Theoretical Prediction of Dynamic Axial Crushing on a Square Tube with Eight Holes Used as a Crush Initiator

Thin-walled square steel tubes are part of automobile structures, used as impact energy absorbers on crush boxes. Under axial crushing, such tubes sometimes produces unpredictable folding forms. There are three modes of dynamic axial crushing folding forms: the symmetric collapse mode; the asymmetric mixed collapse mode A; and the asymmetric mixed collapse mode B. The objective of this paper is to develop a theoretical prediction for the three modes on a thin-walled square steel tube with eight holes used as crush initiator. The basic folding mechanism is used to predict the dynamic axial crushing analysis on the tube. These theoretical analyses can also can be implemented in different crush initiator positions.

Two types of cross section (c/h) were used in this study: specimen A (c/h = 60.42) and specimen B (c/h = 45.69). Thirty-six experimental drop test studies were conducted on the thin wall square tube. In addition, the results of a previous drop test were compared to the results of the theoretical prediction. The results show that the theoretical analysis has good agreement with the experimental drop test study. This study proposes nine formulas to predict average force, peak force and energy absorption of the dynamic axial crushing on the thin-walled square steel tube with eight holes of crush initiator. The limitation of these formulas, however, is that they are unable to show the relationship between force and time in a graph.

Crush initiators; Dynamic axial crushing; Square tube

2017; Ding et al., 2018); and hybrid square tubes (Zhang et al., 2019), have been researched in detail. From these studies, it was well known that the most severe plastic deformation occurs near sectional corners of tubes, which could dissipate a great amount of energy due to membrane and bending deformation along the bending hinge lines (Hu et al., 2019). For example, Shena et al. (2017) investigated the crashworthiness of rectangular single-, double- and unequal triple-cell tubes made of aluminium alloy. An analytical formula for predicting the mean crushing force (MCF) of unequal triple-cell tubes was first derived. Quasi-static crushing experiments and finite element analysis (FEA) were then conducted in the axial direction for the three di?erent tubes. The results from numerical simulations were compared in detail with data from experimental tests and theoretical predictions. It was found that the triple-cell tube exhibited the best crash performance, followed by its double- and single-cell counterparts. A multi-objective optimization design was employed to investigate the effect of the thickness of each plate and the arrangements of internal ribs on specific energy absorption (SEA) and MCF. An experimental study on the crashworthiness of star-shaped tubes under axial compression was conducted by Deng et al. (2018). A star-shaped tube with twelve corners produced global buckling in quasi-static compression, while such a tube with eight corners had a better energy absorption capacity than tubes with six and twelve corners. The deformation modes of quasi-static and dynamic impact were very similar. According to Ge et al. (2018), the elliptical cross-section becomes narrower compared to the circle one, which is also helpful in increasing the energy absorption of the thin-walled tube.

To enhance the energy absorption capability and reduce the initial peak crushing force of thin-walled tubes, the attention of many researchers has also been drawn to corrugated tubes (Mahbod & Asgari, 2018), functionally graded thickness (FGT) tubes (Sun et al., 2017), foam-?lled tubes (Xiong et al., 2018; Googarchin et al., 2019) and metal/composite hybrid tubes (Zhang et al., 2019). For instance, Mahbod and Asgari (2018) conducted a theoretical study to investigate the behavior of corrugated composite tubes under axial and oblique crushing. The corrugated model Cr6n16 improved CFE by about 343% in comparison to the straight tube, without an sharp reduction in the SEA in axial crushing. The corrugated tube Cr6n18 increased the SEA by about 18%, with the same value of CFE as the straight tube in oblique crushing. Sun et al. (2017) proposed uniform thickness (UT), axial functionally graded thickness (AFGT) and lateral functionally graded thickness (LFGT) for thin-walled square structures, and then investigated their crushing characteristics through theoretical, numerical and experimental approaches under axial crushing loads. These primary outcomes demonstrate that FGT structures have considerable potential as energy absorption devices for axial impact. In terms of the methods for improving energy absorption capacity, a theoretical and numerical investigation of the energy absorption behavior of foam-?lled tapered multi-cell tubes was made by Googarchin et al. (2019). Their analysis indicates that the SEA in the crushing of the foam-?lled tubes with three cell rows in the cross section and a taper angle equal to 7 degrees would be five times greater than that in the crushing of the reference tube.

In particular, for the energy absorption structures of bio-inspired design strategies, natural structural hierarchies were fruitful learning resources. Based on their results from experimental testing and numerical modelling, Zhang et al. (2018) concluded that a 2nd-order hierarchical circular tube (HCT) offers signi?cantly greater energy absorption than non-hierarchical structures (0th order HCT). Sub-circle diameters and wall thickness have a signi?cant effect on the crashworthiness indictors of 2nd order HCT. A bio-inspired bionic honeycomb tubular nested structure (BHTNS) was proposed and the energy absorption characteristics of this structure under axial crushing conditions were also investigated by systematic experiments, numerical simulations, and theoretical analysis (Hu et al., 2019). Finally, a theoretical model was developed to predict the Pmean of BHTNS. It was observed that the Pmean of the theoretical model was in good agreement with that from the numerical simulation.

There are some limitations in conducting experimental drop test studies. They need a high tower, rig, load cell, high-speed camera, LabVIEW software, and computer. Meanwhile, numerical analysis requires software such as ANSYS LS DYNA, LS DYNA, ABAQUS or Pam Crash. Significant costs are involved in setting up the two methods, which is the main reason why development of theoretical analysis is needed.

As mentioned above, this study is related to the theoretical predictions of Zhang et al. (2019), Tran et al. (2017), Ding et al. (2018), Sun et al. (2017) and Xie et al. (2017). However, the most common theoretical predictions for rectangular tubes are suggested by Abramowicz and Jones (1984) and Nguyen et al. (2013). In terms of the number of crush initiators, Nguyen et al. used one or two, while Abramowicz and Jones did not use any. From the point of view of the basic folding mechanism, Nguyen et al. employed the symmetric collapse mode, whereas Abramowicz and Jones used the asymmetric mixed collapse modes A and B, together with the symmetric collapse mode. Abramowicz and Jones suggested the prediction of an average force equation, while Nguyen et al. proposed prediction of average force and peak force equations. Nguyen et al. also used a coefficient of geometry to peak force equation. This study proposes a new concept for predicting dynamic axial crushing performance, including the use of an eight crush initiator, asymmetric mixed collapse modes A and B and symmetric collapse mode for the basic folding mechanism, and equations of average force, peak force and energy absorption. These prediction equations consider the coefficients of peak force and stress concentration. Finally, the objective of this study is to develop a theoretical prediction of dynamic axial crushing on a thin-walled square steel tube with eight holes used as a crush initiator.

This work has presented a comparison between theoretical and experimental analysis of the behavior of a thin-walled square tube structure under axial crushing. For the symmetric collapse mode, the formula for predicting mean dynamic force is shown as Equation 23; for peak crushing force as Equation 24; and for energy absorption as Equation 25. For asymmetric mixed collapse mode A, the formula for predicting mean dynamic force is shown as Equation 35; for peak crushing force as Equation 36; and for energy absorption as Equation 37. For asymmetric mixed collapse mode B, the formula for predicting mean dynamic force is shown as Equation 47; for peak crushing force as Equation 48; and for energy absorption as Equation 49. The theoretical analysis shows close agreement with the experimental drop test study.

The authors greatly appreciate the Departemen Riset dan Pengabdian Masyarakat Universitas Indonesia for the research grant Year 2015 by Hibah Riset Pascasarjana 2015.

Abramowicz, W., Jones, N., 1984. Dynamic Axial Crushing of Square Tubes. International Journal of Impact Engineering, Volume 2(2), pp. 179–208

Deng, X., Liu, W., Lin, Z., 2018. Experimental and Theoretical Study on Crashworthiness of Star-shaped Tubes under Axial Compression. Thin-Walled Structures, Volume 130, pp. 321–331

Ding, X., Zeqi, T., Liu, Y., Liu, S., 2018. Dynamic Axial Crush Analysis and Design Optimization of a Square Multi-cell Thin-walled Tube with Lateral Variable Thickness. International Journal of Mechanical Sciences, Volume 140, pp. 13–26

Dionisius, F., Istiyanto, J., Suliono., Rohmat, Y.R., 2017. Pengembangan Pengujian Crashworthiness dengan Simulasi Numerik Menggunakan Impact Transferability (Development of Crashworthiness Testing with Numerical Simulation using Impact Transferability). Jurnal Teknologi Terapan, Volume 3(1), pp. 12–17

Ge, C-Q., Gao, Q., Wang, L., 2018. Theoretical and Numerical Analysis of Crashworthiness of Elliptical Thin-walled Tube. International Journal of Mechanical Sciences, Volume 148, pp. 467–474

Googarchin, H.S., Pasandidehpoor, M., Mahmoodi, A., Shojaeefard, M.H., 2019. Energy Absorption Analysis for Tapered Multi-cell Tubes Improved by Foams: Theoretical Development and Numerical Simulation. Composite Structures, Volume 207, pp. 213–222

Hu, D., Wanga, Y., Songa, B., Danga, L., Zhang, Z., 2019. Energy-absorption Characteristics of a Bionic Honeycomb Tubular Nested Structure Inspired by Bamboo under Axial Crushing. Composites Part B: Engineering, Volume 162, pp. 21–32

Mahbod, M., Asgari, M., 2018. Energy Absorption Analysis of a Novel Foam-?lled Corrugated Composite Tube under Axial and Oblique Loadings. Thin-Walled Structures, Volume 129, pp. 58–73

Malawat, M., Istiyanto, J., Sumarsono, D.A., 2017. Effects of Wall Thickness and Crush Initiators Position under Experimental Drop Test on Square Tube. Applied Mechanics and Material, Volume 865, pp. 612–618

Malawat, M., Sumarsono, D.A., 2017. Optimization of Impact Energy Absorber Parameters for Automobile Crush Box using Response Surface Method. In: Proceedings of Quality in Research 2017, Bali, July 24-27^th, Volume 15, pp. 333–339

Najafi, A., Rais-Rohani, M., 2011. Mechanics of Axial Plastic Collapse in Multi-cell, Multi-Corner Crush Tubes. Thin-Walled Structures, Volume 49(1), pp. 1–12

Nguyen, C.N., Dirgantara, T., Gunawan, L., Putra, I.S., Ly, H.A., 2013. Analytical Prediction of Square Crash Box Structure with Holes Due to Impact Loading. In: Regional Conference on Mechanical and Aerospace Technology 2013, Kuala Lumpur, November 25-26^th

Palomba, G., Epasto, G., Crupi, V., Guglielmino, E., 2018. Single and Double-layer Honeycomb Sandwich Panels under Impact Loading. International Journal of Impact Engineering, Volume 121, pp. 77–90

Shena, W., Gu, X., Jiang, P., Hu, J., Lv, X., Qian, L., 2017. Crushing Analysis and Multi Objective Optimization Design for Rectangular Unequal Triple-cell Tubes Subjected to Axial Loading. Thin-Walled Structures, Volume 117, pp. 190–198

Sumarsono, D.A., Istiyanto, J., Malawat, M., 2015. Pengembangan Impact Energy Absorber Dengan Pengaturan Jarak Crash Initiators (Development of Impact Energy Absorber with Distance Management of Crash Initiators). In: Prosiding Seminar Nasional Tahunan Teknik Mesin XIV 2015, Banjarmasin, Volume 15

Sun, G., Pang, T., Xu, C., Zheng, G., Song, J., 2017. Energy Absorption Mechanics for Variable Thickness Thin-walled Structures. Thin-Walled Structures, Volume 118, pp. 214–228

Tran, T.N., 2017. Crushing and Theoretical Analysis of Multi-cell Thin-walled Triangle Tube under Lateral Loading. Thin-Walled Structures, Volume 115, pp. 205–214

Xie, S., Yang, W., Wang, N., Li, H., 2017. Crashworthiness Analysis of Multi-cell Square Tubes under Axial Loads. International Journal of Mechanical Sciences, Volume 121, pp. 106–118

Xiong, F., Wang, D., Yin, S., 2018. Optimization Analysis of Novel Foam-?lled Elliptical Columns under Multiple Oblique Impact Loading. Materials and Design, Volume 156, pp. 198–214

Zhang, J., Lu, B., Zheng, D., Li, Z., 2019. Axial Crushing Theory of Metal-FRP Hybrid Square Tubes Wrapped with Antisymmetric Angle-ply. Thin-Walled Structures, Volume 137, pp. 367–376

Zhang, Y., Xu, X., Wang, J., Tengteng, C., Wang, C.H., 2018. Crushing Analysis for Novel Bio-inspired Hierarchical Circular Structures Subjected to Axial Load. International Journal of Mechanical Sciences, Volume 140, pp. 407–431