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Approximate method for estimation of collapse loads of steel cable stayed bridges Hoon Yoo a Ho Sung Nab Dong Ho Choib a R fax 82 2 2220 4322 E mail addresses hoonyoo hdec co kr H Yoo saintna hanyang ac kr H S Na csmile hanyang ac kr D H Choi 0143 974X see front matter 2011 Elsevier Ltd All rights reserved doi 10 1016 j jcsr 2011 12 003 Contents lists available at SciVerse ScienceDirect Journal of Constructional Steel Research the need for a simpler and faster method to estimate the collapse load of these special bridges has been reasonably claimed by many engineers without sacrifi cing much in accuracy Consequently a simple alternative for the complex nonlinear inelastic analysis was proposed and studied by several researchers in order to approximately obtain the collapse load of a steel cable stayed bridge 10 11 17 These studies adopted the inelastic buckling analysis which utilized the column strength curve in design specifi cations in order to take into account the material inelasticity of structural members Iterative eigenvalue computations were adopted in the method for determining the tangent modulus of each structural member The inelastic buckling analysis could only partially consider the geometrical nonlinearities of a cable stayed bridge According to the corresponding references 10 14 16 18 however the ultimate behavior of a steel cable stayed bridge is mainly affected by material nonlinearity rather than by geometrical nonlinearities even near the collapse state of the bridge system The results of these studies seem to be evidence of the validity of the inelastic buckling analysis Of course there is also the criticism that the method of inelastic buckling analysis is not valid for obtaining the collapse load of a cable stayed bridge due to its failure in capturing geometrical nonlinearities of the bridge system 19 This paper proposes a new method for simple estimation of the collapse loads of steel cable stayed bridges that can be effectively used for checking many candidate designs with various load cases in the preliminary design stage Based on the fundamental concept of the inelastic buckling analysis previously established by the authors 9 10 18 we widen the range of application for the method by suggesting a new criterion of each structural member in the bridge system The proposed method determines the tangent stiffness of each structural member in the bridge system by iterative eigenvalue computations with the classical tangent modulus theory In addition an improved convergence criterion for girder and tower members is proposed to take into account the beam column interactions After summarizing theoretical approaches we analyze the two example bridges representing medium and long span models with different girder depths To show the validity and applicability of the method the results of the proposed method are compared with those of the establishedinelasticbucklinganalysisandnonlinearinelasticanalysis Some discussionsare alsomadeabout the failure modes oftheexample bridges 2 Nonlinear inelastic analysis Exact approach 2 1 Beam column element for girder and tower members Since the large axial force and bending moments occur in girder and tower members in a typical cable stayed bridge under service loads these structural members are usually considered to be beam column members It is well known that the beam column interaction called the second order effect is observed in a beam column member 8 20 23 In general the second order effect can be conveniently taken intoaccount by using thestability functions 8 20 Frommanip ulation of the slope defl ection equation for a beam column member which is based on the assumption of Euler Bernoulli beam theory and Saint Vernant torsion the tangent stiffness of a three dimensional prismatic beam column element shown in Fig 1 may be described as 8 9 uavawa xa ya zaubvbwb xb yb zb ktan abz by000 a bzby000 cz000dz bz cz000gz cy0 dy000 cy0 gz0 e00000 e00 fy000dy0hy0 fz0 dz000hz abz by000 cz000 gz Symm cy0gy0 e00 iy0 iz 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 where the terms are defi ned by a EtA L by Mya Myb L2 bz Mza Mzb L2 cy kiiy 2kijy kjjy L2 P L cz kiiz 2kijz kjjz L2 P L dy kiiy kijy L dz kiiz kijz L e GJ L fy kiiy fz kiiz gy kijy kjjy L gz kijz kjjz L hy kijy hz kijz iy kjjy and iz kjjz In the tangent stiffness in Eq 1 the terms Et G and J indicate the tan gent modulus shear modulus and torsional constant respectively The terms P Myand Mzare the axial force and bending moments of the y and z axes at the current state of loadings respectively The tangent modulus Etof a member is simply approximated from the column strength curve with respect to the magnitude of axial force of each member at the current state of loadings as described in Refs 8 9 The stiffness terms of k are defi ned with the stability functions for geometric nonlinearities and gradual yielding parameters for material yielding as given in Table 1 In Table 1 the terms S1 S2 S3and S4represent the conventional stability functions which consider the second moment effect for a beam column member The explicit form of these functions was omitted here for brevity and can be found in Refs 8 9 18 23 The terms aand bin Table 1 are scalar parameters that allow for gradual yielding of an element associated with plasticization at the end of nodes These terms are equal to 1 0 when a member is fully elastic and zero when a plastic hinge is formed in a member The terms aand bare assumed to vary according to the parabolic function that is derived with the form of the column strength curve as 8 9 23 1 0when b 0 39 2 7243 ln when 0 39 2 where is a force state parameter that measures the magnitude of axial force and bending moment at the end of the member The term may be described in AISC LRFD 24 as P Py 8 9 My Mpy Mz Mpz when P Py 2 9 My Mpy Mz Mpz P 2Py My Mpy Mz Mpz when P Py b 2 9 My Mpy Mz Mpz 8 3 where Py Mpyand Mpzare theyield load and full plastic moments of the y and z axes of an element respectively The stability functions and material yielding parameters are changed at a certain incremental step of applied loads in the incremental load analysis Fig 1 Degrees of freedom of a beam column element Table 1 The stiffness terms of k in the tangent stiffness of a prismatic beam column y axisz axis kiiy a S1 S2 2 S1 1 b EtIy L kiiz a S3 S4 2 S3 1 b EtIz L kijy a bS2EtIy Lkijz a bS4EtIz L kjjy b S1 S2 2 S1 1 a EtIy L kjjz b S3 S4 2 S3 1 a EtIz L 144H Yoo et al Journal of Constructional Steel Research 72 2012 143 154 2 2 Cable element When a cable is suspended between the girder and towermembers it sags into the shape of a catenary due to its own weight 15 25 The peculiar nonlinear behavior in cables called the sag effect results from this phenomenon The tensile axial force in a cable is affected by not only the deformation of the cable as a usual tension member but also the cable sag Therefore the axial stiffness of the cable varies non linearly as a function of the amount of the cable sag as well as the changeinthelengthofcables Ifatensileforceinacableislargeenough to neglect the amount of the cable sag the cable acts as a general tension member However as a tensile force in the cable decreases the cable sag increases considerably and the axial stiffness of the cable is dramatically reduced Furthermore the cable is only capable of resisting a tensile axial force and there is no stiffness against the axial compressionforce Toconsiderthesenonlinearbehaviors anequivalent bar element incorporating the equivalent tangent modulus is usually used for modeling cables in a cable stayed bridge system 26 27 The tangent stiffness matrix of cables may be written as uavawaubvbwb ktan a00 a00 cz00 cz0 cy00 cy a00 Symm cz0 cy 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 4 where the terms are defi ned by a Et cAc Lcand cy cz P Lc The terms Et c Ac and Lcare the equivalent tangent modulus sectional area and the length of a cable member respectively The equivalent tangent modulus of a cable may be given as 18 26 28 Et c Ec 1 w Lch 2AcEc 12T3 c when 0 b c y c 0when cb 0 or c y c 8 5 whereEc w and Tcare theoriginalmodulusofelasticity theunit weight and theaxialtensionof a cable respectively ThetermLchis a horizontal projected length of a cable The terms cand y care the current cable stress including initial stress and the yield stress of a cable member respectively 2 3 Incremental load analysis considering large displacements Since a cable stayed bridge system is inherently fl exible and undergoes large displacements under service loads the global tangent stiffness of the system changes as external loads are applied Therefore a step by step incremental load analysis should be conducted to trace the response of the bridge system The global tangent stiffness of the bridge system is required to be updated at each load level throughout the incremental load analysis By performing conventional matrix manipulations in fi nite element procedures the element tangent stiffness matrices of girder tower and cable members in Eqs 1 and 4 may be combined into the global tangent stiffness matrix Ktan The basic system equation of the incre mental load analysis is then described as 9 18 25 26 Ktan d fg dPfg 6 where d and dP are the incremental displacement vector and the incremental force vector respectively A detailed derivation of Eq 6 and the procedures of incremental load analysis can be found in Refs 8 9 18 23 The collapse load of a bridge system is determined at the maximum load point where the tangent stiffness matrix in Eq 6 is nearly indefi nite i e the slope of a load displacement path in the global bridge system is nearly zero as given in Eq 7 The critical load factor CLF is calculated as the ratio of the collapse load to the service load initially applied to the bridge models det Ktan jj 0 7 3 Proposed inelastic buckling analysis Approximate approach 3 1 Criterion for a column member The tangent modulus theory proposed by Engesser 29 considers theinelasticity of a column asthe tangent modulus that is the gradient ataspecifi c pointofthestress straincurve Theinelastic criticalloadPcr may be written as Pcr 2EtI L2 e Et E Pe 8 where Le and Peare the effective length and the elastic critical load respectively The modulus of elasticity of a column is unique for a given material of a column and the elastic critical load may be easily calculated by conventional eigenvalue computation Therefore we can approximate the inelastic critical load of a column without tracing a complex path of material inelastic behavior if we know the accurate tangent modulus of a column As expected the tangent modulus cannot be obtained by a one step analysis because the accurate inelastic critical load of a column is also an unknown value in Eq 8 Therefore a simple iterative scheme is needed to obtain the tangent modulus and the inelastic critical load of a column simultaneously Eq 8 can be transformed into E Et Pe Pcr 9 Iterative eigenvalue computations are the modifi cation process of the tangent modulus such that the elastic critical load Pe the Euler buckling load is equalized with the inelastic critical load Pcrfrom the column strength curve In other words the ratio of the current tangent modulus to the previous tangent modulus of a column approaches the value of 1 0 at the end of the iterationsteps of iterative eigenvalue computations as Ei 1 t Eit Pie Picr 1 0 10 where superscript i denotes the number of iterations In Eq 10 the elastic critical load of a column is obtained by conventional eigenvalue analysis in conjunction with the theoretical formula as Pe i iP0 where is the eigenvalue at the i th iteration and P0is the axial force of each member determined from a linear stress analysis The inelastic critical load of a column is approximated from the column strength curve in combination with the slenderness ratio calculated from the elastic critical load Eq 11 derived by substitutingPe i iP0intoEq 10 isthecriterionforacolumnmember used in the established inelastic buckling analysis described in Refs 10 11 17 19 Eit Picr iP0 Ei 1 t 11 The eigenvalue computation should be repeated in order to deter mine the tangent modulus and the inelastic critical load of a column until Eq 10 is satisfi ed for all members in a bridge system After the convergenceiscompleted theconvergedeigenvalue convmayrepresent 145H Yoo et al Journal of Constructional Steel Research 72 2012 143 154 theinelastic criticalload factor of the system The critical load factormay be accepted as the index of the collapse load of a cable stayed bridge system provided that the bridge system fails due to the buckling instability ThecriterionforacolumnmemberinEq 11 wasalreadyappliedto some cable stayed bridge models by several researchers 10 11 17 19 In these papers they insisted that the inelastic buckling analysis with thecriterionforacolumninEq 11 gavesatisfactoryresultstoapprox imately determine the collapse load of long span steel cable stayed bridges 10 11 17 but others did not 19 In theory it is certain that the convergence criterion of Eq 11 is suffi cient only for the analysis of the column dominated structural system such as the isolated tower or the high rise frames For a complex bridge system such as a cable stayed bridge the validity of Eq 11 is questionable because this crite rioncannottakeintoaccounttheeffectoftheprimarybendingmoment in each member The convergence criterion of Eq 11 may need some modifi cation for the analysis of the complex bridge systems 3 2 Proposed criterion for a beam column member For a typical beam column member the stability of a member is usually checked by the axial fl exural interaction equation Eq 12 which is a fundamental form of Eq 3 20 22 P Pn My Mpy Mz Mpz 1 0 12 where the terms Pnrepresent the axial resistance of a beam column member After some modifi cations we may use Eq 12 as a new criterion for the inelastic buckling analysis instead of using Eq 11 At a specifi c load state the terms of axial force and moments of a member in Eq 12 may be written with the eigenvalue from a conventional eigenvalue analysis as P iP0 My iM0 yand Mz iM0 z 13 where the terms P0 My 0 and Mz 0 are the member force and moments that are calculated from a linear stress analysis The axial resistance Pnof a member is also described as Pn Pcr i where the term Pcr i is the inelasticcriticalloadofamembercalculatedfromthecolumnstrength curveatthei thiteration Weassumedthatthefullplasticmomentsof a member are not affected by the iterations in iterative eigenvalue computations By substituting Eq 13 and the term of the axial resistance into Eq 12 and comparing it with the case for a column memberofEq 10 weobtainthecriterionforabeam columnmember as Ei 1 t Eit iP0 Picr iM0 y Mpy iM0 z Mpz 1 0 14 By reversing the numerator terms and denominator terms of Eq 14 the tangent modulus of a beam column member can be obtained at i th iteration steps as Eit PicrMpyMpz iP0MpyMpz Picr iM0 yMpz PicrMpy iM0z Ei 1 t 15 Fig 2 Flow chart of inelastic buckling analysis 146H Yoo et al Journal of Constructional Steel Research 72 2012 143 154 It can be seen that the criterion of Eq 15 for a beam column is equivalent to the criterion of a column member when the moments are not exerted on a member Therefore we can say that Eq 15 is the general convergence criterion for a column and beam column member in inelastic buckling analysis and may be used for structural members in a steel cable stayed bridge system without any theoretical discrepancy 3 3 Procedures of the inelastic buckling analysis The inelastic buckling analysis proposed in this paper utilizes iterative eigenvalue computations based on the bifurcation point stability concept The basic equation of the inelastic buckling analysis is similar to that of conventional elastic buckling analysis 10 18 except for the elastic stiffness matrix term of the structure which is det KeEt Kg 0 16 where Ke Et and Kg are the modifi ed stiffness matrix and geometric stiffness matrix of the global bridge system corresponding to eigen values of respectively Eq 16 is conceptually placed along the same line with Eq 7 of the nonlinear inelastic analysis By the same defi nitions of the degrees of freedom in nonlinear inelastic analysis Fig 1 the modifi ed stiffness matrix ke Et for a beam column element may be explicitly given as 9 18 uavawa xa ya zaubvbwb xb yb zb keEt a00000 a00000 bz000cz0 bz000cz by0 cy000 by0 cy0 d00000 d00 ey000cy0fy0 ez0 cz000fz a00000 bz0000 Symm by0cy0 d00 ey0 ez 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 17 where the individual terms are defi ned by a EtA L by 12EtIy L3 bz 12EtIz L3 cy 6EtIy L2 cz 6EtIz L2 d GJ L ey 4EtIy L ez 4EtIz L fy 2EtIy L and fz 2EtIz L The geometric stiffness matrix for a beam column element kg is also given as uavawa xa ya zaubvbwb xb yb zb kg hi 0bzby0000 bzby000 a0cyade bz a0cyb de acza edby0 aczb e d fgz gy0 cya cza f gzgy h00 de gz ij h0 e dgy j i 0bz by000 a0 cybd e Symm a czbed fgz gy h0 h 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 18 where the terms in Eq 18 are defi ned by a 6P 5L by Mya Myb L2 bz Mza Mzb L2 cya Mya L cyb Myb L cza Mza L czb Mzb L d Mx L e P 10 f PJ AL gy Mya Myb 6 gz Mza Mzb 6 h 2PL 15 i PL 30 and j Mx 2 For a cable element the element and geometric stiffness matrices areextractedfromthetangentstiffnessofEq 4 innonlinearinelastic analysis By splitting Eq 4 into conventional material terms and 120m 14 20m 280m 120m 14 20m 280m 14 20m 280m14 20m 280m 600m 40m 120m 68m 13 4m 52m SC1 SC2 160m160m x z a Center span length of 600 m 300m 29 20m 580m 280m 29 20m 580m 29 20m 580m29 20m 580m 1200m 40m 240m 128m 28 4m 112m SC1 SC2 280m300m x z b Center span length of 1200 m Fig 3 Numerical models of the example cable stayed bridges Table 2 Cross sections and geometric properties of girder members in example cable stayed bridges Cross sectionDepth H m Area