...- ants. The case of constant temperature is considered in this paper so the temperature dependences are not analysed. Under constant stresses, thed-,a, t£ parameters are also cons- tant so the equation (6) is integrated and resulted in the fo~ h'i = a. (i /( (- ilh (7) According to (6) and (7), the t=tf...

... probability distribution with thick tails (high kurtosis) If the equations of motion were quasi-static, the high kurtosis of the exciting force would pass unabated into the response. When the dynamics is taken into account, it is found that the motion becomes more Gaussian. This is beneficial in terms...

... formulation of the problem is justi- fied. Let us suppose that the wavelength of ice cover flexure is much greater than the thickness of ice. In this case the change of strain energy of ice in process of crack growth, e.g. the energy needed for fracture, may be calculated by means of flexure equation...

... bi-frequency domain sum frequency frequency monochromatic wave mooring system equation upstream oil & gas surge response experiment dual wave subsea system coefficient drift force drift motion wave drift amplitude surge drift motion prediction excitation slow drift motion...

...]. (Except that the sign convention is opposite.) Eq.(5) was derived above by consid~ring longitudinal equilibrium of a piece of the tube including the end cap, see Fig.1d. Still, the equation applies to any tube (at any point on it) whether it has end caps or not, and irrespective of support conditions...

... an dy (A3) To satisfy moment equilibrium on the element, VTbdl+1f od-Y)dY] (A4) Substituting for an and 'tnt from equations (AI) and (A2) into equilibrium equations (A3) and (A4) and equating the two expressions gives a formula which upon back substitution into either equilibrium equation gives...

.... It is hoped that such analyses will serve to help understand the force/time history of ice structure interactions, and to help de- velop theoretical foundations for safe design of structures in the ice-infested waters. MATHEMATICAL FORMULATION Basic Equations The propagation of stress and the evolution...

...<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 equilibrium equations of the moored body are indeterminate if the number of mooring legs is more than the number of unconstrained degrees of free- dom being considered at the moored body. In the pres- ence of spatially varlable sub-surface currents, it is generally not appropriate to approximate the cable...

... are incorporated into the model in section 3. They are expressed in terms of the empirical drag coefficient and the tri-chromatic relative velocity, providing viscous damping and viscous excitation forces. In section 5 the non-linear equations of motion are derived to second-order. Results are presented for a loth...

... properties have revealed complex behavior including bifurcations, coexistence of attractors, and sensitivity to initial conditions. Examples of such systems are the biased hardening Duffing equation [Ueda 1980] and single equilibrium systems in which the restoring force is described by a power series...

... is to introduce a consistent procedure for the introduction of both elastic and plastic multi-axial stress-strain effects into the design process. allowable hoop stress plasticity axial stress calculation equation equivalent strain upheaval hoop stress submarine pipeline specification thickness...

...) In order to reduce the amount of computational effort involved in computing the kinematics at the displaced position of the platform at each instant of time, the direct FFT of 71(0, t) is performed only once and the summations in equations (12) to (16) are done directly at each time step. The equations...

... of Naval Architecture and Marine Engineering of the National Technical University of Athens. Preliminary results from these experiments, as well as comparison of the obtained data with numerical predictions are also briefly discussed. GOVERNING EQUATiONS The cable is modelle4 as a slender rod...

... are free to slip, there will not be any interference between wire layers which could increase bending stiffness. PART II - DEGRADATION MODEL This part includes discussions of the equation for wire slip during pipe bending, the criteria for the onset of slip, and public-domain wear data that will be used...

... is based on the Morison equation and/or two-dimensional potential flow theory, and global structural analysis IS executed by using spatial framework model, e.g., see [Akita et al , 1978). Recent remarkable progress of electronic computers, however, has enabled three-dimensional diffraction/radiation...

.... variation interaction strength equation adhesion structure building intensity ice strength granulated ice strength variation spray cone method air temperature calculation foundation ice building ice structure cross-section experimental result show upstream oil &amp; gas P, oceedmgs...

... with constant panel methods and HOB EMs that are employed in conjunction with the hybrid boundary integral equation procedure, for a variety of structural configurations. It was concluded from the study that HOB EM has several important features: it uses many fewer boundary elements and much less computer time...

... Theoretical Background The equilibrium equation is derived from the principle of virtual velocities: (1) where Eext is the rate of external work done on the shell and Emt is the rate of internal energy in the shell. 154 For s~ells crushed by a thin flat transverse punch load, as shown in Figure 1, the rate...

... real time system prediction estimation drift force convolution upstream oil &amp; gas equation inverse filter time history Peedmgs of the FlIst (1990) EUiopean OffshOle Me(han{(s Sympo"um TlOndhelm. NO/way. 20-22 Augu't 1990 CopvlIght © 1990 by The IntflnallOnal Souely ofOffshOle and Polm...