# Durability Assessment Method of Hollow Thin-Walled Bridge Piers under Rockfall Impact Based on Damage Response Surface

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rockfall-HTWBP Impact Model

#### 2.1. FE Model of the Bridge and Rockfall

#### 2.2. Material Model

^{3}[1]. Moreover, previous studies found that the rockfall only appears with slightly less damage compared with the concrete pier [21,22]. Therefore, the rockfall is regarded as a rigid material and the material parameters are listed in Table 2.

#### 2.3. Verification

#### 2.3.1. Verification against Two Experiments

#### 2.3.2. Mesh Independence Verification

## 3. Rockfall Impact Resistance

**s**, is 30, 40, and 50 MPa, respectively. The rockfall size is determined according to two previous research [1,2]. In the study by Xie et al., 2020, three rock diameters, i.e., 1.2 m, 1.5 m, and 1.8 m, are considered [1]. In another reference [2], the rockfall diameters range from 0.4 m to 4 m. However, the rockfall with a diameter of lower than 1.0 m is considered to have limited influence on the pier. Table 4 summarizes the case studies used in this section.

#### 3.1. Impact Force Characteristics

^{3}kN, 33.4 × 10

^{3}kN, 31.3 × 10

^{3}kN, and 29.4 × 10

^{3}kN, respectively. When the impact height increases from 10 m to 40 m, the peak impact force decreases by 15.5%. The possible reason for this phenomenon is that when the impact position is closer to the pile cap, the pier is constrained more strongly, and the corresponding peak impact force is larger. As shown in Figure 11b–d, the peak impact force increases with the increase in impact velocity, rockfall diameter, and concrete strength. When the impact velocity increases from 10 m/s to 30 m/s, the peak impact force increases by 170.1%. When the rockfall diameter increases from 1.5 m to 4.0 m, the peak impact force increases by 100.0%. This is because the larger the velocity and diameter of the rockfall, the larger the corresponding kinetic energy, leading to an increase in the impact force. When the concrete strength grade increases from 30 MPa to 50 MPa, the peak impact force only increases by approximately 5%. This indicates that the impact force mainly depends on the stiffness of the rockfall, while the stiffness of the bridge pier plays a secondary role, which is consistent with the research results reported in [1].

#### 3.2. Damage Characteristics

#### 3.3. Deformation Characteristics

## 4. Damage Assessment Criterion Based on the Response Surface Model

#### 4.1. Definition of Damage Index

_{e}calculated by the CSCM, the volume damage index d

_{v}of the HTWBP volume is defined as follows:

_{v}is normalized, and the corresponding damage index D can be expressed as follows:

#### 4.2. Establishment of the Response Surface Model

^{4}= 81 FE simulations need to be involved. Hence, the Box–Behnken design is adopted to reduce the number of FE simulations and improve the computational efficiency of sensitivity analysis. Table 5 shows the factors and levels of Box–Behnken design in this section. According to the Box–Behnken design in Table 5, the corresponding cases and damage results are listed in Table 6. According to the data in Table 6, the response surface model can be expressed as Equation (13).

^{2}, and relative error (RE) are used to evaluate the accuracy of the model. As listed in Table 7 and shown in Figure 19, the maximum difference between the FE model and predicated by the RSM is only 4.8%. The determination coefficient R

^{2}value is 0.9610, which is close to 1.0, and the average RE is only 4.5%. Therefore, the response surface model established expressed in Equation (13) shows high competence in predicting the damage of HTWBP after rockfall impact.

#### 4.3. Damage Assessment Method

## 5. Discussion

## 6. Conclusions

- (1)
- The impact force of rockfalls has a considerable impulse characteristic, and the duration of the impulse load is approximately 0.01 s. When the impact position is closer to the pile cap, the pier is constrained more strongly, and the corresponding peak impact force is larger. The peak impact force increases with the increase in impact velocity, rockfall diameter, and concrete strength. When the concrete strength grade increases from 30 MPa to 50 MPa, the peak impact force only increases by approximately 5%.
- (2)
- The impacted surface is dominated by the final elliptic damage with the conical and strip damage areas as the symmetry axis. At the side surface, the damage develops from a triangular damage area into a radial damage area across the side surface of the pier. The cross-sectional damage mode is compression failure in the impact area and shear failure at the corner.
- (3)
- The displacement of the top and bottom of the pier is significantly lower than that of other cross-sections due to constraints. The time when the maximum displacement occurs of the cross-sections below the impact position is earlier than the cross-sections upper the impact position because the upper part of the pier has greater flexibility and is more prone to deformation.
- (4)
- The maximum displacement occurs in the middle height of the pier. This is because the HTWBP has a smaller cross-section at the top and a larger cross-section at the bottom. The lower part of the pier has a larger stiffness and a stronger ability to resist deformation. The maximum displacement increases with impact height, impact velocity, and rockfall diameter and decreases with the uniaxial compressive strength of the concrete.
- (5)
- The initial impact velocity and diameter of the rockfall are the most remarkable parameters affecting the damage indices. The damage assessment method with a damage zoning diagram based on the response surface method is established. With the proposed damage assessment method with a damage zoning diagram, the fast assessment of the damage level of impacted HTWBP can be realized according to rockfall diameter and velocity after rockfall impact.

## 7. Limitations

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**FE model of the bridge and rockfall. (

**a**) bridge; (

**b**) reinforcement; (

**c**) cross-section; (

**d**) rockfalls.

**Figure 5.**The drop weight test [23].

**Figure 9.**Comparison between experiment [23] and FE analysis: (

**a**) mid-span displacement; (

**b**) impact force.

**Figure 10.**Three-dimensional view of the experiment setup and reinforcement of the column [24]. (

**a**) experiment setup; (

**b**) reinforcement of the column.

**Figure 11.**Comparison between the experiment in [24] and FE analysis: (

**a**) damage pattern; (

**b**) impact force.

**Figure 13.**Impact force of rockfall-HTWBP impact. (

**a**) impact height; (

**b**) impact velocity; (

**c**) rockfall diameter; (

**d**) strength grade.

**Figure 17.**Maximum displacement of the HTWBP. (

**a**) impact height; (

**b**) impact velocity; (

**c**) rockfall diameter; (

**d**) strength grade.

**Figure 18.**Damage response surfaces with different parameters. (

**a**) impact height and impact velocity; (

**b**) impact height and rockfall diameter; (

**c**) impact height and strength grade; (

**d**) impact velocity and rockfall diameter; (

**e**) impact velocity and strength grade; (

**f**) rockfall diameter and strength grade.

**Figure 20.**Response surface of the damage index under different impact velocities and rockfall diameter.

**Figure 22.**Comparison between a single rockfall and double-rockfall impact: (

**a**) front view; (

**b**) side view.

Component | Element Type | Grid Size (mm) |
---|---|---|

Impacted pier | Solid | 50 |

Other piers | Solid | 100 |

Pile cap | Solid | 200 |

Material Type | Keyword in LS-DYNA | Parameter | Value |
---|---|---|---|

Transverse reinforcement | * MAT_PLASTIC_KINEMATIC | Elastic modulus (GPa) | 210 |

Density (g/cm^{3}) | 7.85 | ||

Poisson’s ratio | 0.3 | ||

Failure strain | 0.25 | ||

Yield strain (MPa) | 300 | ||

Longitudinal reinforcement | * MAT_PLASTIC_KINEMATIC | Elastic modulus (GPa) | 210 |

Density (g/cm^{3}) | 7.85 | ||

Poisson’s ratio | 0.3 | ||

Failure strain | 0.2 | ||

Yield strain (MPa) | 400 | ||

Rockfall | * MAT_RIGID | Elastic modulus (GPa) | 26.0 |

Density (g/cm^{3}) | 2.6 | ||

Poisson’s ratio | 0.22 |

Impact Height | Peak Impact Force (kN) | Mid-Span Displacement (mm) | ||||
---|---|---|---|---|---|---|

Experiment [23] | FE Analysis | Difference | Experiment [23] | FE Analysis | Difference | |

0.15 m | 124.3 | 125.8 | 1.2% | 6.1 | 6.2 | 1.6% |

0.30 m | 182.7 | 180.9 | 1.0% | 10.9 | 10.6 | 2.8% |

0.60 m | 243.8 | 250.7 | 2.8% | 20 | 20.2 | 1.0% |

1.20 m | 308.4 | 310.9 | 0.8% | 36.6 | 36.5 | 0.3% |

Case Study | h (m) | v (m/s) | d (m) | s (MPa) |
---|---|---|---|---|

C1 (Reference case study) | 40 | 20 | 1.5 | 30 |

C2 | 10 | 20 | 1.5 | 30 |

C3 | 20 | 20 | 1.5 | 30 |

C4 | 30 | 20 | 1.5 | 30 |

C5 | 40 | 10 | 1.5 | 30 |

C6 | 40 | 30 | 1.5 | 30 |

C7 | 40 | 20 | 2 | 30 |

C8 | 40 | 20 | 2.5 | 30 |

C9 | 40 | 20 | 3.0 | 30 |

C10 | 40 | 20 | 3.5 | 30 |

C11 | 40 | 20 | 4.0 | 30 |

C12 | 40 | 20 | 1.5 | 40 |

C13 | 40 | 20 | 1.5 | 50 |

Factor | Level | ||
---|---|---|---|

−1 | 0 | 1 | |

h (m) | 20 | 40 | 60 |

v (m/s) | 10 | 20 | 30 |

d (m) | 1.5 | 2.0 | 2.5 |

s (MPa) | 30 | 40 | 50 |

Std | Run | h (m) | v (m/s) | d (m) | s (MPa) | d_{v} (×10^{−3}) | D |
---|---|---|---|---|---|---|---|

1 | 24 | 20 | 10 | 2 | 40 | 0.89 | 0.156 |

2 | 28 | 60 | 10 | 2 | 40 | 0.67 | 0.117 |

3 | 17 | 20 | 30 | 2 | 40 | 1.49 | 0.260 |

4 | 9 | 60 | 30 | 2 | 40 | 1.61 | 0.281 |

5 | 16 | 40 | 20 | 1.5 | 30 | 1.12 | 0.196 |

6 | 7 | 40 | 20 | 2.5 | 30 | 1.32 | 0.231 |

7 | 12 | 40 | 20 | 1.5 | 50 | 0.89 | 0.156 |

8 | 23 | 40 | 20 | 2.5 | 50 | 1.63 | 0.285 |

9 | 4 | 20 | 20 | 2 | 30 | 1.20 | 0.210 |

10 | 22 | 60 | 20 | 2 | 30 | 1.14 | 0.199 |

11 | 18 | 20 | 20 | 2 | 50 | 1.13 | 0.198 |

12 | 5 | 60 | 20 | 2 | 50 | 0.93 | 0.162 |

13 | 21 | 40 | 10 | 1.5 | 40 | 0.78 | 0.136 |

14 | 8 | 40 | 30 | 1.5 | 40 | 1.16 | 0.203 |

15 | 29 | 40 | 10 | 2.5 | 40 | 1.16 | 0.203 |

16 | 15 | 40 | 30 | 2.5 | 40 | 1.95 | 0.341 |

17 | 6 | 20 | 20 | 1.5 | 40 | 1.00 | 0.175 |

18 | 2 | 60 | 20 | 1.5 | 40 | 1.00 | 0.175 |

19 | 27 | 20 | 20 | 2.5 | 40 | 1.62 | 0.283 |

20 | 14 | 60 | 20 | 2.5 | 40 | 1.39 | 0.243 |

21 | 1 | 40 | 10 | 2 | 30 | 0.93 | 0.163 |

22 | 11 | 40 | 30 | 2 | 30 | 1.48 | 0.259 |

23 | 10 | 40 | 10 | 2 | 50 | 1.08 | 0.189 |

24 | 26 | 40 | 30 | 2 | 50 | 1.51 | 0.264 |

25 | 3 | 40 | 20 | 2 | 40 | 1.02 | 0.178 |

26 | 25 | 40 | 20 | 2 | 40 | 0.99 | 0.173 |

27 | 20 | 40 | 20 | 2 | 40 | 1.11 | 0.194 |

28 | 19 | 40 | 20 | 2 | 40 | 1.01 | 0.177 |

29 | 13 | 40 | 20 | 2 | 40 | 0.94 | 0.164 |

Run | h (m) | v (m/s) | d (m) | s (MPa) | D | Difference | |
---|---|---|---|---|---|---|---|

FE Results | RSM Results | ||||||

30 | 40 | 20 | 1.5 | 50 | 0.16 | 0.15 | 4.8% |

31 | 40 | 20 | 1.5 | 40 | 0.16 | 0.16 | 1.2% |

32 | 40 | 20 | 2 | 30 | 0.19 | 0.19 | 1.6% |

33 | 40 | 15 | 1.5 | 30 | 0.19 | 0.18 | 4.7% |

34 | 40 | 25 | 2.5 | 30 | 0.35 | 0.34 | 2.7% |

35 | 60 | 20 | 2.5 | 30 | 0.24 | 0.24 | 3.2% |

Factor | Level | ||||
---|---|---|---|---|---|

v (m/s) | 10 | 20 | 30 | 40 | 50 |

d (m) | 1.0 | 1.375 | 1.75 | 2.125 | 2.5 |

Std | Run | V (m/s) | D (m) | d_{v} (×10^{−3}) | D |
---|---|---|---|---|---|

1 | 2 | 10 | 1 | 0.0476 | 0.01 |

2 | 8 | 50 | 1 | 2.61 | 0.46 |

3 | 9 | 10 | 2.5 | 1.24 | 0.22 |

4 | 6 | 50 | 2.5 | 5.72 | 1.00 |

5 | 3 | 20 | 1.75 | 2.18 | 0.38 |

6 | 5 | 40 | 1.75 | 3.84 | 0.67 |

7 | 1 | 30 | 1.375 | 2.43 | 0.42 |

8 | 7 | 30 | 2.125 | 3.49 | 0.61 |

9 | 4 | 30 | 1.75 | 2.81 | 0.49 |

10 | 4 | 30 | 1.75 | 3.14 | 0.55 |

11 | 8 | 30 | 1.75 | 2.94 | 0.51 |

12 | 13 | 30 | 1.75 | 2.92 | 0.51 |

13 | 3 | 30 | 1.75 | 3.26 | 0.57 |

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**MDPI and ACS Style**

Li, F.; Liu, Y.; Yang, J.
Durability Assessment Method of Hollow Thin-Walled Bridge Piers under Rockfall Impact Based on Damage Response Surface. *Sustainability* **2022**, *14*, 12196.
https://doi.org/10.3390/su141912196

**AMA Style**

Li F, Liu Y, Yang J.
Durability Assessment Method of Hollow Thin-Walled Bridge Piers under Rockfall Impact Based on Damage Response Surface. *Sustainability*. 2022; 14(19):12196.
https://doi.org/10.3390/su141912196

**Chicago/Turabian Style**

Li, Fei, Yikang Liu, and Jian Yang.
2022. "Durability Assessment Method of Hollow Thin-Walled Bridge Piers under Rockfall Impact Based on Damage Response Surface" *Sustainability* 14, no. 19: 12196.
https://doi.org/10.3390/su141912196