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.     

Experimental and numerical crashworthiness investigation of empty and foam-filled end-capped conical tubes

Foam-filled thin-wall structures exhibit significant advantages in light weight and high energy absorption. They have been widely applied in automotive, aerospace, transportation and defense industries. Quasi-static tests were done to investigate the crash behavior of the empty and polyurethane foam-filled end-capped conical tubes. Non-linear dynamic finite element analyses were carried out to simulate the quasi-static tests. The predicted numerical crushing force and fold pattern were found to be in good agreement with the experimental results. The energy absorption capacities of the filled tubes were compared with the empty end-capped conical tubes. The results showed that the energy absorption capability of foam-filled tube is somewhat higher than that of the combined effect of the empty tube and the foam alone. Finally, the crash performance of the empty and foam filled conical and cylindrical tubes were compared. Results from this study can assist aerospace industry to design sounding rocket carrier payload based on foam-filled conical tubes.     

Crashworthiness design of functionally graded foam-filled multi-cell thin-walled structures

Foam-filled thin-walled structure has recently gained attention due to its excellent crashworthiness. Based on the previous study, a new kind of foam-filled thin-walled structure called as functionally graded foam-filled thin-walled structure has more excellent crashworthiness than the traditional uniform foam-filled thin-walled structure. Moreover, as far as we know multi-cell thin-walled structure has more excellent crashworthiness than the traditional single-cell thin-walled structure. As an integrator of the above two kinds of excellent thin-walled structures, functionally graded foam-filled multi-cell thin-walled structure (FGFMTS) may has extremely excellent crashworthiness. Based on our study, the crashworthiness of the FGFMTSs is significantly affected by the design parameter of the graded functional parameter m. Thus, in order to obtain the optimal design parameters, the FGFMTSs with different cross sections and different wall materials are optimized using the multiobjective particle swarm optimization (MOPSO) algorithm to achieve maximum specific energy absorption (SEA) capacity and minimum peak crushing force (PCF). At the same time, the corresponding uniform foam-filled multi-cell thin-walled structures (UFMTS) which have the same weight as these FGFMTSs are also optimized in our study. In the multiobjective design optimization (MDO) process, polynomial functional metamodels of SEA and PCF of FGFMTSs are used to reduce the computational cost of crash simulations by finite element method. The MDO results show that the FGFMTS with PCF in the initial period of its crash not only has better crashworthiness than the traditional UFMTS with the same weight but also performs superior balance of crashing stability. Thus, the optimal design of the FGFMTS with PCF occurring in the initial crash is an extremely excellent energy absorber and can be used in the practical engineering.     

Quasi-static response and multi-objective crashworthiness optimization of oblong tube under lateral loading

This paper addresses the energy absorption responses and crashworthiness optimization of thin-walled oblong tubes under quasi-static lateral loading. The oblong tubes were experimentally compressed using three various forms of indenters named as the flat plate, cylindrical and a point load indenter. The oblong tubes were subjected to inclined and vertical constraints to increase the energy absorption capacity of these structures. The variation in responses due to these indenters and external constraints were demonstrated. Various indicators which describe the effectiveness of energy absorbing systems were used as a marker to compare the various systems. It was found that unconstrained oblong tube (FIU) exhibited an almost ideal response when a flat plate indenter was used. The design information for such oblong tubes as energy absorbers can be generated through performing parametric study. To this end, the response surface methodology (RSM) for the design of experiments (DOE) was employed along with finite element modeling (FEM) to explore the effects of geometrical parameters on the responses of oblong tubes and to construct models for the specific energy absorption capacity (SEA) and collapse load (F) as functions of geometrical parameters. The FE model of the oblong tube was constructed and experimentally calibrated. In addition, based on the developed models of the SEA and F, multi-objective optimization design (MOD) of the oblong tube system is carried out by adopting a desirability approach to achieve maximum SEA capacity and minimum F. It is found that the optimal design of FIU can be achieved if the tube diameter and tube width are set at their minimum limits and the maximum tube thickness is chosen.     

Design optimization of cross-sectional configuration of rib-reinforced thin-walled beam

In this paper, a rib-reinforced thin-walled hollow tube-like beam (named as rib-reinforced beam) is presented for potential application in vehicle bumper. Through numerical simulation of the bending behavior under impact loads, the rib-reinforced beam is compared with thin-walled hollow tube-like beams filled with and without foam materials (empty beam and foam-filled beam) in crashworthiness. The effects of the shape of the reinforced rib are investigated and the shape optimization design is performed for increasing energy absorption and reducing the initial peak force. A multi-objective crashworthiness optimization formulation including maximum energy absorption, maximum specific energy absorption and minimum initial peak force is constructed based on the ideal point method (IPM). The optimum configuration of the reinforced rib is given with a normalized cubic spline function. Numerical example results show that, compared with the empty and foam-filled beams with same weights, the optimized rib-reinforced beam has higher energy absorption performance and lower initial crash force. It is found that for the rib-reinforced beam little rumple is formed around the compressed indention, which helps to retard the collapse of the side wall and means more energy absorption.     

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.     

碰撞试验中泡沫填充铝管的最优化设计

对泡沫填充空心方铝管进行轴向撞击粉碎试验。此外，为获得更多有关撞击过程的信息，也对试验进行了有限元模拟分析。为找到更有效轻便的撞击减震器，并达到吸收最多能量的目的，在方矩形管的优化设计中采用了多元设计优化方法（MDO）。基于管的最佳几何尺寸考虑将具有最轻重量并且吸收能量最多作为设计目标。前期研究表明，使用高密度蜂窝材料填充会使管吸收更多能量，但重量不是最轻[Zarei HR，Kroger M．Optimum honey-comb filled crash absorber design．Mater Des 2007，29：193-204]。因此，为了解采用不同密度的泡沫填充管的撞击性能，进行了全面的研究，。采用MDO方法寻找一种优化填充管，使其吸收的能量与最优空心管吸收的能量一样多。     

Theoretical prediction and crashworthiness optimization of multi-cell triangular tubes

The triangular tubes with multi-cell were first studied on the aspects of theoretical prediction and crashworthiness optimization design under the impact loading. The tubes׳ profiles were divided into 2-, 3-, T-shapes, 4-, and 6-panel angle elements. The Simplified Super Folding Element theory was utilized to estimate the energy dissipation of angle elements. Based on the estimation, theoretical expressions of the mean crushing force were developed for three types of tubes under dynamic loading. When taking the inertia effects into account, the dynamic enhancement coefficient was also considered. In the process of multiobjective crashworthiness optimization, Deb and Gupta method was utilized to find out the knee points from the Pareto solutions space. Finally, the theoretical prediction showed an excellent coincidence with the numerical optimal results, and also validated the efficiency of the crashworthiness optimization design method based on surrogate models.     

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.     

Computer simulation and energy absorption of tapered thin-walled rectangular tubes

Tapered thin-walled tubes have been considered desirable energy absorbers under axial loading due to their relatively stable crush load and deformation response compared with straight tubes. This paper compares the energy absorption response of straight and tapered thin-walled rectangular tubes under quasi-static axial loading, for variations in their wall thickness, taper angle and number of tapered sides. Overall the study highlights the advantages of using tapered tubes as energy absorbers. In particular, the peak load required to crush the tubes decreases with the introduction of a taper, and as the taper angle increases. This is desirable for minimising the impact loads transmitted to the protected structure. The practical outcome of the study is design information for the use of tapered thin-walled rectangular tubes as energy absorbers in impact loading applications. Analysis has been undertaken using a finite element model, validated using existing theoretical and numerical models.     

Axial crushing analysis of end-capped circular tubes

The paper investigates collapse mechanisms and energy absorption capacity during the axial compression of the end-capped thin-walled circular aluminum tubes which are hollow or filled with polyurethane foam. An experimental technique is used to evaluate the crushing behavior of the circular tubes under compressive quasi-static strain rate. A numerical model is presented based on finite element analysis to simulate the crushing of circular tubes considering nonlinear response due to material behavior, contact boundary conditions and large deformation. The validated model using existing experimental results is used to evaluate the dynamic response in order to determine the dynamic amplification factor relating the quasi-static results to dynamic response. The experimental and numerical results are used to determine energy absorption capacity due to the plastic deformation of thin-wall tube and crushable foam. The performance of end-capped tubes is compared with non-capped tubes and it is found that maximum initial peak load can be controlled and convenient crash protection systems can be obtained using end-capped circular tubes.     

空心和泡沫填充且端部封闭锥形筒的耐撞性试验和数值分析

泡沫填充的薄壁结构的优点是质量轻和高吸能性能,广泛地应用于汽车、航空、运输和防护等行业。采用拟静力试验,分析空心和泡沫填充的端部封闭的锥形筒的耐撞性能,采用非线性动力有限元模拟拟静力试验,数值分析的抗冲撞力与试验数据很吻合。对未填充的锥形筒和填充的锥形筒的耗能能力进行对比,结果显示,泡沫填充的锥形筒的耗能能力比较好。最后,比较了空心、泡沫填充端部封闭的锥形筒和圆筒的耐撞性。     

Axial crushing and optimal design of square tubes with graded thickness

Introducing thickness gradient in cross-section is a quite promising approach to increase the energy absorption efficiency and crashworthiness performance of thin-walled structures. This paper addresses the deformation mode and energy absorption of square tubes with graded thickness during axial loading. Experimental study is firstly carried out for square tubes with two types of thickness distributions and numerical analyses are then conducted to simulate the experiment. Both experimental and numerical results show that the introduction of graded thickness in cross-section can lead to up to 30–35% increase in energy absorption efficiency (specific energy absorption) without the increase of the initial peak force. In addition, structural optimization of the cross-section of a square tube with graded thickness is solved by response surface method and the optimization results validate that increasing the material in the corner regions can indeed increase the energy absorption efficiency of a square tube.     

Analysis and optimization of externally stiffened crush tubes

Nonlinear finite element analysis is used to investigate the quasi-static axial collapse response of cylindrical tubes which are externally stiffened by multiple identical rings. The rings divide the long tube into a series of short thin-walled tubes. It is assumed that the size and shape of integral stiffeners are controlled through a machining process. The effects of various geometric parameters such as wall thickness, ring spacing, ring thickness and width on the collapse response, crush force and energy absorption of monolithic, integrally stiffened steel tubes are studied and used as a general framework for a design optimization study. Through design and analysis of computer experiments, global metamodels are developed for the mean crush force and energy absorption, using the radial basis function approximation technique. Using both single- and multi-objective design optimization formulations, optimum designs for different response characteristics are found. The crush mode in the form of progressive collapse or buckling is found to heavily depend on the ratio of stiffener spacing to stiffener height as well as the ratio of wall thickness to stiffener thickness. The optimization results show the viability of externally stiffened tubes as efficient energy absorbers.     

Effects of buckling initiators on mechanical behavior of thin-walled square tubes subjected to oblique loading

Thin-walled structures usually collapse in Eulerian buckling mode under oblique loads. Energy absorption capacity and crush force efficiency of the structure in this type of collapse are low. Collapse initiators are used to improve these properties. In this research, effect of collapse initiators on energy absorption characteristics of square tubes under oblique quasi-static loads is investigated both experimentally and numerically. Initiators are in the form of cuttings on the tube corners. Results show that collapse initiators in most of the specimens change deformation mode from general buckling to progressive buckling and decrease considerably the peak load; therefore increase crush force efficiency. Furthermore, effect of location and number of initiators is studied. There is good agreement between the numerical results and data from experiments.     

钢管混凝土截面形状的比选研究

不同截面形状的钢管混凝土在不同的压弯条件下所表现出来的承载力性质是不同的,当给定压弯设计荷载后不同截面形状的钢管混凝土具有不同的经济性。通过计算和分析不同形状钢管混凝土的承载能力,找出了一种实用的优选截面形状的方法,使构件在满足承载力要求的前提下经济性最佳,对钢管混凝土截面选型具有参考价值。     

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.     

Quasi-static axial compression behavior of constraint hexagonal and square-packed empty and aluminum foam-filled aluminum multi-tubes

The axial crushing behavior of empty and Al close-cell foam-filled single Al tubes and Al multi-tube designs (hexagonal and square) were investigated through quasi-static compression testing. The effects of foam filling on the deformation mode and the crushing and average crushing loads of single tubes and multi-tube designs were determined. The foam filling was found to shift the deformation mode of empty single tube and empty multi-tube designs from diamond into concertina. In multi-tube designs the constraint effects and the frictional forces were found to increase the average crushing loads over those of single tubes. It was also found that foam filling induced a higher strengthening coefficient in multi-tube than single tubes. Although foam filling increased the energy absorption in single tubes and multi-tube designs, it was not effective in increasing the specific absorbed energy over that of the empty tubes. However, multi-tube designs were found to be energetically more effective than single tubes at similar foam-filler densities, proving a higher interaction effect in multi-tube designs.     

Experimental optimization of composite collapsible tubular energy absorber device

A four-phase program to improve the specific energy absorbed by axially crushed composite collapsible tubular energy absorber devices was undertaken. In the first phase, examining of the crushing behaviour of non-triggered tubes was carried out. The second phase is aimed at obtaining the best position for the triggered wall. The third phase focuses on the effects of material sizing in order to understand the influence of triggered wall length on the responses of composite circular tubes to the axial crushing load. The results of these three phases of the study contribute to the fourth whose objective is to optimize the shape geometry of the cross-section area to further improving in tube energy absorption capability. The experimental results demonstrated the strong potential benefits of optimizing the material distribution. The sizing and shape optimization of composite collapsible tubes exhibited a pronounced effect on their capability to absorb high specific energy under axial compressive load.