TOE STABILITY IN VERY SHALLOW WATER COMBINED WITH STEEP SEA BOTTOM

It is common to construct a rock toe berm of three to four rocks wide when concrete armor units are placed in the armor layer. This toe berm is a relevant element, especially in very shallow waters combined with steep sea bottoms, where waves directly attack the toe berm and the lowest part of the armor. Several formulas are available to estimate the damage to rock toe berms. In this paper, these formulas are compared for different design conditions within their range of application. Most of these formulas use the damage parameter Nod. However, there are often situations in which wider toe berms are required for a safe design, and the damage parameter Nod is not recommended. A methodology is thus proposed to design wider toe berms based on the damage to the nominal toe berm of three rocks wide (Nod*), considered as the most shoreward part of the toe berm which effectively supports the armor layer.


INTRODUCTION
When concrete armor units are used, a toe berm of three or four rocks wide (Bt=3-4Dn50) is usually placed on the seafloor to provide support to the armor layer.Fig. 1 shows the cross section of a conventional rock toe berm.The design of the toe berm depends mainly on the bottom slope (m), the water depth at the toe (hs), and the design wave storm (Hs and Tp).USACE (1984) proposed estimating the mass of the toe berm rocks as one order of magnitude lower than the mass of the armor units.However, this design criterion is not valid for depth-limited breaking wave conditions, especially, when breakwaters are placed on rocky coastlines with steep seafloors and shallow waters.In these conditions, mound breakwaters may require emerged toe berms, and the rock size may exceed the armor unit size (see Herrera and Medina, 2015).
This paper focuses attention on the design of rock toe berms, the relevance of the sea bottom slope on the hydraulic stability of toe berms, and the use of wider than conventional rock toe berms to reduce the size of the required rocks.Other design alternatives, not analyzed in this paper, to increase the toe berm hydraulic stability may involve using concrete units in the toe berm (see Burchart andLiu 1995, or Van Gent andVan der Werf 2014), or excavating trenches in the sea bottom to support the rocks in the toe berm (USACE 2006).

DESIGN OF ROCK TOE BERM
Different formulas have been proposed to design rock toe berms.Most of them use the stability number (Ns=Hs/∆Dn50), where Hs is the significant wave height measured at the toe of the structure, Δ=(ρr−ρw)/ρw is the relative submerged mass density, ρr and ρw are the mass densities of the rocks and the sea water, respectively, and Dn50 is the nominal diameter of the rocks in the toe berm.Gerding (1993) proposed Eq. 1 to estimate toe berm damage using the dimensionless damage parameter Nod=N/BDn50, where N is the number of rocks displaced from the toe, and B is the width of the reference area.Eq. 1 was based on small-scale tests conducted with a bottom slope m=1/20 and water depths at the toe hs(cm)=30, 40, and 50.Two wave steepnesses (s0p=2πHsg/gTp 2 =0.02 and 0.04) were tested with increasing significant wave height at the wave generating zone (Hsg(cm)=15, 20 and 25).Rock toe berms with Dn50(cm)=1.7, 2.5, 3.5 and 4.0 were tested with relative toe berm widths within the range 3Dn50≤Bt≤12Dn50 , and relative toe berm thicknesses of 2.3Dn50≤tt≤8.8Dn50;however, Bt was not explicitly introduced as an explanatory parameter for toe berm damage.

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where ht is the water depth above the toe berm.Van der Meer (1998) proposed Eq. 2 to estimate toe berm damage, based on Gerding's tests.This author introduced the ratio ht/hs as an explanatory parameter, instead of the relative water depth above the toe, ht/Dn50.
Ebbens ( 2009) conducted small-scale tests with three bottom slopes m=1/10, 1/20 and 1/50 and water depths at the toe in the range 7≤hs(cm)≤25.He measured toe berm damage using the damage parameter Nod, and also introduced the damage number N%=NDn50/Vtot (1-nv), where Vtot is the total volume of the toe berm, and nv is the void porosity.Tests were conducted with three wave steepnesses (s0p=2πHsg/gTp 2 =0.02, 0.03 0.04) and four significant wave heights at the wave generating zone (Hsg(cm)=6, 8, 10 and 12).Toe berms with Dn50(cm)=1.88, 2.15, and 2.68 were considered with a toe berm width of Bt (cm)=10 and toe berm thickness of tt (cm)=6.According to this author, the bottom slope (m) had a significant influence on toe berm damage (see Fig. 4); thus, it was introduced as an explanatory parameter in the proposed equation (Eq.3), where L0p=gTp 2 /2π is the wave length in deep water conditions and Tp is the peak period.By contrast, hs, Bt and tt were not included in Eq. 3. (3) Muttray (2013) proposed Eq. 4 to estimate Nod based on tests conducted by different authors, including as Gerding (1993) and Ebbens (2009):

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More recently, Van Gent and Van der Werf (2014) proposed Eq. 5 to estimate rock armor damage after conducting small-scale tests with m=1/30 and hs(cm)=20, 30 and 40.Two wave steepnesses (s0p=2πHsg/gTp 2 =0.0015 and 0.04) were tested with increasing significant wave height up to Hsg(cm)=28.Toe berms with Dn50(cm)=1.46 and 2.33 were considered with toe berm widths of Bt=3Dn50 and 9Dn50, and toe berm thicknesses of tt=2Dn50 and 4Dn50.In this case, Bt and tt were introduced as explanatory parameters for toe berm damage.For wide toe berms (3Dn50<Bt≤9Dn50), these authors proposed multiplying the Nod obtained with Eq. 5 by the factor fb=(nt/3) 1/2 , where nt is the number of rock rows placed on the upper layer of the toe berm.
and Tm-1,0 is the spectral wave period.Herrera and Medina (2015) analyzed the hydraulic stability of emerged and submerged rock toe berms, with m=1/10 and very shallow waters (-2≤hs(cm)≤20).2D small-scale tests were conducted in the wave flume of the Laboratory of Ports and Coasts at the Universitat Politècnica de València (LPC-UPV).Five peak periods were tested: Tp=1.2, 1.5, 1.8, 2.2 and 2.4s.For each Tp, irregular waves were generated from no damage to waves high enough to break in the wave generating zone (8≤Hsg(cm)<22).Toe berms with Dn50(cm)=3.99 and 5.12 were considered with a toe berm width of Bt= 3Dn50, and a toe berm thickness of tt= 2Dn50.Toe berm damage was measured after each test run using the damage parameter Nod, and cumulative damage was considered during each test series defined by hs (35 to 40 tests per series).Based on test results, Eq. 6 was proposed to estimate Nod, where Hs0 is the significant wave height in deep water conditions.Herrera et al. (2016) analyzed the influence of toe berm width (Bt=ntDn50) on the hydraulic stability.Small-scale tests were conducted in the LPC-UPV wave flume with m=1/10 and 7.7≤hs(cm)≤10.6.The same wave periods and significant wave heights considered in Herrera and Medina (2015) were tested (1.2≤Tp(s)≤2.4 and 8≤Hsg(cm)<22).Toe berms with Dn50(cm)=3.04, 3.99 and 5.12 were studied for toe berm widths of Bt=3Dn50, 5Dn50 and 12Dn50, and fixed toe berm thickness tt= 2Dn50.To characterize the damage to wide toe berms (Bt>3Dn50), the authors introduced two new concepts (see Fig. 2): 1. Nominal toe berm: the 3Dn50-wide most shoreward part of the toe berm, necessary to support the armor layer.2. Sacrificial toe berm: the most seaward part of the toe berm, which protects the nominal toe berm.After each test, two damage parameters were measured: (1) Nod, corresponding to the total damage to the toe berm, and (2) Nod* corresponding only to the damage to the 3Dn50-wide nominal toe berm.Based on the Nod* measured in tests, Eq. 6 was modified to take into account the toe berm width (Bt=ntDn50) for toe berm design, where Dn50,nt is the nominal diameter required for a ntDn50wide toe berm, and hss is the water depth at the toe of the nominal berm.
Table 1 summarizes the test conditions and the range of parameters used by different authors.a Refers to measurements at the wave generating zone.
Eqs. 1 and 2 can be used to estimate rock toe berm damage caused by a single storm of 1000 waves, using Hs measured at the toe of the structure.Eq. 3 considers the cumulative damage of each test series defined by a water depth, and Eq. 5 considers the cumulative damage of each test series defined by a wave steepness.For Eqs. 6 and 7, the model was rebuilt after 35-40 tests with the same water depth, taking into account that breakwaters must withstand several wave storms of less energy than the design storm.Eqs. 3, 5, 6 and 7 also consider the wave period Tp or Tm-1,0 to characterize the wave storm.
Eqs. 1 to 5 are valid for submerged toe berms (ht>>0) and relatively gentle sea bottoms, while Eq. 6 is valid for emerged and submerged toe berms placed on steep sea bottoms (m=1/10).Most of the formulas were obtained from laboratory tests with different toe berm geometries; the toe berm width (Bt) and thickness (tt) were not usually used as explanatory variables for the observed toe berm damage; only Eq. 5 considers Bt and tt as explanatory variables in the design of submerged toe berms placed on a m=1/30 sea bottom.When considering wider and/or higher toe berms, common damage parameters (Nod, N%) are not recommended since larger toe berms require more rocks displaced from the toe (N) to be damaged; rocks situated in the most seaward part of the toe berm do not contribute to support the armor, but only to protect the most shoreward part of the toe berm.Thus, Herrera et al. (2016) proposed a new methodology to design wide toe berms (3Dn50≤Bt≤12Dn50) placed on a m=1/10 sea bottom, based on the damage to the nominal toe berm (Nod*) which actually supports the armor layer.When designing with Nod*, common values for acceptable damage (Nod*<0.5-nodamage; Nod*≈1−significant movements; Nod*≈2−moderate damage; Nod*>4−failure) can be directly applied.Fig. 5 shows the estimations of Nod reported by Herrera and Medina (2015) compared to the dimensionless variable (Hs0 L0p) 1/2 /∆Dn50, for three relative water depths at the toe.hs/Dn50=0 corresponds to an emerged toe berm; hs/Dn50=4 corresponds to a submerged toe berm, and hs/Dn50=2 corresponds to the still water level (SWL) just at the top of the toe berm.Nod increased with (Hs0 L0p) 1/2 /∆Dn50 for the three relative water depths at the toe (hs/Dn50).Note that Eqs. 1 to 5 require knowing Hs at the toe of the structure, while Eq. 6 given by Herrera and Medina (2015) uses the wave parameters in deep water wave conditions (Hs0, Tp).The influence of the relative water depth at the toe (hs/Dn50) on toe berm damage was also analyzed for standard toe berms having Bt=3Dn50 and tt=2Dn50.Fig. 6 shows the Nod estimated by Eqs. 1 to 5 as a function of hs/Dn50, for a typical storm in the Alboran Sea (Hs(m)=5, Tp(s)=11) and rocks of 6-tonnes in weight with a mass density of ρr(t/m 3 )=2.7.Each equation was represented within its range of application: 7.5≤hs/Dn50≤29 for Eqs. 1 and 2; 2.7≤hs/Dn50≤19 for Eq.3; ht/Hs<3 for Eq.4; and 8.6≤hs/Dn50≤27.4 for Eq. 5.For Eqs. 1, 2, 4 and 5, Nod decreased when increasing hs/Dn50.For Eq. 3, Nod did not vary for m=1/10 and m=1/20 when increasing or reducing hs/Dn50.Fig. 7 shows the estimations of Nod reported by Herrera and Medina (2015) compared to hs/Dn50, when considering Hs0(m)=5, Tp(s)=11 in deep water conditions, and rocks of 6-tonnes in weight with a mass density of ρr(t/m 3 )=2.7.Nod was highest when the SWL was close to the top of the toe berm (hs/Dn50=3).From hs/Dn50=3, Nod decreased when increasing hs/Dn50.For the rock size and wave storm considered in this example, there was a range of relative water depths for which Nod was excessive.In order to reduce toe berm damage, Eq. 6 and Fig. 7 can be used to determine a more stable position if the toe of the structure is moved shoreward or seaward in the range -0.5≤hs/Dn50≤5.0.When it is not feasible to move the toe position of the structure due to environmental, economic or operational constraints, a sacrificial toe berm can be considered following the methodology proposed by Herrera et al. (2016).

Sacrificial toe berms
As mentioned before, when considering wider toe berms, common damage numbers (Nod or N%) are not suitable to characterize toe berm stability.When increasing the toe berm width (Bt=ntDn50>3Dn50), it is necessary to analyze the behavior of the sacrificial and nominal toe berms.Fig. 8 provides the total toe berm damage (Nod) and the nominal toe berm damage (Nod*) measured in the laboratory tests conducted by Herrera et al. (2016) with m=1/10, rocks of Dn50(cm)=3.99 and nt=3, 5 and 12, as a function of the variable (Hs0L0p) 1/2 .The total toe berm damage (Nod) increased when the toe berm width (nt) was increased, while the nominal toe berm damage (Nod*) was less when the toe berm width was increased.Thus, given a rock size (Dn50), a wider toe berm reduces Nod* although the Nod increases.As a result, Nod is not the best estimator of toe berm damage for wide toe berms (Bt >3Dn50); Nod* is better and should be considered for design purposes.Considering the nominal toe berm damage (Nod*), Herrera et al. (2006) proposed a methodology to reduce the rock size required for a toe berm placed on a m=1/10 sea bottom, when it is not possible to find this rock size or when the required rock size is not available near the construction site.Given a design wave storm, Eq. 6, proposed by Herrera and Medina (2015), can be first applied to design a standard 3Dn50-wide and 2Dn50-high rock toe berm within the ranges 0.02≤s0p≤0.07,−0.15≤hs/Hs0≤1.5,and −0.5≤hs/Dn50≤5.0.Eq. 8 can be used later to reduce the required rock size in the toe berm by increasing the toe berm width (Bt=ntDn50 and nt≤12), following a 0.4-power relationship.Fig. 9 shows a sketch of this process.
where Dn50,3 is the nominal diameter required for a nominal toe berm measuring 3Dn50 in width, and Dn50,nt is the nominal diameter required for a ntDn50,nt-wide toe berm.Fig. 10 depicts the reduction in rock mass (M50) given by Eq, 8 when increasing the toe berm width up to nt=12.For this example, the rock mass required for a standard toe berm (nt=3) placed on m=1/10 sea bottoms, was calculated first using Eq.6 with the recommended design value of Nod=Nod*=1.M50(t)=30-tonne rocks (ρr(t/m 3 )=2.7) is necessary for the toe berm, if the water depth at the nominal toe berm is hs(m)=hss(m)=4.5 for the wave storm Hs0(m)=5 and Tp(s)=11.When increasing the toe berm width to nt=6, 9 and 12 following Eq.8, the required rock mass is reduced to M50(t)=13, 8.7 and 5, respectively.As described in Section 2, other authors, such as Gerding (1993) and Van Gent and Van der Werf (2014), conducted laboratory tests with different toe berm widths.Gerding (1993) did not include Bt as an explanatory parameter for toe berm damage.In contrast, Van Gent and Van der Werf (2014) did include it but only total damage Nod was considered; these authors proposed multiplying the design Nod by the factor fb=(nt/3) 1/2 .Following the methodology described by Herrera et al. (2016), Eq. 5 proposed by Van Gent and Van der Werf (2014) was re-written in terms of Dn50,nt/ Dn50,3.The ratio Dn50,nt/ Dn50,3 also followed the relationship given by Eq. 8 but elevated to the 2/17-power (Eq.9).Eq. 9 can be applied to rock toe berms placed on m=1/30 sea bottoms with 3Dn50≤Bt≤9Dn50 and 2Dn50≤tt≤4Dn50, within the ranges 0.012≤s0p≤0.042,1.2≤hs/Hs≤4.5, and 8.6≤hs/Dn50≤27.4.
Fig. 11 portrays the reduction in rock mass (M50) given by Eq. 9, when increasing berm width up to nt=9.For this example, the rock mass required for a nominal toe berm (nt=3), placed on m=1/30 sea bottoms, was calculated using Eq. 5 given by Van Gent and Van der Werf (2014) with the design value of Nod=Nod*=1.If the water depth at the nominal toe berm is hs(m)=hss(m)=4.5,Hs (m)=5 and Tp(s)=11 at the toe, one-tonne rocks (ρr(t/m 3 )=2.7)are necessary to build the toe berm.
When increasing the toe berm width to nt=5, 7 and 9 following Eq.9, the required rock mass is reduced to M50(t)=0.84,0.74 and 0.68, respectively.Note that the estimations of the required rock size differ significantly when using Eq. 8 or Eq.9; toe berms behave differently when dealing with shallow waters and waves breaking on m=1/10 steep sea bottoms (Herrera et al. 2016), than when dealing with less severe wave breaking on m=1/30 gentle sea bottoms (Van Gent and Van der Werf, 2014).

CONCLUSIONS
The hydraulic stability of toe berms depends mainly on (1) design wave storm (Hs and Tp), (2) water depth at the toe (hs), and (3) bottom slope (m) existing at the construction site.
Most formulas to design rock toe berms are based on laboratory tests with gentle sea bottom slopes and toe berms below the SWL (hs>>0); in these conditions, toe berm damage usually decreases with increasing hs.However, on rocky coastlines with steep sea bottoms, sea defenses may require emerged toe berms with heavier rocks.According to Herrera and Medina (2015), the worst situation corresponds to SWL close to the top of the toe berm.The standard rock toe berm has Bt=3Dn50 and tt=2Dn50; this corresponds to a relative water depth at the toe of hs/Dn50=3.In this situation, for certain design storms, the required rock size may be so large that the construction of toe berms is not feasible with rocks available at quarries.To solve this problem, the toe position may be moved to shallower or deeper waters in order to increase the toe berm stability, or the toe berm width may be enlarged using smaller rocks.
When increasing the toe berm width (Bt>3Dn50), common damage parameters (Nod or N%) are not recommended, since wider toe berms lead to more damage despite the better performance.Thus, two parts must be distinguished in rock toe berms when Bt>3Dn50: (1) nominal toe berm and (2) sacrificial toe berm.

Figure 1 .
Figure 1.Schematic cross section of conventional rock toe berm.