## Abstract

This paper presents an investigation of the crashworthiness of expanding circular tube energy absorbers with a cylindrical anti-clamber under eccentric loading for subway vehicles. A finite-element model of energy absorbers is created and validated by two quasi-static experiments, and the strain rate sensitivity is evaluated using standard coefficients of the Cowper-Symonds constitutive model. A parametric study is performed using the validated finite-element models. It is found that the eccentric collision of two energy absorbers has a significant effect on their crashworthiness and leads to a decrease in the specific energy absorption (*SEA*) and an increase in the peak crushing force (*PCF*). The structural parameters of the energy absorber, including the wall thickness *t*, conical angle *α*, and punch radius *r*_{p}, have a significant effect on its crashworthiness under different eccentric distances. Sobol sensitivity analysis is employed to analyze the effects of the design parameters (*t*, *α*, and *r*_{p}) on the objective responses (*EA* and *PCF*). The results show that the design variables *t* and *α* have a greater effect on the sensitivity of the *EA* and *PCF* functions for the eccentric distance of *h* = 0 and 40 mm. The interaction between punch angle *α* and punch radius *r*_{p} has significant effect on *EA.* To improve the crashworthiness of the expanding circular tubes, a multi-objective optimization is applied to achieve the maximum *EA* and minimum *PCF* values, and weight factors are introduced to investigate the crashworthiness optimization under single-eccentric loading and multiple-eccentric loading. The optimization solutions of the expanding circular tubes under different single-eccentric loadings have considerable implications. For multiple-eccentric loading, the results show that the crashworthiness of the expanding circular tube energy absorber can be better balanced.

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## Funding

The authors gratefully acknowledge financial support from the National Key R&D Program of China (2016YFB1200403), the National Natural Science Foundation of China (U1334208), the Strategic Leading Science and Technology Project of Central South University (ZLXD2017002), and the Fundamental Research Funds for the Central Universities of Central South University (2017gczd009 and 2018zzts026).

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## Appendix: the quartic response model expressions

### Appendix: the quartic response model expressions

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Guan, W., Gao, G., Yu, Y. *et al.* Crashworthiness analysis and multi-objective optimization of expanding circular tube energy absorbers with cylindrical anti-clamber under eccentric loading for subway vehicles.
*Struct Multidisc Optim* **61, **1711–1729 (2020). https://doi.org/10.1007/s00158-019-02427-z

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### Keywords

- Crashworthiness
- Eccentric
- Expanding
- Sensitivity analysis
- Multi-objective optimization