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Crushing analysis and multiobjective crashworthiness optimization of combined shrinking circular tubes under impact loading

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Abstract

This paper proposes a new type of energy absorber with combined shrinking circular tubes for a high-speed train. An impact experiment is conducted to investigate the dynamic crushing performance of this energy absorber with combined shrinking circular tubes. The results show that the combined tubes experience steady dynamic shrinking deformation. Finite element (FE) models of the energy absorber are then developed, and the dynamic crushing forces are in good agreement with the impact test. A theoretical solution for the dynamic shrinking crushing load is derived. Based on the validated FE models, the effects of the friction coefficient, wall thickness and die radius on the dynamic crushing force and energy absorption are investigated. An increase in the wall thickness leads to a substantial growth in the maximum crushing force (Fmax) and specific energy absorption (SEA), but the growth rate of Fmax is much larger than that of the SEA as the wall thickness increases. In addition, comparing theoretical and FE results demonstrates that predictions of the dynamic steady-state forces for the combined shrinking circular tubes with different friction coefficients, wall thicknesses (t) and die radius (Rdie) are satisfactory. Finally, to improve the crashworthiness of the expanding circular tubes, Sobol’ sensitivity analysis is employed to analyze the effects of the design parameters (t and Rdie) on the objective responses (SEA and Fmax) using the Kriging model. A Pareto front of double optimization objective SEA and Fmax was obtained after being optimized by multiobjective particle swarm optimization (MOPSO). The results show that SEA and Fmax are positively correlated, and a balance between the SEA and Fmax was obtained at optimal point C (SEA = 13.09 kJ/kg, Fmax = 581.11 kN).

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Abbreviations

L :

Tube length

R s :

Radius of the tube

t :

Wall thickness of the tube

v 0 :

Impact velocity

σ d :

Dynamic stress

σ 0 :

Initial yield stress

C,P :

Strain rate parameters

β :

Strain hardening parameter

\( {\varepsilon}_P^{eff} \) :

Effective plastic strain

E :

Elastic modulus

E p :

Plastic hardening modulus

E t :

Tangent modulus

σ t :

True stress

ε t :

True strain

σ e :

Engineering stress

ε e :

Engineering strain

ρ :

Material density

v :

Poisson’s ratio

R 0 :

Midline radius

R 0 :

Midline radius of the tube

R 1 :

Tube radius during deformation

R 2 :

Tube radius after deformation

R die :

radius of the die

r die :

Radius of the arc AB

α :

Tangential angle of point B

φ :

Angle of arc contact surface

u :

Friction coefficient

r 1 :

Curvature of the tube in regions AB

r 2 :

Curvature of the tube in regions BC

F :

Dynamic steady-state force

\( \dot{W} \) :

Energy dissipation rate

\( {\dot{E}}_{\theta } \) :

Energy dissipation rate of plastic stretching

\( {\dot{E}}_l \) :

Energy dissipation rate of plastic bending

\( {\dot{E}}_f \) :

Energy dissipation rate of friction

ε θ :

Circumferential strain

ε l :

Meridional strain

ε t :

Thickness strain

κ :

Curvature of contact surface B1B2

M p :

Fully plastic bending moment

N p :

Fully plastic stretching moment

\( {\dot{\varepsilon}}_{AB},{\dot{\varepsilon}}_{BC} \) :

Strain rate

T :

Time

κ AB :

Curvature of contact surface AB

κ AB :

Curvature of contact surface BC

N :

Normal contact force

F f :

Friction force

EA :

Energy absorption

SEA :

Specific energy absorption

F max :

Maximal axial crushing force

RMSE :

Root mean square error

ME :

Maximum error

R 2 :

Coefficient of determination

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Funding

The authors gratefully acknowledge the financial support from the National Key R & D Program of China (2016YFB1200403), the Fundamental Research Funds for the Central Universities of Central South University (2017gczd009 and 2018zzts026) and the National Natural Science Foundation of China (51975588).

Author information

Authors and Affiliations

  1. Key Laboratory of Traffic Safety on the Track (Central South University), Ministry of Education, Changsha, 410075, China

    Weiyuan Guan, Guangjun Gao & Yao Yu

  2. Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Central South University, Changsha, 410075, China

    Weiyuan Guan, Guangjun Gao & Yao Yu

  3. National & Local Joint Engineering Research Center of Safety Technology for Rail Traffic Vehicle, Central South University, Changsha, 410075, China

    Weiyuan Guan, Guangjun Gao & Yao Yu

Authors
  1. Weiyuan Guan
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  2. Guangjun Gao
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  3. Yao Yu
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Corresponding author

Correspondence to Guangjun Gao.

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The authors declare that they have no conflict of interest.

Replication of results

All the presented results can be replicated, and all the necessary data sets for the problems are present in the manuscript. The nonlinear finite-element code LS-DYNA version SMP R7.1.1 is used to solve the FEM problems in this paper. The impact test trolley components are modelled using shell elements; the weight of the trolley body is 21.28 t, which is approximative to the mass of a subway vehicle. The trolley equipped with an energy-absorbing structure impacts the rigid body at velocity v0. The length, wall thickness and tube radius of the shrinking circular tube are 600, 12 and 53 mm, respectively. Three element sizes, including 4 × 4 × 4, 2 × 2 × 2 and 1 × 1 × 1 mm, are modelled. The shrinking circular tube energy absorbers meshed with SOLID164 elements. The eight-node solid elements are utilized in the simulation process, and the fully integrated methods are adopted to avoid hourglass energy. The sensitivity analysis is implemented in the global sensitivity indices for nonlinear mathematical models (Sobol 2001).

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Guan, W., Gao, G. & Yu, Y. Crushing analysis and multiobjective crashworthiness optimization of combined shrinking circular tubes under impact loading. Struct Multidisc Optim 64, 1649–1667 (2021). https://doi.org/10.1007/s00158-021-02938-8

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  • DOI: https://doi.org/10.1007/s00158-021-02938-8

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