# Dynamic Behaviour of Bridge Girders with Trapezoidal Profiled Webs Subjected to Moving Loads

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## Abstract

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## 1. Introduction

_{c}= 36.8°, as shown in Figure 1b. The corresponding web height-thickness ratio is 17.78. All configuration and geometric details of the main girder as well as related vehicle moving loads were accounted in the dynamic analysis of this study. The finite element model has been developed using ANSYS 15.0 software (ANSYS Inc. Canonsburg, WA, USA). Please consider this suggested change.

## 2. Modelling and Dynamic Analysis

_{s}= 206,000 MPa and υ = 0.3 respectively.

_{k}to 4 P

_{k}(i.e., the nodal assembly of four forces applied along the longitudinal direction of the girder is shown in Figure 3). Conversely, as the vehicle leaving the bridge, the number of nodal forces is decreased with the decrement of 4 P

_{k}to 0. The time for moving load to arrive ith node is x

_{i}/v, where x

_{i}is the node location. The maximum and minimum stresses caused by the dynamic loading are then compared against the allowable fatigue stresses. Five loading cases are considered for the modelling the dynamic behaviour of the girders under different fatigue load travelling velocity and increased vehicle load as listed in Table 2. The edge of the top flange is laterally restrained to exclude torsion and out-of-plane deflection. For simplicity, the bridge damping, road roughness, and the interaction between the vehicle and bridge are ignored and the dynamic moving vehicle load is directly applied on the nodes of the top flange of the girder. According to the position of the moving load, three phases of nodal force assembly are considered in proper sequence as: front wheel load only, front wheel load + rear wheel load (as examplified in Figure 3), and rear wheel load only. Assuming the location of ith node from the support is x

_{i}and the moving velocity is v, the time for the load to arrive ith node is t

_{i}= x

_{i}/v. Using ANSYS parameter design language (APDL), a program for dynamic modelling is compiled in the following steps: (1) developing finite model with proper mesh discretization adapting to the moving load location; (2) checking whether the instant load location of the front wheel and the rear wheel are within the span of the girder or not; (3) reading the proper loading case and apply the nodal forces on the girder model using transient analysis in ANSYS.

_{l}and b

_{i}are the lengths of longitudinal fold and inclined fold respectively. θ

_{c}is the corrugation angle.

_{0}is the correction coefficient converted from original form for the cross section of box girder as

_{m}and b

_{0}are the shear modulus and breadth of the flange respectively. A

_{w}is the cross section area of the web and l

_{0}is the span of the beam.

## 3. Results and Discussion

#### 3.1. Numerical Verification

#### 3.2. Modal Analysis

#### 3.3. Dynamic Analytical Results Concerning Varied Vehicle Velocity

_{0}), as shown in Figure 14. It is evident that the increase in vehicle travel speed has a limited effect on the increase in peak stress. This can be explained as the load passing through the girder in a very short time so that the local connecting part of the structure is less stressed.

#### 3.4. Dynamic Analytical Results Concerning Overload Vehicle Condition

_{k}and 2P

_{k}. Given high stresses at the intersection of the longitudinal fold and the inclined fold of the trapezoidal web as mentioned previously, the dynamic stresses of point B under three loading cases are compared in Figure 17. Much greater amplification of dynamic peak stresses is observed with the increase of the applied vehicle load in contrast to the increase of the vehicle travel speed. This suggests that the critical dynamic behaviour of the bridge girders with trapezoidal profiled webs is more sensitive to the overload vehicle in contrast to overspeed vehicle.

## 4. Fatigue Life Prediction Based Fracture Mechanics Method

_{f}and a

_{0}are the steel flange plate thickness and initial crack depth as 0.12 mm [25]. C and m are the material constants which can be taken as 1.3 × 10

^{−12}and 3 respectively [32]. The stress intensity factor, ΔK, can be defined by assuming that a semi-elliptical crack propagation in depth at the weld toes as

_{S}is a free surface correction factor, F

_{E}is a crack shape correction factor, F

_{T}is a finite thickness correction factor, F

_{G}is a geometry correction factor, Δσ is an applied stress range. The ratio of the crack depth, a, to the crack length, c, is assumed as equal to 0.75. θ is the integration parameter. Each correction factor can be obtained from the following expressions:

## 5. Concluding Remarks

- The results of the developed finite element model incorporating proper boundary condition and corrugation details agree well with referred excitation test and theoretical results in terms of natural frequencies and vibration amplitudes. Apart from the deflection induced by the bending moment, the trapezoidal profiled web connecting part at the midspan of the bottom tension flange is sensitive to vibration in the form of localized deflection and crumpling.
- The dynamic deflection and nodal velocity are notably increased with the increase in vehicle travel speed. The dynamic stress is low as the front wheel load is applied on the girder and subsequently increased until the rear wheel load approaching the end of the support. Due to very short time of load passing, the peak stress at the tension flange is marginally affected by the variation of the vehicle travel speed. The resultant stresses are relatively high at the intersection of the trapezoidal profiled web and located at a quarter wavelength of the trapezoidal web away from the midspan of the girder.
- The overload vehicle has adverse effect on the girder dynamic behaviour subjected to moving loads since its resultant dynamic stress at the intersection of the longitudinal fold and the inclined fold of the trapezoidal web is almost linearly amplified. Moreover, the resultant fatigue life is greatly reduced in contrast to that caused by the effect of the vehicle travel speed from above the detail category 100 to no more than the detail category 90; thus the limitation of the maximum required load of the vehicle is required when the maintenance of such bridge girders is concerned. Additionally, above understanding could be taken as a basis for the assessment of the service life of the bridge providing the desired level of performance or functionality with any required level of repair and maintenance as a following up study.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Diagram of moving load model (unit for length: m) [14].

**Figure 7.**Summary of typical four mode shapes of vibration. (

**a**) 1st mode shape of vibration. (

**b**) 2nd mode shape of vibration. (

**c**) 3rd mode shape of vibration. (

**d**) 4th mode shape of vibration.

**Figure 11.**Comparison of dynamic stress distribution of the corrugation near midspan centreline (v = 40 km/h).

**Figure 12.**Comparison of dynamic stress distribution of the corrugation near midspan centreline (v = 90 km/h).

**Figure 13.**Comparison of dynamic stress distribution of the corrugation near midspan centreline (v = 140 km/h).

**Figure 15.**Comparison of dynamic deflection with varied overload cases (Case referring to Table 2).

**Figure 16.**Comparison of nodal dynamic velocity with varied overload cases (Case referring to Table 2).

**Figure 17.**Comparison of dynamic stress of point B with varied overload cases (Case referring to Table 2).

Component | Chemical Composition (%) | Mechanical Properties | ||||||
---|---|---|---|---|---|---|---|---|

C | Si | Mn | P | S | Yield Stress [MPa] | Ultimate Stress [MPa] | Elongation/% | |

Flange | 0.16 | 0.33 | 1.36 | 0.035 | 0.022 | 480 | 565 | 25 |

Web | 0.14 | 0.32 | 0.48 | 0.045 | 0.04 | 410 | 495 | 22 |

Load Case | P_{k} [kN] | v [km·h^{−1}] | Passage Time [s] | Overload |
---|---|---|---|---|

1 | 120 | 40 | 2.25 | × |

2 | 120 | 90 | 1.00 | × |

3 | 120 | 140 | 0.64 | × |

4 | 180 | 90 | 1.00 | √ |

5 | 240 | 90 | 1.00 | √ |

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**MDPI and ACS Style**

Wang, Z.; Shi, Y.; You, X.; Jiang, R.; Gai, W.
Dynamic Behaviour of Bridge Girders with Trapezoidal Profiled Webs Subjected to Moving Loads. *Materials* **2021**, *14*, 38.
https://doi.org/10.3390/ma14010038

**AMA Style**

Wang Z, Shi Y, You X, Jiang R, Gai W.
Dynamic Behaviour of Bridge Girders with Trapezoidal Profiled Webs Subjected to Moving Loads. *Materials*. 2021; 14(1):38.
https://doi.org/10.3390/ma14010038

**Chicago/Turabian Style**

Wang, Zhiyu, Yunzhong Shi, Xiang You, Ruijuan Jiang, and Weiming Gai.
2021. "Dynamic Behaviour of Bridge Girders with Trapezoidal Profiled Webs Subjected to Moving Loads" *Materials* 14, no. 1: 38.
https://doi.org/10.3390/ma14010038