research article upper bound solution of safety...

Research ArticleUpper Bound Solution of Safety Factor for ShallowTunnels Face Using a Nonlinear Failure Criterion andShear Strength Reduction Technique

Fu Huang Zai-lan Li and Tong-hua Ling

School of Civil Engineering and Architecture Changsha University of Science and Technology Changsha Hunan 410114 China

Correspondence should be addressed to Fu Huang hfcsu0001163com

Received 5 February 2016 Accepted 26 April 2016

Academic Editor Yakov Strelniker

Copyright copy 2016 Fu Huang et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A method to evaluate the stability of tunnel face is proposed in the framework of upper bound theorem The safety factor whichis widely applied in slope stability analysis is introduced to estimate the stability of tunnel face using the upper bound theoremof limit analysis in conjunction with a strength reduction technique Considering almost all geomaterials following a nonlinearfailure criterion a generalized tangential technique is used to calculate the external work and internal energy dissipation in thekinematically admissible velocity fieldTheupper bound solution of safety factor is obtained by optimization calculation To evaluatethe validity of the method proposed in this paper the safety factor is compared with those calculated by limit equilibriummethodThe comparison shows the solutions derived from these two methods match each other well which shows the method proposedin this paper can be considered as effective

1 Introduction

Shallow tunnels are very common nowadays in municipalengineering construction as they make traveling more con-venient and also reduce engineering costs Though shallowtunnels have a number of advantages accidents due toinstability of tunnels face would occur if the supportingpressure cannot resist the earth pressure Thus to avoid theoccurrence of such accidents it is necessary to determine thesuitable retaining pressure on the tunnel face The stabilityanalysis of the face for a tunnel excavated in shallow stratahas drawn the attention of many investigators [1ndash4] Onthe other hand the retaining pressure on the tunnel face isconstant in practical engineering when the earth pressurebalance of tunnel face is achieved by using compressed airor bentonite slurry Therefore it is necessary to establisha method to evaluate the stability of tunnel face when theretaining pressure is known The safety factor (FOS) is avalid index to evaluate the stability of geotechnical structurewhich is widely used in geotechnical engineering Presentlythe safety factor used in geotechnical stability analysis ismostly calculated by the finite element method with shear

strength reduction technique (SSRFEM) [5ndash9] Based on thismethod the safety factor and critical failure surface can beobtained by reducing the actual shear strength parametersuntil geotechnical material fails So the accuracy of FOSderived from SSRFEM relies strongly on the determinationof failure of geotechnical structure However the definition offailure for geotechnical structure is a disputed issue Griffithsand Lane [10] chose the nonconvergence of algorithm as theindicator of failure for homogeneous slope Snitbhan andChen [11] regarded bulging or progressive loss of groundalong the vertical cut as the failure of vertical slope As thereis divergence of definition for failure in the geotechnical engi-neering field the different definition of failure would inducedifferent analysis result which confines the application ofSSRFEM in geotechnical engineering

Large numbers of theoretical researches and practicalprojects illustrate that the failure of shallow tunnel facecan be divided into collapse and blow-out failure mecha-nisms The collapse failure mechanism is used to describethe collapse of soil in front of the tunnel face which iscaused by the insufficient retaining pressure on the tunnelface Contrarily blow-out failure mechanism occurs when

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4832097 8 pageshttpdxdoiorg10115520164832097

2 Mathematical Problems in Engineering

the retaining pressure is so great that soil is heaved in frontof the tunnel face Based on the failure mode and kinemati-cally admissible velocity characteristic of the shallow tunnelface Leca and Dormieux [12] proposed a three-dimensionalfailure mechanism composed of solid conical blocks Due tothe slide between the solid conical blocks and surroundingsoil the plastic flow occurs along the velocity discontinuitysurface Using the energy dissipation rate along the surfaceand the rate of work caused by external force Leca andDormieux [12] derived the upper bound solution of retainingpressure for tunnel face As the mechanism proposed by LecaandDormieux [12] is supported by centrifugemodel tests andwell reflects the mechanical characteristics of failure modefor shallow tunnel face many scholars used this mechanismto analyze the stability of shallow tunnel face under variousconditions [13ndash16]

These literatures mentioned above all used linear Mohr-Coulomb criterion However the stress-strain relation ofsoils and rocks is nonlinear This viewpoint has been sup-ported by experiments and some scholars used nonlinearfailure criterion to study the stability of tunnel face andother geotechnical structures [17 18] Based on the failuremechanism proposed by Leca andDormieux [12] Huang andYang [19] calculated the upper bound solution of retainingpressure on the tunnel face using the upper bound theoremin conjunction with nonlinear failure criterion Senent et al[20] studied the face stability of circular tunnels excavatedin heavily fractured rock masses which are subjected tothe nonlinear Hoek-Brown failure criterion According totheir study the critical retaining pressures computed withlimit analysis are very similar to those obtained with thenumericalmodel which proves theirmethod is validThoughtheir studies reflect the influence of nonlinearity on thecritical retaining pressures they failed to propose a methodto evaluate the stability of tunnel face when the retainingpressure is known

In this work upper bound theorem combined with shearstrength reduction technique is used to estimate the facestability of a tunnel excavated in shallow strata followingthe nonlinear failure criterion On the basis of upper boundtheorem the rate of external work and the internal energyrate of dissipation for the failure mechanism are calculatedBased on the relationship between the rate of external workand the internal energy rate of dissipation the convergenceof iteration in the strength reduction calculation can becontrolled which avoids the selection of the definition offailure that occurs in the SSRFEM calculation process Tovalidate the new methodology the FOS of tunnel face iscompared with the result computed by limit equilibriummethod Furthermore the influence of nonlinear parameteron the stability of tunnel face is investigated

2 Upper Bound Theorem with ShearStrength Reduction Technique

Shear strength reduction technique was proposed by Bishop[22] whose core content is the reduction of soil shear strengthparameters until the soil fails To achieve this reduction an

important concept is introduced the shear strength reductionfactor 119865

119904 When the actual shear strength parameters 119888 and

120601 are divided by the shear strength reduction factor the soilstrength parameters 119888

119891and 120601

119891used in upper bound analysis

are obtained

119888119891=

119888

119865119904

120601119891= arctan(tan

120601

119865119904

)

(1)

While shear strength reduction factor 119865119904increases incremen-

tally newly reduced soil strength parameters are obtainedThe iterative process continues until failure occurs

As mentioned above the selection of a suitable definitionof failure is a problem in SSRFEMTo overcome this difficultythe upper bound theorem is used to control the convergenceof iterative operation in the strength reduction techniqueTheupper bound theorem states that when the velocity boundarycondition is satisfied the load derived by equating the rateof external work to the rate of the energy dissipation in anykinematically admissible velocity field is no less than theactual collapse load Therefore if the reduced soil strengthparameters are introduced in the energy dissipation calcula-tion the shear strength reduction factor can be obtained onthe basis of the relationship between the rate of external workand the rate of the energy dissipation

3 Upper Bound Solution of FOS forShallow Tunnel Face

In this work the failure mechanisms proposed by Leca andDormieux [12] are used to calculate the rate of external workand the rate of the energy dissipation in the frameworkof the upper bound theorem of limit analysis The failuremechanism of Leca and Dormieux [12] is composed ofcollapse and blow-out failure mechanisms which is shownin Figure 1 For instance we use failure mechanism III toillustrate the computational process of the upper boundsolution of FOS As Leca and Dormieux [12] have computedthe rate of external work 119875

119890for mechanism III the expression

of 119875119890can be written as

119875119890=1205871198632

4[minus

119877119861119877119862

2

cos120601120590119904

minus1

3sin120572

(119877119861119877119862 sin120572)3 minus (119877

119860 cos120572)3

sin 2120601 cos120601120574119863]119881

(2)

where119863 is the tunnel diameter 120574 is the unit weight of the soil120601 is the friction angle of the soil 120590

119904is surcharge loading 119881 is

the velocity of conical block120572 is the angle between symmetryaxis of conical block and centre line of the tunnel and theparameters 119877

119860 119877119861 119877119862are expressed as

119877119860= cos120572radiccos (120572 + 120601) cos (120572 minus 120601)

119877119861= sin120572radicsin (120572 + 120601) sin (120572 minus 120601)

119877119862=sin 2120572 + (2119862119863 + 1) sin 2120601

cos 2120601 minus cos 2120572

(3)

Mathematical Problems in Engineering 3

O Z

y

B

B

H C

D

Z1

Z2

Δ

Δ1

Δ2

120590s

V1

V2

h1

h2

120590T

X2

(120587)

x1

Y2

Σ12

2r2

2b12

120573

2a12

120572

120572

e

ey e

2a1=D

Σ1

Σ2

2b1

eX1

eX

eZ2

120573 minus 120572

y1

X

(120587998400)

Δ998400

x998400

y998400

x998400 120601998400

120601998400

Ω2

Ω1Z998400

y998400

Z998400

eX9984001

ey1

ey2

(a) Failure mechanism II

O

y

Z

H

C

D

y

h

V

z

eyey

eXeX

2a

2b

eZ

eZ

Δ

120590s 120590s

120590T

Σ

120572

x

x 120577

Ω

h998400

120601998400 2a998400=D 2b998400

Σ998400

(b) Failure mechanism III

Figure 1 Failure mechanism of tunnel face proposed by Leca and Dormieux [12]


H

BB

L

G

TP

NC

N

E

F

A

M

120572

120572

120590v

Figure 2 Wedge stability model of tunnel face proposed by Anagnostou and Kovari [21]

where 119862 is the tunnel depth Moreover the rate of the energydissipation produced in kinematically admissible velocityfield is

119875119881=1205871198632

4[119877119861119877119862

2

minus 119877119860]

119888119881

sin120601 (4)

where 119888 is the cohesion of soil Based on the upper boundtheorem the expression of retaining pressure 120590

119879is obtained

by equaling the rate of external work to the rate of the energydissipation

120590119879=

119875119881minus 119875119890

(12058711986321198771198604 cos120601)119881

(5)

To calculate the upper bound solution of FOS the initialcohesion 119888 and friction angle 120601 of soil are substituted into (1)to obtain the reduced strength parameters 119888

119891and 120601

119891 Then

the reduced strength parameters 119888119891and 120601

119891are substituted

into (2) and (4) to derive the upper bound solution of retain-ing pressure 120590

119879 Finally by equaling the practical retaining

pressure to the retaining pressure 120590119879expressed in (5) the

objective function of safety factor 119891(120572 119865119904) which includes an

angle variable is obtained However the objective function119891(120572 119865

119904) is just an expression of numerous upper bound

solutions According to the upper bound theorem the min-imum value of objective function 119891(120572 119865

119904) is the real upper

bound solution of FOS Therefore a sequential quadraticprogramming is employed to search the minimum value ofobjective function 119891(120572 119865

119904) As some compatibility relations

of velocity should be satisfied in the kinematically admissiblevelocity field the optimization calculation is achieved whencorresponding constraint conditions are satisfied Therefore

the expression of mathematical planning for the problem canbe written as

min 119865119904= 119865119904(120572 119865119904)

st 119875120574+ 119875119879= 119875119881 120601 lt 120572 lt

120587

2minus 120601

(6)

where 119875119879is the power of practical retaining pressure

4 Comparison with the Results Computed byLimit Equilibrium Method

To evaluate the validity of the method proposed in this workthe FOS of tunnel face is calculated by limit equilibriummethod and the upper bound theorem with shear strengthreduction technique Numerical results for these two meth-ods with different parameters are presented and comparedwith each other

Based on the silo theory a three-dimensional failuremode of the tunnel face composed of wedges as shown inFigure 2 is proposed by Anagnostou and Kovari [21] in theframework of limit equilibriummethod According to Bishop[22] the FOS for slope can be defined as the ratio of theavailable shear strength of the soil to that required tomaintainequilibrium Therefore the FOS of tunnel face derived fromthewedgemodel presented in Figure 2 is expressed as follows

FOS =119865119903

119865119894

(7)

where 119865119903is the shearing resistance force and119865

119894is the shearing

force acting on the wedge base The detailed calculatingprocedure of 119865

119903and 119865

119894can be seen in the appendix


M

Tangent

Envelope for nonlinear failure criterion

1205901 1205903

120590n

120591

120601t

ct

Figure 3 Generalized tangential technique for a nonlinear failure criterion

Table 1 Comparison of FOS for tunnel face

Number 119862119863 120574 (kNm3) 120601 (∘) 120590119879(Pa) FOS

LEM LAM1 04 20 20 24960 10442354 100856972 04 20 25 18120 10395553 101895463 04 20 30 12960 10307114 100945964 04 20 35 10560 1031479 100745995 04 20 40 7920 10314455 100986536 04 20 45 6000 10327247 10112345

Dawson et al [23] pointed out that if the actual height ofslope is equal to the critical height computed by the upperbound theorem the FOS of the slope is exactly 10 Similarlyif the actual retaining pressure applied on the tunnel face isequal to the upper solution of retaining pressure the FOS oftunnel face calculated by this method should also be 10

Table 1 presents FOS of tunnel face calculated by LEMand LAM for 119862119863 = 04 and 120574 = 20 kNm3 with the 120601

varying from 20∘ to 45∘ The retaining pressures 120590119879presented

in Table 1 are the upper solutions computed by Leca andDormieux [12] It is found fromTable 1 that the FOS of tunnelface computed by themethodproposed in this paper is almostequal to those using the limit equilibriummethodThereforethe good agreement between these two methods shows thatthe method proposed in this paper is an effective method forcalculating the FOS of shallow tunnel face

5 Nonlinear Failure Criterion and GeneralizedTangential Technique

Zhang and Chen [24] adopted a nonlinear expression todescribe the relationship between the normal and shear stresswhen a plastic flow of geotechnical materials occurs which isexpressed as

120591 = 1198880(1 +

120590119899

120590119905

)

1119898

(8)

where120590119899and 120591 are the normal and shear stresses on the failure

surface respectively and 1198880120590119905 and119898 arematerial parameters

determined by geotechnical test As the strength envelopeof this nonlinear failure criterion is curve the strengthparameter of geotechnical materials cannot be determinedlike linear Mohr-Coulomb failure criterion To overcomethis difficulty Yang and Yin [25] proposed a generalizedtangential technique which uses a tangential line on thenonlinear failure criterion at point 119872 to determine thestrength parameter The tangential line on the curve at thelocation of tangency point 119872 as shown in Figure 3 can bewritten as

120591 = 119888119905+ 120590119899tan120601119905 (9)

where 120601119905is a tangential frictional angle and 119888

119905is the intercept

of the straight line on the 120591-axis 119888119905and 120601

119905can be obtained by

the following two expressions

119888119905=119898 minus 1

119898sdot 1198880(119898 sdot 120590119905sdot tan120601

119905

1198880

)

1(1minus119898)

+ 120590119905

sdot tan120601119905

tan120601119905=

1198880

119898120590119905

(1 +120590119899

120590119905

)

(1minus119898)119898

(10)

Since the tangential line in Figure 3 is random 120601119905is regarded

as a variable to calculate the rate of external work and energydissipation Using sequential quadratic programming theupper bound solution of objective function and correspond-ing value of 120601

119905are obtained Obviously the tangential line is

determined by optimization calculation which indicates thatnonlinear failure criterion represented by the tangential linewill provide the optimum upper bound of actual load for thegeotechnical material

6 Upper Bound Solution of FOS withNonlinear Failure Criterion

To study the effect of the nonlinear parameter on the FOSof tunnel face upper bound solutions of FOS are calcu-lated by the method proposed in this paper with different


Fs

m = 1

m = 125m = 15

m = 175m = 2

04 06 08 102CD

1

15

2

25

3

35

4

45

(a)

CD = 08CD = 10

CD = 06CD = 04

CD = 02

Fs

125 15 175 21m

1

15

2

25

3

35

4

45

5

(b)

Figure 4 Effect of119898 and119862119863 on the FOS for collapse failure mechanism (120590119879= 5 kPa 120574 = 20 kNm3 119888

0= 10 kPa 120590

119905= 2473 kPa119862119863 = 02sim1

119898 = 10sim20)

Fs

m = 1

m = 125m = 15

m = 175m = 2

2

3

4

5

6

7

8

9

10

11

15 2 25 31CD

(a)

CD = 25CD = 3

CD = 2CD = 15

CD = 1

Fs

2

3

4

5

6

7

8

9

10

11

12

125 15 175 21m

(b)

Figure 5 Effect of 119898 and 119862119863 on the FOS for blow-out failure mechanism (120590119879= 800 kPa 120574 = 20 kNm3 119888

0= 90 kPa 120590

119905= 2473 kPa 119862119863 =

1sim3119898 = 10sim20)

parameters As nonlinear coefficient 119898 which controls theshape of strength envelope for nonlinear failure criterionhas significant influence on the other parameters this paperfocuses on the effect of 119898 on FOS Using the collapse failuremechanism the upper bound solutions of FOS with differentnonlinear coefficient119898 are plotted in Figure 4with120590

119879=5 kPa

120574= 20 kNm3 1198880= 10 kPa120590

119905= 2473 kPa and depth ratio119862119863

= 02sim1 It can be found from Figure 4 that the FOS decreaseswith the increase of 119898 and 119862119863 for collapse failure mech-anism Similarly Figure 5 shows the change law of FOS forvarying 119898 and 119862119863 with parameters corresponding to 120590

119879=

800 kPa 120574 = 20 kNm3 1198880= 90 kPa and 120590

119905= 2473 kPa on


the basis of blow-out failure mechanism Figure 5 shows thatthe values of FOS tend to increase directlywith the depth ratio119862119863 and inversely with119898

It can be seen that the change trend of depth ratio 119862119863

on the FOS for collapse failure mechanism is just oppositeto that of the blow-out failure mechanismThis phenomenonis caused by the different velocity of solid conical blocks inkinematically admissible velocity field for these two failuremechanisms In the kinematically admissible velocity fieldof collapse failure mechanism the velocity component ofsolid conical block in front of the tunnel face moves in theopposite direction of the tunnel excavation On the contraryin kinematically admissible velocity field of blow-out failuremechanism induced by the huge retaining pressure on tunnelface the velocity component of solid conical block in frontof the tunnel face moves in the direction of the tunnelexcavation As the velocities of solid conical block in thesetwo failure mechanisms are opposite the effects of powerof the soil weight on the energy dissipation calculation arealso opposite Therefore when collapse failure occurs on thetunnel face low value of depth ratio 119862119863 will contribute tothe tunnel stability However when blow-out failure occurson the tunnel face high value of depth ratio 119862119863 willcontribute to the tunnel stability On the other hand withthe increase of nonlinear coefficient119898 the values of 119888

119905and 120601

119905

obtained by generalized tangential technique both decreaseAs 119888119905and 120601

119905are used to calculate the energy dissipation and

the energy dissipation is independent of velocity of solidconical blocks the change laws of nonlinear coefficient119898 onFOS for these two failure mechanisms are the same

7 Conclusion

The upper bound theorem combined with shear strengthreduction technique is adopted to calculate the FOS ofshallow tunnel face in the framework of nonlinear failurecriterion Upper bound solutions of FOS are derived fromcollapse and blow-out failure mechanisms proposed by Lecaand Dormieux [12] Using generalized tangential techniquethe nonlinear failure criterion is introduced in the energydissipation calculation

The upper bound solutions of FOS are compared with theresults calculated by limit equilibriummethodThe solutionsof FOS derived in this paper are almost equal to thosecalculated by limit equilibriummethodwhich proves that themethod proposed in this paper is effective

Based on the motion features of solid conical block inkinematically admissible velocity field for these two failuremechanisms the reason for different effect of depth ratio onFOS is explained By parameter study it is found that theparameters 119898 and 119862119863 have large influence on the FOS ofshallow tunnel face

Appendix

According to Qiao et al [26] the detailed calculating proce-dure of 119865

119903and 119865119894can be illustrated as follows On the basis of

Terzaghi relation soil pressure theory the three-dimensionalrelation soil pressure is

120590V =120574 minus 120582 sdot 119888

120582 sdot 1198960sdot tan120601

sdot [1 minus 119890minus120582sdot1198960 sdot119885sdottan120601] + 119875

0

sdot 119890minus120582sdot1198960 sdot119885sdottan120601

(A1)

where 1198960is lateral pressure coefficient 120574 is the unit weight

of the soil 119888 is the cohesion of soil 120601 is the friction angle ofthe soil 119885 is the distance between tunnel roof and groundsurface 119875

0is surcharge pressure and 120582 can be calculated by

the following expression

120582 =2 (tan120572 minus 1 (tan120573 + 1 sin120573))

119861 (1 minus 1 tan120572) (A2)

where 120572 120573 and 119861 are parameters which can be seen inFigure 2 The soil pressure is

119875V = 120590V sdot 119860119862119873119864119865 (A3)

where119860119862119873119864119865

is the area of119862119873119864119865 The weight of the wedge is

119866 =1198632

2sdot cot120572 sdot 119861 sdot 120574 (A4)

where 119863 is tunnel diameter Using static equilibrium equa-tion the pressure119873 is obtained

119873 = 119875 sin120572 + (119866 + 119875V) cos120572 (A5)

where 119875 is retaining pressure Based on the Mohr-Coulombfailure criterion the friction resistance 119879 of slip surface119860119872119864119865 in the wedge is derived

119879 = 119888 sdot 119860119860119872119864119865

+ 119873 sdot tan120601 (A6)

where 119860119860119872119864119865

is the area of slip surface 119860119872119864119865 in the wedgeThus the shearing resistance force 119865

119903and the shearing force

119865119894can be written as

119865119903= 119879 + 119875 sdot cos120572

119865119894= (119866 + 119875V) sdot sin120572

(A7)

Competing Interests

The authors declare that the mentioned received funding intheAcknowledgments did not lead to any competing interestsregarding the publication of this paper

Acknowledgments

The preparation of this paper received financial support fromthe National Natural Science Foundation of China (nos51308072 and 51278071) Educational Commission of HunanProvince of China (no 15C0052) and Innovation ResearchProject of Priority Key Disciplines at Changsha Universityof Science and Technology (no 15ZDXK13) The financialsupports are greatly appreciated


References

[1] M Ahmed andM Iskander ldquoEvaluation of tunnel face stabilityby transparent soil modelsrdquo Tunnelling and Underground SpaceTechnology vol 27 no 1 pp 101ndash110 2012

[2] R P Chen J Li L G Kong and L J Tang ldquoExperimentalstudy on face instability of shield tunnel in sandrdquoTunnelling andUnderground Space Technology vol 33 no 1 pp 12ndash21 2013

[3] S H Kim and F Tonon ldquoFace stability and required supportpressure for TBM driven tunnels with ideal face membranemdashDrained caserdquo Tunnelling and Underground Space Technologyvol 25 no 5 pp 526ndash542 2010

[4] P Oreste ldquoFace stabilization of deep tunnels using longitudinalfibreglass dowelsrdquo International Journal of Rock Mechanics andMining Sciences vol 58 pp 127ndash140 2013

[5] H Zheng G H Sun and D F Liu ldquoA practical procedure forsearching critical slip surfaces of slopes based on the strengthreduction techniquerdquo Computers and Geotechnics vol 36 no1-2 pp 1ndash5 2009

[6] T Wang H G Wu Y Li et al ldquoStability analysis of the slopearound flood discharge tunnel under inner water exosmosis atYangqu hydropower stationrdquo Computers and Geotechnics vol51 pp 1ndash11 2013

[7] W B Wei and Y M Cheng ldquoStrength reduction analysisfor slope reinforced with one row of pilesrdquo Computers andGeotechnics vol 36 no 7 pp 1176ndash1185 2009

[8] M S Huang and C Q Jia ldquoStrength reduction FEM instability analysis of soil slopes subjected to transient unsaturatedseepagerdquoComputers and Geotechnics vol 36 no 1-2 pp 93ndash1012009

[9] S M Marandi M Anvar and M Bahrami ldquoUncertaintyanalysis of safety factor of embankment built on stone columnimproved soft soil using fuzzy logic 120572-cut techniquerdquo Comput-ers and Geotechnics vol 75 pp 135ndash144 2016

[10] D V Griffiths and P A Lane ldquoSlope stability analysis by finiteelementsrdquo Geotechnique vol 49 no 3 pp 387ndash403 1999

[11] N Snitbhan and W-F Chen ldquoElastic-plastic large deformationanalysis of soil slopesrdquo Computers and Structures vol 9 no 6pp 567ndash577 1978

[12] E Leca and L Dormieux ldquoUpper and lower bound solutionsfor the face stability of shallow circular tunnels in frictionalmaterialrdquo Geotechnique vol 40 no 4 pp 581ndash606 1990

[13] I M Lee J S Lee and S W Nam ldquoEffect of seepage force ontunnel face stability reinforced with multi-step pipe groutingrdquoTunnelling and Underground Space Technology vol 19 no 6 pp551ndash565 2004

[14] Y Li F Emeriault R Kastner and Z X Zhang ldquoStability anal-ysis of large slurry shield-driven tunnel in soft clayrdquo Tunnellingand Underground Space Technology vol 24 no 4 pp 472ndash4812009

[15] I M Lee and S W Nam ldquoThe study of seepage forces acting onthe tunnel lining and tunnel face in shallow tunnelsrdquo Tunnellingand Underground Space Technology vol 16 no 1 pp 31ndash402001

[16] H T Wang and J Q Jia ldquoFace stability analysis of tunnelwith pipe roof reinforcement based on limit analysisrdquo ElectronicJournal of Geotechnical Engineering vol 14 pp 1ndash15 2009

[17] X L Yang and Z X Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[18] X L Yang J S Xu Y X Li et al ldquoCollapse mechanism oftunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[19] F Huang and X L Yang ldquoUpper bound solutions for theface stability of shallow circular tunnels subjected to nonlinearfailure criterionrdquo in Proceedings of the GeoShangai InternationalConference Deep and Underground Excavations pp 251ndash256Shangai China 2010

[20] S Senent G Mollon and R Jimenez ldquoTunnel face stabilityin heavily fractured rock masses that follow the HoekndashBrownfailure criterionrdquo International Journal of Rock Mechanics andMining Sciences vol 60 pp 440ndash451 2013

[21] G Anagnostou and K Kovari ldquoFace stability conditions withEarth-Pressure-Balanced shieldsrdquo Tunnelling and UndergroundSpace Technology vol 11 no 2 pp 165ndash173 1996

[22] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth slopesrdquo Geotechnique vol 5 no 1 pp 7ndash17 1954

[23] E M Dawson W H Roth and A Drescher ldquoSlope stabilityanalysis by strength reductionrdquo Geotechnique vol 49 no 6 pp835ndash840 1999

[24] X J Zhang and W F Chen ldquoStability analysis of slopes withgeneral nonlinear failure criterionrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 11 no1 pp 33ndash50 1987

[25] X L Yang and J H Yin ldquoSlope stability analysis with nonlinearfailure criterionrdquo Journal of Engineering Mechanics vol 130 no3 pp 267ndash273 2004

[26] J L Qiao Y T Zhang J Gao et al ldquoApplication of strengthreduction method to stability analysis of shield tunnel facerdquoJournal of Tianjin University vol 43 no 1 pp 14ndash20 2010

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Stochastic AnalysisInternational Journal of


the retaining pressure is so great that soil is heaved in frontof the tunnel face Based on the failure mode and kinemati-cally admissible velocity characteristic of the shallow tunnelface Leca and Dormieux [12] proposed a three-dimensionalfailure mechanism composed of solid conical blocks Due tothe slide between the solid conical blocks and surroundingsoil the plastic flow occurs along the velocity discontinuitysurface Using the energy dissipation rate along the surfaceand the rate of work caused by external force Leca andDormieux [12] derived the upper bound solution of retainingpressure for tunnel face As the mechanism proposed by LecaandDormieux [12] is supported by centrifugemodel tests andwell reflects the mechanical characteristics of failure modefor shallow tunnel face many scholars used this mechanismto analyze the stability of shallow tunnel face under variousconditions [13ndash16]

These literatures mentioned above all used linear Mohr-Coulomb criterion However the stress-strain relation ofsoils and rocks is nonlinear This viewpoint has been sup-ported by experiments and some scholars used nonlinearfailure criterion to study the stability of tunnel face andother geotechnical structures [17 18] Based on the failuremechanism proposed by Leca andDormieux [12] Huang andYang [19] calculated the upper bound solution of retainingpressure on the tunnel face using the upper bound theoremin conjunction with nonlinear failure criterion Senent et al[20] studied the face stability of circular tunnels excavatedin heavily fractured rock masses which are subjected tothe nonlinear Hoek-Brown failure criterion According totheir study the critical retaining pressures computed withlimit analysis are very similar to those obtained with thenumericalmodel which proves theirmethod is validThoughtheir studies reflect the influence of nonlinearity on thecritical retaining pressures they failed to propose a methodto evaluate the stability of tunnel face when the retainingpressure is known

In this work upper bound theorem combined with shearstrength reduction technique is used to estimate the facestability of a tunnel excavated in shallow strata followingthe nonlinear failure criterion On the basis of upper boundtheorem the rate of external work and the internal energyrate of dissipation for the failure mechanism are calculatedBased on the relationship between the rate of external workand the internal energy rate of dissipation the convergenceof iteration in the strength reduction calculation can becontrolled which avoids the selection of the definition offailure that occurs in the SSRFEM calculation process Tovalidate the new methodology the FOS of tunnel face iscompared with the result computed by limit equilibriummethod Furthermore the influence of nonlinear parameteron the stability of tunnel face is investigated

2 Upper Bound Theorem with ShearStrength Reduction Technique

Shear strength reduction technique was proposed by Bishop[22] whose core content is the reduction of soil shear strengthparameters until the soil fails To achieve this reduction an

important concept is introduced the shear strength reductionfactor 119865

119904 When the actual shear strength parameters 119888 and

120601 are divided by the shear strength reduction factor the soilstrength parameters 119888

119891and 120601

119891used in upper bound analysis

are obtained

119888119891=

119888

119865119904

120601119891= arctan(tan

120601

119865119904

)

(1)

While shear strength reduction factor 119865119904increases incremen-

tally newly reduced soil strength parameters are obtainedThe iterative process continues until failure occurs

As mentioned above the selection of a suitable definitionof failure is a problem in SSRFEMTo overcome this difficultythe upper bound theorem is used to control the convergenceof iterative operation in the strength reduction techniqueTheupper bound theorem states that when the velocity boundarycondition is satisfied the load derived by equating the rateof external work to the rate of the energy dissipation in anykinematically admissible velocity field is no less than theactual collapse load Therefore if the reduced soil strengthparameters are introduced in the energy dissipation calcula-tion the shear strength reduction factor can be obtained onthe basis of the relationship between the rate of external workand the rate of the energy dissipation

3 Upper Bound Solution of FOS forShallow Tunnel Face

In this work the failure mechanisms proposed by Leca andDormieux [12] are used to calculate the rate of external workand the rate of the energy dissipation in the frameworkof the upper bound theorem of limit analysis The failuremechanism of Leca and Dormieux [12] is composed ofcollapse and blow-out failure mechanisms which is shownin Figure 1 For instance we use failure mechanism III toillustrate the computational process of the upper boundsolution of FOS As Leca and Dormieux [12] have computedthe rate of external work 119875

119890for mechanism III the expression

of 119875119890can be written as

119875119890=1205871198632

4[minus

119877119861119877119862

2

cos120601120590119904

minus1

3sin120572

(119877119861119877119862 sin120572)3 minus (119877

119860 cos120572)3

sin 2120601 cos120601120574119863]119881

(2)

where119863 is the tunnel diameter 120574 is the unit weight of the soil120601 is the friction angle of the soil 120590

119904is surcharge loading 119881 is

the velocity of conical block120572 is the angle between symmetryaxis of conical block and centre line of the tunnel and theparameters 119877

119860 119877119861 119877119862are expressed as

119877119860= cos120572radiccos (120572 + 120601) cos (120572 minus 120601)

119877119861= sin120572radicsin (120572 + 120601) sin (120572 minus 120601)

119877119862=sin 2120572 + (2119862119863 + 1) sin 2120601

cos 2120601 minus cos 2120572

(3)


O Z

y

B

B

H C

D

Z1

Z2

Δ

Δ1

Δ2

120590s

V1

V2

h1

h2

120590T

X2

(120587)

x1

Y2

Σ12

2r2

2b12

120573

2a12

120572

120572

e

ey e

2a1=D

Σ1

Σ2

2b1

eX1

eX

eZ2

120573 minus 120572

y1

X

(120587998400)

Δ998400

x998400

y998400

x998400 120601998400

120601998400

Ω2

Ω1Z998400

y998400

Z998400

eX9984001

ey1

ey2


O

y

Z

H

C

D

y

h

V

z

eyey

eXeX

2a

2b

eZ

eZ

Δ

120590s 120590s

120590T

Σ

120572

x

x 120577

Ω

h998400

120601998400 2a998400=D 2b998400

Σ998400




H

BB

L

G

TP

NC

N

E

F

A

M

120572

120572

120590v



119875119881=1205871198632

4[119877119861119877119862

2

minus 119877119860]

119888119881

sin120601 (4)


119879is obtained


120590119879=

119875119881minus 119875119890

(12058711986321198771198604 cos120601)119881

(5)


119891and 120601

119891 Then















min 119865119904= 119865119904(120572 119865119904)

st 119875120574+ 119875119879= 119875119881 120601 lt 120572 lt

120587

2minus 120601

(6)





FOS =119865119903

119865119894

(7)




119903and 119865



M

Tangent


1205901 1205903

120590n

120591

120601t

ct



Number 119862119863 120574 (kNm3) 120601 (∘) 120590119879(Pa) FOS

LEM LAM1 04 20 20 24960 10442354 100856972 04 20 25 18120 10395553 101895463 04 20 30 12960 10307114 100945964 04 20 35 10560 1031479 100745995 04 20 40 7920 10314455 100986536 04 20 45 6000 10327247 10112345







120591 = 1198880(1 +

120590119899

120590119905

)

1119898

(8)




120591 = 119888119905+ 120590119899tan120601119905 (9)






119888119905=119898 minus 1


119905

1198880

)

1(1minus119898)

+ 120590119905

sdot tan120601119905

tan120601119905=

1198880

119898120590119905

(1 +120590119899

120590119905

)

(1minus119898)119898

(10)








Fs

m = 1

m = 125m = 15

m = 175m = 2

04 06 08 102CD

1

15

2

25

3

35

4

45

(a)

CD = 08CD = 10

CD = 06CD = 04

CD = 02

Fs

125 15 175 21m

1

15

2

25

3

35

4

45

5

(b)


0= 10 kPa 120590

119905= 2473 kPa119862119863 = 02sim1

119898 = 10sim20)

Fs

m = 1

m = 125m = 15

m = 175m = 2

2

3

4

5

6

7

8

9

10

11

15 2 25 31CD

(a)

CD = 25CD = 3

CD = 2CD = 15

CD = 1

Fs

2

3

4

5

6

7

8

9

10

11

12

125 15 175 21m

(b)


0= 90 kPa 120590

119905= 2473 kPa 119862119863 =

1sim3119898 = 10sim20)


119879=5 kPa

120574= 20 kNm3 1198880= 10 kPa120590



119879=

800 kPa 120574 = 20 kNm3 1198880= 90 kPa and 120590

119905= 2473 kPa on





119905and 120601

119905




7 Conclusion




Appendix




120590V =120574 minus 120582 sdot 119888

120582 sdot 1198960sdot tan120601


0


(A1)





120582 =2 (tan120572 minus 1 (tan120573 + 1 sin120573))

119861 (1 minus 1 tan120572) (A2)


119875V = 120590V sdot 119860119862119873119864119865 (A3)

where119860119862119873119864119865


119866 =1198632



119873 = 119875 sin120572 + (119866 + 119875V) cos120572 (A5)


119879 = 119888 sdot 119860119860119872119864119865

+ 119873 sdot tan120601 (A6)

where 119860119860119872119864119865




119865119903= 119879 + 119875 sdot cos120572

119865119894= (119866 + 119875V) sdot sin120572

(A7)

Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










O Z

y

B

B

H C

D

Z1

Z2

Δ

Δ1

Δ2

120590s

V1

V2

h1

h2

120590T

X2

(120587)

x1

Y2

Σ12

2r2

2b12

120573

2a12

120572

120572

e

ey e

2a1=D

Σ1

Σ2

2b1

eX1

eX

eZ2

120573 minus 120572

y1

X

(120587998400)

Δ998400

x998400

y998400

x998400 120601998400

120601998400

Ω2

Ω1Z998400

y998400

Z998400

eX9984001

ey1

ey2


O

y

Z

H

C

D

y

h

V

z

eyey

eXeX

2a

2b

eZ

eZ

Δ

120590s 120590s

120590T

Σ

120572

x

x 120577

Ω

h998400

120601998400 2a998400=D 2b998400

Σ998400




H

BB

L

G

TP

NC

N

E

F

A

M

120572

120572

120590v



119875119881=1205871198632

4[119877119861119877119862

2

minus 119877119860]

119888119881

sin120601 (4)


119879is obtained


120590119879=

119875119881minus 119875119890

(12058711986321198771198604 cos120601)119881

(5)


119891and 120601

119891 Then















min 119865119904= 119865119904(120572 119865119904)

st 119875120574+ 119875119879= 119875119881 120601 lt 120572 lt

120587

2minus 120601

(6)





FOS =119865119903

119865119894

(7)




119903and 119865



M

Tangent


1205901 1205903

120590n

120591

120601t

ct



Number 119862119863 120574 (kNm3) 120601 (∘) 120590119879(Pa) FOS

LEM LAM1 04 20 20 24960 10442354 100856972 04 20 25 18120 10395553 101895463 04 20 30 12960 10307114 100945964 04 20 35 10560 1031479 100745995 04 20 40 7920 10314455 100986536 04 20 45 6000 10327247 10112345







120591 = 1198880(1 +

120590119899

120590119905

)

1119898

(8)




120591 = 119888119905+ 120590119899tan120601119905 (9)






119888119905=119898 minus 1


119905

1198880

)

1(1minus119898)

+ 120590119905

sdot tan120601119905

tan120601119905=

1198880

119898120590119905

(1 +120590119899

120590119905

)

(1minus119898)119898

(10)








Fs

m = 1

m = 125m = 15

m = 175m = 2

04 06 08 102CD

1

15

2

25

3

35

4

45

(a)

CD = 08CD = 10

CD = 06CD = 04

CD = 02

Fs

125 15 175 21m

1

15

2

25

3

35

4

45

5

(b)


0= 10 kPa 120590

119905= 2473 kPa119862119863 = 02sim1

119898 = 10sim20)

Fs

m = 1

m = 125m = 15

m = 175m = 2

2

3

4

5

6

7

8

9

10

11

15 2 25 31CD

(a)

CD = 25CD = 3

CD = 2CD = 15

CD = 1

Fs

2

3

4

5

6

7

8

9

10

11

12

125 15 175 21m

(b)


0= 90 kPa 120590

119905= 2473 kPa 119862119863 =

1sim3119898 = 10sim20)


119879=5 kPa

120574= 20 kNm3 1198880= 10 kPa120590



119879=

800 kPa 120574 = 20 kNm3 1198880= 90 kPa and 120590

119905= 2473 kPa on





119905and 120601

119905




7 Conclusion




Appendix




120590V =120574 minus 120582 sdot 119888

120582 sdot 1198960sdot tan120601


0


(A1)





120582 =2 (tan120572 minus 1 (tan120573 + 1 sin120573))

119861 (1 minus 1 tan120572) (A2)


119875V = 120590V sdot 119860119862119873119864119865 (A3)

where119860119862119873119864119865


119866 =1198632



119873 = 119875 sin120572 + (119866 + 119875V) cos120572 (A5)


119879 = 119888 sdot 119860119860119872119864119865

+ 119873 sdot tan120601 (A6)

where 119860119860119872119864119865




119865119903= 119879 + 119875 sdot cos120572

119865119894= (119866 + 119875V) sdot sin120572

(A7)

Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H

BB

L

G

TP

NC

N

E

F

A

M

120572

120572

120590v



119875119881=1205871198632

4[119877119861119877119862

2

minus 119877119860]

119888119881

sin120601 (4)


119879is obtained


120590119879=

119875119881minus 119875119890

(12058711986321198771198604 cos120601)119881

(5)


119891and 120601

119891 Then















min 119865119904= 119865119904(120572 119865119904)

st 119875120574+ 119875119879= 119875119881 120601 lt 120572 lt

120587

2minus 120601

(6)





FOS =119865119903

119865119894

(7)




119903and 119865



M

Tangent


1205901 1205903

120590n

120591

120601t

ct



Number 119862119863 120574 (kNm3) 120601 (∘) 120590119879(Pa) FOS

LEM LAM1 04 20 20 24960 10442354 100856972 04 20 25 18120 10395553 101895463 04 20 30 12960 10307114 100945964 04 20 35 10560 1031479 100745995 04 20 40 7920 10314455 100986536 04 20 45 6000 10327247 10112345







120591 = 1198880(1 +

120590119899

120590119905

)

1119898

(8)




120591 = 119888119905+ 120590119899tan120601119905 (9)






119888119905=119898 minus 1


119905

1198880

)

1(1minus119898)

+ 120590119905

sdot tan120601119905

tan120601119905=

1198880

119898120590119905

(1 +120590119899

120590119905

)

(1minus119898)119898

(10)








Fs

m = 1

m = 125m = 15

m = 175m = 2

04 06 08 102CD

1

15

2

25

3

35

4

45

(a)

CD = 08CD = 10

CD = 06CD = 04

CD = 02

Fs

125 15 175 21m

1

15

2

25

3

35

4

45

5

(b)


0= 10 kPa 120590

119905= 2473 kPa119862119863 = 02sim1

119898 = 10sim20)

Fs

m = 1

m = 125m = 15

m = 175m = 2

2

3

4

5

6

7

8

9

10

11

15 2 25 31CD

(a)

CD = 25CD = 3

CD = 2CD = 15

CD = 1

Fs

2

3

4

5

6

7

8

9

10

11

12

125 15 175 21m

(b)


0= 90 kPa 120590

119905= 2473 kPa 119862119863 =

1sim3119898 = 10sim20)


119879=5 kPa

120574= 20 kNm3 1198880= 10 kPa120590



119879=

800 kPa 120574 = 20 kNm3 1198880= 90 kPa and 120590

119905= 2473 kPa on





119905and 120601

119905




7 Conclusion




Appendix




120590V =120574 minus 120582 sdot 119888

120582 sdot 1198960sdot tan120601


0


(A1)





120582 =2 (tan120572 minus 1 (tan120573 + 1 sin120573))

119861 (1 minus 1 tan120572) (A2)


119875V = 120590V sdot 119860119862119873119864119865 (A3)

where119860119862119873119864119865


119866 =1198632



119873 = 119875 sin120572 + (119866 + 119875V) cos120572 (A5)


119879 = 119888 sdot 119860119860119872119864119865

+ 119873 sdot tan120601 (A6)

where 119860119860119872119864119865




119865119903= 119879 + 119875 sdot cos120572

119865119894= (119866 + 119875V) sdot sin120572

(A7)

Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M

Tangent


1205901 1205903

120590n

120591

120601t

ct



Number 119862119863 120574 (kNm3) 120601 (∘) 120590119879(Pa) FOS

LEM LAM1 04 20 20 24960 10442354 100856972 04 20 25 18120 10395553 101895463 04 20 30 12960 10307114 100945964 04 20 35 10560 1031479 100745995 04 20 40 7920 10314455 100986536 04 20 45 6000 10327247 10112345







120591 = 1198880(1 +

120590119899

120590119905

)

1119898

(8)




120591 = 119888119905+ 120590119899tan120601119905 (9)






119888119905=119898 minus 1


119905

1198880

)

1(1minus119898)

+ 120590119905

sdot tan120601119905

tan120601119905=

1198880

119898120590119905

(1 +120590119899

120590119905

)

(1minus119898)119898

(10)








Fs

m = 1

m = 125m = 15

m = 175m = 2

04 06 08 102CD

1

15

2

25

3

35

4

45

(a)

CD = 08CD = 10

CD = 06CD = 04

CD = 02

Fs

125 15 175 21m

1

15

2

25

3

35

4

45

5

(b)


0= 10 kPa 120590

119905= 2473 kPa119862119863 = 02sim1

119898 = 10sim20)

Fs

m = 1

m = 125m = 15

m = 175m = 2

2

3

4

5

6

7

8

9

10

11

15 2 25 31CD

(a)

CD = 25CD = 3

CD = 2CD = 15

CD = 1

Fs

2

3

4

5

6

7

8

9

10

11

12

125 15 175 21m

(b)


0= 90 kPa 120590

119905= 2473 kPa 119862119863 =

1sim3119898 = 10sim20)


119879=5 kPa

120574= 20 kNm3 1198880= 10 kPa120590



119879=

800 kPa 120574 = 20 kNm3 1198880= 90 kPa and 120590

119905= 2473 kPa on





119905and 120601

119905




7 Conclusion




Appendix




120590V =120574 minus 120582 sdot 119888

120582 sdot 1198960sdot tan120601


0


(A1)





120582 =2 (tan120572 minus 1 (tan120573 + 1 sin120573))

119861 (1 minus 1 tan120572) (A2)


119875V = 120590V sdot 119860119862119873119864119865 (A3)

where119860119862119873119864119865


119866 =1198632



119873 = 119875 sin120572 + (119866 + 119875V) cos120572 (A5)


119879 = 119888 sdot 119860119860119872119864119865

+ 119873 sdot tan120601 (A6)

where 119860119860119872119864119865




119865119903= 119879 + 119875 sdot cos120572

119865119894= (119866 + 119875V) sdot sin120572

(A7)

Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Fs

m = 1

m = 125m = 15

m = 175m = 2

04 06 08 102CD

1

15

2

25

3

35

4

45

(a)

CD = 08CD = 10

CD = 06CD = 04

CD = 02

Fs

125 15 175 21m

1

15

2

25

3

35

4

45

5

(b)


0= 10 kPa 120590

119905= 2473 kPa119862119863 = 02sim1

119898 = 10sim20)

Fs

m = 1

m = 125m = 15

m = 175m = 2

2

3

4

5

6

7

8

9

10

11

15 2 25 31CD

(a)

CD = 25CD = 3

CD = 2CD = 15

CD = 1

Fs

2

3

4

5

6

7

8

9

10

11

12

125 15 175 21m

(b)


0= 90 kPa 120590

119905= 2473 kPa 119862119863 =

1sim3119898 = 10sim20)


119879=5 kPa

120574= 20 kNm3 1198880= 10 kPa120590



119879=

800 kPa 120574 = 20 kNm3 1198880= 90 kPa and 120590

119905= 2473 kPa on





119905and 120601

119905




7 Conclusion




Appendix




120590V =120574 minus 120582 sdot 119888

120582 sdot 1198960sdot tan120601


0


(A1)





120582 =2 (tan120572 minus 1 (tan120573 + 1 sin120573))

119861 (1 minus 1 tan120572) (A2)


119875V = 120590V sdot 119860119862119873119864119865 (A3)

where119860119862119873119864119865


119866 =1198632



119873 = 119875 sin120572 + (119866 + 119875V) cos120572 (A5)


119879 = 119888 sdot 119860119860119872119864119865

+ 119873 sdot tan120601 (A6)

where 119860119860119872119864119865




119865119903= 119879 + 119875 sdot cos120572

119865119894= (119866 + 119875V) sdot sin120572

(A7)

Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119905and 120601

119905




7 Conclusion




Appendix




120590V =120574 minus 120582 sdot 119888

120582 sdot 1198960sdot tan120601


0


(A1)





120582 =2 (tan120572 minus 1 (tan120573 + 1 sin120573))

119861 (1 minus 1 tan120572) (A2)


119875V = 120590V sdot 119860119862119873119864119865 (A3)

where119860119862119873119864119865


119866 =1198632



119873 = 119875 sin120572 + (119866 + 119875V) cos120572 (A5)


119879 = 119888 sdot 119860119860119872119864119865

+ 119873 sdot tan120601 (A6)

where 119860119860119872119864119865




119865119903= 119879 + 119875 sdot cos120572

119865119894= (119866 + 119875V) sdot sin120572

(A7)

Competing Interests


Acknowledgments



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article upper bound solution of safety...

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