short: Short Column Function

Description

The short column function is defined by $$f_{\rm short}(x) = 1 - \frac{4M}{b h^2 Y} - \frac{P^2}{b^2 h^2 Y^2}$$ with $x = (Y, M, P)$.

Usage

short(x, b = 5, h = 15)
shortGrad(x, b = 5, h = 15)

Value

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

shortGrad returns the gradient of short column function at x.

Arguments

x: a numeric vector of length 3 or a numeric matrix with n rows and 3 columns.
b: width of the cross-section in $\rm mm$ of the short column. Default is 5.
h: depth of the cross-section in $\rm mm$ of the short column. Default is 15.

Author

Carmen van Meegen

Details

The short column function describes the limite state function of a short column with uncertain material properties and loads.

Input	Distribution	Mean	Standard deviation	Description	$Y$
$\mathcal{LN}$	$5$	$0.5$	yield stress in $\rm MPa$	$M$	$\mathcal{N}$
$2000$	$400$	bending moment in $\rm MNm$	$P$	$\mathcal{N}$	$500$

The bending moment and the axial force are correlated with $\rm{Cor}$$(M, P) = 0.5$. Note, $\mathcal{N}$ represents the normal distribution and $\mathcal{LN}$ is the log-normal 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	\(Y\)
\(\mathcal{LN}\)	\(5\)	\(0.5\)	yield stress in \(\rm MPa\)	\(M\)	\(\mathcal{N}\)
\(2000\)	\(400\)	bending moment in \(\rm MNm\)	\(P\)	\(\mathcal{N}\)	\(500\)