A Pragmatic Approach to Rock Toe Stability

A Pragmatic Approach to Rock Toe Stability Markus Muttray Delta Marine Consultants, H.J. Nederhorststraat 1, P.O. Box 268, 2800 AG Gouda, The Netherlands Corresponding Author: M. Muttray, Email: [email protected], Tel.: +31 182 590 689, Mob.: +31 6 1588 5432 Abstract A new stability formula is proposed for the toe protection of rubble mound structures. The new approach is based on a critical stability number that refers to the initial movement of armour stones on the toe berm. Damage progression is subsequently described by the ratio of actual and critical stability numbers. The ratio of toe berm submergence and incoming wave height has been identified as the governing parameter for the toe stability. Other parameters including sea bed gradient, wave steepness and toe berm geometry were found to be of minor importance. The new toe stability formula has been validated against model test results and has been compared with the widely used formula of van der Meer (1998). The new approach appears more accurate and more physically meaningful. The practical application of the new formula for toe berm design is briefly described. Keywords: Rubble mound breakwater, toe protection, damage progression, critical stability number, toe stability formula Introduction Rubble mound breakwaters have a quarrystone toe berm to protect the toe of the primary armour layer. This toe berm is placed on a bedding layer that acts as a filter and scour protection (see Figure 1). Toe failure is one of the main reasons for breakwater damage or failure. The hydraulic stability of a toe berm is commonly assessed by empirical formulae that are similar to the design formulae for the main armour. The toe stability is primarily determined by the stone weight and density and by the submergence of the toe berm. The latter complicates the prediction of toe stability compared to the armour layer stability. Design formulae for toe berms are characterised by large scatter when plotted against model test results. Most formulae use dimensionless parameter combinations, the implications of which are lacking transparency. Therefore, the existing toe stability formulae are considered vague and unreliable by designing engineers. The stability of a toe berm has been re-examined; the experimental results of Markle (1989), Gerding (1993) and Ebbens (2009) have been re-analysed. Starting point was a simple analytical description of those processes, which are primarily relevant for the toe stability. This analytical approach to the toe berm stability has been empirically validated and where necessary adjusted. The objective was a rational toe stability formula that includes the effect of designated processes while the potential effect of other, secondary processes is neglected. The latter should be assessed by engineering judgement and should be considered in the final evaluation of the calculated toe stability. Earlier Work The hydraulic stability of a toe berm was investigated in various experimental studies (see Table 1). All toe stability formulae include the stability number, , i.e. rock diameter and wave height are linearly related, when all other parameters are kept constant. Moreover, all formulae in Table 1 include the depth of the toe berm below the water line, .

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Stability of Rock Toes

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The toe stability formulae for rubble mound breakwaters and composite breakwaters (caisson breakwaters on a rubble foundation) in Table 1 have a similar structure. The stability number is a function of the water depth above the toe, ; the latter is normalised either by the water depth, , the wave height, or the nominal stone diameter, . More recent formulae include a damage level, that refers to the number of displaced stones within a strip with width . The water depth ratio, was identified by Brebner and Donelly (1962) as governing parameter for the stability of rock berms in front of vertical breakwaters. This approach was applied to the toe stability of rubble mound breakwaters by Markle (1989) and was further developed by Gerding (1993) who included the damage level, . Gerding’s design formula was refined by van der Meer (1998). A toe berm stability formula for vertical breakwaters that includes the damage level, was proposed by Madrigal and Valdés (1995). The ratio of water depth on toe berm and nominal rock diameter, was applied by van der Meer at al. (1995) as governing parameter for the toe stability. Burcharth et al. (1995) replaced by in van der Meer’s formula to assess the stability of concrete armour units on a toe berm. The ratio of water depth and wave height, was found to be the main parameter for the toe berm stability of vertical breakwaters by Tanimoto et al (1982). Their toe stability formula included further the berm width, and the relative water depth . Burcharth et al. (1995) proposed a second toe stability formula for rubble mound breakwaters that includes and the damage level, by re-arranging their first formula that is based on . Table 1: Overview of toe stability formulae Author (Year)

Structure type

Brebner Donelly (1962)

Vertical

Tanimoto et al (1982)

Vertical

Markle (1989)

lower bound central estimate

Rubble mound

Madrigal Valdés (1995)

Vertical

van der Meer et al. (1995) Burcharth et al. (1995), 1st Burcharth et al. (1995), 2nd van der Meer (1998)

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Rubble mound Rubble mound Rubble mound Rubble mound

⁄

1.64

18%

)

1.46

35%

( )

0.86

28%

0.98

28%

1.01

31%

0.82

37%

( ) (

Rubble mound

Gerding (1993)

̅̅̅̅̅̅̅̅̅̅̅̅̅ ( )

Stability number

( ) ( ) (

)

(

)

1.04

13%

(

)

1.04

13%

1.22

49%

1.09

16%

(

) (

( )


)

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Incoming Wave

Breakwater Armour Layer

H

SWL

d

dt ht Seabed

B

Underlayer Core Material

tb Toe Berm

Bedding Layer

Figure 1: Definition sketch Toe berm stability numbers as determined in the model tests of Gerding (1993) and Ebbens (2009) have been compared with those predicted by the various toe stability formulae. Tests with seabed slope 1:20 and damage numbers of either (formulae including ) or (i.e. acceptable damage if is not included) were considered. The uncertainties of the design formulae in Table 1 are quantified by two parameters, bias (average ⁄ of ) and scatter (relative standard deviation, whereby the bias was compensated). It appears from Table 1 that toe stability formulae for vertical breakwaters (Brebner and Donelly, 1962, Tanimoto et al, 1982 and Madrigal and Valdés, 1995) are associated with large uncertainties (bias 0.82 to 1.64, scatter 18% to 37%) when applied for rubble mound structures. As expected the formulae of van der Meer et al. (1995) [vdM95] and the first formula of Burcharth et al. (1995) [B95-1] turn out to be of similar accuracy. Surprisingly, much larger uncertainties were found for the second formula of Burcharth et al. (1995) [B95-2], although B95-2 was derived by re-arranging B95-1. However, when comparing the stone diameters, from the model tests of Gerding (1993) and Ebbens (2009) with stone diameters calculated by vdM95, B95-1 and B95-2 the uncertainties are very much the same; the bias of all three formulae is about 0.95, the scatter is about 28%. When normalising the water depth on the toe berm, by (or ) the same divisor is applied on both sides of the stability formula. This is apparently an efficient way of hiding scatter. The scatter however will return when the equation is re-arranged in a way that appears only on one side of the equation, for example when predicting the required stone diameter. The contrary happens when is normalised by . When applying the wave height as dividend, ⁄ on one side and as divisor, on the other side of the equation, uncertainties are increased. Re-arranging the stability equation in a way that will appear only on one side (for example when predicting ) will reduce bias and scatter significantly. Thus, using a relative water depth on the toe berm, ⁄ in a toe berm stability formula (vdM95 and B95-1) is mere window dressing. Stability formulae that are based on ⁄ are similar accurate, no matter if they are used to predict a stability number or a stone diameter. When comparing for example stone diameters calculated by the formula of van der Meer (1998) [vdM98] with the model tests of Gerding (1993) and Ebbens (2009) the uncertainties are similar to those in Table 1, the bias is about 0.89, the scatter is about 15%. The relative water depth, is commonly presented as a parameter that refers to the toe berm geometry and is sometimes called the relative height of the toe berm. It is obvious that the stability of a toe berm will increase with increasing submergence (i.e. with increasing ). It is less obvious why the stability should decrease with increasing water depth unless the incoming waves are depth limited. And even then, the toe stability would be determined by the wave height and not the water depth. The water depth, has apparently little influence on the toe berm stability, is mostly relative invariable in hydraulic M. Muttray 29/03/2013


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model tests and seems thus the obvious counterpart for the water depth on the toe berm, However, this approach is rather easy than physical meaningful.

.

The basic idea behind the stability formula of Brebner and Donelly (1962), the stability number shall be at least equal to 2 (if and is further increasing with increasing submergence of the toe berm, appears sensible. Most of the later toe stability formulae follow this concept. The reasoning behind the formulae of Gerding (1993) and Madrigal and Valdés (1995) is less clear; both formulae are not applicable for toe berms that are close to the water line (i.e. ⁄ has to be larger than 0.3). Gerding (1993) suggested a directly proportional relation between damage number, and stability number, (if ⁄ is constant). In Figure 2 it can be seen that an upper limit of measured damage can be described quite accurately by a linear function (dashed line in Figure 2, right). The fit function (cause variable to the power , solid line in Figure 2, right) as proposed by Gerding appears arbitrary. Nonetheless, this concept has been picked up by vdM95, B95-1+2 and by vdM98. From the above it appears that the toe berm stability (quantified by the stability number, ) is primarily determined by the water depth on the toe berm, . It is not clear how the toe berm stability can be described by a dimensional correct approach. It is further not clear how toe damage (quantified by the damage number, ) is progressing with increasing wave height or stability number. Experimental Data Results of wave flume tests on toe berm stability by Markle (1989), Gerding (1993) and Ebbens (2009) have been re-analysed; key parameters of these studies are summarised in Table 2. Markle (1989) refers to model tests of the US Army Engineer Waterways Experiment Station (WES), performed in wave flumes of the Coastal Engineering Research Center (CERC), Vicksburg, Mississippi (Table 2). Rubble mound breakwaters with slope gradients of 1:1.5 or 1:2 were tested. The tests were performed with regular waves; the seabed in front of the structure had a gradient of 1:10. Toe berms of height 2 and width 3 nominal stone diameters, were installed on a bedding layer of thickness 1.2 cm. Four different stone gradations were tested ( ). The weight of individual stones varied by less than ±30% from the average weight, . Markle specified a rock density of 2.64 kg/m3 for all stone gradations. Test series were performed with increasing wave height, constant wave period ( ) and constant water depth ( ). Damage level are not specified; the reported results refer to acceptable damage, which is defined as “some stone movement, showing that the toe berm was not over designed but the amount of movement was minor and acceptable”. Results of tests with either no or excessive movement are not reported. Gerding (1993) performed model tests in the Scheldt flume of Delft Hydraulics in Delft, The Netherlands (Table 2). A rubble mound breakwater with a 1:1.5 slope was tested; the seabed slope in front of the structure was 1:20. The tests were performed with irregular waves (JONSWAP wave spectrum, 1,000 waves per tests). The toe berm was installed on a bedding layer of thickness 2.0 cm. Toe berms of width 12, 20 and 30 cm with a constant height of 15 cm (including bedding layer) and of height 8, 15 and 22 cm with a constant width of 12 cm were tested. Four different stone gradations were applied ( ); the ratio varied from 1.15 to 1.3; the rock density was constant (2.68 kg/m3). Test series were performed with increasing wave height (incident wave height at the wave paddle, ), constant wave steepness of either 0.02 or 0.04 (at the wave paddle; some additional tests were conducted with wave steepness 0.03) and constant water depth ( ). Damage numbers, are reported for all tests. M. Muttray 29/03/2013


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Ebbens (2009) performed model tests in the laboratory of Delta Marine Consultants (DMC) in Utrecht, The Netherlands (Table 2). The test set-up was similar to the tests of Gerding (1993); a rubble mound breakwaters with slope gradient 1:1.5 was installed in the flume. Seabed slopes of 1:10, 1:20 and 1:50 were investigated. The tests were performed with irregular waves (JONSWAP wave spectrum, 1,000 waves per tests). Toe berms of height 8 cm (including bedding layer) and of width 10 cm were installed on a bedding layer of thickness 2.0 cm. Three different stone gradations were applied ( with rock density , respectively); the ratio was about 1.5. Test series were performed with increasing wave height (incident wave height at the wave paddle, ), constant wave steepness of either 0.02, 0.03 or 0.04 (at the wave paddle) and constant water depth. Ebbens tested with seven different water levels; tests with deeper water ( ) are similar to the test conditions of Gerding (1993). The main emphasis of Ebben’s test programme was on shallow water condition ( ). Damage numbers, are specified for all tests. Damage numbers as determined by Gerding (1993) and Ebbens (2009) are plotted in Figure 2 (left) against the stability number (based on actual rock parameters and incident wave height in front of the breakwater). Little correlation can be seen between and . It becomes clear that toe berm damage cannot be predicted solely by the stability number. The same damage numbers are plotted in Figure 2 (right) against the damage prediction of vdM98 (see Table 1), a widely used design formula. Start of damage ( ) refers according to vdM98 ⁄ to stability numbers of (if ). As stated above an upper limit of the measured damage can be described quite accurately by a linear function (dashed line in Figure 2). Based on this upper limit toe berm damage may start if (with ⁄ submergence ). The damage prediction by vdM98 as plotted in Figure 2 (right) appears a bit arbitrary and may not necessarily be suitable for design. Table 2: Key parameters of wave flume experiments Item Markle (1989) Seabed slope, [–] 1:10 Wave conditions Regular Significant wave height, [cm] 6.3 – 9.33) Rock size, [cm] 2.6 – 4.0 Water depth at structure, [cm] 12.2 – 27.4 Water depth on toe berm, [cm] 2.9 – 21 Toe berm width, [cm] 7.7 – 12.3 1) Toe berm height, [cm] 6.3 – 9.3 Relative wave height, [–] 0.61 – 1.123) Relative berm height, [–] 0.24 – 0.77 Relative berm height, [–] 0.21 – 1.253) 2) Wave steepness, [–] 0.009 – 0.0683) Stability number, [–] 1.84 – 4.403) Damage number, [–] – Height above seabed (including bedding layer) Wave steepness at the toe of the structure, 3) Refers to average wave height, and wave period,

Gerding (1993) Ebbens (2009) 1:20 1:10, 1:20, 1:50 Irregular Irregular 14.1 – 24.4 5.0 – 12.9 1.7 – 4.4 1.88 – 2.68 30 – 50 7.3 – 33.9 15 – 42 -0.7 – 25.9 12 – 30 10 8 – 22 8 0.30 – 0.61 0.17 – 1.40 0.45 – 0.84 -0.10 – 0.76 0.82 – 2.78 -0.14 – 4.63 0.010 – 0.044 0.008 – 0.042 2.10 – 8.37 1.07 – 4.16 ≤ 9.21 ≤ 4.37

1) 2)

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⁄


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10

10 Gerding (1993)

9

Ebbens (2009)

8 Damage Number Nod [-]

8 Damage Number Nod [-]

1:10 1:20 1:50 vdM (1998)

9

7 6 5

4 3

7 6 5

4 3

2

2

1

1

0

0 0

1

2 3 4 5 6 7 Stability Number Ns [-]

8

9

0.4

0.8 1.2 1.6 Ns /[2 + 6.2 (dt/d)2.7] [-]

Figure 2: Toe berm damage observed in model tests of Gerding (1993) and Ebbens (2009) plotted against stability number (left) and predicted damage by vdM98 (right) Toe Berm Stability Analysis The hydraulic stability of stones can be assessed from the ratio of driving forces (flow induced drag force, ) and retaining forces (weight, ) on a stone. Stones of a given size start moving when a critical flow velocity, is reached. Izbash (1936) introduced an empirical stability coefficient, for assessing the hydraulic stability of stones:

(1)

The surface area, is approximated by ; the volume, by . The relative density of a submerged stone is and refers to the gravitational acceleration. Izbash characterised the rock material by the average diameter and derived stability coefficients, of 0.86 (movement of loose stones) and 1.2 (movement of embedded stones). Replacing the average rock diameter, by the average nominal rock diameter (ca. ) leads to stability coefficients, of 0.96 and 1.34 for movement of loose stones and embedded stones, respectively. The critical velocity (start of stone movement) is thus: (2) The hydraulic stability of stones that are exposed to breaking waves can also be assessed by equation 1. In this case the flow velocity refers to the water particle velocity in a breaking wave, which is approximately equal to the wave celerity (here described by linear wave theory, shallow water approximation) at the breaking point: √

M. Muttray 29/03/2013

√


(3)

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The water depth at the breaking point is closely related to the breaker height, . The ratio of driving forces and retaining forces according to equation 1 turns into a stability number, when inserting the breaker velocity according to equation 3: (4) The stability number, refers to the initial movement of stones due to a breaking wave of height, of 0.96 and 1.34 (as determined by Izbash, 1936) correspond . Stability coefficients to stability numbers, of 1.8 and 3.6 for the initial movement of loose and embedded stones. The stability number seems to be the obvious approach to the stability of armour stones on a toe berm that is sufficiently close to the water line, i.e. where the wave induced flow velocity is close to the breaker velocity. However, lowering the toe berm will reduce the wave forces on the stones. In other words, the wave induced flow velocity decreases with increasing water depth, on the toe berm. A simple, dimensional correct approach to the wave induced flow velocity on a toe berm is: √

√

(

)

(5)

The wave height, refers to an individual wave that is breaking at or on the toe berm; is the water depth above the toe berm. The empirical coefficient, describes the decay of the flow velocity with depth. This approach is only applicable if . Replacing the critical flow velocity in equation 1 by the wave induced flow velocity on a toe berm according to equation 5 leads to the following toe stability formula for start of damage:

(6)

The stability number, is plotted against in Figure 3 (left) using the results of regular wave tests by Markle (1989). The stability numbers determined by Markle do not refer to initial damage but to acceptable damage ( around 1 to 2). Nonetheless, the relation between and as described by equation 6 is largely confirmed by Markle’s results.

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0.8

0.8

Stability Number Ns-1 [-]

0.6 Ns-1 = -0.3 dt/Hs + 0.57 0.5 0.4 0.3 0.2 Damage Number Nod ca. 1.5

0.1

Stability Number Ns-1 [-]

Markle (1989)

0.7

0.7

Gerding (1993) Ebbing (2009)

0.6

Ns-1 = -0.17 dt/Hs + 0.57

0.5 0.4 0.3 0.2 Damage Number 0.2 < Nod < 0.8

0.1

0

0 0

0.25 0.5 0.75 1 1.25 1.5 Relative Water Depth on Toe Berm dt/Hs [-]

-0.5

0 0.5 1 1.5 2 2.5 3 Relative Water Depth on Toe Berm dt/Hs [-]

Figure 3: Stability number plotted against : Acceptable damage observed in regular waves tests (left) and initial damage observed in irregular wave tests (right) In a natural sea state the wave height, refers to a larger wave in a storm that causes the initial movement of stones on a toe berm. Natural sea states are commonly characterised by the significant wave height, and the stability number is based on this wave height parameter. The toe stability formula for initial damage as defined by equation 6 can also be written as follows:

(7)

Stability numbers from irregular wave tests (Gerding, 1993 and Ebbens, 2009) with damage numbers around 0.5 (initial damage) have been plotted against in Figure 3 (right). The ⁄ empirical fit function indicates that and ⁄ . Tanimoto et al (1982) compared toe berm damage from regular and irregular wave tests and found similar damage for regular waves of height, and for irregular waves of height . ⁄ Applying a ratio leads to a coefficients and and thus to the right expression in equation 7, a toe berm stability number describing the initial movement of stones on a toe berm. Damage can be quantified by the damage number . (i.e. the number of displaced stones in a strip of width ). Start of damage ( ) can be expected when the stability number, reaches a critical value, (see Figure 3, damage numbers of are closely approximated by equation 7). It is obvious that stability numbers in excess of will lead to more damage. A 10% increase in significant wave height may increase the number of supercritical waves (i.e. waves that are larger than ) by a factor of about 1.4 to 3.3 (assuming Rayleigh distributed wave heights). This and the involved magnification of wave loads will lead to progressing damage. The deformation of the toe berm may lead to a stabilisation, when moving stones find eventually a stable position. The processes that are involved in the damage progression can hardly be described analytically. Therefore, the damage progression has been M. Muttray 29/03/2013


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approximated by an empirical approach assuming that it is primarily determined by the ratio of actual and critical stability number, . Damage as observed in the model tests of Gerding (1993) and Ebbens (2009) is plotted in Figure 4 against the ratio of actual and critical stability number. Virtually no stones are displaced if ⁄ . The number of displaced stones is gradually increasing when ⁄ approaches 1; the damage is increasing rapidly if exceeds . The scatter, especially at lower damage numbers, is significant. Damage numbers, of about 1 are associated with stability ratios, varying from 0.6 to 1.7. At start of damage (i.e. if ) the damage number, varies from 0 to 1 (and may even reach 2) with an average around 0.5. For stability ratios larger than 1 the damage progression can be approximated by a linear function (dashed line in Figure 4) predicting a damage number of 0.5 at start of damage. This function is in line with the above considerations and, following the law of parsimony, it may be the obvious choice with respect to the uncertainties of the damage progression. However, for practical applications it may be convenient to have a damage function that includes the range of marginal damage (i.e. if ⁄ ) and provides a slightly conservative estimate of the damage numbers around the start of damage. Approximating the damage progression by a cubic function (solid line in Figure 4) appears thus more favourable from a practical point of view. The cubic trend line is defined by: (8)

√

Equation 8 suggests a damage number of 1 at start of damage, which is a typical upper limit; the average damage is around 0.5. The empirical damage function is only applicable for the range of tested parameters ( ⁄ ; the stability of a re-shaped and possibly dynamically stable toe berm may follow a different trend. It should be noted that a cubic approach according to equation 8 does not necessarily provide a more meaningful description of the physics involved in the toe damage progression than a linear or a polynomial approach. 10

Stability Number 4 - 6 8 Damage Number Nod [-]

(Ns/Ns,cr)3

Stability Number 3 - 4 Stability Number 2 - 3 Stability Number 1 - 2

6 1/2 + 7 (Ns/Ns,cr - 1)

4 Seabed Slope 1:20 2

0 0

0.5

1 1.5 Stability Ratio Ns/Ns,cr [-]

Figure 4: Damage number, number

2

2.5

plotted against the ratio of actual and critical stability

Some of the scatter in Figure 4 can be possibly deduced to parameters that have been varied in the tests of Gerding (1993) and Ebbens (2009) but are not considered in equation 8. These M. Muttray 29/03/2013


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parameters include the seabed slope, the water depth in front of the structure, the height and width of the toe berm and the wave period. The possible influence of these parameters is briefly discussed hereunder. Ebbens (2009) reported increased toe berm damage in tests with steep seabed slope. A steep seabed gradient may affect the breaker height and the width of the surf zone. The wave height is explicitly addressed in equations 7 and 8; the possible effect of a narrow surf zone, with an increased number of waves breaking closely to the toe berm, is not yet considered. The toe berm stability according to equation 7 is determined by three parameters, the Izbash coefficient, the damping water layer on the toe berm and the ratio of critical and significant wave height. A steep seabed slope might affect the wave height ratio, , while it is hard to imagine that the other two parameters would be influenced by the seabed slope. Wave height ratios as calculated by equations 7 and 8 from model test results of Ebbens (2009), the only study with varying seabed slope are plotted in Figure 5 (left) against the seabed gradient. 2.2

1.5

Damage Number 0.5 < Nod < 1.5

2

1.6

1.4

1.4

1.2 1 0.8

0.6 0.4

Measured wave height

0.2

Corrected wave height

0

Izbash Coefficient β [-]

Wave Height Ratio Hcr/Hs [-]

1.8 1.25

1

Gerding (1993)

0.75

Ebbens (2009)

0.5 0

0.04 0.08 0.12 Seabed Slope tan(α) [-]

0

5 10 15 20 Toe Berm Size (Bt + ht)/Dn50 [-]

Figure 5: Effect of parameters that are not included in equation 8: Effect of seabed slope on critical wave height (left) and effect of toe berm size on damage progression (right) When applying the incident wave height at the structure as reported by Ebbens (referred to as “measured wave height” in Figure 5), the wave height ratio is increasing with increasing seabed gradient (see Figure 5, left); the scatter is large but the trend is clear. Larger wave height ratios correspond to smaller toe berm stability numbers (see equation 7), in other words the toe berm damage increases on a steeper seabed, as reported by Ebbens (2009). However, Ebbens measured the incoming waves in front of the structure in tests with different seabed slope at a position with constant seabed level. The water depth at this location, was 4.2 cm larger than the water depth at the toe of the structure, ; the distance to the structure varied for each seabed slope. The wave heights measured by Ebbens may differ from the incoming wave heights at the toe of the structure. The latter have been assessed by the following pragmatic approach. ⁄ In tests without wave breaking ( where refers to the incoming wave height at the paddle) it has been assumed that the wave height at water depth and would be ⁄ identical. In tests with wave breaking ( ) the measured breaker index, was ⁄ and the breaker index at the toe of the structure, plotted against (and thus the incoming wave height at water depth, ) was read from the trend line of measured breaker indices. M. Muttray 29/03/2013


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The wave heights measured by Ebbens have been replaced by the wave heights at the toe of the structure according to above approach; the resulting wave height ratio, has been plotted against the seabed gradient (referred to as “corrected wave height” in Figure 5, left). The scatter is large and it may be arguable whether the wave height ratio is constant or not. However, there is no evidence that wave height ratio and thus toe berm stability are influenced significantly by the seabed slope gradient. A constant wave height ratio, of about 1.4 appears to be a reasonable estimate for seabed slopes of 1:50 to 1:10. This value will most probably also be applicable for more gentle seabed slopes. In case of steep seabed slopes (steeper than 1:10) the toe stability may be reduced and confirmative model tests may be required for assessing the toe stability. The toe stability may be further affected by the geometry of the toe berm, which can be simply described by the sum of toe berm height and width, normalised by the nominal stone diameter, . A typical toe berm, a double layer of stones with a berm width of 3 to 5 stones would have a size, of 5 – 7 stones. Izbash coefficients, as recalculated by equations 7 and 8 from model test results (Gerding, 1993 and Ebbens, 2009, tests with and seabed slope 1:20) are plotted in Figure 5 (right) against the toe berm size. The results of Ebbens indicate an unexpected increase of toe berm stability with increasing toe berm size. The results of Gerding suggest a gradual decay for toe berms larger than 10 stones. Altogether, there is no evidence that the toe berm size has a significant influence on toe stability. Further to seabed gradient and toe berm geometry the effect of water depth, (in front of the structure), peak wave period, and local wave length has been evaluated. The effect of wave steepness, , relative wave height and relative water depth , , and on the toe berm stability, i.e. on start of damage or on the damage progression was found to be insignificant. The scatter in Figure 4 cannot be deduced to parameters that have been varied systematically in the model tests but are not considered in equations 7 and 8. This scatter is apparently caused by influences that are beyond the control of an experimenter or a designing engineer. Toe berm damage is apparently to some extent a random process that cannot be described in explicit detail by a deterministic approach. Practical Application Toe berm damage is largely independent of the toe berm size and geometry (see Figure 5) and is thus properly described by a damage number, that refers to the number of displaced stones within a strip of width . However, a certain number of displaced stones that may be acceptable for a larger toe berm, may be totally unacceptable for a small toe berm. A damage number of may refer either to 5% or to 20% displaced stones, depending on the toe berm size. The percentage of displaced stones, defined by the ratio of displaced stone volume and total stone volume, appears to be a more suitable damage parameter for practical applications:

̅

(9)

The total stone volume per strip of width is determined by the layer thickness of the toe armouring, , the average width, ̅ (averaged over the layer thickness) and the porosity, of the toe berm ( is typically about 40%). About 5% to 10% displaced stones are commonly considered as start of damage. The damage is insignificant if less than 5% of stones are displaced. Flattening of the toe berm starts when about 20% of stones are displaced. A toe berm with about 40% displaced stones is largely flattened but may be still functional. With 50% damage the functionality of the toe berm becomes uncertain. These percentages refer to the place with the largest damage and not to the average damage of a longer toe section. M. Muttray 29/03/2013


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A damage number, of 1 or less may be a sensible starting point for the toe berm design. The submergence of the toe berm has to be estimated. The toe stability number can then be assessed from equation 7; the minimum required average stone weight is: (

)

(10)

After selecting a suitable toe armour stone gradation the toe berm geometry can be defined. The minimum width of a toe berm is about three stone diameter ( ); the minimum layer thickness is two stone diameter ( ). The percentage of displaced stones can be calculated subsequently by equation 9 (with . The damage may be unacceptable and may require adjustments to toe berm geometry or stone size. The assessment of the required stone weight for damage numbers is straightforward; the outcome of equation 10, can be used and equation 8 simplifies to: (11) Equations 10 and 11 can be combined in a toe stability formula of the same type as the equations in Table 1. The new toe stability formula reads:

(12)

Model test results by Markle (1989), Gerdes (1993) and Ebbens (2009) have been plotted against the new toe stability formula in Figure 6. All tests with damage levels of have been considered. A damage level of has been assumed for the regular wave tests of Markle; a factor of 1.4 has been applied for the conversion between irregular and regular wave heights. The new stability formula provides a reasonable estimate of the toe berm stability as observed in model tests. It should be notes that the predicted toe stability is not a central estimate of the experimental results but a slightly conservative estimate. This may be arguable from an academic point of view, for practical applications this seems to be a sensible way of dealing with the inevitable uncertainties of the toe berm stability.

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Calculated Stability Number Nod1/3/Ns,cr [-]

7

Nod1/3/Ns,cr = Ns

1:10 6

1:20 1:50

5

1:20 (regular) 4 3

2 1

Damage Number Nod > 0.1

0 0

1

2 3 4 5 Measured Stability Number Ns [-]

6

7

Figure 6: Comparison of measured and calculated stability numbers The conservatism of the new toe stability formula may have implications for design. The results of the toe stability formula of van der Meer (1998), a widely used design formula have been compared with the new approach. Stability numbers and the required average rock weight have been determined by both methods; four different sections of a typical breakwater toe have been considered (Table 3). A damage level of has been applied for all sections. The first section is located in deeper water (10 m water depth) and has an elevated toe of 2.3 m height consisting of a bedding layer and a rock berm. The seabed drops by 2 m in the second section. This drop is compensated by the bedding layer; the top level of the toe berm remains constant. The other sections are located in the surf zone; water depth and wave height are reduced. The third section has an elevated toe of 2.3 m height; the fourth section has an embedded toe. Table 3: Comparison of new toe stability formula with vdM98 Location Deeper water Surf zone Elevated Elevated Elevated Embedded Rock toe (2.3 m) (4.3 m) (2.3 m) (0 m) Water depth, [m] 10.0 12.0 5.0 3.0 Water depth above toe, [m] 7.7 7.7 2.7 3.0 Significant wave height, [m] 4.0 4.0 2.8 2.2 New approach 4.8 4.8 3.0 3.5 Stability number vdM98 5.6 4.3 3.5 9.1 [-] 1) Measured 3.9 – 5.7 3.9 – 5.7 2.6 – 3.2 3.4 – 4.5 New approach 390 390 550 170 Rock weight [kg] vdM98 250 550 350 10 3 3 Other parameters: Rock density 2,650 kg/m , sea water density 1,030 kg/m , seabed slope 1:20, damage number 1) Model tests with similar toe geometry ( and ) and with damage numbers of

In the first breakwater section the application of the new toe stability formula results in a 15% lower stability number and in about 50% increased rock weight as compared to vdM98. These M. Muttray 29/03/2013


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differences can be put down to the conservatism of the new approach (see Figure 6). The wave conditions and the submergence of the toe berm in first and second section are identical, the only difference is the toe geometry. As demonstrated above, the toe berm geometry has little influence on the toe stability (see Figure 5, right). Surprisingly, the required rock weight in the second section according to vdM98 is more than doubled as compared to the first section, while the new approach requires the same stone size in both sections. The third section is located in the surf zone. Although the wave height is reduced, the required rock weight according to both approaches is about 40% larger than in the first section. As in the first section the new toe stability formula requires about 50% heavier stones as compared to vdM98. In the final section the toe berm is embedded and less exposed; the wave height is further reduced and the submergence is slightly larger than in the third section. The required rock weight according to the new approach is reduced by about 60% as compared to the first section. According to vdM98 the required rock weight would be reduced by 96%; this appears optimistic. The stability numbers according to the new approach and vdM98 have been further compared with stability numbers as determined in the model tests of Gerding (1993) and Ebbens (2009) with similar toe geometry, i.e. similar values of and and with damage numbers varying from 1.5 to 2.5 (see Table 3). In all four sections the stability numbers according to the new approach fall inside the range of measured stability numbers. The stability numbers according to vdM98 are close to the upper and lower bound of measured stability numbers in the first and second section, respectively. They predicted stability by vdM98 is slightly larger than the measured stability in the third section and significantly larger in the final section. It appears from the above that the new toe stability formula is more accurate for toe berms in deeper water with larger bedding layers. The new formula appears further more meaningful in the surf zone, especially for embedded or partly embedded toes. The main limitations of the new toe stability formula are twofold: The toe stability may be overestimated in a situation with seabed slopes stepper than 1:10. A steep seabed may require an embedded toe; the toe stability should be preferably confirmed in model tests. The new formula is further only applicable if the water depth on the toe berm is less than 3 times the incoming wave height (i.e. if ). Otherwise, the bottom velocity under partial standing waves may be larger than the reduced breaker velocity according to equation 5 and should thus replace the breaker velocity in the toe stability formula. This however cannot be verified by the available experimental data. Acknowledgements This study would not have been possible without the support of Delta Marine Consultants. My thanks and appreciations also go to Reinder Ebbens who provided valuable details of his study and to my colleague who have willingly helped me out wherever they could. References Baart, S.; Ebbens, R.; Nammuni-Krohn, J.; Verhagen, H.J., 2010: Toe Rock Stability for Rubble Mound Breakwaters. Proc. 32nd Int. Conf. Coastal Eng., Shanghai, China, ASCE, pp. 13. Brebner, A., Donnelly, P. , 1962: Laboratory Study of Rubble Foundations for Vertical Breakwaters. Proc. 8th Int. Conf. Coastal Eng., Mexico City, Mexico, ASCE, pp. 408 - 429. Burcharth, H. F.; P. Frigaard; J. Uzcanga; M. Berenguer; B.G. Madrigal; J. Villanueva, 1995: Design of the Ciervana Breakwater, Bilbao. Proc. of the Advances in Coastal Structures and Breakwaters Conference, Institution of Civil Engineers, Thomas Telford, London, UK, pp. 26 – 43. Ebbens, R.E., 2009: Toe Structures of Rubble Mound Breakwaters – Stability in Depth Limited Conditions. Master’s Thesis, Delft University of Technology, pp. 84.

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Gerding, E., 1993: Toe Structure Stability of Rubble Mound Breakwaters. Master’s Thesis, Delft University of Technology, Delft Hydraulics Report H1874. Izbash, S.V. (1936): Construction of Dams by Depositing Rock in Running Water, Transaction, Second Congress on Large Dams, Vol. 5, pp. 123-126. Madrigal, B.G.; J.M. Valdés, 1995: Study of Rubble Mound Foundation Stability. Proc. of Final Workshop, MAST II, MCS-Project. Markle, D.G., 1989: Stability of Toe Berm Armor Stone and Toe Buttressing Stone on RubbleMound Breakwaters and Jetties, Physical Model Investigation. Technical Report REMRCO-12, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Tanimoto, K.; T. Yagyu, Y. Goda, 1982: Irregular Wave Tests for Composite Breakwater Foundations. Proc. 18th Int. Conf. Coastal Eng., Cape Town, South Africa, ASCE, pp. 2144 – 2163. Van der Meer, J. W.; K. d'Angremond; E. Gerding, 1995: Toe Structure Stability of Rubble Mound Breakwaters. Proceedings of the Advances in Coastal Structures and Breakwaters Conference, Institution of Civil Engineers, Thomas Telford Publishing, London, UK, pp 308-321. Van der Meer, J. W., 1998: Geometrical Design of Coastal Structures. Chapter 9 in: "Seawalls, Dikes and Revetments", Editor K.W. Pilarczyk, Balkema, Rotterdam.

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