Parametric design of multi-cell thin-walled structures
for improved crashworthiness with stable progressive
buckling mode
Daneesha Kenyona,∗, Yi Shua, Xingchen Fana, Sekhar Reddyb, GuangDongb, Adrian J. Lewa
aDepartment of Mechanical Engineering, Stanford University, Stanford, CA 94305,United States
bTesla, Inc., Palo Alto, CA 94304, United States
Abstract
Thin-walled single and multi-cell structures are an ongoing topic of interestin the field of crashworthiness, due to their wide range of applications in au-tomotive and aerospace industry as lightweight energy-absorbing structuresin crash environments. This work presents a new five-cell cross-section thatmerges high performance multi-cell and twelve-edge cross-sections from pre-vious research, and compares its performance to four- and nine-cell squarecross-sections. Super Folding Element (SFE) theory and Finite ElementAnalysis (FEA) in LS-DYNA are used to analyze cross-sections and found tohave good agreement. The LS-DYNA environment is validated with physicaltesting. The geometry of the cross-sections is varied in order to find maximalvalues of the performance parameters specific energy absorption (SEA) andcrush force efficiency (CFE) under stable progressive buckling mode andconstraints for manufacturability. The nine- and five-cell cross-sections ulti-mately out-perform the four-cell cross-section, with the nine-cell having thehighest SEA and CFE, though the five-cell design has a significantly lower(47%) mean crush force (Pm) for only an 11% and 14% loss in SEA and CFErespectively. As a final refinement, the geometry was varied across these twohigh-performing cross-sections to create equivalent mean crush forces to thefour-cell cross-section, which showed the five-cell cross-section to have an im-
∗Corresponding authorEmail address: [email protected] (Daneesha Kenyon)
Preprint submitted to Thin-Walled Structures June 21, 2018
proved SEA and better mass efficiency over the nine-cell under a mean crushforce constraint.
Keywords: Multi-cell, Crashworthiness, Energy absorption, Thin-walledstructures, Axial crushing, Collapse modes
1. Introduction1
Thin-walled single and multi-cell structures have generated significant2
interest due to their high energy-absorbing characteristics, low weight, in-3
expensive manufacturing, and crashworthiness applications within the auto-4
motive and aerospace industry. In particular these tube-shaped structures5
have been implemented widely in the structural frame of vehicles as frontal6
impact energy absorbers. The value of these structures as energy absorbers7
was first explored by Alexander [1] as well as Pugsley and Macaulay [2] and8
Magee and Thornton [3]; Alexander in particular proposed the first theo-9
retical model for characterizing the plastic collapse of thin-walled structures10
with one folding wave and stationary hinge line in [1]. Alexander’s theory11
was expanded by Abramowicz and Jones [4] and by Weirzbicki and Bhat12
[5, 6] which added moving hinge lines to the model. Abramowicz expanded13
on this theory by also introducing the concept of effective crushing distance14
[7]. Much of this early research focused on the collapse of circular tubes,15
which was later expanded to square tubes through the work of Abramowicz16
and Jones [4, 8, 9]. The addition of dynamic effects to this model was later17
developed by Hanssen [10]. Notably, these folding theories and subsequent18
models rely on the observed behavior of stable progressive buckling, whereby19
a structure undergoes periodic “folding” in order to maximally absorb energy.20
The next theoretical advancement, Super Folding Element (SFE) theory,21
was proposed by Abramowicz and Weirzbicki in [11, 12] and advanced in [13].22
This theory, which combines concepts from plasticity with moving hinge23
line collapse theories, is foundational for modern theoretical models used24
to predict the collapse parameters (mean crush force, energy absorption,25
and folding wavelength) of structures with geometries consisting of single-26
cell cornered cross-sections (e.g. square, hexagonal). Notably, this theory27
was limited to two-flange corner elements, until expanded by Najafi and28
Rais-Rohani to include three flange “T” shaped elements meeting at various29
angles; this approach was based on observed behavior in numerical simulation30
[14, 15]. A simplified folding element model using a reduced number of energy31
2
mechanisms from standard SFE was proposed by Chen [16] and expanded by32
Zhang [17] in order to also model “T” and “criss-cross” elements with three33
or four-flange elements. Preliminary work from Zheng [18] also presented34
SFE models for “T” and “criss-cross” elements, as well as derived equations35
to support variable wall thicknesses. The work by Chen, Zheng, Zhang, and36
Najafi has enabled the advanced modeling of more geometrically complex37
cross-sections consisting of multiple cells.38
As the theory to predict the behavior of multi-cell cross-sections has39
advanced, so have the processes that enable the manufacture of multi-cell40
structures. Early research was limited to welded steel square and “top-hat”41
cross-sections as well as extruded shapes [4, 8, 9, 19, 20, 21], though later42
works looked at more complex welded cross-sections [22]. More recently43
the ability to extrude geometrically complex single and multi-cell aluminum44
structures has increased interest in refining the design of these more complex45
structures. This manufacturing process also provides some limitations on46
the design space, as, for example, conventional machining guidelines do not47
recommend extruding aluminum shells with wall thickness less than 1 mm48
(0.040 in) [23]. Furthermore, more complex geometry like multi-cell cross-49
sections tend to have further restrictions on wall thickness in order to allow50
for proper flow of material. Cross-sections that utilize variable wall thickness,51
either axially or laterally, have been proposed and tested as in [18, 24] using52
either tailor rolled blank (TRB) technology as initially proposed in [25], or53
Electrical Discharge machining (EDM). While TRB technology is promis-54
ing and opens new avenues for design, it is limited in application to single-55
celled cross-sections at the moment. EDM, however, can produce multi-celled56
cross-sections, but is used primarily as a prototyping method rather than a57
mass manufacturing method, rendering it less suitable for automotive ap-58
plications. Comparatively, extrusion is a widely available process common59
to mass manufacturing, capable of producing both single and multi-celled60
designs, and therefore a desirable process to use for the immediate imple-61
mentation of new structural elements. Fang considered the use of extrusion62
to produce cross-sections with functionally graded thickness in [26], but was63
unable to produce cross-sections with non-uniform wall thickness, resulting64
in the inability perform physical testing on high-performing cross-sections65
found through numerical simulations. When evaluating these structures for66
applications in automotive collisions, manufacturing presents constraints on67
what cross-sections can be practically implemented and should be duly con-68
sidered.69
3
However, in addition to advancing manufacturing techniques, more work70
has been done on generally improving the comparative performance of thin-71
walled energy-absorbing structures. Early investigation into designing op-72
timal multi-cell aluminum cross-sections by Kim [27] focused on improving73
single and multi-cell structures by maximizing specific energy absorption74
(SEA). Kim focused on single, double, triple, and four-cell cross-sections,75
as well as a new five-cell cross-section with square corners, and compared76
them to demonstrate the superior performance of multi-cell cross-sections.77
Furthermore, Kim addressed the problem of inducing the stable progres-78
sive buckling mode addressed by previous theoretical models through the79
introduction of triggers. Tang [28] proposed even more complex cylindrical80
multi-cell cross-sections and again demonstrated the improvements over con-81
ventional square and circular cross-sections. Chen and Masuda expanded82
this work to include multi-cell hexagon sections [29]. Additional compara-83
tive work on multi-cell cross-sections was performed by Song in [30], who84
also produced windowed structures to reduce peak force. Wu [31] expanded85
upon Kim’s work in optimizing five-cell cross-sections, and demonstrated86
their good performance in comparison to four-cell and single-cell structures87
through numerical simulation and physical testing. Chen introduced a fur-88
ther variation on five-cell cross-section geometry by combining circular cor-89
ners and orthogonal internal webs [32], noting that corner cell geometry can90
have a large impact on the structure performance. Alternate optimization91
techniques have also been used to develop the relationship between common92
crashworthiness performance parameters and structural geometry, such as93
shell thickness and cross-section width, as can be seen in [31, 33, 34]. Fur-94
ther works explored number of cells in multi-cell cross-sections for square,95
hexagonal, and hierarchical honeycomb structures under axial and oblique96
loading cases [35, 36, 37], with the general finding that multi-celled cross-97
sections outperform single-cell cross-sections, and higher numbers of cells also98
improve results. Topology optimization using finite element analysis and var-99
ious algorithms was also performed on square and hexagonal cross-sections in100
[38, 39, 40, 41], under axial, lateral, and oblique loading conditions, though101
the resulting structures were not compared with SFE models or physical tests102
or evaluated for manufacturability.103
While research into multi-cell cross-sections progressed, Abbasi and Reddy104
investigated higher-order single-cell structures in [42, 43]. These works com-105
pared the performance of square, hexagonal, octagonal, and a newly intro-106
duced twelve-edge cross-section through SFE modeling, numerical simula-107
4
tion, and physical testing. Furthermore, the reliance of the stability of the108
buckling mode on corner angle was investigated, and bounds for stability109
were defined based on numerical simulation. Generally, single-cell cross-110
sections with more corners tend to perform better [44], and the twelve-edge111
cross-section reflected that result by out-performing other tested single-cell112
cross-sections with fewer corners. The twelve-edge cross-section was further113
studied by Sun in [45], under the name “criss-cross configuration”, where114
further parametric studies were performed and the effect of rounding corners115
with spline curves was explored. In these works, the twelve-edge cross-section116
buckling mode was found to be strongly influenced by corner geometry.117
This work proposes a new cross-section synthesizing the findings from118
Abbasi and Reddy [42, 43] as well as the numerous works on optimal five-cell119
cross-sections [18, 27, 31, 32, 36, 40] by introducing a five-cell multi-corner120
cross-section that adds four connecting webs to the twelve-edge cross-section121
investigated by Abbasi and Reddy. This new cross-section is compared to a122
previously studied nine-cell square cross-section that was principally investi-123
gated by Zhang and others [14, 17, 18, 36, 39], as well as a four-cell model that124
has been studied in numerous works on multi-cell geometry [17, 28, 31, 33].125
These three cross-sections are analyzed using a combined approach for mod-126
eling through Super Folding Element theory and LS-DYNA. Additional in-127
vestigations into the “criss-cross” corner element SFE model are made, given128
the preliminary work in [18], and shown to have reasonable accuracy com-129
pared to the LS-DYNA models. Physical testing of the four-cell cross-section130
is performed to establish baseline confidence in LS-DYNA simulation envi-131
ronment.132
The four-cell, nine-cell, and five-cell cross-sections of interest are analyzed133
through a parametric sensitivity study and evaluation of the SFE model,134
with the goal of distinguishing high-performing cross-sections. Identifying135
the buckling mode in the sensitivity study in LS-DYNA was critical, given136
that the performance parameters change monotonically with variation in137
geometry and high-performing cross-sections are found at the boundary of138
the stable buckling region, rather than on the interior of the design space.139
Through the parametric sensitivity study, the buckling mode transition point140
from stable progressive collapse to global bending [46] was used in this way141
as a key limiting parameter, given that energy-absorbing performance de-142
creases drastically after this transition point. Notably, this transition point143
cannot be predicted for multi-cell structures using SFE models, and is gen-144
erally demonstrated using simulation or physical testing, such as that done145
5
by Abramowicz and Jones in [47] for square and circular steel columns. In146
this study, the buckling transition point was mapped and defined for all147
cross-sections of interest, providing an upper-bound on potential geometries.148
Manufacturing, as stated previously, is a further limitation upon the poten-149
tial design space, and provided a lower bound for some parameters, such as150
thickness of the cross-section. With the design space appropriately bounded,151
comparison of performance within the regime of stable progressive buckling152
became possible. Thus, the four-cell, nine-cell, and five-cell cross-sections153
were compared in order to determine the highest performing structure based154
on current standard metrics within the field, such as crush force efficiency155
(CFE) and specific energy absorption (SEA).156
Finally, in order to prepare these structures for practical application in157
automotive industry, a further refinement study was performed based on158
mean crush force. Given that automotive structural frames are designed to159
undergo a targeted mean crush force depending on their size (e.g. a sedan or160
a truck), this parameter can provide greater insight into design performance161
than SEA and CFE alone. By targeting a specific mean crush force, a162
proposal for both nine-cell and five-cell cross-sections was developed within163
the stable buckling regimes mapped in the parametric sensitivity study. This164
method demonstrates the advantages and disadvantages of both the five-cell165
and nine-cell cross-sections as components in a vehicle body frame, and allows166
for appropriate selection of a structure to meet practical automotive design167
constraints.168
2. Problem description169
To address the design of these multi-cell structures, some basic terminol-170
ogy will be introduced in the following section, in addition to the constraints171
and the performance metrics of this design space. One of the most critical172
factors in identifying a successful structure is that the correct buckling mode173
occurs during collapse; an ideal structure will naturally exhibit, at various174
impact speeds, the desired mode of progressive top-down buckling behavior.175
When considering the buckling mode, there are a number of qualitatively176
different observable behaviors for which the following terms are used: stable,177
transitional, and unstable buckling. Stable buckling is the desired form of178
progressive top-down buckling. Unstable buckling is seen when alternate,179
undesirable buckling modes appear, typically global bending as can be seen180
in Figure 1. Finally, transitional buckling is when a structure begins to181
6
Figure 1: Examples of different buckling modes. The buckling of the undeformed structureon the right can be stable (left), transitional (center left), or unstable (center right).The center left structure bowed slightly yet still buckles progressively, whereas the stablestructure exhibits no lateral bending and the unstable structure exhibits no top-downbuckling. Geometric parameter length (L) shown for convenience on undeformed structure.For later reference, the wall impacting the structure to emulate a crash event is also shown.
bend globally, but recovers to a progressive, stable buckling mode; struc-182
tures with this mode define the transition point between the stable and un-183
stable buckling modes. Ultimately in this design space, only cross-sections184
exhibiting stable buckling are deemed acceptable. Cross-sections that ex-185
hibit predictable buckling regimes from stable, to transitional, to unstable186
buckling are also desirable.187
Therefore, the first consideration when approaching these structures is188
whether the necessary buckling mode is observed. Following that, the per-189
formance of a particular cross-section can be fine tuned depending on the190
constraints in effect in order to improve the final performance. In this work,191
the design space is constrained to three particular gross cross-sectional ge-192
ometries: four-cell, five-cell, and nine-cell. A maximum space coupon and193
the material were determined based on existing industry applications. In or-194
der to optimize the performance of these three cross-sections with the given195
space coupon and material, the geometric factors of wall thickness (t), cell196
width (C), and corner angle (φ) were varied. These geometric parameters197
of interest, as well as the cross-sectional geometries, are detailed in Figure198
7
2. The cross-section width (w) is also shown in Figure 2 for reference, which199
is a variable dependent upon C. The constraint on maximum space coupon
Figure 2: Four-cell cross-section (left), five-cell multi-corner cross-section (center) andnine-cell cross-section (right) shown with parameters of interest labeled.
200
was set as follows: each cross-section was limited to a maximum width (w)201
of 114 mm (measured from wall centerline) and an initial length (L) of 450202
mm (parameter L is shown in Figure 1). One further constraint on geometry203
is the manufacturing process: in this case, extrusion. For this type of cross-204
section, obtained by extrusion, a minimum wall thickness for aluminum is205
1mm [23]. Furthermore, the ratio of cell width (leg length) to wall thickness206
is also significant to manufacturability, again limiting the design space [23],207
though this constraint was not considered when varying parameters. These208
constraints on manufacturability, while limiting, enable the production of a209
design feasible for immediate implementation in industry, as extrusion is a210
technologically mature and widely adopted process in mass manufacturing.211
Finally, with the buckling mode and constraints considered, performance212
parameters based on the variation of geometric parameters were studied.213
Wall thickness, cell width, and corner angle (where relevant) were varied214
as a part of a parametric study in order to observe their effect upon the215
performance parameters of specific energy absorption (SEA) and crush force216
efficiency (CFE). SEA is defined as the total amount of energy absorbed217
during the crushing process via plastic and elastic deformation (E) over the218
mass of the structure (m)219
SEA =E
m. (1)
8
The second parameter, CFE, is defined as220
CFE =Pm
Pmax
, (2)
where Pmax is the peak force experienced and Pm is the mean (time-averaged)221
force experienced during crushing after the initial peak force. Notably, a222
structure undergoing progressive buckling tends to experience a high initial223
force, visible as a sharp spike on a force-displacement diagram, before pro-224
ceeding to periodic folding behavior. Mean crush force excludes this initial225
peak behavior, in lieu of late periodic behavior, as can be seen in Figure 3.
Figure 3: An example diagram of forces experienced by a structure during the crushingprocess with mean and peak forces labeled.
226
A crucial parameter that affects the buckling behavior of the structure is227
the aspect ratio C/t, which has been used to develop predictions of the be-228
havior of various structures [46, 47]. Given that observing the desired stable229
(progressive top-down) buckling behavior is a principal concern, identifying230
successful aspect ratios is an important design consideration. Therefore, the231
9
performance parameters were further related to the aspect ratio (C/t) as a232
part of this study.233
An ideal structure in this crushing environment would have a high SEA, a234
CFE of 1, and an aspect ratio that corresponds to the correct buckling mode.235
Maximizing SEA for structures results in an efficient lightweight design that236
absorbs the maximum amount of energy for the amount of material used,237
and serves as a good metric for comparing different cross-sectional designs238
with the same outer dimensions. Furthermore, in regards to CFE, the initial239
peak force experienced during crushing is the principal deceleration felt by240
the passenger in a vehicle, followed by lower amplitude periodic decelerations241
as the structure progressively folds. If the peak force is closer to the mean242
force, high initial decelerations are not experienced by the passenger, and the243
CFE is 1.244
Some studies, in order to ensure the correct mode of buckling is observed,245
add “triggers” to their test coupons or numerical simulations, usually in the246
form of small beads or indentations at the free end of the hollow structure247
to which load is applied. These triggers, while introducing more reliable248
folding behavior, also reduce the peak force experienced during the crushing249
process. Increased reliability and decreased peak force are desirable features250
in automotive industry, which has popularized triggers as a structural feature.251
Triggers can also offset structural defects generated during manufacturing252
that might otherwise change the folding behavior of a sample and reduce253
attrition at mass manufacturing scale. However, while triggers may improve254
the performance of a cross-section, it is still valuable to study the natural255
(un-triggered) behavior of a new cross-section to understand what factors256
influence its buckling behavior. Structures that buckle progressively without257
triggers will continue to present this behavior with triggers added. While258
triggers would be added to the proposed structures in this work at a later259
design stage, this study only considers the natural, untriggered modes of260
buckling of structures in both simulation and physical testing.261
3. Approximate analysis of multi-cell structures262
3.1. Super Folding Element Theory263
Super Folding Element (SFE) theory is a method for estimating the mean264
crush force and total energy absorbed by a thin-walled structure when it265
undergoes a crushing process. Readers unfamiliar with the details of the266
theory can consult [11, 13]. In the following section the main ideas behind267
10
SFE are explained and introduced. SFE theory assumes that the overall268
buckling mode is stable buckling as described in Section 2. Stable buckling269
involves periodic regular folding of the structure, which SFE theory takes270
advantage of by first assuming that the regular folds in the structure will271
be identical to each other. Therefore, in order to define the mean force for272
an entire columnar structure, one only needs to determine the mean crush273
force of a characteristic fold, or segment along the length of the structure.274
This characteristic segment is further broken down into corner elements, as275
corners are the main energy-absorbing component and also determine the276
mean crush force. For example, a four-cell cross-section would be broken277
down into three types of corner elements: two-flange elements, T-elements,278
and criss-cross elements, as shown in Figure 4.279
Figure 4: A thin-walled four-cell structure decomposed into one criss-cross, four T, andfour two-flange corner elements.
These corner elements are modeled in terms of several energy-absorbing280
plastic deformation mechanisms, which are used in turn to compute the mean281
crush force of the characteristic segment. These mechanisms are sketched in282
Figure 5 and are explained in more detail in [11, 13]. Flanges modeled in283
SFE have uniform thickness (t), cover half of the cell width (C), and have284
height 2H, where H is the half folding wavelength. Notably, the initial angle285
between flanges of a corner element can be either acute, obtuse or orthogonal,286
allowing more complex cross-sections like hexagonal and octagonal to be287
modeled. In this theory all corners are sharp and fillets are not modeled.288
There are three unknown parameters used to determine mean crush force Pm:289
half folding wavelength H, rolling radius r and switching angle α. Since the290
11
Figure 5: A two-flange element in SFE theory: 1) Deformation of a toroidal surface (inex-tensional) 2) Bending along horizontal hinge lines (inextensional) 3) Rolling deformation(inextensional) 4) Opening of conical surfaces (extensional) 5) Bending along inclined andhorizontal hinge lines after the locking of traveling hinge lines (extensional) [13]
.
crush process is assumed to be defined by the principle of maximum energy291
dissipation, the unknowns can be determined through numerical optimization292
[11]. Angle α is a time-like variable indicating the progress of deformation,293
which increases from 0 to 90 degrees throughout the entire crush process for294
a single cross-section. There is a switching angle α∗ where the transition295
from asymmetric to symmetric mode occurs, and some changes to the rate296
of energy absorption occur, as further detailed in [11]. Energy absorption297
is determined using the bending moment and a series of derived kinematic298
integrals, as well as the flow stress of the material, as detailed in [7, 11, 43].299
The original SFE theory addresses the two-flange corner element shown in300
Figures 4 and 5. This work, and previous works, have expanded the model301
to include T- and criss-cross elements, which use the same energy-absorbing302
mechanisms of Figure 5 in different combinations.303
12
3.2. Super Folding Element theory with criss-cross elements304
The original SFE theory models a two-flange corner element as shown305
in Figure 5 [11]. As previously stated, this theory has been extended to in-306
clude T-shaped elements in [14, 15]. The extension to criss-cross elements307
had been largely limited to a version of the SFE theory consisting of only308
extensional triangular elements and stationary hinge lines as energy absorb-309
ing mechanisms, rather than including other mechanisms from SFE theory310
such as toroidal surfaces and moving hinge lines [16, 17]. More recently, [18]311
presented preliminary work that regards a criss-cross corner element as two312
two-flange elements and used the standard two-flange SFE model for each.313
Given that there is little experience with the criss-cross corner element, we314
include a comparison between the SFE model and results in LS-DYNA.315
Investigation was performed by modeling a singular criss-cross element in316
LS-DYNA (See Section 4) and observing the energy mechanisms in a similar317
approach to [14, 15]. The energy mechanisms observed are highlighted in318
Figure 6. The joint deformation of both two-flange elements is accomplished319
by assuming that they have the same values for Pm, H, r, and α. Energy320
dissipation via stretching of the material near the centerline of the criss-cross321
element was ignored, as well as the compatibility of the displacements be-322
tween the two two-flange elements, as only a very simple model was desired.323
The results from this energy mechanisms survey were found to be consistent324
with the results of [18]. This model for a criss-cross corner element was com-325
pared with LS-DYNA simulations (see Section 5) and was shown to capture326
the basic kinematics of the deformation for at least some regimes (see Figure327
6).328
The summary of mechanisms to be considered, Najafi’s model for the T-329
shaped element [14, 15], and the summary of mechanisms observed for the330
criss-cross corner element are shown in Table 1. Given that the previous
Corner Ele-ment
ToroidalSurface
HorizontalHinge Line
InclinedHinge Line
ConicalSurface
Two-flange 1 2 2 2T 1 3 2 2Criss-Cross 2 4 4 4
Table 1: Folding mechanism breakdowns for various folding elements.
331
energy mechanisms from SFE theory can be used to describe the criss-cross332
13
Figure 6: Breakdown of folding mechanisms in a criss-cross element shown in isometricview (top) and top view (bottom). More detailed explanations of these mechanisms canbe found in [11, 13, 14, 15].
corner element, the equations and derivations of [11, 13] are applied in this333
work to compute results. Further supporting energy equation derivations by334
Najafi and Zheng can be found in [14, 15, 18] and are not supplied here.335
The two two-flange model of a criss-cross corner element enabled full336
SFE modeling of four-cell and nine-cell cross-sections, as well as the novel337
14
five-cell cross-section presented in this work. The number of criss-cross, two-338
flange, and T-shaped corner elements for each cross-section are detailed in339
Table 2. The elements interact with each other through common values of
Cross-Section Two-flange T Criss-CrossFour-cell 4 4 1Five-cell 8 0 4Nine-cell 4 8 4
Table 2: Element breakdowns for various cross-sectional geometries.
340
H, r, and α such that the overall mean crush force and energy absorption341
is determined by the particular combination of elements (and subsequent342
energy absorbing mechanisms) contained by a cross-sectional geometry. The343
optimization technique to find the maximum energy dissipation for a two-344
flange element can also be applied to other types of elements.345
The evaluated SFE model for criss-cross elements, as well as the more ex-346
tensively researched two-flange and T-shaped SFE models, were implemented347
in MATLAB in order to predict the performance of different cross-sections348
through variation of parameters. Predictions were compared with numerical349
simulations as shown in Section 5. Following this comparison, the SFE model350
was used to determine optimization trends and performance parameter im-351
provements with regards to geometric (thickness, cell width, corner angle)352
variations.353
4. Numerical model setup354
A numerical model was constructed using the nonlinear explicit finite el-355
ement software LS-DYNA, which is widely adopted by authors [17, 20, 31]356
in the numerical simulation of crushing of thin-walled structures. Boundary357
and loading conditions were chosen to reflect a typical physical testing envi-358
ronment of these structures, with the base of the structure fixed to a test bed359
and the top of the structure unconstrained. The unconstrained end of the360
structure is later impacted by a wall moving according to a designated veloc-361
ity profile, which collides with the structure to replicate a crash environment,362
see Figure 1. In simulation terms, the base of the structure is constrained363
across all degrees of freedom and unconstrained at the top. In turn, the mov-364
ing rigid wall compresses the structure in the axial direction beginning from365
15
the free, top end of the structure. The wall and the structure are always in366
contact both in the physical testing environment and the simulation.367
Current general practice for selecting a wall velocity is to either use a368
constant compressive velocity ranging from 1 m/s to 11 m/s [17, 29, 31] or a369
ramped profile [20]. However, quasi-static testing environments tend to use370
lower velocities, on the order of 1-2 m/s, to determine buckling behavior.371
Given that the material model for these simulations is aluminum, which is372
not sensitive to strain rate, no detrimental impact of these assumptions on373
the predictions of structural collapse has been found [17]. Therefore, for374
these simulations a ramped velocity profile was applied to the rigid wall,375
moving from 0 to 8 m/s in 80 ms with constant acceleration. This ramped376
velocity profile enables a more gradual loading of the structure than a direct377
impact, better approximating a quasi-static environment, while triggering378
the initial buckling behavior. Higher velocities using both constant velocity379
and constant acceleration profiles produced more stable progressive buckling380
rather than transitional or unstable behavior, and were not as instructive in381
differentiating cross-sectional performance. The chosen parameters (0 to 8382
m/s in 80 ms) were the most illustrative of buckling behavior, while producing383
computationally reasonable results within the LS-DYNA explicit modeling384
framework.385
The structure is modeled with a uniform cross-section, with no triggers386
added in order to observe the natural behavior of the structures. In this way387
structures that are more likely to exhibit stable progressive buckling can be388
identified, per Section 2. This structure is partitioned into fully integrated389
shell elements with hourglass control (SHELL 16), as also adopted in [42, 43].390
A mesh size of 4 mm × 4 mm was selected based on a convergence study,391
and is comparable to the mesh size used in other works [29]. Elements that392
have a negative volume are deleted as they arise.393
The constitutive model for the material of the structure is an elasto-plastic394
material with J2 plasticity and piecewise linear isotropic hardening (MAT 24)395
for an aluminum alloy. A Von Mises yield condition was used, as the material396
had very good ductility and no material fracture was observed in physical397
testing; therefore material failure conditions were not further considered.398
The wall is modeled as a rigid body with the material properties of steel399
to compute the contact penalty parameters. Contact phenomena within400
the structure is separated into two classes, self contact within the structure401
(automatic single-surface contact) and contact between the structure and the402
impacting wall (automatic surface to surface contact). The friction coefficient403
16
µk is chosen as 0.2 as suggested by Chen [29].404
In order to validate the numerical model setup, a comparison was made405
between physical test results and the LS-DYNA results. Quasi-static testing406
of three coupons was performed using procedures described in [43]. As this407
work studied cross-sectional designs without triggers, the physical models408
did not use triggers during experimentation either. A four-cell cross-section409
was used for the comparison, with parameters as follows: thickness (t) of410
2.1 mm, cell width (C) of 57 mm, and length (L) of 450 mm. SEA and411
mean crush force (Pm) were used as the parameters for comparison, with the412
Pm results normalized by dividing by the averaged mean crush force from413
physical testing. The results of this comparison are detailed in Table 3. The
SEA SEA Normalized Mean[kJ/kg] Error Crush Force
Physical TestingResults (Averaged) 22.96 N/A 1.00Simulation Results 24.53 6.8% 1.15
Table 3: Comparison of LS-DYNA and quasi-static physical testing results.
414
LS-DYNA results deviate from the physical testing results by about 7% and415
15%, which is an acceptable margin. The LS-DYNA environment detailed416
above was therefore considered sufficient for further study.417
5. Model comparison418
In order to test the performance of the SFE model for the criss-cross cor-419
ner element, a single criss-cross element was modeled in LS-DYNA, as shown420
in Figure 6. The results of this model were then compared to those from421
the SFE model. In particular, the predicted values of SEA and half folding422
wavelength H stemming from both models were compared. The criss-cross423
had a thickness of 3 mm, and a width (w) of 114 mm, as well as the veloc-424
ity and boundary conditions described in Section 4. The results from these425
comparisons are shown in Table 4. The SEA predictions are comparable and426
provide a reasonable estimate. The values of H are not matched as well. H427
was measured as the distance between the two horizontal hinges lines in the428
folded configuration in LS-DYNA assuming some overlapping of higher and429
lower flanges. The validity of this assumption needs further examination and430
only observations are reported here.431
17
Model SEA [kJ/kg] SEA Error H [mm] H ErrorSFE 22.51 N/A 33.1 N/ALS-DYNA 23.52 4.5% 40.2 17.6%
Table 4: Criss-cross model comparison.
To further compare the LS-DYNA and SFE models, the comparisons of432
SEA and H were performed for all cross-sections, as shown in Figure 7.433
These models used all types of corner elements described previously (two-434
flange, T, criss-cross) rather than a single isolated criss-cross element. Geo-
Four-Cell Five-Cell Nine-Cell0
5
10
15
20
25
30SFE and LS-DYNA Performance Comparison: t=3mm, w=114mm, =90deg (five-cell)
SE
A [kJ/
kg]
SFELS-DYNA
Four-Cell Five-Cell Nine-Cell0
5
10
15
20
25
30
35
40SFE and LS-DYNA Performance Comparison: t=2mm, w=114mm, =90deg (five-cell)
SE
A [kJ/
kg]
SFE
LS-DYNA
Figure 7: SEA (left) and H (right) comparison of SFE and LS-DYNA models for differentcross-sections, with listed geometry.
435
metric parameters (t, C, φ) had values indicated in Figure 7. The deviations436
for SEA are 4%, 8%, and 14% for the four-cell, five-cell, and nine-cell cross-437
sections respectively. These deviations are sufficiently small to suggest that438
it is possible to use both SFE and LS-DYNA for continued design space439
analysis. Furthermore, this model showed decent agreement for half-folding440
wavelength, varying in the range 10-23% across different cross-sectional ge-441
ometries. Notably, the nine-cell cross-section shows the greatest error com-442
pared to the other two models. One likely source of this error is in the SFE443
model’s prediction of asymmetric folding, whereas the LS-DYNA simulations444
show some symmetric folding in the nine-cell cross-section. This symmetric445
folding was not witnessed in the five- or four-celled cross-sections. Given446
that these folding modes have different energy-absorbing capabilities, some447
disagreement is expected as a result.448
18
The SFE extension chosen for the criss-cross element is very simple, con-449
sisting only of two two-flange elements put together, though these initial450
results are positive. The approach is approximate, and further refinement451
of the kinematics of the criss-cross SFE theory are needed, particularly at452
the centerline juncture of the two-flange corner elements. In actuality in LS-453
DYNA the centerline was shown to curve back and forward in an “S” shape,454
which should be addressed in future works as an alternate energy absorption455
mechanism.456
6. Parametric sensitivity study results and discussion457
Next, a series of sensitivity studies were performed by sweeping the458
parametrized design space and analyzing the performance parameters SEA459
and CFE. Initially, for each of the three cross-sections, a study using the460
SFE model was performed to determine general trends in behavior. SFE is461
used as a preliminary investigative tool, as the computation resources and462
time needed for this model are significantly smaller than running numerous463
LS-DYNA models. However, while the SFE model can demonstrate trends464
in SEA, it does not provide information about the buckling mode or CFE.465
These parameters can only be estimated from the LS-DYNA model. All466
numerical values computed from LS-DYNA simulations and depicted in the467
figures below are included in Appendix A.468
6.1. Nine-cell cross-section469
To begin, the nine-cell cross-section was analyzed. Parameters varied470
include wall thickness (t) and cell width (C). The SFE model results are471
shown in Figure 8. Two clear relationships can be seen in this figure. First,472
an increase in thickness results in an increase in SEA. Furthermore, as the473
cell width decreases, SEA increases. These relationships are monotonic in474
SEA in both arguments and hence optimal structures can be found only on475
the boundaries of the design space, either at the smallest cell width or the476
largest thickness. In contrast, the LS-DYNA models provide a richer picture477
and can be seen in Figure 9. The stability behavior is determined by visually478
checking LS-DYNA simulation results. Similar trends in SEA are visible in479
the LS-DYNA results, with increases in SEA for increasing thickness and480
decreases in SEA for increasing cell width until instability occurs, at which481
point the performance declines. These results in LS-DYNA and using SFE482
models are consistent with other works [31, 34].483
19
1 1.5 2 2.5 3 3.5 415
20
25
30
35
40
45
50
55
60
65Nine-Cell SFE Model Analysis
t [mm]
SE
A [
kJ/k
g]
C=26mm
C=29mm
C=32mm
C=35mm
C=38mm
Figure 8: Effect of thickness and cell width on SEA for nine-cell cross-section using SFEmodel.
Figure 9: Effect of thickness and cell width on SEA and CFE for nine-cell cross-sectionusing LS-DYNA. Green data point markers indicate stable behavior, orange data pointmarkers indicate transitional behavior, and red data point markers indicate unstable be-havior.
To further illustrate the relationship between cell width C, thickness t,484
and buckling mode, a phase diagram is provided in Figure 10. This figure485
maps the buckling mode across aspect ratio and the ratio of structure length486
20
Figure 10: Buckling mode diagram for nine-cell cross-section. Red indicates unstable be-havior, orange indicates transitional behavior, and green represents stable behavior. Greenshading indicates the approximate contours of SEA values, with darker color representinghigher values to show the general performance trend.
(L) to cell width (C), a phase mapping also used in [47]. Stable, unstable,487
and transitional regions are marked in different colors, and clear regions are488
visible based on geometry and buckling behavior. The regions, as drawn489
in red, yellow, and shades of green, are approximate and supplied only to490
illustrate general trends in buckling behavior and performance and should491
not be interpreted as the actual boundary of that phase behavior.492
6.2. Four-cell cross-section493
The four-cell cross-section was analyzed in the same manner as the nine-494
cell cross-section. Parameters varied include wall thickness (t) and cell width495
(C). The SFE model results are shown in Figure 11, showing similar mono-496
tonic behavior to that observed in the nine-cell. Proceeding to the LS-DYNA497
results, however, shows that the four-cell cross-section exhibits different be-498
havior from the nine-cell. While monotonicity is partially preserved for499
SEA, similar to the nine-cell cross-section, the trends in stability behavior500
are less clearly defined. The monotonicity for the CFE of this cross-section501
bears even less resemblance to the results from the nine-cell cross-section.502
Given that the stability behavior of this cross-section is less well defined, the503
four-cell is considered less attractive overall in terms of performance. This504
21
1 1.5 2 2.5 3 3.5 415
20
25
30
35
40
45
50
55
60
t [mm]
Four−Cell SFE Model Analysis
C=30mm
C=38mm
C=48mm
C=57mm
Figure 11: Effect of thickness and cell width on SEA for four-cell cross-section using SFEmodel.
Figure 12: Effect of thickness and cell width on SEA and CFE for four-cell cross-sectionusing LS-DYNA. Green data point markers indicate stable behavior, orange data pointmarkers indicate transitional behavior and red data point markers indicate unstable be-havior.
behavior is particularly clear in the phase diagram in Figure 13, though the505
regions as drawn are only approximate. For this cross-section, unlike the506
preceding nine-cell cross-section, there is no clear transition between differ-507
ent buckling phases, further emphasizing its apparent limited stability. The508
22
Figure 13: Buckling mode diagram for four-cell cross-section. Red indicates unstable be-havior, orange indicates transitional behavior, and green represents stable behavior. Greenshading indicates the approximate contours of SEA values, with darker color representinghigher values to show the general performance trend.
four-cell cross-section also produced greater numerical issues than either the509
five- or nine-cell in LS-DYNA. Therefore, the conclusions and results for the510
four-cell could be related to numerical issues, and the use of a different tool511
might provide better clarity. However, the four-cell cross-section performs512
less well than the five and nine-cell for both CFE and SEA overall, as can513
be seen in Table 5, and thus was not investigated further in the interest of514
finding superior performance.515
6.3. Five-cell cross-section516
Finally, the five-cell cross-section was investigated using the same modal-517
ity of the previous two cases, with the added variation of geometric parameter518
corner angle. The corner angle was varied from 90 to 95 degrees to determine519
its effect on performance, with range chosen based on previous literature re-520
lated to multi-corner sections [42, 43]. The output graphs from this analysis521
are shown in Figure 14. In the plot on the left, the corner angle was frozen522
at ninety degrees, whereas in the plot on the right the cell width was frozen523
at 38 mm. The same behavior with respect to C and t as observed in earlier524
cross-sections is observed here. Notably, the corner angle has a markedly525
smaller impact on the output performance in SEA.526
23
1 1.5 2 2.5 3 3.5 420
30
40
50
60
70
80Five-Cell SFE Model Analysis: = 90o
t [mm]
SE
A [
kJ/k
g]
C=26mm
C=29mm
C=32mm
C=35mm
C=38mm
1 1.5 2 2.5 3 3.5 420
25
30
35
40
45
50
55
60Five-Cell SFE Model Analysis: C=38mm
t [mm]
SE
A [
kJ/k
g]
= 90o
= 91o
= 92o
= 93o
= 94o
= 95o
Figure 14: Effect of thickness, corner angle, cell width on SEA for five-cell cross-sectionusing SFE model.
The LS-DYNA results for the stability of each cross-section for a spe-527
cific corner angle are shown in Figure 15. The monotonic trends in behavior
Figure 15: Effect of thickness and cell width on SEA for five-cell cross-section usingLS-DYNA with stability indicated.
528
24
observed in previous cross-sections are still present. Furthermore, a clear up-529
per bound on SEA and CFE is visible based on the stability behavior: the530
performance parameters continue to increase until they encounter a tran-531
sitional region, after which they decrease drastically at unstable behavior.532
This trend was observed across all three corner angles tested (90,92.5, 95),533
however only one set is displayed. The effects of corner angle are shown in534
Figure 16. For a fixed cell width, the stability behavior is identical for a
Figure 16: Effect of thickness and corner angle on SEA for five-cell cross-section usingLS-DYNA. Green data point markers indicate stable behavior, orange data point markersindicate transitional behavior, and red data point markers indicate unstable behavior.
535
particular thickness and there is little differentiation between performance536
parameters. This is the same low differentiation that was seen in the SFE537
model. However, given that stability behavior does not change with corner538
angle, corner angle was determined to be a less critical geometric parameter.539
Stability results are summarized in a phase diagram in Figure 17 for a sin-540
gle corner angle, given that the stability behavior is identical across corner541
angles. This figure is consistent with the findings for the nine-cell model.542
There is an unstable regime, followed by a transitional regime, leading to a543
region of stable behavior. Furthermore, once again the highest performing544
cross-sections are in the bottom left corner of the stable regime. This phase545
behavior makes the five-cell cross-section a higher performing cross-section546
than the four-cell, comparable to the nine-cell. However, regions as drawn547
are still approximate, and meant to illustrate general trends in behavior for548
25
Figure 17: Buckling mode diagram for five-cell cross-section. Red indicates unstable be-havior, orange indicates transitional behavior, and green represents stable behavior. Greenshading indicates the approximate contours of SEA values, with darker color representinghigher values to show the general performance trend.
investigated cross-sections.549
6.4. Comparison of the three cross-sections550
Overall, this sensitivity study focused on varying wall thickness, cell551
width, and corner angle to map out the stability behavior, SEA, and CFE552
for promising cross-sections. Some key parameters were held constant, such553
as length L and material in order to focus the scope of this exploration,554
though these represent potentially informative avenues for investigation. The555
LS-DYNA results also show that stability is a limiting factor; as the geome-556
try is varied and the SEA and CFE move towards their maximum values,557
the buckling behavior moves away from the desired stable mode, through a558
transitional mode, to a fully unstable mode. While these transitional mode559
variations on the nine- and five-cell (not four-cell) cross-sections appear to560
be the highest performing, their proximity to unstable behavior makes them561
undesirable. The general ubiquity of monotonic performance behavior in re-562
sponse to geometric variation suggests that locating the transitional regime563
for a given set of constraints is an efficient way to select a cross-section, given564
that top performing stable cross-sections are immediately proximate to this565
regime, i.e. at the boundary of acceptable buckling behavior rather than566
26
interior to the stable buckling region. Furthermore, as shall be considered567
in the following section, the tuning of geometry to produce particular loads568
is also desirable, for which mapping the boundary and interior of the sta-569
ble regime is also a useful starting point. Given that the nine- and five-cell570
cross-sections exhibited distinct regions of stability and instability, unlike the571
four-cell cross-section, they were considered more desirable cross-sections and572
thus given greater focus in the subsequent geometric tuning analysis.573
As a final part of the parametric study, the highest performing case from574
each cross-sectional type was selected and compared. Using the stability575
regime as the limiting factor on top performing cross-sections, and disallow-576
ing transitional behavior, allows for the top performing cross-sections to be577
identified. The best performing cross-sections are listed in Table 5. The
Geometry Parameters PerformanceFour-Cell Thickness = 4 mm SEA = 38.58 kJ/kg
Cell Width = 57 mm CFE = 0.4862Corner Angle = 90o E = 128 kJ
Mass = 3.32 kgPm = 389 kN
Nine-Cell Thickness = 4 mm SEA = 49.54 kJ/kgCell Width = 38 mm CFE = 0.6162Corner Angle = 90o E = 220. kJ
Mass = 4.43 kgPm = 674 kN
Five-Cell Thickness = 3 mm SEA = 43.44 kJ/kgCell Width = 38 mm CFE = 0.5546Corner Angle = 92.5o E = 93.2 kJ
Mass = 2.04 kgPm = 357 kN
Table 5: Parameters for top performing cross-sections, where E represents the totalamount of energy absorbed during the crushing process.
578
nine-cell and five-cell have the highest performance across SEA and CFE579
for all cross-sections surveyed, with the nine-cell out-performing the five-cell580
throughout the same parameter variation. One important distinction be-581
tween the five- and nine-cell is in the mean crush force. While the nine-cell582
absorbs significantly more energy, it also experiences a significantly higher583
mean crush force (almost double the five-cell) for only a 14% gain in SEA and584
27
11% gain in CFE, in addition to having more than twice the mass. While585
the nine-cell objectively maximizes the performance metrics, in practical ap-586
plications in automotive design the mass of the structure must also be taken587
into account, as well as the mean crush force, which would be transmitted588
to a vehicle occupant during a crash event. Therefore, further refinement of589
these cross-sections can be performed in order to better quantify the trade-590
off between mean crush force and mass in the interest of practical design591
implementation.592
7. Design refinement via mean crush force results and discussion593
In order to prepare these cross-sections for practical implementation in an594
automotive front end structure, a further design refinement was performed.595
Generally, these thin-walled structures are mounted behind the front bumper596
of vehicles in order to absorb energy during frontal impact crash events. This597
structure is highlighted in an overall vehicle body frame in Figure 18. Given598
the mounted position of these structures, the vehicle structural frame is de-599
signed to handle a given mean crush force (Pm), depending on the weight600
of the vehicle and the crush length availability in the given package space.601
Therefore, there can be a mean crush force constraint applied when iden-602
tifying the optimal cross-section by maximizing CFE and SEA for vehicle603
applications. This constraint is applied to the top performing cross-sections,604
five-cell and nine-cell, in order to fine tune the top performing cross-sections605
for applications in the front end structure. The baseline values for the target606
parameters (based on a vehicle using a four-cell cross-section) are detailed in607
the top data row of Table 6.608
In this analysis, the geometry parameters of the nine-cell and five-cell609
cross-sections will be varied within the bounded design space of the sensi-610
tivity study to match the mean crush force in the baseline. The cell width611
was frozen at the maximum dimension (38 mm for nine and five-cell) given612
the desired stability behavior and performance illustrated in the previous613
investigation. For the baseline simulation, the elasto-plastic aluminum alloy614
material model described earlier in Section 4 was used. The results for this615
parameter variation are detailed in Table 6. The geometry that meets the tar-616
geted mean crush force of 154 kN is bolded for each cross-sectional geometry.617
As in the previous study, the five-cell and nine-cell cross-sections outper-618
form the four-cell in terms of SEA, though they are now normalized around619
mean crush force. Both cross-sections are also more mass efficient than the620
28
Figure 18: Structures shown mounted in front of the main rails and behind the frontbumper of a vehicle, CAD models supplied from [48].
four-cell for equivalent mean crush force. In contrast however, the five-cell621
cross-section is now shown to marginally outperform the nine-cell in terms622
of SEA, with the large advantage of energy absorption of the nine-cell being623
completely eliminated. Furthermore, the five-cell design achieves the same624
mean crush force at a lower mass than the nine-cell cross-section, showing625
a 10% advantage. These results show that while the nine-cell cross-sections626
performs better globally when comparing across geometry, for specific appli-627
cations (such as targeting a mean crush force) the five-cell proves a better628
design choice. This finding is significant in light of other works, which gener-629
ally find that higher numbers of cells result in higher performing structures,630
as for example in [35, 36, 37].631
29
Geometry Thickness Mass Mean Force Mean Force/ Energy SEA(t) [mm] [kg] (Pm) [kN] Mass [kN/kg] [kJ] [kJ/kg]
Four-Cell 2.1 1.74 154 88.51 41.7 23.99
Five-Cell 3.0 2.20 357 162.3 112 50.732.0 1.47 170 115.7 55.1 37.481.9 1.40 154 110.0 49.9 35.631.8 1.33 141 106.0 44.1 33.181.6 1.18 115 97.46 35.5 30.051.0 0.739 52.0 70.37 15.6 21.16
Nine-Cell 2.0 2.20 275 125.0 85.0 38.661.6 1.77 190 107.3 53.7 30.361.4 1.55 154 99.35 48.6 31.381.2 1.33 120 90.23 35.9 27.031.0 1.10 90 81.82 27.0 24.54
Table 6: Nine and five-cell performance comparison to baseline cross-section for variedparameters
8. Summary632
A new five-cell cross-section was analyzed and compared to existing four-633
and nine-cell cross-sections to assess its performance in crashworthiness. A634
parametric sensitivity study was performed on all three cross-sections to pro-635
vide bounds on the design space and illustrate the relationship between ge-636
ometry and performance. Parameter variation was limited by manufactura-637
bility considerations, amongst other constraints. The sensitivity study had638
two major components, an SFE model implemented in MATLAB, and a large639
body of simulation work in LS-DYNA. Good agreement with the LS-DYNA640
model was shown for the SFE model for a single criss-cross element, and good641
global agreement across all three cross-sections. This comparison suggested642
the LS-DYNA simulation environment and SFE model were an acceptable643
method for investigation, and allowed for more in-depth study. Physical644
test results were found to have good agreement with LS-DYNA results and645
established confidence in the simulation environment as well. The paramet-646
ric sensitivity study illustrated trends in behavior correlating performance647
and geometric factors, as well as bounded the design space by identifying648
acceptable regions of stable progressive buckling behavior. Mapping this de-649
30
sign space allowed for general comparison of all three cross-sections, where650
the nine-cell marginally outperformed the five-cell in terms of CFE, SEA,651
and significantly for total energy absorption, with both outperforming the652
four-cell cross-section. To prepare these cross-sections for industrial appli-653
cations, a design investigation was performed comparing five- and nine-cell654
cross-sections to a single four-cell with a constraint on mean crush force.655
Both the five-cell and nine-cell outperformed the four-cell cross-section un-656
der this constraint in terms of mass efficiency, with the five-cell performing657
best by a 10% margin. Normalizing on mean crush force also eliminated658
the previous advantages in energy absorption and SEA seen for the nine-cell659
cross-section, showing that for certain applications the five-cell is actually660
the higher-performing cross-section.661
Acknowledgements662
The authors would like to acknowledge and thank Jean-claude Angles663
(Tesla) for study guidance and advisor-ship, as well as Zaifeng Zheng (Stan-664
ford) for providing assistance with simulation and theory development.665
References666
[1] J. Alexander, An approximate analysis of the collapse of thin cylindrical667
shells under axial loading, The Quarterly Journal of Mechanics and Ap-668
plied Mathematics 13 (1) (1960) 10–15. doi:10.1093/qjmam/13.1.10.669
[2] A. Pugsley, M. Macaulay, The large-scale crumpling of thin cylindrical670
columns, The Quarterly Journal of Mechanics and Applied Mathematics671
13 (1) (1960) 1–9. doi:10.1093/qjmam/13.1.1.672
[3] C. Magee, P. Thornton, Design considerations in energy absorption by673
structural collapse, Tech. rep., SAE Technical Paper (1978).674
[4] W. Abramowicz, N. Jones, Dynamic progressive buckling of circular and675
square tubes, International Journal of Impact Engineering 4 (4) (1986)676
243–270. doi:10.1016/0734-743X(86)90017-5.677
[5] T. Wierzbicki, S. U. Bhat, W. Abramowicz, D. Brodkin, Alexander678
revisited - a two folding elements model of progressive crushing of tubes,679
International Journal of Solids and Structures 29 (24) (1992) 3269–3288.680
doi:10.1016/0020-7683(92)90040-Z.681
31
[6] T. Wierzbicki, S. Bhat, A moving hinge solution for axisymmetric crush-682
ing of tubes, International journal of mechanical sciences 28 (3) (1986)683
135–151. doi:10.1016/0020-7403(86)90033-0.684
[7] W. Abramowicz, The effective crushing distance in axially compressed685
thin-walled metal columns, International Journal of Impact Engineering686
1 (3) (1983) 309–317. doi:10.1016/0734-743X(83)90025-8.687
[8] W. Abramowicz, N. Jones, Dynamic axial crushing of square tubes,688
International Journal of Impact Engineering 2 (2) (1984) 179–208. doi:689
10.1016/0734-743X(84)90005-8.690
[9] W. Abramowicz, N. Jones, Dynamic axial crushing of circular tubes,691
International Journal of Impact Engineering 2 (3) (1984) 263–281. doi:692
10.1016/0734-743X(84)90010-1.693
[10] A. Hanssen, M. Langseth, O. S. Hopperstad, Static and dynamic crush-694
ing of square aluminium extrusions with aluminium foam filler, Inter-695
national Journal of Impact Engineering 24 (4) (2000) 347–383. doi:696
10.1016/S0734-743X(99)00169-4.697
[11] T. Wierzbicki, W. Abramowicz, Axial crushing of multicorner sheet698
metal columns, Journal of Applied Mechanics 56 (1989) 113–120. doi:699
10.1115/1.3176030.700
[12] T. Wierzbicki, W. Abramowicz, On the crushing mechanics of thin-701
walled structures, Journal of Applied mechanics 50 (4a) (1983) 727–734.702
doi:10.1115/1.3167137.703
[13] W. Abramowicz, Thin-walled structures as impact energy absorbers,704
Thin-Walled Structures 41 (2) (2003) 91–107. doi:10.1016/705
S0263-8231(02)00082-4.706
[14] A. Najafi, M. Rais-Rohani, Influence of cross-sectional geometry707
on crush characteristics of multi-cell prismatic columns, in: 49th708
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and709
Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures710
Conference, 10th AIAA Non-Deterministic Approaches Conference, 9th711
AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design712
Optimization Specialists Conference, 2008, p. 2014. doi:10.2514/6.713
2008-2014.714
32
[15] A. Najafi, M. Rais-Rohani, Mechanics of axial plastic collapse in multi-715
cell, multi-corner crush tubes, Thin-Walled Structures 49 (1) (2011)716
1–12. doi:10.1016/j.tws.2010.07.002.717
[16] W. Chen, T. Wierzbicki, Relative merits of single-cell, multi-cell and718
foam-filled thin-walled structures in energy absorption, Thin-Walled719
Structures 39 (4) (2001) 287–306. doi:10.1016/S0263-8231(01)720
00006-4.721
[17] X. Zhang, G. Cheng, H. Zhang, Theoretical prediction and numeri-722
cal simulation of multi-cell square thin-walled structures, Thin-Walled723
Structures 44 (11) (2006) 1185–1191. doi:10.1016/j.tws.2006.09.724
002.725
[18] G. Zheng, T. Pang, G. Sun, S. Wu, Q. Li, Theoretical, numerical, and726
experimental study on laterally variable thickness (lvt) multi-cell tubes727
for crashworthiness, International Journal of Mechanical Sciences 118728
(2016) 283–297. doi:10.1016/j.ijmecsci.2016.09.015.729
[19] M. White, N. Jones, Experimental quasi-static axial crushing of top-730
hat and double-hat thin-walled sections, International Journal of Me-731
chanical Sciences 41 (2) (1999) 179–208. doi:10.1016/S0020-7403(98)732
00047-2.733
[20] V. Tarigopula, M. Langseth, O. S. Hopperstad, A. H. Clausen, Ax-734
ial crushing of thin-walled high-strength steel sections, International735
Journal of Impact Engineering 32 (5) (2006) 847–882. doi:10.1016/736
j.ijimpeng.2005.07.010.737
[21] Y. Tai, M. Huang, H. Hu, Axial compression and energy absorption738
characteristics of high-strength thin-walled cylinders under impact load,739
Theoretical and applied fracture mechanics 53 (1) (2010) 1–8. doi:740
10.1016/j.tafmec.2009.12.001.741
[22] V. J. Shahi, J. Marzbanrad, Analytical and experimental studies on742
quasi-static axial crush behavior of thin-walled tailor-made aluminum743
tubes, Thin-Walled Structures 60 (2012) 24–37. doi:10.1016/j.tws.744
2012.05.015.745
[23] E. Oberg, F. D. Jones, H. L. Horton, H. H. Ryffel, J. H. Geronimo,746
Machinery’s Handbook, Vol. 200, Industrial Press New York, 2004.747
33
[24] G. Sun, T. Pang, C. Xu, G. Zheng, J. Song, Energy absorption mechan-748
ics for variable thickness thin-walled structures, Thin-Walled Structures749
118 (2017) 214–228. doi:10.1016/j.tws.2017.04.004.750
[25] R.-J. Yang, Y. Fu, G. Li, Application of tailor rolled blank in vehicle751
front end for frontal impact, Tech. rep., SAE Technical Paper (2007).752
doi:10.4271/2007-01-0675.753
[26] J. Fang, Y. Gao, G. Sun, G. Zheng, Q. Li, Dynamic crashing behavior754
of new extrudable multi-cell tubes with a functionally graded thickness,755
International Journal of Mechanical Sciences 103 (2015) 63–73. doi:756
10.1016/j.ijmecsci.2015.08.029.757
[27] H.-S. Kim, New extruded multi-cell aluminum profile for maximum758
crash energy absorption and weight efficiency, Thin-Walled Structures759
40 (4) (2002) 311–327. doi:10.1016/S0263-8231(01)00069-6.760
[28] Z. Tang, S. Liu, Z. Zhang, Analysis of energy absorption characteristics761
of cylindrical multi-cell columns, Thin-Walled Structures 62 (2013) 75–762
84. doi:10.1016/j.tws.2012.05.019.763
[29] D.-H. Chen, K. Masuda, Crushing behavior of thin-walled hexagonal764
tubes with partition plates, ISRN Mechanical Engineering 2011. doi:765
10.5402/2011/503973.766
[30] J. Song, F. Guo, A comparative study on the windowed and multi-cell767
square tubes under axial and oblique loading, Thin-Walled Structures768
66 (2013) 9–14. doi:10.1016/j.tws.2013.02.002.769
[31] S. Wu, G. Zheng, G. Sun, Q. Liu, G. Li, Q. Li, On design of multi-cell770
thin-wall structures for crashworthiness, International Journal of Impact771
Engineering 88 (2016) 102–117. doi:10.1016/j.ijimpeng.2015.09.772
003.773
[32] S. Chen, H. Yu, J. Fang, A novel multi-cell tubal structure with circular774
corners for crashworthiness, Thin-Walled Structures 122 (2018) 329–343.775
doi:10.1016/j.tws.2017.10.026.776
[33] C. Qi, S. Yang, F. Dong, Crushing analysis and multiobjective crash-777
worthiness optimization of tapered square tubes under oblique impact778
34
loading, Thin-Walled Structures 59 (2012) 103–119. doi:10.1016/j.779
tws.2012.05.008.780
[34] S. Hou, Q. Li, S. Long, X. Yang, W. Li, Multiobjective optimization781
of multi-cell sections for the crashworthiness design, International Jour-782
nal of Impact Engineering 35 (11) (2008) 1355–1367. doi:10.1016/j.783
ijimpeng.2007.09.003.784
[35] N. Qiu, Y. Gao, J. Fang, Z. Feng, G. Sun, Q. Li, Crashworthiness anal-785
ysis and design of multi-cell hexagonal columns under multiple load-786
ing cases, Finite Elements in Analysis and Design 104 (2015) 89–101.787
doi:10.1016/10.1016/j.finel.2015.06.004.788
[36] J. Fang, Y. Gao, G. Sun, N. Qiu, Q. Li, On design of multi-cell tubes789
under axial and oblique impact loads, Thin-Walled Structures 95 (2015)790
115–126. doi:10.1016/j.tws.2015.07.002.791
[37] J. Fang, G. Sun, N. Qiu, T. Pang, S. Li, Q. Li, On hierarchical honey-792
combs under out-of-plane crushing, International Journal of Solids and793
Structuresdoi:10.1016/j.ijsolstr.2017.08.013.794
[38] J. Fang, G. Sun, N. Qiu, G. P. Steven, Q. Li, Topology optimization795
of multicell tubes under out-of-plane crushing using a modified artifi-796
cial bee colony algorithm, Journal of Mechanical Design 139 (7) (2017)797
071403. doi:10.1115/1.4036561.798
[39] G. Sun, T. Liu, X. Huang, G. Zhen, Q. Li, Topological configuration799
analysis and design for foam filled multi-cell tubes, Engineering Struc-800
tures 155 (2018) 235–250. doi:10.1016/j.engstruct.2017.10.063.801
[40] G. Sun, T. Liu, J. Fang, G. P. Steven, Q. Li, Configurational op-802
timization of multi-cell topologies for multiple oblique loads, Struc-803
tural and Multidisciplinary Optimization 57 (2) (2018) 469–488. doi:804
10.1007/s00158-017-1839-5.805
[41] N. Qiu, Y. Gao, J. Fang, G. Sun, N. H. Kim, Topological design of806
multi-cell hexagonal tubes under axial and lateral loading cases using a807
modified particle swarm algorithm, Applied Mathematical Modelling 53808
(2018) 567–583. doi:doi.org/10.1016/j.apm.2017.08.017.809
35
[42] M. Abbasi, S. Reddy, A. Ghafari-Nazari, M. Fard, Multiobjective crash-810
worthiness optimization of multi-cornered thin-walled sheet metal mem-811
bers, Thin-Walled Structures 89 (2015) 31–41. doi:10.1016/j.tws.812
2014.12.009.813
[43] B. R. S. Reddy, Multi cornered thin wall sections for crashworthiness814
and occupant protection, Ph.D. thesis, RMIT University (2015).815
[44] M. Yamashita, M. Gotoh, Y. Sawairi, Axial crush of hollow cylindrical816
structures with various polygonal cross-sections: Numerical simulation817
and experiment, Journal of Materials Processing Technology 140 (1-3)818
(2003) 59–64. doi:10.1016/S0924-0136(03)00821-5.819
[45] G. Sun, T. Pang, J. Fang, G. Li, Q. Li, Parameterization of criss-820
cross configurations for multiobjective crashworthiness optimization, In-821
ternational Journal of Mechanical Sciences 124 (2017) 145–157. doi:822
10.1016/j.ijmecsci.2017.02.027.823
[46] D.-H. Chen, Crush Mechanics of Thin-Walled Tubes, CRC Press, 2015.824
[47] W. Abramowicz, N. Jones, Transition from initial global bending to825
progressive buckling of tubes loaded statically and dynamically, In-826
ternational Journal of Impact Engineering 19 (5-6) (1997) 415–437.827
doi:10.1016/S0734-743X(96)00052-8.828
[48] 2012 Toyota Camry Detailed Finite Element Model, Center for Collision829
Safety and Analysis (2016).830
URL https://www.ccsa.gmu.edu/models/2012-toyota-camry/831
36
Appendix A. Supplemental data tables832
The data used to generate the LS-DYNA plots for the five-, nine-, and833
four-cell cross-sections are provided below. All values of CFE are unitless834
and all SEA values are given in units of kJ/kg. Furthermore, the observed835
stability behavior is also supplied in a tabular format.836
Corner Angle Item Cell Width 26mm Cell Width 32mm Cell Width 38 mm90 SEA 31.11 27.17 23.82
CFE 0.4338 0.3860 0.379392.5 SEA 31.58 26.92 23.01
CFE 0.4342 0.3825 0.369695 SEA 31.41 26.74 22.17
CFE 0.4403 0.3817 0.3775
Table A.7: Five-cell SEA and CFE for t = 1mm.
Corner Angle Item Cell Width 26mm Cell Width 32mm Cell Width 38 mm90 SEA 45.23 39.75 33.09
CFE 0.5973 0.5208 0.462192.5 SEA 44.46 39.15 34.24
CFE 0.5901 0.5155 0.454795 SEA 45.52 38.37 33.55
CFE 0.5988 0.5083 0.4562
Table A.8: Five-cell SEA and CFE for t = 2mm.
Corner Angle Item Cell Width 26mm Cell Width 32mm Cell Width 38 mm90 SEA 14.73 49.5 43.34
CFE 0.1268 0.6487 0.550792.5 SEA 15.10 49.15 43.44
CFE 0.1323 0.6406 0.554695 SEA 16.11 19.37 41.34
CFE 0.1464 0.1877 0.5468
Table A.9: Five-cell SEA and CFE for t = 3mm.
37
Corner Angle Item Cell Width 26mm Cell Width 32mm Cell Width 38 mm90 SEA 16.11 19.37 49.38
CFE 0.1466 0.1877 0.663792.5 SEA 16.74 20.13 48.92
CFE 0.1558 0.1959 0.652795 SEA 17.50 36.89 47.77
CFE 0.1648 0.5217 0.6329
Table A.10: Five-cell SEA and CFE for t = 4mm.
Thickness Item Cell Width 26mm Cell Width 32mm Cell Width 38 mm1mm SEA 32.23 27.79 24.28
CFE 0.4454 0.3972 0.37222mm SEA 44.63 39.28 36.22
CFE 0.5828 0.5216 0.48303mm SEA 54.04 47.63 42.78
CFE 0.6954 0.6208 0.54304mm SEA 25.31 56.22 49.54
CFE 0.2861 0.7220 0.6162
Table A.11: Buckling behavior for numerical simulation study of nine-cell.
Thickness Item Cell Width 38mm Cell Width 48mm Cell Width 57 mm1mm SEA 24.04 20.07 17.44
CFE 0.4115 0.4742 0.48582mm SEA 33.91 28.86 25.10
CFE 0.4593 0.3905 0.35373mm SEA 41.75 36.90 33.04
CFE 0.5492 0.4805 0.42884mm SEA 49.32 43.05 38.58
CFE 0.6504 0.5573 0.4862
Table A.12: Buckling behavior for numerical simulation study of four-cell.
38
Thickness Cell Width 38mm Cell Width 48mm Cell Width 57mm1 T S T (rotation)2 T S US (middle)3 T S S4 T T S
Table A.13: Buckling behavior for numerical simulation study of four-cell. S: Stablebuckling, US: Unstable buckling (global bending) T: transitional buckling (transitionalbehaviors)
Thickness Cell Width 26mm Cell Width 32mm Cell Width 38mm1 S S S2 T S S3 T T S4 US T S
Table A.14: Buckling behavior for numerical simulation study of nine-cell. S: Stablebuckling, US: Unstable buckling (global bending) T: transitional buckling (transitionalbehaviors)
Corner Angle Thickness Cell Width Cell Width Cell Width26mm 32mm 38mm
90 1 S S S2 T S S3 US T S4 US US T
92.5 1 S S S2 T S S3 US T S4 US US T
95 1 S S S2 T S S3 US T S4 US US T
Table A.15: Buckling behavior for numerical simulation study of five-cell. S: Stablebuckling, US: Unstable buckling (global bending) T: transitional buckling (transitionalbehaviors)
39