Active Earth Pressure of Limited C-Φ Soil Based on Improved Soil Arching Eﬀect

applied sciences

Article Active Earth Pressure of Limited C-ϕ Soil Based on Improved Soil Arching Eﬀect

Meilin Liu 1,2,* , Xiangsheng Chen 2,3, Zhenzhong Hu 1,4 and Shuya Liu 2

1 Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China; [email protected] 2 Shenzhen Metro Group Co., Ltd., Shenzhen 518026, China; [email protected] (X.C.); [email protected] (S.L.) 3 College of Civil and Transportation Engineering, Shenzhen University, Shenzhen 518060, China 4 Department of Civil Engineering, Tsinghua University, Beijing 100084, China * Correspondence: [email protected]; Tel.: +86-1312-039-9806

Received: 14 April 2020; Accepted: 4 May 2020; Published: 7 May 2020

Featured Application: This study extends the previous active earth pressure studies to include c-ϕ soil of limited width in a dense underground environment by considering the soil-wall interaction and the ground surface overload.

Abstract: For c-ϕ soil formation (cohesive soil) of limited width with ground surface overload behind a deep retaining structure, a modified active earth pressure calculation model is established in this study. And three key issues are addressed through improved soil arching effect. First, the soil-wall interaction mechanism is determined by considering the soil arching effect. The slip surface of a limited soil is proved to be a double-fold line passing through the retaining wall toe and intersecting the side wall of the existing underground structure until it reaches the ground surface along the existing side wall. Second, the limited width boundary is explicated. And third, the variation in the active earth pressure from parameters of limited c-ϕ soil is determined. The lateral active earth pressure coefficient is nonlinear distributed based on the improved soil arching effect of the symmetric catenary curve. Furthermore, the active earth pressure distribution, the tension crack at the top of the retaining wall and the resultant force and its action point were obtained. By comparing with the existing analytical methods, such as the Rankine method, it demonstrates that the model proposed in this study is much closer to the measured and numerical results. Ignoring the influence of soil cohesion and the limited width will exponentially reduce the overall stability of the retaining structure and increase the risk of accidents.

Keywords: active earth pressure; c-ϕ soil; limited width; limit equilibrium method; soil arching eﬀect

1. Introduction As the development of construction technology and density of the underground environment, the surrounding soil behind a deep retaining wall is often constrained by many underground structures. For example, a large number of underground projects, including but not limited to deep foundation pits, slope protection, and underground pipe corridors, occur near existing large-rigid underground structures, such as an existing deep basement or an operating subway station, as shown in Figure1. The force and deformation of soil behind the deep retaining structure cannot pass through the side wall of an adjacent existing large rigid underground structure. Thus, a soil mass with a limited width is formed. The stress and strain of the limited soil are significantly different from those of a semi-infinite soil [1–4]. In this case, a reasonable soil of limited width between the newly constructed deep retaining

Appl. Sci. 2020, 10, 3243; doi:10.3390/app10093243 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 18

Appl. Sci. 2020, 10, 3243 2 of 17 from those of a semi-infinite soil [1–4]. In this case, a reasonable soil of limited width between the newly constructed deep retaining wall and the existing underground structure must be scientifically considered.wall and the And existing the undergroundtraditional Rankine structure method must beor scientiﬁcallyCoulomb method considered. is not Andsuitable. the traditional It is still unclearRankine whether method orthe Coulomb effect of methodthe existing is not large-ri suitable.gid It underground is still unclear structure whether the on etheﬀect clipped of the existing soil of limitedlarge-rigid width—and underground thus on structure the retaining on the structure—is clipped soil positive of limited or width—andnegative, especially thus on when the retaining the soil conditionstructure—is is not positive simply or dry negative, sand. especially when the soil condition is not simply dry sand.

FigureFigure 1. 1. AA deep deep retaining retaining wall wall besides besides a a la large-rigidrge-rigid underground underground structure.

Many studies investigated the the active active earth pressure distribution behind a a deep retaining wall with limited c-φϕ soil (cohesive soil) [[1–17]1–17] throughthrough multiplemultiple kindskinds ofof methods.methods. Centrifuge model teststests and and field field measurements [5–10] [5–10] have been conducted to analyze the active earth pressure distribution of of different different limited soil widths, provid providinging a a great great deal of valuable data for the future analysis. Numerical Numerical solutions solutions [11–14] [11–14] have have also also been implemented due to their comprehensiveness inin considering multiple multiple influencing influencing factors factors if if a su suitableitable mechanical model and boundary conditions are considered carefully. Theoretical Theoretical methods methods [15– [15–19]19] have have been been introduced introduced to to ensure ensure that that the resultsresults ofof the the field field measurements, measurements, model model tests, tests, and numerical and numerical simulations simulations are scientifically are scientifically expressed expressedand conveniently and conveniently applied. Scholars applied. [ 7Scholars,15,17–19 [7,15,17–19]] have incorporated have incorporated the limited the soil limited width soil into width their intocalculations their calculations of the active of the earth active pressure. earth Thepressure active. The earth active pressure earth gradually pressure gradually increases with increases increasing with increasingof limited soilof limited width. However,soil width. these However, methods these do not methods consider do the not influence consider of the soil influence cohesion [of20 –soil26] cohesionand ground [20–26] surface and overload ground [surface27–29], overload which are [27–29], important which factors are inimportant the design factors of a retaining in the design structure. of a retainingFurther, these structure. methods Further, did not these specify methods theexplicit did notrange specify of the limited explicit soil range width. of limited soil width. A proper analytical method is needed for calcul calculatingating the the active earth pressure acting on the retainingretaining wall. The The limit limit equilibrium equilibrium method method of of a a horizontal horizontal soil soil strip strip [30–35] [30–35] has has been widely used because of itsits convenienceconvenience and and briefness briefness in in practical practical applications. applications. And And the the methods methods for for calculating calculating the thelateral lateral coe fficoefficientcient of active of active earth earth pressure pressure include include moment moment equilibrium equilibrium method method [36, 37[36,37]] and and the soilthe soilarching arching method method [38–40 [38–40].]. The moment The moment equilibrium equilibrium method doesmethod not considerdoes not theconsider force transmitted the force transmittedby the soil arching, by the soil which arching, leads towhich the calculationleads to the results calculation being results inconsistent beingwith inconsistent the actual with values. the actualThe soil values. arching The eff ectsoil is arching a common effect phenomenon is a common caused phenomenon by the transfer caused of by internal the transfer stress, suchof internal as the stress,earth pressure such as the decreasing earth pressure in the yield decreasing region andin the increasing yield region in the and adjacent increasing region in the [41]. adjacent And it isregion more [41].notable And in it the is more soil of notable limited in width the soil between of limited two width wall faces. between Thus, two the wall soil faces. arching Thus, method the soil is used arching and methodimproved is inused the limitand equilibriumimproved in method the limit of aequilibrium horizontal soilmethod strip forof a calculating horizontal the soil active strip earth for calculatingpressure of the limited active soil earth in this pressure study. of limited soil in this study. Two main main key key problems problems about about stress stress redistribution redistribution in the in soil the arching soil arching method method include include the stress the directionstress direction deflection deflection and the and stress the stress transfer transfer path path [39], [ 39and], and they they are are all all reflected reflected by by the the uneven deformation. The keykey problemsproblems are are determined determined by by the the slip slip surface surface shape shape and theand small the small principal principal stress stresstrajectory trajectory [9,42]. Di[9,42].fferent Different types of types slip surface of slip shape surface have shape been proposedhave been by proposed theoretical by assumptions theoretical assumptionsand model tests, and suchmodel as planetests, such [43,44 as], cycloidplane [43,44], [45], logarithmic cycloid [45], spirals logarithmic [46], and spirals polylines [46], [15 and,16]. polylinesRankine plane [15,16]. slip Rankine surface plane [1,29,43 slip,44 surface] is widely [1,29,43,44] used in theis widely traditional used soil in the arching traditional method. soil However, arching method.the Rankine However, plane slip the surfaceRankine is onlyplane suitable slip surface for soil is ofonly semi-infinite suitable for width soil of and semi-infinite for a smooth width retaining and forwall. a Moreover,smooth retaining the formulas wall. of Moreover, cycloid and the logarithmic formulas spiralsof cycloid are tooand complex logarithmic to be usedspirals in practice.are too complexMany kinds to be of used arching in practice. curves for Many simulating kinds of the arching small principalcurves for stress simulating trajectory, the suchsmall as principal arc [39], stresscatenary trajectory, [17], and such parabola as arc [ 47[39],] have catenary been proposed.[17], and parabola There is still[47] no have evidence been proposed. to suggest There which is kinds still Appl. Sci. 2020, 10, 3243 3 of 17 of slip surface shape and small principal stress trajectory are closer to the stress redistribution of the limited c-ϕ soil behind a retaining wall. There are still no systematic studies to evaluate the active earth pressure for the limited c-ϕ soil between a deep retaining wall and an existing large-rigid underground structure when there is ground surface overload. The purpose of this study is to extend the previous studies to include the limited width of c-ϕ soil in a dense underground environment and propose an improved soil arching method. In addition, explore the minimum and maximum boundary of limited soil width and to clarify the variation in the active earth pressure with the properties of limited c-ϕ soil and the soil-wall interaction mechanism. It is necessary to account for various factors as comprehensively as possible, especially when the ground conditions are poor, and the safety level of the existing structures is high. The results can be used to guide the design of a deep retaining structure in dense underground environment, which would lower costs and increase safety.

2. Theoretical Methods

2.1. Basic Assumptions It is common for a deep retaining wall to be constructed near an existing large rigid underground structure in a dense underground environment. For the convenience of theoretical analysis and focus on active earth pressure of limited c-ϕ soil, the following assumptions were made in this study.

1. The calculation model is suitable for plane strain conditions. 2. Saturated unit weight of soil is used to consider the influence of underground water. 3. Both retaining wall surface next to the soil and the existing wall surface are vertical and the ground surface is horizontal. 4. The retaining wall movement is translational and the soil behind it is in the active limit state. 5. The isotropic, homogeneous soil obeys the Mohr-Coulomb failure criterion. 6. The rigidness of the existing underground structure is sufficiently large. 7. The interaction between the retaining wall and soil is the same as that between the existing side wall and soil. 8. The burial depth of the existing underground structure is deeper than that of the influence range of retaining wall, i.e., the influence of the basal pressure of existing underground structure can be neglected.

2.2. Slip Surface For soil of limited width, there are two kinds of horizontal soil strips between the retaining wall and the existing underground structure. AB is the horizontal soil strip from the retaining wall to the side wall of existing underground structure, and CD is the horizontal soil strip from the retaining wall to the soil slip surface. AB and CD with an extremely small thickness dz are shown in Figure2, which are behind the retaining wall at an arbitrary burial depth z. AB disappears for semi-inﬁnite soil or when the slip surface does not intersect the existing structure. The width of AB is B and the width of CD depends on the slip surface shape and its position. And they can be expressed by the superposition of the inclined angle β of the slip surface. The parameter t in Figure2 indicates the ratio of limited soil width. t equals to Btanβ/h when the slip surface is a plane, in which β is the slip surface angle with respect to the horizontal, and h is the depth of retaining wall. βz equals to β when the slip surface is a plane. However, βz changes with the burial depth z when the slip surface is curved. In this case, t needs to be calculated using the integral of cotβz. This study mainly focuses on 0 t 1. ≤ ≤ As the distance x of an arbitrary point E on the horizontal soil strip AB or CD from the retaining wall increases, the angle between the principal stress direction and the horizontal direction θ gradually increases from θw at the retaining wall surface (point A and C) to 90◦. Then it gradually decreases to Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 18

Appl. Sci. 2020, 10, 3243 4 of 17

θw at the existing side wall at point B because of the same soil-wall friction angle with the retaining wall. When x tends to B, θ tends to θs for horizontal soil strip CD. θs equals to θw to make the integral ofAppl. the Sci. horizontal 2020, 10, x FOR shear PEER stress REVIEW to be 0. 4 of 18

Figure 2. Stress analysis of horizontal soil strip within the slip wedge.

The parameter t in Figure 2 indicates the ratio of limited soil width. t equals to Btanβ/h when the slip surface is a plane, in which β is the slip surface angle with respect to the horizontal, and h is the depth of retaining wall. βz equals to β when the slip surface is a plane. However, βz changes with the burial depth z when the slip surface is curved. In this case, t needs to be calculated using the integral of cotβz. This study mainly focuses on 0 ≤ t ≤ 1. As the distance x of an arbitrary point E on the horizontal soil strip AB or CD from the retaining wall increases, the angle between the principal stress direction and the horizontal direction θ gradually increases from θw at the retaining wall surface (point A and C) to 90°. Then it gradually decreases to θw at the existing side wall at point B because of the same soil-wall friction angle with the retaining wall. WhenFigure x 2. tendsStress analysisto B, θ oftends horizontal to θs soilsoforil strip horizontal within thethe soil slipslip strip wedge.wedge. CD. θs equals to θw to make the integral of the horizontal shear stress to be 0. TheTheThe analysis analysisparameter of of any t any in pointFigure point E E2 onindicates on the the horizontal the ratio soilofsoil limited stripstrip AB ABsoil and andwidth. CD CD t using equalsusing the thetoMohr B Mohrtanβ stress/h stresswhen circle thecircle waswasslip implemented, surface implemented, is a plane, as as shown shown in which in in Figure Figureβ is the 33. .slip TheThe surface stressesstresses angle σσ andand withτ τalso respectalso transform transform to the horizontal, gradually gradually fromand from h A is toA the to B B andanddepth from from of C retaining Cto to D, D, respectively. respectively. wall. βz equals There There to β is when an angleangle the slip ofofθ θsurfaceββ betweenbetween is a the plane.the principal principal However, stress stress βz directionchanges direction with and and thethe the verticalverticalburial ofdepth of slip slip zsurface when surface the at at slipthe the surfaceactive state. state.is curved. θθββ equalsequals In this to case, ((ππ//4+4 t+ needsϕφ//2)2) accordingaccording to be calculated to to the the Mohrusing Mohr stressthe stress integral circle circle analysis.of cotβz. This From study Figure mainly2 we canfocuses obtain on that 0 ≤ t θ≤s 1. θβ = π/2 βz. analysis. From Figure 2 we can obtain that θs − θβ = π/2 −− βz. As the distance x of an arbitrary point E on the horizontal soil strip AB or CD from the retaining wall increases, the angle between the principal stress direction and the horizontal direction θ gradually increases from θw at the retaining wall surface (point A and C) to 90°. Then it gradually decreases to θw at the existing side wall at point B because of the same soil-wall friction angle with the retaining wall. When x tends to B, θ tends to θs for horizontal soil strip CD. θs equals to θw to make the integral of the horizontal shear stress to be 0. The analysis of any point E on the horizontal soil strip AB and CD using the Mohr stress circle was implemented, as shown in Figure 3. The stresses σ and τ also transform gradually from A to B and from C to D, respectively. There is an angle of θβ between the principal stress direction and the vertical of slip surface at the active state. θβ equals to (π/4+φ/2) according to the Mohr stress circle analysis. From Figure 2 we can obtain that θs − θβ = π/2 − βz. FigureFigure 3. MohrMohr stress circlecircle analysis.analysis.

σ (σ = γz + q) and σ (σ = K σ ) are separately the major and the minor principal stress in σ1 (1σ1 =1 γz+q) and σ3 (σ3 =3 Ka3σ1) area separately1 the major and the minor principal stress in Figure 2 Figures2 and3, where γ is the soil saturated unit weight, q is the ground surface overload, Ka is the and 3, where γ is the soil saturated unit weight, q is the ground surface overload, Ka is the coefficient coefficient of the active earth pressure, and ϕ is the soil internal friction angle. The deflection angle of of the active earth pressure, and φ is the soil internal friction angle. The deflection angle of the minor the minor principal stress direction θw at depth z of the c-ϕ soil is deduced from Figure3 as: principal stress direction θw at depth z of the c-φ soil is deduced from Figure 3 as: [ + + ( + )] ππδδ 1 sin δ γγϕz +−q c cos ϕ +cw cot δ ()1 +sinϕϕ θw = + 1 arcsin sinzqc− cos cw cot 1 sin , (1) θ =+−2 2 − 2arcsin sin ϕ(γz + q + c cot ϕ) , (1) w 222 sinϕγ()zqc++ cot ϕ where c is the soil cohesion, cw is the soil-wall cohesion, and δ is the soil-wall friction angle. When where c is the soil cohesion, cw is the soil-wall cohesion, and δ is the soil-wall friction angle. When δ = δ = 0, θw is equal to π/2, and theFigure soil arching 3. Mohr e ffstressect disappears. circle analysis. In general, if cwcotδ = ccotϕ, then 0, θθww =is [equalπ + δ toarcsin(sin π/2, andδ /thesinϕ soil)]/2, arching which is effect consistent disappears. with Rao’s In [general,26] results. if cwcotδ = ccotφ, then θw = − [π+δ-arcsin(sinAccordingσ1 (σ1 = γδz/sin+ toq) and Figureφ)]/2, σ3 3which( σand3 = K Equationa σis1 )consistent are separately (1), the with slip the Rao’s surface major [26] angleand results. theβ ofminor c-ϕ soil principal at depth stressz of in horizontal Figure 2 soiland strip3, where CD isγ is the soil saturated unit weight, q is the ground surface overload, Ka is the coefficient of the active earth pressure, and φ is the soil internal friction angle. The deflection angle of the minor π ϕ δ 1 sin δ[(γz + q) c cos ϕ + cw cot δ(1 + sin ϕ)] principal stressβ directionz = + θw− at +deptharcsin z of the c-φ soil is− deduced from Figure 3 as: , (2) 4 2 2 sin ϕ(γz + q + c cot ϕ) πδ δ γϕ+− +δ () + ϕ 1 sinzqc cos cw cot 1 sin θ =+−arcsin , (1) w 222 sinϕγ()zqc++ cot ϕ

where c is the soil cohesion, cw is the soil-wall cohesion, and δ is the soil-wall friction angle. When δ = 0, θw is equal to π/2, and the soil arching effect disappears. In general, if cwcotδ = ccotφ, then θw = [π+δ-arcsin(sinδ/sinφ)]/2, which is consistent with Rao’s [26] results. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 18

According to Figure 3 and Equation (1), the slip surface angle β of c-φ soil at depth z of horizontal soil strip CD is

πϕδ− δ ()γϕ+− +δ ( + ϕ ) 1 sinzq c cos cw cot 1 sin β =+ +arcsin , (2) z 422 sinϕγ()zqc++ cot ϕ thus,Appl. β Sci.z is2020 not, 10 affected, 3243 by B, but increases with increasing γ, q, and φ. βz is a function of φ and5 ofδ 17and the slip surface is a plane when cw = ccotφ/cotδ. βz is equal to (π/4 + φ/2) when δ = 0, which is consistent with Rankine's slip surface angle, and the soil arching effect disappears. Furthermore, βz is thus, βz is not affected by B, but increases with increasing γ, q, and ϕ. βz is a function of ϕ and δ and the φ/2 and the slip surface coincide with the retaining wall when δ = φ. slip surface is a plane when cw = ccotϕ/cotδ. βz is equal to (π/4 + ϕ/2) when δ = 0, which is consistent The relationship between cw and δ are discussed to get clear the soil-wall interaction with Rankine’s slip surface angle, and the soil arching effect disappears. Furthermore, βz is ϕ/2 and the mechanism.slip surface Here coincide we assume with the that retaining cw and wall δ are when independentδ = ϕ. from each other; thus, the influence of cw and δ on β is shown in Figure 4a,b, respectively, in which γ = 19 kN/m3, q = 0, φ = 37°, δ = 25°, and c = The relationship between cw and δ are discussed to get clear the soil-wall interaction mechanism. 30 kPa, then c×cotφ/cotδ = 18.564 kPa. When β = 90°, cw = 23.118 kPa. Comparison of Figure 4a,b, it Here we assume that cw and δ are independent from each other; thus, the influence of cw and δ on β is shows that the influence of δ on the curvature of the 3slip surface can offset the effect of cw and β shown in Figure4a,b, respectively, in which γ = 19 kN/m , q = 0, ϕ = 37◦, δ = 25◦, and c = 30 kPa, then independentsc cotϕ/cotδ of= 18.564z. Then kPa. the Whenslip surfaceβ = 90 is, caw plane,= 23.118 such kPa. as Comparisoncw = 18.564 kPa of Figurein Figure4a,b, 4a it and shows δ = that 25° in × ◦ Figurethe influence 4b. of δ on the curvature of the slip surface can offset the effect of cw and β independents of z. Then the slip surface is a plane, such as cw = 18.564 kPa in Figure4a and δ = 25◦ in Figure4b.

(a) (b)

FigureFigure 4. InfluenceInﬂuence ofof ((a)a) ccww and ((b)b) δδ onon thethe slipslip surface. surface.

ForFor the the limit limit equilibrium equilibrium method method of a horizontal soilsoil strip,strip, the the horizontal horizontal force force is is not not disturbed disturbed whenwhen the the burial burial depth depth zz isis deeper deeper than thethe wallwall depthdepthh h.. AfterAfter this,this, the the slip slip surface surface needs needs to to pass pass through the wall toe. Furthermore, when c cotδ = ccotϕ, the double-fold line slip surface for limited through the wall toe. Furthermore, when cwwcotδ = ccotφ, the double-fold line slip surface for limited soilsoil is is aa planeplane passing passing through through the wallthe toewall and toe intersecting and intersecting the side wall the of side the existingwall of underground the existing undergroundstructure, then structure, till the ground then till surface the ground along the surface existing along wall. the The existing inclination wall. angle The inclination of the double-fold angle of line slip surface for the limited c-ϕ soil can be simpliﬁed as: the double-fold line slip surface for the limited c-φ soil can be simplified as:   ππ ; (0 z (h B tan β))  2 ()≤≤() − β β =  π ϕ δ ≤ 1 ≤ − sin δ ; 0zhB tan . (3)  2 + − + arcsin ; ((h B tan β) < z h) β =  4 2 2 sin ϕ − ≤ . (3) πϕδ− 1sin δ  ++arcsin ;()()hB − tan β <≤ z h Figure5 shows the comparison 422 of the double-fold sinϕ line slip surface from Equation (3) with the results obtained from centrifuge model tests [6]. The results of this study are consistent with those obtainedFigure from 5 shows centrifuge the comparison model tests. of Therefore, the double-fold by setting linecw =slipc surfacetanδ/tan fromϕ, the Equation soil-wall interaction(3) with the × resultscan be obtained determined from by centrifugeδ and the model calculation tests of[6]. active The results earth pressure of this study distribution are consistent based on with the slipthose obtainedsurface isfrom simpliﬁed. centrifugecw =modelc tan tests.δ/tan Therefore,ϕ is assumed by insetting the following cw = c×tan calculationδ/tanφ, the procedure. soil-wall interaction × can be determined by δ and the calculation of active earth pressure distribution based on the slip surface is simplified. cw = c×tanδ/tanφ is assumed in the following calculation procedure. Appl. Sci. 2020, 10, 3243 6 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 18

Figure 5. Comparison of slip surface with centrifuge model tests.

2.3. Soil Arching Analysis Based onon thethe aboveabove research,research, the the trapezoidal trapezoidal limited limitedc- ϕc-φsoil soil slip slip wedge wedge and and the the stresses stresses acting acting on theon the horizontal horizontal soil soil strip strip at depth at depthz are z are shown shown in Figure in Figure2. The 2. The soil slipsoil slip wedge wedge is surrounded is surrounded by the by red the linesred lines delimiting delimiting the ground the ground surface surface ab, the existingab, the sideexisting wall side segment wall bc,segment the slip bc, surface the slip segment surface cd, andsegment the retaining cd, and the wall retaining segment wall ad. segment The external ad. The forces exte actingrnal forces on the acting horizontal on the soilhorizontal strip within soil strip the slip wedge are the horizontal normal stress σ and shear stress τ near the retaining wall and the within the slip wedge are the horizontal normalwh stress σwh and shearw stress τw near the retaining wall existing side wall; the vertical stress σ and shear stress τ near the slip surface cd; and the vertical and the existing side wall; the verticals stress σs and shears stress τs near the slip surface cd; and the stress σ and (σ + dσ ) and shear stress τ and (τ + dτ ) above and below the horizontal soil strip, verticalv stress σv and (vσv + dσv) and shear stressx τx xand (τxx + dτx) above and below the horizontal soil respectively.strip, respectively. Based Based on the on Mohr the stressMohr circle stress analysis circle analysis as shown as inshown Figure in3 ,Figure the stresses 3, the at stresses any distance at any xdistancefrom the x from retaining the retaining wall within wall the within slip wedge the slip are wedge deduced are as:deduced as:  +  1cos2sin+ = θϕ1 cos 2θ sin ϕ ( + c ) c σσ=+− σh 1+sin ϕ()σ1 ϕϕcot ϕ cot ϕ  h  1 cccot cot−  1sin+ 1 ϕcos 2θ sin ϕ   σv = − + (σ1 + c cot ϕ) c cot ϕ , (4)  1 sin ϕ −  1cos2sin− sinθϕ 2θ sin ϕ σσ=+− τx = + ()(σ1 +ccccotcotϕϕϕ) cot ,  v + 1 ϕsin ϕ 1 (4)  1sin in which θ corresponds to xsin. Equation 2θϕ sin (4) corresponds to the stress state at the wall surface (points A, τσϕ=+()c cot x + ϕ 1 B, C) when θ = θw, whereas it1sin corresponds to the stress state at soil slip surface (point D) when θ = θs. The ranges of θ for soil strips AB and CD are [θw, π θw] and [θw, π θs], respectively. − − in whichIn order θ corresponds to satisfy the to x shear. Equation stress 4 integral corresponds of the to horizontal the stress soilstate strip at the to bewall zero surface within (points the slip A, R B /2 wedge,B, C) when i.e., θ =z θw,τ whereas(x)dx = 0it, corresponds the stress arch to of the the stress horizontal state at soil soil strip slip mustsurface be (point bilaterally D) when symmetric, θ = θs. Bz/2 The ranges of− θ for soil strips AB and CD are [θw, π-θw] and [θw, π-θs], respectively. in which Bz is the soil width at depth z. Bz = B when 0 z (h-Btanβ), and the stress arch is identical In order to satisfy the shear stress integral of the ≤horizontal≤ soil strip to be zero within the slip everywhere. However, Bz = (h-z) cotβ when (h-Btanβ) < z h, and the arc length decreases as the Bz /2 ≤ burialwedge, depth i.e., increases.τ ()xdx The= stress0 , the state stress of the arch horizontal of the soilhorizontal strip, in thesoil active strip state,must is be like bilaterally that of an − Bz /2 inverted arch. In order to satisfy the self-support of the inverted arch, and the traits unchanged after symmetric, in which Bz is the soil width at depth z. Bz = B when 0 ≤ z ≤ (h-Btanβ), and the stress arch is compressing or stretching [17,48,49], the symmetric catenary curve is proposed to describe the stress identical everywhere. However, Bz = (h-z) cotβ when (h-Btanβ) < z ≤ h, and the arc length decreases as arc of limited soil in active state and its mathematical expression is: the burial depth increases. The stress state of the horizontal soil strip, in the active state, is like that of an inverted arch. In ordera to xsatisfy the xself-support B of the invertedB Barch, andB the traits unchanged y = exp + exp exp z exp z z x z , (5) after compressing or stretching2 a [17,48,49],− a the− symmetric2a − catenary− 2a cu− rve2 ≤ is proposed≤ 2 to describe the stress arc of limited soil in active state and its mathematical expression is: where Bz is the horizontal soil strip width at depth z, a is the undetermined coeﬃcient, which indicates  the size of the catenaryax opening and  is x a positiveBBBBzzzz value in  the  active state, and x is the horizontal yx=+−−−−−≤≤exp exp   exp exp   , (5) distance from the vertex.22222 Whenaaax = Bz/2, y’(x = Bz/2) = cot aθw, and thus: −   − −   B where Bz is the horizontal soil strip width at depthz z, a is the undetermined coefficient, which a = p , (6) 2 indicates the size of the catenary opening2 ln cot andθw is+ a positivecot θw +value1 in the active state, and x is the horizontal distance from the vertex. When x = -Bz/2, y’ (x = -Bz/2) = -cotθw, and thus: Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 18

B a = z , ( θθ++2 ) (6) 2ln cotww cot 1 Appl. Sci. 2020, 10, 3243 7 of 17 1 xx   cotθ =− exp − exp  − . (7) 2 1 aax  x  cot θ = exp exp . (7) −2 a − − a The stress values at any position on the horizontal soil strip within the slip wedge can be obtainedThe stressby substituting values at any Equation position on(7) theinto horizontal Equation soil (4), strip which within is therepresented slip wedge by can a besymmetric obtained bycatenary substituting arch curve. Equation (7) into Equation (4), which is represented by a symmetric catenary arch curve. The comparison between the catenary arching curvecurve in Equation (5) and the arc archingarching curvecurve obtained byby Rao Rao [26 [26]] is is shown shown in Figurein Figure6, in 6, which in whichϕ is 30 φ◦ isand 30°B isand 10 m.B is As 10 can m. be As seen, canthe be curvatureseen, the ofcurvature the arch of curve the arch increases curve with increases increasing with δincreasing, and the curvatureδ, and the of curvature the catenary of the arch catenary is larger. arch The is resultslarger. fromThe results the catenary from the curve catenary tend tocurve be consistent tend to be with consistent the arc with when theδ is arc small. when The δ is larger small.δ ,The the morelarger obvious δ, the more the diobviousﬀerence the becomes. difference becomes.

Figure 6. Comparison of minor principal stress trajectory.

2.4. Lateral Active Earth Pressure CoeCoefficientfficient K The laterallateral active active earth eart pressureh pressure coe fficoefficientcient aw isKaw the is key the to key calculating to calculating the active the earth active pressure. earth Thepressure. soil arching The soil effect, arching which effect, deflects which the stress deflects trajectory the stress direction, trajectory has a significantdirection, has influence a significant on Kaw. The average vertical stress of the horizontal soil strip σv calculated from the catenary arching curve influence on Kaw. The average vertical stress of the horizontal soil strip σ calculated from the obtained in Section 2.3 is: v catenary arching curve obtained in Section 2.3 is: 0 0 R σvdx B σ  − z Bz  vdx 1exp1 exp  Bz/2 γϕ++γz + q + c cot ϕ a a  −B /2− zqccot  a a−  σ σ==v =z =  −−ϕϕ1 sin ϕ 4 sin ϕ  −  ϕc cot ϕ. (8) v Bz/2 1 + sin1ϕ sin − 4sin− Bz cB cotz  −. (8) BB/2 1+ sinϕ 1B+ exp a zz+ z 1exp a When δ = 0, σv = γz + q. Thus, the lateral active earth pressure coefficient Kaw is: σγ=+ When δ = 0, zq. Thus, the lateral active earth pressureB coefficient!! Kaw is: v γz+q+c cot ϕ 8 exp( z ) + − a c 1+sin ϕ 1 sin ϕ 1 B 2 cot ϕ − (exp( z )+1) − σwh − a Kaw = = B !! . (9) −Bz z σv γz+q+c cot ϕ 8expexp(a ) 1 γϕ++ 1 sin ϕ1 + 4a − − c cot ϕ zqc1+sincotϕ Bz Bz a+ 1+−− sinϕϕ 1exp( a ) 1 − −c cot + ϕ − 2 1sin B exp−+z 1 It is the general expression of Kaw obtained by considering the influence of soil cohesion. σ a K = f (c, ϕ, δ, γ,==z, whq) and is non-linearly distributed along the wall depth. When δ (9)= 0, aw Kaw . 1 sin ϕ 2c cot ϕ σsin ϕ B Kaw = − v . When ϕ = 0, Kaw = 1 2c/(γz−−+ zq), which is the earth pressure 1+sin ϕ − γz+q 1+sin ϕ − exp 1 γϕzqc++cotc K 4Ka aB a K coefficient under static conditions. When = δ1sin1−+= 0, ϕaw= a. Since ( z/ ) is only −c co at functionϕ of θw, aw is not affected by the presence of1sin limited+ ϕ c-ϕsoil width.B B z exp−+z 1  a 2.5. Active Earth Pressure Distribution

The lateral active earth pressure coeﬃcient Kaw obtained in Equation (9) by considering the inﬂuence of soil cohesion c is a function of the burial depth z. Thus, Kaw must be synchronously integrated when the limit equilibrium method of the horizontal soil strip is used to raise the active Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 18

It is the general expression of Kaw obtained by considering the influence of soil cohesion. Kaw = f (c, φ, δ, γ, z, q) and is non-linearly distributed along the wall depth. When δ = 0, 1sin− ϕϕϕ 2cotsinc K =− . When φ = 0, Kaw = 1 − 2c/(γz+q), which is the earth pressure aw 1sin+++ϕγzq 1sin ϕ coefficient under static conditions. When c = δ = 0, Kaw = Ka. Since (Bz/a) is only a function of θw, Kaw is not affected by the presence of limited c-φ soil width.

2.5. Active Earth Pressure Distribution

The lateral active earth pressure coefficient Kaw obtained in Equation (9) by considering the influence of soil cohesion c is a function of the burial depth z. Thus, Kaw must be synchronously integrated when the limit equilibrium method of the horizontal soil strip is used to raise the active earth pressure model and makes the calculation process more complicated. To simplify the calculation process, the coordinate translation method is used to convert Kaw of the cohesion soil into

aw ()σϕσϕ+=+() the K ’ of sandy soil. From Figure 3 we know that Kcaw vcot v cK cot aw ' , and thus:

Appl. Sci. 2020, 10, 3243 BB  8 of 17 exp−+zz exp  − 6 aa  1sin+ ϕ BB  earth pressure model and makes the calculation processexp more−+ complicated.zz exp  To + 2 simplify the calculation σσ'cot+ c ϕ aa process, the coordinate== translationwh wh method is = used to convertKaw of the  cohesion soil into the Kaw’ of Kaw ' , (10) sandy soil. From Figure3 σσwe'cot know+ thatc ( ϕKawσv + c cot ϕ) = (σv + c cotϕB)Kaw0, and thus: vv 1exp− z 4a a 1sin1−+ϕ exp( Bz )+exp( Bz ) 6 1 + sinϕB − a B a − σ + c cot ϕ expz (+ Bz )+expz( Bz )+2 σwh0 wh 1exp− a a Kaw0 = = = , (10)  a Bz ! σv0 σv + c cot ϕ 1 exp( ) 1 sin ϕ 1 + 4a − a Bz 1+exp( Bz ) Kaw’ is independent from z. − a B is in the range of (0, hcotβ) when the existing large rigid underground structure is in the Kaw’ is independent from z. influence range of the retaining wall, and the soil width is limited. The refined boundary of the c-φ B is in the range of (0, hcotβ) when the existing large rigid underground structure is in the inﬂuence soil limited width will be discussed later. The horizontal soil strip of the limited c-φ soil behind the range of the retaining wall, and the soil width is limited. The reﬁned boundary of the c-ϕ soil limited retaining wall can be divided into two main parts along the wall depth: the rectangular region width will be discussed later. The horizontal soil strip of the limited c-ϕ soil behind the retaining (case1) in which (0 ≤ z ≤ (h-Btanβ)), and the trapezoidal region (case2) in which ((h-Btanβ) < z ≤ h), as wall can be divided into two main parts along the wall depth: the rectangular region (case1) in which shown in Figure 7a,b. The saturated weight of the horizontal soil strip is dG. (0 z (h-Btanβ)), and the trapezoidal region (case2) in which ((h-Btanβ) < z h), as shown in ≤ ≤ ≤ Figure7a,b. The saturated weight of the horizontal soil strip is dG.

(a) (b)

Figure 7. ForceForce analysis analysis of of horizontal horizontal soil soil strip strip of ( a)) rectangular rectangular type and ( b) trapezoidal type. 2.5.1. Case1: 0 z (h-Btanβ) 2.5.1. Case1: 0 ≤≤ z ≤≤ (h-Btanβ) From the stress equilibrium of the horizontal soil strip in the rectangular region in Figure7a, we From the stress equilibrium of the horizontal soil strip in the rectangular region in Figure 7a, we can obtain that can obtain that dσ 2K tan δ γ = v0 + aw0 σ . (11) dz B v0 dKσδ'2 'tan γ =+vaw σ ' . (11) When z = 0, σ = q; thus: v v0 dz B ! γ ηz γ σ = q e− + , (12) v10 − η η

2Kaw0 tan δ where η = B . When δ = 0, σv10 = q + γz. Thus, when 0 z (h-Btanβ), the active earth pressure for the limited c-ϕ soil is: ≤ ≤ " ! # γ ηz γ σ = Kaw0 q e− + c cot ϕ. (13) wh1 − η η −

γB When q = c = 0, σ = 1 exp 2Kaw0 tan δ z , which is consistent with Handy’s [17] results, wh1 2 tan δ − − B except for the lateral active earth pressure coeﬃcient.

2.5.2. Case2: (h-Btanβ) < z h ≤ From the horizontal and vertical stress equilibrium of the horizontal soil strip in the trapezoidal region in Figure7b, we can obtain that: ! dσv0 1 τw0 + τs0 = γ + σv σs0 , (14) dz h z 0 − − cot β − Appl. Sci. 2020, 10, 3243 9 of 17

σ 0 + τs0 cot β σs0 = 0. (15) wh − When B > hcotβ, case1 is negligible and (h-Btanβ) = (1 t) h = 0. Otherwise, 0 < t < 1, and γ η(1 t)h γ − σv (z = (1 t)h) = σ (z = (1 t)h) = q e + . Thus: 20 − v10 − − η − − η

" ! # !χKaw0 1 γ η(1 t)h γ γth h z − γ(h z) σv20 = q e− − + + − − , (16) − η η 2 χKaw th − (2 χKaw ) − 0 − 0 tan β cos(β ϕ δ) = − − where χ cos δ sin(β ϕ) . When (h-Btanβ)−< z h, the active earth pressure for the limited c-ϕ soil is: ≤   " ! # !χKaw0 1 ( )  γ η(1 t)h γ γth h z − γ h z  σwh2 = Kaw0 q e− − + + − −  c cot ϕ. (17)  − η η 2 χKaw th − (2 χKaw )  − − 0 − 0

When δ = 0, σ = Ka(q + γz) c cot ϕ. wh − Equations (13) and (17) form the calculate model for the active earth pressure distribution of limited c-ϕ soil between a deep retaining wall and a large rigid underground structure in this study. It is also suitable for a semi-inﬁnite c-ϕ soil (t = 1).

2.6. Tension Crack Depth in the Top Wall

There are tension cracks within a certain depth z0 near the top of wall for a c-ϕ soil in active state. This indicates that the soil in this depth (0 < z < z0) is separated from the back of retaining wall and then σwh = 0. When σwh = 0 in Equation (13), the burial depth z0 is:

B c cot ϕ tan δ γB/2 z0 = ln − . (18) −2Kaw tan δ qKaw tan δ γB/2 0 0 −

Equation (18) shows that z0 is aﬀected by B, c, ϕ, δ, γ and q. z0 increases as c and B increase, but c cot ϕ 1+sin ϕ q decreases as increasing ϕ, δ and q. z = while δ = 0. Furthermore z = 2c/ γ √Ka 0 γ 1 sin ϕ − γ 0 while δ = 0 and q = 0; this is consistent with the− classical Rankine method [1].

2.7. Boundary Values of the Limited Width

Due to the ground surface overload q, z0 may not necessarily occur in the active state. In this case, z0 must be analyzed additionally. According to Equation (18), when z0 < 0, z0 = 0. When c cot ϕ tan δ γB/2 2c cot ϕ 2c cot ϕ z > 0, 0 − 1. Thus, B , while ccotϕ qK ’. However, B while ccotϕ 0 qKaw tan δ γB/2 γ cot δ aw γ cot δ ≤ 0 − ≤ ≥ ≥ ≤ < qKaw’. Therefore, the boundary for c-ϕ soil limited width B is:

 c  2 cot ϕ B h ( c K q)  γ cot δ cot β when cot ϕ aw0  2c cot≤ ϕ ≤ ≥ . (19)  B and B h cot β (when c cot ϕ < Kaw q) ≤ γ cot δ ≤ 0 Otherwise, the c-ϕ soil width is semi-inﬁnite or beyond the calculation scope of this study for B < 2ccotϕ/(γcotδ) while ccotϕ qKaw’. ≥ 2.8. Resultant Force and Its Action Point From the integral of Equations (13) and (17), the horizontal component of active earth pressure resultant force acting on the retaining wall Ewh is:

h B tan β h −Z Z Ewh = σwh1dz + σwh2dz. (20)

z0 h B tan β − Appl. Sci. 2020, 10, 3243 10 of 17

The action point of the resultant force hw is:

h B tan β h − R R σ (h z)dz + σ (h z)dz wh1 − wh2 − z0 h B tan β − hw = . (21) Ewh

When δ = 0: 1 sin ϕ E = − q(h z ) + γ h2 z 2 /2 c cot ϕ(h z ), (22) wh 1 + sin ϕ − 0 − 0 − − 0

2 1 sin ϕ h 2 i (h z ) − 2 2 3 3 0 1+sin ϕ q(h z0) /2 + γh h z0 /2 γ h z0 /3 c cot ϕ −2 h = − − − − − . (23) w 1 sin ϕ − [q(h z ) + γ(h2 z 2)/2] c cot ϕ(h z ) 1+sin ϕ − 0 − 0 − − 0 2 1 sin ϕ γh h γh+3q = − + = Furthermore, when c = 0, Ewh 1+sin ϕ qh 2 and hw 3 γh+2q ; and when c = q = 0, 2 1 sin ϕ γh h − Ewh = 1+sin ϕ 2 and hw = 3 , which is coincident with Rankine’s solution [1].

3.Appl. Comparison Sci. 2020, 10, andx FOR Veriﬁcation PEER REVIEW 11 of 18

3.1. NumericalA numerical Analysis simulation method was used to clarify the limited c-φ soil mechanism between a deep retaining wall and an existing large rigid underground structure. The finite difference A numerical simulation method was used to clarify the limited c-ϕ soil mechanism between a deep numerical analysis model established in FLAC3D (Fast Lagrangian Analysis of three-dimensional retaining wall and an existing large rigid underground structure. The finite difference numerical analysis Continua) [50–53] is shown in Figure 8. In which the boundary conditions are: the thickness of the model established in FLAC3D (Fast Lagrangian Analysis of three-dimensional Continua) [50–53] is mesh was only 1 m in order to simulate the plane strain state; the depth of the mesh, and the width at shown in Figure8. In which the boundary conditions are: the thickness of the mesh was only 1 m in the right side of the large rigid underground structure was four times the retaining wall depth [13]. order to simulate the plane strain state; the depth of the mesh, and the width at the right side of the The bottom of the mesh was fixed; the surrounding side can slide vertically; and the top surface was large rigid underground structure was four times the retaining wall depth [13]. The bottom of the free. mesh was fixed; the surrounding side can slide vertically; and the top surface was free.

Retaining Limited Existing wall soil underground structure

Interface element

(a) (b)

FigureFigure 8. 8.Numerical Numerical calculation calculation model model of of ( a(a)) model model mesh mesh and and ( b(b)) interface interface element. element.

The burial depth of the large rigid existing underground structure and of the retaining wall were The burial depth of the large rigid existing underground structure and of the retaining wall both 12 m. The width between the retaining wall and the existing wall was determined by the limited were both 12 m. The width between the retaining wall and the existing wall was determined by the soil width, which were separately 1, 2, 4, 6, and 8 m. The thickness of the retaining wall was 0.8 m at limited soil width, which were separately 1, 2, 4, 6, and 8 m. The thickness of the retaining wall was the top and 3.8 m at the bottom. Its back in contact with soil was vertical. The soil-wall interaction was 0.8 m at the top and 3.8 m at the bottom. Its back in contact with soil was vertical. The soil-wall simulatedinteraction using was an simulated interface using element an (theinterface rigidness elemen of thet (the interface rigidness element of the was interface 10 times element that of the was soil 10 c rigidness),times that andof the the soil constitutive rigidness), model and the was constitu used thetive Coulomb model was shear used model. the Coulomb The -ϕ soil shear was model. simulated The using a Mohr Coulomb model (γ = 24.5 kN/m3; E = 15 MPa; µ = 0.35; c = 30 kPa; ϕ = 25 ). The c-φ soil was simulated using a Mohr Coulomb model (γ = 24.5 kN/m3; E = 15 MPa; μ = 0.35;◦ c = 30 simulationkPa; φ = 25°). was The carried simulation out for was both carried the retaining out for structure both theand retaining the existing structure large and rigid the underground existing large rigid underground structure using an elastic element (μ = 0.2; γ = 30 kN/m3; E = 35 GPa), where E was the soil elastic modulus, and μ was the Poisson's ratio. The min and max limited soil width were separately 0.76 and 6.04 m, according to Equation (19). Generally, the retaining wall displacement which forces the soil into the active state is about 0.2 × 10-2h, in which h is the retaining wall depth [54]. By giving a leftward displacement of about 24 mm to the retaining wall, the calculated cloud map of displacement for different limited c-φ soil width is shown in Figure 9. The slope of the inclined plane is approximately the same for all the cases. The inclined plane intersects the side wall of the existing large rigid underground structure when the limited c-φ soil width is less than 6.04 m. The results verify that the theoretical result for the double-fold line slip surface of the limited c-φ soil obtained in Equation (3) is reasonable.

(a) (b) (c) (d) (e)

Figure 9. Displacement cloud map for c-φ soil (cohesive soil) of limited width of (a) 1 m; (b) 2 m; (c) 4 m; (d) 6 m; and (e) 8 m. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 18

A numerical simulation method was used to clarify the limited c-φ soil mechanism between a deep retaining wall and an existing large rigid underground structure. The finite difference numerical analysis model established in FLAC3D (Fast Lagrangian Analysis of three-dimensional Continua) [50–53] is shown in Figure 8. In which the boundary conditions are: the thickness of the mesh was only 1 m in order to simulate the plane strain state; the depth of the mesh, and the width at the right side of the large rigid underground structure was four times the retaining wall depth [13]. The bottom of the mesh was fixed; the surrounding side can slide vertically; and the top surface was free.

Retaining Limited Existing wall soil underground structure

Interface element

(a) (b)

Figure 8. Numerical calculation model of (a) model mesh and (b) interface element.

The burial depth of the large rigid existing underground structure and of the retaining wall were both 12 m. The width between the retaining wall and the existing wall was determined by the limited soil width, which were separately 1, 2, 4, 6, and 8 m. The thickness of the retaining wall was 0.8 m at the top and 3.8 m at the bottom. Its back in contact with soil was vertical. The soil-wall interaction was simulated using an interface element (the rigidness of the interface element was 10 timesAppl. Sci. that2020 of, 10 the, 3243 soil rigidness), and the constitutive model was used the Coulomb shear model.11 The of 17 c-φ soil was simulated using a Mohr Coulomb model (γ = 24.5 kN/m3; E = 15 MPa; μ = 0.35; c = 30 kPa; φ = 25°). The simulation was carried out for both the retaining structure and the existing large 3 rigidstructure underground using an elasticstructure element using ( µan= elastic0.2; γ =element30 kN /m(μ =; E0.2;= 35γ = GPa), 30 kN/m where3; EE =was 35 GPa), the soil where elastic E wasmodulus, the soil and elasticµ was modulus, the Poisson’s and μ ratio. was the The Poisson's min and ratio. max limited The min soil and width max were limited separately soil width 0.76 were and separately6.04 m, according 0.76 and to 6.04 Equation m, according (19). to Equation (19). Generally, the the retaining retaining wall wall displacement displacement which which forces forces the the soil soil into into the the active active state state is about is about 0.2 0.2 10 2 h, in which h is the retaining wall depth [54]. By giving a leftward displacement of about × 10×-2h, in− which h is the retaining wall depth [54]. By giving a leftward displacement of about 24 mm to24 the mm retaining to the retaining wall, the wall, calculated the calculated cloud map cloud of displacement map of displacement for different for dilimitedﬀerent c- limitedφ soil widthc-ϕ soil is shownwidth isin shownFigure in9. FigureThe slope9. The of the slope inclined of the plan inclinede is approximately plane is approximately the same for the all same the forcases. all The the inclinedcases. The plane inclined intersects plane the intersects side wall the of side the wallexisting of the large existing rigid largeunderground rigid underground structure when structure the limitedwhen the c- limitedφ soil widthc-ϕ soil is width less than is less 6.04 than m. 6.04 The m. result The resultss verify verify that that the the theoretical theoretical result result for for the the double-fold line slip surface of the limited c-φϕ soil obtained in Equation (3) is reasonable.

(a) (b) (c) (d) (e)

Figure 9. Displacement cloud map for c-ϕ soil (cohesive soil) of limited width of (a) 1 m; (b) 2 m; Appl.Figure Sci. 2020 9., 10Displacement, x FOR PEER REVIEWcloud map for c-φ soil (cohesive soil) of limited width of (a) 1 m; (b) 2 m; (c) 412 of 18 m;(c) ( 4d m;) 6 (m;d) and 6 m; ( ande) 8 (m.e) 8 m. The numerical results of the active earth pressure distribution were obtained for B/h of 0.083, The numerical results of the active earth pressure distribution were obtained for B/h of 0.083, 0.167, 0.167, 0.333, 0.500, and 0.667, as shown in Figure 10. The results are also compared with the 0.333, 0.500, and 0.667, as shown in Figure 10. The results are also compared with the theoretical results theoretical results obtained in this study and by other scholars. The distribution of the active earth obtained in this study and by other scholars. The distribution of the active earth pressure is large in the pressure is large in the middle and small at both ends. The active earth pressure decreases as the middle and small at both ends. The active earth pressure decreases as the decreasing of limited soil decreasing of limited soil width in the numerical results and this study. The numerical results are width in the numerical results and this study. The numerical results are close to the values both from close to the values both from this study and other scholars when the limited soil width is large. this study and other scholars when the limited soil width is large. However, it is closer to the results by However, it is closer to the results by this study when the limited soil width is small. This this study when the limited soil width is small. This demonstrates that the theory developed in this demonstrates that the theory developed in this study reflects the actual situation well. study reﬂects the actual situation well.

(a) (b) (c) (d) (e)

FigureFigure 10.10. Comparison of the active active eart earthh pressure pressure distribution distribution for for (a ()a )BB/h/ h= =0.083;0.083; (b ()b B)/hB /=h 0.167;= 0.167; (c) (Bc)/hB /=h 0.333;= 0.333; (d) ( dB)/hB =/h 0.500;= 0.500; and and (e) (Be/)hB =/h 0.667.= 0.667.

3.2.3.2. CentrifugeCentrifuge ModelModel TestsTests TheThe resultsresults of of the the active active earth earth pressure pressure distribution distribution and itsand standard its standard values values for c-ϕ forsoil c of-φlimited soil of widthlimited were width compared were compared with the with centrifuge the centrifuge model test model results test of results Frydman of Frydman and Keissar and (1987)Keissar [6 (1987)] and of[6] Take and andof Take Valsangkar and Valsangkar (2001) [7 ].(2001) They [7]. were They also were compared also compared with the theoreticalwith the theoretical results of results multiple of scholars,multiple scholars, as shown as in shown Figure in 11 Figure. The 11. parameters The parame areters the are same the same as those as thos usede used in Take’s in Take’s (2001) (2001) [7] 3 q c h centrifuge[7] centrifuge model model tests, tests, i.e., γi.e.,= 16.2 γ = 16.2 kN/m kN/m, =3, q= = 0,c =ϕ 0,= φ29 =◦ ,29°,δ = δ23 =◦ ,23°,= h5.25 = 5.25 m (equalsm (equals to 140 to 140 mm mm on theon centrifugethe centrifuge model model scale), scale), and andB = B0.525 = 0.525 m, 1.4175m, 1.4175 m, 2.625m, 2.625 m, andm, and 6.825 6.825 m. m. The maximum width of limited soil is 1.421 m according to Equation (19). The soil width set by the centrifuge model tests exceed the maximum boundary of limited soil width in this study except for B = 0.525 m and 1.4175 m. However, B = 1.4175 m is like the max limited soil width 1.421 m. the active earth pressure distribution is shown in Figure 11(a). The black, red, and blue lines in Figure 11(a) are almost overlapped because they represent the active earth pressure for B = 6.825 m, 2.625 m, and 1.4175 m. Figure 11b gives out the normalized active earth pressure distribution, which is obtained through standardizing the active earth pressure σh with the vertical gravity stress γz and standardizing the burial depth z with the limited soil width B. By comparing the theoretical results with those of other scholars, it shows that the results obtained in this study are almost the same as the solution by Handy (1985) [17] and Chen (2017) [19] when the burial depth is small. However, Handy’s (1985) [17] and Chen’s (2017) [19] solution does not transit from top to bottom smoothly and Chen’s (2017) [19] solution overrated σwh while the burial depth is large. Although the soil cohesion is considered in Rao’s (2016) [26] method, the influence of the limited width cannot be considered. Both soil cohesion and the limited width are considered in this study. Figure 11 shows that the results in this study are the closest to the results of centrifuge model tests by comparing with the results of other theoretical methods. Appl. Sci. 2020, 10, 3243 12 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18

(a) (b)

FigureFigure 11. 11. ComparisonComparison of of (a ()a )the the active active earth earth pressure pressure andand ((bb)) itsits normalizednormalized distribution distribution with with centrifugecentrifuge model model tests tests and and theoretical theoretical results results by other scholars.scholars.

4. ParametricThe maximum Study width of limited soil is 1.421 m according to Equation (19). The soil width set by the centrifuge model tests exceed the maximum boundary of limited soil width in this study except forTheB = 0.525active m earth and 1.4175pressure m. distribution, However, B =as1.4175 well as m the is like resultant the max force limited and soilits action width point, 1.421 m. are the key controllingactive earth factors pressure in the distribution design of is a showndeep retaining in Figure structure.11a. The black, In this red, study, and blueit can lines be found in Figure that 11 thea mainare almostinfluencing overlapped factors because are the theysoil cohesion represent c the, the active ground earth surface pressure overload for B = q6.825, the m,wall 2.625 depth m, andh, the distance1.4175 m.between the retaining wall and the existing large rigid underground structure B, the soil internalFigure friction 11b angle gives outφ, and the normalizedthe soil-wall active friction earth angle pressure δ. Unless distribution, otherwise which stated, is obtained the parameters through 3 usedstandardizing in the following the active computations earth pressure are γσ h=15.8with kN/m the vertical; h = 8.5 gravity m; q =0 stress kPa;γ cz =and 0 kPa; standardizing φ = 36°; δ = the 25°; andburial B = 1 depth m, whichz with are the consistent limited soil with width Fan’sB. By(2010) comparing [13] numerical the theoretical simulation. results In with this thosecase, ofthe other factor ofscholars, the limited it shows width that of c the-φ soil results is t obtained= 0.4. The in boundary this study valu are almoste for the the maximum same as the limited solution c-φ by soil Handy width is (1985)Bmax = [2.5217] and m; Chenand the (2017) maximum [19] when soil the cohesion burialdepth is 12.3 is kPa small. when However, B = 1 m Handy’s according (1985) to [the17] above and calculationChen’s (2017) parameters. [19] solution does not transit from top to bottom smoothly and Chen’s (2017) [19] solution overrated σwh while the burial depth is large. Although the soil cohesion is considered in Rao’s 4.1.(2016) Soil Internal [26] method, Griction the inﬂuenceAngle and of Soil-Wall the limited Friction width Angle cannot be considered. Both soil cohesion and the limited width are considered in this study. Figure 11 shows that the results in this study are the closest The influence of the soil inner friction angle φ and the soil-wall friction angle δ on the active to the results of centrifuge model tests by comparing with the results of other theoretical methods. earth pressure distribution σwh according to Equation (13) and (17) is shown in Figure 12. Different values4. Parametric of φ varying Study from 5° to 55° and different values of δ varying from 3.6° to 35.64° were used. σwh and Bmax decrease as φ and δ increase. The burial depth of the maximum active earth pressure The active earth pressure distribution, as well as the resultant force and its action point, are key increases significantly with increasing δ. This indicates that increasing the wall friction can reduce controlling factors in the design of a deep retaining structure. In this study, it can be found that the the active earth pressure acting on the wall. main inﬂuencing factors are the soil cohesion c, the ground surface overload q, the wall depth h, the distance between the retaining wall and the existing large rigid underground structure B, the soil internal friction angle ϕ, and the soil-wall friction angle δ. Unless otherwise stated, the parameters 3 used in the following computations are γ = 15.8 kN/m ; h = 8.5 m; q = 0 kPa; c = 0 kPa; ϕ = 36◦; δ = 25◦; and B = 1 m, which are consistent with Fan’s (2010) [13] numerical simulation. In this case, the factor of the limited width of c-ϕ soil is t = 0.4. The boundary value for the maximum limited c-ϕ soil width is Bmax = 2.52 m; and the maximum soil cohesion is 12.3 kPa when B = 1 m according to the above calculation parameters.

4.1. Soil Internal Griction Angle and Soil-Wall Friction Angle The influence of the soil inner friction angle ϕ and the soil-wall friction angle δ on the active earth pressure distribution σwh according to Equations (13) and (17) is shown in Figure 12. Different values of ϕ varying from 5◦ to 55◦ and(a) different values of δ varying from 3.6(b◦ )to 35.64◦ were used. σwh and Bmax decrease as ϕ and δ increase. The burial depth of the maximum active earth pressure increases Figure 12. Influence of (a) φ and (b) δ on σwh.

4.2. Soil Cohesion and Ground Surface Overload The influence of soil cohesion c and the ground surface overload q on the active earth pressure distribution σwh is shown in Figure 13. Both of c and q vary from 0 to 25 kPa. σwh decreases with Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18

(a) (b)

Figure 11. Comparison of (a) the active earth pressure and (b) its normalized distribution with centrifuge model tests and theoretical results by other scholars.

4. Parametric Study The active earth pressure distribution, as well as the resultant force and its action point, are key controlling factors in the design of a deep retaining structure. In this study, it can be found that the main influencing factors are the soil cohesion c, the ground surface overload q, the wall depth h, the distance between the retaining wall and the existing large rigid underground structure B, the soil internal friction angle φ, and the soil-wall friction angle δ. Unless otherwise stated, the parameters used in the following computations are γ =15.8 kN/m3; h = 8.5 m; q =0 kPa; c = 0 kPa; φ = 36°; δ = 25°; and B = 1 m, which are consistent with Fan’s (2010) [13] numerical simulation. In this case, the factor of the limited width of c-φ soil is t = 0.4. The boundary value for the maximum limited c-φ soil width is Bmax = 2.52 m; and the maximum soil cohesion is 12.3 kPa when B = 1 m according to the above calculation parameters.

4.1. Soil Internal Griction Angle and Soil-Wall Friction Angle The influence of the soil inner friction angle φ and the soil-wall friction angle δ on the active

Appl.earth Sci. pressure2020, 10, 3243distribution σwh according to Equation (13) and (17) is shown in Figure 12. Different13 of 17 values of φ varying from 5° to 55° and different values of δ varying from 3.6° to 35.64° were used. σwh and Bmax decrease as φ and δ increase. The burial depth of the maximum active earth pressure signiﬁcantlyincreases significantly with increasing with increasingδ. This indicates δ. This thatindicates increasing that increasing the wall friction the wall can friction reduce can the reduce active earththe active pressure earth acting pressure on theacting wall. on the wall.

(a) (b)

FigureFigure 12.12. InﬂuenceInfluence ofof ((aa)) ϕφ andand ((bb)) δδ onon σσwhwh..

4.2. Soil Cohesion and Gr Groundound Surface Overload Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 18 The influenceinfluence of soil cohesion c and the ground surface overload q on the active earth pressure distribution σwh is shown in Figure 13. Both of c and q vary from 0 to 25 kPa. σwh decreases with increasingdistribution c. σwh increasesis shown within Figure increasing 13. Both q when of c theand burial q vary depth from is 0 small.to 25 kPa.It is worthσwh decreases noting that with a increasing c. σ increases with increasing q when the burial depth is small. It is worth noting that a larger soil cohesionwh may lead to tensile stress of the active earth pressure behind the retaining wall, larger soil cohesion may lead to tensile stress of the active earth pressure behind the retaining wall, which will affect the overall stability of the deep retaining wall. According to Equation (19), the soil which will affect the overall stability of the deep retaining wall. According to Equation (19), the width in Figure 13b all exceed the boundary value of the maximum limited soil width. That is, the soil width in Figure 13b all exceed the boundary value of the maximum limited soil width. That calculation results are based on a semi-infinite soil. The burial depth of the maximum active earth is, the calculation results are based on a semi-infinite soil. The burial depth of the maximum active pressure increases with increasing q. Figure 13b shows that it is necessary to implement larger earth pressure increases with increasing q. Figure 13b shows that it is necessary to implement larger reinforcement measures within the top half of the retaining wall when the ground surface is reinforcement measures within the top half of the retaining wall when the ground surface is overloaded. overloaded.

(a) (b)

Figure 13. InﬂuenceInfluence of (a) c and (b)) q on σwh..

4.3. Soil’s Soil’s Limited Width and Wall Height The influence inﬂuence of the spatial effect eﬀect of the limited soil width B and the wall height h on the active earth pressure distribution σ is shown in Figure 14, in which B vary from 0.4 to 1.9 m and h vary earth pressure distribution σwh is shown in Figure 14, in which B vary from 0.4 to 1.9 m and h vary from 1 to 10 m. σ increases with increasing B and h. The smaller the limited soil width, the greater from 1 to 10 m. σwh increases with increasing B and h. The smaller the limited soil width, the greater its impact on the active earth pressure distribution. σ cannot be solved using the method developed its impact on the active earth pressure distribution.wh σwh cannot be solved using the method developedin this study in whenthis studyB is lesswhen than B is the less minimum than the widthminimum calculated width calculated using Equation using (19). Equation The boundary (19). The boundaryvalue for the value maximum for the maximum limited soil limited width soil increases width with increases increasing with increasing wall depth. wall depth.

(a) (b)

Figure 14. Influence of (a) B and (b) h on σwh.

4.4. Comparison After Normalization In order to compare the effects of each parameter (φ, δ, c, q, B, h) in Figures 12, 13, and 14, this study standardized the parameters to i, the ratio of the difference between the current value and the minimum value of the parameter to the range, that is (current value – minimum value)/(maximum value – minimum value). The standard values of the resultant force (Ewh/(γh2/2)) and its standard action point (hw/h) calculated according to Equations (20) and (21) change with the parameter i, as shown in Figure 15. (Ewh/(γh2/2)) increases as B and q increase but decreases as c, φ, δ, and h increase. Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 18 increasing c. σwh increases with increasing q when the burial depth is small. It is worth noting that a larger soil cohesion may lead to tensile stress of the active earth pressure behind the retaining wall, which will affect the overall stability of the deep retaining wall. According to Equation (19), the soil width in Figure 13b all exceed the boundary value of the maximum limited soil width. That is, the calculation results are based on a semi-infinite soil. The burial depth of the maximum active earth pressure increases with increasing q. Figure 13b shows that it is necessary to implement larger reinforcement measures within the top half of the retaining wall when the ground surface is overloaded.

(a) (b)

Figure 13. Influence of (a) c and (b) q on σwh.

4.3. Soil’s Limited Width and Wall Height The influence of the spatial effect of the limited soil width B and the wall height h on the active earth pressure distribution σwh is shown in Figure 14, in which B vary from 0.4 to 1.9 m and h vary from 1 to 10 m. σwh increases with increasing B and h. The smaller the limited soil width, the greater its impact on the active earth pressure distribution. σwh cannot be solved using the method Appl.developed Sci. 2020 ,in10 this, 3243 study when B is less than the minimum width calculated using Equation (19).14 ofThe 17 boundary value for the maximum limited soil width increases with increasing wall depth.

(a) (b)

FigureFigure 14.14. InﬂuenceInfluence ofof ((aa))B Band and ((bb))h hon onσ σwhwh.

4.4.4.4. Comparison afterAfter Normalization Normalization InIn orderorder toto compare the effects effects of each parameter parameter ( (φϕ,, δδ,, cc, ,qq, ,BB, ,hh) )in in Figures Figures 12, 12 –13,14 ,and this 14, study this standardizedstudy standardized the parameters the parameters to i, the ratioto i, the of the ratio di ffoference the difference between thebetween current the value current and value the minimum and the valueminimum of the value parameter of the parameter to the range, to the that range, is (current that is value(current – minimumvalue – minimum value)/(maximum value)/(maximum value – 2 minimumvalue – minimum value). Thevalue). standard The standard values ofvalues the resultant of the resultant force (E whforce/(γ h(E/wh2))/(γ andh2/2)) its and standard its standard action point (h /h) calculated according to Equations (20) and (21) change with the parameter i, as shown Appl.action Sci. point 2020w , 10 (h, xw /FORh) calculated PEER REVIEW according to Equations (20) and (21) change with the parameter15 ofi, as18 2 inshown Figure in 15Figure.(Ewh 15./(γ (hEwh/2))/(γ increasesh2/2)) increases as B and as Bq andincrease q increase but decreases but decreases as c, ϕ as, δ c,, andφ, δ,h andincrease. h increase. The 2 impactsThe impacts of the of parameterthe parameter on ( Eonwh (/E(whγh/(γ/2))h2/2)) from from largest largest to smallestto smallest are are as follows:as follows:ϕ, φδ, δc,, Bc,, Bq, q and, andh. (hh.w (h/hw)/h increases) increases with with increasing increasingϕ, δφ,, andδ, andq but q but decreases decreases with with increasing increasingB, c ,B and, c, andh. The h. The impacts impacts of the of parametersthe parameters on ( honw/ (hh)w from/h) from largest largest to smallest to smallest are asare follows: as follows:c, δ, cq,, δϕ, ,qB, ,φ and, B, andh. Controlling h. Controlling of the of keythe influencingkey influencing factors, factors, i.e., increasing i.e., increasing the soil-wall the soil-wall friction friction and decreasing and decreasing the limited the soillimited width soil (not width less than(not less the minimumthan the minimum boundary boundary calculated calculated using Equation using Equation (19)), will (19)), increase will the increase safety the and safety economic and eeconomicfficiency ofefficiency the retaining of the structure retaining in structure a dense in underground a dense underground environment environment significantly. significantly.

(a) (b)

Figure 15. Comparison of each parameters’ e ffectsﬀects after standardization on (a) the standard resultant force andand ((b)) itsits standardstandard actionaction point.point.

5. Conclusions In order toto obtainobtain thethe activeactive earthearth pressurepressure andand itsits actionaction lawlaw onon thethe retainingretaining structurestructure underunder the limited c-ϕφ soil condition,condition, and toto clarifyclarify thethe soil-wallsoil-wall interactioninteraction mechanism and the limited c-ϕφ soil boundary width when the soil arching eeffectﬀect is signiﬁcant,significant, the theoretical method on the basis of previous studiesstudies waswas improvedimproved inin thisthis study.study. FourFour mainmain conclusionsconclusions areare summarizedsummarized asas below:below: • The effect of soil-wall cohesion can be expressed by the soil-wall friction angle. A double-fold line slip surface that considers the soil internal friction angle and the soil-wall friction angle was determined to be suitable for calculating the active earth pressure of a limited c-φ soil, and a symmetrical catenary arching curve was proposed to simulate the minor principal stress trajectory of a horizontal soil strip. • An active earth pressure calculation model for a limited c-φ soil that considers ground surface overload was proposed based on the limit equilibrium method of a horizontal soil strip and the improved soil arching effect theory. The model proposed in this study can be simplified into the classic active earth pressure theory, and it reflects the situation for a limited c-φ soil better than other theoretical methods. The laterally active earth pressure coefficient of a limited c-φ soil is a function of burial depth, but it is independent of the limited soil width. The proposed active earth pressure distribution is large in middle and small at both ends, which is consistent with the results of the numerical simulation and the centrifuge model tests. • An explicit boundary width for a limited c-φ soil during the active earth pressure calculation was determined. When the distance between the existing large rigid underground structure and the retaining wall of the foundation pit is within the boundary of the limited c-φ soil width, the active earth pressure model proposed in this study is needed for the calculation. However, the method proposed in this study is not suitable for situations where B < (2ccotφ)/(γcotδ) and ccotφ ≥ qKaw’. • The changes in the active earth pressure due to the limited c-φ soil parameters were obtained. The factors that have the greatest influence on the resultant force and its action point are the soil internal friction angle and the soil cohesion separately. Thus, the influence of soil cohesion cannot be ignored. The smaller of the limited soil width, the more sensitive the active earth pressure to it. Increasing the wall friction and reducing the limited soil width can reduce the Appl. Sci. 2020, 10, 3243 15 of 17

The effect of soil-wall cohesion can be expressed by the soil-wall friction angle. A double-fold • line slip surface that considers the soil internal friction angle and the soil-wall friction angle was determined to be suitable for calculating the active earth pressure of a limited c-ϕ soil, and a symmetrical catenary arching curve was proposed to simulate the minor principal stress trajectory of a horizontal soil strip. An active earth pressure calculation model for a limited c-ϕ soil that considers ground surface • overload was proposed based on the limit equilibrium method of a horizontal soil strip and the improved soil arching effect theory. The model proposed in this study can be simplified into the classic active earth pressure theory, and it reflects the situation for a limited c-ϕ soil better than other theoretical methods. The laterally active earth pressure coefficient of a limited c-ϕ soil is a function of burial depth, but it is independent of the limited soil width. The proposed active earth pressure distribution is large in middle and small at both ends, which is consistent with the results of the numerical simulation and the centrifuge model tests. An explicit boundary width for a limited c-ϕ soil during the active earth pressure calculation was • determined. When the distance between the existing large rigid underground structure and the retaining wall of the foundation pit is within the boundary of the limited c-ϕ soil width, the active earth pressure model proposed in this study is needed for the calculation. However, the method proposed in this study is not suitable for situations where B < (2ccotϕ)/(γcotδ) and ccotϕ qKaw’. ≥ The changes in the active earth pressure due to the limited c-ϕ soil parameters were obtained. • The factors that have the greatest influence on the resultant force and its action point are the soil internal friction angle and the soil cohesion separately. Thus, the influence of soil cohesion cannot be ignored. The smaller of the limited soil width, the more sensitive the active earth pressure to it. Increasing the wall friction and reducing the limited soil width can reduce the active earth pressure acting on the retaining wall. Reinforced measures should be taken on the upper part of the retaining wall when the ground surface is overloaded.

Author Contributions: Data curation, M.L.; methodology, X.C.; writing, M.L., X.C. and Z.H.; formal analysis, S.L. All authors have read and agreed to the published version of the manuscript. Funding: This research is supported by the National Natural Science Foundation of China (no. 51938008 and no. L1924061), the China Postdoctoral Science Foundation (no. 2019M660653) and the Special Project on the Integration of Industry, Education and Research of Guangdong Province (no. 192019071811500001). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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