Influence of ovalisation on the plastic collapse of thick cylindrical tubes under uniform bending

An accurate assessment of the bending resistance of thick cylindrical metal tubes is necessary for the safe and efficient design of pipelines, piles, pressure vessels, circular hollow sections and other common tubular structures. Bending tests continue to be widely performed as part of many engineering research programmes, but despite their ubiquity they often generate results that are difficult to interpret. Discrepancies from the attainment of the classical full plastic moment are common and often attributed to a mixture of ovalisation, local buckling, imperfections and strain hardening. However, the effects of these phenomena are yet to be quantified in isolation, even for a system as classical as a cylinder under uniform bending. The goal of this computational study is to quantify the extent to which geometrically nonlinear effects, specifically ovalisation and bifurcation buckling, may depress the resistance of a thick perfect cylinder under uniform bending that would otherwise be expected to attain the full plastic moment. Simulations are performed using two- and three-dimensional finite element models with a simple ideal elastic-plastic material law that excludes the influence of strain hardening. Additionally, the study aims to arm designers of test programmes on the bending of tubulars with ‘rules of thumb’ to approximately quantify the likely influence of tube length on their results, recently shown to be an important parameter controlling geometric nonlinearity. For thick tubes, ovalisation at the plastic limit state under bending is found to be almost negligible. Three-dimensional models using 3D shell finite elements were employed, but an additional set of analyses was performed using a 2D generalised plane strain formulation to investigate the asymptotic behaviour of ‘infinitely’ long tubes. In all cases, an elastic modulus of E = 200 GPa and Poisson ratio of ν = 0.3 were assumed.


Introduction
Research into the response of tubulars under bending continues to be widely conducted despite the apparent simplicity of the cylindrical geometry and the apparent ease with which this allows a first estimate of the elastic buckling and plastic collapse resistances.
A significant database of test results has been accumulated over several decades, supported by an increasingly rich body of research based on theoretical and computational studies. An illustration of 141 test results for tubes under bending gathered from the literature is shown in Fig. 1, where the dimensionless characteristic resistance M k / M pl of each tube is plotted against its dimensionless length Ω = (L/r)√(t/r) and dimensionless cross-sectional slenderness d/(tε 2 ). Here L, r and t are the cylinder length, mid-surface radius and thickness respectively, and σ y is the yield stress such that ε 2 = 235/σ y . The full plastic moment M pl is given by: The measured values of the control test 0.2% proof stress σ y were used in constructing the illustration in Fig. 1, but only nominal values for the geometry since measured values are usually not available. The data is taken from publications from research programmes on carbon steel tubes under four-point bending tests both historical (Korol, 1979;Stephens et al., 1982;Gresnigt and van Foeken, 2001) and recent (i.e. RFCS Combitube (2014) and follow-up tests in Rotterdam; Peters et al., 2017). The reader is invited to consult the reference list of Buchanan et al. (2016) for the sources of the studies shown in Fig. 1 but not referenced here for brevity.
The parameter Ω has a long history, being closely related to the well-known Batdorf (1947) parameter Z = L 2 /(rt)√(1-ν 2 ) (ν is the Poisson ratio), but the current notation was coined by Calladine (1983). It was later confirmed by Rotter et al. (2014) that Ω is the relevant dimensionless group that controls pre-buckling ovalisation in cylinders under uniform bending. Ovalisation was found to be entirely absent from the elastic response of cylinders shorter than Ω = 0.5 but plays an increasingly important role for cylinders longer than this, becoming fully-developed (invariant with further increases in length) beyond Ω ≈ 7 where it causes a reduction in the bifurcation buckling moment of close to M A N U S C R I P T

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3 50% under elastic conditions. It should be stressed that ovalisation occurs on the fundamental equilibrium path and precedes the inelastic response. However, while its effect on the elastic buckling capacity of thin cylinders is now well understood (Brazier, 1927;Karamanos, 2002;Rotter et al., 2014), its influence on the plastic limit moment in a thick tube is significantly less so. length Ω = (L/r)√(t/r) and dimensionless cross-section slenderness d/(tε 2 ), assuming a measured yield stress but nominal geometry.
It may be seen in Fig. 1 that virtually all tests exhibit Ω > 0.5. Consequently, it is to be expected that their fundamental response may be, to a greater or lesser extent, affected by geometrically nonlinear effects such as ovalisation. Yet despite this, a significant portion of tests of 'stocky' tubes with d/(tε 2 ) < ~70 (approximately the internationallyaccepted upper slenderness limit for 'Class 1 or 2' sections; see Table 13 in Gresnigt and Karamanos;2008) attain and exceed the full plastic moment even for Ω >> 7. This suggests that ovalisation is likely to only have a negligible effect on the plastic limit moment of thick tubes. This hypothesis is explored computationally in this study.

2.
Modelling with finite elements

Scope of the study
This study seeks to quantify computationally the probable influence of ovalisation on the moment capacity of unpressurised thick cylindrical tubes under uniform bending when dominated by plasticity. A comprehensive set of finite element analyses was performed using the ABAQUS software (Dassault Systèmes, 2017) to investigate the following: • Cylinders with a range of lengths between those that are short enough to be unaffected by ovalisation but long enough to be free of boundary restraint effects (5r/t < Ω < 0.5) to those long enough to experience fully-developed ovalisation before elastic bifurcation buckling (Ω > 7).
Two sources of great ambiguity, the influences of imperfections and strain hardening, were purposefully omitted to allow the influence of geometrical nonlinearity to be studied in isolation. For each combination of parameters, the following computational analyses were performed (EN 1993-1-6, 2007): • Materially nonlinear but geometrically linear analysis (MNA) to obtain the classical plastic limit moment of the perfect tube.
• Geometrically nonlinear but materially linear analysis (GNA) to obtain the nonlinear elastic bifurcation buckling resistance of the perfect tube.
• Geometrically and materially nonlinear analysis (GMNA) to obtain the characteristic nonlinear elastic-plastic resistance of the perfect tube.
Three-dimensional models using 3D shell finite elements were employed, but an additional set of analyses was performed using a 2D generalised plane strain formulation to investigate the asymptotic behaviour of 'infinitely' long tubes. In all cases, an elastic modulus of E = 200 GPa and Poisson ratio of ν = 0.3 were assumed.

Conventional 3D shell model
A previous study by the authors (Sadowski and Rotter, 2013) showed that shear-flexible 3D shell elements such as the reduced-integration linear S4R element are capable of accurately predicting the response of tubes as thick as d/t = 20 (r/t = 10) under uniform bending efficiently, with thick shell elements both predicting the same response as solidcontinuum elements and reproducing test results with high fidelity at only a fraction of the computational cost. Consequently, the S4R element was used in all 3D models in this study, with details of load application and boundary conditions illustrated in the schematic in Fig. 2a. In the interest of brevity, the reader may to consult the authors' previous publications for modelling details (Sadowski and Rotter, 2013;Rotter et al., 2014;Xu et al., 2017) which employed the same arrangement and meshing schemes.

Generalised 2D plane strain model
It is known that a long elastic perfect cylinder under uniform bending will fail by local bifurcation buckling on the flattened compressed side at a moment approximately 5% below the Brazier elastic limit moment M Braz (Eq. 2; Karamanos, 2002;Rotter et al., 2014;Xu et al., 2017), and far below the elastic critical bifurcation buckling moment M cr (Eq. 3). This reduction is due to an increased local circumferential radius of curvature reducing the critical buckling stress and lever arm at the most ovalisation cross-section. Both ovalisation and local bifurcation buckling thus inevitably afflict a buckling prediction from a 3D analysis. By contrast, a 2D generalised plane strain model allows the direct simulation of the cross-section of an infinitely-long tube by enforcing a constant longitudinal curvature and thus truly uniform bending (Fig. 2b), with only limit point snap-through buckling possible under a loss of cross-sectional stability. This allows an accurate evaluation of the asymptotic behaviour as Ω → ∞, unachievable with 3D models. The 4-node bilinear quadrilateral reduced-integration CPEG4R element was used here, with at least 4 elements through the thickness.

Comparison between 3D and 2D finite element models and reference theory
A brief illustration of the validity of the predictions for a tube with d/t = 200 (r/t = 100) by the above two modelling approaches is presented in Fig. 3. Here, 2D GNA predictions of the nonlinear elastic moment-curvature relationships (made dimensionless using Eqs 3 and 4, where φ cr is the mean longitudinal curvature at buckling as predicted by linear bending theory) are compared with 3D GNA predictions for a perfect cylinder at Ω = 7. Both GNAs are additionally compared with the algebraic moment-curvature relationship of Brazier (1927) as reported in Table 1 of Karamanos (2002), It is shown that a 2D generalised plane strain model accurately reproduces the Brazier relationship and undergoes limit point buckling at a moment very close to the Brazier reference result M Braz (Eq. 2). It also produces an almost identical response to the 2D ovalisation analysis of Karamanos (2002). In both cases, the Brazier moment M Braz is attainable only in very thin shells, so neither 2D analysis comes closer than within 2% of it. By contrast, although the 3D model follows almost the same initial nonlinear path, and thus correctly captures the fully-developed pre-buckling ovalisation response when Ω = 7 or beyond as the 2D model, it only reaches a moment approximately 6% below M Braz (or 49% below M cr ) before bifurcating onto a distinct post-buckling path.

Computed relationship between the buckling moment and length
The predict a very similar ultimate moment M k (Fig. 5, top-left). For very long and thick tubes susceptible to ovalisation (Ω > 7), the Brazier moment M Braz will also significantly exceed M pl , which then also controls as the limiting resistance. Further, as the initial portion of the Brazier equilibrium path is very linear (Fig. 3), the response would have to proceed very far up this path in order to reach the nonlinear portion where ovalisation begins to noticeably degrade the pre-buckling stiffness. The response of a thick tube under cannot bending cannot achieve this, being limited by M pl << M cr or M Braz , and thus it remains confined to a linear region of the equilibrium path that remains ovalisationfree such that MNA ≈ GMNA for all Ω (Fig. 5, top-left). However, local buckling may still cause a GMNA prediction to fall slightly beneath that of a MNA (Fig. 4, top-right). The effect of making the tube more slender by increasing the d/t ratio or the yield strength σ y is to permit a convergence of M pl to either M cr or M Braz such that the tube's response begins to cover an increasingly large portion of the theoretical elastic equilibrium path (Fig. 4, bottom). In the case Ω → 7, this leads to an increasing involvement of the upper nonlinear portions of the path as M → M Braz . Thus, as Ω, d/t and σ y increase, Fig. 5 shows a growing divergence between MNA and GMNA, with the GMNA predictions becoming increasingly affected by pre-buckling ovalisation. The modern tendency to use increasingly higher-strength steels in circular hollow sections M A N U S C R I P T

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11 should be seen in this context, as their response to bending will involve a larger portion of the nonlinear fundamental elastic path at any Ω and thus risks being more affected by ovalisation than a tube of a lower grade of steel. In all cases, the theoretical GNA prediction (dotted lines in Fig. 5) acts as an upper bound to the GMNA predictions but not the MNA ones, and for very thin cylinders GNA ≈ GMNA.

Computed relationship between the buckling moment and crosssection slenderness
These results consider only tubulars that are long enough to exhibit the most nonlinear possible fundamental elastic path corresponding to 'fully-developed' ovalisation (i.e. GMNA are presented in Fig. 6. The 3D GMNA system was analysed at Ω = 7, while the 2D plane strain GMNA system corresponds to an 'infinitely' long tube. Although the focus is nominally on thick tubes with low d/(tε 2 ), the relationship is extended to very slender cylinders for completeness. In the slenderness domain widely accepted as 'stocky' (d/(tε 2 ) ≤ ~70), the difference between MNA and both types of GMNA analyses is almost negligible and thus the tube fails essentially by plastic collapse at a moment very close to M pl with very little influence of any geometrically nonlinear effect. This is entirely consistent with the tendency observed in test results (Fig. 1). With increasing d/(tε 2 ), the 2D GMNA predictions approach M Braz asymptotically, whereas the 3D GMNA predictions settle on a moment approximately 5% below this moment.
The same data is presented in an alternative format in terms of M k / M pl against d/(tε 2 ) in Fig. 7. The GMNA predictions suggest that M k may never quite reach M pl as d/(tε 2 ) → 0, a consequence of no strain hardening in the assumed material law (see Figure 1 in Sadowski et al., 2015). Real steels exhibit strain hardening, and thus real tubes attain M pl at a finite value of d/(tε 2 ). The ideal elastic-rigid plastic assumptions employed here thus provides an upper bound on the damaging influence of ovalisation on M k , since any strain hardening would cause an increase in M k bringing it closer to M pl . Under this conservative assumption, Fig. 7 shows that the reduction on M k due to pre-buckling geometric nonlinearity is less than 5% for tubes with d/(tε 2 ) below 150. At large section M A N U S C R I P T A C C E P T E D slendernesses, however, the deleterious effect of pre-buckling geometric nonlinearity increases steadily. At d/(tε 2 ) ≈ 250 (roughly the upper limit of test specimens in the past; Fig. 1), the reduction in M k from M pl due to geometric nonlinearity is approximately 20%, although only 10% is attributable to ovalisation with the remainder due to bifurcation buckling on the compressed side.
The authors stress that these predictions relate to the very conservative condition of fully-developed ovalisation (Ω > 7) under uniform bending. Many of the test results conducted in the past (Fig. 1) are significantly shorter than this, thus the influence of ovalisation on the moment capacity may be expected to be milder. Additionally, conditions of truly uniform bending are rare and the effect of a moment gradient is to mollify the nonlinearity of the fundamental path due to ovalisation (Wang and Sadowski, 2018), reducing its detrimental effect on thick tubes yet further.

Conclusions
This paper has presented a detailed computational investigation that attempted to quantify the extent to which ovalisation may be expected to decrease elastic-plastic moment resistance of tubes under bending. This was achieved by a comprehensive parametric study of idealised perfect tubes with an elastic-plastic material law and uniform bending through a selection of different types of computational analyses. It is hoped that this study will provide guidance to designers of test programmes and drafters of structural Standards on the potential for ovalisation to affect the plastic resistance behaviour of thick tubulars and hollow sections under bending loads.
It is found that even under the most conservative conditions of no strain hardening, no pressurisation, uniform bending and 'infinite' length, ovalisation is likely to be responsible for only a negligible reduction in the reference full plastic moment capacity M A N U S C R I P T

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14 in a thick tubular. For thick tubes with slenderness up to d/(tε 2 ) up to ~100, the reduction is less than 5%. Any significant negative discrepancies of tested thick tubulars with the full plastic moment in this slenderness domain are therefore likely attributable to other reasons, most likely imperfections in the tube or errors in the testing process. The influence of geometric imperfections is currently being investigated by the authors and will be quantified in an upcoming study.

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