Classification: Internal Status: Draft Predicting characteristic loads for jackets and jack-ups Shallow water location – quasi-static approach Problem definition Needed environmental information Design wave properties Deeper water location – dynamic effects are important Problem definition Needed environmental information Time domain solution of equation of motion (steps of the analysis) Simple example of tie domain solution technique Sverre Haver January 2008

2 Method when structures are in shallow and moderate water depths What is to be understood by shallow and moderate depths will not be very precise figures. A proper limiting depth will depend on structural layout and soil-structure solution.

3 Problem definition For depths considered here the largest natural period should stay below – say - 2s. This means that natural periods are far away from energentic wave exitation and we can neglect effects of dynamics. For these structures, there is a very strong corrrelation between the annual exceedance probability of the wave height passing the structure and the response level experienced in the structure. This means that a design wave method will typically be used for obtaining proper characteristic responses. As a design wave, a Stokes 5th order wave profile is most often adopted. P1 P2 P4 P3 P5 r 0.01 = ? r(Pn) – wave in various positions relative to structure P1 P2 P3 P4 P – leg response

4 Needed environmental information – annual probability wave height: Conservative estimate for the wave period to be associated with the wave height: Can be estimated by establishing the joint distribution of H and T. If conditional mean and 90% band for the spectral period given Hs 0.01 is known, corresponding values for T to be associated with H 0.01 can be found by multiplying the t p -value with Can be predicted after establishing the long term analysis of wave height. Short term distribution is reasonably well modelled by Rayleigh with parameter given by Arvid Næss. Simplified approach: If annual probability Hs is known, h 0.01 can be estimated by 1.92*h s0.01

5 Design Waves – Illustration of profile, particle speed, particle acceleration and instantaneous load picture for 80m depth – annual probability wave for a North Sea location is given by: H = 28.5m, T_mean = 15.3s and 90% period band = 13.6s – 17.1s Observations: Skewed profile vertically. Symmetric profile horizontally around crest position. Crest is narrower than trough.

6 Particle speed and acceleration Depth: 80m

7 Instantaneous load picture

8 Dynamic amplification 1-DOF system

9 For which depth are dynamics expected to be important? Depths below may be typical depths – not THE LIMITING DEPTH for all jackets and jack-ups. For some cases the design wave method may be accurate also for these depths and for other cases dynamics may be important for a water depth smaller than the numbers indicated below. d > 150m d > 100m

10 Problem definition Largest natural period, T0, is now so close to the energetic wave period band, that a considerable loading around T0 is present – either directly or indirectly through super-harmonic exitation.  Solution method is solving equation of motion in time domain! This solution method is fast for linear mechanical systems. (For a non-linearity in e.g. stiffness and/or damping where these quantities depend on the response one may have to iterate for each time step and that process may increase computational time considerably.) Kvitebjørn: Quasistatic versus full dynamic solution Kvitebjørn: Simulated versus measured deck displacement.

11 Time domain analysis Steps involved in solving the equation of motion in the time domain: i) From a given wave spectrum, s  (f), a realization of the surface field is generated,  (x,y,t). ii) From the surface field,  (x,y,t), particle speed and particle acceleration are calculated, u(x,y,z,t) and a(x,y,z,t). iii) Loads on the various submerged structural members are calculated as function of time. Loads are given at each submerged nodal point of structure: F(t i ) = { f1(t i ), f2(t i ), ….., fn(t i )}, i=1, 2, …, imax. A fine resolution over a structure is needed close to the surface. This means that we know the load vector for the wanted lengt of the time history! (Important: A sufficient frequency resolution must be ensured under step i). iv) The equation of motion is solved for the given force vector using some step by step procedure. Result are time histories of all nodal point displacements (or nodal structural forces). v) From the – say – 3-hour time history for a given response quantity, the distribution function for response maxima and for the 3-hour extreme value can be estimated. This is short term analysis of the response for a non-linear response problem.

12 Time domain analysis Example of response, x(t), to a transient load episode, F(t).

13 The solution for x i+1 in terms of known quantities: Expressions for speed and motion: and Assuming acceleration within a step of length h to be constant and equal to the average value: (One of several possible choices.) One can show (See App. D): and Dynamic eqilibrium at step i+1: After some manipulations we find: Mission completed! With x i+1 known, we can find the derivative and the acceleration at step i+1, i.e. we are ready for the next step!

14 Numerical example Initial conditions: , here = 0. System characteristics:  T 0 = 3s, lambda = 0.10

15 Response for reduced duration of load relative the natural period

16 Response for increased duration of load relative the natural period

17 Effect of damping