Reviewof SoilMechanicsI.Terzaghi's theory of one-dimensional conditionCite=Zm=（(sinm1)exp（-M2T）(3)Tod2constSlopesiTwo sides are20free drainageHalfclosedlayeraOpen layerui= constant alongdepthT=0.3T,=0.3T=0.05T,=0.7T,=0.8Tv=0.0588全=0=Od2d008HalfclosedlayerOpenlayerOnesideis freedrainage,anothersideisTwo sides are free drainageimpermeableui is triangular distributionu; is triangular distribution11
11 𝑢𝑒 = σ𝑚=0 𝑚=∞ 2𝑢𝑖 𝑀 (sin 𝑀𝑧 𝑑 ) exp −𝑀2𝑇𝑣 (3) 𝑇𝑣 = 𝐶𝑣 𝑡 𝑑 2 𝑢𝑖 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 along depth 𝑢𝑖 Two sides are free drainage One side is free drainage, another side is Two sides are free drainage impermeable is triangular distribution 𝑢𝑖 is triangular distribution Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional condition
Review of SoilMechanicsI.Terzaghi's theory of one-dimensional conditionDegreeofconsolidationThe degree ofconsolidationatdepthzandtimetcanbe obtained asm=α2MzC,tue = 1-NU,=1.-) exp(-M2T,)(sinMuid2m=0The average degree of consolidation (U) over the depth of the layer as a whole is of interestfor ui = constant along depth12drhr2d mv(oe - 00)dzJ,'(ui - ue)dzuedzJ" EvedzuedzSt2dJoU=2duiuiSfJ" uidzevidzJ" mv(o -)dznm=002Z:U=1Mzexp(- M2T))m=0foru;isnotconstant(uj - ue)dzd2StU=Sfouidzu;dzGood approximate solution for U (curve 1 in next slide):for U0.60, Tv==u2forU>0.60,Tv=-0.933log(1-U)-0.08512
12 Degree of consolidation 𝑈𝑧 = 1 − 𝑢𝑒 𝑢𝑖 = 1 − 𝑚=0 𝑚=∞ 2 𝑀 (sin 𝑀𝑧 𝑑 ) exp −𝑀2𝑇𝑣 The degree of consolidation at depth z and time t can be obtained as: The average degree of consolidation (U) over the depth of the layer as a whole is of interest 𝑓𝑜𝑟 𝑢𝑖 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 along depth 𝑈 = 𝑠𝑡 𝑠𝑓 = 0 𝐻 𝜀𝑣𝑒𝑑𝑧 0 𝐻 𝜀𝑣1𝑑𝑧 = 0 𝐻 𝑚𝑣(𝜎𝑒 ′ − 𝜎0 ′ )𝑑𝑧 0 𝐻 𝑚𝑣(𝜎1 ′ − 𝜎0 ′ )𝑑𝑧 = 0 𝐻 (𝑢𝑖 − 𝑢𝑒)𝑑𝑧 0 𝐻 𝑢𝑖𝑑𝑧 = 1 − 0 2𝑑 𝑢𝑒𝑑𝑧 2𝑑𝑢𝑖 = 1 − 1 2𝑑 0 2𝑑 𝑢𝑒𝑑𝑧 𝑢𝑖 𝑈 = 𝑠𝑡 𝑠𝑓 = 0 𝐻 (𝑢𝑖 − 𝑢𝑒)𝑑𝑧 0 𝐻 𝑢𝑖𝑑𝑧 = 1 − 0 2𝑑 𝑢𝑒𝑑𝑧 0 2𝑑 𝑢𝑖𝑑𝑧 ∴ 𝑈 = 1 − 𝑚=0 𝑚=∞ 2 𝑀2 exp( − 𝑀2𝑇𝑣) Good approximate solution for U (curve 1 in next slide): 𝑓𝑜𝑟 𝑈 < 0.60, 𝑇𝑉= 𝜋 4 𝑈 2 𝑓𝑜𝑟 𝑈 > 0.60, 𝑇𝑉 = −0.933 log 1 − 𝑈 − 0.085 𝑓𝑜𝑟 𝑢𝑖 is not constant Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional condition 𝑇𝑣 = 𝐶𝑣 𝑡 𝑑 2
Reviewof SoilMechanicsI.Terzaghi'stheoryofone-dimensional conditionDegree of consolidationTwo sidesarefreeBottomsides2ddrainageimpermeableCurve (2)Curve (1)Curve (3)Curve (1)Curve (1)Curve (1)(b)Half-closed layers(a)OpenlayersInitialvariationsofexcessporewaterpressure010020030210.40U0500600.70C,t0.80Tud20.90000001001C,tT:d213Relationshipsbetweenaveragedegreeofconsolidationandtimefactor
13 Degree of consolidation Relationships between average degree of consolidation and time factor Two sides are free drainage Bottom side is impermeable Initial variations of excess pore water pressure Review of Soil Mechanics I. Terzaghi’s theory of one-dimensional condition 𝑇𝑣 = 𝐶𝑣 𝑡 𝑑 2
Reviewof SoilMechanicsIl.DeterminationofcoefficientofconsolidationkT,d?CtCuC1d2mYwtWhenthesizeofground ismorethan3-5timeslargerthanthethicknessofsoil layer,thesettlement of soil canbe regardedas 1-D compression condition.Wenormallyadopt oedometertest in laboratoryto investigatethe behaviourofsoil.??64O> The log time method (due to Casagrande)?d:half of average height of0.196d2speciment5otso: the time needed for 50%degreeofconsolidationConventionaloedometercell> The root time method (due to Taylor)①Loading capd: half of average height ofPorousstone0.848d2specimenClampingnutTopflangeCvtso:thetime needed for90%to③Externalbox@Oedometerringdegreeof consolidation14
14 𝑐𝑣 = 𝑘 𝑚𝑣𝛾𝑤 ∵ 𝑇𝑣 = 𝑐𝑣𝑡 𝑑 2 ; ∴ 𝑐𝑣 = 𝑇𝑣𝑑 2 𝑡 ➢ The log time method (due to Casagrande) ➢ The root time method (due to Taylor) 𝑐𝑣 = 0.196𝑑 2 𝑡50 𝑐𝑣 = 0.848𝑑 2 𝑡90 When the size of ground is more than 3-5 times larger than the thickness of soil layer, the settlement of soil can be regarded as 1-D compression condition. We normally adopt oedometer test in laboratory to investigate the behaviour of soil. d: half of average height of specimen t50: the time needed for 50% degree of consolidation d: half of average height of specimen t50: the time needed for 90% degree of consolidation Review of Soil Mechanics II. Determination of coefficient of consolidation
The log time methodOr5.00-a0Theoretical curveInitialcompression-asAdU4.50△d1/2=4:tgit.104.00logTvPrimarydceaoeconsolidation1/2主固结3.50AhtEcreep109ehotEOP,lab3.00a100FSecondarycompressionat2.50次固结/蠕(creep)t502.0010.0001000101000.1logt(min)tEOP,labt500.196d2A1-△eCae_ is secondary coefficientCvCae=Alogt=1+e.Alogtt50=1+eo15
tB :tA=4:1 Dd Dd 1/2 t50 1/2 主固結 次固結/蠕變(creep) 𝜀𝑐𝑟𝑒𝑒𝑝 = Δℎ ℎ𝑜 = 𝐶𝛼𝜀 log 𝑡 𝑡𝐸𝑂𝑃,𝑙𝑎𝑏 𝑡𝐸𝑂𝑃,𝑙𝑎𝑏 𝐶𝛼𝜀 = Δ𝜀 Δ log 𝑡 = 1 1 + 𝑒0 −Δ𝑒 Δ log 𝑡 = 𝐶𝛼𝑒 1 + 𝑒0 is secondary coefficient 15 The log time method 𝑐𝑣 = 0.196𝑑 2 𝑡50