# Short Column Function

The 3-dimensional short column function models a short column with a rectangular cross-section (width b and depth h) having uncertain material properties (yield stress Y_C) and being subjected to uncertain loads (bending moment M_C and axial force P_C). The function is used as a test function in metamodeling (Eldred and Burkardt, 2009; Eldred et al., 2008) and reliability-based design optimization (Kirjner-Neto et al., 1995; Kuschel and Rackwitz, 1997; Eldred et al., 2007). The test problem features a correlated pair of inputs.

## Description

The analytical limit state function is given as:

f(\mathbf{x},\mathbf{p}) = 1 - \frac{4 M_c}{b h^2 Y_C} - \frac{P_C^2}{b^2 h^2 Y_C^2},

where \mathbf{x} = \{Y_C, M_C, P_C\} is the vector of uncertain input variables and \mathbf{p} = \{ b, h\} is the vector of parameters.

The failure event and the failure probability are defined as f(\mathbf{x},\mathbf{p}) \leq 0 and P_f = \mathbb{P}[f(\mathbf{x},\mathbf{p}) \leq 0], respectively.

For reliability-based design optimization (RBDO), the cost to be minimized is the weight of the column given as:

C(\mathbf{p}) = b h, \;\;\; b \in [5, 15], \; h \in [15,25],

with a target value for the failure probability P_f of less than 0.00621 (or reliability index \beta of more than 2.5).

## Inputs

The input variables are modeled as random variables with distributions and their parameters given in the table below. The variables M_C and P_C are correlated.

No Variable Distribution Parameters Description
1 Y_C Lognormal \mu_{Y_C} = 5,
\sigma_{Y_C} = 0.5
Yield stress
2 M_C Gaussian \mu_{M_C} = 2 \times 10^3,
\sigma_{M_C} = 400,
\rho_{M_C, Y_C} = 0.5
Bending moment
3 P_C Gaussian mu_{P_C} = 500,
\sigma_{P_C} = 100,
\rho_{Y_C, M_C} = 0.5
Axial force

## Parameters

The nominal values for the parameters b (width) and h (depth) are 5 and 15, respectively. For the RBDO problem, these parameters are the design variables with their domains specified as (Kuschel and Rackwitz, 1997): b \in [5,15] and h \in [15,25].

## Reference values

Some reference values for the optimal b and h under reliability-based design optimization are shown in the table below.

Source b^* h^*
Kirjner-Neto et al. (1995) 8.668 25.0
Kuschel and Rackwitz (1997) 8.668 25.0
Eldred et al. (2007) 8.68 25.0

## Resources

The vectorized implementation of the short column function in MATLAB as well as the script file with the model and probabilistic inputs definitions for the function in UQLab can be downloaded below:

uq_shortColumn.zip (2.7 KB)

Filename Description
uq_shortColumn.m vectorized implementation of the short column function in MATLAB
uq_Example_shortColumn.m definitions for the model and probabilistic inputs in UQLab