Dynamic analysis of a cable-stayed bridge subjected to a continuous sequence of moving forces

Abstract

In this work, an eigenfunction expansion approach is used to study the dynamic response of a cable-stayed bridge excited by a continuous sequence of identical, equally spaced moving forces. The nonlinear dynamic response of the cable-stayed bridge is obtained by simultaneously solving nonlinear and linear partial differential equations that govern transverse and longitudinal vibrations of stay cables and transverse vibrations of segments of the deck beam, respectively, along with their boundary and matching conditions. Orthogonality conditions of exact mode shapes of the linearized cable-stayed bridge model are employed to convert the coupled nonlinear partial differential equations of the original nonlinear model to a set of ordinary differential equations by using the Galerkin method. The dynamic response of the cable-stayed bridge is numerically solved. Convergence of the dynamic response from the Galerkin method is investigated. Effects of close natural frequencies, mode localization, the distance between any two neighboring forces, and geometric nonlinearities of stay cables on the forced dynamic response of the cable-stayed bridge are captured using a convergent modal truncation.

Keywords

Cable-stayed bridge continuous sequence of moving forces Galerkin method dynamic analysis convergence

Introduction

Dynamic vehicle–bridge interaction (VBI) problems have attracted much attention of investigators for over a century.^1–4 Researchers focused on the development of models of vehicle–bridge systems and computational methods, which were mainly dependent on the power of computers at that time. Most earlier studies associated with VBI problems focused on very basic and simple cases, where moving force or moving mass models were considered with basic beam models.^1–4 The basic beam models were described by partial differential equations, and their analytical or closed-form solutions are available.

In the past decades, cable-stayed bridges have been widely used all over the world due to their economic advantages and aesthetic qualities. Many studies have focused on VBI problems of cable-stayed bridges since this type of bridges can exhibit interesting dynamic behaviors, such as energy transfer between different vibration modes and the beating phenomenon.⁵ With the advent of high-speed computers, computationally efficient numerical techniques have been developed to model and analyze VBI problems of complex cable-stayed bridges. The finite element method is one of the most versatile numerical methods used by most researchers. However, the finite element method is time-consuming and inconvenient for parameter studies due to large numbers of elements and amount of input data. Hence, the modal superposition method was used to study dynamic behaviors of VBI problems formulated by finite element models, which truncated higher vibration modes and considerably reduced numbers of equations.^4,6–9 For instance, Madrazo-Aguirre et al.⁹ employed this method to conduct dynamic analysis of an under-deck cable-stayed bridge with steel-concrete composite decks subjected to a moving load. On the other hand, due to much computational time and model complexity, new strategies such as a substructure method were developed to model complex bridge structures.^10–14

Close natural frequencies and mode localization often occur in complex structures that are composed of several components. Pierre¹⁵ found that mode shapes are strongly localized in nearly periodic structures with weak coupling. Wei and Pierre^16,17 found the same phenomenon in shrouded blade assemblies with cyclic symmetry. Natsiavas¹⁸ used a perturbation method to investigate the mode localization phenomenon in a 2 degree-of-freedom damped linear oscillator. Further investigations of similar systems were performed in Balmes.¹⁹ Cao et al.²⁰ studied the free vibration of a cable-stayed bridge that consists of a simply supported four-cable-stayed deck beam and two towers for various symmetrical and non-symmetrical bridge cases with regard to sizes of components of the bridge and initial sags of cables and found that there are very close natural frequencies when the bridge model is symmetrical and/or partially symmetrical, and mode shapes tend to be more localized when the bridge model is less symmetrical.

As mentioned by Wei and Pierre,¹⁷ neglecting the localization phenomenon caused by small mistuning may lead to qualitative errors in the forced response of a periodic structure. However, to the authors’ best knowledge, effects of closed natural frequencies and mode localization on the vehicle-induced vibration of the cable-stayed bridge have not been investigated. Dynamic analysis of a beam excited by a sequence of moving mass loads has attracted many researchers^21–24 since it has many practical applications such as a train traveling on a railroad track.

In this work, an eigenfunction expansion approach is used to study the dynamic response of a cable-stayed bridge subjected to a continuous sequence of identical, equally spaced vehicles traveling at a constant speed across the bridge deck. For simplicity, the vehicles are modeled as moving forces. Geometric nonlinearities of stay cables are taken into consideration. Nonlinear and linear partial differential equations that govern transverse and longitudinal vibrations of the stay cables and transverse vibrations of segments of the deck beam, respectively, are derived, along with their boundary and matching conditions. The exact natural frequencies and mode shapes of the linearized undamped cable-stayed bridge model are used to spatially discretize partial differential equations of the original nonlinear cable-stayed bridge model with damping. The dynamic response of the cable-stayed bridge is obtained by solving the resulting nonlinear ordinary differential equations (ODEs) via a numerical method. Convergence of the Galerkin method for the dynamic response of the cable-stayed bridge subjected to a continuous sequence of moving forces is investigated. Effects of closed natural frequencies, mode localization, the distance between any two neighboring forces, and geometric nonlinearities of stay cables on the forced dynamic response of the cable-stayed bridge are captured using a convergent modal truncation.

Problem formulation

As shown in Figure 1, a cable-stayed bridge consists of a simply supported four-cable-stayed deck beam and two towers subjected to a continuous sequence of moving concentrated forces. The deck beam consists of seven segments separated by its junctions with the stay cables and towers. The following assumptions are made in the formulation of the vibration problem of the cable-stayed bridge model subjected to a continuous sequence of moving forces: (1) the cable-stayed bridge is modeled as a planar system. (2) Both the cables and the deck beam are considered as a homogeneous one-dimensional continuum having linear elastic behaviors. (3) The two towers, to which the stay cables are attached, are built on a hard rock foundation and can be assumed to be rigid;^20,25 they are connected to the deck beam through roller supports. (4) The concentrated forces are equally spaced and have the same amplitude, moving from the left end of the deck beam to its right end; the initial displacement and velocity of the stay cables and deck beam are all zeros.

Figure 1.

Schematic of a cable-stayed bridge that consists of a simply supported four-cable-stayed deck beam and two rigid towers, subjected to a continuous sequence of identical, equally spaced moving forces.

The four stay cables are anchored to the deck beam at junctions S₁, S₃, S₄, and S₆, and the two towers are connected to the deck beam at junctions S₂ and S₅. The junctions S₁, S₂, …, S₆ divide the deck beam into seven segments b₁, b₂, …, b₇. The length, mass per unit length, elastic modulus, and cross-sectional area of the stay cable $c_{i}$ are denoted by $L_{c_{i}}$ , $m_{c_{i}}$ , $E_{c_{i}}$ , and $A_{c_{i}}$ , respectively. The length, mass per unit length, elastic modulus, and area moment of inertia of the segment b_j of the deck beam are denoted by $2 L_{b_{j}}$ , $m_{b_{j}}$ , $E_{b_{j}}$ , and $I_{b_{j}}$ , respectively.

Let $(X_{c_{i}}, Y_{c_{i}})$ be local coordinates of the stay cable c_i in the vertical plane, with the origin located at the point C for the stay cables c₁ and c₂ and at the point D for the stay cables c₃ and c₄. Let $(X_{b_{j}}, Y_{b_{j}})$ be local coordinates of the segment b_j of the deck beam in the vertical plane, with the origin located in the middle of the segment b_j of the deck beam. Under the assumption of a small ratio of the sag $D_{c_{i}}$ to the length $L_{c_{i}}$ , the static equilibrium of the stay cable c_i can be described through a parabolic function $Y_{c_{i}} (X_{c_{i}}) = 4 D_{c_{i}} [X_{c_{i}} / L_{c_{i}} - (X_{c_{i}} / L_{c_{i}})^{2}]$ in its domain, while the static deflection of the deck beam is assumed to be negligible. With respect to the above equilibrium configuration, the dynamic configuration of the cable-stayed bridge model is completely described by longitudinal and transverse displacements of the stay cables $U_{c_{i}} (X_{c_{i}}, t)$ and $V_{c_{i}} (X_{c_{i}}, t)$ , respectively, and transverse displacements of the segments of the deck beam $V_{b_{j}} (X_{b_{j}}, t)$ .

The following nondimensional variables are introduced

{\begin{matrix} x_{c_{i}} = \frac{X_{c_{i}}}{L}, x_{b_{j}} = \frac{X_{b_{j}}}{L}, l_{c_{i}} = \frac{L_{c_{i}}}{L}, l_{b_{j}} = \frac{L_{b_{j}}}{L}, y_{c_{i}} = \frac{Y_{c_{i}}}{Δ_{st}} \\ u_{c_{i}} = \frac{U_{c_{i}}}{Δ_{st}}, v_{c_{i}} = \frac{V_{c_{i}}}{Δ_{st}}, v_{b_{j}} = \frac{V_{b_{j}}}{Δ_{st}}, d_{c_{i}} = \frac{D_{c_{i}}}{Δ_{st}}, τ = ε t \end{matrix}

(1)

where $L = min {L_{b_{1}}, L_{b_{2}}, \dots, L_{b_{7}}}$ , $ε = (1 / L_{b_{1}}^{2}) \sqrt{(E_{b_{1}} I_{b_{1}}) / m_{b_{1}}}$ , and $Δ_{st} = (5 F_{1} L_{b_{1}}^{3}) / (48 E_{b_{1}} I_{b_{1}})$ . Moreover, the following nondimensional parameters need to be introduced to furnish a complete definition of elastodynamic properties of the cable-stayed bridge model

μ_{c_{i}} = \frac{E_{c_{i}} A_{c_{i}}}{H_{c_{i}}}, η_{b_{j}} = \frac{E_{b_{j}} I_{b_{j}}}{E_{b_{1}} I_{b_{1}}}, χ_{c_{i}} = \frac{L^{2} H_{c_{i}}}{E_{b_{1}} I_{b_{1}}}, κ = \frac{Δ_{st}}{L}

(2)

where $H_{c_{i}}$ is the tension in the stay cable c_i on which its initial sag is dependent, that is, $D_{c_{i}} = (m_{c_{i}} {gL}_{c_{i}}^{2} \cos θ_{i}) / (8 H_{c_{i}})$ , in which g is the acceleration of gravity. Transverse damping per unit length of the stay cable $c_{i}$ and segment $b_{j}$ of the deck beam are denoted by $C_{c_{i}}$ and $C_{b_{j}}$ , respectively, and their nondimensional parameters are defined by

ξ_{c_{i}} = \frac{C_{c_{i}} ε L^{2}}{H_{c_{i}}}, ξ_{b_{j}} = \frac{C_{b_{j}} ε L^{4}}{E_{b_{j}} I_{b_{j}}}

(3)

respectively.

The Newtonian method is used to derive both the nonlinear equations of motion of the cable-stayed bridge model and a full set of boundary and matching conditions. Assuming that cable longitudinal inertial forces $(m_{c_{i}} {\overset{\cdot\cdot}{U}}_{c_{i}})$ are negligible in the prevalent low-frequency transverse vibration of the cable-stayed bridge, the longitudinal cable displacement $U_{c_{i}}$ can be statically condensed, leading to coupled nonlinear dynamic equations in terms of only the transverse cable and deck beam displacements $V_{c_{i}}$ and $V_{b_{j}}$ , respectively. The equations of motion governing the transverse vibration of the cable-stayed bridge model read²⁰

β_{c_{i}}^{2} {\overset{\cdot\cdot}{v}}_{c_{i}} + ξ_{c_{i}} {\overset{\cdot}{v}}_{c_{i}} - v_{c i}^{″} - μ_{c_{i}} e_{c_{i}} (τ) (v_{c i}^{″} + y_{c i}^{″}) = 0, x_{c_{i}} \in [0, l_{c_{i}}]

(4)

β_{b_{j}}^{4} {\overset{\cdot\cdot}{v}}_{b_{j}} + ξ_{b_{j}} {\overset{\cdot}{v}}_{b_{j}} + v_{b j}^{″ ″} = \sum_{α = 1}^{\infty} f_{b_{j α}} (τ) δ (x_{b_{j}} - {\bar{x}}_{b_{j α}}), x_{b_{j}} \in [- l_{b_{j}}, l_{b_{j}}]

(5)

where a prime and dot denote differentiation with respect to nondimensional local abscissae $x_{c_{i}}$ and $x_{b_{j}}$ and the time $τ$ , respectively; $δ$ is Dirac delta function; $e_{c_{i}} (τ)$ represents the uniform dynamic elongation given by

e_{c_{i}} (τ) = \frac{κ}{l_{c_{i}}} v_{c_{i}} (l_{c_{i}}, τ) \tan θ_{i} + \frac{κ^{2}}{l_{c_{i}}} \int_{0}^{l_{c_{i}}} (y'_{c_{i}} v'_{c_{i}} + \frac{1}{2} v'_{c_{i}}^{2}) d x_{c_{i}}

(6)

\begin{array}{l} β_{c_{i}} = L ε {(\frac{m_{c_{i}}}{H_{c_{i}}})}^{1 / 2}, β_{b_{j}} = L {(\frac{m_{b_{j}} ε^{2}}{E_{b_{j}} I_{b_{j}}})}^{1 / 4}, \\ f_{b_{j α}} = \frac{L^{3} F_{α}}{E_{b_{j}} I_{b_{j}} Δ_{s t}}, {\bar{x}}_{b_{j α}} = {\bar{X}}_{b_{j α}} / L \end{array}

(7)

in which

{\bar{X}}_{b_{j α}} = V_{F} t - (α - 1) L_{F} - 2 \sum_{\bar{j} = 1}^{j} L_{b_{\bar{j}}} + L_{b_{j}}, j = 1, 2, \dots, 7

(8)

The functions $v_{c_{i}}$ and $v_{b_{j}}$ satisfy the following geometric boundary conditions for any value of $τ$

A : v_{b_{1}} (- l_{b_{1}}) = 0, v_{b 1}^{″} (- l_{b_{1}}) = 0

(9)

B : v_{b_{7}} (l_{b_{7}}) = 0, v_{b 7}^{″} (l_{b_{7}}) = 0

(10)

C : v_{c_{1}} (0) = 0, v_{c_{2}} (0) = 0

(11)

D : v_{c_{3}} (0) = 0, v_{c_{4}} (0) = 0

(12)

The matching conditions at the junction $S_{k}$ , where $k = 1, 3, 4, 6$ , which involve the stay cable $c_{i}$ , are

v_{b_{k}} (l_{b_{k}}) = v_{b_{k + 1}} (- l_{b_{k + 1}})

(13)

v'_{b_{k}} (l_{b_{k}}) = v'_{b_{k + 1}} (- l_{b_{k + 1}})

(14)

η_{k} v_{b k}^{″} (l_{b_{k}}) = η_{k + 1} v_{b k + 1}^{″} (- l_{b_{k + 1}})

(15)

\begin{matrix} η_{b_{k}} v ″'_{b_{k}} (l_{b_{k}}) - η_{b_{k + 1}} v ″'_{b_{k + 1}} (- l_{b_{k + 1}}) = \frac{χ_{c_{i}} μ_{c_{i}}}{κ} e_{c_{i}} \sin θ_{i} \\ + χ_{c_{i}} [v'_{c_{i}} (l_{c_{i}}) + μ_{c_{i}} e_{c_{i}} (τ) (v'_{c_{i}} (l_{c_{i}}) + y'_{c_{i}} (l_{c_{i}}))] \cos θ_{i} \end{matrix}

(16)

The matching conditions at the junction $S_{k}$ , where $k = 2, 5$ , with a roller support are

v_{b_{k}} (l_{b_{k}}) = 0, v_{b_{k + 1}} (- l_{b_{k + 1}}) = 0

(17)

v'_{b_{k}} (l_{b_{k}}) = v'_{b_{k + 1}} (- l_{b_{k + 1}})

(18)

η_{b_{k}} v_{b k}^{″} (l_{b_{k}}) = η_{b_{k + 1}} v_{b k + 1}^{″} (- l_{b_{k + 1}})

(19)

Equations (4) and (5) together with the boundary and matching conditions in equations (9)–(19) describe the nonlinear forced vibration of the cable-stayed bridge. The equations governing the small amplitude free vibration of the cable-stayed bridge can be obtained by linearizing equations (4)–(19) and neglecting the structural damping and external loadings in the neighborhood of the equilibrium configuration. The rth undamped natural frequency $ω_{r}$ and its corresponding eigenfunction of the linearized cable-stayed bridge model can be numerically solved using the methodology in Cao et al.²⁰ The rth undamped eigenfunction of the transverse vibration of the linearized cable-stayed bridge model can be expressed by $ϕ^{(r)} = {[ϕ_{c_{1}}^{(r)} \dots ϕ_{c_{4}}^{(r)} ϕ_{b_{1}}^{(r)} \dots ϕ_{b_{7}}^{(r)}]}^{T}$ , where $ϕ_{c_{i}}^{(r)}$ and $ϕ_{b_{j}}^{(r)}$ are the rth undamped eigenfunction of the transverse vibration of the stay cable c_i and the segment b_j of the deck beam, respectively.

Solution method

The Galerkin method is used to analyze the vibration of the cable-stayed bridge. The dynamic response of the stay cable $c_{i}$ and segment $b_{j}$ of deck beam are expressed by

v_{c_{i}} (τ, x_{c_{i}}) = \sum_{r = 1}^{N} q_{r} (τ) ϕ_{c_{i}}^{(r)} (x_{c_{i}})

(20)

v_{b_{j}} (τ, x_{b_{j}}) = \sum_{r = 1}^{N} q_{r} (τ) ϕ_{b_{j}}^{(r)} (x_{b_{j}})

(21)

where $q_{r} (τ)$ are generalized coordinates. Substituting equation (20) into equation (4) yields

\begin{matrix} β_{c_{i}}^{2} (\sum_{r = 1}^{N} {\overset{\cdot\cdot}{q}}_{r} ϕ_{c_{i}}^{(r)}) + ξ_{c_{i}} (\sum_{r = 1}^{N} {\overset{\cdot}{q}}_{r} ϕ_{c_{i}}^{(r)}) - (\sum_{r = 1}^{N} q_{r} {ϕ_{c_{i}}^{(r)}}^{''}) \\ - \frac{μ_{c_{i}} κ}{l_{c_{i}}} [\tan θ_{i} (\sum_{r = 1}^{N} q_{r} ϕ_{c_{i}}^{(r)} (l_{c_{i}})) + κ \int_{0}^{l_{c_{i}}} y'_{c_{i}} (\sum_{r = 1}^{N} q_{r} {ϕ_{c_{i}}^{(r)}}^{'}) d x_{c_{i}} + \frac{κ}{2} \int_{0}^{l_{c_{i}}} {(\sum_{r = 1}^{N} q_{r} {ϕ_{c_{i}}^{(r)}}^{'})}^{2} d x_{c_{i}}] \\ \times [(\sum_{r = 1}^{N} q_{r} {ϕ_{c_{i}}^{(r)}}^{''}) + y_{c i}^{″}] = 0 \end{matrix}

(22)

Substituting equation (21) into equation (5) and recalling equations (1) and (7)–(8), one has

\begin{array}{l} β_{b_{j}}^{4} (\sum_{r = 1}^{N} {\overset{\cdot\cdot}{q}}_{r} ϕ_{b_{j}}^{(r)}) + ξ_{b_{j}} (\sum_{r = 1}^{N} {\dot{q}}_{r} ϕ_{b_{j}}^{(r)}) + (\sum_{r = 1}^{N} q_{r} ϕ_{b_{j}}^{(r) ″ ″}) \\ = \sum_{α = 1}^{\infty} f_{b_{j α}} δ (x_{b_{j}} - {\bar{x}}_{b_{j α}}) \end{array}

(23)

Multiplying equation (22) by $χ_{c_{i}} ϕ_{c_{i}}^{(s)} (x_{c_{i}})$ and integrating the resulting equation with respect to $x_{c_{i}}$ from 0 to $l_{c_{i}}$ , multiplying equation (23) by $η_{b_{j}} ϕ_{b_{j}}^{(s)} (x_{b_{j}})$ and integrating the resulting equation with respect to $x_{b_{j}}$ from $- l_{b_{j}}$ to $l_{b_{j}}$ , adding all the resulting equations, and using the following orthogonality relations of eigenfunctions of the linearized cable-stayed bridge model²⁰

\begin{array}{l} \sum_{i = 1}^{4} χ_{c_{i}} β_{c_{i}}^{2} \int_{0}^{l_{c_{i}}} ϕ_{c_{i}}^{(r)} (x_{c_{i}}) ϕ_{c_{i}}^{(s)} (x_{c_{i}}) d x_{c_{i}} \\ + \sum_{j = 1}^{7} η_{b_{j}} β_{b_{j}}^{4} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} (x_{b_{j}}) ϕ_{b_{j}}^{(s)} (x_{b_{j}}) d x_{b_{j}} = M_{s} δ_{s r} \end{array}

(24)

\begin{matrix} - \sum_{i = 1}^{4} χ_{c_{i}} [\int_{0}^{l_{c_{i}}} ϕ_{c_{i}}^{(r)} (x_{c_{i}}) {ϕ_{c_{i}}^{(s)}}^{''} (x_{c_{i}}) d x_{c_{i}} - 8 μ_{c_{i}} {\hat{e}}_{c_{i}}^{(s)} \frac{d_{c_{i}}}{l_{c_{i}}^{2}} \int_{0}^{l_{c_{i}}} ϕ_{c_{i}}^{(r)} (x_{c_{i}}) d x_{c_{i}}] \\ + \sum_{j = 1}^{7} η_{b_{j}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} (x_{b_{j}}) {ϕ_{b_{j}}^{(s)}}^{''''} (x_{b_{j}}) d x_{b_{j}} = ω_{s}^{2} M_{s} δ_{sr} \end{matrix}

(25)

where ${\hat{e}}_{c_{i}}^{(s)} = κ \frac{ϕ_{c_{i}}^{(s)} (l_{c_{i}})}{l_{c_{i}}} \tan θ_{i} + κ^{2} \frac{1}{l_{c_{i}}} \int_{0}^{l_{c_{i}}} y'_{c_{i}} ϕ'_{c_{i}}^{(s)} d x_{c_{i}}$ , $M_{s}$ are positive constants, and $δ_{sr}$ is Kronecker delta, one obtains spatially discretized equations of the cable-stayed bridge

[M] \overset{\cdot\cdot}{q} (τ) + [C] \overset{\cdot}{q} (τ) + [K] q (τ) + N^{Q} (τ) + N^{C} (τ) = F^{MF} (τ)

(26)

where $q (τ) = {[q_{1} (τ), q_{2} (τ), \dots, q_{N} (τ)]}^{T}$ ; $[M]$ , $[C]$ , and $[K]$ are $N \times N$ real symmetric mass, damping, and stiffness matrices, respectively; and $N^{Q}$ and $N^{C}$ are N-dimensional vectors whose entries are quadratic and cubic nonlinear terms, respectively; and $F^{MF}$ is an N-dimensional modal force vector. Entries of the matrices $[M]$ , $[C]$ , and $[K]$ , and the vectors $N^{Q}$ , $N^{C}$ , and $F^{MF}$ are

M_{rs} = M_{s} δ_{sr}

(27)

C_{rs} = \sum_{i = 1}^{4} χ_{c_{i}} ξ_{c_{i}} \int_{0}^{l_{c_{i}}} ϕ_{c_{i}}^{(s)} ϕ_{c_{i}}^{(r)} d x_{c_{i}} + \sum_{j = 1}^{7} η_{b_{j}} ξ_{b_{j}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(s)} ϕ_{b_{j}}^{(r)} d x_{b_{j}}

(28)

K_{rs} = ω_{s}^{2} M_{rs}

(29)

N_{r}^{Q} = \sum_{i = 1}^{4} \sum_{m = 1}^{N} \sum_{n = 1}^{N} χ_{c_{i}} μ_{c_{i}} q_{m} q_{n} (4 \frac{d_{c_{i}} κ^{2}}{l_{c_{i}}^{3}} \int_{0}^{l_{c_{i}}} {ϕ_{c_{i}}^{(m)}}^{'} {ϕ_{c_{i}}^{(n)}}^{'} d x_{c_{i}} \int_{0}^{l_{c_{i}}} ϕ_{c_{i}}^{(r)} d x_{c_{i}} - {\hat{e}}_{c_{i}}^{(m)} \int_{0}^{l_{c_{i}}} ϕ_{c_{i}}^{(r)} {ϕ_{c_{i}}^{(n)}}^{''} d x_{c_{i}})

(30)

N_{r}^{C} = - κ^{2} \sum_{i = 1}^{4} \sum_{m = 1}^{N} \sum_{n = 1}^{N} \sum_{o = 1}^{N} \frac{χ_{c_{i}} μ_{c_{i}}}{2 l_{c_{i}}} q_{m} q_{n} q_{o} \int_{0}^{l_{c_{i}}} {ϕ_{c_{i}}^{(m)}}^{'} {ϕ_{c_{i}}^{(n)}}^{'} d x_{c_{i}} \int_{0}^{l_{c_{i}}} ϕ_{c_{i}}^{(r)} {ϕ_{c_{i}}^{(o)}}^{''} d x_{c_{i}}

(31)

F_{r}^{MF} = \sum_{α = 1}^{\infty} \sum_{j = 1}^{7} η_{b_{j}} f_{b_{j α}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} δ (x_{b_{j}} - {\bar{x}}_{b_{j α}}) d x_{b_{j}}

(32)

respectively. Equation (26) is a set of nonlinear, coupled second-order ODEs. If one ignores the geometric nonlinearities of the stay cables, then equations (30) and (31) become $N_{r}^{Q} = 0$ and $N_{r}^{C} = 0$ , respectively, and equation (26) consequently becomes a set of uncoupled ODEs.

Numerical results and discussion

Geometric and physical parameters of a cable-stayed bridge are listed in Table 1. The first 20 natural frequencies of the cable-stayed bridge and their corresponding mode shapes are calculated using the method in Cao et al.²⁰ and listed in Figure 2. It can be seen that symmetrical and anti-symmetrical mode shapes alternatively occur since the bridge model is symmetrical. An index $Γ_{r}$ is introduced as in Cao et al.²⁰ to measure the degree of closeness of the rth natural frequency to its neighboring natural frequencies

{\begin{array}{l} Γ_{r} = \frac{ω_{r + 1} - ω_{r}}{ω_{r}} \times 100 % i f r = 1 \\ Γ_{r} = \min {\frac{ω_{r} - ω_{r - 1}}{ω_{r - 1}}, \frac{ω_{r + 1} - ω_{r}}{ω_{r}}} \times 100 % i f r \neq 1 \end{array}

(33)

Table 1.

Geometric and physical parameters of the cable-stayed bridge.

Parameter		Unit	Value
Deck beam	Mass per unit length of the deck beam, $m_{b_{j}}$	kg/m	16,940
	Elastic modulus of the deck beam, $E_{b_{j}}$	N/m²	2.0 × 10¹¹
	Area moment of inertia of the deck beam, $I_{b_{j}}$	m⁴	1.20
	Length of the segment b₁ of the deck beam, $2 L_{b_{1}}$	m	35
	Length of the segment b₂ of the deck beam, $2 L_{b_{2}}$	m	40
	Length of the segment b₃ of the deck beam, $2 L_{b_{3}}$	m	50
	Length of the segment b₄ of the deck beam, $2 L_{b_{4}}$	m	50
	Length of the segment b₅ of the deck beam, $2 L_{b_{5}}$	m	50
	Length of the segment b₆ of the deck beam, $2 L_{b_{6}}$	m	40
	Length of the segment b₇ of the deck beam, $2 L_{b_{7}}$	m	35
Stay cables	Mass per unit length of the stay cables, $m_{c_{i}}$	kg/m	286
	Elastic modulus of the stay cables, $E_{c_{i}}$	N/m²	2.0 × 10¹¹
	Cross-sectional area of the stay cables, $A_{c_{i}}$	m²	0.0362
	Length of the stay cable c₁, $L_{c_{1}}$	m	52
	Length of the stay cable c₂, $L_{c_{2}}$	m	60
	Length of the stay cable c₃, $L_{c_{3}}$	m	60
	Length of the stay cable c₄, $L_{c_{4}}$	m	52
	Sag-to-span ratios of the stay cables, ${\bar{d}}_{c_{i}} = D_{c_{i}} / L_{c_{i}}$		0.01

Figure 2.

The first 20 natural frequencies of the cable-stayed bridge and their corresponding mode shapes.

An index $Λ_{ri}$ ^20,26 is introduced to measure the degree of localization of the rth mode shape of the stay cable c_i

Λ_{r i} = \frac{m_{c_{i}} \int_{0}^{l_{c_{i}}} ({ϕ_{c_{i}}^{(r)}}^{2} (x_{c_{i}}) + {φ_{c_{i}}^{(r)}}^{2} (x_{c_{i}})) d x_{i}}{\sum_{i = 1}^{4} m_{c_{i}} \int_{0}^{l_{c_{i}}} ({ϕ_{c_{i}}^{(r)}}^{2} (x_{c_{i}}) + {φ_{c_{i}}^{(r)}}^{2} (x_{c_{i}})) d x_{c_{i}} + \sum_{j = 1}^{7} m_{b_{j}} \int_{- l_{b_{j}}}^{l_{b_{j}}} {ϕ_{b_{j}}^{(r)}}^{2} (x_{b_{j}}) d x_{b_{j}}} \times 100 %

(34)

The degree of localization of the rth mode shape of the deck beam can be measured by $1 - \sum_{i = 1}^{4} Λ_{ri}$ . The closeness degrees of the natural frequencies and the degree of localization of the mode shapes of the stay cables and the deck beam are shown in Table 2. A natural frequency is said to be close to its neighboring natural frequencies if its closeness degree is larger than zero and less than 0.05%, and a mode shape is said to be localized at stay cables when the degree of localization of the deck beam is less than 5.00%. It is observed that mode localization occurs on the stay cables when there are close natural frequencies, but the converse may not be true (mode localization occurs for mode 11, but the natural frequencies are not close). With zero initial conditions, the dynamic response of the cable-stayed bridge can be calculated from equation (26) using MATLAB’s hybrid fourth/fifth-order variable-step Runge–Kutta solver, ode45. It should be noted that $F_{α} = 0.6 m_{b} g$ , and the modal damping ratio $C_{rs}$ in equation (28) is always equal to $0.1 δ_{rs}$ for all the following calculation.

Table 2.

The degrees of localization of the mode shapes of different components and the closeness degrees of the natural frequencies.

Mode	$Λ_{r 1} (%)$	$Λ_{r 2} (%)$	$Λ_{r 3} (%)$	$Λ_{r 4} (%)$	$1 - \sum_{i = 1}^{4} Λ_{r i} (%)$	$Γ_{r} (%)$
1	1.2438	21.6019	21.6019	1.2438	54.3086	47.9402
2	19.8862	15.6383	15.6383	19.8862	28.9510	10.1223
3	37.8115	1.9823	1.9823	37.8115	20.4124	6.3825
4	18.6717	25.5319	25.5319	18.6717	11.5928	6.3825
5	0.0024	49.9431	49.9431	0.0024	0.1090	0.0226
6	0.0021	49.9885	49.9885	0.0021	0.0188	0.0226
7	0.6643	27.3293	27.3293	0.6643	44.0128	10.4698
8	6.5361	10.7282	10.7282	6.5361	65.4714	13.0723
9	31.8058	0.4892	0.4892	31.8058	35.4100	2.0451
10	18.9553	25.6454	25.6454	18.9553	10.7986	1.3942
11	0.3224	48.6503	48.6503	0.3224	2.0546	1.3942
12	21.8307	21.4331	21.4331	21.8307	13.4724	1.8651
13	27.4705	0.0143	0.0143	27.4705	45.0304	6.2182
14	14.3402	1.0894	1.0894	14.3402	69.1408	6.2182
15	0.0002	49.9778	49.9778	0.0002	0.0440	0.0111
16	0.0042	49.9307	49.9307	0.0042	0.1302	0.0111
17	49.8283	0.0003	0.0003	49.8283	0.3428	0.0251
18	49.9181	0.0010	0.0010	49.9181	0.1618	0.0251
19	1.0456	0.0559	0.0559	1.0456	97.7970	5.4377
20	0.0003	49.9839	49.9839	0.0003	0.0316	0.0011

When the distance between any two neighboring forces (L_F) is 30 m, the transverse displacement of the mid-point of the deck beam with two different velocities (V_F) of the moving forces, namely, V_F = 17 m/s and V_F = 100 m/s are shown in Figure 3(a) and (b), respectively. The number of Galerkin truncation terms is set to 10, 20, and 50. The results presented in Figure 3 and other results for a large number of cases that are not reported here for the sake of brevity show that Galerkin truncation with 20 terms yields rather accurate results for the dynamic response of the cable-stayed bridge subjected to a continuous sequence of moving forces when the value of L_F is larger than 30 and the value of V_F is smaller than 100. In the following numerical examples, the first 20 modes of the linearized undamped cable-stayed bridge model are used in the Galerkin method.

Figure 3.

Dynamic response of the mid-point of the deck beam for different numbers of Galerkin truncation terms when L_F = 30 m: (a) V_F = 17 m/s and (b) V_F = 100 m/s.

For $1 m / s \leq V_{F} \leq 100 m / s$ , Figure 4 plots the amplitude of the steady-state response of the mid-point of the deck beam versus the velocity of the moving forces with L_F = 30 m as an example in both cases of considering and neglecting the geometric nonlinearities (the linear and nonlinear solutions). Paying attention to the fact from equation (32)

\begin{matrix} F_{r}^{MF} & = \sum_{α = 1}^{\infty} \sum_{j = 1}^{7} f_{b_{j α}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} δ (x_{b_{j}} - {\bar{x}}_{b_{j α}} (τ)) d x_{b_{j}} \\ = \sum_{α = 1}^{\infty} \sum_{j = 1}^{7} f_{b_{j α}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} δ (x_{b_{j}} - (\frac{V_{F} τ}{L ε} - (α - 1) l_{F} - 2 \sum_{\bar{j} = 1}^{j} l_{b_{\bar{j}}} + l_{b_{j}})) d x_{b_{j}} \\ = \sum_{α = 1}^{\infty} \sum_{j = 1}^{7} f_{b_{j α}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} δ (x_{b_{j}} - (\frac{V_{F} τ}{L ε} - (α - 2) l_{F} - 2 \sum_{\bar{j} = 1}^{j} l_{b_{\bar{j}}} + l_{b_{j}})) d x_{b_{j}} \\ = \sum_{α = 1}^{\infty} \sum_{j = 1}^{7} f_{b_{j α}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} δ (x_{b_{j}} - (\frac{V_{F}}{L ε} (τ + \frac{ε L_{F}}{V_{F}}) - (α - 1) l_{F} - 2 \sum_{\bar{j} = 1}^{j} l_{b_{\bar{j}}} + l_{b_{j}})) d x_{b_{j}} \\ = \sum_{α = 1}^{\infty} \sum_{j = 1}^{7} f_{b_{j α}} \int_{- l_{b_{j}}}^{l_{b_{j}}} ϕ_{b_{j}}^{(r)} δ (x_{b_{j}} - {\bar{x}}_{b_{j α}} (τ + \frac{ε L_{F}}{V_{F}})) d x_{b_{j}} \end{matrix}

(35)

a continuous sequence of identical, equally spaced moving forces can be regarded as a periodic force with a nondimensional period T that equals to εL_F/V_F. Hence, resonance may occur if the moving velocity of the forces is $V_{ω r} = (L_{F} ω_{r}) / 2 π$ or is close V_ωr by linear or nonlinear analysis. The velocity V_ωr is called the rth resonant velocity in this work. The first 20 resonant velocities when L_F = 30 m are listed in Table 3. However, as shown from the linear solutions in Figure 4, there are not peaks on the curve when the moving velocity of the forces is a resonant velocity except the 1st, 3rd, 7th, 9th, 13th, and 19th resonant velocities. The reasons for this lie in the following two facts: (1) the mid-point of the deck beam happens to be a node of the mode shapes for modes 2k (k = 1, 2, 3, …) that are anti-symmetric, which results in a zero dynamic displacement for the mid-point of the deck beam when the velocity of the moving forces is the 2kth resonant velocities. (2) The mode shapes are localized on the stay cables for modes 5, 11, 15, and 17; hence, the moving forces cannot excite the vibration of the cable-stayed bridge when their velocity is the 5th, 11th, 15th, and 17th resonant velocities, which can be seen from equation (32).

Figure 4.

Amplitude of the steady-state response of the mid-point of the deck beam versus the velocity of the moving forces; L_F = 30 m. A vertical dashed line indicates that resonance occurs when the velocity of the moving forces is that shown in the expression near it there.

Table 3.

Values of the first 20 resonant velocities when L_F = 30 m.

$V_{ω r} = \frac{L_{F} ω_{r}}{2 π}$	V_ω ₁ = 16.94 m/s	V_ω ₂ = 25.06 m/s	V_ω ₃ = 27.60 m/s	V_ω ₄ = 29.36 m/s
	V_ω ₅ = 39.11 m/s	V_ω ₆ = 39.12 m/s	V_ω ₇ = 43.22 m/s	V_ω ₈ = 51.34 m/s
	V_ω ₉ = 58.05 m/s	V_ω ₁₀ = 59.23 m/s	V_ω ₁₁ = 60.06 m/s	V_ω ₁₂ = 61.18 m/s
	V_ω ₁₃ = 65.90 m/s	V_ω ₁₄ = 70.00 m/s	V_ω ₁₅ = 78.27 m/s	V_ω ₁₆ = 78.28 m/s
	V_ω ₁₇ = 80.77 m/s	V_ω ₁₈ = 80.79 m/s	V_ω ₁₉ = 85.18 m/s	V_ω ₂₀ = 98.02 m/s

Resonance can also occur when the velocity of the moving forces is the product of a unit fraction and a resonant velocity, which can be seen from the peaks when the velocity of the moving forces is V_ω₁/2, V_ω₁/3, V_ω₁₃/2, and V_ω₁₉/4 on the linear solutions in Figure 4. The reason for this lies in the fact that there are higher harmonics with frequencies equal to integer multiples of the fundamental frequency 2π/T in the Fourier series

\begin{matrix} F_{r}^{MF} (τ) = \frac{1}{T} \int_{- T / 2}^{T / 2} F_{r}^{MF} d τ + \frac{2}{T} \sum_{n = 1}^{\infty} [\cos (n \frac{2 π}{T} τ) \int_{- T / 2}^{T / 2} F_{r}^{MF} \cos (n \frac{2 π}{T} τ) d τ] \\ + \frac{2}{T} \sum_{n = 1}^{\infty} [\sin (n \frac{2 π}{T} τ) \int_{- T / 2}^{T / 2} F_{r}^{MF} \sin (n \frac{2 π}{T} τ) d τ] \end{matrix}

(36)

for the periodic force $F_{r}^{MF}$ in equation (32). In addition, comparing the linear and nonlinear solutions in Figure 4, one can see that the geometric nonlinearities of the stay cables have some influence on the dynamic response of the cable-stayed bridge. Generally, the peaks of the curves do not exactly occur at the undamped natural frequencies of the linearized model of the cable-stayed bridge. However, the linear and nonlinear solutions almost coincide with each other when the velocity of the moving forces is around the 19th resonant velocity. The reason for this lies in the fact that the deck beam, which is a linear component, has a major contribution to the dynamic response of the cable-stayed bridge since the mode shape for mode 19 is localized on the deck beam.

Figure 5 shows the amplitude of the steady-state response of the mid-point of the deck beam versus the distance between any two neighboring forces when the velocity of moving forces is the first resonant velocity. It can be seen that there are a critical distance L_F = 193 m. In the region L_F < 193 m, the amplitude of the steady-state response of the mid-point of the deck beam generally increases with L_F and in the region L_F > 193 m decreases with L_F. However, there are fluctuations at some L_F. It can also be seen from Figure 5 that the geometric nonlinearities of the stay cables have some influence on the dynamic response of the cable-stayed bridge.

Figure 5.

Amplitude of the dynamic response of the mid-point of the deck beam versus the distance between any two neighboring forces.

Conclusion

The dynamic behavior of a cable-stayed bridge that consists of a simply supported four-cable-stayed deck beam and two rigid towers subjected to a continuous sequence of identical, equally spaced forces moving at a constant velocity has been investigated. The dynamic response of the cable-stayed bridge is calculated using the Galerkin method via MATLAB. Some conclusions from the numerical simulation are as follows: (1) the moving forces with a resonant velocity corresponding to a mode of the cable-stayed bridge, whose natural frequency is close to those of its neighboring modes and whose mode shape is localized on the stay cables, cannot excite the vibration of that mode. This means that modes with close natural frequencies, whose mode shapes are localized on stay cables, play a minor role in the dynamic response of the cable-stayed bridge. (2) The geometric nonlinearities of stay cables can generally have some influence on the dynamic response of the cable-stayed bridge, but have almost no influence on that when the velocity of the moving forces is around the resonant velocity corresponding to a mode whose mode shape is localized on the deck beam. (3) Resonance can also occur when the velocity of the moving forces is the product of a unit fraction and a resonant velocity. (4) There is a critical distance between any two neighboring forces below which transverse displacements of the cable-stayed bridge generally increase with the distance.

Footnotes

Academic Editor: Nikolaos Nikitas

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 work was supported by the National Natural Science Foundation of China under Grant Nos 11302087 and 11442006, Natural Science Foundation of Jiangsu Province under Grant No. BK20130479, Research Foundation for Advanced Talents of Jiangsu University under Grant No. 13JDG068, and National Science Foundation under Grant No. CMMI-1000830.

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