Simplified seismic resistant design of base isolated single pylon cable-stayed bridge

Abstract

The seismic vulnerability of single pylon cable-stayed bridges under strong ground motions in the transverse direction is of great concern to earthquake engineering researchers and bridge engineers. Introduction of base isolation to cable-stayed bridges has been proved very effective in reducing seismic forces in the bridges in previous studies. This paper proposes a direct displacement based seismic design (DDBD) procedure for base isolated cable-stayed bridge under transverse seismic excitation. One of the key aspects of the DDBD is the realization of a uniform transverse target displacement of the deck under seismic excitation, which is achieved by appropriate design of the isolator stiffness at the bottom of the pylon and the ends of the deck. The proposed DDBD procedure is applied in this paper to the seismic design of a single pylon cable-stayed bridge isolated by friction pendulum bearings. The effectiveness and the accuracy of the resulting design are checked by nonlinear time history analyses. The numerical results indicate that the proposed DDBD procedure can predict the deck displacement profile and amplitudes, as well as the base shear within a reasonable degree of accuracy. The case study demonstrates that the proposed DDBD procedure is sufficiently accurate and practical for the seismic design of base isolated single pylon cable-stayed bridges.

1 Introduction

In the last few decades, single pylon cable-stayed bridges have found wide applications due to their appealing esthetics, excellent spanning capacity, rapid and simple construction, and efficient utilization of structural materials (Petrangeli 2004; Casciati et al. 2008; Ali and Abdel-Ghaffar 1995). Cable-stayed bridges play an important role in the modern transport networks. Therefore, the main structural components of the bridge should not have notable damage and the bridges must remain operational to provide necessary emergency access under the design ground motions (JTG/T D65-01-2007 2007; Calvi et al. 2010). The dynamic responses of a single pylon cable-stayed bridge are largely dependent on the method that the bridge deck is connected to the pylon. The current trend in the design of single pylon cable-stayed bridges with short to medium span is to integrally connect the pylon, the deck, and the column, in order to reduce the deformation of the superstructure under regular loads and wind actions (Li 2006). However, Camara (Camara and Efthymiou 2016) and Martínez (Martínez-Rodrigo and Filiatrault 2015) pointed out that rigid connection of deck and pylon significantly increases the transverse shear force and bending moment, resulting in higher seismic demands of the structural components in the transverse direction than in the longitudinal direction. Severe earthquake damage was reported in the Chi-Lu Bridge, a cable-stayed bridge with the aforementioned configurations. Flexural hinges formed at the pylon above the bridge deck in the transverse direction and the cracks extended upward to the height of the lowest cables. The traditional seismic design strategy relies on dissipation of seismic energy by inelastic deformation of the structural components, such as the development of plastic hinge zones. However, the requirements for the traditional seismic design become more stringent under the strong ground motions. Seismic isolation strategy has been proven as an effective and economical method for seismic design and retrofit of structures in seismic active regions. For the cable-stayed bridge, the seismic isolation devices could decouple the deck-pylon system from the substructure to reduce the seismic inertia forces and effectively reduce the inelastic deformation in the structural components. Therefore, seismic isolation could be used as a promising strategy to improve the seismic performance of the single pylon cable-stayed bridges in the transverse direction.

Numerical studies were performed by previous researchers to investigate the effectiveness of the isolation system for protecting the cable-stayed bridge in the last two decades. Nonlinear time history (NLTH) analyses were performed by Chadwell (Chadwell 2003), Atmaca et al. (Atmaca et al. 2014), and Javanmardi et al. (Javanmardi et al. 2017) to study the seismic performance of the cable stayed bridge isolated by lead rubber bearings (LRB) or friction pendulum bearings (FPB) at the base of the pylon. Soneji and Jangid (2008, 2010) and Wesolowsky and Wilson (2003) considered to place different isolation devices between the pylon and the deck to protect the superstructure from the seismic damages during earthquakes. These studies have demonstrated that isolations in the single pylon cable-stayed bridge could substantially reduce the bending moment and shear force in the pylon, but significantly increase the displacements in the superstructure. Although nonlinear time-history analyses are most widely used techniques for seismic response analyses of structures, simplified seismic design procedure is needed for engineering practice (Şadan et al. 2013). However, the investigation of simplified design procedure was relatively limited in the above mentioned studies. Wesolowsky and Wilson (2003) proposed the design procedure for isolated cable-stayed bridge in the longitudinal direction. The approach is first to specify a design displacement and an estimated design period, and then determine the longitudinal stiffness of the isolated bearings. Chadwell (2003) developed the required stiffness relationship between the ends of the deck and the base of the pylon to optimize the effectiveness of the isolation system.

It is widely recognized that the force-based design (FBD) approach is insufficient to implement the concepts of the performance-based seismic design (PBSD) (Bertero and Bertero 2002). Performance levels, using displacements as damage indicators, are better correlated to the maximum material strain or the plastic hinge rotation at the base of piers than strength (Kowalsky 2002). Consequently, a direct displacement based design (DDBD) concept was proposed (Priestley 1993), which is consistent with the concept of PBSD. The primary aim of DDBD is to design a structure, which will achieve a prescribed limit state when subjected to a prescribed earthquake. The fundamental concepts of the DDBD procedure are presented in Fig. 1, using a continuous bridge as an example. The same approach can also be applied to the seismic design of the cable stayed bridge in this study. The DDBD procedure starts with identifying the initial design displacement profile, as shown in Fig. 1a. Step 1 involves determining a displacement profile and the ultimate limit state displacement capacities of the vulnerable components. In Step 2, as shown in Fig. 1b, the effective stiffness (K_e), the effective mass (m_e) and the equivalent viscous damping (ξ_e) of a substitute structure consisting of a single-degree of freedom (S-DOF) are determined to replace the nonlinear multi-degree of freedom (M-DOF) model. In Step 3, since the equivalent properties of the substitute structure are elastic, a set of elastic displacement response spectra is used to determine the effective period, T_e, of the elastic substitute structure associated to the design displacement, Δ_d, and the corresponding damping ratio, as shown in Fig. 1c. Finally, the design base shear is calculated and equivalently assigned as the inertia forces to the masses of the M-DOF structure. The described DDBD procedure has been successfully applied to the design of both seismic isolated buildings (Cardone et al. 2010) and bridge structures (Cardone et al. 2009). Numerous studies have been concentrated on the evaluation of the displacement method by comparing with the NLTH analyses. The previous studies have employed the characteristics of earthquake motions, including far-field (Pant et al. 2013; Kircher and Lashkari 1989; Theodossiou and Constantinou 1991; Winters and Constantinou 1993) and near-fault (Pant et al. 2013; Warn and Whittaker 2004; Ozdemir and Constantinou 2010) ground motions. These studies demonstrated that the accuracy of the displacement method significantly relies on the methodologies used in selecting and scaling ground motions. Although unidirectional displacement method could predict the seismic responses quite well, several studies (Winters and Constantinou 1993; Ozdemir and Constantinou 2010) pointed out that the previous procedure may be not applicable to the bidirectional excitations. The results of these studies demonstrated that the simplified displacement method can over-predict or under-predict the average results of response history analysis. The results primarily depend on the selection and scaling of motions to represent the target response spectrum.

Fig. 1

Fundamentals of DDBD (Cardone et al. 2009). a SDOF simulation, b effective stiffness and damping, and c design displacement spectrum

A similar seismic design procedure for base-isolated cable-stayed bridges in the transverse direction is developed in this study as an extension of the aforementioned DDBD procedure. The displacement profile is presented and a comprehensive design algorithm is developed for the base isolated cable-stayed bridge in this paper. NLTH analyses were also performed on the base isolated cable-stayed bridge designed using the DDBD procedure to verify the accuracy and feasibility of the proposed DDBD procedure.

2 Substitute structure for base isolated cable stayed bridge

The first step of the DDBD procedure is to identify the target displacement profile of the bridge deck, which is commonly determined by the first inelastic deformation shape (Adhikari et al. 2010), before the transformation of the bridge model to an equivalent S-DOF. In a base isolated cable-stayed bridge, isolation bearings are commonly installed at the pylon base and the abutments of the bridges, as shown in Fig. 2a. The shape of the transverse vibration modes of the bridge deck is closely dependent on the relative flexibility between the isolation system placed at the base of the pylon and at the ends of the deck. For the case when the transverse stiffness of the isolation system placed at the ends of the deck is significantly larger than that of the isolation system placed at the base of the pylon, the fundamental transverse vibration mode is illustrated in Fig. 2b. Figure 2c presents the fundamental transverse vibration mode of the bridge for the opposite case, in which the transverse stiffness of the isolation system placed at the ends of the deck is negligible compared to that of the isolation system placed at the base of the pylon. The other vibration modes are bounded between these two modes. To create an effective isolation system in the transverse direction, the stiffness of the isolation system placed at the base of the pylon should be tuned to the stiffness of the isolation system placed at the ends of the deck of the bridge so as to achieve uniform translation of the bridge as shown in Fig. 2d. Otherwise, the bending of the deck along the strong axis (bending around the z-axis, as shown in Fig. 2a) may result in severe damage at the connection of the deck and the pylon.

Fig. 2

Transverse modes of vibration: a bridge model, b end span support restrained, c pylon base support restrained, and d target displacement profile

To determine the target stiffness ratio between the isolation system placed at the base of the pylon and the ends of the deck of the bridges that minimize transverse bending of the deck, a simplified two-degree of freedom (2-DOF) model was built in the generalized coordinates, which can be described by two shape functions, as shown in Fig. 3a, b. In the first shape function, φ_1a, assuming that the stiffness of the bearings at the abutments is significantly smaller than that of the bearings under the pylon, the corresponding shape function is the bending of deck along the strong axis. On the contrary, in the second shape function, φ_1b, considering the rigid bearings at the abutments, the pylon and the deck move along the transverse direction of the bridge. It is important to note that, in the shape function, the pylon is assumed to be rigid.

Fig. 3

Generalized function and interpolation of superstructure (Chadwell 2003)

The characteristic equation of the 2-DOF model is given by:

$$ \varvec{K\varPhi }_{n} = \omega_{n}^{2} M\varvec{\varPhi}_{n} $$

(1-a)

$$ \left( {\begin{array}{*{20}c} {k_{11} } & {k_{12} } \\ {k_{21} } & {k_{22} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\varphi_{1a} } \\ {\varphi_{1b} } \\ \end{array} } \right) = \omega_{n}^{2} \left( {\begin{array}{*{20}c} {m_{11} } & {m_{12} } \\ {m_{21} } & {m_{22} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\varphi_{1a} } \\ {\varphi_{1b} } \\ \end{array} } \right) $$

(1-b)

where K and M are the stiffness and mass matrices; ω_n and φ_n are the natural frequencies and modes. The stiffness and mass matrix elements are defined as follows:

$$ k_{ij}^{e} = \int_{0}^{l} {EI\left( x \right)} \varphi_{1i}^{\prime\prime} \left( x \right)\varphi_{1j}^{\prime\prime} \left( x \right)dx $$

(2)

$$ m_{ij}^{e} = \int_{0}^{l} {m\left( x \right)} \varphi_{1i} \left( x \right)\varphi_{1j} \left( x \right)dx $$

(3)

The shape functions are given as (Chadwell 2003):

$$ \varphi_{1a} \left( x \right) = \frac{3}{2}\left( {\frac{x}{L}} \right)^{2} - \frac{1}{2}\left( {\frac{x}{L}} \right)^{3} $$

(4)

$$ \varphi_{1b} \left( x \right) = 1 - \left[ {\frac{3}{2}\left( {\frac{x}{L}} \right)^{2} - \frac{1}{2}\left( {\frac{x}{L}} \right)^{3} } \right] $$

(5)

where L is the length of one span.

Therefore, the elements in the stiffness and mass matrices in Eq. (1-b) are derived as:

$$ \begin{aligned} k_{11} & = \frac{{6EI_{s} }}{{L^{3} }} + 2k_{e} ,\quad k_{12} = k_{21} = - \frac{{6EI_{s} }}{{L^{3} }},\quad k_{22} = \frac{{6EI_{s} }}{{L^{3} }} + k_{p} \\ m_{11} & = \frac{33}{70}m_{s} L,\quad m_{12} = \frac{39}{140}m_{s} L,\quad m_{22} = \frac{34}{35}m_{s} L + M_{p} \\ \end{aligned} $$

(6)

where M_p is the total mass of the pylon; m_s is the mass of the superstructure (per length); k_p and k_e are the stiffness under the pylon and the stiffness at the end of the deck.

Given that the optimized deformation shape is the uniform translation of the cable-stayed bridge, the modal coordinates of the generalized shape functions are φ_1a = φ_1b = 1. The Eq. (1-a) can be expanded to the following equations:

$$ \left\{ \begin{aligned} k_{11} + k_{12} = \omega_{n}^{2} \left( {m_{11} + m_{12} } \right) \hfill \\ k_{21} + k_{22} = \omega_{n}^{2} \left( {m_{21} + m_{22} } \right) \hfill \\ \end{aligned} \right. $$

(7)

Substitution of Eq. (6) into Eq. (7), the relationship of the stiffness of the bearings under the pylon and the bearings at the end of the deck can be calculated as follows:

$$ \frac{{k_{p} }}{{k_{e} }} = \frac{10}{3} + \frac{16}{3}\left( {\frac{{M_{p} }}{{M_{s} }}} \right) $$

(8)

where M_s is the total mass of the superstructure.

In the derivation of Eq. (8), the pylon is assumed to be rigid. The flexibility of the pylon will be considered in the following steps (Fig. 3c). In order to take the stiffness of the pylon into consideration, another simplified 2-DOF system is built, as shown in Fig. 4. DOF 1 represents the uniform translation of the cable-stayed bridge and DOF 2 indicates the bending deformation of the pylon. The effective mass and stiffness of the pylon can be computed by:

$$ \begin{aligned} m_{eff} & = \frac{1}{{U_{m}^{2} }}\sum\limits_{j}^{{}} {P_{j} U_{j}^{2} } \\ k_{eff} & = \frac{g}{{U_{m}^{2} }}\sum\limits_{j}^{{}} {P_{j} U_{j} } \\ \end{aligned} $$

(9)

where P_j is the mass of the jth node in the discretized model; U_j is the displacement of the jth node in the discretized model; g is the gravitational acceleration; U_m is the transverse displacement of the node at the top of the pylon.

Fig. 4

A simplified 2-DOF model of pylon

The stiffness and mass elements in the characteristic equation of the 2-DOF system, Eq. (1-a), for the new 2-DOF system are derived as:

$$ \left( {\begin{array}{*{20}c} {k_{p} + 2k_{e} + k_{eff} } & -{ k_{eff} } \\ -{ k_{eff} } & {k_{eff} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\varphi_{1a} } \\ {\varphi_{1b} } \\ \end{array} } \right) = \omega_{n}^{2} \left( {\begin{array}{*{20}c} {M_{p} + M_{s} } & 0 \\ 0 & {m_{eff} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\varphi_{1a} } \\ {\varphi_{1b} } \\ \end{array} } \right) $$

(10)

Combining Eq. (10) and Eq. (8), the following equation can be obtained:

$$ \left( {k_{eff} - \omega_{n}^{2} m_{eff} } \right)\left[ {k_{p} \left( {\frac{{8M_{p} + 8M_{s} }}{{5M_{p} + 8M_{s} }}} \right) + k_{eff} - \omega_{n}^{2} \left( {M_{p} + M_{s} } \right)} \right] = k_{eff}^{2} $$

(11)

The relation between the stiffness of the pylon base and the target period can be derived from Eq. (11):

$$ k_{p} = \frac{1}{2}\frac{{\pi^{2} }}{{T^{2} - T_{1}^{2} }}\left[ {\left( {1 + \frac{{m_{eff} }}{{M_{p} + M_{s} }}} \right) - \left( {\frac{{T_{1} }}{T}} \right)^{2} } \right]\left( {5M_{s} + 8M_{p} } \right) $$

(12)

where T is the first period of vibration; T₁ can be calculated using the following equation:

$$ T_{1} = 2\pi \sqrt {\frac{{m_{eff} }}{{k_{eff} }}} $$

(13)

In the DDBD procedure, it is required to convert the nonlinear M-DOF model of the structure into an equivalent linear S-DOF system, which involves the assumption of a target displacement profile. As shown in Fig. 4, the transverse displacement profile of the single pylon cable-stayed bridge includes the uniform translation of the whole bridge (DoF 1) and transverse bending of the pylon (DoF 2). It is found that the effective mass of the DoF 2 is significantly smaller than the DoF 1. Therefore, the effects of the pylon in transverse bending are ignored in the proposed DDBD procedure, which greatly simplifies the design procedure. In general, the base isolated cable stayed-bridge can be transferred into an S-DOF equivalent structural model with characteristic equation given by Eq. (11).

3 Design algorithm

The primary objective of the DDBD methodology is to determine the critical characteristics of the isolators, which deforms at the target displacements under given intensity of ground motions. Figure 5 presents the flowchart of the procedure for DDBD of the base isolated cable-stayed bridge. The basic input data are determined in Step1, including the geometry, the mass, the type and the design displacement of the isolators. Step 2 includes the selection of the design displacement of the deck, which is calculated as follows:

$$ \Delta_{d} = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {m_{i} \cdot \Delta_{i}^{2} } \right)} }}{{\sum\nolimits_{i = 1}^{n} {\left( {m_{i} \cdot \Delta_{i} } \right)} }} $$

(14)

where m_i and Δ_i are the masses and displacements of the ith significant mass location respectively.

Fig. 5

Design algorithm of DDBD

As discussed in the previous section, the contribution of the pylon to the deformation is insignificant, Eq. (14) can then be reduced to Δ_d = Δ_i. Thus, the target displacement is determined by the design displacement of the isolation devices.

Steps 3 and 4 include the preliminary selections of the design parameters of the isolation devices. It should be noted that the stiffness of the isolators must satisfy Eq. (8) to ensure the target displacement profile, namely, uniform translation of the deck.

In Step 5, the equivalent stiffness, K_e, and the equivalent damping, ξ_e, of the isolation system, are obtained using Eqs. (15) and (16):

$$ K_{e} = \sum {K_{i} } $$

(15)

$$ \xi_{e} = \frac{{\sum {K_{i} \xi_{i} } }}{{K_{e} }} $$

(16)

where K_i and ξ_i are the equivalent stiffness and damping for the ith isolator, respectively.

In Step 6, the design displacement spectrum is determined for the equivalent S-DOF structure at a specific site with the given equivalent damping (ξ_e) of and the soil type. In the DDBD procedure, the Damping Reduction Factor (CEN ENV-1-1 European Committee for Standardisation 1998) has been utilized to derive high damping (larger than 5%) response spectra using the following expression:

$$ DRF = \sqrt {\frac{10}{{5 + \xi_{e} }}} $$

(17)

Then, the target displacement of the S-DOF system (Δ_d) is read directly from the spectrum at a given effective period T_e, as shown in Fig. 1c. A new equivalent stiffness (K’_e) of the system corresponding to the target displacement can be calculated by:

$$ K'_{e} = 4\pi^{2} \cdot m_{e} /T_{e}^{2} $$

(18)

where m_e is the effective mass of the equivalent system.

If the absolute value between the new equivalent stiffness (K’_e) and the previous one in Step 5 is more than the specified tolerance, iterations are needed to update the parameters of the isolators (Step 3) as shown in Fig. 5, until the convergence is reached. The design base shear is then calculated by multiplying the effective stiffness by the target displacement:

$$ V_{b} = K_{e} \Delta_{d} $$

(19)

Finally, as suggested by Cardone et al. (2009, 2010) and seismic design codes, in the procedure of DDBD for base isolated structures, the limit values of isolation ratio are still needed, which is defined as the ratio between the fundamental period of the isolated and non-isolated structure as estimated by Eq. (12). The effective periods of base isolated structures are defined between 2T_fb (T_fb is the fundamental periods of non-isolated structure) and 4 s (Cardone et al. 2010).

4 Numerical validation

4.1 Bridge prototype and FE model

The prototype of the bridge model used in this study is the Longwan Bridge over the Jian River in Guizhou Province, China. The bridge is a semi-harp type cable-stayed bridge with equal spans of 114 m, as shown in Fig. 6. The single RC pylon is approximately 62 m high above the deck and supports the superstructure with 16 pairs of steel stay cables. The deck is rigidly connected to the pylon to limit the displacement of the deck under regular loads. Each end of the Longwan bridge is simply supported on an abutment.

Fig. 6

General view of the bridge structure and cross sections under study

The pylon below the deck has a 19.9 m by 7.4 m rectangular section, which consists of two-cell reinforced concrete boxes with web thickness of 1 and 1.2 m, respectively. The deck of the cable-stayed bridge consists of a five-cell pre-stressed concrete box girder, which is approximately 32.0 m wide by 3.16 m deep. The stay cables are made of 7 mm diameter galvanized wires. The cross-sectional areas of the cable number from 1 to 14 are taken as 0.0108 m², and the number 15 to 16 are taken as 0.115 m². In this study, FPBs are considered as the base isolation system to apply into the Longwan Bridge engineering practice. The details of the bridge is shown in Fig. 6.

Three dimensional (3D) FE model of the Bridge was developed in OpenSees platform (Mazzoni et al. 2006), an object oriented, open source FE analysis framework. The deck and pylon are modeled by 3D linear elastic beam-column elements. In this study, the values of Young’s modulus, mass density and Poisson’s ratio for the deck and pylon are taken as 34500 MPa, 2300 kg/m³, and 0.2, respectively. Table 1 shows the masses, moments of inertia, and areas for the pylon and deck sections.

Table 1 Geometric properties of the cable-stayed bridge

The pre-stressed cables are simulated by the truss elements with the initial stress. The Giuffré-Menegotto-Pinto model (Filippou et al. 1983), namely uniaxial Steel02 material in OpenSees, which is described by bi-linear kinematic hardening, were assigned to the truss elements. The Young’s modulus and ultimate strength for the steel material are 195 GPa and 1860 MPa. The axial stiffness of inclined cables exhibits geometrical nonlinearities caused by cable sag. The geometrical nonlinearities can be idealized as the equivalent modulus of elasticity (E_eq) for different cables, E_eq is defined as (Ernst 1965):

$$ E_{eq} = \frac{{E_{c} }}{{\left[ {\frac{{(WL)^{2} A_{c} E_{c} }}{{12F_{T}^{3} }}} \right] + 1}} $$

(20)

where E_c is the material modulus of elasticity for the straight cable; L is the horizontal projected length of the cable; W is the cable weight per unit length; A_c is the cross-section area of the cable; F_T is the pretension force.

Friction pendulum bearings (FPB) (Zayas et al. 1990) are used as the sliding systems to achieve base isolation of the cable-stayed bridge. The classical force–displacement behavior of FPB is described by the following equation:

$$ F = \frac{Wd}{{R_{1} }} + \mu_{FR} W\text{sgn} \left( {\dot{\theta }} \right) $$

(21)

where F is the restoring force; μ_FR is the coefficient of friction; W is the vertical load; R₁ is the radius of the upper spherical surface; d is the relative displacement of the articulated slider and the upper spherical surface; sgn(·) is the signum function, and the value returns − 1 or + 1 depending upon the negative or positive of the velocity ${\dot{\theta }}$.

As for FPB, the effective damping ratio is obtained as (Cardone et al. 2009):

$$ \xi_{IS} = \frac{2}{{\pi \left( {1 + \varLambda } \right)}} $$

(22)

In Eq. (22), Λ is defined as:

$$ \varLambda = \frac{D}{{R\mu_{FR} }} $$

where D is the maximum expected horizontal displacement.

4.2 Verification study

A series of NLTH analyses are performed on the base isolated Longwan cable stayed bridge designed using the proposed DDBD procedure. The numerical results are subsequently compared with the results predicted by DDBD method to verify the accuracy of the proposed DDBD procedure. Based on the site survey report, a suite of six ground motion records are artificially generated to match the design response spectrum, as shown in Fig. 7. The calculation method is based on the periodic function. The artificial seismic signal can be decomposed as a series of harmonic signals:

$$ \ddot{x}_{g} \left( x \right) = F\left( t \right)\sum\limits_{i = 1}^{{N_{m} }} {\left( { - 1} \right)^{i} Sa_{i} \sin \left( {2\pi f_{i} t + \varphi_{p} } \right)} $$

(23)

where F(t) is the modulating functions to describe the envelope of ground motions; N_m is the number of vibration modes included in the signal generation. S_ai and f_i are the spectral acceleration associated with i mode and its frequency. φ_p is the phase angle, randomly varied to make synthetic accelerograms. The same envelop function as recommended by Priestley et al. (2000) and EC8 (CEN ENV-1-1 European Committee for Standardisation 1998) were used in this study. Iterations are performed to obtain acceptable records matching the target code spectra. The mean errors of the average spectrum resulting from the accelerograms should be less than 10%. The average of the displacement response spectrum corresponding to the selected accelerograms is compared with the design response spectrum. It is found that the average spectrum well agrees with the design spectrum, with some small discrepancies at the long period.

Fig. 7

a Acceleration time histories employed in the verification study and b comparison between the design spectrum and the average response spectrum

The bridge is excited in its transverse direction using the six ground motion records, and the results of NLTH analyses are presented in terms of (1) displacement profile of the deck at the moment when the deck reached its peak displacement; (2) maximum displacements of the deck; (3) maximum base shear.

Figure 8 compares the target displacement profiles of the deck obtained from DDBD procedure with the peak displacements recorded in the NLTH analyses along the transverse direction of the bridge. For the base isolated cable-stayed bridge, a range of possible target displacements (120–200 mm) is considered. Priestley et al. (2007) and Cardone et al. (2009) recommended that the design displacement of the isolation devices can be determined by the re-centering capability of the FPB. According to the research of Tsopelas and Constantinou (1994) and Constantinou et al. (2011), the re-centering capability can be determined as follows. The force at the design displacement D_d minus the force at D_d/2 is great than 0.025 WD_R/D_d, where D_R is the permanent displacement. The requirement can be expressed by K_dD ≥ (0.05μ)^0.5W, or T ≤ 28(0.05/μ)^0.25 (D_d/g)^0.5, where μ is the coefficient of friction; T is the period of isolation system. It should be noted that the coefficient of friction should not be less than 0.5 times the value under high speed motion conditions (Constantinou et al. 2011). According to the research of Priestley et al. (2007), the ratio of D_d/R should be lower than 0.15, in order to avoid excessive vertical displacements. Therefore, the range for target displacements can be approximately estimated through aforementioned method. It should be noted that because of the symmetry of the cable-stayed bridge about the pylon in the longitudinal direction, the results are listed for only the left half of the bridge. It can be seen from Fig. 8 that the displacement profiles of the deck obtained from the DDBD are in close agreement with the mean values of NLTH analyses. It indicates that the isolation systems of the bridge designed following the proposed procedure is able to achieve the desired uniform rigid translation of the deck in the transverse direction of the bridge. The results in Fig. 8 also indicate that the proposed DDBD procedure is capable of capturing the peak deck displacements.

Fig. 8

Comparison of deck transverse displacements obtained from DDBD and NTHA. a 100 mm, b 120 mm, c 140 mm, d 160 mm, e 180 mm, and f 200 mm

As shown the displacement profiles estimated by DDBD method are in general smaller than the average values of the 6 nonlinear time history analyses, and the differences between the DDBD estimated displacement profile and NLTH analyses decrease with the increase of the target displacement response. The differences between the displacement profiles of the deck of NLTH and the target profile are listed in Table 2, labeled as Profile Error. The profile errors are defined as the absolute difference of the peak displacement between the two ends and the center of the deck normalized by the peak displacement at the end of the deck. Well agreement of displacement profiles can be seen from the comparison of the results obtained using NLTH and DDBD with all errors below 5.00%.

Table 2 Comparison between NLTH and design objectives for the base isolated cable-stayed bridge

The displacement error is defined as Eq. (24) labeled with Displacement Error in Table 2:

$$ \hat{u}_{\text{max,NLTH}}^{DDBD} = \frac{{\left| {u_{\hbox{max} ,NLTH} - u_{DDBD} } \right|}}{{\left| {u_{DDBD} } \right|}} $$

(24)

where $\hat{u}_{\text{max,NLTH}}^{DDBD}$ is the absolute difference between the maximum displacement of NLTH and DDBD results normalized by NLTH results. It can also be observed from Table 2 that the maximum deck displacement is underestimated by DDBD procedure. Obviously, the errors increase as the damping ratios of the isolators increase and the maximum displacement error is 11.73%.

The maximum base shear force is also reported in Table 2 labeled with Shear Force Error. Similarly, the shear force error is defined by Eq. (25):

$$ \hat{R}_{\text{ave,NLTH}}^{DDBD} = \frac{{\left| {R_{ave,NLTH} - R_{DDBD} } \right|}}{{\left| {R_{DDBD} } \right|}} $$

(25)

where $\hat{R}_{\text{max,NLTH}}^{DDBD}$ is the absolute value of the difference between the average base shear force of NLTH and DDBD results normalized by NLTH results. The shear force is underestimated by DDBD procedure compared to NLTH with a maximum error of 8.60%.

Moreover, modal analyses are performed on series of bridge models designed using proposed DDBD method with target isolation ratios ranging from 1.596 to 2.659 to examine the accuracy of the proposed method in predicting the isolation ratio. As observed from Table 3, the results obtained from the modal analyses are in close agreement with that obtained from analytical predictions of the 2-DOF model and the errors are less than 2%.

Table 3 Comparison between modal analysis and design objectives for the base isolated cable-stayed bridge

The discrepancies between the results from NLTH and DDBD can be attributed to:

• 1. The DDBD design procedure assume the structure deforms in the shape of the fundamental mode without accounting for the contributions from higher modes. Although most of the cumulative mass participation is contributed by the first mode for the base isolated bridge, it is commonly known that cable-stayed bridges typically have many closely-spaced higher modes (Wesolowsky and Wilson 2003). The amplification of the higher modes’ contributions to the dynamic response of the bridge is particularly significant for the hysteretic damping of the bridge (Politopoulos 2008).

• 2. The discrepancies are also attributed to the inaccurate estimations of the damping reduction factor as shown in Eq. 17 in the DDBD design. Detailed investigation of the effects of the damping reduction factor can be found in Cardone et al. (2009) and Priestley et al. (2007).

• 3. The contribution of the effective mass of the pylon bending to the dynamic response of the bridge is generally insignificant (nearly 3%) compared with the mass of uniform translation of the pylon. Therefore, it was neglected in the DDBD procedure. However, the effective mass of the pylon bending was taken into consideration during NLTH analyses.

The effects of bi-directional earthquake excitations should be taken account in the estimation of the isolator displacement demand (Cilsalar and Constantinou 2017) for base isolated cable-stayed bridge. It should be noted that the dynamic characteristics of the base-isolated cable-stayed bridge are significantly different in its longitudinal and transverse directions. In current practice, the deck of the cable-stayed bridge is disconnect from the pylon in the longitudinal direction and dampers were installed between the deck and pylon to minimize the seismic response. But the deck and the pylon are connected in the transverse direction of the bridge in order to reduce the lateral movement of the deck under service loads (Camara et al. 2017). Moreover, the bidirectional interactions of hysteretic behavior of the bearings are needed to properly consider in the analyses. Therefore, the current simplified design method is difficult to apply to the bidirectional excitations. This study is aimed at reducing the transverse responses of the cable-stayed bridge through base isolation.

In order to estimate the displacement demand of the bridge under bidirectional excitations, as proposed by Ryan and Chopra (2004), Pant et al. (2013), and International Building Code (2000), the 100% + 30% rule is generally used as shown in Eq. (26). This estimation is based on the assumption that bidirectional grounds motions have the same intensity. The ratio of peak displacements due to biaxial and uniaxial excitations is calculated by:

$$ \frac{{u_{bo} }}{{u_{bo(uniaxial)} }} = \sqrt {1^{2} + 0.3^{2} } = 1.044 $$

(26)

The predictions of the displacement demand using the DDBD procedure are compared with the results from NLTH analyses considering bidirectional excitations, as shown in Fig. 9. In the figure, the blue solid line is the design displacement predicted by the DDBD, and the red solid line is the results from NLTH analyses considering bidirectional excitations. It is found that compared to NLTH, the average peak displacement is underestimated by DDBD procedure with a maximum error of 20%. Further studies are needed to fully account for the effects of bidirectional excitation in DDBD procedure.

Fig. 9

Comparison of displacement of DDBD and NTHA considering bidirectional excitations

5 Conclusions

In this paper, a modified DDBD procedure was developed for the base-isolated single pylon cable-stayed bridge in its transverse direction. The nonlinear M-DOF model of the cable-stayed bridge was converted into an equivalent linear S-DOF system based on the assumption of the uniform transverse displacement profile of the cable-stayed bridge. It is achieved by appropriate design of the isolator stiffness at the pylon base and the two ends of the bridge deck. The proposed procedure was applied to a cable-stayed bridge with the deck and the pylon supported on FPBs. The predictions of the DDBD procedure were verified against the NLTH analyses performed in OpenSees platform, using a set of six spectrum compatible accelerograms. The main conclusions of this study are drawn as follows:

• 1. The NLTH results indicate that the base-isolated cable-stayed bridge designed according to the proposed DDBD procedure exhibits the target displacement profile, namely uniform translation of the deck, in the transverse direction of the bridge under different earthquake loadings.

• 2. The dynamic response of the bridge in terms of the target displacement and the base shear predicted by the DDBD method is slightly underestimated compared to the results obtained from NLTH. The differences between the DDBD results and NLTH results increase with the increase of the damping.

• 3. The isolation ratio can be well predicted by the simplified numerical model. Modal analyses are performed on a range of target isolation ratio, and the results are in close agreement with those obtained from analytical solutions.

Generally, the proposed DDBD procedure for the single pylon base isolated cable-stayed bridge in transverse direction is simple and straightforward to be implemented in engineering practice.