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The cable-stayed bridge (CSB) is often used to span over the large rivers on the highway with a high-level navigational clearance; however, CSB is very sensitive to live load. Most of the previous studies on vibration analysis of CSB that focus on complex traffic loading and vehicle dynamic interaction as well as on the bridge deck do not consider braking effects thoroughly. In this paper, the finite element method (FEM) is used to investigate the dynamic response of CSB due to a three-axle vehicle considering braking effects. Vertical reaction forces of axles that change with time make bending vibration of the bridge deck increase significantly. The braking in a span is able to create response in other spans, towers, and cables. In addition, the impact factors are investigated on both FEM and experiment with a case study of Pho Nam bridge (Danang city, Central Vietnam). The results of this study provide an improved understanding of the CSB dynamic behaviors, and they can be used as useful references for bridge codes by practicing engineers.

Researchers have studied the response of bridges subjected to vehicles moving since the 50s of the 19th century. Recently, the previous study has analyzed the complex problem of oscillations with the model interaction between vehicle and bridge which is closer to the reality. Most studies focus on the dynamic behavior of bridge subjected to vehicles moving with constant speeds. Only a few research directions towards the dynamic behavior of the bridge under the effect of vehicle with variable velocities (due to many reasons like brakes, incident on the bridge, etc.). Fryba [

Although some good researches as discussed above can be found in overall bridge dynamics, very few researches have been conducted related to the dynamic interaction in the cable-stayed bridge with considering braking effects. This paper presents the results of the dynamic response of cable-stayed bridge subjected to dynamic wheel loads by FEM analysis and experimental investigation. The vibration of the cable-stayed bridge was analyzed with various vehicle speeds considering the braking effects varied to assess the amount of interaction between the vehicle and the cable-stayed bridge.

The main objective of the test is to validate a calculation procedure for determination of the Impact Factor (IF) of cable-stayed bridge to dynamic wheel loads due to vehicle speed and braking effects. There are many definitions for IF or (

Dynamic and static displacement under a vehicle moving.

The cable-stayed bridge used in this study is the Pho Nam bridge located in Danang, Vietnam, which is three-span steel cable-stayed bridge with a main span of 80 m and a side span of 35.7 m. The three-axle vehicle moving on the bridge is shown in Figure

A three-axle vehicle move on the Pho Nam bridge.

Stayed cables were modeled as cable elements with considering tensional force and deflection of the cables. The tower structures of the cable-stayed bridge were modeled as frame elements which can be found in Zienkiewicz and Taylor [

The dynamic interaction model between a three-axle vehicle and beam element.

Inertial forces, damping forces, elastic forces, stimulating forces, and braking forces of system are shown in Figure

The following assumptions are adopted:

The mass of the vehicle, excluding the mass of the axles, is transferred to the mass center of the system. It is equivalent to the mass

The mass of the 1st axle is

The chassis is assumed to be absolutely rigid. The materials of beam are linear elastic stage. The bridge surface has the homogeneous friction coefficient over the entire bridge surface.

Braking forces of the axles of vehicle are assumed to occur simultaneously. The forces direction between bridge surface and tires is assumed to be in the opposite direction of the moving vehicle as shown in Figure

When the vehicle is suddenly braked, the friction forces

Based on the calculation model and assumptions in Section

According to the study Ray and Joseph [

The differential equation of longitudinal motion for a beam element due to uniform loading

The Galerkin method and Green theory are applied to (

Apply the FEM and the algorithm of the FEM can be found in Zienkiewicz and Taylor [

After imposing boundary and initial conditions on (

^{2}; number 2, 3, 8, 9, 12, 13, 18, and 19: ^{2}; No. 1, 10, 11, 20: ^{2}.

The cables plane of the Pho Nam bridge.

^{9} kN/m^{2}, ^{4}, ^{2}, ^{2},

^{9} kN/m^{2}, ^{4}, ^{2}, and

Consider the part T2: ^{9} kN/m^{2}, ^{4}, ^{2}, and

The three-axle vehicle used in the FEM investigation is KAMAZ-55111 (Russia) dumper truck as shown in Figure

The three-axle vehicle dimensional parameters.

The (

The FEM model of the Pho Nam bridge.

Variation of IF in terms of vertical deflections at 18 km/h vehicle speed considering braking.

Variation of IF in terms of vertical deflections at 36 km/h vehicle speed considering braking.

Variation of IF in terms of vertical deflections at 54 km/h vehicle speed considering braking.

Variation of IF in terms of vertical deflections at 72 km/h vehicle speed considering braking.

Figure

Figure

Figure

Figure

The IFs are evaluated at various points at nodes 4, 7, 8, 9, 24, 29, 39, and 40 in terms of axial displacement for vehicle speeds of 18 km/h, 36 km/h, 54 km/h, and 72 km/h and used sudden braking that are shown in Figures

Variation of IF in terms of axial displacements at 18 km/h vehicle speed considering braking.

Variation of IF in terms of axial displacements at 36 km/h vehicle speed considering braking.

Variation of IF in terms of axial displacements at 54 km/h vehicle speed considering braking.

Variation of IF in terms of axial displacements at 72 km/h vehicle speed considering braking.

Figure

Figure

Figure

Figure

The IFs are evaluated at various points at nodes 4, 7, 8, 9, 24, 29, 39, and 40 in terms of rotation angle for vehicle speeds of 18 km/h, 36 km/h, 54 km/h, and 72 km/h and the results in cases of sudden brakes are used that are shown in Figures

Variation of IF in terms of angular displacements at 18 km/h vehicle speed considering braking.

Variation of IF in terms of angular displacements at 36 km/h vehicle speed considering braking.

Variation of IF in terms of angular displacements at 54 km/h vehicle speed considering braking.

Figure

Figure

Figure

Figure

Variation of IF in terms of angular displacements at 72 km/h vehicle speed considering braking.

In most cases, the IFs increase with the increments in vehicle speeds. The maximum value of the average IFs is 1.44 at the vehicle speed of 72 km/h with braking effect in terms of element vertical deflection.

The Pho Nam bridge, a one lane bridge over the Cu De river, is shown in Figure

Detail of the Pho Nam bridge.

Pho Nam bridge

Elevation plan

Cross-section of double I-girder

Instrumentation for measuring deflections was installed at specified locations prior to testing. The vertical deflections were measured, with linear variable displacement transducers (LVDTs), designed to provide displacement measurements. The LVDTs used for the dynamic tests were CDP–50 and CDP–100 gages manufactured by Tokyo Sokki Kenkyujo Co., Ltd., Japan, and shown in Figure

The instruments of testing.

The data record system

CDP–50 and CDP–100 displacement transducer

The test vehicle for the Pho Nam bridge is a KAMAZ-55111 dumper truck with three axles and the vehicle weight is 8560 kg with leaf spring suspension on the steering axle and the tandem rear axle (Figure

The KAMAZ-55111 dumper truck.

The experimental results of IFs at points 1, 2, 3, and 4 on the Pho Nam bridge (Figure

IF versus velocity at sudden braking.

IF versus velocity, sudden braking at point 1

IF versus velocity, sudden braking at point 2

IF versus velocity, sudden braking at point 3

IF versus velocity, sudden braking at point 4

IF versus velocity, sudden braking at all points 1, 2, 3, and 4

IF versus braking position.

IF versus braking position,

IF versus braking position,

IF versus braking position,

IF versus braking position,

IF versus braking position,

IF versus braking position,

IF versus braking position,

IF versus braking position,

Through experimental results and FEM analysis of IFs at positions 1, 2, 3, and 4 of the Pho Nam bridge shown in Figures

The uptrend and the downtrend of the IFs are found in experimental investigation and FEM analysis is similarity.

In velocity range of 10–40 km/h, the IF increases with an increase in velocity at sudden braking; the variations of IF with velocity at sudden braking are shown in Figures

In the limits of velocity, the IF decreases with increasing the distance from the bearing location; the variations of IF with braking positions are shown in Figures

In this paper, the FEM is used to investigate the dynamic response of CSB due to a three-axle vehicle considering braking effects. The overall findings with higher vehicle speed offer to take a standard IF or (

In addition, the impact factors are investigated on both FEM analysis and experiment study on the Pho Nam bridge. In velocity range of 10–40 km/h, the IF increases with an increase in velocity at sudden braking and the IF decreases with increasing the distance from the bearing location. Accordingly, the authors recommend that in bridge design, engineers should take into account the dynamic interaction caused by the vehicle moving on bridge and the sudden braking.

The authors declare that there are no conflicts of interest regarding the publication of this article.

The research described in this paper was financially supported by the fund for scientific research of Vietnam’s Ministry of Education and Training.