... the time domain approach. The ocean environment is represented by a number of sea states, and the line tension response statistics in each sea states is studied. The study addresses the non-Gaussian response characteristics. It is shown that, due to the nonlinearity, the line tension response is non...

...<f 1990 COPYllghr © 1990 hy The Intel natIOnal Souery of Otf,hOl e and Polm Engmeel S ISBN 0-9626104-4-5 STRESS-RESPONSE OF OFFSHORE STRUCTURES BY EQUIVALENT POLYNOMIAL EXPANSION TECHNIQUES G SlgUldsson and S R K Nielsen Department of BuildIng Technology and Structural EngIneenng UnIversIty of Aalborg Aalborg, Denmark Abstract This paper concerns an investigation of the effects of nonlinearity of drag loading on offshore structures excited by 2D wave fields, where the nonlinear term in the Morison equation is replaced by an equivalent cubic expansion. The equivalent cubic expansion coefficients for the equivalent drag model are obtained using the least mean square procedure. Numerical results are given. The displacement response and the stress response processes obtained using the above loading model are compared with simulation results and those obtained from equivalent linearization of the drag term. 1. Introduction The loading imposed on structural members of an offshore struc- ture subjected to wave action represents one of the major steps in design of deepwater bottom-supported structures. The wave loading is normally estimated using the well-known Morison equa- tion for a member with dimensions such that the presence of the member does not significantly disturb the wave field. This paper concerns an investigation of the effects of nonlinear- ity of drag loading on offshore structures excited by irregular 2D wave fields, where the nonlinear term in the Morison equation is replaced by an equivalent cubic expansion. The structural system is modelled by a linear system with a finite number of degrees of freedom. A system reduction based on an eigenmode expansion is applied, where the frequency response matrix of the system is expressed in two terms, corresponding to the quasi-static con- tribution and the dynamic contribution, respectively. The first order wave theory is applied to relate the surface elevation with the local kinematics of water particles. The influence of the ve- locity of the structure is ignored in the drag term. It is assumed 134 that the sea surface can be considered as a realization of a sta- tionary zero-mean Gaussian process, which is also homogeneous in the horizontal space parameters. The response processes of the system are determined based on a spectral approach. The equiv- alent cubic expansion coefficients for the equivalent drag model are obtained using the least mean square procedure. The vari- ance of the displacement response and the stress response pro- cesses obtained using the above loading model are compared with simulation results and with results obtained by using two differ- ent equivalent linearization methods of the drag term, namely by using the least mean square procedure and by the require- ment that the variance of the original and the equivalent linear drag loading is alike. Several papers in recent years have dealt with methods for estimation of the displacement and stress re- sponse, obtained using polynomial expansion of the drag term in Morison's equation, e.g. (Burrows, 1977), (Burrows, 1983), (Burrows, 1986) and (Bruce, 1985). However, a very important limitation of these approaches has been that they only deal with a quasi-static response, or that the structure is considered one- dimensional vertical, ignoring the horisontal spatial correlation of the wave loading. 2. Short-Term Model of the Sea States The observed sea elevation, '17( x, t) at the fixed location x = (x, y) at a time t, can be considered as a realization of a non-stationary stochastic process, whose characteristic parameters vary slowly with time. Further, it is assumed that for short-term periods (a few hours) the sea surface 'I7(x, t) can be considered as a realisa- tion of a stationary stochastic process, which is also homogenous in the horizontal space parameters. This process is assumed to be a zero-mean Gaussian process. A consequence of these sim- plifying assumptions is that within the short-term time scale the sea surface elevation is completely defined by the cross-covariance function Ix, r), defined as Ix, r) = E[T/(x, t) T/(x + ~x, t + r)] (1) where ~x = (Xl - X2,Yl - Y2), r = tl -t2. (XX,YI) and (X2,Y2) are the spatial coordinates of two points at the sea surface. In structural analysis it may be more convenient to use spectral densities than correlation functions. Applying linear wave the- ory and assuming long crested waves the corresponding spectral density can be obtained as SX,w) = exp( -zk(wx cosB + ~Y sinB)) Sw) (2) where w is the frequency (rad/sec), ~x = (Xl - X2), ~Y = (YI - Y2). B is the angle from the x-axis to the direction of wave propagation of the 2D sea state in counter-clockwise direction. z = A and k(w) is the wave number obtained as w 2 = kg tanh(kh) w 0, k 0 (3) where 9 is the acceleration of gravity and h is the water depth. Sw) is the double sided auto-spectral density function, and Sx,w) is the double sided cross-spectral density of the sea surface. For negative frequencies the wave number should be defined from the asymmetry condition k(-w) = -k(w) (4) In most practical applications a standard formula involving a few sea state characteristics is used for Sw). Over the last 30 years many spectral expressions have been suggested. A common feature of most spectral models is that they are of a unimodal form and mainly meant to characterise a pure wind driven sea. Here, the JONSWAP spectrum is adopted as·a model for wind sea. This spectrum can be written, (Hasselmann et al., 1973) Sw) 2 -5 (5( W )-4) expIIL.-I)/tI)2) 0: 9 w exp -- - i "'p 4 w, where 0: equilibrium range parameter w, spectral peak frequency (= 27T /T, ) i spectral peak parameter (T spectral peak width parameter The mean values from the JONSWAP measurements are usually adopted for (T, i.e. (T = 0.07 for w ~ w, and (T = 0.09 for w > w,. Here (T is chosen as 0.08 for all frequencies and all sea states. For a sea state with a given value of the significant wave height Ha the remaining parameters (0:, i and w,) are related to each other through the following equation, (Haver, 1985) i = exp(3.484(1- 0.19750H (6) Eq. (5) is expected...

... the linear damage accumulation (Palmgren-Miner's rule) is used with a multi-linear SoN fatigue model. This model can also be used for a nonlinear fatigue model applying a piecewise linear approximation. Formulation of the mean damages in short-term and long-term sea states are presented in general...

... of the deep water jacket, a stepwise linearization is recommended, i.e. using different lineari- zations for different sea state levels. It should be noted that the resulting effect of introducing short crested sea is not accu- rately indicated in the present study. A con- cistent demonstration of this effect...

... The marginal (long term) distribution function for a given response quantity is then estimated as a weIghted sum of the short term distributions for all possible sea states. Thus the probabIlities of exceedance from the various sea states are prop- erly accounted for and the problem of selectmg a consistent...

... irregular sea states described as operational and design conditions. The operational condition was represented by a significant wave height Hs = 6.0m, mean zero-upcrossing period Tz - 8.5s, and mean design condition was T = l3.5s and u z c current speed Uc represented by 0.5m/s. The Hs - lS.5m, 1.0m/s...

... ' and peak wave perlod, Tp ' are applied deterministically for a given short term condition and assumed to be constant during the sea state. Wave directionality and spreading should in general be accounted for in the short term description. The steady current velocity, VC' and dlrection at the sea bed...

... that can be combmed with rehability methods IS reqUired when several environmental vanables are to be combmed. Two alternative pro- cedures for this estimation of extremes are considered as described below. Method 1) : Long term statistics by mdependent sea states. This approach was formulated m [18...