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