**3. Experimental Results and Discussion**

The impact tests were carried out using a 100 kg impact module with an adjustable impact speed and impact cushion to apply a range of impact loading to the specimens. A total of nine tests were carried out on four specimens. For tests 1 to 5, planar rubber thicknesses ranging from 20 to 100 mm, and impact velocities of approximately 15 m/s were selected. Tests 6 to 9 used pyramidal rubber in a 5 × 5 arrangement, with impact velocities of 10 to 25 m/s. Tests 10 and 11 were repeat control tests of Test 3. Table 2 summarizes the impact velocity, contact force and loading time for each test and provides the results for the pressure and shock volumes. The calculation of the parameters and the analysis of the test results are presented below.

**Table 2.** Experimental results (force, time, peak pressure, and impulse).


Notes: *h*—Planar rubber thickness; *v*—Velocity; *F*—Force; *T*—Time; *P*—Peak pressure; *I*—Impulse.

Based on the assumption of uniform loading, we do not want the specimen to experience significant deformation and damage, as this would not be conducive to the repeatability of our measurement results. As you can see, this is reflected in the test speed, which is controlled below 25 m/s, far below the equipment's speed limit, making the specimen less prone to damage. Figure 5 shows the damage to the specimen before and after a single impact, with several cracks appearing but with no significant deformation occurring. Due to the distance of the specimen's bottom from the buffer bar being only about 100 mm, it was not possible to effectively place the displacement sensors to obtain the mid-span deflection value of the specimen.

**Figure 5.** Damage condition of the specimen after a single impact. (**a**) Before the test; and (**b**) After the test.

#### *3.1. Methodologies for Calculated Parameters*

#### 3.1.1. Load

The accelerometers and load cells were strategically placed to capture test data, which was then used to calculate pressure and impulse. Assuming that the VMLH Blast Simulator applies a uniform load to the surface of the specimen, the accelerometer data was converted to force by multiplying it with the weight of the impact module, and to pressure by dividing it by the impact area. Similarly, load cell data was converted directly to pressure by dividing it by the impact area. For instance, the pressure and impulse data obtained from Test 8 are illustrated in Figure 6. The impact loads obtained from both methods were found to be very similar, which is why the load cell data was primarily used for subsequent analysis.

**Figure 6.** Impact loading measured using accelerometers and load cells. (**a**) Pressure profile; and (**b**) Impulse profile.

#### 3.1.2. Velocity

A high-speed camera was used to record the impact test procedure. Figure 7 illustrates the four distinct stages of compression, acceleration, separation, and impact. Following the separation from the brittle bolt, the impacting module is still 0.9 m away from the specimen surface and begins to fall freely. As a result, the velocity of the impact module at the moment of impact with the specimen can be determined by calculating the final velocity of the displacement sensor.

**Figure 7.** High-speed video of impact test. (**a**) Inflation; (**b**) Acceleration; (**c**) Separation; and (**d**) Impact.

#### *3.2. Comparison of Impact Loading and Blast Loading*

A typical blast scenario is shown in Figure 8, which includes a spherical charge of TNT weight, W, at a standoff distance, R, away from a structure [19,20]. The detonation of the explosive creates a shock wave that forms a reflected wave when it reaches the surface of the structure. Under these conditions, an example of a typical reflected pressure profile at a point on the structure is also shown in Figure 8, where *Pr* is the peak reflected overpressure, and *Tp* is the positive phase duration. The area under the pressure–time history is the specific impulse (hereafter simply referred to as impulse). As the value of the negative pressure is much smaller than the positive pressure [21], this study will only focus on the positive phase of the impulse, *Ir*. Various methods have been used to evaluate the true values of *Pr*, *Tp* and *Ir* [22].

**Figure 8.** Blast scenario with representative pressure profile.

Close-in charges, such as roadside car bombs, last between 2 and 4 ms and have an impulse maximum of about 11 Mpa·ms to 15 Mpa·ms [19,23,24]. As long as the characteristic

response time of the specimen is greater than four times the duration of the impulse, the impulse will dominate the response of the specimen, regardless of the exact shape of the pressure–time history [25]. Civil structures, including individual elements, such as beams and slabs, meet this condition. Therefore, this paper discusses two equivalence criteria for simulating blast loading, one that considers only the impulse force without regard to the exact shape of the pressure curve, and the other that considers both the impulse and the pressure–time curves. An example of equivalent conversions for Test 4 is given in the Table 3, where the parameters of blast environment have been obtained using graphical methods in TM5-1300 [24]. The data in the TM5-1300 manual is based on real test data and empirical formulas, and has been verified and applied multiple times, and widely cited in a series of studies [26–29]. In Figure 9, the pressure–time history of Test 6 is compared to the corresponding ideal blast profile, which is calculated using ConWep [30]. The ConWep algorithm is an empirical formula for calculating explosive loads. By inputting parameters, such as the type, mass, initiation method, distance, and height of the explosive, various aspects of the explosive load can be calculated [31]. The pressure–impulse criterion is used to evaluate the blast loading, and it is found that the blast loading closely matches the pressure and impulse of the impact loading. This comparison shows that both equivalence criteria are suitable for simulating blast loading. However, when using the pressure–impulse criterion for conversion, the resulting impact is equivalent to a blast condition; when using the impulse criterion, the results are not unique and the charge must be assumed before the corresponding blast parameters can be calculated.

**Table 3.** Equivalent conversion of impact loading and blast loading.


Note: *Q*—Criteria for equivalence; *v*—Velocity; *T*—Time; *P*—Peak pressure; *I*—Impulse; *W*—Spherical charge of weight; *R*—Standoff distance; *Z*—Scaled distance; -<sup>1</sup> —Pressure–impulse criterion; -2 —Impulse criterion.

**Figure 9.** Comparison of impact loading and ideal blast loading. (**a**) Pressure–impulse criterion; (**b**) Impulse criterion.

#### *3.3. Analysis and Discussion*

The results of tests 1 to 5 are shown in Figure 10. As the thickness of the planar rubber sheet increases, the peak pressure decreases and the impulse also tends to increase gradually. However, there is a significant difference in the curve profile between the impact loading and the ideal blast loading. For example, at a cushion thickness of 50 mm, the

impact loading first rises rapidly, then falls rapidly to zero, then rises again to around 2 MPa and finally falls slowly. This phenomenon is due to the oscillations of the rubber. The phenomenon of secondary peaks may be related to the compression of the rubber. After the initial contact, the rubber is compressed and removed from the specimen surface, at which point the contact force is almost zero; as the rubber reaches the densification stage, the load increases again and the curve becomes smoother. Therefore, the shock loads generated under the above conditions cannot be converted to blast loading using the pressure–impulse criterion, thus, the impulse criterion should be used.

**Figure 10.** Pressure–time history versus thickness.

The results of tests 6 to 9 are shown in Figure 11. The shape of the pressure profile is characterized by a steep increase in pressure followed by a rapid decay for a duration of approximately 3 to 5 s, which is normal for the equivalent blast loading. From the peak trend, it is evident that pressure and impulse increase as the shock velocity increases. The impact loading is smoother due to the pyramidal shape of the rubber. Compared to the planar rubber case, the pressure tends to fall more gently after reaching its peak, although the difference is not significant initially. This makes it possible to apply both equivalence criteria to simulate blast loading when pyramidal rubber is used as the impact cushion.

Table 4 presents the impact force, loading time, pressure, and impulse for three repeated tests, along with the average and variance of the data. In addition, Figure 12 shows the pressure and impulse time history curves of the three sets of repeated experiments. The results indicate that the equipment has high loading accuracy and the data collection reliability of the test is also high, meeting the requirements of load repeatability for mechanical impact simulation explosion tests.

**Table 4.** Experimental results of three repeated tests.


Notes: *F*—Force; *T*—Time; *P*—Peak pressure; *I*—Impulse.

**Figure 11.** Pressure–time history versus impact velocity.

**Figure 12.** Comparison of impact loading of three repeated tests (**a**) Pressure–time curve; and (**b**) Impulse–time curve.

#### **4. Numerical Simulations**

In this study, we utilized the non-linear dynamic analysis software LS-DYAN to simulate the impact loading caused by RC plates when subjected to a blast simulator. Through a comparison of the numerical simulations and experimental test data, we were able to verify the accuracy of the numerical model and the reliability of the test method. Additionally, we investigated the effect of rubber shape, the impact velocity, the bottom thickness, and the upper thickness on the impact loading.

#### *4.1. Material Models*

#### 4.1.1. Concrete

The CSCM CONCRETE (MAT\_159) material model, which is available in LS-DYNA, is used to simulate the dynamic performance of reinforced concrete protection structures during vehicle collisions [32]. This material model was developed by the Federal Highway Administration and its parameters are defined based on the results of cubic compression tests. Table 5 shows the parameters of this material model that are used in the present study to model concrete. It is important to note that this material model is specifically designed

to simulate the behavior of roadside reinforced concrete protection structures and has been validated for this purpose.

#### 4.1.2. Steel

The steel of the slabs in the present study is modeled using the material model Plastic Kinematic (MAT\_003) in Ls-Dyna [33], which is an elastic-plastic model with kinematic and isotropic hardening. Reports of material property tests provided by steel producers are used in the numerical simulation of test cases. The expression for the dynamic yield strength of steel, taking into account the effect of strain rate on the intrinsic structure relationship of the material, is as follows:

$$
\sigma\_{\mathcal{Y}} = \left[1 + \left(\dot{\varepsilon}/\mathbb{C}\right)^{1/P}\right] \left(\sigma\_0 + \beta E\_P \varepsilon\_P^{eff}\right) \tag{1}
$$

where, *<sup>σ</sup><sup>y</sup>* is the dynamic yield strength of the steel, . *ε* is the strain rate, *C* and *P* are the parameters of the strain rate, *σ*<sup>0</sup> is the initial yield strength of the steel, *β* is the hardening parameter, *EP* is the hardening modulus, and *ε<sup>P</sup> eff* is the effective plastic strain. The input material parameters of steel in the current study are tabulated in Table 5. The Plastic Kinematic material model in Ls-Dyna is capable of accurately capturing the complex material behavior of steel under such extreme loading conditions.

#### 4.1.3. Rubber

Blatz–Ko rubber is a combination of Blatz and Ko [34] defined by a hyper-elastic rubber model using type II Piola–Kirchoff stresses. The Blatz–Ko strain energy density function is a powerful tool for modeling compressible types of rubber, and it can be expressed in a precise mathematical form.

$$\mathcal{W} = \frac{1}{2}\mathcal{G}\left(\frac{I\_2}{I\_3} + 2\sqrt{I\_3} - 5\right) \tag{2}$$

where, *G* is the shear modulus at infinitesimal deformation, *E* is the Young's modulus of elasticity and υ is the Poisson's ratio. *I*(*n* = 1, 2, 3) is the invariant of the Cauchy–Green deformation tensor. Equation (2) contains only one material constant, *G*. The material parameters are shown in Table 5.

**Table 5.** Input parameters for concrete, steel and rubber material models.

