Multiobjective crashworthiness optimization of hollow and conical tubes for multiple load cases

Much attention of current design analysis and optimization of crashworthy structures have been largely paid to the scenarios with single load case in literature. Nevertheless the designed structures may often have to be operated in other load conditions, thus raising a critical issue of optimality. This paper aims to understand and optimize the dynamic responses and energy absorption of foam-filled conical thin-walled tubes under oblique impact loading conditions by using multiobjective optimization method. The crashworthiness criteria, namely specific energy absorption (SEA) and crushing force efficiency (CFE), are related to loading parameters and design variables by using D-optimal design of experiments (DoE) and Kriging model. To obtain the optimal Pareto solutions of hollow and foam-filled conical tubes, design optimization is first performed under different loading case (DLC) using multiobjective particle swarm optimization (MOPSO) algorithm separately. The optimal designs indicate that hollow tube has better crashing performance than the foam-filled tube under relatively high impacting velocity and great loading angle. To combine multiple load cases (MLC) for multiobjective optimization, a double weight factor technique is then adopted. It is found that the optimal foam-filled tube has better crashing performance than empty conical tube under any of overall oblique loading cases concerned. The study gains insights in deriving multiobjective optimization for multiple load cases, providing a guideline for design of energy absorber under multiple oblique loading.     

Multiobjective optimization for tapered circular tubes

As more and more new functional requirements are placed, some novel development of sectional configurations of the structural members has been increasingly introduced. This paper presents the optimal design for tapered tubes of three different configurations, namely hollow single, foam-filled single and collinear double tubes. To represent complex crashworthiness objective functions, a surrogate model method, more specifically, response surface method (RSM), was adopted in this study. The design of experiments (DoEs) of the factorial design and Latin Hypercube Sampling techniques is employed to construct the response surface models of specific energy absorption (SEA) and the maximum impact load (MaxL), respectively. In this paper, the linearly weighted average, geometrical average and particle swarm optimization methods are utilized in the multiobjective optimization for these three different tapered tube cases, respectively. A comparison is made among the different tapered profiles with the different optimization algorithms, and the crashworthiness merits of foam-filled tapered tubes are identified.     

Crushing analysis and multiobjective crashworthiness optimization of tapered square tubes under oblique impact loading

In this paper, a class of axisymmetric thin-walled square (ATS) tubes with two types of geometries (straight and tapered) and two kinds of cross-sections (single-cell and multi-cell) are considered as energy absorbing components under oblique impact loading. The crash behavior of the four types of ATS tubes, namely single-cell straight (SCS), single-cell tapered (SCT), multi-cell straight (MCS) and multi-cell tapered (MCT), are first investigated by nonlinear finite element analysis through LS-DYNA. It is found that the MCT tube has the best crashworthiness performance under oblique impact regarding both specific energy absorption (SEA) and peak crushing force (PCF). Sampling designs of the MCT tube are created based on a four-level full factorial design of experiments (DoE) method. Parametric studies are performed using the DoE results to investigate the influences of the geometric parameters on the crash performance of such MCT tubes under oblique impact loading. In addition, multiobjective optimization design (MOD) of the MCT tube is performed by adopting multiobjective particle swarm optimization (MOPSO) algorithm to achieve maximum SEA capacity and minimum PCF with and without considering load angle uncertainty effect. During the MOD process, accurate surrogate models, more specifically, response surface (RS) models of SEA and PCF of the MCT tubes are established to reduce the computational cost of crash simulations by finite element method. It is found that the optimal designs of the MCT tubes are different under different load angles. It is also found that the weighting factors for different load angles are critical in the MOD of the MCT tubes with load angle uncertainty.     

Crashworthiness optimization design for foam-filled multi-cell thin-walled structures

Foam-filled thin-walled structure and multi-cell thin-walled structure both have recently gained attentions for their excellent energy absorption capacity. As an integrator of the above two kinds of thin-walled structures, foam-filled multi-cell thin-walled structure (FMTS) may have extremely excellent energy absorption capacity. This paper firstly investigates the energy absorption characteristics of FMTSs by nonlinear finite element analysis through LS-DYNA. Based on the numerical results, it can be found that the FMTS with nine cells has the most excellent crashworthiness characteristics in our considered cases. Thus, the FMTSs with cell number n=9 are then optimized by adopting a multi-objective particle swarm optimization (MOPSO) algorithm to achieve maximum specific energy absorption (SEA) capacity and minimum peak crushing force (PCF). During the process of multi-objective optimization design (MOD), four kinds of commonly used metamodels, namely polynomial response surface (PRS), radial basis function (RBF), Kriging (KRG) and support vector regression (SVR) for SEA and PCF, are established to reduce the computational cost of crash simulations by the finite element method. In order to choose the best metamodel for optimization, the accuracies of these four kinds of metamodels are compared by employing the error evaluation indicators of the relative error (RE) and the root mean square error (RMSE). The optimal design of FMTSs with nine cells is an extremely excellent energy absorber and can be used in the future vehicle body.     

Effects of boundary conditions on the energy absorption of thin-walled polymer composite tubes under axial crushing

Polymer composite tubes can be designed to absorb high levels of impact energy by progressive crushing. When a tube is crushed onto a flat platen, energy is absorbed by bending failure of the plies, delamination and friction mechanisms. In the present work, significant increases in energy absorption are shown when a shear mode of failure is initiated by crushing the tube onto a radiused plug (or initiator). A study of plug radius, R, normalised with respect to the tube wall thickness, t, in the range of 0R/t5 for circular tube diameter/thickness ratios of 10<D/t<33 was undertaken with continuous filament random mat glass/polyester composite. Different radii plugs lead to significantly different deformed shapes and crush zone morphologies. Large radius initiators (R/t>2) cause the tubes to split and energy is absorbed primarily through friction and axial splitting. As the initiator radius decreases, the amount of through-thickness shear damage in the fronds increases along with specific energy absorption (SEA). When the plug radius becomes small compared to the wall thickness (R/t<0.75) a debris wedge forms between the initiator and the tube and acts like a larger radius initiator. The highest energy absorption was seen to occur at R/t1 when through-thickness shear damage was induced. In this range, under static loading conditions, SEA was seen to be higher than that for tubes crushed onto a flat platen.     

Analysis of nested tube type energy absorbers with different indenters and exterior constraints

The present work presents both numerically and experimentally the quasi-static lateral compression of nested systems with vertical and inclined side constraints. The force–deflection response of mild steel short tubes compressed using two types of indenters is examined. The variation in response due to these indenters and external constraints and how these can contribute to an increase in the energy absorbing capacity of such systems are illustrated. The implicit version of the Finite Element code via ANSYS is used to simulate these nested systems and comparison of results is made with those obtained in experiments and were found to be in good agreement.     

A new method to investigate the energy absorption characteristics of thin-walled metal circular tube using finite element analysis

A numerical study is made to investigate the energy absorbing rule of thin-walled metal circular tube made of three different materials (steel, copper, aluminum) by using response surface methodology (RSM). At the same time, the application prospect of RSM in terms of the research on the energy absorption rule of energy absorption structure can be explored. The test result shows that, the compression process of thin-walled metal circular tube can be divided into three stages: elastic stage, yielding plateau stage, compact stage; To get the greatest value of average plateau force (APF) , a tube with a shorter height and thicker wall should be adopted; To get the greatest length energy absorption (LEA), a tube with thicker wall should be adopted and the ratio of its height and diameter should be as big as possible; To get the greatest specific energy absorption (SEA), a tube with a thicker wall should be adopted and the ratio between its height and diameter should be as big as possible. Thus, it can be seen that, RSM is an advanced experiment design method, and it can be widely used in the research on the energy absorption characteristics of thin-walled metal circular tube and has a promising application prospect in the development of new energy absorbing material and structure.     

Quasi-static and dynamic crushing behaviors of aluminum and steel tubes with a cutout

Tubular members are commonly used as an energy absorber in engineering structures and many such members have a cutout. In this study, the crushing behaviors of tubes with a cutout are characterized and the effects of cutout on the energy absorption capabilities of these tubes are quantified. Systematic parametric studies were carried out to study the effect of material properties, including yield and ultimate strength of material, strain rate effect, location of cutout, tube length and impact speed on the crushing behaviors and energy absorption capacity of aluminum and steel tubes. First, a numerical model was constructed with a commercial explicit finite element code. It will be first proven that the numerical simulation can produce sufficiently accurate results in an economic manner. Subsequently, the crushing behavior of aluminum and steel tubes with a cutout was experimentally characterized and their energy absorption capacity was evaluated in terms of mean crushing force, peak crushing force and specific energy absorption (SEA). Tubes of various lengths with a cutout located at different locations, subject to both quasi-static and dynamic impact loadings were considered. For steel tubes, the numerical simulation investigated the influence of the strain rate effect and variation in strain hardening ratio of the material. Empirical equations describing the mean and peak crushing forces of aluminum and steel tubes with a cutout were developed using linear and nonlinear regression methods applied to the results obtained from the numerical and experimental studies.     

Multi-objective crashworthiness optimization of tapered thin-walled tubes with axisymmetric indentations

In this paper, the effects of tapering and introducing axisymmetric indentations on the crash performances of thin-walled tubes are investigated. The crash performances of the tubes are evaluated using two metrics: the crush force efficiency (CFE, the ratio of the average crushing load to the peak load), and the specific energy absorption (SEA, absorbed energy per unit mass). The optimum values of the number of the axisymmetric indentations, the radius of the indentations, the taper angle and the tube thickness are sought for maximum CFE and maximum SEA using surrogate based optimization. In addition, multi-objective optimization of the tubes is performed by maximizing a composite objective function that provides a compromise between CFE and SEA. The CFE and SEA values at the training points of surrogate models (metamodels) are computed using the finite element analysis code LS-DYNA. Polynomial response surfaces, radial basis functions, and Kriging are the different surrogate models used in this study. Surrogate based optimization of the tubes showed that the tubes with indentations have better crush performance than tubes without indentations. It is found that maximum CFE requires large number of indentations with high radius, small thickness, and medium taper angle, while maximum SEA requires small number of indentations with low radius, large thickness and small taper angle. It is also found that the globally most accurate surrogate model does not necessarily lead to the optimum.     

Lateral compression of a square or rectangular tube between two parallel narrow width indenters placed non-orthogonally

Collapse behavior of aluminum tubes of square and rectangular cross-sections, when compressed between two identical narrow width indenters placed symmetrically in parallel alignment, is examined. Experiments were performed wherein the angle between the axes of tube and indenters was varied from 0° to 90°. Load compression curves and deformation histories of typical specimens are presented. The collapse of the tubes was seen to be generally symmetrical, though asymmetries were observed in some tubes. Considering only the symmetrical mode of deformation, an analysis is presented for constructing the load compression curves as well as the shape of the deforming tube. The analysis considers the energy, absorbed in stationary and rolling plastic hinges which are formed in the collapsing tube. Computed results thus obtained compare well with the experiments.     

Experimental study on crashworthiness of tailor-welded blank (TWB) thin-walled high-strength steel (HSS) tubular structures

This paper aims to investigate the crushing behaviors of tailor-welded blank (TWB) thin-walled structures. A series of TWB high-strength steel (HSS) square tubes with different weld line locations is used to perform the crushing tests for evaluating the effects of different TWB parameters, such as weld line locations and material combinations, on crushing characteristics. These TWB specimens are fabricated through the laser welding process to ensure sufficiently narrow weld line. In the study, the center edge of TWB square specimens is not welded so that such special TWB structures have open cross section. The crushing test results exhibit excellent repeatability. The collapse modes and force–displacement relationships are compared with each other. It is found that the crushing behaviors of different material combinations are fairly significant for a given weld line location. Such key crushing characteristics as specific energy absorption (SEA), average crush force (F_avg), peak force (F_max) are also evaluated for understanding crashworthiness of these TWB structures. The experimental results provide us with some insightful guidance to crashworthiness design of TWB thin-walled HSS structures.     

Multiobjective crashworthiness optimization of multi-cornered thin-walled sheet metal members

Plastic deformation of structures absorbs substantial kinetic energy when impact occurs. Therefore, energy-absorbing components have been extensively used in structural designs to intentionally absorb a large portion of crash energy. On the other hand, high peak crushing force, especially with regard to mean crushing force, may lead to a certain extent and indicate the risk of structural integrity. Thus, maximizing energy absorption and minimizing peak to mean force ratio by seeking for the optimal design of these components are of great significance. Along with this analysis, the collapse behavior of square, hexagonal, and octagonal cross-sections as the baseline for designing a newly introduced 12-edge section for stable collapse with high energy absorption capacity was characterized. Inherent dissipation of the energy from severe deformations at the corners of a section under axial collapse formed the basis of this study, in which multi-cornered thin-walled sections was focused on. Sampling designs of the sections using design of experiments (DOE) based on Taguchi method along with CAE simulations was performed to evaluate the responses over a range of steels grades starting from low end mild steels to high end strength. The optimization process with the target of maximizing both specific energy absorption (SEA) and crush force efficiency (CFE), as the ratio of mean crushing load to peak load, was carried out by nonlinear finite element analysis through LS-DYNA. Based on single-objective and multi-objective optimizations, it was found that octagonal and 12-edge sections had the best crashworthiness performance in terms of maximum SEA and CFE.     

An analytical model for axial crushing of a thin-walled cylindrical shell with a hollow foam core

A thin-walled tube filled with light-weighted foam has wide engineering applications because of its excellent energy absorption capacity. When the structure is axially crushed, the interaction between the tube and foam core plays an important role in its energy absorption performance. Previous theoretical studies so far have largely been concerned with fully in-filled tubes. In this paper, a theoretical model is proposed to predict the axi-symmetric crushing behaviour of such structures but with a partial infill. Using a modified model for shell and considering the volume reduction for the foam core, the mean crushing force is predicted by the energy balance. The proposed formula agrees well with previous results reported in literature. A parametric study is carried out to examine the contribution of foam core plateau stress (σ_f), amount of filling and shell's radius-to-thickness ratio (R/h) on the axial crushing behaviour of the structure. This study can give valuable design guidelines in using thin-walled structures as an energy absorber.     

Comparative analysis of energy absorption and deformations of thin walled tubes with various section geometries

In this paper, deformations and energy absorption capacity of thin walled tubes with various section shapes (circular, square, rectangular, hexagonal, triangular, pyramidal and conical) are investigated both experimentally and numerically. The tubes have the same volume, height, average section area, thickness and material and are subjected under axial quasi static loading. The results of simulations are in good agreement with the experimental data and show that the section geometry has considerable effect on the energy absorption. The circular tube has the most energy absorption capacity and the most average force among all investigated sections. Since the maximum force is concerned in impact events, pyramidal and conical tubes are recommended, due to their uniform load–displacement curves and therefore, less difference between the maximum and the average forces.     

Plastic collapse analysis of thin-walled circular tubes subjected to bending

Circular tubes have been widely used as structural members in many engineering applications. Therefore, its collapse behavior has been studied for many decades, focusing on its energy absorption characteristics and collapse mechanism. In order to predict the collapse behavior of members, one could rely on the use of finite element codes or experiments. These tools are helpful and have high accuracy but are costly and require extensive running time. Therefore, an approximate model of tubes collapse mechanism is an alternative especially for the early step of design. This paper is also aimed to develop a closed-form solution to predict the moment–rotation response of circular tube subjected to pure bending. The model was derived based on the principle of energy rate conservation. The collapse mechanism was divided into three phases. New analytical model of ovalisation plateau in phase 2 was derived to determine the ultimate moment. In phase 3, the Elchalakani et al. model [Int. J. Mech. Sci. 2002; 44:1117–1143] was developed to include the rate of energy dissipation on rolling hinge in the circumferential direction. The 3-D geometrical collapse mechanism was analyzed by adding the oblique hinge lines along the longitudinal tube within the length of the plastically deformed zone. Then, the rates of internal energy dissipation were calculated for each of the hinge lines which were defined in terms of velocity field. Inextensional deformation and perfect plastic material behavior were assumed in the derivation of deformation energy rate. In order to compare, the experiment was conducted with a number of tubes having various D/t ratios. Good agreement was found between the theoretical prediction and experimental results.     

Energy absorption of axially compressed thin-walled square tubes with patterns

The paper suggests the introduction of patterns to the surface of conventional thin-walled square tubes to improve the energy absorption capacity under axial compressive loads. A quasi-static axial crushing analysis has been conducted numerically by the nonlinear explicit finite element code LS-DYNA. Two types of patterns constructed using the basic pyramid elements were introduced. Type A pattern was aimed at triggering the extensional mode for relatively thin square tubes whereas type B pattern was intended to develop new collapse mode capable of absorbing more energy during collapse. A total of 30 tubes with a length of 120 mm, thickness 1.2 mm and widths of 40 or 60 mm were simulated. Numerical results showed that all tubes with type A patterns developed the extensional collapse mode instead of the symmetric collapse mode and absorbed about 15–32.5% more energy than conventional thin-walled square tubes with a mass increase less than 5%. Meanwhile, a new collapse mode named octagonal collapse mode was observed for tubes with type B pattern and the energy absorption of tubes developing this mode increased by 54–93% compared with the conventional tube. The influence of various configurations of the patterns on the deformation and energy absorption of the tubes was also discussed. The paper opens up a new avenue in design of high energy absorption components.     

Analytical and experimental studies on quasi-static axial crush behavior of thin-walled tailor-made aluminum tubes

In this paper, the crush behavior of segmented circular tubes, made of aluminum alloy 6061 and subjected to quasi-static axial loading, has been analytically and experimentally investigated. Crush behavior of these tubes was modeled by integrating available analytical models and superposition principle. In the certain overall length of segmented circular tubes, effects of changing the wall thickness and length of each segment on the energy absorption characteristics have been evaluated. One successful approach toward obtaining lightweight energy absorbers with high energy absorption capacity is the use of thin-walled Tailor-Made Tubes (TMTs). In these tubes, the thickness and mechanical properties of the wall vary along the length of the tube. Applying these tubes; crush force can be controlled by changing the length and thickness of each tube segment, improving the performance of energy absorbing systems. Results of this research showed that Tailor-made tubes have higher energy absorption capacity at identical crush lengths, and they can absorb more energy per unit weight compared to simple tubes with constant wall thickness and mechanical properties. Moreover, for the same specific energy absorption, the TMTs exhibit a considerable reduction in the magnitude of the mean and initial maximum crush forces. With the use of TMTs, the maximum crush force shifts to the end of the crush range, reducing the exerted deceleration on occupants and equipments. Comparing mean crush force and specific energy absorption obtained by analytical and experimental approaches, it was observed that combining current analytical models with superposition principle can prepare a set of analytical formulations to predict TMTs crush characteristics within an acceptable proximity.     

准静态轴向和侧向挤压下对径向波纹管的分析

介绍了波纹几何形状对破坏性能、能量吸收、失效机理和玻璃纤维方格布／环氧复合材料管失效模式的影响。对承受轴向和侧向压力的3个具有不同几何形状的复合材料管进行试验研究。同时对相同受力条件下的径向波纹复合管、圆柱型复合管、圆柱管环绕的波纹管进行试验,以了解波纹几何形状的影响。结果表明,轴向挤压中波纹几何形状会显著影响管的承载能力。然而,侧向挤压中并没有发现波纹几何形状的影响。试验中绘制了荷载一位移曲线,因此可以对各种不同几何形状的构件进行清晰的对比。研究同时发现,径向波纹可以稳定并有效地吸收能量。     

Optimization of the foam-filled aluminum tubes for crush box application

Axial impact crush tests on empty and foam-filled square aluminum tubes have been performed. Furthermore, in order to find more details about the crush processes, finite element simulations of the experiments have been done. In terms of finding more efficient and lighter crush absorber and achieving maximum energy absorption, multidesign optimization (MDO) technique has been applied for optimizing the square rectangular tubes. Based on practical requirements the optimum tube geometry, which absorbs maximum energy and has a minimum weight, has been determined. Results of previous work indicated that using high density honeycomb for filling the tubes will results more energy absorption but the weight efficiency has been lost [Zarei HR, Kröger M. Optimum honeycomb filled crash absorber design. Mater Des 2007;29:193–204]. Therefore, a comprehensive study has been performed in order to find out the crush behavior of tube filled with foam with different densities. The MDO procedure has been implemented to find an optimum filled tube that absorbed the same energy as an optimum empty tube can absorb.     

泡沫填充的薄壁圆柱壳体轴向压碎的分析模型

由于填充轻型泡沫的薄壁钢管有良好的耗能能力而被广泛地应用于工程中。当结构轴心受压时,钢管与泡沫芯之间的作用对吸收能量起到关键作用。已有理论研究的大部分对象为完全填充的钢管。本文提出了一个理论模型,用于分析部分填充的钢管的轴向对称压碎性能。采用改进的模型分析壳体,并考虑了泡沫芯的作用。由能量平衡原理得到极限压力的平均值。建议公式得到的结果与先前文献中的结果吻合。参数分析用于研究泡沫芯稳定时期的压力值(σf),以及填充的比例和壳体的径厚比对结构轴向受压性能的影响。为薄壁结构的吸能性能方面的设计提供指导。