&REP_SELECT  -OPTIONS

and the options applicable here are:

     -SLA_COEFFICIENT, S_COE

     -SLA_FIXITY, S_FIXITY

     -SLA_DAF, S_DAF

     -SLA_CDAMP, S_CDAMP

     -SLA_MULTIPLIER S_VEL(1), S_MUL(1), ... S_VEL(n), S_MUL(n)

The data will be discussed below, and only beams that are out of the water in the mean position and of which are allowed to have forces due to added inertia will be considered for beam fatigue.

First, notice that here we do not have the normal frequency domain phenomenon. Instead, we have an occasional impulsive load which acts on the beam, and when the load is removed, then beam experiences a free vibration decay. Thus, according to DNV, the force per unit length on a beam due to slamming is:

     W = .5 * rho * D * S_COE * abs ( V * Vv )

Where rho is the water density, D is the tube diameter, and the option -SLA_COEFFICIENT defines S_COE is the slam coefficient (nominally 3), and V is the relative velocity between the beam and the water, and Vv is the relative velocity vertically. Now, the maximum bending moment in the beam, assuming that the load is uniformly applied over the length, can be written as

     M = S_FIXITY * W * L * L

Here the -SLA_FIXITY is used to define S_FIXITY which is a number that depends on the fixity of the beam. It is 1/8 if the beam is simply supported and 1/12 if it is built in. This, in turn gives a maximum stress in the beam of

     s = S_DAF * SCF * R * M / I

Where R is the beam radius, I is the section inertia, SCF is the stress concentration factor, and the option -SLA_DAF defines S_DAF is the dynamic amplification factor which is nominally 2.

After the impulse is applied and released, the beam will freely vibrate with decreasing amplitude

     s(k) = s(k-1) * exp ( -200*pi/S_CDAMP)

Here S_CDAMP is the percentage of critical damping defined with the -SLA_CDAMP option. Now, the total damage due to a single impulse is

     CDR  = SUM [ 1./ N(s(k)) ]

Here, N is the number of allowable cycles at the stress s(k), and the sum continues until there is no further damage.

There remain two unanswered questions: the first is how many slam events do we have and what are the velocities associated with each slam event. The first question is easily answered if one assumes that the slam events are Raleigh distributed. In this case, one simply computes the number of times in a given time that the probable motion will exceed the mean. This gives us a slam velocity or number of slams per hour. The second question is not so clear. Suppose that we write the velocity as a multiple of the RMS.

     V = f * Fmax * Vrms

Here V is the velocity which will be used, Fmax is the multiplier which gives the "maximum" event from the RMS, and f is a factor. If the slam velocity is "low" then the slam events are associated with extreme events and the multiplier f should be 1 so that the velocity used is the "maximum" velocity. If on the other hand the slam velocity is high, then slams are not rare events and the multiplier should be one appropriate to a normal event such as 1/Fmax. MOSES uses an linear interpolation for f with the points in the table defined by the -SLA_MULTIPLIER option. This option defines a table of slam velocities, S_VEL(k), vs. multipliers, S_MUL(k). By default, the table consists of three points: If the slams per hour are <= 1 then f is 1, if the slams per hour are 10, then f = 1/1.86, and if they are >= 360, then f = 1/3.72.