RELIABILITY ANALYSIS OF WAVE RUN-UP by F U A D CURI B. E., The University of Cartagena, Colombia, 1995 M . Eng., The University of Los Andes, Colombia, 1996 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES DEPARTMENT 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 March, 2002 © Fuad Curi, 2002 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 Assessing the run-up arising from ocean waves as they reach a coastal or offshore structure is an important aspect of engineering projects. By combining formulations for calculating the run-up of regular waves with wave statistical descriptions, a procedure is implemented to assess wave run-up in random sea conditions. This procedure uses the First and Second Order Reliability Methods. It is applied to four cases representing common coastal structures: vertical wall, vertical cylinder, smooth impermeable slope and rough permeable slope. The results of the reliability methods are compared with two other methodologies commonly used in coastal and ocean engineering. The first methodology (Method I) consists of calculating the significant wave height for a given return period from the long-term distribution of storms. Then, the maximum wave height corresponding to the significant wave height, and its corresponding wave run-up are obtained. Method II consists of obtaining a long-term distribution of individual wave heights, so that the maximum wave height for a given return period can be directly calculated. Then, the corresponding wave run-up is determined as in Method I, by applying regular wave formulations to the maximum wave height. Method I provides lower values of wave run-up than all other methods, while Method II provides the highest values of wave run-up, and proves to be dependent on the severity factor, a parameter characterizing the long-term distribution of storms. Results show that both the First and Second Order Reliability Methods correspond with Method II when the values of the severity factor are low, but approach Method I as the value of the severity factor increases. The wave run-up is found to depend on the characteristics of the structure in place, such as cylinder radius, slope angle and surface permeability, as well as on such sea conditions as severity factor and the duration and frequency of storms. The wave run-up to significant i i wave height ratio is independent of the return period for the four cases studied here. This ratio results in a non-dimensional form of wave run-up that appears to provide useful means for describing the phenomenon. 111 TABLE OF CONTENTS page ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES viii LIST OF SYMBOLS x ACKNOWLEDGMENTS xii 1.INTRODUCTION 1 1.1. GENERAL 1 1.2. LITERATURE REVIEW 2 1.3. SCOPE OF THIS STUDY 6 2. METHODOLOGY 7 2.1. WAVE RUN-UP 7 2.1.1. Vertical Wall 7 2.1.2. Vertical Cylinder 8 2.1.3. Smooth Impermeable Slope 9 2.1.3.1. Dimensional Analysis 9 2.1.3.2. Regular Waves 11 2.1.3.3. Random Waves., 12 2.1.4. Rough Permeable Slope 13 2.1.4.1. Regular Waves 13 iv page 2.1.4.2. Random Waves 14 2.2. RELIABILITY ANALYSIS 15 2.2.1.Statistics in Wave Problems 16 2.2.2. Reliability Analysis in Wave Problems 17 2.2.3. Reliability Analysis of Wave Run-up 22 2.2.3.1. Vertical Wall 23 2.2.3.2. Vertical Cylinder 24 2.2.3.3. Smooth Impermeable Slope 26 2.2.3.4. Rough Permeable Slope 27 3. RESULTS 29 3.1. G E N E R A L 29 3.1.1. Dependent Variables 30 3.1.2. Variables Defining the Structure Geometry 30 3.1.3. Variables Describing Sea Conditions 31 3.1.3.1. Wave Height 31 3.-1.3.2. Wave Period 31 3.1.3.3. Duration and Frequency of Storms 32 3.2. COMPARISON OF SOLUTION METHODS 33 3.3. F O R M A N D SORM RESULTS FOR THE V E R T I C A L W A L L C A S E 34 3.4. F O R M A N D SORM RESULTS FOR THE VERTICAL C Y L I N D E R C A S E 35 3.5. F O R M A N D SORM RESULTS FOR THE SMOOTH I M P E R M E A B L E SLOPE C A S E 35 v page 3.6. FORM AND SORM RESULTS FOR THE ROUGH PERMEABLE SLOPE CASE 36 4. EXAMPLE APPLICATION 37 4.1. ANALYSIS PROBLEM 37 4.2. DESIGN PROBLEM 38 5. CONCLUSIONS AND RECOMMENDATIONS 40 6. REFERENCES 43 v i LIST O F T A B L E S page Table 2.1. Parameters of the Sea State Distributions Used in this Study 45 Table 2.2. Summary of Case Studies, Variables and Assumptions 45 Table 3.1. Non-dimensional Wave Run-up of Vertical Wall 46 Table 3.2. Non-dimensional Wave Run-up of Vertical Cylinder for Constant Wave Period T=12 seconds 47 Table 3.3. Non-dimensional Wave Run-up of Vertical Cylinder for T=4.43(HS) I / 2 (Constant Wave Steepness) 48 Table 3.4. Non-dimensional Wave Run-up of Smooth Impermeable Slope for Constant Wave Period T=12 seconds 49 Table 3.5. Non-dimensional Wave Run-up of Rough Permeable Slope for T=4.43(H s) l / 2 (Constant Wave Steepness) 50 Table 3.6. Non-dimensional Wave Run-up of Smooth Impermeable Slope for Constant Wave Period T=12 seconds 51 Table 3.7. Non-dimensional Wave Run-up of Rough Permeable Slope for T=4.43(HS)1 / 2 (Constant Wave Steepness) 52 V l l LIST OF FIGURES page Figure 2.1. Vertical Wall Geometry 53 Figure 2.2. Vertical Cylinder Geometry ; 53 Figure 2.3. Smooth Impermeable Slope Geometry 53 Figure 2.4. Rough Permeable Slope Geometry ..54 Figure 2.5. Maximum Wave Run-up of Vertical Cylinder 54 Figure 2.6 Procedure Scheme for the Analysis Problem 55 Figure 2.7 Procedure Scheme for the Design Problem 56 Figure 3.1. Comparison of Calculation Methods for Non-dimensional Wave Run-up of Vertical Wall. TR=30 years and Constant Wave Period T=12 sec 57 Figure 3.2. Comparison of Calculation Methods for Non-dimensional Wave Run-up of Vertical Cylinder. TR=30 years and Constant Wave Period T=12 sec .... 57 Figure 3.3. Comparison of Calculation Methods for Non-dimensional Wave Run-up of Smooth Impermeable Slopes. TR=30 years and Constant Wave Period T= 12 sec 58 Figure 3.4. Comparison of Calculation Methods for Non-dimensional Wave Run-up of Rough Permeable Slope. TR=30 years and Constant Wave Period T=12sec 58 Figure 3.5. S O R M Results for Non-dimensional Wave Run-up on Vertical Wall as a Function of Return Period for Various Values of Severity Factor and Constant Wave Period T=12 sec 59 Figure 3.6. SORM Results for Non-dimensional Wave Run-up on Vertical Wall as a Function of Return Period for Various Values of Severity Factor and Wave Period T=4.43(HS) I / 2 (Constant Wave Steepness) 59 Vlll page Figure 3.7. Effect of the Cylinder Radius on the Non-dimensional Wave Run-up of Vertical Cylinders, based on SORM Results. Various Return Periods, Constant Wave Period T=12 sec and Severity Factors of 1.05 and 1.30 60 Figure 3.8. Effect of the Period Assumption on the Non-dimensional Wave Run-up of Smooth Impermeable Slopes as a Function of the Severity Factor, Based on SORM Results, for Slopes of 1:3.5 and 1:6 and Return Period of 50 Years 60 Figure 3.9. Effect of the Slope on the Non-dimensional Wave Run-up of Smooth Impermeable Slopes for Constant Wave Period T=12 sec, Based on SORM Results, for Severity Factors of 1.05 and 1.3 and Various Return Periods 61 Figure 3.10. Effect of the Roughness and Permeability on the Non-dimensional Wave Run-up of Sloping Structures for Constant Wave Period T=12 sec, Based on SORM Results for Slopes of 1:3.5 and 1:6, Various Severity Factors and Return Period of 30 Years 61 Figure 3.11. Effect of the Severity Factor on the Probability Distribution Storms in Terms of the Return Period for an Extreme Type III (Weibull) Distribution, Assuming H s = 7.34 m for a One-year Return Period 62 Figure 4.1. S O R M Results for Non-dimensional Wave Run-up of Rough Permeable Slopes, Assuming Constant Wave Period T=12 sec, Severity Factor of 1.05 and Return Period of 50 Years 62 ix LIST OF SYMBOLS a cylinder radius a, b parameters defining the long-term probability distribution of storms bi, b2 parameters defining roughness and permeability of a structure's surface 8 acceleration of gravity H deep water individual wave height Hm(x) Hankel function of the argument x Hs significant wave height Ir Iribarren number k wave number L deep water wave length rim maximum water level P(x) cumulative probability distribution of the argument x Q(x) cumulative probability distribution of the argument x being exceeded R wave run-up Rm maximum wave run-up Rs wave run-up corresponding to the significant wave height T wave period TR return period (0 wave angular frequency a slope of the structure P reliability index X 6 angular coordinate measured from the back of cylinder (6= 180° at front) yw specific weight of water 7 non-dimensional wave run-up v kinematic viscosity <fj surf-similarity parameter xi ACKNOWLEDGMENTS Any personal achievement requires the participation of many individuals. I want to express my gratitude to many people who have helped me both personally and professionally: Dr. Michael Isaacson, my supervisor, whose guidance, encouragement and support made possible this thesis and my graduate studies at The University of British Columbia; Dr. Ricardo Foschi, who patiently shared his knowledge and warm conversations; Members of the Faculty of Graduate Studies and the Department of Civil Engineering for all their support; My friends at U B C -students, professors and staff- with whom I shared experiences during the past two years, in particular Vicki Dimopoulos and John Baldwin, who kindly reviewed early drafts of this work, as well as Jeannette Smith; My parents, Lina and Fuad, to whom I dedicate this thesis, and other members of my family, who have always supported and encouraged me; my wife, Karina, and daughter, Hannia, who are truly co-authors of this and all my work; And, especially, to the One who makes possible our curiosity and our will to understand. GRACIAS TOT ALES xii 1. INTRODUCTION 1.1. G E N E R A L Wave run-up associated with ocean waves reaching a structure or the coastline is an important feature that engineers need to evaluate. Waves climb the face of a structure reaching a higher level than the crest elevation in open water, a phenomenon dominated by different mechanisms, depending on the characteristics of the particular structure. The maximum wave run-up adds to the still water level to produce the maximum level that water reaches on an offshore or coastal structure. The SWL, in turn, fluctuates as it is affected by tides, wave set-up and storm surge. Calculating the wave run-up, which is measured from the SWL to the transformed wave crest, is, therefore, a necessary step for evaluating the risk of the structure being overtopped as well as the forces it endures. In the case of a dike, for example, wave run-up is related to the flooding of the hinterland and to resulting damage caused to people, to their property, and to the dike itself. Similarly, for breakwaters, wave run-up is associated with overtopping, and damage to the structure and facilities located on it. In the case of offshore structures, wave run-up affects both the wave-structure interaction and the operation of the facilities. Theoretical and experimental studies of wave run-up have been conducted in the past, frequently considering uniform wave conditions. However, to take into account the non-homogeneous nature of the ocean, a different approach is needed. For these cases, probabilistic methods, such as reliability analysis, seem to be appropriate. In addition, the reliability analysis provides valuable tools for the decision making process. In fact, assessing and quantifying the risk is a main step when making a decision that involves important budgets, in particular when the consequences of an eventual failure include losses in property and even threats to human lives. For this reason, reliability-based 1 designs have been in place for offshore and coastal structures and are expected to be of greater use in the future. The purpose of this thesis is to develop a methodology for assessing the reliability of a structure subjected to wave run-up. Such methodology could be applied in coastal engineering designs or for evaluating the risk of failure of an existing structure. Since most of the equations used here were empirically obtained, their range of applicability is often restricted to some specific conditions enforced during experimental or prototype studies. Implementing a reliability analysis often requires the use of a computer program in order to test a system's behavior under the random conditions it is expected to undergo. In this study, the First and Second Order Reliability Methods (FORM and SORM) were applied by the use of the R E L A N program. This program was developed at The University of British Columbia and has been extensively used in studies related to reliability analysis. 1.2. LITERATURE REVIEW The maximum run-up generated by a wave approaching a structure is strongly affected by the type of structure at issue. The wave-structure interaction differs from one structure to another, and so does the feature controlling the phenomena. For instance, wave reflection is dominant in waves reaching a vertical wall, whereas diffraction plays the most important role in the case of a large vertical cylinder. In the case of sloping structures, such as breakwaters, wave run-up is controlled by wave breaking and in most cases is affected by the degree of roughness and permeability of the structure. The simplest case of a vertical smooth wall, causing perfect reflection, generates standing waves whose amplitude, from the SWL to the crest, renders the maximum wave run-up. For vertical cylinders, with a diameter larger than one-fifth the wavelength (D/L > 0.2), Sarpkaya and Isaacson (1981) register the findings of MacCamy and Fuchs. Isaacson (1978) developed an approximate formula for the maximum wave run-up in large cylinders limited to values of ka up to 0.4, where k is the wave number and a is the cylinder radius. These formulations were obtained from potential flow theory, applying linear wave refraction. Isaacson (1978) also presents a solution based on cnoidal wave theory. 2 In the case of sloping structures, the way waves break as they reach the structure, determines the resulting wave run-up. Bruun (1985) provides an extensive review of studies devoted to this type of structure. He registers the work of Iribarren and Nogales, who developed a parameter for determining whether a wave would break or not. This parameter, which includes the wave period, the wave-height and the slope of the structure, has been extensively used by other authors for establishing the type of breaker generated by the wave. In this respect, Bruun (1985) refers to the work of Battjes and Gtinbak. Based on Iribarren's parameter (also known as surf-similarity parameter), Hunt developed a formulation for wave run-up on smooth impermeable slopes. Hunt's formula is as follows (Bruun, 1985): = £ (far ^< 2.3) (1.1) where R is the wave run-up, H is the wave height and t, is the surf-similarity parameter. Experiments conducted by Gtinbak on smooth surfaces (plexiglass) and slopes that ranged from 1:1.5 to 1:3 are in agreement with Hunt's formula. Bruun (1985) refers to the experimental results of Gtinbak and Bruun, and Inoue, to show that the wave run-up is independent of the depth at the structure toe, d, if such depth is more than three times the wave height (d/H>3). Addressing the same issue, the Shore Protection Manual (U.S. Army, Corps of Engineers, 1984) presents a set of figures applicable for different values of d/H]. These figures are in agreement with Hunt's formula within its range of applicability (d/H>=3.0, % <2.3, tana<l/3, a being the slope angle of the structure). Saville, Hunt and Hebrich, as presented by Bruun (1985), studied wave run-up for structures with composite slopes. For rough permeable slopes, Bruun (1985) presents formulations obtained by Dai and Kamel, Jackson, Giinbak, CERC and Stoa, as well as experimental results obtained by Dai 1 These figures were obtained from experiments by Saville. The Shore Protection Manual includes correction for model scales and graphs that can be used for some rough permeable surfaces. 3 and Kamel, Ahrens and McCartney, and Losada and Gimenez-Curto. These experiments cover a wide range of surfaces, including rip rap, stone armor, dolos block and tetrapods. The above studies apply to regular wave conditions. For assessing the wave run-up resulting from random waves, some additional considerations are necessary. Saville proposed an 'hypothesis of equivalency' for solving this problem. According to this hypothesis, the probability distribution of wave run-up can be obtained by assigning to each individual wave the run-up value of a periodic wave train of corresponding height and period (Battjes, 1971). Van Oorschot and Angremond, and Giinbak (Bruun, 1985) as well as Battjes (1971), used the hypothesis of equivalence for obtaining the run-up of random waves on sloping structures with smooth and rough surfaces. This approach requires combining a long-term joint probability distribution of wave heights and periods with a short-term distribution of individual wave run-up. For the latter, a Rayleigh distribution is usually adopted, which is similar to the distribution of individual wave heights during a sea state. Van Oorshot and Angremond (Bruun, 1985) obtained the following formulation for random waves in breakwaters: where R„% is the wave run-up corresponding to a certain percentage of exceedance, n; C „% is a coefficient related to the percentage of exceedance, which depends on the spectral width ; Hs is the significant wave height; and Tp is the peak period of the spectrum. The same formulation is presented by Pilarczyk and Zeidler (1996) as: (1.2) (1.3) where 2 values of C2% for different cases can be found in Pilarczyk and Zeidler (1996, p.98) and Bruun (1985,p. 99). 4 lanu, lanix tana tana H, fgT^ 2K Equation 1.3 applies when tan a is less than one third (%p<2.0)3 and has been used to a large extent for designing sea structures exposed to random seas. By using C2%-0.70 and Hs/Lp=5% (accepted values for the North Sea Coast), the "Old Delft Formula", commonly used for the design of Dutch sea dikes, is obtained: R2%= 8 Hstan a (1.5) or ^ = 1.75^ (1.6) Equation 1.6 is common design practice in the Netherlands and provides wave run-up values of about 3.2 times the significant wave height. Several methods have been used in the past for studying wave run-up on rough permeable slopes, under random sea conditions. Pilarczyk and Zeidler (1996) present one method based on the £ parameter. Bruun (1985), on the other hand, proposes a methodology that combines the 'hypothesis of equivalence' with a long-term joint probability distribution of wave heights and wave periods and a formulation for the run-up of regular waves. Most of the formulations presented thus far were obtained from empirical results. Theoretical approaches to wave run-up have also been in place. These include the research of Miche on perturbation theories for Stokes-type waves (Mei, 1983); the work of Buhr and Hansen on Cnoidal theory (Bruun, 1985); studies by Longuet-Higgins and Hwang and Divoky, who integrated the momentum flux equation assuming Cnoidal waves (Van Dorn, 1976); and models based on non-linear shallow water equations, from Hibber and Peregrine, Kobayashy, Mase, Titov and Synolakis, and Dodd (Dodd, 1998). 3 Some authors suggest %,,<2.5. This range of L, values corresponds to plunging breakers. 5 Reliability analysis, on the other hand, has been in place for solving coastal and ocean engineering problems. Isaacson and Foschi studied the maximum wave height for a given return period, based on a reliability model (Isaacson and Foschi, 1996; Isaacson and Foschi, 2000). Ganapathy (1997) studied alternative methods for determining design wave forces, both in slender and large cylinders. Foschi et al (1996; 1998) applied reliability analysis for studying the combined effect of waves and iceberg loading on offshore structures. 1.3. S C O P E OF THIS S T U D Y The objective of this study is to develop a methodology, based on reliability analysis, for assessing the maximum wave run-up when designing breakwaters and marine structures. There are two potential problems to be solved: analysis and design of a structure. The first consists of evaluating a given structure's risk of being overtopped under certain sea conditions. Its solution can be expressed in terms of annual risk or return period. The second problem arises when designing a structure for the wave run-up, corresponding to a certain risk level, return period or probability of failure. The solution to this problem consists of finding the structural characteristics that suit the desired risk level. Both analysis and design approaches are applied to four cases, representing four different types of offshore and coastal structures: vertical wall, vertical large cylinder, smooth impermeable slope and rough permeable slope. The wave run-up is the height reached by the waves from the still water level (SWL) to their crest. Other features, including wave set-up, tide fluctuation and storm surge, modify the position of the SWL and also affect the total height reached by the water in front of the structure at issue. For adding these features to the reliability analysis, some extra considerations are needed. This study refers only to wave run-up, leaving tide variations and other factors affecting the SWL for further studies. 6 2. M E T H O D O L O G Y 2.1 W A V E R U N - U P From the still water level to the crest, the maximum vertical distance that a wave reaches on a coastal structure constitutes the wave run-up. It is affected by characteristics of the structure, such as its geometry, surface roughness and porosity, as well as properties of incoming waves, such as wave height, period and direction. For the purpose of this study, the wave run-up of four possible structure configurations is considered. These are vertical wall, vertical cylinder, smooth impermeable slope and rough permeable slope. 2.1.1. Vertical Wal l The simplest case consists of a vertical wall located on a horizontal seabed. For this structure, perfect reflection is expected, with not much dissipation of energy. In this scenario, standing waves would develop whose total height results from superimposing the incident and the reflected waves. Such superposition is possible for small amplitude waves, under the assumption of linearity. The resulting standing wave doubles the height of the incoming waves. Therefore, the run-up, which is half the standing wave height in this case, equals the incident wave height, as presented in Figure 2.1: R , — = 1 (2.1) H where R is the wave run-up and H is the incident wave height in deep water conditions. 7 2.1.2. Vertical Cylinder In the case of a vertical cylinder, wave-structure interaction has been extensively studied. In a slender cylinder, flow separation is expected. Otherwise, when the horizontal dimension of the structure is comparable with the incoming wavelength, the structure is considered large, and linear wave diffraction theory is applied. The vertical cylinder structures studied here are only those regarded as large, which means that the ratio of the diameter (D) to the deep-water wavelength (L) is greater than one fifth (D/L>0.2). In the case of large vertical cylinders (Figure 2.2), diffraction plays a dominant role. Consequently, the resulting wave run-up is controlled by the ratio of the cylinder radius to the deep-water wavelength. This ratio is represented by the diffraction parameter, ka, in which k is the wave number (k=2Tt/L) and a is the cylinder radius. MacCamy and Fuchs derived the following expression for the wave run-up of regular waves on this type of structure: R(e) H Y ipmcos(md) -imt £0Kk<iHlly(ka)e (2.2) in which f3„, = 1 for m=0 or j3,„ = 2i"' for m > 1; Hm(l> is the derivative of the Hankel function of the first kind of order ra with respect to the argument (ka in this case), and the wave run-up is a function of 6, which is the angular coordinate measured about the cylinder axis from the direction of wave propagation, so that 0 =0 corresponds to the rear of the cylinder. At the front of the cylinder (0 =180") Equation 2.2 becomes: flB(-lw) R H I %Kk2uHf{ka) (2.3) The wave run-up obtained from Equation 2.3 corresponds to its maximum value around a vertical cylinder. It is a function of the diffraction parameter, ka. By applying the method of least squares, a polynomial approximation to MacCamy and Fuchs' formula can be obtained. One such approximation, which combines two quadratic 8 expressions while enforcing the same value of the two polynomials and their first derivatives at the intercept, was found to be: A = 0.4396 + 0.7362(&a)-0.398 l(fca)2 for ka<0.5 (2.4.a) — = 0.5052 + 0.474l(fca)-0.1360(fca)2 for ka>0.5 (2.4.b) H Due to its simplicity, Equation 2.4 was used in this study instead of Equation 2.3. Figure 2.5 shows a comparison of the two formulations. 2.1.3. Smooth Impermeable Slope In the case of structures that present a slope to the waves, several factors are involved in the resulting wave run-up. These include water depth at the toe of the structure, geometry, roughness and permeability of the structure, as well as parameters related to the incoming waves. The geometry corresponding to the structures studied in this section has been simplified in Figure 2.3, and roughness and permeability have not been considered. However, these will be studied in upcoming sections. Among the factors affecting the wave run-up of smooth impermeable sloping structures, the way in which waves break on the slope is considered to play the most dominating role. In order to demonstrate this, while getting some insight into the variables involved in the phenomena, dimensional analysis constitutes an appropriate tool. 2.1.3.1. Dimensional Analysis The variables physically involved in the phenomena can be outlined as: Specific weight of water, yw Kinematic viscosity, v Acceleration of gravity, g Wave height, H 9 Wave period, T Wave approach angle, ji Depth at the toe of the structure, d Slope angle of the structure, a The dependent variable, wave run-up (R), can be generally expressed as: R=f(7», v, g, H, T, B, d, a) (2.5) Assuming little influence of viscosity and density and perpendicular approach angle of the waves, this formula can be simplified to: R=f( g,H,T,d,a) (2.6) By dimensional analysis, the number of variables can be reduced to three non-dimensional parameters: H H d —,—,a L H (2.7) where gf2 has been replaced by L (wavelength), using the dispersion relation for deep water conditions. It is generally accepted that, if the water depth at the toe of the structure, d, is large enough (d/H>3.0), its effect on the wave run-up can be neglected. In this case, only two parameters remain in Equation 2.7. These two parameters are usually combined into one for describing breaking wave features, constituting the. Iribarren number or surf-similarity parameter. In 1949, Iribarren outlined the importance of a factor relating the slope of a beach or structure, tan a, to the height and the period of the incoming waves, to determine whether or not those waves would break. He defined this factor as: T \ Ir = -r=J-$-tona (2.8) where, g represents the acceleration of gravity, and the characteristics of the wave refer to deep water conditions. 10 The Iribarren number was soon reformulated in terms of the wavelength, L, or more precisely, the deep-water wave steepness (WL), as: (2.9) This number was also found useful in establishing the type of breaker a wave train would develop on a structure or a beach. With regard to this, Bruun (1985) and Mei (1983) present a classification of breakers based on this factor, more generally called surf-similarity parameter or £, parameter. 2.1.3.2. Regular Waves For assessing the wave run-up of smooth impermeable slopes, Hunt derived a simple formulation based on the £, parameter, assuming that the depth at the toe of the structure was greater than one-third (d/H>3.0): 4 = £ (for Z< 2.3) (2.10) Experimental results showed that the ratio R/H reaches a maximum within the stated range of | values, and remains constant or even decreases for higher values of the parameter4. Examples of experimental results in agreement with Equation 2.10 are those from Gtinbak and the Dutch Technical Advisory Committee, cited in Bruun (1985). The limited range of applicability of the formula in terms of the % parameter refers to the fact that it is only valid for waves that break on the structure. For larger values of the t, parameter, the waves do not break but surge on the face of the structure. In these cases, the ratio of the wave run-up to the deep-water wave height remains constant. It is important The limit for this range is not exactly defined and depends on the real roughness of the structure in place. Other experiments locate the maximum wave run-up at £,= 3.0 or £,= 4.0. Note that this maximum corresponds to the transition from breaking to non-breaking wave conditions on the slope. 11 to note that, by limiting the value of the slope of the structure is also restricted. Thus, under normal conditions, tan a must be less than one-third for this formula to be applied. Some procedures have been developed for taking into account multiple angles on the slope or the inclusion of a berm. These variations in the structure's geometry affect the resulting wave run-up, as do changes in the angle of incidence of the incoming wave. These factors are not considered in this study, where the geometry was simplified, as in Figure 2.3. 2.1.3.3. Random Waves The previous section applies to regular waves, as they can be reproduced in laboratory conditions. In reality, structures are exposed to random seas where different types of waves are combined. Different approaches have been attempted for addressing this situation. Saville proposed the 'hypothesis of equivalency' where, as commonly used for determining random wave heights, the wave run-up is calculated for a 'significant' wave that represents the non-uniform conditions of a sea state. The individual wave run-up for a certain probability of exceedance can be obtained from a known distribution of individual wave run-ups relative to the run-up of the significant wave. This relationship corresponds to a Rayleigh conditional probability distribution. Pilarczyk and Zeidler (1996) present an analogous formulation to Equation 2.10 for addressing wave run-up in random wave conditions: where Rn% is the wave run-up corresponding to a certain percentage of exceedance and C„% is a coefficient, related to that percentage of exceedance, which depends on the spectral spectrum, Tp, are used for calculating the corresponding 5 Values of C2% for different cases can be found in Pilarczyk and Zeidler (1996, p.98) and Bruun (1985,p. 99). (2.11) width5. Additionally, the significant wave height, Hs, and the peak period of the wave 12 tana tana (2.12) Equation 2.11 is applicable for values of tan a less than one-third. In this study, reliability analysis is used for addressing random wave conditions. 2.1.4. Rough Permeable Slope The preceding section refers to structures with smooth impermeable surface and constant slope, which are normally oriented to the direction of the waves. Pilarczyk and Zeidler (1996) present a more general formula for other conditions: R=RsvrvBVp (2.13) where Rs is the corresponding wave run-up for smooth surfaces, and v, , VB , and Vp are reduction coefficients for considering roughness and permeability, berm and oblique wave attack. In reality, these coefficients are not constant but depend on the value of the £ parameter. The tables providing such coefficients refer to average values or to experimental results bound to some specific conditions6. Roughness and permeability are studied in this section, assuming the structure is presented as in Figure 2.4. 2.1.4.1. Regular Waves For this type of structure, the wave run-up is expected to be lower than that obtained for smooth impermeable slopes. This is because the roughness and permeability of the surface provide new mechanisms that favor the absorption and transmission of energy from the wave and, therefore, dampen the wave run-up. In the case of rough permeable slopes, the 6 Values of the roughness and permeability coefficient can be found in Pilarczyk and Zeidler (1996, p. 100) and Bruun (1985,p. 105). 13 ratio R/H increases gently for small values of the t, parameter, until t, reaches a value of 4.0 or 5.0. For larger values of £ , the ratio R/H remains constant. To represent such behavior, Losada and Gimenez-Curto formulated7: 4 = A(l - e x p ( * § ) ) (2.14)° where A and B are coefficients used for adjusting the formulas to experimental results. In a similar way, Giinbak obtained the following formulation: R b£ H l + b£ (2.15) where the values of b\ and b2 are combined to represent the roughness and permeability of the structure at issue. Gtinbak found that b,=0.8 and b2=0.5 suited his experimental results (Bruun, 1985). 2.1.4.2. Random Waves Several approaches have been used to address the more realistic case of random waves on a rough permeable structure. Pilarczyk and Zeidler (1996), as well as Bruun (1985), present methods based on the <f; parameter and Saville's hypothesis of equivalence. In particular, the method suggested by Bruun (1985) combines the 'hypothesis of equivalence' with an appropriate probability distribution of wave heights and periods (derived by Overvik) and Giinbak's formulation (Equation 2.15). The method uses the probability that the surf-similarity parameter be less than a value which is the cumulative probability distribution P( P(Z) = exp(-2(l-^)^4^ -4) (2.16.a) where In this case, however, the maximum wave run-up occurs at about ^ =6.0 (Brunn, 1985, pp 35-37). Losada and Gimenez-Curto used Ir instead of i;, referring to the work of Iribarren. 14 1 = t a n a (2.16.b) and (2.16.C) Hs being the significant wave height, Tz the average zero-crossing period and e the spectral width parameter, in combination with the formulation for wave run-up of regular waves: Alternatively, for individual wave run-up, a Rayleigh distribution is commonly assumed: for which Rp is the wave run-up associated with a certain probability of exceedance, p, and Rs is the wave run up corresponding to the significant wave height, Hs. A similar set of elements was used in this study to develop a reliability analysis model for wave run-up. 2.2 RELIABILITY ANALYSIS Probabilistic and statistical methods have been developed for the purpose of studying wave related phenomena, including maximum individual wave height and forces exerted by waves on structures. These methods are suitable for studying the non-uniform conditions of random seas. Among the available methods, the First and Second Order Reliability Methods (FORM and SORM) provide a way to evaluate the probability of failure of any given system undergoing certain external conditions. To represent the system, these methods use a performance function that combines its capacity with the demands of the conditions it endures. R _ 0.8£ H ~ l + 0.5£ (2.J6.d) (2.17) 15 The performance function (G (x)) must be expressed in terms of a set of variables (xh x2, x3... x„) related to the capacity and/or the demand of the system. These variables can be random, in which case they are defined by their probability distributions, or deterministic parameters. The reliability methods F O R M and SORM calculate the chances of the demand of the system exceeding its capacity, and render the probability of failure of the system, which is the probability of the G function being negative. In the n-dimensional space (each one of the variables, x,, representing a dimension of the space), the reliability methods define a "failure surface" for which the performance function equals zero. Then, a point P is found as the closest point to the origin within the "failure surface". Point P gives the combination of variables, x„ that is most likely to make the system fail. The distance from the origin to P is called the reliability index /3, and is used to estimate the corresponding probability of failure of the system. F O R M uses a linear approximation of the "failure surface" in the proximity of P. SORM, instead, uses a quadratic approximation, which in some cases, depending on the type of performance function, provides better results. The accuracy of these methods depends on the non-linearity involved in the performance function. In general, F O R M and SORM demand less time and fewer resources than a traditional Montecarlo simulation process. 2.2.1.Statistics in Wave Problems In wave statistics, the sea-state or storm corresponding to a given return period is usually represented by the significant wave height. It is the average of the highest one-third of the waves in that sea-state. Additionally, individual wave heights within the storm are described by a short-term conditional probability that is known as the Rayleigh distribution. It is common coastal engineering practice to obtain the significant wave height for a certain return period and, subsequently, to define the design wave height, from the Rayleigh distribution, as the maximum wave height given within the selected storm or sea-state. Isaacson and Foschi (1996) demonstrated that the maximum individual wave height for a given return period can be greater than the maximum individual wave height that occurs 16 within the storm corresponding to that return period. Their findings point out that the above-described traditional method is not always conservative. In the case of wave run-up, the formulations presented in previous sections combine several random and deterministic elements, including wave height. Isaacson and Foschi's findings suggest that the individual wave run-up for a certain return period may differ from the run-up of the maximum wave of the sea-state corresponding to that return period. Therefore, it is advisable to study the possibility of implementing alternative methods, such as reliability analysis, for evaluating the wave run-up in random conditions. 2.2.2. Reliability Analysis in Wave Problems In their reliability analysis of maximum wave heights, Isaacson and Foschi (1996; 2000) defined a performance function of the form: where the capacity, H, is a specific wave height level whose exceedence probability, Q(H), needs to be determined. The demand, Hm, corresponds to the maximum individual wave height for a given return period. Hm is a function of the variables Hs and U . H, Hs and U constitute the vector or set of variables (x) defining the performance function. In this case, H is introduced as deterministic values of the capacity, whereas Hs and U are stochastic variables. Of these, U accounts for the Rayleigh distribution of individual wave heights with respect to Hs, the significant wave height, which, in turn, accounts for the long-term distribution of storms. The method requires defining probability distributions for the two stochastic variables in the vector (x). The significant wave height, Hs, is assumed to follow an Extreme Type III (Weibull) distribution defined as: for which Q(HS), the long-term exceedence probability of Hs, is related to its cumulative probability P(HS) by: G(x) = H-H,n(Hs, U) (2.18) P(HS) = l-exp(-aHsb) (2.19) Q(HS) = 1- P(HS) (2.20) 17 The parameters a and b, in Isaacson and Foschi (1996; 2000), are constants that define the Weibull probability distribution of storms. These parameters can be obtained from two points in the distribution, Hst and Hs2, corresponding to two specified return periods TR/ and TR2, by using: a -and Hi In b = l n f e j In for which TRi and TR2 are related to a recording interval, r9, by:. T T -r T T — R 1 (2.2 La) (2.21. b) (2.21.c) (2.21.d) where r is the time interval, average if appropriate, between successive storms, given in years. The probability distribution of the stochastic variable, Hs, included in the analysis for representing the long-term distribution of storms, is thus defined by the parameters a and b. In order to obtain appropriate values for these two parameters, representing the sea conditions of a particular site, Isaacson and Foschi selected a representative wave height (in this case the significant wave for a return period of one year HJTR=/)10 ) and a single dimensionless parameter, defined as the ratio HS(TR=W)/Hs(TR=i). This parameter was called It is important to note here that the selected value of r affects the resulting probability distribution of storms, represented by Hs. A value of r = 3 hours was used in this study. 10 HS(TR=!) = 7.34 m was used in this study. 18 severity factor, s." By choosing a fixed value of HS(TR=I) (H, for brevity) and a representative r, the parameters a and b can be found for any value of s, from Equations 2.21. 12 The use of the severity factor, a non-dimensional parameter, for describing the long-term distribution of storms proved favorable to the analysis. However, the selection of a particular value for the representative wave height has undesirable scaling effects on the results. Therefore, Isaacson and Foschi referred their results to a non-dimensional form of the maximum wave height for a given return period, TR, obtained as its ratio to the significant wave height for the same return period: Y(TR)= (2.22) Since this last variable is non-dimensional, the results are independent of the selected representative wave height and are only affected by the severity factor, s. The second stochastic variable, U, is taken as uniformly distributed from 0 to 1, and corresponds to the short-term Rayleigh distribution for individual waves with respect to Hs: U=P(Hm)=l-cxp -2-H 2 A H: (2.23) Using Equation 2.23, the final form of the demand, H,„, in the performance function (Equation 2.18) becomes: Hm=Hj--ln 2 ( 1^ l-U" (2.24) 13 J " The severity factor was defined as the ratio of the significant wave height for a return period o f 10 years to the significant wave height for a return period of 1 year. The values used in this study range from s =1.05 lo .v = 1.40. 1 2 Table 2.1 presents values used in this study for the parameters characterizing the long-term distribution o f storms. 19 The inclusion of n, the number of waves within a storm, ensures that the results are related to the duration of a storm. The value of n is calculated as the ratio of the storm duration, T, to the wave period, T.14 The probability of the performance function being negative, calculated by this method, represents the probability of any particular value of H, input as the capacity, being exceeded by individual waves, during a storm. By applying the method to a set of values of H, a correspondent set of values of probabilities of failure, Q(H), is obtained, resulting in a long-term exceedance probability distribution of individual wave heights within a storm'\ In order to relate these values of probability of exceedance with their correspondent return period, an approximate formula was used in Isaacson and Foschi (1996; 2000): TR(H) 1 Q(H) (2.25) This formula includes the recording interval, r, which is given in years, meaning the time interval, average if appropriate, between successive storms. Equation 2.25 is accurate for small values of Q(H). A more general expression for relating probability of exceedance and return period is: TR(H) = ( 8760 A (2.26) 1 - (l-Q{H))-where r is given in hours and 8760 is the total number of hours in a year. The return period, TR(H), is given in years in both Equations 2.25 and 2.26.16 1 J This equation is similar to Equation 2.19, which defines a Rayleigh distribution for individual wave run-up with respect to the run-up for the significant wave height. 1 4 Note that H,„ is affected by the duration of the sea states, T. A value of T = 3 hours was used in this study. 1 3 Therefore, it would be more accurately represented by Qi\H). This notation will be used in Chapter 4 l u differentiate the probability of exceedance of wave run-up during a storm, Q£R), from its corresponding annual risk (QA(R) =1/Tr). 1 6 Note that (8760/r) is the average number of storms within a year. 20 In order to be more precise, the calculation of TR(H), as well as the definition of the long-term distribution of storms (Equation 2.19), would require using a joint long-term probability distribution for wave heights, H, and wave periods, T. This requirement would increase the difficulty of the procedure. Instead, some assumptions are used to associate H and T. In this regard, Isaacson and Foschi (2000) and Ganapathy (1997) implemented two approaches for relating the wave period to the significant wave height, Hs. The first consists of assuming a constant value for T[1, independent of the wave height. The second approach is given as: where T is in seconds and Hs in meters.18 Neu found this last relationship suitable for the conditions of Canadian Atlantic waters (Foschi et al. (1998)). By applying the wave dispersion relation, it becomes apparent that the second assumption corresponds to a constant ratio of the significant wave height to the deep-water wavelength. Therefore, in the following sections, this approach is regarded as constant wave steepness assumption. Using the above two assumptions for wave period, Ganapathy was able to calculate wave-generated forces on slender and large cylinders from the reliability methods F O R M and SORM. Her performance functions were based on Morison's and MacCamy and Fuchs' formulations. Ganapathy compared the F O R M and SORM results with those obtained from traditional methods. Ganapathy, as well as Isaacson and Foschi, showed that the results based on calculating the significant wave height for a certain return period, and, from this, the maximum wave height, are rather unconservative compared to those obtained from reliability analysis methods. An alternative method consists of calculating the maximum wave height from an integration that combines the two distributions: the long-term distribution of storms, represented by Hs, and the Rayleigh distribution of individual wave 17 T = 12 sec was used. 18 lfHs is replaced by 7.34 m (TR = 1 year), T = 12 sec is obtained. (2.27) 21 heights within a storm. The results from this last method are also higher than those previously described (Isaacson and Foschi, 1996, 2000; Ganapathy, 1997). 2.2.3. Reliability Analysis of Wave Run-up When applying reliability analysis to the study of wave run-up in structures, two types of problems need to be considered: evaluating structures exposed to certain known conditions or, inversely, designing a structure, given the conditions it is expected to endure. In this study, both analysis and design problems are addressed by applying F O R M and S O R M to the four types of structures presented in Section 2.1. The R E L A N program is used in all cases. For each structure, it is necessary to formulate a performance function and to define its correspondent variables. Both performance function and variables depend on sea conditions and on structural characteristics, as stated in previous sections. Also, each of these variables must be defined by a probability distribution or by a constant value, in the case of stochastic parameters. The performance function has two components: capacity and demand. The calculated maximum wave run-up (Rm), with all its variables and formulations, is incorporated as the demand. For the capacity, deterministic values of wave run-up (R), whose probability of failure needs to be obtained, are input, one at a time. The results from this procedure provide the probability of failure within a storm (Q(R)) related to the correspondent values of wave run-up used as the capacity. This probability of failure can be alternatively expressed in terms of the return period or annual risk, by using Equations 2.25 or 2.26. Values used in this study for the parameters characterizing the long-term distribution of storms (distribution of Hs) are presented in Table 2.1. Two types of distributions are considered, namely Extreme Type III (Weibull) and Extreme Type I (Gumbel). Table 2.2 summarizes the case studies considered here. Figures 2.6 and '2.7 illustrate the procedures for assessing analysis and design problems. 22 2.2.3.1. Vertical Wall This first case represents full reflection conditions on a vertical wall. The performance function is obtained by combining Equations 2.1, 2.18 and 2.24: G(x) = R-RM(Hs,U) (2.28) where RM = H - - I n l-U" (2.29) Since the reliability analysis renders the probability of failure corresponding to different values of R during a storm, which, in turn, is related to the return period, Equation 2.28 can be interpreted as:19 R =f(TR, H S , U) (2.30.a) Equation 2.30.a is appropriate for addressing the above design problem. The same equation can be presented in a suitable form for the analysis problem as: TR =f(R, H S , U) (2.30.b) It is worth noting that, however general, this equation holds important simplifications. In order to be more accurate, the incoming wave period, T, should be included in Equation 2.30. To do so, determining a joint probability distribution of HS and T is required. For the purpose of this study, instead, two simplifications for the wave period are alternatively put in place, namely constant wave period (T = 12 sec) and constant wave steepness (Equation 2.27). Since U only accounts for the Rayleigh distribution of individual wave heights for a given significant wave height, it does not introduce new variables into the problem2 0. U is input in I y More precisely, R =f(TR, Hs, U, n, r), acknowledging the effect of the number of waves during a storm and the number of storms during a year (see section 3.1.3.3.). The effect of the last two variables is not considered in this study, where both have been fixed to a constant value. 2 0 The Rayleigh distribution is a conditional probability of H\HS. The parameters describing the distribution are all known, as presented in Equation 2.23. 23 the analysis as a uniform distribution from 0 to 1. Consequently, the wave run-up, represented by Equation 2.30, remains only a function of the return period and the significant wave height, and can be affected by the selected assumption for the wave period. For describing the long-term probability distribution of storms, P(HS), two parameters are needed. These are, as presented in section 2.2.2, a representative wave height HS(TR=I), H/ for brevity, and the severity factor, s. In this study, the first parameter, Hh is fixed to a value of 7.34 meters, so any change in the long-term probability distribution of storms is addressed by modifying the severity factor. Applying these last considerations, Equation 2.30 becomes: R=f(TR,H1>S) (2.31) Although the wave period is not explicitly included as a separate variable, the two above-noted assumptions need to be considered. As in the case of individual wave heights (Equation 2.22), a non-dimensional form of the wave run-up is independent of the selected value of Hi. 7 = f(TR, s) (2.32) where /represents the non-dimensional wave run-up, defined as: rW = | ^ l (2.33) This non-dimensional wave run-up is the ratio of the maximum wave run-up for a given return period to the correspondent significant wave height for the same return period. This parameter is similar to the one used in Issacson and Foschi (2000) and results very suitable for the analysis. Two sets of tests in Table 2.2 (case studies land 2) refer to the vertical wall structure. 2.2.3.2. Vertical Cylinder In the case of a large vertical cylinder, the resulting wave run-up is controlled by the diffraction parameter, ka, presented in section 2.1.3. The performance function is obtained by combining Equations 2.4, 2.18 and 2.24: 24 G(x) = R-Rm(Hs, a, U) R,n = |).4396 + 0.7362(A;a)-0.398 i(/:a)2}fYs -/».! ]-U" V R„, =|)..5052 + 0.474l(A-a)--0.1360(^a)2^ \-~ln ( J ^ \-U" (2.34) forka<0.5 (2.35.a) forka>0.5 (2.35.b) where k, the wave number, is obtained for the two wave period assumptions, by using the wave dispersion relation. It is, for the constant wave steepness assumption (from Equation 2.27): (2Kf 4.4V gHs Alternatively, assuming constant period \T = 12 seconds): 144c (2.36.a) (2.36. b) In general, Equations 2.3 and 2.4, for regular wave run-up on large vertical cylinders, can be expressed as: - = /(*aj H (2.37) Similarly, Equation 2.34, applying the considerations for T, U and Hs presented in section 2.2.3.1, becomes: R =/(TR, Hh S, a) (2.38) where k is either .constant or a function of the significant wave height as presented in Equations 2.36. The non-dimensional wave run-up for the vertical cylinder case is also independent of///: 7 =/(TR, s, a) (2.39) 25 Four sets of tests in Table 2.2 (case studies 3 to 6) refer to this structure. The design problem is considered while studying the effects of the cylinder radius on the wave run-up (case studies 5 and 6). 2.2.3.3. Smooth Impermeable Slope In the case of smooth impermeable slopes, the E, parameter contains the influences of the structure slope and the incoming wave characteristics. The corresponding performance function is obtained from Equations 2.10 (Hunt's formula), 2.18 and 2.24. G(x) = R-R,n(Hs, a,U) (2.40) Rm = Hs^\-Un \-U" \ i (for 2.3) (2.41) Under the constant wave steepness assumption (using Equation 2.27 and the wave dispersion relation) the resulting £ parameter is: tana \2KHS \gT2 (2.42.a) | =4.43 tan a Otherwise, assuming constant wave period (T = 12 seconds): £ =12tana M— (2.42.b) (2.42.c) In general, Equation 2.10, which applies to regular wave cases, can be expressed as: R H = / (£) (2.43) Applying the considerations for T, U and Hs presented in section 2.2.3.1, Equation 2.40 becomes: R =f(TR, Hj, s, a) (2.44) 26 In this case, the two assumptions for the wave period affect the definition of which becomes a function of the slope angle and the significant wave height. The non-dimensional wave run-up for smooth impermeable slopes is independent of///: 7 =/(TR, s, a) (2.45) Four sets of tests in Table 2.2 (case studies 7 to 10) refer to this structure. The design problem is considered while studying the effects of the slope angle on the resulting wave run-up (case studies 9 and 10). 2.2.3.4. Rough Permeable Slope The performance function for the wave run-up on rough permeable slopes is obtained from Equations 2.15 (Giinbak's formula), 2.18 and 2.24. G(x) = R-R,n(Hs,a,U) (2.46) (for $ < 4.0) (2.47) -,1-U" 1 + 0 . 5 ^ 2 where £ can be obtained from Equation 2.42 for the two wave period assumptions. Assuming that the effect of the surface conditions is properly represented by the coefficients bj and b2 in Equation 2.15, and applying the same considerations for T, U and Hs presented in the previous cases, the general form of Equation 2.46 is: R =f(TR, H,, s, a) (2.48) Also, the non-dimensional wave run-up for rough permeable slopes can be expressed as: 7 =f(TR, s, a) (2.49) The last two general equations for this type of structure are the same as those obtained for the smooth impermeable slope case. However, the particular form of the equation differs from one case to the other, as can be seen by comparing Equations 2.41 and 2.47. 27 Four sets of tests in Table 2.2 (case studies 11 to 14) refer to this structure. The design problem is considered while studying the effects of the slope angle on the wave run-up (case studies 13 and 14). 28 3. RESULTS 3.1. GENERAL In the previous chapter, formulations were presented for calculating the wave run-up of regular waves on four types of structures: vertical wall, vertical cylinder, smooth impermeable slope and rough permeable slope. Based on these formulations, and on statistical descriptions of ocean waves, a reliability analysis model was developed to assess wave run-up in random sea conditions. To this end, the First and Second Order Reliability Methods (FORM and SORM) were put in place. The results of applying these methods, using the R E L A N program, are presented in this section and compared with two other commonly used methods, referred to here as Method I and Method II. Method I consists of calculating the significant wave height for a given return period from the long-term distribution of storms. Then, the maximum wave height, corresponding to that significant wave height, is obtained from the Rayleigh distribution of individual waves within a storm (currently about 1.8 to 2.0 times the significant wave height). The resulting maximum wave height is then put into an appropriate formulation for obtaining the wave run-up (see Equations 2.1, 2.3, 2.4, 2.10, 2.14 and 2.15) Method II consists of obtaining a long-term distribution of individual wave heights, by combining the distribution of significant wave heights with the Rayleigh distribution of individual waves within a storm. From the resulting probability distribution, the maximum individual wave height for a given return period can be directly calculated. Then, the corresponding wave run-up is determined as in Method I, by applying regular wave formulations to that maximum wave height. As shown in Figures 2.6 and 2.7, the procedures can be applied to analysis and design problems. 29 Before presenting the results, the variables and simplifications used in this study are summarized. 3.1.1. Dependent Variables The wave run-up, R, and the return period, TR, are taken alternatively as dependent variables or deterministic parameters during the study. In the case of the analysis problem, TR is the dependent variable, whereas in the design problem, the return period is specified in advance and its corresponding wave run-up, R, is the dependent variable. The return period is directly related to the annual risk and the probability of failure of the 2 E structure . Values of the return period of 30, 50 and 100 years are used in this study. In the following analysis, R is frequently replaced by its non-dimensional form, y. This non-dimensional wave run-up, for a given return period, results from dividing it by the corresponding significant wave height, as presented in Equation 2.33. 3.1.2. Variables Defining the Structure Geometry A variable is needed for defining the geometry of the problem. The radius, a, is used for this purpose in the large vertical cylinder case, while in the sloping structure cases, the slope angle, a, is in place. For rough permeable slopes, it is assumed that the factors b\ and b2 in Equation 2.15 account for the roughness and permeability of the structure. The values of 0.8 and 0.5 have been adopted here for these factors, which in real cases vary from one type of surface to another. In the case of vertical wall, the analysis does not require a geometric variable. 2 1 See Equations 2.25 and 2.26. 30 3.1.3. Variables Describing Sea Conditions For the purpose of this study, describing the wave height in random conditions is of main concern. In addition, as previously explained, the wave period must be taken into account for properly representing sea conditions. Finally, the duration and frequency of storms are characteristics of a particular site that have an effect on the resulting wave run-up. 3.1.3.1 .Wave Height Two variables are necessary for describing the long-term probability distribution of storms, represented by the significant wave height, HS. These are, as presented in section 2.2.2, a representative wave height, HS(TR=I), HI for brevity, and the severity factor, s. In this study, the representative wave height is fixed to 7.34 meters, so changes in the long-term probability distribution of storms, P(HS), are addressed by modifying the severity factor, s.22 The selection of a particular value for the representative wave height can affect the results in an undesirable way. To avoid this effect, the resulting wave run-up is presented in its non-dimensional form. Additionally, the short-term probability distribution of individual wave heights within a storm is defined by the use of a stochastic variable, U, uniformly distributed from 0 to 1, which accounts for the Rayleigh distribution of individual waves (see Equations 2.23 and 2.24). 3.1.3.2. Wave Period For an accurate representation of random sea conditions, defining a joint probability distribution of wave height and wave period is necessary. Unfortunately, finding such probability distribution brings additional difficulties to the study, which are avoided here by using two simple assumptions for the wave period. These assumptions are used 31 alternatively, and consist of constant wave period of 12 seconds and constant wave steepness parameter (HJgT2), defined in Equation 2.27. 3.1.3.3. Duration and Frequency of Storms Despite not being explicitly included in Equations 2.32, 2.39, 2.45 and 2.49, two variables play an important role in the resulting wave run-up for a given return period. These are storm duration, x, and recording interval, r. Their selection requires special care when studying the wave run-up of a particular structure located on a particular site. The storm duration, T, in conjunction with the wave period, T, determines the number of waves in a storm. This number of waves, n, is included in the performance functions used in this study (see Equations 2.29, 2.35, 2.41 and 2.47). In Method I, for instance, the maximum individual wave height is directly related to nP The recording interval, r, is a measure of the frequency of storms for a site. It determines the number of storms in a year, and, therefore, affects the value of the return period or annual risk (see Equations 2.25 and 2.26). In addition, it affects the parameters used for describing the long-term probability distribution of storms, P(HS) (see Equations 2.2l.c and 2.21 .d). For this study, both T and r are taken to equal 3 hours. This allows the comparison of different methods of calculation as long as the two variables are used consistently. However, while the relative trends of the results did not change regardless of their value, rand r proved to affect the absolute values of the wave run-up.24 In addition, a proper selection of r ensures the accuracy of combining wave run-up with other phenomena, tides for example. In such cases, r represents the arrival time of storms for the eventual application of Poisson's processes. Z 2 Table 2.1 presents how the parameters characterizing the probability distribution of storms are obtained from Ht and s. 2 3 For Method I: Hm = Hs (0.5 In (n))in. 2 4 Isaacson and Foschi (2000) refer to the findings of Battjes and Nolte, and provide a practical rule, recommending that r must be taken long enough to ensure independent measurements of Hs. 32 The results of the study are presented in Tables 3.1 to 3.7 and Figures 3.1 to 3.10 for the assumption of an Extreme Type III (Weibull) probability distribution of storms. These results differ little from those obtained when using an Extreme Type I (Gumbel) probability distribution of storms. 3.2. C O M P A R I S O N OF S O L U T I O N M E T H O D S The results from the sets of tests presented in Table 2.2 allow us to compare the methods for calculating wave run-up -Method I (M I), Method II (M II), F O R M and SORM- on the four structures studied here. In general, Method I provides lower values for maximum wave run-up than the other methods, as can be seen in Figures 3.1 to 3.4. These figures also show that F O R M and S O R M give very similar results. Method II provides higher values than the other methods when the severity factor is high. This tendency increases with the severity factor, giving very high values to the non-dimensional wave run-up, y, when the severity equals 1.4. Method II provides a minimum value of the non-dimensional wave run-up for intermediate severity factors (s from 1.1 to 1.2). Up to this minimum value, F O R M and SORM results coincide with Method II. The results from F O R M and SORM are in agreement with Method II for low values of the severity factor (up to 1.1 in the case of the vertical wall and the vertical cylinder). For higher values of the severity factor, F O R M and SORM results differ from Method II and approach the results from Method I. Since the severity factor is a measure of how widespread the probability distribution of Hs is (Figure 3.11), it can be said that y, as obtained by F O R M and SORM, has greater variability when Hs tends to be uniform (low severity factor). On the other hand, when Hs differs drastically from one return period to another (high severity factor), F O R M and SORM results for y tend to be constant. In other words, for these higher values of the severity factor, the wave run-up associated with a return period, R, changes just as Hs does, and the non-dimensional wave run-up (y=R/H) gets closer to the constant value provided by Method I. 33 Although Figures 3.1 to 3.4 correspond to specific values of the return period, TR, the same results of the non-dimensional run-up apply to other values of TR considered in this study. This is because y does not change significantly with TR, as can be seen in Figures 3.5, 3.6, 3.7 and 3.9. The results for the vertical wall and the cylinder present similar tendencies, as if they were shifted up or down, depending on the cylinder radius, and reaching maximum values for the seawall (Figures 3.1 and 3.2). On the other hand, in the cases of smooth and rough slopes, for low severity factors, the results from F O R M and SORM are higher than those obtained from Method II. When the severity increases, F O R M and SORM results approach the values from Method I, as in the seawall and the cylinder. For the sloping structures, however, y, calculated from Method I, is not constant as in the other two cases but decreases when the severity factor increases (Figures 3.3 and 3.4). 3.3. FORM AND SORM RESULTS FOR THE VERTICAL WALL CASE For the vertical wall, the non-dimensional wave run-up obtained with F O R M and SORM is almost constant despite changes in the return period, and decreases when the severity factor increases (see Table 3.1 and Figures 3.5 and 3.6). The two assumptions for the wave period provide similar results. However, the small changes in y that occur within different return periods are more noticeable for the constant wave steepness assumption (Figure 3.6) than for the constant wave period assumption (Figure 3.5). The effect of the severity factor on the non-dimensional wave run-up is higher for severity factors up to 1.2. As the severity increases, y tends to become uniform. In general y ranges from about 1.94 (for s=1.4) to about 2.39 (for 5=1.05) (Table 3.1). 34 3.4. FORM AND SORM RESULTS FOR THE VERTICAL CYLINDER CASE In the case of the vertical cylinder, as in the vertical wall, the return period is not an important factor in the resulting non-dimensional wave run-up, as presented in Tables 3.2 and 3.3. Also in this case, ydecreases when the severity increases. Figure 3.7 shows the effect of the radius of the cylinder, a, in the non-dimensional wave run-up for the constant wave period assumption. The wave run-up is higher for larger radiuses, as expected. Changes in the wave run-up with the cylinder radius are very similar for all the calculation methods (M I, M II, F O R M and SORM) and for different values of the severity factor (Tables 3.2 and 3.3). Figure 3.7 is an appropriate tool for solving the design problem. It provides, for a selected severity factor (corresponding to the conditions at a particular site) and a given return period, the possible combinations of cylinder radius and cylinder heights. This last can be obtained from y since Hs is known. Because only two cylinder radiuses were used in this study, the accuracy of this figure is limited. However, a graph of this sort can be improved by the addition of more data. 3.5. FORM AND SORM RESULTS FOR THE SMOOTH IMPERMEABLE SLOPE CASE Tables 3.4 and 3.5 present the resulting non-dimensional wave run-up of smooth impermeable slopes. Figure 3.8 shows how yis affected by changes in the severity factor, for two different values of the slope angle (1:3.5 and 1:6.0), and for the two wave period assumptions. As can be seen in this figure, a gentler slope cause less wave run-up than a steeper slope, but the effect of the severity factor on the non-dimensional wave run-up is very similar for both slopes. This effect, on the other hand, is more dramatic for the constant wave period assumption than for the constant wave steepness assumption, which is an expected outcome, since the second assumption implies a constant value of the surf-similarity parameter, b,. Figure 3.9 shows no important influence of the return period on the resulting non-dimensional wave run-up. The severity factor, on the other hand, does have an effect, 35 resulting in the different values of y and the different slopes of the lines presented in Figure 3.9. This figure shows that the slope angle affects y more dramatically when the severity factor is low. Figure 3.9 is an appropriate tool for solving the design problem in the case of smooth slopes. It provides combinations of slope angle and structure height that satisfy a certain return period for a selected severity factor. As in the case of the cylinder, the accuracy of this figure can be improved by adding more slope angles to the study. 3.6. FORM AND SORM RESULTS FOR THE ROUGH PERMEABLE SLOPE CASE As in the previous cases, the return period is not a factor affecting the non-dimensional wave run-up of rough permeable slopes. Furthermore, the results appear to be affected in the same manner by the two wave period assumptions (Tables 3.6 and 3.7). In this case, however, the effect of the severity factor on the non-dimensional wave run-up is less notorious than in smooth impermeable slopes. In addition, y does not change significantly with changes in the slope angle. It seems that the effect of the roughness and permeability itself diminishes the influences from other factors (Figure 3.10). Graphs of the same sort as Figure 3.9, for smooth impermeable slopes, can be developed to assess the design problem in the case of rough permeable slopes. An example is given in the next chapter. However, care must be taken in choosing appropriate values for the coefficients representing the structure's roughness and permeability, which appear to play a determinant role in the results. For this reason, physical models and prototype studies are commonly implemented. 36 4. E X A M P L E A P P L I C A T I O N 4.1. A N A L Y S I S P R O B L E M The reliability analysis method, presented in previous chapters for addressing wave run-up in random sea conditions, can be put into practice in the solution of a sample problem. Figure 2.6 illustrates the procedure for the analysis case. Let us consider the evaluation of three breakwaters of slope 1:3.5 and heights: 12.5 m, 12.75 m and 13.00 m above SWL, in order to find their respective annual risk. These structures correspond to a rough permeable slope case. Let us assume that Gunbak's formula (Equation 2.15), with bi=0.8 and b2=0.5 representing roughness and permeability, is appropriate for the three structures. Following Figure 2.6, the first step in solving this problem is to define the site sea conditions. For this example, these conditions are assumed as: • T = r = 3 hours • HS(TR=1)= 7.34 m • s = l.l • T= 12 sec • a = cot"1(3.5) (slope 1:3.5) • R, = 12.5 m; R2 = 12.75 m; R3 = 13.00 m25 The R E L A N program renders the probability of exceedance during a storm. Using SORM the results are: Note that in this chapter wave run-up, R, and structure's height above S W L are considered equal. 37 • QJR)i= 1.73 x Iff5; Qr(R)2= 1.09x Iff5; QT(R)J= 6.91 x Iff6 Using Equation 2.2626, the resulting annual risks and return periods are: • QA(R), = 0.049; QA(R)2= 0.031; QA(R)s= 0.020 • TRI= 20.3 years; TR2= 32.0 years; TR3= 50.1 years The result for the third structure corresponds to Table 3.6. Usually the storms are not as close to each other. Let us see what happens if the recording interval, r, is changed to 30 hours. It would modify the distribution for HS (Equation 2.19) and the relationship QT(R)-QA(R) (Equation 2.26). The results from R E L A N , using SORM, are, for the probability of exceedance during a storm: • Q m = L46x Iff4; Q^R)2= 9.28 x IO'5; QJR)3= 5.95 x Iff5 The correspondent annual risks and return periods are: • QA(R)J= 0.042; QA(R)2= 0.027; QA(R)3= 0.017 • TR/= 24.0 years; TR?= 37.4 years; TR3= 58.1 years 4.2. DESIGN PROBLEM For the design problem, let us change the severity factor of the example to 1.05, and find the characteristics of a breakwater designed for a return period of 50 years. Figure 2.7 illustrates the procedure for the design problem. The conditions are: • T = r = 3 hours • HS(TR=I)= 7.34 m • s = 1.05 Note that Qr(R).\s the probability of R being exceeded during a storm. It can be related to the annual risk by using the number of storms within a year, as in Equation 2.26, in which the return period is the inverse of the annual risk (QA(R) =1/Tr) 38 • T= 12 sec • TR = 50 years From Table 3.6 it can be seen that a combination of slope 1:3.5 and R = 13.47 m 2 7 or slope 1:6 and R = 9.70 m satisfies the imposed conditions. However, let us study two other alternatives, applying SORM. These are: Slope = 1:4 and Slope = 1:5 Following Figure 2.7, and using R E L A N , the corresponding wave run-ups for a return period of 50 years are: • For Slope = 1:4, R = 12.50 m • For Slope = 1:5, R = 10.92 m These results are plotted on Figure 4.1, using Hs = 7.93 m from Table 3.6. Depending on other technical, practical and functional criteria, one of the four alternatives (slope 1:3.5 -R = 13.47 m, slope 1:4 -R = 12.50 m, slope 1:5 -R = 10.92 m, slope 1:6 - R = 9.70 m) could be selected. Note that in this chapter wave run-up, R, and structure's height above SWL are considered equal. 39 5. CONCLUSIONS AND RECOMMENDATIONS Using reliability analysis, this study addresses the maximum wave run-up that can be expected within a certain return period in four types of coastal and ocean structures. In the study, the First and Second Order Reliability Methods have been compared with two other methodologies that are commonly used in coastal engineering, referred to here as Method I and Method II. The first method consists of calculating the significant wave height for a given return period from the long-term distribution of storms; then, the maximum wave height corresponding to the significant wave height, and its resultant wave run-up are obtained. Method II consists of obtaining a long-term distribution of individual wave heights, so that the maximum individual wave height for a given return period can be directly calculated. The corresponding wave run-up is then determined as in Method I by applying regular wave formulations to that maximum wave height. In addition, the effects of structural characteristics and parameters representing sea conditions in the wave run-up have been investigated. The interpretation of the results led to the following conclusions and recommendations. The ratio of the wave run-up to the significant wave height is independent of the return period for the four cases studied here. Thus, this ratio, regarded here as non-dimensional wave run-up, provides useful means for studying the phenomenon. The severity factor, characterizing the long-term probability distribution of storms, plays an important role in the resulting wave run-up. However, its effect varies, depending on the method of solution adopted. For the procedure here called Method I, the non-dimensional wave run-up of vertical wall and vertical cylinder structures is constant whatever the return period or the severity factor. In fact, in this case, it is a direct result of the Rayleigh distribution of individual wave heights during a storm, only affected by the number of waves in the storm. For sloping 40 structures, the non-dimensional wave run-up obtained by Method I decreases at a constant rate when the severity increases. In all cases, the results from Method I are lower than those obtained with the other methods. The results from Method II are higher than those from other methods. Moreover, Method II is highly affected by the severity factor, s, providing results for the non-dimensional wave run-up that decrease when increasing the severity, approximately until s equals 1.2. At this point, the non-dimensional wave run-up reaches a minimum and thereafter it increases with the severity factor. This method provides very high values in the non-dimensional wave run-up for larger values of the severity. The results of F O R M and SORM are similar to each other and depend on the severity factor. For low severity factors, the results from F O R M and SORM are in accordance with Method II, up until s equals approximately 1.1. For larger values of the severity factor, however, F O R M and S O R M differ from Method II and their results approach those from Method I. The results for the vertical wall and the vertical cylinder cases follow relatively similar trends. The absolute values of the non-dimensional wave run-up, however, are different, being higher for the wall and lower for the cylinder. In this last case, the wave run-up declines as the cylinder radius decreases. In the case of sloping structures, for low values of the severity factor, F O R M and SORM results are higher than those from Method II. In these structures, the severity factor has an influence on the way the slope angle affects the resulting wave run-up. For rough permeable slopes, severity factor and slope angle affect'the results. However, it is the roughness and permeability of the structure that play the most important role in the resulting non-dimensional wave run-up. A proper selection of storm duration and storm time arrival is crucial for attaining an accurate application of the reliability method to a real structure situation. This is even more important when combining other features with the wave run-up, since, in those cases, assessing the arrival of the different phenomena and their probability of occurring at the same time is fundamental. 41 Incorporating other phenomena into the study, such as tides, set-up and storm surge, is advisable for a proper assessment of the risk of a structure being overtopped or the maximum absolute level that the water reaches on the structure 42 6. REFERENCES Battjes, J. A. , (1971). "Run-up Distributions ofWaves Breaking on Slopes", Journal of the Waterways, Harbors and Coastal Engineering Division, ASCE, 97 (1), pp. 91-114. Bruun, P., (1985). "Design and Construction of Mounds for Breakwaters and Coastal Protection", Elsevier Science Publishing Company Inc, Amsterdam. Dodd, N . , (1998). "Numerical Model of Wave Run-Up, Overtopping, and Regeneration", Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE, 124 (2), pp. 73-81. Foschi, R., Isaacson, M . , Allyn, N . , and Yee, S., (1996). "Combined Wave-Iceberg Loading on Offshore Structures", Canadian Journal of Civi l Engineering, 23 (5), pp. 1099-1110. Foschi, R., Isaacson, M . , Allyn, N . , and Saudy, I., (1998). "Assessment of the Wave-Iceberg Load Combination Factor", International Journal of Offshore and Polar Engineering, 8(1), pp. 1-8. Ganapathy, S., (1997). "On the Estimation of Design Waves Forces", M . A. Sc. Thesis, The University of British Columbia, Vancouver. Isaacson, M . (1978). "Wave Run-up Around Large Circular Cylinder", Journal of the Waterway Port Coastal and Ocean Division, ASCE, 104 (1), pp. 69-79. Isaacson, M . and Foschi, R. (1996). "On the Return Period of Design Wave Heights", International Conference in Ocean Engineering COE'96, Madras, India, pp. 491-495. Isaacson, M . and Foschi, R. (2000). "On the Selection of Design Wave Conditions", International Journal of Offshore and Polar Engineering, 10 (2), pp. 99-106. Mei, C , (1983), "The Applied Dynamics of Ocean Surface Waves", John Wiley & Sons, New York. Pilarczyk, K. and Zeidler, B., (1996), "Offshore Breakwaters and Shore Evolution Control", A. A . Balkema, Rotterdam. 43 Roos, A . and Battjes, J., (1976). "Characteristics of Flow in Run-Up of Periodic Waves", Proceedings of the Fifteenth Coastal Engineering Conference, ASCE, Honolulu, Hawaii, pp. 781-795. Department of the Army, Waterways Experiment Station, Corps of Engineers, U.S. Army Coastal Engineering Research Center, (1984), "Shore Protection Manual", (2), Fourth Edition, Fort Belvoir, Virginia. Sarpkaya, T. and Isaacson, M . , (1981), "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold Company, New York. Sorensen, R., (1997). "Basic Coastal Engineering", Chapman & Hall, New York. Van Dorn, W., (1976). "Set-Up and Run-Up in Shoaling Breakers", Proceedings of the Fifteenth Coastal Engineering Conference, ASCE, Honolulu, Hawaii, pp. 738-751. 44 Table 2.1. Parameters of the Sea State Distributions Used in this Study"' H1 (m) 7.34 7.34 7.34 7.34 7.34 7.34 s 1.05 1.1 1.3 1.05 1.1 1.3 H10 (m) 7.71 8.07 9.54 7.71 8.07 9.54 Type of Distribution for H s Type III Type I a 2.53 x 10"4 3.97 x 10"2 1.16 6.27 3.14 1.05 b 5.20 2.66 0.97 -38.08 -15.05 0.30 m 4.92 3.36 0.86 6.27 3.14 1.05 k 5.20 2.66 0.97 6.07 4.80 -0.29 Table 2.2. Summary of Case Studies, Variables and Assumptions29 Case Study Describing T Describing H s Describing Structure Description constant wave period constant wave steepness H, s a a 1 On Off F V Vertical Wall 2 Off On F V 3 On Off F V F Vertical Cylinder / varying severity 4 Off On F V F 5 On Off F F V Vertical Cylinder / effect of diameter 6 Off On F F V 7 On Off F V F Smooth Impermeable Slopes 8 Off On F V F 9 On Off F F V Smooth Impermeable Slopes / effect of slope angle 10 Off On F F V 11 On Off F V F Rough Permeable Slopes 12 Off On F V F 13 On Off F F V Rough Permeable Slopes / effect of slope angle 14 Off On F F V z s Note that r = 3 hours, m and k correspond to the scale and shape parameters used for describing the Weibull distribution. Location = 0 was also used. For the Gumbel distribution case (Type I), m and k correspond to the parameters A and B characterizing the distribution. Some adjustments are made for the distribution in R E L A N , where the Type III is defined slightly differently than in equation 2.19. This table uses Equation 2.21 for obtaining a and b. 29 V stands for 'variable', meaning that more than one value of the referred variable is used. F stands for 'fixed', meaning that just one value of the referred variable is used. 45 Table 3 .1. Non-dimensional Wave Run-up of Vertical Wall T (sec) s I R (years) Y H s (m) M I M II F O R M S O R M 12 1.05 30 1.84 2.33 2.34 2.32 7.86 50 1.84 2.35 2.36 2.34 7.93 100 1.84 2.38 2.39 2.37 8.01 12 1.1 30 1.84 2.14 2.14 2.14 8.39 50 1.84 2.15 2.16 2.14 8.53 100 1.84 2.17 2.18 2.16 8.71 12 1.2 30 1.84 2.07 2.01 2.02 9.48 50 1.84 2.07 2.02 2.03 9.78 100 1.84 2.07 2.03 2.04 10.19 12 1.3 30 1.84 2.13 1.97 1.98 10.60 50 1.84 2.12 1.97 1.99 11.09 100 1.84 2.10 1.98 1.99 11.76 12 1.4 30 1.84 2.25 1.95 1.96 11.76 50 1.84 2.22 1.95 1.97 12.46 100 1.84 2.20 1.96 1.97 13.44 4.43(Hs) 1' 2 1.05 30 1.84 2.33 2.34 2.32 7.86 50 1.84 2.35 2.36 2.34 7.93 100 1.84 2.38 2.39 2.39 8.01 4.43(Hs) 1 / l ! 1.1 30 1.84 2.14 2.14 2.13 8.39 50 1.83 2.15 2.15 2.15 8.53 100 1.83 2.17 2.17 2.18 8.71 4.43(Hs) 1 / 2 1.2 30 1.83 2.05 2.00 2.00 9.48 50 1.82 2.05 2.01 2.02 9.78 100 1.82 2.05 2.02 2.02 10.19 4.43(Hs) 1' 2 1.3 30 1.82 2.08 1.95 1.96 10.60 50 1.82 2.07 1.95 1.96 11.09 100 1.81 2.06 1.96 1.97 11.76 4.43(Hs) 1 / i ; 1.4 30 1.81 2.16 1.92 1.93 11.76 50 1.81 2.14 1.92 1.94 12.46 100 1.80 2.11 1.92 1.94 13.44 46 Table 3.2. Non-dimensional Wave Run-up of Vertical Cylinder for Constant Wave Period T=12 seconds s Radius (m) I R (years) Y H S (m) M I M II FORM SORM 1.05 30 30 1.49 1.88 1.89 1.87 7.86 50 1.49 1.90 1.90 1.89 7.93 100 1.49 1.92 1.93 1.91 8.01 1.1 30 30 1.49 1.73 1.73 1.72 8.39 50 1.49 1.74 1.74 1.73 8.53 100 1.49 1.75 1.76 1.74 8.71 1.2 30 . 30 1.49 1.67 1.62 1.63 9.48 50 1.49 1.67 1.63 1.64 9.78 100 1.49 1.67 1.64 1.65 10.19 1.3 30 30 1.49 1.72 1.59 1.60 10.60 50 1.49 1.71 1.59 1.60 11.09 100 1.49 1.70 1.60 1.61 11.76 1.4 30 30 1.49 1.81 1.57 1.58 11.76 50 1.49 1.79 1.58 1.59 12.46 100 1.49 1.77 1.58 1.59 13.44 1.05 60 30 1.69 2.13 2.14 2.13 7.86 50 1.69 2.16 2.17 2.15 7.93 100 1.69 2.18 2.19 2.17 8.01 1.1 60 30 1.69 1.96 1.96 1.96 8.39 50 1.69 1.98 1.98 1.96 8.53 100 1.69 1.99 2.00 1.98 8.71 1.2 60 30 1.69 1.90 1.84 1.85 9.48 50 1.69 1.90 1.85 1.86 9.78 100 1.69 1.90 1.86 1.87 10.19 1.3 60 30 1.69 1.95 1.80 1.82 10.60 50 1.69 1.94 1.81 1.82 11.09 100 1.69 1.93 1.82 1.83 11.76 1.4 60 30 1.69 2.06 1.79 1.80 11.76 50 1.69 2.04 1.79 1.81 12.46 100 1.69 2.02 1.80 1.81 13.44 47 Table 3.3. Non-dimensional Wave Run-up of Vertical Cylinder for T=4.43(HS) (Constant Wave Steepness) s Radius (m) I R (years) Y H s (m) M I M II FORM SORM 1.05 30 30 1.46 1.85 1.93 1.92 7.86 50 1.46 1.86 1.95 1.94 7.93 100 1.45 1.88 1.97 1.96 8.01 1.1 30 30 1.43 1.67 1.73 1.73 8.39 50 1.42 1.67 1.74 1.74 8.53 100 1.42 1.68 1.75 1.74 8.71 1.2 30 30 1.38 1.55 1.55 1.55 9.48 50 1.37 1.53 1.55 1.55 9.78 100 1.35 1.52 1.54 1.54 10.19 1.3 30 30 1.34 1.53 1.46 1.46 10.60 50 1.32 1.50 1.44 1.45 11.09 100 1.30 1.47 1.43 1.44 11.76 1.4 30 30 1.30 1.55 1.39 1.40 11.76 50 1.28 1.51 1.38 1.39 12.46 100 1.25 1.46 1.35 1.36 13.44 1.05 60 30 1.68 2.13 2.15 2.12 7.86 50 1.68 2.15 2.17 2.14 7.93 100 1.68 2.17 2.20 2.17 8.01 1.1 60 30 1.67 1.94 1.96 1.96 8.39 50 1.66 1.95 1.98 1.98 8.53 100 1.66 1.96 1.99 1.97 8.71 1.2 60 30 1.63 1.83 1.81 1.81 9.48 50 1.62 1.81 1.81 1.81 9.78 100 1.60 1.80 1.81 1.80 10.19 1.3 60 30 1.59 1.82 1.72 1.73 10.60 50 1.57 1.79 1.71 1.71 11.09 100 1.54 1.75 1.70 1.70 11.76 1.4 60 30 1.54 1.84 1.66 1.66 11.76 50 1.52 1.80 1.64 1.64 12.46 100 1.49 1.74 1.61 1.62 13.44 48 Table 3.4. Non-dimensional Wave Run-up of Smooth Impermeable Slope for Constant Wave Period T=12 seconds s I an a I R (years) Y H s (m) M I M II FORM SORM 1.05 1/3.5 30 2.82 3.55 3.90 3.88 7.86 50 2.81 3.57 3.92 3.91 7.93 100 2.79 3.60 3.96 3.95 8.01 1.1 1/3.5 30 2.73 3.17 3.47 3.44 8.39 50 2.71 3.16 3.47 3.44 8.53 100 2.68 3.15 3.48 3.45 8.71 1.2 1/3.5 30 2.57 2.88 2.99 2.98 9.48 50 2.53 2.83 2.97 2.95 9.78 100 2.48 2.78 2.94 2.93 10.19 1.3 1/3.5 30 2.43 2.80 2.72 2.72 10.60 50 2.37 2.72 2.67 2.67 11.09 100 2.30 2.63 2.61 2.62 11.76 1.4 1/3.5 30 2.30 2.81 2.53 2.54 11.76 50 2.24 2.70 2.46 2.47 12.46 100 2.16 2.57 2.39 2.40 13.44 1.05 1/6 30 1.64 2.07 2.27 2.26 7.86 50 1.64 2.08 2.29 2.28 7.93 100 1.63 2.10 2.31 2.30 8.01 1.1 1/6 30 1.59 1.85 2.02 2.01 8.39 50 1.58 1.84 2.03 2.01 8.53 100 1.56 1.84 2.03 2.01 8.71 1.2 1/6 30 1.50 1.68 1.75 1.74 9.48 50 1.47 1.65 1.73 1.72 9.78 100 1.44 1.62 1.71 1.71 10.19 1.3 1/6 30 1.42 1.63 1.59 1.59 10.60 50 1.38 1.59 1.56 1.56 11.09 100 1.34 1.53 1.50 1.50 11.76 1.4 1/6 30 1.34 1.64 1.47 1.48 11.76 50 1.31 1.57 1.44 1.44 12.46 100 1.26 1.50 1.39 1.40 13.44 49 Table 3.5. Non-dimensional Wave Run-up of Rough Permeable Slope for T=4.43(H S) 1 / 2 (Constant Wave Steepness) s Ian a I R (years) Y H s (m) M I M II FORM SORM 1.05 1/3.5 30 2.91 3.68 3.70 3.67 7.86 50 2.91 3.72 3.74 3.70 7.93 100 2.91 3.76 3.79 3.77 8.01 1.1 1/3.5 30 2.90 3.38 3.38 3.37 8.39 50 2.90 3.40 3.41 3.40 8.53 100 2.90 3.43 3.44 3.44 8.71 1.2 1/3.5 30 2.89 3.24 3.16 3.17 9.48 50 2.89 3.24 3.17 3.19 9.78 100 2.88 3.24 3.19 3.20 10.19 1.3 1/3.5 30 2.88 3.29 3.08 3.10 10.60 50 2.87 3.27 3.09 3.11 11.09 100 2.87 3.25 3.09 3.11 11.76 1.4 1/3.5 30 2.87 3.42 3.04 3.06 11.76 50 2.86 3.38 3.04 3.06 12.46 100 2.85 3.34 3.04 3.06 13.44 1.05 1/6 30 1.70 2.15 2.16 2.14 7.86 50 1.70 2.17 2.18 2.16 7.93 100 1.70 2.20 2.21 2.20 8.01 1.1 1/6 30 1.69 1.97 1.97 1.96 8.39 50 1.69 1.98 1.99 1.98 8.53 100 1.69 2.00 2.01 2.01 8.71 1.2 1/6 30 1.69 1.89 1.84 1.85 9.48 50 1.68 1.89 1.85 1.86 9.78 100 1.68 1.89 1.86 1.87 10.19 1.3 1/6 30 1.68 1.92 1.80 1.81 10.60 50 1.68 1.91 1.80 1.81 11.09 100 1.67 1.90 1.80 1.82 11.76 1.4 1/6 30 1.67 2.00 1.77 1.78 11.76 50 1.67 1.97 1.77 1.79 12.46 100 1.66 1.95 1.77 1.79 13.44 50 Table 3.6. Non-dimensional Wave Run-up of Smooth Impermeable Slope for Constant Wave Period T=12 seconds s I an a I R (years) ' Y H S (m) M I M II F O R M S O R M 1.05 1/3.5 30 1.28 1.61 1.69 1.68 7.86 50 1.28 1.62 1.70 1.70 7.93 100 1.27 1.64 1.72 1.72 8.01 1.1 1/3.5 30 1.25 1.46 1.52 1.52 8.39 50 1.25 1.46 1.53 1.52 8.53 100 1.24 1.46 1.53 1.54 8.71 1-2 1/3.5 30 1.21 1.36 1.36 1.36 9.48 50 1.20 1.34 1.35 1.35 9.78 100 1.19 1.33 1.35 1.35 10.19 1.3 1/3.5 30 1.17 1.35 1.27 1.28 10.60 50 1.15 1.33 1.26 1.27 11.09 100 1.13 1.29 1.25 1.25 11.76 1.4 1/3.5 30 1.13 1.38 1.22 1.22 11.76 50 1.11 1.34 1.20 1.21 12.46 100 1.09 1.30 1.18 1.18 13.44 1.05 1/6 30 0.91 1.15 1.22 1.21 7.86 50 0.91 1.16 1.23 1.22 7.93 100 0.90 1.17 1.24 1.24 8.01 1.1 1/6 30 0.89 1.03 1.09 1.09 8.39 50 0.88 1.03 1.09 1.09 8.53 100 0.88 1.03 1.10 1.09 8.71 1.2 1/6 30 0.85 0.95 0.96 0.96 9.48 50 0.84 0.94 0.96 0.96 9.78 100 0.83 0.93 0.95 0.95 10.19 1.3 1/6 30 0.82 0.94 0.90 0.90 10.60 50 0.81 0.92 0.89 0.89 11.09 100 0.79 0.90 0.87 0.88 11.76 1.4 1/6 30 0.79 0.96 0.85 0.85 11.76 50 0.77 0.93 0.83 0.84 12.46 100 0.75 0.89 0.82 0.82 13.44 51 Table 3.7. Non-dimensional Wave Run-up of Rough Permeable Slope for T=4.43(HS)1/2 (Constant Wave Steepness) s I an a I R (years) Y H s (m) M I M II FORM SORM 1.05 1/3.5 30 1.30 1.64 1.65 1.64 7.86 50 1.30 1.66 1.67 1.65 7.93 100 1.30 1.68 1.69 1.69 8.01 1.1 1/3.5 30 1.30 1.51 1.51 1.50 8.39 50 1.30 1.52 1.52 1.52 8.53 100 1.29 1.53 1.54 1.54 8.71 1.2 1/3.5 30 1.29 1.45 1.41 1.41 9.48 50 1.29 1.45 1.42 1.42 9.78 100 1.29 1.45 1.42 1.43 10.19 1.3 1/3.5 30 1.29 1.47 1.38 1.39 10.60 50 1.28 1.46 1.38 1.39 11.09 100 1.28 1.45 1.38 1.39 11.76 1.4 1/3.5 30 1.28 1.53 1.36 1.37 11.76 50 1.28 1.51 1.36 1.37 12.46 100 1.27 1.49 1.36 1.37 13.44 1.05 1/6 30 0.93 1.18 1.18 1.17 7.86 50 0.93 1.19 1.19 1.18 7.93 100 0.93 1.20 1.21 1.21 8.01 1.1 1/6 30 0.93 1.08 1.08 1.08 8.39 50 0.93 1.09 1.09 1.08 8.53 100 0.93 1.10 1.10 1.10 8.71 1.2 1/6 30 0.92 1.03 1.01 1.01 9.48 50 0.92 1.03 1.02 1.02 9.78 100 0.92 1.03 1.02 1.03 10.19 1.3 1/6 30 0.92 1.05 0.98 0.99 10.60 50 0.92 1.05 0.99 0.99 11.09 100 0.92 1.04 0.99 0.99 11.76 1.4 1/6 30 0.92 1.09 0.97 0.98 11.76 50 0.91 1.08 0.97 0.98 12.46 100 0.91 1.07 0.97 0.98 13.44 52 Figure 2.1. Vertical Wall Geometry 53 Figure 2.4. Rough Permeable Slope Geometry 5 Q. o 5 Q. o z 5 LU DC 8?I ° = 5>l 5? "8 ' 1 — J3 _ 1- O O QI Q LU O < 1 < o I— > Q IL 2 < Z Q O < Q_ < O I— > Figure 3.1. Comparison of Calculation Methods for Non-dimensional Wave Run-of Vertical Wall. TR=30 years and Constant Wave Period T=12 sec Figure 3.2. Comparison of Calculation Methods for Non-dimensional Wave Run-of Vertical Cylinder. TR=30 years and Constant Wave Period T=I2 sec 57 4.0 3.5 3.0 2.5 2.0 1.5 1.0 1.0 1.1 1.2 1.3 severity factor 1.4 1.5 Figure 3.3. Comparison of Calculation Methods for Non-dimensional Wave Run-up of Smooth Impermeable Slopes. TR=30 years and Constant Wave Period T=12 sec. Figure 3.4. Comparison of Calculation Methods for Non-dimensional Wave Run-up of Rough Permeable Slope. TR=30 years and Constant Wave Period T=12 sec 58 2.4 2.3 2.2 2.1 2.0 1.9 20 -X-40 60 80 TR (years) 4 100 - e — s = 1.05 - 6 —s = 1.10 -X—s = 1.20 -B— s = 1.30 S = 1.40 120 Figure 3.5. SORM Results for Non-dimensional Wave Run-up on Vertical Wall as a Function of Return Period for Various Values of Severity Factor and Constant Wave Period T=12 sec 2.4 2.3 2.2 2.1 2.0 1.9 20 B--X-- s -40 60 80 T R (years) 100 -O— s = 1.05 -&— S = 1.10. -X— s = 1.20 -a— s = 1.30 -3K— s = 1.40 120 Figure 3.6. SORM Results for Non-dimensional Wave Run-up on Vertical Wall as a Function of Return Period for Various Values of Severity Factor and Wave Period T=4.43(H S) 1 / 2 (Constant Wave Steepness) 59 2.2 2.1 2.0 1.9 1.8 1.7 1.6 25 s=1.05 s=1.30 35 45 Radius (m) - - * --30y — B - -50y - 100 y 55 65 Figure 3.7. Effect of the Cylinder Radius on the Non-dimensional Wave Run-up of Vertical Cylinders, based on SORM Results. Various Return Periods, Constant Wave Period T=12 sec and Severity Factors of 1.05 and 1.30 1 1 slope: 1 :3.5 | — H — Constant Period — A — Constant Steepness slope: 1:6.0 ^—• 1 3 L \ 1.0 1.1 1.2 1.3 1.4 1.5 severity factor Figure 3.8. Effect of the Period Assumption on the Non-dimensional Wave Run-up of Smooth Impermeable Slopes as a Function of the Severity Factor, Based on S O R M Results, for Slopes of 1:3.5 and 1:6 and Return Period of 50 Years. 60 Figure 3.9. Effect of the Slope on the Non-dimensional Wave Run-up of Smooth Impermeable Slopes for Constant Wave Period T=12 sec, Based on S O R M Results, for Severity Factors of 1.05 and 1.3 and Various Return Periods. -e—s=l.09 Cot a Figure 3.10. Effect of the Roughness and Permeability on the Non-dimensional Wave Run-up of Sloping Structures for Constant Wave Period T=12 sec, Based on S O R M Results for Slopes of 1:3.5 and 1:6, Various Severity Factors and Return Period of 30 Years. 61 Figure 3.11. Effect of the Severity Factor on the Probability Distribution Storms in Terms of the Return Period for an Extreme Type III (Weibull) Distribution, Assuming H s = 7.34 m for a One-year Return Period 1.9 1.8 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Cot a Figure 4.1.SORM Results for Non-dimensional Wave Run-up of Rough Permeable Slopes, Assuming Constant Wave Period T=12 sec, Severity Factor of 1.05 and Return Period of 50 Years. 62
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Reliability analysis of wave run-up Curi, Fuad 2002
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Title | Reliability analysis of wave run-up |
Creator |
Curi, Fuad |
Date Issued | 2002 |
Description | Assessing the run-up arising from ocean waves as they reach a coastal or offshore structure is an important aspect of engineering projects. By combining formulations for calculating the run-up of regular waves with wave statistical descriptions, a procedure is implemented to assess wave run-up in random sea conditions. This procedure uses the First and Second Order Reliability Methods. It is applied to four cases representing common coastal structures: vertical wall, vertical cylinder, smooth impermeable slope and rough permeable slope. The results of the reliability methods are compared with two other methodologies commonly used in coastal and ocean engineering. The first methodology (Method I) consists of calculating the significant wave height for a given return period from the long-term distribution of storms. Then, the maximum wave height corresponding to the significant wave height, and its corresponding wave run-up are obtained. Method II consists of obtaining a long-term distribution of individual wave heights, so that the maximum wave height for a given return period can be directly calculated. Then, the corresponding wave run-up is determined as in Method I, by applying regular wave formulations to the maximum wave height. Method I provides lower values of wave run-up than all other methods, while Method II provides the highest values of wave run-up, and proves to be dependent on the severity factor, a parameter characterizing the long-term distribution of storms. Results show that both the First and Second Order Reliability Methods correspond with Method II when the values of the severity factor are low, but approach Method I as the value of the severity factor increases. The wave run-up is found to depend on the characteristics of the structure in place, such as cylinder radius, slope angle and surface permeability, as well as on such sea conditions as severity factor and the duration and frequency of storms. The wave run-up to significant wave height ratio is independent of the return period for the four cases studied here. This ratio results in a non-dimensional form of wave run-up that appears to provide useful means for describing the phenomenon. |
Extent | 2956341 bytes |
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Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-08-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0063497 |
URI | http://hdl.handle.net/2429/12054 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2002-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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