The hydrostatic stiffness matrix is used to calculate the hydrostatic varying force and moment due to the small movement away from the equilibrium position.

The translational motions of the internal tank center of floatation do not generate any linear restoring force and moment with respect to the LTA origin. The linear hydrostatic restoring force on the internal tank with respect to the LTA origin induced by the rotational movement of the tank is given by:

(3–32)

The linear hydrodynamic restoring moment on the internal tank with respect to the LTA origin is

(3–33)

From the above equation,

(3–34)

By using the general matrix form, the hydrostatic restoring force and moment with respect to the LTA origin is

(3–35)

which is of the 6x6 matrix form,

(3–36)

where

To calculate the hydrostatic stiffness with respect to the combined COG in equilibrium, the liquid in each of the tanks is considered as a rigid part of the whole structure. The fictitious rigid body motion of the liquid in each tank must be taken into account, which gives

(3–37)

is the position of center of gravity of the liquid in the single tank, which is the same as the tank buoyancy center in the LTA. Converting Equation 3–37 into the matrix form,

(3–38)

where .

Combining Equation 3–35 and Equation 3–38, the total force and moment with respect to the LTA origin is

(3–39)

where

Employing Equation 3–26, Equation 3–27, and Equation 3–39, the hydrostatic stiffness of the fluid in the tank with respect to the LSA origin can be derived:

(3–40)

From Equation 3–40, it is found that the hydrostatic stiffness matrix with respect to the LSA origin is

(3–41)

Together with the hydrostatic stiffness due to external fluid buoyant force and moment variation given by Equation 3–19, the total hydrostatic stiffness with respect to the LSA origin is

(3–42)

Note that when there is any internal tank, the structure center of gravity in Equation 3–19 should be replaced by the LSA origin defined in the internal tank section.