Static Behaviors of a Long-Span Cable-Stayed Bridge with a Floating Tower Under Dead Loads

Journal of Marine Science and Engineering

Article Static Behaviors of a Long-Span Cable-Stayed Bridge with a Floating Tower under Dead Loads

Minseo Jang 1, Yunwoo Lee 1, Deokhee Won 2 , Young-Jong Kang 1 and Seungjun Kim 1,*

1 School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 02841, Korea; [email protected] (M.J.); [email protected] (Y.L.); [email protected] (Y.-J.K.) 2 Coastal and Ocean Engineering Division, Korea Institute of Ocean Science and Technology, Busan 49111, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3290-4868

Received: 14 September 2020; Accepted: 15 October 2020; Published: 20 October 2020

Abstract: Owing to the structural characteristics of floating-type structures, they can be effectively applied to overcome the limitation of conventional long-span bridges in deep water. Unlike cable-supported bridges with fixed towers, floating cable-supported bridges show relatively large displacements and rotations under the same load because of floating towers; moreover, the difference in the support stiffness causes differences in the behavior of the superstructures. In addition, the risk of overturning is greater than in conventional floating offshore structures because the center of gravity of the tower is located above the buoyancy center of the floater. A floating cable-supported bridge in which the tether supports the floating main tower is directly influenced by the tether arrangement, which is very important for the stability of the entire structure. In this study, according to the inclined tether arrangement, the outer diameter of the floater, and the buoyancy vertical load ratio (BVR), the static behavioral characteristics of the long-span cable-stayed bridges with floating tower are evaluated through nonlinear finite-element analysis. When the intersection of the tension line of the tether and a pivot point of the tower coincide, the tethers can no longer resist the tower’s rotation. For this reason, a large displacement occurs to equilibrate the structure, and further increases as it approaches the specific slope, even if it is not exactly the specific tether slope. The analytical model of this study indicates that, in terms of increasing the rotational stiffness of the main tower, it is advantageous to increase the floater diameter until a BVR of 1.8 is reached and to increase the axial stiffness of the tether from a BVR of 2.0 or higher.

Keywords: cable-stayed bridge; ﬂoating bridge; global static performance; nonlinear analysis; ocean and shore technology; tether arrangement; Ocean and Shore Technolog

1. Introduction A traditional sea-crossing bridge is generally designed as a long-span bridge to secure the driving route of ships and reduce the cost and construction period of piers. The span length of the bridge is determined by the tower, foundation, and ground conditions. However, there are technical and economical limitations to the construction of sea-crossing bridges in deep water. A cable-supported bridge with a floating tower does not require a fixed pier on the seabed and is a structural type that can overcome the limitations in deep water because the main tower of the cable bridge is supported by the buoyancy produced by a floating body. The current floating bridge is a multi-span continuous bridge with a very short span. So far, no floating bridges with very long spans have been built. The superstructure of a cable-supported bridge with a floating tower, as shown in Figure1, is the same as that of a general fixed cable-supported bridge, and the tower supporting the cable and girder

J. Mar. Sci. Eng. 2020, 8, 816; doi:10.3390/jmse8100816 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 816 2 of 21 isJ. Mar. supported Sci. Eng. 2020 by, a 8, floating x FOR PEER body. REVIEW Through many previous studies, the fixed cable type of the upper2 of 22 structure has been verified, and the floating type of the offshore structure is already used in the tension upper structure has been verified, and the floating type of the offshore structure is already used in leg platform (TLP) type at a depth of 1000 m or more. The two structural types have been sufficiently the tension leg platform (TLP) type at a depth of 1000 m or more. The two structural types have been verified, and if the safety and stability of the tower supported by the floating body are secured, it can sufficiently verified, and if the safety and stability of the tower supported by the floating body are be sufficiently used as a floating long-span cable bridge. The proposed long-span cable-stayed bridge secured, it can be sufficiently used as a floating long-span cable bridge. The proposed long-span with floating towers can be constructed with well-known technology, which will be the most suitable cable-stayed bridge with floating towers can be constructed with well-known technology, which will long-span sea-crossing bridge technology in deep depths. be the most suitable long-span sea-crossing bridge technology in deep depths.

Figure 1. The long-span bridge with floating tower. Figure 1. The long-span bridge with floating tower. The TLP type is mainly a floating support type for offshore structures including offshore oil The TLP type is mainly a floating support type for offshore structures including offshore oil and and gas platforms and submerged floating tunnels [1–3]. Vertical and buoyant forces acting on a gas platforms and submerged floating tunnels [1–3]. Vertical and buoyant forces acting on a structure structure form a balance of forces by using the buoyancy of a floating body submerged in water. form a balance of forces by using the buoyancy of a floating body submerged in water. A cable- A cable-supported bridge with a floating tower has been proposed to control the displacement of supported bridge with a floating tower has been proposed to control the displacement of cable-stayed cable-stayed bridges by tethers connected to the seabed using TLP type and to compensate for the bridges by tethers connected to the seabed using TLP type and to compensate for the disadvantages disadvantages of sea-crossing bridges [4]. Unlike fixed tower bridges or small span floating bridges [5], of sea-crossing bridges [4]. Unlike fixed tower bridges or small span floating bridges [5], cable-stayed cable-stayed bridges with floating towers additionally moor the floater connected to the towers, bridges with floating towers additionally moor the floater connected to the towers, tending to show tending to show a more complex nonlinear structural behavior due to the tether–tower–cable–girder a more complex nonlinear structural behavior due to the tether–tower–cable–girder interaction. For interaction. For the purpose of transportation, cable-stayed bridges with floating towers require the purpose of transportation, cable-stayed bridges with floating towers require additional horizontal additional horizontal displacement control, as well as vertical displacement, which is important for displacement control, as well as vertical displacement, which is important for conventional floating conventional floating structures. Unlike the conventional TLP-type floating structure, it is important structures. Unlike the conventional TLP-type floating structure, it is important to analyze the overall to analyze the overall behavior of the structure according to the tether arrangement, for additional behavior of the structure according to the tether arrangement, for additional horizontal displacement horizontal displacement control [6], and there have been no results of parameter analysis for various control [6], and there have been no results of parameter analysis for various tether arrangements. tether arrangements. Various analysis techniques and analytical studies have been conducted on Various analysis techniques and analytical studies have been conducted on floating bridges. floating bridges. However, most of these [7,8] have mainly focused on floating bridges in the form of However, most of these [7,8] have mainly focused on floating bridges in the form of short-span short-span continuous bridges that ships cannot pass. continuous bridges that ships cannot pass. Papinutti [9] used time-domain tools to analyze the coupled wind and wave load responses of a Papinutti [9] used time-domain tools to analyze the coupled wind and wave load responses of a long-span floating suspension bridge. For the initial screening analysis, a frequency-domain analysis long-span floating suspension bridge. For the initial screening analysis, a frequency-domain analysis was conducted to investigate the critical combination of wind events of a TLP suspension bridge [10]. was conducted to investigate the critical combination of wind events of a TLP suspension bridge [10]. In addition, the effects of extreme and irregular wind and wave loads on the dynamic behaviors of the In addition, the effects of extreme and irregular wind and wave loads on the dynamic behaviors of cable supported bridges with floating pylons were studied by numerical simulation in frequency-and the cable supported bridges with floating pylons were studied by numerical simulation in frequency- time-domain [11,12]. In the study centered on the E39 project in Norway, the floating suspension and time-domain [11,12]. In the study centered on the E39 project in Norway, the floating suspension bridge model was studied and the environmental conditions in the E39 region were found to be relatively mild. Kim et al. [13] provided an overview of the main hydrodynamic analysis techniques for cable-supported bridges with floating towers. Kim et al. [14] studied a tether’s short-term fatigue

J. Mar. Sci. Eng. 2020, 8, 816 3 of 21 bridge model was studied and the environmental conditions in the E39 region were found to be relatively mild. Kim et al. [13] provided an overview of the main hydrodynamic analysis techniques for cable-supported bridges with floating towers. Kim et al. [14] studied a tether’s short-term fatigue under severe environmental conditions according to various tether slopes. Studies on such floating cable bridges have been actively conducted in recent years. However, research on various parameters and floating cable-stayed bridges using the tether’s slope arrangement is still insufficient. In this study, a nonlinear analysis is conducted to investigate the static behavioral characteristics of a cable-stayed bridge with floating towers, considering various initial inclination arrangements of tethers, buoyancy-vertical load ratios, and outer diameters (ODs) of floaters. A total of 755 nonlinear analyses are performed using the parameters such as OD of the floating body, tether slope, and initial buoyancy ratio parameters to confirm the complex nonlinear global system interaction behavior of the cable-stayed bridge with floating towers.

2. Analysis Method for Investigating the Structural Behavior of Floating Cable-Stayed Bridges This section presents details of the analysis method for investigating the static structural behavior of floating cable-stayed bridges. The prototype models of the floating cable-stayed bridge were modeled refer to the cable-stayed bridge models studied by Kim et al. [14]. To consider various geometric nonlinearities of the bridges, a nonlinear finite-element analysis was mainly conducted. To model the main structural members of the structure, such as girders, towers, floaters, stay cables, and tethers, nonlinear beam truss elements were used. For the nonlinear analysis, an incremental-iterative analysis based on the Newton–Rapshon method was conducted. For the analytical study, dead loads estimated by the structure’s self-weight were applied, and the analytical model was assumed to be a six-lane round-trip bridge with reference to the Korean Highway Bridge Design Code [15]. The analysis was performed using ABAQUS V2018 [16].

2.1. Initial Shape Analysis Before proceeding with the parametric analysis, the initial shape analysis was performed to select an appropriate cable and tether tension. In general, nonlinearity, such as the effect of a sag, the beam-column effect, and a large deformation effect, should be considered in a cable. To accurately represent cable behavior, Irvine [17] presented the theory of a self-weighted two-dimensional elastic suspension line, based on which the elastic suspension cable element has been extensively used to model the main cables of the cable-stayed and suspension bridges. The initial shape determination methods studied to date include the trial-and-error [18], initial member force [19], and TCUD (target configuration under dead load) [20] methods. In the trial-and-error method, the designer changes tension forces of stay cables to determine the optimum initial tension of them; however, it takes considerable time to perform iterative calculation and it is difficult to implement a relatively accurate initial shape. In the first step of the analysis, the initial member force method performs nonlinear analysis in the absence of the initial member force, and from the subsequent steps, assumes a member force obtained through the previous step’s nonlinear analysis as the initial member force, introduces it to the analysis model, and repeats the nonlinear analysis. This means that the analysis method repeats the nonlinear analysis until the desired initial shape is obtained, under the assumed initial member force. However, this method is sensitive to the initial tension. On the other hand, the TCUD analysis method uses elastic suspension cable elements, and beam-column elements. This method, unlike the initial member force method, performs an initial shape analysis by adding the non-stress length of the cable as a variable and constraining the node displacement corresponding to the number [20]. The TCUD analysis method is not sensitive to the initial tension, but in the case of cable-stayed bridges, the displacement caused by the compression force of the tower cannot be controlled. Under the same conditions, the lower part of the tower of the floating cable bridge has a larger displacement than that of the fixed cable bridge, as shown in Figure2a. In this study, the initial J. Mar. Sci. Eng. 2020, 8, 816 4 of 21

memberJ. J.Mar. Mar. Sci. Sci. forceEng. Eng. 2020 2020 method, 8, ,8 x, xFOR FOR (Figure PEER PEER 3REVIEW ),REVIEW which is most suitable for a ﬂoating cable bridge with relatively large4 4of of 22 22 displacement of the tower, was used as the initial shape analysis method

(a)(a)

(b)(b)

FigureFigure 2.2. 2. GlobalGlobal Global deformation deformation deformation shape shape shape (scale (scale (scale factor factor factor= 5). = = (5). a5).) Comparison( a()a )Comparison Comparison of fixed of of fixed andfixed floating and and floating floating cable-stayed cable- cable- bridgesstayedstayed bridges without bridges without pre-tension without pre-tension pre-tension (fixed: — (fixed:, floating:(fixed: — —, —,floating: floating:); (b) floating — —);); ( b( cable-stayedb) )floating floating cable-stayed cable-stayed bridge (without bridge bridge pre-tension: (without (without —pre-tension:pre-tension:, with pre-tension: — —, with, with —pre-tension: pre-tension:). — —).).

FigureFigure 3. 3. Initial Initial member member force force method. method.

FigureFigure4 4 shows4 shows shows thethe the iterationiteration iteration resultsresults results ofof of the the the initial initial initial shape shape shape analysis. analysis. analysis. FigureFigure Figure5 5 shows5 shows shows the the the result result result of of of tensiontension distributiondistribution distribution of of one-sideone-side one-side cablescables cables withwith with andand and withoutwithout without pre-tension.pre-tension. pre-tension. Cabl CablesCableses 1 1 and and 40 40 were were cablescables cables arrangedarranged atat at bothboth both ends.ends. ends. EachEach Each towertower tower was was was located located located between be betweentween cables cables cables 10 10 and10 and and 11 11 and11 and and between between between cables cables cables 30 and30 30 and and 41. 41.41. TheThe maximum maximum vertical vertical displacement displacement of of 0 0 iterat iterationion girders girders without without pre-tension pre-tension was was 1.36 1.36 m m and and thethe maximum maximum vertical vertical displacement displacement of of eight-iterat eight-iterationion girders girders applying applying the the tension tension result result of of seven seven iterationsiterations was was 0.24 0.24 m; m; thus, thus, the the vertical vertical displa displacementcement of of the the girders girders was was reduced reduced by by 82.4%. 82.4%. The The KoreanKorean Highway Highway Bridge Bridge Design Design Code Code [15] [15] is is presented presented as as l/1000 l/1000 (l: (l: span span length) length) as as a a conservative conservative deflectiondeflection standard. standard. As As shown shown in in Figure Figure 4(a), 4(a), the the initial initial tension tension must must be be applied applied through through the the initial initial shapeshape analysis analysis to to satisfy satisfy the the deflection deflection criteria criteria of of the the girder. girder.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 5 of 22

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 5 of 22 To reduce the influence of initial tension magnitude in the initial member force method, the numberTo of reduce iterations the ofinfluence the initial of shapeinitial analysistension magnitude was determined in the asinitial four, member with a tolerance force method, of less the than 5%.number In this of study, iterations the parametric of the initial analysis shape analysis models was used determined the tension as four,results with of threea tolerance iterative of less nonlinear than analysesJ. Mar.5%. Sci.In thisas Eng. the study,2020 initial, 8, the 816 tensio parametricn of the analysis cable-stayed models bridge.used the tension results of three iterative nonlinear5 of 21 analyses as the initial tension of the cable-stayed bridge.

(a) (b) (a) (b) Figure 4. Iteration results of initial shape analysis about each tether inclination. (a) Maximum vertical FigureFigure 4. 4.Iteration Iteration results results ofof initialinitial shapeshape analys analysisis about about each each tether tether inclination. inclination. (a) ( aMaximum) Maximum vertical vertical displacement of a girder; (b) displacement reduction by iterations. displacementdisplacement of of a a girder; girder; ( b(b)) displacement displacement reduction by by iterations. iterations.

FigureFigure 5. Cable tension tension distribution. distribution. Figure 5. Cable tension distribution. 2.2.The Analysis maximum Model vertical displacement of 0 iteration girders without pre-tension was 1.36 m and 2.2. Analysis Model the maximumThe analytical vertical model displacement for the global of eight-iteration static behavior girders analysis applying of the cable-stayed the tension bridge result with of seven a iterationsfloatingThe analytical wastower 0.24 is shown m; model thus, in for theFigure the vertical 6.global This displacement bridgestatic behaviorhad a of total the analysis girderslength of wasof 920 the reduced m, cable-stayed a center by 82.4%.span bridge of The480 Koreanm,with a a floatingHighwaytower towerheight Bridge isof shown Design128 m (toin Code Figurethe top [15 6.] of is This tower presented bridge to girder ashad l/), 1000a andtotal (l: free length span water length) of surface-to-girder920 m, as aa conservativecenter span spacing of deflection 480 of 25 m, a towerstandard.m. The height water As shownof depth 128 m in of (to Figure the the analytical 4topa, the of initialtowermodel tensionto was girder assumed must), and be to free appliedbe 500 water m. through surface-to-girder the initial shape spacing analysis of 25 m.to satisfyThe water the deflectiondepth of the criteria analytical of the model girder. was assumed to be 500 m. To reduce the influence of initial tension magnitude in the initial member force method, the number of iterations of the initial shape analysis was determined as four, with a tolerance of less than 5%. In this study, the parametric analysis models used the tension results of three iterative nonlinear analyses as the initial tension of the cable-stayed bridge.

2.2. Analysis Model

The analytical model for the global static behavior analysis of the cable-stayed bridge with a ﬂoating tower is shown in Figure6. This bridge had a total length of 920 m, a center span of 480 m, J. Mar. Sci. Eng. 2020, 8, 816 6 of 21 aJ. Mar. tower Sci. height Eng. 2020 of, 1288, x FOR m (to PEER the REVIEW top oftower to girder), and free water surface-to-girder spacing of6 25 of m.22 TheJ. waterMar. Sci. depth Eng. 2020 of, the8, x FOR analytical PEER REVIEW model was assumed to be 500 m. 6 of 22

Figure 6. Fan-type analysis model. FigureFigure 6. 6. Fan-type Fan-type analysis analysis model.model. The tower of the analytical model is shown in Figure7a. The cables showed a two-sided TheThe tower tower of ofthe the analytical analytical model model is is shown shown inin FigureFigure 7(a).7(a). TheThe cablescables showedshowed a atwo-sided two-sided arrangement in the H-type tower of a single box type, as indicated in Figure7b. The tower and girder arrangementarrangement in inthe the H-type H-type tower tower of of a asingle single box box type,type, asas indicatedindicated inin FigureFigure 7(b).7(b). The The tower tower and and used the beam element B31, and the girder was assumed to be mounted on the cross beam of the tower girdergirder used used the the beam beam element element B31, B31, and and the the girder girder was was assumedassumed to be mountedmounted on on the the cross cross beam beam of of and connected using the ABAQUS [16] MPC-link option. In addition, a hinge connection was assumed the thetower tower and and connected connected using using the the ABAQUS ABAQUS [16] [16] MPC- MPC-linklink option.option. InIn addition,addition, a a hinge hinge connection connection between the girder and the cross beam of the tower. waswas assumed assumed between between the the girder girder and and the the cross cross beam beam of of thethe tower.tower.

(a) (a)

Figure 7. Cont.

J. Mar. Sci. Eng. 2020, 8, 816 7 of 21 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 7 of 22

(b) (c)

(d)

FigureFigure 7. Analysis 7. Analysis model. model. (a) Fan-type (a) Fan-type two-sided two-sided arrangement arrangement of cables; of cables; (b) cross (b) crosssection section of tower; of tower; (c) cross(c) crosssection section of girder; of girder; (d) tether (d) tether attachment attachment position. position.

2 TheThe cross-sectional cross-sectional area area of of all all cables cables was was 0.01 0.01 m m2 andand supported supported the the bridge girder byby 8080 cablescableson onthe the two two sides. sides. As As there there are are no no floating floating long-span long-span bridges bridges yet yet built, built, the girderthe girder section section of references of references [21,22 ], [21,22],which which studied studied the ultimate the ultimate behavior behavior of a cable-stayed of a cable-stayed bridge by bridge analyzing by analyzing its various its parameters, various parameters,was used. was As shown used. As in Figureshown7 c,in the Figure girder 7(c), was the assumed girder was to be assumed a four-cell to boxbe a section,four-cell which box section, assumed whichsuffi assumedcient stiff enerssufficient and stiffeners ribs to prevent and ribs local to prevent buckling. local Each buckling. submerged Each floater submerged was a floater beam elementwas a beam(B31), element formed (B31), of steel formed (API of X65 steel grade) (API andX65 mooredgrade) and with moored 12 tethers, with as 12 shown tethers, in as Figure shown7d. in The Figure cable 7(d).and The tether cable were and modeledtether were as trussmodeled elements as truss (T3D2) elements that (T3D2) could not that receive could not compressive receive compressive and bending andforce. bending The floaterforce. ofThe the floater analytical of the model analytical was assumed model was to be assumed a watertight to be 0.04-m-thick a watertight cylinder, 0.04-m-thick with its cylinder,diameter with and its draft diameter set as and the analysisdraft set variables.as the analysis variables. ToTo take take into into account account the the buoyancy buoyancy of of all all memb membersers submerged inin thethe water, water, the the buoyancy buoyancy force force was wasmodeled modeled by by modifying modifying the the magnitude magnitude and and direction direction ofthe of the member’s member’s gravitational gravitational acceleration acceleration in the in analysis.the analysis. This This can becan used be used for further for further study study in which in which the mass the mass does notdoes change not change in the in dynamic the dynamic analysis, analysis,and the and buoyancy the buoyancy always always acts opposite acts opposite to the to gravitational the gravitational acceleration. acceleration. By calculating By calculating the sum the of sumthe of force the offorce the of dead the loaddead and load the and buoyancy the buoyancy considering considering the submerged the submerged volume, volume, the unit the weight unit of weightthe submerged of the submerged member member was fixed, was and fixed, the buoyancy and the buoyancy force was force considered was considered by reflecting by the reflecting corrected theacceleration corrected acceleration of gravity in of the gravity analysis. in the For anal membersysis. For with members greater with buoyancy greater than buoyancy the dead than load, the the deadbuoyancy load, the force buoyancy was applied force was as the applied dead as load the by dead inverting load by the inverting direction the of direction acceleration of acceleration of gravity in of thegravity analysis. in the analysis. TheThe rational rational approach approach to totake take into into account account the the restoring restoring moment moment induced induced by bythe the change change of ofthe the buoyancybuoyancy center center of ofthe the floater floater in inthe the heeled heeled cond conditionition is isrequired. required. One One effective effective way way to totake take into into accountaccount the the restoring restoring moment moment is to is touse use the the rotational rotational spring spring element element wi withth a rotational a rotational stiffness stiffness for for the the submerged members. In that case, the rotational stiffness of the submerged cylinder can be

J. Mar. Sci. Eng. 2020, 8, 816 8 of 21

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 8 of 22 submerged members. In that case, the rotational stiﬀness of the submerged cylinder can be determined determined by the equation of the restoring moment of the submerged cylinder in the heeled by the equation of the restoring moment of the submerged cylinder in the heeled condition as below: condition as below:

Mr=M ρ-gIρ∅gI∅ (1)(1) r− where 𝑀 =restoring moment of the submerged cylinder in the heeled condition, 𝜌=water density, where Mr= restoring moment of the submerged cylinder in the heeled condition, ρ=water density, 𝑔 𝐼 nd ∅ g=gravity=gravity acceleration, acceleration,I= 2=2nd moment moment of of area area of of the the water water plane, plane,∅ ==angle angle of of heel heel (roll (roll/pitch)/pitch) However,However, it it is is well well known known that that the the restoring restoring moment moment of of the the ﬂoater floater moored moored by by tethers tethers is is generally generally inducedinduced by by the the tethers. tethers. For For the the numerical numerical models models in in this this study, study, the the restoring restoring moment moment induced induced by by the the fourfour tethers tethers shown shown in in Figure Figure8 can8 can be be evaluated evaluated as as below: below: 𝑀 ∅∙𝑂𝐷∙𝑐𝑜𝑠𝜃 for small angle of heel (2) _ EA 2 Mr_tether = ∅ (OD cosθ) for small angle of heel (2) L · · where 𝑀_ =restoring moment induced by the symmetric tethers (the four tethers shown in whereFigureM 8), ==axialrestoring stiffness moment of induceda tether, by 𝑂𝐷 the=outer symmetric diameter tethers of (the the four cylinder, tethers shown∅ = angle in Figure of heel8), r_tether EA L(roll/pitch)=axial sti ﬀness of a tether, OD=outer diameter of the cylinder, ∅= angle of heel (roll/pitch)

Figure 8. The floater and tethers in the heeled condition. Figure 8. The floater and tethers in the heeled condition. Table1 shows that the contribution of the tethers on the rotational sti ffness. Based on the comparison result, the rotational spring for considering the restoring moment due to the change of the Table 1 shows that the contribution of the tethers on the rotational stiffness. Based on the buoyancy center of the floater was not included in the numerical models. comparison result, the rotational spring for considering the restoring moment due to the change of the buoyancy center of the floater was not included in the numerical models. Table 1. Comparison of the rotational stiffnesses.

Table 1. ComparisonRotational of the Sti ffrotationalness (kNm stiffnesses./rad) OD [m] Change of the Buoyancy Center 12 TethersRotational stiffness (kNm/rad) in the Heeled Condition OD [m] Change of the buoyancy center 12 tethers6 2 20.0 1.08 10 1.97 in10 the heeled condition × 7 × 2 30.0 2.21 10 4.44 10 20.0 1.08× 10 × 1.97 10 40.0 3.81 107 7.90 102 × × 30.0 50.0 2.215.87 10 107 1.23 103 4.44 10 × × 40.0 60.0 3.818.40 10 107 1.78 103 7.90 10 50.0 5.87× 10 × 1.23 10 60.0 8.40 10 1.78 10 Table2 shows the main section specifications of the analytical model. Various geometrical parameters were considered to identify the nonlinear global behavior of a fan-type cable-stayed bridge Table 2 shows the main section specifications of the analytical model. Various geometrical with floating towers parameters were considered to identify the nonlinear global behavior of a fan-type cable-stayed bridge with floating towers

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 9 of 22

J. Mar. Sci. Eng. 2020, 8, 816 9 of 21

Table 2. Section properties. Table 2. Section properties. E (N/m2) A (m2) I (m4) 𝜸 (kN/m3) Compression Girder 2.1E (N /m 102) A0.75(m2 ) 1.45I (m4 ) γ77.01(kN/ m3) Compression Y GirderTower 2.1 10 1011 0.370.75 3.14 1.45 77.01 77.01 Y Y × CrossTower beam 2.1 10 1011 0.370.37 3.14 3.14 77.01 77.01 Y Y × CrossFloater beam 2.1 10 1011 2.510.37 ~ 7.54 125 ~ 3.14 3392 77.01 77.01 Y Y × 11 FloaterCable 2.1 10 10 2.51~7.540.01 125~3392- 77.01 77.01 N Y × 11 Cable 2.1 10 0.01 - 77.01 N Tether 2.1× 10 0.0364 - 77.01 N Tether 2.1 1011 0.0364 - 77.01 N × 2.3. Considered Parameters 2.3. Considered Parameters In this study, the effects of the OD on the structural behavior under dead loads were In this study, the e ects of the OD on the structural behavior under dead loads were investigated. investigated. It can be ffassumed that the OD directly affects to the rotational stiffness induced by tethers.It can beSo, assumedthe rational that range the of OD the directlyparameter aff wasects designed to the rotational for the case sti study.ffness The induced floater by thickness tethers. andSo, thesubmerged rational volume range of were the fixed parameter for equal was buoyancy, designed and for the the OD case was study. set in Thethe range floater of thickness20–60 m. Itand was submerged modeled to volume have equal were submerged fixed for equal volume buoyancy, and buoyancy, and the OD by wasadjusting set in the the submerged range of 20–60 depth m. accordingIt was modeled to the tochange have equalin the submergedfloater diameter, volume as andshown buoyancy, in Figure by 9. adjusting the submerged depth according to the change in the floater diameter, as shown in Figure9.

(a) (b) (c)

Figure 9. OuterOuter diameter diameter (OD) (OD) of of the the fl ﬂoateroater with with equal equal buoyancy: buoyancy: (a) ( aOD) OD = 20= 20m; m;(b) (ODb) OD = 40= m;40 ( m;c) (ODc) OD = 60= m.60 m.

Unlike the conventionalconventional TLP-typeTLP-type floatingfloating sea-crossing sea-crossing structures structures with with a verticala vertical arrangement arrangement of oftethers, tethers, where where vertical vertical displacement displacement control control is is mainly mainly considered, considered, the the cable-supported cable-supported bridge bridge with with a afloating floating tower tower is a transportationis a transportation structure structure that requires that properrequires control proper of thecontrol horizontal of the displacement. horizontal Thedisplacement. initial vertical The arrangementinitial vertical of tethersarrangement causes aof relatively tethers causes large horizontal a relatively displacement, large horizontal as the displacement,horizontal stiff asness the of horizontal the tether stiffness occurs only of the when teth theer occurs horizontal only displacement when the horizontal of the floater displacement occurs. of theAs floater another occurs. analysis parameter, a tether arrangement with various inclinations of the cable-supportedAs another bridgeanalysis with parameter, a floating a tower tether was arrang selected.ement A parametricwith various analysis inclinations of 60–120 of◦ theslope cable- was performedsupported withbridge a 90with◦ slope a floating when the tower tether was arrangement selected. A was parametric vertical. Toanalysis evaluate of the60–120° global slope behavior was performedof the structure with accordinga 90° slope towhen the inclinedthe tether arrangement arrangement of was the vertical. tethers, To which evaluate serves the as global the boundary behavior ofcondition the structure of the superstructure,according to the the inclined initial inclinedarrangement arrangement of the tethers, of the tetherwhich was serves modeled as the at boundary the same depth as that shown in Figure 10. J. Mar. Sci. Eng. 2020,, 8,, xx FORFOR PEERPEER REVIEWREVIEW 10 10 ofof 2222 conditionJ. Mar. Sci. Eng. of 2020the ,superstructure,8, 816 the initial inclined arrangement of the tether was modeled 10at ofthe 21 same depth as that shown in Figure 10.

(a) 80° (b) 90° (c) 100°

FigureFigure 10.10. InclinedInclined arrangementarrangement ofof tethers:tethers: (a) 80°; 80◦;( (b) 90°; 90◦;( (cc)) 100°. 100◦.

The analysis was was performed performed by by selecting selecting the the buoyancy-vertical buoyancy-vertical load load ratio ratio (BVR) (BVR) as asa variable. a variable. A Aparametric parametric analysis analysis of the of the BVR BVR was was performed performed by byvarying varying the thesubmerged submerged volume volume of the of thefloater. floater. At Atthis this time, time, the the diameter diameter and and thickness thickness ofof the the float floaterer were were fixed, fixed, and and the the buoyancy buoyancy ratio ratio was was selected as an analysis variable ranging from 1.6 toto 2.42.4 byby adjustingadjusting thethe floater’sfloater’s draft, asas shownshown inin FigureFigure 1111..

(a) BVR = 1.6 (b) BVR = 2.0 (c) BVR = 2.4

FigureFigure 11.11. Buoyancybuoyancy verticalvertical loadload ratioratio (BVR)(BVR) parameters:parameters: ((aa)) BVRBVR == 1.6; (b)) BVRBVR = 2.0; ( c)) BVR BVR = 2.4.2.4.

Table3 3 shows shows allall thethe parametersparameters consideredconsidered inin thisthis paper.paper. TheThe numbernumber ofof consideredconsidered BVR,BVR, OD,OD, and inclinedinclined arrangementarrangement of of tethers tethers parameters parameters was was 5, 5, and5, and 31, 31, and and geometric geometric nonlinear nonlinear analysis analysis was wasconducted conducted for all for the all 775 the cases.775 cases.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 11 of 22

J. Mar. Sci. Eng. 2020, 8, 816 Table 3. Considered parameters. 11 of 21

Inclined BVR Table 3. ConsideredOD parameters.Arrangement of BVR OD Inclined Arrangement ofTether Tether 60˚ 1.6 20 m 60 1.6 20 m ◦ 62˚ 62◦ 64 64˚ 1.8 1.8 30 m 30 m ◦ 66◦ 66˚ .  2.0 2.0 40 m 40 m . 114 114˚ 2.2 2.2 50 m 50 m ◦ 116◦ 116˚ 118 2.4 60 m ◦ 118˚ 2.4 60 m 120 ◦ 120˚

3. 3.Static Static Behavior Behavior of ofFloating Floating Cable-Stayed Cable-Stayed Bridges Bridges under under Dead Dead Loads Loads 3.1. Displacement of the Floater 3.1. Displacement of the Floater The measuring point of the displacement is shown in Figure 12. The horizontal displacement of the The measuring point of the displacement is shown in Figure 12. The horizontal displacement of lower part of the floater according to the tether slope and the diameter–buoyancy ratio of the floater is the lower part of the floater according to the tether slope and the diameter–buoyancy ratio of the shown in Figure 13. As with the top horizontal displacement of the tower, the horizontal displacement floater is shown in Figure 13. As with the top horizontal displacement of the tower, the horizontal of the bottom of the floater was also found to be sensitively affected by the slope of a specific tether displacement of the bottom of the floater was also found to be sensitively affected by the slope of a according to the floater OD. Under the horizontal displacement of the floater, a clear trend of behavior specific tether according to the floater OD. Under the horizontal displacement of the floater, a clear was confirmed. In the case of horizontal displacement of the floating body, the amount of initial tension trend of behavior was confirmed. In the case of horizontal displacement of the floating body, the acting on the tether also increased with the buoyancy, and thus, the amount of change caused by the amount of initial tension acting on the tether also increased with the buoyancy, and thus, the amount buoyancy did not appear much at the position where the tether was attached. of change caused by the buoyancy did not appear much at the position where the tether was attached.

FigureFigure 12. 12. Floater-BotFloater-Bot horizontal horizontal displacement; displacement; BVR BVR = 2.4,= 2.4, OD OD = 20= 20m, m,84 84deg, deg, Scale Scale Factor Factor = 100.= 100.

3.2. Tower Displacement The horizontal displacement of the top of the tower depended on the tether slope, floater diameter, and buoyancy ratio. The measuring point of the displacement is shown in Figure 14. It was confirmed that, under the floating body ODs of 20 m and 30 m, the structure behaved sensitively at a particular slope of the tether arrangement. In addition, as buoyancy increased, the horizontal displacement tended to decrease at the abovementioned ODs, as shown in Figure 15.

J. Mar. Sci.J. Mar. Eng. Sci.2020 Eng., 8 2020, 816, 8, x FOR PEER REVIEW 12 of 22 12 of 21

(a) (b)

‘

(e)

FigureFigure 13. Floater-Bot 13. Floater-Bot horizontal horizontal displacement: displacement: ( a(a)) BVR BVR == 1.6; ( (bb)) BVR BVR = =1.8;1.8; (c) ( cBVR) BVR = 2.0;= 2.0;(d) BVR (d) BVR = = 2.2; (e) BVR2.2;= 2.4.(e) BVR = 2.4. J. Mar.3.2. Sci.Tower Eng. Displacement 2020, 8, x FOR PEER REVIEW 13 of 22 The horizontal displacement of the top of the tower depended on the tether slope, floater diameter, and buoyancy ratio. The measuring point of the displacement is shown in Figure 14. It was confirmed that, under the floating body ODs of 20 m and 30 m, the structure behaved sensitively at a particular slope of the tether arrangement. In addition, as buoyancy increased, the horizontal displacement tended to decrease at the abovementioned ODs, as shown in Figure 15.

FigureFigure 14. Tower-top 14. Tower-top horizontal horizontal displacement;displacement; BVR BVR = 2.4,= 2.4, OD OD= 20 m,= 2084 deg, m, 84 scale deg, factor scale = 100. factor = 100.

(a) (b)

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 13 of 22

J. Mar. Sci.Figure Eng. 202014. Tower-top, 8, 816 horizontal displacement; BVR = 2.4, OD = 20 m, 84 deg, scale factor = 100. 13 of 21

(a) (b)

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 22 (c) (d)

(e)

Figure 15. Tower-top horizontalhorizontal displacement: displacement: (a ()a BVR) BVR= 1.6;= 1.6; (b ()b BVR) BVR= 1.8;= 1.8; (c) ( BVRc) BVR= 2.0; = 2.0; (d) ( BVRd) BVR= 2.2; = (2.2;e) BVR(e) BVR= 2.4. = 2.4.

3.3. Girder Displacement 3.3. Girder Displacement The maximum displacement of of the girder center according to the tether slope and floater floater diameter–buoyancy ratioratio isis shownshown in in Figure Figure 16 16.. In In the the case case of of girder girder displacement, displacement, it wasit was connected connected to tothe the tower tower by by the the cable, cable, and and when when the the tower tower was was supported supported by by tethers,tethers, itit waswas interactivelyinteractively affected affected by the vertical and horizontal displacements of th thee floaterfloater and tower. The unstable behavior of the girder displacement appeared appeared at at sens sensitiveitive parts parts of of the the tether tether slopes slopes inin the the horizontal horizontal displacement displacement of theof the towers towers and and floaters. floaters. In addition, In addition, at floater at floater OD ODss of 20 of m 20 and m and 30 m, 30 as m, the as thebuoyancy buoyancy increased, increased, the horizontalthe horizontal and andvertical vertical displacements displacements of ofthe the tower tower top top decreased, decreased, which which in in turn turn decreased decreased the maximum deflection deflection of the girder.

(a) (b)

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 22

(e)

Figure 15. Tower-top horizontal displacement: (a) BVR = 1.6; (b) BVR = 1.8; (c) BVR = 2.0; (d) BVR = 2.2; (e) BVR = 2.4.

3.3. Girder Displacement The maximum displacement of the girder center according to the tether slope and floater diameter–buoyancy ratio is shown in Figure 16. In the case of girder displacement, it was connected to the tower by the cable, and when the tower was supported by tethers, it was interactively affected by the vertical and horizontal displacements of the floater and tower. The unstable behavior of the girder displacement appeared at sensitive parts of the tether slopes in the horizontal displacement of the towers and floaters. In addition, at floater ODs of 20 m and 30 m, as the buoyancy increased, the horizontal and vertical displacements of the tower top decreased, which in turn decreased the J. Mar. Sci. Eng. 2020, 8, 816 14 of 21 maximum deflection of the girder.

(a) (b)

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 22 (c) (d)

(e)

FigureFigure 16. 16.Maximum Maximum verticalvertical displacement displacement ofof the the girder: girder: ( a(a)) BVR BVR= =1.6; 1.6;( b(b)) BVR BVR= =1.8; 1.8; (c(c)) BVR BVR= = 2.0;2.0; (d(d)) BVR BVR= =2.2; 2.2; ((ee)) BVRBVR= = 2.4.2.4. 3.4. Effect of Design Parameters on the Structural Behavior under Dead Loads 3.4. Effect of Design Parameters on the Structural Behavior under Dead Loads 3.4.1. Effect of Tether Arrangement on the Static Behavior 3.4.1. Effect of Tether Arrangement on the Static Behavior As the tether arrangement becomes inclined from the vertical arrangement, if the pre-tension As the tether arrangement becomes inclined from the vertical arrangement, if the pre-tension of of the tether is the same, the horizontal stiffness produced by it increases and the vertical stiffness the tether is the same, the horizontal stiffness produced by it increases and the vertical stiffness decreases. As shown in Figure 17, the inclinations of 60◦ and 120◦ acted as the same support form decreases. As shown in Figure 17, the inclinations of 60° and 120° acted as the same support form in in terms of the horizontal and vertical force components. However, although the magnitudes of the terms of the horizontal and vertical force components. However, although the magnitudes of the force component of the two forces were the same, the locations of the point of action of the tension and force component of the two forces were the same, the locations of the point of action of the tension its intersection differed. The graph of the structural analysis results shows that the results were not and its intersection differed. The graph of the structural analysis results shows that the results were not symmetrical with respect to the vertical arrangement (90°) of the tether. The difference in the tension intersection is confirmed by the fact that the analysis results did not show symmetry.

Figure 17. Force component produced by tension of tether, OD = 60 m.

As shown in Figure 13, Figure 15, and Figure 16, when the displacements of the floater, tower, and girder became unstable, the intersection of the tether tension direction gathered at the same specific point according to each floater diameter and tether slope.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 22

(e)

Figure 16. Maximum vertical displacement of the girder: (a) BVR = 1.6; (b) BVR = 1.8; (c) BVR = 2.0; (d) BVR = 2.2; (e) BVR = 2.4.

3.4. Effect of Design Parameters on the Structural Behavior under Dead Loads

3.4.1. Effect of Tether Arrangement on the Static Behavior As the tether arrangement becomes inclined from the vertical arrangement, if the pre-tension of the tether is the same, the horizontal stiffness produced by it increases and the vertical stiffness decreases. As shown in Figure 17, the inclinations of 60° and 120° acted as the same support form in J.terms Mar. Sci. of Eng.the 2020horizontal, 8, 816 and vertical force components. However, although the magnitudes 15of ofthe 21 force component of the two forces were the same, the locations of the point of action of the tension and its intersection differed. The graph of the structural analysis results shows that the results were symmetricalnot symmetrical with with respect respect to the to vertical the vertical arrangement arrangement (90◦) of(90°) the tether.of the tether. The di ﬀTheerence difference in the tension in the intersectiontension intersection is conﬁrmed is confirmed by the fact by thatthe fact the analysisthat the analysis results did results not showdid not symmetry. show symmetry.

Figure 17. Force component produced by tension of tether, OD = 60 m. Figure 17. Force component produced by tension of tether, OD = 60 m. As shown in Figure 13, Figure 15, and Figure 16, when the displacements of the floater, tower, and girderAs shown became in Figure unstable, 13, the Figure intersection 15, and ofFigure the tether 16, when tension the direction displacements gathered of at the the floater, same specific tower, pointand girder according became to each unstable, floater the diameter intersection and tether of the slope. tether tension direction gathered at the same specificThe point displacements according ofto each the structure floater diameter were sensitive and tether when slope. the intersection of the tether tension directions was gathered at the tower–girder intersection. As the tower and girder indicated a hinged connection, they were connected by the pivot point of the tower. When the tension direction of the tether was gathered at the tower’s pivot point, without a large displacement, the tension created by the tether could no longer resist the tower’s rotation. As a result, the tower exhibited unstable behavior with respect to rotation, and its instability affected the global behavior of floating cable-stayed bridges. Figure 18 shows an analytical model at BVR = 2.4, where the intersection of the tether’s tensile force is closest to the tower–girder connection, according to the floater diameter. Table4 indicates the calculation result showing the slope when the intersection of the tether tension exactly matches the girder, according to the BVR and floater diameter. From the analysis results of Figure 13, Figure 15, and Figure 16, the tether slope at which the global behavior is sensitive is similar to each slope shown in Table4. The phenomenon that did not appear at slopes over 90 ◦, where the force components were the same, and appeared when the intersection of tensions coincided with that of the girder. The unstable behavior of the structure due to the tether slope appeared when the tether was no longer resistant to the tower rotation at certain slopes.

3.4.2. Effect of BVR and Floater OD on the Static Behavior To exclude the rotational instability of the tower caused by the inclined arrangement of the tether, the behavior evaluation by BVR and OD was compared based on the vertical (90◦) arrangement of the tethers. As the BVR increased, the required buoyancy, as well as the submerged volume of the floater, increased. As the floater had the same diameter, the buoyancy increased by adjusting the submerged depth and the length of tether attached to the bottom of the floater decreased with increasing BVR. Because the submerged floater depth varied significantly with the buoyancy ratio at a small OD, this tendency induced the largest change in tether length, as shown Figures 19 and 20. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 16 of 22

The displacements of the structure were sensitive when the intersection of the tether tension directions was gathered at the tower–girder intersection. As the tower and girder indicated a hinged connection, they were connected by the pivot point of the tower. When the tension direction of the tether was gathered at the tower’s pivot point, without a large displacement, the tension created by the tether could no longer resist the tower’s rotation. As a result, the tower exhibited unstable behavior with respect to rotation, and its instability affected the global behavior of floating cable- stayed bridges. Figure 18 shows an analytical model at BVR = 2.4, where the intersection of the tether’s tensile force is closest to the tower–girder connection, according to the floater diameter. Table 4 indicates the calculation result showing the slope when the intersection of the tether tension exactly matches the girder, according to the BVR and floater diameter. From the analysis results of Figure 13, Figure 15, and Figure 16, the tether slope at which the global behavior is sensitive is similar to each slope shown in Table 4. The phenomenon that did not appear at slopes over 90°, where the force components were the same, and appeared when the intersection of tensions coincided with that of J. Mar. Sci. Eng.the 2020girder., 8, The 816 unstable behavior of the structure due to the tether slope appeared when the tether 16 of 21 was no longer resistant to the tower rotation at certain slopes.

(a) (b) (c)

Figure 18.FigureIntersection 18. Intersection of tether of tether tension tension according according to to OD ODof of thethe floater ﬂoater and and teth tetherer inclined, inclined, at BVR at = BVR = 2.4, in the analysis2.4, in model:the analysis (a) model: OD = (20a) OD m, = 84 20◦ m,;( b84°;) OD (b) OD= 30 = 30 m, m, 74 74°;◦;( (c)) OD OD = =40 40m, 64°. m, 64◦.

Table 4. InclinationTable 4. Inclination of tether of tether where where the the intersection intersection of of tension tension exactl exactlyy matches matches the girder. the girder. Floater Tether BVR Floater Tether BVR OD [m] Draft [m] Inclination [deg] OD [m]20 Draft60.13 [ m] 81.8 Inclination [deg] 30 26.73 71.1 20 60.13 81.8 1.6 40 15.03 60.0 30 26.73 71.1 50 9.62 50.4 1.6 40 15.03 60.0 60 6.68 42.9 50 9.62 50.4 20 67.65 82.5 1.860 6.68 42.9 30 30.07 72.2 20 67.65 82.5 30 30.07 72.2 1.8 40 16.91 61.1 50 10.82 51.4 60 7.52 43.6 20 75.16 83.0 30 33.41 73.2 2.0 40 18.79 62.2 50 12.03 52.4 60 8.35 44.4 20 82.68 83.5 30 36.75 74.1 2.2 40 20.67 63.3 50 13.23 53.3 60 9.19 45.2 20 90.20 84.0 30 40.09 74.9 2.4 40 22.55 64.2 50 14.43 54.2 60 10.02 45.9 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 17 of 22

40 16.91 61.1 50 10.82 51.4 60 7.52 43.6 20 75.16 83.0 30 33.41 73.2 2.0 40 18.79 62.2 50 12.03 52.4 60 8.35 44.4 20 82.68 83.5 30 36.75 74.1 2.2 40 20.67 63.3 50 13.23 53.3 60 9.19 45.2 20 90.20 84.0 30 40.09 74.9 2.4 40 22.55 64.2 50 14.43 54.2 60 10.02 45.9

3.4.2. Effect of BVR and Floater OD on the Static Behavior To exclude the rotational instability of the tower caused by the inclined arrangement of the tether, the behavior evaluation by BVR and OD was compared based on the vertical (90°) arrangement of the tethers. As the BVR increased, the required buoyancy, as well as the submerged volume of the floater, increased. As the floater had the same diameter, the buoyancy increased by adjusting the submerged depth and the length of tether attached to the bottom of the floater decreased with increasing BVR. Because the submerged floater depth varied significantly with the J.buoyancy Mar. Sci. Eng. ratio2020 at, 8 ,a 816 small OD, this tendency induced the largest change in tether length, as shown17 of 21 Figures 19 and 20.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 22

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 22 (a)

(b)

(b) Figure 19. Comparison of tether lengths: (a) OD = 60 m; (b) OD = 20 m.

FigureFigure 19.19. Comparison of of tether tether lengths: lengths: (a () aOD) OD = 60= 60m; m;(b) (ODb) OD = 20= m.20 m.

FigureFigure 20. 20.LengthLength of tether of tether by BVR. by BVR. Figure 20. Length of tether by BVR. Although the rigidity of the tether was the same, the length decreased as the BVR increased, Although the rigidity of the tether was the same, the length decreased as the BVR increased, and Although the rigidityETA ofT the tether was the same, the length decreased as the BVR increased, and and the axial stiﬀness of the tether increased in inverse proportion to the length, as shown in the axial stiffness ofL Tthe tether increased in inverse proportion to the length, as shown in the axial stiffness of the tether increased in inverse proportion to the length, as shown in Figure 21. Figure 21. Figure 21.

FigureFigure 21. 21. Axial Axial stiffness stiffness of of tether tether by by BVR. BVR.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 22

(b)

Figure 19. Comparison of tether lengths: (a) OD = 60 m; (b) OD = 20 m.

Figure 20. Length of tether by BVR.

Although the rigidity of the tether was the same, the length decreased as the BVR increased, and the axial stiffness of the tether increased in inverse proportion to the length, as shown in FigureJ. Mar. 21. Sci. Eng. 2020, 8, 816 18 of 21

J. Mar. Sci. Eng. 2020, 8, x FOR PEERFigure REVIEWFigure 21. Axial 21. Axial stiffness stiﬀness of tether of tether by BVR. by BVR. 19 of 22

AccordingAccording to to the the result result graphgraph shown in in Figure Figure 22, 22 the, the displacement displacement of the of tower the tower and girder and girder decreased as the BVR increased (increased axial stiffness of the tether). Figure 22(c) shows that, at decreased as the BVR increased (increased axial stiffness of the tether). Figure 22c shows that, at BVRBVR= = 1.8,1.8, the effect effect of of increase increase in in the the axial axial stiffne stiffssness of the of thetether tether was wasgreater greater than that than of that the ofincrease the increase in the floater OD, due to an increase in the rotational stiffness produced by the tether. At a small BVR in the floater OD, due to an increase in the rotational stiffness produced by the tether. At a small BVR (below 1.8), the vertical displacement of the girder was reduced due to the increase in OD, while at a (belowlarger 1.8), BVR the (over vertical 2.0), it displacement decreased with of the the OD girder due to was an reducedincrease in due the toaxial the stiffness increase of inthe OD, tether. while at a larger BVR (over 2.0), it decreased with the OD due to an increase in the axial stiffness of the tether.

78.4 122.2 OD=20 OD=20 OD=30 OD=30 [mm] [mm] OD=40 OD=40 76.8 OD=50 OD=50 H.displ V.displ 119.6 OD=60 OD=60

75.2 117.0 Tower-Top Tower-Top 73.6 114.4

1.61.82.02.22.4 1.6 1.8 2.0 2.2 2.4 BVR BVR

(a) (b)

9.4 338.0 OD=20 OD=20 OD=30 OD=30 Rad] -4

OD=40 [mm] OD=40 332.8 9.2 OD=50 OD=50 [10 OD=60 OD=60 φ Max.displ 9.0 327.6 Girder 8.8 322.4 Tower Rotaion

8.6 1.6 1.8 2.0 2.2 2.4 1.6 1.8 2.0 2.2 2.4 BVR BVR

FigureFigure 22. 22.Displacement Displacement and and rotation rotation by byBVR BVR and andOD: ( OD:a) vertical (a) vertical displacement displacement of the tower; of the (b) tower; (b)horizontal displacement displacement of ofthe the tower; tower; (c) rotational (c) rotational angle angle of the oftower; the tower;(d) maximum (d) maximum displacement displacement of the girder. of the girder. At a small BVR, an increase in OD is advantageous for an increase in the rotational stiffness, but at a large BVR, an increase in the axial stiffness due to a decrease in the tether length is more effective in increasing the rotational stiffness. However, an increase in BVR is accompanied by that in the pre- tension acting on the tether (especially in the case of an inclined arrangement), leading to a decrease in the safety margin of the tether as shown in Figure 23.

J. Mar. Sci. Eng. 2020, 8, 816 19 of 21

At a small BVR, an increase in OD is advantageous for an increase in the rotational stiffness, but at a large BVR, an increase in the axial stiffness due to a decrease in the tether length is more effective in increasing the rotational stiffness. However, an increase in BVR is accompanied by that in the pre-tension acting on the tether (especially in the case of an inclined arrangement), leading to a J. Mar.decrease Sci. Eng. in 2020 the, 8 safety, x FOR marginPEER REVIEW of the tether as shown in Figure 23. 20 of 22

1.0 BVR=1.6 BVR=1.8 0.8 BVR=2.0 BVR=2.2 y BVR=2.4

/ f 0.6 Pre T 0.4

0.2

0.0 60 70 80 90 100 110 120 Tether Inclination [deg]

FigureFigure 23. 23. RequiredRequired pre-tension pre-tension of the of the tether tether by byinclined inclined tethers. tethers.

4. Conclusions4. Conclusions In Inthis this study, study, the the geometrical geometrical static static behaviors behaviors of ofcable-stayed cable-stayed bridges bridges with with floating floating towers towers were were analyzedanalyzed by by conducting conducting a geometri a geometricc nonlinear nonlinear finite-element finite-element analysis analysis.. The The cable-stayed cable-stayed bridge bridge with with floatingfloating tower, tower, even even under under the the dead dead load load only, only, co confirmednfirmed the the following following behavior behavior characteristics characteristics that that diddid not not appear appear in inthe the fixed fixed cable-stayed cable-stayed bridge. bridge. (1) The displacement of the structure sensitively responds to the specific inclination of the (1) The displacement of the structure sensitively responds to the specific inclination of the tether tether according to each OD. This tendency is particularly evident at the lower horizontal according to each OD. This tendency is particularly evident at the lower horizontal displacement displacement of the floater, as well as in the displacement of the tower and girder. The of the floater, as well as in the displacement of the tower and girder. The difference between the difference between the tether's obtuse and acute angle arrangements is the intersection of tether’s obtuse and acute angle arrangements is the intersection of the tether’s tension, and it is the tether’s tension, and it is confirmed that the structure’s behavior becomes more sensitive confirmed that the structure’s behavior becomes more sensitive as the intersection approaches as the intersection approaches the connection between the main tower and the girder. the connection between the main tower and the girder. (2) Without the large displacement of the cable-stayed bridge with a floating tower, the tether (2) Without the large displacement of the cable-stayed bridge with a floating tower, the tether tensions will no longer resist the tower rotation when the intersection of the tether tensions tensions will no longer resist the tower rotation when the intersection of the tether tensions coincides with the rotation center of the tower. Unlike the conventional floating sea-crossing coincides with the rotation center of the tower. Unlike the conventional floating sea-crossing structures, the cable-stayed bridge with a floating tower plays a very important role in structures, the cable-stayed bridge with a floating tower plays a very important role in securing securing stability against overturning of the structure because the center of gravity of the stability against overturning of the structure because the center of gravity of the superstructure superstructure is located above the center of buoyancy. The exact center of rotation of the is located above the center of buoyancy. The exact center of rotation of the towers should be towers should be clearly interpreted and confirmed, and the design of the tether clearly interpreted and confirmed, and the design of the tether arrangement must be performed arrangement must be performed considering the intersection of tether tensions. considering the intersection of tether tensions. (3) The larger the floater diameter, the lesser the change in the static behavior of the structure, (3) The larger the floater diameter, the lesser the change in the static behavior of the structure, according to the buoyancy ratio and tether slope. When the floater diameter is 60 m and the according to the buoyancy ratio and tether slope. When the floater diameter is 60 m and the buoyancy ratio is 2.0, the sensitive behavior exhibited by the inclined tether will also appear buoyancy ratio is 2.0, the sensitive behavior exhibited by the inclined tether will also appear at 44.4°, where the tether tension intersects the rotational center of the tower. On the other at 44.4 , where the tether tension intersects the rotational center of the tower. On the other hand,◦ if the floater diameter is small, the sensitive behavior of the structure occurs only with hand, if the floater diameter is small, the sensitive behavior of the structure occurs only with a a very small inclination arrangement of the tether, and the overall displacement decreases very small inclination arrangement of the tether, and the overall displacement decreases as the as the BVR increases. BVR increases. (4) As the BVR increases, the floater depth of the same diameter increases, in order to create a (4) Aslarger the BVR buoyancy. increases, The the tether floater attached depth of to the the same floater diameter bottom increases, becomes in shorter order to at create the same a larger buoyancy.depth as Thethe tethersubmerged attached depth to theof floaterthe floater bottom increases. becomes This shorter decrease at the in same tether depth length as the submergedincreases depthits axial of thestiffness. floater In increases. addition, This the decrease smaller inthe tether floater length diameter, increases the its larger axial stiis fftheness. reduction in length and increase in axial stiffness. That is, as the BVR increases, the static behavior of the structure decreases due to an increase in the axial stiffness of the tether. However, a larger BVR leads to a larger required pre-tension of the tether (even if the tether is inclined, additional pre-tension is required). (5) The rotational stiffness produced by the tether is determined by the length of the rotating arm resulting from the floater diameter and the axial stiffness of the tether. In the analytical model of this study, until the BVR reaches 1.8, the larger the diameter of the floating body,

J. Mar. Sci. Eng. 2020, 8, 816 20 of 21

In addition, the smaller the floater diameter, the larger is the reduction in length and increase in axial stiffness. That is, as the BVR increases, the static behavior of the structure decreases due to an increase in the axial stiffness of the tether. However, a larger BVR leads to a larger required pre-tension of the tether (even if the tether is inclined, additional pre-tension is required). (5) The rotational stiffness produced by the tether is determined by the length of the rotating arm resulting from the floater diameter and the axial stiffness of the tether. In the analytical model of this study, until the BVR reaches 1.8, the larger the diameter of the floating body, the lesser is the rotational displacement of the tower. After BVR 2.0, the shorter the tether length, the larger is its axial stiffness and the smaller is the rotational displacement.

Author Contributions: Conceptualization, M.J., Y.-J.K., and S.K.; methodology, S.K.; structural analysis, M.J.; validation, M.J., and Y.L.; investigation, M.J., Y.L, D.W., and S.K.; resources, Y.-J.K.; data curation, M.J. and Y.L; writing—original draft preparation, M.J.; writing—review and editing, S.K.; visualization, M.J.; supervision, S.K.; project administration, S.K.; funding acquisition, S.K. All authors have read and agreed to the published version of the manuscript. Funding: The research was funded by a Korea University Grant (K1923211). Acknowledgments: Authors appreciate kind supports from Korea University. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Won, D.; Seo, J.; Kim, S.; Park, W.-S. Hydrodynamic Behavior of Submerged Floating Tunnels with Suspension Cables and Towers under Irregular Waves. Appl. Sci. 2019, 9, 5494. [CrossRef] 2. Won, D.; Kim, S. Feasibility Study of Submerged Floating Tunnels Moored by an Inclined Tendon System. Int. J. Steel Struct. 2018, 18, 1191–1199. [CrossRef] 3. Kim, S.; Won, D.; Seo, J.; Jeong, W.-M.; Kang, Y.-J. Hydrodynamic Behavior of Submerged Floating Pipeline under Regular Waves. J. Pipeline Syst. Eng. Pract. 2020, 11, 04020017. [CrossRef] 4. Ellevset, O. Norwegian coastal highway route E39-Status overview. In Proceedings of the Bridges Annual Conference, Oslo, Norway, 3–4 November 2014. 5. Wan, L.; Jiang, D.Q.; Dai, J. Numerical Modelling and Dynamic Response Analysis of Curved Floating Bridges with a Small Rise-Span Ratio. J. Mar. Sci. Eng. 2020, 8, 467. [CrossRef] 6. Cifuentes, C.; Kim, S.; Kim, M.; Park, W. Numerical simulation of the coupled dynamic response of a submerged floating tunnel with mooring lines in regular waves. Ocean Syst. Eng. 2015, 5, 109–123. [CrossRef] 7. Kvåle, K.A.; Sigbjörnsson, R.; Øiseth, O. Modelling the stochastic dynamic behaviour of a pontoon bridge: A case study. Comput. Struct. 2016, 165, 123–135. [CrossRef] 8. Shixiao, F.; Weicheng, C.; Xujun, C.; Cong, W. Hydroelastic analysis of a nonlinearly connected floating bridge subjected to moving loads. Mar. Struct. 2005, 18, 85–107. [CrossRef] 9. Papinutti, M.; Bruer, A.; Marley, M.H.; Kvaleid, K.J.; Hatami, A.; Pathak, A.; Bhide, S. A frequency domain tool for investigation of wind response of TLP suspension bridges. In Proceedings of the IABSE Symposium 2017, Vancouver, BC, Canada, 19–23 September 2017; Engineering the Future: Vancouver, BC, Canada, 2017; pp. 198–205. 10. Papinutti, M.; Aas-Jakobsen, K.; Kaasa, L.H.; Bruer, A.; Marley, M.H.; Veie, J.; Holtberget, S.H. Coupled Wind and Wave Load Analyses of Multi-span Suspension Bridge Supported by Floating Foundations. In Proceedings of the IABSE Symposium, Vancouver, BC, Canada, 19–23 September 2017; Engineering the Future: Vancouver, BC, Canada, 2017; pp. 2052–5039. 11. Xu, Y.; Øiseth, O.; Moan, T.; Naess, A. Prediction of long-term extreme load effects due to wave and wind actions for cable-supported bridges with floating pylons. Eng. Struct. 2018, 172, 321–333. [CrossRef] 12. Xu, Y.; Øiseth, O.; Moan, T. Time domain simulations of wind- and wave-induced load effects on a three-span suspension bridge with two floating pylons. Mar. Struct. 2018, 58, 434–452. [CrossRef] 13. Kim, S.; Won, D.; Kang, J.S. Dynamic Behavior of Cable-stayed Bridges with Floating Towers under Waves. J. Korean Soc. Steel Constr. 2018, 30, 205–216. [CrossRef] J. Mar. Sci. Eng. 2020, 8, 816 21 of 21

14. Kim, S.; Won, D.H.; Jang, M.; Lee, Y.; Kang, Y.J. Short-term Fatigue Damage of the Tendons for Cable Supported Bridgeswith Floating Towers under the Severe Wave Condition. J. Korean Soc. Steel Constr. 2019, 31, 211–221. [CrossRef] 15. Korea Road & Transportation Association. Korean Hightway Bridge Design Code; Korea Road & Transportation Association: Seoul, Korea, 2016. 16. SIMULIA. ABAQUS/Standard V2018; Dassault Systems Simulia Corp; Simulia: Johnston, RI, USA, 2018. 17. Irvine, H.M. Cable Structures; MIT Press: Cambridge, MA, USA, 1981; p. 259. 18. Karoumi, R. Some modeling aspects in the nonlinear ﬁnite element analysis of cable supported bridges. Comput. Struct. 1999, 71, 397–412. [CrossRef] 19. Wang, H.P.; Tseng, T.C.; Yang, C.G. Initial shape of cable-stayed bridges. Comput. Struct. 1993, 44, 1095–1106. [CrossRef] 20. Kim, K.-S.; Lee, H.S. Analysis of target conﬁgurations under dead loads for cable-supported bridges. Comput. Struct. 2001, 79, 2681–2692. [CrossRef] 21. Kim, S.; Won, D.; Kang, Y.J. Ultimate behavior of steel cable-stayed bridges-I. Rational ultimate analysis method. Int. J. Steel Struct. 2016, 16, 601–624. [CrossRef] 22. Kim, S.; Won, D.; Kang, Y.J. Ultimate behavior of steel cable-stayed bridges-II. Parametric study. Int. J. Steel Struct. 2016, 16, 625–636. [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).