土的固结理论（讲稿）Consolidation of soils

THEHONGKONGDEPARTMENTOFCIVILANDENVIRONMENTALENGINEERINGPOLYTECHNICUNIVERSITY土木及崋境工程系香港理工大學Consolidationof soils(1-D, 2-D and 3-D)Prof. Jian-Hua YINRm ZS909, Tel: 2766 6065, Email: cejhyin@polyu.edu.hkHomepage:http://www.zn903.com/cejhyin/Dr.Wen-BoCHENRmZS969,Tel:34008075,Email:wenbochen@polyu.edu.hk

Consolidation of soils (1-D, 2-D and 3-D) Prof. Jian-Hua YIN Rm ZS909, Tel: 2766 6065, Email: cejhyin@polyu.edu.hk Homepage: http://www.zn903.com/cejhyin/ Dr. Wen-Bo CHEN Rm ZS969, Tel: 3400 8075, Email: wenbochen@polyu.edu.hk

THEHONGKONGDEPARTMENTOFCIVILANDENVIRONMENTALENGINEERINGPOLYTECHNICUNIVERSITY土木及璨境工程季系香港理工大學>EssentialReferencesLecture notesCraig,R.F.(2004). Soil Mechanics, 7th edition, Spon Press,Londonand New York (ISBN 04-415-32702-2) (or more updatedversion).Das, Braja M.(2007).Principles of Foundation Engineering (6thedition),Thomson,UnitedStates(ISBN0-534-40752-8)(ormoreupdated version).2

➢ Essential References • Lecture notes • Craig, R.F. (2004). Soil Mechanics, 7th edition, Spon Press, London and New York (ISBN 04-415-32702-2) (or more updated version). • Das, Braja M. (2007). Principles of Foundation Engineering (6th edition), Thomson, United States (ISBN 0-534-40752-8) (or more updated version). 2

THEHONGKONGDEPARTMENTOFCIVILANDENVIRONMENTALENGINEERINGPOLYTECHNICUNIVERSITY土木及璟境工程学系香港理工大學>Learningoutcomes ofthis lectureUnderstandthederivationprocess of Terzaghi'stheoryofonedimensional consolidation and its 8 assumptions;Understand the meaning of coefficient of consolidation, cy, andwhat soil properties it includes;Masterthecalculationmethodforconsolidationsettlementofsoilswithout or with vertical drains;Understandthemechanismof soilcreep,andmasterthesimplifiedmethod for calculation of consolidation settlement of clayey soilswith creep.3

➢ Learning outcomes of this lecture • Understand the derivation process of Terzaghi’s theory of one￾dimensional consolidation and its 8 assumptions; • Understand the meaning of coefficient of consolidation, cv , and what soil properties it includes; • Master the calculation method for consolidation settlement of soils without or with vertical drains; • Understand the mechanism of soil creep, and master the simplified method for calculation of consolidation settlement of clayey soils with creep. 3

Reviewof SoilMechanicsI.Terzaghi'stheoryofone-dimensional consolidationReviewonconsolidationprocessforsaturatedsoilsChange in total stress (Ao) generates excess pore pressure, ue, because soilis undrained under initial loading.Overtime waterflows outof soil inresponseto change intotal stressAs water flows out, soil undergoesa.a decrease in magnitude ofub.anincreasein o'asu.decreases(principle ofeffective stress:'=g-ue)adecreaseinvolumegeneratingsettlement,sC.Rate of consolidation will depend ona.soilproperties:compressibility&hydraulicconductivityb.distributionofinitialexcessporepressure,ueC.drainageboundaryconditionsAs soil consolidates, ue +, o'↑, and S ↑4

4 Review on consolidation process for saturated soils • Change in total stress (Δσ) generates excess pore pressure, ue , because soil is undrained under initial loading • Over time water flows out of soil in response to change in total stress • As water flows out, soil undergoes a. a decrease in magnitude of ue b. an increase in σ’ as ue decreases (principle of effective stress: σ’ = σ- ue ) c. a decrease in volume generating settlement, S • Rate of consolidation will depend on a. soil properties: compressibility & hydraulic conductivity b. distribution of initial excess pore pressure, ue c. drainage boundary conditions • As soil consolidates, ue ↓, σ’ ↑, and S ↑ Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional consolidation

ReviewofSoilMechanicsI.Terzaghi's theory of one-dimensional consolidationReview on consolidation process forsaturated soilssurchargegroundAo0.ut5

5 Review on consolidation process for saturated soils t t t t Δσz σz u σ’z surcharge ground Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional consolidation

ReviewofSoilMechanicsI.Terzaghi'stheoryof one-dimensional consolidation8assumptionsforTerzaghi'stheoryof one-dimensional consolidation1. The soil is homogeneous (ok for each layer)2.The soil isfully saturated (okforsoil underwater)3. The solid particles and water are incompressible (ok)4. Compression andflow are one-dimensional (vertical) (ok)5. Strains are small (ok for most civil engineering problems)6. Darcy's law is valid at all hydraulic gradients (ok for water in common civilengineeringproblems)7. The permeability k and volume compressibility my are constants through theprocess(approximateforalayerandsmallpressure)8. There is a unique relationship, independent of time, between void ratio andeffectivestress(approximateforalayerand siltysoils,nogoodforsoftclay)6

6 8 assumptions for Terzaghi’s theory of one-dimensional consolidation 1. The soil is homogeneous (ok for each layer) 2. The soil is fully saturated (ok for soil underwater) 3. The solid particles and water are incompressible (ok) 4. Compression and flow are one-dimensional (vertical) (ok) 5. Strains are small (ok for most civil engineering problems) 6. Darcy’s law is valid at all hydraulic gradients (ok for water in common civil engineering problems) 7. The permeability k and volume compressibility mv are constants through the process (approximate for a layer and small pressure) 8. There is a unique relationship, independent of time, between void ratio and effective stress (approximate for a layer and silty soils, no good for soft clay) Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional consolidation

Reviewof SoilMechanicsI.Terzaghi's theory of one-dimensional consolidationa(z + us + ue)VzkaueahYwVz =kiz =-kazazYwazus = hs = constant (static total head)dy:z+YwavzkauekauedzdlVz + dvz = Vz +azaz2YwazYwThe condition of continuitydxnetwatercoming out rate=volume compressionrateOv.dzV,+dyOzka2ueaavdxdy(vz+dvz)out-dxdy(vz)inxdydzYwaz2atTotal volume is:dg'aVa(e,dxdydz)a(mvo')Dedxdydz :dxdydz=mydxdydzV= dxdydzatatatatat[α - (us + ue)]aueV.dxdydzdxdydz ==my-mvatatflow-inrate of waterka?ueavkaueaue dxdydzV, +dv.:xdydzdydzmvYw0z2atYw0z2atflow-outrateofwaterka'ueduek a2ueaueCv刀moatYw 0z2ataz2mvYw7Cy is the coefficient of consolidation

7 Total volume is: V = dxdydz vz : flow-in rate of water vz +dvz : flow-out rate of water 𝑣𝑧 = 𝑘𝑖𝑧 = −𝑘 𝜕ℎ 𝜕𝑧 = −𝑘 𝜕(𝑧 + 𝑢𝑠 + 𝑢𝑒 𝛾𝑤 ) 𝜕𝑧 = − 𝑘 𝛾𝑤 𝜕𝑢𝑒 𝜕𝑧 ∵ 𝑧 + 𝑢𝑠 𝛾𝑤 = ℎ𝑠 = constant (static total head) 𝑣𝑧 + 𝑑𝑣𝑧 = 𝑣𝑧 + 𝜕𝑣𝑧 𝜕𝑧 𝑑𝑧 = − 𝑘 𝛾𝑤 𝜕𝑢𝑒 𝜕𝑧 − 𝑘 𝛾𝑤 𝜕 2𝑢𝑒 𝜕𝑧 2 𝑑𝑧 The condition of continuity: net water coming out rate = volume compression rate 𝑑𝑥𝑑𝑦 𝑣𝑧 + 𝑑𝑣𝑧 𝑜𝑢𝑡 − 𝑑𝑥𝑑𝑦 𝑣𝑧 𝑖𝑛 = − 𝑘 𝛾𝑤 𝜕 2𝑢𝑒 𝜕𝑧 2 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝜕Δ𝑉 𝜕𝑡 𝜕Δ𝑉 𝜕𝑡 = 𝜕(𝜀𝑣𝑑𝑥𝑑𝑦𝑑𝑧) 𝜕𝑡 = 𝜕𝜀𝑣 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝜕 𝑚𝑣𝜎 ′ 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝑚𝑣 𝜕𝜎 ′ 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝑚𝑣 𝜕[𝜎 − (𝑢𝑠 + 𝑢𝑒 )] 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 = −𝑚𝑣 𝜕𝑢𝑒 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 ∴ − 𝑘 𝛾𝑤 𝜕 2𝑢𝑒 𝜕𝑧 2 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝜕𝑉 𝜕𝑡 ⇒ − 𝑘 𝛾𝑤 𝜕 2𝑢𝑒 𝜕𝑧 2 𝑑𝑥𝑑𝑦𝑑𝑧 = −𝑚𝑣 𝜕𝑢𝑒 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 𝑚𝑣 𝜕𝑢𝑒 𝜕𝑡 = 𝑘 𝛾𝑤 𝜕 2𝑢𝑒 𝜕𝑧 2 ⇒ 𝜕𝑢𝑒 𝜕𝑡 = 𝑐𝑣 𝜕 2𝑢𝑒 𝜕𝑧 2 𝑐𝑣 = 𝑘 𝑚𝑣𝛾𝑤 cv is the coefficient of consolidation Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional consolidation

ReviewofSoilMechanicsI.Terzaghi's theory of one-dimensional conditiona2ueaueTosolvethegoverningequation:0z2atOpenlayerTheinitialvalueofexcessporewaterpressure(initialcondition)is(Free drainage)for0≤z≤2dwhent=0ue=uiTheboundary conditions ofexcessporewaterpressureare:zforz=owhent>0u= 0for z =2dwhen t > ou=02dUsingmethod of"separation of variables",the solutionfortheexcesspore waterpressureatdepthz aftertime tis:n元cutc2d Jo"ui sin nz)ue = Zn=(1)X4d2Openlayerwhered=lengthofthe longestdrainagepathandu=initialexcessporewater(Freedrainage)pressure,ingeneralafunctionofz.d:length of longestdrainage pathFora particular case in which ui is constantthroughout the clay layer:z:depthtothetop surfacen'元'Cutue = H=P (1 - cosn)(sn (2)Dexp4d22dWhenniseven,(1-cosnπ)=0andwhennisodd,(1-cosn)=2.Onlyoddvalues of n are therefore relevant, and it is convenientto make the substitutions8

8 To solve the governing equation: 𝜕𝑢𝑒 𝜕𝑡 = 𝑐𝑣 𝜕 2𝑢𝑒 𝜕𝑧 2 The initial value of excess pore water pressure (initial condition) is: for 0 ≤ 𝑧 ≤ 2𝑑 when 𝑡 = 0 𝑢𝑒 = 𝑢𝑖 The boundary conditions of excess pore water pressure are: for 𝑧 = 0 when 𝑡 > 0 𝑢𝑒 = 0 for 𝑧 = 2𝑑 when 𝑡 > 0 𝑢𝑒 = 0 Using method of “separation of variables”, the solution for the excess pore water pressure at depth z after time t is: 𝑢𝑒 = σ𝑛=1 𝑛=∞ 1 𝑑 0׬ 2𝑑 𝑢𝑖 sin 𝑛𝜋𝑧 2𝑑 exp − 𝑛 2𝜋 2 𝑐𝑣𝑡 4𝑑2 (1) where d = length of the longest drainage path and 𝑢𝑖 = initial excess pore water pressure, in general a function of z. For a particular case in which 𝑢𝑖 is constant throughout the clay layer: 𝑢𝑒 = σ𝑛=1 𝑛=∞ 2𝑢𝑖 𝑛𝜋 (1 − cos𝑛𝜋)(sin 𝑛𝜋𝑧 2𝑑 ) exp − 𝑛 2𝜋 2 𝑐𝑣𝑡 4𝑑2 (2) When n is even, 1 − cos𝑛𝜋 = 0 and when n is odd, 1 − cos𝑛𝜋 = 2 . Only odd values of n are therefore relevant, and it is convenient to make the substitutions Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional condition Open layer (Free drainage) Open layer (Free drainage) 2d 𝑧 d: length of longest drainage path z: depth to the top surface

Review of SoilMechanicsI.Terzaghi's theory of one-dimensional conditionOpenlayern元ctue = En= 2ui (1 - cosnn)(sin 2(2)(Freedrainage)n=14d22dnTEquation(2)thenbecomesue=Zm=2（sin）exp(-M2T）(3)Z2dSetn=2m+1C,tTu =(adimensionlessnumbercalledthe TimeFactor)d2TOpenlayerM :-(2m+ 1)2(Free drainage)2uem=00exp(-M(4)(sinZm=0MduzThreedimensionlessCutueTdparameters:12d2ui9

9 𝑢𝑒 = σ𝑛=1 𝑛=∞ 2𝑢𝑖 𝑛𝜋 (1 − cos𝑛𝜋)(sin 𝑛𝜋𝑧 2𝑑 ) exp − 𝑛 2𝜋 2 𝑐𝑣𝑡 4𝑑2 (2) 𝑛 = 2𝑚 + 1 𝑇𝑣 = 𝐶𝑣 𝑡 𝑑 2 (a dimensionless number called the Time Factor) 𝑀 = 𝜋 2 (2𝑚 + 1) Set Equation (2) then becomes 𝑢𝑒 = σ𝑚=0 𝑚=∞ 2𝑢𝑖 𝑀 (sin 𝑀𝑧 𝑑 ) exp −𝑀2𝑇𝑣 (3) Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional condition Open layer (Free drainage) Open layer (Free drainage) 2d 𝑧 𝑢𝑒 𝑢𝑖 = σ𝑚=0 𝑚=∞ 2 𝑀 (sin 𝑀𝑧 𝑑 ) exp −𝑀2𝑇𝑣 (4) Three dimensionless parameters: 𝑢𝑒 𝑢𝑖 𝑧 𝑑 𝑇𝑣 = 𝐶𝑣 𝑡 𝑑 2

ReviewofSoilMechanicsI.Terzaghi'stheory of one-dimensional condition0OpenlayerCut(Freedrainage)TV=d20.550n2dFa1.0N220+36%1TEOITLKKK1中We1.5OOpenlayer1R(Freedrainage)2.00.51.0uelui10

10 Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional condition Open layer (Free drainage) Open layer (Free drainage) 2d 𝑢𝑒