Analytical solution for assessing continuum buckling in sedimentary rock slopes based on the tangent-modulus theory

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

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Analytical solution for assessing continuum buckling in sedimentary rock slopes based on the tangent-modulus theory Sergio Esteban Rosales Garzon

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Drummond Ltd., Mina Pribbenow km 31 vía San Roque - Bosconia, Cesar, Colombia, South America

A R T I C L E I N F O Keywords: Stratified rock Surface coal mine Slope stability Buckling failure mode

1. Introduction Slopes in stratified rock masses cut or naturally parallel to bedding form a non-daylighted slab structure. When these slopes contain a continuous weak bedding plane or a persisting discontinuity, and the rock mass lacks any relevant cross-jointing able to favor wedge, planar or ploughing failure, then the kinematic potential for buckling emerges. Buckling collapse hazard in a rock mass slope1–6 suppose an absolute instability where the failure is often rapid and without previous easily recognizable warnings. When the slope reaches a critical height, any extra height, pore pressure increase or small external load could cause it to buckle. Whether in the case of a footwall slope of a surface coal mine or in a natural slope, that potential risk demands a rigorous assessment of the slope stability for proper design or mitigation. The use of the limit equilibrium technique in combination with Euler's buckling theory has resulted in some closed solutions.7–12 However, as stated by others1,6,13,14 and from the author’s own experience, these approaches have proven unable to consistently explain the buckling failure mode. This article suggests a formula that can be used to calculate the length of the passive segment and the use of the tangentmodulus theory to identify the mode of failure and evaluate slope stability. Nine real buckling cases were analyzed with acceptable results. 2. Length of the passive segment In a slope under buckling potential, the overall slope length, Ls, can be divided into two different regions: the “passive segment”, which is at the toe of the slope, and the active region, which is the section above the passive segment and is called the “driving segment” (see Fig. 1).

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During failure, the “passive segment” starts bulging upward and progressively releases the “driving segment” along the dip direction in a concomitant translational failure mode. A one span beam-column with a pin-roller support on its left end and a pin support on its right end is used to model the passive segment (see Fig. 2). The bedding parallel to the slope face is modeled as an orthogonal fracture system with a flat overall geometry. Since the beam-column is assumed to be straight, it is necessary to recreate this same natural condition in the model by introducing an auxiliary eccentric compressive force, T, at the level of the pinned supports. The magnitude of this eccentric force shall be exactly the force needed to compensate the downward deflection due to self-weight and thus keep the beam-column straight. Beam-column cannot deflect downwards due to natural constraints, nor can they deflect upwards assuming negligible tensile strength (as is common for major surface stratified rock masses). Therefore, in this compression-only setting, the critical point from which equilibrium is lost can be defined as the moment when the beam-column begins to deflect upward. Before this critical point, the beam-column is still straight and can develop compressive strength, and after this critical point, the beam-column fails by losing its compression-only status. The word “straight” is used here to mean a beam-column in a compressiononly setting; consequently, curved beam-columns while working in a compression-only setting, in essence, also adhere to this meaning. Based on this failure mechanism, at the critical point, the length of the passive segment,ℓ , shall be such that the total midspan deflection in the beam-column model is zero. In addition, the driving force, P, is evaluated by isolating the potential driving segment as illustrated in Fig. 3. By summing the forces along dip, we can obtain the expression (1) for the resultant driving force P. In this expression, rock mass unit

Correspondence address: Cra 82A #50A 83 Calasanz, Medellín, Antioquia, Colombia, South América E-mail addresses: [email protected], [email protected].

http://dx.doi.org/10.1016/j.ijrmms.2016.10.002 Received 28 March 2016; Received in revised form 29 August 2016; Accepted 11 October 2016 1365-1609/ © 2016 Elsevier Ltd. All rights reserved.

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

that the force P shall be equal to the force T to guarantee the overall equilibrium of the slope. This equilibrium exercise implies rigid blocks; nevertheless, for the purpose of surface sliding analysis where the stress is relatively low, it is reasonable to assume that the blocks are rigid.2 Using deflection formulas derived from the Euler-Bernoulli beam equation15,16 we can evaluate the midspan deflection. If only the weight of the beam is considered in the beam-column model (T =0 ), then the maximum downward elastic deflection (ymax ↓)16 (p. 193) is:

ymax ↓ = 5qℓ 4/384EI

(2)

If only the eccentric force is considered in the beam-column model (q=0 ), then the maximum upward elastic deflection (ymax ↑)16 (p. 194) is:

ymax ↑ = Teℓ 2/8EI

(3)

Fig. 1. Sketch of buckling failure mode in slopes of stratified rock.

Therefore, to obtain zero midspan total deflection, through the use of the principle of superposition and setting T equal to P, we find at the critical point, the downward deflection from Eq. (2) shall be equal to the upward deflection from Eq. (3):

5qℓ 4/384EI = Peℓ 2/8EI

(4)

Substituting P as defined in Eq. (1), inserting the eccentricity, e, equal to d /2 and then simplifying the results for Eq. (4), we obtain the quadratic equation for ℓ :

ℓ 2 + [9.6(γd sin α − C − γd cos α tan φ)/ q]ℓ Fig. 2. Model for the passive segment (eccentrically loaded beam-column with selfweight).

− [9.6(γd sin α − C − γd cos α tan φ) Ls / q] = 0

(5)

The quadratic Eq. (5) has the following simplified solution for its positive root; in this equation, the constant C1 and the beam-column self-weight, q, are known:

ℓ=

C12 + 2C1 Ls − C1

(6)

C1 = 4.8(γd sin α − C − γd cos α tan φ)(d /2)/ q

q = dγ cos α In conclusion, for any given slope, the length of the passive segment can be calculated using Eq. (6) as a function of the overall slope length along the dip. When the toe slope contains a nose or a natural imperfection, the passive segment length is no longer the one identified above. In this case, it is believed that passive segment length must be the same as the nose length or controlled by the imperfection itself.2,3,6 3. Compressive strength The tangent-modulus theory by Shanley for column buckling concluded that the lateral deflection starts very near the tangentmodulus load, Pt, that is, the concept of tangential modulus predicts the maximum load that can be applied to a column before it undergoes lateral deformation.17 Given that we previously defined the straight state as the needed requirement for the beam-column model to be in equilibrium, it is reasonable to adopt the tangent-modulus theory. To properly investigate the buckling strength of columns loaded in the inelastic range using tangent-modulus theory, it is necessary to know the real stress-strain diagram of the rock mass uniaxial state of stress. Using previous stress-strain diagram, we can deduce the corresponding tangent-modulus, Et. Finally, by introducing this tangent-modulus into Engesser's formula, we can obtain the buckling formulas (7) and (8) needed to assess, respectively, the compressive load and the compressive stress in the inelastic domain.18

Fig. 3. Equilibrium of forces for the driving segment.

weight, γ, slab thickness, d, friction angle of the failure surface, ɸ, cohesion of the failure surface, C, slope angle, α, and dip overall slope length,Ls , are known, and the length of the passive segment along the dip, ℓ , is unknown. Under certain conditions of slope angle, depth of the potential failure surface and sliding surface friction and cohesion, it is possible to obtain a negative driving force. This negative result means that the buckling failure mode could be not a real concern in such a setting.

P = (γd sin α − C − γd cos α tan φ)(Ls − ℓ)

(1)

The two segments, active and passive, have been evaluated independently. To integrate both together, we need to ensure the equilibrium condition. The driving force, P, in the driving segment acts along dip direction as well as axially, assuming that the center of gravity matches the geometric center of the slab. As explained before, the eccentric force and the self-weight in the model for the passive segment cancel bending moments each other, leaving only the axial force along dip, T. Consequently, and only at the critical point, it is true

Pt = π 2Et I /ℓ 2

(7)

σt = π 2Et /(ℓ/ r )2

(8)

On the other hand, the buckling force of a column in the elastic domain is obtained from Euler's buckling formulas (9) and (10) which 54

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

bench formation). Both ends of the failure terminate at a joint perpendicular to bedding that intersects the 370 RL bench. This end joint offers a side release mechanism that may also contribute to the character of toe-buckle fold. Jointing perpendicular to bedding occurs frequently in the pit, with several joints separated by up to 3 cm of orange gouge. Daily June rainfalls never reached more than 8 mm and there was no measurable rainfall from 26 June to the time of the toebuckle failure, therefore, rainfall may not be a direct contributing factor. The relatively rapid unloading of the slope during batter formation was believed to cause both pore pressure effects and strength-lowering to residual friction angles of the failure surface material by wetting up. Scoble4 described that an extensive but thin slab in the pavement of the Westfield Shale coal (“P” coal) failed catastrophically at 10:30 pm on 14 January 1974 at the western wall of the mine's Phase II excavation, see photo on the right in Fig. 4. A bulldozer at the base of the slope was engulfed and the driver killed. Illumination of the slope at the time was confined to the toe area but two witnesses in a tractor at the northern toe area stated that the middle of the face first bulged and then slid down over the lower part, breaking into fragments as it fell. No signs of distress were observed in the slope prior to failure. It had been standing for four years. The possibility exists that an initial displacement of the lower slab, caused by buckling at its toe, sliding on a cross-over joint or fracture through intact mudstone may have occurred, thus allowing the more noticeable and devastating overriding movement of the upper slab. Groundwater pressures are not directly indicated to have contributed significantly to failure. No blasting occurred nearby the pavement during the period immediately preceding the failure. Quintette Operating Corporation owned an open pit coal mine located near the town of Tumbler Ridge in northeastern British Columbia, Canada. At this property, hosted in stratified sedimentary rock situated on the eastern slope of the Rocky Mountains, Wang et al.2 described a buckling failure event. The footwall with sandstone slabs potentially subject to sliding was exposed from a crest elevation of approximately 1650 m to the pit floor at approximately 1455 m. The P2 Pit footwall had been gradually exposed since startup without significant problems. In August 1997, when the pit floor reached an elevation of 1530 m, two thin slabs, each approximately 0.75 m thick, slid over a wedge-shaped area, see Fig. 5. The failure extended approximately 90 m above the pit floor and was approximately 150 m wide. The unloading during the end footwall slope formation was the only factor believed to have caused this failure event. Choquet et al.3 and Soukatchoff et al.21 described the analysis of the strata buckling mechanism at the Grand-Baume coal mine in France. At this mine, a 250-meter-high footwall with an adverse bedding dip of 40° was planned. The open pit of Grand-Baume (Gard-France) allows access to the Stephanian coal deposits within; the northwest area of this open pit comprises an overthrust fold. The coal is extracted in the reverse side wall of the fold and the slope of the open pit, which has an angle of approximately 40°. In this slope, planar failures between benches have been observed, see Fig. 6. These phenomena are explained by a buckling effect. Choquete et al.3 analyzed the slope using a combination of physical modeling (friction table) and threehinge buckling limit equilibrium analysis. Assuming a passive region of concave curvature toward the pit extending approximately 60% of the overall slope length, friction table analysis yielded a critical height of 115 m for a layer 0.5 m thick, a critical height of 170 m for a layer 1 m thick, and a critical height of 240 m for a layer 1.5 m thick. Bawang Mountain landslide is a natural slope failing through buckling described by Yang.5 The Bawang Mountain landslide, with a slope height of 940 m and a slope angle of 40°, occurred in 1983 (see schematic graph on the top of the Fig. 7). The slabs of this slope are composed of Denying limestone strata. The total thickness of the failed slabs is 10 m, and 3 separate slabs were involved in the failure. The slabs, approximately parallel with the slope surface, are separated from

provide, respectively, the compressive load and the compressive stress in the elastic domain.

Pe = π 2EI /ℓ 2

σe = π 2E /(ℓ/ r )2

(9) (10)

Euler's formulas (9) and (10) are valid only when the compressive stress, σe, is less than the proportional limit, σp. By introducing the expression for σe from (10) in the inequality σe < σp and solving for slenderness ratio (λ = ℓ/ r ) we obtain the valid limit (11) of formulas (9) and (10).

λ > π E / σp

(11)

By definition, the ultimate uniaxial compressive stress σu of the column material defines the upper-bound of the buckling stress range according to tangent-modulus theory. In other words, compressive stress shall be always equal to or less than the ultimate compressive stress,σ ≤ σu . As such, we assumed that the ultimate compressive strength of the rock column corresponds to the uniaxial compressive strength of the rock mass, UCS, because, according to Hoek-Brown failure criterion, failure initiates at the boundary of an excavation when UCS is exceeded by the stress induced on that boundary.19 Once Et is known, the rest of the variables in formulas (7)–(11) can also be deduced. The length of the column, ℓ , can be estimated as explained earlier in Section 1. Because the bedding is parallel to the driving compressive force, the deformation modulus of the rock mass, E, can be set equal to of the deformation modulus of the intact rock, Ei. The radius of gyration r = I / A is given by the geometry of the problem. The column's effective length factor k = 1.0 is used to account for the column with pinned ends as modeled in the previous section. This end condition is very often encountered in practical applications13 and is also the fundamental case from which Euler’s formula was derived. 4. Critical slope height The critical slope height Hcr is defined as the slope height for which the ratio between the critical force of the column Pcr and the driving force P acting on it (1) is 1.0. Pcr could be defined as either Pt or Pe, depending on the case. 5. Case study results Nine buckling failures reported in the literature were selected to conduct an evaluation of the above approach for buckling assessment. Three of them describe different buckling failures at the surface coal mines of Malvern Hills1, Quintette2,20 and Westfield4. Other three come from physical models (friction table) developed to understand buckling failures at Grand-Baume coal mine.3,21 To complete the sample, other three cases were selected to account for buckling in natural slopes namely in Bawang Mountain5,22 and Lavini di Marco6. Next, we describe the failure mechanism and the factors that could or not influence the instability for each case study. Seale1 described a large slab toe-buckle footwall failure that occurred overnight on 1 July 2004, when the excavator operator arrived to extract the three-meter-thick coal seam that was exposed the day before. Sometime during the night, a two-meter-thick slab had translated 6.2 m down dip and buckled, developing a broadly cylindrical fold structure at the toe of the slab partially covering the main seam coal, see the photo on the left in Fig. 4. The failure extended 85 m along the length of the 355 RL bench. To achieve an economical pit design and minimize waste in the coal-bearing units that are steeply dipping (45 ± 5°), the footwall batters were cut to parallel the dip of the bedding. The footwall design has 5 m wide benches cut at 15 m vertical intervals, creating batter dip slopes of 21 m in length. The failure occurred immediately following the rapid excavation of the toe (355 RL 55

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

Fig. 4. On the left a photo of the large toe-buckle, footwall failure that occurred on 1 July 2004 at the Malvern Hills surface coal mine, inland Canterbury, New Zealand indicatig bench levels and the toe-buckle slab dimensions (From Seale1). On the right a photo of pavement slab failure occurred on 14 January 1974 at Westfield surface coal mine in the United Kingdom (From Scoble4).

direction. Sliding of the pack of layers produced a gaping tension crack which results from the combination of two joint sets. The second, which is referred as “flexural buckling” (case 6) is observed in an area located at a lower elevation, downhill from the change of curvature marking the axis of the “three-hinge buckling”. In this area the lower part exhibits an apparent concavity. “flexural buckling” involves a thick pack (5–6 m) of layers that are significantly thinner than those displaced by the “three-hinge buckling” (thickness ranging between 10 cm and 20 cm). See schematic graph of these two cases on the bottom of Fig. 7. Deformations develop according to cylindrical symmetry parallel folds. All the area was monitored between 1992 and 2002 using a Synthetic Aperture Radar Interferometry (INSAR) resulting in an active downward displacement of average velocity of about 3 mm/year. During the autumn 2000 a period of intense rainfall lasting some weeks triggered important landslides in the Trento province but it was no detectable in the displacement-time curves. Such behavior suggests that the relationship, if any, between buckling deformations and hydraulic conditions in not simple. Furthermore, after numerical analysis the buckling is only explained by introducing significant uplift pressures; however such water pressure induces

one another and from an underlying weak interlayer composed of broken marlite, formed by relative sliding between interfaces. The stratum below the weak interlayer consists of massive limestone. Under the long-term gravitational force the slabs near the toe of the slope buckled gradually and the slabs close to the slope top were torn away. Tommasi et al.6 described a buckling instability in a high natural slope known as the case of Lavini di Marco. For approximately 3 km south of the town of Rovereto (Trento, Italy) the left flank of the Adige river valley is formed by well stratified limestones with clayey and marly interbeds. The slope surface created by the youngest and largest of these slides (Lavini di Marco) is wrinkled by buckle folds which were first noticed in 1969 as a sliding of rock layers after their progressive “heave”. Two nearby but independent buckling areas are reported. The first, which is referred as “three-hinge buckling” (case 5) is located on the Lavini slide surface between elevations 700 and 750. In this area layer dip progressively increases proceeding downhill, giving the slope surface an overall gentle convexity. Buckling involves a 2.7 m thick pack of several layers which have a thickness ranging between 0.5 m and 1 m, and appear subdivided by the subvertical joint sets. The rotation axis is parallel to the slope strike and normal to the sliding

Fig. 5. Photograph showing buckling failure surface occurred on August 1997 at Quintette coal mine in northeastern British Columbia, Canada (From Wang et al.2).

56

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

Fig. 6. Left photograph shows the Sandstone layers at elevation 250 m of West Wall in Grand-Baume coal mine in France (note the concave curvature of layers towards inside of pit) and on the right the corresponding graphical interpretation of the failure mechanism (From Choquet et al.3).

ties for each case. Data constraining the geometric parameters for each case, including the thickness of the unstable rock slab, d, and its corresponding slope angle, α, have been compiled from the literature. Additionally, the rock mass unit weight, γ; the strength properties of the failure surface, ɸ, C, the uniaxial compressive strength of the intact rock material, σci, and the modulus of deformation of the rock mass, E, are also sourced from authors’ reports. 5.1. Ultimate compressive strength UCS The rock mass uniaxial compressive strength, UCS, is determined on a case-by-case using the Hoek-Brown failure criterion.19 With this, we need three parameters to derive UCS; uniaxial compressive strength of the intact rock material, σci, disturbance factor, D, and the geological strength index, GSI. The value of σci is known, see Table 1. In cases 4, 5 and 6, D is equal to zero to account for the undisturbed rock mass of natural slopes. The end footwalls at coal mines (cases 1, 2, 3, 7, 8 and 9) are composed of the remaining in-situ material after the recovery of the last recoverable coal seam; because the coal is brittle and soft, excavation can be usually carried out by ripping and dozing (mechanical excavation) so the degree of damage to the slope correlates well with a disturbance factor of D=0.7. To determine GSI, we correlated

Fig. 7. The top schematic diagram shows the failure process of Bawang Mountain landslide (From Yang5); a) Prior to deformation; b) During deformation. And the bottom schematic diagram shows the profile of the left flank of the Adige valley at south of Rovereto, North-Easters Italy (From Tommasi et al.6).

noticeable displacements normal to the bedding that are not observed in situ. Table 1 summarizes the reported parameters and material proper-

Table 1 Physical and mechanical properties of the rock mass and failure surface materials. #

1 2 3 4 5 6 7 8 9

Case

α (°)

Rock mass

Name

Kind of slope

Type

σci (MPa)

E (GPa)

γ (kN/ m3)

Westfield Malvern Hills Quintette Bawang Lavini di Marco

Coal mine footwall

Mudstone Mudstone/Siltstone Sandstone Limestone Limestone

42 25 50 80 55

20.5 22 27 27 27

Sandstone/ Siltstone

51.5

11 0.61 30 50 30 20 30

25

Grand-Baume

Natural

Physical model (friction table)

57

Failure surface Material

d (m)

Ø (°)

C (kPa)

35 42 44 40 22

Thin clay band Thin coal seam Carbonaceous parting Marlite layer Clayey-marly interbed Planar clean contact

22 3 25 17 18 15 25

0 0 0 40 0

40

3.5 2 1.5 10 2.7 5.5 0.5 1 1.5

0

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

Table 2 Selection of the Geological Strength Index, GSI, based on descriptive inputs sourced from the cases’ literature, and the required parameters to apply the Hoek-Brown failure criterion and thus obtain the uniaxial compressive strength of the rock mass, UCS, for each case.

agrees with the stress-strain diagram.24 After applying some boundary conditions Ylinen provides these three parameters on the basis of the modulus of elasticity, E, proportional limit, σp, and compressive strength, UCS, of the material:

the rock types and the descriptive inputs on structure and surface condition with the existing charts for GSI selection.23 Coal mine cases are heterogeneous rocks, so accordingly, we used the GSI chart “Geological Strength Index for heterogeneous rocks such as flysch” provided by Marinos and Hoek,23 while for the homogeneous rock of cases 4, 5 and 6, we used the “General chart for GSI estimates from geological observations” provided by Marinos and Hoek23. Table 2 summarizes the corresponding parameters used to apply the HoekBrown failure criterion and the resulting UCS for each case. The columns “Rock Type” and “Descriptive Input” of Table 2 contain the descriptive inputs sourced from forensic descriptions, pictures and field observations, reported in the literature and earlier described in this section for each case study.

a = UCS b = UCS / E c′ = c / E

c = 1 + (Eδp /UCS )/(σp /UCS ) + ln(1 − σp / UCS ) In addition, the parameter c is affected by the quantity δp , which indicates the allowable deviation from Hooke's law at the proportional limit. In all study cases, the required δp to calibrate the stress-strain diagrams and capture the expected behavior for each rock type was 0.0000005; this figure also conforms to the acceptable range of δp since it is less than the maximum values allowed between 0.00002 and 0.0002. 24 Deere and Miller classified the uniaxial stress-strain curves into six types.25 We considered the first three types of behavior, given that they correspond better with the rock types of the nine selected cases. With this, we defined the values for the proportional limit, in terms of the ratioσp / UCS , that better describe the type of curve of stress-strain behavior, after matching and grouping all case studies by rock type. Table 3 summarizes previous analyses and shows the resultant parameter c and proportional limit for each type of behavior. By integrating the expression (12) and applying additional boundary conditions, Ylinen demonstrates the expression (13) as the stress-

5.2. Tangent-modulus Et The stress-strain diagrams are not reported for the case studies. Nevertheless, by using some rock mechanics concepts and the method of determining the buckling stress as developed by Ylinen 24, we can estimate a solution. For the approximation of the stress-strain diagram, Ylinen proposes the expression (12) in terms of the first derivate with respect to the strain.

∂σ /∂ε = (a − σ )/(b − c′σ )

(12)

where σ is the stress, ϵ is the strain and a, b, c′ are three free parameters, the values of which should be determined so that the stress-strain function deduced from Eq. (12) by integration suitably 58

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

Table 3 Correlation between the Rock Type for each case study and the types of curve of stress-strain behavior according Deere and Miller25, and Bell26.

Fig. 8. Dimensionless stress-strain diagrams for each case study applying Ylinen's method24 and the types of curve of stress-strain behavior according Deere and Miller25, and Bell26.

strain function.24 By introducing the required parameters for each case study into this expression, we can develop a stress-strain function for our cases. To obtain a general idea of the form of the stress-strain curves in an advantageous comparative form, Fig. 8 shows the dimensionless diagram of σ / UCS plotted against E ϵ/ UCS . We see that the more inelastic the rock is, the greater is the deviation of the stressstrain curve from the Hooke's line.

ε = 1/E [cσ − (1 − c ) UCS⋅ ln(1 − σ / UCS )]

equation for buckling stress σt as a continuous function of the slenderness ratio λ valid for the entire domain (elastic and inelastic):

σt = (π 2E + UCSλ2 )/2cλ2 −

(π 2E + UCSλ2 )2 − 4π 2EcUCSλ2 /2cλ2

(14)

In Fig. 9, the ratio σt / UCS is plotted against the dimensionless quantity λ / π σp / E . At great values of the slenderness ratio, when the buckling stress is low, the value of the tangent-modulus is constant and the buckling stress diagrams coincide with Euler's hyperbola. When the slenderness ratio decreases, the buckling stress increases, the value of the tangent-modulus decreases and the buckling stress diagrams deviate from Euler's hyperbola and approach the ultimate compressive

(13)

Finally, by introducing the tangent-modulus function (12) into Engesser's formula (8) and taking into consideration that σ = σt when inelastic buckling occurs, Ylinen's method provides the following 59

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

Fig. 9. Buckling stress diagrams for each case study in dimensionless form according to formula (14).

The determination of these ratios usually depends upon individual author criteria or upon base friction modeling.27 The ratios appear remarkably varied in the literature with values from 0.36 to 0.46,1 0.5,9 or even as high as 0.66.21 In contrast, the proposed method is based on the deterministic calculation of the length of buckle through a formula that uses the geometry of the slope and the strength of the potential failure surface. Although, in practice, a direct measurement of this length is rare due to the immediate collapse of the structure after failure, the Malvern Hills case reports an estimated buckle segment length of 12 m, which agrees with the length of 11 m calculated by Eq. (6) for this case. Additionally, the length of the buckle as calculated herein has proven valid to the extent that the critical height prediction for all cases was satisfactory. The mean relative error for the predicted critical height, Hcr, according to this method, is 2%, with a standard deviation of 7%. This can be contrasted with the solution applying the approach developed by Cavers 9, for which we obtained a relative error for the predicted critical height, Hc, of 211%, with a standard deviation of 249%. Although Cavers’ approach fails in the overall prediction, it seems to approach better to the results when the case is closer to the elastic mode. This is reasonable because this approach is based on Euler's buckling theory. Analysis of the results also highlights the relevance of the passive segment slenderness ratio (ℓ/r) and, to select the proper stability criteria for each case, the necessity of establishing if the mode of failure is brittle, elastic buckling or inelastic buckling. When stability is controlled by elastic buckling, the slope follows the Eulerian behavior which is only based on the geometry of the rock column and the modulus of elasticity of the material. On the contrary, when brittle failure controls, the slope stability appears to depend exclusively on the rock mass uniaxial compressive strength of the material forming the slab. In between, under inelastic buckling, the slope stability obeys to a combination between material strength and buckling. The review of the cases shows that the buckle geometry sometimes could appear in the

strength of the material, UCS. In terms of the mode of failure, when λ approaches zero, stress approaches UCS and failure is brittle. When λ satisfies the limit stablished by expression (11), the failure is elastic. In between, buckling occurs after the stress in the column exceeds the proportional limit of the column material and before the stress reaches the ultimate strength. This last situation is called inelastic buckling. 5.3. Results Table 4 summarizes the results; in Fig. 9, the dots labeled with a number indicate the exact point in which the failure in each case occurred. It is important to notice that cases 7 and 8 fall in the zone of pure elastic behavior, case 9 falls in the zone of inelastic behavior close to the elastic zone, and cases 1, 2, 3, 4, 5 and 6 behave brittlely since they plot in the top left area of the stress diagrams, where failure is linked to the ultimate compressive strength of the column material and is not related to its geometry. This brittle behavior appears when the stress equals or is very near to the UCS; although there is no defined limit for this extreme, a sensitivity analysis shows that a ratio of σt / UCS ≈ 0.98 overestimates the critical height by approximately 20%. Table 4 also shows that the actual failure height, Hf, for each case is satisfactorily predicted compared with the predicted critical height Hcr. The critical slenderness ratio λcr =π E / σp is compared with the slenderness ratio for each case to define the elastic behavior, as explained in Section 3. Finally, the critical stress or stress at failure, σf, is presented to highlight that for the elastic or inelastic behaviors the stress is always lower than UCS. 6. Discussion Other methods use empirical ratios of the length of the buckle to the total length of the slope (ℓ/ Ls ) to determine the length of the buckle.

Table 4 Results for the length of the passive segment, ℓ ; critical and at-failure slenderness ratios λcr andλ; compressive strength and at-failure stress UCS and σf ;and critical height predicted by this method Hcr contrasted with the actual height of failure Hf and the critical height predicted by Caver's approach Hc.9 #

Case

Rock type

σf (kPa)

UCS (kPa)

ℓ (m)

λ

λcr

Mode of failure

Hf (m)

Hcr (m)

Hc (m)

1 2 3 4 5 6 7 8 9

Westfield coal mine Malvern Hills coal mine Quintette coal mine Bawang Mountain landslide Lavini di Marco

Mudstone Mudstone/siltstone Sandstone Limestone Limestone Sandstone/siltstone

612 197 1076 9959 1651 796 1342

25 11 20 152 29 29 109 172 223

24 19 46 53 38 18 754 595 515

596 247 587 235 446 525 525

Brittle

Grand-Baume coal mine

612 197 1076 9958 1651 796 516 818 1061

92 15 90 940 170 95 115 170 240

84 17 92 1005 171 88 117 184 239

403 68 225 1119 582 817 127 202 265

60

Elastic Inelastic

International Journal of Rock Mechanics & Mining Sciences 90 (2016) 53–61

S.E. Rosales Garzon

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brittle mode, for example in the cases 2, 4, 5 and 6. In these cases, the buckle form observed on field after collapse could become a confounder when identifying the specific mode of failure. The most plausible explanation is a progressive failure related to the non-uniform stress distribution in the slab; failure initiates at the most crucial point close to the bottom of the slab and propagates gradually to ultimate failure. In the process, the rock mass at the bottom of the slab crushes and distorts relatively faster, resulting in the heaving (in the form of buckle) of the relatively less affected rock mass at the top of the slab. Additionally, the results also reveal that buckling can be explained only by gravity without invoking other extreme factors like significant water pressures, which usually are difficult to substantiate for near surface fractured rock masses. 7. Conclusion A formula to calculate the passive segment length was derived by treating this passive segment as a beam-column stability problem and guaranteeing equilibrium in elastic range. Then, by considering the known extension of the passive segment and the tangent-modulus theory, the article provides a conceptual understanding of the buckling stability of stratified slopes, yielding a proper and workable tool for evaluating whether the mode of failure is brittle, elastic buckling or inelastic buckling. If the failure mode is brittle, the critical load depends exclusively on the uniaxial compressive strength of the rock mass. If the failure mode is through elastic buckling, the critical load can be calculated applying Euler's theory. Finally, if the failure mode is through inelastic buckling, the critical load can be assessed using the tangent-modulus according to Shanley's theory. The stability analysis on nine reported cases using the suggested method shows good correlation between the predicted critical height and the actual height of failure in each case. Conflict of interest I wish to confirm that there are no known conflicts of interest associated with this publication and there has been no financial support for this work that could have influenced its outcome. References 1 Seale J. An Engineering Geological Investigation of Footwall Toe-Buckle Instability at the Malvern Hills Opencast Coal Mine (M.Sc. thesis), Christchurch, New Zealand: University of Canterbury; 2006. 2 Wang B, Cavers D, Wong B. Surface buckling failure study and support design at the Quintette coal mine, Canada. In: Taylor & Francis Group, editor. Landslides: Evaluation and Stabilization/Glissement de Terrain: Evaluation et Stabilisation, Set of 2 Volumes. London: CRC Press; 2004:475–480.

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