# Feasibility Study of Tractor-Test Vehicle Technique for Practical Structural Condition Assessment of Beam-Like Bridge Deck

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Tractor-Test Vehicle Technique (TTVT) of Non-Destructive Testing

#### 2.1. Indirect Measurement of Mode Shape of Bridge Structures

_{v,2}and m

_{v,1}, supported on a dashpot of damping coefficient c

_{v2}, c

_{v1}and a spring of stiffness k

_{v2}, and k

_{v1}, respectively. The bridge is simply-supported with length L, mass m

^{*}per unit length, and bending stiffness EI with a damping ratio of the n

^{th}mode ${\xi}_{n}$, there the bending stiffness EI here includes the contribution of non-structural members such as bridge railings and bridge deck pavements in actual bridges, and the mass per unit length m* can be obtained from the actual bridge design data or preliminary estimated by the area of cross-section. The bridge is assumed at rest prior to the arrival of the test vehicle.

_{v1}, m

_{v2}is much less than that of the deck, i.e., ${m}_{v1}\ll {m}^{\ast}L$ and ${m}_{v2}\ll {m}^{\ast}L$. This assumption is easy to implement for actual bridges. By substituting Equation (6) into Equation (3), multiplying by $\mathrm{sin}(n\pi x/L)$, integrating $x$ from 0 to L, and according to the orthogonal conditions of the sinusoidal function, the n

^{th}modal equation of equilibrium of the structure can be written as:

^{th}modal angular frequency of bridge vibration, given by:

^{th}modal frequency of the bridge can be separated from the response of the test vehicle [29] by a band-pass filter with limits ${\omega}_{n}\sqrt{1-{\xi}_{n}^{2}}-n\pi v/L$ and ${\omega}_{n}\sqrt{1-{\xi}_{n}^{2}}+n\pi v/L$, due to the different coupling degree of vehicle and bridge considered in the theoretical solution, the filtering range will be adjusted appropriately in the following section of numerical simulation [15]. The resulting signal is the transient response from the n

^{th}vibration mode of the bridge structure, which is directly related to the last term on the right-hand-side (RHS) of Equation (10).

^{th}mode shape of the deck is obtained as:

_{1}of the test vehicle associated with the 1st vibration mode of the bridge deck can also be written as:

_{1}to A

_{2}can be determined by comparing terms in Equations (9) and (10) as:

#### 2.2. Element Bending Stiffness of Beam from the Improved DSC Method

#### 2.3. Reducing the Effect of Road Surface Roughness

^{th}modal equation of the deck in Equation (7) can be modified as:

^{th}mode shape of the deck can be expressed as:

^{th}mode shape of the deck can also be deduced as:

#### 2.4. Effect of Ambient Temperature

#### 2.5. Analysis Procedure of Tractor-Test Vehicle Technique for Non-Destructive Testing

- (1)
- Record the acceleration responses of the two test vehicles in the undamaged state.
- (2)
- Identify the first modal frequency of the bridge structure from spectra of the residual displacement $\Delta u(t)$ and corresponding residual acceleration $\Delta \ddot{u}(t)$, which indicate a vehicle response without the effect ofroad surface roughness based on tractor-test vehicle technique, in both the undamaged and damaged states of the structure. If the ambient temperature is different from the reference temperature, adjust the identified frequencies in both states to that at the reference temperature with Equation (36).
- (3)
- Normalize the acceleration response components ${\ddot{R}}_{1}$ in Equation(16) for the first vibration mode of the deck with ${e}^{-{\xi}_{1}{\omega}_{1}t}$ to remove the transient effect. After the 1st bridge frequency ${\omega}_{1}$ is made available, one can extract the acceleration response components ${\ddot{R}}_{1}$ and the damping ratio of the 1st vibration mode of the deckassociated with ${\omega}_{1}$ from the corresponding residual accelerationby any feasible signal processing tools.
- (4)
- Then, obtaining the instantaneous amplitude history of the bridge component response for the 1
^{st}vibration mode shape. Performing the Hilbert transform to the decomposed bridge component response ${\ddot{R}}_{11}$ in Equation (18) yields its transform pair $H[{\ddot{R}}_{11}]$ in Equation (19). Then, one can obtain the instantaneous amplitude history $A(t)$ of the bridge component response using Equations (20) and (21). - (5)
- Construct the first vibration mode shape from of the component response by Equation (21), the curve of the instantaneous amplitude function $A(t)$ is representative of the mode shape of the bridge in absolute value. The sign of the 1st mode shape can be decided according to engineers’ judgment or experience [9].
- (6)
- Calculate the modal curvature $\mathsf{\kappa}$ of the beam using the mode shape $\phi $ obtained above by the central difference method with due correction described in Section 3.
- (7)
- Calculate the bending moment M of the beam with Equation (23) for each monitored cross-section of the deck.
- (8)
- Calculate the bending stiffness EI of the beam using Equation (22) for each monitored cross-section of the deck.

## 3. Numerical Study

^{2}and 0.2422 m

^{4}, respectively. The elastic modulus of concrete of the bridge is 29 GPa and the material density is 2400 kg/m

^{3}. The 30 m span of the bridge is treated as a beam-like structure and it is discretized into 10 Euler-Bernoulli beamelements as shown in Figure 5. The element number is shown in circles and the numbers below the beam denote the node numbers. The time step of analysis is 0.01s, the EI value of the deck is calculated as $8.52\times {10}^{10}N\cdot {m}^{2}$ with the first modal frequency at 3.67 Hz and a damping ratio of 0.01. Class C [50] road surface roughness is considered. All these parameters are adopted in the following studies unless otherwise stated.

_{v}is much less than those of the bridge in deriving Equation (7).

#### 3.1. Ratio of Vehicle Parameters

#### 3.2. Effect of Constant Vehicle Speed

#### 3.3. Effect of Non-Constant Vehicle Speed

#### 3.4. Effect of Vehicle Modal Frequency

#### 3.5. Effect of Road Surface Roughness

^{th}circle spatial frequency considered, ${\theta}_{i}$ the random phase angle of the i

^{th}cosine function, and d

_{i}the amplitude for each class of roughness selected, defined as ${d}_{i}=\sqrt{2{G}_{d}\left({n}_{i}\right)\Delta n}$. Three classes of road surface roughness [50] are considered with the functional ${G}_{d}\left({n}_{0}\right)$ in the PSD function of the road surface roughness defined as ${G}_{d}\left({n}_{i}\right)={G}_{d}\left({n}_{0}\right){\left(\frac{{n}_{i}}{{n}_{0}}\right)}^{-w}$, where n

_{i}denotes the i

^{th}spatial frequency per meter, w is a constant equal to 2, ${n}_{0}$ = 0.1 cycle/m. ${G}_{d}\left({n}_{0}\right)$ is related to the class of roughness as given in ISO 8608 [50] and it equals to $16\times {10}^{-6}{\mathrm{m}}^{3}$, $64\times {10}^{-6}{\mathrm{m}}^{3}$ and $256\times {10}^{-6}{\mathrm{m}}^{3}$ for Classes A, B, and C road surface roughness, respectively. It is noted that each case of adding road surface roughness is different even in the same Class level because of randomness.

#### 3.6. Effect of Bridge Damping Ratio

#### 3.7. Effect of Measurement Noise

^{th}sampling time point, ${\sigma}_{i}$ is the noise value at the i

^{th}sampling time point, and SNR is the signal-to-noise ratio whose unit is db. It is noted that the SNR decreases when the noise level increases.

## 4. Damage Scenarios Studied

#### 4.1. Case 1

#### 4.2. Case 2

#### 4.3. Case 3

#### 4.4. Case 4

#### 4.5. Discussion on the Boundary Effects

#### 4.6. Discussion on the Variation of Bridge Bearing

## 5. Field Test Study

^{4}, and the elastic modulus is 3.25×10

^{10}N/m

^{2}. The cross-section of the bridge has a total width of 12m with 12 hollow slabs, as shown in Figure 27. The bridge consists of two lanes in each direction and serves traffic with a maximum speed of 30 km/h. The traffic flow on the bridge is very limited due to its remote location so that although not newly built, it is still structurally sound and ingood condition.

## 6. Conclusions

- (1)
- Two test vehicles are designed to have identical modal frequency and damping ratio, but the No.2 test vehicle has a mass, stiffness and damping coefficient proportional to those of the No.1 test vehicle. This technique can help to generate a response from an equivalent vehicle of a single vehicle-bridge system that is free from the effect of road surface roughness.
- (2)
- The first modal frequency and mode shape of the deck structure can be accurately estimated from the response of the equivalent vehicle with consideration of damping of the vehicle-tractor-bridge system, non-uniform test vehicle speed, measurement noise, and different ambient temperatures in the measurements.
- (3)
- The bending stiffness EI of the bridge deck can be better estimated with improvements proposed for the DSC method. For locations such as simply-supported ends of the beam, the improved DSC method can be used to obtain the stiffness by extrapolating the mode shape and using a refined model (or denser data measurements) near these locations.
- (4)
- The tractor-test vehicle technique of non-destructive testing with the proposed modifications has been demonstrated to be feasible for practical application to regular monitoring and evaluation of the structural health condition of a beam-like bridge deck with the advantages of simplicity, mobility, and ease of implementation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Member forces and modal displacements: (

**a**) Sign convention of internal forces; (

**b**) 1st mode shape extrapolation.

**Figure 6.**Comparison ofa single vehicle-bridge system (without tractor and roughness) and tractor-test vehicle technique (Class C road surface roughness): (

**a**) Displacement; (

**b**) Acceleration.

**Figure 7.**Identified resultsfor different vehicle speeds: (

**a**) Mode shape and; (

**b**) Element bending stiffness EI.

**Figure 9.**Identified results for non-constant speed: (

**a**) Mode shape and; (

**b**) Element bending stiffness EI.

**Figure 10.**Identified results for different vehicle frequencies with bridge damping ratio ${\xi}_{n}=0.005$: (

**a**) Mode shape; (

**b**) Element bending stiffness EI.

**Figure 11.**Identified results for different vehicle frequencies with bridge damping ratio ${\xi}_{n}=0.01$: (

**a**) Mode shape; (

**b**) Element bending stiffness EI.

**Figure 12.**Identified resultsfor three classes of road surface roughness with bridge damping ratio ${\xi}_{n}=0.01$: (

**a**) Mode shape; (

**b**) Element bending stiffness EI.

**Figure 13.**Contrast diagram of different damping ratios of the bridge with undamaged conditions for Class C of road surface roughness: (

**a**) Mode shape; (

**b**) Element bending stiffness EI.

**Figure 14.**Identified results for different signal-to-noise ratio: (

**a**) Mode shape; (

**b**) Element bending stiffness EI.

**Figure 15.**Estimated mode shape of undamped bridge for Case 1: (

**a**) Scenario A; (

**b**) Scenario B; (

**c**) Scenario C; (

**d**) Scenario D.

**Figure 16.**Estimated element bending stiffness EI of the undamped bridge for Case 1: (

**a**) Scenario A; (

**b**) Scenario B; (

**c**) Scenario C; (

**d**) Scenario D.

**Figure 17.**Estimated element bending stiffness EI of bridge for Case 2: (

**a**) Scenario A; (

**b**) Scenario B; (

**c**) Scenario C; (

**d**) Scenario D.

**Figure 18.**Estimated element bending stiffness EI of the bridge for Case 3 with Class C road surface roughness: (

**a**) Scenario A; (

**b**) Scenario B; (

**c**) Scenario C; (

**d**) Scenario D.

**Figure 19.**Estimated element bending stiffness EI of the bridge for Case 3 with Class D road surface roughness: (

**a**) Scenario A; (

**b**) Scenario B; (

**c**) Scenario C; (

**d**) Scenario D.

**Figure 21.**Estimated element bending stiffness EI from measurements at different ambient temperature.

**Figure 22.**Identified element bending stiffness EI from measurements at different ambient temperatures: (

**a**) Scenario A, D

_{2}= 30%; (

**b**) Scenario B, D

_{6}= 30%; (

**c**) Scenario C, D

_{4}= 30% and D

_{7}= 30%; (

**d**) Scenario D, D

_{5}= 30% and D

_{6}= 30%.

**Figure 23.**Identified element bending stiffness EI from the combined effect analysis: undamaged scenario & Scenario C, D

_{4}= 30% and D

_{7}= 30%.

**Figure 25.**Identified results for different left bearing stiffness reduction: (

**a**) Mode shape and; (

**b**) element bending stiffness EI.

**Figure 35.**Identified results of the 4th span for field test: (

**a**) Mode shape; (

**b**) Bending stiffness EI.

**Figure 36.**Identified results of the 3rd span for field test: (

**a**) Mode shape; (

**b**) Bending stiffness EI.

Properties of Test Vehicle | No.1 | No.2 |
---|---|---|

Frequency ${\omega}_{v}$ (Hz) | 0.503 | 0.503 |

Mass ${m}_{v}$ (kg) | 5000 | 10,000 |

Stiffness ${k}_{v}$ (kN/m) | 50 | 100 |

Damping coefficient ${c}_{v}$ $\text{}\mathrm{kN}\xb7\mathrm{s}/\mathrm{m}$ | 1 | 2 |

Speed $v$ (m/s) | 1 | 1 |

Properties of the No.1 Test Vehicle | No.1-1 | No.1-2 | No.1-3 | No.1-4 | No.1-5 | No.1-6 |
---|---|---|---|---|---|---|

Frequency ${\omega}_{v}$ (Hz) | 0.503 | 0.650 | 1.125 | 1.592 | 2.251 | 2.757 |

Mass ${m}_{v}$ (Kg) | 5000 | 3000 | 1000 | 500 | 1000 | 1000 |

$\mathrm{Stiffness}\text{}{k}_{v}$ (kN/m) | 50 | 50 | 50 | 50 | 200 | 300 |

Damping coefficient ${c}_{v}$ ($\mathrm{kN}\xb7\mathrm{s}/\mathrm{m}$) | 1 | 1 | 1 | 1 | 1 | 1 |

Speed $v$ (m/s) | 1 | 1 | 1 | 1 | 1 | 1 |

Cases | Vehicle | Bridge | Road Surface Roughness | Temperature (°C) |
---|---|---|---|---|

1 | damped | undamped | -- | - |

2 | damped | damped | -- | - |

3 | damped | damped | Class C, D | - |

4 | damped | damped | Class C | −20, 0, 20 & 40 |

Damage Scenario | Damage Element(s) | Related Node Numbers | Reduction in Element Stiffness (%) | |||
---|---|---|---|---|---|---|

(a) | (b) | (c) | (d) | |||

A | D_{2} | 2, 3 | D_{2} = 0 | D_{2} = 15 | D_{2} = 30 | D_{2} = 50 |

B | D_{6} | 6, 7 | D_{6} = 0 | D_{6} = 15 | D_{6} = 30 | D_{6} = 50 |

C | D_{4}, D_{7} | 4, 5 &7, 8 | D_{4} = 0D _{7} = 0 | D_{4} = 15D _{7} = 15 | D_{4} = 30D _{7} = 30 | D_{4} = 50D _{7} = 50 |

D | D_{5}, D_{6} | 5, 6, 7 | D_{5} = 0D _{6} = 0 | D_{5} = 15D _{6} = 15 | D_{5} = 30D _{6} = 30 | D_{5} = 50D _{6} = 50 |

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**MDPI and ACS Style**

Yang, Y.; Cheng, Q.; Zhu, Y.; Wang, L.; Jin, R. Feasibility Study of Tractor-Test Vehicle Technique for Practical Structural Condition Assessment of Beam-Like Bridge Deck. *Remote Sens.* **2020**, *12*, 114.
https://doi.org/10.3390/rs12010114

**AMA Style**

Yang Y, Cheng Q, Zhu Y, Wang L, Jin R. Feasibility Study of Tractor-Test Vehicle Technique for Practical Structural Condition Assessment of Beam-Like Bridge Deck. *Remote Sensing*. 2020; 12(1):114.
https://doi.org/10.3390/rs12010114

**Chicago/Turabian Style**

Yang, Yang, Quan Cheng, Yuanhao Zhu, Lilei Wang, and Ruoyu Jin. 2020. "Feasibility Study of Tractor-Test Vehicle Technique for Practical Structural Condition Assessment of Beam-Like Bridge Deck" *Remote Sensing* 12, no. 1: 114.
https://doi.org/10.3390/rs12010114