# Sensitivity Analysis of Factors Influencing Blast-like Loading on Reinforced Concrete Slabs Based on Grey Correlation Degree

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## Abstract

**:**

## 1. Introduction

## 2. Grey Correlation Analysis Method

- (1)
- Determination of the sequence matrix

_{1}, and ellipsoid rubber thickness h

_{2}) are selected as the influence factor subsequence X, X = (X X

_{12}...X)

_{i}

^{T}, the corresponding peak load pressure as the parent sequence Y, Y = (Y Y

_{12}...Y)

_{i}

^{T}, and the load impulse as the parent sequence Z, Z = (Z Z

_{12}...Z)

_{i}

^{T}. Each factor of series X, series Y, and series Z has j values and the matrix form

- (2)
- Matrix dimensionless

- (3)
- Differential sequence matrix

- (4)
- Grey correlation coefficient matrix

- (5)
- Solve for the grey correlation G

## 3. Modelling of Blast Loading

#### 3.1. The Basics of Blast Loading

#### 3.2. Blast Similarity Law

_{EXP}is the energy of the explosives, Q

_{TNT}is the energy of the TNT, W is the mass of the TNT charge, and W

_{EXP}is the mass of the charge.

#### 3.3. Blast Load Parameters

## 4. Finite Element Modelling

#### 4.1. Calibrated Model

#### 4.2. Improved Model

#### 4.3. Material Modelling and Parameters

#### 4.3.1. Concrete

#### 4.3.2. Steel

_{0}is the initial yield strength of steel; β is the hardening parameter; ${E}_{P}$ is the hardening modulus; and ${\epsilon}_{P}^{eff}$ is the effective plastic strain.

#### 4.3.3. Rubber

_{n}(where n is equal to 1, 2, or 3), is also incorporated into the equation. Notably, Equation (15) contains the sole material constant, G, as show in Table 2.

#### 4.3.4. Supporting Structure

Material | Parameter | Value | Comments |
---|---|---|---|

Concrete | RO (Density) | 2400 kg/m^{3} | Material test data |

FPC (Uniaxial compression strength) | 45.6 MPa | ||

NPLOT | 1 | According to [23,25,26] | |

INCRE | 0 | ||

IRATE (Rate effects options) | 1 | ||

Elements erode | 1.1 | ||

DAGG (Maximum aggregate size) | 24 mm | ||

UNITS (Units options) | 4 | ||

Steel | Density | 7800 kg/m^{3} | Material test data |

Young’s modulus | 2.09 × 10^{5} MPa | ||

Poisson’s ratio | 0.3 | ||

Yield stress | 435.3 or 450.1 MPa | ||

Rubber | Density | 1.27 kg/m^{3} | According to [24,27] |

Poisson’s ratio | 0.463 | ||

Shear modulus | 24 MPa | ||

Supporting structure | Density | 7800 kg/m^{3} | According to [27] |

Young’s modulus | 2.09 × 10^{5} MPa | ||

Poisson’s ratio | 0.3 |

#### 4.4. Parameter Setting

#### 4.5. Defining Outputs

## 5. Study of Impact Load Characteristics

#### 5.1. The Shape of the Impact Module

#### 5.2. Effect of Flat Rubber Thickness

_{1}and impact loading, the flat rubber thickness h

_{1}in the impact module was adjusted and set in the range of 10 mm to 100 mm. At the same time, ellipsoid rubber thickness was kept at 40 mm, and the thickness of the counterweight steel plate was changed accordingly to ensure that the total mass of the impact module remained constant, as shown in Figure 9. According to this design scheme, six impact modules with different flat rubber thicknesses were designed and they were applied to the RC plate separately using the same velocity of 20 m/s to investigate the correlation between the flat rubber thickness and the impact loading.

_{1}increases. It is worth noting that this trend starts to diminish when the flat rubber thickness h

_{1}reaches 60 mm. Overall, the relationship between the thickness of the screed rubber and the loading characteristics can be expressed as Equations (16) and (17).

#### 5.3. Effect of Ellipsoid Rubber Thickness

_{2}and the impact loading, the ellipsoid rubber thickness h

_{2}in the impact module was adjusted to a range of 10 mm to 100 mm, and flat rubber thickness h

_{1}was kept at 20 mm, while the thickness of the counterweight steel plate was changed accordingly to ensure that the total mass of the impact module remained constant. Based on this principle, six different ellipsoid thicknesses of the impact module were designed, see Figure 11, and they were applied to the RC plate using the same velocity of 20 m/s to examine the correlation between the ellipsoid rubber thickness and the impact loading.

_{2}the peak pressure gradually decreases, but the impact loading time is prolonged and, at the same time, the impulse also increases accordingly. It is worth noting that when the ellipsoid thickness h

_{2}reaches 60 mm the peak pressure decreases, and the impulse begins to show a decreasing trend. This correlation can be illustrated by Equations (18) and (19).

#### 5.4. Effect of Impact Velocity

#### 5.5. Effect of Impact Module Quality

## 6. Sensitivity Analyses of Influencing Factors

#### 6.1. Relevance Calculation

_{P}is obtained as

_{I}is

#### 6.2. Sensitivity Assessment

_{1}, ellipsoid rubber thickness h

_{2}, impact velocity V, and impact modulus mass M. Unlike pressure, impulse is more sensitive to changes in flat rubber thickness h

_{1}than in ellipsoid rubber thickness h

_{2}; load impulse, in descending order of sensitivity, is impact modulus mass m and impact velocity v. The sensitivities are impact modulus mass m and impact velocity v. The sensitivities are impact modulus mass m and impact velocity v, flat rubber thickness h

_{1}and ellipsoid rubber thickness h

_{2}.

## 7. Summary and Conclusions

- (1)
- Changes in ellipsoid rubber thickness had a more positive effect on impact loading than flat layer rubber thickness. It is worth noting that, when the ellipsoid thickness increased from 10 mm to 100 mm, the peak pressure showed a maximum decrease of 29%, and the average decrease was maintained at 21%. The impulse also increased from the initial growth, and showed a decreasing trend at the later stage; however, during the growth of the flat layer rubber thickness from 10 mm to 100 mm, the peak pressure showed a maximum decrease of 19% and the average decrease was only 8%, and the impulse increased gradually and the average increase was 5%.
- (2)
- The impact velocity has a significantly greater effect on the peak pressure of the impact load than the impulse. When the velocity increases from 10 m/s to 15 m/s the peak pressure increases by 68% and the impulse increases by about 40%. However, after that, when the speed increases every 5 m/s the peak pressure can still keep a high increase of 46%. It maintains this increase until the speed reaches 40 m/s, where it can still maintain a minimum increase of 22%. The increase in impulse is rapidly reduced to 18%. When the speed increases to 40 m/s, the impulse only increased by 8%.
- (3)
- Unlike impact velocity, the mass of the impact module had a significantly greater effect on the impact load impulse than the peak pressure. When the mass increased from 100 kg to 200 kg the peak pressure increased by 54% and the impulse increased by about 108%. However, thereafter, as the mass increases by 100 kg, the increase in peak pressure decreases to 27%. It continues to decrease until the mass increases to 600 kg, where the increase is only 6%. Meanwhile, the increase in impulse decreases rapidly but still retains an increase of 32%. When the mass increases to 600 kg, the impulse also maintains a minimum increase of 18%.
- (4)
- When the four factors of impact module mass, impact velocity, ellipsoid rubber thickness, and flat rubber thickness are changed the peak load pressure and impulse can be affected. Peak load pressure and impulse are sensitive to changes in these factors, but there are differences in the degree of sensitivity to the thickness of the ellipsoid rubber and flat rubber thickness. Specifically, both peak load pressure and impulse are most sensitive to the mass of the impact module. The difference is that peak pressure is more sensitive to changes in ellipsoid rubber thickness than flat rubber thickness, while the opposite is true for impulse.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Spherical TNT airburst shock wave: (

**a**) Reflected overpressure–time-course curve; (

**b**) Positive reflected overpressure peaks.

**Figure 3.**Spherical TNT airburst shock wave: (

**a**) Positively reflected impulse time-course curve; (

**b**) Proportional impulse peaks.

**Figure 4.**Comparison of numerical and experimental results: (

**a**) Pressure curve (v = 23.41 m/s); (

**b**) Peak pressure and impulse [20].

**Figure 7.**Schematic design of four impact cushion shapes: (

**a**) Ellipsoidal; (

**b**) Prismatic conical; (

**c**) Prismatic; (

**d**) Prismatic.

**Figure 10.**Influence of flat rubber thickness on the loading characteristics: (

**a**) Pressure–time-course curve; (

**b**) Peak pressure and impulse.

**Figure 12.**Effect of ellipsoid rubber thickness on loading characteristics: (

**a**) Pressure–time-course curve; (

**b**) Peak pressure and impulse.

**Figure 13.**Effect of impact velocity on load characteristics: (

**a**) Pressure–time-course curve; (

**b**) Peak pressure and impulse.

**Figure 15.**Effect of impact mass on load characteristics: (

**a**) Pressure–time-course curve; (

**b**) Peak pressure and impulse.

T = 0 | $0\text{}\mathbf{T}\text{}{\mathit{T}}_{\mathit{i}}^{-}$ | $\mathbf{T}={\mathit{T}}_{\mathit{i}}^{-}$ | $\mathbf{T}={\mathit{T}}_{\mathit{i}}$ |
---|---|---|---|

Initial | Acceleration | Separation | Impact |

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**MDPI and ACS Style**

Xiong, Z.; Wang, W.; Wu, Y.; Liu, W.
Sensitivity Analysis of Factors Influencing Blast-like Loading on Reinforced Concrete Slabs Based on Grey Correlation Degree. *Materials* **2023**, *16*, 5678.
https://doi.org/10.3390/ma16165678

**AMA Style**

Xiong Z, Wang W, Wu Y, Liu W.
Sensitivity Analysis of Factors Influencing Blast-like Loading on Reinforced Concrete Slabs Based on Grey Correlation Degree. *Materials*. 2023; 16(16):5678.
https://doi.org/10.3390/ma16165678

**Chicago/Turabian Style**

Xiong, Zhixiang, Wei Wang, Yangyong Wu, and Wei Liu.
2023. "Sensitivity Analysis of Factors Influencing Blast-like Loading on Reinforced Concrete Slabs Based on Grey Correlation Degree" *Materials* 16, no. 16: 5678.
https://doi.org/10.3390/ma16165678