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

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Thin-Walled Structures 44 (2006) 872–885

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Analysis of nested tube type energy absorbers with different indentersand exterior constraints

Edmund Morris, A.G. Olabi�, M.S.J. Hashmi

School of Mechanical and Manufacturing Engineering, Dublin City University, Dublin, Ireland

Received 7 October 2005; received in revised form 31 July 2006; accepted 7 August 2006

Available online 9 October 2006

Abstract

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.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Tubes; Energy absorption; Finite element; Plastic deformation; Lateral compression; Quasi-static; ANSYS

1. Introduction

In the study of energy absorbing systems and impactattenuation devices, rings/tubes have received a largeamount of research due to its adaptability. An energyabsorbing system may consist of a tube or system of tubesthat are compressed laterally, which absorbs the kineticenergy upon impact and hence dissipating it as plasticwork. It is usually ideal in the design of these devices toabsorb as much energy per unit mass. This can be done byintroducing some form of side constraints which causesmore plastic hinges to form on the component in questionand hence an increase in the compression force.

Mutchler [1] analysed the energy absorbed by analuminium tube compressed between rigid platens. Thetheoretical model was developed using a numericalintegration scheme. The deformed contour consists oftwo circular arcs which are reduced as compressionproceeds. These two arcs are connected by flattenedsegments which maintain their shape throughout thecompression stroke.

e front matter r 2006 Elsevier Ltd. All rights reserved.

s.2006.08.014

ing author. Tel.: +353 1 700 7718; fax: +353 1 700 5345.

ess: [email protected] (A.G. Olabi).

DeRuntz and Hodge [2] also analysed the compressionof a tube using limit analysis. A rigid perfectly plasticmaterial model was used to predict the load deformationresponse. The geometrical component of stiffening wasaccounted for in the theoretical model. However, the rateof increase was underestimated, this was due to thematerial strain hardening phenomena. The deformedcontour consists of four circular arcs which maintaintheir original radius and plastic deformation occurs in thehinges only.Burton and Craig [3] examined this problem also and

managed to predict a more rapid stiffening of the tube thanproposed by DeRuntz and Hodge. The deformed shape issimilar to that suggested by Mutchler.Redwood [4] endeavoured to include the effects of strain

hardening using the Burton and Craig model. A rigid linearstrain hardening material model as opposed to a rigidperfectly plastic model was used to predict the force andenergy absorption response.The effect of strain hardening was further examined by

Reid [5]. The theoretical model produced by the authors isbased on a rigid linearly strain hardening material modeland is the most accurate one to date. An improvement inthe strain hardening prediction was achieved by replacingthe localised hinges with an arc whose length changes with

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Nomenclature

A Area under the force–deflection curveeE Energy efficiencyeg Geometric or crush efficiency

Lo Original length (outer diameter of outer tube)Pd Peak loadScs Specific energy absorption

Weff Weight effectiveness

E. Morris et al. / Thin-Walled Structures 44 (2006) 872–885 873

deflection. Hence this model accounts for both thegeometrical and material strain hardening effect. Animportant dimensionless parameter was developed whichgoverns the shape of the force–deflection curve. Thisparameter is defined as mR and is a function of the yieldstress in tension, the mean radius R, the strain hardeningmodulus Ep and the thickness t. According to the author itmay be possible to maximise the energy absorbing capacityby choosing appropriate tube dimensions such that the mRvalue is minimised since this is a function of tube geometry.

Reddy [6] examined both theoretically and experimen-tally the quasi-static lateral compression of a tubeconstrained so that its horizontal diameter is preventedfrom increasing. This is a way of increasing the specificenergy absorption capacity of the tube by introducing moreplastic hinges into the structure. Also the relationshipbetween a single tube and a system of tubes with differentconfigurations was investigated. It was found that theenergy absorbed by a closed system (side constraints) is 3times more than that of an open system (no constraints);however, the maximum deflection of the former is less thanthat of an open system. Overall it can be concluded that theintroduction of side constraints and creating a closedsystem is a feasible method of increasing its energyabsorbing efficiency.

The introduction of inclined constraints as opposed tovertical ones can also increase the energy absorbingcapacity of tubes/rings. Reid [7] examined the effect ofvarying the block angle for the lateral compression oftubes. It was shown how varying this angle also varies thecharacter of the load deflection response.

The work described up to now involved the compressionof tubes between rigid platens. However, another way ofcompressing these systems is through point-load indentersas shown by Reid [8]. The load deflection for this type ofcompression tends to be unstable once the collapse loadhas occurred. This behaviour is termed deformation-softening as opposed to deformation-hardening (rigidplates). The author examined the influence of strainhardening on the deformation of thin rings subjected toopposed concentrated loads. It was observed by the authorthat the role of strain hardening is important in the postcollapse stage. During this type of compression asdeformation proceeds, the moment arm increases, there-fore resulting in a decreasing load. This is termed anunstable response. However, this instability can becontrolled through the inclusion of material strain hard-ening; this is achieved by modifying the shape of the loaddeflection curve such that the value of the dimensionless

parameter is less than 10. Fortunately it was recognised bythe author that this instability vanishes for mR values (afunction of tube dimensions) commonly encounteredwithin energy absorbing devices.Morris [9] analysed and presented some work on the post

collapse response of nested tube systems with sideconstraints. The aim of this work was to show how theintroduction of external constraints allows more volume ofmaterial within the structure to deform plastically in thepost collapse stage of compression thereby absorbing moreenergy. Nested systems consist of ‘short tubes’ of varyingdiameter placed within each other with an eccentricconfiguration. To ensure that dynamic effects are notsignificant, a low velocity is applied to crush these systems.Both experimental and finite element techniques were usedto analyse the response of such systems. Numerical resultsgenerated were in good agreement to those of experiment.

2. Experimental procedure

The experiments were carried out on the Instron Model4204 testing Instrument for measuring the mechanicalproperties of materials and structures. A highly sensitiveload is based in the moving crosshead of the loading frameand this measures the compression force of the specimen.The 4204 series software is used to obtain the correspond-ing load deflection response. To stimulate quasi-staticconditions the Instron crosshead had an applied velocity of3mm/min, thus taken approximately 40min to compressthe nested systems.

2.1. Experimental results: flat plate compression

Tubing can be classified as rings if its length is notgreater than a few thicknesses or classified as a tube if itslength is not less than its mean diameter. Hence the systemsanalysed here may be called ‘short tubes’; however, forconvenience all specimens will be referred to as ‘tube’throughout the paper.As already mentioned, Reddy [6] conducted experiments

using external constraints as a means of increasingthe energy absorbing capacity, and Reid [7] analysed thecompression of a tube using point indenters; hence thefollowing analysis is an extension of their work in that bothplate and point-load indenters in conjunction with externalconstraints are used.Fig. 1a shows the double y-axis plot of crush force per

metre and the specific energy absorbed versus stroke lengthfor a three-tube system with no external constraints

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Fig. 1. (a) Load–displacement plot of an assembly of three tubes under flat plate compression. (b) Load–displacement plot of an assembly of three tubes

under flat plate compression with inclined constraints.

E. Morris et al. / Thin-Walled Structures 44 (2006) 872–885874

imposed on it. Fig. 9a depicts how three tubes of varyingdiameter and each of length 60mm were placed within eachother to form a nested system of eccentric configuration.Due to this arrangement, an initial spacing of 17 and21mm exist between the tubes and it is when these gapshave established contact a rise in force will be observed.This rise in force is due to a combination of the two outertubes stiffening and elastic deformation of the inner tube.Finally, as the inner tube collapses the whole system beginsto strain harden up to a maximum deflection of 116mm.As shown in Fig. 9a the upper and lower halves of the innertube have begun to touch. It is possible to increase thestroke length; however, this may be seen as an overloadingof the system and fracture may also occur. Also from thesame figure, notice how only separation of the inner tubeoccurs while the remaining tubes stay in contact. (Note:units of displacement for all graphs are converted to mmfor clarity.)

Fig. 1b also exhibits a similar force deflection response;however, the specific energy has increased due to theintroduction of external inclined constrains of angle 151.These constraints cause more volume of material to deformparticularly for the outer and central tube. Due to the

design of the constraints, the compression stroke wasstopped at 95mm since the system began to deflect over theedges of the constraints as shown in Fig. 9b. This can beavoided by increasing the length of the blocks. However, itshould be noticed that little benefit would be achieved sincefracture may occur around this magnitude of compressiondue to the sharp kinks that are formed.Fig. 2a offers a plot consisting of the crush force and

specific energy absorbed for a three-tube system com-pressed between rigid plates and vertical side constraints.Photos consisting of the initial and final stages aredisplayed in Fig. 10a. The introduction of side constraints,which prevents an increase in the horizontal diameterof the outer tube during compression, causes more volumeof material to deform plastically as already establishedby Reddy [6]. Notice how at approximately 55mmdeflection there is a rapid rise in force; this behaviour isdue to the stiffening of the outer tube since it hasconformed to the shape of the vertical side constraintswhile at the same time the central tube prevents it fromcollapsing inward.A crush force and specific energy response graph for a

system compressed between rigid plates with both vertical

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Fig. 2. (a) Load–displacement plot of an assembly of three tubes under flat plate compression with vertical side constraints. (b) Load–displacement plot of

an assembly of three tubes under flat plate compression with vertical and inclined constraints.


and side constraints is shown in Fig. 2b. The resultingcrush force has again increased as expected due tointroduction of extra constraints. A rapid rise in forceoccurs at approximately 48mm and this is also due to thehorizontal diameter of the outer tube prevented fromdeflecting in either direction. Fig. 10b depicts the variousstages of compression for this system.

2.2. Experimental results: point-load indenter

Fig. 11a provides illustrations of the various stages of athree-tube system compressed with a point-load indenterwithout any external constraints. Notice that symmetry islost between the upper and lower halves of all tubes due tocompression between a single rigid plate and single pointindenter. The corresponding force and energy response isdepicted in Fig. 3a. The collapse force of each tube issimilar in magnitude to that of the three-tube systemcompressed between rigid plates. Once the inner tube hascollapsed the whole system begins to soften as opposed tostrain hardening, thus there appears to be instability withinthe system and this is due to an increase in the momentarm; this results in a reduction in the force required tomaintain the deformation [8]. At approx. 116mm deflec-tion onwards there appears to be a rise in force; this is dueto the adjacent faces of the indenter obstructing the outertube. This can be confirmed in Fig. 11a.

Fig. 3b offers the crush force and specific energy responseof a three-tube system with 151 inclined constraints. Theresulting force has increased slightly due to introduction ofside constraints. In the post collapse stages of the inner tube(at approximately 48mm) the system is unstable for theremaining deflection. Compression of this system is illu-strated in Fig. 11b. Notice how for the final stage ofdeformation the inclined constraints cause a greater inter-ference of the adjacent sides of the indenter with the tube.Figs. 4a and 12a show results of a three-tube system

compressed with vertical side constraints. Upon inspectionof Fig. 4a we see how the central tube begins to softenbefore it establishes contact with the inner tube. Thissoftening appears to be more excessive in contrast to all ofthe above-mentioned systems. Once collapse is initiated inthe inner tube the system softens for the remainder ofdeflection. Again a rise in force is observed due to thepresence of the indenter obstructing the outer tube.The final experiment consists of a three-tube system with

both vertical and inclined constraints as shown in Fig. 4b.Fig. 12b shows the corresponding force and specific energyresponse for this system. At approximately 115mmdeflection a rapid rise in force is observed due to thereasons mentioned in the preceding paragraph.Note that for all systems compressed with a point-load

indenter the crush efficiency can be maintained atapproximately 70–80% and possibly more provided a

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System F

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Fig. 3. (a) Load–displacement plot of an assembly of three tubes under point load compression with no constraints. (b) Load–displacement plot of an

assembly of three tubes under point-load compression with inclined constraints.


narrower point indenter is used in order to avoidinterference with the outer tubes.

3. The finite element method

For the finite element model the implicit time integrationscheme as opposed to the explicit method was used since alltesting was carried out under static conditions. Elementsused were three-dimensional 20 node structural solidelements of tetrahedral shape. All models contain threenon-linearities, namely (1) material non-linearity, (2) contactstatus and (3) large deflection large strain deformation. Abilinear isotropic hardening model was used to capture thestrain hardening effects in the plastic stages of deformation.Two contact classifications were used: rigid to flexible andflexible to flexible. The former was used to simulate contactbetween the rigid plate/point indenter surface and the outertube while the latter was used to simulate the remaining twocontact pairs. All contact pair models were of surface tosurface orientation. Since deflections are approximately60–80%, the tube systems will experience changes in itsvolume, hence this must be accounted for as a geometricnon-linearity in the finite element code. The variousindenters used to compress the systems were defined as

rigid bodies and constrained to translate vertically over apredefined displacement. Table 1 shows the geometric andmaterial information of the tubes used in both theexperiments and numerical models. Symmetry boundaryconditions were imposed on all models to speed up solutionand hence the corresponding reactive forces were multipliedby their appropriate factor. Plastic modulus of 1.5GPa wasused to represent the plastic region of the stress–strain curvefor the three different tubes analysed. This value was alsoused by Reid [10]. One possible way of improving orconfirming present numerical results is to cut a samplespecimen from the tube stock and perform tensile test toobtain the stress–strain curve. This curve can be convertedto a true stress strain plot and input to the finite elementcode as a ‘multilinear’ material model. It should be notedthat although the bilinear material model is only anapproximation, the results obtained are very satisfactory.

4. Comparison of numerical results with those of

experiments

Figs. 5a, b, 6a, b, 7a, b, 8a, b show the comparison ofresults between the numerical solutions offered by ANSYSand those observed by experiment. For all results except

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Fig. 4. (a) Load–displacement plot of an assembly of three tubes under point load compression with vertical side constraints. (b) Load–displacement plot

of an assembly of three tubes under point-load compression with inclined and vertical side constraints.

Table 1

Geometric and mechanical properties of the short tubes used

Outer diameter (mm) Thickness (mm) Mass (kg) Yield stress (Mpa) Elastic modulus (Gpa) Plastic modulus (Gpa)

150 3 0.65 400 200 1.5

127 2.11 0.39 450 200 1.5

101.6 3.25 0.47 450 200 1.5


Fig. 6a, b the numerical solution tends to underpredict thecollapse force followed by an overprediction for theremainder of the stroke. This overprediction is due to theuse of a bilinear material model which assumes the flowstress to increase indefinitely as the strain increases;however, this does not happen in real conditions. Fig. 6a,6b shows excellent predictions of the actual response forthese types of systems.

Figs. 13a, b, 14a, b, 15a, b, 16a, b show the graphic plotsof the various tube systems compressed. These can becompared where relevant to Figs. 9a, b, 10a, b, 11a, b,12a, b. Notice how ANSYS, in Fig. 13a, b accuratelypredicts the separation of the inner tube from itssurrounding environment while the remaining two tubesconform to the shape of the external constraints. Also in

Fig. 14b the actual separation of the outer tube from therigid plate is accurately predicted. It should be noted,however, that upon comparison of the graphic plots tothose of experiment, in the final stages of deformation, thehinges predicted by ANSYS are quite curved as opposed toa slight kink as observed in the experiments. This slightdifference is due to the bilinear material model which doesnot consist of any fracture parameters and hence ANSYSpredicts excessive ‘ductile’ deformation to occur withoutany incidence of fracture. (Figs. 15 and 16)

5. Energy absorption characteristics

It is usual to describe the behaviour of energy absorbingsystems in terms of performance indicators. Thornton [11]

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Fig. 5. (a) Experimental results and numerical solutions for a three-tube assembly. Compressed between rigid plates and no constraints. (b) Experimental

results and numerical solutions for a three-tube assembly compressed between rigid plates and 151 inclined constraints.


suggested indicators such as crush efficiency, energyefficiency, specific energy absorption capacity and weighteffectiveness that may be useful in describing the perfor-mance of absorbing systems. Table 2 gives an outline of thevarious performance indicators used to describe theresponse of the various systems tested. For systems A–Dit can be seen how the crush efficiency decreases due toexistence of external constraints. For systems E–H (point-load indenter), it is possible to maintain a crush efficiencybetween 70–80%, therefore it can be observed that the useof external constraints has much less of an impact on thecrush efficiency. The energy efficiency of the varioussystems compressed is also shown. Energy efficiency asdescribed by Thornton [11] is given as

eE ¼A

Pd � L0, (1)

where A is the area under the force deflection curve, Pd isthe peak load observed and Lo the original length. Uponexamination of this equation, it can be seen that to achieve100% energy efficiency a rectangular force–deflection curveor a rigid perfectly plastic force–deflection response isrequired. From a practical point of view, to achievemaximum energy efficiency, the force -deflection responseneeds to be adjusted so the peak force will occur in theearly stages of deflection and preferably remain at this

magnitude for the remainder of the stroke. This is an idealrequirement in the design of energy absorbing systemssince the deceleration force is desired to be kept at aminimum. By examining systems A–D, it can be seen thatthe energy efficiency is quite low particularly for C and D.This is because the peak force occurs at the final stage ofdeflection as opposed to the early stages. Systems E–Hshow higher energy efficiencies; this is due to the peak forceremaining constant or decreasing for the remaining strokelength. Although the systems E–H show better crush andenergy efficiencies, two points need to be addressed. Firstly,the specific energy absorption capacity of the systemscompressed with a point-load indenter is smaller since lessvolume of material is exposed to plastic deformation.Secondly, and more importantly, point-load compressioncauses a softening to occur as already explained previously,thereby reducing the system’s energy absorbing ability.A final performance indicator that may be used to

describe the behaviour of such systems is the weighteffectiveness, which is given as

W eff ¼ eg � Scs, (2)

where eg is the crush efficiency and Scs the specific energy

absorption capacity. It is a useful indicator since therelative energy absorption capacity of the system is used inconjunction with an important variable, the stroke length.

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Fig. 6. (a) Experimental results and numerical solutions for a three-tube assembly. Compressed between rigid plate and vertical side constraints.

(b) Experimental results and numerical solutions for a three-tube assembly compressed between rigid plates, vertical and inclined constraints.

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Fig. 7. (a) Experimental results and numerical solutions for a three-tube assembly. Compressed by a point indenter and no constraints. (b) Experimental

results and numerical solutions for a three-tube assembly compressed by a point indenter and 151 inclined constraints.


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Fig. 8. (a) Experimental results and numerical solutions for a three-tube assembly. Compressed by a point indenter and vertical constraints.

(b) Experimental results and numerical solutions for a three-tube assembly compressed between by a point indenter, vertical and inclined constraints.

Fig. 9. (a) Initial and final stages of compression for a three-tube assembly with no constraints. (b) Initial and final stages of crushing for a three-tube

system with 151 inclined constraints.


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Fig. 10. (a) Initial and final stages of system compressed between rigid platens and vertical side constraints. (b) Initial and final stages of compression for

three-tube system with vertical and inclined constraints.

Fig. 11. (a) Initial and final stages of compression for a three-tube system with no constraints. (b) Initial and final stages of compression for a three-tube

system with 151 inclined constraints.


From Table 2 it can be seen how systems A–D (flatplate compression) result in higher specific energy absorp-tion than systems E–H (point-load indenter); however, due

to their low crush efficiencies, the overall weight effective-ness of the former systems is somewhat reduced signifi-cantly. For the remaining systems E–H, the overall

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Fig. 13. (a) Graphic display of the initial and final stages of compression of deformation for a three-tube system compressed between rigid plates using

finite element methods. (b) Graphic display of the initial and final stages of compression for a three-tube system compressed between rigid plates and 151

inclined constraints.

Fig. 12. (a) Initial and final stages of compression for a three-tube system with vertical side constraints. (b) Initial and final stages of compression for a

three-tube system with 151 inclined constraints.


effectiveness remains close in magnitude to those com-pressed under a rigid flat plate. This is due to the high crushefficiencies that are achievable when using a point-loadindenter.

6. Conclusion

Experimental investigations into the compression ofnested systems under two different loading conditions

ARTICLE IN PRESS

Fig. 15. (a) Graphic display of the various stages of deformation for a three-tube system compressed with a point indenter and no constraints. (b) Graphic

display of the various stages of deformation for a three-tube system compressed between rigid plates and 151 inclined constraints.

Fig. 14. (a) Graphic display of the initial and final stages of compression for a three-tube system compressed between rigid plates and vertical side

constraints using finite element methods. (b) Graphic display of the initial and final stages of compression for a three-tube system compressed between

rigid plates and vertical and inclined side constraint.


ARTICLE IN PRESS

Table 2

Various performance indicators used to describe the behaviour of the nested systems

System type Geometric efficiency eg (%) Energy efficiency eE (%) Specific energy at final stage Scs (Nm/kg) Weight effectiveness Weff

A 77 39 995@116mm 766

B 63 29 882@95mm 555

C 46 12 930@69mm 427

D 41 8 1120@61mm 459

E 80 70 769@120mm 615

F 69 60 682@103mm 470

G 75 67 781@113mm 586

H 70 62 735@105mm 515

Fig. 16. (a) Graphic display of the various stages of deformation for a three-tube system compressed with a point indenter and vertical constraints. (b)

Graphic display of the various stages of deformation for a three-tube system compressed with a point indenter and 151 inclined and vertical constraints.


have been conducted. The configuration of these systemsdescribed previously can prove to be advantageous in suchcases where space is a very important design parameter aswell as maintaining maximum energy absorption.

Different variations of external constraints have beenused as a means of increasing the energy absorbingcapacity of these nested systems. Although the constraintsincrease the specific energy of both systems A–D and E–H,the crush efficiency of the former suffers as a consequenceand hence the weight effectiveness is somewhat reduced. Itis possible to achieve reasonably high crush efficiencieswhen a point-load indenter is used. However, the designershould be aware of the softening phenomenon that occurswith this type of system, since this may not be a desirablefeature in the design of energy absorbing systems.

Comparison of the results from both experiment andfinite element methods has been made. The numerical

results were found to be well behaved and quite satisfactoryin comparison to those of experiment. Such energyabsorbing devices may find application where the structur-al body in question is subjected to compressive loadsduring impact, examples being bumpers attached to carsand trucks. Other applications such as crash barriers,aircraft fuselages, oil tankers also require energy absorbingdevices for obvious safety reasons.

References

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[3] Burton RH, Craig JM. An investigation into the energy absorbing

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

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