Static Catenary Analysis
1. General
This section will give an introduction to the 2D static catenary analysis offered as a supplement to the more time consuming finite element analysis for standard systems SA, SB, SC, CA and CB. The following features are included:

2D catenary equilibrium calculation of all parts consisting of flexible pipes or cables

Rigidbody equilibrium of articulated elements

Clump weights and buoys

Inplane current forces

Elastic elongation, linear or nonlinear axial stiffness characteristics

Seafloor contact
It should, however, be noted that bending stiffness is not included in the analysis. The catenary analysis will therefore serve as an approximation for systems significantly influenced by bending stiffness.
The basic problem in catenary analysis is to compute the static equilibrium configuration of a composite single line with boundary conditions specified at both ends. The boundary conditions at both ends can be divided into specified boundary conditions, which are known prior to the analysis, and unknown boundary conditions, which are determined by the static analysis.
A typical example is that the end positions are specified while force components are unknown. This problem is in mathematical terms denoted a twopoint boundary value problem. The twopoint boundary value problem can be solved very efficiently by application of the socalled "shooting method" or iteration on unknown boundary conditions at one end in order to satisfy specified boundary conditions at the other end.
It is important to observe that the catenary configuration is uniquely determined when all boundary conditions are specified at one end. The problem is in this situation reduced to an initial value problem and the static configuration can be found by catenary computations, element by element, starting at the end with all boundary conditions specified.
2. Catenary Element Equilibrium Equations
The purpose of catenary element analysis is to compute the static equilibrium configuration of a uniform cable with all boundary conditions specified at the first end for a uniform loading condition, e.g. solution of an initial value problem.
The zaxis of the local element coordinate system is opposite the direction of the resulting loading, i.e. loading is similar to selfweight in the local system, see Figure 1. The local element system coincides with the global coordinate system when no current loading is present.
The classic catenary equations can therefore be applied to compute the coordinates and force components at the second element end in the local element system, see for instance Peyrot and Goulois, 1979:
where

\(\mathrm {T_1=\sqrt{F^2_{X1}+F^2_{Z1}}\quad }\)  Tension at the first end

\(\mathrm {T_2=\sqrt{F^2_{X2}+F^2_{Z2}}\quad }\)  Tension at the second end

\(\mathrm {F_{X1},\:F_{Z1}\quad \quad \quad \quad \quad }\)  Force components at the first end

\(\mathrm {F_{X2},\:F_{Z2}\quad \quad \quad \quad \quad }\)  Force components at the second end

\(\mathrm {l\quad \quad \quad \quad \quad \quad \quad \:\:\quad }\)  Unstretched cable length

\(\mathrm {l_d\quad \quad \quad \quad \quad \quad \:\:\:\:\quad }\)  Stretched cable length

\(\mathrm {H,V\quad \quad \quad \quad \quad \quad \:\:\:}\)  Coordinates of the second cable end

\(\mathrm {q \quad \quad \quad \quad \quad \quad \quad \:\:\:\:\:}\)  Resulting loading along the cable
3. Catenary Equilibrium of MultiSegment Single Lines
3.1. The Shooting Method
The shooting method for solving twopoint boundary value problems for composite lines can be formulated in the following main steps:

Guess (estimate) initial values of unknown boundary conditions at the first end.

Compute configuration, element by element starting at the first end.

Compare computed boundary conditions at the second end with specified values.

If result is satisfactory, go to 6.

Compute improved estimate of unknown boundary conditions at the first end. Go to 2.

Computations completed.
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).
3.2. Solution of the Initial Value Problem
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 zaxis 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 zaxis 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:

Assume that the initial chord direction is equal to the direction of the cable tension at the first element end.

Compute the drag forces based on present estimate of the chord direction.

Compute coordinates, forces and updated chord direction by use of the catenary equations.

If the change in the chord direction is
small
then stop, solution is accepted. Otherwise, go to 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).
4. Application of Catenary Analysis
4.1. General
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.

The catenary analysis will give the exact static solution for single line cables (i.e. no bending stiffness) when there is no current loading present.

The catenary solution is for all practical purposes identical to finite element results for single line cables exposed to inplane current.

The catenary analysis may become numerically unstable for systems totally dominated by current loading.

The catenary solution will represent an approximation for branched systems, systems significantly influenced by bending stiffness and systems exposed to outofplane current loading.

Bending stiffness will normally have only a minor effect on the overall static configuration of commonly used flexible riser configurations in normal operating conditions. The catenary solution will therefore represent a good approximation to the overall static configurations of such systems in the case of no current or inplane current loading.

For computational convenience, the same element mesh is applied in both catenary and finite element analysis. This is in particular convenient for combined catenary/finite element analysis where the catenary solution is applied as the initial solution for the finite element analysis.
4.2. Catenary Analysis of SASystems
Boundary conditions of SAsystems are specified in terms of specified positions of supports at the seafloor and support vessel.
Single line SAsystems 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 SAsystems it is necessary to define a
socalled 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 multisegment 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 SAsystems 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.
4.3. Catenary Analysis of SBSystems
The analyses of single line SBsystems and SBsystems with free end branches are similar to analysis of SAsystems. Catenary analyses of SBsystem 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 xdirection but restricted by the anchor line length in the zdirection. 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 stressfree configuration.
4.4. Catenary Analysis of SCSystems
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.
5. References
Fylling, I.J. (1978): Cabsta  A Program for Static Analysis of Partly Submerged Composite Cables, User’s Manual, The Ship Research Institute of Norway, Trondheim.
Peyrot, A.H. and Goulois, A.M. (1979): Analysis of Cable Structures, Computers and Structures, Vol. 10, pp. 805813.
Peyrot, A.H. (1980): Marine Cable Structures, Journal of the Structural Division, ASCE, Vol. 106, No. 12, pp. 23912403.
Sødahl, N. (1991): Design and analysis of flexible risers, Dr.ing. Thesis, Division of Marine Structures, the Norwegian Institute of Technology, Trondheim.
Sødahl, N. (1995): Solution of TwoPoint Boundary Value Problems for Marine Cables , Marintek Memo, Interntnotat001168, Trondheim