THE EFFECTS OF FORESHORE SLOPE ON
BREAKWATER ARMOUR UNIT STABILITY
Student: Imraan Bux
Graduation Committee
Professor H J Verhagen DUT, Faculty of Civil Engineering: Coastal Engineering
Professor D Roelvink UNESCO-IHE, Coastal Engineering, Port Development
THE EFFECTS OF FORESHORE SLOPE ON
BREAKWATER ARMOUR UNIT STABILITY
Master of Science Thesis
Imraan Bux
Supervisor
Professor H J Verhagen (TU Delft)
Internal Supervisor
Professor D Roelvink (UNESCO-IHE)
This research is done for the partial fulfilment of requirements for the Master of Science degree at the UNESCO-IHE Institute for Water Education, Delft, the Netherlands
i
DEDICATION
ii
ABSTRACT
Experience has shown that the stability parameters relating to breakwater armour unit stability vary considerably for steep foreshores when compared to milder foreshore slope. Furthermore, the change from the deep water spectra (Tm0) to the shallow water wave spectra (Tm-1.0) does not fully
explain the difference. Leading on from recent laboratory tests that were undertaken at the Research Flume at the Delft University of Technology indicate that the stability of breakwater armour unit is further dependant on parameters that are not described by the wave spectrum at the toe of the breakwater. The skewness of the wave is one such parameter.
From recent tests that were conducted at TU Delft’s Laboratory it was found that for equal deep water waves, the damage to breakwater armour unit was more for a steep foreshore slope than that of a milder slope. It was also found that the wave spectra are identical but the damage to the armour unit vary considerably, implying that the damage to the breakwater has to depend on a wave parameter that is not represented in the shallow water wave energy spectrum. This parameter has to be cognisant of waves approaching over different foreshore slopes.
iii
differed to that which was used in the study, made such comparisons difficult. Wave peakness was also found to be an important parameter contributing to stability together with stone density and porosity, obliqueness and angularity together with a proper definition of damage area.
This study proved the importance of foreshore slope on breakwater armour unit stability and for the importance of further research in understanding the other contributing parameters.
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ACKNOWLEDGEMENTS
I would sincerely like to acknowledge the following organisations and individuals:
The Blom Foundation for having sponsored my studies;
UNESCO-IHE for having afforded me this incredible study opportunity;
My supervisor, Professor Verhagen for his invaluable guidance in bringing this study to fruition;
Professor Roelvink for his internal supervision;
v
TABLE OF CONTENTS
DEDICATION ... i ABSTRACT ... ii ACKNOWLEDGEMENTS ... iv TABLE OF CONTENTS ... v LIST OF TABLES ... viChapter 1: Problem Definition and Research Methodology ... 10
1.1 Functional Requirements... 12
1.2 Problem Definition ... 12
1.3 Research Objective ... 14
1.4 Report Outline ... 14
Chapter 2: Theory... 16
2.1 Stability formula of Iribarren... 16
2.2 Stability formula of Hudson ... 18
2.3 Stability formula of van der Meer ... 19
Chapter 3: Experimental Set-up ... 21
3.1 General experimental set-up... 21
3.2 Breakwater... 22
Chapter 4: Analysis ... 29
4.1 Data Analysis... 29
4.2 Description ... 34
4.3 Damage area and void ratio calculation ... 35
Chapter 5: Results... 44
5.1 Results of Laboratory tests ... 44
5.2 Comparisons based on erosion area... 46
Chapter 6: Conclusion and Recommendation ... 51
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LIST OF TABLES
Table 1: Laboratory measurement results
Table 2: Test parameters relating to program number
Table 3: Final datasheet of test results
vii
LIST OF FIGURES
Figure 1: Typical section through a rubble mound breakwater structure
Figure 2: Definition of foreshore slope
Figure 3: Breakwater layout
Figure 4: Breakwater Cross-section
Figure 5: Side elevation of the front side of the breakwater
Figure 6: Definition of damage area on breakwater slope
Figure 7: Rip-Rap armour after testing
Figure 8: Description of damage level on breakwater slope
Figure 9: Initial plot of the actual damage of laboratory results
Figure 10: Damage level in terms of S
Figure 11: Van der Meer formula
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LIST OF SYMBOLS
β Bottom steepness (slope angle) [-]
ξ Iribarren (breaker) parameter [-]
∆ Relative mass density [-]
α
Slope angle of breakwater [-]Ae Damage area [m2]
Ae erosion area [m2]
B width of flume [m]
D Stone diameter [m]
d stone thickness or axial breadth [m]
D15 sieve size where 15% of the stones fits through [m]
D50 sieve size where 50% of the stones fits through [m]
D85 sieve size where 85% of the stones fits through [m]
Dn50 Nominal mean diameter of stone [m]
E(f) spectral density [m2/Hz]
f frequency [Hz]
fm peak frequency [Hz]
g acceleration of gravity [m/s2]
H Wave height [m]
Hm0,0 Hm0 at deep water [m]
Hs Significant wave height [m]
KD Breakwater stability co-efficient (Hudson) [-]
l maximum axial stone length [m]
L wavelength [m]
L0 deep-water wavelength [m]
M stone mass [kg]
mD dry mass of a stone [g]
mU underwater mass of a stone [g]
N number of waves [-]
Nod number of displaced stones (normalised) [-]
NS number of displaced stones [-]
P Permeability factor [-]
S Damage level [-]
ρs Density of stone [g/cm3]
ρw Density of water [kg/m3]
s wave steepness [-]
Tm-1,0 wave period (Van Gent) [-]
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Vs volume of a stone [cm3]
Vs volume of a stone [m3]
W stone weight [N]
z sieve size (side of smallest square hole) [m]
α scaling factor (Pierson-Moskowitz) [-]
γ0 scaling factor (Jonswap peak enhancement factor) [-]
10
Chapter 1: Problem Definition and Research Methodology
This introductory chapter will focus on the role and functional requirements of rubble mound breakwaters together with the problem definition and research methodology adopted. The proposed outline of the study will also be explained in this chapter.
1.0
General Introduction
The most important functional requirement of all coastal structures is their ability to withstand forces caused by wave attack. In the case of ports the breakwaters play a vital role in this process. Breakwaters1 are structures placed offshore to dissipate the energy of incoming waves. They are designed to typically absorb the energy of the approaching waves which can be done by either using mass (e.g. with caissons) or by using a revetment slope (e.g. with rock or concrete armour units). Many breakwaters in the world presently are rubble mound structures which use the voids in the structure to dissipate the wave energy with the armour units absorbing most of the incoming energy. The main strength parameter for most breakwaters is the type and mass of the armour units. The interlocking abilities of the armour units to the breakwater are also important and contribute significantly to its overall stability. Rubble mound breakwaters are structures specifically built of quarried rock or other stone material. Generally, the larger rock armourstones are used for the outer layer which must withstand forces due to wave attack. The rocks or stones placed in the outer layer is normally undertaken with more care to obtain better interlocking and consequently better stability. Figure 1 shows a typical section through a rubble mound breakwater structure.
11
Figure 1: Typical section through a rubble mound breakwater structure
Rubble mound structures typically consist of a core usually made of small stone or rocks, a filter layer and an armour layer. As stated above, larger stones are normally used for the armour layer (outer layer) in order to withstand forces due to wave attack. A toe is constructed on both sides of the structure which contributes to the overall stability. A heavy toe built of rock or stone is constructed on the seaward side of the breakwater. A concrete crown wall or splash wall is then placed at the top of the breakwater.
The wave forces acting on a rubble mound breakwater slope can cause armour unit movement which is termed as hydraulic instability. According to van der Meer (1988), rubble mound structures can be classified according to their
D H
∆ parameter, where H is the wave height, ∆is the relative mass density and D is the characteristic diameter of the structure armour unit. Statically stable structures are those were little or no damage is allowed under design conditions. Damage is referred to the displacement of the armour units under wave attack. The mass of the individual rock units within the armour layer must be large enough to withstand wave forces during design conditions. Traditionally designed rubble mound breakwaters are statically stable structures with an =1− 4
∆D H
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1.1
Functional Requirements
Breakwaters generally serve to provide for safe navigable access for vessels by delineating the harbour entrance channel and limiting the amount of energy penetration into the harbour basin. The strong currents and wave forces are limited by the breakwaters thus allowing for the safe berthing and mooring of vessels.
Other functional requirements of breakwaters could include namely:
• Protection of the ports access channel and turning circle for safe stopping and manoeuvring;
• Reduction of wave disturbance within the channel thus allowing for tugs to operate;
• Morphological function by blocking the longshore transport of sediments that could silt up within the entrance channel thus increasing dredging costs.
1.2
Problem Definition
Many methods for the prediction of stability of armour units based on physical model tests have been proposed. However, these tests were conducted with deep water wave conditions at the toe of the structure and most breakwaters are generally located in shallow water. Furthermore, the design formulae do not take into account directly the effects of the foreshore slope on the stability of the breakwater. Breakwaters that are constructed on relatively steep foreshores are problematic from a stability point of view with significant damage to the armour units.
13
(Hovestad, 2005) confirmed that for equal deep-water waves the damage caused by steep foreshores is greater than that for milder slopes. It is for this reason that many guidelines particularly the CUR report recommends a heavier class of rock in this situation. Manufactures of single armour units like Accropode, Core-Loc and Xbloc recommend a lower KD value in the case of steep foreshores.
A first attempt into a more systematic analysis of the problem into shallow water and steep foreshores was undertaken by Van Gent et al (2003). The results of the research revealed that the load on the structure can better be described by a shallow water spectrum at the toe of the structure and because the longer periods are more relevant than the shorter periods more attention should be given to the low frequency part of the spectrum. The Tm-1.0 should be used instead of the Tp
or Tm0 in the stability formula as it yields more reliable results according to Van
Gent et al. Van Gent following a statement of Van der Meer (1998) stated that for shallow water conditions the 2% wave height should be used instead of the significant wave height and the value of ξ should be calculated with the spectral wave period Tm-1.0.
From engineering practice it is supposed that the steepness of the sea bottom in front of a breakwater or sea defense, as can be seen in Figure 2, denoted as the angle β may have influence on the stability of the armour units on these structures.
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The stability of breakwaters on steep foreshores is still a problem which resulted in a series of experiments being undertaken at the research flume of the Fluid Mechanics Laboratory at Delft University of Technology. As was expected, the tests revealed for equal deep water waves, the damage caused to the armour unit was more significant for steep foreshores when compared to shallow foreshores.
1.3
Research Objective
Recent studies undertaken at TU Delft (Hovestad, 2005), indicate that the wave spectra are identical but the damage to the armour units is clearly not identical. This implies that the damage to the breakwater is also dependant on a wave parameter which is not represented by the shallow water wave energy spectrum and has to consider waves approaching over different foreshore slopes.
What is the parameter not included in the wave spectrum which seems to be relevant for breakwater armour unit stability on steep foreshores? What is the
physical relationship between steep foreshores and armour unit stability?
The research shall not only focus on the statistical relationship of the problem (curve fitting) but will endeavour to establish the physical relation between steep foreshores and armour unit stability which is presently unclear.
1.4
Report Outline
15
16
Chapter 2: Theory
This chapter will review the existing theories that are relevant to breakwater armour unit stability in shallow water. The most common are that of van der Meer, Hudson and Iribarren. The theories shall be examined in order to establish which theory would support the research objectives the closest.
Many prediction methods for the stability of armour units have been proposed. Most of these tests were conducted with deep water conditions at the toe of the structure.
2.1
Stability formula of Iribarren
The Iribarren number also known as the surf similarity parameter was introduced by Iribarren in 1938. This parameter was introduced by Iribarren and Nogales as an indicator whether breaking would occur on a plane slope. As discussed by Battjes (1974)-, the derivation of the Iribarren formula suggests the parameter gives the ratio of the structure slope steepness to the square root of the wave steepness as defined by the local wave height at the toe of the structure divided by the deep water wave length.
Battjes (1974)-, popularised the Iribarren number (surf similarity number) and showed its application to a number of surf zone processes, namely, wave breaking, differentiation of breaker types, wave set-up, wave up and run-down and wave reflection
17
Assuming that the velocity in a breaking wave or wave breaking on a slope is proportional to the wave celerity in shallow water with the wave height as being representative of the water depth:
µ
∝ gH , Iribarren proposed the following formula in 1938:(
ρ ρ)
(
φ α α)
ρwgHd2 ∝ s− w gd3 tan cos ±sin (2.7)
“drag” force resisting force slope correction
Where:
w
ρ
Density of water [kg/m3]s
ρ
Density of stone [kg/m3]g Acceleration due to gravity [m/s2] u Velocity parallel to the breakwater [m/s]
H Wave height [m]
d Diameter of stone [m]
φ
Angle of internal friction [-]α
Slope angle of the structure [-]By raising all the terms in the above formula to the third power and considering the mass of the stone
(
M ∝ρ
sd3)
the Iribarren formula becomes:(
)
3 3 3 sin cos tanφ
α
α
ρ
± ∆ ≥ H M s (2.8) Where: M Mass of stone [kg]∆ Relative buoyant density of the material [-]
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parameters related to armour unit stability of breakwaters on steep foreshores.
2.2
Stability formula of Hudson
Extensive research has been carried out on the stability of stones on a slope in the case of breakwaters. In general, the linear wave theory is used in the calculation of wave heights. Linear wave theory can calculate wave heights at any given depth as a function of a given off-shore wave height and a wave period. However, it may be limited in that it does not consider the steepness of the bottom slope.
The Hudson formula which is based on the Irribarren formula (the formula as proposed by Irribarren is discussed later on in section 2.3) is well known and was commonly used for the prediction of rock armour size for structures under direct wave attack. After extensive tests Hudson proposed the following formula: W =
α
ρ
cot * * * 3 3 ∆ D c K H (2.1)Re-writing the above equation in terms of weight reveals:
3 1 50 ) cot ( D
α
n s K D H = ∆ (2.2)The Hudson formula was based on waves that did not break at the toe of the structure and no overtopping was considered. The KD value is more like a
“dustbin” co-efficient taking into consideration all other variables pertaining to stability. The Coastal Engineering Manual provides values for KD based on
specific circumstances and conditions. Various KD values have been derived for
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However, the Hudson formula has limitations in that:
• Potential scale effects due to small scales under which most of the tests were conducted;
• It uses regular waves;
• No consideration is given to wave period and storm duration; • It does not describe the damage level;
• It does not consider wave overtopping and is related to permeable structures only;
• It does not take into consideration foreshore slope.
The use of KD and cot
α
does not always best describe the effect of theforeshore slope.
2.3
Stability formula of van der Meer
Based on the earlier work of Thompson and Shuttler (1975), van der Meer undertook a series of extensive tests at the Delft Hydraulics Laboratory (1987). The tests included a wide range of structures for different core and underlayer permeabilities over a wide range of wave conditions. Van der Meer’s research was based on a large data set and considered deep water conditions at the toe of the structure for plunging and surging waves. Finally van der Meer proposed the following formulas:
5 . 0 2 . 0 18 . 0 50 2 . 6 − = ∆ N
ξ
S P D H n sPlunging waves
ξ
<ξ
transition (2.3)p n s N S P D H
ξ
α
cot 0 . 1 2 . 0 13 . 0 50 =∆ − Surging waves
ξ
>ξ
transition (2.4)20
(
)
0.5 1 31 . 0 tan 2 . 6 + = P transition Pα
ξ
(2.6) Where:Hs Significant wave height [m]
Dn50 Nominal mean diameter [m]
P Permeability factor [-]
S Damage level [-]
N Number of waves [-]
ξ
Surf similarity parameter [-]The reliability of these formulas depends on the difference due to random behaviour of rock slopes, accuracy of measuring damage and curve fitting. P is a measure of the permeability of the structure, S a measure of the damage and N is the number of waves. The van der Meer formula seems to be more involved when compared to the Hudson formula in that more parameters are included like the Iribarren number (wave steepness), the porosity (permeability) of the structure and the damage level, however it is still limited in the case of steep and shallow foreshores. Furthermore, it considers deep water conditions at the toe of the structure.
The slopes considered by Hudson and van der Meer were in the range 1:15 to 1:6 which can be considered as mild slopes. It can then be deduced that the van der Meer formula would not be able to satisfy the research objectives of this study were steep foreshore slopes and shallow water conditions are considered. However, the van der Meer formula for the plunging case which is based on a zero foreshore slope will be used later on in the analysis when making comparisons to the laboratory experiments.
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Chapter 3: Experimental Set-up
For the purpose of completion, this chapter shall illustrate the experimental process that was undertaken at TU Delft’s laboratory. The experiment as described by Hovestad (2005) shall be used as the analysis in this study is based on his data set.
3.1
General experimental set-up
The dataset is based on experiments that focussed on the different influences of armour unit stability considering foreshore slope. As mentioned earlier, the design of breakwaters on steep foreshores is problematic and the design formulas do not take into account the effects of foreshore slope on armour unit stability. The main experimental questions were to investigate whether armour unit stability is affected by foreshore slope and if so, how large are these influences?
In the experiments two foreshore slopes were considered, namely a 1:302 and a 1:8 foreshore slope. The values of the foreshore slope steepness were based on practical considerations. The values had to also be as far apart as possible in order to create the biggest possible effect. The 1:30 slope was chosen as the mild case as the floor to the flume in the laboratory is permanently fixed to this inclination. The 1:8 slope was a result of optimisation, it needed to be as steep as possible, but making it too steep would result in the horizontal length of the slope becoming too short in relation to the wave lengths to be used. If the horizontal length became too short, this could impact on the ability of the wave to shoal correctly. Thus the steepness was chosen in such a way that the length of the slope was approximately twice the wave length (the wave length in this case defined at the deepest part of the flume, based on the peak period). This process resulted in the 1:8 foreshore slope.
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3.2
Breakwater
The tests on the breakwater were conducted in the “Lange Speurwerkgoot”-(Long Research Flume) of the Fluids Mechanics Laboratory of the Faculty of Civil Engineering and Geosciences at Delft University of Technology.
The dimensions of the flume were 80cm wide 42m long and had a maximal depth of 100cm. The dimensions of the flume especially the width will be considered later on when the damage level on the breakwater slope will have to be expressed in terms of damage level S. The damage level in these tests were expressed in terms of damage level N and, within the context of this study damage level will be expressed in terms of damage level S in order to make a comparison with the van der Meer equation.
Figure 3: Breakwater layout
The layout of the breakwater in the flume can be seen in figure 3. The breakwater was in all series constructed at 50cm above the reference level with the top of the breakwater just reaching the chain rail thus making it 45cm high. The front slope was constructed at a 1:2 gradient and the rear slope to a 1:1.5 grade. The crest was 25cm wide. As can be seen from figure 4, the rock armour layer did not extend all the way to the top of the breakwater as this was not
23
necessary as most of the wave impact occurred exclusively in the lower zone. Figure 4 and figure 5 show the cross-section of the breakwater in the flume.
Figure 4: Breakwater Cross-section
Figure 5: Side elevation of the front side of the breakwater
The selection of the stones was done by measuring a sample of stones with a slide gauge. The definition of the stone dimensions was according to the CUR manual, 1994, where:
z sieve size (stone that fits through the small sieve hole) l maximum axial length
d thickness or axial breadth 183 16 45 25 1:2 1:1.5 Toe Armour Core Filter
24
The definition was also used in the selection process were the CUR manual advises to limit the amount of stones with a ratio of l/d >3 to an amount of 3 to 5 percent.
After an iterative process the D50 of the stones (ie. the sieve size through which
50% of the stone passes) was found to be 2.02cm and the D85 was 2.22cm. The
D15 was 1.78cm thus yielding a D85/D15 ratio of 1.25cm.
From the above weights the volume of the stones were then calculated. The Dn50
and the stone density were calculated according to the following expression:
U D D w s m m m − =ρ ρ (3.1) and
(
)
3 w U D n m m Dρ
− = (3.2)ρw was taken at a constant 1000 kg/m3
where:
Vs volume of a stone (cm3)
ρs density of the stone (g/cm3)
ρw density of water (g/cm3)
mD dry mass of the stone (g)
mU underwater mass of the stone (g)
The stones that were chosen, had a Dn50 (i.e. a median Dn) of 1,57cm.
In the calculation of the damage area on the breakwater slope, the area will be expressed in terms of the Dn50 of the stone. Stone densities varied from
25
Due to the differences in densities, the damage estimation may be affected as the stability of the stones is heavily influenced by the density. The thickness of the armour layer was according to CUR guidelines which recommend a layer of at least 2dn50. As large numbers of damage was expected in these tests the armour
layer was doubled resulting in a layer thickness of 6cm.
For the purposes of completion it is worth noting that in the wave height calculation used in this study which is based on the experiment conducted, irregular waves were used. This was done in order to simulate a “real” sea state in front of the breakwater so that the results could be translated to prototype situations. The waves were generated according to the standard Jonswap-spectrum which describes a young sea state as opposed to the Pierson-Moskowitz spectrum which describes a fully developed sea state which would hardly occur in nature.
The standard Jonswap-spectrum is described by:
− − − − − − = m m f f f m f f f g f E
α
π
γ
2 σ 1 exp 0 4 5 4 2 4 5 exp ) 2 ( ) ( (3.3) with:E spectral energy density [m2/Hz]
α scaling parameter (Pierson-Moskowitz) [-]
f frequency [Hz]
fm peak frequency [Hz]
γ0 scaling parameter (Jonswap peak-enhancement factor) [-]
σ scaling parameter (Jonswap peak enhancement factor) [-]
26
The value of σ changes according to the frequency:
σ = σa if f < fm
σ = σb if f > fm
For the standard Jonswap spectrum, the values in the peak-enhancement factor are:
γ0=3,3
σa=0,07
σb=0,09
These values were used in all the experiments. The theory on wave spectra is explained according to BATTJES (1992).
The wave steepness was defined as
L H s= with: s wave steepness [-] H wave height [m] L wave length [m]
From the above analysis a deep water wave height of 12cm was used based on different wave steepness’s. The measured wave heights at the toe and in deep water together with the wave steepness shall be used later in the calculation of the damage area to the breakwater on different foreshore slopes.
27 2 50 n e d A S= (3.4) With: S damage level [-] Ae Erosion area [m2]
It is imperative to understand this definition within the context of this study as the results of the experiment based on a 1:30 and 1:8 foreshore slope shall be compared to the van der Meer formula. As can be seen from figure 6 if a stone is removed from its original band but remained within the erosion area it will be counted as damage if just the displacements of the stones are counted. However, according to the definition of damage by van der Meer, which is based on the
erosion area of the cross-section, this will amount to no damage. The counting
of the stones method will thus over-estimate the damage level when compared to the van der Meer formula. A comparison will be made in the following chapter in order to see the relationship based on the experimental tests and that of the van der Meer formula.
Also important to note is that according to the van der Meer and van Gent formula the damage should grow with the 5th power of the wave height. Both formulas have a termH ∝S0.2 or inverselyS∝H5. This expression becomes important in the next chapter when the van der Meer equation has to be expressed in terms of damage level S.
28
All the tests conducted in the experiment were undertaken using N=1000 waves which is common in breakwater research. It is also important to note at this point, as this will influence the comparisons being made in the following chapters: the tests that were conducted on the stone armour layer (rip-rap) showed a strong influence on the wave steepness and the target wave height was used enabling the stones that were moved from their original band to be counted.
The normal displacement of stones that would be expected to occur under calm wave conditions did not occur resulting in a higher damage level when compared to the van der Meer formula. The definition of damage is thus critical in understanding the comparisons being made. Figure 7 shows the damage on the armour layer after being subjected to testing. The damage can be seen on the lower part of the breakwater slope.
Figure 7: Rip-Rap armour after testing
The majority of the tests were undertaken to make comparisons on steep foreshores. Four different wave steepness’s were used: so=0.030, 0.044, 0.058
and 0.086 respectively. The wave height at the board varied between Hmo=11.4cm and Hmo=12.1cm. These values will be used later when calculating
29
Chapter 4: Analysis
This chapter shall focus on the analysis of the results undertaken in the laboratory when compared to the van der Meer formula for breakwater armour unit stability, in order to determine what correlation exists between damage level and foreshore slope.
4.1
Data Analysis
During the laboratory testing undertaken by Hovestad (2005), large amounts of data were available which was undertaken for damage of breakwater armour layer based on the two different foreshore slopes mentioned earlier. The focus of this research was to find that parameter not included in the wave spectrum which seems to be relevant for armour unit stability. As previously mentioned, in the case of coasts with steep foreshores, coastal structures experience more damage than can be normally expected from given boundary conditions at deep water.
It is primarily for this reason that most design guidelines and manufactures of armour units suggest using a heavier class of rock or a lower Kd value in the
case of steep foreshores. As stated in the introduction a first step into the more systematic approach to addressing the problem associated with shallow water and steep foreshores was introduced by Van Gent et al (2003), were the results revealed that the load on the structure can best be described by a shallow water wave spectrum at the toe of the structure. Furthermore, since the longer period waves are more significant in this case than the shorter period waves, more care should be given to the lower frequency part of the spectrum.
30
performed were tabulated in terms of time series and frequency series with their associated program number. As a first step each test number was sorted to match its corresponding times series and frequency series. The test results included the test number, the program number and the damage level expressed in terms of Ns and Nod. Table 1 shows a typical result summary table indicating
all the relevant parameters. The values of tan α, Dn50, the flume width B etc
31
dn50 0.0157 (m) tanα 0.50
B 0.8 (m) ∆ 1.78
P 0.6
N 1000 (T01040/41/42: 2000)
Testnumber Program Wh Rd Bl Pk Ye Gr Rd+ Ye+ Gr+ Bk NS Nod
T00010 M05D66 4 15 15 8 4 46 0.90 T00011 M06D66 6 24 51 41 23 1 146 2.87 T00012 M07D66 8 56 100 74 43 6 287 5.63 T00013 M08D66 15 73 138 123 113 35 20 517 10.15 T00014 M09D66 18 125 192 185 209 150 23 902 17.70 T00015 M10D66 20 238 224 211 285 261 57 1296 25.43 T00016 M11D66 31 324 344 225 298 324 55 30 1631 32.01 T00020 M05D66 2 19 23 13 1 58 1.14 T00021 M06D66 6 35 59 37 10 1 148 2.90 T00022 M07D66 10 55 108 82 59 4 318 6.24 T00023 M08D66 13 86 158 147 123 40 1 568 11.15 T00024 M09D66 16 116 191 192 186 127 11 839 16.47 T00025 M10D66 24 223 224 227 286 211 42 21 7 1265 24.83 T00026 M11D66 36 273 257 258 425 303 54 29 13 1648 32.34 T00100 M05D66 9 26 43 9 87 1.71 T00101 M06D66 14 49 96 82 8 249 4.89 T00102 M07D66 26 80 164 183 82 6 541 10.62 T00103 M08D66 33 123 249 346 266 75 1092 21.43 T00104 M09D66 39 148 295 415 391 284 9 1581 31.03 T00110 M05D66 1 18 38 7 64 1.26 T00111 M06D66 9 51 83 51 9 203 3.98 T00112 M07D66 15 94 190 207 101 7 614 12.05 T00113 M08D66 22 115 238 311 231 126 1043 20.47 T00114 M09D66 30 156 286 386 364 254 3 1 1480 29.05 T00120 M05D66 5 36 31 5 77 1.51 T00121 M06D66 9 70 81 39 1 200 3.93 T00122 M07D66 18 131 164 162 60 3 538 10.56 T00123 M08D66 30 196 237 303 231 78 1075 21.10 T00124 M09D66 37 228 296 410 387 256 3 1617 31.73 T00130 M08D66 10 145 214 260 210 30 869 17.05 T00131 M08D66 10 138 200 262 162 19 791 15.52 T00132 M08D66 12 155 240 255 163 6 831 16.31 T00140 M08B66 18 175 357 261 339 258 6 1414 27.75 T00141 M08B66 7 176 258 371 337 275 1 1425 27.97 T00142 M08B66 14 163 251 371 364 292 10 1465 28.75 T00150 M08C66 15 171 239 323 271 122 1141 22.39 T00151 M08C66 21 207 221 274 214 74 1011 19.84 T00152 M08C66 10 215 229 329 268 113 1164 22.84 T00160 M08E66 14 162 160 165 37 1 539 10.58
Table 1: Laboratory measurement results
32
study. Wh (white), Rd (red), Ye (yellow) all refer to the colours of the stones that were removed from that band during testing. As can also be seen the damage level is expressed in terms of Ns and Nod.
For the purposes of this study the damage level taken from table 1 shall be transformed and expressed in terms of damage level, S. Table 2 shows a typical layout of the tabulation of the test results corresponding to their program numbers. Relating each test to their relevant test number and program numbers, the dataset could thus be completed.
Program M05D66 M06D66 M07D66 M08D66 M09D66 M10D66 M11D66 M08E66 M08F66 T (s) 0.89 0.98 1.06 1.13 1.2 1.27 1.33 1.01 0.92 Hm0,b (m) 0.071 0.084 0.098 0.111 0.124 0.137 0.151 0.112 0.113 hb (m) 0.66 0.66 0.66 0.66 0.66 0.66 0.66 0.66 0.66 Hm0,0 (m) 0.071 0.086 0.101 0.116 0.131 0.147 0.162 0.114 0.114 s0 (-) 0.057 0.057 0.057 0.058 0.058 0.058 0.059 0.072 0.086
Table 2: Test parameters relating to program number
33
Time Series Frequency
(File name) (File name) Toe Deep water 1:30 1:8
1 T00010 0.071 0.071 0.89 46 2 T00011 0.084 0.086 0.98 146 3 T00012 0.098 0.101 1.06 287 4 T00013 0.111 0.116 1.13 517 5 T00014 0.124 0.131 1.2 902 6 T00015 0.137 0.147 1.27 1296 7 T00016 0.137 0.147 1.27 1631 8 T00020 0.151 0.162 1.33 58 9 T00021 0.084 0.086 0.98 148 10 T00022 0.098 0.101 1.06 318 11 T00023 0.111 0.116 1.13 568 12 T00024 0.124 0.131 1.2 839 13 T00025 0.137 0.147 1.27 1265 14 T00026 0.151 0.162 1.33 1648 15 T00100 R00100DN S00100DN 0.071 0.071 0.89 87 16 T00101 R00101DN S00101DN 0.084 0.086 0.98 249 17 T00102 R00102SN S00102DN 0.098 0.101 1.06 541 18 T00103 R00103DN S00103DN 0.111 0.116 1.13 1092 19 T00104 R00104DN S00104DN 0.124 0.131 1.2 1581 20 T00110 R00110DN S00110DN 0.071 0.071 0.89 64 21 T00111 R00111DN S00111DN 0.084 0.086 0.98 203 22 T00112 R00112DN S00112DN 0.098 0.101 1.06 614 23 T00113 R00113DN S00113DN 0.111 0.116 1.13 1043 24 T00114 R00114DN S00114DN 0.124 0.131 1.2 1480 25 T00120 R00120DN S00120DN 0.071 0.071 0.89 77 26 T00121 R00121DN S00121DN 0.084 0.086 0.98 200 27 T00122 R00122DN S00122DN 0.098 0.101 1.06 538 28 T00123 R00123DN S00123DN 0.111 0.116 1.13 1075 29 T00124 R00124DN S00124DN 0.124 0.131 1.2 1617 30 T00130 0.111 0.116 1.13 869 31 T00131 0.111 0.116 1.13 791 32 T00132 0.111 0.116 1.13 831 33 T00140 0.110 0.120 1.6 1414 34 T00141 0.110 0.120 1.6 1425 35 T00142 0.110 0.120 1.6 1465 36 T00150 0.110 0.118 1.31 1141 37 T00151 0.110 0.118 1.31 1011 38 T00152 0.110 0.118 1.31 1164 39 T00160 0.112 0.114 1.01 539 40 T01000 R01000DN S01000DN 0.110 0.120 1.60 1482 41 T01001 R01001DN S01001DN 0.110 0.120 1.60 1417 42 T01002 R01002DN S01002DN 0.110 0.120 1.60 1386 43 T01010 R01010DN S01010DN 0.110 0.118 1.31 948 44 T01011 R01011DN S01011DN 0.110 0.118 1.31 990 45 T01012 R01012DN S01012DN 0.110 0.118 1.31 1011 46 T01020 R01020DN S01020DN 0.111 0.116 1.13 653 47 T01021 R01021DN S01021DN 0.111 0.116 1.13 664 48 T01022 R01022DN S01022DN 1.13 717 49 T01023 R01023DN S01023DN 0.111 0.116 1.13 694 50 T01030 R01030DN S01030DN 0.113 0.114 0.92 284 51 T01031 R01031DN S01031DN 0.113 0.114 0.92 295 52 T01032 R01032DN S01032DN 0.113 0.114 0.92 277 53 T01040 R01040DN S01040DN 1.13 1072 54 T01041 R01041DN S01041DN 1.13 1056 55 T01042 R01042DN S01042DN 1.13 1005 56 T02000 R02000DN S02000DN 0.110 0.12 1.6 1786 57 T02010 R02010DN S02010DN 0.110 0.118 1.31 1455 58 T02011 R02011DN S02011DN 0.110 0.118 1.31 1394 59 T02020 R02020DN S02020DN 0.111 0.116 1.13 1102 60 T02030 0.113 0.114 0.92 632 61 T02050 R02050DN S02050DN 0.096 0.102 1.22 973 62 T02051 R02051DN S02051DN 0.096 0.102 1.22 947 63 T02060 0.105 0.112 1.31 1311 64 T02070 R02070DN S02070DN 0.096 0.104 1.31 1082 65 T02071 R02071DN S02071DN 0.096 0.104 1.31 1106 66 T02080 R02080DN S02080DN 0.098 0.102 1.13 915 67 T02081 R02081DN S02081DN 0.098 0.102 1.13 890 68 T02090 R02090DN S02090DN 0.096 0.105 1.6 1390 69 T02091 R02091DN S02091DN 0.096 0.105 1.6 1466 70 T02100 0.099 0.1 0.92 458 71 T02110 S02110SN 0.094 0.095 0.92 395 72 T02111 0.094 0.095 0.92 359
Table 3: Final datasheet of test results
Damage level Ns Number
Wave height Foreshore slope
Test number
34
Tests numbers that begin with 00 indicate preparatory tests that were undertaken to obtain the best configuration and layout of the breakwater. Test numbers that begin with the 01 indicate measurements that were undertaken for a mild foreshore slope and tests that begin with the numbers 02 indicate tests on a steep foreshore slope. After this was completed and the results tabulated in table 3, the tests that have complete records that would be beneficial to this study can be seen. The tests that were used in this analysis for the different foreshore slope is ticked in table 3. In table 3 a link was created between the time series, the frequency series and the test data in order to obtain a complete picture of the testing procedure and data. Once this picture of complete and relevant records was obtained the analysis in terms of damage level and comparisons can now be undertaken.
4.2
Description
In order to obtain a more reliable and accurate correlation between the measured laboratory results and the van der Meer equation which is based on a zero foreshore slope, the void ratio of the stones had to be calculated and the damage level on the breakwater slope expressed in terms of dn50. The erosion area had to
be calculated and expressed in dn50 considering the width of the flume and the
van der Meer equation re-written in terms of damage level, S.
35
4.3
Damage area and void ratio calculation
The damage level as defined by van der Meer is given by the expression:
2 50 n e d A S= (4.1)
The damage of the armour layer can be given as a percentage of the displaced rocks related to a certain area (the whole or part of the layer). In general it is difficult to compare various structures as the damage figures are related to different totals for each structure and test condition. Another possibility is to describe the damage by the erosion area around the still water level. When this erosion area is related to the dn50 of the rock, a dimensionless damage level can
be obtained which is independent of the size, slope angle and height of the structure. This damage level is defined by equation 4.1 and the extent of Ae can
be seen in figure 8.
Figure 8: Description of damage level on breakwater slope
36
The damage should consider both the settlement and displacement of the rocks. Another description of S is the number of cubic stones with a side of Dn50 that is
eroded within a Dn50 wide strip of the structure, the approach that will be
adopted in this study. The actual number of stones eroded will be dependant on the porosity, the stone grading and the obliqueness of the stone (shape of the stone).
The erosion area on the breakwater slope will now be calculated in order to apply equation 4.1 and express the damage level in terms of S. All the test results will then be factored by this and a new plot generated in terms of wave period and damage level, S. The outputs of the different plots shall be discussed in the next chapter entitled results.
Considering the erosion area Ae in figure 8 and expressing in terms of dn50
reveals:
Since the stones are spherical and not all the stones have the same shape, the erosion area shall be expressed in terms of dn.
Volume of sphere = 3 3 4 r π W ≡ρx dn503
Calculating in terms of dn, volume equals
37
The value of d will have to be incorporated into the damage level area calculation in order for it to be expressed in terms of S.
The void ratio of the stones will now have to be considered.
VT = Va + Vs Void ratio e = s a V V Va = air + water.
In the laboratory experiment the layer thickness was taken as 6cm which is double when compared to the Manual on the use of rock in coastal and shoreline engineering (CUR, 1991) which advises a layer thickness of at least 2dn50. The
layer thickness was doubled in the experiment as large scale damage was anticipated.
Va
Vs
38
In the calculation of the void ratio, the dimensions of the flume especially the width (80cm) used in the laboratory testing will have to be considered.
The total volume of rock in the armour layer is given by:
L
t ×0.8× (see illustrative diagram above)
Now the number of stones per metre width of flume = stones
d 37 8 . 0 50 = Volume of cube V = dn3
dn = 0.81 d50 (from expression 4.2 above)
Substituting dn = 0.81 d50 into the volume of the cube we get:
39 V = dn3
= (0.81 d50)3 This is the volume of 1 stone.
An assumption is made that the stones are spherical but in reality this is not the case, hence the co-efficient will be somewhat greater than 0.81.
Number of stones = 2 x 37 x 50
n d
L
(the armour layer consisted of 2 layers)
Total volume of stones = number of stones x volume of 1 stone
Vstones =
(
)
3 50 50 81 . 0 37 2 x d d l x x n (4.3)In order to calculate the total volume we need the volume of voids as the total volume is equal to the volume of stone + volume of voids. VT = Vv + Vs.
We now need to calculate the volume of the space (voids).
Vspace = B x t x L
Volume of space Vspace = 0.8 x 0.06 x L (4.4)
40 Total volume VT = Vv + Vs
= (Vspace x e) + Vs
= Vs (1 + e) volume of the erosion area on the breakwater
Vv = Vspace x void ratio
Having worked out this expression we now need to calculate the void ratio e in order to compute the total volume.
stone stone space s v V V V V V e ratio Void = = − (4.5)
Vspace = B x t x L from 4.4 above
Vstone =
(
)
3 50 50 81 . 0 37 2 x d d L x x n from 4.3 aboveSubstituting Vspace and Vstone into expression 4.5 reveals:
41
From expression 4.6 we can now calculate the void ratio as we have the d50 and
L.
VT = Vs (1 +e)
VT = (0.81 d50)3 x N (1+e)
Equating the (erosion area x the width of the flume) to VT we get:
Width of flume x A = volume of 1 stone x number of stones + the void ratio.
This reveals,
(0.8 x A) = (0.81 d50)3 x N (1+e)
Re-writing this expression in terms of A
A= d e N B (1 ) 81 . 0 3 50 + A = d (1 e) N 8 . 0 81 . 0 3 50 + (4.7)
In the above expression B equals the width of the flume, d50 the stone size, e is
the void ratio (expression 4.6) and N is the damage level from the laboratory experiments.
We have now arrived at an expression for the damage area A and equation 4.7 will now be used to calculate the areas and then expressed in terms of damage level, S. We know the value of N (from laboratory tests), the value of d50 and
42
values of S will be plotted against wave period and the various output graphs shall be discussed in the following chapter.
We have now expressed all the laboratory tests in terms of damage level, S and in order to make a comparison to the van der Meer formula we need to express the van der Meer formula in terms of S. Van der Meer proposed the following formula for the plunging case. This formula will have to be re-written in terms of S. Van der Meer described conditions for the surging and plunging case, within the context of this study the plunging case will only be considered as the laboratory experiments were conducted for the plunging case.
5 . 0 2 . 0 18 . 0 50 2 . 6 − = ∆ N
ξ
S P D H n sPlunging waves
ξ
<ξ
transition (4.8)where:
Hs significant wave height ≡ H1/3 [m]
dn50 nominal diameter 3 50
s
M
ρ
≡ [m]
M50 median stone mass [kg]
P notional permeability [-] S damage number 2 50 n e d A ≡ [-]
Ae cross section of the erosion area [m2]
N number of waves [-]
ξα0 Iribarren-parameter using deep water steepness and breakwater slope
43
Re-writing expression 4.8 in terms of S reveals:
5 . 0 2 . 0 18 . 0 50 2 . 6 − = ∆ N
ξ
S P D H n s 2 . 0 18 . 0 5 . 0 50 6.2 = ∆ N S P x D H n sξ
N S P x D H n s = ∆ 5 18 . 0 5 . 0 50 6.2ξ
S N x P x D H n s = ∆ 5 18 . 0 5 . 0 50 6.2ξ
S =( )
5 18 . 0 2 . 0 5 . 0 50 6.2 ∆ P N D H n sξ
(4.9)This derivation supports the theory of van der Meer who states that the damage should grow with the 5th power of the wave height.
44
Chapter 5: Results
The focus of this chapter shall be on the results of the analysis undertaken in the previous chapters. Comparisons will be made with the laboratory tests on different foreshore slopes when compared to the van der Meer formula which is based on a zero foreshore slope.
5.1
Results of Laboratory tests
Table 4: Summary of relevant tests
Table 4 is a summary of all the relevant tests that have complete records. The test numbers in table 4 are extracted from the comprehensive dataset in table 3. All the comparisons in this study were based on the dataset as reflected in table 3. A first step would be to plot a graph showing the exact damage in terms of damage level Ns against wave period without calculating the “damage area” and
factoring the Ns values. Figure 9 shows the relationship between damage level
Ns from the laboratory experiments and wave period.
45 Laboratory tests 200 400 600 800 1000 1200 1400 1600 1800 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Tp (s) N u m b e r o f s to n e s m o v e d N s 1:30 slope 1:8 slope
Figure 9: Initial plot of the actual damage of laboratory results
Figure 9 shows the actual damage as calculated in the laboratory experiments when plotted against wave period. The damage in the laboratory experiments were based on Ns which is the total number of stones moved.
The basic settings for this experiment were:
Wave height (at the wave board) Hm0,0 : 12.1 cm
Wave period Tp : 1.6 seconds
Water depth : 66 cm
Rock size Dn50 : 1.57 cm
ρ : 2780 kg/m3
In order to simulate actual conditions and make reliable comparisons, in the first series of tests the wave height at the wave maker was kept identical and in the second series the wave at the toe was kept identical. This was done in order to keep the spectra the same were the incoming spectrum at the toe of the breakwater was exactly the same for both the 1:8 and 1:30 foreshore slope.
46
Furthermore, with the tests that had identical waves at the toe of the breakwater the results indicated a significantly higher damage for the steeper slope especially when the wave steepness is considered.
This yields interesting results: because the spectra are identical but the damage is clearly not identical when comparing the cases of the different foreshore slopes means that there is a parameter which is not represented by the shallow water wave energy spectrum for waves approaching over different foreshore slopes. One observation is clearly evident that the damage level is greater for a steep foreshore slope when compared to a milder slope even when the damage level is based on the actual number of stones displaced.
It is also important to note at this point that when a plot of
ξ
(surf-similarity parameter) is generated against increase in damage, one notices the damage level increasing with increasing alpha values. This means that higherξ
values yield lower damage and vice versa. Furthermore, important to note is that most graphs are generally plotted in terms ofξ
, but the graphs in this study are plotted in terms of wave period Tp. The primary reason for this is that the slopeis constant in this case and Hm,0 is kept the same at the toe and deep water,
therefore, plotting the graphs in terms of period is the same as plotting them in terms of
ξ
.5.2
Comparisons based on erosion area
47
New damage in terms of S
0 20 40 60 80 100 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Period Tp (s) D a m a g e l e v e l S 1:30 slope 1:8 slope
Figure 10: Damage level in terms of S
Figure 10 shows the damage level of the laboratory experiments expressed in terms of S. This leads to the conclusion that if the wave heights at the toe are equal, the trend is the same for a given bottom steepness. Equation 4.7 was used to transform the damage level from the laboratory experiments expressed in terms of N as can be seen in figure 9 and, equation 3.4 was used to express them in terms of S as shown in figure 10.
In order to make a comparison with the van der Meer formula, expression 4.9 was used to generate the van der Meer curve as can be seen in figure 11. The surf similarity parameter was based on the dataset as shown in table 4.
van der M eer in terms of S
0 30 60 90 120 150 1.1 1.2 1.3 1.4 1.5 1.6 Wave Period Tp D a m a g e L e v e l (S )
van der Meer Linear (van der Meer)
48
The blue line in figure 11 is a linear baseline fit to the van der Meer formula. When comparing figure 10 with figure 11 there seems to be no correlation between the measured results for a 1:8 slope and a 1:30 slope when compared to the van der Meer formula. One expected the damage to be less significant. This could be as a result of the following:
1. In the calculation of the damage area on the breakwater slope, the porosity of the stone together with the void ratio is ignored.
2. Furthermore, it is assumed that all the stones have the same shape and are packed one next to each other with the same packing density.
Also important to note is the definition of damage according to van der Meer. Van der Meer describes damage as the area of the erosion in the cross-section of the breakwater divided by the square of the stone size. The damage in the case of the experiments were based on the actual number of stones moved, which might not necessarily be out of the “erosion area” as defined according to van der Meer. This means that the counting of stones method structurally overestimates the damage level hence the van der Meer equation cannot be applied easily. Furthermore, an important consideration is that during the testing conducted by Hovestad (2005), the target wave height was immediately used in order to reduce the gradual build up of wave heights. This was done to aid the counting process and save on time, which resulted in the initial settlement of stone that usually take place under calm wave conditions, not taking place, so a higher damage level could be expected especially when comparing to the van der Meer formula.
49
All damage curves
0 30 60 90 120 150 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Wave Period Tp D a m a g e L e v e l (S ) 1:30 slope-old 1:8 slope-old
Linear (van der Meer) Linear (1:8 slope-New)
Linear (1:30 slope-New)
Figure 12: Graph of all damage levels
Figure 12 shows all the test results superimposed onto the same system of axis and the original test results based on actual measured damage has been included to complete the picture. The blue and yellow dots are the original laboratory experiments. Expression 4.9 was used to obtain the damage in terms of S, the van der Meer formula denoted as the pink line in figure 12. Expressions 4.7 and 4.1 were used to transform the laboratory results based on the different foreshore slopes denoted as the green and blue lines in figure 12. The blue, green and pink lines in figure 12 are a baseline fitted through the points in order to see the differences clearer.
50
51
Chapter 6: Conclusion and Recommendation
This research has clearly shown the importance of foreshore slope on breakwater armour unit stability. There exists a clear correlation between damage level and foreshore slope: increasing foreshore slope resulting in greater armour unit damage for equal off-shore conditions. When comparing wave heights on different sea bottoms, average waves on a steep sea bottom can become higher when compared to milder foreshores. This is as a result of shoaling before the waves actually break. The larger wave heights will consequently result in a greater amount of damage to the armour units of breakwaters. The longer period waves seem to reflect more damage over both foreshore slopes.
The research has further showed that there could be other parameters affecting armour unit stability namely:
• Wave skewness or peakedness; • Porosity and Density of the stones; • Void ratio;
• Angularity and elongation of the stones;
• Velocity and acceleration of the waves on a slope.
52
slopes. As a preliminary result, the value of 50
n s
D H
∆ in the van der Meer
formula has to be reduced by a “factor” when considering structures on steep foreshores.
In general structures that are to be constructed on steep foreshores should be designed with more consideration given to armour unit stability. In the absence of practical design guidelines relating specifically to steep foreshores, extensive physical model studies should be undertaken for structures on steep foreshores.
Recommendations
• More research should be conducted on breakwaters with steep foreshores; • Investigation into other parameters other than foreshore slope, like wave
peakness, velocity and acceleration;
• Specific definition of damage area for armour unit stability;
• As a possible recommendation when calculating the damage area for the van der Meer equation to be expressed in terms of S, the amount N can be normalised to the number of displaced units by the following expression: Nod = B d Ns n50 Where:
Nod number of displaced units [-]
NS number of stones that were removed from their original band [-]
B width of the measured section [m]
Theoretically, the volume of the erosion can then be derived using Ve = Ae B
and with Ae Dn502, Ve = SBdn50. The porosity has to be considered for reliable
53
BIBLIOGRAPHY
Battjes, J.A., “Surf-Similarity”, Proc. of the 14th conference on coastal engineering, Copenhagen, Denmark, 1974
CUR, “Manual on the use of Rock in Hydraulic Engineering”, Cur Publication 169, 1994
Hovestad, M., “Breakwaters on steep foreshores”, MSc Thesis, Delft University of Technology, 2005
Terrile, E., “The threshold of motion of coarse sediment particles by regular non-breaking waves”, MSc Thesis, Delft University of Technology, 2004
TROMP, M., “Influences of Fluid Accelerations on the Threshold of Motion”, MSc Thesis, Delft University of Technology, 1994
USACE, “Coastal Engineering Manual”, Fundamentals of Design, U.S Army Corps of Engineers, 2005
USACE, “Shore Protection Manual”, Fourth edition, U.S. Army Engineers Waterways Experiments Station, Coastal Engineering Research Center, 1984
Van der Meer J.W., “Rock slopes and gravel beaches under wave attack”, Doctoral Thesis, Delft University of Technology, 1988
Van Gent, M.R.A., Smale, A., Kuiper, C., “Stability of rock slopes with shallow foreshores”, Portland, ASCE, 2003
