For a pipeline of constant rigidity, horizontal tension, and submerged weight per unit length, there is a maximum value that the maximum sag-bend moment attains as the distance from the free end to the sea floor increases.

This paper presents the results of a parametric study analyzing a pipeline displaced from the sea floor to some pipeline construction configuration. The pipeline iE modeled as a continuous beam having constant submerged weight per unit length and a constant flexural rigidity throughout the length of the beam. The beam is taken as being under a constant horizontal tension or force with one end supported vertically by the sea floor., The governing differential equations are solved in terms of dimensionless combinations of the three characteristic variables - the submerged weight per unit length, the flexural rigidity, and the constant horizontal tension. This allows any possible pipeline configuration meeting stated constraints to be analyzed.


The equations for the unconstrained pipeline are easily derived from the analysis of the free-body diagram of a section of pipeline. The basic assumptions of the derivations are that the pipeline is uniformly continuous and has a constant value of submerged weight per unit length (q), a constant value of flexural rigidity (EI), and a constant value of horizontal tension (T). Summing the vertical forces on the differential element of pipe shown in Fig. 1, one arrives at Eq. 1, which simplifies to Eq. 2.

(Equation available in full paper)

By summing moments at the lower position of the pipe, one arrives at Eq. 3, which simplifies to Eq. 4, assuming that the variation in V is sufficiently small to neglect the dV term. The last term in the equation is second order and therefore may be neglected.

(Equation available in full paper)

The interesting characteristic about this set of finite-element equations is that, instead of having three characteristic variables El, T, and q, the equations have only one ~ the loading number (q). This dimensionless number is a combination of the three characteristic variables, but for the solution of Eqs. 27 through 30 only one numerical value, the loading number, is required along with four boundary conditions.

Next, an analysis of the pipeline on the sea floor is shown in Fig. 2. For any pipeline resting on the sea floor, it is known that the moment at the sea floor MSF = O. It is further known that the angle at the sea floor is fixed by the geometry of the sea floor and can be defined as zero or some constant number. For the purposes of this study, 8 SF = O. The vertical displacement is also defined as zero and positive in the upward direction. These sea-floor boundary conditions are easily calculated into dimensionless-variable-type boundary conditions via Eqs. 21 and 22.

(Equation available in full paper)

All of the sea-floor variables except one are now known - VSF, the dimensionless seafloor vertical reaction.

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