Two-dimensional (2D) Cantilever Beam Reliability Problem
Contents
Two-dimensional (2D) Cantilever Beam Reliability Problem#
import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf
The 2D cantilever beam problem is a reliability test function from [RE93]. This is an often revisited problem in reliability analysis (see, for instance, [LGG+18]).
The plots of the function are shown below. The left plot shows the surface plot of the performance function, the center plot shows the contour plot with a single contour line at function value of \(0.0\) (the limit-state surface), and the right plot shows the same plot with \(10^6\) sample points overlaid.
Test function instance#
To create a default instance of the test function:
my_testfun = uqtf.CantileverBeam2D()
Check if it has been correctly instantiated:
print(my_testfun)
Name : CantileverBeam2D
Spatial dimension : 2
Description : Cantilever beam reliability problem from Rajashekhar and Ellington (1993)
Description#
The problem consists of a cantilever beam with a rectangular cross-section subjected to a uniformly distributed loading. The maximum deflection at the free end is taken to be the performance criterion and the performance function reads1:
where \(I\), the moment inertia of the cross-section, is given as follows:
By plugging in the above expression to the performance function, the following expression for the performance function is obtained:
where \(\boldsymbol{x} = \{ w, h \}\) is the two-dimensional vector of input variables, namely the load per unit area and the depth of the cross-section. These inputs are probabilistically defined further below.
The parameters of the test function \(\boldsymbol{p} = \{ E, l \}\), namely the beam’s modulus of elasticity (\(E\)) and the span of the beam (\(l\)) are set to \(2.6 \times 10^{4} \; \mathrm{[MPa]}\) and \(6.0 \; \mathrm{[m]}\), respectively.
The failure state and the failure probability are defined as \(g(\boldsymbol{x}; \boldsymbol{p}) \leq 0\) and \(\mathbb{P}[g(\boldsymbol{X}; \boldsymbol{p}) \leq 0]\), respectively.
Probabilistic input#
Based on [RE93], the probabilistic input model for the test function consists of two independent standard normal random variables (see the table below).
my_testfun.prob_input
Name: Cantilever2D-Rajashekhar1993
Spatial Dimension: 2
Description: Input model for the cantilever beam problem from Rajashekhar and Ellingwood (1993)
Marginals:
| No. | Name | Distribution | Parameters | Description |
|---|---|---|---|---|
| 1 | W | normal | [1000. 200.] | Load per unit area [N/m^2] |
| 2 | H | normal | [250. 37.5] | Depth of the cross-section [mm] |
Copulas: None
Reference results#
This section provides several reference results of typical UQ analyses involving the test function.
Sample histogram#
Shown below is the histogram of the output based on \(10^6\) random points:
def is_outlier(points, thresh=3.5):
"""
Returns a boolean array with True if points are outliers and False
otherwise.
This is taken from:
https://stackoverflow.com/questions/11882393/matplotlib-disregard-outliers-when-plotting
Parameters:
-----------
points : An numobservations by numdimensions array of observations
thresh : The modified z-score to use as a threshold. Observations with
a modified z-score (based on the median absolute deviation) greater
than this value will be classified as outliers.
Returns:
--------
mask : A numobservations-length boolean array.
References:
----------
Boris Iglewicz and David Hoaglin (1993), "Volume 16: How to Detect and
Handle Outliers", The ASQC Basic References in Quality Control:
Statistical Techniques, Edward F. Mykytka, Ph.D., Editor.
"""
if len(points.shape) == 1:
points = points[:,None]
median = np.median(points, axis=0)
diff = np.sum((points - median)**2, axis=-1)
diff = np.sqrt(diff)
med_abs_deviation = np.median(diff)
modified_z_score = 0.6745 * diff / med_abs_deviation
return modified_z_score > thresh
xx_test = my_testfun.prob_input.get_sample(1000000)
yy_test = my_testfun(xx_test)
yy_test = yy_test[~is_outlier(yy_test, thresh=10)]
idx_pos = yy_test > 0
idx_neg = yy_test <= 0
hist_pos = plt.hist(yy_test, bins="auto", color="#0571b0")
plt.hist(yy_test[idx_neg], bins=hist_pos[1], color="#ca0020")
plt.axvline(0, linewidth=1.0, color="#ca0020")
plt.grid()
plt.ylabel("Counts [-]")
plt.xlabel("$\mathcal{M}(\mathbf{X})$")
plt.gcf().set_dpi(150);
Failure probability (\(P_f\))#
Some reference values for the failure probability \(P_f\) from the literature are summarized in the table below.
Method |
\(N\) |
\(\hat{P}_f\) |
\(\mathrm{CoV}[\hat{P}_f]\) |
Source |
Remark |
|---|---|---|---|---|---|
\(10^6\) |
\(9.594 \times 10^{-3}\) |
— |
[LGG+18] |
— |
|
\(27\) |
\(9.88 \times 10^{-3}\) |
— |
[LGG+18] |
— |
|
— |
\(9.9031 \times 10^{-3}\) |
— |
[RE93] |
— |
|
\(32\) |
\(9.88 \times 10^{-3}\) |
— |
[LGG+18] |
— |
|
\(10^3\) |
\(9.6071 \times 10^{-3}\) |
— |
[RE93] |
Importance Sampling (IS) |
|
\(9'312\) |
\(1.00 \times 10^{-2}\) |
— |
[SG05] |
— |
|
IS + RS |
\(2'192\) |
\(9.00 \times 10^{-3}\) |
— |
[SG05] |
IS + Response Surface (RS) |
IS + SP |
\(358\) |
\(1.00 \times 10^{-2}\) |
— |
[SG05] |
IS + Splines (SP) |
IS + NN |
\(63\) |
\(1.20 \times 10^{-2}\) |
— |
[SG05] |
IS + Neural Networks (NN) |
DS |
\(551\) |
\(1.000 \times 10^{-2}\) |
— |
[SG05] |
Directional sampling (DS) |
DS + RS |
\(60\) |
\(6.00 \times 10^{-3}\) |
— |
[SG05] |
DS + Response Surface (RS) |
DS + SP |
\(57\) |
\(7.00 \times 10^{-3}\) |
— |
[SG05] |
DS + Splines (SP |
DS + NN |
\(40\) |
\(8.00 \times 10^{-3}\) |
— |
[SG05] |
DS + Neural Networks (NN) |
SSRM |
\(18\) |
\(9.499 \times 10^{-3}\) |
— |
[LGG+18] |
Sequential surrogate reliability method |
Bucher’s |
— |
\(1.37538 \times 10^{-2}\) |
— |
[RE93] |
— |
Approach A-0 |
— |
\(9.5410 \times 10^{-3}\) |
— |
[RE93] |
— |
Approach A-1 |
— |
\(9.6398 \times 10^{-3}\) |
— |
[RE93] |
— |
Approach A-2 |
— |
\(1.11508 \times 10^{-2}\) |
— |
[RE93] |
— |
Approach A-3 |
— |
\(9.5410 \times 10^{-3}\) |
— |
[RE93] |
— |
References#
- LGG+18(1,2,3,4,5)
Xu Li, Chunlin Gong, Liangxian Gu, Wenkun Gao, Zhao Jing, and Hua Su. A sequential surrogate method for reliability analysis based on radial basis function. Structural Safety, 73:42–53, 2018. doi:10.1016/j.strusafe.2018.02.005.
- RE93(1,2,3,4,5,6,7,8,9,10)
Malur R. Rajashekhar and Bruce R. Ellingwood. A new look at the response surface approach for reliability analysis. Structural Safety, 12(3):205–220, 1993. doi:10.1016/0167-4730(93)90003-j.
- SG05(1,2,3,4,5,6,7,8)
Luc Schueremans and Dionys Van Gemert. Benefit of splines and neural networks in simulation based structural reliability analysis. Structural Safety, 27(3):246–261, 2005. doi:10.1016/j.strusafe.2004.11.001.