Geometry and Maximum Width of a Stable Slope Considering

Journal of Earth Science, Vol. 31, No. 6, p. 1087–1096, December 2020 ISSN 1674-487X Printed in China https://doi.org/10.1007/s12583-020-1052-0

Fang, K., Tang, H. M., Su, X. X., et al., 2020. Geometry and Maximum Width of a Stable Slope Considering the Arching Effect. Journal of Earth Science, 31(6): 1087–1096. https://doi.org/10.1007/s12583-020-1052-0. http://en.earth-science.net

Geometry and Maximum Width of a Stable Slope Considering the Arching Effect

Kun Fang 1, Huiming Tang *1, Xuexue Su1, Wentao Shang2, Shenglong Jia3

1. Faculty of Engineering, China University of Geosciences, Wuhan 430074, China 2. School of Civil Engineering, Shandong Jianzhu University, Jinan 250101, China

3. Faculty of Engineering, University of Alberta, Edmonton ABT6G2R3, Canada Kun Fang: https://orcid.org/0000-0002-8866-4076; Huiming Tang: https://orcid.org/0000-0002-0385-8155

ABSTRACT: The stability of an arching slope in deformable materials above strong rocks strongly de-pends on the shape and width of the span. Equations for a free surface problem that incorporate these two parameters were derived using a simplified two-dimensional arching slope model, and were vali-dated using physical model tests under 1g and centrifugal conditions. The results are used to estimate the maximum excavation width for a weak claystone slope in a lignite mine, for which we calculate a safety factor of 1.31. KEY WORDS: slope failure, geological engineering, cracked soil slope, large-scale model test, deforma-tion mechanism, arching slope.

0 INTRODUCTION

The arching effect, defined as the stress transformation be-tween deformable soils or rocks and adjacent non-deformable sections (Terzaghi, 1943, 1936), is important to the stability of both natural areas and excavated slopes. Based on the similar yielding part and unyielding part (Terzaghi, 1943), these slopes can be defined as arching slopes. Arching slopes have the fol-lowing characteristics: (1) there is non-uniform or relative dis-placement between the yielding and unyielding parts of slope; (2) the unyielding part is supported; (3) the locally yielding part is due to the loss of support in the slope; (4) the slope lies on a bed-ding plane with a lower friction. The characteristics (1)–(3) of arching slope are similar to the condition of a trapdoor. For the third aspect, the loss of support in the slope can be produced by some cases, such as mining excavation in open-pit mining, foun-dation excavation in house-building, road construction and so on. For the fourth characteristics, if the base resistance in the slope is much larger than the lateral resistance resulted from arching ef-fect, the analysis of arch action can be ignored. Thus, the low interface friction in the slope is considered. A schematic figure of the arching slope is shown in Fig. 1. A better understanding of arching slopes is vitally required in the work and studies of engi-neers and scholars.

With regard to arching effect in slopes, past studies have mainly focused on the soil arching among slope stabilization piles (Li et al., 2019b; Chen and Martin, 2002; Ito et al., 1981; Wang and Yen, 1974). Specifically, Ito et al. (1981) derived the *Corresponding author: [email protected] © China University of Geosciences (Wuhan) and Springer-Verlag GmbH Germany, Part of Springer Nature 2020 Manuscript received December 10, 2019. Manuscript accepted July 26, 2020.

equations of lateral stress located on a row of piles and consid-ered six important factors including row position, pile diameter, pile thickness, slope angle, and pile head in stabilizing pile design. Bosscher and Gray (1986) examined the effect of arch action in pile retaining walls and provided some information about conservative pile design. Chen and Martin (2002) applied a fast Lagrangian analysis of continua method to investigate the arching area within pile groups. However, the study of the arching effect on a whole slope is limited.

A few studies have investigated the arch affecting inside slopes instead of pile-slope interactions. Pipatpongsa et al. (2009) reported that an unstable rock undercut slope had a potential failure after the lignite was removed at the toe of the slope in mining excavation. Zhang (2015) presented an engineering prob-lem of a slope in house-building. The slope was close to the con-structed building during the process of foundation excavation. In addition, similar arch-shaped failures can be found in road con-struction near reservoirs (Li et al., 2019a; Tang et al., 2019; Yao et al., 2019; Wang et al., 2014; Huang et al., 2013).

The engineering challenges of unstable slopes in Mae Moh mine, as typical arching slopes, are of concern. In the northeast pit of the Mae Moh lignite mine (Pipatpongsa et al., 2009; EGAT, 1985), a shearing zone is located within the overburden layers, which causes a few unstable areas. A cross-section of a rock slope in area 4.1, one of the unstable areas in Mae Moh mine is shown in Fig. 2. A part of lignite (300–500 m in length, 80–120 m in height) located at seams Q and K supports the upper rock mass. The rock mass lies on bedding planes filled with clay seams G1 and G2 (Pipatpongsa et al., 2016). The engineering challenges in area 4.1 appear after the excavation of the lignite. The upper rock mass may slide along the bedding plane, which results in loss of mineral resources, schedule de-lays, and casualties. Considering arching effect in the slope, after the part of the lignite is excavated, the lateral load transfer

Kun Fang, Huiming Tang, Xuexue Su, Wentao Shang and Shenglong Jia

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Figure 1. Schematic illustration of an arching slope.

Figure 2. A cross section figure of the unstable grey claystone slope in area 4.1 (modified from Pipatpongsa et al., 2009).

mechanism can protect the slope in some ranges. As a result, the maximum excavation width or the spacing between the unexcavated areas are crucial indexes in the construction, par-ticularly in some large open-pit slope projects. Moreover, it is essential to design the slope protection method and to monitor the potential slope failure and the failure shape of the arching slope under the maximum excavation width. Hence, it is nec-essary to investigate the failure shape and critical width of a stable arch in an arching slope.

Aiming at solving engineering problems in Mae Moh mine, Khosravi et al. (2011) presented a simplified physical model of a slope. The undercut slope consisted of a slope part and a base part that was cut. The stable arch was observed in the 1g physi-cal model until the base part was cut to the maximum width. In the centrifugal-scale model, experimental evidence regarding arch failure was observable in 50g slope models (Khosravi et al., 2016). Therefore, the physical model tests of arching slope in 1g and centrifugal scale were suitable for achieving the oc-currence of the stable arch and validating the analytical results.

Several studies have been done on the stable arch forming a free surface. The parabolic shape could be found in the condition where a chain of spherical particles is applied (Sakaguchi et al., 1993). McCue and Hill (2005) provided a continuum mechanics derivation about a surface problem in Mohr-Coulomb criteria and formulated the shape of the stable arch in a hopper. Guo and Zhou (2013) further extended the theory in trapdoor test with two-dimensional analysis. However, previous research has paid little attention to the relevant three-dimensional arching slope with such free surface problems.

In this study, the shape of the stable arch and maximum

excavation width in an arching slope are investigated based on free surface analysis. The stress distribution in a simplified two-dimensional arching slope model is also examined. De-pending on the solid boundaries and resulting acting force, theoretical formulations have been developed for estimating the failure arching shape and the critical width in arching slope. Two physical model tests in 1g and centrifugal conditions are used for the validation of the proposed equations. Moreover, the esti-mated maximum excavation width is applied for stability analysis of a grey claystone slope at the site.

1 THEORETICAL ANALYSIS

The solution of the shape of a stable arch in two dimensions based on the Mohr-Coulomb criterion was reviewed. More de-tails about hoppers and trapdoors were presented in McCue and Hill (2005) and Guo and Zhou (2013). The maximum excavation width in the arching slope was deduced with the consideration of the base friction. The analysis of the slope was related to a three-dimensional problem. The three-dimensional arching slope was simplified in a two-dimensional problem by considering an acceleration on a friction slope along the slope direction instead of the acceleration of gravity in the trapdoor test along the vertical direction. 1.1 Shape of a Stable Arch 1.1.1 Governing equation

The geo-mechanics sign convention is applied throughout the text. The values of compression and tension in the conven-tion are defined by positive and negative, respectively. To solve the symmetry of the excavation problem in an arching slope,


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the rectangular coordinate system is first adopted. In the coor-dinate system, compressive stress components σxx and σzz as well as shear stress component σxz are expressed as (Davis and Selvadurai, 2005; McCue and Hill, 2005; Hill, 1954)

0xx xzσ σ

x z

(1)

xz zzσ σρa

x z

(2)

where σij denotes the components of the Cauchy stress tensor in the coordinate system, ρ denotes the bulk density of the geo-material, and a is the acceleration acting along the slope direc-tion (the details on the resultant acceleration and forces are explained in the section on the maximum excavation width).

In the Mohr circle (Fig. 3), the mean stress p and the devia-toric stress q defined as the centre and radius in the circle, respec-tively, are expressed with major σ1 and minor σ3 principal stresses

1 3( ) 2p (3)

1 3( ) 2q (4)

By introducing the angle of the minor principal axis ϕ, the stress components are expressed as follows, which are also shown in the Mohr circle.

cos2xx p q (5)

cos2zz p q (6)

sin 2xz q (7)

A relation between the shear stress and the normal stress with the angle of the minor principal axis is obtained by re-moving the stress invariants p and q.

tan 2 2 / ( )xz xx zz (8)

The Mohr-Coulomb yield criterion as a constitutive law is applied (Coulomb, 1781, 1776). In the geomaterial, the stress components must satisfy Coulomb’s law within the form of an inequality condition.

2sin 1 sinq p c (9)

τ

Mohr circle

p q

σ3=0 Mohr-Coulomb

envelope

ϕ

ϕ

c

σ3 σ

( , )σ σzz xz

σ1

( , )σ σxx xz

2ψ

Figure 3. State of stress confined within Coulomb’s failure criterion in

Mohr’s stress circle.

where ϕ denotes the angle of internal friction of geomaterial and c denotes the cohesion of geomaterial.

The inequality indicates that the stress field is at the limit or under the yielding level. When the material is at the yield con-dition, Eq. (9) can be expressed as

sin = cosq p c (10)

By combining Eqs. (5)–(7) and Eq. (10), the equation of equilibrium in Eq. (1) is changed with regard to stress invariant q and the angle of minor principal stress ϕ.

2

2 sin 2

sin2 (sin cos2 )+

cossin sin 2

qx

qq

x z

(11)

2

2 (sin cos2 )

sin2 sin 2 +1

cossin cos2

qx

qq

z z

(12)

1.1.2 Free surface conditions in the stable arch

The unit normal of an element surface is defined as n=nxi+nzj. The traction vector on the surface is t(n)=nxi+nzj (McCue and Hill, 2005). Therefore, the states on a free surface are shown as follows.

=0xx x xz zn n (13)

=0xz x zz zn n (14)

We rewrite Eqs. (13)–(14) in the equivalent form on a free surface.

x xz zz

z xx xz

n

n

(15)

2xz xx zz (16)

Combining Eqs. (5)–(7), we have the following equations on every point of the free surface.

q p (17)

z

dcot

dxz n

x n (18)

Since t(n)=0 in a stress-free condition, the minor principle stress must be zero from Eqs. (13)–(14). In addition, due to the obedience of the Mohr-Coulomb yield criterion for geomaterial from Eq. (10) and Eq. (17), the terms are expressed in the gen-eral case where the friction angle of geomaterial is ϕ and the cohesion is c. This is also shown in the Mohr circle.

πtan

4 2q p c

(19)

1.1.3 Intersection between yielding and unyielding part A free surface OB and a solid boundary OA intersect at


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Figure 4. Geometry and boundary conditions in an arching slope. (a) The selection of the direction of the axis in the arching slope; (b) an intersection between the

free surface and the boundary in a half-width symmetric side.

point O, as shown in Fig. 4b. In geotechnical engineering, the solid boundary is considered a rigid wall used for supporting soil laterally or a stagnant edge of the geomaterial itself. Thus, the friction angle at the boundary depends on the interface friction angle ϕw or internal friction angle ϕ. Due to the geometry and boundary condition of the arch, the states of stress (σrr, σθθ, σrθ) expressed in polar coordinates are presented. These components of stress also obey static equilibrium conditions in the coordinate direction with a downward-acting force along the slope direction. With the assumption that stress at point O is not a singularity of stress, the equilibrium equation in the polar system is expressed (McCue and Hill, 2005; Hill, 1954). Here, θ is measured anti-clockwise from the x-axis.

d0

dr

rr

(20)

d2 0

d r

(21)

Which obeyed

0 (1 cos 2 )rr q (22)

0 sin 2r q (23)

0 (1 cos2 )q (24)

where q0 denotes the stress invariant q at point O. Alternatively,

sin 2 πtan

1 cos2 2r

(25)

As mentioned above, the solid boundary with regard to the arching slope is unyielding geo-material. To investigate the geometry of the stable arch, the boundary between the yielding part and the unyielding part needs to be clarified. Based on the Mohr-Coulomb criterion, the angle of solid boundary OA can be expressed as

0

π=

4 2

(26)

The angle α measured from horizontal to the free surface is

0

π=π =

4 2

(27)

where ω denotes the wedge angle between the free surface and solid boundary in Fig. 4. The value is π/2–ϕ.

1.1.4 Geometrical shape of a stable arch The geometrical shape of the free-surface can be solved

according to the derivation of the governing equations, free surface conditions and boundary conditions. In the arching slope, the maximum excavation width or critical excavation width Bmax is defined. When the excavation width B is less than Bmax, the slope is considered safe. Note that the complicated conditions at sites such as stress concentration are ignored.

Given the critical condition B=Bmax that geomaterial near the free surface yields, the solution of the shape of a stable arch is solved by Eqs. (11)–(12) with the summarised boundary conditions (McCue and Hill, 2005). The polar coordinate sys-tem is applied as shown in Fig. 5.

Sokolovskii (1965) proposed stress invariant to be a linear function of radius r in the polar system. Stress states change linearly with depth. The problems regarding the arching effect on the equilibrium in the trapdoor test are also investigated (McCue and Hill, 2005; Jenike, 1964). Therefore, the form of stress in the arching slope is also compatible with an angle of minor principal stress in the polar system.

( )q ra (28)

= ( ) (29)

where q denotes the stress invariant, θ denotes the angle meas-ured from the x-axis in Fig. 5, χ denotes a nondimensional scaled

Figure 5. Schematic figure of a stable arch in an arching slope.


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stress variable that depends only on the angle θ (Sokolovskii, 1965), and ϕ denotes the angle of minor principle stress which is a function of the angle θ alone.

The differential equations are expressed by combining Eqs. (28)–(29)

d sin 2( ) sin cos(2 )

d sin cos2( )

(30)

2 2d cos sin sin 2( ) sin sin

d 2 sin cos2( ) sin

(31)

In the limiting equation, the friction angle of the geomaterial is constant throughout the arching slope. By adding the boundary conditions, the two differential equations can be solved. It is clear that shear stress is zero at the centre in the symmetry condition for the one boundary condition as shown in Fig. 5. Another boundary condition is the solid boundary between the yielding and un-yielding geo-material in the static equilibrium condition. Here, subscripts c and o indicate the location along the central plane and solid boundary plane, respectively.

π π, where

2 2cc c

(32)

π π, where

2 4 2oo c

(33)

The solutions of the stress system and angle of minor principle stress are obtained from Eqs. (28)–(31). Therefore, according to the information about the free boundary description in Eq. (18), the numerical solution of the shape of the stable arch is determined. 1.2 Maximum Excavation Width in the Arching Slope

The maximum excavation width in the arching slope can be calculated by considering the solved stress variable from the last section. The problem in the arching slope is a three-dimensional question due to base friction between the bedding plane and sliding slope. A simplified two-dimensional arching slope model is proposed by considering a resultant acceleration on a friction slope. The diagram of an arching slope lying on a bedding plane is shown schematically in Fig. 6. The acceleration of the slope on a bedding plane is obtained from the resultant forces and mass. The forces the along slope direction are the driving force Fd and resisting force at the bedding plane fb.

W BTL g (34)

Figure 6. The acting stress along the slope direction in an arching slope.

cos tanbf W c A (35)

sindF W (36)

Therefore, the acceleration of the arching slope on bedding plane a is expressed.

sin( )

cosd bF f c

a gM gT

(37)

where W and M denote the weight and mass of the slopes, re-spectively, A denotes the base area between the slope and bed-ding plane (A=LB), β denotes the slope angle, and ϕ′ and c′ denote the interface friction angle and apparent adhesion be-tween the slope and bedding plane, respectively.

The maximum excavation width in the slope is solved with the boundary conditions. Taking the boundary condition q0=r0ρaχ(θ0) in Eqs. (30)–(31), the maximum excavation width is shown as follows

0 0max 0 0

0

2 cos2 cos

( )

qB r

a

(38)

By adding q0 obtained by Eq. (19) and θ0 calculated by Eq. (26) to Eq. (38), it becomes

max0

π2 sin

4 2( )

cB

a

(39)

With the estimation of 1/χ(θ0)=2.57(tanh 3ϕ)3.5 in a friction angle range of 20º to 80º (Guo and Zhou, 2013) and Eq. (37), the maximum excavation width in the arching slope is expressed as follows. Note that this maximum excavation width only applies in cohesive geomaterials.

3.5

max

5.14(tan 3 ) πsin

4 2sin( )

cos

h cB

c

T

(40)

2 VALIDATION OF THE PROPOSED EQUATIONS BASED ON PHYSICAL MODEL IN 1G AND CEN-TRIFUGAL CONDITIONS

In this part, a 1g physical model and a centrifugal model of arching slopes were used to check the geometrical shape of stable arches and the maximum excavation width. 2.1 1g Physical Model Test

A 1g physical model of the arching slope containing a slope part and a base part is shown in Fig. 7a (Fang, 2019). To present a complete arching failure, the length of the slope part (80 cm) and the base part (40 cm) are selected according to a set of pre-liminary tests. The width and thickness of both parts are 130 and 6 cm, respectively. The sand slope model with a 38º slope angle lying on a Teflon sheet is constructed by using compacted silica sand No. 6. Silica sand No. 6 is a poorly graded sand (Khosravi et al., 2013). The benefits of using silica sand No. 6 include the unified standards from factories, recycling in slope model prepa-ration, and the suitable color of the sand for PIV (particle image velocimetry) analysis. Therefore, silica sand No. 6 is chosen in


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Figure 7. Physical model of the 1g arching slope. (a) Physical model of the arching slope in the 1g condition; and (b) failure of the slope at the maximum exca-

vation width in the 1g test (after Fang, 2019).

the 1g model test. The unit weight of silica sand No. 6 with 10% water content is 13.68 kN/m3. The friction angle and the apparent cohesion from the direct shear test are 41.5º and 8.0 kN/m2, re-spectively (Khosravi et al., 2011). The high apparent cohesion results from the suction in unsaturated soil (Ohta et al., 2010). Typically, the range of the stress level for measuring the geoma-terial strength properties and the scope of stress change of this material in the physical model tests need to be set closely. In the conducted direct shear test, the range of stress level is from 0 to 100 kPa. However, in the 1g arching slope physical test, stress changes are only from 0 kPa to a maximum of approximately 10 kPa. Therefore, the apparent cohesion in the 1g test from the direct shear test may not be suitable. In this study, the apparent cohesion from the unconfined compressive strength and friction angle σc=2c(1+sinϕ)/cosϕ are estimated. The unconfined com-pressive strength measured in 1g condition is 2.1 kN/m2 (Fang, 2019). The calculated apparent cohesion of 0.47 kPa is used to maintain close stress levels. Moreover, the interface friction angle and cohesion between silica sand No. 6 and the Teflon sheet are 18.5º and 0.1 kPa, respectively. The detailed parameters of silica sand No. 6 are shown in Table 1 and Fig. 8.

The instruments including earth pressure cells, a continuous- shooting camera for PIV analysis, and a high-speed camera cap-turing the velocity during failure are applied. The lens of the digital camera and the high-speed video camera are perpendicular to the slope part. The process of the 1g arching slope test generally includes preparation, excavation, and data recording. First, in construction, the main target is to compact the sand evenly and carefully in every layer and to prevent stress concentration or inhomogeneous stress inside the slope. The compaction of the slope part is prepared every 5 cm along the slope direction, as shown in Fig. 7a. Second, the physical excavation using a long knife starts from the centerline at the base part. After the center of the base is cut in 5 cm, the right remaining base is first cut in 2.5 cm, and then the left base is excavated in 2.5 cm to maintain the symmetrical excavation process. This process continues until slope failure occurs. To avoid vibration during manual cutting, a finely excavated operation is needed. Before the next cutting, it takes one min to wait for potential failures. Finally, all data is recorded in a timely manner by the high-speed camera.

The result of the arching slope is shown in Fig. 7b. The an-gles between the stress-free surface are 64º±2º on the right side

Table 1 Basic properties of the geomaterial in the tests (after Fang, 2019;

Fang et al., 2018)

Parameter Silica sand No. 6 Edosaki sand

Water content (w) 10% 10%

Specific gravity (Gs) 2.65 2.69

Bulk unit weight (γ) 13.7 kN/m3 15.3 kN/m3

Relative density (Dr)※ 10% 90.4%

Maximum void ratio (emax) 1.132 1.37

Minimum void ratio (emin) 0.711 0.85

Average particle diameter (D50) 0.33 mm 0.20 mm

Friction angle (ϕ) 41.5º 41.1º

Cohesion (c) 0.47 kPa 3.0 kPa

Interface friction angle (ϕ') 18.5º 17.0º

Interface adhesion (c') 0.1 kPa 0 kPa

※. maxr

max min

= .e e

De e

100

Edosaki sand

50

Silica sand No. 6

01010.10.010.001

Per

cent

finer

by w

eight

(%)

Grain size (mm)

Figure 8. Grain size distributions of silica sand No. 6 and Edosaki sand (after

Fang, 2019).

and 55º±2º on the left side, respectively. The right part is very close to the theoretical value of 65.8º. The slope fails at 67.5 cm when the excavation length of the right base is larger than the length at the left base of 2.5 cm. Therefore, unsymmetrical cut-ting may cause a small stress concentration on the left. The dif-ference in the left part results from the small stress concentration during the excavation process. The stable arch forms when the base part is excavated by 67.5 cm in the experiment. The calcu-lated maximum excavation width is 64.3 cm based on Eq. (40)


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30

20

25

10

15

dir

ecti

on (

cm)

z

P io nts tracked from 1 testg

0

5

0

Numerical calculation

10 20 30 40 50 60 70 80

x direction (cm)

Figure 9. Comparison of the shape of the stable arch in the 1g arching slope.

and the parameters in Table 1. The estimation error of the critical width obtained by the proposed equation is 4.7%, which means that the equation can reasonably estimate the critical width. The calculated value is less than the experimental result, which indicates that the calculated value is a conservative estimation. In practice, using the proposed equation produces a considerable undercut estimation on the safe side. Moreover, the selection of the apparent cohesion of the sand needs more attention.

To compare the shape of the stable arch with the theoretical analysis in Eq. (18), the points of the shape are tracked in the physical model, as shown in Fig. 9. The configuration from the numerical calculation generally fits with the stable arch in the 1g test. The two sides of the arch in both the model and the analysis are close. However, the larger differences in the curve are located at the top of the arch. In the physical model, the failure line at the top shows a nearly straight line, while a curved edge is obtained from the theoretical analysis. These differences are probably due to the preparation of the physical arching slope model. In the compaction of the slope part, the sand slope is compacted along the slope direction every 5 cm. Although the surface of every layer along the slope orientation is scratched in the compaction process, some discontinuities may remain between the two layers. This causes the straight line near the top area of the arch.

2.2 Centrifugal Model Test A centrifugal model test was also conducted and checked

by using a centrifugal machine (Fang et al., 2018). The detailed information of the device is described by Higo et al. (2015). On the centrifugal scale, the manual excavation process cannot be achieved due to the flight of the device during the test. Thus, the pre-excavation method is applied. Unlike the slope failure that occurred by cutting the base part in the 1g arching slope test, the collapse of the slope is due to the gradual increase in centrifugal force in the g-up process.

A cross section of the slope model lying on a Teflon sheet is shown in Fig. 10. A soil chamber consists of three rigid walls and a visible front side made of transparent glass. To enhance the arching effect in the arching slope, two pieces of sandpaper were used between the sand slope and soil chamber walls. The width and thickness of the slope are 20 and 5 cm, respectively. The lengths of the slope part and base part are set as 22.5 and 17.5 cm, respectively, based on the dimensions of the soil chamber and the hardwood bedding plane. The inclination of the slope is 30º according to the widely used slope angle in centrifugal slope model tests (Beddoe and Take, 2015; Take et al., 2004). The soil

used in the experiment was selected as Edosaki sand, a well-graded sand was recommended in the centrifugal model (Eab et al., 2014), with 10% water content to prevent water leakage and to maintain unchanged water content during the high centrifugal acceleration. The main parameters are as fol-lows: the unit weight is 15.3 kN/m3; the friction angle and ap-parent cohesion are 41.1º and 3.0 kPa, respectively; the interface friction angle and cohesion between Edosaki sand and Teflon sheet are 17.0º and 0 kPa, respectively. The basic properties of Edosaki sand are shown in Table 1.

The main procedures and detailed preparation for the slope model are as follows.

(1) Dry Edosaki sand and water with a 10 : 1 mass ratio are mixed and compacted every 1 cm for all base parts with support plates. Due to the location of arching failure in the slope part, the compaction process needs more care. The total length of the slope part is 22.5 cm. The length is first divided into four sec-tions including the lower three parts (5 cm) and the last part (7.5 cm). In every part mentioned above, the slope is perpendicularly compacted every 1 cm layer five times with supports. A total of 20 compactions are conducted for the slope part. Noting that scratching the surface of each soil layer after compaction is needed to avoid discontinuity between two layers.

(2) After removing the support plates, the base part is cut as a pre-excavation simulation. Since the vertical cut may cause the collapse at the edge of the base part in the centrifugal model, inclined cutting is used to make an average of 11 cm at the base part (top cutting Bt is 12 cm, and bottom cutting Bb is 10 cm, as shown in Fig. 11a).

(3) In the last procedure, the soil chamber with the prepared slope is assembled in the centrifugal machine. Centrifugal accel-eration acting on the slope starts to increase until arching failure occurs. The acceleration at the slope failure is recorded. The in-strumentation for stress analysis can be found in Fang et al. (2018).

The result of the arching slope is shown in Fig. 11b. The stable arch forms when the acceleration increases to 28.5g in the test. In the theoretical analysis, Eq. (40) is modified to consider the centrifuge acceleration as follows.

3.5

max

5.14(tan 3 ) πsin

4 2sin( )

cos

h cB

cng

ngT

(41)

where ng denotes centrifugal acceleration (n is the multiple of acceleration of gravity).

The calculated maximum excavation width is 11.9 cm, which is based on Eq. (41) and the parameters in Table 1. The

Figure 10. Schematic figure of the centrifugal arching slope model (modi-

fied from Fang et al., 2018).


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Figure 11. Centrifugal model of the arching slope. (a) Outline of the model test of the arching slope after preparation; and (b) failure of the slope in the centrifugal

test (modified from Fang et al., 2018).

4.0

4.5

3.0

3.5

zd

irec

tio

n(c

m)

1.5

2.0

2.5

Points tracked from centrifugal test

0.5

1.0

Numerical calculation (11 cm)

Numerical calculation (10 cm)

0.0210186420

x direction (cm)

Figure 12. Comparison of the shape of the stable arch in the centrifugal

arching slope.

estimation error between the estimated critical width and aver-age excavation width (11 cm) is 8.1%. However, under centri-fuge conditions, the estimated value is larger than the experi-mental results. This is because the inclined cutting process is applied in centrifugal model. The upper layer of the base part with 12 cm causes earlier failure.

The points of the shape from the physical model are also depicted in Fig. 12. To consider the method of cutting, the shapes of a stable arch at 11 and 10 cm are both added for comparison. The physical model points fit inside two predicted lines. The discontinuities occurring in the 1g test are less obvious on the centrifugal scale due to the better compaction process.

In summary, the theoretical analysis of both the maximum

excavation width and the geometry of the stable arch is in ac-ceptable agreement with the experimental results of the 1g and centrifugal arching slope model.

3 APPLICATION OF THE PROPOSED EQUATION IN THE STABILITY ANALYSIS OF AN UNDERCUT SLOPE AT THE SITE

An undercut slope in area 4.1 of the Mae Moe mine is chosen for a case investigation. A diagrammatic view of the slope with the unstable rock mass and the bedding plane is also shown in Fig. 13. Similar to the simplified arching slope physical model, the slope in area 4.1 can also be considered as the slope part (potentially unstable rock mass) and the base part (the lignite that will be excavated or is already partly excavated). The unstable slope, which has an inclined angle β=18º and a thickness of T=33 m, is composed of weathered grey claystone (EGAT, 1985). The clay seam G1 between the grey claystone in the bedding plane has an angle of shearing resistance ϕ′=12º. According to some investiga-tions at the site (Pipatpongsa et al., 2016; EGAT, 1985), the dry unit weight and saturated unit weight of the grey claystone are γd=16.47 kN/m3 and γs=20.13 kN/m3, respectively. The uncon-fined compressive strength of the grey claystone measured from a hydraulic jack is σu=0.75 MPa. The peak strength value (ϕ=45º and c=0.932 MPa) and residual shearing value (ϕ=43º and c=0.071 MPa) have been reported (EGAT, 1985). It can be found that the cohesion values in different tests produce considerable differences. Considering the similar value of the friction angle, the cohesion from the unconfined compressive strength and friction angle are also used. The calculated cohesion of the grey claystone


1095

Figure 13. Sketch of a rock slope in area 4.1 with a potential sliding plane.

is c=0.159 MPa. The detailed parameters of the undercut slope in area 4.1 are shown in Pipatpongsa et al. (2009). In the report, the width B=130 m of lignite was excavated. According to all the information above, the maximum excavation width in area 4.1 and the factor of safety based on the proposed equations are examined.

The maximum excavation width of lignite in the dry condi-tion is assessed using Eq. (40). The critical width is 399.0 m, and the safety factor of the slope is 3.1 by the following equation.

maxBF

B (42)

The safety factor is high in dry conditions, which means that the slope is stable. However, the most dangerous condition is the rainy season. The fully saturated condition with seepage due to heavy rainfall may cause the collapse of the slope. Therefore, slope stability analysis with regard to rainy conditions is necessary in area 4.1. By considering the extreme condition of the fully saturated slope part with seepage (Das and Sivakugan, 2016), the equation of maximum excavation width is shown here.

3.5

max

5.14(tan 3 )

(sin (1 / ) tan cos )

πsin

4 2

sat w sat

h cB

(43)

The critical width is 169.9 m and the safety factor of the slope in the fully saturated condition is 1.31 by Eq. (43). In accordance with the analyzing results, the slope is in a safe condition and approaches failure. Thus, the excavation process needs more consideration. If further excavation is planned, excavation with a suitable protection method can be applied, such as constructing compacted wastelands to protect the ex-cavated part or applying a multiple excavation method. More-over, the predicted failure arch shape can be extended in the application of the effective slope monitoring area in landslide monitoring and the selection of the method of landslide treat-ment. Those topics related to the shape of the arch in the arching slope can be further investigated in practice. On the other hand, these equations can also be applied to slope problems in reser-voir areas (Li et al., 2019a; Tang et al., 2019) and slope analysis in seismic conditions (Ma and Xu, 2019; Tian et al., 2019). Generally, the proposed equations make it possible to obtain the maximum excavation width and to carry out stability analysis of the arching slope based on free surface analysis. 4 CONCLUSIONS

The aim of this study is to investigate the geometrical shape of a stable arch and the maximum excavation width in arching slopes. The three-dimensional question of the slope is simplified as a two-dimensional problem by considering a resultant accel-eration along the slope direction. Based on a stress-free surface

analysis in a hopper, the theoretical equations for the curve of the stable arch and critical excavation width in the arching slope are derived according to the continuum theory.

A 1g physical model and a centrifugal model were used to validate the proposed formulas. In the physical model tests, the stable arch forms due to the simulated manual excavation process in the 1g condition. Arch failure in the centrifugal arching slope with a pre-excavation process is achieved by increasing centrifugal acceleration. The analytical solutions have acceptable agreement with the experimental measurements in both 1g and centrifugal conditions.

Finally, a safety factor (1.31) of the slope in the fully saturated condition was calculated in the stability analysis of a grey claystone slope at the site, and alternative methods for further excavation in area 4.1 were suggested. Moreover, the shape of a stable arch can be applied in the monitoring and treatment of arching slopes. ACKNOWLEDGMENTS

This study was supported by the National Key Research and Development Program of China (No. 2017YFC1501305) and the National Major Scientific Instruments and Equipment Development Projects of China (No. 41827808). The first author would like to gratitude to Associate Prof.

Thirapong Pipatpongsa for his guidance. The authors sincerely thank Prof. Changdong Li for his valuable suggestions and great support in this research and Ms. Man Zheng for her careful paper modification, and the anonymous reviewers for their useful suggestions. The final publication is available at Springer via https://doi.org/10.1007/s12583-020-1052-0.

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Geometry and Maximum Width of a Stable Slope Considering