NOTE / NOTE

Formulation of a three dimensional analyticalsolution to evaluate stresses in backfilled verticalnarrow openings

Li Li, Michel Aubertin, and Tikou Belem

Abstract: The mechanical response of backfill in narrow openings is significantly influenced by its interaction with thesurrounding walls. Previous work conducted on backfilled trenches and mining stopes indicates that the theory of arch-ing can be used to estimate earth pressures in narrow, vertical backfilled openings. In this paper, a 3D analytical solu-tion is proposed to evaluate the state of stress along the boundaries of the openings. The proposed solution, based on ageneralized version of the Marston approach, is compared with numerical modeling and laboratory experimental resultstaken from the literature. A discussion follows on some particular features and limitations of the analytical solutions.

Key words: backfill, earth pressure, 3D openings, analytical solutions, trenches, mining stopes.

Résumé : La réponse mécanique du remblai placé dans des ouvertures étroites est largement influencée par son inte-raction avec les paroies adjacentes. Les travaux antérieurs portant sur les tranchées et les chantiers remblayés montrentque la théorie de l’effet d’arche peut être utilisée pour estimer la pression dans les ouvertures étroites. Dans cet article,une solution analytique tridimensionnelle est proposée pour évaluer l’état de contraintes dans le remblai le long des pa-rois de l’ouverture. La solution, basée sur une généralisation de l’approche de Marston, est comparée avec des résultatsnumériques et expérimentaux de laboratoire tirés de la littérature. L’article se termine par une discussion sur certainescaractéristioques et limitations de la solution analytique.

Mots clés : remblai, pression, ouvertures 3D, solutions analytiques, tranchées, chantiers miniers. Li et al. 1717

Introduction

Many situations encountered in geotechnique require aquantitative evaluation of the loading conditions induced bybackfill placed in confined narrow openings. Examples in-clude load on conduits in trenches (e.g., Spangler and Handy1984; McCarthy 1988), lateral stress on retaining walls(Frydman and Keissar 1987), vertical stress above tunnels(Iglesia et al. 1999), and pressure in backfilled mined stopes(e.g., Hustrulid et al. 1989; Aubertin et al. 2003; Harvey2004; Belem et al. 2004). The latter applications are of par-ticular interest to the authors as the practice of stope back-filling is increasingly important in mining around the world,for reasons of ground stability (e.g., Thomas et al. 1979;

Singh and Hedley 1981; Hassani and Archibald 1998; Kump2001; Jung and Biswas 2002) and also for improved minewaste management (Aubertin et al. 2002).

The backfill material placed in openings is usually muchsofter than that of the surrounding walls, which are oftenmade of concrete or rock. As backfill tends to settle underits own weight, the two types of media interact in a complexmanner. The characteristics of the fill material and of thewalls must therefore be taken into account to evaluate the in-duced stress distribution in and around the openings. Numer-ical modeling tools constitute powerful means to investigatethe response of such systems, as they can consider variousfactors, such as natural stress conditions, excavation andplacement sequence and discontinuities (e.g., Pariseau 1981;Hustrulid et al. 1989; Brummer et al. 1996; Brechtel et al.1999; Li et al. 2003). Nevertheless, analytical methods mayalso provide rapid, low cost, and valuable solutions for eval-uating the behavior of backfilled openings (e.g., Knutsson1981; Mitchell 1983; Aubertin et al. 2003; Belem et al.2004; James et al. 2004). Previous work on this issue, in-cluding some recent studies conducted by the authors, indi-cates that the theory of arching may be well suited forestimating the earth pressures in narrow vertical openings(e.g., Handy 1985; Aubertin 1999; Take and Valsangkar2001; Aubertin et al. 2003; Li et al. 2003, 2005). However,most existing arching solutions are based on 2D limit equi-

Can. Geotech. J. 42: 1705–1717 (2005) doi: 10.1139/T05-084 © 2005 NRC Canada

1705

Received 11 August 2004. Accepted 29 July 2005. Publishedon the NRC Research Press Web site at http://cgj.nrc.ca on7 December 2005.

L. Li and M. Aubertin.1 Department of Civil, Geologicaland Mining Engineering, École Polytechnique de Montréal,C.P. 6079, Succursale Centre-ville, Montréal, QC H3C 3A7,Canada.T. Belem. Department of Applied Sciences, Université duQuébec en Abitibi-Témiscamingue, 445 boulevard del’Université, Rouyn-Noranda, QC J9X 5E4, Canada.

1Corresponding author (e-mail: Michel.Aubertin@polymtl.ca).

librium analysis, in which the two long walls of the openingare identical. In some practical cases, such solutions may beincomplete, especially when the backfilled opening has alimited length or when the walls have different characteris-tics.

In this paper, a general 3D analytical solution is proposed.Its validity is demonstrated, at least in part, using numericalmodeling and representative experimental results. A discus-sion follows on the main assumptions and limitations of theproposed approach.

Arching concept and solutionsWhen a frictional particulate material is placed in a con-

fined narrow opening, the fill tends to yield as it movesdownward. The surrounding rigid walls then hold the yield-ing material by shear forces along the interfaces. Part of theload due to the weight of the material is thus transferred tothe walls, so the resulting vertical stress in the fill is re-duced. This type of phenomenon is known as arching (e.g.,Richmond and Gardner 1962; Handy 1985; Hunt 1986).

Arching effects have been observed in many situations,particularly within silos and bins used to store particulatematerials. Handling materials, such as powder and grain, re-quires an estimate of the minimum span needed to avoid theformation of a stable arch in the containers (e.g., Richards1966; Cowin 1977; Blight 1986a, 1986b). The basic archingtheory, initially proposed by Janssen (1895), is often used toanalyze this type of situation. Arching was later introducedto geotechnical engineering by Marston (1930) and his team,for calculating pressures on underground conduits placed inditches (see also Handy 1985; McCarthy 1988), and byTerzaghi (1936, 1943) to evaluate the stress distributionabove tunnels (e.g., Ladanyi and Hoyaux 1969; Atkinson etal. 1974). Arching theory has also been applied to retainingwalls (Frydman and Keissar 1987; Take and Valsangkar2001) and to dams (Kutzner 1997).

In geotechnique, many of the above-mentioned types ofstructure have a dimension that is much larger than the othertwo, so most practical problems are treated with 2D (planestrain) models (e.g., Hustrulid et al. 1989; Iglesia et al. 1999;Aubertin et al. 2003). Nevertheless, there are exceptions, in-

cluding the solution given by Van Horn (1964) who used theMarston theory to obtain three dimensional loads on under-ground structures. Three dimensional considerations havealso been given to cases where one of the retaining wallsaround the opening is removed (e.g., Mitchell 1983).

In the powder and grain industry on the other hand, con-tainers have finite dimensions on all sides, so 3D modelshave typically been adopted (e.g., Richmond and Gardner1962; Richards 1966; Cowin 1977; Blight 1986a, 1986b;Williams et al. 1987); these solutions are often axisym-metric.

With most existing (2D and 3D) solutions, a unique fill-wall friction angle is used. In practice, it is not uncommonthat the walls around vertical openings have different charac-teristics.

In the following, a set of 3D equations is developed toevaluate the stress along vertical walls of narrow backfilledopenings having different properties; the proposed equationsare extensions of those recently proposed by Li et al. (2004).

Proposed 3D analytical solutions

Theoretical developmentFigure 1 shows schematically a vertical, narrow backfilled

opening, with the various forces on a layer element, basedon uniformly distributed stresses on the isolated layer. Thisrepresentation is based on the model proposed by Marston(1930) for 2D conditions (see also Handy 1985; McCarthy1988). In Figure 1, H is the backfill height, B is the openingwidth, and L is its length. The four walls are identified as 1for left wall (LW), 2 for front wall (FW), 3 for right wall(RW), and 4 for back wall (BW). At position h, the horizon-tal layer element is subjected to lateral compressive forcesCi (i = 1–4), vertical and horizontal fill-wall shearing forcesSi and Ti (i = 1–4), backfill material longitudinal and trans-verse internal shearing forces SL (and SL + dSL) and SB (andSB + dSB), and internal vertical forces V (and V + dV). Basedon this representation, general 3D equations can be devel-oped, as described below.

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Fig. 1. A vertical backfilled opening with acting forces on an isolated layer element.

The weight of the backfill W in the thin layer element isgiven by

[1] W = γ B L dh

where γ is the unit weight of the backfill, and dh is the thick-ness of the layer element.

The vertical force V is obtained by assuming a uniformvertical stress distribution along the horizontal plane (thisand other assumptions will be discussed later in the paper).The vertical force can then be expressed as

[2] V = σvh B L

where σvh is the vertical stress at position h.The longitudinal (SL) and transverse (SB) shearing forces

are expressed as

[3a] SL = τL B L

[3b] SB = τB B L

where τL and τB are the longitudinal and transverse shearstresses, respectively. These are also assumed to be uni-formly distributed on the horizontal plane.

Using a linear relationship between the vertical stress σvhand the horizontal stress σhhi (i = 1–4) in the backfill layerelement, the lateral compressive force Ci (i = 1–4) on eachwall can be expressed as

[4a] Cj = dh L σhhj = dh L Kcj σvh = dh Kcj V/B

for j = 1, 3

[4b] Ck = dh B σhhk = dh B Kck σvh = dh Kck V/L

for k = 2, 4

where the coefficient of lateral pressure Kci (i = j, k) is theratio of the horizontal stress to vertical stress at the ith fill-wall interface

[5] Kci = σhhi /σvh

The usual forms used for the coefficient of lateral pressureKci (i = 1–4) can be written in a unified format as (see Li etal. 2005)

[6] K Kc

i ih

icv

= + 2σ

αtan

The value for Ki and αi in eq. [6] is given in Table 1; c ineq. [6] and φ in Table 1 are cohesion and friction angle ofthe fill material, respectively. For typical backfill problemswith (quasi) fixed walls, the actual value of Ki is expected tobe between the at rest (K0) and active (Ka) states, dependingon the position in the backfilled opening, as observed byFrydman and Keissar (1987) (see also Discussion below).Hence, calculations shown in the following will include bothvalues (i.e., Ka and K0) of the coefficient of lateral pressure.

The Coulomb criterion is used to define the shearingforces Si (i = 1–4) along the walls

[7a] Sj = (σhhj tan δj + cj) L dh

= (Kcj σvh tan δj + cj) L dh for j = 1, 3

[7b] Sk = (σhhk tan δk + ck) B dh

= (Kck σvh tan δk + ck) B dh for k = 2, 4

where δi and ci (i = j, k) are the friction angle and cohesionof the ith fill-wall interface, respectively.

Static equilibrium of the layer element in the vertical (z)direction implies that

[8] W V Sii

= +=∑d

1

4

Introducing eqs. [1], [2], [6], and [7] into eq. [8], one canthen obtain

[9]dd

vv

σ σ λ λ κ κ γ4 4hh

h B L B L+ +⎛

⎝⎜

⎞⎠⎟ + +⎛

⎝⎜

⎞⎠⎟ − =13 2 13 2 0

where

[10] λ13 = K1 tan δ1 + K3 tan δ3

λ24 = K2 tan δ2 + K4 tan δ4

and

[11] κ13 = c1 + c3 + 2c(tan α1 tan δ1 + tan α3 tan δ3)

κ24 = c2 + c4 + 2c(tan α2 tan δ2 + tan α4 tan δ4)

with δi ≤ φ and ci ≤ c (i = 1–4). Note that for δi > φ and (or)ci > c along the interface, yielding is expected to take placein the fill (rather than directly along the fill-wall interface),so the values of φ and c are taken for δi and ci, respectively.

From eq. [9], the vertical stress acting across the horizon-tal plane at position h is deduced as follows:

[12] σ γ κ κλ λ

λvhB L

B Lh B= − +

+− −

− −

− −( )

( ){ exp[ (13

124

1

131

241 131 −1

+ −λ241L )]}

With this equation, it can be shown that the value of the ver-tical stress tends toward the overburden pressure (γh) whenthe size of the opening (B, L) is large enough (i.e., whenarching effects become negligible).

The horizontal stress σhhi can be obtained from eqs. [5],[6], and [12]

[13] σhhi = Ki σvh + 2c tan αi

On the other hand, by considering the moment equilib-rium around axes x and y (and neglecting the very small sec-ond order terms: dSBdh and dSLdh), one obtains

[14a] SL = L (S4 – S2)/(2dh)

[14b] SB = B (S3 – S1)/(2dh)

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Li et al. 1707

Fill condition Ki αi

At rest (K0) 1 – sin φ 0°

Active (Ka) (1 – sin φ )/(1 + sin φ ) (φ /2) – 45°

Passive (Kp) (1 + sin φ )/(1 – sin φ ) 45° + (φ /2)

Table 1. Definition of Ki and αi (in eq. [6]) for different fillpressure conditions.

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1708 Can. Geotech. J. Vol. 42, 2005

Introducing eqs. [3], [6], and [7] into these last two equa-tions, the longitudinal (τL) and transverse (τB) shear stressescan be deduced as

[15a] τ δ δ σL v= −K Kh

4 4 2 2

2tan tan

+ − + −c c c4 2 4 4 2 222

(tan tan tan tan )α δ α δ

[15b] τ δ δ σB v= −K Kh

3 3 1 1

2tan tan

+ − + −c c c3 1 3 3 1 122

(tan tan tan tan )α δ α δ

Equations [12], [13], and [15] represent the proposed 3D

general solution for the stresses in vertical backfilled open-ings; more details on the developments leading to theseequations are given in Li et al. (2005).

Special casesThe general solutions presented above can be simplified

to address particular cases where some simplifications canbe made for their application. A few of these situations arepresented here.(i) The case of opposite walls that have the same properties

may be encountered in practical situation. For instance,in backfilled mine stopes, the hanging and foot walls areoften made of one rock type, while the two side (lateral)walls are made of another rock type (i.e., δ1 = δ3, δ2 =δ4, c1 = c3, and c2 = c4). In this particular case, eq. [12]becomes

[16] σ γ α δ α δvh

c c B c c L= − + + +− −2 2 21 1 11

2 2 21[( tan tan ) ( tan tan ) ]

21 2

11 1

12

11 1

1

( tan tan ){ exp[ ( tan

B K L Kh B K L K− −

− −

+− − +

δ δδ

22 2tan )]}δ

(ii) When the four walls around the opening are composed of a single material (i.e., δ1 = δ2 = δ3 = δ4 = δ and c1 = c2 = c3 =c4 = c); eq. [12] becomes

[17] σ γ α δ α δvh

c B L

B= − + + +− −

−2 1 2 1 2

21

12

1

1

[( tan tan ) ( tan tan ) ]( K L K

h B K L K1

12

11

121 2

+− − +−

− −

) tan{ exp[ ( ) tan ]}

δδ

(iii) If the four walls react in the same manner, K1 = K2 = K3 = K4 = K, and α1 = α2 = α3 = α4 = α. Equation [17] then be-comes

[18] σ γ α δδvh

B L c

KhK= + − + − −

− − −( ) ( tan tan )tan

{ exp[ (1 1 1 2 1 2

21 2 B L− −+1 1) tan ]}δ

This would be equivalent to the Van Horn’s (1964) solu-tion if the fill is in an at rest state (α = 0°) or for acohesionless material (c = 0).(iv) If the opening length is significantly larger than its

width (i.e., L >> B), eq. [18] reduces to a 2D solution:

[19] σvh =γ α δ

δδB c

KKhB

− + − − −2 1 22

1 2 1( tan tan )tan

[ exp( tan )]

(v) For a friction angle of the fill equal to or smaller thanthe fill-wall friction angle (i.e., φ ≤ δ), one can thenwrite

[20] σvh =γ α φ

φφB c

KKhB

− + − − −2 1 22

1 2 1( tan tan )tan

[ exp( tan )]

(vi) For a cohesionless fill, eq. [20] reduces to one ofMarston’s equations (McCarthy 1988; Aubertin et al.2003) as

[21] σ γ φφvh B

KhB

K= − −⎡

⎣⎢

⎤

⎦⎥

−1 22

1exp( tan )tan

Other special cases are presented in Li et al. (2005).

Graphical representationFigure 2 shows the vertical (Fig. 2a), horizontal (Fig. 2b),

and shear (Fig. 2c) stress variation in a backfilled openingusing the general 3D solution (eqs. [12], [13], and [15]) withthe fill and walls at rest (K1 = K2 = K3 = K4 = K = 1 – sinφand α1 = α2 = α3 = α4 = α = 0) (the cases of active and pas-sive states were given in Li et al. 2005); a single fill-wall in-terface cohesion is considered (c1 = c2 = c3 = c4 = c =1 kPa). For this illustrative example, the opening size is B =5 m and L = 10 m. The fill internal friction angle, φ, is 35°,while the fill-wall friction angles are δ1 = 10°, δ2 = 20°, δ3 =30°, and δ4 = 35°. The arching effect can be clearly observedfor each case. All of the graphs in Fig. 2 show that the stressmagnitude in the fill is reduced, compared with overburdenstresses, when depth increases. It can also be seen that theinternal shear stresses τB and τL are not nil (Fig. 2c), aswould be the case for a single side wall friction angle δ. Itcan equally be seen that the vertical (Fig. 2a) and horizontal(Fig. 2b) stresses would be overestimated (for δ1 = δ2 = δ3 =δ4 = 10°; upper limit) or underestimated (for δ1 = δ2 = δ3 =δ4 = 35°; lower limit) if a single value of the fill-wall frictionangle had been used.

The influence of cohesion on the vertical (Fig. 3a), hori-zontal (Fig. 3b), and shear (Figs. 3c and 3d) stresses isshown for the case where the fill is at rest state (K1 = K2 =K3 = K4 = K = 1 – sinφ and α1 = α2 = α3 = α4 = α = 0). Onecan see that increasing the internal cohesion of fill decreases

the stress magnitude; this is in accordance with field obser-vations (e.g., Grice 1989).

Figure 4 illustrates the influence of the third dimension(the ratio of opening length to width L:B = 5:3) on the verti-cal stress σvh (Fig. 4a) and horizontal stress σhh (Fig. 4b).The stresses calculated with the 3D solution (eq. [18]) arecompared with those obtained with the 2D solution(eq. [19]). It shows that both the vertical and horizontalstress magnitudes can be significantly overestimated by the2D solution, and the overestimation increases with depth.The difference between the stress value provided by the twosolutions depends on material properties (c, φ) and dimen-sions of the opening (B, L, h) (see Discussion below).

Comparison with numerical modeling resultsNumerical modeling results obtained by Li et al. (2003)

with FLAC-2D (Itasca 2002) and a few new cases are intro-duced for comparison purposes. These results are used tohelp assess the validity of the proposed analytical solutionsfor plane strain conditions, where the 2D solutions havebeen shown to represent fairly well the actual behavior oflong backfilled openings (e.g., Knutsson 1981; Hustrulid etal. 1989; Aubertin et al. 2005). The geometry of the open-

ing, as well as the natural field stress state, and the rockmass and fill properties are shown in Fig. 5. The rock massis linearly elastic while the fill is modeled as a nonlinearelastoplastic Coulomb material. A one step mining and onestep backfilling sequence is considered; hence, wall conver-gence occurs before the backfill is put in place.

For this 2D problem, the following equations are used tocompare the analytical solution with some of the numericalmodeling results:

[22] σ γ κλ

λvhB

h B= − − − −13

1313

11[ exp( )]

with

[23] λ13 = K1(tan δ1 + tan δ3)

κ13 = c1 + c3 + 2c(tan δ1 + tan δ3)tan α1

In this sample case, the granular fill and walls have differ-ent friction properties: c1 = c3 = c = 0 kPa, δ1 = 25°, δ3 = φ =35°. As seen in Fig. 6, the proposed equations (with K = K0and Ka) appear suitable for this case. An interesting featureof the new solution (Fig. 6c) is that it accounts for the inter-

© 2005 NRC Canada

Li et al. 1709

Fig. 2. Calculated values of vertical (a), horizontal (b), and shear (c) stresses versus elevation h; B = 5 m, L = 10 m, c1 = c2 = c3 =c4 = c = 1 kPa, φ = 35°, γ = 20 kN/m3; the fill is at rest (K1 = K2 = K3 = K4 = K = 1 – sinφ and α1 = α2 = α3 = α4 = α = 0); threecases are concerned: a base case (δ1 = 10°, δ2 = 20°, δ3 = 30°, δ4 = 35°), the upper limit (δ1 = δ2 = δ3 = δ4 = 10°) and the lower limit(δ1 = δ2 = δ3 = δ4 = 35°).

nal shear stress developing in the fill when different proper-ties are used for the walls (which is confirmed by thenumerical results); other existing solutions do not includethis particular aspect of backfill response. Although the nu-

merical and analytical results shown in Fig. 6 (and others,not presented here) are not identical, they show the sametendencies with about the same stress magnitudes (additionalcomparisons are presented in Li et al. 2005). Considering

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1710 Can. Geotech. J. Vol. 42, 2005

Fig. 3. Calculated values of vertical (a), horizontal (b), transverse (c), and longitudinal shear (d) stresses versus elevation h; B = 5 m,L = 10 m, φ = 35°, γ = 20 kN/m3, δ1 = 10°, δ2 = 20°, δ3 = 30°, and δ4 = 35°; the fill is in an at rest state (see Table 1).

Fig. 4. Calculated values of vertical (a) and horizontal (b) stresses versus the ratio h/B obtained using 2D (eq. [19]) and 3D (eq. [18])solutions with the fill in an at rest state (see Table 1); B = 6 m, L = 10 m, c = 1 kPa, δ = φ = 30°, K = 1 – sin φ = 0.5, γ = 20 kN/m3.

the basic assumptions and simplifications made with the nu-merical modeling (e.g., Aubertin et al. 2003; Li et al. 2003),the correspondence between the two sets of results isdeemed satisfactory for this preliminary assessment.

Application to laboratory dataCentrifuge test results were reported by Take and

Valsangkar (2001). In these experiments, earth pressure cellswere mounted inside a backfilled box with aluminum walls.To investigate the effect of boundaries with dissimilar fric-tional characteristics, one of the aluminum surfaces was cov-ered with a sheet of 120A-grit sandpaper (Take 1998). Thebackfill used in these tests was a poorly graded sand with lit-tle or no fines. The maximum and minimum dry densitieswere 1.62 and 1.34 g/cm3, respectively. Angles of internalfriction (φ) and interface friction (δ) angles are given in Ta-ble 2. The centrifuge experiments were performed at an ac-celeration of 35.7g (g is the gravity acceleration) to simulatea 5 m high retaining wall prototype. With this acceleration,the equivalent unit weight of the dense backfill (having a79% relative density) is 0.554 MN/m3, while that of theloose backfill (34% relative density) becomes 0.508 MN/m3.Table 3 gives the conditions and parameters for the differentcases. Adopting a criterion based on a difference of 10% be-tween the stress values calculated with the 2D and 3D equa-tions introduced above, it can be showed that most of theexperimental conditions from these tests pertain to 3D situa-tions (i.e., all tests except for those with B = 15 mm). Hence,these experimental results can be used to assess the 3D na-ture of the newly developed solution.

A first series of tests was conducted on dense backfill.The walls were not covered, so φ = 36°, δ1 = δ2 = δ4 = δ3 =25°. Figure 7 shows the comparison between the experimen-

tal results and the proposed analytical solution (eqs. [13] and[18]). At rest (K0) and active (Ka) states are considered here(K1 = K2 = K3 = K4 = K and α1 = α2 = α3 = α4 = α; see Ta-ble 1 for values of K and α). The horizontal stress based onthe overburden calculation is also plotted for the four geom-etries considered (L = 25.4 cm and B = 1.5, 3.8, 7.5, and18.4 cm; see Table 3). The proposed analytical solution de-scribes the experimental results quite well, and confirms theexistence of stress transfer to the abutment (due to the arch-ing effect), which reduces the pressure on the walls (com-pared with that obtained from the overburden weight, givenby the dark straight lines in Fig. 7).

Another series of tests was performed by Take andValsangkar (2001) with this same dense backfill, with wall 2covered with sandpaper to simulate a rough surface. In thiscase, φ = 36°, δ1 = δ3 = δ4 = 25°, δ2 = 36°. There were no so-lutions available to analyze this type of test, which includestwo different fill-wall friction angles. Take and Valsangkar(2001) used 2D calculations with upper and lower parameterbounds to compare with their experimental results. Here, theproposed general analytical solution (eqs. [12] and [13]) isused to evaluate this situation in more detail. Figure 8 showsa comparison between the experimental results and the pro-posed solution for at rest and active states (K1 = K2 = K3 =K4 = K and α1 = α2 = α3 = α4 = α; see Table 1 for values ofK and α). Again, the newly developed equations describe theexperimental results well (with the parameters given in Ta-ble 3), thus showing the magnitude of the arching effect(i.e., difference between overburden and nonlinear solu-tions).

Take and Valsangkar (2001) have also conducted a seriesof tests with loose backfill (34% relative density). Themodel rock face (wall 2) was again covered with sandpaperto simulate a rough surface. The friction angles are φ = 30°,δ1 = δ3 = δ4 = 23°, δ2 = 32°. As indicated previously, whenthe fill-wall friction angle is larger than the fill material fric-tion angle φ, the latter is used for the interface, so δ2 = φ =30° is adopted here (see Table 3). Again, the proposed gen-eral solution (eqs. [12] and [13]) with an active or an at reststate (K1 = K2 = K3 = K4 = K and α1 = α2 = α3 = α4 = α; seeTable 1 for values of K and α) gives a fairly good predictionof the experimental results (Fig. 9).

These results tend to confirm the validity of the proposed3D solution, which may be used to evaluate the stress statein vertical narrow backfilled openings. The results obtainedare not nearly perfect (by any means), but the new equationsnevertheless provide a useful approximation of the pressure(or pressure range) that can develop under typical situations.This type of analytical approach is practical for a prelimi-nary assessment, especially in the early stages of a project,and also for comparing various design scenarios (with vari-ous configurations and (or) material properties).

Discussion

The general 3D analytical solution presented in this paperhas been developed based on the arching theory, using theMarston (1930) solution as a starting point. The extendedsolution has been shown in the previous sections to providerepresentative estimates for the earth pressures in backfilledopenings with walls that can have different interface proper-

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Li et al. 1711

Fig. 5. Numerical model geometry of a narrow backfilled stope(not to scale) for FLAC-2D (Itasca 2002); the main propertiesfor the rock mass and backfill are given using classicalgeomechanical notations.

ties. However, the potential user should keep in mind thatthis analytical solution is based on some simplifying as-sumptions. Some key aspects are recalled below.

The main equations (eqs. [12], [13], and [15]) are basedon a solution that divides the narrow, vertical backfilledopening into thin horizontal layer elements on which acts auniform state of stress. The contact stress between the filland walls is expressed as a function of the coefficient of lat-eral earth pressure Kci (i = 1–4). When the wall does notmove, an “at rest” condition can theoretically be adopted todefine Kci (= K0, see eq. [6]). However, for the many caseswhere an outward displacement is expected to occur at theinterface, the value of Kci has become a topic of investiga-tion and discussion for many arching problems (e.g.,

Krynine 1945; Handy 1985, 2004; Blight 1986a; Frydmanand Keissar 1987; Harrop-Williams 1989; Iglesia et al.1999).

In many practical applications of arching equations ensu-ing from Janssen’s (1895) basic theory, problems with out-ward movement of the walls have been linked to active earthpressure conditions, hence the Rankine active earth pressurecoefficient is often used (i.e., Kci = Ka). The use of Ka is,however, known to be strictly (theoretically) incorrect as theRankine active coefficient defines the ratio of the principalstresses σ3/σ1 (where σ3 and σ1 are the minor and majorprincipal stress, respectively). For an opening backfilledwith a soft, yielding material, shearing along the interfaceinduces a local rotation of the stresses. Thus, the horizontaland vertical stresses acting on the wall are not the principalstresses if a shear stress is induced (as is expected to occurwith frictional materials).

Both analytical solutions and numerical modeling resultsshow that there is indeed a rotation of the principal stresses,and that the stresses are not distributed uniformly in the hor-izontal layer element (e.g., Handy 1985; Li et al. 2003).

Data presented by Frydman and Keissar (1987) show thatthe value of the σh/(γh) ratio (horizontal stress over verticaloverburden pressure) is typically close to K0 near the surfaceof the fill, and progressively decreases with depth to values

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Fig. 6. Comparison between the numerical model and the analytical solutions for vertical (a), horizontal (b), and shear (c) stresses ver-sus the opening height, with c1 = c3 = c = 0 kPa, δ1 = 25°, δ3 = φ = 35°. LW, left wall of the opening; VCL, vertical central line.

Loose backfill* Dense backfill†

φ backfill (°) 30 36δ backfill and aluminium (°) 23 25δ backfill and sandpaper (°) 32 36

*34% relative density, density 1.42 g/cm3.†79% relative density, density 1.55 g/cm3.

Table 2. Internal (φ) and interface (δ) friction angles for thebackfill (after Take and Valsangkar 2001).

below Ka (down to about 1/3 to 1/2 Ka). This would tend toindicate that adopting Kci = Ka would overestimate σh. How-ever, with the approach proposed here, Kci has been definedas the ratio of the horizontal (σhh) to vertical (σvh) stress at agiven position h. In this definition of K ic , the value of σvh isobtained from the arching equation, so it can be muchsmaller than the overburden weight of the fill column. Com-parisons made with both numerical and experimental results

tend to indicate that the calculated value of σhh is generallyclose to the expected range of value. Hence, the simplifyingassumptions adopted to develop the analytical solution andto define Kci (with the common definitions for Ko and Ka)are seen to provide realistic estimates of the earth pressureon the opening boundaries.

Considering the other uncertainties encountered in suchtypes of geotechnical applications (including opening geom-

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Li et al. 1713

Fig. 7. Comparison between the proposed solution (eqs. [13] and [18]) and experimental results obtained on 3D physical model back-filled with dense sand (data taken from Take and Valsangkar 2001; see Table 3 for details).

Geometry Fill parameters Wall-fill friction angle* (°)

Figure Backfill B (cm) × L (cm) c γ† φ (°) δ1 δ2 δ3 δ4

7a Dense 18.4×25.4 0 554 36 25 25 25 257b Dense 7.5×25.4 0 554 36 25 25 25 257c Dense 3.8×25.4 0 554 36 25 25 25 257d Dense 1.5×25.4 0 554 36 25 25 25 258a Dense 18.4×25.4 0 554 36 25 36 25 258b Dense 7.5×25.4 0 554 36 25 36 25 258c Dense 3.8×25.4 0 554 36 25 36 25 258d Dense 1.5×25.4 0 554 36 25 36 25 259a Loose 3.8×25.4 0 508 30 23 32* 23 239b Loose 1.5×25.4 0 508 30 23 32* 23 23

*The backfill friction angle, φ, is used when the wall-fill friction angle δ is higher than φ.†Values obtained by multiplying the backfill density by the centrifuge acceleration (in kN/m3).

Table 3. Parameters used in the calculation results presented in Figs. 7–9, based on data taken fromTake and Valsangkar (2001).

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1714 Can. Geotech. J. Vol. 42, 2005

etry and material properties), the proposed solution is henceconsidered an acceptable compromise with more rigorousdefinitions of earth pressure (when K0 and Ka are used forKci).

A similar argument could be made for conditions where

there is an inward displacement of the walls. This could bethe case, for example, with cut and fill stopes excavated andbackfilled progressively. In such situations, depending onthe amount of wall convergence and backfill compression,the confined fill can even be submitted to a horizontal pres-

Fig. 8. Comparison between the proposed general solution (eqs. [12] and [13]) and the experimental results obtained on a 3D physicalmodel backfilled with dense sand (data taken from Take and Valsangkar 2001; see Table 3 for details).

Fig. 9. Comparison between the proposed solution (eqs. [12] and [13]) and the experimental results obtained on a 3D physical modelbackfilled with loose sand (data taken from Take and Valsangkar 2001; see Table 3 for details).

sure that exceeds the value obtained from a Rankine passivecoefficient Kp (e.g., Aubertin et al. 2003). This aspect, how-ever, requires more work before it can be assessed morecompletely, as wall sizes, interface properties, and filling se-quence are known to have a large influence on such passiveearth pressure (e.g., Bransby and Smith 1975; Li et al. 2003).

Another aspect that will require more work is the influ-ence of pore pressure under saturated and unsaturated condi-tions, and backfill material response in confined openings.

Finally, even if the solution proposed here was developedfor vertical openings, it could also be applicable for sub-vertical openings when the stresses on each side have aboutthe same magnitude (e.g., Knutsson 1981). A more completeextension to inclined openings is nevertheless needed forgeneral applications and has been the object of other publi-cations (e.g., James et al. 2004).

Conclusion

The stress components in backfilled openings are affectednot only by the weight of the fill, but also by interaction be-tween the fill and the opening walls. Some of the stress inthe backfill is thus transferred to the adjacent walls throughshearing forces between the fill and walls. This leads to adecrease of stress in the fill compared to the overburdenstress. This phenomenon is known as the arching effect.

Based on the arching theory, a 3D analytical solution wasdeveloped using the Marston method as a starting point toevaluate the earth pressures in narrow, vertical backfilledopenings. Both the 3D opening geometry and the fill-wallinterface properties were taken into account. It has beenshown that pressure estimations, which ignore differentshear properties along the four walls of the opening, maygenerate some significant errors. The proposed solution can,however, provide representative estimates for the earth pres-sures in backfilled openings with four walls having differentinterface properties. It has also been shown that the openinglength has a significant influence on the earth pressure in thebackfill. The length to width ratio at which the 2D solutionbecomes significantly different from the 3D solution is afunction of opening size and material property. The versatil-ity and descriptive capability of the proposed solution weredemonstrated by comparison with a numerical model andwith experimental results taken from the literature. The pro-posed analytical solution can thus be used to estimate therange of earth pressure in narrow, vertical backfilled open-ings, including mining stopes and trenches.

Acknowledgement

The authors acknowledge the financial support from theInstitut de Recherche Robert-Sauvé en Santé et en Sécuritédu Travail du Québec (IRSST) and from the participants ofthe Industrial NSERC Polytechnique-UQAT Chair (http://www.polymtl.ca/enviro-geremi/). The authors thank JohnMolson and Michael James for their review of this manu-script.

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List of symbols

B opening widthc fill cohesionci cohesion of the ith fill-wall interface (i = 1–4)Ci lateral compressive forces on the ith wall (i = 1–4)dh thickness of layer element

g the gravity accelerationh elevation in backfillH backfill heightKi reaction coefficient for cohesionless material on the ith

wall (i = 1–4)K0 at rest reaction coefficient for cohesionless materialKa active reaction coefficient for cohesionless materialKci coefficient of lateral pressure (i = 1–4) for cohesive or

cohesionless fill materialKp passive reaction coefficient for cohesionless material

L opening length

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SB (and SB + dSB) backfill material transverse internalshearing force

Si vertical fill-wall shearing force on the ith wall (i = 1–4)SL (and SL + dSL) backfill material longitudinal internal

shearing forceTi horizontal fill-wall shearing force on the ith wall (i = 1–4)V (and V + dV) internal vertical forceW weight of the backfillz depth of a modeled stope

αi backfill state angle on the ith wall (i = 1–4)δ a unique value of wall-fill friction angle

δi the friction angle of the ith fill-wall interface (i = 1–4)φ fill friction angleγ unit weight of material (backfill or rock mass)

κ13, κ24 parameters defined in eq. [10]λ13, λ24 parameters defined in eq. [11]

ν Poisson’s ratioσhhi horizontal stress on the ith wall (i = 1–4)σvh vertical stress at position hτB transverse shear stressτL longitudinal shear stress

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