The test system developed in this study consists of three parts: rockfall device, protection device, and measuring unit. As shown in Fig. 1a, the rockfall device takes a bracket structure to withstand the slideway, which is a smooth steel U-shaped channel 7 m long and 30 cm wide inside. Both ends of the slideway are placed on scaffolding brackets. An alternative device (Fig. 1b) is composed of an upper slideway, lower slideway, and a platform in-between. The length of the platform is adjustable. During tests the inner sides of the slideway and the platform were fully lubricated with a mineral oil to minimize the friction.

The protection device, being about 1.4 m high, consists of a baffle plate, a bottom plate, and four lifting outriggers (Fig. 2a). The baffle plate is made of steel and is 1.2 m long, 0.8 m wide, and 15 mm thick. The bottom plate is a 1.3 m long, 0.9 m wide, and 10 mm thick steel plate. The inclining angle of the baffle plate can be adjusted by lifting/lowering the outriggers to mold different incident angles. The measuring device includes four force transducers seated on the bottom plate and passing through the holes in corners of the baffle (Fig. 2a and b). During the test, the falling fragments were directed by the slideway to impact one of the transducers. As the tips of the force transducers were clear of the baffle by 5 mm (Fig. 2b), the transducer could grasp the full impact force without participation by the baffle. The impact forces collected by the transducer were then transmitted and stored in a data log system. In the cases where buffering effects of cushion layer were considered, the cushion materials were evenly placed on the baffle plate at a certain thickness. The test set-up simulated the impaction of rockfall on a protection measure, as shown in Fig. 3.

Three types (sphere, cube, and rectangular cuboid in shape) of samples were adopted in the experiments (Fig. 4). Each type of samples had three specimens with the weight of 4, 5, and 6 kg.

Points A, B, and C marked on the slideway were the starting points to slide (Fig. 1), representing different drop heights, i.e. 4.0, 3.5, and 3.0 m respectively. The impact incident angles were set to be 30, 60, and 90∘. The buffer platform on the slideway was adjusted to be 30, 60, and 90 cm long. The cushion materials on the baffle plate were gravel, sand, or clay. The physical and mechanical parameters of cushion materials are listed in Table 1.

In total, 109 tests were conducted for encompassing the possible permutation and combination of factors listed in Table 2. A test with the identifier of S-6-4-90 denotes a spherical sample with the weight of 6 kg falling from 4 m and impacting the baffle plate at an incident angle of 90∘. Similarly, a test with the identifier of C-5-3-90-2S-30 represents a cubic sample with the weight of 5 kg falling from 3 m, travelling a platform of 90 cm long, and impacting a sand buffer layer of 2 cm thick at an incident angle of 30∘. A speedometer was employed in tests to capture the instantaneous velocity of rockfall immediately approaching the baffle. It was mounted on the lower end of the slideway. Real-time measurements of the rockfall velocity can be stored in a flash memory card and displayed on a LED screen of the data log system.

The maximum impact force (47.7 kN) occurred in the case where a 6 kg weight spherical sample falling from 4.0 m height impacts the baffle plate at the incident angle (α) of 90∘ (sample no. S-6-4-90). Generally speaking, the impact force increases with incident angle regardless of sample shape (sh), weight (w), and drop height (h; Fig. 5). The average impact force at incident angle of 90∘ is about 15 % higher than that at 60∘, which in turn is about 14 % higher than that at 30∘. However, the impact force increases with drop height regardless of other factors. On average, the impact force of rockfall from 4 m height is about 11 % greater than that from 3.5 m height, which in turn is about 12 % greater than that from 3.0 m height. The observation here is in good agreement with what was concluded by Pichler et al. (2005) and Tetsuya et al. (2004).

As shown in Fig. 6, the impact force increases with sample weight. The average impact force for 6 kg weight samples is about 10 % greater than that for 5 kg weight samples, which in turn is about 16 % greater than that for samples of 4 kg weight. However, sample shape is found to impose strong effects on the measured impact forces. The spherical block had a higher rotation rate in its motion and less energy dissipated during its impact (Labiouse and Heidenreich, 2009). As shown in Fig. 7, spherical samples generate greater impact forces than cubes and elongated cuboids.

The embankment is often used as a protective measure against rockfalls, and it is commonly composed of a core wall and cushion layer made of buffer material. In this study, gravel, sand, and clay are chosen as cushion materials and were evenly placed on the baffle with a thickness (t) of 2 or 4 cm. Spherical samples with a weight of 6 kg are used as the rockfall. Previous work (Schellenberg and Volkwein, 2008) has shown that the reaction forces could be reduced substantially with a cushion system. As shown in Fig. 8, such a thin cushion layer shows a significant effect in reducing impact forces. In general, clay cushion layers exhibit the strongest reduction effect among these three, while the effect by gravel cushion layer is the minimum, as the measured impact force reduced by a clay cushion layer is about half of that by a gravel cushion layer. However, the thickness of the cushion layer influences the extent of impact force reduction. The measured impact force after reduction by a 4 cm thick cushion layer is about half of that after reduction by a 2 cm thick cushion layer.

In the case of right collision (α=90∘), the impact force of sample S-6-4-2C-90 (a spherical 6 kg weight sample falling from 4.0 m height onto the 2 cm thick clay cushion layer at an incident angle of 90∘) was 2.9 kN, which is only about 11 % of that from direct impact (sample S-6-4-90, without cushion layer). The impact force is reduced down to 1.6 kN by a 4 cm thick clay cushion layer (sample S-6-4-4C-90). A 2 cm thick gravel cushion layer reduces the impact force from 38 down to 5 kN, making an reduction of about 86 %. In addition, the incident angle is found to influence the reduction of impact force by cushion layer. At an incident angle of 30∘, the impact force of sample S-6-4-2C-30 is 1.4 kN, which is about half of sample S-6-4-2C-90 at the incident angle of 90∘.

For simulating the reduction effects by a platform on natural slope, a set of tests were conducted with platform lengths of 30, 60, and 90 cm (see Fig. 1b for test set-up). Samples of different shapes and weights fell from a certain drop height h of 4.0 m. The incident angle was set to be 90∘.

As shown in Fig. 9, a platform of 30 cm long may reduce the impact force by about 10 %. The 60 and 90 cm platforms can even reduce the impact force by 18 and 30 % respectively. The longer the platform, the less the impact force measured. Another observation is that the reduction effect of platform is more obvious for rockfalls of cube and elongated cuboid than cubic ones, as the gradient of the trend line for cubic samples is the least in Fig. 9.

Based on the test results, several findings can be drawn: (1) the incident angle and drop height positively affect the impact force; (2) spherical rockfalls introduce higher impact force than cubes and elongated cuboids; (3) the impact force increases with weight of rockfall; (4) the cushion layer made of gravel, sand, or clay may significantly reduce the impact force, and the thicker the cushion layer, the greater the extent of reduction; and (5) a flat platform on the slideway can lead to a reduction of impact force, and the longer the platform, the more the impact force is reduced.

According to the theorem of momentum, the rockfall impact force can come out from the following equation (Johnson, 1985; Han et al., 2004): F=mv/Δt, where m is the mass of rockfall (kg), v is the instantaneous velocity of rockfall immediately approaching the baffle (m s-1), F is the impact force (N), and Δt is the time duration of the impact process (s). As the time duration (Δt) is instant and Δt are nearly consistent for all tests, Eq. (1) indicates positive correlation between weight and instantaneous velocity of rockfall and the impact force. This is consistent with the observation in Fig. 6, where the measured impact force increases with weight of rockfall.

The instantaneous velocity (v) of rockfall immediately approaching the baffle plate was measured by means of a speedometer for all tests. As shown in Fig. 10, the measured impact velocity (v) increases with drop height (h) for a certain shape of rockfalls. The average impact velocity of 6 kg spherical samples falling from 4 m height is about 1.3 times as much as that from 3.0 m height. In addition, the impact velocity (v) is found to be influenced by the shape of rockfalls. The impact velocity of 6 kg spherical rockfalls falling form 4.0 m height was 6.36 m s-1, while that of cubes and elongated cuboids from the same height was 3.71 and 3.48 m s-1 respectively. According to the regression of test results (Fig. 10), relationships between impact velocity (v) and drop height (h) are obtained as follows for different shapes of rockfall: sphere:v=2.186h0.778(R=0.92),cube:v=0.958h1.031(R=0.93),elongated cuboid:v=0.636h1.164(R=0.96).

The above equations indicate a positive exponential correlation between drop height and impact velocity of rockfalls, which is consistent with the theoretical formula deriving falling velocity (vo) of a object from a certain height (h): vo=2gh=4.43h0.5, where g is the gravity acceleration.

Figure 11 shows the normalized velocity (λ), which is defined here as the ratio of impact velocity (v) by Eq. (2) to falling velocity (vo) by Eq. (3). It is found that with increase in drop height, the impact velocity (v) is approaching the theoretical falling velocity (vo), regardless of the shape of rockfalls. However, at small drop heights, there are big differences between them. In addition, a spherical rockfall exhibits the highest normalized velocity, indicating its approximation of the theoretical values especially at a big drop height. In the cases of cubes and elongated cuboids, the low values of λ indicate the influences of shape of rockfall on the impact velocity. Among the considered three types, the elongated cuboid contributes the most to reduction of impact velocity.

Nonetheless, the instantaneous impact velocity is significantly reduced by a buffer platform on the slideway. The impact velocity of 6 kg spherical rockfalls falling form 4.0 m height was 5.19, 4.68, and 4.11 m s-1 when the buffer platform is 30, 60, and 90 cm long respectively. It is seen in Fig. 12 that the longer the buffer platform, the lower the impact velocity. This is in good agreements with previous researches. Huang et al. (2010) and Okura et al. (2000), for instance, carried out field trials of rockfall travelling through a slideway with platform of different lengths and indicated that the platform length reduces the impact force of rockfalls.

Figure 13 demonstrates the change process of rockfall velocity during the falling process. At point O (the start point), the velocity (v) is equal to 0. It gets greater when the rockfall runs towards point A due to the gravity. At point A, the component in vertical direction (vy) becomes 0 due to the upward counterforce by the platform, leaving the component in the horizontal direction (vx) alone. vx may get smaller due to energy dissipation by friction along the platform. After point B, vy starts increasing from 0 due to the gravity acceleration, leading to increase in v and vx. The process above indicates that the platform works as a barrier to eliminate the vertical component vy of the rockfall velocity v and dissipate the kinetic energy of rockfall by friction, which leads to an overall reduction of rockfall velocity.

Regression analysis taking the measured impact velocity (v) and weight of rockfall (w) as independent variables and the measured impact force as dependent gives the following nonlinear relationship (Eq. 4): F=11.2w0.216v0.502(R=0.87).

The above exponential equation indicates that the impact force is positive to weight and impact velocity of rockfall. Thus, the impact velocity is dependent not only upon the drop height but also upon the shape of rockfall and the platform length. However, in the present engineering practice, both rockfall shape and the platform are not taken into account during designing of protection measures; instead an equivalent spherical object is normally used. This is thought to overestimate the impact force.

According to the law of energy conservation, the kinetic energy (E‾) of the rockfall, which is equal to 1/2mv2, is transferred into the strain energy (U) of the buffer layer during the impacting process. The strain energy can be calculated according to the following theoretical formula: E‾=U=∭Vu0εijdV, where u0(εij) is the density of strain energy i.e. the strain energy per unit volume.

According to the Green formula, ∂u0(εij)∂εij=σij, by integral there are ∫0u(ε)du0=∫0εσijdεij=u0εij-u0(0), where u0(εij) and u0(0) is the strain energy density after and before the deformation respectively. Taking u0(0) as 0, u0(εij)=∫0εσijdεij.

Considering σ‾ as the average stress of cushion material during deformation by impaction of rockfall, Eq. (8) can be simplified and reformed as u0=12σ‾ε=σ‾22E, where ε and E are the strain and elastic modulus of cushion material respectively.

The total strain energy (Eq. 5) of the cushion layer therefore can be expressed as Eq. (10): U=h⋅S⋅σ‾22E, where h is the thickness of the cushion layer and S is the rockfall–cushion layer contact area.

Combining Eqs. (5) and (10) gives σ‾=2EUhS=2EE‾hS=Emv2hS.

According to Eq. (11), for a certain impaction a thick cushion layer made of material with low elastic modulus would introduce a relatively low stress in the cushion layer. The above derivation explains the greater buffering effect by a layer of clay than that by gravel and explains the contribution of cushion layer thickness (Fig. 8). However, the contact area changes with the shape of rockfall. A spherical rockfall minimizes the contact area, which maximizes the stress and therefore the measured impact force.