Keywords: offshore structures, flexible riser, cable, hybrid finite element, dynamic, non-linear analysis.

In offshore oil and gas exploration activities, different alternatives are available to convey fluids from the wellhead on the sea bottom to a moored floating platform on the sea surface. The simplest and more common solution employed in floating production systems are flexible risers (a kind of pipe comprised by several steel and plastic layers) in a free-hanging configuration.

Flexible risers present severe geometric non-linear behaviour under dynamic excitation. Complex motions take place, mainly around the touchdown point (TDP) region - the portion of the pipeline that cyclically bumps on the seabed before lying down. Therefore, care must be taken in the generation of finite element models for the spatial discretization of risers; for the global analysis, geometrically non-linear frame elements are usually employed.

Through this work, the mixed finite element formulation concepts are applied in the implementation of a hybrid geometric non-linear three dimensional frame element. Geometric non-linear problems are dealt with aid of a co-rotational formulation. This hybrid frame element has both displacements and axial forces as unknowns of the equilibrium equations, which improves computational efficiency in problems involving dynamic analysis of highly slender structures such as free- hanging flexible risers and cables.

Being extremely slender structures, the axial stiffness may be several orders of magnitude greater than the bending stiffness. Such physical properties yield small strains even in the presence of high axial tension. This can lead to an ill-conditioned global stiffness matrix, then to numerical instability and lack of convergence in the solution procedures. In these circumstances, a standard frame element formulation, which is based only on displacements, may greatly enlarge even small errors in displacement and strain evaluation during stress calculations.

These facts have motivated the development and implementation of the non- linear hybrid frame element described here. In this formulation, the axial force is evaluated independently of the axial strain. This procedure overcomes the numerical problems associated with high values of axial stiffness. Therefore, besides the nodal displacements and rotations, the axial force is treated as an independent variable of the system of equilibrium equations.

The frame hybrid element formulation presented here is based on a twelve degrees of freedom frame element (corresponding to the spatial nodal displacements and rotations). A suitable approach [1,2] introduces the axial force as new variable. In the present work, the interpolation thereof is linear, which demands two additional degrees of freedom, and requires continuity across adjacent elements.

Through the development of the variational-based integral expressions presented in this work, the linear and geometric stiffness matrices may be obtained [3]. In order to better consider the displacements that actually yield deformations, the internal forces vector is given in terms of a co-rotational formulation [4].

Moreover, since the interpolation of the axial force and the axial displacement influences the convergence of the non-linear solution technique [5], special care must be taken while choosing the respective number of degrees of freedom. Therefore, a straightforward and effective criterion is presented.

Numerical simulations are used to assess the behaviour of the hybrid frame element, and to compare their results with those provided by standard beam elements. A classical example (the Euler's Column) and offshore applications, like free-hanging flexible pipelines, are analysed. It is seen how the increase of slen- derness ratio affects the performance of the analyses based on the standard frame element, and how the hybrid formulation may overcome convergence difficulties in such circumstances. These examples revealed additional benefits from the use of the hybrid element: the analyses of more slender structures implied less computational cost, due to either less number of iterations or the use of coarser meshes.

As oil and gas exploration goes towards ultra deep water, pipelines will become even more slender. Certainly hybrid formulation may represent a profitable tool for the design of production systems in these new scenarios.

1

McNamara, J.F., O'Brien, P.J., Gilroy, S.G., "Non-linear Analysis of Flexible Risers Using Hybrid Finite Elements", in "Proceedings of the 5th OMAE Symposium", Tokyo, Japan, 3, 371-377, April, 1986.

2

Mathisen, K.M., "Large Displacement Analysis of Flexible and Rigid Systems Considering Displacement-Dependent Loads and Non-linear Constraints", D.Sc. Thesis, Division of Structural Engineering/The Norwegian Institute of Technology, Trondheim, Norwegian, June, 1990.

3

Kayser Jr., D.L., "Análise Dinâmica de Linhas Flexíveis com Elemento de Pórtico Não-Linear Geométrico Híbrido", M.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro, Brasil, July, 2003.

4

Mourelle, M.M., "Análise Dinâmica de Sistemas Estruturais Constituídos por Linhas Marítimas", D.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro, Brasil, December, 1993.

5

Hughes, T.J.R., "The Finite Element Method", Prentice-Hall, New Jersey, US, 1987.

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