<< Click to Display Table of Contents >> Navigation:  Flexcom > Theory > Fundamentals >
Finite Element Formulation

Introduction

Flexcom’s advanced computational technique provides supreme confidence in the engineering design. The software uses an industry-proven finite element formulation, incorporating a hybrid beam-column element with fully coupled axial, bending and torque forces. Up to 10 integration points are used within each finite element to ensure a precise distribution of applied forces. Third order shape functions are used to predict solution variable (e.g. moment, curvature) variations within each element.

A truss element has also been added in more recent versions of Flexcom. This is essentially a simplified version of the traditional beam element (it does not solve for nodal rotations) which is better suited to structures with very low structural bending stiffness such as mooring chains.

Hybrid Beam-Column Element

In order for Flexcom to be capable of analysing flexible risers, flowlines and mooring lines, in addition to more traditional structures such rigid risers, TLP tethers and pipelines, a numerical solution scheme is required which caters for (i) bending stiffnesses which are much lower than both axial stiffness and torsional stiffness values, and (ii) arbitrarily large and nonlinear displacements and rotations in three dimensions.

To accommodate the low or zero bending stiffness problem, early versions of Flexcom were based on a 2D hybrid beam-column element (McNamara et al., 1988). In this approach the axial force appeared as an explicit nodal solution variable, and was interpolated independently of the axial strain. The stress-strain compatibility relationship was applied outside of the virtual work statement by means of a Lagrangian constraint. This proved an accurate approach, and was subsequently extended to three dimensions (O'Brien & McNamara, 1988). Further developments resulted in the addition of an extra Lagrangian constraint on the torque degree of freedom, in order to make the scheme more robust and accurate when this variable plays a significant role in the solution (O'Brien. et al., 1991). This leads to a 14-DOF hybrid finite element with two end nodes, where the axial force and torque are added to the usual form of a three-dimensional beam element. The fundamental finite element equations for this element are derived in the following sections.

14-DOF Hybrid Finite Element

TRUSS ELEMENT

The truss element has 3 translational degrees of freedom at each node, and deforms only in the axial direction (it does not deform in bending or torsion). As it does not solve for nodal rotations, the connection at each node is essentially a pure hinge. The axial force penalty term is retained making the truss element a 7-DOF hybrid finite element with two end nodes. The truss element is designed specifically for modelling structures which have very low levels of structural bending stiffness (such as mooring chains) and is essentially a simplified version of the standard beam-column element employed by Flexcom. Refer to Truss Element for further information.

Convected Axis System

Flexcom uses detailed kinematics for finite rotations in three dimensions. This is based on the use of convected coordinate axes in developing the equations of motion, a technique also developed initially in 2D and subsequently extended to 3D (O'Brien and McNamara, 1988), (O'Brien. et al., 1988), (McNamara and O'Brien, 1986). Each element of the finite element discretisation has a convected axis system associated with it, which moves with the element as it displaces in space. The convected system is denoted in the following as the system. (By contrast the global axes, which are common for all elements and are the system in which the equilibrium equations are assembled and solved, is denoted the system) The instantaneous orientation of the convected axes for an element is found as follows, with reference to the below. The convected axis always joins the first local node of the element to the second node; the and axes are oriented in such a way that at the first node the local twist rotation of the deformed element relative to the convected axes is zero (O'Brien. et al., 1988).

 

Convected Axis System

The finite element development in the following is based on the assumption that strains are small and elastic under arbitrarily large three-dimensional displacements and rotations. The internal and external virtual work statements are written in the convected system; deformations along the element relative to this system are assumed to be moderate. Further details are provided in the following sections, and are also available in Chaudhuri. et al., (1987), O'Brien, and McNamara, (1989), Karve et al., (1988), O’Brien et al., (2002), O’Brien et al., (2003).

Further Information

Further information on this topic is contained in the following sections:

•Nomenclature explains the matrix algebra notation used extensively throughout this chapter.

•Internal Virtual Work Statement derives the internal incremental virtual work statement for the hybrid beam formulation.

•Finite Element Model describes the nodal solution vector.

•Strain Increments describes the strain vector, in terms of linear and nonlinear strain terms.

•External Virtual Work Statement presents the external virtual work for the beam element.

•Equations of Motions presents the finite element equations of motion, based on equating internal and external virtual work expressions.

•Finite Rotation Kinematics presents the Flexcom kinematics for finite rotations in three dimensions.

•Truss Element discusses the truss element in more detail.

Relevant Keywords

•*PRINT is used to request additional printed output to the main output file. Specifically, the OUTPUT=CONVECTED ELEMENT AXES option is used to request the local convected axes for each element as a function of time.