9. Stability of Monolithic breakwaters¶
The implemented formula to determine the stability of a monolithic breakwater is the extended Goda formula. In figure 9.1 the definition of the used parameters can be seen.
Figure 9.1: Definition of the used parameters in the extended Goda formula
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class
breakwater.core.goda.Goda(Hs, Hmax, h, d, h_acc, hc, Bm, T, beta, rho, slope_foreshore, B=None, lambda_=[1, 1, 1], logger=None)[source]¶ Compute wave pressure with the extended Goda formula (Takahasi, 2002)
Goda (1992) analysed many of the successful and unsuccessful monolithic breakwaters and developed a practical formula that can be used to analyse the stability of monolithic breakwaters. The formula developed by Goda was not meant to compute the pressures for breaking waves (impulsive conditions), therefore Takahasi (2002) included an impulsive pressure coefficient in the formula.
Warning
Goda (1992) advices to avoid impulsive pressures when designing monolithic breakwaters.
Parameters: - Hs (float) – mean of the highest 1/3 of the wave heights [m].
- Hmax (float) – design wave height, equal to the mean of the highest 1/250 of the wave heights [m].
- h (float) – water depth [m]
- d (float) – water depth in front of the caisson, on top of the foundation [m]
- h_acc (float) – submerged depth of the caisson [m]
- hc (float) – height of the caisson above the water line [m]
- Bm (float) – width of the berm [m]
- T (float) – wave period, Goda (2000) advises to use \(T_{1/3}\) [s]
- beta (float) – angle between direction of wave approach and a line normal to the breakwater [rad]
- rho (float) – density of water [kg/m³]
- slope_foreshore (float) – slope of the foreshore [rad]
- B (float, optional, default: None) – width of the monolithic breakwater [m]
- lambda (list, optional, default: [1, 1, 1]) – modification factors of Takahasi (2002) for alternative monolithic breakwater. Input must be lambda_= [\(\lambda_1, \lambda_2, \lambda_3\)].
- logger (dict, optional, default: None) – dict to log messages, must have keys ‘INFO’ and ‘WARNINGS’
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hb¶ offshore water depth at a distance of five times Hs (=H13) [m]
Type: float
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L¶ wave length computed with the dispersion relation [m]
Type: float
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eta_star¶ the elevation to which the wave pressure is exerted [m]
Type: float
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p1, p3, p4 representative wave pressure intensities [Pa]
Type: floats
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pu¶ uplift pressure [Pa]
Type: float
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h_c_star¶ elevation to which the wave pressure is exerted on the caisson, minimum value of hc and eta_star [m]
Type: float
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B¶ width of the monolithic breakwater [m]. None by default so it can be computed with
required_width()Type: float
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Ma()[source]¶ Compute the moment around the center of the caisson
Warning
This method assumes a symmetric caisson
Returns: float – moment around the center of the caisson [Nm]
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Mp()[source]¶ Compute moment at the heel due to the pressure
Returns: float – moment around the heel due to the horizontal pressures [Nm]
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Mu()[source]¶ Compute moment at the heel due to the uplift
Returns: float – moment around the heel due to the uplift pressure [Nm]
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P()[source]¶ Compute horizontal force due to the pressure
Returns: float – horizontal force due to the pressures [Pa]
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bearing_pressure(Pc, rho_c, rho_fill, t=0.5, B=None)[source]¶ compute the bearing pressure at the heel
Method to compute the bearing pressure at the heel of the caisson.
\[\begin{split}p_{e}=\left\{\begin{array}{ll} \frac{2 W_{e}}{3 t_{e}} & : t_{e} \leq \frac{1}{3} B \\ \frac{2 W_{e}}{B}\left(2-3 \frac{t_{e}}{B}\right) & : t_{e}>\frac{1}{3} B \end{array}\right.\end{split}\]in which:
\[t_{e}=\frac{M_{e}}{W_{e}}, \quad M_{e}=M g t-M_{U}-M_{p}, \quad W_{e}=M g-U\]Parameters: - Pc (float) – contribution of concrete to the total mass of the caisson. value between 0 and 1
- rho_c (float) – density of concrete [kg/m³]
- rho_f (float) – density of the fill material, for instance sand [kg/m³]
- t (scalar, optional, default: 0.5) – horizontal distance to the centre of gravity [m]
- B (float, optional, default: None) – width of the monolithic breakwater [m], used with
bearing_pressure_width()to compute the required width to satisfy the maximum bearing pressure.
Returns: pe (float) – the bearing pressure at the heel of the caisson [Pa]
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bearing_pressure_width(B1, Pc, rho_c, rho_fill, pe_max, t=0.5)[source]¶ Compute the required width for the bearing pressure
Method uses fsolve from scipy to compute the width that satisfy the maximum bearing pressure of the foundation.
Parameters: - B1 (float) – first estimate for the width [m]
- Pc (float) – contribution of concrete to the total mass of the caisson. value between 0 and 1
- rho_c (float) – density of concrete [kg/m³]
- rho_f (float) – density of the fill material, for instance sand [kg/m³]
- pe_max (float) – maximum value of the bearing pressure at the heel of the caisson. Goda (2000) advises a value between 400 and 500 kPa.
- t (scalar, optional, default: 0.5) – horizontal distance to the centre of gravity [m]
Returns: float – required width for the maximum bearing pressure [m]
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eccentricity(M)[source]¶ Compute the eccentricity of the net vertical force
Parameters: M (float) – mass of the caisson [kg] Returns: float – eccentricity of the net vertical force [m]
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effective_width(M)[source]¶ Compute the effective width
The effective width must be used for geotechnical computations, due to the fact that the net vertical force of the caisson is eccentric.
Parameters: M (float) – mass of the caisson [kg] Returns: float – the effective width [m]
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mass(Pc, rho_c, rho_fill)[source]¶ Compute the mass of the caisson
Parameters: - Pc (float) – contribution of concrete to the total mass of the caisson. value between 0 and 1
- rho_c (float) – density of concrete [kg/m³]
- rho_f (float) – density of the fill material, for instance sand [kg/m³]
Returns: m (float) – minimal required mass per meter length to satisfy the safety factors [kg/m]
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plot()[source]¶ Plot pressure distribution
Plots the pressure distribution together with the dimensions of the monolithic breakwater.
Warning
Do not read the dimensions of the monolithic breakwater from the axes of the figure. The correct dimensions of the monolithic breakwater can be read from the figure.
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required_mass(mu, t=0.5, SF_sliding=1.2, SF_turning=1.2, logger=None)[source]¶ Compute required mass of the monolithic breakwater
Compute the minimal required mass of the monolithic breakwater based on the failure mechanisms sliding and overturning.
\[M_{sliding} = \frac{P SF_{sliding}}{g \mu} + \frac{U}{g} + \rho B h'\]\[M_{turning} = \frac{M_p SF_{turning}}{g t} + \frac{M_u}{g t} + \rho B h'\]Parameters: - mu (float) – friction factor between the caisson and the foundation [-]
- t (scalar, optional, default: 0.5) – horizontal distance to the centre of gravity [m]
- SF_sliding (float, optional, default: 1.2) – safety factor against sliding. Default value according to Goda (2000)
- SF_turning (float, optional, default: 1.2) – safety factor against sliding. Default value according to Goda (2000)
- logger (dict, optional, default: None) – dict to log messages, must have keys ‘INFO’ and ‘WARNINGS’
Returns: mass (float) – minimal required mass per meter length to satisfy the safety factors [kg/m]
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required_width(Pc, rho_c, rho_f, rho_w, mu, t=0.5, SF_sliding=1.2, SF_turning=1.2, logger=None)[source]¶ Compute the required width of the monolithic breakwater
Compute the minimal required width of the monolithic breakwater based on the failure mechanisms sliding and overturning.
Parameters: - Pc (float) – contribution of concrete to the total mass of the caisson. value between 0 and 1
- rho_c (float) – density of concrete [kg/m³]
- rho_f (float) – density of the fill material, for instance sand [kg/m³]
- rho_w (float) – density of water [kg/m³]
- mu (float) – friction factor between the caisson and the foundation [-]
- t (scalar, optional, default: 0.5) – horizontal distance to the centre of gravity [m]
- SF_sliding (float, optional, default: 1.2) – safety factor against sliding. Default value according to Goda (2000)
- SF_turning (float, optional, default: 1.2) – safety factor against sliding. Default value according to Goda (2000)
- logger (dict, optional, default: None) – dict to log messages, must have keys ‘INFO’ and ‘WARNINGS’
Returns: B (float) – minimal required width to satisfy the safety factors [m]