Modified Probability Density Evolution Method for the Solution of Multi-Degree-of-Freedom Nonlinear Stochastic Dynamical Systems

A probability density evolution based exponential polynomial regression (PDEM-EPR) method to calculate the probability density function (PDF) of the high dimensional nonlinear stochastic dynamical systems is presented in this paper. Several typical examples, such as linear oscillator and Duffing oscillator are solved by PDEM-EPR method, and the results fit well with the analytical solutions. An engineering practice problem of ship nonlinear random roll in the beam waves is involved in this paper. The results obtained by PDEM-EPR is compared with those obtained by path integral method. The later results were given by Chai Wei [3]. It shows that PDEM-EPR method has the advantages as following: 1) The multi-degree-of-freedom (MDOF) nonlinear stochastic dynamical problems can be solved by PDEM-EPR method; 2) High efficiency can be obtained by PDEM-EPR method.

A Method for Fatigue Evaluation of Trimaran Cross Structure With the Influence of Slamming

Slamming is a highly non-linear phenomenon between hull structure and wave. Due to the special structure of trimaran, the slamming mode is extremely different from that of traditional vessel. Besides bow emergence and enter, the slamming phenomenon of the out shell at the cross structure is also obvious. In conventional hull structure fatigue strength evaluation, the slamming load is usually not considered. However, the slamming problem is unavoidable at danger load cases, and the stress concentration of the trimaran cross structure is serious. So it is dangerous to ignore the existence of slamming in serious load cases when evaluating the structural fatigue strength. Therefore, it is necessary to study the contribution of slamming load to fatigue damage. In this paper, a practical method for calculating and analyzing is presented to consider the effect of slamming on the fatigue strength of the trimaran cross structure to ensure that the fatigue life of the structure is closer to the true value. According to the linear theory, the relative motion and relative speed of the hull in wave and the stress response of the wave load on the structure are calculated firstly. Then, the stress response of the non-linear out shell slamming force is calculated. The linear response and non-linear response are combined. And the stress response time history under the combined action of slamming and wave load are obtained. Finally, the fatigue damage of the structure under dangerous operating conditions is calculated by the rain flow counting method. And the contribution value of the slamming load to the structural damage degree is calculated. The paper will put forward some reference suggestions for fatigue study calculation and evaluation of Trimaran cross structure with the influence of slamming.

Concrete Modeling for Extreme Wave Slam Events

Experience from model tests has initiated a growing attention towards extreme wave slam as a critical load situation for offshore large volume structures. Most of the problem is related to the local slam pressure, which may go up to several MPa’s for 100-year and 10 000-year waves. The paper deals with modeling techniques for marine concrete structures under extreme slam loading from waves where dynamic effects together with material softening play a major role for the response. Different analysis approaches for ultimate limit state (ULS) and accidental limit state (ALS) controls are discussed in view of reliability philosophy as basis for conventional design approach. The present paper is devoted to the local impact scenario and the alternative approaches for response and capacity control involving non-linear time domain analyses. Conventional design schemes as based on linear elastic models for response calculation together with code specified capacity control often come out more conservative than non-linear approach. The paper demonstrates by case studies how softening of the structure in general reduces the response in terms of section forces. A key issue when going from conventional linear approaches into non-linear techniques is to still keep an acceptable reliability level on the capacity control. Load and material factors are normally based on structures with limited non-linearity where linear response modeling is representative. Implementing non-linear material model in time domain analysis has a major challenge in limiting the sensitivity in response and capacity calculation. The paper demonstrates the way material model of concrete affects the section forces to go into local capacity control, and concludes on needed sensitivity analyses. Practical approaches on the concrete slam problem together with resulting utilizations from the control are demonstrated. The full non-linear technique by response and capacity control in one analysis is also handled, using average material parameters and justifying safety factors for the effect of implementing characteristic lower strength of concrete in the capacity. The paper ends up in a recommendation on non-linear time domain analysis procedure for typically slam problems. A discussion is also given on applicable design codes with attention to non-linear analysis.

Wave-in-Deck Impact Loads in Relation With Wave Kinematics

Breaking waves have been studied for many decades and are still of interest as these waves contribute significantly to the dynamics and loading of offshore structures. In current MARIN research this awareness has led to the setup of an experiment to determine the kinematics of breaking waves using Particle Image Velocimetry (PIV). The purpose of the measurement campaign is to determine the evolution of the kinematics of breaking focussed waves. In addition to the PIV measurements in waves, small scale wave-in-deck impact load measurements on a fixed deck box were carried out in the same wave conditions. To investigate the link between wave kinematics and wave-in-deck impact loads, simplified loading models for estimating horizontal deck impact loads were applied and compared to the measured impact loads. In this paper, the comparison of the model test data to estimated loads is presented.

Numerical Modeling of Dynamic Response of Water Tank in Collision

A simplified model is developed to analyse the interaction between the liquid motion and structural dynamic response of the side plate of a water tank. A mathematical model is established to simulate a knife-edged indenter impacting the side plate of a partially-filled water tank. The minimum potential energy principle is used to simulate the structural deformation and the kinematic equation is established to describe the two-dimensional ideal-fluid motion in the water tank. Considering the structural displacement as a connection between the water motion and dynamic response of the side plate, there is a displacement-pressure exchange between the water and the side plate for every time step. With increase of time, the water will finally become still and the plate will arrive at its final deformation. The numerical results based on the present simplified model are compared with the results from numerical simulations of an empty tank under the same impact condition, so as to investigate the effect of the water motion on the structural dynamic response of the side plate.

Study on Ship Manoeuvering in Adverse Sea State

IMO introduced Energy Efficiency Design Index (EEDI) to regulate the greenhouse gas (GHG) emissions from ships. The cheapest and easiest way to fulfil the EEDI requirement is to reduce installed power for most ships. Therefore, it has raised serious concerns that some ship designers might choose to lower the installed power to achieve EEDI requirements and not consider ship safety in a satisfactory way. This could induce ship manoeuvrability and safety problems in adverse seas, which needs urgent investigations on minimum power to maintain ship manoeuvrability in adverse sea. A time domain code ‘Waqum’ has been developed based on the force superposition of unified theory to study the minimum required power for maintaining ship manoeuvring ability in adverse sea states. The code combines sea-keeping and maneuvering equations, together with an engine model to predict ship responses in waves. The code can help us to study ship responses in transit situation and give us better insight into ship maneuvering ability in adverse sea states. In order to improve the simulation speed, the time domain code does not calculate all the hydrodynamic forces directly. Thus, some precalculations should be done for some force components before launching the simulation for a new ship. Therefore, the methodology and accuracy of each force component will influence the accuracy of the manoeuvring code. The methodology for determining each force component will be discussed, especially the identification of maneuvering derivatives based on CFD simulations. The code has been improved recently, and another rudder model has been implemented. Further, the the code with new rudder model is verified in calm water. The code’s ability to capture ship maneuvering in waves is also demonstrated.

Fatigue Damage Estimation of Welded Joints Considering Mechanochemical Interaction

The high stress region around weld joints accelerates corrosion and may induce non-uniform corrosion. In this study, the effect of loading on corrosion behavior of the steel in NaCl solution was investigated. The relationship between the corrosion rate and applied loading was deduced based on the electrochemical theory. Electrochemical experiments were carried out to investigate the interaction between loading and corrosion rate on Q235 steel. A butt weld joint of ship deck structure was selected as a case study. Time-dependent stress concentration factor of welded joint as a function of the corrosion deterioration was analyzed, and the iterative process of stress and corrosion degeneration of plate thickness was used to simulate the coupling effect based on results of the experiment. The hot spot stress approach was adopted to calculate the fatigue damage.

Static Stability of Floating Units in Operational Conditions: A Physics-Driven Approach

When designing a new floating unit concept, static stability computations are performed in order to check stability criteria defined in regulations. Calculations for design conditions generally include the estimation of buoyancy force, gravity force and wind force acting on the floater for a given condition and a desired axis of rotation. However, when studying the stability of a floating platform in operational conditions, all external forces acting on the unit should be comprised in the assessment in order to get a more realistic — and even physically admissible — picture of the platform stability. Those forces include among others wind, current and anchor line system forces. In addition, limiting the study to one axis of rotation may not provide a complete picture of the floater stability, especially when the hull is of a semi-submersible type. Following this physical approach, a numerical tool has been developed based on the SINTEF Ocean’s SIMA software package. The latter package initially includes a time domain simulator of complex multibody systems for marine operations. The developed tool provides accurate physical models for each force component that may have effects on the stability. It opens the possibility to study the operational stability of a floater without restraining the study to one axis of rotation. It also allows the analysis of damaged conditions with large inclination angles. This paper describes the model implemented in this numerical tool. Validation work is presented for simple geometries. Results from an operational stability study of a semi-submersible are discussed. Finally, possible further work is discussed.

Wave Load Prediction on a Stationary Bergy Bit Near a Fixed Offshore Platform

Offshore oil and gas operations conducted in harsh environments such offshore Newfoundland may pose additional risks due to collision of smaller ice pieces and bergy bits with the offshore structures, including their topsides in the case of gravity based structures particularly in extreme waves. In this paper, CFD (Computational Fluid Dynamics) prediction for wave loads acting on a bergy bit around a fixed offshore platform is presented. Often the vertical column of a gravity based structure is designed against ice collisions, if operating in such an environment. In practices, topsides are usually protected by being placed sufficiently high from the still water level, away from the reach of the bergy bits. This vertical clearance between the still water level and the topside deck is known an air gap. Hence, the amount of the air gap planned for such an offshore structure is an important factor for the safety of the topsides at a given location. In this study a CFD method is applied to estimate the dynamic response of the bergy bit and provide a reliable air gap to reduce the potential risk of the bergy bit collision. In advance of more complex collision simulations using a free-floating ice for the airgap design, CFD analysis of wave load prediction on a stationary bergy bit is carried out and reported in this paper. In the experiments and CFD simulations, the location of the bergy bit is changed to quantify the change of wave load due to the hydrodynamic interaction between the bergy bit and the platform. Finally, the results of the CFD simulations are compared with the relevant experiment results to confirm the simulation performance prior to the free floating bergy bit simulations.