1. 3.6 LOADS – Sources and consequences of non-linearities
Sinusoidal wave and sylinder structure: z
h(t) =H/2 sinwt H x l d D
* l/D > 5 and H/D = 1  Mass term of Morison dominates * linear wave theory: f(z,t;x=0) = p cm D2 (H/2) gk cosh k(d+z)/cosh kd coswt Kinematics defined to still water level Total load obtained by integrating to still water surface * Total load
 Force frequency = wave frequency, mean force = 0

2. For irregular sea: - Can be written as a sum of sinusoidals
For irregular sea: - Can be written as a sum of sinusoidals. - Linear wave theory, linear load and linear mechanical system. The load – and response properties can be characterized by the transfer function: HhF(w) = F0(w)/h0(w), Hhx(w) = x0(w)/h0(w) If assumptions above sufficiently accurate relflect loads and responses, the problem can be solved in the frequency domain, i.e.:
Variance: 
Short term distribution 
Long term distribution: 

3. What are the approximations
What are the approximations? * Integration is carried out to still water level ( mean wetted surface for floaters). * Boundary conditions are fulfilled at the still water surface. * Viscous forces are neglected. If all these terms are accounted for (to some order), the total load will consist of the following terms: F(t) = Fmean +FLF(t) + FWF(t) + FHF(t)
Slowly varying force, frequency wj – wi T > 30s
Wave frequency force, frequency wj 25s >T > 4s
High frequency force, frequency wj + wi T < 4s
Mean force
Typically: FLF and FHF an order of magnitude smaller than FWF  These load processes will primarily be of concern if they hit lightly damped eigenmodes

4. Dynamic amplification Load: F(t) = F0coswt  steady state motion: u(t) = u0 cos(wt – q) where:
Static response
Dynamic amplification, DAF
l = 0.01  DAF= 50 l=0.1  DAF = 5
For w=w0, DAF = 1/(2l) 
Fixed platforms: FWF generally most important, FHF may be important if it hits a natural period. Floating platforms: Fmean, FLF, FWF generally most important, FHF can be important for TLP (springing) and ship (whipping).

5. Load on a drag dominated pile
f(z,t) = 0.5r cDDu(t)|u(t)|
Kinematics above mean free surface is a challenge. 1. One can select a regular Stokes 5th order wave.
2. If a sinusoidal wave is used, Wheeler stretching Is most common. (Will underestimate speed for steep waves.
Let us adopt a sinusoidal wave and look at f(z,t) well below the still water surface.
sinwt
sinwt|sinwt|

6. A Fourier expansion of sinwt|sinwt| will show: sinwt|sinwt| = 0
A Fourier expansion of sinwt|sinwt| will show: sinwt|sinwt| = 0.85sinwt+0.17sin3wt -0.02sin5wt+ .. (Inertial term: proportinal to coswt)
As drag is important, integration has to be carried out to the exact surface:
For the inertial term:
 Remember that for slender structures, loads will In addition to the wave frequency also appear at multiples of the wave frequency:

7. Design of drag dominated structures (jackets and jack-ups)
Up to 200m in the North Sea
Tn = 1 – 4s
Up to 100m in the North Sea Tn = 2-6s
Pinned or fixed, important for natural period.
If Tn < 2s (ca), platform behaves quasistatially, and a design wave approach is convenient for obtaining design loads.
If Tn > 2s more detailed analysis may have to be done in order to properly assess the dynamic behaviour.

8. Design wave approach: a) A proper 10-2 – load is obtained by exposing the structure to a regular wave with height equal to the 10-2 height and a conservative associated wave period (often most unfavourable of 90% band). b) A Stoke 5th order is most often adopted as the regular wave model. c) A conservative way of accounting for current, is to utilize a 10-1 current profile. d) Effects of dynamics can be approximated by estimating the DAF given by:
(Example: T=15s, Tn=1.5s, l=0.015  DAF=1.003)
e) Quastistatic 10-2 load found as worst load as the Stoke 5th wave is stepped through the structure.

9. Time domain analysis If an accurate assessment of dynamics is to be obtained, the equation of motion has to be solved in the time domain.
It is not easy to find a very precise way to perform such an analysis. Most experience is gathered for structures essentially behaving quasistatic. Norsok N-003 gives some guidance. For the Kvitebjørn jacket (Tn = 5s in design calculations), the following approach was followed: 1) 10-2 quasistatic load was estimated using Stoke 5th order wave and 10-2 wave height. 2) A number of 3-hour simulations speed and acceleration set equal to zero were performed for the worst sea state along the 10-2 contour line for hs and tp.

10. 3) The 3-hour maxima were identified, and a Gumbel distribution was fitted to the data. The percentile correponding to the extreme obtained by the Stokes 5th was estimated. A value of 95% was found. 4) The same 3-hour simulations were repeated, but now the structure was permitted to move, i.e. a full dynamic analysis was carried out. 5) The 3-hour maxima were again selected and a Gumbel model was fitted to the data. From the fitted model the dynamic extreme value at a percentile of 95% was estimated. 6) The dynamic amplification was estimated by: DAF = x_3hrmax_dyn_0.95/x_3hrmax_qstat_ ) Characteristic design loads are calculated using the values obtained by Stoke 5th wave multiplied by dynamic amplications factor.