Keywords: riser, nonlinear finite element, pipe elbow element, fluid dynamics.

Flexible risers are slender marine structures that are widely used in offshore production to convey fluids between the well-head and the surface unit. In deep-water applications, because of the low bending stiffness when compared to axial and torsional stiffness, a flexible pipe can suffer large displacements which demands special geometrically nonlinear analysis.

In this paper, a nonlinear finite element formulation is derived for large displacement analysis of the flexible risers using pipe elbow elements. The proposed element consists of four nodes with twenty-four degrees of freedom. In the three-dimensional space, three vectors are used to define the geometry of this pipe elbow element. One vector expresses the configuration of the pipe centre line and two other vectors, which are usually called the unit normal vectors, will express any position through the pipe cross section.

In geometrically nonlinear deformations as a result of the continuously changing in configuration of the body an incremental procedure is required to solve the equations of equilibrium [1]. Actually, these equations are expressed by the principle of virtual work in a current configuration of structure [2]. Since the current configuration is unknown, it is necessary to write the principle of virtual work for the body at a known equilibrium configuration. When the reference solution is chosen at the starting point of the deformation process, the solution is called total Lagrangian; when it is taken at the previous configuration which precedes the current configuration, the formulation is known as updated Lagrangian formulation. In both cases, strongly nonlinear equations will be obtained which should be linearized [3]. Also, the updated Lagrangian formulation involves little numerical effort [1,4].

Using the principle of virtual displacements as well as the updated Lagrangian incremental analysis approach the equilibrium equations of the deforming body are achieved for the pipe elbow element. The 2ed Piola-Kirchoff and the Green-Lagrange strain tensors, because of their independency from rigid body rotations, are used in driving these formulations. In order to proceed in an incremental level, incremental decompositions of the 2nd Piola-Kirchoff and the Green-Lagrange strain tensors are also conducted.

The nonlinear finite element formulation has been implemented in a finite element code using a proper nonlinear algorithm. This code is then used to model and analyze a typical pipe system subject to an assumed current loading. In order to estimate the hydrostatic forces due to the current, the steady state flow around the cylinder is solved at various sections. For this purpose, the continuity and Navier-Stokes equations which govern a two dimensional incompressible viscous Newtonian fluid flow are solved using Galerkin Least Square method by considering a four node element. Some case studies also are conducted in order to shown the capability of the proposed formulation and algorithm.

1

K.J. Bathe, "Finite Element Procedures in Engineering Analysis", Prentice Hall, 1982.

2

L.E. Malvern, "Introduction to Mechanics of a Continues Medium", New-York, Prentice-Hall, 1969.

3

L.C. Jiang, W. Michael and G. Pegg Neil, "Corotational Updated Lagrangian Formulation for Geometrically Non-linear Finite Element Analysis of Shell Structures", Finite Element in Analysis and Design. 18, 129-140, 1994. doi:10.1016/0168-874X(94)90097-3

4

T.Y. Chang, K. Sawamiphakdi, "Large deforamtion analysis of laminated shells by finite element method", Computer and Structures, 13, 331-340, 1981. doi:10.1016/0045-7949(81)90141-3

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