Static Solution |
 
|
Static Solution |
|
of the total solution vector 
. The total stiffness matrix on the left-hand side of Mathematical Background (Eq.9) (which also appears on the left-hand side of Mathematical Background (Eq.8) is the sum of two components, the ordinary linear stiffness matrix 
and the geometric stiffness matrix 
. This latter term includes the effect of the distribution of effective tension in the structure on the structure bending stiffness. This effective tension distribution is a function of the loading on the structure due to, for example, gravity, buoyancy and any user-specified point or distributed loads (such as a riser top tension). Applied constant displacements (for example a mean vessel surge) also influence the effective tension distribution. It is a basic premise of the frequency domain development that the dynamic variation in the geometric stiffness matrix is negligible and can be ignored.So the first step in a frequency domain analysis is to evaluate the total stiffness matrix 
. As mentioned previously, the frequency domain analysis is always preceded by a separate initial static analysis (this is logically termed the initial static solution), from which 
is determined without any further computation.
Current forces in the initial static solution may not be entirely accurate. The reason why this is the case is related to the drag linearisation algorithm, which was mentioned previously in Mathematical Background and is further discussed later in the section. For now, it is only necessary to note that linearised drag forces due to current, in an analysis with waves and current, are a function of the dynamic response due to wave action, whether a regular wave or random sea. So the current forces cannot be evaluated with complete accuracy until the dynamic response has been found.
What this means is that an analysis with waves and current is a three step sequence with first an initial static analysis (to evaluate 
); then the dynamic analysis, to find the response to a single harmonic (regular wave) or a discretised spectrum (random sea); followed finally by a so-called full static solution, which involves the assembly and solution of Mathematical Background (Eq.9) with accurate current forces included in the (total) static force vector at this stage. Like the initial static solution this last analysis phase is iterative, because large displacements can be considered, and so the full static solution is nonlinear. In this context it is worth noting that nonlinear effects can include sections of riser moving onto or off the seabed. This point is elaborated in Seabed Interaction.
Ideally you should include current in the initial static analysis, even though the current loading can be based only on the current velocity distribution as the dynamic response is as yet unknown. The rationale for this is that it provides the dynamic phase with an accurate an estimate as possible of the mean or static riser configuration. A good example is the fatigue analysis of SCRs, where an accurate assessment of the effect of current on the riser touchdown point can be important.