Rock Mechanics for Natural Resources and Infrastructure SBMR 2014 – ISRM Specialized Conference 09-13 September, Goiania, Brazil © CBMR/ABMS and ISRM, 2014

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Application of the Discrete Element Method for Modelling the Block and Flexural Toppling Mechanisms in Rock Slopes Fredy Alvaro Elorrieta Agramonte PUC-Rio, Rio de Janeiro, Brazil, fredyelorrieta@gmail.com Eurípedes do Amaral Vargas Junior PUC-Rio, Rio de Janeiro, Brasil, vargas@puc-rio.br Rodrigo Pelucci de Figueiredo UFOP, Ouro Preto, Brazil, rpfigueiredo@yahoo.com.br Luis Arnaldo Mejia Camones PUC-Rio, Rio de Janeiro, Brazil, luisarnaldo@puc-rio.br

SUMMARY: The toppling failure mode is a mechanical process in rock slopes with regularly spaced layers or foliation, which involves block overturning and bending of columnar structures. As those processes develop, the failing of internal structures in the slope may occur, leading the whole system to its collapse. The Discrete Element Method based on circular discrete elements was used to analyze two modes of toppling: the block and flexure toppling modes. The method of the study was structured by means of a progressive modelling of the phenomena, and the subsequent validation of the results through a comparison with analytical and semi-analytical approaches. The strain and strength of rock structures subjected to bending process were modeled using a new contact model between every discrete element, which resulted in a better response than other conventional contact models. Subsequently, in the numerical simulation of a physical model, the strains and strength behavior of the experimental slope are reproduced by the numerical calibration of the rock and the joint mechanical properties. The result of this process shows a significant dependence on the stiffness and frictional joint components rather than the properties of the rock itself. That was a different but an acceptable conclusion among other similar works, which aim the rock properties and the joint frictional angle the main factors that control the slope stability in toppling process.

KEYWORDS: Flexural toppling, block toppling, joint stiffness, rock slope stability, discrete element method. 1 INTRODUCTION

Defined by several authors, the toppling failure modes in rock slopes are one of the most common and hazardous events in the industry as, hydroelectric, mining, highways construction and also in natural slopes. The authors in the literature continue by describing the various types of toppling modes as primary and secondary (Goodman and Bray, 1976; Cruden, 1989; Cruden and Varnes, 1996;

Benko, 1997). Blocks and columns structures in rock slopes prone to toppling are usually created by the orthogonal intersection of at least three joint sets, the principal joint set is comprised by continuous joints and with a direction nearly parallel to the slope face, but dipping against it. Since the capability of differential slippage between columns is important to produce flexion which leads the rock to tensile fracture mode, experimental and numerical studies have found the tensile strength and the joint friction

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angle, main factors controlling the slope stability (Adhikary and Dyskin, 2007; Alzo’ubi et al, 2010; Abdullah et al, 2012). However, no previous work specifically performed by the authors listed above has considered the joint stiffness as an important factor in their simulations. Nevertheless, the simulation based on the discrete element method (PFC2d) was able to consider the joint mechanical properties in order to establish the effect in the behavior of the numerical analysis.

2 TOPPLING STUDIES

The stability of a single block or a blocky column prone to toppling or sliding under their own weight was studied by Ashby (1971), and Hoek and Bray (1974), and then by Goodman and Bray (1976). These studies were based on the normal and shear components of the weight developed on the basal plane and then solved assuming the equilibrium in the force distribution. The following relationships were suggested by Goodman and Bray (1976): • Cannot topple if: ....(1) • Cannot slide if:

…..(2)

Where ∆x is the block base length, y is the height of the block, α is the angle of the basal plane, and φ, µ are the frictional components. Sagaseta (1986), proposed a relationship for the stability of a single block based on a pseudo-static formulation and then Aydan et al. (1989), based on the previous work, developed four conditions of instability for block and blocky columns: toppling after sliding was started, sliding only, toppling only, and sliding after toppling was started. Perhaps the most accepted method for Block toppling is one the proposed by Hoek and Bray (1976), based on equilibrium of statics. Later Aydan et al. (1989) proposed an analytical approach based on pseudo-statics force distribution. The two methods are in good agreement in evaluating the safety factor for

rock slopes. Flexural toppling analytical model was later approached by Aydan and Kawamoto (1992), which considered a series of superimposed cantilever beams in their theoretical model. These beams may fail by bending stresses produced by the sum of forces acting among them. The main assumptions made at this point are: the bending just only for those columns portions that lie above the orthogonal plane to the joints; and this orthogonal plane arises from the slope toe.

3 NUMERICAL MODELLING OF ROCK

TOPPLING

3.1 The discrete element method.

The DEM was introduced for the analysis of rock mechanics problems (Potyondy and Cundall, 2004), and it was later applied to soils by Cundall and Strack (1979). In this method, the discrete elements can have finite displacements and rotations, as new contacts are automatically recognized while the calculation progresses. Newton's second law is used to describe the individual motion of each element, and the interaction between elements is defined by a force-displacement law applied to each contact (Mejía et al, 2013).

3.2 Flat-jointed contact model.

The flat-joint contact model was proposed in order to avoid relative rotations between particles even if the contact has been broken due to overcoming stresses. The contact idealizes two notional bonded surfaces, each of them belonging rigidly to each particle. This bonded state between surfaces may evolve to a frictional state after the tensile or shear strength of the bond has been reached. This event could be seen as a contact breakage. After this has occurred, the capability of the contact model to resists relative particles rotations remains because the notional surfaces are not removed and frictional forces begin to act among them.

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3.3 Smooth-joint contact model. The smooth-joint contact model allows a relative displacement between two particles in contact, without producing the over-riding between them. This contact model is created when two particles meet at a single point described by the intersection of their interfaces. At this contact point, two finite parallel notional surfaces are idealized with the same direction of the discontinuity intended to be modeled. Then any further force calculation is done considering a set of mechanical properties among those surfaces. This allows the modelling of a discontinuity, by assigning this contact model to all contacts amid particles that lie on opposite sides of the joint.

3.4 The Response of toppling and sliding

phenomena in PFC2D.

In order to reproduce the toppling and sliding phenomena for one single rock block in PFC2D, a symmetrical arrangement of disk-shape particles was used. These particles were bonded together with parallel and contact bonds and with high elastics constants in order to have no influence of body deformations.

Figure 1. Single block stability subject to sliding and toppling by its own weight.

The basal plane with a friction angle of 38.155° degrees, was represented by a group of single smooth joint contacts aligned to the basal plane Since PFC considers the gravitational acceleration forces, the stability condition suggested by Goodman and Bray (1976) for a single block with, arctan(∆x/y) = 31° and α = 30°, is not followed. As presented in figure 1, the velocity of the upper left particle is incremented by each step. Therefore, the numerical analysis fits better with the pseudo-static approach developed by Sagaseta (1986).

3.5 Flexure of a rock column.

The strain and strength capabilities of the synthetic material comprising the rock columns were the first factors considered for idealizing the numerical flexural toppling model used, since those characteristics have a significant role in the mechanical behavior of the slope.

Figure 2. Behaviour in bending strain (BPM, FJ-BPM, vs Timoshenko analytical approach). In doing so, the Parallel Bonded Particle Material (BPM) versus the Flat-Jointed Bonded Particle Material (FJ-BPM) behavior under flexion was contrasted against the analytical formulation based on the elasticity theory, to study the behavior of a cantilever beam loaded at the end (Timoshenko and Goodier, 1970).

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Figure 2 shows a better approximation of the Flat-Jointed BPM to the analytical curve than the BPM material. The elastic properties of the material evaluated were 4.0e9 Pa and 0.27, for Young’s and Poisson’s moduli respectively. The latest feature and the best relation of the material between USC and Brazilian tensile strength showed by Potyondy (2012), suggests the Flat-Jointed BPM to be the best option to use in the flexural toppling simulations of the present work. 3.6 Response of the numerical model for

bending strength of rock columns.

Caused by the external forces acting on the column faces of the slope, bending stresses could bring the synthetic rock to its failure by tensile modes. In order to establish the relation between bending and direct tensile strength, a series of three point (TPBT), and four point bending tests (FPBT), was performed in PFC2D. Four different samples, with the same micro and macro properties were loaded in direct tensile and bending modes to reach the rupture.

Figure 3. Peak strength of the same material (Flat-jointed BPM) for Direct Tension Test, Three Point and Four Point Bending Test. The results showed a good approximation between mean strength values and standard deviations, (see figure 3). For the material being used in this work, figure 3 suggests that the failure in bending

mode occurring in flexural toppling process, takes place when the tensile strength is reached. 3.7 Modelling of block toppling on a stair-

shaped base.

Goodman and Bray (1976), suggested an analytical ideal model for the solution of block toppling problems solving the resultant forces in the system through the analysis of rigid body. To emulate that, a symmetrical distribution of disk-shaped particles with identical elastic properties between contacts was considered in PFC2D, (see figure 4).

Figure 4. Ideal block toppling problem modeled in PFC.

High values of Young’s modulus were used in the contacts, to avoid the influence of body strains. The bases where every block of the system lies, are formed by fixed particles; it est., no translation or rotation in particles is allowed. The force distribution in every basal plane is presented in figure 5. The resultant force in each block face is determined by the sum of each force component of every smooth joint contact model. The velocity measured in the block 1, situated in the slope toe (figure 4), confirms the required friction angle of φ = 38.182° to reach a stable condition in the model, which contrasts with the friction angle of φ = 38.155°, suggested by the analytical formulation. The last statement is accepted since the resulting velocity of the toe block tends toward zero when the last angle of friction is used (see figure 6).

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Figure 5. Numerical force distribution in basal planes

vs. analytical approach.

A safety factor proposed by the limit equilibrium approach is presented in the equation 3. Here, assuming the “available” value the numerical friction angle of stability, and the “required” value the stable friction angle found in the analytical method, the safety factor calculation results in 1.001. This shows a good approximation between the methods evaluated and an appropriate response of the numerical model.

..................(3)

3.8 Numerical approach of flexural toppling

on rock slopes.

A “step by step” analytical solve method was proposed by Aydan and Kawamoto (1992); based on a series of experimental tests and case histories, (Kawamoto, 1982; Aydan and Kawamoto, 1992). According to the works mentioned, the limit equilibrium condition of a slope with the characteristics listed in table 1 is met by a slope angle of 80°. This means that any other higher value in the slope angle will end in collapse (where the slope angle and slope high are defined by Ψ and H respectively, and the distance between joints is denoted by “t” ). In order to represent the slope first outlined, a random distribution of particles was generated. The particles were bounded by walls

with stiffness values equals to the average stiffness in the system. Table 1. Slope properties and geometry. Kawamoto, 1982.

σt (KPa)

σc (KPa)

γγγγ (kN/m3)

E (MPa)

ν

4,0 11,4 12,4 3,02 0.35

t (mm)

H (mm)

φφφφ' (°)

Ψ (°)

30 150 40 80

The tensions produced by the final overlapping between particles, as a result of the packaging, were reduced to a 1% the contact strength. After, the geometry of the slope was generated by eliminating the upper wall and by erasing the particles sequentially from the top to the bottom of the slope, until it gets its final geometry. Between every event, the model was allowed to reach the equilibrium of forces (Jaramillo, 2013).

Figure 6. Velocity in the block toe with different joints’ friction angles against time step. On the other hand, it is well known that the relative displacement between two surfaces in contact is controlled by the friction angle and their stiffnesses, (Bandis et al, 1983). As well as the stability of slopes in discontinuous rock masses, is usually governed by the geometrical distribution and mechanical properties of

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discontinuities rather than those of the rock itself (Aydan et al, 1989).

Slope angle Ψ=80°

(a) kn=2e8 ks=6.67e5 (b) kn=1e8 ks=3.33e5

Slope angle Ψ=81°

(c) kn=2e8 ks=6.67e5 (d) kn=1e8 ks=3.33e5

Figure 7. Response of the numerical model with different stiffness values in smooth-joint contacts. Following those concepts, since no joint stiffness were defined in the related work, several relationships of micro-parameters kn/ks of singular smooth-joint contacts were tested (50, 100, 200 and 300). In doing so, the best relationship seems to be kn/ks =300. In figure 7, in agreement with the stability analytical results, the values of kn=2e8 Pa/m and ks=6.67e5 Pa/m, in the smooth-joint contact model, follows the condition of stability and instability for slope angles of 80° and 81° respectively, (see figures 7.a and 7.b). Instead, the stability condition is not reached for stiffness values of kn=1e8 Pa/m, ks=3.33e5 Pa/m. 3.9 Modelling of a centrifugal test with

gravity force increments.

Adhikary et al. (1997), using a model manufactured in laboratory, recorded the deformation of slopes tested on a centrifugal

device. The model was constructed by casting horizontal layers of a mixture consisting of ilmenite, sand, and 15% of gypsum then superimposed to form a block. After, it was cut to form the exact slope model geometry. The slope characteristics reported by Adhikary et al. (1997) are listed in table 2. Table 2. Adhikary’s slope properties and geometry.

σt (KPa)

σc (KPa)

γγγγ (kN/m3)

E (MPa)

ν

1.1 7.0 23.8 2.4 0.35

t (mm)

H (mm)

φφφφ' (°)

Ψ (°)

10 330 24 61

Simulating the increase of gravity loads, the model was subjected to a series of increases in the velocity of the centrifugal machine until it reached the collapse. To model this condition in PFC2D, the gravity acceleration was increased gradually in 2 g’s increments and the displacements were monitored.

Figure 8. Physical model of slope subjected to a centrifugal test, with monitoring points “A” and “B” Adhikary et al. (1997).

As it was expected from the last example, the joint stiffness played a significant role in the numerical simulation. Both micro-parameters, kn and ks were tested and modified in order to fit the horizontal deformation in two points of the physical model represented in figures 8 and 10 (Adhikary et al, 1997).

The strain and gravity level of the centrifugal test reported in the related work, which

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precedes the slope fracture was reproduced with smooth-joint stiffnesses values of kn=1e12 Pa/m and ks=0.9e12 Pa/m, (figure 9). Afterwards, the micro-parameters encountered were tested in a joint direct shear test performed in PFC2D in order to obtain the stifnesses of the modeled joint. According to that, the resulting values of the macro stiffnesses of the discontinuity were Kn=90.9 GPa/m and Ks=0.3 GPa/m (as can be noticed here, the stiffnesses of the joint are denote with capital letters). The stiffnesses found for the discontinuities of the numerical model are consistent with the values showed in the works of Barton (1971) and Bandis et al. (1983). It was observed that kn and ks controls the behavior in terms of deformability and failure in a certain gravity level for the numerical model. Higher values of smooth-joint normal stiffness seem to produce greater normal intercolumn forces, which may restrain the slippage between columns and, therefore, the resultant deformation in the slope could be decreased. On the contrary, it is reasonable to establish that lower values in the shear stiffness of smooth-joint contacts, can allow bending strains and the subsequent development of tractive stresses that may bring the columns to its rupture. Figure 9 shows the horizontal displacements measured in the physical and the numerical model. Point A represents the top of the slope and point B the middle point of its height (see figure 8). As it can be notice here, the maximum slope strain prior to failure in the numerical model has a good approximation with the experimental results. However, in earliest stages it presents a slight difference. The transitional deformation patterns occurred under 40 g’s was not reproduced by the present numerical simulation. It is reasonable to consider that the full closure of the discontinuities in the experimental model happens in this range, accumulating deformations between columns that are not shown in the control points until the above mentioned “g” level has been overcome. Further studies have to be done in order to assert this likely explanation, since that topic exceeds the

scope of the present article. However, as mentioned previously, Bandis et al. (1983) has shown the dependence of the joint shear stiffness with the confinement stress level, which could be considered as a way to explain the changing behavior of the gradients in figure 9. Another feature shown is the rupture surface that lies below the surface showed in the experimental model.

Figure 9. Comparison between numerical and physical horizontal displacements in point “A” and “B”.

Figure 10. Numerical flexural toppling model, comparison of the numerical rupture surface and the experimental rupture surface. Red lines represents the tensile stresses in the rock columns, and the black lines the compression stresses.

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4 CONCLUSIONS Block and flexural toppling failure were investigated using the discrete element method through program PFC2D. Analytical and semi-analytical methodologies were also used for comparisons in order to validate the models proposed in the present article. The analyses showed that the calculations done in PFC2D fits better with the results obtained by means of the pseudo-static formulation proposed by Sagaseta (1987). On the other hand, the flat-joint contact model seems to show a better response in the bending deformation as compared with the Timoshenko analytical approach. The tensile strength of flat-jointed BPM obtained by means of direct tensile test, was reproduced with good approximation via the three and four points flexural tests. After the tensile strength of the numerical synthetic material, the numerical simulations showed a significant dependence of Kn and Ks in the results. The normal and shear joint stiffnesses controlled the deformability and failure gravity level for the numerical model. This suggests that, once the slippage between columns is allowed to occur, the bending stresses produced by flexural event may lead the synthetic rock to its failure in tensile mode. The present work provides some insight into the mechanical process of toppling failures not observed in other methods. ACKNOWLEDGEMENTS We would like to acknowledge the financial contribution of the CAPES, and the Brazilian Ministry of Education. REFERENCES Adhikary, D., Dyskin, R., Jewell, R., & Stewart, D.

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