A Numerical Procedure for the Fictitious Support Pressure in the Application of the Convergence–Confinement Method for Circular Tunnel Design

7242019 A Numerical Procedure for the Fictitious Support Pressure in the Application of the ConvergencendashConfinement Methellip

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A numerical procedure for the 1047297ctitious support pressure in theapplication of the convergencendashcon1047297nement method for circulartunnel design

Lan Cui Jun-Jie Zheng n Rong-Jun Zhang Han-Jiang LaiInstitute of Geotechnical and Underground Engineering Huazhong University of Science and Technology Wuhan 430074 China

a r t i c l e i n f o

Article historyReceived 6 February 2015Received in revised form25 June 2015Accepted 3 July 2015

Keywords

Fictitious support pressureCircular openingTunnel face effectFinite difference methodStrain-softening behaviour

a b s t r a c t

In the vicinity of a tunnel face the 1047297ctitious support pressure represents the support pressure acting onthe periphery of the tunnel provided by the rock mass itself It is an indicator of the self-supportingcapacity of the rock massThis paper aims at solving the 1047297ctitious support pressure on a circular tunnelwith hydrostatic initial stress 1047297eld A numerical procedure for the 1047297ctitious support pressure is proposedfor the elastic-perfectly-plastic elastic-brittle and strain-softening rock masses The procedure is com-posed of two steps 1047297rst the ground reaction curve (GRC) and the longitudinal deformation pro1047297le (LDP)are solved by a modi1047297ed numerical approach in this step the 1047297nite difference method is utilised toderive the strain components the stress components and the radius of the plastic zone then by cou-pling the GRC and LDP a 1047297ctitious support pressure is obtained by a simpli1047297ed approach By using theproposed procedure the in1047298uencing factors of the 1047297ctitious support pressure such as the critical plasticsoftening parameter the rock mass quality the dilatancy angle and the initial stress condition are studiedindividually The results indicate that the effect of the critical plastic softening parameter on the 1047297ctitioussupport pressure especially for the rock mass with good quality is obvious For a rock mass with strongdilatancy behaviour the 1047297ctitious support pressure ahead of the tunnel face decreases rapidly The elasticassumption of the rock mass behaviour will underestimate the stress relief factor by a large extent

amp 2015 Elsevier Ltd All rights reserved

1 Introduction

The convergencendashcon1047297nement method (CCM) is a widely usedtool to estimate the load imposed on the support It suggests thatin the vicinity of a tunnel face part of load is carried by the faceitself this is called the tunnel face effect1 The 1047297ctitious supportpressure is a surrogate for the presence of the tunnel face effect

Based on whether the 1047297ctitious support pressure is consideredor not the existing studies deal with the CCM mainly by two ap-proaches By the traditional approach2ndash6 the intersection of thesupport characteristic curve (SCC) and the ground reaction curve(GRC) is regarded as the equilibrium point of the ground-supportsystem The GRC can be estimated by the analytical approach forthe elastic-perfectly-plastic and elastic-brittle rock mass 7-9 andthe numerical approach for the strain-softening rock mass10-12The SCC can be obtained for the shotcrete steel set and ungroutedbolt by the empirical equations presented in Ref 1 and the com-posite support in Ref 4 The traditional approach provides a

convenient tool to optimise the support design However thedissipation of the tunnel face effect or the decrease of the 1047297ctitioussupport pressure is not re1047298ected by the traditional approachMoreover the support structures and rock reinforcement inpractical cases can be fairly complex In most cases the rock boltsare non-uniformly distributed and different types of support areinstalled In this aspect the elastic stiffness of SCC is dif 1047297cult torepresent the support properties

The improved approach can overcome the above limitations It

introduces the 1047297

ctitious support pressure and analyses the me-chanical behaviour of the support-ground system morerealistically13-19 Before the support installation the rock mass isassumed to be supported by the 1047297ctitious support pressure Afterthe support installation as the tunnel face moves forwards theprevious 1047297ctitious support pressure is gradually unloaded and thesupport or reinforcement deforms and carries the load This stagecan be realised by the numerical method A wide range of supporttypes can be simulated

For the studies with the improved approach the 1047297ctitioussupport pressure at the tunnel face was usually simpli1047297ed as 50ndash

70 of the initial stress based on the assumption that the rock

Contents lists available at ScienceDirect

journal homepage wwwelseviercomlocateijrmms

International Journal of Rock Mechanics amp Mining Sciences

httpdxdoiorg101016jijrmms2015070011365-1609amp 2015 Elsevier Ltd All rights reserved

n Corresponding authorE-mail address zhengjjhusteducn (J-J Zheng)

International Journal of Rock Mechanics amp Mining Sciences 78 (2015) 336 ndash349



mass is an elastic material and undergoes a displacement of ap-proximately 30ndash50 of the 1047297nal displacement13-17 Actually the1047297ctitious support pressure is found to be in1047298uenced by the me-chanical characteristics of the rock mass such as the plastic soft-ening behaviour the dilatancy behaviour the initial stress condi-tions and rock mass quality The 1047297ctitious support pressure at acertain distance to the tunnel face should be predicted as it di-rectly relates to the load on the support during the tunnel ex-cavation However it is regrettable that the 1047297ctitious supportpressure in the aforementioned studies is estimated in an overlysimpli1047297ed way

The objective of this paper is to bridge the gaps between themethods It mainly covers three parts to develop a numericalprocedure for the 1047297ctitious support pressure of a circular tunnelto validate the proposed procedure by comparing with the nu-merical results of other studies and to study the in1047298uencingfactors of the 1047297ctitious support pressure such as the critical plasticsoftening parameter the rock mass quality the dilatancy angle andthe initial stress condition

2 Problem statement

21 Improved approach to apply the CCM (convergencendashcon 1047297nement

method)

The interaction between the rock mass and support by theimproved approach is shown in Fig 1 The longitudinal deforma-tion pro1047297le (LDP) is the radial displacement which occurs along

the longitudinal direction of an unsupported excavation Theground reaction curve (GRC) is de1047297ned as the decreasing internalpressure and increasing radial displacement at the periphery of the circular tunnel The derivation of GRC is a two-dimensionalproblem under the plain strain condition The cross-section AndashAprime isused to discuss GRC As illustrated in Fig 1 before the initial stagethe pressure on the rock mass is the 1047297ctitious support pressure pf and p f at a certain location x can be solved by coupling LDP andGRC For intermediate and 1047297nal stages the pressure on the rockmass is provided by the tunnel face and the support which is thesum of the 1047297ctitious support pressure p f ( pfint and pf1047297n) and thesupport pressure ps ( psint and ps1047297n) The ground-support systemreaches equilibrium for every excavation step Thus pf and ps arevariable in the process When the tunnel face effect disappears at

Fig 1 Interaction between rock mass and support by improved approach

L Cui et al International Journal of Rock Mechanics amp Mining Sciences 78 (2015) 336 ndash 349 337



the 1047297nal stage pf decreases to zero and the interaction processceases

The pf along the tunnel axis represents the tunnel face effectfrom a quantitative standpoint The schematic diagram to solve pf

is presented in Fig 2 As shown in Fig 2 u0 at a given distance x of the LDP corresponds to the pf in the GRC Hence prior to pre-dicting pf the LDP and GRC should be presented 1047297rst

22 Basic assumptions

Before solving the LDP and GRC the following assumptions areto be employed

The opening is circular The initial stress 1047297eld is hydrostatic andaxisymmetric In the plane perpendicular to the axis of the tunnela plane strain condition is postulated sr and sθ represent theminor principal stress s3 (ie the con1047297ning stress) and majorprincipal stress s1 respectively

Good to very good quality rock mass (GSI475 where GSI is theGeological Strength Index) the average quality rock mass(25oGSIo75) and the low quality rock mass present the elasticndash

brittlendashplastic strain-softening and elastic-perfectly-plastic beha-

viours respectively20

Based on this elastic-perfectly-plastic (EPP)strain-softening (SS) and elastic-brittle (EB) rock masses are analysedhere The rock mass is isotropic continuous in1047297nite and initiallyelastic

According to the above assumptions Fig 3 presents the dis-tribution of the plastic zones in EPP SS and EB rock masses andcorresponding stressndashstrain relationships A hydrostatic stress1047297eld s0 exists prior to the excavation The radius of the circularopening is R0 sr2 and sθ2 are the radial and tangential stresses atthe elasto-plastic boundary The radius of the plastic zone for theEPP rock mass and the radius of the plastic residual zone for the EBrock mass are denoted by Rp For the SS rock mass the radii of theplastic softening and residual zones are denoted by Rp and Rr theradial and tangential stresses at the plastic softening-residual

boundary are sr1 and sθ1 An internal pressure pi is uniformly

distributed along the excavation boundary Before the supportinstallation pi is equal to pf During the interaction process of theground-support system pi is composed of the 1047297ctitious supportpressure pf and the support pressure ps which are provided by thetunnel face effect and the installed support respectively

23 Failure criterion and the 1047298ow rule

According to the theory of plasticity21-22 the deformationprocess is characterised by a failure criterion f and a plastic po-tential g f and g depend not only on the stress tensor sij but theplastic softening parameter η The failure criterion is de1047297ned asfollows

f 0 1ijσ η( ) = ( )

In the existing studies η is often assumed to be the differencebetween the major and minor principal plastic strains under aplane strain condition ie2310-12

21p

3pη ε ε= minus ( )

For an axisymmetric condition Eq (2) can be rewritten as

3prpη ε ε= minus ( )θ

231 Failure criterion

The widely used MohrndashCoulomb (MndashC) and HoekndashBrown (Hndash

B) failure criteria are accommodated here The MndashC failure cri-terion is expressed as follows

f K C K 2 0 41 3 1 3σ σ η σ σ ( ) = minus minus = ( )φ φ

where K φ and C are the friction coef 1047297cient and cohesion of therock mass respectively K φ is equal to 1 sin 1 sinφ φ( + ) ( minus ) andφ is the friction angle

The latest version of Hndash

B failure criterion23

is written as

Fig 2 Schematic diagram for solving 1047297ctitious support pressure pf

L Cui et al International Journal of Rock Mechanics amp Mining Sciences 78 (2015) 336 ndash 349338



f m s 5a

1 3 1 3 ci b 3 ci( )σ σ η σ σ σ σ σ ( ) = minus minus + ( )

in which sci is the uniaxial compression strength of the rock massin the intact state mb s and a are the strength parameters

The HndashB and MndashC failure criteria for EPP SS EB rock masses canbe rewritten as

⎡⎣ ⎤⎦ f m s

H B failure criterion 6a

a

r r ci b r ci( )σ σ η σ σ σ η σ σ η= minus minus ( ) + ( )

( minus ) ( )

η

θ θ

( )

f K C K 2

M C failure criterion 6b

r r( )σ σ η σ η σ η η= minus ( ) minus ( ) ( )

( minus ) ( )

φ φθ θ

Nowω can represent any one of the strength parameters (mb sa φ or C ) and the relation between η and ω can be expressed asfollows

EPP rock mass 7apeakω η ω( ) = ( ) ( )

⎧

⎨⎪

⎩

⎪

0

SS rock mass

7b

peak peak res

res

ω ηω ω ω

η

ηη η

ω η η

( ) =minus ( minus )

lt lt

ge ( )

( )

EB rock mass 7cresω η ω( ) = ( ) ( )

where η is the critical plastic softening parameter The paramterη

is in1047297nitely large for the EPP rock mass and it is equal to 0 forthe EB rock mass The parameters ωpeak and ωres are the peak andresidual values of a given strength parameter For the SS rock massω reduces linearly with the increase in η during the plastic soft-ening stage and remain constant beyond the critical value η (iein the plastic residual stage) For the EPP and EB rock masses ω is

constant in the plastic or plastic residual zone

232 Flow rule

The MndashC type of criterion is selected as the plastic potentialfunction g which is written as

g K 8r rσ σ η σ σ ( ) = minus ( )ψ θ θ

where K ψ is the dilatancy coef 1047297cient ie

K 1 sin

1 sin 9

ψ

ψ =

+

minus ( )ψ

and ψ is the dilatancy angle

Fig 3 Distribution of plastic zone and corresponding stressndashstrain relationship (a) for EPP rock mass (b) for SS rock mass (c) for EB rock mass




3 Determination of GRC

The closed-form solution of the GRC for EPP and EB rockmasses can be obtained because the strength parameters in theplastic (or plastic residual) zone are constant Relevant solutionscan be found in Refs 7924-2635 The derivation of the GRC forthe SS rock mass seems to be more complicated which should besolved by numerical methods810-1225 For instance Lee and

Pietruszczak12 divided the potential plastic zone into a 1047297nitenumber of concentric rings and calculated the increments of stress and strain for each ring in a successive manner Carranzandash

Torres8 and Alonso et al10 analysed the strain-softening rock massby the self-similarity method In this section a modi1047297ed numericalapproach is proposed to solve the GRC in a simpler way

31 Finite difference method

The 1047297nite difference method (FDM) proposed in Refs 12 and25 is utilised to cope with this issue The plastic zone (also in-cluding the plastic softening zone and plastic residual zone) isdivided into a set of concentric annuli where r (i) and r (i-1) are the

radii of the inner and outer boundaries of the ith annulus At theouter boundary of the plastic zone sr(0) and sθ(0) are equal to sr2

and sθ2 at the elasto-plastic boundary A constant radial stressincrement r Δσ is assumed for each annulus ie

p

n 10i r i r ir 1r2 i

σ σ σ σ

Δ = minus =minus

( )( ) ( minus ) ( )

where n is the total number of the concentric annuli and sr(i)

denotes the radial stress at r r i= ( ) (i frac14 0 1 hellip n)

32 Modi 1047297ed numerical approach

321 Radial stress at the elasto-plastic boundary

According to Ref 12 the radial stress at the elastio-plasticboundary sr2 can be solved by the following equations

m s 2 2 0

H B failure criterion 11

a

ci bpeak

r2 cipeak

r2 0

peak

( )σ σ σ σ σ + + minus =

( minus ) ( )

C K K 2 2 1

M C failure criterion 12

r2 0peak peak peak( ) ( )σ σ = minus +

( minus ) ( )

φ φ

m s 2 2 0 H B failure criteriona

ci bres

r2 ci res

r2 0

res

( )σ σ σ σ σ + + minus = ( minus )

C K K 2 2 1 M C failure criterionr2 0 res res resσ σ = ( minus ) ( + ) ( minus )φ φ

The term sr2 in Eqs (11) could be obtained by the Newtonndash

Raphson methodThe relation between r (i) and r (i-1) in Ref 12 was derived as

r

r

H

H

2

2 13

i

i

i

i1

r r

r r

σ σ

σ σ =

( ) + Δ

( ) minus Δ ( )

( )

( minus )

( )

( )

where ir 2i ir r 1( )σ = σ σ

( ) +( ) ( minus ) and

⎧

⎨⎪⎪

⎩⎪⎪

H m s

C K K

H B failure criterion

1 M C failure criterion

ii i i

a

i i i i

rci 1 r ci 1

1 1 r 1

i 1

( ) ( )σ

σ σ σ

σ

= + ( minus )

minus ( minus )( minus )φ φ

( )( minus ) ( ) ( minus )

( minus ) ( minus ) ( ) ( minus )

( minus )

-

values of the strength parameters (m(i-1) s(i-1) a(i-1)) are calculated

by Eq (7)

322 Stress and strain components in the elastic and plastic zones

In the elastic zone the closed-form solutions of the stresscomponents and strain components have been proposed by Ref 25According to Hookes law the elastic strain increments in theplastic zone are

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭E

1 1

1 14

r i

i

r i

i

e

e

ε

ε

μ μ μ

μ μ

σ

σ

Δ

Δ=

+ minus minus

minus minus

Δ

Δ( )

( )

θ( )

( )

θ( )

in which E and μ are the Youngs modulus and Poissons ratio of the rock mass i

eεΔ θ( ) and ireεΔ ( ) are the tangential and radial elastic

strain increments at the ith annulusThe increment of plastic softening parameter iηΔ ( ) at the ith

annulus can be described as (referring to Eq (3))

15i i r ip pη ε εΔ = Δ minus Δ ( )( ) θ( ) ( )

where ipεΔ θ( ) and r i

pεΔ ( )

are the tangential and radial plastic strainincrement at the ith annulus

In accordance with the non-associated 1047298ow rule the relationbetween r i

pεΔ ( ) and i

pεΔ θ( ) is

K 16r i ip pε εΔ = minus Δ ( )ψ ( ) θ( )

In the plastic zone the strain increment includes the plasticstrain increment and elastic strain increment then the equationcan be written as

K K K

K 17

r i i r i i r i i r i

i

1 1 e e p

p

ε ε ε ε ε ε ε

ε

+ = + + Δ + Δ + Δ

+ Δ ( )

ψ ψ ψ

ψ

( ) θ ( ) ( minus ) θ ( minus ) ( ) θ( ) ( )

θ( )

In terms of the small strain case the displacement compat-ibility is

du

dr

u

r

18r ε ε= =( )θ

where u is the radial displacement of the rock mass and r is theradial distance to the centre of the opening In order to solve the

strain components Eq (18) can be rewritten asu

r

u

r

19i

i

ii

i

irε ε=

Δ

Δ=

( )( )

( )

( )θ( )

( )

( )

where u(i) is the radial displacement at r frac14 r (i)By combining Eqs (14) (17) and (19) the radial displacement

u i( ) can be expressed as

u A r r r u r

r K r r 20i

i i i i i i

i i i i

1 1 1

1

( )

( )=

minus +

+ minus ( )ψ

( )

( minus ) ( ) ( ) ( minus ) ( minus ) ( )

( ) ( ) ( ) ( minus )

where

⎡⎣

⎤⎦

A K

K B K K 1

i r i i i

E i i i i i

1 1 1

1

r 1( ) ( )

ε ε

σ μ μ μ μ

= +

+ Δ minus minus + minus minus

ψ

ν

ψ ψ ψ

( minus ) ( minus ) ( ) θ( minus )

( + )

( ) ( ) ( minus ) ( ) ( )

and B H H i i i1 r r r 1σ σ σ = minus Δ + ( ) minus ( )( minus ) ( ) ( minus )

Substituting Eq (20) into Eq (19) εr(i) and εθ(i) can be obtained

u

r

A r r u r

r r K r r

1

1 21i

i

i

i i i i i

i i i i i

1 1 1 1

1 1

( )

( )ε = =

minus +

+ minus ( )ψ

θ( )

( )

( )

( minus ) ( ) ( minus ) ( minus ) ( minus )

( ) ( minus ) ( ) ( ) ( minus )

u

r K A

r r

r r

1

1 22i

i

ii i i

i i

i ir 1

1

1

ε ε=Δ

Δ= minus + sdot

minus

minus ( )ψ ( )

( )

( )( ) θ( ) ( minus )

( ) ( minus )

( minus ) ( )

As illustrated in Eqs(6) (10) (21) (22) sr(i) sθ(i) εr(i) and εθ(i)

can be determined provided that the geological characteristics (s0η

ωpeak ωres sci) are given If the selected Δsr is small enoughand the calculation accuracy is guaranteed the stress components

(sr(i) sθ(i)) and the strain components (εr(i) εθ(i)) are in one-to-one




correspondence

323 Radius of the plastic zone

The relation between R0 and Rp for EPP SS and EB rock massescan be derived as

R R

23

i

n H

H

p0

1

2

2

i

i

r r

r r

=

prod

( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )

where n is the number of the annulus at the internal boundary of the circular tunnel

Likewise Rr of the SS rock mass can be written as

R R

24i

j H

H

r0

1

2

2

i

i

r r

r r

=

prod( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )

where j is the number of the annulus immediately outside theplastic softening-residual boundary

325 Discussion on the critical support pressure

Eqs (11) and (12) reveal that sr2 of the EPP SS and EB rockmasses are calculated by s0 ωpeak and sci Then sr2 remainsconstant while s0 ωpeak and sci are determined For the SS rockmass if s0 η ωpeak ωres sci are de1047297ned and the calculation ac-curacy is reached the stress components (sθ(i) sr(i)) and straincomponents (εr(i) εθ(i)) refer to the one-to-one correspondenceAs long as the plastic residual zone exists when the accumulatedΔη(i) is equal to η the plastic softening-residual boundary will bereached and sr2 will decrease to sr1 As Δη(i) is the differencebetween Δεθ(i) and Δεr(i) then η corresponds to sr1Thereforesr1 remains constant while s0 η ωpeak ωres sci are determined Itis observed in the calculation process that if the support pressure

pi osr1 both the plastic softening and residual zones are formedif sr1o piosr2 only plastic softening zone is formed if pi4sr2 noplastic zone is formed Therefore for given geological and geo-metrical conditions sr2 and sr1 are regarded as the critical support

pressures for the appearances of plastic softening and residualzones respectively For the EPP and EB rock masses the criticalsupport pressure for the appearance of the plastic zone or theplastic residual zone is equal to sr2

Although the proposed numerical approach is presented on thebasis of Ref 12 the approach is different from that proposed inRef 12 in the following way Firstly in Ref 12 a parameter ρ isintroduced to indicate the ratio of r (i) to Rp ρ should be solvedbefore obtaining r (i) In the proposed approach r (i) can be directlyobtained by Eq (13) secondly the compatibility equation used inRef 12 is more complicated than Eq(18) or (19) presented here Onthis basis solutions of εθ(i) εr(i) and ur(i) by their approach shouldbe solved step by step whereas εθ(i) εr(i) and ur(i) by the proposedapproach are explicitly shown in Eqs (20)ndash(22) thirdly the

solutions of Rr and sr1 are not mentioned in Ref 12 Actually Rr

and sr1 are fairly important parameters in predicting the occur-rence of plastic residual zone Rr and sr1 are discussed by theproposed approach

4 Determination of LDP

At present the empirical and numerical approaches have beenproposed for predicting LDPs Panet et al27ndash29 and Chern30 de-veloped the empirical equations based on in-place measurementsfor different tunnels Unlu and Gecek31 and Panet28 derived arelationship for the LDP through the elastic analysis New ap-proaches for EPP rock masses have recently been proposed based

on data 1047297tting in 3D numerical models3233

and 2D axisymmetric

models34 By the 2D axisymmetric model Vlachopoulos andDiederichs34 obtained the equations of the LDP for EPP rock massie

⎧

⎨

⎪⎪

⎩

⎪⎪

u

u e forX

u e forX

u u

u e

0

1 1 0

1

3 25

X

X

R

R

0

0

3

2

00

0max

015

( ) =

sdot lt

minus minus sdot ge

= =( )

minus

minus

where R is the normalised plastic radius de1047297ned as the ratiobetween the maximum plastic radius Rp

max and the tunnel radiusR0 Rp

max and u0max are the maximum radius of the plastic zone and

the maximum radial displacement when the internal pressure pi is0 X is the normalised distance to the tunnel face de1047297ned as theratio between the distance to the tunnel face x and the radius of

the opening R0 u is the normalised radial displacement at X u0

is the normalised radial displacement at the tunnel faceThe rock mass reveals different softening behaviours It seems

to be incomprehensive to evaluate the LDP by regarding therock mass as either elastic2831 or elastic-perfectly-plasticmaterials32-34 Based on the work of Ref 34 Alejano et al3 pro-posed the numerical and analytical approaches to obtain the LDPfor the SS rock mass For the analytical approach in Ref 3 Rp

max ispredicted by the regression 1047297tting analysis of a large number of numerical results from Ref 10 Then the LDP of the EPP and SSrock masses can be solved by substituting Rp

max into Eq(25) Bycomparing the results of numerical approach (ie FLAC2D and 3D)the analytical approach is veri1047297ed for solving the LDP This meansthat Eq (25) of Ref 34 is applicable for the SS rock mass Hencethe proposed study follows Eq (25) to predict the LDP of the EPPand SS rock masses It should be emphasised that Rp

max of theproposed approach differs from that of Ref 3 in which Rp

max wasobtained by the 1047297tting equation whereas Rp

max of the proposed

approach is obtained by Eq (23) Eq (23) is found to be moresimpli1047297ed Moreover Rpmax by the proposed approach is proved to

1047297t better with the numerical results by FLAC2D and 3D codes inRef 3 and this will be discussed later

5 Simpli1047297ed approach for 1047297ctitious support pressure pf

Due to the fact that the LDP relates u0 and x and the GRC relates pi (or pf ) and u0 the variation of pf versus x can be predicted bycoupling the GRC and LDP However the above approach for theGRC simply gives the solution of u0 by use of a certain p i (or p f )The solution of pf needs the reverse calculation process of the GRCAlthough the iteration approach can obtain pf the implementation

of the procedure seems to be complicatedHere a simpli1047297ed approach to obtain pf is proposed The dia-gram for evaluating pf is proposed in Fig 4 As shown in Fig4 if s0η

ωpeak ωres sci are de1047297ned pf is 0 and the calculation accuracyis reached sr(i) εθ(i) and Rp

max can be determined by the GRC u0 at

certain X can be obtained on basis of the LDP solution u0 shouldcorrespond to the tangential strain εθ(i) at a certain annulus r r i= ( ) This is because εθ(i) can be treated as the ratio of u0 to R0 More-over it should be noticed that εθ(i) correlates with sr(i) Conse-quently u0 corresponds to sr(i) at r r i= ( ) Since u0 is the radialdisplacement at the tunnel surface sr(i) can represent the pressure

pf acted at the internal boundary of the circular tunnelIn summary the sequences of solving pf are assume pi (or pf ) is

0 solve the GRC to determine sr(i) εθ(i) and Rpmax obtain the LDP

by Rpmax

and then obtain pf by correlating u0 (and εθ(i)) to sr(i)




It should be mentioned that the concept of pf is different fromthat of the LDP As discussed before to deal with the CCM thetraditional and improved approaches are classi1047297ed Essentially pf

represents the improved method whereas the LDP represents thetraditional method In the improved approach pf can guide thesupport design by determining the load acted on the supportwhich has been referred by several researchers13-17 However inthe traditional approach the support design cannot be guided bythe LDP itself It should be conducted by combining the LDP GRCand SCC Moreover the SCC is restricted to a few support types bythe traditional approach whereas a variety of the support typescan be realised by the improved approach with pf

6 Veri1047297cation

61 Veri 1047297cation of LDP and GRC solutions

For the veri1047297cation of the LDP solution Table 1 lists 1047297ve ana-

lysis conditions from the examples in Ref 3 Cases A1 to D1

represent the SS rock mass Case E1 represents the EPP rock massFor the veri1047297cation of the GRC solution Table 2 lists four analysisconditions from Refs 7 and 11 Cases A2 and B2 represent the EPPand EB rock masses Case C2 and D2 represent the SS rock mass

GSImin and GSImax indicate the values of GSI in the plastic residualzone and in the elastic zone respectively The HndashB strengthparameter a is equal to 05 for each case The number of the annulin is selected as 1500 to con1047297rm the calculation accuracy

611 GRC

Fig 5(a) and (b) plot distributions of dimensionless radial dis-placement by the proposed approach and the closed-formapproach7 for the EPP and EB rock masses Fig 5(c) and (d) plot theGRCs by the proposed approach and the multi-step brittle plasticapproach11 for the SS rock mass As shown in Fig 5 the dis-placement distribution by the proposed approach shows a perfectagreement with the closed-form solution7 for the EPP and EB rockmasses A very good matching of the GRC by the two approaches

for the SS rock mass is observed Therefore the proposed approach

Fig 4 Schematic diagram for pf evaluation

Table 1

Parameters of rock masses for veri1047297cation of LDP

A13 B13 C13 D13 E13

GSImax 75 60 50 40 40

GSImin 40 35 30 27 ndash

mp 287 168 117 0821 0821

sp103 622 117 39 13 13

mr 0821 0687 0575 0516 ndash

sr103 13 07 04 03 ndash

pφ deg 2952 2568 2313 2064 2064

C p MPa 3637 2673 2242 1878 1878rφ deg 2064 1942 1821 1749 ndash

C r MPa 1878 1707 1536 1432 ndash

ψ deg 738 449 289 155 ndash

K pψ ndash ndash ndash ndash ndash

K rψ ndash ndash ndash ndash ndash

η 103 108 622 288 119 ndash

E GPa 365 154 866 487 487 piMPa 0 0 0 0 0

μ 025 025 025 025 025R0m 25 25 25 25 25

cip

σ MPa 35 35 35 35 35

cirσ MPa 35 35 35 35 35

σ MPa 375 375 375 375 375

Table 2

Parameters of rock masses for veri1047297cation of GRC

A211 B211 C27 D27

SImax ndash ndash ndash ndash

SImin ndash ndash ndash ndash

mp 75 17 2 05

sp10-3 100 39 4 10

mr 1 1 06 01

sr10-3 10 0 2 05pφ deg ndash ndash ndash ndash

C pMPa ndash ndash ndash ndash

rφ deg ndash ndash ndash ndash

C rMPa ndash ndash ndash ndash

ψ deg ndash ndash ndash ndash

K pψ ndash ndash 1698 1698

K rψ ndash ndash 1191 1191

η 103 ndash ndash 10 125

E GPa 40 55 57 138 piMPa 0 5 ndash ndash

μ 02 025 025 025R0m 4 5 3 3

cip

σ MPa 300 30 30 275

cirσ MPa 300 30 25 275

σ MPa 108 30 15 331




for the GRC is practicable for the three types of rock mass

612 LDP

In Ref 3 numerical and analytical approaches were proposedto obtain the LDP The numerical approach was conducted by useof FLAC2D code with Neumann boundary condition (appliedstress) and FLAC3D code with Dirichlet boundary condition (1047297xeddisplacement) For the analytical approach Rp

maxof the EPP rock

mass was obtained according to the solution in Ref 8 Rpmax of the

SS rock mass was estimated according to the minimum square1047297tting processes of statistical study for 400 tunnels Then the LDPof the EPP and SS rock masses can be solved by substituting Rp

max

into Eq (25) The result of the analytical approach shows goodagreement with that of the numerical approach (FLAC2D and 3Dcodes3) In order to validate the accuracy of the proposed ap-proach on the basis of cases D1-E1 the calculated LDPs are com-pared with those obtained by the numerical and analytical solu-tions in Ref 3

Fig 6 shows the comparison of the LDP solution of EPP and SSrock masses by the proposed approach and the approach taken inRef 3 with HndashB and MndashC failure criteria respectively As shown inFig 6 for the EPP rock mass the results by the proposed approachare highly consistent with those by FLAC2D and 3D codes3 For the

SS rock mass the proposed LDP solution 1047297ts well with the FLAC3D

solution3 by HndashB failure criterion Compared with the analyticalsolution for the MndashC failure criterion the proposed solution iscloser to the FLAC2D solution3

In order to show the advantage of the proposed approach overthe analytical approach3 Fig 7 plots the values of Rp

max for the SSrock mass by several approaches The proposed approach thenumerical approach (FLAC2D and 3D) and the analytical approach3

are involved The self-similar approach for MndashC failure criterionfrom Ref 10 is included to make comparison Fig 7 shows that theproposed solution tends to be closer to the numerical solutions by

Refs 3 and 10 With low values of GSI the analytical solutionoverestimates Rp

max to some extent On the whole in contrast tothe analytical approach the results by the proposed approach 1047297tbetter with the numerical results by FLAC2D and 3D This meansthe accuracy of the proposed approach for the LDP is acceptable

62 Veri 1047297cation of the 1047297ctitious support pressure solution

In fact the simpli1047297ed solution of pf has been deduced from atheoretical standpoint The objective for verifying the simpli1047297edsolution is to check whether the stress and the strain componentsrefer to the one-to-one correspondence for a given condition If the stress and strain components remain constant when Δsr isvery small the simpli1047297ed solution will be veri1047297ed Therefore in

this section the solution of pf is validated by discussing the

Fig 5 Comparison of radial displacement (a) u E R0 0 0σ versus r R 0 with case A2 (b) u E R0 0 0σ versus r R 0 with case B2 (c) GRC with case C2 (d) GRC with case D2




accuracy of the GRC solutionHere cases A3 and B3 with different qualities of the rock mass

are considered The parameters are listed in Table 3 The equiva-

lent friction angle φ and cohesion C of Mndash

C failure criterion are

calculated by the strength constants mb s and a of HndashB failurecriterion according to the method introduced in Ref 23

Based on cases A3 and B3 η is regarded as in1047297nitely large 001

0 to represent the EPP SS and EB rock masses respectively The

Fig 6 Comparison of LDP solution (a) case E1 for EPP rock mass with H ndashB failure criterion (b) case E1 for EPP rock mass with M ndashC failure criterion (c) case D1 for SS rock

mass with HndashB failure criterion (d) case D1 for SS rock mass with M ndashC failure criterion

Fig 7 Comparison of Rpmaxsolution for SS rock mass (a) M ndashC failure criterion (b) HndashB failure criterion




HndashB and MndashC failure criteria are utilised with K ψ regarded as 1

The values of the maximum radial displacement u0max (when the

internal pressure pi is 0) at the tunnel surface with different n

(number of the annuli) for EPP SS and EB rock masses are listed inTable 4 n ranges from 25 to 5000

As displayed in Table 4 the decreasing rate of u0max is re-

markably reduced as n increases When n is larger than 3000 thevalues of u0

max are basically constant in most cases Some cases willnot converge to a certain value when n reaches to 5000 whereas itis acceptable since the decreasing rates of these cases are very

small In a word the values in Table 4 validate the accuracy of thesimpli1047297ed approach for pf

7 Discussion

The in1047298uences of the critical plastic softening parameter therock mass quality the initial stress condition s0 and the dilatancycoef 1047297cient K ψ on the 1047297ctitious support pressure pf and the tunnelface effect are discussed It should be noticed that the tunnel faceeffect is re1047298ected by three typical distances X 1 X 2 and X 3

Speci-1047297cally X 1 represents the distance to the tunnel face when thetunnel face effect disappears It is evaluated as an integer value forsake of simplicity X 2 and X 3

represent the distances when the

plastic softening (or plastic) and plastic residual zones appearrespectively Essentially X 1 means the duration of the tunnel faceeffect It correlates with decreasing rate of the pf X 2 and X 3

in-dicate the stability of the rock mass during tunnelling

71 In 1047298uence of the plastic softening parameter η

Tables 5 and 6 list eight values of η for cases A3 and B3 (theEPP and EB rock masses included) respectively The dilatancycoef 1047297cient K ψ is regarded as 113

711 Variation law of pf versus X

Figs 8 and 9 plot the variation law of pf versus X for analysisconditions ①-⑧ with cases A3 and B3 respectively It should bementioned that Figs 8(b) and 9(b) indicate p f behind the tunnel

face It is observed that the rock mass with a larger η provides a

higher p f for a certain X Moreover by comparing the results of cases A3 and B3 it is observed that the rock mass with betterquality reveals a greater pf behind the tunnel face This means that

the rock mass with higher η and better quality leads to a morestable rock mass behind the tunnel face Table 7 lists the percen-

tages of pf s0 at X 0 = for cases A3 and B3 It is found that the

percentage of pf s0 at X 0 = decreases for 4252 from η = infin(the

EPP rock mass) to 0η = (the EB rock mass) in the case A3 whereasit decreases for 2066 in the case B3 This means that pf of the

rock mass with better quality tends to be affected by η more

signi1047297cantly Therefore the in1047298uence of η

on the pf especially for

Table 3

Parameters of rock masses for cases A3 and B3

A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50

ciσ MPa 110 80σ MPa 35 35

Table 4Values of u0

max with different n (a) for EPP rock mass (b) for SS rock mass (c) for EBrock mass

case A3(H-B) case A3(M-C) case B3(H-B) case B3(M-C)

(a)n u0

max(mm) u0max(mm) u0

max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5

Different values of η for case A3

A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB

Table 6Different values of η for case B3

B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB




the rock mass with good quality should be highlighted

712 Correlation between p f 0σ and u u0 0max

On the assumption that the rock mass is elasticu u u0

max0 0

max( )minus is solved as identical to p f 0σ For example in

Ref 36 it was argued that 30 of the u0max will be achieved when

the internal support pressure p i is taken to be equal to 0σ multi-plied by 70 for the researchers concerning the numerical si-mulation in tunnelling 131419 it is assumed that the stress relievefactor f s ( f p1 s f 0σ = minus ) is equal to u u0 0

max In fact the rock massbehind the tunnel face reveals the plastic behaviour in most casesFig 10 plots the relation between p f 0σ and u u0 0

max for analysisconditions ①ndash⑧ in cases A3 and B3 The elastic condition is in-cluded to make comparison As shown in Fig 10 while the same

value of u u0 0max is determined pf for analysis conditions ①ndash

⑧ is

smaller than that for the elastic condition and a higher η givesrise to a larger value of pf Consequently for a given u u0 0

max(in thepractical tunnel engineering u u0 0

max is commonly estimated bythe 1047297eld test data) the stress relieve factor f s will be under-

estimated with the elastic condition or with a η higher than thereality From a practical standpoint the support design with theseconditions tends to become unsafe

72 In 1047298uence of the initial stress 0σ

It is postulated that s0 varies from 5 MPa to 65 MPa with15 MPa in intervals The case B3 is analysed in which K ψ and η

areregarded as 113 and 001 respectively


Fig 11 plots the variation law of pf s0 versus X for different s0

with the case B3 It shows that as s0 increases pf s0 decreasesBehind the tunnel face the value of pf s0 for s0 is 5 MPa which isremarkably greater than other conditions This is because theplastic softening zone appears behind the tunnel face (when X is09963) for s0 is 5 MPa whereas this zone appears ahead of thetunnel face for other conditions The appearance of the plastic

softening zone gives rise to a fast reduction of the pf s0 As a result

Fig 8 Variation law of p f versus X with case A3 (a) X ranges from 10 to 10 (b) enlarged view of (a)

Fig 9 Variation law of pf versus X with case B3 (a) X ranges from 10 to 10 (b) enlarged view of (a)

Table 7

Percentage of p f s0 at X 0 = for cases A3 and B3

pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415




values of pf s0 with higher initial stress conditions are lowerMoreover Fig 11(b) indicates that for different high initial stressconditions (s0 is 25MPa 35 MPa 50 MPa 60 MPa) the stress re-lieve factor f s ( p1 f 0σ minus ) are basically identical and the stress re-lease due to the excavation at the tunnel face is signi1047297cant

As indicated in Figs 9 and 11 the higher s0 and the weaker rock

mass give rise to relatively small decreasing rate of pf The de-formation of high s0 and weak rock mass near the tunnel faceincreases signi1047297cantly as pf is small As a result in order to preventthe rock mass from squeezing the support the stress near thetunnel face can be released prior to the interaction of the rockmass and support This conclusion can be validated by many casehistories of tunnels and mines37-40 In these cases while con-

fronted with the squeezing problems for tunnels excavated in thesoft rock with the high initial stress condition the workers install1047298exible or yielding support or allow the deformation to relieve thehigh stress The purpose is to avoid the support buckling orbreaking down when suffering from the heavy load and largedeformation

722 X 1 X 2 and X 3

The values of X 1 X 2 and X 3 for different s0 are displayed in

Table 8 It is revealed that when s0 is 50 MPa or 65 MPa theplastic softening and residual zones emerge far ahead of the tun-nel face Meanwhile X 1 develops with the increase in s0 Weaker

rock mass also leads to a higher X 1

Fig 10 Relationship between p f 0σ and u u0 0max (a) case A3 (b) case B3

Fig 11 Variation law of p f with X for case B3 (a) X ranges from 10 to 10 (b) enlarged view of (a)

Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9

Five values of ψ and K ψ

0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282




73 In 1047298uence of dilatancy coef 1047297cient K ψ

Five values of the dilatancy angle ψ are selected to representdifferent dilatancy behaviours of the rock mass The case B3 isanalysed It is postulated that ψ is equal to 0 φ8 φ4 φ2 φrespectively For the sake of simplicity the friction angle φ istreated as the average value of the peak friction angle φpeak andthe residual friction angle φres in the case B3 η is regarded as001 respectively The calculated ψ and K ψ are listed in Table 9


Fig12 plots the variation law of pf versus X for different ψ It isobserved in Fig 12 that pf decreases as ψ increases Fig 12 revealsthat pf decreases rapidly far away ahead of the tunnel face when ψ reaches φ The area will become unstable due to the remarkablereduction of pf

732 X 1 X 2 and X 3

The values of X 1 X 2 and X 3 for different ψ are displayed in

Table 10 It is found that X 1 remains constant until ψ increases toφ Consequently ψ tends to have negligible impact on the durationof the tunnel face effect In contrast the in1047298uence of ψ on X 2

and

X 3 are more obvious For instance the absolute values of X 2 and X 3

for ψ φ= are 283 and 268 which are 337 and 865 times as largeas those for 0ψ = respectively As listed in Table 8 the absolutevalues of X 2 and X 3 for s0 frac14 60 MPa are 198 and 171 respectivelyThis indicates that for the rock mass with high initial stress con-dition or strong dilatancy behaviour the plastic softening andresidual zones emerge far ahead of the tunnel face In this case therock mass in vicinity of tunnel tends to be unstable In an effort toprevent the rock mass from collapse the pre-reinforcementshould be installed This allows the rock mass to deform in a

controlled manner and mobilise its strength

Conclusions

In this paper for the circular tunnel subjected to a hydrostaticcondition a numerical procedure to solve the 1047297ctitious supportpressure is presented The procedure is mainly composed of twoparts a new numerical approach to solve the GRC and LDP and asimpli1047297ed approach to solve the 1047297ctitious support pressure Thenby comparing the calculated results with those in other studies itis indicated that the proposed procedure is capable of providingreasonable estimation In the end a series of parametric studies iscarried out and the following conclusions are drawn

The in1047298uence of the critical plastic softening parameter on the1047297ctitious support pressure especially for the rock mass with goodquality is obvious The in1047298uence of the dilatancy behaviour on the1047297ctitious support pressure is trivial

The elastic assumption of the rock mass behaviour will un-

derestimate the stress relieve factor during tunnelling Thus thesupport design with elastic conditions tends to become unsafe Inaddition for different high initial stress conditions the stress re-lieve factors are signi1047297cant and basically identical

While the weak and soft rock mass is excavated in the highinitial stress 1047297eld the 1047297ctitious support pressure and its decreasingrate along the axial direction is relatively small In order to preventthe rock mass from squeezing the support the high stress near thetunnel face can be appropriately released prior to the interactionof ground-support system

For the rock mass with high initial stress condition and strongdilatancy behaviour the plastic softening and residual zonesemerge far ahead of the tunnel face In this case the pre-re-inforcement should be installed allowing the rock mass to deform

in a controlled manner and mobilise its strength

Acknowledgement

The authors would like to express their gratitude to the Na-tional Natural Science Foundation of China (51278216 and51478201) for the 1047297nancial support and to Professor Phil Dight forproviding assistance with English usage

References

1 Carranze-Torres C Fairhurst C Application of the convergence-con1047297nement

method of tunnel design to rock masses that satisfy the HoekndashBrown failure

Fig 12 Variation law of pf with X for different ψ (a) X ranges from 10 to 10 (b) enlarged view of (a)

Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268




criterion Tunn Undergr Space Technol 20 0015(2)187ndash2132 Alejano LR Alonso E Rodriguez-Dono A Application of the convergence-con-

1047297nement method to tunnels in rock masses exhibiting Hoek ndashBrown strain-softening behaviour Int J Rock Mech Min Sci 201047150ndash160

3 Alejano LR Rodriguez-Dono A Veiga M Plastic radii and longitudinal de-formation pro1047297les of tunnels excavated in strain-softening rock masses TunnUndergr Space Technol 201230169ndash182

4 Oreste PP Analysis of structural interaction in tunnels using the convergence-con1047297nement approach Tunn Undergr Space Technol 20 0318347ndash363

5 Gonzaacutelez-Niciezaa C Aacutelvarez-Vigilb AE Meneacutendez-Diacuteazc A Gonzaacutelez-PalacioaC In1047298uence of the depth and shape of a tunnel in the application of the con-

vergence-con1047297nement method Tunn Undergr Space Technol 20082325ndash376 Eisenstein Z Branco P Convergence-con1047297nement method in shallow tunnels

Tunn Undergr Space Technol 19916(3)343ndash3467 Carranza-Torres C Fairhurst C The elasto-plastic response of underground ex-

cavations in rock masses that satisfy the HoekndashBrown failure criterion Int J RockMech Min Sci Geomech Abstr 199936(6)777ndash809

8 CarranzahyndashTorres C Self -similarity Analysis of the Elastoplastic Response of Un-derground Openings in Rock and Effects of Practical Variables University of Min-nesota Minneapolis 1998[PhD thesis]

9 Sharan SK Exact and approximate solutions for displacements around circularopenings in elasto-brittle-plastic HoekndashBrown rock Int J Rock Mech Min Sci200542542ndash549

10 Alonso E Alejano LR Varas F Fdez-Manin G Carranza-Torres C Ground re-sponse curves for rock masses exhibiting strain-softening behaviour Int J Numer

Anal Method Geomech 2003271153ndash118511 Wang SL Yin XT Tang H Ge XR A new approach for analyzing circular tunnel in

strain-softening rock masses Int J Rock Mech Min Sci 201047170ndash17812 Lee YK Pietruszczak S A new numerical procedure for elasto-plastic analysis of

a circular opening excavated in a strain-softening rock mass Tunn Undergr SpaceTechnol 200823588ndash59913 Carranza-Torres C Rysdahl B Kasim M On the elastic analysis of a circular lined

tunnel considering the delayed installation of the support Int J Rock Mech MinSci 2013657ndash85

14 Carranza-Torres C Diederichs M Mechanical analysis of circular liners withparticular reference to composite supports For example liners consisting of shotcrete and steel sets Tunn Undergr Space Technol 200924506ndash532

15 Schwartz CW Einstein HH Simpli1047297ed analysis for tunnel supports J Geotech Eng Div ASCE 1979104(4)499ndash518

16 Einstein HH Schwartz CW Discussion of the article simpli1047297ed analysis fortunnel supports J Geotech Eng Div ASCE 1980106(7)835ndash838

17 Schwartz CW Einstein HH Simpli1047297ed analysis for ground-structure interactionin tunnelling In Proceedings of the 21st Symposium on Rock Mechanics RollaUniversity of Missouri 1980 787ndash796

18 Osgoui RR Oreste P Elasto-plastic analytical model for the design of groutedbolts in a HoekndashBrown medium Int J Numer Anal Method Geomech2010341651ndash1686

19 Fahimifar A Ranjbarnia M Analytical approach for the design of active grouted

rockbolts in tunnel stability based on convergence-con1047297nement method TunnUndergr Space Technol 200924363ndash375

20 Hoek E Brown ET Practical estimates of rock mass strength Int J Rock Mech Min

Sci 199734(8)1165ndash118621 Hill R The Mathematical Theory of Plasticity New York Oxford University Press

195022 Kaliszky S Plasticity Theory and Engineering Applications Amsterdam Elsevier

198923 Hoek E Carranza-Torres C Corkum B HoekndashBrown failure criterionndash2002 edi-

tion In Proceedings of the 5th North American Rock Mechanics Symposium and17th Tunneling Association of Canada Conference 2002 267ndash273

24 Serrano A Olalla C Reig I Convergence of circular tunnels in elastoplastic rockmasses with non-linear failure criteria and non-associated 1047298ow laws Int J RockMech Min Sci 201148878ndash887

25 Brown ET Bray JW Ladanyi B Hoek E Ground response curves for rock tunnels J Eng Mech ASCE 198310915ndash39

26 Wang Y Ground response of circular tunnel in poorly consolidated rock J Eng Mech ASCE 1996122703ndash708

27 Panet M Understanding deformations in tunnels In Hudson JA editor Com- prehensive Rock Engineering vol 1 Oxford Pergamon 1993

28 Panet M Calcul des Tunnels par la Meacutethode de Convergence-Con 1047297nement ParisPress de lrsquoeacutecole Nationale des Ponts et Chausseacutees 1995

29 Panet M Guenot A Analysis of convergence behind the face of a tunnel LondonInstitute of Mining and Metallurgy 1982 p 197ndash204

30 Chern JC Shiao FY Yu CW An empirical safety criterion for tunnel constructionIn Proceedings of the Regional Symposium on Sedimentary Rock Engineering 1998 222ndash227

31 Unlu T Gercek H Effect of Possion rsquos ratio on the normalized radial displace-ments occurring around the face of a circular tunnel Tunnell Undergr SpaceTechnol 200318547ndash553

32 Basarir H Genis M Ozarslan A The analysis of radial displacements occurringnear the face of a circular opening in weak rock masses Int J Rock Mech Min Sci

201040771ndash

78333 Pilgerstorfer T Schubert W Forward prediction of spatial displacement devel-opment Rock Engineering in dif 1047297cult ground conditions-soft rocks and KarstIn Proceedings of Europe Rock Mechanics Symposium 2009 495ndash500

34 Vlachopoulos N Diederichs MS Improved longitudinal displacement pro1047297lesfor convergence con1047297nement analysis of deep tunnels Rock Mech Rock Eng200942131ndash146

35 Sharan SK Elasticndashbrittlendashplastic analysis of circular opening in Hoek ndashBrownmedia Int J Rock Mech Min Sci 200340817ndash824

36 Carranza-Torres C Analytical and numerical study of the mechanics of rockboltreinforcement around tunnels in rock masses Rock Mech Rock Eng200942175ndash228

37 Wang KZ Li ZK Wang YP Zhang ZZ Liu YR Study of strong 1047298exible supportingmechanism and deformation characteristics for fracture zone in large under-ground caverns Chin J Rock Mech Eng 201332(12)2455ndash2462

38 Liu CX Wang L Liu ZH Huang DC Zhang XL Time effects of 1047298exible invertedarch with composite structures to control stability of chamber adjoining withsoft rock masses Chin J Geotech Eng 201234(8)1464ndash1468

39 He MC Jing HH Sun XM Engineering Mechanics of the Soft Rock Beijing Science

Press 200240 Zhao XF The Temporal and Spatial Effect in Construction and Control of LargeDeformation of Tunnels Tongji University Shanghai [PhD thesis]2007




mass is an elastic material and undergoes a displacement of ap-proximately 30ndash50 of the 1047297nal displacement13-17 Actually the1047297ctitious support pressure is found to be in1047298uenced by the me-chanical characteristics of the rock mass such as the plastic soft-ening behaviour the dilatancy behaviour the initial stress condi-tions and rock mass quality The 1047297ctitious support pressure at acertain distance to the tunnel face should be predicted as it di-rectly relates to the load on the support during the tunnel ex-cavation However it is regrettable that the 1047297ctitious supportpressure in the aforementioned studies is estimated in an overlysimpli1047297ed way

The objective of this paper is to bridge the gaps between themethods It mainly covers three parts to develop a numericalprocedure for the 1047297ctitious support pressure of a circular tunnelto validate the proposed procedure by comparing with the nu-merical results of other studies and to study the in1047298uencingfactors of the 1047297ctitious support pressure such as the critical plasticsoftening parameter the rock mass quality the dilatancy angle andthe initial stress condition

2 Problem statement

21 Improved approach to apply the CCM (convergencendashcon 1047297nement

method)

The interaction between the rock mass and support by theimproved approach is shown in Fig 1 The longitudinal deforma-tion pro1047297le (LDP) is the radial displacement which occurs along

the longitudinal direction of an unsupported excavation Theground reaction curve (GRC) is de1047297ned as the decreasing internalpressure and increasing radial displacement at the periphery of the circular tunnel The derivation of GRC is a two-dimensionalproblem under the plain strain condition The cross-section AndashAprime isused to discuss GRC As illustrated in Fig 1 before the initial stagethe pressure on the rock mass is the 1047297ctitious support pressure pf and p f at a certain location x can be solved by coupling LDP andGRC For intermediate and 1047297nal stages the pressure on the rockmass is provided by the tunnel face and the support which is thesum of the 1047297ctitious support pressure p f ( pfint and pf1047297n) and thesupport pressure ps ( psint and ps1047297n) The ground-support systemreaches equilibrium for every excavation step Thus pf and ps arevariable in the process When the tunnel face effect disappears at

Fig 1 Interaction between rock mass and support by improved approach



















f 0 1ijσ η( ) = ( )


21p











is written as





f m s 5a




⎡⎣ ⎤⎦ f m s


a


( minus ) ( )

η

θ θ

( )

f K C K 2



( minus ) ( )

φ φθ θ



⎧

⎨⎪

⎩

⎪

0

SS rock mass

7b

peak peak res

res

ω ηω ω ω

η

ηη η

ω η η


lt lt

ge ( )

( )





232 Flow rule




K 1 sin

1 sin 9

ψ

ψ =

+

minus ( )ψ














p


σ σ σ σ

Δ = minus =minus

( )( ) ( minus ) ( )






m s 2 2 0


a

ci bpeak

r2 cipeak

r2 0

peak


( minus ) ( )

C K K 2 2 1



( minus ) ( )

φ φ


ci bres

r2 ci res

r2 0

res





r

r

H

H

2

2 13

i

i

i

i1

r r

r r

σ σ

σ σ =

( ) + Δ

( ) minus Δ ( )

( )

( minus )

( )

( )


( ) +( ) ( minus ) and

⎧

⎨⎪⎪

⎩⎪⎪

H m s

C K K



ii i i

a

i i i i

rci 1 r ci 1

1 1 r 1

i 1

( ) ( )σ

σ σ σ

σ

= + ( minus )


( )( minus ) ( ) ( minus )


( minus )

-


by Eq (7)



⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭E

1 1

1 14

r i

i

r i

i

e

e

ε

ε

μ μ μ

μ μ

σ

σ

Δ

Δ=

+ minus minus

minus minus

Δ

Δ( )

( )

θ( )

( )

θ( )







pεΔ ( )



pεΔ ( ) and i

pεΔ θ( ) is



K K K

K 17


i

1 1 e e p

p


ε

+ = + + Δ + Δ + Δ

+ Δ ( )

ψ ψ ψ

ψ

( ) θ ( ) ( minus ) θ ( minus ) ( ) θ( ) ( )

θ( )


du

dr

u

r

18r ε ε= =( )θ



r

u

r

19i

i

ii

i

irε ε=

Δ

Δ=

( )( )

( )

( )θ( )

( )

( )



u A r r r u r

r K r r 20i

i i i i i i

i i i i

1 1 1

1

( )

( )=

minus +

+ minus ( )ψ

( )


( ) ( ) ( ) ( minus )

where

⎡⎣

⎤⎦

A K

K B K K 1

i r i i i

E i i i i i

1 1 1

1

r 1( ) ( )

ε ε

σ μ μ μ μ

= +


ψ

ν

ψ ψ ψ


( + )

( ) ( ) ( minus ) ( ) ( )



u

r

A r r u r

r r K r r

1

1 21i

i

i

i i i i i

i i i i i

1 1 1 1

1 1

( )

( )ε = =

minus +

+ minus ( )ψ

θ( )

( )

( )


( ) ( minus ) ( ) ( ) ( minus )

u

r K A

r r

r r

1

1 22i

i

ii i i

i i

i ir 1

1

1

ε ε=Δ

Δ= minus + sdot

minus

minus ( )ψ ( )

( )

( )( ) θ( ) ( minus )

( ) ( minus )

( minus ) ( )








correspondence



R R

23

i

n H

H

p0

1

2

2

i

i

r r

r r

=

prod

( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )



R R

24i

j H

H

r0

1

2

2

i

i

r r

r r

=

prod( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )












and 2D axisymmetric


⎧

⎨

⎪⎪

⎩

⎪⎪

u

u e forX

u e forX

u u

u e

0

1 1 0

1

3 25

X

X

R

R

0

0

3

2

00

0max

015

( ) =

sdot lt

minus minus sdot ge

= =( )

minus

minus












max of the proposed











by Rpmax







6 Veri1047297cation






611 GRC




Table 1


A13 B13 C13 D13 E13

GSImax 75 60 50 40 40


mp 287 168 117 0821 0821

sp103 622 117 39 13 13

mr 0821 0687 0575 0516 ndash

sr103 13 07 04 03 ndash

pφ deg 2952 2568 2313 2064 2064


C r MPa 1878 1707 1536 1432 ndash

ψ deg 738 449 289 155 ndash



η 103 108 622 288 119 ndash

E GPa 365 154 866 487 487 piMPa 0 0 0 0 0

μ 025 025 025 025 025R0m 25 25 25 25 25

cip

σ MPa 35 35 35 35 35

cirσ MPa 35 35 35 35 35

σ MPa 375 375 375 375 375

Table 2


A211 B211 C27 D27



mp 75 17 2 05

sp10-3 100 39 4 10

mr 1 1 06 01










μ 02 025 025 025R0m 4 5 3 3

cip

σ MPa 300 30 30 275

cirσ MPa 300 30 25 275

σ MPa 108 30 15 331





612 LDP


maxof the EPP rock



max






































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash



























f 0 1ijσ η( ) = ( )


21p











is written as





f m s 5a




⎡⎣ ⎤⎦ f m s


a


( minus ) ( )

η

θ θ

( )

f K C K 2



( minus ) ( )

φ φθ θ



⎧

⎨⎪

⎩

⎪

0

SS rock mass

7b

peak peak res

res

ω ηω ω ω

η

ηη η

ω η η


lt lt

ge ( )

( )





232 Flow rule




K 1 sin

1 sin 9

ψ

ψ =

+

minus ( )ψ














p


σ σ σ σ

Δ = minus =minus

( )( ) ( minus ) ( )






m s 2 2 0


a

ci bpeak

r2 cipeak

r2 0

peak


( minus ) ( )

C K K 2 2 1



( minus ) ( )

φ φ


ci bres

r2 ci res

r2 0

res





r

r

H

H

2

2 13

i

i

i

i1

r r

r r

σ σ

σ σ =

( ) + Δ

( ) minus Δ ( )

( )

( minus )

( )

( )


( ) +( ) ( minus ) and

⎧

⎨⎪⎪

⎩⎪⎪

H m s

C K K



ii i i

a

i i i i

rci 1 r ci 1

1 1 r 1

i 1

( ) ( )σ

σ σ σ

σ

= + ( minus )


( )( minus ) ( ) ( minus )


( minus )

-


by Eq (7)



⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭E

1 1

1 14

r i

i

r i

i

e

e

ε

ε

μ μ μ

μ μ

σ

σ

Δ

Δ=

+ minus minus

minus minus

Δ

Δ( )

( )

θ( )

( )

θ( )







pεΔ ( )



pεΔ ( ) and i

pεΔ θ( ) is



K K K

K 17


i

1 1 e e p

p


ε

+ = + + Δ + Δ + Δ

+ Δ ( )

ψ ψ ψ

ψ

( ) θ ( ) ( minus ) θ ( minus ) ( ) θ( ) ( )

θ( )


du

dr

u

r

18r ε ε= =( )θ



r

u

r

19i

i

ii

i

irε ε=

Δ

Δ=

( )( )

( )

( )θ( )

( )

( )



u A r r r u r

r K r r 20i

i i i i i i

i i i i

1 1 1

1

( )

( )=

minus +

+ minus ( )ψ

( )


( ) ( ) ( ) ( minus )

where

⎡⎣

⎤⎦

A K

K B K K 1

i r i i i

E i i i i i

1 1 1

1

r 1( ) ( )

ε ε

σ μ μ μ μ

= +


ψ

ν

ψ ψ ψ


( + )

( ) ( ) ( minus ) ( ) ( )



u

r

A r r u r

r r K r r

1

1 21i

i

i

i i i i i

i i i i i

1 1 1 1

1 1

( )

( )ε = =

minus +

+ minus ( )ψ

θ( )

( )

( )


( ) ( minus ) ( ) ( ) ( minus )

u

r K A

r r

r r

1

1 22i

i

ii i i

i i

i ir 1

1

1

ε ε=Δ

Δ= minus + sdot

minus

minus ( )ψ ( )

( )

( )( ) θ( ) ( minus )

( ) ( minus )

( minus ) ( )








correspondence



R R

23

i

n H

H

p0

1

2

2

i

i

r r

r r

=

prod

( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )



R R

24i

j H

H

r0

1

2

2

i

i

r r

r r

=

prod( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )












and 2D axisymmetric


⎧

⎨

⎪⎪

⎩

⎪⎪

u

u e forX

u e forX

u u

u e

0

1 1 0

1

3 25

X

X

R

R

0

0

3

2

00

0max

015

( ) =

sdot lt

minus minus sdot ge

= =( )

minus

minus












max of the proposed











by Rpmax







6 Veri1047297cation






611 GRC




Table 1


A13 B13 C13 D13 E13

GSImax 75 60 50 40 40


mp 287 168 117 0821 0821

sp103 622 117 39 13 13

mr 0821 0687 0575 0516 ndash

sr103 13 07 04 03 ndash

pφ deg 2952 2568 2313 2064 2064


C r MPa 1878 1707 1536 1432 ndash

ψ deg 738 449 289 155 ndash



η 103 108 622 288 119 ndash

E GPa 365 154 866 487 487 piMPa 0 0 0 0 0

μ 025 025 025 025 025R0m 25 25 25 25 25

cip

σ MPa 35 35 35 35 35

cirσ MPa 35 35 35 35 35

σ MPa 375 375 375 375 375

Table 2


A211 B211 C27 D27



mp 75 17 2 05

sp10-3 100 39 4 10

mr 1 1 06 01










μ 02 025 025 025R0m 4 5 3 3

cip

σ MPa 300 30 30 275

cirσ MPa 300 30 25 275

σ MPa 108 30 15 331





612 LDP


maxof the EPP rock



max






































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash












f m s 5a




⎡⎣ ⎤⎦ f m s


a


( minus ) ( )

η

θ θ

( )

f K C K 2



( minus ) ( )

φ φθ θ



⎧

⎨⎪

⎩

⎪

0

SS rock mass

7b

peak peak res

res

ω ηω ω ω

η

ηη η

ω η η


lt lt

ge ( )

( )





232 Flow rule




K 1 sin

1 sin 9

ψ

ψ =

+

minus ( )ψ














p


σ σ σ σ

Δ = minus =minus

( )( ) ( minus ) ( )






m s 2 2 0


a

ci bpeak

r2 cipeak

r2 0

peak


( minus ) ( )

C K K 2 2 1



( minus ) ( )

φ φ


ci bres

r2 ci res

r2 0

res





r

r

H

H

2

2 13

i

i

i

i1

r r

r r

σ σ

σ σ =

( ) + Δ

( ) minus Δ ( )

( )

( minus )

( )

( )


( ) +( ) ( minus ) and

⎧

⎨⎪⎪

⎩⎪⎪

H m s

C K K



ii i i

a

i i i i

rci 1 r ci 1

1 1 r 1

i 1

( ) ( )σ

σ σ σ

σ

= + ( minus )


( )( minus ) ( ) ( minus )


( minus )

-


by Eq (7)



⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭E

1 1

1 14

r i

i

r i

i

e

e

ε

ε

μ μ μ

μ μ

σ

σ

Δ

Δ=

+ minus minus

minus minus

Δ

Δ( )

( )

θ( )

( )

θ( )







pεΔ ( )



pεΔ ( ) and i

pεΔ θ( ) is



K K K

K 17


i

1 1 e e p

p


ε

+ = + + Δ + Δ + Δ

+ Δ ( )

ψ ψ ψ

ψ

( ) θ ( ) ( minus ) θ ( minus ) ( ) θ( ) ( )

θ( )


du

dr

u

r

18r ε ε= =( )θ



r

u

r

19i

i

ii

i

irε ε=

Δ

Δ=

( )( )

( )

( )θ( )

( )

( )



u A r r r u r

r K r r 20i

i i i i i i

i i i i

1 1 1

1

( )

( )=

minus +

+ minus ( )ψ

( )


( ) ( ) ( ) ( minus )

where

⎡⎣

⎤⎦

A K

K B K K 1

i r i i i

E i i i i i

1 1 1

1

r 1( ) ( )

ε ε

σ μ μ μ μ

= +


ψ

ν

ψ ψ ψ


( + )

( ) ( ) ( minus ) ( ) ( )



u

r

A r r u r

r r K r r

1

1 21i

i

i

i i i i i

i i i i i

1 1 1 1

1 1

( )

( )ε = =

minus +

+ minus ( )ψ

θ( )

( )

( )


( ) ( minus ) ( ) ( ) ( minus )

u

r K A

r r

r r

1

1 22i

i

ii i i

i i

i ir 1

1

1

ε ε=Δ

Δ= minus + sdot

minus

minus ( )ψ ( )

( )

( )( ) θ( ) ( minus )

( ) ( minus )

( minus ) ( )








correspondence



R R

23

i

n H

H

p0

1

2

2

i

i

r r

r r

=

prod

( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )



R R

24i

j H

H

r0

1

2

2

i

i

r r

r r

=

prod( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )












and 2D axisymmetric


⎧

⎨

⎪⎪

⎩

⎪⎪

u

u e forX

u e forX

u u

u e

0

1 1 0

1

3 25

X

X

R

R

0

0

3

2

00

0max

015

( ) =

sdot lt

minus minus sdot ge

= =( )

minus

minus












max of the proposed











by Rpmax







6 Veri1047297cation






611 GRC




Table 1


A13 B13 C13 D13 E13

GSImax 75 60 50 40 40


mp 287 168 117 0821 0821

sp103 622 117 39 13 13

mr 0821 0687 0575 0516 ndash

sr103 13 07 04 03 ndash

pφ deg 2952 2568 2313 2064 2064


C r MPa 1878 1707 1536 1432 ndash

ψ deg 738 449 289 155 ndash



η 103 108 622 288 119 ndash

E GPa 365 154 866 487 487 piMPa 0 0 0 0 0

μ 025 025 025 025 025R0m 25 25 25 25 25

cip

σ MPa 35 35 35 35 35

cirσ MPa 35 35 35 35 35

σ MPa 375 375 375 375 375

Table 2


A211 B211 C27 D27



mp 75 17 2 05

sp10-3 100 39 4 10

mr 1 1 06 01










μ 02 025 025 025R0m 4 5 3 3

cip

σ MPa 300 30 30 275

cirσ MPa 300 30 25 275

σ MPa 108 30 15 331





612 LDP


maxof the EPP rock



max






































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash




















p


σ σ σ σ

Δ = minus =minus

( )( ) ( minus ) ( )






m s 2 2 0


a

ci bpeak

r2 cipeak

r2 0

peak


( minus ) ( )

C K K 2 2 1



( minus ) ( )

φ φ


ci bres

r2 ci res

r2 0

res





r

r

H

H

2

2 13

i

i

i

i1

r r

r r

σ σ

σ σ =

( ) + Δ

( ) minus Δ ( )

( )

( minus )

( )

( )


( ) +( ) ( minus ) and

⎧

⎨⎪⎪

⎩⎪⎪

H m s

C K K



ii i i

a

i i i i

rci 1 r ci 1

1 1 r 1

i 1

( ) ( )σ

σ σ σ

σ

= + ( minus )


( )( minus ) ( ) ( minus )


( minus )

-


by Eq (7)



⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥⎧⎨⎩

⎫⎬⎭E

1 1

1 14

r i

i

r i

i

e

e

ε

ε

μ μ μ

μ μ

σ

σ

Δ

Δ=

+ minus minus

minus minus

Δ

Δ( )

( )

θ( )

( )

θ( )







pεΔ ( )



pεΔ ( ) and i

pεΔ θ( ) is



K K K

K 17


i

1 1 e e p

p


ε

+ = + + Δ + Δ + Δ

+ Δ ( )

ψ ψ ψ

ψ

( ) θ ( ) ( minus ) θ ( minus ) ( ) θ( ) ( )

θ( )


du

dr

u

r

18r ε ε= =( )θ



r

u

r

19i

i

ii

i

irε ε=

Δ

Δ=

( )( )

( )

( )θ( )

( )

( )



u A r r r u r

r K r r 20i

i i i i i i

i i i i

1 1 1

1

( )

( )=

minus +

+ minus ( )ψ

( )


( ) ( ) ( ) ( minus )

where

⎡⎣

⎤⎦

A K

K B K K 1

i r i i i

E i i i i i

1 1 1

1

r 1( ) ( )

ε ε

σ μ μ μ μ

= +


ψ

ν

ψ ψ ψ


( + )

( ) ( ) ( minus ) ( ) ( )



u

r

A r r u r

r r K r r

1

1 21i

i

i

i i i i i

i i i i i

1 1 1 1

1 1

( )

( )ε = =

minus +

+ minus ( )ψ

θ( )

( )

( )


( ) ( minus ) ( ) ( ) ( minus )

u

r K A

r r

r r

1

1 22i

i

ii i i

i i

i ir 1

1

1

ε ε=Δ

Δ= minus + sdot

minus

minus ( )ψ ( )

( )

( )( ) θ( ) ( minus )

( ) ( minus )

( minus ) ( )








correspondence



R R

23

i

n H

H

p0

1

2

2

i

i

r r

r r

=

prod

( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )



R R

24i

j H

H

r0

1

2

2

i

i

r r

r r

=

prod( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )












and 2D axisymmetric


⎧

⎨

⎪⎪

⎩

⎪⎪

u

u e forX

u e forX

u u

u e

0

1 1 0

1

3 25

X

X

R

R

0

0

3

2

00

0max

015

( ) =

sdot lt

minus minus sdot ge

= =( )

minus

minus












max of the proposed











by Rpmax







6 Veri1047297cation






611 GRC




Table 1


A13 B13 C13 D13 E13

GSImax 75 60 50 40 40


mp 287 168 117 0821 0821

sp103 622 117 39 13 13

mr 0821 0687 0575 0516 ndash

sr103 13 07 04 03 ndash

pφ deg 2952 2568 2313 2064 2064


C r MPa 1878 1707 1536 1432 ndash

ψ deg 738 449 289 155 ndash



η 103 108 622 288 119 ndash

E GPa 365 154 866 487 487 piMPa 0 0 0 0 0

μ 025 025 025 025 025R0m 25 25 25 25 25

cip

σ MPa 35 35 35 35 35

cirσ MPa 35 35 35 35 35

σ MPa 375 375 375 375 375

Table 2


A211 B211 C27 D27



mp 75 17 2 05

sp10-3 100 39 4 10

mr 1 1 06 01










μ 02 025 025 025R0m 4 5 3 3

cip

σ MPa 300 30 30 275

cirσ MPa 300 30 25 275

σ MPa 108 30 15 331





612 LDP


maxof the EPP rock



max






































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash












correspondence



R R

23

i

n H

H

p0

1

2

2

i

i

r r

r r

=

prod

( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )



R R

24i

j H

H

r0

1

2

2

i

i

r r

r r

=

prod( )

σ σ

σ σ =

( ) + Δ

( ) minus Δ

( )

( )












and 2D axisymmetric


⎧

⎨

⎪⎪

⎩

⎪⎪

u

u e forX

u e forX

u u

u e

0

1 1 0

1

3 25

X

X

R

R

0

0

3

2

00

0max

015

( ) =

sdot lt

minus minus sdot ge

= =( )

minus

minus












max of the proposed











by Rpmax







6 Veri1047297cation






611 GRC




Table 1


A13 B13 C13 D13 E13

GSImax 75 60 50 40 40


mp 287 168 117 0821 0821

sp103 622 117 39 13 13

mr 0821 0687 0575 0516 ndash

sr103 13 07 04 03 ndash

pφ deg 2952 2568 2313 2064 2064


C r MPa 1878 1707 1536 1432 ndash

ψ deg 738 449 289 155 ndash



η 103 108 622 288 119 ndash

E GPa 365 154 866 487 487 piMPa 0 0 0 0 0

μ 025 025 025 025 025R0m 25 25 25 25 25

cip

σ MPa 35 35 35 35 35

cirσ MPa 35 35 35 35 35

σ MPa 375 375 375 375 375

Table 2


A211 B211 C27 D27



mp 75 17 2 05

sp10-3 100 39 4 10

mr 1 1 06 01










μ 02 025 025 025R0m 4 5 3 3

cip

σ MPa 300 30 30 275

cirσ MPa 300 30 25 275

σ MPa 108 30 15 331





612 LDP


maxof the EPP rock



max






































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash














6 Veri1047297cation






611 GRC




Table 1


A13 B13 C13 D13 E13

GSImax 75 60 50 40 40


mp 287 168 117 0821 0821

sp103 622 117 39 13 13

mr 0821 0687 0575 0516 ndash

sr103 13 07 04 03 ndash

pφ deg 2952 2568 2313 2064 2064


C r MPa 1878 1707 1536 1432 ndash

ψ deg 738 449 289 155 ndash



η 103 108 622 288 119 ndash

E GPa 365 154 866 487 487 piMPa 0 0 0 0 0

μ 025 025 025 025 025R0m 25 25 25 25 25

cip

σ MPa 35 35 35 35 35

cirσ MPa 35 35 35 35 35

σ MPa 375 375 375 375 375

Table 2


A211 B211 C27 D27



mp 75 17 2 05

sp10-3 100 39 4 10

mr 1 1 06 01










μ 02 025 025 025R0m 4 5 3 3

cip

σ MPa 300 30 30 275

cirσ MPa 300 30 25 275

σ MPa 108 30 15 331





612 LDP


maxof the EPP rock



max






































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash













612 LDP


maxof the EPP rock



max






































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash

































7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash




















7 Discussion



















Table 3


A3 B3

SImax 75 50

GSImin 27 26mi 1630 1200

mp 6044 1650

sp103 50987 25996

ap 0501 0506

mr 0916 0626

sr103 0178 0142

ar 0527 0530pφ deg 45359 324739

C p MPa 6216 3077rφ deg 29827 24405

C rMPa 2489 1896

E GPa 300 90 μ 025 025R0m 50 50


Table 4Values of u0



(a)n u0


max(mm) u0max(mm)

25 81572 83088 62108 5817750 81491 8299 61532 5743775 81464 82957 61308 57194100 81450 82940 61185 57073125 81442 82930 61107 57000250 81426 82911 60940 56856500 81417 82901 60849 56784750 81415 82897 60818 567581500 81412 82894 60788 56736

2000 81411 82893 60778 567303000 81411 82892 60770 567245000 81411 82892 60764 56719(b)25 84241 83629 15105 1049450 84241 83538 15280 1017475 84240 83508 15399 10048100 84240 83493 15469 10006125 84239 83484 15541 10008250 84239 83465 15632 99817500 84238 83456 15663 99740750 84238 83453 15677 997001500 84238 8345 15645 996712000 84238 83449 15659 996723000 84238 83448 15663 996535000 84238 83448 15659 99644(c)

25 16887 15120 12554 1440950 16872 15042 12287 1395475 16863 15017 12204 13809100 16856 15004 12164 13738125 1685 14996 12139 13696250 16835 14981 12082 13613500 16824 14973 12045 13572750 1682 14971 12030 135581500 16814 14968 12012 135442000 16814 14967 12007 135413000 16814 14967 12002 135385000 16814 14967 12002 13535

Table 5


A3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 0005 0003 0002 EB


B3 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

ηn EPP 5 05 005 003 001 0001 EB







max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash















max0 0







⑧ is














Table 7


pf s0 at X 0 = () ① ② ③ ④ ⑤ ⑥ ⑦ ⑧

A3 6830 6830 6828 6812 5309 3338 319 2578B3 3481 3481 3411 2499 1509 1316 1445 1415








722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash
















722 X 1 X 2 and X 3






Table 8

X 1 X 2 and X 3

for different σ

s0MPa 5 20 35 50 65

X 1 5 6 6 7 8

X 2 010 016 098 154 196

X 3 042 050 121 171

Table 9


0 φ8 φ4 φ2 φ

ψ 0 356 711 1423 2845K ψ 1 113 128 165 282








732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash
















732 X 1 X 2 and X 3



and




Conclusions








Acknowledgement


References




Table 10

X 1 X 2 and X 3

for different ψ

ψ 0 φ8 φ4 φ2 φ

X 1 6 6 6 6 5

X 2 084 098 115 155 283

X 3 031 050 071 121 268








































201040771ndash
















































201040771ndash










Documents

A Numerical Procedure for the Fictitious Support Pressure in the Application of the Convergence–Confinement Method for Circular Tunnel Design