# Dynamic Amplification Factor of Continuous versus Simply Supported Bridges Due to the Action of a Moving Vehicle

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Element Modelling of Vehicle-Bridge Interaction

#### 2.1. Vehicle Model

- The 2-axle model represents a rigid truck with four Degrees of Freedom (DOFs) corresponding to axle hop displacements (y
_{u}_{1}, y_{u}_{2}) of the two axle masses (un-sprung masses, m_{u}_{1}and m_{u}_{2}), bounce displacement, y_{s}_{1}, as well as the pitch rotation, θ_{T}_{1}, of the body mass (sprung mass, m_{s}). The two axle masses are linked to the road surface by means of linear springs of stiffness (K_{t}_{1}and K_{t}_{2}) and damping elements (C_{t}_{1}and C_{t}_{2}) representing the tires. The body mass is linked to the two axle masses with the help of springs of stiffness K_{s}_{1}and K_{s}_{2}that have linear viscous dampers, with values of C_{s}_{1}and C_{s}_{2}respectively, representing the suspensions. - The 5-axle model is a truck comprising two major bodies, truck and trailer, with a total of 9 DOFs. Four of these DOFs are located in the tractor and they correspond to axle hop displacements (y
_{u}_{1}, and y_{u}_{2}) of the two axle masses (un-sprung masses, m_{u}_{1}and m_{u}_{2}), bounce displacement, y_{s}_{1}, as well as the pitch rotation, θ_{T}_{1}, of the body mass (sprung mass, m_{s}_{1}). The two axle masses are linked to the road surface by means of linear springs of stiffness (K_{t}_{1}and K_{t}_{2}) and damping elements (C_{t}_{1}and C_{t}_{2}) representing the tires. The body mass is linked to the axle masses with the help of springs of stiffness K_{s}_{1}and K_{s}_{2}that have linear viscous dampers, with values of C_{s}_{1}and C_{s}_{2}respectively, representing the suspensions. Another 5 DOFs are located in the trailer, and they correspond to the axle hop displacements (y_{ui}(i = 3 to 5) of each axle mass (un-sprung masses m_{ui}with i = 3 to 5)), bounce displacement, y_{s}_{2}, and pitch rotation, θ_{T}_{2}, of body mass (sprung mass, m_{s}_{2}). The same description of the tire and suspensions elements of the tractor apply to the trailer. Tire elements are labelled K_{ti}(i = 3 to 5) and C_{ti}(i = 3 to 5), and suspension elements K_{si}(i = 3 to 5) and C_{si}(i = 3 to 5).

_{v}], [C

_{v}] and [K

_{v}] are mass, damping and stiffness matrices of the vehicle, respectively and $\left\{\ddot{{y}_{v}}\right\}$, $\left\{{\dot{y}}_{v}\right\}$ and $\left\{{y}_{v}\right\}$ are the respective vectors of nodal acceleration, velocity and displacement. $\left\{{f}_{int}\right\}$ is the time-varying dynamic interaction force vector applied to the vehicle’s DOFs.

#### 2.2. Bridge Model

^{−1}kg, modulus of elasticity, E, of 35 GPa, and second moment of area, I, of 0.5273 m

^{4}. The first natural frequency of the continuous and simply supported bridge models is 5.65 Hz.

_{b}], [C

_{b}] and [K

_{b}] are the respective mass, damping and stiffness matrices of the beam model and $\left\{\ddot{{y}_{b}}\right\}$, $\left\{{\dot{y}}_{b}\right\}$ and $\left\{{y}_{b}\right\}$ are vectors of nodal bridge acceleration, velocity and displacement, respectively. Rayleigh damping is used here, which is given by:

#### 2.3. Road Profile

^{3}/cycle and $4.95\times {10}^{-6}$ for class “A” and $82.5\times {10}^{-6}$ m

^{3}/cycle and $31.05\times {10}^{-6}$ for class “B” [30]. A moving average filter is applied to the generated road profile heights over a distance of 0.24 m to simulate the attenuation of short wavelength disturbances by the tire contact patch [25,31]. Furthermore, a road approach that spans over 100 m is added before the bridge to induce initial conditions of dynamic equilibrium in the vehicle. Figure 2 shows an instance of class “A” and of class “B” road profiles of 130 m length, including the 100 m approach.

#### 2.4. Coupling of the VBI system

_{g}] is the combined system mass matrix, and [C

_{g}] and [K

_{g}] are the coupled time-varying system damping and stiffness matrices, respectively. The vector $\left\{u\right\}=\left\{\begin{array}{c}\left\{{y}_{v}\right\}\\ \text{}\left\{{y}_{b}\right\}\end{array}\right\}$ is the displacement vector of the system. $\left\{f\right\}$ is the coupled force vector system. In this paper, the coupled force vector $\left\{f\right\}$ has different formulae depending on the vehicle system; whether a truck (Equation (6)) or a truck-trailer (Equation (7)) is employed:

- the zero elements in the force vectors correspond to pitching and heaving DOFs for the trucks and trailers.
- $\left[{N}_{b}\right]$ is an (n × n
_{f}) matrix which distributes the n_{f}applied interaction forces on beam elements to equivalent forces acting on the nodes (i.e., n_{f}is equal to 2 and 5 for the 2- and 5-axle vehicles respectively) and n is the total number of DOFs of the beam (i.e., n is equal to 60 and 120 for the single span and two-span beam FE models respectively). - P
_{i}is the static axle weight corresponding to axle i of the vehicle. - r
_{i}is the road profile displacement under axle i.

## 3. Dynamic Amplification Factors

#### 3.1. Simply Supported Beam

#### 3.2. Continuous Beam

#### 3.2.1. DAF of Sagging Moments

#### 3.2.2. DAF of Hogging Moments

## 4. Discussion

## 5. Summary and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Road irregularities for two random profiles with different road class: (

**a**) Class “A”; (

**b**) Class “B”.

**Figure 3.**Static and total Bending Moment (BM) response of continuous beam due to the effect of the two-axle truck: (

**a**) sagging moment at mid-length of the 1st span; (

**b**) hogging moment over the internal support.

**Figure 6.**Location of maximum total bending moment (TBM) along the simply supported beam due to the crossing of: (

**a**) 2-axle vehicle; (

**b**) 5-axle vehicle.

**Figure 10.**HDAF/FHDAF due to the 2-axle vehicle on road class “B”: (

**a**) mean; (

**b**) standard deviation.

**Figure 11.**HDAF/FHDAF due to the 5-axle vehicle on road class “B”: (

**a**) mean; (

**b**) standard deviation.

**Figure 12.**Location of maximum total bending moment along the continuous beam versus velocity for 2-axle vehicle: (

**a**) Sagging (TSBM); (

**b**) hogging (THBM).

**Figure 13.**Location of maximum total bending moment (along the continuous beam versus velocity for 5-axle vehicle: (

**a**) sagging (TSBM); (

**b**) hogging (THBM).

Property | Symbol | Value | Unit |
---|---|---|---|

Body Mass | m_{s} | 26,750 | kg |

Axle1 mass | m_{u}_{1} | 700 | kg |

Axle2 mass | m_{u}_{2} | 1100 | kg |

Suspension stiffness | K_{s}_{1} | 4 × 10^{5} | N m^{−1} |

K_{s}_{2} | 10 × 10^{5} | N m^{−1} | |

Suspension Damping | C_{s}_{1} | 10 × 10^{3} | Ns m^{−1} |

C_{s}_{2} | 20 × 10^{3} | Ns m^{−1} | |

Tire Stiffness | K_{t}_{1} | 1.75 × 10^{6} | N m^{−1} |

K_{t}_{2} | 3.5 × 10^{6} | N m^{−1} | |

Tire Damping | C_{t}_{1} | 3 × 10^{3} | Ns m^{−1} |

C_{t}_{2} | 5 × 10^{3} | Ns m^{−1} | |

Moment of Inertia | I_{s} | 154,320 | kg m^{2} |

Body bounce frequency | f_{bounce} | 0.86 | Hz |

Body pitch frequency | f_{pitch} | 1.02 | Hz |

Axle1 hop frequency | f_{axle}_{1} | 8.83 | Hz |

Axle2 hop frequency | f_{axle}_{2} | 10.19 | Hz |

Property | Symbol | Value | Unit |
---|---|---|---|

Body Mass 1 | m_{s}_{1} | 25,200 | kg |

Body Mass 2 | m_{s}_{2} | 30,700 | kg |

Axle1 mass | m_{u}_{1} | 700 | kg |

Axle2 mass | m_{u}_{2} | 1100 | kg |

Axle3 mass | m_{u}_{3} | 1100 | kg |

Axle4 mass | m_{u}_{4} | 1100 | kg |

Axle5 mass | m_{u}_{5} | 1100 | kg |

Suspension stiffness | K_{s}_{1} | 4 × 10^{5} | N m^{−1} |

K_{s}_{2}, K_{s}_{3}, K_{s}_{4}, K_{s}_{5} | 10 × 10^{5} | N m^{−1} | |

Suspension Damping | C_{s}_{1} | 10 × 10^{3} | Ns m^{−1} |

C_{s}_{2}, C_{s}_{3}, C_{s}_{4}, C_{s}_{5} | 20 × 10^{3} | Ns m^{−1} | |

Tire Stiffness | K_{t1} | 1.75 × 10^{6} | N m^{−1} |

K_{t}_{2}, K_{t}_{3}, K_{t}_{4}, K_{t}_{5} | 3.5 × 10^{6} | N m^{−1} | |

Tire Damping | C_{t}_{1} | 3 × 10^{3} | Ns m^{−1} |

C_{t}_{2}, C_{t}_{3}, C_{t}_{4}, C_{t}_{5} | 5 × 10^{3} | Ns m^{−1} | |

Moment of Inertia 1 | I_{s}_{1} | 86,410 | kg m^{2} |

Moment of Inertia 2 | I_{s}_{2} | 112,440 | kg m^{2} |

Body 1 bounce frequency | f_{1bounce} | 1.56 | Hz |

Body 1 pitch frequency | f_{1pitch} | 2.39 | Hz |

Axle1 hop frequency | f_{axle}_{1} | 9.97 | Hz |

Axle2 hop frequency | f_{axle}_{2} | 8.77 | Hz |

Body 2 bounce frequency | f_{2bounce} | 2.12 | Hz |

Body 2 pitch frequency | f_{2pitch} | 2.33 | Hz |

Axle3 hop frequency | f_{axle}_{3} | 10.03 | Hz |

Axle4 hop frequency | f_{axle}_{4} | 10.15 | Hz |

Axle5 hop frequency | f_{axle}_{5} | 10.17 | Hz |

**Table 3.**Highest mean value (µ) and standard deviation (σ) of DAF in the simply supported beam with road classes “A” and “B”.

Vehicle Type | Class Type of Road Surface | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Class “A” | Class “B” | |||||||||||

DAF | FDAF | DAF | FDAF | |||||||||

µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | |

2-axle | 1.12 | 0.036 | 85.32 | 1.12 | 0.036 | 85.32 | 1.12 | 0.065 | 85.32 | 1.12 | 0.065 | 85.32 |

5-axle | 1.12 | 0.04 | 93.6 | 1.14 | 0.04 | 91.8 | 1.14 | 0.07 | 91.8 | 1.17 | 0.07 | 92.88 |

**Table 4.**Highest mean value (µ) and standard deviation (σ) of FSDAF/SDAF in the continuous beam with road class “A”.

Vehicle Type | SDAF1 | SDAF2 | FSDAF1 | FSDAF2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | |

2-axle | 1.1 | 0.04 | 83.11 | 1.08 | 0.04 | 105.8 | 1.13 | 0.06 | 120 | 1.1 | 0.04 | 108 |

5-axle | 1.09 | 0.03 | 85.32 | 1.05 | 0.04 | 101.5 | 1.1 | 0.03 | 84.24 | 1.14 | 0.04 | 90.72 |

**Table 5.**Highest mean value (µ) and standard deviation (σ) of FSDAF/SDAF in the continuous beam with road class “B”.

Vehicle Type | SDAF1 | SDAF2 | FSDAF1 | FSDAF2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | |

2-axle | 1.09 | 0.07 | 84.24 | 1.06 | 0.07 | 104.8 | 1.14 | 0.09 | 120 | 1.09 | 0.07 | 108 |

5-axle | 1.12 | 0.06 | 87.48 | 1.1 | 0.07 | 88.56 | 1.12 | 0.06 | 85.32 | 1.19 | 0.07 | 90.72 |

**Table 6.**Highest mean value (µ) and standard deviation (σ) in FHDAF/HDAF of the continuous beam with road classes “A” and “B”.

Vehicle Type | Class “A” | Class “B” | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

HDAF | FHDAF | HDAF | FHDAF | |||||||||

µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | µ | σ | Velocity km/h | |

2-axle | 1.11 | 0.06 | 120 | 1.15 | 0.07 | 120 | 1.2 | 0.11 | 120 | 1.25 | 0.11 | 120 |

5-axle | 1.06 | 0.03 | 36.76 | 1.06 | 36.72 | 0.03 | 1.13 | 0.07 | 62.64 | 1.14 | 0.07 | 62.64 |

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**MDPI and ACS Style**

González, A.; Mohammed, O.
Dynamic Amplification Factor of Continuous versus Simply Supported Bridges Due to the Action of a Moving Vehicle. *Infrastructures* **2018**, *3*, 12.
https://doi.org/10.3390/infrastructures3020012

**AMA Style**

González A, Mohammed O.
Dynamic Amplification Factor of Continuous versus Simply Supported Bridges Due to the Action of a Moving Vehicle. *Infrastructures*. 2018; 3(2):12.
https://doi.org/10.3390/infrastructures3020012

**Chicago/Turabian Style**

González, Arturo, and Omar Mohammed.
2018. "Dynamic Amplification Factor of Continuous versus Simply Supported Bridges Due to the Action of a Moving Vehicle" *Infrastructures* 3, no. 2: 12.
https://doi.org/10.3390/infrastructures3020012