EXPERIMENTAL AND NUMERICAL ANALYSIS OF CIRCULAR TUBE
SYSTEMS UNDER QUASI-STATIC AND DYNAMIC LOADING.
Edmund Morris1, A.G. Olabi1, M.S.J. Hashmi1, M.D Gilchrist2
1.
School of Mechanical and Manufacturing Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland.
2. School of Electrical, Electronic and Mechanical Engineering, University College Dublin, Belfield, Dublin 4, Ireland.
ABSTRACT.
In the study of impact attenuation devices, rings/tubes have received a large amount of research due to their adaptability,
i.e. they are low in cost and are readily available for selection in the design process. They also exhibit desirable forcedeflection responses which is important in the design of energy absorbing devices. The function of such a device is to
bring a moving mass to a controlled stop and ideally cause the occupant ride down deceleration to be within acceptable
limits so as to avoid injuries or to protect delicate structures.
In this work, the quasi-static and dynamic analysis of nested circular tube energy absorbers was examined using
experimental and numerical techniques. Although these devices are usually exposed to much higher velocities, it is
common to analyse the quasi-static response first, since the same pre-dominant geometrical effects will also occur under
dynamic loading conditions.
In this investigation, quasi-static tests were performed via a Universal Instron machine with an applied velocity of 35mm/min whilst dynamic tests were performed with velocities ranging between 5m/s and 7 m/s. The various nested tube
systems consisted of one standard and one optimised design. Their crushing behaviour and energy absorption capabilities
were obtained and analysed both experimentally and numerically.
KEYWORDS: ENERGY ABSORBERS, LATERAL CRUSHING, NESTED SYSTEM.
INTRODUCTION.
Energy absorption through material deformation has been extensively studied over the last three decades, particularly in
the form of tubular systems. The lateral compression of a single circular tube and its strain hardening phenomena have
been analysed both experimentally and analytically by various authors such as Burton and Craig [1], DeRuntz and Hodge
[2], and Redwood [3]. These authors were among the first to analyse such problems and each one of them proposed a
slightly different deformation mechanism for the compression of a tube between rigid flat platens.
The effect of strain hardening was further examined by Reid and Reddy [4] who developed a theoretical model based on a
rigid linearly strain hardening material model which appears to be the most accurate one to date. The authors improved
the strain hardening prediction by replacing the localised hinges with an arc in which its length changes with deflection.
Hence this theoretical model accounts for both the geometric and material strain hardening effect. An important
dimensionless parameter which was developed governs the shape of the force-deflection curve. This parameter was
defined as ‘mR’ and is a function of the yield stress in tension, the mean radius R of the tube, the strain hardening
modulus Ep and the thickness t. According to Reid and Reddy it may be possible to maximise the energy absorbing
capacity by choosing appropriate tube dimensions such that the ‘mR’ value is minimised since this is a function of tube
geometry.
Avalle and Goglio [5] examined the strain field generated during the lateral compression of aluminium tubes and
proceeded to verify the known theoretical models. Of the three known theoretical models proposed by the various authors
[1, 2, 3], it was found that the latter accounted for all the main features observed experimentally, hence this model seems
the most realistic in describing the actual behaviour of the tube both qualitatively and quantitatively. Reddy and Reid [6]
proposed a method to calculate a more realistic force-deflection curve using a rigid linear work hardening material model.
These tubes were also compressed laterally between rigid platens. It was suggested that an average value of strain
hardening modulus could be used to calculate the parameter ‘mR’, therefore these two parameters would be considered
constant throughout the deflection range. However, it has been further proposed that if the variation of strain hardening
modulus with strain is known, this could be used to update ‘mR’ at each load step or load increment and thus obtain a
more realistic load-deflection characteristic. It was suggested that the method described above could be used as a basis
for obtaining some of the material properties from a ring compression test.
Nested systems in the form of a line of rings subjected to end impact loading were examined by Reid and Reddy [7]. The
authors were principally concerned with identifying the main mechanism which controls the deformation of such systems.
Upon experimentation, the main parameters were identified and varied, thereby leading to a suggestion for the
construction of a mathematical model of the system. It was found that in low speed impact testing on tube systems, the
effect of inertia was secondary; therefore the design of energy absorbing systems could be achieved provided that the
material strain rate was taken into account. Reddy et al [8] described experiments in which a variety of one dimensional
systems with free distal ends, as opposed to fixed ends, were subjected to lateral impact by a rigid projectile. An elastic–
plastic structural shock wave theory, which employs a bilinear material model to describe the collapse behaviour of the
rings, was used to analyse the deformation of typical ring chain systems.
A nested system analysed by Shrive et al [9] consisted of two concentric rings with a layer of smaller tubes between them,
the axis of all tubes been parallel. Tack welding was used to attach the rings to the concentric tubes. It was found that
increases in system stiffness, maximum load and energy absorption was apparent as the level of tack welding increased.
From the impact loading experiment, it was found that full deformation did not occur but maximum opposing forces similar
to the quasi-static case were achieved. Shim et al [10] analysed the lateral crushing of thin walled tubes using cylindrical
indenters and side constraints. A complete range of indenter radii have been used varying from zero curvature (flat
platens) to infinite radius (point load indenters) to crush these thin walled tubes and to examine their responses. It was
discovered that, depending on the radius of curvature of the indenters, the post collapse behaviour of laterally compressed
tubes can be either stable (deformation- hardening) or unstable (deformation-softening).
Morris et al [11] analysed the quasi-static lateral compression of nested tube systems between rigid platens both
experimentally and numerically. These energy absorbers consisted of both two and three tube systems which were
assembled ‘In Plane’. Such a term describes two or more tubes of varying diameter being placed within each other and
their axes being parallel. This type of energy absorber was compressed under flat rigid platens at a velocity of 3mm/min to
ensure no dynamic effects were present. It was demonstrated how such a system is well suited to applications where
space or volume restrictions are an important design consideration without compromising energy absorbing requirements.
A related work [12] modified the ‘In Plane’ system by rotating the central tube 90 degrees to define an ‘Out of Plane
System’. In doing so, the force-deflection response changed from a non-monotonically increasing response to one that
increased monotonically without sacrificing its energy absorbing capabilities. The quasi-static lateral compression was
achieved using three different devices such as flat rigid platens, cylindrical and point-load indenters and their
corresponding force-deflection response were compared.
Morris et al [13] also studied nested tubes crushed laterally between rigid platens at two different velocities. The first
category of energy absorber consisted of an ‘In Plane’ system; the second device consisted of an ‘Out of Plane’ system.
Material used was cold finished, drawn over mandrel (DIN 2393 ST 37-2) mild steel. The Cowper-Symonds relation was
used to predict the dynamic yield stress of the rings and this was included in the FE material model.
It can be seen that extensive research has been conducted by the aforementioned authors in the study of both singular
tubes and externally stacked nested systems subjected to lateral loading. However it appears that no investigation has
been conducted on the quasi-static or dynamic crushing of an internally stacked system in which a cluster of circular are
assembled as described by Morris et al [11]. Therefore, this paper is an attempt to consider the effectiveness of this new
form of nested system as an energy absorber. In addition, the paper demonstrates how such systems can be optimised so
that a desirable force – deflection response can be obtained by introducing a simple mechanism (CIPDS). Experimental
analysis of these devices was achieved with the aid of a universal materials testing machine. The loading and response
was analysed via the finite element method and the predicted force-deflection response was compared against that
measured experimentally.
EXPERIMENTAL COMPRESSION PROCEDURE OF THE CIRCULAR TUBE ENERGY ABSORBERS.
Eighteen tube pieces of length 25mm were cut from three different tube stocks of outside diameter 127mm, 101.6mm and
76.2mm respectively. Two different energy absorbers were experimentally analysed (CIPSS and the CIPDS) with three
test specimens representing each, resulting in a total of six samples to be tested under quasi-static conditions. Each test
specimen consisted of a 127mm O.D., 101.6mm O.D. and a 76.2mm O.D. cut tubular piece. Figure 1 illustrates a
schematic of the CIPSS (Circular In Plane Standard System) and the CIPDS (Circular In Plane Damped System) which is
the optimised design compressed laterally by a rigid indenter.
The quasi-static testing of the respective samples were carried out on Instron Model 4204 testing instrument which was
used for measuring the mechanical properties of materials. The maximum capacity of the loading frame attached to the
table mounted unit was 50kN. This loading frame consisted of two vertical lead screws, a moving crosshead and an upper
and lower bearing plate which bears the load of the lead screws. Various parameters such as loading, displacement, strain
and energy can be calculated.
Typically, all specimens were loaded in compression by placing them between the mounting table and a custom built
platen attached to the moving crosshead. The custom built platen was a square plate of dimensions 160mm by 160mm
and thickness 15mm. This platen was made of high carbon steel and was used to compress the said specimens laterally
in the Instron Testing machine. To stimulate quasi-static conditions and to ensure no dynamic effect were present; a
velocity of 5mm/min was applied to the moving crosshead of the Instron machine. Velocities between 0.5mm/min and
15mm/min have been applied by various researchers [14,15,16,17,18] in the quasi-static lateral compression of tubes
between various indenters.
Figure 1. Schematic illustrating the CIPSS and CIPDS compressed laterally by a rigid indenter.
MATERIAL PROPERTIES.
The tubes used in this work were made of mild steel which was cold finished, drawn over mandrel by the manufacturer
according to DIN standards (DIN 2393 ST 37.2) and containing approximately 0.15% carbon. Between five and seven dog
bone samples for each of the three different sized tubes were analysed in order to ensure consistent results. The dog
bone samples were machined from cut-out specimens obtained from the acquired tube stock. The true static stress-strain
curve was obtained using a tensile test based on ASTM standards. Since the tubes were cold worked by the tube
manufacturer, the true stress - strain curves obtained from the samples exhibited a deformation characteristic in which
necking occurred immediately after yielding followed by a geometrical softening stage. This is to be expected since the
cold rolling process, in addition to increasing the yield strength, decreases the ductility due to the concentrated
dislocations in the material.
Therefore, it was decided to approximate the material property of the three tubes using a bilinear stress-strain curve. In
doing so, the yield stress of 470MPa was carefully obtained from the experimental stress-strain curve at 0.2% strain. A
very small non-zero value of 1500 MPa for the plastic modulus was assigned to closely approximate the plastic portion of
the stress-strain curve. It should be noted that slope (plastic modulus) value of 1500MPa reasonably approximates the
plastic portion of the stress-strain curve in Figure 2 from the point of yielding. The yield stress is validated according to DIN
standards which claim the yield stress of this material to be within the range of 450 MPa to 525 MPa.
Stress [N/mm2]
True Stress-Strain
450
300
True Stress -SP1
True Stress -SP2
150
True Stress -SP3
0
0
0.02
0.04
Strain [mm/mm]
0.06
0.08
0.1
Figure 2. A true stress - strain curve obtained from three tensile specimens.
EXPERIMENTAL RESULTS AND DISCUSSION.
Figure 3 shows the response of a CIPSS (Circular In-Plane Standard System) compressed under quasi-static conditions.
Three specimens were tested to represent this type of absorber. As can be seen, results for each sample were identical.
For this system there was an initial gap of approximately 17mm and 19mm between the tubes before crushing was
initiated. These two gaps allowed all three components to deform sequentially as loading proceeded, hence the reason for
the non-monotonic rise in force throughout the deformation stroke as illustrated by point A in Figure 3. Also illustrated in
this figure are the stages at which each tube began to yield in series (point B) and how the whole systems strain hardened
from approximately 41mm deflection. It can be seen that the energy response becomes linear from this stage of deflection
for the remainder of the displacement stroke. The initial and final stages of deformation for this CIPSS are illustrated in
Figure 4.
Experimental Results - CIPSS.
8.00E+05
2000
Compressive force [N/m]
1600
6.00E+05
1400
5.00E+05
1200
4.00E+05
B
3.00E+05
2.00E+05
Force - test1.
1000
Force - test2.
800
Force - test3.
600
Energy - test1.
A
Energy - test2.
1.00E+05
400
Energy absorption [Nm/Kg]
1800
7.00E+05
200
Energy - test3.
0.00E+00
0
20
40
60
80
100
0
120
Displacement [mm]
Figure 3. A typical force, energy-deflection response for a CIPSS.
Figure 4. Initial and final stages of compression for a CIPSS.
ANALYSIS OF THE CIRCULAR IN-PLANE DAMPED SYSTEM (CIPDS).
The force and energy response of a CIPDS (Circular In-Plane Damped System) which is an optimised design is shown in
Figure 5 with the initial and final stages of displacement for the three samples in Figure 6, Figure 7 and Figure 8. Upon
examination of these figures, there exist two cylindrical spacers inserted between the gaps of the three tubes. These
spacers served two purposes. Firstly, to dampen out the abrupt rise in force as contact was establish between tubes as
deformation proceeded, resulting in a rectangular shaped force-deflection response and secondly, to reduce the rate of
strain hardening due to the radius of curvature of the spacers. With reference made to [10] in which the author illustrated
that varying the radius of the indenter will change the force-deflection response of a tube under lateral compression. From
this, it can be noted that the radius of curvature of the spacers inserted between the three tubes in this work serve as an
intermediate condition between the limiting cases of a flat plate and point load-indenter. This generated a force-deflection
response that is approximately rectangular in shape as shown in Figure 5.
Upon observation of this figure, there is a slight rise in force at approximately 60mm displacement for the three samples;
this was due to the ‘bottoming out’ of each sample. This can be avoided by simple applying a slightly shorter displacement
stroke. As a result of obtaining this rectangular shaped response, the corresponding energy absorption was quite linear for
the entire deflection stroke. Note that for Sample 1 (Figure 6), how the deformation process has become non-symmetric.
This is due to the sensitivity of geometric imperfections within the structure. Such imperfections may comprise of how
accurate the tubes and spacers are placed centrally with respect to one another. Although upon observation of the initial
stages prior to deformation the structure appears symmetric to the naked eye. As deformation proceeds, the spacers will
shift in the direction of least resistance resulting in a non-symmetric mode of deformation.
In an attempt to counteract this problem, a mild steel dowel was placed into the structure connecting the upper halves of
the tubes and the two spacers. A spot weld was used to fuse both ends of the dowel to the structure and in doing so it was
hoped that some symmetry would be retained as the absorber was compressed; it can be seen that this was achieved as
depicted in Figure 7 and Figure 8. Despite the asymmetric behaviour of Sample 1 during lateral deformation, it appears
that its corresponding force-deflection result still exhibits a desirable response. Note how the energy absorption of each
sample is quite linear as a result of their corresponding rectangular force-deflection behaviour.
Experimental results - CIPDS.
5.00E+05
1200
4.50E+05
Compressive force
[N/m]
3.50E+05
800
3.00E+05
2.50E+05
600
Force - test1.
Force - test2.
Force - test3.
Energy - test2.
Energy - test1.
Energy - test3.
2.00E+05
1.50E+05
1.00E+05
5.00E+04
400
Energy absorption
[Nm/Kg]
1000
4.00E+05
200
0.00E+00
0
0
10
20
30
40
50
Displacement [mm]
60
70
80
Figure 5. A typical force-and energy-deflection response for a CIPDS.
Figure 6. Initial and final stages of compression for a CIPDS. [Sample 1]
Figure 7. Initial and final stages of compression for a CIPDS. [Sample 2]
Figure 8. Initial and final stages of compression for a CIPDS. [Sample 3]
The authors Morris et al [19] conducted an impact analysis using both experimental and numerical methods on both the
CIPSS and the CIPDS. It can be seen from Figure 9 and Figure 10 that a similar mode of deformation response occurred
in comparison their quasi-static counterparts shown in Figure 3 and Figure 5 respectively. In addition, Figure 10 and
Figure 12 show the dynamic displacement evolutions of the CIPSS and CIPDS respectively subjected to a drop weight.
The authors concluded that the desired rectangular force-deflection response of the CIPDS can be maintained when
subjected to a dynamic loading condition. A more detailed description of the dynamic analysis on these energy absorbers
which used both experimental and numerical techniques can be obtained from reference [19].
Filtered - Unfiltered data. SP 5
6000
Force [N]
5000
4000
Frame 10.
3ms
3000
2000
CIPSS - Filtered Data.
Frame 14
7ms
1000
CIPSS - Unfiltered Data.
0
0
0.005
0.01
Time [s]
0.015
0.02
Figure 9. Sample 5: Filtered-unfiltered data. [19]
Frame 1: 0ms.
Frame 10: 3ms.
Frame 14: 7ms.
Frame 34: 19ms.
Figure 10. Experimental displacement evolution of sample 5. [19]
Force [N]
Filtered - Unfiltered data. SP 8
7000
6000
5000
4000
3000
2000
1000
0
Frame 14.
7ms
Frame 10.
3ms.
0
0.002
CIPDS - Unfiltered Data.
CIPDS - Filtered Data.
0.004
0.006
0.008
0.01
0.012
Time [s]
0.014
0.016
0.018
0.02
Figure 11. Sample 8: Filtered-unfiltered data. [19]
Frame 1: 0ms.
Frame 10: 3ms.
Frame 14: 7ms
Frame 29: 19ms
Figure 12. Experimental displacement evolution of sample 8. [19]
NUMERICAL PROCEDURE.
For the numerical analysis of these energy absorbers under static loading, ANSYS, an implicit finite element code was
used to simulate the quasi static responses. Since the loading rate was in the static range and no dynamic effects were
present, the implicit version of the code was deemed a suitable choice. The numerical models contain three non-linear
phenomena and are as follows: Material non-linearity in which a bilinear isotropic hardening material model was used to
capture the strain hardening effects in the plastic stages of deformation. The values for the yield stress and plastic
modulus as outlined in the material properties section were incorporated into this material model. It should be noted that a
bilinear material model was commonly used by various researchers, Ruan et al [20], Wu and Carney [21], Jing and Barton
[22], Kadaras and Lu [23], Reid and Harrigan [24,25]. Good agreement was found by the researchers between the
numerical results and those obtained by experiments.
An augmented Lagrangian penalty method defines the contact algorithm used to capture the models changing contact
status throughout the deformation stroke. This was achieved using surface to surface contact pairs. The various indenters
used to compress the systems were defined as rigid bodies and constrained to translate vertically over a predefined
displacement. Finally, the third non-linearity being large strain-large deformation, this feature must be included since
changes in volume will occur due to the large displacements that are applied. Elements used were 3-dimensional 8 node
brick elements which have large strain, large deflection and plasticity capabilities. Due to symmetrical loading conditions,
only one quarter of the energy absorbers were created and the resulting reactive forces were increased by a factor of four
for analysis.
NUMERICAL RESUTLS EVALUATION OF THE CIPSS AND THE CIPDS.
The numerical and experimental force, energy-deflection response of a CIPSS is shown in Figure 13. A slight under
prediction of the collapse load is observed for each of the three tubes followed by a slight over estimation from
approximately 70mm of displacement. The numerical energy response is also less in magnitude than that observed in
experiment since this quantity was derived from the integral of the force over its displacement stroke. The initial and final
displacement plots produced by ANSYS are depicted in Figure 14 which can be compared with that in Figure 4 and
likewise for the remaining absorbers to be analysed.
The numerical response of a CIPDS is illustrated in Figure 15 with the corresponding displacement plots in Figure 16. It
appears the numerical code under predicted the force to a greater a degree than for the CIPSS, however this under
prediction was a culmination of force since the tubes were been displaced at the same time. A linear strain hardening
response by the numerical method was observed for the remainder of the displacement stroke. The numerical results for
the energy absorbers described previously have shown reasonable agreement with those of experiment considering that
only a simple bilinear material model was used to represent elastic-plastic stress-strain curve for mild steel. The overprediction in the later stages of deformation observed earlier is due to the material model used which assumes the stress
to increase indefinitely as the strain increases. This implies that when large deformations ensue and hence large strains,
the resulting force and corresponding stress will tend to be larger than actual observed values.
Figure 17 illustrates the convergence plot of three different mesh densities. It can be seen that a convergence solution
was achieved when three or five elements were used to mesh the thickness of the tubes. Hence all subsequent models
consisted of a mesh density involving three elements through the thickness of each tube as represented by the inlay in
Figure 17.
Numerical and Experimental results - CIPSS.
2500
Experimental force - test1.
1.00E+06
Numerical force.
2000
Experimental energy - test1.
8.00E+05
1500
Numerical energy.
6.00E+05
1000
4.00E+05
500
2.00E+05
0.00E+00
0
20
40
60
80
Displacement [mm]
100
0
120
Figure 13. Experimental and numerical comparison of the CIPSS.
Figure 14. Initial and final stages of displacement of a CIPSS as produced by ANSYS.
Energy absorption
[Nm/Kg]
Compressive force
[N/m]
1.20E+06
Experimental Results - CIPDS.
1200
4.50E+05
1000
3.50E+05
3.00E+05
800
2.50E+05
600
2.00E+05
Experimental force - test1.
Numerical force.
Experimental energy - test1.
Numerical energy.
1.50E+05
1.00E+05
5.00E+04
0.00E+00
0
10
20
30
40
50
Displacement [mm]
60
70
400
200
Energy absorption
[Nm/Kg]
Compressive force
[N/m]
4.00E+05
0
80
Figure 15. Experimental and numerical comparison of the CIPDS.
Figure 16. Initial and final stages of displacement of a CIPDS produced by ANSYS.
Force [N]
Mesh refinement - Convergence plot.
5879 elements 5 elements through thickness.
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
4553 elements 3 elements through thickness.
1649 elements 1 element through thickness.
0
1000
2000
3000
4000
Quantity of Brick Elements
5000
6000
Figure 17. A force convergence plot of three different mesh densities.
CONCLUSION.
Experimental and numerical investigations have been made into the static lateral compression of circular tube energy
absorbers with the aid of rigid flat platens. For the CIPDS it was possible to maintain symmetry using a simple mechanism
consisting of steel dowel to connect the two cylindrical spacers inserted between the tubes.
The numerical code via ANSYS reasonably predicted the quasi-static force-deflection response of the CIPSS and CIPDS
using a simple bilinear material model. An optimised force deflection response has been obtained. The optimised energy
absorber CIPDS are seen as an improvement on the CIPSS since the insertion of cylindrical rods caused the crushing
force to be relatively constant once the collapse load was reached, which is a desirable feature in the design of energy
absorbers.
Such energy absorbers may find application where it is subjected to a compressive load under impact with the aim of
bringing the moving mass to a controlled stop. Such practical cases may consist of energy absorbers in the aircraft,
automobile and spacecraft industries, nuclear reactors, steel silos and tanks for the safe transportation of solids and
liquids.
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