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:
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:
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 |
| LICENSE | license for the function (BSD 3-Clause) |
References
- C. Kirjner-Neto, E. Polak, and A. Der Kiureghian, “Algorithms for reliability-based optimal design,” In Reliability and Optimization of Structural Systems. IFIP — The International Federation for Information Processing, Springer, pp. 144–152, 1995. DOI:10.1007/978-0-387-34866-7_13
- N. Kuschel and R. Rackwitz, “Two basic problems in reliability-based structural optimization,” Mathematical Methods of Operations Research, vol. 46, no. 3, pp. 309–333, 1997. DOI:10.1007/BF01194859
- M. S. Eldred, H. Agarwal, V. M. Perez, S. F. Wojtkiewicz Jr., and J. E. Renaud, “Investigation of reliability method formulations in DAKOTA/UQ,” Structure and Infrastructure Engineering, vol. 3, no. 3, pp. 199–213, 2007. DOI:10.1080/15732470500254618
- M. S. Eldred and J. Burkardt, “Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification,” In the 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, January 5–8, 2009. DOI:10.2514/6.2009-976
- M. S. Eldred, C. G. Webster, and P. G. Constantine, “Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos,” In the 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Material Conference, Schaumburg, Illinois, April 7–10, 2008. DOI:10.2514/6.2008-1892