compressibility of soil

10Compressibility of Soil

A stress increase caused by thc construction of foundations or other loacls comprcssessoi l layers. The compression is caused by (a) deformat ion of so i l par t ic les. (b) re loca-tions of soil particles, and (c) expulsion of water or air from thc void sperct;s. In gcn-eral, the soil settlement caused by loads may be divided into three brclad catcgorics:

l. Immediute settlemcnt (or elastic settlemcnt). which is causcd by thc elastic de-format ion of dry soi l and of moist and saturated soi ls wi thout any change inthe moisture content . Immcdiatc set t lement calculat ions are gcneral ly basedon equations derived from the thcory of elasticity.

2. Primarv consolidation settlement, which is the result of a volume change insaturated cohesive soi ls because of expuls ion o l ' thc water that occupies thevoid spaces.

3. Secondury consoliclatknt settlemanL which is observcd in saturatcd cohesivesoi ls and is the resul t o f the p last ic adjustment of so i l fabr ics. I t is an addi t ionalform of compression that occurs at constant effective strcss.

This chapter presents the fundamcntal pr inc ip lcs for cst imat ing thc immcdiateand consolidation settlemenls of soil laycrs under supcrimposed loaclings.

The total settlement of a foundation can then be given as

5 7 : S , + S . + S ,

where Sr : total settlementS,. : primary consolidation settlement,S. : secondary consolidation settlemcntS,, : immediate scttlement

When foundations are constructed on very compressible clays, the consolidationsettlement can be several t imes greater than the immediate scttlement.

IMMEDIATE SETTLEMENT

Contact Pressure and Settlement Profile

Immediate, or elastic, settlement of foundations (S") occurs directly after the appli-cation of a load, without a change in the moisture content of the soil. The masnitude

10.1

Chapter 10 Compressibility of Soil

Table 10.1 Inf luence Factors for Foundations [Eq. (10.2)]

Iftl

CircleRectanglc I

1 . 523-5

l 020-s0

l(x)

1 .001 . 1 21 .361. .531 .782 . 1 0

2.993.574.01

0.640.560.680.770.u91.0-51.271 .49l . u2.0

0.790.it81.0'71 . 2 11 .421 .702 . 1 02.463.0-) . +- l

Table 10. 1 gives the influcnce factors for rigid and flexible foundations. Repre-

sentative values of the modulus of elasticity and Poisson's ratio for different types of

soi ls arc g iven in Tables 10.2 and 10.3. respect ivc ly .Note that Eq. ( 1 t ) . I ) is bascd on the assumpt ion that the prcssure Ao is appl ied

at the ground surl 'ercc. In practicc, l 'oundations arc placed at a ccrtain depth below

the ground surlarcc. Deepcr foundation embedmcnt tends to reduce the magnitude

o[ t lre foundertion settlentent S,.. However, if Eq. (lt). l) is uscd to calculate settle-

mcnt" i t resul ts in a conservat ive est imate.

Table 10.2 Rcprcsentativc Values ol the Modulus ol'Elasticity of Soil

Soil type kN /m2 lb / i n . 2

E,

Soft clayHard clayLoclse sandDense sand

I ,t3(X) -3.-5(X)6,(XD 14,000

r 0.000,28,0(x)35.0(X) -70,000

250 -500t350 -2,(XX)

1 ,-s00 - 4,000-5.000 - 10.(xx)

Table 10.3 Reoresentative Values of Poisson's Ratio

Type of soil Poisson's ratio, p,

Loose sandMedium sandDense sandSil ty sandSoft clayMcdium clay

0.2,0.40.25 - 0.40.3 - 0.4-50.2-0.4

0.1-5 -0.250.2 0.5

10.3 lmproved Relationship for lmmediate Settlement

Example 10.1

Estimate the immediate settlement of a column footing 1.5 m in diameter that isconstructed on an unsaturated clay layer, given that the total load carried by thecolumn footing = 150 kN, E. : 7000 kN/m2, and p, :0.25. Assume the footingto be r igid.

SolutionUsing Eq. (10.1), we have

s" - AoB 1J

In, E

(1s0)[ , s - : \ - " ' t : 8 4 . 9 k N / m r

[o sYFrom Table 10.1, for a circular rigid foundation,l,, : 0.79, so

I r * n r s l Is" : (84,9)(1 5)l -

7000- lto.zo) : 0.013-s m : L3.5 mm r

10.3 Improved Relationship for lmmediate Settlement

Mayne and Poulos (1999) recent ly prescnted an improvccl rc lat ionship for ca lculat -ing the immediate set t lement o l I 'oundat ions. ' l 'h is

re lat ionship takes in to accountthe r ig id i ty of the loundat ion, the dcpth of cmbedment o[ the loundat ion. the in-creasc in the modulus of c last ic i ty of so i l wi th dcpth, and the locat ion o l ' r ig id layersat l imi ted depth. In orc ler to use th is re lat ionship, onc needs to dctermine the cquiv-a lent d iameter of a rectangular foundat ion. which is

tiBLB , - , i

\l rr

where B - width of foundationL : length of foundation

For circular foundations,

8 , , : B

( I { ) .3a )

(10 .3b)where B : diameter of foundation.

Figure 10.3 shows a foundation having an equivalent diameter of 8,. located ata depth Dlbelow the ground surface. Let the thickness of the foundation be r and themodulus of elasticity of the foundation material be Er. A rigid layer is located at adepth ft below the bottom of the foundation. The modulus of elasticity of the com-pressible soil layer can be given as

E,: E,, * kz. ( 10.4)


R +

AOt l l l

I Y Y Y Y

tL/ '...

A 'I

Compressib le soi l layer

Ilt

"f" ' '"'"

II

FIt

IIIv

' . " ' -"

, : .Rigid laycr

Figure 70.3 lmprovcd relat ionship for inrmcdiate sctt lcment

., Eo, -

kB,,

Figure 10.4 Yariation of 1.r with B

hlB,, = 0.2

70.3 rmproved Retationship for rmmediate setttement 26s

with the preceding parameters dellned. the immediate settlement can be given as

s,: LoB*!!t(r - pi) (ro.s)where 1,; : influence t'actor fbr the variation of E, with clepth : f (L:,,, k, 8", anc] h)1e : foundation rigidity correction factor

./o : foundation embedment correction factorFigure l0.4showsthevar iat ionof 1. ,wi th B: E, , lkB, ,ancl ,h lB, , .Thcfouni la t ionr ig id_ity correction factor can be expressed as

, - T ,t r . - t' 4 46r ) (

#t - )wSimilarly, the embedmcnt crtrrection facl.or is

Ir - I -

3.5 exp(t.22p",

10.-5 and 10.6 show thc var iat ions ol

( 10 .6)

( 10 .7)

by Eqs . (10 .6) andFigures( r0 .7) .

/ u \- 0 . 4 ) l r , * , 0 , , |

/0, and 1r. exprcssed

1 . 0

0.9

+ o.t{5

0.lt

0.10.001 2

Figure 10.5

4 6 8 0 . 0 1 10.0 I 00.0Kt..

Variation of rigidity correction f'actor, 1r, with flcxibility factor, Ko.. [Eq. (rr.r.6)J

0 . I

r=(-+ \ / : : \' \ u , , * + ^ ) \ t )

= Flexib i l i ty f : rcror


0.9

-s 0.1t5

0.tt

0 .75

t \ .10 5 r 0 1 5

!,.,8,,

Figure 70.6 Vrriat ion ol 'crnbcdmcnt corrcct ion faclor. / , . IEq. ( l {) .7)]

Example 10.2

Refer to Figure 10.3. For a shallow foundation supported by a silty clay, the fol-lowing are given:

L e n g t h : L : 1 . 5 mW i d t h : B - 1 mDepth of foundation : D/ : 1 illThickness of foundation : / : 0.23 mLoad per unit area : Ao : 190 kN/m2Er :15 x 106 kN/m2

The silty clay soil had the following properties:

h - 2 m&, : 0'3E = 9000 kN/m2k : 500 kN/m2/m

Estimate the immediate settlement of the foundation.

10.3 lmproved Relationship for lmmediate Settlement

the equivalent diameter is

1.38 m

267

SolutionFrom Eq. (10.3a),

From Figure 10.4,from Eq. (10.6),

I r : L +' 4

From Eq. (10.7),

Ao : 190 kN/m2

_ E, 90001 1 : _ - _ , , - ^ ^ : 1 3 . 0 . 1'- kB,. (sooxt.38)

h 2: 1 4 5

B" 1.39

for B :13 .04 and h /8 , , :1 .45 , the va lue o f 1 r , . :0 .74 . Thus ,

^^."(;T1{#)'4

- : 0 .781

[ 0 , , t , - ( f ) r r * r ]L r38 l

I r : 1 *

3.5 exp(\.22p., * ,.r(*I

+ 1 6 )

- t _- I = 0.907

From Eq. (10.5),

^ L,oB, I r ; l1 ls ,_' E n

= 0.014 m * 14 mm

3.5 expi(1 .22)(0.3) - o of (f + 1.6)

jr( re0)(1.38x0.74X0.787)(0.90f)

r I _ n rz)9 0 0 0 \ - " ' " l

268 Chapter 10 Compressibility of Soil

CONSOLI DATION SETTLEMENT

10.4 Fundamentals of Consolidation

When a saturated soil layer is subjected to a stress increase. the pore water pressureis suddenly increased. In szrndy soils that are highly permeable. the drainage causedby the increase in the pore water pressure is completed immcdiately. Pore waterdrainage is accompanied by a reduction in thc volumc of thc soil mass, which resultsin settlement. Because of rapid drainage of thc porc watcr in sandy scli ls, immediateset t lement and consol idat ion occur s imul tancously.

When a saturated compressible clay lzryer is subjected to a strcss incrcasc, elas-tic settlement occurs immcdiately. Because the hydraulic conductivity of clay is sig-nificzrntly smaller than that of sand, thc cxcess pore water pressure generated by load-ing gradual ly d iss ipatcs over a long per iod. Thus. the associated volume change ( thatis , thc consol idat ion) in the c lay may cont inue long af ter the immediatc sct t lcment .Thc settlement causecl by consolidation in clay mary bc scvcral t imcs grcater than theimmcd i l r l c se l l l emcn t .

Thc t imc-dependent del i r rmat ion of saturated c layey soi l can bcst bc under-stood by considcr ing a s imple model that c i lns is ts o[ a cy l inder wi th a spr ing at i tscenter . Let thc ins ido arr : a of the cross scct ion of the cy l inder be equal to A.

' l 'he cy l in-

der is f i l led wi th watcr and has a f r ic t ion lcss water t ight p is ton and valve as shown inFigurc 10.7a. At th is t imc. i [ 'we p lace a load Pon the p is ton (F igure 10.7b) and keepthc valvc c losecl , thc ent i re load wi l l be taken by the water in the cy l indcr bccausewatcr is intrtmprassible. The spring wil l not go through zrny dcf'ornraticln. Thc cxcesshydrostatic pressure at this tirne can be given as

( 10.8)

This value can be obscrvcd ir.r thc prcssurc gauge attached to the cylinder.I n genc ra l . wc c l rn w r i t c

P : P,- l P, , (10.9)

where P, : load carricd by the spring and P,,, - lclad carried by the water.From the preceding discussion, we celn see that when the valve is closed after

the placement of the load I

P, : 0 and P, , , : P

Now, if the valve is opened, the water wil l f low outward (Figure 10.7c). This flow wil lbe accompanied by a reduction of the excess hydrostatic pressure and an increase inthe compression of thc spring. So, at this time" Eq. ( lt).9) wil l hold. However,

P . > 0 a n d P , , , 1 P (that is, L,tt < PIA)

After some time, the excess hydrostatic pressure wil l become zero and the systemwill reach a state of equil ibrium, as shown in Figure 10.7d. Now we can write

P

A

P,: P and P," : t )

10.4 Fundamentals of Consolidation

Valve

I c losed

:fi{s:s

/A a = 0 . p

Art = a-A ( D )( a )

P. . . . :

' t

rit;,i;ir:i;ri:ii:::iri;*:ii:i:lriii:i:

Vr lvc

I ---*-l oPcll

::t;it!ii:!:ii:ri:$l:j.itit;.*t*lii:i:ilr:{; SXIljllti. '

-.J''

Att---

( e )

Figure 10. 7 Spring-cylinclcr modcl

and

P : P , t P , , ,

Wi th th is in mind, we can analyze the st ra in of a saturatec l c lay layer subjectedto a stress incrcase (Figure 10.8a). Consider the case wherc a layer of saturated clayo f t h i ckness H tha t i scon f i nedbe twecn two laye rso f sanc l i sbe ingsub jec ted toaninstantaneous increase of tolul r ' /re.r.r o[ Ao. This incremental total stress wil l betransmitted to the pore water and the soil solids. This means that the total stress, Ao,wil l be divided in some proportion betwcen effectivc stress and pore water pressure.The behavior of the cffective strcss change wil l be similar to that of the spring in Fig-ure 10'7, and the behavior of the pore water pressure change wil l be similar to thatof the excess hydrostatic pressurc in Figure 10.7. From the principle of effectivestress (Chapter tt). i t follows that

A o : A o ' * A r l

where Arr' : increase in thc effective stressAlz : incrcase in the pore water pressure

Vrlvcopcn

::1: lr:ltlil

ilrlili$i!i

i:.;

ff'l , # :

(10 .10)

AO

I

Depth

Total stress increasc

( r )

ElI'ectivcstross increrse

TH

i

IH

( c ) A t t i m e 0 < r < -

)<tre wttcrTotal stress increase pressure increase

Eflective\tress increase

1H

IDepth Depth Depth

(d) At t ime t = -

Figure 10.8 Variation of total stress. pore water pressure, and effective stress in a clay layer

drained at top and bottom as the result of an added stress, A<r

Pore waterplcssure lncrelse-

'fotal stress incrcasc

270

10.5 One-Dimensional Laboratory Consolidation Test 271

Because clay has a very low hydraulic conductivity and water is incompressibleas compared with the soil skeleton, at t ime r : 0, the entire incremental stiess. Ao,wil l be carried by water (A,r : A,) at all depths (Figure 10.gb). None wil l be carriedby the soil skeleton - that is, incremental effective itress (A,ri) : (t.

After the application of incremental stress. -\u. to thc clay layer, the water inthe void spaces wil l start to be squeezecl out anci wil l drain in botl directions into thesand layers. By this process, the cxcess pore water pressure at any depth in the claylayer wil lgradually decrease. and the stress carried by the soil solids (eiTective stress)wil l increase. Thus, at t ime 0 < r < m.

A o : A r r ' * A r r ( A o ' > 0 a n d A r r < A r r )

10.5

However' thc magnitudes of Arr' and Arr at various depths wil l change (Figurc 1O.gc),depending on the minimum disternce of thc drainage path to eithe r rhe top 9r bettomsand layer.

.. Theoretically' at t imc 1 : oo, the entirc excess pore water prcssure wcluld bcdissipated by drainage from erll points of the clay layer; thus, A4 : 0. Now the totalstress increasc, A.', wil l be carricd by thc soil structurc (Figurc 10.ltcl). Hcncc,

Arr : Arr'This gradual process o1 'dra inage under an ac ld i t ional load appl icat ion and the

associated transfcr of excess pore water prcssure to efTective slress cilusc thc time-dependent set t lement in thc c lay soi l layer .

one'Dimensional Laboratory consolidation Test

The one-dimensional consolidation lesting procedure was first suggested by Terza-ghi. This test is perlbrmed in a consolidometer (sornctimes refcrred to as eln oe-dometer). The schematic diagrant of a consolidometer is shown in Figure 10.9a. Fig-ure 10.9b. shows a photograph of a consolidomcter. The soil ,p".i-"n is placJiinside a metal ring with two porous stones, onc at the top of the specimen and an_other at the bot tom. The specimcns are usual ly 64 mm ( :2.5 in . ) in d iameter and 2-5mm. ( : I in . ) th ick. The load on the spccimen is appl icc l through a lever arm, andcompression is mcasured by a micrometer dizrl gauge. Thc spccimcn is kept unilcrwater during the test. Each load is usually kept for 24 hours. After that, the load isusually doublcd, which doubles the pressure on the specimen. and the compressionmeasurement is continued. At thc end of the test, the dry weight of the test spccimenis determined. Figure 10.9c shows a consolidation test in p.o!res, (right-hand sicle).

The general shape of the plot of cleformation of the specimen against t ime fora given load increment is shown in Figure 10. 10. From the piot, we can observe threedistinct stages, which may be described as follows:

stage I Init ial compression, which is caused mostry by preloacring.stage II Primary consoliclation, during which exceir po." water pressure is

gradually transferred into effective stress because of the expulsionof pore water.

Stage III Secondary consolidation, which occurs after complete dissipation ofthe excess pore water pressure, when some deformation of the speci-men takes place because of the plastic readjustment of soil fabric.


Figure 10.9(a) Schematic cl iagram o[ a consolidonrcter;(b ) photograph o l a conso l idornc tc r , (c ) a con-so l ida l ion tcs t in p rogrcss ( r igh t -hanc l s idc)

l'

-

:1 i* ' . : Sncclnl tn

i$ilr.' ,;ng4:!.:lM!*si!;ittritii

:i!g!:ti:i.:i!it!:t.;tj

.2

c

€

I

10.6 Void Ratio-Pressure Plots

Secondary consol idat ion Figure 10.10Time-deformation plot duringconsolidation for a given loadincrementTime ( log scale)

Void Ratio-Pressure Plots

After the time-dcformation plots for various loadings are obtained in the labora-tory, it is necessary to study the change in the void ratio of the specimen with pres-sure. Following is a step-by-step procedure for doing so:

1. Calculatc the height of solids, H., in the soil specimen (Figure 10.11) using theequat ion

H . - w '

- J t4 ,

' AG,7,,, AG.,P*

where lV. : dry weight of the specimenM, : dry mass of the specimenA : area of the specimenG, : specific gravity of soil solids

7,,, : unit weight of waterp,,, - density of water

2. Calculate the init ial height of voids as

( 1 0 . 1 1 )

H, , : H - H , (10 .12)

where H : initial height of the specimen.

3. Calculate the initial void ratio, e,r, of the specimen, using the equation

V , H , A H ,r r t :, . V , H , A H ,

10.6

(10 .13)


TII

In i t ia lheight ofspeclmen

= H

III+

Figure

tI

H , = H H ,

II

. --Lu," '

- , lC ,y , ,

t10.1 1 C.hange of height of spccimcn in onc-dimensional consol idation test

4. For the first incremenlal loading, o1 (total load/unit area o[ specimen), whichcauses a deformat ion AH,. ca lculate the chernge in the void rat io as

L I I ,LC I :

H*( r0 .14 )

10.7

(AH, is obta ined f rom thc in i t ia l and the l lna l d ia l readings lor the loading) .

It is important to note that. at the end of consolidation, total stress rrl is equal to ef-fective strcss rri.

5. Clalculate the new void ratio aftcr consoliclation caused by the pressure incre-ment as

c r - ae A r , ( 10 .15 )

Fcrr the next loading, o1 (note: o2 cquals thc cumulative load per unit area ofspecimen), which causes addi t ional deformat ion AHr, the void rat io at the end ofconsolidation can be calculatecl as

( t : c r - L I 1 2

H , ( 1 0 ' 1 6 )

At this time. o2 : cffcctive stress, rr!. Proceeding in a similar manner, one can ob-tain the void ratios at the end of the consolidation for all load increments.

The efTectivc strcss o' and the corresponding void ratios (c) at the end of con-solidation are plotted on semilogarithmic graph paper. The typical shape of such aplot is shown in F igure 10.12.

Normally Consolidated and Overconsolidated Clays

Figure 10.12 shows that the upper part of the e-log a' plot is somewhat curved witha flat slope, followed by a linear relationship for the void ratio with log o' having asteeper slope. This phenomenon can be explained in the following manner:

A soil in the fleld at some depth has been subjected to a certain maximum ef-fective past pressure in its geologic history. This maximum effective past pressuremay be equal to or less than the existing effective overburden pressure at the time ofsampling. The reduction of effective pressure in the field may be caused by natural

10.7 Normally Consolidated and Overconsolidated Clavs 275

Figure 10.12Typical plot of c against log o'

geologic processes or humern processes. During the soil sampling, the existing effec-tive overburdcn pressure is also released, which results in some expansion. When thisspecimen is subjected to et consolidation test, a small amount of compression (that is,a small change in v<tid ratio) wil l occur when thc effective pressure applied is less thanthe maximum cffective overburden prcssure in the field to which lhe soil has beensubjected in the past. When the effective pressure on the specimen becomes greaterthan the merximum efTective past pressure, the change in the void ratio is much larger,and the e-log o' relationship is practically l inear with a steeper slope.

This relationship can be veril lcd in the laboratory by loading the specimen toexceed the maximum effective overburden pressure, and then unloading and reload-ing again. The e-log ' ' plot for such cases is shown in Figure 10. 13, in which cd rep-resents unloading and dfg represents the reloading process.

Figure 10.13Plot of e against log a' showing loading,unloading, and reloading branches

0 1 o ' r

E l lec t i vc p rcssurc . o ' ( log sca lc )

d.E

!

o

Elfective pressurer o'(log scale)


Prcssurc. o' (log scalc)

Figure 10.14 Graphic procedurc l 'or detelmining preconsolidation pressure

This leads us to thc two basic dcfinit ions of clay based on stress history:

l. Normally utrtsolidated, whose present effective overburden pressure is themaximum pressure that the soil was subjected to in the past.

2. Overconsolidaterl, whose present effective overburden pressure is less thanthat which the soil experienced in the past. The maximum effcctivc past pres-sure is called the preutnsolklatk.tn pressure.

Casagrande (1936) suggested a simple graphic construction to determine thepreconsolidation pressure rri. from the labclratory e-log o' plot. The procedure is asfo l lows (see Figure 10.14) :

By visual observation, establish point a, at which the e-log rr' plot has a mini-mum radius of curvature.Draw a horizontal line ob.Draw the l ine ac tangent at d.Draw thc l ine ad, which is the bisector of the angle bnc.Project the straight-l ine portion gh of the e-logo' plot back to intersect l ine adat I The abscissa of point./ is the preconsolidation pressure, aj..

The overconsolidation ratio (OCR) for a soil can now be defined as

OCR

where oi : preconsolidation pressure of a speclmeno' : Dresent effective vertical Dressure

o'E

!

o

)3.4.5.

: n L(r' (10.17)

o-

F

III

I

10.8

10.8 Effect of Disturbance on Void Ratio-Pressure Relationship 277

Effect of Disturbance on VoidRati o - Pressure Re I ati o n sh i p

A soil specimen wil l be remolded when it is subjected to some degree of disturbance.This remolding wil l result in some deviation of the e_log o' plot as observed in thelaboratory from the actual behavior in the field. The field e-log a' plot can be re-constructed from the laboratory test results in the manner described in this section(Terzaghi and Peck, 1967).

Normally Consolidated Clay of Low to Medium Plasticity (Figure tO.IS)

1. ln Figure 10. l-5, curve 2 is the laboratory e-log o' plot. From this plot, deter-mine the prcconsolidation pressure (rrj) : o',, (that is, the present effectiveovcrburden pressure). Knowing where c', : ol1, draw verticall ine ab.

2. Calculate the void rat io in the f ic ld . e, , [Scct ion 10.6, Eq. (10.13) ] . Draw hor i -zontal l ine c'r1.

3. Calculat.e 0.4c, and draw line e.f. (Note: . l ' is the point of intersection of the l inewi th curve 2. )

4. Join points.f and g. Notc that g is the point of intcrsection of l ines ab and cd.This is thc virgin cornpression curvc.

I t is important to point out th t t t i f a soi l is complete ly rcmolded, the general po-s i t ion of the c- log r . r ' p lot wi l l bc as reprcssnted by curve 3.

I V t rg tn

-- consolidation

curve; slope = C,

Consolidationcurve forrernolded specimen

Latloratoryconsolidationcurve

o'o = o', 'Pressure, o' ( log scale)

Figure 10. 15 Consolidation characteristics of normally consolidated clay of low tomedium sensit ivi ty


O.4eq1

6'o o ' .

Pressure. o ' ( log scale)

Figure 70. 16 Consol idat ion character is t ics of overconsol ic lated c lay of low tcr

nredium scnsi t iv i ty

Overconsolidated CIay of Low to Medium Plasticity (Figure 10.16)

1. In F igure 10. 16, curvc 2 is thc laboratory e- logo' p lot ( loading) , and curve 3 isthc laboratory unloading, or rebound, curve. From curve 2, determine the pre-consolidation pressure rrj.. Draw the vertical l ine ab.

2. Determine the field effective overburden pressure oi;. Draw vertical l ine cd.3. Determine the void ratio in the field. er. Draw the horizontal l ine fg. The point

of intersection of l ines /! and cd is h.4. Draw a l ine hi, which is parallel to curve 3 (which is practically a straight l ine).

The point of intersection of l ines hi and ab is j.5. Join points 7 and k. Point k is on curve 2, and its ordinate is 0.4e6.

The field consolidation plot wil l take a path hjk. The recompression path in thefiefd is hj and is parallel to the laboratory rebound curve (Schmertmann, 1953).

Example 10.3

Following are the results of a laboratory consolidation test on a soil specimenobtained from the field: Dry mass of specimen = L28 g, height of specimen atthe beginning of the test : 2.54 cm, G,:2.75, and area of the specimen =30,68 cm2.

c

-5Laboratoryrebound curve;slope = C" =

swell index

10.8 Effect of Disturbance on Void Ratio-Pressure Relationshio

Final height ofEffeetive specimen at the

pressure, o' end of consolidation(ton /ft2) (cm)

2.5402.4882.4652.4312.389z , -12+

2.2252.115

(.)0.5I248

1 6- )L

Make necessary calculations and draw an e vs. log o, curve.

SolutionFrom Eq. (10 .11)

w, M, 128 gH . :' ' AG,T,, AC.,p,, (30.68 cmr)(2.75)(1g/cm3)

Now the following table can be prepared:

Effective Height at rhe endpressure, rr' of consolidation, H Hn = H - H"

(ton /ft2l (cm) (cml

- 1 .52 cm

e = HulH"

o0..51248

l 6J L

2.5402.4882.4652.4312.389z--) /+

2.2252.175

1.020.9680.94-s0 .9110.ti690.8040.7050.595

0.6710.6370.6220.5990.5720.5290.4640.390

The e vs. log o' plot is shown in Figure 10.17.

0 . 3 I 3 l 0 3 0Ellective pressure. o'(ton/ft2) log scale

Figure 10,17Variation of void ratio witheffect ive Dressure

r t,..\!a

3 l 0 3 0

Cross-sectional area =


Heighl

I)

1

LV,, - L,eV,

where Ae - change of void ratio. But

A { VotumeA y l

l ll t Ai, v. I

l l Li t l

I+H

A y l

1 lv'r, ulr,t lt lY Y

Tv.i

10.9

Figure 10. 18 Sett lement caused by onc-dimcnsional consol idation

Calculation of Settlement fromOne-Dimensional Pri mary Consolidation

With the knowledgc gaincd from thc analysis of consolidation test results, we cannow proceed to calculate thc probablc scttlement caused by primary consolidationin the { ie ld, assuming one-dimensional consol idat ion.

Let us consider a saturatcd clay layer of thickness H and cross-sectional area.4under an existing averagc cfTcctivc overburden pressure oir. Because of ern increaseof effective pressurc, Arr', lct the primary settlement be S,.. Thus, the change in vol-ume (Figure 10.1t3) can bc given by

L .V : V11 - V t : HA - (H - S . )A : S ,A ( 10 .18)

whcre l/,, and I/1 are the init ial and final volumes, rcspectivcly. However, the changein the total volume is equal to the change in thc volumc of voids, Atr/,,. Hence,

LV : S,A : V,s - V,,, : LV,, (10.19)

where /,,1; and V,,, are the init ial and linal vclid volumes, rcspectively. From thedefinit ion of void ratio. it follows that

Vuv . : : '' ' 7 t e , ,

A H

( 10.20)

( 10.21)l * e r 1

where eo : init ial void ratio at volume V,,. Thus, from Eqs (10.18) through (10.21),

LV - S,A = LeV,: ,Tr, ,O,

A.eS , . : H' I l e , l

(r0.22)

10.10 Compression lndex (C") and Swell lndex (C") 281

For normally consolidated clays that exhibit a l inear e-logo' relationship (seeFigure 10.15) .

Ae - C,.[ log(o'o + L,o') - log oi,] (10.23)where c. : slope of the e-log o' plot and is defined as the compression index. Sub-stitution of Eq. (10.23) into Eq. (10.22) gives

(t0.24)

In overconsolidatcd clays (sce Figure 10. 16), for o|, + Ao' < oj , f ield e-logo,variation wil l be along the l ine hj, the slope of which wil l be approximately equal tothat for the laboratory rebound curve. The slclpc of the rebound curve C" is referredto as the swell index; so

Ac : C , !og (o l1+ L , c ' ) l ogo i , , )

From Eqs. ( l t . l5) and ( lJ . l8) , we obrain

( lo.2s)

',*#*'"-(4#)

s.:#' .r(n#) (10.26)

l [ t r ' , , + L t r ' - r r j . thcn

10.10

(10.27)

Howevcr, if the e-log rr' curve is givcn, one can simply pick Ae off the plot for theappropriate range of pressurcs. This number may be substituted into Eq. (I0.22) forthe calculat ion of set t lement . S. .

Compressron lndex (C") and Swell lndex (C,)

The compression index for the calculation of f ield settlement caused by consolida-tion can be determined by graphic construction (as shown in Figure 10.15) after oneobtains the laboratory test results for void ratio and pressure.

Skempton (1944) suggested the following empirical expression for the com-pression index for undisturbed clays:

C " : 0 . 0 0 9 ( L l * 1 0 ) (10.28)

where LL : l iquid l imit.Several other correlations for the compression index are also available. They

have been developed by tests on various clays. Some of these correlations are givenin Table 10.4.

'.:ffi ̂ r#.#r'"-(tn;*)

Chapter l0 Compressibility of Soil

Table 10.4 Correlations for Compression Index. C.*

Equation Reference Region of appl icabi l i ty

c, - 0.0()'7(LL 7)C, : 0.01u,,yC , . : l . 15 (e , , - 0 .27 )C , :0 .30 (eo -0 .27 )

C, - 0.01l5r.r. ' 'c , : 0 . 0 0 4 6 ( L L - 9 )C , : 0 . 7 5 ( c o - 0 . - 5 )C . . : 0 .208c r2+0 .0083C, = 0. 156c,, + (.).0107

Skempton (1944)

Nish ida (1956)Hough (1957)

Remolded claysChicago claysAll claysInorganic cohesivc soi l : si l t , si l ty clay, clayOrganic soi ls, peats, organic si l t , and clayBrazi l ian claysSoils with low plastici tyChicago claysAll clays

'k Alter Renclon-Hcrrero ( 1980)Note: a71 -- in silu void ratio: toa : in.ri l tt water contcnt.

On thc basis of observel t ions on several natura l c lays, Rendon-Herrero (1983)

gave the rclationship for thc compression index in tlre form

. - / | + c , , \ 2 3 r 1c . : 0 . r4 lc i ' [ q , )

Nagaraj and Murty (19u5) cxpressecl the compression index as

C , : l r o * c ,Thc swell index was cxpressed by Nagaraj and Murty (l9lt5) as

I L L ( v " \ lc , _ { ) .04631

l ( ) ( ) ]c ,

( r 0.29)

I L L ( % \ lc, : 0.23431 ,ou

.1", ( l0.30)

' Ihc swell index is apprcciably smurller in rnagnitude than thc compression index and

can general lv be dctcrmined l rom labt t ratorv tcsts. In most cases,

( 10.3 r )

(10.32)

Example 10.4

A soil profile is shown in Figure 10^19. If a uniformly diSributed load, Ao, is ap-plied at the ground surface, what is the settlement of theplay layer caused by pri-mary consol idat ion i f

a. The clay is normally consolidatedb. The preconsolidation pressure (oL) :190 kN/m2c' o ' , : 170 kN/m2

U s e C . : I C , .

10.10 Compression lndex (C") and Swell lndex (C,)

ao = 100 kN/mj

{ , } tY,iry = l4 kN/ml

1' Cround water table..-

Sand

%*r = 18 kN/m3

' ' ' , ' : : r " . . ' ' : l : . : : ' . ' . i C l a y " " . ' . i

; " " . ' '

Figure 10.19

Solutiona. The average effective stress at the middle of the clay layer is

ab:2Tarv + 4lTruqsonay * y,o] * z4 l7sar1"roy; -* TrrJ

OT

os : (z)Qa) + 4(18 * e.81) + 2(19 - 9.81) : 79.1,4 kN/mzFrom Eq. (10.24),

s , : -c :H ron(ob * A ' ' )

' l * e , , " \ o 6 )

From Eq. (10.28),

C" : 0.009(Lr * 10) = 0.009(40 * I0) :0.27

So

^ (0.27)(4) . / 79.14 + 100 \r . :

I + o s l o g \

7 e . 1 4 ) : r r l 3 m : 2 l 3 m m

b. os + Lo' :79.74 + L00 = 179.14 kN/mzo',: 190 kNlmz

Because a'6 * A,cr' ) ol, use Eq. (10.26) to get i

^ C " H / o l , * A o ' 1 ;s . : l + e o , " t ( ? )

^ c" 0.27c , : ; : ; : 0 . 0 4 s

^ (0.04sX4) . ( tg.t+ + 100 \s : il 0*:

ros( ,q11 )

= 0.036 m : 36 mm


c . o b : 7 9 . 1 4 k N / m 2o ' s i Ao ' : I 79 .14kN/m2

o ' , : 170 kN /m2

Because vb < oL < (rb + Ao', use Eq. (10.27),C"H o ' - C, H / 6 ' , , + L,o ' \

s ' : 1 * " , l " c . b * r + " r l o e ( 4 /

(0.045X4) . ( t70 \ (0.27)(4) . ( t7e.14\: * tos[ff i/ + rs '"t[ '^ /

: 0.0468 m

: 46.8 mm

Example 10.5

The laboratory consolidation data for an undisturbed clay specimen are as follows:

e t : l . l o ' t : 95 kN /m2

ez : 0.9 oz : 475 kN/mz

What wi l l be the void rat io for a pressure of 600 kN/m2I (Note: o i < 9.5 kN/m2.)

6

-' E

,175 60095 ,+75 600

Pressure, o ' ( log scale) (kN/ml)

$olutionFrom Figure L0.20,

C,

€ t - € z

€3

( t - e z

Figure 10.20

1.1 - 0.9 ;= logd? * logdl: log 475 - log$. - - :

: C.(log 600 - log 95)

600: € t C loe=" 9 5

= 1.1 - 0.286 log S : o.rt

: 0.286

1 0.1 1 Secondary Consolidation Settlement 285

loJ I Seco n da ry Co n so I i dati o n Settl e m ent

Section 10.5 showed that at the end of primary consolidation (that is, after completedissipation of excess pore water pressure) some settlement is observed because ofthe plastic adjustment of soil fabrics. This stage of consolidation is called secondaryconsolidation. During secondary consolidation the plot of deformation against thelog of t ime is practically l inear (see Figure 10.10). The variation of the void ratio, e,with time t for a given load increment wil l be similar to that shown in Figure 10.10.This variation is shown in Figure 10.21. From Figure 10.21, the secondary compres-sion index can be defined as

log 12 - log r,Ae

l"g(t/,)( 10.3-1)

where C,, : secondary compression indexAe : change of void ratio

/p, /2 - t ime

The magnitude of the secondary consolidation can be calculated as

s" : clH ( r0 .34)

Time, r (log scale)

Figure 10.21 Yariatton of e with log / under a given load increment and definition of sec-ondary consolidation index

A,eco

/ t , \'"r(. i/

!

s


where

Cc'-: Th: (10'3s)

e,, : void ratio at the end of primary consolidation (see Figure 10.21)H : thickness of clay layer

The general magnitudes of Ci as observed in various natural deposits are as follows:

. Overconsolidated clays : 0.001 or less

. Normally consolidated clays : 0.005 to 0.03

. Organic soil : 0.04 or more

Secondary consolidation settlement is more important than primary consolida-tion in organic and highly compressible inorganic soils. In overconsolidated inorganicclays, the secondary compression index is very small and of less practical significance.

Example 10.6

For a normally consolidated clay layer in the field, the following values are given:

r thickness of clay layer : 8.5 ft. Void ratio (es) : 0.8r Compression index (C.) : 0.2So Average effective pressure on the clay layer (ob) :2650lblf9r Atr ' :970lbl f t2r Secondary compression index (C") : 0,42

What is the total consolidation settlement of the clay layer five years after thecompletion of primary consolidation settlement? (Note: Time for completion ofprimary settlement = 1.5 years.)

SolutionFrom Eq. (10.35),

C'* =co

1 * e ,

;

The value of eocan be calculated as. ep : €o Aepri_ury

10.12 Time Rate of Consolidation 287

Combining Eqs. ( 10.22) and (10.23). we f ind rhat"

Le : C, rcn( ob *

,oo') : o.r , ron( 26s9,1,e70 )- , - "o\

os / " . - ' . " " \ 2650 )

: 0.038

Primary consolidatior, s" : ++ - (0'03qx8'l J( 12) : 2.15 in.' 7 * e o l + 0 . 8

It is given lhat es - 0.8, and thus,

e o * 0 . 8 * 0 . 0 3 8 : 0 . 7 6 2

Hence,

c - : o 9 - 2 , , : o . o 1 r-a

| + 0.162

From Eq. (10.34),

Total consolidation settlement : primary consolidation (S.) + secondary settle-ment (S"). So

total consolidation settlement :2.15 + 0.59 : 2.74in. r

10.12 Time Rate of Consolidation

The total settlement caused by primary consolidation resulting from an increase inthe stress on a soil laycr can be calculated by the use of one of the three equations -(10.24), (10.26), or (10.27) givcn in Section 10.9. However, they do nor provide anyinformation regarding the ratc of primary consolidation. Terzaghi (1925) proposedthe first theory to consider the rate of one-dimensional consolidation for saturatedclay soils. The mathematical derivations are based on the following six assumptions(also see Taylor, 1948):

1. The clay-water system is homogeneous.2. Saturation is complete.3. Compressibil i ty of water is negligible.4. Compressibil i ty of soil grains is negligible (but soil grains rearrange).5. The flow of water is in one direction only (that is, in the direction of compression).6. Darcy's law is valid.

Figure 10.22a shows a layer of clay of thickness 2H,,,thatis located between twohighly permeable sand layers. If the clay layer is subjected to an increased pressureof Aa, the pore water pressure at any point,4 in the clay layer will increase. For one-dimensional consolidation, water wil l be squeezed out in the vertical direction towardthe sand laver.

s, : c"H'"r(f) = (0.011)(8.s x 12) ros(*) = o.rn,n


r 1 :"ir j .

, lt )

. . . u |

dz.) d,r d)'

dr

(b)

Figure I022b shows the flow of water throughthe soil element shown,

Rate of outflow Rate of inflowof water o[ water

Figure 10.22(a) Clay layer undergoingconsolidation; (b) flowof water at -4 duringconsolidation

a prismatic element at,4. For

Rate ofvolume change

/ 6 u . \l u . + : d z . l\ 3 2 /

avd x d y - D . d x d y : -

dt

Thus,

r10.12 Time Rate of Consolidation

volume of the soil elementvelocitv of f low in z direction

289

where

or

V :

oa, av;t7

dx dl ' lz - 7

Using Darcy's law, wc have

, dh k itrrl ' ; - - K t - - *

; , r . - - y , , . i

where u : excess pore water pressure caused by the increasc of stress.From Eqs. (10.36) and (10.37) ,

K d-u. l a v

(10.36)

(10 .37)

( 10.38)

soi l element is equal

( r 0.3e)

- y , , .o r . t

: ax l y az a t

During consol idat ion, the raLe of change in the volume of theto the rate of changc in the volurne of voids. Thus.

dv : ilv, : !Uti:y) !lv' + V .Y + ,dV'a t a tat al i)t At

wherc / , : vo lume of 's<l i l so l ids7,, - volume ctl 'voicls

But (assuming that so i l so l ids arc incompressib le)

! ! r : udt

and

V dx dv dz.V,

Substitution for itV.litt and V, in

where d(Aa' )o, ,

l l a , , l + e , , ,

Eq. ( 10.39) yic lds

aV _ dx dy dz ;te( 10.40)

a t l l e 6 a t

whcre co : in i t ia l vo id rat io .Combining Eqs. (10.38) and (10.40) g ives

K ( t -L l | ( le

y,, oz2 | + eo at( 1 0 . 4 1 )

The change in the void ratio is caused by the increase of effective stress (i.e.,a decrease of excess pore water pressure). Assuming that they are l inearly related,we have

i)e - tr,.O(L,o, ) : -o,1u

change in effective pressurecoefficient of compressibil i ty (a,. can be consideredconstant for a narrow range of pressure increase)

(r0.42)


where

or.

wht:re

Thus.

Combining Eqs. (10.a1) and (10.42) givcs

k i)2Lt 0,, i lu

y , , d * : l + q , i : - n ' '

r.t,, : coefficient of volumc compressibil i ty - a,,l( l + e7)

(l- l l

o1.

c, , : ccre l l ic icnt o l 'consol idat ion : k l (y , , ,m, . )

i)tt

dt

Eq. ( l { ) .aa) is thc basic d i l l 'e rcnt ia l cquat ion of 'Tcrzaghi 's consol idat ion theory

and can bc solved wi t l - r the fo l lowing boundary condi t ions:

z - 0 . r r - 0

7 , : 2 1 t , 1 , , u : 0

/ : 0 , L t : u 1 1

1'hc solution yiclcls

- M ? t , ( 10.47)

w h e r e m : a n i n t e g c rM : Q r l 2 ) ( 2 m + t )Ilo : initial excess pore watel prcssure

,):l :dt

, -'TlT""(#,)1'

f , , t n , ( u , \Y " \ r + c ; /

r . : :+ : t ime fac lor' Hit,

( 10.43)

( 10.44)

( ro.4s)

( 10.46)

( r0.48)

The time factor is a nondimensional number.Because consolidation progresses by thc dissipation of excess pore water pres-

sure. the desree of consolidation at a distance z at anv time r is

Llrt - Ll- Ll-u " - ' - l - -

- Llo uo

where u: : excess pore water pressure at t ime 1.

(10.4e)

10.12 Time Rate of Consolidation

0 (). I O.2 0.3 0. ' t 0.5 0.(r O.1I)egrec of consol idal ion. t /-

Figure 10.23 Yariation of U- with 7',. and z./l!,1,

Equat ions ( l1 .4 l ) and (10.49) can be combined to obta in the dcgree of con-sol idat ion at any depth z. This is shown in F igure 10.23.

The average dcgree of consolidation for the entire depth of thc clay laycr atany t ime / can be wr i t ten f ront Eq. (10.49) as

291

U : ?/ r \ f tn,,,l . - l l u , d z

_ 1 \ z H d , / J our)

where IJ : average degree of consolidationS,1,y : settlement of the layer at t imc t

S,. : ult imate settlement of the layer from primary consolidation

Substitution of the expression for excess pore water pressure a_ given inEq. (10.47) into Eq. (10.50) gives

( r 0.s0)

(10.-5 r ), - | ' f 2 - " , , ,,7:r, M'

The variation in the average degree of consolidation with the nondimensional t imefactor, I,,, is given in Figure 10.24, which represents the case where r.r., is the same forthe entire depth of the consolidating layer.


0..1 0.6Tinre lactor, 7',.

Di f}'erent typcs of' drainagewith &., constilnt

Figure 10.24 Yariation of avcrage degrcc of consolidation with time factor, 7',, (r2,, constant

wi th depth)

The values of thc time ferctor and their corrcsponding average degrees of con-

solidation for the case presented in Figure 10.24 may also be approximated by the

following simple rclationship:

> o li : b o lr E l; , .= |

A ; l

6 i l l

F F I

O ! Y

b\

; 2 0'E

=e 4 0

o Ot,

!

8 1 8 0o

0.8

AI

H ,

IY

1'., :

: 'l

IH,r,

IIV

* / t t o t ^ \ 2

F o r U : 0 t o 6 0 % , 7 , . : + ( = )' 4 \ 1 0 0 /

For IJ 7 60o/o, T, * 1.78I * 0.933 log(100 * U"/.)

(10.s2)

(10.s3)

10.13

Table 10.5 gives the variation of ?',, with U on the basis of Eqs. (10.52) and (10.53).

Coefficient of Co nsolidatio n

The coef{icient of consolidation cu generally decreases as the l iquid l imit of soil in-

creases. The range of variation of cu for a given l iquid l imit of soil is wide.For a given load increment on a specimen, two graphical methods are commonly

used for determining cu from laboratory one-dimensional consolidation tests. The

first is the logarithm-of-time methodproposed by Casagrande and Fadum (1940), and

the other is the square-root-of-time method given by Taylor (1942). More recently, at

least two other methods were proposed. They are the hyperbola method (Sridharan

and Prakash, 1985) and the early stage log-t method (Robinson and Allam, 1996). Thegeneral procedures for obtaininB cu by these methods are described in this section.

10.13 Coefficient of Consolidation 293

Table 10.5 Variation of f,, with U

u (%l r" U lo/"1 T,Tu u(%l0I2315676

9l 0l tt 2l 3l 4l-5l 6l 7I t iI 9202 l22L- )

242526272u29303 t- )z

33

-)+

3-5--to

373tt39404 lA ' )

43414.546414u49-s05 t) t

53-54-s556575f,i.59606 l6263646.566

67

6869707 l72

747576717t3l()u0u l82lJ3n4|.i.5u6ti7uttllc)

909 l9293949-s96o?

9u99r(x)

0.3770.3900.4030.4110.4310.4460.4610.4770.4930.-51 I0.5290.5470..s670.5t3u0 .6 I00.6330.65u0.61J40 .7120.7420.1740.u090.t3480.139I0.93u0.9931.055| .129t .2191 .3361.500I . 7 u l

00.000080.00030.000710.(n 1260.00I 960.002830.003u.50.00-5020.006360.007ft.50.(x)9-50 .01 l 30.0I 330.01-540 .0 t770.02010.02270.02.540.02rJ30 .03 I40.03460.03t300.041.50.04520.04910.053I0.05720.06 l-s0.06600.07('t70.07540.011030.08-5-5

0.09070.09620.1 020.1 070 . 1 l 30 . 1 l 90 . 1 2 60.1320 .1 380 .1450.1.520. t -590.1 660 . 1 7 30. t t3 l0 . I 8 [ t0. t970.2040 . 2 1 20.2210.2300.2390.24tt0.2570.2610.2760.2u60.2910.3070 .3 r80.3290.3040.3-520.364

Lo ga rith m -of-Ti me M eth od

For a given incremental loading of the laboratory test, the specimen deformationagainst log-of-time plot is shown in Figure 10.2-5. The following consrrucrlons areneeded to determine c,,:

1. Extend the straight-l ine portions of primary and secondary consolidations tointersect at ,4. The ordinate of ,4 is represented by d,uu - that is, the deforma-tion at the end of 100% primary consoliclation.

2. The initial curved portion of the plot of deformation versus log r is approxi-mated to be a parabola on the natural scale. Select t imes /, and /, on the curvedportion such that t2: 4t1. Let the difference of specimen deformation durinstime (1, - r1) be equal to x.


t I t ) I 5 t )

- l ' i rne ( log scalc)

Figure 10.25 Logztr i lhrn-of-t irnc method l irr dctcrmining coefl icicnt ol 'consol idation

Draw a horizontal l ine DE such thal the verticirl distancc BD is equal tor. The

deformat ion corrcsponding to the l inc DE is r / , , ( that is , dcformat ion at 0%consol idat ion ) .The clrdinatc of point f on thc consoliclation curvc rcpresents the deformationat 50% primary consolidation, and its abscissa rcprcsents the correspondingt ime ( r r , , ) .For -50'/o average clegree of consolidation, I,, : 0.191 (see Table 10.5), so,

,,,,:Eo.tel H:t ,

4.,, :

,,

where 11,1,. : avcragc longest drainage path during consolidation.

For specimens draincd at both top and bottom, H,1, equals one-half the aver-age height of the specimen during consolidation. For specimens drained on only oneside, H,,, equirls the avcrage height of the specimen during consolidation.

Squ a re-Ro ot-of -Ti m e M eth od

In the square-root-of-time method, a plot of deformation against the square root oftime is made for the incremental loading (Figure 10.26). Other graphic constructionsrequired are as follows:

1. Draw aline AB through the early po419n of the curve.2. Draw aline AC such that OC : 1.1508. The abscissa of point D, which is the

intersection of AC and the consolidation curve, gives the square root of t imefor 90o/o consol idat ion 1f ,4, ) .

aa

E, a

o

()r

( 10.54)


E

a

B C'tr--\ I iltt(t /

Figure 10.26Scluare-root-ol-t ime fitt ing method

: 0.1't4U (sce Table 10.-5), so

7i,, : {).t{4tt: !+fI ,,,

3. For 907o consolidation, f,rn

0.848H?t,

tgl

H,1, in Eq. ( I0.-5-5) is detcrmincd in a manncr sinti lar to thatof - t ime method.

Hyperbola Method

ln the hyperbola method, thc following procedure is recommencled for the determi-nat ion of c , , :

l. obtain the time l and the specimen deformation (AH) from the laboratoryconso l i da t i on t es t .

2. Plot the graph of t lL,H against / as shown in Figure 10.27.

Figure 10.27Hyperbola method for determination of c.,

( 10.ss)

in the logar i thm-

{D

t


3. Identify the straight-l ine portion hc and project it back to point d. Determine

the intercept D.4. Determine the slope m of the l ine bc.

5. Calculate c,, as

. ̂ / m I I ) , \c , , : 0 3 ( ; J ( 1 0 s 6 )

Note that because the unit of D is t ime/length and the unit of m is (t ime/length)/

t ime - 1i length. thc unit of c,, is

(#*)""ngth)2

/ t ime \

\ l eng th /

( length)2

t ime

The hyperbola method is fairly simple to use. and it gives good results for U =

60"/" to 90"/".

Early Stage log-t Method

The early stage log-r method, an extcnsion of the logarithm-of-time method, is based

on specimcn deformation against log-of-time plot as shown in Figure 10.2U. Accord-

ing to this methocl, follow steps 2 and 3 describcd for the logarithm-of-time method

to determinc r1,,. Draw a horizontal l ine DE through d,,. Thcn draw a tangent through

thc point of inflection, f, The tangent intersects l ine DE at point Ci. Determine the

time 1 corresponding to G, which is the time at U - 22.14"/" . Sct

o.038sH:1, ( Io.-s7)c , , :t22Jq

ln most cases, for a given soil and pressure range, the magnitude of c,. determined

by using the logarithm-o,f-time method provides ktwe st value.The highest value is ob'

tained frcrm the early stoge lctg-t ntethod. The primary reason is because the early stage

log-/ method uses the earlier part of the consolidation curve, whereas the logarithm-

of-time method uscs the lower portion of the consolidation curve. When the lower

Figure 10.28Early stage log-r method

EI; lE IE I€ lo

Time, t (log scale)


portion of the consolidation curve is taken into account, the effect of secondary con-solidation plays a role in the magnitude of c,,. This fact is demonstrated for severalsoils in Table 10.6.

Several investigators have also reported that the c,, value obtained from thefield is substantially higher than that obtained from laboratory tests conducted byusing conventional testing methods (that is, logarithm-of-time and square-root-of-time methods). Hence, the early stage log-/ method may provide a more realisticvalue of f ieldwork.

Tahle 10.6 Comparison of c,. Obtaincd from Various Methods*

cu x 104 cm2/sec

Range ofpressure o'

(kN/m'zl

Logarithm- Square-root-of-time of-time Early stagemethod method log t method

l l l i te

Red earth

Brown soi l

Black cotton soi l

Bentonite

Chicago clay(Taylor, 1948)

25 -50-50 t(x)

t(x) -200200 - 400400-80025 -5050 -100

100 -20t)200 - 4004(X)-U002-5 --5050 -100

100 -2(x)200 - 4(x)400 - 80025 -50.50 I(X)

l (n-200200-'100400 - 80t)2-s -.5050 -100

100-200200 - 4004(X) 800r2.5-25

25 -5050 -100

100 -20t)200-400400 - 800800 -1 600

4.636.43

t3.1 4u . I 03.13 t3.022.862.091 .305.073.062.00l . l 50.-561 .661 .342.203 . l 54. l -50.0630.0460.0440.0210.01-5

25.1020.70t3;703 .184.566.0.s'7.09

.5.4-51.989.99

10.90I 1 .994.453.773.4t)2 .211.4-s6.-5-53.692.501 .570.642.253 . l 33 . l 8,1.595.820 .1300.1000.0520.0220 .017

45.5023.9017.404.714.406.448.62

6 . 1 29.00

I 1 .43t2.56t 2.ttO

) . 4 1

3.n03.522.74t .369 ;734.78- i .4 )

2.030 ;792.50- ) . ) L

3.6-55 . 1 46.450 . t620 .1300.0810.0400.()22

46.0031.-5020.204.974.917.419.09

* After a table from "Determination of Coefficient of Consolidation from Early Stage of

Log r Plot," by R. G. Robinson and M. M. Allam, L996, Geotechnical Testing Journal, 19(3)pp. 3 16 -320. Copyright O 1 996 American Society for Testing and Materials. Reprinted withpermission.


Example 10.7

The time required for 50% consolidation of a 25-mm-thick clay layer (drained atboth top and bottom) in the laboratory is 2 min. 20 sec. How long (in days) will ittake for a 3-m-thick clay layer of the same clay in the field under the same pres-sure increment to reach 50% consolidation? In the field, there is a rock layer atthe bottom of the clay.

Solution

L ' , . / loh ( ' r l t i . l , l

' ) r r L J z u 2r r d r ( la l ' ) r , , / r l t ie ld )

flrb _

HtarQort)

140 sec

7on2s;T:t _ l\ z /

t *' f ie ld -

I"l?n(,,,tu\

/fi"r,r

(3 rn)t

8,064,000 sec : 93.33 days

' f ic ld

Example 10.8

Refer to Example 10.7. How long (in days) will it take in the field for 30% primaryconsolidation to occur? Use Eq. (10.52).

SolutionFrom Eq. (10.52),

So

c r J i i . t , t : T " , x l J 2

H'artlah)

t x ( J z

\ : U it2 Ui

93.33 days 502T^ 302

/e * 33.6 days

10.13 Coefficient of Consolidation 2gg

Example 10.9

A 3-in.-thick layer (double drainage) of saturated clay under a surcharge loadingunderwent 90% primary consolidation in 75 days. Find the coefficient of consoli_dation of clay for the pressure range.

$olution

Tun : c "t!'.,

H rd,

P:"1u1" the clay layer has two-way drainage, Ha, : 3 m/2 = 1.5 m. Also. Z*, =0.848 (see Table 10.5). So

0.848 - c"(75 x 24 x 6o x 6o)

(1.s x 100)2

",, : s1lli?lrioo75 x 24 x 60 x OO

= 0.00294cm2/sec r

Example 10.10

For a normally consolidatecl laboratory clay specimen drained on both sides, thefbllowing are given:

n i2 :30001b/ f t2 € = e71- l . l .o'r, * Ao' : 60001b/ft2 e : 0.gThickness of clay specimen : 1 in.Time for 50% consolidation : 2 min

a. Determine the hydraulic conductivity (frlmin) of the clay for the loadingrange.

b. How long (in days) will it take for a 6-ft clay layer in the field (drained onone side) to reach 60% consolidation?

SolutionPart AThe coefficient of compressibility is

/ A u \Q , , \ G ' /

t n . . : - :" 7 * e o . , , 1 , + e r , )

A e = L . 1 - 0 . 9 : 0 . 2

Ac' = 6000 - 3000 : 30001b/ft,

1 . 1 + 0 . 9e r r : * _ ; - x 1 . 0


So0.2

3000m, :

f f i : 3 .33 x 10 5 f t2 / l b

From Table 10.5, for U : 504/", Z, : 0.197; thus,

(0.1e?) (r*)'c u :

k: c.m,y,,,- ( l . i \ x 10-1ft2/min)(3.33 x 10-sft?lb)(6z.4lbltf)

: 3.55 x l0 7ftlmin

Part BL ^ l a' f - - t ' r n ,

l A t r -

H!,,

, - TnuHl,,t60 *

c.,.

From Table 10.5, for U : 60o/o and 76u = 0.2t16,

{0.286)(6) lteo:

, r , * f f i :60.211min :41.8days

Calculation of ConsolidationSettlement under a Foundation

: 1 .71x l 0 4 f r2 lm in

10.14

Chapter 9 showed that the increase in the vertical stress in soil caused by a load ap-plied ovcr a l imited area decreases with depth z measured from the ground surfacedownward. Hence to estimate the one-dimensional settlement of a foundation. wecan use Eq. ( lU.2a), ( 10.26), or ( 10.27). However, the increase of effective stress, Aa',in these equations should be the average increase in the pressure below the center ofthe foundation. The values can be determined by using the procedure described inChapter 9.

Assuming that the pressure increase varies parabolically, using Simpson's rule,we can estimate the value of Aoj,,. as

LrvLuA,rr ' ,+4Acr '^* Ltr '6

( 10.s8)

where L,o',, Lrr',,, and Aoi, represent the increase in the effective pressure at the top,middle, and bottom of the layer, respectively.

10.14 Calculation of Consolidation Settlement under a Foundation

Example 10.11

Calculate the settlement of the 10-ft-thick clay layer (Figure 1A.29) that will resultfrom the load carried by a S-ft-square footing. Tire ciay-is normaliy consolidated.use the weighted average method [Eq. (10.5s)] to ca]culate rhe average increaseof effective pressure in the clay layer.

ilr I, +

r l+ '+

_ l lDrY sand | 5 {t eootins sizey d r y = l o o p c t

, o n I s r r x i t iI

t -------v-g':T1itiyy1-- * - 1 - - - - €

I Sand%at = 120 Pcf

Figure 1A.29

$olutionFor normally consolidated clay, from Eq. (70.24),

n C,,H o'0 + L,o'out " :

l + e o r r g o b

where

C. = 0.009(Lr - 10) = 0.009(40 - 1.0) = g.y1

1 1 : 1 0 X 1 2 : 1 2 0 i n . iitrf$e o : 1 . 0

ob: lof t x ydry(sand) + l0f t l7,o,1,u"d) - 62.4] + fgly,u, i . ruy )- 62.4]

: l 0 x 1 0 0 + 1 0 ( t 2 0 - 6 2 . 4 ) + 5 ( l t 0 - 6 2 . 4 )

: 1814lblft2


From Eq. (f0.58),

Lr'L,Lo ' , *4Lo ' ^ * Lo '6

A{r',, L.o'^,and Aaibelow the centerof thefootingcan be obtainedfrom Eq. (9.33).Now we can prepare the following table (note: LIB = 5/5 = 1):

L,c' = qlt

{kip/lt'?lz b= B l2

fttl (ft) (ft) ht = zlbcl

{kip/ft'?} t1

0.051 0.408 - Aol

0.232 =0.152 =

Lo'^LoL

0.o290.019

2005 x 588

r l s 2 . 5 6

t 2 0 2 . 5 81 25 2.5 10

10.15

So

LoL,:

Hence,

0.408 + (4)(0.232) + 0.152: 0.248 kip/ft2 :24Blblft2

t. - q1+P,o*s1r*1#€: o.ein

Methods for Accelerating Consolidation Settlement

In many instances, santl drains and prefubricated verticul drains are used in the field

to acceleratc consolidation settlement in soft, normally consolidated clay layers andto achievc precompression before the construction of a desired foundation. Sanddrains are constructed by dril l ing holes through the clay layer(s) in the field at regu-

lar intervals. The holes are then backfi l led with sand. This can be achieved by several

means, such as (a) rotary dri l l ing and then backfi l l ing with sand; (b) dri l l ing by con-

tinuous fl ight auger with hollow stem and backfi l l ing with sand (through the hollow

stem); and (c) driving hollow steel piles. The soil inside the pile is then jetted out, and

the hole is backfi l led with sand. Figure 10.30 shows a schematic diagram of sand

drains. After back{il l ing the dril l holes with sand, a surcharge is applied at the ground

surface. This surcharge wil l increase the pore water pressure in the clay. The excesspore water pressure in the clay wil l be dissipated by drainage -both vertically and

radially to the sand drains - which accelerates settlement of the clay layer. In Figure

10.30a, note that the radius of the sand drains is r,,,. Figure 10.30b shows the plan of

the layout of the sand drains. The effective zone from which the radial drainage will

be directed toward a given sand drain is approximately cylindrical, with a diameter

of d". The surcharge that needs to be applied at the ground surface and the length of

time it has to be maintained to achieve the desired degree of consolidation will be a

function of r,,,, d", and other soil parameters. Figure 10.31 shows a sand drain instal-

lation in progress.

10.15 Methods for Accelerating Consolidation Settlement

I ISurcharge

t t t lt t t lr t t t

brouno

-g-I|aleltlble**

. t : 'nuverilcxr dralnage

AI

II

. Radial ,drainage :

IIIt

, ,Ver1!9q!.dra!ng9Sant l

(a ) Scc t ion

t lt l

Y Y

. . ' - , - . ' , . . . :i Sand drain;I radius = r,n.

: Radial(rrarnage

' j . . : " . t . ' : . " . " '

l l q a| *t'l *

Hc

II ClavI tuv.t

. 1 . .- : , i : V , . : . - .

Figure 70.30 Sand drains

(b) l ' lan

Figure 10.31Sand drain installation irrprogress (courtesy of E. C. Shin,University of Inchon,South Korea)

<jruS;;;1J1:;Jg;j:j;;;1:g:;r

:-jj:tt;Sr:,:ij::.:r:s:t


Polypropylenccore

Ceotext i lelitbric

Figure 10.32 Prctabricated vert ical drain (PVD)

Figure 10.33In ta l la t ion o1 'PVDs in

progrcss (courtesy of E. C.

Sh in . l . ln ivers i ty o [ Inchon,

South Korea)

10.16

prelabricatcd vertical drains (PVDs). which are also referred to as wlck or strip

tlruins, were originally developed as et substitute for thc commonly used sand drain'

With the advent of msterials scicnce, these drains are manufactured from synthetic

polymcrs such as polypropylene ancl high-density polyethylene. PVDs are normally

manufactured with a corrugated or chernneled synthetic core enclosed by a geotex-

ti le f i l ter, as shown schematically in Figure 10.32. lnstallation rates reported in the

literature are on the order of 0.1 to 0.3 m/s, excluding equipment mobil ization and

setup time. PVDs have been used extensively in the past for expedient consolidation

of low permeabil ity soils under surface surcharge. The main advantage of PVDs over

sand diains is that they do not require dri l l ing and, thus, installation is much faster.

Figure 10.33 shows the installation of PVDs in the field.

Summary and General Comments

In this chapter we discussed the fundamental concepts and theories for estimating

elastic and consolidation (primary and secondary) settlement. Elastic settlement of

a foundation is primarily a function of the size and rigidity of the foundation, the

modulus of elasiicity, the Poisson's ratio of the soil, and the intensity of load on the

foundation.Consolidation is a time-dependent process of settlement of saturated clay lay-

ers located below the ground water table by extrusion of excess water pressure gen-

erated by application of load on the foundation. Total consolidation settlemerr.clay fbundation is a function of compression index (C.), swell index (C"), init ial vo-ratio, (e,,) and the average stress increase in the clay layer. The degree of consolida-tion for a given soil layer at a certain time after the load application depends on itscoefficient of consolidation (c,,) and also on the length of the minimum drainagepath. Installation of sand drains and wick drains helps reduce the time for accom-plishing the desired degree of consolidation for a given construction project.

There are several case histories in the l iterature for which the fundamental prin-ciples of soil compressibil i ty have been used to predict and compare the actual totalsettlement and the time ratc of settlement of soil profi les under superimposed load-ing. In some cases. the actual and predicted maximum settlements agree remarkably;in many others, the predicted settlements deviate to a large extent from the actualsettlements observed. The disagreement in the latter cases may have several causes:

1. Improper evaluation of soil properties2. Nonhomogeneity and irrcgularity of soil profi les3. Error in thc evaluat ion of thc nct s t ress increase wi th denth. which induces

settlement

The variation betwcen the predictcd and clbserved timc rate o1 settlement mavalso be due tcr

a. Improper evaluat ion of c , . (see Sect ion 10. l3)b. Presence of irrcgular sandy seams within the clay laycr, which reduces the

length of the maximum drerinage path, H,1,.

Problemsl0.l Estimate the immcdiate settlemcnt of a column footing 4.-5 it in diameter

that is constructed on an unsaturated clay layer. The column carries a load of20 tons, and i t is g iven that E, : 1500 lb/ in .2 and g. , : 0 .25. Assume the foot-ing to be r ig id. [Use Eq. ( 10.1) . ]

l l).2 Refer to Figure 10.3. For a square foundation measuring 3 m X 3 m in plansupported by a layer of sand and givcn thal D, - 1.5 m, t : 0.25 tn, E,, :

16,000 kN/m2, k : 400 kN/m2/m, / r , : 0 . .1, n - Z0 m. Er : 15 x 106 kN/m2,and Ao : 100 kN/m2. c i t lcu late the immediate set t lement .

10.3 Following are the -'J,;jdjonsolidation test:

1 . 11.0851.0551 . 0 10.940.'790.63

0.2-50..5l . t )2.04.08.0

16.0

a. Plot the e-log o' curve.b. Using Casagrande's method, determine the preconsolidation pressure.c. Calculate the compression index C. from the laboratory e-log a' curve.


10.4 Repeat Problem 10.3. using the following values:

Pressure, a'e (kN/m2)

r . 2 l1 .1951 .1 -51.060.980.925

2550

100200400500

10.5 A soil profi le is shown in Figure 10.34. Thc uniformly distributed load on the

ground surface is Arr. Estimate the primary settlement of the normally con-

solidated clay layer, given thatH r : 4 t t , H , : 6 f t , H . - 4 1 1For sand, c - 0.-58, G, - 2.67For c lay, c : l . l , G, : 2 .12. LL : 45

A a - l f t ( X ) l h / l t r10.6 Repeat Problem 10.-5, using thc following data:

I 1 1 : 2 ' 5 m ' Hz : 2 ' 5 m ' H1 : 3 rnFor sand, a : 0 .64. G, - 2 .65For c l ay , c - 0 .9 , G , - 2 .75 , LL : 55

A t r - 100 kN i mr10.7 Repeat Problem 10.-5, using the l 'ollowing data:

Arr - 9() kN/nrrI I ' : 2m ' I 12 : 2m ' I I l : l " 5 m

For sand.7,1,u - 14.6 kN/mr, y , , , r : 17.3 kN/ml

For c lay,7,u, : 19.-1 kN/mr. LL : 38,e - 0 '15

t lt l+ +

:!:]:;:;ill:.]:il;ni::i..-

A o rIIJ

IIIIv

Sand

Figure 10.34

Problems

1 1 1: : : 1 : : : : : [ : '

l0 frIf

Dry sande = 0 . 6

C,=2.65 y Croundwater tabfe""--" tI Sand

e = 0 . 6l0 ft

Figure 10.35

10.8 A soil profi le is shown in Figure 10.35. The preconsolidation pressure of theclay is 3400 lb/ft2. Estimate the primary consolidation settlement that wil ltake place as the result of a surcharge equal to2200 lb/ft2. Assume C. : JC,.

10.9 Refer to Problem 10.6. Given that c,, : 2.8 X 10 6 m2/min, how long wil l i ttake for 60% primary consolidation to take place'?

10.10 The coordinates of two points on a virgin compression curve are as follows:er : l . l t2 u ' t - 200 kN/m2c; - 1.54 o'2 = 400 kN/m2

a. Determine the coefficient of volume compressibil i ty for the pressurerange stated above.

b. Given that c,, : ().003 cmr/sec, determine k in cm/sec corresponding to theaverage void ratio.

10.11 For the virgin curve stated in Problem 10.10, what would be the effectivepressure o' corresponding to e : 7.7?

10.12 For the virgin curve stated in Problem 10.10, what would be the void ratiocorresponding to an effective pressure o' that is equal to 500 kN/m2?

10.13 Following are the relationships of e and o' for a clay soil:

e r' (ton/ft2l

1 . 00.9'70.8-5o.'75

0.20.51 . 83 . 2

For this clay soil in the field, the following values are given: H : 4.5 ft, o'u :

0.1 tonlft2,ando6 * Ao' :2tonlft2. Calculate the expected settlementcaused by primary consolidation.

10.14 During a laboratory consolidation test, the time and dial gauge readings ob-tained from an increase in pressure on the specimen from 50 to 100 kN/m2are given in the following table:


Dial gaugeTime reading(min) (mm)

Dial gaugeTime reading(min) (mm)

00 .100.250.601.02.04.08.0

3.984.084 . 1 04 .134.r7/ 1 1

4.304.42

16.t)30.060.0

120.0240.0480.0960.0

t440.0

4.574.744.925.08-s.2 I5.285.335.39

a. Find the time for 50% primary consolidation (rr,,) using the logarithm-of-time method.

b. Find the time for 90% primary consolidation (rr,,) using the square-root-of-time method.

c. If the average height of the specimen during consolidation caused by thisincremental loading was22 mm and it was drained at both the top and thebottom, calculate the coefficient of consolidation using /r,, and /r,, obtainedfrom parts (a) and (b).

10.15 Refer to the laboratory tcst rcsults givcn in Problem 10. 14. Using the hyper-bola method, determine r:,.. The average height of the spccimen during con-solidation was22 mm, and it was drained at the top and bottom.

10.16 The time for 50% consolidation of a 2-5-mm-thick clay layer (drained at topand bottom) in the laboratory is l-50 sec. How long (in days) wil l i t take for a3-m-thick laycr of thc samc clay in the field under the same pressure incre-ment to reach 50% consolidation'l There is a rock layer at the bottom of theclay in the field.

10.17 For a normally consolidated clay, the following values are given:o ' , t : 2 t t l n / f t r ( - ( . - l . 2 l

6 ' , ,1 L. r r ' - 4 t r ln / l ' t2 r , - 0 .96The hydraulic conductivity k of the clay for the preceding loading range is1.8 x 10-4 f t lday.a. How long (in days) wil l i t take for a 9-ft-thick clay layer (drained on one

side) in the held to reach 60% consolidation'?b. What is the settlement at that t ime (i.e., at 60"/" consolidation)?

10.18 A 1O-ft-thick layer (two-way drainage) of saturated clay under a surchargeloading underwent 90% primary consolidation in 100 days.a. Find the coefficient of consolidation of clay for the pressure range.b. For a 1-in-thick undisturbed clay specimen, how long wil l i t take to

undergo 90% consolidation in the laboratory for a similar consolidationpressure range? The laboratory tests's specimen will have two-waydrainage.

L0.19 Laboratory tests on a 25-mm-thick clay specimen drained at the top onlyshow 50% consolidation takes place in 11 min.a. How long will it take for a similar clay layer in the field, 4 m thick and

drained at the top and bottom, to undergo 50% consolidation?b. Find the time required for the clay layer in the field, as described in part

(a), to reach 10"/" consolidation.

!r.:.'

References 309

Load = 0

II

V

10.20 For a laboratory consolidation test on a clay specimen (drained on bothsides), the following results were obtained:

Thickness of the clay soil - 2-5 mmrr i : '511 kN/m) t '1 - 0 '92tr '2 - l20 kN/m: c2 - { ) .713Time for -507o consolidation : 2.5 min

Determine the hydraulic conductivity of the clay fbr the loading range.10.21 Refer to Figure 10.36. Given that B : 1.5 m, L - 2.5 m, and O : 120 kN,

calculate thc primary consolidation settlement of the foundation.10.22 Redo Problem 10.21 wi th B : I m, L : 3 m, and Q: 110 kN.

ReferencesCnsecn'rNoe, A. (1936). "Determination of the Preconsolidation Load and Its Practical

Significance." Proceedings, 1st International Conference on Soil Mechanics and Foun-dation Engineering, Cambridge, Mass., Vol. 3, 60-64.

casac;naNoE, A. and Faouv, R. E. (1940). "Notes on Soil resting for Engineering pur-poses," Harvard University Graduate School of Engineering publication No. 8.

Houcs, B. K. (19-57). Basic Sctils Engineering, Ronald Press, New york.MavNE, P. w., and Pour-os, H. G. (1999). "Approximate Displacement Influence Factors for

Elastic Shallow Foundations," Journal of Geotechnical and Geoenvironmental Engi-neering, ASCE, Vol. 125, No. 6, 453-460.

Sand%at' l8 kf{/lu3

Sand

3 1 0 Chapter 10 Compressibility of Soil

Na.canar, T., and MuRrv, B. R. S. (1985). "Predict ion of the Preconsolidation Pressure andRecompression Index of Soils," Geotechnical Testing Journal, Vol. 8, No. 4, 199-202.

Ntsstoe, Y. (1956). "A Brief Note on Compression Index of Soils," Journal of the Soil Me-chanics and Foundations Division, ASCE, Vol. 82, No. SM3, 1027-1-1027-I4.

RENooN-HEnneno, O. (1983). "Universal Compression Index Equation," Drscusslon, Jour-nal of Geotechnical Engineering, ASCE, Vol. 109, No. 10, 1349.

ReNpoN-HERRe,no, O. (1980). "Universal Compression Index Equation," Journal of the Geo-technical Enginee ring D ivision, ASCE, Vol. 1 06, No. GT1 1, 1779 -1200.

RoerNSON, R. G., and Ar.t-err, M. M. (1996). "Determination of Coeff lcient of Consolidationfrom Early Stage of log r Plot," Geotechnical Testing Journal, ASTM. Vol. 19, No. 3,316-320.

ScnrEtc rspR,F. (1926) . "ZurTheor iedesBaugrundes, " Buu ing ,en ieur ,Yo l .T .93 l -935,949-952.

ScHvEnrnrnNN, J. H. (1953). "Undisturbed Consolidation Behavior of Clay," Transactions,ASCE, Vo l . 120, 1201.

Sre naproN, A. W ( 1944). "Notes on the Comprcssibility of Clays." Quarterly Jotrrnal of theGeological Society o.f' London, Vol. 100, I 19-13-5.

SRIosa.RaN, A., and PnRrnsu, K. (1985). "Improved Rcctangular Hyperbola Method forthe Determination of Cocfficient of Consolidalion," Geotechnical Testing Journal,ASTM. Vo l .8 , No. 1 ,37-40.

' frryr-on, D. W. (1942). "Research on Consolid:rt ion of Clays," Serial No.82, Department of

Civi l and Sanitary Engincering, Massachusetts lnst i tute of Technology, Cambridge,Mass.

TnyLon, D. W. ( l94u). Fundamentals of Soil Mechanlcl, Wiley, New York.TI-:Rz-I.<;tl, K. (192-5). Erdbattmechanik auf Bodenphysikalischer Grundlager, Deuticke,

Vienn:r.T'enz-ncur, K., and Pr,.c rc, R. B. (1967). Soi l Mechanics in Enginaering Pructice,2nd cd., Wi-

ley, New York.

compressibility of soil

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