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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