*4.2. Model Calibration and Validation*

#### 4.2.1. Numerical Model

The test results clearly indicate that using flat rubber as an impact cushion leads to a secondary peak in the impact load, which is distinct from blast loading and thus unsuitable for simulating them. Therefore, a parametric analysis of pyramidal rubber was conducted to investigate the impact loading characteristics, taking into account the effects of impact velocity and rubber thickness. Figure 13 illustrates the numerical model used in this study. Initially, the original design was a 500 mm long and 500 mm wide pyramidal rubber during the experiment, which required a large mold for processing. Considering the cost and processing time, it was divided into 25 small pyramidal rubbers measuring 100 mm in length and width, each requiring only a small mold. In the numerical calculations, since the dispersed small pyramidal shapes needed to be considered for contact, we simplified the model accordingly. Based on the comparison between the numerical and experimental results, this simplification was found to be feasible.

**Figure 13.** Finite element model.

The model comprises an RC plate, rubber, steel impact module, and fixture. Meshing was performed using 8-node solid hexahedral cells, and mesh convergence analysis was conducted to determine the appropriate cell size. Based on the convergence analysis, a concrete mesh size of 7.5 mm was used, which was doubled outside the range of ±600 mm from the center of the slab to reduce calculation time. The impact module had a cell size of 10 mm in both the side length and thickness direction. The upper half of the rubber was pyramidal and tangentially treated to achieve a mesh size of approximately 10 mm. The grid division details are presented in Figure 14. Further mesh refinement was found to yield similar simulation results, but would significantly increase calculation time. Details of the mesh refinement analysis will be presented in Section 4.2.3.

## 4.2.2. Boundary Conditions

Reasonable boundary conditions are crucial for obtaining accurate numerical results. In this study, we assume the supporting structure of the RC slab to be a rigid body that is fixed, and thus surface-to-surface contact was used, which uses a penalty function to determine the contact force to define the contact between the specimens and the supporting structure to constrain the specimens. To simplify the calculation and save computing resources, we directly define the impact velocity using \*INITIAL\_VELOCITY\_GENERATION, applying a downward vertical initial velocity to the impact module. Additionally, a surfaceto-surface contact was used to define the contact between the impact module, rubber, and concrete.

#### 4.2.3. Grid Refinement Analysis

The numerical model's grid size was determined by conducting five analyses with varying grid resolutions, and the results are presented in Table 6. The grid convergence tests involved five cell sizes, namely 5 mm, 7.5 mm, 10 mm, 20 mm, and 30 mm. The peak pressures calculated for the models with the five grid sizes were found to be very similar, with a maximum error of only 3.41%. This indicates that reducing the grid size has little impact on the numerical results, but it significantly increases the computational time. Therefore, a grid size of 7.5 mm was selected for this study to strike a balance between accuracy and computational efficiency.

**Figure 14.** Detail of the grid division. (**a**) Impact module; (**b**) Rubber; (**c**) and RC slab.



4.2.4. Comparison of Experimental and Numerical Results

The numerical model was calibrated by comparing the results of numerical calculations with the experimental test results. In Figure 15a,b, the pressure–time histories of Test 9 and the peak pressure and impulse of Tests 6 to 9 are presented, respectively. Additionally, the percentage differences between the experimental results and the numerical results are presented in Table 7. The comparison results indicate that the pressure curve obtained via numerical simulation is in good agreement with the measurement results.

**Figure 15.** Comparison of numerical and experimental results. (**a**) Pressure profile; and (**b**) Peak pressure and impulse.


**Table 7.** Comparison of numerical calculations with experimental test.

#### *4.3. Parametric Studies*

#### 4.3.1. Effect of Rubber Shapes

The effect of rubber shape on the impact loading was investigated by comparing the results of six rubber shapes that are shown in Figure 16. The impact velocity was 20 m/s and the values of *h*<sup>1</sup> and *h*<sup>2</sup> are 20 mm and 30 mm, respectively. Figure 17a shows the impact response of six rubber shapes of rubber as an impact cushion for a reinforced concrete slab with a thickness of 180 mm. As expected, the pressure–time curve is smoothest when the upper side *la* is equal to 0. As *la* gradually increases, the pressure–time curve begins to oscillate and reaches a maximum when *la* is 100 mm. In addition, Figure 15b, highlights the peak pressure and impulse of the impact response for *la* from 0 to 90 mm and the fitted curve, excluding the case where the upper side *la* is 100 mm. The peak pressure and impulse of the upper side between 0 and 90 mm can be calculated according to the following equation.

$$P\_{la} = 22.124 + 4.409 \times \sin(\pi \times (l\_a - 9.254)/9.72), \ 0 \le l\_a \le 90 \text{ mm} \tag{3}$$

$$I\_{l\_a} = 22.668 + 2.543 \times \sin(\pi \times (l\_a + 0.498)/6.49), \\ 0 \le l\_a \le 90 \text{ mm} \tag{4}$$

#### 4.3.2. Effect of Impact Velocities

To further characterize the effect of velocity on the impact loading characteristics, velocities ranging from 10 to 50 m/s were set, and numerical calculations were carried out. Figure 18 shows the load profile and the relationship between peak pressure and impulse versus velocity. As the speed increases, the peak pressure gradually increases, while the time of the load decreases accordingly, a connection can be established using

Equation (5). The impulse also increases with velocity, unlike the pressure, which increases at a progressively slower rate and can be described using Equation (6).

$$P\_v = -21.275 + 17.135e^{-v/28.489} \tag{5}$$

$$I\_{\upsilon} = 565.256 - 13.415e^{\left(-\upsilon/9.804\right)} - 555.419e^{\left(-\upsilon/1972.268\right)}\tag{6}$$

**Figure 16.** Schematic of rubber shapes variation of the upper side.

**Figure 17.** Influence of rubber shapes on the impact loading. (**a**) Pressure–time history; and (**b**) Peak pressure and impulse.

**Figure 18.** Influence of impact velocity on the impact loading. (**a**) Pressure–time history; and (**b**) Peak pressure and impulse.

#### 4.3.3. Effect of Bottom Thicknesses

The effect of bottom thickness *h*<sup>1</sup> on the impact loading was investigated by comparing the results of ten bottom thicknesses. At an impact velocity of 20 m/s, the *h*<sup>2</sup> value was held constant at 30 mm while the *h*<sup>1</sup> varied from 20 mm to 170 mm, as illustrated in Figure 19. The peak pressure initially decreases significantly as *h*<sup>1</sup> increases, and then gradually decreases while impulse increases linearly with the *h*1, as shown in Figure 20. To simplify the calculations, the results are fitted with the peak pressure calculated as shown in Equation (7) and with the impulse calculated using Equation (8).

$$P\_{h\_1} = 5.454 + 10.794e^{-h\_1/4.983}, \text{ 2 cm} \le h\_1 \le 17 \text{ cm} \tag{7}$$

$$I\_{h\_1} = 61.479 - 48.185e^{-h\_1/310.853}, \text{ 2 cm} \le h\_1 \le 17 \text{ cm} \tag{8}$$

**Figure 19.** Schematic of the variation of the rubber bottom thickness *h*1.

**Figure 20.** Influence of bottom thickness *h*<sup>1</sup> on the impact loading. (**a**) Pressure–time history; and (**b**) Peak pressure and impulse.

#### 4.3.4. Effect of Upper Thicknesses

Similarly, the effect of upper thickness on the impact loading was investigated by comparing the results of nine upper thicknesses of pyramidal rubber. At an impact velocity of 20 m/s, the bottom thickness *h*<sup>1</sup> was held constant at 20 mm while the *h*<sup>2</sup> thickness varied from 10 mm to 130 mm, as detailed in Figure 21. The impact response of nine upper thicknesses of rubber as impact cushion for a reinforced concrete slab with a thickness of 180 mm is shown in Figure 22.

**Figure 21.** Schematic of the variation of the rubber upper thickness *h*2.

It can be seen that the pressure gradually decreases while the time of load and impulse increase accordingly as *h*<sup>2</sup> gradually increases. It is worth noting that when *h*<sup>2</sup> is between 70 mm and 130 mm, the impulse is almost unaffected, accompanied by a significant reduction as *h*<sup>2</sup> continues to increase. To simplify the calculations, the results are fitted with the peak pressure calculated as shown in Equation (9) and with the impulse calculated using Equation (10).

$$P\_{h\_2} = 2.664 + 45.495e^{-h\_2/1.14} + 9.17e^{-h\_2/10.12}, \text{ 1 cm} \le h\_2 \le 17 \text{ cm} \tag{9}$$

$$I\_{h\_2} = 7.811 - 2.927 \times h\_2 - 0.41 \times h\_2^2 + 0.026 \times h\_2^3 - 6.441^{-4} \times h\_2^4, \text{ } 1 \text{ cm} \le h\_2 \le 13 \text{ cm} \tag{10}$$

**Figure 22.** Influence of upper thickness *h*<sup>2</sup> on the impact loading. (**a**) Pressure–time history; (**b**) Peak pressure and impulse.

#### **5. Summary and Conclusions**

In this study, experimental tests were performed on four specimens to demonstrate the feasibility of the VMLH Blast Simulator for simulating blast loading. A numerical model was also developed to predict the impact loading using LS-DYNA. The validity of the model has been calibrated against experimental test results. Using the calibrated model, further studies are carried out to investigate the effect of different parameters on the impact loading of RC plates. The parameters investigated within the scope of this study were rubber shapes, impact velocities and the bottom thicknesses and the top thicknesses of the pyramidal rubber. The following conclusions can be drawn from the detailed experimental and numerical studies presented in this paper.

(1) The use of pyramidal rubber with a 0 mm upper side was more effective in regulating the peak pressure and impulse of impact loading compared to planar rubber with a 100 mm upper side. This was evident in the pressure–time curve, which closely resembled ideal blast loading. However, when *la* was between 40 mm and 100 mm, the pressure profile oscillated significantly, making it unsuitable for the pressure–impulse criterion.

(2) The impact velocity was found to have a significant effect on the pressure and impulse of impact loading. For a pyramidal rubber thickness of 50 mm, both pressure and impulse increased rapidly with increasing velocity. When the speed increases from 12.76 m/s to 23.41 m/s, the corresponding range of peak pressure is from 6.457 to 17.108 MPa, with an increase of 164.22%. The corresponding range of impulse is from 8.573 to 14.151 MPa·ms, with an increase of 65.07%.

(3) Variations in the upper thickness of the pyramidal rubber have a more positive effect on the impact loading than the bottom thickness. Notably, increasing the top thickness from 30 mm to 130 mm resulted in a 59.01% decrease in peak pressure and a 16.64% increase in impulse. Conversely, increasing the bottom thickness from 30 mm to 130 mm resulted in a 44.59% decrease in peak pressure and an 11.01% increase in impulse. As the bottom thickness increases, it takes longer for the rubber to compress and become compressed. When bottom thickness is 110 mm, the pressure reaches its peak and then barely changes in a time of approximately 2 ms, which is unsuitable for using the pressure–impulse criterion for modeling blast loading. Increasing the upper thickness only allows the pressure to rise and fall more smoothly without changing its shape characteristics, making it possible to adopt both criteria for simulating blast loading.

(4) The impact loading and blast loading can be converted using the "pressure-impulse" and " impulse" criterion. By obtaining the peak pressure and impulse of the impact loading, a corresponding explosive environment can always be found. These criteria can be widely applied in simulating blast loading using non-explosive methods.

**Author Contributions:** Conceptualization, G.Y. and L.W.; methodology, W.W.; investigation, J.M.; resources, G.Y.; data curation, W.Z.; writing—original draft preparation, Z.X.; writing—review and editing, G.Y. and W.W.; supervision, L.W.; project administration, G.Y.; funding acquisition, G.Y. and L.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.
