Journal of Marine Science and Engineering

Article Numerical Modelling and Dynamic Response Analysis of Curved Floating with a Small Rise-Span Ratio

Ling Wan 1,2,3, Dongqi Jiang 4,* and Jian Dai 5

1 The State Key Laboratory of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin 300072, 2 Faculty of Science Agriculture and Engineering, Newcastle University, Singapore 599493, Singapore; [email protected] 3 Newcastle Research & Innovation Institute Pte Ltd. (NewRIIS), Singapore 609607, Singapore 4 Department of Civil Engineering, School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 5 Department of Marine Technology, Norwegian University of Science and Technology, 7491 Trondheim, Norway; [email protected] * Correspondence: [email protected]

 Received: 9 May 2020; Accepted: 17 June 2020; Published: 24 June 2020 

Abstract: As a potential option for transportation applications in coastal areas, curved floating bridges with a same small specified rise to span ratio of 0.134, supported by multiple pontoons, are investigated in this paper. Two conceptual curved bridges are proposed following a circular arc shape with different span lengths (500 and 1000 m). Both bridges are end-connected to the shoreline without any underwater mooring system, while the end-connections can be either all six degrees of freedom (D.O.F) fixed or two rotational D.O.F released. Eigen value analysis is carried out to identify the modal parameters of the floating system. Static and dynamic analysis under extreme environmental conditions are performed to study the pontoon motions as well as structural responses of the bridge deck. Deflections and internal forces (axial forces, shear forces, and bending moment) are thoroughly studied with the variation of the span length and end support conditions in terms of the same specified small rise-span ratio. The ratio of axial force to horizontal bending moment are presented. From the study, it is found that the current parameters for the bridge are relatively reasonable regarding responses. However, the small rise-span does not provide enough arch effects. A higher rise-span ratio or stiffer bridge cross-sectional property is preferred, especially for the long bridge. In addition, the flexible end connections are preferred considering the structural responses at the end regions.

Keywords: floating bridge; integrated hydro-structural analysis; dynamic response; curved bridge; bridge structural design

1. Introduction Due to the large density of the population in coastal cities, sustaining and developing coastal regions is getting preferable to exploit more space in offshore areas. To accommodate the expanding industrial needs and transportation facilities, research has been carried out for coastal areas [1–6]. Bridges on rivers and offshore areas are important infrastructures for transportation connections. However, when water is getting deep and/or seabed condition is extremely soft, conventional supporting piers of bridges become too expensive or even unaffordable. On this occasion, a floating bridge is a more economical and attractive alternative, as the self-weight of bridge and vehicle loads are supported

J. Mar. Sci. Eng. 2020, 8, 467; doi:10.3390/jmse8060467 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 467 2 of 19 by the buoyancy forces from pontoons or floaters. In addition, a floating bridge allows easy removal by tugboats when relocation is needed. There has been a long history of application of floating bridges. The first floating bridge can be dated back to about 480 BC, when two rows of floating bridges were constructed, with each consisting of about 300 laid side by side, for military crossing. In recent years, there has been many modern floating bridges built, including Lacey V. Murrow Bridge (2020 m, 1940, USA), Evergreen Point Bridge (2350 m, 1963, USA), Homer Hadley Bridge (1772 m, 1989, USA) built on Lake , and Bridge (2398 m, 1961, USA) also in the Seattle area, USA., Bergsøysund Bridge (931 m, 1992, Norway), (1614 m, 1994, Norway) built in Norway on the fjords. Yumemai Bridge (876 m, 2000, Japan) built in Japan, etc. [7,8]. In general, floating bridges are more suitable to be deployed under the following conditions: Long span and deep-water conditions, such as Norwegian Fjords, and very soft seabed conditions, and conditions where the seabed marine environment needs to be protected. There are mainly two types of pontoon-supported floating bridges: The continuous pontoon type and discrete pontoon type. Various analysis methods have been developed for the analysis of continuous and discrete pontoon type floating bridges. The modal expansion method and direct calculation method [9,10] are two frequency domain analysis methods for continuous pontoon type floating bridges, while the time domain method can also be used. Hydroelasticity should also be considered [11,12] for bridges in a large scale and especially for cases when the eigen frequency of the continuous pontoon is located in the environmental excitation frequency regions. For the discrete type, single pontoon hydrodynamic properties should be considered, together with the finite element method (FEM) applied for the bridge deck and other structural components. Hydrodynamic interaction among the pontoons and shore should be taken into consideration [13] if the pontoons are close to each other or close to the shore. A time domain method to simulate the dynamic behavior of a three-span with two floating pylons was developed in [14]. A sensitivity-based FEM updating method was proposed and applied in the analysis of Bergsøysund bridge in [15]. A model test on Bergsøysund bridge was carried out in the ocean basin at MARINTEK (Now Sintef Ocean) in 1989 and the results were compared with the numerical model, and it was shown that the calculation results based on potential flow theory can have good agreements [16] with the model test. Generally, floating bridges are supported vertically by pontoons. In addition, there are also a variety of mooring systems designed to retain pontoon motions of the floating bridges, e.g., anchor chain, marine dolphin, and abutment, located beneath each pontoon and/or at the ends, depending on the sit conditions and design requirements. There are also curved bridge designs with no mooring system, which can reduce the cost. This kind of design makes use of the horizontal stiffness provided by the curvature shape and the end connections, replacing the restoring forces by traditional mooring. The Bergsøysund bridge [17] and Nord-Hordland bridge are of this kind and no mooring system underwater is applied. The Bergsøysund Bridge, built in Norway in 1990s, is the first free floating pontoon bridge with a cost-effective horizontally arch curved design. It is well known that an arch is a much stronger member than a straight beam when subjected to in-plane loadings as it transfers the loadings to axial rather than to bending as in the case of a straight beam. However, a drawback of using a laterally curved configuration is the increased reaction at the supports. In Bergsøysund Bridge, the transmission of internal forces was managed by the utilization of a flexible steel rod from the bridge structure to the bridge abutment. Many aspects need to be considered in the design of floating bridges. The structural eigen properties of floating bridges are affected by several parameters, including the bridge’s cross-section properties, end connections, pontoon types and number of pontoons, and the curvature of the curved bridges, etc. The dynamic responses of the bridge are excited by the environmental loads, including the wind, wave, current, and tidal forces, and various floating bridge eigen modes may be excited under the different types of loads. An important issue is the inhomogeneous environmental conditions [18–20], especially for floating bridges in Fjord of Norway, and the inhomogeneity induces different dynamic J. Mar. Sci. Eng. 2020, 8, 467 3 of 19 response properties compared with homogeneous conditions. In the case of a tsunami or extreme wave impact loads from the environment, fluid–structure interaction is especially critical, and springing or ringing effects may occur due to higher-order wave loads and large eigen frequencies of the bridge structure. Tsunami loads on a bridge and the connections were found to be significantly affected by structural flexibility [21]. A hybrid test was carried out regarding the tsunami loads on a prestressed and the connections, and damage was found in the specimen, indicating the importance of the design and investigation of the bridge’s structural capability [22]. Ship collision is also one important issue of consideration of the design. Ship collision on the pontoon wall and bridge deck girder have been investigated [23,24]. Various limit states, i.e., the fatigue limit state (FLS), ultimate limit state (ULS), serviceability limit state (SLS), and accidental limit state (ALS), should be investigated for the floating bridge design. These limit states define various design criteria to ensure proper service states under operational conditions and survivability under extreme or accidental conditions [25,26].

2. Research Significance The development of curved floating bridges refers to traditional onshore arch bridges that are defined as a vertically curved and axially compressed structural system. As a curved structure, the rise is a critical parameter that is as important as the span length. Moreover, the rise-to-span ratio is a key indicator of the structural properties of a curved bridge. A perfect arch structure theoretically induces only compressive forces along the centroid line. Although it is practically impossible due to complex load combinations, it is viable to explore a preferable rise-to-span ratio to achieve a desirable internal force distribution. As for floating bridges, the rise-to-span ratio cannot be designed too large due to site constraints. However, a shallow arch is considered to cause significant horizontal movement of the abutments and has the disadvantages of large deformation and low rigidity. It remains unknown whether such disadvantages also exist for horizontally curved floating bridges when subjected to wave actions. In this paper, a relatively small rise-to-span ratio is selected for the curved floating bridge, and the response and internal forces are investigated by varying the boundary conditions (B.Cs) and the span length. The deployment of the floating bridge is assumed to be in the coastal waters of Singapore. The water depth is quite shallow; consequently, the wave force properties are also different with deep water floating bridges. In addition, the bridge deck cross-section is developed to take high traffic flow requirements (three lanes in each way), and the bridge span is relatively short. Under such circumstances, the bridge’s horizontal rigidities are expected to be large. Therefore, a small rise-span ratio is studied in the initial stage to investigate whether the arch effect can be achieved. These new features of the floating bridges are relatively new compared with previous research and thus worthy of investigation.

3. Floating Bridge Description The deployment location of the floating bridge in this study is the coastal waters of Singapore, which are characterized as relatively shallow water with a water depth of 20 m and mild environmental conditions. Extreme sea conditions with a 100-year return period is taken as Hs = 2 m, Tp = 5.8 s. Long crested waves are studied. The water depth is 30 m. Since the focus is to find out different structural configuration effects and response properties. Other environmental load effects are not considered in this study. Single curved bridges with two bridge total spans of 500 and 1000 m are the focus, with the same curve radius of 1000 m. The two bridges have the same rise-to-span ratio of 0.134. Discrete pontoons are used to support the bridge, with each pontoon supporting approximately 100 m of the bridge deck. The short bridge (500 m) includes four pontoons, while the long bridge (1000 m) has nine pontoons in total. The bridge deck is designed to have two ways, with three lanes in each way. The bridge cross-section is designed to be a truss configuration. The steel truss work forms the key structural component of the bridge. Figure1 shows the two bridge configurations. The global coordinate system, J. Mar. Sci. Eng. 2020, 8, 467 4 of 19

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 4 of 19 J.local Mar. coordinateSci. Eng. 2020, system, 8, x FOR PEER and pontoonREVIEW positions are also shown. The local Y axis represents the 4weak of 19 axis of the bridge, while the local Z axis is the strong bridge axis. The two ends of the bridge are of the bridge are connected to the shore. Two types of end connections are investigated, i.e., rigid BC ofconnected the bridge to are the connected shore. Two to types the shore. of end Two connections types of end are connections investigated, are i.e., inve rigidstigated, BC with i.e., all rigid the BC six with all the six D.O.F fixed and flexible BC with rotational D.O.F regarding local Y and local Z free. withD.O.F all fixed the six and D.O.F flexible fixed BC and with flexible rotational BC with D.O.F rotational regarding D.O.F local regarding Y and local local Z free. Y and local Z free.

Figure 1. Floating bridge configurations. Figure 1. FloatingFloating bridge bridge configurations. configurations. The pontoon supporting the bridges has an elliptic cylinder shape and 6 D.O.F rigid body The pontoonpontoon supportingsupporting the the bridges bridges has has an elliptican ellip cylindertic cylinder shape shape and 6 D.O.Fand 6 rigidD.O.F body rigid motions body motions as shown in Figure 2. The pontoons have uniform dimensions with the major axis length of motionsas shown as in shown Figure in2 .Figure The pontoons 2. The pontoons have uniform have un dimensionsiform dimensions with the with major the axismajor length axis length of 60 m, of 60 m, minor axis length of 22 m, and height of 9 m. The elliptic shape is expected to reduce the drag 60minor m, minor axis length axis length of 22 m,of and22 m, height and height of 9 m. of The 9 m. elliptic The elliptic shape shape is expected is expected to reduce to reduce the drag the forces drag forces applied on the pontoon. These pontoons are evenly distributed along the bridge length, and forcesapplied applied on the on pontoon. the pontoon. These These pontoons pontoons are evenly are evenly distributed distributed along along the bridgethe bridge length, length, and and the the pontoon major axis is pointing to the global x direction. The center of gravity (C.O.G) of the thepontoon pontoon major major axis isaxis pointing is pointing to the to global the global x direction. x direction. The center The ofcenter gravity of (C.O.G)gravity (C.O.G) of the pontoon of the pontoonis 2 m fromis −2 them from still waterthe still line water (SWL), line and (SWL), the C.O.G and the of C.O.G the bridge of the deck bridge is 5 mdeck from is the5 m SWL.from Thethe pontoon− is −2 m from the still water line (SWL), and the C.O.G of the bridge deck is 5 m from the SWL. The bridge deck weight distribution is estimated to be 3.99 Mkg/100 m. The stability of the SWL.bridge The deck bridge weight deck distribution weight distribution is estimated is to esti bemated 3.99 Mkg to /be100 3.99 m. TheMkg/100 stability m. ofThe the stability pontoons of wasthe pontoons was studied previously [27]. The dimensions and weight parameters of the pontoons and pontoonsstudied previously was studied [27 ].previously The dimensions [27]. The and dimensions weight parameters and weight of the parameters pontoons of and the bridge pontoons deck and are bridge deck are listed in Table 1. bridgelisted in deck Table are1. listed in Table 1.

FigureFigure 2. 2. EllipticElliptic cylinder cylinder pontoon pontoon supporting supporting the the bridge bridge,, and and the the modes modes of of the the rigid rigid body motion. Figure 2. Elliptic cylinder pontoon supporting the bridge, and the modes of the rigid body motion. Table 1. Pontoon and bridge deck parameters. Table 1. Pontoon and bridge deck parameters. Table 1. Pontoon and bridge deck parameters. Bridge Total Span L 500/1000 (m) Bridge Total Span L 500/1000 (m) Bridge Total Span L 500/1000 (m) PontoonPontoon Number Number 4/ 4/99 Pontoon OverallPontoon Dimension Number (Elliptic Cylinder) 60 22 4/99 (m) Pontoon Overall Dimension (Elliptic Cylinder) 60× × 22× × 9 (m) Pontoon OverallPontoon Dimension Draft (Elliptic Cylinder) 60 × 6 (m)22 × 9 (m) PontoonPontoon C.O.G Draft (0, 0, 6 2)(m) (m) Pontoon Draft − 6 (m) GM (HorizontalPontoon and Longitudinal) C.O.G 3 and(0, 0, 35.5 −2) (m) (m) PontoonPontoon mass C.O.G 2.319(0, 0, (Mkg) −2) (m) GM (Horizontal and Longitudinal) 3 and 35.5 (m) 2 PontoonGM rotational(Horizontal Inertia and I xxLongitudinal),Iyy and Izz 145.39, 660.29 3 and and 35.5 721.03 (m) (Mkg m ) Pontoon mass 2.319 (Mkg) BridgePontoon Deck C.O.G mass (0, 2.319 0, 5) (Mkg) (m) PontoonBridge rotational Deck WeightInertia Ixx, Iyy and Izz 145.39, 660.29 3.990 (Mkg and/100 721.03 m) (Mkg m2) Pontoon rotational Inertia Ixx, Iyy and Izz 145.39, 660.29 and 721.03 (Mkg m2) Bridge Deck C.O.G (0, 0, 5) (m) Bridge Deck C.O.G (0, 0, 5) (m) Bridge Deck Weight 3.990 (Mkg/100 m) A truss configurationBridge is Deck used Weight in the bridge cross-section, resembling 3.990 (Mkg/100 the Bergsøysund m) Bridge in Norway [28] but with different dimensions and bridge curvatures. To evaluate the bridge deck flexural A truss configuration is used in the bridge cross-section, resembling the Bergsøysund Bridge in A truss configuration is used in the bridge cross-section, resembling the Bergsøysund Bridge in Norway [28] but with different dimensions and bridge curvatures. To evaluate the bridge deck Norway [28] but with different dimensions and bridge curvatures. To evaluate the bridge deck flexural rigidity EIy and EIz, and torsional rigidity GJ, which is about the local y, z and x axis, flexural rigidity EIy and EIz, and torsional rigidity GJ, which is about the local y, z and x axis, respectively, finite element models of truss work segments with various configurations and structural respectively, finite element models of truss work segments with various configurations and structural

J. Mar. Sci. Eng. 2020, 8, 467 5 of 19

rigidityJ.J. Mar. Mar. Sci. Sci. EIy Eng. Eng. and 2020 2020 EIz,, ,8 8, ,x andx FOR FOR torsional PEER PEER REVIEW REVIEW rigidity GJ, which is about the local y, z and x axis, respectively, finite 5 5 of of 19 19 element models of truss work segments with various configurations and structural member sizes were developedmembermember sizessizes using werewere commercial developeddeveloped software usingusing SAP2000 commercialcommercial [29], assoftwaresoftware shown SAP2000 inSAP2000 Figure 3 [29],.[29], Among asas shownshown these parameters, inin FigureFigure 3.3. EAmongAmong is the Young’s thesethese parameters,parameters, modulus, E AE isis thethe Young’sYoung’s bridge’s modulus,modulus, cross-sectional AA isis thethe area, bridge’sbridge’s I is the cross-sectionalcross-sectional moment of inertia area,area, I ofI isis the thethe cross-sectionalmomentmoment of of inertia inertia area, of of G the isthe the cross-sectional cross-sectional shear modulus, area, area, and G G Jisis is the the the shear shear torsional modulus, modulus, constant and and of J J theisis the the cross-section. torsional torsional constant constant Shear modulusofof thethe cross-section.cross-section. and Young’s ShearShear modulus modulusmodulus have andand the Young’srelationshipYoung’s modulusmodulus of G = havehaveE/2/ (1 thethe+ relationshipνrelationship), with Poisson ofof GG ratio == E/2/(1E/2/(1ν = +0.3.+ νν),), Gwithwith= 76.92 Poisson Poisson GPa, ratio ratio E = 200ν ν = = 0.3. GPa.0.3. G GIn = = 76.92 the76.92 end, GPa, GPa, steel E E = circular= 200 200 GPa. GPa. hollow In In the the sections end, end, steel steel with circular circular a diameter hollow hollow of 1200sectionssections mm with with for theaa diameter maindiameter chord ofof members 12001200 mmmm and forfor 800 thethe mm mainmain for diagonalchordchord meme membersmbersmbers andand are considered 800800 mmmm forfor in thisdiagonaldiagonal study, members withmembers EIy, EIz, areare andconsideredconsidered GJ values in in of this this 1.0 study, study,107 MNmwith with EIy, 2EIy,, 2.0 EIz, EIz,10 and 7andMNm GJ GJ values 2values, and 1.0of of 1.0 1.010 × ×7 10 10MNm77 MNm MNm2/rad,22, ,2.0 2.0 respectively. × × 10 1077 MNm MNm EA2,2 ,and and is taken 1.0 1.0 × × 7 2 × × 6 × as1010 17 MNm MNm106 MN.2/rad,/rad, Structural respectively. respectively. damping EA EA is is istaken taken also as consideredas 1 1 × × 10 106 MN. MN. and Structural Structural is calculated damping damping by Rayleigh is is also also damping considered considered with and and the is is × masscalculatedcalculated and sti byffbyness RayleighRayleigh proportional dampingdamping coeffi withwithcients, thethe then massmass the an dampingandd stiffnessstiffness ratio proportionalproportional is calculated ascoefficients,coefficients,ξ = 0.5(µ/ω thenthen+ λ ωthethe). Withdampingdamping the variation ratio ratio is is calculated ofcalculated the coeffi as ascients ξ ξ = = 0.5( 0.5(µ andμμ//ωω λ+ + fromλ λ ω ω).). 0.01With With to the the 0.1, variation variation the damping of of the the ratio coefficients coefficients is plotted μ μ inand and Figure λ λ from from4. It0.010.01 is seen toto 0.1, that0.1, the thethe dampingdamping ratioratioratio varies isis plottedplotted significantly. inin FigureFigure Focusing 4.4. ItIt isis on seenseen the criticalthatthat thethe wave dampingdamping periods, ratioratio which variesvaries is aroundsignificantly.significantly. 2 s–10 Focusing sFocusing (0.628–3.14 on on the radthe critical /criticals) in this wave wave study, periods, periods, the parameters which which is is around aroundµ = 0.02 2 2 s–10 ands–10 λs s (0.628–3.14 =(0.628–3.140.02 induce rad/s) rad/s) damping in in this this ratiosstudy,study, generally the the parameters parameters below 3%μ μ = = based 0.02 0.02 and and on theλ λ = = estimation.0.02 0.02 induce induce This damping damping may be ratios ratios regarded generally generally as a reasonablebelow below 3% 3% based assumptionbased on on the the consideringestimation.estimation. thatThisThis othermay may be viscousbe regarded regarded effects, as as a fora reasonable reasonable example, assumption fromassumption the wind considering considering and current, that that are other other not viscous viscous considered effects, effects, in theforfor numericalexample, example, from model.from the the wind wind and and current, current, are are not not considered considered in in the the numerical numerical model. model.

FigureFigureFigure 3. 3. 3.Numerical Numerical Numerical modelmodel model ofof of variousvarious various bridgebridge bridge girdergirder girder cross-sections. cross-sections. cross-sections.

FigureFigureFigure 4. 4. 4.Rayleigh Rayleigh Rayleigh dampingdamping damping ratiosratios ratios forfor for variousvarious various angularangular angular frequenciesfrequencies frequencies underunder under didifferent differentfferent mass mass mass proportional proportional proportional coecoefficientscoefficientsfficients µ μ μand and and sti stiffness stiffnessffness proportional proportional proportional coe coefficients coefficientsfficients λ .λ λ. .

4.4.4. NumericalNumerical Numerical ModelModel Model NumericalNumericalNumerical modelling modelling modelling of of of the the the floatingfloating floating bridgebridge bridge involvesinvolves involves bothboth both hydrodynamichydrodynamic hydrodynamic and and and structuralstructural structural dynamicdynamic dynamic models,models,models, and and and these these these two two two models models models are are are coupled coupled coupled in in in one one one integratedintegrated integrated model. model. model. HydrodynamicHydrodynamic Hydrodynamic propertiesproperties properties of of of the the the pontoonspontoonspontoons as asas well wellwell as asas the thethe structural structuralstructural properties propertiesproperties of ofofthe thethebridge bridgebridgedeck deckdeckand andandthe thetheend endendconnections connectionsconnections are arearetaken takentaken intointointo consideration. consideration. consideration. ConsideringConsidering Considering nonlinearnonlinear nonlinear features,features, features, such such such asas as viscousviscous viscous eeffects ffeffectsectsand and andstructural structural structuralgeometrical geometrical geometrical nonlinearity,nonlinearity,nonlinearity, time time time domain domain domain analysis analysis analysis is is is needed. needed. needed. FrequencyFrequency domain domain hydrodynamic hydrodynamic properties properties are are calculated calculated using using the the boundary boundary element element method method [30][30] basedbased onon potentialpotential flowflow theory,theory, followedfollowed byby ththee transferringtransferring ofof thesethese propertiesproperties intointo thethe timetime domain.domain. This This method method is is called called th thee hybrid hybrid time time and and frequency frequency domain domain method method [31,32], [31,32], and and is is widely widely usedused inin differentdifferent offshoreoffshore applicationapplication studiesstudies [4,33–40].[4,33–40]. InIn thethe casecase ofof stronglystrongly nonlinearnonlinear

J. Mar. Sci. Eng. 2020, 8, 467 6 of 19

Frequency domain hydrodynamic properties are calculated using the boundary element method [30] based on potential flow theory, followed by the transferring of these properties into the time domain. This method is called the hybrid time and frequency domain method [31,32], and is widely used in different offshore application studies [4,33–40]. In the case of strongly nonlinear hydrodynamic phenomenon, such as the slamming or green water problem [41], the linear hydrodynamic model is not suitable anymore. However, in this study, the linear hydrodynamic model is good enough, since there . is no water entry and exit phenomena of the pontoons expected to occur. In the time domain, if rPT, rPT, .. and rPT are the displacement, velocity, and acceleration vectors of a pontoon, then the hydrodynamic forces applied on the pontoon can be expressed as:

Z t ...... Fhyd = Fext(t) A( )rPT(t) B( )rPT(t) k(t τ)rPT(τ)dτ CrPT(t) rPT(t) KrPT(t) (1) − ∞ − ∞ − 0 − − − where on the right hand side of the equation, the first term is the excitation force including Froude–Kryloff force due to the wave pressure distribution and diffraction force of the pontoon; the second to the fourth terms are forces due to radiation, in which the third term is the force due to the wave memory effect (retardation function force); the fifth term is due to the wave viscous effect, which is not considered in potential flow theory but is supposed to be considered in the time domain model; and the last term is the restoring force due to the pontoons’ vertical positions. More explanation is provided in the following. The finite element method is used for the structural modelling of the bridge deck. The global dynamic equilibrium of the structural finite element formulation in time domain is expressed as [42]:

.. . . RI(r, r, t) + RD(r, r, t) + RS(r, t) = RE(r, r, t), (2)

. .. where r, r, and r are the displacement, velocity, and acceleration vectors for all the nodes in the .. FEM model; RI(r, r, t) is the inertia force vector of the nodes. It includes the bridge structural mass and pontoon mass. In addition, the added mass of the pontoons at infinite frequency as shown in .. Equation (1) can be considered. Therefore, the pontoon inertia term is expressed as (M + A( ))r (t), ∞ PT where M is the pontoon structural mass matrix; A( ) is the added mass matrix at infinite frequency; .. ∞ and rPT(t) is the pontoon acceleration, which is also the acceleration of the node, through which bridge element and pontoons are connected. The pontoon inertia forces are added in the inertia terms of the connecting node. RS(r, t) is the stiffness force vector, including the structural internal stiffness of the bridge deck. In addition, the hydrostatic stiffness forces of the pontoons, like the inertia terms, are added to the stiffness terms of the connecting node. The hydrostatic stiffness of the pontoon is expressed as KrPT(t), where K is the hydrostatic stiffness matrix, and rPT(t) is the pontoon displacement, which is also the displacement of the connecting node. The end connections of the bridge are also specified in the stiffness force vector. . RD(r, r, t) is the damping force vector, including the structural internal damping and infinite . . frequency wave radiation damping of the pontoons B( )r (t); and rPT(t) is the velocity of the ∞ PT pontoon, which also represents the velocity of the connecting node. It is noted that for floating structures with zero forward speed, B( ) = 0. . ∞ RE(r, r, t) is the external force vector. This term also includes forces that are not specified in the previous terms. In this study, those forces include the wave excitation forces, quadratic viscous forces, current forces, tidal variation-induced forces, and retardation forces. All these external applied forces . . are exerted on the pontoons. Note that the quadratic viscous forces are expressed as CrPT(t) rPT(t) , where C is the quadratic viscous damping coefficient matrix, in which the coefficients are set to 0.9 for surge and sway D.O.Fs of the pontoons, and 1.2 for the heave D.O.F [43]. The retardation force, as R t . shown in Equation (1), is expressed by the convolution integral k(t τ)rPT(τ)dτ, where k(τ) is the 0 − retardation function matrix, and is expressed as: J. Mar. Sci. Eng. 2020, 8, 467 7 of 19

Z Z 2 ∞ 2 ∞ k(t) = [B(ω) B( )] cos(ωt)dω = ω[A(ω) A( )] sin(ωt)dω, (3) π 0 − ∞ π 0 − − ∞ which tends to vanish with time. In this study, the JONSWAP spectrum is used to describe the irregular wave condition or sea state. The JONSWAP spectrum is expressed as:

2 ( " #) αg 5ωp 4 1 ω 2 S(ω) = exp + ln γ exp 1 , (4) ω5 −4 ω −2σ2 ω −

2 where g = 9.81 ms− ; ωp is the peak frequency of the sea state; γ and σ are the spectral shape parameters, in which γ = 3.3 is used and σ = 0.07 when ω ωp, σ = 0.09 when ω > ωp; and α is determined ≤ from the following equation: !2 H α = 5.06 s (1 0.287 ln γ), (5) Tp − where Hs is the significant wave height, and Tp is the spectral peak period. Once the sea state is determined, the JONSWAP spectrum can be used to generate the wave force spectrum from the wave force response amplitude operator; then, wave force time series are generated based on the superposition principle. Several cases are specified and used in this investigation. The simulation cases are summarized in Table2, where RS (Rigid B.Cs with Short bridge) and FS (Flexible B.Cs with Short bridge) represent the short 500-m bridge with rigid and flexible end connections, respectively, and RL (Rigid B.Cs with Long bridge) and FL (Flexible B.Cs with Long bridge) represent the 1000-m-long bridge. Rigid connection is fully rigid on the end connection, which represents the strong connections in the real structures. Flexible connection means the rotation D.O.F along the y and z directions (Ry and Rz) in the local coordinate system are set free.

Table 2. Cases for the bridge deck’s cross-sectional rigidity and boundary conditions.

Total EIy EIzv EA GJ Cases Span ( 106 ( 106 ( 106 ( 106 Boundary Conditions (B.Cs) (m) MNm× 2) MNm× 2) MN)× MNm× 2/rad) RS 500 10 20 1 10 Fixed at 6 D.O.Fs RL 1000 10 20 1 10 FS 500 10 20 1 10 Fixed at Translational D.O.F Free at Ry and Rz, fixed at Rx FL 1000 10 20 1 10

The time domain calculation is carried out in SIMO-RIFLEX code, which was developed by Sintef Ocean (previously MARINTEK). SIMO [44] is a code to simulate marine operations involving various bodies in the time domain. RIFLEX [42] is a code for analysis of slender marine structures. It is a nonlinear time domain program with a finite element formulation that can handle large displacement and rotations. The frequency domain hydrodynamic properties are calculated for the pontoon through the penal method in WADAM [45], generating the hydrodynamic excitation forces and added mass, potential damping, etc. Then, these data are transferred to the time domain through SIMO. The bridge deck girders are modelled in RIFLEX using beam elements. The pontoons are connected to the FEM model in RIFLEX through the connecting nodes [42]. Then, the wave loads are transferred to the FEM model. The FEM calculation is carried out in RIFLEX in the time domain, providing pontoon motion parameters, then these parameters are transferred to SIMO to update the wave loads applied on pontoons. Therefore, the model is the coupled hydro-structural dynamic model, in which the pontoon motion and bridge deck responses are calculated and updated at each time step. However, the J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 8 of 19

pontoons. Therefore, the model is the coupled hydro-structural dynamic model, in which the pontoon motion and bridge deck responses are calculated and updated at each time step. However, J. Mar.the Sci. pontoons Eng. 2020 are, 8, 467 assumed to be rigid, and no hydroelasticity on the pontoons is considered, which8 of 19 means the hydrodynamic properties in the frequency domain are kept linear and unchanged in the coupled model. The calculation time step of 0.1 s is applied, and the Newmark method is used for pontoons are assumed to be rigid, and no hydroelasticity on the pontoons is considered, which means numerical integration to solve the differential equation. Under irregular wave conditions, a time thelength hydrodynamic of 4000 s propertiesof calculation in theis performed, frequency domainand 3600 are s (1 kept h) of linear steady-state and unchanged simulation in results the coupled are model.used Thefor the calculation analysis. time step of 0.1 s is applied, and the Newmark method is used for numerical integration to solve the differential equation. Under irregular wave conditions, a time length of 4000 s of calculation5. Numerical is performed,Results and 3600 s (1 h) of steady-state simulation results are used for the analysis.

5. Numerical5.1. Static Responses Results

5.1. StaticFigures Responses 5 and 6 present the static structural responses of the curved floating bridges with different B.Cs and span lengths. As for the vertical bending moment (Figure 5a), negative moment valuesFigures were5 and observed6 present at both the static ends structuralfor bridges responses with rigid of supports, the curved and floating the magnitude bridges is with much di largerfferent B.Csthan and that span of the lengths. positive As moment for the vertical at the mid-span. bending momentCompared (Figure to the5 short-spana), negative bridge moment cases, values the long- were observedspan floating at both bridges ends for own bridges the same with rigidend negative supports, moments and the magnitudebut smaller ispositive much largermoments than in that theof themiddle, positive which moment are atbeneficial the mid-span. for future Compared structural to de thesigns. short-span In terms bridgeof the horizontal cases, the bending long-span moment floating bridges(Figure own 5b), the its samemagnitude end negative is much momentssmaller than but that smaller of thepositive vertical moment, moments and in thethe middle,long-span which bridge are beneficialgenerally for took future less structural moment designs.than the In short-span terms of thebridge. horizontal Axial bendingforces (Figure moment 6a) (Figure show 5someb), its magnitudedifferences is muchfor the smaller four cases. than The that reasons of the verticalare due to moment, different and B.Cs the and long-span the uneven bridge load generally distribution took lessalong moment the thanbridge the length short-span (differences bridge. between Axial forces the bridge (Figure deck6a) showweight some distribution di fferences and for pontoon the four cases.buoyancy The reasons force distribution). are due to di ffTheerent torsional B.Cs and moment the uneven values load are larger distribution for bridge along cases the with bridge flexible length (diffendserences (Figure between 6b). the bridge deck weight distribution and pontoon buoyancy force distribution). The torsional moment values are larger for bridge cases with flexible ends (Figure6b).

(a) (b)

FigureFigure 5. Static5. Static vertical vertical (a ()a and) and horizontal horizontal ( b(b)) bending bending momentmoment under the the bridge’s bridge’s self-weight self-weight for for J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 9 of 19 diffdifferenterent cases. cases.

(a) (b)

FigureFigure 6. Static6. Static axial axial force force (a) and(a) and torsional torsional moment moment (b) under (b) under the bridge’s the bridge’s self-weight self-weight for di forfferent different cases. cases.

5.2. Eigen Value Analysis Eigen value analysis was carried out to identify the eigen properties of the bridge with different B.Cs and total lengths. It is noted that only the added mass at infinite frequency (constant value) rather than its variation with frequency was considered. This may have introduced limited uncertainties. The uncertainties due to this simplification were addressed as follows: For the 4600-m- long floating bridge with discrete pontoons [38], the largest error with and without considering the frequency-dependent added mass is 10% for the first eigen period of around 57 s, while for the subsequent eigen period, the error varies from around 0.1% to 7%. In this study, the largest first eigen period is around 10s (shown later), which means the bridge is much stiffer than the 4600-m bridge. In this case, the mass effect on the eigen period is significantly less, indicating the uncertainties due to neglecting the added mass variation with frequency are quite limited. The first 15 eigen periods are listed in Table 3. It is obvious that the largest eigen period corresponds to the softest bridge configuration, which is case FL. The value for the first eigen mode is 9.63 s for case FL, followed by 7.77 s for case RL, 7.36 s for case FS, and 6.03 s for case RS. The following eigen modes generally have the same trends. This trend shows that the softness of the bridge is most significantly influenced by the length for these two bridges. The data also show that for the long bridge, the eigen values decrease much slower than that for the short bridge, which means more eigen modes will be excited for the long bridge under the same excitation spectrum. It is noted that the Rayleigh damping is used in this study, so the damping ratios vary with the eigen periods. Considering that the first several eigen modes contribute significantly to the dynamic responses, damping ratios based on the first five eigen periods were calculated to be in the range of 2.2%–4.3%, with mass and stiffness proportional coefficients of 0.02, which is in a reasonable range. This damping level will not affect the dynamic responses significantly, which indicate negligible uncertainties due to damping coefficients.

Table 3. The first 15 eigen periods (unit: s) for the floating bridge under different B.Cs and total lengths.

Length Cases 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1000 RL 7.77 6.84 6.13 5.18 4.28 3.67 2.82 2.63 2.61 1.96 1.90 1.54 1.44 1.28 1.27 (m) FL 9.63 7.92 7.44 6.05 4.75 4.37 3.25 3.09 2.63 2.39 2.24 1.71 1.53 1.44 1.36 RS 6.03 3.10 1.82 1.74 1.55 1.42 1.22 0.79 0.78 0.57 0.54 0.48 0.40 0.31 0.31 500 (m) FS 7.46 4.38 2.42 2.25 1.85 1.42 1.36 1.05 0.78 0.62 0.57 0.49 0.44 0.38 0.35

The first five eigen mode shapes are plotted in Figure 7 and Figure 8. The first mode shapes are like simply supported and fixed beams, and except for case FL, the first mode shape is in the horizontal plane (X-Y plane). The other mode shapes are also like that of simple beams, as expected.

J. Mar. Sci. Eng. 2020, 8, 467 9 of 19

5.2. Eigen Value Analysis Eigen value analysis was carried out to identify the eigen properties of the bridge with different B.Cs and total lengths. It is noted that only the added mass at infinite frequency (constant value) rather than its variation with frequency was considered. This may have introduced limited uncertainties. The uncertainties due to this simplification were addressed as follows: For the 4600-m-long floating bridge with discrete pontoons [38], the largest error with and without considering the frequency-dependent added mass is 10% for the first eigen period of around 57 s, while for the subsequent eigen period, the error varies from around 0.1% to 7%. In this study, the largest first eigen period is around 10s (shown later), which means the bridge is much stiffer than the 4600-m bridge. In this case, the mass effect on the eigen period is significantly less, indicating the uncertainties due to neglecting the added mass variation with frequency are quite limited. The first 15 eigen periods are listed in Table3. It is obvious that the largest eigen period corresponds to the softest bridge configuration, which is case FL. The value for the first eigen mode is 9.63 s for case FL, followed by 7.77 s for case RL, 7.36 s for case FS, and 6.03 s for case RS. The following eigen modes generally have the same trends. This trend shows that the softness of the bridge is most significantly influenced by the length for these two bridges. The data also show that for the long bridge, the eigen values decrease much slower than that for the short bridge, which means more eigen modes will be excited for the long bridge under the same excitation spectrum. It is noted that the Rayleigh damping is used in this study, so the damping ratios vary with the eigen periods. Considering that the first several eigen modes contribute significantly to the dynamic responses, damping ratios based on the first five eigen periods were calculated to be in the range of 2.2%–4.3%, with mass and stiffness proportional coefficients of 0.02, which is in a reasonable range. This damping level will not affect the dynamic responses significantly, which indicate negligible uncertainties due to damping coefficients.

Table 3. The first 15 eigen periods (unit: s) for the floating bridge under different B.Cs and total lengths.

Length Cases 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 RL 7.77 6.84 6.13 5.18 4.28 3.67 2.82 2.63 2.61 1.96 1.90 1.54 1.44 1.28 1.27 1000(m) FL 9.63 7.92 7.44 6.05 4.75 4.37 3.25 3.09 2.63 2.39 2.24 1.71 1.53 1.44 1.36 RS 6.03 3.10 1.82 1.74 1.55 1.42 1.22 0.79 0.78 0.57 0.54 0.48 0.40 0.31 0.31 500 (m) FS 7.46 4.38 2.42 2.25 1.85 1.42 1.36 1.05 0.78 0.62 0.57 0.49 0.44 0.38 0.35

J. Mar.The Sci. firstEng. 2020 five, 8 eigen, x FORmode PEER REVIEW shapes are plotted in Figures7 and8. The first mode shapes are 10 like of 19 simply supported and fixed beams, and except for case FL, the first mode shape is in the horizontal planeThe dynamic (X-Y plane). responses The other will mode be the shapes superposition are also like of thatthe first of simple dominant beams, mode as expected. shapes with The dynamicdifferent responsesweighting will factors. be the superposition of the first dominant mode shapes with different weighting factors.

Figure 7. First five eigen mode shapes for the floating bridge for RS (a) and FS (b) cases. (RS: Rigid Figure 7. First five eigen mode shapes for the floating bridge for RS (a) and FS (b) cases. (RS: Rigid B.Cs with Short Bridge; FS: Flexible B.Cs with Short bridge.). B.Cs with Short Bridge; FS: Flexible B.Cs with Short bridge.).

Figure 8. First five eigen mode shapes for the floating bridge for RL (a) and FL (b) cases. (RL: Rigid B.Cs with Long Bridge; FL: Flexible B.Cs with Long bridge.).

5.3. Irregular Wave Analysis Bridge responses under irregular wave actions can to some extent reflect the real performance. In this study, the current and wind effects were considered marginally important, so were not included. Homogeneous wave conditions were assumed, which means the irregular wave forces applied on all the pontoons are the same. The 100-year return period storm with Hs = 2m and Tp = 5.8 s was used in the irregular wave analysis. There was no repetition of the simulation, as only one seed number was used, and the wave was coming in one direction as shown in Figure 2. By irregular wave analysis, the pontoon motion, bridge deck structural response distribution along the bridge length, and response properties could be investigated. The results show the real performance of the bridge under extreme conditions. Based on 1-h of steady-state simulation, the statistical values and spectral results are presented. Pontoon motions and accelerations directly affect the driving comfort and serviceability of the floating bridges. Under operational conditions, the wave-induced deflection should not exceed ±0.3 m in both the surge and heave, and ±0.5 degree in pitch, while the acceleration induced by the wave should not exceed ±0.5 m/s2, ±0.5 m/s2, and ±0.05 rad/s2 (±2.87 degree/s2) in the surge, heave, and pitch, respectively [25]. Under survival conditions, the structural responses should not exceed the capacity of the structure. The pontoon motion Maximum (MAX) and Standard Deviation (STD) values along the bridge length are shown in Figure 9 and Figure 10. The pontoon acceleration MAX values along the bridge are shown in Figure 11. The spectral results of the surge, heave, and pitch motions of pontoon number

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 19

The dynamic responses will be the superposition of the first dominant mode shapes with different weighting factors.

J. Mar. Sci.Figure Eng. 7.2020 First, 8 ,five 467 eigen mode shapes for the floating bridge for RS (a) and FS (b) cases. (RS: Rigid 10 of 19 B.Cs with Short Bridge; FS: Flexible B.Cs with Short bridge.).

Figure 8. First five eigen mode shapes for the floating bridge for RL (a) and FL (b) cases. (RL: Rigid B.CsFigure with 8. LongFirst five Bridge; eigen FL: mode Flexible shapes B.Cs for the with floating Long bridge bridge.). for RL (a) and FL (b) cases. (RL: Rigid B.Cs with Long Bridge; FL: Flexible B.Cs with Long bridge.). 5.3. Irregular Wave Analysis 5.3. Irregular Wave Analysis Bridge responses under irregular wave actions can to some extent reflect the real performance. In this study,Bridge the responses current andunder wind irregular effects wave were actions considered can to some marginally extent reflect important, the real so performance. were not included. In this study, the current and wind effects were considered marginally important, so were not Homogeneous wave conditions were assumed, which means the irregular wave forces applied on all included. Homogeneous wave conditions were assumed, which means the irregular wave forces theapplied pontoons on all are the the pontoons same. are the same. TheThe 100-year 100-year returnreturn period period storm storm with with Hs Hs = 2m= 2mand andTp = Tp 5.8= s 5.8was s used was in used the inirregular the irregular wave wave analysis.analysis. There There waswas no repetition repetition of of the the simulation simulation,, as only as only one seed one seednumber number was used, was and used, theand wave the wave waswas coming coming in in oneone directiondirection as as shown shown in inFigure Figure 2. 2By. Byirregular irregular wave wave analysis, analysis, the pontoon the pontoon motion, motion, bridgebridge deck deck structural structural responseresponse distribution alon alongg the the bridge bridge length, length, and and response response properties properties could could be investigated.be investigated. The The results results show show the the real real performance performance of of the the bridge bridge under under extreme extreme conditions. conditions. Based Based on 1-hon of 1-h steady-state of steady-state simulation, simulation, the the statistical statistical values values and spectral spectral results results are are presented. presented. PontoonPontoon motionsmotions and accelerations accelerations directly directly affect aff ectthe thedriving driving comfort comfort and serviceability and serviceability of the of the floating bridges. Under operational conditions, the wave-induced deflection should not exceed ±0.3 floating bridges. Under operational conditions, the wave-induced deflection should not exceed 0.3 m m in both the surge and heave, and ±0.5 degree in pitch, while the acceleration induced by the wave ± in both the surge and heave, and 0.5 degree in pitch, while the acceleration induced by the wave should not exceed ±0.5 m/s2, ±0.5 m/s±2, and ±0.05 rad/s2 (±2.87 degree/s2) in the surge, heave, and pitch, should not exceed 0.5 m/s2, 0.5 m/s2, and 0.05 rad/s2 ( 2.87 degree/s2) in the surge, heave, and respectively [25]. Under± survival± conditions, the± structural responses± should not exceed the capacity pitch,of the respectively structure. [25]. Under survival conditions, the structural responses should not exceed the capacityThe of pontoon the structure. motion Maximum (MAX) and Standard Deviation (STD) values along the bridge lengthThe are pontoon shown motionin Figure Maximum 9 and Figure (MAX) 10. The and pontoon Standard acceleration Deviation MAX (STD) values values along the along bridge the bridge lengthare shown are shown in Figure in 11. Figures The spectral9 and results10. The of the pontoon surge, heave, acceleration and pitch MAX motions values of pontoon along number the bridge are shown in Figure 11. The spectral results of the surge, heave, and pitch motions of pontoon number 4 (PT4) for the long bridge and pontoon number 2 (PT2) for the short bridge (both located at 0.4 L position) are shown in Figure 12. From Figures9 and 10, both surge and pitch responses for cases FL and RL are significantly larger than that for cases FS and RS. It is noted that the maximum values are also affected by the static deflection under bridge deck weight, so STD values can more properly reflect the dynamic effects. It can also be observed that for the RL case, the surge motion is significantly reduced compared with the FL case, while for the FS and RS cases, the surge STDs are similar, which means the BC strongly affects the surge motions for the 1000-m bridge. The reason could be due to the modes that are excited, because the fifth eigen mode of FL is 4.75 s (1.32 rad/s), which is close to the excitation periods and is in the X-Y plane, while eigen periods of 1.82 s (3.47 rad/s, RS) and 2.42 s (2.60 rad/s, FS) correspond to the horizontal eigen modes of the short bridges far away from excitation. This can also be observed in the surge spectra in Figure 12. The heave motion responses are similar for the four simulation cases. By looking at the eigen periods, they all have vertical modes (global Y-Z plane) located in the wave region, which is also reflected in Figure 12. For the pitch motion, the effects from BC are not so significant for long bridges but are important for short bridges, inducing a larger pitch for FS. There are also significant influences from the bridge length to pitch motion. From the pitch spectra, it is seen that several eigen modes are involved, especially the modes close to 0.9 rad/s. The acceleration plots in Figure 11 also show similar J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 11 of 19

4 (PT4) for the long bridge and pontoon number 2 (PT2) for the short bridge (both located at 0.4 L position) are shown in Figure 12. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 11 of 19 J. Mar. Sci. Eng. 2020, 8, 467 11 of 19

4 (PT4) for the long bridge and pontoon number 2 (PT2) for the short bridge (both located at 0.4 L position)trends are with shown Figure in 10 Figure, and the 12. acceleration MAX values reflect the driving comfort. Even under the extreme sea state, the acceleration MAX values mostly do not exceed the guideline values.

(a) (b)

(a) (b)

(c)

Figure 9. Pontoon surge (a), heave (b), and pitch (c) motion maximum values’ distribution along the bridge length.

(c)

Figure Figure 9. Pontoon 9. Pontoon surge surge (a), (a heave), heave (b (b),), and and pitch ( c()c) motion motion maximum maximum values’ values’ distribution distribution along the along the bridgebridge length. length.

(a) (b)

(a) (b)

(c)

FigureFigure 10.10. PontoonPontoon surge surge (a ),(a heave), heave (b), ( andb), and pitch pitch (c) motion (c) motion standard standard deviation deviation (STD) distribution (STD) distribution alongalong thethe bridgebridge length. length.

(c)

Figure 10. Pontoon surge (a), heave (b), and pitch (c) motion standard deviation (STD) distribution along the bridge length.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 12 of 19 J. Mar. Sci. Eng. 2020, 8, 467 12 of 19 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 12 of 19

(a) (b) (a) (b)

(c) (c) Figure 11. Pontoon surge (a), heave (b), and pitch (c) acceleration maximum values’ distribution along theFigureFigure bridge 11. 11. length. PontoonPontoon surgesurge ( a(),a), heave heave (b (),b and), and pitch pitch (c) acceleration(c) acceleration maximum maximum values’ values’ distribution distribution along along thethe bridge bridge length. length.

(a) (b) (a) (b)

(c) (c) FigureFigure 12. 12. SpectralSpectral response response of the surgesurge ( a(),a), heave heave (b ),(b and), and pitch pitch (c) motions(c) motions as well as aswell the as wave the wave spectrumFigurespectrum 12. for forSpectral different different response cases. cases. of the surge (a), heave (b), and pitch (c) motions as well as the wave spectrum for different cases. From Figure 9 and Figure 10, both surge and pitch responses for cases FL and RL are significantly larger Fromthan that Figure for 9cases and FSFigure and 10,RS. both It is notedsurge andthat pitcthe hmaximum responses values for cases are FLalso and affected RL are by significantly the static deflectionlarger than under that bridgefor cases deck FS andweight, RS. Itso is STD noted values that thecan maximum more properly values reflect are also the affecteddynamic by effects. the static It candeflection also be observedunder bridge that deck for the weight, RL case, so STD the surgevalues motion can more is significantly properly reflect reduced the dynamic compared effects. with It thecan FL also case, be while observed for the that FS for and the RS RL cases, case, the the su surgerge STDs motion are issimilar, significantly which reducedmeans the compared BC strongly with the FL case, while for the FS and RS cases, the surge STDs are similar, which means the BC strongly

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 13 of 19 affects the surge motions for the 1000-m bridge. The reason could be due to the modes that are excited, because the fifth eigen mode of FL is 4.75 s (1.32 rad/s), which is close to the excitation periods and is in the X-Y plane, while eigen periods of 1.82 s (3.47 rad/s, RS) and 2.42 s (2.60 rad/s, FS) correspond to the horizontal eigen modes of the short bridges far away from excitation. This can also be observed in the surge spectra in Figure 12. The heave motion responses are similar for the four simulation cases. By looking at the eigen periods, they all have vertical modes (global Y-Z plane) located in the wave region, which is also reflected in Figure 12. For the pitch motion, the effects from BC are not so significant for long bridges but are important for short bridges, inducing a larger pitch for FS. There are also significant influences from the bridge length to pitch motion. From the pitch spectra, it is seen that several eigen modes are involved, especially the modes close to 0.9 rad/s. The acceleration plots in Figure 11 also show similar trends with Figure 10, and the acceleration MAX values reflect the driving comfort. Even under the extreme sea state, the acceleration MAX values mostly do not exceed the guideline values. Figures 13–15 show the STD, MAX, and Minimum(MIN) values of the internal forces along the bridge length, including the axial force, vertical bending moment, and horizontal bending moment. Compared with the static results, the axial forces under wave actions are significantly larger. While being similar to the static results, the negative vertical bending moments under wave actions occur J. Mar. Sci. Eng. 2020, 8, 467 13 of 19 at ends of RL and RS, and the magnitudes of the negative moments are larger than the positive moment values in the mid-span location, which should be handled with caution in the structural Figures 13–15 show the STD, MAX, and Minimum(MIN) values of the internal forces along the design. In bridgeaddition, length, a includingshorter thespan axial length force, vertical (RS and bending FS) moment, usually and results horizontal in bendingrelatively moment. larger positive vertical momentsCompared (middle with the of static Figure results, 14 the and axial middle forces under of Figure wave actions 15), areand significantly rigid-end larger. connections While (RL and RS) also introducebeing similar large to the end static values. results, the On negative the verticalcontrary, bending much moments more under significant wave actions horizontal occur at bending ends of RL and RS, and the magnitudes of the negative moments are larger than the positive moment moment valuesvalues are in the obtained mid-span for location, long-span which should bridges be handled (FL and with RL) caution as compared in the structural to short design. spans In (right of Figure 14 andaddition, right a shorterof Figure span length15), in (RS terms and FS) of usually STD, results MAX, in relativelyand MIN. larger This positive indicates vertical momentsthat the arch effect (external loads(middle transfer of Figure to 14 theand middleaxial force of Figure rather 15), and than rigid-end bending connections moment) (RL and is RS)not also so introduceobvious, especially large end values. On the contrary, much more significant horizontal bending moment values are for the longobtained bridge. for In long-span further bridges structural (FL and design RL) as compared and optimization, to short spans (right it is ofsuggested Figure 14 and to right increase of the rise- span ratio orFigure increase 15), in termsthe horizontal of STD, MAX, stiffness and MIN. of This lo indicatesng-span that bridge the arch cases effect while (external the loads vertical transfer stiffness can be slightly toreduced. the axial force rather than bending moment) is not so obvious, especially for the long bridge. In further structural design and optimization, it is suggested to increase the rise-span ratio or increase the horizontal stiffness of long-span bridge cases while the vertical stiffness can be slightly reduced.

(a) (b)

(c)

Figure 13. AxialFigure 13.forceAxial (a force), vertical (a), vertical moment moment (b (b),), and horizontal horizont momental moment (c) Standard (c) Standard Deviation (STD)Deviation (STD) distribution along the bridge length. distribution along the bridge length.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 19

J. Mar.J. Mar. Sci. Sci. Eng. Eng. 20202020, 8, 8, ,x 467 FOR PEER REVIEW 14 of 19 14 of 19

(a) (b)

(a) (b)

(c)

Figure 14. Axial force (a), vertical moment (b), and horizontal moment (c) maximum values’ distribution along the bridge length.

(c)

Figure Figure 14. 14. AxialAxial force force (a), vertical(a), vertical moment moment (b), and horizontal(b), and moment horizontal (c) maximum moment values’ (c) distributionmaximum values’ distributionalong the bridge along length. the bridge length.

(a) (b)

(a) (b)

(c)

FigureFigure 15. AxialAxial force force (a), vertical(a), vertical moment moment (b) and horizontal (b) and moment horizontal (c) minimum moment values (c) distribution minimum values distributionalong bridge along length. bridge length.

(c)

Figure 15. Axial force (a), vertical moment (b) and horizontal moment (c) minimum values distribution along bridge length.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 19 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 19 Figure 16 and Figure 17 present the correlation between the axial force and horizontal bending momentsFigure (represented 16 and Figure by the17 present1-h time the series correlatio of variousn between positions the alongaxial forcethe bridge and horizontal length, e.g., bending 0.1 L, 0.2moments L, etc.) for(represented curved bridges by the with 1-h differenttime series span of variouslengths andpositions B.Cs. alongIt is noted the bridgethat only length, the half-length e.g., 0.1 L, of0.2 the L, etc.)bridge for was curved analyzed bridges due with to thedifferent symmetrical span lengths properties. and B.Cs. Different It is noted trends that of onlythe axial the half-length force (N)- horizontalof the bridge bending was analyzed moment due (M toz) couplethe symmetrical effects were properties. observed Different at different trends sections of the axialof the force bridges (N)- z withhorizontal the varying bending span moment length (Mand) coupleB.Cs. Fo effectsr instance, were the observed distribution at different of the sectionsN-Mz points of the of bridgesthe FL bridgewith the changes varying from span a horizontallength and line B.Cs. to Foinclinedr instance, lines the as thedistribution distance offrom the the N-M endz points support of theto the FL sectionbridge increaseschanges from (left ofa horizontalFigure 16 andline leftto inclined of Figure lines 17), aswhile the andistance opposite from trend the isend observed support for to thethe J. Mar. Sci. Eng. 2020, 8, 467 15 of 19 RLsection bridge increases (right of (left Figure of Figure 16 and 16 right and ofleft Figure of Figu 17).re 17), while an opposite trend is observed for the RL bridgeThe N-M (rightz coupling of Figure trends16 and indicateright of Figurethe arch 17). effect in the horizontal curved bridge when subjectedFiguresThe N-Mto 16 wave andz coupling 17 loads. present Ittrends theis observed correlation indicate that the between thearch ratio theeffect axialof thein force themaximum andhorizontal horizontal bending curved bending moment bridge moments towhen the (representedmaximumsubjected toaxial by wave the force 1-h loads. timeis larger seriesIt is for observed of the various long that positionsbridge the than ratio along the of the shortthe bridge maximum bridge, length, indicating bending e.g., 0.1 the L, moment 0.2 arch L, etc.)effect to forthe is curvedlessmaximum significant bridges axialwith for force the diff islongerent larger bridge span for lengths thecompared long and bridge with B.Cs. thanthe It is short notedthe short one. that Thebridge, only maximum the indicating half-length horizontal the of arch the moment bridgeeffect is waschangesless analyzed significant at different due for to the locations the long symmetrical bridge while compared the properties. maximum with Di axialthefferent short force trends one. almost The of remains themaximum axial constant. force horizontal (N)-horizontal Although moment N- bendingMchangesz points moment at scatter different (Min zdifferent) locations couple equadrants,ff whileects were the fairlymaximum observed clear at boundariesaxial different force sections almost are seen ofremains in the Figure bridges constant. 16 with and Although theFigure varying 17 N-as z spanleftM pointsand length right scatter and vertical B.Cs. in different Forlines, instance, and quadrants, the the critical distribution fairly N-M clearz point of boundaries the can N-M be zutilized pointsare seen offor in the further Figure FL bridge section 16 and changes designs.Figure from 17 as aleft horizontal and right line vertical to inclined lines, linesand the as thecritical distance N-Mz frompoint thecan end be utilized support for to further the section section increases designs. (left of Figure 16 and left of Figure 17), while an opposite trend is observed for the RL bridge (right of Figure 16 and right of Figure 17).

(a) (b) (a) (b) Figure 16. Correlation between the axial force and horizontal bending moment for the long-span FigurecurvedFigure 16. bridge16.Correlation Correlation cases FL between (betweena) and RL thethe (b axial axial), represented force force and and by horizontal hori thezontal 1-h time bending bending series moment momentof various for for positions the the long-span long-span along curvedthecurved bridge bridge bridge length cases cases (half-length FL FL (a )(a and) and RLdue RL (b to ),( bsymmetry). represented), represented by theby the 1-h 1-h time time series series of various of various positions positions along along the bridgethe bridge length length (half-length (half-length dueto due symmetry). to symmetry).

(a) (b) (a) (b) FigureFigure 17. 17. CorrelationCorrelation betweenbetween thethe axialaxial forceforce andand horizontalhorizontal bendingbending momentmoment forfor thethe short-span short-span curvedcurvedFigure bridge bridge17. Correlation cases cases FS FS ( a(betweena)) and and RS RS the ( b(b), ),axial represented represented force and by by hori the the 1-hzontal 1-h time time bending series series of momentof various various for points points the alongshort-span along the the bridgebridgecurved length length bridge (half-length (half-length cases FS (a due )due and to to RS symmetry). symmetry). (b), represented by the 1-h time series of various points along the bridge length (half-length due to symmetry). The N-Mz coupling trends indicate the arch effect in the horizontal curved bridge when subjected to wave loads. It is observed that the ratio of the maximum bending moment to the maximum axial force is larger for the long bridge than the short bridge, indicating the arch effect is less significant for the long bridge compared with the short one. The maximum horizontal moment changes at different locations while the maximum axial force almost remains constant. Although N-Mz points scatter in different quadrants, fairly clear boundaries are seen in Figures 16 and 17 as left and right vertical lines, and the critical N-Mz point can be utilized for further section designs. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 16 of 19

The spectralJ. Mar. Sci. Eng. results2020, 8, 467for the axial force, and horizontal and vertical bending moments16 of 19 at the bridge deck position of 0.4 L are shown in Figure 18. The axial forces and horizontal bending moments for theThe spectralshort bridge results forare the much axial force,smaller and horizontalthan the andlong vertical bridge, bending which moments is confirmed at the bridge in Figures 13–15, anddeck differences position of between 0.4 L are shown FL and in FigureRL are 18 mainly. The axial due forces to various and horizontal eigen bending modes. moments It is also for seen that most responsethe short peaks bridge are are located much smaller at the than wave the region long bridge,, which which is isdue confirmed to the infirst Figures several 13–15 eigen, and periods that are closedifferences to the betweenspectral FL period. and RL are mainly due to various eigen modes. It is also seen that most response peaks are located at the wave region, which is due to the first several eigen periods that are close to the spectral period.

(a) (b)

(c)

Figure 18.Figure Spectral 18. Spectral response response of the of axial the axial force force (a (),a), and and vertical vertical (b ()b and) and horizontal horizontal bending bending moment moment (c) (c) at the 0.4 Lat position, the 0.4 L position, as well as as well th ase wave the wave spectrum spectrum for for didifferentfferent cases. cases. 6. Conclusions and Discussion 6. Conclusions and Discussion This paper investigated curved floating bridges with a same small rise-span ratio. Different bridge This lengthspaper ofinvestigated 500 and 1000 mcurved were studied. floating Varied bridges B.Cs were with also a investigated.same small Static, rise-span eigen value, ratio. and Different dynamic analysis were carried out for the four different cases. Pontoon motions and structural reaction bridge lengths of 500 and 1000 m were studied. Varied B.Cs were also investigated. Static, eigen value, forces and moments were presented and analyzed. The motion responses indicated the satisfaction of and dynamicthe survivability analysis were criteria carried for the out bridge for configuration the four different under the cases. operational Pontoon and extrememotions sea and states. structural reaction forcesIn this study,and moments only the extreme were sea presented state was studiedand analyzed. to focus on The the structural motion responseresponses properties indicated the satisfactionunder of dithefferent survivability bridge configurations. criteria Thefor structuralthe bridge reaction configuration results showed under that although the operational the same and extreme searise-span states. ratio In wasthis applied, study, bridges only the with extrem differente lengths sea state have was various studied response to properties,focus on and the the structural short bridge provides a prominent arch effect compared to the long bridge. In addition, the response response propertiesproperties provide under insights different into furtherbridge structural configurations. design. Based The on structural the investigation, reaction the following results showed that althoughconclusions the same can be rise-span drawn. ratio was applied, bridges with different lengths have various response properties,Eigen value and analysisthe short considers bridge provides the infinite a frequencyprominent added arch masseffect rathercompared than to the the long bridge. Infrequency-dependent addition, the response added pr mass,operties which provide involves insights uncertainties, into but further these uncertaintiesstructural design. are quite Based on limited. The eigen periods showed that the softest bridge configuration is FL, followed by RL, FS, and the investigation, the following conclusions can be drawn. Eigen value analysis considers the infinite frequency added mass rather than the frequency- dependent added mass, which involves uncertainties, but these uncertainties are quite limited. The eigen periods showed that the softest bridge configuration is FL, followed by RL, FS, and RS. The eigen values were varied from 0.3 s to around 10 s. For the long bridge, eigen values decreased much slower than that for the short bridge, which means more eigen modes will be excited for the long bridge under the same excitation spectrum. Negative end moments were induced for the RL and RS bridge cases in static conditions, and the negative vertical moment value was much larger than the

J. Mar. Sci. Eng. 2020, 8, 467 17 of 19

RS. The eigen values were varied from 0.3 s to around 10 s. For the long bridge, eigen values decreased much slower than that for the short bridge, which means more eigen modes will be excited for the long bridge under the same excitation spectrum. Negative end moments were induced for the RL and RS bridge cases in static conditions, and the negative vertical moment value was much larger than the horizontal one. When compared to short-span bridge cases (FS and RS), a smaller positive moment was observed in the middle span of FL and RL, which is beneficial for structural design. Irregular wave analysis was performed under the extreme sea state. The pontoon motions showed that BC strongly affected the surge motions for the 1000-m bridge, because it varied the eigen periods, i.e., the resonant region; another reason may be due to the relatively flexible properties of the long bridge, which highlights the importance of the BC. The heave motion responses were similar for the four simulation cases and frequencies were in the wave region. For pitch motions, there were significant influences from the bridge length. The acceleration MAX values mostly did not exceed the guideline values of the operational condition even under the extreme sea state, which indicates that the motion of the bridges under wave action is acceptable. From the structural responses, it can be concluded that flexible end connections are better, considering the vertical and horizontal end bending moment. Short bridges are better in terms of the axial forces and horizontal bending moment, while long bridges have relatively lower vertical bending moments. In further structural design and optimization, it is suggested to increase the horizontal stiffness of long-span bridge cases while the vertical stiffness can be slightly reduced. The bending effect was apparently observed in the current research study on the shallow curved floating bridge with a small rise-to-span ratio (0.134). It is not a preferable force state for an geometry, where the axial compression effect is still prominent. However, the short bridge showed a greater arch effect than the long bridge. Therefore, larger rise-to-span ratios are to be considered in a companion paper to explore the optimal rise-to-span ratios for curved floating bridges and the coupling N-M effect will be further studied.

Author Contributions: Conceptualization, L.W., D.J. and J.D.; Methodology, L.W., D.J. and J.D.; Software, L.W.; Validation, L.W., D.J. and J.D.; Formal analysis, L.W., D.J.; Investigation, L.W., D.J.; Resources, L.W., D.J. and J.D.; Data curation, L.W., D.J. and J.D.; Writing—original draft preparation, L.W., D.J.; Writing—review and editing, L.W., D.J. and J.D.; Visualization, L.W.; Supervision, L.W., D.J. and J.D.; Project administration, L.W. and D.J.; Funding acquisition, L.W., D.J. All authors have read and agreed to the published version of the manuscript Funding: This research was funded by Natural Science Foundation of Jiangsu Province, grant No. BK20180487; National Natural Science Foundation of China (NSFC), grant No.51808292; Research funds from Newcastle University through Newcastle Research & Innovation Institute Pte Ltd. (NewRIIS), Singapore; And research funds from State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Grant number: HESS-1709. The aforementioned funds are gratefully acknowledged. Conflicts of Interest: The authors declare no conflict of interest.

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