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Article

Theoretical and Numerical Analyses of Earth Pressure

Coefﬁcient along the Centerline of Vertical Openings

with Granular Fills

Pengyu Yang 1,2 , Li Li 2,* and Michel Aubertin 2

1School of Architecture and Civil Engineering, Xi’an University of Science and Technology, 58 Yanta Rd.,

Xi’an 710054, China; pengyu.yang@polymtl.ca

2Research Institute on Mines and Environment, Department of Civil, Geological and Mining Engineering,

École Polytechnique de Montréal, C.P. 6079, Succursale Centre-Ville, Montréal, QC H3C 3A7, Canada;

michel.aubertin@polymtl.ca

*Correspondence: li.li@polymtl.ca; Tel.: +1-514-340-4711 (ext. 2408)

Received: 18 July 2018; Accepted: 13 September 2018; Published: 22 September 2018

Abstract:

Granular ﬁlling materials are placed into conﬁning structures for various purposes,

including but not limited to silos, trenches, mine stopes, and retaining walls for backﬁll. Stresses in

these backﬁlled openings are commonly estimated using theoretical arching models, with equations

that often involve the earth pressure coefﬁcient K(=

σ

’

h

/

σ

’

v

). Such stress estimation can be

dramatically impacted by the magnitude of K, but its value remains debatable. Along the centerline

of vertical openings with granular cohesionless ﬁlls, the value of Kis sometimes obtained by Jaky’s

earth pressure coefﬁcient at rest K

0

, based on the assumption of ﬁxed conﬁning walls, whereas

Rankine’s active earth pressure coefﬁcient K

a

is regarded more suitable for Kas claimed by some

others. Recent numerical analyses from the authors have shown that the state of stress close to

the center of backﬁlled openings cannot be solely related to wall movement. It was also shown

that the Kvalue can vary between K

a

and K

0

in backﬁlled openings with ﬁxed (immobile) walls,

depending on the locations and respective values of ﬁll internal friction angle

ϕ

’and Poisson’s ratio

ν

.

However, none of the existing works have addressed the mechanisms and answered this fundamental

but critical question: which value of coefﬁcient K(K

0

,K

a

, or other) should be used with analytical

solutions to assess the stresses in backﬁlled openings (and why)? After assessing the state of the

ﬁll placed in a conﬁned opening, theoretical relationships and speciﬁc mechanisms are proposed,

for the ﬁrst time, to evaluate critical values of

ν

and

ϕ

’for deﬁning the at-rest and active states in

ﬁlls. The approach indicates that when

ν

or

ϕ

’are smaller than or equal to critical values, the value

of Knear the center line of a backﬁll opening should be close to K

a

; otherwise, Ktends to approach

K

0

deﬁned from

ν

. The theoretical analysis is complemented and validated (in part) by numerical

simulations. The results also demonstrate that Poisson’s ratio can play a major role on the stress

distribution within cohesionless ﬁlls, and should thus be accurately evaluated.

Keywords:

backﬁlled openings; stress state; earth pressure coefﬁcient; Poisson’s ratio; internal

friction angle; theoretical model; numerical modeling

1. Introduction

Granular ﬁlling materials are commonly placed inside conﬁning structures for various engineering

purposes; such applications include but are not limited to silos, trenches, mine stopes, and backﬁll

retaining walls [

1

–

11

]. Upon placement, the ﬁlls settle downward due to self-weight, whereas the stiff

conﬁning walls tend to keep ﬁlls by limiting vertical displacements. Shear stresses develop between

the softer ﬁlls and stiffer walls, inducing a transfer of stress to the latter. The resulting vertical stress

Appl. Sci. 2018,8, 1721; doi:10.3390/app8101721 www.mdpi.com/journal/applsci

Appl. Sci. 2018,8, 1721 2 of 11

reduction (compared to those due to overburden) in the backﬁll is commonly known as arching.

Theoretical methods for evaluating stresses in backﬁlled openings mostly stem from the arching theory

of Janssen [

1

] proposed to estimate stresses within silos. Marston [

3

] applied Janssen’s arching theory

in geotechnique for calculating loads exerted on buried conduits in backﬁlled trenches. Based on

Marston’s [

3

] theory, the vertical

σ

’

v

and horizontal (lateral)

σ

’

h

effective stresses within cohesionless

ﬁlls placed in openings can be estimated as follows:

σ0v=Bγ

2Ktan δ01−e−2hK tan δ0/B(1a)

σ0h=Kσ0v(1b)

where

γ

(kN/m

3

) represents the ﬁll’s unit weight; B(m) denotes the opening width; h(m) represents the

depth from the backﬁll top surface;

δ

’(

◦

) denotes the effective internal friction angle of the backﬁll–wall

interfaces. The actual value of

δ

’is usually regarded as equal to the effective internal friction angle of

the ﬁll

ϕ

’(

◦

) in the case of mine stopes, because the ﬁll yield tends to occur within ﬁlls as rock walls

can have very stiff and rough surfaces following production blasting [5,12].

In Equation (1), Krepresents the earth pressure coefﬁcient that is deﬁned by the ratio of

the horizontal over the vertical effective stresses (K=

σ

’

h

/

σ

’

v

). The Kvalue is not an intrinsic

material property but serves as a convenient parameter introduced in simpliﬁed analyses of the

2D (two-dimensional) stress state in soils and ﬁlls.

To estimate stresses in mineﬁll stopes, Marston’s [

3

] solution has been widely used (with or

without modiﬁcations) [

5

,

10

,

13

]. Applying this type of solution requires knowledge of the Kvalue,

which largely affects the estimated stress state in various backﬁlled openings (silo, trench, mine stope,

backﬁll retaining wall, etc.). However, the actual values of Kin such openings are still debatable.

When the backﬁll is deposited after a stope is entirely mined out in the case of isolated stopes,

the movement of the wall can be regarded as negligible. Accordingly, the value of Kin backﬁlled stopes

is sometimes associated with the earth pressure coefﬁcient at-rest K

0

[

14

–

20

]. However, some numerical

simulations and their comparisons with arching solutions have shown that Rankine’s active earth

pressure coefﬁcient K

a

is sometimes more appropriate, especially along the centerline [

12

,

13

,

21

–

24

].

Some of these numerical analyses have been supported (in part) by laboratory testing results [

16

,

24

,

25

].

Other numerical results, including those obtained by Caceres [

26

], Jahanbakhshzadeh et al. [

27

], and

Yang et al. [

28

,

29

], indicate that the Kvalue across backﬁlled stopes tends to vary, typically between K

a

and Jaky’s K

0

, depending on the location and values of

ϕ

’and

ν

. However, none of these investigations

addressed the mechanisms controlling the backﬁll state in openings with ﬁxed conﬁning walls. There

is thus a need to establish more speciﬁcally why and how the Kvalue varies with the ﬁll properties,

speciﬁcally

ϕ

’and

ν

, and deﬁne which coefﬁcient (K

0

,K

a

, or another value) should be used with

analytical solutions to estimate the stresses in backﬁlled openings. This work aims to clarify this

aspect (at least in part) based on theoretical and numerical analyses of stresses along the centerline of

openings ﬁlled with granular cohesionless materials.

In this paper, the origins and deﬁnitions of the at-rest (K

0

) and active (K

a

) earth pressure

coefﬁcients are ﬁrst brieﬂy revisited, emphasizing the distinctions between the natural soil state,

the states behind retaining walls and in backﬁlled openings. The state of stress along the vertical

centerline (VCL) of backﬁlled openings is then analyzed by considering its elasto-plastic behavior.

This leads to an explicit theoretical relationship and corresponding mechanisms which can be used to

identify the state that may prevail within ﬁlls. The results of this analysis are compared and partly

validated using simulations conducted with FLAC (Fast Lagrangian Analysis of Continua) [30].

Appl. Sci. 2018,8, 1721 3 of 11

2. Theoretical Analyses

2.1. State of Stress in Natural Soils and behind Retaining Walls

In soil mechanics, the state of stress in the soil mass (or ﬁll material) is usually associated with

the lateral translations of retaining walls and related to the vertical stress. Three distinct states

and corresponding earth pressure coefﬁcients are typically applied in geotechnical engineering for

cohesionless soils (ﬁlls) to calculate the horizontal stresses [31–33].

The situation is typically described using the conceptual model of a stiff and very thin wall

(to minimize disturbance) which is theoretically introduced into an initially at-rest (natural) soil,

and then removing the soil on one side of the wall and applying a horizontal force (pressure) to

keep the vertical smooth wall in place. When the retaining wall is pushed (sufﬁciently) against the

compressed soil, it may yield and fully mobilize its frictional strength to resist the horizontal movement.

This corresponds to a passive state; such condition is not frequently encountered in backﬁlled openings.

When the wall is allowed to move outward, the horizontal stress on this wall is reduced by shear

yielding of the soil; the corresponding plastic equilibrium leads to an active state. For cohesionless ﬁlls

having ﬂat horizontal top surface, the coefﬁcient Kcan then be represented by Rankine’s active earth

pressure coefﬁcient Ka[7,31–33]:

Ka=1−sin ϕ0

1+sin ϕ0(2)

Another (theoretical) situation arises when the retaining wall is ﬁxed (immobile), without

horizontal strain in the soil (fill). The stress state then remains unchanged from that of the natural soil deposit.

This corresponds to the at-rest state, with K=K

0

. For loose granular soils, the following semi-empirical

relationship is widely used to express the corresponding earth pressure coefﬁcient [34,35]:

(K0)ϕ0=1−sin ϕ0(3)

Alternatively, the value of K

0

can be estimated, in a more fundamental manner, from Poisson’s

ratio νusing Hooke’s law for a linear-elastic, isotropic, and homogeneous material [36,37]:

(K0)ν=ν

1−ν(4)

2.2. Fill State in Backﬁlled Openings

In the precedent theoretical (idealized) cases, the soil is initially at rest, and would remain at-rest

if the retaining wall is immobile. The motion (displacement) of the wall then controls the stress state in

the soil. The situation is different in backﬁlled openings (trenches, silos, mine stopes, and behind ﬁll

retaining walls), in which the ﬁll is placed into a preexisting rigid conﬁning structure.

Numerical analyses performed by Sobhi et al. [

24

] showed that ﬁlls near the center of vertical

openings can reach an active state because of some local yielding, even if the walls displacement upon

ﬁlling is negligible. Simulation results reported by Yang et al. [

28

,

29

] revealed that the stresses

along the VCL of similar openings can be close to an active or at-rest state, depending on the

relationships (independent or related) between

ϕ0

and

ν

.Near the opening walls, the numerical

results of Yang et al. [

28

] showed that the ratios of the minor over major principal effective stresses

approach Rankine’s active earth pressure coefﬁcient K

a

, even though the conﬁning walls remain

unmoved. Thus, the stress state in such cases cannot be solely linked to the movement of the conﬁning

wall(s). However, the mechanism behind this behavior has not been explicitly addressed.

In the following, the state of ﬁlls placed in a vertical opening is analyzed based on an elasto-plastic

model with varying values of

ν

and

ϕ

’. It will be seen that the backﬁll can approach active or at-rest

state along the VCL of a backﬁlled opening, depending on the relations between the respective values

of ϕ’and ν(even for ﬁxed conﬁning walls).

Appl. Sci. 2018,8, 1721 4 of 11

When obeying the Mohr–Coulomb elastoplastic criterion, the ﬁll exhibits a linear elastic behavior

under relatively small stresses before reaching a yield state. Upon placement, the major

σ

’

1

and minor

σ

’

3

principal stresses along the VCL of the opening, for an elastic state, are

σ

’

v

and (K

0

)

νσ

’

v

, respectively.

As shown in Figure 1a, two situations can arise for a given friction angle

ϕ

’, considering the limit stress

state (

σ

’

v

,K

aσ

’

v

), with the Mohr circle becoming tangential to the Coulomb yield envelope (

τ

=

σ

tan

ϕ

’, where

τ

is the maximum shear stress for a given normal stress

σ

). When (K

0

)

νσ

’

v≤

K

aσ

’

v

, the

corresponding Mohr circle (dotted line in Figure 1) theoretically exceeds the yield envelope. Since

this is not allowed, yielding with plastic strain occurs and the horizontal stress is controlled by the

circle tangent to the yield envelope. Such stress state corresponds to an active state in the ﬁll, and can

be associated with Rankine’s active coefﬁcient K

a

. On the other hand, if (K

0

)

νσ

’

v

>K

aσ

’

v

, the Mohr

circle (solid line) remains below the Coulomb yield envelope. The backﬁll along the VCL can then be

expected to be near the at-rest (elastic) state (with K= (K

0

)

ν

). In Figure 1,

ϕ

’

c

and

νc

denote the critical

values of the ﬁll’s internal friction angle ϕ’and Poisson’s ratio ν(deﬁned below).

Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 11

When obeying the Mohr–Coulomb elastoplastic criterion, the fill exhibits a linear elastic

behavior under relatively small stresses before reaching a yield state. Upon placement, the major σ’1

and minor σ’3 principal stresses along the VCL of the opening, for an elastic state, are σ’v and (K0)νσ’v,

respectively. As shown in Figure 1a, two situations can arise for a given friction angle ϕ’, considering

the limit stress state (σ’v, Kaσ’v), with the Mohr circle becoming tangential to the Coulomb yield

envelope (τ = σ tanϕ’, where τ is the maximum shear stress for a given normal stress σ). When (K0)νσ’v

≤ Kaσ’v, the corresponding Mohr circle (dotted line in Figure 1) theoretically exceeds the yield

envelope. Since this is not allowed, yielding with plastic strain occurs and the horizontal stress is

controlled by the circle tangent to the yield envelope. Such stress state corresponds to an active state

in the fill, and can be associated with Rankine’s active coefficient Ka. On the other hand, if (K0)νσ’v >

Kaσ’v, the Mohr circle (solid line) remains below the Coulomb yield envelope. The backfill along the

VCL can then be expected to be near the at-rest (elastic) state (with K = (K0)ν). In Figure 1, ϕ’c and νc

denote the critical values of the fill’s internal friction angle ϕ’ and Poisson’s ratio ν (defined below).

(a) (b)

Figure 1. Stress states in backfills along the vertical centerline (VCL) of the opening illustrated by

Mohr–Coulomb planes, with constant (a) internal friction angle ϕ’ and (b) Poisson’s ratio ν. The

dotted line circle indicates an active state while the solid line circle indicates an at-rest state.

Similarly, Figure 1b illustrates that, for a given Poisson’s ratio ν, the resulting Mohr circle based

on the initial elastic stress state (σ’v, (K0)νσ’v) may or may not theoretically exceed the Coulomb yield

envelope, depending on the value of ϕ’. When ϕ’ is small enough, (K0)νσ’v ≤ Kaσ’v, and the stress state

will be active (with K = Ka). As the value of ϕ’ increases, (K0)νσ’v > Kaσ’v, and the backfill remains in an

at-rest (elastic) state (with K = (K0)ν).

For both cases illustrated in Figure 1a,b the theoretical condition for distinguishing between the

at-rest and active states is given by (K0)νσ’v = Kaσ’v (or (K0)ν = Ka). Combining Equations (2) and (4)

leads to the following relationship:

1 sin '

1 1 sin '

(5)

For a given friction angle ϕ’, the critical value of Poisson’s ratio νc of the fill can be given by

rearranging Equation (5):

1 sin '

2

c

(6a)

For a given Poisson’s ratio ν, the critical value of the internal friction angle ϕ’c becomes:

Figure 1.

Stress states in backﬁlls along the vertical centerline (VCL) of the opening illustrated by

Mohr–Coulomb planes, with constant (

a

) internal friction angle

ϕ

’and (

b

) Poisson’s ratio

ν

.The dotted

line circle indicates an active state while the solid line circle indicates an at-rest state.

Similarly, Figure 1b illustrates that, for a given Poisson’s ratio

ν

, the resulting Mohr circle based

on the initial elastic stress state (

σ

’

v

, (K

0

)

νσ

’

v

) may or may not theoretically exceed the Coulomb yield

envelope, depending on the value of

ϕ

’. When

ϕ

’is small enough, (K

0

)

νσ

’

v≤

K

aσ

’

v

, and the stress

state will be active (with K=K

a

). As the value of

ϕ

’increases, (K

0

)

νσ

’

v

>K

aσ

’

v

, and the backﬁll remains

in an at-rest (elastic) state (with K= (K0)ν).

For both cases illustrated in Figure 1a,b the theoretical condition for distinguishing between the

at-rest and active states is given by (K

0

)

νσ

’

v

=K

aσ

’

v

(or (K

0

)

ν

=K

a

). Combining Equations (2) and (4)

leads to the following relationship:

ν

1−ν=1−sin ϕ0

1+sin ϕ0(5)

For a given friction angle

ϕ

’, the critical value of Poisson’s ratio

νc

of the ﬁll can be given by

rearranging Equation (5):

νc=1−sin ϕ0

2(6a)

For a given Poisson’s ratio ν, the critical value of the internal friction angle ϕ’cbecomes:

ϕ0v=sin−1(1−2ν)(6b)

Appl. Sci. 2018,8, 1721 5 of 11

Equation (6) can thus be used to calculate the critical values of

ϕ

’or

ν

that deﬁne the transition

between an at-rest state and active state near the center of backﬁlled openings with rigid conﬁning

walls. This analysis indicates that when

ν≤νc

or

ϕ

’

≤ϕ

’

c

, the backﬁll is in an active state because the

stress state induces yielding; otherwise, an at-rest state prevails (as illustrated in Figure 1).

Sometimes, the values of angle

ϕ

’and Poisson’s ratio

ν

can be linked through a unique

(and consistent) value of K

0

(i.e., (K

0

)

ϕ’

= (K

0

)

ν

), based on Equations (3) and (4) for an elastoplastic

model. This would lead to a special situation, as shown in the following relationship [27,37–41]:

ν=1−sin ϕ0

2−sin ϕ0(7)

As the

ν

value given by this equation always exceeds that of Equation (6a), the ﬁll state associated

with Equation (7) corresponds to an at-rest condition (with K= (K0)ν= (K0)ϕ’) [42,43].

The validity of this theoretical analysis is further assessed below using numerical simulations

conducted with FLAC 2D.

3. Numerical Simulations and Comparisons

3.1. Numerical Model

Figure 2a shows a two-dimensional (plane strain conditions) backﬁlled opening and Figure 2b

illustrates the corresponding FLAC model [

30

]. The rock mass is a linear elastic material characterized

by a Young’s modulus E

r

of 30 GPa, a unit weight

γr

of 27 kN/m

3

, and a Poisson’s ratio

νr

of 0.25.

The granular cohesionless ﬁll behaves as a Mohr–Coulomb elastoplastic material with a dry unit weight

γ

of 18 kN/m

3

, zero effective cohesion c’, and zero dilation angle

ψ

(ultimate state with non-associated

ﬂow rule). The values of the ﬁll Poisson’s ratio

ν

, internal friction angle

ϕ

’, and Young’s modulus

Evary within a range of values given in Table 1(which also gives other parameters and identiﬁes

simulated cases).

Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 11

1

' sin 1 2

c

(6b)

Equation (6) can thus be used to calculate the critical values of ϕ’ or ν that define the transition

between an at-rest state and active state near the center of backfilled openings with rigid confining

walls. This analysis indicates that when ν ≤ νc or ϕ’ ≤ ϕ’c, the backfill is in an active state because the

stress state induces yielding; otherwise, an at-rest state prevails (as illustrated in Figure 1).

Sometimes, the values of angle ϕ’ and Poisson’s ratio ν can be linked through a unique (and

consistent) value of K0 (i.e., (K0)ϕ’ = (K0)ν), based on Equations (3) and (4) for an elastoplastic model.

This would lead to a special situation, as shown in the following relationship [27,37–41]:

1 sin '

2 sin '

(7)

As the ν value given by this equation always exceeds that of Equation (6a), the fill state associated

with Equation (7) corresponds to an at-rest condition (with K = (K0)ν = (K0)ϕ’) [42,43].

The validity of this theoretical analysis is further assessed below using numerical simulations

conducted with FLAC 2D.

3. Numerical Simulations and Comparisons

3.1. Numerical Model

Figure 2a shows a two-dimensional (plane strain conditions) backfilled opening and Figure 2b

illustrates the corresponding FLAC model [30]. The rock mass is a linear elastic material characterized

by a Young’s modulus Er of 30 GPa, a unit weight γr of 27 kN/m3, and a Poisson’s ratio νr of 0.25. The

granular cohesionless fill behaves as a Mohr–Coulomb elastoplastic material with a dry unit weight

γ of 18 kN/m3, zero effective cohesion c’, and zero dilation angle ψ (ultimate state with non-associated

flow rule). The values of the fill Poisson’s ratio ν, internal friction angle ϕ’, and Young’s modulus E

vary within a range of values given in Table 1 (which also gives other parameters and identifies

simulated cases).

Figure 2. (a) A backfilled opening and (b) the corresponding numerical model.

Figure 2. (a) A backﬁlled opening and (b) the corresponding numerical model.

Appl. Sci. 2018,8, 1721 6 of 11

Table 1.

Geometric and mechanical characteristics of the numerical models simulating the backﬁlled

stopes.

Cases Figure Reference B(m) E(MPa) Peak ϕ’(◦)ν(-)

1 Figure 3a 4 to 20 300 30 0.2

2 Figure 3b 8 10 to 1000 30 0.2

3 Figure 3c 8 300 10 to 45 0.2

4 Figure 3d 8 300 30 0.001 to 0.499

5 * Figure 48 300 10◦to 40◦0.452 to 0.263

* The values of ϕ’and νare related through Equation (7).

In Figure 2b, the two side boundaries of the rock mass are ﬁxed horizontally, while the

displacements are restricted in all directions at the base. The opening is ﬁrst created instantaneously.

After the induced displacements are reset to zero, backﬁlling is performed layer-by-layer (1 m per layer)

to a height (H) of 40 m, with an open and horizontal top surface. This progressive ﬁlling sequence

reduces the effect of the added layers (due to momentum) on the simulated stresses and earth pressure

coefﬁcient (see details in Yang [

40

]). Hence, there is no rock wall closure on the backﬁll (and no stress

imposed by the rock walls to the backﬁll). This approach is valid for the delayed ﬁlling of a single

opening, after convergence for stiff rock walls [12].

Sensitivity analyses of the mesh indicate that the 0.2

×

0.2 m square mesh elements can be

used to model the ﬁlls. Interface elements are not used between the walls and backﬁll, which are

considered to be very rough due to blasting (as is typically the case in underground mine stopes) [

5

,

44

].

Furthermore, numerical results presented by Yang et al. [

28

,

29

] have indicated that the roughness of

interface elements tends to alter the magnitude of the vertical and horizontal stresses along the vertical

centerline of backﬁlled openings, but it has little impact on the corresponding values of K.

3.2. Theoretical Predictions

For Cases 1 and 2, the critical Poisson’s ratio

νc

is 0.25 based on Equation (6a) with

ϕ

’= 30

◦

; the

critical friction angle

ϕ

’

c

is 36.9

◦

for

ν

= 0.2 according to Equation (6b). In these two cases,

ν≤νc

or

ϕ

’<

ϕ

’

c

, so the backﬁll along the VCL of the openings can be expected to yield and reach an active

state with Kclose to Ka= 0.33 (Equation (2) with ϕ’= 30◦).

For Cases 3 with

ϕ

’

≤ϕ

’

c

= 36.9

◦

(from Equation (6b) with

ν

= 0.2), an active state is anticipated

within the yielding ﬁll along the VCL (i.e., Knear K

a

deﬁned by Equation (2)). For

ϕ

’>

ϕ

’

c

= 36.9

◦

,

an at-rest state can be expected, and the Kvalue should then approach (K

0

)

ν

= 0.25 (Equation (4) with

ν= 0.2).

For Cases 4 with

ν≤νc

= 0.25 (Equation (6a) with

ϕ

’= 30

◦

), an active state is also expected along

the VCL of the backﬁlled opening (i.e., Kis close to K

a

= 0.33, with

ϕ

’= 30

◦

). For

ν

>

νc

= 0.25, a Kvalue

close to (K0)ν(Equation (4)) is anticipated within the ﬁll.

For Cases 5, values of the backﬁll Poisson’s ratio

ν

and internal friction angle

ϕ

’are related

according to Equation (7). The theoretical analysis presented above shows that the backﬁll state should

be at-rest, with K= (K0)ν= (K0)ϕ’.

3.3. Numerical Results

In all simulated cases, the Kvalues were obtained from the ratios of numerical horizontal and

vertical stresses. Figure 3shows the simulated variations of Kalong the VCL of backﬁll openings,

considering different values for opening width B(Figure 3a; Cases 1), Young’s modulus E(Figure 3b;

Cases 2), friction angle ϕ’(Figure 3c; Cases 3), and Poisson’s ratios νof the ﬁlls (Figure 3d; Cases 4).

Appl. Sci. 2018,8, 1721 7 of 11

Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 11

For Cases 2, Figure 3b illustrates that K along the VCL remains almost constant as the fill

modulus E changes between 10 MPa and 1 GPa. The stress state near the opening center is known to

be almost insensitive to the fill modulus, which also has little effect on the K value (for a range of E

varying from 10 MPa to 1 GPa). The K values obtained from the numerical modeling are close to Ka,

which is consistent with the respective values of the fill Poisson’s ratio and internal friction angle (as

predicted above from Equation (6)).

Figure 3c indicates that as the value of internal friction angle ϕ’ changes from 10° to 35° (Cases

3), the K values near the backfill opening center tend to decrease, but remain fairly around Ka, except

near the fill top. As the values of ϕ’ increase to 40° and 45°, the ratio K continues to decrease slightly

and then stays close to a constant value given by (K0)ν (with ν = 0.2). These numerical results are also

consistent with the theoretical analyses presented above.

Figure 3d shows that the K values near the backfilled opening center keep almost unchanged

(near Ka), when Poisson’s ratio ν varies between 0.001 and 0.2, except near the top surface of the

backfilled opening (Cases 4). When the value of Poisson’s ratio ν ≥ 0.3, the K value at the center of

vertical openings is seen to be a function of Poisson’s ratio, with values near (K0)ν (Equation (4)),

except near the top surface and bottom of the backfill. Figure 3d also illustrates that the K value

approaches unity as the value of ν goes close to 0.5 (i.e., 0.499), indicating a hydrostatic (isotropic)

state. Again, these numerical results agree with the theoretical analyses presented above.

Figure 3. Distribution of K along the VCL of backfilled openings as a function of: (a) B (Cases 1); (b)

E (Cases 2); (c) ϕ’ (Cases 3); and (d) ν (Cases 4) (see Table 1 for details).

Figure 4 illustrates that, for Cases 5 (ν and ϕ’ related through Equation (7)), the simulated K

values along the VCL are reduced when ϕ’ is increased (and ν is decreased). These simulated values

correlate well with K0 as the internal friction angle ϕ’ rises from 10° to 40° (with Poisson’s ratio ν

ranging between 0.452 and 0.263). These simulation results correspond well with those predicted by

the theoretical analyses presented above.

Figure 3.

Distribution of Kalong the VCL of backﬁlled openings as a function of: (

a

)B(Cases 1); (

b

)E

(Cases 2); (c)ϕ’(Cases 3); and (d)ν(Cases 4) (see Table 1for details).

For Cases 1 (Figure 3a), it is seen that the stress ratio Kin the ﬁll along the VCL appears almost

unchanged with the variation of the width of opening B(ranging between 4 and 20 m). As expected,

the Kvalues near the center line of the backﬁll opening are close to Rankine’s active coefﬁcient K

a

,

for Bvarying between 4 and 20 m (except near the ﬁll surface).

For Cases 2, Figure 3b illustrates that Kalong the VCL remains almost constant as the ﬁll modulus

Echanges between 10 MPa and 1 GPa. The stress state near the opening center is known to be almost

insensitive to the ﬁll modulus, which also has little effect on the Kvalue (for a range of Evarying

from 10 MPa to 1 GPa). The Kvalues obtained from the numerical modeling are close to Ka, which is

consistent with the respective values of the ﬁll Poisson’s ratio and internal friction angle (as predicted

above from Equation (6)).

Figure 3c indicates that as the value of internal friction angle

ϕ

’changes from 10

◦

to 35

◦

(Cases 3),

the Kvalues near the backﬁll opening center tend to decrease, but remain fairly around K

a

, except

near the ﬁll top. As the values of

ϕ

’increase to 40

◦

and 45

◦

, the ratio Kcontinues to decrease slightly

and then stays close to a constant value given by (K

0

)

ν

(with

ν

=0.2). These numerical results are also

consistent with the theoretical analyses presented above.

Figure 3d shows that the Kvalues near the backﬁlled opening center keep almost unchanged

(near K

a

), when Poisson’s ratio

ν

varies between 0.001 and 0.2, except near the top surface of the

backﬁlled opening (Cases 4). When the value of Poisson’s ratio

ν≥

0.3, the Kvalue at the center of

vertical openings is seen to be a function of Poisson’s ratio, with values near (K

0

)

ν

(Equation (4)), except

near the top surface and bottom of the backﬁll. Figure 3d also illustrates that the Kvalue approaches

unity as the value of

ν

goes close to 0.5 (i.e., 0.499), indicating a hydrostatic (isotropic) state. Again,

these numerical results agree with the theoretical analyses presented above.

Figure 4illustrates that, for Cases 5 (

ν

and

ϕ

’related through Equation (7)), the simulated K

values along the VCL are reduced when

ϕ

’is increased (and

ν

is decreased). These simulated values

correlate well with K

0

as the internal friction angle

ϕ

’rises from 10

◦

to 40

◦

(with Poisson’s ratio

ν

Appl. Sci. 2018,8, 1721 8 of 11

ranging between 0.452 and 0.263). These simulation results correspond well with those predicted by

the theoretical analyses presented above.

Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 11

Figure 4. Distribution of K along the VCL of backfilled openings as a function of ϕ’ and ν (Cases 5 in

Table 1).

4. Discussion

In soil mechanics, the stress state behind a retaining wall is commonly associated with its

movement [33]. The soil is initially assumed to be at rest before introducing a (stiff and very thin)

retaining wall. An immobile wall is thus sometimes regarded as a necessary and sufficient condition

to maintain an at-rest state in the soil. However, it has been shown above that the stress ratio K (and

corresponding state) in backfilled openings is not solely associated with the wall movement (see also

simulation results from Sobhi et al. [24] and Yang et al. [28,29]). In such openings, yielding of the fill

may occur even if the walls are immobile. The theoretical analyses show that the state of stress of

backfill at the center of openings also depends on the respective values of ϕ’ and ν. This has been

confirmed by the numerical results presented above.

The theoretical and numerical analyses presented above would benefit from further validation,

based on specific experimental data that comprise all the required parameters, including the vertical

and horizontal normal stresses and backfill properties (ϕ’ and ν). To the authors’ knowledge, such a

complete data set does not yet exist. A few results, such as the in-situ measurements performed by

Thompson et al. [45], nonetheless indicate that the K value may approach Ka or K0, depending on the

local conditions in the backfilled stope. These results appear to follow the trends identified by the

theoretical and numerical results shown above, particularly for a relatively large ϕ’ value (=37°), as

shown for Cases 3 in Figure 3c. However, a more complete assessment would require additional

testing results to specifically determine the backfill Poisson’s ratio. In addition, the complex

conditions in the field may influence the stress state and related results due to additional factors not

considered here (i.e., complex geometry, water pressures, cementation and curing, etc.).

All these indicate that more effort should be made to determine the backfill Poisson’s ratio,

despite the difficulties raised by accurate measurements. Ongoing study is also being performed by

the authors to help obtain the value of ν from indirect means.

The numerical analyses performed here consider dry cohesionless filling materials. Previous

studies have shown that the generation and dissipation of pore water pressure, cohesion, and cement

curing could affect the state of stress developed in backfilled mine stopes [25,43,45]. It has also been

shown that a nonzero dilation angle would alter the simulated stress state in backfill openings

[17,25,40]. Additional work is urged to assess the impact of these additional aspects.

The simulations conducted here assume constant values for Poisson’s ratio and Young’s

modulus of the backfill. In practice, these parameters can vary in the opening due to the changes in

falling height or loading associated with the addition of fill layers. It would thus be useful to assess

the effect of a spatial variation of the fill properties using numerical simulations and experimental

tests.

In order to analyze the effect of parameters of interest, the parameter of interest was treated as

a variable while other parameters were kept constant (see Cases 1–4 in Table 1). The values of ν and

ϕ’ were varied simultaneously in Cases 5. Additional work is required to check the validity of the

simulated outcomes when two or more parameters are varied at the same time.

Figure 4.

Distribution of Kalong the VCL of backﬁlled openings as a function of

ϕ

’and

ν

(Cases 5 in

Table 1).

4. Discussion

In soil mechanics, the stress state behind a retaining wall is commonly associated with its

movement [

33

]. The soil is initially assumed to be at rest before introducing a (stiff and very thin)

retaining wall. An immobile wall is thus sometimes regarded as a necessary and sufﬁcient condition

to maintain an at-rest state in the soil. However, it has been shown above that the stress ratio K(and

corresponding state) in backﬁlled openings is not solely associated with the wall movement (see also

simulation results from Sobhi et al. [

24

] and Yang et al. [

28

,

29

]). In such openings, yielding of the ﬁll

may occur even if the walls are immobile. The theoretical analyses show that the state of stress of

backﬁll at the center of openings also depends on the respective values of

ϕ

’and

ν

.This has been

conﬁrmed by the numerical results presented above.

The theoretical and numerical analyses presented above would beneﬁt from further validation,

based on speciﬁc experimental data that comprise all the required parameters, including the vertical

and horizontal normal stresses and backﬁll properties (

ϕ

’and

ν

). To the authors’ knowledge, such

a complete data set does not yet exist. A few results, such as the in-situ measurements performed

by Thompson et al. [

45

], nonetheless indicate that the Kvalue may approach K

a

or K

0

, depending on

the local conditions in the backﬁlled stope. These results appear to follow the trends identiﬁed by

the theoretical and numerical results shown above, particularly for a relatively large

ϕ

’value (=37

◦

),

as shown for Cases 3 in Figure 3c. However, a more complete assessment would require additional

testing results to speciﬁcally determine the backﬁll Poisson’s ratio. In addition, the complex conditions

in the ﬁeld may inﬂuence the stress state and related results due to additional factors not considered

here (i.e., complex geometry, water pressures, cementation and curing, etc.).

All these indicate that more effort should be made to determine the backﬁll Poisson’s ratio, despite

the difﬁculties raised by accurate measurements. Ongoing study is also being performed by the authors

to help obtain the value of νfrom indirect means.

The numerical analyses performed here consider dry cohesionless ﬁlling materials. Previous

studies have shown that the generation and dissipation of pore water pressure, cohesion, and cement

curing could affect the state of stress developed in backﬁlled mine stopes [

25

,

43

,

45

]. It has also been

shown that a nonzero dilation angle would alter the simulated stress state in backﬁll openings [

17

,

25

,

40

].

Additional work is urged to assess the impact of these additional aspects.

The simulations conducted here assume constant values for Poisson’s ratio and Young’s modulus

of the backﬁll. In practice, these parameters can vary in the opening due to the changes in falling

height or loading associated with the addition of ﬁll layers. It would thus be useful to assess the effect

of a spatial variation of the ﬁll properties using numerical simulations and experimental tests.

Appl. Sci. 2018,8, 1721 9 of 11

In order to analyze the effect of parameters of interest, the parameter of interest was treated as a

variable while other parameters were kept constant (see Cases 1–4 in Table 1). The values of

ν

and

ϕ

’were varied simultaneously in Cases 5. Additional work is required to check the validity of the

simulated outcomes when two or more parameters are varied at the same time.

The results presented above are based on plane strain conditions, thus corresponding to openings

with very long third dimensions. It is known that the magnitude of arching may be under-predicted

by plane strain models. Additional work is needed to evaluate the effect of the third dimension,

to complement the recent work of Jahanbakhshzadeh [43].

It should be noted that the stress paths and the intermediate state of ﬁlls are not considered here.

Subsequently, the ﬁll state depends solely on the relationship between

ν

(an elastic constant) and

ϕ

’

(a yield parameter) in an elasto-plastic Mohr–Coulomb framework. More representative models are

required to better represent the characteristics of ﬁlls placed in various openings [46].

The numerical results indicate that the value of the stress ratio Kalong the centerline can be

different near the very top and bottom of backﬁlled openings. This is related to the abrupt change

of the local characteristics (material properties and stress state), which may also cause numerical

instability close to these boundaries. Additional work, such as laboratory tests, is required to better

investigate this aspect.

As mentioned previously, yield can occur due to the deviatoric stresses when the values of

ϕ

’or

ν

are smaller than or equal to their critical values

ϕ

’

c

or

νc

, respectively. The corresponding Kcan then

be expressed using Rankine’s active earth pressure coefﬁcient K

a

(as a function of

ϕ

’). When the values

of

ϕ

’or

ν

are greater than their critical values, Kis then linked to the Poisson’s ratio

ν

through (K

0

)

ν

.

An exception is when ϕ’or νare related to each other through Equation (7) (K= (K0)ν= (K0)ϕ’).

Finally, it is worth recalling that the internal friction angle

ϕ

’of soils (and ﬁlling materials) is

commonly determined from usual testing, while their Poisson’s ratio

ν

is difﬁcult to measure. Several

expressions (somewhat similar to Equation (7)) have thus been proposed to estimate the value of

ν

from that of

ϕ

’through their respective connections with K

0

(Equations (3) and (4)) [

37

,

38

,

40

,

41

,

47

].

Despite some superﬁcial similarities, these expressions are fundamentally different from the proposed

Equations (5) and (6), which serve to explain why and when the value of Kshould be equal to K

0

or K

a

to assess stresses in backﬁlled openings from analytical solutions.

5. Conclusions

For granular cohesionless ﬁlls, the analytical and numerical analyses presented above show that

either an active (K

a

) or an at-rest (K

0

) stress state may exist at the center of backﬁll openings with stiff

and ﬁxed conﬁning walls, associated with the respective values of ﬁll Poisson’s ratio

ν

and internal

friction angle

ϕ

’. An original relationship (Equation (6)) between

ϕ

’and

ν

, and the corresponding

mechanisms, are developed to deﬁne when K

a

or K

0

should be used (based on the Mohr–Coulomb

elasto-plastic criterion). Theoretically, the earth pressure coefﬁcient Knear the centerline of a vertical

backﬁll opening can be expected to be near Rankine’s active coefﬁcient K

a

when

ϕ

’and

ν

are smaller

than or equal to critical values (given by Equations (6a) and (6b), respectively). Otherwise, Kshould be

close to the earth pressure coefﬁcient at-rest K

0

(Equation (4)). The theoretical analysis was validated,

in part, by numerical modeling performed with FLAC. The results presented here also highlight the

importance of an accurate evaluation of the Poisson’s ratio for granular cohesionless ﬁlls.

Author Contributions:

Funding acquisition, L.L., M.A., and P.Y.; Investigation, P.Y., L.L., and M.A.; Methodology,

P.Y., L.L., and M.A.; Project administration, L.L.; Software, P.Y.; Supervision, L.L. and M.A.; Validation, P.Y.;

Writing—original draft, P.Y.; Writing—review & editing, P.Y., L.L., and M.A.

Funding:

This research was funded by the Natural Sciences and Engineering Research Council of Canada

(402318), Institut de recherche Robert-Sauvéen santéet en sécuritédu travail (2013-0029), Fonds de recherche

du Québec—Nature et Technologies (2015-MI-191676), Mitacs Elevate Postdoctoral Fellowship (IT08484), and

industrial partners of the Research Institute on Mines and Environment (RIME UQAT-Polytechnique; http:

//rime-irme.ca/).

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appl. Sci. 2018,8, 1721 10 of 11

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