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New insights into the collapsing of cylindrical thin-walled tubes871In order to use the introduced parameter in applieddesign of energy absorbers, two methods are suggested to activate the axisymmetric plastic buckling mode: cutting an initial circumferential edgegroove outside the tube and using one- and twocircumferential stiffeners. Performing FE simulations,the extension of the axisymmetric collapsing regionafter using these methods is shown in the L/D–D/tdiagram.Almost all of the reported works on designingthe cylindrical tubes as impact energy absorbers arejust carried out to satisfy the imposed crashworthiness requirements, whereas, nowadays, optimizationbecomes a necessary part of designing procedure. Theoptimization of the structure under crashworthinessrequirements is very complex and expensive from thecomputational point of view because the objectivefunction is often non-smooth and highly non-linearin terms of design variables. The optimization procedure requires repetitive and iterative validation of theselected objective function for various values of designvariables. To avoid the calculation of the objectivefunction in each iteration by computationally costlyFE simulations, approximated functions may be usedto simulate the crush behaviour of the structure. Theresponse surface methodology (RMS) [11], and neuralnetwork systems [12] are the most common devicesto reproduce the crush behaviour of the structure. Inthe present study, a system of parallel neural networksis developed to reproduce the structural behaviourduring the crush phenomenon. A limited numberof FE simulations are carried out to train and testthe neural network systems. The remarkable preference of the presented work to most similar studies isusing different neural network systems to return eachcrashworthiness parameter instead of using a globalnetwork to simulate all the parameters. After training the neural network systems, the response surfaceof selected crashworthiness parameters against thedesign variables are calculated and shown in graphicalform. Finally, using the response surfaces developedby the neural network systems, the genetic algorithm(GA) is implemented to find the optimal configurationof tube dimensions for both single-objective (SO) andmulti-objective (MO) optimizations.is used. Four-noded shell elements with reduced formulation (S4R), suitable for large strain analysis areused to model all the analysed tubes. Three integrationpoints are used through the shell thickness to modelbending. The shell thickness is set to t 2 mm for allthe specimens except in the cases that another thickness is mentioned. After convergence, an element sizeof 3 mm is found to produce suitable results. Two rigidwalls are fixed to the ends of the tube. For simulatingthe impact load, a point mass (m 250 kg) is attachedto the upper rigid wall and an initial downward velocity (V0 ) is defined for the wall just before the impactevent. The quasi-static load is simulated by movingthe upper plate with a constant velocity downward.The tube is tied to the lower rigid wall and free at theother end. The lower plate is constrained in all degreesof freedom and the upper plate is fixed in all translational and rotational degrees of freedom except theaxial displacement in order to avoid the twisting of theimpactor plate. The contact between the tube and rigidwalls is assumed to be frictionless, but a friction coefficient equal to 0.1 is used to model the self-contact ofthe inner and the outer surfaces of the shell.The material properties are defined as linear elasticfollowed by non-linear work hardening in the plasticregion. The true static stress–strain curve of a typicalaluminium alloy obtained by a standard tensile test asshown in Fig. 1 is used to introduce the approximatedtrue stress-plastic strain data points in numerical simulations. These points are shown in Table 1. Theelastic modulus of this material is 70 Gpa, the density is ρ 2700 kg/m3 and the Poisson ratio is ν 0.3.The material is assumed to have only isotropic strainhardening and strain rate effects on the yield strengthare neglected.2Fig. 1 True stress–true strain characteristic of thealuminium alloy (experimental)2.1NUMERICAL MODEL AND VERIFICATIONDescription of the FE model and materialpropertiesNumerical simulations of axial crushing of tubes underimpact loading are carried out using the FE codeABAQUS/Explicit. In order to calculate the plasticbuckling mode shapes, the FE code ABAQUS/StandardJMES562 IMechE 2007Table 1 True stress–plastic strain data points used foraluminium in numerical simulationsσ (N/mm2 )175185192200205210εp00.010.020.030.040.05Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

872M Shakeri, R Mirzaeifar, and S SalehghaffariFig. 2Collapsing of a tube under axial quasi-static load obtained from the experimental [5] andnumerical results. (a) The deformed shape, (b) the absorbed energy, and (c) reaction forceagainst the axial deformation2.2 Verification of the FE simulation usingthe previously reported experimental resultsCrushing simulation of a tube under quasi-static axialloading is carried out and the results are comparedwith the experimental results of Hsu and Jones [5]in order to evaluate the accuracy of the FE simulation in predicting the absorbed energy and the crushforce as well as the deformation mode of the tube. Thetube has a nominal outer diameter of 50.8 mm (2 in),length of 250 mm, and a wall thickness of 1.53 mm. Thetube is made of 6063-T6 aluminium alloy. The materialproperties of this alloy are described in reference [5].The specimen is sandwiched between two parallelhigh-strength steel plates and the upper plate movesdownward with a constant speed of 2 mm/min. Themaximum axial deflection is set to 150 mm. Figure 2(a)shows the deformed shape obtained from the experimental and numerical results. The absorbed energyand crush force against the axial deformation of thetube obtained by the numerical simulation and experimental test are compared in Figs 2(b) and (c), respectively. As it is shown, the numerical simulation predictsclosely the deformed shape as well as the absorbedenergy and crush force.test, an aluminium alloy tube of external diameter75.6 mm, length of 151.2 mm, and a wall thicknessof 1.4 mm is loaded quasi-statically in axial directionby using a compression testing machine at a nominalcross head-speed of 5 mm/min. In order to obtain thematerial data, a quasi-static material test is performedon a strip cut from a shell using a standard tensiletest machine and the resulting stress–strain curve isused to introduce the approximated true stress-plasticstrain data points in the numerical simulation. Thespecimen is placed between parallel steel plates ofthe test machine without any additional fixing. Themaximum axial deflection is set to 105 mm. Figure 3compares the experimental and numerical results ofthe deformed shape for the tube. Figure 4 shows theaxial load against the axial deformation obtained fromthe experimental and numerical results. It is obvious that the numerical method simulates the crushingbehaviour of the tube with sufficient accuracy.3 THE INFLUENCE OF THE PLASTIC BUCKLINGMODES ON THE COLLAPSING SHAPE OF THETUBE3.1 The influence of initial imperfection on thecollapsing modes classification chart2.3Quasi-static crush test to verify the FEsimulationA quasi-static crush test is carried out to verify theaccuracy of the numerical simulation results. In thisInasmuch as the tubes deformation under axial loadwill involve buckling, it is necessary to perturb theinitial geometry of the tube in the crushing analysisproportional to the buckling modes. By ignoring theProc. IMechE Vol. 221 Part C: J. Mechanical Engineering ScienceJMES562 IMechE 2007

New insights into the collapsing of cylindrical thin-walled tubes873Fig. 3 The deformed shape of the tube under axial quasi-static load obtained from theexperimental and numerical resultsFig. 4 The reaction force against the axial deformationof the tube under quasi-static load obtained fromthe experimental and numerical resultsgeometrical perturbation, numerical methods onlypredict the axisymmetric collapsing mode for all casestudies because the geometrical model and load condition are both axisymmetric in collapsing of tubesunder axial load. However, the experimental tests showthat concertina collapsing happens only for a narrowrange of tube dimensions. This phenomenon can beexplained by considering an instantaneous bucklingjust before the crushing of tube. The effect of thisinitial buckling can be introduced in numerical simulation by applying an initial imperfection proportionalto the buckling modes on the tube in crushing analysis.Experimental results show that the best geometrical imperfection for applying on the tube model innumerical simulation is a linear combination of someof the first buckling modes, for example the first tenplastic buckling modes [13]. Typically, the magnitudeof the perturbation used for each eigenmode is afunction of the relevant structural dimension, such asshell thickness and the magnitude of the corresponding eigenvalue. Since the lowest eigenmodes are mostpertinent to the crushing behaviour of the structure,appropriate magnitudes may be found by obtaininga mesh imperfection of a few percent of the shellthickness for the first eigenmode and a decreasingJMES562 IMechE 2007percentage as the corresponding eigenvalue of modesincreases. The magnitude of these imperfections arefound by a trial and error procedure and comparingthe results of numerical simulation with the experimental results. Note that the magnitudes related tothe modes change proportional to the change of eacheigenvalue related to the first eigenvalue, so the onlyunknown in each attempt of trial and error procedure is the imperfection proportional to the first mode.In this study, the magnitude for the first mode is setto 2 per cent of shell thickness that was previouslyreported in the literature too [13].In order to perform the procedure of applying theseimperfections on the structure, in each numericalsimulation, the first ten buckling modes and theircorresponding eigenvalues of the tube are obtainedby running an eigenvalue buckling analysis usingABAQUS/Standard. As a sample, the first four bucklingmode shapes of a tube of D 120 mm, L 216 mm,and t 2 mm are shown in Fig. 5. These modes arerelated to buckling of the tube placed between twoFig. 5 The first four mode shapes of a tube subjected toaxial compressionProc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

874M Shakeri, R Mirzaeifar, and S Salehghaffaririgid walls (the rigid walls are eliminated in Fig. 5 inorder to improve the clarity of picture), the back wallis constrained and an axial force is applied to the fronttube. The results of buckling analysis are stored, and inthe next step, the IMPERFECTION keyword is used inABAQUS/Explicit to read the buckling modes from thestored data, scale them by the defined magnitudes,and perturb the nodal coordinates of the FE modelbefore the crushing analysis.All the previous studies in the literature concentrateon finding an initial geometric imperfection in numerical simulations in order to simulate the experimentalcollapsing shape of tube with a good accuracy. Thecontribution of this initial imperfection on the collapsing mode of tube and finding applied methods basedon this imperfection to control the collapsing modeis not studied yet. In this paper, initial imperfectionproportional to the plastic buckling modes is introduced as a new parameter that controls the collapsingshape of tube under axial impact load. By performing numerical simulations for tubes of various L/Dand D/t ratios for the impact velocity V0 7 m/s, thelimits of the region in which the concertina crushing mode is guaranteed in the L/D–D/t diagram isobtained and shown in Fig. 6. The results of an experimental test in the case of quasi-static loading [1]are shown in this figure too. As it is shown, the limits of the concertina collapsing region for the crushsimulations are similar to the results of quasi-statictest. It is evident that, in both conditions of loading, only the tubes with relatively small L/D andD/t ratios that have a lower ability in dissipating theenergy, compare to larger tubes, collapse in concertinamode.In the next step, the same numerical simulationsof tube crushing under axial impact load are performed, but solely the initial geometric imperfectionof the axisymmetric buckling mode (like mode 1in Fig. 5) is applied on the structure at the beginningof the crushing analysis. As it is shown in Fig. 6the limits of the concertina collapsing mode regionextend remarkably in the L/D–D/t diagram usingthis method, except for the tubes of relatively greatdiameters. This phenomenon reveals that by activating the axisymmetric buckling mode of the tubeunder axial compression, the concertina collapsing mode takes place for a wider range of dimensions. The following sections, describe applied designmethods in order to activate the axisymmetric buckling mode.3.2Six quasi-static crushing tests are performed in orderto verify the classification chart presented in Fig. 6.According to this chart, numerical simulations predictconcertina collapsing mode for tubes of any 20 D/t 75 and L/D of two or less and predict mixedmode for L/D of two or greater in this range of D/t.The main purpose of this section is evaluating thenumerical results for the classification chart in therange of 20 D/t 75, so the tests are carried outfor two different D/t ratios (D/t 54 and 72) andthree L/D ratios for each D/t value. The test conditions and material properties are the same as the testin section 2.3. The dimensions of tubes and the axialdeflection for each test are shown in Table 2, note thatthe thickness of tube is set to 1.4 mm for all specimens.Fig. 6 The limits of the concertina collapsing regionTable 2Quasi-static crush tests to verify the collapsingmodes classification chartResults from the quasi-static experimental tests and numerical simulationsTest 0519085140200Fmax (kN)ModeaFmean 17.617.419.820.120/3CM3M3CM4M4CM3M3CM4M4a C, concertina; M3, mixed mode with three circumferential lobes; M4, mixed mode with four circumferential lobes.b Exp, experimental; Num, numerical.Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering ScienceJMES562 IMechE 2007

New insights into the collapsing of cylindrical thin-walled tubesFig. 7875Comparison of the results for tubes collapsing mode under axial quasi-static load obtainedfrom experimental tests and numerical simulationsThe collapsed shape of tube obtained by FE simulation and experimental tests are compared in Fig. 7 forthese tests. It is obvious from the presented resultsthat numerical method can simulate the collapsingshape of tube with sufficient accuracy. Table 2 showsthe values of the maximum and mean collapsing forceobtained from FE simulation and experimental tests.It is obvious from Fig. 7 and Table 2 that the numericalsimulation can predict the collapsing shape and thecrashworthiness parameters with a great accuracy.4 TUBES WITH A CIRCUMFERENTIAL EDGEGROOVEJMES562 IMechE 2007Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering ScienceAs it is shown in Fig. 5, in the axisymmetric buckling mode the edge ring of the tube deforms outwards. An applied design to activate this mode isweakening the edge ring of the tube by cutting a circumferential edge groove outside the tubes to forcethis ring to deform outwards at the beginning of thecrushing.

876M Shakeri, R Mirzaeifar, and S SalehghaffariMost of the previous reported works on designing the grooved tubes in order to control the collapsed shape under axial loading are restricted tothe tubes with grooves alternately cut outside [14] orinside and outside [15] the tube or the spirally slotted tubes [16]. The common weakness of all thesedesigns is in the reduction of the amount of materialparticipating in the plastic deformation and energyabsorption. However, in the presented design, as itwill be shown in the following sections, the amount ofmaterial that does not contribute in energy absorptionis so low that the groove have a negligible influence on the energy absorption capacity of the structure, on the other hand the presented design have aremarkable influence on the collapsed shape of thetube.The edge grooves are of W 3 mm wide and d 1 mm depth. The details of the specimen design aregiven in Fig. 8. In order to model the specimen, theshell thickness of the first row of elements in theFE model is changed to t 1 mm and the OFFSETparameter is used in the SHELL SECTION keyword,to adjust the position of this row of elements likeFig. 8. The procedure of extracting the buckling modeshapes and their corresponding eigenvalues, and thecrush analysis is like before. Numerical simulationsfor tubes in the range of 20 D/t 100 and 1 L/D 6, before and after cutting the circumferentialgroove are performed and the limits of the concertinacollapsing region are obtained. Figure 9 shows theremarkable extension of the axisymmetric collapsingregion after cutting a circumferential edge groove onthe tubes.Fig. 8Details of the circumferential groove designProc. IMechE Vol. 221 Part C: J. Mechanical Engineering ScienceFig. 9 The limits of the concertina collapsing regionbefore and after cutting the circumferentialgroove5QUASI-STATIC EXPERIMENTAL TESTS ONTUBES WITH A CIRCUMFERENTIAL EDGEGROOVEAs explained in the previous section, cutting acircumferential edge groove on the tube activatesthe axisymmetric buckling mode and increases theprobability of forming axisymmetric folds in contrastwith diamond folds. In order to verify the numerical results, 16 experimental tests are carried out.These tests are performed on tubes with D/t 49,t 2 mm and four different L/D ratios. The material properties and test conditions are similar to thosegiven in section 3.2. For each L/D and D/t ratio, twospecimens with and without grooves are prepared.Figure 10 shows a specimen with the edge groove.The dimensions of groove are similar to that givenFig. 10A specimen with circumferential edge grooveJMES562 IMechE 2007

New insights into the collapsing of cylindrical thin-walled tubesin Fig. 8. Table 3 contains the properties of theseeight specimens. In order to guaranty the accuracyof the experimental tests, each test was carried outtwo times. Figure 11 compares the collapsing modeof tubes with edge groove and the initial tubes. It isobvious that cutting the edge groove increases thenumber of axisymmetric folds. In all the tests fortubes with circumferential edge groove, after forming two or three axisymmetric folds the collapsingmode changes to diamond. This phenomenon may bebecause of non-symmetric deflections that are generated on the tube during cutting the edge groove. Asshown in Table 3, cutting the edge groove decreasesthe value of maximum reaction force that is a greatadvantage in designing tubes as mechanical shockabsorbers.Table 3Details of specimens used for studying theinfluence of cutting a circumferential edgegroove on the collapsing mode (as shown inFig. 11)Test #L/DFmax (kN)Number ofaxisymmetric 152.7131212138776 TUBES WITH ONE- ANDTWO-CIRCUMFERENTIAL STIFFENERSAs another design for broadening the concertina collapsing region, the circumferential stiffeners may beadded to the tube. In the first step, it is assumed thata stiffener ring is attached to the top ring of the tubeas shown in Fig. 12. In order to model the specimenwith the stiffener ring, a boundary condition thatconstrains the radial displacement of nodes on theedge of the tube is used. In the range of 20 D/t 100and 1 L/D 6, the collapsing mode classificationchart for the tubes before and after using the stiffenerring is depicted in Fig. 13. It is evident that the stiffenerring extends the concertina mode remarkably withoutinfluencing the energy absorption of th

fection of plastic buckling modes in the postbuckling analysis is introduced as a new factor that can extend theconcertinacollapsingregion.Theprevailingtheory on the postbuckling analysis of tubes under axial load-ing is applying an initial imperfection proportional to a linear combination of all the plastic buckling modes