Efficient estimation of extreme long-term stresses by considering a combination of longitudinal bending stresses

Longitudinal stresses due to combined horizontal and vertical bending moments in ships, corresponding to a return period of 20 years, are estimated by linear response analysis. In principle, the stress should be obtained by combining the stress in all sea states that can occur over a long-term period. A method to determine the desired long-term extreme stress by considering only a few short-term sea states is presented. The sea states have a certain probability of occurrence, and are each identified by a contour line in the (H _s, T _p)-plane. This approach makes it possible to estimate the extreme loads on the vessel in a practical and accurate manner. Moreover, it is shown that the long-term stress can be estimated by combining the individual long-term extreme stresses due to vertical and horizontal bending moments by using the sum-of-squares approach and accounting for the correlation between stresses. It was found that the correlation coefficient can be taken as the largest of the ones calculated along the contour line. It is shown that this correlation coefficient can even be approximated by the normalized phase angle at the wave length where the dominant response has its peak value. A comparison with the results obtained using well-known combination rules is presented. While linear analysis has been used here, it is believed that the approach can be generalized to stresses with nonlinear behavior, and hence represent a significant improvement in calculation efficiency. Received: September 18, 2001 / Accepted: December 18, 2001     

Numerical and experimental investigations into the application of response conditioned waves for long-term nonlinear analyses

The coefficient of contribution method, in which the extreme response is determined by considering only the few most important sea states, is an efficient way to do nonlinear long-term load analyses. To furthermore efficiently find the nonlinear short-term probability distributions of the vessel responses in these sea states, response conditioned wave methods can be used. Several researchers have studied the accuracy of response conditioned wave methods for this purpose. However, further investigations are necessary before these can become established tools. In this paper we investigate the accuracy by comparing the short-term probability distributions obtained from random irregular waves with those from response conditioned waves. We furthermore show how response conditioned wave methods can be fitted into a long-term response analysis. The numerical and experimental investigations were performed using a container vessel with a length between perpendiculars of 281 m. Numerical simulations were done with a nonlinear hydroelastic time domain code. Experiments were carried out with a flexible model of the vessel in the towing tank at the Marine Technology Centre in Trondheim. The focus was on the probability distributions of the midship vertical hogging bending moments in the sea states contributing most to the hogging moments with a mean return period of 20 years and 10 000 years. We found that the response conditioned wave methods can very efficiently be used to accurately determine the nonlinear short-term probability distributions for rigid hulls, but either accuracy or efficiency is to a large effect lost for flexible hulls, when slamming induced whipping responses are accounted for.     

深水悬链复合锚泊线疲劳损伤计算

以某座Spar平台的锚泊系统为研究对象,首先利用三维绕射理论计算Spar平台主体波浪力,得到平台的总体运动响应时程。再建立复合锚泊线的二维非线性有限元动力分析模型,基于DelVecchio(1992)提出的经验公式,采用迭代的方法计算复合锚泊线的刚度。锚泊线和海床之间的接触作用基于刚性海床假定,基于Morrison公式计算锚泊线的惯性力和拖曳力荷载,根据计算得到的平台主体运动响应时程作为锚泊线顶端输入条件,在时域范围内进行复合锚泊线的动力分析。计算得到中国南海某海域各短期海况条件下复合锚泊线应力的时间历程曲线,采用雨流法对其计数得到对应于各短期海况条件下的疲劳载荷谱。最后根据Miner线性累积损伤模型,对复合锚泊线在长期海况条件下的疲劳损伤进行比较计算。     

Comparing different contour methods with response-based methods for extreme ship response analysis

Environmental contours are often applied in probabilistic structural reliability analysis to identify extreme environmental conditions that may give rise to extreme loads and responses. They facilitate approximate long term analysis of critical structural responses in situations where computationally heavy and time-consuming response calculations makes full long-term analysis infeasible. The environmental contour method identifies extreme environmental conditions that are expected to give rise to extreme structural response of marine structures. The extreme responses can then be estimated by performing response calculations for environmental conditions along the contours.Response-based analysis is an alternative, where extreme value analysis is performed on the actual response rather than on the environmental conditions. For complex structures, this is often not practical due to computationally heavy response calculations. However, by establishing statistical emulators of the response, using machine learning techniques, one may obtain long time-series of the structural response and use this to estimate extreme responses.In this paper, various contour methods will be compared to response-based estimation of extreme vertical bending moment for a tanker. A response emulator based on Gaussian processes regression with adaptive sampling has been established based on response calculations from a hydrodynamic model. Long time-series of sea-state parameters such as significant wave height and wave period are used to construct N-year environmental contours and the extreme N-year response is estimated from numerical calculations for identified sea states. At the same time, the response emulator is applied on the time series to provide long time-series of structural response, in this case vertical bending moment of a tanker. Extreme value analysis is then performed directly on the responses to estimate the N-year extreme response. The results from either method will then be compared, and it is possible to evaluate the accuracy of the environmental contour method in estimating the response. Moreover, different contour methods will be compared.     

Long-term correlation structure of wave loads using simulation

This paper proposes a new method for combining the lifetime wave-induced sectional forces and moments that are acting on the ship structure. The method is based on load simulation and can be used to determine the exceedance probabilities of any linear and nonlinear long-term load combination. It can also be used to determine the long-term correlation structure between these loads in the form of the long-term correlation coefficients. They are essential part of the load combination procedures in design and strength evaluations as well as in the fatigue and reliability analysis of ship structures.The simulation method treats the non-stationary wave elevations during the ship’s entire life (long-term) as a sequence of different stationary Gaussian stochastic processes. It uses the rejection sampling technique for the sea state generation, depending on the ship’s current position and the season. Ship’s operational profile is then determined conditional on the current sea state and the ship’s position along its route. The sampling technique significantly reduces the number of sea state-operational profile combinations required for achieving the convergence of the long-term statistical properties of the loads. This technique can even be used in combination with the existing long-term methods in order to reduce the number of required weightings of the short-term CDFs. The simulation method does, however, rely on the assumption that the ship is a linear system, but no assumptions are needed regarding the short-term CDF of the load peaks.The load time series are simulated from the load spectra in each sea state, taking into account the effects of loading condition, heading, speed, seasonality, voluntary as well as involuntary speed reduction in severe sea states and the short-crested nature of the ocean waves. During the simulation procedure, special care has been given to maintaining the correct phase relation between all the loads. Therefore, time series of various load combinations, including the nonlinear ones, can be obtained and their correlation structure examined. The simulation time can be significantly reduced (to the order of minutes rather than hours and days) by introducing the seasonal variations of the ocean waves into a single voyage simulation. The estimate of the long-term correlation coefficient, obtained by simulating only a single voyage with the correct representation of seasonality, approaches the true correlation coefficient in probability. This method can be applied to any ship and any route, or multiple routes as long as the percentage of the ship’s total lifetime spent in each of them is known.A study has been conducted to investigate the effects of ship type, route and the longitudinal position of the loads on the values of the correlation coefficients between six different sectional loads; vertical, horizontal and twisting moments, as well as shear, horizontal and axial forces. Three ocean-going ship types have been considered; bulk carrier, containership and tanker, all navigating on one of the three busy ship routes; North America-Europe, Asia-North America and Asia-Europe. Finally, the correlation coefficient estimates have been calculated for five different positions along the ship’s length to investigate the longitudinal variation of the correlation coefficient.     

非线性混合海况下近海风机塔筒底部载荷的动力响应

本文主要研究在非线性混合海况(即风浪和涌浪组合海况)下,以NREL 5MW_Baseline Monopile近海风机为研究对象,对其塔筒底部(基线)所受到的剪力和弯矩载荷的动力响应进行仿真。在近海风机的时域仿真中,选用了Ochi-Hubble六参数波浪谱,并编制了该谱的程序嵌入到FAST中进行编译。计算过程中,共进行了20次10 min的仿真分析。对于得到的短期载荷,给出了波高程,塔筒底部首尾向剪力和弯矩在线性与非线性不规则波作用下的时程曲线对比图。采用分块最大值法对每一次的短期载荷提取极值,并基于20次仿真所得的极值,给出了塔筒底部首尾向剪力与弯矩在线性与非线性不规则波作用下的超越概率曲线对比图。研究表明,在非线性混合海况下进行近海风机塔筒底部载荷的动力响应研究,计算结果对工程实际应用具有指导意义。     

Stochastic modeling of long term wave induced responses of ship structures

This paper presents a procedure for evaluating the long term response distribution in ship structures. A specified travel route through given geographical areas during the lifetime of the ship is defined. The individual scatter diagrams, describing the relative occurrence of different sea states for the geographical areas, are weighted with respect to the relative time occupancy of the ship. A two-dimensional analytical distribution is fitted to the weighted scatter diagram over the lifetime.

Frequency response functions for all the sectorial forces are calculated for different ship speeds, heading angles and loading conditions, and fitted to a bi-cubic spline. The wave spectrum is defined by applying a gamma spectrum. The response spectrum is found using a linear frequency domain analysis. From the response spectrum, the peak distribution of the response in each short term period, conditioned on the loading condition, the ship speed and the wave heading direction is computed. The peak distribution over the lifetime is estimated by unconditioning with respect to sea states, maneuvering and loading condition.     

Time series prediction based on LSTM neural network for top tension response of umbilical cables

The umbilical cable is an essential component of offshore oil and gas extraction systems. The severe marine environment poses a high challenge to the safety of the umbilical cable structure during operation. The analysis of an umbilical cable requires complex and resource-demanding finite element time-domain simulations to obtain their nonlinear dynamic response. Therefore, in order to solve the problem of structural safety monitoring and real-time assessment of remaining life of umbilical cables under extreme sea states, there is a great need to predict the dynamic response of umbilical cables quickly and accurately during operation, for ease of making fast decisions for system operation and maintenance before the arrival of extreme sea states. Given the strong nonlinear function-approximation ability of the neural network, this study proposes an efficient method for the prediction of the time series of umbilical cable top tension response based on LSTM (long short-term memory) neural network. We use LSTM neural network and ARIMA (autoregressive integrated moving average) model in a real engineering case for time series prediction of the top tension response of the umbilical cable, and the results of the two models are analyzed and compared, and the efficiency and accuracy of the LSTM neural network model are verified. Furthermore, the hyperparameter, dataset and generalization ability of LSTM model are discussed. The results indicate that feasibility of the tension response prediction of umbilical cables under dynamic load in complex marine environments.     

Response spectrum method for extreme wave loading with higher order components of drag force

Response spectra of fixed offshore structures impacted by extreme waves are investigated based on the higher order components of the nonlinear drag force. In this way, steel jacket platforms are simplified as a mass attached to a light cantilever cylinder and their corresponding deformation response spectra are estimated by utilizing a generalized single degree of freedom system. Based on the wave data recorded in the Persian Gulf region, extreme wave loading conditions corresponding to different return periods are exerted on the offshore structures. Accordingly, the effect of the higher order components of the drag force is considered and compared to the linearized state for different sea surface levels. When the fundamental period of the offshore structure is about one third of the main period of wave loading, the results indicate the linearized drag term is not capable of achieving a reliable deformation response spectrum.     

Sequential sampling method using Gaussian process regression for estimating extreme structural response

A methodology for estimating extreme response statistics for marine structures, that takes both the long-term variability of the metocean environment and the short-term variability of response into account is presented. The proposed methodology uses Gaussian process regression to estimate parameters of the short-term response distribution, based on output from computationally expensive hydrodynamic simulations. We present an adaptive design strategy for sequential updating of the model, focusing on the metocean conditions that contribute the most to the long-term extreme. With this approach, only a limited number of hydrodynamic simulations are needed.The suggested approach is demonstrated on the problem of estimating the 25-year extreme vertical bending moment on a ship. We show that a relatively small number of iterations (full hydrodynamic simulations) are needed to converge toward the “exact” results obtained by running a large number of simulations covering the entire range of sea states.The results suggest that the proposed method can be used as an alternative to contour-based methods or other methods that consider a few sea states using accurate numerical simulations, with little or no added complexity or computational effort.     

考虑高阶拖曳力的导管架平台极端载荷响应谱方法（英文）

Response spectra of fixed offshore structures impacted by extreme waves are investigated based on the higher order components of the nonlinear drag force. In this way, steel jacket platforms are simplified as a mass attached to a light cantilever cylinder and their corresponding deformation response spectra are estimated by utilizing a generalized single degree of freedom system. Based on the wave data recorded in the Persian Gulf region, extreme wave loading conditions corresponding to different return periods are exerted on the offshore structures. Accordingly, the effect of the higher order components of the drag force is considered and compared to the linearized state for different sea surface levels. When the fundamental period of the offshore structure is about one third of the main period of wave loading, the results indicate the linearized drag term is not capable of achieving a reliable deformation response spectrum.     

Efficient long-term fatigue analysis of deepwater risers in the time domain including wave directionality

Accurate fatigue assessment is a challenging and crucial aspect of riser design. The prediction of the long-term fatigue damage must account for numerous sea states of different wave heights, periods, and directions. Each sea state entails a dynamic analysis, often performed in the time domain owing to the significant nonlinearities. Because of the short-term uncertainties from irregular waves, the simulation duration must be sufficiently long for results to converge. To alleviate the hefty computational cost of long-term fatigue analysis, researchers have proposed efficient methods, but these are not without drawbacks; in particular, wave directionality is commonly neglected. This paper presents an efficient method for long-term fatigue analysis based on time domain simulation, considering wave directionality among other things. The proposed method is based on an enhanced version of control variates to reduce the variance in Monte Carlo simulations (MCS). The control function is constructed by training artificial neural network (ANN) models using existing MCS data. Here, a customized scheme is developed to allow for the situation that the training data and ANN prediction cases have different wave directions. The proposed method is unbiased and provides an error estimate. Simulations are performed on a floating system, and the proposed method is found to improve the efficiency of MCS significantly. Different scenarios such as fixed and random wave directions are compared, confirming that wave directionality is critical and should be included in a long-term fatigue assessment.     

大型集装箱船舷侧外飘砰击特性研究

基于概率随机理论研究大型集装箱船舷侧外飘砰击特性,预报不同海况和航速情况下船舶非线性运动响应时历,采用Chuang理论计算船艏舷侧区域的外飘砰击压力,对随机海浪下集装箱船外飘砰击概率、砰击压力极值沿船长和高度方向空间分布规律进行分析,在直接预报的基础上给出载荷的确定方法。     

Iterative methodology for response based analysis of an FPSO mooring system

Response based analysis (RBA) has been developed for prediction of extreme N-year return period responses and design metocean conditions of offshore structures. For applying the RBA, the behaviour of the offshore system subjected to a long history of metocean conditions needs to be predicted, and then, the probabilistic analysis is applied to estimate its long-term responses. Due to the large number of analysis cases required, the structural simulation is usually performed either by simplifying the structural model or by using computationally efficient tools, such as frequency-domain (FD) analysis. These approaches usually decrease the accuracy of predictions mainly when they are utilized for nonlinear systems. On the other hand, employing time-domain (TD) simulations leads to more accurate results but it is computationally expensive. Application of RBA for a weathervaning FPSO, which is the subject of the present study, makes TD analysis an essential requirement because of a highly nonlinear behaviour of the system. In the present study, an efficient methodology is proposed that aims at reducing the computational efforts of RBA by joint application of TD and FD simulations in combining the structural and statistical analyses through a single process, such that the number of time-consuming TD simulations is minimized. After initial screening using the results from FD simulations, the methodology identifies the response events (storms) that contribute the most to the N-year response and sets out an iterative process in which only those events that are most important are analysed by fully-coupled TD simulations. Within such events, a similar approach is also applied to intervals (sea states) where only the most contributing intervals are analysed in TD, and the remaining intervals are left for a less accurate FD analysis without sacrificing the overall accuracy. The proposed methodology provides a robust framework for distinguishing between “mild” and “severe” response events, without specifying any predefined limits for the metocean parameters or making a subjective judgement. Although it is developed for the mooring system of a weathervaning FPSO, it should also be applicable to any type of offshore structure and any structural response. This paper is the first part of the study and concentrates on the development of the efficient methodology to optimize the application of RBA to FPSO mooring systems, whilst its detailed application is subject of the second part of the study.     

Combining low- and high-frequency loads acting on large floating structures

Loads acting on large floating structures usually consist of high-frequency and low-frequency loads. The high-frequency loads are associated with the hydroelastic behavior of the structure and excitation of the natural frequency modes. The low-frequency loads are associated with the body motion of the structure and the wave profile. In design analysis, extreme values of these loads must be combined taking into consideration the correlation between them. This paper discusses a methodology for combining the extreme loads, and proposes a simple formulation suitable for use in reliability analysis. A proposed load combination factor K was found to depend on the correlation coefficient of the two loads, the ratio of their standard deviations and the frequency content of the processes from which the loads are determined. The correlation coefficient was found to depend on the complex frequency response functions of the loads and the input wave spectrum. The paper also discusses characteristic extreme values of slightly nonlinear loads acting on large floating structures.Extreme loads may be based on a storm condition with a specified return period. Since very large floating structures are expected to have a long operational lifetime, the return period must be selected carefully. The paper discusses a method for selecting return periods based on the expected operational life of the structure and encounter probability.     

Tail-Equivalent Linearization Method in frequency domain and application to marine structures

An extension of the Tail-Equivalent Linearization Method (TELM) to the frequency domain is presented. The extension defines the Tail-Equivalent Linear System in terms of its frequency-response function. This function is obtained by matching the design point of the nonlinear response with that of the linearized response, thus guaranteeing the equivalence of the tail probability of the latter and the first-order approximation of the tail probability of the nonlinear response. The proposed approach is particularly suitable when the input and response processes are stationary, as is usually the case in the analysis of marine structures. When linear waves are considered, the Tail-Equivalent Linear System possesses a number of important properties, such as the capability to account for multi-support excitations and invariance with respect to scaling of the excitation. The latter property significantly enhances the computational efficiency of TELM for analysis with variable sea states. Additionally, the frequency-response function of the Tail-Equivalent Linear System offers insights into the geometry of random vibrations discretized in the frequency domain and into the physical nature of the response process. The proposed approach is applied to the analysis of point-in-time and first-passage statistics of the random sway displacement of a simplified jack-up rig model.     

A regression and beam theory based approach for fatigue assessment of containership structures including bending and torsion contributions

Container shipping has been expanding dramatically during the last decade. Due to their special structural characteristics, such as the wide breadth and large hatch openings, horizontal bending and torsion play an important role to the fatigue safety of containerships. In this study the fatigue contributions from vertical bending, horizontal bending and torsion are investigated using full-scale measurements of strain records on two containerships. Further, these contributions are compared to results from direct calculations where a nonlinear 3D panel method is used to compute wave loads in time domain. It is concluded that both bending and torsion have significant impacts on the fatigue assessment of containerships. The stresses caused by these loads could be correctly computed by full-ship finite element analysis. However, this requires large computational effort, since for fatigue assessment purposes the FE analysis needs to be carried out for all encountered sea states and operational conditions with sufficient time steps for each condition. In this paper, a new procedure is proposed to run the structure finite element analysis under only one sea condition for only a few time steps. Then, these results are used to obtain a relationship between wave loads and structural stresses through a linear regression analysis. This relation can be further used to compute stresses for arbitrary sea states and operational conditions using the computed wave loads (bending and torsion moments) as input. Based on this proposed method for structure stress analysis, an efficient procedure is formulated and found to be in very good agreement with the full-ship finite element analysis. In addition it is several orders of magnitude more time efficient for fatigue assessment of containership structures.     

Ultimate load analysis of a 10 MW offshore monopile wind turbine incorporating fully nonlinear irregular wave kinematics

Loads from storm waves can in some cases be dimensioning for offshore wind turbine substructures. Accurate determination of nonlinear wave loads is therefore important for a safe, yet economic design. In this paper, the fully nonlinear waves, realized by a fully nonlinear potential wave solver OceanWave3D, are incorporated into coupled aero-servo-hydro-elastic simulations for a reduced set of wave-sensitive design load cases, in comparison with the widely used linear and constrained waves. The coupled aero-elastic simulations are performed for the DTU 10 MW reference wind turbine on a large monopile at 33 m water depth using the aero-elastic code HAWC2. Effect of the wave nonlinearity is investigated in terms of the ultimate sectional moments at tower bottom and monopile mudline. Higher ultimate moments, 5% at tower bottom and 13% at monopile mudline as maximum, are predicated when the nonlinear waves are used. It could be explained by the fact that the extreme nonlinear waves, that are close to the breaking limit, can induce resonant ringing-type responses, and hereby dominate the ultimate load responses. However, the constrained wave approach shows marginal difference compared to the standard linear wave approach. It can be concluded at least for the present configuration that the industry standard approaches (linear and constrained wave approach) underestimate the ultimate load responses on offshore wind turbines in severe sea states.