Das Elastic Settlement Estimation P

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ABSTRACT: Developments in major procedures available in the literature relating to elasticsettlement of shallow foundations supported by granular soil are presented and compared. Thediscrepancies between the observed and the predicted settlement are primarily due to the inabilityto estimate the modulus of elasticity of soil using the results of the standard penetration tests and/orcone penetration tests. Based on the procedures available at this time, recommendations have beenmade for the best estimation of settlement of foundations

KEY WORDS: Cone penetration test, elastic settlement, granular soil, shallow foundation,standard penetration test

1 INTRODUCTION

The estimation of settlement of shallow foundations is an important topic in the design andconstruction of buildings and other related structures. In general, settlement of a foundationconsists of two major componentselastic settlement ( S e) and consolidation settlement ( S c). Inturn, the consolidation settlement of a submerged clay layer has two parts; that is, the contributionof primary consolidation settlement ( S p) and that due to secondary consolidation ( S s). For afoundation supported by granular soil within the zone of influence of stress distribution, the elastic

settlement is the only component that needs consideration. This paper is a general overview ofvarious aspects of the elastic settlement of shallow foundations supported by granular soil deposits.During the last fifty years or so, a number of procedures have been developed to predict elasticsettlement; however, there is a lack of a reliable standardized procedure.

2 ELASTIC SETTLEMENT CALCULATION PROCEDURESGENERAL

Various methods to calculate the elastic settlement available at the present time can be divided intotwo general categories. They are as follows:

1. Methods Based on Observed Settlement of Structures and Full Scale Prototypes. These

methods are empirical or semi-empirical in nature and are correlated with the results of thestandard in situ tests such as the standard penetration test (SPT), the cone penetration test

Developments in elastic settlement estimation procedures forshallow foundations on granular soil

Braja M. DasDean Emeritus, California State University, SacramentoHenderson, Nevada, U.S.A.


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(CPT), the flat dilatometer test, and the pressurementer test (PMT). The procedures usuallyreferred to in practice now are those developed by Terzaghi and Peck (1948, 1967), Meyerhof(1956, 1965), DeBeer and Martens (1957), Hough (1969), Peck and Bazaraa (1969),Schmertmann (1970), Schmertmann et al. (1978), Burland and Burbidge (1985), Briaud (2007),and Lee et al. (2008).

2. Methods Based on Theoretical Relationships Derived from the Theory of Elasticity. Therelationships for settlement calculation available in this category contain the term modulus ofelasticity ( E s).

The general outline for some of these methods is given in the following sections.

METHODS BASED ON OBSERVED SETTLEMENT

3 TERZAGHI AND PECKS METHOD

Terzaghi and Peck (1948) proposed the following empirical relationship between the settlement(S e) of a prototype foundation measuring B B in plan and the settlement of a test plate [ S e(1)]

measuring B1 B1 loaded to the same intensity

+

=2

1)1( 1

4

B BS

S

e

e (1)

Although a full-sized footing can be used for a load test, the normal practice is to employ a plate ofthe order of 0.3 m to 1 m. Bjerrum and Eggestad (1963) provided the results of 14 sets of loadsettlement tests. This is shown in Figure 1 along with the plot of Eq. (1). For these tests, B1 was0.35 m for circular plates and 0.32 m for square plates. It is obvious from Figure 1 that, althoughthe general trend is correct, Eq. (1) represents approximately the lower limit of the field test results.

Bazaraa (1967) also provided several field test results. Figure 2 shows the plot of S e/S e(1) versus B/ B1 for all tests results provide by Bjerrum and Eggestad (1963) and Bazaraa (1967) as compiled by DAppolonia et al. (1970). The overall results with the expanded data base are similar to thosein Figure 1 as they relate to Eq. (1).

Terzaghi and Peck (1948, 1967) proposed a correlation for the allowable bearing capacity,standard penetration number ( N 60), and the width of the foundation ( B) corresponding to a 25 -mmsettlement based on the observation given by Eq. (1). This correlation is shown in Figure 3. Thecurves shown in Figure 3 can be approximated by the relation

2

60 303

(mm)

+=

. B B

N q

S e (2)

where q = bearing pressure in kN/m 2 B = width of foundation (m)

If corrections for ground water table location and depth of embedment are included, then Eq. (2)takes the form

2

60 303

+=

. B B

N q

C C S DW e (3)

where C W = ground water table correctionC D = correction for depth of embedment = 1 ( D f /4 B)

D f = depth of embedment


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Figure 3. Terzaghi and Pecks (1948, 1967) recommendation for allowable bearing capacity for 25-mmsettlement variation with B and N 60.

Jayapalan and Boehm (1986) and Papadopoulos (1992) summarized the case histories of 79foundations. Sivakugan et al (1998) used those case histories to compare with the settlement

predicted by the Terzaghi and Peck method. This comparison is shown in Figure 4. It can be seen

from this figure that, in general, the predicted settlements were significantly higher than thoseobserved. The average value of S e(predicted) /S e(observed) 2.18.

Similar observations were also made by Bazaraa (1967). With B1 = 0.3 m, Eq. (1) can berewritten as

2

)1( 304

+=

. B B

S S

e

e

or

=

+ )1(

2

41

30 ee

S S

. B B

(4)

Combining Eqs. (2) and (4)

=

)1(60 413

e

ee S

S N

qS

or

750

60

)1( .

N

S

q

e

= (5)


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Figure 4. Sivakugan et al.s (1998) comparison of predicted with observed settlement for 79 foundations predicted settlement based on Terzaghi and Peck method (1948, 1967).

Figure 5. Bazaraas plate load test resultsplot of q/S e(1) versus N 60.

Bazaraa (1967) plotted a large number of plate load test results ( B1 = 0.3 m) in the form of q/S e(1) versus N 60 as shown in Figure 5. It can be seen that the relationship given by Eq. (5) is veryconservative. In fact, q/S e(1) versus N 60/0.5 will more closely represent the lower limiting condition.


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4 MEYERHOFS METHOD

In 1956, Meyerhof proposed relationships for the elastic settlement of foundations on granular soilsimilar to Eq. (2). In 1965 he compared the predicted (by the relationships proposed in 1956) andobserved settlements of eight structures and suggested that the allowable pressure ( q) for a desiredmagnitude of S e can be increased by 50% compared to what he recommended in 1956. The revisedrelationships including the correction factors for water table location ( C W ) and depth of embedment(C D) can be expressed as

m)1.22(for251

60

= B N

q.C C S DW e (6)

and

m)1.22(for30

22

60

>

+= B

. B B

N q

C C S DW e (7)

0.1=W C (8)

and

B

D.C f D 401 = (9)

If these equations are used to predict the settlement of the 79 foundations shown in Figure 4, thenwe will obtain S e(predicted) /S e(observed) 1.46. Hence, the predicted settlements will overestimate theobserved values by about 50% on the average.

Table 1 shows the comparison of the maximum observed settlements of mat foundationsconsidered by Meyerhof (1965) and the settlements predicted by Eq. (7). The ratios of the predictedto observed settlements are generally in the range of 0.8 to 2. This is also what Meyerhof concludedin his 1965 paper.

Table 1. Comparison of observed maximum settlements provided by Meyerhof (1965) for eight matfoundations with those predicted by Eq. (7)

Structure B(m)

Average N 60

q(kN/m 2)

MaximumS e(observed) (mm)

S e(predicted) by Eq. (7)(mm) )observed(

predicted)(

e

e

S

S

T. Edison, Sao PauloBanco do Brasil, Sao PauloIparanga, Sao PauloC.B.I. Esplanada, Sao PauloRiscala, Sao PauloThyssen, DusseldorfMinistry, DusseldorfChimney, Cologne

18.322.99.1514.63.9622.615.920.4

151892220252010

229.8239.4220.2383.0229.8239.4220.4172.4

15.2427.9435.5627.9412.7024.1321.5910.16

29.6625.7445.8833.4319.8618.6521.2333.49

1.950.991.291.201.560.770.983.30

Average 1.5

5 DE BEER AND MARTENS METHOD

DeBeer and Martens (1957) and DeBeer (1965) proposed the following relationship to estimate theelastic settlement of a foundation


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H

C .

S o

oe

+

= 10log32

(10)

where C = a constant of proportionalityo = effective overburden pressure at the depth considered

= increase in pressure at that depth due to foundation loading H = thickness of the layer considered

The value of C can be approximated as

o

c

q

.C

51 (11)

where qc = cone penetration resistance.Equation (10) is essentially in the form of the relationship for estimating the consolidation

settlement of normally consolidated clay. We can rewrite Eq. (10) as

+

+=

o

o

o

ce

H e

C S 10log1 (12)

where

=

+ co

o

c

q

eC

5.11

(13)

C c = compression indexeo = in situ void ratio

For the field cases considered by DeBeer and Martens (1957), the average ratio of predicted toobserved settlement was about 1.9. DeBeer (1965) further observed that the above stated methodonly applies to normally consolidated sands. For overconsolidated sand, a reduction factor needs to

be applied which can be obtained from cyclic loading tests carried out in an oedometer. Hough(1969) expressed C c in Eq. (12) as

)( beaC oc = (14)

Approximate values of a and b are given in Table 2.

Table 2. Values of a and b from Eq. (14) (based on Hough, 1969)

Type of soilValue of constant

a b

Uniform cohesionless material (uniformity coefficient C u 2)Clean gravelCoarse sandMedium sandFine sandInorganic silt

0.050.060.070.080.10

0.500.500.500.500.50

Well-graded cohesionless soilSilty sand and gravelClean, coarse to fine sandCoarse to fine silty sandSandy silt (inorganic)

0.090.120.150.18

0.200.350.250.25

* The value of the constant b should be taken as emin whenever the latter is known orcan conveniently be determined. Otherwise, use tablulated values as a roughapproximation.


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6 THE METHOD OF PECK AND BAZARAA

Peck and Bazaraa (1969) recognized that the original Terzaghi and Peck method in Section 3 wasoverly conservative and revised Eq. (3) to the following form

2

601 30)(2

+=

. B B

N q

C C S DW e (15)

where S e is in mm, q is in kN/m2, and B is in m

( N 1)60 = corrected standard penetration number

foundationtheof bottom below the0.5atfoundationtheof bottom below the0.5at

B B

C o

oW

= (16)

o = total overburden pressure

o = effective overburden pressure

50

4001.

f D q

D..C

= (17)

= unit weight of soilThe relationships for ( N 1)60 are as follow:

)kN/m75(for0401

4)( 260601 +

= oo

.

N N (18)

and

)kN/m75(for010253

4)( 260601 >+

= oo

..

N N (19)

where o is the effective overburden pressure (kN/m2)

DAppolonia et al. (1970) compared the observed settlement of several shallow foundationsfrom several structures in Indiana (USA) with those estimated using the Peck and Bazaraa method,and this is shown in Figure 6. It can be seen from this figure that the calculated settlement fromtheory greatly overestimates the observed settlement. It appears that this solution will providenearly the level of settlement that was obtained from Meyerhofs revised relationships (Section 5).

Figure 6 Plot of measured versus predicted settlement based on Peck and Bazaraas method (adapted from

DAppolonia et al., 1970).


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7 STRAIN INFLUENCE FACTOR METHOD

Based on the theory of elasticity, the equation for vertical strain z at a depth below the center of aflexible circular load of diameter B, can be given as

[ ] B A E

q s

s

s z +

+= )21()1(

or

[ ] B A q E

I s s s z

z ++== )21()1( (20)

where A' and B' = f ( z / B)q = load per unit area

E s = modulus of elasticity s = Poissons ratio I z = strain influence factor

Figure 7 shows the variation of I z with depth based on Eq. (20) for s = 0.4 and 0.5. Theexperimental results of Eggestad (1963) for variation of I z are also given in this figure. Considering

both the theoretical and experimental results cited in Figure 7, Schmertmann (1970) proposed asimplified distribution of I z with depth that is generally referred to as 2 B 0.6 I z distribution and it isalso shown in Figure 7. According to the simplified method,

z E I

qC C S B

o s

z e =

2

21 (21)

where q = net effective pressure applied at the level of the foundation

C 1 = correction factor for embedment of foundation =qq

. o501 (22)

qo = effective overburden pressure at the level of the foundation

Figure 7 Theoretical and experimental distribution of vertical strain influence factor below the center of acircular loaded area (based on Schmertmann, 1970).


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C 2 = correction factor to account for creep in soil =

+10

log201.t

. (23)

t = time, in yearsFor use in Eq. (21) and the strain influence factor shown in Figure 7, it was recommended that

cS q E 2= (24)

where qc = cone penetration resistanceSivakugan et al. (1998) used the case histories of the 79 foundations given in Figure 4 and

compared those with the settlements obtained using the strain influence factor shown in Figure 7and Eq. (21), and this is shown in Figure 8. From this figure, it can be seen that se(predicted) /S e(observed) 3.39.

Schmertmann et al. (1978) modified the strain influence factor variation (2 B 0.6 I z ) shown inFigure 7. The revised distribution is shown in Figure 9 for use in Eqs. (21)(23). According to this,

Figure 8 Sivakugan et al.s comparison (1998) of predicted and observed settlements from 79 foundations predicted settlement based on 2 B0.6 I z procedure.

Figure 9 Revised strain influence factor diagram suggested by Schmertmann et al. (1978).


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For square or circular foundation: I z = 0.1 at z = 0 I z (peak) at z = z p = 0.5 B I z = 0 at z = z o = 2 B

For foundation with L/ B 10: I z = 0.2 at z = 0 I z (peak) at z = z p = B I z = 0 at z = z o = 4 B

where L = length of foundation. For L/ B between 1 and 10, interpolation can be done. Also

5.0

) peak ( 1.05.0

+=o

z

q I

(25)

The value of o in Eq. (25) is the effective overburden pressure at a depth where I z (peak) occurs.Salgado (2008) gave the following interpolation for I z at z = 0, z p, and z o (for L/ B = 1 to L/ B 10.

2.00111.01.0)0at(

+== B L I z z (26)

110555.05.0

+= B L

B

z p (27)

41222.02

+= B L

B z o (28)

Noting that stiffness is about 40% larger for plane strain compared to axisymmetric loading,Schmertmann et al. (1978) recommended that.

s)foundationcircularandsquare(for5.2 c s q E = (29)

and

)foundationstrip(for5.3 c s q E = (30)With the modified strain-influence factor diagram,

z E I

C C S o z z

z s

z e =

=

=021 (31)

The modified strain influence factor and Eqs. (29) and (30) will definitely reduce the average ratioof predicted to observed settlement. However, it may still overestimate the actual elastic settlementin the field.

8 RECENT MODIFICATIONS IN STRAIN-INFLUENCE FACTOR DIAGRAMS

More recently some modifications have been proposed to the strain-influence factor diagramsuggested by Schmertmann et al. (1978). Two of these suggestions are discussed below.


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8.1 Modification Suggested by Terzaghi, Peck and Mesri (1996)

The modification suggested by Terzaghi et al. (1996) is shown in Figure 10. For this case, forsurface foundation condition (that is, D f / B = 0)

I z = 0.2 at z = 0 I z = I z (peak) = 0.6 at z = z p = 0.5B I z = 0 at z = z o

Figure 10 Strain influence diagram suggested by Terzaghi et al. (1996).

4log12

+= B L

z o (32)

For D f / B > 0, I z should be modified to z I . Figure 11 shows the variation of z z I I / with D f / B.The end of construction settlement can be estimated as

z E I

qS o z z

z s

z e

==

=0 (33)

The settlement due to creep can be calculated as

=

day1log

1.0 dayscreep

t z

qS o

c

(34)

where cq = weighted mean value of measured qc values of sublayers between z = 0 and z = z o (MN/m 2)

It has also been suggested that


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Figure 11 Variation of z z I I / with D f / B (after Terzaghi et al. 1996).

4.1log4.01)1/(

)/(

+== B

L E

E

B L s

B L s (35)

where c B L s q E 5.3)1/( == (36)

Figure 12 shows the plot of E s versus qc from 81 foundations and 92 plate load tests on whichEq. (36) has been established. The magnitude of E s recommended by Eq. (36) is about 40% higherthan that obtained from Eq. (29). Figure 13 shows a comparison of the end-of-construction

predicted [using Eqs. (33), (35) and (36)] and measured settlement of foundations on sand andgravelly soils (Terzaghi et al., 1996).

Figure 12 Correlation between E s and qc for square and circularly loaded areas [adapted from Terzaghi et al.(1996)].


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Figure 13 Comparison of end of construction predicted and measured S e of foundations on sand and gravellysoils based on Eqs. (33), (35) and (36) [adapted from Terzaghi et al. (1996)].

8.2 Modification Suggested by Lee et al. (2008)

Based on finite element analysis, Lee et al. (2008) suggested the following modifications to thestrain influence factor diagram suggested by Schmertmann et al. (1978). This assumes that I z (peak) and I z at z = 0 is the same as given by Eqs. (25) and (26). However Eqs. (27) and (28) are modifiedas

6at1of maximumawith111.05.0 =

+=

B L

B L

B z p (37)

6atmaximumawith315

cos95.0 =+

= B L

B L

B z o (38)

With these modifications, the elastic settlement can be calculated using Eq. (21).

9 METHOD OF BURLAND AND BURBIDGE (1985)

Burland and Burbidge (1985) proposed a method for calculating the elastic settlement of sandy soilusing the field standard penetration number N 60. The method can be summarized as follows:

9.1 Determination of Variation of Standard Penetration Number with Depth

Obtain the field penetration numbers ( N 60) with depth at the location of the foundation. Thefollowing adjustments of N 60 may be necessary, depending on the field conditions:For gravel or sandy gravel,

6060(a) 25.1 N N (39)

For fine sand or silty sand below the ground water table and N 60 > 15,


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C. For overconsolidated soil ( q > c )

[ ]

+

=a

c

.

R.

(a) R

e

p

.q

B

B

B L.

B L

.

N or N

..

B

S 670

250

251570

14070

2

416060

(45)

Sivakugan and Johnson (2004) used a probabilistic approach to compare the predictedsettlements obtained by the methods of Terzaghi and Peck (1948, 1967), Schmertmann et al.(1970), and Burland and Burbidge (1985). Table 3 gives a summary of their studythat is,

predicted settlement versus the probability of exceeding 25 mm settlement in the field. This showsthat the method of Burland and Burbidge (1985), although conservative, is a substantially improvedtechnique to estimate elastic settlement.

Table 3. Probability of exceeding 25 mm settlement in the field

Predictedsettlement

(mm)

Probability of exceeding 25 mm settlement in fieldTerzaghi and Peck

(1948, 1967)Schmertmann et al.

(1970)Burland and

Burbidge (1985)15

10152025303540

0.000.000.000.090.200.260.310.350.387

0.000.000.020.130.200.270.320.370.42

0.000.030.150.250.340.420.490.550.61

Compiled from Sivakugan and Johnson (2004)

10 LOAD-SETTLEMENT CURVE APPROACH BASED ON PRESSUREMETER TESTS(PMT)

Briaud (2007) presented a method based on field Pressuremeter tests to develop a load-settlementcurve for a given foundation from which the elastic settlement at a given load intensity can beestimated. This takes into account the foundation load eccentricity, load inclination, and thelocation of the foundation on a slope (Figure 14). Following is a step-by-step procedure of the

procedure suggested by Briaud (2007).1. Conduct several Pressuremeter tests at the site at various depths.2. Plot the PMT curves as pressure p p on the cavity wall versus relative increase in cavity radius

R/ Ro. Extend the straight line part of the PMT curve to zero pressure and shift the vertical axisto the value of R/Ro where that strain line portion intersects the horizontal axis (Figure 15).

3. Plot the strain influence factor diagram proposed by Schmertmann et al. (1978) for thefoundation. Based on the p p versus R/Ro diagrams (Step 2) and the location of the depth of thetests, develop a mean plot of p p versus R/Ro as shown in Figure 16.The mean p p for a given R/Ro can be given as

. . . )3(3

)2(2

)1(1

)( +++= p p pmean p p A A

p A A

p A A

p (46)

where A1, A2, A3, . . . are the areas tributary to each test under the influence diagram A = total area of the strain-influence factor diagram


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Figure 14 Pressuremeter test to obtain load-settlement curve.

Figure 15 Adjustment of field Pressuremeter test plot of p p versus R/ Ro.

4. Convert the plot of p p(mean) versus R/ Ro plot to q versus S e/ B plot using the followingequations.

)(,/ ))(( mean pd e B L p f f f f q = (47)

o

e

R R

BS = 24.0 (48)

where = Gamma function linking q and p p(mean) (see Figure 17)

+== L B

f L B 2.08.0factor shape/ (49)

(center) 33.01factor tyeccentriciload

== Be

f e (50)


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Figure 16 Development of the mean p p versus R/ Ro plot.

Figure 17 Variation of function.

(edge) 15.0

= Be

f e (51)

(center) 90

(degrees)1factor ninclinatio

== f (52)

(edge) 360(degrees)

1

0.5= f (53)

slope)1:(3 18.0factor slope0.1

,

+== B

d f d (54)

slope)1:(2 17.00.15

,

+= B

d f d (55)

5. Based on the load-settlement diagram developed in Step 4, obtain the actual S e(maximum) whichcorresponds to the actual intensity of load q to which the foundation will be subjected.

6. To account for creep over the life-span of the structure,


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)(1

101 A A F += (59)

21

2 tan

2

A

n F = (60)

( )( )11

11ln

22

222

0+++

+++=nmm

nmmm A (61)

1

11ln

22

22

1+++

+++=

nmm

nmm A (62)

1222

+++=

nmn

m A (63)

== B

L B

D f I s

f f and,,1948)(Fox,factordepth (64)

' = a factor that depends on the location below the foundation where settlement is beingcalculated

To calculate settlement at the center of the foundation, we use

4= (65)

B L

m = (66)

and

=

2 B H

n (67)

To calculate settlement at a corner of the foundation,

1= (68)

B L

m =

and

B H

n =

The variations of F 1 and F 2 with m and n are given Tables 4 and 5. Based on the works of Fox(1948), the variations of depth factor I f for s = 0.3 and 0.4 and L/ B have been determined byBowles (1987) and are given in Table 6. Note that I f is not a function of H / B.


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Table 4. Variation of F 1 with m and n

nm

1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.00.250.500.751.001.251.501.752.002.252.502.753.003.25

3.503.754.004.254.504.755.005.255.505.756.006.256.506.757.007.257.507.758.008.258.508.759.009.259.50

9.7510.0020.0050.00

100.00

0.0140.0490.0950.1420.1860.2240.2570.2850.3090.3300.3480.3630.376

0.3880.3990.4080.4170.4240.4310.4370.4430.4480.4530.4570.4610.4650.4680.4710.4740.4770.4800.4820.4850.4870.4890.4910.4930.495

0.4960.4980.5290.5480.555

0.0130.0460.0900.1380.1830.2240.2590.2900.3170.3410.3610.3790.394

0.4080.4200.4310.4400.4500.4580.4650.4720.4780.4830.4890.4930.4980.5020.5060.5090.5130.5160.5190.5220.5240.5270.5290.5310.533

0.5360.5370.5750.5980.605

0.0120.0440.0870.1340.1790.2220.2590.2920.3210.3470.3690.3890.406

0.4220.4360.4480.4580.4690.4780.4870.4940.5010.5080.5140.5190.5240.5290.5330.5380.5410.5450.5490.5520.5550.5580.5600.5630.565

0.5680.5700.6140.6400.649

0.0110.0420.0840.1300.1760.2190.2580.2920.3230.3500.3740.3960.415

0.4310.4470.4600.4720.4840.4940.5030.5120.5200.5270.5340.5400.5460.5510.5560.5610.5650.5690.5730.5770.5800.5830.5870.5890.592

0.5950.5970.6470.6780.688

0.0110.0410.0820.1270.1730.2160.2550.2910.3230.3510.3770.4000.420

0.4380.4540.4690.4810.4950.5060.5160.5260.5340.5420.5500.5570.5630.5690.5750.5800.5850.5890.5940.5980.6010.6050.6090.6120.615

0.6180.6210.6770.7110.722

0.0110.0400.0800.1250.1700.2130.2530.2890.3220.3510.3780.4020.423

0.4420.4600.4760.4840.5030.5150.5260.5370.5460.5550.5630.5700.5770.5840.5900.5960.6010.6060.6110.6150.6190.6230.6270.6310.634

0.6380.6410.7020.7400.753

0.0100.0380.0770.1210.1650.2070.2470.2840.3170.3480.3770.4020.426

0.4470.4670.4840.4950.5160.5300.5430.5550.5660.5760.5850.5940.6030.6100.6180.6250.6310.6370.6430.6480.6530.6580.6630.6670.671

0.6750.6790.7560.8030.819

0.0100.0380.0760.1180.1610.2030.2420.2790.3130.3440.3730.4000.424

0.4470.4580.4870.5140.5210.5360.5510.5640.5760.5880.5980.6090.6180.6270.6350.6430.6500.6580.6640.6700.6760.6820.6870.6930.697

0.7020.7070.7970.8530.872

0.0100.0370.0740.1160.1580.1990.2380.2750.3080.3400.3690.3960.421

0.4440.4660.4860.5150.52205390.5540.5680.5810.5940.6060.6170.6270.6370.6460.6550.6630.6710.6780.6850.6920.6980.7050.7100.716

0.7210.7260.8300.8950.918

0.0100.0370.0740.1150.1570.1970.2350.2710.3050.3360.3650.3920.418

0.4410.4640.4840.5150.5220.5390.5540.5690.5840.5970.6090.6210.6320.6430.6530.6620.6710.6800.6880.6950.7030.7100.7160.7230.719

0.7350.7400.8580.9310.956


22/33

Table 4. (Continued)

nm

4.5 5.0 6.0 7.8 8.0 9.0 10.0 25.0 50.0 100.00.250.500.751.001.251.501.752.002.252.502.753.003.25

3.503.754.004.254.504.755.005.255.505.756.006.256.506.757.007.257.507.758.008.258.508.759.009.259.50

9.7510.0020.0050.00

100.00

0.0100.0360.0730.1140.1550.1950.2330.2690.3020.3330.3620.3890.415

0.4380.4610.4820.5160.5200.5370.5540.5690.5840.5970.6110.6230.6350.6460.6560.6660.6760.6850.6940.7020.7100.7170.7250.7310.738

0.7440.7500.8780.9620.990

0.0100.0360.0730.1130.1540.1940.2320.2670.3000.3310.3590.3860.412

0.4350.4580.4790.4960.5170.5350.5520.5680.5830.5970.6100.6230.6350.6470.6580.6690.6790.6880.6970.7060.7140.7220.7300.7370.744

0.7510.7580.8960.9891.020

0.0100.0360.0720.1120.1530.1920.2290.2640.2960.3270.3550.3820.407

0.4300.4530.4740.4840.5130.5300.5480.5640.5790.5940.6080.6210.6340.6460.6580.6690.6800.6900.7000.7100.7190.7270.7360.7440.752

0.7590.7660.9251.0341.072

0.0100.0360.0720.1120.1520.1910.2280.2620.2940.3240.3520.3780.403

0.4270.4490.4700.4730.5080.5260.5430.5600.5750.5900.6040.6180.6310.6440.6560.6680.6790.6890.7000.7100.7190.7280.7370.7460.754

0.7620.7700.9451.0701.114

0.0100.0360.0720.1120.1520.1900.2270.2610.2930.3220.3500.3760.401

0.4240.4460.4660.4710.5050.5230.5400.5560.5710.5860.6010.6150.6280.6410.6530.6650.6760.6870.6980.7080.7180.7270.7360.7450.754

0.7620.7700.9591.1001.150

0.0100.0360.0720.1110.1510.1900.2260.2600.2910.3210.3480.3740.399

0.4210.4430.4640.4710.5020.5190.5360.5530.5680.5830.5980.6110.6250.6370.6500.6620.6730.6840.6950.7050.7150.7250.7350.7440.753

0.7610.7700.9691.1251.182

0.0100.0360.0710.1110.1510.1890.2250.2590.2910.3200.3470.3730.397

0.4200.4410.4620.4700.4990.5170.5340.5500.5850.5800.5950.6080.6220.6340.6470.6590.6700.6810.6920.7030.7130.7230.7320.7420.751

0.7590.7680.9771.1461.209

0.0100.0360.0710.1100.1500.1880.2230.2570.2870.3160.3430.3680.391

0.4130.4330.4530.4680.4890.5060.5220.5370.5510.5650.5790.5920.6050.6170.6280.6400.6510.6610.6720.6820.6920.7010.7100.7190.728

0.7370.7450.9821.2651.408

0.0100.0360.0710.1100.1500.1880.2230.2560.2870.3150.3420.3670.390

0.4120.4320.4510.4620.4870.5040.5190.5340.5490.5830.5760.5890.6010.6130.6240.6350.6460.6560.6660.6760.6860.6950.7040.7130.721

0.7290.7380.9651.2791.489

0.0100.0360.0710.1100.1500.1880.2230.2560.2870.3150.3420.3670.390

0.4110.4320.4510.4600.4870.5030.5190.5340.5480.5620.5750.5880.6000.6120.6230.6340.6450.6550.6650.6750.6840.6930.7020.7110.719

0.7270.7350.9571.2611.499


23/33

Table 5. Variation of F 2 with m and n

nm

1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.00.250.500.751.001.251.501.752.002.252.502.753.003.25

3.503.754.004.254.504.755.005.255.505.756.006.256.506.757.007.257.507.758.008.258.508.759.009.259.50

9.7510.0020.0050.00

100.00

0.0490.0740.0830.0830.0800.0750.0690.0640.0590.0550.0510.0480.045

0.0420.0400.0370.0360.0340.0320.0310.0290.0280.0270.0260.0250.0240.0230.0220.0220.0210.0200.0200.0190.0180.0180.0170.0170.017

0.0160.0160.0080.0030.002

0.0500.0770.0890.0910.0890.0840.0790.0740.0690.0640.0600.0560.053

0.0500.0470.0440.0420.0400.0380.0360.0350.0330.0320.0310.0300.0290.0280.0270.0260.0250.0240.0230.0230.0220.0210.0210.0200.020

0.0190.0190.0100.0040.002

0.0510.0800.0930.0980.0960.0930.0880.0830.0770.0730.0680.0640.060

0.0570.0540.0510.0490.0460.0440.0420.0400.0390.0370.0360.0340.0330.0320.0310.0300.0290.0280.0270.0260.0260.0250.0240.0240.023

0.0230.0220.0110.0040.002

0.0510.0810.0970.1020.1020.0990.0950.0900.0850.0800.0760.0710.067

0.0680.0600.0570.0550.0520.0500.0480.0460.0440.0420.0400.0390.0380.0360.0350.0340.0330.0320.0310.0300.0290.0280.0280.0270.026

0.0260.0250.0130.0050.003

0.0510.0830.0990.1060.1070.1050.1010.0970.0920.0870.0820.0780.074

0.0700.0670.0630.0610.0580.0550.0530.0510.0490.0470.0450.0440.0420.0410.0390.0380.0370.0360.0350.0340.0330.0320.0310.0300.029

0.0290.0280.0140.0060.003

0.0520.0840.1010.1090.1110.1100.1070.1020.0980.0930.0890.0840.080

0.0760.0730.0690.0660.0630.0610.0580.0560.0540.0520.0500.0480.0460.0450.0430.0420.0410.0390.0380.0370.0360.0350.0340.0330.033

0.0320.0310.0160.0060.003

0.0520.0860.1040.1140.1180.1180.1170.1140.1100.1060.1020.0970.093

0.0890.0860.0820.0790.0760.0730.0700.0670.0650.0630.0600.0580.0560.0550.0530.0510.0500.0480.0470.0460.0450.0430.0420.0410.040

0.0390.0380.0200.0080.004

0.0520.0860.1060.1170.1220.1240.1230.1210.1190.1150.1110.1080.104

0.1000.0960.0930.0900.0860.0830.0800.0780.0750.0730.0700.0680.0660.0640.0620.0600.0590.0570.0550.0540.0530.0510.0500.0490.048

0.0470.0460.0240.0100.005

0.0520.0870.1070.1190.1250.1280.1280.1270.1250.1220.1190.1160.112

0.1090.1050.1020.0990.0960.0930.0900.0870.0840.0820.0790.0770.0750.0730.0710.0690.0670.0650.0630.0620.0600.0590.0570.0560.055

0.0540.0520.0270.0110.006

0.0520.0870.1080.1200.1270.1300.1310.1310.1300.1270.1250.1220.119

0.1160.1130.1100.1070.1040.1010.0980.0950.0920.0900.0870.0850.0830.0800.0780.0760.0740.0720.0710.0690.0670.0660.0640.0630.061

0.0600.0590.0310.0130.006


24/33

Table 5. (continued)

nm

4.5 5.0 6.0 7.0 8.0 9.0 10.0 25.0 50.0 100.00.250.500.751.001.251.501.752.002.252.502.753.003.25

3.503.754.004.254.504.755.005.255.505.756.006.256.506.757.007.257.507.758.008.258.508.759.009.259.50

9.7510.0020.0050.00

100.00

0.0530.0870.1090.1210.1280.1320.1340.1340.1330.1320.1300.1270.125

0.1220.1190.1160.1130.1100.1070.1050.1020.0990.0970.0940.0920.0900.0870.0850.0830.0810.0790.0770.0760.0740.0720.0710.0690.068

0.0660.0650.0350.0140.007

0.0530.0870.1090.1220.1300.1340.1360.1360.1360.1350.1330.1310.129

0.1260.1240.1210.1190.1160.1130.1110.1080.1060.1030.1010.0980.0960.0940.0920.0900.0880.0860.0840.0820.0800.0780.0770.0750.074

0.0720.0710.0390.0160.008

0.0530.0880.1090.1230.1310.1360.1380.1390.1400.1390.1380.1370.135

0.1330.1310.1290.1270.1250.1230.1200.1180.1160.1130.1110.1090.1070.1050.1030.1010.0990.0970.0950.0930.0910.0890.8880.0860.085

0.0830.0820.0460.0190.010

0.0530.0880.1100.1230.1320.1370.1400.1410.1420.1420.1420.1410.140

0.1380.1370.1350.1330.1310.1300.1280.1260.1240.1220.1200.1180.1160.1140.1120.1100.1080.1060.1040.1020.1010.0990.0970.0960.094

0.0920.0910.0530.0220.011

0.0530.0880.1100.1240.1320.1380.1410.1430.1440.1440.1440.1440.143

0.1420.1410.1390.1380.1360.1350.1330.1310.1300.1280.1260.1240.1220.1210.1190.1170.1150.1140.1120.1100.1080.1070.1050.1040.102

0.1000.0990.0590.0250.013

0.0530.0880.1100.1240.1330.1380.1420.1440.1450.1460.1460.1450.145

0.1440.1430.1420.1410.1400.1390.1370.1360.1340.1330.1310.1290.1280.1260.1250.1230.1210.1200.1180.1170.1150.1140.1120.1100.109

0.1070.1060.0650.0280.014

0.0530.0880.1100.1240.1330.1390.1420.1450.1460.1470.1470.1470.147

0.1460.1450.1450.1440.1430.1420.1400.1390.1380.1360.1350.1340.1320.1310.1290.1280.1260.1250.1240.1220.1210.1190.1180.1160.115

0.1130.1120.0710.0310.016

0.0530.0880.1110.1250.1340.1400.1440.1470.1490.1510.1520.1520.153

0.1530.1540.1540.1540.1540.1540.1540.1540.1540.1540.1530.1530.1530.1530.1520.1520.1520.1510.1510.1500.1500.1500.1490.1490.148

0.1480.1470.1240.0710.039

0.0530.0880.1110.1250.1340.1400.1440.1470.1500.1510.1520.1530.154

0.1550.1550.1550.1560.1560.1560.1560.1560.1560.1570.1570.1570.1570.1570.1570.1570.1560.1560.1560.1560.1560.1560.1560.1560.156

0.1560.1560.1480.1130.071

0.0530.0880.1110.1250.1340.1400.1450.1480.1500.1510.1530.1540.154

0.1550.1550.1560.1560.1560.1570.1570.1570.1570.1570.1570.1580.1580.1580.1580.1580.1580.1580.1580.1580.1580.1580.1580.1580.158

0.1580.1580.1560.1420.113


25/33

Table 6. Variation of I f (Fox, 1948)*

D / B L/ B

1.0 1.2 1.4 1.6 1.8 2.0 5.0Poissons ratio s = 0.30

0.050.100.200.400.600.801.002.00

0.9790.9540.9020.8080.7380.6870.6500.562

0.9810.9580.9110.8230.7540.7030.6650.571

0.9820.9620.9170.8340.7670.7160.6780.580

0.9830.9640.9230.8430.7780.7280.6890.588

0.9840.9660.9270.8510.7880.7380.7000.596

0.9850.9680.9300.8570.7960.7470.7090.603

0.9900.9770.9510.8990.8520.8130.7800.675

Poissons ratio s = 0.400.050.100.200.400.600.801.002.00

0.9890.9730.9320.8480.7790.7270.6890.596

0.9900.9760.9400.8620.7950.7430.7040.606

0.9910.9780.9450.8720.8080.7570.7180.615

0.9920.9800.9490.8810.8190.7690.7300.624

0.9920.9810.9520.8870.8280.7790.7400.632

0.9930.9820.9550.8930.8360.7880.7490.640

0.9950.9880.9700.9270.8860.8490.8180.714

Adapted from Bowles (1987)

Due to the non-homogeneous nature of a soil deposit, the magnitude of E s may vary with depth.For that reason, Bowles (1987) recommended

z

z E E i s s

= )( (69)

where E s(i) = soil modulus within the depth z z = 5 B or H (if H < 5 B)

Bowles (1987) also recommended that

260 kN/m)15(500 += N E s (70)

The elastic settlement of a rigid foundation can be estimated as

center)(flexible,)rigid( 93.0 ee S S (71)

Bowles (1987) compared this theory with 12 case histories that provided reasonable good results.

12 ANALYSIS OF MAYNE AND POULOS BASED ON THEORY OF ELASTICITY

Mayne and Poulos (1999) presented an improved formula for calculating the elastic settlement offoundations. The formula takes into account the rigidity of the foundation, the depth of embedmentof the foundation, the increase in the modulus of elasticity of the soil with depth, and the location ofrigid layers at a limited depth. To use the equation of Mayne and Poulos, one needs to determinethe equivalent diameter Be of a rectangular foundation, or

BL

Be4

= (72)


26/33


27/33

Figure 20 Variation of I G with .

Figure 21 Variation of I R with K F .

Figure 22 Variation of I E with s and D f / Be.

13 BERARDI AND LANCELLOTTAS METHOD

Berardi and Lancellotta (1991) proposed a method to estimate the elastic settlement that takes intoaccount the variation of the modulus of elasticity of soil with the strain level. This method is also

described by Berardi et al. (1991). According to this procedure,


28/33


29/33

Figure 23 Variation of K E with D r and N 60 (adapted from Berardi and Lancellotta, 1991).

Figure 24 Plot of %)10/()/( / . BS E BS E ee K K = with S e/ B (adapted from Berardi and Lancellotta, 1991).

3. Determine the average corrected blow count from standard penetration tests 601 )( N and hencethe average relative density as

5.0

1

60

= N D r (83)

4. With a known value of D r , determine %)10/( . BS E e K = from Figure 23 and hence E s from Eq. (81)for S e/ B = 0.1%

5. With the known value of E s (Step 4), the magnitude of S e can be calculated from Eq. (78).6. If the calculated S e/ B is not the same as the assumed value, then use the calculated value of S e/ B

from Step 5 and Figure 24 to estimate a revised )/( BS E e K . This value can now be used in Eqs.

(81) and (78) to obtain a revised S e. The iterative procedures can be continued until theassumed and calculated values are the same.


30/33

Based on a probabilistic study conducted by Sivakugan and Johnson (2004), the probability ofexceeding 25 mm settlement in the field for various predicted settlement levels using the iteration

procedure of Berardi and Lancellotta (1991) is shown in Table 8. When compared with Table 3,this shows a promise of improved prediction in elastic settlement.

Table 8. Probability of exceeding 25 mm settlement in the field procedure of Berardi and Lancellotta (1991)

(based on Sivakugan and Johnson, 2004)

Predictedsettlement

(mm)

Probability of exceeding25 mm in the field

(%)15

10152025

303540

61932435260

667277

14 GENERAL COMMENTS AND CONCLUSIONS

A general review of the major developments over the last sixty years for estimating elasticsettlement of shallow foundations on granular soil is presented. Based on the above review, thefollowing general observations can be made.1. Meyerhofs relationship (1965) is fairly simple to use. It will probably yield predicted

settlements that are 50% higher on the average than those observed in the field. Peck andBazaraas method (1969) provides results that are almost similar to those obtained fromMeyerhofs method (1965).

2. Burland and Burbidges solution (1985) will provide more reasonable estimations of S e thanthose obtained from the solution of Meyerhof (1965). However it will be difficult to determinethe overconsolidation ratio and the preconsolidation pressure for granular soils from fieldexploration.

3. The modified strain influence factor diagrams presented by Schmertmann et al. (1978),Terzaghi et al. (1996), and Lee et al. (2008) will all provide reasonable estimations of theelastic settlement provided a more realistic value of E s is assumed in the calculation. Theauthors feel that the empirical relationships for E s provided by Eqs. (35) and (36) are morereasonable.

4. The relationships for E s provided by Eqs. (35) and (36) are based on the field cone penetrationresistance. These equations can be converted to expressions in terms of N 60 and D50 (meangrain size). Figure 25 shows some of the relationships available in the literature. Based on thedata of Burland and Burbidge et al. (1985)

305.050

60

8 D N

pq

a

c

=

(84)

Based on the data of Robertson and Campanella (1983) and Seed and DeAlba (1986)


31/33

Figure 25 Variation of ( qc/ p a)/ N 60 with D50. (a) Adapted from Terzaghi et al. (1996); (b) Adapted fromAnagnostopoulos, 2003).

228.050

60

6 D N

pq

a

c

=

(85)

Based on the data of Anagnostopoulos et al. (2003)

26.050

60

6429.7 D N

pq

a

c

=

(86)

where pa = atmospheric pressure (same unit as qc) D50 = mean grain size, in mm.

5. The procedure for developing the load-settlement plot based on pressuremeter tests is aversatile technique; however, the cost effectiveness should be taken into account.

6. Relationships for elastic settlement using the theory of elasticity will be equally as good as theother methods, provided a realistic value of E s is adopted. This can be accomplished using the


32/33

iteration method suggested by Berardi and Lancellotta (1991). In lieu of that, the E s relationshipgiven by Terzaghi et al. (1996) can be used.

In his landmark paper in 1927 entitled The Science of Foundations, Karl Terzaghi wroteFoundation problems, throughout, are of such character that a strictly theoretical mathematicaltreatment will always be impossible. The only way to handle them efficiently consists of findingout, first, what has happened on preceding jobs of a similar character; next, the kind of soil onwhich the operations were performed; and, finally, why the operations have lead to certain results.By systematically accumulating such knowledge, the empirical data being well defined by theresults of adequate soil investigations, foundation engineering could be developed into a semi-empirical science, . . . .

What is presented in this paper is a systematic accumulation of knowledge and data over the past sixty years. In summary, the parameters for comparing settlement prediction methods areaccuracy and reliability. Reliability is the probability that the actual settlement would be less thanthat computed by a specific method. In choosing a method for design, it all comes down to keepinga critical balance between reliability and accuracy which can be difficult at times knowing the non-homogeneous nature of soil in general. We cannot be over-conservative but, at the same time, not

be accurate. We need to keep in mind what Karl Terzaghi said in the 45th James Forrest Lecture at

the Institute of Civil Engineers in London: Foundation failures that occur are not longer an act ofGod.

REFERENCES

Anagostopoulos, A., Kourkis, G., Sabatakakis, N. & Tsiambaos, G. 2003. Empirical correlation of soil parameters based on cone penetration tests (CPT) for Greek soils. Geotechnical and GeologicalEngineering, 21(4): 377-387.

Bazaraa, A.R.S.S. 1967. Use of the standard penetration test for estimating settlements of shallowfoundations on sand. Ph.D. Thesis, University of Illinois, Champaign-Urbana, Illinois.

Berardi, R., Jamiolkowski, M. & Lancellotta, R. 1991. Settlement of shallow foundations in sands: selectionof stiffness on the basis of penetration resistance. Geotechnical Engineering Congress 1991, Geotechnical

Special Publication 27, ASCE, 185-200.Berardi, R. & Lancellotta, R. 1991. Stiffness of granular soil from field performance. Geotechnique, 41(1):149-157.

Bjerrum, L. & Eggestad, A. 1963. Interpretation of load test on sand. Proceedings, European Conference onSoil Mechanics and Foundation Engineering, Weisbaden, West Germany, 1: 199.

Bowles, J.E. 1987. Elastic foundation settlement on sand deposits. Journal of Geotechnical Engineering,ASCE, 113(8): 846-860.

Briaud, J.L. 2007. Spread footing on sand: load settlement curve approach. Journal of Geotechnical andGeoenvironmental Engineering, ASCE, 133(8): 905-920.

Burland, J.B. & Burbidge, M.C. 1985. Settlement of foundations on sand and gravel. Proceedings, Institutionof Civil Engineers, 78(1): 1325-1381.

DAppolonia, D.J., DAppolonia, E. & Brissette, R.F. 1970. Settlement of spread footings on sand: closure.Journal of the Soil Mechanics and Foundations Division, ASCE, 96(2): 754-762.

DeBeer, E.E. 1965. Bearing capacity and settlement of shallow foundations on sand. Proceedings,Symposium on Bearing Capacity Settlement of Foundations, Duke University, Durham, N.C., 15-33.

DeBeer, E. & Martens, A. 1957. Method of computation of an upper limit for the influence of heterogeneityof sand layers in the settlement of bridges. Proceedings, 4 th International Conference on Soil Mechanicsand Foundation Engineering, London, 1: 275-281.

Eggestad, A. 1963. Deformation measurements below a model footing on the surface of dry sand.Proceedings, European Conference on Soil Mechanics and Foundation Engineering, Weisbaden, 1: 233-239.

Fox, E.N. 1948. The mean elastic settlement of a uniformly loaded area at a depth below the ground surface.Proceedings, 2 nd International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, 1:129-132.

Hough, B.K. 1969. Basic Soils Engineering, Ronald Press, New York.Janbu, N. 1963. Soil compressibility as determined from oedometer and triaxial tests. Proceedings, European

Conference on Soil Mechanics and Foundation Engineering, Weisbaden, 1: 19-24.


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Jeyapalan, J.K. & Boehm, R. 1986. Procedures for predicting settlements in sands. In W. O. Martin (ed.),Settlements of Shallow Foundations on Cohesionless Soils: Design and Performance, ASCE, Seattle, 1-22.

Lee, J., Eun, J., Prezzi, M. & Salgado, R. 2008. Strain influence diagrams for settlement estimation of bothisolated and multiple footings in sand. Journal of Geotechnical and Geoenvironmental Engineering,

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