Static Catenary Analysis

The shooting method for solving two-point boundary value problems for composite lines can be formulated in the following main steps:

This approach reflects the basic principle of the shooting methods: iterative correction of unknown boundary conditions at the first end to satisfy specified boundary conditions at the second end by repeated solution of the initial value problem (step 2).

A detailed description of the iteration process including methods for generation of start values is given by Sødahl (1991, 1995).

In the case of no current loading, the initial value problem can be solved by catenary equilibrium calculations, segment by segment, which will give the exact catenary solution with a minimum of computational effort.

An approximate solution of the initial value problem can be found when current loading is present. Each segment is in this case subdivided into elements to obtain an adequate representation of the current loading. The initial problem can then be solved by application of catenary equilibrium calculations element by element assuming constant current loading in tangential and normal chord directions for each element, see Figure 2. The static configuration of each element is found by application of the catenary equations Equation (2) - Equation (4) in a local system x, z where the local z-axis is opposite to the resulting load vector governed by weight, buoyancy and drag forces. Local force components and coordinates can then be transformed to the global system by standard rules for transformation. An iterative approach is needed since the chord direction is not known prior to the catenary computations.

The direction of the local z-axis is opposite to the resulting load direction governed by weight and buoyancy, \(\mathrm {q_w}\),

and drag loads tangential and normal to the cable chord denoted \(\mathrm {q^h_t}\) and \(\mathrm {q^h_n}\), respectively.

The following recalculation procedure has been used for calculation of current loading for each element, see Figure 2:

Experience has shown that one recalculation cycle is sufficient to obtain an accurate solution for most purposes. Several cycles may be required to obtain stable numerical solutions in the case of strong current or when long elements have been specified.

Concentrated forces can easily be applied at the segment intersections during the solution of the initial value problem as shown in Figure 3. Such forces can be applied to account for buoys, clump weights and possible rigid segments.

\(\mathrm {F_{X1}}\) and \(\mathrm {F_{Z1}}\) are computed by catenary equations for segment 1. The forces at the first end of segment 2, \(\mathrm {F_{X2}}\) and \(\mathrm {F_{Z2}}\) , are computed by considering equilibrium at the intersection point, i.e. \(\mathrm {F_{X2}+F_{X}+F_{X1}=0}\) and \(\mathrm {F_{Z2}+F_{Z}+F_{Z1}=0}\).

The catenary configuration can also be modified during solution of the initial value problem to account for bottom contact, a description is given by Sødahl (1995).

The catenary analysis provides an efficient tool for analysis of standard systems and can also be used to establish a good initial solution for nonlinear static finite element analysis. See Static Finite Element Analysis for a discussion of combined catenary/finite element analysis. The most important general properties of catenary analysis are summarized below. Special considerations for each particular standard system are discussed in Section 4.2 - Section 4.4.

Boundary conditions of SA-systems are specified in terms of specified positions of supports at the seafloor and support vessel.

Single line SA-systems can therefore be analysed by direct application of the technique described in Section 3. The shooting method for this problem is formulated in terms of iteration on tension and tension angle (e.g. force components) at the upper end until the lower end position is as specified. The initial value problem is therefore solved element by element starting at the upper end.

The acceptance criteria used for termination of the iteration are given by

where

\(\mathrm {\varepsilon _x}\) and \(\mathrm {\varepsilon _z}\) are relative error measures, \(\mathrm {l}\) is total line length and \(\mathrm {\Delta x}\) , \(\mathrm {\Delta z}\) are absolute differences from the specified lower end \(\mathrm {x}\) and \(\mathrm {z}\) coordinates, respectively.

For catenary analysis of branched SA-systems it is necessary to define a so-called main system line which can be identified as the direct seafloor to vessel structural connection excluding branches. The catenary analysis can then be carried out similarly to single line systems treating the main system line as one multi-segment line. Free end branches are assumed vertical and applied as equivalent buoys/clump weights during solution of the initial value problem. The catenary solution for branched SA-systems is exact for branched cable systems when there is no current loading present.

In other situations, the finite element analysis using the catenary configuration as start solution should always be used to find the final static solution.

The analyses of single line SB-systems and SB-systems with free end branches are similar to analysis of SA-systems. Catenary analyses of SB-system with an additional anchor line connected to the seafloor will, however, call for special attention.

Two separate catenary analyses are applied for the lower and upper part relative to the anchor line branching point. In both analyses, the anchor line branching point is free to move in x-direction but restricted by the anchor line length in the z-direction. System equilibrium is obtained by requiring force equilibrium at the anchor line branching point.

It is important to note that the horizontal position of the anchor line support is determined by the analysis.

It is, however, also possible to specify the horizontal anchor line support position by application of finite element analysis starting from the stress-free configuration.

The shooting method for this problem is formulated by iteration of vertical position on the lower end until the vertical position on the upper end is as specified. The initial value problem is solved element by element starting at lower end. Special algorithms have been implemented to handle zero tension at the lower node, e.g. free hanging cables.