Research Article Stability of Three-Dimensional Slurry Trenches with Inclined Ground Surface: A Theoretical Study

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1 Advances in Materias Science and Engineering Voume 015, Artice ID 36160, 7 pages Research Artice Stabiity of Three-Dimensiona Surry Trenches with Incined Ground Surface: A Theoretica Study Xiao-Fei Jin, 1 Shu-Ting Liang, 1 and Xiao-Jun Zhu 1 Schoo of Civi Engineering, Southeast University, Nanjing 10096, China Architectura Design and Research Institute, Southeast University, Nanjing 10096, China Correspondence shoud be addressed to Shu-Ting Liang; stiang@seu.edu.cn Received 9 Apri 015; Accepted 4 May 015 Academic Editor: João M. P. Q. Degado Copyright 015 Xiao-Fei Jin et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Stabiity of surry trenches is an important issue during the construction of the groundwater cutoff was and diaphragm was, and thus graduay draws attention. In this paper, a theoretica method for a three-dimensiona trench mode with an incined ground was proposed. Based on the Couomb-type force equiibrium, a safety factor assessing the stabiity was derived. The resuts showed that the existing two-dimensiona mode was conservative compared to the present three-dimensiona mode; concretey, a greater incined ange of the incined ground and trench ength decreased the safety factor. This work coud be used to assess the trench stabiity for both D and 3D cases with incined ground surfaces. 1. Introduction Surry trenches are often used as a hydrauic barrier to prevent the groundwater from fowing into the trenches during the construction of groundwater cutoff was and diaphragm was, and the surry in the excavated trenches pays an important roe in providing a atera supporting force to the trench was before backfiing. Therefore, surry trench stabiity is a major concern. However, most of the existing studies focused on the ground movements and stress reief induced by diaphragm wa construction: Poh and Wong concuded that evauated aspects of performance incude atera and vertica soi movements during the construction of the test wa pane [1]; Ng and Yan confirmed the horizonta arching and downward oad transfer mechanisms during diaphragm wa instaation using the finite difference method []; Gourvenec and Powrie investigated the impact of three-dimensiona effects and pane ength on horizonta ground movements and changes in atera stress during the sequentia instaation of a number of diaphragm wa panes [3]; Ng and Lei derived an anaytica soution for cacuating horizonta stress changes and dispacements caused by the excavation for a diaphragm wa pane [4]; Schäfer and Triantafyidis investigated the infuence of a diaphragm wa construction on the stress fied in a soft cayey soi [5]; Arai et a. conducted to examine ground movement and stress after the instaation of circuar diaphragm was and soi excavation within the was [6]; Conti et a. studied themechanismsofoadtransferandthedeformationsof the ground during surry trenching and concreting in dry sand [7]; Comodromos et a. proposed a new approach for simuating the excavation and construction of subsequent panes to investigate the effects from the instaation of diaphragm was on the surrounding and adjacent buidings [8]; Lei et a. proposed an approximate anaytica soution to predict ground surface settements aong the centre-ine perpendicuar to a surry-supported diaphragm wa pane [9]. Unti now, the probem of the surry trench stabiity graduay receives much attention in underground engineering. In fact, in the eary stage, a two-dimensiona imit equiibrium method for trench stabiity based on a simpe Couomb wedge for dry soi conditions was deveoped [10]; ater, the method was extended to account for infuences of different eves of surry in the trench and groundwater in the cohesioness soi [11]. Moreover, based on the Rankin theorem, the trench stabiity was assessed by considering the pressures from sois, hydrostatic surry, and groundwater [1]. However, these mentioned methods treated the trench stabiity as

2 Advances in Materias Science and Engineering a two-dimensiona probem and negected stabiizing forces or shear forces acting on both ends of the faiure mass, which may produce conservative resuts. Indeed, considering the rea case incuding contribution of the shear forces on the both ends of the faiure wedge, three-dimensiona method derived from the two-dimensiona imit equiibrium theory was aready used to anayze the trench stabiity; Prater [13] and Washbourne [14] proposed panar sides to enhanced existing two-dimensiona modes. Nonpanar sides aso were proposed [15, 16] and may yied a coser approximationto the curved geometry observed for trench faiure surfaces [14, 17]; Fox presented Couomb-type force equiibrium anayses for genera three-dimensiona stabiity of a surry-supported [18]. Stabiity anayses using the methods from the abovementioned works have aready taken into account the infuences of severa primary design parameters, incuding trench ength and depth, surry depth and density, groundwater depth, and tension crack. However, if cutoff was ocate beow an incined ground, it is necessary to consider the effect of the incined ground surface on the trench stabiity. In this regard, Li et a. investigated the infuence of incination ange on trench stabiity by a two-dimensiona mode, and they showed that it was unconservative to negect ground surface incination when anayzing trench stabiity [19]. Here, extending the two-dimensiona mode of Li et a. [19] to three-dimensiona case, we presented an anaytica soution of the safety factor and critica faiure ange for a surry-supported trench with an incined ground, and an exampe was finay discussed to iustrate the variation of the safety factor and critica faiure ange with different incined ange, trench ength, and groundwater depth.. Theory A three-dimensiona mode of a faiure wedge with ength and force anaysis of the mode are shown in Figure 1. In this mode, the top surface of surry is assumed to be higher than that of the groundwater surface (i.e., h s >h w ), and the trench was were considered to be impermeabe after the excavation, because there existed a ayer of graduay thickened fiter cake-ike bentonite, which resuts in independency between the pore pressure in the soi and surry pressure in the trench. It is worth mentioning that the shear strength due to soi suctions above the groundwater surface can be incuded by specifying an appropriate vaue for c 1, which is caed effective stress cohesion intercept above groundwater surface, according to the tota cohesion method [18, 0]. Besides, the wedge was assumed to be rigid, and temporary oads were considered uniformy distributed, and the geometric parameters 1,,a,b,andc in Figure 1 satisfy 1 =(cosβ/ sin(θ β))h, =h w cscθ, a= 1 cosθ hcotθ, b= 1 cosθ,andc=hcotθ,inwhichθ and β are anges made by the faiure pane and incined ground with respect to the horizonta pane, respectivey, and h is the depth of the trench. Considering the three-dimensiona case different from the two-dimensionacase by Li et a. [19], the shear resistance force S acting on end panes of the wedge is treated to be parae to the faiure pane (see Figures 1(b) and 1(c)). The shear resistance force is assumed to be uniformy distributed aong the faiure pane, and its magnitude proportionay varies;thus,theprogressivefaiureeffectcanbenegectedand the safety factor with respect to shear faiure of each end pane isequatothatofthefaiurepane[18]. The tota shear force on each end pane of the wedge is cacuated as S=S 1 +S, (1) where S 1 and S are shear resistant forces acting on two portions of each end pane of the wedge, respectivey, that is, above (I) and beow (II and III) the horizonta pane; see Figure 1(c). For the portion (I) above the horizonta pane of the wedge, through the force anaysis, S 1 can be expressed as S 1 = 1 b 0 F ( (c 1 +σ h1,1 tan φ )dzdx 0 tan βx b c 0 (c 1 +σ h,1 tan φ )dzdx) ( b tan β/a)(x c) = h F (c 1 cot θ tan β tan θ tan β Kγ 1 tan φ (tan β) cot θ h (tan θ tan β) +Kqtan 1 φ ( tan θ tan β cot θ)), where q is the uniformy distributed temporary oad on the incined ground surface, F is the safety factor, γ 1 and c 1 are the unit weight and effective stress cohesion intercept of the soi above the groundwater surface, respectivey, φ is the effective stress friction ange of the entire soi profie, K taken as the at-rest atera earth pressure coefficient K 0 = 1 sinφ is the average atera earth pressure coefficient for the end panes of the faiure wedge, and the horizonta effective stresses are σ h1,1 =K(q+γ x 1 (z + x tan β)), b x [0,b], z [ tan βx, 0], σ h,1 =K(q+γ x c b tan β (3) 1 (z + (x c))), a a b tan β x [c, b], z [ (x c), 0]. a For the shear resistance force S acting on the portion (II and III) beow the horizonta pane, it can be cacuated as S = 1 F ( z w h + = 0 cz/h+c z w 0 cot θ F cz/h+c (c 1 +σ h1, tan φ )dxdz 0 {K tan φ 3 (c +σ h, tan φ )dxdz) [3qh +γ h 3 ()

3 Advances in Materias Science and Engineering 3 x Q q z y Incined ground surface z s β Incined ground surface Horizonta pane β γ 1 End pane c 1 z w h h s 1 h w Faiure pane Top surface of surry Groundwater surface γ c θ θ Trench with fied surry (a) (b) Q β II I S 1 P S W III S N U T x β dx c b I a θ (c) Figure 1: Three-dimensiona faiure wedge. (a) Schematic mode of faiure wedge, (b) definitions of geometric parameters, and (c) force anaysis. +z w (γ 1 γ )(z w 3hz w + 3h )] + z w (c 1 c ) (h z w )+c h }, where γ and c are the unit weight and effective stress cohesion intercept of the soi beow the groundwater surface, respectivey, z w is the distance between the horizonta pane and groundwater surface, and the horizonta effective stresses are σ h1, =K(q+γ x tan β 1 +γ b 1 z), z [0,z w ], (4) Substituting ()and (4) into (1), the tota shear resistance forceappiedoneachendpaneis where S= Φ F, (6) Φ= 1 tan cot θ βh (tan θ tan β) [c 1 (tan θ tan β) Kγ 1 tan φ tan βh] + +γ h 3 cot θ K tan φ 6 3qh [ tan θ tan β (7) σ h, =K(q+γ x tan β 1 +γ b 1 z w +γ (z z w )), z [z w,h]. (5) +z w (γ 1 γ )(z w 3hz w + 3h ) cot θ] + cot θ [z w (c 1 c )(h z w)+c h ].

4 4 Advances in Materias Science and Engineering For the entire wedge, the force equiibrium in the directions norma and tangentia to the faiure pane yieds F N =N +U (W+Q) cos θ P S sin θ=0, F T = S+T (W+Q) sin θ+p S cos θ=0, where W is the soi weight, Q is the equivaent concentrated force by q, P S is the atera force by fied surry, U is the hydrostatic groundwater force, N istheeffectivenorma force, and T istheshearforceonthefaiurepane.through the cacuationsofthe forcesin (8), they are expressed W+Q T U P S =Γ= + (γ γ 1 )h w, tanθ = 1 F (Ψ + N tan φ h tan θ tan β (q + hγ 1 ) )= 1 F (c 1 h cos β sin (θ β) +(c c 1 )h w csc θ+ N tan φ ), =Λ= γ w csc θh w, =Ω= γ sh s in the expression of P S /, andγ s and h s are unit weight and theheightofthefiedsurry,respectivey. Soving (8) for the safety factor of the wedge, the foowing is obtained: F= Φ (Γ sin θ Ωcos θ) + Ψ Γ sin θ Ωcos θ Γ +( Γ tan θ Ω + Ω Γ Ωcot θ Λ Γ sin θ Ωcos θ ) tan φ. (8) (9) (10) The critica ange θ cr of the faiure pane that corresponds to the minimum safety factor F S for the faiure wedge is found by taking df/dθ = 0,andtheequationcanbesovedbyan iterative method. It is noted that the soution of θ cr shoud ocate in the range 45 θ cr 90 ;otherwisef S wi be ess than zero if θ cr < 45 [18]. Considering a pecuiar case of cohesioness sois (i.e., c 1 = =0), the expression of F reduces to c F=f(θ) tan φ, (11) where f (θ) = Π (Γ sin θ Ωcos θ) + Γ Γ tan θ Ω Ω + Γ Ωcot θ Λ Γ sin θ Ωcos θ, Π=Φcot φ, Π= 1 8 Kγ 1 (tan β) h 3 cot θ (tan θ tan β) qh K[ tan θ tan β +γ h 3 cot θ +z w (γ 1 γ )(z w 3hz w + 3h ) cot θ]. (1) Equation (11) indicates that θ cr is independent of φ,whichis consistent with the anaytica resut for the horizonta ground by Fox [18]. The proposed anaytica method is aso appicabe when the ground is soping away from the trench (i.e., β<0). Under this condition, the intersection point of the incined groundsurfaceandthefaiurepaneshoudbeabovethe groundwater surface (i.e., 1 > ). 3. Exampe Here, we investigate the infuences of different ground incinations and trench engths on the trench stabiity. The geometric and physica parameters of the trench, soi, and surry are from Fox [1], namey, h = 0m, h s = 0m, z w =3m, γ s = 11.8 kn/m 3, γ 1 = 19.0 kn/m 3, c =0kPa, γ = 0.0 kn/m 3, φ =37,andq = 0kN/m.Moreover, c 1 =10kPaisusedforthesoiabovethegroundwatersurface to consider the soi suction effect. Figure shows the reationships of the minimum safety factor F S or critica ange θ cr versus incined ange β for the two-dimensiona case. In Figure (a), wecanseethatf S is negativey reevant with β,andf S decreases from 1.48 to 1.19 as β varies from 0 to 15. This indicates that the assumption of a horizonta ground (i.e., 0 ) for the incined ground resuts in an overestimation of the trench stabiity, which may resut in siding faiure of the excavated pane. When the incined ange β comesdownintonegativevaueforaspecificcase β = 10, F S equas 1.74, which is greater than 1.48 for β = 0. And more, the present prediction is comparabe to that based on Rankin s earth pressure theory (the dashed ine in Figure (a), MorgensternandAmir-Tahmasseb,1965 [11]), which vaidates our derived theory, and is aso equa to the cacuated vaues with the method proposed by Li et a. [19]. With the given θ cr, engineers can cacuate the size of the potentia faiure mass, and reinforcement design can be made if the trench stabiity is not satisfactory; for exampe, when β = 0 and θ cr = 58.56, the area of the end crosssection is.6% greater than that for θ cr = 45 +φ (= 63.5 ). In Figure (b), it is readiy seen that θ cr decreases and then

5 Advances in Materias Science and Engineering 5 Factor of safety, F S Critica faiure ange, θ cr β ( ) β ( ) F S for θ=45 +φ / F S with β c1 =0 c1 =10 (a) (b) Figure : Infuences of the incined ange β on (a) F S and (b) θ cr. Factor of safety, F S Trench ength, L (m) Critica faiure ange, θ cr Trench ength, L (m) c 1 =10kPa,β= 0 c1 =0, β=0 c c 1 =10kPa,β= 5 1 =0, β=5 c 1 =0, β=5 c1 =0, β=0 c1 =10kPa,β= 5 c 1 =10kPa,β= 0 (a) (b) Figure 3: Infuences of the trench ength L on (a) F S and (b) θ cr. increases as β changes from 10 to 35, and the minimum vaue θ cr =56 is obtained when β=5.thisagainshows that it is necessary to determine θ cr instead of using θ cr = 45 +φ.pus,figure (b) aso shows that increasing c 1 tends to a reduced vaue of θ cr. The infuences of the trench ength L on F S and θ cr are potted in Figure 3. ItshowsthatF S is obviousy affected by the trench ength L, and the resuts of the present threedimensiona case move toward the two-dimensiona case as L tends to infinity. Figure 3(a) shows that increasing c 1 and β resuts in increasing and decreasing F S,respectivey.For this exampe, vaues of the trench ength L which correspond to F S equaing1.8are39.3m,34.8m,9.3m,and.6mfor c 1 =10kPa (β =0), c 1 =0kPa (β =0), c 1 =10kPa (β =5), and c 1 =0kPa (β =5), respectivey, whereas, corresponding to c 1 and β, F S is cacuated as 1.48, 1.44, 1.37, and 1.3 for the two-dimensiona case, respectivey. Figure 3(b) shows that θ cr decreases with increasing trench ength L, and increasing both c 1 and β resuts in decreasing θ cr, andθ cr is apparenty affected by β. For this exampe, vaues of θ cr that correspond to L = 40 are59.8m,58.4m,60.1m,and58.5mforc 1 = 10 kpa (β =0), c 1 =10kPa (β =5), c 1 =0kPa (β =0), and c 1 =0kPa (β =5), respectivey. Finay,basedontheparametersL=40mandβ=5, the infuence of the rising groundwater on the safety factor is shown in Figure 4.Itisnoticedthatargerz w represents ower

6 6 Advances in Materias Science and Engineering Factor of safety, F S c 1 =10 c 1 = c 1 =0, F S with z w c1 =10, F S with z w L=40m, β=5 Depth to groundwater, z w (m) Figure 4: Infuence of the depth of groundwater z w on F S. groundwater surface. Then, Figure 4 shows that increasing both z w and c 1 produces an increasing safety, and this can be easiy understood. 4. Concusions In this paper, we have presented an anaytica soution of the safety factor for the three-dimensiona mode of a surry trench with an incined ground surface. The soution incudes severa parameters, such as the incined ange of the incined ground surface, trench ength and depth, surry depth, temporary oad, and groundwater surface eevation. The resuts showed that increasing incined ange β and trench ength L resuts in decreasing the safety factor, but increasing the depth to groundwater z w and suction effect c 1 produces an increase in the safety factor; meanwhie, the existing two-dimensiona mode was conservative compared tothepresentthree-dimensionamode.thestudycoudbe usefu to assess the trench stabiity for both D and 3D cases with incined ground surface. Confict of Interests The authors decare that there is no confict of interests regarding the pubication of this paper. Acknowedgments The financia support received from Nationa Natura Science Foundation of China (NSFC), Grant no , and Key Projects in the Nationa Science & Technoogy Piar Program during the Twefth Five-Year Pan Period, Grant no. 011BAJ10B08, is gratefuy acknowedged. The authors aso thank Professor Dr. Qiang Chen, Southeast University, for the Engish correction. References [1] T. Y. Poh and I. H. Wong, Effects of construction of diaphragm wa panes on adjacent ground: fied tria, Geotechnica and Geoenvironmenta Engineering, vo.14,no.8,pp , [] C. W. W. Ng and R. W. M. Yan, Three-dimensiona modeing of a diaphragm wa construction sequence, Geotechnique, vo. 49,no.6,pp ,1999. [3] S. M. Gourvenec and W. Powrie, Three-dimensiona finiteeement anaysis of diaphragm wa instaation, Geotechnique, vo. 49, no. 6, pp , [4] C.W.W.NgandG.H.Lei, Anexpicitanayticasoutionfor cacuating horizonta stress changes and dispacements around an excavated diaphragm wa pane, Canadian Geotechnica Journa,vo.40,no.4,pp ,003. [5] R. Schäfer and T. Triantafyidis, Modeing of earth and water pressure deveopment during diaphragm wa construction in soft cay, Internationa Journa for Numerica and Anaytica Methods in Geomechanics, vo. 8, no. 13, pp , 004. [6] Y.Arai,O.Kusakabe,O.Murata,andS.Konishi, Anumerica study on ground dispacement and stress during and after the instaation of deep circuar diaphragm was and soi excavation, Computers and Geotechnics,vo.35,no.5,pp , 008. [7]R.Conti,L.deSanctis,andG.M.B.Viggiani, Numerica modeing of instaation effects for diaphragm was in sand, Acta Geotechnica,vo.7,no.3,pp.19 37,01. [8] E. M. Comodromos, M. C. Papadopouou, and G. K. Konstantinidis, Effects from diaphragm wa instaation to surrounding soi and adjacent buidings, Computers and Geotechnics,vo. 53, pp , 013. [9]G.H.Lei,H.S.Sun,andC.W.W.Ng, Anapproximate anaytica soution for cacuating ground surface settements due to diaphragm waing, Computers and Geotechnics,vo.61, pp , 014. [10]J.K.T.L.NashandG.K.Jones, Thesupportoftrenches using fuid mud, in Proceedings of the Symposium on Grouts and Driing Muds in Engineering Practice, pp ,Butterworths, London, UK, [11] N. Morgenstern and I. Amir-Tahmasseb, The stabiity of a surry trench in cohesioness sois, Géotechnique,vo.15,no.4, pp , [1] P. P. Xanthakos, SurryWasasStructuraSystems,McGraw- Hi,NewYork,NY,USA,1994. [13] E. G. Prater, Die Gewöbewirkung der Schitzwände, Der Bauingenieur,vo.48,no.4,pp ,1973. [14] J. Washbourne, The three-dimensiona stabiity anaysis of diaphragm wa excavations, Ground Engineering,vo.17,no.4, pp. 4 9, [15] A. Piaskowski and Z. Kowaewski, Appications of thixotropic cay suspensions for stabiity of vertica sides of deep trenches without strutting, in Proceedings of the 6th Internationa ConferenceonSoiMechanicsandFoundationEngineering,vo.,pp , [16] J.-S. Tsai and J.-C. Chang, Three-dimensiona stabiity anaysis for surry-fied trench wa in cohesioness soi, Canadian Geotechnica Journa, vo. 33, no. 5, pp , [17] J.-S. Tsai, L.-D. Jou, and H.-S. Hsieh, A fu-scae stabiity experiment on a diaphragm wa trench, Canadian Geotechnica Journa,vo.37,no.,pp ,000.

7 Advances in Materias Science and Engineering 7 [18] P. J. Fox, Anaytica soutions for stabiity of surry trench, Geotechnica and Geoenvironmenta Engineering,vo. 130, no. 7, pp , 004. [19] Y.-C. Li, Q. Pan, and Y.-M. Chen, Stabiity of surry trenches with incined ground surface, Geotechnica and Geoenvironmenta Engineering, vo.139,no.9,pp , 013. [0] D. G. Fredund and H. Rahardjo, Soi Mechanics for Unsaturated Sois, Wiey, New York, NY, USA, [1] P. J. Fox, Discussion of stabiity of ong trenches in sand supported by bentonite-water surry by George M. Fiz, Tiffany Adams, and Richard R. Davidson, Geotechnica and Geoenvironmenta Engineering,vo.13,no.5,p.666,006.

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