Probing the stability of thin-shell space structures under bending

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Fabien Royer, John W. Hutchinson, Sergio Pellegrino

PII:

S0020-7683(22)00291-8

DOI:

https://doi.org/10.1016/j.ijsolstr.2022.111806

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SAS 111806

To appear in:

International Journal of Solids and Structures

Received date : 21 December 2021

Revised date :

9 May 2022

Accepted date : 13 June 2022

Please cite this article as: F. Royer, J.W. Hutchinson and S. Pellegrino, Probing the stability of

thin-shell space structures under bending.

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Probing the Stability of Thin-Shell Space Structures

Under Bending

Fabien Royer

, John W. Hutchinson

, Sergio Pellegrino

∗

Graduate Aerospace Laboratories, California Institute of Technology,

1200 E California Blvd.,Pasadena, CA 91125, USA

Currently at: Department of Aeronautics and Astronautics,

Massachusetts Institute of Technology, Cambridge MA 02139-4307, USA

John A. Paulson School of Engineering and Applied Sciences, Harvard University,

29 Oxford St, Cambridge, MA 02138, USA

Graduate Aerospace Laboratories, California Institute of Technology,

1200 E California Blvd.,Pasadena, CA 91125, USA

Abstract

The stability of lightweight space structures composed of longitudinal thin-

shell elements connected transversely by thin rods is investigated, extending

recent work on the stability of cylindrical and spherical shells. The role of

localization in the buckling of these structures is investigated and early tran-

sitions into the post-buckling regime are unveiled using a probe that locally

displaces the structure. Multiple probe locations are studied and the probe

force versus probe displacement curves are analyzed and plotted to assess

the structure’s stability. The probing method enables the computation of

the energy input needed to transition early into a post-buckling state, which

is central to determining the critical buckling mechanism for the structure.

A stability landscape is finally plotted for the critical buckling mechanism.

It gives insight into the post-buckling stability of the structure and the exis-

∗

Corresponding author: sergiop@caltech.edu

Preprint submitted to Journal of Solids and Structures

May 9, 2022

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tence of localized post-buckling states in the close vicinity of the fundamental

equilibrium path.

Keywords:

thin shells, buckling, stability, buckling localization, probing

1. Introduction

1

Thin-shell structures are used extensively in engineering applications. In

2

the aerospace sector, they are a key enabler of lightweight air and space

3

vehicles. While the use of thin-shell structures dramatically reduces the

4

structural mass, their mode of failure is often governed by buckling, which

5

is hard to predict. Buckling of thin-shell structures is characterized by a

6

sub-critical bifurcation, which means that the structure exhibits a falling

7

unstable post-buckling path right after the bifurcation point is reached. This

8

sudden drop in load-carrying capabilities leads to a dramatic collapse if the

9

post-buckling path never regains stability. Buckling is to be avoided at all

10

cost in these cases. However, in recent adaptive structures and materials,

11

buckling is no longer seen as failure but as a key shape-changing mechanism,

12

which enables switching among multiple functional configurations (Hu and

13

Burgue ̃

no, 2015; Medina et al., 2020). Whether buckling is used or to be

14

avoided, understanding its cause and predicting its occurrence is crucial, and

15

this has been the subject of numerous research studies over the past one

16

hundred years.

17

From the early 1920s, many shell buckling experiments were conducted,

18

and experimental buckling loads were consistently observed to be lower than

19

linearized classical buckling predictions. This discrepancy was later linked to

20

the presence of initial imperfections in the shell geometry (Von Karman and

21

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Tsien, 1941; Donnell and Wan, 1950; Koiter, 1945). Indeed, for sub-critical

22

bifurcations, there exists a range of loading for which the structure’s fun-

23

damental (unbuckled) state is meta-stable, which makes the transition into

24

post-buckling extremely sensitive to imperfections and disturbances. On the

25

upside, this can also offer opportunities to build complex meta-stable struc-

26

tures (Zareei et al., 2020) by using buckled thin-shells as the main build-

27

ing blocks. In order to deal with the extremely sensitive buckling behav-

28

ior in engineering applications, the design process relies heavily on buckling

29

knockdown factors applied to the classical buckling load. Determining the

30

adequate knockdown factor, unique for each structure/load combination, is

31

of utter importance. It led to the NASA space vehicle design criteria for

32

the buckling of thin-walled circular cylinders (NASA, 1965). These crite-

33

ria, widely seen as very conservative, have been revisited by NASA’s Shell

34

Buckling Knockdown Factor (SBKF) Project, which has focused on testing

35

shells with known imperfections and non-uniformities in loading and bound-

36

ary conditions (Hilburger, 2012). It has been shown that knowing accurately

37

the structure’s initial geometry enables the accurate prediction of the buck-

38

ling event (Lee et al., 2016). However, in many applications, measuring the

39

shape of the structure before use can be both expensive and in some cases

40

impossible, and the traditional buckling and post-buckling predictions rely

41

on seeding a linear combination of the first buckling modes as imperfections

42

(Riks, 1979; Rahman and Jansen, 2010).

43

Another complication arising from unstable bifurcations is the localiza-

44

tion of buckling deformations. This is observed for instance for beams on an

45

elastic foundation (Wadee et al., 1997) and more importantly for thin-shell

46

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structures such as the compressed cylindrical shell (Hunt and Neto, 1991) as

47

well as the spherical shell under pressure (Hutchinson, 2016). The nature of

48

localization itself generates a large number of post-buckling solutions even

49

for a small set of classical buckling modes, since the deformations can localize

50

at many different locations on the structure. This is referred to as spatial

51

chaos (Thompson and Virgin, 1988). Localization can arise on post-buckling

52

branches determined by the buckling modes, as observed in the spherical shell

53

under pressure (Audoly and Hutchinson, 2020; Hutchinson and Thompson,

54

2017). In addition, localization can also appear on post-buckling paths dis-

55

connected from the fundamental path while running asymptotically close to

56

it (Groh and Pirrera, 2019). In both cases, localized buckling can be trig-

57

gered earlier than the first buckling load if a small amount of energy is input

58

into the structure. It has been shown, for the compressed cylindrical shell,

59

that a single localized dimple forming in the middle of the structure consti-

60

tutes the lowest escape into buckling (Hor ́

ak et al., 2006) and may therefore

61

be the critical buckling mechanism. This mode is not a bifurcation per se,

62

but rather a mode ”broken away” from the fundamental path. The single

63

dimple state sits on a ridge in the total energy of the system between the

64

pre-buckling well and the local post-buckling well and corresponds to the low-

65

est mountain pass between these two states in the energy landscape (Hor ́

66

et al., 2006). For the cylinder, the single dimple can evolve to more and more

67

complex post-buckling deformations through a series of destabilizations and

68

restabilizations, until the cylinder is fully populated by dimples (Kreilos and

69

Schneider, 2017; Groh and Pirrera, 2019). This process is called snaking and

70

adds additional complexity to the full post-buckling sequence resolution.

71

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For all of the reasons mentioned above, predicting buckling is extremely

72

difficult for shell structures and often relies on a case by case approach.

73

Recent work has focused on the sensitivity of the buckling phenomenon to

74

disturbances in thin cylindrical and spherical shells. A non-destructive ex-

75

perimental method, first proposed in

2015

to study the meta-stability of

76

the fundamental path, focuses on determining the energy barrier separating

77

the fundamental path and critical localized post-buckling states (Thompson,

78

2015; Thompson and Sieber, 2016; Hutchinson and Thompson, 2017). The

79

search for the critical buckling mechanism is carried out by imposing a lo-

80

cal radial displacement in the middle of the structure using a probe. This

81

method effectively quantifies the resistance of a shell buckling in the single

82

dimple mode mentioned earlier. The method has been successfully applied

83

to cylindrical shells (Virot et al., 2017) and pressurized hemispherical shells

84

(Marthelot et al., 2017). These experiments quantified in particular the on-

85

set of meta-stability, often referred to as ”shock sensitivity” (Thompson and

86

van der Heijden, 2014) and a comparison with historical test data has shown

87

that this specific loading can serve as an accurate lower bound for experi-

88

mental buckling loads (Groh and Pirrera, 2019; Gerasimidis et al., 2018).

89

More recent work has investigated the interaction between probing and

90

geometric defects in cylindrical (Yadav et al., 2021) and spherical shells (Ab-

91

basi et al., 2021). These experiments showed that a specific probing strategy,

92

called ridge tracking (Abramian et al., 2020), enables the non-destructive de-

93

termination of the actual buckling load of an imperfect shell. Probing in the

94

immediate vicinity of the dominant imperfection is required. Finally, a sim-

95

ilar probing methodology has been applied to circular arches (Shen et al.,

96

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2021a), cylindrical shell roofs (Shen et al., 2021b), and prestressed stayed

97

columns (Shen et al., 2022), and the use of multiple probes has enabled the

98

exploration of the complete unstable behavior of these structures, beyond

99

limit and branching points.

100

The present paper applies these recent breakthroughs to more complex

101

thin-shell structures, and is inspired by recently proposed spacecraft struc-

102

tures that use thin-shell components to build large space systems. In partic-

103

ular, modular structural architectures for ultralight, coilable space structures

104

suitable for large, deployable, flat spacecraft (Goel et al., 2017; Arya et al.,

105

2016) are being investigated in the Space-based Solar Power Project (SSPP)

106

at Caltech. In the deployed configuration, each spacecraft measures up to

107

60 m

60 m in size and is loaded by solar pressure. The main building

108

block is a ladder-type structure made of two triangular rollable and collapsi-

109

ble (TRAC) longerons (Murphey and Banik, 2011), connected transversely

110

by rods (battens). Scaled laboratory prototypes of this structure have been

111

built (Gdoutos et al., 2020, 2019), and analysis has shown that local buck-

112

ling plays a key role in its behavior (Royer and Pellegrino, 2020). The size

113

of the structure, together with the complexity of its components and the

114

distributed nature of the loading, would make it very challenging to conduct

115

experimental studies.

116

In order to address these limitations, a simpler structure is proposed in the

117

present paper and its behavior under pure bending is studied. This structure,

118

shown in Figure 1, is made of longerons and battens like the SSPP structures,

119

but the longeron’s complex original cross-section has been replaced by a

120

circular-arc cross-section. While this structure and loading are different from

121

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the specific structures of interest for the above-described space application,

122

it enables us to draw general conclusions on the buckling of space structures

123

with thin-shell open cross-sections. The computational analysis presented

124

here investigates the buckling behavior of such a structure and assesses if and

125

when early transitions into post-buckling can occur, using the novel probing

126

methodology. It also serves as a proof of concept for the experimental study

127

in Royer (2021).

128

The paper is structured as follows. Section 2 describes in more detail the

129

structure and the problem. Following a classical buckling analysis, Section 3

130

highlights the importance of localization and spatial chaos and justifies the

131

use of the newly-introduced probing methodology. In Section 4, probing is

132

applied along the entire structure to determine the location at which local

133

buckling can appear, and a critical probing scheme is identified. The analysis

134

is then generalized in Section 5 to more complex probing scenarios exhibit-

135

ing instabilities, and leads to an energy map from which the critical buckling

136

mechanism is identified. Finally a stability landscape of shell buckling is

137

computed in Section 6 to highlight key characteristics of the critical buckling

138

mechanism. It shows qualitative agreement with landscapes previously con-

139

structed for cylindrical and spherical shells, and for ladder-type structures

140

containing TRAC longerons (Royer and Pellegrino, 2020, 2022).

141

2. Computational model of strip structure

142

2.1. Geometry and material properties

143

The analysis presented in this paper is restricted to the single geometry

144

shown in Figure 1. The dimensions were chosen on the basis of a future

145

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experiment that will use an existing experimental apparatus.

146

The structure, referred as a strip, is composed of two thin-shell longerons

147

of length 0

4 m and with circular-arc cross section. The opening angle is

148

60 deg, the arc radius is 10 mm, and the shell thickness is 0

1 mm, which

149

correspond to a bending stiffness comparable to the SSPP structures. The

150

two longerons are connected by six regularly spaced transverse circular rods

151

called battens. The batten spacing is 80 mm, which ensures that several

152

battens connecting the two longerons. The batten length is 50 mm, and the

153

batten cross-section radius is 1 mm.

154

A finite element model of the structure is built using the Abaqus 2019

155

commercial software. The longerons are modeled with 4-node reduced inte-

156

gration shell elements (S4R) and the battens with linear 3D beam elements

157

(B31). An isotropic material with Young’s modulus

= 130 GPa, and

158

Poisson’s ratio

35 is considered for both battens and longerons.

159

Figure 1: Strip structure composed of two thin-shell longerons connected by battens.

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2.2. Finite element analysis

160

The end battens and the longeron end cross-sections are made unde-

161

formable and fully coupled to reference points R1 and R2, as shown in Figure

162

2. The boundary conditions and loading are applied to these reference points.

163

The structure is simply supported at both ends: one reference point is pinned

164

(all translations blocked) at one end while the

-translation is allowed for the

165

reference point at the other end. Two equal and opposite moments of mag-

166

nitude

are applied at the reference points, and an arc-length solver (Riks

167

solver in Abaqus standard) is used to statically deform the structure and ex-

168

tract the overall moment/rotation curve. In addition, in Section 4, for each

169

value of the moment, the top edge of the longeron will be probed by apply-

170

ing a transverse nodal displacement

x

at location

, and the probe reaction

171

force will be extracted. The two control parameters in these calculations are

172

thus the end moment and the probe displacement.

173

This strip structure has nonlinear pre-buckling behavior

, meaning that

174

the computed buckling eigenmodes change as the structure approaches the

175

buckling limit. This type of nonlinearity was previously reported for thin

176

shell structures (Leclerc and Pellegrino, 2020). Hence,

we will need to distin-

177

guish between two types of bifurcation buckling analyses and their associated

178

modes. We will use the standard terminology, classical buckling loads and

179

modes, for results in which the pre-buckling state used in the eigenvalue

180

analysis has been linearized, either about the condition at zero load or at

181

a non-zero load. Our approach will be making use of these eigenloads and

182

eigenmodes to gain insight into the buckling behavior of the strip. How-

183

ever, most references to buckling load and modes throughout the paper will

184

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Figure 2: Schematic representation of finite element model. The end battens and cross-

sections (green) are undeformable. R1 is allowed to translate along the

-axis and to rotate

along all 3 axes, R2 is pinned and is free to rotate. Two equal and opposite moments are

applied at the reference points. For a probing simulation (Section 4), a probe is applied

to the top edge of the longeron (longeron and z location determined by probing scheme).

It consists in an applied displacement on the probe node directed along the

-axis.

be to ”exact” buckling loads and modes computed by analyzing the bifur-

185

cation from the nonlinear pre-buckling state. We will mostly refer to the

186

”exact” analysis and its outcome with the brief terminology: buckling anal-

187

ysis, buckling loads, and buckling modes. However, if there is any ambiguity

188

the additional terminology, linearized or nonlinear pre-buckling state, will be

189

appended.

190

3. Localization and spatial chaos

191

3.1. Buckling modes and limit points

192

The first step in assessing the buckling behavior of the strip is to carry

193

out a classical eigenvalue analysis to determine a sequence of the applied mo-

194

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ments and associated modes at which buckling bifurcations from the perfect

195

strip occur. This information gives a picture of not only the lowest buckling

196

load and associated mode but also of the bifurcation modes lurking above

197

the lowest critical mode. Such information gives insight into potentially im-

198

portant imperfection shapes and to ”nearby paths” which might play a role

199

in the post-buckling behavior.

200

The computation of the ”exact” bifurcation moments and modes is itself

201

an iterative procedure because the pre-buckling behavior is nonlinear. To

202

obtain first estimates of the bifurcation points, the pre-buckling nonlinearity

203

is neglected using the ground-state linearity to compute a sequence of the

204

lowest bifurcation eigenvalues (ABAQUS and other structural codes have op-

205

tions for making such eigenvalue evaluations). These bifurcation estimates

206

are then used to guide the search for the bifurcations computed accounting

207

for nonlinear pre-buckling behavior. With the full pre-buckling nonlinearity

208

accounted for, the strip is then loaded by a moment below the first eigen-

209

value, the nonlinear pre-buckling problem is solved, and new estimates of the

210

sequence of bifurcation points are computed by linearizing about that state.

211

This iterative process is repeated with an increasing applied moment in each

212

iteration until the bifurcation moments converge. For the strip, nine bifur-

213

cation points are determined in the loading interval before the strip attains

214

a limit moment on the fundamental pre-buckling path. As noted earlier,

215

to distinguish between a buckling load of the perfect strip computed using

216

ground state linearity (traditionally called a ”classical buckling load”) and

217

the buckling load computed accounting for pre-buckling nonlinearity, we will

218

briefly refer to the latter as the ”buckling load” and is associated eigenmode

219

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as the ”buckling mode”. The results of this analysis are shown in Figure 3.

220

Figure 3: Nine buckling modes with associated buckling moments found on the strip

fundamental path. For each mode, the deformations of both longerons are concentrated

along the longerons’ top edge (edge in compression). These deformations involve both

inward (towards the strip center) and outward displacements. The battens do not exhibit

any appreciable deformation.

Both a classical Newton-Raphson solver and the Riks solver are used to

221

trace the response of the structure in its unbuckled configuration. The Riks

222

method uses the load magnitude as an additional unknown and solves simul-

223

taneously for loads and displacements. The simulation progresses by incre-

224

menting the arc-length along the static equilibrium path in load-displacement

225

space, enabling the resolution of unstable responses. The Newton-Raphson

226

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solver reaches a limit point at

464

2 Nmm, while the Riks solver

227

bifurcates from the fundamental path to a secondary branch at

435

228

Nmm. Note that this moment magnitude is between the first and second

229

buckling moments in Figure 3.

230

3.2. Localization and post-buckling paths

231

We wish to trace the post-buckling paths corresponding to several of

232

the lowest buckling eigenmoments and study the evolution of the structure’s

233

shape along these paths. Of primary interest is the moment/rotation relation

234

for the strip when equal and opposite moments are applied at the strip ends

235

and the rotation corresponds to the rotation around the

-axis of the end

236

located at

= 0 (c.f., Figure 2).

237

As a first step, a standard method is used to trace the post-buckling paths

238

associated with the first three buckling modes as described next. Each mode

239

is seeded in the structure’s initial geometry as a geometric imperfection. The

240

maximum amplitude of this initial imperfection is taken between 1% and 10%

241

of the shell thickness,

. The Riks solver is used to trace the post-buckling

242

response of the imperfect structure.

243

The computed paths are shown in Figure 4, and the corresponding de-

244

formed shapes are shown in Figure 5. For the second buckling mode, two

245

imperfection amplitudes have been used, yielding the two post-buckling paths

246

shown.

247

The main observation is that, contrary to the bifurcation buckling modes,

248

the deformed shapes for all the paths exhibit highly localized deformations.

249

For the first and second mode branches, the post-buckling shapes are quite

250

different from the initial imperfection. These shapes only exhibit inward

251

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0.6

0.7

0.8

0.9

1.1

End rotation (deg)

0.8

0.9

1.1

1.2

1.3

1.4

Bending moment (Nmm)

Fundamental path

1st mode branch

2nd mode branch

2nd mode branch, alternate

3rd mode branch

Figure 4: Moment vs. rotation curves for the strip. The fundamental path (black) stops at

the limit point

464

2 Nmm. The first buckling mode branch (blue) is obtained by

seeding the first mode as imperfection with an amplitude of 8%

. The second branch (red)

is obtained for the second mode imperfection with an amplitude of 8%

. The alternate

second branch (green) is obtained for the second mode imperfection with an amplitude

of 10%

. The third branch (purple) is obtained for the third mode imperfection with an

amplitude of 8%

buckling deformations, whereas the buckling modes also exhibit outward

252

deformations. For the second mode branch, even a slight variation in im-

253

perfection amplitude changes the buckling location. For the second mode

254

and third mode, the post-buckling paths undergo destabilization and resta-

255

bilization. This phenomenon is referred to as homoclinic snaking and is also

256

observed in axially compressed cylindrical shells (Groh and Pirrera, 2019).

257

It physically corresponds to the sequential formation of buckles leading to a

258

fully buckled shell. Snaking may occur also in the remaining localized paths

259

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Figure 5: Deformed shapes with magnification of 15X, obtained at the end of the four

post-buckling paths of Figure 4. They consist in localized longeron deformations and

differ from the previously computed buckling modes. All deformations are inward, and

the localization location differs between longerons for the mode 1 branch (labeled 1) and

mode 2 branch (labeled 2).

if the analysis is pushed further. It is interesting to note that it was possible

260

to resolve the post-buckling path for the third buckling mode without seeding

261

any imperfection in the initial geometry.

262

For mode 1 and mode 2, the localization process initiates on the im-

263

perfect structure’s fundamental path, before reaching the falling unstable

264

post-buckling path. The initial deformation grows proportionally to the ini-

265

tial imperfection and then is followed by a transition to a localized mode

266

shape before attaining a limit point. At this point, the location of maximum

267

deformation has already been determined and, on the falling unstable path,

268

the local deformation increases in amplitude without changing location. It is

269

important to emphasize that the limit point for the imperfect structure is off-

270

set from the perfect structure’s fundamental path, although extremely close

271

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to it, due to the eroding effect of the imperfection on the initial stiffness. In

272

addition, these limit points appear at values of applied moment lower than

273

the first buckling moment which reveals the structure’s imperfection-sensitive

274

nature.

275

Figure 6 highlights the localization process for each of the first two buck-

276

ling modes. The displacement of the longeron top edge in the

plane is

277

plotted at the limit point, as well as at the first post-buckling restabilization

278

point and at the end of the post-buckling path. The normalized buckling

279

mode of the perfect strip is also reported as a dashed line, for comparison.

280

For mode 1, localization occurs on two levels. At the structure’s scale,

281

local deformations only arise in longeron 1, while for longeron 2, the global

282

deformation tends to cancel the undulations associated with the initial im-

283

perfection away for the point of localization. At the longeron scale, the

284

deformed shape goes from a smooth hill to a sharp peak for longeron 2.

285

In addition, the localization process is not unique. Different localization

286

mechanisms are observed for buckling mode 2, depending on the imperfection

287

amplitude, as seen in the deformed shape comparison of Figure 5. The local-

288

ization of mode 2 for an imperfection amplitude of 8%

is shown in Figure

289

6c-d. It highlights the sequential formation of the longeron 1 and longeron

290

2 buckle, characteristic of the snaking process. In the case of buckling mode

291

3, the buckling mode shape is relatively localized and resembles the shape

292

observed in Figure 5 for the two central buckles. Therefore, no further local-

293

ization is observed on the post-buckling path before the snaking process is

294

triggered, and four highly localized buckles are formed closer to the longeron

295

ends.

296

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To conclude this section, we re-emphasize that multiple post-buckling

297

paths have been shown to have initially unstable behavior, and in some cases

298

the paths re-stabilized at lower loads. Four different imperfections based on

299

the first three buckling modes have been considered here; other imperfections

300

or linear combinations of buckling modes would give rise to different paths.

301

Seeding different imperfections has highlighted qualitatively the importance

302

of localization for this thin-shell structure and the fact its deformation can

303

easily localize at many different locations. This multiplicity of buckling and

304

post-buckling solutions is referred to as ”spatial chaos.” However, not all

305

possible localized paths have been considered, and hence it is not known

306

which path constitutes the easiest escape into post-buckling. Based on these

307

qualitative observations, the next section searches for the critical localized

308

path using the probing methodology introduced.

309

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100

200

300

400

(

)

Top edge Ux (mm)

Mode 1

Limit point

Stabilization

End

(a)

100

200

300

400

(

)

Top edge Ux (mm)

Mode 1

Limit point

Stabilization

End

(b)

100

200

300

400

(

)

-1

Top edge Ux (mm)

Mode 2

Limit point

Stabilization

End

(c)

100

200

300

400

(

)

-1

Top edge Ux (mm)

Mode 2

Limit point

Stabilization

End

(d)

Figure 6: (a-b) Localization process for (a) longeron 1 and (b) longeron 2, on the first

mode post-buckling path, for an imperfection amplitude of 8%

. The longeron top edge

displacement in the

-direction is plotted as a function of the

location. The normalized

buckling mode is shown as a dashed line. The evolution of the longeron top edge defor-

mation is reported at the limit point, where the post-buckling path first stabilizes, and at

the end of the post-buckling path. (c-d) Localization process on the second mode post-

buckling path, for an imperfection amplitude of 8%

for (c) longeron 1 and (d) longeron

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4. Probing along the strip length

310

4.1. Probing methodology

311

The previous section has shown that buckling localization can lead to a

312

large number of post-buckling paths. Hence, the focus in the rest of this

313

paper is on finding the critical buckling mechanism. Here ”critical” means

314

finding the easiest way the structure can buckle or, in other words, finding

315

how early the transition into buckling can occur and which deformed shape

316

is most likely to arise.

317

Two situations may be encountered when end-moments are applied on

318

a strip. The first corresponds to an early transition to a path that inter-

319

sects the fundamental path, and for which the deformation matches one of

320

the buckling modes (at least at the bifurcation point). This situation may

321

arise for buckling mode 3, for which no imperfection is needed to resolve the

322

post-buckling path. The second situation corresponds to a transition to a

323

disconnected equilibrium path, running in close vicinity of the fundamental

324

path but without intersecting it (Hunt and Neto, 1991). In both cases, a finite

325

input of energy into the system is required to make the structure transition

326

early to a secondary equilibrium path. Note that here, ”early transition”

327

means that the transition to post-buckling occurs before reaching the first

328

buckling moment. A key assumption made here is that the critical buckling

329

mechanism will exhibit highly localized deformations. This is generally the

330

case for thin-shell structures for which buckling is a sub-critical bifurcation

331

and is motivated by the observations made in the previous section.

332

The probing method, which uses a probe that displaces the structure

333

locally, is used to quantify the amount of disturbance needed to trigger early

334

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localized buckling. In this paper, the probing method is explored numerically

335

and consists in applying a displacement directed along the

-axis to a node

336

on the top edge of the longeron (the probed node), as illustrated in Figure 2.

337

The top edge is chosen because it corresponds to the location of the largest

338

compressive stress when bending moments are applied to the structure.

339

The analysis goes as follows. Two end moments are applied on the perfect

340

structure. When the desired moment magnitude is reached, the moment is

341

kept constant and the probe displacement is increased. During probing, the

342

probe reaction force is computed. This process is repeated for a range of

343

moments, up to the first buckling moment, and for various probe locations

344

along the longeron’s top edge. The Abaqus static general solver (Newton-

345

Raphson) is used for both the bending and probing steps. The analysis

346

presented in this section is restricted to probing paths for which the probe

347

displacement is monotonic.

348

Two features are of particular interest. The first corresponds to the range

349

of applied moments for which buckled equilibrium states exist. An equilib-

350

rium state is found when the probe reaction force falls to zero. When such

351

a situation is encountered, there exist at least two equilibrium configura-

352

tions for a given moment and therefore the fundamental path is meta-stable.

353

Above the moment for which negative probe forces are first encountered, a

354

disturbance may trigger early buckling. The second important feature is the

355

critical amount of energy that needs to be provided to the system to reach

356

the buckled equilibria. It indicates the level of disturbance needed for the

357

structure to transition early into these states.

358

Inspired by the types of deformations seen in the buckling modes, and

359

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restricting the study to at most a single probe per longeron, five probing

360

schemes have been investigated: double outward probing, double inward

361

probing, alternate probing, single outward probing, and single inward prob-

362

ing, as illustrated in Figure 7.

These probing schemes were chosen such that

363

it would be possible to trigger the localized buckling modes of Figure 5.

364

By characterizing the onset of meta-stability and the critical probe work

365

needed to trigger buckling, we will be using probing as an efficient tool to

366

navigate through the spatial chaos and to find the structure’s critical buckling

367

mechanism.

368

Probe

Double Inward

Double Outward

Alternate

Single Inward

Single Outward

Figure 7: Five probing schemes considered in this paper, with arrows representing the

transverse probe displacement.

Journal Pre-proof

4.2. Double inward probing scheme

369

The double inward probing scheme is considered first. In this case, con-

370

vergence is hard to achieve for probing with applied moments of around

371

100 Nmm, because instabilities are encountered. These instabilities

372

are analyzed in detail in the next section.

373

For moments under 1

000 Nmm, the probing forces remain positive and

374

the contours of constant probe force exhibit local extrema in the probe lo-

375

cation / displacement plane. The probe force for two values of the moment

376

has been plotted in Figure 8 as a function of the probe location along the

377

longeron edge (

-axis) and of the probe displacement. Figure 8a shows the

378

probing map for

= 800 Nmm. The probe force is shown as a function of

379

the probe displacement along the

-axis (

x

) and the probe location along

380

the top of the longeron (

-axis). For ease of visualization, the regions cor-

381

responding to probe locations between 0 mm and 50 mm as well as between

382

350 mm and 400 mm are not shown since they exhibit large probe forces.

383

In these two regions, the probe force vs. probe displacement curve is almost

384

linear. For all other probe locations, the probe force increases monotoni-

385

cally as the probe displacement increases. However, the map exhibits many

386

features, such as regularly spaced local minima of probe force for a given

387

probe displacement. The lowest local minimum is attained in the middle of

388

the structure (200 mm). The probe force is positive for all values of probe

389

displacement. Figure 8(b) shows the probing map for

040 Nmm. For

390

probe locations ranging from 0 mm to 60 mm and from 340 mm to 40 mm,

391

the probe force increases monotonically as the probe displacement increases.

392

For all other probe locations, the probe force increases and then decreases.

393

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Regularly spaced local minima of probe force appear, and negative values

394

are reached in the middle (200 mm).

The spacing between local minima

395

corresponds to the batten spacing.

396

100

200

300

Probe location (mm)

0.5

1.5

Probe Ux (mm)

(a)

100

200

300

Probe location (mm)

0.5

1.5

-0.1

0.1

0.2

0.3

Probe force (N)

(b)

Figure 8: Double inward probing map for (a)

= 800 Nmm and (b)

040 Nmm.

The spacing between contours is 0.05 N.

In fact, additional simulations showed that the probe force at the center

397

first falls to zero for

015

5 Nmm. This critical load corresponds to

398

the onset of meta-stability, at which early transition into buckling becomes

399

possible. Based on the probing scheme, the associated post-buckling shape

400

consists of an inward local buckle in the middle of each longeron. This shape

401

resembles the third non-linear buckling mode found in Section 3.1.

402

4.3. Single inward probing scheme

403

The single inward probing scheme is considered next. The probing maps

404

for four values of the applied moment are shown in Figure 9.

405

Figure 9a shows the probing map for

= 800 Nmm. As the probe

406

displacement increases, the probe force increases monotonically, except near

407

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the middle, where a basin of local minima appears (probe displacement of

408

2 mm). The probe force is positive everywhere.

409

Figure 9b shows the probing map for

040 Nmm. Local maxima

410

of probe force appear and form a hill separating the fundamental path from

411

regions with local minima. The local minima are negative near the middle of

412

the strip, whereas at other locations they are positive, although very close to

413

zero. This map resembles the map obtained for the double inward probing

414

scheme.

415

Figure 9c shows the probing map for

200 Nmm, which resembles

416

qualitatively Figure 9b. A local minimum of probe force appears for a probe

417

displacement of 0.2 mm, before reaching a second minimum at 0.35 mm, at

418

the center of a region of negative probe forces. However, when probing at

419

locations other than the middle, the probing path encounters instabilities as

420

the probe force decreases after the peak, and the Newton-Raphson solver

421

aborts. It leaves the probing map incomplete. The probe displacement for

422

which local minima of probe force are attained decreases as the moment

423

increases.

424

Figure 9d shows the probing map for

350 Nmm. The probe

425

instabilities appear as early as 0

1 mm of probe displacement and cause

426

a severe truncation of the map. The probing path for the mid-point of

427

the structure exhibits negative probe forces for displacements of 0

075 mm

428

and 0

14 mm, indicating the existence of two adjacent buckled equilibrium

429

states. However, the overarching goal of the probing method

which

430

to compute the minimum energy input needed to trigger early buckling for

431

every probe locations, it is not yet possible due to the probe instabilities. At

432

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the locations where the probing sequence suddenly stops it is impossible to

433

draw any conclusions regarding the structure’s meta-stability. It is therefore

434

necessary to resolve probing sequences past these instabilities, and this is the

435

subject of Section 5.

436

100

200

300

Probe location (mm)

0.5

1.5

Probe Ux (mm)

(a)

100

200

300

Probe location (mm)

0.2

0.4

0.6

0.8

0.1

0.2

Probe force (N)

(b)

100

200

300

Probe location (mm)

0.1

0.2

0.3

Probe Ux (mm)

(c)

100

200

300

Probe location (mm)

0.05

0.1

0.15

-0.02

0.02

0.04

0.06

Probe force (N)

(d)

Figure 9: Single inward probing maps for (a)

= 800 Nmm, (b)

040, (c)

200 Nmm, and (d)

350 Nmm.

The spacing between contours is 0.02 N for

(a) and (b), and 0.005 N for (c) and (d).

An important observation is that meta-stability appears earlier for this

437

type of probing than for the double inward probing scheme. For higher

438

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moment magnitudes, the minimum of probe force is still achieved at the mid-

439

point of the structure, with regions of negative probe force spreading over a

440

larger portion of the structure. Therefore, there exist multiple locations at

441

which buckled equilibrium states are found. This supports the observations

442

of Section 3 where we saw that localization for the second mode imperfection

443

can occur at multiple locations. However, we see qualitatively that the hill

444

of probe force separating the unbuckled and buckled states is lowest at the

445

mid-point, which signifies that the minimum energy input required to form

446

an inward buckle is also achieved in the middle of each longeron.

447

4.4. Outward and alternate probing schemes

448

For the double outward probing scheme it is found that there is no value

449

of the moment for which the probe forces decreases to 0 N. Instead, as the

450

longeron is locally displaced outwards under constant applied moments, the

451

probe force always increases monotonically. Typically, the probe force reaches

452

1 N for a probe displacement of about 1 mm, which is an order of magnitude

453

higher than the probe force obtained with the double inward probing scheme.

454

Probing does not reveal any buckled equilibria in this case.

455

The alternate probing scheme involves an inward probe on longeron 1 and

456

an outward probe on longeron 2. The outward probe force increases monoton-

457

ically, as this case is similar to the double outward probing scheme. However

458

the inward probe force in the center becomes negative for all probe dis-

459

placements, above a certain moment magnitude. Although the disturbance

460

introduced by probing can be transferred between longerons, the outward

461

probe force never falls to 0 N and hence no buckled equilibria are found.

462

Similar behavior is observed for the single inward probing scheme. When

463

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the outward probe displacement is increased, the probe force monotonically

464

increases, while an inward buckle forms in the unprobed longeron. Similarly

465

to the alternate probing scheme, no equilibrium configurations are encoun-

466

tered, but the probing path is truncated before the prescribed end displace-

467

ment is reached, due to instabilities. These instabilities are analyzed in

468

Section 5 and it is shown that buckled equilibria exist if probing is extended

469

past instabilities.

470

4.5. Critical probe work and initial comparison of probing schemes

471

In order to find the critical buckling mechanism for the strip structure,

472

the probing schemes presented above need to be compared. The critical

473

buckling mechanism corresponds to the minimum amount of energy needed

474

to reach buckled equilibria, but special care has to be taken when computing

475

the energy barrier to buckling and the critical probe work.

476

In previous buckling and probing studies, the energy barrier refers to

477

the difference in total potential energy between the unbuckled state and the

478

unstable buckled state. As explained in the introduction, the unstable buck-

479

led state corresponds to a saddle point (also called mountain pass point) in

480

the energy landscape and is attained for a critical value of the probe dis-

481

placement, when the zero threshold in probe force is reached. If the main

482

loading is kept constant, the probe work reaches a local maximum at this

483

critical displacement. We will use the terminology ”critical probe work” to

484

refer to this local maximum of the probe work. When the probe displace-

485

ment is monotonic during probing (i.e., no folding of the path), and for

486

a displacement-controlled main loading, the critical probe work is equal to

487

the energy barrier. This scenario is for instance encountered for the probed

488

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cylinder under constant end shortening (Virot et al., 2017). However in the

489

present study, the energy barrier and the critical probe work can be different

490

for two reasons:

491

•

Moment-controlled loading implies that probing occurs under a con-

492

stant value of the end-moment. During probing, the ends of the strip

493

rotate and hence the end-moments do work. As a result, the energy

494

barrier is greater than the critical probe work since it accounts for

495

the end-moments’ additional contribution to the energy of the system.

496

However, the constant moments are part of the known conditions the

497

structure is subjected to during operation and, since the contribution

498

of an unknown disturbance is only represented by the probe, the quan-

499

tity of interest is the critical probe work. The study has been repeated

500

for a rotation-controlled loading and the results are presented in Ap-

501

pendix A. In the latter case, the probe work only contributes to the

502

total external work of the system.

503

•

For unstable probing sequences, a vertical tangent can be reached, be-

504

yond which the probing path can fold. In such cases, snap-buckling

505

can be triggered before the zero probe force threshold is attained, and

506

the value of the critical probe work is computed at the point of vertical

507

tangent rather than at the first buckled equilibrium. Such cases are

508

presented and analyzed further in Section 5.

509

Next, the critical probe work for the two inward probing schemes is dis-

510

cussed. Since the probing path does not exhibit any instabilities in the middle

511

of the structure, for both schemes, the critical probe work required to reach

512

Journal Pre-proof

the buckled equilibrium states can be computed. The critical probe work

513

obtained for a central probe location and for both probing schemes is shown

514

in Figure 10.

515

1.1

1.2

1.3

1.4

Bendin

moment

(

)

0.02

0.04

0.06

0.08

Critical probe work (mJ)

Double inward

Single inward

Figure 10: Critical probe work as a function of the applied bending moment, for both

single and double inward probing schemes. It is smallest for the single inward probing

scheme.

The single inward probing scheme gives a lower critical probe work than

516

the double inward probing scheme for the entire range of moments considered.

517

As a result, if buckling is triggered early, it will likely consist of a single

518

buckle in the middle of one of the longerons rather than in both longerons.

519

When comparing the local maximum of probe force obtained for both probing

520

schemes, we also see that it is lowest for the single inward probing scheme,

521

regardless of the probe location. It seems therefore that if meta-stability is

522

detected at a specific probe location, the single inward probing scheme would

523

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also give the lowest critical probe work at this specific location.

524

Finally, it has been shown in this section that buckled equilibrium states

525

appear for lower values of moments for the single inward probing scheme. As

526

snaking appears to play a prominent role for this structure, we would expect a

527

sequential formation of single buckles which supports the energy comparison

528

between the two probing schemes. For all of these reasons, the rest of the

529

paper will focus only on the single inward / outward probing schemes.

530

5. Unstable probing sequences

531

5.1. Single inward probing

532

This section extends the probing simulations to cases in which instabil-

533

ities are encountered. The probing displacement is applied similarly to the

534

previous part of the study, but an arc-length solver (Riks solver) is now used,

535

which allows probing to continue after a vertical tangency (fold) in the probe

536

force vs. probe displacement plane has been reached. Additional probing

537

sequences are computed for the single inward probing scheme and for all

538

probing locations, and the two main types of path instabilities encountered

539

are analyzed.

540

The results of the analysis for a probe located at 100 mm from the end of

541

the structure are shown in Figure 11. For

050 Nmm, the probing path

542

is stable and the probe force exhibits a local maximum and local minimum.

543

However, the probe force is always positive and no locally buckled equilibrium

544

solutions exist. For

050 Nmm, a vertical tangent is encountered and

545

the path folds. The path eventually restabilizes for a value of probe force

546

of about

−

1 N. However, the restabilized path is short and does not reach

547

Journal Pre-proof

positive probe forces. This suggests that another bifurcation is encountered

548

for a probe displacement of about 0

2 mm. This behavior is also encountered

549

for higher values of moments, although the corresponding probing paths do

550

not restabilize for positive values of probe displacement. Figure 12a shows

551

the probing path for

050 Nmm with four points 1-4 marking key

552

stages of the probing sequence.

553

0.2

0.4

0.6

0.8

1.2

Probe dis

lacement

(

)

-0.1

-0.05

0.05

0.1

Probe force (N)

M = 1000 Nmm

M = 1050 Nmm

M = 1200 Nmm

M = 1385 Nmm

Figure 11: Probe force vs. probe displacement for a probe located at

= 100 mm and for

four values of applied moment. The loop formed by the folded path becomes smaller as

the moment magnitude increases until it folds on itself for

385 Nmm.

The deformed shapes corresponding to these four points are shown in Fig-

554

ure 12b. On the stable part of the path (before reaching point 2), displacing

555

the probe results in an increase of the local buckle amplitude. After point

556

2, the probing path becomes unstable. As the probe displacement decreases,

557

the probe force increases until it reaches point 3 and then decreases to 0 N at

558

Journal Pre-proof

point 4, which corresponds to a buckled equilibrium solution. This unstable

559

path corresponds to the change of location of the buckle formed during the

560

stable part of the path. At point 4, the structure is in a buckled equilib-

561

rium configuration, but the final buckle location does not correspond to the

562

probing location.

563

Note that the probe force vs. probe displacement curve has a positive

564

slope at point 4 which means that the equilibrium is stable. The critical

565

probe work required to reach the localized buckled configuration at point

566

4 corresponds to the shaded area in Figure 12a. It is important to point

567

out that this area does not correspond to the energy barrier, as explained

568

in Section 4.5. In order to compute the energy barrier, i.e. the difference in

569

total potential energy between the unbuckled state and the buckled state at

570

point 4, the area enclosed by the probing path would have to be considered.

571

The area under the curve formed by points 2, 3 and 4 would have to be

572

subtracted from the shaded area, and the work done by the end-moments

573

would have to be added.

574

Path folding has also been encountered in compressed spherical shells

575

probed at the apex, under rigid volume control (Thompson and Sieber, 2016),

576

and all of the bifurcations that can arise and disrupt a probing sequence

577

have been described (Thompson et al., 2017). Two approaches have been

578

proposed to explore experimentally these unstable probing sequences. The

579

first one consists in introducing feedback control (Thompson et al., 2017).

580

If the probe displacement and probe force are chosen as inputs, it is then

581

possible to resolve vertical tangents. It is also possible to navigate around

582

the fold and avoid unstable probing paths by using the moment and probe

583