Research Article An Analytical Solution for Lateral Buckling Critical Load Calculation of Leaning-Type Arch Bridge

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1 Mathematical Problems in Engineering, Article ID , 4 pages Research Article An Analytical Soltion for Lateral Bckling Critical Load Calclation of Leaning-Type Arch Bridge Ai-rong Li, Yong-hi Hang, Qi-cai Y, and Ri Rao Gangzho University-Tamkang University Joint Research Center for Engineering Strctre Disaster Prevention and Control, Gangzho University, Gangzho 56, China Correspondence shold be addressed to Yong-hi Hang; knll@foxmail.com Received 4 March 24; Revised May 24; Accepted May 24; Pblished 25 May 24 Academic Editor: Sarp Adali Copyright 24 Ai-rong Li et al. This is an open access article distribted nder the Creative Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited. An analytical soltion for lateral bckling critical load of leaning-type arch bridge was presented in this paper. New tangential and radial bckling models of the transverse brace between the main and stable arch ribs are established. Based on the Ritz method, the analytical soltion for lateral bckling critical load of the leaning-type arch bridge with different central angles of main arch ribs and leaning arch ribs nder different bondary conditions is derived for the first time. Comparison between the analytical reslts and the FEM calclated reslts shows that the analytical soltion presented in this paper is sfficiently accrate. The parametric analysis reslts show that the lateral bckling critical load of the arch bridge with fixed bondary conditions is abot.4 to.6 times as large as that of the arch bridge with hinged bondary condition. The lateral bckling critical load increases by approximately 3.5% to 4.2% when stable arch ribs are added, and the critical load increases as the inclined angle of stable arch rib increases. The differences in the center angles of the main arch rib and the stable arch rib have little effect on the lateral bckling critical load.. Introdction Leaning-type arch bridge is a relatively new type of spatial tied-arch bridge developed from x-type arch bridge. It is composed of two load-bearing ribs (called main arch ribs) which are perpendiclar to the bridge deck and two leaning arch ribs (called stable arch ribs) on the sides of the main arch ribs. These two types of arch ribs constitte a space stable system when connected by transverse braces between them. This type of bridge is one of the most competitive rban bridges becaseitisstylishandniqe,givesanopenandclear view for drivers, and is economically efficient. As the vertical stiffness of the main arch rib is mch higher than that of the stablearchrib,themajorportionofthedeadandliveloads isspportedbythemainarchribs.stablearchribspports only a small portion of the live load, and its main fnction is to assre the lateral stability of the main arch rib. Becase there are no transverse braces placed between the main arch ribs, and the stable arch rib s contribtion to the improvement of the lateral stability of bridge is very limited, the lateral stability often becomes a key factor deciding the bridge s safety []. The world s first leaning-type arch bridge named BacdeRoadBridgewasbiltin992inBarcelona.Itwas designed by the Spanish architect, Santiago Calatrava. This bridge is 52 m long, 25.8 m wide, and the inclined angle of stable arch rib is 3. The vertical main arch ribs are hinged and the inclined stable arch ribs are fixed at both ends [2]. Over the past decades, leaning-type arch bridges have developed rapidly. At present, more than 2 leaning-type arch bridges have been bilt arond the world, most of them in China. Usally, fixed bondary conditions are adopted for both the main and stable arch ribs, sch as the Kangf Bridge in Yiyang city, China, Hanjiang North Bridge in Chaozho city, China, and Shengli Bridge in Jiangmen city, China (Figre ). However, nder special circmstances, in order to redce main arch rib s internal force and horizontal thrst or adapt to the local environmental conditions [3], hinged bondary condition is applied at the arch ends of the main arch, sch as Yfeng Bridge in Knshan city, China, and Danxi Bridge in Yiw city, China. At present, the derivation of the analytical soltion of the arch strctre s bckling problem is mainly focsed on the individal arch rib or conventional arch bridges with symmetrical arch ribs. Closed-form soltions for ot-of-plane

2 2 Mathematical Problems in Engineering (a) (b) Figre : Shenli Bridge in Jiangmen City of China. bckling of arches, sbjected to niform bending or niform compression, were obtained by Lim and Kang [4]. Pi et al. [5, 6] investigated the flexral torsional bckling of shallow arches with an open thin-walled section, sbjected to a radial load niformly distribted arond the arch axis. In-plane static and dynamic bckling of shallow pin-ended parabolic arches with a horizontal cable was investigated by Chen and Feng [7]. La et al. [8] presented an experimental investigation of the elastic-plastic ot-of-plane bckling response of roller bent circlar steel arches sbjected to a single force appliedtothecrown.bradfordandpi[9] derived a new nified analytical soltion for the lateral-torsional bckling load of pin-ended arches by acconting for the combined bending and axial compressive action. Do et al. [] investigated the sectional rigidities of trsses and the ot-ofplane bckling loads of pin-ended circlar steel tblar trss arches in niform axial compression and in niform bending. In addition, Jin and Zhao [] derived the lateral bckling critical load for X-type twin ribbed arch braced with transverse beams. Moreover, Li and L [2] derived the analytical soltion of the lateral bckling critical load for the conventional tied-arch bridge with transverse braces and discssed the effect of strctral parameters on lateral stability. Additionally, Xiang [3] presented a formla of the lateral bckling critical load for a half-throgh bridge at servicestageandtheeffectofbeamlocationforvehiclelane on the stability of this type of bridge was frther stdied. Compared with individal arch rib or conventional arch bridges with symmetrical arch ribs, the leaning-type arch rib system is different in several aspects inclding an inclined angle between the main and stable arch ribs, different stiffness of main and stable arch ribs, more significant spatial effect, and more complicated loading conditions. These factors reslt in difficlties in deriving the analytical soltion of the lateral bckling critical load. Especially that the mechanical model of transverse brace is very different from those of the conventional arch bridge when a lateral bckling occrs. Presently, researchers have condcted preliminary stdies on the lateral stability of leaning-type arch bridge based on finite element method (FEM) [4]. The derivation of the analytical soltion formla of the lateral bckling critical load of leaning-type arch ribs system has been preliminarily stdied by Li et al. [5]. However, in their stdies the inflence of the bridge deck system and hanger tensions on leaning-type arch bridge was not taken into consideration. The central angles of the main and stable arch ribs were assmed to be the same in their stdies, bt in fact they are different in some conditions. And in their stdies the inclined angles of stable arch ribs shold be less than 5 degrees, while the inclined angles larger than 5 degrees; the precision of the soltion is bad. Compared with the FEM calclation reslts, the relationship among different design parameters of the bridgecanbeclearlyrevealedbytheanalyticalsoltionand the optimization of strctre design is made easier; the complicated process of constrcting the FE model can also be simplified. In order to consider the inflence of the components of leaning-type arch bridge comprehensively, the global transverse brace deformation parameter β is considered and the central angles of the main arch ribs and stable arch ribs are assmed different, and the tangential and radial mechanical models of the transverse brace between the main and stable arch ribs are established, then the analytical soltion of the lateral bckling critical load for leaning-type arch rib system is derived in this paper. Compared with the analytical soltion derived by Li et al. [5], the one derived in this paper has a wider scope of applications. The deformation energy of the main arch rib, stable arch rib, and transverse braces between them are constrcted; and the potential energy cased by the hangers tensions are also established for both the main arch rib and stable arch rib nder fixed and hinged bondary conditions. And the total potential energy of the bridge in a bckling process is obtained thereafter. Based on the stationary energy principle, the analytical soltion for the lateral bckling critical load of leaning-type arch bridge is obtained. In the end, parametric analysis is carried ot in order to investigate how changes in certain design parameters wold affect the critical load of the leaning-type arch bridges, which coldleadtoanoptimmdesignofthistypeofbridge strctres. 2. Lateral Bckling Critical Load nder the Fixed Bondary Condition of Main Arch Rib 2.. Calclation Model of Leaning-Type Arch Bridge. The following assmptions are made in the derivation process:

3 Mathematical Problems in Engineering 3 z y φ w v d h L R q x φ b(x) z x d b Transverse brace Stable arch rib Main arch rib Bridge deck (a) Elevation view (b) Side view (c) Plan view Figre 2: Calclation sketch of leaning-type arch bridge. () the main arch ribs and the stable arch ribs are fixed in their ends; (2) the axis of main and stable arch ribs are arcshaped crves; (3) the stiffness of arch ribs and transverse braces are constants; (4) the axial deformation of arch rib in the bckling process is neglected; (5) the external load is evenly and vertically distribted along the bridge deck and is transmitted to the arch rib via the hangers; and (6) the force acting on arch ribs satisfies the film tension assmption. The simplified calclation model of a leaning-type arch bridge is shown in Figre 2. In the figre, x-y-z is the global coordinate system and -V-w is the local coordinate system., V, andw represent the lateral, radial, and tangential displacement of arch ribs, respectively, φ represents the angle between the main arch rib and the stable arch rib, φ is the angle of a position in the arch ribs, is the central angle of the main arch ribs, isthecentralangleofthestable arch rib, and R istheradisofthemainandstablearchribs. b(x) is the distance between the main arch rib and the stable arch rib. In the arch crown position the distance is b and b(x) = b +2Rsin φ ( cos φ). The transverse braces between the main and stable arch ribs are eqidistant with a distance of d; the hangers are also arranged eqidistantly with a distance of d h. According to the basic assmption and calclation scheme, the relationship between the lateral displacement I, II 2 and the radial displacement VI, VII 2 ofthemainarchribs and the stable arch ribs nder the local coordinate is given, respectively, as I =, V I = y y b (x) β, +y 2 II 2 = y 2 y b (x) β sin φ + cos φ +y 2 V II 2 = y 2 y b (x) β cos φ + sin φ, +y 2 where represents the lateral displacement of the arch axis ofthemainarch,β represents the global torsional angle of the transverse brace cased by deformation of arch rib in radial plane, b(x) isthedistancebetweenthemainarchrib,andy and y 2 represent the distance from the contraflexre point of () φ M b 2 E II I II b(x) y2 y E II b III b θ 2 θ c θ c θ 2 θ M b E I I I Figre 3: Radial bending deformation of transverse brace. transverse brace s radial deformation to the main and stable arch ribs, respectively (shown in Figre 3). When there is a lateral deformation in the arch rib system, axial strain of the main arch is given as ε= VI R + dwi ds [(di ds ) + ( dvi 2 ds ) ], (2) where w I represents the tangential displacement of the main arch ribs nder the local coordinate. According to the basic assmption ε=, dw I ds = VI R 2 2 [(di ds ) +( dvi 2 ds ) ]. (3) Becase the arches are fixed at both ends, s dw I =,so ( VI s R )ds= 2 ( di 2 s ds ) ds + 2 ( dvi 2 s ds ) ds. (4) v

4 4 Mathematical Problems in Engineering Assming the ends of the main arch rib and the stable arch rib are perfectly fixed, the torsional angle and the lateral displacements of arch axis of the main and stable arch ribs are given, respectively, as θ =C ( cos 2πφ ) (5a) θ 2 =C 3 ( cos 2πφ ), (5b) =C 2 ( cos 2πφ ) (6a) 2 =C 2 ( cos 2πφ ). (6b) The lateral displacement of bridge deck system is given as d =C 4 ( cos 2πφ ). (7) The global torsional angle of the transverse brace cased by deformation of arch rib in radial plane is given as β=c 5 ( cos 2πφ ). (8) The above eqations satisfy the following displacement bondary conditions. () When φ=and φ=, θ =, θ =, = d = β=,and = d =β =. (2) When φ=and φ=, θ 2 =, θ 2 =, 2 =,and 2 =. The lateral deflection crvatres along the V axis of the main and stable arch ribs are given as [6] 2.2. Energy Eqations. The lateral deformation energy of a leaning-type arch bridge can be written as W=U I +UII +UI w +UII w +UI c +UII c +U bh +U bv +V H +U d +V, () where U I, UII, UI w, UII w, UI c,anduii c represent the total lateral bending deformation energy, torsional deformation energy, and local bending deformation energy of the main andstablearchribs,respectively;u bv and U bh are the bending deformation energy of the transverse braces in radial and tangential directions along the main arch rib s axis; V H is the elastic potential energy cased by the horizontal part of the tension of the hangers; U d represents the elastic potential energy of bridge deck system; and V represents the potential energy of external loading applied to the arch bridge. The total lateral bending deformation energy of the main and stable arch ribs are given in U I +UII = 2 EI I I s (K I )2 ds + 2 EII I II s 2 (K II )2 ds = 2 EI I I [C2 A + C2 2 R 3 A 2 + C C 2 R 2 A 3 ] + 2 EII I II [C2 2 R 3 A 2 + C2 3 R A 22 + C2 5 R A 23 + C 2C 3 R 2 A 24 + C 2C 5 R 2 A 25 + C 3C 5 R A 26], (2) K I = θ R d2 I ds 2, K II = θ 2 R d2 II 2 ds 2, where K I V and KII V represent the lateral deflection crvatre along V axis of the main and stable arch ribs respectively. The torsional deflection crvatres along w axes are also given as [6] K I w = dθ ds + d 2 I R ds, K II w = dθ 2 ds + d II 2 R ds, (9) () where K I w and KII w represent the torsional deflection crvatres along w axes of the main and stable arch ribs, respectively. where E I I I and EII I II are the lateral bending stiffness of the main and stable arch ribs. Consider the following: A = ( cos 2πφ 2 ) A 2 = ( 2π 4 ) cos 2 2πφ A 3 = 2( 2π 2 ) ( cos 2πφ A 2 = cos φ ( 2π 4 ) cos 2 2πφ A 22 = ( cos 2πφ 2 ) ) cos 2πφ

5 Mathematical Problems in Engineering 5 A 23 = 4 R 2 (2π b(x) 2 sin 2 φ ) ( + e 2 ) 2 cos 2 2πφ A 24 = 2 cos φ ( 2π 2 ) cos 2πφ 2b (x) sin φ A 25 = cos φ R(+e 2 ) ( 2π 2 ) ( 2π 2 ) cos 2πφ A 26 = 2b (x) sin φ 2 R(+e 2 ) (2π ) cos 2πφ ( cos 2πφ ) cos 2πφ ( cos 2πφ ). (3) The torsional deformation energy of the main and stable arch ribs is given in U I w +UII w = 2 GI T I s (K I w )2 ds + 2 GII T II s 2 (K II w )2 ds = 2 GI T I [ C2 R B + C2 2 R 3 B 2 + C C 2 R 2 B 3 ] + 2 GII T II [ C2 2 R 3 B 2 + C2 3 R B 22 + C2 5 R B 23 + C 2C 3 R 2 B 24 + C 2C 5 R 2 B 25 + C 3C 5 R B 26], (4) where G I T I and G II T II are the torsional stiffness of the main and stable arch ribs. Consider the following: B = ( 2πφ 2 ) sin 2 2πφ B 2 = ( 2π 2 ) sin 2 2πφ B 3 = 2( 2π 2 ) sin 2 2πφ B 2 = cos 2 φ ( 2π 4 ) sin 2 2πφ B 22 = ( 2π 2 ) sin 2 2πφ B 23 = 2 R 2 (2π b(x) 2 sin 2 φ ) ( + e 2 ) 2 sin 2 2πφ B 24 = 2 cos φ ( 2π 2 ) sin 2 2πφ b(x) γ M c2 γ 2 γ M c y γ γ 3 γ 4 γ 3 E b I bh γ E II I II E I I I M bh M bh2 Figre 4: Bending deformation of arch ribs and transverse braces. 2b (x) sin φ B 25 = cos φ 2π 2π sin 2πφ R(+e 2 ) 2b (x) sin φ B 26 = 2π 2π sin 2πφ R(+e 2 ) sin 2πφ. y 2 y b sin 2πφ (5) Transverse brace bending deformation occrs in the tangential direction along arch axis when there is a lateral bckling in the arch rib system as shown in Figre 4.Inthiscase, the tangential bending deformation energy of the single transverse brace is given as U bh = 2E b I bh y M 2 bh dy + 2E b I bh y 2 M 2 bh2 dy, (6) where E b I bh is transverse brace s bending stiffness along tangential direction of arch rib and y and y 2 represent the distances from the contraflexre point of transverse brace tangential deformation to the main and stable arch ribs, respectively. Therefore, the length of transverse brace is b(x) = y +y 2. From Figre 4, the tangential bending moment of transverse brace near the main and stable arch ribs is given by the sperposition principle. M bh = 4E bi bh b M bh2 = 2E bi bh b γ + 2E bi bh γ b 3, γ + 4E bi bh γ b 3, (7) where γ represents the tangential anglar rotation of the transverse brace at the intersection points between the main arch rib and the transverse brace and γ 3 represents the tangentialanglarrotationofthetransversebraceattheintersection points between the stable arch rib and the transverse brace. It can be frther derived as y = 2γ +γ 3 b (x), 3(γ +γ 3 ) y 2 = γ +2γ 3 b (x), 3(γ +γ 3 ) (8)

6 6 Mathematical Problems in Engineering b(x) canbereplacedbyaconstanth, which is the distance between the main and stable arch ribs at qarter span for calclation simplification. Althogh the length of transverse brace b(x) is not a constant, the length of the transverse brace h at qarter span is very close to the average length of the transverse braces of the entire bridge. Sch simplification can make the derivation process become simple. The vales of y and y 2 are closely related to the bending stiffness of arch ribs, panel length, bending stiffness, and length of transverse braces. De to the inclined angle and the different stiffness of leaning-type arch bridge s main and stable arch ribs, the tangential and radial deformation of the transverse brace between the main and stable arch ribs differ from that of the conventional arch bridges. It is not a simple S shape. For leaning-type arch bridges, bending stiffness of the main arch rib is larger than that of the stable arch rib, so y >y 2,asshowninFigre 4, as an exceptional case, for conventional arch bridges, y =y 2. In order to obtain the relationship between y, y 2, a tangential mechanical model of transverse brace along the arch axis is established when alateralbcklingoccrs,asshowninfigre 5(a); itscorresponding bending moment is shown in Figre 5(b), from which (9)canbegivenas M bh M bh2 = E I E b I I I bhd+6e I E II I I III b (x) cos φ 6E I E II I I III b (x) cos φ +EII E b I II I bhd cos φ =e, (9) where e is a constant. From (7)and(9), it can be obtainedthat M bh M bh2 = 2γ +γ 3 γ +2γ 3 =e. (2) When the tangential local deformation occrs in the main and stable arch ribs, assming the bending moments of the main and stable arch ribs along the radial direction are M c and M c2, respectively, the single-panel section arch rib s local bending energy of the main and stable arch ribs can be expressed as U I c + UII c = 2E I I I M 2 c dx + d = 6EI I I d 2E II I II M 2 c2 dx d (24) γ EIII II γ 2 4 d. If the local bending energy of each single-panel section arch rib is the same, the fll-arch-rib local bending deformation energy can be written as U I c +UII c = U I c s d ds + U II c s d ds = s 3E b I bh dh γ γ 2 ds + s It cold be obtained from Figre 4 that 3E b I bh dh γ 3γ 4 ds. (25) 2M c =M bh, 2M c2 =M bh2. (26) Therefore, from (23), (25), and (26), the local bending energy of arch rib and the tangential bending energy of transverse brace are obtained as Assming γ 3 =a γ,then a = γ 3 γ = 2 e 2e ; (2) ths, from (8)and(2), the relationship of y and y 2 can be obtained as y = 2+a +2a y 2. (22) Sbstitting (7)and(8)into(6), the eqation of tangential bending energy of the transverse braces in the fll arch rib range is given as U U bh = bh s d ds = 2E bi bh 9dh [ 2γ +γ 3 (4γ 2 s γ +γ +4γ γ 3 +γ 2 3 )ds 3 γ + +2γ 3 (γ 2 s 2 γ +γ +4γ γ 3 +4γ 2 3 )ds]. 3 (23) U bh +U I c +UII c = 3E bi bh (N dh η +η 2 )η γ 2 ds s where D = 3n D 2 = 3n + 3E bi bh dh s 2 (N 2 η 3 +η 4 )η 3 γ 2 ds =E b I bh C 2 2 R 3 D +E b I bh C 2 2 R 3 D 2, α bs (N η +η 2 )η α bs (N 2 2η 3 +η 4 )η 3 γ =γ +γ 2, γ =γ 3 +γ 4, R 2 ( 2π 2 ) R 2 ( 2π 2 ) η = γ γ = +(de b I bh ) / (2hE I I I ), sin 2 2πφ, sin 2 2πφ, (27)

7 Mathematical Problems in Engineering 7 d/2 E I I I q M bh E b I bh b E II I II φ M bh2 φ (a) Mechanical model (b) Bending moment diagram Figre 5: Mechanical model and bending moment diagram in tangential direction. η 2 = γ 2 γ = + (2hE I I I )/(de bi bh ), η 3 = γ 3 γ = +(de b I bh )/(2hE II I II ), η 4 = γ 4 γ = + (2hE II I II )/(de bi bh ), N = 2 27 (a +2) (a +) (4 + 4a +a 2 ), N 2 = 2 27 (/a +2) (/a +) ( a ). a (28) The mechanical model and moment diagram of arch ribs and transverse brace in radial direction are shown in Figre 3. And the radial deformation energy of a single transverse brace can be written as U bv = 2E b I b y + 2E b I bv ( M bv y y 2dy ) y 2 ( M bv2 y2 y 2dy ), (29) where E b I bv is the transverse brace s bending stiffness along radial direction of arch rib and y and y 2 represent the distance from the contraflexre point of transverse brace s radial deformation to the main and stable arch ribs, respectively. The vales of y and y 2 are closely related to the bending stiffness of arch ribs, the length of the arch rib section, the bending stiffness, and length of the transverse braces. For conventional arch bridges the bending stiffness andinclinedangleoftwomainarchribsarethesame,so y = y 2.However,forleaning-typearchbridges,asthe bending stiffness of main arch rib is far larger than that of stable arch rib, so y >y 2,asshowninFigre 3. In order to obtain the vales of y and y 2,radialmechanicalmodel of transverse brace along the arch central axis is established when a lateral bckling occrs, as shown in Figre 6(a). The bending moment cased by the transverse braces radial deflection can be obtained by Castigliano s theorem of material mechanics, as shown in Figre 6(b),and(3)canbegiven as e 2 = M bv = (E bi bh d+2bcos φ G II T II )G I T I M bv2 (E b I bh d+2bcos φ G I T I )G II T. (3) II According to the principle of similar triangles, we arrived at y = e 2 +e 2 b, y 2 = +e 2 b. (3) The radial deformation energy of transverse brace can be expressed as U bv = 6E bi bv db + 6E bi bv db e 2 (β θ +e ) 2 ds 2 s =E b I bv (C 5 C ) 2 (β θ +e 2 ) 2 ds 2 s R D 3 +E b I bv C 2 3 R D 4 +E b I bv C 2 5 R D 4 +E b I bv C 3 C 5 R D 6, (32)

8 8 Mathematical Problems in Engineering b b q d/2 E I I I E b I bv d/2 φ M bv M bv2 E II I II φ (a) Mechanical model (b) Bending model diagram Figre 6: Mechanical model and moment diagram in radial direction. where H H D 3 = 6n bs e 2 +e 2 ( cos 2πφ 2 ) R 2 D 4 = 6n bs D 5 = 6n bs D 6 = 6n bs +e 2 +e 2 +e 2 ( cos 2πφ 2 ) R 2 ( cos 2πφ 2 ) R 2, 2 ( cos 2πφ )( cos 2πφ )R 2. (33) Assming the distance between hangers of the main arch rib is d h and its corresponding arc length of arch rib is d h, d h is approximately eqal to d h as it is previosly assmed that the distance between hangers is small, and the distance between bridge deck and main arch rib is y (φ) = R [cos (φ 2 ) cos ]. (34) 2 As shown in Figre 7 the tension of the hanger is T=qd h. (35) The horizontal component of hanger tensions is H=qd h sin φ. (36) As the lateral displacement is sfficiently small, one arrives at where y(φ) is the length of the hanger. sin φ =φ = d y (φ), (37) φ φ d T Hanger Bridge deck y(φ) Figre 7: Schematic diagram of the horizontal component of hanger tensions with lateral bckling. The elastic potential energy of arch ribs and bridge deck system cased by the horizontal part of the tension of the hanger is given in where V H = 2 L L E = 2 H( d ) d x b =q(c 2 C 4 ) 2 E, (38) 2 R 2πφ cos φ( cos ), (39) 2f f is the rise of main arch rib, and in order to simplify its integral, a conservative assmption of y(φ) = f is adopted. d T φ

9 Mathematical Problems in Engineering 9 The lateral bending deformation energy of the bridge deck system is given in U d = 2 E d I d ( d )2 dl = L 2 E d I d cos φ( d 2ds ) s =E d I d C 2 4 R 3 E 2, (4) where E d I d is the lateral bending stiffness of the bridge deck system E 2 = α 2 ( 2π 4 ) cos φcos 2 2πφ R 4. (4) Combined with (4), the potential energy of the external loading is where V= 2 Vqds= q(c2 2 F +C 2 5 F 2R 2 ), (42) F = ( 2π 2 ) sin 2 2πφ F 2 = ( 2π 2 e b 2πφ ) ( + e 2 ) 2 R 2 sin2. (43) 2.3. The Analytical Soltion of Lateral Critical Bckling Loading. The total potential energy of the leaning-type arch bridge can be obtained based on (2), (4), (27), (32), and (38) (42). According to the principle of stationary potential energy, the vales of C i minimizing the fnction W(C i ) shold therefore satisfy the algebraic eqations: (W) = (i=,2,...,5). (44) C i The existence of nontrivial soltions of (44) forc i reqires that the determinant of its coefficient matrix be eqal to zero, then we obtained where H = 64F F 2 S S 3 E H λ 3 cr +H 2λ 2 cr +H 3λ cr +H 4 =, (45) H 2 =8S 2 5 S 3E F 2 32S S 3 S 4 F E 32S S 2 S 3 E F S S 3 E 2 F F 2 k 6 +8S S 2 7 E F 2 +8S S 2 9 E F +8S 3 S 2 6 E F 64S S 3 E E 2 F 2 k 6 H 4 =4S S 2 S 2 9 E 2k 6 +S 2 5 S2 9 E 2k 6 4S S 4 S 2 7 E 2k 6 2S 5 S 6 S 7 S 9 E 2 k 6 4S S 3 S 2 8 E 2k 6 +4S 3 S 5 S 6 S 8 E 2 k 6 +4S S 7 S 8 S 9 E 2 k 6 4S 2 S 3 S 2 6 E 2k 6 +S 2 6 S2 7 E 2k 6 + 6S S 2 S 3 S 3 E 2 k 6 4S 3 S 4 S 2 5 E 2k 6. (46) The lateral bckling critical load coefficient was obtained by solving (45), and then the lateral critical bckling load of leaning-type arch bridge is q cr =λ cr E I I I R 3. (47) 3. Lateral Bckling Critical Load nder the Hinged Bondary Condition of Main Arch Rib The following assmptions are made in the derivation process: the main arch ribs are hinged, the stable arch ribs are fixed, and the other assmptions are the same as those stated in Section 2.. The variables withot special explanation are thesameasaforementioned. The torsional angle of arch axis of the main and stable arch ribs is shown as θ =C sin πφ, (48a) θ 2 =C 3 ( cos 2πφ ). (48b) The lateral displacements of arch axis of the main and stable arch ribs are given as =C 2 sin πφ, (49) as 2 =C 2 sin πφ. (5) The lateral displacements of bridge deck system is given d =C 4 sin πφ. (5) The global torsional angle of the transverse brace in radial planecasedbyarchrib sdeflectionis β=c 5 sin πφ, (52) where C, C 2, C 3, C 4,andC 5 are all constants. The above displacement fnctions shold satisfy the following bondary conditions: () when φ=and φ=, θ =β=, = d =, θ =β =,and = d = (2) when φ=and φ=, θ 2 =, 2 =, θ 2 =,and 2 =. The derivation method of the lateral bckling critical load nder the hinged bondary condition of main arch ribs is the same as stated above. De to the limitation of the paper length, the derivation process is omitted; only the calclated reslts are discssed in the following section.

10 Mathematical Problems in Engineering Figre 8: The FE model nder the fixed bondary condition. 4. Verification Example The leaning-type arch bridge, Shengli Bridge, with a span of 75minJiangmencityofChinaissedtoverifytheaccracy of the derived analytical soltion presented in this paper. A three-dimensional finite element model is established by sing the Midas/Civil FEM software to calclate arch bridge s lateralbcklingcriticalloadforcomparison.themainand stable arch ribs of this bridge are both fixed at the arch ends. The FE model is shown in Figre 8. There are 284 elements and 27 nodes in this FE model. Spatial beam element with 6 degrees of freedom at each node is sed to simlate the arch rib, transverse brace, girder, and transverse girder. Spatial trss element with 3 degrees of freedom at each node is sed to simlate the hanger. The calclation parameters of the leaning-type bridge are listed in Table. Nmerical analysis is carried ot as the following steps. () N/mniformloadisappliedtothemiddleof the transverse girder of the bridge deck system. (2) By sing Midas/Civil s bckling eigenvale solver, the eigenvale λ of the bridge is obtained which indicates the lateral bckling critical load of the bridge. Comparison of the FEM reslts and the analytical reslts of the leaning-type arch bridge nder fixed and hinged bondary conditions when the stable rib inclined angle is 5, 7,9,,3,5,7,9,2,24,27,and3,areshownin Tables 2 and 3. The contrastive reslts show that the analytical reslts agree well with the FEM reslts with the relative error no more than 3.47%, 3.2% nder fixed bondary condition and hinged bondary condition, respectively, which indicate the accracy of the analytical soltion for the stable critical load of the leaning-type arch bridge presented in this paper. 5. Parametric Analysis 5.. Effect of Main Arch Rib s Bondary Condition on the Critical Bckling Load. Figre 9 shows the lateral bckling critical load of a leaning-type arch bridge for both cases of fixed-end main arch ribs and hinged-end main arch ribs. It can be seen from this figre that the lateral bckling critical load of the leaning-type arch bridge with fixed main arch ribs is approximately.4 to.6 times that of the leaningtype arch bridge with hinged main arch ribs. As the inclined angle increases from 5 to 3,thecriticalloadq cr increases by approximately.8% for cases of fixed-end main arch ribs q cr (MN/m) Inclined angle φ ( ) Fixed end Hinged end Figre 9: Comparison of critical bckling load nder different bondary conditions. and the critical load q cr increases by approximately 9.57% for cases of hinged-end main arch ribs Effect of the Central Angle on the Critical Bckling Load. Figre shows the critical load vale q cr when the central angle of the main arch rib and the stable arch rib are the same (the central angle of the main arch rib and the stable arch rib is 87.2 in this case), and different (the central angle of the main arch rib is 87.2 andthecentralangleofthestablearchribis 4 in this case) as the inclined angle increases from 5 to 3. ItcanbeseenfromFigre that the difference of the critical load q cr between same central angle model and the different centralanglemodelissmallenoghtobeneglected,theformer s lateral bckling critical load is only.2 to.3 times of that of the latter. It indicates that the central angle of the stable arch has relatively less effect on the lateral bckling critical loadofthearchbridge,andthecentralanglesofthemain arch rib and the stable arch rib can be considered to be the same. 5.3.EffectofHangerTensionsandBridgeDeckonCritical Bckling Load. Figre shows the critical load vale with or withot considering hanger tensions and bridge deck as the inclined angle increases from 5 to 3. From this figre it can be seen that if the hanger tensions and bridge deck are considered, the critical load is MN/m and 7.98 MN/m when inclined angle is φ =5 and 3,respectively.However,ifthe hanger tensions and the bridge deck are neglected, the critical load is 2.7 MN/m and 2.45 MN/m when inclined angle is φ = 5 and 3, respectively. The critical load increases by 2.94 and 3.7 times, respectively, as compared with that of neglecting the hanger tensions and bridge deck. The reslts indicate that the hanger tensions and bridge deck can greatly improve the lateral stability of the leaning-type arch bridge.

11 Mathematical Problems in Engineering Table : Calclation parameters sed in the FE model. Span (m) 75 Rise-span ratio /4 The central angle of main arch ( ) 87.2 Thecentralangleofstablearch( ) 4 Inclined angle of stable arch rib ( ) 5,7,9,,3,5,7,9,2,24,27and3 Transverse brace length on arch crown (m).55 Nmber of transverse brace 6 Lateral bending stiffness of main arch rib (MN m 2 ).44 4 Torsionalstiffnessofmainarchrib(MN m 2 ). 3 Lateral bending stiffness of stable arch rib (MN m 2 ).79 4 Torsionalstiffnessofmainarchrib(MN m 2 ) Transverse brace s bending stiffness along radial direction of arch rib (MN m 2 ) Transverse brace s bending stiffness along tangential direction of arch rib (MN m 2 ).58 2 Lateral bending stiffness of girder (MN m 2 ) Vertical bending stiffness of girder (MN m 2 ) Lateral bending stiffness of transverse beam (MN m 2 ).5 4 Vertical bending stiffness of transverse beam (MN m 2 ) The cross section area of hangers (m 2 ) Poisson s ratio.2 Table 2: Comparison between the analytical soltion and FEM reslts nder fixed bondary condition. Inclined angle Analytical (MN/m) FEM (MN/m) Error (%) q cr (MN/m) Inclined angle φ ( ) Different center angle Same center angle Figre:Comparisonofcriticalbcklingloadnderthesameand different central angle. q cr (MN/m) Inclined angle φ ( ) Considering the effect of hanger tensions Neglect the effect of hanger tensions Figre : Comparison of critical bckling load with and withot hanger tensions effect Effect of the Stable Arch Rib on Critical Bckling Load. Figre 2 shows the critical load q cr of the models with or withot stable arch ribs as the inclined angle increases from 5 to 3. From this figre it can be seen that the critical load of the models with stable arch ribs is always larger than that of the models withot stable arch ribs and the percentage of increase is from 3.5% and 42.% as the inclined angle increased. The reslts indicate that the effect of stable arch rib on critical load is significant.

12 2 Mathematical Problems in Engineering Table 3: Comparison between analytical soltion and FEM reslts nder hinged bondary condition. Inclined angle Analytical (MN/m) FEM (MN/m) Error (%) q cr (MN/m) Inclined angle φ ( ) With stable arch rib Withot stable arch rib Figre 2: Comparison of critical bckling load with and withot stable arch rib. 6. Conclsions This paper has derived an analytical soltion for lateral bckling critical load of leaning-type arch bridge based on theritzmethod,andtheaccracyofthissoltionhasbeen verified throgh a nmerical example. Moreover, parametric analysis is carried ot in order to investigate how changes in certain design parameters wold affect the critical load of the leaning-type arch bridges by sing the analytical soltion presented in this paper. The main conclsions are as follows. () The analytical soltion present in this paper can be sed to calclate the lateral bckling critical load of the leaning-type arch bridges in different cases, inclding the central angles of the main arch rib and stable arch rib which are different; both the main arch ribs and the stable arch ribs are fixed in their ends, the main arch ribs with hinged bondary condition while the stable arch ribs with fixed bondary conditions. From the comparison of the analytical reslts and the FEM reslts, the analytical soltion presented in this paper is verified to be sfficiently accrate. (2) The lateral bckling critical load nder fixed bondary condition is approximately.4 to.6 times as large as that nder hinged bondary conditions, which indicate that the lateral stability of the former is better than that of the latter. (3) The critical load with the same central angles is slightly bigger than the one with different central angles, and the former s lateral bckling critical load is.2 to.3 times as big as that of the latter. It indicates that the central angle of the stable arch has relatively less inflence on the lateral bckling critical load of theleaning-typearchbridge,andthereforethecentral anglesofthemainarchandthestablearchcanbe considered to be the same for convenience. (4) Stable arch rib can significantly increase the lateral bckling critical load q cr of leaning-type arch bridge by 3.5% to 42.% when stable arch rib is considered nder the fixed bondary condition, where the vale of q cr increases as the inclined angle of stable arch rib increases. (5) The hanger tensions and bridge deck have significant effect on the critical load, and when considering the effect of hanger tensions and bridge deck, the critical load can improve by 2.94 to 3.7 times. Notations θ, θ 2 : Thetorsionalangleofthemainandstable arch ribs, 2 : The lateral displacement of the main and stable arch ribs nder the global coordinate I, II : The lateral displacement of the main and stable arch ribs nder the local coordinate d : The lateral displacement of bridge deck system φ : Theanglebetweenthemainarchribandthe stable arch rib φ: Theangleofapositioninthearchribs, : Thecentralangleofthemainandstablearch ribs β: The global torsional angle of the transverse brace cased by deformation of arch rib in radial plane R: Theradisofthemainandthestablearch ribs b(x): Thedistancebetweenthemainarchriband the stable arch rib b : The length of the transverse brace at the arch crown h: Thedistancebetweenthemainandstable arch ribs at qarter span d: Thedistancebetweenthetransversebraces d h : Thedistancebetweenthehangers K I, KII : The lateral deflection crvatre of the main and stable arch ribs respectively K I w, KII w : The torsional deflection crvatres of the main and stable arch ribs, respectively

13 Mathematical Problems in Engineering 3 U I, UII : U I w, UII w : U I c, UII c : U bv : U bh : V H : V d : V: U I c, UII c : U bh : U bv : The lateral bending deformation energy of the main and stable arch ribs, respectively The torsional deformation energy of the main and stable arch ribs, respectively The local bending deformation energy of the main and stable arch ribs, respectively The bending deformation energy of the transverse braces in radial directions The bending deformation energy of the transverse braces in tangential directions Theelasticpotentialenergyofthearchribs and the bridge deck system nder the horizontal component of the hanger tensions Theelasticpotentialenergyofbridgedeck system The potential energy of external loading The single-panel arch rib local bending energy of main and stable arch ribs The tangential bending deformation energy of the single transverse brace The radial deformation energy of a single transverse brace E I I I, EII I II : The lateral bending stiffness of the main and stable arch ribs G I T I, G II T II : The torsional stiffness of the main and stable arch ribs E b I bh : E b I bv : M c, M c2 : M bh, M bh2 : M bv, M bv2 : y, y 2 : y, y 2 : γ, γ 3 : γ 2, γ 4 : Conflict of Interests The bending stiffness of transverse brace along tangential direction of arch rib The bending stiffness of transverse brace along radial direction of arch rib The bending moments of main and stable arch ribs along radial direction The tangential bending moment of transverse bracenearthemainandstablearchribs The vertical bending moment on both ends of transverse brace near the main and stable arch ribs The distances from the contraflexre point of transverse brace tangential deformation to main and stable arch ribs, respectively The distance from the vertical contraflexre point of transverse brace s radial deformation to main and stable arch ribs, respectively The tangential anglar rotation of the transverse brace at the intersection points betweenthemainandstablearchribandthe transverse brace, respectively The tangential anglar rotation of the main and stable arch rib at the intersection points betweenthemainandstablearchribandthe transverse brace, respectively. The athors declare that there is no conflict of interests regarding the pblication of this paper. Acknowledgments This stdy was sponsored by the National Natral Science Fondation of China (nos , , and 52823), the Science and Technology Planning Major Project of Gangzho City (no. 2Y2-6), the Key Technological Innovation Program of Gangdong Ministry Edcation (no. 22CXZD28), the Key Project spported by the Natral Science Fondation of Gangdong Province (no. S2328), and the Talent Introdction Project spported by the Higher Edcation Department of Gangdong Province in 22. References [] A.-R.Li,Q.-C.Y,R.Song,andJ.-P.Zhang, Dynamicstability of leaning-type arch bridge nder earthqake, Jornal of Shenzhen University Science and Engineering, vol.27,no.3,pp , 2 (Chinese). [2] A. C. Franciso, Acro Color Thematic Architectre, Acro Editorial, 989. [3] R. C. Xiao, H. T. Sn, and L. J. 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