Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges

Yan Xu; Zeng Zeng; Cunyu Cui; Shijie Zeng

doi:10.3390/app9142857

Xu, Yan;Zeng, Zeng;Cui, Cunyu;Zeng, Shijie

2019-07-17 00:00:00

applied sciences Article Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges 1 , 1 1 2 Yan Xu *, Zeng Zeng , Cunyu Cui and Shijie Zeng State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China China Three Gorges Project Development Co., Ltd., Chengdu 610000, China * Correspondence: yanxu@tongji.edu.cn Received: 29 April 2019; Accepted: 15 July 2019; Published: 17 July 2019 Featured Application: A time ecient method for determining the initial yielding strength of the yielding steel damper applied in cable stayed bridges is proposed. Abstract: Restrained transversal tower/pier–girder connections of cable-stayed bridges may lead to high seismic demands for tower columns when subject to earthquake excitations; however, free transversal tower/pier–girder connections may cause large relative displacement. Using an energy dissipation system can eectively control the bending moment of tower columns and the relative tower/pier-girder displacement simultaneously, but repeated time history analyses are needed to determine reasonable design parameters, such as yield strength. In order to improve design eciency, a practical design method is demanded. Therefore, the inﬂuence of yielding strength at dierent locations is studied by using comprehensive and parametric time history analyses at ﬁrst. The results indicate that yielding steel dampers can signiﬁcantly reduce the bending moment at tower columns and the relative pier–girder displacement due to the system switch mechanism during the vibration. Meanwhile, the yielding steel damper shows its general eect on reducing relative displacement between all piers/tower columns and the main girder as well, with only a localized eect on controlling seismic induced forces. Furthermore, a practical design method is proposed for engineering practices to determine key parameters of the yielding steel damper. Keywords: cable-stayed bridge; transversal direction; yielding steel damper; design method; earthquake 1. Introduction Cable-stayed bridges with a ﬂoating system (i.e., there is no connection between the tower column and main girder) are naturally seismically isolated structures in the longitudinal direction; hence, velocity-related viscous ﬂuid dampers (VFDs) are commonly installed to restrain the longitudinal displacement during strong earthquake vibration and to release the temperature stress under service load. In the transversal direction, however, traditional tower–girder connections that restrain the movement of the girder by using a transversally ﬁxed but longitudinally free friction plate bearing are commonly designed to meet the requirement of service load, such as wind and vehicles, in China. Therefore, such structural measures may lead to a higher seismic demand on the tower columns and the substructure under transversal intensive ground motions. According to the performance objectives that most current seismic design codes have speciﬁed [1–3], one solution is to increase the reinforcement ratio of tower columns by a great amount in addition to meeting its static loading demand, which causes engineering ineciency as well as a rising seismic demand for the substructure, such as the pile foundation. The other is to adopt an energy dissipation system to reduce the seismic demands on the main structure [4–7]. Appl. Sci. 2019, 9, 2857; doi:10.3390/app9142857 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 2857 2 of 24 Over the past decades, many researchers have studied the energy dissipation system and the vibration control of cable-stayed bridges under earthquake excitations, which includes passive control [5–15], active control [8,9,14] and semi-active control [7–10,16–18]. As we all know, passive control does not need external energy. In comparison, active control requires external energy; on the basis of structural response and feedback, the optimal control would be achieved by the optimal control algorithm, and then the control force would be imposed on the controlled structure relying on external energy. It is therefore dicult to ensure that a large amount of external energy can be provided during an earthquake if active control is applied to the structure. Instead, passive control may be a reasonable option. In engineering practice, the elastomeric bearings and VFDs are usually used alone or simultaneously to control the force and displacement responses, and their eectiveness has already been veriﬁed in the transversal direction. However, considering that the cable-stayed bridge may experience large displacement in a longitudinal direction under strong earthquakes, the VFD, though proven to be eective, is dicult to use in the transversal direction [19]. The yielding steel damper (YSD) is another kind of dissipation device, which has the advantages of a lower cost, reliable hysteretic characteristic and a higher initial stiness that can meet the demand of service load. It has various shapes that can adapt to dierent structure forms and thus has gained increasing application in recent years, not only in building structures but also in bridge structures [20–25]. Generally, a YSD’s speciﬁc form can be classiﬁed as E shape [26,27], C shape [28,29], X shape [30,31], triangle shape [19–21] and pipe shape [24,25] etc. Some new types of YSD have also been developed in recent years [22–25,32], and their eectiveness in reducing the seismic demands of several types of bridges has also been studied [19,21,26,30,31,33,34]. In particular, the application of YSDs in cable-stayed bridges are investigated [19–21,28,35], and one of them has also been veriﬁed through the shake table test [19,28]. However, the seismic design of YSDs installed in the transverse direction of cable-stayed bridges requires a time-consuming time history analysis. Additionally, the eect of their design parameters (such as the initial yielding strength of YSD) on the seismic reduction of bridges has not been well studied. Additionally, there exists a conﬂict between reducing seismically induced displacement and seismically induced force simultaneously for cable stayed bridges [12,13,26] when the energy dissipation system is adopted. This paper aims to obtain a simple and practical method to design YSDs based on the inﬂuence of their yielding strength (F ) on the seismic responses of the bridge at dierent locations longitudinally (auxiliary pier, transition pier and tower column). Primarily, a ﬁnite element (FE) model should be established based on a real cable-stayed bridge in China. Furthermore, it is necessary to verify the FE model by comparing with the results of shake table test. Then, the FE model would be used to carry out a comprehensive parametric analysis to investigate the inﬂuence of F on the seismic responses of the bridge. The theoretically appropriate YSD design would be achieved according to a complicated multivariate function analysis or a simpliﬁed single variable function analysis in terms of dierent designed objectives. In order to develop a practical design method, the YSD working mechanism on the cable-stayed bridge is further investigated, and ﬁnally a method is proposed to quickly determine the yield strength of YSDs based on the investigation in this paper. 2. The Analytical Model and Input Ground Motions 2.1. FE Modelling of the Cable-Stayed Bridge The bridge has two H-shaped reinforced concrete towers with a height of 93 m, and the total span is 640 m with a main span of 380 m and two side spans of 70 m and 60 m, respectively. The elevation of the bridge is shown in Figure 1. The dimension and section of the tower/pier are shown in Figure 2. The major materials and their characteristics are listed in Table 1. Appl. Sci. 2019, 9, 2857 3 of 24 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 25 Figure 1. Elevation of the cable-stayed bridge (unit: m). Figure 1. Elevation of the cable-stayed bridge (unit: m). Figure 1. Elevation of the cable-stayed bridge (unit: m). Figure 2. Dimensions of the major structure and their sections (unit: cm). Figure Figure 2. 2. Dim Dimensions ensions of of the m the major ajor stru structur cture e a and nd their their sections sections (u (unit: nit: cm cm). ). Table 1. Main materials of the bridge. Table 1. Main materials of the bridge. Table 1. Main materials of the bridge. Material and Component Standard Strength (MPa) Young’s Modulus (MPa) Material and Component Standard Strength (MPa) Young’s Modulus (MPa) Material and Component Standard Strength (MPa) Young’s Modulus (MPa) Concrete (tower) 32.4 3.45 × 10 4 4 Concrete (towe Concrete (tower) r) 32.4 32.4 3.45 3.45 × 10 10 Concrete (transition pier) 26.8 3.25 × 10 Concrete (transition pier) 26.8 Concrete (transition pier) 26.8 3.25 3.25 × 10 10 Concrete (auxiliary pier) 26.8 3.25 × 10 Concrete (auxiliary pier) 26.8 3.25 10 Concrete (auxiliary pier) 26.8 3.25 × 10 5 5 Cable Cable 1770 1770 2.05 × 105 2.05 10 Cable 1770 2.05 × 10 Rebar Rebar 400 400 2.06 2.06 × 10 105 Rebar 400 2.06 × 10 Note: 1. The standard strength of concrete is the axial compressive strength of concrete measured with Note: 1. The standard strength of concrete is the axial compressive strength of concrete measured with a 150 mm Note: 1. The standard strength of concrete is the axial compressive strength of concrete measured with 150 mm 300 mm prism as the standard specimen, and the standard strength means the strength characteristic a 150 mm × 150 mm × 300 mm prism as the standard specimen, and the standard strength means the a 150 mm × 150 mm × 300 mm prism as the standard specimen, and the standard strength means the value with a certain guarantee rate (95%). 2. Statistical analysis shows that the test value of the rebar and cable strength characteristic value with a certain guarantee rate (95%). 2. Statistical analysis shows that the strength characteristic value with a certain guarantee rate (95%). 2. Statistical analysis shows that the strength conforms to normal distribution, and the standard strength of is the strength characteristic value with a certain guarantee rate (97.73%). test value of the rebar and cable strength conforms to normal distribution, and the standard strength test value of the rebar and cable strength conforms to normal distribution, and the standard strength of is the strength characteristic value with a certain guarantee rate (97.73%). of is the strength characteristic value with a certain guarantee rate (97.73%). A three-dimensional FE model had been developed in a Structural Analysis Program 2000 version A three-dimensional FE model had been developed in a Structural Analysis Program 2000 A three-dimensional FE model had been developed in a Structural Analysis Program 2000 19.0.0 (SAP2000 v 19.0.0) [36,37]. The simulation of a cable-stayed bridge consists of four parts, version 19.0.0 (SAP2000 v 19.0.0) [36,37]. The simulation of a cable-stayed bridge consists of four version 19.0.0 (SAP2000 v 19.0.0) [36,37]. The simulation of a cable-stayed bridge consists of four namely the girder, the tower/pier, the cable and the boundary conditions. Generally, the girder is parts, namely the girder, the tower/pier, the cable and the boundary conditions. Generally, the girder parts, namely the girder, the tower/pier, the cable and the boundary conditions. Generally, the girder equipped with a closed box section with large free torsional stiness, and a single girder model could is equipped with a closed box section with large free torsional stiffness, and a single girder model is equipped with a closed box section with large free torsional stiffness, and a single girder model be adopted considering the axial stiness, bending stiness, torsional stiness, distributed mass and could be adopted considering the axial stiffness, bending stiffness, torsional stiffness, distributed could be adopted considering the axial stiffness, bending stiffness, torsional stiffness, distributed mass moment of inertia concentrated on the central axis. In addition, the girder would be elastic even mass and mass moment of inertia concentrated on the central axis. In addition, the girder would be mass and mass moment of inertia concentrated on the central axis. In addition, the girder would be under occasionally occurring earthquakes. Simultaneously, the tower/pier is also elastic due to the elastic even under occasionally occurring earthquakes. Simultaneously, the tower/pier is also elastic elastic even under occasionally occurring earthquakes. Simultaneously, the tower/pier is also elastic energy dissipation system. Therefore, the girder and tower/pier could be modeled by an elastic frame due to the energy dissipation system. Therefore, the girder and tower/pier could be modeled by an due to the energy dissipation system. Therefore, the girder and tower/pier could be modeled by an element which includes the eects of biaxial bending, torsion, axial deformation, and biaxial shear elastic frame element which includes the effects of biaxial bending, torsion, axial deformation, and elastic frame element which includes the effects of biaxial bending, torsion, axial deformation, and deformations [38]. Obviously, cables should be simulated by a truss element, and the modulus of biaxial shear deformations [38]. Obviously, cables should be simulated by a truss element, and the biaxial shear deformations [38]. Obviously, cables should be simulated by a truss element, and the elasticity modiﬁed by the Ernst formula [39], considering the mechanical properties and the sag eect. modulus of elasticity modified by the Ernst formula [39], considering the mechanical properties and modulus of elasticity modified by the Ernst formula [39], considering the mechanical properties and For boundary conditions, since only the overall response of the structure is considered rather than the sag effect. For boundary conditions, since only the overall response of the structure is considered the sag effect. For boundary conditions, since only the overall response of the structure is considered Appl. Sci. 2019, 9, 2857 4 of 24 the pile–soil eect, the node constraint of SAP2000 can be directly used to simulate the boundary conditions; speciﬁcally, both the tower bottom and pier bottom are ﬁxed. In order to simulate a double-cable-stayed bridge, a rigid arm can be used to connect the girder and the cable; thus, a spine model should be adopted. In the longitudinal and transverse directions, the connections between the girder and the tower/pier are shown in Table 2. The FE model of SAP 2000 in this paper is shown in Figure 3. The girder, tower and pier are divided into around 4-m sections as a unit, and the model has a total of 607 nodes and 447 units. Table 2. The boundary condition of the bridge. Degree of Freedom Degree of Freedom Location (without YSDs) (with YSDs) UX UY UZ RX RY RZ UX UY UZ RX RY RZ Tower 0 1 1 0 0 0 0 YSD (2) 1 0 0 0 Auxiliary pier 0 0 1 0 0 0 0 YSD (2) 1 0 0 0 Transition pier 0 0 1 0 0 0 0 YSD (2) 1 0 0 0 Note: 1. “1” means ﬁxed, “0” means free; yielding steel dampers (YSDs) are installed between the girder and the substructure, and numbers in () are the quantities of YSDs used on each side of the bridge. 2. X means the longitudinal direction, Y means the transverse direction, and Z means the vertical direction. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 25 Figure 3. Finite element (FE) model. Figure 3. Finite element (FE) model. It is necessary to note that ordinary constraints and steel dampers are simulated in dierent ways. An elastic link element with large stiness would be employed when the girder is ﬁxed in the transverse direction at the tower locations. By contrast, the simulation of steel damper is relatively complex. As shown in Figure 4, the YSD is connected at the bottom of the deck through the sliding groove to accommodate the deck movement longitudinally. The YSDs are arranged symmetrically on the two sides of the bridge, and their connections at longitudinal locations on the bridge are shown in Figures 4 and 5. The YSD is designed according to the design criterion for steel dampers [29,40,41]. For the wind load and normal trac loads, the YSD remains elastic and provides enough transverse stiness. During Figure 4. C-shaped steel damper group. the earthquake, the YSD starts to deform and dissipate seismic energy inelastically, thus protecting the tower columns and piers. Oh et al. [42] and Domaneschi et al. [7] performed the cyclic tests of the YSD, showing that it has stable hysteretic behavior, and the hysteretic curve is assumed as the bilinear response (Figure 6), regardless of the speciﬁc form [19–21,24–26,30,32]. Therefore, the YSD is simulated Figure 5. One applied YSD and its connection. K =αK 1 0 Experiment Bilinear Δ Δ Figure 6. Simplified model of the YSD. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 25 Appl. Sci. 2019, 9, 2857 5 of 24 by Plastic Wen units in SAP2000, which are bilinear constitutive models. There are four parameters in damper design, namely the yielding strength F , yielding displacement D , the ratio of post-yielding y y stiness to pre-yielding stiness and ultimate deformation D , while the ultimate strength F is not an u u independent parameter. Normally, the D is set to 10 mm and the ratio of post-yielding stiness to pre-yielding stiness is set as 0.6% (less than 5% depending on the dierent shapes of products), which can provide good hysteretic performance according to related studies [19–21,24–26,30,32]. Therefore, F is the key parameter which needs to be determined in this bilinear model. It is notable that the ultimate deformation D reaches a certain amount in a fabricated YSD product; however, the maximum deformation of the bilinear model is assumed to be unlimited in the analysis so that the inﬂuence of F Figure 3. Finite element (FE) model. Figure 3. Finite element (FE) model. Figure 3. Finite element (FE) model. on the relative displacement between the super- and sub-structures can be observed. Figure 4. C-shaped steel damper group. Figure 4. C-shaped steel damper group. Figure Figure4 4. . C-sha C-shaped ped st steel eel damper group. damper group. Figure 5. One applied YSD and its connection. Figure 5. One applied YSD and its connection. Figure Figure5 5. . One One applied Y applied YSD SD a and nd its its connec connection. tion. Fu Fy K =αK K1=αK0 1 0 K =αK 1 0 K0 Experiment Experiment Experiment Bilinear Bilinear Bilinear Δ Δu Δy Δ y u Δ Δ Figure 6. Simplified model of the YSD. Figure 6. Simpliﬁed model of the YSD. Figure 6. Simplified model of the YSD. Figure 6. Simplified model of the YSD. In addition, it is more accurate to use complex a FE model such as shell elements or ﬁber elements to better consider the dynamic mechanical behavior of the girder instead of a simpliﬁed spine model. However, according to some recent studies [4,28], acceptable dynamic behavior and corresponding seismic responses can be obtained, and we use this model to carry out our comprehensive parametric study of the proper yielding strength of the applied steel dampers for time eciency. 2.2. FE Model Validation Additionally, a shake table test of the bridge model with a scale factor of 1/20 was conducted [27]. Before the parameter analysis of the yield strength of the YSD, it is necessary to prove the correctness of the model without YSDs; therefore, the numerical results were compared with the shaking table test results. Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 25 2.2. FE Model Validation 2.2. FE Model Validation Additionally, a shake table test of the bridge model with a scale factor of 1/20 was conducted [27]. Before the parameter analysis of the yield strength of the YSD, it is necessary to prove the Additionally, a shake table test of the bridge model with a scale factor of 1/20 was conducted correctness of the model without YSDs; therefore, the numerical results were compared with the [27]. Before the parameter analysis of the yield strength of the YSD, it is necessary to prove the shaking table test results. Appl. Sci. 2019, 9, 2857 6 of 24 correctness of the model without YSDs; therefore, the numerical results were compared with the As is well known, whether a dynamic analysis or static analysis of cable-stayed bridges is shaking table test results. conducted, nonlinear geometric effects must be taken into consideration, including the cable-sag As is well known, whether a dynamic analysis or static analysis of cable-stayed bridges is As is well known, whether a dynamic analysis or static analysis of cable-stayed bridges is effect, P-Δ effect, and large displacement effect [43]. The gravity-nonlinear static analysis should be conducted, nonlinear geometric effects must be taken into consideration, including the cable-sag conducted, nonlinear geometric eects must be taken into consideration, including the cable-sag eect, conducted before dynamic analysis, because there would be large tension forces in the cables due to effect, P-Δ effect, and large displacement effect [43]. The gravity-nonlinear static analysis should be P-D eect, and large displacement eect [43]. The gravity-nonlinear static analysis should be conducted dead load, which would prevent the cables from becoming slack under seismic loads and influence conducted before dynamic analysis, because there would be large tension forces in the cables due to before dynamic analysis, because there would be large tension forces in the cables due to dead load, the structural stiffness. Nonlinear direct integration with the Newmark iterative method [44] has been dead load, which would prevent the cables from becoming slack under seismic loads and influence which would prevent the cables from becoming slack under seismic loads and inﬂuence the structural employed in nonlinear time-history analysis. Two key parameters for this method, gamma and beta, the structural stiffness. Nonlinear direct integration with the Newmark iterative method [44] has been stiness. Nonlinear direct integration with the Newmark iterative method [44] has been employed in are 0.5 and 0.25, respectively. In addition, Rayleigh damping [44] has been adopted to consider the employed in nonlinear time-history analysis. Two key parameters for this method, gamma and beta, nonlinear time-history analysis. Two key parameters for this method, gamma and beta, are 0.5 and structural damping, and the mass proportional coefficient and stiffness proportional coefficient are are 0.5 and 0.25, respectively. In addition, Rayleigh damping [44] has been adopted to consider the 0.25, respectively. In addition, Rayleigh damping [44] has been adopted to consider the structural 0.082 and 0.0084, respectively. structural damping, and the mass proportional coefficient and stiffness proportional coefficient are damping, and the mass proportional coecient and stiness proportional coecient are 0.082 and Figure 7 shows the seismic time history used in the numerical calculation and shaking table test, 0.082 and 0.0084, respectively. 0.0084, respectively. while Figures 8–10 show the displacement response time history of key location of the bridge. Figure 7 shows the seismic time history used in the numerical calculation and shaking table test, Figure 7 shows the seismic time history used in the numerical calculation and shaking table Obviously, they coincide with each other, but there is a large deviation in the range of 8 s~10 s, which while Figures 8–10 show the displacement response time history of key location of the bridge. test, while Figures 8–10 show the displacement response time history of key location of the bridge. is probably due to the strong nonlinear effect caused by the structural damage, making it difficult for Obviously, they coincide with each other, but there is a large deviation in the range of 8 s~10 s, which Obviously, they coincide with each other, but there is a large deviation in the range of 8 s~10 s, which the numerical calculation to be accurate. Generally, this shake-table test verified the correctness of is probably due to the strong nonlinear effect caused by the structural damage, making it difficult for is probably due to the strong nonlinear eect caused by the structural damage, making it dicult for the FE model. the numerical calculation to be accurate. Generally, this shake-table test verified the correctness of the numerical calculation to be accurate. Generally, this shake-table test veriﬁed the correctness of the the FE model. FE model. 0.15 0.10 0.15 0.05 0.10 0.00 0.05 -0.05 0.00 -0.10 -0.05 -0.15 -0.10 02468 10 Time[s] -0.15 02468 10 Time[s] Figure 7. The seismic time history. Figure 7. The seismic time history. Figure 7. The seismic time history. -5 Measured -10 -5 Numerical Measured -1 -1 50 02468 10 Numerical -15 Time[s] 02468 10 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 25 Time[s] Figure 8. The displacement of the middle span. Figure 8. The displacement of the middle span. Figure 8. The displacement of the middle span. -5 Measured -10 Numerical -15 02468 10 Time[s] Figure 9. The displacement of the top tower. Figure 9. The displacement of the top tower. -5 Measured -10 Numerical -15 02468 10 Time[s] Figure 10. The displacement of the middle tower. 2.3. Input Ground Motions The peak ground acceleration (PGA) of the design level ground motions is 0.3 g according to the site earthquake risk assessment, and seven artificial ground motions are produced from the design spectrum specified in bridge seismic codes, as the input. Figure 11 shows the acceleration response spectra with a 5% damping ratio. The following study uses the averaged result of the seven artificial ground motions. 1.2 Design Spectrum Artificial-1 1.0 Artificial-2 Artificial-3 Artificial-4 Artificial-5 0.8 Artificial-6 Artificial-7 0.6 0.4 0.2 0.0 01 2345 678 9 Period [s] Figure 11. Acceleration response spectra of the artificial ground motions. 3. The Influence of the Fy of YSD In order to select the Fy of YSD at different locations—the YSD yielding strength installed at the auxiliary pier (Fya), the transition pier (Fyp) and the tower (Fyt)—it is necessary to determine the effect of all these parameters, resulting in the structural response as a ternary function; it is therefore necessary to carry out a detailed three-variable function analysis. Acceleration[g] Acceleration[g] Displacement[mm] Displacement[mm] Displacement[mm] Acceleration [g] Displacement[mm] Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 25 -5 Measured -5 -10 Measured Numerical -10 -15 Numerical 02468 10 -15 Time[s] 02468 10 Time[s] Figure 9. The displacement of the top tower. Appl. Sci. 2019, 9, 2857 7 of 24 Figure 9. The displacement of the top tower. -5 Measured -5 -10 Measured Numerical -10 -15 Numerical 02468 10 -15 Time[s] 02468 10 Time[s] Figure Figure 10. 10. The The displa displacement cement o of f the middle the middle to tower wer. . Figure 10. The displacement of the middle tower. 2.3. Input Ground Motions 2.3. Input Ground Motions 2.3. In The put Gro peak und gr ound Motions acceleration (PGA) of the design level ground motions is 0.3 g according to the The peak ground acceleration (PGA) of the design level ground motions is 0.3 g according to the site earthquake risk assessment, and seven artiﬁcial ground motions are produced from the design site earthquake risk assessment, and seven artificial ground motions are produced from the design The peak ground acceleration (PGA) of the design level ground motions is 0.3 g according to the spectrum speciﬁed in bridge seismic codes, as the input. Figure 11 shows the acceleration response spectrum specified in bridge seismic codes, as the input. Figure 11 shows the acceleration response site earthquake risk assessment, and seven artificial ground motions are produced from the design spectra with a 5% damping ratio. The following study uses the averaged result of the seven artiﬁcial spectra with a 5% damping ratio. The following study uses the averaged result of the seven artificial spectrum specified in bridge seismic codes, as the input. Figure 11 shows the acceleration response ground motions. ground motions. spectra with a 5% damping ratio. The following study uses the averaged result of the seven artificial ground motions. 1.2 Design Spectrum Artificial-1 1.2 1.0 Artificial-2 Artificial-3 Design Spectrum Artificial-1 1.0 Artificial-4 Artificial-5 0.8 Artificial-2 Artificial-3 Artificial-6 Artificial-7 Artificial-4 Artificial-5 0.8 0.6 Artificial-6 Artificial-7 0.6 0.4 0.4 0.2 0.2 0.0 01 2345 678 9 0.0 Period [s] 01 2345 678 9 Period [s] Figure 11. Acceleration response spectra of the artificial ground motions. Figure 11. Acceleration response spectra of the artiﬁcial ground motions. Figure 11. Acceleration response spectra of the artificial ground motions. 3. The Inﬂuence of the F of YSD 3. The Influence of the Fy of YSD 3. The Influence of the Fy of YSD In order to select the F of YSD at dierent locations—the YSD yielding strength installed at In order to select the Fy of YSD at different locations—the YSD yielding strength installed at the the auxiliary pier (F ), the transition pier (F ) and the tower (F )—it is necessary to determine the ya yp yt auxiliary pier (Fya), the transition pier (Fyp) and the tower (Fyt)—it is necessary to determine the effect In order to select the Fy of YSD at different locations—the YSD yielding strength installed at the eect of all these parameters, resulting in the structural response as a ternary function; it is therefore of all these parameters, resulting in the structural response as a ternary function; it is therefore auxiliary pier (Fya), the transition pier (Fyp) and the tower (Fyt)—it is necessary to determine the effect necessary to carry out a detailed three-variable function analysis. necessary to carry out a detailed three-variable function analysis. of all these parameters, resulting in the structural response as a ternary function; it is therefore necessary to carry out a detailed three-variable function analysis. 3.1. Determining the Scope of Analysis The range of F should be considered ﬁrstly when a multivariate function is used. In fact, the seismically induced force of the substructure includes two parts: one is transmitted by the inertial force of the girder, and the other is generated by the self-vibration. Therefore, as shown by the above formula, the value of Fy should be determined according to the designed section capacity (M ) and self-vibration (M ) as well. sv Taking the bridge prototype in this paper as an example (Figure 12), on the one hand, the M is ut around 500,000 kNm, and M (or M ) is around 300,000 kNm, while the H is around 40 m. Therefore, up ua F could be 12,500 kN at maximum, and F (or F ) could be 7500 kN at maximum. On the other y1t y1p y1a hand, is around 0.3, which can be veriﬁed in the chart of subsequent calculation. Therefore, the upper limits of the YSD yielding strength F set at the auxiliary pier, the transition pier and the tower are ymax 5000 kN, 5000 kN, and 9000 kN, respectively. Displacement[mm] Displacement[mm] Acceleration [g] Displacement[mm] Displacement[mm] Acceleration [g] Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 25 3.1. Determining the Scope of Analysis The range of Fy should be considered firstly when a multivariate function is used. In fact, the seismically induced force of the substructure includes two parts: one is transmitted by the inertial force of the girder, and the other is generated by the self-vibration. Therefore, as shown by the above formula, the value of Fy should be determined according to the designed section capacity (Mu) and self-vibration (Msv) as well. Taking the bridge prototype in this paper as an example (Figure 12), on the one hand, the Mut is around 500,000 kN·m, and Mup (or Mua) is around 300,000 kN·m, while the H is around 40 m. Therefore, Fy1t could be 12,500 kN at maximum, and Fy1p (or Fy1a) could be 7500 kN at maximum. On the other hand, α is around 0.3, which can be verified in the chart of subsequent calculation. Therefore, the upper limits of the YSD yielding strength Fymax set at the auxiliary pier, the transition Appl. Sci. 2019, 9, 2857 8 of 24 pier and the tower are 5000 kN, 5000 kN, and 9000 kN, respectively. MM usv F= ,F = ,M =αM,F = F - F =() 1-α F y1 y2 sv u ymax y1 y2 y1 HH Mass of superstructure H —height of tower / pier M —designed section capacity F —the maximum transmissible force y1 M —the moment due to the self -vibration Tower/Pier sv F —the equivalent force du e to the self -vibration y2 F —the upper limit of the YSD yielding strength ymax Bottom of tower/pier α—the proportion of the self -vibration response to the total Figure 12. Mechanical diagram. Figure 12. Mechanical diagram. In addition, the lower limits of Fya, Fyp and Fyt are 0 kN, 0 kN and 1000 kN, respectively, because In addition, the lower limits of F , F and F are 0 kN, 0 kN and 1000 kN, respectively, because ya yp yt the girder–pier can be free but the girder–tower must be restrained to meet the service load function. the girder–pier can be free but the girder–tower must be restrained to meet the service load function. Thus, 441 cases in total are analyzed in order to find the most appropriate design value of Fy at each Thus, 441 cases in total are analyzed in order to ﬁnd the most appropriate design value of F at each location; i.e., Fya varies from 0 kN to 5000 kN, Fyp varies from 0 kN to 5000 kN and Fyt varies from 1000 location; i.e., F varies from 0 kN to 5000 kN, F varies from 0 kN to 5000 kN and F varies from ya yp yt kN to 9000 kN with an interval of 1000 kN, respectively. 1000 kN to 9000 kN with an interval of 1000 kN, respectively. 3.2. Multivariate Function Analysis 3.2. Multivariate Function Analysis Figures 13a, 14a and 15a show the eect of the yielding strength F on the relative displacement Figures 13a–15a show the effect of the yielding strength Fy on the rel y ative displacement (RD) of (RD) of the auxiliary pier, the transition pier and the bridge tower with the increase of other two the auxiliary pier, the transition pier and the bridge tower with the increase of other two yielding yielding strengths, which share the same characteristics. On one hand, there are distinct layers between strengths, which share the same characteristics. On one hand, there are distinct layers between the the RD surfaces at each location, indicating that the RD decreases rapidly to a stable small value with RD surfaces at each location, indicating that the RD decreases rapidly to a stable small value with the the increase of the yielding strength of the YSD at its own location. On the other hand, each surface is a increase of the yielding strength of the YSD at its own location. On the other hand, each surface is a subduction surface, which indicates that the yielding strengths of the YSDs at the other two locations subduction surface, which indicates that the yielding strengths of the YSDs at the other two locations have a signiﬁcant impact on the RD only when their values are small. Therefore, in order to use steel have a significant impact on the RD only when their values are small. Therefore, in order to use steel dampers eciently, the yielding strength of YSDs at each location should not be too large. dampers efficiently, the yielding strength of YSDs at each location should not be too large. It can be seen from Figures 13b, 14b and 15b that there are obvious layers between the ratio of capacity to demand (RCD) surfaces, and each RCD surface is an approximate horizontal plane, which means that the RCD at each location is only aected by the yielding strength of the YSD at its own location but not by the yielding strength of YSDs at the other two locations. Therefore, the yielding strength of YSDs at each location can be decided independently according to this observation. As for RCD, there is a common feature at each location: it increases at ﬁrst and then decreases with the increasing yielding strength of YSD at its own location. The reason for this is the coupling eect of the installation of YSDs, which restrains the self-vibration of piers and produces the inertial force of the main girder. The former eect is signiﬁcant when the yielding strength is relatively small, while the latter will become dominant with the increase of yielding strength. Based on the above analysis, two designed objectives—i.e., the RCD is large and uniform or the RD is small and uniform—are set to determine the YSD yielding strength at each location. To achieve the two designed objectives above, the standard deviations of the RCD at dierent locations need to be calculated. As shown in Figures 16 and 17, the minimum of standard deviation is obtained when F = 4000 kN, F = 4000 kN and F = 1000 kN (layout 1 in Table 3). ya yp yt H Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 25 Appl. Sci. 2019, 9, 2857 9 of 24 (a) (a) (b) (b) Figure 13. Responses of the auxiliary pier with varying Fya and Fyt. (a) Relative displacement, (b) ratio Figure 13. Responses of the auxiliary pier with varying Fya and Fyt. (a) Relative displacement, (b) ratio Figure 13. Responses of the auxiliary pier with varying F and F . (a) Relative displacement, (b) ratio ya yt of capacity to demand (RCD). (a) (a) Figure 14. Cont. Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 25 Appl. Sci. 2019, 9, 2857 10 of 24 Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 25 (b) (b) Figure 14. Responses of the transition pier with varying Fyp and Fyt. (a) Relative displacement, (b) ratio of capacity to demand. Figure 14. Responses of the transition pier with varying Fyp and Fyt. (a) Relative displacement, (b) ratio Figure 14. Responses of the transition pier with varying F and F . (a) Relative displacement, (b) yp yt of capacity to demand. ratio of capacity to demand. (a) (a) (b) Figure 15. Responses of the tower column with varying F and F . (a) Relative displacement; (b) ratio ya yp (b) of capacity to demand. Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 25 Figure 15. Responses of the tower column with varying Fya and Fyp. (a) Relative displacement; (b) ratio of capacity to demand. Figure 15. Responses of the tower column with varying Fya and Fyp. (a) Relative displacement; (b) ratio of capacity to demand. It can be seen from Figures 13b–15b that there are obvious layers between the ratio of capacity to demand (RCD) surfaces, and each RCD surface is an approximate horizontal plane, which means It can be seen from Figures 13b–15b that there are obvious layers between the ratio of capacity that the RCD at each location is only affected by the yielding strength of the YSD at its own location to demand (RCD) surfaces, and each RCD surface is an approximate horizontal plane, which means but not by the yielding strength of YSDs at the other two locations. Therefore, the yielding strength that the RCD at each location is only affected by the yielding strength of the YSD at its own location of YSDs at each location can be decided independently according to this observation. but not by the yielding strength of YSDs at the other two locations. Therefore, the yielding strength As for RCD, there is a common feature at each location: it increases at first and then decreases of YSDs at each location can be decided independently according to this observation. with the increasing yielding strength of YSD at its own location. The reason for this is the coupling As for RCD, there is a common feature at each location: it increases at first and then decreases effect of the installation of YSDs, which restrains the self-vibration of piers and produces the inertial with the increasing yielding strength of YSD at its own location. The reason for this is the coupling force of the main girder. The former effect is significant when the yielding strength is relatively small, effect of the installation of YSDs, which restrains the self-vibration of piers and produces the inertial while the latter will become dominant with the increase of yielding strength. force of the main girder. The former effect is significant when the yielding strength is relatively small, Based on the above analysis, two designed objectives—i.e., the RCD is large and uniform or the while the latter will become dominant with the increase of yielding strength. RD is small and uniform—are set to determine the YSD yielding strength at each location. Based on the above analysis, two designed objectives—i.e., the RCD is large and uniform or the To achieve the two designed objectives above, the standard deviations of the RCD at different RD is small and uniform—are set to determine the YSD yielding strength at each location. locations need to be calculated. As shown in Figures 16 and 17, the minimum of standard deviation To achieve the two designed objectives above, the standard deviations of the RCD at different is obtained when Fya = 4000 kN, Fyp = 4000 kN and Fyt = 1000 kN (layout 1 in Table 3). locations need to be calculated. As shown in Figures 16 and 17, the minimum of standard deviation Appl. Sci. 2019, 9, 2857 11 of 24 is obtained when Fya = 4000 kN, Fyp = 4000 kN and Fyt = 1000 kN (layout 1 in Table 3). Figure 16. Standard deviation of RCD. Figure 16. Standard deviation of RCD. Figure 16. Standard deviation of RCD. Figure 17. Contour of standard deviation of RCD (F = 4000 kN). ya Figure 17. Contour of standard deviation of RCD (Fya = 4000 kN). Table 3. Key response comparison. Figure 17. Contour of standard deviation of RCD (Fya = 4000 kN). Bridge YSD Layout RCD at RCD at RCD at RD at RD at RD at Tower Mid-Span System Plan Tower TP AP Tower TP AP TD Lateral Drift A Without YSDs 0.59 2.04 1.99 0 417 235 173 515 Designed 1.21 1.19 1.19 189 108 69 174 312 layout 1 (M) Designed 1.17 1.02 1.99 118 125 125 168 227 layout 2 (M) Designed 1.25 1.15 1.24 126 106 53 169 235 layout 3 (S) Designed 1.18 1.14 2.02 125 135 131 164 209 layout 4 (S) Note: 1. TP means the transition pier, AP means the auxiliary pier, TD means top drift, M means multi-variable analysis, and S means single variable analysis. Designed layout 1: F = 4000 kN, F = 4000 kN and F = 1000 kN ya yp yt (designed for a large and uniform RCD), designed layout 2: F = 1000 kN, F = 5000 kN and F = 4000 kN ya yp yt (designed for a small and uniform RD), designed layout 3: F = 4000 kN, F = 4000 kN and F = 2000 kN (designed ya yp yt for a large and uniform RCD), designed layout 4: F = 1000 kN, F = 4000 kN and F = 4000 kN (designed for a ya yp yt small and uniform RD). 2. The unit of displacement is mm, and RCD is a dimensionless constant. In the same way, as shown in Figures 18 and 19, the minimum of the standard deviation of RD is obtained when F = 1000 kN, F = 2000 kN and F = 3000 kN or F = 1000 kN, F = 5000 kN and ya yp yt ya yp F = 4000 kN. Recalling Figures 13–15, the RD of the latter is smaller, so F = 1000 kN, F = 5000 kN yt ya yp and F = 4000 kN is preferable (layout 2 in Table 3). yt Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 25 In the same way, as shown in Figures 18 and 19, the minimum of the standard deviation of RD is obtained when Fya = 1000 kN, Fyp = 2000 kN and Fyt = 3000 kN or Fya = 1000 kN, Fyp = 5000 kN and In the same way, as shown in Figures 18 and 19, the minimum of the standard deviation of RD Fyt = 4000 kN. Recalling Figures 13–15, the RD of the latter is smaller, so Fya = 1000 kN, Fyp = 5000 kN is obtained when Fya = 1000 kN, Fyp = 2000 kN and Fyt = 3000 kN or Fya = 1000 kN, Fyp = 5000 kN and and Fyt = 4000 kN is preferable (layout 2 in Table 3). Fyt = 4000 kN. Recalling Figures 13–15, the RD of the latter is smaller, so Fya = 1000 kN, Fyp = 5000 kN Appl. Sci. 2019, 9, 2857 12 of 24 and Fyt = 4000 kN is preferable (layout 2 in Table 3). Figure 18. Standard deviation of relative displacement (RD). Figure 18. Standard deviation of relative displacement (RD). Figure 18. Standard deviation of relative displacement (RD). Figure 19. Contour of standard deviation of RD (F = 1000 kN). ya Figure 19. Contour of standard deviation of RD (Fya = 1000 kN). 3.3. Single Variable Function Analysis Figure 19. Contour of standard deviation of RD (Fya = 1000 kN). 3.3. Single Variable Function Analysis As shown above, although we can achieve the designed layout of the bridge system by carrying As shown above, although we can achieve the designed layout of the bridge system by carrying out 3.3. a Si compr ngle Var ehensive iable Functi parametric on Analysi analysis, s it is time-consuming. Considering the result that the RD out a comprehensive parametric analysis, it is time-consuming. Considering the result that the RD at at each location decreases with the increase of the yielding strengths at any location, but the RCD As shown above, although we can achieve the designed layout of the bridge system by carrying each location decreases with the increase of the yielding strengths at any location, but the RCD is is only inﬂuenced by the yielding strength of its own location, we can make the analysis easier by out a comprehensive parametric analysis, it is time-consuming. Considering the result that the RD at only influenced by the yielding strength of its own location, we can make the analysis easier by only only using a one-variable function; that is, two of the variables (F and F or F and F or F yt yp yt ya yp each location decreases with the increase of the yielding strengths at any location, but the RCD is using a one-variable function; that is, two of the variables (Fyt and Fyp or Fyt and Fya or Fyp and Fya) can and F ) can remain unchanged to study the inﬂuence of the other variable (F or F or F ) on the ya yt yp ya only influenced by the yielding strength of its own location, we can make the analysis easier by only remain unchanged to study the influence of the other variable (Fyt or Fyp or Fya) on the structure’s structure’s seismic response. Consequently, three series of cases are analyzed: (1) F varies from 0 kN ya using a one-variable function; that is, two of the variables (Fyt and Fyp or Fyt and Fya or Fyp and Fya) can seismic response. Consequently, three series of cases are analyzed: (1) Fya varies from 0 kN to 5000 to 5000 kN, while F and F remain constant; (2) F varies from 0 kN to 5000 kN, while F and F yt yp yp ya yt remain unchanged to study the influence of the other variable (Fyt or Fyp or Fya) on the structure’s kN, while Fyt and Fyp remain constant; (2) Fyp varies from 0 kN to 5000 kN, while Fya and Fyt remain remain constant; and (3) F varies from 1000 kN to 9000 kN, while F and F remain constant. In fact, yt ya yp seismic response. Consequently, three series of cases are analyzed: (1) Fya varies from 0 kN to 5000 constant; and (3) Fyt varies from 1000 kN to 9000 kN, while Fya and Fyp remain constant. In fact, it is it is required that the others should remain when analyzing the inﬂuence of yielding strength of the kN, while Fyt and Fyp remain constant; (2) Fyp varies from 0 kN to 5000 kN, while Fya and Fyt remain required that the others should remain when analyzing the influence of yielding strength of the speciﬁed location. For illustration purposes, 2000 kN was set. constant; and (3) Fyt varies from 1000 kN to 9000 kN, while Fya and Fyp remain constant. In fact, it is specified location. For illustration purposes, 2000 kN was set. It can be seen from Figures 20a, 21a and 22a that when the yielding strength is relatively small, required that the others should remain when analyzing the influence of yielding strength of the with the increase of F , the girder–pier/column RD at the auxiliary pier, transition pier and tower specified location. For illustration purposes, 2000 kN was set. column decrease signiﬁcantly. However, when the yielding strength is relatively large, the F only aects the displacement at its own location without a signiﬁcant change at the other two locations. As shown in Figures 20b, 21b and 22b, with the increase of F , there is a signiﬁcant ﬂuctuation of the RCD curve at its own location, but the RCD curves at the other two locations are approximately ﬂat. In addition, the RCD curve at each location increases ﬁrst and then decreases, indicating that a peak Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 25 It can be seen from Figures 20a–22a that when the yielding strength is relatively small, with the increase of Fy, the girder–pier/column RD at the auxiliary pier, transition pier and tower column decrease significantly. However, when the yielding strength is relatively large, the Fy only affects the displacement at its own location without a significant change at the other two locations. As shown in Figures 20b–22b, with the increase of Fy, there is a significant fluctuation of the RCD curve at its own location, but the RCD curves at the other two locations are approximately flat. In addition, the RCD Appl. Sci. 2019, 9, 2857 13 of 24 curve at each location increases first and then decreases, indicating that a peak must exist. Obviously, this result is consistent with that of the multivariate function analysis but much more concise. In must exist. Obviously, this result is consistent with that of the multivariate function analysis but much addition, according to Figures 20b–22b, the following Table 4 can be obtained to verify the correctness more concise. In addition, according to Figures 20b, 21b and 22b, the following Table 4 can be obtained of the proportion of the above natural vibration response. to verify the correctness of the proportion of the above natural vibration response. Auxiliary Pier 250 Transition Pier Tower 0 1000 2000 3000 4000 5000 Fya[kN] (a) 3.00 Auxiliary Pier 2.50 Transition Pier Tower 2.00 1.50 1.00 0.50 0.00 0 1000 2000 3000 4000 5000 Fya[kN] (b) Figure 20. Seismic responses with varying Fya. (a) Relative displacement (unit: mm); (b) ratio of Figure 20. Seismic responses with varying F . (a) Relative displacement (unit: mm); (b) ratio of ya capacity to demand. capacity to demand. As a matter of fact, the overall shear force transferred to the column end consists of two parts: one is from the inertial force developed by the girder and the other is from the self-vibration of the column Auxiliary Pier itself. In order to determine the scope of parameter analysis, it is necessary to determine the proportion 250 Transition Pier that the self-vibration response would take. Figures 16–18 can be obtained after the possible range Tower of this proportion is determined, 200 and one of the cases is taken as an example to illustrate that this self-vibration ratio is indeed around 0.3. It can be approximated that the increase of F generally constrains the RDs of piers or towers, and only has a signiﬁcant impact on its own location compared to the other two. Therefore, it can be considered that the selection of F , F and F is independent and not coupled. ya yp yt According to Figures 20b, 21b and 22b, the tower has the minimal RCD curve compared with 0 1000 2000 3000 4000 5000 other piers; to achieve one of the designed objectives as mentioned in multivariable analysis (large Fyp[kN] and uniform RCDs), F should be set to its peak value of 2000 kN as shown in Figure 22b. Then, yt on the grounds of the intersection of the RCD curves of the auxiliary pier and of the tower in Figure 20b, F should be 4000 kN, and F should be approximately 4000 kN. Therefore, F = 4000 kN, ya yp ya (a) F = 4000 kN and F = 2000 kN (layout 3 in Table 3) is one of the reasonable solutions obtained yp yt through single variable function analysis. Relative Displacement[mm] Ratio of Capacity to Demand Relative Displacement[mm] Appl. Sci. 2019, 9, 2857 14 of 24 Table 4. Proportion of the above natural vibration responses. M of of F M M of Tower F /F M M of TP M of AP M of TP M of AP of TP of AP yt 1 sv yp ya 2 sv sv Tower Tower 9000 360000 508542 148542 0.29 5000 200000 272719 256396 72719 56396 0.27 0.22 Note: 1. M means the moment of the tower due to the YSD, M means the moment of the pier due to the YSD, M means the moment due to self-vibration, means the proportion of the sv 1 2 self-vibration response to the total, TP means the transition pier, and AP means the auxiliary pier. 2. The unit of bending moment is kNm, and is a dimensionless constant. Appl. Sci. Appl. 2019 Sci. , 2019 9, x FO , 9, 2857 R PEER REVIEW 15 of 24 14 of 25 Auxiliary Pier 250 Transition Pier Tower 0 1000 2000 3000 4000 5000 Fyp[kN] (a) 3.00 Auxiliary Pier 2.50 Transition Pier Tower 2.00 1.50 1.00 0.50 0.00 0 1000 2000 3000 4000 5000 Fyp[kN] (b) Figure 21. Figure 21. Seism Seismic ic response responses s with vary with varying ing FF yp. ( . a) Relati (a) Relative ve di displacement; splacement; ( (b b)) rat ratio io of of capacity capacity to yp to demand. demand. In terms of Figures 20a, 21a and 22a, it is dicult to obtain a set of reasonable yielding strengths directly. Therefore, it is necessary to draw the standard deviation curve varying with each yielding strength and assume that the three curves are independent of each other. As shown in Figure 23, Auxiliary Pier the reasonable range of F , F and F is 1000 kN~2000 kN, 1000 kN~4000 kN and 2000 kN~4000 kN, ya yp yt 250 Transition Pier respectively. In order to achieve the other designed objective (i.e., a small and uniform RD), the yielding Tower strengths should be as large as possible; therefore, F = 1000 kN, F = 4000 kN and F = 4000 kN is ya yp yt the other reasonable solution (layout 4 in Table 3). 3.4. Comparison of the Designed Results Based on the above analysis, we can obtain four designed layouts as shown in Table 3 accordingly. We can observe that the designed result is dierent according to dierent designed objectives. 1000 2000 3000 4000 5000 6000 7000 8000 9000 Comparing the results of the two analysis methods, it is obvious that the results are quite similar; the Fyt[kN] one-variable analysis results only dier slightly from those of the multi-variable function analysis. The reason is that one-variable analysis cannot fully consider the inﬂuence of three yielding strengths on structural response and assumes that the three variables are independent from each other. Therefore, (a) they are not exactly the same but generally consistent. However, as seen in Table 3, it is easy to 3.00 determine that using YSDs can eectively improve the bridge’s seismic performance at key locations Auxiliary Pier overall, irrespective of which designed objective is used. 2.50 Transition Pier Tower 2.00 1.50 1.00 0.50 0.00 1000 2000 3000 4000 5000 6000 7000 8000 9000 Fyt[kN] Ratio of Capacity to Demand Relative Displacement[mm] Ratio of Capacity to Demand Relative Displacement[mm] Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 25 3.00 Auxiliary Pier 2.50 Transition Pier Tower 2.00 1.50 1.00 0.50 0.00 0 1000 2000 3000 4000 5000 Fyp[kN] (b) Figure 21. Seismic responses with varying Fyp. (a) Relative displacement; (b) ratio of capacity to Appl. demSci. and. 2019 , 9, 2857 16 of 24 Auxiliary Pier Appl. Sci. 2019, 9, x FOR PEER RE 25 VIEW 0 17 of 25 Transition Pier Tower As a matter of fact, the overall shear force transferred to the column end consists of two parts: one is from the inertial force developed by the girder and the other is from the self-vibration of the column itself. In order to determine the scope of parameter analysis, it is necessary to determine the proportion that the self-vibration response would take. Figures 16b–18b can be obtained after the possible range of this proportion is determined, and one of the cases is taken as an example to 1000 2000 3000 4000 5000 6000 7000 8000 9000 illustrate that this self-vibration ratio is indeed around 0.3. Fyt[kN] It can be approximated that the increase of Fy generally constrains the RDs of piers or towers, and only has a significant impact on its own location compared to the other two. Therefore, it can be (a) considered that the selection of Fya, Fyp and Fyt is independent and not coupled. According to Figures 20b–22b, the tower has the minimal RCD curve compared with other piers; 3.00 Auxiliary Pier to achieve one of the designed objectives as mentioned in multivariable analysis (large and uniform 2.50 Transition Pier RCDs), Fyt should be set to its peak value of 2000 kN as shown in Figure 22b. Then, on the grounds Tower 2.00 of the intersection of the RCD curves of the auxiliary pier and of the tower in Figure 20b, Fya should 1.50 be 4000 kN, and Fyp should be approximately 4000 kN. Therefore, Fya = 4000 kN, Fyp = 4000 kN and Fyt 1.00 = 2000 kN (layout 3 in Table 3) is one of the reasonable solutions obtained through single variable function analysis. 0.50 In terms of Figures 20a–22a, it is difficult to obtain a set of reasonable yielding strengths directly. 0.00 1000 2000 3000 4000 5000 6000 7000 8000 9000 Therefore, it is necessary to draw the standard deviation curve varying with each yielding strength Fyt[kN] and assume that the three curves are independent of each other. As shown in Figure 23, the reasonable range of Fya, Fyp and Fyt is 1000 kN~2000 kN, 1000 kN~4000 kN and 2000 kN~4000 kN, (b) respectively. In order to achieve the other designed objective (i.e., a small and uniform RD), the yielding strengths should be as large as possible; therefore, Fya = 1000 kN, Fyp = 4000 kN and Fyt = 4000 Figure 22. Figure 22. Seism Seismic ic response responses s with vary with varying ing Fyt F. ( . a) Relat (a) Relative ive di displacement; splacement; ( (b b)) Ratio Ratio of capacity of capacity to yt kN is the other reasonable solution (layout 4 in Table 3). to demand. demand. Table 3. Key response comparison. Standard Deviation of Relative Displacement Bridge YSD Layout RCD at RCD at RCD RD at RD at RD at Tower Mid-Span Standard Deviation Curve (Changing Fya) System Plan Tower TP at AP Tower TP AP TD Lateral Drift Standard Deviation Curve (Changing Fyp) A Without YSDs 0.59 2.04 1.99 0 417 235 173 515 Standard Deviation Curve (Changing Fyt) Designed 1.21 1.19 1.19 189 108 69 174 312 layout 1 (M) Designed B 1.17 1.02 1.99 118 125 125 168 227 layout 2 (M) Designed 1.25 1.15 1.24 126 106 53 169 235 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 layout 3 (S) Fy[kN] Figure 23. Standard deviation of RD (changing Fya or Fyp or Fyt). Figure 23. Standard deviation of RD (changing F or F or F ). ya yp yt It is obvious that the designed layouts 2 and 4 can obtain a better seismic performance overall if 3.4. Comparison of the Designed Results other factors, such as tower top drift and girder displacement, are included. It is worth noting that Based on the above analysis, we can obtain four designed layouts as shown in Table 3 the comparable large RCD at the auxiliary pier from layouts 2 and 4 is almost the same as that of the accordingly. We can observe that the designed result is different according to different designed bridge system without YSDs, which indicates the seismic design capacity of the auxiliary pier can essentially be decreased for cost eciency. objectives. Comparing the results of the two analysis methods, it is obvious that the results are quite similar; the one-variable analysis results only differ slightly from those of the multi-variable function analysis. The reason is that one-variable analysis cannot fully consider the influence of three yielding strengths on structural response and assumes that the three variables are independent from each other. Therefore, they are not exactly the same but generally consistent. However, as seen in Table 3, it is easy to determine that using YSDs can effectively improve the bridge’s seismic performance at key locations overall, irrespective of which designed objective is used. It is obvious that the designed layouts 2 and 4 can obtain a better seismic performance overall if other factors, such as tower top drift and girder displacement, are included. It is worth noting that the comparable large RCD at the auxiliary pier from layouts 2 and 4 is almost the same as that of the bridge system without YSDs, which indicates the seismic design capacity of the auxiliary pier can essentially be decreased for cost efficiency. Ratio of Capacity to Demand Relative Displacement[mm] Ratio of Capacity to Demand Standard Deviation Appl. Sci. 2019, 9, 2857 17 of 24 4. Dynamic Vibration of Bridge Systems with and without YSDs As mentioned above, Table 3 shows the averaged maximum response comparison of the two bridge systems with and without YSDs. Firstly, the RCD of the tower column signiﬁcantly jumps to around 1.2 from 0.59, which means that the internal force of the tower column is signiﬁcantly reduced and thus meets seismic performance after using the YSDs. Secondly, the RDs were reduced by 59%~75% and 35%~77% at the transition pier and auxiliary pier, respectively. Meanwhile, the yielding strength of each YSD can be designed, making the RCD large and uniform or the RD small and uniform. This phenomenon can be explained by the system switch during the vibration, which is caused by the yielding of YSDs when the peak acceleration of the ground motion occurs. By investigating the ﬁrst two dominant transversal modes of the two systems (Table 5), it can be seen that there are three main characteristics of the system switch: (1) the lateral deformation of the girder becomes uniform, which avoids the extremely large lateral drift in the mid-span and girder ’s two ends; (2) the uncoupled girder–tower deformation, which reduces the inertia force transmitted from the deck, and consequently decreases the bending moment of tower below the deck and the tower top drift; and (3) the delay of the period of post-yielding dominant transversal mode (transversal swing) until around 7.41s, resulting away from the predominant periods (3–4 s) of the ground motion, which restrains the girder displacement after yielding. Therefore, this dynamic behavior also shows why a better seismic performance will be achieved when we adopt the designed result based on the designed objective RD. Appl. Sci. 2019, 9, 2857 18 of 24 Appl. Appl. Sci. Sci. 2019 2019 , 9 , ,9 x FO , x FO R P R P EER EER RE RE VIEW VIEW 19 of 19 of 25 25 Appl. Appl. Sci. Sci. 2019 2019, , 9 9,, x FO x FOR P R PEER EER RE REVIEW VIEW 19 of 19 of 25 25 Appl. Appl. Sci. Sci. 2019 2019 , 9, , x FO 9, x FO R P R P EER EER RE RE VIEW VIEW 19 of 19 of 25 25 Appl. Appl. Sci. Sci. 2019 2019 , 9,, x FO 9, x FO R P R P EER EER RE RE VIEW VIEW 19 of 19 of 25 25 Table 5. Dominant transversal modes of the cable-stayed bridge. Table 5. Table 5. Dom Dom inant transvers inant transvers aa l m l m oode des of s of the the cable cable -stay -stay ed ed b b ridg ridg e.e. Table 5. Table 5. Dom Domiinant transvers nant transversa al m l mo ode des of s of the the cable cable--stay stayed ed b brridg idge e.. Table 5. Dominant transversal modes of the cable-stayed bridge. Table 5. Table 5. Table 5. Dom Dom Dom inant transvers inant transvers inant transvers aa l m l m al m oo de de os of de s of s of the the the cable cable cable -stay -stay -stay ed ed b ed b r b idg ridg ridg ee . .e . YSD System (After Yielding) Conventional System YY SS D Sys D Sys tem tem (Af (Af ter ter Y Y ield ield ing) ing) YSD System (After Yielding) Y Y S SD Sys D Sys tt em em (Af (Af tter er Y Y ii eld eld ii ng) ng) Y Y S Y S D Sys D Sys SD Sys tem tem tem (Af (Af (Af ter ter t Y er Y i Y eld ield ield ing) ing) ing) layout 3 layout 4 Conventional System Con Con ven ven tion tion ala Sys l Sys tem tem Con Con v ven en ti ti on on aa ll Sys Sys tt em em Con Con vv en en ti ti on on aa l Sys l Sys tem tem Conventional System Top View Side View lay lay oo ut ut 3 3 lay lay oo ut ut 4 4 lay lay lay o o out ut ut 3 3 3 lay lay lay o o out ut ut 4 4 4 layout 3 layout 4 lay lay out o 3 ut 3 lay lay out o 4 ut 4 TT oo p V p V iew iew Sid Sid ee V V iew iew Top View Side View MPMR Period Top View Side View MPMR Period MPMR Period T T o o p V p V ii ew ew Sid Sid e e V V ii ew ew T T oo T p V p V op V iew iew iew Sid Sid Sid ee V e V iV ew iew iew Top View Side View MPMR Period MPMR Period MPMR Period Top V Top V iew iew Sid Sid e e V V iew iew MPMR MPMR Per Per iod iod MPMR MPMR Per Per iod iod MPMR MPMR Per Per iod iod T T o o p V p V ii ew ew Sid Sid e e V V ii ew ew MPMR MPMR Per Per ii od od MPMR MPMR Per Per ii od od MPMR MPMR Per Per ii od od T T oo T p V p V op V iew iew iew Sid Sid Sid ee V e V iV ew iew iew MPMR MPMR MPMR Per Per Per iod iod iod MPMR MPMR MPMR Per Per Per iod iod iod MPMR MPMR MPMR Per Per Per iod iod iod 1010 10.8%.8%.8% 3. 3.4242 3.42s s s 48 48.0% 48 .0% .0% 7. 7.30 7. 30 30 s s s 48 48 .4% 48.4% .4% 7. 7. 52 7.52 52 s s s 10.8% 3.42 s 48.0% 7.30 s 48.4% 7.52 s 1010.8%.8% 3. 3.4242 ss 48 48 .0% .0% 7. 7. 30 30 s s 48 48 .4% .4% 7. 7. 52 52 s s 101010.8%.8%.8% 3. 3. 3.424242 s s s 48 48 48 .0% .0% .0% 7. 7. 30 7. 30 30 s s s 48 48 48 .4% .4% .4% 7. 7. 52 7. 52 52 s s s 53.5% 1.18 s 19.0% 0.95 s 19.0% 0.95 s 5353.5%.5% 1. 1.1818 s s 19 19 .0% .0% 0.0. 95 95 s s 19 19 .0% .0% 0.0. 95 95 s s 5353.5%.5% 1. 1.1818 ss 19 19 .0% .0% 0. 0. 95 95 s s 19 19 .0% .0% 0. 0. 95 95 s s 5353 53.5%.5%.5% 1. 1.1818 1.18 s s s 19.0% 19 19 .0% .0% 0. 0.95 0. 95 95 s s s 19 19 19.0% .0% .0% 0. 0. 0.95 95 95 s s s 53.5% 1.18 s 19.0% 0.95 s 19.0% 0.95 s Note: MPMR means the modal participating mass ratio in the transversal direction. Note: Note: MPMR MPMR m m eans the m eans the m od oa d l partic al partic ipating ipating m m ass rat ass rat io iin t o in t he transversal he transversal directi directi on. on. Note: Note: MPMR MPMR m m ee ans the m ans the m o o d d aa l partic l partic ipating ipating m m aa ss rat ss rat ii o o in t in t h h e transversal e transversal directi directi o o n. n. Note: Note: MPMR MPMR m m ee ans the m ans the m oo dd aa l partic l partic ipating ipating m m aa ss rat ss rat io io in t in t hh e transversal e transversal directi directi oo n. n. Note: MPMR means the modal participating mass ratio in the transversal direction. Note: MPMR means the modal participating mass ratio in the transversal direction. Appl. Sci. 2019, 9, x FOR PEER REVIEW 20 of 25 Appl. Sci. 2019, 9, 2857 19 of 24 5. Equal Yielding Strength Analysis 5. Equal Yielding Strength Analysis According to the above analysis, designing by using one-variable analysis is relatively simple, According to the above analysis, designing by using one-variable analysis is relatively simple, but but the yielding strengths of the YSDs at each location are often unequal, which means various the yielding strengths of the YSDs at each location are often unequal, which means various speciﬁcations specifications of YSDs are needed. Meanwhile, the system switch actually demonstrates the of YSDs are needed. Meanwhile, the system switch actually demonstrates the eectiveness of the effectiveness of the YSDs is according to the delay of the period of the post-yielding dominant YSDs is according to the delay of the period of the post-yielding dominant transversal mode, which transversal mode, which restrains the girder displacement after yielding. Therefore, an equal yielding restrains the girder displacement after yielding. Therefore, an equal yielding strength, which means strength, which means Fya = Fyp = Fyt may be reasonable for engineering practice. Additionally, the F = F = F may be reasonable for engineering practice. Additionally, the calculated cases were ya yp yt calculated cases were greatly reduced from 441 to 5. greatly reduced from 441 to 5. Figure 24 and table 6 show that the RCD can be large and uniform when the yielding strength is Figure 24 and Table 6 show that the RCD can be large and uniform when the yielding strength is 3000 kN, and the RD can be small and uniform when the yielding strength is 2000 kN. 3000 kN, and the RD can be small and uniform when the yielding strength is 2000 kN. Auxiliary Pier Transition Pier Tower 1000 2000 3000 4000 5000 Fy[kN] (a) 3.50 Auxiliary Pier 3.00 Transition Pier 2.50 Tower 2.00 1.50 1.00 0.50 0.00 1000 2000 3000 4000 5000 Fy[kN] (b) Figure 24. Seismic responses with varying Fy. (a) Relative displacement; (b) ratio of capacity to Figure 24. Seismic responses with varying F . (a) Relative displacement; (b) ratio of capacity to demand. demand. Table 6. Key response comparison (Units: mm). Table 6. Key response comparison (Units: mm). Bridge YSD Layout RCD of RCD of RCD of RD of RD of RD of Tower Mid-Span System Plan Tower TP AP Tower TP AP TD Lateral Drift Bridge YSD Layout RCD of RCD of RCD RD of RD of RD of Tower Mid-Span Without System Plan Tower TP of AP Tower TP AP TD Lateral Drift A 0.59 2.04 1.99 0 417 235 173 515 YSDs A Without YSDs 0.59 2.04 1.99 0 417 235 173 515 Designed 1.24 1.30 1.44 124 130 80 164 198 Designed layout B layout 5 1.24 1.30 1.44 124 130 80 164 198 Designed 1.28 1.70 1.66 189 170 152 165 245 Designed layout layout 6 1.28 1.70 1.66 189 170 152 165 245 Note: TP means the transition pier, AP means the auxiliary pier, and TD means top drift. Designed layout 5: Note: F = T 3000 P m kN, eans the F = transi 3000 kN tion and pier F , = AP 3000 me kN ans the (designed auxiliary for a lar pier, ge and and uniform TD means top RCD), designed drift. De layout signed 6: ya yp yt F = 2000 kN, F = 2000 kN and F = 2000 kN (designed for a small and uniform RD). ya yp yt layout 5: Fya = 3000 kN, Fyp = 3000 kN and Fyt = 3000 kN (designed for a large and uniform RCD), designed layout 6: Fya = 2000 kN, Fyp = 2000 kN and Fyt = 2000 kN (designed for a small and uniform Compared with Table 3, the equal yielding strength can also achieve a signiﬁcant eect in terms of RD). reducing the seismic responses of the bridge in terms of RCD or RD. RD values are reduced by 59% Ratio of Capacity to Demand Relative Displacement[mm] Appl. Sci. 2019, 9, x FOR PEER REVIEW 21 of 25 Compared with Table 3, the equal yielding strength can also achieve a significant effect in terms of reducing the seismic responses of the bridge in terms of RCD or RD. RD values are reduced by 59% and 35% at the transition pier and auxiliary pier for layout 6, respectively, whereas they are reduced by 68% and 44% for layout 4. Meanwhile, the RCD of the tower column increases to around 1.25 from 0.59 for both layouts 5 and 6, which agrees well with the results of the layouts 1 and 3 by Appl. Sci. 2019, 9, 2857 20 of 24 comprehensive multi-variable and one-variable analysis in Table 3. Remembering layout 3 in Table 3, Fyt = 2000 kN is also the designed result by single-variable analysis in terms of RCD at the tower– girder location, and so layout 6 is preferable. and 35% at the transition pier and auxiliary pier for layout 6, respectively, whereas they are reduced by In fact, the yielding strength of YSDs is closely related to the mass of the superstructure, so the 68% and 44% for layout 4. Meanwhile, the RCD of the tower column increases to around 1.25 from 0.59 approximate yielding strength can be quickly determined according to the mass of the for both layouts 5 and 6, which agrees well with the results of the layouts 1 and 3 by comprehensive superstructure. The results (Figure 25 and Table 7) show that there is an approximately constant ratio multi-variable and one-variable analysis in Table 3. Remembering layout 3 in Table 3, F = 2000 kN yt relationship between the total yielding strength of YSDs and the weight of the superstructure, at is also the designed result by single-variable analysis in terms of RCD at the tower–girder location, around 10%. and so layout 6 is preferable. Therefore, in the preliminary analysis, the yielding strength of YSD can be determined by the In fact, the yielding strength of YSDs is closely related to the mass of the superstructure, so the weight of the superstructure, and the RCD is large and uniform or the RD is small and uniform. At approximate yielding strength can be quickly determined according to the mass of the superstructure. this time, the calculated cases were greatly reduced from 5 to 1, which greatly improves the efficiency The results (Figure 25 and Table 7) show that there is an approximately constant ratio relationship of bridge design. between the total yielding strength of YSDs and the weight of the superstructure, at around 10%. Standard Deviation of Ratio of Capacity to Demand 0.7 0.6 1.0mass 1.5mass 2.0mass 0.5 0.4 0.3 0.2 0.1 1000 2000 3000 4000 5000 Fy[kN] (a) Standard Deviation of Relative Displacement 60 1.0mass 1.5mass 2.0mass 1000 2000 3000 4000 5000 Fy[kN] (b) Figure 25. Standard deviation with varying Fy. (a) Standard deviation of RCD; (b) standard deviation Figure 25. Standard deviation with varying F . (a) Standard deviation of RCD; (b) standard deviation of RD. Note: 1.0 mass means the original weight of the main girder, 1.5 mass means 1.5 times the of RD. Note: 1.0 mass means the original weight of the main girder, 1.5 mass means 1.5 times the original weight, 2.0 mass means 2.0 times the original weight. original weight, 2.0 mass means 2.0 times the original weight. Table 7. The ratio between the yield strength of YSDs and the weight of the superstructure (unit: kN). Table 7. The ratio between the yield strength of YSDs and the weight of the superstructure (unit: kN). For For a a L Large arge aand nd Unif Uniform orm RCD RCD For F aoSmall r a Sma and ll and Unif Uniform orm RD RD Yield Strength Weight Ratio Yield Strength Weight Ratio Yield Strength Weight Ratio Yield Strength Weight Ratio 1.0 mass 12 × 3000 280,000 13% 12 × 2000 280,000 9% 1.0 mass 12 3000 280,000 13% 12 2000 280,000 9% 1.5 mass 12 × 4000 420,000 11% 12 × 3000 420,000 9% 1.5 mass 12 4000 420,000 11% 12 3000 420,000 9% 2.0 mass 12 × 4000 560,000 9% 12 × 4000 560,000 9% 2.0 mass 12 4000 560,000 9% 12 4000 560,000 9% Therefore, in the preliminary analysis, the yielding strength of YSD can be determined by the weight of the superstructure, and the RCD is large and uniform or the RD is small and uniform. At this time, the calculated cases were greatly reduced from 5 to 1, which greatly improves the eciency of bridge design. Standard Deviation Standard Deviation Appl. Sci. 2019, 9, 2857 21 of 24 Appl. Sci. 2019, 9, x FOR PEER REVIEW 22 of 25 Appl. Sci. 2019, 9, x FOR PEER REVIEW 22 of 25 For illustration, the time histories of the bending moment at the bottom of the tower and the RD For illustration, the time histories of the bending moment at the bottom of the tower and the RD For illustration, the time histories of the bending moment at the bottom of the tower and the RD between the girder and the transition pier (under the artiﬁcial-1) are illustrated in Figures 26 and 27. between the girder and the transition pier (under the artificial-1) are illustrated in Figures 26 and 27. between the girder and the transition pier (under the artificial-1) are illustrated in Figures 26 and 27. This shows that the application of YSDs with equal yielding strengths of 2000 kN does reduce the This shows that the application of YSDs with equal yielding strengths of 2000 kN does reduce the This shows that the application of YSDs with equal yielding strengths of 2000 kN does reduce the moment at the tower bottom and the girder–pier RD eectively, especially during the time steps when moment at the tower bottom and the girder–pier RD effectively, especially during the time steps moment at the tower bottom and the girder–pier RD effectively, especially during the time steps the ground motions become intensive. when the ground motions become intensive. when the ground motions become intensive. System with YSD System with YSD System without YSD System without YSD -200000 -200000 -400000 -400000 -600000 -600000 0 10 2030 405060 70 0 10 2030 405060 70 Time[s] Time[s] Figure 26. Time history of the tower bottom moment. Figure Figure 26. 26. Tim Time e history of history of the the tower bottom tower bottom moment. moment. 0.3 0.3 System with YSD System with YSD 0.2 0.2 System without YSD System without YSD 0.1 0.1 0.0 0.0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 0 102030 40506070 0 102030 40506070 Time[s] Time[s] Figure 27. Time history of the relative displacement between the transition pier and girder. Figure 27. Time history of the relative displacement between the transition pier and girder. Figure 27. Time history of the relative displacement between the transition pier and girder. 6. Practical Method of YSD Design 6. Practical Method of YSD Design 6. Practical Method of YSD Design Based Based on the a on the above bove a analysis, nalysis, a a mor more pra e practical cticalmethod method t too determine determine the yi the yieldi eldi ng ng st strength rength of of YSD YSD is Based on the above analysis, a more practical method to determine the yielding strength of YSD proposed (Figure 28). Obviously, unequal yielding strength analysis is accurate but complicated, while is proposed (Figure 28). Obviously, unequal yielding strength analysis is accurate but complicated, is proposed (Figure 28). Obviously, unequal yielding strength analysis is accurate but complicated, equal while e yielding qual yiel strding engthst analysis rength ana canlnot ysis only can not achieve only the achieve t design goal he desi but is gn goa also rel l a but tively is a simple, lso relat and iveso ly while equal yielding strength analysis can not only achieve the design goal but is also relatively the latter is more suitable for engineering practice. In addition, according to the calculation, there is a simple, and so the latter is more suitable for engineering practice. In addition, according to the simple, and so the latter is more suitable for engineering practice. In addition, according to the ratio calcula relationship tion, there is between a ratio rela the reasonable tionship between yielding str the reasona ength of YSDs ble yiand elding strength of the weight of the YSDs a mainn gir d the der, calculation, there is a ratio relationship between the reasonable yielding strength of YSDs and the which is around 10% for a medium-span cable-stayed bridge. Therefore, the total yielding strength of weight of the main girder, which is around 10% for a medium-span cable-stayed bridge. Therefore, weight of the main girder, which is around 10% for a medium-span cable-stayed bridge. Therefore, YSDs the tota can l yi be eldi obtained ng strength of quickly, YSDs ca the yielding n be obta strength ined of q YSD uickis lyequally , the yieapplied lding stin ren each gth o location, f YSD is and equ then ally the total yielding strength of YSDs can be obtained quickly, the yielding strength of YSD is equally the key indicators are checked, such as RD and RCD. If it is necessary to adjust the yielding strength, applied in each location, and then the key indicators are checked, such as RD and RCD. If it is applied in each location, and then the key indicators are checked, such as RD and RCD. If it is the necessa yielding ry to str adength just the yiel analysis ding strength, the yi range can be determined elding strength analysis using the method range can be determined in Section 3.1, and then necessary to adjust the yielding strength, the yielding strength analysis range can be determined the equal yielding strength analysis can be carried out with the method in Section 5 to determine the using the method in Section 3.1, and then the equal yielding strength analysis can be carried out with using the method in Section 3.1, and then the equal yielding strength analysis can be carried out with yielding the method in Secti strength of o YSD. n 5 tThe o determi ﬂowchart ne th of e yi this eldi practical ng strength of YSD. The flow design method is shown cha in rt of Figur this p e 28.ractical the method in Section 5 to determine the yielding strength of YSD. The flowchart of this practical design method is shown in Figure 28. design method is shown in Figure 28. M M o o me me n n tt [[ k k N N ·· m] m] Relative displacement[m] Relative displacement[m] Appl. Sci. 2019, 9, 2857 22 of 24 Appl. Sci. 2019, 9, x FOR PEER REVIEW 23 of 25 Preliminary Calculate the weight of The total yield strength of YSDs is YSDs are arranged with Calculate the Check RD bridge design the main girder 10% of the weight of the main girder equal yield strength section capability and RCD No Yes Calculate the maximum The self-vibration response The upper limit of yield Yield strength of YSD transmissible force calculated by time history strength of YSD Try a few cases is determined according Finished F =M /H F =M /H, M =aM F=F -F to RD and RCD y1 u y2 sv sv u y1 y2 Figure 28. Practical design method. Figure 28. Practical design method. 7. Conclusions 7. Conclusions Using YSDs can eectively improve a bridge’s seismic performance at key locations overall. Using YSDs can effectively improve a bridge’s seismic performance at key locations overall. The The inﬂuence of the yielding strength on the seismic responses of a medium-span cable-stayed bridge is influence of the yielding strength on the seismic responses of a medium-span cable-stayed bridge is investigated through comprehensive parametric analysis, and reasonable layouts of YSDs are achieved investigated through comprehensive parametric analysis, and reasonable layouts of YSDs are according to two dierent design objectives by unequal yielding strength analysis and equal yielding achieved according to two different design objectives by unequal yielding strength analysis and equal strength analysis. Then, a more practical method to quickly determine the eective yielding steel yielding strength analysis. Then, a more practical method to quickly determine the effective yielding damper parameter is proposed for engineering practice. Conclusions can be drawn as follows: steel damper parameter is proposed for engineering practice. Conclusions can be drawn as follows: 1 The increase of the yielding strength can eectively reduce the RD at each location, but its 1 The increase of the yielding strength can effectively reduce the RD at each location, but its eciency decreases with the increase of the yielding strength. For current engineering practice, efficiency decreases with the increase of the yielding strength. For current engineering practice, the seismic design capacity of the auxiliary pier can essentially be decreased for cost eciency. the seismic design capacity of the auxiliary pier can essentially be decreased for cost efficiency. 2 2 Equal Equal yie yielding lding st strength rengthanalysis analysis isis m much uch si simpler mpler than thanunequal unequal yi yielding elding strstrength a ength analysis; nalysis; however, it may not be the most appropriate result theoretically, but it is eective enough for however, it may not be the most appropriate result theoretically, but it is effective enough for engineering engineering practices. practices. 3 3 The The practic practical a design l design meth method using od using equal equal y yielding ieldi strength ng strength an analysis can alysis c greatly an greatly reduce th reduce the analysis e analysis cases and hence improve the efficiency of seismic analysis. cases and hence improve the eciency of seismic analysis. It should be noted that the yielding strengths achieved in this study might not work for other It should be noted that the yielding strengths achieved in this study might not work for other cases. cases. H However owev,ewe r, we prov provideid ae a pr practical actic pr al ocedur procedure to e to quickly quickly achieve achie the ve the proper yie proper yielding l str din ength g stre and ngth layout of steel dampers (not only the shape used in the study but also other shapes) by simply running and layout of steel dampers (not only the shape used in the study but also other shapes) by simply a few running a few time history time analysis histor cases y ana instead lysis ca of secarrying s instead o outf compr carrying ehens out ive comprehensi parametric v study e par , a which metric might study, cause less theoretically accurate values but is time-ecient in engineering practice. which might cause less theoretically accurate values but is time-efficient in engineering practice. Author Contributions: Conceptualization, methodology and writing—review and editing, Y.X.; validation, Author Contributions: Conceptualization, methodology and writing—review and editing, Y.X.; validation, analysis and writing—original draft preparation, Z.Z.; software and investigation, C.C.; investigation and data analysis and writing—original draft preparation, Z.Z.; software and investigation, C.C.; investigation and data curation, S.Z. curation, S.Z. Funding: This research was funded by National Key Research and Development Plan, China, grant number Funding: This research was funded by National Key Research and Development Plan, China, grant number 2017YFC1500702 and the National Science Foundation of China, grant number 51878492. 2017YFC1500702 and the National Science Foundation of China, grant number 51878492. Conﬂicts of Interest: The authors declare no conﬂict of interest. Conflicts of Interest: The authors declare no conflict of interest. References Reference 1. 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Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges

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Publisher: Multidisciplinary Digital Publishing Institute
ISSN: 2076-3417
DOI: 10.3390/app9142857
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Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges

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Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges

Practical Design Method of Yielding Steel Dampers in Concrete Cable-Stayed Bridges

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