INVESTIGATIONS ON QUARRY STONE TOE BERM STABILITY

Model test results from four experimental studies have been compiled in a data set with 687 test results. Three widely used or recently developed toe stability formulae have been validated against these model test results. It was found that all three formulae suffer from a lack of accuracy and general validity. This is caused by shortcomings of the underlying wave flume studies. More precisely, the wave height and the water depth above the toe berm are two main influence parameters for the toe berm stability and they are not independent in these studies. Testing the interdependence of parameters is therefore recommended for wave flume studies. An alternative toe berm stability formula was developed by a step-wise approach starting with a simple case with a minimum number of influencing factors and followed by more complex cases. The new formula is believed to provide a more meaningful description of the toe berm stability than existing formulae; this however requires further substantiation. The new approach is proposed as a working hypothesis for further studies and is recommended as benchmark for toe berm testing and design.


INTRODUCTION
The toe of most rubble mound breakwaters is protected by a quarry stone toe berm that is placed on a bedding layer.The hydraulic stability of this toe berm is commonly assessed by empirical formulae; the approach of van der Meer (1998) [vdM98] is widely used.It combines the toe stability formulae of Brebner and Donelly (1962) and of Gerding (1993).However, vdM98 tends to overestimate the stability of embedded toes and underestimates toe berms on a thick bedding layer.The formula was therefore modified by Muttray (2013) [M13].An alternative toe stability formula that includes the size of the toe berm and a fictitious flow velocity on the toe berm was proposed recently by van Gent and van der Werf (2014) [GW14].
Existing toe stability formulae are characterised by large scatter when plotted against model test results (Muttray, 2013).This is indeed a cause of worry for designers.M13 is aiming at practical applications.The problem of the inherent uncertainties was bypassed by providing a somewhat conservative estimate of the toe stability.In contrast to all other formulae, model test are closely reproduced by GW14 indicating that this approach would be a major advancement.
The initial aim of this paper was refining M13 and extending the application range to deeper water.This objective was changed after publication of GW14 to an analysis of strength and weaknesses of existing formulae in order to provide some guidance for design.With this in mind a data base of toe stability tests has been set up covering a wide range of toe structures and test conditions.The evaluation focused on vdM98, M13 and GW14; a review of earlier formulae can be found in Muttray (2013).Significant shortcomings have been identified for all three formulae.The underlying reasons have been investigated and based on these findings an alternative approach has been developed.The various steps of this analysis are presented in this paper.

INFLUENCE PARAMETERS OF THE TOE BERM STABILITY
The stability of individual stones on a toe berm is commonly described by the stability number, combining significant wave height, with median nominal rock diameter, and relative density of submerged stones, ⁄ .The latter includes the density of rock, and water, .The stability number can be considered as a simplified description of the driving and resisting forces on an individual stone (Muttray, 2013) and is in this regard comparable to the factor of safety that is used in geotechnical engineering for slope stability analysis.
Besides wave height, rock size and relative density ( , and ) the submergence of the toe berm, was identified by Brebner and Donelly (1962) as key parameter for the toe stability.All toe stability formulae include these four parameters.More recent formulae include further the damage number, defined by the number of displaced stones in a toe section of width .It is arguable if should be considered as a governing parameter for the toe stability.It could also be considered as a data analysis strategy that allows the inclusion of test results with little or excessive damage in the analysis.
A toe stability formula that includes the obvious parameters ( , , and ) and the damage number, should have the following form: (1) At least one more parameter of unit length has to be added to arrive at a dimensionally correct equation.Potential influence parameters of unit length are the water depth and wave length in front of the structure, and as well as the width and height of the toe berm, and .Other parameters that might affect the toe berm stability are the seabed slope, (referring to a gradient of ), the breakwater slope, (referring to ) and the front slope of the toe berm, (referring to ).These dimensionless slope parameters and the dimensions of the toe berm are defined in Fig. 1.Gerding (1993) performed toe stability tests in the Scheldt flume of Delft Hydraulics with five different toe berm geometries.The toe width varied from 12 to 30 cm (with a constant height of 15 cm) and the toe height varied from 8 to 22 cm (with constant width of 12 cm).Docters van Leeuwen (1996) conducted wave flume tests at Delft University.Worth noting about these tests is the rock material.Toe berms of basalt, porphyry and crushed brick with a specific density varying from 1,900 to 2,850 kg/m 3 were investigated.The damage was less than in other studies; it may be for this reason that these findings got little attention.Ebbens (2009) carried out model tests in the laboratory of Delta Marine Consultants (DMC) with seven different water levels.Tests with a water depth of 16 to 26 cm at the toe of the structure are similar to the conditions tested by Gerding and Docters van Leeuwen.These tests were complemented by shallow water tests with a water depth of 8 to 14 cm.Moreover, the experiments were performed with different seabed slopes.The toe berm was placed on a bedding layer of thickness 2.0 cm; in the three other studies the toe berm was placed on the seabed.Van Gent and van der Werf (2014) performed model tests in a wave flume at Deltares.Remarkable about these tests are the gentle breakwater slope and the sophisticated damage measurements.The number of displaced stones, has been validated against independent measurements of the eroded toe profile and refers to stones that moved more than one stone diameter from their initial position.In the other studies only stones that were washed away from the toe berm had been counted.
In all four studies JONSWAP spectra were applied; the test duration was 1,000 waves except the tests of Docters van Leeuwen with 2,000 waves.A series of tests was performed with stepwise increasing wave height and all other conditions kept constant until the toe berm was severely damaged or until the limits of the wave generator were reached.The wave period was varied in these test series with the aim of having a constant wave steepness, ⁄ .This wave steepness may differ significantly from the actual wave steepness at the structure as it is commonly based on the deep water wave length and on the target wave height at the wave paddle.Ebbens however applied the wave length at the paddle; the wave periods in his tests are therefore somewhat longer.
Damage in the test of Gerding and Docters van Leeuwen refers to the number of displaced stones in a single test.The toe berm was repaired after each test.The cumulative damage in a series of tests was recorded by Ebbens and by van Gent and van der Werf.In these studies the toe berms were repaired when the maximum wave height in a series of tests had been reached and before continuing testing with lower wave heights.

REVIEW OF EXISTING FORMULAE
The validity of three toe toe stability formulae by van der Meer (1998), Muttray (2013) and van Gent and van der Werf (2014) has been verified against model test results.The vdM98 approach (Eq.2) is widely used and was therefore selected.The other two formulae, M13 (Eq. 3) and GW14 (Eq.4) are recent developments and claim to be an improvement over vdM98.All three are predictive equations for the damage number, being a function of damage level, toe geometry and wave conditions. (3) Predictive equations for the damage number, (for given rock properties, toe geometry and wave conditions) or for the required rock size, (for given rock density, wave conditions and toe geometry) are obtained by rearranging Eqs. 2, 3 and 4. They were validated by means of the model test results by Gerding (1993), Docters van Leeuwen (1996), Ebbens (2009) and van Gent and van der Werf (2014) (see Table 1).The stability number, rock size and damage were calculated by the three toe stability formulae (Eqs.2, 3 and 4, rearranged if necessary) based on the reported test conditions.In other words, the reported test conditions were applied on the right hand side of the predictive equations; the predicted left hand side (calculated value) was then compared with the left hand side as observed in the experiments (measured value).
The measured and calculated values of ⁄ , and are presented in Fig. 2 for vdM98, in Fig. 3 for M13 and in Fig. 4 for GW14.Different marker types are used for each of the four experimental studies.Model test results that had been used in the original derivation of the respective toe stability formula are indicated by grey-shaded markers.Transparent markers refer to other model tests results.
When comparing the stability numbers, ⁄ according to vdM98 with model test results (Fig. 2, bottom) the 90% confidence interval (CI) is given by a factor 1.5 (i.e.ranging from prediction divided by 1.5 to prediction times 1.5).This factor reduces to 1.3 when considering only the test results of Gerding (1993), which were used in the derivation of vdM98.The same can be observed for the required rock size, (Fig. 2, top right).The CI factor for all tests is 1.7 and reduces to 1.3 when only looking at the tests of Gerding.When predicting damage numbers, (Fig. 2, top left) uncertainties become much larger.A CI factor of 2.0 was found for the tests of Gerding (considering only tests with ).When considering all tests, there is virtually no correlation between measured and predicted damage.Different from the two other formulae M13 is proposed as a design formula providing a somewhat conservative estimate of damage, required rock size and toe stability.Therefore the 90% upper confidence band (CU) was applied for the evaluation of M13.The CU of the stability numbers, ⁄ according to M13 (Fig. 3, bottom), of the required rock size, (Fig. 3, top right) and of the damage numbers, (Fig. 3, top left) are given by factors 1.4, 1.3 and 1.6, respectively.These numbers refer to all tests; damage numbers have only been determined for tests with .These CU factors do not reflect significant deviations of predicted stability numbers and rock diameters from the model test results of Docters van Leeuwen (1996).The same applies for predicted damage numbers when compared with model test results of Doctors van Leeuwen and of van Gent and van der Werf.
Stability numbers, ⁄ according to GW14 (Fig. 4, bottom) have a 90% confidence interval (CI) that is defined by a factor 1.6 for all tests results and by a factor of 1.3 when considering only the test results of van Gent and van der Werf (2014).When calculating the required rock size, (Fig. 2, top right) the CI factor increases to 1.9 for all tests; a CI factor of 1.3 was found for the tests of van Gent and van der Werf.Similar to vdM98 and M13 the prediction of damage numbers, is vague (Fig. 2, top left).When only looking at the test results of van Gent and van der Werf the CI factor is 2.0 (for all tests with ) and reduces to 1.5 for tests with .When considering all tests, there is little correlation between measured and predicted damage.All three toe stability formulae (Eqs.2, 3 and 4) provide a more or less rough estimate of the actual stability numbers and required rock diameters.The accuracy of all three formulae is significantly improved when they are applied to the data sets that had been used in the derivation of the respective formula.In this case the uncertainties of the predicted and are of order ±30%.These uncertainties increase to about ±60% when considering all model tests.
When toe stability formulae are used to predict damage, the results are of limited value, indicative at most.Even when the formulae are applied only to those test results that had been used in the derivation of the respective formula the uncertainties are close to ±100%.When considering results from all tests, extremely poor damage predictions are found.
It appears from the above that the most recent (M13 and GW14) and the most accepted (vdM98) toe stability formulae suffer from a lack of accuracy and general validity.Lack of accuracy refers to uncertainties of 30% to 60% in predicting stability numbers and rock diameters; damage number predictions bear even larger uncertainties.Lack of general validity refers to the finding that all three formulae perform significantly worse when applied to new data sets, despite the fact that parameters are well within the range of validity of the respective formula.In other words, the applicability of the empirical toe stability formulae to parameter combinations that deviate from those in the developer's own data sets is limited.This lack of general validity is considered as the main shortcoming of the three toe stability formulae (Eq.2, 3 and 4).This has -to the authors' knowledgenot been noted earlier; the underlying reasons have to be investigated to clear the way towards a better toe stability formulae.

THE PROBLEM WITH DEPENDENT VARIABLES
The experimental studies that have been applied for this review are not fully consistent.The most obvious difference is the definition of damage, which may refer either to the accumulated number of displaced stones in a series of tests or to the number of displaced stones in a single test.On top of this a stone may be considered as displaced if it moved more than one stone diameter from its initial position or if it rolled away from the toe berm.Other differences between the experimental studies are for example the seabed slope, the breakwater slope or the presence of a bedding layer under the toe berm (see Table 1).Besides these reported differences between the toe berm experiments there are unassigned differences, the so-called model effects.The tests were performed by different experimenters in different labs, they used different control signals for the wave generator and they constructed and repaired the toe berms slightly different.All this may affect the outcome of the experiments and thus the agreement between predictive formulae and experimental results.
The disagreement that can be seen in Figs. 2, 3 and 4 is too large to expect it to just be model effects or the result of slight differences in the experimental studies.There seems to be a more fundamental problem with the existing toe stability formulae; they might for example miss out an important aspect of toe berm stability.
In order to confirm the appropriateness of the experimental data, the interdependencies of the basic dimensional parameters were analysed.As expected, most parameters show little or no correlation and thus can be considered as independent parameters.However two of the main parameters, the water depth above the toe, and the wave height, are not independent.The correlation of and is plotted in Fig. 5  and could be the result of depth limited wave conditions.This however is not the case; the relative wave height ⁄ (using here the water depth in front of the structure) of the same set of tests is plotted in Fig. 5 (right) and varies from 0.3 to 1.3 indicating that these tests cover a wide range of test conditions with and without depth limited waves.
An almost constant ratio of and was found, when the toe damage is close to .This ratio is largely independent of (see Fig. 5, left).If we would draw conclusions on toe stability from this observation, we would arrive at ridiculous results.This is a direct consequence of the interdependencies of and in all four model studies, which has apparently nothing to do with toe stability.Both parameters, and are influenced by the experimenters' choices when defining the model dimensions and setting up the test programme.The close correlation between and might be the result of these choice.And if so, similar choices have apparently been made in the selection of test conditions in all four model studies.Two of the main parameters for the toe berm stability, the wave height, and the water depth above the toe berm are not independent.This applies to all four model studies and might be the result of choices that have been made in setting up the test programmes.All attempts to improve the accuracy and predictive power of an empirical toe stability will inevitably make use of this close correlation between and .However, as this correlation has nothing to do with toe stability, it will result in a biased and thus not generally-applicable toe stability formula.This explains the severe shortcomings of Eq. 2 (vdM98), Eq. 3 (M13) and Eq. 4 (GW14) as illustrated in Figs. 2, 3 and 4. Most of the earlier toe stability formulae are likely to face the same problems as they are similar to Eq. 2 (see Muttray, 2013).
When performing wave flume experiments, the independence of test parameters should be routinely tested.However in practice the test parameters are combined in dimensionless numbers without checking for dependencies.There might be more empirical formulae in coastal engineering, which have been derived from wave flume tests and which are based on data sets with interdependent parameters.In consequence, these formulae may also be biased or may lack generality.

STEP-BY-STEP DEVELOPMENT OF A TOE STABILITY FORMULA
The wave height, and the water depth above the toe berm, are both relevant parameters for the toe berm stability.Excluding one of these parameters is thus not sensible and by the way, this would not necessarily overcome the problem of dependent parameters.A theoretical approach would be more promising; an empirical approach with physical meaningful parameter combinations might work as well.The latter route was selected for the step-wise development of an alternative toe berm Outside Surf Zone stability formula.We will start with a simple case with a minimum number of influencing factors and proceed then with more complex cases taking into account an increasing number of influencing parameters.
The first case is a toe berm with zero submergence; hence the effect of is excluded.The influence of the damage number, is also excluded by selecting tests with damage numbers close to one (i.e. ).In this case two of the five most influential parameters can be neglected; the remaining three parameters are , and .This toe berm configuration is sketched in Fig. 6 (right), the observed stability numbers are plotted in Fig. 6 (left).If the toe berm is close to the water line, i.e. if ⁄ is close to zero, the stability numbers vary between 1.4 to 2.4, are in average about 1.8 and are largely constant.A toe berm stability formula of the following form would be the obvious choice in this situation: (5) As expected the stability of a toe berm that extends to the water line is quite similar to the stability of the main armour.The Hudson formula predicts the same stability for the main armour as Eq. 5 for the toe berm when using the toe berm slope of the model tests (1:1.5) and a Hudson stability coefficient, of 4: It may be arguable whether the Hudson formula refers to damage numbers, of about 1. Nonetheless, the general agreement between Eqs. 5 and 6 provides some confidence in the validity of Eq. 5.
The next case refers to a submerged toe berm; the effect of is thus included.This is a critical step as the interdependency of and may affect the outcome.In order to avoid further complicating factors the influence of the damage number, is excluded as for the previous case.The following descriptive model (see also Fig. 7, right) has been applied:  The wave induced water movements above the toe berm have an oscillating pattern;  The flow velocities are related to the wave orbital motions even though the actual particle velocities may differ significantly from a progressive wave due to the wave reflection and due to the influence of the toe berm;  Similar to wave orbital motions the motion amplitudes of the wave induced flow above the toe berm are decreasing over depth;  This decay is in analogy to wave orbital motions related to the ratio of local water depth, and peak wave length in front of the structure, .The observed stability numbers are plotted in Fig. 7 (left) against .In line with the above model the toe berm stability is increasing with increasing submergence and decreasing for long waves.This variation is approximately linear; a toe berm stability formula of the following form may be considered: (7) The experimental results are reasonably reproduced by a coefficients of about 1.8 (see also Eq. 5) and a coefficient of order 20 to 30.Eq. 7 is consistent with Eq. 5 and is in line with the above descriptive model.Therefore, Eq. 7 appears to be a plausible toe stability approach.Nonetheless it must be noted that and are numerators on both sides of the equation; this may affect the general validity of this stability formula.Further substantiation by additional model test results or by a sound theoretical analysis would be required.Eq. 10 could not be confirmed against model tests.Therefore results of Eq. 10 with toe berm slopes different from 1:1.5 should be considered as indicative.Further substantiation by model test or by a theoretical analysis will be required.The measured and calculated values of ⁄ , and according to Eq. 10 are presented in Fig. 10 in the same format as vdM98, M13 and GW14 in Figs. 2, 3 and 4. When comparing the stability numbers, ⁄ according to Eq. 10 with model test results (Fig. 10, top right) the 90% confidence interval (CI) is defined by a factor 1.4.In other words, the observed stability is typically 1/1.4 to 1.4 times the predicted value.The CI of the required rock size, (Fig. 10, top left) is defined by a factor 1.5.When predicting damage numbers, (Fig. 10, bottom) the CI factor increases to 3.0.

CONCLUSIONS AND RECOMMENDATIONS
Three widely used (vdM98) and recently developed toe stability formulae (M13 and GW14) have been validated against a comprehensive dataset with 687 model test results.The empirical formulae predict those model test results, which had been used for the derivation of the particular formula, reasonably well.The predictions are significantly less reliable when applied to other data sets.This lack of general validity is probably the result shortcomings in the underlying wave flume studies.In all data sets that have been considered in this study two main influence parameters for the toe berm stability, the wave height and the water depth above the toe berm, are not independent.Interdependence of parameters may easily lead to erroneous conclusions in empirical studies.This is the case for the three toe stability formulae that have been reviewed here and it is most probably also the case for many of the earlier toe stability formulae.It would be surprising if this issue would affect only toe stability; other empirical formulae that are based on wave flume experiments may be affected as well.
An alternative toe berm stability formula (Eq.10) was developed by a step-wise approach starting with a simple case with a minimum number of influencing factors.More complex cases with increasing number of influence parameters were investigated subsequently.The following conclusions have been drawn:  The toe berm stability, is the sum of two terms (see Eqs. 7 and 10). The first term refers to the minimum stability of a toe berm with virtually zero submergence and is nearly constant (i.e. is not significantly influenced by any of the parameters that have been varied in the tests). The second term describes the increased stability of a submerged toe berm and is thus proportional to . The effect of submergence is probably a function of .This hypothesis, although supported by the available data and by a simple descriptive model, cannot yet be confirmed due to the interdependence of and . The second term is further influenced by the seabed slope.The toe stability is reduced significantly on a steep seabed.However, the effect of seabed slope can only be determined without doubt when the uncertainties regarding the effect of submergence (previous item) have been clarified. The toe berm stability is likely to be increased for gentle toe berm slopes.The effect of slope gradient was included in the toe berm stability formula by a Hudson-style approach.The validity of this approach cannot be confirmed by the available data. The prediction of toe berm damage, bears large uncertainties and is thus hardly advisable for practical applications.Many aspects of the proposed toe stability formula (Eq.10) have not yet been proved beyond doubt.Therefore this formula is far from being a design formula and should be considered as a working hypothesis.The new formula is believed to provide a more meaningful description of the toe berm stability than existing formulae.In view of the limitations of existing formulae, Eq. 10 is recommended as a benchmark for toe berm testing and design.

ACKNOWLEDGMENTS
This study would not have been possible without the support of Delta Marine Consultants.Our thanks and appreciations go to our co-workers who run and maintain the testing facilities of Delta Marine Consultants and assisted in this study.We wish to say thanks for the encouragement by many during the discussion and in personal conversations at the ICCE conference.

Figure 1 :
Figure 1: Definition of toe berm geometry and adjacent slopes THE DATA SET Model test results from wave flume experiments by Gerding (1993), Docters van Leeuwen (1996), Ebbens (2009) and van Gent and van der Werf (2014) have been compiled; the data set comprises 687 model tests.Key parameters of these tests are summarised in Table1.

Figure 2 :
Figure 2: Predicted damage number (top, left), required nominal stone diameter (top right) and stability number (bottom) according to Eq. 2 (van der Meer, 1998) plotted against experimental results

Figure 4 :
Figure 4: Predicted damage number (top, left), required nominal stone diameter (top right) and stability number (bottom) according to Eq. 4 (van Gent and van der Werf, 2014) plotted against experimental results

Figure 5 :
Figure 5: Relation of wave height and water depth on the toe berm: Interrelation in tests with moderate damage (left), occurrence of depth limited wave conditions in these tests (right)

Figure 6 :
Figure 6: Stability of a toe berm with zero submergence (damage numbers close to one)

Figure 7 :
Figure 7: Stability of a submerged toe berm (damage numbers close to one)

Figure 8 :
Figure 8: Effect of seabed slope on the toe berm stability

Figure 9 :
Figure 9: Progression of toe berm damage

Figure 10 :
Figure 10: Predicted damage number (top, left), required nominal stone diameter (top right) and stability number (bottom) according to Eq. 10 plotted against experimental results