steel: Steel Column Function

Description

The steel column function is defined by $$f_{\rm steel}(x) = F_S - P \left(\frac{1}{2BD} + \frac{F_0 E_b}{BDH(E_b - P)} \right),$$ with $P = P_1 + P_2 + P_3$, $E_b = \frac{\pi^2 EBDH^2}{2L^2}$ and $x = (F_S, P_1, P_2, P_3, B, D, H, F_0, E)$.

Usage

steel(x, L = 7500)
steelGrad(x, L = 7500)

Value

steel returns the function value of steel column function at x.

steelGrad returns the gradient of steel column function at x.

Arguments

x: a numeric vector of length 9 or a numeric matrix with n rows and 9 columns.
L: length in $\rm mm$ of the steel column. Default is 7500.

Author

Carmen van Meegen

Details

The steel column function describes the limite state function of a steel column with uncertain parameters.

Input	Distribution	Mean	Standard Deviation	Description	$F_S$
$\mathcal{LN}$	$400$	$35$	yield stress in $\rm MPa$	$P_1$	$\mathcal{N}$
$500000$	$50000$	dead weight load in $\rm N$	$P_2$	$\mathcal{G}$	$600000$
$90000$	variable load in $\rm N$	$P_3$	$\mathcal{G}$	$600000$	$90000$
variable load in $\rm N$	$B$	$\mathcal{LN}$	$b$	$3$	flange breadth in $\rm mm$
$D$	$\mathcal{LN}$	$t$	$2$	flange thickness in $\rm mm$	$H$
$\mathcal{LN}$	$h$	$5$	profile height in $\rm mm$	$F_0$	$\mathcal{N}$
$30$	$10$	initial deflection in $\rm mm$	$E$	$\mathcal{W}$	$210000$

Here, $\mathcal{N}$ is the normal distribution and $\mathcal{LN}$ is the log-normal distribution. Further, $\mathcal{G}$ represents the Gumbel distribution and $\mathcal{W}$ denotes the Weibull distribution.

References

Kuschel, N. and Rackwitz, R. (1997). Two Basic Problems in Reliability-Based Structural Optimization. Mathematical Methods of Operations Research, 46(3):309--333.

Surjanovic, S. and Bingham, D. (2013). Virtual Library of Simulation Experiments: Test Functions and Datasets. https://www.sfu.ca/~ssurjano/ (retrieved January 19, 2024).

Input	Distribution	Mean	Standard Deviation	Description	\(F_S\)
\(\mathcal{LN}\)	\(400\)	\(35\)	yield stress in \(\rm MPa\)	\(P_1\)	\(\mathcal{N}\)
\(500000\)	\(50000\)	dead weight load in \(\rm N\)	\(P_2\)	\(\mathcal{G}\)	\(600000\)
\(90000\)	variable load in \(\rm N\)	\(P_3\)	\(\mathcal{G}\)	\(600000\)	\(90000\)
variable load in \(\rm N\)	\(B\)	\(\mathcal{LN}\)	\(b\)	\(3\)	flange breadth in \(\rm mm\)
\(D\)	\(\mathcal{LN}\)	\(t\)	\(2\)	flange thickness in \(\rm mm\)	\(H\)
\(\mathcal{LN}\)	\(h\)	\(5\)	profile height in \(\rm mm\)	\(F_0\)	\(\mathcal{N}\)
\(30\)	\(10\)	initial deflection in \(\rm mm\)	\(E\)	\(\mathcal{W}\)	\(210000\)