Investigation on the vehicle-bridge dynamic interaction considering the vertical curvature of bridge and environmental thermal effects

Abstract

River-crossing bridges often utilise curved design to account for the highway profile change, distribute traffic loads and manage water flow. The vehicle-curved bridge interaction (VBI) is complicated, especially combining with environmental thermal effects. These thermo-mechanical interactions differ substantially from those in straight bridges and warrant detailed study. In this study, a curved bridge is modeled as a Timoshenko beam (TB) with pinned supports at both ends. Governing equations are derived using the four-node isoparametric beam element and cubic polynomial shape function. The vehicle is modelled as a half-car model to account for its bouncing and pitching motions. Thermal effects are represented by a linear temperature gradient through the beam depth, which is converted into equivalent thermal loads. An analytical framework that superimposes moving vehicle loads and thermal loads within the coupled motion equations is developed. The dynamic response of the VBI system is analyzed using a zero-order displacement and velocity overshoot (U₀-V₀) iterative method. The parametric study reveals that: (1) the shear deformation significantly affects high vibration modes; (2) the bridge stiffness increases when the radius of its curvature reduces, which induces the reduction of bridge and vehicle displacement responses; (3) the thermal bending and expansion of the bridge have a large effect on dynamic amplification factor (DAF), while the impact of its curvature on dynamic load coefficient (DLC) becomes dominant for high vehicle speeds. These findings provide valuable insights for the design, analysis, and maintenance of curved bridges operating under combined vehicular and thermal loading conditions.

Keywords

vehicle-curved bridge interaction Timoshenko beam element bridge curvature thermal effects dynamic impact

Introduction

Vertical curved beams are widely used in bridge engineering, and their vibration characteristics have received significant attention due to their outstanding performance in long-span and high load-bearing capacity (Mohamed et al., 2024; Zhong et al., 2021; Zhou et al., 2024), such as the arch bridges (Ding et al., 2023) and slightly curved beam bridges (Ye et al., 2020). The dynamic interaction between vehicles and bridges plays a crucial role in modern bridge engineering, particularly in applications such as load rating in bridge design, structural condition monitoring, and ride comfort assessment (Li and Feng, 2023). Initial studies primarily utilized Euler–Bernoulli (EB) beam theory to model bridge dynamics under moving vehicle loads (Deng et al., 2023; Li et al., 2022a; Liu et al., 2013; MacLeod and Arjomandi, 2023). Although EB theory is effective for slender structures, it assumes plane sections remain plane and neglects shear deformation, limiting its accuracy for short- to mid-span highway bridges with small aspect ratios and thick decks (Zhang et al., 2020). To overcome these limitations, researchers have employed Timoshenko beam (TB) theory, which incorporates both shear deformation and rotary inertia, providing a more realistic representation of bridge dynamics under vehicular loading (Gao et al., 2022). Several studies have examined Timoshenko beams subjected to moving forces or moving masses. Wang (1997) analyzed the dynamic behavior of multi-span Timoshenko beams under constant moving loads. Yavari et al. (2002) adopted a discrete element technique to study moving loads on bridge. Kocatürk and Şimşek (2006) used polynomial-based trial functions to capture beam deflection and rotation under harmonic loads. Others, such as Attar et al. (2017), explored the effects of cracks and flexible foundations, and Esen (2020) incorporated size-dependent behavior using two-node finite elements. The slope-inertia-based TB theory has also been developed to enhance model fidelity for thick beam under dynamic loading (Lei et al., 2022). The Timoshenko beam under moving masses has been studied (Esmailzadeh and Ghorashi, 1997). Lou et al. (2006) presents a finite-element formulation of a Timoshenko beam subjected to a moving mass. The time-dependent matrices for the equation of motion were derived using the variational approach. Pirmoradian et al. (2015) investigated the instability and resonance of a Timoshenko beam excited by a sequence of moving masses using the incremental harmonic balance method. The stability and resonance behaviors of the Timoshenko beam employing more sophisticated deformation theories were different from Euler-Bernoulli beam. While the moving force and mass models capture different aspects of vehicle loading, these models do not fully account for the dynamic coupling between the vehicle and the bridge (Green and Cebon, 1997). Moghaddas et al. (2009) addressed this limitation by formulating the weak form of the coupled VBI problem using TB theory. Similarly, Wang et al. (2020) used a state-space approach to analyze a coupled system with a single-degree-of-freedom quarter-car model. These works underline the importance of VBI coupling and road surface effects. As above, bridge geometries are typically assumed as straight, and the curvature of the bridge has not been considered (Lu et al., 2019).

Bridge structures often feature vertical or horizontal curvature due to terrain constraints or aesthetic considerations. Some studies have been conducted on the dynamic behavior of such curved structures under moving loads (Zhai et al., 2023; Zhang et al., 2022). Wu and Chiang (2003) analyzed radial and out-of-plane vibration of circular arches under moving loads using curved beam elements. Rostam et al. (2015) studied vibration suppression in curved beams subjected to off-center moving loads, while others examined the effects of crack location, load speed, and structural geometry (Wu and Chiang, 2004). These investigations have demonstrated that the curvature can significantly influence the dynamic response by altering stiffness distribution and mode shapes. The implications for VBI modeling are profound, as ignoring curvature may lead to inaccurate predictions in real-world bridge systems (Poojary and Roy, 2021).

In addition to mechanical loads, environmental thermal effects can significantly impact the dynamic response of bridge structures. Changes in environmental temperature induce thermal expansion and bending of the bridge, altering its stress distributions and dynamic characteristics. Several studies have explored the interaction between thermal effects and dynamic loading. Paganelli (2014) analyzed beams and plates under combined thermal and moving loads, while Zhong et al. (2015) and Caddemi et al. (2017) incorporated prestress and axial forces in their dynamic analyses. Al Rjoub and Hamad (2020) studied the forced vibration response of axially loaded Euler-Bernoulli and Timoshenko beams using a modal expansion method. Li et al. (2020) investigated the dynamic response of an axially loaded Timoshenko beam under a moving load. The thermal expansion under temperature variation was considered as the axial load. Further investigations have examined functionally graded materials and micro-scale behaviors (Wang and Wu, 2016). Abdelrahman et al. (2021) developed a finite element method for Timoshenko microbeams subjected to thermal and moving mass effects. Liu et al. (2018) reported that thermal loads can significantly alter bridge responses under dynamic loads. Despite these advancements, most existing work considers straight beams and simplified thermal fields, such as uniform or axial gradients.

While some studies have investigated either the effects of bridge curvature or thermal loading independently, their combined impact on dynamic vehicle-bridge interaction remains largely unexplored. This gap is particularly relevant for real-world bridge systems exposed to environmental temperature variations and designed with vertical curvature for functional or aesthetic reasons. The novelty of this work lies in the development and analysis of a vertically curved Timoshenko beam bridge model subjected to both moving vehicle loads and thermal effects, capturing the coupled dynamic behavior that is often oversimplified in prior research. This integrated approach provides insights critical for the design and maintenance of modern highway bridges.

Formulation for VBI and thermal loads

Linear temperature rise model along the beam depth

Consider a beam with rectangular cross section of width b and depth h, the temperatures at the top and bottom surfaces of the beam are T_t and T_b, respectively. As shown in Figure 1, the temperature distribution in the beam is assumed to be linear through the depth of the beam. Considering the reference temperature $T_{0}$ at which the thermal stress is free, the temperature rise at z along the depth of the beam is defined as (Fazzolari and Carrera, 2014)

Δ T (z) = \frac{T_{t} + T_{b}}{2} + \frac{T_{t} - T_{b}}{h} z - T_{0} (- h / 2 \leq z \leq h / 2)

(1)

where α is the coefficient of thermal expansion in the figure.

Figure 1.

Linear temperature distribution along the cross-section.

While the linear temperature gradient assumption is commonly adopted for analytical tractability, it represents a simplification of actual thermal behavior in bridge components. In real-world conditions, such as during intense solar exposure, rapid environmental temperature changes, or asymmetric heating, the through-thickness temperature distribution may become nonlinear. These nonlinear gradients can introduce additional thermal stresses and affect deformation patterns, potentially altering the dynamic response of the structure. Therefore, while the linear gradient is appropriate for moderate thermal conditions, its application should be approached with caution under extreme environments. Future studies may consider incorporating nonlinear thermal profiles to improve the accuracy and robustness of dynamic analyses involving complex thermal loading.

Governing equation of the vehicle model

The vehicle and bridge models employed for dynamic analysis are shown in Figure 2. The bridge is represented by a pinned-pinned curved beam with a span length L. The pinned supports are widely used to the planar curved beams that is critical to study the thermal behaviors of the beams (Rostam et al., 2015; Zhou et al., 2024).

Figure 2.

Vehicle-bridge interaction model.

The vehicle is considered as a half-car model, which contains two independent degrees of freedom corresponding to the bounce y_s and pitch rotation $θ_{s}$ of the vehicle body. The sprung mass and moment of inertia of the vehicle are m_v and I_v respectively. The vehicle body is supported by two sets of systems with a combination of spring and damper. The stiffness of spring is k_i and the damping coefficient is c_i representing the combined tyre and suspension components for the front and rear axles (i = 1,2) of the vehicle. The distances of front and rear axles to the vehicle’s centre of gravity are denoted as d₁ and d₂, respectively. The point-wise coupling between the supporting structure and moving subsystem modeled as a spring-damper element provides sufficient fidelity for vehicles (Yang et al., 2019). The equation of motion for the half-car model is given by

[M_{v}] {{\ddot{Z}}_{v}} + [C_{v}] {{\dot{Z}}_{v}} + [K_{v}] {Z_{v}} = {F_{v}}

(2)

where,

{F_{v}} = {f_{v 1}, f_{v 2}}^{T}

{f_{v 1} = k}_{1} w_{v, 1} + c_{1} {\dot{w}}_{v, 1} + k_{2} w_{v, 2} + c_{2} {\dot{w}}_{v, 2}

and

f_{v 2} = (k_{1} w_{v, 1} + c_{1} {\dot{w}}_{v, 1}) d_{1} - (k_{2} w_{v, 2} + c_{2} {\dot{w}}_{v, 2}) d_{2}

w_{v, i}

is the total displacement under the vehicle wheel i, including the road roughness r_i and the bridge displacement under the wheel,

w_{b, i}

w_{v, i} = w_{b, i} + r_{i}; i = 1, 2

(3)

The mass, damping and stiffness matrices are as,

[M_{v}] = d i a g {m_{v}, I_{v}}, [C_{v}] = [\begin{array}{c} c_{1} + c_{2} & c_{1} d_{1} - c_{2} d_{2} \\ c_{1} d_{1} - c_{2} d_{2} & c_{1} d_{1}^{2} + c_{2} d_{2}^{2} \end{array}], [K_{v}] = [\begin{array}{c} k_{1} + k_{2} & k_{1} d_{1} - k_{2} d_{2} \\ k_{1} d_{1} - k_{2} d_{2} & k_{1} d_{1}^{2} + k_{2} d_{2}^{2} \end{array}]

(4)

The total contact-point force between vehicle wheel and bridge represented by $F_{i}$ , can be expressed as:

F_{i} = P_{i} - k_{i} (y_{s} - {(- 1)}^{i} d_{i} θ_{s} - w_{v, i}) + c_{i} ({\dot{y}}_{s} - {(- 1)}^{i} d_{i} {\dot{θ}}_{s} - {\dot{w}}_{v, i}); i = 1, 2

(5)

where

P_{1} = m_{v} d_{2} / (d_{1} + d_{2})

and

P_{2} = m_{v} d_{1} / (d_{1} + d_{2})

are static front and rear axle loads of the vehicle, respectively. The work done by the moving vehicular loads to the beam system can be written as (Zhai et al., 2023),

W = \int_{t_{1}}^{t_{2}} F_{i} (t) w_{v, i} (x = x_{i}, t) d t

Governing equation of vertically curved Timoshenko beam bridges

The curved beam is discretised as a sequence of straight beam elements (Yang et al., 2024). If $u (x, t)$ , $w (x, t)$ and $ϕ (x, t)$ are the axial displacement in neutral plane, transverse displacement and bending rotation of a section at a distance x in the coordinate system, the axial strain $ε_{x x}$ and shear strain $γ_{x z}$ are given as (Abdelrahman et al., 2021; Yang et al., 2002),

ε_{x x} = \frac{\partial u (x, t)}{\partial x} - z \frac{\partial ϕ (x, t)}{\partial x} - α Δ T (z)

(6a)

γ_{x z} = \frac{\partial w (x, t)}{\partial x} - ϕ (x, t)

(6b)

The stress constitutive equations based on Hooke’s law can be written as,

σ_{x x} = E ε_{x x}

(7a)

τ_{x z} = κ G γ_{x z}

(7b)

The potential energy includes contributions from the axial deformation, shear deformation, bending deformation, and thermal effects as,

U = \frac{1}{2} \int_{0}^{L} \iint_{A} (σ_{x x} ε_{x x} + τ_{x z} γ_{x z}) d A d x

= \frac{1}{2} \int_{0}^{L} \int_{- \frac{h}{2}}^{\frac{h}{2}} [E {(u_{, x} - z ϕ_{, x} - α Δ T (z))}^{2} + κ G {(w_{, x} - ϕ)}^{2}] b d z d x

(8)

The kinetic energy

T

of the beam is obtained as,

T = \frac{1}{2} \int_{0}^{L} \iint_{A} ρ ({\dot{u}}^{2} + {\dot{w}}^{2}) d A d x = \frac{1}{2} \int_{0}^{L} \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ ({\dot{u}}^{2} + {\dot{w}}^{2}) b d z d x

= \frac{1}{2} \int_{0}^{L} [ρ A {(\frac{\partial u (x, t)}{\partial t})}^{2} + ρ I {(\frac{\partial ϕ (x, t)}{\partial t})}^{2} + ρ A {(\frac{\partial w (x, t)}{\partial t})}^{2}] d x

(9)

By Hamilton’s principle, the variation of the integral over time of the difference between kinetic and potential energies should be zero,

δ \int_{t_{1}}^{t_{2}} (T - U) d t + δ W = 0

(10)

Based on Hamilton’s principle and the Euler-Lagrange equations, the following equations of motion are obtained,

ρ A \frac{\partial^{2} u (x, t)}{\partial t^{2}} - E A \frac{\partial^{2} u (x, t)}{\partial x^{2}} + E A α (\frac{T_{t} + T_{b}}{2} - T_{0}) = 0

(11a)

ρ A \frac{\partial^{2} w (x, t)}{\partial t^{2}} - κ G A (\frac{\partial^{2} w (x, t)}{\partial x^{2}} - \frac{\partial ϕ (x, t)}{\partial x}) - F_{v, i} (t) δ (x = x_{i}, t) = 0

(11b)

ρ I \frac{\partial^{2} ϕ (x, t)}{\partial t^{2}} - κ G A (\frac{\partial w (x, t)}{\partial x} - ϕ (x, t)) - E I \frac{\partial^{2} ϕ (x, t)}{\partial x^{2}} + E I α \frac{T_{b} - T_{t}}{h} = 0

(11c)

where E, G, I, A, κ, ρ are the Young’s modulus, shear modulus, second moment of area, cross-sectional area, shear correction factor, and the mass density of the beam, respectively. t is the time. These equations lead to a more comprehensive descriptions of the beam’s transverse displacement, rotation, and axial deformation due to external loads and environmental temperature changes.

In practice, the time for a vehicle to cross the bridge is significantly shorter than that required for environmental temperature changes to affect the bridge. In this study, environmental thermal effects are considered as equivalent static loads derived from the linear temperature gradient. These loads induce initial displacements and internal stresses in the structure prior to the vehicle entry. When the vehicle enters the bridge, the dynamic load interacts with this thermal state. Although thermal effects are considered as time-invariant under slowly varying environmental conditions, they affect the structural stiffness, boundary force distribution, and initial deformation profile. As a result, dynamic responses by the moving vehicle differ from that of a non-thermally loaded structure. Therefore, the interplay between thermal-induced quasi-static deformations and time-varying vehicular excitations is essential to accurately capture the dynamic behavior of the bridge-vehicle system.

Road surface roughness

As listed in equation (3), the road surface roughness has a significant effect on dynamic behaviour of the VBI system. The road surface roughness is described as (Li et al., 2022b),

r (x) = \sum_{i}^{M} \sqrt{2 G_{d} (n_{0}) {(\frac{n_{i}}{n_{0}})}^{- w} Δ n} \cos (2 π n_{i} x + φ_{i})

(12)

where M is the size of the road profile vector with M = 1+L_total/L_step, that L_total and L_step are the total length and step length, respectively.

G_{d} (n_{0})

is the power spectrum density for a reference spatial frequency

n_{0} = 0.1 cycle / m

. In this study,

G_{d} (n_{0}) = 16 \times 10^{- 6} m^{3} / cycle

is selected to simulate a class ‘A’ road profile defined in ISO-8608 (1995).

n_{i}

is the spatial frequency varying from

n_{\min}

n_{\max}

in increments of

Δ n = (n_{\max} - n_{\min}) / (M - 1)

φ_{i}

is the phase angle with a uniform probabilistic distribution between 0 and 2π, and w = 2. Besides, a linear interpolation is used for vdt between two adjacent points with the interval L_step. dt is the sampling interval.

Solution procedures

Finite element modeling

The bridge is discretised into beam finite elements. To derive the elemental stiffness and mass matrices, the principle of virtual work is applied. Consider a four-node isoparametric TB element of length l, the nodal variables are the longitudinal deflection $u$ , the transverse deflection $w$ and the bending slope $ϕ$ . This results in an element with 12 degrees of freedom, and the element nodal displacements can be given as,

{q}^{e} = {[u_{1}, w_{1}, ϕ_{1}, u_{2}, w_{2}, ϕ_{2}, u_{3}, w_{3}, ϕ_{3}, u_{4}, w_{4}, ϕ_{4}]}^{T}

(13)

where subscripts 1-4 denote the local node number on each beam element. Assuming that u, w and

ϕ

are completely independent variables, they can be obtained as,

u = [N_{1} (ξ), N_{2} (ξ), N_{3} (ξ), N_{4} (ξ)] {\begin{cases} \begin{array}{l} u_{1} \\ u_{2} \end{array} \\ \begin{array}{l} u_{3} \\ u_{4} \end{array} \end{cases}} = \sum_{i = 1}^{4} N_{i} (ξ) u_{i}

(14a)

w = [N_{1} (ξ), N_{2} (ξ), N_{3} (ξ), N_{4} (ξ)] {\begin{cases} \begin{array}{l} w_{1} \\ w_{2} \end{array} \\ \begin{array}{l} w_{3} \\ w_{4} \end{array} \end{cases}} = \sum_{i = 1}^{4} N_{i} (ξ) w_{i}

(14b)

ϕ = [N_{1} (ξ), N_{2} (ξ), N_{3} (ξ), N_{4} (ξ)] {\begin{cases} \begin{array}{l} ϕ_{1} \\ ϕ_{2} \end{array} \\ \begin{array}{l} ϕ_{3} \\ ϕ_{4} \end{array} \end{cases}} = \sum_{i = 1}^{4} N_{i} (ξ) ϕ_{i}

(14c)

where

ξ = x / l (- 1 \leq ξ \leq 1)

is a dimensionless variable.

N_{1} (ξ), N_{2} (ξ), N_{3} (ξ)

and

N_{4} (ξ)

are the cubic interpolation functions as,

{\begin{cases} N_{1} (ξ) = - \frac{9}{16} (ξ - 1) (ξ^{2} - \frac{1}{9}) \\ N_{2} (ξ) = \frac{27}{16} (ξ^{2} - 1) (ξ - \frac{1}{3}) \\ N_{3} (ξ) = - \frac{27}{16} (ξ^{2} - 1) (ξ + \frac{1}{3}) \\ N_{4} (ξ) = \frac{9}{16} (ξ + 1) (ξ^{2} - \frac{1}{9}) \end{cases}

(15)

The Jacobian matrix is defined as $[J] = \frac{d x}{d ξ} = [\frac{d N_{1}}{d ξ} \frac{d N_{2}}{d ξ} \frac{d N_{3}}{d ξ} \frac{d N_{4}}{d ξ}]$ . For equally spaced points on the beam element, $[J] = l / 2$ .

By interpolation functions, the stiffness and mass matrices of the beam element can be derived from strain energy and kinetic energy expressions by substituting equation (14) to equation (11). The local axial ${[K_{a}]}^{e}$ , bending ${[K_{b}]}^{e}$ and shear stiffness ${[K_{s}]}^{e}$ matrices are defined as

{[K_{a}]}^{e} = \int_{- 1}^{1} {[B_{a}]}^{T} E A [B_{a}] \det | J | d ξ

(16a)

{[K_{b}]}^{e} = \int_{- 1}^{1} {[B_{b}]}^{T} E I [B_{b}] \det | J | d ξ

(16b)

{[K_{s}]}^{e} = \int_{- 1}^{1} {[B_{s}]}^{T} κ G A [B_{s}] \det | J | d ξ

(16c)

and the elemental mass matrix is defined as

{[M]}^{e} = \int_{- 1}^{1} {[N_{u}]}^{T} ρ A [N_{u}] d e t | J | d ξ + \int_{- 1}^{1} {[N_{w}]}^{T} ρ A [N_{w}] \det | J | d ξ + \int_{- 1}^{1} {[N_{ϕ}]}^{T} ρ I [N_{ϕ}] \det | J | d ξ

(16d)

where the bending strain-displacement matrix

[B_{b}]

, the shear strain-displacement matrix

[B_{s}]

and axial strain-displacement matrix

[B_{a}]

are given as,

[B_{a}] = [\frac{d N_{1}}{d ξ} \frac{d ξ}{d x} 0 0 \frac{d N_{2}}{d ξ} \frac{d ξ}{d x} 0 0 \frac{d N_{3}}{d ξ} \frac{d ξ}{d x} 0 0 \frac{d N_{4}}{d ξ} \frac{d ξ}{d x} 0 0]

(17a)

[B_{b}] = [0 0 \frac{d N_{1}}{d ξ} \frac{d ξ}{d x} 0 0 \frac{d N_{2}}{d ξ} \frac{d ξ}{d x} 0 0 \frac{d N_{3}}{d ξ} \frac{d ξ}{d x} 0 0 \frac{d N_{4}}{d ξ} \frac{d ξ}{d x}]

(17b)

[B_{s}] = [0 \frac{d N_{1}}{d ξ} \frac{d ξ}{d x} {- N}_{1} 0 \frac{d N_{2}}{d ξ} \frac{d ξ}{d x} {- N}_{2} 0 \frac{d N_{3}}{d ξ} \frac{d ξ}{d x} {- N}_{3} 0 \frac{d N_{4}}{d ξ} \frac{d ξ}{d x} {- N}_{4}]

(17c)

and

[N_{u}] = [N_{1} (ξ) 0 0 N_{2} (ξ) 0 0 N_{3} (ξ) 0 0 N_{4} (ξ) 0 0]

(18a)

[N_{w}] = [0 N_{1} (ξ) 0 0 N_{2} (ξ) 0 0 N_{3} (ξ) 0 0 N_{4} (ξ) 0]

(18b)

[N_{ϕ}] = [0 0 N_{1} (ξ) 0 0 N_{2} (ξ) 0 0 N_{3} (ξ) 0 0 N_{4} (ξ)]

(18c)

The numerical integration in equation (16) is performed by using Gaussian quadrature. For the exact numerical integration of matrices ${[M]}^{e}$ and the flexural stiffness matrix ${[K_{b}]}^{e}$ , four-point Gauss quadrature is used. For the approximation of the shear stiffness matrix ${[K_{s}]}^{e}$ , a selective reduced integration technique using three-point quadrature is selected to avoid shear locking issue (Hughes et al., 1978). These local element stiffness and mass matrices of arbitrarily oriented in space at an angle $θ$ relative to x-axis can be transformed to the global coordinate system by $K^{e} = T^{T} {[K]}^{e} T$ and $M^{e} = T^{T} {[M]}^{e} T$ . The transformation matrix is defined as

T = b l o c k d i a g o n a l (T_{u n i t}, T_{u n i t}, T_{u n i t}, T_{u n i t})

with

T_{u n i t} = [\begin{array}{c} \cos θ & \sin θ & 0 \\ - \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{array}]

The element stiffness matrix

K^{e}

and mass matrix

M^{e}

are then assembled to the global stiffness matrix

K_{b}

and mass matrix

M_{b}

, respectively. Assuming Rayleigh damping, the motion equation of the beam can be written as

M_{b} {\ddot{x}}_{b} + C_{b} {\dot{x}}_{b} + K_{b} x_{b} = NF

(19)

Numerical calculation of governing equations

A zero-order displacement and velocity overshoot (U0-V0) technique, which is one of the generalized single step single solve (GSSSS) algorithms (Papazafeiropoulos and Plevris, 2018; Zhou and Tamma, 2004), is used for the dynamic analysis. The mid-point rule (MPR) a-form, that is a particular case of optimal numerical dissipation and dispersion U0-V0 method, is used in this study. Considering the equation of motion for the bridge, given $x_{b} (t), {\dot{x}}_{b} (t)$ and ${\ddot{x}}_{b} (t)$ , $x_{b} (t + 1), {\dot{x}}_{b} (t + 1)$ and ${\ddot{x}}_{b} (t + 1)$ are calculated from

(\frac{1}{2} M_{b} + \frac{1}{4} C_{b} Δ t + \frac{1}{8} K_{b} {Δ t}^{2}) Δ a = - M_{b} {\ddot{x}}_{b} (t) - C_{b} {{\dot{x}}_{b} (t) + \frac{1}{2} {\ddot{x}}_{b} (t) Δ t}

- K_{b} {x_{b} (t) + \frac{1}{2} {\dot{x}}_{b} (t) Δ t + \frac{1}{4} {\ddot{x}}_{b} (t) {Δ t}^{2}} + \frac{1}{2} {Φ (t) F (t) + Φ (t + 1) F (t + 1)}

(20a)

x_{b} (t + 1) = x_{b} (t) + {\dot{x}}_{b} (t) Δ t + \frac{1}{2} {\ddot{x}}_{b} (t) {Δ t}^{2} + \frac{1}{4} Δ a {Δ t}^{2}

(20b)

{\dot{x}}_{b} (t + 1) = {\dot{x}}_{b} (t) + {\ddot{x}}_{b} (t) Δ t + \frac{1}{2} Δ a Δ t

(20c)

{\ddot{x}}_{b} (t + 1) = {\ddot{x}}_{b} (t) + Δ a

(20d)

where

Δ t

is time interval, which is set as 0.005 second in this study.

Iterative procedure for the VBI analysis

As shown in equations (2) and (19), the VBI system includes two uncoupled equations for the vehicle and bridge subsystems respectively. These two uncoupled equations could be solved individually and the analysis of dynamic interaction between the vehicle and bridge is implemented in an iterative procedure. The first step is setting initial parameters including vehicle moving speed, time interval for the calculation, number of beam element and assumed static initial conditions of VBI system at time step IT = 0; The second step is to calculate and assemble matrices, i.e., $M_{v}$ , $C_{v}$ , $K_{v}$ , $M_{b}$ , $K_{b}$ , $C_{b}$ and interpolation functions; At last, the detail procedure for solving the equations (2) and (19) at each time step (IT = 1,2,3,4…) in an iterative uncoupled way is shown in Figure 3.

Figure 3.

Iterative procedure for the VBI analysis.

Verification of the proposed method

Transient responses of the VBI system

To verify the proposed VBI model, the dynamic analysis of a Timoshenko beam under a moving vehicle in (Gao et al., 2022) is adopted for comparison without the thermal load. The vehicle as shown in Figure 1 is supported by two sets of spring-damper elements characterized by stiffness coefficient k_i and damping coefficient c_i. The Rayleigh damping ratio is set to zero for the bridge. The parameters of the vehicle model and Timoshenko beam are listed in the Table 1. The displacement responses at the midspan of the beam bridge are considered when the vehicle moves over it at a speed of 10 m/s and 20 m/s, respectively. In the reference, analytical eigenfunctions of the beam and generalized assumed-mode method were used to approximate the beam’s motion which was solved using Runge-Kutta scheme. The bridge displacements at midspan with different vehicle speeds by the proposed method are compared with those from (Gao et al., 2022), as shown in Figure 4. From the figure, the bridge displacements by the proposed method are in good agreement with that from reference considering the given vehicle moving speeds. The results show that the proposed method is effective and accurate for dynamic analysis of the Timoshenko beam subject a moving vehicle.

Table 1.

Parameters of vehicle and bridge model.

Notation	Description	Value	Notation	Description	Value
Vehicle			Timoshenko beam
m _v	Mass of vehicle	2500 kg	L	Length of bridge	10 m
d₁/d₂	Distance of front/rear axle to center of mass	1.7/1.3 m	E	Elastic modulus	35 GPa
d₁/d₂			I	Area moment of inertia	0.05 m⁴
I _v	Moment of inertia	2300 kg⋅m²	A	Cross-sectional area	0.6 m²
I _v			G	Shear modulus	21 GPa
k ₁ /k ₂	Stiffness of the front/rear axle	2.3/1.8 × 10⁵ N/m	ρ	Density	2400 kg/m³
c ₁ /c ₂	Damping of the front/rear axle	0/0 Ns/m	κ′	Shear coefficient	0.4

Figure 4.

Displacements at midspan of a bridge subjected to a moving vehicle. (a) For vehicle speed v = 10 m/s (b) For vehicle speed v = 20 m/s.

Eigenvalue analysis considering the thermal axial load to the beam bridge

In this section, the eigenvalue analysis of the axial loaded Timoshenko beam is made by the proposed method. Considering a straight beam bridge with an thermal axial load N_x, the length of the bridge is L = 15 m, Young’s modulus is E = 3×10¹⁰ N/m², shear modulus is G = 1.1×10¹⁰ N/m², mass density is ρ = 2400 kg/m³, cross-sectional area is A = 0.3 m², depth of the bridge is 0.8 m, second moment of inertia is I = 0.025 m⁴, shear coefficient is κ = 5/6 and thermal expansion coefficient is

α = 54 \times 10^{- 6} ℃^{- 1}

. The first five natural frequencies of the beam bridge with different thermal loads, corresponding to the compressive axial forces induced by the thermal expansion N_x = 0, −2000 kN and −6000 kN, are obtained by the proposed method and the results are compared with that of Reference (Jiang and Ye, 2010). As shown in Table 2, the results show that the frequency decreases with increase of the compressive axial force induced by the thermal load. The frequencies with different axial forces by the proposed method are very close to those corresponding values in (Jiang and Ye, 2010). The results in Figure 4 and Table 2 confirm the correctness and accuracy of the current model and analysis method. The proposed method is validated.

Table 2.

Natural frequencies with different thermal loads (rad/sec).

Mode no.	N_x = 0		N_x = −2000 kN		N_x = −6000 kN
Mode no.	Proposed	Reference	Proposed	Reference	Proposed	Reference
1	44.42	44.42	43.03	43.03	40.11	40.11
2	173.78	173.74	172.39	172.35	169.58	169.53
3	377.80	377.57	376.38	376.15	373.53	373.30
4	643.03	642.41	641.58	640.95	638.66	638.03
5	956.04	954.74	954.53	953.23	951.50	950.20

Numerical investigation

Dynamic responses of bridges using different beam theories

Dynamic responses of the beam bridge under a moving vehicle using TB and EB theories are compared in this section. The bridge parameters are same as those used for eigenvalue analysis except that the bridge damping ratio is set as 2% without loss of generality. The vehicle parameters are listed in Table 1. Figure 5 shows displacement responses at midspan of the bridge subjected to a moving vehicle at a speed of 10 m/s and 20 m/s, respectively. In the figure, there is a slight difference in bridge displacement responses using TB and EB theories. The maximum beam deflection at midspan using TB is larger than that with EB. The moving speed changing from 10 m/s to 20 m/s only increases the maximum deflection of the beam slightly. The first six natural frequencies of the bridge are obtained using these two beam theories, listed in Table 3. From the results, the difference of the first two modes using two theories is less than 5% and it is increased with the mode order. The frequency difference reaches 18.57 % for the sixth mode. The results indicate that the shear deformation considered using TB theory has more significant impacts to high vibration modes.

Figure 5.

Bridge responses with different vehicle speeds using EB and TB theories. (a) For vehicle speed v = 10 m/s (b) For vehicle speed v = 20 m/s.

Table 3.

Natural frequencies corresponding to different axial load (rad/sec).

Frequency	Mode 1	Mode 2	Mode 3	Mode 4	Mode 5	Mode 6
TB	44.42	173.78	377.80	643.03	956.04	1305.07
EB	44.68	177.87	398.38	707.26	1107.84	1602.65
Difference (%)	0.58	2.30	5.17	9.08	13.70	18.57

Effect of bridge curvatures on dynamic responses of VBI systems

Exemplary planar curved beam bridges with uniform cross-section similar to the beam models in references (Mohamed et al., 2024; Paganelli, 2014; Ye et al., 2020; Zhong et al., 2021) are adopted to study the effect of bridge curvatures on dynamic responses of VBI systems. Four beam bridge cases with different curvature radius are considered, i.e., R0 = ∞ (Straight beam), R1 = 200 m, R2 = 100 m, R3 = 50 m, respectively. Figure 6 shows mid-span displacements of the bridges with different curvature radius subject to a vehicle with the moving speed of 5 m/s and 10 m/s, respectively. The results show that the maximum mid-span displacement of the bridge decreases with the curvature radius. The dynamic component of the bridge deflection is increased with the vehicle speed. Figure 7 shows the vertical displacement of the vehicle for different speeds and different curvature radius of beams. The vehicle displacement response is reduced with the decrease of bridge curvature radius. The local fluctuations of the vehicle responses are suppressed with a high vehicle speed.

Figure 6.

Bridge mid-span displacements considering different curvature radius. (a) For vehicle speed v = 5 m/s (b) for vehicle speed v = 10 m/s.

Figure 7.

Vehicle displacements considering different curvature radius. (a) For vehicle speed v = 5 m/s (b) for vehicle speed v = 10 m/s.

The observed reduction in bridge and vehicle displacement with decreasing curvature radius can be attributed primarily to the geometric stiffening effect introduced by vertical curvature. In a vertically curved beam, axial forces are coupled with bending deformation, resulting in an increased overall stiffness as curvature becomes more pronounced. This axial-bending coupling mechanism effectively enhances resistance to vertical deflection, especially near the mid-span where bending effects dominate. The influence of geometric nonlinearities is not explicitly captured in the current linear formulation, but the results suggest that curvature-induced stiffness plays a key role in moderating displacement responses.

Effect of temperature changes on dynamic responses of VBI systems

The thermal effects considering different temperatures on the vehicle-bridge dynamics are studied when the bridge curvature radius of 100 m is considered. The vehicle speed is set as 5 m/s and four thermal cases are studied. The first is the baseline study with no thermal loads. The second case is thermal expansion only. The temperatures at the top and bottom of the beam are the same as $T_{t} = T_{b} = 25 ℃$ , and the reference temperature is $30 ℃$ . The third case is thermal bending only. The temperatures at the top and bottom of the beam are $T_{t} = 20 ℃$ and $T_{b} = 25 ℃$ , respectively. The reference temperature is equal to the average temperature through the height of the beam cross-section that is $T_{0} = 22.5 ℃$ . The fourth case is the combined thermal expansion and thermal bending strain. The temperatures at the top and bottom of the beam are $T_{t} = 20 ℃$ and $T_{b} = 25 ℃$ , respectively, and the reference temperature is $T_{0} = 25 ℃$ . Figure 8(a) shows the bridge mid-span displacement responses under a moving vehicle and different thermal effects. The thermal effect due to the temperature change would cause the increase of the bridge mid-span displacement compared to the baseline case. The thermal expansion and combined thermal expansion and bending have more prominent effects on the bridge displacement responses. Figure 8(b) presents the bridge displacement due to thermal effects and Figure 8(c) shows the dynamic components of the bridge response due to vehicle load. From the figures, it can be seen that the thermal effects cause significant changes to bridge responses and there is only a slightly difference between the dynamic components of the bridge displacements. Figure 9 presents the vehicle displacement responses considering different thermal effects. In Figure 9, it is found that the features of thermal effects on the vehicle response are similar to that of the bridge response as shown in Figure 8(a).

Figure 8.

Bridge mid-span displacement considering different thermal effects. (a) Total displacement. (b) Displacement due to thermal effects (c) Displacement due to dynamic vehicle load.

Figure 9.

Vehicle displacement considering different thermal effects.

The numerical results reveal that thermal bending induces a pronounced increase in the bridge displacement and vehicle response. This distinction has important implications for bridge design and maintenance. Current bridge design codes typically address thermal effects primarily through expansion allowances, such as bearing movements, without explicitly accounting for differential thermal gradients that can cause bending. However, the results here indicate that thermal bending can lead to higher mid-span deflections and dynamic amplification, potentially affecting serviceability and fatigue performance. These findings highlight the need for bridge codes to consider both thermal expansion and bending effects in evaluating displacement limits and load factors, particularly for structures exposed to significant temperature gradients.

Effect of road surface roughness

A set of road surface roughness is presented in Figure 10. The road surface roughness is incorporated into the VBI model to study its effects on the system dynamics. Figure 11 presents the bridge mid-span displacement when the curvature radius of 100 m and combined thermal effect are considered. It can be seen that the road surface roughness causes local fluctuations of the displacement response compared that with smooth road surface. Figure 12 shows that there are large fluctuations in the vehicle response for the bridge with road surface roughness.

Figure 10.

A set of road surface roughness.

Figure 11.

Bridge mid-span displacement considering road surface roughness.

Figure 12.

Vehicle displacement considering road surface roughness.

Dynamic load coefficient and dynamic amplification factor

Dynamic impact of moving vehicles on bridges is an important and long-standing concern in the design and evaluation of bridges (Deng et al., 2015; Ding et al., 2009). Effects of the bridge curvature radius, the speed of the moving vehicle and the thermal strain on the dynamic amplification factor (DAF) and dynamic load coefficient (DLC) are analyzed. The DAF is defined in terms of the dynamic response of the bridge, as the ratio of the maximum bridge deflection $δ_{dyn}$ to the maximum static deflection $δ_{st}$ under static vehicle load, as $DAF = δ_{dyn} / δ_{st}$ . The DLC is defined in terms of the dynamic axle load, as the ratio of the maximum value of the axle load $P_{dyn}$ to the static axle load $P_{st}$ , as $DLC = P_{dyn} / P_{st}$ .The vehicle speed is changed from 5 to 30 m/s with an increment of 5 m/s. The curvature radius is selected from ∞ (Straight beam), R2 = 200 m, R3 = 100 m, and R4 = 50 m, and four thermal cases are considered. Figure 13 shows the DLC and Figure 14 shows the DAF considering the effects of curvature radius, vehicle speed and thermal loads. When no thermal load is considered, the DLC mainly increases along with the vehicle speed and the largest DLC value occurs when the curvature radius is the lowest. The DLC value decreases with increasing bridge curvature radius when only thermal expansion is considered, but increases when thermal bending is considered in the high speed range (15 m/s ∼ 30 m/s). However, when the combined thermal expansion and bending effects are considered, the DLC value is smaller when the beam is straight or the curvature radius is small (50 m), and it is larger when the curvature radius is 100 m and 200 m. When no thermal load is considered, the DAF value is fluctuated along with the vehicle speed and shows slight decrease when the vehicle speed is large. When the curvature radius of the bridge is 50 m, the DAF values are the lowest. When the thermal load is considered, the DAF value is changed with the curvature radius. The thermal expansion deflection of a beam with curvature of 1/50 is 20 times greater than that of a straight beam. On the contrary, when only the thermal bending is considered, the DAF value decreases with the curvature radius. When the thermal expansion or both thermal expansion and bending are considered, the DAF value increases when the curvature radius decreases.

Figure 13.

Thermal effects on DLC with different vehicle speeds.

Figure 14.

Thermal effects on DAF with different vehicle speeds.

It is important to note that the displacements and DAF values are computed based on a chosen reference temperature. This temperature serves as the baseline for evaluating deviations in thermal loading. As a result, the magnitude and direction of thermal expansion or bending are directly influenced by this reference point. If different reference temperatures were used (e.g., based on daily mean, seasonal average, or construction time conditions), the initial deformation state of the bridge could change, potentially altering the dynamic amplification characteristics. This highlights the need for considering the reference temperature in thermal analysis, especially for serviceability assessments and fatigue-sensitive design.

In the present study, pinned-pinned boundary conditions are employed to represent typical support configurations in simply supported bridges and to allow direct comparison with existing literature. While this assumption is commonly accepted for analytical purposes, actual bridge supports often display partial rotational restraint and flexibility due to elastomeric bearings. To incorporate more realistic boundary conditions, a sensitivity analysis is conducted using elastic supports modeled by transverse and rotational springs at both beam ends. The stiffness values for the vertical and rotational springs are adopted as $k_{s} = 2 \times 10^{8} N / m$ and $k_{θ} = 1 \times 10^{4} N / m$ , respectively (Ni and Hua, 2018). Two scenarios are considered: (1) elastic supports with only vertical springs, and (2) elastic supports with both vertical and rotational springs. Figure 15 presents the corresponding DAF values under combined thermal expansion and bending effects. The results show that the inclusion of vertical support springs generally leads to a reduction in DAF values, reflecting increased energy dissipation. The influence of rotational springs at the supports is found to be minor. Furthermore, the effect of elastic support conditions diminishes as the bridge curvature radius decreases, indicating that curvature-induced stiffness plays a dominant role in such configurations.

Figure 15.

Thermal effects on DAF considering elastic beam supports.

Conclusions

An advanced VBI model for vertically curved beam bridges is developed to include thermal effects through a linear temperature gradient. The bridge is modeled using Timoshenko beam theory to capture shear deformation and rotary inertia, while the vehicle is represented by a two-axle half-car. The uncoupled equations are solved iteratively via the zero-order displacement and velocity overshoot method. The existing results have been used to validate the proposed method. Some conclusions can be obtained as below.

(1) Numerical results highlight that the shear deformation has a significant effect on the vibration mode. Compared with the EB theory, the frequency difference is increased with the mode order.

(2) The results show that the vehicle and mid-span bridge displacements increases with the bridge curvature radius. This is due to the reduction of the bridge curvature radius induces the increase of the bridge stiffness. The thermal loads induce the bridge initial deflection and the dynamic deflection of the bridge under the moving vehicle does not have the obvious change.

(3) Thermal loads and the bridge curvature have significant effects on both DAF and DLC. The DAF value of the curved bridge due to the thermal expansion could be 20 times larger than that of the straight bridge. These findings provide practical insights for bridge design and maintenance under combined thermal and dynamic loading.

(4) The linear temperature gradient along the depth of the cross-section is considered in the study. Further study is needed to consider the extreme temperature environment with the nonlinear temperature gradient and the effect of the reference temperature is also needed to be considered.

Footnotes

Acknowledgments

The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No.:52108288) and China Postdoctoral Science Foundation (Grant No.: 2023M733407).

ORCID iDs

Jiantao Li

Jie Tan

Xinqun Zhu

Weidong Ruan

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by National Natural Science Foundation of China (Grant No.:52108288) and China Postdoctoral Science Foundation (Grant No.: 2023M733407).

Declaration of conflicting interests

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

References

Abdelrahman

Esen

Eltaher

(2021) Vibration response of timoshenko perforated microbeams under accelerating load and thermal environment. Applied Mathematics and Computation 407: 126307.

Al Rjoub

Hamad

(2020) Forced vibration of axially-loaded, multi-cracked euler-bernoulli and timoshenko beams. Structures 25: 370–385.

Attar

Karrech

Regenauer-Lieb

(2017) Dynamic response of cracked timoshenko beams on elastic foundations under moving harmonic loads. Journal of Vibration and Control 23(3): 432–457.

Caddemi

Calio

Cannizzaro

(2017) The dynamic stiffness matrix (DSM) of axially loaded multi-cracked frames. Mechanics Research Communications 84: 90–97.

Deng

Zou

, et al. (2015) State-of-the-art review of dynamic impact factors of highway bridges. Journal of Bridge Engineering 20(5): 04014080.

Deng

, et al. (2023) Genuine influence line and influence surface identification from measured bridge response considering vehicular wheel loads. Journal of Bridge Engineering 28(2): 04022145.

Ding

Hao

Zhu

(2009) Evaluation of dynamic vehicle axle loads on bridges with different surface conditions. Journal of Sound and Vibration 323: 826–848.

Ding

Kang

Zhang

, et al. (2023) Dynamic modeling and analysis on planar free vibration of long-span arch bridges during construction. Applied Mathematical Modelling 121: 843–864.

Esen

(2020) Dynamics of size-dependant timoshenko micro beams subjected to moving loads. International Journal of Mechanical Sciences 175: 105501.

10.

Esmailzadeh

Ghorashi

(1997) Vibration analysis of a timoshenko beam subjected to a travelling mass. Journal of Sound and Vibration 199(4): 615–628.

11.

Fazzolari

Carrera

(2014) Thermal stability of FGM sandwich plates under various through-the-thickness temperature distributions. Journal of Thermal Stresses 37(12): 1449–1481.

12.

Gao

Yang

, et al. (2022) Structure carrying moving subsystems with distributed viscoelastic coupling: part I-modeling and dynamics response. Acta Mechanica 233: 4467–4485.

13.

Green

Cebon

(1997) Dynamic interaction between heavy vehicles and highway bridges. Computers & Structures 62(2): 253–264.

14.

Hughes

TJR

Cohen

Haroun

(1978) Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engineering and Design 46(1): 203–222.

15.

ISO-8608 (1995) Mechanical Vibration-Road Surface Profiles-Reporting of Measured Data. Geneva: International Organization for Standardization (ISO).

16.

Jiang

(2010) Dynamics of a prestressed timoshenko beam subject to arbitrary external load. Journal of Zhejiang University - Science 11(11): 898–907.

17.

Kocatürk

Şimşek

(2006) Dynamic analysis of eccentrically prestressed viscoelastic timoshenko beams under a moving harmonic load. Computers & Structures 84(31): 2113–2127.

18.

Lei

Zheng

, et al. (2022) Dynamic response of slope inertia-based timoshenko beam under a moving load. Applied Sciences 12: 3045.

19.

Feng

(2023) A comparative study of vehicle-bridge interaction dynamics with 2D and 3D vehicle models. Engineering Structures 292: 116493.

20.

Feng

Chen

(2020) Moving load response of an axially loaded timoshenko beam on a multilayered transversely isotropic half-space comprising different media. International Journal for Numerical and Analytical Methods in Geomechanics 44(18): 2501–2523.

21.

Guo

Zhu

, et al. (2022a) Nonlinear characteristics of damaged bridges under moving loads using parameter optimization variational mode decomposition. Journal of Civil Structural Health Monitoring 12(5): 1009–1026.

22.

Zhu

Guo

(2022b) Bridge modal identification based on successive variational mode decomposition using a moving test vehicle. Advances in Structural Engineering 25(11): 2284–2300.

23.

Liu

Gao

Song

, et al. (2013) Interval dynamic response analysis of vehicle-bridge interaction system with uncertainty. Journal of Sound and Vibration 332(13): 3218–3231.

24.

Liu

Wang

Tan

, et al. (2018) Effect of temperature and spring-mass systems on modal properties of timoshenko concrete beam. Structural Engineering & Mechanics 65: 389–400.

25.

Lou

Dai

Zeng

(2006) Finite-element analysis for a timoshenko beam subjected to a moving mass. Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science 220: 669–678.

26.

Brilakis

Middleton

(2019) Detection of structural components in point clouds of existing RC bridges. Computer-Aided Civil and Infrastructure Engineering 34: 191–212.

27.

MacLeod

Arjomandi

(2023) Analytical vehicle-bridge interaction simulation using estimated modal parameters from ambient vibration tests. Journal of Bridge Engineering 28(9): 04023067.

28.

Moghaddas

Sedaghati

Esmailzadeh

, et al. (2009) Finite element analysis of a timoshenko beam traversed by a moving vehicle. Proceedings of the Institution of Mechanical Engineers - Part K: Journal of Multi-body Dynamics 223(3): 231–243.

29.

Mohamed

Eltaher

(2024) Nonlinear forced vibration of curved beam with nonlinear viscoelastic ends. International Journal of Applied Mechanics 16(03): 2450031.

30.

Hua

(2018) Axial-bending coupled vibration analysis of an axially-loaded stepped multi-layered beam with arbitrary boundary conditions. International Journal of Mechanical Sciences 138-139: 187–198.

31.

Paganelli

(2014) Dynamic Response of Structural Elements Undergoing Moving Loads and Thermal Strains Using Finite Elements. (M.Sc Dissertation). University of Minnesota.

32.

Papazafeiropoulos

Plevris

(2018) OpenSeismoMatlab: a new open-source software for strong ground motion data processing. Heliyon 4(9): e00784.

33.

Pirmoradian

Keshmiri

Karimpour

(2015) On the parametric excitation of a timoshenko beam due to intermittent passage of moving masses: instability and resonance analysis. Acta Mechanica 226(4): 1241–1253.

34.

Poojary

Roy

(2021) In plane radial vibration of uncracked and cracked circular curved beams subjected to moving loads. International Journal of Structural Stability and Dynamics 21(10): 2150146.

35.

Rostam

Javid

Esmailzadeh

, et al. (2015) Vibration suppression of curved beams traversed by off-center moving loads. Journal of Sound and Vibration 352(15): 1–15.

36.

Wang

(1997) Vibration of multi-span timoshenko beams to a moving force. Journal of Sound and Vibration 207: 731–742.

37.

Wang

(2016) Thermal effect on the dynamic response of axially functionally graded beam subjected to a moving harmonic load. Acta Astronautica 127: 171–181.

38.

Wang

Pan

Zhang

, et al. (2020) Vehicle–bridge interaction analysis by the state-space method and symplectic orthogonality. Archive of Applied Mechanics 90: 533–557.

39.

Chiang

(2003) Out-of-plane responses of a circular curved timoshenko beam due to a moving load. International Journal of Solids and Structures 40(26): 7425–7448.

40.

Chiang

(2004) Dynamic analysis of an arch due to a moving load. Journal of Sound and Vibration 269: 511–534.

41.

Yang

Chong

ACM

Lam

DCC

, et al. (2002) Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures 39(10): 2731–2743.

42.

Yang

Zhang

Wang

, et al. (2019) Two-axle test vehicle for bridges: theory and applications. International Journal of Mechanical Sciences 152: 51–62.

43.

Yang

Liu

, et al. (2024) Straight-beam approach for vibration analysis of horizontal curved beams. International Journal of Structural Stability and Dynamics 25: 2571002.

44.

Yavari

Nouri

Mofid

(2002) Discrete element analysis of dynamic response of timoshenko beams under moving mass. Advances in Engineering Software 33(3): 143–153.

45.

Mao

Ding

, et al. (2020) Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions. International Journal of Mechanical Sciences 168: 105294.

46.

Zhai

Cai

Zhu

(2023) Implementation of timoshenko curved beam into train-track-bridge dynamics modelling. International Journal of Mechanical Sciences 247: 108158.

47.

Zhang

Thompson

Sheng

(2020) Differences between euler-bernoulli and timoshenko beam formulations for calculating the effects of moving loads on a periodically supported beam. Journal of Sound and Vibration 481: 115432.

48.

Zhang

Chen

, et al. (2022) Development of the dynamic response of curved bridge deck pavement under vehicle-bridge interactions. International Journal of Structural Stability and Dynamics 22(11): 2241003.

49.

Zhong

Yang

Gao

(2015) Dynamic responses of prestressed bridge and vehicle through bridge–vehicle interaction analysis. Engineering Structures 87: 116–125.

50.

Zhong

Liu

, et al. (2021) In-plane dynamic instability of a shallow circular arch under a vertical-periodic uniformly distributed load along the arch axis. International Journal of Mechanical Sciences 189: 105973.

51.

Zhou

Tamma

(2004) Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics. International Journal for Numerical Methods in Engineering 59: 597–668.

52.

Zhou

Ling

Yin

, et al. (2024) Transfer matrix modeling for asymmetrically-nonuniform curved beams by beam-discrete strategies. International Journal of Mechanical Sciences 276: 109425.