International Journal of Mining Science and Technology 24 (2014) 81–85

Contents lists available at ScienceDirect

International Journal of Mining Science and Technology

journal homepage: www.elsevier .com/locate / i jmst

Prediction of plastic zone size around circular tunnels in non-hydrostaticstress field

2095-2686/$ - see front matter � 2014 Published by Elsevier B.V. on behalf of China University of Mining & Technology.http://dx.doi.org/10.1016/j.ijmst.2013.12.014

⇑ Corresponding author. Tel.: +98 936 984 0126.E-mail address: behnambagheri10405@yahoo.com (B. Behnam).

Behnam Bagheri a,⇑, Fazlollah Soltani a, Hamid Mohammadi b

a Geotechnical Department, Faculty of Civil Engineering, Graduate University of Advanced Technology, Kerman 7631133131, Iranb Mining Engineering Department, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman 7618868366, Iran

a r t i c l e i n f o

Article history:Received 5 May 2013Received in revised form 10 June 2013Accepted 20 July 2013Available online 3 January 2014

Keywords:Plastic zone radiusConvergence–confinement methodNon-hydrostatic conditionStress ratio

a b s t r a c t

This paper discusses the calculation of plastic zone properties around circular tunnels to rock-masses thatsatisfy the Hoek–Brown failure criterion in non-hydrostatic condition, and reviews the calculation ofplastic zone and displacement, and the basis of the convergence–confinement method in hydrostatic con-dition. A two-dimensional numerical simulation model was developed to gain understanding of the plas-tic zone shape. Plastic zone radius in any angles around the tunnel is analyzed and measured, usingdifferent values of overburden (four states) and stress ratio (nine states). Plastic zone radius equationswere obtained from fitting curve to data which are dependent on the values of stress ratio, angle andplastic zone radius in hydrostatic condition. Finally validation of this equation indicate that results pre-dict the real plastic zone radius appropriately.

� 2014 Published by Elsevier B.V. on behalf of China University of Mining & Technology.

1. Introduction

Due to tunnel excavation in the rock mass, it occurs redistribu-tion for stresses in surrounding tunnel. By developing the plasticzone around the tunnel, radial convergence sets in, resulting inthe reduction of stresses in rock mass [1,2].

The prediction of the response of rock masses (often in terms ofstresses and displacements) is an essential part of the design of anengineering structure in/on rock. Convergence confinement methodis one of the analysis methods that used for investigating interactionbetween rock displacements within plastic zone around the tunneland the characteristics of support system. Generally, with consider-ing analytical solutions, the convergence confining method hasbeen applied to circular tunnels in a hydrostatic stress field [3–16].

The shape and size of plastic zone have an important role inthe results of convergence confinement method, therefore it mustbe determined with high accuracy. Generally, with consideringthe ratio between horizontal stress and vertical stress equal 1(hydrostatic stress filed), the shape of plastic zone surroundingcircular tunnels is considered as a circular zone, whereas theplastic zone shape around the tunnel changes from the horizontalbutterfly to horizontal parabolic, circular, vertical parabolic andvertical butterfly shapes [16–19]. In reality, the analytical solutionscan not considered the non-hydrostatic stress field, thus there isnot any perfectly analytical formulation for calculating the plasticzone radius.

The main purpose is to present a series of semi-analytical rela-tionships to predict shape and radius of plastic zone around a cir-cular tunnel under the different non-hydrostatic stress fields.Therefore, the relationship between plastic zone radius in non-hydrostatic and hydrostatic condition is analytically obtainedbased on the some assumptions as follows: (1) circular tunnelcross section; (2) two-dimensional plane strain; and (3) rockmasses satisfy the Hoek–Brown failure criterion.

By using two-dimensional elasto-plastic finite element stressanalysis program (Phase 2) for divided cases of stress ratio at eachangles, the non-hydrostatic plastic zone equations have been ex-plored with one independent variables (stress ratio) for eachangles.

2. Calculation of plastic zone and displacement in hydrostaticcondition

The analytical methods were carried out to determine the sizeof the plastic zone, and surrounding a circular tunnel subjectedto a hydrostatic condition is dependent on the kind of failurecriterion that defined by: (1) based on the Mohr–Coulomb failurecriterion; (2) based on the Hoek–Brown failure criterion. Theanalysis that is based on the Mohr–Coulomb failure criterion givesa very simple solution for the progressive failure of the rock masssurrounding a circular tunnel [16,20,21].

In this analysis, Duncan Fama et al. assumed that the surround-ing heavily jointed rock mass behaves as an elastic-perfectly plasticmaterial in which failure involving slip along intersecting

82 B. Behnam et al. / International Journal of Mining Science and Technology 24 (2014) 81–85

discontinuities is assumed to occur with zero plastic volumechange [22,23].

In the present work, an analytical solution derived by Carranza-Torres and Fairhurst is based on the general form of the Hoek–Brown criterion proposed by Londe [20,22]. Assume that a circulartunnel of radius r0 is subjected to hydrostatic stresses p0 and a uni-form internal support pressure pi while the stress ratio and therock mass compressive strength are defined by k and rcm, respec-tively. The uniform internal pressure Pi and far-field stress S0 aregiven by Carranza-Torres et al. [3,9,24]:

Pi ¼pi

mbrciþ s

m2b

ð1Þ

Fig. 1. Two-dimensional numerical mesh.

S0 ¼r0

mbrciþ s

m2b

ð2Þ

The actual (i.e., non-scaled) critical pressure is given by Eq. (3).

Pcri ¼

116

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 16S0

ph i2� S

m2b

� �mbrci ð3Þ

On the condition that pi > pcri , the relationship between the internal

pressure pi in the elastic part of the GRC and radial displacementsuel

r is given by Eq. (4).

Uelr ¼

r0 � pi

2GrmR ð4Þ

where Grm is the shear modulus of the rock-mass. Only if pi < pcri ,

the extent of the plastic region Rpl that develops around the tunnelis Eq. (5).

Rpl ¼ R exp 2ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipcr

i �ffiffiffiffipipqÞ

� �ð5Þ

Eq. (5) shows that by increasing the radius of tunnel and field stress,the plastic zone radius is also increased. Because of hydrostatic con-dition, the effect of stress ratio in expanding or decreasing of theplastic zone radius is not considered.

So for investigating the effect of stress ratio in non-hydrostaticcondition, the tunnel geometry and rock mass properties must beconstant.

Fig. 2. Yielded elements contouring lines.

(a) k=0.5 (b) k=0.8 (c) k=1

Fig. 3. Plastic zone shape.

3. Calculation of plastic radius in non-hydrostatic conditionaround circular tunnels

3.1. Stepwise procedure for calculation of plastic radius

The main objective of this section is to find the equations fordemonstration of relationship between plastic zone radius andstress ratio corresponding to the angles in non-hydrostaticcondition.

The equations obtained from numerical analyses results withPhase 2. Nine different values of stress ratio (k = 0.5, 0.6, 0.7, 0.8,1, 1.25, 1.5, 1.75 and 2) in four significant depths were selectedin this study and then divided stress ratio into two splits: firstly,stress ratio is smaller than one (k 6 1); secondly, stress ratio is lar-ger than one (k P 1). After that the plastic radius is calculatedalong a radius related to the angles which were chosen from therange of 0–90� with constant step angle of 10� for each stress ratio.

Then analyses procedure evaluates coefficient k and averageof factor d in four significant depth interfaces in the figures,in which factor of d and Rp are, respectively, the ratioRp(k–1)/Rp(k=1) and the plastic zone radius of rock mass. So that,the equations will be resulted by fitting the linear or polynomialtrend line to data values.

3.2. Numerical modeling

The elasto-plastic finite element program for calculating stres-ses and displacements around underground openings in 2-dimen-sional (Phase 2, Itasca, 2009) was used in the study. The tunnelsection is circular with 5.64 m radius (R) and was excavatedthrough very poor-quality rock mass with geological strength in-dex (GSI) = 30. The two-dimensional numerical mesh and bound-ary size are shown in Fig. 1. For the number of nodes that wasconsidered for meshing the model was equal to 300 on theexcavation.

The rock mass surrounding the tunnel is assumed to be homo-geneous and isotropic, and obeys Hoek–Brown failure criterion.The parameters used are as follows: GSI = 30; rci = 20 MPa (intactrock); Hoek–Brown constant mi = 8; rock mass compressivestrength rcm = 1.7 MPa; deformation modulus Em = 1400 MPa; fric-tion angle U = 24�; cohesive strength c = 0.55 MPa; and Poisson’sratio m = 0.25; Dilation angle a = 0�. A way to confide the valuesof plastic zone measurements is to consider the average of plasticzone in four significant depths. For this purpose the depths that

B. Behnam et al. / International Journal of Mining Science and Technology 24 (2014) 81–85 83

were considered are 140, 223, 315 and 445 m. This model is not in-cluded the self-weight rock mass, so the acceleration of gravity hasbeen set equal to zero. The state of stress has been applied asfollows:

0.2 0.4 0.6 0.8 1.0 1.200.95

1.00

1.05

1.10

1.15

1.20

k

Fact

or o

f d

y=0.427x2+0.330x+1.097

R2=0.999

Fig. 4. Relation between factor of d and stress ratio for h = 0.

(a) k=1 (b) k=1.5 (c) k=2

Fig. 5. Plastic zone shape.

0.20.40.6

1.25 1.50 2.00k

0.81.01.21.4

0 0.25 0.50 0.75 1.00 1.75 2.25

Fact

or o

fd

y=0.756x2 2.052x+2.296R2=0.995

Fig. 6. Relation between factor of d and stress ratio for h = 0.

Table 1Results of the comparison of the two different cases.

Arbitrary angle (�) Factor of d

k 6 1 0 �0.427k2 + 0.10 �0.2386k2 + 020 0.8376k2 � 1.30 3.2529k2 � 5.40 8.4288k2 � 1350 3.2175k2 � 4.60 2.4158k2 � 3.70 0.6111k2 � 0.80 0.1468k2 + 0.790 0.2208k2 + 0.7

k P 1 0 0.688k2 � 1.910 0.7566k2 � 2.20 1.0411k2 � 2.30 0.8194k2 � 1.40 2.488k2 � 6.150 2.384k2 � 5.360 0.4309k2 � 0.70 �0.2469k2 + 180 �0.0434k2 + 090 0.2171k2 + 1.4

rv ¼ c � z ð6Þ

where c is the unit weight of the rock mass; z the depth below sur-face; and rv the vertical stress. Usually the ratio of the average hor-izontal stress to the vertical stress is denoted by the letter k’’.

k ¼ rh

rvð7Þ

Before excavating the face, for applying non-uniform internal pres-sure that is corresponded with external pressure, core replacement(material softening) technique was used. So a plane strain modelthat sequentially replaces and reduces the modulus of the materialinside the excavation over a number of stages was built.

The final stage, with the material excavated inside the tunnel,will be used to determine plastic zone radius. After analyzing themodel, for displaying the exact interpolated values of data at anypoint on the contour plots, the number of yielded elements con-touring lines is increased to 60 and the length of the plastic zonefor each cases was measured on 28th contour (see Fig. 2).

3.3. Plastic zone radius along a radius corresponding to the angle

In general, the plastic zone radius obtained from the numericalanalysis that is under non-uniform pressure. For non-hydrostaticpressure the plastic zone radius Rp(k–1), can be evaluated in closedform using Eq. (5) if the factor of d is defined from a linearequation.

d ¼ RPðk–1Þ

RPðk¼1Þ! RPðk–1Þ ¼ d� RPðk¼1Þ ð8Þ

However, this equation did not see the other parameter of rockmass but for calculating the hydrostatic plastic zone radius, andthe importance of them will be illustrated. This is normalized ofplastic zone radius that is done for clarity and better understandingof the stress ratio’s constituents on the plastic zone radius devel-oped in the tunnel.

3.3.1. Stress ratio for k 6 1The shape of the plastic zone in rock masses for k 6 1 will be

changed from circular to horizontal Parabola and butterfly (seeFig. 3).

For k = 1, the factor of d is equal to one for any angles because ofcircular plastic zone shape. By decreasing k (0.6 6 k < 1), the factorof d for angles equal to 0� to 30� will get increased but for angles

Rp(k–1)

3301k + 1.097 (�0.427k2 + 0.3301k + 1.097) ⁄ Rp(k=1)

.0695k + 1.1713 (�0.2386k2 + 0.0695 k + 1.1713) ⁄ Rp(k=1)

6415k + 1.8109 (0.8376k2 � 1.6415 k + 1.8109) ⁄ Rp(k=1)

5251k + 3.2861 (3.2529k2 � 5.5251 k + 3.2861) ⁄ Rp(k=1)

.708k + 6.2719 (8.4288k2 � 13.708 k + 6.2719) ⁄ Rp(k=1)

7967k + 2.581 (3.2175k2 � 4.7967 k + 2.581) ⁄ Rp(k=1)

2656k + 1.8514 (2.4158k2 � 3.2656 k + 1.8514) ⁄ Rp(k=1)

1074k + 0.5035 (0.6111k2 � 0.1074 k + 0.5035) ⁄ Rp(k=1)

661k + 0.0953 (0.1468k2 + 0.7661 k + 0.0953) ⁄ Rp(k=1)

282k + 0.0591 (0.2208k2 + 0.7282 k + 0.0591) ⁄ Rp(k=1)

008k + 2.2112 (0.688k2 � 1.9008 k + 2.2112) ⁄ Rp(k=1)

0521k + 2.2961 (0.7566k2 � 2.0521 k + 2.2961) ⁄ Rp(k=1)

7294k + 2.6984 (1.0411k2 � 2.7294 k + 2.6984) ⁄ Rp(k=1)

9883k + 2.1663 (0.8194k2 � 1.9883 k + 2.1663) ⁄ Rp(k=1)

788k + 4.743 (2.488k2 � 6.1788 k + 4.743) ⁄ Rp(k=1)

536k + 3.9698 (2.384k2 � 5.3536 k + 3.9698) ⁄ Rp(k=1)

2226k + 0.7586 (0.4309k2 � 0.2226 k + 0.7586) ⁄ Rp(k=1)

.4902k � 0.271 (�0.2469k2 + 1.4902 k � 0.271) ⁄ Rp(k=1)

.9831k + 0.0575 (�0.0434k2 + 0.9831 k + 0.0575) ⁄ Rp(k=1)

53k � 0.242 (0.2171k2 + 1.453 k � 0.242) ⁄ Rp(k=1)

84 B. Behnam et al. / International Journal of Mining Science and Technology 24 (2014) 81–85

equal to 30� to 90� it will get decreased. It means that plastic zoneshape changes from circular to horizontal Parabola. For k < 0.6, thefactor of d based on horizontal butterfly plastic zone gets changed.

In this section for example for angle equal to 0� the relation be-tween factor of d and stress ratio is plotted and trend line is drawnfor achieving the equation (see Fig. 4).

3.3.1. Stress ratio for k P 1The shape of the plastic zone in rock masses for k P 1 will be

changed from circular to vertical Parabola and butterfly, as shownin Fig. 5. The relationship between factor of d and stress ratio for1 < k 6 1.75 are indicated the procedure of increasing or decreasingthe factor of d is approximately inverse of the procedure for0.6 6 k 6 1. For achieving the plastic zone equations for k P 1,the trend line of the plotted graph in relationship to factor of dand stress ratio is drawn. The graph of factor of d in relationshipto stress ratio for angle equal zero is shown in Fig. 6.

3.3.2. Analysis of the graphsAfter plotting the plastic zone radius in related k, the equations

were obtained. These equations are relying on the angles. Theequations consist of two main parameters: stress ratio and hydro-static plastic zone radius. By increasing the diameter of tunnel, itcan be expected that the equations show the extension of plasticzone. This is demonstrated the effect of hydrostatic plastic zone ra-dius in these equations. Table 1 shows the plastic zone radius alonga radius corresponding to the angle equations for k – 1.

4. Validation

In this section, the results of proposed equations are comparedwith Andrea’s results [25]. In order to show the differences be-tween the proposed equations for the calculation of equivalentplastic zone radius in tunnelling and Andrea’s results, the valuesof factor of d0 which are defined from a linear equation Eq. (9)are shown in Tables 2 and 3.

d0 ¼ DRPðk–1Þ=DRPðk¼1Þ ! d0 ¼ ðRPðk–1Þ � R0Þ=ðRPðk¼1Þ � R0Þ ð9Þ

Table 2Results obtained with Phase 2.

Factor of d0 k = 0.5 k = 0.75 k = 1 k = 1.5

Case A Min 0.00 0.430 1 0.80Max 1.32 1.210 1 3.98

Case B Min 0.11 0.540 1 0.84Max 1.26 1.170 1 3.42

Case C Min 0.21 0.590 1 0.86Max 1.23 1.150 1 3.14

Case D Min 0.30 0.630 1 0.87Max 1.20 1.141 1 2.93

Table 3Andrea’s results.

Factor of d0 k = 0.5 k = 0.75 k = 1 k = 1.5

Case A Min 0.45 0.70 1 0.79Max 1.11 1.04 1 1.47

Case B Min 0.42 0.66 1 0.85Max 1.11 1.04 1 1.47

Case C Min 0.44 0.67 1 0.87Max 1.16 1.07 1 1.57

Case D Min 0.44 0.65 1 0.86Max 1.17 1.08 1 1.50

5. Conclusions

The formulations of the plastic zone radius in non-hydrostaticcondition in terms of the Hoek–Brown criterion have been pre-sented in this study. In this numerical procedure, the effects ofnon-hydrostatic condition and variable angles within the plasticzone radius are considered.

The following main conclusions can be drawn:

(1) In order to validate the proposed equations, the results ofplastic zone radius at the tunnel section are compared withAndrea’s results. However, each of them used two differentnumerical methods whereas both of them are illustrated asimilar procedure for factor of d0.

(2) The effect of field stress increments within the plastic regionshape is investigated by comparing the results of cases. Inthe Hoek–Brown rock mass under non-hydrostatic condi-tion, the shape of the plastic zone in rock masses will bechanged from horizontal to vertical butterfly by increasingthe field stress.

(3) The effect of variable stress ratio within the plastic zoneradius is investigated by relationship between factor of dand variable stress ratio in constant angles.

(4) The effects of variable angle and stress ratio in the plasticzone radius are investigated by using curve fitting to datafrom formulations which were obtained from 2D curvefitting.

References

[1] Rammamurthy T. Engineering in rock for slopes, foundations and tunnel. NewDelhi: PHI Learning Private Limited; 2010.

[2] Golshani A, Oda M, Okui Y, Takemura T, Munkhtogoo E. Numerical simulationof the excavation damaged zone around an opening in brittle rock. Int J RockMech Min Sci 2009;44:835–45.

[3] Carranza-Torres C, Fairhurst C. Application of the convergence–confinementmethod of tunnel design to rock masses that satisfy the Hoek–Brown failurecriterion. Tunn Undergr Space Technol 2000;15(2):187–213.

[4] Oreste PP. Analysis of structural interaction in tunnels using the convergence–confinement approach. Tunn Undergr Space Technol 2003;18(4):347–63.

[5] Corbetta F, Bernaud D, Minh DN. Contribution to the convergence–confinement method by the principle of similitude. Revue Francaise deGeotechnique 1991;54:5–11.

[6] Oreste PP, Pella D. Modeling progressive hardening of shotcrete inconvergence-confinement approach to tunnel design. Tunn Undergr SpaceTechnol 1997;12(3):425–31.

[7] Oreste PP, Peila D. Radial passive roc bolting in tunnelling design with a newconvergence–confinement model. Int J Rock Mech Min Sci Geomech Abstr1996;33(5):443–54.

[8] Gill DE, Leite MH, Labrie D. Designing mine pillars with the convergence-confinement method. Int J Rock Mech Min Sci 1994;31(6):687–98.

[9] Alejano LR, Alonso E, Rodriguez-Dono A, Fernandez-Manin G. Application ofthe convergence-confinement method to tunnels in rock masses exhibitingHoek–Brown strain-softening behavior. Int J Rock Mech Min Sci2010;47(1):150–60.

[10] Gonzalez-Nicieza C, Alvarez-Vigil AE, Menendez-Diaz A, Gonzalez-Palacio C.Influence of the depth and shape of a tunnel in the application of theconvergence-confinement method. Tunn Undergr Space Technol2008;23:25–37.

[11] Gschwandtner GG, Robert G. Input to the application of the convergenceconfinement method with time-dependent material behavior of the support.Tunn Undergr Space Technol 2012;27(1):13–22.

[12] Dias D. Convergence–confinement approach for designing tunnel facereinforcement by horizontal bolting. Tunn Undergr Space Technol2011;26(4):517–23.

[13] Kitagawa T, Kumeta T, Ichizyo T, Sato S, Sota M, Yasukawa M. Application ofconvergence confinement analysis to the study of preceding displacement of asqueezing rock tunnel. Rock Mech Rock Eng 1991;24(1):31–51.

[14] Fahimifar A, Ranjbarnia M. Analytical approach for the design of active groutedrock bolts in tunnel stability based on convergence–confinement method.Tunn Undergr Space Technol 2009;24(4):363–75.

[15] Eisenstein Z, Branco P. Convergence–confinement method in shallow tunnels.Tunn Undergr Space Technol 1991;6(3):343–6.

[16] Panet M. Lecalcul des tunnels par la method convergence–confinement. Paris: Presses de’ ENPC l; 1995.

[17] Daemen JJK. Tunnel support loading caused by rock failure. TwinCities: University of Minnesota; 1975.

B. Behnam et al. / International Journal of Mining Science and Technology 24 (2014) 81–85 85

[18] Wei YG. Perturbation solutions for elasto-plastic analysis of circular tunnelunder unequal compression in two directions. Chin J Geotech Eng1990;12(4):11–20.

[19] Lombardi G, Amberg W, Rechsteiner G. Une methode de calcul elasto-plastique del’etat de tension de deformation autourd’ unecavitèsu uterraine.In: Proceedings of III Congress International de la SIMR. Denver, 1974.

[20] Carranza-Torres C, Fairhurst C. The elasto-plastic response of undergroundexcavations in rock masses that satisfy the Hoek–Brown failure criterion. Int JRock Mech Min Sci 1999;36(6):777–809.

[21] Sofianos AI, Nomikos PP. Equivalent Mohr–Coulomb and generalized Hoek–Brown strength parameters for supported tunnels in plastic or brittle rock. Int JRock Mech Mining Sci 2006;43:683–704.

[22] Londe P. Discussion on the determination of the shear stress failure in rockmasses. ASCE J Geotech Eng Div 1988;114(3):374–6.

[23] Duncan-Fama ME. Numerical modeling of yield zones in weak rocks.Comprehensive rock engineering, vol. 2. Pergamon Press; 1993. p.49–75.

[24] Sharan SK. Elastic–brittle–plastic analysis of circular openings in Hoek–Brownmedia. Int J Rock Mech Min Sci 2003;40:817–24.

[25] Andrea A, Michela C, Pietro C. Convergence-confinement method: limit ofapplication of the closed form solutions compared with numerical models.ECCOMAS Austria: Thematic Conference on Computational Methods inTunnelling, 2007.