Technical paper: Framework for optimised design of rockfall protection embankments reinforced with geosynthetics, part 2

By Pietro Rimoldi, independent civil engineering consultant, and Nicola Brusa, independent civil engineer at Tailor Engineering

The most critical part of the design procedure, and the most under developed, concerns the dynamic analysis of the design impact, with the evaluation of the penetration depth on the hillside and the extrusion length on the valley side.

Referring to the previous sections about the available full scale tests, numerical models, and existing (poor) design methods (GE January/February 2023), the authors propose the following framework for the optimised design of reinforced soil rockfall protection embankments (RS-RPEs) through the dynamic impact modelling.

The framework is based on the following evidence:

Such evidence translates into the following rational assumptions:

E0 = ½ Vm · (γm / g) · vb2 (1)

where Vm is the volume of the boulder (assumed either as a sphere of diameter D or a cube with size D), γm is the unit weight of the boulder, vb is the design impact velocity of the boulder, and g is the gravity acceleration.

The following assumptions are made for the compressed zone at the upstream face:

The values of the load spreading angle α and of the facing coefficient Cg should be evaluated from the results of full scale impact tests on RS-RPE of similar configuration to the one under consideration.

Another way of setting α and Cg is to perform a back calculation of a known impact on the specific system under consideration, using the framework herein presented, where the parameters are modified by trial and error starting from realistic initial values.

If no specific full scale tests or known impact events are available, the default values in Tables 1 and 2 are proposed: in Table 1 the load spreading angle α varies as a function of the reinforcement layout (unreinforced RPE, RS-RPE with transversal reinforcement only, or with both transversal and longitudinal reinforcement), of the number NG of reinforcing layers within the height D of the diffusion cone (see Figure 10 (a)), and of the type of reinforcement (with open mesh allowing soil interlocking like geogrids and steel wire meshes or without open mesh like woven geotextiles or geostrips). In Table 2 the facing coefficient Cg varies as a function of the cushioning capacity of the facing system, where the simple wrap around facing system is assumed with Cg = 1.0.

Note: If there is experimental evidence for the value of the spreading angle α or for the values of the facing coefficient Cf , then the default values in Tables 1 and 2 can be modified.

Taking into account the previously listed assumptions, the penetration depth on the upstream face can be 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, characterised by a viscous damper and a spring (Figure 12), which undergoes a deformative cycle with angular frequency, ω. The lumped mass, m, of the 1-DOF oscillator is the mass ms of the soil contained in the cone as previously identified (see Figure 10 (a) and (b)) plus the mass of the boulder mm. The masses ms and mm are equal to the respective weights Ws and Wm divided by the gravity acceleration, g. The equations for calculating the energy absorbed by soil deformation on the uphill side, Ep and the transmitted energy, Es (which produces the downhill extrusion), as shown in Figure 10 (a), are the following:

E0 = Ep + Es = Ep + E0 · Es / E0 (2)

Es / E0 = mm / (mm + ms) = Wm / (Wm + Ws) (3)

While the weight Wm is an input data from the risk analysis, the weight Ws can be easily calculated from the geometry of the problem (see Figure 10 (a) and (b)).

According to these assumptions, the parameters of the equivalent 1-DOF oscillator depend on the embankment geometry, the geotechnical properties of the embankment fill, the type, properties and layout (namely, the number and vertical spacing of reinforcement layers) of the reinforcement, and the type of uphill facing.

Considering the viscous work of the equivalent 1-DOF oscillator during a deformative cycle, it is possible to calculate the maximum displacement of the 1-DOF oscillator, which is equal to the penetration depth, Lp:

Ep = ¼ (π · ω · Cs · Lp2) (4)

The circular frequency ω is calculated as:

ω = (Ktot / mtot)0.5 = [g · (Ks+ Kg)/(Wm + Ws)]0.5 (5)

The dumping coefficient of the soil Cs can be evaluated from the the dumping ratio ζ of soil:

Cs = 2 · ζ · (Ktot · Ws / g)0.5 (6)

where ζ may be assumed, in this case of a single dynamic cycle and large strains, in the range 0.15 ÷ 0.20 for granular soil, while if the cushioning system on the uphill face includes sand – rubber or gravel – rubber mixtures ζ may be assumed in the range 0.20 ÷ 0.30.

These assumptions easily show the effect of the uphill cushioning system.

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 of the embankment (see Figure 10 (a))

The following rational assumptions are made for the zone between the penetration depth and the downstream face:

τds = fds · σv · tanφs (7)

where fds is the direct shear factor for considering the effect of first detachment friction under fast applied loads, σv is the vertical stress on the considered surface, and φs is the friction angle of the fill.

τpo = 2 · fpo · σv · tanφs (8)

where fpo is the pullout factor for the specific reinforcement with the specific fill, σv is the vertical stress on the considered surface, and φs is the friction angle of the fill. The total pullout force Fpo is obviously the sum of the pullout forces provided by all reinforcement within the height D of the diffusion cone.

Es = (St + Sb + Fpo) · Lv (9)

Following the described above framework, the impact analysis allows the setting of the required geometry of the embankment (see Figure 10 (a)), 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 serviceability limit state (SLS) targets in Section 5, point 7 (GE January/February 2023).

Once established with the dynamic analysis, such geometry and reinforcement layout should be checked for internal stability, external and global stability under dynamic conditions (see Section 5, point 8 in the previous issue), considering the accidental load of the impact force as an equivalent static force, Fimp (kN), applied horizontally in the centre of impact (Figure 13), 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:

Fimp = Fp + Fv = (Ep / Lp) + (Es / Lv) (10)

The full design procedure is summarised in Appendix 1.

The step by step procedure for the dynamic analysis is illustrated in the example in Appendix 2.

A new design procedure for RS-RPEs has been described.

The analysis of available full scale impact tests, numerical analysises and design standards allowed to develop an optimised and rational framework for the dynamic analysis of RS-RPE under rock impacts.

The proposed framework affords to calculate the penetration depth and the extrusion length caused by the impact of the design boulder with its mass, velocity, bounce height and kinetic energy on a given layout of the RS-RPE.

By using the proposed method, the designer can quickly set all the characteristics of the RS-RPE, including the geometry, the facing system on the uphill face, the reinforcement properties and vertical distribution in order to respect all the ultimate limit state (ULS) conditions (collapse of the structure) and SLS conditions (deformations should not affect other structures and should permit an easy reabilitation and repair of the RS-RPE).

Moreover, the designer could also check the benefit given by the inclusion of longitudinal geosynthetic reinforcements and of a cushioning system on the uphill face of the embankment.

To the authors’ knowledge, the proposed framework is, at present, the only design method for RS-RPEs that takes into account all the parameters contributing to the penetration and extrusion resistance. Moreover, it is the only design method that clearly defines and justifies the use of geosynthetic reinforcements inside rockfall protection embankments.

Carotti A, Peila D, Castiglia C, Rimoldi P (2000). Mathematical modelling of geogrid reinforced embankments subject to high energy rock impact. Proc., 2nd European Geosynthetics Conference, Bologna, Italy.

Appendix 1: RS-RPE – Design summary table

A reinforced rockfall embankment should be designed by considering:

Stability analyses

Appendix 2: Example of RS-RPE design

Download appendix 2 below.

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