Example: Montecarlo Simulation

This example shows how the Polynomial L-Flange Segment model implementation can be used to predict a random serie of actualizations of bolt force and bolt moment, based on the random imput of the following parameters.

  • Gap Length, assumed normally distributed with mean value 30° and COV 10%
  • Gap Height, assumed log-normally distributed according to IEC 61400-6 AMD1, section 6.7.5.2
  • Bolt preload, assumed normally distributed with mean value 2876 kN and COV 10%

All the other parameters are assumed to be deterministically known, and defined as follows:

  • Bolt M80 with meterial grade 10.9
  • Distance between inner face of the flange and center of the bolt hole: a = 232.5 mm
  • Distance between center of the bolt hole and center-line of the shell: b = 166.5 mm
  • Shell thickness: s = 72 mm
  • Flange thickness: t = 200 mm
  • Flange outer diameter: D = 7.5 m
  • Bolt hole diameter: Do = 86 mm
  • Washer outer diameter: Dw = 140 mm

Let's first define a function that, given an actualization of gap length, gap height and bolt preload generates the corresponding FlangeSegment object.

# Imports
from math import pi
from pyflange.bolts import StandardMetricBolt, ISOFlatWasher, ISOHexNut
from pyflange.flangesegments import PolynomialLFlangeSegment

# Define some units for the sake of readability
m = 1
mm = 0.001*m
N = 1
kN = 1000*N

# Create the fastener parts
M80_bolt = StandardMetricBolt("M80", "10.9", shank_length=270*mm stud=True)
M80_washer = ISOFlatWasher("M80")
M80_nut    = ISOHexNut("M80")
Nb = 120   # Number of bolts

# Flange Segment Constructor
def create_flange_segment (gap_angle, gap_height, bolt_preload):
    return PolynomialLFlangeSegment(
        a = 232.5*mm,              # distance between inner face of the flange and center of the bolt hole
        b = 166.5*mm,              # distance between center of the bolt hole and center-line of the shell
        s = 72*mm,                 # shell thickness
        t = 200.0*mm,              # flange thickness
        R = 7.5*m / 2,             # shell outer curvature radius
        central_angle = 2*pi/Nb,   # angle subtented by the flange segment arc

        Zg = -15000*kN / Nb,    # load applied to the flange segment shell at rest

        bolt = M80_bolt,        # bolt object created above
        Fv = bolt_preload,      # applied bolt preload

        Do = 86*mm,             # bolt hole diameter
        washer = M80_washer,    # washer object created above
        nut = M80_nut,          # nut object created above

        gap_height = gap_height,   # maximum longitudinal gap height
        gap_angle = gap_angle      # longitudinal gap length
    )

Next, we define the stochastic variables gap_angle, gap_height and bolt_pretension:

# Gap Height Log-Normal Distribution
from pyflange.gap import gap_height_distribution
D = 7.5*m
gap_length = pi/6 * D/2
gap_height_dist = gap_height_distribution(7.5*m, 1.4*mm/m, gap_length)

# Gap angle distribution
from scipy.stats import normal
mean = pi/6
std = 0.10 * mean
gap_angle_dist = norm(loc=mean, scale=std)

# Bolt pretension distribution
mean = 2876*kN
std = 0.10 * mean
bolt_preload_dist = norm(loc=mean, scale=std)

Next we generate random actualizations of the stochastic parameters and evaluate the corresponding values of Fs(Z) and Ms(Z), in discrete form.

# Let's define the discrete domain Z of the Fs(Z) and Ms(Z) functions we
# want to determine. We define Z as an array of 1000 items, linearly
# spaced between -1500 kN and 2100 kN.
import numpy as np 
Z = np.linspace(-1500*kN, 2100*kN, 1000)

# Let's generating 25000 actualizations of Fs(Z) and Ms(Z) and store them in
# two 1000x25000 matrices, where each row is an actualization of the discrete
# image of Z through Fs and Ms.

Fs = np.array([])    # Initialize Fs with an empty matrix
Ms = np.arrat([])    # Initialize Ms with an empty matrix

for i in range(25000):

    # Generate a random gap height
    gap_height = gap_height_dist.rvs()

    # Generate a random gap angle
    gap_angle = gap_angle_dist.rvs()

    # Generate a random bolt pretension
    bolt_preload = bolt_preload_dist.rvs()

    # Generate the corresponding random FlangeSegment actualization, using the
    # factory function defined above
    fseg = create_flange_segment(gap_angle, gap_height, bolt_preload)

    # Generate the Fs image of Z and store it in the Fs matrix
    Fs.append( fseg.bolt_axial_force(Z) ) 

    # Generate the Ms image of Z and store it in the Ms matrix
    Ms.append( fseg.bolt_bending_moment(Z) )

The generated data in Fs and Ms can be then used to fit a distribution to the data.