analysis of soil-compacting effect caused by shield...

Research ArticleAnalysis of Soil-Compacting Effect Caused byShield Tunneling Using Three-Dimensional ElastoplasticSolution of Cylindrical Cavity Expansion

Mengxi Zhang 1 Xiaoqing Zhang 2 Chengyu Hong 2

Lalit Borana3 and Akbar A Javadi4

1Department of Civil Engineering Shanghai University 99 Shangda Road Shanghai 200444 China2Department of Civil Engineering Shanghai University 99 Shangda Road Shanghai 200072 China3Department of Civil and Environmental Engineering The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong4Department of Engineering College of Engineering Mathematics and Physical Sciences University of Exeter ExeterDevon EX4 4QF UK

Correspondence should be addressed to Mengxi Zhang mxzhangishueducn

Received 16 September 2017 Revised 10 March 2018 Accepted 1 April 2018 Published 27 May 2018

Academic Editor Edoardo Artioli

Copyright copy 2018 Mengxi Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Soil squeezing effect and formation disturbance caused by tunnel excavation can be simulated by cylindrical cavity expansiondue to the comparability between tunneling and cavity expansion Although most of the existing theoretical derivation is basedon simple constitutive model of soil foundation not only the relation between principal stress components was simplified in thesolution process but also the stress history initial stress anisotropy and stress-induced anisotropy of structural soil were neglectedThe mechanical characteristics of soil are closely related to its stress history so there is a gap between the above research and theactual engineering conditions A three-dimensional elastoplastic solution of cylindrical cavity expansion is obtained based on thetheory of critical state soil mechanics and engineering characteristics of shield tunneling In order to fully consider the influenceof initial anisotropy and induced anisotropy on the mechanical behavior of soils the soil elastoplastic constitutive relation of cavityexpansion is described in the course of 1198700-based modified Cam-clay (1198700-MCC) model after soil yielding An equation with equalnumber of variables is obtained under the elastic-plastic boundary condition based on the Lagrange multiplier method By solvingthe extreme value of the original function the analytical solution of radial tangential and vertical effective stresses distributionaround the circular tunnel excavation is obtained In addition changes of elastic deformation area and plastic deformation area forsoil during the shield excavation have been analyzed Calculation results are compared with the numerical solutions which usuallyconsider isotropic soil behavior as the basic assumption In this paper a constitutive model which is more consistent with the actualmechanical behavior of the soil and the construction process of the shield tunnel is considered Therefore the numerical solutionsare more realistic and suitable for the shield excavation analysis and can provide theoretical guidance required for design of shieldtunneling

1 Introduction

Theoretical analysis of cylindrical cavity expansion theoryin geotechnical engineering has been widely used in pilesinking shield tunneling static cone penetration test (CPT)pressuremeter test (PMT) and mixing pile constructiondisturbance problems [1ndash6] Hill (1950) first proposed thespherical cavity expansion method and derived a general

solution of stresses and displacements in Tresca material [7]Vesic (1972) obtained the classical solutions considering theplastic zone volume change of ideal elastic-plastic expansionproblem based on Mohr-Coulomb model [8] Cao et al(2001) presented closed form solutions for both sphericaland cylindrical cavities by assuming the ultimate deviatorstress distributions in the plastic zone and taking the in situstress conditions into consideration [9] Alonso et al (2003)

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 8718274 14 pageshttpsdoiorg10115520188718274

2 Mathematical Problems in Engineering

shield

interspaces

soil movement

(a) Excavation stage

driving

interspaces

soil squeezing

(b) Driving stage

shield

initial stress

soil squeezing

(c) Effect of initial stress field

Figure 1 Schematic diagram of compacting soil by shield

obtained the self-similar solution for the circular tunnelin strain-softening rock masses [10] Wheeler et al (2003)and Nakano et al (2005) adjusted yield surface equation onthe basis of 1198700-based modified Cam-clay considering 1198700consolidation caused by induced anisotropy and its evolutionlaw in the loading process [11 12] Yin and Hicher (2008)identified parameters of modified Cam-clay model withviscosity of soil from the cavity expansion [13] The problemof cavity expansion and cavity contraction has attractedmuch attention in geotechnical problems with applicationto the bearing capacity of deep foundations interpretationof pressuremeter tests breakout resistance of anchors piledriving wellbore instability underground excavation andblasting fracturing by explosives

However based on the above-mentioned researches theconventional cylinder expansion theory assumes that initialstress is isotropic (static earth pressure coefficient 1198700 = 10)Due to the sedimentation history mineralogical characteris-tics and other several factors the initial stress of soil layeris usually anisotropic For example the vertical stress andhorizontal stress of the cylinder bore are not the same inthe horizontal tunneling construction of the undergroundpipe laying tunnel engineering As a result the traditionaltheory of cylindrical expansion generally used in practicalengineering fails in explaining some practical phenomenonAlso the traditional cylindrical expansion theory assumesthat the cylindrical boundary condition is controlled bydisplacement which presents around the cylinder in the formof symmetrical distribution as a function of distance Thetraditional control theory can be applied in geotechnicalproblems analysis such as cylindrical pile and static conepenetration test But in the practical engineering of shieldtunnel boundary condition of the cylinder expansion iscontrolled by stress and displacement of the cylinder whichwas unable to satisfy the condition of axial symmetryHence it may not be reasonable to explain such geotechnicalproblems considering traditional theory of cylindrical expan-sion

In this paper a three-dimensional elastoplastic solution ofcylindrical cavity expansion is obtained based on the theory

of critical state soilmechanics and engineering characteristicsof shield tunneling to calculate the undrained cylindricalcavity expansion considered anisotropic clay of 1198700 Thisresearch work not only contributes to the theoretical valuesin field of geotechnical engineering but also has an importantpractical engineering significance

2 Definition of Soil-Compacting Effect andMechanical Model

21 Definition of Soil-Compacting Effect Shield tunnelingis a typical three-dimensional problem Soil deformation isclosely related to the relative position of the shield machineIn actual shield construction earth pressure and supportpressure ahead of excavation face are not completely balanced[15] When support pressure is greater than passive earthpressure the soil is extruded and then soil-compactingeffect presents And the free surface may collapse if supportpressure is less than active earth pressure This paper mainlyaims to study the soil-compacting effect produced when thesupport pressure is larger than passive earth pressure Soil-compacting effect of shield tunneling is shown in Figure 1

(a) Along with the shield approach the soil which issqueezed by the excavation face will be pushed around theshield And then the soil is output by the screw conveyor toform a voidThe surrounding soil is excavated and unloadedthe stress is released to cause soil expansion and the front soilmoves to the void (Figure 1(a))

(b) As the shield machine moves forward the soil will besqueezed around the shield and excavation face In the planeperpendicular to the axis of the tunnel the soil is assumesubjected to uniform radial extrusion force which generatesan equal amount of outward radial movement of the tunnel(Figure 1(b))

(c) The extruded soil moves outwards of the shieldmachine and is also affected by the friction force arise fromthe driving process of shield machine Radial deformationalong the outer side of the tunnel finally formed under theinfluence of initial stress field (Figure 1(c))

Mathematical Problems in Engineering 3

a

o

r

y

x

Elastic zone

Plastic zone

rp

a0

a

rxrx0 rp0 ℎ K0=

Figure 2 Expansion of a cylindrical cavity in 1198700 consolidated anisotropic clays

It can be seen that the squeezing effect of shield con-struction makes the surface subsidence or uplift and layeredsoil movement soil stress water content pore water pres-sure elastic modulus Poissonrsquos ratio strength and bearingcapacity of the physical and mechanical properties are likelyto change It is feasible to simulate the soil-compacting effectof shield tunneling by using cylindrical cavity expansiontheory

22 Mechanical Model Figure 2 shows a cylindrical cavityexpansion in an infinite 1198700 consolidated saturated clay withan in-plane initial horizontal pressure stress 120590ℎ = 1198700120590V andvertical pressure stress 120590V at infinity as well as the hydrostaticpressure 1199060 Note that cylindrical polar coordinate system isadopted for present analysis and the occurring compressivestresses and strains are taken as positive As average internalpressure gradually increases from120590ℎ to 120590119886 the soil adjacent tothe cavity wall yields first Further increase in average internalcavity wall pressure will lead to current cavity radius to theelastic-plastic (EP) boundary 119903119901 with a further increasedinternal cavity pressure with the continuation of the shieldexcavation The symbol 1199031199010 represents the initial position ofthe soil when the soil becomes plastic state

Under undrained condition relationship between cur-rent radius 119903119909 initial radius 1199031199090 of a material particle andcurrent radius 119886 and initial radius 1198860 of the cylindrical cavityis as follows 1199032119909 minus 11990321199090 = 1198862 minus 11988620 (1)

At any stage of the shield excavation any soil elementwithin the surrounding soil mass satisfies the followingequilibrium equation in both elastic and plastic regions(effective stress form)

1205971205901015840119903120597119903 + 120597119906120597119903 + 1205901015840119903 minus 1205901015840120579119903 = 0 (2)

or alternatively in the total stress form

120597120590119903120597119903 + 120590119903 minus 120590120579119903 = 0 (3)

where 1205901015840119903 and 1205901015840120579 are effective radial and tangential stressesrespectively 120590119903 and 120590120579 are total radial and tangential stressesrespectively 119906 is pore water pressure

23 Constitutive Relations The 1198700 consolidated clay isassumed to be linearly elastic and infinitesimal deformationuntil the onset of yield Thus according to the Hookrsquos Lawthe elastic stress-strain relationship can be expressed in theincrement form as follows

119889120576119890119894119895 = 1 + ]119864 119889120590119894119895 minus ]119864119889120590119898119898120575119894119895 (4)


CSL

Op

q

1

1

M=12K0-line

radicM2 + 20

0

K0=05

(a)

pO

q

K0=05

K0=10

(b)

Figure 3 Yield curves of anisotropic elastoplastic model

where119864 and119866 are the elasticmodulus and the shearmodulusrespectively which are defined in the 1198700-MCC model asfollows [6] 119864 = 2119866 (1 + ])

119866 = 3 (1 minus 2]) 12059211990110158402 (1 + ]) 120581 (5)

where 120581 is the slope of swelling line 120592 is the specific volumeand ] is the Poissonrsquos ratio

After yielding elastoplastic behavior of soil can bedescribed by large deformation theory and 1198700-MCC modelwhere a relative stress ratio 120578 lowast (= 120578 minus 1205780) is adopted insteadof the stress ratio 120578(= 1199021199011015840) as used in the MCC model toconsider the effect of initially stress-induced anisotropy onmechanical behavior of 1198700 consolidated clay Yield function(119891) for associative plasticity also serves as a plastic potentialfunction (119892) of the 1198700-MCC model can be expressed asfollows [6 11 12]119891 = (119902 minus 12057801199011015840)2 + 11987221199011015840 (1199011015840 minus 1199011015840119888) = 0 (6)

where 1205780 = |3(1 minus 1198700)(21198700 + 1)| is stress ratio under 1198700consolidated condition 119872 = 6 sin1205931015840(3 minus sin1205931015840) is slope ofcritical state line 1205931015840 is effective friction angle 1199011015840119888 is effectiveyield pressure and stress parameters 1199011015840 and 119902 are definedrespectively as follows

1199011015840 = 131205901015840119894119894 (7)

119902 = radic32 (1205901015840119894119895 minus 1199011015840120575119894119895) (1205901015840119894119895 minus 1199011015840120575119894119895) (8)

where 120575119894119895 is Kroneckerrsquos deltaAfter taking strain hardening rule into consideration

yield function or plastic potential function can be given asfollows

119891 = 120582 minus 1205811 + 1198900 ln 119901101584011990110158400 + 120582 minus 1205811 + 1198900 ln(1 + (119902 minus 12057801199011015840)2119872211990110158402 ) minus 120576119901V= 0

(9)

where 120582 and 120581 are slopes of consolidation and swelling linesrespectively 1198900 is initial void ratio of soil for 1199011015840 = 11990110158400 120576119901Vis plastic volumetric strain which is used as a hardeningparameter

As Figure 3 shows plastic potential surface of 1198700-MCCmodel in1199011015840minus119902 space is a rotated ellipse and degree of rotationportrays the extent of anisotropy When shearing starts fromisotropic stress 120578lowast = 120578 = 1199021199011015840 because 1205780 = 0 the 1198700-MCCmodel becomes the MCC model The yield surface rotationcan also be found in more detail in Yin et al (2011 2015) [1617]

For elastoplastic constitutivemodel the total strain incre-ments 119889120576119894119895 are decomposed into elastic and plastic strainincrements ie

119889120576119894119895 = 119889120576119890119894119895 + 119889120576119901119894119895 (10)

where elastic strain increments 119889120576119901119894119895 can be still calculatedusing (3) and plastic strain increments 119889120576119901119894119895 in the 1198700-MCCmodel can be derived by taking account the associated plasticflow rule as follows

119889120576119901119894119895 = Ω 1205971198911205971205901015840119894119895 (11)

where the scalar multiplier Ω is given as follows

Ω = minus(1205971198911205971199011015840) 1198891199011015840 + (120597119891120597119902) 119889119902(120597119891120597120576119901V ) (1205971198911205971205901015840119894119894) (12)

where

1205971198911205971199011015840 = 120582 minus 1205811 + 1198900 11199011015840119872211990110158402 minus (1199022 minus 1205782011990110158402)119872211990110158402 + (119902 minus 12057801199011015840)2 (13)


120597119891120597119902 = 120582 minus 1205811 + 1198900 2 (119902 minus 12057801199011015840)119872211990110158402 + (119902 minus 12057801199011015840)2 (14)

12059711990110158401205971205901015840119894119895 = 1205751198941198953 (15)

1205971199021205971205901015840119894119895 = 3 (1205901015840119894119895 minus 1199011015840120575119894119895)2119902 (16)

120597119891120597120576119901V = minus1 (17)

1205971198911205971205901015840119894119895 = 1205971198911205971199011015840 12059711990110158401205971205901015840119894119895 + 120597119891120597119902 1205971199021205971205901015840119894119895 (18)

Substituting (13)ndash(16) into (18) yields the following equa-tion

1205971198911205971205901015840119894119895 = 119888119901 [[119872211990110158402 minus (1199022 minus 1205782011990110158402)119872211990110158402 + (119902 minus 12057801199011015840)2

12057511989411989531199011015840+ 3 (1205901015840119894119895 minus 1199011015840120575119894119895)119872211990110158402 + (119902 minus 12057801199011015840)2 119902 minus 12057801199011015840119902 ]]

(19)

where 119888119901 = (120582 minus 120581)(1 + 1198900)Substituting (18) and (13)ndash(18) into (11) the scalar multi-

plier Ω can be obtained as follows

Ω = 1198891199011015840 + 2 (120578 minus 1205780)1198722 minus (1205782 minus 12057820)119889119902 (20)

Then detailed relationship between the plastic strainincrement 119889120576119901119894119895 and the stress increment 119889120590119894119895 can be expressedas follows

119889120576119901119894119895 = Ω11988811990111990110158402 [1198722 + (120578 minus 1205780)2] [[1198722 minus (1205782 minus 12057820)] 11990110158401205751198941198953+ 3 (120590119894 minus 1199011015840120575119894119895) (1 minus 1205780120578)]

(21)

3 Exact Numerical 3D Solution

31 Elastic Zone Based on the small-strain theory and(2) and (3) stress and displacement in the spherical

coordinates of the elastic region can be obtained as follows[9 14 18]

120590119903 = 120590ℎ + (120590119901 minus 120590ℎ) (119903119901119903 )2 (22)

120590120579 = 120590ℎ minus (120590119901 minus 120590ℎ) (119903119901119903 )2 (23)

120590119911 = 120590V (24)

119880119903 = 120590119901 minus 120590ℎ21198660 1199032119901119903 (25)

Δ119906 = 0 (26)

where 120590119901 is the total radial stress in the phase plane atEP boundary It is worth noticing that there is no excesspore water pressure in plastic region because effective meanstress is constant under undrained condition (119889120592 = 3(1 minus2V)1198891199011015840119864 = 0) and total mean stress 119901 also keeps constantin elastic zone

32 Elastoplastic Zone For the problem of shield excavationsoil position around the tunnel cavity can be described as119903 120579 and 119911 in cylindrical polar coordinate system Corre-spondingly there are only variables of principal stress andprincipal strain without the deviator stress and deviatorstrain induced during shield excavation process Thereforeelastoplastic constitutive relation of the 1198700-MCC model canbe simplified to a much simple matrix equation

119889120576119903119889120576120579119889120576119911

= [[[[[[[

1119864 + 1198871198862119903 minus V119864 + 119887119886119903119886120579 minus V119864 + 119887119886119903119886119911minus V119864 + 119887119886119903119886120579 1119864 + 1198871198862120579 minus V119864 + 119887119886120579119886119911minus V119864 + 119887119886119911119886119903 minus V119864 + 119887119886119911119886120579 1119864 + 1198871198862119911]]]]]]]

119889120590101584011990311988912059010158401205791198891205901015840119911

(27)

where

119886119903 = 1199011015840 [1198722 minus (1205782 minus 12057820)]3 + 3 (1205901015840119903 minus 1199011015840) (1 minus 1205780120578 )119886120579 = 1199011015840 [1198722 minus (1205782 minus 12057820)]3 + 3 (1205901015840120579 minus 1199011015840) (1 minus 1205780120578 )119886119911 = 1199011015840 [1198722 minus (1205782 minus 12057820)]3 + 3 (1205901015840119911 minus 1199011015840) (1 minus 1205780120578 )119887 = 11988811990111990110158403 [1198722 + (120578 minus 1205780)2] [1198722 minus (1205782 minus 12057820)]

(28)


Taking the inverse calculation to (27) gives

119889120590101584011990311988912059010158401205791198891205901015840119911

= 1120585 [[[

11988811 11988812 1198881311988821 11988822 1198882311988831 11988832 11988833]]]

119889120576119903119889120576120579119889120576119911

(29)

where

11988811 = 11198642 (1 minus V2 + 1198641198862120579119887 + 2119864V119886120579119886119911119887 + 1198641198862119911119887)11988812 = 11198642 [minus119864119886119903 (119886120579 + V119886119911) 119887

+ V (1 + V minus 119864119886120579119886119911119887 + 1198641198862119911119887)]11988813 = 11198642 [minus119864119886119903 (V119886120579 + 119886119911) 119887

+ V (1 + V minus 119864119886120579119886119911119887 + 1198641198862120579119887)]11988822 = 11198642 (1 minus V2 + 1198641198862119903119887 + 2119864V119886119903119886119911119887 + 1198641198862119911119887)11988823 = 11198642 [V + V2 + 119864V1198862119903119887 minus 119864119886120579119886119911119887

minus 119864V119886119903 (119886120579 + 119886119911) 119887]11988833 = 11198642 (1 minus V2 + 1198641198862119903119887 + 2119864V119886119903119886120579119887 + 1198641198862120579119887)11988821 = 1198881211988823 = 11988832

(30)

120585 = minus1 + V1198643 [(minus1 + V + 2V2)+ 119864119887 (minus1 + V) (1198862119903 + 1198862120579 + 1198862119911)minus 2119864V119887 (119886120579119886119911 + 119886120579119886119903 + 119886120579119886119911)]

(31)

For undrained cylindrical cavity expansion 119889120576V = 119889120576119911thus according to the large deformation theory radial andtangential strains can be given in incremental form as follows

119889120576119903 = minus119889120576120579 = 119889119903119903 (32)

Substituting (32) into (29) constitutive matrix can bereduced to a set of first-order ordinary differential equationsas follows

1198891205901015840119903119889119903 minus 11988711 minus 11988712120585119903 = 0 (33)

1198891205901015840120579119889119903 minus 11988712 minus 11988722120585119903 = 0 (34)

1198891205901015840119911119889119903 minus 11988731 minus 11988732120585119903 = 0 (35)

which can be solved by Lagrangian analysis method as aninitial value problem with 119903 starting at 119903119909119901 provided thatinitial values of 1205901015840119903(119903119909119901) 1205901015840120579(119903119909119901) and 1205901015840119911(119903119909119901) are given Here119903119909119901 denotes the position of a specific particle which comesinto plastic state instantly All of the initial values required forsolving the differential equations can be derived from the EPboundary conditions which are further analyzed as follows

33 Elastic-Plastic Boundary Conditions As shown in (22)(23) (24) and (26) the change of mean effective stress Δ1199011015840is equal to zero in elastic region which is still valid at the EPboundary ie

1199011015840119901 = 11990110158400 (36)

where 1199011015840119901 denotes mean effective stress at the EP boundarySubstituting (36) into (6) the deviator stress at the EP

boundary 119902119901 can be obtained as follows

119902119901 = 120578011990110158400 + 11987211990110158400radicOCR minus 1 (37)

where OCR is overconsolidation ratio defined as 1199011015840119888011990110158400 andpressure 11990110158401198880 is the maximummean preconsolidation stress

According to (7) the deviator stress at the EP boundarycan also be written as

119902119901= 1radic2radic(1205901015840119903119901 minus 1205901015840

120579119901)2 + (1205901015840119911119901 minus 1205901015840

120579119901)2 + (1205901015840119903119901 minus 1205901015840119911119901)2 (38)

where 1205901015840119903119901 1205901015840120579119901 and 1205901015840119911119901 are effective radial tangential andvertical stresses respectively at the EP boundary

From (22)ndash(24) and (36) one obtains

1205901015840119903119901 + 1205901015840120579119901 = 12059010158401199030 + 12059010158401205790 = 21205901015840ℎ (39)

1205901015840119911119901 = 12059010158401199110 = 1205901015840V0 (40)

where 12059010158401199030 12059010158401205790 and 12059010158401199110 are initial effective radial tangentialand vertical stresses respectively

For the1198700 consolidated clay relationship between in situhorizontal effective stress 12059010158401199030 (or 12059010158401205790) and in situ verticalstress 12059010158401199110 can be given as follows

12059010158401199030 = 12059010158401205790 = 119870012059010158401199110 (41)

where symbol 1198700 denotes the anisotropic degree of claySubstituting (39) (40) and (41) into (38) effective radial

and tangential stresses at the EP boundary can be derived as

1205901015840119903119901 = 12059010158401199030 + 1radic3radic1199022119901 minus (1 minus 1198700)2 1205901015840211991101205901015840120579119901 = 12059010158401199030 minus 1radic3radic1199022119901 minus (1 minus 1198700)2 120590101584021199110 (42)


The position 119903119909119901 where the specific particle becomesplasticity can be derived by taking the displacement at EPboundary 119880119903119901 and the undrained condition into consider-ation According to (25) one can obtain

119880119903119901 = 119903119909119901 minus 1199031199090 = 1205901015840119903119901 minus 1205901015840119903021198660 119903119909119901 (43)

Substituting (1) into (43) gives [14]

119903119909119901 = 2119866011988621198660 minus (1205901015840119903119901 minus 12059010158401199030)radic(119903119909119886 )2 + (1198860119886 )2 minus 1 (44)

Furthermore current location of the EP boundary 119903119901which is required for full analysis of stress distributionsaround the cavity can be obtained by equating both 119903119909119901 and119903119909 to 119903119901 [18]

119903119901119886 = radic (1198860119886)2 minus 1((1205901015840119903119901 minus 12059010158401199030) 21198660 minus 1)2 minus 1 (45)

So far (40) (42) and (44) provide initial conditions forsolving the first-order ordinary differential equations (33)(34) and (35)

After solving these first-order ordinary differential equa-tions by numerical method excess pore water pressure at anarbitrary location in the plastic zone Δ119906119903119909 can be computedby integrating (2) from the EP boundary 119903119901 to the point 119903119909

Δ119906119903119909 = 1205901015840119903119901 minus 1205901015840119903119909 + int119903119901119903119909

1205901015840119903 minus 1205901015840120579119903 119889119903 (46)

4 Closed Form 3D Analytical Solution

In the front section (refer to Figure 3) the semianalyticalsolution of expansion of the cylindrical hole has been pre-sented with the limitation of high computation cost In orderto reduce the computational cost and achieve the purposeof practical application this paper deduces a closed 3D totalstress solution for cylindrical cavity undrained expansion in1198700 consolidated clay

Under the undrained condition the total volume of thesoil is equal to zero That is

12058111988911990110158401205921199011015840 + (120582 minus 120581) 11988911990110158401198881205921199011015840119888 = 0 (47)

Taking EP boundary conditions into consideration in(47) the following equation can be found

1199011015840119888 = 1199011015840OCR(119901101584011990110158400)minus120581(120582minus120581) (48)

According to (37) and (48) the relationship betweenaverage effective stress and the plastic zone can be presentedas follows

119902 = 12057801199011015840 + 1198721199011015840 [OCR(119901101584011990110158400)minus1Λ minus 1]12 (49)

where Λ is the plastic volumetric strain ratio defined as (1 minus120581120582)For the 1198700-MCC model critical state condition can be

expressed as follows

120578 = radic1198722 + 12057820 (50)

Substituting (50) into (49) ultimate effective mean stress1199011015840119891 and ultimate deviator stress 119902119891 can be obtained asfollows

1199011015840119891 = 11990110158400 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(51)

119902119891 = 11990110158400radic1198722 + 12057820 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(52)

On the other hand the closed form 3D solutions in elasticregionhave been already derived as shown in (22)ndash(26)Onlythe solutions in plastic region are derived in this study Totalradial stresses at an arbitrary location in plastic region 120590119903119909can be obtained by integrating (4) from the EP boundary 119903119901to the point 119903119909

120590119903119909 = 120590119903119901 minus 2radic3 int119903119909119903119901

119902119903 119889119903 (53)

where 120590119903119901 is the total radial stress at the EP boundary Thedeviator stress 119902 is related to the location 119903119909 so that (53)cannot be integrated directlyTherefore in order to obtain theclosed form 3D solution the following two assumptions haveto be made at first

(a) Relationship between Deviator Stress and Radial Distancein the Plastic Region The deviator stress 119902 in the plastic zoneis assumed to be equal to the ultimate deviator stress 119902119891which is similar to 119902(b) Relationship between and among Total Radial Tangentialand Vertical Stresses in the Plastic Region The vertical totalstress 120590119911 is assumed to be the mean of radial stress 120590119903 andtangential stress 120590120579 after soil yields

Although these assumptions seem to be bold and strongthe approximate analytical results match the exact numericalresults fairlywell in the plastic regionwhichwill be comparedand discussed in detail later in the paper

According to assumption (a) substituting (25) into (45)and ignoring higher order terms of radic1199022

119891minus (1 minus 1198700)21205902V0119866


position of the EP boundary which is related to the currentcavity radius can be written as

(119903119901119886 )2 = radic3119866radic1199022119891minus (1 minus 1198700)2 1205902V0 [1 minus (1198860119886 )2] (54)

Then taking assumption (a) into consideration radialstress in the plastic zone can be derived from (53) and canbe written as follows

120590119903119909 = 120590ℎ0 + radic1199022119891minus (1 minus 1198700)2 12059010158402V0radic3 [[[1

+ lnradic3119866radic1199022

119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)

According to the previous two assumptions substituting(55) into (8) and taking (52) into consideration total tangen-tial and vertical stresses in the plastic region can be obtained

120590120579 = 120590ℎ0+ radic1199022119891minus (1 minus 1198700)2 12059010158402V0radic3 [[[ln

radic3119866radic1199022119891minus (1 minus 1198700)2 12059010158402V0

sdot (1198862 minus 119886201199032119909 ) minus 1]]]120590119911 = 120590ℎ0 + radic1199022

119891minus (1 minus 1198700)2 12059010158402V0radic3

sdot ln radic3119866radic1199022119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )

(56)

Furthermore excess pore water pressure in plastic regioncan be derived by effective stress principle

Δ119906119903119909 = (1198700 minus 1) 119901101584001 + 21198700 + radic1199022119891minus (1 minus 1198700)2 12059010158402V0radic3


1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)

Equating 119903119909 to 119886 pressure-expansion relationship andexcess pore water pressure at cavity wall can be derived from(55) and (57) respectively as follows



sdot (1 minus 119886201198862) + 1]]]

(58)


sdot ln radic3119866radic1199022119891minus (1 minus 1198700)2 12059010158402V0 (1 minus 119886201198862) + 11990110158400 [[[[

1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)

From (58) as the cavity continues to expand to infinity(1198861198860 rarr infin) the limit cavity wall pressure 120590119903119906119897119905 and excesspore water pressure Δ119906119886119906119897119905 for undrained cavity expansionin 1198700 consolidated anisotropic clay can be obtained

120590119903119906119897119905 = 120590ℎ0+ radic1199022119891minus (1 minus 1198700)2 12059010158402V0radic3 (ln


+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)


Table 1 Physical and mechanical parameters of soil [14]

119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100

It is interesting to note here that takingradic1199022119891minus (1 minus 1198700)212059010158402V0 as 119902119891 the present solutions are similar

to previous MCC model based solutions obtained by Caoet al [9] Especially for isotropic clay (1198700 = 1) the closedform 3D solutions presented in this paper reduce to thesolutions obtained by Cao et al [9] which demonstratesthat the solution obtained by Cao et al [9] is just a specialcase of the presented solution Actually for nature clay thehorizontal in situ stress is different from the vertical oneand the soil is initial anisotropy (ie 1198700 = 1) Note thatthe term of radic1199022

119891minus (1 minus 1198700)212059010158402V0 in the presented solution

reflects the effect of initially stress-induced anisotropy onshield excavation Therefore presented solutions can yieldmore reasonable results for cylindrical cavity expansion inanisotropy clay In addition the term of (1198700 minus 1)11990110158400(1 + 21198700)in (57) (59) and (61) implies that initial anisotropy also hasnotable effects on excess pore water pressure during shieldexcavation process

5 Calculation Result Analysis and Discussion

In order to verify the accuracy and suitability of this methodconsidering the influencing factors of initial anisotropy andinduced anisotropy calculation results of presentmethod andChen and Abousleimanrsquos solution [18] using MCC constitu-tive model results were respectively validated in this paper

Physical and mechanical parameters of soil used inthis paper are taken from Chen and Abousleiman [18]Table 1 shows the properties of three typical 1198700 consolidatedclays

51 Pressure-Expansion Relationship As shown in Figure 4comparison of pressure-expansion relationship around shieldtunnel is calculated by the proposed 3D closed form solu-tion and Chenrsquos solution All calculated stresses have beennormalized Effective radial stress and vertical stress havebeen overestimated while the tangential stress is noted to beunderestimated in the plastic zone based on Chenrsquos solutionapart from 1198700 = 1 Therefore the anisotropy inducedby initial stress has a significant influence on the stressdistribution in the plastic zone If the tunnel is in the isotropicstress conditions that is to say 1198700 = 1 1198700-MCC constitutivemodel will degenerate into a MCC constitutive model inthis case the results of the two calculations are in completeagreement

It is also interesting to note in Figure 4 that vertical stressis always equal to the average of radial and tangential stresses

in critical state zone for all cases which can be proved exactlyas follows Combining (33)ndash(35) gives

119889 (1205901015840119903 + 1205901015840120579 minus 21205901015840119911)119889119903= minus3 (1 minus 2V) (119886119903 minus 119886120579) (1205901015840119903 + 1205901015840120579 minus 21205901015840119911) 119887119864120585119903 (1 minus 1205780120578 ) (62)

Since the stress components keep constant in the criticalregion therefore the left side of (62) is equal to zero thereby(62) can be reformulated as

1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)

Equation (63) indicates that the second assumption forapproximate closed form 3D solution holds in the criticalregion

It can also be seen from Figure 4(d) that excess porewater pressure is underestimated near the cavity wall butoverestimated as the radial distance increases Besides theexcess pore water pressure keeps constant in critical regionand thus the second assumption for the approximate closedform 3D solution still holds well in the critical region Whenthe shield is compacted to a certain degree excess porewater pressure in the soil squeezing effect zone is graduallystabilized and the same level of pressure can be achievedregardless of the value of 119870052 Shield Excavation Process Analysis According to cylin-drical cavity expansion theory effect of soil compactioncaused by shield tunneling is mainly revealed in dynamicchange of the normalized excess pore water pressure Δ11990611988611990110158400the soil pressure inside the cavity 12059011988611990110158400 and the cavity radius119903119901119886 Figure 5 shows variations of normalized cavity pressuresoil pressure and excess pore water pressure with cavityradius during shield excavation process Results computedby the three different typical values of 1198700 (0625 1 and2) [14] as well as the results from the MCC model basedsolutions are presented in the same figures for comparisonSimilar to previous analysis all calculation results have beennormalized

It can be seen from Figure 5 that excess pore waterpressure and internal cavity pressure are affected by the initialstress-induced anisotropy significantly when 1198861198860 gt 2 Inthe process of gradual expansion of inner wall of the tunnelthe result of MCC constitutive model is smaller than that ofthe 1198700-MCC constitutive model Stress values at the innerwall of the tunnel and excess pore water pressure around thetunnel increase dramatically when 1198861198860 lt 2 Both parametersgradually achieve a stable level as the shield continues toadvance Compared to Figures 5(a) 5(b) and 5(c) it is clearthat as the1198700 value increases internal cavity pressure causedby shield tunneling is effectively suppressed while the impactof excess pore water pressure and plastic radius is not veryobviousThis is because at the initial stage of pore expansionnegative pore pressure first occurs at the hole wall Thenwith the increase of pore size the excess pore pressure will


1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0

(a) 1205901015840119911ndash expansion relationship

1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0

(b) 1205901015840119903ndash expansion relationship

1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0

(c) 1205901015840120579ndash expansion relationship

1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0

(d) Δ119906ndash expansion relationship

Figure 4 Comparison of pressure-expansion relationships around shield tunnel calculated by 3D closed form solution and Chenrsquos solution

turn to a positive value from negative to a critical stateFurthermore when the surrounding soil is in isotropic initialstress state condition (1198700 = 1) the 1198700-MCC constitutivemodel is reduced to the MCC constitutive model and theresults are consistent with numerical results However effectof anisotropy degree on discrepancy of excess pore waterpressure calculated from the two solutions is not obviouswhich implies that the OCR also has a pronounced effecton excess pore water pressure at cavity wall during cavityexpansion process

Evolution of normalized plastic radius 119903119901119886 with nor-malized instant cavity radius 1198861198860 for three different valuesof 1198700 and OCR together with the MCC model based resultsis also plotted in Figure 5 for comparison It is seen thatfor normally consolidated clay with 1198700 = 0625 plasticradius during whole cavity expansion process is significantlyaffected by initially stress-induced anisotropy However forheavily overconsolidated clay discrepancy between presented

solution and MCC model based solution is not apparentbecause OCR also plays a major role in plastic radius forheavily consolidated clay during cavity expansion process

53 Effective Stress Path in 1199011015840-119902 Plane Figure 6 showseffective stress path (ESP) in 1199011015840-119902 plane for a soil particlelocated at cavity wall in three typical 1198700 consolidated clayswhere two stress parameters 1199011015840 and 119902 have been normalizedby 11990110158400 Note that points O P and F in Figure 6 denotethe in situ stress point yield stress point and failure stresspoint respectively It is also clear that initial yield loci arerotational ellipses for anisotropic clays (1198700 = 1) and degreeof rotation reflects degree of anisotropic For isotropic clay(1198700 = 1) 1198700 line overlaps with 1199011015840 axis and yield locus issymmetric to 1199011015840 axis and thus 1198700-MCC model reduces toMCC model For anisotropic clays yield locus and CSL ofthe 1198700-MCC model are higher than that of MCC modelTherefore yield and failure stresses during cavity expansion


rp

ultimate stateslowlychange state

rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp

ultimate state slowlychange state

rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2

Figure 5 Evolution of cavity pressure and excess pore water pressure with cavity radius during shield excavation process

process are underestimated by previous MCC model basedsolutions

For normally (OCR = 1) consolidated clay with 1198700 =0625 in Figure 6 yielding occurs immediately after cavityexpansion so that in situ stress point O overlaps the yieldstress point P The corresponding ESP starts form point Owhich is located in 1198700 line and moves up to the left untilESP reaches critical state line (CSL) at point F For lightly(OCR = 3) and heavily overconsolidated clay (OCR = 10)ESP firstly moves vertically and soil keeps elastic until theESP approaches initial yield locus at point P Then soil yieldsand the ESP turns up to the right until reaching CSL atpoint F

54 Stress Distribution of Shield Tunneling under Different1198700 Conditions Figure 7 shows stress distribution of shieldtunneling under different 1198700 conditions calculated by theapproximate closed form 3D solution where normalized

vertical total stress 12059011991111990110158400 mean of radial stress 12059011990311990110158400 tan-gential stress 12059012057911990110158400 and excess pore water pressure Δ11990611988611990110158400are plotted against normalized radial distance 1199031198860 It is seenwhen 119903119886 lt 2 the rise of 1198700 leads to a significant change ofstress distribution which will reach a relatively stable stateafter EP boundary presents Besides the value of 1198700 hasno effect on the final stress except the vertical total stress12059011991111990110158400 Stress values under different 1198700 values finally reachthe same magnitude near the cavity wall It also indirectlyverifies that the two basic assumptions of the closed form 3Dsolution is feasible and reasonableTherefore the closed form3D solution is exact enough to interpret and solve generalgeotechnical problems

6 Conclusions

This paper aims to study the soil-compacting effect producedby shield tunneling by a precise semianalytical solution


Initial yield surface

CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2

Figure 6 ESP in 1199011015840-119902 plane for a soil particle located at cavity wall during shield excavation process

Particular emphasis is given to develop an approximate closed3D solution of cylindrical cavity undrained expansion insaturated anisotropic clay based on the 1198700-based modifiedCam-clay model Furthermore efforts have been made toexamine (a) the effects of the initial anisotropic and initiallystress-induced anisotropy on shield excavation and (b) stressdistribution of shield tunneling under different1198700 conditionscalculated by the approximate closed form 3D solution Someof key observation and findings from the study are as follows

(a) Both numerical solution and closed from 3D solutionpresented in this study improve the understanding of con-ventional solutions by considering anisotropic properties ofnatural clay and reducing MCC model based solutions when1198700 = 1 Therefore presented solutions can yield a morerealistic stress field induced by cylindrical cavity expansionin natural clay

(b) Vertical total stress 120590119911 radial stress 120590119903 and tangentialstress 120590120579 become intermediate principal stress and majorand minor principal stress respectively when soil came intocritical state and vertical total stress 120590119911 is equal to themean ofradial stress 120590119903 and tangential stress 120590120579 in critical state regionaround the cavity for all cases

(c) Initially stress-induced anisotropy has significantinfluence on stress field and plastic radius around the cavityespecially in normally consolidated clay Excess pore waterpressure is not only affected by the initially stress-inducedanisotropy but also by the initial anisotropy of clayThe stressstate of the soil around the borehole wall changes greatly dueto the expansion of the column hole in the 1198700 consolidatedclay and the stress around the hole decays logarithmicallywith the increase of the radial distance and tends to the stressstate at the time of departure

(d) The initial stress anisotropy and stress historyhave obvious influence on the expansion process of cylin-drical cavity especially for the ultimate expansion pres-sure of cylindrical cavity expansion process Stress fieldcaused by shield excavation is overestimated whereas plasticradius during cavity expansion process is underestimated byMCC constitutive model based on solutions in anisotropicclays

Conflicts of Interest

The authors declare that they have no conflicts of interest


1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=

EP boundary (K0=2)EP boundary (K0=1)EP boundary (K0=0625)

M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906

Figure 7 Stress distribution of shield tunneling under different 1198700 conditions calculated by the approximate closed form 3D solution

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 41172238) China Railway 20 BureauGroup Co Ltd has also given a lot of practice guidance inthe study of the paper

References

[1] D Durban ldquoLarge strain solution for pressurized elastoplastictubesrdquo Journal of Applied Mechanics vol 46 no 1 pp 228ndash2301979

[2] A G F De Sousa Coutinho ldquoRadial expansion of cylindricalcavities in sandy soils application to pressuremeter testsrdquoCanadian Geotechnical Journal vol 27 no 6 pp 737ndash7481990

[3] B Ladanyi ldquoIn-situ determination of undrained stress-strain behavior of sensitive clays with the pressuremeterrdquo

Canadian Geotechnical Journal vol 9 no 3 pp 313ndash3191972

[4] A C Palmer ldquoUndrained plane-strain expansion of a cylindri-cal cavity in clay A simple interpretation of the pressuremetertestrdquo Geotechnique vol 22 no 3 pp 451ndash457 1972

[5] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[6] M F Chang C I Teh and L F Cao ldquoUndrained cavity expan-sion in modified Cam clay II Application to the interpretationof the piezocone testrdquo Geotechnique vol 51 no 4 pp 335ndash3502001

[7] R HillTheMathematical Theoy of Plasticity Oxford UniversityPress London UK 1950

[8] A C Vesic ldquoExpansion of cavities in infinite soil massrdquo Journalof the Soil Mechanics and Foundation Division ASCE vol 98pp 265-90 1972


[9] L F Cao C I Teh and M F Chang ldquoUndrained cavity expan-sion inmodified Cam clay I theoretical analysisrdquoGeotechniquevol 51 no 4 pp 323ndash334 2001

[10] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[11] S J Wheeler A Naatanen M Karstunen and M LojanderldquoAn anisotropic elastoplastic model for soft claysrdquo CanadianGeotechnical Journal vol 40 no 2 pp 403ndash418 2003

[12] M Nakano K Nakai T Noda and A Asaoka ldquoSimulation ofshear and one-dimensional compression behavior of naturallydeposited clays by supersubloading Yield Surface Cam-claymodelrdquo Soils and Foundations vol 45 no 1 pp 141ndash151 2005

[13] Z-Y Yin and P-Y Hicher ldquoIdentifying parameters controllingsoil delayed behaviour from laboratory and in situ pressureme-ter testingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 32 no 12 pp 1515ndash1535 2008

[14] S L Chen and Y N Abousleiman ldquoExact undrained elasto-plastic solution for cylindrical cavity expansion inmodified camclay soilrdquo Geotechnique vol 62 no 5 pp 447ndash456 2012

[15] A M Marshall ldquoTunnel-pile interaction analysis using cavityexpansion methodsrdquo Journal of Geotechnical and Geoenviron-mental Engineering vol 138 no 10 pp 1237ndash1246 2012

[16] Z-Y Yin M Karstunen C S Chang M Koskinen and MLojander ldquoModeling time-dependent behavior of soft sensitiveclayrdquo Journal of Geotechnical and Geoenvironmental Engineer-ing vol 137 no 11 pp 1103ndash1113 2011

[17] Z-Y Yin J-H Yin and H-W Huang ldquoRate-Dependent andLong-Term Yield Stress and Strength of Soft Wenzhou MarineClay Experiments and Modelingrdquo Marine Georesources ampGeotechnology vol 33 no 1 pp 79ndash91 2015

[18] S L Chen and Y N Abousleiman ldquoExact drained solutionfor cylindrical cavity expansion in modified cam clay soilrdquoGeotechnique vol 63 no 6 pp 510ndash517 2013

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shield

interspaces

soil movement

(a) Excavation stage

driving

interspaces

soil squeezing

(b) Driving stage

shield

initial stress

soil squeezing

(c) Effect of initial stress field

Figure 1 Schematic diagram of compacting soil by shield

obtained the self-similar solution for the circular tunnelin strain-softening rock masses [10] Wheeler et al (2003)and Nakano et al (2005) adjusted yield surface equation onthe basis of 1198700-based modified Cam-clay considering 1198700consolidation caused by induced anisotropy and its evolutionlaw in the loading process [11 12] Yin and Hicher (2008)identified parameters of modified Cam-clay model withviscosity of soil from the cavity expansion [13] The problemof cavity expansion and cavity contraction has attractedmuch attention in geotechnical problems with applicationto the bearing capacity of deep foundations interpretationof pressuremeter tests breakout resistance of anchors piledriving wellbore instability underground excavation andblasting fracturing by explosives

However based on the above-mentioned researches theconventional cylinder expansion theory assumes that initialstress is isotropic (static earth pressure coefficient 1198700 = 10)Due to the sedimentation history mineralogical characteris-tics and other several factors the initial stress of soil layeris usually anisotropic For example the vertical stress andhorizontal stress of the cylinder bore are not the same inthe horizontal tunneling construction of the undergroundpipe laying tunnel engineering As a result the traditionaltheory of cylindrical expansion generally used in practicalengineering fails in explaining some practical phenomenonAlso the traditional cylindrical expansion theory assumesthat the cylindrical boundary condition is controlled bydisplacement which presents around the cylinder in the formof symmetrical distribution as a function of distance Thetraditional control theory can be applied in geotechnicalproblems analysis such as cylindrical pile and static conepenetration test But in the practical engineering of shieldtunnel boundary condition of the cylinder expansion iscontrolled by stress and displacement of the cylinder whichwas unable to satisfy the condition of axial symmetryHence it may not be reasonable to explain such geotechnicalproblems considering traditional theory of cylindrical expan-sion

In this paper a three-dimensional elastoplastic solution ofcylindrical cavity expansion is obtained based on the theory

of critical state soilmechanics and engineering characteristicsof shield tunneling to calculate the undrained cylindricalcavity expansion considered anisotropic clay of 1198700 Thisresearch work not only contributes to the theoretical valuesin field of geotechnical engineering but also has an importantpractical engineering significance

2 Definition of Soil-Compacting Effect andMechanical Model

21 Definition of Soil-Compacting Effect Shield tunnelingis a typical three-dimensional problem Soil deformation isclosely related to the relative position of the shield machineIn actual shield construction earth pressure and supportpressure ahead of excavation face are not completely balanced[15] When support pressure is greater than passive earthpressure the soil is extruded and then soil-compactingeffect presents And the free surface may collapse if supportpressure is less than active earth pressure This paper mainlyaims to study the soil-compacting effect produced when thesupport pressure is larger than passive earth pressure Soil-compacting effect of shield tunneling is shown in Figure 1

(a) Along with the shield approach the soil which issqueezed by the excavation face will be pushed around theshield And then the soil is output by the screw conveyor toform a voidThe surrounding soil is excavated and unloadedthe stress is released to cause soil expansion and the front soilmoves to the void (Figure 1(a))

(b) As the shield machine moves forward the soil will besqueezed around the shield and excavation face In the planeperpendicular to the axis of the tunnel the soil is assumesubjected to uniform radial extrusion force which generatesan equal amount of outward radial movement of the tunnel(Figure 1(b))

(c) The extruded soil moves outwards of the shieldmachine and is also affected by the friction force arise fromthe driving process of shield machine Radial deformationalong the outer side of the tunnel finally formed under theinfluence of initial stress field (Figure 1(c))


a

o

r

y

x

Elastic zone

Plastic zone

rp

a0

a

rxrx0 rp0 ℎ K0=






1205971205901015840119903120597119903 + 120597119906120597119903 + 1205901015840119903 minus 1205901015840120579119903 = 0 (2)


120597120590119903120597119903 + 120590119903 minus 120590120579119903 = 0 (3)



119889120576119890119894119895 = 1 + ]119864 119889120590119894119895 minus ]119864119889120590119898119898120575119894119895 (4)


CSL

Op

q

1

1

M=12K0-line

radicM2 + 20

0

K0=05

(a)

pO

q

K0=05

K0=10

(b)



119866 = 3 (1 minus 2]) 12059211990110158402 (1 + ]) 120581 (5)




1199011015840 = 131205901015840119894119894 (7)





(9)




119889120576119894119895 = 119889120576119890119894119895 + 119889120576119901119894119895 (10)


119889120576119901119894119895 = Ω 1205971198911205971205901015840119894119895 (11)


Ω = minus(1205971198911205971199011015840) 1198891199011015840 + (120597119891120597119902) 119889119902(120597119891120597120576119901V ) (1205971198911205971205901015840119894119894) (12)

where




12059711990110158401205971205901015840119894119895 = 1205751198941198953 (15)

1205971199021205971205901015840119894119895 = 3 (1205901015840119894119895 minus 1199011015840120575119894119895)2119902 (16)

120597119891120597120576119901V = minus1 (17)

1205971198911205971205901015840119894119895 = 1205971198911205971199011015840 12059711990110158401205971205901015840119894119895 + 120597119891120597119902 1205971199021205971205901015840119894119895 (18)




(19)






(21)




120590119903 = 120590ℎ + (120590119901 minus 120590ℎ) (119903119901119903 )2 (22)

120590120579 = 120590ℎ minus (120590119901 minus 120590ℎ) (119903119901119903 )2 (23)

120590119911 = 120590V (24)

119880119903 = 120590119901 minus 120590ℎ21198660 1199032119901119903 (25)

Δ119906 = 0 (26)



119889120576119903119889120576120579119889120576119911

= [[[[[[[


119889120590101584011990311988912059010158401205791198891205901015840119911

(27)

where


(28)



119889120590101584011990311988912059010158401205791198891205901015840119911

= 1120585 [[[

11988811 11988812 1198881311988821 11988822 1198882311988831 11988832 11988833]]]

119889120576119903119889120576120579119889120576119911

(29)

where

11988811 = 11198642 (1 minus V2 + 1198641198862120579119887 + 2119864V119886120579119886119911119887 + 1198641198862119911119887)11988812 = 11198642 [minus119864119886119903 (119886120579 + V119886119911) 119887

+ V (1 + V minus 119864119886120579119886119911119887 + 1198641198862119911119887)]11988813 = 11198642 [minus119864119886119903 (V119886120579 + 119886119911) 119887


minus 119864V119886119903 (119886120579 + 119886119911) 119887]11988833 = 11198642 (1 minus V2 + 1198641198862119903119887 + 2119864V119886119903119886120579119887 + 1198641198862120579119887)11988821 = 1198881211988823 = 11988832

(30)


(31)


119889120576119903 = minus119889120576120579 = 119889119903119903 (32)


1198891205901015840119903119889119903 minus 11988711 minus 11988712120585119903 = 0 (33)

1198891205901015840120579119889119903 minus 11988712 minus 11988722120585119903 = 0 (34)

1198891205901015840119911119889119903 minus 11988731 minus 11988732120585119903 = 0 (35)



1199011015840119901 = 11990110158400 (36)



119902119901 = 120578011990110158400 + 11987211990110158400radicOCR minus 1 (37)




120579119901)2 + (1205901015840119911119901 minus 1205901015840

120579119901)2 + (1205901015840119903119901 minus 1205901015840119911119901)2 (38)



1205901015840119903119901 + 1205901015840120579119901 = 12059010158401199030 + 12059010158401205790 = 21205901015840ℎ (39)

1205901015840119911119901 = 12059010158401199110 = 1205901015840V0 (40)



12059010158401199030 = 12059010158401205790 = 119870012059010158401199110 (41)






119880119903119901 = 119903119909119901 minus 1199031199090 = 1205901015840119903119901 minus 1205901015840119903021198660 119903119909119901 (43)







Δ119906119903119909 = 1205901015840119903119901 minus 1205901015840119903119909 + int119903119901119903119909

1205901015840119903 minus 1205901015840120579119903 119889119903 (46)




12058111988911990110158401205921199011015840 + (120582 minus 120581) 11988911990110158401198881205921199011015840119888 = 0 (47)


1199011015840119888 = 1199011015840OCR(119901101584011990110158400)minus120581(120582minus120581) (48)


119902 = 12057801199011015840 + 1198721199011015840 [OCR(119901101584011990110158400)minus1Λ minus 1]12 (49)



120578 = radic1198722 + 12057820 (50)


1199011015840119891 = 11990110158400 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(51)

119902119891 = 11990110158400radic1198722 + 12057820 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(52)


120590119903119909 = 120590119903119901 minus 2radic3 int119903119909119903119901

119902119903 119889119903 (53)





119891minus (1 minus 1198700)21205902V0119866







119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)







1198862 minus 119886201199032119909 )

(56)




1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)




sdot (1 minus 119886201198862) + 1]]]

(58)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)




+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)



119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









a

o

r

y

x

Elastic zone

Plastic zone

rp

a0

a

rxrx0 rp0 ℎ K0=






1205971205901015840119903120597119903 + 120597119906120597119903 + 1205901015840119903 minus 1205901015840120579119903 = 0 (2)


120597120590119903120597119903 + 120590119903 minus 120590120579119903 = 0 (3)



119889120576119890119894119895 = 1 + ]119864 119889120590119894119895 minus ]119864119889120590119898119898120575119894119895 (4)


CSL

Op

q

1

1

M=12K0-line

radicM2 + 20

0

K0=05

(a)

pO

q

K0=05

K0=10

(b)



119866 = 3 (1 minus 2]) 12059211990110158402 (1 + ]) 120581 (5)




1199011015840 = 131205901015840119894119894 (7)





(9)




119889120576119894119895 = 119889120576119890119894119895 + 119889120576119901119894119895 (10)


119889120576119901119894119895 = Ω 1205971198911205971205901015840119894119895 (11)


Ω = minus(1205971198911205971199011015840) 1198891199011015840 + (120597119891120597119902) 119889119902(120597119891120597120576119901V ) (1205971198911205971205901015840119894119894) (12)

where




12059711990110158401205971205901015840119894119895 = 1205751198941198953 (15)

1205971199021205971205901015840119894119895 = 3 (1205901015840119894119895 minus 1199011015840120575119894119895)2119902 (16)

120597119891120597120576119901V = minus1 (17)

1205971198911205971205901015840119894119895 = 1205971198911205971199011015840 12059711990110158401205971205901015840119894119895 + 120597119891120597119902 1205971199021205971205901015840119894119895 (18)




(19)






(21)




120590119903 = 120590ℎ + (120590119901 minus 120590ℎ) (119903119901119903 )2 (22)

120590120579 = 120590ℎ minus (120590119901 minus 120590ℎ) (119903119901119903 )2 (23)

120590119911 = 120590V (24)

119880119903 = 120590119901 minus 120590ℎ21198660 1199032119901119903 (25)

Δ119906 = 0 (26)



119889120576119903119889120576120579119889120576119911

= [[[[[[[


119889120590101584011990311988912059010158401205791198891205901015840119911

(27)

where


(28)



119889120590101584011990311988912059010158401205791198891205901015840119911

= 1120585 [[[

11988811 11988812 1198881311988821 11988822 1198882311988831 11988832 11988833]]]

119889120576119903119889120576120579119889120576119911

(29)

where

11988811 = 11198642 (1 minus V2 + 1198641198862120579119887 + 2119864V119886120579119886119911119887 + 1198641198862119911119887)11988812 = 11198642 [minus119864119886119903 (119886120579 + V119886119911) 119887

+ V (1 + V minus 119864119886120579119886119911119887 + 1198641198862119911119887)]11988813 = 11198642 [minus119864119886119903 (V119886120579 + 119886119911) 119887


minus 119864V119886119903 (119886120579 + 119886119911) 119887]11988833 = 11198642 (1 minus V2 + 1198641198862119903119887 + 2119864V119886119903119886120579119887 + 1198641198862120579119887)11988821 = 1198881211988823 = 11988832

(30)


(31)


119889120576119903 = minus119889120576120579 = 119889119903119903 (32)


1198891205901015840119903119889119903 minus 11988711 minus 11988712120585119903 = 0 (33)

1198891205901015840120579119889119903 minus 11988712 minus 11988722120585119903 = 0 (34)

1198891205901015840119911119889119903 minus 11988731 minus 11988732120585119903 = 0 (35)



1199011015840119901 = 11990110158400 (36)



119902119901 = 120578011990110158400 + 11987211990110158400radicOCR minus 1 (37)




120579119901)2 + (1205901015840119911119901 minus 1205901015840

120579119901)2 + (1205901015840119903119901 minus 1205901015840119911119901)2 (38)



1205901015840119903119901 + 1205901015840120579119901 = 12059010158401199030 + 12059010158401205790 = 21205901015840ℎ (39)

1205901015840119911119901 = 12059010158401199110 = 1205901015840V0 (40)



12059010158401199030 = 12059010158401205790 = 119870012059010158401199110 (41)






119880119903119901 = 119903119909119901 minus 1199031199090 = 1205901015840119903119901 minus 1205901015840119903021198660 119903119909119901 (43)







Δ119906119903119909 = 1205901015840119903119901 minus 1205901015840119903119909 + int119903119901119903119909

1205901015840119903 minus 1205901015840120579119903 119889119903 (46)




12058111988911990110158401205921199011015840 + (120582 minus 120581) 11988911990110158401198881205921199011015840119888 = 0 (47)


1199011015840119888 = 1199011015840OCR(119901101584011990110158400)minus120581(120582minus120581) (48)


119902 = 12057801199011015840 + 1198721199011015840 [OCR(119901101584011990110158400)minus1Λ minus 1]12 (49)



120578 = radic1198722 + 12057820 (50)


1199011015840119891 = 11990110158400 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(51)

119902119891 = 11990110158400radic1198722 + 12057820 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(52)


120590119903119909 = 120590119903119901 minus 2radic3 int119903119909119903119901

119902119903 119889119903 (53)





119891minus (1 minus 1198700)21205902V0119866







119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)







1198862 minus 119886201199032119909 )

(56)




1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)




sdot (1 minus 119886201198862) + 1]]]

(58)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)




+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)



119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









CSL

Op

q

1

1

M=12K0-line

radicM2 + 20

0

K0=05

(a)

pO

q

K0=05

K0=10

(b)



119866 = 3 (1 minus 2]) 12059211990110158402 (1 + ]) 120581 (5)




1199011015840 = 131205901015840119894119894 (7)





(9)




119889120576119894119895 = 119889120576119890119894119895 + 119889120576119901119894119895 (10)


119889120576119901119894119895 = Ω 1205971198911205971205901015840119894119895 (11)


Ω = minus(1205971198911205971199011015840) 1198891199011015840 + (120597119891120597119902) 119889119902(120597119891120597120576119901V ) (1205971198911205971205901015840119894119894) (12)

where




12059711990110158401205971205901015840119894119895 = 1205751198941198953 (15)

1205971199021205971205901015840119894119895 = 3 (1205901015840119894119895 minus 1199011015840120575119894119895)2119902 (16)

120597119891120597120576119901V = minus1 (17)

1205971198911205971205901015840119894119895 = 1205971198911205971199011015840 12059711990110158401205971205901015840119894119895 + 120597119891120597119902 1205971199021205971205901015840119894119895 (18)




(19)






(21)




120590119903 = 120590ℎ + (120590119901 minus 120590ℎ) (119903119901119903 )2 (22)

120590120579 = 120590ℎ minus (120590119901 minus 120590ℎ) (119903119901119903 )2 (23)

120590119911 = 120590V (24)

119880119903 = 120590119901 minus 120590ℎ21198660 1199032119901119903 (25)

Δ119906 = 0 (26)



119889120576119903119889120576120579119889120576119911

= [[[[[[[


119889120590101584011990311988912059010158401205791198891205901015840119911

(27)

where


(28)



119889120590101584011990311988912059010158401205791198891205901015840119911

= 1120585 [[[

11988811 11988812 1198881311988821 11988822 1198882311988831 11988832 11988833]]]

119889120576119903119889120576120579119889120576119911

(29)

where

11988811 = 11198642 (1 minus V2 + 1198641198862120579119887 + 2119864V119886120579119886119911119887 + 1198641198862119911119887)11988812 = 11198642 [minus119864119886119903 (119886120579 + V119886119911) 119887

+ V (1 + V minus 119864119886120579119886119911119887 + 1198641198862119911119887)]11988813 = 11198642 [minus119864119886119903 (V119886120579 + 119886119911) 119887


minus 119864V119886119903 (119886120579 + 119886119911) 119887]11988833 = 11198642 (1 minus V2 + 1198641198862119903119887 + 2119864V119886119903119886120579119887 + 1198641198862120579119887)11988821 = 1198881211988823 = 11988832

(30)


(31)


119889120576119903 = minus119889120576120579 = 119889119903119903 (32)


1198891205901015840119903119889119903 minus 11988711 minus 11988712120585119903 = 0 (33)

1198891205901015840120579119889119903 minus 11988712 minus 11988722120585119903 = 0 (34)

1198891205901015840119911119889119903 minus 11988731 minus 11988732120585119903 = 0 (35)



1199011015840119901 = 11990110158400 (36)



119902119901 = 120578011990110158400 + 11987211990110158400radicOCR minus 1 (37)




120579119901)2 + (1205901015840119911119901 minus 1205901015840

120579119901)2 + (1205901015840119903119901 minus 1205901015840119911119901)2 (38)



1205901015840119903119901 + 1205901015840120579119901 = 12059010158401199030 + 12059010158401205790 = 21205901015840ℎ (39)

1205901015840119911119901 = 12059010158401199110 = 1205901015840V0 (40)



12059010158401199030 = 12059010158401205790 = 119870012059010158401199110 (41)






119880119903119901 = 119903119909119901 minus 1199031199090 = 1205901015840119903119901 minus 1205901015840119903021198660 119903119909119901 (43)







Δ119906119903119909 = 1205901015840119903119901 minus 1205901015840119903119909 + int119903119901119903119909

1205901015840119903 minus 1205901015840120579119903 119889119903 (46)




12058111988911990110158401205921199011015840 + (120582 minus 120581) 11988911990110158401198881205921199011015840119888 = 0 (47)


1199011015840119888 = 1199011015840OCR(119901101584011990110158400)minus120581(120582minus120581) (48)


119902 = 12057801199011015840 + 1198721199011015840 [OCR(119901101584011990110158400)minus1Λ minus 1]12 (49)



120578 = radic1198722 + 12057820 (50)


1199011015840119891 = 11990110158400 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(51)

119902119891 = 11990110158400radic1198722 + 12057820 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(52)


120590119903119909 = 120590119903119901 minus 2radic3 int119903119909119903119901

119902119903 119889119903 (53)





119891minus (1 minus 1198700)21205902V0119866







119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)







1198862 minus 119886201199032119909 )

(56)




1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)




sdot (1 minus 119886201198862) + 1]]]

(58)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)




+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)



119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018










12059711990110158401205971205901015840119894119895 = 1205751198941198953 (15)

1205971199021205971205901015840119894119895 = 3 (1205901015840119894119895 minus 1199011015840120575119894119895)2119902 (16)

120597119891120597120576119901V = minus1 (17)

1205971198911205971205901015840119894119895 = 1205971198911205971199011015840 12059711990110158401205971205901015840119894119895 + 120597119891120597119902 1205971199021205971205901015840119894119895 (18)




(19)






(21)




120590119903 = 120590ℎ + (120590119901 minus 120590ℎ) (119903119901119903 )2 (22)

120590120579 = 120590ℎ minus (120590119901 minus 120590ℎ) (119903119901119903 )2 (23)

120590119911 = 120590V (24)

119880119903 = 120590119901 minus 120590ℎ21198660 1199032119901119903 (25)

Δ119906 = 0 (26)



119889120576119903119889120576120579119889120576119911

= [[[[[[[


119889120590101584011990311988912059010158401205791198891205901015840119911

(27)

where


(28)



119889120590101584011990311988912059010158401205791198891205901015840119911

= 1120585 [[[

11988811 11988812 1198881311988821 11988822 1198882311988831 11988832 11988833]]]

119889120576119903119889120576120579119889120576119911

(29)

where

11988811 = 11198642 (1 minus V2 + 1198641198862120579119887 + 2119864V119886120579119886119911119887 + 1198641198862119911119887)11988812 = 11198642 [minus119864119886119903 (119886120579 + V119886119911) 119887

+ V (1 + V minus 119864119886120579119886119911119887 + 1198641198862119911119887)]11988813 = 11198642 [minus119864119886119903 (V119886120579 + 119886119911) 119887


minus 119864V119886119903 (119886120579 + 119886119911) 119887]11988833 = 11198642 (1 minus V2 + 1198641198862119903119887 + 2119864V119886119903119886120579119887 + 1198641198862120579119887)11988821 = 1198881211988823 = 11988832

(30)


(31)


119889120576119903 = minus119889120576120579 = 119889119903119903 (32)


1198891205901015840119903119889119903 minus 11988711 minus 11988712120585119903 = 0 (33)

1198891205901015840120579119889119903 minus 11988712 minus 11988722120585119903 = 0 (34)

1198891205901015840119911119889119903 minus 11988731 minus 11988732120585119903 = 0 (35)



1199011015840119901 = 11990110158400 (36)



119902119901 = 120578011990110158400 + 11987211990110158400radicOCR minus 1 (37)




120579119901)2 + (1205901015840119911119901 minus 1205901015840

120579119901)2 + (1205901015840119903119901 minus 1205901015840119911119901)2 (38)



1205901015840119903119901 + 1205901015840120579119901 = 12059010158401199030 + 12059010158401205790 = 21205901015840ℎ (39)

1205901015840119911119901 = 12059010158401199110 = 1205901015840V0 (40)



12059010158401199030 = 12059010158401205790 = 119870012059010158401199110 (41)






119880119903119901 = 119903119909119901 minus 1199031199090 = 1205901015840119903119901 minus 1205901015840119903021198660 119903119909119901 (43)







Δ119906119903119909 = 1205901015840119903119901 minus 1205901015840119903119909 + int119903119901119903119909

1205901015840119903 minus 1205901015840120579119903 119889119903 (46)




12058111988911990110158401205921199011015840 + (120582 minus 120581) 11988911990110158401198881205921199011015840119888 = 0 (47)


1199011015840119888 = 1199011015840OCR(119901101584011990110158400)minus120581(120582minus120581) (48)


119902 = 12057801199011015840 + 1198721199011015840 [OCR(119901101584011990110158400)minus1Λ minus 1]12 (49)



120578 = radic1198722 + 12057820 (50)


1199011015840119891 = 11990110158400 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(51)

119902119891 = 11990110158400radic1198722 + 12057820 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(52)


120590119903119909 = 120590119903119901 minus 2radic3 int119903119909119903119901

119902119903 119889119903 (53)





119891minus (1 minus 1198700)21205902V0119866







119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)







1198862 minus 119886201199032119909 )

(56)




1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)




sdot (1 minus 119886201198862) + 1]]]

(58)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)




+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)



119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018










119889120590101584011990311988912059010158401205791198891205901015840119911

= 1120585 [[[

11988811 11988812 1198881311988821 11988822 1198882311988831 11988832 11988833]]]

119889120576119903119889120576120579119889120576119911

(29)

where

11988811 = 11198642 (1 minus V2 + 1198641198862120579119887 + 2119864V119886120579119886119911119887 + 1198641198862119911119887)11988812 = 11198642 [minus119864119886119903 (119886120579 + V119886119911) 119887

+ V (1 + V minus 119864119886120579119886119911119887 + 1198641198862119911119887)]11988813 = 11198642 [minus119864119886119903 (V119886120579 + 119886119911) 119887


minus 119864V119886119903 (119886120579 + 119886119911) 119887]11988833 = 11198642 (1 minus V2 + 1198641198862119903119887 + 2119864V119886119903119886120579119887 + 1198641198862120579119887)11988821 = 1198881211988823 = 11988832

(30)


(31)


119889120576119903 = minus119889120576120579 = 119889119903119903 (32)


1198891205901015840119903119889119903 minus 11988711 minus 11988712120585119903 = 0 (33)

1198891205901015840120579119889119903 minus 11988712 minus 11988722120585119903 = 0 (34)

1198891205901015840119911119889119903 minus 11988731 minus 11988732120585119903 = 0 (35)



1199011015840119901 = 11990110158400 (36)



119902119901 = 120578011990110158400 + 11987211990110158400radicOCR minus 1 (37)




120579119901)2 + (1205901015840119911119901 minus 1205901015840

120579119901)2 + (1205901015840119903119901 minus 1205901015840119911119901)2 (38)



1205901015840119903119901 + 1205901015840120579119901 = 12059010158401199030 + 12059010158401205790 = 21205901015840ℎ (39)

1205901015840119911119901 = 12059010158401199110 = 1205901015840V0 (40)



12059010158401199030 = 12059010158401205790 = 119870012059010158401199110 (41)






119880119903119901 = 119903119909119901 minus 1199031199090 = 1205901015840119903119901 minus 1205901015840119903021198660 119903119909119901 (43)







Δ119906119903119909 = 1205901015840119903119901 minus 1205901015840119903119909 + int119903119901119903119909

1205901015840119903 minus 1205901015840120579119903 119889119903 (46)




12058111988911990110158401205921199011015840 + (120582 minus 120581) 11988911990110158401198881205921199011015840119888 = 0 (47)


1199011015840119888 = 1199011015840OCR(119901101584011990110158400)minus120581(120582minus120581) (48)


119902 = 12057801199011015840 + 1198721199011015840 [OCR(119901101584011990110158400)minus1Λ minus 1]12 (49)



120578 = radic1198722 + 12057820 (50)


1199011015840119891 = 11990110158400 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(51)

119902119891 = 11990110158400radic1198722 + 12057820 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(52)


120590119903119909 = 120590119903119901 minus 2radic3 int119903119909119903119901

119902119903 119889119903 (53)





119891minus (1 minus 1198700)21205902V0119866







119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)







1198862 minus 119886201199032119909 )

(56)




1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)




sdot (1 minus 119886201198862) + 1]]]

(58)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)




+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)



119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018










119880119903119901 = 119903119909119901 minus 1199031199090 = 1205901015840119903119901 minus 1205901015840119903021198660 119903119909119901 (43)







Δ119906119903119909 = 1205901015840119903119901 minus 1205901015840119903119909 + int119903119901119903119909

1205901015840119903 minus 1205901015840120579119903 119889119903 (46)




12058111988911990110158401205921199011015840 + (120582 minus 120581) 11988911990110158401198881205921199011015840119888 = 0 (47)


1199011015840119888 = 1199011015840OCR(119901101584011990110158400)minus120581(120582minus120581) (48)


119902 = 12057801199011015840 + 1198721199011015840 [OCR(119901101584011990110158400)minus1Λ minus 1]12 (49)



120578 = radic1198722 + 12057820 (50)


1199011015840119891 = 11990110158400 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(51)

119902119891 = 11990110158400radic1198722 + 12057820 [[[[1198722OCR

(radic1198722 + 12057820 minus 1205780)2 + 1198722]]]]Λ

(52)


120590119903119909 = 120590119903119901 minus 2radic3 int119903119909119903119901

119902119903 119889119903 (53)





119891minus (1 minus 1198700)21205902V0119866







119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)







1198862 minus 119886201199032119909 )

(56)




1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)




sdot (1 minus 119886201198862) + 1]]]

(58)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)




+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)



119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018














119891minus (1 minus 1198700)2 12059010158402V0 (

1198862 minus 119886201199032119909 )]]] (55)







1198862 minus 119886201199032119909 )

(56)




1198862 minus 119886201199032119909 )

+ 11990110158400 [[[[1 minus ( 1198722OCRradic1198722 + 12057820 + 1198722)

Λ]]]]

(57)




sdot (1 minus 119886201198862) + 1]]]

(58)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(59)




+ 1)(60)



1

minus ( 1198722OCRradic1198722 + 12057820 + 1198722)Λ]]]]

(61)



119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018










119872 = 12 120582 = 015 120581 = 003 V = 02781198700 OCR 120590V0 (kPa) 1198660 (kPa) 1205920 1199060 (kPa)0625 1 160 4348 209 1001 3 120 4113 197 1002 10 72 3756 18 100













1205901015840119911 = 12 (1205901015840119903 + 1205901015840120579) (63)





1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









1 10 1000

1

2

3

4

5

6

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

z p0


1 10 1000

2

4

6

8

10

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

K0=1

K0=2

K0=0625

K0=1

K0=2

K0=0625

r p0


1 10 100

Critical value

Maximum value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

20

15

10

05

00

minus05

minus10

minus15

K0=1K0=2

K0=0625

K0=1

K0=2

K0=0625

p0


1 10 100

0

1

2

3

4

Critical value

MCC

Elastic region

Plastic region

Critical region

K0-MCC

ra

minus1

minus2

minus3

K0=1

K0=2

K0=0625

Δup

0








rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

25

30

35

40

MCCK0-MCC

aa0

a

v

Δua

o

ry

x

Λ=08 ]=03M=12 Gp

0=35

(K0=0625)

ℎ K0=

Δuap

0

ap

0r p

a

(a) 1198700 = 0625

rp


rapidlychangestate

2 4 6 8 100

5

10

15

20

MCCK0-MCC

aa0

a

v

Δua

o

ry

xΛ=08 ]=03M=12 Gp

0=35(K0=1)

ℎ K0=

Δuap

0

ap

0r p

a

(b) 1198700 = 1

rp


rapidlychangestate

2 4 6 8 10

0

4

8

12

MCCK0-MCC

aa0

a

v

Δua

Λ=08 ]=03M=12 Gp

0=35

(K0=2)

o

ry

x

Δuap

0

ap

0r p

a

minus4

ℎ K0=

(c) 1198700 = 2






6 Conclusions




CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018










CSLF

O(P)

1

1

K0 line

00

02

04

06

qp

0

08

10

12

02 04 06 08 10 1200pp0

radicM2 + 20

0

OCR=1Λ=08K0=0625]=03 M=12

(a) OCR = 11198700 = 0625

00

05

10

15

20

25

O


CSL

F

1

P

qp

0

04 08 12 16 20 24 28 3200pp0

radicM2 + 20 (M)

K0 line

OCR=3Λ=08K0=1]=03 M=12

(b) OCR = 31198700 = 1

0

2

4

6

8

10

12

O


F 1

1

P

qp

0

K0 line

2 4 6 8 10 120pp0

radicM2 + 20

0

OCR=10Λ=075K0=2]=03 M=12

(c) OCR = 101198700 = 2










1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018









1 10 100

OCR=1

OCR=3

OCR=10v

o

ry

x

0

3

6

9

12

15zp

0

ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(a) 120590119911

OCR=1

OCR=3

OCR=10

1 10 100

0

3

6

9

12

rp

0

ra

v

o

ry

xℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(b) 120590119903

0

3

6

OCR=1

OCR=3

OCR=10

v

o

ry

x

p

0

minus3

1 10 100ra

ℎ K0=


M=12

Gp0=36

K0=0625

K0=1

K0=2

(c) 120590120579

OCR=10

OCR=3OCR=10

Δuap

0

v

ℎ K0=

o

ry

x


M=12

Gp0=36

0

3

6

minus3

1 10 100

K0=0625

K0=1

K0=2

ra

(d) Δ119906


Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018


























Journal of












Journal of







Volume 2018






Volume 2018








analysis of soil-compacting effect caused by shield...

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