8. Stability of Rock and Armour Units

The stability of the breakwater is divided into three sections, the stability of the armour layer, the toe and the substructure of the breakwater. With the functions in substructure the required nominal diameter of the first underlayer and filter layer can be computed.

8.1. Armour Layer

The armour layer is the outermost layer of the breakwater, the stability of this layer is thus important for the overall stability of the breakwater. The stability of this layer can be computed with either the Hudson formula or the Van der Meer formula

8.1.1. Hudson

breakwater.core.stability.hudson(H, Kd, Delta, alpha)[source]

Hudson formula for armour layer stability (Hudson, 1959)

The Hudson formula is based on experimental research on the stability of rock and armour units (Tetrapods and Dolose) on a slope. The formula is derived by fitting a line through the data points. All unknown factors are included in the ‘dustbin’ factor \(k_D\) (Van den Bos and Verhagen, 2018). The Hudson formula expressed in the form with the stability number is:

\[\frac{H}{\Delta D_{n 50}}=\sqrt[3]{k_{D} \cot \alpha}\]

Warning

The formula is based on regular waves (Hudson, 1959), therefore the wave height H to use in case of irregular waves is not defined. CERC (1984, p. 7-203) advised in the Shore Protection Manual to use \(H_{1/10}\), the average of the highest 10% of the waves.

Note

Although the Hudson formula is derived for the stability of rock and armour units, Van den Bos and Verhagen (2018) recommend to use the Van der Meer formula for the stability of rock. As the Van der Meer formula includes the effects of storm duration, wave period, the structures permeability and a clearly defined damage level (CIRIA, CUR, CETMEF, 2007, p. 567).

Parameters:
  • H (float) – The design wave height [m]
  • Kd (int) – Stability coefficient [-]
  • Delta (float) – Relative buoyant density [-]
  • alpha (float) – Angle of the front slope [rad]
Returns:

Dn50 (float) – the nominal diameter [m]

8.1.2. Van der Meer

Three functions are implemented for the Van der Meer equations, one for deep water conditions, one for shallow water conditions and a full functions. The latter function takes into account the range in which the functions are applicable. The applicability range is determined by the ratio of the water depth over the significant wave height, see table 8.1 for these ranges.

range of applicability of the Van der Meer equations for deep and shallow water

Table 8.1: Range of applicability of the Van der Meer equations for deep and shallow water conditions (CIRIA, CUR, CETMEF, 2007, p. 579)

breakwater.core.stability.vandermeer(LimitState, Delta, P, N, alpha, slope_foreshore, val='max', safety=1, logger=None)[source]

Van der Meer formulae for deep and shallow water conditions

Implementation of Van der Meer formulae for deep and shallow water conditions. The function first determines if the water depth and wave height are in range of the shallow or deep water formulae, and then uses the correct formulae to compute the nominal diameter of the armour layer. In case the input is in the range of the deep and shallow water formulae both formulae are used to compute the diameter, the largest diameter of the two is then returned.

Parameters:
  • LimitState (LimitState) – ULS, SLS or another limit state defined with LimitState
  • Delta (float) – Relative buoyant density [-]
  • P (float) – Notional permeability of the structure [-]
  • N (int) – Number of incident waves at the toe of the structure [-]
  • alpha (float) – Angle of the front slope [rad]
  • slope_foreshore (float) – slope of the foreshore [rad]
  • val ({min, max, avg}, optional, default: max) – value to return in case both the deep and shallow water formula are valid. min for the lowest value, max for the highest value and avg for the average value, default is max.
  • safety (float, optional, default: 1) – With this parameter the model constants, Cpl and Cs, can be decreased to increase the safety. This is implemented as \(C_{pl} = \mu - safety \cdot \sigma\).
  • logger (dict, optional, default: None) – dict to log messages, must have keys ‘INFO’ and ‘WARNINGS’
Returns:

Dn50 (float) – the nominal diameter of the armourstone [m]
breakwater.core.stability.vandermeer_deep(Hs, Delta, P, Sd, N, xi_m, alpha, safety=1)[source]

Van der Meer formulae for deep water conditions

For deep water conditions Van der Meer (1988) derived formulae to predict the stability of armourstone on uniform armourstone slopes with crests above the maximum run-up level. These formulae are only valid in deep water conditions, which is defined as \(h > 3 \cdot H_{s-toe}\), where \(h\) is the water depth on the toe of the structure and \(H_{s-toe}\) the significant wave height at the toe. The Van der Meer formulae are given by:

\[ \begin{align}\begin{aligned}\frac{H_{s}}{\Delta D_{n 50}}=c_{p l} P^{0.18}\left(\frac{S} {\sqrt{N}}\right)^{0.2} \xi_{m}^{-0.5} \; \; \text{for plunging waves}\\\frac{H_{s}}{\Delta D_{n 50}}=c_{s} P^{-0.13}\left(\frac{S} {\sqrt{N}}\right)^{0.2} \sqrt{\cot \alpha} \, \xi_{m}^{P} \; \; \text{for surging waves}\end{aligned}\end{align} \]

As the formulae are based on a large amount of experiments, there is a certain reliability for the model constants. For \(C_{pl}\) this is given by \(6.2 \pm 0.4\) and \(1 \pm 0.08\) for \(C_{s}\). Furthermore, each parameter has its own range of validity. The range of validity of parameters is presented in the table below.

Parameter Symbol Range
Slope angle tan(alpha) 1:6 - 1:1.5
Number of waves N < 7500
Fictitious wave steepness s_om 0.01 - 0.06
Surf similarity parameter xi_m 0.7 - 7
Relative buoyant density of armour Delta 1 - 2.1
Relative water depth at toe h/H_s-toe > 3
Notional permeability parameter P 0.1 - 0.6
Armour gradation Dn85/Dn15 < 2.5
Damage-storm duration ratio Sd/sqrt(N) < 0.9
Stability number Hs/(Delta Dn50) 1 - 4
Damage level parameter Sd 1 - 20
Parameters:
  • Hs (float) – The significant wave height, \(H_{1/3}\) of the incident waves at the toe [m]
  • Delta (float) – Relative buoyant density [-]
  • P (float) – Notional permeability of the structure [-]
  • Sd (float) – Damage level parameter [-]
  • N (int) – Number of incident waves at the toe of the structure [-]
  • xi_m (float) – \(\xi_m\), the surf-similarity parameter computed with the mean wave period \(T_m\) [-]
  • alpha (float) – Angle of the front slope [rad]
  • safety (float, optional, default: 1) – With this parameter the model constants, Cpl and Cs, can be decreased to increase the safety. This is implemented as \(C_{pl} = \mu - safety \cdot \sigma\).
Returns:

Dn50 (float) – the nominal diameter of the armourstone [m]
breakwater.core.stability.vandermeer_shallow(Hs, H2, Delta, P, Sd, N, xi_s_min_1, alpha, safety=1)[source]

Van der Meer formulae for shallow water conditions

Based on the analysis of the stability of rock-armoured slopes in many conditions, mainly focussed on conditions with shallow foreshores, it was proposed in Van Gent et al (2003) to modify the formulae of Van der Meer (1988) to extend its field of application. The Van der Meer formulae for shallow water are:

\[ \begin{align}\begin{aligned}\frac{H_{s}}{\Delta D_{n 50}}=c_{p l} P^{0.18}\left(\frac{S} {\sqrt{N}}\right)^{0.2}\left(\frac{H_{s}}{H_{2 \%}}\right) \, \xi_{m-1,0}^{-0.5} \; \; \text{for plunging waves}\\\frac{H_{s}}{\Delta D_{n 50}}=c_{s} P^{-0.13}\left(\frac{S} {\sqrt{N}}\right)^{0.2}\left(\frac{H_{s}}{H_{2 \%}}\right) \sqrt{\cot \alpha} \, \xi_{m-1,0}^{P} \; \; \text{for surging waves}\end{aligned}\end{align} \]

As the formulae are based on a large amount of experiments, there is a certain reliability for the model constants. For \(C_{pl}\) this is given by \(8.4 \pm 0.7\) and \(1.3 \pm 0.15\) for \(C_{s}\). Furthermore, each parameter has its own range of validity. The range of validity of parameters is presented in the table below.

Parameter Symbol Range
Slope angle tan(alpha) 1:4 - 1:2
Number of waves N < 3000
Fictitious wave steepness s_om 0.01 - 0.06
Surf similarity parameter xi_m 1 - 5
Surf similarity parameter xi_m_min_1 1.3 - 6.5
Wave height ratio H2%/Hs 1.2 - 1.4
Deep-water H over h at the toe Hso/h 0.25 - 1.5
Armour gradation Dn85/Dn15 1.4 - 2.0
Core material - armour ratio Dn50-core/Dn50 0 - 0.3
Stability number Hs/(Delta Dn50) 0.5 - 4.5
Damage level parameter Sd < 30
Parameters:
  • Hs (float) – The significant wave height, \(H_{1/3}\) of the incident waves at the toe [m]
  • H2 (float) – \(H_{2\%}\), wave height exceeded by 2% of the incident waves at the toe [m]
  • Delta (float) – Relative buoyant density [-]
  • P (float) – Notional permeability of the structure [-]
  • Sd (float) – Damage level parameter [-]
  • N (int) – Number of incident waves at the toe of the structure [-]
  • xi_m_min_1 (float) – \(\xi_{s-1.0}\), the surf-similarity parameter computed with the energy wave period \(T_{m-1.0}\) [-]
  • alpha (float) – Angle of the front slope [rad]
  • safety (float, optional, default: 1) – With this parameter the model constants, Cpl and Cs, can be decreased to increase the safety. This is implemented as \(C_{pl} = \mu - safety \cdot \sigma\).
Returns:

Dn50 (float) – the nominal diameter of the armourstone [m]

8.2. Toe

breakwater.core.toe.toe_stability(Hs, h, ht, Delta, Nod)[source]

Toe stability formula of Van der Meer (1998)

The armour layer should is supported by a toe, the formula of Van der Meer (1998) computes the minimal required nominal diameter of the stones necessary. The formula is given as:

\[\frac{H_{s}}{\Delta D_{n50}}=\left(6.2 \frac{h_{t}}{h}+2 \right) N_{o d}^{0.15}\]

The formula is based on experiments and the range of validity of the parameters can be seen in the table below.

Parameter Symbol Range
toe depth over water depth ht/h 0.4 - 0.9
toe depth over nominal diameter ht/Dn50 3 - 25
Parameters:
  • Hs (float) – The significant wave height of the incident waves [m]
  • h (float) – The water depth in front of the toe [m]
  • ht (float) – The water depth on top of the toe [m]
  • Delta (float) – Relative buoyant density [-]
  • Nod (float) – Damage number [-]
Returns:

Dn50 (float) – Nominal diameter of the armourstones in the toe [m]
breakwater.core.toe.toe_berm_stability(Hs, T, d, Bm, Delta, beta=0, alpha_s=0.45)[source]

Berm protection to caisson or vertical wall breakwaters

Compute the nominal diameter of the armourstone on the berm of a caisson or vertical wall breakwaters with the modified Tanimoto formula from Takahasi (2002).

\[D_{n50} = \frac{H_{s}}{\Delta N_{s}}\]

in which:

\[N_{s}=\max \left\{1.8,\left(1.3 \frac{1-\kappa}{\kappa^{1 / 3}} \frac{h^{\prime}}{H_{s}}+1.8 \exp \left[-1.5 \frac{ (1-\kappa)^{2}}{\kappa^{1 / 3}} \frac{h^{\prime}}{H_{s}} \right]\right)\right\}\]

where:

\[\kappa=\frac{4 \pi h^{\prime} / L^{\prime}}{\sinh \left(4 \pi h^{\prime} / L^{\prime}\right)} \kappa_{2}\]

and:

\[\kappa_{2}=\max \left\{\alpha_{S} \sin ^{2} \beta \cos ^{2} \left(\frac{2 \pi x}{L^{\prime}} \cos \beta\right)\right., \left.\cos ^{2} \beta \sin ^{2}\left(\frac{2 \pi x} {L^{\prime}} \cos \beta\right)\right\}\]
Parameters:
  • Hs (float) – mean of the highest 1/3 of the wave heights [m]
  • T (float) – wave period (s)
  • d (float) – water depth at which the armour is placed [m]
  • Bm (float) – width of the berm [m]
  • Delta (float) – Relative buoyant density [-]
  • beta (float, optional, default: 0) – angle between direction of wave approach and a line normal to the breakwater [rad]
  • alpha_s (float, optional, default: 0.45) – factor the include the effect of the slope, the value of 0.45 is given by measures data (Goda, 2000)
Returns:

Dn50 (float) – Nominal diameter of the armourstones on the berm [m]

8.3. Substructure

breakwater.core.substructure.underlayer(Dn_armour, armour_layer, rho, rho_rock=2650)[source]

Design first underlayer of a rubble mound breakwater

Design the layer of rock immediately below the armour layer, the nominal diameter of this layer (the underlayer) is determined using one of the following rules:

  • Rock (CIRIA, CUR, CETMEF, 2007, p. 630)
    \(M_{50u} = \frac{M_{50a}}{15}\) to \(\frac{M_{50a}}{10}\)
  • Xbloc (Delta Marine Consultants, 2018)
    \(M_{50u} = \frac{M_{50a}}{15}\) to \(\frac{M_{50a}}{6}\)
  • XblocPlus (Delta Marine Consultants, 2018)
    \(M_{50u} = \frac{M_{50a}}{20}\) to \(\frac{M_{50a}}{8}\)
Parameters:
  • Dn_armour (float) – nominal diameter of the armour layer [m]
  • armour_layer ({'Rock', 'Xbloc', 'XblocPlus'}) – type of the armour layer
  • rho (float, optional, default: 2650) – density of the armourstone [kg/m³]
  • rho_rock (float, optional, default: 2650) – density of the rock used in the underlayer [kg/m³]
Returns:

[float, float] – list with the upper bound and lower bound limit of the Dn50 of the underlayer
breakwater.core.substructure.filter_layers(Dn, rho=2650)[source]

Design filter layer of a rubble mound breakwater

Design a layer of rock below the first underlayer, or subsequent layers of rock. The nominal diameter of this layer (the filter layer) is given by the following rule:

  • Rock (Van den Bos and Verhagen, 2018, p. 142)
    \(M_{50l} = \frac{M_{50u}}{25}\) to \(\frac{M_{50u}}{10}\)
Parameters:
  • Dn (float) – nominal diameter of the layer above the filter layer [m]
  • rho (float, optional, default: 2650) – density of the armourstone [kg/m³]
Returns:

[float, float] – list with the upper bound and lower bound limit of the Dn50 of the filter layer
breakwater.core.substructure.layer_coefficient(material, layers=None, placement=None)[source]

Get the layer thickness coefficient

Determine the layer thickness coefficient, \(k_{t}\). In Table 1 and 2 the included layer thickness coefficients can be found.

Table 1: \(k_{t}\) values for Rock (CIRIA, CUR, CETMEF, 2007, p. 126)

Layers Placement Value
1 Dense 0.84
2 Standard 0.91
2 Dense 0.91

Table 2: \(k_{t}\) values for Armour Units (CIRIA, CUR, CETMEF, 2007, p. 260)

Armour Unit Layers Value
Cubes 2 1.1
Tetrapods 2 1.02
Dolos 2 0.94
Accropode 1 1.29
CoreLoc 1 1.52
Xbloc 1 1.4
XblocPlus 1 1.33
Parameters:
  • material (str) – material of the layer
  • layers (int, optional, default: None) – number of layers, required if the material is rock. For armour units the number of layers is not required, but a warning will be shown if the specified layer is different from the table
  • placement ({Standard, Dense}, optional, default: None) – placement of the material, required if the material is rock
Returns:

kt (float) – the layer thickness coefficient