Accepted for publication in ASCE Journal of Engineering Mechanics.

http://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0001571

1 Former MSc student in structural engineering, now PhD student at Zhejiang University.

2 Senior Lecturer in structural engineering

3 Professor Emeritus of structural engineering (F.ASCE)

Ovalization restraint in four-point bending tests of tubes 1

Qing Liu1, Adam J. Sadowski

2 & J. Michael Rotter

3 2

Department of Civil and Environmental Engineering, Imperial College London, UK 3

4

Abstract 5

Four-point bending tests have been a staple in many structural engineering experiments as a 6

reliable way of assessing the bending resistance of circular hollow sections, tubes and 7

cylindrical shells, and they continue to be widely performed. However, relatively little 8

attention appears to have been paid to quantify the effects of different boundary conditions 9

on the test outcome. In particular, the restraint or freedom given to the cross-section at the 10

ends of the specimen to ovalize can have a significant impact when the specimen is in an 11

appropriate length range. Ovalization is an elastic geometrically nonlinear phenomenon that 12

is known to reduce the elastic bending resistance by as much as half in long tubes or 13

cylinders. 14

15

This paper presents a short distillation of some recent advances in understanding the 16

buckling of cylindrical shells under uniform bending, identifying the strong influence of the 17

cylinder length on cross-section ovalization. A sample set of three-dimensional load 18

application arrangements used in existing four-point bending tests was simulated using 19

finite elements, allowing an assessment of the differences caused by pre-buckling 20

ovalization and its effect on the tested bending resistance. The study is limited to elastic 21

behaviour to identify the effect of ovalization alone in reducing the stiffness without 22

material nonlinearity. The outcomes demonstrate that maintaining circularity at the inner 23

load application points by appropriate stiffening has a significant effect. With freedom to 24

ovalize, a significant reduction in stiffness occurs, leading to much lower bending 25

resistance at buckling than may be achievable in practical applications. 26

27

Keywords 28

Four-point bending; tubular forms; nonlinear elastic buckling; ovalization; length effect; 29

rigid restraint. 30

31

2

1. Introduction 32

Cylindrical shells and tubes are widely used in structural engineering due to their optimized 33

geometry and ease of fabrication, with applications as structural members, piles, pipelines, 34

chimneys, wind towers and other structures where bending is the dominant condition. 35

Although structural members are rarely subject to uniform bending without a moment 36

gradient (Fig. 1a), uniform bending is still widely used as the reference condition to 37

establish the bending resistance. However, uniform bending throughout the length of a test 38

specimen is difficult to achieve experimentally. The classical test is in four-point bending, 39

leading to a uniform bending moment region with constant mean curvature between shear 40

zones at the ends (Figs 1b-d). 41

42

43

Fig. 1 – Different loading arrangements for two- and four-point bending tests. 44

45

Following a review of the literature, three broad classes of four-point loading test 46

arrangements were found. In the first, loads are applied directly onto the tube surface 47

through stiff loading heads or rollers (Fig 1e; Elchalakani et al., 2002a,b,c; Jiao & Zhao, 48

2004) without any local reinforcement of the cross-section. In the second, the potential for 49

local stress concentrations is alleviated with stiff mounting rings that transfer the loads 50

3

uniformly into the tube (Fig. 1f; Kiymaz, 2005; Guo et al., 2013). In the third, the loads are 51

introduced by flexible straps of finite width wound half way around the tube (Fig. 1g; 52

Gresnigt & van Foeken, 2001; van Es et al., 2016). The two-point loading arrangement of 53

Fig. 1a has also been used (Reinke et al., 2014). This paper explores the consequences of 54

these different loading arrangements on the pre-buckling ovalization and changes in 55

buckling resistance using computational tools. The constitutive law is kept as linear elastic 56

to ensure that only the fundamental geometric response and loss of stiffness affect the 57

outcome, which are important in thinner tubes and higher strength materials. 58

59

2. Relevant dimensionless groups 60

The computational study of Rotter et al. (2014) on the reference ‘two-point’ system (Fig. 61

1a), in which the ends were held rigidly circular and loaded by a moment, confirmed the 62

analytical predictions of previous authors (e.g. Calladine, 1983) that pre-buckling 63

ovalization in cylinders is governed by the dimensionless group Ωu: 64

uu

L t

r rΩ = (1) 65

in which Lu is the length of the uniform moment region while r and t are the midsurface 66

radius and thickness respectively. Since other test arrangements have a total length Ltot, this 67

loading arrangement may be defined as Lu = Ltot. Rotter et al. (2014) found that ovalization 68

affects the buckling resistance in cylinders longer than Ωu ≈ 0.5, ultimately reaching the 69

maximum achievable for bifurcation buckling and remaining mostly invariant at buckling 70

when the length exceeds Ωu ≈ 4. In a four-point bending test (Fig. 1b), the reduced length 71

of the uniform moment region is Lu = (1 – 2a)Ltot. One goal of this study was to find 72

whether the above ovalization length limits in terms of Ωu also apply to other loading 73

arrangements. The description follows the common test situation where the total length Ltot 74

is limited by space requirements. 75

76

The distribution of membrane stresses in the central and outer regions may be determined 77

simply from the membrane theory of shells. The shear membrane stress resultant Nzθ in the 78

moment gradient region (of length Lg = aLtot, with 0 ≤ a ≤ 0.5) and the axial membrane 79

stress resultant Nz in the uniform moment region are given by: 80

sinz

FN

rθ θ

π= and

2cos

g

z

FLN

rθ

π= such that ,maxz

FN

rθ

π= and

,max 2

g

z

FLN

rπ= (2) 81

4

Ignoring boundary effects, the classical elastic critical buckling values for these membrane 82

stress resultants are (EN 1993-1-6, 2007): 83

4, 20.75

z cl

g

t rtN Et

r Lθ

=

and

,0.605

z cl

tN Et

r

=

(3) 84

If the end zones are short, shear buckling is expected to occur before axial buckling in the 85

uniform moment zone. To avoid this possibility, the required minimum length of the outer 86

zones Lg = aLtot may be estimated by equating Nzθ,max / Nzθ,cl = Nz,max / Nz,cl., which leads to 87

the conclusion that the dimensionless parameter a should be greater than: 88

0.65z

tot

aτ − ≈Ω

where tottot

L t

r rΩ = (4) 89

Alternatively, for any defined value of a, the dimensionless length of the uniform moment 90

region Ωu should be greater than: 91

,

1 20.65

u z

a

aτ −

− Ω ≈

where

,

,

u z

u z

L t

r r

τ

τ

−

−Ω = (5) 92

93

3. Finite element models of four-point loading arrangements 94

High-fidelity finite element analyses were designed to explore the possible effects of 95

different potential boundary conditions on the elastic but geometrically nonlinear response 96

of cylindrical tubes of varying length using the ABAQUS (Simulia, 2017) commercial 97

software. The tubes were assumed to have a unit thickness t, a radius to thickness (r/t) ratio 98

of 100 and be made of isotropic steel with an elastic modulus of E = 205 GPa and a 99

Poisson’s ratio of ν = 0.30. Details of the reference ‘two-point’ system previously modeled 100

by Rotter et al. (2014) and used by Reinke et al. (2014) are presented in Fig. 2, where a 101

moment of magnitude Mcl (Eq. 6) was applied directly at a reference point that was rigidly 102

coupled to the circumferential edge of the cylinder, thus maintaining circularity of this 103

boundary throughout the analysis. The reference point was restrained against displacements 104

transverse to the cylinder axis (ux = uz = 0, assuming the axis convention shown in Fig. 2), 105

and against rotations perpendicular to the axis of bending as well as torsion (βx = βy = 0). A 106

plane of meridional symmetry at mid-span and a plane of circumferential symmetry in the 107

x-y direction were exploited for a computationally efficient quarter-shell model, common 108

practice for shells without torsional deformations and local bifurcation buckling modes 109

(Teng and Song, 2001; Song et al., 2004; Limam et al., 2010; Rotter et al, 2014; Xu et al., 110

5

2017; Wang et al., 2018). Careful mesh refinement was applied in the vicinity of the loaded 111

edge to correctly capture the local bending deformations that arise due to compatibility 112

there, and at mid-span to capture the local bifurcation buckling deformations that develop 113

on the compressed side. The mesh was considered ‘refined’ if the element size was smaller 114

than ~0.2√(rt), or about 10% of both the linear bending half-wavelength λ ≈ 2.444√(rt) and 115

the half-wavelength of a typical axisymmetric buckle 1.728√(rt) under meridional 116

compression (Rotter et al., 2014). The ‘two-point’ model was load-controlled using the 117

‘Riks’ arc-length path-tracing algorithm and employed the four-node reduced-integration 118

S4R thick shell element, widely used in similar computational shell buckling studies (see 119

above references). Small load increments of at most 0.02Mcl were used to facilitate in 120

convergence during the computation of the equilibrium path, and a check for negative 121

eigenvalues in the was made at every converged increment in the global tangent stiffness 122

matrix to ensure that no bifurcation was missed. Readers are invited to consult Rotter et al. 123

(2014) and Wang et al. (2018) for verification of the Riks procedure against standard 124

results for cylinder bending from the literature. 125

22 2

,2

1.813 1.9011

cl z cl

ErtM r N Ertπ

ν= ≈ ≈

− for ν = 0.3 (6) 126

Axis of meridional symmetry:

plane through Ltot / 2

Outline of the portion of tube

analysed with finite elements

(quarter shell model)

Reference point of load

application:

ux = uz = βx = βy = 0

Mz = M

z

x

y

Rigid body coupling

between reference

point and tube edge

M

r

Ltot / 2

Tension side

Compression side

2λ 4λ

Extra mesh refinement 2λ near

loaded edge and 4λ on either side of

axis of symmetry (λ ≈ 2.444√(rt))

127 Fig. 2 – Details of numerical models for the Rotter et al. (2014) ‘two-point’ system, with 128

only one half of the doubly-symmetric setup shown for compactness. 129

130

The four-point ‘rigid ring’ loading was modeled in a manner similar to the ‘two-point’ 131

system, as illustrated in Fig. 3. The outer and inner load application locations were modeled 132

as reference points, rigidly linked to the circumference of the tube at that meridional 133

location. The inner reference point was loaded by an enforced displacement U which 134

introduced a force into the cylinder transverse to its long axis. The outer reference point 135

was restrained against displacement and developed an equal and opposite reaction, thus 136

producing a four-point bending loading as shown in Fig. 1b. This arrangement is 137

6

representative of test setups where the load is transferred into the tube through very stiff 138

rings or plates (Fig. 1f) to maintain the circularity of the cylinder at these locations. A 139

quarter-shell model was again used for efficiency, with mesh refinement in the vicinity of 140

the loaded locations and at mid-span. To improve convergence, displacement-controlled 141

analyses were performed using an implicit dynamic solver with settings for ‘quasi-static’ 142

conditions, including Rayleigh damping parameters at a 5% damping ratio on sufficient 143

vibration modes to activate 90% of the mass in the x and y directions. The S8R thick shell 144

element was used due to its higher-order interpolation field allowing an improved 145

discretization of both shear and local meridional compression buckling deformations. 146

Axis of meridional symmetry:

plane through Ltot / 2

Outline of the portion of tube

analysed with finite elements

(quarter shell model)

z

x

y

Rigid body coupling

between both reference

points and tube wall Extra mesh refinement 2λ on

either side of loaded cross-section

and 4λ on either side of axis of

symmetry (λ ≈ 2.444√(rt))

r

Lg

U

Tension side

Compression side

Lu / 2

2nd reference point of

enforced displacement:

uz = βx = βy = 0; ux = U

Shear region

1st reference point of

load application:

ux = uz = βx = βy = 0

2λ 2λ 2λ 4λ

147 Fig. 3 – Details of numerical models for four-point bending ‘rigid ring’ loading, with only 148

one half of the doubly-symmetric setup shown for compactness. 149

150

By contrast, in the four-point ‘flexible strap’ loading shown in Fig. 4, the load was 151

introduced through flexible steel straps in frictionless contact with the tube surface around 152

half of the cylinder’s circumference. The straps are very thin curved plates, providing a 153

restraint to cross-sectional distortion only in proportion to their very low bending stiffness. 154

The outer strap was assigned a width of r while the inner strap was given a wider width of 155

2r to minimize stress concentrations near loading points. The mesh was refined in the 156

vicinity of the contact area of the tube with the straps to aid convergence, as well as at 157

mid-span to allow an accurate modeling of the local bifurcation buckling deformations. The 158

analyses were performed as ‘damped’ quasi-static with displacement control, found to be 159

beneficial for convergence when contact modeling is employed (see Kobayashi et al., 160

2012). A very small damping factor of 10-8

was adopted to minimize violations of 161

equilibrium and both S4R and S8R elements were used depending on the modeled length. 162

In Fig. 4, the lengths Lg and Lu are defined relative to the locations of the reference points 163

of load application for consistency with Fig. 1b. Lastly, the ‘roller’ arrangement (Fig. 1e) 164

was not modeled as it is really only useful for very thick tubes. Elchalakani et al. 165

7

(2002a,b,c) tested tubes with r/t ratios from ~7 to ~50) whose elastic fundamental path was 166

very short and the response was dominated by plasticity. 167

Axis of meridional symmetry:

plane through Ltot / 2

1st reference point of

restrained displacements:

ux = uy = uz = 0

r

Lg

Displacement

U

Tension side

Compression side Lu / 2

r

2r

2nd reference point of

enforced displacement:

uy = uz = 0; ux = U

Beam-type constraint between

reference point and strap edge: extra

mesh refinement 2λ on either side of

a strap and 4λ on either side of axis

of symmetry (λ ≈ 2.444√(rt))

2λ

2λ

2λ

‘Slave’ contact surfaces

on tube, ‘master’ contact

surfaces on straps

Shear

region

z

x

y

Rigid body coupling

between either reference

point and tube wall

4λ

Outline of the portion of tube

analysed with finite elements

(quarter shell model)

168 Fig. 4 – Details of numerical models for the four-point ‘flexible strap’ loading, with only 169

one half of the doubly-symmetric setup shown for compactness. 170

171

4. Exploration of model predictions for four-point loading with a = 0.25 172

A selection of equilibrium diagrams illustrating the relationship between the mean 173

cross-sectional moment and curvature within the uniform bending region are presented here 174

for a ‘four-point’ bending loading with load introduction points placed such that a = 0.25 175

(Fig. 1b). The applied mean cross-sectional moment was deduced from simple equilibrium 176

considerations as M = F.Lg (Fig. 1d) and was normalized by the classical elastic critical 177

buckling moment Mcl (Eq. 6) from linear theory which assumes that local bifurcation 178

buckling occurs when the most compressed fiber reaches Nz,cl (Eq. 3). The mean 179

cross-sectional curvature was deduced from the rotation βz of the inner reference points for 180

both the ‘rigid ring’ (Fig. 3) and ‘flexible strap’ (Fig. 4) loadings as ϕ = 2βz/Lu and was 181

normalized by the mean cross-sectional curvature at buckling predicted by linear theory ϕcl 182

(Eq. 7; Rotter et al., 2014). According to the simple membrane theory analysis, a 183

shear-dominated response should be expected in the outer moment gradient region if the 184

length of the uniform moment region Ωu is shorter than Ωu,τ-z ≈ 1.3 (Eq. 5) for a = 0.25. 185

( )2

2 20.605

3 1cl

t t

rrφ

ν= ≈

− for ν = 0.3 (7) 186

The computed equilibrium relationships for the ‘rigid ring’ loading with a = 0.25 are shown 187

in Figs 5a and 5b for cylinders with Ωu less than and greater than Ωu,τ-z ≈ 1.3 respectively. 188

8

As the focus in these analyses is on the elastic stiffness of the pre-buckling path and the 189

first buckling event, only an initial fragment of the post-buckling relationship is shown 190

(which is known to be difficult to follow computationally; see Kobayashi et al., 2012; 191

Sadowski and Rotter, 2014). A representative selection of corresponding buckled shapes is 192

shown in Fig. 6 together with annotations showing the positions of the load application 193

points. The equilibrium relationships and buckled shapes for cylinders at risk of shear 194

buckling within the moment gradient region (Ωu < Ωu,τ-z ≈ 1.3; Fig. 5a) illustrate that the 195

simple estimate for Ωu,τ-z acts as an upper bound, as buckling under meridional compression 196

in the uniform moment region may begin to dominate at lengths as short as Ωu ≈ 0.8, 197

significantly shorter than 1.3. All cylinders with Ωu > Ωu,τ-z ≈ 1.3 were found to buckle in 198

the uniform moment region (Fig. 5b). 199

200

The computational study of Rotter et al. (2014) on the reference ‘two-point’ system showed 201

that under elastic but geometrically nonlinear conditions, a cylinder that is too short to be 202

affected by ovalization buckles at a moment approximately 0.95Mcl and exhibits a very 203

linear pre-buckling equilibrium path. However, ovalization begins to significantly influence 204

the moment resistance for Ωu > ~0.5, manifest by a gradual reduction in the predicted 205

bifurcation buckling moment relative to Mcl and an increasingly nonlinear pre-buckling 206

equilibrium path as Ωu → ~4. The current predictions on the four-point ‘rigid ring’ loading 207

are consistent with this behavior: cylinders with Ωu < Ωu,τ-z exhibit a very linear 208

pre-buckling path regardless of buckling mode, while those with Ωu > Ωu,τ-z exhibit an 209

increasingly nonlinear pre-buckling path. For cylinders with ‘rigid ring’ loading and Ωu ≥ 210

~4, the equilibrium path closely follows that of the Brazier cross-sectional ovalization 211

response (curved dashed lines in Fig. 5b) to snap-through or limit point buckling in a long 212

circular hollow section (see Table 1 in Karamanos, 2002). This limit point is almost 50% 213

lower than the classical elastic critical moment Mcl, due to both the reduced lever arm 214

(raising the stress) and the increased local radius of curvature at the most ovalized 215

cross-section. Ovalization does not stop at the limit point but continues in the post-buckling 216

path. However, local bifurcation buckling intervenes at a moment ~5% lower than the 217

Brazier limit point moment MBraz (Eq. 8), so that the ovalization at the Brazier moment is 218

never quite reached. This limiting ovalization is here termed the bifurcation ovalization. 219

22

2

2 21.035 0.544

9 1Braz cl

E rtM Ert M

π

ν= ≈ ≈

− for ν = 0.3 (8) 220

9

221

222

Fig. 5 – Computed equilibrium relationships for the ‘rigid ring’ loading with a = 0.25 223

showing the mean bending moment in the uniform region normalized by the elastic critical 224

buckling moment Mcl against the mean cross-sectional curvature in the uniform region 225

normalized by its reference value at buckling ϕcl. a) Ωu < Ωu,τ-z and b) Ωu > Ωu,τ-z. 226

10

227

Fig. 6 – Buckled shapes for the ‘rigid ring’ loading at selected lengths (a = 0.25) – these are 228

shown with compression on the top side of the cylinder. 229

230

The computed equilibrium relationships for the ‘flexible strap’ loading with a = 0.25, 231

shown in Figs 7a and 7b for cylinders with Ωu less than and greater than Ωu,τ-z ≈ 1.3 232

respectively, indicate a very different behavior. The weak cross-sectional restraint offered 233

by the loading straps is predicted to be unable to prevent severe shear distortions (Fig. 8a) 234

from developing in very short cylinders loaded in this arrangement (e.g. Ωu < ~1), although 235

these lengths are admittedly outside the range of geometries known to the authors to have 236

been tested in this manner. The fundamental response of these very short cylinders is 237

predicted to be so dominated by shear distortions in the moment gradient region that these 238

do not exhibit beam-like behavior at all, and the mean moment-curvature equilibrium 239

relations are non-linear and do not even have the expected initial slope relative to the 240

flexural rigidity EI (Fig. 7a). Remarkably, the simple prediction of Ωu,τ-z (Eq. 5) still 241

appears to offer an upper bound estimate to the length of the uniform moment region Ωu at 242

which the qualitative behavior is likely to change from predominantly shear buckling under 243

the moment gradient to predominantly meridional compression buckling in the uniform 244

moment region. 245

11

246

247

Fig. 7 – Computed equilibrium relationships for the ‘flexible strap’ loading with a = 0.25 248

showing the mean bending moment in the uniform region normalized by the elastic critical 249

buckling moment Mcl against the mean cross-sectional curvature in the uniform region 250

normalized by its reference value at buckling ϕcl. a) Ωu < Ωu,τ-z and b) Ωu > Ωu,τ-z. 251

252

12

Cylinders with Ωu > Ωu,τ-z with ‘flexible strap’ loading are long enough to develop 253

beam-like behavior and exhibit a mean moment-curvature relationship that is initially linear 254

as expected for the flexural rigidity EI (Fig. 7b). However, the equilibrium paths rapidly 255

become nonlinear and follow the theoretical Brazier ovalization relationship for an 256

asymptotically long cylinder, with every cylinder experiencing bifurcation buckling at a 257

moment approximately 5% below MBraz without approaching the linear Mcl prediction (Eq. 258

6). A loading arrangement that cannot guarantee circularity of the load application points 259

must therefore be assumed to produce a fundamental elastic response that may exaggerate 260

the effect of ovalization. Thick cylinders, however, may begin to yield before significant 261

cross-sectional distortion occurs as the Brazier path is initially very linear. 262

263

264

Fig. 8 – Buckled shapes for the ‘flexible strap’ loading at selected lengths (a = 0.25) – these 265

are shown with compression on the top side of the cylinder. 266

267

13

5. Variation of the predicted buckling moment with length 268

A representative range of total cylinder dimensionless lengths Ωtot was investigated in 269

greater detail to allow the length of the uniform moment region to be varied from Ωu = 0.5 270

to 7 under different conditions, covering the full range of uniform region lengths identified 271

by Rotter et al. (2014) as lengths where ovalization may be expected. The placement of the 272

load application points was maintained at a = 0.25 for the ‘rigid ring’ loading, but a number 273

of different placements were investigated for the ‘flexible strap’ loading. These ranged 274

from placement near the outer load point (a → 0), giving a steep moment gradient and 275

more probable shear buckling, to those placed close to the mid-span (a → 0.5) for lower 276

shear and greater certainty of meridional buckling, namely a = 0.1, 0.15, 0.2 and 0.25. At 277

all values of a and Ωu, the full cylinder length is given by Ωtot = Ωu/(1 – 2a). 278

279

The relationships between the computed nonlinear elastic bifurcation buckling moment Mk 280

(normalized by Mcl, Eq. 6) and the dimensionless length of the uniform moment region Ωu 281

are shown in Fig. 9. The predictions for the reference ‘two-point’ loading from Rotter et al. 282

(2014) are shown for comparison. These reference findings show that under fully ‘uniform’ 283

bending, buckling occurs close to the classical prediction Mcl when Ωu < ~0.5, but in longer 284

cylinders ovalization increasingly depresses the moment until it reaches the full reduction 285

of Mcl,long ≈ 0.516Mcl beyond Ωu ≈ 4. Interpreting the new results by taking 286

Ωu = (1 – 2a)Ωtot, the four-point ‘rigid ring’ loading (shown for a = 0.25) reproduces this 287

reference ‘two-point’ relationship remarkably well, but shear buckling intercedes in shorter 288

cylinders (Ωu < ~0.7; Fig. 5a). By contrast, when the four-point ‘flexible strap’ loading is 289

used (shown for a = 0.1, 0.15, 0.2 and 0.25), the buckling moment Mk never rises above the 290

value ~0.516Mcl corresponding to buckling at the bifurcation ovalization (Fig. 7b). This 291

again indicates that flexible strap loading is not effective in restricting ovalization during a 292

bending test. 293

294

Recalling the possible limits identified above to ensure that the test does fail by buckling 295

under meridional compression in the uniform moment region instead of shear buckling in 296

the end zones, the data in Fig. 10 also reveals interesting differences between the different 297

loading arrangements. The value of Ωu,τ-z below which buckling is controlled by shear in 298

the moment gradient region is consistently represented conservatively by Eq. 5. Thus, for a 299

given total cylinder length Ωtot where meridional buckling is the required failure mode, the 300

14

location of the inner load application point defined by a = Lg / Ltot should be chosen to be 301

significantly greater than au,τ-z = 0.65 / Ωtot (Eq. 4). 302

303

304

Fig. 9 – Computed relationship between the elastic bifurcation buckling moment Mk / Mcl 305

and the dimensionless length of the uniform moment region Ωu. 306

307

A more important finding concerning flexible straps and similar four-point loading 308

arrangements is that ovalization is not effectively restricted at the load application points. 309

Even where buckling occurs by meridional compression (i.e. under uniform bending alone), 310

the buckling resistance is always less than the reference value of MBraz, regardless of length. 311

This indicates that all such tests lead to over-conservative estimates of the buckling 312

resistance that would be found in practical structures, where the joints and fixtures at the 313

ends of cylinders frequently provide a strong restraint against ovalization and lead to 314

resistances that are closer to those of ‘medium’ length cylinders where Ωu < ~0.5 (Rotter et 315

al., 2014). The flexible strap loading arrangement should therefore be recognized as 316

providing test data that may be unrepresentative and inappropriate for the development of 317

general design rules for cylinders under bending. 318

15

6. Variation of predicted ovalization with length for a = 0.25 319

The calculated deformed geometries at the instant before bifurcation of portions of the 320

cylinders within the uniform moment region were processed further to obtain distributions 321

of ovalization at bifurcation buckling. This was quantified using the ‘out of roundness’ 322

parameter U defined by EN 1993-1-6 (2007): 323

max min

nom nom

D DDU

D D

−∆= = (9) 324

where Dmax, Dmin and Dnom are the largest deformed, smallest deformed and original 325

undeformed diameters at the same section. Xu et al. (2017) used the assumptions of Brazier 326

(1927) and Karamanos (2002) to show that this parameter should attain a maximum value 327

of Umax ≈ 0.34 at the bifurcation ovalization. The distributions of U at a = 0.25 for different 328

Ωu for the ‘rigid ring’ loading (Fig. 10a) confirm the findings of Rotter et al. (2014) and Xu 329

et al. (2017) that there is negligible ovalization (U ≈ 0) in cylinders with a uniform moment 330

region shorter than Ωu < ~0.5. Ovalization attains the bifurcation value (U ≈ 0.34) by Ωu ≈ 331

5, corresponding to the length of the uniform moment region where the predicted bending 332

resistance falls to ~0.516Mcl (Fig. 9). This simple illustration further shows that if the end 333

boundaries of the uniform moment region are maintained rigidly circular in the four-point 334

arrangement (U → 0 at these locations), then the effect of ovalization in four-point bending 335

follows the same well-quantified relationship with the length of the uniform moment region 336

Ωu as that in the reference ‘two-point’ system. 337

338

However, as the ‘flexible strap’ loading offers only a very weak restraint against 339

cross-sectional distortion, cylinders are found to undergo significant ovalization at the 340

location of the innermost straps where the load is introduced (Fig. 10b). It is also 341

remarkable that this arrangement leads to a mid-span ovalization of U ≈ 0.44 that is very 342

much larger than the apparent ‘maximum’ of U ≈ 0.34 identified previously as the 343

bifurcation value. Thus, where the boundaries of the uniform moment region are not 344

maintained rigidly circular in a four-point bending test, there is a risk that the cylinder may 345

ovalize more than the bifurcation ovalization, regardless of total cylinder length. It is also 346

reasonable to surmise that such a high level of ovalization may have a more damaging 347

effect in cylinders where failure is in the elastic-plastic region. Where the outer zones are 348

too short, the cylinder may still suffer shear buckling in the moment gradient region for any 349

four-point loading arrangement. 350

16

351

352

Fig. 10 – Distribution of the ovalization parameter U along half of the uniform moment 353

region and its variation with Ωu for a) ‘rigid ring’ and b) ‘flexible strap’ loading. 354

355

356

357

17

7. Discussion 358

The finite element simulations presented in this paper on different four-point bending test 359

arrangements for cylindrical shells suggest that if circularity is maintained at the inner load 360

application points, either by using rigid loading plates or stiffening rings, the relationship 361

between the extent of pre-buckling cross-sectional ovalization and elastic buckling 362

resistance corresponding to the length of the uniform moment region is well understood and 363

follows that of simple uniform bending enforced by an end moment application, as used in 364

some tests. For such test arrangements, ovalization is fully restrained if the uniform 365

moment region is sufficiently short, although in cylinders loaded through rigid plates or 366

rings, shear buckling may occur in the outer zones if they are too short. 367

368

By contrast, where the ends of the uniform moment region are not rigidly retained as 369

circular, the cylinder is free to distort and to ovalize to its maximum extent at any length. 370

The consequences of this lack of restraint against distortion may include an exaggeration of 371

the effects of ovalization, leading to test resistances that may well be below and 372

unrepresentative of practical construction. In practical applications of circular hollow 373

sections, tubular piles, pipelines and other tubular structures subject to bending loads, the 374

situation is arguably closer to the fully-restrained end condition, achieved by encasement in 375

concrete or welded connections to diaphragms or stiffening rings. 376

377

The authors wish to be clear that they do not hold a preference for a particular four-point 378

bending test arrangement. It can be argued that a test arrangement that produces lower 379

bending resistances offers a valuable lower-bound upon which to establish conservative 380

design recommendations. However, if this reasoning is to be used, the authors believe that 381

the choice should be made consciously and with full awareness of the consequences of the 382

chosen boundary conditions on the test outcome. 383

384

8. Conclusions 385

This paper has presented a computational illustration of elastic cylindrical tubes and shells 386

under uniform bending, exploring the effect of four test loading arrangements that are 387

commonly used to determine the buckling resistance. The calculations identify the 388

consequences of different test arrangements in different buckling strengths. Under each 389

loading condition, the nonlinear elastic buckling resistance for both bending and shear was 390

18

explored, as well as the degree of ovalization that occurs before buckling for each test 391

length and each loading point location. The outcomes have been presented in dimensionless 392

form so that they may be applied to any geometry and any size of test. The chief focus has 393

been on ‘four-point’ bending tests in which two outer zones are subject to shear (moment 394

gradient), and a central zone between the load application points is under uniform bending. 395

396

This study should give useful guidance to the developers of test rigs for tubular members 397

and shells in choosing arrangements that either eliminate unwanted or excessive 398

out-of-round distortions, or permit these effects to be identified and quantified. Tests 399

should be arranged, as far as possible, to represent the conditions of practical structural 400

applications. The increasingly widespread use of thinner and higher strength cylindrical 401

tubes makes this conclusion particularly important, since the elastic and mildly plastic 402

behavior has a significant effect on the buckling strength in these cases. 403

404

Acknowledgements 405

The partial sponsorship of the China Scholarship Council is gratefully acknowledged. 406

407

References 408

1) Brazier L.G. (1927) “On the flexure of thin cylindrical shells and other ‘thin’ sections” 409

Proceedings of the Royal Society Series A, 116, 104-114. 410

2) Calladine C. (1983) “Theory of shell structures” Cambridge University Press. 411

3) Elchalakani M., Zhao X. & Grzebieta R. (2002a) “Bending tests to determine slenderness limits 412

for cold-formed circular hollow sections” Journal of Constructional Steel Research, 58(11), 413

1407-1430. 414

4) Elchalakani M., Zhao X. & Grzebieta R. (2002b) “Plastic slenderness limits for cold-formed 415

circular hollow sections” Australian Journal of Structural Engineering, 3(3), 127-141. 416

5) Elchalakani M., Zhao X. & Grzebieta R. (2002c) “Plastic mechanism analysis of circular tubes 417

under pure bending” International Journal of Mechanical Sciences, 44, 1117-1143. 418

6) EN 1993-1-6 (2007) “Eurocode 3, Part 1-6: Strength and stability of shell structures” Comité 419

Européen de Normalisation, Brussels. 420

7) van Es S.H.J., Gresnigt A.M., Vasilikis D. & Karamanos S.A. (2016) “Ultimate bending 421

capacity of spiral-welded steel tubes – Part I: Experiments” Thin-Walled Structures, 102, 422

286-304. 423

19

8) Gresnigt A.M. & van Foeken R.J. (2001) “Local buckling of UOE and seamless steel pipes” 424

Proc. 11th International Offshore and Polar Engineering Conference, Stavanger, Norway, June 425

17-22, pp. 131-142. 426

9) Guo L., Yang S. & Jiao H. (2013) “Behaviour of thin-walled circular hollow section tubes 427

subjected to bending” Thin-Walled Structures, 73, 281-289. 428

10) Jiao H. & Zhao X.-L. (2004) “Section slenderness limits of very high strength circular steel 429

tubes in bending” Thin-Walled Structures, 42, 1257-1271. 430

11) Karmanos S. (2002) “Bending instabilities of elastic tubes” International Journal of Solids and 431

Structures, 39, 2059-2085. 432

12) Kiymaz G. (2005) “Strength and stability criteria for thin-walled stainless steel circular hollow 433

section members under bending” Thin-Walled Structures, 43, 1534-1549. 434

13) Kobayashi T., Mihara Y. & Fujii F. (2012) “Path-tracing analysis for post-buckling process of 435

elastic cylindrical shells under axial compression” Thin-Walled Structures, 61, 180-187. 436

14) Limam A., Lee L.H. & Kyriakides S. (2010) “On the collapse of dented tubes under combined 437

bending and internal pressure” International Journal of Mechanical Sciences, 55, 1-12. 438

15) Reinke T., Sadowski A.J., Ummenhofer T. & Rotter J.M. (2014) “Large scale bending tests of 439

spiral welded steel tubes” Proc. Eurosteel 2014, Sep. 10-12, Naples, Italy. 440

16) Rotter J.M., Sadowski A.J. & Chen L. (2014) “Nonlinear stability of thin elastic cylinders of 441

different length under global bending” International Journal of Solids and Structures, 34(12), 442

1419-1440. 443

17) Sadowski A.J. & Rotter J.M. (2014) “Modeling and behaviour of cylindrical shell structures 444

with helical features” Computers and Structures, 133, 90-102. 445

18) Simulia Corporation (2017) “ABAQUS 2017” Commercial finite element software, Dassault 446

Systèmes, USA. 447

19) Song C.Y., Teng J.G. & Rotter J.M. (2004) “Imperfection sensitivity of thin elastic cylindrical 448

shells subject to partial axial compression” International Journal of Solids and Structures, 41, 449

7155-7180. 450

20) Teng J.G. & Song C.Y. (2001) “Numerical models for nonlinear analysis of elastic shells with 451

eigenmode-affine imperfections” International Journal of Solids and Structures, 38, 452

3263-3280. 453

21) Wang J., Sadowski A.J. & Rotter J.M. (2018) “Influence of ovalisation on the plastic collapse 454

of thick cylindrical tubes under uniform bending” International Journal of Pressure Vessels 455

and Piping, 165. 456

22) Xu Z., Gardner L. & Sadowski A.J. (2017) “Nonlinear stability of elastic elliptical cylindrical 457

shells under uniform bending” International Journal of Mechanical Sciences, 128-129, 458

593-606. 459