Reliability problems

The structural reliability benchmarking framework is an on-line documentation platform that contains all kind of reliability problems with their descriptions, implementation files, and visualization of the performance functions. By making this information publicly available, it is intended to stimulate a systematic and reproducible assessment of the accuracy and efficiency of currently available reliability methods and those to come in the future. Furthermore, the platform is envisaged as a dynamic environment in which you are encouraged to contribute and add new, interesting problems.

Note

For most problems the failure probabilities are calculated with numerical methods that converge to the exact solution. Due to the applied stopping criteria of these numerical methods you may attain slightly different results. The difference should be a small difference in the second decimal of the normal form (scientific notation) representation of the failure probability. The reliability index (\(\beta\)) is calculated from the failure probability (\(P_\mathrm{f}\)) the following way: \(\beta=-\Phi^{-1}(P_\mathrm{f})\).

RP8

Six-dimensional hyperplane.

Table 9 – Tutorial set.
set_id problem_id
-1 1

Overview

Category Value
Type symbolic
Number of random variables 6
Failure probability, \(P_\mathrm{f}\) \(7.84\cdot10^{-4}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.16
Number of performance functions 1
Reference [Xu2018]

Performance function

(1)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = X_1 + 2X_2 + 2X_3 + X_4 - 5X_5 - 5X_6 \cr & {g_{\mathrm{sys}}}({\bf X}) = g_{\mathrm{comp}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Lognormal 4.783 0.09975     120.0 12.0
\(X_2\) NA Lognormal 4.783 0.09975     120.0 12.0
\(X_3\) NA Lognormal 4.783 0.09975     120.0 12.0
\(X_4\) NA Lognormal 4.783 0.09975     120.0 12.0
\(X_5\) NA Lognormal 3.892 0.198     50.0 10.0
\(X_6\) NA Lognormal 3.669 0.198     40.0 8.0

The random variables are mutually independent.

Visualization

_images/rp_8_matrix.png

Implementation

Python

gfun_8(x)[source]

Performance function for reliability problem 8.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

Matlab

See gfun_8.m on GitLab.


RP14

Table 10 – Challenge set 1.
set_id problem_id
1 1

Overview

Category Value
Type Symbolic
Number of random variables 5
Failure probability, \(P_\mathrm{f}\) \(7.52\cdot10^{-3}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.42
Number of performance functions 1
Reference [Schueller2004]

Performance function

(2)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = {X_1} - {{32} \over {\pi X_2^3}} \cdot \root 2 \of {{{X_3^2X_4^2} \over {16}} + X_5^2} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Uniform 70.0 80.0     75.0 2.887
\(X_2\) NA Normal 39.0 0.1     39.0 0.1
\(X_3\) NA Gumbel-max 1342.0 272.9     1500.0 350.0
\(X_4\) NA Normal 400.0 0.1     400.0 0.1
\(X_5\) NA Normal 250000.0 35000.0     250000.0 35000.0

The random variables are mutually independent.

Visualization

_images/rp_14_matrix.png

Implementation

Python

gfun_14(x)[source]

Performance function for reliability problem 14.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP22

Quadratic function with mixed term, convex.

Table 11 – Tutorial set.
set_id problem_id
-1 2

Overview

Category Value
Type symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(4.16\cdot10^{-3}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.64
Number of performance functions 1
Reference [Grooteman2011]

Performance function

(3)\[ \eqalign{ & {g({\bf X})} = 2.5-\frac{(X_1 + X_2)}{\sqrt{2}} + 0.1\cdot (X_1 - X_2)^2 \cr & {g_{\mathrm{sys}}}({\bf X}) = g_{\mathrm{comp}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(X_2\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_22_matrix.png

Implementation

Python

gfun_22(x)[source]

Performance function for reliability problem 22.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

Matlab

See gfun_22.m on GitLab.


RP24

Table 12 – Challenge set 1.
set_id problem_id
1 2

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(2.86\cdot10^{-3}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.76
Number of performance functions 1
Reference [Dai2016]

Performance function

(4)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = 2.5 - 0.2357 \cdot ({X_1} - {X_2}) + 0.00463 \cdot {({X_1} + {X_2} - 20)^4} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 10.0 3.0     10.0 3.0
\(X_2\) NA Normal 10.0 3.0     10.0 3.0

The random variables are mutually independent.

Visualization

_images/rp_24_matrix.png

Implementation

Python

gfun_24(x)[source]

Performance function for reliability problem 22.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP25

Table 13 – Challenge set 2.
set_id problem_id
2 1

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(6.14\cdot10^{-6}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 4.36
Number of performance functions 2
Reference [Dai2016]

Performance function

(5)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = X_1^2 - 8 \cdot {X_2} + 16 \cr & {g_{\mathrm{comp},2}}({\bf X}) = - 16{X_1} + {X_2} + 32 \cr & {g_{\mathrm{sys}}}({\bf X}) = \max \left\{ \matrix{ {g_{\mathrm{comp},1}}{\bf X}) \hfill \cr {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_25_matrix.png

Implementation

Python

gfun_111(x)[source]

Performance function for reliability problem 111.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP28

Table 14 – Challenge set 1.
set_id problem_id
1 3

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(1.46\cdot10^{-7}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 5.11
Number of performance functions 1
Reference [Dai2016]

Performance function

(6)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = {X_1} \cdot {X_2} - 146.14 \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 78064.0 11710.0     78064.0 11710.0
\(X_2\) NA Normal 0.0104 0.00156     0.0104 0.00156

The random variables are mutually independent.

Visualization

_images/rp_28_matrix.png

Implementation

Python

gfun_28(x)[source]

Performance function for reliability problem 28.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP31

Table 15 – Challenge set 1.
set_id problem_id
1 4

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(1.80\cdot10^{-4}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.58
Number of performance functions 1
Reference [Schueller2004]

Performance function

(7)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = 2 - {X_2} + {(4 \cdot {X_1})^4} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(X_2\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_31_matrix.png

Implementation

Python

gfun_31(x)[source]

Performance function for reliability problem 31.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP33

Table 16 – Tutorial set.; – Challenge set 2.
set_id problem_id
-1 3
2 2

Overview

Category Value
Type Symbolic
Number of random variables 3
Failure probability, \(P_\mathrm{f}\) \(2.57\cdot10^{-3}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.80
Number of performance functions 2
Reference [Schueller2004]

Performance function

(8)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = - {X_1} - {X_2} - {X_3} + 3 \cdot \root 2 \of 3 \cr & {g_{\mathrm{comp},2}}\left( {\bf X} \right) = - {X_3} + 3 \cr & {g_{\mathrm{sys}}}({\bf X}) = \min \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr {g_{\mathrm{comp},2}}\left( {\bf X} \right) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{3}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_33_matrix.png

Implementation

gfun_33(x)[source]

Performance function for reliability problem 33.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP35

Table 17 – Challenge set 2.
set_id problem_id
2 3

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(3.54\cdot10^{-3}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.70
Number of performance functions 2
Reference [Dai2016]

Performance function

(9)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = 2 - {X_2} + \exp ( - 0.1 \cdot X_1^2) + {(0.2 \cdot {X_1})^4} \cr & {g_{\mathrm{comp},2}}({\bf X}) = 4.5 - {X_1} \cdot {X_2} \cr & {g_{\mathrm{sys}}}({\bf X}) = \min \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_35_matrix.png

Implementation

gfun_35(x)[source]

Performance function for reliability problem 35.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP38

Table 18 – Challenge set 1.
set_id problem_id
1 5

Overview

Category Value
Type Symbolic
Number of random variables 7
Failure probability, \(P_\mathrm{f}\) \(8.10\cdot10^{-3}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.48
Number of performance functions 1
Reference [Schueller2004]

Performance function

(10)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = 15.59 \cdot {10^4} - {{{X_1} \cdot X_2^3} \over {2 \cdot X_2^3}} \cdot {{X_4^2 - 4 \cdot {X_5} \cdot {X_6} \cdot X_7^2 + {X_4} \cdot ({X_6} + 4 \cdot {X_5} + 2 \cdot {X_6} \cdot {X_7})} \over {{X_4} \cdot {X_5} \cdot ({X_4} + {X_6} + 2 \cdot {X_6} \cdot {X_7})}} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 350.0 35.0     350.0 35.0
\(X_2\) NA Normal 50.8 5.08     50.8 5.08
\(X_3\) NA Normal 3.81 0.381     3.81 0.381
\(X_4\) NA Normal 173.0 17.3     173.0 17.3
\(X_5\) NA Normal 9.38 0.938     9.38 0.938
\(X_6\) NA Normal 33.1 3.31     33.1 3.31
\(X_7\) NA Normal 0.036 0.0036     0.036 0.0036

The random variables are mutually independent.

Visualization

_images/rp_38_matrix.png

Implementation

gfun_38(x)[source]

Performance function for reliability problem 38.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP53

Table 19 – Challenge set 1.
set_id problem_id
1 6

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(3.13\cdot10^{-2}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 1.86
Number of performance functions 1
Reference [Schueller2004]

Performance function

(11)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = \sin \left( {{{5 \cdot {X_1}} \over 2}} \right) + 2 - {{(X_1^2 + 4) \cdot ({X_2} - 1)} \over {20}} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 1.5 1.0     1.5 1.0
\(X_2\) NA Normal 2.5 1.0     2.5 1.0

The random variables are mutually independent.

Visualization

_images/rp_53_matrix.png

Implementation

Python

gfun_53(x)[source]

Performance function for reliability problem 53.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP54

Table 20 – Challenge set 1.
set_id problem_id
1 7

Overview

Category Value
Type Symbolic
Number of random variables 20
Failure probability, \(P_\mathrm{f}\) \(9.98\cdot10^{-4}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.09
Number of performance functions 1
Reference [Dai2016]

Performance function

(12)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = \sum\limits_{i = 1}^{20} {{X_i} - 8.951} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Exponential 1.0       1.0 1.0
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)
\(X_{20}\) NA Exponential 1.0       1.0 1.0

The random variables are identically distributed and mutually independent.

Visualization

_images/rp_54_matrix.png

Implementation

gfun_54(x)[source]

Performance function for reliability problem 54.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP55

Table 21 – Challenge set 2.
set_id problem_id
2 4

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(3.60\cdot10^{-1}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) -0.15
Number of performance functions 4
Reference [Schueller2004]

Performance function

(13)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = 0.2 + 0.6 \cdot {({X_1} - {X_2})^4} - {{{X_1} - {X_2}} \over {\root 2 \of 2 }} \cr & {g_{\mathrm{comp},2}}({\bf X}) = 0.2 + 0.6 \cdot {({X_1} - {X_2})^4} + {{{X_1} - {X_2}} \over {\root 2 \of 2 }} \cr & {g_{\mathrm{comp},3}}({\bf X}) = {X_1} - {X_2} + {5 \over {\root 2 \of 2 }} - 2.2 \cr & {g_{\mathrm{comp},4}}({\bf X}) = {X_2} - {X_1} + {5 \over {\root 2 \of 2 }} - 2.2 \cr & {g_{\mathrm{sys}}}({\bf X}) = \min \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr {g_{\mathrm{comp},3}}({\bf X}) \hfill \cr {g_{\mathrm{comp},4}}({\bf X}) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Uniform -1.0 1.0     0.0 0.5774
\(X_{2}\) NA Uniform -1.0 1.0     0.0 0.5774

The random variables are mutually independent.

Visualization

_images/rp_55_matrix.png

Implementation

Python

gfun_55(x)[source]

Performance function for reliability problem 55.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP57

Table 22 – Challenge set 2.
set_id problem_id
2 5

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(2.84\cdot10^{-2}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 1.91
Number of performance functions 3
Reference [Schueller2004]

Performance function

(14)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = - X_1^2 + X_2^3 + 3 \cr & {g_{\mathrm{comp},2}}({\bf X}) = 2 - {X_1} - 8 \cdot {X_2} \cr & {g_{\mathrm{comp},3}}({\bf X}) = {({X_1} + 3)^2} + {({X_2} + 3)^2} - 4 \cr & {g_{\mathrm{sys}}}({\bf X}) = \min \left\{ \matrix{ \max \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr} \right. \hfill \cr {g_{\mathrm{comp},3}}({\bf X}) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_57_matrix.png

Implementation

Python

gfun_57(x)[source]

Performance function for reliability problem 57.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP60

Table 23 – Challenge set 2.
set_id problem_id
2 6

Overview

Category Value
Type Symbolic
Number of random variables 5
Failure probability, \(P_\mathrm{f}\) \(4.56\cdot10^{-2}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 1.70
Number of performance functions 7
Reference [Schueller2004]

Performance function

(15)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = {X_1} - {X_5} \cr & {g_{\mathrm{comp},2}}({\bf X}) = {X_2} - {{{X_5}} \over 2} \cr & {g_{\mathrm{comp},3}}({\bf X}) = {X_3} - {{{X_5}} \over 2} \cr & {g_{\mathrm{comp},4}}({\bf X}) = {X_4} - {{{X_5}} \over 2} \cr & {g_{\mathrm{comp},5}}({\bf X}) = {X_2} - {X_5} \cr & {g_{\mathrm{comp},6}}({\bf X}) = {X_3} - {X_5} \cr & {g_{\mathrm{comp},7}}({\bf X}) = {X_4} - {X_5} \cr & {g_{\mathrm{sys}}}({\bf X}) = \min \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr \max \left\{ \matrix{ \min \left\{ \matrix{ {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr {g_{\mathrm{comp},3}}({\bf X}) \hfill \cr {g_{\mathrm{comp},4}}({\bf X}) \hfill \cr} \right. \hfill \cr \max \left\{ \matrix{ \min \left\{ \matrix{ {g_{\mathrm{comp},5}}({\bf X}) \hfill \cr {g_{\mathrm{comp},6}}({\bf X}) \hfill \cr} \right. \hfill \cr {g_{\mathrm{comp},7}}({\bf X}) \hfill \cr} \right. \hfill \cr} \right. \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Lognormal 7.691 0.09975     2200.0 220.0
\(X_{2}\) NA Lognormal 7.645 0.09975     2100.0 210.0
\(X_{3}\) NA Lognormal 7.736 0.09975     2300.0 230.0
\(X_{4}\) NA Lognormal 7.596 0.09975     2000.0 200.0
\(X_{5}\) NA Lognormal 7.016 0.3853     1200.0 480.0

The random variables are mutually independent.

Visualization

_images/rp_60_matrix.png

Implementation

Python

gfun_60(x)[source]

Performance function for reliability problem 60.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP63

Table 24 – Challenge set 1.
set_id problem_id
1 8

Overview

Category Value
Type Symbolic
Number of random variables 100
Failure probability, \(P_\mathrm{f}\) \(3.79\cdot10^{-4}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.36
Number of performance functions 1
Reference [Schueller2004]

Performance function

(16)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = 0.1 \cdot \sum\limits_{i = 2}^{100} {X_i^2} - {X_1} - 4.5 \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)
\(X_{100}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_63_matrix.png

Implementation

Python

gfun_63(x)[source]

Performance function for reliability problem 63.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP75

Table 25 – Challenge set 1.
set_id problem_id
1 9

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(1.07\cdot10^{-2}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.33
Number of performance functions 1
Reference [Schueller2004]

Performance function

(17)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = 3 - {X_1} \cdot {X_2} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(X_2\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_75_matrix.png

Implementation

Python

gfun_75(x)[source]

Performance function for reliability problem 75.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP77

Table 26 – Challenge set 2.
set_id problem_id
2 7

Overview

Category Value
Type Symbolic
Number of random variables 3
Failure probability, \(P_\mathrm{f}\) \(2.87\cdot10^{-7}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 5.00
Number of performance functions 2
Reference [Schueller2004]

Performance function

(18)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = \left\{ \matrix{ {X_1} - {X_2} - {X_3}{\rm{ , }}{X_3} \le 5 \hfill \cr {{\rm{X}}_3}{\rm{ - }}{{\rm{X}}_2}{\rm{ , }}{X_3} > 5{\rm{ }} \hfill \cr} \right. \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 10.0 0.5     10.0 0.5
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{3}\) NA Normal 4.0 1.0     4.0 1.0

The random variables are mutually independent.

Visualization

Python

_images/rp_77_matrix.png

Implementation

gfun_77(x)[source]

Performance function for reliability problem 77.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP89

Table 27 – Challenge set 2.
set_id problem_id
2 8

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(5.43\cdot10^{-3}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 2.55
Number of performance functions 2
Reference [Schueller2004]

Performance function

(19)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = - (X_1^2 + {X_2} - 8) \cr & {g_{\mathrm{comp},2}}({\bf X}) = - \left( {{{{X_1}} \over 5} + {X_2} - 6} \right) \cr & {g_{\mathrm{sys}}}({\bf X}) = \min \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_89_matrix.png

Implementation

Python

gfun_89(x)[source]

Performance function for reliability problem 89.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP91

Table 28 – Challenge set 2.
set_id problem_id
2 9

Overview

Category Value
Type Symbolic
Number of random variables 5
Failure probability, \(P_\mathrm{f}\) \(6.97\cdot10^{-4}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.19
Number of performance functions 3
Reference [Schueller2004]

Performance function

(20)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = 0.847 + 0.96 \cdot {X_2}{\rm{ + }}0.986 \cdot {X_3} - 0.216 \cdot {X_4} + {\rm{ }}0.077 \cdot X_2^2 + 0.11 \cdot X_3^2 + {{0.007 \cdot X_{_4}^2} \over {0.378}} - {\rm{ }}{X_2} \cdot {\rm{ }}{X_3} - 0.106{\rm{ }} \cdot {X_2} \cdot {X_4} - {\rm{ }}0.11 \cdot {X_3} \cdot {X_4} \cr & {g_{\mathrm{comp},2}}({\bf X}) = {{8400 \cdot {X_1}} \over {\root 2 \of {X_3^2 + X_4^2 - {X_3} \cdot {X_4} + 3 \cdot X_5^2} }} - 1 \cr & {g_{\mathrm{comp},3}}({\bf X}) = {{8400 \cdot {X_1}} \over {\left| {{X_4}} \right|}} - 1 \cr & g({X}) = \min \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr {g_{\mathrm{comp},3}}({\bf X}) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.07433 0.005     0.07433 0.005
\(X_{2}\) NA Normal 0.1 0.01     0.1 0.01
\(X_{3}\) NA Normal 13.0 60.0     13.0 60.0
\(X_{4}\) NA Normal 4751.0 48.0     4751.0 48.0
\(X_{5}\) NA Normal -684.0 11.0     -684.0 11.0

The random variables are mutually independent.

Visualization

_images/rp_91_matrix.png

Implementation

Python

gfun_91(x)[source]

Performance function for reliability problem 91.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP107

Table 29 – Challenge set 1.
set_id problem_id
1 10

Overview

Category Value
Type Symbolic
Number of random variables 10
Failure probability, \(P_\mathrm{f}\) \(2.92\cdot10^{-7}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 5.0
Number of performance functions 1
Reference [Schueller2004]

Performance function

(21)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = 5 \cdot \root 2 \of {10} - \sum\limits_{i = 1}^{10} {{X_i}} \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)
\(X_{10}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are identically distributed and mutually independent.

Visualization

_images/rp_107_matrix.png

Implementation

Python

gfun_107(x)[source]

Performance function for reliability problem 107.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP110

Table 30 – Challenge set 2.
set_id problem_id
2 10

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(3.19\cdot10^{-5}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 4.00
Number of performance functions 2
Reference [Schueller2004]

Performance function

(22)\[ \eqalign{ & {g_{\mathrm{comp},1}}({\bf X}) = \left\{ \matrix{ 4 - {X_1}{\rm{ , }}{X_1} > 3.5{\rm{ }} \hfill \cr 0.85 - 0.1 \cdot {X_1}{\rm{ , }}{X_1} \le 3.5 \hfill \cr} \right. \cr & {g_{\mathrm{comp},2}}({\bf X}) = \left\{ \matrix{ 0.5 - 0.1 \cdot {X_2}{\rm{ , }}{X_2} > 2 \hfill \cr 2.3 - {X_2}{\rm{ , }}{X_2} \le 2 \hfill \cr} \right. \cr & g({\bf X}) = \min \left\{ \matrix{ {g_{\mathrm{comp},1}}({\bf X}) \hfill \cr {g_{\mathrm{comp},2}}({\bf X}) \hfill \cr} \right. \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_110_matrix.png

Implementation

Python

gfun_110(x)[source]

Performance function for reliability problem 25.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP111

Table 31 – Challenge set 1.
set_id problem_id
1 11

Overview

Category Value
Type Symbolic
Number of random variables 2
Failure probability, \(P_\mathrm{f}\) \(7.65\cdot10^{-7}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 4.81
Number of performance functions 1
Reference [Schueller2004]

Performance function

(23)\[ \eqalign{ & {g_{\mathrm{comp}}}({\bf X}) = {{{5^2}} \over 2} - \left| {{X_1} \cdot {X_2}} \right| \cr & {g_{\mathrm{sys}}}({\bf X}) = {g_{\mathrm{comp}}}({\bf X}) \cr}\]

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(X_2\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_111_matrix.png

Implementation

Python

gfun_111(x)[source]

Performance function for reliability problem 111.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP201

Table 32 – Challenge set 1.
set_id problem_id
1 12

Overview

Category Value
Type Numerical (finite element analysis)
Number of random variables 21
Failure probability, \(P_\mathrm{f}\) \(1.05\cdot10^{-4}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.7
Number of performance functions 1
Reference [Blatman2010]

Performance function

Too complex to be written here, see the implementation in the code repository.

Random variables

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)
\(X_{21}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_201_matrix.png

Implementation

Python

gfun_201(x)[source]

Performance function for reliability problem 201.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP202

Table 33 – Challenge set 2.
set_id problem_id
2 11

Overview

Category Value
Type Numerical (finite element analysis)
Number of random variables 225
Failure probability, \(P_\mathrm{f}\) \(6.03\cdot10^{-5}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.43
Number of performance functions 2
Reference [Schueller2007]

Performance function

Too complex to be written here, see the implementation in the code repository.

Random variables

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)
\(X_{225}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

Implementation

Python

gfun_202(x)[source]

Performance function for reliability problem 202.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP203

Table 34 – Challenge set 1.
set_id problem_id
1 13

Overview

Category Value
Type Numerical (finite element analysis surrogate)
Number of random variables 4
Failure probability, \(P_\mathrm{f}\) \(4.33\cdot10^{-7}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 4.92
Number of performance functions 1
Reference [Eijnden2019]

Performance function

Too complex to be written here, see the implementation in the code repository.

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{2}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{3}\) NA Normal 0.0 1.0     0.0 1.0
\(X_{4}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_203_matrix.png

Implementation

Python

gfun_203(x)[source]

Performance function for reliability problem 203. Bram’s geotechnical problem.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP213

Table 35 – Challenge set 1.
set_id problem_id
1 14

Overview

Category Value
Type Numeric (finite element analysis)
Number of random variables 13
Failure probability, \(P_\mathrm{f}\) \(2.84\cdot10^{-4}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.45
Number of performance functions 1
Reference [Blatman2010]

Performance function

Too complex to be written here, see the implementation in the code repository.

Random variables

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) NA Normal 0.0 1.0     0.0 1.0
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)
\(X_{13}\) NA Normal 0.0 1.0     0.0 1.0

The random variables are mutually independent.

Visualization

_images/rp_213_matrix.png

Implementation

Python

gfun_213_le_frame(x)[source]

Performance function for reliability problem 213.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP300

Table 36 – Challenge set 1.
set_id problem_id
1 15

Overview

Category Value
Type Numerical
Number of random variables 14
Failure probability, \(P_\mathrm{f}\) \(5.22\cdot10^{-5}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.88
Number of performance functions 1
Reference [Sepulveda2019]

Performance function

Too complex to be written here, see the implementation in the code repository.

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_1\) DACC Normal 0 1        
\(X_2\) DACC Normal 0 1        
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)        
\(X_{14}\) DACC Normal 0 1        

The random variables are mutually independent.

Visualization

_images/rp_300_matrix.png

Implementation

Python

gfun_300(x)[source]

Performance function for reliability problem 300.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.

RP301

Table 37 – Challenge set 1.
set_id problem_id
1 16

Overview

Category Value
Type Numerical (finite element analysis surrogate)
Number of random variables 12
Failure probability, \(P_\mathrm{f}\) \(6.9\cdot10^{-5}\)
Reliability index, \(\beta=-\Phi^{-1}(P_\mathrm{f})\) 3.81
Number of performance functions 1
Reference [Slobbe2019]

Performance function

Too complex to be written here, see the implementation in the code repository.

Random variables

The parametrization of distributions follows that of in Distributions.

Variable Description Distribution \(\theta_1\) \(\theta_2\) \(\theta_3\) \(\theta_4\) Mean Std
\(X_{1}\) NA Lognormal 4.476 0.05677     88.0 5.0
\(X_{2}\) NA Lognormal 6.084 0.0681     440.0 30.0
\(X_{3}\) NA Lognormal 6.18 0.0681     484.0 33.0
\(X_{4}\) NA Lognormal -2.663 0.08982     0.07 0.0063
\(X_{5}\) NA Lognormal 6.378 0.06941     590.0 41.0
\(X_{6}\) NA Lognormal 6.473 0.06925     649.0 45.0
\(X_{7}\) NA Lognormal -2.663 0.08982     0.07 0.0063
\(X_{8}\) NA Normal 590.0 59.0     590.0 59.0
\(X_{9}\) NA Gumbel-max 268.4 46.0     295.0 59.0
\(X_{10}\) NA Lognormal 0.03797 0.04997     1.04 0.052
\(X_{11}\) NA Lognormal -0.004975 0.09975     1.0 0.1
\(X_{12}\) NA Lognormal -0.004975 0.09975     1.0 0.1

The random variables are mutually independent.

Visualization

_images/rp_301_matrix.png

Implementation

Python

gfun_301(x)[source]

Performance function for reliability problem 301. High strength RC deep beam.

Parameters:x (numpy.array of float(s)) – Values of independent variables: columns are the different parameters/random variables (x1, x2,…xn) and rows are different parameter/random variables sets for different calls.
Returns:
  • g_val_sys (numpy.array of float(s)) – Performance function value for the system.
  • g_val_comp (numpy.array of float(s)) – Performance function value for each component.
  • msg (str) – Accompanying diagnostic message, e.g. warning.