华科高等土力学consolidation

1、Consolidation theory and analysis共四十七页Flow in a soil mass共四十七页Water head and hydraulic gradientHere we only focus on flow of water in saturated soilFlow in a soil mass also has to follow the fundamental principles of Fluid MechanicsFrom Fluid Mechanics, it is known that the total head at a specific

2、position in a flow can be described by Bernoullis Equation:VelocityheadPressureheadElevationheadh = total headu = water pressurevw = water flow velocityrw = unit weight of waterg = acceleration due to gravity共四十七页For water flow in a soil mass, the water velocity is usuallysmall, so the term of veloc

3、ity head can be neglected:This is also the measurable water head by piezometersThe loss of head between Point A and Point B:The hydraulic gradient:共四十七页Darcys lawIn 1856, in his experiment Darcys found that the discharge velocity of water flow through a saturated soil mass is proportional to the hyd

4、raulic gradient:v=discharge velocity, the quantity of water flowing in unit time through a unit gross cross-sectional area of soil at right angles to the direction of flowk=hydraulic conductivity (or, coefficient of permeability)Important note: the discharge velocity is NOT the actual velocity vw of

5、 water in the soil pores. It is based on the gross cross-sectional area of the soil. It is SMALLER than the actual velocity vw.Area of water-filled pores AvArea of soil ATheir relationship ?共四十七页Soil permeabilityThe hydraulic conductivity of saturated soil is affected byThe intrinsic characteristics

6、 of soilThe characteristics of pore waterThe intrinsic characteristics of soilGrain-size distributionPore-size distributionVoid ratioRoughness of mineral particlesSoil fabric and structureSaturated hydraulic conductivity kClean gravel (102100cm/s) Course sand (10010-2cm/s) Fine sand (10-210-3cm/s) S

7、ilty clay (10-310-5cm/s) Clay (10-5cm/s)The characteristics of pore waterViscosityUnit weight共四十七页Laplaces equation of continuityConsider a 2-D problem of steady flow. An Representative VolumeElement (RVE) of dv=dx*dz*1 is taken from the soil massThe water flow into the RVE:The water flow out of the

8、 RVE:Since water can be assumed to be incompressible, the condition of continuity states thatHence:So共四十七页Now, the hydraulic gradients:And So the equation of continuity becomesIf the soil is isotropic with respect to permeability, the Laplaces equation of continuitybecomes: Because 共四十七页Fundamentals

9、 of consolidationRecall the definition of effective stressExternal load applied on a saturated soil strataWho would carry the load?The pore water, the soil skeleton, or both?The portion of load carried by the pore water is known as the excess pore pressure ueThe portion carried by the soil skeleton

10、is the increase in the effective stress The increase in the soil skeleton will result in the decrease of the soil volume, i.e, the decrease of the void ratioThe pore water will be expulsed from the soil mass (water can be considered to be incompressible)共四十七页pTotal stress: z=pEx. pore press.: u=z=pE

11、ffective stress :z=0Spring-cylinder analogy model for soil consolidation共四十七页Total stress:z=pEx. pore press.: u 0Spring-cylinder analogy model for soil consolidation共四十七页Total stress:z=pEx. pore press.: u =0Effective stress:z=pSpring-cylinder analogy model for soil consolidation共四十七页ImperviouslayerS

12、aturatedcohesive soilz=pp0 z H:u=pz=0: u=0z=H: uz 0 z H: u=0zChange of excess pore pressure u during consolidation 共四十七页Important issues in soil consolidationCoefficient of permeability (“size of valve”)Compressibility of soil skeletonTimeDistance of dissipation (消散(xiosn)Gravel or sandLarge coeffic

13、ient of permeabilityExcess pore pressure can be dissipated momentarilyThe deformation could be considered as immediate deformationClaySmall coefficient of permeabilityExcess pore pressure dissipates slowlySettlement increases graduallyImmediate settlement and consolidation settlementSilt?共四十七页One-di

14、mensional compression testOne-dimensional compression test (侧限压缩试验(shyn) and time-deformation plotApparatus: Oedometer (一维固结仪)Dial gageSpecimenPorous stoneRing共四十七页Time-deformation plotInitial compression (初始瞬时(shn sh)压缩)Primary consolidation (主固结( ji)Secondary consolidation (次固结)共四十七页Compression cu

15、rveFrom oedometer testSelect several pressure levelsDetermine the initial void ratio e0 of the saturated soil specimenStarting from the smallest pressure, read the deformation H1 when it stabilizes (end of primary consolidation)Increase the pressure step by step, read the corresponding deformation H

16、iCalculate the void ratio ei at each pressure level共四十七页Compression curve:eWhy smaller slope?Compression curve of a natural soil specimen obtained from the field(Log scale)(压缩(y su)指数)共四十七页Expansion curveThe smaller slope is due to the pre-consolidation pressure applied on theon the soil specimen in

17、 its history! If unloaded during compressione(log)dUnloading and reloading:expansion curve(回弹(hu dn)指数)共四十七页Isotropic compression testApparatus: triaxial compression deviceNo axial load is neededStress state共四十七页Time-deformation plotInitial compression (初始瞬时(shn sh)压缩)Primary consolidation (主固结( ji)

18、Secondary consolidation (次固结)共四十七页Compression curve:eCompression curve of a natural soil specimen obtained from the field(Log scale)共四十七页Two pre-consolidation pressures1D stress state:The maximum vertical effective stress ever experienced bythe soil in its history!Hydrostatic stress state:The maximu

19、m mean effective stress ever experienced bythe soil in its history!共四十七页Normally consolidated clay and overconsolidated clayNormally consolidated, the effective pressure is the maximum pressure in the soils historyOverconsolidated, the effective pressure is less than that which the soil experienced

20、in the past, i.e.,The overconsolidation ratio (超固结( ji)比):There are two definitions of overconsolidation ratio!共四十七页Terzaghis theory of 1D consolidationThe basic assumptions of the Terzaghi consolidation theoryThe soil is homogeneous;The soil is fully saturated;The pore water is assumed to be incomp

21、ressible;Pore water only flow in the direction of compression;Darcys law is valid.These assumptions are generally valid for most soils under one-dimensionalconsolidation.共四十七页Consider a saturated soil elementThe rate of water flow into the element:The rate of water flow out of it: The change in the

22、water volume in the element,which is also the change of the soil volume:V=volume of the soil elementvz=flow velocity in z directionSo,Now according to Darcys law of flow,h=pore water headue=excess pore water pressureK=coefficient of permeabilityTerzaghis 1-D consolidation theory共四十七页Now we know that

23、We also know that the volume of the solid particles of the soil Vs does not changeTherefore,But the void volume can be related to the volume of the soil solids and the initialvolume of the soil element as Hence,Finally, the partial differential equation becomes共四十七页The change of the void ratio can b

24、e related to the change in the effective stressthrough a parameter, namely coefficient of compressibility of soil av: the change in the effective stress, since the total stress =constant, The coefficient of compressibility for a soil can be considered constant fora narrow range of pressure changeNow

25、 the partial differential equation becomesmv: coefficient of volume compressibility = Or,cv: coefficient of consolidation = 共四十七页For the 1-D consolidation equation:IfThe solution yieldsWe define the degree of consolidation (固结( ji)度) at a depth z asDegree of consolidationue =0 zue = 0dd(时间(shjin)因数)

26、共四十七页Free drainageseepagezFree DrainagedTv=0Tv=0.05Tv=0.2Tv=0.7Tv=u0=pFree drainage on just one side?dseepage共四十七页The average degree of consolidation can be integrated from the degree ofconsolidation at each point, for example, for the problem aboveRelationship between U and the time factor Tv:共四十七页

27、Summary of 1D consolidationBasic equations:Stress strain relationship:Mass balance of fluid:Darcys law:In essence: coupling of soil deformation and seepage!共四十七页ExampleQ: What is the driving force ofsettlement?A: Increase of effective stress due to water pumping.Q: What is the increase of effectives

28、tress?共四十七页The final settlement of the clay layer:The increase of effective stress at the center of the claylayer.共四十七页The settlement with time:共四十七页Vertical drainsThe degree of consolidation is largely determined by the length of drainage path.Vertical drains can be installed in thick clay layer to introduce horizontal drainage, expediting the consolidation.共四十七页Simplified analytical modelGoverning equation (assuming axi-symmetric condition):ch: coefficient of consolidation in the horizontal direction共四十七页Two average degrees of consolidation: vertical degree of consolidation: Radia