5. dilatometer (dmt) tests - civil engineeringbartlett/cveen6340/dmtests.pdf · 5-1 5. dilatometer...

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5-1 5. DILATOMETER (DMT) TESTS The flat dilatometer test (DMT) was developed in Italy by Silvano Marchetti (1980). It was initially introduced in North America and Europe in 1980 and is currently used in over 40 countries. 5.1. DMT Test Apparatus and Procedure As seen in Figure 5.1, the flat dilatometer is a stainless steel blade having a flat, circular steel membrane mounted flush on one side. The blade is advanced into the ground using push rods. The general layout of the dilatometer test is shown in Fig. 5.2. Figure 5.1 Flat Dilatometer The blade is connected to a control unit by a pneumatic-electrical tube which runs through the insertion rods. A gas tank, connected to the control unit, supplies the gas pressure required to expand the membrane.

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5-1

5. DILATOMETER (DMT) TESTS

The flat dilatometer test (DMT) was developed in Italy by Silvano Marchetti

(1980). It was initially introduced in North America and Europe in 1980 and is currently

used in over 40 countries.

5.1. DMT Test Apparatus and Procedure

As seen in Figure 5.1, the flat dilatometer is a stainless steel blade having a flat,

circular steel membrane mounted flush on one side. The blade is advanced into the

ground using push rods. The general layout of the dilatometer test is shown in Fig. 5.2.

Figure 5.1 Flat Dilatometer

The blade is connected to a control unit by a pneumatic-electrical tube which runs

through the insertion rods. A gas tank, connected to the control unit, supplies the gas

pressure required to expand the membrane.

5-2

1. DMT Blade 2. Push Rods 3. Pneumatic-Electric Cable 4. Control Box 5. Pneumatic Cable 6. Gas Tank 7. Expansion of the membrane

Figure 5.2 General layout of the Flat Dilatometer Test (DMT)

The test started by pushing the dilatometer into the ground. When the test depth is

reached, the membrane is inflated using the control unit and A and B pressure readings

are taken. These raw pressure readings are then corrected by using values of A, B, and

mz . These factors take into account the membrane stiffness and are subsequently

converted into p0, p1 where:

The A-pressure is the pressure required to just initiate movement of the membrane

against the soil ("lift-off pressure").

The B-pressure is the pressure required to move the center of the membrane a

distance of 1.1 mm against the soil.

The A pressure is the vacuum pressure required to keep the membrane in contact

with its seating.

Push Force

5-3

The B pressure is the air pressure required to deflect the membrane 1.1 mm in

air.

mz is the gauge pressure deviation from zero when vented to atmospheric

pressure.

5.2 DMT Data Reduction (Marchetti, 1980)

Corrected oP and 1P readings are calculated from:

105.005.1 PAzAP mo (5.1)

BzBP m 1 (5.2)

From oP and 1P , DMT index parameters are calculated from:

oo

oD uP

PPI

1 (5.3)

'vo

ooD

uPK

(5.4)

oD PPE 17.34 (5.5)

where DI = material index, DK = horizontal stress index, DE = dilatometer modulus,

ou = hydrostatic pore water pressure and 'vo = in-situ effective vertical stress.

The overconsolidation ratio (OCR) is defined as:

56.15.0 DKOCR for 0.2 DI 2.0 (5.6)

The in-situ coefficient of earth pressure at rest oK is defined as;

47.0

5.1

D

o

KK - 0.6 (5.7)

5-4

The undrained shear strength us is calculated from:

25.1' 5.022.0 Dvou Ks for 2.1DI (5.8)

and the vertical drained constrained modulus M is found from:

DM ERM (5.9)

where: If 6.0DI DM KR log36.214.0 (5.9a)

If 0.3DI DM KR log25.0 (5.9b)

If 0.36.0 DI DoMoMM KRRR log5.2 ,, with (5.9c)

6.015.014.0, DoM IR (5.9d)

If 10DK DM KR log18.232.0 (5.9e)

Always 85.0MR (5.9f)

The compression ratio CR can be calculated from:

CRC

eM p

c

op

3.210ln

1 ''

(5.10)

and

MCR p

'3.2 . (5.11)

5.3 DMT Results

For each site, DMT results are presented as diagrams of ,oP 1P and calculated

DMT parameters ,, DD KI and DE versus depth. Results for the South Temple site

(DMT-2) are shown in Figure 5.3. As shown in Figure 5.3, oP and 1P increase linearly

with depth for the upper Bonneville Clay, but for the lower Bonneville Clay, 1P did not

5-5

follow the same trend. Also in the upper Bonneville Clay, oP and 1P values are very

close to each other. (This might be attributed to very small DI values, which is an index

of relative spacing between oP and 1P . DI values ranged from 0.22 to 0.4 for this zone).

The horizontal stress index, DK , is almost constant both for the upper Bonneville Clay

having an average value of 3.667 and for the lower Bonneville Clay the average value is

3.045. The dilatometer modulus, DE , is almost constant for the upper Bonneville Clay,

except for a silty clay layer at the middle of this zone. Values of DE linearly increase

with depth in the lower Bonneville Clay. Values of 1, PPo , ,, DD KI and DE versus depth

relations for the other research sites are found in Appendix G. Table 5.1 also summarized

the average DD KI , and DE values for the Lake Bonneville Clays at the three different

research sites.

Table 5.1 Summary of DMT results (Average values of DDD EKI ,, for Bonneville Clay)

DMT Average

DDD EKI ,, Upper

Bonneville Clay

Lower Bonneville

Clay

DMT-1 N. Temple

DI 0.468 0.249

DK 3.040 3.031

DE (Bar) 44.100 31.770

DMT-2 S. Temple

DI 0.430 0.330

DK 3.667 3.045

DE (Bar) 43.730 57.450

DMT-3 S. Temple

Embankment

DI 0.434 -

DK 1.846 -

DE (Bar) 110.35 -

5-6

DMT-2 S. Temple Po, P1, and sv' vs.

Elevation

1268.00

1270.00

1272.00

1274.00

1276.00

1278.00

1280.00

1282.00

1284.00

1286.00

1288.00

1290.00

0 5 10 15

Po, P1, and sv' (Bar)

Ele

vatio

n (

me

ters

)

Po P1 sv'

DMT-2 S. Temple ID vs.

Elevation

1268.00

1270.00

1272.00

1274.00

1276.00

1278.00

1280.00

1282.00

1284.00

1286.00

1288.00

1290.00

0.00 0.50 1.00 1.50 2.00

ID

Ele

vati

on

(m

eter

s)

ID

DMT-2 S. Temple KD vs.

Elevation

1268.00

1270.00

1272.00

1274.00

1276.00

1278.00

1280.00

1282.00

1284.00

1286.00

1288.00

1290.00

0 20 40

KD (Hor. Stress Index)

Ele

vati

on

(m

eter

s)

KD

DMT-2 S. Temple ED vs.

Elevation

1268.00

1270.00

1272.00

1274.00

1276.00

1278.00

1280.00

1282.00

1284.00

1286.00

1288.00

1290.00

0 100 200 300

ED (DMT Modulus, Bar)

Ele

vati

on

(m

eter

s)

ED

Figure 5.3 DMT-2 Test Results (S. Temple)

5-7

5.3.1 OCR and 'p Correlations

The OCR defined by Marchetti (1980) is given in Equation 5.6. From this, the

pre-consolidation stress can be calculated as:

''vop OCR . (5.12)

A comparison of the calculated OCR and 'p values using Marchetti’s method and from

the consolidation tests (CRS and IL) are shown in Figures 5.4 and 5.5, respectively. As

seen in Figure 5.4, Marchetti’s method underestimates the OCR and 'p values for most

of the consolidation tests for the North and South Temple sites. However, calculated

OCR and 'p values at the South Temple embankment site were close to that found by

the CRS consolidation tests, except for the first three tests.

As we can see from Equation 5.6, Marchetti (1980) proposed a functional form

for determining the OCR that includes DK . Values of DK from the DMT are plotted

against laboratory determined OCR and 'p values in Figure 5.6. As seen, statistical

relations correlating OCR and 'p relations with DK have relatively low 2R values of

0.4581 and 0.5257, respectively.

To improve the predictive performance of the Marchetti equation, additional

regression analyses were carried to find additional variables that might improve the

predicted behavior. Details of these regression analyses were presented in Chapter

5.3.1.a.

5-8

DMT-1 N. Temple OCR vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 1 2 3

OCR

Ele

va

tio

n (

me

ters

)

DMT-1 (Marchetti,1980)CRS TestsIL Tests

DMT-3 S. Temple Embankment

OCR vs. Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 1 2 3

OCR

Ele

va

tio

n (

me

ter)

DMT-3 (Marchetti, 1980)CRS Tests

DMT-2 S. Temple OCR vs. Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 1 2 3

OCR

Ele

va

tio

n (

me

ter)

DMT-2 (Marchetti, 1980)CRS TestsIL Tests

Figure 5.4 Comparison of OCR values with Marchetti’s Method

5-9

DMT-1 N. Temple p' vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 100 200 300 400

p' (kPa)

Ele

va

tio

n (

me

ters

)

DMT-1 (Marchetti, 1980)CRS TestsIL Tests

DMT-2 S. Temple p' vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 100 200 300 400

p' (kPa)

Ele

va

tio

n (

me

ters

)

DMT-2 (Marchetti, 1980)CRS TestsIL Tests

DMT-3 S. Temple Embankment p'

vs. Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 200 400 600

p' (kPa)

Ele

va

tio

n (

me

ters

)

DMT-3 (Marchetti, 1980)CRS

Figure 5.5 Comparison of 'p values with Marchetti’s Method

5-10

KD vs. OCR

y = 0.7956e0.2075x

R2 = 0.4581

0.000

0.500

1.000

1.500

2.000

2.500

0.000 1.000 2.000 3.000 4.000 5.000

KD

OC

R

KD vs. p'

y = 805.9e-0.4609x

R2 = 0.5257

0

100

200

300

400

500

600

0.000 1.000 2.000 3.000 4.000 5.000

KD

p

' (kP

a)

Figure 5.6 (a) Dilatometer KD vs. Laboratory Determined OCR (b) Dilatometer KD vs. Laboratory Determined '

p .

5.3.1.a Development of Multiple Linear Regression Model for Preconsolidation Stress for the DMT

Multiple linear regression analysis (MLR) is a statistical method to estimate the

linear relationship between the dependent variable, which denoted by a symbol y , and

independent variables which are denoted by a symbols nxxx ,......,, 21 . The main objective

of regression analysis is to develop a regression model that will enable us to describe and

predict the dependent variable from the independent variables.

(a)

(b)

5-11

In applying multiple linear regression analysis, the true response, ∩, is expressed

in terms of unknown parameters, sB , that accompany the sx . The sB are partial slopes

or partial derivatives that accompany the sx and are estimated by the regression process.

∩ = nn BBBxxx ,.....,,;,.....,, 2121 (5.13)

For example, as seen in the following figures, the preconsolidation stress is

related to the difference between dilatometer contact stress and hydrostatic pore water

pressure, oo uP , and the difference between dilatometer expansion stress and the

hydrostatic pore water pressure, ouP 1 . These independent variables are measured by

the dilatometer test (DMT) and are related to the total overburden stress, vo .

∩ = voouo

BBBuPuP uPuPvoooo ,,;,,11 (5.14)

where:

voouoBBB uPuP ,,

1 unknown parameters corresponding to ooo uPuP 1, and vo .

For example, as seen from Figures 5.7a, b and c, the simple linear regression

models given in Equation 5.14 have better 2R values than the Marchetti’s correlation, as

seen in Figure 5.6.b for the preconsolidation stress of the Bonneville Clay.

In reality, the true response is seldom completely predicted (i.e., 12 R ) due to

the presence of other uncontrolled and unmeasured factors that affect the true response.

The deviation of the observed response, y , from ∩ is called experimental error, . The

experimental error results from measurement error and other factors which prohibit the

exact measurement of ∩.

5-12

y = 84.955e0.002x

R2 = 0.8444

0.00

100.00

200.00

300.00

400.00

500.00

600.00

0 200 400 600 800 1000

Po-uo (kPa)

p

' (kP

a) (

fro

m C

RS

an

d I

L T

ests

)

y = 85.354e0.0015x

R2 = 0.8718

0.00

100.00

200.00

300.00

400.00

500.00

600.00

0 200 400 600 800 1000 1200 1400

P1-uo (kPa)

p

' (kP

a) (

fro

m C

RS

an

d I

L t

ests

)

y = 81.378e0.0033x

R2 = 0.9257

0.00

100.00

200.00

300.00

400.00

500.00

600.00

0.00 100.00 200.00 300.00 400.00 500.00 600.00

vo (kPa)

p

' (kP

a) (

fro

m C

RS

an

d I

L T

ests

)

Figure 5.7 (a) Dilatometer oo uP vs. Laboratory Determined 'p (b) Dilatometer

ouP 1 vs. Laboratory Determined 'p (c) Total overburden stress vs. '

p

(a)

(b)

(c)

5-13

∩ - y = (5.15)

In multiple linear regression, the values of y and the sx are often transformed

(e.g., x1 , xlog , xe , etc.) in order to produce a linear form.

exbxbxbby nno .....2211 (5.16)

A linear form is required in order to perform the regression or best-fit estimate. The fitted

regression coefficients, no bbb ,....,, 1 are best-fit estimates of the sB , and e is a best-fit

estimate of . Linear regression uses the method of least squares to estimate the sB by

minimizing the error sum of squares, eS :

2eSe (5.17)

where: e is the difference between the measured response, y , and the response predicted by the

regression equation, y , i.e.,

yye ˆ . (5.18)

As shown in Figure 5.8, standardized residual plots are commonly used to evaluate the

validity of the linear regression model’s assumptions. The standardized residual, se , for

each observation is calculated from:

se = e /(standard deviation of e ). (5.19)

The performance of regression models is judged by the coefficient of determination, 2R

n

yy

n

ySy

Re

22

22

2 (5.20)

5-14

where n is the sample size. The value of 2R ranges from 0 to 1 and measures the

proportion of the variability of y being explained by the sx .

Figure 5.8 Examples of Standard Residual Plots. (a) Satisfactory residual plot gives overall impression of horizontal box centered on zero line. (b) A plot showing non-constant variance. (c) A plot showing a linear trend suggesting that the residuals are not independent and that another variable is needed in the model. (d) A plot illustrating the need for a transformation or a higher order term to alleviate curvature in the residuals.

Draper and Smith (1981) proposed two balancing criteria of selecting the

parameters include in the regression equation:

(1) To make the equation useful for predictive purposes, we want the model to

include as many as x ’s possible so that reliable prediction can be made.

5-15

(2) Because of the costs involved in obtaining information on a large number of

x ’s, and subsequently monitoring them, we want the equation to be as efficient as

possible by including as few of the x ’s as needed.

For these analyses, a stepwise regression procedure, which is improved version of

the forward selection procedure, was used. This procedure starts with searching the set of

possible independent x variables for the variable most highly correlated with y . In the

next step, the next most highly correlated variable enters into the regression. The process

of adding new variables continues till no additional variables can be found that

significantly improve the coefficient of determination, 2R of the linear regression model.

Also, at each step, all x variables in the current model are re-examined to verify that they

are still contributing to the model. Sometimes variable introduced at earlier steps no

longer are statistically significant as new variables are introduced into the model. This

sometimes happens due to the cross-correlation between the x variables.

As given in Equation 5.6, Marchetti (1980) proposed a functional form for OCR

that includes the horizontal stress index, DK :

1BDo KBOCR . (5.21)

This functional form was tested and the results are presented in Figures 5.6.a and b. As

seen from Figures 5.7a, b and c, values of ooo uPuP 1, and vo show good

correlation with the laboratory-determined preconsolidation stresses. Thus, a MLR model

was set up for 'p by dividing those factors correlated with '

p into seven different

models, which can be seen in Table 5.2. For an application standpoint, a regression model

5-16

should not depend on the stress units, so all variables were divided by aP (1 aP = 101.325

kPa = 1.01325 Bar) which is atmospheric pressure, to make the variables dimensionless.

Table 5.2 Data variables sets for preconsolidation pressure

Data Set Variables in Equation 2R

A

a

o

a

p

P

uPfPy 1

' %0247.882 R

B

a

oo

a

p

P

uPfPy

' %603.832 R

C

a

vo

a

p

PfPy '

%894.852 R

D

a

oo

a

o

a

p

P

uP

P

uPfPy ,1

' %006.892 R

E

a

vo

a

o

a

p

PP

uPfPy

,1

'

%221.892 R

F

a

vo

a

oo

a

p

PP

uPfPy

,

'

%554.882 R

G

a

vo

a

oo

a

o

a

p

PP

uP

P

uPfPy

,,1

'

%197.872 R

All the above regression analyses were carried out using the MLR statistical options in

Microsoft EXCEL. It was observed that model E, which has a

vo

a

o

Px

P

uPx

2

11 , as

independent variables, gave the highest 2R value. This model has the general form:

2121 xxy o (5.22)

5-17

This can be expressed in a linear form for multiple regression using:

2211 loglogloglog xxy o (5.23)

Table 5.3 gives the regression summary of the Equation 5.23, which includes the

logarithmic transformation of ,'

a

p

P

a

o

P

uP 1 and a

vo

P

.

Table 5.3 Linear regression output using log ofa

o

P

uP 1 as 1x anda

vo

P

as 2x and

y = log of a

p

P

'

Regression Statistics Multiple R 0.94756886 R Square 0.89788675 Adjusted R Square 0.89221379 Standard Error 0.06558324 Observations 39 ANOVA

df SS MS F Significance F Regression 2 1.3615315 0.6807658 158.27487 1.45706E-18Residual 36 0.1548418 0.0043012 Total 38 1.5163733

Coefficients Standard

Error t Stat P-value Lower 95%

Intercept -0.2771291 0.0730006-

3.7962598 0.00054389 -0.425180867X Variable 1 0.60937626 0.172906 3.5243205 0.00117624 0.258707039X Variable 2 0.35232128 0.1558951 2.2599887 0.02996872 0.036151682

From the above model and regression output, the linear regression can be back

transformed to:

3523.06094.0

1

'

5283.0

a

vo

a

o

a

p

PP

uP

P

(5.24)

5-18

Regression models were also attempted using total overburden stress instead of 1

atmospheric pressure in the denominator of the Equation (5.24). The model has the form:

vo

oo

vo

p uP

11

'

logloglog (5.25)

The 2R value of the regression analysis of Equation 5.25 was only 5.57% which

is considerably lower than 89.22% for Equation 5.24. Thus this model is not

recommended because if its poor predictive performance. Also, as seen from Figures

5.9a, b and c, laboratory OCR values plotted against ooo uPuP 1, and vo yielded

lower 2R values than the preconsolidation stress correlations shown in Figures 5.7a, b

and c. Therefore the model given in Equation 5.24 was chosen as the final model for

prediction of preconsolidation pressure for the Bonneville Clay.

y = 2.0215e-0.0008x

R2 = 0.5787

0.00

0.50

1.00

1.50

2.00

2.50

0 200 400 600 800 1000

Po-uo (kPa)

OC

R (

La

bo

rato

ry)

(a)

5-19

y = 2.0584e-0.0006x

R2 = 0.6689

0.00

0.50

1.00

1.50

2.00

2.50

0 200 400 600 800 1000 1200 1400

P1-uo (kPa)

OC

R (

Lab

ora

tory

)

y = 2.1047e-0.0014x

R2 = 0.721

0.00

0.50

1.00

1.50

2.00

2.50

0.00 100.00 200.00 300.00 400.00 500.00 600.00

vo' (kPa)

OC

R (

La

bo

rato

ry)

Figure 5.9(a) Dilatometer oo uP vs. Laboratory Determined OCR (b) Dilatometer

ouP 1 vs. Laboratory Determined OCR (c) Total overburden stress vs. OCR

Residual plot for Equation 5.24 can be seen in Figure 5.10. As it illustrated in Figure 5.8,

an acceptable residual plot gives an overall impression of horizontal box with the data

centered on the zero line. This type of plot suggests an acceptable residual behavior.

Figure 5.10 shows a similar behavior, thus equation 5.24 appears to be satisfactory

model.

(c)

(b)

5-20

-80.0

-60.0

-40.0

-20.0

0.0

20.0

40.0

60.0

80.0

1268.0 1270.0 1272.0 1274.0 1276.0 1278.0 1280.0 1282.0 1284.0

Elevation (meters)

e

p' C

RS

-

p' D

MT (

kPa)

Figure 5.10 Residual plot of Equation 5.25

A comparison of the preconsolidation pressure predicted from Equation 5.24 with

that of Marchetti’s model and the laboratory results can be seen in Figure 5.11. Equation

5.24 shows a better prediction of the laboratory values than Marchetti’s (1980) model for

Bonneville Clay. Thus, Equation 5.24 is recommending for use for these deposits.

5.3.2 Compression Ratio (CR) and Constrained Modulus (M) Correlations

The constrained modulus, M, defined by Marchetti (1980) for the DMT is given

in Equations 5.9a, b, c, d, e, and f. From this, Equations 5.10 and 5.11 can be used to

calculate the CR. Comparison of the calculated CR values from DMT results, using the

method proposed by Marchetti (1980), with the laboratory CR values can be seen in

Figure 5.12. As seen in this figure, Marchetti’s model considerably underestimates the

CR values for the Bonneville Clay.

5-21

DMT-1 N. Temple p' vs.

Elavation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 100 200 300

p' (kPa)

Ele

vati

on

(m

eter

s)

Marchetti, 1980CRS TestsIL TestsEq. 5.24

DMT-2 S. Temple p' vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 100 200 300 400

p' (kPa)

Ele

vati

on

(m

eter

s)

Marchetti, 1980CRS TestsIL TestsEq. 5.24

DMT-3 S. Temple Embankment p'

vs. Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 200 400 600

p' (kPa)

Ele

vati

on

(m

eter

s)

Marchetti, 1980CRSEq. 5.24

Figure 5.11 Comparison of Preconsolidation Stress

5-22

As seen from Equations 5.9 and 5.10, Marchetti also proposed a model to

determine CR from DK . The dilatometer DK results plotted against laboratory

determined CR values are shown in Figure 5.13. As seen in this figure, the correlation

between laboratory CR values and DK values is very low ( 2R =5.29%). This is also

explains why Marchetti’s model does not agree with the laboratory determined CR

values, as shown in Figure 5.12.

To improve the predictive performance, additional regression analyses were

carried to find out if other variables might improve the predicted behavior of the model.

As shown in Figures 5.14a, b and c, laboratory determined CR values are plotted against

oo uPuP 10 , and vo . With these newly included variables, the 2R values

improved, but they are still relatively low (i.e., around 20%).

As given in Equation 5.10, one can also back-calculate CR values from the

constrained modulus, M. Because very low 2R values were obtained for the CR

correlations, we decided to investigate possible correlations between the DMT and

laboratory determined M values. As seen in Figures 5.15a, b and c, laboratory determined

M values plotted against values of oo uPuP 10 , and vo produced better correlation.

The 2R values significantly improved to about 77 to 84 percent.

5-23

DMT-1 N. Temple CR vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 0.2 0.4 0.6

CR

Ele

va

tio

n (

me

ters

)

Marchetti, 1980CRS TestsIL Tests

DMT-2 S. Temple CR vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 0.2 0.4 0.6

CR

Ele

va

tio

ns

(m

ete

rs)

Marchetti, 1980CRS TestsIL Tests

DMT-3 S. Temple

Embankment CR vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 0.2 0.4 0.6

CR

Ele

va

tio

ns

(m

ete

rs)

Marchetti, 1980CRS Tests

Figure 5.12 Comparison of the laboratory CR values with Marchetti (1980)

5-24

y = 0.0298x + 0.2141

R2 = 0.0529

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.000 1.000 2.000 3.000 4.000 5.000

KD

CR

(la

bo

rato

ry)

Figure 5.13 DK vs. CR

Details of the multiple linear regression model were explained in Chapter 5.3.1.a.

Here using the same procedures, additional models were developed to predict M values

based on the DMT’s oo uPuP 10 , and the total vertical overburden stress, vo . As

done for the preconsolidation pressure in the previous chapter, independent variables

were divided into seven different models and regression analyses were carried out.

Potential MLR models for the constrained modulus are given in Table 5.4.

5-25

y = 1.7191x-0.2963

R2 = 0.2021

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0 200 400 600 800 1000

Po-uo (kPa)

CR

y = 1.8121x-0.2909

R2 = 0.1939

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0 200 400 600 800 1000 1200 1400

P1-uo (kPa)

CR

y = 1.5597x-0.3055

R2 = 0.263

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.00 100.00 200.00 300.00 400.00 500.00 600.00

vo (kPa)

CR

Figure 5.14 (a) Dilatometer oo uP vs. Laboratory Determined CR(b) Dilatometer

ouP 1 vs. Laboratory Determined CR (c) Total overburden stress vs. CR

(a)

(c)

(b)

5-26

y = 0.8904x1.2495

R2 = 0.7723

0.0

1000.0

2000.0

3000.0

4000.0

5000.0

6000.0

0 200 400 600 800 1000

Po-uo (kPa)

M (

kPa)

y = 0.5412x1.2704

R2 = 0.7948

0.0

1000.0

2000.0

3000.0

4000.0

5000.0

6000.0

0 200 400 600 800 1000 1200 1400

(P1-u0)

M (

kPa)

y = 2.4657x1.1782

R2 = 0.8409

0.0

1000.0

2000.0

3000.0

4000.0

5000.0

6000.0

0.00 100.00 200.00 300.00 400.00 500.00 600.00

vo (kPa)

M (

kPa)

Figure 5.15 (a) Dilatometer oo uP vs. Laboratory Determined M (b) Dilatometer

ouP 1 vs. Laboratory Determined M (c) Total overburden stress vs. M

(a)

(b)

(c)

5-27

Table 5.4 Data variables sets for constrained modulus

Data Set Variables in Equation 2R

A

a

o

a P

uPfP

My 1 %925.782 R

B

a

oo

a P

uPfP

My %616.762 R

C

a

vo

a PfP

My

%662.832 R

D

a

oo

a

o

a P

uP

P

uPfP

My ,1 %167.802 R

E

a

vo

a

o

a PP

uPfP

My

,1 %751.832 R

F

a

vo

a

oo

a PP

uPfP

My

, %348.842 R

G

a

vo

a

oo

a

o

a PP

uP

P

uPfP

My

,,1 %946.832 R

It was observed that model F, which has a

vo

a

oo

Px

P

uPx

21 , as independent

variables, gave the highest 2R value. From the MLR analysis of model F, it was

experienced that independent variable 1x is not significant in the model (P-value of 1x is

11.42%). The same problem encountered in models D, E and G. Independent variable 2x

in models D and E has a high P-value of 7.69% thus it is not very significant in the

model. Independent variables 1x and 2x in model G have also high P-value of 75.47%

5-28

and 23.88%, respectively. Model C has total overburden pressure as an independent

variable, has the highest 2R value after models D, E, F, and G. Constrained modulus, M,

is the modulus at the preconsolidation stress level (Equation 5.10). As shown in Figure

3.18, OCR values at the research sites have relatively constant behavior over the depth.

Another word, since the total over burden stress increasing with depth, preconsolidation

pressure is also increasing proportional to the total stress. Since the constrained modulus

is the modulus at the preconsolidation stress level, it was expected to get high correlation

coefficient in model C depends on the total stress. Model C has the general form:

11 xy o . (5.26)

This can be expressed in a linear form for multiple linear regression using:

11 logloglog xy o . (5.27)

Table 5.5 gives the regression summary of the Equation 5.27, which includes the

logarithmic transformation of aP

Mand

a

vo

P

.

5-29

Table 5.5 Linear regression output using log of a

vo

P

as 1x ; y = log of

aP

M

Regression Statistics Multiple R 0.917017381 R Square 0.840920877 Adjusted R Square 0.836621441 Standard Error 0.110408532 Observations 39 ANOVA

df SS MS F Significance

F Regression 1 2.384234353 2.384234353 195.5886599 2.40043E-16Residual 37 0.451031625 0.012190044 Total 38 2.835265977

Coefficients Standard

Error t Stat P-value Lower 95% Intercept 0.749140953 0.037680796 19.88123991 2.41924E-21 0.672792482X Variable 1 1.178158546 0.084242627 13.98530157 2.40043E-16 1.007466937

From the above model and the regression output, the linear model can be back

transformed to:

17816.1

6123.5

a

vo

a PP

M (5.28)

Residual plot of the Equation 5.28 can be seen in Figure 5. 16. Residual plot for

M values give an overall impression of a horizontal box centered on zero line, so the

model is deemed satisfactory.

Equation 5.28 does not use any DMT result; it is just depend on the total

overburden stress. As an alternative to Equation 5.28 model A from Table 5.4 which has

a

o

P

uPx

1

1 as independent variable analyzed in order to develop a relationship between

constrained modulus at the preconsolidation level and DMT results.

5-30

-800.0

-600.0

-400.0

-200.0

0.0

200.0

400.0

600.0

800.0

1268.0 1270.0 1272.0 1274.0 1276.0 1278.0 1280.0 1282.0 1284.0

Elevation (meters)

e=M

lab

-MD

MT (

kPa)

Figure 5.16 Residual plot of Equation 5.28

Model A has the same general form with model C:

11 xy o . (5.29)

This can be expressed in a linear form for multiple linear regression using:

11 logloglog xy o . (5.30)

Table 5.5gives the regression summary of the Equation 5.30, which includes the

logarithmic transformation of aP

Mand

a

o

P

uP 1 .

5-31

Table 5.6 Linear regression output using log of a

o

P

uP 1 as 1x ; y = log of aP

M

Regression Statistics Multiple R 0.891516885 R Square 0.794802357 Adjusted R Square 0.789256475 Standard Error 0.125395579 Observations 39 ANOVA

df SS MS F Significance

F Regression 1 2.253476082 2.253476082 143.313962 2.73611E-14Residual 37 0.581789896 0.015724051 Total 38 2.835265977

Coefficients Standard

Error t Stat P-value Lower 95% Intercept 0.27550499 0.080967 3.402682426 0.001615999 0.111450423X Variable 1 1.270378933 0.106117994 11.97138096 2.73611E-14 1.055363663

From the above model and the regression output, the linear model can be back

transformed to:

27037.1

18858.1

a

o

a P

uP

P

M (5.31)

Residual plot of the Equation 5.31 can be seen in Figure 5. 17. Residual plot for

M values give an overall impression of a horizontal box centered on zero line, so the

model is deemed satisfactory.

5-32

-800.0

-600.0

-400.0

-200.0

0.0

200.0

400.0

600.0

800.0

1268.0 1270.0 1272.0 1274.0 1276.0 1278.0 1280.0 1282.0 1284.0

Elevation (meters)

e=M

lab

-MD

MT (

kPa)

Figure 5.17 Residual plot of Equation 5.31

One can also back-calculate CR values from M, using the definition of M from Equation

5.10:

31.5.

3.2 '

fromEqMCR vo

DMT

(5.32)

Comparison of M from Equations 5.28 and 5.31 the back-calculated CR from Equation

5.32 with the laboratory results is shown in Figures 5.18 and 5.19, respectively. Note that

Equation 5.32 is used to back-calculate CR from M for overconsolidated clays.

However, if the clay is normally consolidated, then '

ov should be substituted into 'p in

Equation 5.32.

5-33

N. Temple DMT1 CR vs

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 0.5 1

CR

Ele

va

tio

n (

me

ters

)

Eq.5.31 CRSIL Eq.5.28

S. Temple DMT2 CR vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 1 2

CR

Ele

va

tio

n

Eq.5.31 CRSIL Eq.5.28

S. Temple Embankment DMT3 CR vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 0.2 0.4

CR

Ele

va

tio

n (

me

ters

)

Eq. 5.31 CRS Eq.5.28

Figure 5.18 Comparison of the Compression Ratio

5-34

N. Temple DMT1 M vs.

Elevation

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 1000 2000 3000

M (kPa)

Ele

va

tio

n (

me

ters

)

Eq.5.31 CRS IL Eq.5.28

S. Temple DMT2 M vs.

Elevation

1266

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 1000 2000 3000

M (kPa)

Ele

va

tio

n (

me

ters

)

Eq.5.31 CRSIL Eq.5.28

S. Temp. Embankment DMT3 M vs.

Elevation

1266

1268

1270

1272

1274

1276

1278

1280

1282

1284

0 5000 10000

M (kPa)

Ele

va

tio

n (

me

ters

)

Eq.5.31 CRS Eq.5.28

Figure 5.19 Comparison of the Constrained Modulus

5-35

As seen from Figures 5.18 and 5.19, calculated values of M from Equations 5.28

and 5.31 and back-calculated CR values from Equation 5.32 closely approximate

laboratory values. The residual plots for CR (Figure 5.20 and 21) also show a horizontal

box centered around the zero center line, thus these models has desirable statistical

qualities. We thus recommend that Equation 5.28, 5.31 and 5.32 be used to determine the

compressibility of Bonneville Clay from the DMT results.

-0.1500

-0.1000

-0.0500

0.0000

0.0500

0.1000

0.1500

1268.0 1270.0 1272.0 1274.0 1276.0 1278.0 1280.0 1282.0 1284.0

Elevation (meters)

e=C

RC

RS-

CR

DM

T

Figure 5.20 Residual plot of Equation 5.32 using Equation 5.28

-0.1500

-0.1000

-0.0500

0.0000

0.0500

0.1000

0.1500

1268.0 1270.0 1272.0 1274.0 1276.0 1278.0 1280.0 1282.0 1284.0

Elevation (meters)

e=C

RC

RS-

CR

DM

T

Figure 5.21 Residual plot of Equation 5.32 using Equation 5.32

5-1