An experimental study on the parameterization of reshaped seaward profile of berm breakwaters

Coastal Engineering 91 (2014) 123–139

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An experimental study on the parameterization of reshaped seaward profile of berm breakwaters Mehdi Shafieefar ⁎, Mohammad Reza Shekari Department of Civil and Environmental Engineering, Tarbiat Modares University, Iran

a r t i c l e

i n f o

Article history: Received 25 May 2013 Received in revised form 27 May 2014 Accepted 28 May 2014 Available online xxxx Keywords: Berm breakwater Reshaped seaward profile Experimental study Key parameters

a b s t r a c t Berm breakwaters are rubble mound structures in which the seaward slope of the initial profile may be reshaped to become more stable under severe wave attack. The stones in the seaward slope move from the initial slope to an equilibrium profile. A 2D experimental study has been carried out in a wave flume at a hydraulic laboratory of Tarbiat Modares University to study the effects of sea state and structural parameters on the reshaped profile parameters of such breakwaters. A series of 287 tests have been performed to cover the effect of various sea state conditions such as wave height, wave period, number of waves and water depth at the toe of the structure, and structural parameters such as berm width, berm elevation above still water level and armor stone size. All the tests have been done employing irregular waves with a JONSWAP spectrum. In this paper, first the reshaped profiles are schematized, and then the key parameters of the reshaped seaward profiles such as step height, step length and depth of intersection point of initial and reshaped profile are investigated, using results of this experimental work. Eventually, formulae that include some sea state and structural parameters are derived for estimation of the reshaped profile parameters. To assess the validity of the proposed formulae, comparisons are made between the estimated parameters of reshaped profiles by these formulae and earlier formulae given by other researchers. The comparisons show that the estimation procedure foretells reshaping parameters well and with less scatter according to the present data and also other experimental results within the range of parameters tested. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Berm breakwaters are rubble mound structures primarily built with a large porous berm at or above still water level at the seaward side. During severe wave attack, seaward slope of such structure may typically reshape into an S-shaped profile which is more stable than the originally built profile. Compared with common types of rubble mound breakwaters, the berm breakwaters have proven to be a more stable structure with easier and cost-effective repair methods. Besides, a berm breakwater is usually designed to make an optimum use of the available quarry material and rather wide stone gradation so that, up to 100% utilization of the quarry yield can be attained. Fig. 1 exhibits the typical original and reshaped profile of the cross sectional and deformation parameters of a berm breakwater. An important and simple method for evaluation the reshaping of a homogeneous berm breakwater is the recession of the berm (Rec) as defined by Burcharth and Frigaard (1988), cf. Fig. 1. Failure of a berm breakwater typically happens when Rec N B, where B is berm width ⁎ Corresponding author. Tel.: +98 21 88011001; fax: +98 21 88005040. E-mail addresses: shafi[email protected] (M. Shafieefar), [email protected] (M.R. Shekari).

http://dx.doi.org/10.1016/j.coastaleng.2014.05.009 0378-3839/© 2014 Elsevier B.V. All rights reserved.

(PIANC, 2003). A number of researchers have worked on stability of berm breakwaters under various sea states and different structural patterns and proposed some experimental formulae for estimating the reshaped profile parameters (Hall and Kao (1991), Tørum (1998), Tørum et al. (1999), Tørum and Krogh (2000), and Moghim et al. (2011)). Several commonly used parameters exist in relation to the stability of berm breakwaters, which the most ordinarily used dimensionless parameter is the stability parameter H0, denoting a relation between the armor layer and the impact of the incoming wave height as: H0 ¼

Hs Δ  Dn50

ð1Þ

where Hs denotes the incident significant wave height in front of the structures resulted from frequency domain analysis, Δ is the relative reduced mass density (Δ = ρs/ρw − 1) containing ρs as the mass density of armor unit and ρw as the mass density of water, and Dn50 reveals the median stone diameter (M50/ρs)1/3, where M50 is median stone mass. The foregoing linear relation between (Hs) and the relative reduced mass density (Δ) is assumed convincing although the relation appears non-linear, Helgason and Burcharth (2005). The effect of the wave

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Fig. 1. Usual initial and reshaped profile and deformation parameters of a berm breakwater.

period on the breakwater stability can be introduced in various ways, but the Van der Meer (1988) dimensionless parameter H0T0 as defined in Eq. (2) is very often employed. rffiffiffiffiffiffiffiffiffiffi Hs g  TZ  H0 T 0 ¼ Δ  Dn50 Dn50

ð2Þ

where Tz is the mean zero up-crossing wave period. Berm breakwaters are categorized into statically stable non-reshaped (H0 b 1.5–2, H0T0 b 20–40), statically stable reshaped (H0 = 1.5 to 2.7, H0T0 = 40 to 70) and dynamically stable reshaped (H0 N 2.7, H0T0 N 70), cf. Tørum et pal., ffiffiffiffiffiffi (2003). In the present research the dimensionless parameter H 0 T 0 is used which has already been introduced pffiffiffiffiffiffi by Moghim (2009). Dimensionless parameters H0, H0T0 and H 0 T 0 are cited as stability indices in some reference, e.g., Moghim et al. (2011). Van der Meer (1988) has performed an extensive two dimensional experimental research on dynamically stable slopes, including berm breakwaters confined to rock and gravel shores. The aim of the tests was to investigate the effect of some parameters on the reshaping of such structures. Afterwards Van der Meer (1992) focusing on the berm breakwaters, considered the sea conditions and armor stone parameters, leading to a computer program for estimation of the behavior of the seaward slope of berm breakwaters. He proposed the following formulae to calculate the reshaped profile parameters, i.e. step height, hs, and step length, ls (see Fig. 1): ðhS =H S Þ  N

0:07

−0:3

¼ 0:22  sm

  0:07 1:3 H 0 T 0 ¼ 3:8 lS =Dn50 N

ð3aÞ

ð3bÞ

where sm is the wave steepness and N is the number of incident waves. Lykke Andersen and Burcharth (2010) studied the front slope stability of breakwaters with a homogeneous berm in a large number of two dimensional model tests and presented a formula for calculating the step height. The formula includes various sea state and structural parameters as follows: −0:3

hS ¼ 0:65  H S  som  f β  f N

ð4Þ

where fβ and fN are dimensionless parameters for the wave direction (β) and, the number of waves (N). Tørum et al. (2003) introduced an

empirical formula for estimating the depth of intersection of initial and reshaped profile:   hf d þ 0:5 ¼ 0:2 Dn50 Dn50

12:5b

d b25 Dn50

ð5Þ

where d is the water depth in front of the structure. Recently, Lykke Andersen et al. (2012) studied the reshaping of berm breakwaters and concluded that the formula of Tørum underestimates hf on the basis of the analyzed data of experimental work. They performed a simple correction in Tørums formula, and replaced the coefficient 0.2 by 0.3 to improve the strong bias, i.e.   hf d þ 0:5 ¼ 0:3 Dn50 Dn50

12:5b

d b25 Dn50

ð6Þ

The berm recession is the most important parameter for describing the reshaping profile. This parameter has been the subject of many research works and various formulae have proposed for estimation of recession, e.g., Lykke Andersen et al. (2012), Tørum et al. (2012) and Moghim et al. (2011). Shekari and Shafieefar (2013) has recently presented a formula for estimation of berm recession which takes into account the influence of wave height and period, storm duration, berm width and elevation variations on the stability of berm with different stone sizes, as follows: Rec ¼ Dn50

   pffiffiffiffiffiffi2 pffiffiffiffiffiffi −0:016 H 0 T 0 þ 1:59H0 T 0 −9:86     B −0:15 hb −0:21  f1:72− exp½−2:19  ðN=3000Þg HS Dn50

ð7Þ

where N = number of waves, B = berm width and hb = berm elevation. The behavior of reshaped profile parameters is however expected to be much more complicated than expressed in the above equations. A review of existing experimental results showed that there is a significant scatter between the results of different formulae for the same sea state condition and structural parameters. The main intention of the present study is to systematically scrutinize the effects of a number of parameters related to reshaping process in order to set up a more reliable regression formula for each key parameter, considering the characteristics of the reshaped profile. To gain this goal, a set of 2D experimental modeling has been performed in a wave flume. Effects of structural parameters such as the berm width and elevation, the stone

Fig. 2. Longitudinal view of the wave flume and setup of wave gauges.

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Fig. 3. Cross-section geometry of the berm breakwater model.

weight, and the sea state conditions such as wave height and period, storm duration and water depth on the reshaped profile parameters, i.e. step height and length, and depth of intersection of reshaped and initial profile have been investigated. Subsequently using the analyzed data of present experimental work, formulae are derived for estimating the reshaped profile parameters that cover the effect of different wave and structural parameters considering different armor stone sizes. 2. Experimental set-up and tested structure designs The tests have been carried out in a 1 m wide, 1 m deep and 16 m long wave channel at the Hydraulic Laboratory of Civil and Environmental Engineering Faculty of Tarbiat Modares University. All the length of the flume is equipped with glass panels for easier observations and photography, cf. Fig. 2. A series of four resistance type wave gauges were placed along the channel according to the Mansard and Funke (1980) pattern for calculation of the wave spectrum. Fig. 3 shows the tested cross-section of the berm breakwater which was constructed at the end of the flume. Several stone sizes were employed in the berm in order to cover both statically and dynamically stable berm breakwaters. Table 1 shows the properties of the materials related to different armor and filter layers, and Fig. 4 reveals their grain size distribution curves. A total of 287 tests have been performed to investigate the effect of berm width and elevation, sea state (wave height and period), stone size and water depth. In all the present tests the slopes were 1:1.25, which was near to the natural angle of repose for the armor stones tested. Table 2 reveals the test program for various sea state conditions and structural parameters. This table gives the number of test cases for variation of different parameters in the experimental set up. For example 18 combinations of wave conditions, 6 different berm widths and 5 water depths were tested for the test cases of the armor with Dn50 = 1.7 cm. The range of dimensional and non-dimensional parameters covered in the tests is shown in Table 3. Results of the tests show that armor 1 corresponds to nearly all tests to a dynamically stable reshaped berm breakwater, while the largest armor stones, i.e. armor 3, corresponds to a statically stable berm breakwater, cf. Table 3. The experimental model was rebuilt for each experiment and the failure mode of the breakwater was considered when the filter layer was revealed. In order to investigate shoaling effects at the toe of the structure the recommendation by Goda and Suzuki (1976) was obeyed and placed one wave gauge at least one wave length from the intersection of still water level with the slope of the structure. This wave gauge indicated the superposition of the incident and reflected wave. Goda and Suzuki

(1976) suggested that the incident significant wave height could be estimated by the following equation in case of no phase locking: H s;iþr Hs;i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ C 2r

ð8Þ

where Hs,i is the incident significant wave height, Hs,i + r is the recorded significant wave height at the structure and Cr represents the reflection coefficient as obtained from the three wave gauges in deeper water. For measurement the reshaped profiles of the structure, a vertical point gauge fixed on a wave profiler system (see Fig. 2) was used in three lines manually. One profiling line was in the middle of the structure and the other lateral lines aligned with 20 cm distance from the middle one. To enable the movement of the wave profiler system along the flume, the flume is supplied with iron railings on the top of the channel longitudinal walls. After each test, the reshaped profile was measured in a lateral spacing of 1 cm along the profiling line. To calculate the reshaped profile parameters, the mean of three profiles has been used as the median of profiles. All the tests were carried out with irregular waves to observe the behavior of the structure using JONSWAP spectrum with a peak enhancement factor γ of 3.3. Fig. 5 shows the tested sea state ranges, i.e. significant wave height and steepness (som) of various wave periods. The armor stone Reynolds number given by Eq. (9) is employed to evaluate viscous scale effects in rubble mound breakwater model tests.

Re ¼

pffiffiffiffiffiffiffiffiffiffiffiffi g  Hs  Dn50 ν

ð9Þ

pffiffiffiffiffiffiffiffiffiffiffiffi where g  H s and Dn50 are assumed as characteristic velocity and length, respectively, and ν is the kinematic viscosity. According to the Van der Meer (1988) researches, to minimize the scale effect on armor stability the Reynolds number should be larger than 1 to 4 × 104. The

Table 1 Material properties.

Mass density ρs (kg/m3) Wn50 (kg) Dn50 (m) fg = Dn85/Dn15

Armor 1

Armor 2

Armor 3

Filter

2700 0.014 0.017 1.50

2700 0.025 0.021 1.50

2700 0.042 0.025 1.50

2800 0.0014 0.007 1.33

Fig. 4. Stone size distribution curves.

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Table 2 Number of test cases for variation of various sea states and structural parameters in the experimental set up. Parameter

Hs, Tp B hbr N d Total of tests

Dn50 = 0.017

Dn50 = 0.021

Dn50 = 0.025

Number of tests

Hs, Tp

B

hbr

N

d

Hs, Tp

B

hbr

N

d

Hs, Tp

B

hbr

N

d

18 6 5 3 5 99

1 3 1 1 1

1 1 4 1 1

1 1 1 6 1

1 1 1 1 5

18 6 5 3 4 94

1 3 1 1 1

1 1 4 1 1

1 1 1 6 1

1 1 1 1 5

18 6 5 3 4 94

1 3 1 1 1

1 1 4 1 1

1 1 1 6 1

1 1 1 1 5

54 54 60 54 65 287

Table 3 Range of dimensional and non-dimensional parameters. Parameter

Expression

Range

Wave height at toe of structure (m) Spectral peak wave period (s) Berm width (m) Berm elevation above still water level (m) Water depth at toe of structure (m) Stability index Front slope Rearward slope Reynolds number for berm stones

Hs Tp B hbr d H0T0 Cot (α) Cot (αr) Re

Stability number

H0

Stability index

H0T0

0.045 to 0.12 1 to 1.54 0.35 to 0.5 0.01 to 0.07 0.24 to 0.28 31.6 to 143.9 1.25 1.25 Armor 1: 1.12 · 104 to 1.74 · 104 Armor 2: 1.53 · 104 to 2.27 · 104 Armor 3: 2.01 · 104 to 2.71 · 104 Armor 1: 1.57 to 3.86 Armor 2: 1.57 to 3.46 Armor 3: 1.59 to 2.88 Armor 1: 33.99 to 158.26 Armor 2: 33.99 to 115.27 Armor 3: 31.59 to 87.26

lowest Reynolds number value for the present experiments with the lowest wave height is (Re b 1.12 × 104), cf. Table 3. 3. Experiments and profile parameterization In this section, new formulae are developed to calculate the key parameters of reshaped profile. The formulae are based on some observations regarding the influence of reshaped seaward profile characteristics, water depth and berm width and elevation. Furthermore, the effects of stone size and number of waves are included in the formula. It is noticeable that the berm breakwater in all experiments is considered as non-overtopped structure. 3.1. Definition of key parameters In order to study the reshaping berm breakwater profile, the profile has to be schematized into profile parameters. Based on the present

experimental observations, the reshaped profile ordinarily has the following characteristics: ■ Along the reshaped profile and below still water level a step-shaped area will be formed in all the tests, where the slope of the reshaped profile, i.e. cot (αd2), considerably approaches to natural angle of repose, see Fig. 6. The vertical distance from SWL to this transition is called step height hs, Van der Meer (1992). ■ Between the transition point and near still water level the profile is slightly curved being steeper as it moves higher. In this part the average slope is around 1:4.5, being comparatively steeper for more stable profiles, see dashed line in Fig. 6. Observations show that reshaped profile moves up exponentially and intersects the initial profile at a single point. The vertical distance from SWL to this point is called depth of intersection point hf. Also, horizontal distance between the step and intersection point is called step length Ls, Van der Meer (1992). ■ The seaward slope from near still water level up to the horizontal berm approaches the natural angle of repose, i.e. cot (αd1), showing a transition to steeper part of the reshaped profile. This is not utterly a precise estimate below still water level but quite good up the still water level. Vertical distance between the transition and berm level is called transition height, ht. ■ The erosion of the berm or the so called recession, i.e. Rec, is the most important parameter for profile schematization. According to the above observations, relationships between governing variables and profile parameters are investigated. 3.2. Estimation of seaward slope

Fig. 5. Sea state ranges.

To investigate the effects of various wave and structural parameters on the up and bed slopes, i.e. cot (αd1) and cot (αd2) respectively, of the reshaped seaward profile, Fig. 7 is shown. Fig. 7a and b reveals the effects of water depth at the toe of the structure and berm elevation above still water level on the seaward slope for different wave

M. Shafieefar, M.R. Shekari / Coastal Engineering 91 (2014) 123–139

127

Fig. 6. Seaward slope deformation.

Fig. 7. Effect of sea state and structural parameters on seaward slope deformation. a) Effect of water depth b) Effect of berm elevation. c) Effect of stability number (H0) d) Effect of H 0

combinations and stone sizes, respectively. It is observed from these figures that water depth and berm elevation have no considerable influence on the slope of the armor layer, and the ultimate slope of armor layer is almost independent of such parameters. Fig. 7c and d plots the variation of seaward with stability number (H0) and dimension pslope ffiffiffiffiffiffi less parameter H 0 T 0 for different stone diameters, wave heights and wave periods, respectively. It is noticed that there is some signifipffiffiffiffiffiffi cant scatter between the data using H 0 T 0 parameter, while for H0 parameter some negligible scatter is remained. One can conclude that, as the wave height and period increase the amount of the seaward slope increases, but the wave height has a very important role for the ultimate slope in comparison with the wave period. In order to investigate the change in seaward slope due to the change in stability number, different algebraic functions are employed and finally the linear function is chosen for the bed and up slopes as: cotðα d Þ ¼ 1:27 þ 0:06 H 0

transmitted to DWG format using AutoCAD software. The point where the slope becomes flatter than 1:1.5 has been chosen for defining the step height. In this section, the effect of sea state and structural parameters on the step height is considered using the present experimental data and then a formula is derived which takes into account some effective parameters. Amount of the step height for a test case is based on the

ð10Þ

3.3. Estimation of step height The step height is not always as simple to define as revealed in Fig. 6. In order to determine step height hs the reshaped profiles were

pffiffiffiffiffiffi T 0.

Fig. 8. Effects of wave height on deposition and step height.

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Fig. 9. Variation of step height versus wave height considering different armors. a) Armor 1 b) Armor 2 c) Armor 3.

average of three initial and reshaped profiles that have been measured during that test. 3.3.1. Effect of wave height on the step height Fig. 8 shows the reshaped profile of a berm breakwater for five different wave heights (HS = 4.45, 5.45, 6.55, 7.45 and 8.55 cm) with a 1 second peak period and after 3000 waves. It is clear that the wave height has a significant influence on the amount of the deposition and step height. It is observed that the step height increases linearly by increasing wave height. It can be concluded that the amount of wave power will increase with the increase in wave height so that it may cause the increase in the wave force on the stones. Furthermore, a larger deposition volume is needed for lager wave heights leading to increasing the step height as illustrated by Lykke Andersen (2006). Fig. 9 exhibits the amount of step height based on different wave heights. In these tests, the water depth is 24 cm, the berm width and elevation are 24 and 4 cm respectively, and the number of waves is 3000. Investigating the figures shows that the step height increases with increase in wave height.

3.3.2. Effect of water depth on the step height Water depth is a significant parameter concerning reshaping of a berm breakwater. Fig. 10a shows the reshaped profile step of a berm breakwater for five different water depths at the toe of the structure (d = 20, 22, 24, 26 and 28 cm). In these test series, the number of waves is 3000. It should be mentioned that for assessing the effect of water depth, the berm elevation above the bottom of the structure was changed to retain a constant berm elevation. It is considered that the water depth affects considerably on the amount of the deposition and step height. To formulate the step height, several dimensionless parameters were examined. Fig. 10b exhibits the variation of dimensionless step height (hs/Hs) with dimensionless water depth (d/Dn50), considering different wave combinations, i.e. wave height and wave period, armor stone sizes and water depths. It is clear that, as the water depth at the toe of the structure – with the same wave height and period – increases, the wave momentum flux will increase so that, it may cause the increase in the wave force on the stones. Furthermore, a larger deposition volume is needed for lager water depths leading to decreasing the step

Fig. 10. Effect of water depth on deposition and step height. a) Reshaped profiles b) Variation of (hs/Hs) versus (d/Dn50).

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129

pffiffiffiffiffiffi Fig. 11. Influence of number of waves on step height. a) Effect of storm duration (N) on step height. b) Trend of coefficient values against H0T0. c) Trend of coefficient values against H0 T 0.

height. In order to investigate the water depth influence on step height, different algebraic functions are employed and eventually the following power function is chosen:   hs d −0:23 ¼ a : Hs Dn50

ð11Þ

The value of a is the function of various parameters such as of wave height, wave period and armor stone size. 3.3.3. Effect of storm duration on the step height Process of reshaping profiles revealed that the step height develops with increment in the number of waves. This parameter is generally introduced by the number of waves (N) in a storm. The design philosophy

of a reshaping structure is that the damage of such a structure is expected to increase during the wave attack until it reaches a stable profile (Sshaped profile). Based on the observations of Lykke Andersen and Burcharth (2010), after 500 waves only minor changes in the profiles occur when the profile is dynamically stable while for more stable profiles the reshaping continues longer. Also up to 3000 waves, most of the reshaping profile takes place. A series of tests were organized to investigate the storm duration influence on step height development. According to the present experimental results, the ultimate reshaping of step height depends on the stability index (H0,T0). To appraise the effect of number of waves on the step height, Fig. 11a is exposed. Considering 90% of ultimate reshaping as a criterion of reshaping, one can conclude from this figure that most of the reshaping profile occurs before N = 3000. Van der Meer (1988, 1992) found that the reshaped profile

pffiffiffiffiffiffi Fig. 12. Trend of dimensionless parameter f (H0,T0) for various stability indices. a) f (H0,T0) vs. H0T0. b) f (H0,T0) vs. H 0 T 0 .

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M. Shafieefar, M.R. Shekari / Coastal Engineering 91 (2014) 123–139

parameters in the rest of the profile is proportional to f (N) = N0.07. In the present study, to probe the effect of number of waves N on reshaped profile step height, a power function is chosen as:  l hS N ¼k : HS 3000

ð12Þ

Trend variation for step height in different wave combinations show that, the coefficients k and l are dependent to the stability of the structure. Fig. 11b and c plot the variation of the coefficients k and l versus H0T0 pffiffiffiffiffiffi and H0 T 0 respectively, for different wave combinations and stone pffiffiffiffiffiffi sizes. The preference of the dimensionless parameter H0 T 0 is appraised by employing the square of the correlation factor (R2) for which a squared correlation factor close to 1 denotes full agreement. It is calculated according to: X X  X N XY− X Y 2 R ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h X X ih X X i 2 2 2 2 Y − XÞ N YÞ N X −

ð13Þ

where X is the calculated value and Y is the observation data. Results show that, assuming stability index H0T0, the square of the correlation factors for the coefficients k andp L ffiffiffiffiffi are ffi 91% and 93% respectively, similarly, assuming stability index H0 T 0 , the corresponding factors are 93% and 96% respectively. Ultimately, the following equation will be used as an appropriate pattern to consider the effect of number of waves on step height.  0:26−0:005H 0 hS N ¼k HS 3000

T0

¼ k  f ðNÞ:

ð14Þ

3.3.4. Proposed formula to estimate the step height There are some coefficients (i.e. a and k) depending on sea state conditions such as wave height and period. By employing the foregoing equations, the dimensionless step height is equal to a product of two functions, which could be written as:   0:26−0:005H pffiffiffiffi T0  0 hS N d −0:23  ¼ f ðH0 ; T 0 Þ  : HS 3000 Dn50

ð15Þ

In fact, the coefficients are substituted by the function of f (H0,T0). The value of this function is appraised by using Eq. (15): 

hS HS

By calculating the right hand side of Eq. (16) for each experiment and plotting these values versus the analogous stability index, the variation of f (H0,T0) is estimated, as revealed in Fig. 12. Fig. 12a shows the effect of H0T0 on f (H0,T0) for different wave combinations and stone sizes. It is viewed that there is some scatter between the data for differpffiffiffiffiffiffi ent wave periods. Fig. 12b reveals f (H0,T0) vs. H 0 T 0. It can be concludpffiffiffiffiffiffi ed that the data scatter with H0 T 0 is less than H0T0. pffiffiffiffiffiffi Results show that R2 is 96% forH 0 T 0, while the corresponding value for H0T0 is 90%. Curve fitting to the present experiments, Eq. (17) is resulted as a good function for estimating f(H0, T0) as follows:  pffiffiffiffiffiffi0:3 : f ðH0 ; T 0 Þ ¼ 1:36  H 0 T 0

ð17Þ

pffiffiffiffi

The difference between the above formula and the formula proposed by Lykke Andersen and Burcharth (2010), i.e. f (N) = (N/ 3000)− 0.046Ho + 0.3, is that it considers the effect of wave period too.

f ðH0 ; T 0 Þ ¼

Fig. 14. Effect of initial berm width on step length.

#  0:26−0:005H pffiffiffiffi T0  0 N d −0:23  : 3000 Dn50

,"

ð16Þ

Eventually, a summary of the proposed step height formula and the included parameters is obtained as follows:  pffiffiffiffiffiffi0:3  N 0:26−0:005H0 hS ¼ 1:36  H0 T 0 HS 3000

pffiffiffiffi   T0 d −0:23  : Dn50

ð18Þ

3.4. Estimation of step length Results of the present tests show that the step length (LS) is influenced by various sea state conditions and structural parameters. In this part a formula is derived which takes into account some effective parameters influencing the step length, according to the measurements for each reshaped profile. According to reshaped profiles, it is observe that the step length is affected by a number of parameters such as the stability index (H0 and T0), the armor stone sizes (Dn50), the berm elevation and width (B and hb) and the water depth at the toe of the structure (d). 3.4.1. Effect of initial berm width on the step length The effect of initial berm width on reshaped profile and step length is shown in Fig. 13. In order to compare the reshaped profiles, various berm widths are compared in this figure while the other sea state and structural parameters are the same. With respect to the reshaped Table 4 Coefficient values of Eqs. (19) and (21) for various wave combinations.

Fig. 13. Reshaping profiles for various berm widths.

H0T0

a

b

c

d

35.77 41.04 52.14 56.09 59.09 81.02 98.65 112.1

2.18 3.79 5.84 9.09 9.16 16.58 21.66 24.62

−0.16 −0.14 −0.17 −0.16 −0.15 −0.15 −0.17 −0.17

0.36 0.55 1.1 1.45 1.47 2.38 2.9 3.36

0.48 0.53 0.52 0.49 0.5 0.5 0.52 0.54

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Fig. 15. Water depth influence on step length. a) Variation of LS versus d. b) Variation of LS/Dn50 versus d/Dn50.

profiles it is observed that the berm width has an influence on the amount of step length indicating that, if the berm width increases the step length will decrease. The reason is probably that due to the extension of the structure berm width, the wave encounters a large porous berm above or at still water level at the seaward side so that, the wave dissipation will increase. Also it is distinguished that the variation of berm width has no significant influence on step height so that, such a parameter has not been appeared in Eq. (18). Fig .14 reveals dimensionless berm width (B/Dn50) versus dimensionless step length (LS/Dn50) for various wave combinations and stone sizes. In order to investigate the berm width effect on step length, a power function is introduced as follows: 

LS B ¼a Dn50 Dn50

b

:

ð19Þ

To determine the coefficients of Eq. (19) regression analysis is employed. Table 4 denotes some values for a and b for corresponding

wave combination (H0T0). It is observed that there are small discrepancies in the coefficient b for various wave combinations, that the average values are used, i.e. b = −0.16. The value of depends on wave conditions. The following equation is used as a suitable form to consider the effect of the berm width on step length.   LS B −0:16 ¼a Dn50 Dn50

ð20Þ

3.4.2. Effect of water depth on the step length A series of tests have been carried out to investigate the water depth influence on the step length. As revealed in Fig. 10a, it is noticed from reshaped profiles that the water depth has a considerable effect on the amount of reshaped profile step length of berm breakwaters. Results of the tests show, as the water depth increases the amount of step length will increase, see Fig. 15a.

Fig. 16. Berm elevation effect on step length. a) Reshaped profiles. b) Variation of LS versus hb. c) Variation of LS/Dn50 versus hb/HS.

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Fig. 17. Coefficient values of Eq. (23) for each wave combination.

Fig. 15b shows dimensionless water depth (d/Dn50) versus dimensionless step length (LS/Dn50). In order to scrutinize the water depth influence on reshaped profile step length, a power function is chosen as: 

LS d ¼c Dn50 Dn50

d

:

employed as a proper model to consider the effect of the water depth on the step length:   LS d 0:51 ¼c : Dn50 Dn50

ð22Þ

ð21Þ

To determine the coefficients of Eq. (21) regression analysis is used. Table 4 shows the variation of the coefficients c and d versus H0T0 for different wave combinations and stone sizes. From Table 4 it is distinguished that there are small differences in the coefficient d for different wave combinations that may be related to experimental scatter. So, the average values are used, i.e. d = 0.51. The following formula can be

3.4.3. Effect of berm elevation on the step length Results of the tests show that the variation of berm elevation has a considerable influence on the amount of eroded depth. Fig. 16a exhibits the reshaped profiles for various berm elevations (hbr = 1, 2.5, 4, 5.5, 7 cm) and invariable wave height and period. It is observed that the berm elevation has an influence on the amount of step length but the step height is nearly independent of the berm elevation. If the

pffiffiffiffiffiffi Fig. 18. Number of waves influence on step length. a) Reshaped profiles. b) Effect of storm duration (N) on step length. c) Trend of coefficient values against H0 T 0 .

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pffiffiffiffiffiffi Fig. 19. Effect of stability indices on f (H0,T0). a) f (H0,T0) vs. H0T0. b) f (H0,T0) vs. H 0 T 0 .

berm elevation increases the amount of step length will decrease, see Fig. 16b. Fig. 16c shows dimensionless berm elevation (hbr/HS) versus dimensionless step length (LS/Dn50) for different wave combinations and stone sizes. In order to investigate the berm elevation influence on reshaped profile step length, a power function is chosen as:  f LS hbr ¼e : Dn50 HS

ð23Þ

Fig. 17 plots the variation of the coefficients e and f versus H0T0 for different wave combinations and stone sizes. As seen, e is a function of sea state conditions such as wave height and period, while the value of f is nearly independent of wave conditions. In order to confirm the values for the coefficient f, the average values are used, i.e. f = − 0.14. Eventually, the following formula could be employed as a suitable pattern to consider the effect of berm elevation on step length.  −0:14 LS hbr ¼e Dn50 HS

ð24Þ

regression statistics, the following equation will be used as a proper pattern to consider the effect of the number of waves N on step length:  0:22−0:006H pffiffiffiffi T0 0 LS N : ¼k Dn50 3000

3.4.5. Proposed formula to estimate the step length As explained in the previous sections, there are some coefficients (i.e. a, c, e and k) depending on sea state conditions such as wave height and period. By employing the equations from the previous sections, the dimensionless step length is equal to a product of four functions, which could be written as: LS ¼ f ðH0 ; T0 Þ  Dn50

 l LS N ¼k : Dn50 3000

"

N 3000

#  0:22−0:006H pffiffiffiffi      T0 0 B −0:16 d 0:51 hbr −0:14   :  Dn50 Dn50 HS

ð27Þ To assess the value of the variable f(H0, T0), Eq. (27) is rewritten as follows: 

3.4.4. Effect of number of waves on the step length Fig. 18a shows the reshaped profile of the breakwater for six different wave durations (N = 500, 1000, 2000, 3000, 4000 and 6000) with the same wave height. In these tests the water depth is 24 cm, and the berm elevation above still water level is 4 cm. With respect to reshaped profiles, it is observed that the number of waves has a considerable effect on the reshaped profile and the step length. If the number of waves increases, the amount of step length will increase (Fig. 18b). As seen, up to 90% of ultimate amount of step length occurs before N = 3000 and such a storm duration can be used as a threshold for estimating the step length, as observed in Section 3.3.3. According to the trend of step length with different sea state conditions, i.e. wave height and wave period, it is ascertained that a power function is a good description of the step length during a storm for all wave conditions as:

ð26Þ

f ðH0 ; T0 Þ ¼

LS Dn50

#  #      0:26−0:005H pffiffiffiffi T0 0 N B −0:16 d 0:51 hbr −0:14 : : : : 3000 Dn50 Dn50 HS

,""

ð28Þ By computing the right hand side of Eq. (28) for each experiment and plotting these values versus the analogous stability index the variation of f (H0,T0) is estimated, as shown in Fig. 19. Fig. 19a reveals the variation of f (H0,T0) with stability number (H0T0), which has been

ð25Þ

Investigation of trend variation for the step length in different wave combinations show that, both coefficients k and l are dependent to the stability of the structure. pffiffiffiffiffiffi Fig. 18c reveals the variation of the coefficients k and L versus H0 T 0 for different wave combinations and stone sizes. According to nonlinear

pffiffiffiffiffiffi Fig. 20. Variation of f (H0,T0) with H 0 T 0 according to the present experimental data.

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Fig. 21. Effect of wave height and period on depth of intersection point. a) Reshaped profiles due to wave height. b) Variation of (Hs) versus (hf). c) Reshaped profiles due to wave period.

introduced by Van der Meer (1988), for different stone sizes, berm widths and elevations and wave conditions. It is distinguished that there is some scatter between the data for various wave periods. Fig. 19b the variation of such a parameter versus stability  reveals pffiffiffiffiffiffi index H0 T 0 . It is obvious that comparatively the data scatter with this parameter is less than other dimensionless parameter. Results indicate that the square of the correlation factor is 93% for pffiffiffiffiffiffi H 0 T 0 , while the analogous value for H0T0 is 90%. Fig. 20 shows the pffiffiffiffiffiffi trend of f(H 0 , T 0) against the corresponding H 0 T 0 according to pffiffiffiffiffiffi present experimental data, implying that H 0 T 0 ¼ 20 acts as a transition point. Employing regression analysis, Eq. (29) is resulted as a good function for estimating f(H0, T0) as follows: f ðH 0 ; T 0 Þ ¼

pffiffiffiffiffiffi 0:328:H 0 pT ffiffiffiffiffi0ffi−2:12 0:124:H 0 T 0 þ 1:95

pffiffiffiffiffiffi H 0 pT ffiffiffiffiffi0ffi ≤20 : H 0 T 0 N20

ð29Þ

3.5. Estimating the depth of intersection point Whereas, estimation of the extension of the reshaped profile is necessary to evaluate the amount of erosion and stability of the berm

breakwater, it is necessary to locate some auxiliary points considering various sea state and structural parameters, e.g. step and intersection points. Based on the foregoing mentioned observations of the reshaped profiles, determining the depth of intersection of reshaped and initial profile is presented in this section. As illustrated above, results of experiments show that reshaped profile moves up exponentially and intersects the initial profile at a single point, called intersection point (hf). The influence of sea state and structural parameters on the depth of intersection is investigated with present experimental data and then a formula is derived which considers some effective parameters. One can conclude that based on the present formula, determination of the depth of intersection is much more complicated than that of Tørum et al. (2003) formula. 3.5.1. Effects of wave height and period In order to investigate the effect of wave height on reshaped profile and depth of intersection Fig. 21a is shown. To compare the reshaped profiles, different wave heights are shown in this figure for the same wave period. Fig. 21b shows the variation of the amount of depth of intersection based on different wave heights. One can observe from these figures that there is no important difference in the reshaping of the

Fig. 22. Effect of water depth on depth of intersection point. a) Reshaped profiles. b) Variation of (hf/Dn50) versus (d/Dn50).

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Fig. 23. Effect of berm elevation on depth of intersection point. a) Reshaped profiles. b) Variation of (hf/Dn50) versus (hbr/Dn50).

armor layer and intersection point location with respect to the wave height. To assess the effect of wave period on reshaped profile and depth of intersection Fig. 21c is plotted, considering three different wave periods with the same wave height. It is obvious that comparatively the wave period has not significant effect on the intersection point location with respect to the wave period. Also observation of Fig. 21b reveals that, if the wave period with the same wave height increases the amount of depth of intersection does not change considerably. 3.5.2. Effects of water depth at the toe of the structure In order to investigate the effect of water depth on the intersection point location Fig. 22a is shown, for five different water depths (d = 20, 22, 24, 26 and 28 cm) and the number of waves of 3000. Considering reshaped profiles, it is observed that the water depth significantly affects the intersection point location of the berm breakwaters, and by increasing the water depth, the depth of intersection point will increase. In fact in larger water depths the amount of eroded volume and depth of intersection point will increase due to the wave force on the armor stones. Fig. 22b plots dimensionless water depth (d/Dn50) versus dimensionless depth of intersection point (hf/Dn50) for different stone sizes. To the water depth effect on depth of intersection point, among different algebraic functions a power function is selected as: 

hf d ¼a Dn50 Dn50

1:38

:

ð30Þ

3.5.4. Estimation of depth of intersection point As revealed in the previous sections, there are some coefficients (i.e. a and c) that should be specified. Finally, dimensionless depth of intersection point may be written as a function of following non-dimensional parameters:     hf d 1:38 hbr −0:29 ¼f : : Dn50 Dn50 Dn50

ð32Þ

To appraise the value of the variable f, Eq. (37) is rewritten as follows:  f ¼

hf Dn50

, 

1:38    hbr −0:29 : : Dn50 Dn50 d

ð33Þ

By computing the right-hand side of Eq. (33) for each experiment pffiffiffiffiffiffi and plotting these values versus the equivalent H0 T 0 the variation of f is estimated, as shown in Fig. 24. The variation of f is nearly independent of the sea state conditions. In order to confirm the values for the coefficient f, the average values are used, i.e. d = 0.21. Eventually, the following equation could be used as a suitable pattern to estimate the depth of intersection point:     hf d 1:38 hbr −0:29 ¼ 0:21   : Dn50 Dn50 Dn50

ð34Þ

3.6. Estimation of berm recession 3.5.3. Effects of berm elevation The effect of berm elevation on reshaped profile and depth of intersection point is exhibited in Fig. 23a. To facilitate the assessment of intersection point location according to variations in berm elevations (hbr = 1, 2.5, 4, 5.5 and 7 cm), we have shown the reshaped profiles in this figure. With respect to the reshaped profiles, it can be concluded that the berm elevation affects on the amount of depth of intersection point. In fact by increasing the berm elevation above still water level, the amount of depth of intersection point will decrease. So, it may result in a smaller deposition volume and berm recession (as demonstrated by Moghim et al. (2011)) that the intersection point comparatively shifts up. Fig. 23b plots dimensionless berm elevation above still water level (hbr/Dn50) versus dimensionless depth of intersection point (hf/Dn50) for different stone sizes. According to the trend of depth of intersection point it is ascertained that a power function is a good description to modify such a parameter as:   hf hbr −0:29 ¼c : Dn50 Dn50

ð31Þ

Using results of present data set, variations of berm recession have been investigated by Shekari and Shafieefar (2013). They proposed formula (7) for estimation of berm recession which takes into account the

pffiffiffiffiffiffi Fig. 24. Variation of variable f with H 0 T 0 according to present experimental data.

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Fig. 25. Comparison of measured and calculated step height for two different estimation methods. a) For the present data. b) For the partial Moghim data.

influence of wave height and period, storm duration, berm width and elevation variations. 4. Proposed reshaped profile key parameters A summary of the proposed formulae for estimating the key parameters of reshaped profile and the related factors is given below. pffiffiffiffi  pffiffiffiffiffiffi0:3  N 0:26−0:005H0 T 0  d −0:23 hS  ¼ 1:36  H0 T 0 HS 3000 Dn50

lS ¼ f ðH0 ; T0 Þ  Dn50

ð35Þ

#  0:22−0:006H pffiffiffiffi      T0 0 N B −0:16 d 0:51 hbr −0:14    3000 Dn50 Dn50 HS

"

Tørum et al. (2003) and, Lykke Andersen and Burcharth (2010, 2012), as shown in Fig. 25. All the dimensions in the figure, i.e. length and height, are scaled by the armor stone size (Dn50). Fig. 25a displays a comparison of observed and calculated step height from different formulae on the basis of the present experimental data set. One can conclude that there is a comparatively good agreement between the present formula and experimental data. The performance of the present estimation method is appraised using square of the correlation factor (Eq. (13)) and standard deviation parameter of dimensionless step height predictions for the present study and other researchers, where standard deviation close to zero denotes full agreement, computed as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N  2 u 1 X ðh =D Þ s n50 meas −ðhs =Dn50 Þcalc : σ ¼t Dn50 N i¼1 i

ð39Þ

where f ðH0 ; T0 Þ ¼

pffiffiffiffiffiffi 0:328  H0 pT ffiffiffiffiffi0ffi−2:12 0:124  H0 T 0 þ 1:95

pffiffiffiffiffiffi H0 pTffiffiffiffiffi0ffi ≤20 H0 T 0 N20 ð36Þ

    hf d 1:38 hbr −0:29 ¼ 0:21   ; Dn50 Dn50 Dn50

Rec ¼ Dn50

   pffiffiffiffiffiffi2 pffiffiffiffiffiffi −0:016 H 0 T 0 þ 1:59H 0 T 0 −9:86     B −0:15 hb −0:21 :  f1:72− exp½−2:19  ðN=3000Þg HS Dn50

ð37Þ

ð38Þ

According to the present experimental work limitations, the above equations are valid for: 8 pffiffiffiffiffiffi > 7:09bH0 T 0 b23:52 > > > < 500bNb6000 : 0:1bhbr =H S b1:57 > > > 8bd=D b16:47 > n50 : 12bB=Dn50 b26:47

Table 5 shows the values of the validation indices according to the present data set for different formulae. One can observe that R2 when using Eq. (35) is comparatively 11% more than that of the Lykke Andersen formula. This means that the present step height formula fits the data better than the other model. To investigate the prediction capability of the present formula in comparison with the Lykke Andersen formula, Moghim (2009) data are used. The data extracted from the Moghim data set are only those in the range of the present studies (sea state and structural limitations as given by Eq. (38)). As seen in Table 5, the proposed formula comparatively gives less standard deviation and relatively better correlation in comparison with the other formula. Fig. 25b reveals a comparison of observed and calculated berm step height from different formulae for the partial Moghim data satisfying Eq. (35). The figure shows that the present step height formula fits the data very well and relatively better than the other three estimation formulae. For all the data sets, it is evident that the Lykke Andersen formula overestimates the step height of the reshaped profile.

Table 5 Validation indices between calculated and measured dimensionless step height for different formulas and different data sets. Data

Validity index

Present formula

Lykke Andersen and Burcharth (2009)

5. Validity evaluation of the proposed formulae

Present data

The performance of the present estimation formulae has been compared to the performance of the formulae by Van der Meer (1992),

Moghim (2009)

R2 σ R2 σ

92.1 0.53 92.3 0.59

81.6 1.07 83.4 1.04

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Fig. 26. Comparison of measured and calculated step length for two different estimation methods. a) For the present data. b) For the partial Moghim data.

Table 6 Validation indices between calculated and measured dimensionless step length for different formulas and different data sets. Data

Validity index

Present formula

Van der Meer (1992)

Present data

R2 σ R2 σ

97.3 0.54 95.6 0.72

88.7 1.82 89.6 1.68

Moghim (2009)

This could be due to a different definition made by Lykke Andersen. He defines hs as the point where the slope gets flatter than 1:2 while in this research we use 1:1.5. This means Lykke Andersen methodology provides slightly smaller hs than this research. Also one may argue that the scatter between the calculated hs by Lykke Andersen formula and measured values in this research might be due to omitting d and Dn50 in Lykke Andersen formula. To evaluate the validity of the present formula to predict the step length in homogeneous reshaping berm breakwaters, a comparison is made between the estimated step lengths by this formula and formula given by Van der Meer (1992), c.f. Fig. 26. Fig. 26a exhibits a comparison of observed and calculated step length from different formulae using the present experimental data. It is observed that there is a moderately good agreement between the present formula and experimental data.

Table 6 shows the standard deviation and the square of the correlation factor of the dimensionless step length predictions for the two formulae tested. It is obvious that the proposed formula comparatively performs better than the other formula. The Van der Meer formula and the present proposed formula are appraised against not only the present data, but also the partial Moghim (2009) data satisfying Eq. (36). The comparison to the data of Moghim is given in Fig. 26b and Table 6. One can conclude that the present formula predicts the step length well and with less standard deviation, and somehow better correlation comparing with the other formulae. Fig. 26 shows that, the formula of Van der Meer (1992) underestimates the step length for the main part of the present data and Moghim data. This might partly be due to the fact that a number of key parameters, i.e. berm width and elevation, and water depth, are not included in that formula. The other reason is that, in the present formula the wave height has more effect on the step length than the wave period, whereas in the Van der Meer formula the orders of wave height and period are the same. To assess the validity of the present formula to estimate the depth of intersection point, for each test the measured intersection depth was plotted together with the analogous amount calculated by the formulae of Tørum et al. (2003), Lykke Andersen et al. (2012) and the present proposed formula. Examples of such plots are shown in Fig. 27. Fig. 27a exhibits a comparison of observed and calculated depth of intersection point from different formulae for the present experimental data. It is obvious that other formulae underestimate the intersection depth that might be because of neglecting the effect of berm elevation,

Fig. 27. Comparison of measured and calculated depth of intersection depth for different estimation methods. a) For the present data. b) For the partial Moghim data.

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Table 7 Validation indices between calculated and measured dimensionless depth of intersection point for different formulas and different data sets. Data

Validity index

Present formula

Lykke Andersen et al. (2012)

Van der Meer (1992)

Present data

R2 σ R2 σ

92.1 0.45 89.3 0.55

73.7 1.68 73.9 1.76

61.7 2.78 69.2 2.93

Moghim (2009)

and using a linear relation between the depth of intersection point (hf) and dimensionless water depth (d/Dn50). The comparison to the data of Moghim (2009) is given in Fig. 27b. It is observed that the estimations made by the present formula are in most of the cases in good agreement with the measurements of Moghim (2009). Table 7 exhibits the standard deviation of the dimensionless intersection depth estimations for the three formulae tested and for different data sets. One can observe that the proposed formula predicts the intersection depth better than the existing formulae in the range of the present experimental work. The validity of the present formula in other conditions was investigated using partial Lykke Andersen data which include the data that deviate from the conditions used in the present work. Comparisons between the measured Lykke Anderson data and the calculated step height, step length and depth of intersection using Eq. (35) are presented in Fig. (28). Results indicate that the R-squared values are 82% for step height, 90% for step length and 85% for the depth of intersection. Even though these values are less than those given in Table (6), however one may conclude that the present formula predicts the parameters

of reshaped profile relatively acceptable in other ranges and could be applicable outside the studied conditions. Nevertheless this needs to be improved by using a wide range of effective parameters in order to arrive at formulae that cover all practical conditions. 6. Conclusions The present research deals with the parameterization of Reshaped Seaward profile of berm breakwaters based on experimental data. In this direction, first the reshaped profiles are schematized and then the key parameters of such profiles, i.e. step height, step length and depth of intersection point of initial and reshaped profiles, are formulated considering various sea state and structural parameters. A total of 287 tests have been performed to cover a wide range of parameters. The following conclusions can be drawn from the present experimental data exploration: 1. Results of this data exploration reveal that the behavior of reshaped profile parameters is complicated but can be formulated. To parameterize the reshaped profile, the effects of various sea state conditions such as wave height and period, number of waves and water depth, and structural parameters such as berm width and elevation, and armor stone size are separately investigated on each key parameter. 2. Lykke Andersen (2006) proposed a formula for estimating the step height (hS) that seems to be independent of the water depth and stability condition. Results of the present research show that the step height depends on the stability index and water depth at the toe of the structure. It seems that there is no significant difference in the step height with respect to the berm width and elevation.

Fig. 28. Comparisons of measured and calculated key parameters for the partial Lykke Andersen data.

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3. Parameterization of the step length (LS) reveals that, various sea state and structural parameters affect on this parameter. Also considering different wave combinations and stone sizes, the variation of step length versus number of waves has a different trend compared to step height. 4. One can observe that the performance of the proposed formula for intersection depth is relatively better than existing formulae. It might be partly stated by considering the effect of berm elevation, and using a non-linear relation between the depth of intersection point (hf) and dimensionless water depth (d/Dn50). 5. Comparisons show that the estimation method foretells reshaped profile key parameters well and with less scatter according to the present data and also other experimental results within the range of parameters tested. References Burcharth, H.F., Frigaard, P., 1988. On the stability of berm breakwater roundhead and trunk erosion in oblique waves. Proc. ASCE Seminar on: Unconventional Rubble Mound Breakwaters, Ottawa, Canada. Goda, Y., Suzuki, Y., 1976. Estimation of incident and reflected wave in random wave experiments. Proc.15th International Conf, pp. 828–844 (Hawaii). Hall, K.R., Kao, S., 1991. A study of the stability of dynamically stable breakwaters. Can. J. Civ. Eng. 18, 916–925. Helgason, E., Burcharth, H.F., 2005. On the use of high-density rock in rubble mound breakwaters. International Coastal Symposium, Höfn, Iceland. Lykke Andersen, T., 2006. Hydraulic response of rubble mound breakwaters. (PhD Thesis) Scale effects — berm breakwatersDepartment of Civil Engineering, Aalborg University, Denmark.

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Lykke Andersen, T., Burcharth, H.F., 2010. A new formula for front slope recession of berm breakwaters. Coast. Eng. J. 57 (4), 359–374 (Elsevier). Lykke Andersen, T., Van der Meer, J.W., Burcharth, H.F., Sigurdarson, S., 2012. Stability of hardly reshaping berm breakwaters. Proceedings of 33rd Conference on Coastal Engineering (Santander, Spain). Mansard, E.P.D., Funke, E.R., 1980. The measurement of incident and reflected spectra using a least squares method. Proc. 17th Coastal Engineering Conference (Sydney, Australia). Moghim, M.N., 2009. Experimental study of hydraulic stability of reshaping berm breakwaters. (PhD thesis) Tarbiat Modares University, Tehran, Iran. Moghim, M.N., Shafieefar, M., Tørum, A., Chegini, V., 2011. A new formula for the sea state and structural parameters influencing the stability of homogeneous reshaping berm breakwaters. Coast. Eng. 58, 706–721. PIANC, 2003. PIANC MarCom Report of Working Group No 40. State-of-the-art of designing and constructing berm breakwaters. International Navigation Association. PIANC General Secretariat, Brussels, Belgium. Shekari, M.R., Shafieefar, M., 2013. An experimental study on the reshaping of berm breakwaters under irregular wave attacks. Appl. Ocean Res. 42, 16–23. Tørum, A., 1998. On the stability of berm breakwaters in shallow and deep water. Proc. 26th International Conference on Coastal Engineering. ASCE, Copenhagen, Denmark, pp. 1435–1448. Tørum, A., Krogh, S.R., 2000. Berm breakwaters. Stone quality. SINTEF Report No. STF22 A00207SINTEF, Civil and Environmental Engineering. Tørum, A., Krogh, S.R., Bjørdal, S., 1999. Design criteria and design procedures for berm breakwaters. Proc. of Coastal Structures' 99. Tørum, A., Kuhnen, F., Menze, A., 2003. On berm breakwaters stability, scour, overtopping. Coast. Eng. 49, 209–238 (Amsterdam, September Issue). Tørum, A., Moghim, M.N., Westeng, K., Hidayati, N., Arntsen, O., 2012. On berm breakwaters: recession, crown wall forces, reliability. Coast. Eng. 40, 299–318. Van der Meer, J.W., 1988. Rock slopes and gravel beaches under wave attack. Delft Hydraulics Communication No. 396. Van der Meer, J.W., 1992. Stability of seaward slope of berm breakwaters. Coast. Eng. 16, 205–234.