Modelling the impact of hailstones on flat steel roofing membranes for residential buildings

Usually, placeholder hailstones of different shapes, sizes and masses are thrown or dropped at different velocities. The fall velocity of natural hail was most recently estimated by Heymsfield et al.19 The hailstones were formed with 12% polyvinyl acetate (PVA) and nitrogen. A mixture of liquid nitrogen and demineralized water was used to produce artificial hailstones that were solid, in what is considered to be a new method. The following is an explanation of the equipment and processes used to produce artificial hailstones. A tank was required to transport liquid nitrogen without harming the surroundings. This method of containing liquid nitrogen protects it from evaporation and limits dangerous conditions. Since the liquid nitrogen is transferred from the tank to the Dewar container, it can be used conveniently every time an experiment is carried out. Pure water was stored in tanks to prevent decomposition. To maintain consistent dipping, the liquid nitrogen is immersed in water tanks in the Dewar vessel. The embryo was exposed to demineralized water droplets or sand particles. The embryo was quickly immersed in liquid nitrogen in a Dewar vessel. The water in the container then suddenly froze. Artificial hailstones made with PVA have sizes of 37.5, 45 and 50.8 mm (see Fig. 1a). The tools required to produce these artificial hailstones are shown in Fig. 1b and c. The steel panels were tested six times with artificial hailstones of different sizes. In this study, 226 impact tests were conducted. The validity of the equation obtained in this study was assessed using hailstones made of PVA and liquid nitrogen.

Figure 1
figure 1

Photos of materials to make hailstones of different sizes9,20.

The projectile integrity in this study was classified into four cases: intact, partly intact, major fragmentation and shattered. In this study, the impact velocity of hailstones was measured by sensors and cameras at different sheet thicknesses. For hailstones with different diameters and densities, the final velocity can be determined by adjusting the pressure in the gas tank. Ten different plates were used in this study. As reported by the supplier, all plates have a yield strength of 300 MPa and measure 1 m × 1 m in size.

In this study, ten sheets were used with thicknesses of 0.3, 0.45, 0.6, 0.7, 0.8, and 1.0 mm. Each test was conducted at perpendicular angles. In this study, only the depths of dents induced by nitrogen ice balls and PVA that did not break were examined.

Dynamic testing equipment

The main components of the test setup are a hail cannon, a protective housing, and a measuring device. Figure 2 shows the setup. Prior to loading the artificial hailstones into the hail cannon, it is necessary to determine the mass, volume, and density of the hailstones. The hail cannon is shown in Fig. 2a.

Figure 2
figure 2

Setup using (a) a hail cannon and (b) a camera and velocity probes.

Observation and recording of the impact are possible due to the large glass window in the protection unit. There are five holes in two rows in the protection unit, which can be used to aim the hail cannon at different areas of the steel sheet. The speed sensors are mounted on rails attached to the protection unit. A high-speed camera was used to detect and measure the speed of hailstones (see Fig. 2b). To make efficient use of a high-speed camera, a 185-W lamp LED was used, powered by DC rather than AC, which reduced the flicker in the videos taken at high frames per second. To accurately measure the trajectory of the artificial hailstones, a steel frame ruler was placed along the trajectory from the barrel. Using trial and error, a frame rate of 1000 frames per second was used based on the quality of the image and the time it took to capture it. The impact velocity of an artificial hailstone can be determined using the scale as shown in Fig. 3.

Figure 3
figure 3

Images of hailstones made of PVA with a diameter of 38 mm (a) before impact and (b) after impact.

In addition, high-speed cameras can be used to observe and assess hailstones upon impact. The camera’s speed measurements were confirmed using MATLAB. A rail system was equipped with two laser sensors placed at a specific distance from one another. The projectile is detected by each sensor in turn. A time duration is displayed when the Arduino board detects the first and second sensors. The test board is replaced after ten shots, and the depth and diameter of the dent are measured. To define the diameter of the dent, the distance between the edges of each axis was evaluated with a Vernier caliper. Currently, the configuration and angle of steel sheets are not considered, as this is outside the scope of the study.

Dent depth equation

The impact energy of an intact artificial hailstone is mainly converted to the plastic deformation energy of the dented sheet, the rebounded energy, and the flexural vibration energy as other forms of energy loss such as heat and sound are negligible. According to Patil and Higgs22,the collision damage equation can be written as Eq. (1).

$${E}_{Impact}={E}_{Vibration}+{E}_{Plastic}+{E}_{Rebound}$$

(1)

As \({E}_{Rebound}\) converges to zero \(\left({E}_{Rebound}\cong 0\right)\), it is considered an insignificant variable, while the vibration energy \({E}_{vibration}\) represents the largest proportion of the impact energy. During impact, the plastic energy \(\left({E}_{Plastic}\right)\) of a material is related to its yield stress and volume change. As the thickness of the material remains constant during impact, the plastic energy can be provided by Eq. (2):

$${E}_{Plastic}={\sigma }_{y}t\Delta A$$

(2)

It is possible to calculate the area changed after impact according to Eq. (3). A simplified depressed area is signified by the radius r before impact, and the radius of the hailstone is represented by \(R\). The indentation depth is denoted by \(D\). The initial (original) depressed area is given as follows.

$$\Delta A={{A}_{f}-A}_{0}$$

(3)

$${A}_{0}=\pi {r}^{2}=\pi \left({R}^{2}-{\left(R-D\right)}^{2}\right)=\pi \left(2RD-{D}^{2}\right)$$

(4)

Equation (5) gives the distorted area after the impact. Equations (6) and (7) are used to calculate the modified area and plastic energy, respectively.

$${A}_{f}={\int }_{\theta }^{\frac{\pi }{2}}2\pi rR{d}_{\theta }=2\pi RD$$

(5)

$$\Delta A=\pi {D}^{2}$$

(6)

$${E}_{Plastic}=\pi {D}^{2}{\sigma }_{y}t$$

(7)

It is assumed that the external vibration and the external frequency \(\left(w\right)\) of the hailstone produced by the impact are zero since the impact is almost instantaneous. In this case, due to the stability of the system, the pressure gap can be expressed by Eq. (8) for a sinusoidal steady-state response.

$$x={u}_{st}\left|{H}_{jw}\right|\mathrm{cos}(\omega t+\Phi )$$

(8)

When \(\omega =0\),

$$\left|{H}_{jw}\right|=\sqrt{\frac{1}{{\left(1-\left(\frac{\omega }{{\omega }_{n}}\right)\right)}^{2}+{\left(2\delta \left(\frac{\omega }{{\omega }_{n}}\right)\right)}^{2}}}=1$$

$$\mathrm{tan}\left(\Phi \right)=\frac{2\delta \left(\frac{\omega }{{\omega }_{n}}\right)}{1-{\left(\frac{\omega }{{\omega }_{n}}\right)}^{2}}=0 \to {\Phi }^{o}={0}^{o}$$

$$\left|{H}_{jw}\right|\mathrm{cos}(\omega t+\Phi )\cong 1$$

(9)

where \(\left|{H}_{jw}\right|\) in Eq. (9) represents the dynamic amplification factor. This means that the vibration energy equals:

$${E}_{Vibration}=\frac{k}{2}.\frac{{F}^{2}}{{k}^{2}}=\frac{{F}^{2}}{2k}$$

(10)

Flat steel sheets have a flexural stiffness \(\left(k\right)\) proportional to their thickness \(\left(t\right)\) and the length \(\left(h\right)\) cubed but inversely proportional to the spacing between their battens cubed.

$$k=\frac{{Eth}^{3}}{{l}^{3}}$$

(11)

The symbol \(\varphi\) symbolizes the diameter of a hailstone (2R). To explain the vibration energy, Eq. (12) can be substituted into Eq. (10).

$$F=\sqrt{\frac{{{\sigma }_{y}}^{2}{\pi }^{2}{\varphi }^{4}}{16}}$$

(12)

$${E}_{Vibration}=\frac{{{\sigma }_{y}}^{2}{\pi }^{2}{\varphi }^{4}{l}^{3}}{{32Eth}^{3}}$$

(13)

The final step is to rewrite Eq. (1) to obtain dent depth:

$$D=\sqrt{\frac{m{V}^{2}}{2t\pi {\sigma }_{y}}-\frac{{\sigma }_{y}\pi {\varphi }^{4}{l}^{3}}{32E{t}^{2}{h}^{3}}}$$

(14)

To optimize the dent depth in Eq. (14), the rebounded energy and the compressive area can be considered by using the following equation:

$$D=\sqrt{\alpha \times \frac{m{V}^{2}}{2t\pi {\sigma }_{y}}-\beta \times \frac{{\sigma }_{y}\pi {\varphi }^{4}{l}^{3}}{32E{t}^{2}{h}^{3}}}$$

(15)

where \(\alpha\) and β denote the rebound energy and the pressure area, respectively. The coefficient of restitution (COR) is usually referred to as the cause of energy dissipation during an impact. The COR is calculated as follows:

$$COR=\frac{Rebound\,\, Velocity}{Impact \,\,Velocity}$$

(16)

\(\alpha\) in Eq. (15) is obtained as follows when COR = 0 for a broken or smashed hailstone during impact:

$$\alpha ={(1-COR)}^{2}$$

(17)

Although in the present experimental study the dent diameter is visible after the collision on steel plates, it is challenging to gauge the exact value of the dent diameter (\({D}_{d}\)) manually without a digital 3D scanner. The compressive area determines the coefficient \(\frac{2{R}_{c}}{\varphi }\). As shown in Table 1, the β coefficient is derived from the distribution of the steel plate thickness and the hailstone diameter. A steel plate’s compressive radius is defined as \({R}_{c}\), while its elliptical dent radius is r.

Table 1 β coefficients based on hail diameter and board thickness.

Knud Thomsen’s formula is used to approximate the surface of an ellipsoid based on the lengths of the semiaxes in Eq. (7) to correct the plastic energy. With Knud Thomsen’s formula, the flat area (\({A}_{0}=\pi (2RD-{D}^{2})\)) remains the same before impact, and the deformed area (\({A}_{f})\) resembles that of a flattened spheroid described as follows:

$${A}_{f}=4\pi {\left(\frac{(a{b)}^{p}+(a{c)}^{p}+(b{c)}^{p}}{3}\right)}^\frac{1}{p}$$

(18)

where \(p=1.6075\), \(a=b=r\) and \(c=D\). Assuming a half-ellipsoid, the elliptical dented radius (r) is computed as \(r=\sqrt{{R}^{2}-{(R-D)}^{2}}\), where R and D are the radius of the artificial hailstone and the dent depth, respectively. Thus, Eq. (18) can be rewritten as follows:

$${A}_{f}=2\pi {\left(\frac{{r}^{3.215}+2{(rD)}^{1.6075}}{3}\right)}^{1/1.6075}$$

(19)

The revised area after the collision is

$${\Delta A}_{r}={A}_{f}-{A}_{0}$$

(20)

$$\Delta {A}_{r}=2\pi {\left(\frac{{r}^{3.215}+2{(rD)}^{1.6075}}{3}\right)}^{1/1.6075}-\pi {r}^{2}$$

(21)

Hence, ϰ \(= \frac{{\Delta A}_{r}}{{\Delta A}_{i}}\) and \({\Delta A}_{i}=\pi {D}^{2}\). The revised plastic energy equation is as follows:

$${E}_{P}=\varkappa \pi {D}^{2}{\sigma }_{Y}t$$

(22)

Dent depths of the specimens in Table 3 were predicted using the equation proposed by Uz et al.23 as follows:

$$D=\sqrt{\frac{\alpha }{\varkappa }\times \frac{m{V}^{2}}{2t\pi {\sigma }_{y}}-\frac{\beta }{\varkappa }\times \frac{{\sigma }_{y}\pi {\varphi }^{4}{l}^{3}}{32E{t}^{2}{h}^{3}}}$$

(23)

The yield stress \({(\sigma }_{Y})\), Young’s modulus (E), average effective length (l), and transversal length (h) are kept constant under laboratory conditions (\({\sigma }_{Y}=320\mathrm{ MPa}\), E = 200 GPa, l = 148.7 mm, h = 1 m). The coefficient ϰ is dependent on the ratio between the radius of the artificial hailstone and the elliptical radius of the damaged area. Accordingly, as the ratio of the radiuses increases, so does the coefficient determined by the ratio of \({\mathrm{\Delta A}}_{\mathrm{r}}\) and \({\mathrm{\Delta A}}_{\mathrm{i}}\). The ϰ coefficient is taken from the study of Uz et al.20 Experimental tests show that the relationship between \({R}_{c}\) and r is \(r=0.58{R}_{c}\), with a tolerance of 0.2 mm. The following formula represents the relationship between the ϰ coefficient and the ratio between the radius of the artificial hailstone and the radius of the obtained indented ellipse:

$$\varkappa =1.3923{\left(\frac{R}{r}\right)}^{0.6304}$$

(24)

For a particular impact energy, Johnson and Schaffnit24 suggested that the dent depth is inversely related to the square of the panel thickness. In Eq. (23), the dent depth can be calculated by combining the plastic energy and the rebounding energy as well as the vibration energy and the compressive area.

Hail impact model validation

The experimental test results obtained by Carney et al.25, which investigated the behavior of ice during impact, are used as a new validation of the FE models in this study. Cylindrical ice projectiles with diameters of 17.5 mm and lengths of 42.2 mm were used for impact tests applied on a rigid plate. To measure the force over time for each test, a force transducer device was placed behind the rigid plate. To obtain consistent data from the experimental test, the plate is set up with a similar geometry and rigidity using a 3D analytic rigid shell. With the help of three discrete spring elements, as shown in Fig. 4, the target plate of each test condition is fully modeled based on the modal characteristics. The spring elements were described by following the line of action axis in the FE models. In the boundary condition of the FE models presented in this study, the target plate is unrestrained to move only through the spring elements during impact.

Figure 4
figure 4

Schematic representation of the steel cylinder (load cell) and supporting springs.

The impact forces recorded over time from the force transducer device are matched with those extracted from FE models. As shown in Fig. 5, compared to the impact test from Carney et al.25 with 62.5 mm diameter hail at 152.4 m/s, the FE model captures the system well based on the trend and maximum impact force. The test setup for capturing the impulse in the test of Carney et al.25 is compared with the FE model given in Fig. 6. The material property values for hailstones are 9.38 GPa for Young’s modulus, 0.33 for Poisson’s ratio, 5.2 MPa for yield stress and 0.517 MPa for hydrostatic cutoff stress. The compressive strength of artificial hailstones does not noticeably affect the prediction of the maximum dent depth of steel plates by assuming that the energy conservation (vibration and plastic energy) of hailstones is negligible21.

Figure 5
figure 5

Time histories of impact force obtained by this study in comparison with the one performed by Carney et al.25.

Figure 6
figure 6

Comparison of the impact captured by experimental results obtained by Carney et al.25 and FE models.

The tabular representation of the strain sensitivity of the hailstone used in the current study corresponds to the data in the study by Uz et al.21 The strain rate determines the failure of compression.

https://www.nature.com/articles/s41598-022-24375-3