (2.7)  2  m  Equation 2.5 describes the time variation of the force over each wave cycle. The maximum force F is given as: m  J3_  A +4A  for2A>B  B  for2A<B  (2.8)  5  Chapter 2 Theoretical Background  2.2 Wave Force Formulation - Large Cylinder When the structure diameter is no longer small relative to the wave length, generally taken as D/L > 0.2, the wave train is scattered or diffracted by the structure, and one of the assumptions on which the Morison equation is based is then no longer valid. On the other hand, for a large structure, flow separation effects are generally unimportant so that the resulting wave field may be described on the basis of a potential flow and so described by linear wave diffraction theory.  The solution to the linear diffraction problem applied to a large vertical cylinder extending from the seabed to the water surface was given by MacCamy and Fuchs (1954). On this basis, the total wave force on a large cylinder of radius a is given by:  _ . , A(ka) tanh(kd) F = 2pgHad — j ^ - ^ cos(cot - 8) T T  (2.9)  where  A(ka) =  1 V i ' (ka) + Y / (ka) J  2  2  8 = -tan- [Y '(ka)/J '(ka)] 1  1  1  Here, Ji(ka) and Yi(ka) are the Bessel function of the first and second kinds respectively of order 1 and argument ka, and a prime denotes a derivative with respect to argument.  6  Chapter 2 Theoretical Background The maximum force is then given as:  A(ka) tanh(kd)  F  m  . 2 p g H a d - ^ i ^ ^  (2.10)  2.3 Exceedence Probability and Return Period Attention is now focused on the long-term variations of the wave height and force. Consideration is given in turn to the significant wave height, H , defined as the average s  height of the highest one-third waves in a sea state; the individual wave height H ; and the force amplitude F corresponding to H . Initially, though, the following summary relates return period to the probability distribution of a discrete random variable X , where, in the present context, X may refer to the significant wave height H , the individual wave s  height H, or the wave force amplitude F.  The random variable X is assumed to possess a known probability distribution, described by the cumulative probability P(X). For example, in case of H , P(X) may generally be s  estimated on the basis of a series of H  s  values corresponding to a specified recording  interval r, which is defined as the time interval (averaged if appropriate) between successive data points. In such cases, it is assumed that successive data points are independent.  [This assumption may be questionable when r is unduly short such that  there is more than one data point in any one storm (e.g. Battjes, 1970, and Nolte, 1973)].  7  Chapter 2 Theoretical Background The return period TR is defined as the average duration between successive occurrences of the argument X being exceeded. If this occurs on average once every n data points, then the corresponding return period is T = nr, while the exceedence probability of X is R  approximately Q(X) ~ 1/n. Thus, the return period is related to the exceedence Q(X) by:  T (X) R  1  2.4 Wave Height Estimation The calculation of the wave force with a specified return period requires the estimation of a design wave height with a specified return period from recorded or hindcast wave data. The present section describes the various forms of wave data that may be available, and then considers in turn the long-term distribution of the significant wave height, the shortterm distribution of the individual wave height, and the long-term distribution of the individual wave height.  2.4.1 Description of Wave Data Wave data for a probability analysis may derive from three different sources: predictions from meteorological observations; ship-based visual observations; and instrumental observations.  Each of these data sources has its own particular limitations.  Data  collected from instrumental observations, such as from wave-rider buoys or ship-borne wave recorders, generally extend over a relatively short duration of up to a few years. 8  Chapter 2 Theoretical Background Hence, the prediction of wave conditions corresponding to long return periods may suffer from a large amount of uncertainty.  Visual observations of wave heights gathered by ships of opportunity are irregularly spaced in both space and time. They may suffer from a fair weather bias, since a ship may deviate its path to avoid severe sea states, and therefore may contain quite large errors. Also, wave data over very large areas are grouped together, resulting in a limited application of data to a specified area.  However, they do provide some valuable  information about the general nature of wave climate (Hogben and Lumb, 1967).  The other major source of wave data arrives from predictions based on wave hindcasting. Here, the required wave parameters are forecast or hindcast from available wind data. Of particular concern in assessing the utility of hindcast studies is the size of the grid used, the resolution of adjacent shorelines, and the time step size used.  Wave height  predictions based on careful hindcast studies can agree to within 1 m with those based on buoy measurements. Hence there is no significant difference in accuracy between the two methods.  In the present study, wave data available in the form of a scatter diagram obtained from three-hourly observations for the Seven Stones station in the United Kingdom, reproduced by Burrows and Salih (1986), is utilized.  9  Chapter 2 Theoretical Background 2.4.2 Long-term Distribution of Significant Wave Height The accurate knowledge of the long-term wave height corresponding to a specified return period is an essential prerequisite for the structural design of any coastal or offshore installation. In such studies, we are interested in the characteristics of the extreme values which are distantly located from the mean. Descriptions of the estimation of extreme waves include those given by Sarpkaya and Isaacson (1981), Isaacson and MacKenzie (1981), Muir and El-Shaarwai (1986), and Gran (1991).  The expected largest wave  height with a specified return period can be obtained from the long-term distribution of significant wave heights. This involves selecting and fitting a suitable probability distribution to significant wave height data, and extrapolating this to locate a suitable design value. This is generally carried out in the following stages;  •  Data consisting of wave heights and periods are collected over a long time (e.g., a few years) at the site of interest. Alternatively, a hindcasting technique may be used to provide wave height data over a much longer time span (say 50 years or more).  •  A plotting formula is used to reduce the data to a set of points describing the longterm distribution of wave heights.  •  These points are plotted on an extreme value probability paper corresponding to a chosen probability distribution function.  •  A straight line or a smooth curve is fitted through the data points.  •  The line is then extrapolated to locate a design sea state ( H ) corresponding to a s  chosen return period TR, or a chosen encounter probability, Q( H ) . s  10  Chapter 2 Theoretical Background •  Once this design sea state is known, an estimate is then made of the maximum individual wave height within this sea state.  Several probability distributions have been proposed to describe the long-term distribution of significant wave heights. These include the log normal distribution, and the Extreme Types I, II and HI distributions introduced by Fisher and Tippett (1928). The Extreme Types I, II and m are also termed the Gumbel, Fretchet and Weibull distributions respectively. The main criterion in choosing a distribution is the goodness of fit to the available data.  Having selected one distribution as a likely model, it remains to estimate the parameter values which will provide the best empirical fit between the distribution and data. The best fit may be derived either by (i) the method of moments (ii) the method of least squares, or (iii) the method of maximum likelihood (e.g. Isaacson and MacKenzie, 1981).  The Extreme Type I (Gumbel distribution) and the two parameter lower-bound Extreme Type HI distribution (Weibull distribution) are given respectively (e.g. Sarpkaya and Isaacson, 1981) as:  Q ( H ) = 1 - exp{-exp[-(aH +b)]} S  s  Q(H ) = exp(-aH )  (2.13)  b  s  (2.12)  s  11  Chapter 2 Theoretical Background where a and b are constants of the distributions. The constants a and b may be related to characteristic values of each distribution, such as the mean and standard deviation, or the slope and intercept of straight line plots of the distributions.  2.4.3 Short-term Distribution of Individual Wave Height The estimation of the maximum individual wave height within a sea state of a specified return period, during which the significant wave height is assumed constant, requires a knowledge of the short-term distribution of wave heights, and this aspect is now considered. There are many forms of distribution available to describe the short-term distribution of individual wave heights within a sea state, during which the water surface elevation is considered to form a stationary random process (e.g. Longuet-Higgins, 1952; Gluhovski, 1968; Ibrageemov, 1972; Goda, 1975; Foristall, 1978).  However, the  Rayleigh distribution (Longuet-Higgins, 1952) is most often used for this purpose because of its consistency with measured heights and because of its ease of application. The Rayleigh distribution P (H) may be expressed approximately in terms of H as: r  s  ^-2H ^ P (H) = 1 - exp V s J 2  forH>0  r  H  2  (2.14)  The expected or most probable value of maximum individual wave height, denoted H , m  occurring within a duration % may be derived from this distribution and is given approximately by the well-known result:  ,  12  Chapter 2 Theoretical Background  (2.15)  Where T here is the zero-crossing period of the sea state. In fact, the above equation does not vary strongly with x/T; and often H  m  ~ 1.8 - 2.0 H is adopted. s  2.4.4 Long-term Distribution of Individual Wave Height Alternative approaches to developing the long-term probability distribution of individual heights are now considered. 2.4.4.1 Analytical Distributions of Significant Wave Height The long-term distribution of individual wave heights may be developed by suitably weighting the short-term distribution of individual wave heights within a sea state, assumed to be given by the Rayleigh distribution with H taken as constant, by the longs  term distribution of H itself. Battjes (1970) emphasized that this weighting should take s  suitable account of the influence of wave period.  The corresponding expression  developed by Battjes is given by:  1  1  Q(H) = 7  J Q (HIH )p(H ,T)dH dT  J 0  o  0  T  r  s  s  (2.16)  s  (2.17) 0  13  Chapter 2 Theoretical Background where Q ( H I H ) r  and P ( H I H )  s  r  s  correspond respectively to the exceedence  and  cumulative probability of H based on the Rayleigh distribution for H , p ( H , T ) is the s  joint long-term probability density of H and T, and f is the long-term average number s  of waves per unit time, which is given as:  (2.18) o  o  The application of the above equations depends on the description of the long-term distribution of H and T that is available. s  In order to make the application of Eq. 2.16 reasonably tractable, it is convenient to adopt a simple assumption for the relationship between wave period and significant wave height. Two alternative assumptions relating to the wave period are considered here. The first assumption is that the wave period T is constant and independent of significant wave height and the second assumption is that T is fully correlated with significant wave height, being proportional to the square root of the significant wave height. The later assumption is equivalent to taking the wave steepness parameter  H /gT s  2  to be  independent of height. The later case with a proportionality factor of 4.43 (with T in sec and H in m) has been found to be suitable for conditions in Canadian Atlantic Waters (Neu, 1982).  14  Chapter 2 Theoretical Background Adopting the above two assumptions in turn, and using Eq. 2.14 for the Rayleigh distribution, Eqs. 2.16 and 2.18 give rise to the following expressions for Q(H) and f ;  T Constant ^-2H ^ 2  Q(H)= J p(H )exp s  V  H  s  dH  (2.19)  c  J  2  f' = T  (2.20)  ^-2H ^ 2  Q(H) = ^ U o 1  p(H )exp s  1  V  f = J-p(H )dH s  The analytical distributions for H  s  H  s  2  dH,  J  (2.21)  (2.22)  s  given by Eqs. 2.12 and 2.13, may be applied to Eqs.  2.19 and 2.21, and the resulting integrals are then evaluated numerically for any selected value of individual wave height H to provide the corresponding exceedence probability, Q(H). Once Q(H) is known, then Eq. 2.11 may be used to determine the return period the specified individual wave height, using a recording period r ' = 1/f'.  Alternatively, the  value of H corresponding to any selected value Q(H) may be obtained by iteration or interpolation.  15  Chapter 2 Theoretical Background This maximum individual wave height  with a certain exceedence  probability  corresponding to a return period could be significantly different from the expected maximum individual wave height in a sea state with the same return period, obtained as described in Section 2.4.2 (Isaacson and Foschi, 1996).  2.4.4.2 Scatter Diagram When the wave data is in the form of scatter diagram, as is considered in the present study, Q(H) can be obtained directly from a discretized form of Eq. 2.16. It is convenient to work with a scatter diagram with entries W y which are normalized to a time span of one year (by multiplying each original entry by the appropriate time span ratio). Figure 2.2 provides a sketch of a scatter diagram indicating the scatter diagram entries and corresponding numbers of individual waves. The long-term distribution of individual waves is then given by:  Q( ) = ^ S N e x p H  (- 2 H  i  V  H  Si  (2.23) J  where N is the total number of waves per year, and is given by:  N=IN  (2.24)  i  Ni is the number of waves per year for the i-th H range, and is given by: s  16  Chapter 2 Theoretical Background N =IN j i  (2.25)  i j  Njj is the number of waves per year for the i-th H range and j-th T range, and is given s  by:  Nij =  3600 rw:: >-  (2.26)  j  and r is the recording interval (in hours). Also, the long-term average number of waves per sec, f , which is required to calculate r ' and hence Q(H) corresponding to a specified return period T , is given as: R  N  f  =  (2.27) ^  365x24x3,600  ;  Q(H) may thereby be obtained numerically for any selected value of H . As before, Eq. 2.11 may then be used to determine the return period for the specified individual wave height; and, as before, the value of H corresponding to any selected value of Q(H) may be obtained by iteration or interpolation.  17  Chapter 2 Theoretical Background 2.5 Reliability Method An alternative approach based on the First or Second Order Reliability Methods (e.g. Madsen et al., 1988), denoted F O R M / S O R M and commonly used for predicting structural reliability, may also be used to estimate the long-term probability distribution of individual wave heights and force levels corresponding to specified probability exceedence levels.  2.5.1 Description of the Method In the reliability method, the performance of a system is formed in terms of basic variables influencing the system.  For the purpose of a generalized formulation, the  performance function is described mathematically as follows (Thoft-Christensen, and Baker, 1982);  G(x) = G(x,,x , 2  where x = (xi,X2, system.  ,XN) is  ,x„)  (2.28)  a n-dimensional vector of variables influencing the  Because most of these variables are random, they need to be described  statistically.  The probability of failure associated with an event or particular combination of variables, x, is given by the probability that the corresponding performance function G(x) is negative, i.e. G(x) < 0. The "failure surface" corresponding to G(x) = 0 is constructed in 18  Chapter 2 Theoretical Background n-dimensional space in which the n components of x are taken as axes. Thus, when n=2, the failure surface corresponds to a curve plotted on a plane. The value of vector x corresponding to the point P on the failure surface which is closest to the origin O is then found. The point P gives the most probable failure combination of the variables in the system and the distance OP gives the "reliability index" denoted (3, which is used to estimate the probability of the event G(x) < 0.  Figure 2.3 illustrates a geometric  representation of the F O R M / S O R M reliability index calculation for the use of two variables xi and X2. In F O R M , the estimate is made by assuming that the failure surface G(x) = 0 can be approximated as linear in the neighborhood of P. In order to improve accuracy, S O R M instead is based on the assumption that the surface G(x) = 0 can be approximated as a quadratic surface in the neighborhood of P.  Then, the probability of failure is given by;  Pf = <K-P)  (2-29)  where 0 is the standard normal probability distribution function.  The estimation of probability of failure using the F O R M / S O R M procedures will be exact if all the intervening variables are normally distributed and uncorrelated, and if they combine linearly in the performance function G(x). Generally the variables are not normal and uncorrelated, and the performance function is nonlinear.  The F O R M /  S O R M procedures then introduce an appropriate transformation to convert all variables 19  Chapter 2 Theoretical Background to normal and to eliminate correlation if present, and hence the approximation of the probability of failure, Pf, is influenced solely by the nonlinearity in G(x).  2.5.2 Calculation of the Reliability Index using the Program R E L A N The calculation of the reliability index P requires an iterative computer algorithm (Hasofer et al., 1974) which requires a description of the performance function and its first and second order derivatives with respect to the intervening random variables. R E L A N , which is a general F O R M / S O R M F O R T R A N program, developed by Foschi, Folz and Yao (1988) has been used here. In R E L A N , it is sufficient to describe the function G(x) for each specific problem, and first and second order derivatives are obtained numerically by the program itself. Also, the algorithm in R E L A N adjusts for the case where the random variables are not normally distributed, and also where the variables are correlated.  2.6 Application of Reliability Method to Wave Height Estimation Based on the above description, the wave height exceedence level may be estimated in the present context by selecting the performance function G(x) as follows:  G(x) = H - H ( x )  (2.30)  m  where H  m  is a specified wave height level whose exceedence probability Q(H) is  required, H(x) is the maximum individual wave height, and x denotes a set of specified 20  Chapter 2 Theoretical Background random variables which determine H. The probability of the event G(x) < 0 corresponds to the required probability Q(H) that the height level H is exceeded. m  The calculation of the maximum height H  m  needed in Eq. 2.30 requires a suitable  selection and specification of the random variables x, and a formulation of H in terms of these variables x.  In the present context, the vector x includes as components the  significant wave height, H , which has a specified probability distribution as already s  described; and a variable U which is uniformly distributed between 0 and 1 and used to account for the Rayleigh distribution for the individual wave height H  m  conditional on  H . On the basis of the Rayleigh distribution, U is defined by: s  1/2  H=H  S  jln(l-U)  Once the variables H  s  (2.31)  and U are defined statistically, for a known exceedence  probability, the maximum individual wave height H can be determined from Eq. 2.30 m  for G(x) and then using the program R E L A N .  2.6 Application of Reliability Method to Wave Force Estimation The reliability method may also be applied to estimating wave force corresponding to a certain exceedence level. Once more, this requires the formulation of an appropriate performance / failure function, in terms of the variables of the system.  21  Chapter 2 Theoretical Background In order to obtain a force exceedence level in the present context, the performance function G(x) is written as follows:  G(X) = F - F ( x )  (2.32)  m  Here, the F , is the applied load level and, F (x) is the maximum wave force amplitude m  which is a function of a specified set of random variables x.  In the case of a slender pile, the maximum wave force amplitude is given by Eq. 2.8, which is repeated here for convenience:  F =  j3_ A +4A  for2A>B  B  for2A<B  (2.33)  where  A=-pgH DC 2  K  d  1+  2kd sin h(2kd)  B= -pgHD C tanh(kd)  (2.34)  (2.35)  z  m  In the case of a large diameter cylinder, the maximum wave force amplitude is instead given by Eq. 2.10, which is repeated here for convenience:  22  Chapter 2 Theoretical Background ^ „ , A(ka) tanh(kd) F = 2 p g H „ a d - ^ ^ T T  (2.36)  1  While applying the reliability method for wave height and wave force estimation, the two assumptions with regard to the wave period T (T constant and T proportional to the square root of the significant wave height) are maintained by considering the random variables as wave height H  s  alone and as the wave height H  s  and the period T,  respectively. A l l other parameters are deterministic.  2.7 Alternative Approaches to Design Force Estimation From the preceding descriptions, five alternative approaches to design wave force estimation have been identified and are described below.  Method A This is the most common and the simplest to use. The significant wave height H , s  corresponding to specified return period T is obtained by fitting a suitable probability R  distribution to H data contained in a wave scatter diagram. The maximum wave height s  H  m  is then obtained from this value of H by the relation given in Eq. 2.15. This value s  of H may then be applied to an appropriate force formulation to obtain the maximum m  wave force corresponding to the specified return period.  23  Chapter 2 Theoretical Background Method B In the second approach, Method B , the maximum wave height H  m  corresponding to a  specified return period TR is obtained directly from the long-term distribution of individual wave heights, as described in Section 2.4.4. This height H is then applied to m  the force formulation (either the Morison equation for a slender cylinder, or the MacCamy and Fuchs solution for a large cylinder) to obtain the wave force.  Method C In Method C, the wave force corresponding to a specified return period is obtained from the joint distribution of the wave heights and wave periods, which is itself derived from the specified wave scatter diagram. This approach is outlined below.  The wave height and period intervals in the scatter diagram are denoted A H and AT respectively and, as indicated earlier, each entry in the scatter diagram normalized to one year as in Section 2.3.4.2 is denoted as W j j . The number of waves Ny per year corresponding to the i-th H range and j-th T range and corresponding to an entry w - is s  y  given by  3600 rw N  4j  =  s  (2.37)  1  24  Chapter 2 Theoretical Background For each such entry, the significant wave height H  s  and period T are taken as constant,  and the individual wave heights are assumed to follow the Rayleigh distribution so that the joint probability density p(H,T) is given as:  2H ^  f  2  p(H,T) = 2 - ^ - exp V  H  (2.38)  J  s  Therefore, the number of waves n  ijk  with heights within the range H ± AH/2 and periods k  within the range Tj + AT/2 and associated with the entry Ny is given approximately by:  njjk =  2H k H< Y expV  AH N , H  si  (2.39)  J  2  Considering the contributions from all entries with the same wave period range, the total number of waves n  jk  with heights H and periods Tj is obtained by summing all such k  contributions for all the H ranges in the scatter diagram. Thus: s  njk  _ ^  2  1  H  H  K  S i "  exp V  H  s;  2  j  AH N  (2.40)  ; :  Each such height and period combination gives rise to a particular force F ( H , Tj), which k  may be obtained by the Morison equation or by the MacCamy and Fuchs solution as appropriate. Therefore, the number of occurrences per year of this force level is nj . The k  25  Chapter 2 Theoretical Background force level, and associated number of occurrences njk can thus be obtained for all Tj and Hk values occurring. These may be ranked in decreasing order in order to obtain the number of occurrences of each force level occurring. This can be used to develop the exceedence probability Q(F).  This can in turn be used to obtain the force level  corresponding to any specified return period, on the basis of Eq. 2.11 with a recording interval given by r = 1/f', where f' is given by Eq. 2.18.  Method D Method D corresponds to the First Order Reliability Method (FORM) being applied to obtain the maximum individual wave height H  m  corresponding to a specified return  period. This height is in turn applied to the Morison equation or the MacCamy and Fuchs solution, as appropriate, to obtain the wave force corresponding to the specified return period.  Method E Method E corresponds to F O R M being applied directly to obtain the maximum force corresponding to a specified return period.  As described before, the performance  function for slender cylinders and large cylinders is developed using the Morison equation and the MacCamy and Fuchs diffraction solution respectively.  A l l the above methods require a suitable assumption relating to the wave period T in order to apply the appropriate force formulations. As described in Section 2.4.4.1, two alternative assumptions for the wave period are considered. It is assumed that either T is 26  Chapter 2 Theoretical Background constant or is proportional to ^ / H ^ • When T is constant, Method D and Method E will give the same force values.  27  Chapter 3 Results and Discussions  As mentioned in the Chapter 2, the present investigation describes a comparison of wave force estimates using five different methods. In this chapter, the results and interpretation of such estimates for both a slender cylinder and a large cylinder are presented.  3.1 Problem Definition The wave data used to develop the present results are taken from a scatter diagram obtained from the Seven Stones station in the United Kingdom, and reproduced by Burrows and Salih (1986). The scatter diagram, which is based on a recording interval r = 3 hours and a total N = 20,645 observations, is presented in Fig. 3.1. In the case of force estimates for a large diameter cylinder, the water depth d is taken as 80 m, and the structure diameter D as 100 m. In the case of force estimates for a slender pile, the water depth is taken as 10 m and the cylinder diameter is taken as 1 m. It is unrealistic to adopt the same wave scatter diagram in this case, and instead the scatter diagram used for large cylinder and given in Fig. 3.1 is scaled such that the wave heights are reduced by a factor of 2 and the wave periods are reduced by a factor of V2 . The resulting scatter diagram is given in Fig. 3.2. The drag and inertia coefficients, Cd and C , needed in the Morison m  equation are assumed to be 1.0 and 2.0 respectively.  28  Chapter 3 Results and Discussion The values of the parameters a and b for the Type I and Type HI distribution (see, Eqs. 2.12 and 2.13) determined in the present analyses, for the slender and large cylinder wave data are given are given in Table 3.1 and Table 3.2 respectively.  As indicated previously, the estimation of the maximum wave force for the large and slender cylinders is carried out using two assumptions relating to wave period and two types of analytical distribution for H . In one, T is taken as constant, and equal to 8.5 s  sec for the slender cylinder and 12 sec for the large cylinder, and in the other it is taken as T = 4.43  •  3.2. Slender Cylinder Results of the wave force estimation for the slender cylinder by the alternative methods for various return periods are given in Tables 3.3 - 3.6 and are plotted in Figs. 3.3 - 3.8. The results show that the wave force predictions by the conventional Method A are low compared to the other methods. It is apparent from the figures that Methods D and E predict larger forces, and the predictions by these methods agree close by with Method B for both assumptions of the wave period. The force prediction by the conventional approach, Method A , is the lowest, while the estimates by Method C are slightly higher than Method A.  Figures 3.5 and 3.6 indicate the effect of the wave period assumption on the wave force estimates for the Type I and Type i n H  s  distributions respectively. For the Type I  29  Chapter 3 Results and Discussion distribution, the wave period does not have a significant influence in the case of Methods A B and C. For the other methods, the estimates are higher when using T = 4.43 ^ / H  S  .  For the Type HI distribution, except for Method A , the estimates are slightly higher when T = 4.43  •  Figures 3.7 and 3.8 indicate the effect of the type of distribution used to describe H on S  the force estimates, for the two wave period assumptions that have been made.  It is  apparent from the figures that in both cases, all the methods estimate the force to be lower for the Type HI distribution than for the Type I distribution. This difference is significant for all the methods except Method A .  3.3 Large Cylinder Results of the wave force estimation for the large cylinder by the alternative methods are given in Tables 3.7 - 3.10 and are plotted in Figs. 3.9 - 3.14. The results generally show that the wave forces estimates by Method C are lower than other methods. Figures 3.9 and 3.10 show the wave forces as function of return period for the Type I and Type HI distributions, and for the two wave period assumptions that have been made. For the Type I H distribution, Method C predicts the lowest force for both wave period S  assumptions.  The force estimate is higher by Method D and Method E for the  assumptions T constant and T = 4.43 ^ / H J respectively. For the Type HI distribution and with T constant, there is not much difference in the force estimates by Methods B , D and  30  Chapter 3 Results and Discussion E. Similarly the force estimates by Methods A and C are almost the same, but are much lower than the other methods.  Figures 3.11 and 3.12 indicate the effect of wave period assumptions on the force estimates for the Types I and HI distributions. For the Type I distribution, the wave period does not much influence the force predicted by Method C, whereas for the other methods, the force values are significantly higher when T = 4.43 -JH^ . For the Type HI distribution, the force values are significantly higher for all the methods when T = 4.43 • /H7 • N  Finally, Figs. 3.13 and 3.14 indicate the effect of the type of H  s  distribution for both  wave period assumptions. It is apparent from the figures that all the methods estimate the forces to be less for the Type HI distribution than for the Type I distribution, and this difference is more significant for Methods A , B, D and E.  31  Chapter 4 Conclusions  This thesis describes the estimation of wave forces on a slender and large cylinder with a specified return period, under different treatments of wave data which is specified in the form of a scatter diagram. Analyses have been carried out to investigate the effects of the wave period assumption that is made, and of the type of the significant wave height probability distribution on the force values.  Slender cylinder The results for a slender cylinder indicate that the force estimation by conventional Method A is the smallest when compared to other methods, and that the predictions by Method C are close to Method A. The wave period assumption does not have much of an influence on force values predicted by Methods A , B and C. For the other methods the force is higher when T = 4.43 ^ / H  s  than when it is constant. The force values are lower  when H is described by the Type IU distribution than by the Type I distribution, and this s  difference being more significant for Methods C, D and E.  32  Chapter 4 Conclusions Large cylinder The results for a large cylinder indicate that the force values predicted by Method C, are the lowest and that the Method A predictions are close to those of Method C. The wave period assumption does not have much of an influence on Method C, while for the other methods, the force values are significantly higher when T = 4.43 ^ / H constant. The form of the H  s  s  than when T is  distribution does not have much of an influence on  Method C, and the force values are significantly higher for all the other methods for the Type I distribution than for the Type III distribution.  Differences in the force values by the alternative methods are higher for  the large  cylinder than for the slender cylinder.  For both slender and large cylinder, the estimation of wave force by the Methods B , D and E might be on the conservative side, while by the Methods A and C may be less conservative.  33  References Battjes, J.A., (1970). "Long-Term Distributions at Seven Stations Around the British Isles," National Institute of Oceanography, Godalming, England, Report No. A-44. Burrows, R., and Salih, B . A . (1986). "Statistical Modeling of Long-term Wave Climates," Proceedings, 20-th International Coastal Engineering Conference, Taipei, Taiwan, ASCE, I, pp. 45-56. Fisher, R.A., and Tippett, L . C . H . (1928). "Limiting Forms of the Frequency Distribution of the Largest and Smallest Member of a Sample," Proceedings, Cambridge Philosophical Society, 24, pp. 180-190. Forristal, G.Z., (1978). "On the Statistical Distribution of Wave Heights in a Storm," Journal of Geophysical Research, 83, pp. 2353-2358. Foschi, R.O., Folz, B., and Yao, F. (1988). " R E L A N : User's Manual," Report, Department of Civil Engineering, University of British Columbia, Vancouver, Canada. Glushovski, B . (1968). "Distribution Characteristics of Wave Parameters and Changes in Wave Action with Depth," Report, Saint Institute of Oceanography, Moscow, 931, pp. 98-111.  Goda, Y . (1975). "Irregular Wave Transformation in the Surf Zone," Coastal Engineering in Japan, 18, pp. 13-26. Gran, S. (1991). "Long-term Distributions," Chapter 4.5 of D N V Research Report No. 91-2036, Ocean Engineering, Det Norske Veritas, Hovik, Norway. Hasofer, A . M . , and Lind, N . (1974). " A n Exact and Invariant First Order Reliability Format," Journal of Engineering Mechanics. ASCE, 100, pp. 111-121.  Hogben, N . , and Lumb, F.E. (1967). "Ocean Wave Statistics," Her Majesty's Stationary Office, London, England, pp. 1-7. Ibrageemov, A . M . (1972). "Investigations of the Distribution Functions of Wave Parameters During their Transformation," Oceanology, 13(4), pp. 584-589. Isaacson, M . , and Foschi, R.O. (1996). "On the Return Period of Design Wave Heights," Proceedings, International Conference in Ocean Engineering, Madras, India, pp. 491-495. 34  References Isaacson, M . , and MacKenzie, N . (1981). "Long-term Distribution of Ocean Waves - A Review," Journal of Waterway, Port, Coastal and Ocean Engineering, A S C E , 107, pp. 93-109. Longuet-Higgins, M.S. (1952). "On the Statistical Distribution of the Heights of Sea Waves," Journal of Marine Research, 11(3), pp. 245-266. MacCamy, R.C., and Fuchs, R.A. (1954). "Wave Forces on Piles - A Diffraction Theory," U.S Army Corps of Engineers, Beach Erosion Board, Technical Memorandum No. 69. Madsen, H.O., Krenk, S., and Lind, N . C . (1986). "Methods of Structural Safety," Prentice-Hall Inc., Englewood Cliffs, New Jersey. Muir, L.R., and El-Shaarwai, A . H . (1986). "On the Calculation of Extreme Wave Heights - A Review," Ocean Engineering, 13(1), pp. 93-118. Neu, H.J.A. (1982). "11 year Deep Water Wave Climate of Canada Atlantic Waters," Canadian Technical Report of Hydrography and Ocean Sciences, No. 13, pp. 48. Nolte, K . G . (1973). "Statistical Methods for Determining Extreme Sea States," Proceedings, 2-nd International Conference on Port and Ocean Engineering under Arctic Conditions, University of Iceland, Helsinki, pp. 705-742. Sarpkaya, T., and Isaacson, M . (1981). "Mechanics of Wave Forces on Offshore Structures," Van Nostrand Reinhold, New York. Thoft-Christensen, P., and Baker, M . J . (1982). "Structural Reliability Theory and its Applications," Springer-Verlag, New York.  35  Table 3.1. Parameters a and b for Type I and Type III H distribution for the slender s  cylinder.  H distribution  Slender cylinder, d = 10 m  s  a  b  Mean  Std. deviation  Type I  2.542  -4.187  1.874  0.504  Type HI  0.330  2.086  1.505  0.756  Table 3.2. Parameters a and b for Type I and Type III H distribution for the large s  cylinder.  H distribution  Large cylinder, d = 80m  s  a  b  Mean  Std. deviation  Type I  1.1051  -2.815  3.069  1.160  Typem  0.077  3.399  2.958  1.487  36  Table 3.3. Wave force on slender cylinder (Type I H distribution and T = 8.5 sec). s  T (years)  F (kN)  R  m  Method A  Method B  Method C  Method D  Method E  1  92  118  152  121  121  10  122  159  171  164  164  20  132  174  177  179  179  50  145  194  184  199  199  100  155  210  190  216  216  200  165  228  195  234  234  '  37  Table 3.4. Wave force on slender cylinder (Type I H distribution and T = 4.43 ^/ETJ )• s  T (years)  F (kN)  R  m  Method A  Method B  Method C  Method D  Method E  1  93  118  155  131  133  10  124  162  174  180  182  20  134  177  179  197  199  50  148  199  187  220  223  100  158  216  192  240  242  200  169  234  198  260  262  38  Table 3.5. Wave force on slender cylinder (Type i n H distribution and T = 8.5 sec). s  T (years)  F (kN)  R  m  Method A  Method B  Method C  Method D  Method E  1  86  103  91  107  107  10  107  134  110  136  136  20  114  144  115  146  146  50  122  157  123  159  159  100  129  168  128  169  169  200  135  179  134  179  179  39  Table 3.6. Wave force on slender cylinder (Type i n H distribution and T = 4.43 ^ / H J ). S  T (years)  F (kN)  R  m  Method A  Method B  Method C  Method D  Method E  1  86  113  96  117  117  10  108  148  114  150  150  20  115  159  120  161  160  50  124  174  127  176  174  100  131  186  133  187  186  200  138  198  139  199  198  40  Table 3.7. Wave force on large cylinder (Type I H distribution and T = 12 sec). s  T (years)  F (MN)  R  m  Method A  Method B  Method C  Method D  Method E  1  614  713  562  739  739  10  716  845  656  878  878  20  745  886  685  921  921  50  783  942  722  979  979  100  811  985  751  1024  1024  200  838  1029  779  1069  1069  41  Table 3.8. Wave force on large cylinder (Type I H distribution and T = 4.43 ^/PLJ ). s  T (years)  F (MN)  R  m  Method A  Method B  Method C  Method D  Method E  1  765  912  582  798  1168  10  1011  1238  677  1117  1433  20  1073  1326  705  1216  1515  50  1153  1446  742  1352  1633  100  1210  1535  771  1451  1754  200  1266  1626  799  1552  1870  42  Table 3.9. Wave force on large cylinder (Type III H distribution and T = 12 sec). s  T (years)  F (MN)  R  m  Method A  Method B  Method C  Method D  Method E  1  591  683  562  670  670  10  668  776  656  763  763  20  689  803  685  791  791  50  716  840  722  828  828  100  736  865  751  855  855  200  755  895  779  883  883  43  Table 3.10. Wave force on large cylinder (Type III H distribution and T = 4.43 • / H 7 )• S  T (years)  N  F (MN)  R  m  Method A  Method B  Method C  Method D  Method E  1  734  825  586  868  931  10  824  940  680  983  1096  20  951  1092  709  1141  1146  50  1009  1168  746  1219  1212  100  1051  1226  775  1277  1263  200  1094  1285  803  1337  1370  44  r—  D  s=z+d \ \ \ \ \ ' \ ' \ \ \ \ \ \  Fig. 2.1. Definition sketch.  45  T  \ \ \ \ \ \ \  N  i(i  waves with individual height H_ due to H ., T N waves per year: N,  N = SN.  H  H  N'  Fig. 2.2. Sketch of scatter diagram indicating the calculation of number of waves from an entry w . and N...  46  N.= Z N  Fig. 2.3. Geometric representation of F O R M / S O R M reliability calculation.  47  Hs(m) 10-11  1  9-10 8-9  1  3  4  1  1  8  15  5  2  7-8  2  18  45  36  14  5  1  6-7  3  38  88  81  32  11  2  3  68  284  182  134  43  11  2  5-6 4-5  2  66  276  433  336  184  86  17  3  1  3-4  28  351  845  773  471  269  58  17  4  1  11  298  1,324  1,456  1,142  638  282  73  11  5  2-3 1-2  1  158  1,281  2,251  1,999  1,191  542  147  34  8  1  0-1  4  121  488  767  687  257  82  15  7  2  1  3-4  4-5  5-6  6-7  7-8  8-9  9-10  10-11  11-12  12-13  13-14  2  14-15  T (sec)  Fig. 3.1. Wave scatter diagram (large cylinder, d = 80 m). H (m) s  5.0-5.5  1  4.5-5.0 4.0-4.5  1  3  4  1  1  8  15  5  2  3.5-4.0  2  18  45  36  14  5  1  3.0-3.5  3  38  88  81  32  11  2  3  68  284  182  134  43  11  2  2.5-3.0 2.0-2.5  2  66  276  433  336  184  86  17  3  1  1.5-2.0  28  351  845  773  471  269  58  17  4  1  11  298  1,324  1,456  1,142  638  282  73  11  5  1.0-1.5 0.5-1.0  1  158  1,281  2,251  1,999  1,191  542  147  34  8  1  0-0.5  4  121  488  767  687  257  82  15  7  2  1  2.1-  2.8-  3.5-  4.2-  4.9-  5.7-  6.4-  7.1-  7.8-  8.5-  9.2-  9.9-  2.8  3.5  4.2  4.9  5.7  6.4  7.1  7.8  8.5  9.2  9.9  10.6  T(sec)  Fig.3.2. Wave scatter diagram (slender cylinder, d = 10 m).  48  2  250  — Method  50  100  A  — A --  Method  B  —+ -  Method  C  —X -  Method D , E  150  200  TR (years)  300 (b) T = 4.43VH. 250  200  •  150 T  Method  A  A  Method  B  —+-  Method  C  —x-  Method  D  •  Method  E  50 50  100  150  200  TR (years)  Fig. 3.3. Alternate predictions of wave force on the slender cylinder for a Type I H distribution, (a) T = 8.5 sec. (b) T =4.43 ^ / H J .  49  s  200  200  Fig. 3.4. Alternate predictions of wave force on the slender cylinder for a Type i n H distribution, (a) T = 8.5 sec. (b) T =4.43 JH^ .  50  s  Fig. 3.5. Effect of wave period assumption on force estimates on the slender cylinder (Type I H distribution), (a) Method A . (b) Method B. (c) Method C. (d) Method D. (e) Method E. s  51  100  4 0  1 50 T  R  1  1  100  150  1 200  (years)  Fig. 3.6. Effect of wave period assumption on force estimates on the slender cylinder (Type m H distribution), (a) Method A . (b) Method B . (c) Method C. (d) Method D. (e) Method E. s  52  250  250 (c) M e t h o d  200  C  (d) M e t h o d  --  D  —^ 200  150  -it  |  150 — A -  100  1 50 T  Fig. 3.7.  R  T  s  H  m  50  h  s  h  1  100  150  y p  e  —A—  T y p e III 100 200  s  H  T  s  y p  e  I  T y p e III  i 50  100 T  s  H  — » —  (years)  Effect of Type of H  +  I  R  150  200  (years)  distribution on force estimates on the slender cylinder  (T = 8.5 sec), (a) Method A . (b) Method B. (c) Method C. (e) Method E.  53  (d) Method D.  (e) M e t h o d  E  T  3.8.  R  (years)  Effect of Type of H  S  distribution on force estimates on the slender cylinder  (T=4.43^/H7). (a) Method A. (b) Method B . (c) Method C. (d) Method D. (e) Method E.  54  1250 (a) T = 12 sec  500  -I  1  1  1  0  50  100  150  1 200  TR (years)  Fig. 3.9. Alternate predictions of wave force on the large cylinder for a Type I H distribution, (a) T = 12 sec. (b) T =4.43 ^ / H J .  55  s  1000  750  500  1500  Fig. 3.10. Alternate predictions of wave force on the large cylinder for a Type III H distribution, (a) T = 12 sec. (b) T =4.43 ^ / i L j .  56  s  2000  T  R  (years)  3.11. Effect of wave period assumption on force estimates on the large cylinder (Type I H distribution), (a) Method A . (b) Method B . (c) Method C. (d) Method D. (e) Method E. s  57  1500  T  R  (years)  Fig. 3.12. Effect of wave period assumption on force estimates on the large cylinder (Type m H distribution), (a) Method A . (b) Method B . (c) Method C. (d) Method D. (e) Method E. s  58  Fig. 3.13. Effect of Type of H distribution on force estimates on the large cylinder (T = 12 sec), (a) Method A . (b) Method B . (c) Method C. (d) Method D. (e) Method E. s  59  1500  2000 (a) M e t h o d  A  •  1500  z  ft.  — * - H  it  B  4-  ft.  Type I  s  — » - H  1000  H  1  —1  50  100 T  R  Type I  s  T y p e III  s  H  1  500  (b) M e t h o d  i\  s  T y p e III  1  500  150  200  50  100  (years)  T  1000  R  150  200  (years)  2000 (c) M e t h o d  C  (d) M e t h o d  1500  D  4-  1000 * - H  Type I  s  # — H  T y p e III  s  500  -A—  H  * —  H  Type I  s  s  T y p e III  500 50  100 T  R  150  200  (years)  50  100 T  R  150  200  (years)  2000 (e) M e t h o d  1500  E  4-  1000  A-  h  T  s  ^ _ H  y p  e  I  T y p e III  S  500 50  100 T  R  150  200  (years)  Fig. 3.14. Effect of Type of H  S  distribution on force estimates on the large cylinder  (T=4.43 7H7). (a) Method A . (b) Method B . (c) Method C . (d) Method D . (e) Method E.  60  

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