A Numerical Method to Solve the Three-Dimensional Static Problem of a Riser With Bending Stiffness

Numerical integration of the governing differential equations of three-dimensional riser statics may experiment large stability problems due to the inclusion of bending stiffness, as the leading order term becomes usually very small. These numerical problems may be avoided by working with a perfectly flexible cable model. However, such a model cannot deal with all the boundary conditions, as for an ideal cable there is no continuity of curvature at the touchdown point, at the top and at the points where there is change in the submerged weight. At the touchdown region, for instance, the perfectly flexible cable model overestimates the maximum curvature. To overcome these difficulties, one approach that can be adopted is to use firstly a perfectly flexible cable model and correct later the results with analytical expressions obtained from a boundary layer method. This approach is based on the fact that bending stiffness is relevant only at small boundary layers around the points where the perfectly flexible cable model cannot represent the curvature continuity. For a two-dimensional formulation it was already shown that this approach is very good. For a three-dimensional formulation, however, the analytical expressions become too cumbersome, the authors could not so far find an exact analytical solution and, therefore, the problem must be solved numerically. Another possible approach is to use finite elements method, as many full nonlinear commercial softwares do. However, it is not difficult to face convergence problems. This work presents a numerical method to solve the set of differential equations of the three-dimensional riser statics, including the bending stiffness. The results obtained are compared to a full nonlinear well-known commercial computer code.