Keywords: non-linear spectra, convolution, Hermite polynomials, offshore structures.

The accompanying Part I paper discusses the need for non-linear spectral analysis of certain offshore structures. In that paper possible approaches to this problem were discussed and the perturbation method selected for this research. Details of the finite element model developed were presented and how the drag force term of Morison's was manipulated to provide a constant term and a time varying term. It was shown how system receptances were derived for use in computing the spectral responses.

In this paper details of how the response spectra were derived are presented. In this derivation use is made of the convolution integral which leads to higher order response terms. The new model is then applied to a two dimensional structure subjected to random gravity wave forces as defined by a Pierson Moskowitz spectrum. The same structure and waves loading is modelled using a linear structural model; the results from both models are compared. From the comparisons of the results of both models it is clear that in certain situations it is necessary to use non-linear spectral analysis.

The force term on the right hand side of the equations of motion contains drag force terms that are non-linear with respect to water particle velocities. These non-linear functions are expressed as a series of Hermite polynomials that possess useful orthogonality properties. It has been shown that Hermite polynomials converge rapidly to the exact solution and that the approach can easily be extended to the situation when steady currents prevail.

The non-linear spectral model developed above was applied to perform analyses on a cantilever and plane frame structures subjected to wave loadings. The results from these analyses were compared with the results from applications of a linear model to the same structures; details of these analyses and comparisons are presented in this paper. Because the linear spectral model has previously been well validated by the author the comparisons made between the linear and non-linear models in this chapter provide good validation for the non-linear model.

The main objective in developing the non-linear model is to determine whether or not structure spatial extent has a significant effect on the non-linear structure response during a random storm as suggested by Moe [1]. In order to achieve this objective, a non-linear model of a cantilever was developed and results compared with results from a linear analysis; in this analysis there are no spatial effects and it will serve as an initial check on the results of the non-linear model. Both linear and non-linear models are then applied to a simple offshore portal frame to consider spatial effects; more complex structures would take much more time to analyse and not provide any better insight into the processes.

The non-linear spectral response model developed in this and the accompanying paper and applied in the paper is an extension of research carried out by others in that it incorporates leg spacing phase effects. The model was firstly applied to a cantilever and results obtained showed that the linear and non-linear models predicted similar responses. The non-linear model was then also applied to portal frames for different storm intensities. The results of this analysis show that there is an appreciable difference between the results of the linear and non-linear models. In particular, it is shown that response spectra computed using the non-linear model may be larger than corresponding spectra computed using the linear model. This difference is due to interactions between the higher order force terms and the phase effects caused by the leg spacings of the portal frame. The author is not aware of this phenomenon being observed by previous researchers and it is an important conclusion of this work.

From their research Eatock-Taylor and Rajagopalan [2] concluded that 'The influence of the non-linearities has been demonstrated to be smaller at lower sea-states'. For the structures considered, the current research shows that when leg spacing is included in the analysis then the discrepancies between the linear and non-linear response spectra can be significant at lower sea-states. This is an important conclusion because lower sea-states have a higher frequency of occurrence and thus the fatigue life of structural members may be affected.

Other researchers have concluded that the difference between the linear and non-linear response spectra is negligible around the frequencies of maximum wave energy, but significant at resonance. Significantly, this is not the situation when analysing portal frames; in this situation the main difference occurs close to the frequencies of maximum wave energy due to the higher order effects interacting with the leg spacing phase effects. Thus, not only is it important to avoid resonance problems but the structural spatial extent may also be an important consideration in the design process because of non-linearities arising from drag force terms.

1

Moe, G., "Stochastic Dynamic Analysis of Jacket Type Platforms. Nonlinear Drag Effects and "Effective" Nodal Area", Safety of Structure under Dynamic Loading, TAPIR, Trondheim, Norway, 760-767, 1977

2

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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