Das Settlement Granular

2nd International Conference on New Developments in Soil Mechanics and Geotechnical Engineering, 28-30 May 2009, Near East University, Nicosia, North Cyprus

9

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

ABSTRACT: Developments in major procedures available in the literature relating to elastic settlement of shallow foundations supported by granular soil are presented and compared. The discrepancies between the observed and the predicted settlement are primarily due to the inability to estimate the modulus of elasticity of soil using the results of the standard penetration tests and/or cone penetration tests. Based on the procedures available at this time, recommendations have been made for the best estimation of settlement of foundations.

1 INTRODUCTION

The estimation of settlement of shallow foundations is an important topic in the design and construction of buildings and other related structures. In general, settlement of a foundation consists of two major components—elastic settlement (Se) and consolidation settlement (Sc). In turn, the consolidation settlement of a submerged clay layer has two parts; that is, the contribution of primary consolidation settlement (Sp) and that due to secondary consolidation (Ss). For a foundation 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 of various 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 elastic settlement; however, there is a lack of a reliable standardized procedure.

2 ELASTIC SETTLEMENT CALCULATION PROCEDURES—GENERAL

Various methods to calculate the elastic settlement available at the present time can be divided into two 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 the standard in situ tests such as the standard penetration test (SPT), the cone penetration test (CPT), the flat dilatometer test, and the Pressuremeter test (PMT). The procedures usually referred to in practice now are those developed by Terzaghi and Peck (1948, 1967), Meyerhof (1956, 1965), DeBeer

Developments in elastic settlement estimation procedures for shallow foundations on granular soil

Braja M. Das Dean Emeritus, California State University, Sacramento, U.S.A., [email protected] Cavit Atalar Department of Civil Engineering, Near East University, Nicosia, North Cyprus, [email protected] Eun Chul Shin Dept. Civil & Environmental Engineering, University of Incheon, Incheon, Korea, [email protected]


10

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. The relationships for settlement calculation available in this category contain the term modulus of elasticity (Es).

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

METHODS BASED ON OBSERVED SETTLEMENT

3 TERZAGHI AND PECK’S METHOD

Terzaghi and Peck (1948) proposed the following empirical relationship between the settlement (Se) of a prototype foundation measuring B×B in plan and the settlement of a test plate [Se(1)] measuring B1×B1 loaded to the same intensity

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

=2

1)1( 1

4

BBS

S

e

e (1)

Although a full-sized footing can be used for a load test, the normal practice is to employ a plate of the order of 0.3 m to 1 m. Bjerrum and Eggestad (1963) provided the results of 14 sets of load settlement tests. This is shown in Figure 1 along with the plot of Eq. (1). For these tests, B1 was 0.35 m for circular plates and 0.32 m for square plates. It is obvious from Figure 1 that, although the 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 Se/Se(1) versus B/B1 for all tests results provide by Bjerrum and Eggestad (1963) and Bazaraa (1967) as compiled by D’Appolonia et al. (1970). The overall results with the expanded data base are similar to those in Figure 1 as they relate to Eq. (1). Terzaghi and Peck (1948, 1967) proposed a correlation for the allowable bearing capacity, standard penetration number (N60), and the width of the foundation (B) corresponding to a 25 -mm settlement based on the observation given by Eq. (1). This correlation is shown in Figure 3. The curves shown in Figure 3 can be approximated by the relation

2

60 303(mm) ⎟

⎠⎞⎜

⎝⎛

+=

.BB

NqSe (2)

where q = bearing pressure in kN/m2 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

⎟⎠⎞⎜

⎝⎛

+=

.BB

NqCCS DWe (3)

where CW = ground water table correction CD = correction for depth of embedment = 1 – (Df /4B) Df = depth of embedment The magnitude of CW is equal to 1.0 if the depth of water table is greater than or equal to 2B below the foundation, and it is equal to 2.0 if the depth of water table is less than or equal to B below the foundation. The N60 value that is to be used in Eqs. (2) and (3) should be the average value of N60 up to a depth of about 3B to 4B measured from the bottom of the foundation.

Developments in elastic settlement estimation procedures for shallow foundations on granular soil Das, B.M., Atalar, C. & Shin, E.C.

11

Figure 1 Variation of Se/Se(1) versus B/B1 from the load settlement results of Bjerrum and Eggestad (1963)

(Note: B1 = 0.36 m for circular plates and 0.32 m for square plates).

Figure 2 Variation of Se/Se(1) versus B/B1 based on the data of Bjerrum and Eggestad (1963) and Bazaraa

(1967) (adapted from D’Appolonia et al., 1970). Jayapalan and Boehm (1986) and Papadopoulos (1992) summarized the case histories of 79 foundations. 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 those observed. The average value of Se(predicted)/Se(observed) ≈ 2.18. Similar observations were also made by Bazaraa (1967). With B1 = 0.3 m, Eq. (1) can be rewritten as


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Figure 3. Terzaghi and Peck’s (1948, 1967) recommendation for allowable bearing capacity for 25-mm

settlement variation with B and N60.

2

)1( 304 ⎟

⎠⎞⎜

⎝⎛

+=

.BB

SS

e

e

or

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎠⎞⎜

⎝⎛

+ )1(

2

41

30 e

e

SS

.BB (4)

Combining Eqs. (2) and (4)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

)1(60 413

e

ee S

SN

qS

or

75060

)1( .N

Sqe

= (5)

Bazaraa (1967) plotted a large number of plate load test results (B1 = 0.3 m) in the form of q/Se(1) versus N60 as shown in Figure 5. It can be seen that the relationship given by Eq. (5) is very conservative. In fact, q/Se(1) versus N60/0.5 will more closely represent the lower limiting condition.


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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. Bazaraa’s plate load test results—plot of q/Se(1) versus N60.

4 MEYERHOF’S METHOD

In 1956, Meyerhof proposed relationships for the elastic settlement of foundations on granular soil similar to Eq. (2). In 1965 he compared the predicted (by the relationships proposed in 1956) and observed settlements of eight structures and suggested that the allowable pressure (q) for a desired magnitude of Se can be increased by 50% compared to what he recommended in 1956. The revised relationships including the correction factors for water table location (CW) and depth of embedment (CD) can be expressed as


14

m) 1.22(for 25160

≤= BN

q.CCS DWe (6)

and

m) 1.22(for 30

2 2

60

>⎟⎠⎞⎜

⎝⎛

+= B

.BB

NqCCS DWe (7)

0.1=WC (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, then we will obtain Se(predicted)/Se(observed) ≈ 1.46. Hence, the predicted settlements will overestimate the observed values by about 50% on the average. Table 1 shows the comparison of the maximum observed settlements of mat foundations considered by Meyerhof (1965) and the settlements predicted by Eq. (7). The ratios of the predicted to observed settlements are generally in the range of 0.8 to 2. This is also what Meyerhof concluded in his 1965 paper.

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

Structure B

(m) Average

N60 q

(kN/m2)

Maximum Se(observed)

(mm)

Se(predicted) by Eq. (7)

(mm) )observed(

predicted)(

e

e

SS

T. Edison, Sao Paulo Banco do Brasil, Sao Paulo

Iparanga, Sao Paulo C.B.I. Esplanada, Sao Paulo

Riscala, Sao Paulo Thyssen, Dusseldorf Ministry, Dusseldorf Chimney, Cologne

18.3 22.9 9.15 14.6 3.96 22.6 15.9 20.4

15 18 9

22 20 25 20 10

229.8 239.4 220.2 383.0 229.8 239.4 220.4 172.4

15.24 27.94 35.56 27.94 12.70 24.13 21.59 10.16

29.66 25.74 45.88 33.43 19.86 18.65 21.23 33.49

1.95 0.99 1.29 1.20 1.56 0.77 0.98 3.30

Average ≈1.5

5 DE BEER AND MARTEN’S METHOD

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

Hσ

σσC.S

o

oe ⎟⎟

⎠

⎞⎜⎜⎝

⎛′Δ+′

= 10log32 (10)

where C = a constant of proportionality oσ′ = effective overburden pressure at the depth considered Δσ = increase in pressure at that depth due to foundation loading H = thickness of the layer considered The value of C can be approximated as


15

o

c

σq.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

⎟⎟⎠

⎞⎜⎜⎝

⎛′Δ+′

+=

o

o

o

ce σ

σσHe

CS 10log1

(12)

where ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′=

+ c

o

o

c

qσ

eC 5.1

1 (13)

Cc = compression index eo = in situ void ratio For the field cases considered by DeBeer and Martens (1957), the average ratio of predicted to observed settlement was about 1.9. DeBeer (1965) further observed that the above stated method only 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 Cc 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)

Value of constant Type of soil a b*

Uniform cohesionless material (uniformity coefficient Cu ≤ 2) Clean gravel Coarse sand Medium sand Fine sand Inorganic silt

0.05 0.06 0.07 0.08 0.10

0.50 0.50 0.50 0.50 0.50

Well-graded cohesionless soil Silty sand and gravel Clean, coarse to fine sand Coarse to fine silty sand Sandy silt (inorganic)

0.09 0.12 0.15 0.18

0.20 0.35 0.25 0.25

* The value of the constant b should be taken as emin whenever the latter is known or can conveniently be determined. Otherwise, use tabulated values as a rough approximation.

6 THE METHOD OF PECK AND BAZARAA

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

2

601 30)(2

⎟⎠⎞

⎜⎝⎛

+=

.BB

NqCCS DWe (15)

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


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(N1)60 = corrected standard penetration number

foundation theof bottom below the 0.5at foundation theof bottom below the 0.5at

BσBσC

o

oW ′= (16)

σo = total overburden pressure oσ′ = effective overburden pressure

50

4001.

fD q

γD..C ⎟⎟

⎠

⎞⎜⎜⎝

⎛−= (17)

γ = unit weight of soil The relationships for (N1)60 are as follow:

)kN/m 75(for 0401

4)( 260601 ≤′

′+= o

o

σσ.

NN (18)

and

)kN/m 75(for 010253

4)( 260601 >′

′+= o

o

σσ..

NN (19)

where oσ′ is the effective overburden pressure (kN/m2) D’Appolonia et al. (1970) compared the observed settlement of several shallow foundations from 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 from theory greatly overestimates the observed settlement. It appears that this solution will provide nearly the level of settlement that was obtained from Meyerhof’s revised relationships (Section 5).

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

D’Appolonia et al., 1970).

7 STRAIN INFLUENCE FACTOR METHOD

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

[ ]BAμEμqε s

s

sz ′+′−

+= )21()1(


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or

[ ]BAμμqEεI ss

szz ′+′−+== )21()1( (20)

where A' and B' = f (z/B) q = load per unit area Es = modulus of elasticity μ s = Poisson’s ratio Iz = strain influence factor Figure 7 shows the variation of Iz with depth based on Eq. (20) for μ s = 0.4 and 0.5. The experimental results of Eggestad (1963) for variation of Iz are also given in this figure. Considering both the theoretical and experimental results cited in Figure 7, Schmertmann (1970) proposed a simplified distribution of Iz with depth that is generally referred to as 2B–0.6Iz distribution and it is also shown in Figure 7. According to the simplified method,

zEIqCCS

B

o s

ze Δ∑=

2

21 (21)

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

C1 = correction factor for embedment of foundation = qq. o501− (22)

qo = effective overburden pressure at the level of the foundation

C2 = correction factor to account for creep in soil = ⎟⎠⎞

⎜⎝⎛+

10log201

.t. (23)

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

cS qE 2= (24) where qc = cone penetration resistance

Figure 7 Theoretical and experimental distribution of vertical strain influence factor below the center of a

circular loaded area (based on Schmertmann, 1970).


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Figure 8 Sivakugan et al.’s comparison (1998) of predicted and observed settlements from 79 foundations—

predicted settlement based on 2B−0.6Iz procedure.

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

Sivakugan 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 7 and Eq. (21), and this is shown in Figure 8. From this figure, it can be seen that se(predicted)/Se(observed) ≈ 3.39. Schmertmann et al. (1978) modified the strain influence factor variation (2B–0.6Iz) shown in Figure 7. The revised distribution is shown in Figure 9 for use in Eqs. (21)–(23). According to this, For square or circular foundations: Iz = 0.1 at z = 0 Iz(peak) at z = zp = 0.5B Iz = 0 at z = zo = 2B For foundations with L/B ≥ 10: Iz = 0.2 at z = 0 Iz(peak) at z = zp = B Iz = 0 at z = zo = 4B where L = length of foundation. For L/B between 1 and 10, interpolation can be done. Also


19

5.0

)peak( 1.05.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛′

+=o

zqIσ

(25)

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

2.00111.01.0 )0at ( ≤⎟⎠⎞

⎜⎝⎛+== B

LI zz (26)

110555.05.0 ≤⎟

⎠⎞

⎜⎝⎛ −+=

BL

Bz p (27)

41222.02 ≤⎟

⎠⎞

⎜⎝⎛ −+=

BL

Bzo (28)

Noting that stiffness is about 40% larger for plane strain compared to axisymmetric loading, Schmertmann et al. (1978) recommended that. s)foundationcircular and square(for 5.2 cs qE = (29) and )foundation strip(for 5.3 cs qE = (30) With the modified strain-influence factor diagram,

zEICCS

ozz

z s

ze Δ∑=

=

=021 (31)

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

8 RECENT MODIFICATIONS IN STRAIN-INFLUENCE FACTOR DIAGRAMS

More recently some modifications have been proposed to the strain-influence factor diagram suggested by Schmertmann et al. (1978). Two of these suggestions are discussed below. 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, for surface foundation condition (that is, Df/B = 0) Iz = 0.2 at z = 0 Iz = Iz(peak) = 0.6 at z = zp = 0.5B Iz = 0 at z = zo

4log12 ≤⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

BLzo (32)

For Df/B > 0, Iz should be modified to zI ′ . Figure 11 shows the variation of zz II /′ with Df/B.


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

Figure 11 Variation of zz II /′ with Df/B (after Terzaghi et al. 1996).

The end of construction settlement can be estimated as

zEIqS

ozz

z s

ze Δ∑

′=

=

=0 (33)

The settlement due to creep can be calculated as

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

day 1log1.0 days

creep

tz

qS o

c

(34)


21

where cq = weighted mean value of measured qc values of sublayers between z = 0 and z = zo (MN/m2) It has also been suggested that

4.1log4.01)1/(

)/( ≤⎟⎠⎞

⎜⎝⎛+=

= BL

EE

BLs

BLs (35)

where cBLs qE 5.3)1/( == (36) Figure 12 shows the plot of Es versus qc from 81 foundations and 92 plate load tests on which Eq. (36) has been established. The magnitude of Es recommended by Eq. (36) is about 40% higher than 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 and gravelly soils (Terzaghi et al., 1996).

Figure 12 Correlation between Es and qc for square and circularly loaded areas [adapted from Terzaghi et al.

(1996)].

Figure 13 Comparison of end of construction predicted and measured Se of foundations on sand and gravelly

soils based on Eqs. (33), (35) and (36) [adapted from Terzaghi et al. (1996)].


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8.2 Modification Suggested by Lee et al. (2008)

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

6at 1 of maximuma with111.05.0 =≤⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛+=

BL

BL

Bzp (37)

6at maximuma with315

cos95.0 =≤+⎭⎬⎫

⎩⎨⎧

π−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

π=

BL

BL

Bzo (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 soil using the field standard penetration number N60. The method can be summarized as follows:

9.1 Determination of Variation of Standard Penetration Number with Depth

Obtain the field penetration numbers (N60) with depth at the location of the foundation. The following adjustments of N60 may be necessary, depending on the field conditions: For gravel or sandy gravel, 6060(a) 25.1 NN ≈ (39) For fine sand or silty sand below the ground water table and N60 > 15, )15(5.015 6060(a) −+≈ NN (40) where N60(a) = adjusted N60 value

9.2 Determination of Depth of Stress Influence (z′)

In determining the depth of stress influence, the following three cases may arise: Case I. If N60 [or N60(a)] is approximately constant with depth, calculate z' from

750

41.

RR BB.

Bz

⎟⎟⎠

⎞⎜⎜⎝

⎛=

′ (41)

where BR = reference width = 0.3 m B = width of the actual foundation (m) Case II. If N60 [or N60(a)] is increasing with depth, use Eq. (41) to calculate z'. Case III. If N60 [or N60(a)] is decreasing with depth, calculate z' = 2B and z' = distance from the bottom of the foundation to the bottom of the soft soil layer (= z"). Use z' = 2B or z' = z" (whichever is smaller).


23

9.3 Determination of Depth of Stress Influence Correction Factor α

The correction factor α is given as

12 ≤⎟⎠⎞

⎜⎝⎛

′−

′=

zH

zHα (42)

where H = thickness of the compressible layer

9.4 Calculation of Elastic Settlement

The elastic settlement of the foundation Se can be calculated as: A. For normally consolidated soil

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=a

.

R.

(a)R

e

pq

BB

BL.

BL.

NorN.α.

BS

70

2

416060 250

251711140 (43)

where L = length of the foundation pa = atmospheric pressure (≈ 100 kN/m2) B. For overconsolidated soil (q ≤ cσ ′ ; where cσ ′ = overconsolidation pressure)

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=a

.

R.

(a)R

e

pq

BB

BL.

BL.

N or N.α.

BS

70

2

416060 250

2515700470 (44)

C. For overconsolidated soil (q > cσ ′ )

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=a

c

.

R.

(a)R

e

pσ.q

BB

BL.

BL.

N or N.α.

BS 670

250

251570140

70

2

416060

(45)

Sivakugan and Johnson (2004) used a probabilistic approach to compare the predicted settlements 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 study—that is, predicted settlement versus the probability of exceeding 25 mm settlement in the field. This shows that the method of Burland and Burbidge (1985), although conservative, is a substantially improved technique to estimate elastic settlement.

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

Briaud (2007) presented a method based on field Pressuremeter tests to develop a load-settlement curve for a given foundation from which the elastic settlement at a given load intensity can be estimated. This takes into account the foundation load eccentricity, load inclination, and the location


24

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

Probability of exceeding 25 mm settlement in field Predicted settlement

(mm) Terzaghi and Peck

(1948, 1967) Schmertmann et al.

(1970) Burland and

Burbidge (1985) 1 5

10 15 20 25 30 35 40

0.00 0.00 0.00 0.09 0.20 0.26 0.31 0.35

0.387

0.00 0.00 0.02 0.13 0.20 0.27 0.32 0.37 0.42

0.00 0.03 0.15 0.25 0.34 0.42 0.49 0.55 0.61

Compiled from Sivakugan and Johnson (2004)

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 pp on the cavity wall versus relative increase in cavity radius

ΔR/Ro. Extend the straight line part of the PMT curve to zero pressure and shift the vertical axis to the value of ΔR/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 the foundation. Based on the pp versus ΔR/Ro diagrams (Step 2) and the location of the depth of the tests, develop a mean plot of pp versus ΔR/Ro as shown in Figure 16. The mean pp for a given ΔR/Ro can be given as

. . . )3(3

)2(2

)1(1

)( +++= pppmeanp pAAp

AAp

AAp (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

Figure 14 Pressuremeter test to obtain load-settlement curve.


25

Figure 15 Adjustment of field Pressuremeter test plot of pp versus ΔR/Ro.

Figure 16 Development of the mean pp versus ΔR/Ro plot.

4. Convert the plot of pp(mean) versus ΔR/Ro plot to q versus Se/B plot using the following equations.

)(,/ ))(( meanpdeBL pffffΓq βδ= (47)

o

e

RR

BS Δ

= 24.0 (48)

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

⎟⎠⎞

⎜⎝⎛+==

LBf LB 2.08.0factor shape/ (49)

(center) 33.01factorty eccentrici load ⎟⎠⎞

⎜⎝⎛−==

Befe (50)

(edge) 15.0

⎟⎠⎞

⎜⎝⎛−=

Befe (51)


26

Figure 17 Variation of Г function.

(center) 90

(degrees) 1factor ninclinatio ⎥⎦⎤

⎢⎣⎡δ−==δf (52)

(edge) 360

(degrees) 10.5

⎥⎦⎤

⎢⎣⎡δ−=δf (53)

slope) 1:(3 18.0factor slope0.1

, ⎟⎠⎞

⎜⎝⎛ +==β B

df d (54)

slope) 1:(2 17.00.15

, ⎟⎠⎞

⎜⎝⎛ +=β B

df d (55)

5. Based on the load-settlement diagram developed in Step 4, obtain the actual Se(maximum) which

corresponds 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,

3.0

1(maximum))( ⎟⎟

⎠

⎞⎜⎜⎝

⎛≈

ttStS ee (56)

where Se(t) = settlement after time t Se(maximum) = settlement obtained from Step 5 t = time, in minutes t1 = reference time = 1 minute

SETTLEMENT CALCULATION BASED ON THEORY OF ELASTICITY

11 STEINBRENNER’S (1934) AND FOX’S (1948) THEORY

Based on the observations made on elastic settlement calculation using empirical correlations and the wide range in the predictions obtained, it is desirable to consider alternative solutions based on the theory of elasticity. With that in mind, Figure 18 shows a schematic diagram of the elastic settlement profile for a flexible and rigid foundation. The shallow foundation measures B×L in plan and is located at a depth Df below the ground surface. A rock layer (or a rigid layer) is located at a depth H below the bottom of the foundation.


27

Figure 18 Settlement profile for shallow flexible and rigid foundation.

Theoretically, if the foundation is perfectly flexible (Figure 18), the settlement may be expressed as (see Bowles, 1987)

fss

se II

EμBαqS

21)( −′′= (57)

where q = net applied pressure on the foundation μ s = Poisson’s ratio of soil Es = average modulus of elasticity of the soil under the foundation, measured from z = 0 to about z = 4B B' = B/2 for center of foundation (= B for corner of foundation)

Is = shape factor (Steinbrenner, 1934) = 21 121 FμμFs

s

−−

+ (58)

)(1101 AA

πF += (59)

21

2 tan2

AπnF −= (60)

( )( )11

11ln22

222

0+++

+++=

nmmnmmmA (61)

( )1

11ln22

22

1+++

+++=

nmmnmmA (62)

1222

+++=

nmnmA (63)

⎟⎟⎠

⎞⎜⎜⎝

⎛==

BL

BD

fI sf

f and ,,1948) (Fox,factor depth μ (64)

α' = a factor that depends on the location below the foundation where settlement is being calculated To calculate settlement at the center of the foundation, we use


28

4=′α (65)

BLm = (66)

and

⎟⎠⎞

⎜⎝⎛

=

2BHn (67)

To calculate settlement at a corner of the foundation, 1=′α (68)

BLm =

and

BHn =

The variations of F1 and F2 with m and n are given Tables 4 and 5. Based on the works of Fox (1948), the variations of depth factor If for μ s = 0.3 and 0.4 and L/B have been determined by Bowles (1987) and are given in Table 6. Note that If is not a function of H/B. Due to the non-homogeneous nature of a soil deposit, the magnitude of Es may vary with depth. For that reason, Bowles (1987) recommended

z

zEE is

s

∑ Δ= )( (69)

where Es(i) = soil modulus within the depth Δz z = 5B or H (if H < 5B) Bowles (1987) also recommended that 2

60 kN/m )15(500 += NEs (70) The elastic settlement of a rigid foundation can be estimated as center) (flexible,)rigid( 93.0 ee SS ≈ (71) Bowles (1987) compared this theory with 12 case histories that provided reasonable good results.


29

Table 4. Variation of F1 with m and n

m n 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75

10.00 20.00 50.00

100.00

0.014 0.049 0.095 0.142 0.186 0.224 0.257 0.285 0.309 0.330 0.348 0.363 0.376 0.388 0.399 0.408 0.417 0.424 0.431 0.437 0.443 0.448 0.453 0.457 0.461 0.465 0.468 0.471 0.474 0.477 0.480 0.482 0.485 0.487 0.489 0.491 0.493 0.495 0.496 0.498 0.529 0.548 0.555

0.013 0.046 0.090 0.138 0.183 0.224 0.259 0.290 0.317 0.341 0.361 0.379 0.394 0.408 0.420 0.431 0.440 0.450 0.458 0.465 0.472 0.478 0.483 0.489 0.493 0.498 0.502 0.506 0.509 0.513 0.516 0.519 0.522 0.524 0.527 0.529 0.531 0.533 0.536 0.537 0.575 0.598 0.605

0.012 0.044 0.087 0.134 0.179 0.222 0.259 0.292 0.321 0.347 0.369 0.389 0.406 0.422 0.436 0.448 0.458 0.469 0.478 0.487 0.494 0.501 0.508 0.514 0.519 0.524 0.529 0.533 0.538 0.541 0.545 0.549 0.552 0.555 0.558 0.560 0.563 0.565 0.568 0.570 0.614 0.640 0.649

0.011 0.042 0.084 0.130 0.176 0.219 0.258 0.292 0.323 0.350 0.374 0.396 0.415 0.431 0.447 0.460 0.472 0.484 0.494 0.503 0.512 0.520 0.527 0.534 0.540 0.546 0.551 0.556 0.561 0.565 0.569 0.573 0.577 0.580 0.583 0.587 0.589 0.592 0.595 0.597 0.647 0.678 0.688

0.011 0.041 0.082 0.127 0.173 0.216 0.255 0.291 0.323 0.351 0.377 0.400 0.420 0.438 0.454 0.469 0.481 0.495 0.506 0.516 0.526 0.534 0.542 0.550 0.557 0.563 0.569 0.575 0.580 0.585 0.589 0.594 0.598 0.601 0.605 0.609 0.612 0.615 0.618 0.621 0.677 0.711 0.722

0.011 0.040 0.080 0.125 0.170 0.213 0.253 0.289 0.322 0.351 0.378 0.402 0.423 0.442 0.460 0.476 0.484 0.503 0.515 0.526 0.537 0.546 0.555 0.563 0.570 0.577 0.584 0.590 0.596 0.601 0.606 0.611 0.615 0.619 0.623 0.627 0.631 0.634 0.638 0.641 0.702 0.740 0.753

0.010 0.038 0.077 0.121 0.165 0.207 0.247 0.284 0.317 0.348 0.377 0.402 0.426 0.447 0.467 0.484 0.495 0.516 0.530 0.543 0.555 0.566 0.576 0.585 0.594 0.603 0.610 0.618 0.625 0.631 0.637 0.643 0.648 0.653 0.658 0.663 0.667 0.671 0.675 0.679 0.756 0.803 0.819

0.010 0.038 0.076 0.118 0.161 0.203 0.242 0.279 0.313 0.344 0.373 0.400 0.424 0.447 0.458 0.487 0.514 0.521 0.536 0.551 0.564 0.576 0.588 0.598 0.609 0.618 0.627 0.635 0.643 0.650 0.658 0.664 0.670 0.676 0.682 0.687 0.693 0.697 0.702 0.707 0.797 0.853 0.872

0.010 0.037 0.074 0.116 0.158 0.199 0.238 0.275 0.308 0.340 0.369 0.396 0.421 0.444 0.466 0.486 0.515 0.522 0539 0.554 0.568 0.581 0.594 0.606 0.617 0.627 0.637 0.646 0.655 0.663 0.671 0.678 0.685 0.692 0.698 0.705 0.710 0.716 0.721 0.726 0.830 0.895 0.918

0.010 0.037 0.074 0.115 0.157 0.197 0.235 0.271 0.305 0.336 0.365 0.392 0.418 0.441 0.464 0.484 0.515 0.522 0.539 0.554 0.569 0.584 0.597 0.609 0.621 0.632 0.643 0.653 0.662 0.671 0.680 0.688 0.695 0.703 0.710 0.716 0.723 0.719 0.735 0.740 0.858 0.931 0.956


30

Table 4. (Continued)

m n 4.5 5.0 6.0 7.0 8.0 9.0 10.0 25.0 50.0 100.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75

10.00 20.00 50.00

100.00

0.010 0.036 0.073 0.114 0.155 0.195 0.233 0.269 0.302 0.333 0.362 0.389 0.415 0.438 0.461 0.482 0.516 0.520 0.537 0.554 0.569 0.584 0.597 0.611 0.623 0.635 0.646 0.656 0.666 0.676 0.685 0.694 0.702 0.710 0.717 0.725 0.731 0.738 0.744 0.750 0.878 0.962 0.990

0.010 0.036 0.073 0.113 0.154 0.194 0.232 0.267 0.300 0.331 0.359 0.386 0.412 0.435 0.458 0.479 0.496 0.517 0.535 0.552 0.568 0.583 0.597 0.610 0.623 0.635 0.647 0.658 0.669 0.679 0.688 0.697 0.706 0.714 0.722 0.730 0.737 0.744 0.751 0.758 0.896 0.989 1.020

0.010 0.036 0.072 0.112 0.153 0.192 0.229 0.264 0.296 0.327 0.355 0.382 0.407 0.430 0.453 0.474 0.484 0.513 0.530 0.548 0.564 0.579 0.594 0.608 0.621 0.634 0.646 0.658 0.669 0.680 0.690 0.700 0.710 0.719 0.727 0.736 0.744 0.752 0.759 0.766 0.925 1.034 1.072

0.010 0.036 0.072 0.112 0.152 0.191 0.228 0.262 0.294 0.324 0.352 0.378 0.403 0.427 0.449 0.470 0.473 0.508 0.526 0.543 0.560 0.575 0.590 0.604 0.618 0.631 0.644 0.656 0.668 0.679 0.689 0.700 0.710 0.719 0.728 0.737 0.746 0.754 0.762 0.770 0.945 1.070 1.114

0.010 0.036 0.072 0.112 0.152 0.190 0.227 0.261 0.293 0.322 0.350 0.376 0.401 0.424 0.446 0.466 0.471 0.505 0.523 0.540 0.556 0.571 0.586 0.601 0.615 0.628 0.641 0.653 0.665 0.676 0.687 0.698 0.708 0.718 0.727 0.736 0.745 0.754 0.762 0.770 0.959 1.100 1.150

0.010 0.036 0.072 0.111 0.151 0.190 0.226 0.260 0.291 0.321 0.348 0.374 0.399 0.421 0.443 0.464 0.471 0.502 0.519 0.536 0.553 0.568 0.583 0.598 0.611 0.625 0.637 0.650 0.662 0.673 0.684 0.695 0.705 0.715 0.725 0.735 0.744 0.753 0.761 0.770 0.969 1.125 1.182

0.010 0.036 0.071 0.111 0.151 0.189 0.225 0.259 0.291 0.320 0.347 0.373 0.397 0.420 0.441 0.462 0.470 0.499 0.517 0.534 0.550 0.585 0.580 0.595 0.608 0.622 0.634 0.647 0.659 0.670 0.681 0.692 0.703 0.713 0.723 0.732 0.742 0.751 0.759 0.768 0.977 1.146 1.209

0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.257 0.287 0.316 0.343 0.368 0.391 0.413 0.433 0.453 0.468 0.489 0.506 0.522 0.537 0.551 0.565 0.579 0.592 0.605 0.617 0.628 0.640 0.651 0.661 0.672 0.682 0.692 0.701 0.710 0.719 0.728 0.737 0.745 0.982 1.265 1.408

0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.256 0.287 0.315 0.342 0.367 0.390 0.412 0.432 0.451 0.462 0.487 0.504 0.519 0.534 0.549 0.583 0.576 0.589 0.601 0.613 0.624 0.635 0.646 0.656 0.666 0.676 0.686 0.695 0.704 0.713 0.721 0.729 0.738 0.965 1.279 1.489

0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.256 0.287 0.315 0.342 0.367 0.390 0.411 0.432 0.451 0.460 0.487 0.503 0.519 0.534 0.548 0.562 0.575 0.588 0.600 0.612 0.623 0.634 0.645 0.655 0.665 0.675 0.684 0.693 0.702 0.711 0.719 0.727 0.735 0.957 1.261 1.499


31

Table 5. Variation of F2 with m and n

m n 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00

0.049 0.074 0.083 0.083 0.080 0.075 0.069 0.064 0.059 0.055 0.051 0.048 0.045 0.042 0.040 0.037 0.036 0.034 0.032 0.031 0.029 0.028 0.027 0.026 0.025 0.024 0.023 0.022 0.022 0.021 0.020 0.020 0.019 0.018 0.018 0.017 0.017 0.017 0.016 0.016 0.008 0.003 0.002

0.050 0.077 0.089 0.091 0.089 0.084 0.079 0.074 0.069 0.064 0.060 0.056 0.053 0.050 0.047 0.044 0.042 0.040 0.038 0.036 0.035 0.033 0.032 0.031 0.030 0.029 0.028 0.027 0.026 0.025 0.024 0.023 0.023 0.022 0.021 0.021 0.020 0.020 0.019 0.019 0.010 0.004 0.002

0.051 0.080 0.093 0.098 0.096 0.093 0.088 0.083 0.077 0.073 0.068 0.064 0.060 0.057 0.054 0.051 0.049 0.046 0.044 0.042 0.040 0.039 0.037 0.036 0.034 0.033 0.032 0.031 0.030 0.029 0.028 0.027 0.026 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.011 0.004 0.002

0.051 0.081 0.097 0.102 0.102 0.099 0.095 0.090 0.085 0.080 0.076 0.071 0.067 0.068 0.060 0.057 0.055 0.052 0.050 0.048 0.046 0.044 0.042 0.040 0.039 0.038 0.036 0.035 0.034 0.033 0.032 0.031 0.030 0.029 0.028 0.028 0.027 0.026 0.026 0.025 0.013 0.005 0.003

0.051 0.083 0.099 0.106 0.107 0.105 0.101 0.097 0.092 0.087 0.082 0.078 0.074 0.070 0.067 0.063 0.061 0.058 0.055 0.053 0.051 0.049 0.047 0.045 0.044 0.042 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.032 0.031 0.030 0.029 0.029 0.028 0.014 0.006 0.003

0.052 0.084 0.101 0.109 0.111 0.110 0.107 0.102 0.098 0.093 0.089 0.084 0.080 0.076 0.073 0.069 0.066 0.063 0.061 0.058 0.056 0.054 0.052 0.050 0.048 0.046 0.045 0.043 0.042 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.033 0.032 0.031 0.016 0.006 0.003

0.052 0.086 0.104 0.114 0.118 0.118 0.117 0.114 0.110 0.106 0.102 0.097 0.093 0.089 0.086 0.082 0.079 0.076 0.073 0.070 0.067 0.065 0.063 0.060 0.058 0.056 0.055 0.053 0.051 0.050 0.048 0.047 0.046 0.045 0.043 0.042 0.041 0.040 0.039 0.038 0.020 0.008 0.004

0.052 0.086 0.106 0.117 0.122 0.124 0.123 0.121 0.119 0.115 0.111 0.108 0.104 0.100 0.096 0.093 0.090 0.086 0.083 0.080 0.078 0.075 0.073 0.070 0.068 0.066 0.064 0.062 0.060 0.059 0.057 0.055 0.054 0.053 0.051 0.050 0.049 0.048 0.047 0.046 0.024 0.010 0.005

0.052 0.087 0.107 0.119 0.125 0.128 0.128 0.127 0.125 0.122 0.119 0.116 0.112 0.109 0.105 0.102 0.099 0.096 0.093 0.090 0.087 0.084 0.082 0.079 0.077 0.075 0.073 0.071 0.069 0.067 0.065 0.063 0.062 0.060 0.059 0.057 0.056 0.055 0.054 0.052 0.027 0.011 0.006

0.052 0.087 0.108 0.120 0.127 0.130 0.131 0.131 0.130 0.127 0.125 0.122 0.119 0.116 0.113 0.110 0.107 0.104 0.101 0.098 0.095 0.092 0.090 0.087 0.085 0.083 0.080 0.078 0.076 0.074 0.072 0.071 0.069 0.067 0.066 0.064 0.063 0.061 0.060 0.059 0.031 0.013 0.006


32

Table 5. (continued)

m n 4.5 5.0 6.0 7.0 8.0 9.0 10.0 25.0 50.0 100.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75

10.00 20.00 50.00

100.00

0.053 0.087 0.109 0.121 0.128 0.132 0.134 0.134 0.133 0.132 0.130 0.127 0.125 0.122 0.119 0.116 0.113 0.110 0.107 0.105 0.102 0.099 0.097 0.094 0.092 0.090 0.087 0.085 0.083 0.081 0.079 0.077 0.076 0.074 0.072 0.071 0.069 0.068 0.066 0.065 0.035 0.014 0.007

0.053 0.087 0.109 0.122 0.130 0.134 0.136 0.136 0.136 0.135 0.133 0.131 0.129 0.126 0.124 0.121 0.119 0.116 0.113 0.111 0.108 0.106 0.103 0.101 0.098 0.096 0.094 0.092 0.090 0.088 0.086 0.084 0.082 0.080 0.078 0.077 0.075 0.074 0.072 0.071 0.039 0.016 0.008

0.053 0.088 0.109 0.123 0.131 0.136 0.138 0.139 0.140 0.139 0.138 0.137 0.135 0.133 0.131 0.129 0.127 0.125 0.123 0.120 0.118 0.116 0.113 0.111 0.109 0.107 0.105 0.103 0.101 0.099 0.097 0.095 0.093 0.091 0.089 0.888 0.086 0.085 0.083 0.082 0.046 0.019 0.010

0.053 0.088 0.110 0.123 0.132 0.137 0.140 0.141 0.142 0.142 0.142 0.141 0.140 0.138 0.137 0.135 0.133 0.131 0.130 0.128 0.126 0.124 0.122 0.120 0.118 0.116 0.114 0.112 0.110 0.108 0.106 0.104 0.102 0.101 0.099 0.097 0.096 0.094 0.092 0.091 0.053 0.022 0.011

0.053 0.088 0.110 0.124 0.132 0.138 0.141 0.143 0.144 0.144 0.144 0.144 0.143 0.142 0.141 0.139 0.138 0.136 0.135 0.133 0.131 0.130 0.128 0.126 0.124 0.122 0.121 0.119 0.117 0.115 0.114 0.112 0.110 0.108 0.107 0.105 0.104 0.102 0.100 0.099 0.059 0.025 0.013

0.053 0.088 0.110 0.124 0.133 0.138 0.142 0.144 0.145 0.146 0.146 0.145 0.145 0.144 0.143 0.142 0.141 0.140 0.139 0.137 0.136 0.134 0.133 0.131 0.129 0.128 0.126 0.125 0.123 0.121 0.120 0.118 0.117 0.115 0.114 0.112 0.110 0.109 0.107 0.106 0.065 0.028 0.014

0.053 0.088 0.110 0.124 0.133 0.139 0.142 0.145 0.146 0.147 0.147 0.147 0.147 0.146 0.145 0.145 0.144 0.143 0.142 0.140 0.139 0.138 0.136 0.135 0.134 0.132 0.131 0.129 0.128 0.126 0.125 0.124 0.122 0.121 0.119 0.118 0.116 0.115 0.113 0.112 0.071 0.031 0.016

0.053 0.088 0.111 0.125 0.134 0.140 0.144 0.147 0.149 0.151 0.152 0.152 0.153 0.153 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.153 0.153 0.153 0.153 0.152 0.152 0.152 0.151 0.151 0.150 0.150 0.150 0.149 0.149 0.148 0.148 0.147 0.124 0.071 0.039

0.053 0.088 0.111 0.125 0.134 0.140 0.144 0.147 0.150 0.151 0.152 0.153 0.154 0.155 0.155 0.155 0.156 0.156 0.156 0.156 0.156 0.156 0.157 0.157 0.157 0.157 0.157 0.157 0.157 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.148 0.113 0.071

0.053 0.088 0.111 0.125 0.134 0.140 0.145 0.148 0.150 0.151 0.153 0.154 0.154 0.155 0.155 0.156 0.156 0.156 0.157 0.157 0.157 0.157 0.157 0.157 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.156 0.142 0.113


33

Table 6. Variation of If (Fox, 1948)*

L/B Df/B 1.0 1.2 1.4 1.6 1.8 2.0 5.0

Poisson’s ratio μs = 0.30 0.05 0.10 0.20 0.40 0.60 0.80 1.00 2.00

0.979 0.954 0.902 0.808 0.738 0.687 0.650 0.562

0.981 0.958 0.911 0.823 0.754 0.703 0.665 0.571

0.982 0.962 0.917 0.834 0.767 0.716 0.678 0.580

0.983 0.964 0.923 0.843 0.778 0.728 0.689 0.588

0.984 0.966 0.927 0.851 0.788 0.738 0.700 0.596

0.985 0.968 0.930 0.857 0.796 0.747 0.709 0.603

0.990 0.977 0.951 0.899 0.852 0.813 0.780 0.675

Poisson’s ratio μs = 0.40 0.05 0.10 0.20 0.40 0.60 0.80 1.00 2.00

0.989 0.973 0.932 0.848 0.779 0.727 0.689 0.596

0.990 0.976 0.940 0.862 0.795 0.743 0.704 0.606

0.991 0.978 0.945 0.872 0.808 0.757 0.718 0.615

0.992 0.980 0.949 0.881 0.819 0.769 0.730 0.624

0.992 0.981 0.952 0.887 0.828 0.779 0.740 0.632

0.993 0.982 0.955 0.893 0.836 0.788 0.749 0.640

0.995 0.988 0.970 0.927 0.886 0.849 0.818 0.714

* Adapted from Bowles (1987)

12 ANALYSIS OF MAYNE AND POULOS BASED ON THEORY OF ELASTICITY

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

πBLBe

4= (72)

For circular foundations, BBe = (73) where B = diameter of foundation Figure 19 shows a foundation with an equivalent diameter Be located at a depth Df below the ground surface. Let the thickness of the foundation be t and the modulus of elasticity of the foundation material be Ef. A rigid layer is located at a depth H below the bottom of the foundation.

Figure 19 Mayne and Poulos’ procedure (1999) for settlement calculation.


34

The modulus of elasticity of the compressible soil layer can be given as kzEE os += (74) where k = rate of increase in Es with depth (kN/m2/m) With the preceding parameters defined, the elastic settlement below the center of the foundation is

( )21 so

ERGee μ

EIIIqBS −= (75)

where ⎟⎟⎠

⎞⎜⎜⎝

⎛== =

eBH

, ekBoE

βfsG EI depth with of variationfor thefactor influence

IR = foundation rigidity correction factor IE = foundation embedment correction factor Figure 20 shows the variation of IG with β = Eo/kBe and H/Be. The foundation rigidity correction factor can be expressed as

3

2

2

1064

14

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

++

+=

eeo

f

R

Bt

kBE

E.

πI (76)

Figure 20 Variation of IG with β.

Similarly, the embedment correction factor is

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=

61)40221(exp53

11.

DB.μ..

I

f

es

E (77)

Figures 21 and 22 show the variation of IR with IE as a function of the terms expressed in Eqs. (76) and (77).


35

Figure 21 Variation of IR with KF.

Figure 22 Variation of IE with μs and Df/Be.

13 BERARDI AND LANCELLOTTA’S METHOD

Berardi and Lancellotta (1991) proposed a method to estimate the elastic settlement that takes into account 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,

s

se EqBIS = (78)

where Is = influence factor for a rigid foundation (Tsytovich, 1951) Es = modulus of elasticity of soil The variation of Is (Tsytovich, 1951) with Poisson’s ratio μs = 0.15 is given in Table 7.

Table 7. Variation of Is

Depth of influence HI /B L/B 0.5 1.0 1.5 2.0 1 2 3 5 10

0.35 0.39 0.40 0.41 0.42

0.56 0.65 0.67 0.68 0.71

0.63 0.76 0.81 0.84 0.89

0.69 0.88 0.96 0.99 1.06


36

Using analytical and numerical evaluations, Berardi and Lancellotta (1991) have shown that, for a circular foundation, BH )3.1 to8.0(25 = (79) For plane strain condition (that is, L/B ≥ 10) (circle)2525 )1.7 to1.5( HH = (80) where H25 = depth from the bottom of the foundation below which the residual settlement is 25% of the total settlement The above implies that H25 ≤ 2.5B for practically all foundations. Thus the depth of influence HI can be taken to be H25. The modulus of elasticity Es in Eq. (78) can be evaluated as (Janbu, 1963)

50

50.

a

oaEs p

σ.σpKE ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′Δ+′= (81)

where pa = atmospheric pressure oσ ′ and Δσ' = effective overburden pressure and net effective stress increase due to the foundation loading, respectively, at a depth B/2 below the foundation KE = dimensionless modulus number After reanalyzing the performance of 130 structures foundations on predominantly silica sand as reported by Burland and Burbidge (1985), Berardi and Lancellotta (1991) obtained the variation of KE with the relative density Dr at Se/B = 0.1% and KE at varying strain levels. Figures 23 and 24 show the average variation of KE with Dr at Se/B = 0.1% and %)10/()/( / .BSEBSE ee

KK = with Se/B. In order to estimate the elastic settlement of the foundation, an iterative procedure is suggested

which can be described as follows: 1. Determine the variation of the blow count N60 from standard penetration tests within the zone of

influence, that is H25. 2. Determine the corrected blow count (N1)60 as

⎟⎟⎠

⎞⎜⎜⎝

⎛′+

=oσ

NN01.01

2)( 60601 (82)

where oσ ′ = vertical effective stress in kN/m2 3. Determine the average corrected blow count from standard penetration tests 601)(N and hence the

average relative density as

5.0

1

60 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

NDr (83)

4. With a known value of Dr, determine %)10/( .BSE e

K = from Figure 23 and hence Es from Eq. (81) for Se/B = 0.1%

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

from Step 5 and Figure 24 to estimate a revised )/( BSE eK . This value can now be used in Eqs. (81)

and (78) to obtain a revised Se. The iterative procedures can be continued until the assumed and calculated values are the same.


37

Figure 23 Variation of KE with Dr and N60 (adapted from Berardi and Lancellotta, 1991).

Figure 24 Plot of %)10/()/( / .BSEBSE ee

KK = with Se/B (adapted from Berardi and Lancellotta, 1991).

Based on a probabilistic study conducted by Sivakugan and Johnson (2004), the probability of exceeding 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)

Predicted settlement (mm)

Probability of exceeding 25 mm in the field (%)

1 5 10 15 20 25 30 35 40

6 19 32 43 52 60 66 72 77


38

14 GENERAL COMMENTS AND CONCLUSIONS

A general review of the major developments over the last sixty years for estimating elastic settlement of shallow foundations on granular soil is presented. Based on the above review, the following general observations can be made. 1. Meyerhof’s 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 and Bazaraa’s method (1969) provides results that are almost similar to those obtained from Meyerhof’s method (1965).

2. Burland and Burbidge’s solution (1985) will provide more reasonable estimations of Se than those obtained from the solution of Meyerhof (1965). However it will be difficult to determine the overconsolidation ratio and the preconsolidation pressure for granular soils from field exploration.

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 the elastic settlement provided a more realistic value of Es is assumed in the calculation. The authors feel that the empirical relationships for Es provided by Eqs. (35) and (36) are more reasonable.

4. The relationships for Es provided by Eqs. (35) and (36) are based on the field cone penetration resistance. These equations can be converted to expressions in terms of N60 and D50 (mean grain size). Figure 25 shows some of the relationships available in the literature. Based on the data of Burland and Burbidge et al. (1985)

305.050

60

8DNpq

a

c

=⎟⎟⎠

⎞⎜⎜⎝

⎛

(84)

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

228.050

60

6DNpq

a

c

=⎟⎟⎠

⎞⎜⎜⎝

⎛

(85)

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

26.050

60

6429.7 DNpq

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 a versatile

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 the

other methods, provided a realistic value of Es is adopted. This can be accomplished using the iteration method suggested by Berardi and Lancellotta (1991). In lieu of that, the Es relationship given by Terzaghi et al. (1996) can be used.


39

Figure 25 Variation of (qc/pa)/N60 with D50. (a) Adapted from Terzaghi et al. (1996); (b) Adapted from

Anagnostopoulos, 2003).

In his landmark paper in 1927 entitled “The Science of Foundations,” Karl Terzaghi wrote “Foundation problems, throughout, are of such character that a strictly theoretical mathematical treatment will always be impossible. The only way to handle them efficiently consists of finding out, first, what has happened on preceding jobs of a similar character; next, the kind of soil on which 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 the results 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 are accuracy and reliability. Reliability is the probability that the actual settlement would be less than that computed by a specific method. In choosing a method for design, it all comes down to keeping a 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 of God’.”


40

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 Geological Engineering, 21(4): 377-387.

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

Berardi, R., Jamiolkowski, M. & Lancellotta, R. 1991. Settlement of shallow foundations in sands: selection of 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 on Soil 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 and Geoenvironmental Engineering, ASCE, 133(8): 905-920.

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

D’Appolonia, D.J., D’Appolonia, 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 heterogeneity of sand layers in the settlement of bridges. Proceedings, 4th International Conference on Soil Mechanics and 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, 2nd 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. 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 both

isolated and multiple footings in sand. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 134(4): 417-427.

Mayne, P.W. & Poulos, H.G. 1999. Approximate displacement influence factors for elastic shallow foundations. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 125(6): 453-460.

Meyerhof, G.G. 1956. Penetration tests and bearing capacity of cohesionless soils. Journal of the Soil Mechanics and Foundations Division, ASCE, 82(1): 1-19.

Meyerhof, G.G. 1965. Shallow foundations. Journal of the Soil Mechanics and Foundations Division, ASCE, 91(2): 21-31.

Papadopoulos, B.P. 1992. Settlements of shallow foundations on cohesionless soils. Journal of Geotechnical Engineering, ASCE, 118(3): 377-393.

Peck, R.B. & Bazaraa, A.R.S.S. 1969. Discussion of paper by D’Appolonia et al, Journal of the Soil Mechanics and Foundations Division, ASCE, 95(3): 305-309.

Robertson, P.K. & Campanella, R.G. 1983. Interpretation of cone penetration tests: part I: sand. Canadian Geotechnical Journal, 29(4): 718-733.

Salgado, R. 2008. The Engineering of Foundations, McGraw-Hill, New York. Schmertmann, J.H. 1970. Static cone to compute static settlement over sand. Journal of the Soil Mechanics and

Foundations Division, ASCE, 96(3): 1011-1043. Schmertmann, J.H., Hartmann, J.P. & Brown, P.R. 1978. Improved strain influence factor diagrams. Journal of

the Geotechnical Engineering Division, ASCE, 104(8): 1131-1135. Seed, H.B. & DeAlba, P. 1986. Use of SPT and CPT tests for evaluating the liquefaction resistance of sands.

Proceedings, ASCE Specialty Conference of Use of In Situ Testing in Geotechnical Engineering, Geotechnical Special Publication 6, Blacksburg, 281-302.

Sivakugan, N., Eckersley, J.D. & Li, H. 1998. Settlement predictions using neural networks. Australian Civil Engineering Transactions, CE40: 49-52


41

Sivakugan, N. & Johnson, K. 2004. Settlement prediction in granular soils: a probabilistic approach. Geotechnique, 54(7): 499-502.

Skempton, A.W. 1985. Standard penetration test procedures and the effect in sands of overburden pressure, relative density, particle size, aging and overconsolidation. Geotechnique, 36(3): 425-447.

Steinbrenner, W. 1934. Tafeln zur setzungsberschnung. Die Strasse, 1: 121-124. Terzaghi, K. & Peck, R.B. 1948. Soil Mechanics in Engineering Practice, 1st Edition, John Wiley and Sons,

New York. Terzaghi, K. & Peck, R.B. 1967. Soil Mechanics in Engineering Practice, 2nd Edition, John Wiley and Sons,

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