Reliability based design method for coastal structures in shallow seas

Transcription

1 Indian Journal of Geo-Marine Sciences Vol. 39(4), December 2010, pp Reliability based design method for coastal structures in shallow seas B Prasad Kumar Department of Ocean Engineering & Naval Architecture, Indian Institute of Technology, Kharagpur , India [ pkbhaskaran@naval.iitkgp.ernet.in] Received 20 August 2010; revised 24 December 2010 Reliability based approach is considered an important component in risk assessment of coastal related projects. Such an approach is mandatory during the initial planning stage of coastal projects thereby providing information on the failure of structures. The damage to a structure can result from various environmental forcing and moreover depends on where the structure is located. As a case study, an existing coastal jetty at Andaman Sea was chosen which was subjected to extreme waves during a cyclonic episode prevalent in the Bay of Bengal. Wave information encompassing this coastal jetty was investigated using a very high resolution state-of-art SWAN wave model run in non-stationary mode. Wave forces are estimated using Sainflou and Miche-Rundgren methods along structural length of this jetty. In addition, wave run-up and wave overtopping were estimated using SWAN model generated wave information. A Level-II reliability analysis was carried out for this coastal jetty, subjected to various levels of assumed uncertainty. Further, lifetime and encounter probability under various degrees of uncertainty in load and resistance for this coastal jetty has been investigated and analyzed. Based on analysis, it is believed the significance of this work can lead vital information for assessment studies in integrated coastal zone management. [Keywords: Coastal structures, Extreme events, Design method, Reliability analysis, Wave forces] Introduction Planning, design, construction and operation of marine activities, offshore and coastal protection of marine structures requires precise knowledge of surface waves and their characteristics. The structural life and probability of failure for a coastal structure can be assessed based on environmental loadings resulting from catastrophic events in the marine environment. Hence reliability based approach in planning and construction of coastal structures will be of immense importance in the economic point of view. It should be noted that there are design codes in the field of coastal engineering and true solutions are site specific. One can find details of resistance and stress due to hydraulic loads and reliability aspects in the published work of Mai and Von Lieberman 1. The various components involved in determination of reliability aspects are shown in Figure-1. There exists various design formula to estimate wave forces on coastal structures. Some of the popular formulations are Sainflou 2, Goda 3 and Minikin 4. For instance, the formula by Goda 3 is quite popular and used as the design standard in Japan, whereas Sainflou 2 formula is widely used in USA. These formulas have been skill assessed for case of vertical sea walls. The problem definition in the present work is a case where wave breaking does not occur on the vertical sea wall, such that Sainflou 2 formula can be used with a good degree of confidence. Model experiments for steep waves show that Sainflou 2 method has a tendency to over-estimate wave forces, which can eventually be corrected using Miche-Rundgren approach 5. Hence it could be advocated that Sainflou 2 formula is best suited for long and less steep waves in absence of structural Fig. 1 Flow chart of the various components involved in Reliability based Design System.

2 606 INDIAN J. MAR. SCI., VOL. 39, NO. 4, DECEMBER 2010 over-topping and wave run-up past the structure. In case of extreme events, the possibility of wave run-up and structural over-topping is quite common which depend on magnitude of wave breaking. This can affect the structure and result in severe failure mechanism. One can find information about such failures of coastal structures reported in European countries, documented in the work of Bleck et al. 6. It should be mentioned the best design practice as of today can only suggest minimizing wave overtopping but not completely avoid it. Site specific model testing is not practical for preliminary design due to time and cost involved. Hence to a large extent engineers depend on well tested formula to estimate overtopping discharges. In Table-1 the listing of various models to estimate wave over-topping is shown based on the work by Burchartch and Hughes 7. Discharge (m 3 s -1 m -1 ) One needs to be careful in estimating the wave runup over the coastal structure. Over-estimation can lead to significant cost factor during the construction stage of the structure. The bathymetric features surrounding the coastal jetty (structural aspects shown in Table-2) consist of a mild slope as one advance offshore from the structural face (shown in Figure-2). For a mild slope the formula by Ahrens and Titus 8 can be used with a good degree of confidence. To assess the impact of extreme waves on this coastal jetty, synthetic track of cyclone Fanoos which occurred during December 2005 has been used. The track and other relevant details of this cyclone were obtained from the best track data of the Joint Typhoon Warning Centre (JTWC) records. Replication of the 2005 synthetic track of cyclone Fanoos was assumed for the period of July 2006 Table 1 Criteria for critical overtopping discharge (from Burchartch and Hughes, 2003) Safety of traffic Structural Safety 10 0 seawalls Vehicles Pedestrians Building Embankment Grass sea dikes Revetment seawalls 10-1 Unsafe at any speed Very dangerous Structural Damage Damage even if fully protected Damage Damage even for paved promenade Damage if promenade not paved 10-2 Damage if back stop not protected No damage Unsafe parking on horizontal composite breakwater Unsafe parking on vertical wall breakwater Dangerous on grass sea dikes Dangerous on vertical wall breakwater Uncomfortable but not dangerous Damage if crest not protected No damage Start of damage No damage Unsafe driving at high speed Wet, but comfortable Minor damage to fittings, sign post, etc Safe driving at all speed No damage

3 KUMAR: RELIABILITY BASED DESIGN METHOD FOR COASTAL STRUCTURES IN SHALLLOW SEAS 607 Fig. 2 Bathymetric features surrounding the coastal structure (a) Coarse bathymetry, (b) Fine bathymetry. Table 2 Structural details of the coastal jetty Si. No. Description Value 1 Dimensions of approach wall Length: 106 m Width: 10.5 m 2 Dimensions of main wall Length: 115 m Width: 10.0 m 3 Freeboard 1.5 m 4 of rubble 0.75 m matching with availability of coastal buoy data. This provided an opportunity to critically evaluate wave forces, overtopping discharges and wave run-up due to extreme waves and in turn calculation of failure probability for the coastal jetty. The subsequent section described below deals with the materials and methods (Section-2) describing mathematical formulations used in the estimation of wave forces, over-topping discharges, wave run-up; the results and discussion (Section-3) followed by conclusions (Section-4). Materials and Methods The synthetic track of cyclone and location of a moored buoy AN-05 which provided the real-time wind data in close vicinity to the structural location is shown in Figure-3. During pre- and post cyclonic periods representative wind data surrounding coastal structure was taken from the moored buoy AN-05. During the cyclonic phase, cyclostrophic wind fields were estimated using the Holland s parametric model. The wave model SWAN was used to estimate wave parameters. One can find a detailed description of SWAN wave model in the published work of Booij et al 9. SWAN was run in a nested mode, where the boundary spectrum of coarser grid is input to the finer grid, which being the domain of interest. The Fig. 3 Actual track of Cyclone Fanoos based on JTWC records during December bathymetric information was obtained from National Geophysical Data Centre (NGDC) of National Oceanic and Atmospheric Administration (NOAA), USA. The coarse grid domain covers a geographical area of 60 Km 60 Km having horizontal spacing of 300 m 300 m. Likewise, the fine grid domain surrounding the coastal structure covers an area of 300 m 300 m having horizontal spacing of 5 m 5 m. The coordinates bounded by N N; E E covers the geographic domain of fine grid, which is region of interest in the present study. Calculation of Wave s The wave pressure distribution sketch based on Sainflou s 2 formula is shown in Figure-4. In a simplified form the pressure distribution has a trapezoidal shape where the pressure intensities and quantity of water level rise ( H ) can be expressed as:

4 608 INDIAN J. MAR. SCI., VOL. 39, NO. 4, DECEMBER 2010 generated using parametric wind field model (cyclonic period). The wave information generated for this period was used in the reliability calculations for the coastal jetty. The profile of time varying computed wave forces at eight locations along geometry of coastal structure are shown in the Tables 3(a-c). Fig. 4 Wave pressure distribution based on Sainflou s formula. 2 kh H = (1) 2tanh( kd) h w = H + (2) H ρ gh P2 = (3) cosh( kd) hw P1 = ( P2 + ρ gd) hw + d (4) where, k is the wave-number, d the water depth, H the wave height, P 1 and P 2 are the pressure intensity at the mean water level and respectively. The horizontal dynamic force per unit length of the coastal structure is the horizontal force resulting from waves and water levels (F h ) and this can be expressed as: F h P1 + P2 Ph 1 = d w (5) Assuming perfect wave reflection occurs off the structural face, pressure of a standing wave of height H on the sea-ward side of the structure at a depth (d) can be expressed as: H P = ρg d + H + cosh(kd) (6) Hence the static force per unit length of the structure (F s ) is: Pd Fs = (7) 2 The SWAN model was run for the full month of July 2006 using wind fields from the moored buoy (preand post cyclonic period) and cyclostrophic wind field Estimation of wave overtopping In estimation of wave over-topping, the formulation resulting from an experimental simulation of 2D (two dimension) and 3D (three dimension) random waves as proposed by Franco and Franco 10 have been used. From literature, it was evident this over-topping is well accepted and widely used among the scientific community. For accurate estimation of wave overtopping the freeboard height of coastal structure is of paramount importance. Discharges due to overtopping occurs when run-up levels exceeds the freeboard height. In most of the design practice, wave information is generally taken from wave climatology. In the realistic sense, the sea-state is non-linear in both space and time scales. Hence, the information from state-of-art models like SWAN will be of significant importance for realistic estimates. As mentioned, the freeboard height is a critical factor to assess the wave over-topping discharges. This has been pointed out by several researchers like Owen 11, De Waal and Van der Meer 12. It is known that parameters such as relative freeboard height (F), significant wave height (H s ), and the average discharge per unit length of the structure (q) are important in the determination of the net discharge due to over-topping. The work by Franco and Franco 10 has connected this parameter in the form: Q = aexp( bf) (8) where, Q is the non-dimensional discharge, the relative freeboard height is the ratio between freeboard height (F c ) still water level (in meters) to the significant wave height (in meters). The coefficients a and b in Eqn. [8] depends on structural shape and free surface behaviour at the seaward face of the coastal breakwater. Based on extensive field data support, the values for coefficients a and b referred to in Eqn. [8] as postulated by Franco and Franco 10 are and -3.0 respectively. Using these values for a and b, Eqn.[8] can be written as:

5 KUMAR: RELIABILITY BASED DESIGN METHOD FOR COASTAL STRUCTURES IN SHALLLOW SEAS 609 Location Water Depth Table 3a Wave calculations using Miche-Rundgren and Sainflou methods for all the eight locations during the pre-storm period of 11 July, 2006 (06 h) Incident Wave ht Wave period (s) (m ) Miche-Rundgren Sainflou Crest Trough Crest Trough (m ) (m ) (m ) A B C D E F G H Location Table 3b Wave calculations using Miche-Rundgren and Sainflou methods for all the eight locations during the storm period of 13 July, 2006 (00 h) Water Depth Incident Wave ht Wave period (s) Miche-Rundgren Sainflou Crest Trough Crest Trough A B C D E F G H Location Table 3c Wave calculations using Miche-Rundgren and Sainflou methods for all the eight locations during the post-storm period of 28 July, 2006 (18 h) Water Depth Incident Wave ht Wave period (s) Miche-Rundgren Sainflou Crest Trough Crest Trough A B C D E F G H Q = 0.082exp(-3.0 F) (9) The pioneering work by Fenton 13 had suggested that maximum wave force per unit structural length is caused by oblique incident waves rather than pure standing waves. Franco and Franco 10 have considered this correction factor in multi-directional seas and the modified form of the equation can be expressed as: -3.0 Q = 0.082exp F γ βγ s (10)

6 610 INDIAN J. MAR. SCI., VOL. 39, NO. 4, DECEMBER 2010 where the term γ β is the reduction factor for the influence of angle of incidence (β) of waves and γ s is the reduction factor for influence of structural slope. Hence the average discharge per unit length of the coastal breakwater can be written in the form: q -3.0F = exp γ γ 3 2gH 0 β s (11) The term in the left hand side (LHS) of Eqn.[11] is the non-dimensional discharge (Q) expressed in Eqn.(8). Based on extensive experimental data, Franco and Franco 10 have connected the reduction factor as a function of the incident angle (β) both for short and long crested waves. The limiting condition for long crested waves can be expressed in the form: γ β = cos (β); 0 β 37 (12a) γ β = 0.79; β > 37 (12b) In case of short-crested waves: γ β = 0.83; 0 β 20 (13c) γ β = 0.83 cos(20 -β); β > 20 (13d) For vertical walls the reduction factor having structure slope γ s is taken to be unity. Estimation of wave run-up The various factors which contribute to estimate wave run-up includes porosity of the structure; face slope geometry; incident wave characteristics; toe water depth of structure and sea-bed slope of structural location. More details on these parameters and their characteristic values can be found in the work of Battjes 14. Considering the structural characteristics and mild slope in the present study, formulation of Ahrens and Titus 8 best fits to estimate the wave run-up. The run-up equation based on their work can be expressed in the form: R = kh i (14) The coefficient (k) is characterized by the surf parameter (ξ) which is popularly termed the Iribarren number or the self similarity number which is a function of wave characteristics (plunging, transitional or non-breaking). The following set of conditions express the functional dependence: (a) Under plunging condition (ξ 2): k = ξ (15) p (b) Under a non-breaking condition (ξ 3.5): k nb π η c = exp θ Hi (c) For transitional case (2<ξ<3.5): 3.5 ξ ξ 2 k = k + k t p nb (16) (17) The modified form of Eqn.[17] in terms of Goda nonlinearity parameter can be expressed (Ahrens and Titus, 1985) as: k nb π = Π (18) 2θ where, Π is the Goda nonlinear parameter which can be expressed in terms of incident wave characteristics. Reliability analysis A Level-II design reliability analysis was carried out to assess the failure mechanism for coastal jetty subjected to extreme waves. The results at location-d of the coastal structure using Level-II reliability analysis is shown in Table-4a. The formula of Hudson was used for this purpose. It was assumed that uncertainty in wave height wasσ = 0.15 ; the H M uncertainty in armor mass of rock wasσ = 0.25 ; and the variables ρ cotθ 0.1 σ = ; KD σ = 0.15 ; σ = 0.25 ; σ = 0.05 are the uncertainties associated with frontslope angle of jetty, damage coefficient, relative under-water armor density and armor unit density respectively. The values of uncertainty in variables defined are quite within the reasonable limits to perform reliability analysis. Based on, the uncertainty in resistance parameter according to Kamphuis 15 can be written as: σ cotθ + σ KD + σ ρ + σ M σ r σ = + (19) 3

7 KUMAR: RELIABILITY BASED DESIGN METHOD FOR COASTAL STRUCTURES IN SHALLLOW SEAS 611 The design equation and mass of armor unit can be written as: 1/3 1 Rch KM M a G = γ ssch = γ e γ r γ eγ r (20) 1/3 γ sγ eγ r ( H des ) γ s ( Hdes ) = 0; Ma = K The values of the mean (µ s ) and standard deviation (σ s ) of waves based on SWAN computations were and respectively. As the highest waves for the one month computational period occurred at index point 35, the reliability calculations for entire structural length of jetty was performed based on the wave data at this location. This information can be used as a test for reliability and failure probability of the structural system. It was intended to achieve a lifetime probability of failure P L = 0.1 which is otherwise a failure expectancy of 10%. Based on literature review, the slope factor would generally lie in between 1.5 and 2.0, and in this study the value of 1.5 was taken. As the material used in armor protection for this case was rock, the rock density of 2650 Kg m -3 was used for calculation and this corresponds to an empirical damage coefficient of K D =4. The density of seawater was taken to be Also assuming the occurrence of periodic breaking waves near the structural face, the encounter probability P E can be taken as 1. Under the assumption that P E = 1, the lifetime probability (P L ) will be equal to the failure probability (P F ). In a practical sense one cannot reduce the value of P E to design a safe structure. This means one necessarily needs to decrease the value of P F to achieve the desired value of lifetime probability (P L ). The M Permanent International Association of Navigation Congresses (PIANC) 16 in their manual used Level-II analysis to derive values of partial safety coefficients. The failure probability (P F ) as a function of γ s and partial safety coefficients (γ e γ r ) based on PIANC 16 report is shown in Table-4b. With the valid assumption of encounter probability P E = 1, and using PIANC 16 (Table-4b) for this case and when P L = P F = 0.1 yields the value γ e γ r = 1.38 and γ s = 1.08 shown as Case-B in Table-4a. The calculations to arrive at P L = 0.1 leads to an estimated value of the uncertainty in standard deviation ( σ s ) of waves around 0.085, and this is shown by Case-C in the Table-4a. Results and Discussion Based on the synthetic track data for the cyclone assumed in the month of July 2006, one can categorize the wind fields as: (i) fair weather period (03 July, 03 h), cyclonic (11-13 July) and postcyclone period (28 July, 18h) as shown in Figures-5. Based on the categorization, the cyclone crossed the region of interest from 13 July, 18h onwards. In the study area significant wave heights during the fair weather period was less than 0.5 m, which increased to almost 3.5 m as the cyclone approached towards the structure (11 July, 2006). During post-cyclone period significant wave heights are of comparable magnitude as that during the fair weather period. The maximum significant wave height computed by SWAN was m in the wind-ward side close to the breakwater head. Two locations were selected along the structural length shown in Figure-6a. The time series information of two-dimensional wave spectrum has been computed by SWAN at these two locations. The Case Table 4 (a) Reliability based on Level-II calculation at location-d, and (b) Partial coefficients based on PIANC σ s σ r (a) γ e γ r γ s µ r M a β P F P L A B C (b) Lifetime Probability (P F ) γ e γ r γ s

8 612 INDIAN J. MAR. SCI., VOL. 39, NO. 4, DECEMBER 2010 Fig. 5 Wind field associated with the synthetic track. Locations-1 and 2 are representative of the wind-ward and lee-ward side of the coastal breakwater respectively. To assess wave impact along the structural length, eight points were selected along various segments of this structure (Figure-6b). Four each of these points are aligned along wind-ward and lee-ward side of the structure. The structural geometry of this breakwater is not straight along its length axis; hence one can expect varying wave forces along cross-sectional lengths due to wave transformation process. Hence the eight points selected along the length axis would provide vital information on the variability of wave forces by this extreme event. The lee-ward side is sheltered from direct impact of waves owing to shoreline connected coastal geometry. The computed energy level during cyclonic period is almost two orders of 4.5 times the energy level as the case during fair weather period. Implication is quite significant on lifetime reliability aspects of this coastal jetty. During post-cyclonic phase the variation of energy levels at Location-1 is marginal as that during the fair weather period. Peak wave period was 4 s with dominance of wind generated waves. During this period at the lee-ward side of breakwater, the prevailing sea-state is a mixture of both short period as well long period swells. As seen, the maximum wave forces are evident at wind-ward side of the breakwater (locations from A to D). The computed maximum wave force along leeward side is an order less compared to the wind-ward side. In the region of breakwater head (location-d) maximum computed wave force was about 265 KNm - 1 during the cyclonic period. During the pre-cyclonic phase corresponding to fair weather period, wave forces are quite small as compared to the cyclonic period. The mean wave force integrated along

9 KUMAR: RELIABILITY BASED DESIGN METHOD FOR COASTAL STRUCTURES IN SHALLLOW SEAS 613 Fig. 6 (a) Location of points 1, 2 for wave spectrum calculation, and (b) index points for estimation of wave forces. structural length at lee-ward side was less for the entire period. Corresponding to the time level of 270 h, when cyclone was closest to the region of interest computed wave forces were the maximum. In Figure-7a, b the wave overtopping discharge (m 3 s -1 m -1 ) and wave run-up are presented respectively at selected locations as a function of time. The maximum overtopping discharge seen from Figure-7a was about 0.1 m 3 s -1 m -1. It should be noted here that wave overtops the breakwater crest from the windward side with discharge occurring along the lee-ward side. As seen in Figure-7a, the average overtopping discharge values along all the locations of breakwater are within the safe limits. However, the eventuality of an intense cyclone as used in this case study can lead to average overtopping discharge that reach critical limit possessing high risk to safety of the structure. As seen from Table-2, the design freeboard height of this breakwater was assumed to be 1.5 m, and based on the computed discharge values for this cyclonic event high risk exists both for vehicular safety as well structural safety (shown in Table-1). Also at this discharge rate the structural safety and Fig. 7 (a) Computed wave overtopping and (b) Wave run-up as function of time for selected locations along the structural length. failure probability of breakwater is very high even if the lee-ward slope is still protected. Finally the probability of failure for the coastal jetty was performed based on the computed wave parameters estimated using the state-of-art SWAN model. To list there are many failure mechanisms that can lead to the overall estimation in the failure probability of coastal structures. This can be due to partial failure probabilities associated with hydraulic instability, geo-technical instability, seismic activities and subsidence of armor cap due to loading from wave attacks. Conclusion Knowledge on wave forces, wave overtopping and run-up are paramount components to be considered in design aspects of a coastal structure. Coastal structure design practice based on a proper scientific

10 614 INDIAN J. MAR. SCI., VOL. 39, NO. 4, DECEMBER 2010 assessment play important role in activities related to integrated coastal zone management. Hydrodynamic forces which are ubiquitous and encountered by coastal breakwaters can result from both breaking and non-breaking waves. The role of freeboard or crest of breakwaters is important in preventing wave overtopping and thereby maintaining tranquility within ports and harbor basins for smooth operations. The design freeboard is maintained at optimum height still water level based on statistics of wave climatology and other environmental parameters. Hence a design freeboard based on proper scientific calculations can help reduce cost factor and lifetime failure probability of any coastal breakwater. Wave models based on advanced physical parameterization have evolved to describe the nonlinear behaviour of ocean waves based on the concept of energy balance. The state-of-art third generation SWAN wave model was used for the present study to compute wave parameters in the region of interest. The wave model was run in a nested non-stationary mode using boundary spectral information from a previous coarse grid run. The fine grid has a spatial resolution of 5 m 5 m which covers the region of interest the coastal breakwater is located. As a case study, the actual track of cyclone Fanoos which occurred during December 2005 in the Andaman Sea was replicated as synthetic track for the present analysis where the moored buoy AN-05 was located. The wind field used to force SWAN model comprises in-situ measured buoy winds during fair weather and post-cyclonic periods. At closest distance of the cyclone at the region of interest, wind field using a parametric wind field model was used to force SWAN model. This means the composite wind field data used for one month study comprises both in-situ buoy winds blended on to the parametric wind field. The significant wave height computed by SWAN was fairly very calm before the onset of the cyclone. The impact of cyclone has generated high waves surrounding this location reaching maximum wave height of about 2.5 m. This provided an opportunity to estimate wave forces, overtopping and wave run-up for the coastal breakwater as a case study. Wave force computations were performed for eight locations covering the structural length of the breakwater. The maximum wave force computed about 155 KNm -1 was found close to the breakwater head in the windward side as the cyclone moved closest vicinity to this structure. The maximum wave force at the sheltered end of the breakwater was an order less compared to the wind-ward side. Analysis of wave overtopping and wave run-up calculations show that during the cyclonic period, the discharge level is dangerous and exceptionally very high when compared with reported critical safety factor. The design freeboard height of coastal breakwater assumed for this case study was 1.5 m still water level. The wave run-up estimated was almost double the design freeboard height. The exceptionally high wave overtopping discharges can lead to excessive flooding in the hinterland possessing risk to vehicle safety as well structural safety of the breakwater. It is believed that the scientific findings from this case study incorporated in a coastal engineering analysis system can benefit the design practice, construction and maintenance of existing coastal breakwaters in a sustainable manner. A Level-II design reliability analysis was performed to evaluate the structural response to extreme wave attacks. This study is however limited to the hydraulic stability of sea-ward facing armor layer. Further in this study, various levels of assumptions concerning uncertainty in resistance parameters are also considered. In this case study, the Level-II approach gives a more realistic approximation of the failure risk and also gives an insight in the contribution of all involved stochastic variables. We believe such an approach may help to define the priorities in design studies leading to a balanced approach. References 1 Mai S & Von Lieberman N, Internet based tools for risk assessment for coastal areas, (Proc. Of the 4 th Int. Conf. on Hydro-informatics, Iowa, USA) 2000, pp Sainflou M, Treatise on vertical breakwaters, (Annales des Ponts et Chaussees, Paris) 1928, 98(11), pp Goda Y, New wave pressure formulae for composite breakers, (Proc. 14 th Int. Conf. Coastal Eng., Copenhagen) 1974, pp Minikin R R, Wind, Waves and Maritime Structures, (Charles Griffin and Company Ltd) 1950, pp Shore Protection Manual, Army Waterway Experiment Station, (US Government Printing Office, Washington D.C, USA) 1984, pp Bleck M, Oumeraci H & Schuttrumpf H, Combined wave overtopping and overflow of dikes and seawalls, (Proc. 27 th Int. Conf. on Coastal Engg., Poster Session, Sydney, Australia) Burchartch H F & Hughes S A, Coastal Engineering Manual, Fundamentals of design, Chap. 5, Part VI, (Coastal Engineering Research Centre, WES, VA, USA) 2003, pp. 76.

11 KUMAR: RELIABILITY BASED DESIGN METHOD FOR COASTAL STRUCTURES IN SHALLLOW SEAS Ahrens, J.P. & Titus, M.F., Wave runup formulas for smooth slopes, Jour. Waterway, Port, Coastal and Ocean Engg., 111(1)(1985) Booij, N., Ris, R.C. & Holthuijsen, L.H., A third-generation wave model for coastal regions, Part I: Model description and validation, J. Geophys. Res., 104 (C4) (1999) Franco, C. & Franco, L., Overtopping formulas for caisson breakwaters with non-breaking 3D waves, Jour. Waterway, Port, Coastal and Ocean Engg., 125 (2) (1999) Owen M W, Design of seawalls allowing for wave overtopping, (Report EX 924, HR Wallingford, U.K) 1980, pp De Waal J P & Van der Meer J W, Wave runup and overtopping on coastal structures, (Proc. 23 rd Int. Conf. on Coast. Engg. Vol 2, ASCE, New York) 1992, pp Fenton, J.D., Wave forces on vertical walls, Jour. Waterway, Port, Coastal and Ocean Engg., 111(4)(1985) Battjes J A, Surf similarity, (Proc. 14 th Coastal Engg. Conf., Copenhagen, Denmark) 1974, pp Kamphuis J W, Introduction to coastal engineering and management, (Advanced Series on Ocean Engineering) 2000, 16, pp PIANC, Analysis of rubble mound breakwaters, (Permanent International Association of Navigation Congresses, Rep. of Working Group, No. 12) 1992, pp. 46.