Form-Finding Analysis of the Rail Cable Shifting System of Long-Span Suspension Bridges

applied sciences

Article Form-Finding Analysis of the Rail Cable Shifting System of Long-Span Suspension Bridges

Quan Pan, Donghuang Yan * and Zhuangpeng Yi School of Civil Engineering, Changsha University of Science & Technology, Changsha 410114, Hunan, China; [email protected] (Q.P.); [email protected] (Z.Y.) * Correspondence: [email protected]

Received: 26 September 2018; Accepted: 19 October 2018; Published: 24 October 2018

Abstract: The determination of the non-loading condition of the rail cable shifting (RCS) system, which consists of the main cables, hangers, and rail cables, is the premise of girder erection for long-span suspension bridges. An analytical form-finding analysis model of the shifting system is established according to the basic assumptions of flexible cable structures. Herein, the rail cable is discretized into segmental linear cable elements and the main cable is discretized into segmental catenary elements. Moreover, the calculation and analysis equations of each member and their iterative solutions are derived by taking the elastic elongation of the sling into account. In addition, by taking the girder construction of the Aizhai suspension bridge as the engineering background, a global scale model of the RCS system is designed and manufactured. The test system and working conditions are also established. The comparison between the test results and analytical results shows the presented analytical method is correct and effective. The process is simplified in the analytical method, and the computational results and precision satisfy practical engineering requirements. In addition, the proposed method is suitable for application in the computation analysis of similar structures.

Keywords: suspension bridge; girder construction; RCS process; form-ﬁnding analysis; model test

1. Introduction Suspension bridges [1–7] are widely used to cross long-span barriers in engineering; their mechanical behaviors during construction have attracted the special attention of researchers [8–15]. Girder erection is the key and difficult step for the construction of the long-span suspension bridges in mountainous areas. To facilitate transportation and erection, structural types of girders are often designed as steel trusses. Mature construction methods include the deck crane construction method [16], and the cable crane construction method, among others [8,9]. Rail cable shifting (RCS) is a new, recently invented method, and it was firstly used in the successful erection of the girder of the Aizhai Bridge [17–23]. The erection of the 1000.5-m steel truss girder required only two and a half months [16]. With RCS, the construction period is greatly shortened, and the amount of steel used for the steel truss girder is significantly reduced. However, this technology is still new in terms of practical use, and its widespread promotion requires the accumulation of more experience. Meanwhile, knowledge and research on this technology remains to be further developed, for example to learn whether the analysis process of RCS system can be further simplified, whether the technological process can be further optimized, and whether the equipment and operations can be standardized. All of these restrict the widespread promotion and application of RCS technology. In the presented technology, the shifting system is a double-layer flexible suspended cable system that consists of the main cables, hangers, saddles, and rail cables. With the tension of the track cable and the cable system under stress, the system will undergo obvious deformation, through which the load will be distributed according to the stiffness in the cable system [24–32]. For the long and

Appl. Sci. 2018, 8, 2033; doi:10.3390/app8112033 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 2033 2 of 13

Appl. Sci. 2018, 8, x FOR PEER REVIEW 2 of 13 flexible sling structure, deviation and elongation will appear in the process of the rail cable tensioning, andthrough variations which the should load notwill bebe ignored.distributed The according rail cable to forcethe stiffness during in the thetensioning cable system process [24–32]. of For the rope the long and flexible sling structure, deviation and elongation will appear in the process of the rail sling is unknown, and an implicit solution needs to be derived with respect to the deformation cable tensioning, and variations should not be ignored. The rail cable force during the tensioning coordination relation of the system. These issues all these bring difficulties and complexity to the process of the rope sling is unknown, and an implicit solution needs to be derived with respect to the static analysis. During the process of the research and development, the finite element method is deformation coordination relation of the system. These issues all these bring difficulties and usedcomplexity to compute to thethe static main analysis. cable andDuring the the rail proc cableess that of arethe regardedresearch and as segmentaldevelopment, catenaries, the finite and the computationalelement method process is used isto complicated.compute the main Moreover, cable and the the rail rail cable cable has that a largeare regarded pretension as segmental force, which is differentcatenaries, from andthe the force computational of the suspension process is cable complicated. segment obtainedMoreover, when the rail only cable the has weight a large force and slingpretension force onforce, both which ends is are different considered. from the force of the suspension cable segment obtained when onlyIn the this weight paper, force to simplifyand sling theforce form-finding on both ends [33 are] analysisconsidered. of the RCS process under the non-loading condition,In this an paper, analytical to simplify method the is developedform-finding to establish[33] analysis the globalof the mechanicalRCS process model under of the the main cable,non-loading sling, and condition, rail cable an andanalytical to solve method the equation, is developed under to the establish condition the thatglobal the mechanical sling force model is unknown. Here,of the themain basic cable, assumptions sling, and rail of the cable mechanical and to solve analysis the equation, of flexible under cables theare condition introduced that the and sling the elastic elongationforce is unknown. of the sling Here, is considered.the basic assumptions The comparisons of the mechanical among the analyticalanalysis of results, flexible the cables finite are element results,introduced and and model the test elastic results elongation show the of presentedthe sling analyticalis considered. method The iscomparisons correct and among it can bethe used in analytical results, the finite element results, and model test results show the presented analytical the computation and analysis of similar structures. method is correct and it can be used in the computation and analysis of similar structures. 2. Mechanical Characteristics of the RCS System under the Non-Loading Condition 2. Mechanical Characteristics of the RCS System under the Non-Loading Condition 2.1. Brief Introduction to RCS Technology 2.1. Brief Introduction to RCS Technology In RCS (shown in Figure1), the main cable and permanent sling are taken as the supports. In RCS (shown in Figure 1), the main cable and permanent sling are taken as the supports. A A shifting cableway with a horizontal pretension force is set below the sling, which is connected shifting cableway with a horizontal pretension force is set below the sling, which is connected with with the sling through the saddle. The steel truss girder segments are separately assembled on both the sling through the saddle. The steel truss girder segments are separately assembled on both sides. sides.A single A singlesegment segment is transported is transported longitudinally longitudinally via girder via girdertransporting transporting vehicles vehicles along the along girder the girder transportingtransporting cableway cableway to to the the position position beneath beneath the permanent the permanent sling. sling.The steel The truss steel girder truss segment girder segmentis islifted lifted by by a cable a cable crane crane or orother other hoisting hoisting equipment equipment to exit to exitthe vehi thecle. vehicle. Subsequently, Subsequently, the segments the segments areare dockeddocked and pin-connected pin-connected to to the the sling. sling. A symme A symmetricaltrical construction construction segment segment by segment by segment then is then is performedperformed from the the mid-span mid-span to to the the two two sides sides until until the the full full bridge bridge is connected. is connected.

main cable sling

rail cable girder transporting vehicles

saddle

steel truss girder

Figure 1. The rail cable shifting system. Figure 1. The rail cable shifting system. The technology allows for the transportation of the entire stiffening girder segment, reduces the The technology allows for the transportation of the entire stiffening girder segment, reduces the aerial work of the girder segment, shortens the construction period, reduces the construction costs, aerialimproves work the of construction the girdersegment, quality and shortens safety, theand constructionprovides an advanced, period, reduces economical, the construction and highly costs, improves the construction quality and safety, and provides an advanced, economical, and highly Appl. Sci. 2018, 8, 2033 3 of 13 efﬁcient construction for the extra-long-span suspension bridge in mountainous areas. A difﬁcult problem in the construction of the stiffening girder for the extra-long-span suspension bridge in mountainous areas is solved. This method is a major breakthrough for the construction of long-span suspension bridges in mountainous areas. Additionally, construction machines and tools are conventional, and are easy to operate and control. Moreover, it is also suitable for the girder construction of the half-through and the through arch bridges, and thus involves great adaptability and promotion value.

2.2. Mechanical Model of the RCS System under Non-Loading Condition During the computation of the conventional suspension bridge construction process, the parameters of main cable are usually solved by the segmental catenary method. The sling force is determined from the completed bridge state, with backward disassembly to the cable finish state. The internal force, line shape, unstressed cable length, and sling length of the unloaded cable can be directly solved through iteration [34]. In the computation and analysis of conventional double-layer cable systems and cable dome structures, the hanger is often considered as being of rigid handling, or it is assumed to have infinite sling stiffness to ensure that the vertical displacements of the load-bearing cable and the stabilization cable are the same. This is inconsistent with the actual situation. In the finite element analysis of single-layer flexible cables, the two-node straight element, the two-node curve element [35], and the multi-node isoparametric element [36], among others, are usually used. Although there are certain limitations and deviations in the computation accuracy, computational efficiency, and applicable range of each element, the computational results in respective adaptive range can reach high accuracy and can be applied to engineering practice [37]. To simplify the form-finding analysis of the RCS process under the non-loading condition, the analytical method is used to establish the global mechanical model of the main cable, sling, and rail cable and to solve the equation, under the condition that the sling force is unknown. The basic assumptions of the mechanical analysis of flexible cables are introduced and the elastic elongation of the sling is considered. The main cable segment is simulated by a segmental catenary element, and the rail cable segment is simulated by a two-node straight pole element. The analytical results are compared with the finite element results and model test results. The non-loading condition is defined as the status after the erection of the main cable and sling, the installation of the saddle support and the rail cable, and the tensioning of the rail cable are completed. To simplify the computational model for the system under the non-loading condition, the following assumptions are introduced [34]:

(1) The ﬂexible cable can only be tensed but not be pressed or bent. (2) The stress-strain of the ﬂexible cable is consistent with Hooke’s theorem. (3) The cross-sectional area before deformation is used for the calculation of the tensile stiffness of the main cable, rail cable, and sling before and after stress. (4) The friction between the rail cable and the saddle is ignored, and the sling does not tilt after the rail cable tension is completed.

Then, a mechanical analytical model for the shifting system under the non-loading condition is established as shown in Figure2, where z represents the main cable, g represents the rail cable, and d represents the sling. To describe the analysis process, an inter-segmental model is extracted for analysis, as shown in Figure3. Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 13 Appl. Sci. 2018, 8, 2033 4 of 13 Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 13 Lz1 Lzi Lzi+1 Lzn

V z1 Lz1 Lzi Lzi+1 Lzn H z1 q1 V z1 z1 V zn z H z1 sz1 h A q1 szn H zn Az2 AVznzn z Az1 sz1 Azn-1 h szn H zn Az2 Azi Azi+1 Azi+2 szi+1 Azn-1 Azn Azi Azi+1 Azi+2 Y szi+1

Y sd1 sdi-1 sdi sdi+1 sdm X q3 q3 q3 q3 q3 sd1 sdi-1 sdi sdi+1 sdm X q3 q3 q3 q3 q3 V gm sgm V g1 m

g1 h q2 s V gm Bgi Bgi+1 Bgi+2 Bgm-1sgm

g1 H gm m H V g1 Bg2 Bgm q2 sg1 h Bg1 Bgi Bgi+1 Bgi+2 g1 Lg1 Lgi Lgi+1 Bgm-1Lgm H gm H Bg2 Bgm Bg1 Lg1 Lgi Lgi+1 Lgm Figure 2. Mechanical analytical model of the rail cable shifting (RCS) system under the non-loading Figure 2. Mechanical analytical model of the rail cable shifting (RCS) system under the Figurecondition. 2. Mechanical analytical model of the rail cable shifting (RCS) system under the non-loading non-loading condition. condition. Lzi Lzi+1

zi Lzi+1 zi+1 L V zi+1 zi-1 zi V h szi+1 h zi zi+1 1 s A zi+1H zi+1 q V zi+1 zi-1

zi zi+2 V A h szi+1 h zi szi Azi+1 H zi-1 A q1 H zi H zi+1 Azi+2 Pzi Azi+1 V zi zi-1 Azi q3 H H zi Pzi Azi+1 sdi V zi Y q3 sdi V gi Y X V gi H gi gi X gi+1 h q2 sgi Bgi+2 h H gi gi gi gi+1 B gi+1 h q2 sgi Bgi+1 Bgi+2H h Bgi+1 H gi-1 Pgi Bgi V gi+1 V gi-1 Bgi+1 H gi+1 Bgi+1 H gi-1 Lgi Lgi+1 Pgi V gi-1 V gi+1 Lgi Lgi+1 Figure 3. Mechanical discrete model of the RCS system under the non-loading condition. Figure 3. Mechanical discrete model of the RCS system under the non-loading condition. 2.2.1. ForceFigure Analysis 3. Mechanical of the Main discrete Cable model of the RCS system under the non-loading condition. 2.2.1. Force Analysis of the Main Cable As shown in Figure3, the main cable segment is discretized into two segments: Azi-Azi+1 and 2.2.1.As Force shown Analysis in Figure of the 3, Main the mainCable cable segment is discretized into two segments: Azi-Azi+1 and Azi+1-Azi+2 under the non-loading condition. In addition, the corresponding segmental catenary Azi+1-Azi+2 under the non-loading condition. In addition, the corresponding segmental catenary elementAs shown equations in Figure [34] can 3, bethe established main cable and segment solved is to discretized obtain lzi, h ziinto, lzi+1 two, hzi +1segments:, Szi: Azi-Azi+1 and element equations [34] can be established and solved to obtain lzi, hzi, lzi+1, hzi+1, Szi: Azi+1-Azi+2 under the non-loading condition. In addition, the corresponding segmental catenary q Hziszi Hzi q 2 element equationsl =H s[34]− canH be{ lnestablished[V + H and2 + solvedV2 ] − ln to[V obtain− q lszi, h+zi, lzi+1H, 2hzi+1+, ( VSzi: − q s ) ]} (1) zi = EziAzi − q zi zi + zi2 + zi2 − zi − 1 zi + zi2 + zi − 1 zi 2 lzi 1 1 1 {ln[Vzi H zi Vzi ] ln[Vzi q1szi H zi (Vzi q1szi ) ]} E A q (1) = H zi1 s1zi − H1zi + 2 + 2 − − + 2 + − 2 lzi q s{ln[2 −V2ziV s H zi qVzi ] ln[Vzi qq1szi H zi (Vzi q1szi ) ]} E A q 1 zi zi zi 1 2 2 2 2 (1) 1 1hzi = 1 − [ Hzi + Vzi − Hzi + (Vzi − q1szi) ] (2) 2E1 A1 q1 2 − q1szi 2Vzi szi 1 2 2 2 2 h = − lzi [= HXzi −+VXzi−−1 H + (V − q s ) ] (2)(3) zi 22−E A q zi zi zi zi 1 zi = q1szi 12V1zi szi − 11 2 + 2 − 2 + − 2 hzi hzi[ =HYzi −VYzi −1 H zi (Vzi q1szi ) ] (2)(4) 2E1 A1 q1 = − lzi X zi X zi−1 (3) where E1 and A1 are elastic modulus and cross-sectional area of the main cable, respectively; q1 is the l = X − X self weight per unit length of the main cable;zi S=zi iszi the− segmentalzi−1 cable length after the sling installed(3) hzi Yzi Yzi−1 (4) to the unloaded cable; and i = 1~n. Other parameters= − are shown in Figure2. hzi Yzi Yzi−1 (4) Appl. Sci. 2018, 8, 2033 5 of 13

Based on the equilibrium state of Azi node and the sling, two additional equations can be established Hzi−1 = Hzi (5)

Vzi = Vzi−1 − Pzi − q1szi (6)

where Pzi is the force of the sling at the Azi node.

2.2.2. Force Analysis of the Rail Cable The diameter, strength grade and type of the rail cable can be determined mainly from its stress level, supporting condition, safety factor, and climbing angle of shifting. As the support of rail cable lies on the saddle connected to the lower anchor point of the sling, the linear sag under non-loading conditions is small. In addition, due to the large pretension force of the rail cable, the inﬂuence of its self weight on the line shape of the cable segment is limited. Meanwhile, the assumption that each segment is in line after completion of the rail cable tension is made, which simpliﬁes the force analysis of subsequent shifting process. Therefore, the rail cable can be discretized into the connection of multiple straight poles. A two-node straight pole element is used to simulate the force for the rail cable segment under the non-loading state, and an analytical diagram shown in Figure4 is obtained. Furthermore, the internal forces of the straight cable segment can be directly obtained by solving the following static equilibrium equations and geometric equations,

Hgi−1 lgi = Sgi (7) Tgi−1

Vgi−1 hgi = Sgi (8) Tgi−1

TgiSgi ∆(Sgi) = (9) E2 A2

q 2 2 Tgi = Vgi + Hgi (10)

Hgi−1 = Hgi (11)

q 2 2 Sgi = lgi + hgi (12)

Vgi = Vgi−1 − Pgi + q2sgi (13)

lgi = Xgi − Xgi−1 (14)

hgi = Ygi − Ygi−1 (15)

where E2 and A2 are respectively the elastic modulus and cross-sectional area of the rail cable; q2 is the self-weight per unit length of the rail cable; Sgi is the cable length of the rail cable segment; Pgi is the Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 13 sling force increment; and i = 1~m. Other parameters are shown in Figure4.

V gi Tgi Bgi+1 sgi q 2 H gi gi H gi-1 Bgi h

V gi-1 Tgi-1 lgi

Figure 4. Discretization diagram for two-node straight pole element of the rail cable. Figure 4. Discretization diagram for two-node straight pole element of the rail cable.

2.2.3. Force Analysis of the Sling The force of sling is relatively simple. According to Assumption 4, the tilting of the sling is not taken into account under the non-loading condition. In an actual construction, a secondary adjustment can be applied after the pre-biasing and the tensioning of the rail cable to keep the sling being vertical after the rail cable is tensioned. The coordination relationships between the sling force and the deformation are as follows: = + + Pzi Pgi q3sdi W , (16)

P − = gi + Yzi Ygi sdi , (17) E3 A3 where W is the constant weight of the saddle; E3 and A3 are the elastic modulus and cross-sectional area of the sling, respectively; q3 is the self-weight per unit length of the sling; and Sdi is the unstressed cable length of the sling. Other parameters are shown in Figure 2.

2.3. Solution Analysis of the System

Before the tensioning of the rail cable, the longitudinal position lg of the two anchorage points, the pretension force Tg0 and the unstressed cable length of the sling are known. After the installation of the sling, the cable force, line shape, cable tower distance lz, and the height difference hz can be computed according to actual conditions. On this basis, other unknown variables can be solved by an iterative method. The iterative scheme is shown below:

(1) Assume the horizontal force Hg0 and the elevation hg0 at the left support of the rail cable; the horizontal force Hz0 at the left support of the main cable, and the increment of the cable force of the sling Pg1; (2) Obtain Hgi, Hzi from Equations (4) and (11); (3) Substitute them into Equations (10), (13), (7), (8), and (9); then Vg0, Vg1, lg1, hg1, and Tg1 are obtained; (4) From the coordination relation of deformation, Yg1, Yz1 and hz1 are obtained by combing with Equations (15), (17), and (4), respectively; (5) Substitute Yg1, Yz1 and hz1 into Equations (1) and (2); Vz0 and lz1 can then be obtained; (6) Substitute Vz0 and lz1 into Equations (16), (6), and (13), Pz1; Vz1 and Vg2 can then be obtained; (7) Substitute Pz1; Vz1 and Vg2 the results of step 6 into Equations (2), (1), (10), (7), and (8); hz2, lz2, Tg2, lg2, and hg2 can be acquired, respectively; (8) Substitute them into Equations (4), (15), and (17); Yz2, Yg2, and Pg2are then obtained; (9) Following the similar iterative manner, all values of lgi, lzi, hgi and hzi are obtained; n − ≤ ε ε −6 (10) Check convergence conditions Δz1 = i=1 lzi lz ( = 1.0 × 10 is the given error n − ≤ ε m − ≤ ε m − ≤ ε limit), Δz2 = i=1 hzi hz , Δg1 = i=1 lgi lg , and Δg2 = i=1 hgi hg 0 . If they are Appl. Sci. 2018, 8, 2033 6 of 13

Pzi = Pgi + q3sdi + W, (16)

Pgi Yzi − Ygi = + sdi, (17) E3 A3 where W is the constant weight of the saddle; E3 and A3 are the elastic modulus and cross-sectional area of the sling, respectively; q3 is the self-weight per unit length of the sling; and Sdi is the unstressed cable length of the sling. Other parameters are shown in Figure2.

2.3. Solution Analysis of the System

(3) Substitute them into Equations (10), (13), (7), (8), and (9); then Vg0, Vg1, lg1, hg1, and Tg1 are obtained;

(4) From the coordination relation of deformation, Yg1, Yz1 and hz1 are obtained by combing with Equations (15), (17), and (4), respectively;

(5) Substitute Yg1, Yz1 and hz1 into Equations (1) and (2); Vz0 and lz1 can then be obtained; (6) Substitute Vz0 and lz1 into Equations (16), (6), and (13), Pz1; Vz1 and Vg2 can then be obtained; (7) Substitute Pz1; Vz1 and Vg2 the results of step 6 into Equations (2), (1), (10), (7), and (8); hz2, lz2, Tg2, lg2, and hg2 can be acquired, respectively; (8) Substitute them into Equations (4), (15), and (17); Yz2, Yg2, and Pg2 are then obtained; (9) Following the similar iterative manner, all values of lgi, lzi, hgi and hzi are obtained; n −6 (10) Check convergence conditions ∆z1 = |∑i=1 lzi| − lz ≤ ε (ε = 1.0 × 10 is the given error limit), n m m ∆z2 = |∑i=1 hzi| − hz ≤ ε, ∆g1 = ∑i=1 lgi − lg ≤ ε, and ∆g2 = ∑i=1 hgi − hg0 ≤ ε. If they are satisﬁed, go to the next step. If not, let Hz0 = Hz0 + ∆z1/DHz, Pg1 = Pg1 + ∆z2/DPg, Hg0 = Hg0 + ∆g1/DHg and hg0 = hg0 + ∆g2/Dhg, and return to step 1, iteratively recalculating, where DHz, DPg, DHg, and Dhg are respectively the ﬁrst derivatives of Hz0, Pg1, Hg0, and hg0 [38].

Then, all the sling forces and the coordinates of each node are determined. Substituting the results into Equations (9) and (12), the unstressed cable length and the anchorage point elevation of the rail cable can be obtained. A ﬂow diagram for the calculation is shown in Figure5. Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 13 satisfied, go to the next step. If not, let Hz0 = Hz0 + Δz1/DHz, Pg1 = Pg1 + Δz2/DPg, Hg0 = Hg0 + Δg1/DHg and hg0 = hg0+Δg2/Dhg, and return to step 1, iteratively recalculating, where DHz, DPg, DHg, and Dhg are respectively the first derivatives of Hz0, Pg1, Hg0, and hg0 [38]. Then, all the sling forces and the coordinates of each node are determined. Substituting the resultsAppl. Sci. into2018 ,Equations8, 2033 (9) and (12), the unstressed cable length and the anchorage point elevation7 of of 13 the rail cable can be obtained. A flow diagram for the calculation is shown in Figure 5.

assume Hg0,hg0,Hz0,Pg1

obtain Hzi,Hgj（i=1~n, j=1~m）

analysis Vg0,Vg1,lg1,hg1,Tg1

obtain Yg1,Yz1,hz1

obtain Vz0,lz1,Pz1,Vz1,Vg2

i=i+1,j=j+1 obtain hz2,lz2,Tg2,hg2,lg2

Incremental step cycle

obtain Yz2,Yg2,Pg2

obtain lgi,lzi,hgi,hzi

n n m N check Δ= − ≤ε , Δ= − ≤ε , Δ= − ≤ε lziz1 ll hziz1  hh ggig1  hh i=1 i=1 i=1 Y

finish

Figure 5. TheThe flow ﬂow diagram diagram of calculations.

3. Global Global Model Model Test Test of the Moving Girder by Rail Cable The AizhaiAizhai suspensionsuspension bridge bridge (shown (shown in Figurein Figure6) is 6) a super-longis a super-long steelgirder steel suspensiongirder suspension bridge bridgewith a mainwith a cable main span cable of span 242 + of 1176 242 + + 116 1176 m, + which116 m, built which in thebuilt mountainous in the mountainous area of Hunan area of province, Hunan province,China. The China. bridge The spans bridge the u-shapedspans the Dehangu-shaped Grand Dehang Canyon, Grand more Canyon, than 500more m than deep. 500 The m middle deep. Thespan middle main cable span adoptsmain cable the vertical adopts spanthe vertical ratio of sp 1/9.6,an ratio the centralof 1/9.6, distance the central of main distance cable of is 27main m, cableand the is plane27 m, cableand the layout plane is adopted.cable layout The is bridge adopte adoptsd. The 71 bridge pairs ofadopts slings 71 with pairs the of standard slings with spacing the standardof slings beingspacing 14.5 of m.slings The being two main14.5 m. cable The towers two main adopt cable the towers reinforced adopt concrete the reinforced tower structures. concrete towerThe gravity structures. anchor The is usedgravity for anchor the anchor is used at the for Jishou the anchor side, while at the the Jishou tunnel side, anchor while is used the fortunnel the anchor at the Chadong side. There are 69 main girder sections for the whole bridge, which are 7.5 m in height, 14.5 m in length, 27 m in width, and 124.5 t in weight. Appl.Appl. Sci. Sci. 2018 2018, 8, ,8 x, xFOR FOR PEER PEER REVIEW REVIEW 8 8of of 13 13

Appl.anchoranchor Sci. is2018 is used used, 8, 2033 for for the the anchor anchor at at the the Chadong Chadong side side. .There There are are 69 69 main main girder girder sections sections for for the the whole 8whole of 13 bridge,bridge, which which are are 7.5 7.5 m m in in height, height, 14.5 14.5 m m in in length, length, 27 27 m m in in width, width, and and 124.5 124.5 t tin in weight. weight.

(a(a) )Side Side view view of of global global bridge bridge (b (b) )layout layout of of bridge bridge deck deck

FigureFigure 6. 6. The The schematic schematic views views of of Aizhai Aizhai long-span long-span suspension suspension bridge. bridge.

TheThe rail railrail cable cablecable launching launchinglaunching method method is is is the the the first first ﬁrst used used used in in inthe the the Aizhai Aizhai Aizhai suspension suspension suspension bridge bridge bridge to to realize torealize realize the the horizontalthehorizontal horizontal movementmovement movement ofof the ofthe the girdergirder girder segment.segment. segment. TheThe The horizontalhorizontal horizontal cablewaycableway utilizes utilizes eight eighteight relatively relatively independentindependent 60-ZZZ-1570-type 60-ZZZ-1570-type60-ZZZ-1570-type sealed sealedsealed steel steelsteel wire wirewire rope ropesropes sas as rail rail cable. cable. One OneOne side sideside of ofof the thethe railrail cable cable is is anchoredanchored by by thethe the OVM-CPSOVM-CPS OVM-CPS shearshear shear forceforce force dispersing dispersing dispersing the the the type type type rock rock rock anchor anchor anchor system, system, system, while while while the the the other other other side side side is isanchoredis anchoredanchored in balance inin balancebalance weight. weight.weight. The pretension TheThe pretensionpretension force of forcforc ae single e ofof aa rail singlesingle cable railrail is 105 cablecable t. The isis 105 global105 t.t. The laboratoryThe globalglobal laboratoryexperimentallaboratory experimental experimental model was model designedmodel was was and designed designed manufactured and and manufactured manufactured with a geometrical with with a a reduced-scalegeometrical geometrical reduced-scale reduced-scale ratio of 1:33. Theratioratio span of of 1:33. 1:33. size The ofThe the span span main size size cable of of ofthe the the main main model cable cable is 7.333of of the the + 35.636model model +is is3.515 7.333 7.333 m, + + its35.636 35.636 center + + 3.515 spacing3.515 m, m, isits its 818 center center mm, spacingandspacing the longitudinalis is 818 818 mm, mm, and and spacing the the longitudinal longitudinal of the sling spacing is spacing 439.4 mm,of of the the as sling sling shown is is 439.4 in439.4 Figure mm, mm,7 as. as shown shown in in Figure Figure 7. 7.

FigureFigure 7. 7. The The global global model model of of the the Aizhai Aizhai suspension suspension bridge bridge (unit: (unit: mm). mm). The test model [39] mainly consists of 2 anchorage systems, 2 main cables, 68 pairs of slings and TheThe test test model model [39] [39] mainly mainly consists consists of of 2 2 anchorag anchoragee systems, systems, 2 2 main main cables, cables, 68 68 pairs pairs of of slings slings and and cable clamps, 3 pairs of rock anchor cables, 2 cable towers and cable towers buttresses, rail cables, rail cablecable clamps, clamps, 3 3 pairs pairs of of rock rock anchor anchor cables, cables, 2 2 cabl cable etowers towers and and cable cable towers towers buttresses, buttresses, rail rail cables, cables, tensioning devices, and test systems. According to the principle of stress equivalence and similar axial railrail tensioning tensioning devices, devices, and and test test systems. systems. Accordin Accordingg to to the the principle principle of of stress stress equivalence equivalence and and similar similar stiffness, the parameters of the rail cable are designed. Four 2.5-mm-diameter high-strength galvanized axialaxial stiffness, stiffness, the the parameters parameters of of the the rail rail cabl cablee are are designed. designed. Four Four 2.5- 2.5-mm-diametermm-diameter high-strength high-strength steel wire ropes are selected to replace rail cables. The rail cable tensioning device is designed to adjust galvanizedgalvanized steel steel wire wire ropes ropes are are selected selected to to replac replacee rail rail cables. cables. The The rail rail cable cable tensioning tensioning device device is is the relative movement of the screws and nuts to achieve the tensioning of rail cables. Due to the fact designeddesigned to to adjust adjust the the relative relative movement movement of of the the screws screws and and nuts nuts to to achieve achieve the the tensioning tensioning of of rail rail that in suspended and cable-stayed bridges the cable forces are responsible for the strong variations cables.cables. Due Due to to the the fact fact that that in in suspended suspended and and cablecable-stayedstayed bridges bridges the the cable cable forces forces are are responsible responsible for for in the static and dynamic performance of the system [40], a pressure sensor is used to monitor the thethe strong strong variations variations in in the the static static and and dynamic dynamic perf performanceormance of of the the system system [40], [40], a a pressure pressure sensor sensor is is changes of their cable forces in real time. usedused to to monitor monitor the the changes changes of of their their cable cable forces forces in in real real time. time. On the basis of the global model test design, the finite element software MIDAS Civil is used to OnOn the the basis basis of of the the global global model model test test design, design, the the finite finite element element software software MIDAS MIDAS Civil Civil is is used used to to establish the full bridge numerical modeling on the initial state. In the test process, the no-load stage establishestablish the the full full bridge bridge numerical numerical modeling modeling on on the the initial initial state. state. In In the the test test process, process, the the no-load no-load stage stage of the cable is ensured to be consistent with the finite element analysis model by adjusting the no-load ofof the the cable cable is is ensured ensured to to be be consistent consistent with with th the efinite finite element element analysis analysis model model by by adjusting adjusting the the stage of the test model. After the cable clamps, slings and saddles are installed, four rail cables are no-loadno-load stage stage of of the the test test model. model. After After the the cable cable clamps, clamps, slings slings and and saddles saddles are are installed, installed, four four rail rail tensioned, and the slings are in a vertical working condition. Then, the main cable force and line shape, cablescables are are tensioned, tensioned, and and the the slings slings are are in in a a vertical vertical working working condition. condition. Then, Then, the the main main cable cable force force rock anchor cable force, rail cable line shape, and sling cable under the tension working condition are tested to find out the changes. Appl. Sci. 2018, 8, 2033 9 of 13

The detailed contents of the MIDAS Civil modeling of the bridge are as follows. Firstly, the towers on both sides are simulated by using variable section beam element. Secondly, the main cable adopts the simulation of segmental catenary cable element, the cable clip is taken as the segmented node, the main cable between each adjacent node adopts the simulation of one catenary cable element, each sling adopts the simulation of one two-node direct cable element, and the sling connects with the main beam through the simulation of the beam element. Thirdly, the track rope is simulated by segmental straight cable element, and the saddle is taken as the segmentation node. The main cable between each adjacent node is simulated by one catenary cable element. Fourthly, the supporting rail rope of the sling saddle is connected with the sling, and its function is not considered in the model, only the load is added to the sling. Fifthly, the shear point of the main cable and the saddle is simulated with a rigid arm, and the change of the shear point is simulated by changing the position of the rigid arm. Sixthly, according to the initial state data of the empty cable tested by the overall model, the state model of the empty cable is established. Analysis and calculations of the bridge are carried out for assembly, and then for inversion to the state of the empty cable. It is checked whether the initial state of empty cable is convergent, and whether the relative displacement difference of the main cable in mid-span is small enough and meets the requirements of the accuracy of the overall model test. Seventhly, the load increment method is combined with the Newton–Raphson method to solve the problem.

4. Testing Results of Model Test After the completion of the rail cable tensioning, the ﬁnite element results, experimental results and the analytical results are compared. The responses of several representative locations are compared in Table1, and the cable forces at the anchor points of the main cable are shown in Table2.

Table 1. Effects of the working condition of rail cable tensioning on the systematical line shape.

Main Cable Displacement Main Cable Displacement Main Cable Displacement at 1/4 Span on the at 1/4 Span on the Items at Mid-Span Jishou Side Chadong Side X Direction/ Y Direction/ X Direction/ Y Direction/ X Direction/ Y Direction/ mm mm mm mm mm mm Finite element solution 1.4 28.6 −0.6 9.0 3.6 7.7 Model test measured value 1.5 27.5 −0.5 8.9 3.5 7.5 Analytical calculation results 1.6 26.3 −0.6 8.7 3.7 7.2 Deviation of the ﬁnite element solution 6.67 4.00 20.00 1.12 2.86 2.67 from the measured value (%) Deviation of the analytical method 6.67 4.36 20.00 2.25 5.71 4.00 from the measured value (%) Note: The measured value is the average of the upstream and downstream measuring points. The value is negative when the X-direction measuring point moves to Jishou side and when the Y-direction measuring point moves to the upstream, and vice versa.

Table 2. Effects of the working condition of rail cable tensioning on the force of main cable.

Model Test Deviation of the Finite Deviation of the Analytical Main Cable Force at Finite Element Analytical Measured Element Solution from Method from the Measured the Anchor Point Solution/kN Results/kN Value/kN the Measured Value/% Value/% Jishou side 61.7 59.9 61.3 3.01 2.34 Chadong side 58.5 57.1 58.2 2.45 1.93

From Table1, it can be seen that the deviation of the results gained by the analytical method from measured values and finite element values in the Y direction is small, the maximum deviation values are 4.36% and 4.00%, respectively. It is shown in Table2 that the deviation of the main cable force at the anchorage point is small, with a maximum of 2.34%. The deviations of the main cable displacement of the mid-span, the 1/4 span on the Jishou side, and the 1/4 span on the Chadong side in X-direction are relatively large, mainly due to the small deformation value in the X direction and the test system accuracy. The comparison between the test results and the analytical results shows that the presented method can be applied to the form-finding computational analysis of the RCS system Appl. Sci. 2018, 8, 2033 10 of 13 under the non-loading condition, which can simplify the analysis of the RCS system. The deviations of the simplified analytical results with the finite element results and model test results are less than 5%, which validates that the proposed method can be used in engineering practice.

5. Conclusions With the application of the basic assumptions of ﬂexible cables, an analytical form-ﬁnding analysis model of the rail cable shifting system is proposed. The rail cable and main cable are seen as segmental linear cable elements and catenary elements, respectively. The elastic elongation of the sling is considered to determine the analytical equation of each member and the iterative solutions. Besides the experimental data of the girder construction of the Aizhai suspension bridge, the compared results verify the accuracy and effectiveness of the presented method. The main conclusions are drawn as follows.

(1) The global mechanical analytical model for the main cable, slings, and rail cables of the RCS system under the non-loading condition is established through the theoretical derivation. The equilibrium equation established from the position after structural deformation can be used to perform the geometrical nonlinear analysis of the structural large deformation. The simplified calculation and analysis process of the RCS system achieve sufficient accuracy, which thus can be applied in engineering. (2) Under the condition that the initial geometric state of the RCS system is unknown, the true internal force and geometry of the rail cable can be iteratively computed by assuming the initial force or configuration of the rail cable. (3) The designed rail cable tensioning and testing device meets the requirements of the model test. The measured results of the main cable line shape and main cable force in the model test are in good agreement with the theoretical results.

Author Contributions: Q.P.: Conceptualization, Data curation, Formal analysis; D.Y.: Funding Acquisition; Z.Y.: Software, Writing-Review and Editing. Funding: The National Natural Science Foundation of China (Grant no. 51678069); the Key Discipline Fund Project of Civil Engineering of Changsha University of Sciences and Technology (Grant no. 15KB03). Acknowledgments: The model test of Aizhai suspension bridge is performed in the Changsha University of Science and Technology, the workplace supported by the key laboratory for safety control of bridge engineering, ministry of education oh Hunan province is acknowledged. Conﬂicts of Interest: The authors declared that they have no conﬂicts of interest to this work.

Glossary

A1 cross-sectional area of the main cable A2 cross-sectional area of the rail cable A3 cross-sectional area of the sling Azi node locations of main cable segment E1 elastic modulus of the main cable E2 elastic modulus of the rail cable E3 elastic modulus of the sling Hg0 horizontal force pretension force of the rail cable at the left support hg0 elevation at the left support of the rail cable hg1 vertical height between left anchorage points of rail cable and node Ag2 hi height difference of cable tower Hz0 horizontal force at the left support of the main cable hz1 height difference between the top of left cable tower and node Az2 hzi vertical length of main cable segment lg longitudinal position of the two anchorage points lg1 horizontal length between left anchorage points of rail cable and node Ag2 Appl. Sci. 2018, 8, 2033 11 of 13

lz cable tower distance lzi horizontal length of main cable segment Pg1 increment of the cable force of the sling Pgi sling force increment Pzi force of the sling at Azi node q1 self-weight per unit length of the main cable q2 self-weight per unit length of the rail cable q3 self-weight per unit length of the sling Sdi unstressed cable length of the sling Sgi cable length of the rail cable segment Szi segmental cable length after sling installed to unloaded cable Tg0 pretension force of the rail cable at the left support Tg1 vertical force at the left support of the rail cable Vg0 vertical force pretension force of the rail cable at the left support Vg1 vertical force between left anchorage points of rail cable and node Ag2 W constant weight of the saddle Yg1 vertical displacement between left anchorage points of rail cable and node Ag2 Yz1 vertical displacement between the top of left cable tower and node Az2 ∆g1 convergence condition ∆g2 convergence condition ∆z1 convergence condition ∆z2 convergence condition ε given error limit

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