Keywords

1 Introduction

DLA is one of the important aspects of the application research of vehicle bridge coupling vibration analysis. There are many factors affecting the DLA of bridges, and the stress mechanism is very complex, so it has attracted more and more attention of researchers.

The DLA of the bridge structure was first a model test conducted by British and French engineers in 1844 to study the dynamic performance and bridge carrying capacity of the vehicle-bridge system. Many experts and scholars have done a lot of studies from many aspects: G G.Stokes [1], K.P.Chatterjee [2], V Kolousek [3] analyzed the coupling response of the bridge under the moving load, and derive the vehicle-bridge coupling vibration equation of the vehicle load when the vehicle load passes through the bridge structure; Song Yifan [4] found that the road roughness of bridge deck is the most significant factor affecting the dynamic action of vehicles based on analysis method of vehicle vibration response caused by pavement roughness; Shi Shangwei [5] analyzed the trend of difference and causes between the measured value and standard value of the DLA of the girder bridge. Jiang Peiwen [6] calculated the time history response of long-span continuous girder bridge under the action of multiple vehicles, and summarized the variation laws of displacement DLA and bending moment DLA based on the vehicle bridge coupling calculation method in ANSYS single environment; Zhou Yongjun [7] comprehensively considered the impact effect of vehicle load on the whole bridge, weighted the DLA at all peaks on the time history curve, and fully included the impact effect at multiple positions of the bridge structure; Denglu [8] found that the strain dynamic amplification factor is basically less than the deflection dynamic amplification factor by using the methods of theoretical derivation and numerical simulation.

In conclusion, many experts and scholars have carried out many researches on the DLA of bridges, and have made fruitful research results. But most of them are only for displacement DLA, and few researches on bending moment DLA are involved. In this paper, the coupling vibration model of the vehicle and bridge is established by ANSYS. The displacement time history curve and bending moment time curve of the bridge under different working conditions are analyzed. The change law of displacement DLA and bending moment DLA of continuous girder bridge under different vehicle speed, vehicle weight and road surface road roughness are studied.

2 Solution Method of Vehicle-Bridge Interaction

Through the coordination relationship of the action force and displacement at the contact point, the equation of vehicle-bridge interaction is established as follows [9]:

$$ \left[ {\begin{array}{*{20}l} {M_{\text{b}} } \hfill & \, \hfill \\ \, \hfill & {M_{\text{v}} } \hfill \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\ddot{y}_{\text{b}} } \\ {\ddot{y}_{\text{v}} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}l} {C_{\text{b}} } \hfill & {C_{{\text{bv}}} } \hfill \\ {C_{{\text{vb}}} } \hfill & {C_{\text{v}} } \hfill \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\dot{y}_{\text{b}} } \\ {\dot{y}_{\text{v}} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}l} {K_{\text{b}} } \hfill & {K_{{\text{bv}}} } \hfill \\ {K_{{\text{vb}}} } \hfill & {K_{\text{v}} } \hfill \\ \end{array} } \right]\left\{ {\begin{array}{*{20}l} {y_{\text{b}} } \hfill \\ {y_{\text{v}} } \hfill \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {F_{{\text{br}}} + F_{{\text{vg}}} } \\ {F_{{\text{vr}}} } \\ \end{array} } \right\} $$
(1)

Where \({{\varvec{M}}},{{\varvec{C}}},{{\varvec{K}}}\) are the quality, damping and stiffness matrix of the vehicle and bridge respectively; \({{\varvec{d}}}\) is the vertical displacement, \({{\varvec{b}}}\,{\text{and}}\,{{\varvec{v}}}\) are the bridge and vehicle respectively; \({{\varvec{F}}}_{{{\varvec{br}}}}\) and \({{\varvec{F}}}_{{{\varvec{vr}}}}\) are the interaction force of the vehicle-bridge system respectively, the subscript \({{\varvec{r}}}\) represents the road roughness, and \({{\varvec{F}}}_{{\bf{vg}}}\) represents the vehicle gravity.

Based on the above principle, the time-variation equation is solved by the Newmark-\(\beta\) method. Using the transient analysis function of large-scale finite element program ANSYS, the vehicle bridge coupling vibration model is established by APDL language and displacement contact method, and the time-varying equation is solved by direct integration method.

3 Bridge, Vehicle and Road Roughness Model

3.1 Bridge Model

In order to study the displacement DLA and bending moment DLA of continuous girder bridge, 2 × 30 m continuous girder is selected the research object, the material is C50 concrete, and its section form is shown in Fig. 1.

Fig. 1.
figure 1

Cross section of bridge (unit: cm)

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3.2 Vehicle Model

1/2 vehicle model is selected. The vehicle layout diagram is shown in the Fig. 2, detailed parameters reference literature [10].

Fig. 2.
figure 2

1/2 Vehicle model

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3.3 Road Roughness Model

Road roughness is a very important excitation source of vehicle bridge coupling vibration. In practice, the statistical characteristics of road roughness are usually described by power spectrum, and the power spectral density of pavement roughness can be fitted as

$$ G_x (n) = G_x (n_0 )\left( {\frac{n}{n_0 }} \right)^{ - \omega } $$
(2)

Where \(n\) is the spatial frequency; \({{\varvec{n}}}_0\) is the reference spatial frequency; \({{\varvec{G}}}_{{\varvec{x}}} \left( {{\varvec{n}}} \right)\) is the displacement power spectral density value, \({{\varvec{G}}}_{{\varvec{x}}} \left( {{{\varvec{n}}}_0 } \right)\) is the coefficient of road roughness; \({{\varvec{\omega}}}\) is the frequency index, \({{\varvec{\omega}}}\) = 2.

4 The Method of DLA Calculation and the Working Condition Layout

4.1 Working Condition Layout

2 × 30 m continuous girder bridge is used for analysis. The effects of vehicle speed, vehicle weight and road roughness on the DLA of continuous girder bridge are considered. Among them, five speeds, five vehicle weights and three grades of road roughness are considered respectively. The specific working conditions are shown in the Table 1:

Table 1. Working conditions
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4.2 Calculation Method of the DLA

The DLA \(\left( \mu \right)\) is defined under the current specification:

$$ \mu = \frac{{f_{d.\max } }}{{f_{s.\max } }} - 1 $$
(3)

Where \({{\varvec{f}}}_{{{\varvec{d}}}.{{\varvec{max}}}}\) is the maximum dynamic response of the vehicle when passing the bridge; \({{\varvec{f}}}_{{{\varvec{s}}}.{{\varvec{max}}}}\) is the maximum static response corresponding to the bridge structure of the same vehicle. \({{\varvec{\mu}}}_{{\varvec{d}}}\) is the displacement DLA, and \({{\varvec{\mu}}}_{{\varvec{M}}}\) is the bending moment DLA.

5 Results Analysis

Based on the established calculation model and considering various influencing factors in the Table 1, the vehicle-bridge interaction calculation is carried out, and the effects of different grades of road roughness, vehicle speed and vehicle body weight on the displacement DLA and bending moment DLA in the mid-span of the first span of continuous girder bridge are analyzed. The relevant results are analyzed as follows.

5.1 Influence of Vehicle Speed on the Displacement DLA and Bending Moment DLA of the Continuous Girder Bridge

In order to study the influence of vehicle speed on the displacement DLA and bending moment, the road roughness is not considered in the analysis. The analysis results are shown in the following Figs. 3, 4 and 5:

Fig. 3.
figure 3

Displacement curve at different speeds of smooth pavement

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Fig. 4.
figure 4

Bending moment curve at different speeds of smooth pavement

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Fig. 5.
figure 5

Comparison of displacement DLA and bending moment DLA at different speeds

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The vehicle speed has a significant impact on the displacement DLA and the bending moment DLA of the continuous girder bridge. With the increase of the vehicle speed, the change rules of the displacement DLA and the bending moment DLA are different. And the maximum value of the bending moment DLA is less than the maximum value of the displacement DLA. At the same time, the difference between the value of the displacement DLA and the value of the bending moment DLA changes with the increase of the vehicle speed. And the value of the former is about double that of the latter.

5.2 Influence of the Vehicle Body Weight on the Displacement DLA and Bending Moment DLA of the Continuous Girder Bridge

In order to study the influence of vehicle body weight on the displacement DLA and bending moment DLA of continuous girder bridge, the influence of road roughness is not considered in the analysis process, and the vehicle speed is set as v = 20 m/s. The analysis results are shown in the Fig. 6:

The weight of vehicle body has a significant impact on the displacement DLA and bending moment DLA of continuous girder bridge. With the increase of vehicle body weight, the variation law of displacement DLA and bending moment DLA is the same, showing a gradual increasing trend, and the displacement DLA is greater than the bending moment DLA. And the value of the former is about triple that of the latter.

Fig. 6.
figure 6

Comparison between displacement DLA and bending moment DLA under different vehicle body weight

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Fig. 7.
figure 7

Comparison between displacement DLA and bending moment DLA under different road roughness grades

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5.3 Influence of Road Roughness on the Displacement DLA and Bending Moment DLA of the Continuous Girder Bridge

In order to study the impact of road roughness on displacement DLA and bending moment DLA of continuous girder bridge, the vehicle speed v = 20 m/s during the analysis. In order to avoid the influence of the randomness of the road roughness samples on the results, the DAL under 15 randomly generated road roughness samples is calculated for each working condition, and then the average value is calculated. The analysis results are shown in the Fig. 7.

The road roughness has a significant influence on the displacement DLA and bending moment DLA of the continuous girder bridge. Compared with grade A road roughness, the value of DLA under grade C road roughness is about 9 times that under grade A road roughness. With the increase of the grade of road roughness, the displacement DLA and bending moment DLA both increase gradually. At the same time, displacement DLA is slightly larger than bending moment DLA for the same grade of road roughness. And the difference between them increases with the increase of the grade of road roughness. The maximum difference between them is ∆ = 0.016.

6 Conclusion

Through the calculation and analysis of the finite element model, the influence of vehicle speed, vehicle body weight and road roughness on the displacement DLA and bending moment DLA of the continuous girder bridge are studied. The conclusion of comparative analysis as follows:

  1. (1)

    The vehicle speed has a certain impact on the displacement DLA and bending moment DLA of the continuous girder bridge. With the increase of the vehicle speed, the displacement DLA gradually increases, and bending moment DLA first increases and then decreases.

  2. (2)

    The vehicle body weight has a certain impact on the displacement DLA and bending moment DLA of the continuous girder bridge. With the increase of the weight of the vehicle body, the displacement DLA and bending moment DLA gradually increase.

  3. (3)

    The road roughness has a significant impact on the displacement DLA and bending moment DLA of continuous girder bridge. With the increase of the grade of road roughness, the displacement DLA and bending moment DLA gradually increases.

  4. (4)

    Compared with the displacement DLA and bending moment DLA, there is a numerical difference between them. And the value of the displacement DLA is slightly larger than the value of the bending moment DLA. It is suggested to distinguish the displacement DLA and bending moment DLA in engineering design and dynamic load test.