Analysis of the overall collapse mechanism of cable-stayed bridges with different cable layouts

Engineering Structures 29 (2007) 2133 2142 www.elsevier.com/locate/engstruct Analysis of the overall collapse mechanism of cable-stayed bridges with different cable layouts Weon-Keun Song a,, Seung-Eock Kim b a R & D Division, Korea Infrastructure Safety and Technology Corporation, Kyung-Gi Province 411-758, South Korea b Civil & Environmental Eng., Construction Tech. Research Institute, Sejong Univ., Seoul, South Korea Received 23 March 2006; received in revised form 1 November 2006; accepted 4 November 2006 Available online 18 December 2006 Abstract The bifurcation point instability approach based on the nonlinear inelastic buckling model, and the limit point instability approach based on the softening plastic-hinge model, are proposed for analyzing the in-plane overall collapse mechanisms of steel cable-stayed bridges under static loads. The example bridges considered are supported by different cable layouts such as fan-type, semi harp-type and harp-type. This paper discusses the problem of the analysis of the in-plane overall collapse mechanisms, and the evaluation of the ultimate load-carrying capacities of example bridges with various cable layouts using the bifurcation point instability approach. The limit point instability approach, which considers both geometric and material nonlinearities, is deemed appropriate for evaluating the ultimate load-carrying capacity of the example bridges. c 2006 Elsevier Ltd. All rights reserved. Keywords: Steel cable-stayed bridge; Softening plastic-hinge model; Fiber element model; Nonlinear inelastic buckling model; Overall collapse mechanism; Ultimate load-carrying capacity 1. Introduction The stability of structural systems is lost due to the existence of the singular points in equilibrium paths that are referred to as critical points. There are two different kinds of critical point: bifurcation point and limit point. A structure is assumed to be suitable for bifurcation point instability analysis based on the elastic buckling model if it corresponds to an eigenvalue problem in terms of the mathematics, whereas limit point instability analysis corresponds to a mathematical boundary value problem. Over many years, research efforts have been devoted to the development of the bifurcation point instability and the limit point instability approaches for cable-stayed bridges, which consider both their geometric and material nonlinearities. High nonlinearities such as the sag effect due to the self-weight of cable stays, the axial force bending moment interaction, the large displacement that is produced by the geometric change of structure, and the nonlinear stress strain Corresponding author. Tel.: +82 31 910 4128; fax: +82 31 910 4181. E-mail address: bauman98@hanmail.net (W.-K. Song). behavior of each structural component (including yielding) have been considered in the limit point instability analysis of cable-stayed bridges. Geometric nonlinearities come from the cable sag, the axial force bending moment interaction, and the large displacement. Material nonlinearities arise when one or more bridge elements exceed their elastic limits due to the member forces (axial forces) or undergo plastification (combined axial forces and bending moments). Many researchers have dealt with the nonlinear inelasticity problem by using the limit point instability approach. Some have disregarded all sources of the nonlinearities (e.g., Krishna et al. [1]), whereas others have included one or more of these sources. Most nonlinear analyses of cable-stayed bridges have focused on plane (Fleming [2]) or space (Kanok-Nukulchai and Guan [3]) geometric nonlinear behavior. But some analyses (Nakai et al. [4]; Seif and Dilger [5]) involving both geometric and material nonlinearities have revealed that the material nonlinearities were dominant in the nonlinear static behavior of cable-stayed bridges. These research undertakings were focused on concrete cable-stayed bridges. Nagai et al. [6] and Nogami et al. [7] have developed the bifurcation point instability approach using the ultimate column strength curve. 0141-0296/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2006.11.005

2134 W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 Cable stays, cross beams, girders and pylons are the main structural components of cable-stayed bridges. The behavior of cross beams, girders and pylons can be described using beam column members in limit point instability analysis. The nonlinear inelastic models for describing beam column members may be grouped into several categories: plastichinge, fiber element and plastic-zone models. The plastichinge model has been developed by Orbison [8], and Liew and Tang [9], while some investigators (Seif and Dilger [5]; Ren [10]) have presented different analysis concepts involving the fiber element model to deal with cable-stayed bridges. El- Zanaty and Murray [11] has developed the second-order plasticzone model considering the spread of the inelastic zone along the member length as well as the gradual yielding of a cross section. In this paper, the ultimate load-carrying capacity and the inplane overall collapse mechanism of the example bridges (see Fig. 4), which are supported by different cable layouts such as fan-type, semi harp-type and harp-type, are discussed based on the bifurcation point instability and the limit point instability approaches. The limit point instability analysis is conducted based on the softening plastic-hinge model, considering both geometric and material nonlinearities. The ultimate behavior of the example bridges predicted by this model is compared with the one predicted by the fiber element model using the commercial code ABAQUS in order to verify the feasibility of this study. The bifurcation point instability approach based on the nonlinear inelastic buckling model considers some structural imperfections (initial imperfection and residual stress) using the standard ultimate column strength curve referred to as the KSHB curve, which is stipulated in the Specifications for Highway Bridges in Korea (KSHB [12]). Shu and Wang [13] stated that the harp-type bridge is a better design than the semi harp-type bridge with regard to ultimate strength in their critical load analysis based on the bifurcation point instability approach, whereas according to Seif and Dilger [5], who used the limit point instability approach, the fan-type bridge collapsed under a load of 253 kn/m, while the semi harp-type and the harp-type bridges resisted, respectively, up to 240 kn/m and 176 kn/m individual loads. Their study suggests that the semi harp-type bridge is superior to the harptype bridge with regard to ultimate strength. Therefore, there is room for argument because of the conflicting results obtained by the investigations mentioned above, and it is for this reason that the comparative analysis of the ultimate behavior of the example bridges using the proposed approaches is essential. 2. Bifurcation point instability approach The nonlinear inelastic buckling model utilizes the effective tangent modulus (E t,eff ) concept in the Design Code of Tower for Suspension Bridges of the Honshu-Shikoku Bridge Authority (HSBA) [14], and it approximately considers not only geometric nonlinearity, but also material nonlinearity, such as the initial imperfection and residual stress, using the KSHB curve (see Eq. (1)). The KSHB curve depends on the profile form and product procedure, and the unexpected eccentricity is assumed to be L/1000. L denotes member length. This curve is used for the main members, excluding the cable stays, of the example bridges in this study. σ e = σ cu σ y = 1.0 for λ e 0.2 (1a) = 1.109 0.545λ e for 0.2 < λ e 1.0 (1b) = 1.0 0.773 + λ 2 e for 1.0 < λ e. (1c) The nonlinear inelastic buckling analysis procedure is described in the following: (1) after the small displacement analysis, in which a cable stay is modeled by using a straight truss element, a geometric stiffness matrix is composed considering the axial member forces ( P) only; (2) eigenvalue analysis K E (Et,eff i ) + κ K G(P i ) = 0, in which K E and K G are the elastic and the geometric stiffness matrices, respectively, provides the critical axial force Pcr i = κ P i and the critical strength σcr i = Pi cr A for each member when the minimum eigenvalue κ is κ min, where A is the cross section of each member; (3) the ultimate strength σcu i for each member, corresponding to the effective slenderness ratio (λ i e = L i e σy rπ Et,eff i ) in terms of the effective buckling length (L i e = E π t,eff I i Z ), should be determined on the KSHB curve, where Pcr i r is the radius of rotation, σ y is the axial yield strength and I Z is the moment of inertia about the z axis; (4) the calculation stops when the given convergence criterion is satisfied, in which E i 1 t,eff Et,eff i Ei 1 t,eff Et,eff i 1 < ε is the value of previous step and the value of current step is calculated from Et,eff i = Ei 1 t,eff, σcr i while the calculation should be returned to the 2nd step if the the convergence criterion is not met. 3. Limit point instability approach 3.1. Softening plastic-hinge model based on beam column approach 3.1.1. Stability functions To capture geometric nonlinearity, stability functions, which date back to Fleming [2], are used to minimize modeling and solution time. Others (Kim and Chen, [15,16]) have used these stability functions, which can be represented by Eq. (2), in a number of more recent publications for three-dimensional beam column elements. For P > 0 S 1 = π 2 ρ y cosh(π ρ y ) π ρ y sinh(π ρ y ) 2 2 cosh(π ρ y ) + π ρ y sinh(π ρ y ) S 2 = π ρ y sinh(π ρ y ) π 2 ρ y 2 2 cosh(π ρ y ) + π ρ y sinh(π ρ y ) S 3 = π 2 ρ cosh(π ρ z ) π ρ z sinh(π ρ z ) 2 2 cosh(π ρ z ) + π ρ z sinh(π ρ z ) σ i cu (2a) (2b) (2c)

W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 2135 S 4 = for P < 0 π ρ z sinh(π ρ z ) π 2 ρ z 2 2 cosh(π ρ z ) + π ρ z sinh(π ρ z ) S 1 = π ρ y sin(π ρ y ) π 2 ρ y cos(π ρ y ) 2 2 cos(π ρ y ) π ρ y sin(π ρ y ) π 2 ρ y π ρ y sin(π ρ y ) S 2 = 2 2 cos(π ρ y ) π ρ y sin(π ρ y ) S 3 = π ρ z sin(π ρ z ) π 2 ρ z cos(π ρ z ) 2 2 cos(π ρ z ) π ρ z sin(π ρ z ) π 2 ρ z π ρ z sin(π ρ z ) S 4 = 2 2 cos(π ρ z ) π ρ z sin(π ρ z ) (2d) (2e) (2f) (2g) (2h) where, ρ y = P/(π 2 E I y /L 2 ), ρ z = P/(π 2 E I z /L 2 ), and the axial force P is positive in tension; E is Young s modulus; I y and I Z are the moments of inertia about the y and z axes, respectively; L is the element length. The numerical solutions obtained from Eqs. (2a) (2h) are indeterminate when the axial force is zero. The stability function approach uses only one element per member, and maintains accuracy in element stiffness terms and in the recovery of element end forces for all ranges of axial loads. 3.1.2. Gradual yielding due to axial forces The softening plastic-hinge model is used to account for gradual yielding (due to residual stresses) along the length of axially loaded beam column members between plastic-hinges with the CRC (Column Research Council) tangent modulus (E t ) concept. In this concept, different members with different residual stresses can be incorporated because the effect of geometrical imperfections is considered. The elastic modulus E, instead of moment of inertia I, of a cross-section gradually decreases because the reduction of E is easier to implement than that of moment of inertia for a different section. From Chen and Lui [17], the CRC tangent modulus can be written as: E t = 1.0E for P 0.5P y ; (3a) E t = 4 P ) E (1 PPy for P > 0.5P y (3b) P y where, P y is the axial yield force. 3.1.3. Gradual plastification due to combined axial forces and bending moments The CRC tangent modulus concept is suitable for beam column members subjected to axial forces, but it is not adequate for cases involving both axial forces and bending moments. A gradual stiffness degradation model for a plastichinge is required to represent the partial plastification effects associated with bending. We shall introduce the parabolic function (η) to represent the transition from elastic to zero stiffness associated with a developing hinge. The parabolic function is expressed as: η = 1.0 for α 0.5; (4a) η = 4α(1 α) for α > 0.5 (4b) Fig. 1. Integration points of B33 element. where α is the force-state parameter that measures the magnitude of axial force and bending moment at the element end. The term α was expressed by AISC-LRFD [18]. Based on the AISC-LRFD bilinear interaction equation, the cross section plastic strength of the beam column member may be expressed as: α = P P y + 8 9 α = M y M yp + 8 9 M z M zp for P P y 2 9 M y M yp + 2 9 P + M y + M z for P < 2 M y + 2 2P y M yp M zp P y 9 M yp 9 M z M zp M z M zp (5a) (5b) where, M y, M z, M yp and M zp are the bending moments and the plastic bending moments about y and z axes, respectively. Initial yielding is assumed to occur based on a yield surface that has the same shape as the full plastification surface, and with the force-state parameter denoted as α 0 = 0.5. If the forces change so that the force point moves inside or along the initial yield surface, the element is assumed to remain fully elastic with no stiffness reduction. If the force point moves beyond the initial yield surface, the element stiffness is reduced to account for the effect of plastification at the element end. 3.2. Fiber element model If we consider a system in the domain of an integration point (IP: see Fig. 1), then the strain nodal displacement relationship may be expressed using a shape function matrix. After defining the strain for each fiber element (see Fig. 2) in the local coordinate system, the corresponding stress should be calculated with the stress strain curve of the given material. The equilibrium state on a cross section produces the total member forces at a given integration point in the local coordinate system. Finally, we can obtain the tangent stiffness matrix that defines the relationship between the total member forces and the strain components in the set of equilibrium equations for an integration point. The matrix reflects the materials nonlinearity by defining the material properties of the fiber elements at every iterative step. 3.3. Cable sag effect The cable stays are assumed to be perfectly flexible and to possess tension stiffness only. It is well known that inclined cable stays of cable-stayed bridges will sag into a catenary shape due to their own weight. The tension stiffness of a cable stay varies depending on the sag, which can be conveniently

2136 W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 Fig. 2. Fiber elements in a cross section at a given integration point. modeled by using an equivalent straight truss element with an equivalent modulus of elasticity. The element can well describe the catenary action of cable stay. The tangent value of the equivalent modulus of elasticity was first proposed by Ernst [19] and then used by Fleming [2]. In this paper, the cable sag effect is considered by using the secant value of the equivalent modulus of elasticity. This concept is applicable independent of the cable layouts of the cable-stayed bridges. The secant value of the equivalent modulus of elasticity, which includes the tension in the cable element changed from Tc 0 to T c 1 during the application of a certain load increment, was given by Gimsing [20] as follows: E c E eq = 1 + (wl c l c) 2 (T 0 c +T c 1) 24(Tc 0)2 (T 1 c )2 E c A c (6) in which E eq is the equivalent modulus of cable; E c is the Young s modulus of cable; l c is the horizontal projected length of cable; w L c is the weight per unit length of cable; A c is the cross sectional area of cable; T c is the cable tension. 3.4. Analysis procedure The limit point instability analysis for the example bridges is done starting with the initial shapes and the initial cable tensions obtained from the initial shape analysis. The initial cable tensions are used for the secant value of the equivalent modulus of elasticity of the cable members that describes the sag effect. Fig. 3 shows the procedure of the nonlinear initial shape analysis. At the first shape iterative step, the nonlinear initial shape analysis is performed under the dead loads. The cable tensions are not considered at this step. At the subsequent shape iterative steps, the analysis is performed considering the cable tensions and the updated structural configuration, using the reverse displacements captured from the previous shape iterative steps. The secant values of the equivalent cable modulus of elasticity increase as the cable tensions change. The shape differences between the updated structural configurations and the configuration before the deformation are gradually Fig. 3. Initial shape analysis procedure. eliminated at every shape iterative step. Finally, the updated structural configurations converge toward the target shape (the initial shape) during the iteration procedure. The sub-iteration procedure at every shape iterative step continues until the sum of total nodal displacement (SOD) becomes less than the given convergence tolerance (ε in ) (see Eq. (7)). When the SOD is calculated, the reference coordinates are the ones in the configuration before the deformation of the example bridges. SOD = N k=1 U 2 k,x + U 2 k,y + U 2 k,z ε in (7) where, N is the number of nodes; U k,x, U k,y, and U k,z are the displacements at the node k along X, Y, and Z global axes, respectively. The iteration procedure for the initial shape determination is continued until the SOD equals or is less than 1.0 m at the 3rd shape iterative step in this study. The configurations and cable tensions of the example bridges should be initiated at the first iterative step in the limit point instability analysis based on the result of the initial shape analysis. The softening plastic-hinge model utilizes the direct iteration method. In this algorithm, the effective load increment size near the limit point of the load displacement curve should be determined to minimize the error when the element stiffness is changed. The element stiffness formation accounts for the stiffness reduction due to gradual yielding, or the presence of a plastic-hinge with incremental loads. The algorithm calculates the incremental displacements corresponding to the applied incremental loads at every iterative step, and new plastichinges can be developed prior to the full application of each incremental load. The algorithm informs the users of the overall structural collapse via the nonpositive definiteness of the structural stiffness matrix at the limit point. The tangent stiffness matrix is recomposed considering the geometric change of the example bridges at every iterative step.

W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 2137 Fig. 4. FE modeling of the example bridges. The fiber element model utilizes the arc-length method. This model establishes the instantaneous properties of a cross section subjected to both axial forces and bending moments in the iterative process. In this process, the cross section is considered to consist of some fiber elements, each subjected to different stresses. Therefore, it is possible for each fiber element to have a different stiffness. When a fiber element at a given integration point exceeds the yield limit of the material on a given elastic perfectly plastic curve, the element stiffness matrix should be gradually revised to form an inelastic stiffness matrix. The large displacement is considered by means of a geometric stiffness matrix, and the geometric change effect in the total structure is reflected by recomposing the tangent stiffness matrix at every iterative step. The applied factored loads are increased by same ratio in the initial shape and the limit point instability analyses. 4. Finite element modeling of the example bridges This section introduces the finite element three-dimensional modeling of the example bridges. Fig. 4 shows the configuration for the example bridges and the numberings of nodes and of cable elements in the global coordinate system. The girders, which have the central span length of 122.0 m, are

2138 W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 supported by a series of cables aligned in a fan-type, a semi harp-type and a harp-type structure respectively. The girders, the cross beams and the pylons are modeled using a number of beam column members. Each cable stay consists of an equivalent straight truss element, and is connected with pins at the girders and the pylons. The behavior of cable-stayed bridge depends highly on the manner in which the girders are connected to the pylons and other piers. The floating system, which has no restraint between the girders and the pylons, is applied to the example bridges. It permits the girders of the example bridges to swing freely between the pylons. All other supports of the side piers to the girders are simply supported by using hinges and rollers. The stress strain curve for beam column members is assumed to be elastic perfectly plastic with an initial elastic modulus of 207 GPa and a yield stress of 248 MPa. The stress strain relationship for cable members should be valid within the elastic limit of the material, with an elastic modulus of 158.6 GPa and a yield stress of 1103 MPa in the limit point instability analysis. The weight per unit volume of the girders, the cross beams and the pylons is 76.82 kn/m 3. One of the cable stays is 60.5 kn/m 3. The dead, the live and the impact loads specified in AASHTO-LRFD [21], which have load factors of 1.25, 1.75 and 0.33, respectively, are used in the instability analyses. The nodal force vectors, including the selfweights of the main members, are applied on the girders and the pylons, while the nodal force vectors due to the live load or the impact load are applied on the girders only in the analyses. The B33 beam element (see Fig. 1) of the commercial code ABAQUS, which has three integration points in an element, is applied to the cross beams, the girders and the pylons for the fiber element model. 5. Verification of the limit point instability analysis results The ultimate behaviors of the example bridges predicted by the softening plastic-hinge and the fiber element models are compared with each other in order to verify the feasibility of this investigation. The difference between the load multipliers (see Table 1) of the fan-type bridge obtained at the limit point by the proposed models is within 3.04%. The corresponding difference in the semi harp-type bridge is within 3.57%, and that in the harp-type bridge is within 2.46%. Fig. 5 presents the load displacement curves of the fan-type bridge having a girder depth of 1.0 m obtained by the proposed models. The analyses from the softening plastic-hinge model and the fiber element model produce displacements of 2.000 m and 2.120 m at the middle point of the central span, respectively, at the overall collapse of the example bridge. The difference between them is within 5.66%. The analyses produce the displacements of 0.491 m and 0.510 m, respectively, at the top in the left pylon and they produce displacements of 0.599 m and 0.646 m, respectively, at the top in the right pylon. Thus, the difference between them observed at the top in the right pylon is within 7.28%. The FE mesh density, which has 181 elements for the example bridges (see Fig. 4), ensures sufficient accuracy of the numerical solution. This indicates that the different approaches Fig. 5. Load displacement curves of the fan-type bridge. cause the difference between the two results for the example bridges. The load displacement curves are almost in straight lines before the first full plastic-hinges are developed at the nodes of 20 and 59. It implies that not many geometrical nonlinear effects of the main members, including the cable sag effect, are in existence in the early ultimate behavior of the example bridges. 6. Analysis of the in-plane overall collapse mechanism This section analyzes the in-plane overall collapse mechanism of the example bridges. The vertical component of the cable reactions acting on the right pylon, which is always larger than the horizontal component of the cable reactions acting on the girders for cable-stayed bridges without regard to the cable layouts, is dominant in the failure of the example bridges in the bifurcation point instability analysis. The vertical component of the cable reactions acting on the right pylon is smallest when the example bridge is supported by a harp-type cable layout, while the fan-type bridge produces the largest one due to the geometry (inclination) of the cable layout. Therefore, the harp-type bridge is superior and the fan-type bridge is inferior with regard to the ultimate load-carrying capacity (see Table 1) based on the bifurcation point instability analysis. The critical axial forces, which are observed at the lower part of the right pylon, are 34,066 kn, 39,511 kn and 41,253 kn when the example bridges are supported by a fan-type, semi harp-type and harp-type cable layouts respectively. Fig. 6 shows the variation of the sag observed at the exterior cable (No. 12) and the interior cable (No. 15) stays during the load increment in the limit point instability analysis. E eq E c = 1.0 means the absolute disappearance of the cable sag. The large part of the sag disappears at the exterior cable stays after the load multiplier reaches 0.5, so that they simultaneously fail at the overall collapse of the example bridges. By contrast, as shown in Figs. 6 and 7, the axial force at the interior cable stay reaches the elastic limit of the material at the overall collapse of the harp-type bridge, since this cable stay of the harptype bridge recovers a straight line prior to the other example bridges. As a result, the relatively small cable reactions (see Fig. 7) of the fan-type bridge produce the smallest bending moment (see Fig. 8) at the lower part of the right pylon, while

Table 1 Load multipliers of the example bridges Bridge type Fan-type Semi harp-type Harp-type (A) (B) (A) 100 (%) (B) 100 (%) (A) (B) (A) 100 (%) (B) 100 (%) (A) (B) (A) 100 (%) L.M 3.18 2.23 2.30 38.26 3.04 3.70 2.16 2.24 65.18 3.57 3.88 1.98 2.03 91.13 2.46 (A), (B) and are the load multipliers obtained by the nonlinear inelastic buckling, the softening plastic-hinge and the fiber element models, respectively. L.M. denotes the load multiplier. (B) 100 (%) W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 2139

2140 W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 Fig. 6. Variation of the cable sag during the load increment. buckling model does not consider the bending moment effect. The limit point instability approach is found to be appropriate in accounting for the conservative and reasonable evaluation of the ultimate load-carrying capacity of the example bridges involving the cable layouts. 7. Conclusions Fig. 7. Cable axial forces at the overall collapse of the example bridges. the relatively large cable reactions (see Fig. 7) of the harp-type bridge, which come from the inefficient cable layout, produce the largest bending moment (see Fig. 8) at the lower part of the right pylon. As seen in Figs. 8 and 9, the multi-plastification at the high level is observed at the middle point on the central span and the lower part of the right pylon in the limit point instability analysis. The bending moment is dominant in the plastification, which defines the ultimate load-carrying capacity of the example bridges at the lower part of the right pylon. The large bending moment observed at the lower part of the right pylon due to the excessive cable reactions does not permit the harp-type bridge to effectively resist the applied load until the margin of the ultimate resistance capacity of the girders is exhausted (see Fig. 9). As a result, the deflection of the harp-type bridge girder is smallest (see Fig. 10). The result of this investigation shows that the example bridges supported by fan-type, semi harp-type and a harp-type cable layouts resist, respectively, up to 2.23, 2.16 and 1.98 times the applied load. From an engineering point of view, the relative errors (see Table 1) between the load multipliers of (A) and (B) for the example bridges are not acceptable. The nonlinear inelastic buckling model not only overestimates but also unreasonably estimates the ultimate load-carrying capacity of the example bridges involving the cable layouts. The absurdity comes from the in-plane overall collapse mechanism, where the failure strongly depends on the vertical component of the cable reactions acting on the right pylon, as the nonlinear inelastic The work is focused on analyzing the in-plane overall collapse mechanism of the example bridges, which are supported by different cable layouts, using the bifurcation point instability and the limit point instability approaches. The following are the findings and conclusions. (1) The nonlinear inelastic buckling model developed for bifurcation point instability analysis can approximately consider not only geometric nonlinearity but also material nonlinearity, such as initial imperfection and residual stress using the KSHB curve, whereas the softening plastic-hinge and the fiber element models developed for limit point instability analysis can account for the following key factors influencing the behavior of cablestayed bridges: the sag effect for cable members; the axial force bending moment interaction for girders and pylons; the large displacement; the material nonlinear behavior of each structural component (including gradual yielding and plastification). (2) Identical solutions are obtained for the results involved with the softening plastic-hinge and the fiber element models. The maximum difference between the load multipliers using the proposed models is observed to be within 3.57% at the limit point when the example bridge is supported by a semi harp-type cable layout. The maximum difference between the displacements is observed to be within 7.28% at the top in the right pylon of the fan-type bridge. (3) The bifurcation point instability approach is not valid for evaluating the ultimate load-carrying capacity of the example bridges with different cable layouts; this is due to the distorted in-plane collapse mechanism, where the vertical component of the cable reactions acting on the right pylon strongly defines the ultimate load-carrying

W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 2141 Fig. 8. Nondimensional member forces of the example bridges at overall collapse. Fig. 9. Level and range of the plastification of the example bridges at overall collapse. Fig. 10. Deformed shape of the example bridges at overall collapse.

2142 W.-K. Song, S.-E. Kim / Engineering Structures 29 (2007) 2133 2142 capacity of the example bridges. Comparatively, the limit point instability approach is found to be appropriate in accounting for the conservative and reasonable evaluation of the ultimate load-carrying capacity of the example bridges with the cable layouts considered. The local failure caused by the full plastification of some critical members defines the ultimate load-carrying capacity of the example bridges, so they do not buckle in the limit point instability analysis. (4) The results of the limit point instability analysis indicate that with regard to the ultimate load-carrying capacity, the fan-type cable-stayed bridge is superior and the harp-type bridge is the most inferior one. References [1] Krishna P, Arya S, Agrawal TP. Effect of cable stiffness on cable-stayed bridges. J Struct Eng, ASCE 1985;111(9):2008 20. [2] Fleming JF. Nonlinear static analysis of cable-stayed bridge structures. Comput Struct 1979;10(4):621 35. [3] Kanok-Nukulchai W, Guan H. Nonlinear modeling of cable-stayed bridges. J Struct Steel Res 1993;26:249 66. [4] Nakai H, Kitada T, Ohminami R, Nishimura T. Elasto-plastic and finite displacement analysis of cable-stayed bridges. Mem Fac Eng, Osaka University 1985;26:251 71. [5] Seif SP, Dilger WH. Nonlinear analysis and collapse load of P/C cablestayed bridges. J Struct Eng, ASCE 1990;116(3):829 49. [6] Nagai M, Asano K, Watanabe K. Applicability of the E f method and design method for evaluating the load-carrying capacity of girders in cable-stayed bridges. J Struct Eng, JSCE 1995;41(A):221 8 [in Japanese]. [7] Nogami K, Nagai M, Kinoshita H, Yamomoto K, Fujino Y. Load-carrying capacity of cable-stayed girders on multiple ultimate strength curves and its stability check. J Struct Eng, JSCE 1997;43(A):253 61 [in Japanese]. [8] Orbison JG. Nonlinear static analysis of three-dimensional steel frames. Report no. 82-6. Ithaca (New York): Department of Structural Engineering, Cornell University; 1982. [9] Liew JY, Tang LK. Nonlinear refined plastic hinge analysis of space frame structures. Research report no. CE027/98. Singapore: Department of Civil Engineering, National University of Singapore; 1998. [10] Ren WX. Ultimate behavior of long-span cable-stayed bridges. J Bridge Eng 1999;4(1):30 7. [11] El-Zanaty MH, Murray DW. Nonlinear finite element analysis of steel frames. J Struct Eng, ASCE 1983;109(2):353 68. [12] Specifications for Highway Bridges in Korea (KSHB). Korean Society of Civil Engineers. 1996 [in Korean]. [13] Shu HS, Wang YC. Stability analysis of box-girder cable-stayed bridges. J Bridge Eng 2001;6(1):63 8. [14] Honshu-Shikoku Bridge Authority (HSBA). Design code of tower for suspension bridges. 1988 [in Japanese]. [15] Kim SE, Chen WF. Practical advanced analysis for braced steel frame design. J Struct Eng, ASCE 1996;122(11):1266 74. [16] Kim SE, Chen WF. Practical advanced analysis for unbraced steel frame design. J Struct Eng, ASCE 1996;122(11):1259 65. [17] Chen WF, Lui EM. Stability design of steel frames. Boca Raton: CRC Press; 1992. [18] American Institute of Steel Construction (AISC). Load and resistance factor design specification. 3rd ed. Chicago: AISC; 2001. [19] Ernst HJ. Der E-Modul von Seilen unter Beruecksichtigung des Durchhanges. Der Bauingenieur 1965;40:52 5. [20] Gimsing NJ. Cable supported bridges: Concepts and design. John Wiley & Sons; 1983. [21] AASHTO. AASHTO LRFD bridge design specification. AASHTO; 1998.