Subgroup D Report

PIANC PTC II Working Group 28

Breakwaters with vertical and inclined concrete walls

Report of Sub-group D

IMPLEMENTATION OF SAFETY IN THE DESIGN J. Dalsgaard S
rensen H.F. Burcharth 2001

Aalborg University, Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark 1

Contents 1. 2. 3. 4.

5.

6. 7.

8.

9.

10.

11.

Terms of reference for sub-group D : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 Members of sub-group D : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 Failure Modes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 4.1 Sliding failure 4 4.2 Overturning failure 4 4.3 Foundation failure modes 4 4.4 Armour layer failure modes 5 4.5 Scour 5 Uncertainties and Statistical Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 5.1 Wave modelling 7 5.2 Soil strength modelling 8 5.3 Model uncertainties 9 5.4 Wave load modelling 10 First Order Reliabilty Methods (level II reliability analysis) : : : : : : : : : : : 13 Partial safety factors (level I reliability code) : : : : : : : : : : : : : : : : : : : : : : : : : 14 7.1 General aspects of code calibration 14 7.2 Estimation of partial safety factors for one failure mode 15 7.3 General procedure for estimating partial safety factors 16 Format for partial safety factors : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 8.1 EUROCODE format 20 8.2 PIANC format (Working Group 12 on Rubble Mound Breakwaters) 21 8.3 Discussion of existing code formats and selection of new format 22 Limit state functions and design equations : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 9.1 Sliding between structure and rubble foundation 24 9.2 Failure by overturning 25 9.3 Foundation failure modes with sand subsoil 25 9.4 Foundation failure modes with clay subsoil 33 9.5 Scour failure 37 9.6 Hydraulic instability of armour layer 38 Partial Safety Factors : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39 10.1 Calibration results 39 10.2 Verication and quality of calibrated partial safety factors 44 10.3 Examples of reliability calculations 49 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51 2

1. Terms of reference for sub-group D

Implementation of safety in the design, see item g in draft TOR.

2. Members of sub-group D

Prof. dr.techn. H.F. Burcharth, Aalborg University, Denmark (chairman) Mr. J. Juhl, Danish Hydraulic Institute, Dennmark Dr. J.W. van der Meer, Delft Hydraulics, The Netherlands Prof. Dr. L. Franco, Politecnico di Milano, Italy Prof. N.W.H. Allsop, HR Wallingford Limited, UK Prof. Dr. H. Oumeraci, Technische Universitat Braunschweig, Germany Mr. V. Morin, Canada Invited specialist: Dr. J. Dalsgaard S
rensen, Aalborg University, Denmark

3

3. Introduction

In this report safety aspects in relation to vertical wall breakwaters are discussed. Breakwater structures such as vertical wall breakwaters are used under quite different conditions. The expected lifetime can be from 5 years (interim structure) to 100 years (permanent structure) and the accepted level of probability of failure in the expected lifetime can vary from a very small number, e.g. 10;4 if failure of the breakwater results in signicant damage to large probabilities, e.g. 0.5 if the consequences are insignicant. This has to be taken into account when discussing safety aspects and possible level I and II code formats. In the following rst the most important failure modes are described, see section 4. In section 5 uncertainties related to the parameters used in the limit state functions and to the mathematical models are discussed and statistical models are suggested. Further wave load models are also described in section 4. In order to evaluate the safety of a given structure and to design new structures two levels of reliability methods are considered, namely level II methods where the reliability is evaluated using First Order Reliability Methods (FORM, see Madsen et al. 1]) and level I methods where reliability is introduced through partial safety factors in a code of practice. First Order Reliability Methods are described in section 6 and in section 7 it is shown how partial safety factors can be introduced. The format of a level I code for design and analysis of vertical wall breakwaters is discussed in section 8. For each failure mode the limit state function and the corresponding design equation are described in section 9 which also contains the statistical parameters used to calibrate the partial safety factors. In section 10 the calibrated partial safety factors are presented.

4. Failure Modes

The various failure modes and the related equations are presented in the Subgroup A report. The failure modes considered in relation to the development of partial safety factors are the following:

4.1 Sliding failure 4.2 Overturning failure 4.3 Foundation failure modes

The following foundation failure modes are assumed to be the most important, see gure 1: 4.3.1 Sliding between structure and bedding layer / rubble foundation, see 1) in gure 1. 4

4.3.2 Failure in rubble mound, see 2) in gure 1. 4.3.3 Failure in rubble and sliding between rubble and clay / sand, see 3) in gure 1. 4.3.4 Failure in rubble mound, see 4) in gure 1. 4.3.5 Failure in rubble and sand, see 5) in gure 1. 4.3.6 Failure in rubble and sand, see 6) in gure 1. 4.3.7 Failure in rubble and clay, see 7) in gure 1. 4.3.8 Failure in rubble and clay (circular), see 8) in gure 1.

4.4 Scour

4.4.1 Scour at the foot of caisson roundheads.

4.5 Armour layer failure modes

4.5.1 Hydraulic instability of the armour layer in rubble foundation

5

5. Uncertainties and Statistical Models

Uncertainties in relation to vertical wall breakwaters can be divided in uncertainties related to the following three groups: Loads (wave modelling) Strengths (modelling of e.g. soil parameters, density and weight) Models (wave load models, models for bearing capacity of the foundation, hydraulic response, scour and armour layer failures)

5.1 Wave modelling

The uncertainty related to environmental data, especially the description of sea states and water levels has been considered in PIANC WG 12, subgroup 12, see Burcharth 2]. Based on that report, statistical models which include statistical uncertainties and model uncertainties, are obtained for the signicant wave height. It is assumed that measured/observed signicant wave heights can be modelled by a Weibull distribution, see eq. (5.1) below. For calibration of partial safety factors wave data from 4 quite dierent geographical locations are selected. In the following table hs is the water depth, N is number of samples and
is the number of observations per year. N






HS

hs

0

Bilbao 50 4.17 1.39 1.06 4.9 29 Sines 15 1.25 1.78 2.53 7.1 35 Tripoli 15 0.75 1.83 3.24 2.9 27 Follonica 46 5.94 1.14 0.58 2.69 10 Table 1. Wave data from dierent locations tted to a Weibull distribution.  , HS and hs are in meters. 0

The wave data from Bilbao, Sines and Tripoli correspond to deep water waves while the wave data from Follonica correspond to shallow water waves. The maximum signicant wave height in T years is denoted HST and can be modelled by the extreme Weibull distribution function:

i
T FHST (hS ) = 1 ; exp ; hS ; HS h

0

(5:1)

In order to model the statistical uncertainty  and  are modelled as independent normal distributed variables (N( ) indicates a normal distribution with expected value  and standard deviation ) : r

 : N ( N1 )

(5:2) 7

s

+ 2=) ; 1)  : N ( p ;;(1 2 (1 + 1=) N

(5:3)

where ;(:) is the Gamme function. The model uncertainty related to the quality of the measured wave data is modelled by a multiplicative stochastic variable ZHS which is assumed to be normal distributed with expected value 1 and standard deviation Z0 HS . Good and poor wave data could be represented by Z0 HS = 0.05 and 0.2, respectively. The water level set-up due to storm wind and waves (storm surge) is dicult to estimate except for simple conditions (straight coastline and constant slope of sea bed). The uncertainty related to storm surge varies considerably with the environmental conditions. In the calibration of the partial safety factors is included an uncertainty on the storm surge water level corresponding to a standard deviation of  = 0:05HS . The motivation is that the storm surge is approximately proportional to HS and the maximum value of the wave generated set-up is approximately 0:2HS . The storm surge water level variation is assumed normal distributed. The bias related to storm surge is assumed to be included in the mean water level.

5.2 Soil strength modelling

Statistical modelling of the strength of the soil (sand and/or clay) is generally dicult and only few models which can be used for practical reliability calculations are available in the literature, see 3]-4] and 21]-29]. In general the material characteristics of the soil have to be modelled as a stochastic eld. The parameters describing the stochastic eld have to be determined on the basis of the measurements which are usually performed to characterize the soil characteristics. Since these measurements are only performed in a few points also statistical uncertainty due to the few data points is introduced and have to be included in the statistical model. Further the uncertainty in the determination of the soil properties and the measurement uncertainty have to be included in the statistical model. In the literature the undrained shear strength of clay is often modelled by a logGaussian distributed stochastic eld fcu(x z)g where z and x are vertical and horisontal coordinates, respectively. The expected value function E cu(x z)] and the covariance function Covcu(x1  z1 ) cu(x2  z2 )] can typically be written, see e.g. Keaveny et al. 3] and Andersen et al. 4].

E cu(x z)] = E cu(z)] Covcu(x1  z1 ) cu(x2  z2 )] = Covcu(x1 ; x2  z1 ; z2)]

(5.4) (5.5)

where (x1  z1 ) and (x2  z2 ) are two points in the soil. E cu(x z)] gives the expected value in depth z of the undrained shear strength of clay. Covcu(x1  z1) cu (x2  z2)] 8

gives the covariance between cu at position (x1  z1) and cu at position (x2  z2). V arcu (x1  z1 )] = Covcu(x1  z1 ) cu(x1  z1 )] is the variance of cu at position (x1  z1 ). It is seen that the expected value depends on the depth and the covariance depends on the vertical and horisontal distances. Generally the correlation lengths in horisontal and vertical direction will be dierent due to the soil stratication. The mean value function and covariance function are in this report assumed to be modelled by

E cu(x z)] = cu0 + cu1z
; 
Covcu(x1  z1 ) cu(x2  z2 )] = c2u exp ;j(z1 ; z2 )j exp ;  (x1 ; x2 ) 2 where cu0 and cu1 model the expected value, cu is the standard deviation and  and  model the correlation. In practical calculations the stochastic eld can be discretized taking into account the correlation lengths of the eld or if an integral over some domain is used, the expected value and the standard deviation of this integral can be evaluated numerically. Since the breakwater foundation is made of friction material and it is assumed that foundation failure modes can develope both in the rubble mound and in sand subsoil, statistical models for the eective friction angle and the angle of dilation are needed for the rubble material and the sand subsoil. Usually these angles are modelled by Normal or Lognormal stochastic variables, i.e. the spatial variation is not taken into account. Here Lognormal stochastic variables are used.

5.3 Model uncertainties

In general model uncertainties related to a given mathematical model can be evaluated on the basis of:
comparisons between experimental tests / measurements and numerical model calculations
comparisons between numerical calculations with the given mathematical model and a more advanced/complex model.
expert opinions
information from the literature For most of the failure modes related to hydraulic response, hydraulic instability of the armour layer and scour many laboratory experiments have been performed. Based on test results the model uncertainty can be estimated. The model uncertainty connected with extrapolation from laboratory to a real structure can be judged on the basis of scale eect analysis, expert opinions, information from the literature and observations of similar existing structures. 9

 = 0:75(1 + cos ) Hdesign p1 = 0:5(1 + cos )(1 + 2cos2  ) w g Hdesign
( 1 ; hc p1 for  > hc p2 = 0 for   hc p3 = 3 p1

(5.6) (5.7) (5.8)




where 

Hdesign

(5.9)

angle of incidence of waves (angle between wave crest and front of structure) design wave height dened as the highest wave in the design sea state at a location just in front of the breakwater. If seaward of a surf zone a value of 1.8 HSTL might be used corresponding to the 0.1% exceedence value for Rayleigh distributed wave heights. If within a surf zone Hdesign is taken as the highest of the random breaking waves at a distance 5HSTL seaward of the structure. "

#2



4 h L 1 = 0:6 + 12 sinh ;4s h L s  2 H h ; d design b 2 = the smallest of 3 h d b "

3 = 1 ; hwh; hc 1 ; s

1   cosh 2 hs L ;

(5.10) and #

2d

Hdesign

(5.11) (5.12)

wave length corresponding to that of the signicant wave Ts ' 1:1Tm , where Tm is the average period. hb water depth at a distance of 5HS seaward of the breakwater front wall. HSTL maximum signicant wave heigt within structure lifetime TL . Although the wave induced uplift pressure, pu, at the front edge of the base plate is equal to p3 it is suggested by Goda to use a somewhat reduced value (5:13) pu = 12 (1 + cos ) 1 3 w g Hdesign

L

This is because analyses of the behaviour of Japanese breakwaters revealed that the use of pu = p3 together with an assumed triangular distribution of the uplift 11

pressure gave too conservative results. Modi cation of Goda's formulae, considering impulsive breaking wave forces Goda's formulae do not consider frequent wave breaking close to and at the vertical breakwater. Therefore the extra eect of larger impulsive forces from breaking waves has been investigated and incorporated in Goda's formulae by Takahashi et al. 17]. The modication of Goda's formula concerns the formula for the pressure p1 at the water surface, eq (5.7) and is a replacement of 2 coecient with a impulsive pressure coecient .

p1 = 0:5(1 + cos )(1 +  cos2 ) w g Hdesign

(5:14)

where  can be expressed as follows

 = maxf2 I g

(5:15)

where 2 is derived from Goda's formulae, eq (5.11), and I is a non-dimensional impulsive pressure coecient being the product of I 0 and I 1, where I 0 represents the eect on the design wave heigth and I 1 represents the shape of the rubble mound.

Hdesign =1:8d Hdesign=1:8d  2 (5.16) 2:0 Hdesign=1:8d > 2 where d is the water depth at the crest of the rubble mound berm in front of the caisson. I 0 =

( cos 2 cosh 1

2  0 (5.17) 1 > 0 1 2 cosh 1 (cosh 2 ) 2 where 1 and 2 are coecients which depend on the structural dimensions of the rubble mound in front of the caisson, cf. g. 2, on the incident wave. I 1 =



20  1 1 1 1  0 15 1 1 1 1 > 0  h ; d B s m 1 1 = 0:93 L ; 0:12 + 0:36 h ; 0:6 s

4:9  2 2 2 2  0 2 = 3   2 2 > 0 22  B h ; d m s 2 2 = ;0:36 L ; 0:12 + 0:93 h ; 0:6

1 =

s

(5.18) (5.19) (5.20) (5.21) 12

where Bm berm width of the rubble mound foundation in front of the caisson breakwater L wave length corresponding to that of the signicant wave period Ts ' 1:1Tm , where Tm is the average period. The term I reaches a maximum value of 2, when Bm =L = 0:12, d=hs = 0:4 and Hdesign =1:8d  2. When the term d=hs > 0:7, then impulsive pressures rarely occur and I is close to zero and smaller than 2.

6. First Order Reliabilty Methods (level II reliability analysis)

In this section a brief introduction to FORM is given. A detailed description can be found in Madsen et al. 1] and Burcharth 33]. The limit state functions modelling the failure modes described in section 4 are assumed to be written

g(x p) = 0

(6:1)

where x = (x1  ::: xn) are realisations of stochastic variables X = (X1  ::: Xn). p = (p1  ::: pm ) are deterministic constants. A transformation from X -variables to normalized and normal distributed U -variables is dened by X = T(U). The reliability index is dened and estimated iteratively as described in Madsen et al. 1]. The  -point in the normalized and normal distributed u-space is denoted u . If the safety margin Z dened by

Z = g(T(U) p)

(6:2)

is linearized in the  -point then

Z  ; TU +  where the elements in the

(6:3) vector are given by

 @g i = ui =  ;1  @u rug i

(6:4)

rug is the gradient of g with respect to u in the  -point u . 13

If the limit state function is not too nonlinear the probability of failure Pf can with good accuracy be determined from

Pf  (; )

(6:5)

where () is the standard normal distribution function. The components in the -vector can be considered as measures of the relative importance of the uncertainty in the corresponding stochastic variable on the reliablity index. However, it should be noted that for dependent (correlated) basic variables the components in the -vector cannot be linked to a specic basic variable. An important sensitivity measure is the reliability elasticity coecient dened by p ep = d (6:6) dp  where p is a parameter in a distribution function (e.g. the expected value or the standard deviation) or p is a constant in the failure function. It is seen that if the parameter p is changed by 1% then the reliability index  is changed by ep %. d dp can be obtained from d =  1  @g (6:7) dp  rug  @p where @g @p is evaluated in the  -point.

7. Partial safety factors (level I reliability code) 7.1 General aspects of code calibration

During the last two decades calibration of partial safety factors in level 1 codes for structural systems has been performed on a probabilistic basis in a number of codes of practice, see e.g. OHBDC (Ontario Highway Bridge Design Code) 6] and Rosenblueth & Esteva 7]. The calibration is generally performed for a given class of structures, materials and/or loads in such a way that the reliability measured by the rst order reliability index  estimated on the basis of structures designed using the new calibrated partial safety factors are as close as possible to the reliability indices estimated using existing design methods. Procedures to perform this type of calibration of partial safety factors are described in for example Ravindra & Lind 8], Thoft-Christensen & Baker 9]. A code calibration procedure usually includes the following basic steps, see e.g. Nowak 10]: - denition of scope of the code, 14

- denition of the code objective, - selection of code format, - selection of target reliability index levels, - calculation of calibrated partial safety factors and - verication of the system of partial safety factors. A rst guess of the partial safety factors is obtained by solving an optimization problem where the objective is to minimize the dierence between the reliability for the dierent structures in the class considered and a target reliability level. In order to ensure that all the structures in the class considered have a satisfactory reliability, constraints are imposed on the reliability for the whole range of structures.

7.2 Estimation of partial safety factors for one failure mode

In code calibration based on rst order reliability methods (FORM) it is assumed that the limit state function can be written g(x p z) = 0 (7:1) where x = (x1  . . .  xn) is a realization of X = (X1  . . .  Xn ). External loads (e.g. wave), strength parameters and model uncertainty variables are examples of uncertain quantities. p = (p1  . . .  pM ) are M deterministic parameters, for example well dened geometrical quantities. z = (z1  . . .  zN ) are N design variables which are used to design the actual structure. Realizations x of X where g(x p z)  0 corresponds to failure states, while g(x p z) > 0 corresponds to safe states. As an example consider sliding failure where the limit state function can be written g = (FG ; FU ) tan ' ; FH (H ) (7:2) where FG ; FU is the weight of the structure minus the wave induced uplift, FH is the horisontal force from waves with height H and ' is the eective friction angle. If the number of design variables is N = 1 then the design (modelled by z) can be determined from the design equation G(xc  p z
)  0 (7:3)

xc = (xc1  . . .  xcn) are characteristic values corresponding to the stochastic variables X.
= (1  . . .  m ) are m partial safety factors. The partial safety factors
are usually dened such that i  1 i = 1 ::: m. In the most simple case m = n. The design equation corresponding to (7.2) can be written G = (FGc ; FUc (H H c )) 1 tan 'c ; FH (H H c) Z

(7:4) 15

where Z and H are partial safety factors. The design equation is closely connected to the limit state function (7.1). In most cases the only dierence is that the variables x are exchanged by design values xd obtained from the characteristic values xc and the partial safety factors
. The characteristic values are for load variables usually the 90 %, 95 % or 98 % fractiles of the distribution function of the stochastic variables, e.g.

xci = FX;i1(0:98) where FXi is the distribution function for Xi . The design values for load variables are then obtained from

xdi = xci i

(7:5)

The characteristic values are for strength variables usually the 10 %, 5 % or 2 % fractiles of the distribution function of the stochastic variables. The design values for strength variables are then obtained from c

xdi = xi

(7:6)

i

For geometrical variables usually the median (50 % fractile) is used and the design values are

xdi = xci i

(7:7)

A reliability analysis by FORM with the limit state function (7.1) gives the reliability index  and the  -point x . Partial safety factors can then be obtained from c i = xxi for strength variables i  i = xxic for load variables i

7.3 General procedure for estimating partial safety factors

In general, the target reliability index to be used in a code calibration can be determined by calibration to the reliability level of existing similar structures. Alternatively or supplementary the target reliability indices can be selected on the basis of the recommneded minimum reliability indices specied in e.g. NKB (the Nordic Committee on Building Regulations) 11] or EUROCODES 13]. 16

In NKB 11] the maximum probability of failure (or equivalently the minimum reliability) are related to the consequences of failure specied by safety classes and failure types. The following safety classes are considered : Less serious, Serious and Very serious. The following failure types are considered : Failure type I (Ductile failures where it is required that there is an extra carrying capacity beyond the dened resistance), Failure type II (Ductile failures without an extra carrying capacity) and Failure type III (Britle failure). For ultimate limit states NKB recommend the following maximum probabilities of failure based on a reference period of 1 year: Safety class Failure type I Less serious 10;3 Serious 10;4 Very serious 10;5 Table 2. Maximum probabilities of failure.

Failure type II 10;4 10;5 10;6

Failure type III 10;5 10;6 10;7

The minimum reliability indices corresponding to the maximum probabilities in table 2 are Safety class Failure type I Failure type II Failure type III Less serious 3.1 3.7 4.3 Serious 3.7 4.3 4.7 Very serious 4.3 4.7 5.2 Table 3. Target (minimum) reliability indices. In EUROCODES 13] Reliability dierentation are introduced. The dierent levels of reliability may depend on:
the cause and mode of failure,
the possible consequences of failure (risk to life, injury, economic losses and social inconvenience),
the expenses necessary to reduce the risk of failure and
dierent degrees of reliability can be required at national, regional or local level. 17

The following indicative values for the target reliability index are given:

T = TL T = 1 year  = 3.8  = 4.7 (Pf = 0:0001) (Pf = 0:000001) Fatigue  = 1.5 - 3.8 (Pf = 0:07 ; 0:0001) Serviceability  = 1.5  = 3.0 (Pf = 0:07) (Pf = 0:0013) Table 4. Target (minimum) reliability indices  and related probabilities of failure Pf . T is the considered time interval and TL is the design life time. Limit state Ultimate

For breakwaters the probability of failure within the design lifetime is typically 0.01-0.4, see e.g. 2]. As explained above calibration of partial safety factors is generally performed for a given class of structures, materials or loads in such a way that the reliability measured by the rst order reliability index  estimated on the basis of structures designed using the new calibrated partial safety factors is as close as possible to the target reliability index or to the reliability indices estimated using existing design methods. On the basis of the limit state function in (7.1) the reliability index  can be determined using FORM (First Order Reliability Methods). If the number of design variables is N = 1 then the design can be determined from the design equation, see (7.3)

G(xc  p z
)  0

(7:8)

If the number of design variables is N > 1 then a design optimization problem can be formulated: min C (z) s:t: ci (z) = 0  i = 1 ::: me ci (z)  0  i = me + 1 ::: m zil  zi  ziu  i = 1 ::: N

(7.9) (7.10) (7.11) (7.12)

C is the objective function and ci  i = 1 2 ::: m are the constraints. The objective function C is often chosen as the construction cost of the structure. The me equality constraints in (7.10) can be used to model design requirements (e.g. constraints on the geometrical quantities) and to relate the load on the structure to the response. The inequality constraints in (7.11) ensure that response characteristics do not exceed codied critical values as expressed by the design equations (7.8). The inequality constraints may also include general design requirements for 18

the design variables. The constraints in (7.12) are so-called simple bounds. zil and ziu are lower and upper bounds to zi . Generally the optimization problem (7.5) (7.8) is non-linear and non-convex. The application area for the code is described by the set I of L dierent vectors pi i = 1 . . .  L. The set I may e.g. contain dierent geometrical forms of the structure, dierent parameters for the stochastic variables and dierent statistical models for the stochastic variables. The partial safety factors
are calibrated such that the reliability indices corresponding to the L p-vectors are as close as possible to a target probability of failure Pft or equivalently a target reliability index t = ;;1 (Pft ). This is formulated by the following optimization problem min
W (
) =

L X j =1

wj (j (
) ; t)2

(7:13) P

where wj  j = 1 . . .  L are weighting factors ( Lj=1 wj = 1) indicating the relative frequency of appearance of the dierent design situations. Instead of using the reliability indices in (7.13) to measure the deviation from the target for example the probabilities of failure can be used. Also, a nonlinear objective function giving relatively more weight to reliability indices smaller than the target compared to those larger than the target can be used. j (
) is the reliability index for combination j obtained as described below. In (7.13) the deviation from the target reliability index is measured by the squared distance. The reliability index j (
) for combination j is obtained as follows. First, for given
the optimal design is determined by solving the design equation (7.3) if N = 1 or by solving the design optimization problem (7.5)-(7.8) if N > 1. Next, the reliability index j (
) is estimated by FORM on the basis of (7.1). It should be noted that, following the procedure described above for estimating the partial safety factors two (or more) partial safety factors are not always uniquely determined. They can be functionally dependent, in the simplest case as a product, which has to be equal to a constant. In the above procedure there is no lower limit on the reliability. An improved procedure which has a constraint on the reliability and which takes the non-uniqeness problem into account can be formulated by the optimization problem min
W (
) =

L X j =1



wj (j (
) ; t )2 +

(
)   min

m X i=1

(i ; ji )2

(7.14)

s:t: i  i = 1 . . .  L (7.15) t il  i  iu  i = 1 . . .  m (7.16) P where wj  j = 1 . . .  L are weighting factors ( Lj=1 wj = 1). is a factor specifying the relative importance of the two terms. j (
) is the reliability index for 19

combination j obtained as described above. ji is an estimate of the partial safety factor obtained by considering combination j in isolation. The second term in the objective function (7.14) is added due to the non-uniqueness-problem and has the eect that the partial safety factors are forced in the direction of the "simple" denition of partial safety factors. For load variables :  = xxc . If only one combination is considered then ji = xxjicji where xji is the design point. Experience with this formulation has shown that the factor should be chosen to be of magnitude one and that the calibrated partial safety factors are not very sensitive to the exact value of . The constraints (7.15) have the eect that no combination has a reliability index smaller than tmin and the constraints in (7.16) are simple bounds on the partial safety factors. This type of code calibration has been used in Burcharth 12] for code calibration of rubble mound breakwater designs. As discussed above a rst guess of the partial safety factors is obtained by solving these optimization problems. Next, the nal partial safety factors are determined taking into account current engineering judgement and tradition.





8. Format for partial safety factors

In this section two formats for partial safety factors for vertical wall breakwaters are discussed. The rst format is similar to the format used in EUROCODE 7 for Geotechnical design. The second format is similar to the format proposed for rubble mound breakwaters by PIANC working group 12, see 12]. The code entry is proposed to be:
the design structure lifetime TL
the acceptable probability of failure Pf (e.g. = 0,01, 0,05, 0,10, 0,20 or 0,40) corresponding to the target reliability indices T (= 2,33, 1,65, 1,28, 0,84 or 0,25)
the coecient of variation ZHS of a model uncertainty variable ZHS modelling the uncertainty on the signicant wave height data. ZHS is modelled as a factor to HSTL . E.g. ZHS = (0,05 and 0.20). 0

0

8.1 EUROCODE format

In the EUROCODES it is not possible to specify target probabilities of failure (or target reliability indices) and it is not possible to specify the design lifetime. The target probabilty of failure used in the EUROCODES is discussed in section 7.3 and the design lifetime is TL =50 years. Three partial safety factors are used in the EUROCODES, namely 20

- a strength safety factor R to be divided to the characteristic value of the strength parameters (one partial safety factor for each of the four ground properties: eective friction angle, cohesion, undrained shear strength and compressive strength of soil/rock). - a load partial safety factor P to be multiplied to the characteristic value of the permanent load (a distinction can be made between favorable and unfavorable permanent loads). - a load partial safety factor W to be multiplied to the characteristic value of the variable load, here the wave load. In EUROCODES the following values are 'recommended' in relation to design and analysis of structures where foundation failures are included: permanent unfavorable load P = 1.00 permanent favorable load P = 1.00 variable load W = 1.30 tangent to the eective friction angle R = 1.25 cohesion R = 1.6 undrained shear strength R = 1.4 compressive strength of soil/rock R = 1.4 The characteristic value for the wave load will typically be based on the TL -year signicant wave height corresponding to the (1 ; 1=TL ) fractile of the distribution function for the yearly signicant wave height. The characteristic values for other variable actions are generally selected as 98% fractiles. For permanent loads the mean values are used as characteristic values. The characteristic values for strength parameters are generally chosen as the 5% fractiles.

8.2 PIANC format (Working Group 12 on Rubble Mound Breakwaters)

The signicance of the PIANC WG12 format is that the designer can specify the probability of failure within the structure lifetime and then calculate the related partial safety factors. Three partial safety factors are specied : - a strength safety factor R and - a load partial safety factor P (=1) to be multiplied to the permanent load. - a partial safety factor for the signicant wave height, HS to be multiplied to 21

the signicant wave height. The following tting formulae for R and HS are used, see 12] :

R = 1 ; k lnPf

(8:1) ; H^ S3TL

(1+ ^ STPf H HS = ^ TL + ZHS HS 0

^ TL H

S

 ;1 k Pf )

+ p ks Pf N

(8:2)

where k  k and ks are constants to be tted and

TPf = (1 ; (1 ; Pf ) TL );1 1

(8:3)

H^ ST is the central estimate of the T -year return period value for HS and N is the number of HS data used for tting the extreme distributions. k and k will depend on the failure mode while ks are common for all failure modes. The factors P , k , k and ks are calibrated for typical applications of rubble mound breakwaters by solving the optimization problem (7.13) or (7.14) - (7.16). HS is multiplied to HSTL and R is divided into the mean value of the product of the strength variables. If the application area is specied in the form of 'usual' values of the parameters in the design equation, partial safety factors can be calibrated using the computer program BWCODE (BreakWater CODE calibration program) developed at Aalborg University.

8.3 Discussion of existing code formats and selection of new format

A comparison between the EUROCODE and the PIANC WG12 code formats shows the following important dierences: In EUROCODES the partial safety factors are given as a small number of factors whereas in the PIANC WG12 format safety factors are given by formulae. In EUROCODES it is not possible to specity the target probability of failure and the design liftime. This is possible in the PIANC WG12 format. In EUROCODES the partial safety factors related to the load are multiplied to the load whereas in the PIANC WG12 format the partial safety factors are multiplied to the signicant wave height. 22

Taking into account the general acceptance of the EUROCODE format in design codes, that the EUROCODE format is easier to apply in practice than the PIANC WG12 format and that the possibility of specing a target probability of failure and a design liftime as included in the PIANC WG12 format should be kept, the following code format is selected: Code entry:
the design structure lifetime TL
the acceptable probability of failure Pf (= 0.01, 0.05, 0.10, 0.20 or 0,40) corresponding to the target reliability indices T (= 2.33, 1.65, 1.28, 0.84 or 0.25)
the coecient of variation ZHS = (0.05 and 0.20).
Deep or shallow water conditions.
Hydraulic model test or not. Partial safety factors: - a load partial safety factor P to be multiplied to the permanent load (= 1). - a load partial safety factor H to be multiplied to H^ STL (the central estimate of the signicant wave height which in average is exceeded once every TL years, where TL is the structure lifetime). - a safety factor Z to be used with friction materials in rubble mound and/or subsoil (tangent to the mean value of the friction angle is divided by Z ). - a safety factor C to be used with the undrained shear strength of clay materials in the subsoil (the mean value of the undrained shear strength is divided by C ). Characteristic values: The characteristic values for permanent and variable loads are chosen as described in section 8.1 for the EUROCODE format. For the soil strengths the characteristic values are assumed to be the mean values. The reason that e.g. a 5 % fractile as in the EUROCODE format is not used to dene the characteristic values is that the reliability levels are small compared with e.g. building structures and therefore here would result in partial safety factors smaller than 1 and this is not wanted. However, a sensitivity analysis will be performed to illustrate the importance of the choice of the charactristic value. 0

9. Limit state functions and design equations

In this section is given a brief description of the failure modes used. A more detailed description and the related formulae are given in the Subgroup A report, see Burcharth 30]. 23

HS is the signicant wave height. In practical use of the partial safety factors wave data from the actual location is used to obtain the parameters in a Weibull distribution function for HS . For calibration of the partial safety factors presented in this report the data in table 1 is used. The statistical uncertainty is taken into account by modelling the parameters  and  in the Weibull distribution as stochastic variables, see section 5.1. H^ ST is the central estimate of the signicant wave height which in average is exceeded once every T years. X^ is the characteristic value of a stochastic variable X . The characteristic value is assumed to be the expected value, except for the eective friction angle, the dilatation angle, and the undrained shear strength of clay where the mean value is used. z is the design parameter which here is assumed to be the width of the caisson, i.e. z = Br . p denes the range of application of the code for the considered failure mode. In the following D denotes a deterministic variable, N(, ) denotes a normal distribution with expected value  and standard deviation  and LN(, ) denotes a lognormal distribution.

9.1 Sliding between structure and rubble foundation Limit state function :

g = g(HS ZHS  c  ZFH  ZFV   f Br ) = (FG ; ZFV FU (HSTL ))f ; ZFH FH (HSTL ) where HSTL ZHS c  ZFH ZFV f Br FG FU FH

signicant wave height model uncertainty related to HSTL density of the caisson tidal elevation model uncertainty on horisontal wave load model uncertainty on vertical wave load friction coecient Width of caisson reduced weight of caisson under water wave induced uplift force horisontal wave force 24

Design equation: G = G(H H^STL  ^c Z^FH  Z^FV  ^ 1 f^ Br ) Z 1 = (F^G ; Z^FV F^U (H H^ STL ))  f^ ; Z^FH F^H (H H^ STL ) Z where f^ is the mean value of f .

9.2 Failure by overturning Limit state function :

g = g(HSTL ZHS  ZMH  ZMV   Br ) = (MG ; ZMV MU ) ; ZMH MH where HSTL ZHS ZMH ZMV  MG

MU MH Br

signicant wave height model uncertainty related to HSTL model uncertainty on horisontal moment load model uncertainty on vertical moment load tidal elevation moment around the heel induced by the weight of the caisson reduced for boyancy moment around the heel from wave induced uplift moment around the heel from horisontal wave force Width of caisson

Design equation : G = G(H H^ ST  ^c Z^MH  Z^MV  ^ Br ) = (M^ G ; Z^MV MU (H H^ STL )) ; ZMH MH (H H^ STL )

9.3 Foundation failure modes with rubble foundation and sand subsoil 6 failure modes related to foundation failure are taken into account:

Failure mode 1: sliding between structure and rubble foundation, see (1) in g. 1 Limit state function : based on sliding failure in the rubble, see 30] g1 = g1(HSTL ZHS  c ZFH  ZFV   tan 'd1  Br ) = (FG ; ZFV FU ) tan 'd1 ; ZFH FH

25

where

'0r cos r tan 'd1 = 1 ;sinsin '0r sin r and HSTL ZHS c  ZFH ZFV r '0r Br FG FU FH

signicant wave height model uncertainty related to HSTL density of the caisson tidal elevation model uncertainty on horisontal wave load model uncertainty on vertical wave load dilation angle of the rubble mound material eective friction angle of the rubble mound material Width of caisson reduced weight of caisson under water wave induced uplift force horisontal wave force

Design equation :

G1 = G1(H H^ ST  ^c Z^FH  Z^FV  ^ 1 tan^'d1  B1) Z

Failure mode 2: rupture in rubble mound, see (2) in gure 1 Limit state function : see 30]

g2 = g2(HSTL ZHS  c ZFH  ZFV  ZMH  ZMV   tan 'd1  Z Br ) where

'0r cos r tan 'd1 = 1 ;sinsin '0r sin r and 26

HSTL ZHS c ZFH ZFV ZMH ZMV  r '0r Z Br

signicant wave height model uncertainty related to HSTL density of the caisson model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load tidal elevation dilation angle of the rubble mound material eective friction angle of the rubble mound material model uncertainty on geotechnical failure mode Width of caisson

Design equation : ^ B2) G2 = G2(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'d1  Z Z

Failure mode 3: rupture in rubble mound, see (4) in gure 1 Limit state function : see 30]

g3 =g3(HSTL ZHS  c ZFH  ZFV  ZMH  ZMV   tan 'd1  Z Br ) where

'0r cos r tan 'd1 = 1 ;sinsin '0r sin r and HSTL ZHS c

signicant wave height model uncertainty related to HSTL density of the caisson 27

ZFH ZFV ZMH ZMV  r '0r Z Br

model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load tidal elevation dilation angle of the rubble mound material eective friction angle of the rubble mound material model uncertainty on geotechnical failure mode Width of caisson

Design equation :

G3 =G3(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'  Z d1 ^ B3 )  Z

Failure mode 4: rupture in rubble mound and subsoil, see (5) in gure 1 Limit state function : see 30]

g4 =g4(HSTL ZHS  c ZFH  ZFV  ZMH  ZMV   tan 'd1  tan 'd2  Z Br ) where

'0r cos r tan 'd1 = 1 ;sinsin '0r sin r sin '0s cos s tan 'd2 = 1 ; sin '0 sin  s

and HSTL ZHS

s

signicant wave height model uncertainty related to HSTL 28

c ZFH ZFV ZMH ZMV  r '0r s '0s Z Br

density of the caisson model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load tidal elevation dilation angle of the rubble mound material eective friction angle of the rubble material dilation angle of the subsoil friction material eective friction angle of the subsoil friction material model uncertainty on geotechnical failure mode Width of caisson

Design equation :

G4 =G4(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'  1 tan^'  Z d1  d2 ^ B4 ) Z Z Failure mode 5: rupture in rubble mound and subsoil, see (6) in gure 1 Limit state function : see 30]

g5 =g5(HSTL ZHS  c ZFH  ZFV  ZMH  ZMV   tan 'd1  tan 'd2  Z Br ) where

'0r cos r tan 'd1 = 1 ;sinsin '0r sin r

sin '0s cos s tan 'd2 = 1 ; sin '0s sin s and 29

HSTL ZHS c ZFH ZFV ZMH ZMV  r '0r s '0s Z Br

signicant wave height model uncertainty related to HSTL density of the caisson model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load tidal elevation dilation angle of the rubble mound material eective friction angle of the rubble material dilation angle of the subsoil friction material eective friction angle of the subsoil friction material model uncertainty on geotechnical failure mode Width of caisson

Design equation :

G5 =G5(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'  1 tan^'  Z d1  d2 ^ B5 ) Z Z Failure mode 6: rupture in rubble and sliding between rubble and sand subsoil, see (3) in gure 1 Limit state function : see 30] g6 =g6(HSTL ZHS  c ZFH  ZFV  ZMH  ZMV   tan 'd1  tan 'd2  Z Br ) where '0r cos r tan 'd1 = 1 ;sinsin '0r sin r

sin '0s cos s tan 'd2 = 1 ; sin '0 sin  s

s

30

and HSTL ZHS c ZFH ZFV ZMH ZMV  r '0r s '0s Z Br

signicant wave height model uncertainty related to HSTL density of the caisson model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load tidal elevation dilation angle of the rubble mound material eective friction angle of the rubble mound material dilation angle of the subsoil friction material eective friction angle of the subsoil friction material model uncertainty on geotechnical failure mode Width of caisson

Design equation :

G6 =G6(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'  1 tan^'  Z d1  d2 ^ B6 )  Z

Z

The reliability index for foundation failure with sand subsoil is determined as minimum of the reliability indices corresponding to the nine failure modes:  = min(1  2  3 4  5  6). For calibration of the partial safety factors the parameters for the stochastic variables shown in table 5 are used. The correlation coecient between ZFH and ZMH and between ZFV and ZMV are estimated roughly to 0.9. The tidal elevation  is modelled as a stochastic variable with distribution function
1  F ( ) =  arccos ;  (9.1) 0 31

where 0 is the maximum tidal height. 0 = 0.75 m is used in this report. distribution variation of p reference c N(2.1, 0.1075) 33] ZFH N(0.90, 0.25) 36] ZFV N(0.77, 0.25) 36] ZMH N(0.81, 0.40) 36] ZMV N(0.72, 0.37) 36] r LN(0.43, 0.043) 0 'r LN(0.61, 0.061) s LN(0.35, 0.035) 0 's LN(0.52, 0.052) Z N(1, 0.1) f N(0.636, 0.0954) 34]  see eq. (9.1) 0=0.75 m 34] HS ex Weibull see table 1 ZHS N(1 Z0 HS ) Table 5. Statistical parameters for calibration of partial safety factors for foundation failure with sand subsoil. The design value of the width of the caisson is selected as the maximum design width corresponding to all failure modes: Br = max(B1 B2  B3 B4 B5  B6). This value of Br is used in the limit state functions to estimate the reliability indices 1:::6. The four vertical breakwaters shown in table 6 are placed on high rubble mounds. The following data is used for the designs: average density of the caisson is taken as c= 2.3 t/m3, wave steepnes sm = 0.035, the slope of the foreshore is taken as zero, friction coecient equal to 0.6, angle of dilatation in the rubble mound and subsoil are !r =0.43 and !s = 0.35, the eective friction angle in the rubble mound and the subsoil are '0r = 0.61 and '0s=0.52 and the shear strength of the clay subsoil is taken as cu= 150 kN/m2 .

hs hc d hw h0 h2 Brm Bilbao 29 5 17 24 17 7 10 Sines 35 9 25 36 25 7 14 Tripoli 27 6 15 23 15 7 12 Follonica 10 4 6 12 8 2 6 Table 6. Design cases. hs: water depth, hc0: crest height, d: water depth in front of the caisson, hw : height of the caisson, h : water level in caisson, h2: height of rubble mound and Brm : rear berm width. All values are in meters. 32

9.4 Foundation failure modes with rubble foundation and clay subsoil

Failure mode 1,2 and 3 in section 9.3 is also used here. Further 4 failure modes related to foundation failure in the clay are taken into account: Failure mode 11: failure in rubble and sliding between rubble and clay subsoil, see (3) in gure 1 Limit state function , see 30] g11 =g11(HSTL ZHS  c  ZFH  ZFV  ZMH  ZMV   tan 'd1  Ucu  cu0 Z Br ) where '0r cos r tan 'd1 = 1 ;sinsin '0 sin  r

r

and HSTL ZHS c  ZFH ZFV ZMH ZMV r '0r Ucu cu0

Z Br

signicant wave height model uncertainty related to HSTL density of the caisson tidal elevation model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load dilation angle of the rubble mound material eective friction angle of the rubble mound material undrained shear strength in clay subsoil, see section 5.2 expected value of undrained shear strength in clay subsoil, see section 5.2 model uncertainty on geotechnical failure mode Width of caisson

Design equation :

G11 =G11(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'  1 U^  c  Z d1  cu u0 ^ B11 )  Z

C

33

Failure mode 12: failure in rubble and clay subsoil, see (7) in gure 1 Limit state function , see 30]

g12 =g12(HSTL ZHS  c  ZFH  ZFV  ZMH  ZMV   tan 'd1  Ucu  cu0 Z Br ) where

'0r cos r tan 'd1 = 1 ;sinsin '0r sin r and HSTL ZHS c  ZFH ZFV ZMH ZMV r '0r Ucu cu0

Z Br

signicant wave height model uncertainty related to HSTL density of the caisson tidal elevation model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load dilation angle of the rubble mound material eective friction angle of the rubble mound material undrained shear strength in clay subsoil, see section 5.2 expected value of undrained shear strength in clay subsoil, see section 5.2 model uncertainty on geotechnical failure mode Width of caisson

Design equation :

G12 =G12(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'  1 U^  c  Z d1  cu u0 ^ B12 )  Z

C

34

Failure mode 13: failure in rubble and clay subsoil (circular), see (8) in gure 1 Limit state function , see 30]

g13 =g13(HSTL ZHS  c  ZFH  ZFV  ZMH  ZMV   tan 'd1  Ucu  cu0 Z Br ) where '0r cos r tan 'd1 = 1 ;sinsin '0 sin 

r

r

and HSTL ZHS c  ZFH ZFV ZMH ZMV r '0r Ucu cu0

signicant wave height model uncertainty related to HSTL density of the caisson tidal elevation model uncertainty on horisontal wave load model uncertainty on vertical wave load model uncertainty on horisontal moment load model uncertainty on vertical moment load dilation angle of the rubble mound material eective friction angle of the rubble mound material undrained shear strength in clay subsoil, see section 5.2 expected value of undrained shear strength in clay subsoil, see section 5.2 Z model uncertainty on geotechnical failure mode Br Width of caisson Design equation :

G13 =G13(H H^ ST  ^c Z^FH  Z^FV  Z^MH  Z^MV  ^ 1 tan^'  1 U^  c  Z d1  cu u0 ^ B13 )  Z

C

The reliability index for foundation failure with clay subsoil is determined as minimum of the reliability indices corresponding to the nine failure modes :  = min(1  2  4 11  12 13 ). 35

For calibration of the partial safety factors the following parameters for the stochastic variables are used: distribution variation of p reference c N(2.1, 0.1075) 33] ZFH N(0.90, 0.25) 36] ZFV N(0.77, 0.25) 36] ZMH N(0.81, 0.40) 36] ZMV N(0.72, 0.37) 36] r LN(0.43, 0.043) '0r LN(0.61, 0.061) Z N(1, 0.1) Ucu N(0, 1) cu0 p p=(150, 200) kPa cu1 0 cu D(37.5 kPa)  D(0.33) see section 5.2  D(0.033) see section 5.2  see eq. (9.1) 0 =0.75 m 34] HS ex Weibull see table 1 0 ZHS N(1 ZHS ) Table 7. Statistical parameters for calibration of partial safety factors for foundation failure with clay subsoil (no model tests have been performed to determine the wave forces). If model tests have been performed to estimate the wave forces the following model uncertainties can be used: distribution variation of p reference ZFH N(0.90, 0.05) 31] ZFV N(0.77, 0.05) 31] ZMH N(0.81, 0.10) 31] ZMV N(0.72, 0.10) 31] Table 8. Statistical parameters for model uncertainties when wave forces are determined on the basis of model tests. The design value of the width of the caisson is selected as the maximum design width corresponding to all failure modes: B = max(B1  B2 B4 B11 B12 B13 ). This value of Br is used in the limit state functions to estimate the reliability indices 1  ::: 11 :::13. The design cases in table 6 are used. 36

9.5 Scour failure for circular roundheads on sand

Limit state function, see Sumer et al. 35] (no rubble foundation): ;  g = BS ; 0:5A 1 ; exp(;0:175(KC ; 1)) r

where S is the scour depth and Br is the diameter of vertical wall roundhead. KC and the max wave generated velocity of water particles at the undisturbed sea bed, Um are determined from

KC = UBm Tp r

TL Z 1 H S HS Um = T sinh(2h0s=Lp) p

The wave peak period Tp is a stochastic variable with expected value Tp0 determined from

Tp0 =

s

ZHS HSTL 2 sp g

The wave length Lp is determined from

Tp2 Lp = g 2 tanh(2h0s=Lp) where sp is the wave steepness and h0s is a stochastic vaiable with expected value hs, see table 1. Parameters for stochastic variables :

S A Tp sp h0s HS ZHS

distribution D(0.5 m) N(1, 0.6) N (Tp0  0:1Tp0 ) N(0.025, 0.005) N(hs 0:05hs) ex Weibull N(1 Z0 HS )

variation of p

hs : see table 1 see table 1

Design equation : ^ ; ^ (H H^ STL ) ; 1)) G = 1 S^ ; 0:5 1 ; exp(;0:175(KC ZB 37

9.6 Hydraulic instability of foundation rubble mound armour layer Limit state function, see Madrigal et al. 32] :

0 g = A"Dn(5:8 hh ; 0:60)Nod0:19 ; (HSTL ZHS ) s

Parameters for stochastic variables : distribution N(p1  p2 ) N(z pz) D(p) D(p) ex Weibull N(1 Z0 HS ) N(1:0 0:2)

" Dn Nod

h hs HS ZHS 0

A Table 9.

variation of p (p1  p2)=(1.4,0.03), (1.6,0.06) p=(0.01, 0.05) p= (0.5 , 2) p= (0.5 , 0.8) see table 1

It is noted that the standard deviation of the model uncertainty variable A is to be veried. Design equation : 0 G = 1 "^ D^ nA^(5:8 hh ; 0:60)Nod0:19 ; H H^ ST Z

s

38

10. Partial Safety Factors 10.1 Calibration results

The computer programme CODEBW is used to calibrate the partial safety factors by solving the optimization problem (7.13). The code format described in section 8.3 is used and the weight factors in (7.13) are chosen to wj = L1 . The partial safety factors have been selected on the basis of the solution to the optimization problem (7.13) and the minimum reliability levels obtained for the example structures. The results shown below are based on mean values as characteristic values for the strength variables. In deterministic design of the breakwater the following bias values for the forces and moments are to be used, see e.g. section 9.2 and 9.3: value ^ ZFH 0.90 ^ ZFV 0.77 ^ ZMH 0.81 ^ 0.72 ZMV Table 10. Values of model uncertainties to be used in deterministic design.

Foundation failure - sand subsoil: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.4 1.4 1.4 1.4 0.05 (1.65) 1.3 1.3 1.3 1.4 0.10 (1.28) 1.2 1.3 1.2 1.3 0.20 (0.84) 1.1 1.2 1.1 1.2 0.40 (0.25) 1.1 1.1 1.1 1.1 Table 11. Partial safety factors for foundation failure - sand subsoil - deep water - no model tests performed. Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.3 1.3 1.4 1.3 0.05 (1.65) 1.3 1.2 1.4 1.2 0.10 (1.28) 1.2 1.2 1.3 1.2 0.20 (0.84) 1.1 1.2 1.1 1.2 0.40 (0.25) 1.1 1.1 1.1 1.1 Table 12. Partial safety factors for foundation failure - sand subsoil - deep water - model tests performed. 39

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.5 1.4 1.3 1.5 0.05 (1.65) 1.4 1.3 1.3 1.4 0.10 (1.28) 1.3 1.2 1.2 1.3 0.20 (0.84) 1.2 1.1 1.1 1.3 0.40 (0.25) 1.1 1.0 1.1 1.1 Table 13. Partial safety factors for foundation failure - sand subsoil - shallow water - no model tests performed.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.3 1.3 1.4 1.3 0.05 (1.65) 1.3 1.3 1.4 1.3 0.10 (1.28) 1.2 1.2 1.3 1.2 0.20 (0.84) 1.1 1.1 1.1 1.1 0.40 (0.25) 1.1 1.1 1.1 1.1 Table 14. Partial safety factors for foundation failure - sand subsoil - shallow water - model tests performed.

Foundation failure - clay subsoil: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z C H Z C 0.01 (2.33) 1.3 1.4 1.4 1.4 1.4 1.4 0.05 (1.65) 1.2 1.3 1.3 1.3 1.3 1.3 0.10 (1.28) 1.1 1.2 1.3 1.2 1.2 1.3 0.20 (0.84) 1.0 1.1 1.2 1.0 1.1 1.2 0.40 (0.25) 1.0 1.1 1.1 1.0 1.0 1.1 Table 15. Partial safety factors for foundation failure - clay subsoil - deep water no model tests performed. 40

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z C H Z C 0.01 (2.33) 1.2 1.4 1.4 1.3 1.4 1.4 0.05 (1.65) 1.1 1.3 1.3 1.2 1.3 1.3 0.10 (1.28) 1.0 1.2 1.2 1.1 1.2 1.3 0.20 (0.84) 1.0 1.1 1.1 1.0 1.1 1.2 0.40 (0.25) 1.0 1.0 1.1 1.0 1.0 1.1 Table 16. Partial safety factors for foundation failure - clay subsoil - deep water model tests performed.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z C H Z C 0.01 (2.33) 1.2 1.7 1.5 1.3 1.7 1.5 0.05 (1.65) 1.1 1.5 1.4 1.2 1.5 1.4 0.10 (1.28) 1.1 1.4 1.3 1.2 1.4 1.4 0.20 (0.84) 1.0 1.3 1.2 1.1 1.3 1.3 0.40 (0.25) 1.0 1.2 1.2 1.1 1.2 1.2 Table 17. Partial safety factors for foundation failure - clay subsoil - shallow water - no model tests performed.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z C H Z C 0.01 (2.33) 1.2 1.5 1.4 1.3 1.5 1.4 0.05 (1.65) 1.1 1.4 1.3 1.2 1.4 1.3 0.10 (1.28) 1.1 1.3 1.3 1.1 1.3 1.3 0.20 (0.84) 1.0 1.2 1.2 1.1 1.2 1.2 0.40 (0.25) 1.0 1.1 1.1 1.1 1.1 1.1 Table 18. Partial safety factors for foundation failure - clay subsoil - shallow water - model tests performed. 41

Sliding failure: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.4 2.0 1.5 2.0 0.05 (1.65) 1.3 1.7 1.4 1.7 0.10 (1.28) 1.3 1.5 1.4 1.5 0.20 (0.84) 1.3 1.2 1.3 1.2 0.40 (0.25) 1.1 1.1 1.1 1.1 Table 19. Partial safety factors for sliding failure - deep water - no model tests performed. Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.3 1.7 1.4 1.7 0.05 (1.65) 1.2 1.6 1.3 1.6 0.10 (1.28) 1.2 1.4 1.3 1.4 0.20 (0.84) 1.2 1.2 1.2 1.2 0.40 (0.25) 1.1 1.2 1.1 1.1 Table 20. Partial safety factors for sliding failure - deep water - model tests performed. Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.3 2.2 1.4 2.2 0.05 (1.65) 1.2 1.9 1.3 1.9 0.10 (1.28) 1.2 1.7 1.3 1.7 0.20 (0.84) 1.2 1.3 1.2 1.3 0.40 (0.25) 1.0 1.2 1.0 1.2 Table 21. Partial safety factors for sliding failure - shallow water - no model tests performed. Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.2 1.7 1.3 1.6 0.05 (1.65) 1.1 1.5 1.2 1.5 0.10 (1.28) 1.1 1.3 1.2 1.3 0.20 (0.84) 1.1 1.2 1.1 1.2 0.40 (0.25) 1.0 1.1 1.0 1.1 Table 22. Partial safety factors for sliding failure - shallow water - model tests performed. 42

Scour failure: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 2.0 2.4 2.0 2.4 0.05 (1.65) 2.0 2.0 2.0 2.0 0.10 (1.28) 2.0 1.8 2.0 1.8 0.20 (0.84) 2.0 1.5 2.0 1.5 0.40 (0.25) 2.0 1.2 2.0 1.2 Table 23. Partial safety factors for scour failure for circular roundheads - deep water. Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 2.0 2.4 2.0 2.4 0.05 (1.65) 2.0 2.0 2.0 2.0 0.10 (1.28) 2.0 1.8 2.0 1.8 0.20 (0.84) 2.0 1.5 2.0 1.5 0.40 (0.25) 2.0 1.2 2.0 1.2 Table 24. Partial safety factors for scour failure for circular roundheads - shallow water.

Armour layer failure:

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.6 1.3 1.7 1.3 0.05 (1.65) 1.4 1.2 1.5 1.2 0.10 (1.28) 1.3 1.2 1.4 1.2 0.20 (0.84) 1.2 1.1 1.3 1.1 0.40 (0.25) 1.1 1.0 1.2 1.0 Table 25. Partial safety factors for armour layer failure - deep water. Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) H Z H Z 0.01 (2.33) 1.5 1.5 1.6 1.5 0.05 (1.65) 1.3 1.3 1.4 1.3 0.10 (1.28) 1.2 1.2 1.3 1.2 0.20 (0.84) 1.1 1.2 1.2 1.2 0.40 (0.25) 1.1 1.0 1.2 1.0 Table 26. Partial safety factors for armour layer failure - shallow water. 43

10.2 Verication and quality of calibrated partial safety factors Foundation failure - sand subsoil: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 2.25 0.25 1.90 2.24 0.18 1.88 0.05 (1.65) 1.73 0.16 1.51 2.10 0.09 1.98 0.10 (1.28) 1.60 0.12 1.43 1.53 0.10 1.42 0.20 (0.84) 0.96 0.10 0.83 0.92 0.15 0.71 0.40 (0.25) 0.45 0.08 0.32 0.50 0.09 0.36 Table 27. Reliability indices for calibration of foundation failure - sand subsoil - deep water - no model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 2.46 0.13 2.23 2.50 0.17 2.20 0.05 (1.65) 1.84 0.20 1.55 1.77 0.17 1.55 0.10 (1.28) 1.62 0.12 1.40 1.69 0.06 1.62 0.20 (0.84) 1.35 0.16 1.01 1.27 0.26 0.94 0.40 (0.25) 0.70 0.12 0.48 0.63 0.19 0.44 Table 28. Reliability indices for calibration of foundation failure - sand subsoil deep water - model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 2.13 0.17 2.01 2.29 0.09 2.20 0.05 (1.65) 1.73 0.09 1.61 1.78 0.04 1.74 0.10 (1.28) 1.31 0.04 1.25 1.35 0.04 1.32 0.20 (0.84) 0.83 0.02 0.82 1.27 0.02 1.26 0.40 (0.25) 0.40 0.02 0.38 0.48 0.04 0.44 Table 29. Reliability indices for calibration of foundation failure - sand subsoil shallow water - no model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index. 44

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 2.39 0.09 2.25 2.48 0.13 2.36 0.05 (1.65) 2.39 0.09 2.25 2.48 0.13 2.36 0.10 (1.28) 1.63 0.04 1.58 1.71 0.06 1.66 0.20 (0.84) 0.83 0.06 0.75 0.90 0.09 0.81 0.40 (0.25) 0.83 0.06 0.75 0.90 0.09 0.81 Table 30. Reliability indices for calibration of foundation failure - sand subsoil - shallow water - model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index.

Foundation failure - clay subsoil: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 2.49 0.17 2.29 2.60 0.20 2.32 0.05 (1.65) 1.95 0.17 1.72 2.03 0.22 1.76 0.10 (1.28) 1.75 0.19 1.53 1.85 0.19 1.63 0.20 (0.84) 0.93 0.19 0.73 0.92 0.24 0.66 0.40 (0.25) 0.40 0.14 0.26 0.33 0.16 0.16 Table 31. Reliability indices for calibration of foundation failure - clay subsoil - deep water - no model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 3.33 0.71 2.65 3.19 0.58 2.58 0.05 (1.65) 2.26 0.11 2.07 2.39 0.21 2.09 0.10 (1.28) 1.31 0.20 0.99 2.10 0.26 1.76 0.20 (0.84) 0.51 0.16 0.29 1.13 0.24 0.87 0.40 (0.25) 0.35 0.16 0.16 0.37 0.19 0.18 Table 32. Reliability indices for calibration of foundation failure - clay subsoil deep water - model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index. 45

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 1.98 0.04 1.94 1.91 0.08 1.81 0.05 (1.65) 1.53 0.03 1.50 1.50 0.02 1.48 0.10 (1.28) 1.19 0.06 1.13 1.28 0.05 1.24 0.20 (0.84) 0.88 0.02 0.86 1.00 0.05 0.93 0.40 (0.25) 0.55 0.02 0.54 0.64 0.02 0.64 Table 33. Reliability indices for calibration of foundation failure - clay subsoil shallow water - no model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ]  ] min E  ]   ] min 0.01 (2.33) 2.55 0.24 2.33 2.59 0.11 2.51 0.05 (1.65) 2.09 0.08 2.01 2.22 0.04 2.20 0.10 (1.28) 1.62 0.07 1.51 1.70 0.04 1.64 0.20 (0.84) 1.07 0.06 1.01 1.19 0.10 1.09 0.40 (0.25) 0.51 0.03 0.48 0.62 0.02 0.59 Table 34. Reliability indices for calibration of foundation failure - clay subsoil - shallow water - model tests performed. E  ]: average reliability index,  ]: standard deviation of reliability index, min : minimum reliability index.

Sliding failure: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.39 2.00 2.28 1.96 0.05 (1.65) 1.74 1.52 1.80 1.62 0.10 (1.28) 1.48 1.31 1.51 1.24 0.20 (0.84) 0.86 0.64 0.86 0.69 0.40 (0.25) 0.29 0.16 0.31 0.15 Table 35. Reliability indices for calibration of sliding failure - deep water - no model tests performed. E  ]: average reliability index, min : minimum reliability index. 46

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.40 2.01 2.31 2.08 0.05 (1.65) 1.91 1.81 1.97 1.82 0.10 (1.28) 1.44 1.28 1.54 1.45 0.20 (0.84) 0.91 0.66 0.85 0.70 0.40 (0.25) 0.65 0.51 0.36 0.18 Table 36. Reliability indices for calibration of sliding failure - deep water - model tests performed. E  ]: average reliability index, min: minimum reliability index.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.47 2.35 2.54 2.49 0.05 (1.65) 1.94 1.91 2.10 2.03 0.10 (1.28) 1.58 1.51 1.70 1.61 0.20 (0.84) 0.84 0.74 0.88 0.84 0.40 (0.25) 0.54 0.49 0.63 0.60 Table 37. Reliability indices for calibration of sliding failure - shallow water - no model tests performed. E  ]: average reliability index, min : minimum reliability index.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.32 2.22 2.19 2.16 0.05 (1.65) 1.95 1.86 1.96 1.86 0.10 (1.28) 1.21 1.19 1.36 1.31 0.20 (0.84) 0.86 0.83 0.96 0.92 0.40 (0.25) 0.38 0.34 0.49 0.44 Table 38. Reliability indices for calibration of sliding failure - shallow water model tests performed. E  ]: average reliability index, min : minimum reliability index. 47

Scour failure: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.33 2.33 2.33 2.33 0.05 (1.65) 1.67 1.67 1.67 1.67 0.10 (1.28) 1.33 1.33 1.33 1.33 0.20 (0.84) 0.83 0.83 0.83 0.83 0.40 (0.25) 0.34 0.33 0.34 0.33 Table 39. Reliability indices for calibration of scour failure for circular roundheads - deep water. E  ]: average reliability index, min: minimum reliability index.

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.33 2.33 2.33 2.33 0.05 (1.65) 1.67 1.67 1.67 1.67 0.10 (1.28) 1.33 1.33 1.33 1.33 0.20 (0.84) 0.83 0.83 0.83 0.83 0.40 (0.25) 0.34 0.34 0.34 0.34 Table 40. Reliability indices for calibration of scour failure for circular roundheads - shallow water. E  ]: average reliability index, min : minimum reliability index.

Armour layer failure: Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.33 2.14 2.25 2.08 0.05 (1.65) 1.73 1.54 1.70 1.54 0.10 (1.28) 1.32 1.50 1.51 1.35 0.20 (0.84) 0.94 0.79 1.03 0.90 0.40 (0.25) 0.27 0.17 0.49 0.40 Table 41. Partial safety factors for armour layer failure - deep water. E  ]: average reliability index, min : minimum reliability index. 48

Z0 HS = 0:05 Z0 HS = 0:2 Pf (t ) E  ] min E  ] min 0.01 (2.33) 2.41 2.38 2.35 2.33 0.05 (1.65) 1.65 1.64 1.66 1.64 0.10 (1.28) 1.17 1.16 1.23 1.22 0.20 (0.84) 0.89 0.88 1.00 1.00 0.40 (0.25) 0.25 0.24 0.46 0.46 Table 42. Partial safety factors for armour layer failure - shallow water. E  ]: average reliability index, min : minimum reliability index.

10.3 Examples of reliability calculations

In this section the reliability indices and -vectors are shown for the case where the design lifetime is TL =50 years, the target probability of failure is Pf =0.1 (reliability index = 1.28), Bilbao waves are used (deep water), no model tests and Z0 HS = 0:05. The statistical data in section 9 are used for the stochastic variables. Geotechnical failure modes with sand subsoil, see section 9.3 : failure mode 1 2 3 4 5 6 rel. index 2.44 1.75 1.82 1.60 2.92 1.84 -vector 1 2 4 5 6 3  -0.47 -0.39 -0.35 -0.37 -0.53 -0.25  0.24 0.24 0.22 0.24 0.24 0.15 T L HS 0.49 0.42 0.38 0.40 0.53 0.27  0.15 0.17 0.16 0.16 0.14 0.06 c -0.25 -0.22 -0.14 -0.11 -0.20 -0.14 ZHS 0.22 0.24 0.22 0.24 0.21 0.06 ZFH 0.50 0.58 0.58 0.63 0.45 0.48 ZFV 0.17 0.17 0.21 0.17 0.19 0.55 ZMH 0.0 0.06 0.17 0.14 0.05 0.23 ZMV 0.0 0.05 0.14 0.12 0.04 0.14 r -0.02 -0.02 -0.03 -0.02 -0.01 0.00 '0r -0.27 -0.32 -0.41 -0.21 -0.18 -0.12 s 0.0 0.0 0.0 -0.01 -0.00 -0.02 '0s 0.0 0.0 0.0 -0.21 -0.04 -0.44 Z 0.00 0.00 0.00 0.00 0.00 -0.06 Table 43. Reliability indices and alpha-vectors for failure modes with sand subsoil. 49

The correlation coecient matrix  is (element ij = Ti 2 1 0:986 0:943 0:932 0:988 1 0:980 0:958 0:964 6 0:986 6 0:980 1 0:958 0:964  = 666 00::943 932 0:958 0:954 1 0:964 4 0:988 0:964 0:922 0:927 1 0:707 0:734 0:764 0:838 0:720

j)

0:707 3 0:734 77 0:764 77 0:838 7 0:720 5 1

The series system reliability index is determined to s = 1:46(Pf = 0:073) It is seen that all reliability indices for the individual failure modes and the systems reliability index are larger than the minimum reliability index (=1.28).

Geotechnical failure modes with clay subsoil, see section 9.4 : failure mode 1 2 3 11 12 13 rel. index 2.53 1.88 1.99 1.92 2.53 2.41 -vector 1 2 4 3 8 9  -0.48 -0.40 -0.36 -0.30 -0.46 -0.35  0.24 0.24 0.22 0.19 0.24 0.20 HSTL 0.50 0.43 0.39 0.33 0.48 0.37  0.15 0.17 0.16 0.12 0.14 0.16 c -0.24 -0.22 -0.14 -0.06 -0.08 -0.07 ZHS 0.22 0.24 0.22 0.20 0.22 0.20 ZFH 0.49 0.57 0.56 0.53 0.55 0.55 ZFV 0.17 0.18 0.22 0.10 0.16 0.20 ZMH 0.0 0.06 0.16 0.08 0.12 0.27 ZMV 0.0 0.05 0.15 0.07 0.11 0.19 r -0.02 -0.02 -0.02 -0.01 -0.00 0.00 0 'r -0.27 -0.32 -0.40 -0.08 -0.05 -0.06 Ucu 0.0 0.0 0.0 -0.63 -0.26 -0.41 Z 0.00 0.00 0.00 0.00 0.00 0.00 Table 44. Reliability indices and alpha-vectors for failure modes with clay subsoil. The correlation coecient matrix  is (element ij = Ti 2 1 0:986 0:876 0:727 0:887 0 : 986 1 0:917 0:742 0:880 6 6 0:917 1 0:707 0:825  = 666 00::876 727 0:742 0:707 1 0:893 4 0:887 0:880 0:825 0:893 1 0:805 0:833 0:827 0:942 0:954

j)

0:805 3 0:833 77 0:827 77 0:942 75 0:954 1 50

The series system reliability index is determined to s = 1:63(Pf = 0:052) It is seen that all reliability indices for the individual failure modes and the systems reliability index are larger than the minimum reliability index (=1.28).

Sliding failure, see section 9.1 : failure mode sliding Takayami reliability index 1.41 -vector  -0.34  0.23 T L HS 0.38  0.15 c -0.24 ZHS 0.22 ZFH 0.49 ZFV 0.17 ZMH 0.0 ZMV 0.0 r -0.02 '0r -0.27 Ucu 0.0 Z 0.00 Table 45. Reliability index and alpha-vector for sliding failure mode. It is seen that the reliability index is larger than the minimum reliability index (=1.28).

11. References

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formula. Delft Hydraulics Report H 1903, MAST II contract MAS2-CT920047, 1994.

54