Research ArticleAn Analytical Solution for Lateral Buckling Critical LoadCalculation of Leaning-Type Arch Bridge

Ai-rong Liu Yong-hui Huang Qi-cai Yu and Rui Rao

Guangzhou University-Tamkang University Joint Research Center for Engineering Structure Disaster Prevention and ControlGuangzhou University Guangzhou 510006 China

Correspondence should be addressed to Yong-hui Huang kunullfoxmailcom

Received 14 March 2014 Revised 1 May 2014 Accepted 1 May 2014 Published 25 May 2014

Academic Editor Sarp Adali

Copyright copy 2014 Ai-rong Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An analytical solution for lateral buckling critical load of leaning-type arch bridge was presented in this paper New tangential andradial buckling models of the transverse brace between the main and stable arch ribs are established Based on the Ritz method theanalytical solution for lateral buckling critical load of the leaning-type arch bridge with different central angles of main arch ribsand leaning arch ribs under different boundary conditions is derived for the first time Comparison between the analytical resultsand the FEM calculated results shows that the analytical solution presented in this paper is sufficiently accurate The parametricanalysis results show that the lateral buckling critical load of the arch bridgewith fixed boundary conditions is about 114 to 116 timesas large as that of the arch bridge with hinged boundary condition The lateral buckling critical load increases by approximately315 to 412 when stable arch ribs are added and the critical load increases as the inclined angle of stable arch rib increases Thedifferences in the center angles of the main arch rib and the stable arch rib have little effect on the lateral buckling critical load

1 Introduction

Leaning-type arch bridge is a relatively new type of spatialtied-arch bridge developed fromx-type arch bridge It is com-posed of two load-bearing ribs (called main arch ribs) whichare perpendicular to the bridge deck and two leaning archribs (called stable arch ribs) on the sides of themain arch ribsThese two types of arch ribs constitute a space stable systemwhen connected by transverse braces between them Thistype of bridge is one of the most competitive urban bridgesbecause it is stylish and unique gives an open and clearview for drivers and is economically efficient As the verticalstiffness of the main arch rib is much higher than that of thestable arch rib the major portion of the dead and live loadsis supported by the main arch ribs Stable arch rib supportsonly a small portion of the live load and its main functionis to assure the lateral stability of the main arch rib Becausethere are no transverse braces placed between the main archribs and the stable arch ribrsquos contribution to the improvementof the lateral stability of bridge is very limited the lateralstability often becomes a key factor deciding the bridgersquossafety [1] The worldrsquos first leaning-type arch bridge named

Bacde Road Bridge was built in 1992 in Barcelona It wasdesigned by the Spanish architect Santiago Calatrava Thisbridge is 52m long 258m wide and the inclined angleof stable arch rib is 30∘ The vertical main arch ribs arehinged and the inclined stable arch ribs are fixed at bothends [2] Over the past decades leaning-type arch bridgeshave developed rapidly At presentmore than 20 leaning-typearch bridges have been built around the world most of themin China Usually fixed boundary conditions are adoptedfor both the main and stable arch ribs such as the KangfuBridge in Yiyang city China Hanjiang North Bridge inChaozhou city China and Shengli Bridge in Jiangmen cityChina (Figure 1) However under special circumstances inorder to reduce main arch ribrsquos internal force and horizontalthrust or adapt to the local environmental conditions [3]hinged boundary condition is applied at the arch ends of themain arch such as Yufeng Bridge in Kunshan city China andDanxi Bridge in Yiwu city China

At present the derivation of the analytical solution of thearch structurersquos buckling problem is mainly focused on theindividual arch rib or conventional arch bridges with sym-metrical arch ribs Closed-form solutions for out-of-plane

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 578473 14 pageshttpdxdoiorg1011552014578473

2 Mathematical Problems in Engineering

(a) (b)

Figure 1 Shenli Bridge in Jiangmen City of China

buckling of arches subjected to uniform bending or uniformcompression were obtained by Lim and Kang [4] Pi et al[5 6] investigated the flexural torsional buckling of shallowarches with an open thin-walled section subjected to a radialload uniformly distributed around the arch axis In-planestatic and dynamic buckling of shallow pin-ended parabolicarches with a horizontal cable was investigated by Chen andFeng [7] La et al [8] presented an experimental investiga-tion of the elastic-plastic out-of-plane buckling response ofroller bent circular steel arches subjected to a single forceapplied to the crown Bradford and Pi [9] derived a newunified analytical solution for the lateral-torsional bucklingload of pin-ended arches by accounting for the combinedbending and axial compressive action Dou et al [10] inves-tigated the sectional rigidities of trusses and the out-of-plane buckling loads of pin-ended circular steel tubulartruss arches in uniform axial compression and in uniformbending In addition Jin and Zhao [11] derived the lateralbuckling critical load for X-type twin ribbed arch bracedwith transverse beams Moreover Liu and Lu [12] derivedthe analytical solution of the lateral buckling critical loadfor the conventional tied-arch bridge with transverse bracesand discussed the effect of structural parameters on lateralstability Additionally Xiang [13] presented a formula of thelateral buckling critical load for a half-through bridge atservice stage and the effect of beam location for vehicle laneon the stability of this type of bridge was further studied

Compared with individual arch rib or conventional archbridges with symmetrical arch ribs the leaning-type arch ribsystem is different in several aspects including an inclinedangle between themain and stable arch ribs different stiffnessof main and stable arch ribs more significant spatial effectand more complicated loading conditions These factorsresult in difficulties in deriving the analytical solution of thelateral buckling critical load Especially that the mechanicalmodel of transverse brace is very different from those ofthe conventional arch bridge when a lateral buckling occursPresently researchers have conducted preliminary studieson the lateral stability of leaning-type arch bridge based onfinite element method (FEM) [14] The derivation of theanalytical solution formula of the lateral buckling criticalload of leaning-type arch ribs system has been preliminarilystudied by Liu et al [15] However in their studies the

influence of the bridge deck system and hanger tensions onleaning-type arch bridge was not taken into considerationThe central angles of the main and stable arch ribs wereassumed to be the same in their studies but in fact they aredifferent in some conditions And in their studies the inclinedangles of stable arch ribs should be less than 15 degrees whilethe inclined angles larger than 15 degrees the precision of thesolution is bad Compared with the FEM calculation resultsthe relationship among different design parameters of thebridge can be clearly revealed by the analytical solution andthe optimization of structure design is made easier thecomplicated process of constructing the FE model can alsobe simplified

In order to consider the influence of the componentsof leaning-type arch bridge comprehensively the globaltransverse brace deformation parameter 120573 is considered andthe central angles of themain arch ribs and stable arch ribs areassumed different and the tangential and radial mechanicalmodels of the transverse brace between the main and stablearch ribs are established then the analytical solution of thelateral buckling critical load for leaning-type arch rib systemis derived in this paper Compared with the analytical solu-tion derived by Liu et al [15] the one derived in this paper hasa wider scope of applications The deformation energy of themain arch rib stable arch rib and transverse braces betweenthem are constructed and the potential energy caused bythe hangers tensions are also established for both the mainarch rib and stable arch rib under fixed and hinged boundaryconditions And the total potential energy of the bridge ina buckling process is obtained thereafter Based on the sta-tionary energy principle the analytical solution for the lateralbuckling critical load of leaning-type arch bridge is obtainedIn the end parametric analysis is carried out in order toinvestigate how changes in certain design parameters wouldaffect the critical load of the leaning-type arch bridges whichcould lead to an optimum design of this type of bridgestructures

2 Lateral Buckling Critical Load under theFixed Boundary Condition of Main Arch Rib

21 Calculation Model of Leaning-Type Arch Bridge Thefollowing assumptions are made in the derivation process

Mathematical Problems in Engineering 3

x

y

R

q

Lz

vu

w

1205721

1205722

dh

120593

(a) Elevation view

1206010

(b)Sideview

Stable arch rib

Main arch rib

Bridge deck

Transverse brace

d

x

z

b(x)

b 0

(c) Plan view

Figure 2 Calculation sketch of leaning-type arch bridge

(1) the main arch ribs and the stable arch ribs are fixed intheir ends (2) the axis of main and stable arch ribs are arc-shaped curves (3) the stiffness of arch ribs and transversebraces are constants (4) the axial deformation of arch ribin the buckling process is neglected (5) the external load isevenly and vertically distributed along the bridge deck and istransmitted to the arch rib via the hangers and (6) the forceacting on arch ribs satisfies the film tension assumption

The simplified calculation model of a leaning-type archbridge is shown in Figure 2 In the figure 119909-119910-119911 is theglobal coordinate system and 119906-V-119908 is the local coordinatesystem 119906 V and119908 represent the lateral radial and tangentialdisplacement of arch ribs respectively120601

0represents the angle

between the main arch rib and the stable arch rib 120593 is theangle of a position in the arch ribs 120572

1is the central angle

of the main arch ribs 1205722is the central angle of the stable

arch rib and 119877 is the radius of the main and stable arch ribs119887(119909) is the distance between the main arch rib and the stablearch rib In the arch crown position the distance is 119887

0and

119887(119909) = 1198870+2119877 sin120601

0(1minuscos120593)The transverse braces between

themain and stable arch ribs are equidistantwith a distance of119889 the hangers are also arranged equidistantly with a distanceof 119889ℎAccording to the basic assumption and calculation

scheme the relationship between the lateral displacement 1199061198681

1199061198681198682

and the radial displacement V1198681

V1198681198682

of the main arch ribsand the stable arch ribs under the local coordinate is givenrespectively as

1199061198681

= 1199061 119906119868119868

2

=119910101584002

119910101584001

+ 119910101584002

119887 (119909) 120573 sin1205930+ 1199061cos1205930

V1198681

=119910101584001

119910101584001

+ 119910101584002

119887 (119909) 120573

V1198681198682

= minus119910101584002

119910101584001

+ 119910101584002

119887 (119909) 120573 cos1205930+ 1199061sin1205930

(1)

where 1199061represents the lateral displacement of the arch axis

of themain arch 120573 represents the global torsional angle of thetransverse brace caused by deformation of arch rib in radialplane 119887(119909) is the distance between the main arch rib and 1199101015840

01

and 119910101584002

represent the distance from the contraflexure point of

y99840002 y99840001

u

b(x)

v

Mb1

Mb2

EIIb I

IIb 120579c

120579c

EIIIIIu

EIIIu

1205792

12057921205791

1205930

Figure 3 Radial bending deformation of transverse brace

transverse bracersquos radial deformation to the main and stablearch ribs respectively (shown in Figure 3)

When there is a lateral deformation in the arch rib systemaxial strain of the main arch is given as

120576 = minusV1198681

119877+

1198891199081198681

119889119904+

1

2[(

1198891199061198681

119889119904)

2

+ (119889V1198681

119889119904)

2

] (2)

where 1199081198681

represents the tangential displacement of the mainarch ribs under the local coordinate

According to the basic assumption 120576 = 0

1198891199081198681

119889119904=V1198681

119877minus

1

2[(

1198891199061198681

119889119904)

2

+ (119889V1198681

119889119904)

2

] (3)

Because the arches are fixed at both ends int119904

1198891199081198681

= 0 so

int119904

(V1198681

119877)119889119904 =

1

2int119904

(1198891199061198681

119889119904)

2

119889119904 +1

2int119904

(119889V1198681

119889119904)

2

119889119904 (4)

4 Mathematical Problems in Engineering

Assuming the ends of the main arch rib and the stablearch rib are perfectly fixed the torsional angle and the lateraldisplacements of arch axis of themain and stable arch ribs aregiven respectively as

1205791= 1198621(1 minus cos

2120587120593

1205721

) (5a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (5b)

1199061= 1198622(1 minus cos

2120587120593

1205721

) (6a)

1199062= 1198622(1 minus cos

2120587120593

1205722

) (6b)

The lateral displacement of bridge deck system is given as

119906119889= 1198624(1 minus cos

2120587120593

1205721

) (7)

The global torsional angle of the transverse brace causedby deformation of arch rib in radial plane is given as

120573 = 1198625(1 minus cos

2120587120593

1205721

) (8)

The above equations satisfy the following displacementboundary conditions

(1) When 120593 = 0 and 120593 = 1205721 1205791= 0 1205791015840

1

= 0 1199061= 119906119889=

120573 = 0 and 11990610158401

= 1199061015840119889

= 1205731015840 = 0(2) When 120593 = 0 and 120593 = 120572

2 1205792= 0 1205791015840

2

= 0 1199062= 0 and

11990610158402

= 0

The lateral deflection curvatures along the V axis of themain and stable arch ribs are given as [16]

119870119868119906

=1205791

119877minus

11988921199061198681

1198891199042

119870119868119868119906

=1205792

119877minus

11988921199061198681198682

1198891199042

(9)

where 119870119868V and 119870119868119868V represent the lateral deflection curvaturealong V axis of the main and stable arch ribs respectively

The torsional deflection curvatures along 119908 axes are alsogiven as [16]

119870119868119908

=1198891205791

119889119904+

1

119877

11988921199061198681

119889119904

119870119868119868119908

=1198891205792

119889119904+

1

119877

1198891199061198681198682

119889119904

(10)

where 119870119868119908

and 119870119868119868119908

represent the torsional deflection cur-vatures along 119908 axes of the main and stable arch ribsrespectively

22 Energy Equations The lateral deformation energy of aleaning-type arch bridge can be written as

119882 = 119880119868119906

+ 119880119868119868119906

+ 119880119868119908

+ 119880119868119868119908

+ 119880119868119888119906

+ 119880119868119868119888119906

+ 119880119887119867

+ 119880119887V + 119881

119867+ 119880119889+ 119881

(11)

where 119880119868119906

119880119868119868119906

119880119868119908

119880119868119868119908

119880119868119888119906

and 119880119868119868119888119906

represent the totallateral bending deformation energy torsional deformationenergy and local bending deformation energy of the mainand stable arch ribs respectively119880

119887V and119880119887119867

are the bendingdeformation energy of the transverse braces in radial andtangential directions along the main arch ribrsquos axis 119881

119867is

the elastic potential energy caused by the horizontal part ofthe tension of the hangers 119880

119889represents the elastic potential

energy of bridge deck system and 119881 represents the potentialenergy of external loading applied to the arch bridge

The total lateral bending deformation energy of the mainand stable arch ribs are given in

119880119868119906

+ 119880119868119868119906

=1

2119864119868119868119868119906

int1199041

(119870119868119906

)2

119889119904 +1

2119864119868119868119868119868119868119906

int1199042

(119870119868119868119906

)2

119889119904

=1

2119864119868119868119868119906

[11986221

11986011

+11986222

119877311986012

+11986211198622

119877211986013]

+1

2119864119868119868119868119868119868119906

[11986222

119877311986021

+11986223

11987711986022

+11986225

11987711986023

+11986221198623

119877211986024+11986221198625

119877211986025+11986231198625

11987711986026]

(12)

where 119864119868119868119868119906

and 119864119868119868119868119868119868119906

are the lateral bending stiffness of themain and stable arch ribs Consider the following

11986011

= int1205721

0

(1 minus cos2120587120593

1205721

)2

119889120593

11986012

= int1205721

0

(2120587

1205721

)4

cos22120587120593

1205721

119889120593

11986013

= int1205721

0

minus2(2120587

1205721

)2

(1 minus cos2120587120593

1205721

) cos2120587120593

1205721

119889120593

11986021

= int1205722

0

cos1206010(2120587

1205722

)4

cos22120587120593

1205721

119889120593

11986022

= int1205722

0

(1 minus cos2120587120593

1205722

)2

119889120593

Mathematical Problems in Engineering 5

11986023

= int1205722

0

1

1198772(2120587

1205721

)4 119887(119909)2sin2120601

0

(1 + 1198902)2

cos22120587120593

1205721

119889120593

11986024

= int1205722

0

minus2 cos1206010(2120587

1205722

)2

cos2120587120593

1205722

(1 minus cos2120587120593

1205722

)119889120593

11986025

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

times (2120587

1205721

)2

(2120587

1205722

)2

cos2120587120593

1205721

cos2120587120593

1205722

119889120593

11986026= int1205722

0

minus2119887 (119909) sin120601

0

119877 (1 + 1198902)

(2120587

1205721

)2

cos2120587120593

1205721

(1minuscos2120587120593

1205722

)119889120593

(13)

The torsional deformation energy of the main and stablearch ribs is given in

119880119868119908

+ 119880119868119868119908

=1

2119866119868119879119868 int

1199041

(119870119868119908

)2

119889119904 +1

2119866119868119868119879119868119868 int

1199042

(119870119868119868119908

)2

119889119904

=1

2119866119868119879119868 [

11986221

11987711986111

+11986222

119877311986112

+11986211198622

119877211986113]

+1

2119866119868119868119879119868119868 [

11986222

119877311986121

+11986223

11987711986122

+11986225

11987711986123

+11986221198623

119877211986124+11986221198625

119877211986125+11986231198625

11987711986126]

(14)

where 119866119868119879119868 and 119866119868119868119879119868119868 are the torsional stiffness of the mainand stable arch ribs Consider the following

11986111

= int1205721

0

(2120587120593

1205721

)2

sin22120587120593

1205721

119889120593

11986112

= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986113

= int1205721

0

2(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986121

= int1205722

0

cos21206010(2120587

1205722

)4

sin22120587120593

1205721

119889120593

11986122

= int1205722

0

(2120587

1205722

)2

sin22120587120593

1205722

119889120593

11986123

= int1205722

0

1

1198772(2120587

1205721

)2 119887(119909)2sin2120601

0

(1 + 1198902)2

sin22120587120593

1205721

119889120593

11986124

= int1205722

0

2 cos1206010(2120587

1205722

)2

sin22120587120593

1205722

119889120593

y02

y0

y01

b(x) b

Mcu1

Mcu2

Mbh2

Mbh1

EbIbh

EIIIIIu

EIIIu

120574998400

120574998400998400

1205741

1205741

1205742

1205743

1205743

1205744

Figure 4 Bending deformation of arch ribs and transverse braces

11986125

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

11986126

= int1205722

0

2119887 (119909) sin1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

(15)

Transverse brace bending deformation occurs in the tan-gential direction along arch axis when there is a lateral buck-ling in the arch rib system as shown in Figure 4 In this casethe tangential bending deformation energy of the singletransverse brace is given as

119880119887119867

=1

2119864119887119868119887ℎ

int11991001

0

1198722119887ℎ1

1198891199100+

1

2119864119887119868119887ℎ

int11991002

0

1198722119887ℎ2

1198891199100 (16)

where 119864119887119868119887ℎ

is transverse bracersquos bending stiffness alongtangential direction of arch rib and 119910

01and 119910

02represent

the distances from the contraflexure point of transversebrace tangential deformation to the main and stable archribs respectively Therefore the length of transverse brace is119887(119909) = 119910

01+ 11991002

From Figure 4 the tangential bending moment of trans-verse brace near the main and stable arch ribs is given by thesuperposition principle

119872119887ℎ1

=4119864119887119868119887ℎ

1198871205741+

2119864119887119868119887ℎ

1198871205743

119872119887ℎ2

=2119864119887119868119887ℎ

1198871205741+

4119864119887119868119887ℎ

1198871205743

(17)

where 1205741represents the tangential angular rotation of the

transverse brace at the intersection points between the mainarch rib and the transverse brace and 120574

3represents the

tangential angular rotation of the transverse brace at the inter-section points between the stable arch rib and the transversebrace

It can be further derived as

11991001

=21205741+ 1205743

3 (1205741+ 1205743)119887 (119909)

11991002

=1205741+ 21205743

3 (1205741+ 1205743)119887 (119909)

(18)

6 Mathematical Problems in Engineering

119887(119909) can be replaced by a constant ℎ which is the distancebetween the main and stable arch ribs at quarter span forcalculation simplification Although the length of transversebrace 119887(119909) is not a constant the length of the transverse braceℎ at quarter span is very close to the average length of thetransverse braces of the entire bridge Such simplification canmake the derivation process become simple

The values of11991001and11991002are closely related to the bending

stiffness of arch ribs panel length bending stiffness andlength of transverse braces Due to the inclined angle andthe different stiffness of leaning-type arch bridgersquos main andstable arch ribs the tangential and radial deformation of thetransverse brace between the main and stable arch ribs differfrom that of the conventional arch bridges It is not a simpleldquoSrdquo shape For leaning-type arch bridges bending stiffness ofthe main arch rib is larger than that of the stable arch rib so11991001

gt 11991002 as shown in Figure 4 as an exceptional case for

conventional arch bridges 11991001

= 11991002 In order to obtain the

relationship between 11991001 11991002 a tangential mechanical model

of transverse brace along the arch axis is established whena lateral buckling occurs as shown in Figure 5(a) its cor-responding bending moment is shown in Figure 5(b) fromwhich (19) can be given as

119872119887ℎ1

119872119887ℎ2

=119864119868119864119887119868119868119906

119868119887ℎ119889 + 6119864119868119864119868119868119868119868

119906

119868119868119868119906

119887 (119909) cos1206010

6119864119868119864119868119868119868119868119906

119868119868119868119906

119887 (119909) cos1206010

+ 119864119868119868119864119887119868119868119868119906

119868119887ℎ119889 cos120601

0

= 1198901

(19)

where 1198901is a constant

From (17) and (19) it can be obtained that

119872119887ℎ1

119872119887ℎ2

=21205741+ 1205743

1205741+ 21205743

= 1198901 (20)

Assuming 1205743= 11988611205741 then

1198861=

1205743

1205741

=2 minus 1198901

21198901minus 1

(21)

thus from (18) and (21) the relationship of 11991001and 11991002can be

obtained as

11991001

=2 + 1198861

1 + 21198861

11991002 (22)

Substituting (17) and (18) into (16) the equation of tangentialbending energy of the transverse braces in the full arch ribrange is given as

119880119887119867

= int119904

119880119887119867

119889119889119904

=2119864119887119868119887ℎ

9119889ℎ[int1199041

21205741+ 1205743

1205741+ 1205743

(412057421

+ 412057411205743+ 12057423

) 119889119904

+int1199042

1205741+ 21205743

1205741+ 1205743

(12057421

+ 412057411205743+ 412057423

) 119889119904]

(23)

When the tangential local deformation occurs in themainand stable arch ribs assuming the bending moments of themain and stable arch ribs along the radial direction are 119872

1198881199061

and1198721198881199062

respectively the single-panel section arch ribrsquos localbending energy of the main and stable arch ribs can beexpressed as

119880119868

119888119906

+ 119880119868119868

119888119906

=1

2119864119868119868119868119906

int119889

11987221198881199061

1198891199090+

1

2119864119868119868119868119868119868119906

int119889

11987221198881199062

1198891199090

=6119864119868119868119868119906

11988912057422

+6119864119868119868119868119868119868119906

11988912057424

(24)

If the local bending energy of each single-panel sectionarch rib is the same the full-arch-rib local bending deforma-tion energy can be written as

119880119868119888119906

+ 119880119868119868119888119906

= int119904

119880119868

119888119906

119889119889119904 + int

119904

119880119868119868

119888119906

119889119889119904

= int119904

3119864119887119868119887ℎ

119889ℎ12057411205742119889119904 + int

119904

3119864119887119868119887ℎ

119889ℎ12057431205744119889119904

(25)

It could be obtained from Figure 4 that

21198721198881199061

= 119872119887ℎ1

21198721198881199062

= 119872119887ℎ2

(26)

Therefore from (23) (25) and (26) the local bendingenergy of arch rib and the tangential bending energy oftransverse brace are obtained as

119880119887119867

+ 119880119868119888119906

+ 119880119868119868119888119906

=3119864119887119868119887ℎ

119889ℎint1199041

(11987311205781+ 1205782) 120578112057410158402119889119904

+3119864119887119868119887ℎ

119889ℎint1199042

(11987321205783+ 1205784) 1205783120574101584010158402119889119904

= 119864119887119868119887ℎ

11986222

11987731198631+ 119864119887119868119887ℎ

11986222

11987731198632

(27)

where

1198631=

3119899

119887119904(11987311205781+ 1205782) 1205781int1205721

0

1198772(2120587

1205722

)2

sin22120587120593

1205721

119889120593

1198632=

3119899

119887119904(11987321205783+ 1205784) 1205783int1205722

0

1198772(2120587

1205722

)2

sin22120587120593

1205722

119889120593

1205741015840 = 1205741+ 1205742

12057410158401015840 = 1205743+ 1205744

1205781=

1205741

1205741015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868

119906

)

Mathematical Problems in Engineering 7

d2

EbIbhb1

EIIIIIu

1205930

EIIIu

q998400

(a) Mechanical model

1205930

Mbh2

Mbh1

(b) Bending moment diagram

Figure 5 Mechanical model and bending moment diagram in tangential direction

1205782=

1205742

1205741015840=

1

1 + (2ℎ119864119868119868119868119906

) (119889119864119887119868119887ℎ)

1205783=

1205743

12057410158401015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868119868119868

119906

)

1205784=

1205744

12057410158401015840=

1

1 + (2ℎ119864119868119868119868119868119868119906

) (119889119864119887119868119887ℎ)

1198731=

2

27sdot(1198861+ 2)

(1198861+ 1)

(4 + 41198861+ 11988621

)

1198732=

2

27sdot(11198861+ 2)

(11198861+ 1)

(1

11988621

+4

1198861

+ 4)

(28)

The mechanical model and moment diagram of arch ribsand transverse brace in radial direction are shown in Figure 3And the radial deformation energy of a single transversebrace can be written as

119880119887V =

1

2119864119887119868119887

int119910

1015840

01

0

(119872119887V1

119910101584001

11991010158400

)

2

11988911991010158400

+1

2119864119887119868119887Vint119910

1015840

02

0

(119872119887V2

119910101584002

11991010158400

)

2

11988911991010158400

(29)

where 119864119887119868119887V is the transverse bracersquos bending stiffness along

radial direction of arch rib and 119910101584001

and 119910101584002

represent the dis-tance from the contraflexure point of transverse bracersquos radialdeformation to the main and stable arch ribs respectively

The values of 119910101584001

and 119910101584002

are closely related to thebending stiffness of arch ribs the length of the arch ribsection the bending stiffness and length of the transversebraces For conventional arch bridges the bending stiffnessand inclined angle of two main arch ribs are the same so119910101584001

= 119910101584002

However for leaning-type arch bridges as the

bending stiffness of main arch rib is far larger than that ofstable arch rib so 1199101015840

01

gt 119910101584002

as shown in Figure 3 In orderto obtain the values of 1199101015840

01

and 119910101584002

radial mechanical modelof transverse brace along the arch central axis is establishedwhen a lateral buckling occurs as shown in Figure 6(a)The bending moment caused by the transverse bracesrsquo radialdeflection can be obtained by Castiglianorsquos theorem of mate-rialmechanics as shown in Figure 6(b) and (30) can be givenas

1198902=

119872119887V1

119872119887V2

=(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119868119879119868119868)119866119868119879119868

(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119879119868) 119866119868119868119879119868119868 (30)

According to the principle of similar triangles we arrived at

119910101584001

=1198902

1 + 1198902

119887 119910101584002

=1

1 + 1198902

119887 (31)

The radial deformation energy of transverse brace can beexpressed as

119880119887V =

6119864119887119868119887V

119889119887

1198902

1 + 1198902

int119904

(120573 minus 1205791)2

119889119904

+6119864119887119868119887V

119889119887

1

1 + 1198902

int119904

(120573 minus 1205792)2

119889119904

= 119864119887119868119887V(1198625minus 1198621)2

1198771198633+ 119864119887119868119887V11986223

1198771198634

+ 119864119887119868119887V11986225

1198771198634+ 119864119887119868119887V11986231198625

1198771198636

(32)

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

(a) (b)

Figure 1 Shenli Bridge in Jiangmen City of China

buckling of arches subjected to uniform bending or uniformcompression were obtained by Lim and Kang [4] Pi et al[5 6] investigated the flexural torsional buckling of shallowarches with an open thin-walled section subjected to a radialload uniformly distributed around the arch axis In-planestatic and dynamic buckling of shallow pin-ended parabolicarches with a horizontal cable was investigated by Chen andFeng [7] La et al [8] presented an experimental investiga-tion of the elastic-plastic out-of-plane buckling response ofroller bent circular steel arches subjected to a single forceapplied to the crown Bradford and Pi [9] derived a newunified analytical solution for the lateral-torsional bucklingload of pin-ended arches by accounting for the combinedbending and axial compressive action Dou et al [10] inves-tigated the sectional rigidities of trusses and the out-of-plane buckling loads of pin-ended circular steel tubulartruss arches in uniform axial compression and in uniformbending In addition Jin and Zhao [11] derived the lateralbuckling critical load for X-type twin ribbed arch bracedwith transverse beams Moreover Liu and Lu [12] derivedthe analytical solution of the lateral buckling critical loadfor the conventional tied-arch bridge with transverse bracesand discussed the effect of structural parameters on lateralstability Additionally Xiang [13] presented a formula of thelateral buckling critical load for a half-through bridge atservice stage and the effect of beam location for vehicle laneon the stability of this type of bridge was further studied

Compared with individual arch rib or conventional archbridges with symmetrical arch ribs the leaning-type arch ribsystem is different in several aspects including an inclinedangle between themain and stable arch ribs different stiffnessof main and stable arch ribs more significant spatial effectand more complicated loading conditions These factorsresult in difficulties in deriving the analytical solution of thelateral buckling critical load Especially that the mechanicalmodel of transverse brace is very different from those ofthe conventional arch bridge when a lateral buckling occursPresently researchers have conducted preliminary studieson the lateral stability of leaning-type arch bridge based onfinite element method (FEM) [14] The derivation of theanalytical solution formula of the lateral buckling criticalload of leaning-type arch ribs system has been preliminarilystudied by Liu et al [15] However in their studies the

influence of the bridge deck system and hanger tensions onleaning-type arch bridge was not taken into considerationThe central angles of the main and stable arch ribs wereassumed to be the same in their studies but in fact they aredifferent in some conditions And in their studies the inclinedangles of stable arch ribs should be less than 15 degrees whilethe inclined angles larger than 15 degrees the precision of thesolution is bad Compared with the FEM calculation resultsthe relationship among different design parameters of thebridge can be clearly revealed by the analytical solution andthe optimization of structure design is made easier thecomplicated process of constructing the FE model can alsobe simplified

In order to consider the influence of the componentsof leaning-type arch bridge comprehensively the globaltransverse brace deformation parameter 120573 is considered andthe central angles of themain arch ribs and stable arch ribs areassumed different and the tangential and radial mechanicalmodels of the transverse brace between the main and stablearch ribs are established then the analytical solution of thelateral buckling critical load for leaning-type arch rib systemis derived in this paper Compared with the analytical solu-tion derived by Liu et al [15] the one derived in this paper hasa wider scope of applications The deformation energy of themain arch rib stable arch rib and transverse braces betweenthem are constructed and the potential energy caused bythe hangers tensions are also established for both the mainarch rib and stable arch rib under fixed and hinged boundaryconditions And the total potential energy of the bridge ina buckling process is obtained thereafter Based on the sta-tionary energy principle the analytical solution for the lateralbuckling critical load of leaning-type arch bridge is obtainedIn the end parametric analysis is carried out in order toinvestigate how changes in certain design parameters wouldaffect the critical load of the leaning-type arch bridges whichcould lead to an optimum design of this type of bridgestructures

2 Lateral Buckling Critical Load under theFixed Boundary Condition of Main Arch Rib

21 Calculation Model of Leaning-Type Arch Bridge Thefollowing assumptions are made in the derivation process

Mathematical Problems in Engineering 3

x

y

R

q

Lz

vu

w

1205721

1205722

dh

120593

(a) Elevation view

1206010

(b)Sideview

Stable arch rib

Main arch rib

Bridge deck

Transverse brace

d

x

z

b(x)

b 0

(c) Plan view

Figure 2 Calculation sketch of leaning-type arch bridge

(1) the main arch ribs and the stable arch ribs are fixed intheir ends (2) the axis of main and stable arch ribs are arc-shaped curves (3) the stiffness of arch ribs and transversebraces are constants (4) the axial deformation of arch ribin the buckling process is neglected (5) the external load isevenly and vertically distributed along the bridge deck and istransmitted to the arch rib via the hangers and (6) the forceacting on arch ribs satisfies the film tension assumption

The simplified calculation model of a leaning-type archbridge is shown in Figure 2 In the figure 119909-119910-119911 is theglobal coordinate system and 119906-V-119908 is the local coordinatesystem 119906 V and119908 represent the lateral radial and tangentialdisplacement of arch ribs respectively120601

0represents the angle

between the main arch rib and the stable arch rib 120593 is theangle of a position in the arch ribs 120572

1is the central angle

of the main arch ribs 1205722is the central angle of the stable

arch rib and 119877 is the radius of the main and stable arch ribs119887(119909) is the distance between the main arch rib and the stablearch rib In the arch crown position the distance is 119887

0and

119887(119909) = 1198870+2119877 sin120601

0(1minuscos120593)The transverse braces between

themain and stable arch ribs are equidistantwith a distance of119889 the hangers are also arranged equidistantly with a distanceof 119889ℎAccording to the basic assumption and calculation

scheme the relationship between the lateral displacement 1199061198681

1199061198681198682

and the radial displacement V1198681

V1198681198682

of the main arch ribsand the stable arch ribs under the local coordinate is givenrespectively as

1199061198681

= 1199061 119906119868119868

2

=119910101584002

119910101584001

+ 119910101584002

119887 (119909) 120573 sin1205930+ 1199061cos1205930

V1198681

=119910101584001

119910101584001

+ 119910101584002

119887 (119909) 120573

V1198681198682

= minus119910101584002

119910101584001

+ 119910101584002

119887 (119909) 120573 cos1205930+ 1199061sin1205930

(1)

where 1199061represents the lateral displacement of the arch axis

of themain arch 120573 represents the global torsional angle of thetransverse brace caused by deformation of arch rib in radialplane 119887(119909) is the distance between the main arch rib and 1199101015840

01

and 119910101584002

represent the distance from the contraflexure point of

y99840002 y99840001

u

b(x)

v

Mb1

Mb2

EIIb I

IIb 120579c

120579c

EIIIIIu

EIIIu

1205792

12057921205791

1205930

Figure 3 Radial bending deformation of transverse brace

transverse bracersquos radial deformation to the main and stablearch ribs respectively (shown in Figure 3)

When there is a lateral deformation in the arch rib systemaxial strain of the main arch is given as

120576 = minusV1198681

119877+

1198891199081198681

119889119904+

1

2[(

1198891199061198681

119889119904)

2

+ (119889V1198681

119889119904)

2

] (2)

where 1199081198681

represents the tangential displacement of the mainarch ribs under the local coordinate

According to the basic assumption 120576 = 0

1198891199081198681

119889119904=V1198681

119877minus

1

2[(

1198891199061198681

119889119904)

2

+ (119889V1198681

119889119904)

2

] (3)

Because the arches are fixed at both ends int119904

1198891199081198681

= 0 so

int119904

(V1198681

119877)119889119904 =

1

2int119904

(1198891199061198681

119889119904)

2

119889119904 +1

2int119904

(119889V1198681

119889119904)

2

119889119904 (4)

4 Mathematical Problems in Engineering

Assuming the ends of the main arch rib and the stablearch rib are perfectly fixed the torsional angle and the lateraldisplacements of arch axis of themain and stable arch ribs aregiven respectively as

1205791= 1198621(1 minus cos

2120587120593

1205721

) (5a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (5b)

1199061= 1198622(1 minus cos

2120587120593

1205721

) (6a)

1199062= 1198622(1 minus cos

2120587120593

1205722

) (6b)

The lateral displacement of bridge deck system is given as

119906119889= 1198624(1 minus cos

2120587120593

1205721

) (7)

The global torsional angle of the transverse brace causedby deformation of arch rib in radial plane is given as

120573 = 1198625(1 minus cos

2120587120593

1205721

) (8)

The above equations satisfy the following displacementboundary conditions

(1) When 120593 = 0 and 120593 = 1205721 1205791= 0 1205791015840

1

= 0 1199061= 119906119889=

120573 = 0 and 11990610158401

= 1199061015840119889

= 1205731015840 = 0(2) When 120593 = 0 and 120593 = 120572

2 1205792= 0 1205791015840

2

= 0 1199062= 0 and

11990610158402

= 0

The lateral deflection curvatures along the V axis of themain and stable arch ribs are given as [16]

119870119868119906

=1205791

119877minus

11988921199061198681

1198891199042

119870119868119868119906

=1205792

119877minus

11988921199061198681198682

1198891199042

(9)

where 119870119868V and 119870119868119868V represent the lateral deflection curvaturealong V axis of the main and stable arch ribs respectively

The torsional deflection curvatures along 119908 axes are alsogiven as [16]

119870119868119908

=1198891205791

119889119904+

1

119877

11988921199061198681

119889119904

119870119868119868119908

=1198891205792

119889119904+

1

119877

1198891199061198681198682

119889119904

(10)

where 119870119868119908

and 119870119868119868119908

represent the torsional deflection cur-vatures along 119908 axes of the main and stable arch ribsrespectively

22 Energy Equations The lateral deformation energy of aleaning-type arch bridge can be written as

119882 = 119880119868119906

+ 119880119868119868119906

+ 119880119868119908

+ 119880119868119868119908

+ 119880119868119888119906

+ 119880119868119868119888119906

+ 119880119887119867

+ 119880119887V + 119881

119867+ 119880119889+ 119881

(11)

where 119880119868119906

119880119868119868119906

119880119868119908

119880119868119868119908

119880119868119888119906

and 119880119868119868119888119906

represent the totallateral bending deformation energy torsional deformationenergy and local bending deformation energy of the mainand stable arch ribs respectively119880

119887V and119880119887119867

are the bendingdeformation energy of the transverse braces in radial andtangential directions along the main arch ribrsquos axis 119881

119867is

the elastic potential energy caused by the horizontal part ofthe tension of the hangers 119880

119889represents the elastic potential

energy of bridge deck system and 119881 represents the potentialenergy of external loading applied to the arch bridge

The total lateral bending deformation energy of the mainand stable arch ribs are given in

119880119868119906

+ 119880119868119868119906

=1

2119864119868119868119868119906

int1199041

(119870119868119906

)2

119889119904 +1

2119864119868119868119868119868119868119906

int1199042

(119870119868119868119906

)2

119889119904

=1

2119864119868119868119868119906

[11986221

11986011

+11986222

119877311986012

+11986211198622

119877211986013]

+1

2119864119868119868119868119868119868119906

[11986222

119877311986021

+11986223

11987711986022

+11986225

11987711986023

+11986221198623

119877211986024+11986221198625

119877211986025+11986231198625

11987711986026]

(12)

where 119864119868119868119868119906

and 119864119868119868119868119868119868119906

are the lateral bending stiffness of themain and stable arch ribs Consider the following

11986011

= int1205721

0

(1 minus cos2120587120593

1205721

)2

119889120593

11986012

= int1205721

0

(2120587

1205721

)4

cos22120587120593

1205721

119889120593

11986013

= int1205721

0

minus2(2120587

1205721

)2

(1 minus cos2120587120593

1205721

) cos2120587120593

1205721

119889120593

11986021

= int1205722

0

cos1206010(2120587

1205722

)4

cos22120587120593

1205721

119889120593

11986022

= int1205722

0

(1 minus cos2120587120593

1205722

)2

119889120593

Mathematical Problems in Engineering 5

11986023

= int1205722

0

1

1198772(2120587

1205721

)4 119887(119909)2sin2120601

0

(1 + 1198902)2

cos22120587120593

1205721

119889120593

11986024

= int1205722

0

minus2 cos1206010(2120587

1205722

)2

cos2120587120593

1205722

(1 minus cos2120587120593

1205722

)119889120593

11986025

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

times (2120587

1205721

)2

(2120587

1205722

)2

cos2120587120593

1205721

cos2120587120593

1205722

119889120593

11986026= int1205722

0

minus2119887 (119909) sin120601

0

119877 (1 + 1198902)

(2120587

1205721

)2

cos2120587120593

1205721

(1minuscos2120587120593

1205722

)119889120593

(13)

The torsional deformation energy of the main and stablearch ribs is given in

119880119868119908

+ 119880119868119868119908

=1

2119866119868119879119868 int

1199041

(119870119868119908

)2

119889119904 +1

2119866119868119868119879119868119868 int

1199042

(119870119868119868119908

)2

119889119904

=1

2119866119868119879119868 [

11986221

11987711986111

+11986222

119877311986112

+11986211198622

119877211986113]

+1

2119866119868119868119879119868119868 [

11986222

119877311986121

+11986223

11987711986122

+11986225

11987711986123

+11986221198623

119877211986124+11986221198625

119877211986125+11986231198625

11987711986126]

(14)

where 119866119868119879119868 and 119866119868119868119879119868119868 are the torsional stiffness of the mainand stable arch ribs Consider the following

11986111

= int1205721

0

(2120587120593

1205721

)2

sin22120587120593

1205721

119889120593

11986112

= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986113

= int1205721

0

2(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986121

= int1205722

0

cos21206010(2120587

1205722

)4

sin22120587120593

1205721

119889120593

11986122

= int1205722

0

(2120587

1205722

)2

sin22120587120593

1205722

119889120593

11986123

= int1205722

0

1

1198772(2120587

1205721

)2 119887(119909)2sin2120601

0

(1 + 1198902)2

sin22120587120593

1205721

119889120593

11986124

= int1205722

0

2 cos1206010(2120587

1205722

)2

sin22120587120593

1205722

119889120593

y02

y0

y01

b(x) b

Mcu1

Mcu2

Mbh2

Mbh1

EbIbh

EIIIIIu

EIIIu

120574998400

120574998400998400

1205741

1205741

1205742

1205743

1205743

1205744

Figure 4 Bending deformation of arch ribs and transverse braces

11986125

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

11986126

= int1205722

0

2119887 (119909) sin1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

(15)

Transverse brace bending deformation occurs in the tan-gential direction along arch axis when there is a lateral buck-ling in the arch rib system as shown in Figure 4 In this casethe tangential bending deformation energy of the singletransverse brace is given as

119880119887119867

=1

2119864119887119868119887ℎ

int11991001

0

1198722119887ℎ1

1198891199100+

1

2119864119887119868119887ℎ

int11991002

0

1198722119887ℎ2

1198891199100 (16)

where 119864119887119868119887ℎ

is transverse bracersquos bending stiffness alongtangential direction of arch rib and 119910

01and 119910

02represent

the distances from the contraflexure point of transversebrace tangential deformation to the main and stable archribs respectively Therefore the length of transverse brace is119887(119909) = 119910

01+ 11991002

From Figure 4 the tangential bending moment of trans-verse brace near the main and stable arch ribs is given by thesuperposition principle

119872119887ℎ1

=4119864119887119868119887ℎ

1198871205741+

2119864119887119868119887ℎ

1198871205743

119872119887ℎ2

=2119864119887119868119887ℎ

1198871205741+

4119864119887119868119887ℎ

1198871205743

(17)

where 1205741represents the tangential angular rotation of the

transverse brace at the intersection points between the mainarch rib and the transverse brace and 120574

3represents the

tangential angular rotation of the transverse brace at the inter-section points between the stable arch rib and the transversebrace

It can be further derived as

11991001

=21205741+ 1205743

3 (1205741+ 1205743)119887 (119909)

11991002

=1205741+ 21205743

3 (1205741+ 1205743)119887 (119909)

(18)

6 Mathematical Problems in Engineering

119887(119909) can be replaced by a constant ℎ which is the distancebetween the main and stable arch ribs at quarter span forcalculation simplification Although the length of transversebrace 119887(119909) is not a constant the length of the transverse braceℎ at quarter span is very close to the average length of thetransverse braces of the entire bridge Such simplification canmake the derivation process become simple

The values of11991001and11991002are closely related to the bending

stiffness of arch ribs panel length bending stiffness andlength of transverse braces Due to the inclined angle andthe different stiffness of leaning-type arch bridgersquos main andstable arch ribs the tangential and radial deformation of thetransverse brace between the main and stable arch ribs differfrom that of the conventional arch bridges It is not a simpleldquoSrdquo shape For leaning-type arch bridges bending stiffness ofthe main arch rib is larger than that of the stable arch rib so11991001

gt 11991002 as shown in Figure 4 as an exceptional case for

conventional arch bridges 11991001

= 11991002 In order to obtain the

relationship between 11991001 11991002 a tangential mechanical model

of transverse brace along the arch axis is established whena lateral buckling occurs as shown in Figure 5(a) its cor-responding bending moment is shown in Figure 5(b) fromwhich (19) can be given as

119872119887ℎ1

119872119887ℎ2

=119864119868119864119887119868119868119906

119868119887ℎ119889 + 6119864119868119864119868119868119868119868

119906

119868119868119868119906

119887 (119909) cos1206010

6119864119868119864119868119868119868119868119906

119868119868119868119906

119887 (119909) cos1206010

+ 119864119868119868119864119887119868119868119868119906

119868119887ℎ119889 cos120601

0

= 1198901

(19)

where 1198901is a constant

From (17) and (19) it can be obtained that

119872119887ℎ1

119872119887ℎ2

=21205741+ 1205743

1205741+ 21205743

= 1198901 (20)

Assuming 1205743= 11988611205741 then

1198861=

1205743

1205741

=2 minus 1198901

21198901minus 1

(21)

thus from (18) and (21) the relationship of 11991001and 11991002can be

obtained as

11991001

=2 + 1198861

1 + 21198861

11991002 (22)

Substituting (17) and (18) into (16) the equation of tangentialbending energy of the transverse braces in the full arch ribrange is given as

119880119887119867

= int119904

119880119887119867

119889119889119904

=2119864119887119868119887ℎ

9119889ℎ[int1199041

21205741+ 1205743

1205741+ 1205743

(412057421

+ 412057411205743+ 12057423

) 119889119904

+int1199042

1205741+ 21205743

1205741+ 1205743

(12057421

+ 412057411205743+ 412057423

) 119889119904]

(23)

When the tangential local deformation occurs in themainand stable arch ribs assuming the bending moments of themain and stable arch ribs along the radial direction are 119872

1198881199061

and1198721198881199062

respectively the single-panel section arch ribrsquos localbending energy of the main and stable arch ribs can beexpressed as

119880119868

119888119906

+ 119880119868119868

119888119906

=1

2119864119868119868119868119906

int119889

11987221198881199061

1198891199090+

1

2119864119868119868119868119868119868119906

int119889

11987221198881199062

1198891199090

=6119864119868119868119868119906

11988912057422

+6119864119868119868119868119868119868119906

11988912057424

(24)

If the local bending energy of each single-panel sectionarch rib is the same the full-arch-rib local bending deforma-tion energy can be written as

119880119868119888119906

+ 119880119868119868119888119906

= int119904

119880119868

119888119906

119889119889119904 + int

119904

119880119868119868

119888119906

119889119889119904

= int119904

3119864119887119868119887ℎ

119889ℎ12057411205742119889119904 + int

119904

3119864119887119868119887ℎ

119889ℎ12057431205744119889119904

(25)

It could be obtained from Figure 4 that

21198721198881199061

= 119872119887ℎ1

21198721198881199062

= 119872119887ℎ2

(26)

Therefore from (23) (25) and (26) the local bendingenergy of arch rib and the tangential bending energy oftransverse brace are obtained as

119880119887119867

+ 119880119868119888119906

+ 119880119868119868119888119906

=3119864119887119868119887ℎ

119889ℎint1199041

(11987311205781+ 1205782) 120578112057410158402119889119904

+3119864119887119868119887ℎ

119889ℎint1199042

(11987321205783+ 1205784) 1205783120574101584010158402119889119904

= 119864119887119868119887ℎ

11986222

11987731198631+ 119864119887119868119887ℎ

11986222

11987731198632

(27)

where

1198631=

3119899

119887119904(11987311205781+ 1205782) 1205781int1205721

0

1198772(2120587

1205722

)2

sin22120587120593

1205721

119889120593

1198632=

3119899

119887119904(11987321205783+ 1205784) 1205783int1205722

0

1198772(2120587

1205722

)2

sin22120587120593

1205722

119889120593

1205741015840 = 1205741+ 1205742

12057410158401015840 = 1205743+ 1205744

1205781=

1205741

1205741015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868

119906

)

Mathematical Problems in Engineering 7

d2

EbIbhb1

EIIIIIu

1205930

EIIIu

q998400

(a) Mechanical model

1205930

Mbh2

Mbh1

(b) Bending moment diagram

Figure 5 Mechanical model and bending moment diagram in tangential direction

1205782=

1205742

1205741015840=

1

1 + (2ℎ119864119868119868119868119906

) (119889119864119887119868119887ℎ)

1205783=

1205743

12057410158401015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868119868119868

119906

)

1205784=

1205744

12057410158401015840=

1

1 + (2ℎ119864119868119868119868119868119868119906

) (119889119864119887119868119887ℎ)

1198731=

2

27sdot(1198861+ 2)

(1198861+ 1)

(4 + 41198861+ 11988621

)

1198732=

2

27sdot(11198861+ 2)

(11198861+ 1)

(1

11988621

+4

1198861

+ 4)

(28)

The mechanical model and moment diagram of arch ribsand transverse brace in radial direction are shown in Figure 3And the radial deformation energy of a single transversebrace can be written as

119880119887V =

1

2119864119887119868119887

int119910

1015840

01

0

(119872119887V1

119910101584001

11991010158400

)

2

11988911991010158400

+1

2119864119887119868119887Vint119910

1015840

02

0

(119872119887V2

119910101584002

11991010158400

)

2

11988911991010158400

(29)

where 119864119887119868119887V is the transverse bracersquos bending stiffness along

radial direction of arch rib and 119910101584001

and 119910101584002

represent the dis-tance from the contraflexure point of transverse bracersquos radialdeformation to the main and stable arch ribs respectively

The values of 119910101584001

and 119910101584002

are closely related to thebending stiffness of arch ribs the length of the arch ribsection the bending stiffness and length of the transversebraces For conventional arch bridges the bending stiffnessand inclined angle of two main arch ribs are the same so119910101584001

= 119910101584002

However for leaning-type arch bridges as the

bending stiffness of main arch rib is far larger than that ofstable arch rib so 1199101015840

01

gt 119910101584002

as shown in Figure 3 In orderto obtain the values of 1199101015840

01

and 119910101584002

radial mechanical modelof transverse brace along the arch central axis is establishedwhen a lateral buckling occurs as shown in Figure 6(a)The bending moment caused by the transverse bracesrsquo radialdeflection can be obtained by Castiglianorsquos theorem of mate-rialmechanics as shown in Figure 6(b) and (30) can be givenas

1198902=

119872119887V1

119872119887V2

=(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119868119879119868119868)119866119868119879119868

(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119879119868) 119866119868119868119879119868119868 (30)

According to the principle of similar triangles we arrived at

119910101584001

=1198902

1 + 1198902

119887 119910101584002

=1

1 + 1198902

119887 (31)

The radial deformation energy of transverse brace can beexpressed as

119880119887V =

6119864119887119868119887V

119889119887

1198902

1 + 1198902

int119904

(120573 minus 1205791)2

119889119904

+6119864119887119868119887V

119889119887

1

1 + 1198902

int119904

(120573 minus 1205792)2

119889119904

= 119864119887119868119887V(1198625minus 1198621)2

1198771198633+ 119864119887119868119887V11986223

1198771198634

+ 119864119887119868119887V11986225

1198771198634+ 119864119887119868119887V11986231198625

1198771198636

(32)

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

x

y

R

q

Lz

vu

w

1205721

1205722

dh

120593

(a) Elevation view

1206010

(b)Sideview

Stable arch rib

Main arch rib

Bridge deck

Transverse brace

d

x

z

b(x)

b 0

(c) Plan view

Figure 2 Calculation sketch of leaning-type arch bridge

(1) the main arch ribs and the stable arch ribs are fixed intheir ends (2) the axis of main and stable arch ribs are arc-shaped curves (3) the stiffness of arch ribs and transversebraces are constants (4) the axial deformation of arch ribin the buckling process is neglected (5) the external load isevenly and vertically distributed along the bridge deck and istransmitted to the arch rib via the hangers and (6) the forceacting on arch ribs satisfies the film tension assumption

The simplified calculation model of a leaning-type archbridge is shown in Figure 2 In the figure 119909-119910-119911 is theglobal coordinate system and 119906-V-119908 is the local coordinatesystem 119906 V and119908 represent the lateral radial and tangentialdisplacement of arch ribs respectively120601

0represents the angle

between the main arch rib and the stable arch rib 120593 is theangle of a position in the arch ribs 120572

1is the central angle

of the main arch ribs 1205722is the central angle of the stable

arch rib and 119877 is the radius of the main and stable arch ribs119887(119909) is the distance between the main arch rib and the stablearch rib In the arch crown position the distance is 119887

0and

119887(119909) = 1198870+2119877 sin120601

0(1minuscos120593)The transverse braces between

themain and stable arch ribs are equidistantwith a distance of119889 the hangers are also arranged equidistantly with a distanceof 119889ℎAccording to the basic assumption and calculation

scheme the relationship between the lateral displacement 1199061198681

1199061198681198682

and the radial displacement V1198681

V1198681198682

of the main arch ribsand the stable arch ribs under the local coordinate is givenrespectively as

1199061198681

= 1199061 119906119868119868

2

=119910101584002

119910101584001

+ 119910101584002

119887 (119909) 120573 sin1205930+ 1199061cos1205930

V1198681

=119910101584001

119910101584001

+ 119910101584002

119887 (119909) 120573

V1198681198682

= minus119910101584002

119910101584001

+ 119910101584002

119887 (119909) 120573 cos1205930+ 1199061sin1205930

(1)

where 1199061represents the lateral displacement of the arch axis

of themain arch 120573 represents the global torsional angle of thetransverse brace caused by deformation of arch rib in radialplane 119887(119909) is the distance between the main arch rib and 1199101015840

01

and 119910101584002

represent the distance from the contraflexure point of

y99840002 y99840001

u

b(x)

v

Mb1

Mb2

EIIb I

IIb 120579c

120579c

EIIIIIu

EIIIu

1205792

12057921205791

1205930

Figure 3 Radial bending deformation of transverse brace

transverse bracersquos radial deformation to the main and stablearch ribs respectively (shown in Figure 3)

When there is a lateral deformation in the arch rib systemaxial strain of the main arch is given as

120576 = minusV1198681

119877+

1198891199081198681

119889119904+

1

2[(

1198891199061198681

119889119904)

2

+ (119889V1198681

119889119904)

2

] (2)

where 1199081198681

represents the tangential displacement of the mainarch ribs under the local coordinate

According to the basic assumption 120576 = 0

1198891199081198681

119889119904=V1198681

119877minus

1

2[(

1198891199061198681

119889119904)

2

+ (119889V1198681

119889119904)

2

] (3)

Because the arches are fixed at both ends int119904

1198891199081198681

= 0 so

int119904

(V1198681

119877)119889119904 =

1

2int119904

(1198891199061198681

119889119904)

2

119889119904 +1

2int119904

(119889V1198681

119889119904)

2

119889119904 (4)

4 Mathematical Problems in Engineering

Assuming the ends of the main arch rib and the stablearch rib are perfectly fixed the torsional angle and the lateraldisplacements of arch axis of themain and stable arch ribs aregiven respectively as

1205791= 1198621(1 minus cos

2120587120593

1205721

) (5a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (5b)

1199061= 1198622(1 minus cos

2120587120593

1205721

) (6a)

1199062= 1198622(1 minus cos

2120587120593

1205722

) (6b)

The lateral displacement of bridge deck system is given as

119906119889= 1198624(1 minus cos

2120587120593

1205721

) (7)

The global torsional angle of the transverse brace causedby deformation of arch rib in radial plane is given as

120573 = 1198625(1 minus cos

2120587120593

1205721

) (8)

The above equations satisfy the following displacementboundary conditions

(1) When 120593 = 0 and 120593 = 1205721 1205791= 0 1205791015840

1

= 0 1199061= 119906119889=

120573 = 0 and 11990610158401

= 1199061015840119889

= 1205731015840 = 0(2) When 120593 = 0 and 120593 = 120572

2 1205792= 0 1205791015840

2

= 0 1199062= 0 and

11990610158402

= 0

The lateral deflection curvatures along the V axis of themain and stable arch ribs are given as [16]

119870119868119906

=1205791

119877minus

11988921199061198681

1198891199042

119870119868119868119906

=1205792

119877minus

11988921199061198681198682

1198891199042

(9)

where 119870119868V and 119870119868119868V represent the lateral deflection curvaturealong V axis of the main and stable arch ribs respectively

The torsional deflection curvatures along 119908 axes are alsogiven as [16]

119870119868119908

=1198891205791

119889119904+

1

119877

11988921199061198681

119889119904

119870119868119868119908

=1198891205792

119889119904+

1

119877

1198891199061198681198682

119889119904

(10)

where 119870119868119908

and 119870119868119868119908

represent the torsional deflection cur-vatures along 119908 axes of the main and stable arch ribsrespectively

22 Energy Equations The lateral deformation energy of aleaning-type arch bridge can be written as

119882 = 119880119868119906

+ 119880119868119868119906

+ 119880119868119908

+ 119880119868119868119908

+ 119880119868119888119906

+ 119880119868119868119888119906

+ 119880119887119867

+ 119880119887V + 119881

119867+ 119880119889+ 119881

(11)

where 119880119868119906

119880119868119868119906

119880119868119908

119880119868119868119908

119880119868119888119906

and 119880119868119868119888119906

represent the totallateral bending deformation energy torsional deformationenergy and local bending deformation energy of the mainand stable arch ribs respectively119880

119887V and119880119887119867

are the bendingdeformation energy of the transverse braces in radial andtangential directions along the main arch ribrsquos axis 119881

119867is

the elastic potential energy caused by the horizontal part ofthe tension of the hangers 119880

119889represents the elastic potential

energy of bridge deck system and 119881 represents the potentialenergy of external loading applied to the arch bridge

The total lateral bending deformation energy of the mainand stable arch ribs are given in

119880119868119906

+ 119880119868119868119906

=1

2119864119868119868119868119906

int1199041

(119870119868119906

)2

119889119904 +1

2119864119868119868119868119868119868119906

int1199042

(119870119868119868119906

)2

119889119904

=1

2119864119868119868119868119906

[11986221

11986011

+11986222

119877311986012

+11986211198622

119877211986013]

+1

2119864119868119868119868119868119868119906

[11986222

119877311986021

+11986223

11987711986022

+11986225

11987711986023

+11986221198623

119877211986024+11986221198625

119877211986025+11986231198625

11987711986026]

(12)

where 119864119868119868119868119906

and 119864119868119868119868119868119868119906

are the lateral bending stiffness of themain and stable arch ribs Consider the following

11986011

= int1205721

0

(1 minus cos2120587120593

1205721

)2

119889120593

11986012

= int1205721

0

(2120587

1205721

)4

cos22120587120593

1205721

119889120593

11986013

= int1205721

0

minus2(2120587

1205721

)2

(1 minus cos2120587120593

1205721

) cos2120587120593

1205721

119889120593

11986021

= int1205722

0

cos1206010(2120587

1205722

)4

cos22120587120593

1205721

119889120593

11986022

= int1205722

0

(1 minus cos2120587120593

1205722

)2

119889120593

Mathematical Problems in Engineering 5

11986023

= int1205722

0

1

1198772(2120587

1205721

)4 119887(119909)2sin2120601

0

(1 + 1198902)2

cos22120587120593

1205721

119889120593

11986024

= int1205722

0

minus2 cos1206010(2120587

1205722

)2

cos2120587120593

1205722

(1 minus cos2120587120593

1205722

)119889120593

11986025

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

times (2120587

1205721

)2

(2120587

1205722

)2

cos2120587120593

1205721

cos2120587120593

1205722

119889120593

11986026= int1205722

0

minus2119887 (119909) sin120601

0

119877 (1 + 1198902)

(2120587

1205721

)2

cos2120587120593

1205721

(1minuscos2120587120593

1205722

)119889120593

(13)

The torsional deformation energy of the main and stablearch ribs is given in

119880119868119908

+ 119880119868119868119908

=1

2119866119868119879119868 int

1199041

(119870119868119908

)2

119889119904 +1

2119866119868119868119879119868119868 int

1199042

(119870119868119868119908

)2

119889119904

=1

2119866119868119879119868 [

11986221

11987711986111

+11986222

119877311986112

+11986211198622

119877211986113]

+1

2119866119868119868119879119868119868 [

11986222

119877311986121

+11986223

11987711986122

+11986225

11987711986123

+11986221198623

119877211986124+11986221198625

119877211986125+11986231198625

11987711986126]

(14)

where 119866119868119879119868 and 119866119868119868119879119868119868 are the torsional stiffness of the mainand stable arch ribs Consider the following

11986111

= int1205721

0

(2120587120593

1205721

)2

sin22120587120593

1205721

119889120593

11986112

= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986113

= int1205721

0

2(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986121

= int1205722

0

cos21206010(2120587

1205722

)4

sin22120587120593

1205721

119889120593

11986122

= int1205722

0

(2120587

1205722

)2

sin22120587120593

1205722

119889120593

11986123

= int1205722

0

1

1198772(2120587

1205721

)2 119887(119909)2sin2120601

0

(1 + 1198902)2

sin22120587120593

1205721

119889120593

11986124

= int1205722

0

2 cos1206010(2120587

1205722

)2

sin22120587120593

1205722

119889120593

y02

y0

y01

b(x) b

Mcu1

Mcu2

Mbh2

Mbh1

EbIbh

EIIIIIu

EIIIu

120574998400

120574998400998400

1205741

1205741

1205742

1205743

1205743

1205744

Figure 4 Bending deformation of arch ribs and transverse braces

11986125

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

11986126

= int1205722

0

2119887 (119909) sin1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

(15)

Transverse brace bending deformation occurs in the tan-gential direction along arch axis when there is a lateral buck-ling in the arch rib system as shown in Figure 4 In this casethe tangential bending deformation energy of the singletransverse brace is given as

119880119887119867

=1

2119864119887119868119887ℎ

int11991001

0

1198722119887ℎ1

1198891199100+

1

2119864119887119868119887ℎ

int11991002

0

1198722119887ℎ2

1198891199100 (16)

where 119864119887119868119887ℎ

is transverse bracersquos bending stiffness alongtangential direction of arch rib and 119910

01and 119910

02represent

the distances from the contraflexure point of transversebrace tangential deformation to the main and stable archribs respectively Therefore the length of transverse brace is119887(119909) = 119910

01+ 11991002

From Figure 4 the tangential bending moment of trans-verse brace near the main and stable arch ribs is given by thesuperposition principle

119872119887ℎ1

=4119864119887119868119887ℎ

1198871205741+

2119864119887119868119887ℎ

1198871205743

119872119887ℎ2

=2119864119887119868119887ℎ

1198871205741+

4119864119887119868119887ℎ

1198871205743

(17)

where 1205741represents the tangential angular rotation of the

transverse brace at the intersection points between the mainarch rib and the transverse brace and 120574

3represents the

tangential angular rotation of the transverse brace at the inter-section points between the stable arch rib and the transversebrace

It can be further derived as

11991001

=21205741+ 1205743

3 (1205741+ 1205743)119887 (119909)

11991002

=1205741+ 21205743

3 (1205741+ 1205743)119887 (119909)

(18)

6 Mathematical Problems in Engineering

119887(119909) can be replaced by a constant ℎ which is the distancebetween the main and stable arch ribs at quarter span forcalculation simplification Although the length of transversebrace 119887(119909) is not a constant the length of the transverse braceℎ at quarter span is very close to the average length of thetransverse braces of the entire bridge Such simplification canmake the derivation process become simple

The values of11991001and11991002are closely related to the bending

stiffness of arch ribs panel length bending stiffness andlength of transverse braces Due to the inclined angle andthe different stiffness of leaning-type arch bridgersquos main andstable arch ribs the tangential and radial deformation of thetransverse brace between the main and stable arch ribs differfrom that of the conventional arch bridges It is not a simpleldquoSrdquo shape For leaning-type arch bridges bending stiffness ofthe main arch rib is larger than that of the stable arch rib so11991001

gt 11991002 as shown in Figure 4 as an exceptional case for

conventional arch bridges 11991001

= 11991002 In order to obtain the

relationship between 11991001 11991002 a tangential mechanical model

of transverse brace along the arch axis is established whena lateral buckling occurs as shown in Figure 5(a) its cor-responding bending moment is shown in Figure 5(b) fromwhich (19) can be given as

119872119887ℎ1

119872119887ℎ2

=119864119868119864119887119868119868119906

119868119887ℎ119889 + 6119864119868119864119868119868119868119868

119906

119868119868119868119906

119887 (119909) cos1206010

6119864119868119864119868119868119868119868119906

119868119868119868119906

119887 (119909) cos1206010

+ 119864119868119868119864119887119868119868119868119906

119868119887ℎ119889 cos120601

0

= 1198901

(19)

where 1198901is a constant

From (17) and (19) it can be obtained that

119872119887ℎ1

119872119887ℎ2

=21205741+ 1205743

1205741+ 21205743

= 1198901 (20)

Assuming 1205743= 11988611205741 then

1198861=

1205743

1205741

=2 minus 1198901

21198901minus 1

(21)

thus from (18) and (21) the relationship of 11991001and 11991002can be

obtained as

11991001

=2 + 1198861

1 + 21198861

11991002 (22)

Substituting (17) and (18) into (16) the equation of tangentialbending energy of the transverse braces in the full arch ribrange is given as

119880119887119867

= int119904

119880119887119867

119889119889119904

=2119864119887119868119887ℎ

9119889ℎ[int1199041

21205741+ 1205743

1205741+ 1205743

(412057421

+ 412057411205743+ 12057423

) 119889119904

+int1199042

1205741+ 21205743

1205741+ 1205743

(12057421

+ 412057411205743+ 412057423

) 119889119904]

(23)

When the tangential local deformation occurs in themainand stable arch ribs assuming the bending moments of themain and stable arch ribs along the radial direction are 119872

1198881199061

and1198721198881199062

respectively the single-panel section arch ribrsquos localbending energy of the main and stable arch ribs can beexpressed as

119880119868

119888119906

+ 119880119868119868

119888119906

=1

2119864119868119868119868119906

int119889

11987221198881199061

1198891199090+

1

2119864119868119868119868119868119868119906

int119889

11987221198881199062

1198891199090

=6119864119868119868119868119906

11988912057422

+6119864119868119868119868119868119868119906

11988912057424

(24)

If the local bending energy of each single-panel sectionarch rib is the same the full-arch-rib local bending deforma-tion energy can be written as

119880119868119888119906

+ 119880119868119868119888119906

= int119904

119880119868

119888119906

119889119889119904 + int

119904

119880119868119868

119888119906

119889119889119904

= int119904

3119864119887119868119887ℎ

119889ℎ12057411205742119889119904 + int

119904

3119864119887119868119887ℎ

119889ℎ12057431205744119889119904

(25)

It could be obtained from Figure 4 that

21198721198881199061

= 119872119887ℎ1

21198721198881199062

= 119872119887ℎ2

(26)

Therefore from (23) (25) and (26) the local bendingenergy of arch rib and the tangential bending energy oftransverse brace are obtained as

119880119887119867

+ 119880119868119888119906

+ 119880119868119868119888119906

=3119864119887119868119887ℎ

119889ℎint1199041

(11987311205781+ 1205782) 120578112057410158402119889119904

+3119864119887119868119887ℎ

119889ℎint1199042

(11987321205783+ 1205784) 1205783120574101584010158402119889119904

= 119864119887119868119887ℎ

11986222

11987731198631+ 119864119887119868119887ℎ

11986222

11987731198632

(27)

where

1198631=

3119899

119887119904(11987311205781+ 1205782) 1205781int1205721

0

1198772(2120587

1205722

)2

sin22120587120593

1205721

119889120593

1198632=

3119899

119887119904(11987321205783+ 1205784) 1205783int1205722

0

1198772(2120587

1205722

)2

sin22120587120593

1205722

119889120593

1205741015840 = 1205741+ 1205742

12057410158401015840 = 1205743+ 1205744

1205781=

1205741

1205741015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868

119906

)

Mathematical Problems in Engineering 7

d2

EbIbhb1

EIIIIIu

1205930

EIIIu

q998400

(a) Mechanical model

1205930

Mbh2

Mbh1

(b) Bending moment diagram

Figure 5 Mechanical model and bending moment diagram in tangential direction

1205782=

1205742

1205741015840=

1

1 + (2ℎ119864119868119868119868119906

) (119889119864119887119868119887ℎ)

1205783=

1205743

12057410158401015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868119868119868

119906

)

1205784=

1205744

12057410158401015840=

1

1 + (2ℎ119864119868119868119868119868119868119906

) (119889119864119887119868119887ℎ)

1198731=

2

27sdot(1198861+ 2)

(1198861+ 1)

(4 + 41198861+ 11988621

)

1198732=

2

27sdot(11198861+ 2)

(11198861+ 1)

(1

11988621

+4

1198861

+ 4)

(28)

The mechanical model and moment diagram of arch ribsand transverse brace in radial direction are shown in Figure 3And the radial deformation energy of a single transversebrace can be written as

119880119887V =

1

2119864119887119868119887

int119910

1015840

01

0

(119872119887V1

119910101584001

11991010158400

)

2

11988911991010158400

+1

2119864119887119868119887Vint119910

1015840

02

0

(119872119887V2

119910101584002

11991010158400

)

2

11988911991010158400

(29)

where 119864119887119868119887V is the transverse bracersquos bending stiffness along

radial direction of arch rib and 119910101584001

and 119910101584002

represent the dis-tance from the contraflexure point of transverse bracersquos radialdeformation to the main and stable arch ribs respectively

The values of 119910101584001

and 119910101584002

are closely related to thebending stiffness of arch ribs the length of the arch ribsection the bending stiffness and length of the transversebraces For conventional arch bridges the bending stiffnessand inclined angle of two main arch ribs are the same so119910101584001

= 119910101584002

However for leaning-type arch bridges as the

bending stiffness of main arch rib is far larger than that ofstable arch rib so 1199101015840

01

gt 119910101584002

as shown in Figure 3 In orderto obtain the values of 1199101015840

01

and 119910101584002

radial mechanical modelof transverse brace along the arch central axis is establishedwhen a lateral buckling occurs as shown in Figure 6(a)The bending moment caused by the transverse bracesrsquo radialdeflection can be obtained by Castiglianorsquos theorem of mate-rialmechanics as shown in Figure 6(b) and (30) can be givenas

1198902=

119872119887V1

119872119887V2

=(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119868119879119868119868)119866119868119879119868

(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119879119868) 119866119868119868119879119868119868 (30)

According to the principle of similar triangles we arrived at

119910101584001

=1198902

1 + 1198902

119887 119910101584002

=1

1 + 1198902

119887 (31)

The radial deformation energy of transverse brace can beexpressed as

119880119887V =

6119864119887119868119887V

119889119887

1198902

1 + 1198902

int119904

(120573 minus 1205791)2

119889119904

+6119864119887119868119887V

119889119887

1

1 + 1198902

int119904

(120573 minus 1205792)2

119889119904

= 119864119887119868119887V(1198625minus 1198621)2

1198771198633+ 119864119887119868119887V11986223

1198771198634

+ 119864119887119868119887V11986225

1198771198634+ 119864119887119868119887V11986231198625

1198771198636

(32)

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Assuming the ends of the main arch rib and the stablearch rib are perfectly fixed the torsional angle and the lateraldisplacements of arch axis of themain and stable arch ribs aregiven respectively as

1205791= 1198621(1 minus cos

2120587120593

1205721

) (5a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (5b)

1199061= 1198622(1 minus cos

2120587120593

1205721

) (6a)

1199062= 1198622(1 minus cos

2120587120593

1205722

) (6b)

The lateral displacement of bridge deck system is given as

119906119889= 1198624(1 minus cos

2120587120593

1205721

) (7)

The global torsional angle of the transverse brace causedby deformation of arch rib in radial plane is given as

120573 = 1198625(1 minus cos

2120587120593

1205721

) (8)

The above equations satisfy the following displacementboundary conditions

(1) When 120593 = 0 and 120593 = 1205721 1205791= 0 1205791015840

1

= 0 1199061= 119906119889=

120573 = 0 and 11990610158401

= 1199061015840119889

= 1205731015840 = 0(2) When 120593 = 0 and 120593 = 120572

2 1205792= 0 1205791015840

2

= 0 1199062= 0 and

11990610158402

= 0

The lateral deflection curvatures along the V axis of themain and stable arch ribs are given as [16]

119870119868119906

=1205791

119877minus

11988921199061198681

1198891199042

119870119868119868119906

=1205792

119877minus

11988921199061198681198682

1198891199042

(9)

where 119870119868V and 119870119868119868V represent the lateral deflection curvaturealong V axis of the main and stable arch ribs respectively

The torsional deflection curvatures along 119908 axes are alsogiven as [16]

119870119868119908

=1198891205791

119889119904+

1

119877

11988921199061198681

119889119904

119870119868119868119908

=1198891205792

119889119904+

1

119877

1198891199061198681198682

119889119904

(10)

where 119870119868119908

and 119870119868119868119908

represent the torsional deflection cur-vatures along 119908 axes of the main and stable arch ribsrespectively

22 Energy Equations The lateral deformation energy of aleaning-type arch bridge can be written as

119882 = 119880119868119906

+ 119880119868119868119906

+ 119880119868119908

+ 119880119868119868119908

+ 119880119868119888119906

+ 119880119868119868119888119906

+ 119880119887119867

+ 119880119887V + 119881

119867+ 119880119889+ 119881

(11)

where 119880119868119906

119880119868119868119906

119880119868119908

119880119868119868119908

119880119868119888119906

and 119880119868119868119888119906

represent the totallateral bending deformation energy torsional deformationenergy and local bending deformation energy of the mainand stable arch ribs respectively119880

119887V and119880119887119867

are the bendingdeformation energy of the transverse braces in radial andtangential directions along the main arch ribrsquos axis 119881

119867is

the elastic potential energy caused by the horizontal part ofthe tension of the hangers 119880

119889represents the elastic potential

energy of bridge deck system and 119881 represents the potentialenergy of external loading applied to the arch bridge

The total lateral bending deformation energy of the mainand stable arch ribs are given in

119880119868119906

+ 119880119868119868119906

=1

2119864119868119868119868119906

int1199041

(119870119868119906

)2

119889119904 +1

2119864119868119868119868119868119868119906

int1199042

(119870119868119868119906

)2

119889119904

=1

2119864119868119868119868119906

[11986221

11986011

+11986222

119877311986012

+11986211198622

119877211986013]

+1

2119864119868119868119868119868119868119906

[11986222

119877311986021

+11986223

11987711986022

+11986225

11987711986023

+11986221198623

119877211986024+11986221198625

119877211986025+11986231198625

11987711986026]

(12)

where 119864119868119868119868119906

and 119864119868119868119868119868119868119906

are the lateral bending stiffness of themain and stable arch ribs Consider the following

11986011

= int1205721

0

(1 minus cos2120587120593

1205721

)2

119889120593

11986012

= int1205721

0

(2120587

1205721

)4

cos22120587120593

1205721

119889120593

11986013

= int1205721

0

minus2(2120587

1205721

)2

(1 minus cos2120587120593

1205721

) cos2120587120593

1205721

119889120593

11986021

= int1205722

0

cos1206010(2120587

1205722

)4

cos22120587120593

1205721

119889120593

11986022

= int1205722

0

(1 minus cos2120587120593

1205722

)2

119889120593

Mathematical Problems in Engineering 5

11986023

= int1205722

0

1

1198772(2120587

1205721

)4 119887(119909)2sin2120601

0

(1 + 1198902)2

cos22120587120593

1205721

119889120593

11986024

= int1205722

0

minus2 cos1206010(2120587

1205722

)2

cos2120587120593

1205722

(1 minus cos2120587120593

1205722

)119889120593

11986025

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

times (2120587

1205721

)2

(2120587

1205722

)2

cos2120587120593

1205721

cos2120587120593

1205722

119889120593

11986026= int1205722

0

minus2119887 (119909) sin120601

0

119877 (1 + 1198902)

(2120587

1205721

)2

cos2120587120593

1205721

(1minuscos2120587120593

1205722

)119889120593

(13)

The torsional deformation energy of the main and stablearch ribs is given in

119880119868119908

+ 119880119868119868119908

=1

2119866119868119879119868 int

1199041

(119870119868119908

)2

119889119904 +1

2119866119868119868119879119868119868 int

1199042

(119870119868119868119908

)2

119889119904

=1

2119866119868119879119868 [

11986221

11987711986111

+11986222

119877311986112

+11986211198622

119877211986113]

+1

2119866119868119868119879119868119868 [

11986222

119877311986121

+11986223

11987711986122

+11986225

11987711986123

+11986221198623

119877211986124+11986221198625

119877211986125+11986231198625

11987711986126]

(14)

where 119866119868119879119868 and 119866119868119868119879119868119868 are the torsional stiffness of the mainand stable arch ribs Consider the following

11986111

= int1205721

0

(2120587120593

1205721

)2

sin22120587120593

1205721

119889120593

11986112

= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986113

= int1205721

0

2(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986121

= int1205722

0

cos21206010(2120587

1205722

)4

sin22120587120593

1205721

119889120593

11986122

= int1205722

0

(2120587

1205722

)2

sin22120587120593

1205722

119889120593

11986123

= int1205722

0

1

1198772(2120587

1205721

)2 119887(119909)2sin2120601

0

(1 + 1198902)2

sin22120587120593

1205721

119889120593

11986124

= int1205722

0

2 cos1206010(2120587

1205722

)2

sin22120587120593

1205722

119889120593

y02

y0

y01

b(x) b

Mcu1

Mcu2

Mbh2

Mbh1

EbIbh

EIIIIIu

EIIIu

120574998400

120574998400998400

1205741

1205741

1205742

1205743

1205743

1205744

Figure 4 Bending deformation of arch ribs and transverse braces

11986125

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

11986126

= int1205722

0

2119887 (119909) sin1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

(15)

Transverse brace bending deformation occurs in the tan-gential direction along arch axis when there is a lateral buck-ling in the arch rib system as shown in Figure 4 In this casethe tangential bending deformation energy of the singletransverse brace is given as

119880119887119867

=1

2119864119887119868119887ℎ

int11991001

0

1198722119887ℎ1

1198891199100+

1

2119864119887119868119887ℎ

int11991002

0

1198722119887ℎ2

1198891199100 (16)

where 119864119887119868119887ℎ

is transverse bracersquos bending stiffness alongtangential direction of arch rib and 119910

01and 119910

02represent

the distances from the contraflexure point of transversebrace tangential deformation to the main and stable archribs respectively Therefore the length of transverse brace is119887(119909) = 119910

01+ 11991002

From Figure 4 the tangential bending moment of trans-verse brace near the main and stable arch ribs is given by thesuperposition principle

119872119887ℎ1

=4119864119887119868119887ℎ

1198871205741+

2119864119887119868119887ℎ

1198871205743

119872119887ℎ2

=2119864119887119868119887ℎ

1198871205741+

4119864119887119868119887ℎ

1198871205743

(17)

where 1205741represents the tangential angular rotation of the

transverse brace at the intersection points between the mainarch rib and the transverse brace and 120574

3represents the

tangential angular rotation of the transverse brace at the inter-section points between the stable arch rib and the transversebrace

It can be further derived as

11991001

=21205741+ 1205743

3 (1205741+ 1205743)119887 (119909)

11991002

=1205741+ 21205743

3 (1205741+ 1205743)119887 (119909)

(18)

6 Mathematical Problems in Engineering

119887(119909) can be replaced by a constant ℎ which is the distancebetween the main and stable arch ribs at quarter span forcalculation simplification Although the length of transversebrace 119887(119909) is not a constant the length of the transverse braceℎ at quarter span is very close to the average length of thetransverse braces of the entire bridge Such simplification canmake the derivation process become simple

The values of11991001and11991002are closely related to the bending

stiffness of arch ribs panel length bending stiffness andlength of transverse braces Due to the inclined angle andthe different stiffness of leaning-type arch bridgersquos main andstable arch ribs the tangential and radial deformation of thetransverse brace between the main and stable arch ribs differfrom that of the conventional arch bridges It is not a simpleldquoSrdquo shape For leaning-type arch bridges bending stiffness ofthe main arch rib is larger than that of the stable arch rib so11991001

gt 11991002 as shown in Figure 4 as an exceptional case for

conventional arch bridges 11991001

= 11991002 In order to obtain the

relationship between 11991001 11991002 a tangential mechanical model

of transverse brace along the arch axis is established whena lateral buckling occurs as shown in Figure 5(a) its cor-responding bending moment is shown in Figure 5(b) fromwhich (19) can be given as

119872119887ℎ1

119872119887ℎ2

=119864119868119864119887119868119868119906

119868119887ℎ119889 + 6119864119868119864119868119868119868119868

119906

119868119868119868119906

119887 (119909) cos1206010

6119864119868119864119868119868119868119868119906

119868119868119868119906

119887 (119909) cos1206010

+ 119864119868119868119864119887119868119868119868119906

119868119887ℎ119889 cos120601

0

= 1198901

(19)

where 1198901is a constant

From (17) and (19) it can be obtained that

119872119887ℎ1

119872119887ℎ2

=21205741+ 1205743

1205741+ 21205743

= 1198901 (20)

Assuming 1205743= 11988611205741 then

1198861=

1205743

1205741

=2 minus 1198901

21198901minus 1

(21)

thus from (18) and (21) the relationship of 11991001and 11991002can be

obtained as

11991001

=2 + 1198861

1 + 21198861

11991002 (22)

Substituting (17) and (18) into (16) the equation of tangentialbending energy of the transverse braces in the full arch ribrange is given as

119880119887119867

= int119904

119880119887119867

119889119889119904

=2119864119887119868119887ℎ

9119889ℎ[int1199041

21205741+ 1205743

1205741+ 1205743

(412057421

+ 412057411205743+ 12057423

) 119889119904

+int1199042

1205741+ 21205743

1205741+ 1205743

(12057421

+ 412057411205743+ 412057423

) 119889119904]

(23)

When the tangential local deformation occurs in themainand stable arch ribs assuming the bending moments of themain and stable arch ribs along the radial direction are 119872

1198881199061

and1198721198881199062

respectively the single-panel section arch ribrsquos localbending energy of the main and stable arch ribs can beexpressed as

119880119868

119888119906

+ 119880119868119868

119888119906

=1

2119864119868119868119868119906

int119889

11987221198881199061

1198891199090+

1

2119864119868119868119868119868119868119906

int119889

11987221198881199062

1198891199090

=6119864119868119868119868119906

11988912057422

+6119864119868119868119868119868119868119906

11988912057424

(24)

If the local bending energy of each single-panel sectionarch rib is the same the full-arch-rib local bending deforma-tion energy can be written as

119880119868119888119906

+ 119880119868119868119888119906

= int119904

119880119868

119888119906

119889119889119904 + int

119904

119880119868119868

119888119906

119889119889119904

= int119904

3119864119887119868119887ℎ

119889ℎ12057411205742119889119904 + int

119904

3119864119887119868119887ℎ

119889ℎ12057431205744119889119904

(25)

It could be obtained from Figure 4 that

21198721198881199061

= 119872119887ℎ1

21198721198881199062

= 119872119887ℎ2

(26)

Therefore from (23) (25) and (26) the local bendingenergy of arch rib and the tangential bending energy oftransverse brace are obtained as

119880119887119867

+ 119880119868119888119906

+ 119880119868119868119888119906

=3119864119887119868119887ℎ

119889ℎint1199041

(11987311205781+ 1205782) 120578112057410158402119889119904

+3119864119887119868119887ℎ

119889ℎint1199042

(11987321205783+ 1205784) 1205783120574101584010158402119889119904

= 119864119887119868119887ℎ

11986222

11987731198631+ 119864119887119868119887ℎ

11986222

11987731198632

(27)

where

1198631=

3119899

119887119904(11987311205781+ 1205782) 1205781int1205721

0

1198772(2120587

1205722

)2

sin22120587120593

1205721

119889120593

1198632=

3119899

119887119904(11987321205783+ 1205784) 1205783int1205722

0

1198772(2120587

1205722

)2

sin22120587120593

1205722

119889120593

1205741015840 = 1205741+ 1205742

12057410158401015840 = 1205743+ 1205744

1205781=

1205741

1205741015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868

119906

)

Mathematical Problems in Engineering 7

d2

EbIbhb1

EIIIIIu

1205930

EIIIu

q998400

(a) Mechanical model

1205930

Mbh2

Mbh1

(b) Bending moment diagram

Figure 5 Mechanical model and bending moment diagram in tangential direction

1205782=

1205742

1205741015840=

1

1 + (2ℎ119864119868119868119868119906

) (119889119864119887119868119887ℎ)

1205783=

1205743

12057410158401015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868119868119868

119906

)

1205784=

1205744

12057410158401015840=

1

1 + (2ℎ119864119868119868119868119868119868119906

) (119889119864119887119868119887ℎ)

1198731=

2

27sdot(1198861+ 2)

(1198861+ 1)

(4 + 41198861+ 11988621

)

1198732=

2

27sdot(11198861+ 2)

(11198861+ 1)

(1

11988621

+4

1198861

+ 4)

(28)

The mechanical model and moment diagram of arch ribsand transverse brace in radial direction are shown in Figure 3And the radial deformation energy of a single transversebrace can be written as

119880119887V =

1

2119864119887119868119887

int119910

1015840

01

0

(119872119887V1

119910101584001

11991010158400

)

2

11988911991010158400

+1

2119864119887119868119887Vint119910

1015840

02

0

(119872119887V2

119910101584002

11991010158400

)

2

11988911991010158400

(29)

where 119864119887119868119887V is the transverse bracersquos bending stiffness along

radial direction of arch rib and 119910101584001

and 119910101584002

represent the dis-tance from the contraflexure point of transverse bracersquos radialdeformation to the main and stable arch ribs respectively

The values of 119910101584001

and 119910101584002

are closely related to thebending stiffness of arch ribs the length of the arch ribsection the bending stiffness and length of the transversebraces For conventional arch bridges the bending stiffnessand inclined angle of two main arch ribs are the same so119910101584001

= 119910101584002

However for leaning-type arch bridges as the

bending stiffness of main arch rib is far larger than that ofstable arch rib so 1199101015840

01

gt 119910101584002

as shown in Figure 3 In orderto obtain the values of 1199101015840

01

and 119910101584002

radial mechanical modelof transverse brace along the arch central axis is establishedwhen a lateral buckling occurs as shown in Figure 6(a)The bending moment caused by the transverse bracesrsquo radialdeflection can be obtained by Castiglianorsquos theorem of mate-rialmechanics as shown in Figure 6(b) and (30) can be givenas

1198902=

119872119887V1

119872119887V2

=(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119868119879119868119868)119866119868119879119868

(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119879119868) 119866119868119868119879119868119868 (30)

According to the principle of similar triangles we arrived at

119910101584001

=1198902

1 + 1198902

119887 119910101584002

=1

1 + 1198902

119887 (31)

The radial deformation energy of transverse brace can beexpressed as

119880119887V =

6119864119887119868119887V

119889119887

1198902

1 + 1198902

int119904

(120573 minus 1205791)2

119889119904

+6119864119887119868119887V

119889119887

1

1 + 1198902

int119904

(120573 minus 1205792)2

119889119904

= 119864119887119868119887V(1198625minus 1198621)2

1198771198633+ 119864119887119868119887V11986223

1198771198634

+ 119864119887119868119887V11986225

1198771198634+ 119864119887119868119887V11986231198625

1198771198636

(32)

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

11986023

= int1205722

0

1

1198772(2120587

1205721

)4 119887(119909)2sin2120601

0

(1 + 1198902)2

cos22120587120593

1205721

119889120593

11986024

= int1205722

0

minus2 cos1206010(2120587

1205722

)2

cos2120587120593

1205722

(1 minus cos2120587120593

1205722

)119889120593

11986025

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

times (2120587

1205721

)2

(2120587

1205722

)2

cos2120587120593

1205721

cos2120587120593

1205722

119889120593

11986026= int1205722

0

minus2119887 (119909) sin120601

0

119877 (1 + 1198902)

(2120587

1205721

)2

cos2120587120593

1205721

(1minuscos2120587120593

1205722

)119889120593

(13)

The torsional deformation energy of the main and stablearch ribs is given in

119880119868119908

+ 119880119868119868119908

=1

2119866119868119879119868 int

1199041

(119870119868119908

)2

119889119904 +1

2119866119868119868119879119868119868 int

1199042

(119870119868119868119908

)2

119889119904

=1

2119866119868119879119868 [

11986221

11987711986111

+11986222

119877311986112

+11986211198622

119877211986113]

+1

2119866119868119868119879119868119868 [

11986222

119877311986121

+11986223

11987711986122

+11986225

11987711986123

+11986221198623

119877211986124+11986221198625

119877211986125+11986231198625

11987711986126]

(14)

where 119866119868119879119868 and 119866119868119868119879119868119868 are the torsional stiffness of the mainand stable arch ribs Consider the following

11986111

= int1205721

0

(2120587120593

1205721

)2

sin22120587120593

1205721

119889120593

11986112

= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986113

= int1205721

0

2(2120587

1205721

)2

sin22120587120593

1205721

119889120593

11986121

= int1205722

0

cos21206010(2120587

1205722

)4

sin22120587120593

1205721

119889120593

11986122

= int1205722

0

(2120587

1205722

)2

sin22120587120593

1205722

119889120593

11986123

= int1205722

0

1

1198772(2120587

1205721

)2 119887(119909)2sin2120601

0

(1 + 1198902)2

sin22120587120593

1205721

119889120593

11986124

= int1205722

0

2 cos1206010(2120587

1205722

)2

sin22120587120593

1205722

119889120593

y02

y0

y01

b(x) b

Mcu1

Mcu2

Mbh2

Mbh1

EbIbh

EIIIIIu

EIIIu

120574998400

120574998400998400

1205741

1205741

1205742

1205743

1205743

1205744

Figure 4 Bending deformation of arch ribs and transverse braces

11986125

= int1205722

0

2119887 (119909) sin1206010cos1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

11986126

= int1205722

0

2119887 (119909) sin1206010

119877 (1 + 1198902)

2120587

1205721

2120587

1205722

sin2120587120593

1205721

sin2120587120593

1205722

119889120593

(15)

Transverse brace bending deformation occurs in the tan-gential direction along arch axis when there is a lateral buck-ling in the arch rib system as shown in Figure 4 In this casethe tangential bending deformation energy of the singletransverse brace is given as

119880119887119867

=1

2119864119887119868119887ℎ

int11991001

0

1198722119887ℎ1

1198891199100+

1

2119864119887119868119887ℎ

int11991002

0

1198722119887ℎ2

1198891199100 (16)

where 119864119887119868119887ℎ

is transverse bracersquos bending stiffness alongtangential direction of arch rib and 119910

01and 119910

02represent

the distances from the contraflexure point of transversebrace tangential deformation to the main and stable archribs respectively Therefore the length of transverse brace is119887(119909) = 119910

01+ 11991002

From Figure 4 the tangential bending moment of trans-verse brace near the main and stable arch ribs is given by thesuperposition principle

119872119887ℎ1

=4119864119887119868119887ℎ

1198871205741+

2119864119887119868119887ℎ

1198871205743

119872119887ℎ2

=2119864119887119868119887ℎ

1198871205741+

4119864119887119868119887ℎ

1198871205743

(17)

where 1205741represents the tangential angular rotation of the

transverse brace at the intersection points between the mainarch rib and the transverse brace and 120574

3represents the

tangential angular rotation of the transverse brace at the inter-section points between the stable arch rib and the transversebrace

It can be further derived as

11991001

=21205741+ 1205743

3 (1205741+ 1205743)119887 (119909)

11991002

=1205741+ 21205743

3 (1205741+ 1205743)119887 (119909)

(18)

6 Mathematical Problems in Engineering

119887(119909) can be replaced by a constant ℎ which is the distancebetween the main and stable arch ribs at quarter span forcalculation simplification Although the length of transversebrace 119887(119909) is not a constant the length of the transverse braceℎ at quarter span is very close to the average length of thetransverse braces of the entire bridge Such simplification canmake the derivation process become simple

The values of11991001and11991002are closely related to the bending

stiffness of arch ribs panel length bending stiffness andlength of transverse braces Due to the inclined angle andthe different stiffness of leaning-type arch bridgersquos main andstable arch ribs the tangential and radial deformation of thetransverse brace between the main and stable arch ribs differfrom that of the conventional arch bridges It is not a simpleldquoSrdquo shape For leaning-type arch bridges bending stiffness ofthe main arch rib is larger than that of the stable arch rib so11991001

gt 11991002 as shown in Figure 4 as an exceptional case for

conventional arch bridges 11991001

= 11991002 In order to obtain the

relationship between 11991001 11991002 a tangential mechanical model

of transverse brace along the arch axis is established whena lateral buckling occurs as shown in Figure 5(a) its cor-responding bending moment is shown in Figure 5(b) fromwhich (19) can be given as

119872119887ℎ1

119872119887ℎ2

=119864119868119864119887119868119868119906

119868119887ℎ119889 + 6119864119868119864119868119868119868119868

119906

119868119868119868119906

119887 (119909) cos1206010

6119864119868119864119868119868119868119868119906

119868119868119868119906

119887 (119909) cos1206010

+ 119864119868119868119864119887119868119868119868119906

119868119887ℎ119889 cos120601

0

= 1198901

(19)

where 1198901is a constant

From (17) and (19) it can be obtained that

119872119887ℎ1

119872119887ℎ2

=21205741+ 1205743

1205741+ 21205743

= 1198901 (20)

Assuming 1205743= 11988611205741 then

1198861=

1205743

1205741

=2 minus 1198901

21198901minus 1

(21)

thus from (18) and (21) the relationship of 11991001and 11991002can be

obtained as

11991001

=2 + 1198861

1 + 21198861

11991002 (22)

Substituting (17) and (18) into (16) the equation of tangentialbending energy of the transverse braces in the full arch ribrange is given as

119880119887119867

= int119904

119880119887119867

119889119889119904

=2119864119887119868119887ℎ

9119889ℎ[int1199041

21205741+ 1205743

1205741+ 1205743

(412057421

+ 412057411205743+ 12057423

) 119889119904

+int1199042

1205741+ 21205743

1205741+ 1205743

(12057421

+ 412057411205743+ 412057423

) 119889119904]

(23)

When the tangential local deformation occurs in themainand stable arch ribs assuming the bending moments of themain and stable arch ribs along the radial direction are 119872

1198881199061

and1198721198881199062

respectively the single-panel section arch ribrsquos localbending energy of the main and stable arch ribs can beexpressed as

119880119868

119888119906

+ 119880119868119868

119888119906

=1

2119864119868119868119868119906

int119889

11987221198881199061

1198891199090+

1

2119864119868119868119868119868119868119906

int119889

11987221198881199062

1198891199090

=6119864119868119868119868119906

11988912057422

+6119864119868119868119868119868119868119906

11988912057424

(24)

If the local bending energy of each single-panel sectionarch rib is the same the full-arch-rib local bending deforma-tion energy can be written as

119880119868119888119906

+ 119880119868119868119888119906

= int119904

119880119868

119888119906

119889119889119904 + int

119904

119880119868119868

119888119906

119889119889119904

= int119904

3119864119887119868119887ℎ

119889ℎ12057411205742119889119904 + int

119904

3119864119887119868119887ℎ

119889ℎ12057431205744119889119904

(25)

It could be obtained from Figure 4 that

21198721198881199061

= 119872119887ℎ1

21198721198881199062

= 119872119887ℎ2

(26)

Therefore from (23) (25) and (26) the local bendingenergy of arch rib and the tangential bending energy oftransverse brace are obtained as

119880119887119867

+ 119880119868119888119906

+ 119880119868119868119888119906

=3119864119887119868119887ℎ

119889ℎint1199041

(11987311205781+ 1205782) 120578112057410158402119889119904

+3119864119887119868119887ℎ

119889ℎint1199042

(11987321205783+ 1205784) 1205783120574101584010158402119889119904

= 119864119887119868119887ℎ

11986222

11987731198631+ 119864119887119868119887ℎ

11986222

11987731198632

(27)

where

1198631=

3119899

119887119904(11987311205781+ 1205782) 1205781int1205721

0

1198772(2120587

1205722

)2

sin22120587120593

1205721

119889120593

1198632=

3119899

119887119904(11987321205783+ 1205784) 1205783int1205722

0

1198772(2120587

1205722

)2

sin22120587120593

1205722

119889120593

1205741015840 = 1205741+ 1205742

12057410158401015840 = 1205743+ 1205744

1205781=

1205741

1205741015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868

119906

)

Mathematical Problems in Engineering 7

d2

EbIbhb1

EIIIIIu

1205930

EIIIu

q998400

(a) Mechanical model

1205930

Mbh2

Mbh1

(b) Bending moment diagram

Figure 5 Mechanical model and bending moment diagram in tangential direction

1205782=

1205742

1205741015840=

1

1 + (2ℎ119864119868119868119868119906

) (119889119864119887119868119887ℎ)

1205783=

1205743

12057410158401015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868119868119868

119906

)

1205784=

1205744

12057410158401015840=

1

1 + (2ℎ119864119868119868119868119868119868119906

) (119889119864119887119868119887ℎ)

1198731=

2

27sdot(1198861+ 2)

(1198861+ 1)

(4 + 41198861+ 11988621

)

1198732=

2

27sdot(11198861+ 2)

(11198861+ 1)

(1

11988621

+4

1198861

+ 4)

(28)

The mechanical model and moment diagram of arch ribsand transverse brace in radial direction are shown in Figure 3And the radial deformation energy of a single transversebrace can be written as

119880119887V =

1

2119864119887119868119887

int119910

1015840

01

0

(119872119887V1

119910101584001

11991010158400

)

2

11988911991010158400

+1

2119864119887119868119887Vint119910

1015840

02

0

(119872119887V2

119910101584002

11991010158400

)

2

11988911991010158400

(29)

where 119864119887119868119887V is the transverse bracersquos bending stiffness along

radial direction of arch rib and 119910101584001

and 119910101584002

represent the dis-tance from the contraflexure point of transverse bracersquos radialdeformation to the main and stable arch ribs respectively

The values of 119910101584001

and 119910101584002

are closely related to thebending stiffness of arch ribs the length of the arch ribsection the bending stiffness and length of the transversebraces For conventional arch bridges the bending stiffnessand inclined angle of two main arch ribs are the same so119910101584001

= 119910101584002

However for leaning-type arch bridges as the

bending stiffness of main arch rib is far larger than that ofstable arch rib so 1199101015840

01

gt 119910101584002

as shown in Figure 3 In orderto obtain the values of 1199101015840

01

and 119910101584002

radial mechanical modelof transverse brace along the arch central axis is establishedwhen a lateral buckling occurs as shown in Figure 6(a)The bending moment caused by the transverse bracesrsquo radialdeflection can be obtained by Castiglianorsquos theorem of mate-rialmechanics as shown in Figure 6(b) and (30) can be givenas

1198902=

119872119887V1

119872119887V2

=(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119868119879119868119868)119866119868119879119868

(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119879119868) 119866119868119868119879119868119868 (30)

According to the principle of similar triangles we arrived at

119910101584001

=1198902

1 + 1198902

119887 119910101584002

=1

1 + 1198902

119887 (31)

The radial deformation energy of transverse brace can beexpressed as

119880119887V =

6119864119887119868119887V

119889119887

1198902

1 + 1198902

int119904

(120573 minus 1205791)2

119889119904

+6119864119887119868119887V

119889119887

1

1 + 1198902

int119904

(120573 minus 1205792)2

119889119904

= 119864119887119868119887V(1198625minus 1198621)2

1198771198633+ 119864119887119868119887V11986223

1198771198634

+ 119864119887119868119887V11986225

1198771198634+ 119864119887119868119887V11986231198625

1198771198636

(32)

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

119887(119909) can be replaced by a constant ℎ which is the distancebetween the main and stable arch ribs at quarter span forcalculation simplification Although the length of transversebrace 119887(119909) is not a constant the length of the transverse braceℎ at quarter span is very close to the average length of thetransverse braces of the entire bridge Such simplification canmake the derivation process become simple

The values of11991001and11991002are closely related to the bending

stiffness of arch ribs panel length bending stiffness andlength of transverse braces Due to the inclined angle andthe different stiffness of leaning-type arch bridgersquos main andstable arch ribs the tangential and radial deformation of thetransverse brace between the main and stable arch ribs differfrom that of the conventional arch bridges It is not a simpleldquoSrdquo shape For leaning-type arch bridges bending stiffness ofthe main arch rib is larger than that of the stable arch rib so11991001

gt 11991002 as shown in Figure 4 as an exceptional case for

conventional arch bridges 11991001

= 11991002 In order to obtain the

relationship between 11991001 11991002 a tangential mechanical model

of transverse brace along the arch axis is established whena lateral buckling occurs as shown in Figure 5(a) its cor-responding bending moment is shown in Figure 5(b) fromwhich (19) can be given as

119872119887ℎ1

119872119887ℎ2

=119864119868119864119887119868119868119906

119868119887ℎ119889 + 6119864119868119864119868119868119868119868

119906

119868119868119868119906

119887 (119909) cos1206010

6119864119868119864119868119868119868119868119906

119868119868119868119906

119887 (119909) cos1206010

+ 119864119868119868119864119887119868119868119868119906

119868119887ℎ119889 cos120601

0

= 1198901

(19)

where 1198901is a constant

From (17) and (19) it can be obtained that

119872119887ℎ1

119872119887ℎ2

=21205741+ 1205743

1205741+ 21205743

= 1198901 (20)

Assuming 1205743= 11988611205741 then

1198861=

1205743

1205741

=2 minus 1198901

21198901minus 1

(21)

thus from (18) and (21) the relationship of 11991001and 11991002can be

obtained as

11991001

=2 + 1198861

1 + 21198861

11991002 (22)

Substituting (17) and (18) into (16) the equation of tangentialbending energy of the transverse braces in the full arch ribrange is given as

119880119887119867

= int119904

119880119887119867

119889119889119904

=2119864119887119868119887ℎ

9119889ℎ[int1199041

21205741+ 1205743

1205741+ 1205743

(412057421

+ 412057411205743+ 12057423

) 119889119904

+int1199042

1205741+ 21205743

1205741+ 1205743

(12057421

+ 412057411205743+ 412057423

) 119889119904]

(23)

When the tangential local deformation occurs in themainand stable arch ribs assuming the bending moments of themain and stable arch ribs along the radial direction are 119872

1198881199061

and1198721198881199062

respectively the single-panel section arch ribrsquos localbending energy of the main and stable arch ribs can beexpressed as

119880119868

119888119906

+ 119880119868119868

119888119906

=1

2119864119868119868119868119906

int119889

11987221198881199061

1198891199090+

1

2119864119868119868119868119868119868119906

int119889

11987221198881199062

1198891199090

=6119864119868119868119868119906

11988912057422

+6119864119868119868119868119868119868119906

11988912057424

(24)

If the local bending energy of each single-panel sectionarch rib is the same the full-arch-rib local bending deforma-tion energy can be written as

119880119868119888119906

+ 119880119868119868119888119906

= int119904

119880119868

119888119906

119889119889119904 + int

119904

119880119868119868

119888119906

119889119889119904

= int119904

3119864119887119868119887ℎ

119889ℎ12057411205742119889119904 + int

119904

3119864119887119868119887ℎ

119889ℎ12057431205744119889119904

(25)

It could be obtained from Figure 4 that

21198721198881199061

= 119872119887ℎ1

21198721198881199062

= 119872119887ℎ2

(26)

Therefore from (23) (25) and (26) the local bendingenergy of arch rib and the tangential bending energy oftransverse brace are obtained as

119880119887119867

+ 119880119868119888119906

+ 119880119868119868119888119906

=3119864119887119868119887ℎ

119889ℎint1199041

(11987311205781+ 1205782) 120578112057410158402119889119904

+3119864119887119868119887ℎ

119889ℎint1199042

(11987321205783+ 1205784) 1205783120574101584010158402119889119904

= 119864119887119868119887ℎ

11986222

11987731198631+ 119864119887119868119887ℎ

11986222

11987731198632

(27)

where

1198631=

3119899

119887119904(11987311205781+ 1205782) 1205781int1205721

0

1198772(2120587

1205722

)2

sin22120587120593

1205721

119889120593

1198632=

3119899

119887119904(11987321205783+ 1205784) 1205783int1205722

0

1198772(2120587

1205722

)2

sin22120587120593

1205722

119889120593

1205741015840 = 1205741+ 1205742

12057410158401015840 = 1205743+ 1205744

1205781=

1205741

1205741015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868

119906

)

Mathematical Problems in Engineering 7

d2

EbIbhb1

EIIIIIu

1205930

EIIIu

q998400

(a) Mechanical model

1205930

Mbh2

Mbh1

(b) Bending moment diagram

Figure 5 Mechanical model and bending moment diagram in tangential direction

1205782=

1205742

1205741015840=

1

1 + (2ℎ119864119868119868119868119906

) (119889119864119887119868119887ℎ)

1205783=

1205743

12057410158401015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868119868119868

119906

)

1205784=

1205744

12057410158401015840=

1

1 + (2ℎ119864119868119868119868119868119868119906

) (119889119864119887119868119887ℎ)

1198731=

2

27sdot(1198861+ 2)

(1198861+ 1)

(4 + 41198861+ 11988621

)

1198732=

2

27sdot(11198861+ 2)

(11198861+ 1)

(1

11988621

+4

1198861

+ 4)

(28)

The mechanical model and moment diagram of arch ribsand transverse brace in radial direction are shown in Figure 3And the radial deformation energy of a single transversebrace can be written as

119880119887V =

1

2119864119887119868119887

int119910

1015840

01

0

(119872119887V1

119910101584001

11991010158400

)

2

11988911991010158400

+1

2119864119887119868119887Vint119910

1015840

02

0

(119872119887V2

119910101584002

11991010158400

)

2

11988911991010158400

(29)

where 119864119887119868119887V is the transverse bracersquos bending stiffness along

radial direction of arch rib and 119910101584001

and 119910101584002

represent the dis-tance from the contraflexure point of transverse bracersquos radialdeformation to the main and stable arch ribs respectively

The values of 119910101584001

and 119910101584002

are closely related to thebending stiffness of arch ribs the length of the arch ribsection the bending stiffness and length of the transversebraces For conventional arch bridges the bending stiffnessand inclined angle of two main arch ribs are the same so119910101584001

= 119910101584002

However for leaning-type arch bridges as the

bending stiffness of main arch rib is far larger than that ofstable arch rib so 1199101015840

01

gt 119910101584002

as shown in Figure 3 In orderto obtain the values of 1199101015840

01

and 119910101584002

radial mechanical modelof transverse brace along the arch central axis is establishedwhen a lateral buckling occurs as shown in Figure 6(a)The bending moment caused by the transverse bracesrsquo radialdeflection can be obtained by Castiglianorsquos theorem of mate-rialmechanics as shown in Figure 6(b) and (30) can be givenas

1198902=

119872119887V1

119872119887V2

=(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119868119879119868119868)119866119868119879119868

(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119879119868) 119866119868119868119879119868119868 (30)

According to the principle of similar triangles we arrived at

119910101584001

=1198902

1 + 1198902

119887 119910101584002

=1

1 + 1198902

119887 (31)

The radial deformation energy of transverse brace can beexpressed as

119880119887V =

6119864119887119868119887V

119889119887

1198902

1 + 1198902

int119904

(120573 minus 1205791)2

119889119904

+6119864119887119868119887V

119889119887

1

1 + 1198902

int119904

(120573 minus 1205792)2

119889119904

= 119864119887119868119887V(1198625minus 1198621)2

1198771198633+ 119864119887119868119887V11986223

1198771198634

+ 119864119887119868119887V11986225

1198771198634+ 119864119887119868119887V11986231198625

1198771198636

(32)

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

d2

EbIbhb1

EIIIIIu

1205930

EIIIu

q998400

(a) Mechanical model

1205930

Mbh2

Mbh1

(b) Bending moment diagram

Figure 5 Mechanical model and bending moment diagram in tangential direction

1205782=

1205742

1205741015840=

1

1 + (2ℎ119864119868119868119868119906

) (119889119864119887119868119887ℎ)

1205783=

1205743

12057410158401015840=

1

1 + (119889119864119887119868119887ℎ) (2ℎ119864119868119868119868119868119868

119906

)

1205784=

1205744

12057410158401015840=

1

1 + (2ℎ119864119868119868119868119868119868119906

) (119889119864119887119868119887ℎ)

1198731=

2

27sdot(1198861+ 2)

(1198861+ 1)

(4 + 41198861+ 11988621

)

1198732=

2

27sdot(11198861+ 2)

(11198861+ 1)

(1

11988621

+4

1198861

+ 4)

(28)

The mechanical model and moment diagram of arch ribsand transverse brace in radial direction are shown in Figure 3And the radial deformation energy of a single transversebrace can be written as

119880119887V =

1

2119864119887119868119887

int119910

1015840

01

0

(119872119887V1

119910101584001

11991010158400

)

2

11988911991010158400

+1

2119864119887119868119887Vint119910

1015840

02

0

(119872119887V2

119910101584002

11991010158400

)

2

11988911991010158400

(29)

where 119864119887119868119887V is the transverse bracersquos bending stiffness along

radial direction of arch rib and 119910101584001

and 119910101584002

represent the dis-tance from the contraflexure point of transverse bracersquos radialdeformation to the main and stable arch ribs respectively

The values of 119910101584001

and 119910101584002

are closely related to thebending stiffness of arch ribs the length of the arch ribsection the bending stiffness and length of the transversebraces For conventional arch bridges the bending stiffnessand inclined angle of two main arch ribs are the same so119910101584001

= 119910101584002

However for leaning-type arch bridges as the

bending stiffness of main arch rib is far larger than that ofstable arch rib so 1199101015840

01

gt 119910101584002

as shown in Figure 3 In orderto obtain the values of 1199101015840

01

and 119910101584002

radial mechanical modelof transverse brace along the arch central axis is establishedwhen a lateral buckling occurs as shown in Figure 6(a)The bending moment caused by the transverse bracesrsquo radialdeflection can be obtained by Castiglianorsquos theorem of mate-rialmechanics as shown in Figure 6(b) and (30) can be givenas

1198902=

119872119887V1

119872119887V2

=(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119868119879119868119868)119866119868119879119868

(119864119887119868119887ℎ119889 + 2119887 cos120601

0

119866119868119879119868) 119866119868119868119879119868119868 (30)

According to the principle of similar triangles we arrived at

119910101584001

=1198902

1 + 1198902

119887 119910101584002

=1

1 + 1198902

119887 (31)

The radial deformation energy of transverse brace can beexpressed as

119880119887V =

6119864119887119868119887V

119889119887

1198902

1 + 1198902

int119904

(120573 minus 1205791)2

119889119904

+6119864119887119868119887V

119889119887

1

1 + 1198902

int119904

(120573 minus 1205792)2

119889119904

= 119864119887119868119887V(1198625minus 1198621)2

1198771198633+ 119864119887119868119887V11986223

1198771198634

+ 119864119887119868119887V11986225

1198771198634+ 119864119887119868119887V11986231198625

1198771198636

(32)

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

d2

b

1205930

q998400

EIIIIIu

EIIIu EbIbv

(a) Mechanical model

b

d2

1205930

Mbv2Mbv1

(b) Bending model diagram

Figure 6 Mechanical model and moment diagram in radial direction

where

1198633=

6119899

119887119904

1198902

1 + 1198902

int1205721

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198634=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205722

)2

1198772119889120593

1198635=

6119899

119887119904

1

1 + 1198902

int1205722

0

(1 minus cos2120587120593

1205721

)2

1198772119889120593

1198636=

6119899

119887119904

1

1 + 1198902

int1205722

0

minus2(1 minus cos2120587120593

1205721

)(1 minus cos2120587120593

1205722

)1198772119889120593

(33)

Assuming the distance between hangers of the main archrib is119889

ℎand its corresponding arc length of arch rib is1198891015840

ℎ

119889ℎis

approximately equal to 1198891015840ℎ

as it is previously assumed that thedistance between hangers is small and the distance betweenbridge deck and main arch rib is

119910 (120593) = 119877 [cos(120593 minus1205721

2) minus cos 1205721

2] (34)

As shown in Figure 7 the tension of the hanger is

119879 = 119902119889ℎ (35)

The horizontal component of hanger tensions is

119867 = 119902119889ℎsin1206011 (36)

As the lateral displacement is sufficiently small one arrives at

sin1206011= 1206011=

119906 minus 119906119889

119910 (120593) (37)

where 119910(120593) is the length of the hanger

Hanger

Bridge deck

H uu H

T

T1205931

1205931

1205930

udud

y(120593)

Figure 7 Schematic diagramof the horizontal component of hangertensions with lateral buckling

The elastic potential energy of arch ribs and bridge decksystem caused by the horizontal part of the tension of thehanger is given in

119881119867

=1

2int119871

minus119871

119867(1199061minus 119906119889)119889119909

1198871

= 119902(1198622minus 1198624)2

1198641 (38)

where

1198641=

1

2int1205721

0

119877

2119891cos120593(1 minus cos

2120587120593

1205721

)2

119889120593 (39)

119891 is the rise of main arch rib and in order to simplify itsintegral a conservative assumption of 119910(120593) = 119891 is adopted

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

The lateral bending deformation energy of the bridgedeck system is given in

119880119889=

1

2int119871

119864119889119868119889(11990610158401015840119889

)2

119889119897 =1

2int1199041

119864119889119868119889cos120593 (11990610158401015840

119889

)2

119889119904

= 119864119889119868119889

11986224

11987731198642

(40)

where 119864119889119868119889is the lateral bending stiffness of the bridge deck

system

1198642=

1

2int1205721

0

(2120587

1205721

)4

cos120593cos22120587120593

1205721

sdot 1198774119889120593 (41)

Combined with (4) the potential energy of the externalloading is

119881 = minus1

2int V119902 119889119904 = minus119902 (1198622

2

1198651+ 11986225

11986521198772) (42)

where

1198651= int1205721

0

(2120587

1205721

)2

sin22120587120593

1205721

119889120593

1198652= int1205721

0

(2120587

1205721

)2 1198902

2

(1 + 1198902)2

1198872

1198772sin2

2120587120593

1205721

119889120593

(43)

23 The Analytical Solution of Lateral Critical Buckling Load-ing Thetotal potential energy of the leaning-type arch bridgecan be obtained based on (12) (14) (27) (32) and (38)ndash(42)According to the principle of stationary potential energy thevalues of 119862

119894minimizing the function119882(119862

119894) should therefore

satisfy the algebraic equations

120597

120597119862119894

(119882) = 0 (119894 = 1 2 5) (44)

The existence of nontrivial solutions of (44) for 119862119894requires

that the determinant of its coefficient matrix be equal to zerothen we obtained

11986711205823cr + 119867

21205822cr + 119867

3120582cr + 119867

4= 0 (45)

where1198671= 64119865

11198652119878111987831198641

1198672= 811987825

119878311986411198652minus 3211987811198783119878411986511198641minus 3211987811198782119878311986411198652

+ 64119878111987831198642119865111986521198966+ 8119878111987827

11986411198652+ 8119878111987829

11986411198651

+ 8119878311987826

11986411198651minus 64119878111987831198641119864211986521198966

1198674= 41198781119878211987829

11986421198966+ 11987825

11987829

11986421198966minus 41198781119878411987827

11986421198966

minus 2119878511987861198787119878911986421198966minus 41198781119878311987828

11986421198966+ 4119878311987851198786119878811986421198966

+ 4119878111987871198788119878911986421198966minus 41198782119878311987826

11986421198966+ 11987826

11987827

11986421198966

+ 16119878111987821198783119878311986421198966minus 41198783119878411987825

11986421198966

(46)

The lateral buckling critical load coefficient was obtained bysolving (45) and then the lateral critical buckling load ofleaning-type arch bridge is

119902cr = 120582cr119864119868119868119868119906

1198773 (47)

3 Lateral Buckling Critical Load underthe Hinged Boundary Condition of MainArch Rib

The following assumptions are made in the derivation pro-cess the main arch ribs are hinged the stable arch ribs arefixed and the other assumptions are the same as those statedin Section 21 The variables without special explanation arethe same as aforementioned

The torsional angle of arch axis of themain and stable archribs is shown as

1205791= 1198621sin

120587120593

1205721

(48a)

1205792= 1198623(1 minus cos

2120587120593

1205722

) (48b)

The lateral displacements of arch axis of the main andstable arch ribs are given as

1199061= 1198622sin

120587120593

1205721

(49)

1199062= 1198622sin

120587120593

1205722

(50)

The lateral displacements of bridge deck system is givenas

119906119889= 1198624sin

120587120593

1205721

(51)

The global torsional angle of the transverse brace in radialplane caused by arch ribrsquos deflection is

120573 = 1198625sin

120587120593

1205721

(52)

where 1198621 1198622 1198623 1198624 and 119862

5are all constants

The above displacement functions should satisfy thefollowing boundary conditions

(1) when 120593 = 0 and 120593 = 1205721 1205791= 120573 = 0 119906

1= 119906119889= 0

120579101584010158401

= 12057310158401015840 = 0 and 119906101584010158401

= 11990610158401015840119889

= 0

(2) when 120593 = 0 and 120593 = 1205722 1205792= 0 119906

2= 0 1205791015840

2

= 0 and119906101584010158402

= 0

The derivationmethod of the lateral buckling critical loadunder the hinged boundary condition of main arch ribs isthe same as stated above Due to the limitation of the paperlength the derivation process is omitted only the calculatedresults are discussed in the following section

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

10

10

Figure 8 The FE model under the fixed boundary condition

4 Verification Example

The leaning-type arch bridge Shengli Bridge with a span of75m in Jiangmen city of China is used to verify the accuracyof the derived analytical solution presented in this paperA three-dimensional finite element model is established byusing theMidasCivil FEM software to calculate arch bridgersquoslateral buckling critical load for comparison The main andstable arch ribs of this bridge are both fixed at the arch endsThe FE model is shown in Figure 8

There are 284 elements and 217 nodes in this FE modelSpatial beam element with 6 degrees of freedom at each nodeis used to simulate the arch rib transverse brace girderand transverse girder Spatial truss element with 3 degreesof freedom at each node is used to simulate the hanger Thecalculation parameters of the leaning-type bridge are listedin Table 1 Numerical analysis is carried out as the followingsteps (1) 1 Nm uniform load is applied to the middle ofthe transverse girder of the bridge deck system (2) By usingMidasCivilrsquos buckling eigenvalue solver the eigenvalue 120582 ofthe bridge is obtained which indicates the lateral bucklingcritical load of the bridge

Comparison of the FEM results and the analytical resultsof the leaning-type arch bridge under fixed and hingedboundary conditions when the stable rib inclined angle is 5∘7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ and 30∘ are shown inTables 2 and 3The contrastive results show that the analyticalresults agree well with the FEM results with the relative errorno more than 347 320 under fixed boundary conditionand hinged boundary condition respectively which indicatethe accuracy of the analytical solution for the stable criticalload of the leaning-type arch bridge presented in this paper

5 Parametric Analysis

51 Effect of Main Arch Ribrsquos Boundary Condition on theCritical Buckling Load Figure 9 shows the lateral bucklingcritical load of a leaning-type arch bridge for both cases offixed-end main arch ribs and hinged-end main arch ribs Itcan be seen from this figure that the lateral buckling criticalload of the leaning-type arch bridge with fixed main archribs is approximately 114 to 116 times that of the leaning-type arch bridge with hinged main arch ribs As the inclinedangle increases from 5∘ to 30∘ the critical load 119902cr increasesby approximately 1118 for cases of fixed-end main arch ribs

50

55

60

65

70

75

80

Fixed end Hinged end

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 9 Comparison of critical buckling load under differentboundary conditions

and the critical load 119902cr increases by approximately 1957 forcases of hinged-end main arch ribs

52 Effect of the Central Angle on the Critical Buckling LoadFigure 10 shows the critical load value 119902cr when the centralangle of the main arch rib and the stable arch rib are the same(the central angle of themain arch rib and the stable arch rib is8721∘ in this case) and different (the central angle of themainarch rib is 8721∘ and the central angle of the stable arch rib is104∘ in this case) as the inclined angle increases from 5∘ to 30∘It can be seen from Figure 10 that the difference of the criticalload 119902cr between same central angle model and the differentcentral angle model is small enough to be neglected the for-merrsquos lateral buckling critical load is only 102 to 103 times ofthat of the latter It indicates that the central angle of the stablearch has relatively less effect on the lateral buckling criticalload of the arch bridge and the central angles of the mainarch rib and the stable arch rib can be considered to be thesame

53 Effect of Hanger Tensions and Bridge Deck on CriticalBuckling Load Figure 11 shows the critical load value with orwithout considering hanger tensions and bridge deck as theinclined angle increases from 5∘ to 30∘ From this figure it canbe seen that if the hanger tensions and bridge deck are consid-ered the critical load is 6664MNm and 7198MNm wheninclined angle is 120601

0= 5∘ and 30∘ respectively However if the

hanger tensions and the bridge deck are neglected the criticalload is 2171MNm and 2451MNm when inclined angle is1206010

= 5∘ and 30∘ respectively The critical load increasesby 294 and 307 times respectively as compared with thatof neglecting the hanger tensions and bridge deck Theresults indicate that the hanger tensions and bridge deck cangreatly improve the lateral stability of the leaning-type archbridge

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 1 Calculation parameters used in the FE model

Span (m) 75Rise-span ratio 14The central angle of main arch (∘) 8721The central angle of stable arch (∘) 104Inclined angle of stable arch rib (∘) 5 7 9 11 13 15 17 19 21 24 27 and 30Transverse brace length on arch crown (m) 155Number of transverse brace 6Lateral bending stiffness of main arch rib (MNsdotm2) 1414 times 104

Torsional stiffness of main arch rib (MNsdotm2) 1111 times 103

Lateral bending stiffness of stable arch rib (MNsdotm2) 1179 times 104

Torsional stiffness of main arch rib (MNsdotm2) 0926 times 103

Transverse bracersquos bending stiffness along radial direction of arch rib (MNsdotm2) 3684Transverse bracersquos bending stiffness along tangential direction of arch rib (MNsdotm2) 1058 times 102

Lateral bending stiffness of girder (MNsdotm2) 23 times 107

Vertical bending stiffness of girder (MNsdotm2) 552 times 104

Lateral bending stiffness of transverse beam (MNsdotm2) 115 times 104

Vertical bending stiffness of transverse beam (MNsdotm2) 7187 times 102

The cross section area of hangers (m2) 125664 times 10minus3

Poissonrsquos ratio 02

Table 2 Comparison between the analytical solution and FEM results under fixed boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 635 642 651 659 666 672 679 685 690 697 702 706FEM (MNm) 642 645 648 651 653 655 659 662 671 674 687 690Error () 109 047 046 123 199 260 303 347 283 341 218 232

60

63

66

69

72

75

Different center angle Same center angle

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 10 Comparison of critical buckling load under the same anddifferent central angle

54 Effect of the Stable Arch Rib on Critical Buckling LoadFigure 12 shows the critical load 119902cr of the models with orwithout stable arch ribs as the inclined angle increases from5∘ to 30∘ From this figure it can be seen that the critical loadof the models with stable arch ribs is always larger than that

00

10

20

30

40

50

60

70

80

Considering the effect of hanger tensions Neglect the effect of hanger tensions

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

q(M

Nm

)cr

Figure 11 Comparison of critical buckling load with and withouthanger tensions effect

of the models without stable arch ribs and the percentageof increase is from 315 and 421 as the inclined angleincreased The results indicate that the effect of stable archrib on critical load is significant

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Table 3 Comparison between analytical solution and FEM results under hinged boundary condition

Inclined angle 5∘ 7∘ 9∘ 11∘ 13∘ 15∘ 17∘ 19∘ 21∘ 24∘ 27∘ 30∘

Analytical (MNm) 557 563 573 586 603 617 627 634 645 652 660 666FEM (MNm) 570 575 587 590 596 600 609 619 625 632 640 647Error () 228 209 238 068 117 283 296 242 320 316 313 293

45

50

55

60

65

70

75

With stable arch rib Without stable arch rib

4 8 12 16 20 24 28 32

Inclined angle 1206010 (∘)

qcr

(MN

m)

Figure 12 Comparison of critical buckling load with and withoutstable arch rib

6 Conclusions

This paper has derived an analytical solution for lateralbuckling critical load of leaning-type arch bridge based onthe Ritz method and the accuracy of this solution has beenverified through a numerical example Moreover parametricanalysis is carried out in order to investigate how changesin certain design parameters would affect the critical load ofthe leaning-type arch bridges by using the analytical solutionpresented in this paper The main conclusions are as follows

(1) The analytical solution present in this paper can beused to calculate the lateral buckling critical loadof the leaning-type arch bridges in different casesincluding the central angles of the main arch rib andstable arch rib which are different both the main archribs and the stable arch ribs are fixed in their ends themain arch ribs with hinged boundary condition whilethe stable arch ribs with fixed boundary conditionsFrom the comparison of the analytical results and theFEM results the analytical solution presented in thispaper is verified to be sufficiently accurate

(2) The lateral buckling critical load under fixed bound-ary condition is approximately 114 to 116 times aslarge as that under hinged boundary conditionswhich indicate that the lateral stability of the formeris better than that of the latter

(3) The critical load with the same central angles isslightly bigger than the one with different central

angles and the formerrsquos lateral buckling critical load is102 to 103 times as big as that of the latter It indicatesthat the central angle of the stable arch has relativelyless influence on the lateral buckling critical load ofthe leaning-type arch bridge and therefore the centralangles of the main arch and the stable arch can beconsidered to be the same for convenience

(4) Stable arch rib can significantly increase the lateralbuckling critical load 119902cr of leaning-type arch bridgeby 315 to 421 when stable arch rib is consideredunder the fixed boundary condition where the valueof 119902cr increases as the inclined angle of stable arch ribincreases

(5) The hanger tensions and bridge deck have significanteffect on the critical load and when considering theeffect of hanger tensions and bridge deck the criticalload can improve by 294 to 307 times

Notations

1205791 1205792 The torsional angle of the main and stable

arch ribs1199061 1199062 The lateral displacement of the main and

stable arch ribs under the global coordinate1199061198681

1199061198681198681

The lateral displacement of the main andstable arch ribs under the local coordinate

119906119889 The lateral displacement of bridge deck

system1206010 The angle between the main arch rib and the

stable arch rib120593 The angle of a position in the arch ribs1205721 1205722 The central angle of the main and stable arch

ribs120573 The global torsional angle of the transverse

brace caused by deformation of arch rib inradial plane

119877 The radius of the main and the stable archribs

119887(119909) The distance between the main arch rib andthe stable arch rib

1198870 The length of the transverse brace at the arch

crownℎ The distance between the main and stable

arch ribs at quarter span119889 The distance between the transverse braces119889ℎ The distance between the hangers

119870119868119906

119870119868119868119906

The lateral deflection curvature of the mainand stable arch ribs respectively

119870119868119908

119870119868119868119908

The torsional deflection curvatures of themain and stable arch ribs respectively

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

119880119868119906

119880119868119868119906

The lateral bending deformation energy ofthe main and stable arch ribs respectively

119880119868119908

119880119868119868119908

The torsional deformation energy of themain and stable arch ribs respectively

119880119868119888119906

119880119868119868119888119906

The local bending deformation energy of themain and stable arch ribs respectively

119880119887V The bending deformation energy of the

transverse braces in radial directions119880119887119867 The bending deformation energy of the

transverse braces in tangential directions119881119867 The elastic potential energy of the arch ribs

and the bridge deck system under thehorizontal component of the hanger tensions

119881119889 The elastic potential energy of bridge deck

system119881 The potential energy of external loading119880119868

119888119906

119880119868119868119888119906

The single-panel arch rib local bendingenergy of main and stable arch ribs

119880119887119867 The tangential bending deformation energy

of the single transverse brace119880119887V The radial deformation energy of a single

transverse brace119864119868119868119868119906

119864119868119868119868119868119868119906

The lateral bending stiffness of the main andstable arch ribs

119866119868119879119868 119866119868119868119879119868119868 The torsional stiffness of the main and stablearch ribs

119864119887119868119887ℎ The bending stiffness of transverse brace

along tangential direction of arch rib119864119887119868119887V The bending stiffness of transverse brace

along radial direction of arch rib1198721198881199061

1198721198881199062

The bending moments of main and stablearch ribs along radial direction

119872119887ℎ1

119872119887ℎ2

The tangential bending moment of transversebrace near the main and stable arch ribs

119872119887V1119872119887V2 The vertical bending moment on both ends

of transverse brace near the main and stablearch ribs

11991001 11991002 The distances from the contraflexure point of

transverse brace tangential deformation tomain and stable arch ribs respectively

119910101584001

119910101584002

The distance from the vertical contraflexurepoint of transverse bracersquos radial deformationto main and stable arch ribs respectively

1205741 1205743 The tangential angular rotation of the

transverse brace at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

1205742 1205744 The tangential angular rotation of the main

and stable arch rib at the intersection pointsbetween the main and stable arch rib and thetransverse brace respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was sponsored by the National Natural ScienceFoundation of China (nos 11272095 51378133 and 51208123)the Science and Technology Planning Major Project ofGuangzhou City (no 2011Y2-00006) the Key Technologi-cal Innovation Program of Guangdong Ministry Education(no 2012CXZD0028) the Key Project supported by theNatural Science Foundation of Guangdong Province (noS2011030002800) and the Talent Introduction Project sup-ported by the Higher Education Department of GuangdongProvince in 2012

References

[1] A-R Liu Q-C Yu R Song and J-P Zhang ldquoDynamic stabilityof leaning-type arch bridge under earthquakerdquo Journal ofShenzhen University Science and Engineering vol 27 no 3 pp286ndash290 2010 (Chinese)

[2] A C Franciso Acro Colour Thematic Architecture Acro Edito-rial 1989

[3] R C Xiao H T Sun and L J Jia ldquoKunshan Yufeng bridge-design of the first long-span leaning-type arch bridge withoutthrustrdquo China Civil Engineering Journal vol 38 no 1 pp 78ndash83 2005

[4] N-H Lim and Y-J Kang ldquoOut of plane stability of circulararchesrdquo International Journal ofMechanical Sciences vol 46 no8 pp 1115ndash1137 2004

[5] Y-L Pi and M A Bradford ldquoEffects of prebuckling deforma-tions on the elastic flexural-torsional buckling of laterally fixedarchesrdquo International Journal ofMechanical Sciences vol 46 no2 pp 321ndash342 2004

[6] Y-L Pi M A Bradford and F Tin-Loi ldquoFlexural-torsionalbuckling of shallow arches with open thin-walled section underuniform radial loadsrdquoThin-Walled Structures vol 45 no 3 pp352ndash362 2007

[7] Y Chen and J Feng ldquoElastic stability of shallow pin-endedparabolic arches subjected to step loadsrdquo Journal of CentralSouth University of Technology vol 17 no 1 pp 156ndash162 2010

[8] P D B La R C Spoorenber H H Sniijder and J C DHoenderkamp ldquoOut-of-plane stability of roller bent arches-an experimental investigationrdquo Journal of Constructional SteelResearch vol 81 no 1 pp 20ndash34 2013

[9] M A Bradford and Y-L Pi ldquoA new analytical solution forlateral-torsional buckling of arches under axial uniform com-pressionrdquo Engineering Structures vol 41 no 1 pp 14ndash23 2012

[10] C Dou Y L Guo S Y Zhao Y L Pi andMA Braford ldquoElasticout-of-plane buckling load of circular steel tubular truss archesincorporating shearing effectsrdquo Engineering Structures vol 52no 7 pp 696ndash706 2013

[11] W Jin andG Zhao ldquoLateral buckling of X-type twin ribbed archbraced with transverse beamsrdquo China Civil Engineering Journalvol 22 no 2 pp 44ndash54 1989 (Chinese)

[12] Z Liu and Z-T Lu ldquoLateral buckling load of tied-arch bridgeswith transverse bracesrdquo EngineeringMechanics vol 21 no 3 pp21ndash54 2004 (Chinese)

[13] Z F Xiang ldquoPractical calculation of the lateral stability ofthe midhight-deck arch bridgerdquo Journal of Chongqing JiaotongInstitute vol 14 no 1 pp 27ndash33 1995 (Chinese)

[14] D Y Gu H Chen Y Wang and F Hu ldquoStability analysisof the Chaozhou Hanjiang River Northen leaning-type arch

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

bridgerdquo Journal of Highway and Transportation Research andDevelopment vol 23 no 3 pp 100ndash103 1995 (Chinese)

[15] A-R Liu F-L Shen J-T Kuang J-P Zhang and Q-CYu ldquoCalculation method for lateral buckling critical load ofleaning-type arch rib systemrdquo Engineering Mechanics vol 28no 12 pp 166ndash172 2011 (Chinese)

[16] H Shafiee M H Naei andM R Eslami ldquoIn-plane and out-of-plane buckling of arches made of FGMrdquo International Journalof Mechanical Sciences vol 48 no 8 pp 907ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of