Hydrostatic Pressure Effects

Hydrostatic pressure effects on marine risers have been discussed by several authors in the open literature, e.g.Morgan and Peret (1974), Sparks (1984) and Mathisen and Bergan (1986). This discussion normally involves the following notions that need to be defined.

  • effective tension: force in the pipe that affects stability. This force is relevant for governing the shape of cables and pipes, including buckling analysis, and is used for calculation of geometric stiffness in the finite element method.

  • axial stress resultant: force found by integrating normal stresses over the cross section. In the presence of external hydrostatic pressure, this force is different from the effective tension.

In principle, two different approaches are possible when analysing a pipe or other slender structure subjected to external and internal hydrostatic pressure.

  1. Volume force model

    1. Use of vertical, conservative forces to represent hydrostatic effects. These forces will be in equilibrium with the effective tension, which means that axial stresses need not necessarily be calculated during an iteration for equilibrium. The theoretical foundation for this way of modelling hydrostatic forces is given by Sparks (1984).

  2. Pressure force model

    1. Calculation of non-conservative hydrostatic forces by considering pressure on the deformed pipe geometry. These forces are in equilibrium with the axial stress resultants. Effective tension needed for calculation of the stiffness matrix must, however, be found by introducing artificial pressure forces in axial direction. This method is described by Mathisen and Bergan (1986).

Both methods are in principle correct, and all nonlinear geometric and material effects can be modelled within both concepts. The choice of method should therefore be made after considering use of computer time and complexity in formulation of the alternatives. For a structure where hydrostatic pressure does not affect the volume or cross section properties the first approach is correct, and it is computationally much simpler. The conclusion is obvious: in the present program hydrostatic effects will be modelled by conservative, vertical forces. No hydrostatic force variation for caused by deformations will be accounted, except for possible increase or decrease of submerged volume and wetted pipe surface at the sea surface.

Hydrostatic pressure is therefore treated in terms of effective weight and effective tension, defined as:

\[w=m_{p}g-A_{e}\rho g+A_{y}\rho _{i}g\]
\[T=T_{p}+A_{e}P_{e}-A_{i}P_{i}-\rho _{i}A_{i}v^2_{i}\]

which may be rearranged to

\[T_{p}=T-A_{e}P_{e}+A_{i}P_{i}+\rho _{i}A_{i}v^2_{i}\]

where

  • \(\mathrm {T\,\qquad}\) : effective tension

  • \(\mathrm {T_{p}\!\qquad}\) : tension in pipe wall, i.e. resulting force from normal stresses

  • \(\mathrm {A_{e},A_{i}}\) : external/internal cross sectional

  • \(\mathrm {P_{e},P_{i}\:}\) : external/internal hydrostatic pressure

  • \(\mathrm {w\:\qquad}\) : effective weight per unit length, i.e. submerged weight of pipe including content

  • \(\mathrm {m_{p}\!\!\qquad}\) : mass of pipe per unit length

  • \(\mathrm {\rho _{i}\,\qquad}\) : density of internal fluid

  • \(\mathrm {\rho \;\,\qquad}\) : water density

  • \(\mathrm {g\;\,\qquad}\) : acceleration of gravity

  • \(\mathrm {v_{i}\;\!\qquad}\) : velocity of internal fluid flow

The basic axial force output from static and dynamic analysis is the effective tension.
The axial pipe wall force can, if required, be calculated by post processing according to Equation (3). The last term in Equation (3), which results from a constant velocity internal fluid flow, is further discussed in Effects from Internal Fluid Flow (Slug Flow).