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Coastal Engineering 37 Z1999. 149­174 www.elsevier.comrlocatercoastaleng

Wave loads on rubble mound breakwater crown walls

Francisco L. Martin

a

a,)

, Miguel A. Losada b, Raul Medina

a

Ocean and Coastal Research Group, UniÕersidad de Cantabria, AÕda. de los Castros sr n, 39005 Santander, Spain b UniÕersidad de Granada, ETSI de Caminos, C.y.P, Campus de la Cartuja sr n, 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. Several methods have been proposed to calculate wave loads on the crown wall, e.g., Iribarren and Nogales wIribarren, R., Nogales, C., 1964. Obras Maritimas. Dossat ZEd.., Madrid, 376 pp.x, ´ Jensen wJensen, O.J., 1984. A Monograph on Rubble Mound Breakwaters. Danish Hydraulic Institutex and Gunbak and Gokce wGunbak, A.R., Gokce, T., 1984. Wave screen stability of ¨ ¨ ¨ ¨ rubble-mound breakwaters. International Symposium of Maritime Structures in the Mediterranean Sea. Athens, Greece, pp. 2.99­2.112x. In this paper, a new method based on those previous results, and on further experimental work, using monochromatic waves, is presented. The application of the new method requires waves breaking on the armour layer; i.e., only broken waves will reach the crown wall. The method is extended to irregular waves via the hypothesis of equivalence introduced by Saville wSaville, T., 1962. An approximation of the wave run-up frequency distribution. Proc. 8th International Conference on Coastal Engineering, Mexico Cityx 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 al. wBurcharth, H.F., Frigaard, P., Berenguer, J.M., Gonzalez, B., Uzcanga, J., Villanueva, J., 1995. Design of the Ciervana breakwater, Bilbao. In: T. Telford ZEd.., Proc. 4th Coastal Structures and Breakwaters, Chap. 3. Institution of Civil Engineersx and Jensen Z1984. 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. PII: S 0 3 7 8 - 3 8 3 9 Z 9 9 . 0 0 0 1 9 - 8

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1. Introduction In Mediterranean countries, mound breakwaters are often built with a concrete parapet resting on the mound layer, and being partially protected by the armour layer. In engineering 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, there are several reasons for topping the breakwater with a crown wall, e.g., Zi. protection of breakwater rear slope if the breakwater is overtopped, Zii. facilitation of some construction procedures, and Ziii. 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 and Nogales Z1964., Jensen Z1984. and Gunbak and Gokce Z1984. are some. However, it ¨ ¨ is known that the first method is pessimistic, yielding conservative design. The second is not reliable since the influence of wave period is not represented adequately, the influence of the armour geometry in reducing wave loading has not been addressed and, therefore, calculated wave forces deviate from measurements up to "30% ZBradbury et al., 1988.. Moreover, Pedersen and Burcharth Z1992. tried to verify Jensen's parameterisation by using experimental measurements from different authors finding a large scatter in the results. The third method is difficult to apply for design purposes. In this paper a new semi-empirical method, based on these previous investigations and on additional experiments using monochromatic waves, is proposed. First, the crown wall problem is discussed from a design point of view. Next, the formulation of the wave pressure on a vertical wall induced by broken waves is presented. After the introduction of the experimental results, the new method is extended to irregular waves via the hypothesis of equivalence introduced by Saville Z1962. and empirically proven by Bruun and Gunbak Z1978. for run-up on rough permeable slopes. ¨ Finally, the method is applied to actual breakwaters and the results are compared to empirical data from Burcharth et al. Z1995. and Jensen Z1984..

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. The procedure for calculating a crown wall usually includes the following steps: Zi. The rate of wave overtopping determines the crest level of the crown wall. Zii. The construction procedure and costs governs the crown wall foundation level. And finally, Ziii. 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 withstand most 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 berm is higher than the maximum wave run-up level, then the design is not dominated by

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wave actions and its overall dimensions are essentially dictated by functional requirements. Among these extreme cases, there are several alternatives ranging from high berm and small crown wall to Zsee Hamilton and Hall, 1992. low berm and large crown wall. A very convenient solution is to build the upper berm high enough so that wave breaking always occurs on the armour layer; i.e., the crown wall will have to withstand only the pressures induced by broken waves. From an engineering point of view, the crown wall problem may be described as follows ZFig. 1.: to determine crown wall geometrical dimensions Zcrest 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 the functional requirements safely and economically. To solve this problem, it is necessary to define: Z1. the geometry of the armour layer which guarantees wave breaking onto the slope, and Z2. the pressure distribution of broken waves on a vertical wall, including uplift pressure. Next, the stability of the upright 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 structures may be found in several papers. Nagai Z1973. analysed wave pressure on structures induced by monochromatic standing waves, partially standing waves and breaking or broken waves. For non-breaking waves, the main feature of the time pressure distribution 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, under different wave steepnesses. For waves with slight steepness reaching the wall, the pressure­time series induced by the standing wave show a sinusoidal shape. Increasing the wave steepness and keeping the wave period constant, the peak pressure at the bottom of the wall fluctuates with twice the wave frequency, Fig. 2a. As the wave steepness is further increased, the fluctuation expands up to the water surface. The

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

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

double peak induced by the standing wave system is symmetric, Fig. 2b. The maximum wave pressure is always around the still water level. Further increasing of the wave steepness, being close to breaking conditions, the double peak of the pressure­time curve becomes asymmetric, with the former being shorter and higher, Fig. 2c. Oumeraci et al. Z1993. pointed out that the asymmetry of the double peak indicates that a transition from a standing wave to a breaking wave system is taking place. Grilli et al. Z1992. described the flow velocities and accelerations for waves close to the breaking conditions. When an incident wave breaks on the wall, the first peak may increase extraordinarily and may even split into two peaks with a very short duration, Fig. 2d. Bagnold Z1939. called it Shock Pressure. This pressure and the effect on the structure stability has been studied by Chan and Melville Z1988., Oumeraci et al. Z1991., Peregrine Z1994. and Topliss Z1994.. The subsequent peak, denoted Secondary Pressure or Reflecting Pressure ZTopliss, 1994. has a relatively slow time variation, and a larger duration than the shock pressure, Fig. 2d. When a broken wave hits the wall, the double peak pattern of the time pressure distribution is still apparent. Their relative magnitude and duration depend on the distance 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 structure Zsee Fenton, 1985 for references.. For the cases where impact forces occur

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Fig. 2d, no theoretical approximation is valid and only impulse methods ZCooker and Peregrine, 1990; Losada et al., 1995. or empirical approximations ZNagai, 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 hitting a wall ZRamsden 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 is produced by the broken wave, and its characteristics depend on the wave evolution after breaking. For the application of the forthcoming equations it is necessary that the waves hit the crown wall as broken waves. If the wave does not break on the slope and it can break onto the crown wall, impulsive forces can occur which are not taken into account in the present method. The most frequently used breakwater slopes are in the range 1.5 - cotan b - 2.5. For rough seas, large wave height and period, or swell conditions, large wave period and moderate wave height, this range of slopes produces essentially collapsing wave breaking. It is well known that waves under swell conditions occur in groups and large waves are followed by other large waves with similar characteristics. Generally, for long period swell waves Z) 15 s. under collapsing wave breaking conditions, the incoming wave generates minor interaction with the run-down of the previous one. Another characteristic of this type of breaker is that wave breaking always occurs around the SWL. Thus, the waves will hit the crown wall as broken waves if the slope extends from the order of the wave height ZAc ) 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 the run-up process. This is a common practice since, in Mediterranean countries, most of the crown walls built were designed following Iribarren recommendations and thus they have the level of the upper berm of the armour layer around Ac s H.

3. Description of model tests Scale model tests were conducted in the 70 m long, 2 m wide, 2 m high wave flume at the Ocean and Coastal Engineering Lab at the University of Cantabria. The test model ZFig. 3. consisted of a 1r90 scale section of the Principe de Asturias breakwater at Port ´ of Gijon ZSpain.. ´ 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 ´ the low tide level Zzero datum., the level of the rubble berm is q13.5 m Zq12.2 m before 1995. and the level of the crown wall top is q18.35 m. The width of the wall is F s 18.72 m and the berm width is B s 3.75 m, which means that the berm is built of one unit of 120 tons. The armour layer slope is 1:1.5. The water depth at the breakwater toe 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 prototype Zq4.0 m.. Monochromatic waves Z36 tests. and random waves Z15 tests. were generated

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Fig. 3. Gijon breakwater cross-section. ´

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by a piston-type wavemaker. The free surface in front of the structure was measured by three capacitance wave gauges and a reflection analysis of the free surface time series was performed. By using this technique it is possible to obtain the incident and the reflected wave time series. The transmitted wave height was measured by one free surface gauge located 1 m from the lee side toe of the breakwater. Four strain-gauge type pressure gauges were installed in the crown wall basement while eight gauges were fixed along the vertical structure front face. Pressures at the front face and under the crown wall were integrated by a rectangular method to obtain the forces induced by waves 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 effect of 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 armour were 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 crown walls is introduced. The method allows the calculation of Zi. wave pressure distribution on the crown wall front face ZSection 4.1. and Zii. uplift pressures ZSection 4.2.. Since a single wave generates two peaks of pressure Zpressure pattern described in Section 4.1.1., there are two loading cases for each of the previous pressure distributions Zfront and uplift. called dynamic and reflecting loads. Therefore, Sections 4.1 and 4.2 have three sub-sections: Zi. observed characteristics, where the overall patterns of the pressure distributions are described; Zii. first peak: dynamic pressures, where the methodology to calculate dynamic pressure distributions is given; and Ziii. second peak: reflecting pressures, similar to Zii. for the reflecting pressure case. In Section 4.3 the method is verified for monochromatic waves by comparison of calculated forces to empirical measurements. Finally, a practical application of the method is shown and the results from the proposed method and from other calculating methods are compared. 4.1. Horizontal pressure distribution at crown wall 4.1.1. ObserÕed characteristics From the results of the experimental study conducted at the Ocean and Coastal Engineering Laboratory of the University of Cantabria, ZLosada 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 the bore front due to the crown wall, while the second peak occurs after the instant of maximum 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. 4 where a force time series recorded in the lab is given and the pressure distributions at the wall at the times A and B, are shown. For the pressure distribution produced by the first peak ZA., dynamic pressure, two regions may be distinguished: the upper one, not

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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-mound layer. In both regions, the pressure is almost constant, but higher in the upper region. The pressure profile due to the secondary peak ZB., reflecting pressure, linearly increases downwards. The crown wall and the armour layer are functionally-dependent elements: Zi. The hydrodynamics of the running-up water is modified by the presence of the crown wall. This modification of the flux affects the resulting forces on the armour units; Zii. the armour layer characteristics Zslope, permeability, roughness, berm width, etc.. determine the characteristics of the run-up water tongue which hits the wall. Therefore, it is obvious that the design of both structures must be related. In the proposed method, the relation between the armour layers Zgeometry, porosity, etc.. and the resulting pressures on the crown wall is taken into account. Following other authors, a semi-empirical approach to formulate the pressure distribution during the first and second peak is developed. The crown wall is partially protected by the armour layer, Fig. 1. Thus, for the pressure distribution, two regions may be distinguished. In the upper region, run-up water hits the wall directly. In the lower region, protected by the armour units, waves reach the wall after flowing through the porous armour layer. The lower region extends from the wall foundation level, z s wf , up to the berm level, z s Ac, where z is the vertical coordinate measured from the still water level, positive upwards. The upper region extends from z s Ac, up to the wall crest level, z s wc . In Fig. 5 the proposed pressure distributions are schematically summarised. 4.1.2. First peak: dynamic pressures Following Nagai Z1973., Jensen Z1984. and Tanimoto and Takahashi Z1994. it may be established that the dynamic pressure, PS0 at the berm crest level z s Ac, in the non-protected region, is linearly related to the water tongue thickness at that level, SZ z s Ac. s S0 , and may be evaluated by the following expression ZMartin, 1995.: PS0 s a r g S0

Z 1.

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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 is the water density and g is the gravity acceleration. In Fig. 5, the geometrical definition of S0 is shown. Because of the experimental evidence of the constancy of the dynamic pressures in positive z-direction ZJensen, 1984; Martin, 1995. the pressures profile Pd Z z ., can be represented by, Pd Z z . s PS0 for z ) A c . Z 2. Moreover, the dynamic pressures in the lower region are also almost vertically constant, Fig. 4, and it was experimentally verified ZMartin, 1995. that it can be related to PS0 through an empirical parameter Z l., analysed further in this section, where l is smaller than one, for wf - z - A c . Pd Z z . s l PS0 Z 3. The determination of S0 and a is based on the experimental evidence that the dynamic and reflecting pressures occur shortly before and shortly after the instant of maximum run-up on the wall, respectively. Assuming that: 1. S0 and a may be evaluated during the occurrence of the maximum run-up, and that 2. for collapsing breakers, the water tongue kinematics and dynamics on the edge of the berm during the maximum run-up event is approximately the same as for slopes with or without a crown wall. Then, it may be concluded that the values of S0 and a depend only on the maximum run-up, Ru, on an infinite slope and on the water particle velocity at the jet tip. Losada and Gimenez-Curto Z1981., based on experimental work under monochromatic waves and normal incidence, proposed the following expression for Ru on an infinite slope: Ru s Au 1 y e Z Bu Ir . Z 4. H

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where, Au and Bu ZFig. 6, after Losada Z1992.. are empirical coefficients, which depend on the type of armour unit and Ir is the Iribarren number defined by, Ir s tan b

(

H L0

Z 5.

where b is the slope angle, H is the local wave height and L0 is the deep water wavelength. In order to verify the applicability of Eq. Z4., an experimental evaluation of Ru on a mound breakwater with a crown wall was carried out. The armour layer was built of concrete parallelepipedic blocks Z a = a = 1.25a where a s 3.8 cm. with a 1:1.5 slope and the berm was built of 2 units. Because of experimental uncertainty, the same test Zsame structure and same waves. was repeated three times. In Fig. 7 the best fit curve to the 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 Desire Z1984. ZAu s 1.2, Bu s y0.65, for parallelepipedic blocks., giving support to assumption Z2.. Gunbak and Gokce Z1984. and Yamamoto and Horikawa Z1992. 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.

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

Zas indicated in Fig. 5.. Then, the water tongue thickness, S, of the impinging bore at the level Z z . may be evaluated by: z S Z z . s Sw 1 y Z 6. Ru where S w is the water tongue thickness at the SWL. For 1.5 - cotan b - 2.5, and following the theoretical and experimental work of Yamamoto and Horikawa Z1992., it may be assumed without significant error that the water tongue thickness at the SWL is on the same order of the wave height, S w ; H. Hence, the thickness, S0 , of the bore at the berm crest, z s Ac, is given by, Ac S0 s H 1 y . Z 7. Ru

z

/

z

/

Assuming no energy loss due to friction above Ac level, the alongslope bore tip celerity can be calculated from gravity acceleration as: C b Z z . s 2 Z Ru y z . g .

(

Z 8.

Therefore, the horizontal component of the bore tip celerity, C bx , at any level z G Ac, may be approximated by the following expression ZMartin, 1995.: C bx Z z . s 2 Z Ru y z . g cos b .

(

Z 9.

Next, the averaged horizontal velocity of the water particles near the bore front, Õx , can be considered to be equal to the celerity of the tip Zsee Ramsden and Raichlen, 1990.. Thus, Õx at the berm crest level Z z s Ac. can be obtained by Eq. Z9.. For the calculation of a , results from Cross Z1967. and Wiegel Z1970. for the maximum pressure induced by a bore hitting a vertical wall Z pmax . are considered: Pmax s r Cf Nf2

2 C bx

2

.

Z 10 .

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Thus, the maximum pressure is defined as Cf Nf2 times the stagnation pressure due to the bore celerity, where Nf is a dimensionless parameter defined as: Nf s Cb

'gS

Z 11 .

where S is the bore thickness, and Cf is a coefficient that depends only on the bore front geometry ZCumberbatch, 1960.. Using Eqs. Z7. and Z8. into Eq. Z11. for z s Ac, the resulting value for Nf is Nf s

(

2

Ru H

.

Z 12 .

By using Eqs. Z1., Z7., Z9., Z10. and Z12., for z s Ac, the dimensionless parameter a is given by:

a s 2Cf

Ru H

2

cos b

.

Z 13 .

Note that Eq. Z13. is valid if the horizontal component of the bore celerity Z C bx . is not affected by the berm width. This assumption has been experimentally verified by the authors for berm widths up to BrL s 0.1 ZMartin et al., 1998.. Cf represents the short-duration pressure oscillations induced by the impact of the water tongue front to the vertical wall. Cumberbatch Z1960. showed that it depends only on the bore wedge angle, Q . For wedge angles of 22.58 and 458, Cumberbatch Z1960. found Cf to be 1.4 and 2.1, respectively. Cross Z1967. computed the coefficient Cf for wedge angles between 08 and 758 and found the following relation agreed with his results: Cf s 1 q Z tan u .

1.2

.

Z 14 .

Fig. 8 shows the values of Cf calculated from the maximum pressures PS0 measured in the tests described in Section 3, using Eqs. Z1., Z7. and Z13.. If Cf is calculated from pressures of which the persistence is larger than Tr100 Zinstead of maximum pressures., the dispersion of the calculated Cf is smaller than the dispersion shown in Fig. 8, and the best fit value for Cf is 1.0. To define Cf for design purposes the dynamic response of the structure must be taken into account. For small structures Zlow inertia. andror rigid foundations, maximum pressures must be taken as the design pressures and a value of Cf s 1.45 ZQ s 278. is proposed Zsee Fig. 8.. Finally, a can be calculated as:

a s 2.9

Ru H

2

cos b

.

Z 15 .

For large structures Zlarge inertia. andror elastic foundations, Cf s 1.0 could represent better the equivalent static loading situation for design purposes. Further research on dynamic response of crown walls and on scale effects is required for better assessment of the design value of Cf .

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Fig. 8. Values of Cf calculated from PS0 measured and Eqs. Z1., Z7. and Z13..

The parameter l was evaluated experimentally from monochromatic wave tests in the flume. A description of the experimental set up is given in Section 2. The experimental values obtained for l are shown in Fig. 9. The range of the measured l values Z0.25 - l - 0.65., is in agreement with those given by Jensen Z1984. and Gunbak and ¨ Gokce Z1984.. Notice, that the experimental wave steepness range is 0.03 - HrL Zat ¨ breakwater toe. - 0.075. The best fit curve to the empirical results is:

l s 0.8ey1 0.9 Br L .

Z 16 .

4.1.3. Second peak: reflecting pressures The 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-

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

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tions. Thus, the resulting pressure field around the crown wall may be considered almost hydrostatic. Furthermore, the pressure distribution recorded on the wall, at the instant of the occurrence of the peak, is continuous along the protected and unprotected regions. Then, the reflecting pressure, Pr , may be evaluated by the following linear expression: Pr Z z . s mr g Z S0 q Ac y z . for wf - z - Ac q S0

Z 17 .

where the dimensionless parameter, m F 1, was evaluated experimentally from monochromatic wave tests. From the tests, it is clear that m depends on the wave steepness Z HrL. and on the non-dimensional berm width, Brle. The parameter le is the equivalent size of the rubble units, and is calculated by, le s

(

3

W

gr

Z 18 .

where W and gr are the total weight and the specific weight of the armour unit, respectively. The experimental values for m are shown in Fig. 10. For wave steepness, HrL - 0.02, the reflecting pressures are r gz Z m s 1., decaying to 0.5 r gz Z m s 0.5., approximately, for HrL ; 0.04. By increasing the wave steepness to 0.075, an asymptotic trend is obtained, which depends on the number of units building the berm. m takes values of 0.45, 0.37 and 0.3 for one, two and three armour units on the berm, respectively.

Fig. 10. Experimental variation of m vs. Hr L and Brle as a parameter Z H and L measured at the toe of the breakwater..

F.L. Martin et al.r Coastal Engineering 37 (1999) 149­174 Table 1 Brle 1 2 3 a 0.446 0.362 0.296 b 0.068 0.069 0.073 c 259.0 357.1 383.1

163

The trend of experimental values for m can be well represented by an exponential 2 curve of the type m s ae cZ H r Lyb. . The best fit parameters for these curves are shown in Table 1: The experiments in which l and m are determined were performed with large armour units Z1200 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 core permeability on these parameters. Meanwhile, the proposed values of l and m can be used as a first approach for design purposes. 4.2. Uplift pressure distribution at crown wall 4.2.1. ObserÕed characteristics It is common practice in maritime engineering ZIribarren and Nogales, 1964. to consider a linear variation of wave pressure under the crown wall. Losada et al. Z1993., applying linear wave theory, obtained a parabolic pressure distribution under an impermeable crown wall resting on a porous media, with porosity ranging from 20% to 40%. However, their findings do not differ significantly from the linear trend. In this paper, the linear law is assumed. To define this linear distribution, the pressures at the toe and at the heel of the crown wall were experimentally recorded with the same experimental set-up described in Section 3. 4.2.2. First peak: dynamic pressures Dynamic pressure beneath the seaward edge of the structure is approximately equal to l PS0 . At the heel dynamic pressures are negligible. 4.2.3. Second peak: reflecting pressures Reflecting 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 is founded below the transmitted wave amplitude. Therefore, heel pressures depend on the wave transmission process. Thus, the following values are adopted Zsee Fig. 5.: Ø Seaward edge: ( dynamic pressures l PS0 Z z s wf . ( reflecting pressures Pr Z z s wf . s Pre Ø Heel: ( dynamic pressure negligible, Pra s 0 ( reflecting pressures Pra , from Losada et al. Z1993..

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Fig. 11. Reflecting pressure at the heel vs. Fr L Z L calculated at the toe of the breakwater.. Dots are experimental data from Gijon prototype measured on 10 February 1996 Z ns 0.4 approximately.. ´

Fig. 11 shows the reflecting pressure at the heel, Pra , non-dimensionalised with the reflecting pressure at the seaward edge, Pre , vs. FrL, where F is the crown wall width. Each plotted curve corresponds to a different mound porosity, n. The numerical model employed to generate the curves ZLosada et al., 1993. was applied for a single overall porosity of the rubble mound. For design purposes the porosity selected must represent the 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. This method for calculating uplift pressures must be regarded as a first engineering approach to the problem since: Zi. lines in Fig. 11 were obtained from linear theory, and Zii. the experimental values were obtained from a low crown wall, where air entrainment is very low. Additional experiments for high crown walls based on less porous core are required to complete the method. Some results from the pressure gauges placed under Gijon's breakwater crown wall ´ Zprototype. are included Zdots. Zsee Fig. 11.. This breakwater is built of an armour of 120 tons resting on a core of 90-ton parallelepipedic blocks and the overall porosity is approximately n s 0.4. 4.3. Application and Õerification 4.3.1. Comparison to test data As 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 comparison of maximum forces measured in the lab for regular waves Ztests described in Section 3. and those calculated by the proposed method Z Cf s 1.45.. Experimental forces per linear meter of structure are integrated from measured pressures on the wall. Furthermore, the values of the forces were obtained as average values of the three highest measured forces in each test for monochromatic waves.

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

In Fig. 12b the dynamic pressures Z PS0 . measured in the lab for random waves are compared to the calculated dynamic pressures for each individual single wave. Single waves Z H,T . are identified from measured free surface time series by zero-upcrossing. The method is applied to each individual wave. PS0 is calculated and compared to the

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correspondent vertically averaged measured dynamic pressure in the unprotected region of the crown wall. The averaging procedure is as follows: Zi. the pressure time series are integrated by a rectangular method to force time series, Zii. the maximum dynamic force and maximum dynamic force time are identified for each individual wave, Ziii. the pressure measurements at this time in the unprotected zone is integrated by a rectangular method Zfor each individual wave., Ziv. the resulting force in the unprotected region is divided by the unprotected region height to give a vertically-averaged pressure in this zone. 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 present method when severe storm data become available. Some preliminary results of the measurements performed for uplift pressures were presented in Section 4.2.3. 4.3.2. Comparison to other methods In this section, forces on the crown wall of the Gijon Breakwater calculated by ´ Iribarren and Nogales Z1964., Gunbak and Gokce Z1984. and the present method are ¨ ¨ compared. An individual wave height of H s 12.0 m, a period of T s 16.0 s and a high tide level Zq4.0 over LLWL. is adopted. The proposed method is applied with the following values of the main parameters: Ru s 13.2 m Zcalculated by Eq. Z4. for Au s 1.2 and Bu s y0.65., Cf s 1.45, BrL s 0.016, l s 0.66, ZFig. 9., HrL s 0.052, Brle s 1, m s 0.48, ZFig. 10., FrL s 0.08, n s 0.4, Pra rPre s 0.375 ZFig. 11. and Ac s 9.5 m. A value of 0.6 was assumed for the friction coefficient between the crown wall and the core material. The weight of the crown wall when the SWL is 4.0 m above LLWL is 3460 kNrm. The resulting value for S0 is 3.37 m and for a is 2.43. In Table 2, the net horizontal force per unit length as well as the uplift forces are given. The sliding and overturning safety coefficients were investigated and for all cases the lower safety factors occurred for the sliding of the crown wall. The safety conditions for sliding calculated from experiments on the wave flume are also presented in Table 2. The pressure distribution in the reflecting pressure condition is linear, growing downwards, thus maximum loads occur at the bottom of the crown wall. Therefore the lower the crown wall foundation is, the larger the pressures at the bottom and the uplift pressures are. Then, the large uplift force in the reflecting pressure condition in this

Table 2 Method Iribarren Gunbak ¨ Present method Dynamic pressure condition Reflecting pressure condition Horizontal force ZkNrm. 2285 1231.5 1044 695 Uplift force ZkNrm. 1919 1259 512 1055 Calculated safety coefficient 0.41 1.08 1.69 2.08 Lab. safety coefficient ­ ­ 1.72 2.01

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particular case is due to the low foundation level of the Gijon breakwater's crown wall ´ Z4 m below SWL.. By comparing the calculated and measured forces and safety coefficients it can be concluded 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 10 years show that storms with significant wave heights greater than 8 m and maximum wave heights greater than 12 m have attacked the breakwater several times. After a visual inspection of the breakwater it can be stated that the crown wall along the breakwater trunk has not experienced displacement.

5. Force distribution for random waves The extension of the previous method to irregular waves is based on the following considerations: 1. The reference parameter for the application of the formulae for calculating the dynamic pressures, see Eqs. Z1., Z7. and Z15., is the run-up on a straight slope. 2. The hypothesis of equivalence introduced by Saville Z1962. 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 the dynamic and reflecting pressures may be considered random variables which can give different values for each individual wave Z 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 same phenomenon value which would be produced by a periodic train of the same wave height and period. This hypothesis was empirically proven by Bruun and Johannesson Z1977. and Bruun and Gunbak Z1978. 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 a periodic 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 necessarily imply that each individual wave produces the same forces as the equivalent regular wave, 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 wall under an irregular sea state is as follows. Z1. Given Hs, Tz, the water depth at the toe of the breakwater Z h., and the spectral shape parameter, a theoretical TMA spectrum is computed. Z2. From this spectrum a synthetic free surface time series is generated.

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Fig. 13. Ciervana breakwater cross-section.

Z3. The following breaking criteria are applied to each individual wave: H - 0.142 L tanh Z 2p drL . Z Miche criterion. .

Z 19 . Z 20 .

Z Hrd . max s 0.55 q 0.88 exp Z y0.012 cot Z w . .

where w is the bed slope angle. Since results will be compared to laboratory data, the breaking criterion ZEq. Z19.. is established for waves in the laboratory with mild bottom slopes ZNelson, 1997.. For steeper slopes andror field conditions there are several formulations in the literature Ze.g., Komar and Gaughan, 1972.. Z4. Waves which break due to limited depth are regenerated with a wave height equal to half the water depth, upper limit of values proposed by Dally et al. Z1985.. Z5. Eqs. Z1., Z3., Z7., Z15. and Z17. and uplift pressures are calculated for each individual wave height and, after integration, the total force per unit length of wall produced by the dynamic and reflecting pressures are obtained. Z6. The force sample data are analysed statistically and the best distribution function is 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 the laboratory: Ciervana Breakwater in Bilbao's Harbour ZSpain. tested by Burcharth et al. Z1995. and a model test carried out by Jensen Z1984..

Table 3 Sea state parameters for Ciervana breakwater Hs Zm. 8 9 10 11 12 Tp Zs. 15 16 17 19 20

F.L. Martin et al.r Coastal Engineering 37 (1999) 149­174 Table 4 Simulation parameters Peak enhancement factor Number of waves generated 1.4 3000

169

5.2.1. CierÕana breakwater The present method is applied to Ciervana Breakwater crown wall ZBilbao, Spain., following the procedure described in Section 5.1. The need of space to support new developments in Bilbao harbour led to the building of the Ciervana Breakwater to protect the land reclamation between the inner port and Punta Lucero Breakwater. It was built south-east of the famous Punta Lucero Breakwater Z150-ton armour units. and it is partially protected by this breakwater from NW storms. Its total length is 3.15 km, and the armour layer is built of parallelepipedic 100-ton blocks Z a = a = 1.25a. in a 1:2 slope. The crown wall base level is q1.5 m over the zero datum, the crest level of the rubble berm is q14.0 m and the crown wall top level is q18.0 m Zsee Fig. 13.. The berm width is B s 9.0 m and the width of the wall is F s 29.0 m. The water depth at the breakwater toe is 26.0 m at LLWL, and the maximum tidal range in the zone is 4.5 m. The significant wave heights ZHs. and peak periods ZTp. and the characteristics of the wave simulation for the test are given in Tables 3 and 4, respectively. Fig. 14 shows the wave height distributions: Zi. the distribution obtained by the computation described in Section 5.1, Zii. the distribution measured in the laboratory ZBurcharth et al., 1995. and Ziii. the Rayleigh distribution. In Fig. 15 the 0.1% exceedance probability force obtained from the present method is compared to the experimental results described by Burcharth et al. Z1995.. The agreement is good.

Fig. 14. Wave height distribution computed by the present method, measured in the lab by Burcharth et al. Z1995. and Rayleigh distribution.

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Fig. 15. The 0.1% probability horizontal force, FH , measured by Burcharth et al. Z1995., and computed from present method.

Moreover, the 0.1% exceedance probability pressure measured at level q1.5 m at the foot of the crown wall for the case Hs s 11 m, is 120 kPa while the computed pressure is 132 kPa. Finally, from measurements performed by CEDEX ZMadrid. for the case Hs s 11 m, also described by Burcharth et al. Z1995., the centre of application of the total horizontal forces corresponding to the 0.3% probability is estimated to be between 11.5 and 12.5 m above SWL. The present method locates the application point 12.4 m above SWL. 5.2.2. Comparison to the data of Jensen (1984) Jensen Z1984. 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 full distribution of forces which makes a comparison possible. The cross-section of the breakwater is shown in Fig. 16. The armour layer is built of rectangular blocks 2.9 = 2.9 = 4.2 m Z82 ton. in a 1:2 slope, the berm width is 6.0 m Z2 units. and the berm height is Ac s 10.9 m, the crown wall foundation is wf s 4.30 m, the crown wall height

Fig. 16. Cross-section of breakwater tested by Jensen Z1984..

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171

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. Three significant wave heights were tested: Hs s 8, 11 and 14 m. For the calculation, a TMA spectrum Zpeak enhancement factor s 2. is used, following the procedure defined in Section 5.1. For the calculation of run-up, Eq. Z4. was used for Au s 1.2 and Bu s y0.7. The comparison of results is shown in Fig. 17. Differences in the maximum values of the forces for Hs s 14 m could be due to the breaking criterion 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 on new experiments, carried out under monochromatic waves, is presented. The method is extended to irregular waves via the hypothesis of equivalence and checked against two sets of lab tests by: Zi. Burcharth et al. Z1995. and Zii. Jensen Z1984.. The comparison is fairly 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 time pressure evolution on the wall has two peaks: the first pressure peak which is usually the largest and the secondary peak which is smaller but lasts longer. In some cases Z 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 Z B . and berm height ZAc.. The proposed dynamic and reflecting pressure distribution

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depends on two parameters, l and m , which are experimentally evaluated and depend on the relative berm width Z BrL., local wave steepness Z HrL. and the number of armour units constituting the berm Znb s Brle.. 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 this instant, which depend on the bore front wedge characteristics Zparameter Cf .. These oscillations could be filtered by the dynamic response of the crown wall and its foundation. Moreover, scale effects could affect these oscillations due to aeration, saltrfresh water testing, water compressibility, etc. Further research on dynamic response of crown walls and on scale effects is being carried out by the authors for better assessment of the design value of Cf for design purposes.

Acknowledgements This research was partially funded by the European Community Research Programme, MAST III ZMarine Science and Technology., Project PROVERBS ZProbabilistic Design Tools for Vertical Breakwaters. under EU contract MAS3-CT95-0041. This financial support is very much appreciated. The first author wishes to thank the Ministerio de Educacion y Ciencia for his funding during part of the research ZF.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 the review of the paper.

References

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