Random Wave Loading Research Articles

Triangular Configuration Tension Leg Platform behaviour under random sea wave loads

This study investigates the dynamic response of a Triangular Configuration Tension Leg Platform (TLP) under random sea wave loads. The random wave has been generated synthetically using the Monte-Carlo simulation with the Peirson–Moskowitz (P–M) spectrum. Diffraction effects and second-order wave forces have not been considered. The evaluation of hydrodynamic forces is carried out using the modified Morison equation with water particle kinematics evaluated using Airy's linear wave theory. Wave forces are taken to be acting in the surge degree-of-freedom. The effect of coupling of various structural degrees-of-freedom (surge, sway, heave, roll, pitch and yaw) on the dynamic response of the TLP under random wave loads is studied. Parametric studies for random waves with different H s and T z under the presence of current have also been carried out. For the orientation of the TLP, surge, heave and pitch degrees-of-freedom responses are influenced significantly. The surge power spectral density function (PSDF) indicates that the mean square response is affected by the amplification at the natural frequency of the surge degree-of-freedom and also at the peak frequency of the wave loading. The PSDF of the heave response shows higher peak values near the surge frequency and near the peak frequency of the wave loading. Surge response, therefore, influences heave response to the maximum. Variable submergence seems to be a major source of nonlinearity and significantly enhances the responses in surge, heave and pitch degrees-of-freedom. In the presence of current, the response behaviour of the TLP is altered significantly introducing a non-zero mean response in all degrees-of-freedom.

Response of a tension leg platform to stochastic wave forces

The equations of motion and the response of a Tension Leg Platform with a single tendon undergoing planar motion are presented. The hull is represented by a rigid cylindrical body and the tendon as a nonlinear elastic beam. Tendons have often been modeled as massless springs, which neglects contribution to the response by the wave forces on the tendons and its varying stiffness due to changes in its tension. The structure is subjected to random wave loading. Linear wave theory is applied and the random wave height power spectrum is transformed into a time history using Borgman's method. The surge and pitch responses for the hull, and the surge response along the tendon are presented for two cases. Case 1 represents a tendon with a hull mass two orders of magnitude smaller than the tendon's mass. In Case 2, the hull mass is two orders of magnitude greater than the tendon. Inclusion of tendon forces were found to significantly increase the amplitude of the surge response for Case 1 but not for Case 2.

Frequency-Domain Analysis of Offshore Platform in Non-Gaussian Seas

A frequency-domain solution approach for the response of a system whose inputs are nonlinear transformations of non-Gaussian (nonlinear) wave kinematic processes is introduced. Particularly, this paper compares the probabilistic response characteristics of jacket-type platforms in deep water that are subjected to both Gaussian and non-Gaussian random wave loadings. Unlike earlier analytical treatments of this class of system, a statistical description of the wave forces is first developed to reflect nonlinearities and associated non-Gaussianity in the wave field kinematics. The kinematics are derived from Laplace's equation and nonlinear boundary conditions using a second-order Stokes' perturbation expansion. The deck response resulting particularly because of the effects of the second-order contribution to the loads on an idealized platform is computed. Consideration is given to the importance of the spacing of the legs to the response of the structure. The impact of swell in addition to locally wind-gen...

Stochastic response of a two DOF articulated tower

In a previous paper [P. Bar-Avi and H. Benaroya, Journal of Sound Vibration 190, 77–103 (1996)], the response of an articulated tower in the ocean subjected to deterministic and random wave loading was investigated. The tower was modeled as an upright rigid pendulum with a concentrated mass at the top and having one angular degree of freedom about a hinge with coulomb damping. In this paper, which is an extension of the previous one, the tower is modeled as a spherical pendulum having two angular degrees of freedom. The tower is subjected to wave, current, and vortex shedding loads. Geometrical non-linearities as well as non-linearities due to wave drag force, which is assumed to be proportional to the square of the relative velocity between the tower and the waves, were considered. The governing coupled differential equations of motion are highly non-linear, and have time-dependent coefficients. The tower's average response is evaluated for uniformly distributed random fluid constants C D, C M, C L, friction coefficient μ and current direction α. This is accomplished computationally via Monte-Carlo simulations with the use of ‘ACSL’ software. The influence on the tower's response of different parameter values is investigated.

Polynomial Approximation of Morison Wave Loading

Abstract For offshore structures with slender elements, the modeling of random wave loads by the Morison equation yields an equation of motion which has no analytical solution for response moments except in a few limiting cases. If polynomial approximations of the Morison drag loads are introduced, some procedures are available to obtain the stationary moments of the approximate response. If the response process is fitted by non-Gaussian models such as proposed by Winterstein (1988), the first four statistical moments of the response are necessary. The paper investigates how many terms should be included in the polynomial approximation of the Morison drag loading to accurately estimate the first four response moments. It is shown that a cubic approximation of the drag loading is necessary to accurately predict the response variance for any excitation. For the fit of the first four response moments, at least a fifth-order approximation appears necessary.

Response of a two degrees of freedom articulated tower to different environmental conditions

In a previous paper [ J. Sound Vibr. (1996)], the response of an articulated tower in the ocean subjected to deterministic and random wave loading was investigated. The tower was modeled as an upright rigid pendulum with a concentrated mass at the top and having one angular degree of freedom about a hinge with coulomb damping. In this paper, which is an extension of the previous one, the tower is modeled as a spherical pendulum having two angular degrees of freedom. The tower is subjected to wave, current, vortex shedding, wind and Coriolis acceleration loads. Geometrical non-linearities as well as non-linearities due to wave drag force, which is assumed to be proportional to the square of the relative velocity between the tower and the waves, were considered. The governing coupled differential equations of motion are highly non-linear, and have time-dependent coefficients. The tower's response to the external forces is found, that is, equilibrium position due to wind and current and response to wave excitation. Resonances (harmonic and subharmonic) and also chaotic response are investigated.

Extreme Jack-Up Response: Simulation and Nonlinear Analysis Methods

Abstract The nonlinear dynamic response of a jack-up structure under random wave loads is considered. For a simplified jack-up model, average behavior and variability in extreme forces and responses are found from simulation over many 6-h seastates. Weibull and Hermite analytical models of response extremes are also presented and evaluated. These models use shorter, less expensive simulations to estimate a limited number of response statistics, such as moments or parameters of the response peak distribution, and fit analytical models to estimate global extremes. Necessary simulation lengths are established both for direct simulation of extremes, and for analytical extreme models.

The Influence of Alternative Wave Loading and Support Modeling Assumptions on Jack-Up Rig Response Extremes

Abstract A study is conducted of the response of a jack-up rig to random wave loading. Steady current and wind load effects are also included. The effects of varying the relative motion assumption (in the Morison equation) and of varying the bottom fixity assumptions are investigated. One “fixity” model employs nonlinear soil springs. Time domain simulations are performed using linearized as well as fully nonlinear models for the jack-up rig. Comparisons of response statistics are made for two seastates. Hydrodynamic damping causes the rms response to be lower in the relative Morison case. The absence of this source of damping in the absolute Morison force model gives rise to larger resonance/dynamic effects—this tends to “Gaussianize” the response. Hence, the relative Morison model leads to stronger non-Gaussian behavior than the absolute Morison model. This is reflected in moments as well as extremes. The different support conditions studied are seen to significantly influence extreme response estimates. In general, stiffer models predict smaller rms response estimates, but also exhibit stronger non-Gaussian behavior. The choice of the Morison force modeling assumption (i.e., the relative versus the absolute motion formulation) is seen to have at least a secondary role in influencing response moments and extremes.

NON-LINEAR DYNAMICS OF AN ARTICULATED TOWER IN THE OCEAN

This paper presents studies on the response of an articulated tower in the ocean subjected to deterministic and random wave loading. The tower is modeled as an upright rigid pendulum with a concentrated mass at the top, having one angular-degree-of-freedom (planar motion) about a hinge with Coulomb friction, and viscous structural damping. In the derivation of the differential equation of motion, non-linear terms due to geometric (large angle) and fluid forces (drag and inertia) are included. The wave loading is derived using a modified Morison's equation to include current velocity, in which the velocity and acceleration of the fluid are determined along the instantaneous position of the tower, causing the equation of motion to be highly non-linear. Furthermore, since the differential equation's coefficients are time dependent (periodic), parametric instability can occur depending on the system parameters such as wave height and frequency, buoyancy, and drag coefficient. The non-linear differential equation is then solved numerically using `ACSL' software. The response of the tower to deterministic wave loading is investigated and a stability analysis is performed (harmonic, subharmonic and superharmonic resonance). To solve the equation for random loading, the Pierson-Moskowitz power spectrum, describing the wave height, is first transformed into an approximate time history using Borgman's method with slight modification. The equation of motion is then solved as buoyancy, initial conditions, wave height and frequency, and current velocity and direction, is investigated.

Stochastic Response of Offshore Platforms by Statistical Cubicization

In this study, a method of statistical cubicization is proposed to predict stochastic response of offshore platform to a Morison-type nonlinear random wave loading. The nonlinearity is replaced by an equivalent polynomial expansion up to cubic order. The Volterra series method is used to solve the resulting nonlinear equivalent system in the frequency domain. Complete expressions to evaluate response central moments up to the fourth order are obtained for a three-term Volterra system driven by a stationary Gaussian input. To make the developed method efficient and practical, an efficient solution scheme is proposed to implement the Volterra series method. Results (i.e., response mean, variance, skewness, and kurtosis) are presented for the cases of fixed offshore structures and compared with those obtained by simulation, equivalent linearization, and quadratization methods. The statistical cubicization method is superior to both the statistical linearization and quadratization method.

Volterra Models of Ocean Structures: Extreme and Fatigue Reliability

A new method is developed to estimate the reliability of nonlinear structural systems, modeled with second‐order Volterra series, against both extreme and fatigue failures. It includes non‐Gaussian response effects for a given input spectrum as well as long‐term variations in the parameters of this input spectrum. Moment influence coefficients are introduced to give response moments for arbitrarily scaled input spectra. Rates of extremes and fatigue damage are then estimated analytically from a non‐Gaussian (Hermite) random process model. The method is applied to study the pitch and heave “springing” motions of a tension leg platform (TLP) under random wave loads. State‐of‐the‐art nonlinear diffraction results are used to model the second‐order loads on the TLP. These forces are then applied to a coupled six‐degree‐of‐freedom (6DOF) linear structural model of the TLP. Statistics of tether tensions, platform motions, and accelerations are estimated. These results are weighted over probabilistic models of the long‐term wave climate to estimate reliability against extreme and fatigue limit states.

Dynamic amplification of offshore steel platform responses due to non-Gaussian wave loads

The dynamic response of offshore steel platforms subjected to random non-normal wave loads is considered. Two procedures are described. The first is a simplified quasi-static approach based on the classical one degree-of-freedom dynamic amplification factor. In the second procedure the statistical moments of the non-normal dynamic response are calculated without approximations using the theory of diffusion processes and moment closure techniques. An example jack-up platform is considered. It is found that the quasi-static approach is rather accurate for extreme value predictions, but not applicable for fatigue analysis where the second procedure should be applied. It is also observed that the dynamic amplification changes the statistical behaviour of the response significantly towards a Gaussian process.

Recent development in the fast corrosion fatigue analysis of offshore structures subject to random wave loading

In many areas of engineering, such as the offshore industry, welded steel joints are widely used as standard structural components. These joints are usually subject to long-term random wave loading and are therefore susceptible to fatigue damage. In many cases, the complex service loading is described by stress (or strain) power spectra, each representing a stationary sea state. These power spectra are obtained from hydrodynamic analysis or in situ monitoring. These will then be used in design calculations, feasibility studies or in-service assessment of fatigue damage on the structures. Usually, the power spectra will have to be realized into real-time histories and then counted before fatigue analysis can be carried out. On many occasions where a large number of design options or joints need to be analysed, this process becomes very time consuming and expensive. The situation is further complicated by the calculations involving corrosion fatigue for joints in sea water. The paper will start with a brief presentation of the fatigue analysis procedures for offshore welded joints and several existing models that were derived to bypass the laborious load history analysis mentioned above. More effort, however, will be concentrated on presenting the development of a new model. This model not only provides a more consistent and accurate prediction, but has also been adopted successfully for corrosion fatigue analysis.

Dynamic Response Statistics of Linear System Under Morison-Type Wave Force

Abstract The primary objective of this study is to establish the non-normal relationship between Morison-type random wave loading and structural response via repeated Monte Carlo simulations. A standardized equation of motion will be developed, such that the numerical results can apply to various structures and sea states, instead of a particular structure or sea state. The reliability of the Monte Carlo simulation technique will also be investigated, analytically and experimentally.

Stochastic Dynamic Response to Nonlinear Wave Loading: Fourth‐Moment Analysis

This study presents an analytical method for calculating the fourth‐moment properties of the response of linear systems to a Morison‐type nonlinear random wave loading. The method is applicable to general cases with no restriction that the structure behaves in a quasi‐static way or that the response is completely dominated by resonance. The developed method is efficient and practical. According to a comparison with the results obtained from repeated simulations, it is found that the proposed method is superior to the other two existing methods, i.e., the Δ‐correlated method and the separable method.

Fatigue reliability assessment of offshore structures: Kam, J.C.P. Qual. Reliab. Eng. Int. Jan.–Mar. 1988 4, (1), 41–48

In many areas of engineering, such as the offshore industry, welded steel joints are widely used as standard structural components. These joints are usually subject to long-term random wave loading and are therefore susceptible to fatigue damage. In many cases, the complex service loading is described by stress (or strain) power spectra, each representing a stationary sea state. These power spectra are obtained from hydrodynamic analysis or in situ monitoring. These will then be used in design calculations, feasibility studies or in-service assessment of fatigue damage on the structures.Usually, the power spectra will have to be realized into real-time histories and then counted before fatigue analysis can be carried out. On many occasions where a large number of design options or joints need to be analysed, this process becomes very time consuming and expensive. The situation is further complicated by the calculations involving corrosion fatigue for joints in sea water.The paper will start with a brief presentation of the fatigue analysis procedures for offshore welded joints and several existing models that were derived to bypass the laborious load history analysis mentioned above. More effort, however, will be concentrated on presenting the development of a new model. This model not only provides a more consistent and accurate prediction, but has also been adopted successfully for corrosion fatigue analysis.

Considerations of probability-based fatigue design for marine structures

Fatigue is a major failure modelin marine structures which respond dynamically to random wave and wind loading. Oscillatory stresses produce fatigue at points of stress concentration, typically the welded joints. Because all fatigue design factors are subject to significant uncertainty, a reliability approach to management of such uncertainty seems appropriate. This article summarizes some studies in fatigue reliability research and demonstrates how reliability methods can be effectively utilized by designers to avoid fatigue in marine structural components. Included are (1) a description of fatigue damage under variable amplitude stresses employing the characteristics S-N and fracture mechanics models, (2) models for reliability assessment relative to fatigue and the use of these models to derive design criteria, and (3) an elementary application of system fatigue reliability analysis to establish component design criteria.

S- N curve vs fracture mechanics approach to fatigue analysis of steel jacket platforms

The fatigue life of a joint in an offshore structure can be estimated by either using the conventional S- N curves or the fracture mechanics approach. In this paper both procedures are applied to obtain fatigue life results for a steel jacket model subjected to random wave loading. The results are obtained using different S- N curves and a fracture mechanics approach. The effect of the choice of various S- N curves on the fatigue life estimate is found to be significant. The fracture mechanics approach is found to yield a much lower fatigue life as compared to the S- N curve approach.

Non-Gaussian stochastic response of nonlinear compliant platforms

A moment function method is presented to estimate the stochastic response of compliant offshore platforms with nonlinearity in stiffness based on non-Gaussian closure. For guyed towers with clump weight, the nonlinearity in stiffness is of the softening type. The random wave loading is expressed in terms of a rational spectrum, making the system Markovian. Using Ito's rule for stochastic differentiation, differential equations for moments up to the fourth order are developed. The system of equations is closed by considering the fifth and sixth cumulants to be zero. For stationary response, differential equations become algebraic equations. The moments are obtained by solving the system of nonlinear algebraic equations. It is observed that the Gaussian closure method is inadequate for defining the complete probabilistic characteristics of the response.

Fatigue analysis of offshore platforms with uncertainty in foundation conditions

Foundation flexibility is an important consideration, in designing offshore structures against fatigue, since it has a significant effect on the system's dynamic response. However, a considerable amount of uncertainty is expected in the estimation of foundation flexibility due to the natural nonhomogeneity of the soil as well as laboratory and in-situ testing errors. The proposed method explicitly uses of the statistical nature of uncertainty in the soil stiffness to estimate the variability in the instanteneous and fatigue response using the First-Order Second Moment technique, with emphasis on uncertainties in the dynamic shear modulus along with random wave loading. Uncertainties in the response are estimated considering the randomness in the eigen values, eigen vectors, and the mechanical transfer function. Fatigue life is estimated at different levels of uncertainty for the dynamic shear modulus. It is observed that the uncertainty in the dynamic shear modulus of the soil has more significant effect on the fatigue life of the joints close to the base. The variability in the fatigue response increases with a reduction in the mean dynamic shear modulus. The response is more sensitive to changes in the dynamic shear modulus at its lower values.