Dynamic response of free span pipelines via linear and nonlinear stability analyses
Introduction
Free spans are commonly encountered in underwater pipelines that lay on the ocean bottom due to a frequently uneven seabed. This particular configuration in association with ocean currents and soil stiffness can greatly accentuate fatigue failure. As a result, frequency analyses become a crucial part of pipeline design for ensuring pipeline integrity and preventing expensive maintenance. Nowadays, the DNV-RP-F105 standard (DNV-RP-F105, 2006) has been used as major industry guideline in configurations involving free spanning pipelines. In spite of the relevance of this international standard, it is limited to a number of free span scenarios, and in many cases alternative methods are required.
One common approach to derive the dynamic response of free spanning offshore pipelines is to use approximate analytical methods. These are often based on a simplified beam theory and consequently of easy application (Fyrileiv and Mork, 2002). Approximate methods are still in use, as presented in (Sollund et al., 2015a), where closed-form analytical expressions for the fundamental frequency of short free spans were obtained using dimensional analysis. The inherent simplicity of these type of methods has drawbacks, since their approximate character can lead to inaccurate solutions, limiting their application to simpler problems.
On the other end of the spectrum, traditional numerical schemes such as Finite Element Methods (FEM) (Huges, 1987; Zienkiewicz et al., 2013), are capable of solving extremely complex problems while still providing good accuracy. Consequently, these methods have been widely used for analyzing subsea pipeline dynamics (Forbes and Reda, 2013; Santos et al., 2014). Naturally, these methods often require a significant amount of CPU time to meet strict accuracy requirements.
Combining the advantages of numerical and analytical approaches, semi-analytical methods arise as a powerful alternative to the former ones. They can provide improved accuracy for complex nonlinear problems, as compared to analytical methods, while still consuming a reasonably small amount of CPU time when compared to traditional numerical schemes. When looking into free span problems, there are some literature studies that employ methods of this kind. (Vedeld et al., 2013) presented a semi-analytical solution for the harmonic response of pipelines, based on the Rayleigh-Ritz approach in combination with a displacement field taken as a Fourier series. In subsequent studies, the dynamic response of both free spanning pipelines (Sollund et al., 2015b), as well as multi-span pipelines (Sollund et al., 2014), was analyzed using the same methodology.
Two well-established semi-analytical methods that are particularly interesting for the analysis of underwater pipeline dynamics are the Linear Stability Analysis (LSA) and the Generalized Integral Transform Technique (GITT). The LSA methodology provides the asymptotic response of dynamical systems when their steady-states are subjected to small disturbances. This is achieved by transforming the original non-linear governing equations into a simpler linear and modal form that is then numerically solved. In spite of this simplification the method can still provide highly accurate solutions and be applied to complex problems. Most LSA applications involve fluids mechanics problems, such as convective and absolute asymptotic instabilities (Huerre and Monkewitz, 1990), transient disturbance growth due to non-monotonic dynamic behavior (Schmid, 2007) and the evaluation of the stability of dynamical systems with two and three-dimensional steady-states (Theofilis, 2011). There are, however, studies related to fluid-structure interaction problems as well, as seen in (Dowell and Hall, 2001).
While LSA involves the solution of a simpler model derived from the original problem formulation, the GITT (Cotta, 1993) addresses the original model in its complete form, in which the sough solutions are expressed as series of orthogonal eigenfunctions. This hybrid analytical-numerical technique is akin to the FEM, as the solutions are written in terms of basis functions. However, the usage of orthogonal bases leads to solutions with higher accuracy. Furthermore, it requires no domain discretization, many times leading to smaller CPU times. The GITT has found numerous applications in heat and fluid flow over the past decades. More recently, it has also been applied to a variety of dynamical systems, such as wind-induced vibrations in overhead conductors (Matt, 2009), dynamics of axially moving beams (An and Su, 2011, 2014a), dynamics of damaged structures (Matt, 2013), vibration in orthotropic plates (An and Su, 2014b), pipe dynamics in the presence of two-phase flow (Gu et al., 2013; An and Su, 2015) and general one-dimensional distributed systems (Matt, 2015).
It is important to highlight that the combined application of Integral Transforms with Linear Stability Analysis has been previously employed. This was first presented in studies focused on the onset of natural convection in porous cavities (Alves et al., 2002) and in superposed fluid and porous layers (Hirata et al., 2006). The former employed a linearization of the integral-transformed system, whereas the latter linearized the original model prior to the integral transformation processes. This second alternative, which involves solving the differential eigenvalue problem resulting from the application of LSA using the GITT, has also become common in recent years (Sphaier and Barletta, 2014; Sphaier et al., 2015; Alves and Barletta, 2015).
Although there are a few literature studies that employ generalized integral transforms to problems involving mechanical systems, there are no studies related to the application of the GITT to free spans in submerged pipelines, let alone combined GITT-LSA applications to these types of problems. In this context, the main purpose of this work can be summarized as two-fold: propose LSA as an alternative to approximate analytical methods capable of providing accurate solutions to complex problems while maintaining simplicity of use and small CPU time requirements, and propose GITT as an alternative to traditional discretization methods that can simulate nonlinear dynamical systems with lower CPU time requirements while still providing highly accurate solutions to complex problems. These are demonstrated as feasible alternatives through the analysis of the dynamic response of a free span pipeline on an uneven seabed that is subjected to axial force, soil stiffness and seawater movement. Numerical results are then compared to the DNV-RP-F105 standard (DNV-RP-F105, 2006), and new data for configurations not covered by this standard are provided.
Section snippets
Mathematical model
Consider a pipeline laying on a seabed subjected to soil conditions and weak ocean currents, as sketched in Fig. 1 (a), and the corresponding physical model illustrated in Fig. 1(b), where T is the axial force, L is the pipeline length, k is the soil stiffness and are the spatial coordinates.
The governing equation for the considered model is given bywhere E is Young's modulus, I is the moment of inertia, is
Linear stability analysis
Dynamical systems subject to supercritical bifurcations can be analyzed through Linear Stability Analysis, where it is assumed that any deviation from a known steady-state of this system is caused by small disturbances. Such an assumption allows the derivation of a linearized disturbance governing equation. These equations can be further simplified by assuming that the disturbances take the form of normal modes in homogeneous coordinates. According to these steps, the pipeline dimensionless
Generalized Integral Transform Technique
The original nonlinear problem defined by Eq. (3) can be solved with the Generalized Integral Transform Technique (Cotta, 1993). In the context of the Linear Stability Analysis presented in the previous sections, GITT can also be understood as a technique for a nonlinear stability analysis of the same model. The method seeks a solution for this problem in terms of an orthogonal eigenfunction expansion and an associated transformation relation, known as the inverse-transform pair:
Results and discussion
Both methodologies proposed in this paper for the study of pipeline dynamics must be verified and validated. Verification is performed by direct comparison to numerical results generated with a Finite Element Method (FEM) implementation. Since the LSA and GITT implementations were coded within the Mathematica environment (Wolfram, 2003), a FEM code was also built using Mathematica. As a result, proper CPU time comparisons between all three methods become possible. The domain used by the FEM is
Conclusions
The most commonly used methods found in the recent literature to solve differential equation models for pipeline dynamics have important limitations. Approximate analytical methods are one example. They are easily employed, but their results often have low accuracy and their utilization is often restricted to simple problems. Finite Element Methods (FEM) are another example. They can be very accurate and also be used in complex problems, but generating solutions is often time consuming. Two
Acknowledgements
The authors would like to acknowledge the financial support by the Brazilian funding agencies CAPES, CNPq and FAPERJ.
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