A Strength Design Method of Cemented Backfill with a High...

Research ArticleA Strength Design Method of Cemented Backfill with a HighAspect Ratio

Zhi-zhihui Wang ,1,2 Ai-xiang Wu,1,2 and Hong-jiang Wang1,2

1School of Civil & Resources Engineering, University of Science & Technology Beijing, Beijing 100083, China2Key Laboratory of High-Efficient Mining and Safety of Metal Mines of the Ministry of Education,University of Science & Technology Beijing, Beijing 100083, China

Correspondence should be addressed to Zhi-zhihui Wang; [email protected]

Received 6 September 2019; Accepted 17 January 2020; Published 19 February 2020

Academic Editor: Paolo Castaldo

Copyright © 2020 Zhi-zhihui Wang et al. ,is is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

To calculate the required strength of a cemented backfill with high aspect ratio, the confirmation of lateral pressure is fundamentaland needs to be determined first. As for the backfill with a high aspect ratio of height to length, the shape of the slip surface is notstraight when in the active state due to the limited space, which is different from the general backfill. For this reason, a formulationof the slip surface with a curved shape and a lateral pressure calculation method based on this curved slip surface were proposed.,e proposed equation of the slip surface is affected by the geometry parameters of the backfill, internal friction angle of thebackfill, and the friction angle of the backfill-rock interface. ,en, by the combination of the minor principal stress trajectorymethod and the horizontal slice method, an ordinary differential equation of stresses was established and then solved numerically.Finally, the method based on Mitchell’s three-dimensional limit equilibrium model was used to calculate the required strength ofthe cemented backfill. ,e calculated results were compared with previous studies and validated with numerical models. ,eresults showed good consistency for the backfills with high aspect ratios.

1. Introduction

Cemented backfill mining is widely used to deal with tailingsefficiently and ensure safety of underground stopes [1, 2].,e performance of the cemented backfill is influenced bymany factors (such as cement type, addition of reinforce-ment, loading rate, multiphysical consolidation process[3–5], chemical shrinkage, and temperature) after itsplacement. And all these factors can influence the stability ofthe cemented backfill. But, from the mechanical view, theeffects of these factors can all be combined in the strengthparameters of cohesion and friction angle. Besides that, thebackfill-rock interface is also important. Due to theroughness of the rock face [6, 7], the cohesion and frictiondeveloped at the backfill-rock interface function as shearresistance preventing the failure of the backfill [8, 9].Considering the common two-step mining method ofextracting and then backfilling, the cemented backfill body

of first-step stope will have a free face when extracting thesecond-step stope. In a series of methods for designing therequired strength of the filling body by the limit equilibriummethod, the shear resistance force from lateral rock mass is amajor concern in the process of evaluating the stability of thebackfill body [10–12]. ,us, the lateral pressure force whichcontributes to the friction resistance has an important in-fluence on the safety factor of the backfill body.

After determining the cohesion and friction angle of thebackfill-rock interface, how to determine the lateral pressureforce of the filling body remains an important problem. ,eproduct of the pressure force and the friction coefficient ofthe backfill-rock interface is part of shear resistance forcepreventing the filling body from sliding to the open space.Overestimating this pressure can lead to overestimation ofthe stability of the filling body, which may result in a lowerstrength of the backfill. ,is may pose a potential hazard,such as the collapse of the backfill to the adjacent mining

HindawiAdvances in Civil EngineeringVolume 2020, Article ID 7159208, 11 pageshttps://doi.org/10.1155/2020/7159208

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area, the liquidation resulted by the insufficiency of strengthof the backfill, and the collapse of barricade. Ting et al. [13]simulated the backfilling body of vertical and inclined stopethrough laboratory experiments and considered that theearth pressure calculated from the active earth pressurestate is closer to the experimental results compared with thestatic earth pressure state. Sobhi et al. [14] also pointed outthat when calculating the lateral pressure, the active earthpressure state is more practical than the static earthpressure state which assumes a very large stiffness of thesurrounding rock. ,e calculation of the lateral pressure ofthe filling body by taking the active earth pressure state isalso adopted in [7, 11, 14]. ,ere are two kinds of slipsurface when calculating the design strength of thecemented backfill. One is the slip surface used to determinethe pressure on the lateral rock. Another is the slip surfaceused for calculating the safety factor of the cementedbackfill when in the limit equilibrium state. And these twoare different.

However, in the abovementioned literature, the lateralpressure of the filling body is calculated based on the linearslip surface. In fact, for the filling body with high aspect ratio(the ratio of the height of free face to the length of freeface> tan (45° +φ/2)), Rankine’s slip surface (namely astraight face with an inclination angle of π/4 +φ/2) cannotdevelop under active earth pressure state, and often a curvedslip surface will occur [15–19]. In the case where a linear slipsurface cannot be developed, how to obtain a usable lateralpressure force calculation method remains a very importantproblem. ,us, this paper focuses on the rarely studiedbackfills with high aspect ratios. ,e core objective of thepresent study is to develop a strength design method forcemented backfills with high aspect ratios. To achieve this, asuitable calculation method for the lateral pressure of thefilling body with high aspect ratio is required first. And then,a strength design formula for filling body in this case isdeveloped.

2. Calculation of Lateral Pressure Force

As shown in Figure 1, a vertical backfill with geometric sizesof Lb × W × H is considered. ,e physical experimentsperformed by Mitchell et al. [20] showed that the slidingbody is a wedge on the slip surface when the filling body isunstable. And the backfill-rock pressure is the lateralpressure of the filling body in the unstable state. As men-tioned above, the two-dimensional active earth pressuremodel is generally used to calculate the lateral pressure. ,emonitored data showed that the earth pressure distributionat the midline and the edge is very close [16, 21], so thedetermined earth pressure is generally considered to be theearth pressure distribution along the midline of lateral face.Moreover, for conservative design and simplicity, it isgenerally considered that the filling body is cohesionlesswhen calculating the side pressure [12]. In fact, the difficultyin calculating earth pressure of cohesive filling is the same asthat of a cohesionless one and can be eliminated by thetranslation of the normal stress coordinate of the Mohrstress circle.

2.1. Determination of the Shape of the Slip Surface. ,eformulation of a curved slip surface involves the specificgeometry of the filling body, which means that it is moretargeted to the filling body and closer to the actual situation.However, the disadvantage is that it needs to be calculatedfor each project, and the formulation of the curved slipsurface is more complex than that of a straight line resultingin difficulty to obtain an analytical solution for earthpressure distribution. But, with the development of nu-merical calculation methods, it is very easy to numericallysolve out the earth pressure. ,erefore, using a curved slipsurface to calculate the lateral pressure should be consideredas a refined design method.

,e slip surface shown in Figure 2 is under the activeearth pressure state. Inspired by the experiments of Tsagareli[15] and Yang and Tang [19], an exponential expression ofthe curved slip surface was selected in this paper.,e specificconstraints of the formula are as follows: the slip surfacepasses through the bottom point and its diagonal vertex, thatis, the points with coordinates of (0, 0) and (Lb, H) in thegiven coordinate system shown in Figure 2.

,e equation of the slip surface is specifically written as

y � C 3.6 ·π4

+φ2

+ 0.1 x/C

− C, (1)

where φ is the internal friction angle of the filling body; C is aconstant associated with the geometry parameters of thegiven filling body, which can be obtained by the above-mentioned constraint conditions of the slip surface. Itshould be noted that an analytical expression for C does not

H

W

Lb

Shear

resistance

Slidingface

W tan β

Gravity Wn

β

H∗

Shear

resistance

x

y

Figure 1: Schematic diagram of limit equilibrium analysis of thefilling body.

2 Advances in Civil Engineering

exist for the complexity of the formula of the slip surface.,us, it is necessary to substitute certain parameters of thegiven filling body to obtain a specific equation of slip surface.

As shown in Figure 3, comparing the slip surfaces ob-tained from the experimental data of Yang and Tang [19], theexponential slip surface proposed in this paper can satis-factorily describe the slip surface morphology of the verticalfilling body under active earth pressure.

,en, the angle of the slip surface changing with heightcan be written as

α(y) � tan− 1 (1.8φ + 0.9π + 0.36)ln(C+y)/ln(1.8φ+0.9π+0.36)

· ln(1.8φ + 0.9π + 0.36).(2)

,e width of the sliding body varying with height can bededuced as

L(y) �C · ln(C + y/C)

ln(1.8φ + 0.9π + 0.36). (3)

2.2. Estimation of Stress State. Handy [22] has long noticedthe existence of friction at the wall-backfill interface and thatthe direction of the principal stress in the filling body willbe deflected at a certain angle anyway. ,en, the method ofcalculating the earth pressure with the minor principalstress trajectory was proposed. As shown in Figure 4, theminor principal stress trajectory method fully considers thedisturbance of the wall-backfill friction to the stress dis-tribution state of the filling body. So, it has been widelyused in the calculation of earth pressure, especially the arc-shaped minor principal stress trajectory [18, 23, 24]. Forthis reason, the horizontal slice method combined with arc-

shaped minor principal stress trajectory is also adopted inthis paper for lateral pressure calculation, as shown inFigures 4 and 5.

For a vertical rock face, the deflection angle of principalstress at the backfill-rock interface is [25]

θw �π2

−12arc sin

sinδsinφ

+δ2. (4)

As for the deflection angle of principal stress at thecurved slip surface, the Mohr-stress circle at the slip surfacedrawn by the Mohr-Coulomb criterion shows that the anglebetween σns and σ1 is always (45° + (ϕ/2)). Meanwhile, if theangle between the slip surface of the horizontal differentiallayer and the horizontal plane is notated as α, that is, theangle between σhs and σ1 is ((π/2) − α), then the followingexpression of the deflection angle of principal stress at theslip surface can be obtained:

Rock

Wall

dy

Backfill

α(y)

x

y

(Lb, H)

y

O

ΔL(y)

L(y)h

Slidingface

Figure 2: Side view of the backfill body.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.0

0.2

0.4

0.6

0.8

1.0

1.2

y (m

)

x (m)

Actual curve for Lb = 0.16mActual curve for Lb = 0.36mActual curve for Lb = 0.56mExponential curve for Lb = 0.16mExponential curve for Lb = 0.36mExponential curve for Lb = 0.56mRankine’s line

Figure 3: Comparison of the actual slip surface and proposedexponential slip surface.

Advances in Civil Engineering 3

θs � π −π2

− α −π4

+φ2

�π4

+ α −φ2

. (5)

,erefore, for a curved slip surface, equations (6) and (7)can be obtained by using the arc-shaped minor principalstress trajectory hypothesis.

σy �θsθwσyR sin θdθ

Ly� σ1 1 −

2 sinφ3(1 + sinφ)

· cos2 θw + cos2 θs + cos θs · cos θw .

(6)

τxy �θsθwτxyR sin θdθ

Ly�

2σ1 sinφ3(1 + sinφ)

·sin3 θw − sin3 θscos θw − cos θs

.

(7)

,us, the following equations are derived:

KAVE �σhwσy

�3 1 + sinφ · cos 2θw(

3(1 + sinφ) − 2 sinφ · cos2 θw + cos2 θs + cos θs · cos θw( ,

Kτ �τxyσy

�2 sin3 θw − sin3 θs( sinφ

3(1 + sinφ) cos θw − cos θs( − 2 cos3 θw − cos3 θs( sinφ.

(8)

2.3. Establishment of Equilibrium Equation. Figure 6 is aschematic diagram of the force analysis of the differentialstrip of the filling body considering the horizontal shearstress. Assuming that the backfill-rock interface and the slipsurface are both in the limit equilibrium state defined by theMohr–Coulomb criterion, the following force analysis canbe performed.

Ty and Vy are the shear force and vertical force in thegiven height y, respectively. Tw and Nw are the shear forceand normal force acting on the backfill-rock interface in thegiven height y, respectively. And Nw � σhw · dy; Tw � σhw·dy · tan δ. Ts and Ns are the shear force and normal forceacting on the slip surface in the given height y, respectively.

And Ts � σns · tanφ · dy · csc α; Ns � σns · dy · csc α. Andthe expressions for differential forces in Figure 6 can be listedas follows:

dG � c ·dy · dy2 tan α

+ c · dy · Ly,

dTy � dτxy Ly + dy cot α + τxydy · cot α,

dVy � σydy · cot α + dσy Ly + dy cot α .

(9)

According to the force balance in horizontal direction,equation (10) can be obtained.

Rock wall

θw

dy

CR

σ3

σ1

σ1 + dσ1

Backfill

α

π – θsy

(Lb, H)

yTrajectory of minor principal

stress

O

σ1

σ1 + dσ1

ΔLyLy

Figure 4: Schematic diagram of the minor principal stresstrajectory.

Rock wall

θw

dθ

σhwτw

dy

CR

σ3

σ1

σ1 + dσ1

x

σ3

σ1

Backfill

dx

θσhw

τw

σ3

σ1

dy

θw

α

π – θs

Ly

Figure 5: Horizontal slice method combined with arc-shapedminor principal stress trajectory.


σhw · dy + σns · dy · cot α · tanφ + dτxy Ly + dy cot α

+ τxydy · cot α � σns · dy.

(10)

Dividing equation (10) by dy and eliminating the higher-order infinitesimal leads to the following equation:

dτxydy

+σhw + σns · cot α · tanφ + τxy · cot α

Ly�σnsLy

. (11)

Similarly, the equilibrium equation in vertical directioncan be obtained as

dσydy

+σy · cot α

Ly� − c +

σhw tan δ + σns · (tanφ + cot α)Ly

.

(12)

After introducing σhw � KAVE · σy and τxy � Kτ · σy,equation (10) becomes

dτxydy

+σhw + σns · cot α · tanφ + Kτσy · cot α

Ly�σnsLy

.

(13)

By dτxy � dKτ · σy + Kτ · dσy, equation (13) is furtherwritten as

dKτdy

·σy +dσydy

· Kτ

+KAVEσy + σns · cot α · tanφ + Kτσy · cot α

Ly�σnsLy

.

(14)

Similarly, the vertical balance equation can be rewrittenas

dσydy

+σy · cot α

Ly� − c +

KAVE tan δ · σy + σns · (tanφ + cot α)Ly

.

(15)

,en, the ordinary differential equation for averagevertical stress in the backfill can be obtained as equation (16)by combining equations (14) and (15).

dσydy

1 − Kτ · cot(α − φ) + σy ·cot α − KAVE tan δ − cot(α − φ)KAVE − cot(α − φ)Kτ · cot α

Ly−dKτdy

cot(α − φ) + c � 0.

(16)

,e use of circular minor stress trajectory may also haveother substitutes because the method of minor stress tra-jectory is just a semi-analytical method. Furthermore, theestablished ordinary differential equation cannot be solvedanalytically because several transcendental functions arewithin this function.

3. Strength Design of the Cemented Backfill

3.1. Calculation of Lateral Pressure Stress. ,e specific ex-pressions of L (y) and α (y) in equation (16) are determinedby the formula of slip surface; the expressions of KAVE (y)and Kτ (y) are determined by the stress distribution de-scribed by the arc-shaped minor principal stress trajectory.,e distribution of average vertical stress with height can beobtained by solving equation (16) numerically. ,en, thesolution can be used to obtain the distribution of the lateralpressure.

3.2. Analysis on the Limit Equilibrium Model. As shown inFigure 1, the weight of the sliding body is

Wn � W · H∗

cLb − 2Cb( − 2P tan δ, (17)

where Cb is the cohesion of the backfill-rock interface;H∗ � H − W tan β/2; P is the lateral pressure force acting onthe filling body; and the sliding angle β is generally taken asβ � 45° + φ/2, according to Mitchell’s experiments [20].

,e safety factor of the sliding body is then calculated as

FS �tanφtan β

+2cLb

H∗cLb − 2P tan δ − 2H∗Cb( sin 2β, (18)

where c is the apparent cohesion along the slip surface. If it isconsidered that FS� 1, Cb � c, the cohesion c required forany given internal friction angle φ is

c �cLH − 0.5 · cLW tan β − 2P tan δ

2H − W tan β − 2L(tan βsecφ/ tanφ − tan β). (19)

Ty + dTy

Ts

Ns

Nw

Tw

α

dG

Vy

x

yL(y)

dy

Vy + dVy

Ty

Figure 6: Force analysis of the differential horizontal backfill strip.


From the abovementioned expression, it can be seen thatwhen other parameters are given and fixed, the larger the P,the smaller the required cohesion. To ensure a conservativedesign, it is necessary to find a P value that is accurate enoughbut little less than the actual value to achieve a safe design.

3.3. Process for Designing the Required Strength. As men-tioned above, the calculation of the lateral pressure in thecase of curved siding face is specific to the parameters ofgiven filling body, that is, the geometry of filling body andthe friction angle of the backfill-rock interface and the in-ternal friction angle of the filling body. ,ese data can bedirectly selected from typical data, such as studies of Liu et al.[21], Chen et al. [26], Falaknaz et al. [27], Jahanbakhshzadehet al. [28], and Yang et al. [29]. And it is recommended toobtain approximate parameters in advance according to theresults of the proportioning test in a laboratory. Also, onecan select parameters within a certain range and then takethe mean value of the lateral pressure force. If a conservativecalculation is needed, it is suggested to take smaller values ofφ and δ.

If the abovementioned lateral pressure calculationmethod is used to obtain a dimensionless lateral pressuredistribution of g(y) � σy(y)KAVE(y)/cH, the lateralpressure force is calculated as

P � cH · W

0

H

x· tan βg(y)dy dx. (20)

,e abovementioned integrals also need to be numeri-cally integrated, and the lateral pressure force P is obtainedand then substituted into equation (19) to obtain the re-quired strength design parameters.

4. Verification and Discussion of theProposed Method

In order to verify the reliability of the new calculationmethod for lateral pressure and the required strength of thebackfill, the calculated lateral pressure and the designedstrength of the filling body were verified. Lateral pressure onsides of the backfill was calculated and then submitted intoequation (19) to evaluate the required cohesion. After that,the calculated cohesions were compared with those ofprevious studies and then used as cohesion parameters inFLAC3D models to evaluate the stability of the backfill.

4.1.Computation of Lateral Pressure. ,e solid data points inFigure 7 are the data from [29] and also [19].,e hollow datapoints are calculated using the proposed method. It can beseen from the comparison that the proposed method canpredict the lateral pressure well. When the buried depth isrelatively large, the calculated results are kind of differentfrom the measured data, which can be viewed from thefollowing perspectives.

Firstly, Wu et al. [30] pointed out that the experimentaldata given by Yang et al. [29] is consistent with similarexperimental data in the shallow part, but there is a problem

of larger earth pressure when the depth is large, which is notconsistent with similar experimental data. ,en, Yang [31]replied that this may be due to the fact that the experimentalconditions are not fully guaranteed according to theoreticalanalysis, including the failure to ensure that the wall iscompletely in translational move and the fact that thefriction of wall-backfill is complicatedly distributed alongthe wall. ,us, the actual experiment does not completelycorrespond to the experimental design that is theoreticallyhypothesized and desired. Secondly, as mentioned above,the purpose of this paper is to find a pressure that is close tothe actual lateral pressure but less than the actual sidepressure. ,is will not overestimate the lateral friction andantisliding force of the filling body and can guarantee a safedesign as well as certain excess safety factor for dynamicdisturbance.

Figures 8 and 9 depict the results of lateral pressure forceand lateral friction calculated with the proposed methodunder the condition that δ �φ, respectively.

It is showed that the lateral pressure increases at anincreasing rate with the increase in aspect ratio (H/Lb), andthe smaller the friction angle, the larger the side pressure.

,e reason why a smaller friction angle will lead to alarger side pressure under the same geometric conditions isthat the friction angles of the filling body and the backfill-rock interface is synchronously reduced, and when a lowervalue is taken, the backfill is closer to the hydrostaticpressure state, and thus the surrounding rock is subjected tothe maximum pressure in the hydrostatic pressure state.Although the lateral pressure P contributing to frictionalresistance becomes larger under smaller values of φ and δ,the overall instability resistance decreases. ,is can be seenfrom the increase in the cohesion required for the backfill inthe following section, which fully demonstrates that thestability of the backfill is not simply linear. If the δ used in thecalculation of lateral pressure is larger than φ, a complexvalue with an imaginary part of P may occur, which meansthat δ should be no more than φ in the physical sense.Similarly, if the friction angle used to calculate the minimalcohesion is greater than the friction angle used to calculatethe side pressure, then the resulting cohesion will also bewrong and there may exist a negative value, which indicatesthat the friction angle has an important influence on thestability of the filling body and need to be identified carefully.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0

1

2

3

4

Late

ral p

ress

ure (

kPa)

Depth from top of backfill (m)

Calculated data for Lb = 0.16m



Test data for Lb = 0.16m



Figure 7: Measured and calculated lateral pressure versus depth.


4.2. Computation of the Required Cohesion. ,e calculatedresults of the required cohesion under the condition thatδ �φ, FS� 1 and fb � 1 are shown in Figure 10.

,e minimal cohesion increases with the increase inaspect ratio. Under the same geometric parameters, thelarger the friction angle is, the smaller the required cohesionis. Similarly, under the same cohesion, a larger friction anglewould allow for a larger aspect ratio. After the aspect ratioexceeds a certain threshold, the cohesion required seems todecrease slightly. However, the abovementioned results arebased on the fact that the damage of the backfill is close to thewedge-shaped failure; for the filling body with a larger aspectratio, whether the failure mode is still the wedge sliding or itis toppling over or of some other forms, need to be identified

with the experiment, and then it is decided whether to usethe calculated cohesion value or not.

4.3. Comparison of the Required Cohesion. In the case thatthe safety factor is 1 and the cohesion and friction angle ofthe rock-backfill interface are equal to those of the backfill,the calculated cohesion were compared with results fromprevious methods by Mitchell [20], Li and Aubertin [32], Li[9], and Li and Aubertin [33]. ,e comparison results areshown in Figure 11.

,e curves show that the cohesions calculated with thesame given parameters are lower than the results given byMitchell et al. [20], Li and Aubertin [32], and Li [9]. Andwhen the aspect ratio is less than 4, the calculated results arelarger than those of Li and Aubertin, [33], but when theaspect ratio is greater than 4, the calculated results areslightly smaller than those given by Li and Aubertin [33].,ese comparisons show that the design of the backfillstrength proposed in this paper is effective and slightlyconservative, but it has been greatly improved comparedwith the traditional method.

4.4. Verification with Numerical Modeling. ,e effectivestress in the cemented backfill is evolutive [6, 34], whichmeans that the strength parameters of the cemented backfillwill develop from zero to a peak and then decline gradually.Furthermore, the stress boundary condition of the backfill ischanging too [7]. ,ese dynamic changes may come frommany factors, such as disturbance from blasting or otherdynamic loads, multiphysical consolidation process, and theattenuation effect from corrosive underground water andsulphate. However, the research on the influence of thesefactors is still ongoing [3, 4, 35] and cannot be quantifiedeasily in laboratory modelling and numerical modelling.And to assess the validation of the proposed method, nu-merical modelling was proved to be an effective way

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

15

20

25

30

35

Coh

esio

n ne

eded

for F

S =

1 (k

Pa)

Ratio of H/Lb

φ = δ = 25°φ = δ = 30°

φ = δ = 35°φ = δ = 40°

Figure 10: Calculated minimal cohesion versus aspect ratio.

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0

5

10

15

20

25

30

35

40

45

Late

ral p

ress

ure (

MN

)

Ratio of H/Lb

φ = δ = 25°φ = δ = 30°

φ = δ = 35°φ = δ = 40°

Figure 8: Calculated lateral pressure force versus aspect ratio.

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0

5

10

15

20

Late

ral f

rictio

nal r

esist

ance

(MN

)

Ratio of H/Lb

φ = δ = 25°φ = δ = 30°

φ = δ = 35°φ = δ = 40°

Figure 9: Calculated lateral resistance force versus aspect ratio.


[2, 11, 12, 21, 33, 36–41]. To assess the stability of the backfillusing 3D numerical simulations was proved to be effective[2, 28–32]. Geometric and mechanical parameters used innumerical modeling were taken as typical values fromprevious studies [7, 8, 11, 12, 20, 36, 38]. ,e width andlength of the backfill are fixed to 10m; the heights of thebackfill were taken as 16m, 20m, 24m, 30m, 40m, and50m, a total of 6 different values. ,e size of the wholenumerical model is varying from 330m× 350m× 400m to330m× 350 m× 500m according to the height of thebackfill to avoid the boundary effect. Four different frictionangle values of 25°, 30°, 35°, and 40° were used for each aspectratio. Other parameters of the filling body are unit weightc � 18 kN/m3, Young’s modulus E� 300MPa, and Poisson’sratio ]� 0.3. Mechanical parameters of rock were taken as

typical values from previous studies [7, 8, 11, 12, 36, 38]: unitweight c � 27 kN/m3, Young’s modulus E� 30GPa, andPoisson’s ratio ]� 0.25. ,e four vertical boundaries ofmeshes were constricted using roller support. ,e bottom ofmeshes was fixed, and the top of the meshes was set as freeface.

Some representative results of these 24 cases were se-lected and shown in Figures 12 and 13. Figure 12 depicts thecontours of displacement to reveal the potential slip surface.Figure 13 illustrates the contours of strength to stress ratio toassess the safety factor of the backfill.

White-colored parts in Figure 12 stand for the backfillwith relatively large displacement towards the void space.From its outline, the general shape of the sliding body maybe obtained.

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5152025303540455055606570

Coh

esio

n ne

eded

for F

S =

1 (k

Pa)

Ratio of H/Lb

Mitchell et al. [20]Li and Aubertin [32]Li [41]

Proposed methodLi and Aubertin [33]

φ = δ = 25°

(a)C

ohes

ion

need

ed fo

r FS

= 1

(kPa

)

Ratio of H/Lb1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

1520253035404550556065



φ = δ = 30°

(b)

Coh

esio

n ne

eded

for F

S =

1 (k

Pa)

Ratio of H/Lb1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

101520253035404550556065



φ = δ = 35°

(c)

Coh

esio

n ne

eded

for F

S =

1 (k

Pa)

Ratio of H/Lb1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

51015202530354045505560



φ = δ = 40°

(d)

Figure 11: Comparison of the proposed method and previous methods: (a) φ� δ� 25°; (b) φ� δ � 30°; (c) φ� δ� 35°; and (d) φ� δ � 40°.


Figure 12 shows that the shape of potential sliding bodyis close to the wedge used in limit equilibrium analysis,which is similar to results of physical experiments [20]. InFigure 13, zones with a color closer to white indicate a largersafety factor. And if matched with the analytical methodstrictly, the safety factor of the backfill should be one.Figure 13 shows that the backfills are in the stable state in thecases calculated since theminimum of strength-stress ratio isone, a totally black region. In most regions of the backfill, theequivalent safety factor (strength-stress ratio) is greater than1 but less than 3. ,e safety factor of a small part of thebackfill is relatively large, and the maximum value can reachabout 10. ,is shows that the safety factor of the major areais enough, but the margin is not large, which effectivelyreduces the cost of the cement on the basis of ensuring safety.

However, the stress state and boundary condition of thebackfill are evolutive. And such evolution is resulted frommany factors, including the development of the hydrationprocess, the evolution of pore water pressure, dynamicloading on the backfill, and the evolutive boundary condi-tions of the backfill (flow of groundwater, attenuation ofcohesion and friction between rock-backfill interface, andchanging temperature). Furthermore, the strength param-eters of the backfill can be weakened by corrosion of hy-dration products under the influence of corrosiveunderground water or sulphate. But, the quantification ofthese influences still needs further study. So, a safety factor

usually larger than 1.5 was adopted in practice, which meansthe adopted strength data will not be in direct correspon-dence with the designed parameters. So, for a more preciseand economical design, a more complicated analysis (takingtime factor, dynamic factor, and multiphysical couplingeffects into consideration) is about to be developed.

5. Conclusions

,is paper focuses on the calculation method intended forthe lateral pressure and strength design of the backfillingbody with high aspect ratio (the ratio of the height of freeface to the length of free face> tan (45° +φ/2)) for mines.,ecore objective of the present study is to develop a strengthdesign method for cemented backfills with high aspectratios.

A formulation of the slip surface and a lateral pressurecalculation method based on the curved slip surface wasproposed. ,e calculation method of the lateral pressure ofthe filling body with high aspect ratio was obtained. ,elateral pressure can be calculated from given parameters ofthe filling body, including the height and length of thebackfill, the friction angles of the backfill, and the backfill-rock interface. ,en, the method based on Mitchell’s three-dimensional limit equilibrium model was used to calculatethe required strength of the cemented backfill.,e calculatedresults were compared with previous studies and validated

4.2500E – 034.4333E – 03

4.0000E – 033.7500E – 033.5000E – 033.2500E – 033.0000E – 032.7500E – 03

2.2500E – 032.0000E – 03

1.5000E – 031.2500E – 031.0000E – 037.5000E – 045.0000E – 042.5000E – 042.1694E – 04

1.7500E – 03

2.5000E – 03

8.3995E – 038.0000E – 037.5000E – 037.0000E – 036.5000E – 036.0000E – 035.5000E – 035.0000E – 034.5000E – 034.0000E – 03

3.0000E – 032.5000E – 032.0000E – 031.5000E – 03

5.0000E – 043.0494E – 04

1.0000E – 03

3.5000E – 03

9.0000E – 03

7.5000E – 037.0000E – 036.5000E – 036.0000E – 035.5000E – 035.0000E – 034.5000E – 034.0000E – 03

3.0000E – 032.5000E – 032.0000E – 03

1.0000E – 031.5000E – 03

5.0000E – 042.5299E – 04

3.5000E – 03

8.0000E – 03

9.5000E – 039.6229E – 03

8.5000E – 03

5.4771E – 03

5.0000E – 03

4.5000E – 03

4.0000E – 03

3.5000E – 03

3.0000E – 03

2.5000E – 03

2.0000E – 03

1.5000E – 03

1.0000E – 03

5.0000E – 04

1.6541E – 04

Figure 12: Typical contours of displacement of the exposed backfill. (a) H� 50m, φ� δ � 40° (b) H� 50m, φ� δ � 25°, (c) H� 40m,φ� δ� 25°, and (d) H� 50m, φ� δ � 40°.

9.5060E + 009.5000E + 009.0000E + 008.5000E + 008.0000E + 007.5000E + 00

6.5000E + 006.0000E + 005.5000E + 005.0000E + 004.5000E + 004.0000E + 00

3.0000E + 002.5000E + 002.0000E + 001.5000E + 001.0000E + 00

3.5000E + 00

7.0000E + 00

9.8887E + 009.5000E + 009.0000E + 008.5000E + 008.0000E + 007.5000E + 00

6.5000E + 006.0000E + 005.5000E + 005.0000E + 004.5000E + 004.0000E + 00

3.0000E + 002.5000E + 002.0000E + 001.5000E + 001.0000E + 00

3.5000E + 00

7.0000E + 00

9.8836E + 009.5000E + 009.0000E + 008.5000E + 008.0000E + 007.5000E + 00

6.5000E + 006.0000E + 005.5000E + 005.0000E + 004.5000E + 004.0000E + 00

3.0000E + 002.5000E + 002.0000E + 001.5000E + 001.0000E + 00

3.5000E + 00

7.0000E + 00

9.5548E + 009.5000E + 009.0000E + 008.5000E + 008.0000E + 007.5000E + 00

6.5000E + 006.0000E + 005.5000E + 005.0000E + 004.5000E + 004.0000E + 00

3.0000E + 002.5000E + 002.0000E + 001.5000E + 001.0000E + 00

3.5000E + 00

7.0000E + 00

Figure 13: Typical contours of strength-stress ratio of the exposed backfill. (a) H� 50m, φ� δ � 25°, (b) H� 50m, φ� δ� 40°, (c) H� 40m,φ� δ� 25°, and (d) H� 40m, φ� δ � 40°.


with the numerical models. ,e results showed good con-sistency for the backfills with high aspect ratios.

Abbreviations

C: Constant used in the exponential slip surfaceC: Cohesion of the backfillCb: Cohesion of the backfill-rock interfaceG: GravityH: Total height of the backfillh: Buried depth from the top of the backfillKAVE(y): Ratio of σhw to σyKτ(y): Ratio of τxy to σyLb: Length of the filling bodyLy: L(y) �width of sliding body at a given height yNw: Normal stress at the rock-backfill interfaceNs: Normal stress at the slip surfaceP: Lateral pressure force of the backfillR: Radius of circle of minor principal stressTw: Tangential stress at the rock-backfill interfaceTs: Tangential stress at the slip surfaceVy: Vertical force on the horizontal backfill strip at a

given height of yW: Width of the backfillWn: Gravity of the sliding bodyy: Height of the horizontal backfill stripα : Inclination of the horizontal backfill stripβ : Sliding angle when the backfill is about to failc: Unit weight of the backfillδ: Friction angle of the rock-backfill interfaceθs: Rotation angle of major principal stress at the slip

surfaceθw: Rotation angle of major principal stress at the

rock-backfill interfaceσhs: Horizontal stress at the slip surfaceσhw: Horizontal stress at the rock-backfill interfaceσns: Normal stress at the slip surfaceσnw: Normal stress at the rock-backfill interfaceσy: Average vertical stress on the horizontal backfill

at a given height of yσ1: Major principal stressσ3: Minor principal stressτxy: Average horizontal shear stress on the backfill

layer at a given height of yφ: Internal friction angle of the backfill.

Data Availability

,e data will be available once required.

Conflicts of Interest

,e authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

,is research was funded by the National Key R&D Programof China, grant no. 2017YFC0602903 and the State Key

Program of the National Natural Science Foundation ofChina, grant no. 51834001.

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