Non-linear vibration of free spanning subsea pipelines with multi-dimensional mid-plane stretching

In the absence of full-length support, subsea pipelines can experience significant oscillations due to flow induced vortex shedding that may result in the accumulation of fatigue damage over time. Such oscillations reflect the intricate interplay between the structural characteristics of the pipeline and the unsteady flow field, which can generate periodic shedding of vortices and pulsating forces. Conventional finite element models and/or empirical relations (e.g. codes of practice) are usually sufficient to predict the frequencies and modeshapes of vibrating free-spanning pipelines, which facilitates the assessment of their fatigue damage. However, the effects of multi-dimensional mid-plane stretching (i.e. coupled non-linear deformations) on the stability and performance of pipelines are still unknown. Here, we formulate the governing equations of free spanning pipelines experiencing lateral and transversal non-linear deformation that apply to in-line and cross-flow vortex induced vibration (VIV), investigate the effect of lateral mid-plane stretching on the response of the structure when it is partially supported by seabed shoulders and constrained by uneven natural supports or buckle initiators, and demonstrate that transversal mid-plane stretching transforms the pipeline into a non-linear oscillator with hardening stiffness and single well potential energy (i.e. energy potential resembling a well-shaped curve). We found that lateral mid-plane stretching reduces the overall deflection of a free spanning pipeline, increases the frequency of vibration for various axial forces and seabed/support stiffnesses, and reduces the effective length at various seabed stiffnesses and axial forces. In addition, considering the transversal mid-plane stretching and solving the problem using Galerkin's method showed that the frequency of resonance increases with the amplitude of vibration and the increase is a function of damping and non-linear stiffness. Our results prove that beyond a critical level of forced oscillation, the pipeline experiences a period doubling bifurcation, which has been overlooked in existing predictive tools and codes of practice.