Non-Linear Static and Modal Analysis of Three Types of Cable- Stayed Bridges

Transcription

1 No-Liear Static ad Modal Aalysis of Three Types of Cable- Stayed Bridges B. O., Rageh *,A. M., Masleikov Departmet of Structural Mechaics, St.-Petersburg state Uiversity of Architecture ad Civil Egieerig, 4, 2d Krasoarmeiskaya Street, St. Petersburg , Russia Federatio. * of the correspodig author: eg_basem85@mail.ru Abstract This paper is cocered about the study of cable-stayed bridges havig three spas with double plae of cables. Three commo types of bridge arragemet are cosidered - harp, fa ad radiatig shapes. The ifluece of the arragemets of cables o the bridge deformatio is examied. The relatioship betwee the sag ad the other parameters of cables for each type is preseted. I the static aalysis, the eergy method, based o the miimizatio of the total potetial eergy of structural elemets, via cojugate gradiet techique is used. A computer program i FORTRAN laguage is used i the aalysis of bridge model. Natural frequecies for the cosidered types of cable-stayed bridge are calculated, usig the SAP2000 program. Keywords: No-liear static aalysis, cable structures, cable stayed bridges, miimizatio of potetial eergy, cojugate gradiet method. 1. Itroductio. Cable stayed bridges have acquired popularity aroud the world as a viable bridge desig for medium to log spa crossigs. Their aesthetic form ad the fast mode of costructio are some reasos behid their icreasig popularity. Cable stayed bridges have greater flexibility which may occur deflectio ca t be expected ad the experiece shortage i desig of this types of structures. This type of bridges cosist of three mai parts, the pylos, the deck ad the stayed cables, each differig i iertial ad stiffess properties. These three sub-systems are desiged to iteract to produce a efficiet system of load trasfer withi the bridge. Smith [12] proposed a mixed displacemet method to aalyses sigle-plae cable-stayed bridge. The simultaeous equatios were developed by iitially assumig the girder to be supported by rigid cables the the flexibility coefficiets are modified to take the extesibility of the cables, rotatio ad shorteig of the tower ito cosideratio. Smith exteded this method to double-pla system. Liear ad oliear aalysis was studied by Lazar [10] usig the stiffess matrix method. Lazar cosidered the oliear effect due to large deformatios ad due to cateary effect of cables. Kajita ad Cheug [8] applied the fiite elemet method o a three- dimesioal cable-stayed bridge system. Baro ad Lie [1] studied the liear ad oliear behavior of the souther crossig bay bridge, i the Uited States as a three-dimesioal structure. The cable was cosidered as a cateary supportig its ow weight. The aalysis was carried out usig the oliear theory based o the taget stiffess matrix which had bee developed by Baro ad Vekaste [2]. Brow [3] preseted a method for the aalysis of cable-stayed bridges usig the potetial eergy method of a stayed girder with a iterative procedure ad the girder deflectio was calculated usig Fourier series, eglectig the axial deformatio of the girder. A combied icremetal ad iterative approach for the oliear aalysis of plae cable-stayed bridges was preseted by Flemig [6]. Krisha ad Agarwal [9] studied the liear behavior of the three-spa radiatig cable-stayed bridge with differet middle spa arragemets. Troistsky [15] reported that cable oliearity ca be igored by liearizatio of the cable stiffess usig the method of equivalet modulus of elasticity. A lot of research were doe o cable-stayed bridge by researches. Some of these research are give i refereces [7,11]. I this paper, a cable stayed bridge has a differet logitudial layout as harp, fa ad radiatig is aalyzed usig the eergy method [4] based o the miimizatio of the total potetial eergy of structure elemets, via cojugate gradiet techique [13,14] i order to study the ifluece of cable arragemet o the deformatios of bridge. I the harp patter, the cables are parallel ad cross each other at a costat agle i the eye of the views gives the structure a most acceptable appearace. I the radiatig type, all the stays together to the top of pylos. A itermediate solutio, betwee the extremes of harp ad fa patters by spreadig the stays i the upper part of the pylo called fa system. Axial stiffess of a cable stay i cable stayed bridges is affected by the cable sag associated with tesio i cable, also this paper presets a proposed equatio to calculate the relatio betwee the sag ad the other mai parameters of cable for harp ad fa patters ad fially, presets the behavior of the bridge model with differet arragemet of cables i the case of free vibratio. 2. Descriptio of the Bridge The bridge with the total legth of 800 m comprises three spas: a 400 m cetral spa betwee two pylos, ad two side spas each with a legth of 200 m. The bridge deck is a steel box-girder 20.8 m wide ad 3.2 m high. 92

2 The vertical momet of iertia (I x ), the trasverse momet of iertia (I y ), the cross-sectio area (A d ), ad the modulus of elasticity (E d ) of the deck are m 4, m 4, m 2, ad kn/m 2, respectively. The self-weight(w d ) ad the equivalet live load which trasferred for each plae of cables are kn/m ad 46 kn/m, respectively. The pylos cosist of two parts: The lower part uder the deck level cosists of two legs 7.3 m 5 m with thickess 0.7 m ad 40 m height, the upper part 6.7 m 3 m with thickess 0.5 m ad 80 m height above the deck level. The modulus of elasticity for the cocrete (E c ) is kn/m 2. The cables have costat cross-sectioal areas(a c ) of m 2. Numberig of cables are illustrated i Fig.1. The modulus of elasticity (E cs ), the breakig load (T ult ), ad the weight per uit legth (w cs ) of the cables are kn/cm 2, kn, ad kn/m, respectively. Spacig betwee cables alog the upper part of pylo are 7 m, 2 m for harp ad fa system, respectively. Figure 1. Numberig of cable for three types of cable arragemet (harp, fa ad radiatig). 3. Methodology Eergy method is a uifyig approach to the aalysis of both liear ad o-liear structures. It is a idirect method of aalysis ad valid for both small ad large structures. The eergy method is applied to the aalysis of geeral pi-eded truss ad cable structures. The solutio of a problem is by determiig the structure equilibrium as a iterative process of miimizig the total potetial eergy (TPE); the equilibrium achieved whe the TPE is miimum. The total potetial eergy is obtaied by summig the eergy cotributios from each member ad formulated i terms of the displacemet of the odes or joits of the structure kowig that the stable equilibrium cofiguratio is the global miimum of the TPE fuctio. Miimizatio techiques are employed to umerically locate the miimum. The total potetial eergy of the bridge model is writte as: W = f = 1s= 1r= 1 1 xsk 2 sr x r + p = 1 U EA + Toe+ 2L Where; f the umber of flexure members, x s or x r the elemet of displacemet vector of a flexural member icludig effect of the pretesio i the cables, K sr stiffess matrix i global co-ordiates of a flexural member, p the umber of pi-joited members ad cable liks, U o iitial strai eergy i a pi-joited member or cable lik due to pretesio, T o iitial force i a pi-joited member or cable lik due to pretesio, e elogatio of pi-joited members or cable liks due to applied load oly, E modulus of elasticity, A area of the pi-joited member or cable lik, F elemet i applied load vector, x Elemet i displacemet vector due to applied load oly, N total umber of degrees of freedom of all joits. o 2 e (1) o N = 1 F x. Figure 2. Coordiates ad force for lik (j), Pretesioed ad loaded cable assembly at stage of the iterative process where the extesio of the lik (j) is e j. Usig fig.2, we ca get the expressio of (e) i the followig equatio ; 93

3 L oto ej = eo + e= ( Xi X ji)( xi xji) + ( xi xji) +. (2) L 2 o i= 1 EA The above equatio gives the total elogatio due to applied load ad due to pretesio for the cable segmet j. The miimizatio ca take place alog defied descet vector usig cojugate gradiet method. The poit of miimum W defies the equilibrium positio of the loaded structure. Mathematically, the equilibrium coditio i the i directio at joit j may be expressed by: W = [ gji] = 0 x (3) ji i= 1,2,3 Where; x ji the displacemet of joit j correspodig to a particular ji degree of freedom, directio, i, g ji the correspodig gradiet of the eergy surface. The locatio of miimum W is achieved by movig dow the eergy surface alog a descet vector (v) a distace Sv util W is a miimum i that directio. The displacemet vector at the (k+1)th iteratio as: [ x ] k+1 = [ x] k + Skvk (4) Where; v k the descet vector at the k-th iteratio from x k i displacemet space, ad S k the step legth determiig the distace alog v k to the poit of W mi. 4. Results ad discussio The bridge is aalyzed usig the eergy method, based o the miimizatio of the total potetial eergy of structural elemets, theoretically the miimizatio ca take place alog defied descet vector usig cojugate gradiet method. Calculatio of the gradiet vector of the total potetial eergy with respect to displacemet is carried out at the begiig of each iteratio. Iitial tesio i cables depedig o a iterative scheme was discussed i details i literature [17]. The vertical deflectio of the deck ad lateral deflectios of the pylos' tops are illustrated i Fig. 3,4. Figure 3. Vertical deflectio of bridge deck. Figure 4. Lateral deflectios of the pylos' tops. As show i Fig.2, the deflectio values i the middle deck for harp, fa ad radiatig are m, m, m, respectively. I fig.3, the lateral displacemet values for harp, fa ad radiatig are m, m, 94

4 0.089 m, respectively. The cables are the most importat part of cable stayed bridges, which brigs a cosiderable differece betwee the cable stayed ad other types of the bridges. Cables cotribute a sigificat part to the overall stiffess of a cable-stayed bridge. More importatly, a cable exhibits strog oliear behavior due to the proouced saggig effect uder its ow gravity weight. For cables, uder the actio of its ow weight ad axial tesile force, a cable supported at its ed will sag ito a cateary shape. The axial stiffess of a cable will chage oliearly with cable tesio ad saggig. The equivalet modulus approach developed by Erst [5] is the most adopted method for cable modelig i cable-stayed bridges. The followig equatio obtaied from the aalysis usig differet coefficiets to calculate the cable sag for harp ad fa system. E A (5) cs c δ = A l б. 2 ( w l) A E l cs c cs T c Coefficiets for harp ad fa type are A=0.274, б=2.341, A=0.533,б=4.152, respectively. Fig.5 shows the variatio of cable sag with the differet legths of cables up to 500 m depeds o the fial stresses as a percetage from 20% to 50% of the breakig stress of cables. Cable forces obtaied at the ed of o-liear static aalysis o this study s models are give i Table.1. Figure 5. Sag of cables for harp ad fa type. Table 1. Cables forces obtaied at the ed of o-liear static aalysis (kn) Cable.No Harp Fa Radiatig Cable.No Harp Radiatig Fa system (left side) system system system (right side) system system CL CR CL CR CL CR CL CR CL CR CL CR CL CR CL CR CL CR CL CR Accordig to Table 1, betwee the least forced cables, radiatig system were less tha the others. I free vibratio aalysis, the bridge is discredited as 3-D fiite elemet model, usig the SAP2000 program. icludig two types of elemet: cateary cable ad frame elemets, the deck is modeled usig spie passig through its shear ceter. The traslatioal ad rotatioal stiffess of the deck are calculated ad assiged to the frame elemets of the spie. The cable achorages ad deck spie are coected by massless, horizotal rigid liks to 95

5 achieve the proper offset of the cables from the ceterlie of the deck [16]. Mode shapes ad correspodig frequecies are illustrated i Table. 2. Table 2. The first 5 mode shapes of three types of cable stayed bridge model Harp system Fa system Radiatig system Mode 1 (Vertical) F= cyc/sec Mode 1 (Vertical) F= cyc/sec Mode 1 (Vertical) F= cyc/sec) Mode 2 (Horizotal) F= cyc/sec Mode 2 (Horizotal) F= cyc/sec Mode 2 (Horizotal) F= cyc/sec Mode 3 (Vertical) F= cyc/sec Mode 3 (Vertical) F= cyc/sec Mode 3 (Vertical) F= cyc/sec Mode 4 (Vertical) F= cyc/sec Mode 4 (Vertical) F= cyc/sec Mode 4 (Vertical) F= cyc/sec Mode 5 (Torsio) F= cyc/sec Mode5 (Torsio) F= cyc/sec Mode 5 (Torsio) F= cyc/sec 5. Coclusio The result of o-liear static aalysis usig eergy method, showed that the smallest deflectio value of deck was i radiatig type ad the biggest oe was i harp type model. For pylos' tops lateral displacemet, the smallest value was i radiatig type ad the biggest oe was i harp. It was observed that the most forced cables were the log oes i the middle spa ad edge spas. The biggest cable forces were i harp type, which 96

6 cotributes to larger ormal axial force i the deck ad smallest forces were i radiatig type. The fidig suggests that the harp cable arragemet appears less suitable tha the fa ad radiatig cable arragemet especially for log-spa bridges. A equatio is proposed for calculatig the relatioship betwee the sag ad the other mai parameters of cable for harp ad fa type usig coefficiets A=0.274, б=2.341 ad A=0.533, б=4.152, respectively. The result of free vibratio aalysis of the first twety (20) modes examied shows that the fudametal mode is pure vertical motio of the deck for the three types of bridge. The first torsioal mode occurs at the fifth mode for all types. For lower modes from 3 to 5, the atural frequecies i harp type are lower tha i fa ad radiatig type, the harp type appears to have more torsioal modes for lower frequecies. Fially, vibratio of cable-stayed bridges become harmful whe a predomiat excitatio frequecy is i the viciity of the atural frequecy of the structural system, therefore, the frequecy estimated ca be used to determie the frequecy rage to be covered i a detailed dyamic aalysis for wid egieerig. Refereces [1] Baro, F., Lie, S.Y. (1971). Aalytical studies of Souther Crossig cable-stayed bridge, Struc. Eg. Deprt., Uiversity of Califoria, Berkeley, Report No. SESM [2] Baro, F., Vekatesa, M. S. (1971). Noliear formulatio of beam- colum effects, J. Stru. Div., ASCE. [3] Brow, C.D. (1965) Desig ad Costructio of the George Street Bridge over the River USK, at Newport, Momouthshire, Proceedigs of the Istitutio of Civil Egieers. [4] Buchholdt, H.A. (1985). A itroductio to cable roof structures, Lodo: Cambridge Uiversity press: [5] Erst J.H. (1965). Der E-Modul vo Seile uter berucksichtigug des Durchhages. Der Bauigeieur, 40(2), 52-5 (i Germa). [6] Flemig, J.F. (1974). Latest developmet of cable-stayed bridges, Eg. Joural, ASCE. [7] Freire, A.M.S, Negrao, J.H.O, ad Lopes, A.V. (2006). Geometrical Noliearities o the Static Aalysis of Highly Flexible Steel Cable Stayed Bridges. Egieerig Structures, 84, [8] Kajita, T., Cheug, Y.K. (1973) Fiite elemet aalysis of cable-stayed bridges, Ite. Assoc. for bridges ad Struc. Eg., Publ. 33-II. [9] Krisha, P., Agarwal, T. P. (1985). Effect of cable stiffess o cable- stayed bridges, J. Stru. E, ASCE, Vol. 111, No. 9. [10] Lazar, B. E. (1972) Stiffess aalysis of cable-stayed girder bridges, Struc. Div., ASCE, Vol. 98, No. ST7. [11] Re, W.X., Peg X.L. & Li, Y.Q. (2005). Experimetal ad Aalytical Studies o Dyamic Characteristics of a Large Spa Cable Stayed Bridge, Egieerig Structures, 27, [12] Smith, B. S. (1967). The sigle plae cable-stayed girder bridge: a method of aalysis suitable for computer use. Proc. Ist. Civ. Eg, 37, [13] Stefaou, G. D, Moossavi, E., Bishop, S. & Koliopoulos, P. (1993). Cojugate gradiets for calculatig the respose of large cable ets to static loads, Computers & Structures:49(5) [14] Stefaou, G. D & Nejad S. E. M. (1995) Geeral method for the aalysis of cable assemblies with fixed ad flexibe elastic boudaries, Computers & Structures, 55(5), [15] Troitsky, M. S (1989). Cable- stayed bridges, Theory ad desig, Crosby Lockwood Staples, ed. Lodo, 2 d. [16] Wilso J.C & Gravelle W. (1991). Modellig of a Cable-stayed Bridge for Dyamic Aalysis. J Earthquake Eg & Struct Dy 20: [17] Рагех, Б.O. (2012). Энергетический подход при анализе вантовых схем мостов системы «АРФА». Вестник гражданских инженеров, 6 (35): 61-6 (i Russia). 97

7 This academic article was published by The Iteratioal Istitute for Sciece, Techology ad Educatio (IISTE). The IISTE is a pioeer i the Ope Access Publishig service based i the U.S. ad Europe. The aim of the istitute is Acceleratig Global Kowledge Sharig. More iformatio about the publisher ca be foud i the IISTE s homepage: CALL FOR JOURNAL PAPERS The IISTE is curretly hostig more tha 30 peer-reviewed academic jourals ad collaboratig with academic istitutios aroud the world. There s o deadlie for submissio. Prospective authors of IISTE jourals ca fid the submissio istructio o the followig page: The IISTE editorial team promises to the review ad publish all the qualified submissios i a fast maer. All the jourals articles are available olie to the readers all over the world without fiacial, legal, or techical barriers other tha those iseparable from gaiig access to the iteret itself. Prited versio of the jourals is also available upo request of readers ad authors. MORE RESOURCES Book publicatio iformatio: Recet cofereces: IISTE Kowledge Sharig Parters EBSCO, Idex Copericus, Ulrich's Periodicals Directory, JouralTOCS, PKP Ope Archives Harvester, Bielefeld Academic Search Egie, Elektroische Zeitschriftebibliothek EZB, Ope J-Gate, OCLC WorldCat, Uiverse Digtial Library, NewJour, Google Scholar