I have seen that when center of gravity of the model remains ABOVE center of buoyancy, a negative sign is seen in K44 and K55 terms (ref 3.1.3 AQWA Line User Manual pg 14 of 136).
Is it correct analysis to have negative values on the principal diagonal of a stiffness matrix?
Does the software take Zgb term as negative if the center of gravity of the model remains ABOVE center of buoyancy?
When the center of buoyancy is below the center of gravity, it means that for roll and pitch motion, the so called restoring force actually will enhance the roll and pitch motion instead of preventing the motion trendency. Thus in such cases, the corresponding stiffness term will be negative
If the structure you are modelling is freely floating, a negative term in the principal diagonal of the hydrostatic stiffness matrix means that your vessel is unstable and will not maintain its position if you run a time-domain analysis.
The fact that your centre of buoyancy is placed below the centre of gravity is quite common and cannot be the only reason, by itself, to determine an unstable structure. If you look at the equations you reported, a large contribution to the hydrostatic stiffness terms comes from the moment of the waterplane area. So, in your case, this result might be an indication that your design has no sufficient waterplane area to ensure stability.
Another way to increase the stability might be by adding other restoring forces to your system. This is what you would achieve if you included, for example, stiff tendons like the ones used in TLPs. If indeed what you are modelling is a TLP, then the possibility of having a negative hydrostatic restoring coefficient in roll and pitch might be acceptable.
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