Recent research indicates that dynamic amplification of tension in lines and risers can be of the same order of magnitude as the so-called quasistatic values. Also, dynamic effects in the mooring system can affect low-frequency motions of the structure by increasing virtual stiffness and damping.

This paper presents three-dimensional (3D) computational procedures that describe the motion, tension, and bending moment along a flexible pipe or mooring line. The paper also discusses correlations with model test data.


The increasing number of mooring concepts for offshore structures and trends toward cheaper technology because of low oil prices put high demands on the design of the mooring arrangement. Important parameters are the large displacement of the structure, deep and hostile waters, and the need for year-round workability.

The wide variety of mooring systems may be illustrated by the existence of shallow and deepwater single-point moorings with temporarily or permanently moored tankers. clump weight systems used for guyed towers, and wire moorings for semisubmersible crane vessels.

The current design procedures include mainly a dynamic motion analysis of the moored object. This provides extreme positions of the structure. The mooring line tensions at these extreme positions can be found from the static-load excursion characteristics of the mooring system.

In this quasistatic mooring analysis, all other phenomena that affect the maximum line load are taken into account in an overall safety factor. as required by certifying authorities. Typical values are three for operational and two for survival conditions.

Experimental and theoretical research has shown that high-frequency oscillations (in the wave-frequency range) of the upper end of a mooring line can generate significant dynamic amplification of line loads.

These dynamic effects depend on

  1. frequency of oscillation,

  2. amplitude of oscillation,

  3. specific line mass.

  4. pretension, and

  5. hydromechanic line properties.

From a systematic series of forced oscillation model tests, van Sluijs and Blok found that the ratios of maximum dynamic tension and maximum quasistatic tension strongly depend on the frequency of oscillation. This ratio is enhanced by increasing oscillation amplitude, increasing pre-tension, and reduction of line mass.

Knowing the importance of dynamics for mooring systems, a similar behavior is expected for related "line-type" configurations such as flexible risers, pipe bundles, etc. The additional parameters concerning dynamic effects in these cases are the direct wave force and the bending stiffness. The traditional theoretical approach to solve the dynamic behavior of cable/riser systems is based on semi-analytical techniques. Geometrical nonlinearities are neglected to reduce the equations to differential equations that can be solved. Perturbation techniques are applied with success. but are restricted to certain areas.

A more general approach to the problem is discretization techniques. The line is assumed to be composed of a limited number of discrete elements. These elements can have physical properties of their own. Thus, the formed system of partial differential equations describing the variables along the line can be replaced by equations of motion in an earth-bound system of coordinates.

The most successful methods are widely known as the lumped mass method (LMM) and the finite-element method.

This paper presents an LMM technique and the resultant algorithm (applied in the DYNFLX computer program). This approach was validated with results of an extensive study conducted by the Maritime Research Inst, Netherlands (MARIN) on behalf of the Netherlands Marine Technological Research (MATS) program (Fig. 1).


Basic Approach. The mathematical model for simulation of 3D behavior of flexible lines is an extension of the LMM technique used for mooring chains and wires. The spacewise discretization of the line is obtained by lumping the mass and all forces to a finite number of nodes.

To derive the governing equations of motions for the jth lumped mass, Newton's law is written in global coordinates (Fig. 2):

{[Mj] + [mj(t)]} ..........................(1)

The added inertia matrix can be derived from the normal and tangential fluid inertia coefficients by directional transformations.

[mj(t)]=nnj ...........................(2)

where anj and ayj represent the normal and tangential added masses, respectively.


atj=p(KIt-1) and

[ nj], [ tj]=directional transformation matrices.

The nodal force vector ->F j contains the internal and external force components

  1. segment tension F sigma(t),

  2. shear forces caused by bending rigidity Fsno(t),

  3. fluid forces Ff (t),

  4. seafloor reactive forces Fr(t),

  5. buoyancy and weight Fw,

  6. buoy forces FB(t), and

  7. tether forces F sigma T(t).

Because the tangential stiffness of the line (represented by its modulus of elasticity, EA) is an order of magnitude higher than the stiffness in normal direction, the tension is taken into account in the solution procedure.

The tension vector on the jth node results from the tension and orientation of the adjacent line segments.




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