Keywords: non-linear, spectral analysis, offshore platforms, finite elements.

Surface water waves incident on an offshore structure induce complex forces distributed along the structure. These waves are generally random functions of time, thus, the resulting forces are also random in nature. Offshore structures subjected to these random forces must, therefore, be analysed to determine the dynamic response of the structure and the interactions between the structure and the surrounding water. In particular, the greater the depth of water in which a jacket platform is deployed, the more important it is to consider the dynamic behaviour of the structure. In the 1970s jacket platforms were regularly located in water depths of about 100m, such as the Kinsale Head Gas Production Platform off the South Irish Coast. However, recent finds of oil and gas deposits in deeper waters have necessitated the deployment of platforms, such as the Bullwinkle jacket, in nearly 500m of water in the Gulf of Mexico. The possibility of using jacket type structures in water depths of approximately 1,000m is currently being investigated. Because of the tendency to deploy structures in deeper water, it is imperative to perform more accurate dynamic response analyses of these structures.

Different types of dynamic analyses are performed on offshore structures depending on the type of structure in question. For large structures such as gravity platforms and buoyancy modules of tension leg platforms, where it is considered that the structure interferes with the flow field, a diffraction analysis is necessary to determine the distribution of pressure around the structure. For slender structures, such as jacket platforms it is assumed that, because the structural members are of small diameter relative to wavelengths, that they do not interfere with the flow field and thus characteristics of the undisturbed flow field are used to compute the incident forces. In this thesis only slender structures are considered.

The dynamic analysis of a jacket platform can be carried out either in the time domain or in the frequency domain. However, because the wave loadings on a platform are random in nature, it is often useful to present the results of the analysis in a statistical framework. This can be achieved most succinctly in the frequency domain using spectral analysis techniques. This is the approach adopted herein.

Previously researchers have addressed non-linear spectral response in a one-dimensional manner. However, the spatial extent of real offshore lattice platforms can induce interactions between the higher order frequencies in the non-linear force spectrum and leg spacings of the structure. These interactions can lead to the calculation of different displacement responses than are calculated using linear spectral theory.

A non-linear model is developed which extends the work of previous researchers by including leg spacing effects. Details of the development of a novel numerical model for non-linear spectral analysis of jacket structures are presented below and in the accompanying Part II paper. The finite element method is used to model the structural system as three-dimensional beam elements. The drag force term in Morison's equation is developed such that a system of equations with time-varying damping coefficients is developed. A solution to this system is obtained by employing a perturbation procedure that transforms the original system of equations into a set of recursive equations. The non-linear drag force terms are expressed as a series of Hermite polynomials such that the spectra of wave force on the structure include terms involving convolutions of spectra of water surface elevations.

Details are presented of a non-linear spectral model using a perturbation technique. In this development some of the non-linear dependencies of excitation on wave kinematics through the drag force term are maintained; linear Airy wave theory is used to describe the wave kinematics. The methodology is based on the modified Morison equation [1], incorporating the effects of relative velocities and accelerations and the hydrodynamic damping terms are time varying. However, the latter terms are assumed small and thus the equations of motion are formulated as a system of successive linear time invariant equations for each successive term of a perturbation expansion.

In particular, this development is an extension of research carried out by Lipsett [2] and Eatock-Taylor and Rajagopalan [3] who considered the effects of non-linearities on single degree of freedom systems and lumped mass systems respectively. The non-linear model described herein has been developed to model structures in two dimensions and is used to investigate the effects of leg spacing on spectral response due to the higher order harmonics in the force term.

1

Hartnett, M., "An Analysis of the Effects of Leg Spacing on Spectral Response of Offshore Structures", Journal of Advances in Engineering Software, 31, 991-998, 2000. doi:10.1016/S0965-9978(00)00065-X

2

Lipsett, A.W., "Nonlinear Response of Structures in Regular and Random Waves", Ph.D. Thesis, Department of Civil Engineering, The University of British Columbia, Canada, 1985.

3

Eatock-Taylor, R. and Rajagopalan, A., "Dynamics of Offshore Structures, Part I: Perturbation Analysis", Journal of Sound and Vibration, 83(3), 410-416, 1982.

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