By Pietro Rimoldi and Nicola Brusa
Rapid and extreme climate change is putting our planet under stress. As a consequence, in mountain and hilly regions, infrastructure and people are more often threatened by quick and unpredictable rockfall events. Falling boulders can have extremely high speeds, exceeding 11 mph (5 m/s) and up to 67 mph (30 m/s); these events involve a complex pattern of movement (e.g., detachment, fall, rolling, sliding, bouncing) of one or more rock fragments.
Rockfall protection embankments (RPEs) have proven to be a safe measure for protecting people, structures and infrastructure from rockfall events (Figure 1). These mitigation measures can be done as unreinforced or reinforced earth embankments and designed to absorb medium- to high-energy impacts (1,000–30,000 kJ). Depending on their construction, these structures may also withstand multiple impacts.
RPEs can be constructed in various shapes and sizes to suit the site, and with a range of internal reinforcing elements and facing materials (soil, riprap, cushion facing). They work through a combination of deformation and internal compaction to absorb the energy imparted by falling rock blocks. They have been used since the 1970s, and many research programs have been executed with a focus on improving their design and ability to resist the impact.
RPEs play an important protective role, but their design is based on simplistic approaches, and, even if specific and rather complex research has been performed, the vast majority of existing structures has been designed with basic approaches, considering dynamics only to a minor extent (Figure 2). In addition, there are no existing guidelines on the performance of reinforced RPEs. Reinforced RPEs could provide further advantages, such as limiting the footprint of the embankment and the ability to resist higher impacts.
Academics and engineers have attempted modeling the behavior of RPEs in order to provide design guidance; however, due to the variability of geometry, fill material, embankment construction, various reinforcement options, interactions between soil and reinforcement, and the dynamic behavior of the soil, a consistent design procedure has not been developed.
The most comprehensive guidelines available worldwide for RPEs are the Italian ones (UNI 11211-4:2018) and the Austrian ones (ONR 24810:2020). In these standards, RPEs can be reinforced either with steel or geosynthetics, and it is clear in both standards that reinforcement layers significantly improve the ability of an embankment to withstand impacts.
However, neither standard specifies nor considers the reinforcement performance. As a consequence, without clear instructions, engineers seek support by reinforcement manufacturers or suppliers capable of providing data or numerical models coming from real-scale test results.
For example, the Italian standard UNI 11211-4:2018 provides recommendations for the input data required for the design of RPEs. But it is not specified how to use the input data for verifying the structure response to a given impact. It is nevertheless indicated that the block penetration should be compared with the RPE width. The design corresponds to the case of a given single block volume in a release area. The design of the RPE relies on the impact height and the kinetic energy of the design block as obtained from statistical elaboration of trajectory simulations (Figure 3).
Numerical models, either done with finite element method (FEM) or discrete element method (DEM), can be useful to evaluate the effects of block impacts on RPEs. FEM and DEM often represent the reinforced soil embankment with the inclusion of geosynthetics as a soil mass with an increased rigidity. But the geosynthetic choice and performance are still not addressed.
In addition, FEM and DEM approaches are less accessible to design engineers in general, as they require specific skills and backgrounds. Even if, at first glance, models developed based on commercially available software seem easy to use, satisfactorily modeling the impact response of RPEs requires highly specific expertise. These concerns are not only related to the constitutive laws and mechanical characteristics of the model itself, but also require a comprehensive knowledge of the different numerical methods and their limits. For this reason, numerical tools are out of reach for most of the companies concerned with the design of RPEs. Also, the computation time is very long for some of the existing numerical models (Lambert and Kister 2017).
Analytical methods are more commonly used, as they are much more accessible and rapid to use. These methods were developed over the years based on block penetration or impact force to provide design engineers with easy-to-use tools. Nevertheless, their applicability can be limited due to the uncertainty associated with their assumptions and calculations, which are not necessarily related to geotechnics or rock mechanics. These methods can then lead the engineer to use formulas derived from research performed in specific contexts that differ from the impact of a block on a RPE. For example, the method proposed by Kar (1978) is based on the penetration depth of bombs and missiles on soil shelter/protection structures.
As mentioned in the research of Lambert and Kister (2017), Kar’s model was developed for cohesive nonfrictional soils impacted by ogive-nose projectiles at a minimum velocity of 671 mph (300 m/s). Mayne and Jones’s model (1983) was developed for heavy soil tamping, with a hammer having a flat tip. Labiouse’s model (Labiouse et al. 1996) was developed for granular strata, a 3.3-foot (1-m) maximum thickness, lying on a rigid support and exposed to impact by a spherical object producing less than 100 kJ in energy.
That said, RPEs are generally 9.8–26.2 feet (3–8 m) in thickness and made by frictional, noncohesive material (Figure 4); they are subject to block impact with different shapes (cubic or spherical), which generate energy from 1,000 kJ to tens of megajoules.
Rimoldi-Brusa RPEs method
The authors of this article, being in the geotechnics and geosynthetics industry for many years (almost 50 years combined), have developed a new analytic approach methodology. Both authors welcome comments from any geosynthetic suppliers and industry experts in order to explore and expand the proposed method. The present article introduces the basics of the proposed method, while the method will be fully presented in future conferences and journals.
The Rimoldi-Brusa method is derived from the Italian standard (UNI 11211-4:2018), so it is based on the resistance capacity of the embankment to impact, and it allows the designer to estimate uphill block penetration and downhill extrusion of the geosynthetic-reinforced RPE.
The aim of the authors is to develop an engineering design method that is a calculation model based on rational physical assumptions derived from available research and that can consider the effect of almost all the variables playing a role in the resistance to penetration on the upward face and the resistance to extrusion on the downward face to finally compute approximate, yet consistent, values of the penetration depth and of the extrusion length.
An engineering design method for RPE should allow the designer to understand:
- How do the geosynthetics perform under impact?
- Do they “stretch”?
- Do they “slide” or “pull out”?
- What is the geosynthetic strain limit to choose?
- What is the impact of load spreading afforded by the inclusion of the reinforcement, especially if the impact is not perpendicular as it is on real-scale tests (Figure 5)?
- What is the reinforcement design strength to consider?
- What is the benefit given by close reinforcement spacing?
- What about the inclusion of a longitudinal reinforcement layer?
- What is the effect of different types of upstream and downstream facing?
- What is the effect of thinner or thicker cross sections?
- What is the effect of soils with different compressive resistance and friction angles?
The Rimoldi-Brusa method answers all these questions; hence, the designer can quickly try different solutions and finally select the best combination of design variables that respect all design limits and factors of safety.
Reinforcement layers, made of geogrids or geotextiles, can spread the impact load along the embankment axes in both transversal and longitudinal ways.
By using this method, the designer could also understand the benefit given by the inclusion of a longitudinal geosynthetic reinforcement. The longitudinal reinforcement can help in dissipating the energy in case of non-RPE-perpendicular block impact and repeated impacts, and in avoiding punching-type failure in favor of a larger envelope-type failure.
The Rimoldi-Brusa method is based on the following assumptions (derived from Peila et al. 2002):
- Being that the impact is an impulsive action in nature, the falling rock transfers to the embankment the momentum and the angular momentum of the motion. The impulsive force that is instantly applied to the embankment depends on the mass, the shape and the velocity of the rock, and on the mass, geometry, geotechnical properties and construction method of the embankment.
- Moreover, the response depends on the number, position, layout and technical characteristics of the reinforcing geosynthetics.
- The energy wave generated by the impulsive force applied to the embankment propagates in a quasispherical shape.
- This means that below the rock imprinting in the front face, the soil is highly compressed, while above the imprinting, the soil is subject to upward vertical stresses, which would lift the mass of soil above if reinforcement is not present.
- The soil mass adjacent to the rock imprinting results in compression, while after a certain distance from the first contact point, the acceleration applied to the soil mass produces an outward horizontal movement equivalent to a tensile force being applied to the soil; at the limit between the compressed and the “tensioned” zone, cracks are formed; the soil mass is therefore separated into two parts. The tensile and pullout resistance of the reinforcement can contribute to avoid the failure due to outward displacement.
- Available full-scale tests, FEM and DEM models clearly show that geogrids are able to “guide” the energy, so that the initial spherical shape is soon converted into a horizontal cone; outward displacements occur within this cone, leaving the remainder of the soil mass in place.
- Available full-scale tests, FEM and DEM models also show that the stresses in the reinforcements have an important component in the direction parallel to the length of the embankment, beyond the compressed zone at the upstream face: this justifies the design assumption of also putting geogrid layers in such direction.
- Both tests results and numerical models confirm that geosynthetic reinforcements are subject to high tensile forces, in many cases almost reaching the tensile resistance of the reinforcement used. But the impulsive nature of the active force doesn’t consider any creep effect and considers a higher dynamic tensile modulus of the geosynthetics than the static one.
- Repeated impacts produce larger and larger outward displacements, which can bring it to the pullout failure of the wrapping length of the reinforcement. Hence, this length shall be designed to be much longer than in static conditions; connection of the frontface and backface reinforcement, that is a back-to-back design, is highly recommended.
In the Rimoldi-Brusa method these assumptions translate into the following:
- The downward vertical component of the impact load is resisted by the embankment fill and may have an effect only on the bearing capacity of the foundation and the global stability of the structure and the downhill slope, while the upward vertical component of the impact load is resisted by the fill confined by reinforcement layers.
- Hence the soil mass involved in the impact-extrusion mechanisms is limited by the horizontal planes tangent at the top and bottom points of the impacting boulder (Figure 6), and laterally by two planes diverging from the boulder lateral limits according to a load-spreading angle, α.
Moreover, the following rational assumptions are made for the compressed zone at the upstream face:
- Geosynthetic reinforcements cannot work in compression; hence, reinforcement placed in the direction of the impact (“transversal reinforcements”) may not contribute to penetration resistance.
- Only reinforcements placed along the axis of the embankment (“longitudinal reinforcements”), if present, can contribute to resist the impact load through the tensioned-membrane effect generated by the deformation of the reinforcement like the cord of an arch. This effect can be considered as an increase of the load-spreading angle, α.
- The soil resists the impact load through its coefficient of viscous damping, C, and its elastic passive resistance with an overall elastic coefficient, K, which is given by the sum of the elastic coefficient, Ks, of the soil and of the elastic coefficient, Kg, of the geosynthetic reinforcements (as said, only by the longitudinal ones, if present).
- The elastic coefficient, Ks, of the soil is assumed to be proportional to the Winkler modulus, Mw, of the compacted fill; while the elastic coefficient, Kg, provided by the longitudinal reinforcements is assumed to be proportional to their ultimate tensile strength, Tu.
- The upstream-facing system contributes to the elastic passive resistance through a coefficient, Cg, which modifies the overall elastic coefficient, K: the higher the facing rigidity and the higher the coefficient, Cg.
- The coefficient of viscous damping, C, is assumed to be proportional to the damping ratio, ξs, of the compacted embankment fill.
The penetration depth on the up-stream face is then computed, according to the method presented by Carotti et al. (2000), based on the theory of totally anelastic impact, through the lumped mass model made up by a 1-DOF (one degree of freedom) oscillator, characterized by a viscous damper and a spring (Figure 7), which undergoes a deformative cycle with angular frequency, ω. The lumped mass, m, of the 1-DOF oscillator is the mass of the soil contained in the cone as previously identified.
According to these assumptions, the parameters of the equivalent 1-DOF oscillator depend on the geotechnical properties of the embankment fill, the type and properties of the reinforcement, and its distribution in the embankment (namely, the number and vertical spacing of reinforcement layers); the embankment geometry; and the type of upstream facing (Figure 8).
Considering the viscous work during a deformative cycle, it is possible to calculate the maximum displacement of the 1-DOF oscillator, which is equal to the penetration length, Lp.
The 1-DOF oscillator model allows a calculation of the part, Ep, of the impact energy, Eo, which is dissipated to stop the boulder through deformation, while the residual energy, Es, is assumed to spread downstream of the penetration depth, generating the tensioned zone that produces the extrusion on the valley side face of the embankment.
The following rational assumptions are made for the tensioned zone between the penetration length and the downstream face:
- The fill resists the extrusion movement through its frictional stresses developed on the top and bottom horizontal surfaces of the extrusion cone.
- The geosynthetic reinforcements confine the fill and increase the load-spreading angle, α.
- The increase of the load-spreading angle, α, depends on the type of reinforcement (geogrid, woven geotextile, geostrip, woven wire mesh, etc.); on the number of reinforcements within the height of the extruded cone; and on the presence or not of the longitudinal reinforcements.
- In the tensioned zone, the transversal reinforcement resists extrusion by pullout resistance between the downstream facing and the penetration depth. Pullout resistance can be activated only if reinforcement is properly wrapped around the downstream face with adequate wrapping length or connected to facing elements, which can transfer the pullout force to the whole fill thickness between two consecutive reinforcement layers.
- Since pullout resistance cannot increase indefinitely and the extrusion length strongly depends on the reinforcement deformation, the pullout force shall be produced with limited tensile elongation; hence, the pullout resistance is assumed to be limited to the tensile strength at 2% elongation, T2%, of the transversal reinforcements.
- Given the impulsive nature of the impact load and the consequent impulsive state of tension in the extrusion zone, a reduction factor for impulsive conditions, RFimp, is applied to the pullout force, which is calculated with the pullout factor, fpo, valid in static conditions.
- The residual energy, Es, is assumed to be equal to the work done by the friction and pullout forces, which allows a calculation of the extrusion length, Lv, on the valley side of the embankment.
The impact analysis allows the setting of the required geometry of the embankment, such as the height, H, the crest width, Lu, the slope angles on the mountain side, βm, and on the valley side, βv; and the required layout of reinforcement (type, strength, vertical spacing in transversal and longitudinal directions), by considering the following design targets:
- The maximum allowable penetration length shall be less than half of the embankment width at the bounce height of the impacting boulder, in order for the impacted embankment to be repaired with simple maintenance works.
- The maximum allowable extrusion length shall be less than 20% of the embankment width at the bounce height of the impacting boulder, in order to avoid instability of the whole embankment.
Then such geometry and re-inforcement layout shall be checked for internal stability with the usual analysis, and for external and global stability considering the accidental load of the impact force as an equivalent static force, Fimp (kN), applied horizontally in the center of impact, which can be calculated as the sum of the equivalent penetration force, Fp (kN), and of the equivalent extrusion force, Fv (kN), simply evaluated as energy/movement.
To the authors’ knowledge, the Rimoldi-Brusa method is, at present, the only design method for RPEs that takes into account all the parameters contributing to the penetration and extrusion resistance.
Accordingly, rockfall embankments should be designed by considering Table 1.
Pietro Rimoldi, P.E., is an independent geosynthetic consultant engineer based in Milan, Italy.
Nicola Brusa, CEng, is an independent geosynthetic consultant engineer at Tailor Engineering, U.K.
All figures courtesy of the authors unless otherwise noted
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