Aqwa computes all of the stiffness contributions directly from analytical expressions for the load/displacement derivatives, or through the use of numerical differentiation.

The global stiffness matrix is nonlinear and is composed of hydrostatic restoring stiffness, mooring stiffness, and 'stiffness' due to the heading variation in wind, current and wave drifting forces and moments.

The cut water-plane area, together with the locations of the center of buoyancy and the center of gravity of the body, determine the hydrostatic stiffness matrix. As each body is moved towards equilibrium, the hydrostatic properties are recalculated at each iterative step based on the new submerged volume. However, there are instances where a detailed geometry of the bodies is not available or not required. You may therefore directly input a hydrodynamic stiffness matrix, which will be assumed to be constant throughout the analysis.

The hydrostatic stiffness components and will be zero and the stiffness matrix will be symmetric if the center of buoyancy and the center of gravity are located on the same vertical line. For a free-floating body in equilibrium, this is automatically the case. However, if the body is in equilibrium under the influence of mooring lines and/or articulations, the center of buoyancy and the center of gravity will not necessarily be located on the same vertical line. In this case, the hydrostatic stiffness matrix will be asymmetric, although the global system stiffness matrix will still be symmetric.

Steady wind, current and wave drift forces are functions of the heading angle only, and their stiffness contributions are therefore found only in changes in the yaw coordinate (i.e. components and of the stiffness matrix).

The fixed reference axes (FRA) are used for the equilibrium analysis of the floating system. If force/moment vectors and stiffness matrices are initially evaluated in the local structure axis system (LSA), they will be transformed into the FRA prior to the calculation of equilibrium. As an example, a thruster force (defined in the LSA of the body on which the thruster is acting) is transformed into a force/moment about the structure center of gravity in the FRA by Equation 11–6.

The formulation of a vector translation may be applied directly to translate the stiffness matrix, , from the point of definition to the center of gravity. It should be noted, however, that if the stiffness matrix is defined in a fixed axis system that does not rotate with the structure, an additional stiffness term is required to account for the change of moment created by a constant force applied at a point when the structure is rotated.

As an example, at an intermediate position in the iterative process, denote as a 3×3 structure-anchor mooring line stiffness matrix corresponding to the translational movements of the attachment point at on the structure in the FRA. We also denote as the X, Y and Z components of the tension in the mooring line at . The full 6×6 stiffness matrix for each mooring line, relating displacements of the center of gravity of the structure, located at , to the change in forces and moments acting on the center of gravity, is therefore given by

(11–7)

where

For a mooring line joining two structures this causes a fully-coupled stiffness matrix, where the displacement of one structure causes a force on the other. This stiffness matrix may be obtained simply by considering that the displacement of the attachment point on one structure is equivalent to a negative displacement of the attachment point on the other structure. Extending the definitions in Equation 11–7, the alternative 12×12 stiffness matrix is given by

(11–8)

where

in which are the coordinates of the attachment point on the second structure with its center of gravity located at , and are the X, Y and Z components of the tension in the mooring line at the attachment point on the second structure.