ELASTIC-PLASTIC SEISMIC BEHAVIOR OF LONG SPAN CABLE-STAYED BRIDGES

Transcription

1 ELASTIC-PLASTIC SEISMIC BEHAVIOR OF LONG SPAN CABLE-STAYED BRIDGES By Wei-Xin Ren 1 and Makoto Obata 2 ABSTRACT: This aer investigates the elastic-lastic seismic behavior of long san cable-stayed steel bridges through the lane finite-element model. Both geometric and material nonlinearities are involved in the analysis. The geometric nonlinearities come from the stay cable sag effect, axial force-bending moment interaction, and large dislacements. Material nonlinearity arises when the stiffening steel girder yields. The examle bridge is a cable-stayed bridge with a central san length of 605 m. The seismic resonse analyses have been conducted from the deformed equilibrium configuration due to dead loads. Three strong earthquake records of the Great Hanshin earthquake of 1995 in Jaan are used in the analysis. These earthquake records are inut in the bridge longitudinal direction, vertical direction, and combined longitudinal and vertical directions. To evaluate the residual elastic-lastic seismic resonse, a new kind of seismic damage index called the maximum equivalent lastic strain ratio is roosed. The results show that the elastic-lastic effect tends to reduce the seismic resonse of long san cable-stayed steel bridges. The elastic and elastic-lastic seismic resonse behavior deends highly on the characteristics of inut earthquake records. The earthquake record with the largest eak ground acceleration value does not necessarily induce the greatest elastic-lastic seismic damage. INTRODUCTION The reconstruction of bridges in Euroe, rimarily Germany, destroyed during World War II rovided bridge engineers with the oortunity to aly new technology to an old concet in bridge design, the cable-stayed bridge, although the concet of suorting a bridge girder by inclined tension stays can be traced back to the seventh century (Podolny and Fleming 1972; Troitsky 1977; Podolny and Scalzi 1978). The increasing oularity of contemorary cable-stayed bridges among bridge engineers can be attributed to the aealing aesthetics; the full and efficient utilization of structural materials; the increased stiffness over susension bridges; the efficient and fast mode of construction; and (5) the relatively small size of the bridge elements. Over the ast 40 years, raid develoments have been made on modern cable-stayed bridges. Cable-stayed bridges are now entering a new era, reaching a central san lengths ranging from 400 to 1,000 m or longer. The raid rogress is largely due to the develoment of box-girders with orthotroic late decks; the manufacturing techniques of high-strength wires that can be used for cables; the use of electronic comuters in structural analysis and design; and the advances in restressed concrete structures. It is well known that the increase in the center san length of cable-stayed bridges makes nonlinear analysis inevitable. For this secial tye of flexible, long san cable-suorted bridge, nonlinear analysis is essential for evaluating the stresses and deformations induced not only by static loads but also by dynamic loads, such as vehicular traffic, wind, and earthquakes. When the center san length increases, a ronounced nonlinearity in the resonse may be exected, which will result in a considerable increase in the dislacement and deformations of the bridge under strong shaking. In this case it is essential to understand the behavior from those dynamic loadings realistically. A long san cable-stayed bridge exhibits nonlinear charac- 1 Prof., Det. of Civ. Engrg., Changsha Railway Univ., Changsha, , Hunan Province, Peole s Reublic of China. 2 Assoc. Prof., Det. of Civ. Engrg., Nagoya Institute of Technology, Gokiso-Cho, Showa-ku, Nagoya 466, Jaan. Note. Discussion oen until January 1, To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscrit for this aer was submitted for review and ossible ublication on December 22, This aer is art of the Journal of Bridge Engineering, Vol. 4, No. 3, August, ASCE, ISSN /99/ /$8.00 $.50 er age. Paer No teristics under any load level. These nonlinear sources may come from The sag effect of inclined cable stays The combined axial load and bending moment interaction effect of the girder and towers The large dislacement effect The nonlinear stress-strain behavior of materials (material nonlinearity) Many investigations have studied the dynamic behavior and seismic resonses of this highly nonlinear structure. Some researchers disregarded all sources of nonlinearities, whereas others included one or more of these sources. The 2D geometric nonlinear seismic resonses of cable-stayed bridges were studied by Fleming and Egeseli (1980), whereas other investigators (Nazmy and Abdel-Ghaffar 1990a,b; Abdel- Ghaffar and Nazmy 1991) studied the 3D geometric nonlinear seismic resonses of long san cable-stayed bridges. Dumanoglu and Stevern (1989) analyzed the earthquake resonse of modern susension bridges subjected to asynchronous longitudinal and lateral ground motion with the lane and linear finite-element model. Betti et al. (1993) dealt with the kinematic soil-structural interaction for long san cable-suorted bridges. Jones and Sartz (1990) resented the results of a study that attemted to searate and estimate the mechanical and aerodynamic daming of a rototye bridge structure using full-scale ambient vibration and wind measurements, finite-element analysis of the bridge structure, and a wind tunnel model of the bridge deck section. Wilson and Gravelle (1991a) and Wilson and Liu (1991b) studied the dynamic behavior of cable-stayed bridges through a linear elastic 3D finite-element model. In their studies the modal behavior redicted by the finite-element model is comared to measured ambient vibrations of the full-scale cable-stayed bridge. Until now, however, the elastic-lastic seismic resonse analysis of long san cablestayed bridges has seldom been done. In fact, large member stresses may be induced under strong ground motions. Another imortant feature of long san bridges is the effects of the dead load. The dead load of a long san cable-stayed bridge may contribute 80 90% to total bridge loads. Dead loads are alied before the earthquake so that nonlinear seismic analysis should start from the deformed equilibrium configuration due to dead loads. This aer is focused on the lane elastic and elastic-lastic seismic resonse behavior of a long san cable-stayed steel bridge with a center san length of 605 m. Both geometric 194 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999

2 and material nonlinearities are considered. Three strong earthquake records of the Great Hanshin earthquake of 1995 in Jaan are used in the analysis. These earthquake records are inut in the longitudinal direction [north-south (N-S) comonent], vertical direction [u-down (U-D) comonent], and combined longitudinal and vertical directions. The seismic resonse comutations start from the deformed equilibrium configuration due to dead load. Geometric nonlinear effects are included. To evaluate the elastic-lastic seismic resonses, a new kind of seismic damage index called the maximum equivalent lastic strain ratio is roosed. NONLINEAR CONSIDERATIONS A long san cable-stayed bridge is a nonlinear structural system in which the main girder is suorted elastically at oints along its length by inclined cable stays. Although the behavior of the material is linearly elastic, the overall loaddislacement resonse may be nonlinear under normal design loads (Fleming and Egeseli 1980; Nazmy and Abdel-Ghaffar 1990b). The geometric nonlinear sources of long san cablestayed bridges arise from the stay cable sag effect, the axial force-bending moment interaction of the suerstructure, and large dislacements. A oular aroach to account for the sagging of inclined cables is to consider a straight chord member with an equivalent modulus of elasticity first suggested by Ernst (1965). If the cable stress changes from 1 to 2 during a certain load increment, the secant value of equivalent elastic modulus over the load increment can then be used as follows (Nazmy and Abdel-Ghaffar 1990c): E E eq = 2 (L 0 ) ( 1 2) E 24 where E eq = equivalent elastic modulus of inclined cables; E = cable material elastic modulus; L 0 = horizontal rojected length of the cable; = weight er unit volume of the cable; and = cable tensile stress. Similar to all nonlinear structural analysis roblems, the nonlinear analysis of a long san cable-stayed bridge finally reduces to forming the nonlinear incremental equilibrium equations of the system and to solving these equations. The nonlinear finite-element method (NFEM), with the concet of continuum body discretization, has been a oular way to study the nonlinear behavior of long san cable-stayed bridges. Based on the characteristics of the geometric nonlinear sources described above, some NFEM formulations have been roosed. In the lane formulation resented by Fleming and Egeseli (1980) or the sace formulation resented by Nazmy and Abdel-Ghaffar (1990a), for examle, the sag effect of cables was accounted for by the Ernst s equivalent elastic modulus concet. The udated structural geometry due to large dislacements was included by udating each set of node coordinates, whereas the axial force-bending moment interaction effect was considered by introducing the stability beam functions based on structural stability analysis. By using the Ernst s equivalent elastic modulus concet, the 1 2 secant stiffness matrix of an inclined cable stay is simly equal to the stiffness matrix of a truss element with length L and cross-sectional area A. Truss elements can therefore be sufficiently used to model the inclined cables. Both bending and axial force members such as the stiffening girder and towers are suitably modeled by nonlinear beam elements. It is acceted in ractice to assume that the geometrical deformations of structural members in a long san cablestayed bridge are characterized by large dislacements and large rotations but small strains. With the develoments of the finite-deformation theory and nonlinear FEM techniques with comuters, researchers are more likely to use the finitedeformation theory to study the nonlinear behavior of long san cable-stayed bridges. There are several rigorous NFEM formulations available such as total Lagrangian formulation, udated Lagrangian formulation, and corotational formulation. Any of these formulations can include large dislacements, large rotations, and small strains. Therefore, the geometric nonlinear sources of a long san cable-stayed bridge, such as the interaction between axial force and bending moment as well as large dislacements, can be accurately considered. A cable-stayed bridge is usually comosed of three main structural arts (or bridge comonents), namely, inclined cable stays, suerstructure, and towers. These structural arts may be made of different materials. The material nonlinear analysis of a long san cable-stayed bridge deends on the nonlinear stress-strain behavior of individual materials for the structural comonents. When some oints (integration oints) of an element exceed the yielding limit of the material, the stiffness matrix of the element should be revised to form the elasticlastic stiffness matrix through the incremental lastic theory. After the long san cable-stayed bridge is comleted, and before the earthquake loading is alied, the bridge has sustained large dead load deformations and stresses in each member, which should be included. To consider the effect of dead load, the seismic resonse analysis should involve two stes: The static analysis under dead loading to form the deformed equilibrium configuration; and the seismic resonse analysis starting from the deformed configuration due to dead load. In the seismic resonse analysis, the nonlinear differential equations of motion are directly integrated by the Newmark -method. The integration constants = 1/4 and = 1/2 are chosen, which corresond to an unconditionally stable scheme. In each time ste the Newton-Rahson iteration technique is used to obtain the solution that satisfies both equilibrium equations and material constitutive relationshis. MODELING OF EXAMPLE CABLE-STAYED BRIDGE Descrition of Examle Bridge The examle bridge studied here is the Ming River long san cable-stayed steel bridge. The bridge central san length is 605 m, which is one of the longest central san cable-stayed bridge design in China. The bridge crosses the Ming River, Fuzhou, the caital of the Fujian Province and reresents an imortant link in the highway between Fuzhou city and the new airort of Fuzhou. The bridge san arrangements are 90 FIG. 1. Elevation View of Examle Bridge JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999 / 195

3 TABLE 1. Material Data of Examle Bridge Material E (GPa) y (MPa) b (MPa) (5) (kg/m 3 ) (6) Girder ,850.0 Cables , ,600.0 Towers ,625.0 FIG. 2. Section of Steel Boxed Girder TABLE 2. Cable Data Cable number Number of wires Section area (cm 2 ) Density (kg/m 3 ) C 18, C ,206.0 C 18, C 17, C 16, C 15; C 16, C ,227.0 C 14, C 13; C 14, C ,254.0 C 12, C 11, C 10; C 12, C 11, C ,105.0 C, 9 C, 8 C ; 7 C 9, C 8, C ,482.0 C, 6 C ; 5 C 6, C ,218.0 C, 4 C ; 3 C 4, C ,488.0 C, 2 C ; 1 C 2, C ,318.0 C i = ith outer cables; C i = ith inner cables. FIG. 3. Reinforced Concrete Tower m. The elevation view of the bridge is shown in Fig. 1. The deck cross section is an aerodynamically shaed closedbox steel girder, 25.1 m wide and 2.8 m high, as deicted in Fig. 2. The bridge towers are A-shaed steel reinforced concrete towers, m high, as shown in Fig. 3. The stay cable arrangement is a two-vertical-lane system. The eight grous of cables are comosed of high-strength 7-mm wires. The design live load of the bridge is the Truck-suer 20 (Highway Bridge Design Secification of China). There are four traffic lanes. Material Data The three bridge comonents, namely, stiffening girder, stay cables, and towers, of the examle bridge are comosed of three different materials. The main stiffening girder is structural steel, while the towers are reinforced concrete. The stay cables are comosed of high-strength wires. Material data of the examle bridge are listed in Table 1. The cable data is summarized in Table 2, where the weight er unit volume of each stay cable deends on the number of wires in individual stay cables. In the elastic-lastic seismic resonse analysis, the elasticlastic behavior of the steel girder is only considered. In other words, stay cables and towers are assumed to remain elastic throughout. This assumtion is acceted according to the numerical results of the examle bridge. The elastic-lastic constitutive model of the steel girder under cycle loading used in the analysis is the common bilinear hysteric model of structural mild steel as shown in Fig. 4, where 1 = 235 MPa; ε 1 = 0.0 (2a) 2 = 300 MPa; ε 2 = (2b) FIG. 4. Bilinear Hysteretic Model It should be noted above that the yield limit of the steel girder used in the elastic-lastic seismic analysis is lower than its actual yield stress given in Table 1. Finite-Element Modeling The finite-element modeling of the examle bridge is almost the same as Ren (1997; 1999). The stiffening girder is divided into 76 lane beam elements, and each tower is divided into 32 lane beam elements. Each cable is treated as a lane truss element with the concet of equivalent elastic modulus. The bridge dead loads are alied in the first analysis ste. The total dead loads of the examle bridge are the uniformly distributed stiffening girder dead load q d = 170 kn/m, which is its own weight, and the weight of the reinforced concrete towers that can be obtained by the mass body density of the towers. The boundary conditions are as follows: The joint between the stiffening girder and left tower is a fixed hinge, whereas another joint between the stiffening girder and right tower is a movable hinge. All side iers are movable hinge (roller) suorts. Inut Ground Motions To evaluate the effects of different strong ground motions on the seismic resonse behavior of long san cable-stayed bridges, three earthquake records are used in the analysis. They were all recorded during the Great Hanshin earthquake of 1995 in Jaan. These are the Kobe record measured at the 196 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999

4 Comonent TABLE 3. PGAs of Each Comonent (cm/s 2 ) Kobe Record Takatori Higashi Kobe N-S U-D Jaan Meteorological Agency Kobe Observatory, the Takatori record measured at Jaanese Railway Takatori station, and the Higashi Kobe recorded at Higashi Kobe Bridge. The bridge longitudinal direction refers to the N-S comonent, and the vertical direction refers to the U-D comonent of each record. The eak ground accelerations (PGA) of each comonent are listed in Table 3. In the numerical comutations of the seismic resonses, a total time of 30 s of each record are used. The time ste is 0.01 s for the Takatori and Higashi Kobe records, whereas the 0.02 s time ste is used for the Kobe record. It is found that those time stes are aroriate. NATURAL FREQUENCIES The natural frequencies and mode shaes of the examle bridge are first calculated. To investigate the effect of the dead load on the natural frequencies of a long san cable-stayed bridge, two cases are comuted. One is the case without the dead load where the dynamic comutation is starting from the undeformed configuration, and another is the case with dead loads where the comutation is starting from the dead load deformed configuration. The first 10 natural frequencies of the examle bridge are listed in Table 4. It is shown that the effects of the dead load decrease the bridge natural frequencies somewhat. Fig. 5 rovides the first mode shae of the examle bridge with the dead load included. The deformed configuration due to dead load may be formed by linear static analysis and geometrically nonlinear static analysis. The natural frequencies starting from those two deformed configurations are comuted resectively. The results are almost the same so that small deformation theory is accetable to form the deformed configuration due to dead load. ELASTIC-PLASTIC SEISMIC BEHAVIOR The elastic and elastic-lastic lane seismic time-history resonses of the long san cable-stayed bridge under three different earthquake records are studied. The first analysis ste is the static comutation under dead load, as mentioned earlier. In this static analysis ste two cases are considered. One case is the linear analysis under dead load where there is no equilibrium iteration. Another case is the geometric nonlinear analysis where the equilibrium iterations are needed. The second analysis ste is the seismic resonse analysis. Three kinds of strong ground motions recorded during the Great Hanshin earthquake of 1995 in Jaan are inut along the bridge longitudinal direction (N-S comonent), vertical direction (U-D comonent), and combined longitudinal and vertical directions, resectively. Again, to investigate the effect of the dead load on the seismic resonse behavior of a long san cable-stayed bridge, the seismic resonse comutations starting from the undeformed configuration and starting from the deformed equilibrium configuration due to dead loads are studied searately and comared. To show the effects of geometric nonlinearity and material nonlinearity, the following four cases are investigated: 1. Small deformation and linear elastic material seismic behavior 2. Geometric nonlinearity and linear elastic material seismic behavior 3. Small deformation and elastic-lastic material seismic behavior (steel girder) 4. Geometric nonlinearity and elastic-lastic material seismic behavior (steel girder) In each seismic comutation case, the following timehistory seismic resonses are recorded: Dislacement time-history resonses the horizontal and vertical dislacements at the san center, as well as the horizontal deartment at the to of the left tower Reaction time-history resonses at the bottom of the left tower the shear force, axial force, and bending moment Cable stress time-history resonse of inner cable No. C 18 Stiffening girder internal force resonses at the joints between the stiffening girder and the left tower the axial force, shear force, and bending moment Only arts of the results are listed in this aer. Table 5 gives the comarison between elastic and elastic-lastic maximum dislacement resonses under three earthquake records inut along the combined longitudinal and vertical directions. Table 6 summarizes the comarison between elastic and elastic-lastic maximum reaction resonses as well as maximum cable stress resonses under three earthquake records inut along the combined longitudinal and vertical directions. The comarisons between elastic and elastic-lastic seismic timehistory resonses under three earthquake records inut along the combined longitudinal and vertical directions are shown in Figs. 6 8, resectively. The seismic time-history resonses include dislacement resonses at the san center, bending moments at the bottom of the left tower, and C 18 cable stresses. All results of Tables 5 and 6 and Figs. 6 8 are obtained starting from the deformed equilibrium configuration due to dead loads. TABLE 4. Natural Frequencies (Hz) Mode Frequency Without dead loads With dead loads FIG. 5. First-Order Vibration Mode Shae (f 1 = Hz) JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999 / 197

5 TABLE 5. Earthquake records Maximum Dislacement (m) Resonses Dislacements at Center of San Horizontal Kobe ( 0.691) (0.317) Takatori ( 0.320) (0.806) Higashi Kobe ( 7.87) (0.062) Vertical ( 6.390) (0.0) ( 4.540) (0.924) ( 3.786) (0.0) Horizontal dislacements at to of left tower ( 0.179) (1.104) ( ) (2.382) ( 8.448) (1.849) Note: Values in brackets are corresonding elastic seismic resonse values. From the results obtained in the elastic and elastic-lastic seismic resonse analyses of a long san cable-stayed bridge under three strong earthquake records, the following can be observed: The comarisons between seismic resonses starting from the undeformed initial configuration and seismic resonses starting from the deformed equilibrium configuration due to dead load have shown that the initial equilibrium configuration is essential to start the seismic resonse analysis (Ren 1997). Both linear and nonlinear seismic resonse analyses of long san cable-stayed bridges should start from the deformed equilibrium configuration due to dead loads. Regarding the equilibrium configuration under dead loads, the seismic time-history resonses starting from the linearly deformed equilibrium configuration and starting from the nonlinearly deformed equilibrium configuration have shown that the difference is so small that linearly elastic analysis is accetable when forming the deformed equilibrium configuration due to the dead loads (Ren 1997). Starting from the deformed equilibrium configuration due to dead loads, it is found that there is only a small difference between linear and geometrically nonlinear seismic time-history analysis under strong ground motions, although the resent center san length of the examle cable-stayed bridge is u to 600 m. The geometric nonlinearity has little influence on the lane seismic resonse behavior of the examle bridge. Therefore, small deformation seismic resonse analysis is accetable for the seismic resonse analysis of long san cable-stayed bridges as was concluded by Fleming and Egeseli (1980) and Nazmy and Abdel-Ghaffar (1990b), if starting from the deformed equilibrium configuration due to dead load. Comared with seismic time-history resonse results under the longitudinal direction inuts, vertical direction inuts, and combined longitudinal and vertical inuts, it is found that the suerosition rincile is not suitable to the seismic resonse of long san cable-stayed bridges even in the linear seismic resonse case. Seismic resonses under combined longitudinal and vertical inuts are not the sum of the individual longitudinal and vertical direction inuts due to the couling effect that exists between longitudinal and vertical seismic resonses. This is robably caused by the comlicated structural dynamic characteristics of such long san cable-suorted structures. From both elastic and elastic-lastic seismic maximum resonses (Tables 5 and 6), it can been seen that the maximum horizontal dislacement resonses at the center of the san and at the to of the left tower under the Higashi Kobe record inut are the largest, although the PGA of the Higashi Kobe record is the lowest among the three earthquake records. The largest maximum vertical dislacement resonse at the san center is caused by the Kobe record, whereas the largest maximum C 18 cable stress resonse and the largest bending moment reaction resonse at the bottom of the left tower are caused by the Takatori record inut. Therefore, both elastic and elasticlastic seismic resonse behaviors of long san cablestayed bridges are strongly deendent on the characteristics of ground motions. The earthquake record with the largest PGA value does not necessarily induce the greatest maximum resonse. Comarisons between elastic and elastic-lastic seismic resonses (Tables 5 and 6 and Figs. 6 8) have shown that the elastic-lastic dynamic deformation of the stiffening steel girder tends to reduce the resonse because the elastic-lastic deformation of the girder absorbs art of the energy. Even under the combined longitudinal and vertical seismic inut, the cable stress resonses are all below the material strength limit so that the stay cables are still elastic for the examle bridge. EVALUATION OF ELASTIC-PLASTIC SEISMIC RESPONSES The ductility caacity and energy absortion caacity of a structural system are robably the most significant design features that affect the ability of a structure to withstand strong seismic ground motions. One of the most imortant roblems in the elastic-lastic seismic resonses is how to evaluate the intensity of the dynamic lastic deformation or the damage of the structure. Actually, engineers are more concerned with the residual deformation (lastic deformation) of the structure from severe earthquakes. This kind of residual lastic deformation is also used to judge the level of earthquake hazard. Engineers decide whether the structure can be reaired or not according to the residual deformation intensity of the structures. Ductility indices or damage indices are commonly used to characterize the elastic-lastic seismic resonses of structures. Many seismic damage arameters or damage indices have been roosed. Some of them may correlate with others. Very imortant feature involves the selection of aroriate damage arameters that are to be determined during a resonse inves- TABLE 6. Maximum Reaction and Cable Stress Resonses Earthquake records Q (MN) Kobe ( 316.5) (275.3) Takatori ( 293.3) (299.5) Higashi Kobe ( 240.9) (179.8) Reactions at Bottom of Left Tower N (MN) (0.0) (651.4) (0.0) (612.9) (0.0) (534.5) Note: Values in brackets are corresonding elastic seismic resonse values. M (MN m) 6, ,552.0 ( 8,780.9) (12,379.0) 7, ,108.0 ( 11,537.0) (13,356.0) 6, ,567.0 ( 7,862.9) (12,804.0) C 18 cable stress (MPa) (5) ,007.8 ( 284.2) (1,118.0) ,222.8 ( 709.9) (1,341.9) ( 313.7) (1,008.7) 198 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999

6 FIG. 6. Comarison between Elastic and Elastic-Plastic Resonses Under Kobe Record JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999 / 199

7 FIG. 7. Comarison between Elastic and Elastic-Plastic Resonses Under Takatori Record 200 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999

8 FIG. 8. Comarison between Elastic and Elastic-Plastic Resonses Under Higashi Kobe Record JOURNAL OF BRIDGE ENGINEERING / AUGUST 1999 / 201

9 tigation. Overall dislacement and overall energy dissiation are oularly used as seismic resonse damage arameters. They are both quantities that have a direct hysical meaning. Various damage indices have been develoed (Loh and Ho 1990). There are global and local damage indices deending on the characteristics of the damage arameters used. Most damage indices are based on a single-degree-of-freedom system or exeriments in which the cycle numbers are limited. Actually, in the numerical comutation of the elastic-lastic seismic resonses for comlicated structures, a large number of degrees of freedom and cycle numbers are used. Therefore, further research is needed to characterize the elastic-lastic seismic damage, which can be easily determined in the NFEM numerical comutation of structural elastic-lastic seismic resonse. Seismic damage indices should take into account both the maximum deformation and the cyclic effects. As a suitable seismic damage arameter, it should ossess the cumulative characteristics during cyclic loading that can be conveniently determined. Plastic strains ossess cumulative characteristics during cycle loading, and it is art of the direct outut of the elastic-lastic seismic resonse comutation with numerical methods such as the NFEM. The maximum equivalent lastic strain of the structural elements may therefore be a suit- able arameter to evaluate the inelastic deformation intensity or the damage intensity of the structural elastic-lastic seismic resonse. Similar to the maximum dislacement ductility ratio where the ductility index is the ratio of the maximum dislacement x max to the yield dislacement x y, based on the lastic strain of structural elements, a new kind of the seismic damage index called the maximum equivalent lastic strain ratio ε is roosed, which is defined as ε max where = total maximum equivalent lastic strain accu- mulated in the structural elements. The denominator ε y is used to normalize the damage index. It may be an aroriate strain as a material constant such as the yielding strain. In the following comutation, this strain is chosen to be the lastic strain ε of defined in the bilinear hysteretic model of the steel girder. Obviously, the maximum equivalent lastic strain ratio ε is a kind of local damage index because the strain is a local arameter. The maximum equivalent lastic strain ratio ε of the examle long san cable-stayed bridge under three strong ground motions, inut along the combined longitudinal and vertical directions, are listed in Table 7. It can be found that ε max TABLE 7. Strain Maximum equivalent lastic strain Maximum Plastic Strain Ductility Ratio Kobe ε max = ε ε y Under the Kobe and Takatori earthquake record inuts, the steel girder element with the maximum equivalent lastic strain is close to the san center, whereas the element with the maximum equivalent lastic strain is located at the left auxiliary ier under the Higashi Kobe earthquake record. Under the Higashi Kobe earthquake record, maximum equivalent lastic strain ratio ε is the largest, although the PGA of the Higashi Kobe record is not the largest among the three earthquake records. Under the Kobe Earthquake Record Takatori Higashi Kobe ε max Ductility ratio ε earthquake record inut with the largest GPA value, maximum equivalent lastic strain ratio ε becomes the least. Therefore, the elastic-lastic seismic behavior or elasticlastic seismic damage of long san cable-stayed bridges deends highly on the characteristics of inut earthquake records and the dynamic roerties of the structures. The earthquake record with the largest PGA value does not necessarily induce the greatest elastic-lastic seismic damage. CONCLUSIONS Several observations can be made from the results of the receding analysis: 1. The effect of the dead load must be considered in the seismic resonse analysis of long san cable-stayed bridges. Both linear and nonlinear seismic analyses should start from the deformed equilibrium configuration due to dead load. 2. The geometric nonlinearity has little influence on the seismic resonse behavior, even under strong earthquake record inuts. Small deformation seismic analysis is adequate as was concluded by Fleming and Egeseli (1980) or Nazmy and Abdel-Ghaffar (1990b). 3. Both elastic and elastic-lastic seismic behavior of long san cable-stayed bridges deend highly on the characteristics of earthquake records and the dynamic roerties of the bridge. The earthquake record with the largest PGA value does not necessarily induce the greatest maximum resonses and vice versa. 4. The elastic-lastic effects tend to reduce the seismic resonses. 5. The elastic-lastic seismic damage index called the maximum equivalent lastic strain ratio roosed in this aer, which is a local value of the damage index, rovides a measure of elastic-lastic seismic damage or ductility at the element level. It can be used in the evaluation of damage during elastic-lastic seismic resonse analyses. The index is articularly suitable when the NFEM is used in the elastic-lastic seismic analysis. 6. The elastic-lastic damage of long san cable-stayed bridges caused by strong ground motions is associated with their own structural dynamic roerties and the characteristics of earthquake records. The earthquake record with the largest PGA value also does not induce the greatest elastic-lastic seismic damage. APPENDIX. REFERENCES Abdel-Ghaffar, A. M., and Nazmy, A. S. 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