FEM ANALYSES OF CUTTING OF ANISOTROPIC DENSELY COMPACTED AND SATURATED SAND

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1 FEM ANALYSES OF CUTTING OF ANISOTROPIC DENSELY COMPACTED AND SATURATED SAND Jisong He 1, W.J. Vlasblom 2 and S. A. Miedema 3 ABSTRACT Te literature studies sow tat until now, te existing investigations on te utting of densely ompated and saturated sand are involved only in te range of isotropi soil wit regard to permeability. Te permeability of sand is one of te most important parameters in saturated sand utting. Te properties of te isotropi soil tat are responsible for te resistane to flow are independent of te diretion. However, in many soil deposits te resistane to flow in te vertial diretion is onsiderably larger tan te resistane to orizontal flow, due to te presene of te layered struture in te soil, generated by its geologial istory. It is neessary to investigate te utting of anisotropi densely ompated and saturated sand. Tis paper builds up a matematial modeling of te utting of anisotropi densely ompated and saturated sand and performs finite element analysis of saturated sand utting wit te ratio / v1, 2, 4, 6, 8, 10 of permeability of soil in te orizontal diretion to tat in te vertial diretion, and wit various sear angles ranging from 15 degree to 35 degree. Te results sow te utting fores required for anisotropi densely ompated and saturated sand will inrease wit te ratio /. Te utting fore for a soil wit / 10 is about 18 % larger tan tat of an isotropi sand v v Keywords: finite element metod, soil, dilatany, pore water pressure, permeability, INTRODUCTION Wen te utting of soil is investigated, te omogeneous soil as been almost all assumed in order to simplify te problems (Os, 1976 and 1987; Miedema, 1985, 1987). However, generally speaing, soil in te real world is anisotropi. Wat is te differene wen te anisotropi soil is used instead of omogeneous soil? Te utting of saturated anisotropi soil is a very ompliated subjet beause it as effets on te strengt and permeability and oter parameters of soil. However, dilatany is more important in te utting of saturated sand. In te paper te effets of anisotropi permeability of densely ompated and saturated sand on te utting fores will be disussed, beause pore water pressure plays a most important role in te utting of densely ompated and saturated sand and permeability as strong effets on te ange of te pore water pressure. MATHEMATICAL MODEL OF CUTTING OF ANISOTROPIC SATURATED SOIL Te literature studies sow tat te investigations on te utting of densely ompated and saturated sand are involved in te range of isotropi soil wit regard to permeability. Te properties of tis type of te soil tat are responsible for te resistane to flow are independent of te diretion. However, in many soil deposits te resistane to flow in te vertial diretion is onsiderably larger tan te resistane to orizontal flow, due to te presene of a layered struture in te soil, generated by its geologial istory. Only te two-dimensional utting of anisotropi soil is onsidered in tis paper. For two-dimensional anisotropi soil, Dary s law an be expressed: 1 Engineer, HL engineering ompany, Oeverwallaan 37, Den Haag, 2498 BV, Te Neterlands, T: , Fax: , . jisonge@otmail.nl 2 Professor, Delft University of Tenology, Faulty of Meanial Engineering, Meelweg 2,2628 CD Delft, Te Neterlands, T: ,Fax: , . W.J.Vlasblom@wbmt.tudelft.nl 3 Dr. Ir, Delft University of Tenology, Faulty of Meanial Engineering, Meelweg 2,2628 CD Delft, Te Neterlands T: ,Fax: , . S.A.Miedema@wbmt.tudelft.nl 43

2 p qx xy q y xy p Tis matrix expresses te most general linear relationsip between te speifi disarge vetor and te gradient of te pore pressure. Te permeability matrix is a symmetri matrix, so tere exist two mutually ortogonal diretions, te so-alled prinipal diretions of permeability, in wi te ross-omponents disappear. Wen te soil is ortogonal, equation 1 beomes: p qx 0 q y 0 p Pysially speaing, tis means tat a gradient of te pore water pressure in one of tese diretions leads to a flow in te same diretion. Te soil to be ut onsists of two ortogonal pore annels wit different ross setions and wit different resistanes, see Figure 1. Te prinipal diretions oinide wit te diretion of te pore annels (1) (2) pore annels sand Figure 1. Two-dimensional model If te pore annels in te x-diretion are wider tan tose in te y-diretion, te permeability will be greater tan. Now we alulate te values of permeability in anoter oordinate system ( ξ, η ) wi rotates an angle α wit te oordinate system ( x, y ). Tere is a relationsip between tese two oordinate systems: ξ osα sinα x η sinα osα y A vetor q wit omponents x y qξ osα sinα qx q sin os q η α α y q and q an be deomposed into omponents Substituting equation 1 into te matrix above, one obtains: qξ and qη : (3) (4) 44

3 p qξ osα sinα xy q sin os xy p η α α (5) Te partial derivative p and p p an be expressed by, and ξ p wit equation (3): η p p ξ p η p x + ξ η osα sinα ξ p p ξ p η sinα osα p y + ξ η η Substituting te equation above into equation 5, one obtains: p qξ osα sinα 0 osα sinα ξ q sin os 0 η α α sinα osα p η Te speifi arge qξ, qη an also be written as: (6) (7) p qξ ξξ ξη ξ q ξη η ηη p η From te above two equations, one obtains: ξξ ξη osα sinα 0 osα sinα ξη ηη sinα osα 0 sinα osα Tey are expanded as: + ξξ os α + sin α os 2α + ηη os α + sin α + os 2α ξη ( )sinαosα sin 2α 2 Te equation above indiates wen te soil is isotropi namely, or wen te oordinate system ( ξη, ) oinides wit oordinate system (x, y), ξη 0. Te equation above desribes a general flow using te oordinate system ( ( ξη, ). It sows tat a gradient of pore water pressure in ξ diretion not only leads to a flow in its diretion but also to a flow in η -diretion. Te annels in x-diretion will pysially transport mu more water tan te narrow annels in y-diretion beause is greater tan. In general te resultant flow will always ave a tendeny towards te most permeable diretion. Te anisotropi law sould be of te form of equation 3. In engineering pratie te ortogonal situation is usually aeptable to distinguis only between te permeability in vertial diretion and te one in orizontal diretion, assuming tat tis differene as been reated during te geologial proess of deposition of te soil. It is assumed tat te x, y diretions of permeability are its prinipal diretions. (8) (9) (10) 45

4 In te utting of saturated sand, storage equation is one of te basi equations, wi is from Biot s teory of onsolidation (Biot, 1941). Tis equation expresses tat volumetri deformations of te soil must be aompanied by a ompression or expulsion of te pore water. Storage equation of soil utting is: e q ( x q y + ) t (11) Te volumetri strain rate e/ t an be expressed in te ange of porosity, using te assumption tat te soil partiles are inompressible. e 1 n (12) t 1 n t Substituting te equation above and equation 1 into equation 13, one obtains: q q x y 1 n ( + ) (13) 1 n t p p p n + + ρ g xy n t ( 2 ) 2 2 xy w 1 (14) In ontinuous utting proess, it is onvenient to introdue a moving oordinate system, wit ζ x vt (15) Were v is te veloity of te utting blade. Equation 14 beomes: p p p v n + + ρ g ζ y n ζ 2 1 ( 2 ) ζζ ζ y w ζ 1 (16) For te saturated soil utting, te soil failure is onsidered to our in sear areas R, so tat te following equations desribe te utting problem: ρ p p p v n ρ 2 1 p p p ( xy ) 0 (x,y) R wg x y 2 1 ( 2 ) + + xy (x,y) R wg x y 1 n x (17) Wen oordinate system (x, y) is te prinipal oordinate system of te permeability, te equations above beome: ρ p p v n ρ 1 p p ( ) 0 (x,y) R wg 1 ( ) + (x,y) R wg 1 n x Using te following relationsips, one an mae te dimensionless of te equation 18. (18) 46

5 1 q q ρw g ev / x' x/ y' y/ max (19) It an be rewritten as: p p + 0 (x', y') R max ' max ' p p + (x', y') R max ' max ' x MODELING AND ANALYSES OF FEM Beause te alulation formulae of te utting fores for te anisotropi saturated soil are te same as tose for isotropi saturated soil. Only te effets of te pore water pressure are onsidered beause of te anisotropi permeability of te soil. Figure 2 is a FEM model of te utting of anisotropi saturated soil. In te model, te permeability of te initial saturated soil is 0.25 and / 1, 2, 4, 6, 8, 10. Te permeability of te ut soil in te rigid wedge is isotropi: max 1. Te utting angle of te blade is 60 degree; searing angle β is respetively 15, 20, 25, 30 and 35 degrees. Ratio of te vertial eigt of te blade to te utting dept is 3. Te results of te alulations of dimensionless average pore pressures distributed bot on te utting blade (pblade) and on te sear surfae (p-sear), and of orizontal-utting fores for different soil wit internal frition angles of 20 to 45 degrees are listed in table 1. Beause te alulations are dimensionless, te following equation is valid: p ( ρ g e v / ) p (21) w max al (20) p 0 p 0 max v p 0 p 0 p 0 n n α q 2 i i l q 1 p 0 n n p 0 n n p 0 n n Figure 2: FEM model of saturated sand utting 47

6 Table 1: Results of FEM analyses of utting of anisotropi saturated soil Pore pressure F β / p-blade p-sear ϕ 20º ϕ 25º ϕ 30º ϕ 35º ϕ 30º ϕ 45º From table 1, we obtain te relationsip urves between te dimensionless pore water pressures and various ratio of te orizontal omponent to vertial omponent of permeability of te initial soil. Figure 3a expresses te relationsips between te pore pressures on te sear plane and on te utting blade and te ratio of te omponents to omponent of te permeability at sear angles 25 degrees and at a utting angle of 60 degree. Figure 3b express te relationsips between te utting fore F and te ratio of te omponents to te omponent of te permeability for different soil wit internal frition angles from 20 ~ 45 degrees at sear angles from 15 ~ 35 degrees. 48

7 pore pressure VS Kx / Ky at beta20 F VS Kx /Ky at beta20 deg pore pressure p-blade p-sear F Pi20 deg Pi25 deg Pi30 deg Pi35 deg Pi40 deg Pi45 deg Kx / Ky Kx /Ky Figure 3. Cutting fores F vs / at α60 º, β25 º Fi / F1 VS Kx / Ky Fi / F Kx / Ky beta15 deg beta20 deg beta25 deg beta30 deg beta35 deg Figure 4. Ratio of utting fores F F 1 vs at α60 ºFigure 5 Figure 4 expresses te relationsip between te ratio of te utting fore from te anisotropi soil to tat from te isotropi soil and te ratio of te omponent to te omponent of te permeability of te soil. From tis figure, wen te ratio of te omponent to omponent of te permeability of te soil is less tan 2, te error of using te alulation results from te isotropi models representing tose from te anisotropi model is less tan 7 %. However as te anisotropi degree inreases, te error will also inreases. Figure 5 expresses te distribution of te exess pore water pressures for te model of te utting of anisotropi soil wile figure 6 expresses te distribution of te exess pore water pressures for te model of te utting of isotropi soil. After omparing tese two figures, it an be found tat te distribution of exess pore water pressures for isotropi soil appears vertial ellipse wile te distribution for anisotropi soil appears a flatter ellipse and tat te values of te exess pore water pressures from te utting model of anisotropi soil are iger tan tose from te utting model of isotropi soil. Tese sow tat for anisotropi model pore water flows into sear zones mainly from orizontal diretion during soil utting, and tat te resistane of pore water flowing into sear zones is larger tan tat of te isotropi model. 49

8 Figure 6: Distribution of te exess pore water pressures of anisotropi soil model 50

9 Figure 7: Distribution of te exess pore water pressures of isotropi soil model 51

10 CONCLUSIONS From te analyses above, te following onlusions an be made: In general speaing, te utting fores will inrease as te ratio of te omponent to omponent of te permeability of te soil. Wen tis ration is less tan 2, te error of using alulation results from isotropi soil models to represent tose from anisotropi soil models is less tan 7%. Wen tis ration is 10, te error of using alulation results from isotropi soil models to represent tose from anisotropi soil models is less tan 20%. In general speaing, te models of te utting of saturated omogeneous soil are preise enoug for te pratial situations as is onerned te permeability of te anisotropi soil. Beause of anisotropi property of soil, during te utting of densely ompated and saturated anisotropi soil, pore water flows into te sear zones mainly in te orizontally diretion. REFERENCES Biot, M.A. (1941): General teory of tree dimensional onsolidation. Journal of Applied Pysis, 12, He, Jisong (1997): Failure Patterns of te 2-D Soil Cutting in Tunneling. Report number: 97.3 VG.4949, TU Delft, te Neterlands Kesteren, W. M.G. van, Steegs, H. J.M.G. and Mastbergen, D.R. (1992): Pore water beaviour in dredging proesses. Proeedings WODCON XIV, Bombay, India Leussen, W. van and Niewenuis J.K. (1984): Soil meanis aspets of dredging. Geotenique 34(3), Miedema, S.A. (1985A): Matematial modeling of te utting of densely ompated sand under water. Journal of Dredging & Port Constrution, July, Miedema, S.A. (1985B): Derivation of te differential equation for sand pore pressures. Journal of Dredging & Port Constrution, September, 35 Miedema, S.A. (1987): Calulation of te utting fores wen utting water saturated sand, basi teory and appliation for 3-D blade movements and periodially varying veloities in dredging ommonly used exavating means, PD dissertation, Delft University of Tenology, te Neterlands Os, A.G. van (1976): Beaviour of soil wen exavated underwater. International Course Modern Dredging, Te Hague, te Neterlands Os, A.G. van and Leussen, W. van (1987): Basi resear on utting fores in saturated sand. Journal of te Geotenial Engineering Division, 102, No. GT4, Verruijt, A. (1970): Teory of groundwater flow, London: Mamillan Zieniewiz, O.C. and Taylor, R. L. (1989): Te finite element metod, New Yor: MGraw-Hill Boo Company 52