Classification: Statoil internal Status: Draft WIND LOADS Wind phenomenology Wind speed is experienced essentially at two different time scales: 1: A slowly varying mean wind level; V m. This wind component can often be considered as constant for a short term period, say 3 hours. 2: A ”rapidly ”” fluctuating wind component, V t, riding on the mean wind speed. The period of fluctuations will be from second to some few minutes. YearFew daysHour1 min.1 sek Storm spectrum Power spectrum Speed Time Turbulence Mean wind speed

Classification: Statoil internal Status: Draft 2 At a given point, the resulting wind speed can be written: V(t) = V m (t) + V t (t) The mean wind speed is typically the largest. Terrain roughness govern the ratio. The ratio between standard deviation of Vt and Vm is called turbulence intencity. Typical turbulence intencity over ocean with storm waves is about The wind speed varies with height, see eq(1a – 1d) in Statoil metocean report. For engineering purposes, the mean wind speed is described by a distribution function often close to a Rayleigh distribution. The mean direction is described by a probability mass function for direction sectors (often) of 30 deg. width. The mean wind speed corresponds to a given length of averaging. Standard meteorological averaging is 10 min.., in design the length of averaging is often taken to be 1 hour. Wind speed will increase with decreasing length of averaging. The ration between a 15sek average and a 1-hour average 10m above sea level is 1.37, see table in Statoil report for other examples. Example of a wind description for design purposes is shown by Statoil Metocean report.

Classification: Statoil internal Status: Draft 3 For structures or structural components where the turbulent wind may cause a dynamic behaviour, the frequency spectrum for wind speed is given by Eq. (2a and 2b) of Statoil Report or Norsok N-003. This wind spectrum is deduced from wind measurements at Frøya. The turbulent wind is not fully correlated over the size of structures. Coherence spectrum between two points are given in Statoil report or Norsok N-003. The loads on structures not exposed to dynamic behaviour can be calculated considering the wind as static: If structural dimensions are less than 50m, 3s gust should be used. If structures are larger, 15s gust can be used. For structures which are exposed to simultaneous actions from wind and waves, and where the wave loading is dominating, the length of averaging of wind gust may be taken to be 1 minute. Check with coming editions of Norsok for possible changes.

Classification: Statoil internal Status: Draft Wind forces The wind force is proportinal to the wind speed squared: F = k * (V m + V t ) 2 = V m 2 + 2V m V t + V t 2 =(ca) V m 2 +2V m V t The mean wind gives a constant force on the structure, while the turbulent wind yields a force proportional the the turbulent wind speed. Practical problems: Offset and mooring line forces for ships and floating platforms. The natural periods of the surge, sway and yaw are often in the order of 1-2 minute, i.e. a period band where the wind frequncy spectrum has a considerable power density. For this sort of problem, a dynamic analysis has to be carried out involving the wind power spectrum. Wind loading on flare towers and drilling towers. A quasi- static analysis is often possible accounting for some dynamics by a proper dynamic amplification factor. The wind loading on complicated structures are determined by means of tests in wind tunel. Remember that in all cases mentioned above, one will also have to include the simultaneous affect of waves.

Classification: Statoil internal Status: Draft 5 The static wind force on a structural member or surface acting normal to the member or surface is given by: F W = ½  C A V 2 sin  C – shape coefficient, see DNV 30.5  – density of air (= kg/m 3 for dry air) A – projected area of member normal to force direction  – angle between wind and axis of the exposed member For the wind load of a plane truss, the load can be calculated by using A as the enclosed area of truss if an effective shape parameter is used, C=Ce, and the transparancy of the truss is accounted for by multiplying the area with the solidity ratio  C e is found in DNV If more than one member or truss are located behind each other, shielding effect can be accounte for by multiplying loads given above by the shielding factor, . Values for  are given in DNV Regarding the shape coefficicent, it is recommended that DNV 30.5 or an similar reference are consulted. NB: For structural sides not facing the wind, a considerable suction force can occurr, see Fig b in kompendium, Moan (2004).

Classification: Statoil internal Status: Draft 6 If structure or structural component can be exposed to wind induced dynamics, the variability of the wind force is to be accounted for: F W (t,z) = ½ C  A (V m (z) 2 + 2* V m (z)*V t (z,t)) * sin  It is seen that load is linear with respect to wind speed (since V t 2 term is neglected). If the wind induced response is linear function of load, the wind response may be obtained using frequency domain analysis, i.e. the cross spectral density for the dynamic wind load is multiplied by response transfer function in order to obtain response spectra for dynamic wind induced response. Alternatively, wind histories for a number of load points may be simulated from wind spectrum and corresponding time histories for the response found by solving the equation of motion in time domain. The total extreme wind induced response can be found by: F 3h-max (V m ) = F m + g *  (V m ) g – extreme value factor for 3-hour maximum dyn. response  – standard deviation of wind response under consideration

Classification: Statoil internal Status: Draft 7 Vortex Induced Vibrations (brief introduction) Vortex shedding frequency in steady flow is given by: f = St * V/D St is the Strouhal number, V is wind speed and D is structural diameter. A critical velocity is defined as the velocity giving vortex shedding frequencies equal to the natural frequency of the structural member: V C = 1/St * f N * D f N – natural frequency of structural member. St is a function of the Reynolds number, Re = VD/, where is the kinematic viscosity of air, (= 1.45*10 -5 m 2 /s at 15 o and standard atmospheric pressure. St is given in Fig. 7.1 in DNV A state of quasi-resonant vibriations of a member may take place if wind velocity is in the range: K 1 *V C < V < K 2 *V C If possible, one should require: V C > 1/K 1 *V max Not always possible and maybe unnecessary strict criterium.

Classification: Statoil internal Status: Draft Wave and Current loads Water levels: Mean still water level Lowest astronomical tide (LAT) Highest astronomical tide (HAT) Tidal range Maximum still water level Positive storm surge Minimum still water level Negative storm surge Classification of structures * hydrodynamic transparent (slender structures, Morison loading) * hydrodynamic compact (large volume structures, diffraction analysis)

Classification: Statoil internal Status: Draft 9 Bølger og strøm viktig Bølger viktigst When to account for which effects?

Classification: Statoil internal Status: Draft Current velocity field For most design work, current profiles (speed versus depth) are established from current measurements. Measurements are typically made at a number of depths. Up to now extremes are typically estimated for each depth separately. Linear interpolation between depths. It is likely that such profiles are concervative for most cases, but not necessarily for all – design profile Presently work is going on regarding developing more adequat design profile: * Current is described as a sum of empirical orthogonal functions. * Family of profiles with speed at one depth and associated values at other depths.

Classification: Statoil internal Status: Draft 11 Current components: * Tidal current. * Background current. * Wind driven current * Meanders or vortex current If data are not avvailable, current field may be taken as the sum of the tidal current (constant through water column) and the wind driven current (= 1-2% of mean wind speed at the surface decaying linearly to zero at about 50m. In connection with loads on structures, the current is considered as a slowly varying phenomenon, i.e. the current speed is kept constant for a short term sea state. Typical surface current speed North Sea (no eddies present): current: 1m/s

Classification: Statoil internal Status: Draft Waves and descrition of waves The sea surface is of an irregular nature, but it can to a first appoximation be written as a sum of sinusoidal with different amplitude, different frequency, (different direction) and different phase. For practical application, the long term variation of the sea surface elevation process is consider as i piecewise stationary (and homogeneous) stochastic process (field). If the sea surface elevation can be modelled as a Gaussian process, each stationary sea state is in a statistical sense completely characterized by the directional wave spectrum: S h (f,  ) = s h (f)*d(  ) Frequency Spectrum Spreading function Several models are proposed for the frequency spectrum: * ISSC (Generalized Pierson Moskowitz, fully developed wind sea) * JONSWAP (pure wind sea, may be growing) * Torsethaugen (combined sea, wind sea + swell) Common for all models is that they are parameterized in terms of significant wave height, h s, and spectral peak period, t p.

Classification: Statoil internal Status: Draft 13 Long term modelling of sea states In view of what is said above, one can conclude that a short term sea state is for practical purposes described in terms of significant wave height, spectral peak period and direction of propagation. The long term description of wave conditions can be done by establishing a joint probability density function for H s, T p and  : f Hs,Tp,  (h,t,  ) = f Hs,Tp|  (h,t|  )*f  (  ) where: approximated by prob. mass function f Hs,Tp (h,t) = f Hs (h) * f Tp|Hs (t|h) 3-p Weibull or LonoWe Fitted to available data Log-normal or Weibull fitted to data for each hs - class

Classification: Statoil internal Status: Draft 14 Short term modelling of sea states It is most common to use long crested sea. This may be non-conservative for ships heading into sea with respect to assess ship rolling. Select spectral model in view of problem to be considered.

Classification: Statoil internal Status: Draft 15 Torsethaugen versus JONSWAP For a further illustration of how to descibe sea states in short and long term for practical applications, see e.g. the Statoil Metocean report.

Classification: Statoil internal Status: Draft 16 Prediction of extreme sea states and waves Sea states Various approaches are used: * All sea state approach (based on modelling Hs and Tp for all 3-hour stationary sea states. Most commonly adopted approach in Norwegian waters * Peak over threshold approach (based on modelling merely storm peak characteristics. Most common in areas of a mixed population wave climate, i.e. a distinct difference between normal conditions and stom conditions.) In the following we will stay with the first approach: The number of 3-hour sea states per year is 2920 if all directions and seasons are pooled together: The – probability significant wave height is then estimated by: 1 – F Hs (h_10 -2 ) = 1/ (2920*100) Corresponding peak period:

Classification: Statoil internal Status: Draft 17 If structural response is rather sensitive to the peak period, it is not necessarily the highest sea state that is the most critical. In order to cover these needs lines of hs and tp corresponding to a constant probability of exceedance are often provided: It will be indicated later how such a contour can be used for design load calculations.

Classification: Statoil internal Status: Draft 18 Prediction of extreme individual waves For a number of response problems, the sea surface process can for practical applications be modelled as a Gaussian process. If the purpose of the assessment is to predict accurate extreme individual waves, this should not be done. Regarding crest height, a proper model is the distribution recommended by Forristall. Where: Short term extreme value: F C (c 3hmax ) = 1- F C (c 3hmax ) = 1/m 3h Characteristic maximum No. of crest heights in 3 hours.

Classification: Statoil internal Status: Draft 19 Forristall model is based on fitting a Weibull model to a huge number of simulations of second order surfaces for various conditions. Regarding wave heights, the model proposed by Arvid Næss (1985) is recommended. This model is a bandwidth corrected Rayleigh distribution.

Classification: Statoil internal Status: Draft 20 Regarding a prediction of long term extremes, a long term analysis should be carried out: The 10_2 crest height is then given by: 1- F X3hour (x_10 -2 ) = 1/ = / 2920 Alternatively, one may use the environmental contour principle. This includes the following steps: a) Find worst sea state along 10-2 contour line. b) Establish the distribution function for the 3-hour maximum crest height. c) An estimate for the 10-2 crest height is obtained by adopting the 90-percentile of this distribution.

Classification: Statoil internal Status: Draft 21 A so far unsolved question regarding waves, is the possible existence of freak waves. (From: BBC Horizon)

Classification: Statoil internal Status: Draft 22

Classification: Statoil internal Status: Draft 23 What can be the problem if freak waves exist? The above scenario is completely unacceptable, In particular for manned structures, it is therefore a good idea to ensure a reasonable robustness against unexpectedly large crest heights. For a given site freak waves if they exist are most probably so rare that they will not effect our 10-2 and 10-4 predictions. But if a possible freak wave mechanism will occur more frequently in sea states beyond thos thatr are observed so far, this conclusion may have to be adjusted.

Classification: Statoil internal Status: Draft 24 Wave loads For the purpose of this course, we will assume these loads to be linear and characterized by a transfer function. In the following we will focus on these type of loads.

Classification: Statoil internal Status: Draft Large volume structures Fixed platforms Most important load is wave frequency load. The load is Most often approximated very well by a linear potential theory. The loadis typically described by the transfer function or response amplitude operator: H(  ) = x(  )/  (  ) where  (  ) is a harmonic wave,  (  ) =  0 exp(-i  t) and x(  ) is the response due to this harmonic wave, x(  ) = x 0 exp(-i(  t+  ))  H(  ) = x 0 /h 0 exp(-i  ) = |H(  )| exp(-i  ) A closed for solution for H(  ) can be found for some few cases, most of the time H(  ) is found for the various frequencies using numerical methods. In addition to the wave frequency load, there will also be a slowly varying force on the platform corresponding to difference frequencies and a high frequency load corresponding to the sum frequencies. These are much lower than the wave frequency load, but the high frequency load may for some structures hit the largest natural period and thus cause som resonant response. This is referred to as ringing.

Classification: Statoil internal Status: Draft 26 Floating structures The same can be said as for the fixed platforms. However, the slowly varying load can now hit the natural period of the horisontal modes of motion and cause rather large offset motiona and mooring line forces. For a TLP the sum frequency term can hit the natural period of the vertical modes of motions Response calculations Select a sea state, hs and tp. The wave spectrum is then known. As the transfer function has been calculated, the response spectrum is given by: s x (  ) = | H(  )| 2 s  (  ) The variance and zero-up-crossing frequency can be found from the spectral moments, m 0 and m 2. THe response process can often be assumed to be Gaussian, this means that the global maxima of the response process is desribed by the Rayleigh distribution. F Xmax | Hs,Tp (x | hs,tp) = 1 – exp(-0.5 (x /  X ) 2 ),  X =  X (h , t p )

Classification: Statoil internal Status: Draft 27 Long term distribution of Xmax is given by: response maximum is given by: Alternatively, the environmental contour approach can be adopted. However, for linear systems one may just as well do a classical long term analysis as indicated above.