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Hydrostatic

For some applications, the buoyancy of the beam/bar elements is a driving paramater of the global system equilibrium and a specific attention shall be paid to correctly evaluate their imersed volumes.

This is the case for instance:

  • for floaters or buoys modelled as series of beam elements connected to each other with diameters much higher than classical pipelines or risers;
  • for floating hoses which are tangent to the sea level.

The main difficulty consists in evaluating the immersed volume through an integration process which is classically based on the strip theory for slender elements (beams or bars). As illustrated below, for a given arc length along an element, a specific procedure is followed to calculate the intersection points \(A_1\) and \(A_2\) with the free surface, if any.

From this procedure are also derived the submerged area and the position of the buoyancy center with respect to the section center. The angle "phi" is computed and the hydrostatic forces are transposed from the center of buoyancy to the center of cross section.

For a column crossing the free surface, the leading idea to correctly evaluate the water plane area. To do so, all beam elements are browsed and all the couples \((A_1, A_2)\) are detected. The water plane contour is finally composed by all these couples of points \(A_i(x,y)\).

From the positions of points \(A_i\), the area and the inertia of the water plane are calculated thanks to the Green Theorem which transforms the following surfacic integrations into integrations over the contour as such:

\[ S_{wp} = \int\int dx dy \]
\[ I_{xx~~wp} = \int\int x^2 dx dy \]
\[ I_{yy~~wp} = \int\int y^2 dx dy \]
\[ I_{xy~~wp} = \int\int xy dx dy \]

With the Green theorem, it comes:

\[ \int _{C^+} = Pdx + Qdy = \int\int_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)dxdy \]

With \(P(x,y)=-y\) and \(Q(x,y)= 0\), it comes: \(S_{xx~~wp} = \int\int dx dy = \oint -ydx\)

With \(P(x,y)= 0\) and \(Q(x,y)= xy\), it comes: \(S_{xxx~~wp} = \int\int y dx dy = \oint xydx\)

With \(P(x,y)=xy\) and \(Q(x,y)=0\), it comes: \(S_{xxy~~wp} = - \int\int x dx dy = \oint xydx\)

With \(P(x,y)=-x^2y\) and \(Q(x,y)= 0\), it comes: \(I_{xx~~wp} = \int\int x^2 dx dy = \oint -x^2ydx\)

With \(P(x,y)= 0\) and \(Q(x,y)= xy^2\), it comes: \(I_{yy~~wp} = \int\int y^2 dx dy = \oint y^2xdy\)

With \(P(x,y)=0\) and \(Q(x,y)=0.5*x^2y\), it comes: \(I_{xy~~wp} = \int\int xy dx dy = \oint \frac 1 2 x^2 ydy\)

At the end, the contributions of all elements are summed up. Depending on the meshing, the contribution of every element may be quite specific according to the global vertical angle and the position with respect to the free surface of the element. The figures below illustrate how the integration is performed for a global column divided into ten beams:

To do so, for every beam element crossing the sea surface, the number of Gaussian points is modified as such:

By default, the number of Gauss points is the one defined by the user \(N_{def}\) ; in the intermediate part, this number is automatically switched to \(10\).

The figure below shows the very good correlation between the numerical results and theory for a column of length \(L=31.5m\) and an outer diameter \(OD= 10m\).

However, as the integration is numerical, the result may slightly differ from theoretical value.

For a vertical pipe, a difference of 1.7% is reached with ten points of integration.

The figure below shows the different couples of points \(A_i\) for different angles of the column:

Note

The process described hereabove is used to compute the contribution of the buoyancy efforts to the global system equilibrium. Nevertheless, for the moment the derivative of these efforts, namely the hydrostatic stiffness, is not automatically included in the Newton-Raphson iterations and this may lead to divergence problems, especially in static. In this case, we recommend to identify the groups of elements (referred as RISER in the LOG file) that cross the MSL (or to create new groups with *GROUP) and to define them with the keyword *BUOY in the "user defined keywords" space.

In version V5R6, corrective actions have been implemented for an automatic introduction of the hydrostatic stiffness. At that stage, the default procedure is kept unchanged but this specific procedure may be activated by adding the keyword HYDROSTIFF as a "user defined keywords".