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Appendix B Determination of Hyperbolic Parameter Values for Soils B1

Appendix BDetermination of HyperbolicParameter Values of Soils

This appendix describes the procedure suggested by Duncan et al. (1980)1

for the determination of hyperbolic parameter values for soils from results oftriaxial tests. Hyperbolic parameter values of Density Sand and Light CastleSand are determined based on the results of the triaxial tests presented inAppendix A. Figures B1 through B16 show the determination of the hyperbolicparameter values for all the sands tested. Figures B1 through B4, correspondingto the determination of parameter values for medium-dense Density Sand, areused in the following description to illustrate the procedure followed. Examplecalculations of hyperbolic parameter values for dense Light Castle Sand arepresented in the last section of this appendix.

B.1 Transformed Plots

The procedure for the determination of hyperbolic parameter values ofmedium-dense Density Sand is illustrated in Figures B1 through B4. The triaxialtest stress-strain data shown in Figure A1 of Appendix A is represented in atransformed plot in Figure B1. In this transformed diagram, the value of axialstrain2 ε measured during the test is divided by the corresponding value ofdeviator stress σ1-σ3 and plotted against the axial strain. If the stress-strainrelationship measured during the triaxial test is hyperbolic, the transformeddiagram is a straight line. The intercept a of this straight line on the ε/(σ1 - σ3)axis is the reciprocal of the initial Young's modulus Ei of the soil specimen. Theslope b of the line is the reciprocal of the asymptotic deviator stress (σ1 - σ3)ult.

The stress-strain relationship of a soil usually differs from a hyperbola.Duncan et al. (1980) indicated that the values of parameters a and b can bedetermined from a straight line passing through the points in the transformed plotthat correspond to 70 and 95 percent of the strength.

The transformed plot in Figure B1 shows the 70 and 95 percent strength datapoints for each of the tests performed on medium-dense Density Sand. Straight

1 References cited in this Appendix are included in the References at the end of the main text2 For convenience, symbols are listed and defined in the Notation (Appendix F)


lines are drawn through each pair of these points. For comparison, the completeset of transformed data from the tests is also shown in the figure. It is seen thatthe lines drawn through the 70 and 95 percent data points match closely thetransformed data sets. Such a comparison may be useful for minimizing errors inthe determination of hyperbolic parameter values that can arise frominconsistencies in the data. Transformation of the entire set of test data is easilyachieved in an electronic spreadsheet. The use of a spreadsheet also facilitatesmodification of the values of parameters a and b to obtain the best possible fit tothe data.

B.2 Hyperbolic Parameter Values

B.2.1 Determination of K and n

The values of the parameters a and b determined from the transformed plotsare presented in the table included in Figure B1. The values of initial Young'smodulus and asymptotic deviator stress are determined using the followingequations:

(B1)

(B2)

The values of Ei and (σ1 - σ3)ult for each of the tests performed on medium-dense Density Sand are presented in the table in Figure B1. It can be seen thatthe values of initial Young's modulus Ei increase with increasing confiningpressure σ3'. Janbu (1963) suggested the following relationship between theinitial Young's modulus and confining pressure:

(B3)

where

K = modulus number

pa = atmospheric pressure

Ei a=

1

( )σ σ1 31

− =ult b

Ei K pa pa

n= ⋅ ⋅

σ'3


n = modulus exponent

Equation B3 can be used for both undrained and drained compression. Forundrained compression, the value of σ3 is used instead of σ'3. This relationshipimplies that there is a linear relationship between the logarithm of the initialYoung's modulus and the logarithm of the confining pressure. Figure B2 is alogarithmic diagram showing the values of normalized Young's modulus, Ei/pa

determined from the table in Figure B1, represented against the values ofnormalized confining stress σ3'/pa. A best-fit straight line is drawn through thedata points. The value of the modulus number K is equal to the value ofnormalized Young's modulus given by this best-fit line for a confining stress ofone atmosphere. The slope of the line is the modulus exponent n.

B.2.2 Determination of Rf

The table in Figure B1 shows the values of deviator stress at failure (σ1-σ3)f

determined from the stress-strain plots of the tests, which are presented in FigureA1 of Appendix A. It can be seen in the table that the values of (σ1 - σ3)ult arelarger than the values of (σ1 - σ3)f in all the tests. The value of the failure ratio Rf

for each of the tests is determined from the following expression:

(B4)

The table contains the values of Rf determined for each of the tests. Formodeling, an average value of Rf is determined from the test results as shown atthe bottom of the table. Typical values of Rf range between 0.5 and 0.9 for mostsoils (Duncan et al. 1980).

B.2.3 Determination of Kb and m

In the hyperbolic model, it is assumed that the value of bulk modulus B isindependent of stress level and dependent of confining pressure (Duncan et al.1980). The following expression is used for the calculation of the bulk modulus:

(B5)

where

( )( )Rf

f

ult=

−

−

σ σ

σ σ1 3

1 3

( )B

v=

−⋅

σ σε

1 33


(σ1-σ3) = deviator stress

εv = volumetric strain

In reality, application of deviator stress during a triaxial test induces volumechanges in the soil specimen. Consequently, the value of bulk modulusdetermined from triaxial test data depends on which points on the stress-strainand volumetric strain-axial strain curves are selected for the calculation. Thefollowing criteria are used for the selection of points in the volumetric strain-axial strain data (Duncan et al. 1980):

a. If the volumetric strain-axial strain data plot does not reach a horizontaltangent (zero volume change) before mobilization of 70 percent of thestrength, the points on the stress-strain and volumetric strain-axial straincurves corresponding to a stress level of 70 percent are used for bulkmodulus determination.

b. If the volumetric strain-axial strain curve reaches a horizontal tangentbefore mobilization of 70 percent of the strength, the point where thevolumetric strain-axial strain curve becomes horizontal and thecorresponding point on the stress-strain curve are used for bulk modulusdetermination.

For Density Sand and Light Castle Sand, it was found that dilation wassignificant and that the volumetric strain-axial strain curve presented a horizontaltangent before mobilization of 70 percent of the strength. Therefore, the secondcriterion was used for the determination of the bulk modulus B for all thespecimens tested, as shown in Tables B1 through B4.

Table B1Determination of Bulk Modulus for MediumDense Density Sand

ConfiningStress σσσσ3'kPa

DeviatorStress(σσσσ1 - σσσσ3)kPa

VolumetricStrain εεεεv

%Bulk Modulus B 1

kPa

45 78.6 0.073 36146103 133.7 0.082 54107280 388.5 0.158 82023

1 (((( ))))

Bv

====−−−−σσσσ σσσσεεεε

1 3


Table B2Determination of Bulk Modulus for DenseDensity Sand




%Bulk Modulus B 1

kPa

45 120.8 0.071 56471103 184.0 0.067 91446280 684.2 0.140 163358

1 (((( ))))

Bv


1 3

Table B3Determination of Bulk Modulus for MediumDense Light Castle Sand




%Bulk Modulus B 1

kPa

45 72.8 0.170 14277103 210.7 0.210 33451280 555.9 0.320 57901

1 (((( ))))

Bv


1 3

Table B4Determination of Bulk Modulus for DenseLight Castle Sand




%Bulk Modulus B 1

kPa

45 73.8 0.047 52340103 199.2 0.109 60917280 548.3 0.190 96192

1 (((( ))))

Bv


1 3

It can be seen in Tables B1 through B4 that the values of bulk modulusincrease with confining pressure. The relationship between bulk modulus andconfining pressure is approximated by the following relationship:

(B6)B Kb pa pa

m= ⋅ ⋅

σ'3


where

Kb = bulk modulus number

m = bulk modulus exponent

The values of Kb and m are determined in a logarithmic diagram ofnormalized bulk modulus B/pa versus normalized confining stress σ3'/pa. Thevalues presented in Table B1 for the medium-dense Density Sand were used tocreate the B/pa versus σ3'/pa diagram of Figure B3. A best-fit straight line wasdrawn through the data points. The value of bulk modulus number Kb is equal tothe value of normalized bulk modulus given by this best-fit line for a confiningstress of 1 atm. The slope of the line is the bulk modulus exponent m.

B.3 Comparison of Model to Test Data

Once the hyperbolic parameter values are determined, it is necessary tocompare the model response to the test data. The stress-strain response from themodel is calculated using the following expression:

(B7)

The deviator stress at failure (σ1-σ3)f is calculated from the followingexpression:

(B8)

( )( )

σ σε

σ

εσ σ

1 3 1

1 33

− =

⋅ ⋅

+ ⋅−

K pa pa

nR f

f'

( ) ( )( )σ σ

σ φφ1 3 2 3

1− =

⋅−f' sin

sin


The volumetric strain-axial strain response is calculated using the followingexpression:

(B9)

The stress-strain and volumetric strain-axial strain responses of the sandstested were calculated using Equations B7, B8, and B9, and the hyperbolicparameters determined following the procedure described previously. FiguresB4, B8, B12, and B16 compare the test data and the calculated hyperbolicresponse. In the figures, the stress-strain hyperbolas are interrupted at the valueof deviator stress at failure (σ1-σ3)f. A horizontal stress-strain relationship, i.e.,zero Young's modulus, is used to model the response of the soil at failure.

It can be seen that the hyperbolic model provides an accurate approximationto the stress-strain response measured during the tests. The volumetric strain-axial strain response calculated using the hyperbolic model also provides a goodapproximation to the test data for the initial stages of shear, in whichcompression takes place. It does not model subsequent dilation of the soils.

B.4 Example Calculations of HyperbolicParameter Values

This section presents an example of the determination of hyperbolicparameter values. The data from the CD triaxial tests performed on densespecimens of Light Castle Sand are used for this example. The example followsthe procedure described by Duncan et al. (1980).

The first step in the determination of hyperbolic parameter values ischecking for inconsistencies in the data from the CD triaxial tests. Figure B17shows the results of the tests performed on dense Light Castle Sand. The datashown in the figure are identical to those shown in Figures A4 and B16. Closeexamination of Figure B17 reveals that the data present some minorinconsistencies. To minimize these inconsistencies, a smooth response of the soilto triaxial testing was assumed for the determination of the hyperbolic parametervalues. This assumed response corresponds to the solid lines in Figure B17.

The next step is the determination of the deviator stress at failure (σ1-σ3)f foreach confining stress. The values of (σ1-σ3)f can be determined from the deviator

( )ε

σ σ

σv

Kb pa pa

m=

−

⋅ ⋅ ⋅

1 3

3 3'


stress-axial strain plots of the tests. Column (2) in the table presented in FigureB18 contains the values of deviator stress at failure determined from Figure B17.The values of deviator stress corresponding to 70 and 95 percent of (σ1-σ3)f arecalculated as shown in columns (3) and (6), respectively.

The values of axial strain corresponding to 70 and 95 percent of the strengthare determined from the deviator stress-axial strain plots. Columns (4) and (7) inFigure B18 contain the strain values determined as shown in Figure B17.

The values in columns (2), (3), (4), (6), and (7) are the basis for thedetermination of the values of initial Young's modulus Ei and failure ratio Rf.The sequence of calculations leading to the determination of the values of Ei andRf is shown in Figure B18, and corresponds to the procedure presented in theprevious sections.

It must be noted that in the method presented in Figure B18 the data are notplotted in transformed coordinates. Only the two data points corresponding to 70and 95 percent of the strength are transformed as shown in columns (5) and (8).Although not strictly necessary, it is recommended to plot always the completedata set in transformed coordinates. The transformed plots are useful to checkthe data for inconsistencies and for verifying the values of Young's modulus andfailure ratio determined from the procedure presented in Figure B18.Transforming the data following the procedure described previously in thisappendix can be accomplished easily with electronic spreadsheets.

The value of failure ratio Rf to be used for modeling is the average of thevalues determined in Figure B18. The values of K and n are determined byplotting the normalized values of initial Young's modulus against the normalizedconfining stress in logarithmic scale as shown in Figure B19.

Figures B17 and B18 also illustrate the determination of the values of bulkmodulus B from the triaxial test data. If the volumetric strain-axial strain plotdoes not reach a horizontal tangent before mobilization of 70 percent of thestrain, the volumetric strain corresponding to 70 percent of the strength is usedfor the determination of B. If the volumetric strain-axial strain plot reaches ahorizontal tangent before mobilization of 70 percent of the strength, themaximum value of volumetric strain is used for the determination of B.

The volumetric strain-axial strain plots of dense Light Castle Sand shown inFigure B17 reach a horizontal tangent before mobilization of 70 percent of thestrength. The maximum values of volumetric strain are determined as shown inthe figure. They are copied to column (10) of the table in Figure B18. Thedeviator stress corresponding to the point of maximum volumetric strain is alsodetermined from the figure and copied to column (9) in Figure B18. These twovalues are used to determine the value of B for each of the confining stresses


applied. The values of Kb and m are determined from a logarithmic plot ofnormalized bulk modulus versus normalized confining stress, as shown in FigureB20.

It must be noted that none of the values presented in Figure B18 wasdetermined graphically. They were obtained directly or by interpolation of datain an electronic spreadsheet. Although graphical determination of values ofstress and strain from a figure such as B17 may provide less significant decimalplaces than their numerical determination, the overall precision of the values ofthe hyperbolic parameters is similar using both procedures. The use of anelectronic spreadsheet is recommended, not for increased precision, but for easein the calculations and verification of the results.

Figure B1. Transformed stress-strain plots from triaxial tests data on medium dense Density Sand,

and determination of hyperbolic parameter values

Axial strain, ε

0.000 0.005 0.010 0.015 0.020 0.025

ε / (σ

1 -

σ 3)

(kP

a-1

)

0.0x100

20.0x10-6

40.0x10-6

60.0x10-6

80.0x10-6

100.0x10-6

120.0x10-6

140.0x10-6

160.0x10-6

σ3' = 45 kPa

σ3' = 280 kPa

σ3' = 103 kPa

70 and 95% stress level data points

σ3'

(kPa)a

(kPa)-1Ei

(kPa)

(σ1 - σ3)ult

(kPa)Rf

45

103

280

2.54 x 10-5

9.02 x 10-6

7.84 x 10-6

39407

110847

127560

7.09 x 10-3

2.83 x 10-3

1.24 x 10-3

141.1

353.7

808.9

120.4

306.5

675.3

0.853

0.866

0.835

b(kPa)-1

(σ1 - σ3)f

(kPa)

Average Rf = 0.852

σ'3 / pa

0.1 1

0.1 1E

i / p

a

100

1000

10000

100

1000

10000

K = 777 n = 0.62

Figure B2. Determination of hyperbolic parameters K and n for medium dense Density Sand

σ'3 / pa

0.1 1

0.1 1B

/ p

a

100

1000

10000

100

1000

10000

Kb = 528

m = 0.42

Figure B3. Determination of hyperbolic parameters Kb and m for medium dense Density Sand

Axial strain, ε

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750

De

via

tor

stre

ss, (σ

1-σ

3)

(kP

a)

0

250

500

750

1000

1250

1500

0

250

500

750

1000

1250

1500

a. Stress-strain model response

Figure B4. Hyperbolic model for medium dense Density Sand and comparison to CD triaxial test data

(Continued)

σ'3 = 280 kPa

σ'3 = 45 kPa

σ'3 = 103 kPa

Hyperbolic model

CD tests data

0.000 0.025 0.050 0.075

Vo

lum

etr

ic s

trai

n, ε

v (c

om

pre

ssio

n is

po

sitiv

e)

0.000

0.001

0.002

0.003

0.004

0.005

0.000

0.001

0.002

0.003

0.004

0.005

b. Volumetric strain-axial strain model response

Figure B4. (Concluded)

σ'3 = 45 kPa

σ'3 = 103 kPa

σ'3 = 280 kPa

Axial strain, ε

0.000 0.025 0.050 0.075

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

Hyperbolic model

CD tests data

Figure B5. Transformed stress-strain plots from triaxial tests data on dense Density Sand,


Axial strain, ε

0.000 0.005 0.010 0.015 0.020

ε / (σ

1 -

σ 3)

(kP

a-1

)

0x100

10x10-6

20x10-6

30x10-6

40x10-6

50x10-6

60x10-6

70x10-6

σ3' = 45 kPa

σ3' = 280 kPa

σ3' = 103 kPa


σ3'

(kPa)a

(kPa)-1Ei

(kPa)

(σ1 - σ3)ult

(kPa)Rf

45

103

280

1.22 x 10-5

7.56 x 10-6

3.58 x 10-6

82035

132274

279539

3.86 x 10-3

1.95 x 10-3

7 x 10-4

258.9

511.6

1425.1

222.0

439.1

1187.7

0.858

0.858

0.833

b(kPa)-1

(σ1 - σ3)f

(kPa)

Average Rf = 0.850

σ'3 / pa

0.1 1

0.1 1

Ei /

pa

100

1000

10000

100

1000

10000

K = 1395 n = 0.63

Figure B6. Determination of hyperbolic parameters K and n for dense Density Sand

σ'3 / pa

0.1 1

0.1 1B

/ pa

100

1000

10000

100

1000

10000

Kb = 914

m = 0.55

Figure B7. Determination of hyperbolic parameters Kb and m for dense Density Sand

Axial strain, ε

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750D

evi

ato

r st

ress

, (σ

1-σ

3)

(kP

a)

0

250

500

750

1000

1250

1500

0

250

500

750

1000

1250

1500


Figure B8. Hyperbolic model for dense Density Sand and comparison to CD triaxial test data (Continued)

σ'3 = 280 kPa

σ'3 = 45 kPa

Hyperbolic model

CD tests data

σ'3 = 103 kPa

0.000 0.025 0.050 0.075

Vo

lum

etri

c st

rain

, εv

(co

mpr

ess

ion

is p

osi

tive)

0.000

0.001

0.002

0.003

0.004

0.005

0.000

0.001

0.002

0.003

0.004

0.005



σ'3 = 45 kPa

σ'3 = 103 kPa

σ'3 = 280 kPa

Axial strain, ε

0.000 0.025 0.050 0.075

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

Hyperbolic model

CD tests data

Figure B9. Transformed stress-strain plots from triaxial tests data on medium dense Light Castle Sand,


Axial strain, ε

0.00 0.02 0.04

ε / (

σ 1 -

σ 3)

(kP

a-1

)

0x100

100x10-6

200x10-6

300x10-6

400x10-6

σ3' = 45 kPa

σ3' = 280 kPa

σ3' = 103 kPa


σ3'

(kPa)a

(kPa)-1Ei

(kPa)

(σ1 - σ3)ult

(kPa)Rf

45

103

280

2.99 x 10-5

2.73 x 10-5

1.27 x 10-5

33451

36640

78834

6.86 x 10-3

2.64 x 10-3

1 x 10-3

145.8

378.1

996.2

139.5

326.0

817.2

0.957

0.862

0.820

b(kPa)-1

(σ1 - σ3)f

(kPa)

Average Rf = 0.880

σ'3 / pa

0.1 1

0.1 1

Ei /

pa

100

1000

10000

100

1000

10000

K = 438 n = 0.48

Figure B10. Determination of hyperbolic parameters K and n for medium dense Light Castle Sand

σ'3 / pa

0.1 1

0.1 1

B /

pa

100

1000

10000

100

1000

10000

Kb = 291

m = 0.72

Figure B11. Determination of hyperbolic parameters Kb and m for medium dense Light Castle Sand

Axial strain, ε

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750

Devi

ato

r st

ress

, (σ 1

-σ3)

(kP

a)

0

250

500

750

1000

1250

1500

0

250

500

750

1000

1250

1500


Figure B12. Hyperbolic model for medium dense Light Castle Sand and comparison to CD triaxial test data

(Continued)

σ'3 = 280 kPa

σ'3 = 45 kPa

σ'3 = 103 kPa

Hyperbolic model

CD tests data

0.000 0.025 0.050 0.075

Vo

lum

etr

ic s

trai

n, ε

v (c

om

pre

ssio

n is

po

sitiv

e)

0.000

0.001

0.002

0.003

0.004

0.005

0.000

0.001

0.002

0.003

0.004

0.005



σ'3 = 45 kPa

σ'3 = 103 kPa

σ'3 = 280 kPa

Axial strain, ε

0.000 0.025 0.050 0.075

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

Hyperbolic model

CD tests data

Figure B13. Transformed stress-strain plots from triaxial tests data on dense Light Castle Sand,


Axial strain, ε

0.000 0.005 0.010 0.015 0.020 0.025 0.030

ε / (σ

1 -

σ 3)

(kP

a-1

)

0x100

100x10-6

200x10-6

σ3' = 37.8 kPa

σ3' = 276.1 kPa

σ3' = 106.1 kPa


Average Rf = 0.813

σ3'

(kPa)a

(kPa)-1Ei

(kPa)

(σ1 - σ3)ult

(kPa)Rf

37.8

106.1

276.1

3.28 x 10-5

1.26 x 10-5

6.80 x 10-6

30455

79198

147017

4.41 x 10-3

1.96 x 10-3

8.17 x 10-4

226.8

510.8

1224.3

182.2

416.6

1003.5

0.804

0.816

0.820

b(kPa)-1

(σ1 - σ3)f

(kPa)

σ'3 / pa

0.1 1

0.1 1E

i / p

a

100

1000

10000

100

1000

10000

K = 688

n = 0.79

Figure B14. Determination of hyperbolic parameters K and n for dense Light Castle Sand

σ'3 / pa

0.1 1

0.1 1

B /

pa

100

1000

10000

100

1000

10000

Kb = 659

m = 0.31

Figure B15. Determinationof hyperbolic parameters Kb and m for dense Light Castle Sand

Axial strain, ε

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750

0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750

De

via

tor

stre

ss, (σ

1-σ

3)

(kP

a)

0

250

500

750

1000

1250

1500

0

250

500

750

1000

1250

1500


Figure B16. Hyperbolic model for dense Light Castle Sand and comparison to CD triaxial test data

(Continued)

σ'3 = 276.1 kPa

σ'3 = 37.8 kPa

σ'3 = 106.1 kPa

Hyperbolic model

CD tests data

0.000 0.025 0.050 0.075

Vo

lum

etr

ic s

trai

n, ε

v (c

om

pre

ssio

n is

po

sitiv

e)

0.000

0.001

0.002

0.003

0.004

0.005

0.000

0.001

0.002

0.003

0.004

0.005



σ'3 = 37.8 kPa

σ'3 = 106.1 kPa

σ'3 = 276.1 kPa

Axial strain, ε

0.000 0.025 0.050 0.075

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

-0.070

-0.060

-0.050

-0.040

-0.030

-0.020

-0.010

Hyperbolic model

CD tests data

Axial strain, ε

0.00 0.01 0.02 0.03 0.04 0.05D

evi

ato

r st

ress

, (σ

1-σ

3) (

kPa

)

0

250

500

750

1000

0

250

500

750

1000

Figure B17. Example determination of axial and volumetric strain values at 70 and 95 percent of strength.

Data from CD triaxial tests on dense Light Castle Sand

σ'3 = 280 kPa

σ'3 = 45 kPa

σ'3 = 103 kPa

Axial strain, ε0.00 0.01 0.02 0.03 0.04 0.05

Vo

lum

etr

ic s

trai

n, ε

v (c

om

pre

ssio

n is

po

sitiv

e)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0000

0.0005

0.0010

0.0015

0.0020

0.7 x (σ1-σ3)f = 702.4 kPa

0.95 x (σ1-σ3)f = 953.3 kPa

ε = 0.0112

ε = 0.0293

291.6 kPa

0.0086

395.8 kPa0.0222

127.5 kPa

0.0096

173.1 kPa

0.0240

εv = 0.00190

0.00109

0.000466

Note:The volumetric strain-axial strain diagramshave a horizontal tangent before mobilizationof 70 percent of the strength. Therefore, thevolumetric strain and deviator stress valuecorresponding to the maximum compressionof the specimens are used for determination ofthe bulk modulus

σ'3 = 276.1 kPa

σ'3 = 106.1 kPa

σ'3 = 37.8 kPa

Data for Determination of Hyperbolic Parameters K and n

70% Stress Level 95% Stress Level

Data for Determination of HyperbolicParameters Kb and m

σσσσ'3(kPa)

(1)

(σσσσ1-σσσσ3)f

(kPa)

(2)

(σσσσ1-σσσσ3)(kPa)

(3)

εεεε

(4)

(((( ))))εεεε

σσσσ σσσσ1 3−−−−

(kPa-1)

(5)


(6)

εεεε

(7)

(((( ))))εεεε

σσσσ σσσσ1 3−−−−

(kPa-1)

(8)


(9)

εεεεv

(10)

(((( ))))σσσσ σσσσεεεε

1 33

−−−−⋅⋅⋅⋅ v

(kPa)

(11)

38 182.2 127.5 0.0096 7.53 x 10-5 173.1 0.0240 1.39 x 10-4 73.8 0.00047 52340

106 416.6 291.6 0.0086 2.95 x 10-5 395.8 0.0222 5.61 x 10-5 199.2 0.00109 60917

276 1003.5 702.4 0.0112 1.59 x 10-5 953.3 0.0293 3.07 x 10-5 548.3 0.00190 96192

σσσσ'3pa

(12)

(((( ))))1

1 3σσσσ σσσσ−−−− ult(kPa)-1

(13)

R f

(14)

Eipa

(15)

B

pa

(16)

0.38 4.42 x 10-3 0.805 300 517

1.05 1.96 x 10-3 0.816 782 601

2.72 8.18 x 10-4 0.820 1465 950

(1) Confining stress. Use effective confining stress, σ'3, for effective stress analyses. Use total confining stress, σ3, for total stress analyses(2) Deviator stress at failure. Determined from the deviator stress-axial strain plots of the data of each test.

(3) 70 percent of (σ1-σ3)f (4) Axial strain corresponding to (3) determined as illustrated in Figure B17 (5) ( )( )( )

εσ σ1 3

4

3−=


εσ σ1 3

7

6−=

(9) Deviator stress for determination of bulk modulus, B. Use (3) if the εv-ε curve does not reach a horizontal tangent in compression before (3) is mobilized.Otherwise, use (σ1-σ3) corresponding to the maximum εv (compression) as illustrated in Figure B17

(10) Volumetric strain for determination of B. Use εv corresponding to (3) if the εv-ε curve does not reach a horizontal tangent in compression before (3) ismobilized. Otherwise, use the maximum εv (compression) as illustrated in Figure B17

(11) ( ) ( )

( )σ σ

ε1 33

9

3 10

−⋅

=⋅v

(12) ( )σ'

.3 1

101 3pa kPa= (13) ( )

( ) ( )( ) ( )

1

1 3

8 5

7 4σ σ−=

−−

ult

(14) ( ) ( )Rf = ⋅2 13 (15) ( ) ( ) ( ) ( ) ( )[ ]Eipa pa

=+ − ⋅ +

⋅2

5 8 13 4 7

1

(16) ( )B

pa pa=

11

Notes:• pa = 101.3 kPa• Figure B17 illustrates the procedure for the determination of the values of

deviator stress, axial strain, and volumetric strain for this example• See Appendix B for a complete explanation of the hyperbolic model and

procedure for determination of hyperbolic parameter values• In this example, the organization of calculations is based on Figure 21 of

Duncan et al. (1980). Numerical values correspond to the dense LightCastle Sand used for this investigation

Figure B18. Determination of the normalized values of Ei and B for each of the CD triaxial tests performed on dense Light Castle Sand (adapted from Duncan et al. 1980)

(4) Confining stress. Use effective confining stress, σ'3, for effective stress analyses. Use total confining stress, σ3, for total stress analyses(5) Deviator stress at failure. Determined from the deviator stress-axial strain plots of the data of each test.


εσ σ1 3

4

3−=


εσ σ1 3

7

6−=

(12) Deviator stress for determination of bulk modulus, B. Use (3) if the εv-ε curve does not reach a horizontal tangent in compression before (3) is mobilized.Otherwise, use (σ1-σ3) corresponding to the maximum εv (compression) as illustrated in Figure B17

(13) Volumetric strain for determination of B. Use εv corresponding to (3) if the εv-ε curve does not reach a horizontal tangent in compression before (3) ismobilized. Otherwise, use the maximum εv (compression) as illustrated in Figure B17

(14) ( ) ( )

( )σ σ

ε1 33

9

3 10

−⋅

=⋅v

(12) ( )σ'

.3 1

101 3pa kPa= (13) ( )

( ) ( )( ) ( )

1

1 3

8 5

7 4σ σ−=

−−

ult

(14) ( ) ( )Rf = ⋅2 13 (15) ( ) ( ) ( ) ( ) ( )[ ]Eipa pa

=+ − ⋅ +

⋅2

5 8 13 4 7

1

(16) ( )B

pa pa=

11

σ'3 / pa

0.1 1

0.1 1E

i / p

a

100

1000

10000

100

1000

10000

K = 688

n = 0.79

Figure B19. Determination of hyperbolic parameters K and n from the Ei/pa values determined in

column (15) of Figure B18

σ'3 / pa

0.1 1

0.1 1

B /

pa

100

1000

10000

100

1000

10000

Kb = 659

m = 0.31

Figure B20. Determinationof hyperbolic parameters Kb and m from the B/pa values determined in

column (16) of Figure B18

appendix b determination of hyperbolic parameter values of ...€¦ · appendix b determination of...

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