Consideration of the Maximum Impact Force Design for the Rock-Shed Slab

This study aims at the estimation of the maximum impact force for a rock-shed slab under collision by a rock fall. A DEM program calibrated from small-scale physical experiments is used to model the movement of rock fall clusters and to measure the resultant impact forces. The results obtained from the small-scale experiments show that the maximum impact forces are significantly affected by the mass of the rock fall, the height from which they fall as well as the falling process. Full-scale numerical modeling based upon a field case not only confirms the experimental results but also sheds light upon the influence of the rock fall mode and the contact stiffness between the rock and the rock-shed. This study provides guidelines for the design of rock-shed structures countering large rock fall problems. reveal the behavior of reinforced concrete structures with cushioning layers under vertical rock falls impact [1-3]. For example, Chuman et al. Ong et al. and Ebeltoft et al. [4,5] investigated single block falls in physical model experiments. In addition, single rock falls impacts on the top slab of a rock-shed have been simulated using the finite element method (Philippe et al., 2003; Delhomme et al., 2005) and the discrete element method. However, general rock falls movements can include the collision and spreading behavior of many clusters of falling rock blocks. Okura [6] provided an extensive discussion of the relation between the volume of the falling blocks, the collision interaction, and the run out distance. The fall of a single rock mass or several in sequences as well as the level of disintegration all have an influence on the impact forces on the rock-shed. Therefore, this study discusses the maximum impact force on a rock-shed induced by various types of falling in a cluster of rock falls. The numerical discrete element method (DEM) program-particle flow code (PFC) 3D [7] calibrated with a simplified analytical solution and small-scale physical modeling tests is used to investigate the maximum impact force. The developed numerical method will then be further applied to a field full-scale case study, i.e., the Malin rock-shed. This paper investigates three types of rock falls (Figure 2): (a) the single fall (SF) where a single block of rock falls from a cliff before rolling and bouncing down a slope; (b) the cluster fall (CF) where a cluster of rock blocks falls simultaneously, which is the most common type of rock falls, including free falling, rolling, bouncing, and collision between rock fragments; and (c) the sub-cluster fall in sequence (SCSF) where multiple clusters of rock blocks (separated by joint sets) fall in sequence. The cracking process during collision is not taken into account. The remainder of this paper comprises three parts. First, the Journal of Geography & Natural Disasters J o u r n a l o f G eo grap hy & Nural Disa s t e r s ISSN: 2167-0587 Citation: Lo CM, Lee CF, Lin ML (2016) Consideration of the Maximum Impact Force Design for the Rock-Shed Slab. J Geogr Nat Disast 6: 169. doi:10.4172/2167-0587.1000169


Introduction
Rock fall disasters occur frequently in the mountainous areas of Taiwan triggered by earthquake and storm events. Falling rocks have been known to strike vehicles, demolish roads, and cause accidents. Rock-sheds, reinforced concrete structures covered with cushioning material to absorb the direct impact force, are often constructed in order to mitigate the damage caused by rock falls and to stabilize slopes prone to rock falls and avalanches. However, large quantities of rock can produce great impact which may ultimately exceed the loadbearing capacity of the rock-shed and even punch through the top slab or destroy the beam-column system.
For example, some rock-sheds along Taiwan's Central Cross-Island Highway were severely damaged by rock fall events during the Chi-Chi Earthquake. The Malin No. 3 rock-shed, located to the west of the Malin No. 3 tunnel on the Central Cross-island Highway at Taichung City, Taiwan (from 40k+866 to 40k+925), is one such case. Figure 1a shows the rock mass with the potential to form rock falls (because of several joint sets) above the Malin NO. 3 rock-shed. The rock-shed has the approximate dimensions of 70 m long, 8 m wide, and 8 m high with tires covering the roof to absorb the impact force from falling rocks. The cluster of rock falls during the Chi-Chi earthquake caused severe damage to the columns of the rock-shed and blocked traffic on the road (Figure 1b). A comparison of the topographic maps and aerial photographs shows that the height from the crest of the slope to the river valley is about 320 m. The rock fall source area is about 120 m upslope from the rock-shed. There is a talus deposit about 150 m long, 90 m wide, and 70 m high that originated from a cliff about 120 m above the rock-shed. The rock on the cliff first slid along a gentle sliding surface and then began to fall because of topographic variation, causing severe damage when striking the rock-shed. Different types of rock falls would produce different maximum impact forces and thus different levels of damage to the rock-sheds. In addition, the stiffness of the top slab and the cushioning material would also affect the maximum impact force. These issues are discussed in this study.
In recent years, there has been a dramatic increase in research regarding the process of rock falls impacts on the upper slab of a rock-shed. Numerous field experiments have been conducted to

Methodology
Due to the scaling effect problem, a small-scale model may differ from the actual rock falls behavior. Therefore, the qualitative analyses of physical modeling tests were conducted to establish the relation models in study area. Undeniably, a full-scale model would be the best method for analysis of rock falls impact force but are far too costly and difficult to execute. In recent years, we witnessed an increase in innovative numerical approaches to the study of full-scale landslide simulation. However, to ensure that a numerical model could reveal reasonable predictions, the physical modeling tests and numerical analysis adopted must cross math with each other. The results of physical modeling were compared with those produced by numerical analysis so that the correctness of the numerical simulation could be justified. Subsequently, calibrated numerical methods adopted in the small-scale model were used to simulate the full-scale model. The simulation results should be as close to reality as much as possible.
A flowchart of this research process is shown in Figure 3. First, an analytical solution is derived showing an estimation of the maximum impact force of a single elastic ball of rock on a flat elastic ground. Meanwhile, small-scale models based on the physical modeling experiments are established using a numerical method to simulate the maximum impact force. The results obtained from the analytical solution, physical experiments and numerical methods are compared. A small-scale numerical model is calibrated based on the physical modeling experiments (Step 1 and step 2 in Figure 3) which is subsequently applied to a full-scale numerical model for estimating the maximum impact force on the Malin rock-shed of rock falls event induced during the Chi-Chi Earthquake (Step 3 in Figure 3). Factors related to the maximum impact force, including the rock falls process, amount of falling rock, and the contact stiffness between the falling rock and the rock-shed are also investigated (Step 4 in Figure 3).

Derivation of the maximum impact force
When a falling rock directly collides with the top of a rockshed, the maximum impact force generated at that moment is much larger the weight of the rock itself and thus causes great damage. The maximum impact force under specified conditions is derived based on the following basic assumptions: (a) The problem is simplified as a rock ball vertically colliding with a spring system, which represents the top slab of the rock-shed.      Figure 3: Flowchart of the research framework.
The mass of the rock ball is M, while the mass of the spring system is neglected. The stiffness of the rock ball and the rock shed are k 1 and k 2, which are assumed to be constants and independent on deformation.
(b) The collision between the rock and the rock shed causes only deformation and no fracturing or breakage. The gravitational potential energy is equal to the elastic strain energy.
(c) The maximum impact force appears to be accompanied by the maximum deformation.
According to Newton's second law of motion, the impact force of a falling object on a surface may be determined as the change of momentum in a time interval. However, it is difficult to measure the velocity of an object before and after a collision in a certain time interval. An alternative is to consider the transformation of energy from gravitational potential energy to strain energy ( Figure 4). When an object with mass M released from a height H hits the spring, the deformations of the falling object and the spring are u 1 and u 2 , respectively. The stiffness of the falling object and the spring are assumed to be constants k 1 and k 2 . The potential energy released at the maximum deformation of the system is where F is the maximum impact force; 1 2 is the contact stiffness of the system. The elastic strain energy at the maximum deformation of the system is By equating (1) with (2), we obtain If M, H, and k 1 are considered as constant, the contact stiffness and the maximum impact force are reduced as the stiffness of the rock-shed (k 2 ) decreases. As the stiffness of the rock-shed is much less than the stiffness of the rock, the contact stiffness of the system approximates the stiffness of the rock-shed.

Simplified physical modeling tests
In this work, the analytical solution is compared with the test results of a single falling rock block. Then, maximum impact forces for different types of rock falls are compared. These test results are also utilized for calibration of the numerical modeling. The equipment used consists of a release container, an impact tester, and a measurement apparatus (CCD). The release container is a vertical tube 5.5 cm in diameter and 39 cm high. The impact tester is located below the release container and is composed of a platform fixed on a sensitive spring scale ( Figure 5).
Tests are carried out with various rock masses and different heights to represent different types of rock falls. As the falling rock hits the platform of the impact tester, the force of the impact is measured and appears on the spring scale. Two CCD cameras capable of taking photographs at a rate of up to 30 frames per second are used to capture images of the impact. Spherical rocks and granular pieces of gravel are used in the three types of rock falls tests. The contact stiffness of the system is dominated by the stiffness of the scale because of the much greater stiffness of the rock than that of the scale.

Numerical modeling
Numerical modeling is carried out using a three-dimensional discrete element program, PFC3D (Particle Flow Code) [8]. The basic principles and related parameters used in PFC3D such as material density, stiffness (k n and k s ), friction coefficient (μ) and damping ratio (D n and D s ) are described below.
Basic principles of PFC3D: PFC was implemented through the development of the DEM proposed by Cundall and Strack [9]. This technique is commonly used for the modelling of granular assemblages with purely frictional or bonded circular particles represented by discs in two dimensions. The original version of the DEM was devoted to the modelling of rock-block systems; it was later applied to the modelling   Figure 4: Derivation of maximum impact force, mass, elastic contact, and falling height. (F max is the maximum impact force, M is the mass, H is the drop height, k 1 is the stiffness of the rockfall, and k 2 is the stiffness of the spring system.).  of granular material. With this method, the global behaviour of an assembly of particles connected by a network of contacts is obtained by satisfying the equations of motion of each component for the whole particle set.
In the explicit finite difference method, the system variations of the operation process are controlled through the force-displacement relations. Beginning with the calculation of the particle position, the adjacent overlap and the relative displacement, PFC determines the contact force on the basis of the force-displacement relations. The particle velocity and up-to-date position are derived from Newton's 2 nd low of motion. In general, the fundamental assumptions for the PFC model are as follows: (a) each element is considered a rigid body and a perfect sphere. By means of the combination command "Clump", particles can be formed in various shapes composed of different amount of elements; (b) PFC allows adjacent particles to overlap. The extent of overlap is related to the contact force and stiffness of particles calculated based on the force-displacement relations; (c) the strength of the bond between two particles is exhibited through contact or parallel bonds.
In PFC3D, the parameter of viscous damping, in contrast to the numerical damping factor influences the motion of the particles and reflects energy loss during collisions. Equation (4) expresses the relation between the restitution coefficient (r) and the critical damping ratio (D) observed in drop tests, and are used to estimate the critical damping ratio (D) in rock falls collisions in the laboratory or in the field. (ln( )) ( ) (ln( )) Model parameters: The macroscopic behaviour of a granular media depends on the mechanical properties of the contact between the particles, but there is no straightforward solution to the selection of these parameters. PFC models are reliable simulation tools, but it is necessary to establish reasonable relationships between the numerical parameters and the mechanical characteristics of real problems [8]. The parameters used for modeling in the physical experiments and for the case study for the Malin rock-shed are explained in the following sections.
The parameters of the physical modelling: In the numerical simulations, the area of impact and release container is modeled as planar wall elements and the spherical rocks as ball elements. The container has a radius of 0.055 m and a height of 0.39 m. The rocks are released from the height of 0.2 m. The impact area is 0.35 m by 0.35 m ( Table 1). The unit weight of the rocks is 2600 kg/m 3 . Comparing the results from numerical and laboratory uniaxial tests ( Figure 6), we obtain the stiffness of the rock balls which is 2 × 10 8 to 3 × 10 8 N/m 2 . Through conversion of the particle stiffness and macroscopic elastic modulus relations, the elastic modulus (E as 5.34 Mpa) obtained from uniaxial compression tests will lead to the stiffness of the rock (ball) as 2 × 10 8 to 3 × 10 8 N/m 2 ( Table 2) and the stiffness of the impact area (wall) including both the impact tester and the scale is 8 × 10 7 N/m 2 . The friction coefficients (μ) between the particles and particle-wall contact are determined to be 0.6 and 0.84, respectively, which are equivalent to friction angles of 31° and 40°. The normal and tangential damping ratios are 0.3 and 0.05 as calibrated using the coefficients of restitution obtained from the experimental CCD recording data. In addition, the fall heights of SF and CF as 39.5 cm, while the fall heights of SCSF as 24.9 cm (first time), 34.6 cm (second time), 44.4 cm (third time), 54.5 cm (fourth time) respectively. The volumes of SF and CF as 337 cm 3 , while the volumes of SCSF as 84.25 cm 3 for each time ( Table 2).
The parameters of the case study: In the case study, the upper slab of the rock-shed and the topography are modeled by planar wall elements. The impact forces from the collision of the rock falls with the wall element are measured in the X, Y, and Z directions. The rock mass is modeled as an assembly of bonded ball elements filling up the volume of the source area. The same parameters are used as in the rock fall tests except that the friction coefficients between the particles and the particle-wall contact are determined to be 0.78 assuming a general repose angle for the field talus ranging from 31° to 40°. The damping coefficients for the case study, which are listed in Table 3 are estimated using Equation (4) and the results of restitution coefficient tests in the field taken from Giani [10]. In addition, the stiffness and thickness are considered when studying the efficiency of the dissipation cushion in this case. The dissipation cushion is modeled as a cluster of ball elements 0.5 m in radius with varied dimensions of 7 × 41 × 2, 7 × 41 × 6, 7 × 41 × 10, 7 × 41 × 14, and 7 × 41 × 9. The stiffness of the dissipation cushion is determined to be 1.0E+4 kN/m 2 . In addition, the fall heights of SF and CF as 120 m, while the fall heights of SCSF as 104 m (first time), 112 m (second time), 120 m (third time), 128 m (fourth time), 136 m (fifth time) respectively. The volumes of SF and CF as 28,000 m 3 , while the volumes of SCSF as 5,600 m 3 for each time ( Table 2).

Maximum impact of a single rock falls
The factors influencing the maximum force of impact include the

Case Study of the Malin No. 3 Rock-shed
A numerical method is used to study the case of the Malin rock mass, the drop height, and the contact stiffness between colliding objects. Figure 7 shows the normalized relations between the maximum impact force and the effect of the drop height and contact stiffness. A comparison of the results from laboratory tests and the case study for the Central Cross-island Highway in Taiwan is shown in Figure 7.

The maximum impact forces from a single and a cluster of rock falls
The maximum impact forces for different weights of rock falls but the same drop height of 0.2 m are shown in Figure 8 and Table  4, including the results from the analytical solution, experiments, and numerical modeling. In the single rock falls tests, the spherical rocks have weights of 0.25 N, 0.65 N, and 1.5 N, the same gross weight as a cluster of small size gravel. The impact forces generated by rock falls clusters (3.2 N, 5.3 N, 8.3 N) are clearly less than those of a single rock falls (2.9 N, 4.8 N, 7.3 N). This trend is more significant as the weight of the rock falls becomes greater.

Numerical modeling
As shown in Figure 8, the maximum impact forces of a single rock falls or clusters of rock falls are estimated in the numerical modeling. The process includes placing rock elements in the container, releasing them, free fall, colliding the slab, and successive deposition of talus ( Figure 9). The variations of the impact forces are measured and recorded.     No. 3 rock-shed located on the Central Cross-island Highway between 40k+866 and 40k+925 (Figure 1c). During the 1999 Chi-Chi Earthquake, this rock-shed was destroyed by rock falls; the surrounding rock mass with several sets of joints is shown in Figure 1 and Figure 11b. Interpretation of aerial photographs from 1998 and 1999, topographic analysis and field reconnaissance shows the height of rock falls to be around 120 m (Figure 11a).

Comparison of experimental results with varied types of rock falls
Measurements of several cases of rock falls on the Central Crossisland Highway show the rock masses to have fallen from heights 50 m to 300 m with volumes from 1000 m 3 to 50000 m 3 . Before it was destroyed, there was a layer of tires on top of the Malin No. 3 rock-shed 0.5 m thick. Numerical modeling is used to investigate the effects of the rock joint spacing, type of rock falls, material stiffness of the slab, and thickness of the cushion on the maximum impact force generated by rock falls clusters. The results are helpful to evaluation of the maximum impact force and the design of rock-shed structures.

Numerical modeling
The results of aerial photograph interpretation and topographic analysis show the area of the rock falls to be 56 m in length, 30 m in width and 10 m in height after falling about 120 m. In the model the top slab of the rock-shed is about 70 m long and 8 m wide (Table 1 and Figure 12). Joints in the rock around the Maline No. 3 rock-shed are analyzed based on the interpretation of aerial photos from 1998 to 2006. Two joint sets orientated N80°W/75° NE (Joint A obtained from 13 data sets) and N10°E/85° NW (Joint B obtained from 19 data sets) are modeled in three dimensions (Figure 11c). The modeling process includes three steps. First, the initial state of the rock mass in the source area is modeled. Second, a cluster of rocks separate from the source area and slide under the influence of gravity without considering the effects of earthquake or rainfall. Third, the rocks slide over a convex point where the inclination of the slope changes from 30° to 65° and then strike the roof slab of the rock-shed. The factors discussed herein for calculating the maximum impact forces include the type of rock falls, joint spacing (1 m, 4 m, 7 m, and 10 m), slab stiffness (1 × 10 5 , 1 × 10 6 , 1 × 10 7 , and 1 × 10 9 KN/m), and the thickness of the dissipation cushion (1 m, 3 m, 5 m, 7 m, and 9 m).

Results of the case study
The force of the maximum impact on the rock-shed is influenced by the factors discussed below.

Maximum impact force (N) Error (%) Remarks
Single fall

(1) Types of rock falls
The type of rock falls is critically important to laying the groundwork for understanding how civil engineers can decrease the maximum force of impact and thus the danger of damage. Three types of rock falls are illustrated in Figure 13a-13c. A single huge rock falling directly onto a slab will cause more serious damage than a cluster of falling rocks. The maximum impact of a single rock falling without sliding, as calculated from the analytical solution, is about 1474 times the magnitude of the weight of the falling rock. On the other hand, the maximum impact force of a single rock falling from a convex slope after sliding is only about 974 times the magnitude of the weight of the falling rock. The maximum impact force generated by a cluster of falling rock separated with a joint spacing of 4 m is 191 times the magnitude of weight of the rock, whereas a cluster of debris falling at the same time (with dimensions of about 0.1 m-0.5 m) generates a smaller maximum impact force ( Figure 14 and Table 5).
An examination of Figure 15 shows that the maximum impact force generated by a cluster of falling rock is larger than that generated by sub-clusters falling in sequence. The greater the separation in the sequence, the less the maximum impact force generated, although there is only a very slight change in the maximum impact force when the sub-cluster in the rock falls sequence is separated along a 4 m joint spacing at four times and the sub-cluster in sequence debris fall at twice respectively. The results indicate that the maximum impact force is affected by the viscous and plastic behavior of a cluster of rock and by the rock falls process. Thus if there is danger of a huge rock that      threatens a lower rock-shed, it may be appropriate to first explore the surrounding situation. It could be preferable to blast the rocky mass to break it up and let it fall in sequence rather than all at once.
(2) Joint spacing of the rock mass Rock falls source areas are often rocky masses with fractures, or slopes with undercuts where the matrix has been destroyed. Discontinuities in the rock mass are relevant to rock fall problems [11][12][13][14][15][16]. An examination of Figure 16 shows that the maximum impact forces of a cluster of rock falls are influenced by the joint spacing. The force of the maximum impact increases as the joint spacing increases past 4 m. The results also indicate that when a rock mass has the same volume but is more fragmented, the influence of the collision interaction will be more significant, although the influence of the mass and the drop height still need to be considered.

(3) Stiffness of the top slab of a rock-shed
One common design guideline is to change the stiffness of the slab by varying the material stiffness. The maximum impact force decreases as the stiffness of the slab decreases. In this case study, when the stiffness is lowered by 1 × 10 -3 times, the maximum impact fore will be reduced to 1.7 times (Figure 17). In particular, when the stiffness is lowered by 1 × 10 -4 times, the maximum impact fore will be significantly reduced to 13.3 times. Apparent, the stiffness reducing is importance for the rock-shed slab.

(4) Thickness of the dissipation cushion
Another design guideline is to use a dissipation cushion on the rock-shed. The results show only a slight reduction in the maximum impact force if the thickness of the dissipation cushions is increase Step3=400000 Figure 13c: Impact processes in case study simulated by SCSF models.     Figure 18). Although increasing the dissipation cushion are not significantly efficiency with respect to the entire structure stiffness reduced for reducing maximum force. However, application of used dissipation cushion layer in rock shed can cushion rock falls impact and effectively reduce the maximum impact force. Dissipation cushion layer thickened can slightly improve reduce stiffness and increase energy absorption capacity for the whole structure of the rock-shed [17][18][19][20].

Discussion and Conclusions
This paper focuses on the estimation of the maximum impact force on a rock-shed by clusters of rock falls. The findings are summarized as follows: (1) A comparison of the results obtained from the analytical solution, physical modeling and numerical modeling shows that the major factors influencing the maximum impact force of a collision are the mass and drop height of the rock falls as well as the contact stiffness of the roof slab material. Furthermore, the volume and type of rock falls, the surrounding terrain and the stiffness of the material of the top slab are the main factors of concern when designing a rock-shed. This study offers guidelines for the preliminary design of rock-sheds for protection from rock falls.
(2) The new structurally dissipating rock-sheds (SDR) (Delhomme, 2005 and2007) are recommended. This type of rock-shed includes extra flexible steel pillars between the top slab and the columns, which decreases the stiffness of contact during rock falls collisions and thus effectively reduces the maximum impact force and damage. Our simulation results (the stiffness decreases part) support this engineering design, when the structure stiffness decreases, the maximum impact force by rock falls also are decreased ( Figure 17). Therefore, reducing the stiffness of the whole structure of the rock-shed is the key method for avoid impact failure risk by rock falls.
(3) Rock shed is commonly used in traffic disaster protection. In order to improve rock shed performance, impact cushion and energy absorption measures are usually adopted. It is important to build impact-resistant, durable and efficient auxiliary energy dissipation cushions. Figure 18 shows that the dissipation cushion is increased by 8 m, the maximum impact fore will be reduced to 12%. In practical engineering application, dissipation cushion layer can be set up with multilayer and filled with polyurethane foam, polystyrene, low-density industrial waste, or other lightweight materials with a cell structure in order to improve the rigidity and energy absorption capacity of cushion layer (Sun et al., 2016). In addition, careful consideration of the thickness of the cushions is indispensable, and the repose angle of field talus deposit should also be taken into account for appropriate design of the slope of the top slab.   Figure 15: Relationship of maximum impact force/weight and sub-cluster in sequence of fall type.   (4) Rocks falling from heights of 50 m to 100 m with volumes of 1000 m 3 to 50000 m 3 are common on the Central Cross-island Highway. The maximum impact force of potential rock falls can be estimated and compared to the capacity of the rockshed. It may sometimes be appropriate to blast the rock mass to ensure it falls in sequence. This measure would diminish the rock size and the contact stiffness and thus effectively reduce the maximum impact force of clusters of rock falls.
(5) This study provides valuable insight into the quantification of the maximum impact force of clusters of rock falls. Nevertheless, there are a few limitations. The first concerns the damping effects, which are not considered in this system. The second concerns the behavior of a breach in the rock, which is not considered in the interaction of rock falls collision. The influences of the shape of the falling rock and the contact modes have not yet been considered.