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: dkenyon@alumni.stanford.edu (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

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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