analysis of submerged breakwaters stability design - … · analysis of submerged breakwaters...

Analysis of submerged breakwaters stability design

F. Taveira-Pinto Faculty of Engineering of Porto, Portugal

Abstract

The stability of a conventional low-crested breakwater above still-water level can be related to the stability of a non-overtopped structure, based on the usual stability Hudson’s or Van der Meer’s formulae. The required rock diameter for an overtopping breakwater can then be determined, for instance, by application of a reduction factor for the mass of the armour. Other formulations to the design of such overtopped structures can also be used. Studies with low-crested structures have shown that different sections of the structure (the front slope, the crest and the back slope) have different stability responses for similar sea state conditions, depending on the relative crest height. The behaviour of the total slope protection reflects the stability behaviour of each section component. If one wants to optimize the armour weight for similar security conditions in each part of the breakwater, then the stability curves in each section should be determined. The slope angle has significant influence on non-overtopped structures, but in the case of submerged structures the wave attack is concentrated on the crest and less on the seaward slope. The stability of submerged breakwaters appeared only to be a function of the relative crest height, the damage level and the spectral stability number, according to performed tests. In this paper a general review and comparison of the available formulae for submerged breakwater stability design is done. Design charts are presented as well as evidence of some physical modelling results. This information will be of some aid to designers, who are considering the use of these kinds of structures. Keywords: submerged breakwaters, stability, design formulae.

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 78, www.witpress.com, ISSN 1743-3509 (on-line)

Coastal Engineering 209

1 Introduction

Breakwaters or groins usually serve their purpose of protecting land from erosion and/or enabling safe navigation into harbours and marinas. Offshore breakwaters/reefs can be permanently submerged, permanently exposed or inter-tidal. In each case, the depth, size and position of the structure relatively to the shoreline determine the coastal protection level provided by the structure. The actual understanding of the functional design of these structures may still be insufficient for optimum design but may be just adequate to consider for these structures as serious alternatives for coastal protection. Offshore breakwaters are mainly built to protect the shoreline from wave action, to prevent beach erosion, and to replenish beach sand by interrupting long shore and wave-generated currents. When properly designed, an offshore breakwater will eventually help to form tombola. Offshore breakwaters dissipate incident wave energy through wave reflection and diffraction. They act as a countermeasure against beach erosion and provide a sheltered area for small craft and bathers. The sheltered area serves as a littoral reservoir for materials brought in by diffracted waves by the breakwaters. In the last decades worldwide the number of built offshore breakwaters has increased in a higher proportion than groin-type structures. This shows a strong trend towards the use of offshore breakwaters over groins as means of beach stabilization and protection. Examples of such tendency are Israel, Japan, United Kingdom, South Africa and USA. Submerged rubble mound breakwaters are being considered more often in coastal engineering design applications, especially where more natural and environmentally friendly solutions to shoreline protection problems, and mitigate measures for stabilization and rehabilitation of existing breakwater structures, are required. Although a number of numerical and physical studies have been completed on the performance of submerged breakwaters, there are still relatively few useful practical tools for the design engineer. Design equations to date have focused largely on the effect of the depth of submergence of the structure on the wave transmission. Some efforts have been made to include the influence of crest width and breakwater material characteristics but these were made on the basis of limited test data. For example, as a consequence of the depth-limited wave conditions on the reef, more frequently occurring wave conditions will impose almost the same wave impacts on the structure as face events such as, for example, 25-year design conditions. This means that the damage induced by the 25-year condition outside the reef will also be induced by "normal" wave conditions with a return period of less than one year. Since the damage to the armour is cumulative, it is important to take the consequences of the depth-limited waves into consideration as appropriate design criterion for the damage to the armour (i.e., the number of destructive waves will be larger). Results of a number of 2-D tests, Seabrook and Hall, [11], Taveira-Pinto, [12], demonstrated the effect of depth of submergence, crest width and breakwater slope on the wave transmission characteristics. Seabrook and


210 Coastal Engineering

Hall, [11] used a rubble mound cross-section consisting of an armour layer and core, with a wide range of both irregular and regular incident wave conditions. Results indicate that the depth of submergence and crest width as well as the incident wave height are the most important factors affecting the transmission of irregular waves. Therefore existing design equations do not adequately predict the transmission coefficient for wide-crested structures. Structural design aspects of low-crested structures are relatively well described in a number of publications (Ahrens, [3], Van der Meer, [15], CUR/CIRIA, [6], US Corps, [14], Pilarczyk and Zeidler, [9], Vidal et al., [18, 19, 20], etc). Methods for stability calculation based on the velocity on the crest of the structure are presented by Verhagen et al., [17]. Laboratory experiments for the breaking of solitary waves over breakwaters, Grilli et al., [7], show a variety of behaviours, depending on both breakwater characteristics and incident wave height: for emerged breakwaters, waves may collapse over the crest, or break backward during rundown; and for submerged breakwaters, waves may break forward or backward, downstream of the breakwater. Transmission is larger over submerged breakwaters (55-90%), but over emerged breakwaters may reach 20 to 40%. Computations using a fully nonlinear potential model, agree well with experimental results for the submerged breakwaters, particularly for the smaller waves (H/d<0,4). For emerged breakwaters, computations correctly predict the limit of overtopping, and the backward collapsing during rundown.

2 Stability analyses

Submerged breakwaters will only continue to be effective whilst relatively undamaged and stable under wave action. If not, its performance in respect of, amongst other things, wave transmission, will be impaired; and consequently higher levels of wave activity may occur in the lee of the breakwater. The stability of a rubble mound breakwater is usually described by the "zero damage" design sea state, allowing still some small armour movements if they do not exceed some limits. According to a number of regular wave tests with no overtopping, that stability is a function of the dimensionless Hudson stability number, NS, given by,

S SS 1/3

n5050r

w r

H HNDW1

= =∆ ρ

− ρ ρ

(1)

where HS represents the significant wave height, W50 the median armour stone weight, ρr is the density of armour stone and ρw is the density of water. NS is a dimensionless wave height parameter and thus does not contain the wave period nor the sea steepness term. Furthermore its derivation in tests that allowed no overtopping suggests that it may overestimate the average armour weight required for front face stability on low-crest breakwaters. Allsop [4] conducted some tests to analyse the relation between damage and the NS parameter. (Ahrens et al., [1, 2], approaches are not considered here as the



used structures were not statically stable.) A multilayered rubble mound breakwater was used, giving maximum damage levels closer to 5%, after which value global damage of the structure is generally considered to occur. Damage was defined as the number of units extracted from their original position, Ne, expressed as a percentage of the total number of armour units, Na. Figures 1 present damage, Ne/Na, in terms of Ns for various relative freeboards Rc/d, according to the following expression,

SNeDamage A exp(BN )Na

= = (2)

where A and B are empirically derived coefficients.

0

2

4

6

8

0 1 2 3 4

Ns

A=0.028, B=2.25,Rc/d=0.29; 0.39; 0.57,Long Wave Regime

A=0.008, B=2.31,Rc/d=0.23; 0.38; 0.57,

Figure 1: Damage curves, Powell and Allsop, [10].

The overall trend is that of an increasing number of stones being extracted as the wave attack becomes more severe, especially for NS>2.0. The data suggests that there is a wave period effect, damage increases with peak period, when other factors remain equal. Furthermore, the use of the Hudson stability number also tends to introduce a wave period effect, which increases the spreading of the data. The use of a spectral stability number, NS*, equation 3 may attenuate that effect but can however overestimate the weight of armour stones required for front face stability, on breakwaters that are designed to be overtopped.

23S OP* S S

S3 n50n50 p S3n50

OP

H LH HN

DD s HDL

= = =∆∆

∆

(3)

Trends of damage against NS*, figure 2, are very similar to those observed with the Hudson stability number but still exhibits a wave period effect. Damage occurs more rapidly under wave spectra with greater periods.

Short Wave Regime



0

2

4

6

8

0 2 4 6 8Ns*

A=0.0007, B=1.66,Rc/d=0.29, Long WaveRegime



A=0.0059, B=1.07,Rc/d=0.23; 0.38; 0.57,Short Wave Regime

Figure 2: Damage curves, Powell and Allsop, [10].

It was also observed that back face damage is generally greater than front face damage, for the longer wave periods, and on the contrary for the shorter wave periods. This implies that the apparent wave period effect is in reality a result of the increased overtopping, which occurs under the longer waves. It seems therefore unlikely that a stability number, specifically derived for the case of no overtopping, could be used to adequately account for the stability of low crest breakwaters over a realistic range of wave periods, namely by a dimensionless freeboard parameter, *

pR , equal to

op* C C Sp 2

S S p

sR R HR

H 2 H gT= =

π (4)

where RC is the crest height with respect to the still water level (SWL) and sop is the deep water wave steepness related to the peak period, TP . The presented trends exhibit an apparent RC/d dependency that suggests being a characteristic of long waves (HS/LP< 0.03). The trend of increasing damage with increasing values of RC/d may therefore be partly due to the effect of the water depth, d, on the shoaling of the longer waves. For structures similar to those used in these tests, figures 1 and 2 may be used to obtain a rough estimate of NS or NS* for the permissible level of damage selected. For other structures NS or NS* may be estimated by comparing the structure to those that were tested. It should however be noted that both Ns and Ns* may overestimate the weight of armour required for front face stability and on the contrary may underestimate the weight of crest and back slope armour required to resist overtopping forces. The Van der Meer and Hudson’s formulae were derived for non overtopped slopes. When slopes are overtopped, a certain wave transmission occur which means that not all energy is dissipated on the slope and thus the stability of the



armour stones will increase. For crests above the still water level Van der Meer, [15], found a reduction factor, R, for the mean diameter, Dn50 , according to the following equation, valid for 0 < *

pR < 0.052,

*p

1R1.25 4.8R

=−

(5)

The minimum value of the reduction factor is 0.8 and the maximum is 1.0, figure 3. An average reduction of 0.8 in diameter is obtained for a structure with the crest height at the water level (RC=0). The required mass in that case is a factor 0.83 = 0.51 of that required for a non-overtopped structure.

0,7

0,8

0,9

1,0

1,1

-0,5 0,0 0,5 1,0 1,5 2,0Rc / Hs

Red

uctio

n fa

ctor

sop=0.04

sop=0.03

sop=0.02

sop=0.01

sop=0.005

Figure 3: Reduction factor of diameter Dn50 for the design of conventional low- crested structures above still-water level, Van der Meer, [15].

Equation 5 describes the stability of a statically stable low-crested breakwater with the crest above still-water level simply by application of a reduction factor on the required diameter of a non-overtopped structure. The reliability of the reduction factor depends on the stability formula that is used to calculate the Dn50 for a non-overtopped structure. However, as a result of the wave transmission, armouring has to be heavier on the other side of the breakwater and the question is whether the total damage will reduce or not. One possible approach is to apply the same armour units on both sides. For crests below the still water level, Van der Meer, [15], formulated a different relation for the stability, which can be rewritten as follows:

C 3S ph1N 7 ln s

2.1 0.1S h

= − + (6)



or *S0.14N* C C

Sh h1N 7ln (2.1 0.1 S)e

2.1 0.1S h h−

= − ⇔ = + + (7)

where sp represents the local wave steepness (HS/LOP) instead of the deep water steepness, hC the structure height, h the local water depth and S the damage level. The slope angle has large influence on non-overtopped structures, but in the case of submerged structures the wave attack is concentrated on the crest and less on the seaward slope. Therefore, to exclude the slope angle of submerged structures from the relevant parameter for stability is legitime. The stability of submerged breakwaters is then mostly a function of the relative crest height hC/h, the damage level and the spectral stability number. For fixed crest height, water level, damage level, and wave height and period, the required ∆Dn50 can be calculated from equation 7, finally yielding the required rock weight. Wave height versus damage curves can also be derived from equation 6 or 7, figure 4. Figure 4 also gives the 90% confidence bands for S=2. The scatter is quite large and this should be considered during design of submerged structures. In fact these formulae should be used with caution, and further research is needed in order to establish an overall relation for the stability increase of low crested breakwaters.

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

2 4 6 8 10 12 14 16

hc /

h

S=2

S=2 (+90%)

S=2 (-90%)S=5

S=8

S=12

Figure 4: Design curves for submerged breakwaters (Pilarczyk, [8]).

Usually for submerged structures, the stability at the water level close to the crest level will be the most critical. So, assuming depth limited conditions (HS=0.5h, where h=local depth), the (rule of thumb) stability criterion becomes, Pilarczyk, [8]:

1/3S S 50

n50 n50n50 S

H H M h2 or D or DD 3 6

= = = = ∆ ρ

(8)

N s*



Vidal et al., [18], analysed the behaviour of low-crested structures (including submerged ones), but they divided the armour layer of the structure into several sections: the front slope (FS), the crest (C), the back slope (BS), the front head (FH) and the back head (BH), figure 5. These different sections of the structure have also different stability responses to a sea state condition. The behaviour of the total slope protection (as described by equation 5,) reflects the stability behaviour of each section component. If one wants to optimize the armour weight to obtain a similar security condition in each part of the breakwater, the stability curves of each section should be determined.

BHTS

BS

FSFH

Incident Wave

C

Figure 5: Considered sections of the submerged breakwater, Vidal et al., [18] (1:1.5 slope, B=0.15 m).

Damage curves of each sector can be represented as a function of the non-dimensional freeboard, Fd=F/D, where F is the freeboard (referred also as RC). These curves where fitted with parabolas of the type:

2s d dN = A + BF + CF (9)

where the coefficients A, B and C will be different for each damage level and breakwater sector. Figure 6 gives a comparison of stability between the five sections as a function of the relative crest height. The described damage level, S, for each section, is between 0.5 and 1.5 (initiation of damage). These fittings are valid only for the experimental range 2.01<Fd<2.41, being Fd=2.40 the maximum non-dimensional freeboard for which overtopping conditions can be considered for that damage level. The fitted stability curves can not be used to assess the damage in any submerged breakwater, which in general, will have structural parameters different from those of the model, such as the type of armour units, under layers, core and armour slope angles.



1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5

Fd = F/D

Ns

FS: A=1.831; B=-0.2450; C=0.0119

C: A=1.652; B=0.0182; C=0.1590

BS: A=2.257; B=-0.5400; C=0.1150

BH: A=1.681; B=-0.4740; C=0.1050

TS: A=1.602; B=-0.2592; C=0.0731

Figure 6: Stability design relations for different sections of a submerged breakwater, Vidal et al., [18].

The back head is less stable for relative crest heights of Fd > 0.5. For Fd < -0.2 the crest section is the least stable one. The back slope is the most stable section for Fd < 1.5. For larger values the crest is more stable (although the stability should be similar for non-overtopped structures). If the freeboard is high (RC/Dn50 around 2.5) and the armour on the crest is the same as the back slope, the damage can start at the back slope and disintegrate the crest. It is also interesting to note that for –0.5<Fd<0.5 the stability number for the front slope, the total slope, the back head and the crest are similar. For Fd<-0.5, i.e., for submerged breakwaters the stability number for the front slope, the total slope and the crest are similar. Finally for Fd>0.5 that similarity occurs for the front slope, the total slope and the back head, as the structure behaves as a low-crested one. The curve corresponding to the total slope reflects the basic principle of equation 5 and figure 3: the stability increases as the relative freeboard decreases. Moreover, the stability number for non-overtopped structures in figure 4 amounts to NS=1.4 and for structures with the crest at still-water level to NS=1.6. This gives a similar reduction factor of 0.8 for structures with the crest at the water line. According to Vidal et al., [18], the first step to assess the armour size of the front slope, DFS , of a submerged breakwater with a freeboard F, is to calculate the armour size, D1 , of the non overtopped breakwater with the same characteristics (armour units, core, under layers, slope and damage level), using any of the available formulas. The non- dimensional freeboard will be Fd,1=F/D1 and the front slope stability number, NS,FS(Fd,1), of the submerged breakwater can be obtained using the

BS

FS

C

BH

TS



expression (9) for the front slope. Being NS,MÁX the non-overtopped limit of the stability number of the submerged breakwater front slope, obtained from the expression (9) for Fd=2.4, the front slope armour unit size of the submerged breakwater, DFS , will be:

S,MAXFS 1

S,FS d,1

ND D

N (F )= (10)

To calculate the size of the armour units of the crest, DC, first the size of the units of the front slope of a similar breakwater but with front slope angle 1V/1.5H, D2, if the front slope angle is different from this one, has to be calculated following the previous procedure. The new non-dimensional freeboard will be Fd,2=F/D2 and then the size of the armour units of the crest, DC, will be:

S,FS d,2C 2

S,C d,2

N (F )D D

N (F )= (11)

Back slope stability can be calculated only if the slope angle of that sector is 1V/1.5H. The size of the armour units of the back slope, DBS , can be calculated following the same previous procedure:

S,FS d,2BS 2

S,BS d,2

N (F )D D

N (F )= (12)

Vidal et al., [19], found that the front head sector showed stability behaviour similar to that of the front slope. The size of the armour units of the back head sector, DBH , of a given low crest or submerged breakwater, is based on the size of the armour units, DH,1 that are necessary for a non overtopped breakwater with similar characteristics of the submerged one. The equivalent dimensional freeboard is Fd,H,1=F/DH,1 and the size of the back head armour units will be:

S,BH,MAXBH H,1

S,BH d,H,1

ND D

N (F )= (13)

where NS,BH,MÁX is the stability number of the back head for the experimental non-overtopping limit (Fd=2.4) and NS,BH(Fd,H,1) is the stability number of the back head for the scaled freeboard, both obtained from the expression 9 with the coefficients for the back head. Verhagen et al., [17], established a methodology for the stability analysis of the inner slope of a overtopped breakwater according to a stability parameter, Θ, given as,

( )2char C

n50 n50

u cos( ) Rsin N

g D Dβ − α

Θ = α∆

(14)

where N represents the number of waves in the experiment and uchar is the time averaged velocity during the plunge of the wave (in fact it is the quotient of the maximum instantaneous discharge and the maximum layer thickness of the wave). The meaning of the variables is represented in figure 7.



Figure 7: Variables definition for the analysis of the impact on an inner slope, Verhagen et al., [17].

The results show a good linear relation between the stability parameter and the damage number, according to the following expression. (This expression is not presented in Verhagen et al., [14]. It results from the analysis of the author.)

od120 NΘ (15) So, for a given desired damage number, Nod=1.0 for instance, and estimating uchar and β, according to the incident wave height and the overtopping relations, as ∆, α and RC are known parameters, it is possible to estimate Dn50 for a storm of N waves. The final expression will be,

( )n50

2char C2 u cos( ) R sin N

D120 g

β − α α=

∆ (16)

which leads to a maximum (when α=β) equal to,

char

n50

2C2 u R sin N

D120 g

α

∆ (17)

For common values of N=2000, ∆=1.65, RC=2 m for a 2:1 slope Dn50 ≅ 0.1435 uchar . The design of a submerged breakwater, Anonymous, [5], Vera-Cruz and Carvalho, [16], to protect the entrance of Leixões harbour, Portugal and later on the head of the main breakwater, lead to some interesting conclusions, related with its configuration. The wave propagation studies showed that the stability of the breakwater’s foundation becomes more unstable as the water depth near the structure decreases. The observation of the structure’s behaviour with real storm tests of 8 to 10 hours, allowed to see that: • Low weight foundation blocks (1/2 to 2 t) used in between the heavier

blocks were transported from the seaward slope to the inner slope. The heavier blocks became unstable especially on the upper edge on the seaward slope.



• The foundation blocks of 1/2 t to 4 t that formed the lower berm in the seaward slope, were dragged to the sandy bottom.

• The heavier blocks of the seaward slope slide over the lower berm. Under these conditions, the waves reach the foundation-stones of 2 to 4 t forming the infrastructure of the breakwater.

The lack of stability of the structure in these first tests was not due just to the insufficient weight of the blocks of the mound, but especially due to a defective composition and disposition of the structure, a bad support of the slopes and also a much steeper slope. After these tests, some modifications were suggested: • Suppression of the elements intercalated along with the heavy blocks. • Increasing of the blocks weight. • Smoothing of the breakwaters’ slope. • Suppression of the intermediate layer of elements. • Enhance of the slope stability on the inner slope. • Extension of the lower berm. The solution built is shown in figure 8. It has been working properly without any stability problems for several years.

stones of up to 1 t stones of up to 4 tstones of up to 4 t

concrete blocks of 90 t

Figure 8: Submerged breakwater cross section, Vera-Cruz and Carvalho, [16].

3 Design formulae comparison

After presenting the possible design formulae, it would be interesting to present a brief comparison of those, based on the data of Vidal et al., [18], namely: Design parameters:

Front slope – 2:1 Water density: 1025 kgf/m3 Blocks density: 2500 kgf/m3 Wave height: 5,7m

Design water depth: 7.1 m Crest freeboard: 1.9 m



Wave period: 10 s (short wave regime) Local wavelength: 79.5 m Wave period: 15 s (long wave regime) Local wavelength: 122.5 m Powell and Allsop [10] approach, figure 1: Damage level: 5% Short wave regime – W=7.2 t (all slopes) Long wave regime – W=12.7 t (all slopes) Powell and Allsop [10] approach, figure 2: Damage level: 5% Short wave regime – W=8.7 t (all slopes) Long wave regime – W=14.2 t (all slopes) Van der Meer [15] approach, figure 3: Block weight based on Iribarren formula – W=7.6 t Short wave regime – Reduction factor=0.887 W=6.7 t (all slopes) Long wave regime – Reduction factor=0.856 W=6.5 t (all slopes) Pilarczyk [8] approach, figure 4: Damage level: 10 Short wave regime – W=8.8 t (all slopes) Long wave regime – W=13.6 t (all slopes) Pilarczyk [8] approach, equation 8: Approach 1 – W=19.4 t (all slopes) Approach 2 – W=17.1 t (all slopes) Approach 3 – W=4.1 t (all slopes) Vidal et al. [18] approach, figure 6 and equations 10 to 13: Front Slope: 4.8 t Back Slope: 2.9 t Back and Front Head: 8.3 t Crest: 3.4 t Verhagen et al. [17] approach, equation 17: Number of waves – 2000 Mean velocity discharge (equal to the wave celerity): 8 m/s Back Slope – W=4.3 t The spreading is quite significant, with some similarities between some design methods, but in general the parameters to be considered are different and for that reason its selection leads to different block weights. Some of the methods do not distinguish the different sections of the structure and allowing its application for structures similar of those used in the tests.



4 Final considerations

The stability of low crest breakwaters was presented in relation to both the Hudson's stability number and the spectral stability number. Although the spectral stability number may adequately take into account the wave period, neither of the former parameters is a satisfactory basis for a comprehensive design method, namely to reproduce the mechanisms responsible for front and back face damage. Other methods that take into account the different sections of the structure were also described. However, through a simple design process it was clear that there is no uniformity of the obtained results, thus leading to important uncertainties. Continued research, especially on submerged breakwaters, should further explore improved techniques to predict shore response and methods to optimise the breakwater design. These new efforts will bring future designers closer to more efficient applications and designs of these promising coastal solutions. More intensive monitoring of the existing structures will also help in the verification of new design rules. Little other stability testing of submerged breakwaters in random waves has been documented. Indeed, most designers seem to rely on stability formulae and coefficients derived from regular wave tests only. There are still considerable uncertainties involved in selecting suitable armour block weights to ensure stability of low crest breakwater under the design wave conditions. At present these uncertainties may only be satisfactorily resolved by physical model testing of the stability aspects relevant for breakwater design. It is therefore strongly recommended all designs to be finalised by physical model testing as the stability of the primary armour layers of submerged breakwaters is not adequately represented by a single simple design graph.

References

[1] Ahrens, J.P., Viggosson, G., Zirkle, K.P., 1982, Stability and wave transmission characteristics of reef breakwaters, CERC Report, Vicksburg, USA.

[2] Ahrens, J.P., 1984, Reef type breakwaters, Proc 19th Coastal Eng Conf, Houston, USA.

[3] Ahrens, J., 1987, Characteristics of reef breakwaters, USAE, CERC TR 87-17, Vicksburg, USA.

[4] Allsop, N.W.H., 1983, Low-crest breakwaters, studies in random waves, Proc Coastal Structures 83, ASCE, Arlington, USA.

[5] Anonymous, Contribution a L’étude expérimentale des digues maritimes en enrochements, pp 95-117 and 119-121, Laboratory of Lausanne

[6] CUR/CIRIA, 1991, Manual on use of rock in coastal engineering, Report 154, Centre for Civil Engineering Research and Codes (CUR), The Netherlands.



[7] Grilli, Stephan T., Losada, M.A. and Martin, F., Characteristics of solitary wave breaking induced by breakwaters, Journal of Waterways, Ports, Coastal and Ocean Engineering, 120 (1), 74-92, USA, 1994.

[8] Pilarczyk, Krystian W., Design of low-crested (submerged) structures – an overview, 6th International conference on coastal and port engineering in developing countries, Colombo, Sri Lanka, 2003.

[9] Pilarczyk, Krystian W. and Zeidler, Ryszard B., Offshore breakwaters and shore evolution control, A.A.Balkema Publishers, Rotterdam, Netherlands, 1996.

[10] Powell, K.A. and Allsop, N.W.H., Low-crest breakwaters, hydraulic performance and stability, Report SR 57, Hydraulics Research, Wallingford, UK, 1985.

[11] Seabrook, S.R. and Hall, K.R., Effect of crest width and geometry on submerged breakwater performance, Proceedings of the 1997 Canadian Coastal Conference, pp.58-72.

[12] Taveira-Pinto, F., Velocity fields analysis near submerged breakwaters under wave action (in Portuguese), PhD Thesis in Civil Engineering, Faculty of Engineering, University of Porto, Portugal.

[13] Uda, T., 1998, Function and design methods of artificial reef (in Japanese), Ministry of Construction, Japan.

[14] US Army Corps, 1993, Engineering design guidance for detached breakwaters as shoreline stabilization structures, WES, Technical Report, CERC-93-19.

[15] Van der Meer, J.W., 1990, Low-crested and reef breakwaters, Delft Hydraulics Report H986.

[16] Vera-Cruz, Daniel and Carvalho, José Reis, Maciço Submerso de Pré-Rebentação das Ondas como meio de protecção de obras marítimas. O caso do quebramar de Leixões, Memory No. 796 (LNEC) “” (in Portuguese), Lisbon, 1993.

[17] Verhagen, H.J., VanDijk, B, Nederpel, A., Riprap stability on the inner slopes of medium-height breakwaters. Proc. Coastal Structures, Portland, Or.; ASCE New York , 2003.

[18] Vidal, C., Losada, M.A., Medina, R., Mansard E.P.D. and Gómez-Pina, G., A universal analysis for the stability of both low crested and submerged breakwaters. Proc. 23rd conf. on Coastal Engineering, Venice, pp 1679-1692, ASCE, 1992.

[19] Vidal, Losada & Mansard, Stability of low-crested rubble-mound breakwater’s heads. Journal of Waterway Port, Coastal and Ocean Engineering, ASCE, pp. 114-122, 1995.

[20] Vidal, Medina & Martín, A methodology to assess the armour unit stability of low crested and submerged rubble-mound breakwaters. Proc. Coastal Structures Conference ’99, pp 721-725. Balkema, Rotterdam, 2000.



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