Riprap stability for deep water, shallow water and
steep foreshores
Henk Jan Verhagen, Delft University of Technology, Delft, Netherlands Marcel Mertens, BAM Infraconsult, Gouda, Netherlands
Introduction
In the Rock Manual [2007] two sets of equations for the determination of rock stability in breakwaters armour layers are presented. One set is the original formula presented by Van der Meer [1988], valid for deep water conditions. This set uses the parameters Hs and Tm. The
other set is an adaptation of these formulae, using the parameters H2% and Tm-1,0, and is
recommended for shallow water conditions. Tests by Van Gent et al. [2003] have lead to a calibration coefficient slightly different than the original Van der Meer values.
Recently the second author [Mertens, 2007] has reanalysed the datasets of Van der Meer and Van Gent, corrected some of the numbers, and explained a part of the differences. This paper tried to unify both sets of formulae and come to one single equation.
Some history
Around 1988 Van der Meer did an extensive research to the stability of rubble mound structures. He continued on the basis of tests by Thompson & Shuttler [1975]. The work of Van der Meer resulted in the well-know stability formulae for riprap:
5 0.5 0.18 50
1
for plunging waves
s m pl n H S P c d N (1) 5 0.13 50 1
tan for surging waves
P s m s n H S P c d N (2)
The formulae are based on tests with random waves in small and large scale flumes. The spectra that have been used are a Pierson-Moskowitz spectrum, a wide spectrum and a narrow spectrum. Van der Meer used relatively deep water conditions and a horizontal, deep bottom in front of the structure. Van der Meer found for the coefficients the following values: cplunging
= 6.2 and csurging = 1.0.
In 2003 Van Gent et al. published a paper with results of test in more shallow water (Hs/h =
0.23 – 0.78), and including a slope in front of the structure. In his tests two foreshore slopes were used, 1:100 and 1:30. On the basis of these tests, Van Gent came to the following formulae:
0.5 0.18 2% 5 1,050 1
for plunging waves
s m pl n s H H S P c d H N (3)
0.13 2% 5 1,0 50 1tan for surging waves
P s m s n s H H S P c d H N (4)
Van Gent used a Jonswap spectrum and he came to the following values of for the constants
cplunging = 8.42 and csurging = 1.3.
In order to compare the results, first the values used by Van der Meer (Hs and Tp) have to be
converted to the values H2% and Tm-1,0, as used by Van Gent. This resulted in the Figure 1.
Figure 1. Data of Van Gent et al.[2003] and Van der Meer [1988] (Van Gent et al, 2003)
In the graph there is a clear difference between both datasets. A thorough investigation by Delft Hydraulics showed that there were no systematic differences in the modelling approach (both test series were performed at Delft Hydraulics). In his paper Van Gent suggested to use the shallow water equation also for deep water in order to prevent design errors.
The Rock Manual [2007] recommends using the original Van der Meer formula for deep water, the Van Gent version in shallow water, recommends to be quite careful in the transition zone between both formulae-sets, see figure 2.
foreshore is to the stability of the structure.
Figure 2. Recommendation on the use of the formulae (Rock Manual, 2007)
Review of de Van der Meer formulae
In order to investigate this, a start has been made with a thorough review of the data of Van der Meer. At the time of Van der Meer’s research data processing was done with punch cards, and the data have been copied several times. To be able to review the formula the original data had to be analysed. In the original data it was found that in a number of cases the transition between plunging and surging was not computed correctly. Also it was found that “a standard PM spectrum” was in reality less standard. The analysis showed that what was originally called a “wide spectrum” fits quite will a standard PM-spectrum, and that what was called a “narrow spectrum” where in fact monochromatic waves.
When converting the Tp from the original test to Tm-1,0 by Van Gent standard conversion
factors are used. However, these conversion factors depend strongly on the shape of the spectrum. Because of the ambiguous definition of the spectra in the original work, Van Gent did not use the correct conversion factors.
Also for the conversion from Hs to H2% one should not use standard values, but the values as
really used in the tests. Also here a number of small differences were found. A similar con-version as for the Van der Meer data has been performed for the dataset of Thompson & Shuttler [1975].
Roundness
The roundness of the stones has an influence on the stability. The stones used by Van der Meer did not all have the same roundness. Partly this was caused by abrasion during the tests. Latham [1988] developed a method to correct for the differences in roundness. This cor-rection has been applied to the original data, and especially for the surging waves, this results in a somewhat different figure (see figure 3).
This correction factor (Latham) is 1.0 for standard stones, it has a value of 0.95 or 1.0 for
semi-round stones; 0.95 or 0.80 for very semi-round stones and 1.1 or 1.3 for tabular stones (first value is for plunging, the second for surging breakers). For more details is referred to the Rock Manual [2007], table 5.30.
Figure 3. Effects of influence of the stone-roundness on graphs of Van der Meer (different colours indicate different test conditions) [1988]
Comparison of the improved datasets of Van Gent, Van der Meer
and Thompson & Shuttler
For the dataset of Van Gent [2003] it is assumed that no corrections are necessary. For spectra, etc. this is obvious, because Van Gent specifically did measure H2% and Tm-1,0. However, no
information is available regarding the roundness of the stones used by Van Gent. It is assum-ed that these stones are of the standard stone type (i.e. with a correction factor Latham = 1.0).
Detailed processing of the Van Gent data is also not possible, because in the 2003 paper only results are published, and the original database is still confidential. In figure 4 all data for plunging waves of all datasets are plotted.
Figure 4. Plunging waves: data of Thompson & Shuttler (green/light), Van der Meer (red/dark) and Van Gent (blue/medium)
From Figure 4 follows that the data of Thompson & Shuttler fit very well to the data of Van der Meer, and that the data of Van Gent are somewhat higher than the other two, but there is a large overlap. Also it is clear that Van Gent did include in his dataset data with quite high values of S/N0.5. A value of S/N0.5 = 0.3 usually means the complete destruction of the break-water. So values higher than S/N0.5 = 0.5 are outside the range of static stability. For proper analysis one may disregard these data points. In general one may conclude that in case of a full conversion, differences are small, the spread of the data of Van Gent is somewhat larger and the average observed damage by Van Gent is somewhat larger.
Consistent use of H
sand H
2%In the formulae for shallow water for describing the wave height the parameter H2% is used.
However, in the Iribarren number still the value of Hs is present. This is not very consistent. It
is more consistent to use also in the Iribarren number the H2% number.
This leads to the following sets of equations:
0.2 0.25 0.18 2% 1,0 50cot for plunging waves
pl m n H S c P s d N (5) H2% dn50 csP0.13 S N 0.2 sm1,0
0.25 s1,0
P0.5The transition between plunging waves and surging waves can be calculated using the critical value for the surf similarity parameter:
cr cpl cs P 0.31 tan 1 P0.5 (7)
For s1,0crwaves are plunging and Equation (5) applies; For s1,0crwaves are surging and Equation (6) applies. In these formulae the following parameters are present:
H2% = wave height exceeded by 2% of the waves in the (short term) wave distribution Tm-1,0 = period of the waves, calculated from the first negative moment of the spectrum sm-1,0 = fictitious wave steepness: 2H2%/(gTm21,0)
s-1,0 = surf similarity parameter: tan / sm1,0
dn50 = nominal median block diameter, or equivalent cube size, dn = (M/s)1/3.
= relative mass density (s – w)/w where s is mass density of stone and w is mass
density of water
N = number of waves
S = damage level A/(dn50)2, where A = erosion area in a cross-section
= angle of the seaward slope of a structure
P = notional permeability coefficient
Depending on slope and permeability, the transition lies between s-1,0 = 2.5 and 4.
When converting all available datasets, and plotting them in a figure with on the vertical axis (S/√N)0.2 a linear regression analysis is possible. Figure 5 shows the results for plunging waves.
From this figure one may conclude that the differences between the datasets of Van der Meer and Thompson & Shuttler as well as the dataset of Van Gent are very small, both regression lines nearly coincide. The difference with the dataset of Van Gent is a larger scatter (the cor-relation of the Van Gent dataset is 51%, the corcor-relation of the vdM&T dataset is 60%). The value of cpl is 1/0.1175=8.4, which is the same value as presented by Van Gent in his original
publication.
Design conditions
For design conditions usually a 5% upper boundary is used. Van der Meer has suggested to include this boundary by calculating the standard deviation of cpl and use the 5% boundary for
this value (the mean of cpl is 8.4, the standard deviation of cpl is 0.7, so the 5% exceedance
S=3, N=3000 S=8, N=3000 y = 0,1175x R2 = 0,5968 y = 0,1165x R2 = 0,5091 0,00 0,20 0,40 0,60 0,80 1,00 1,20 1,40 0,00 2,00 4,00 6,00 8,00 10,00 vdM/T vGent 5% design Linear (vdM/T) Linear (vGent) 2% 1,0 0.18 2% , 50 (1 ) m Latham f b H T n H c P d
Figure 5. All datasets (plunging waves) using only H2% and Tm-1,0
For most riprap structures a value of S=2 is considered as light damage. Storms may last for some 3000 waves, so a common critical value for
S/ N
0.2is 0.5. In case of high damage(failure) usually a value of S=8 is assumed. This gives a critical value of 0.7. Both limits are also indicated in figure 5.
Steep foreshores
Many researchers (e.g. Allsop et al, 1998, Hovestad, 2005, Verhagen et al, 2006, Muttray and Reedijk, 2008) have found that steep foreshores result in more damage than gentle foreshore slopes. Therefore some designers advise to increase the required stone diameter dn50 with 10%
The influence of the foreshore can be included by adding a correction factor in the stability formula:
2% 1,0
0.2 0.5 0.18 2% , 50 1 (plunging) m pl f H T n Latham c H S c P d N (8)
0.2
0.25
0.5 0.13 2% 1,0 1,0 50 1 s P (surging) f m s n Latham H c S c s P d N (9)in which is the Iribarren number calculated with the slope of the foreshore, the local H2%
and the local Tm-1,0. The calibration factor cf seems to be in the order of 0.035. In figure 6 this
correction factor has been applied for the Van Gent dataset, however, the effect is very small and not visible in the figure. In fact the slope of the regression line changed from 0.119 to 0.117, which means a change in cpl from 8.4 to 8.5.
Calculations have been made with the one-dimensional version of SWAN (Verhagen et al,
2008). This was done for different configurations with shallow and deep foreshores and varying foreshore slopes. It was found that in case of a very detailed Swan calculation (with very small step size) on steep slopes the local H2% may become quite high. This leads to a
high required stone diameter.
Figure 6 shows some results. The required stone size has been calculated for a 1:3 breakwater on a water depth of 5 m. The slope in front of the breakwater was varied from 1:3 to horizon-tal. It was found that a horizontal bed on a water depth of 5m and a 1:1000 slope starting at the same depth of 5m give the same results. For plotting purposes, the graph ends at the 1:1000 slope. The used wave condition in deep water (50 m deep) was a Hs of 5m, a Tp of 12s,
a Jonswap spectral shape with peak enhancement factor γ =3.3 and an f 5 spectral tail.
For the 1:3 foreshore slope two calculations have been made; a calculation assuming that the 1:3 breakwater starts at -5m on top of a 1:3 foreshore (so including the wave boundary condi-tions at -5 m) and a calculation with a 1:3 breakwater extending to -50 m, and using the wave boundary conditions at -50 m. This has to lead to the same required stone size, because there is no real difference between a breakwater with a 1:3 slope on top of a 1:3 foreshore and a 1:3 breakwater extending to deep water. Applying the above mentioned correction factor cf =
0.035 leads to a value of H2%/dn50 which is 8 % more in case of using the breakwater toe at
-5m. However, at that point the H2% is also 8% higher, so the resulting required stone diameter
Conclusions
An accurate comparison of the data of Thompson & Shuttler [1975], of Van der Meer [1988] and of Van Gent et al. [2003] shows that, in combination with the shallow water wave para-meters H2% and Tm-1,0 there is no need to use different formulae for deep and shallow water.
The effect of the foreshore can be incorporated by adding a correction factor based on the Iribarren number for the foreshore. This implies that formulae (8) and (9) can be used in all cases, provided the wave boundary data at the toe of the breakwater are determined very accurately.
From the above one may conclude that the previously mentioned advice to use a 10% larger stone diameter in case of a steep foreshore is probably caused by inaccurate calculation of the boundary conditions at the toe of the breakwater. From the analyse carried out in this research follows that an accurate calculation of the wave height and wave period, including a precise determination of H2% and Tm-1,0 may lead to an sufficiently safe determination of the required
stone size.
References
Allsop, N.W.H., Durand, N., Hurdle, D.P. [1998] Influence of Steep Sea Bed Slopes on
Breaking Waves for Structure Design, Proc. 26th ICCE, Copenhagen
Hovestad, M [2005] Breakwaters on steep foreshores, M.Sc.-thesis, Delft University of Technology, The Netherlands.
Latham, J.P. [1988] The influence of armour stone shape and rounding on the stability of
armour stone stability. Queens Mary College, University of London, UK
Mertens, M. [2007] Stability of rock slopes under wave attack, M.Sc.-thesis, Delft University of Technology, The Netherlands.
Muttray, M., Reedijk, B. [2008] Reanalysis of breakwater stability with steep foreshore, proc. 31st ICCE 2008, Hamburg 0,0 0,5 1,0 1,5 2,0 1 10 100 1000 slope 1:x sto n e d iam e ter dn (m )
Rock Manual [2007] The use of Rock in Hydraulic Engineering, CIRIA-CUR-CETMEF (C683)
Thompson, D.M., Shuttler, R.M. [1975] Riprap design for wind-wave attack, a laboratory
study in random waves, Hydraulics Research Station, report EX707, Wallingford, UK
Van der Meer, J.W. [1988] Rock slopes and gravel beaches under wave attack, Ph.D.-thesis, Delft University, The Netherlands
Van Gent, M.R.A., Smale, A.J., Kuiper, C. [2003] Stability of rock slopes with shallow
foreshores. Proc. 4th Int. Coastal Structures Conf., Portland; ASCE, Reston VA, USA
Verhagen, H.J., Reedijk, B., Muttray, M. [2006] The effect of foreshore slope on breakwater
stability, Proc. 30th ICCE, San Diego, pp4828-4840
Verhagen, H.J., Van Vledder, G., Eslami Arab, S. [2008] A practical method for design of
