Dynamic Response Analysis of Overpass Ramp Based on Grey System Theory Model

Abstract

An interchange is a pivotal traffic facility that connects highways and controls access. It is necessary to study their dynamic response characteristics to analyze the operational safety of ramp bridges on interchanges. Based on the numerical simulation results of the finite element model of the Fuxing Interchange Bridge, non-destructive measurement techniques were used to conduct field dynamic load tests on the bridge, including ramp strain testing and acceleration testing. These tests aimed to study the dynamic response characteristics of the ramp bridge under moving loads. Due to the design speed limitation of the ramp bridge, the grey prediction GM(1, 1) model was used to predict the maximum dynamic deflection, maximum dynamic strain, and vibration acceleration when the vehicle speed was 60 km/h. Subsequently, finite element software was used to simulate the dynamic deflection under vehicle speeds ranging from 30 to 60 km/h. The simulated value was compared with the predicted value, and the difference between the simulated value and the predicted value was slight. This model can evaluate the operational safety performance of off-ramps at different speeds.

1. Introduction

Against the backdrop of rapid national economic development, the demand for transportation facilities is gradually increasing [1,2]. Interchanges are crucial transportation facilities that serve as hubs for connecting highways and controlling access, occupying an extremely important position in the road network. After many interchanges are built, traffic congestion on interchanges occurs frequently, leading to a high incidence of traffic accidents and prominent hidden safety hazards [3]. In recent decades, structural health monitoring (SHM) has been widely used in bridge engineering [4,5], with research focusing on structural response evaluations, load effect monitoring, and reliability-based assessments, among other aspects [6]. Bridge load testing is one of the important steps that cannot be ignored when assessing the bearing capacity performance of bridges. It involves testing data such as the stress, strain, displacement, and acceleration at specified locations of the bridge structure by utilizing the deflection and vibration of the structure caused by the applied load. Depending on the loading method, it can be divided into static load testing and dynamic load testing [7]. Both newly built bridges and those repaired or strengthened need to undergo bridge load testing to verify whether their bearing capacity meets the requirements before they can be put into use [8].

Zhang [9] conducted a dynamic load test to examine the health status of the bridge span structure and its forced vibration characteristics under the dynamic load of vehicles, thereby assessing the dynamic performance of the bridge span structure. In order to explore the application of dynamic load testing in highway bridge health monitoring, Qi [10] used the Yili River Bridge as the engineering background and analyzed the dynamic performance of the bridge structure under pulsating periodic loads and barrier-free traffic loads. The test results indicated that the bridge span structure was in an elastic working state, and its stiffness and carrying capacity met the design requirements. To improve the reliability of bridge health assessments, Du [11] proposed a method for assessing bridge health based on static and dynamic load testing. This method involves adjusting the assessment and testing control interface, setting up loading vehicles, arranging static and dynamic load measurement points, and conducting separate static and dynamic load test analyses. Finite element simulation analyses are used to obtain test results, ultimately completing the assessment of bridge health through static and dynamic load testing. The results indicate that this method can effectively calculate the health assessment verification coefficient and impact coefficient of the bridge, and complete the finite element assessment analysis according to relevant standards. It demonstrates a high sensitivity and effectively improves the assessment reliability, providing a certain reference for solving issues related to bridge operation and maintenance. To address the issue of the lack of mass normalization in modal shapes during an operational modal analysis (OMA), Carbonari et al. [12] introduced trucks as additional masses during standard static proof load tests for bridges. By incorporating a finite element model, they studied the dynamics of bridges with parked trucks to address the dynamic coupling between bridges and vehicles. Subsequently, a simple, yet effective, two-degrees-of-freedom system was introduced. Finally, numerical applications were conducted to determine the proposed method’s effectiveness and suitability for practical applications. Sheibani et al. [13] proposed utilizing the dynamic effects of traffic vehicles crossing bridges as the required perturbation to the structural modal properties. Various traffic conditions were considered and uncertainty analyses were conducted to evaluate the suitability of loads in different traffic scenarios. A numerical case study was performed using the Hafez Bridge in Tehran, Iran, as an example. Actual traffic conditions were simulated using the traffic micro-simulation method, and vehicle forces and masses were applied to the finite element model of the bridge using the equivalent nodal force method. An operational modal analysis (OMA) was conducted in the time domain, and modal parameters were extracted using the stochastic subspace identification method. Finally, the study achieved a satisfactory accuracy in estimating modal scaling factors while avoiding the practical difficulties encountered by the traditional mass change method in real-world applications.

Grey theory effectively uncovers and summarizes internal laws and makes predictions when data and information are scarce and uncertain factors exist. The existing prediction methods for bridge deflection mainly include grey theory, neural networks, and Kalman filtering. Among them, grey theory has received widespread attention due to its simple principles and ease of implementation. Based on grey system theory, Liu et al. [14] addressed the mismatch between traditional parameter estimation methods and accuracy validation criteria for the GM(1, 1) model in previous construction control applications. He transformed the parameter estimation problem under the criterion of minimizing the average relative error into a linear programming problem, implemented through Python programming. Consequently, a new theoretical model was obtained for deflection predictions. Using a bridge numbered 1 in Shaanxi Province as the engineering background, Liu predicted the deflection values of the beam ends after concrete pouring and prestressing tension. Based on the relevant principles of grey system theory, Yang et al. [15] and his colleagues conducted a modeling analysis using the GM(1, 1) model with mathematical-statistical methods, relying on a bridge currently in operation as the engineering backdrop. They compared and analyzed the model results with the measured data and conducted relevant tests. The results indicated that all of the GM(1, 1) model indicators met the first-level accuracy requirements. Based on grey model theory, Yan [16] established a grey GM(1, 1) model to predict the changes in deflection during bridge construction and compared the predictions with actual measured values. The conclusions indicated that the grey prediction model can effectively predict the changes in deflection during construction, providing a reference for the next construction phase. Yuan et al. [17] applied grey system theory to establish a GM(1, 1) model using the numerical simulation results and field-measured results of deflection and stress under different working conditions during the actual construction process of a continuous rigid frame bridge as the original differential sequence. Using Matlab as the calculation tool, they developed a corresponding computational program to predict the changes in deflection and stress during cantilever construction. The results showed that the predictions agreed with the actual field test results.

The above information mainly discusses load testing for straight bridges, but there are a few research projects on load testing for interchanges, including ramp bridges. The grey prediction model only requires a small amount of data—typically, only four data points are needed for the modeling and prediction—making it particularly suitable for situations where historical data are scarce or difficult to obtain. This model boasts a high prediction accuracy and simple computation and can be integrated with other prediction models (such as neural networks). This study proposes a safety prediction method for interchanges based on grey prediction and conducts experimental research on a modal analysis, displacement, and strain. According to the requirements in the specifications [18], the monitoring sections were determined, and the measuring points were reasonably arranged. The loading test scheme is introduced in detail. The Fuxing Interchange Bridge was selected as an example, and the applicability of the proposed method is illustrated. The results of this study can provide a scientific basis for the safety monitoring research of interchanges.

2. Contents of Dynamic Load Test

2.1. Project Overview

This project intersects with the Ji-Hei Expressway, which is currently under construction near Fuxing Village in Changbao Town, north of Wuchang City. The B ramp bridge consists of two unequal-span steel box girders. The first span of the bridge is 48 m long, the second span is 34 m long, and the total width of the bridge is 10.5 m, with two one-way lanes. The bridge layout is shown in Figure 1.

After conducting a field survey of the B ramp bridge, its overall appearance is shown in Figure 2 below:

2.2. Dynamic Load Test

2.2.1. Purpose of Dynamic Load Test

The bridge structure must not only bear its own weight and the additional static loads placed on it, but also withstand moving loads such as pedestrian traffic and vehicular traffic. Vibrations in bridges can be caused by various factors, such as pedestrian traffic, vehicular traffic, wind dynamics, earthquakes, environmental factors, and the coupling effects of these factors. Bridge vibrations themselves are not dreadful; any structure will vibrate under exciting loads. However, when the frequency of the exciting load matches the natural vibration frequency of the bridge, resonance will occur, which can lead to structural collapse. Therefore, conducting dynamic load testing on bridges is one of the essential tests to assess whether a bridge meets the designed load-bearing capacity requirements [19].

2.2.2. Dynamic Load Test Conditions

The main testing contents of bridge dynamic load testing include the bridge structure’s dynamic response parameters and the bridge structure’s natural vibration characteristic parameters [20]. According to the standard [18], the working conditions of this test are shown in Table 1, including the pulsation test, sports car test, jump test, and brake test. The test vehicle was a 15t three-axle standard vehicle.

2.2.3. Monitoring Section

Analyzing the collected data can reveal the location and number of vulnerable points in the bridge structure. Based on the relevant regulations of the highway bridge load test specifications [18] and the actual situation at the site, the mid-span of the second span was selected as the monitoring section, and sensors were arranged in this section. The location of the monitoring section is shown in Figure 3a.

2.3. Test Equipment

The selected sensors were mainly acceleration sensors and surface strain gauges based on their sensitivity, accuracy, economic practicability, and other factors combined with short-term monitoring and the bridge’s actual requirements.

2.3.1. Strain Gauge

In bridge damage, stress can easily cause cracking in some bridge components, which impacts the bridge’s safety. Therefore, it is necessary to monitor the magnitude of stress constantly. In practical engineering, it is generally not possible to directly measure the stress of materials, so strain is used to further reflect the stress. In this test, four strain sensors were arranged at the mid-span of the second span of the ramp bridge. The selected strain measurement instrument was the JMZX-212HAT surface chord-type strain gauge. The specific arrangement of measuring points for the strain gauges is shown in Figure 3b.

2.3.2. Magnetoelectric Speed Sensor

Changes in a bridge’s dynamic performance reflect alterations in its stiffness. Monitoring the main girder and vibrations can identify the dynamic characteristic parameters of the bridge structure, enabling the recording of the history of fluctuating loads borne by the main girder structure [22]. A magnetoelectric speed sensor can be utilized to monitor the vibration characteristics of the bridges and investigate the impact of vehicle speed on the vibration acceleration of highway bridges. The JMCZ-2081 magnetoelectric velocity sensor was used for ultra-low- or low-frequency vibration measurements, as shown in Figure 4. It was primarily used for pulsating measurements of the ground and structures, general industrial vibration measurements of structures, ultra-low-frequency and large-amplitude measurements of highly flexible structures, and weak vibration measurements. During installation, the mounting baseplate of the sensor was welded onto the surface of the structure to be measured, and then M6 screws were used to fix the sensor and protective cover together onto the mounting baseplate.

2.3.3. Strain Acquisition Module

The steel chord dynamic strain acquisition instrument is an 8-channel automatic synchronous acquisition device, as shown in Figure 5a. It can collect signals from vibrating wire sensors, soil pressure cells, anchor cable meters, and inductance frequency modulation-type sensors. Figure 5b shows its workflow.

2.3.4. Acceleration Acquisition Module

The acceleration signal acquisition system comprises the 8-channel dynamic signal conditioning module and the 8-channel dynamic signal acquisition module, as shown in Figure 6. It can effectively and accurately acquire and process structural vibration data. It is characterized by its small size, ease of use, and convenience for installation.

2.4. Finite Element Model

MIDAS CIVIL NX was used to create an overall finite element model (FEM) of the bridge. This model possesses linear and nonlinear analysis capabilities [23]. The CIVIL finite element model of the two-span B ramp bridge is shown in Figure 7.

The main girder was simulated using beam elements, with 63 elements. The material parameters were determined according to the design drawings, and Q355qE steel was adopted for the main structures, such as the top and bottom plates, webs, diaphragms, stiffeners, and other components of the steel box girder. The top joint of the support was rigidly connected to the main beam joint, the bottom joint of the support was fixedly connected, and an elastic connection was adopted between the top and bottom of the support. The stress analysis of this model was divided into stress conditions under a dead load, live load, temperature load, and combined loads. This paper considered the self-weight and the second-stage dead load to analyze the dead load. The moving vehicle load was simulated using the lane load, with two lanes loaded according to the actual bridge conditions. The temperature load included the overall temperature increase and overall temperature decrease. The self-weight and secondary loads were converted to node masses and the deck pavement, guardrails, and other deck system components were approximated as linear loads for loading without participating in the structural forces. The vehicle loads were arranged according to the specifications for highway bridges. The parameters obtained from the finite element model calculations, such as the mid-span deflection, strain, vibration frequency, and vibration time history of the bridge segment, were used as theoretical values. A comparative analysis was conducted with the measured values from field tests to evaluate the dynamic performance of the tested bridge.

3. Analysis of Experimental Results

3.1. Analysis of Bridge Dynamic Response at Different Speeds

In the test, a 15-ton three-axle standard vehicle was used as a concentrated moving load applied to the highway bridge. The vehicle was driven over the bridge at speeds of 20 km/h, 30 km/h, 40 km/h, and 50 km/h. The dynamic responses of displacement, strain, and vibration acceleration at the mid-span of the second span were analyzed. Field testing measured the dynamic response values at the bridge’s test section at different speeds, as shown in Table 2.

Table 2 shows that, within the driving speed range of 20 to 50 km/h, the dynamic strain and dynamic deflection caused by vehicle-induced vibrations were generally minor. As the vehicle speed gradually increased, there was a clear trend of increasing the dynamic deflection, dynamic strain, and vibration acceleration at the mid-span measurement point. Under the action of moving vehicle loads, the maximum vertical displacement value and strain of the bridge met the design requirements specified in the relevant codes [24]. According to the “Specification for Inspection of Railway Bridges” in China, the maximum vertical vibration acceleration value should not exceed 3.43 m/s², indicating that the vibration acceleration of the bridge met the specification requirements when the speed was not higher than 50 km/h.

3.2. The Influence of Speed on the Impact Coefficient

The impact coefficient is the vertical dynamic amplification factor generated when a vehicle passes over a bridge structure. It is a factor that accounts for the increase in vertical dynamic effects on the bridge structure due to moving vehicles, and it is used to calculate the total vertical vehicle load effect on the bridge structure. The impact coefficient can be obtained through an analysis and a calculation of bridge dynamic load testing. In dynamic load testing, a test vehicle is driven at a constant speed over the bridge to excite the bridge structure being tested. Dynamic change signals, such as dynamic deflection and dynamic strain, generated on the bridge structure are detected using elements such as velocity sensors, strain sensors, and electromechanical dial indicators placed on the bridge. Based on the measured dynamic strain and dynamic deflection curves obtained from field testing, the impact coefficient of dynamic loads on the bridge structure can be analyzed and calculated [25]. The calculation of the impact coefficient is shown in Equation (1) [26].

(1)$\mathsf{\mu }={\displaystyle \frac{f_{d\max }}{f_{j\max }}}$

where f_dmax is the maximum dynamic deflection value or dynamic strain value at the measurement point under dynamic loading, while f_jmax is the maximum deflection value or strain value at the measurement point under the corresponding static loading. The peak value of the centerline trace of the waveform amplitude was obtained or derived through low-pass filtering.

The vehicle’s speed was varied, with a fixed curvature radius of 270 m and a vehicle weight of 15 tons. Under excitation from vehicle speeds of 20 km/h, 30 km/h, 40 km/h, and 50 km/h, dynamic strain data were collected from the bottom of the monitored cross-section of the bridge structure. Using the formula above, the impact coefficient was calculated. The impact coefficients under excitation from different vehicle speeds are shown in Figure 8.

An analysis of the test results for the impact coefficient under excitation from different vehicle speeds reveals that the higher the vehicle speed, the larger the impact coefficient. The measured impact coefficient values above met the specifications for the inspection and assessment of a highway bridge carrying capacity, indicating that the bridge deck has a good smoothness and that the impact effect of vehicles on the structure is relatively small.

3.3. Relationship Between Vehicle Driving Radius and Impact Coefficient

Vehicles traveling at 20 km/h were directed to cross the bridge via Lane 1, the middle of the bridge deck, or Lane 2. The radius of the left lane was 265 m, the radius of the centerline of the bridge deck was 270 m, and the radius of the right lane was 275 m. With a vehicle weight of 15 tons, the dynamic response at the mid-span of the second span was extracted to calculate the impact coefficient. The calculation results are shown in Figure 9.

As the driving radius R increased from 265 m to 275 m, it can be observed from Figure 9 that the vertical deflection impact coefficients at the midpoints of the side spans were roughly symmetrical. When the driving radius was smaller, the impact coefficient was the largest. The above data indicate that, when vehicles are driving on the inner side of the bridge, the impact on the vertical deflection impact coefficient is the greatest. A smaller driving radius of the vehicle may increase the impact on the bridge, increasing the impact coefficient.

3.4. Vibration Response Analysis of Car Jump

During this test, a loaded truck (15 tons) was used to conduct a hop test with its rear wheels at the midpoint of the second span. The test method involved the test vehicle jumping over a 10 cm high bumper plate and performing a vertical vibration assignment analysis and a vibration response analysis on the measurement points. The measured vibration response at the measurement point at the midpoint of the second span during the hop test with the test vehicle’s rear wheels is shown in Figure 10.

The large vibration amplitude during the hop test indicated that the bridge structure has a poor driving capacity when the smoothness of the bridge deck pavement layer is poor.

3.5. Brake Test Analysis

During this test, loaded vehicles were used to perform emergency braking at speeds of 20 km/h, 30 km/h, and 40 km/h at the test section of the bridge. The longitudinal acceleration vibration response of the bridge is shown in Figure 11.

Within the speed range of 20 to 40 km/h, as the vehicle speed gradually increased, there was a clear trend of the dynamic strain and vibration acceleration increasing at the midpoint measurement point, caused by the dynamic excitation of the running vehicle. However, all values remained within the safe range, indicating that the bridge was operating safely.

3.6. Modal Analysis

A structural dynamic analysis was conducted using the MIDAS CIVIL NX finite element analysis software, and the structure’s vibration frequencies and modes were obtained. The calculated results for the bridge’s first three natural frequencies are shown in Figure 12.

Table 3 shows the measured vertical vibration frequency values obtained from the bridge ambient vibration test and the theoretical natural frequency values calculated through a finite element analysis.

When the measured data were compared with the theoretical frequencies, the measured frequencies for the three natural vibration modes were all higher than the theoretical values. This indicates that the overall stiffness of the structure was relatively high, and the dynamic characteristics of the bridge were good. The overall vertical stiffness of the structure met the design requirements, and it had a good energy dissipation performance during structural vibration.

4. Predictive Analysis

4.1. Grey Prediction Model

The grey model GM(n, h) was proposed by Julong Deng in 1982 [27]. The grey prediction model can effectively predict data sequences with very small quantities and low data integrity and reliability. It uses differential equations to fully explore the essence of the data, requires little information for modeling, has a high accuracy, is simple to calculate, is easy to verify, and does not require the consideration of distribution laws or trends. However, the grey prediction model is generally only suitable for short-term predictions, and GM(1, 1) is commonly used for this purpose [28].

The modeling principle of the GM(1, 1) prediction model is to generate a set of new data series with apparent trends using accumulation; build a model according to the growth trend of the new data series for prediction; use the regressive method to perform a reverse calculation; restore the original data series; and then obtain the prediction results [29]. The modeling steps are as follows:

Let ${\mathrm{X}}^{(0)}=\{{\mathrm{X}}^{(0)}(1),{\mathrm{X}}^{(0)}(2),\dots {\mathrm{X}}^{(0)}(\mathrm{n})\}$ be the original series; ${\mathrm{X}}^{(1)}$ is the data sequence generated after a sum of ${\mathrm{X}}^{(0)}$ and ${\mathrm{Z}}^{(1)}$ is the adjacent-mean equally weighted generating sequence of ${\mathrm{X}}^{\mathrm{(}1\mathrm{)}}$.

(2)${\mathrm{X}}^{(0)}=\{{\mathrm{X}}^{(0)}(1),{\mathrm{X}}^{(0)}(2),\dots {\mathrm{X}}^{(0)}(\mathrm{n})\}$

(3) ${\mathrm{X}}^{(1)}=\{{\mathrm{X}}^{(1)}(1),{\mathrm{X}}^{(1)}(2),\dots {\mathrm{X}}^{(1)}(\mathrm{n})\}$

(4) ${\mathrm{Z}}^{(1)}=\{{\mathrm{Z}}^{(1)}(1),{\mathrm{Z}}^{(1)}(2),\dots {\mathrm{Z}}^{(1)}(\mathrm{n})\}$

(5) ${\mathrm{x}}^{(1)}(\mathrm{k})={\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{x}}^{(0)}(\mathrm{i})},\mathrm{k}=1,2,\dots \mathrm{n}$

(6) ${\mathrm{z}}^{(1)}(\mathrm{k})=0.5({\mathrm{x}}^{(1)}(\mathrm{k}-1)+{\mathrm{x}}^{(1)}(\mathrm{k})),\mathrm{k}=1,2,\dots \mathrm{n}$

(7) ${\mathrm{x}}^{(0)}(\mathrm{k})+\mathrm{a}{\mathrm{z}}^{(1)}(\mathrm{k})=\mathrm{b}$

Then, Equation (7) is the GM(1, 1) model [30].

The parameter identification of the GM(1, 1) model is as follows:

(8)$\mathrm{P}={\left[\mathrm{a},\mathrm{b}\right]}^{\mathrm{T}}={({\mathrm{B}}^{\mathrm{T}}\mathrm{B})}^{-1}{\mathrm{B}}^{\mathrm{T}}\mathrm{Y}$

$\mathrm{Among},\mathrm{Y}=\left[\begin{array}{c}{\mathrm{x}}^0(2)\\ {\mathrm{x}}^0(3)\\ ⋮\\ {\mathrm{x}}^0(\mathrm{n})\end{array}\right],\mathrm{B}=\left[\begin{array}{c}-{\mathrm{z}}^1(2),1\\ -{\mathrm{z}}^1(3),1\\ ⋮\\ -{\mathrm{z}}^1(\mathrm{n}),1\end{array}\right]$

The whitening differential equation is constructed as follows:

(9)${\displaystyle \frac{{\mathrm{dx}}_1^{\left(1\right)}}{\mathrm{dt}}}+\mathrm{a}{\mathrm{x}}_1^{\left(1\right)}=\mathrm{b}$

In the equation, “a” is the parameter to be identified, also known as the development coefficient, and “b” is the endogenous variable to be identified, also known as the grey action quantity.

The solution of the whitening equation is as follows:

(10)${\mathrm{x}}^{\left(1\right)}(\mathrm{t})=({\mathrm{x}}^{\left(1\right)}(1)-{\displaystyle \frac{\mathrm{b}}{\mathrm{a}}}){\mathrm{e}}^{-\mathrm{a}(\mathrm{t}-1)}+{\displaystyle \frac{\mathrm{b}}{\mathrm{a}}}$

The time response sequence of the model is as follows:

(11)${\stackrel{\wedge }{\mathrm{x}}}^{(1)}(\mathrm{k}+1)=({\mathrm{x}}^{\left(0\right)}(1)-{\displaystyle \frac{\mathrm{b}}{\mathrm{a}}}){\mathrm{e}}^{-\mathrm{ak}}+{\displaystyle \frac{\mathrm{b}}{\mathrm{a}}}$

Then, the cumulative reduction value is as follows:

(12)${\stackrel{\wedge }{\mathrm{x}}}^{(0)}(\mathrm{k}+1)={\stackrel{\wedge }{\mathrm{x}}}^{(1)}(\mathrm{k}+1)-{\stackrel{\wedge }{\mathrm{x}}}^{(1)}(\mathrm{k})=\mathrm{(}1-{\mathrm{e}}^{\mathrm{a}}\mathrm{)}({\mathrm{x}}^{\left(0\right)}(1)-{\displaystyle \frac{\mathrm{b}}{\mathrm{a}}}){\mathrm{e}}^{-\mathrm{ak}}$

The restored value is the predicted value of the GM(1, 1) model.

4.2. Prediction Result Analysis

The grey prediction model was used to predict the maximum dynamic deflection, maximum dynamic strain, and vibration acceleration of the measuring point when the vehicle speed was 20–60 km/h. The prediction results are shown in Table 4.

Figure 13 shows the dynamic deflection of a vehicle simulated to pass through the mid-span measurement point at a constant speed ranging from 30 to 60 km/h in the finite element software MIDAS CIVIL NX.

Similarly, the maximum dynamic strain and vibration acceleration at speeds ranging from 30 to 60 km/h were simulated in the finite element software, and the simulated values were then compared with the predicted values, as shown in Figure 14.

Figure 14 above shows a slight difference between the values predicted by the grey prediction model and those simulated by the finite element method. This indicates that the simulation results are compelling. The model can simulate vehicle operating conditions at different speeds and can be used when field test conditions are limited.

5. Conclusions

This study used the MIDAS CIVIL NX finite element analysis software to model and conduct numerical simulations on the ramp bridge of the Fuxing Interchange Bridge. Reasonable loading methods and conditions were determined. Based on this, experimental research on the ramp bridge’s displacement, strain, and acceleration was conducted. Finally, the grey prediction model was used to predict the maximum dynamic deflection, maximum dynamic strain, and vibration acceleration at a speed of 60 km/h. The following conclusions were drawn:

By comparing the maximum dynamic deflection, acceleration, and dynamic strain when passing through the monitoring section at different speeds, the dynamic strain and dynamic deflection caused by vehicle excitation were generally minor. As the vehicle speed gradually increased, there was a clear trend of increasing dynamic deflection, dynamic strain, and vibration acceleration at the midpoint measurement point. Under the action of moving vehicle loads, the bridge’s maximum vertical displacement value, strain, and vibration acceleration all met the design requirements specified in the relevant codes.

An analysis of the impact coefficient test results under different vehicle speeds and excitations revealed that the higher the vehicle speed, the greater the impact coefficient. When the vehicle’s driving radius R increased from 265 m to 275 m, the vertical deflection impact coefficient at the midpoint of the side span was approximately symmetrical. When the driving radius was smaller, the impact coefficient was the largest. The experimental data indicate that driving on the inside of the bridge has the greatest impact on the vertical deflection impact coefficient.

Considering the speed limit on the ramp, the maximum dynamic deflection, maximum dynamic strain, and vibration acceleration at the monitoring section were predicted using a grey prediction model when the vehicle speed ranged from 20 to 60 km/h. Subsequently, finite element software was utilized to simulate the maximum dynamic deflection, maximum dynamic strain, and vibration acceleration at the mid-span measurement point as a loaded vehicle passed through at 30 to 60 km/h speeds. The difference between the grey prediction and finite element simulation values was relatively small, indicating a good prediction result.

Author Contributions

Conceptualization, Y.J. and G.L.; methodology, G.L.; software, G.L.; validation, G.L., Y.J., and W.X.; formal analysis, Y.J.; investigation, Y.J.; resources, Y.J.; data curation, G.L.; writing—original draft preparation, Y.J.; writing—review and editing, Y.J.; visualization, Y.J.; supervision, W.X.; project administration, Y.J.; funding acquisition, Y.J. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Our data came from the civil engineering intelligent monitoring cloud network platform provided by Changsha Yijian Information Technology Co., LTD.

Acknowledgments

Thanks are extended to Liao Guangwen for helping to complete the dynamic load test process, and to Xu Wenyuan for reviewing the writing.

Conflicts of Interest

The authors declare no conflicts of interest. The funders collected and interpreted the data in the study design.

Footnotes

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Figures and Tables

Figure 1. Bridge layout: (a) standard cross-sectional drawing; (b) profile of the bridge (unit: mm).

Figure 2. B ramp bridge appearance: (a) front view; (b) side view.

Figure 3. Point arrangement: (a) monitoring section diagram; (b) JMZX-212HAT surface chord-type strain gauge.

Figure 4. JMCZ-2081 magnetoelectric velocity sensor.

Figure 5. Dynamic strain acquisition: (a) steel chord dynamic strain acquisition instrument; (b) workflow of steel chord dynamic strain acquisition.

Figure 6. Schematic diagram of the acceleration acquisition system.

Figure 7. Model diagram of the B ramp bridge.

Figure 8. Shock coefficient under different speed excitation levels.

Figure 9. Relationship between impact coefficient and driving radius.

Figure 10. Jump floor acceleration time history curve.

Figure 11. Brake vibration response at different speeds: (a) braking at 20 km/h; (b) braking at 30 km/h; and (c) braking at 40 km/h.

Figure 11. Brake vibration response at different speeds: (a) braking at 20 km/h; (b) braking at 30 km/h; and (c) braking at 40 km/h.

Figure 12. Mode diagram: (a) first-order mode diagram; (b) second-order mode diagram; and (c) third-order mode diagram.

Figure 13. Simulation of dynamic deflection of measuring points at different speeds (unit: mm): (a) 30 km/h; (b) 40 km/h; (c) 50 km/h; and (d) 60 km/h.

Figure 13. Simulation of dynamic deflection of measuring points at different speeds (unit: mm): (a) 30 km/h; (b) 40 km/h; (c) 50 km/h; and (d) 60 km/h.

Figure 13. Simulation of dynamic deflection of measuring points at different speeds (unit: mm): (a) 30 km/h; (b) 40 km/h; (c) 50 km/h; and (d) 60 km/h.

Figure 14. Comparison of predicted values and simulated values: (a) comparison of maximum dynamic deflections; (b) comparison of maximum dynamic strains; and (c) comparison of vibration accelerations.

Figure 14. Comparison of predicted values and simulated values: (a) comparison of maximum dynamic deflections; (b) comparison of maximum dynamic strains; and (c) comparison of vibration accelerations.

Table 1

Test conditions.

Type of Working Condition	Test Content
Pulsation test	The natural vibration frequency of the bridge structure was obtained by a pulsation test [21].
Sports car test	The test vehicle passed the bridge at speeds of 20, 30, 40, and 50 km/h.
	The test vehicle passed the bridge at a speed of 20 km/h from Lane 1, the middle of the bridge deck, and Lane 2.
Jump-off test	The test car slowly crossed the 10 cm wedge at a speed of 10 km/h and braked.
Braking test	The test vehicle braked at the test section at 20, 30, and 40 km/h.

Table 2

Dynamic response at different speeds.

Speed (km/h)	Maximum Dynamic Strain (uɛ)	Maximum Dynamic Deflection (mm)	Vibration Acceleration (mm/s2)
20	6.4	0.825	3.017
30	7.2	1.047	3.249
40	8.5	1.126	3.288
50	9.0	1.344	3.326

Table 3

Comparison of pulsation test results.

Natural Vibration Characteristic	Theoretical Frequency	Measured Value of Load Test
First-order mode	1.330	1.835
Second-order mode	2.644	3.314
Third-order mode	4.481	4.939

Table 4

Monitoring project projections.

Monitoring Item Speed (km/h)	20	30	40	50	60
Maximum dynamic strain (uɛ)	6.40	7.36	8.19	9.12	10.16
Maximum dynamic deflection (mm)	0.825	1.023	1.164	1.324	1.506
Vibration acceleration (mm/s2)	3.017	3.250	3.287	3.326	3.365

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