Influence of Shield Attitude Change on Shield–Soil Interaction

applied sciences

Article Inﬂuence of Shield Attitude Change on Shield–Soil Interaction

Xiang Shen 1,2,3 , Da-Jun Yuan 1,2,3 and Da-Long Jin 1,2,3,*

1 Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China; [email protected] (X.S.); [email protected] (D.-J.Y.) 2 Tunnel and Underground Engineering Research Center of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China 3 School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China * Correspondence: [email protected] or [email protected]; Tel.: +86-1800-109-1006

Received: 9 April 2019; Accepted: 26 April 2019; Published: 1 May 2019

Featured Application: Shield attitude control and shield tunnel construction risk control.

Abstract: The mechanism of shield–soil interaction and multi-phase equilibrium control theory in shield tunneling process still lack suﬃcient understanding. Aiming at this problem, with the improved calculation method of loose earth pressure, the initial boundary problem of shield attitude calculation was solved. Based on the ground reaction curve, the shield–soil interaction was simulated by the equivalent springs, and the displacement of surrounding soil was calculated during the change of the shield attitude. Then, the theoretical method of surrounding soil load acting on the shield were obtained. In summary, the calculation method of shield attitude was obtained. This method has three main applications in engineering, namely the inversion of shield–soil interaction force, the prediction of pitch angle and the prediction of yawing angle. Finally, combined with Jinan Metro Line R2 shield tunnel project, the shield attitude was monitored in real time and compared with the theoretical value. The results show that the trend of the theoretical values of pitch angle and yawing angle were basically the same as the measured value, but the theoretical value was generally larger than the measured value. The research results can provide a useful reference for the shield attitude adjustment.

Keywords: shield; shield attitude; pitch angle; yawing angle; interaction; advance cylinders

1. Introduction Shield tunneling has been widely used in the construction of urban subway tunnels. Due to the uncertainty of load and stratum during the excavation process, the shield cannot always be consistent with the theoretical design axis [1–4]. The automatic control system used to control the shield attitude is based on existing engineering experience and not supported by appropriate theories [5]. Therefore, how to properly control the advance load is one of the main concerns in the construction of shield tunnels. At present, relevant research mainly focuses on the two directions of advance load modeling and automatic correction. The difficulty in establishing a shield tunneling mechanics model is that the forces around the shield cannot be calculated and expressed accurately. Therefore, many scholars have carried out a series of studies on the establishment of shield tunneling mechanics models and attitude control. Koyama et al. [6] mentioned that it is impossible to deal with the precise control of the shield machine in the case of complex geological structures and sharp curves. Xie et al. [7] applied the automatic trajectory tracking control system to the control of the shield’s route, but these systems also only rely on empirical relationships, without precise theoretical background. Shimizu et al. [8] analyzed the tunneling and motion laws of

Appl. Sci. 2019, 9, 1812; doi:10.3390/app9091812 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 20

Appl. Sci. 2019, 9, 1812 2 of 19 et al. [9] established a mathematical analysis model of shield tunneling movement. Robust theory was applied to shield attitude control by Komiya et al. [10]. Based on the force balance condition of thetunnel shield, construction Sugimoto without et al. [11] the influence determined of nonlinear the relationship factors. Shimizu between et the al. [9advance] established load aand mathematical the earth pressure,analysis model and then of shield connected tunneling the movement. earth pressure Robust wi theoryth the wasshield applied attitude, to shield thus attitude establishing control the by theoreticalKomiya et al.model [10]. Basedof the onshield the force advance balance load. condition Yue et ofal. the [12] shield, and SugimotoSun et al. et[13] al. [proposed11] determined a shield the attituderelationship and between trajectory the automatic advance load control and thesystem earth ba pressure,sed on the and sliding then connected mode robust the earth controller pressure and with a shieldthe shield attitude attitude, dynamic thus establishing coordination the theoreticalcontrol system model based of the on shield the advancedynamic load. model Yue of et al.the [ 12shield] and attitudeSun et al. adjustment [13] proposed process. a shield Ates attitude et al. [14] and believed trajectory that automatic it is crucially control important system based to select on the a proper sliding TBMmode and robust define controller its basic and specifications a shield attitude such dynamic as installed coordination cutter-head control torque system and TBM based thrust on the capacities. dynamic Zhangmodel ofet al. the [15] shield established attitude adjustmenta predicting process. model of Ates the et thrust al. [14 and] believed torque thatfor the it is total crucially load importantthat fully reflectsto select the a proper influences TBM of and geological, define its basicoperating, specifications and structural such as parameters. installed cutter-head In summary, torque the and current TBM researchthrust capacities. on shield Zhang attitude et al.is more [15] established about the aapp predictinglication of model control of thetheory, thrust while and torqueignoring for the the basic total problemsload that fullysuch reflectsas shield–soil the influences interaction of geological, and shield operating, tunneling and mechanics. structural parameters. In summary, the currentWith the research necessity on shieldto accurately attitude predict is more performa about thence application of machines of control in different theory, ground while ignoringconditions, the manybasic problemsresearchers such have as shield–soil worked interactionto develop and new shield prediction tunneling models mechanics. or adjustment factors for the commonWith existing the necessity models to accurately[16]. Acarog predictlu et performanceal. [17] established of machines a model in di toff erentpredict ground specific conditions, energy requirementmany researchers of constant have workedcross-section to develop disc cutters new predictionin the rockmodels cutting orprocess adjustment by using factors fuzzy for logic the method.common Barton existing et modelsal. [18] proposed [16]. Acaroglu a model et named al. [17] Q establishedTBM that uses a modelmany input to predict parameters specific (such energy as RQD,requirement joint condition, of constant Stress cross-section condition, disc intact cutters rock instrength, the rock quartz cutting content process and by TBM using thrust). fuzzy logicThe modifiedmethod. BartonCSM model et al. was [18] proposedadded rock a modelmass properties named Q TBMas inputthat usesparameters many inputinto the parameters model [19]. (such In hardas RQD, rock jointformations, condition, such Stress models condition, have often intact shown rock results strength, that quartz it would content not expand and TBM too thrust).much. However,The modified in the CSM soft model soil layer, was the added interaction rock mass mo propertiesdel for shield as inputtunneling parameters is still in into its infancy. the model [19]. In hardIn view rock of formations, the above such deficiencies, models havethe influence often shown of the results shield that attitude it would parameter not expand on the too shield– much. soilHowever, interaction in the was soft studied. soil layer, The the shield–soil interaction interaction model for was shield represented tunneling by is an still equivalent in its infancy. spring, and the interactionIn view of force the above between deficiencies, the shield the and influence the soil was of the solved shield and attitude analyzed parameter according on theto the shield–soil ground reactioninteraction curve. was Based studied. on Thethis, shield–soilthe theoretical interaction calculation was method represented of shield by an attitude equivalent was spring, proposed and and the comparedinteraction with force the between actual project. the shield and the soil was solved and analyzed according to the ground reaction curve. Based on this, the theoretical calculation method of shield attitude was proposed and 2.compared Methodology with the actual project.

2.1.2. Methodology Model of Load Acting on Shield Based on the shield load model established by Sugimoto [20], the contact relationship between 2.1. Model of Load Acting on Shield the shield and the stratum is blurred. The force of shield–soil interaction is decomposed along the axis direction,Based on thethe shieldnormal load direction, model establishedand the tangen by Sugimototial direction [20], of the the contact shield, relationship and the load between model the is shownshield and in Figure the stratum 1. is blurred. The force of shield–soil interaction is decomposed along the axis direction, the normalThe loads direction, acting and on thethe tangentialshield are: direction f1, shield of self-weight; the shield, and f2, theforce load on model the shield is shown tail; inf3, Figureadvance1. load providedThe loads by acting the jacks; on the f4, shield load acting are: f1 on, shield the cutter self-weight; head; andf2, forcef5, surrounding on the shield soil tail; loadf3 ,acting advance on theload shield provided periphery by the [21]. jacks; f4, load acting on the cutter head; and f5, surrounding soil load acting on the shield periphery [21].

(a) (b) (c) Figure 1. ShieldShield load load model: model: ( (aa)) Section Section A-A; A-A; ( (bb)) Section Section B-B; B-B; and and ( (cc)) Section Section C-C. C-C. Appl.Appl. Sci. 20192019,, 99,, 1812x FOR PEER REVIEW 33 of 1920

If the shield is regarded as a rigid body, the distance between any two points on the shield remainsIf the unchanged shield is regarded during the as a tunneling rigid body, process. the distance Six parameters between any are two required points to on describe the shield the remains shield unchangedattitude: cutterhead during the center tunneling coordinate process. (x0, Sixy0, z parameters0), yawing angle are required α, pitch to angle describe β, and the rolling shield angle attitude: Ω. cutterheadThe attitude center angles coordinate are shown (x0 ,iny 0Figure, z0), yawing 2a. angle α, pitch angle β, and rolling angle Ω. The attitude anglesThe are coordinate shown in Figuresystem2a. is established as shown in Figure 2b: in the geodetic coordinate system CE, theThe x-axis coordinate is vertically system downward is established and the as showny-axis is in on Figure the horizontal2b: in the plane; geodetic in the coordinate shield-structure system E Ccoordinate, the x-axis system is vertically CM, the downward p-axis is vertically and the y-axisdownward is on the and horizontal the r-axis plane; is along in thethe shield-structureshield axis; and M coordinatethe coordinate system systemC , theCM is p-axis rotated is vertically by a certain downward angle around and the the r-axis r-axis is to along obtain the a shield coordinate axis; and system the M MR coordinateCMR. system C is rotated by a certain angle around the r-axis to obtain a coordinate system C .

(a) (b)

FigureFigure 2.2. Attitude angles and coordinatecoordinate system:system: (a) attitude angles;angles; andand ((b)) coordinatecoordinate system.system.

WhenWhen thethe shieldshield isis consideredconsidered to bebe inin aa staticstatic equilibriumequilibrium state duringduring the tunneling process, thethe staticstatic balancebalance equationequation mustmust bebe met:met: 5 b P5 f = 0 f b n = 0 n=1 n n = 1 (1)(1) P5 5 Mb b==0 Mn n 0 n = n1=1 ： Then:Then bbb++++b + b bbb+ b + b = FFFFFF123451eeeeee F2e F3e F4e +=0F5e 0 b b b b b (2) +bb+ + b+ bb= (2) M1e M+++2e M3e M4e M5e 0 MMMMM12345eeeee+=0 wherewhere ee == p,, q,, andand rr representrepresent didifferentﬀerent directionsdirections ofof force;force; andand bb == E, MM,, andand MRMR areare expressedexpressed inin didifferentﬀerent coordinatecoordinate systems.systems

2.2.2.2. Initial Earth Pressure CalculationCalculation OneOne ofof the the keys keys to to establishing establishing a shield–soil a shield–soil interaction interaction model model during during the change the change of shield of attitude shield isattitude the calculation is the calculation of initial earthof initial pressure. earth Loosepressure. earth Loose pressure earth theory pressure is often theory used is for often deformation used for analysisdeformation of segments. analysis Theof segments. initial shield The stress initial state shield should stress also state be should in the non-uniform also be in the state. non-uniform This part solvesstate. This the initial part solves boundary the probleminitial boundary of the shield problem interaction of the model shield by interaction means of relevantmodel by theories. means of relevant theories. 2.2.1. Calculation of Loose Soil Pressure in the Overlaying Soil 2.2.1.Terzaghi Calculation deduced of loose the soil classical pressure theory in the of overlaying loose earth soil pressure through the trap-door test [22]. The analytical expression of the vertical stress σ of the trap-door is: Terzaghi deduced the classical theory of loosev earth pressure through the trap-door test [22]. The ! analytical expression of the vertical Bstressc σv of theK trap-door0 tan ϕH is: K0 tan ϕH 1γ B B σv = − 1 e− K 1tanϕϕHKH+ P e− 1 tan (3) K tanγ −ϕ --000 0Bc1 − BB σ =−+1e11P e (3) v K tanϕ 0 0  Appl. Sci. 2019, 9, 1812 4 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 20 where σσvvis is the the loose loose earth earth pressure; pressure;K0 is K the0 is later the pressurelater pressure coeﬃcient; coefficient;H is the H overlying is the overlying soil thickness; soil Pthickness;0 is the upper P0 is the load; upperc is the load; cohesion; c is the cohesion;ϕ is the internal φ is the friction internal angle; frictionγ is angle; the weight γ is the density; weight and density;B1 is and B1 is the loose band width. the loose band width. ! ππϕ/4/4++ϕ/2 /2 B1 == R cot (4) BR1 cot 2 (4) 2 where R is the shield radius. where R is the shield radius. Li [23] introduced the factor of formation loss rate based on the stress of Terzaghi loose soil, Li [23] introduced the factor of formation loss rate based on the stress of Terzaghi loose soil, and and set the key parameter A1 to correct the theoretical solution of Terzaghi loose soil pressure: set the key parameter A1 to correct the theoretical solution of Terzaghi loose soil pressure: AH! AH B γ c A--1H11A1H Bc1 γ − B B = 1 − 1 BB11+ − 1 σvσ =−+− 11ee P0Pe e (5) v A1A − 0 1  2 1++ Ka tan2 θ (()1 −Ka) tan θθ A = 1tanKa 1tan− Ka (6) 1A =⋅2 2 1 tan22θθθ +++ Ka · 1 + Ka tan θ (6) tanKKaa 1 tan 2 where Ka is the active side pressure coeﬃcient, Ka = tan (45◦- ϕ/2). where Ka is the active side pressure coefficient, Ka = tan2 (45°- φ / 2). According to Figure3 3,, thethe followingfollowing conclusions conclusions can can be be drawn: drawn:

BBs11 s tantanθθ==− (7) 2s − 2B1 22sB1 where ss isis thethe displacementdisplacement ofof thethe dome.dome.

Figure 3. Calculation of tan θ [[23].23].

2.2.2. Initial Force Acting on the Shield Periphery 2.2.2. Initial force acting on the shield periphery The earth pressure around the shield increases with the increase of the depth of the soil, as shown The earth pressure around the shield increases with the increase of the depth of the soil, as in Figure4. η is the angle with the p-axis in the CM coordinate system. To simplify the calculation, shown in Figure 4. η is the angle with the p-axis in the CM coordinate system. To simplify the π π linearcalculation, regression linear method regression is adopted. method In theis adopted. process of Inη fromthe process/2 to 3 of/2, theη from corresponding π/2 to 3π normal/2, the earthcorresponding pressure actingnormal on earth the shieldpressure at anyacting point on isthe linearly shield increased,at any point so is that linearly the normal increased, force so on that the shieldthe normal can be force obtained. on the Theshield normal can be earth obtained. pressure The at normal any point earth in pressure the shell at is shownany point in Figurein the 5shell. is shownTherefore, in Figure the 5. earth pressure on the shield shell can be calculated as follows: Therefore, the earth pressure on thep1 pshield2 shellπ can be calculatedπ as follows:  − η + + p2 0 η  π 2 ≤ ≤ 2  p2 p1 π π 3π pη = − η + p1 < η (8)  π − 2 2 ≤ 2  p1 p2 3π 3π  − η + p < η 2π π − 2 2 2 ≤ Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 20 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 20  pp− ππ  pp12− ηηππ≤≤  12ηη+ +p2 0≤≤ π +22 +p2 0  π 22    pp− πππ3 pp=− pp21− ηηπππ+ <≤3 ppη =− 21π ηη+1 <≤ (8) η  π 2221 (8)  222  pp− 33ππ Appl. Sci. 2019, 9, 1812  pp12− η −<≤33ππ+p ηπ 2 5 of 19  12π η −<≤2 ηπ  22+p2 2  π 22 M 44FFM1Mp =+4F1p1p (9) pp21=+ p2 pp= p1 + π (9)(9) 21ππDLDL where p1 is the normal earth pressure at the top of the shield, calculated according to improved loose wherewhere pp11 is the normal earth pressure at the top of the shield, calculated according to improved looseloose soil pressure formula [23]; p2 is the normal earth pressure at the bottom of the shield; D is the shield soil pressurepressure formulaformula [[23];23]; pp22 is the normal earth pressure at the bottombottom of thethe shield;shield; D isis thethe shieldshield diameter; and L is the shield length. diameter; andand LL isis thethe shieldshield length.length.

Figure 4. Initial shield shell force calculation model. Figure 4. Initial shield shell force calculation model.model.

FigureFigure 5.5. InitialInitial shieldshield shellshell forceforce distributiondistribution curve.curve. Figure 5. Initial shield shell force distribution curve. 2.3. Shield Shell–Soil Interaction Model 2.3. Shield Shell–Soil Interaction Model 2.3. Shield Shell–Soil Interaction Model 2.3.1. Basic Assumption 2.3.1. Basic assumption 2.3.1. Basic assumption The shield is constrained by the soil. Based on the characteristics of the ground reaction curve, The shield is constrained by the soil. Based on the characteristics of the ground reaction curve, the soilThe can shield be approximated is constrained as by an the equivalent soil. Based spring on th withine characteristics a certain range of the (Figure ground6). Thereaction equivalent curve, the soil can be approximated as an equivalent spring within a certain range (Figure 6). The equivalent springthe soil iscan nonlinear be approximated deformation, as an andequivalent the correlation spring within coeffi acient certain can range be obtained (Figure 6). from The the equivalent ground spring is nonlinear deformation, and the correlation coefficient can be obtained from the ground reactionspring is curve nonlinear [24]. Accordingdeformation, to the and deformation the correlation of the coefficient soil spring, can the be interaction obtained forcefrom betweenthe ground the reaction curve [24]. According to the deformation of the soil spring, the interaction force between the shieldreaction and curve the soil[24]. can According be obtained. to the deformation of the soil spring, the interaction force between the shield and the soil can be obtained. shieldThe and center the soil of gravitycan be obtained. of the shield machine is generally not in the geometric centroids, as shown The center of gravity of the shield machine is generally not in the geometric centroids, as shown in FigureThe7 centera. Point of gravityS is the centerof the shield of gravity machine of the is shield, generally Point notO isin thethe geometricgeometric centroid, centroids, and asl showns is the in Figure 7a. Point S is the center of gravity of the shield, Point O is the geometric centroid, and ls is distancein Figure between 7a. Point two S is points. the center of gravity of the shield, Point O is the geometric centroid, and ls is the distance between two points. the distanceWhen the between shield attitudetwo points. is adjusted in the actual project, the yawing angle and pitch angle should When the shield attitude is adjusted in the actual project, the yawing angle and pitch angle be changedWhen the simultaneously. shield attitude However, is adjusted when in thethe time actu intervalal project, of shieldthe yawing attitude angle adjustment and pitch tends angle to should be changed simultaneously. However, when the time interval of shield attitude adjustment infinitesimal,should be changed it can be simultaneously. considered that However, the pitch anglewhen and the yawingtime interval angle areof shield in succession. attitude The adjustment order in tends to infinitesimal, it can be considered that the pitch angle and yawing angle are in succession. whichtends to the infinitesimal, pitch and yawing it can anglesbe considered occur does that not the aff pitchect the angle final and calculation yawing resultangle fromare in the succession. aspect of The order in which the pitch and yawing angles occur does not affect the final calculation result from theThe mechanics order in which analysis. the pitch Assuming and yawing that the angles shield occur is performing does not attitudeaffect the adjustment, final calculation it can result be divided from into four stages, as follows:

(i) Stage 1: Gravity stage. The point of gravity action is moved to the geometric centroid while the gravity eccentric moment MG is generated, and only gravity is considered at this stage. Under the action of gravity, the shield machine produces a displacement of ∆sv in the vertical direction, and ∆sv is positive downward and negative upward, as shown in Figure7b. Appl.Appl. Sci.Sci. 20192019,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 66 ofof 2020 thethe aspectaspect ofof thethe mechanicsmechanics analysis.analysis. AssumingAssuming thatthat thethe shieldshield isis performingperforming attitudeattitude adjustment,adjustment, itit cancan bebe divideddivided intointo fourfour stages,stages, asas follows:follows: (i)(i) StageStage 1:1: GravityGravity stage.stage. TheThe pointpoint ofof gravitygravity actionaction isis movedmoved toto thethe geometricgeometric centroidcentroid whilewhile thethe gravitygravity eccentriceccentric momentmoment MMGG isis generated,generated, andand onlyonly gravitygravity isis consideredconsidered atat thisthis stage.stage. Appl. Sci. 2019UnderUnder, 9, 1812 thethe actionaction ofof gravity,gravity, thethe shieldshield machinemachine producesproduces aa displacementdisplacement ofof ΔΔssvv inin6 of thethe 19 verticalvertical direction,direction, andand ΔΔssvv isis positivepositive downwarddownward andand negativenegative upward,upward, asas shownshown inin FigureFigure 7b.7b. (ii) (ii)(ii)Stage StageStage 2: Gravity 2:2: GravityGravity eccentric eccentriceccentric moment momentmoment stage. stage.stage. The shield ThThee deflectsshieldshield deflectsdeflects due to gravityduedue toto eccentricgravitygravity eccentriceccentric bending moment on the vertical plane. The pitch angle caused by the M is called , as shown in bendingbending momentmoment onon thethe verticalvertical plane.plane. TheThe pitchpitch angleangle causedcausedG byby thethe MMβGG G isis calledcalled ββGG,, asas Figureshownshown7c. The inin angleFigureFigure is 7c.7c. positive TheThe angleangle when isis its positivepositive direction whenwhen is counterclockwise, itsits directiondirection isis counterclockwise,counterclockwise, and the angle is negative andand thethe whenangleangle its direction isis negativenegative is clockwise. whenwhen itsits directiondirection isis clockwise.clockwise. (iii) (iii)(iii)Stage StageStage 3: Vertical 3:3: VerticalVertical bending bendingbending moment momentmoment stage. stage.stage. Upper UpperUpper and andand lower lowerlower partition partitionpartition jacks jacksjacks produce produceproduce deflectiondeflectiondeflection β ββ β ββ momentsM M on the vertical plane. The pitch angle causedM by the M is called βA, as momentsmomentsA onMA theA on vertical the vertical plane. plane. The pitch The angle pitch caused angle bycaused the byA isthe called MAA βisA ,called as shown βA, inas Figureshownshown7c. The inin angleFigureFigure is 7c.7c. positive TheThe angleangle when isis its positivepositive direction whenwhen is counterclockwise, itsits directiondirection isis counterclockwise,counterclockwise, and the angle is negative andand thethe whenangleangle its direction isis negativenegative is clockwise. whenwhen itsits directiondirection isis clockwise.clockwise. (iv) (iv)(iv)Stage StageStage 4: Horizontal 4:4: HorizontalHorizontal bending bendingbending moment momentmoment stage. stage. Understage. UnderUnder the action thethe of actionaction horizontal ofof horizontalhorizontal deflection deflectiondeflection bending bendingα momentα Mαα, α which is called yawing angle is generated on the horizontal plane, momentbendingMA, momentwhich isM calledAA, α which yawing is called angle yawing is generated angle onis ge thenerated horizontal on the plane, horizontal as shown plane, in Figureasas 7 shownshownd. α is inin positive FigureFigure when 7d.7d. αα its isis directionpositivepositive whenwhen is counterclockwise itsits directiondirection isis and counterclockwisecounterclockwise negative when andand its directionnegativenegative is clockwise.whenwhen itsits directiondirection isis clockwise.clockwise.

FigureFigure 6.6. Shield–soilShield–soil interaction interaction model.model.

((aa)) ((bb)) ((cc)) ((dd))

FigureFigureFigure 7.7. ShieldShield 7. Shield attitudeattitude attitude changechange change process:process: process: ((aa)) Stage1;Stage1; (a) Stage1; ((bb)) Stage2;Stage2; (b) Stage2; ((cc)) Stage3;Stage3; (c) Stage3; andand ((dd and)) StageStage (d) Stage4.4. 4. 2.3.2. Geometric Parameters 2.3.2.2.3.2. GeometricGeometric parametersparameters Assuming that the section of the shield at r = L has a displacement ∆sv in the vertical direction AssumingAssuming thatthat thethe sectionsection ofof thethe shieldshield atat rr == LL hashas aa displacementdisplacement ΔΔssvv inin thethe verticalvertical directiondirection and ∆sh in the horizontal direction, the displacement occurring at each point on the shield is calculated. andand ΔΔsshh inin thethe horizontalhorizontal direction,direction, thethe displacementdisplacement occurringoccurring atat eacheach pointpoint onon thethe shieldshield isis As shown in Figure8, D is the shield diameter, r is the shield radius, L is the length of the shield, O is calculated.calculated. AsAs shownshown inin FigureFigure 8,8, DD isis thethe shieldshield diameter,diameter, rr isis thethe shieldshield radius,radius, LL isis thethe lengthlength ofof thethe the initial position of the shield, O’ is the ﬁnal position of the shield, and η is the angle with the p-axis shield,shield, OO isis thethe initialinitial positionposition ofof thethe shield,shield, O'O' isis thethe finalfinal positionposition ofof thethe shield,shield, andand ηη isis thethe angleangle in the CM coordinate system. withwith thethe pp-axis-axis inin thethe CCMM coordinatecoordinate system.system. As shown in Figure8, the following inference can be made: AsAs shownshown inin FigureFigure 8,8, thethe followingfollowing inferenceinference cancan bebe made:made: Circle O: x22+y22=r22 Circle O: Circlex +y =rO : x2 + y2 = r2 22 22 2 (10)(10) Circle O': (x − Δ𝑠) +(y − Δ𝑠) =r 2 2 Circle O': (x − Δ𝑠) +(y − Δ𝑠2) =r 2 (10) Circle O : (x ∆s ) +(y ∆sv = r PolarPolar coordinatecoordinate conversion:conversion: xyxy=−=−=−=−ρρcos(cos(ηη− ) ) ;; h ρρ sin( sin(ηη− ) ).Bring.Bring inin EquationEquation (4)(4) toto get:get: Polar coordinate conversion: x = ρ cos( η); y = ρ sin( η). Bring in Equation (4) to get: − − Circle O : ρ = r (11) Circle O : ρ2 2(cos η∆s sin η∆s )ρ + ∆s2+∆s2= r2 − h − v h v Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 20

Appl. Sci. 2019, 9, 1812 7 of 19 Circle O: ρ=r (11) Circle O': ρ2 − 2(cosηΔs − sinηΔs )ρ+Δs2+Δs2=r2 q h v h v 2 2 22222 2 2 MMMMssrssssr'v==cos ηηη ⋅Δ∆sh −sin sinη ∆ ⋅Δsv + +r − sinsin 2 2 ηη ⋅Δ∆sh ∆ ⋅Δsv −sin sinη η∆ ⋅Δs −cos cosη η∆ ⋅Δs r − (12) 0 · hv− · − · hvh· − · h − · v − v(12) MMMM '0<< 0 indicates thatthat thethe shield shield is is subjected subjected to to active active earth earth pressure, pressure, and andMM MM> 0'0indicates> indicates that 0 0 thethat shield the shield is subjected is subjected to passive to earth passive pressure. earth Therefore, pressure. the Therefore, vertical and the horizontal vertical displacementand horizontal of thedisplacement shield: of the shield:

Uv(UMMUMMθ)()θηθη= =⋅MM 'sin,sin η , U h(θ()) = =⋅MM 'cos cos η (13) vh0· 0· (13)

Figure 8.8. Displacement calculation diagram. 2.3.3. Solution of Vertical and Horizontal Forces of Shield–Soil Interaction 2.3.3. Solution of vertical and horizontal forces of shield–soil interaction Through the ground reaction curve (Figure9), considering the change of the shield attitude, Through the ground reaction curve (Figure 9), considering the change of the shield attitude, the the vertical and horizontal interaction forces between the shield and the soil are deduced and analyzed vertical and horizontal interaction forces between the shield and the soil are deduced and analyzed underAppl. Sci. the 2019 premise, 9, x FOR of PEER small REVIEW deformation. 8 of 20 under the premise of small deformation. The ground reaction curve [24], which shows the relationship between the ground displacement and the earth pressure acting on the shield periphery as shown in Figure 9, can be represented by  aU ()−+≤ii ()  KKio imin tanh K io U i 0 KK− ()= io i min KUii  (14) aU ()KK−+≥tanhii K () U 0  io imax − io i  KKio imax where K is the coefficient of earth pressure, which is defined as the earth pressure σ divided by the initial vertical earth pressure σv0 (Ki=σi/σv0); U is the ground displacement; a is the gradient of functions K(U), which represents the coefficient of subgrade reaction ai; subscripts v and h are the vertical horizontal directions, respectively; and subscripts o, min, and max are the initial, lower limit and upper limit of the coefficient of earth pressure, respectively.

Figure 9. Ground reaction curve [[24].24].

The groundshield is reaction discretized, curve divided [24], which into showsn parts the along relationship the circumferential between the direction, ground displacement and divided andinto them parts earth along pressure the longitudinal acting on the direction, shield periphery as shown as in shown Figure in 10 Figure 9, can be represented by Vertical and horizontal stress of each unit are obtainedh ias follows:  (K K )tanh aiUi + K (U 0)  io imin Kio Kimin io i Ki(Ui) σησσησ= ==KU− (),h − KUi  () ≤ (14) vsij v vij v00 ij hsijaiUi h  hij h ij (15)  (Kio Kimax)tanh K K + Kio (Ui 0) − io− imax ≥ where σv0ij and σh0ij are the initial vertical and horizontal earth pressures, respectively, and the initial earth pressure is calculated as the loose earth pressure. Therefore, the conclusions can be obtained as follows: nm nm MM=⋅=⋅σσ FAFA55p vsij ij, q  hsij ij (16) ij==11 ij == 11 2π L where Aij is the area of each unit of the shield shell, and Aij= ·r· . n m It can be known from Equations (1) and (2): MM+= M = FF51pp0, F 5 q 0 (17)

Figure 10. Shield discretization.

2.3.4. Solution of bending moment of shield–soil interaction

Assuming that the pitch angle of the shield is β (β=βG+βA) and the yawing angle is α, then the vertical and horizontal displacement of each element on the shield are determined as follows: Δ=Δ+' M −L β ssxvvrtan 2 (18) Δ=' M −L α sxhrtan 2 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 20

Figure 9. Ground reaction curve [24].

The shield is discretized, divided into n parts along the circumferential direction, and divided into m parts along the longitudinal direction, as shown in Figure 10 Vertical and horizontal stress of each unit are obtained as follows: σησσησ==()  () vsijKU v vij v00 ij, hsij KU h  hij h ij (15)

Appl.where Sci. σ2019v0ij and, 9, 1812 σh0ij are the initial vertical and horizontal earth pressures, respectively, and the initial8 of 19 earth pressure is calculated as the loose earth pressure. Therefore, the conclusions can be obtained as follows: where K is the coefficient of earth pressure,nm which is defined nm as the earth pressure σ divided by the MM=⋅=⋅σσ initial vertical earth pressure σFAFAv055p(Ki= σi/σv0 vsij); U ij,is the q ground  displacement; hsij ij a is the gradient(16) of ij==11 ij == 11 functions K(U), which represents the coefficient of subgrade reaction ai; subscripts v and h are the 2π L where Aij is the area of each unit of the shield shell, and Aij= ·r· . vertical horizontal directions, respectively; and subscripts o,nminm, and max are the initial, lower limit and upperIt can be limit known of the from coeffi Equationscient of earth (1) and pressure, (2): respectively. The shield is discretized, divided intoMMn parts along M the circumferential direction, and divided FF+=0, F = 0 (17) into m parts along the longitudinal direction,51pp as shown in 5 q Figure 10.

Figure 10. Shield discretization.

2.3.4.Vertical Solution and of bending horizontal moment stress ofof eachshield–soil unit are interaction obtained as follows: h i h i Assuming that the pitchσvsij angle= K vofU thevij( ηshield) σv0ij ,isσ hsijβ (β=βGK+βh AU) hijand(η )theσh 0yawingij angle is α, then (15)the vertical and horizontal displacement of each element on the shield are determined as follows: where σv0ij and σh0ij are the initial verticalΔ=Δ+' and horizontalM − earthL pressures,β respectively, and the initial ssxvvrtan earth pressure is calculated as the loose earth pressure.2 (18) Therefore, the conclusions can beΔ= obtained' M as − follows:L α sxhrtan 2 Xn Xm Xn Xm M M F5p = σvsijAij, F5q = σhsijAij (16) i = 1 j = 1 i = 1 j = 1 where A is the area of each unit of the shield shell, and A = 2π r L . ij ij n · · m It can be known from Equations (1) and (2):

M + M = M = F5p F1p 0, F5q 0 (17)

2.3.4. Solution of Bending Moment of Shield–Soil Interaction

Assuming that the pitch angle of the shield is β (β = βG + βA) and the yawing angle is α, then the vertical and horizontal displacement of each element on the shield are determined as follows:

M L ∆s0 = ∆sv + x tan β v r − 2 (18) ∆s = xM L tan α h0 r − 2 where ∆sv0 and ∆sh0 are the vertical and horizontal displacements of the elements on diﬀerent sections, M M respectively. xr is the value of the r axis in the C coordinate system.

n m n m X X L X X L MM = σ xM A , MM = σ xM A (19) 5q vsij rij − 2 ij 5p hsij rij − 2 ij i = 1 j = 1 i = 1 j = 1

It can be known from Equations (1) and (2):

M + M + β = M + α = M5q M1q MA 0, M5p MA 0 (20) Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 20

' ' where ∆sv and ∆sh are the vertical and horizontal displacements of the elements on different M sections, respectively. 𝑥 is the value of the r axis in the C coordinate system. nm nm MMMM=⋅−⋅=⋅−⋅σσLL  M55q vsijxAM rij ij, p  hsij  xA rij ij (19) ij==1122 ij == 11  It can be known from Equations (1) and (2): Appl. Sci. 2019, 9, 1812 MM++βα M 9 of 19 MM51qq+=0 M A， M 5 p M A=0 (20)

3. Shield3. Shield Pitch Pitch Angle Angle and and Yawing Yawing Angle Angle CalculationCalculation Method

3.1.3.1. Solution Solution Process Process of of Yawing Yawing Angle Angle and andPitch Pitch AngleAngle AccordingAccording to to the the above above calculation calculation theory,theory, thethe calculation method method of of the the pitch pitch angle angle and and yawing yawing angleangle of of the the shield shield can can be be summarized. summarized. TheThe specificspecific calculation process process is is shown shown in in Figure Figure 11. 11 . TheThe first first stage stage is is to to input input the the basic basic parametersparameters such as as soil soil parameters parameters and and shield shield parameters, parameters, andand establish establish the shield–soilthe shield–soil interaction interaction model. model. The relationship The relationship curve between curve thebetween vertical the displacement vertical anddisplacement vertical force and on vertical the shield force (RCDV) on the is shield obtained. (RCDV) Through is obtained. the RCDV, Through the vertical the RCDV, displacement the vertical∆ sv displacement △sv under gravity is obtained. In the second stage, the △sv obtained in the first stage under gravity is obtained. In the second stage, the ∆sv obtained in the first stage and the unknown pitchand angle the unknown are set into pitch Equation angle are (18), set andinto theEquation relationship (18), and curve the relationship between the curve pitch between angle and the moment pitch onangle the vertical and moment plane on (RCPV) the vertical is obtained, plane (RCPV) which is is obtained, obtained which under is theobtained action under of gravity the action eccentric of gravity eccentric bending moment. Through the RCPV, the pitch angle βG generated by the MG and bending moment. Through theβ RCPV, the pitch angle βG generated by the MG and the βA generated by the ββA generated by the MA can be obtained. In the third stage, △sv, βG, and βA are brought into the the M can be obtained. In the third stage, ∆sv, β , and β are brought into the calculation program, calculationA program, and the unknown yawing angleG valueA is continuously given. The relationship and the unknown yawing angle value is continuously given. The relationship curve between the curve between the yawing angle and horizontal bending moment applied by the shield (RCYH) is yawing angle and horizontal bending momentα applied by the shield (RCYH) is obtained. Through the obtained. Through the RCYH, the M is calculated, then the yawing angle α is obtained. Finally, △ α A RCYH, the M is calculated, then the yawing angle α is obtained. Finally, ∆sv, β , β , and α are output. sv, βG, βA, andA α are output. G A

β M A

α M A

FigureFigure 11. 11.Shield Shield attitudeattitude calculation flow ﬂow chart. chart.

3.2.3.2. Calculation Calculation Example Example BasedBased on on the the above above derivation derivation results, results, thethe followingfollowing set set of of typical typical parameters parameters for for calculation calculation was was 3 selected.selected. Suppose Suppose a a shield shield tunnel tunnel is is excavatedexcavated inin a uniform sand sand layer. layer. The The soil soil weight weight γ isγ is19 19kN/m kN/3m, , thethe cohesion cohesionc is c 0is kPa, 0 kPa, and and the the internal internal friction friction angle angleϕ is φ 30 is◦. 30°. The The shield shield diameter diameterD is D 12 is m, 12 the m, lengththe L is 10.8 m, the shield mass Gshield is 3000 t, and the thickness of covering soil is 24 m. The displacement β α of the dome is 5 mm, and the eccentricity of gravity ls is 3.5 m. M = 20 MN m and M = 75 MN m. A · A · The ground reaction curve coeﬃcient is shown in Table1[11]. Through the Matlab programming calculation, the shield attitude parameters could be calculated. As shown in Figure 12a, the weight of the shield machine was 3000 t, and ∆sv was obtained β (∆sv = 8.1 mm). In the second stage, the sum of M and M is 125 MN m, then the pitch angle β could G A · be obtained (Figure 12b) (β = 1.16 ). In the third stage, it was found that Mα = 75 MN m, and then the ◦ A · yawing angle α could be obtained (Figure 12c) (α = 0.7◦). Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 20 length L is 10.8 m, the shield mass Gshield is 3000 t, and the thickness of covering soil is 24 m. The β displacement of the dome is 5 mm, and the eccentricity of gravity ls is 3.5 m. MA = 20 MN · m and α MA = 75 MN · m. The ground reaction curve coefficient is shown in Table 1 [11]. Through the Matlab programming calculation, the shield attitude parameters could be calculated. As shown in Figure 12a, the weight of the shield machine was 3000 t, and △sv was β obtained (△sv=8.1 mm). In the second stage, the sum of MG and MA is 125 MN·m, then the pitch α angleAppl. Sci. β could2019, 9, be 1812 obtained (Figure 12b) (β=1.16°). In the third stage, it was found that MA= 75 MN·m,10 of 19 and then the yawing angle α could be obtained (Figure 12c) (α=0.7°).

TableTable 1. Ground reaction curve coefficient coeﬃcient table [[11].11].

Khmin KhminK hoKho KKhmaxhmax KKhh KvminKvmin Kvo KvoKvmax KvmaxKv Kv 0.30.3 11 5 5 33 MN/m MN/m3 0.30.3 1 1 5 3 5 MN/m33 MN/m3

(a) (b)

(c)

Figure 12. Calculation of shield attitude parameters: ( (aa)) Stage Stage 1, 1, solving solving △∆sv;;( (bb)) Stage Stage 2, 2, solving solving ββ;; and ( cc)) Stage Stage 3, 3, solving solving α..

4. Application Application Engineering Engineering 4.1. Engineering Details 4.1. Engineering Details Jinan is located in the central and western parts of Shandong Province, overlooking the Mount Jinan is located in the central and western parts of Shandong Province, overlooking the Mount Tai to the south and across the Yellow River to the north, as shown in Figure 13. Jinan is famous for Tai to the south and across the Yellow River to the north, as shown in Figure 13. Jinan is famous for its springs with complex groundwater network [25]. The underground spring water is extremely its springs with complex groundwater network [25]. The underground spring water is extremely precious for the city of Jinan, thus the protection of spring water during the construction of the subway precious for the city of Jinan, thus the protection of spring water during the construction of the is very important. Shield attitude control is the key to spring protection during shield construction. subway is very important. Shield attitude control is the key to spring protection during shield Therefore, Jinan Metro Line R2 Line was analyzed and discussed. construction. Therefore, Jinan Metro Line R2 Line was analyzed and discussed. Jinan Metro Line R2 is an east–west city express line. It is a rail transit backbone line that alleviates Jinan Metro Line R2 is an east–west city express line. It is a rail transit backbone line that east–west traﬃc pressure and supports the expansion of strip-shaped urban space. This study selected alleviates east–west traffic pressure and supports the expansion of strip-shaped urban space. This the Wangfuzhuang–Renjiazhuang section of Jinan Metro Line R2. The buried depth of the tunnel study selected the Wangfuzhuang–Renjiazhuang section of Jinan Metro Line R2. The buried depth of varies from 7.9 to 15.2 m. The maximum gradient of the tunnel is 12%, and it passes underneath important regions such as Beijing–Shanghai high-speed railway, Beijing–Fuzhou expressway and Yufu River. The advancement of shield machine usually encountered a silty clay soil and loess during the tunneling process in the studied section. The geological proﬁle of the encountered stratum is displayed in Figure 14. The construction process of Ring 504 segments was selected for analysis and research. The soil characteristic parameters of Ring 504 are shown in Figure 15. The third and fourth layers are the soil parameters of the shield excavation section. Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 20 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 20 the tunnel varies from 7.9 to 15.2 m. The maximum gradient of the tunnel is 12‰, and it passes underneaththe tunnel variesimportant from 7.9regions to 15.2 such m. Theas Beijing–maximumShanghai gradient high-speed of the tunnel railway, is 12‰, Beijing–Fuzhou and it passes expresswayunderneath and important Yufu River. regions The advancementsuch as Beijing– of shieShanghaild machine high-speed usually encountered railway, Beijing–Fuzhou a silty clay soil andexpressway loess during and Yufu the River.tunneling The advancement process in theof shie studiedld machine section. usually The encountered geological aprofile silty clay of soilthe encounteredand loess during stratum the is displayedtunneling inprocess Figure in14. theTh estudied construction section. process The ofgeological Ring 504 profilesegments of wasthe selectedencountered for analysis stratum and is displayedresearch. Thein Figure soil characteristic 14. The construction parameters process of Ring of 504Ring are 504 shown segments in Figure was selected for analysis and research. The soil characteristic parameters of Ring 504 are shown in Figure Appl.15. The Sci. 2019third, 9 ,and 1812 fourth layers are the soil parameters of the shield excavation section. 11 of 19 15. The third and fourth layers are the soil parameters of the shield excavation section.

Figure 13. Plan view of the district division and geology in Jinan [26]. FigureFigure 13. 13.Plan Plan viewview ofof thethe districtdistrict divisiondivision andand geologygeology inin JinanJinan [[26].26].

Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 20 FigureFigure 14.14. Geology Geology section section ofof Wangfuzhuang–RenjiazhuangWangfuzhuang–Renjiazhuang sectionsection ofof JinanJinan MetroMetro LineLine R2.R2.

0 FigureFigure 15. Geotechnical 15. Geotechnical profiles profiles of of Ring Ring 504. 504. γγ,, weight weight density;density;ω ,ω water, water content; content;K0, laterK , later pressure pressure coefficient;coefficient; c, cohesion;c, cohesion; φ,ϕ internal, internal friction friction angle; KKvv,, verticalvertical foundation foundation coefficient; coefficient;Kh, horizontalKh, horizontal foundationfoundation coefficient. coefficient.

4.2. Shield Details An earth pressure balanced shield (EPB) machine of 6.68 m in diameter was used to excavate in Jinan Metro Line R2. The length of shield is 9 m, and its weight is about 500 t. Cutter opening ratio is 35%, as shown in Figure 16a. The shield machine specifications are summarized in Table 2. The shield machine advance system uses 22 sets of cylinders, which are divided into four groups according to the position. The number of cylinders in Groups A and C is 4, and the number of cylinders in Groups B and D is 7. The cylinder section is shown in Figure 16b.

Table 2. Summary of the main specifications for the EPB.

Shield type EPB External diameter (m) 6.68 The length of shield (m) 9.0 Shield weight (t) Approximately 500 Shield eccentricity (m) Approximately 0.5 Cutter opening ratio (%) 35 Maximum total thrust (kN) 40,860 Number of propulsion cylinders 22 Maximum advance speed (mm/min) 80

Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 19

Figure 15. Geotechnical profiles of Ring 504. γ, weight density; ω, water content; K0, later pressure coefficient; c, cohesion; φ, internal friction angle; Kv, vertical foundation coefficient; Kh, horizontal foundation coefficient.

Shield type EPB 4.2. Shield Details External diameter (m) 6.68 An earth pressure balancedThe length shield of (EPB)shield machine(m) of 6.68 m in 9.0 diameter was used to excavate in Shield weight (t) Approximately 500 Jinan Metro Line R2. The length of shield is 9 m, and its weight is about 500 t. Cutter opening ratio is Shield eccentricity (m) Approximately 0.5 35%, as shown in Figure 16a. The shield machine speciﬁcations are summarized in Table2. The shield Cutter opening ratio (%) 35 machine advance system usesMaximum 22 sets total of cylinders, thrust (kN) which are divided 40,860 into four groups according to the position. The number ofNumber cylinders of propulsion in Groups cylinders A and C is 4, and the 22 number of cylinders in Groups B and D is 7. The cylinderMaximum section advance is shown speed in Figure (mm/min) 16b. 80

(a) (b)

FigureFigure 16. 16.Cutterhead Cutterhead and and advance advance system system of of shield: shield: (a )(a shield) shield cutterhead; cutterhead; and and (b )(b shield) shield cylinder cylinder partition. partition. Table 2. Summary of the main speciﬁcations for the EPB. 5. Results and Discussion Shield Type EPB 5.1. Engineering Application I: ExternalInversion diameterCalculation (m) of Soil-Shield Interaction 6.68 Force The length of shield (m) 9.0 The soil layer above the sectionShield of weight Ring 504 (t) is composed Approximatelyof backfill and loess, 500 which is a non- uniform clay stratum. AccordingShield to eccentricity the method (m) of this paper and Approximately in [27], initial 0.5 force acting on the shield periphery can be calculatedCutter in opening the non-uniform ratio (%) clay stratum. As shown 35 in Figure 15, the section of Ring 504 is also a non-uniformMaximum layer. total The thrust loess (kN) layer and silty clay 40,860each account for about half of the entire excavation surfaceNumber andof can propulsion be approximated cylinders by 1:1. 22 Through the guidingMaximum system advance and data speed monitoring (mm/min) system of the EPB, 80 when the shield had been advancing at Ring 504, partition cylinder thrust, pitch angle and yawing angle were recorded every 5. Results1 min. The and time Discussion to excavate Ring 504 was about 50 min, but for various reasons, it was interrupted

5.1. Engineering Application I: Inversion Calculation of Soil-Shield Interaction Force The soil layer above the section of Ring 504 is composed of backﬁll and loess, which is a non-uniform clay stratum. According to the method of this paper and in [27], initial force acting on the shield periphery can be calculated in the non-uniform clay stratum. As shown in Figure 15, the section of Ring 504 is also a non-uniform layer. The loess layer and silty clay each account for about half of the entire excavation surface and can be approximated by 1:1. Through the guiding system and data monitoring system of the EPB, when the shield had been advancing at Ring 504, partition cylinder thrust, pitch angle and yawing angle were recorded every 1 min. The time to excavate Ring 504 was about 50 min, but for various reasons, it was interrupted during the tunneling process. Continuous monitoring data were more valuable and informative. The longest continuous boring time of the ring was about 15 min. To avoid the influence of shield start and stop, this study selected continuous monitoring data of 10 min in the middle of this time. The data for a period were selected for analysis and comparison. The measured data of the pitch angle and the yawing angle are shown in Figure 17. The yawing angle varied greatly, while the pitch angle remained basically stable. Appl. Sci. 2019, 9, 1812 13 of 19

Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 20

FigureFigure 17. Pitch 17. Pitch angle angle and and yawingyawing angle angle measured measured data. data.

The pitch angle and the yawing angle had been known through the guiding system. According to the ﬂow chart of the shield attitude calculation (Figure 11), the force of shield–soil interaction during the tunneling process can be inverted. The calculation results are shown in Figure 18. Figure 18 shows the distribution of the total normal earth pressure around the shield periphery. The height of graphs represents the length of shield, the width of graphs represents the shield periphery from 0◦ to 360◦. Along the line at 180◦, it can be found that the normal pressure is gradually decreasing from the bottom to the top of graphs, indicating that the pitch angle of the shield is positive. The gradient of the normal pressure reduction reﬂects the magnitude of the pitch angle. The stress nephogram on the left or right Figure(a) 17. Pitch angle and yawing angle measured(b) data. of the line at 180◦ shows the direction of the yawing angle of the shield.

(a) (b)

(e) (f)

Figure 18. Cont. Appl. Sci. 2019, 9, 1812 14 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 20

(g) (h)

(i) (j)

Figure 18.FigureShield 18. Shield stress stress nephogram nephogram ateach at each moment: moment: ( a(a)) 1 1 min; min; ((b) 2 min; min; (c (c) )3 3min; min; (d ()d 4) min; 4 min; (e) (5e ) 5 min; (f) 6 min;min; (g )(f 7) 6 min; min; ((hg) 7 8 min; min; (h ()i )8 9min; min; (i) and9 min; (j )and 10 ( min.j) 10 min. 5.2. Engineering Application II: Shield Pitch Angle Prediction The change of the pitch angle of the shield is small, and the fluctuation of the stress nephogram is mainly dueThe to thesecond change engineering of the application yawing angle. is the predicti The stresson of the nephogram pitch angle. Combined can be used with to the preliminarily Jinan Metro Line R2, when the advance force applied by the four cylinder partitions is known, the attitude determine the degree and the direction of deflection on the horizontal plane of the shield. The greater change of the shield can be predicted in advance. The soil mass Wsoil with the same volume as the is the yawingshield is angle, about the600 t, more which severe is larger is than the distortionthe self-weight of the of the stress shield nephogram. machine. Therefore, For example, under the there was a suddenpremise increase of not of applying the yawing any angleexternal between force, there time will= 4be min a phenomenon and time = of5 “upward min, as canfloating”, be seen that from the stress nephogram.is, Δsv < 0. The first step is to calculate Δsv (Δsv =-13 mm), according to the method in Figure 11. β The MA actively applied by the EPB can be calculated by the thrust difference between Group 5.2. EngineeringA and Group Application C of the advance II: Shield system. Pitch AngleThe M PredictionG is 2.5 MN·m, which can be calculated. The change β β curve of the M +MG can be obtained during construction, as shown in Figure 19. When M +MG The second engineeringA application is the prediction of the pitch angle. Combined withA the Jinan and Δsv are known, the theoretical values of the shield pitch angle during the study period can be Metro Line R2, when the advance force applied by the fourβ cylinder partitions is known, the attitude obtained. The theoretical values are calculated by the MA +MG. Therefore, the trend of theoretical change of the shield can be predicted in advance.β The soil mass Wsoil with the same volume as the values should be similar to the trend of the MA +MG. However, the measured values did not have shield is about 600 t, which is largerβ than the self-weight of the shield machine. Therefore, under the this trend, indicating that the MA +MG did not fully act on shield attitude adjustment. premise of notFigure applying 20 is a comparative any external analysis force, of there the meas willured be a and phenomenon predicted values of “upward of the shield floating”, pitch that is, ∆sv < 0.angle The during first step the study is to calculateperiod. It can∆s bev ( seen∆sv in= the13 figure mm), that according the theoretical to the value method is generally in Figure larger 11 . β − ThethanM theactively measured applied value, indicating by the EPB that canthe moment be calculated actively byapplied the thrustby the shield difference through between the jack Group cylinderA cannot fully act on the shield itself. When the advance cylinder acts on the shield, the friction A and Group C of the advance system. The M is 2.5 MN m, which can be calculated. The change at the joint and the characteristics of the shieldG itself may cause· the deflection moment to decrease, β + β + curve ofresulting the MA in aM smallerG can pitch be obtained angle actually during being construction, formed. as shown in Figure 19. When MA MG and ∆sv are known, the theoretical values of the shield pitch angle during the study period can be β + obtained. The theoretical values are calculated by the MA MG. Therefore, the trend of theoretical β + values should be similar to the trend of the MA MG. However, the measured values did not have this β + trend, indicating that the MA MG did not fully act on shield attitude adjustment. Figure 20 is a comparative analysis of the measured and predicted values of the shield pitch angle during the study period. It can be seen in the figure that the theoretical value is generally larger than the measured value, indicating that the moment actively applied by the shield through the jack cylinder cannot fully act on the shield itself. When the advance cylinder acts on the shield, the friction at the joint and the characteristics of the shield itself may cause the deflection moment to decrease, resulting in a smaller pitch angle actually being formed. Appl. Sci. 2019, 9, 1812 15 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 20 Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 20

β Figure 19. The change graph of Mββ +MG. A++ G. FigureFigure 19.19. The change graph of MMAA MG.

Figure 20. Contrast between the measured and the theoretical value of the shield pitch angle. Figure 20. Contrast between the measured and thethe theoreticaltheoretical valuevalue ofof thethe shieldshield pitchpitch angleangle 5.3. Engineering Application III: Shield Yawing Angle Prediction 5.3. Engineering Application III: Shield Yawing Angle Prediction The third engineering application is the prediction of the yawing angle. The ∆sv is calculated The third engineering application is the prediction of the yawing angle. The Δsv is calculated in in theThe first third stage. engineering The theoretical application value is ofthe the prediction pitch angle of the at yawing each moment angle. The is shown Δsv is calculated in Figure 20in. Thethe first known stage.∆s vTheand theoreticalβ are brought value into of thethecalculation pitch angle process at each of moment the third is stageshown to in obtain Figure the 20. RCYH The known Δsv and β αare brought into the calculation process of the third stage to obtain the RCYH (Figure 21).v The M actively applied by the EPB can be calculated by the thrust difference between known Δs and β areA brought into the calculation process of the third stage to obtain the RCYH Group(Figure B21). and The Group MA D actively of the applied advance by system. the EPB The can change be calculated curve ofby the theM thrustα is shown difference in Figure between 22. (Figure 21). The MA actively applied by the EPB can be calculated by the thrustA difference between Group B and Group D of the advance system. The change curve of the MA is shown in Figure 22. Then,Group the B and theoretical Group value D of ofthe the advance yawing system. angle can The be obtained.change curve The comparisonof the MA isbetween shown the in theoreticalFigure 22. valueThen, andthe thetheoretical measured value value of is the shown yawing in Figure angle 23 can. The be trends obtained. of theoretical The comparison and measured between values the aretheoretical both similar value to and the trendthe measured of the Mα .value The M isα showis obtainedn in Figure by measurement. 23. The trends It indicates of theoretical that the andMα theoretical value and the measuredA value isA shown in Figure 23. The trends of theoretical andA measured values are both similar to the trend of the MA. Theβ MA is obtained by measurement. It measuredapplied by values the shield are canboth act similar more to fully the on trend the shieldof the thanMA. MThe. MA is obtained by measurement. It A β β indicates that the MA applied by the shield can act more fully on the shield than MA. indicatesComparing that the the MA measured applied valuesby the withshield the can theoretical act more valuesfully on in the Figure shield 23 ,than it can M beA. found that the trendComparing of the theoretical the measured value is values basically with similar the theoretical to the measured values in value,Figure but 23, theit can theoretical be found valuethat the is trendgenerally of the larger theoretical than the value measured is basically value. similar In the to calculation thethe measuredmeasured of the value,value, theoretical butbut the value, theoretical the deflection value is generallymoment value larger is than the initialthe measured data, and value. the moment In the ca usedlculationlculation forthe ofof thethe shield theoreticaltheoretical correction value,value, is reduced thethe deflectiondeflection due to momentmechanical value loss, is etc., the soinitial that data, the final and yawing the moment angle formedused for by the the shield shield correction is small. is reduced due to mechanical loss, etc., so that the final yawing angle formed by the shield is small. Appl. Sci. 2019, 9, 1812 16 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 17 of 20 Appl. Sci. 2019,, 9,, xx FORFOR PEERPEER REVIEWREVIEW 17 of 20

α Figure 21. The relationship between the yawing angle and MαA. Figure 21. The relationship between thethe yawingyawing angleangle andand MαAα.. Figure 21. The relationship between the yawing angle and MAA.

Figure 22. The change graph of Mα . Figure 22. The change graph of MMααA.. Figure 22. The change graph of MAA .

Figure 23. Contrast between the measured and the theoreticaltheoretical value of the shield yawing angle. Figure 23. Contrast between the measured and the theoretical value of the shield yawing angle. 6. Conclusions 6. Conclusions Appl. Sci. 2019, 9, 1812 17 of 19

6. Conclusions This study revealed the mechanism of the interaction between shield and soil during the change of shield attitude. Through theoretical analysis, programming calculation and ﬁeld monitoring, the calculation method of the pitch angle and yawing angle of the shield was studied and veriﬁed. The main achievements of this study are outlined as follows:

(1) Based on the ground reaction force curve, the interaction between the shield and the soil was simulated by the equivalent spring, and the theoretical calculation method of f5 was obtained in the change of the shield attitude. (2) The improved calculation method of loose earth pressure solved the initial boundary problem of the shield attitude calculation. Combined with the theoretical calculation method of f5, the calculation process of the shield attitude was formed. (3) Based on the monitoring data of Jinan Metro Line R2, the interaction between the shield and the soil during the construction of the shield was inverted. Through the stress nephogram, the stress concentration area of the shield can be judged to guide the next step of attitude adjustment. (4) Based on the measured data of the project, the theoretical calculation method of the pitch angle and yawing angle of the shield was verified. The study found that the theoretical value was close to the measured value, but, since the moment generated by the advance cylinders cannot fully act on the shield itself, the theoretical values of the shield attitude angles were generally greater than the measured values. (5) The model of shield–soil interaction has some benefits for upcoming projects. Firstly, this model can guide the shield operator to perform shield attitude correction. Secondly, it is possible to initially obtain a position where the attitude control of any shield tunnel construction is difficult for upcoming projects.

Author Contributions: X.S., D.-J.Y. and D.-L.J. contributed equally to this work. Funding: The research work described herein was funded by the National Basic Research Program of China (973 Program: 2015CB057800) and Fundamental Research Funds for the Central Universities of China (NO. 2017YJS151). Acknowledgments: The research work described herein was funded by the National Basic Research Program of China. This financial support is gratefully acknowledged. Conflicts of Interest: The authors declare no conflict of interest.

Notations

Symbol Description Unit Symbol Description Unit f 1 Shield self-weight kPa f 2 Shield tail load kPa f 3 Advance load provided by the jacks kPa f 4 Load acting on the cutterhead kPa f 5 Force acting on the shield periphery kPa (x0, y0, z0) cutterhead center coordinates m α Yawing angle ◦ β Pitch angle ◦ Ω Rolling angle ◦ C Coordinate system [-] F Total force kN M Total moment MN m · σv Loose earth pressure kPa K0 Later pressure coefficient [-] H Overlying soil thickness m P0 Upper load kPa c Cohesion kPa ϕ Internal friction angle ◦ 3 γ Weight density kN/m B1 Loose band width m R Radius of the tunnel m A1 Characteristic parameters [-] p1 Initial soil pressure upper value kPa p2 Initial earth pressure lower value kPa D Shield diameter m L Shield length m ls Shield eccentricity m ∆s Shield displacement m M U Shield shell unit displacement m η Angle with the p-axis in the C ◦ K coefficient of earth pressure [-] a the gradient of functions K(U) [-] σ Stress kPa A Area m2 Superscripts E Global coordinate system [-] M Machine coordinate system [-] MR Rotated coordinate system [-] b Different coordinate systems [-] Appl. Sci. 2019, 9, 1812 18 of 19

Subscripts h, v Horizontal and vertical directions [-] e Diﬀerent directions of force [-] A Shield apply [-] max Maximum [-] min Minimum [-] o Origin or initial [-] i, j ith and jth calculation points [-] s Final [-]

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