Shock sensitivity in the localised buckling of a beam on a nonlinear foundation: The case of a trenched subsea pipeline
Introduction
Beams and plates resting on a nonlinear foundation are known to buckle in a localised manner under (uni-)axial loading (Hunt et al., 1989; Kerr, 1978; Pocivavsek et al., 2008). If the foundation force is non-monotonic (of destiffening-restiffening characteristic) then typically at some point under increase of the load the localised buckle stops growing in amplitude and instead starts to spread, thereby creating a periodic pattern that gradually takes up most of the length of the structure (Hunt et al., 2000; Peletier, 2001). The phenomenon is governed by a Maxwell load that marks a ‘phase transition’ from the pre-buckled straight state to the periodic state. Similar behaviour of localisation followed by spreading is found in axially loaded cylinders buckling into a diamond-like pattern (Groh and Pirrera, 2019; Hunt et al., 1999), coiled twisted rods constrained to deform on a cylinder (van der Heijden et al., 2002), wrinkling of a thin film on a substrate (Jin et al., 2015) and folding of geological layers (Hobbs and Ord, 2012).
In most analytical work on these localisation phenomena only lateral resistance (elastic or frictional) is taken into account and the structure is free to slide in the axial direction. This frictionless sliding may be an acceptable approximation for the initial buckling pattern within a small-deflection theory as the amount of axial contraction goes as the deflection squared and is therefore small. However, as the buckling pattern spreads the end shortening can become large even for small deflections by the cumulative effect. Here we consider the case with axial resistance included using the subsea pipeline as a concrete and meaningful example.
Trenched subsea pipelines indeed offer an ideal example to explore the above complicated localisation phenomena in a realistic setting. In long pipelines localised buckling is the natural mode of buckling under the alignment conditions imposed by the longer structure at the ends of the buckle, and with the pipeline free to find its own lateral mobilisation length. In studies of pipeline buckling the effect of axial resistance has traditionally been included (Hobbs, 1984). Thermal effects as a result of axial friction lead to a coupling between axial and lateral deformation and we will explore its consequences for localised buckling.
Lateral foundation forces considered in the literature are often somewhat artificial, especially if they are treated as purely elastic. The pipeline problem offers a natural case for a realistic non-monotonic lateral resistance. Embedment of the pipeline, due to its own weight, produces a softening behaviour after breakout, while the trench walls give rise to restiffening behaviour at larger deflections. On the downside, due to the frictional nature of the resistance the proposed foundation characteristic is only valid in situations in which the displacements grow monotonically. We show that, nevertheless, valuable results can be obtained, by focussing on stability of the trivial state under finite perturbations (shocks).
The increasing global demand for oil and gas pushes the exploitation of hydrocarbon sources into ever deeper water. Long subsea pipelines are consequently becoming increasingly important for the transport of the hydrocarbon products from deep sea to the shore. To prevent solidification of the wax fraction in these products, subsea pipelines are required to operate under high-temperature and high-pressure conditions. This may lead to excessive axial compressive forces and localised lateral buckling is well-known to occur in exposed subsea pipelines (DNV-RP-F110, 2018). During their whole operational life pipelines undergo regular start-up and shut-down cycles. The resulting thermal cycles induce repeated localised buckling, which causes soil berms to accumulate in front of the pipeline's motion. This leads to increased soil resistance (Wang et al., 2017) and after several thermal cycles the pipeline appears to buckle in a trench. Another case is that the pipeline is laid in an open trench without backfill for mitigating hydrodynamic loads (DNV-RP-F109, 2011). Here we investigate the effect of the trench wall on lateral thermal pipeline buckling.
Much of the analytical research on lateral, as well as upheaval, subsea pipeline buckling in the literature is based on Hobbs's work (Hobbs, 1984). In this work the pipeline is modelled as a beam-column and the lateral resistance force is assumed to be constant, independent of the deflection. Based on this approach, Taylor and co-workers derive analytical solutions for ideal submarine pipelines by considering a deformation-dependant (nonlinear) resistance model (Taylor and Gan, 1986a) as well as analytical solutions for lateral and upheaval buckling of pipelines with initial imperfections (Taylor and Gan, 1986b). In Hong et al. (2015) lateral buckling modes of pipelines with imperfection are compared against finite-element solutions. A nonlinear soil resistance model, allowing for partial pipeline embedment, is used in a lateral buckling analysis in Zhang and Guedes Soares (2019).
In all the above pipeline research the buckling mode is taken to be given by a solution of the linearised equations. The buckle profiles consist therefore of a (small) number of essentially sinusoidal lobes. It is good to realise though that localised buckling, with exponentially decaying deflection, is an intrinsic property of perfect elastic structures resting on a nonlinear foundation. This localised buckling is quite different from (Euler) column buckling. It is described by a Hamiltonian-Hopf bifurcation rather than the pitchfork bifurcation of column buckling. An important consequence is that unlike the critical load for column buckling, which depends strongly (quadratically) on the length of the structure, the critical load for localised buckling does not depend on this length (although the structure of course has to be long enough to support a localised buckle). The critical load for localised buckling is in fact lower than that for Euler buckling, although even this load is generally not reached as localised deflection is initiated by imperfections or perturbations.
Genuine localised buckling is considered in Wang and van der Heijden (2017) where we compute bifurcation diagrams of localised solutions (homoclinic orbits) for a partially embedded pipeline on an even seabed. Localised solutions are also considered in Zhu et al. (2015) although the quoted boundary conditions do not maintain localisation as parameters of the system are varied. In Zeng and Duan (2014) a nonlinear and non-monotonic lateral soil resistance is employed to model partial embedment and homoclinic orbits are explicitly computed. Although the authors are not motivated by trenched pipelines, and do not consider thermal buckling, they find some of the homoclinic phenomena that we report on in this paper. However, they give limited physical interpretation of their results and do not investigate the stability implications.
Here we study localisation phenomena for a trenched pipeline with both lateral and axial resistance included. In the absence of axial resistance the pipeline is effectively under a dead compressive load. We identify a critical Maxwell point marking a ‘phase transition’ to a periodic buckling pattern with the pipeline bouncing between the trench walls. Associated with this critical point we find load-displacement curves with high sensitivity of solutions to small perturbations, giving rise to what has been called shock sensitivity (Thompson and van der Heijden, 2014). So, unlike the usual practice of reading load-displacement bifurcation diagrams, with or without imperfections included, as quasi-static processes that might encounter linear instability under infinitesimal perturbations, we are here interested in nonlinear instability phenomena with the pipeline being forced, by external finite disturbances, out of a linearly stable state and into another stable state. In particular, we are interested in the energy barrier, represented by an intermediate unstable ‘mountain pass’ state, to be overcome for such a transition from the straight pre-buckled state to a localised state, and its dependence on the axial resistance.
The organisation of the paper is as follows. In Section 2 we give details of the thermal, elastic and soil modelling of our trenched pipeline and identify the central Hamiltonian-Hopf bifurcation with associated homoclinic orbits describing localised buckling modes. In Section 3 we compute these homoclinic orbits and their bifurcation diagrams (load-displacement curves). By considering energy, in Section 4 we then discuss pipeline stability implications of these diagrams under various types of loading, dead, rigid and thermal. Thermal loading is found to interpolate between dead and rigid loading as the axial resistance increases from zero. In the process, shock sensitivity gradually diminishes. We quantify the shock sensitivity by computing energy barriers as a function of the axial soil resistance. Section 5 briefly discusses the dependence of our results on the breakout coefficient, which controls the non-monotonicity of the lateral resistance, and Section 6 closes the study with a summary and discussion.
Section snippets
Thermal pipeline buckling
We imagine a pipeline laid in a trench and subjected to a total temperature difference T0 between the fluid flowing inside the pipe and the environment. If the ends of the pipe are unrestrained then under an increase of the temperature difference the pipe will expand axially. This expansion will be resisted by friction between pipe and seabed (and surrounding soil). If the soil resistance for axial movement is constant, say fA, then a compressive force will build up in the pipe, which will
Homoclinic solutions and their bifurcations
For P < Pcr, we compute approximate (half) homoclinic solutions as shown in Fig. 5(a) by formulating a shooting problem on a truncated x interval [, 0]. Here L, the half length of the homoclinic solution, is chosen large enough that the solution is well-localised in the sense that it is very nearly decayed to the trivial straight solution j at . For details of the shooting method we refer to Wang and van der Heijden (2017). The (realistic) physical parameters used in this study are listed
Stability under finite disturbances for various types of loading
The complicated intertwining bifurcation behaviour in Figs. 6 and 10 is familiar from previous studies, especially those of the Swift-Hohenberg equation, and is commonly referred to as homoclinic snaking (Avitabile et al., 2010; Burke and Knobloch, 2006, 2007; van der Heijden et al., 2002; Woods and Champneys, 1999). An explanation of the behaviour in terms of the underlying phase-space dynamics as well as a physical interpretation is given in the Appendix. This in particular clarifies why a
Effect of varying μbrk
In this section we briefly investigate the dependence of the various phenomena observed in the previous sections on the trench parameter μbrk. This μbrk, representing the breakout resistance of the soil, is the source of non-monotonicity of the lateral resistance F, so one would expect the complicated homoclinic tower to disappear when μbrk vanishes and F becomes monotonic. However, our soil resistance model, Eq. (17), is such that even for 0 the force F has a nearly horizontal plateau,
Summary and discussion
We have studied localisation in a slender structure resting on a nonlinear foundation that provides both lateral and axial resistance. The lateral resistance is of destiffening-restiffening type known to lead to complex localisation and jump phenomena near a Maxwell critical load. By considering the coupling of lateral and axial deformation we are able to investigate the mitigating effect of axial resistance on these phenomena. To have a concrete and realistic example we consider the case of a
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Author Statement
The listed authors are the sole authors of this paper.
Acknowledgement
ZW acknowledges support by the International Postdoctoral Fellowship Program of the China Postdoctoral Council (No. 20180049).
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