# Truss model

The 23-bar planar truss bridge (Blatman and Sudret, 2011) is a simply supported structure. Bars are categorized into two groups: the first group consists of 11 members, including the upper and lower chords, while the second group comprises 12 diagonals. The vertically oriented static loading is located in all six upper chord nodes. All truss bars are made from identical material.

## Description

The response of the truss model is the mid-span deflection defined as:

f(\mathbf{x}) = - \frac{2\sqrt{2} A_1 P_1 + 6\sqrt{2} A_1 P_2 + 10\sqrt{2} A_1 P_3 }{A_1 A_2 E_2} \\ - \frac{ 10\sqrt{2} A_1 P_4 + 6\sqrt{2} A_1 P_5 + 2\sqrt{2} A_1 P_6}{A_1 A_2 E_2} \\ - \frac{36 A_2 E_2 P_1 + 100 A_2 E_2 P_2 + 140 A_2 E_2 P_3 }{A_1 A_2 E_1 E_2} \\ - \frac{140 A_2 E_2 P_4 + 100 A_2 E_2 P_5 + 36 A_2 E_2 P_6}{A_1 A_2 E_1 E_2},

where \mathbf{x} = \{E_1, E_2, A_1, A_2, P_1, ..., P_6 \} are independent input variables modeled by lognormal and Gumbel distributions.

Figure 1: The truss model, adapted from Dubourg (2011).

## Inputs

For computer experiment purposes, the inputs E_1, E_2, A_1, A_2, P_1, ..., P_6 are modeled as ten independent random variables:

No Variable Description Distribution Statistics
1 E_1 Young’s modulus [Pa] Lognormal \mu_{E_1} = 2.10 \times 10^{11},
\sigma_{E_1} = 2.10 \times 10^{10}
2 E_2 Young’s modulus [Pa] Lognormal \mu_{E_2} = 2.10 \times 10^{11},
\sigma_{E_2} = 2.10 \times 10^{10}
3 A_1 Cross-sectional area [m²] Lognormal \mu_{A_1} = 2.0 \times 10^{-3},
\sigma_{A_1} = 2.0 \times 10^{-4}
4 A_2 Cross-sectional area [m²] Lognormal \mu_{A_2} = 1.0 \times 10^{-3},
\sigma_{A_2} = 1.0 \times 10^{-4}
5-10 P_1 - P_6 Load [N] Gumbel \mu_{P_1-P_6} = 5.0 \times 10^{4},
\sigma_{P_1-P_6} = 7.5 \times 10^{3}

## Resources

The vectorized implementation of the analytical solution of the mid-span deflection of the truss model 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_trussModel.zip (2.4 KB)

The contents of the file are:

Filename Description
uq_trussModelAnalytical.m vectorized implementation of the truss model response in MATLAB
uq_Example_truss.m definitions for the model and probabilistic inputs in UQLab
LICENSE license for the function (BSD 3-Clause)

## Open-access repository

The dataset used in this benchmark study is titled “Benchmark case datasets - Truss model” and is authored by Adéla Hlobilová, Stefano Marelli, and Bruno Sudret. It was published in 2024 and is available on Zenodo. The dataset can be accessed directly via the following DOI link: 10.5281/zenodo.12699396.

The experimental designs include datasets with 100, 200, 300, 400, and 500 samples, each generated using optimized Latin Hypercube Sampling (LHS) with 1,000 iterations to improve the maximin criterion. Each dataset is replicated 20 times. The validation set contains 100,000 samples generated by Monte Carlo simulation. Each dataset contains samples and responses of the computational model.

For citation purposes, please use the following format:
Hlobilová, A., Marelli, S., and Sudret, B. (2024). Benchmark case datasets - Truss model. Zenodo. https://doi.org/10.5281/zenodo.12699396.

This project was supported by the Open Research Data Program of the ETH Board under Grant number EPFL SCR0902285.

## References

• Blatman, G., Sudret, B. “Adaptive sparse polynomial chaos expansion based on least angle regression,” Journal of computational Physics, vol. 230, issue 6 pp.2345–2367, 2011. DOI:10.1016/j.jcp.2010.12.021
• Dubourg, V. “Adaptive surrogate models for reliability analysis and reliability-based design optimization”. Diss. Université Blaise Pascal-Clermont-Ferrand II, 2011. PDF file
• Lüthen, N., Marelli, S., Sudret, B. “Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark,” SIAM/ASA Journal on Uncertainty Quantification, vol. 9, issue 2, pp. 593–649, 2021. DOI:10.1137/20M1315774