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（China Ship Scientific Research Center,Wuxi 214082,China）

Abstract:The rapid expansion of world population,the depletion of inland resources and the requirement of sustainable development of world economy have strengthened the efforts of mankind to increase the capabilities of resource exploitation and space utilization in the ocean.Very Large Floating Structures(VLFS)have attracted much attention in ocean utilization among those marine structures for several decades.Despite the extensive research on VLFS,there still remain many key technologies to be solved,one of which is the ultimate bearing capacity of VLFS.This paper mainly reviews the research development of ultimate strength of VLFS,starting by the presentation of VLFS configuration and illustration of the structural load characteristics in accordance with the configuration,followed by summarization of the calculation methods of ultimate strength and the trial research on ultimate strength of VLFS,and ending with the discussion of the research development of ultimate strength.

Key words：VLFS;ultimate strength;model test;wave load;numerical simulation;progressive collapse method

0 Introduction

The concept of VLFS was initially introduced in 1924 when Edward Armstrong proposed a seadrome as the stepping stones for aircrafts flying across the oceans[1].Because very large floating bodies can better solve the environmental problems of land reclamation,they have attracted the attention of urban planners,engineers and architects[2].During the World War II,by using this concept,the U.S.Marine Corps constructed a large floating barge structure as a flight deck with the dimensions of 552 m×83 m×1.5 m(with a 0.5 m draft)[3].In the 1970s,a large floating airport was proposed in Japan for the new Kansai International Airport.In 1995,a large floating body in Tokyo bay was built to study the hydroelastic characteristics,anchoring system,anti-corrosion system and other functions of Mega-Float.Although the floating airport design was eventually rejected,the VLFS technology was developed rapidly.In order to adapt to the new situation of its global strategy,the concept of Mobile Offshore Base(MOB)was proposed in the 1990s[4]in the United States.MOB is a multifunctional floating logistics base that can be located offshore or in open seas.It is built from self-propelled modules assembled at sea to provide runways for fixed-wing fighters and helicopters.Since 2013,research on VLFS has also systematically carried out in China,from the aspects of the hydroelasticity[5-6],mooring[7],strength[8]and connector[9-11].

The wave load has been the most important and complex external load for the VLFS in the complicated and changeable marine environment for a long time.Under the long-term action of wind,wave and current environment and storm weather conditions,there may be many structural failure modes,which are far beyond the scope of ships and general marine engineering structures.Shuku et al(2001)[12],Watanabe et al(2004)[13]and Suzuki(2005)[14]summarized the types,applications and designs of the VLFS,such as floating airport,floating island,floating oil storage base,clean energy plant and so on.However,two classical structures in VLFS have mainly been studied in the world.VLFSs are classified into two broad categories:the pontoon-type,limited to sheltered waters,and the semi-submersible type,for open-ocean[15].Moreover,the semi-submersible VLFS has many advantages[16].

The classical VLFS is composed of connectors connecting multiple modules with a huge scale.In extreme sea conditions,the connector parts and the whole system will bear a large load in all directions.Although ships also bear complex loads,they mainly bear the vertical load,and the other loads in each direction are 1～2 orders of magnitude less than the vertical load.The calculation of ultimate strength under uniaxial loading is no longer applicable to that of the VLFS.And the ultimate strength of the structure is an important index of the adaptability of the structure to the environment.In order to obtain safe structural design and improve the working ability and survival ability of the VLFS in the limit state,it is necessary to accurately evaluate the ultimate strength of the VLFS.

In this paper,firstly,the configuration of VLFS is given.Secondly,according to that,the load characteristics of the structure are given,because the failure modes of the structure are different if the configurations and load characteristics are different.Thirdly,the calculation methods of ultimate strength are summarized as well as the trial research on ultimate strength of VLFS.Finally,the research development of ultimate strength is discussed.

1 Configuration of semi-submersible VLFS

The basic modules of VLFS are semi-submersibles of various specifications with a length of 150～300 m,a width of 60～100 m and a displacement of 50 000～200 000 t,as shown in Fig.1.When a basic module is moored at the edge of an island,a reef or a beach,it can not only be a dock,but also have the functions of traditional or green power generation,fuel and material storage,seawater desalination,rainwater collection,transit for tourism,marine pasture and logistics base,supporting marine development,rights protection,life and production activities.VLFS is often composed of several basic modules assembled according to functional requirements,which have the ability to migrate,assemble,reconstruct and adapt to harsh marine environments.

Fig.1 Basic module

With a total length of 300 m to thousands of meters,the combination of two or three modules can be used as a large wharf and floating logistics base.Six to eight modules can be connected in series to form a floating airport with a length of 1 800～2 400 m,as shown in Fig.2.

Fig.2 Very large floating structure

There are two typical configurations of a semi-submersible VLFS,namely the transverse buoy configuration and the longitudinal buoy configuration,as shown in Fig.3 and Fig.4.The advantage of the transverse buoy configuration is that the motion performance is better in the case of horizontal and horizontal waves,while the advantage of the longitudinal buoy configuration is that the total longitudinal strength is better.Each type of configuration is composed of an upper box,columns,lower buoys,transverse braces and other components to form a semi-submersible configuration,so as to effectively reduce the impact of the reef wave environment on the platform motion and structural response.

Fig.3 Transverse buoy configuration[23]

Fig.4 Longitudinal buoy configuration

According to the comparative analysis of the two configurations of the VLFS loading conditions,integrity and damage stability,movement and other overall performance,it is found that the rolling natural period of the horizontal type buoy floating structures is less than that of the longitudinal buoy type,while the pitch natural period is greater.But the natural period of heave is close to each other,and all the natural periods are far more than the usual period of wave,which is very beneficial to the floating body movement.

2 Wave and connector loading characteristics of VLFS

The countries with the most extensive and in-depth research on large and medium floating structures are Japan and the United States,where four international conferences on very large floating structures were held(VLFS 1991,1996,1999,2003).Due to much larger dimensions,relatively smaller global rigidities and lower natural frequencies than those of an ordinary ship,VLFS has apparent flexible body responses rather than rigid body motions in waves.The wave loads of VLFS are more complex,so its hydroelastic response is larger.Hence hydroelastic analyses are of great importance in design and safety assessment of VLFS.Extensive researches have been carried out during the last 20 years on the development of prediction methods of hydroelastic responses of VLFS.

Ohmatsu(1997)[17]proposed a new numerical calculation method based on the 3-dimensional eigenfunction expansion method for the diffraction and radiation problems and the modal expansion method for the elastic response representation.The proposed method solves the problem that the traditional method cannot analyze the hydroelastic response of the VLFS.The feasibility of the method is verified by comparison with the experimental data.

Mansour(2000)[18]considered that the loads acting on large floating structures could be divided into loads of high frequency and low frequency.The high-frequency loads are associated with the hydroelastic behavior of the structure and excitation of the natural frequency modes while the lowfrequency loads are associated with the body motion of the structure and the wave profile.In design analysis,extreme values of these loads and the correlation between them must be taken into consideration.

Watanabe et al(2004)[19]summarized the hydroelastic analysis methods of the pontoon-type VLFS and pointed out the direction of the hydroelastic analysis methods in the future,including:(a)the effect of nonlinear waves;(b)non-uniform seabed topography;(c)developing simplified methods of analysis and models for design;(d)the use of smart anti-motion control devices,and so on.Many scholars had done a lot of research on them.Seto et al(2005)[20]proposed a new method for Mega-Float structural design and hydroelastic analysis,which had been successfully applied to two actual Mega-Float models,the Mega-Float Phase II model and the Tokyo Bay prototype model.However,that structural analysis for realistic 3D models was limited to smaller VLFS because of the computational requirements.Kim et al(2007)[21]investigated the characteristics of bending moments,shear forces and stresses at unit connections of tandem-arranged VLFS under wave loads.The influence of connector stiffness,wave frequency and wave direction angle on the response of VLFS was given.Based on the linear shallow-water theory,Sturova(2008)[22]studied the effect of bottom topography on the unsteady behaviour of an elastic plate floating on shallow water,and the results obtained showed that the form of the bottom irregularities can have a considerable effect on the oscillations of an elastic plate floating in shallow water.

Ding et al(2017)[23]used a direct coupled method to analyze the hydroelastic responses of a single module VLFS near islands and reefs.The results showed that the hydroelastic responses of VLFS in waves are one of the major concerns,both in deep open sea and in shallow waters near islands and reefs.Wu et al(2018)[24]and Ding et al(2019)[25]used the same method to analyze the hydroelastic responses of a three-module VLFS near islands and reefs.The load value of the connector along the longitudinal direction(x)and the vertical direction(z)is equivalent,while the transverse direction(y)load is slightly smaller.Yang et al(2019)[26]investigated hydroelastic responses of a 3-module VLFS in the waves influenced by complicated geographic environment,and obtained the same conclusion as Wu,as shown in Fig.5.

Fig.5 Loading of connector[26]

It can be seen from the analysis of the hydroelastic load of VLFS and the connector force by many scholars in recent years that the load of VLFS is relatively complex,the load value of the connection longitudinal direction is large,and the vertical load value is similar.This will lead to a variety of failure modes that may occur when VLFS is subjected to extreme loads.

3 Analysis method and development of ultimate strength

VLFS,connected together by connectors,can be several hundred to several thousand meters long and much wider than conventional ships,so they are different from ordinary ships.The transverse load of VLFS is much larger than that of an ordinary ship,and for a semi-submersible VLFS,there will be a large torque effect that cannot be ignored in the severe sea conditions.At present,the commonly-used methods for the ultimate strength analysis of floating structures can be divided into direct calculation method,progressive collapse analysis method,nonlinear finite element method,ideal structural element method and test method.Most of the research objects are the ultimate strength of panel under complex loads and box girder structures under uniaxial loads.There are few studies on the ultimate strength of VLFS under complex loads.

3.1 Direct calculation method

The direct calculation methods consist of the regression results obtained by theoretical calculation and the empirical formulas derived under certain assumptions.Based on a large number of scholars’research results,Okada et al(2004)[27]developed a simplified method for estimating strength and reliability of stiffened plates for pontoon-type VLFS,which was supposed as a part of studies on reliability based preliminary design systems.Firstly,limit state conditions were shortly presented for the buckling and ultimate collapse strength of stiffened plates under combined compression,shear and lateral pressure in the deck,bulkhead and bottom parts of VLFS,especially,by using a simplified estimation formula.The validity was shown by nonlinear finite element method.The reliability level was investigated through numerical examples for a 5 000 m-class VLFS under a trial design.Fujikubo(2005)[28]carried out buckling checking on the plate lattice model and sandwich plate model of the bottom structure and deck structure of a VLFS under the action of biaxial pressure and shear load according to DNV specification(1994)[29].This method also took the plate lattice structure as the research object,and no relevant experimental technology was used to support it.

Iijima and Fujikubo(2012)[30]developed a mathematical model that could describe the post-ultimate strength of a VLFS,taking into account the effects of hydroelasticity.The whole VLFS was modeled by two beams on an elastic foundation connected via a nonlinear rotational spring,assuming that VLFS was to collapse at midship under severe bending moment,as shown in Fig.6.Finally,a simple formula to describe the collapse extent of VLFS was derived.Based on the previous results[31],Iijima et al(2016)[32]developed a hydro-elastoplastic method for studying the failure characteristics of VLFS under large wave loads,which divided a tandem-arranged VLFS structure into two elastic beams with an elasto-plastic hinge embedded at the connection.The deformation behavior formulated by using finite element method and the hydrodynamic behavior modeled by Rankine source panel method are coupled.The effectiveness of the method is verified by experiments.

Fig.6 Two beams on elastic foundation with an elasto-plastic hinge at the connection

Iijima and Fujikubo(2019)[33]studied the parametric dependencies of the collapse extent of a VLFS under extreme vertical bending moments.The mathematical equation to describe the collapse behavior of VLFS was developed by taking account of the hydroelastic deformation.A simple formula to predict the extent of collapse of the VLFS was developed by solving the equation in an analytical manner.The results of simple formula are well correlated with those of numerical simulation developed by the present authors.And the results show that the characteristic length defined for VLFS plays a key role in the collapse behavior.

Through the research work of a large number of scholars,it can be found that the direct calculation method requires a lot of numerical simulation analysis,and the curve fitting method can be used to obtain the curve of ultimate bearing capacity and the simplified formula.This method can only be used for specific results or the corresponding simple structure,but not applicable to the structure of different forms.Therefore,the direct calculation method generally simplifies the structure and carries out the preliminary calculation of the ultimate strength of the structure,which can be applied to the initial stage of structural design.

3.2 Progressive collapse method

Based on the plane section assumption,Smith(1977)[34]first proposed the incremental curvature method for the progressive failure of structures,and considered the post-buckling effect.Under the action of pure bending moment load,the method can obtain more accurate results,but it is difficult to solve the stress-strain relationship of structures.Therefore,a large number of scholars have improved this method.

Yao(1997)[35]derived the stress-strain relationship between beam elements and plate elements from theoretical analysis.Rigo et al(2001)[36]discussed the influence of the shape of the average stress-strain.Fujikubo et al(2005)[37]focused on the ultimate bearing characteristics of the continuous plate and the continuous stiffened plate under transverse load and uniformly distributed load.Paik et al(2008)[38]calculated the stiffness matrix of the element stress-strain relationship in the pre-and post-ultimate strength stages,and found that the accuracy of the progressive collapse behavior was depended on the constraint condition of the free boundary.Gannon et al(2012)[39]obtained the welding residual stress of the stiffened plate through the finite element method,giving the stress-strain curve of the stiffened plate under the action of axial pressure and the welding residual stress.

Based on the progressive failure method and the torsion theory of thin-walled beam,Zhao(2018)[40]carried out the research on the theoretical method for ultimate strength of the VLFS under complex loads.A method for evaluating the ultimate strength of the structure under the combined pressure and bending moment or torsion moment was proposed.For the ultimate compressive strength of the structure under bending load,the method equates the bending load to the initial stress.For the ultimate compressive strength of structures under torsional load,the effect of torsional load is equivalent to the reduction of yield stress.However,when the torsional load is large,the method is no longer applicable because the torsional load plays a major role in the failure mode.

It can be seen from the studies of the above scholars that at present what most scholars have calculated and researched are focused on the ultimate bearing capacity of stiffened plates,box girders and other structures under uniaxial load,the determination method of the stress-strain relationship of elements,the creation of special elements and the influence of various parameters on ultimate strength.However,the calculation of the ultimate bearing capacity of structures under complex loads has not been developed quickly.

3.3 Nonlinear finite element method

With the progress of science and technology and the rapid development of computer hardware and software,the nonlinear finite element numerical simulation technology has become increasingly mature,such as Abaqus,MSC,Marc,Ansys.For the nonlinear finite element method,as long as the relevant parameters such as load,boundary and material are selected reasonably,generally good results can be obtained.Therefore,relevant research results obtained by conventional ships are referrable for the research on ultimate strength of VLFS.At present,a large number of domestic and foreign scholars have been carrying out researches on the ultimate strength of nonlinear finite element numerical simulation method.

Eldeen et al(2012,2013)[41-42]proposed the failure deformation model prediction criterion and the stress-strain relationship capable of being directly applied to the finite element calculation,including the influence of residual stress and corrosion.Shi et al(2012)[43-44]studied the residual strength of structures with initial defects under different loads by using nonlinear finite element software.Xu et al(2013,2015,2017)[45-47]used nonlinear finite element method to study the influence of boundary conditions on the ultimate strength of stiffened plate structure.Yamada(2014)[48]and Jiang et al(2014)[49]studied the residual ultimate strength after collision through finite element analysis.Glassman et al(2016)[50]used nonlinear finite element method to study the failure mechanism of shear buckling under shear load,and found that the failure mode under axial compression load and shear load were similar.Estefen et al(2016)[51]studied the impact of semi-wave simulation and buckling simulation of initial deformation on ultimate strength by using nonlinear finite element method.Cui et al(2017)[52]studied the influence of corrosion on the ultimate strength of container ship structure based on nonlinear finite element method.

Zhao(2018)[40]used nonlinear finite element software(Abaqus)to study the ultimate strength of structures under complex loads.Taking the simple box girder and strut structure as the research object,the numerical simulation of the ultimate strength of the structure under the complex loads was carried out,the factors such as grid scale,loading rate,material model,initial defects and loading sequence were included and a reasonable and feasible numerical simulation method for ultimate strength of structures under complex loads was established.Finally,with the connector foundation support reinforcing area structure and strut structure of the VLFS taken as the research objects,the correctness of the numerical simulation method was verified by comparison with experiment results,as shown Figs.7-8.Gu et al(2018)[53]and Li et al(2019)[54]used the nonlinear finite element method(Abaqus)to simulate the ultimate strength of the braced structure and the reinforced area structure of the connector base of VLFS.

Fig.7 Collapse mode of connector foundation support reinforcing area structure of VLFS

Fig.8 Collapse mode of brace strut of VLFS

It can be known from the above researches that the initial deformation and welding residual stress of the structure have a great influence on the ultimate strength of the structure,which needs to be taken into account in the nonlinear finite element simulation.The influence of welding residual stress is particularly important for the calculation of ultimate strength of structures under complex loads.

At present,most researches on the calculation of ultimate strength of structures under complex loads are based on plate or stiffened-plate structures,while few researches on numerical simulation of ultimate strength of large complex structures under complex loads have been carried out.

3.4 Idealized structural unit method(ISUM)

Idealized structural unit method(ISUM)mainly uses large structural elements to reduce computation time,and the emphasis is on how to develop efficient and simple ISUM model elements considering yield and buckling effects.A large number of scholars have carried out researches on the ultimate strength of structures based on the ISUM and have made corresponding improvements.For example,Ueda and Rashed(1991)[55]and Abdel(1992)[56]used ISUM method to analyze the gradual collapse of hull structure.Kaeding and Fujikubo(2001)[57]took the structure of VLFS as the research object,based on the ISUM method framework and the typical failure mode of stiffenedplate under longitudinal pressure obtained by the nonlinear finite element analysis method,finally developed a new ISUM element to simulate the failure mode of VLFS,which was composed of large plate element and beam-column element.However,this method does not take into account the welding residual stress and the failure mode of the VLFS under the action of multiple loads.

Fujikubo et al(2003)[58]used ISUM method to evaluate the ultimate strength of pontoon-type VLFS surrounded by a gravity-type breakwater under vertical bending load.And the calculated failure probability for the floating structure is compared with the specified target safety level.The results show that the floating structure is most likely to fail by bending in transverse waves,and that the corresponding failure probability satisfies the target level.

Based on previous research results[59-60],Fujikubo(2005)[28]used ISUM to analyze the ultimate strength of pontoon-type VLFS under abnormal loads.The progressive collapse analysis of global structures was used.Regarding the assessment of structural safety for accidental loads,a simulation of airplane collision on VLFS was presented.

Since 2005,Paik and Kim et al[61-63]have repeatedly applied the ALPS system to the analysis of the ultimate carrying capacity of hull and components.At present,the research on the ISUM is still continuing.Although there are relatively mature ISUM calculation methods and programs,some element types are still under development,and the current algorithm is still unable to simulate for some small element components reasonably.Therefore,the research on the ISUM for the ultimate strength of VLFS has been less in the past 10 years.

3.5 Test method

Experiments are the most effective means to verify theoretical methods and numerical simulation methods.Through tests,the collapse process from local to whole under external load can be directly reflected.However,real scale tests are very expensive,and it is very difficult to carry out the strength test.Therefore,it is more important to carry out hull structure model tests.Compared with a real scale test,a model test can save money,reflect the real failure mode of a structure,and the material nonlinearity,structure nonlinearity and other nonlinear factors.

Offshore structures are formed by a large number of plates and stiffeners,so many scholars have calculated simple frame structural ultimate bearing capacity.But what is more typical is the small scale test of ultimate bearing capacity carried out by Dowling et al(1976)[64]and Recklin et al(1979)[65].These basic experiments laid the foundation for the research of a large number of scholars.However,model tests related to the ultimate strength of VLFS structures have been carried out only in recent years,so there are few relevant studies.

Iijima et al(2016)[32,66]studied the failure characteristics of VLFS by static and dynamic tests(as shown in Figs.9-11).The experimental results are compared with those obtained by the theoretical method to verify the correctness of the theoretical method(as shown in Fig.12).This research provides a certain technical basis for the ultimate strength model test of VLFS under wave loading.

Fig.12 Capacity curve obtained in test

Based on distorted similarity theory,Zhao et al(2018)[67]carried out the experiment study on the ultimate strength of the VLFS connector foundation support reinforcing area structure under complex loads,as shown in Fig.13.This structure model test technology was first established to obtain the ultimate compressive load of the connector foundation support reinforcing area structure under complex loads,which is of great importance to the study and promotion of the ultimate strength of structures under complex loads.

Fig.9 Sketch of collapsing mechanism

Fig.10 Sketch of scaled model

Fig.11 Photo of a collapsed specimen

Fig.13 Loading system of connector foundation

VLFS is subjected to multi-axial loads in complex marine environment,which makes brace strut with weak stiffness easy to damage.Thus,it will affect the safety and reliability of VLFS.Based on the model similarity theory,Zhao et al(2019)[8]carried out model design and test of compressive ultimate strength of the brace strut under torsion loads,as shown in Fig.14.The ultimate bearing capacity and the failure mode of the VLFS bract struts under combined compression-torsion loading are obtained by experimental study.The research results can support the design and safety reliability of VLFS under combined loads.

Fig.14 Loading system of brace strut structure

Although at present,there are many researches on the ultimate strength model test of ship structures,including the study on the ultimate strength of stiffened plate structure,the study on the ultimate strength of ship girder structure,the study on the ultimate bearing capacity of structure such as corrosion and initial deformation,etc.,few structural ultimate strength model tests have been carried out under complex loads,and most of the relevant studies are structural ultimate strength model tests under vertical loads alone.For VLFS,due to the complexity of the marine environment,the structures will be subjected to complex loads.

4 Discussion

With the development of ocean space and energy,the use of VLFS has become an inevitable trend.And its ultimate strength calculation method will become a hot spot.But at present,the research of VLFS mainly focuses on other key technologies such as external load and mooring system.As an important index of structural safety,ultimate strength must be studied extensively.

For VLFS near islands and reefs,due to the complex topography of islands and reefs,VLFS’s super-large geometric size and many other influencing factors,VLFS is subjected to extremely complex environmental loads,with an uneven spatial and temporal distribution.Therefore,under the influence of the complex marine environment load,the interaction between multiple modules of VLFS and the coupling effect of the composite mooring system,the failure mode and the calculation method of ultimate strength of VLFS are greatly different from the existing research results.

However,the simplified method is seldom used in the calculation of the overall ultimate strength of floating structures,and it also lacks more precise calculation methods and model tests.At the same time,due to super large dimensions and unconventional structure form of VLFS,the failure mode of the structure is more diversified.The analysis methods such as ultimate torsional strength of VLFS under combined loads have not been established or verified.Therefore,it is necessary to study the failure mode of VLFS under combined loading and the ultimate strength under corresponding failure mode.

5 Conclusions

In this paper,the configuration and load characteristics of semi-submersible VLFS have been summarized and analyzed.The various estimation methods of ultimate strength of VLFS in uniaxial load and complex load conditions have been reviewed and categorized.The research progress on ultimate strength of semi-submerged VLFS is pointed out.The relevant research methods in this paper will provide technical support for the safety evaluation of VLFS.

By analyzing the development of the ultimate strength of VLFS,the following conclusions are summarized.

(1)The wave loads of VLFS are more complex,so hydroelastic analyses are of great importance in design and safety assessment of VLFS.At the same time,the influence of terrain on the wave load of the structure cannot be ignored.

(2)The loads are transferred between modules of VLFS through connectors,resulting in relatively large load amplitude and motion response at the connectors.These factors need to be fully considered in the structural design of the connector,and the transition area between the connector and the main structure needs to be reinforced accordingly.

(3)VLFS are arranged in an ocean environment for a long time,the influence of extreme environment,such as typhoons and tsunami,should be considered in the design of a mooring system.

(4)The failure modes of VLFS under extreme loads are diversified and complicated,so the influence of different failure modes on the structural damage should be considered in the strength evaluation.

(5)At present,there is no uniform rules for structural design and safety evaluation of VLFS.With the development of research,the rules and strength evaluation methods need to be developed.