Dynamic and Bending for FGT

8/18/2019 Dynamic and Bending for FGT

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Dynamical bending analysis and optimizationdesign for functionally graded thickness (FGT)tube

ARTICLE in INTERNATIONAL JOURNAL OF IMPACT ENGINEERING · APRIL 2015Impact Factor: 2.2 · DOI: 10.1016/j.ijimpeng.2014.12.007

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Guangyong Sun

Hunan University

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Jianguang Fang

University of Sydney

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Fengxiang Xu

Wuhan University of Technology

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X. Huang

RMIT University

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Available from: Guangyong SunRetrieved on: 10 February 2016

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Dynamical bending analysis and optimization design for functionallygraded thickness (FGT) tube

Guangyong Sun a , Xuanyi Tian a, Jianguang Fang b, Fengxiang Xu a , Guangyao Li a , *

,Xiaodong Huang a , ca State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, Chinab School of Automotive Studies, Tongji University, Shanghai 201804, Chinac Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia

a r t i c l e i n f o

Article history:Received 11 July 2014Received in revised form17 November 2014Accepted 19 December 2014Available online 26 December 2014

Keywords:Crashworthiness designMultiobjective optimizationFunctionally graded thicknessThin-walled structuresEnergy absorption

a b s t r a c t

As a relatively new component with a higher ef ciency of material utilization, functionally gradedthickness (FGT) structure with desired varying wall thickness has been becoming more and moreattractive. In order to suf ciently understand the crashworthiness of FGT under lateral impact, rstly, the

nite element (FE) models of thin-walled columns with uniform thickness (UT) and FGT under lateralloading are established and validated by experimental results. It is exhibited that the FE simulations arein good agreement with experimental tests. Then, the crashworthiness of UT tube and the correspondingFGT tube is compared, and the results reveal that the FGT tube can absorb more energy but generatelarger force than UT tube under the same weight. Further, parametric analyses show the gradientexponent, wall thickness range, the tube diameter and yielding stress have signi cant effects on thecrashworthiness of FGT tubes. Finally, a multiobjective particle swarm optimization (MOPSO) method isused to optimally seek for those design parameters, where surrogate modeling methods are adopted toformulate the speci c energy absorption (SEA) and peak crushing force functions. The results yielded

from the optimization demonstrate that the FGT tube is superior to its uniform thickness counterpart inoverall crashing behavior under lateral impact. Therefore, FGT tubes are recommended as potentialabsorbers of crashing energy under lateral loading.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Thin walled structures have the advantages of light weight, lowprice, high strength and stiffness, high reliability, excellent loading-carrying ef ciency and energy absorption capacity, which havebeen widely used as energy absorbers in crashworthiness appli-cations such as automobile, train and aeronautical industries toprotect passenger from severe injury [1 e 3] . To better understandcrashing mechanism of thin-walled structures and further improvethe design quality of energy absorbers, exhaustive studies havebeen conducted by using analytical, numerical and experimentalapproaches for thin-walled structures with various sections, suchas circular [4] , square [5], hexagonal [6,7] , top-hat [8] , multi-cell[9,10] , convex polygon [11] , concave polygon [12] , etc. Wierzbicki

and Abramowicz [13] , Abramowicz and Jones [14,15] derived someapproximate theoretical formula based on the balance of externaland internal work to predict the average crushing force of circularand square tubes under either statically or dynamically loading. Thegeneral characteristics of force e displacement curves for circulartubes and square tubes are similar: the axial force rst reaches aninitial peak, followed by a drop and then uctuates. However, theircollapse modes are very different. The concertina mode, diamondmode, mixed mode andglobal bucking arethe main collapse modesfor a circular tube, while the common collapse mode of squaretubes are symmetric mode, asymmetric mode, extensional modeand global bucking [16] . These different collapse modes directlyaffect the energy absorption ability of thin-walled columns.Further, these theoretical models were validated experimentally byLangseth and Hopperstad [17] , and Abramowicz and Jones [14] .Although these theoretical formula show good agreements withthe experimental results for some simple thin-walled structures, itis still a dif cult task to derive a closed-form formula to predict the

* Corresponding author. Tel.: þ 86 731 8881 1717; fax: þ 86 731 8882 2051.E-mail address: [email protected] (G. Li).

Contents lists available at ScienceDirect

International Journal of Impact Engineering

j ou rna l homepage : www.e l sev i e r. com/ loca t e / i j impeng

http://dx.doi.org/10.1016/j.ijimpeng.2014.12.007

0734-743X/©

2014 Elsevier Ltd. All rights reserved.

International Journal of Impact Engineering 78 (2015) 128 e 137

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protected]://www.sciencedirect.com/science/journal/0734743Xhttp://www.elsevier.com/locate/ijimpenghttp://dx.doi.org/10.1016/j.ijimpeng.2014.12.007http://dx.doi.org/10.1016/j.ijimpeng.2014.12.007http://dx.doi.org/10.1016/j.ijimpeng.2014.12.007https://www.researchgate.net/publication/271883292_Multiobjective_optimization_for_empty_and_foam-filled_square_columns_under_oblique_impact_loading?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/239060430_On_the_Crushing_Mechanics_of_Thin-Walled_Structures?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/222075788_Dynamic_progressive_buckling_of_circular_and_square_tubes?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/257095870_Energy_absorption_of_multi-cell_stub_columns_under_axial_compression?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/256087479_Crashworthiness_of_foam-filled_thin-walled_circular_tubes_under_dynamic_bending?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/222986497_Static_and_dynamic_axial_crushing_of_square_thin-walled_aluminium_extrusions?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/232827107_Thin-walled_curved_hexagonal_beams_in_crashes_-_FEA_and_design?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/222314043_Design_optimization_of_regular_hexagonal_thin-walled_columns_with_crashworthiness_criteria?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/228493998_Theoretical_prediction_and_numerical_simulation_of_multi-cell_square_thin-walled_structures?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/223249222_Dynamic_axial_crushing_of_square_tubes?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/223249222_Dynamic_axial_crushing_of_square_tubes?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/251508657_Quasi-static_axial_compression_of_thin-walled_tubes_with_different_cross-sectional_Shapes?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/222293234_Finite_element_simulation_of_the_axial_collapse_of_metallic_thin-walled_tubes_with_octagonal_cross-section?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/259122583_Crashworthiness_optimization_design_for_foam-filled_multi-cell_thin-walled_structures?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/245390922_Experimental_study_into_the_energy_absorbing_characteristics_of_top-hat_and_double-hat_sections_subjected_to_dynamic_axial_crushing?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==http://dx.doi.org/10.1016/j.ijimpeng.2014.12.007http://dx.doi.org/10.1016/j.ijimpeng.2014.12.007http://dx.doi.org/10.1016/j.ijimpeng.2014.12.007http://www.elsevier.com/locate/ijimpenghttp://www.sciencedirect.com/science/journal/0734743Xhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.ijimpeng.2014.12.007&domain=pdfmailto:[email protected]://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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crashworthiness responses for some complex thin-walled struc-tures up to date. Fortunately, Non-linear explicit nite element (FE)codes such as PAM-CRASH, ABAQUS Explicit and LS-DYNA providepowerful tools available in numerical simulations of highlynonlinear dynamic response of complicate thin-walled structures,which have been successfully applied in the crashworthinessdesign of thin-walled structures [18 e 23] .

The studies mentioned above mainly focused on the crashingbehavior and energy absorption characteristics of thin-walled tubesunder axial loading. In fact, the front, rear and side components of cars often collapse in bending mode, thus bending collapse shouldbe considered when designing vehicle body [24 e 27] . In addition, acrash test study on 81 vehicles showed that up to 90% involvedstructural members failed in bending collapse mode [28] . Thus, ithas a great signi cance to carry out the investigation into thebending behavior of thin-walled structures. The rst comprehen-sive experimental and theoretical study on the bending perfor-mance of square and rectangular prismatic beams was made byKecman [29] . He proposed simple failure mechanisms involvingstationary and moving plastic hinge lines, and found the moment-rotation characteristics in the post-buckling range up to 40 . Theproposed theory has been veri ed by quasi-static bending tests.Wierzbicki et al. [24] extended the concept of a superfoldingelement and derived a closed-form formula for the bending resis-tance of thin-walled box-section beams. Lately, Kim and Reid [25]proposed a more accurate theoretical solution to predict thebending collapse of thin-walled structures based on some existinganalytical relationship [24] . Liu and Day [30] analyzed the bendingcollapse of the thin-walled channel section beams and derivedtheir approximation moment-rotation characteristics. An empiricalformula based on dimensional analysis has also been presented byHuang and Lu [31] .

Generally, the bending collapse of thin-walled beams is local-ized in a plastic hinge, with the remaining parts of beams rotatingas rigid bodies. Since the inward fold of plastic hinge signi cantlyreduces the cross-section area of the crush zone, the plastic

bending resistance drops signi cantly after local sectional collapseoccurs at relatively small rotation angle [32] . A distinctive charac-teristic of such a deformation mechanism is that the plasticdeformation is concentrated over a relatively narrow zone, whichresults in low energy absorption ef ciency [27] . In order to achievehigher bending resistance and weight ef ciency in energy ab-sorption, ultra-light metal llers such aluminum foams as honey-comb introduced into the thin-walled structures has attractedincreasing interest. Santosa and Wierzbicki [33] and Santosa et al.[34] studied the effect of foam lling on the bending resistance of thin-walled prismatic columns through numerical simulations andquasi-static experiments. It was shown that lling of foam couldimprove the load-carrying capacity by offering additional supportfrom inside and increased the absorbed energy, and concluded that

partial lling of foam might be achieve even higher energy ab-sorption to weight ratio of the structure. Chen et al. [35] carried outthe numerical and experimental studies on the bending collapse of thin-walled tubes lled with aluminum foam or aluminum hon-eycomb. They showed the potentials of signi cant weight savingand volume reduction by utilizing ultralight metal ller. Zarei andKroger [32] performed crashworthiness optimization for maxi-mizing speci c energy absorption on foam- lled tubes underbending, and found that the foam- lledtubescan save more weightto absorb the same energy. Recently, Yu and his co-authors[4,36,37] have carried out systematic experimental and numericalinvestigation into the performance of different topological thin-walled structures under three-point bending. Reyes et al. [38]developed a material model including fracture criteria for both

extrusion and foam to study the in uence of material failure on

bending behavior of square aluminum extrusions lled withaluminum foam. Fang et al. [39] investigated and optimized thecrashing behaviors of three types of functionally graded foam(FGF)- lled thin-walled structures subjected to lateral impact, andfound that the FGF- lled tubes can provide better designs than theordinary uniform foam (UF) lled counterpart.

Although the foam- lled thin-walled structures signi cantlyimprove the crashworthiness, the inherent discontinuity inside themetallic foam easily results in fracture at tensile anges and di-minishes the energy absorption [38,40] , which directly limits theapplication of foam- lled thin-walled structures under bendingload to some extent. In addition, the traditional thin-walled tubealways form a localized plastic hinge on the compressed anges forbending collapse, which weakens the impact resistant force andreduces the energy absorption. Therefore, some new thin-walledstructures without foam ller need to be developed. In brief, if we canprevent or delay the formation of the localized plastic hinge,the bending collapse of thin-walled columns could be signi cantlyimproved. Based on this principle, a novel thin-walled structurenamed functionally graded thickness (FGT) tube will be presentedin this paper. Unlike existing thin-walled structures with uniformthickness, the wall thickness of the present FGT tube varies alongthe longitudinal direction with a certain gradient. To the authors'best knowledge, no study has been conducted on the energy ab-sorption response of non-uniform thickness thin-walled structuresunder lateral loads so far. Although fabricating FGTstructures couldbe inherently more sophisticated than making uniform thickness(UT) structures, some advanced technologies e.g. tailor rollingblanks have been available to produce such structures with a givengradient [41,42] . It is the time to make use of FGT tubes for energyabsorption under impact scenarios. To do so, it is essential to un-derstand the energy absorption characteristics in comparison withthose well-studied uniform thickness (UT) thin-walled structuresunder lateral loading. On the other hand, crashworthiness design of thin-walled structures is typically characterized by a number of quality and/or performance indices such as peak crushing force,

speci c energy absorption (SEA) and crash load ef ciency, some of which could con ict with each other. Therefore, it is of particularsigni cance to seek the best possible thickness gradient for optimalcrashworthiness under multiple criteria.

In this paper, the energy absorption characteristic of FGT tubeunder lateral impact loading is studied by using nonlinear elementcode LS-DYNA. After validating nite element model with experi-ments, the effects of various parameters including the gradientexponent, wall thickness range, the tube diameter and yieldingstress on the crashworthiness of FGT tubes are analyzed. Finally, amultiobjective particle swarm optimization method [43] is used tooptimize those parameters, wheresurrogate modeling methods areadopted to formulate the speci c energy absorption (SEA) and peakcrushing force functions. To ef ciently approximate the responses

of interest, several widely-used surrogate modeling techniquessuch as Kriging, response surface method (RSM) and radial basisfunction (RBF) are compared with each other. The results willdemonstrate that the FGT tube provides better crashworthinessperformance than the UT tube.

2. Simulation modeling and experimental validation

2.1. Structural crashworthiness indicators

In order to evaluate the crashworthiness of the thin-walledstructures under lateral impact load, peak crash force ( F max ), en-ergy absorption ( EA), speci c energy absorption ( SEA) and crashforce ef ciency ( CFE ) are usually used as the important indicators

for evaluating the crashworthiness.

G. Sun et al. / International Journal of Impact Engineering 78 (2015) 128 e 137 129

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hickness?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/225395300_Multidisciplinary_design_optimization_on_vehicle_tailor_rolled_blank_design?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/222185319_Validation_of_constitutive_models_applicable_to_aluminium_foams_Int_J_Mech_Sci_44_359-406?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/245149353_Modeling_of_material_failure_in_foam-based_components?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/245149353_Modeling_of_material_failure_in_foam-based_components?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/232950105_Bending_collapse_of_thin-walled_beams_with_channel_cross-section?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/245583181_Bending_collapse_of_rectangular_and_square_section_tubes_Int_J_of_Mech_Sci_259-10_623-636?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/222206276_Bending_collapse_of_thin-walled_rectangular_section_columns?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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The energy absorption of a structure subjected to the lateralimpact load can be expressed as

EAðdÞ ¼Z d

0

F ð xÞdx (1)

where d is a given deformation distance and F(x) denotes the cor-responding crushing force. The speci c energy absorption isde ned as the ratio of the absorbed energy to the mass of thestructure and can be written as [44] .

SEAðdÞ ¼EAðdÞ

M (2)

where M is the total mass of the structure. Apparently, the higherthe SEA, the better the energy absorption capacity of a structure.Another indictor in relation to energy absorption capacity is crashforce ef ciency (CFE), which can be given as [45] .

CFE ¼ F a v g F max

100% (3)

where F max is the maximum impact force during the whole colli-sion progress. F avg is the mean crash force expressed as the totalenergy absorption divided by the corresponding crushingdisplacement, as

F a v g ¼EAðdÞ

d (4)

2.2. Functional graded column

Fig. 1 illustrates the key geometrical features of the circularcolumn under lateral impacting loads. The column lays on twocylindrical supports, and the span and diameters are 230 mm and46 mm, respectively. A cylindrical punch with a diameter of 10 mmand a mass of 14.44 kg impacts on the mid-span of columns and an

initial velocity of v0 ¼ 3.44 m/s. As the local deformation zone is inthe middle, it is reasonable that the middle thickness is set as themaximum value t max and both ends as minimum t min . The corre-sponding pro le is shown in Fig.1(b) . It is assumed that the circularcolumn is fabricated with gradually varying wall thickness alongthe axial direction. The idea is similar to the density gradient of thefunctionally graded foam materials developed by authors [3] . In theaxial gradient case ( x-direction), the thickness variation t f ( x) can bede ned through the following power law function:

t f ð xÞ ¼ t min þ ð t max t min Þ2 xL

m(5)

where x is the distance starting from the left end to punch location(0 x L/2 due to symmetric), and m is the gradient exponent andassumed to vary between 0 and 10 in this study. L represents thetotal length of the column. From Eq. (5) , it is easily found that thewall thickness increases along the length with a gradient functionchanging from convexity to concavity correspondingly when thegradient exponent m value varies from less than 1 to greater than 1,as shown in Fig. 2.

2.3. Numerical models and validation

2.3.1. Finite element modeling The numerical models are developed using explicit non-linear

nite element code LS-DYNA in this study. Fig. 3 shows the niteelement model subjected to lateral impact. Belytschko-Tsayreduced integration shell elements with ve integration pointsthrough the shell thickness were employed to model the wall of column, support and punch. Compared with column, the supportand punch are almost not deformation, thus they are taken as rigidbody. Stiffness-based hourglass control was employed to avoidspurious zero energy deformation modes, and reduced integrationwas used to avoid volumetric locking. The contact between thepunch and the tube column is modeled with a surface-to-surfacecontact algorithm with a friction coef cient of 0.3. The contactbetween the supports and the specimen is also simulated with asurface-to-surface contact with the friction coef cient of 0 in orderto allow sliding rotation [27] . The “ automatic single surface ” con-tact was used to prescribe the column wall itself to avoid inter-penetration of column folding during bending collapse.

Fig.1. Schematic and pro le showing thickness grading patterns in the axial direction:

(a) Front view; (b) Ae

A cross-sectional view.

Fig. 2. Variation of normalized thickness (0 ¼ t min , 1 ¼ t max ) vs normalized distance

(0 ¼

left/right cross-section, 1 ¼

middle cross-section).


https://www.researchgate.net/publication/223294590_New_extruded_multi-cell_aluminum_profile_for_maximum_crash_energy_absorption_and_weight_efficiency?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/223294590_New_extruded_multi-cell_aluminum_profile_for_maximum_crash_energy_absorption_and_weight_efficiency?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/254224572_Crashworthiness_optimization_of_empty_and_filled_aluminum_crash_boxes?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/254224572_Crashworthiness_optimization_of_empty_and_filled_aluminum_crash_boxes?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/254224572_Crashworthiness_optimization_of_empty_and_filled_aluminum_crash_boxes?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==https://www.researchgate.net/publication/223294590_New_extruded_multi-cell_aluminum_profile_for_maximum_crash_energy_absorption_and_weight_efficiency?el=1_x_8&enrichId=rgreq-1f760525-d86a-4e07-b689-e536f28a0691&enrichSource=Y292ZXJQYWdlOzI3MDcwMzMzMztBUzoxODQ2MTAwMDA0ODIzMDRAMTQyMTAyNTg2MjQwNg==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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The material of thin wall column considered herein is aluminumalloy AA6061. The engineering stress e strain curve which is ob-tained by the standard uniaxial tension test depicted in Fig. 4. Themechanical properties are as follow: density r ¼ 2700 kg/m 3,Poisson's ratio n ¼ 0.3, Young's modulus E ¼ 70 GPa. The consti-tutive behavior of the aluminum column was modeled via thepiecewise linear plasticity material model, Mat 24, in LS-DYNA [46] .The strain rate effect was neglected as the aluminum alloy is strainrate insensitive material [17] .

For an ideal continuous variable thickness tube, the columnwallshould be divided to an in nite number of layers to approximatethe actual thickness variation. In the nite element framework, theminimum width of a layer would be equal to the size of each shellelement. In order to determine the optimum element size to bal-ance the accuracy and computational cost, a convergence test with

ve different mesh sizes was carried out. The convergence resultsare listed in Fig. 5. From which, it can be found that the relativeerror between element sizes 2 mm and 3 mm is less than 1% forboth FGT and UT. Further element re nement will not improve theaccuracy signi cantly, while the computational ef ciency will belargely decreased. Therefore, the optimum element size is3 mm 3 mm, which is adopted throughout this paper.

2.3.2. Validation of the numerical modelTo evaluate the effectiveness of the simulation models of FGT

and UT tubes, the UT tube with the 1.5 mm thickness and the FGTtube with t max ¼ 1.5 mm, t min ¼ 0.8 mm and m ¼ 1 were made andtested. The curves of force versus displacement and deformation of the experiments and simulations are given in Fig. 6. From which, itcan be seen that the simulation results agree well with experi-mental results. The validation provides considerable con dence forus in exploring the energy absorption capacity of thin-walled

circular column and optimization using the developed numericalmodel.

3. Parametric analyses

To better understand the crashworthiness characteristics of FGTtubes under lateral impact loading, it is necessary to carry out thecomprehensive parametric analyses. In this section, the parametricanalyses mainly investigate the effects of the gradient exponentparameter, thickness range, diameter and yield stress of the tubeson crashworthiness performance. Because the crashworthinessindicators are essentially related with the given deformation dis-tance, for comparison, the deformation distance is set as 40 mm forall these following parametric analyses and optimization design.

3.1. Effect of thickness variation

In order to compare the energy absorption capacity between UTtubes and FGT tubes under the same weight, numerical simulationsof the FGT tubes with different gradient parameters are performed.For all the FGT tubes, t max and t min are de ned as 1.5 mm and0.8 mm, respectively. To easily calculate the equivalent thickness(t avg ) of UT tube, it is assumed that the length of each layer is thesame and de ned as Le. Thus the t avg can be given as follows,

t a v g ¼ XN

i¼1ðt iLeÞ=ðNLeÞ ¼X

N

i¼1t i=N (6)

where N denotes the total numberof element layers of the FGT tubealong the axial direction, and t i is thickness of the ith layer element.

Fig. 7(a) and (b) compare the change trend of F max and EA of theUT and FGT tubes with increasing gradient parameters, respec-tively. Generally, FGT tubes have larger values of F max and EAcomparing with their UT counterparts. Moreover, FGT tubes canproduce larger SEAthan UT tubes as shown in Fig. 7(c) . Itcan alsobefound from Fig. 7(c) that the SEA of FGT tubes increases upto a peakvalue and then decreases with the increasing of the gradientexponent parameter m. Thus, the FGT tube seems much morepromising than the corresponding UT tube in terms of EA and SEA.The reason mainly lies in that the thickness of FGT tube is highest atthe region of severe deformation (mid-span part of the tube).Meanwhile, Fig. 7(d) shows the similar relationship between CFEand m of the FGT tube and the corresponding UTtube. Note that CFEof the FGT tube is only slightly bigger than that of the corre-

sponding UT tube when m <

4.

Fig. 3. Finite element model.

Fig. 4. Engineering stresse

strain curves.

Fig. 5. Convergence results of element sizes.


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3.2. Effect of thickness range

To explore the effect of wall thickness range ( D p ¼ t max t min )on the crashworthiness of FGT tube, the value of t min was variedwith 0.8, 1.0 and 1.2 mm while the t max ¼ 1.5 mm. Fig. 8(a)-(c)display F max , EA and SEA of FGT tubes with different gradientexponent m. Note that the thickness range ( D p) has a noticeableeffect on crashworthiness performance. Speci cally, larger valuesof D p (smaller values of t min ) can help improving SEA, while it couldhave a negative effect on EA. Besides, it could also reduce the valuesof F max . Meanwhile, it can be found from Fig. 8(d) that the CFE is

decreased with D p increasing.

3.3. Effect of diameter

Four diameters of columns ( d ¼ 40 mm, 44 mm, 48 mm and52 mm) are considered here. Fig. 9(a) and (b) give the F max, and EAof four diameters of FGT tubes with different gradient exponent m.It can be found that F max, and EA of four diameters are all decreasedwith the increasing of m . F max of the FGT tube with a diameter of 40 mm is superior to others. Generally, the larger the diameter, thegreater F max. The rule of F max is completely different with that of EA.EA of the FGT tube with a diameter 55 mm is better than othertubes for 0 m 0.8. For 0.8 < m < 4, the diameter of 48 mm showsthe best EA performance. However, the energy absorption of tube

with a diameter of 44 mm becomes the highest for m 4. In

addition, it can also be found from Fig. 9(a) and (b) that the dif-ferences of F max and EA between four diameters shrink with theincreasing of m, respectively.

Fig. 9(c) gives the tendency of SEA for different diameters withan increase of gradient exponent m. It can be seen that SEAs of FGTtubes increase up to their peak values and then decrease with theincreasing of m. Meanwhile, SEA of the FGT tube with the diameterof 40 mm is larger than other tubes. From Fig. 9(d) , it is observedthat the difference of CFE is not so obvious with the change of di-ameters when m < 1, while the change of CFE is signi cant whenm 1. CFE is largest for the FGT tube with a diameter of 44 mm

when m < 2, while the FGT with the 40 mm diameter is the bestselection when m 2.

3.4. Effect of yielding stress

Fig. 10(a)-(c) depict F max , EA and SEA of FGT tubes with differentyielding stress ( s y ¼ 120, 135, 150 and 165 MPa). It shows that F max ,EA and SEAwill increase with the increasing of yielding stress for allgradient exponents m. However, it can be found from Fig. 10(d) thatthe CFE is decreased with the increasing of yielding stress. Becausethe variation of yielding stress does not change the column mass ateach speci c case of m, the SEA values of different yielding stressfollow the same order of magnitude as the EA values. The only dif-ference is that the SEAvalues rst increase and then decrease while

the EA values decrease monotonically with the increasing of m.

(a)

(b)

Fig. 6. Comparison between experiments and FE simulations (a) FGT tube ( t max ¼ 1.5 mm, t min ¼ 0.8 mm and m ¼ 1); (b) UT tube (t ¼ 1.5 mm).


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It can also be observed that the tubes with s y ¼ 165 MPa and120 MPa perform worst and best for F max and CFE , respectively.Therefore, the columns with lower s y are preferred in terms of these two indicators. However, this conclusion con icts with therequirements of SEAand EA, that is, the lower yield stress will resultinto the smaller SEA and EA.

4. Multiobjective optimization design

4.1. Multiobjective optimization problem

As an energy absorber, the column structure is expected toabsorb as much energy as possible per unit mass. Thus, the SEA

(a) (b)

(c) (d)

Fig. 8. Variation of crashworthiness performance vs thickness range: (a) F max ; (b) EA; (c) SEA; (d) CFE.

(a) (b)

(c) (d)

Fig. 7. Energy absorption characteristics versus gradient exponent m of the FGT tubes and the corresponding UT tubes: (a) F max ; (b) EA; (c) SEA; (d) CFE.



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(a) (b)

(c) (d)

Fig. 9. Comparisons of energy absorption characteristics of FGT tube with different diameters: (a) F max ; (b) EA; (c) SEA; (d) CFE.

(a) (b)

(c) (d)

Fig. 10. Comparisons of energy absorption characteristics of FGT tube with different yielding stress: (a) F max ; (b) EA; (c) SEA; (d) CFE.



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should be an objective function and maximized. Besides, F max isanother indicator for the safety of the occupants, which should beminimized. Therefore, the optimization problem can be formulatedin a multiobjective framework as follows:

8>>>:

Minimize F max m ; d; s y ; SEA m ; d; s y

Subject to 8


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The ideal optimal values of the two single objective functionsSEA and F max are provided in Table 3 . Here, the

“ ideal ” means only asingle objective ( SEA or F max ) is pursued. These four single

“ ideal ”

optimums listed in Table 3 correspond to the four special points inthe Pareto space, which just lie at both ends of the Pareto curves. Itcan be seen from Table 3 that the FGT is superior to UT in Ideal MaxSEA which is 0.83 J/g and 0.68 J/g, respectively. However, the IdealMin F max of FGT (1.88 kN) is worse than that of UT (1.19 kN). If weonly aim to low F max , we can adopt UT tube. Otherwise, we perfectto use the FGT tube.

5. Conclusions

In this paper, a functionally graded thickness (FGT) tube underlateral impact loads has been investigated by employing the vali-dated nite element model. A series of FGT tubes whose gradientexponent varies from 0 to 10 are compared with the uniformthickness (UT) tubes under the same weight. It was found that theFGT tubes would provide better values in the speci c energy ab-sorption ( SEA), energy absorption ( EA) and crash force ef ciency(CFE ) but worse values in maximum impact force ( F max ). Throughthe further parametric study, the gradient exponent, thicknessrange, diameter of tube and yielding stress of wall material couldaffect the crashworthiness performance signi cantly. To achievethe optimum design for the crashworthiness of UT and FGT tubes,the multiobjective optimization problems are formulated, aiming

to simultaneously minimize the F max and maximize the SEA. Theoptimal results show that the FGT and UT tubes have their cons andpros in practical applications. As the SEA is the most importantindicator of an energy absorber for the designers, in this sense, FGTtubes can provide better crashworthiness performance than UTtubes under bending.

Acknowledgment

This work was supported from The National Natural ScienceFoundation of China (11202072, 11102067), The Doctoral Fund of Ministry of Education of China (20120161120005), The HunanProvincial Science Foundation of China (13JJ4036) and the Programof the Methodology Study on Designing Automotive Parts withVariable Thickness Blanks (QYT-KT-2013-003).

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Fig. 11. Comparison of Pareto fronts of the FGT tube and UT tube.

Table 3Ideal optimums of the twosingle objectivefunctions forthe FGTand UT tubes underlateral impact.

Type Single objective m d (mm) s y(MPa) SEA (J/g) F max (kN)

FGT Ideal Max SEA 2.48 40 165 0.83 2.91Ideal Min F max 10.00 40 120 0.65 1.88

UT Ideal Max SEA 0.00 40 165 0.68 3.29Ideal Min F max 8.51 40 120 0.41 1.19


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