Coupled responses of stay cables under the combined rain–wind and support excitations by theoretical analyses

Abstract

Stay cables on several cable-stayed bridges all over the world have been found to experience rain-wind-induced vibrations under the combined action of rain and wind. Meanwhile, the bridge deck might also have obvious oscillation under the wind and/or traffic loads. The coupled responses of a stay cable under the combined rain–wind and support excitations are numerically investigated in this article. The equations of motion of a three-dimensional continuous stay cable are derived by considering the high-order nonlinear components of the dynamic cable tension, together with the equation of motion of the rivulet on the cable surface. The forces induced by rain–wind excitation are determined by the quasi-steady theory, and the support excitation is achieved by the boundary condition. The coupled equations of the cable and the rivulet are numerically solved by using the finite difference method and the fourth-order Runge–Kutta method, respectively. The numerical results show that the high-order nonlinear components of the dynamic cable tension should be taken into account to numerically reproduce the parametric vibration of the stay cable, whereas they hardly have any effects on the rain-wind-induced vibration and the resonance vibration of the stay cable. The responses of stay cable under vertical support oscillation only and the rain–wind excitation only obtained from this study agree well with the literature results. Compared with the results induced by single-source excitation, the cable response amplitude under the combined excitations is smaller than that induced only by support excitation and larger than that induced only by rain–wind excitation. The rivulet is prone to be thrown from the cable surface if the parametric vibration of the stay cable is evoked.

Keywords

high-order nonlinear components numerical method parametric vibration rain-wind-induced vibration resonance vibration stay cable

Introduction

Stay cables on cable-stayed bridges experience many kinds of wind-induced vibrations due to its low frequency and small structural damping, such as rain-wind-induced vibration (RWIV; Hikami and Shiraishi, 1988), dry galloping (Cheng et al., 2008a, 2008b), and vortex-induced vibration (Matsumoto et al., 2001; Zuo et al., 2008). Among these types of wind-induced vibrations, RWIV of stay cable is the most frequently observed and has significant impact on the safety of the cable-stayed bridges (Saul and Hohle, 1995; Watson and Stafford, 1988).

RWIV was first observed on the Meiko Nishi Bridge in Japan in the early 1980s (Hikami and Shiraishi, 1988). Since then on, many investigators have made great efforts to clarify the underlying mechanism of RWIVs of stay cables by means of field observations (Chen et al., 2004; Hikami and Shiraishi, 1988; Matsumoto et al., 2003; Phelan et al., 2006), wind tunnel tests (Bosdogianni and Olivari, 1996; Du et al., 2013; Flamand, 1995; Gu and Du, 2005; Hikami and Shiraishi, 1988; Li et al., 2010, 2016; Yamaguchi, 1990), theoretical analyses (Cao et al., 2003; Cosentino et al., 2003; Gu et al., 2009; Gu and Lu, 2001; Li et al., 2013, 2014; Peil and Nahrath, 2003; Van Der Burgh and Hartono, 2004; Wilde and Witkowski, 2003; Wu et al., 2013; Xu and Wang, 2003; Yamaguchi, 1990), and computational fluid dynamics (CFD) simulations (Li and Gu, 2006; Robertson et al., 2010). The exact features of RWIV could be obtained from field observations, and elaborated parametric studies could be conducted by using wind tunnel tests to evaluate the most sensitive factors for RWIV. However, the most direct way to explain the underlying mechanism of RWIV is to develop accurate analytical models.

The focus of the theoretical analyses for RWIV of stay cable is the forces acting on the rivulet, including the wind force on the rivulet and the surface tension between the cable surface and the rivulet. The early analytical models, which simplified the three-dimensional continuous stay cable as an oscillator, usually made some assumptions to determine the forces on the rivulet. The first analytical model of RWIV was proposed by Yamaguchi (1990) of which the cable and the rivulet were treated as a whole. Therefore, it was not necessary to separate the wind forces acting on the rivulet and the cable, and the surface tension between the cable and the rivulet was not needed. Some researchers (Van Der Burgh and Hartono, 2004; Wilde and Witkowski, 2003; Xu and Wang, 2003) assumed a sinusoidal oscillation of the rivulet for theoretically analyzing RWIV of stay cable. Gu and Lu (2001) established equations of motion for both the cable and the upper rivulet by assuming that the wind force acting on the rivulet was proportional to the torsional force on the cable attached with the rivulet, while the surface tension between the cable surface and the rivulet was set to zero. Peil and Nahrath (2003) put forward an analytical model including the in-plane and out-of-plane degrees-of-freedom of the cable and a tangential degree-of-freedom of the rivulet. They assumed the surface tension between the cable surface and the rivulet to be a linear damping force. Cao et al. (2003) treated the rivulet oscillation as a stochastic process and assumed the surface tension between the cable surface and the rivulet to be a restoring force and a linear damping force. Gu et al. (2009) carried out wind tunnel tests to accurately obtain the wind force acting on the cable and the upper rivulet and established an analytical model, of which the in-plane degree-of-freedom of the cable and the tangential degree-of-freedom of the rivulet are taken into account. In this model, the surface tension between the cable surface and the rivulet was assumed to be the combination of a Coulomb damping force and a linear damping force. Li et al. (2013) extended the model of Gu et al. (2009) by also including the out-of-plane degree-of-freedom of the cable and took the fluctuating wind velocity into account. Recently, some researchers (Gu, 2009; Li et al., 2014) tried to conduct analytical study based on a three-dimensional continuous stay cable, in which the coupled effects between different modes of vibration could be taken into account. In order to simplify the motion equations of the sagged cable, the high-order components in the dynamic cable tension were ignored in Gu (2009) and Li et al. (2014). Jing et al. (2017, 2018) proposed an excitation mechanism and a numerical model of wind forces to explain RWIV of stay cables. They explained that RWIV is caused by the interaction between the upper rivulet, boundary layer, and cable vibration.

However, support vibration of a stay cable at deck or pylon, which might be induced by wind load or traffic load, and so on, might also lead to large amplitude of cable vibration. The support excitation of a stay cable could be classified into two types. Macdonald (2016) classified support excitation into two types: the first is direct excitation induced by components of end motion transverse to the cable and the second is parametric excitation induced by axial component of end motion. The direction excitation could lead to a traditional resonance vibration, and the parametric excitation could lead to a parametric vibration when the oscillation frequency of the support is twice of one of the natural frequencies of stay cable. There are plenty of literatures about parametric vibration of stay cable (Kovacs, 1982; Lilien and Pinto da Costa, 1994; Macdonald, 2016; Pinto da Costa et al., 1996; Takahashi, 1991).

The supports of stay cables may have significant vibration when RWIV occurs. In this study, the coupled response of a stay cable under the combined rain–wind and support excitations is numerically investigated. First, the equations of motion of a three-dimensional continuous stay cable are derived by considering the high-order nonlinear components of the dynamic cable tension, together with the equation of motion of the rivulet. These equations are numerically solved by using the finite difference method (FDM) and the fourth-order Runge–Kutta method. The cable responses under the combined rain–wind and support excitations are obtained, and the coupled effect of the two excitations is investigated.

Equations of motion of the three-dimensional stay cable

The following assumptions are made to simplify the complicated problem: (1) the effect of the bending stiffness of stay cable is ignored; (2) the constitutive relation obeys Hooke’s law and the stress on the cross section of stay cable is uniform; (3) quasi-steady theory could be used to determine the rain-wind-induced forces on the stay cable and the rivulet; (4) the size and shape of the rivulet do not change when it oscillates on the cable surface; (5) the mass of the rivulet is far less than that of the stay cable; (6) the interaction between the rivulet and the cable surface includes the Coulomb and linear damping forces; (7) the effect of the lower rivulet is ignored; and (8) the effect of the axial flow is ignored.

A three-dimensional continuous stay cable with an inclination angle of α is taken into account in this study, as shown in Figure 1. The higher end point of the stay cable, O, is fixed at the bridge pylon, whereas the lower end point, A, is connected to the bridge deck. l is the total length of the stay cable, and its vertical and horizontal projections are represented by R and L, respectively. Two types of coordinate systems are established. One is a rectangular coordinate system Oxyz of which x, y, and z axes are along the in-plane horizontal, in-plane vertical, and out-of-plane horizontal directions, respectively. The other is a curvilinear coordinate s, which represents the length of the stay cable counted from the higher end point O. According to quasi-steady theory, the forces on the stay cable induced by the rain–wind excitation could be expressed as F_x(x, t), F_y(x, t), and F_z(x, t).

Figure 1.

Diagram of a three-dimensional continuous stay cable.

The equations of motion governing the three-dimensional continuous stay cable in x, y, and z directions can be, respectively, expressed as

\frac{\partial}{\partial s} [(T + τ) (\frac{dx}{ds} + \frac{\partial u}{\partial s})] + F_{x} (x, t) = M \frac{\partial^{2} u}{\partial t^{2}} + c_{1} \frac{\partial u}{\partial t}

(1)

\frac{\partial}{\partial s} [(T + τ) (\frac{dy}{ds} + \frac{\partial v}{\partial s})] + F_{y} (x, t) = M \frac{\partial^{2} v}{\partial t^{2}} + c_{1} \frac{\partial v}{\partial t} - Mg

(2)

\frac{\partial}{\partial s} [(T + τ) \frac{\partial w}{\partial s}] + F_{z} (x, t) = M \frac{\partial^{2} w}{\partial t^{2}} + c_{2} \frac{\partial w}{\partial t}

(3)

where T is the static cable tension under the action of the gravity only; τ is the dynamic cable tension under the actions of the rain-wind-induced and/or support excitations; t is the time; u, v, and w are the dynamic displacement components in the x, y, and z directions, respectively; y(x) is the cable static profile; F_x(x, t), F_y(x, t), and F_z(x, t) are the aerodynamic forces per unit length in the x, y, and z directions, respectively; M is the cable mass per unit length; c₁ and c₂ are the in-plane and out-of-plane linear structural damping coefficients of the stay cable, respectively; and g is the gravitational acceleration.

By using the following relationships

\frac{\partial}{\partial s} = \frac{1}{\sqrt{1 + y_{x}^{2}}} \frac{\partial}{\partial x}

(4)

H = T \frac{dx}{ds}

(5)

h = τ \frac{dx}{ds}

(6)

\frac{d}{ds} (T \frac{dy}{ds}) = - Mg

(7)

Equations (1) to (3) could be rewritten as

\frac{1}{\sqrt{1 + y_{x}^{2}}} \frac{\partial}{\partial x} [(H + h) (1 + \frac{\partial u}{\partial x})] + F_{x} (x, t) = M \frac{\partial^{2} u}{\partial t^{2}} + c_{1} \frac{\partial u}{\partial t}

(8)

\frac{1}{\sqrt{1 + y_{x}^{2}}} \frac{\partial}{\partial x} [(H + h) \frac{\partial v}{\partial x} + h y_{x}] + F_{y} (x, t) = M \frac{\partial^{2} v}{\partial t^{2}} + c_{1} \frac{\partial v}{\partial t}

(9)

\frac{1}{\sqrt{1 + y_{x}^{2}}} \frac{\partial}{\partial x} [(H + h) \frac{\partial w}{\partial x}] + F_{z} (x, t) = M \frac{\partial^{2} w}{\partial t^{2}} + c_{2} \frac{\partial w}{\partial t}

(10)

where H is the horizontal component of the static cable tension under the action of the gravity, which does not change along the cable axis; h is the horizontal component of the dynamic cable tension under the action of the rain-wind-induced and/or support excitations; and y_x is the derivative of cable static profile y(x) with respect to x.

The dynamic cable tension τ could be obtained from

τ = EA \frac{d {s'}^{2} - d s^{2}}{2 d s^{2}}

(11)

where E is the elasticity modulus of the stay cable; A is the area of the stay cable cross section; and ds′ and ds are the arc-lengths of the deformed and undeformed cable element, respectively. One have

d s^{2} = d x^{2} + d y^{2}

(12)

d s'^{2} = (dx + \partial u)^{2} + (dy + \partial v)^{2} + \partial w^{2}

(13)

By using equations (6) and (11) to (13), h could be written as

h = \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} {\frac{\partial u}{\partial x} + y_{x} \frac{\partial v}{\partial x} + \frac{1}{2} [{(\frac{\partial u}{\partial x})}^{2} + {(\frac{\partial v}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}]}

(14)

Linear and high-order nonlinear components are both included in equation (14). Neglecting the high-order nonlinear components, equation (14) can be rewritten as

h = \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} (\frac{\partial u}{\partial x} + y_{x} \frac{\partial v}{\partial x})

(15)

Substituting equation (14) into equations (8) to (10), we have

\frac{1}{\sqrt{1 + y_{x}^{2}}} {\begin{matrix} \frac{\partial}{\partial x} H + (H + \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}}) \frac{\partial u}{\partial x} + \frac{EA y_{x}}{{(1 + y_{x}^{2})}^{3 / 2}} \frac{\partial v}{\partial x} + \frac{EA y_{x}}{{(1 + y_{x}^{2})}^{3 / 2}} \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} \\ + \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} [\frac{3}{2} {(\frac{\partial u}{\partial x})}^{2} + \frac{1}{2} {(\frac{\partial v}{\partial x})}^{2} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}] + \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} [\begin{matrix} \frac{1}{2} {(\frac{\partial u}{\partial x})}^{3} + \frac{1}{2} \frac{\partial u}{\partial x} \\ {(\frac{\partial v}{\partial x})}^{2} + \frac{1}{2} \frac{\partial u}{\partial x} {(\frac{\partial w}{\partial x})}^{2} \end{matrix}] \end{matrix}} + F_{x} (x, t) = M \frac{\partial^{2} u}{\partial t^{2}} + c_{1} \frac{\partial u}{\partial t} (16)

(16)

\frac{1}{\sqrt{1 + y_{x}^{2}}} {\begin{matrix} \frac{\partial}{\partial x} (H + \frac{{EAy}_{x}^{2}}{{(1 + y_{x}^{2})}^{3 / 2}}) \frac{\partial v}{\partial x} + \frac{EA y_{x}}{{(1 + y_{x}^{2})}^{3 / 2}} \frac{\partial u}{\partial x} + \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} \\ + \frac{EA y_{x}}{{(1 + y_{x}^{2})}^{3 / 2}} [\frac{1}{2} {(\frac{\partial u}{\partial x})}^{2} + \frac{3}{2} {(\frac{\partial v}{\partial x})}^{2} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}] + \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} [\begin{matrix} \frac{1}{2} {(\frac{\partial u}{\partial x})}^{2} \frac{\partial v}{\partial x} \\ + \frac{1}{2} {(\frac{\partial v}{\partial x})}^{3} + \frac{1}{2} \frac{\partial v}{\partial x} {(\frac{\partial w}{\partial x})}^{2} \end{matrix}] \end{matrix}} + F_{y} (x, t) = M \frac{\partial^{2} v}{\partial t^{2}} + c_{1} \frac{\partial v}{\partial t} (17)

(17)

\frac{1}{\sqrt{1 + y_{x}^{2}}} \frac{\partial}{\partial x} {\begin{matrix} H \frac{\partial w}{\partial x} + \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} \frac{\partial u}{\partial x} \frac{\partial w}{\partial x} + \frac{EA y_{x}}{{(1 + y_{x}^{2})}^{3 / 2}} \frac{\partial v}{\partial x} \frac{\partial w}{\partial x} + \frac{EA}{{(1 + y_{x}^{2})}^{3 / 2}} \\ [\frac{1}{2} {(\frac{\partial u}{\partial x})}^{2} \frac{\partial w}{\partial x} + \frac{1}{2} {(\frac{\partial v}{\partial x})}^{2} \frac{\partial w}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{3}] \end{matrix}} + F_{z} (x, t) = M \frac{\partial^{2} w}{\partial t^{2}} + c_{2} \frac{\partial w}{\partial t} (18)

(18)

The boundary and initial conditions should be given before equations (16) to (18) are solved. The boundary condition could be given as follows

u (x = 0, t) = φ_{u 0} (t) u (x = L, t) = φ_{uL} (t) t \in (0, T)

(19)

v (x = 0, t) = φ_{v 0} (t) v (x = L, t) = φ_{vL} (t) t \in (0, T)

(20)

w (x = 0, t) = φ_{w 0} (t) w (x = L, t) = φ_{wL} (t) t \in (0, T)

(21)

For the cable-stayed bridges, the pylon is far rigid than the deck. In this study, it is assumed that the cable support on the pylon remains at rest, and there is a simple harmonic vibration at the cable support on the deck

φ_{u 0} (t) = φ_{v 0} (t) = φ_{w 0} (t) = 0 t \in (0, T)

(22)

φ_{uL} (t) = D_{u} \sin Ω_{u} t φ_{vL} (t) = D_{v} \sin Ω_{v} t φ_{wL} (t) = D_{w} \sin Ω_{w} t t \in (0, T)

(23)

in which, D_u, D_v, and D_w are the oscillation amplitude of the cable lower support in the x, y, and z directions, respectively; Ω_u, Ω_u, and Ω_u are the oscillation frequency of the cable lower support in the x, y, and z directions, respectively.

The initial displacement and velocity of the stay cable could be given as

u (x, t = 0) = ϕ_{u} (x) \frac{\partial u (x, t = 0)}{\partial t} = ϕ'_{u} (x) x \in (0, L)

(24)

v (x, t = 0) = ϕ_{v} (x) \frac{\partial v (x, t = 0)}{\partial t} = ϕ'_{v} (x) x \in (0, L)

(25)

w (x, t = 0) = ϕ_{w} (x) \frac{\partial w (x, t = 0)}{\partial t} = ϕ'_{w} (x) x \in (0, L)

(26)

In this study, it is assumed that the stay cable remains at rest at t = 0, that is

ϕ_{u} (x) = ϕ'_{u} (x) = ϕ_{v} (x) = ϕ'_{v} (x) = ϕ_{w} (x) = ϕ'_{w} (x) x \in (0, L)

(27)

The wind forces induced by rain-wind-induced excitation, F_x(x, t), F_y(x, t), and F_z(x, t) in equations (16) to (18), could be determined by using the quasi-steady theory

F_{x} (x, t) = \frac{1}{2} ρ D U^{2} [C_{L} (θ) \cos (θ) + C_{D} (θ) \sin θ] \sin α

(28)

F_{y} (x, t) = - \frac{1}{2} ρ D U^{2} [C_{L} (θ) \cos (θ) + C_{D} (θ) \sin θ] \cos α

(29)

F_{z} (x, t) = \frac{1}{2} ρ D U^{2} [- C_{L} (θ) \sin (θ) + C_{D} (θ) \cos θ]

(30)

where ρ is the air density, D is the cable diameter, U is the approaching wind velocity, θ is the rivulet angle which reflects the position of the rivulet on the cable surface (as shown in Figure 2), α is the inclination angle of the stay cable, and C_L and C_D are the mean lift and drag coefficients of the cable with rivulet, respectively. In this study, the data of C_L and C_D for α = 30º and β = 35º (β is the wind attack angle) experimentally measured by Du et al. (2013) are adopted.

Figure 2.

The forces acting on the rivulet.

Equation of motion of the rivulet

There is a key parameter in equations (16) to (18), the rivulet angle θ, which should be determined from the equation governing the motion of the rivulet. The forces acting on the rivulet is shown in Figure 2, and the equation of motion of the rivulet in the tangential direction could be expressed as

mR \overset{\cdot\cdot}{θ} + F_{0} sign (\overset{\cdot}{θ}) + c_{r} R \overset{\cdot}{θ} = f_{τ} + m \overset{\cdot\cdot}{y} \cos θ - m \overset{\cdot\cdot}{z} \sin θ - mg \cos α \cos θ

(31)

where m is the rivulet mass per unit length, R is the cable radius, F₀ is the Coulomb damping force between the cable surface and the rivulet per unit length, c_r is the linear damping coefficient of the rivulet, θ is the rivulet angle, and f_τ is the wind forces on the rivulet in the tangential direction of the cable cross section. The rivulet is assumed to remain at rest at the initial time, and the initial position of the rivulet is set to be θ(t = 0) = 1 rad (57.3°). The detailed rivulet parameters used in this article are the same as Li et al. (2013).

Equations (16) to (18) and (31) are coupled with each other through θ, y, and z. Equations (16) to (18) are numerically solved for the stay cable by using the FDM, and the equation of the rivulet is solved by the fourth-order Runge–Kutta method.

Numerical example

Stay cable A20 of the Second Nanjing Yangtze River Bridge is taken as the background for the numerical example. The total length of this cable is l = 330.4 m, the static cable tension is T = 6403 kN, the mass per unit length is M = 81.167 kg/m, the cable diameter is D = 0.114 m, the elasticity modulus is E = 1.9 × 10¹¹ N/m², the inclination angle is α = 30°, and the first natural frequency of this cable is f₁ = 0.43 Hz. The structural damping ratios for the first mode in the in-plane and out-of-plane directions, ξ₁, are both 0.1%.

Support excitation

The responses induced by the support excitation of the stay cable could be obtained by letting F_x(x, t) = F_y(x, t) = F_z(x, t) = 0. Figure 3(a) presents the time history of the in-plane displacement at the mid-point of the stay cable when D_u = D_w = 0, D_v = 0.03 m, Ω_v = f₁, and ξ₁ = 0. This means that a vertical vibration is given at the lower end point of the stay cable and the oscillation frequency at the end point is equal to the first natural frequency of the stay cable. Figure 3(b) shows the power spectrum density (PSD) of the time history of the in-plane displacement at the quarter-point of the stay cable. It can be found from Figure 3 that maximum cable response amplitude is about 0.7 m, and the cable vibration at this moment is purely resonance response. However, when the oscillation frequency at the cable endpoint is equal to the second natural frequency of the stay cable (Ω_v = f₂), the maximum amplitude of the beating oscillation of the stay cable becomes about 1.2 m, as shown in Figure 4(a). The PSD of the in-plane displacement at the quarter-point, as given in Figure 4(a), indicates that there are two frequency components (f₁ and f₂) in the cable vibration. The component of the first mode shape of the stay cable is induced by parametric vibration and the second mode shape by resonance vibration. The results given in Figures 3 and 4 demonstrate that the features of the vibration induced by support excitation could be well simulated by the present numerical method.

Figure 3.

Time history and PSD of the cable displacement for D_u = D_w = 0, D_v = 0.03 m, Ω_v = f₁, and ξ₁ = 0. (a) Time history of the displacement at mid-point and (b) PSD of time history of the displacement at quarter-point.

Figure 4.

Time history and PSD of the cable displacement for D_u = D_w = 0, D_v = 0.03 m, Ω_v = f₂, and ξ₁ = 0. (a) Time history of the displacement at mid-point and (b) PSD of time history of the displacement at quarter-point.

It should be emphasized that the parametric vibration could not be reproduced if the high-order nonlinear components in equation (14) are ignored. For example, no large amplitude of cable vibration at the mid-point of the stay cable for Ω_v = f₂ is found if the high-order nonlinear components of the dynamic cable tension are ignored, as shown in Figure 5(a). At this moment, only the peak at the second natural frequency, which is induced by resonance vibration, is found in the PSD of the displacement at the quarter-point of the stay cable, as shown in Figure 5(a). This indicates that the high-order nonlinear components of the dynamic cable tension should be taken into account to accurately simulate the parametric vibration because parametric vibration of stay cable is a nonlinear phenomenon in nature. It appears that the high-order nonlinear components have little effects on the resonance responses because the responses of the stay cable nearly have no change when they are ignored for Ω_v = f₁.

Figure 5.

Cable responses as the high-order components are ignored (D_u = D_w = 0, D_v = 0.03 m, Ω = f₂, and ξ₁ = 0). (a) Time history of the displacement at mid-point and (b) PSD of time history of the displacement at quarter-point.

Figures 6 and 7 present the responses of the stay cable for D_u = D_w = 0, D_v = 0.03 m, and Ω_v = f₁ and f₂ when the structural damping ratio of the stay cable ξ₁ is 0.1%. It could be found from Figures 6 and 7 that the vibrations of the stay cable have no beating feature when the structural damping is taken into account. The stable resonance vibration for the support excitation frequency Ω_v = f₁ has a maximum amplitude of 0.35 m. The shapes of the stay cable at several typical time instants show that only the first mode shape of the stay cable is evoked, as indicated in Figure 6(a). The stable amplitudes at the mid-point and the quarter-point of the stay cable are about 1.1 and 0.85 m, respectively, when the support excitation frequency is Ω_v = f₂. The oscillation amplitude at the quarter-point is about 0.5 m at the start time, whereas the amplitude at the mid-point time is almost zero within the time t = 0–1000 s. The cable shape at t = 1000 s, as shown in Figure 7(a), is like the second mode shape of the cable. These indicate that the parameter vibration is evoked before the resonance vibration. It is interesting to find that the stable lower amplitude (minus value in Figure 7(a)) at the quarter-point is larger than the upper amplitude (positive value in Figure 7(a)), which should be caused by the nonlinear feature of the parameter vibration.

Figure 6.

Time history and PSD of the cable displacement at mid-point for D_u = D_w = 0, D_v = 0.03 m, Ω_v = f₁, and ξ₁ = 0.1%. (a) Time history of the displacement at mid-point, (b) PSD of time history of the displacement at mid-point, and (c) cable shapes for several typical time instants.

Figure 7.

Time history and PSD of the cable displacement at mid- and quarter-points for D_u = D_w = 0, D_v = 0.03 m, Ω_v = f₂, and ξ₁ = 0.1%. (a) Time history of the displacement at mid-point, (b) PSD of time history of the displacement at mid-point, (c) time history of the displacement at quarter-point, (d) PSD of time history of the displacement at quarter-point, and (e) cable shapes for several typical times.

Rain–wind excitation

The RWIV of stay cable could be obtained through letting D_u = D_v = D_w = 0. First, the effects of the high-order components of the dynamic cable tension on the RWIVs of stay cables are evaluated. The structural damping ratio of the stay cable of 0.1% is adopted. Figure 8 presents the rain-wind-induced in-plane and out-of-plane displacement at the mid-point of the stay cable for wind velocity at the deck level U_d = 7.45 m/s when the high-order components are ignored. Figure 9 shows the results when the high-order components are taken into account. Compare Figures 8 and 9, it can be found that the high-order components hardly have any effect on the rain-wind-induced responses of the stay cable. This implies that the geometric nonlinearity of the stay cable is rather weak because of the small cable sag (<1%).

Figure 8.

Rain-wind-induced displacement at the mid-point by ignoring high-order components (U_d = 7.45 m/s): (a) out-of-plane and (b) in-plane.

Figure 9.

Rain-wind-induced displacement at the mid-point by considering high-order components (U_d = 7.45 m/s): (a) out-of-plane and (b) in-plane.

Figure 10 shows the variations of the rain-wind-induced in-plane and out-of-plane response amplitudes at the mid-point of the stay cable with the wind velocity at the deck level. The structural damping ratio of 0.1% is used. It can be found from Figure 10 that the RWIVs of the stay cable takes place within the wind velocity range of U_d = 6.3–8.0 m/s, which also agrees well with the numerical results of Gu (2009) by ignoring the effects of high-order components. It appears that the in-plane vibration amplitude is considerably larger than the out-of-plane amplitude. The largest in-plane displacement, which is nearly 0.16 m, occurs at the wind velocity U_d = 7.04 m/s. Furthermore, it should be noted that the cable amplitude evoked by support excitation is significantly higher than that of RWIV.

Figure 10.

Rain-wind-induced in-plane and out-of-plane amplitudes at the mid-point of the stay cable.

Figure 11 presents the distribution of the rivulet balance position (time-average value of θ) along the cable axis at four typical time (t = 300, 500, 800, and 1000 s) under wind velocity U_d = 6.35, 7.04, and 9.94 m/s. It should be noted that the initial position of the rivulet is θ = 57.3° (1 rad). The two solid red horizontal lines in Figure 11 define the sudden decrease region of the mean lift coefficient of the cable attached with the rivulet, which is measured from wind tunnel tests (Du et al., 2013). The region between the two red lines is called as “dangerous region.” The numerical results show that the rivulet will fall down from the cable surface if the wind velocity at the deck level U_d < 6.30 m/s. This is because the wind force on the rivulet, which pushes the rivulet upward on the cable surface, is smaller than the gravity component of the rivulet, which pulls the rivulet downward. For U_d = 6.35 m/s, as given in Figure 11, the rivulet keeps stable oscillation across the dangerous region on almost a half upper part of the stay cable. At this moment, large amplitude of RWIV of the stay cable starts to be evoked. When the wind velocity U_d increases to be 7.04 m/s, the rivulet on almost the entire cable can keep stable oscillation across the dangerous region, which leads to the maximum cable response amplitude. When the wind velocity U_d increases to over 8.0 m/s, the rivulet will remain at rest at the rivulet angle around θ = 70°, which is out of the dangerous region. Accordingly, no large amplitude cable vibration is observed.

Figure 11.

Rivulet balance position along the cable span for typical wind velocity for different stages: (a) t = 300 s, (b) t = 500 s, (c) t = 800 s, and (d) t = 1000 s.

It should be noted that the rivulet amplitude becomes larger when the rain-wind-induced cable vibration occurs. The rivulet is assumed to fall down from the cable surface when the rivulet angle θ is smaller than 0° or larger than 90°. In the computation code of this study, the aerodynamic forces of smooth cable are used if the rivulet falls down from the cable surface. Therefore, the cable amplitude becomes smaller after reaching its maximum amplitude, as shown in Figure 9. However, the water will be continuously supplemented from the rainfall in the real environment of RWIV of stay cable, so the exact responses of RWIV of stay cable will not cease as shown in Figure 9.

Combined rain–wind and support excitations

Figure 12 presents the responses of the stay cable at the mid-point and quarter-point under the combined action of rain–wind and support excitations, together with the rivulet angle θ at the mid-point and quarter-point of the stay cable. The end excitations are D_u = D_w = 0, D_v = 0.03 m, and Ω_v = f₂; the structural damping ratios for the first and second modes (ξ₁ and ξ₂) are both 0.1% and wind velocity of the approaching flow is 7.04 m/s. The results in Figure 7 show that the responses induced by end excitation for Ω_v = f₂ include both parametric and resonance vibrations. It can be found from Figure 12 that the stable in-plane amplitudes at the mid-point and quarter-point are 0.75 and 0.72 m, respectively, which is slightly smaller than those under the support excitation only case (1.1 and 0.8 m) and significantly larger than those under the rain–wind action only case (0.155 and 0.06 m for the in-plane and out-of-plane, respectively). Because of the large response amplitude of the stay cable, it appears that the rivulet is threw away from the cable surface due to the additional inertial forces (the second and third components on the right-hand side of equation (31)) acting on the rivulet induced by the cable vibration, as shown in Figure 12(a) and (f). Therefore, the maximum stable cable vibration shown in Figure 12 takes place actually without the rivulet on the cable surface. Furthermore, it can be found from Figure 12 that the out-of-plane amplitude of the stay cable is much smaller than the in-plane amplitude.

Figure 12.

Responses of the cable and the rivulet under the combined actions of rain–wind and end excitations (D_u = D_w = 0, D_v = 0.03 m, Ω_v = f₂, ξ₁ = ξ₂ = 0.1%, and U_d = 7.04 m/s): (a) in-plane cable displacement at the mid-point, (b) out-of-plane cable displacement at the mid-point, (c) in-plane cable displacement at the quarter-point, (d) out-of-plane cable displacement at the quarter-point, (e) rivulet position at the mid-point, and (f) rivulet position at the quarter-point.

Figure 13 presents the responses of the stay cable at the mid-point and quarter-point under the combined action of rain–wind and support excitations, together with the rivulet angle θ at the mid-point and quarter-point of the stay cable. The frequency of the support excitation is Ω_v = f₁, and other parameters are the same as in Figure 12. The results in Figure 6 show that only the resonance vibration is evoked when Ω_v = f₁. It is found from Figure 13 that the stable in-plane cable amplitudes at the mid-point and the quarter-point are 0.31 and 0.23 m, respectively, slightly smaller than the response under the support excitation only case shown in Figure 6 (0.35 m at the mid-point) and larger than those under the rain–wind excitation only case. The time histories of the displacement of the rivulet, as shown in Figure 13(a) and (f), indicate that the rivulet at the mid-point is threw out when the cable amplitude increases, whereas the rivulet at the quarter-point always stay on the cable surface. Again, it seems that the out-of-plane cable amplitude is much smaller than the in-plane value.

Figure 13.

Responses of the cable and the rivulet under the combined actions of rain–wind and end excitations (D_u = D_w = 0, D_v = 0.03 m, Ω_v = f₁, ξ₁ = ξ₂ = 0.1%, and U_d = 7.04 m/s): (a) in-plane cable displacement at the mid-point, (b) out-of-plane cable displacement at the mid-point, (c) in-plane cable displacement at the quarter-point, (d) out-of-plane cable displacement at the quarter-point, (e) rivulet position at the mid-point, and (f) rivulet position at the quarter-point.

Conclusion

The responses of the stay cable under the combined action of rain–wind and support excitations are analytically investigated based on a three-dimensional continuous stay cable in this study. The equations of motion of the three-dimensional continuous stay cable are derived by considering the high-order nonlinear components in the dynamic cable tension. Rain-wind-induced forces are determined by using the quasi-steady theory, and the support excitation is achieved through the boundary condition. In order to take the coupled effect between the cable and the rivulet into account, the equation of motion of the rivulet in Li et al. (2013) is adopted. Then, the FDM and the fourth-order Runge–Kutta method are used to solve the cable equations and the rivulet equation, respectively. Stay cable A20 of the Second Nanjing Yangtze River Bridge is used as the background for the numerical example. Results show that the high-order nonlinear component of the dynamic cable tension should be taken into account to numerically reproduce the parametric vibration of the stay cable. However, they hardly have any effects on the RWIV and the resonance vibration. The responses of stay cable under only vertical oscillation at the lower end point of the stay cable are obtained, and the results show that the cable amplitude for Ω_v = f₂, which includes the components of parametric and resonance vibration, is larger than that for Ω_v = f₁, which only includes the resonance vibration component. The response under only the rain-wind-induced excitation successfully reproduces the main features of RWIV. It seems that the cable response amplitude induced by support excitation is considerably larger than that induced by rain–wind excitation. Finally, the responses of the stay cable under the combined rain-wind-induced and support excitations are investigated. Compared with the results induced by a single excitation, the cable response amplitude under the combined excitations is smaller than that induced by support excitation only and larger than that induced by only rain-wind-induced excitation. It is found that the rivulet will fall down from the cable surface if the cable amplitude reaches about 0.3 m.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was jointly supported by the National Natural Science Foundation of China (51578234) and the National Basic Research Program of China (973 Program: 2015CB057700 and 2015CB057702), which the authors gratefully appreciate.

ORCID iD

Shouying Li

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