Determination of K
Overview
In order to properly design a mooring system for a Spar structure, its spring constant (K) needs to be determined. Determining K is an iterative process and can thus be a time consuming part of the total design. A more detailed overview of the process can be schematically seen below.
The exciting force of a structure are cause by the combined action of:
Wind
Current
Waves
Wind
Wind speeds are measured at one location. An adequate vertical distribution of the true wind speed z meters above sea level is represented by:
The spar is moored, so the relative wind is equal to the absolute wind. The variations in mean wind speed are small compared to the wave period. Thus the wind will be considered as steady, both in magnitude and direction, resulting in constant forces (vectors).
Wind forces on structures can be approximated by dividing the structure into a number of components, with elementary geometry. The forces on these structures can then be estimated as drag forces on each separated element. The drag coefficients can be based on several, good literature resources.
The drag forces can be represented by the formula:
As the wind velocity varies in height, the drag force also varies. The total force on the element can then be determined by integrating over the total height of the structure.
Current
Spars are structures with deep drafts. Currents can thus be considered as very important loading factors. Due to the deep draft, the magnitude and direction of the current can differ along the draft of the structure. Thus the current distribution as a function of depth needs to be known.
The forces on the spar can be approximated by assuming that the spar is a slender cylinder in a current. The loads on this body can be calculated by:
As said, the magnitude and direction of the current can vary with depth. The total resultant force can thus be determined as the integral along the draft of the spar.
Again this leads to a magnitude and direction
Waves
To determine the motions generated by waves on a moored spar structure, the wave spectrum is needed. This spectrum is based on the 100 year limit state conditions. This is a spectrum with a 100 year significant wave height and significant mean zero crossing period.
Diffraction Method
Considering the complexity of the forces, the diffraction method is used. This method is based on the 3D Potential Theory. DELFRAC is a well known software program which uses this theory, numerous good results were found.
According to linear potential theory theory, the potential of a floating body is a superposition of the potentials due to the undisturbed wave Φw, the potential due to the diffraction of the undisturbed incoming wave on the fixed body Φd and the radiation potentials due to the six body motions Φj:
These potentials have to satisfy some boundary conditions on:
Laplace Equation.
Sea bed boundary condition.
Free surface boundary condition.
Kinematic boundary condition on the oscillating body surface: This boundary condition describes the shape and surface of the spar.
Radiation condition.
Symmetric or anti-symmetric condition: This boundary condition can be used, since a spar is rotation symmetric.
As an input for the diffraction method a transfer function must be determined. This transfer function gives the relation between the first order forces and motion and is dependent on the M and added mass of the spar, the viscous damping of the water and the combined spring constants of the mooring system (K) restoring (hydrodynamic) forces.
The diffraction method uses computer models, it can basically be describe in the following steps:
Determination of the first order forces on the spar while it is subjected to the given wave spectrum and kept in position (zero motions).
Determination of the first order motions. These are based on the first order forces and a transfer function. It must be underlined that (although the spar geometry suggests otherwise) all six first order motions are imported for the spar, since they have great influence on the second order wave forces.
The final and most important step is to determine the second order wave forces by repeating the first step, but then while the body is oscillating with the (in step 2 determined) first order motions.
Details on the diffraction method can be found in the Offshore Hydromechanics Syllabus, Chapter 7.
As said the final result of the diffraction method are the 6 second order wave forces. These forces can be approximated by the following equation:
Unfortunately this hard to solve, but the second order wave force can be described as:
Fmean(2) is known as the mean second order wave drift force. These forces are relatively large in comparison to the first order wave forces. The graph below shows an example of surge motions. The large excitation is the mean second order wave drift motions, the small ripple on top are the first order motion. Thus, the mean second order wave forces are dominant.
Now, with the three calculated forces the following expression can be written:
As can be seen, the amplitude of the mean second order wave force needs to be superimposed with the current and wind drag forces. It should be reminded that all forces are vectors!
Dividing the summation by K gives a displacement. Because of the geometry of the spar these motions will generally be surge and sway excitations.
Whether the assumed K value is correct depends on whether the motions of the spar are within the given limitations. Finding the correct value for K is an iterative process. If the assumed K value leads to motion which stay within limitation, the K value can be used in the design of the mooring system. If this is not the case, all calculations need to be redone and based on a new value for K.
Now that the mooring stiffness has been obtained, the next phase is the design of the mooring system. This however can only be done if some preliminary work has been performed and a concept is ready.