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
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