DRAFT

Risk assessment of revetments by Monte Carlo simulation

C. E. Balas BSc (Hons), MSc, PhD, MASCE, L. Balas BSc (Hons), MSc, PhD, MASCE andA. T.Williams PhD

In risk assessment, the second-order reliability method(SORM) and the conditional expectation Monte Carlo(CEMC) simulation were inter-related in order to analysethe safety of Brean and Newton revetments on the southand north coasts of the Inner Bristol Channel, SouthWales, UK. In the CEMC simulation, storm surge andwave height were generated as correlated random vari-ables with probability distributions describing the phe-nomena. Then, damage probability was predicted byapproximating the limit state surface by a quadratic onehaving the identical curvature at the design point. It wasfound that revetment reliability was sensitive to tidalrange, storm surge and wave data. Brean revetment wasfound to be safe, whereas maintenance works wererequired during the service life of Newton revetment.The model introduced a new technique into the reliabilityassessment of revetments subject to large tidal ranges andits predictions satisfactorily reflected the safety/damageconditions of the structures at the site

NOTATIONa, b, c location, shape and scale parameters of probability

distributionsDn50 nominal diameter of armour rockDL damage level in Hudson equationFTT–1 Fisher–Tippett Type 1 probability distributionG(Z) limit state (performance) functionHs significant wave heightCEMC conditional expectation Monte Carlo simulationKD dimensionless stability coefficientPe exceedance probability of a certain damage levelRp return periodS damage level defined as the number of cubic stones

with a side of nominal stone diameter, eroded aboveand below the water level within a width of onenominal stone

SORM second-order reliability methodZ primary variable vector in the normalised spaceW50 median weight of the armour unita angle of structure slopeb first-order reliability indexki positive curvatures of the concave limit state func-

tionli direction cosinesDr relative density of armour unit

d coefficient of variationm mean values standard deviation valuef standard normal probability density functionF standard normal distribution function.

1. INTRODUCTIONRisk analysis techniques provide a rational basis for hazardmanagement decision-making on a national scale, as well asregionally and locally. For example, Hall et al. developed amethodology for a national scale flood risk assessment appliedin England and Wales,1 and Vrijling introduced a probabilisticdesign methodology for the water defence system in theNetherlands.2 Such techniques are now being adopted incoastal and river engineering projects from high-level planningbased on reliability analysis to detailed designs using high-resolution simulation techniques.3

For the structural risk assessment of revetments, the partialcoefficient system introduced by the Permanent InternationalAssociation of Navigation Congresses (PIANC) has been widelyutilised4 In this system, the first-order mean value approach(FMA) has been applied in the design by using the Hudsonperformance (limit state) function.5 In FMA, the limit statesurface was converted into a line around the expected value.The Hasofer–Lind (HL) second-order reliability index is thecommonly used level II first-order method to compare risklevels of coastal structures.6–8 Goda and Tagaki introduced areliability design method by using Monte Carlo simulationwhere an expected distance has been utilised for the slidingfailure mode of caisson breakwaters.9

The Brean and Newton revetments selected on the south andnorth coasts of the Bristol Channel (Fig. 1) were designed bydeterministic methods using the Hudson equation and con-structed in 1997 by the Environmental Agency of the UK. Inthis study, the safety of these revetments is examined by theapplication of a new reliability method as an improved riskassessment technique developed in this paper. The techniqueinvolves the second-order reliability method (SORM) inter-related with conditional expectation Monte Carlo (CEMC)simulation. In this technique, uncertainties that affect most ofthe variables in the design were incorporated as risk factorsthroughout the lifetime of structures.

Proceedings of the Institution ofCivil EngineersMaritime Engineering 157June 2004 Issue MA2Pages 1^10

Paper 13107Received ??/??/2004Accepted ??/??/2004

Keywords:coastal engineering//risk andprobability analysis/sea defences

Can E. BalasAssociate Professor of Civil and CoastalEngineering, Civil Engineering Department,Engineers, Faculty of Engineering andArchitecture, Gazi University, Ankara, Turkey

Lale BalasAssociate Professor of Civil and CoastalEngineering, Vice Chairperson of Civil EngineeringDepartment, Faculty of Engineering andArchitecture, Gazi University, Ankara, Turkey

Allan T. WilliamsProfessor, Applied ScienceDepartment, University ofGlamorgan, Pontypridd,Wales, , UK

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2. RELIABILITY^RISK ASSESSMENTIn structural safety, the limit state function consists of randomload and strength variables designated by the primary variablevector Z in the normalised space. The functional form of thebasic variables at the limit state is the performance functiondenoted by G(Z).10 The structural safety can be assured bydesignating an admissible probability of achieving the limitstate defined by G (Z) ¼ 0.11 Hence, the acceptable risk level forthe design of rubble-mound structures is given in terms ofexceedance probability of the limit state.12 The serviceability(performance) limit-state was implemented for safety evalua-tions, since the exceeding of a specified damage level maynot result in complete breakdown of the structure, but maycause an interruption in the achievement of its functions.13

Exceedance probabilities of damage levels of rubble-moundrevetments were evaluated by using the limit state equationgenerated from an expression obtained from the Hudsonequation by taking into account the damage level14

G ¼ YDrDn50ðKD cot yÞ1=3Sk � Hs1

where Y is the variable signifying the uncertainty inherentin equation and regression; k is the power regression coeffi-cient; Hs is the significant wave height; y is the structure frontface slope angle; S is the damage level of the structure,S ¼ Ae/Dn50

2; Ae is the erosion area in a cross-section; KD isthe dimensionless stability coefficient; Dn50 is the nominaldiameter of armour rock, Dn50 ¼ (M50/rr) � 1; rr is the massdensity of armour stone; M50 is the 50% value of the massdistribution curve; Dr is the relative density of stone defined byDp ¼ (rr/rw) � 1; rw is the mass density of water.

The Hudson equation was rewritten for the damage-levelparameter of Van der Meer by using a power regressionanalysis, in which the mean of uncertainty variable and thevalue of the regression coefficient were obtained as my ¼ 0·7

Fig. 1. Brean and Newton revetments (! designates the location of revetments)

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and k ¼ 0·15, respectively.15

The (mean) damage level S inequation (1) was defined asthe number of cubic stoneswith a side of nominal stonediameter, eroded above andbelow the water level within awidth of one nominal stonediameter. The no-damage cri-terion of S ¼ 2 was equiva-lent to DL ¼ 2·5% damageand the damage level of S =4 corresponded to DL ¼ 5%damage of the Shore Protec-tion Manual.16

A new reliability approach inwhich the second-order re-liability method (SORM) andthe conditional expectationMonte Carlo (CEMC) simula-tion17 are inter-related wasapplied for safety evaluations.The tidal range of the site,storm surge and wave heightwere generated numeroustimes in CEMC simulation ascorrelated random variableswith probability distributionsdescribing the phenomena.When the generated super-posed sea level exceeded thetoe elevation of the structurein simulation, the reliabilityof the structure was investi-gated by SORM. The prob-ability of exceeding the limit state was predicted in SORM byapproximating the limit state surface by a quadratic one havingidentical curvature at the design point. Then, exceedanceprobability of limit state (Pe) was calculated by equation (2), asdescribed by Zhao and Ono18

Pe �jð�bÞYn�1

i¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2

jðbÞlibFð�bÞ

s 1n� 1

�Xn�1

i¼1

F �b� ðn� 1Þli

1þ 2jðbÞliFð�bÞ

0BB@

1CCA

Fð�bÞ expðn� 1ÞjðbÞli

Fð�bÞ

1þ 2jðbÞliFð�bÞ

0BB@

1CCA

2

where F is the standard normal distribution function, f is thestandard normal probability density function, b is the first-order reliability index, li are the direction cosines defined asli ¼ �0·5ki and ki are the positive curvatures of the concavelimit state function. For each generated load combination in thecomputer, the exceedance probability of the damage level wasevaluated at the limit state in order to reflect the structuralperformance under the wave action correlated with variationsin water level.

For life cycle damage estimation in the model, a computationalapproach was performed. In this approach, the annual damagethat occurred in the revetment was stored in the computer byconsidering the return period of the generated significant waveheight. In other words, the damage was accumulated in thecomputer for the simulation of damage progression in time as afunction of the occurrence probability of loading in the lifecycle, and the temporal variations in the mean damage wasevaluated in each epoch by using the approach recommendedby Melby and Kobayashi.19

SORM and the CEMC simulation were applied together todetermine the safety of Brean and Newton revetments on thesouth and north coasts of Inner Bristol Channel, South Wales,UK (Figs 2 and 3). Computer simulations repetitively repro-duced revetment performance at the limit state condition untila specified standard mean error of convergence (e) wassatisfied.

3. RISK ASSESSMENTOF REVETMENTS

3.1. Brean revetmentThe shore area in Brean is covered by up to 20 m of Holocenesediments (mainly marine and estuarine) covering Lias mud-stone sediments which abut against a Carboniferous limestoneheadland. These sediments form the low-lying northwestern

(a)

(b)

Fig. 2. Brean revetment: (a) general view (b) cross-section

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DRAFTextremity of the Somerset levels. Sand dunes cover the coastalstrip and can reach a height of greater than 10 m and wide(approximately 3 km) tidal flats front the dunes. Dunes areeroding usually due to sliding and toppling of blocks of sand asa result of wave action.

The UK Environment Agency has built approximately 1·2 km ofrock armour frontage, protecting caravan sites/private proper-ties/golf course/agricultural land, and adjacent to the headland,a wave return wall has been built (200 m of reinforced concretewalls plus steel sheet piling and rock armour). It is estimatedthat the rock revetment at Brean (Fig. 2), which was constructedin 1997, has a 200-year standard of protection against overflowwith a reduced standard (50 years) against overtopping due tothe constraint on defence height.20 The revetment was designedand constructed by the UK Environment Agency by usingdeterministic methods in which the Hudson equation wasutilised. The damage level of (0–5)% (Hudson no-damagecriterion) was accepted in the deterministic design for the

armour layer that is made upof randomly placed, single-unit rocks of approximatelyW50 ¼ 2·0 tonnes. The revet-ment has an armour layerslope of 1 :4. The offshorewave climate of the site waspredicted by CarmarthenshireCounty Council21 and thedesign significant waveheight used in deterministicdesign was Hs ¼ 2·2 m con-sidering the effects of refrac-tion, shoaling and breaking.The site has two (diurnal)tidal cycles. The spring tideshave a greater vertical tidalrange than neaps and producehigher tidal flow rates orcurrents. The tidal range is11·2 m at Brean and flood-dominated tidal currentsreach an average of 0·7 m/s.Therefore, tidal range is adominant factor in the designwith the addition of the 1 in50 year storm surge that canincrease sea level by morethan 1·5 m.21 Deterministicdesign values utilised duringthe construction stage of therevetment are listed in Table1. The structural and loadingcharacteristics listed inTable 1 were obtained fromthe shoreline and seadefence management studiesof the EnvironmentAgency.22,23

In the present study, thesafety of the revetments is

examined by the application of a new reliability method. InCEMC–SORM simulations (level III), the probability distribu-tions of random variables (Table 2) were obtained from thestatistical analysis of wind, wave, tide and storm surge dataobtained by the Hydrographer of the Navy,24 oceanographic

(a)

(b)

Fig. 3. Newton revetment: (a) general view (b) cross-section

Design parameter Breanrevetment

Newtonrevetment

Nominal diameter of stone Dn50: m 0˝91 0˝15Weight of stone W50: tonnes 2 0˝009Wave height Hs : m 2˝2 1˝8Tidal range RT: m 7˝1 8˝1Storm surge SS: m 1˝5 1˝5Relative density of stone Dr 1˝64 1˝62Height of structure: m 2˝5 4˝3Structure slope, cot y 4 0˝84

Table 1. Resistance and loading characteristics of revetmentsin deterministic design

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institutions, such as: the Institute of Oceanographic Sciences25

and Proudman Oceanographic Laboratory;26 studies on shore-line management (e.g. Shoreline Management Partnership andHR Wallingford27); environmental assessment studies by theBritish Maritime Technology28 and HR Wallingford;29 waveprediction studies;30–32 and analysis of wave data recorded byScarweather wave-rider buoy (51827’ N, 03855’W)33 andSkomer wave buoy (51848’20’’ N, 05820’00’’ W).34 The bathy-metric data for the sites were obtained from the HydrographicalOffice in Cardiff. In the CEMC simulation, storm surge andwave height were generated as correlated random variableswith their probability distributions, and the exceedance prob-ability of limit state was simulated by using equation (2) foreach generated set of random values considering their correla-tion. Resistance variables were generated from beta distribu-tions, since with the determination of statistical parameters thedistribution could fit a wide variety of frequency shapes.

The cumulative exceedance probabilities of various damagelevels at the Brean revetment are shown in Fig. 4(a). Theexceedance probabilities of no-damage (0–5%) and service-ability limit-state (40–50%) levels in L ¼ 50 years of lifetimewere determined as Pe ¼ 52·3% and Pe ¼ 19·4%, respectively(Fig. 4(a)). The SORM–CEMC simulation of the structuralperformance at the limit-state gave the variation in environ-mental loading conditions as the dispersion of performancefunction values. The variation of the uncertainties as a trendchart of annual probability and as a function of limit state aregiven in Figs 5(a) and 6(a), respectively. Simulation of the limitstate was repeated for 30000 trials performed for an averagecentral processing unit (CPU) time of 2 min 13 s, with astandard mean error of e ¼ 1% by using a portable computerhaving an AMD K6-2+ (3-D) processor. The CPU time isrelatively short for the large number of simulations performedon a portable computer. This enables the efficient utilisation ofthe methodology in the design of coastal structures subject tolarge tidal variations.

The rank correlation coefficients (rij) of random variables35

were determined from the simulated values of performancefunction and the averages of all cases are tabulated in Table 3.Uncertainties involved in the design, such as the reliability ofwave and/or wind data, prediction model, near-shore calcula-tion, statistical methods used, deviation between design andconstruction, all influenced the random generation range ofcorrelated distributions in the simulations. The frequencydistribution of the limit state function was fitted to a Weibulldistribution36 shown in Fig. 6(a). Location, shape and scaleparameters of the distribution were calculated as a ¼ �2·49;b ¼ 3·05 and c ¼ 9·29, respectively (Table 4). In the simulation,

the annual exceedance probability for the (0–5)% damage levelwas calculated as Pe ¼ 1·47%. The scatter range of therandomly generated values was between Gmin ¼ �2·14 m andGmax ¼ 16·41 m, signifying the effect of uncertainties on thelimit-state having a mean value of mG ¼ 5·84 m for the damagelevel of (0–5)% (Table 4).

The sensitivity study carried out by using the contribution tovariance, showed that tidal level mainly influenced thesimulation (46·7%; Fig. 7(a)). Although the reliability ofrevetments is generally regarded as a function of waveconditions, this sensitivity study showed that, it is mainly afunction of water level due to large tidal variations. In thesimulations, when the water level increased, the structure wasexposed to larger waves, because the water depth governed the

Random variable (Zi) Probability distribution Brean revetment Newton revetment

Y Beta a¼ 3; b¼ 2; c¼ 1˝5 a¼ 3; b¼ 2; c¼ 1˝5Dn50: m Beta a¼ 1˝5; b¼ 1˝1; c¼ 1˝5 a¼ 3; b¼ 2˝8; c¼ 0˝25Hs: m Fisher^Tippett Type 1 a¼ 1˝8; c¼ 1˝0 a¼ 1˝8; c¼ 0˝28Tidal range, RT: m Triangular mode¼ 3˝54 mode¼ 4˝05Dr Normal mxi¼ 1˝64; sxi¼ 0˝15 mxi¼ 1˝62; sxi¼ 0˝18Cot y Beta a¼ 5; b¼ 2; c¼ 5 a¼ 3; b¼ 1˝8; c¼ 1˝2

Table 2. Distribution parameters of basic variables used in risk assessment of revetments

Fig. 4. Cumulative exceedance probabilities of damage levels(DL) of (a) Brean and (b) Newton revetments: A, DL = (0^5)%; B, DL = (10^15)%; C, DL = (20^30)%; D, DL = (40^50)%

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breaking phenomena in the surf zone. On the other hand, whenthe water level dropped below the toe level of the structure, thestructure was not exposed to waves during the simulation, as innature. The uncertainty variable (18·2%) and wave height(15·5%) also affected the limit state function. Other variables,namely stone size (8·2%), slope (2·1%) and relative density(3·5%), had less effect on the limit-state.

3.2. Newton revetmentNewton is located on the eastern edge of Porthcawl, animportant residential, coastal town in South Wales, which has asubstantial tourism capacity due to its beaches, sand dunes,fishery harbour, amusement park and one of the largestcaravan sites in Europe. The main shore protection structuresand the harbour are located at the south-east part of the town.The harbour was constructed for the coal export trade and atpresent the inner harbour has been abandoned and used as acar park for tourists. The sand dune system that backs the beachis the relict of a much larger system that swept around the

South Wales coastline. Thissystem represents the easternedge of soft sediments beforeoutcrops of hard rock cliffsform the major portion of theGlamorgan Heritage Coast.Sand was blown into thesystem in the storm period,which encompassed theperiod between 1450 and1660.30 Up to the mid-1950s,much sand extractionoccurred for building pur-poses, but this has nowceased. The system is wellvegetated and stable, butHippophae rhamnoides (seabuckthorn) is excessivelyspreading in the dunes at arate of some 2 hectares perannum.

The major dune field of theregion that extends from theeast of the town to the RiverOgmore is defended by threerock groins, which are spacedsome 50 m apart, have alength of 30 m and a heightof approximately 1·5 m abovethe beach level. In order toprevent flooding of the lowerstretches of the village, theUK Environment Agency hasdesigned and constructed arubble-mound revetment thatextends for some 180 m (Fig.3). Deterministic designvalues utilised during theplanning stage of this revet-ment are listed in Table 1, asobtained from the shorelineand sea defence management

studies of the Environment Agency.22,23

The revetment in Newton encountered damage of approxi-mately 6% during a relatively short period after construction.Therefore, the safety of the revetment was examined by theapplication of inter-related SORM and CEMC simulations.Probability distributions of random variables (Table 2) in thesimulation were obtained from the statistical analysis of wind,wave, tide and storm surge data of the site. The dominant wavedirection for the northern coast of Bristol Channel is the SW–Wquadrant. A narrow band in this sector is exposed to swellwaves approaching from the Atlantic Ocean having a maximumfetch distance of 6500 km. Generally; waves from the sector of180–0 WCB have a fetch distance in excess of 100 km.25 Theextreme value significant wave height of the Fisher–TippettType 1 probability distribution37 and zero crossing periods weredetermined for a return period of Rp ¼ 50 years as Hs ¼ 6·1 mand Tz ¼ 8·6 s, irrespective of direction for an average stormduration of 12 h.27 Several near-shore sand banks including

Per

form

ance

func

tion:

m

Cumulative probability: %

18.00

16.00

14.00

12.00

10.00

8.00

6.00

4.00

2.00

0.000 10 20 30 40 50 60 70 80 90 100

(a)

0 10 20 30 40 50 60 70 80 90 100

Per

form

ance

func

tion:

m

Cumulative probability: %

4.00

3.00

2.00

1.00

0.00

_1.00

_2.00

_3.00

(b)

Fig. 5. Annual cumulative probability as the trend chart of reliability for (a) Brean and(b) Newton

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DRAFTScarweather, Hugo and Nash provide some shelter for Newton.The expected value of significant wave height was determinedas Hs ¼ 1·8 m from the wave transformation study whichincludes shoaling, refraction and breaking, and maximum tide

level was measured at the toe elevation of the structure. Thetidal flow reaches locally up to 3 m/s in front of the Porthcawlharbour breakwater where neap and spring tidal ranges are4·6 m AOD (above ordnance datum) and 8·1 m, respectively.24

Tidal level is enhanced by storm surges in excess ofSs ¼ 1·5 m.29

Utilising SORM–CEMC simulation, the structural performanceof the revetment and the dispersion of the randomly evaluatedlimit state function were simulated. Cumulative exceedanceprobabilities of various damage levels at the Newton revetmentare shown in Fig. 4(b). The exceedance probability of no-damage (0–5%) level in the lifetime of the structure wasdetermined as Pe ¼ 99·24%, signifying the requirement formaintenance works. The annual cumulative probability distri-bution is given in Fig. 5(b), as the trend chart of reliability. Theeffect of uncertainties in environmental loading conditions canbe observed by the frequency distribution of the limit statefunction given in Fig. 6(b). The simulated values of limit statefunction in this figure were fitted to a triangular distributionwith a mode of 0·20. Structural safety was sensitive to theenvironmental loading conditions, namely the variation in tidalrange, storm surge and wave data. The sensitivity (77·2%) ofstructural reliability to tidal level is illustrated in Fig. 7(b).

4. DISCUSSIONSThe UK Environmental Agency carried out the design andconstruction of Brean and Newton revetments on the south andnorth coasts of the Inner Bristol Channel, South Wales, UK, byusing a deterministic approach in which the Hudson equationwas utilised. In this study, the safety of these revetments wasassessed by the application of the new reliability modeldeveloped. This new risk assessment model reflected the effect

of water-level changes on thereliability of revetments bysimulating environmentalloading conditions describedby Kamphuis.38 It was foundthat, the safety of revetmentswas sensitive to variations inthese conditions, namely tidalrange, storm surge and wavedata. Hence, it is suggestedthat the reliability of revet-ments should be determinedby using Monte Carlo simu-lations inter-related with thesecond-order reliabilitymethod (SORM), in which thelimit state surface at thedesign point was approxi-mated by a parabolic surfacehaving its axis along thedirection normal to thedesign point vector.

The revetment in Brean wasconsidered to be safe, as theprobability of exceeding thedamage level of 5% wasevaluated as Pe ¼ 52·3% fromthe model and the actual

0.0

0.6

1.2

1.8

(a)

(b)

Performance function: m

Performance function: m

Occ

uren

ce p

roba

bilit

y (%

) O

ccur

ence

pro

babi

lity:

%

2.4

1.9

1.4

1.0

0.5

0.0

_2.5 1.9 6.3 10.6 15.0

_3.0 _0.8 1.5 3.8 6.0

Fig. 6. Frequency simulation and the fitted theoreticaldistribution of Hudson performance function for: (a) Breanand (b) Newton revetments

Y 1

Dn50 0˝13 1

Hs 0 �0˝53 1

SS 0 0 0˝76 1

Dr 0˝05 0˝16 0 0 1

Cot y 0˝07 0˝19 �0˝21 0 0 1

Variables Y Dn50 Hs SS Dr Cot y

Table 3. Average rank correlation coefficient matrix (rij) of variables in performance function

Revetments Brean Newton

Fitted distribution of G Weibull TriangularDistribution parameters a =�2˝5; b = 3˝1; c = 9˝3 mode = 0˝2Average of G (mG): m 5˝84 1˝42Variation coefficient: d% 51 135Annual Pe (%) of DL = (0^5)% 1˝47 27˝93Minimum value of range �2˝14 �3˝52Maximum value of range 16˝41 6˝21

Table 4. Simulation characteristics of limit state (performance) function (G) with 30000 trials

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DRAFTdamage observed in the revetment was less than 5% in varioussections since its construction (in a period of 6 years). Hence,damage actually suffered by the revetment is in good agree-ment with the model predictions.22 There was no damageencountered in the access road of the beach and the revetmentshowed a safe performance after its construction. The revetmentin Newton required maintenance, as the exceedance probabilityof 5% damage in the lifetime of structure was determined asPe ¼ 99·24%. Notable damage of approximately 6% wasalready visible in the revetment and the Environmental Agencyconsequently carried out some repair work,22 which supportedthe damage level predictions of the new reliability modelintroduced. Alternatives to such hard engineering structures ina Heritage Coast can be pebble beaches that abound in the areaand these are probably the best protection that can be givenboth to nature and structures.

5. CONCLUSIONSReliability assessment has important advantages when com-pared with the conventional methods, since the randombehaviour of load and resistance parameters throughout thelifetime of the structure can be estimated at the preliminarydesign stage.39 This enables the designer to perform a cost-optimisation study, in which the total cost composed of initialand maintenance costs spent over the economical lifetime canbe minimised, and produces an economical design associatedwith low damage risk.40,41

Structural safety is sensitiveto the environmental loadingconditions, namely variationin tidal range, storm surgeand wave data. Thereforeprior to any coastal structuredesign, it is recommendedthat the design parameters aremodelled by representativeprobability distributions attheir limit state. When theuncertainties inherent indesign parameters are exten-sive, to obtain the safety ofcoastal structures it is sug-gested that Monte Carlosimulation inter-related withreliability methods is applied.

In this study, the safety oftwo revetments constructedon the south and north coastsof the Inner Bristol Channel,South Wales, UK was investi-gated by utilizing the riskassessment model developed,in which the CEMC simula-tion42 generated the environ-mental loading conditionsand the SORM determined thestructural safety at the limitstate. Brean revetment wasconsidered to be safe, but therevetment in Newton requiredmaintenance. Site observa-

tions for case studies showed that, the hybrid reliability modelintroduced in this paper satisfactorily predicted the safety levelsof revetments under the effect of wave loading and water-levelchanges. The model application of CEMC–SORM simulations(level III), which is generally performed within a few minutes ofCPU time in fast computers, has the advantage of robustnesswhen compared with the second-order methods such as theHasofer–Lind method, provided that the probability distributionof random variables and their correlation are described with anacceptable accuracy, which can be obtained from the dataaccumulated over a sufficient number of years.

6. ACKNOWLEDGMENTThe authors wish to thank the referees of this paper for theircritical comments and suggestions. The financial support forthis research, which is gratefully acknowledged, was providedin part by the Royal Society of London and by the Scientificand Technical Research Council of Turkey. Thanks are extendedto Environment Agency of Wales, the Universities of Glamor-gan, Cardiff and Bath Spa for providing most of the data.

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Specific density4%Slope

2%Stability coefficient

6%Nominal diameter

8%

Wave height16%

Uncertainty variable18%

Tidal range46%

(a) Sensitivity of design variables to limit state

Tidal range80%

Specific density0.1%

Slope0.1%

Stability coefficient0.5%

Nominal diameter2%

Wave height10%Uncertainty variable

8%

(b) Sensitivity of design variables to limit state

Fig. 7. The contribution (%) of design variables to the variance of limit state: (a) Brean and (b)Newton

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