# Creating a Fragility Curve¶

The main purpose of this example is to demonstrate how to utilize the GEOLib Python package to construct a fragility curve depending on various water levels.

To undertake failure probability studies (or reliability analyses) for levee macrostability Handreiking Faalkansanalyse, the designer should provide fragility curves. A Fragility Curve represents the (conditional) probability of failure as a function of load. In the case of levee macrostability, the water level (h) is usually used for this purpose. These curves will later be used to include the uncertainties in the water level in the reliability measures of the dike.

After establishing the base macrostability model as a .stix file, the designer may manually evaluate a few situations and generate the associated reliability index ($$\beta$$) as fragility points. In this section, we describe an alternative method in which the GEOLib Python package can automate these operations and extract the fragility curve. The syntax has been enhanced to incorporate water level uncertainty into the fragility curve and calculate the dike’s final dependability index.

The user must first prepare the basis model in D-Stability software and install the GEOLib package. The next step is to load the necessary packages.

from pathlib import Path

import numpy as np # for array calculations
import scipy.stats as st # for statisitcal functions
import pandas as pd # for reading excel files
from scipy.interpolate import interp1d # for linear interpolation and extrapolation outside the range
import matplotlib.pyplot as plt #for ploting the results

from scipy.stats import norm
from scipy.stats import gumbel_r #the distribution function of water level

from geolib.models.dstability import DStabilityModel


In order to change the water level, the phreatic line should be find to be later modified. The next line of codes define a function to do this task.

def find_phreatic_line(waternet: Waternet) -> PersistableHeadLine:
"""
Find the phreatic line in a given waternet.

:param waternet: Waternet to search in
:return: Phreatic line
"""
raise ValueError("No phreatic line found")


The function below computes the fragility curve for a given input file and a range of water levels.

def calculate_fragility_curve(input_file, z_start, z_end, z_step) -> pd.DataFrame:
"""

This method will raise the first point of the water level by the given step size.

:param input_file: Path to the input file
:param z_start: Start of the water level range
:param z_end: End of the water level range
:param z_step: Step size of the water level range
:return: Dataframe with the water level and the corresponding reliability index
"""

# Prepare dataframe
df = pd.DataFrame(columns=["Waterlevel", "Beta", "Filename"])

# Parse input file
dm = DStabilityModel()
input_file_path = Path(input_file)
dm.parse(input_file_path)

output_folder = input_file_path.parent / "output"

# Create output folder if not exists
if not output_folder.exists():
output_folder.mkdir()
else:
# Delete old results
for file in output_folder.glob("*.*"):

# Loop over water levels
for i in range(int((z_end - z_start) / z_step)):
# Calculate new water level
new_z = z_start + i * z_step

# Find phreatic line and set new water level
phreatic_line = find_phreatic_line(dm.datastructure.waternets[0])

# Serialize and execute
output_file = output_folder / (
input_file_path.stem + "_" + str(new_z) + input_file_path.suffix
)
dm.serialize(Path(output_file))
dm.execute()

# Get result
result = dm.get_result(0, 0)
print("Result of Z level: " + str(phreatic_line.Points[0].Z))
print("Reliability index: " + str(result.ReliabilityIndex))

df.loc[i] = [
phreatic_line.Points[0].Z,
result.ReliabilityIndex,
output_file,
]

return df


In the following syntax, the address of base model i.e., fc.stix is introduced. The user should also provide the interested water level range through z_start and z_end. The user may also indicate a folder where the outcomes of different scenarios and the graphs shall be saved.

if __name__ == "__main__":
# Define input file and water level range
input_file = "examples\\dstability\\fragility_curve_waterlevel\\fc.stix"
z_start = -9
z_end = 3
z_step = 6

# Calculate fragility curve
df = calculate_fragility_curve(input_file, z_start, z_end, z_step)

# Save dataframe to csv in subfolder of input file
output_folder = Path(input_file).parent / "output"
df.to_csv(output_folder / "fragility_curve.csv")

# Plot fragility curve
plt.plot(df["Waterlevel"], df["Beta"], "o-")
plt.xlabel("Water level [m]")
plt.ylabel("Reliability index")
plt.title("Fragility curve")
plt.grid()
plt.savefig(output_folder / "fragility_curve.png")
plt.show()


This tutorial’s base file fc.stix and source code can be obtained here.

Determining final failure probability (“integrating out” load uncertainty)

As a final step, the reliability measure of the levee is calculated by combining the conditional failure probability (fragility curve) with the load statistics (in this case, water level statistics). This step is often called “integrating out” because the next integral must be solved:

$P_f=\int \Phi[-\beta(h)] f_h(h) d_h$

where $$f_h(h)$$ is the probability density function of the water level (or other load variable) and $$\Phi$$ is the standard normal cumulative probability function. This integral can be quickly solved using numerical integration as follows.

Note: In this example, Gumbel distribution is assigned as the PDF for waterlevel

# Define water level range
range_h = [-10,2]

# Define bin width for integration water level distribution, 𝑑ℎ
delta_h = 0.1

# Define the distribution parameters of the load (water level) uncertainty distribution
mu= -5
std = 0.2

# indicating the obtained fragility points (𝛽 - h) from the above calculations
b = df["Beta"].to_numpy()
h = df["Waterlevel"].to_numpy()

# function to intepolate the FC curve between fragility points
def densify_extrapolate(x, y, xrange, xdelta):
xnew = np.arange(xrange[0], xrange[1] + xdelta, xdelta)
f = interp1d(x, y, kind='linear', bounds_error=False, fill_value='extrapolate')
ynew = f(xnew)
return xnew, ynew

# intepolate the fragility curve
H, B = densify_extrapolate(h, b, range_h, delta_h)

# breaking the probability density function of load into bins with width of 𝑑ℎ

xnew = np.arange(range_h[0], range_h[1]+delta_h, delta_h)

fh = np.empty(len(xnew), dtype=object)

for h in range(len(xnew)):
fh[h]=gumbel_r.pdf(xnew[h], loc=mu, scale=std)   #Gumbel distribution is assumed for 𝑓ℎ(ℎ)

sumFh = sum(fh)*delta_h

# calculating Φ[−𝛽(ℎ)]
P_fh = norm.cdf(-1*B)

Pf = P_fh* fh * delta_h

sumPf = sum(Pf)/sumFh

# evaluating the relevant relaibiltiy index for th eobtained failure probbaility
Beta = -1*norm.ppf(sumPf)

print('Final Beta after integration = ', Beta)