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Theoretical and Numerical Analyses of Earth Pressure Coefficient along the Centerline of Vertical Openings with Granular Fills

Authors:

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 K is sometimes obtained by Jaky’s earth pressure coefficient at rest K0, based on the assumption of fixed confining walls, whereas Rankine’s active earth pressure coefficient Ka is regarded more suitable for K as 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 K value can vary between Ka and K0 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 (K0, Ka, 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 K near the center line of a backfill opening should be close to Ka; otherwise, K tends to approach K0 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.
applied
sciences
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 δ01e2hK 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 [3133].
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[7,3133]:
Ka=1sin 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=1sin 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
1ν(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 ϕand 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 ϕand 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’vKaσ’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 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
1ν=1sin 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=1sin ϕ0
2(6a)
For a given Poisson’s ratio ν, the critical value of the internal friction angle ϕcbecomes:
ϕ0v=sin1(12ν)(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
ννc
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 [27,3741]:
ν=1sin ϕ0
2sin 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 10to 400.452 to 0.263
* The values of ϕand 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,
ννc
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
ννc
= 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 ν0.3, 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 ϕor 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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2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
... Processes 2021, 9, x FOR PEER REVIEW 8 of 32 applied in the numerical simulations while the average normal displacement on the top of platen is recorded to calculate the compressive strain for different applied stresses. In the numerical simulations, the Poisson's ratio of backfill is related to its friction angle through ν = (1 − sinϕ)/(2 − sinϕ) by considering a unique value of at-rest earth pressure coefficient K0 [49,50].The tensile strength T of backfill is taken as 1/10 of its uniaxial compressive strength (UCS) [39]. The initial value of void ratio eini is calculated based on the measured porosity. ...
... These parameters are determined by calibrating the numerical results based on part of laboratory results of Pierce [26] associated with the loading paths 1 and 2 as shown in Figure 4. Table 1 summarizes all parameters applied for different constitutive models. Numerical models with the calibrated parameters are then called the calibrated models, which are further applied to predict the other part of laboratory results associated with loading paths 3 and 4. In the numerical simulations, the Poisson's ratio of backfill is related to its friction angle through ν = (1 − sinφ)/(2 − sinφ) by considering a unique value of at-rest earth pressure coefficient K 0 [49,50].The tensile strength T of backfill is taken as 1/10 of its uniaxial compressive strength (UCS) [39]. The initial value of void ratio e ini is calculated based on the measured porosity. ...
... In numerical simulations, the Poisson's ratio of backfill relates to the friction angle as ν = (1 − sinφ)/(2 − sinφ), which is based on a unique value of at-rest earth pressure coefficient K 0 [49,50]. Such equation is practical in numerical modeling with an elastoplastic model. ...
Article
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The compressibility of mining backfill governs its resistance to the closure of surrounding rock mass, which should be well reflected in numerical modeling. In most numerical simulations of backfill, the Mohr–Coulomb elasto-plastic model is used, but is constantly criticized for its poor representativeness to the mechanical response of geomaterials. Finding an appropriate constitutive model to better represent the compressibility of mining backfill is critical and necessary. In this paper, Mohr–Coulomb elasto-plastic model, double-yield model, and Soft Soil model are briefly recalled. Their applicability to describing the backfill compressibility is then assessed by comparing numerical and experimental results of one-dimensional consolidation and consolidated drained triaxial compression tests made on lowly cemented backfills available in the literature. The comparisons show that the Soft Soil model can be used to properly describe the experimental results while the application of the Mohr–Coulomb model and double-yield model shows poor description on the compressibility of the backfill submitted to large and cycle loading. A further application of the Soft Soil model to the case of a backfilled stope overlying a sill mat shows stress distributions close to those obtained by applying the Mohr–Coulomb model when rock wall closure is absent. After excavating the underlying stope, rock wall closure is generated and exercises compression on the overlying backfill. Compared to the results obtained by applying the Soft Soil model, an application of the Mohr–Coulomb model tends to overestimate the stresses in the backfill when the mine depth is small and underestimate the stresses when the mine depth is large due to the poor description of fill compressibility. The Soft Soil model is recommended to describe the compressibility of uncemented or lightly cemented backfill with small cohesions under external compressions associated with rock wall closure.
... A non-associated flow rule was imposed with φ d = 0 • (dilation angle). It is noted that the values of the Poisson's ratio ν b and internal friction angle φ are interrelated through ν b = (1 − sinφ)/(2 − sinφ) to ensure a consistent at-rest earth pressure coefficient K 0 [28,29,[44][45][46]. This results in a value of Poisson's ratio ν b = (1 − sinφ) / (2 − sinφ) = (1 − sin31 • ) / (2 − sin31 • ) = 0.327 and a Jaky's [47] earth pressure coefficient at rest K 0 = 1 − sinφ = 1 − sin31 • = 0.485; the same value of K 0 is obtained from the equation based on Poisson's ratio. ...
... A non-associated flow rule was imposed with d = 0° (dilation angle). It is noted that the values of the Poisson's ratio νb and internal friction angle  are interrelated through νb = (1−sin)/(2−sin) to ensure a consistent at-rest earth pressure coefficient K0 [28,29,[44][45][46]. This results in a value of Poisson's ratio νb = (1−sin)/(2−sin) = (1−sin31°)/(2−sin31°) = 0.327 and a Jaky's [47] earth pressure coefficient at rest K0 = 1−sin = 1−sin31° = 0.485; the same value of K0 is obtained from the equation based on Poisson's ratio. ...
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The multiaxial Mises-Schleicher and Drucker-Prager unified (MSDPu) criterion has been shown to exhibit several specific features compared to other yield and failure criteria, including a nonlinear mean stress dependency, influence of the Lode angle, use of independent uniaxial compressive and tensile strength values and absence of an apex (singularity) on the envelope surface in the negative stress quadrant. However, MSDPu has been seldom used in practice to solve geotechnical and geomechanical engineering problems mainly because it had not yet been fully implemented into three-dimensional (3D) numerical codes. To fill this gap, a 3D elastoplastic MSDPu formulation is developed and implemented into FLAC3D. The proposed MSDPu elastic-perfectly plastic (EPP) constitutive model is then validated against existing analytical solutions developed for calculating the stress and displacement distributions around cylindrical openings. The FLAC3D MSDPu-EPP model is then applied to evaluate the vertical and horizontal stress distributions in a three-dimensional vertical backfilled stope. The numerical results obtained with the MSDPu-EPP model are compared with those obtained with the Mohr-Coulomb EPP model, to highlight key features of the new formulation.
... In China, the seed-box structure is designed mostly as the intersection-angle type, according to the empirical design method [15][16][17]. The arching problem of particles in the box first appears at the intersection angle of the steelplate welding (bending) [18] due to the bad dynamic conditions of particle flow at the intersection angle, the increased resistance and the long-term retention of materials, so the discharge is stopped to form an arching phenomenon [6,19]. Especially in the field of cup-belt potato planters [20], the seed potato is generally cut from the whole potato, and an irregular seed potato aggravates the arch problem in the seed box [21]. ...
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To reduce the missing rate caused by the arching phenomenon of seed potatoes in the traditional seed boxes of cup-belt potato planters, a conical-shaped seed box comprising a seed-metering box and a reseeding box connected together was proposed. The method in this paper combined the discrete-element-analysis method and response-surface method and took the movement of the seed-potato group in the seed box as the research object. First, by analyzing the force and speed of seed potatoes, the main structural parameters affecting the seed-box seed-metering performance were determined, and the indices for evaluating the seed-box seed-metering performance were established. Additionally, a single-factor simulation test and orthogonal test were carried out for the main structural parameters of the seed box. Using the optimized structural parameters of the seed box, a trial conical seed box was produced, and bench-verification tests were carried out. The results showed that the multiple rate, missing rate and coefficient of the variation of the plant distance for the conical seed box were reduced by 4.76%, 4.0% and 9.18%, respectively. The research results have practical significance as guidance for improving the sowing performance of cup-belt potato planters. At the same time, the research results have reference value for solving the arching problem for granular materials in the box.
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The prediction of active earth pressure was generally implemented under the assumptions of two-dimensional conditions and cohesionless soils. However, in practice, the soils usually display a considerable level of cohesion, and the collapse of retained slopes exhibits a three-dimensional (3D) nature. Considering this fact, this paper intends to predict the 3D active earth pressure in cohesive soils based on the kinematic limit-analysis method and a 3D rotational collapse mechanism. The influence of cracks is considered, including a crack forming before the failure of retained soil masses (open crack) and a crack forming simultaneously with the failure (formation crack). The active earth pressure coefficient is estimated based on the work-energy balance principle. In order to facilitate direct application, several design charts are provided. It is shown that accounting for soil cohesion and 3D effects results in a notable decrease in the active earth pressure, whereas considering the existence of cracks would increase the pressure value. This paper develops the studies on active earth pressure, which considers the presence of cohesion, cracks, and 3D effects together for the first time. The results of this paper can offer references in designs of retaining structures for cohesive slopes.
... Furthermore, the Pareto chart in Fig.4 shows that all variables and interactions between variables (the product of variables) affect K statistically. Like previous theory, the setback of a retaining wall increases, the leverage from course to course rises [7,8,9,10]. ...
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Retaining walls are a relatively common type of protective structure in construction to hold soil behind them. The form of the retaining wall is also relatively diverse with changing setback angle. Design cross-selection of retaining wall virtually ensures the stability of the retaining wall depends on many aspects. It is essential to consider these to bring the overall picture. For this reason, the authors selected a research paper on the influence of the setback angle on the overturning stability of the retaining wall. To evaluate the behavior stability of retaining wall with some key factors having different levels such as setback angle, internal friction angle of the soil, the slope of the backfill is based on the design of the experiment (DOE) with useful statistical analysis tools. These, proposing the necessary technical requirements in choosing significant cross-sections of retaining structure to suit natural terrain and save construction costs, ensure safety for the project.
... Furthermore, the Pareto chart in Fig.4 shows that all variables and interactions between variables (the product of variables) affect K statistically. Like previous theory, the setback of a retaining wall increases, the leverage from course to course rises [7,8,9,10]. ...
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Retaining walls are a relatively common type of protective structure in construction to hold soil behind them. The form of the retaining wall is also relatively diverse with changing setback angle. Design cross-selection of retaining wall virtually ensures the stability of the retaining wall depends on many aspects. It is essential to consider these to bring the overall picture. For this reason, the authors selected a research paper on the influence of the setback angle on the overturning stability of the retaining wall. To evaluate the behavior stability of retaining wall with some key factors having different levels such as setback angle, internal friction angle of the soil, the slope of the backfill is based on the design of the experiment (DOE) with useful statistical analysis tools. These, proposing the necessary technical requirements in choosing significant cross-sections of retaining structure to suit natural terrain and save construction costs, ensure safety for the project.
... The wall motion is caused by the pressure exerted by the granular material against the wall. Although the earth pressure distribution in presence of retaining structures can be, in general, calculated both theoretically and numerically [36], under these kinematic conditions it can be convenient to use a numerical approach [37]. The wall mass and the friction between the wall and the base act together like a regulator, therefore when the force exerted by the granular material on the wall is balanced by the frictional resistance, the wall stops. ...
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The study of processes characterized by impulsive nature (i.e. impacts) can be considered of great interest in both physics and engineering disciplines: in the geotechnical field, for instance, their effect on the interaction between soil and structures need to be investigated. The present work aims at the validation, by means of two-dimensional finite element simulations, of a methodology of force calibration which uses results obtained from three-dimensional discrete element analysis for predicting the stress at the base of a granular bed, retained by a movable wall, arising when the system is hit by a projectile. To approach this problem, the low-velocity impact has been modeled as a punctual impulsive force on a granular packing.
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Silos generally work as storage structures between supply and demand for various goods, and their structural safety has long been of interest to the civil engineering profession. This is especially true for dynamically loaded silos, e.g., in case of seismic excitation. Particularly thin-walled cylindrical silos are highly vulnerable to seismic induced pressures, which can cause critical buckling phenomena of the silo shell. The analysis of silos can be carried out in two different ways. In the first, the seismic loading is modeled through statically equivalent loads acting on the shell. Alternatively, a time history analysis might be carried out, in which nonlinear phenomena due to the filling as well as the interaction between the shell and the granular material are taken into account. The paper presents a comparison of these approaches. The model used for the nonlinear time history analysis considers the granular material by means of the intergranular strain approach for hypoplasticity theory. The interaction effects between the granular material and the shell is represented by contact elements. Additionally, soil-structure interaction effects are taken into account.
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Backfill mining is an effective option to mitigate ground subsidence, especially for mining under surface infrastructure, such as buildings, dams, rivers and railways. To evaluate its performance, continual long-term field monitoring of the deformation of backfilled gob is important to satisfy strict public scrutiny. Based on industrial Ethernet, a real-time monitoring system was established to monitor the deformation of waste-rock-backfilled gob at −700 m depth in the Tangshan coal mine, Hebei Province, China. The designed deformation sensors, based on a resistance transducer mechanism, were placed vertically between the roof and floor. Stress sensors were installed above square steel plates that were anchored to the floor strata. Meanwhile, data cables were protected by steel tubes in case of damage. The developed system continually harvested field data for three months. The results show that industrial Ethernet technology can be reliably used for long-term data transmission in complicated underground mining conditions. The monitoring reveals that the roof subsidence of the backfilled gob area can be categorized into four phases. The bearing load of the backfill developed gradually and simultaneously with the deformation of the roof strata, and started to be almost invariable when the mining face passed 97 m.
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There are several good reasons for using backfill in underground stopes, including a reduction of mine wastes on the surface and the improvement of ground stability. Backfilling is now commonly used in underground operations worldwide, so practical methods are required to assess the stress state in stopes, on the surrounding rock mass and on support structures. The majority of existing analytical solutions for the stresses have been developed for vertical openings. In practice, stopes often have inclined walls, and this affects the stress state. Recent numerical studies have shown how the stresses distribution in inclined backfilled stopes is influenced by stope geometry and backfill strength. It has also been shown that existing analytical solutions do not capture the essential tendencies regarding these influence factors. In this paper, a new solution is proposed for the vertical and horizontal stresses in backfilled stopes with inclined walls. This solution takes into account the variation of the stresses along the opening height and width, including the difference between the hanging wall and footwall, for various inclination angles of the walls. Key results are presented and validated using recently performed numerical simulations.
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Estimation of stresses in backfilled stopes is a critical issue for backfill design in underground mines. This task can be accomplished by numerical modeling. To date, most numerical modelings were performed without considering interface elements between the backfill and rock walls. The few published works using planar fill–wall interfaces showed that the stress states can be different than those obtained by numerical modeling without considering interface elements, especially when the shear strength of the planar interfaces is lower than that of the backfill. This tends to indicate that the stress estimation in backfilled stopes must be made by accounting for the (planar) fill–wall interfaces. In practice, blasting is commonly used in hard rock mines, and the stope walls are seldom regular and planar. Consequently, it is expected to be more appropriate to analyze the mechanical behavior of the backfill by considering nonplanar interfaces. In this paper, the stress distributions along the vertical central line of 2D backfilled stopes are analyzed using FLAC3D by considering nonplanar interface elements. The asperities of nonplanar interfaces are idealized by saw teeth and the roughness of interfaces is characterized by the teeth height and angle. The numerical results show that the stress state within backfilled stopes can be largely dependent on the shear strength and geometry of the interfaces. This indicates that a survey of the actual rock wall faces should be made and taken into account in the numerical modelings to obtain a better estimate of the stresses in the backfilled stopes. Nevertheless, the results further show that the numerical modelings using planar fill–wall contacts without interface elements lead to comparable stress distributions as those obtained by considering nonplanar interface elements when the interfaces are rough enough. More results are presented by considering different stope geometries and mechanical properties of the backfill with nonplanar interface elements.
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The earth pressure coefficient K, defined as the horizontal to vertical normal (effective) stresses ratio (σh/σv), is a key parameter in analytical solutions for estimating the stresses in backfilled stopes. In the case of vertical stopes, the value of K has sometimes been defined using the at-rest earth pressure coefficient K0, while others have applied Rankine’s active earth pressure coefficient Ka. To help clarify this confusing situation, which can lead to significantly different results, the origin and nature of the at-rest and Rankine’s active coefficients are first briefly recalled. The stress state in backfilled stopes is then investigated using numerical simulations. The results indicate that the value of K can be close to Ka for cohesionless backfills along the vertical central line (CL) of vertical stopes, due to sequential placement and partial yielding of the backfill. For inclined stopes, simulations show that the ratio between the minor and major principal stresses (σ3/σ1) along the CL in t...