Time Rate Of Consolidation Settlement - KSU

Time Rate of Consolidation Settlement We know how to evaluate total settlement of primaryconsolidation Sc which will take place in a certain clay layer. However this settlement usually takes place over time, muchlonger than the time of construction. One question one might ask is in how much time thatmagnitude of settlement will take place. Also might beinterested in knowing the value of Sc for a given time, or thetime required for a certain magnitude of settlement. In certain situations, engineers may need to know thefollowings information:1. The amount of settlement Sc(t) at a specific time, t,before the end of consolidation, or2. The time, t, required for a specific settlement amount,before the end of consolidation.

How to get to know the rate of consolidation? From the spring analogy we can see that Sc is directly related to howmuch water has squeezed out of the soil voids. How much water has squeezed out and thus the change in void ratio e isin turn directly proportional to the amount of excess p.w.p that hasdissipated. Therefore, the rate of settlement is directly related to the rate of excessp.w.p. dissipation. What we need is a governing equation that predict the change in p.w.p.with time and hence e, at any point in TIME and SPACE in theconsolidation clay layer. In other words, we need something to tell us how we get from the momentthe load is entirely carried by the water to the point the load is completelysupported by the soil. It is the THEORY OF CONSOLIDATION which tells us that.

Spring Analogy , the increase in total stress remains the same during consolidation,while effective stress ’ increases. u the excess pore-water pressure decreases (due to drainage)transferring the load from water to the soil.Excess pore pressure ( u)is the difference between the current pore pressure (u) and the steady statepore pressure (uo). u u - uo uniformly distributed pressure uq ’ A u ’saturated clay ’q uTime

1-D Theory of Consolidation Terzaghi developed a theory based on the assumption that anincrement of load immediately is transferred to the pore waterto create excess pore water pressure (p.w.p). Then as the pore water squeezed out, the excess p.w.p.relaxes gradually transferring the load to effective stress. He assumed that all drainage of excess pore water is verticaltoward one or two horizontal drainage faces. This is describedas ONE-DIMENSIONAL CONSOLIDATION. 3-D e However 1-D theory is useful and still the one used inpractice, and it tends to overpredict settlement.

ASSUMPTIONS The soil is homogeneous. The soil is fully saturated. The solid particles and water are incompressible. Compression and flow are 1-D (vertical). Darcy’s law is valid at all hydraulic gradients. The coefficient of permeability and the coefficient of volumechange remain constant throughout the process. Strains are small.

Mathematical DerivationRate of outflow of water - Rate of inflow of water Rate of Volume Change v V v z z dz dx dy v z dx dy z t v z zdx dy dz V t(1)

Mathematical Derivationv ki k h k uz z zwFrom (2) and (3) 2z k u k u z z z 2ww z v2 u dxdydz V 2 tw z k V t e a ( ) a uvv(2) (V eV ) V Vss s V e ess t t t t Vs 0 t V V es t tThe one-dimensionalconsolidation equationderived by TerzaghidxdydzV V s1 e1 eoo V dxdydz e t1 e to(3)2k u 1 e 2 1 e tw zo u c 2u t v z 22a k u v u m uv t 21 e t zwo2 u c uv 2 t zc k v mw vk a v w 1 e o

Solution of Terzaghi’s 1-D consolidation equationTerzaghi’s equation is a linear partial differential equation in onedependent variable. It can be solved by one of various methods with thefollowing boundary conditions:The solution yieldsWhereu excess pore water pressureuo initial pore water pressureM p/2 (2m 1) m an integerz depthHdr maximum drainage path(*)

Remarks The theory relates three variables: Excess pore water pressure u The depth z below the top of the clay layer The time t from the moment of application of load u c 2u t v z 2Or it gives u at any depth z at any time t The solution was for doubly drained stratum. Finding degree of consolidation for single drainage is exactly thesame procedure as for double drainage case except here Hd theentire depth of the drainage layer when substituting in equationsor when using the figure of isochrones. Eq. (*) represents the relationship between time, depth, p.w.p forconstant initial pore water pressure u0 . If we know the coefficient of consolidation Cv and the initial p.w.p.distribution along with the layer thickness and boundaryconditions, we can find the value of u at any depth z at any time t.

Degree of ConsolidationoooThe progress of consolidation after sometime t and at any depth z inthe consolidating layer can be related to the void ratio at that time andthe final change in void ratio.This relationship is called the DEGREECONSOLIDATION or CONSOLIDATION RATIO.orPERCENTofBecause consolidation progress by the dissipation of excess porewater pressure, the degree of consolidation at a distance z at any timet is given by: (**)

Degree of ConsolidationSubstituting the expression for excess pore water pressure, i.e.into Eq. (**) yields2Uz 1- (***) The above equation can be used to find the degree of consolidationat depth z at a given time t. At any given time excess pore water pressure uz varies with depth,and hence the degree of consolidation Uz also varies. If we have a situation of one-way drainage Eq. (***) is still be valid,however the length of the drainage path is equal to the totalthickness of the clay layer.

Degree of Consolidation2Uz 1-Permeable layerHHdrHdrTv0.1Variation of Uz with Tv and Z/Hdr

RemarksTv0.1 From this figure it is possible to findthe amount or degree of consolidation(and therefore u and ’) for any realtime after the start of loading and atany point in the consolidating layer.Variation of Uz with Tv and z/Hdr All you need to know is the Cv for the particular soil deposit, the totalthickness of the layer, and boundary drainage conditions. These curves are called isochrones because they are lines of equal times. With the advent of digital computer the value of Uz can be readilyevaluated directly from the equation without resorting to chart.

Length of the drainage path, Hdr During consolidation water escapes from the soil to the surface or to apermeable sub-surface layer above or below (where u 0). The rate of consolidation depends on the longest path taken by a drop ofwater. The length of this longest path is the drainage path length, HdrPermeable layerHdrHdrHClayHdrHdr HdrL Typical cases are:– An open layer, a permeable layer both above and below (Hdr H/2)– A half-closed layer, a permeable layer either above or below (Hdr H)– Vertical sand drains, horizontal drainage (Hdr L/2)

Degree of ConsolidationUz 1-2

ExampleA 12 m thick clay layer is doubly drained (This means that a very pervious layercompared to the clay exists on top of and under the 12 m clay layer. Thecoefficient of consolidation Cv 8.0 X 10-8 m2/s.

61%100%46%61%

Average Degree of Consolidationo In most cases, we are not interested in how much a given point in alayer has consolidated.o Of more practical interest is the average degree or percentconsolidation of the entire layer.o This value, denoted by U or Uav , is a measure of how much theentire layer has consolidated and thus it can be directly related tothe total settlement of the layer at a given time after loading.o Note that U can be expressed as either a decimal or a percentage.o To obtain the average degree of consolidation over the entire layercorresponding to a given time factor we have to find the area underthe Tv curve.

Average Degree of ConsolidationThe average degree of consolidation for the entire depth of clay layer is,uo 1 dr u z dz (1)2 H dr 0 U 1 uo2HSubstituting the expression ofuz given by2 HdrArea under thepore pressurecurveInto Eq. (1) and integrating, yieldsDegree of consolidation

SummaryoBecause consolidation progress by the dissipation of excess pore waterpressure, the degree of consolidation at a distance z at any time t is givenby: 1 dr u z dz2 H dr 0 U 1 uo2H2Uz 1-Degree ofconsolidationAverage Degree ofconsolidation

Average Degree of ConsolidationFig. 11.30 Variation of U with TvSc(t) Settlement at any time, tSc Ultimate primary consolidation settlement of the layer.

Average Degree of Consolidation

Approximate relationships for U vs. TV Many correlations of variation of U with Tv have been proposed. Terzaghi proposed the followings:oror

Approximate relationships for U vs. TVor:NoteThese equations can be applied for all ranges of U value with smallerrors .Error in Tv of less than 1% for 0% U 90% and less than 3% for90% U 100%.

ExampleA soil profile consists of a sand layer 2 m thick, whose top is the ground surface,and a clay layer 3 m thick with an impermeable boundary located at its base. Thewater table is at the ground surface. A widespread load of 100 kPa is applied at theground surface.100 kPa(i) What is the excess water pressure, ucorresponding to: t 0 (i.e. immediately after applying theload)t (very long time after applying theload)SandClay(ii) Determine the time required to reach 50%consolidation if you know that Cv 6.5 m2/year.Impermeable layerImpermeable layer2m3m

Solution(i) Immediately after applying the load, the degree of consolidation Uz 0% and thepore water would carry the entire load:100 kPaat t 0 u0 100 kPaOn contrary, after very long time, the degree ofconsolidation U 100% and the clay particles would carrythe load completely:Sand2mat t u 0(ii) The time required to achieve 50% consolidation canbe calculated from the equation:One-way drain3mClay cv coefficient of consolidation (given) 6.5 m2/yearImpermeable layerHdr the drainage path length height of clay 3m (because the water drain away from the sandImpermeable layerlayer only)Tv is the time factor for U 50%, and can approximately be calculated from: 0.197Can also be obtained from thetheoretical relationship or graphSubstitution of these values in the above equation of t:t 0.27 yeart Hdr2.Tv / cv

ExampleAn open layer of clay 4 m thick is subjected to loading that increases the averageeffective vertical stress from 185 kPa to 310 kPa. Assuming mv 0.00025 m2/kN,Cv 0.75 m2/year, determine:i. The ultimate consolidation settlementii.The settlement at the end of 1 year,iii.The time in days for 50% consolidation,iv.The time in days for 25 mm of settlement to occur.Solution(i) The consolidation settlement for a layer of thickness H can be representedby the coefficient of volume compressibility mv defined by:Sc mv H z 0.00025 * 4 * 125 0.125m 125mm.

Example(ii) The procedure for calculation of the settlement at a specific time includes: Calculate time factor: . 0.1875 Calculate average degree of consolidationUt . 0.49Calculate the consolidation settlement at the specific time (t) from:St Ut . Sc . 61 mm(iii) For 50% consolidation Tv 0.197 . . (vi) For St 25 mm Ut 0.20 . . , therefore fromt 1.05 year 384 days, thereforet 0.1675 year 61 days

ExampleThe time required for 50% consolidation of a 25-mm-thick clay layer(drained at both top and bottom) in the laboratory is 2 min. 20 sec.(i) How long (in days) will it take for a 3-m-thick clay layer of the same clay inthe field under the same pressure increment to reach 50% consolidation?In the field, there is a rock layer at the bottom of the clay.LaboratoryPorous stone(permeable)ClayField25mm(ii) How long (in days) will it take in thefield for 30% primary consolidationto occur? Assuming:GWSandClayRock (impermeable)3m

Solution(i) As the clay in lab and field reached the same consolidation degree (U 50%),time factor in the lab test The time factor for the fieldThus, TheApproach I:Approach II:orFrom Lab.At U 50% . Tv 0.197From Tv Cv t/Hd2 . Cv 2.2 X 10-7 m2/S12.5mm/1000 m3In the field0.197 2.2 X 10-7 * t(3)2t 93.3 days(ii)Tv 3.14 X (0.3)2 0.0714Tv Cv * tHd20.071 2.2X10-7 * t(3)2t 33.5 days

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