Probability models for offshore structural response due to Morison wave loading Part I: Drag-only response

Offshore structures are exposed to random wave loading in the ocean environment and hence the probability distribution of their response to wave loading is a minimum requirement for efficient probabilistic analysis of these structures. Due to nonlinearity of Morison wave loading and also due to intermittency of wave loading on members in the splash zone, the response is often non-Gaussian. Analysis of simulated data has, however, shown that neither of the two probability models proposed in the literature can accurately predict the tails of the response distribution. New probability models are therefore required to overcome this deficiency. This paper is composed of two parts: Part I is devoted to the development and validation of a new probability model for drag-only responses (i.e. responses due to the drag component of Morison wave loading), while Part II is devoted to the development and validation of new probability models for both inertia-only and total responses.     

Application of system identification techniques in efficient modelling of offshore structural response. Part I: Model development

Offshore structures are exposed to random wave loading in the ocean environment and hence the probability distribution of the extreme values of their response to wave loading is of great value in the design of these structures. Wave loading on slender members of bottom-supported jacket or jack-up structures is frequently calculated by Morison’s equation. Due to nonlinearity of the drag component of Morison wave loading and also due to intermittency of wave loading on members in the splash zone, the response is often non-Gaussian; therefore, simple techniques for derivation of their extreme response probability distribution are not available. Finite-memory nonlinear systems (FMNS) are extensively used in establishing a simple relationship between the output and input of complicated nonlinear systems. In this paper, it will be shown how the response of an offshore structure exposed to Morison wave loading can be approximated by the response of an equivalent finite-memory nonlinear system. The approximate models can then be used to determine the probability distribution of response extreme values with great efficiency. Part I of this paper is devoted to the development of an efficient FMNS model for offshore structural response while part II is devoted to the validation of the developed models.     

Application of system identification techniques in efficient modelling of offshore structural response. Part II: Model validation

Finite-memory nonlinear systems (FMNS) are extensively used in establishing a simple relationship between the output and input of complicated nonlinear systems. In Part I of this paper, it was shown how the response of an offshore structure exposed to (random) Morison wave loading can be approximated by the response of an equivalent finite-memory nonlinear system. The approximate FMNS models can then be used to determine, with great efficiency, the probability distribution of response extreme values. Part I of this paper was devoted to the development of FMNS models for offshore structural response. In this part, the validity of the developed models has been investigated by examining the response of three test structures under different environmental conditions. The results are promising.     

Comparisons of internal solitary wave and surface wave actions on marine structures and their responses

In this paper, a time-domain numerical model is established for computing the action of internal solitary wave on marine structures and structure motion responses. For a cylindrical structure, its side and bottom are discretized by pole and surface elements, respectively. The drag and inertial forces in the perpendicular direction of the structure are computed by the Morison equation from the pole elements, and the Froude–Krylov force in the axial direction of the structure due to internal wave motion is computed by integration of the dynamic pressure over the surface elements. The catenary theory is used to analyze the reaction force due to mooring lines, and the motion equation of the marine structure is solved by the fourth-order Runge–Kutta method in the time domain. The model is used to calculate the interaction of the internal solitary wave with a Spar platform with mooring system, and the surface wave action with the platform has also been computed by a frequency-domain boundary element method for comparison. Through the comparison based on a practical internal wave and surface wave states, it can be concluded that the internal wave force on the structure is only 9% of the one due to surface waves. However, the motion response due to the internal wave is much greater than the one due to the surface waves. It shows that the low-frequency effect of internal solitary waves is a great threat to the safety of marine structures.     

The estimation of wave height and wind speed persistence statistics from cumulative probability distributions

Results are summarized of an investigation concerned with the development and validation of a method for estimating persistence statistics from cumulative probability distributions. Primary attention has been devoted to estimations of wave height persistence and the motivation has been the requirement to provide estimates of persistence statistics as an additional output for the wave climate synthesis programme called NMIMET which can provide data on a worldwide basis. The opportunity has also been taken to adapt the methods developed for application to estimation of wind speed persistence.A method due to Graham is taken as the starting point and is modified in the light of detailed examination of a number of measured data sets. The method thus derived is shown to be more reliable and much simpler to apply than Graham's method and to give results in good agreement with a range of measured data sets for both exceedance and non exceedance. The measured data used for validation include 2 sets in areas remote from the sites in UK waters used for most of the development. A comparison is also included between persistence statistics estimated using NMIMET output of wave height probabilities from visual wave data and results from measured data.     

Feedback and Feedforward Optimal Control for Offshore Jacket Platforms

The optimal control is investigated for linear systems affected by external harmonic disturbance and applied to vibration control systems of offshore steel jacket platforms. The wave-induced force is the dominant load that offshore structures are subjected to, and it can be taken as harmonic excitation for the system. The iineafized Morison equation is employed to estimate the wave loading. The main result concerns the existence and design of a realizable optimal regulator, which is proposed to damp the forced oscillation in an optimal fashion. For demonstration of the effectiveness of the control scheme, the platform performance is investigated for different wave states. The simulations axe based on the tuned mass damper (TMD) and the active mass damper (AMD) control devices. It is demonstrated that the control scheme is useful in reducing the displacement response of jacket-type offshore platforms.     

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.     

Accumulated Rotation of A Modified Suction Caisson and Soil Deformation Induced by Cyclic Loading

Modified suction caissons (MSCs) acting as offshore wind turbine foundations will generate the accumulated rotation under cyclic loading resulted from waves. The accumulated rotation and the range of soil deformation around the MSC under long-term cyclic wave loading were studied using 3-D numerical simulations. The Morison equation was adopted to calculate the wave loadings. It was found that the MSC accumulated rotation increases linearly with the increase of the logarithm of cyclic number. The normalized expression was proposed to reflect the relationship between the accumulated rotation and cyclic number. The soil deformation range around the MSC increases when increasing the cyclic number and loading amplitude. It can also be concluded that the accumulated rotation increases rapidly with this change of excess pore pressure in the first 4000 cycles. The responses of the MSC to wave and wind loads were also investigated. Results show that the accumulated rotation of the MSC under both wave and wind loadings is larger than that under the wave loading only.     

Stochastic analysis for offshore structures with added mechanical dampers

Presented in this paper is a method to mitigate the vibration of an offshore structural system in the marine environment when subjected to in-line random wave forces. A stochastic approach by using an alternative complex analytical model for the viscoelastic material in the mechanical damper was used. The viscoelastic materials applied here were tested and verified that they have high energy absorption capacity. In the stochastic analysis, a random type of wave forces derived from the Morison equation for small bodies was applied. Results of the vibration responses for the system with added damping devices were presented and compared to the responses of structures with traditional design. It was observed that in terms of the power spectral density, the effect of the vibration mitigation and the dynamic performance of the offshore structural system were greatly improved when the new damping devices were applied to the offshore structural system.     

内孤立波作用下Spar平台动力响应特性

以三类内孤立波理论(Kd V、e Kd V和MCC)的适用性条件为依据,采用Morison和傅汝德-克雷洛夫公式分别计算Spar平台内孤立波水平力和垂向力,结合时域有限位移运动方程,建立了有限深两层流体中内孤立波与带分段式系泊索Spar平台相互作用的理论模型。以东沙群岛某海域实测内孤立波为对象,数值分析了在内孤立波作用下某经典式Spar平台的内孤立波动态载荷、运动响应及其系泊张力的变化特性。研究表明,内孤立波不仅会对Spar平台产生突发性冲击载荷,使其产生大幅度水平漂移运动,而且还会使其系泊张力显著增大。因此,在Spar平台等深海平台的设计应用中,内孤立波的影响不可忽视。     

Effect of slowly varying drift forces on the motion characteristics of truss spar platforms

Spar platform has been regarded as a competitive floating structure for deep and ultra deep water oil and gas production. In this paper, an efficient methodology has been developed to determine the slow motion responses of slender floating offshore structures due to wave forces. Based on this methodology, a MATLAB program named ‘TRSPAR’ was developed to predict the dynamic responses in time domain and it was used in this study to obtain the numerical results of a typical truss spar platform connected to sea bed using nine taut mooring lines. The difference frequency forces were calculated using the principles of the extension of Morison equation for an inclined cylinder and the wave kinematics were predicted using hyperbolic extrapolation. Mooring lines were modelled as nonlinear springs and their stiffness was obtained by conducting the static offset simulation. Because of the lack of detailed calculations in literature, most of the equations used were derived and presented in this paper. The effects of the different sources of the second order difference frequency forces were compared for inertia and drag forces in terms of response spectra. To validate the TRSPAR code, its results were compared to results of a typical truss spar model test.     

Vibration Control of Multi-Tuned Mass Dampers for An Offshore Oil Platform

The purpose of this study is to investigate the effectiveness of multi-tuned mass dampers (MTMD) on mitigating vi-bration of an offshore oil platform subjected to ocean wave loading. An optimal design method is used to determine the optimal damper parameters under ocean wave loading. The force on the structure is determined by use of the linearized Morison equation. Investigation on the deck motion with and vvithout MTMD on the structure is made under design condi-tions. The results show that MTMD with the optimized parameters suppress the response of each structural mode. The sensitivity of optimum values of MTMD to characteristic wave parameters is also analyzed. it is indicated that a single TMD on the deck of a platform can have the best performance, and the small the damping value of TMD, the betler the vibration control.     

基于实测波浪的单桩式海上风电基础波浪力计算研究

基于实测波面的波浪力获取作为结构动力响应分析以及数字孪生模型建立的必备环节,对海上风电数字化运维至关重要。为了满足更大的装机容量需求,单桩式海上风电基础趋于大型化,其尺度因子D/L也随之增大;并且实际海域均为非规则波,以尺度因子划分波浪力计算理论的方法对非规则波的适用性尚不明确。通过建立数值水槽,依据实际工况对不规则波与桩基的作用进行数值模拟,得到入射波浪场与桩基所受波浪力,在此基础上,基于入射波浪场分别采用Morison方程以及绕射理论求解波浪力并将之与数值模拟结果进行对比,分析了不同波浪力计算理论关于尺度因子的适用性,同时探究了波浪要素对计算精度的影响。结果表明：Morison方程在波高较大时精度下降;相对于Morison方程,绕射理论在该尺度下的精度更高。最后,通过分析实测数据进一步探讨了典型工况下的波浪力特征,以期通过实测波面计算波浪力的方法为实际服役风机波浪力计算提供技术支持。     

Response cumulant analysis of a linear oscillator driven by Morison force

A frequency-domain cumulant spectral analysis method is developed in this study to estimate the higher-order statistics of the linear oscillator responses driven by Morison wave force. The fourth-order cumulant function of the nonlinear drag force is formulated in terms of the autocorrelation functions of water particle velocity. Price's theorem is applied to evaluate the associated higher-order joint moments. Three-dimensional Fourier Transforms are employed to obtain the trispectra of Morison force and oscillator responses. The estimated force and response kurtosis are in good agreement with those obtained from time-domain simulations; while the proposed method is found to be much more efficient. The numerical results also show that the drag force cubicization based on the least square approximation results in an overestimation of the kurtosis values; in addition, it is necessary to include the joint moments of order higher than eighth.     

Probability distribution of random wave forces in weakly nonlinear random waves

Based on the second-order random wave theory, the joint statistical distribution of the horizontal velocity and acceleration is derived using the characteristic function expansion method. From the joint distribution and the Morison equation, the theoretical distributions of drag forces, inertia forces and total random wave forces are determined. The distribution of inertia forces is Gaussian as that derived using the linear wave model, whereas the distributions of drag forces and total random forces deviate slightly from those derived utilizing the linear wave model. It is found that the distribution of wave forces depends solely on the frequency spectrum of sea waves associated with the first order approximation and the second order wave–wave interaction.     

Expected fatigue damage and expected extreme response for Morison-type wave loading

Non-linear probability distributions for Morison-type wave loading are used to indicate the effect of drag forces on the expected fatigue damage and the expected extreme response of quasi-statically responding (members of) offshore structures. Results are compared with those from commonly used equivalent linear methods of analysis. It is found that the expected fatigue damage and the expected extreme response based on non-linear methods are approximately equal to results from linear methods when inertia is the dominant force. However, in the event of the drag forces forming a considerable part of the total wave loading, both fatigue damage and extreme response can significantly exceed those predicted by linear methods. The difference between the two methods is quantified in terms of a drag-inertia parameter, which is directly related to the sea state under consideration.     

In-line response of vertical cylinders in regular and random waves

The in-line response of a vertical flexibly mounted cylinder in regular and random waves is reported.Both theoretical analyses and experimental measurements have been performed.The theoretical predictions are based on the Morison equation which is solved by the incremental harmonic balance method.Experiments are then performed in a wave flume to determine the accuracy of the Morison equation in predicting the in-line response of the cylinder in regular and random waves.The interaction between waves and vibrating cylinders are investigated.     

Response behaviour of triangular tension leg platforms under impact loading

Excellent station keeping characteristics and relative insensitivity with increasing water depth make triangular tension leg platforms (TLPs) a proven concept in deep water oil exploration. TLPs are often subjected to less probable forces which arise due to collision of ships, icebergs or any other huge sea creature. Dynamic analysis of two triangular TLP models at water depths 1200 and 527.8 m is performed under regular waves along with impulse load acting at an angle of 45 degrees at the TLP column. Hydrodynamic forces on these TLPs are evaluated using modified Morison equation, based on water particle kinematics arrived at using Stokes’ fifth order wave theory. Based on numerical studies conducted, it is seen that impulse loading acting on corner column of TLP significantly affect its response while that acting on pontoons dose not affect TLPs behaviour.     

An experimental investigation of hydrodynamic coefficients for a vertical truncated rectangular cylinder due to regular and random waves

The research into hydrodynamic loading on ocean structures has concentrated mostly on circular cross-section members and relatively limited work has been carried out on wave loading on other cross-sections such as rectangular sections. These find applications in many offshore structures as columns and pontoons in semi-submersibles and tension-leg platforms. The present investigation demonstrates the behaviour of rectangular cylinders subject to wave loading and also supplies the hydrodynamic coefficients for the design of these sections.This paper presents the results of wave forces acting on a surface piercing truncated rectangular cylinder set vertically in a towing tank. The experiments are carried out in a water depth of 2.2 m with regular and random waves for low Keulegan–Carpenter number up to 6. The rectangular cylinder is of 2 m length, 0.2 m breadth and 0.4 m width with a submergence depth of 1.45 m from still water level. Based on Morison equation, the relationship between inertia and drag coefficients are evaluated and are presented as a function of KC number for various values of frequency parameter β, for two aspect ratios of cylinders, equals to 1/2 and 2/1. Drag and inertia coefficients obtained through regular wave tests are used for the random wave analysis to compute the in-line force spectrum.The results of the experiments show the drag and inertia coefficients are strongly affected by the variation in the aspect ratios of the cylinder. The drag coefficients decreases and inertia coefficients increases with increase in Keulegan–Carpenter number up to the range of KC number tested. The random wave results show a good correlation between measured and computed force spectrums. The transverse forces in both regular and random waves are found to be small compared to in-line forces.     

The Optimal Design of TMD for Offshore Structures

This paper presents the optimal design procedure of Tuned Mass Damper(TMD)for re-ducing vibration of an actual steel jacket offshore platform excited by random wave loading.In this study,a frequency domain is taken.The force on the structure is determined by use of the linearized Morisonequation for an input Power Spectral Density(PSD)of wave elevation.The sensitivity of optimum valuesof TMD to characteristic parameters of random wave spectrum is analyzed.An optimized TMD designfor the modeled platform is given based on design conditions and the findings of the study.