TECHNICAL REPORT CERC-86-2 COST-EFFECTIVE OPTIMIZATION OF pfctl el RUBBLE-MOUND BREAKWATER CROSS SECTIONS by Orson P. Smith Coastal Engineering Research Center DEPARTMENT OF THE ARMY Waterways Experiment Station, Corps of Engineers PO Box 631, Vicksburg, Mississippi 39180-0631 BRARY graphic Institut” | DATA L Woods Hele ees February 1986 Final Report Approved For Public Release; Distribution Unlimited Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 Under Civil Works Research Work Unit 31234 Destroy this report when no longer needed. Do not return it to the originator. The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. 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Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. l Il 01092 | Libra | | | iii 00301100 TE Unclassified SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE 1. REPORT NUMBER 2. GOVT ACCESSION NO. RECIPIENT'S CATALOG NUMBER Technical Report CERC-86-2 3% 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED COST-EFFECTIVE OPTIMIZATION OF RUBBLE-MOUND BREAKWATER CROSS SECTIONS Final report 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(e) 7. AUTHOR(a) Orson P. Smith 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS Civil Works Research Work Unit 31234 12. REPORT DATE February 1986 13. NUMBER OF PAGES 130 15. SECURITY CLASS. (of thia report) Unclassified 9. PERFORMING ORGANIZATION NAME AND ADDRESS US Army Engineer Waterways Experiment Station Coastal Engineering Research Center PO Box 631, Vicksburg, Mississippi CONTROLLING OFFICE NAME AND ADDRESS DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 . MONITORING AGENCY NAME & AODRESS(/f different from Controlling Office) 39180-0631 MW. 15a. DECL ASSIFICATION/ DOWNGRADING SCHEOULE - DISTRIBUTION STATEMENT (of thia Report) Approved for public release; distribution unlimited. - DISTRIBUTION STATEMENT (of the abstract entered in Block 20, If different from Report) 18. SUPPLEMENTARY NOTES Available from National Technical Information Service, 5285 Port Royal Road, Springfield, Virginia 22161. - KEY WORDS (Continue on reverse side if necessary and identify by block number) Breakwater Optimization Criteria Planning Design Rubble-mound 20. ABSTRACT (Continue em reverse oid ff nmeceweary and identify by block number) This report discusses design criteria, design procedures, and practical considerations involved in planning, design, and construction of rubble-mound breakwaters. Currently available methods for estimating rates of damage to breakwater armor and for predicting wave transmission characteristics are also described. Using information typically available today to most breakwater designers, a step-wise procedure is presented which can identify an optimum breakwater cross section, both in terms of structural integrity and functional performance. FORD ’ ee DD . jam 73 1473 = EorTiow oF 1 Nov 651s OBSOLETE Unclassified SECURITY CLASSIFICATION OF THIS PASE (hen Data Entered) —_ SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) Se SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE The investigations summarized in this report were authorized by the Office, Chief of Engineers (OCE), US Army Corps of Engineers, and performed as a part of Civil Works Research Work Unit 31234, "Developing Functional and Structural Design Criteria." Funds were provided through the Coastal Struc- tures Evaluation and Design Research and Development Program administered by the Coastal Design Branch of the Coastal Engineering Research Center (CERC) at the US Army Engineer Waterways Experiment Station (WES). The author gratefully acknowledges the assistance of the following indi- viduals: Messrs. Robert B. Lund and Perry Holman and Ms. Debra L. Rouse, co-op students in the Coastal Design Branch, for their help in software development; Dr. Michael E. Andrew, of the Prototype Measurement and Analysis Branch, for his assistance in the probability and statistics considerations; the staff of the Wave Research Branch, under the direction of Mr. D. D. Davidson, for their suggestions and review comments; and Ms. Shirley A. J. Hanshaw, of the Publications and Graphic Arts Division, WES, for her editing. The work was conducted under general direction of Dr. Frederick E. Camfield, Chief, Coastal Design Branch; Mr. C. E. Chatham, Jr., Chief, Wave Dynamics Division; Mr. Charles C. Calhoun, Jr., Assistant Chief, CERC; and Dr. James R. Houston, Chief, CERC. COL Allen F. Grum, USA, was Director of WES during the preparation and publication of this report, and Dr. Robert W. Whalin was Technical Director. CONTENTS Page PREEA CHa aici arcnenc rt aree av hot ate home Aree hs ctotalteds: Cilahen orators tangles tetas: stent delat yet remenaperel ens eree ts 1 PART I: ENDRODU GION Bie toe SR Le Bisse Saale Glebe: dcssete ie Sis teww oh ecebsdatieted ones 4 ObjectivesNss MILO PES ES Ut sible conte Suna. Bcleiels, have taalee g 4 COPS ieragerecuiehens tipsters enous waster us waytnesyctts fa velantayralia'aice ye ieee o sigeiecse oul ouerieteyoravesorecategeienoceneus 5 Definition and Purposes of Rubble-Mound Breakwaters..............-. 5 ThesiNeed for sOptimiZatiiomy cs ei. dries cI Gaede, clelebeiavele oe icla osSie oh ahisdceatale 6 OnganiZatilon oh ther REPO mtb iccs craveneuenesoj0r cweyone cop's 10 0)-s4s)rsirs je cheys.cysitel ovays eveye epautssenses T PART II: BASTC DESTGNPERUNCGMBICES eye te cia snc) nut a at-a co ovate lanes ob svenaneneleucuerecsleucnans 9 DesirermeiC rake @ ritcegs2. cc tevae swede enen ays ed sisi eto, sues ah oh eatero se inlioitesMerapaitsi size eres ols Pavlahaveueseevole’ seots fe) The sHudsSoOnBOnMUl arse. egos o stsusl eres! oka he Soe O averenei ene oe eeieebehousnent e Sucreueenet 10 Alternat vet Stab iMiGyrRedaclon Spr sen. silisiy Gistelels ag wo cleceheletoiers: eleseleveiens sc ate 13 PracticalsiConsideratiion's for), Stability y. cic ce cle cetes c sccies «es 6 Sues © eres 18 Bhysacaile Modeluingitormsivalpuslbittc yc ieve creas aucuolcisie aucledeielclcieccncne cienere sini cyenere 27 PART Wis:s cS Sit MAM IENG DAMAGE CRATES cvesscrs stele cere evs icsee age cos oe a eel er cues eu aie) cleterel 35 Dama gesASsSessmenibincswzcete s S:c)c0e1 bie; swage rorsud a mieergraeree eve ag aveteue, #idceOuial aloiecera oe 35 Analytical’ Damage vPred Teton sec ies pst ate lets © op ereleleratece sileleleds Sodicte ohete ctet one 37 PART IV: PS TIMATING WAVE: pDRANSMISS TON one 275 ie, 5 Sis sone cites cles etclateets crere.s 6) etene 45 Wave! Transmission! by sDiP fraction sjccuisictets wile cists i lets ssnetshereseve edie tele aleve o, Me 45 Wav.e)-Eransmission aby OVERCODDIN ge 22. cisteccis oes w 2 es. «410.6 wove) 06 a siw 4 le Sue 45 Wave Transmission Through Permeable Breakwaters.........ccecceccecs 52 PART V: COST=EREECTIVE sOPTIMiInZ AION 215 ctsets ote crocrens sone) o Gste hsce cine ence 54 ThevPrineiple of jOptimaiZat von... cahiece acu sweeten Veorwn vaae nw oes 54 An@idealltiaedW@Ampnoac hiss s cycicts:a-d a.ctetafersievoretele cs, «eG eadia. a Be sc av atetetegareies 54 ASP racileal 7 AD POAC Ml ce serene aey ci ope neloss sue ous pete tenet oie) eneievarci fis’ 41.5 2100 eveus s\/eu ay econ eyeee 57 PART VI: SUMMARY JANDSCONCIUSTIONS i eramicuereisnerecereucheciereiciecrcnereetecenaie ei ciers cr eiareieuente 69 SUMMARY asa ts ants Sete acts Sate oss, Seed erase. ay cis eave aang levee eae Amn atete Saar erne ie 69 COMGINIST ONS coco cncvencveciwieve ou siiee fo.eustie eiloveeueue ere eyee: srepeneve: eveuareueislere aun coetnvakene lorena 10: REBEREN GES in cutie tecey ote: oi crouse tone ores cticecavevetsevlore Ccevevenciste:eiiere se: cheneus 0.5, sieve wie sci euel evens eueue ate 73 APPEND DX@A) COMBULERS BROGRAM BWEOSSileccre ce ceieee se eiciee eciae ie este a cercierecets creel Al Estimation of Economic Losses as a Function of Wave Height......... Al Sample PXECUC TONMANG OU PUIG serene cress caja eters BWDAMAGE Yr sree wes cietcielecciere-clete siete Gere eicie eec sro D1 Estimation of Rubble-Mound Breakwater Armor Layer Expected DAMAR © Siiycr eve. creeronepere ron steuel suersn «penaye tere wekop susie ceriore creer cue Suet siaiewshalssilsvevever evckeionene vatek« D1 Sample interactives S@SSiloMmens sestcieuse oerersrs ners cue eieieiera eaceeree cio eve ares D3 SaMmphe Prop ramps USib ii Gives chews evcses serene coiroreite eis) ciation e onsite Io reusuelietedeus tee vere rouele tava le D5 ARBENDIPX UE 32 NOTATIONS. eeiostete tel oiler chsne; « h sieereto ie ucla tal ere ba tavarereke aie ce Yass ttre Suuanene aes eFeleite E1 COST-EFFECTIVE OPTIMIZATION OF RUBBLE-MOUND BREAKWATER CROSS SECTIONS PART I: INTRODUCTION Objectives 1. The primary objective of this report is to introduce a systematic method by which planners and designers of rubble-mound breakwaters, specifi- cally those in the US Army Corps of Engineers (Corps) District offices, can formulate an optimum cross-section configuration and verify its effectiveness, both in terms of structural integrity and functional performance. Rubble- mound breakwaters, the most common coastal structures worldwide, are built to provide protection from direct wave attack to boat harbors (Figure 1) and to port facilities. Recent advances in coastal oceanography have greatly im- proved the understanding of wave generation, propagation, and transformation into shallow water. These advances, along with greater availability of mea- sured and hindeast wave data, have allowed procedures for design of rubble- mound structures to become much more complex than in previous years. The cee Figure 1. A rubble-mound breakwater protecting a boat harbor guidance available in the Shore Protection Manual (SPM) (1984) provides the basic tools for planning and designing breakwaters. This paper is intended to supplement that guidance by providing a practical perspective to the wide variety of environmental data now available to coastal engineers for rubble- mound breakwater design. Scope 2. A brief review is presented of past and present criteria development procedures, design techniques, and related practical considerations, followed by a more detailed discussion of breakwater damage prediction and estimation of wave transmission characteristics. A systematic procedure is proposed to formulate alternative cross-section designs, evaluate their structural and functional effectiveness, and determine detailed dimensions which realize max- imum net incremental benefits. Definition and Purposes of Rubble-Mound Breakwaters 3. Breakwaters and, to some degree, jetties and groins are designed as barriers to sea waves, providing calmer water in their lees. Wave barriers can be constructed in many different ways, including vertical-sided concrete caissons, sheet-pile walls, wooden crib structures, and floating bodies. The oldest and most common type of wave barrier is the rubble-mound breakwater be- cause of its typical economy and constructibility in harsh coastal conditions. The long history of rubble-mound breakwaters has proven them quite reliable in a wide range of environments (Bruun 1985). A rubble-mound breakwater consists of sloped layers of stone or concrete shapes that are sized to withstand wave attack, excess settlement or loss of fill material, and to prevent scour, as shown in the typical cross section in Figure 2. Their inherent flexibility CREST CREST WIDTH ELEVATION x eae DESIGN SWL = - SWL (MINIMUM) (SECONDARY ARMOR) -H DESIGN WAVE HEIGHT, H w/200 TO w/6000 (CORE) —- cross section tends to prevent catastrophic failure, even in the event of underdesign. The design parameters for rubble-mound breakwaters are rather inexact compared to those of most rigid civil engineering structures; thus, conservative overde- sign is quite common. 4. Rubble-mound breakwaters can have a number of secondary purposes that are related to their primary purpose as a wave barrier. A breakwater protecting a harbor entrance and mooring area from wave attack might serve to divert currents and longshore transport of sediments. Also, it could be designed to provide access by people and equipment to the outer or deeper portions of the harbor. A breakwater protecting port facilities where cargo is being discharged and loaded might have these additional purposes and could even serve as a foundation for the port facilities themselves. This paper concentrates on considerations surrounding the wave barrier function. Fur- thermore, the perspective of the Corps as a public works agency is maintained since, in this case, the owners of the structure and the beneficiaries of its protection are the same (i.e. the taxpayers). The discussion to follow could also easily apply to a rubble-mound breakwater financed by private enterprise for commercial purposes, since tangible public benefits can, in many in- stances, be translated as profits. Many features of the planning and design procedures discussed later in this report can be extrapolated to planning and design of facilities other than rubble-mound breakwaters. The emphasis and most computational aspects will apply specifically to rubble-mound breakwaters intended as wave barriers. The Need for Optimization 5. The construction cost for rubble-mound breakwaters is usually on the order of millions of dollars for smaller harbor or shore protection projects and on the order of tens of millions of dollars for larger harbor or port projects. The consequences of a dramatic structural failure include costs for repair of the breakwater which may approach the order of magnitude of the original construction costs due in part to expensive mobilization. Also, such consequences may include costs from property damage and inconvenience to port and harbor operations which occurred during the storm that damaged the break- water. These latter costs would typically be of a lower order of magnitude than the breakwater construction costs. All of these costs of rubble-mound breakwater failure are minimized by the tendency for this type of structure not to fail catastrophically. Catastrophic failure of flood control struc- tures (dams and levees) causes tremendous adverse consequences for the prop- erty and people in their flood plains, often including loss of life. The costs of these consequences can easily exceed the order of magnitude of the construction costs for the flood protection. This comparison illustrates that, in comparison to some other civil engineering works, a certain small risk of failure for rubble-mound breakwaters can be tolerated. 6. Federal public works agencies in the United States have the statu- tory constraint for project authorization that the tangible benefits realized by the proposed plan must exceed all the life-cycle costs. This constraint has been further defined to apply to the incremental benefits and costs of each major feature of a proposed project. A rubble-mound breakwater built as a part of a federally funded project must "carry its own weight" in terms of its incremental net benefits. Recent administrative policies have provided additional restrictive criteria for federal financing of public works projects by requiring cost sharing with regional or local governments. These policies force planners to carefully consider the financeability of a project as well as its overall economic feasibility. Local sponsors of federally funded navi- gation projects commonly have severe limits on what costs they can share. A proposed breakwater project may be theoretically justified by a wide margin, but if it is not affordable it will not be built. Conversely, a sponsor may have the luxury of ample funding sources for cost sharing, but if a breakwater plan does not achieve enough incremental benefits, federal participation will not be possible. It is therefore critical that rubble-mound breakwaters be designed to provide the optimum trade-off between life-cycle costs and incre- mental benefits. This paper will deal with methods of formulating such an op- timum plan without extending planning schedules and budgets beyond reason. A commitment, both in time and money, is necessary, however, to address enough key questions for systematic optimization to be possible. Organization of the Report 7. This introduction will be followed by a review of design principles for structural stability, including some of the many practical considerations involved in rubble-mound breakwater design. Current references offering more detailed discussions of various specific design considerations are given wherever possible, and readers are urged to consult these works. Review of design procedures is necessary in this paper to place an appropriate perspec- tive on simplifying assumptions made in this and other discussions of optimi- zation procedures. An introduction to a number of methods now in use to pre- dict damages to rubble-mound structures will be presented as tools to estimate future maintenance and repair costs for a breakwater design. Similarly, a discussion of methods to predict the wave transmission characteristics of breakwaters will follow to show how the structure's functional performance may be evaluated. The main paper will be concluded with a procedure to guide planners and designers of rubble-mound breakwaters from the choice of design criteria to determination of final dimensions. Appendixes will document the software available to accomplish some steps of this procedure. PART II: BASIC DESIGN PRINCIPLES Design Criteria 8. There is a well-known tendency for subjective judgments to creep into supposedly systematic project planning endeavors in the earliest phases. A proven method to order your thinking in the conceptual phase of a project is to first thoroughly define the problems and opportunities at the site in terms of desirable goals to be achieved. This has long been the first step in the civil works planning process as practiced by the Corps. Two types of design criteria or "planning objectives," as stated in Corps planning guidance (Board of Engineers for Rivers and Harbors 1985 and Water Resources Council 1983), can be identified at this point relative to the function of a breakwater as a wave barrier. The first, and most familiar, is a criterion which defines the structure's ability to withstand the effects of extreme storms without itself suffering significant damages. This type of criterion can be referred to as the "structural integrity" or "survival" criterion. The second type, referred to as the "functional performance" criterion, deals with the effectiveness of the structure at its intended function which is to provide protection from waves. 9. The structural integrity criterion determines the breakwater's life- cycle costs to the extent that a certain level of investment is necessary to prevent damages from an extreme event. There will always be a finite proba- bility that any storm, no matter how extreme, will be exceeded in intensity, so this criterion also determines the expected repair costs during the proj- ect's life. The most extreme sea state in which a particular breakwater de- sign will suffer no damages cannot, in practice, be precisely defined, as will be discussed later. The statement of a structural integrity criterion should be phrased with this in mind. It should be stated in terms of the desired effect, that is, prevention of breakwater damages (and associated repair costs). An example would be "damages to more than 5 percent of the breakwater armor will occur with less than 2 percent probability per year." There are, of course, numerous complications in achieving such a goal, including defini- tion of the types of possible damages and determination of the combined proba- bility per year of the physical parameters (wave height, wave period, wave direction, water level, storm duration, and others) which could cause them. Nevertheless, this is a workable statement in terms of an objective which is adaptable to more than one means of determining structural dimensions. 10. The functional performance criterion determines the incremental economic benefits of a breakwater design since it defines the structure's level of effectiveness as a wave barrier. It also affects the cost since a certain additional increment of investment may be necessary to achieve a given level of effectiveness. This level of effectiveness can usually be stated in terms of a maximum transmitted wave condition during a given extreme event. The probability of exceedance for this event can in turn be related to property damage and other economic losses. Probability of exceedance is usually stated in terms of any single year, but it can also be stated in terms of all or some portion of the life of the project. A workable statement of a functional performance criterion might be that "10 percent of transmitted waves in any storm will exceed 1 m with less than 5 percent probability per year." This statement assumes that "10 percent of transmitted waves" can be related to some level of unacceptable property damage or operational disruption inside the breakwater. An even more general statement might be that "navigational delays and property damages from transmitted waves shall occur with less than 5 percent probability per year." 11. Criteria of both types need to be defined for each section of the breakwater where either the environment (water depth, wave exposure, or other factors) or the required level of protection significantly differs. These sections can essentially be treated independently until a point when economy of breakwater materials, related constructibility constraints, and the transi- tion requirements become apparent. Usually the breakwater head, any elbows, and one particular section of trunk will take precedence over other sections. Head and trunk designs do not as yet lend themselves to reliable analytical methods and typically require much subjective judgment and extensive physical modeling. Most remarks in the rest of this paper will refer to the critical trunk section, with the understanding that other less critical trunk sections may have different design criteria. The Hudson Formula 12. Investigations into the stability of rubble-mound coastal struc- tures were performed in the decade before the second World War by a Spanish 10 engineer named Cavanilles Iribarren. Iribarren (1938) presented the first widely used empirical formula for estimating a stable armor unit weight, given incident wave height, seaward slope of the structure, density of the sea wa- ter, and certain characteristics of the armor material. He assumed that stones on the outer slope were subject to gravity and wave forces, the latter of which included buoyant, impact, and friction components. The Iribarren formula was intended to predict the minimum weight stone which would remain in place when subject to waves of a given height. This height, defined in scale model tests as the level of "incipient damage," indicated that over the entire slope no more than 1-5 percent of the stones was displaced (d'Angremond 1975). The Iribarren formula is coming back into use in its original and in modified forms and will be discussed again later in this report. 13. During and after World War II, the approach of Iribarren was con- tinued by Robert Hudson, a Corps investigator at the Waterways Experiment Sta- tion (WES) in Vicksburg, Mississippi. Hudson performed a great number of scale model tests on a variety of rubble-mound breakwater configurations. He also published a paper (Hudson 1958) which presented an armor unit weight pre- diction formula with many of the same features and assumptions as those of Iribarren. This formula is still in almost universal use by coastal engineers because of its relative simplicity and the many experimental and prototype tests of its reliability. The Hudson formula is p gH? 3 (1) A Ky cot 6 where W = weight of armor unit at the level of incipient damage* p,, = mass density of the armor material g = acceleration of gravity H = incident wave height AS SAO BD. p,, = mass density of the water Ky = an empirical stability coefficient @ = the angle from horizontal of the seaward slope of the structure * For convenience, symbols and abbreviations are listed in the Notation (Appendix E). 14. Table 7-8 in the SPM (1984) presents the values for Ky recom- mended by the Corps for use in the Hudson formula. Values are presented for a variety of quarried materials and artificial concrete shapes. Each value is associated with a number of factors, including: a. Shape characteristics of the armor units (i.e., smooth, rough, round, or elongated rock). b. Position of units on the trunk or head of the breakwater. c. Wave form (i.e., whether or not the wave is breaking directly on the structure). d. Slope or range of slopes (in some cases). e. Method of placement (random versus special individual placement). f. Number of layers of armor units to be placed on the slope. g. Relative gradation and smoothness (for quarried rock). 15. An important point to note about the Ky, values in the SPM (1984) is that 58 percent of them were derived from monochromatic wave model test re- sults, while the rest are interpolated values. Another factor of importance is that some of the armor unit types for which Ky values are presented have actually been used in only a small number of prototype breakwaters. All of the units lack systematically documented prototype verification of their rela- tive stability, though efforts are currently under way to consolidate histori- cal performance of Corps constructed breakwaters. Uniform rough angular quar- rystone, riprap (graded rough angular quarrystone), and dolosse have been most extensively tested in scale models and currently have the best documentation of prototype experience (Jackson 1968a and Carver 1983). 16. The coefficient Ky» as applied in the Hudson formula with its basis in the assumptions of Iribarren, does not directly account for as many as 20 or more design conditions (Ligteringen and Heijdra 1984) that are now known (in at least a qualitative sense) to affect breakwater stability. Some investigators (Brorsen, Burcharth, and Larsen 1974 and Burcharth 1979) have questioned whether the Hudson formula is reliable for predicting stability of dolosse and other slender concrete armor units. In the future, these units may require variable Ky factors related to slope and other conditions not now inherent in the values presented in Table 7-8 of the SPM (1984). Some of the other conditions of concern include: a. Influence of wave period or the steepness of individual waves (Ahrens and McCartney 1975 and Losada and Gimenez-Curto 1979). Influence of wave groupiness in natural irregular seas (Bur- charth 1979) ec. Effect of the foreshore or the breakwater toe on wave transfor- mation (Bruun 1979 and Kjelstrup 1979) d. Effect of oblique waves (Losada and Gimenez-Curto 1982 and Christensen et al. 1984). e. Interaction of waves with monolithic crest elements or densely packed underlayers, such as resonance of reflected waves with incident waves (Jensen 1983). In f. Friction of outer armor material with underlayers (Hedges 1984) g. Mechanical strength (resistance to tension, compression, im- pact, fatigue, etc.) of individual armor units (Poole et al. 1984 and Groeneveld, Mol, and Zwelsloot 1983). h. Potential settlement, foundation failure, and related geotech- nical problems (Thorpe 1984). i. Seismic stability. 17. The Hudson formula can be applied to interpret scale tests of pro- posed designs to measure the "actual" Ky of an armor unit in a particular breakwater configuration, in which case many of the above factors would be ad- dressed. A series of successive tests on the same configuration with varying monochromatic wave period can determine the critical period when waves of that height would break directly on the face of the armor slope. Likewise, this "sensitivity analysis" approach (vary one parameter while holding others con- stant) can provide estimates for the reliability of the point of incipient damage and damage rates for more severe wave height and period combinations. Tests with irregular waves are also possible and should be considered, even though the procedures involved and interpretation of results in terms of Hud- son formula parameters are less standardized. Physical modeling is an essen- tial step in the cost-effective design of rubble-mound breakwaters and should not be neglected for any except the smallest, most inconsequential structures (Paape and Ligteringen 1980). Some specific techniques for verifying armor stability and damage rates by scale model testing will be discussed later in this report. Alternative Stability Relations 18. The Iribarren formula, as mentioned earlier, has recently been re- ceiving renewed attention worldwide because of some spectacular failures of 13 large rubble-mound breakwaters in the last 10 years (Stickland 1983). The Iribarren formula in its original form appears as follows (d'Angremond 1975): No ust? W = (2) A(u cos @ + sin @)3 where N = empirical coefficient related to the armor material character- istics (comparable to K, in Equation 1) u = coefficient of static friction between individual armor units (equivalent to the tangent of the angle 6 at which armor would slide from gravity alone; values found by Graveson, Jensen, and Sorensen (1980) are presented in Table 1) Table 1 Values for the Coefficient of Static Friction u _Type of Armor ~~ ~+Coefficient u =~=~=~=~~—~—~— Angle of Repose 90 Round seastones 120 45 Quarrystones eet 48 Concrete cubes tee 50 Concrete tetrapods ~1.4 ~55 Concrete dolosse 2a if ~70 19. The original Iribarren formula has only one additional parameter u with N being essentially equivalent to Ky . This additional explicit pa- rameter appears to have little advantage to offer, except that static friction has recently been investigated as a potentially critical factor in the overall stability of complex artificial shapes such as dolosse (Price 1979). It al- lows the Iribarren formula to account for the marginal stability of materials placed at their natural angle of repose. This factor might be used also in the future as a measure of seismic stability of rubble-mound structures. 20. Engineers at the Danish Hydraulics Institute (DHI) have proposed a modification to the Iribarren formula for application with scale model tests using irregular waves (Graveson, Jensen, and Sorensen 1980). This DHI- Iribarren formula is 3he po gy H_L Z Sapa a ate W fae (3) K oC cos @ - sin a)? where H, = significant wave height of the incident irregular waves (average height of highest one-third waves at site) Ly = wave length at the site corresponding to the period of peak energy density for the incident irregular waves K, = alternate stability coefficient = Lp /NH, 21. The principal modification of the original Iribarren formula is the substitution of an alternate stability coefficient on the basis that the orig- inal stability coefficient (N in the numerator of Equation 2) is a function of the wave steepness H,/L A number of other investigators have proposed p° similar stability relations (Rybtchevsky 1964, Jensen 1984, and Ahrens 1984). A similar modification to the Hudson formula could be made by substituting Ky = KqH,/Lp . Ahrens (1984) found that stability by "reef type breakwaters," or low-crested breakwaters without traditional multi-layered cross sections (ba- sically homogeneous rubble-mounds), was reflected with greater confidence us- 2 stp original Hudson formula (Figures 3 and 4). ing a modified Hudson formula with H in the numerator than with the 22. Engineers at Delft Hydraulics Laboratory in The Netherlands re- cently performed an extensive series of scale model tests of the stability of rock slopes under random wave attack (Van der Meer and Pilarezyk 1984). These tests resulted in the formulation of a set of stability formulae for quarry- stone armor of breakwaters and revetments. Their tests also gave information on how to predict damage rates as a function of the number of incident waves. Armor layer gradation was found to have a lesser effect than that found by other investigators (Ahrens and McCartney 1975). Slope angle was found to have an effect on stability similar to that predicted by the Hudson formula. Wave period effect was investigated as a function of the "Iribarren number" or surf parameter as follows: ae tan 6 a 2 Hy (4) Ie fe) where Lo e gTe/2n, based on the average wave period T, 23. The influence of wave period was found to correspond roughly with the traditional distinction between breaking and nonbreaking waves. The ef- fect of variations in the incident wave spectral shape, as measured in various 15 LCBW, SUBSET 1 0 60 : Aa LEGEND o FILE 1, Tp = 1.45 SEC a 30 ao FILE 2, Tp = 2.25 SEC x 4 4 FILE 3, Ty = 2.86 SEC e FILE 4, Tp = 3.58 SEC 40 H, N= s 5 (Wiprg)"/3.0 DIMENSIONLESS DAMAGE, 0’ STABILITY NUMBER, Ns Figure 3. Model data plotted by Hudson stability number LCBW, SUBSET 1 70 a a 60 ; 50 Z 2 a 7 LEGEND : 2 o FILE 1, Tp = 1.45 SEC = 40 o FILE 2, Ty = 2.25 SEC . = 4 FILE 3, T, = 2.86 SEC a @ FILE 4, Ty = 3.58 SEC 8 3 30 nae (H2L,)1/3 “A 2 © (Wiorg)/3.4 i z 2 8 i ° ae Sa a ee) eee eee Dae 0 2 4 6 8 10 12 14 16 18 20 SPECTRAL STABILITY NUMBER, N,* Figure 4. Model data plotted by spectral stability number 16 ways to reflect both irregularity and groupiness, was found to be minimal. This result differs from the conclusions of other tests relative to the influence of wave groupiness on stability (Burcharth 1979). A major influ- ence by core permeability was found. The stability formulae proposed for rubble-mound (quarrystone) structures with permeable cores (D5 armor /De¢ core = 3.2 , as tested) for breaking waves (&— < 2.5 - 3.5) was Os22 H, 33 -0.54 AD = 5\58 73 (5) n50 N or, equivalently, H, p Whe S, 1/3 n50 n50 N The formula proposed for nonbreaking waves (& > 2.5 - 3.5) with cot @ < 3 was H S 1/2 2 Oje = 1.65 (cot 6) — E (7) y/e whereas for nonbreaking waves (& > 2.5 - 3.5) with cot @ > 3 the formula was ils »2 ae 0.1 AD = 2.86 72 E (8) n50 N where H. = the significant wave height of the incident spectrum Dn50 = the nominal diameter, based on the mass of the 50th percentile W50 from the armor material mass distribution curve | = (Weo/0,)'/3 So =a dimensionless damage level, defined as the number of equivalent Dn50 cubes eroded over a width of Dn50 = 2-3 for incipient damage (as with the Hudson formula) = 8 to 17 for armor layer "failure" (significant exposure or underlayers) N = number of incident waves The range of & values from 2.5 to 3.5 for the transition from breaking to nonbreaking wave conditions apparently represents the difficulty in de- scribing an irregular sea state as either breaking or nonbreaking, since both 17 breaking and nonbreaking waves can occur in the same sea state. Others who have investigated breakwater stability as a function of the surf parameter (Equation 4) include Gunbak (1976) and Losada and Gimenez-Curto (1980). 24. A relatively complete list of rubble-mound breakwater stability formulae, proposed by various investigators over the years, was published by the Permanent International Association of Navigation Congresses (PIANC) (1976). The variety of model tests and prototype experience inherent in these formulae and those developed since 1976 is but a small fraction of the many thousands of monochromatic wave tests conducted to determine Hudson formula parameters used to design hundreds of breakwaters all over the world. Use of these other stability relations should, therefore, be applied only in conjunc- tion with traditional procedures using the Hudson formula for comparison. A conservative choice can then be made between the stable armor weights and damage rates predicted by the Hudson formula and these alternate methods. Practical Considerations for Stability 25. The analytical methods available for predicting rubble-mound break- water stability have been shown not to include many important considerations that could cause a structure to fail. Breakwater design has always involved a great deal of subjective judgment and probably always will. Some of the most pertinent practical considerations that must be made in determining rubble- mound breakwater material characteristics and dimensions are reviewed below. Comprehensive review of both practical and analytical considerations is avail- able in the SPM (1984), Angerschou et al. (1983), Institute of Civil Engineers (1984), Jensen (1984), and Bruun (1985). Incident wave conditions 26. The incident wave conditions are traditionally defined as the wave height at the seaward face of the structure with a further distinction as to whether or not the waves are breaking. This breaking versus nonbreaking cri- terion has been argued extensively over the years. The convention remains in practice, however, due to obvious differences in design conditions for rubble- mound structures built in shallow water, where wave heights are depth limited, and in deeper water (depth > ~15 m), where waves have not transformed to the point of breaking in front of the structure. The natural irregularity of sea states can be fairly well represented by a2 single height and period related to 18 some specified exceedance value, but it must be acknowledged that both height and period will vary in any storm. Consequently, some incident waves will be breaking on the structure, and others will not. The succession of high and low waves (wave groupiness) and of breaking and nonbreaking waves can be a critical factor. The potential effects of wave groupiness or multiple con- verging wave trains (multi-peaked spectra) are difficult to assess without a substantial amount of field data and scale model testing with irregular waves. 27. The alternative stability coefficients for the Hudson and Iribarren formulae discussed above which include wave steepness H/L provide one means of making a more explicit description of incident wave conditions. Other de- scriptive parameters that have been investigated include the surf similarity parameter in Equation 4 (Bruun and Gunbak 1978, Burcharth 1979, Losada and Gimenez-Curto 1980, Van der Meer and Pilareczyk 1984, and Bruun 1985) and the Stokes or Ursell parameter HL®/h3 (Carver 1983). Estimated values of these wave form parameters can be used as a more systematic means of classifying individual waves as breaking in the critical plunging mode, as spilling, or as nonbreaking. Irregular sea states require further definition in terms of ei- ther time domain characteristics or spectral (frequency domain) parameters. A number of useful parameters for characterizing irregular waves are discussed by Rye (1977). 28. Wave transformation effects caused by the proposed construction works themselves cannot be neglected. Breakwaters with shallow slopes or with extensive toe development can also change the wave conditions at the waterline on the seaward face by "tripping" the waves. Scale-model tests are necessary to quantify these effects on the armor layer (Jackson 1968b). Relatively steep and impermeable structures may partially reflect incident waves such that resonance of incident and reflected waves causes scour near the toe. De- termination of the sensitivity of a structure to these effects from oblique waves requires scale-model testing in a three-dimensional wave basin. These potential problems make physical modeling critical for reliable estimation of the stability of a proposed rubble-mound breakwater. 29. The duration of a storm at sea is a real world parameter that should be considered in any design effort or laboratory stability analysis. Figure 5 illustrates the time-history of significant wave height, peak spec- tral wave period, and predominant direction of wave propagation for a storm in the Gulf of Alaska simulated from synoptic weather data at 6-hr intervals. 19 8L (QHOE*O go uoqoez e Aq ATdtqTnw ‘suaqow 04 4aaz J4aauo0d OL) Suaqoweued saem Jo Auoysty-ouwty TeotdAy °G aun3tTy 99/62/LL 9S/8Z/LL 9S/LZ/Lt 99/97/LL 99/82/11 al 90 00 8L (a 90 00 BL ZL 90 00 gL Zl 90 00 8L ZL 90 00 NOILWLS LNAWIYSdx4 SAVMYSLVM Y4SNIONZ AWHYY ‘S'N 3HL JO 3SV9 VLVG LSVOGNIH 3HL ONISN G3LVINWIS SV aoe’ VASV1V ‘NVIGOM YVAN VNSV1V 4O 31ND NYSHLYON FHL NI 9S6L ‘YSSWSAON 4O WHOLS oO ° fo) wo ° ool (0) NOILOSHIG SAVM 39DVHSAV @B ° (SL) GOIH4ad IWHLO3dS AVAd Y (SH) LHDISH SAVMLNVOISINDIS @ 00z anu. 334930 ‘6 OL (a vi id ale 938 OL Si 02 SZ oe GE 20 This rise to and fall from peak conditions over many hours, sometimes days, is typical for severe storms in most areas of the world. The peak condition is typically applied in extremal analyses, but the duration of conditions above a threshold related to the stability of a proposed structure is also important. Simulation of many hours (or many thousands of waves) is performed as standard practice for breakwater stability tests by a number of prominent laboratories (Owen and Allsop 1984 and Van der Meer and Pilarcyzk 1984). The effect of duration on breakwater stability is discussed by Graveson, Jensen, and Sorensen (1980), Jensen (1984) and Bruun (1985). Foundation considerations 30. The weight of a rubble-mound breakwater and the hydraulic effects it causes near its foundation are potential factors which can lead to a struc- tural failure. Investigation of gravity related stability problems, such as slip failure of the foundation or excessive (possibly differential) settle- ment, requires the attention of a geotechnical specialist. Hydraulic problems such as scour at the toe must be addressed in the earliest stages of design. The suitability of a natural foundation and the possibilities for preventive measures can ultimately determine the feasibility of constructing an entire breakwater. Excavation of poor foundation materials and replacement with fill or artificial improvement of the strength of natural materials can amount to a substantial fraction of the project cost. The need to place filter materials or other scour protection along a breakwater can also substantially constrain the geometry of the armor and underlayers. Seismic stability analyses in areas subject to earthquakes should be performed. All of these geotechnical considerations require extensive field data consisting of numerous borings supplemented by acoustic surveys and penetrometer tests. Primary armor 31. During the past 40 years many lengthy journal articles, textbook chapters, and conference papers have been written on the subject of armor de- sign for rubble-mound breakwaters. A discussion of the entire multitude of practical considerations applicable to armor design would be beyond the scope of this report. A comprehensive review is available by Baird and Hall (1984) in which many of the most important factors in armor design are discussed. Rubble-mound breakwaters have a tendency to be designed from the top down be- cause the exigencies of design and construction of those portions exposed to direct wave attack tend to constrain all other features. The stability 21 formulae, presented as Equations 1 through 8, apply only to the resistance to displacement of individual armor units. The use of concrete armor units also requires the investigation of mechanical strength related to the interaction betw rn the units in the armor layer and the associated impacts, fatigue, and creep (static) effects that occur. Quarrystone can be subject also to fracturing without displacement, but experience shows rock and the bulkier conerete units (such as plain or modified cubes) develop less of this sort of damage than do more slender concrete units (such as dolosse). A num- ber of proof tests and other quality control procedures have been proposed to account for mechanical strength limitations in concrete armor units (Burcharth 1981 and Price 1979) which should be considered for application in any project involving these units. Large concrete armor units should be designed with the advice of a specialist in concrete engineering, particularly where fiber re- inforcement is contemplated. The availability of existing forms should be investigated before fabrication of expensive specialized concrete forms is undertaken (Owen 1985). Design formulae indicate a minimum size armor unit, but the availability of existing forms and other practical factors may make slightly larger units more economical. 32. Design considerations related to the geometry of the armor layer are of particular interest in discussions of optimization since armor units are typically the most expensive materials used in a rubble-mound breakwater. The extent to which primary armor extends below the still-water level on the seaward face is typically set subjectively at 1.5 to 2.0 design wave heights (Figure 1). A berm of secondary armor or underlayer material at the toe of the primary armor is considered good practice, enhancing both the accuracy of underwater placement of the primary armor units and their resistance to slid- ing failure near the toe. The primary armor is usually extended below the waterline on the leeward side by 0.5 to 1.5 wave heights, depending on the de- gree of overtopping anticipated. If a monolithic wave screen is planned for construction on the crest (as illustrated in Figure 6) and virtually no over- topping is to be allowed, armor on the lee side need only be sized to remain stable in the ambient wave climate on that side of the breakwater. Wave sereens and monolithic crest structures are sensitive and highly specialized features (Jensen 1983) and will not be dealt with in this paper. 33. The allowances above, along with the crest width, crest height, and number of armor units comprising the thickness of the primary armor layer, 22 CREST ELEMENT Pots Figure 6. Typical breakwater cross section with a monolithic crest element determine the total volume of primary armor per unit length of structure. The conditions determining these dimensions will change along the length of a breakwater. Transitions should be gradual with conservative allowances for the limited confidence in the predicted variations in design conditions. All of these considerations must account for both extreme high water conditions and the possibility of a low tide condition which could greatly complicate the stability of features near the toe. 34. The dimensions of the armor layer generally are formulated as func- tions of the primary armor weight. The armor thickness and crest width are related to the weight of the armor unit by the following relation: 1/3 | ES We r= nk, (4) (9) where r = total average layer thickness or crest width n = number of armor units comprising the thickness or width (usually 2 for the thickness and 3 for the crest width) K, = "layer coefficient" (see Table 7-13, SPM 1984), an empirical measure of the thickness compared to tnat of the same number of equivalent cubes 35. The weight of the individual armor units, as determined by the Hudson formula (Equation 1), is a function of the slope, the armor material's density, the K, factor, and the wave height cubed. A small increase in de- sign wave height makes a substantial difference in the armor weight, i.e., a 10 percent increase in H corresponds to a 33 percent increase in W . The armor thickness will increase only 10 percent. The in-place unit price of armor material (both quarrystone and concrete) will vary directly with the total weight of the units relating also to the practicalities of quarry 23 development, concrete unit forming, and difficulties in handling. A reduced slope (increased cot @ in Equation 1) will reduce the armor weight require- ment but will change the runup characteristics of the seaward face in a non- linear manner. Overtopping and the associated transmitted wave characteris- tics are then affected, which in turn affects the required crest elevation for acceptable wave attenuation. The overall volume of a roughly trapezoidal- shaped breakwater (in cross section) increases as the square of the increase in the crest elevation. A significant effort is therefore necessary to deter- mine the most economical combination of slope, armor type, armor weight, and crest elevation for every pair of functional and structural design criteria, even when first cost is the only consideration. Other breakwater features 36. The constraints involved in primary armor layer design can sometimes overshadow other considerations for design of the secondary armor layers, underlayers, core, foundation filters, and scour protection (Fig- ure 2). The terminology of the SPM (1984) refers to a secondary armor layer as material placed on the face of the breakwater below the primary armor layer. An underlayer is placed between the armor on the exposed face and the core in the interior of the structure. Underlayers serve basically three functions: to keep the core in place through filtering action, to further dissipate wave energy that has penetrated through the primary armor, and to act as a foundation for the primary armor. These functions also apply to underlayers between the primary armor and the natural foundation (sea floor). Multiple underlayers may be required to satisfactorily accomplish all these functions. Material with small enough voids to hold finer core material in place may be too fine to stay in place itself under the larger voids in the primary armor layer. The primary armor also needs a relatively rough surface under it to discourage sliding. A coarser underlayer also provides some pro- tection from waves during placement of the primary armor (Hedges 1984). 37. Filtering criteria developed for water quality or seepage control purposes, such as Dis (filter) < 5Dgc (foundation) (Sowers and Sowers 1970), can be restrictive, thus the size gradation should be a consideration in evalua- tion of borrow areas for core material. Efficient use of quarry materials is encouraged in the SPM (1984). Given the practical problems of accurate place- ment of complex underlayers in the field (especially underwater), this goal may not always prove as economical as relaxing gradations of the various 24 layers such that their number, complexity, and associated construction quality control requirements are minimized. Unfortunately, breakwater specialists do not agree on a precise filter criterion for rubble-mound breakwater underlay- ers (Jensen, Graveson, and Kirkegaard 1983), and physical modeling of scour of core material is complicated by scale effects (Hedges 1984). 38. A densely packed core can reflect a significant amount of wave en- ergy back through the underlayers and reduce the stability of the armor or in- crease scour near the toe of the breakwater. A core and underlayer system that reduces wave energy through turbulence and frictional loss is preferred to a more reflective system. A core that is too permeable can transmit waves as much as 80 percent of the incident wave height (Kogami 1978), and it may pass littoral materials. Some useful experiments with wave transmission through porous rubble-mound breakwaters were performed by Madsen and White (1976) and continued by Seelig (1980a). Their methods are helpful in predict- ing wave transmission characteristics and will be discussed again later in this report. The effects of variations in permeability are discussed further in Bruun (1985). Head and elbow construction 39. The inevitable lateral flow across round heads and elbows and the reduced interlocking and compaction in these areas complicate just about every facet of breakwater design. Practical methods to deal with these compli- cations consist primarily of conservative adjustments to analyses as applied to sections of the breakwater trunk. This type of adjustment has limited confidence as evidenced by the frequent need to repair heads and elbows of conventionally designed rubble-mound breakwaters. Model testing in a three- dimensional wave basin is at present the only reliable means of improving this confidence. This is particularly important with slender concrete units (such as dolosse), which may have little or no increased stability over rock or bulky units in lateral flows (Burcharth and Thompson 1982). It is this fact that has caused some investigators to question the reliability of the Hudson formula and the associated K, factors published in the SPM (1984) for use in head or elbow design (Angerschou et al. 1983). The detail design of heads and elbows will very likely remain a highly subjective and empirical process for some time. Toe construction 40. A number of practical problems related to the toe of rubble-mound 25 breakwaters have already been mentioned. This area of transition from a hopefully stable static environment (the breakwater) to the natural, often dynamic, sea floor is critical to the overall stability of the structure. Toe features not only protect the bottom from scouring (which can lead to under- mining) but also support the weight of the armor material above. The need to provide primary armor 1.5 to 2.0 wave heights below the still-water level can conflict with the need to filter foundation sediments at the tow. This is particularly true in high tidal ranges where low water conditions can expose the toe to more extreme wave effects. The support of armor materials is most reliably accomplished with a substantial berm of secondary armor or underlayer material at the toe of the armor slope. This berm should have at least sev- eral units or a minimum 3-m top width. Wide differences in the size of the bottom sediments and the breakwater material near the bottom may require excavation of a trench along the toe to accommodate a toe berm with an ade- quate filtering underlayer, as illustrated in Figure 7. Geotextiles can be used also in some instances to reduce the height of toe features and the as- sociated exposure to more severe wave energy. The concurrent physical mod- eling of armor stability and toe scour is complicated by scale effects, but model tests can reveal trends which could suggest a compromise of either the filtering criteria or the extent of primary armor. One radical concept in toe design is the "wave reducing berm" (Delft Hydraulics Laboratory 1983) which provides artifically shallow depths for dissipation of wave energy. Sugges- tions for design of more conventional toe features are discussed by Eckert (1983) and Jensen (1984). Construction equipment and techniques 41. The constructibility of a rubble-mound breakwater design is an extremely important and practical consideration that can control its over- all feasibility. Smaller breakwaters can often be constructed with con- ventional land-based construction equipment and techniques by building from the shore outward. Detached breakwaters can be constructed in this fashion only if a temporary causeway to the permanent portion is constructed and later removed. Larger or more exposed breakwaters often include features which make construction exclusively with land-based equipment difficult. For example, placement of large armor units in relatively deep water near the toe of a shallow slope (perhaps at the head) may be too far to reach for a mobile crane on the breakwater crest. Another example is the occasional need to build up 26 FILTER BLANKET SEA FLOOR Figure 7. Typical toe trench underwater features using floating equipment, particularly toe berms, prior to placement of core material, underlayers, and primary armor. The sequence of operations, specific placement techniques, and the associated equipment avail- able to perform this work usually constrain the range of alternate breakwater configurations to some degree. No breakwater configuration should be con- ceived without thorough attention to its method of construction. More de- tailed discussions of these considerations are available in Bruun (1979), Kjelstrup (1979), Maquet (1984), and Bruun (1985). Physical Modeling for Stability Guiding principles 42, The specific techniques applied in physical modeling of rubble- mound breakwaters by various hydraulic laboratories differ in detail, but the guiding principles of similitude offer the same basic constraints in all cases. Scale models of breakwaters for hydraulic stability are designed ac- cording to the Froude scaling relation which requires that the Froude number of the model be equal to that of the full-scale prototype in its intended Cat natural setting. This relation is expressed as ve ua gL} ~ \eL eg®) a g D g = acceleration of gravity where V = flow velocity L = a linear dimension associated with the flow These scaling criteria provide that the linear dimensions of the model are all geometrically similar to those of the prototype. Typical rubble-mound break- water model scales range from 1:5 to 1:70. The Froude number theoretically represents the ratio of inertial to gravitational forces, an appropriate mea- sure in situations where gravity is the predominant force. It is widely accepted that this is usually the case for rubble-mound breakwaters (Hudson et al. 1979). 43. Another scaling law sometimes applies, however, which requires that the Reynolds numbers of the model and prototype be equal, or LV LV m p where v is the kinematic viscosity of the fluid. The Reynolds number the- oretically represents the ratio of inertial forces to viscous forces. Viscous forces in the primary armor layer, underlayers, and core are now thought to have greater importance than they did in the pioneering days of rubble-mound breakwater design. The Reynolds criterion conflicts in many instances with the Froude criterion in sizing structural materials for models (particularly in smaller, more economical models), and compromising measures are usually ne- cessary. Other scale effects can come into play when model waves are so short that surface tension has a significant effect (seldom a real problem in prac- tice) or when the mechanical strength of armor units is critical. Dealing with these conflicting criteria makes physical modeling of rubble-mound break- waters a highly specialized practice. Proper execution of a rubble-mound breakwater scale model study requires both specialized equipment and extensive experience available only at a handful of hydraulic laboratories around the world. Operational procedures 44, The representation of the sea state in scale models continues to improve in modeling facilities because of enhancements of wave generating equipment and improved understanding of the physics of water waves. The ear- liest wave generators were capable only of a sinusoidal motion generating monochromatic waves. The last decade has seen these facilities replaced in many laboratories with wave generators capable of producing irregular waves which simulate specified prototype energy spectra or irregular time series. The techniques for application of monochromatic waves are somewhat standard- ized, but presently there are widely differing opinions on the most appro- priate application of irregular waves in scale models of rubble-mound stabil- ity. The transformation of the waves from deep water to shallow water must be arranged to be equivalent to that in the prototype for both monochromatic and irregular waves. The shallow-water waves of interest for stability are usu- ally taken to be those naturally transformed waves that would exist at the site without the structure in place. This convention usually involves a cal- ibration of the model facility before the model structure is placed in a flume or basin. 45. Complications with reflected waves arise after the structure is in place. Techniques are available for analysis of model wave data which resolve incident and reflected waves (Goda and Suzuki 1976). Some facilities are ca- pable of compensating for reflected waves by modified motion of the wave gen- erator. It is necessary in facilities without this capability to reduce wave reflection as much as possible by various other means. Use of irregular model waves can also result in spurious long-period waves (Jensen and Kirkegaard 1985) which must be compensated for by the generator or in the interpretation of measured results. 46. Model breakwater materials must reflect a number of prototype con- ditions, including geometry, density, surface roughness, and orientation in the structure. A number of recent tests have also involved attempts (the re- sults of which remain in question) to estimate mechanical stresses within armor units (Timco 1981 and Delft Hydraulics Laboratory 1985). Geometry, den- sity, and surface roughness are controlled by careful choice of model mate- rials and preparation of the units. Minor density scale effects due to use of fresh water in a model of a saltwater site can usually be compensated for by small adjustments to the weight of the model breakwater units. Orientation 29 in the structure is accomplished with a variety of manual and automatic tech- niques designed to simulate the realities of full-scale field placement. The placement tolerances of model rubble-mound breakwaters often are smaller than their prototype counterparts, however. The hydraulic characteristics of un- derlayers and the core can be especially difficult to model by the Reynolds criteria since the shape of the units, their surface friction (particularly between layers), and the shape of the interstices are critical. Erosion of fine foundation material at the toe of breakwaters is also a problem, gener- ally yielding only qualitative conclusions. An account of these and other scale effects is necessary for reliable interpretation of model results (Jensen and Klinting 1983). 47. Scale modeling operational procedures associated with the design of rubble-mound breakwaters can be classified in three general groups: (a) cross-section design tests run in two-dimensional flumes (Figure 8), == W434-180n) Figure 8. Scale model testing in two-dimensional wave flume (b) tests of heads, elbows, transitions, offshore hydrographic effects, and oblique waves in three-dimensional wave basins, and (c) tests of breakwaters at various stages of construction (in either flumes or wave basins). The first of these is of primary interest to discussions of analytical optimiza- tion, since it is this type of scale model testing which has generated most of the analytical relations used by designers. These tests of proposed cross- section designs are intended to verify the predictions of analytical proce- dures and to refine detailed features of the cross section. They are often 30 more than a fail/no fail "proof test" of a design and should be arranged to provide the maximum information of use for similar future designs. 48. A procedure used for many years to verify the stability of proposed cross-section designs involves subjecting a model breakwater to a short series of monochromatic waves at the design (stability) wave height at various wave periods above and below the design period. The water level is also varied within the range of possible levels predicted for the prototype site. This sensitivity analysis approach is intended to reveal the breakwater's response to the severe condition when plunging breakers are directly impacting the sea- ward face, as seen in Figure 8. Displacement of some fraction of the armor layer is measured by before and after soundings of the model structure. This procedure is relatively economical and provides an indication of the design's resistance to armor unit displacement by a group of waves with a "worst case" combination of period and water level. Some statistical confidence is lost since the design criteria for wave period and water level are not held con- stant in the modeling procedure. Subsequent changes to the cross section in response to unacceptable damages in the model contribute to further departure from initial design criteria and any associated risk analysis. Design cri- teria must then be reformulated and associated analyses repeated with the new criteria. 49. Tests of cross-section designs with irregular waves typically involve a longer series of waves, since a significant number of waves (100 or more) are necessary to adequately resolve a specified energy spectrum. The added test condition parameters related to reconstructing a specific spectral shape in a wave flume discourage the sensitivity analysis method described above. Hydraulic laboratories differ in their approach to tests for the effect of wave groups with irregular waves, however. Some favor manipulation of spectral shape parameters to enhance wave groupiness, while others prefer spectra that are as natural as possible. Recorded spectra are reproduced in some instances to assure a completely natural incident wave condition in sta- bility tests. Durations of individual tests also vary from relatively short tests of around 100 waves (30 to 45 min) to tests of thousands of waves and many hours simulating the growth and decline of a storm, as illustrated in Figure 5. Further discussion of model tests with irregular waves is available in Jensen (1984) and Bruun (1985). 50. Evaluation of damages after a test is a critical step which 31 requires special care and can involve sophisticated techniques and equipment. Color coding armor units in their initial placement is a simple way of illus- trating the degree of overall displacement of the armor layer. Soundings on a small grid before and after a test will measure the overall volume of material which was moved, though net profile changes can hide more drastic gross move- ments which may have occurred during the test. Photographic or video proce- dures have been used to follow actual movements, including rocking in place, of individual units with good success (Delft Hydraulics Laboratory 1985). Detection of rocking is especially important in testing dolosse or other slender concrete armor units since it is known that they experience signifi- cant breakage in place from impacts between individual units. Testing for damage rates of these units is therefore a highly subjective process because the excessive mechanical strength of model units prevents evaluation of the stability of a design after some of the armor units have broken. Reduced strength model units (Timco 1981) may eventually provide a better means to measure stability of slender units, but model units fully similar to their prototype units in mechanical strength are not currently available. Stresses in prototype concrete armor units are far from fully understood, but research in this area is under way at most leading hydraulic laboratories. 51. A number of other characteristics are sometimes measured in con- junction with stability tests of breakwater designs, including reserve sta- bility and wave transmission. Reserve stability refers to the extent of dam- age that occurs when the breakwater is subjected to waves in excess of the design condition, an important consideration in risk analyses. Wave trans- mission characteristics require additional tests to be fully defined, par- ticularly when the functional performance design criterion (in terms of wave transmission) is substantially different from the structural integrity design criteria. Runup is a useful parameter to measure in conjunction with wave transmission tests, since the ratio of runup to freeboard seems to be the most sensitive parameter in analytically predicting wave transmission by overtop- ping. Runup is difficult to gage precisely on rough permeable slopes, and traditional visual methods are still common. Measurement of volumetric over- topping rates is also occasionally of interest, but a special setup with pro- visions for containing overtopped water is necessary. Techniques for measur- ing and evaluating the detailed relationship of. runup, volumetric overtopping, and transmitted waves to incident waves in terms of wave-by-wave effects and 32 time series analysis parameters need a great deal of further development. 52. Tests in wave basins to determine the overall susceptibility of a rubble-mound breakwater to direct and oblique wave attack (with attention to the head and elbows), to transitions between cross sections, and to the hydro- graphic features offshore of the structure, are necessary for most projects. Wave basin facilities are larger and more complex than wave flumes (Figure 9) and therefore are more expensive to use. Tests of this nature not only reveal unique information about breakwater stability and other characteristics but also provide important confirmation of conclusions from flume tests. Basin models are typically at smaller scales than most flume models; thus, Reynolds scale effects are exaggerated. Basin model testing confirms the location of the most critical cross section of which more precise stability tests should be performed in a flume at larger scale. Wave transmission by diffraction through the entrance channel or other breakwater gaps is one of the most im- portant measurements in a basin test. Long-period oscillations resulting from the enclosure of a harbor area by a breakwater are also important to detect. Model tests including tidal fluctuations can reveal circulation pat- terns inside a proposed breakwater (Headquarters, Department of the Army 1984). 53. The last category of breakwater model test is most important for large breakwaters requiring complex construction procedures and many months of construction time. Provisions for interim protection of partially completed breakwaters must be tested to justify what can be a significant additional cost to the project. Wave basin testing is more often appropriate for this work, but flume testing can be quite helpful also. 54, The modeling procedures discussed above are the true basis of virtually all the analytical tools available to rubble-mound breakwater de-, signers. Quantitative measurements of prototype breakwater performance are just now becoming available and have yet to be applied toward reliable ana- lytical design procedures. Each application of analytical procedures is an interpolation or extrapolation of limited prior experience. More often than not, refinements which reduce cost and improve performance result from model tests of a proposed design. Vital confirmation of analytical assumptions, both explicit and implicit, is provided by even the simplest model test. The expense and time are worth it in every case. 33 4804 Tepow aTeos utseq BAEM [TRUOTSUOUT p- aeuu, 6 eun3Tty 34 PART III: ESTIMATING DAMAGE RATES 55. A key step in identification of an optimum among alternative rubble-mound breakwater plans is to estimate the expected damages and life cycle costs of related maintenance and repairs. The concept of designing a rubble-mound breakwater for zero damage is unrealistic because a finite risk always exists for the stability criteria to be exceeded in the life of the structure. The stochastic nature of the physical phenomena affecting coastal engineering structures requires that a probabilistic approach be applied, if these maintenance cost estimates are to be more than guesses. The incident wave climate can be characterized by estimating probability distribution func- tions by a number of relatively well accepted methods (Battjes 1984). The crucial problem for rubble-mound breakwater designs is in relating a given level of damage and associated repair costs to specific incident wave condi- tions. The rate at which this damage accumulates must also be predicted in order to tentatively schedule maintenance and related cash flows. The fol- lowing section will review some techniques proposed for making these predic- tions. Their relative merits will be discussed and areas of ongoing or needed future research identified. Damage Assessment 56. Damages to rubble-mound breakwaters have been quantified in many ways by researchers and field engineers. The current issue surrounding break- age of concrete armor units has led to a number of recent publications pointed at systematic assessment of damages of all kinds. One useful characterization of prototype damages in terms of displaced primary armor units was proposed by Groeneveld, Mol, and Den Boer (1984) and is presented in Table 2. Table 2 Classification of Breakwater Damage Type of Failure Displacement, 7% Description Minor 0-3 A few individual units of top layer dis- placed, but no gaps in top layer larger than 4 units; bottom layer intact (Continued) 35 Table 2 (Concluded) Type of Failure Displacement, b Description Moderate 3-5 No gaps in top layer larger than 6 units; only slight displacements of bottom units Major 5-30 Top layer removed over a large area; bottom layer over not more than 2 units Total Over 30 Primary armor and underlayers removed over a large area with exposure of core material 57. This classification of prototype damages is realistic as far as field reconnaissance of a damaged breakwater is concerned, but it departs somewhat from the convention of detecting incipient damage in model tests. It does not take into account any concrete armor units which have broken in place. This inadequacy is compensated for, in part, by the displacement of intact primary armor units being accompanied, in most instances, by concurrent displacement of broken pieces. It is the exposure and, ultimately, the ero- sion of underlayers and core that spell the actual failure of a rubble-mound breakwater in the functional sense, with the exception of the case when a monolithic crest element has been rendered ineffective. Field investigators should also search for evidence of other modes of failure besides hydraulic ' displacement, including sliding due to toe failure, excessive foundation set- tlement, and seismic displacements. Classification of damages as a function of both cause and effect is discussed in detail in Bruun (1985). 58. Laboratory investigations, as pioneered by Iribarren (1938) and Hudson (1958), typically attempt to identify the point of incipient damage. Kogami (1978) defined this criterion as "...the condition in which the number of armor units clearly recognized to have been moved or rocked on the cover layer surface by wave actions was less than 1% of the total of the units on the forward cover layer...." The account of rocking implies that a precise method of measuring the extent of rocking is available. Another interpreta- tion relates to the point at which displacement has reached a depth in the armor layer equal to the equivalent cube dimension of the armor units (Losada and Gimenez-Curto 1979). Techniques developed by WES in the 1950's for mea- suring model breakwater displacements with before and after soundings have been estimated to have a resolution (repeatability) of +2 percent (Carver 1983). Identification of incipient damage with this commonly used method 36 is therefore only meaningful in the range of 0-3 percent primary armor dis- placement. Nielsen and Burcharth (1983) have indicated that measurements of very low levels of displacement or rocking (0-3 percent) are less reliable than those of higher levels. This trend relates to the resolution of mea- surement techniques as well as the repeatability of the experimental results themselves. 59. Given a relatively consistent and precise method of measuring displacement, Ahrens (1984) has proposed a useful dimensionless parameter for systematic quantification of breakwater damage: 4D Es Ww 2/3 i Where Ap is the average eroded cross-sectional area for a specific length of D' (12) model breakwater (Figure 10). Van der Meer and Pilarezyk (1984) applied the following dimensionless damage parameter Sp in their model tests of quarry- stone, which was mentioned previously (Equations 5 through 8) in the discus- sion of their conclusions regarding stability: An Oe 2 (ia) It is also important to identify erosion of the underlayers or core that may S (13) coincidentally occur with erosion of the armor layer. Analytical Damage Prediction 60. Scale model studies reported by Jackson (1968a) and Carver and Dubose (in preparation) have addressed, to a limited degree, the level of dam- age to breakwater armor layers experienced when the design wave height is ex- ceeded. This information was applied to formulate Table 7-9 in the SPM (1984) which predicts the percent damage %D for various armor types as a function of the design wave exceedance ratio H/Hy where H is a monochromatic inci- dent wave height which is greater than the design wave height Hy - The re- serve stability trends, or tendency for damage levels to increase with design wave exceedance ratio, can also be characterized by a function of the follow- ing form: ai WI ‘WHOdLWId JAOSY NOILWAI13 aZeurep SSOTUOTSUSUTP JOJ yo Jays UOTATUTJeq ‘OL eun3TYy WO “TUNNWHO ONOIW JONVISIO . >> 31140ud">> Wau ywya"” H* zD < SD(Hq) <0 £30), 40; 50, 60, *70, 80; “9021100 20 <30;<40, 50, 60, 70, 80, 9077100 30 <30; 40,5060; 70,80; °90)'"100 HO <305140;1/50,, 60; 70, 80; 90, 100 50 <30, 40, 50, 60, 70, 80, 90, 100 90. Table 7 lists a comprehensive set of potential functional and structural design criteria combinations which may be abbreviated by carefully considered subjective judgments. The first column in Table 7, "functional performance," refers to an exceedance value x% of transmitted wave heights H, greater than some critical wave height H* . H* might conveniently be taken as the Ho value applied in the loss function of Step 2, but this is not necessary. This column includes a range of functional performance design criteria which could be addressed in terms of wave transmission. A wave height of 1m, for example, might be a threshold value for damage to vessels * US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, in preparation," Estimation of Expected Annual Economic Losses Due to Wave Attack--Computer Program BWLOSS1 (MACE-15)," Coastal Engineering Technical Note, Vicksburg, Miss. 60 moored behind the breakwater. The last functional performance criteria would thus be that x%Z of the waves transmitted by the breakwater during a 50-year storm would be in excess of 1m. The return period convention is in keeping with traditional practice, though the phrase "with 2 percent probability per year" would be a more accurate description of the storm of interest. Esti- mated probability per year might be a more appropriate increment in terms of providing even steps of cost between alternatives, but either convention will serve. The value of x% should relate to some consideration of the actual number of waves of H* or greater necessary to cause a measurable effect. A storm whose peak conditions lasted 3 hr with Tp = 10 sec would include roughly 1,080 waves. A small value of x% is appropriate, on the order of 1 percent, which for the example condition would include 10 or 11 waves. These waves would not likely occur in sequence, but a few of them might. 91. The shorter return periods of 20 or <10 years might be too risky for a small boat harbor where relatively fragile vessels and mooring facili- ties are planned immediately on the lee of the breakwater. These criteria are reasonable, however, when losses due to cargo handling inefficiencies or ves- sel transit time are all that is at stake. The 50-year storm is, on the other hand, a very conservative criterion for wave transmission. At least four functional performance criteria should be addressed to assure identification of an optimum design. 92. The second column of Table 7 includes choices for structural in- tegrity criteria in terms of the damage to the armor layer, as might be esti- mated by Equation 14. The hD(Hq) value chosen should be consistent with the incipient damage level, as measured in model experiments pertinent to the breakwater design at hand. Hy, is the wave height applied in analyti- cal stability relations. Return periods of 30 years or less for the storm represented by Hq will plainly involve substantial expected damage and therefore should be investigated only for minor breakwaters where repairs can be easily accomplished or postponed without significant adverse consequences. Long return periods greater than 50 years are important to address, however, since rubble-mound breakwaters require such a tremendous commitment of equip- ment and materials to repair. The risk of affordable quarrystone being un- available 30 or 40 years in the future might be great, even though it may be readily available at present. Repair of breakwaters in remote areas involves high mobilization and demobilization costs, even for small repair efforts. 61 Another important consideration in favor of addressing these longer return periods is the uncertainty of the future funding capacity of local sponsors for repair efforts. 93. The final choice of alternatives should contain a minimum of 15-20 pairs of functional performance and structural integrity criteria pairs. A single pair of these criteria will define each alternative breakwater con- figuration throughout the optimization process. Consistency in application of these criteria in analytical design efforts is critical to maximizing the re- liability of the procedure. New alternatives should not be added without car- rying the new ones through the entire procedure. Step 4-identify apparent optimum com- bination of armor size and type, slope, and crest elevation for each alternative 94. Each pair of design criteria will have several combinations of features that will provide the same performance and stability. An acceptable method of choosing an apparent cost-effective combination for each plan is to consider a standard parameterized cross section, as illustrated in Figure 17. The Hudson formula (Equation 1), the relation of armor thickness and crest width to armor weight (Equation 9), and the wave transmission relations (Equa- tions 26 through 29) can then be used to approximate all the dimensions of this standard cross section for a range of armor type and slope combinations. The relative advantages of armor unit hydraulic stability and of runup dissi- pation are both measured by this approach. The relative cost per unit length of breakwater trunk for each slope and unit type combination can also be esti- mated by incorporating representative unit prices for each armor type and size. 95. This method does not deal with the variation of reserve stability between armor types which would involve a substantial amount of extra input and computational effort. The question of reserve stability is addressed later in this proposed procedure, but at this stage it is neglected as a time-saving measure. "BWCOMP," an interactive computer program, has been developed to estimate the volume and first cost per unit trunk length of the parameterized cross section of Figure 17, given the two design criteria (as incident H, and Tp values and an H* maximum transmitted height) along with the other information discussed above. The program is documented in Appendix B of this report and in a CETN (WES, CERC 1984b). 62 pettdde uotyoes ssouod peztueqjeweueg “*)| aun3ty dWOOMd weasoud ut 3dO 1S GUVMVIS 4O LNIONVLOD OSSD DS NT 3d0 1S GYVMII7 JO LNIONVLOD 3YOO ONV SY3AV1Y3S0NN (YILVM T1LLS JAOGV) LH9OIFH LS3I¥9 LHOIFH JIAVM LNIGIONI L LHOIFH JIAVM GFILLIWSNV EL ——— Se NS QuVvMVas —___ qguvM331 63 Step 5--design detailed cross section for each alternative 96. This is the second highly subjective step in the proposed optimi- zation procedure where coastal engineers should, for each pair of design cri- teria, prepare a cross-section design with all the detailed features appropri- ate for the site conditions and other constraints. Practical considerations discussed in Part II of this report should be incorporated. All the special- ized experience and intuition available should be applied in this step, but it must be applied consistently to each alternative. It is critical that bias be studiously avoided at this stage. An estimate of the construction cost for each alternative detailed cross section should be prepared at the conclusion of this step. Step 6--estimate wave transmission characteristics of each alternative 97. An analytical procedure should be performed at this point to esti- mate the wave transmission characteristics as a function of incident waves H,(H; ) for each alternative. The program MADSEN (Seelig 1980a and WES, CERC 1984a) is useful for this purpose. The program accounts for the relative size and permeability of each layer of the breakwater cross section and the rela- tive runup characteristics of the armor layer. Wave transmission by overtop- ping (Equations 26 through 29) and permeation (Madsen and White 1976) is es- timated. The program is not as well verified for concrete armor units as for quarrystone, but it serves well at this stage for comparative purposes. A range of incident wave conditions should be simulated to obtain a substantial set of H, (H; ) points, including several more severe than the design condi- tion. The incident wave conditions need to correspond to height and period combinations predicted for the site in Step 1. Wave period is a sensitive factor for wave transmission, as applied in the program MADSEN. An appropri- ate wave period (such as the peak spectral period Ty) must therefore be as- sociated with each (significant) incident wave height, as suggested in Step 1. Transmitted waves are not Rayleigh distributed, as discussed in Part IV and Andrew and Smith (in preparation), but can be represented by a single height such as the root mean square wave height ans or H43 54° MADSEN predicts the Hrms Of waves transmitted by the combined effects of both permeation and overtopping. 64 Step 7--estimate economic losses with the breakwater for each alternative 98. The climate of transmitted waves behind the breakwater can now be approximated as a cumulative probability distribution F(H,) given a set of H, (A; ) points from Step 6 and the cumulative distribution of incident waves F(H; ) from Step 1. The loss function estimated in Step 2 can be used to es- timate the expected annual economic losses E{$L'/yr} for each alternative by ' dF(H, ) e{ $"| Siig) SLC) Uda (33) yr dH, t "BWLOSS2," a computer program, has been developed to perform these computa- tions. It has been documented in a CETN*, and it is included in Appendix C of this report. Step 8--estimate expected annual breakwater damages for each alternative 99. The methods discussed in Part III can be applied to relate a damage funetion %D(H/Hq) to each alternative. The incident wave climate defined by F(H) from Step 1 can in turn be applied to estimate the expected annual damages E{$D/yr} given representative unit repair prices $/vol and the volume of the armor layer Vol by adapting Equation 18 as follows: p22 | yore (5) Sa ee (34) lyr vol dH 100. This quantity is useful for comparative purposes, but it does not relat directly to a programmed cash flow for repairs. It is better that Equation 18 be applied to each alternative in its unmodified form to predict the expected annual %D in order to make some judgment if and when a repair project should be scheduled. The average time to reach a threshold level of unacceptable damage %D* can be estimated by simply dividing that value by E{%D/yr} . The return period of %D* could also be estimated by solving for H(ZD*) in the damage function (Equation 14) and applying Equation 21 to determine the asso- ciated return for that particular storm intensity. A computer program titled * US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, in preparation, "Estimation of Expected Annual Economic Losses from Waves Transmitted by a Breakwater--Computer Program BWLOSS2 (MACE-16) ," Coastal Engineering Technical Note, Vicksburg, Miss. 65 "BWDAMAGE" has been developed which applies Equation 14 and the information of Table 3 to estimate E{ZD/yr} , E{$D/yr} , and the repair interval by both methods discussed above. This program has been documented in a CETN,* and it is presented in Appendix D of this report. Once a repair interval and the as- sociated extent of repairs have been estimated for an alternative, discounted cash flow methods can be used to estimate the equivalent annual amount which can be substituted for E{$D/yr} . The damage functions, as stated in Part | III, are currently the least reliable of the analytical tools available for rubble-mound breakwater design and should be used with circumspection. Step 9--tabulate and sum costs for each alternative 101. This is the final analytical step of the proposed procedure, fol- lowed only by laboratory verification of the analytical predictions. The min- imum sum of the three costs identifies the cost-effective optimum alternative, as indicated in the following equation: ' st | Zz S| el g{ $2 (35) yr yr The first cost must be transformed from a present worth value to an equivalent annual amount E{$18* yr} by discounting: prior to the summation. Incremental benefits E{$B/yr} can be estimated by subtracting E{$L'/yr} from E{$L/yr}: ( e{ 8} - oft - of Eh (36) yr yr yr Net benefits E{$B,.,/yr} can in turn be estimated by subtracting E{$18* yr} and E{$D/yr} from E{$B/yr} as follows: oft] = BB) ofA] - of) an This method of estimating benefits may not be appropriate for some projects, however, as discussed at the beginning of Part V. * US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, in preparation, "Estimation of Rubble-Mound Breakwater Expected Damages--Computer Program BWDAMAGE (MACE-18) ," Coastal Engineering Technical Note, Vicksburg, Miss. 66 Step 10--verify by physical modeling damages and wave transmis- sion of apparent optimum alternative 102. This step is necessary to assure that all the compounded assump- tions and analytical inaccuracies are within acceptable limits. This is the case with any analytical design procedure for rubble-mound breakwaters since the empirical relations have been shown to all have limited confidence. Each laboratory test of analytical assumptions applied to a specific design will narrow the confidence limits and improve the reliability of future analytical efforts. A simple proof test with monochromatic waves of varying period con- stitutes a minimum effort in this direction, but it is inadequate to test the accuracy of an optimization procedure such as that proposed above. 103. The damage function %D(H/Hq) must be verified by model testing, including simulation of conditions for incipient motion and a range of more severe conditions. The design conditions should be simulated as accurately as possible in order to include the effects of the numerous physical parameters not explicit in the analytical stability formula that was applied. Wave pe- riod, wave groupiness, storm duration, and static stability, among other fac- tors, should be considered. The static friction factor yw from the Iribarren formula (Equation 2) should be measured by sliding tests, as proposed by Price (1979) and Graveson, Jensen, and Sorensen (1980). 104. The fully described incident’ wave conditions cannot be simulated with monochromatic waves. Either an average (for example JONSWAP) spectral shape or one adjusted to be similar to measured spectra for extreme storms near the site can be applied in flume tests of the apparent optimum cross sec- tion. Simulation of a gradual rise to peak conditions, then 1,000 waves or more at the peak (stability criterion) condition, followed by a gradual de- crease of wave energy, would be most useful for tests to verify damage func- tions. A test or tests at the design condition should be followed by tests at more extreme conditions related to the extremal distribution of wave heights (and periods) derived in Step 1. Enough %D(H/Hq) points must be measured to verify or refine the %D(H/Hy) analytical function that was applied in Step 8. A minimum of three tests would be useful, including the mD(Hq) point and at least two more severe conditions. More stability tests should be conducted if agreement with the predicted damage function is not good. Techniques to detect gross rocking motion should be applied in identifying 67 incipient motion. Actual damage should be measured by before and after sound- ings on a fine grid, but some judgment must be made as to the additional dam- age that might have occurred in prototype from armor unit breakage. 105. Wave transmission characteristics of the apparent optimum cross section must also be verified. Tests of these design conditions simulating the fully described forecast conditions at the site as accurately as possible should be performed for the functional performance criteria and a number of more extreme conditions. Again, these extremes should relate to the F(H, A Ty) derived in Step 1. At least three Hee (Hoa) points should be measured in order to verify or refine the economic loss function derived in Step 7. Operational techniques should include efforts to accurately model reflection and wave transmission by both overtopping and permeation. Transmitted waves should be measured as time series comparable to time series measured of inci- dent waves. Coherence and cross-correlation analyses should be performed for the incident and transmitted time series along with computation of more common spectral parameters. Individual runs of 100 or more waves are recommended for the wave transmission tests in keeping with the widely accepted assumption of stationarity in natural sea states. 106. The measured %D(H/H,) data and H,(H;) data should be applied in Steps 6 through 9 for the apparent optimum cross section. All its associ- ated costs should then be adjusted according to the revised expected damages and economic losses with the breakwater in place. Model tests often make sig- nificant refinements to a design cross section obvious, and any such refine- ments should be incorporated. Drastic changes to the original apparent opti- mum cross section may require similar changes to be made to all the alterna- tives and for Steps 5 through 9 to be repeated for these cross sections as well. If the original apparent optimum is still indicated as the optimum cross section, then no further model testing will be necessary. A new appar- ent optimum should have its #D(H/Hg) and H, (H; ) functions verified in as thorough a manner as the first. 68 PART VI: SUMMARY AND CONCLUSIONS Summary Optimization 107. Optimization has been demonstrated as a systematic process of max- imizing net tangible economic benefits or of minimizing the total costs (in- cluding economic losses) to the beneficiaries of a public works project. Op- timization of rubble-mound breakwaters addresses the incremental net benefits of these structures which are often major features of a larger coastal devel- opment. Federal laws and policies currently require that incremental net ben- efits be positive for all major features of projects proposed for federal funding. Furthermore, cost sharing policies have placed a substantial burden for financing these projects on local and regional governments. Financeabil- ity of civil works projects is now an important question outside that of posi- tive net benefits. Rubble-mound breakwaters must achieve the maximum benefits for the least cost in order to be affordable as well as economically feasible. Arbitrary conservatism in design of rubble-mound breakwaters is no longer affordable, and coastal engineers must use all the tools and information available to assure the optimum alternative has been proposed. Design criteria 108. Alternatives for rubble-mound breakwaters should be optimized ac- cording to two criteria: functional performance and structural integrity. The functional performance criterion refers to the structure's effectiveness as a wave barrier as measured by its wave transmission characteristics. The structural integrity criterion refers to the structure's ability to survive an extreme storm without significant damage and the rate it suffers damage from storms more extreme (less probable) than the structural design event. Analytical design and laboratory verification 109. The analytical tools available to designers of rubble-mound break- waters have been reviewed in some detail. They have all been shown to be the products of a finite set of laboratory experiments, with very little quantita- tive prototype verification. Current research continues to refine the preci- sion of these empirical relations, but this precision is not yet sufficient to warrant construction of rubble-mound breakwaters without verification of 69 analytical predictions by scale model tests. Nevertheless, analytical proce- dures are available for prediction of armor unit hydraulic stability (resis- tance to displacement by waves), armor layer damage rates, and breakwater wave transmission characteristics. These tools, with laboratory verification, can be used to systematically select an optimum alternative. The proposed procedure 110. A systematic optimization procedure has been proposed which makes use of the analytical tools currently available to coastal engineers for rubble-mound breakwater design. The procedure begins with definition of the site conditions and formulation of an ensemble of alternative design criteria pairs. These steps are followed by estimates of first costs, maintenance costs, and user costs with the breakwater in place for each alternative. The concept of statistical expectation is applied to measure the costs of all al- ternatives on the same basis. The process is concluded by physical model tests to verify the analytical predictions for structural stability and wave transmission characteristics of the apparent optimum alternative. The entire procedure is summarized in Table 8, with references to pertinent formulae, software, and documentation. Table 8 Summary of Optimization Procedure Pertinent Available Step Procedure Equations and Tables Software | Define site Equations 19* or 22* WAVDIST (WES, conditions CERC (in preparation) ) 2 Estimate economic Equations 31* and 32* BWLOSS1 (WES, losses without CERC (in breakwater preparation) and Appendix A) 3 Formulate an en- Table 6 -- semble of alterna- tive functional and structural criteria pairs (Continued) Note: * indicates the equations which are applied in the referenced software. 70 Procedure 4 Identify optimum armor, type W cot @ , and crest ele- vation for each alternative 5 Design detailed cross section for each alternative 6 Estimate wave transmission char- acteristics of each alternative if Estimate economic losses with break- water for each alternative 8 Estimate breakwater damages for each alternative 9 Tabulate expected costs for each alternative and identify apparent optimum 10 Verify predicted damage and wave transmission by scale modeling Table 7 (Concluded) Pertinent Equations and Tables Equations 1* (or 2-8), 29* (or 30), and Table 5* or 6 Equations 1-9 Equations 19 or 22, 26*, 27%, 287, and 29* (or 30) and Table 5* or 6 Equation 19% or 22,13 1*; and 33 Equations 19* or 22, 14%, (or (5,5. °65. (7 sxand )8.c0r 23 and 24), 18*, and 34* and Table 3* Equations 35, 36, and 37 Equations 1-8, 10-14, and 26-29 Available Software BWCOMP (WES, CERC 1984b and Ap- pendix B) MADSEN (Seelig 1980a and WES, CERC 1984a) BWLOSS2 (WES, CERC (in prep- paration) and Appendix C) BWDAMAGE (WES, CERC (in prep- aration) and Appendix D) Uthke Conelusions The investigation which was conducted in order to develop the above optimization procedure led to the following conclusions regarding rubble-mound breakwater design: a. A systematic optimization procedure should be applied in any rubble-mound breakwater design to assure that an alternative with maximum cost effectiveness is proposed. 7) Io Ke) Rubble-mound breakwater designs should not be constructed without physical model testing of some kind due to the limited confidence of available analytical methods. The confidence of the key analytical tools for rubble-mound breakwater design would be improved if current research were continuously concentrated in the following specific areas with probabilistic applications in mind: (1) Site conditions--Estimation of the long-term joint proba- bility distribution .F(H:, T , t., d 9.eé¢so0)e0 for vay site should be developed for application in estimating ex- pected breakwater damages and the long-term distribution of transmitted wave characteristics. (2) Armor stability--Standardized methods should be developed for scale model testing of rubble-mound stability in nat- ural irregular sea states. Improved analytical stability prediction should be the goal of tests conducted by these methods, explicitly including the effect of wave period, storm duration, and other factors. Prototype verifica- tion of analytical predictions should be attempted also, particularly for new constructions where the design as- sumptions are most thoroughly documented. (3) Mechanical stength of armor units--Prediction of armor unit breakage by scale model tests should be developed in order that both incipient damage and reserve stability can be more accurately defined. (4) Breakwater damage prediction--The reserve stability of a wide range of rubble-mound breakwater configurations should be comprehensively tested by methods similar to those developed to detect incipient damage. Improved analytical prediction of reserve stability should be the goal of these tests. (5) Runup on rubble-mound breakwaters--Improved instrumen- tation and testing methods need development for measure- ment of irregular runup on rough permeable slopes. A con- certed effort should be made to define runup coefficients for Equations 31 and 32 while concurrently investigating means for improved analytical prediction of irregular runup. The possibility of armor units designed both for enhanced hydraulic stability and for efficient attenua- tion of runup should be explored. (6) Wave transmission--The characteristics of irregular waves transmitted by rubble-mound breakwaters should be investigated. Improved analytical prediction of trans- mitted wave characteristics as a function of incident ir- regular wave characteristics should be the goal of this research. T2 REFERENCES Ahrens, J. 1984. "Reef Type Breakwaters," Proceedings, 19th International Conference on Coastal Engineering, Houston, Tex., pp 2648-2662. Ahrens, J., and McCartney, B. 1975. "Wave Period Effect on the Stability of Riprap," Proceedings, Civil Engineering in the Oceans/III, American Society of Civil Engineers, New York, N.Y., pp 1019-1034. Andrew, M., and Smith, O. In preparation. "Revised Method for Estimating Irregular Runup and Wave Transmission by Overtopping," CERC Miscellaneous Paper, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Andrew, M., Smith, O., and McKee, J. 1985. 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"Stability of Rubble-Mound Breakwaters II," Danish Hydraulic Institute, Horsholm, Denmark. 74 Groeneveld, R., Mol, A., and Zwelsloot, P. 1983. "West Breakwater - Sines, New Aspects of Armor Units," Proceedings, Coastal Structures '83, American So- ciety of Civil Engineers, New York, N.Y., pp 45-56. Groeneveld, R. et al. 1983. “Optimization of Breakwater Lengths in Relation to Wave Penetration," Proceedings, International Conference on Coastal and Port Engineering in Developing Countries, Colombo, Sri Lanka, pp 860-874. Groeneveld, R., Mol, A., and Den Boer, K. 1984. "Rehabilitation Methods for Damaged Breakwaters," Proceedings, 19th International Conference on Coastal Engineering, Houston, Tex. Gunbak, A. 1976. "The Stability of Rubble-Mound Breakwaters in Relation to Wave Breaking and Run-Down Characteristics," Report No. 1/76, Technical Uni- versity of Norway, Trondheim, Norway. Headquarters, Department of the Army. 1984. "Small Boat Navigation Projects: Hydraulic Design," EM 1110-2-1615, Washington, DC. Hedges, T. 1984. "The Core and Underlayers of a Rubble-Mound Structure," Breakwaters - Design and Construction, Institute of Civil Engineers, London, England. Hudson, R. 1958. "Design of Quarrystone Cover Layers for Rubble-Mound Break- waters; Hydraulic Laboratory Investigation," Research Report No. 2-2, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Hudson, R. et al. 1979. "Coastal Hydraulic Models," CERC Special Report-5, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Institute of Civil Engineers. 1984. "Breakwaters - Design and Construction," Proceedings of a Conference Held in London, London, England, p 20. Iribarren, Cavanilles R. 1938. "Una formula para el calculo de los diques de escollera," M. Bermejillo-Pasajes, Madrid, Spain. Jackson, R. 1968a. "Design of Cover Layers for Rubble-Mound Breakwaters Sub- jected to Nonbreaking Waves," WES Research Report 2-11, US Army Engineer Wa- terways Experiment Station, Vicksburg, Miss, Jackson, R. 1968b. "Limiting Heights of Breaking and Nonbreaking Waves on Rubble-Mound Breakwaters," Technical Report H-68-3, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Jensen, O. 1983. "Breakwater Superstructures," Proceedings, Coastal Struc- tures '83, American Society of Civil Engineers, New York, N.Y., pp 272-285. Jensen, 0. 1984. A Monograph on Rubble-Mound Breakwaters, Danish Hydraulic Institute, Horsholm, Denmark. Jensen, 0., Graveson, H., and Kirkegaard, J. 1983. "Breakwater Optimization for Small-Craft Harbors," Proceedings, Conference on Coastal and Port Engi- neering in Developing Countries, Colombo, Sri Lanka, pp 1-14. Jensen, O., and Kirkegaard, J. 1985. "Comparison of Hydraulic Models of Port and Marine Structures with Field Measurements," Preprint for Conference on Nu- merical and Hydraulic Modeling of Ports and Harbours (United Kingdom), Danish Hydraulic Institute, Horsholm, Denmark. Jensen, O., and Klinting, P. 1983. "Evaluation of Scale Effects in Hydraulic Models by Analysis of Laminar and Turbulent Flows," Coastal Engineering, Elsevier Scientific Publishing Co., Amsterdam, Vol 7, pp 319-329. 12) Jensen, O. and Sorensen, T. 1979. "“Overspilling/Overtopping of Rubble-Mound Breakwaters," Coastal Engineering, Elsevier Scientific Publishing Co., Amster- dam, Vol 3, pp 51-65. Keulagan, G. 1973. "Wave Transmission Through Rock Structures," Research Re- port No. H-73-1, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Kjelstrup, S. 1979. "Common Reasons for Damage or Breakdown of Mound Break- waters: Discussion," Coastal Engineering, Elsevier Scientific Publishing Co., Amsterdam, Vol 3, 137-142. Kogami, Y. 1978. "Research on Stability of Rubble-Mound Breakwaters," Coastal Engineering in Japan, Japanese Society of Civil Engineers, Tokyo, Japan, Vol 21, pp 75-93. Ligteringen, H., and Heijdra, G. 1984 (Jul). "Recent Progress in Breakwater Design," The Dock and Harbor Authority, Foxlow Publications, Ltd., London, pp 47-50. Longuet-Higgins, M. 1975. "On the Joint Distribution of the Periods and Amplitudes of Sea Waves," Journal of Geophysical Research, Vol 80, No. 18, pp 2688-2694. Losada, M., and Gimenez-Curto, L. 1979. "The Joint Effect of Wave Height and Period on the Stability of Rubble-Mound Breakwaters Using Iribarren's Number," Coastal Engineering, Elsevier Scientific Publishing Co., Amsterdam, Vol 3, pp 77-96. Losada, M., and Gimenez-Curto, L. 1980. "Mound Breakwaters Under Wave At- tack," Department of Oceanographical and Ports Engineering Report, University of Santander, Santander, Spain. Losada, M., and Gimenez-Curto, L. 1982. "Mound Breakwaters Under Oblique Wave Attack; A Working Hypothesis," Coastal Engineering, Elsevier Scientific Publishing Co., Amsterdam, Vol 6, pp 83-92. Madsen, O., and White, S. 1976. "Reflection and Transmission Characteristics of Porous Rubble-Mound Breakwaters," CERC Miscellaneous Report 76-5, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Maquet, J. 1984. "Construction Methods and Planning," Breakwaters - Design and Construction, Institute of Civil Engineers, London. Nielsen, S., and Burcharth, H. 1983. "Stochastic Design of Rubble-Mound Breakwaters," Aalborg University, Aalborg, Denmark. Ochi, M. 1980. "Stochastic Analysis and Probabilistic Prediction of Random Seas," Report No. UFL/COEL TR1042, University of Florida, Gainesville, Fla. Owen, A. 1985. "Concrete Armor Unit Forms Inventory," Journal of Waterway, Port, Coastal and Ocean Engineering, American Society of Civil Engineers, Vol 111, No. 4, pp 755-758. Owen, M., and Allsop, N. 1984. "Hydraulic Modeling of Rubble-Mound Break- waters," Breakwaters - Design and Construction, Institute of Civil Engineers, London. Paape, A., and Ligteringen, H. 1980. "Model Investigations as a Tool Part of the Design of Rubble-Mound Breakwaters," International Seminar on Criteria for Design and Construction of Breakwaters, Santander, Spain, pp 1-15. 76 Permanent International Association of Navigation Congresses. 1976. "Final Report of the International Commission for the Study of Waves," PIANC Bulle- tin 25, Vol III, Brussels, Belgium. Petraukas, C., and Aagaard, D. 1970. "Extrapolation of Historical Storm Data for Estimating Design Wave Heights," Proceedings, Offshore Technology Confer- ence, Houston, Tex., pp 411-428. Poole, A., et al. 1984. "Durability of Rock in Breakwaters," Breakwaters - Design and Construction, Institute of Civil Engineers, London. Price, W. 1979 (May). 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"Theme Paper: State of the Art," Breakwaters - Design and Construction, Institute of Civil Engineers, London, England. Thorpe, W. 1984. "Foundation Problems," Breakwaters - Design and Construc- Institute of Civil Engineers, London, England. tion, Timco, G. 1981. "The Development, Properties, and Production of Strength- Reduced Model Armour Units," National Research Council, Hydraulics Laboratory, Ottawa, Canada. US Army Engineer Waterways Experiment Station. Coastal Engineering Research Center. 1983. "Availability of Concrete Armor Unit Forms," Coastal Engineer- ing Technical Note III-19, Vicksburg, Miss. i US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center. 1984a. "Computer Program MADSEN for Wave Transmission by Permeable Rubble-Mound Breakwaters," Coastal Engineering Technical Note I-22, Vicksburg, Miss. 1984b. “Comparison of Breakwater Volumes and Costs," Coastal En- gineering Technical Note III-23, Vicksburg, Miss. Van der Meer, J., and Pilarcyzk, K. 1984. "Stability of Rubble-Mound Slopes Under Random Wave Attack," Proceedings, 19th International Conference on Coastal Engineering, Houston, Tex., pp 1-16. Water Resources Council. 1983. "Economic and Environmental Principles and Guidelines for Water and Related Land Resources Implementation Studies," Washington, D.C. 78 APPENDIX A: COMPUTER PROGRAM BWLOSS1 Estimation of Economic Losses as a Function of Wave Height Program purpose 1. The program BWLOSS1 is intended to aid planners of coastal struc- tures which provide protection from wave attack by deriving an empirical math- ematical expression relating a given level of economic losses to the responsi- ble incident significant wave height. This loss function can be used to de- fine the "without-project" condition with respect to the incremental economic benefits provided by artificial wave protection. The program optionally pro- vides an estimate of expected annual economic losses due to wave attack, given the coefficients of an Extremal Type I cumulative probability distribution function of significant wave heights for the site. Program capabilities 2. BWLOSS1 is written in FORTRAN IV as implemented on the Honeywell DPS-8 mainframe system at the US Army Engineer Waterways Experiment Station (WES). A BASIC version written for the IBM PC is also available. The least squares method is applied to historical data on economic losses associated with the significant wave heights of the storms that caused the losses. A loss function is derived from the following form: A(H_-H._) $L(H,) = Li 23 3 Lo? | (A1) $L(H,) = economic losses as a function of significant wave height H, = maximum conceivable economic loss from wave attack (at any intensity) A = site-specific coefficient derived by regression Hp = maximum significant wave height for which economic losses are negligible 3. The regression requires at least one point for H, , $L(H,) » but it can deal with up to 100. The coefficient A is presented along with the nonlinear correlation coefficient and the sum of the square residuals. A ta- ble of residuals is optionally presented. Losses can be optionally predicted, given a specific significant wave height, or the significant wave height Al corresponding to a given level of losses can be predicted. The form of this function is illustrated in Figure 16 in the main text. The program will also apply an Extremal Type I cumulative probability distribution of significant [e-H,/¢] wave heights as follows: F(H_) = e617 (A2) “where F(H,) = cumulative probability distribution of events Where ~H_ < H s Ss e and » = site-specific coefficients derived by regression of historical wave data to estimate the expected annual economic losses by H dF(H_) $L\_ So Ss e\t = a $L(H,) ai. dH, (A3) Lo where A} = the average number of extreme events per year above the threshold H. value originally used to derive e« and 9 (must be input by the user) H = a practical upper limit taken as the H value whose probability of exceedance is 0.0000001 4, This formulation assumes that the number of extreme events per year is random and can be represented by a mean value and is independent of the Significant wave heights representing the intensity of the individual storms. The lower limit of integration is H;, , below which the expected losses are taken as zero. Extrapolation of F(H,) to H, values below the threshold value applied to data used to originally derive e and 9 is probably con- servative, but this question will be the subject of further study. A thresh- old H, value set equal to H;, would presumably resolve any problems if adequate statistical confidence can be maintained. The integration is accom- plished by a numerical application of Simpson's Rule with 100 intervals. 5. The majority of the expected losses statistically occur during storms whose H,; is just above H,, where the probability density is sub- Stantial. The higher H, values occur on the tail of the probability density function and may even be precluded by depth limitations. The program does not deal with depth limitations and assumes the Extremal Type I function fully A2 represents the wave climate at the site. A potential improvement of BWLOSS 1 is the incorporation of period and depth effects for an estimate of $L(H, , Tp , d) given the joint probability distribution FHS P Tp wd)! ov heeurther improvement would also incorporate the storm duration t for an estimate of $L(H, i Ty ; tsa) , given GES ; Ty , t , d) . These enhancements will involve a much more rigorous computation than is now performed by BWLOSS1. The program is now completely interactive and easily adaptable to execution by microcomputer systems. Sample Execution and Output 6. Below is a sample execution and output for computer program "BWLOSS1." INPUT THE MAXIMUM CONCEIVABLE LOSS IN MILLIONS OF DOLLARS =20. INPUT THE MAXIMUM SIGNIFICANT WAVE HEIGHT FOR WHICH LOSSES ARE NEGLIGIBLE - USE CONSISTENT UNITS =2, HOW MANY SIGNIFICANT WAVE HEIGHT VS LOSS DATA POINTS DO YOU HAVE? =4 ENTER SIGNIFICANT WAVE HT.,COMMA,LOSS IN MILLIONS OF DOLLARS AND RETURN FOR EACH POINT 23.,.5 =4.5,1. #605205 winavi 5 DATA ON EXPONENTIAL CURVE... CURVE HAS FORM: $L(Hs)=$Lmax#{1-explA#(Hs-HLo) J} $Lmax= 20. 0000000 HLo= 2.00008 A= -8.8437137 $L(Hs)= LOSSES Hs= SIGNIFICANT WAVE HEIGHT NON-LINEAR CORRELATION IS @.9735292 SUM SQR RESIDUALS..... 1.9485841 A3 PRINT RESIDUAL TABLE(Y/N)? =Y XVALUE YVALUE YEST DIFF 2.0008 Q. 0.2000 0.0000 3.0008 0.5800 0.8554 8.3554 4.5008 1.0008 2.0785 1.0785 6.0080 2.5008 3.2084 0.7084 12.0000 7.5000 7.0823 0.4177 DO YOU WANT TO MAKE SOME LOSS PREDICTIONS FROM SIGNIFICANT WAVE HEIGHT DATA(Y/N)? =y INPUT SIGNIFICANT WAVE HEIGHT =10. PREDICTED LOSS IN MILLIONS OF DOLLARS IS 5.98 DO YOU WISH TO MAKE ANOTHER PREDICTION(Y/N)? =N DO YOU WANT TO PREDICT SIGNIFICANT WAVE HEIGHTS FROM LOSS DATA(Y/N)? zy INPUT LOSS IN MILLIONS OF DOLLARS 215. PREDICTED SIGNIFICANT WAVE HEIGHT IS 33.71 DO YOU WISH TO MAKE ANOTHER PREDICTION(Y/N)? =N DO YOU WANT TO PREDICT EXPECTED ANNUAL LOSSES(Y/N)? =y SELECT A DISTRIBUTION... EXTREMAL TYPE I...1 WETBUL( catdokacaaiee 2 LOG-EXTREMAL......3 SELECT 1, 2, OR 3 =1 INPUT EXTREMAL TYPE I EPSILON, AND PHI =-2,27,3.216 INPUT AVERAGE NUMBER OF EXTREMAL EVENTS PER YEAR, THE POISSON ‘LAMBDA’ PARAMETER =4 EXPECTED ANNUAL LOSS IN MILLIONS OF DOLLARS IS 2.4141522 J AY Program Listing 7. Below is a program listing for computer program BWLOSS1 (FORTRAN version). 10@C PROGRAM "BWLOSSI". 11/85 VERSION 20C DESIGN BRANCH-COASTAL ENGINEERING RESEARCH CENTER 3OC U.S. ARMY ENGINEERS WATERWAY EXPERIMENT STATION 4ec P. 0. BOX 631 98C VICKSBURG, MS 39180-0631 60C FOR FURTHER INFORMATION CONCERNING THE APPLICATION 7OC OF “BWLOSS1", CALL.. BOC ORSON P. SMITH (601)-634-2013 FTS:542-2013 OR 98C ROBERT B. LUND (601)-634-2068 FTS:542-2068 OR 1@@C DOYLE L. JONES (601)-634-2069 FTS:542-2069 11@C 120C FORTRAN 4 HONEYWELL DPS-8 130C REF: "COMPUTER PROGRAM WAVDIST" CETN-I- 140C REF: "PROBABILITY AND STATISTICS" BY MORRIS DEGROOT 158C REF: "COST EFFECTIVE OPTIMIZATION OF RUBBLE-MOUND BREAKWATER 16@C CROSS-SECTIONS" BY ORSON P. SMITH 170C REF: "EXTREMAL STATISTICS IN WAVE CLIMATOLOGY" BY BORGMAN AND RESIO 188C 19@C N THE NUMBER OF DATA POINTS 208C X THE ARRAY OF SIGNIFICANT WAVE HEIGHTS 218C YH THE ARRAY OF LOSSES CORRESPONDING TO EACH SIGNIFICANT WAVE HEIGHT THE TRANSFORMED Y ARRAY USED IN THE METHOD OF LEAST SQUARES Hlo, THE MAXIMUM WAVE HEIGHT FOR WHICH LOSSES ARE NEGLIGBLE $Lmax, THE MAXIMUM CONCEIVABLE LOSS IN MILLONS OF DOLLARS A, THE REGRESSION COEFFICIENT A be ¢ = < on 4 ) ee + iW : Fi =! = - : , - hoy > le f. eo) i ‘ . =, ; ; ws ‘ @ ‘ ‘ a > rs f . 5 ' ' : ” ' 3 3 i= i i i oy - I. 4 ‘ a1 - ’ 1 ' ’ i 7 ' Y 1 4 bes 7 f 4 : x e : ; : ‘ > - . ' “ =~ i a) 4 { -_ = 7 | : : - 1 ” : i. - ‘ 1 = ‘ - { - 7 7 ' ' : ‘ be : a . 7 er = A ‘ ‘ i sp fos ' | - = a a i 7 . = R P 7 7 A ut - he” ‘ - he =— =, é = ra . os = e 4 - F ye - : ‘ ia ‘ eo - 2 “ ad - = 7 = rn - ~ oie. - if, ! _ - ’ = ‘ 0 p = 1” arp VAN m oy v fa oa o ay i, : oi : cn ; j : f | hos ¥ “ . “ Vi oH, : oi ee | Ln a | io ib ey 1 s i - ' ' ni rau a i RC j a 7 v es feed ‘ — sd : a ‘ ‘ 7 ‘ ne wi i, a o ray i , " ° if) Fuca a ce $i ' z rt, 7 , - ee - a aa 1 x , i j sf : 7 i 7 A i rr Hy ay il o 7 _ _ “ne eee caer i ' i f Ny, no y cree \ : ij hoe, ye a ae f “" 1 a ; vi 7 ie 7 nt f i n ant ee i ‘ i ‘ : i i . 1 aa " = » 8 1 1 oo ho (ee ’ a Ama io ai mt 1 ' ‘ ’ 1 ‘ 7 \ ' ne 7 Ty r oe " i a re: i . 1, Ly ‘i : ' 1 4 - . oo . 7 oy Dae 1, 7 i" 5 te ny ye 7