Wave loads on rubble mound breakwater crown walls - …hera.ugr.es/doi/15000345.pdf · Wave loads...

Ž .Coastal Engineering 37 1999 149–174www.elsevier.comrlocatercoastaleng

Wave loads on rubble mound breakwatercrown walls

Francisco L. Martin a,), Miguel A. Losada b, Raul Medina a

a Ocean and Coastal Research Group, UniÕersidad de Cantabria, AÕda. de los Castros srn,39005 Santander, Spain

b UniÕersidad de Granada, ETSI de Caminos, C.y.P, Campus de la Cartuja srn, 18071 Granada, Spain

Received 10 March 1997; received in revised form 11 February 1999; accepted 18 February 1999

Abstract

Crown walls are primarily built to reduce wave overtopping of mound breakwaters. Severalmethods have been proposed to calculate wave loads on the crown wall, e.g., Iribarren and

w Ž . xNogales Iribarren, R., Nogales, C., 1964. Obras Marıtimas. Dossat Ed. , Madrid, 376 pp. ,´wJensen Jensen, O.J., 1984. A Monograph on Rubble Mound Breakwaters. Danish Hydraulicx wInstitute and Gunbak and Gokce Gunbak, A.R., Gokce, T., 1984. Wave screen stability of¨ ¨ ¨ ¨

rubble-mound breakwaters. International Symposium of Maritime Structures in the MediterraneanxSea. Athens, Greece, pp. 2.99–2.112 . In this paper, a new method based on those previous

results, and on further experimental work, using monochromatic waves, is presented. Theapplication of the new method requires waves breaking on the armour layer; i.e., only brokenwaves will reach the crown wall. The method is extended to irregular waves via the hypothesis of

wequivalence introduced by Saville Saville, T., 1962. An approximation of the wave run-upxfrequency distribution. Proc. 8th International Conference on Coastal Engineering, Mexico City

and is applied to the crown walls of Gijon and Bilbao breakwaters in Spain. The comparison of´the probability force distributions obtained by the present method to that measured by Burcharth et

wal. Burcharth, H.F., Frigaard, P., Berenguer, J.M., Gonzalez, B., Uzcanga, J., Villanueva, J.,Ž .1995. Design of the Ciervana breakwater, Bilbao. In: T. Telford Ed. , Proc. 4th Coastal Structures

x Ž .and Breakwaters, Chap. 3. Institution of Civil Engineers and Jensen 1984 is relatively good.q 1999 Elsevier Science B.V. All rights reserved.

Keywords: Wave loads; Crown walls; Mound breakwaters

) Corresponding author. Fax: q34-42-20-18-60; E-mail: [email protected]

0378-3839r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0378-3839 99 00019-8

( )F.L. Martin et al.rCoastal Engineering 37 1999 149–174150

1. Introduction

In Mediterranean countries, mound breakwaters are often built with a concreteparapet resting on the mound layer, and being partially protected by the armour layer. Inengineering practice, this parapet is known as a crown wall, wave wall, wave screen,etc.

Although the primary function of the crown wall is to reduce wave overtopping, thereŽ .are several reasons for topping the breakwater with a crown wall, e.g., i protection of

Ž .breakwater rear slope if the breakwater is overtopped, ii facilitation of some construc-Ž .tion procedures, and iii reduction of required volume of quarry material and thus

reduction of construction costs, etc.There are a few methods for the calculation of wave forces on crown walls: Iribarren

Ž . Ž . Ž .and Nogales 1964 , Jensen 1984 and Gunbak and Gokce 1984 are some. However, it¨ ¨is known that the first method is pessimistic, yielding conservative design. The second isnot reliable since the influence of wave period is not represented adequately, theinfluence of the armour geometry in reducing wave loading has not been addressed and,

Žtherefore, calculated wave forces deviate from measurements up to "30% Bradbury et. Ž .al., 1988 . Moreover, Pedersen and Burcharth 1992 tried to verify Jensen’s parameteri-

sation by using experimental measurements from different authors finding a large scatterin the results. The third method is difficult to apply for design purposes. In this paper anew semi-empirical method, based on these previous investigations and on additionalexperiments using monochromatic waves, is proposed.

First, the crown wall problem is discussed from a design point of view. Next, theformulation of the wave pressure on a vertical wall induced by broken waves ispresented. After the introduction of the experimental results, the new method is extended

Ž .to irregular waves via the hypothesis of equivalence introduced by Saville 1962 andŽ .empirically proven by Bruun and Gunbak 1978 for run-up on rough permeable slopes.¨

Finally, the method is applied to actual breakwaters and the results are compared toŽ . Ž .empirical data from Burcharth et al. 1995 and Jensen 1984 .

2. Definition of the problem

2.1. The crown wall problem

In this section the main factors involved in the design of a crown wall are discussed.Moreover, the physical background for the derivation of the present method is given.

Ž .The procedure for calculating a crown wall usually includes the following steps: iŽ .The rate of wave overtopping determines the crest level of the crown wall. ii The

construction procedure and costs governs the crown wall foundation level. And finally,Ž .iii a stability analysis determines the width and the other dimensions of the crown wall.

If the upper berm of the armour layer is very low, the crown wall has to withstandmost of the wave actions, including those of wave breaking at the wall. Traditionally,this type of structure is denoted ‘composite breakwater’. On the other hand, if the bermis higher than the maximum wave run-up level, then the design is not dominated by

( )F.L. Martin et al.rCoastal Engineering 37 1999 149–174 151

wave actions and its overall dimensions are essentially dictated by functional require-ments. Among these extreme cases, there are several alternatives ranging from high

Ž .berm and small crown wall to see Hamilton and Hall, 1992 low berm and large crownwall.

A very convenient solution is to build the upper berm high enough so that wavebreaking always occurs on the armour layer; i.e., the crown wall will have to withstandonly the pressures induced by broken waves. From an engineering point of view, the

Ž .crown wall problem may be described as follows Fig. 1 :

Žto determine crown wall geometrical dimensions crest elevation, foundation level.and width for a given design water level and wave characteristics as a function of the

height and width of the armour layer upper berm. These dimensions must satisfy thefunctional requirements safely and economically.

Ž .To solve this problem, it is necessary to define: 1 the geometry of the armour layerŽ .which guarantees wave breaking onto the slope, and 2 the pressure distribution of

broken waves on a vertical wall, including uplift pressure. Next, the stability of theupright section has to be verified.

2.2. WaÕe breaking on the slope of rubble mound breakwaters

Descriptions of pressure distribution when waves are impinging on vertical structuresŽ .may be found in several papers. Nagai 1973 analysed wave pressure on structures

induced by monochromatic standing waves, partially standing waves and breaking orbroken waves. For non-breaking waves, the main feature of the time pressure distribu-tion is the occurrence of a symmetrical double peak around the wave crest.

Fig. 2 shows the time evolution of the wave pressure on a vertical wall, underdifferent wave steepnesses. For waves with slight steepness reaching the wall, thepressure–time series induced by the standing wave show a sinusoidal shape. Increasingthe wave steepness and keeping the wave period constant, the peak pressure at thebottom of the wall fluctuates with twice the wave frequency, Fig. 2a. As the wavesteepness is further increased, the fluctuation expands up to the water surface. The

Fig. 1. Overall dimensions of the crown wall problem.


ŽFig. 2. Time evolution of wave pressure distribution on a vertical wall under increasing wave steepness after.Losada et al., 1995 .

double peak induced by the standing wave system is symmetric, Fig. 2b. The maximumwave pressure is always around the still water level. Further increasing of the wavesteepness, being close to breaking conditions, the double peak of the pressure–timecurve becomes asymmetric, with the former being shorter and higher, Fig. 2c. Oumeraci

Ž .et al. 1993 pointed out that the asymmetry of the double peak indicates that a transitionŽ .from a standing wave to a breaking wave system is taking place. Grilli et al. 1992

described the flow velocities and accelerations for waves close to the breaking condi-tions.

When an incident wave breaks on the wall, the first peak may increase extraordinarilyŽ .and may even split into two peaks with a very short duration, Fig. 2d. Bagnold 1939

called it Shock Pressure. This pressure and the effect on the structure stability has beenŽ . Ž . Ž .studied by Chan and Melville 1988 , Oumeraci et al. 1991 , Peregrine 1994 and

Ž .Topliss 1994 . The subsequent peak, denoted Secondary Pressure or Reflecting PressureŽ .Topliss, 1994 has a relatively slow time variation, and a larger duration than the shockpressure, Fig. 2d.

When a broken wave hits the wall, the double peak pattern of the time pressuredistribution is still apparent. Their relative magnitude and duration depend on thedistance between the breaking point and the hit wall, Fig. 2e.

For the cases where the wave does not break directly on the wall, Fig. 2a, b and c,there are several theoretical solutions which provide the pressure and forces on the

Ž .structure see Fenton, 1985 for references . For the cases where impact forces occur


ŽFig. 2d, no theoretical approximation is valid and only impulse methods Cooker. Žand Peregrine, 1990; Losada et al., 1995 or empirical approximations Nagai, 1973;

Goda, 1985, revisited by Takahashi et al., 1992; Tanimoto and Takahashi, 1994;.Oumeraci and Kortenhaus, 1997 are available.

When the wave impinges the wall after breaking, empirical methods for a bore hittingŽ .a wall Ramsden and Raichlen, 1990 may be applied. In the case of a crown wall, the

wave breaks on the armour layer, the wave pressure distribution on the crown wall isproduced by the broken wave, and its characteristics depend on the wave evolution afterbreaking.

For the application of the forthcoming equations it is necessary that the waves hit thecrown wall as broken waves. If the wave does not break on the slope and it can breakonto the crown wall, impulsive forces can occur which are not taken into account in thepresent method.

The most frequently used breakwater slopes are in the range 1.5-cotan b-2.5. Forrough seas, large wave height and period, or swell conditions, large wave period andmoderate wave height, this range of slopes produces essentially collapsing wavebreaking. It is well known that waves under swell conditions occur in groups and largewaves are followed by other large waves with similar characteristics. Generally, for long

Ž .period swell waves )15 s under collapsing wave breaking conditions, the incomingwave generates minor interaction with the run-down of the previous one. Anothercharacteristic of this type of breaker is that wave breaking always occurs around theSWL.

Thus, the waves will hit the crown wall as broken waves if the slope extends from theŽ .order of the wave height Ac)0.8 to 0.9H, see Fig. 1 , measured vertically from the

SWL. In this case, the wave breaks on the slope, and hits the crown wall during therun-up process. This is a common practice since, in Mediterranean countries, most of thecrown walls built were designed following Iribarren recommendations and thus theyhave the level of the upper berm of the armour layer around AcsH.

3. Description of model tests

Scale model tests were conducted in the 70 m long, 2 m wide, 2 m high wave flumeat the Ocean and Coastal Engineering Lab at the University of Cantabria. The test modelŽ .Fig. 3 consisted of a 1r90 scale section of the Prıncipe de Asturias breakwater at Port´

Ž .of Gijon Spain .´The Principe de Asturias breakwater is the main protective structure of El Musel Port,

located at Gijon harbour in the North of Spain. The crown wall base level is 0.0 m over´Ž . Žthe low tide level zero datum , the level of the rubble berm is q13.5 m q12.2 m

.before 1995 and the level of the crown wall top is q18.35 m. The width of the wall isFs18.72 m and the berm width is Bs3.75 m, which means that the berm is built ofone unit of 120 tons. The armour layer slope is 1:1.5. The water depth at the breakwatertoe is 21.0 m at LLWL, and the maximum tidal range in the zone is 4.5 m.

In the tests, the water depth was set to correspond to high tide level in the prototypeŽ . Ž . Ž .q4.0 m . Monochromatic waves 36 tests and random waves 15 tests were generated


Fig.

3.G

ijon

brea

kwat

ercr

oss-

sect

ion.

´


by a piston-type wavemaker. The free surface in front of the structure was measured bythree capacitance wave gauges and a reflection analysis of the free surface time serieswas performed. By using this technique it is possible to obtain the incident and thereflected wave time series. The transmitted wave height was measured by one freesurface gauge located 1 m from the lee side toe of the breakwater. Four strain-gaugetype pressure gauges were installed in the crown wall basement while eight gauges werefixed along the vertical structure front face. Pressures at the front face and under thecrown wall were integrated by a rectangular method to obtain the forces induced bywaves hitting the structure. The logging data rate was 120 Hz.

One of the main targets of the tests was to identify and quantify the protective effectof the berm on the resulting pressures. Therefore, three berm widths were tested,corresponding to the length of 1 mound unit, 2 units and 3 units. Two types of armourwere used corresponding to 90 and 120 ton blocks in the prototype.

4. Semi-empirical procedures for monochromatic waves

In this section, the proposed method to calculate wave-induced pressures on crownŽ .walls is introduced. The method allows the calculation of i wave pressure distribution

Ž . Ž . Ž .on the crown wall front face Section 4.1 and ii uplift pressures Section 4.2 .ŽSince a single wave generates two peaks of pressure pressure pattern described in

.Section 4.1.1 , there are two loading cases for each of the previous pressure distributionsŽ .front and uplift called dynamic and reflecting loads. Therefore, Sections 4.1 and 4.2

Ž .have three sub-sections: i observed characteristics, where the overall patterns of theŽ .pressure distributions are described; ii first peak: dynamic pressures, where the

Ž .methodology to calculate dynamic pressure distributions is given; and iii second peak:Ž .reflecting pressures, similar to ii for the reflecting pressure case.

In Section 4.3 the method is verified for monochromatic waves by comparison ofcalculated forces to empirical measurements. Finally, a practical application of themethod is shown and the results from the proposed method and from other calculatingmethods are compared.

4.1. Horizontal pressure distribution at crown wall

4.1.1. ObserÕed characteristicsFrom the results of the experimental study conducted at the Ocean and Coastal

ŽEngineering Laboratory of the University of Cantabria, Losada et al., 1995; Martin,.1995 , it can be concluded that when the wave impinges the crown wall after breaking in

the armour slope, the first peak is generated during the abrupt change of direction of thebore front due to the crown wall, while the second peak occurs after the instant ofmaximum run-up and is related to the water mass down-rushing the wall.

The distinct nature of these peaks, denoted by A and B, may be observed in Fig. 4where a force time series recorded in the lab is given and the pressure distributions atthe wall at the times A and B, are shown. For the pressure distribution produced by the

Ž .first peak A , dynamic pressure, two regions may be distinguished: the upper one, not


Fig. 4. Experimental dynamic and reflecting pressure distributions for broken waves.

protected by the rubble-mound layer, and the lower one, protected by the rubble-moundlayer. In both regions, the pressure is almost constant, but higher in the upper region.

Ž .The pressure profile due to the secondary peak B , reflecting pressure, linearlyincreases downwards.

Ž .The crown wall and the armour layer are functionally-dependent elements: i Thehydrodynamics of the running-up water is modified by the presence of the crown wall.

Ž .This modification of the flux affects the resulting forces on the armour units; ii theŽ .armour layer characteristics slope, permeability, roughness, berm width, etc. determine

the characteristics of the run-up water tongue which hits the wall. Therefore, it isobvious that the design of both structures must be related. In the proposed method, the

Ž .relation between the armour layers geometry, porosity, etc. and the resulting pressureson the crown wall is taken into account.

Following other authors, a semi-empirical approach to formulate the pressure distri-bution during the first and second peak is developed. The crown wall is partiallyprotected by the armour layer, Fig. 1. Thus, for the pressure distribution, two regionsmay be distinguished. In the upper region, run-up water hits the wall directly. In thelower region, protected by the armour units, waves reach the wall after flowing throughthe porous armour layer. The lower region extends from the wall foundation level,zsw , up to the berm level, zsAc, where z is the vertical coordinate measured fromf

the still water level, positive upwards. The upper region extends from zsAc, up to thewall crest level, zsw . In Fig. 5 the proposed pressure distributions are schematicallyc

summarised.

4.1.2. First peak: dynamic pressuresŽ . Ž . Ž .Following Nagai 1973 , Jensen 1984 and Tanimoto and Takahashi 1994 it may

be established that the dynamic pressure, P at the berm crest level zsAc, in theS0

non-protected region, is linearly related to the water tongue thickness at that level,Ž . Ž .S zsAc sS , and may be evaluated by the following expression Martin, 1995 :0

P sar g S 1Ž .S0 0


Fig. 5. Schematic representation of the pressure distribution on the wave screen.

where a is a non-dimensional parameter that will be analysed further in this paper, r isthe water density and g is the gravity acceleration. In Fig. 5, the geometrical definitionof S is shown.0

Because of the experimental evidence of the constancy of the dynamic pressures inŽ . Ž .positive z-direction Jensen, 1984; Martin, 1995 the pressures profile P z , can bed

represented by,

P z sP for z)A . 2Ž . Ž .d S0 c

Moreover, the dynamic pressures in the lower region are also almost verticallyŽ .constant, Fig. 4, and it was experimentally verified Martin, 1995 that it can be related

Ž .to P through an empirical parameter l , analysed further in this section, where l isS0

smaller than one,P z slP for w -z-A . 3Ž . Ž .d S0 f c

The determination of S and a is based on the experimental evidence that the0

dynamic and reflecting pressures occur shortly before and shortly after the instant ofmaximum run-up on the wall, respectively. Assuming that:1. S and a may be evaluated during the occurrence of the maximum run-up, and that0

2. for collapsing breakers, the water tongue kinematics and dynamics on the edge of theberm during the maximum run-up event is approximately the same as for slopes withor without a crown wall.Then, it may be concluded that the values of S and a depend only on the maximum0

run-up, Ru, on an infinite slope and on the water particle velocity at the jet tip.Ž .Losada and Gimenez-Curto 1981 , based on experimental work under monochro-

matic waves and normal incidence, proposed the following expression for Ru on aninfinite slope:

RuŽ .Bu IrsAu 1ye 4Ž .

H


Ž Ž ..where, Au and Bu Fig. 6, after Losada 1992 are empirical coefficients, which dependon the type of armour unit and Ir is the Iribarren number defined by,

tan bIrs 5Ž .

H( L0

where b is the slope angle, H is the local wave height and L is the deep water0

wavelength.Ž .In order to verify the applicability of Eq. 4 , an experimental evaluation of Ru on a

mound breakwater with a crown wall was carried out. The armour layer was built ofŽ .concrete parallelepipedic blocks a=a=1.25a where as3.8 cm with a 1:1.5 slope

and the berm was built of 2 units. Because of experimental uncertainty, the same testŽ .same structure and same waves was repeated three times. In Fig. 7 the best fit curve tothe values of Ru is shown. The best fit values for Au and Bu are 1.2 and y0.7,respectively. These values are very close to the values proposed by Losada and DesireŽ . Ž .1984 Aus1.2, Busy0.65, for parallelepipedic blocks , giving support to assump-

Ž .tion 2 .Ž . Ž .Gunbak and Gokce 1984 and Yamamoto and Horikawa 1992 showed that the¨ ¨

water surface at the instant of maximum run-up may be approximated by a straight line

Fig. 6. Au and Bu coefficients for run-up calculation as a function of armour layer porosity.


Fig. 7. Comparison of measured and calculated run-up.

Ž .as indicated in Fig. 5 . Then, the water tongue thickness, S, of the impinging bore at theŽ .level z may be evaluated by:

zS z sS 1y 6Ž . Ž .w ž /Ru

where S is the water tongue thickness at the SWL. For 1.5-cotan b-2.5, andwŽ .following the theoretical and experimental work of Yamamoto and Horikawa 1992 , it

may be assumed without significant error that the water tongue thickness at the SWL ison the same order of the wave height, S ;H. Hence, the thickness, S , of the bore atw 0

the berm crest, zsAc, is given by,

AcS sH 1y . 7Ž .0 ž /Ru

Assuming no energy loss due to friction above Ac level, the alongslope bore tipcelerity can be calculated from gravity acceleration as:

(C z s 2 Ruyz g . 8Ž . Ž . Ž .b

Therefore, the horizontal component of the bore tip celerity, C , at any levelbxŽ .zGAc, may be approximated by the following expression Martin, 1995 :

(C z s 2 Ruyz g cos b . 9Ž . Ž . Ž .bx

Next, the averaged horizontal velocity of the water particles near the bore front, Õ ,xŽcan be considered to be equal to the celerity of the tip see Ramsden and Raichlen,

. Ž . Ž .1990 . Thus, Õ at the berm crest level zsAc can be obtained by Eq. 9 .xŽ . Ž .For the calculation of a , results from Cross 1967 and Wiegel 1970 for the

Ž .maximum pressure induced by a bore hitting a vertical wall p are considered:max

C 2bx2P srC N . 10Ž .max f f 2


Thus, the maximum pressure is defined as C N 2 times the stagnation pressure due tof f

the bore celerity, where N is a dimensionless parameter defined as:f

CbN s 11Ž .f 'gS

where S is the bore thickness, and C is a coefficient that depends only on the bore frontfŽ . Ž . Ž . Ž .geometry Cumberbatch, 1960 . Using Eqs. 7 and 8 into Eq. 11 for zsAc, the

resulting value for N isf

RuN s 2 . 12Ž .(f H

Ž . Ž . Ž . Ž . Ž .By using Eqs. 1 , 7 , 9 , 10 and 12 , for zsAc, the dimensionless parameter a

is given by:

2Ruas2C cos b . 13Ž .f H

Ž . Ž .Note that Eq. 13 is valid if the horizontal component of the bore celerity C isbx

not affected by the berm width. This assumption has been experimentally verified by theŽ .authors for berm widths up to BrLs0.1 Martin et al., 1998 .

C represents the short-duration pressure oscillations induced by the impact of thefŽ .water tongue front to the vertical wall. Cumberbatch 1960 showed that it depends only

Ž .on the bore wedge angle, Q . For wedge angles of 22.58 and 458, Cumberbatch 1960Ž .found C to be 1.4 and 2.1, respectively. Cross 1967 computed the coefficient C forf f

wedge angles between 08 and 758 and found the following relation agreed with hisresults:

1.2C s1q tan u . 14Ž . Ž .f

Fig. 8 shows the values of C calculated from the maximum pressures P measuredf S0Ž . Ž . Ž .in the tests described in Section 3, using Eqs. 1 , 7 and 13 . If C is calculated fromf

Ž .pressures of which the persistence is larger than Tr100 instead of maximum pressures ,the dispersion of the calculated C is smaller than the dispersion shown in Fig. 8, andf

the best fit value for C is 1.0.f

To define C for design purposes the dynamic response of the structure must be takenfŽ .into account. For small structures low inertia andror rigid foundations, maximum

Ž .pressures must be taken as the design pressures and a value of C s1.45 Qs278 isfŽ .proposed see Fig. 8 . Finally, a can be calculated as:

2Ruas2.9 cos b . 15Ž .

H

Ž .For large structures large inertia andror elastic foundations, C s1.0 could repre-f

sent better the equivalent static loading situation for design purposes. Further researchon dynamic response of crown walls and on scale effects is required for betterassessment of the design value of C .f


Ž . Ž . Ž .Fig. 8. Values of C calculated from P measured and Eqs. 1 , 7 and 13 .f S0

The parameter l was evaluated experimentally from monochromatic wave tests in theflume. A description of the experimental set up is given in Section 2. The experimentalvalues obtained for l are shown in Fig. 9. The range of the measured l valuesŽ . Ž .0.25-l-0.65 , is in agreement with those given by Jensen 1984 and Gunbak and¨

Ž . ŽGokce 1984 . Notice, that the experimental wave steepness range is 0.03-HrL at¨.breakwater toe -0.075.

The best fit curve to the empirical results is:

ls0.8ey10.9Br L . 16Ž .

4.1.3. Second peak: reflecting pressuresThe second peak occurs shortly after the occurrence of the maximum wave run-up.

The horizontal and vertical velocities at this instant are very small as are the accelera-

Ž .Fig. 9. Experimental variation of l vs. BrL H and L measured at the toe of the breakwater .


tions. Thus, the resulting pressure field around the crown wall may be considered almosthydrostatic. Furthermore, the pressure distribution recorded on the wall, at the instant ofthe occurrence of the peak, is continuous along the protected and unprotected regions.Then, the reflecting pressure, P , may be evaluated by the following linear expression:r

P z smr g S qAcyz for w -z-AcqS 17Ž . Ž . Ž .r 0 f 0

where the dimensionless parameter, mF1, was evaluated experimentally frommonochromatic wave tests. From the tests, it is clear that m depends on the wave

Ž .steepness HrL and on the non-dimensional berm width, Brle. The parameter le is theequivalent size of the rubble units, and is calculated by,

3 Wles 18Ž .( gr

where W and g are the total weight and the specific weight of the armour unit,r

respectively.The experimental values for m are shown in Fig. 10. For wave steepness, HrL-0.02,

Ž . Ž .the reflecting pressures are r gz ms1 , decaying to 0.5 r gz ms0.5 , approximately,for HrL;0.04. By increasing the wave steepness to 0.075, an asymptotic trend isobtained, which depends on the number of units building the berm. m takes values of0.45, 0.37 and 0.3 for one, two and three armour units on the berm, respectively.

ŽFig. 10. Experimental variation of m vs. HrL and Brle as a parameter H and L measured at the toe of the.breakwater .


Table 1

Brle a b c

1 0.446 0.068 259.02 0.362 0.069 357.13 0.296 0.073 383.1

The trend of experimental values for m can be well represented by an exponentialcurve of the type msaecŽHr Lyb.2

. The best fit parameters for these curves are shown inTable 1:

The experiments in which l and m are determined were performed with large armourŽ .units 1200 kN parallelepipedic blocks in the prototype and a large porous core. Further

experiments are needed to analyse in more detail the effect of unit size and corepermeability on these parameters. Meanwhile, the proposed values of l and m can beused as a first approach for design purposes.

4.2. Uplift pressure distribution at crown wall

4.2.1. ObserÕed characteristicsŽ .It is common practice in maritime engineering Iribarren and Nogales, 1964 to

Ž .consider a linear variation of wave pressure under the crown wall. Losada et al. 1993 ,applying linear wave theory, obtained a parabolic pressure distribution under animpermeable crown wall resting on a porous media, with porosity ranging from 20% to40%. However, their findings do not differ significantly from the linear trend. In thispaper, the linear law is assumed. To define this linear distribution, the pressures at thetoe and at the heel of the crown wall were experimentally recorded with the sameexperimental set-up described in Section 3.

4.2.2. First peak: dynamic pressuresDynamic pressure beneath the seaward edge of the structure is approximately equal to

lP . At the heel dynamic pressures are negligible.S0

4.2.3. Second peak: reflecting pressuresReflecting pressure beneath the seaward edge of the structure is equal to the pressure

at the front. Reflecting pressures at the heel are only significant if the crown wall isfounded below the transmitted wave amplitude. Therefore, heel pressures depend on thewave transmission process.

Ž .Thus, the following values are adopted see Fig. 5 :Ø Seaward edge:

Ž .( dynamic pressureslP zswS0 fŽ .( reflecting pressuresP zsw sPr f re

Ø Heel:( dynamic pressure negligible, P s0ra

Ž .( reflecting pressuresP , from Losada et al. 1993 .ra


Ž .Fig. 11. Reflecting pressure at the heel vs. FrL L calculated at the toe of the breakwater . Dots areŽ .experimental data from Gijon prototype measured on 10 February 1996 ns0.4 approximately .´

Fig. 11 shows the reflecting pressure at the heel, P , non-dimensionalised with thera

reflecting pressure at the seaward edge, P , vs. FrL, where F is the crown wall width.re

Each plotted curve corresponds to a different mound porosity, n. The numerical modelŽ .employed to generate the curves Losada et al., 1993 was applied for a single overall

porosity of the rubble mound. For design purposes the porosity selected must representthe porosity of the material on which the crown wall is founded.

The proposed curves neither depends on the wave height nor on the water depth. Thismethod for calculating uplift pressures must be regarded as a first engineering approach

Ž . Ž .to the problem since: i lines in Fig. 11 were obtained from linear theory, and ii theexperimental values were obtained from a low crown wall, where air entrainment is verylow. Additional experiments for high crown walls based on less porous core are requiredto complete the method.

Some results from the pressure gauges placed under Gijon’s breakwater crown wall´Ž . Ž . Ž .prototype are included dots see Fig. 11 . This breakwater is built of an armour of120 tons resting on a core of 90-ton parallelepipedic blocks and the overall porosity isapproximately ns0.4.

4.3. Application and Õerification

4.3.1. Comparison to test dataAs a first verification of the proposed method, experimental pressure and forces on

the scale model of Gijon’s crown wall were compared to their corresponding analytical´values for regular waves and random waves. As an example, Fig. 12a shows the

Žcomparison of maximum forces measured in the lab for regular waves tests described in. Ž .Section 3 and those calculated by the proposed method C s1.45 .f

Experimental forces per linear meter of structure are integrated from measuredpressures on the wall. Furthermore, the values of the forces were obtained as averagevalues of the three highest measured forces in each test for monochromatic waves.


Ž .Fig. 12. a Comparison between experimental and calculated forces. Tests done with monochromatic waves.Ž .b Comparison between experimental and calculated dynamic pressures. Tests done with random waves,calculation done by splitting of single waves by zero-upcrossing.

Ž .In Fig. 12b the dynamic pressures P measured in the lab for random waves areS0

compared to the calculated dynamic pressures for each individual single wave. SingleŽ .waves H,T are identified from measured free surface time series by zero-upcrossing.

The method is applied to each individual wave. P is calculated and compared to theS0


correspondent vertically averaged measured dynamic pressure in the unprotected regionŽ .of the crown wall. The averaging procedure is as follows: i the pressure time series are

Ž .integrated by a rectangular method to force time series, ii the maximum dynamic forceŽ .and maximum dynamic force time are identified for each individual wave, iii the

pressure measurements at this time in the unprotected zone is integrated by a rectangularŽ . Ž .method for each individual wave , iv the resulting force in the unprotected region is

divided by the unprotected region height to give a vertically-averaged pressure in thiszone. The comparison gives relatively good results.

The crown wall of Gijon Breakwater has been instrumented to supply full-scale wave´pressure records under storm conditions. This information will help validate the presentmethod when severe storm data become available. Some preliminary results of themeasurements performed for uplift pressures were presented in Section 4.2.3.

4.3.2. Comparison to other methodsIn this section, forces on the crown wall of the Gijon Breakwater calculated by´

Ž . Ž .Iribarren and Nogales 1964 , Gunbak and Gokce 1984 and the present method are¨ ¨compared. An individual wave height of Hs12.0 m, a period of Ts16.0 s and a high

Ž .tide level q4.0 over LLWL is adopted. The proposed method is applied with theŽ Ž .following values of the main parameters: Rus13.2 m calculated by Eq. 4 for

. Ž .Aus1.2 and Busy0.65 , C s1.45, BrLs0.016, ls0.66, Fig. 9 , HrLs0.052,fŽ . Ž .Brles1, ms0.48, Fig. 10 , FrLs0.08, ns0.4, P rP s0.375 Fig. 11 andra re

Acs9.5 m. A value of 0.6 was assumed for the friction coefficient between the crownwall and the core material. The weight of the crown wall when the SWL is 4.0 m aboveLLWL is 3460 kNrm. The resulting value for S is 3.37 m and for a is 2.43.0

In Table 2, the net horizontal force per unit length as well as the uplift forces aregiven. The sliding and overturning safety coefficients were investigated and for all casesthe lower safety factors occurred for the sliding of the crown wall. The safety conditionsfor sliding calculated from experiments on the wave flume are also presented in Table 2.

The pressure distribution in the reflecting pressure condition is linear, growingdownwards, thus maximum loads occur at the bottom of the crown wall. Therefore thelower the crown wall foundation is, the larger the pressures at the bottom and the upliftpressures are. Then, the large uplift force in the reflecting pressure condition in this

Table 2

Method Horizontal force Uplift force Calculated safety Lab. safetyŽ . Ž .kNrm kNrm coefficient coefficient

Iribarren 2285 1919 0.41 –Gunbak 1231.5 1259 1.08 –¨Present methodDynamic pressure 1044 512 1.69 1.72conditionReflecting pressure 695 1055 2.08 2.01condition


particular case is due to the low foundation level of the Gijon breakwater’s crown wall´Ž .4 m below SWL .

By comparing the calculated and measured forces and safety coefficients it can beconcluded that the Iribarren and Nogales method and the Gunbak method are pessimistic¨when analysing the static stability of the crown wall. Wave records over the last 10years show that storms with significant wave heights greater than 8 m and maximumwave heights greater than 12 m have attacked the breakwater several times. After avisual inspection of the breakwater it can be stated that the crown wall along thebreakwater trunk has not experienced displacement.

5. Force distribution for random waves

The extension of the previous method to irregular waves is based on the followingconsiderations:1. The reference parameter for the application of the formulae for calculating the

Ž . Ž . Ž .dynamic pressures, see Eqs. 1 , 7 and 15 , is the run-up on a straight slope.Ž .2. The hypothesis of equivalence introduced by Saville 1962 can be applied for

computing run-up distribution on a rough, permeable slope.Given a sea state defined by the significant wave height, Hs, the zero crossing

averaged wave period, Tz, the value of the total forces on the wall generated by thedynamic and reflecting pressures may be considered random variables which can give

Ž .different values for each individual wave H,T of the sea state.The hypothesis of equivalence proposes that the distribution function of a random

variable may be obtained by assigning to each individual irregular wave the samephenomenon value which would be produced by a periodic train of the same waveheight and period.

Ž .This hypothesis was empirically proven by Bruun and Johannesson 1977 and BruunŽ .and Gunbak 1978 for run-up on rough, permeable slopes.¨

Thus, the distribution function of the forces on a crown wall, under a given sea state,may be obtained by assigning the same value of the force that would be produced by aperiodic wave train of the same height and period, to each individual irregular wave.

It is important to note the statistical nature of this hypothesis. It does not necessarilyimply that each individual wave produces the same forces as the equivalent regularwave, but is less restrictive as it refers to average rather than to individual values.

5.1. WaÕe force distribution

The procedure to compute the distribution function of the forces on the crown wallunder an irregular sea state is as follows.

Ž . Ž .1 Given Hs, Tz, the water depth at the toe of the breakwater h , and the spectralshape parameter, a theoretical TMA spectrum is computed.

Ž .2 From this spectrum a synthetic free surface time series is generated.


Fig. 13. Ciervana breakwater cross-section.

Ž .3 The following breaking criteria are applied to each individual wave:

H-0.142 L tanh 2p drL Miche criterion . 19Ž . Ž . Ž .

Hrd s0.55q0.88 exp y0.012 cot w 20Ž . Ž . Ž .Ž .max

where w is the bed slope angle. Since results will be compared to laboratory data, theŽ Ž ..breaking criterion Eq. 19 is established for waves in the laboratory with mild bottom

Ž .slopes Nelson, 1997 . For steeper slopes andror field conditions there are severalŽ .formulations in the literature e.g., Komar and Gaughan, 1972 .

Ž .4 Waves which break due to limited depth are regenerated with a wave height equalŽ .to half the water depth, upper limit of values proposed by Dally et al. 1985 .

Ž . Ž . Ž . Ž . Ž . Ž .5 Eqs. 1 , 3 , 7 , 15 and 17 and uplift pressures are calculated for eachindividual wave height and, after integration, the total force per unit length of wallproduced by the dynamic and reflecting pressures are obtained.

Ž .6 The force sample data are analysed statistically and the best distribution functionis ascribed to the forces under the dynamic and reflecting pressure conditions.

5.2. Application for random waÕes and its comparison to data from other authors

The described methodology has been applied to two cases previously tested in theŽ .laboratory: Ciervana Breakwater in Bilbao’s Harbour Spain tested by Burcharth et al.

Ž . Ž .1995 and a model test carried out by Jensen 1984 .

Table 3Sea state parameters for Ciervana breakwater

Ž . Ž .Hs m Tp s

8 159 16

10 1711 1912 20


Table 4Simulation parameters

Peak enhancement factor 1.4Number of waves generated 3000

5.2.1. CierÕana breakwaterŽ .The present method is applied to Ciervana Breakwater crown wall Bilbao, Spain ,

following the procedure described in Section 5.1. The need of space to support newdevelopments in Bilbao harbour led to the building of the Ciervana Breakwater toprotect the land reclamation between the inner port and Punta Lucero Breakwater. It was

Ž .built south-east of the famous Punta Lucero Breakwater 150-ton armour units and it ispartially protected by this breakwater from NW storms. Its total length is 3.15 km, and

Ž .the armour layer is built of parallelepipedic 100-ton blocks a=a=1.25a in a 1:2slope. The crown wall base level is q1.5 m over the zero datum, the crest level of the

Ž .rubble berm is q14.0 m and the crown wall top level is q18.0 m see Fig. 13 . Theberm width is Bs9.0 m and the width of the wall is Fs29.0 m. The water depthat the breakwater toe is 26.0 m at LLWL, and the maximum tidal range in the zone is

Ž . Ž .4.5 m. The significant wave heights Hs and peak periods Tp and the characteristicsof the wave simulation for the test are given in Tables 3 and 4, respectively.

Ž .Fig. 14 shows the wave height distributions: i the distribution obtained by theŽ .computation described in Section 5.1, ii the distribution measured in the laboratory

Ž . Ž .Burcharth et al., 1995 and iii the Rayleigh distribution. In Fig. 15 the 0.1%exceedance probability force obtained from the present method is compared to the

Ž .experimental results described by Burcharth et al. 1995 . The agreement is good.

Fig. 14. Wave height distribution computed by the present method, measured in the lab by Burcharth et al.Ž .1995 and Rayleigh distribution.


Ž .Fig. 15. The 0.1% probability horizontal force, F , measured by Burcharth et al. 1995 , and computed fromH

present method.

Moreover, the 0.1% exceedance probability pressure measured at level q1.5 m at thefoot of the crown wall for the case Hss11 m, is 120 kPa while the computed pressure

Ž .is 132 kPa. Finally, from measurements performed by CEDEX Madrid for the caseŽ .Hss11 m, also described by Burcharth et al. 1995 , the centre of application of the

total horizontal forces corresponding to the 0.3% probability is estimated to be between11.5 and 12.5 m above SWL. The present method locates the application point 12.4 mabove SWL.

( )5.2.2. Comparison to the data of Jensen 1984Ž .Jensen 1984 published a monograph for the design of rubble mound breakwaters.

Data on crown wall forces are also included. For one test the author shows the fulldistribution of forces which makes a comparison possible. The cross-section of thebreakwater is shown in Fig. 16. The armour layer is built of rectangular blocks

Ž . Ž .2.9=2.9=4.2 m 82 ton in a 1:2 slope, the berm width is 6.0 m 2 units and the bermheight is Acs10.9 m, the crown wall foundation is w s4.30 m, the crown wall heightf

Ž .Fig. 16. Cross-section of breakwater tested by Jensen 1984 .


Fig. 17. Comparison of Jensen’s experimental results to calculations from the present method.

is 16.7 m. The SWL in the test is at level q5.3 and the peak period is 18 s. Threesignificant wave heights were tested: Hss8, 11 and 14 m.

Ž .For the calculation, a TMA spectrum peak enhancement factors2 is used, follow-Ž .ing the procedure defined in Section 5.1. For the calculation of run-up, Eq. 4 was used

for Aus1.2 and Busy0.7. The comparison of results is shown in Fig. 17. Differ-ences in the maximum values of the forces for Hss14 m could be due to the breakingcriterion used in the method. The agreement is relatively good.

6. Concluding remarks and future work

A new method to calculate forces on crown walls based on previous works and onnew experiments, carried out under monochromatic waves, is presented. The method isextended to irregular waves via the hypothesis of equivalence and checked against two

Ž . Ž . Ž . Ž .sets of lab tests by: i Burcharth et al. 1995 and ii Jensen 1984 . The comparison isfairly good.

The application of the new method requires that the waves break onto the main slope;i.e., only broken waves will reach the crown wall. Under such conditions, the timepressure evolution on the wall has two peaks: the first pressure peak which is usually the

Žlargest and the secondary peak which is smaller but lasts longer. In some cases HrAc.small andror BrL large the reflecting pressure peak can be larger than the dynamic

pressure peak.The magnitude of these peak are related to the armour berm geometry: berm width

Ž . Ž .B and berm height Ac . The proposed dynamic and reflecting pressure distribution


depends on two parameters, l and m, which are experimentally evaluated and dependŽ . Ž .on the relative berm width BrL , local wave steepness HrL and the number of

Ž .armour units constituting the berm nbsBrle .Dynamic pressures are generated during the abrupt change of direction of the bore

front due to the crown wall. Short duration pressure oscillations are produced in thisŽ .instant, which depend on the bore front wedge characteristics parameter C . Thesef

oscillations could be filtered by the dynamic response of the crown wall and itsfoundation. Moreover, scale effects could affect these oscillations due to aeration,saltrfresh water testing, water compressibility, etc. Further research on dynamic re-sponse of crown walls and on scale effects is being carried out by the authors for betterassessment of the design value of C for design purposes.f

Acknowledgements

This research was partially funded by the European Community Research Pro-Ž . Žgramme, MAST III Marine Science and Technology , Project PROVERBS Probabilis-

.tic Design Tools for Vertical Breakwaters under EU contract MAS3-CT95-0041. Thisfinancial support is very much appreciated. The first author wishes to thank the

ŽMinisterio de Educacion y Ciencia for his funding during part of the research F.P.I.´.Research Grant . The authors want to thank Autoridad Portuaria de Gijon for its´

continuous technical support, Prof. C. Vidal for his unconditional help and Prof.Burcharth and Prof. Oumeraci for their very interesting and useful comments in thereview of the paper.

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