Simplified design of nonlinear damper parameters and seismic responses for long-span cable-stayed bridges with nonlinear viscous dampers

Abstract

Viscous dampers are widely used as passive energy dissipation devices for long-span cable-stayed bridges for mitigation of seismic load-induced vibrations. However, complicated finite element (FE) modeling, together with repetitive and computationally intensive nonlinear time-history analyses (NTHAs) are generally required in conventional design methods. To streamline the preliminary design process, this paper developed a simplified longitudinal double degree of freedom model (DDFM) for single and symmetric twin-tower cable-stayed bridges. Based on the proposed simplified longitudinal DDFM, the analytical equations for the relevant mass- and stiffness-related parameters and longitudinal natural frequencies of the structure were derived by using analytical and energy methods. Modeling of the relationship between the nonlinear viscous damper parameters and the equivalent damping ratio was achieved through the equivalent linearization method. Additionally, the analytical equations of longitudinal seismic responses for long-span cable-stayed bridges with nonlinear viscous dampers were derived. Based on the developed simplified DDFM and suggested analytical equations, this paper proposed a simplified calculation framework to achieve a simplified design method of nonlinear viscous damper parameters. Moreover, the effectiveness and applicability of the developed simplified longitudinal DDFM and the proposed calculation framework were further validated through numerical analysis of a practical cable-stayed bridge. Finally, the results indicated the following. 1) For the obtained fundamental period and longitudinal stiffness, the differences between results of the simplified longitudinal DDFM and numerical analysis were only 2.05% and 1.5%, respectively. 2) Relative calculation errors of the longitudinal girder-end displacement and bending moment of the bottom tower section of the bridge obtained from the simplified longitudinal DDFM were limited to less than 25%. 3) The equivalent damping ratio of nonlinear viscous dampers and the applied loading frequency had significant effects on the longitudinal seismic responses of the bridge. Findings of this study may provide beneficial information for a design office to make a simplified preliminary design scheme to determine the appropriate nonlinear damper parameters and longitudinal seismic responses for long-span cable-stayed bridges.

Graphical abstract

Keywords

cable-stayed bridges / viscous dampers / simplified analytical model / equivalent damping ratio / seismic mitigation

1 Introduction

Earthquakes can cause serious damages to infrastructures, significant economic losses, tremendous social disruption, and most importantly loss of life [1–4]. Due to their beautiful aesthetics, superior spanning ability, and convenient construction, cable-stayed bridges have been widely applied [5–9]. However, since long-span cable-stayed bridges have several distinct characteristics, such as long fundamental period, low damping, great slenderness, and high flexibility, they tend to have many serious dynamic issues under the seismic excitations [7,10–12]. For example, after several famous earthquakes––the 1999 Chi-Chi Earthquake [13], the 2011 Tohoku Earthquake [14], and the 2008 Wenchuan Earthquake [15]––many post-earthquake investigations suggested that the deck-pylon connection could significantly affect the seismic responses of the whole cable-stayed bridge. Three types of deck-pylon connections are popularly employed in practical engineering practices [5,16], including: 1) the fully rigid connection, 2) free deck-pylon connection, and 3) the intermediate solution made by seismic protective devices (e.g., viscous dampers). Among these connections, viscous dampers are demonstrated to be efficient and reliable in mitigating the seismic hazards that are faced by many civil infrastructures [5,17–19].

Although a portion of the input seismic energy to cable-stayed bridges could be dissipated owing to their excellent slenderness and flexibility, they may exhibit large longitudinal girder-end displacements because of the weak longitudinal restrictions. The resultant base bending moments and shear forces on the tower, and longitudinal girder-end displacements, cannot be ignored, because they may result in localized or global structural damages to cable-stayed bridges [11,12,20]. As a consequence, seismic protective devices (e.g., viscous dampers) are generally arranged to mitigate nonlinear dynamic behavior. Such an intermediate solution can also provide excellent ductility, sufficient stiffness, and strength, as well as energy dissipation capacity to help cable-stayed bridges to resist seismic excitations [5,17–19]. For instance, to mitigate the structural damages more effectively, viscous dampers are generally employed as efficient passive control devices to dissipate the input earthquake energy [20–25]. Therefore, viscous dampers are widely employed in cable-stayed bridges to help to reduce seismic responses. Additionally, several previous studies proposed some relevant optimization algorithms to achieve the superior efficiency of nonlinear viscous damping to bridges to help them to work safely [5,17,18,26,27]. These aforementioned studies also suggested that the optimized damper parameters could reduce the seismic responses and mitigate possible seismic damages.

According to the above literature review, although the capability and feasibility of viscous dampers in controlling the dynamic seismic behavior of cable-stayed bridges have been demonstrated, the optimal design of critical damper parameters has not been well investigated. These fundamental damper parameters, such as the damping coefficient (C), velocity exponent (α), and damping ratio (ξ), have significant effects on the energy dissipation ability of viscous dampers. However, it is crucial to optimize these nonlinear damper parameters during the preliminary design state [11,20]. Although the efficiency and capability of nonlinear viscous dampers have been validated by many previous studies [20,21,28–31], the optimal design of nonlinear damper parameters of long-span cable-stayed bridges with nonlinear viscous dampers has not been well investigated. In addition, most of the traditional design methods for determining the optimal nonlinear viscous damper parameters generally require complicated finite element (FE) modeling and repetitive nonlinear time-history analyses (NTHAs), which are usually non-economical and time-consuming [11,24,25,32]. To this end, several studies have suggested some simplified models or design methods for the calculation of, 1) the longitudinal seismic responses [33–35] and 2) viscous damper parameters under the longitudinal seismic excitations [24,25,36–38].

For example, by considering the cable-stayed bridge as a single degree of freedom model (SDFM), Xu et al. [33] suggested equations to calculate the tower top displacement and the bending moment of tower lower section. However, in their suggested simplified SDFM, the pylon was considered as a lumped mass, whereas contributions made by the girder and stay cables to the stiffness and damping coefficient of the bridge were ignored. This may cause inaccurate estimations of the longitudinal seismic responses and natural frequencies of the bridge. Likewise, several previous technical reports, such as NCEER [36] and FEMA-356 [37] recommended some analytical equations to obtain the C of linear fluid viscous dampers. Hwang and Tseng [39] derived the analytical equations of viscous dampers for highway bridges by using the eigenvalue analysis method. Moreover, to streamline the preliminary design process, Xu et al. [24] suggested several analytical equations to compute the nonlinear viscous damper parameters of cable-stayed bridges under pulse-like ground motions. Numerical analysis of a real cable-stayed bridge was also used to validate the accuracy and effectiveness of the simplified design model. More recently, Li et al. [25] proposed a simplified approach to calculate the nonlinear viscous damper parameters and they validated the feasibility of this simplified model through a shaking table test. In summary, the aforementioned studies indicated that viscous dampers are helpful in restricting the dynamic seismic responses of long-span cable-stayed bridges, such as the longitudinal girder-end displacements, resultant base shear forces and bending moments of the pylons. However, in the determination of the appropriate nonlinear damper parameters and seismic responses, most of the previous studies mainly focused on the developments of simplified analytical models, which require complicated numerical modeling and computationally expensive NTHAs. Furthermore, the relevant critical parameters (i.e., mass- and stiffness-related parameters, damping coefficients) for the currently available simplified analytical models in constructing the dynamic motion equations were mainly determined based on the designers’ experience, which is very subjective. Therefore, the rationality and accuracy of the existing simplified analytical models and calculation methods in determining the nonlinear viscous damper parameters and seismic responses of long-span cable-stayed bridges need to be further discussed. Hence, it is of great importance to propose a simplified preliminary design scheme that a design office can use to determine their appropriate nonlinear viscous damper parameters and longitudinal seismic responses.

Thus, to achieve a simple preliminary design procedure for long-span cable-stayed bridges with nonlinear viscous dampers, this paper established a longitudinal double degree of freedom model (DDFM) and proposed a simplified calculation framework to appropriately determine their nonlinear viscous damper parameters and longitudinal seismic responses. Specifically, this study made the following unique contributions to the existing research results. 1) The proposed simplified longitudinal DDFM considered the tower and girder as the independent lumped masses according to the equivalent modal method. Also, this paper established the dynamic equation of motion by considering the contributions made by the tower, girder, and all stay cables to the stiffness and damping coefficient of the bridge. 2) The relevant mass- and stiffness-related parameters were derived in detail by using the analytical and energy methods, and the formulas of longitudinal natural frequencies and modal shapes of the bridge were developed. 3) The relationship between nonlinear viscous damper parameters and the equivalent damping ratio was established. 4) Calculation formulas for the longitudinal seismic responses of the structure under the arbitrary non-periodic seismic excitations were also derived. Finally, effectiveness of the simplified computational framework in determining the nonlinear viscous damper parameters and longitudinal seismic responses of long-span cable-stayed bridges was verified through numerical analysis of a prototype bridge.

2 Development of the simplified analytical model

2.1 Development of the simplified longitudinal double degree of freedom model

Cable-stayed bridges exhibit obvious double degree of freedom characteristics during longitudinal seismic excitations [25,32]. Meanwhile, the bridges’ longitudinal seismic responses under the longitudinal seismic excitations are mainly dominated by the first-order vibration mode, which lays a foundation for simplifying the whole bridge into a simplified longitudinal DDFM [25,32]. For example, Fig.1 and Fig.2 show the schematic diagrams of the simplified longitudinal analytical models for the single cable-stayed bridges and symmetric twin-tower cable-stayed bridges, respectively.

For instance, as seen from Fig.2, for symmetric twin-tower cable-stayed bridges, the required steps in developing the simplified longitudinal DDFM were introduced as follows: 1) by ignoring the longitudinal stiffness and friction forces of the transitional piers, the bridge (Fig.2(a)) could be simplified into the half bridge model (Fig.2(b)) by adding another transitional pier in the center; 2) by considering all stay cables as an equivalent pair of cables and by considering the tower and girder as the independent equivalent lumped masses, the half bridge model could be transferred into the simplified longitudinal DDFM as given in Fig.2(c). As shown in Fig.2, H₀ is the actual height and H is the equivalent height of the tower in the simplified longitudinal DDFM (i.e., H < H₀); the girder can be considered as a lumped mass (m_b) moving longitudinally and the tower can be considered as another lumped mass (m_t) according to the equivalent modal method [24,25,32,40]; k_t is the equivalent longitudinal stiffness of the tower particle; k_b is the equivalent longitudinal stiffness of the girder particle; c_b and c_t are the damping coefficients of the girder and tower particles, respectively. Thus, after these relevant critical parameters (including H, m_t, k_t, m_b, k_b, c_b, and c_t) were determined, the simplified longitudinal DDFM could be developed, and then the longitudinal natural frequencies, viscous damper parameters, and seismic responses of the structure could be further determined. The equation of motion of the simplified longitudinal DDFM could be represented by Eqs. (1) or (2):

(1)[M]\{\ddot{u}\}+[C]\{\dot{u}\}+[K]\{u\}=-[M][I]{\ddot{u}}_{\mathrm{g}},

(2)\begin{array}{rl}& \left[\begin{array}{cc}{m}_{\mathrm{b}}& 0\\ 0& m_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}{\ddot{u}}_{\mathrm{b}}\\ {\ddot{u}}_{\mathrm{t}}\end{array}\right\}+\left[\begin{array}{cc}{c}_{\mathrm{b}}& -c_{\mathrm{b}}\\ -c_{\mathrm{b}}& c_{\mathrm{b}}+c_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}{\dot{u}}_{\mathrm{b}}\\ {\dot{u}}_{\mathrm{t}}\end{array}\right\}\\ +& \left[\begin{array}{cc}{k}_{\mathrm{b}}& -k_{\mathrm{b}}\\ -k_{\mathrm{b}}& k_{\mathrm{b}}+k_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}{u}_{\mathrm{b}}\\ u_{\mathrm{t}}\end{array}\right\}=-\left[\begin{array}{cc}{m}_{\mathrm{b}}& 0\\ 0& m_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}1\\ 1\end{array}\right\}{\ddot{u}}_{\mathrm{g}},\end{array}

where [M], [K], and [C] are the mass, stiffness, and damping matrixes of the system, respectively. [I] is the unit matrix. u_{\mathrm{b}}, {\dot{u}}_{\mathrm{b}}, and {\dot{u}}_{\mathrm{b}} are the displacement, velocity, and acceleration time-histories of the girder relative to the ground, respectively. Similarly, u_{\mathrm{t}}, {\dot{u}}_{\mathrm{t}}, and {\ddot{u}}_{\mathrm{t}} are the displacement, velocity, and acceleration time-histories of the tower relative to the ground, respectively. {\ddot{u}}_{\mathrm{g}} is the acceleration time-history from the input seismic records.

2.2 Determination of the longitudinal stiffness of the girder

Based on the force and deformation characteristics of the structure, and when the tower is assumed to be perfectly rigid, longitudinal stiffness of the girder (k_b) can be calculated as the sum of changes of cable forces in all stay cables along the longitudinal direction when unit longitudinal displacement of the girder occurs. To simplify the calculation of k_b, several additional assumptions should be considered as following: 1) the sag effect of cables is neglected; 2) there is no relative slip between the cables and tower, and the slip between the cables and girder is ignored; 3) the longitudinal displacements of all cable anchorage points on the girder are equal; 4) decrease of the cable forces is significantly less than the initial tension forces for all stay cables under longitudinal seismic excitations; and 5) the longitudinal displacement of the tower is zero when unit longitudinal displacement of the girder occurs.

Fig.3 presents the deformation diagram of the half bridge model of symmetric twin-tower cable-stayed bridges when unit longitudinal displacement of the girder occurs. k_b is equal to the force F required to cause such a unit longitudinal displacement. Thus, according to the law of energy conservation, the work done by the external force F (W_F) is equal to the sum of the vertical bending deformation energy of the girder (W_bv) and the total deformation energy of all stay cables (W_c), which can be represented as

(3)W_{\mathrm{F}}=W_{\mathrm{b}\mathrm{v}}+W_{\mathrm{c}}.

As shown in Fig.3, assuming the vertical bending deformation of the girder can be described by a two-stage sine function and the peak value of vertical bending deformation of the left girder (part l₁) is V_f, so the peak value of the vertical bending deformation of the right girder (part l₂) is (l₂/l₁)·V_f. By taking the centroid position of the girder at the tower in the initial state as the origin of coordinates, the vertical displacement functions of the left and right girder parts can be represented by Eqs. (4) and (5), respectively. Thus, W_bv can be determined by using Eq. (6), where E_b is the material elastic modulus of the girder; and I_b is the vertical cross-sectional bending moment of inertia of the girder.

(4)V_1(x)=-V_{\mathrm{f}}\sin \frac{\text{π}}{l_1}x,x\in \left[-l_1,0\right],

(5)V_2(x)=-\frac{l_2}{l_1}{V}_{\mathrm{f}}\sin \frac{\text{π}}{l_2}x,x\in \left[0,l_2\right],

(6)\begin{array}{rl}{W}_{\text{bv}}=& \frac{1}{2}{{\displaystyle \int }}_L{E}_{\text{b}}{I}_{\text{b}}{\left(V_1^{″}(x)\right)}^2\text{d}x+\frac{1}{2}{{\displaystyle \int }}_L{E}_{\text{b}}{I}_{\text{b}}{\left(V_2^{″}\text{(}x\text{)}\right)}^2\text{d}x\\ =& \frac{{\text{π}}^4{E}_{\text{b}}{I}_{\text{b}}{V}_{\text{f}}\left(l_1+l_2\right)}{4l_1^3{l}_2}.\end{array}

Similarly, when unit longitudinal displacement of the girder occurs, the total deformation energy of all stay cables (W_c) can be calculated by

(7)W_{\text{c}}=\frac{1}{2}\sum _{i=1}^n\mathrm{\Delta }{S}_i\mathrm{\Delta }{l}_i=\frac{E_{\text{c}}}{2}\sum _{i=1}^n{A}_i\frac{\mathrm{\Delta }{l}_i^2}{l_i},

where E_c is the material elastic modulus of the cables; A_i and Δl_i are the cross-sectional area and deformation of ith cable, respectively; and l_i is the initial length of ith cable. In addition, the work done by the external force F (W_F) can be determined by Eq. (8). Based on the equilibrium condition of the girder, the external force F can be calculated by using Eq. (9), where cos θ_i is the projection of ith cable along the longitudinal direction. Thus, based on Eqs. (3)–(9), the longitudinal stiffness of the girder (k_b) can be determined by Eq. (10):

(8)W_{\mathrm{F}}=\frac{1}{2}F\mathrm{\Delta },

(9)F=\sum _{i=1}^n{E}_{\text{c}}{A}_i\frac{\mathrm{\Delta }{l}_i}{l_i}\text{cos}{\theta }_i,

(10)k_{\text{b}}=\frac{F}{\mathrm{\Delta }}=\frac{E_{\text{c}}}{\mathrm{\Delta }}\sum _{i=1}^n{A}_i\frac{\mathrm{\Delta }{l}_i}{l_i}\text{cos}{\theta }_i.

Moreover, for cable-stayed bridges with auxiliary piers, when the vertical displacement of the girder between the transitional pier and auxiliary pier is ignored, the bridge girder part l₁ (as shown in Fig.3) could be taken as the distance between the auxiliary pier and the tower. The aforementioned deduction of k_b is applicable for both floating and semi-floating cable-stayed bridges.

2.3 Determination of the equivalent height and equivalent mass of the tower

According to the equivalent modal method [41], the equivalent mass of the tower (m_t) can be expressed as

(11)m_{\text{t}}=\frac{{\int }_0^{H_0}xm\text{(}x\text{)d}x}{H},

where H₀ is the initial height of the tower, whereas H represents the equivalent height of tower (i.e., H < H₀); x is the distance of the target point to the tower top; and m(x) is the mass of the target tower segment dx. In addition, under the assumption of perfect rigidity of the tower, when longitudinal unit displacement of the girder occurs, as given in Fig.3, variations of the cable force in each stay cable can be determined, and then the equivalent height H of the tower can be determined by

(12)\sum _{i=1}^n\left(H-d_i\right)E_{\text{c}}{A}_i\frac{\mathrm{\Delta }{l}_i}{l_i}\text{cos}{\theta }_i=0,

where d_i is the distance of the anchor point of the ith cable on the tower from the tower base. Thus, after the tower equivalent height is determined, the equivalent mass of tower (m_t) is obtained by Eq. (11).

2.4 Determination of the longitudinal stiffness of the tower

A computation diagram of the longitudinal stiffness of the tower is shown in Fig.4. When a unit force is applied at the tower top section, the resultant longitudinal displacement of each tower segment can be determined by using the principle of force method. Thus, the longitudinal stiffness of tower (k_t) can be represented by

(13)k_{\text{t}}=\frac{1}{h_{\text{n}}}=\frac{3E_{\text{t}}}{\sum _{i=1}^n{\displaystyle \frac{h_i^3}{I_i}}-\sum _{i=1}^{n-1}{\displaystyle \frac{h_i^3}{I_{i+1}}}},

where E_t is the material elastic modulus of the tower; h_i is the height of the ith tower segment; and I_i is the cross-sectional longitudinal bending moment of inertia of the ith tower segment.

2.5 Determination of the longitudinal natural frequencies

According to Eqs. (1) or (2), the undamped natural frequencies and mode shapes of the simplified longitudinal DDFM can be determined by

(14)\left|[K]-{\omega }_n^2[M]\right|=0,

(15)\begin{array}{rl}& {\omega }_{1,2}^2\\ =& \frac{1-\gamma }{R_{\mathrm{s}}}\frac{\left[1+R_{\mathrm{s}}/(1-\gamma )\right]\mp \sqrt{{\left[1+R_{\mathrm{s}}/(1-\gamma )\right]}^2-4\gamma R_{\mathrm{s}}/(1-\gamma )}}{2\gamma }\\ & \cdot {\omega }_{\mathrm{b}}^2,\end{array}

(16)\{\varphi \}=\left\{\begin{array}{l}{\varphi }_{11}\\ {\varphi }_{12}\end{array}\right\}=\left\{\begin{array}{c}1\\ 1-{\alpha }_1\end{array}\right\},

(17)\left\{{\varphi }_2\right\}=\left\{\begin{array}{l}{\varphi }_{21}\\ {\varphi }_{22}\end{array}\right\}=\left\{\begin{array}{c}1\\ 1-{\alpha }_2\end{array}\right\},

where ω₁ and ω₂ are the first and second orders natural frequencies of the simplified longitudinal DDFM, respectively; \{{\varphi }_1\} and \left\{{\varphi }_2\right\} are the first and second normalized mode shape vectors of the structure, respectively. Other related parameters could be calculated by the following formulas: {\omega }_{\mathrm{b}}=\sqrt{k_{\mathrm{b}}/m_{\mathrm{b}}}, {\omega }_{\mathrm{t}}=\sqrt{k_{\mathrm{t}}/\left(m_{\mathrm{b}}+m_{\mathrm{t}}\right)}, \gamma =m_{\mathrm{t}}/\left(m_{\mathrm{b}}+m_{\mathrm{t}}\right), R_{\mathrm{s}}=k_{\mathrm{b}}/k_{\mathrm{t}}, {\alpha }_1={\omega }_1^2/{\omega }_{\mathrm{b}}^2, and {\alpha }_2={\omega }_2^2/{\omega }_{\mathrm{b}}^2.

3 Calculation of the nonlinear viscous damper parameters

When the viscous dampers (c_d) are employed between the tower and girder, the motion equation of the simplified longitudinal DDFM (as shown in Fig.5) can be represented by

(18)\begin{array}{rl}& \left[\begin{array}{cc}{m}_{\mathrm{b}}& 0\\ 0& m_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}{\ddot{u}}_{\mathrm{b}}\\ {\ddot{u}}_{\mathrm{t}}\end{array}\right\}+\left[\begin{array}{cc}{c}_{\mathrm{b}}+c_{\mathrm{d}}& -c_{\mathrm{b}}\\ -c_{\mathrm{b}}& c_{\mathrm{b}}+c_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}{\dot{u}}_{\mathrm{b}}\\ {\dot{u}}_{\mathrm{t}}\end{array}\right\}\\ +& \left[\begin{array}{cc}{k}_{\mathrm{b}}& -k_{\mathrm{b}}\\ -k_{\mathrm{b}}& k_{\mathrm{b}}+k_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}{u}_{\mathrm{b}}\\ u_{\mathrm{t}}\end{array}\right\}=-\left[\begin{array}{cc}{m}_{\mathrm{b}}& 0\\ 0& m_{\mathrm{t}}\end{array}\right]\left\{\begin{array}{l}1\\ 1\end{array}\right\}{\ddot{u}}_{\mathrm{g}},\end{array}

where c_d is the equivalent damping coefficient. According to some previous studies [25,32], under the modal coordinates, the mass and damping matrixes of the system should be diagonalized, indicating that the off-diagonal elements of {\left\{{\varphi }_i^{\mathrm{n}}\right\}}^{\mathrm{T}}[M]\left\{{\varphi }_j^{\mathrm{n}}\right\} and {\left\{{\varphi }_i^{\mathrm{n}}\right\}}^{\mathrm{T}}[C]\left\{{\varphi }_j^{\mathrm{n}}\right\} can be ignored.

Then, the longitudinal natural frequencies of the simplified longitudinal DDFM can be calculated by Eqs. (19) and (20), respectively. In addition, based on these defined characteristic parameters: c_{\mathrm{b}}=2m_{\mathrm{b}}{\xi }_{\mathrm{b}}{\omega }_{\mathrm{b}}, c_{\mathrm{d}}=2m_{\mathrm{b}}{\xi }_{\mathrm{d}}{\omega }_{\mathrm{b}}, and c_{\mathrm{t}}=2\left(m_{\mathrm{b}}+m_{\mathrm{t}}\right){\xi }_{\mathrm{t}}{\omega }_{\mathrm{t}}, Eqs. (19) and (20) can be further developed to Eqs. (21) and (22), respectively.

(19){\xi }_1=\frac{C_1}{2M_1{\omega }_1}=\frac{c_{\mathrm{d}}+{\alpha }_1^2{c}_{\mathrm{b}}+{\left(1-{\alpha }_1\right)}^2{c}_{\mathrm{t}}}{2{\omega }_1\cdot \left[m_{\mathrm{b}}+{\left(1-{\alpha }_1\right)}^2{m}_{\mathrm{t}}\right]},

(20){\xi }_2=\frac{C_2}{2M_2{\omega }_2}=\frac{c_{\mathrm{d}}+{\alpha }_2^2{c}_{\mathrm{b}}+{\left(1-{\alpha }_2\right)}^2{c}_{\mathrm{t}}}{2{\omega }_2\cdot \left[m_{\mathrm{b}}+{\left(1-{\alpha }_2\right)}^2{m}_{\mathrm{t}}\right]},

(21){\xi }_1=\frac{{\xi }_{\mathrm{d}}(1-\gamma )+{\xi }_{\mathrm{b}}{\alpha }_1^2(1-\gamma )+{\xi }_{\mathrm{t}}{\left(1-{\alpha }_1\right)}^2\sqrt{(1-\gamma )/R_{\mathrm{s}}}}{\sqrt{{\alpha }_1}\left[(1-\gamma )+{\left(1-{\alpha }_1\right)}^2\gamma \right]},

(22){\xi }_2=\frac{{\xi }_{\mathrm{d}}(1-\gamma )+{\xi }_{\mathrm{b}}{\alpha }_2^2(1-\gamma )+{\xi }_{\mathrm{t}}{\left(1-{\alpha }_2\right)}^2\sqrt{(1-\gamma )/R_{\mathrm{s}}}}{\sqrt{{\alpha }_2}\left[(1-\gamma )+{\left(1-{\alpha }_2\right)}^2\gamma \right]},

where ξ_b and ξ_t are the damping ratios of the girder and tower, respectively; and ξ_d denotes the equivalent damping ratio. Moreover, compared to that of the first order modal damping, the effect of the second order modal damping on the longitudinal seismic responses of the structure is negligible [25,32], so the equivalent damping ratio (ξ_eq) equals that under the first order natural frequency, which means ξ_eq = ξ_1. Thus, the damping coefficient of linear viscous dampers (c_{d_linear}) can be obtained by

(23)c_{\mathrm{d}\mathrm{\_}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}=2m_{\mathrm{b}}{\omega }_{\mathrm{b}}\cdot \frac{{\xi }_1\sqrt{{\alpha }_1}\left[(1-\gamma )+{\left(1-{\alpha }_1\right)}^2\gamma \right]-{\xi }_{\mathrm{b}}{\alpha }_1^2(1-\gamma )-{\xi }_{\mathrm{t}}{\left(1-{\alpha }_1\right)}^2\sqrt{(1-\gamma )/R_{\mathrm{s}}}}{1-\gamma }.

Damping coefficient of nonlinear viscous dampers (c_{d_nonlinear}) could be determined based on the theory of equivalent linearization. An equivalent relationship of c_{d_linear} and c_{d_nonlinear} was suggested in Huang et al. [32] by using the equivalence principle of energy dissipation. The damping forces of the linear and nonlinear viscous dampers under the harmonic loading u_{\mathrm{d}}=u_0\sin (\overline{\omega }t)can be determined by Eqs. (24) and (25), respectively.

(24)F_{\mathrm{d}}(t)=c_{\mathrm{d}\mathrm{\_}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\cdot v,

(25)F_{\mathrm{d}}(t)=c_{\mathrm{d}\mathrm{\_}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\cdot v^{\alpha },

where α is the velocity exponent. Then, the energy dissipations of a linear and nonlinear viscous damper in a vibration cycle can be represented by Eqs. (26) and (27), respectively.

(26)E_{\text{I}}={\displaystyle \oint F_{\text{d}}\text{d}u}={\int }_0^{2\text{π}/\overline{\omega }}{c}_{\text{d}\mathrm{\_}\text{linear}}\cdot u\text{d}t=\text{π}{c}_{\text{d}\mathrm{\_}\text{linear}}\overline{\omega }{u}_0^2,

(27)\begin{array}{rl}{E}_{\text{f}}=& {\displaystyle \oint F_{\text{d}}\text{d}u}={\int }_0^{2\text{π}/\overline{\omega }}{c}_{\text{d}\mathrm{\_}\text{nonlinear}}\cdot u^{\alpha +1}\text{d}t\\ =& 2^{\alpha +1}{c}_{\text{d}\mathrm{\_}\text{nonlinear}}{u}_0^{\alpha +1}{\overline{\omega }}^{\alpha }\frac{{\mathrm{\Gamma }}^2\text{(}0.5\alpha \text{)}}{\mathrm{\Gamma }\text{(}\alpha +2\text{)}},\end{array}

where \overline{\omega } represents the applied loading frequency and μ₀ represents the maximum relative deformation of the tower and girder. Γ is the gamma function and \mathrm{\Gamma }(n)=(n-1)!. Thus, the equivalent relationship of c_{d_linear} and c_{d_nonlinear} can be represented by

(28)\begin{array}{rl}{c}_{\mathrm{d}\mathrm{\_}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}=& \frac{2^{\alpha +2}}{\text{π}}\cdot c_{\mathrm{d}\mathrm{\_}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}{u}_0^{\alpha -1}{\overline{\omega }}^{\alpha -1}\cdot \frac{{\mathrm{\Gamma }}^2(0.5\alpha +1)}{\mathrm{\Gamma }(\alpha +2)}\\ =& \frac{c_{\mathrm{d}\mathrm{\_}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}{u}_0^{\alpha -1}{\overline{\omega }}^{\alpha -1}}{\text{π}}\cdot \lambda ,\end{array}

(29)\lambda =2^{\alpha +2}\cdot \frac{{\mathrm{\Gamma }}^2(0.5\alpha +1)}{\mathrm{\Gamma }(\alpha +2)}.

Finally, based on Eqs. (23) and (28), c_{d_nonlinear} can be further determined as

(30)c_{\mathrm{d}\mathrm{\_}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}=\frac{\text{π}{c}_{\mathrm{d}\mathrm{\_}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}}{{\overline{\omega }}^{\alpha -1}{u}_0^{\alpha -1}\lambda }=2\text{π}{m}_{\mathrm{b}}\cdot \frac{{\xi }_1\sqrt{{\alpha }_1}\left[(1-\gamma )+{\left(1-{\alpha }_1\right)}^2\gamma \right]-{\xi }_{\mathrm{b}}{\alpha }_1^2(1-\gamma )-{\xi }_{\mathrm{t}}{(1-\alpha )}^2\sqrt{(1-\gamma )/R_{\mathrm{s}}}}{{\omega }_{\mathrm{b}}^{\alpha -2}{D}_{\mathrm{g}}^{\alpha -1}\lambda (1-\gamma )}.

4 Calculation of longitudinal seismic responses

According to the modal superposition method [40], the dynamic motion equation as given in Eq. (1) could be represented by

(31){\ddot{Y}}_n(t)+2{\xi }_n\cdot {\omega }_n{\dot{Y}}_n(t)+{\omega }_n^2{Y}_n(t)=\frac{P_n(t)}{M_n},

where {\xi }_n=C_n/2{\omega }_n{M}_n represents the nth order modal damping ratio; M_n={\left\{{\varphi }_n\right\}}^{\mathrm{T}}[M]\left\{{\varphi }_n\right\}, C_n={\left\{{\varphi }_n\right\}}^{\mathrm{T}}[C]\left\{{\varphi }_n\right\}, and P_n(t)={\left\{{\varphi }_n\right\}}^{\mathrm{T}}\cdot p(t) denote the modal mass, damping, and the applied dynamic load of the nth modal shape, respectively. As suggested by previous studies [25,32], the longitudinal seismic responses of a cable-stayed bridge are mainly dominated by the first order longitudinal mode during longitudinal seismic excitations, so the dynamic motion equation can be represented by Eq. (32). Moreover, the corresponding modal mass, stiffness, and damping of the first modal shape can be determined by Eqs. (33)–(35):

(32){\ddot{Y}}_1(t)+2{\xi }_1\cdot {\omega }_1{\dot{Y}}_1(t)+{\omega }_1^2{Y}_1(t)=\frac{P_1(t)}{M_1},

(33)M_1={\left\{{\varphi }_1\right\}}^{\mathrm{T}}[M]\left\{{\varphi }_1\right\}=m_{\mathrm{b}}+{\left(1-{\alpha }_1\right)}^2{m}_{\mathrm{t}},

(34)K_1={\left\{{\varphi }_1\right\}}^{\mathrm{T}}[K]\left\{{\varphi }_1\right\}={\alpha }_1^2{k}_{\mathrm{b}}+k_{\mathrm{t}}{\left({\alpha }_1-1\right)}^2,

(35)C_1={\left\{{\varphi }_1\right\}}^{\mathrm{T}}[C]\left\{{\varphi }_1\right\}=c_{\mathrm{d}\mathrm{\_}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}+{\alpha }_1^2{c}_{\mathrm{b}}+{\left(1-{\alpha }_1\right)}^2{c}_{\mathrm{t}}.

The acceleration time-history function of a given arbitrary ground motion can be transferred into the sine or cosine functions [38]. Hence, by analyzing the responses of the structures excited by the harmonic loadings, such as those having sine or cosine forms, the simplified calculation of the structural response under arbitrary non-periodic earthquakes can be realized. This is because the acceleration time-history of a given arbitrary non-periodic earthquake record can be transformed into the harmonic functions by using the Fourier transform analysis [38,41]. Thus, this paper presented the longitudinal seismic responses of the structure under the arbitrary non-periodic ground motions through the simplified longitudinal DDFM under harmonic loadings. First, for a sine acceleration time-history, A(t)=A_{\mathrm{p}}\cdot \sin (\overline{\omega }t), the applied dynamic load P_1(t) can be obtained as

(36)P_1(t)=-\left[m_{\mathrm{b}}+\left(1-{\alpha }_1\right)m_{\mathrm{t}}\right]A_{\mathrm{p}}\cdot \sin \overline{\omega }t=P_0\cdot \sin (\overline{\omega }t).

Based on the initial conditions, a solution of Eq. (32) can be obtained as

(37)\begin{array}{rl}{Y}_1(t)=& \left(A_1\cos ({\omega }_{1\mathrm{d}}t)+B_1\sin ({\omega }_{1\mathrm{d}}t)\right)\cdot {\mathrm{e}}^{-{\xi }_1{\omega }_{1}t}\\ & +C_1\sin (\overline{\omega }t)+D_1\sin (\overline{\omega }t),\end{array}

where {\omega }_{1\mathrm{d}} represents the first dynamic damping ratio considering the damping effect and {\omega }_{1\mathrm{d}}={\omega }_1\cdot \sqrt{1-{\xi }_1^2}. The corresponding constant coefficients are given as

(38)\begin{array}{r}{A}_1=-D_1,C_1=\frac{P_0}{K_1}\frac{1-{\beta }_1^2}{{\left(1-{\beta }_1^2\right)}^2+{\left(2{\xi }_1{\beta }_1\right)}^2},\\ B_1=\frac{-D_1{\xi }_1}{\sqrt{1-{\xi }_1^2}}-\frac{C_1{\beta }_1}{\sqrt{1-{\xi }_1^2}},D_1=\frac{P_0}{K_1}\frac{-2{\xi }_1{\beta }_1}{{\left(1-{\beta }_1^2\right)}^2+{\left(2{\xi }_1{\beta }_1\right)}^2},\end{array}

where P_0=-\left[m_{\mathrm{b}}+\left(1-{\alpha }_1\right)m_{\mathrm{t}}\right]A_{\mathrm{p}} and \beta =\text{π}/{\omega }_1. Similarly, for a cosine acceleration time-history, A(t)=A_{\mathrm{p}}\cdot \cos (\overline{\omega }t), the applied dynamic load P_1(t) can be obtained as

(39)P_1(t)=-\left[m_{\mathrm{b}}+\left(1-{\alpha }_1\right)m_{\mathrm{t}}\right]A_{\mathrm{p}}\cdot \cos \overline{\omega }t=P_0\cdot \cos \overline{\omega }t.

Based on the initial conditions, a similar solution can be obtained as given in Eq. (37), but the corresponding constant coefficients are given as

(40)\begin{array}{c}{A}_1=-D_1,C_1={\displaystyle \frac{P_0}{R_1}}{\displaystyle \frac{2\xi {\beta }_1}{{\left(1-{\beta }_1^2\right)}^2+{\left(2\xi {\beta }_1\right)}^2}},\\ B_1={\displaystyle \frac{-D_1{\xi }_1}{\sqrt{1-{\xi }_1^2}}}-{\displaystyle \frac{C_1{\beta }_1}{\sqrt{1-{\xi }_1^2}}},D_1={\displaystyle \frac{P_0}{K_1}}{\displaystyle \frac{1-{\beta }_1^2}{{\left(1-{\beta }_1^2\right)}^2+{\left(2{\xi }_1{\beta }_1\right)}^2}}.\end{array}

Moreover, the longitudinal displacement vector of the simplified longitudinal DDFM under both the sine and cosine loads can be obtained by

(41)\{u(t)\}=\left\{\begin{array}{l}{u}_{\mathrm{b}}(t)\\ u_{\mathrm{t}}(t)\end{array}\right\}=\left\{\begin{array}{l}{\varphi }_{11}\\ {\varphi }_{12}\end{array}\right\}{Y}_1(t),

where u_{\mathrm{b}}(t) and u_{\mathrm{t}}(t) are the displacements of the girder and tower particles in the simplified longitudinal DDFM as shown in Fig.5. Furthermore, based on the obtained u_{\mathrm{b}}(t) and u_{\mathrm{t}}(t), the static equivalent force applied at the tower particle and bending moment at the bottom tower section could be calculated by

(42)f_{\mathrm{s}}(t)=k_{\mathrm{t}}{u}_{\mathrm{t}}(t),

(43)M_0(t)=k_{\mathrm{t}}{u}_{\mathrm{t}}(t)H+c_{\mathrm{d}\mathrm{\_}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}{\dot{u}}_{\mathrm{b}}(t)h.

Based on the established simplified longitudinal DDFM given in Fig.5, this paper proposed a simplified calculation framework as shown in Fig.6 to calculate the required nonlinear viscous damper parameters and estimate the longitudinal seismic responses of long-span cable-stayed bridges.

5 Numerical validation

5.1 Introduction and finite element modeling of the case-study bridge

To validate the effectiveness and applicability of the simplified longitudinal DDFM and the proposed simplified calculation framework, an actual five span double-tower cable-stayed bridge located in China was taken as the case-study for numerical validation. The span arrangement of the bridge is 50 + 95 + 350 + 95 + 50 = 640 m. The girder is 3.41 m high, and 37 m wide steel-reinforced concrete composite. It has twin A-shaped towers each with a height of 125.8 m. Fig.7 presents the schematic diagram of the bridge. To resist longitudinal seismic excitations, there are a total of eight nonlinear viscous dampers arranged between the pylons and girder along the longitudinal direction, while no dampers are installed at the transitional and auxiliary piers.

The three-dimensional FE model of the bridge was developed using Midas Civil as shown in Fig.8. Spatial beam−column elements were used to simulate the towers, steel−concrete composite girders, and the transitional and auxiliary piers. The stay cables were simulated using truss elements and an elastic perfectly plastic material was used to model steel yielding and hardening. The pre-tension of the cable was considered by defining the initial strain of the truss elements. Rigid links were used to connect the cables with the girder and towers. The expansion joints at the girder-end were simulated using the gap elements, and the horizontal and vertical elastic link elements were employed to simulate the girder-pier connections. In addition, the towers and the piers were fixed and the effect of soil-structure interaction was ignored. A Rayleigh damping model was employed and a damping ratio of 3% was selected in the following numerical validations. The modal analysis was performed and the result indicated that the fundamental period of the case-study bridge was 5.819 s. The first order modal shape of the bridge is given in Fig.8(b), and the longitudinal motion of the girder dominated this modal shape.

5.2 Results and discussions

Based on the simplified longitudinal DDFM, the proposed simplified calculation framework was applied to the bridge to obtain the appropriate nonlinear damper parameters and its longitudinal seismic responses. The bridge was a semi-floating continuous cable-stayed bridge with the symmetric twin towers, according to the proposed simplified calculation framework shown in Fig.6. A half bridge model was developed, in which, l₁ = 95 m, l₂ = 175 m, and Δ = 1 cm, as shown in Fig.9, ignoring the vertical deformation of the girder between the transitional and auxiliary piers. Based on the developed FE model and the given design information, the longitudinal stiffness of the girder was calculated to be 291659 kN/m, while the value obtained from the simplified longitudinal DDFM was 296042 kN/m. That is, the values had only 1.5% difference, as shown in Tab.1. Similarly, the fundamental period of the bridge calculated by using the simplified longitudinal DDFM was only 2.05% different from that obtained from numerical analysis. Thus, the effectiveness and reliability of the simplified longitudinal DDFM were validated. Moreover, based on the simplified longitudinal DDFM, the longitudinal stiffness of the tower was 32525 kN/m. Based on the determined mass- and stiffness-related parameters, the required nonlinear damper parameters and longitudinal seismic responses of the bridge were then introduced.

Based on the simplified longitudinal DDFM and the derived equations, the longitudinal seismic responses of the bridge, such as the longitudinal girder-end displacement, bending moment at the bottom tower section, and damping force were calculated when the bridge was excited by longitudinal harmonic loadings. A sine acceleration time-history, A(t)=A_{\mathrm{p}}\cdot \sin \overline{\omega }t, was considered as the applied excitation. Nonlinear damper parameters, such as c_{d_nonlinear} and α in Eq. (30) could be calculated according to the given equivalent damping ratio (ξ_d). Then, these critical parameters were input into the developed FE models and the structural longitudinal responses based on a number of NTHAs were calculated. The numerical seismic responses of the bridge were compared with the results acquired from the derived equations to verify the effectiveness and accuracy of the simplified longitudinal DDFM. For example, for velocity exponent of the viscous dampers (α) of value 0.4, and for ξ_d of the viscous dampers less than 0.5, Fig.10 shows comparisons of the relative errors of the obtained longitudinal girder-end displacements, bending moments at the bottom tower section, and damping forces from the numerical results and the simplified DDFM with respect to the applied loading period (T_p) under different values of ξ_d.

As seen from Fig.10(a), when the applied T_p was very close to the fundamental period of the bridge, the relative errors of the obtained longitudinal girder-end displacements from the numerical results and the simplified DDFM exhibited a diverging trend under different values of ξ_d. This might have been because the longitudinal girder-end displacement that corresponds to this loading period (very close to the fundamental period) was characterized by large amplitude and rapid change. In addition, when T_p was less than the fundamental period, the relative displacement error increased significantly with the increase of ξ_d. However, when T_p was greater than the fundamental period, the relative displacement error was small, and the effect of increasing ξ_d on its change was negligible. Meanwhile, as shown in Fig.10(b), the relative damping force error generally increased with the increase of the loading period. When ξ_d was small, the relative damping force error was very large. However, when ξ_d was greater than 0.15, the maximum relative damping force error was around 25% and it decreased gradually with increase of ξ_d.

Moreover, calculation of the longitudinal bending moment of the bottom tower section is related to the longitudinal girder-end displacement and damping force. As seen from Fig.10(c), when T_p was close to and greater than the fundamental period of the bridge, the relative bending moment error exhibited a similar trend to that of the relative displacement error, as shown in Fig.10(a), and change of this moment error was within 10%. When T_p was less than the fundamental period of the structure, the relative moment error decreased with the increase of ξ_d. This was even significant when T_p was 2 s. This might be because the value of ξ_d corresponding to the minimum bending moment of the bottom tower section in the FE model was relatively small. According to the numerical results, the value of ξ_d corresponding to this minimum bending moment was between 0.2 and 0.25, whereas it was greater than 0.5 when T_p was between 2 and 5 s. Furthermore, as observed from Fig.10, when ξ_d was less than 0.5, the calculation errors of the longitudinal girder-end displacement and bending moment of the bottom tower section from the simplified DDFM and the suggested analytical equations could be restrained within 25%, which was acceptable in the preliminary design stage. In fact, the actual damper parameters (i.e., c_{d_nonlinear} and α) of the installed viscous dampers in the case-study bridge were 3000 and 0.4, and the corresponding ξ_d was 0.15, which is within the range of 0.1–0.3. Such a range of ξ_d is widely employed in the design practice of cable-stayed bridges [42,43]. Therefore, the above analysis demonstrated that the proposed simplified calculation framework in determining the damper parameters and longitudinal seismic responses exhibited good accuracy and applicability.

To further investigate the effects of T_p and ξ_d on the longitudinal responses of the bridge, Fig.11 shows comparisons of the calculated longitudinal girder-end displacements, damping forces, and bending moments at the bottom tower section with respect to T_p under different values of ξ_d. As observed from Fig.11, under the action of harmonic loading, the longitudinal girder-end displacement and bending moment of the bottom tower section decreased with the increase of ξ_d, whereas the damping force increased with the increase of ξ_d. When ξ_d was less than 0.3, the bridge had peak longitudinal girder-end displacement and bending moment as it was close to the fundamental period. However, when T_p was greater than 6 s, the displacement and moment decreased significantly with the increase of ξ_d. When T_p was greater than 25 s, effects of T_p and ξ_d on the displacement and moment were negligible. In addition, as seen from Fig.11(a) and Fig.11(b), selecting an excessively large ξ_d (i.e., ξ_d > 0.5) might have significant influence on the modal analysis and the longitudinal displacement and bending moment of the bridge, such that even no extremum point occurs.

Furthermore, the longitudinal girder-end displacement and bending moment of the bottom tower section can be reduced by increasing ξ_d. As seen from Fig.11, the reduction of the longitudinal displacement and moment decreased with the increase of ξ_d. For instance, as ξ_d increased from 0.05 to 0.10, the peak moment of the bottom tower section decreased from 20340 to 11554 MN·m, the reduction was 43%. However, when ξ_d increased from 0.4 to 0.5, the peak moment decreased from 3229 to 2741 MN·m, the reduction was about 15%. Thus, during the preliminary design of nonlinear damper parameters, it is not necessary to blindly pursue the minimum longitudinal girder-end displacement and bending moment of the bottom tower section, but to apply less than a certain threshold value, so the potential contribution of nonlinear viscous dampers can be fully made while meeting the relevant seismic design requirements.

6 Conclusions

Traditional design methods of the nonlinear damper parameters and longitudinal seismic responses for long-span cable-stayed bridges require complicated numerical modeling, with repetitive and time-consuming NTHAs. To fill this research gap, this study developed a simplified longitudinal DDFM, suggested the analytical equations, and proposed a simplified calculation framework to obtain the nonlinear viscous damper parameters and longitudinal seismic responses of long-span cable-stayed bridges. Subsequently, the effectiveness and accuracy of the proposed simplified longitudinal DDFM and the simplified calculation framework were verified through the numerical analysis of a prototype cable-stayed bridge. Finally, some conclusions are summarized as follows.

1) The developed simplified longitudinal DDFM and the proposed simplified calculation framework in determining the nonlinear damper parameters and longitudinal seismic responses of cable-stayed bridges have good effectiveness and accuracy. The simplified longitudinal DDFM can help a design office to make a simplified preliminary design scheme for long-span cable-stayed bridges with nonlinear viscous dampers.

2) The obtained fundamental period of the case-study bridge and longitudinal stiffness of the girder from the simplified longitudinal DDFM were only 2.05% and 1.5% different from those of the numerical analysis results, respectively.

3) When the equivalent damping ratio of the viscous dampers was less than 0.5, the relative calculation errors of the longitudinal girder-end displacement and bending moment of the bottom tower section from the simplified DDFM could be limited within 25%, which is acceptable during the preliminary design procedure of the bridges.

4) The equivalent damping ratio of the viscous dampers and the applied loading frequency are the crucial factors affecting the longitudinal seismic responses of long-span cable-stayed bridges. The longitudinal seismic responses decreased with the increase of the equivalent damping ratio. Therefore, during the preliminary design of the nonlinear damper parameters, it is necessary to consider a specific seismic response rather than the minimum value as the target control value, so the potential contribution of nonlinear viscous dampers can be fully made while satisfying the relevant seismic design requirements.

Based on the proposed simplified longitudinal DDFM and calculation framework, this paper only conducted some limited theoretical and numerical investigations to determine the nonlinear damper parameters and longitudinal seismic responses for cable-stayed bridges. Analytical equations for the relevant mass- and stiffness-related parameters and longitudinal natural frequencies of the structure were derived, and the relationship between the nonlinear viscous damper parameters and the equivalent damping ratio was determined. Additionally, the analytical equations of the longitudinal seismic responses for long-span cable-stayed bridges with nonlinear viscous dampers under the longitudinal seismic excitations (i.e., the harmonic loadings: sine and cosine acceleration time-histories) were also developed. However, during the numerical validation of the proposed simplified longitudinal DDFM and calculation framework on the case-study bridge, only a sine acceleration time-history (i.e., cosine acceleration time-history is similar) was employed as the applied loading. Although a given arbitrary seismic record could be transferred into the sine or cosine functions by using Fourier transform analysis, it would be more convincing if the verification results for the scenarios and seismic responses of the bridge under the real seismic records could be attained. Thus, more in-depth theoretical and numerical studies for studying the nonlinear damper parameters and longitudinal seismic responses under the practical ground motions should be performed in future. Furthermore, for the numerical validation of the bridge, the applied loading period (T_p) of the input harmonic loading was limited to the range of 2 to 9 s. However, for a given real seismic record, the period may be in the range of 0 to 2 s, which would result in more pronounced seismic responses. Therefore, it is of utmost importance to carry out more comprehensive theoretical and numerical analyses to obtain the nonlinear damper parameters and seismic responses for long-span cable-stayed bridges under the real earthquake records with the applied loading period less than 2 s.

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