JANBU’s SOLUTIONS FOR MODULUS REPRESENTATION AND CONSOLIDATION OF SOILS

Arnfinn Emdal, Norwegian University of Science and Technology, Trondheim, Norway

Written at Utah State University, October 1999

Lecture notes

Reduced version for:

M.Sc. HPD

TABLE OF CONTENTS

1

1 BACKGROUND 2

2 SETTLEMENT CALCULATIONS 3

3 LABORATORY TEST RESULTS AND INTERPRETATION 4

3.1 Oedometer curve plotting. 4

3.2 Load stepping in the oedometer test 6

4 MODULUS AND STRESS DEPENDENCY 6

4.1 What is relevant? 6

4.2 Preconsolidation pressure pc’ 6

4.3 Oedometer and modulus curves for various soils 7

5 DERIVATION OF STRESS AND STRAIN FUNCTIONS FOR THREE SOIL TYPES 8

5.1 General 8

5.2 Constant modulus (a=1) 9

5.3 Granular soils (a=0.5, chosen) 9

5.4 Clays (a=0) 11

6 A FEW EXAMPLES 12

6.1 Wide surface load – primary consolidation settlement 12

6.2 Ground water withdrawal in sand. 13

6.3 Example from Dunn, Anderson and Kiefer 14

7 LOADING AND UNLOADING 15

2

JANBU’s SOLUTIONS FOR MODULUS REPRESENTATION AND CONSOLIDATION OF SOILS Arnfinn Emdal, Norwegian University of Science and Technology, Trondheim, Norway

Written at Utah State University, September 1999

1 Background Janbu’s modulus concept aims at being an engineers tool for prediction of deformations of soil subjected to external load, typically to find settlements as a function of surcharge on soil surface or from ground water drawdown. More specific – to find the effect of any change in effective stresses – whatever the reason for the change. Further, the stress dependency of the soil stiffness is clearly exposed, as the analysis always goes through the estimation of a modulus [Pa] and its variation, the calculation of strains [], including plotting these versus depth, and finally estimating the settlement contribution from the various soil strata.

The approach has two advantages:

1) The intermediate step of estimating the modulus vs effective stress-relation displays the best knowledge of the soils preconsolidation, and also how the specific soil behaves compared to what is implicit in the model.

2) Displaying the strain profile with depth, elucidates where the strain appears in the soil, and its distribution with depth. The latter has clear impact on the consolidation time, as strains deepseated in the soil requires longer consolidation time than shallow strains do.

The Norwegian soils are dominated by clays in the lower regions of the country, in the areas below the marine level. The clays may be very soft and normally consolidated, but may also be strongly overconsolidated due to the presence of the ice sheet that in Scandinavia withdrew appr. 10000 years ago. In many areas, large sand and silt deposits can be found above the clay, with significant variations in density and relative porosity. (The Lake Bonneville sediments may have large similarities with the Norwegian soils – except for the quick clay phenomenon.)

The development of Janbu’s concept was undertaken in the early 60’ies, and published. A full description is in Janbu’s book (in Norwegian) (1970). Here a compact review of the concept will be given.

3

2 Settlement calculations

Figure 1 Principle sketch of settlement calculation

A simplified description of settlement calculation For a soil element, and a soil profile, the process of strain evaluation and settlement becomes:

1) Find initial vertical stress σ0’ Using the best knowledge of the stress distribution versus depth.

2) Determine total stress increase from external load q: ∆σ' = ∆σ After long time: ∆u → 0 ⇒ ∆σ'= ∆σ, i.e. at the end of the primary consolidation.

3) Onedimensional confined modulus, M: dσ' = M ·dε Found from the oedometer test.

4) Strain in the element from ∆σ' becomes: ∫∆+

=''

'

0

0

' σσ

σ

σεMd

5) The settlement contribution from the element: dδ = ε ·dz

6) Total settlement: ∫ ⋅=H

dz0

εδ

In the following the focus will be on the stiffness formulation.

4

3 Laboratory test results and interpretation 3.1 Oedometer curve plotting. From the textbook of Dunn, Anderson and Kiefer an example is taken to show how the test results can be presented. The incremental values of effective stress (dσ’) and strain (dε) is computed, and the relation M= dσ’/dε is found.

Table 1 Oedeometer results. Example from Dunn, Anderson and Kiefer. Page 146.

Effective stress on each load step

[kPa]

Strain at end of each load step

[]

Log(effective stress)

Strain at end of load step

[]

Change in stress between load steps n and n-1

[]

Change in strain between load steps n and n-1

[]

Effective stress value for plotting of M

[kPa]

Modulus

[kpa] 0 0 #NUM! 0

12 0.0025 1.079181 0.0025 12 0.0025 6 4800 24 0.003 1.380211 0.003 12 0.0005 18 24000 48 0.004 1.681241 0.004 24 0.001 36 24000 96 0.005 1.982271 0.005 48 0.001 72 48000

192 0.0125 2.283301 0.0125 96 0.0075 144 12800 384 0.0355 2.584331 0.0355 192 0.023 288 8347 768 0.082 2.885361 0.082 384 0.0465 576 8258

1536 0.1295 3.186391 0.1295 768 0.0475 1152 16168

The oedometer result is plotted in effective stress versus strain (σ’:ε) in arithmetic scale. The e:logp’, or ε:logp’ which is commonly used represents the same data, but may tend to hide some of the information in the results while other information clearly is shown, the logarithmic strain function for clays is an example of this.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 1 2 3 4

log-stress

stra

in

(a) (b)

Figure 2 Oedometer curve in (a) arithmetic scale and (b) in logarithmic scale

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 500 1000 1500 2000

stress

stra

in

5

The derivative of the oedometer curve directly gives the tangent modulus (oedometer modulus or confined modulus, M) of the soil, at the actual stress level. Hence, by plotting the derived M-values versus σ’, a stress dependent soil stiffness is found.

Figure 3 Constrained modulus, M=dσ’/dε, versus effective stress, σ’. All values in kPa.

A closer look at the oedometer curve shows that the stress-strain development in the low stress range is rather linear, meaning that in this range the soil stiffness is constant, ot close to constant.

Figure 4 Close up of the low stress range of the oedometer curve. All stress values in kPa.

0

10000

20000

30000

40000

50000

60000

0 500 1000 1500

stress

Mod

ulus

The drawing clearly shows the difference in stiffness in the overconsolidated area and in the normally consolidated area. It also shows that a stiffness in OC-area from this test is very hard to interprete. The slope inclination in the OC-range in the log-scale does hide this fact.

Modulus number m = slope of NC-line. Here, m = 14.

This corresponds with a C-value of 0.16 (log-scale), as given in Dunn, Anderson, Kiefer.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 50 100 150 200 250

stress

stra

in

This closeup of the low stress part of the curve shows a normal behavior for an oedometer test. The initial load steps are very soft, whereas the stiffness below pc’ is close to constant.

Here this stiffness called MOC = 250/(.009-.0022)= 37,000 kPa.

6

As every settlement problem deals with soil at various depths, the use of stress dependent moduli is important. Again, the logarithmic strain function from conventional analyses of settlements in clay takes care of this.

These two charts (σ’:ε) and (M:σ’) is Norwegian standard for oedometer presentation, and is taken into use also other places in Europe. From these one can find preconsolidation pressure pc’ (or range of), the stiffness of the material in overconsolidated stress range and for stresses above pc’ and, unloading stiffness (if unloading-reloading is done during the test) may be found, a parameter which is important to know in analyses including e.g. excavations/unloading. The relation between the loading stiffness and unloading stiffness is often used in analysis software. In stiff soils, the unloading modulus may take very large values. An example from oedometer tests on sand (Grande/Schantz, Germany shows this. In clays, MOC 3 to 10 times the M at pc’ may be found.

3.2 Load stepping in the oedometer test The general practice in Norway differs little from other countries, but the time on each loading step is lower, compared with the commonly used 24 hour practice. For clays the load step time is usually in the range 30 to 60 minutes. A consolidation test is therefore completed in one working day. The aim is to ensure completion of the primary consolidation before the next step is started, and this is normally assessed during the first loadsteps of the test, by use of Taylor-construction.

There will undoubtedly also occur secondary consolidation during each load step. The amount of secondary consolidation strains generated during the test may be discussed, but, as it is impossible to avoid secondary effects, one choose to stay at the 60 minute line as well as at the 24 hour line in a e-logp’-diagram with creep-curves present, see Bjerrums diagram. The danger of this may be an erratic estimation of the preconsolidation pressure, this is discussed in the Chapter on Time dependency.

4 Modulus and stress dependency 4.1 What is relevant? Settlements due to presence of clay are inevitable unless the design aims at a fully compensated situation. In magnitude, the settlements in clay, and the uncertainties connected to these are usually much larger than what is experienced from strains in sands and overconsolidated material. Nevertheless, the importance of possible deformations will always be given by the involved structures, not by the soil, and stiffness description of sands and OC-clays are therefore also important.

4.2 Preconsolidation pressure pc’ The overconsolidation ratio (OCR) is frequently used, and it is normal to relate other parameters to OCR. OCR does indicate relatively how far below pc’ the insitu effective stress po’ is, but it does not really indicate how the material should be expected to behave. For a heavily overconsolidated soil, the OCR in upper area is very large, (po’ is small), and decreasing whith depth (po’ increases). However, the reason to believe that the behaviour of the soil will be very different in the various depths is little. The importance is in knowing when the effective stress condition passes pc’. This is valid for both sands and clay, as sand also shows clear memory from earlier preconsolidation stresses.

Determination of pc’ in sand is not done in the laboratory, as all native soil structure disappears when the sand is sampled. To find this parameter, insitu measurements are needed, as e.g. with screwplate or similar tools. Still it will be a very hard problem to solve. The best way to estimate this is probably from assessment of the geology of the area.

7

4.3 Oedometer and modulus curves for various soils The figure shows an assembly of results from various soft and stiff soils. (leire=clay)

Figure 5 Oedometer tests on soil with large variation in stiffness.

8

Grossly the modulus curves could be divided in three categories, constant, parabolic, and linear, but with large variation in the inclination of the curves for sand and clay.

Figure 6 Effect of stress exponent.

5 Derivation of stress and strain functions for three soil types 5.1 General The previous chapter states that the soil behavior may grossly be divided in three categories.

1) Constant stiffness with effective stress, valid for any material that behaves like this, typical for overconsolidated clays.

2) Linearly increasing with effective stress. Typical for normally consolidated clays, i.e. clays in the stress range above pc’.

3) Parabolic increase with effective stress. Valid for sands and coarse silts, or any granular material.

Janbu’s general equation for confined modulus is )1(

'a

aamM

−

⋅=

σσσ ,

where m is the modulus number, σ’ is the actual effective stress level, σa is the stress equivalent to one atmosphere (in SI: 100000kPa), and a is called the stress exponent.

(By choosing the correct value for σa one can express the modulus equation in ones own preferred units.)

If a soil element is exposed to a change from stress condition σ0’ to stress condition σ’=σ0’+ ∆σ’, one can derive the following expression for the strain that this stress increment will give:

M-function with m =1 for varying stress exponent a

0

100

200

300

400

500

600

700

800

900

1000

0 1 2 3 4

Effective stress/atm.stress

M

10.50.250-0.5-1

Janbu chose a general formulation to be able to model the various modulus curves that was seen in the results. The stress is made dimensionless by division with athmospheric pressure.

9

−

=

=

=

=

⋅==

∫−

−

−

a

a

a

a

a

a

aa

a

aa

a

aa

a

aa

am

am

dm

dm

d

mddM

σσ

σσε

σσσ

σε

σσσ

σε

σσσ

σε

σσσ

εσ

σ

σ

σ

σ

''1

'1

''1

''1

''

0

'

'

'

'

)1(

)1(

)1(

0

0

5.2 Constant modulus (a=1) The stress exponent a=1 gives constant stiffness

amM σ= ,

and the strain

Mmm aaa

''''1 0 σσσ

σσ

σσε ∆

=∆

=

−=

.

not very surprisingly, and suites OC-materials.

5.3 Granular soils (a=0.5, chosen) Tests on sand show that in most cases a stress exponent close to 0.5 will represent the stress-strain-curve quite well. The stiffness expression becomes

'')5.01(

σσσσσ a

aa mmM =

⋅=

−

,

and the strain

−=

aam σσ

σσε

''2 0 .

10

Table 2 Typical modulus numbers for sand

Material type: Modulus number

Loose, fine sand m < 150

Medium dense sand 150 < m < 250

Dense, coarse m > 250

Modulus numbers and relation to relative porosity and to Ko’ is shown in the following figures from Janbu, 1970, on fully saturated (a) and dry (b) sand.

Figure 7 Modulus numbers for sand, from Janbu, 1970

A simple stiffness and strain-evaluation:

A soil element at depth 5m in a medium dense sand, γ=18kN/m3 ,GW at terrain level can typically have a start stiffness of 250x√(100x5x8)=15,800kPa. A stress increase of e.g. 200kPa will give a new stiffness of 250x√(100x(5x8+200))=38,700kPa. An estimate of the strain from the 200kPa load increase may be found from taking an average value of these: 200/(15800+38700)/2=0.73%, or by using the strain equation (2/250)( √((5x8+200)/100)- √(5x8)/100)) = 0.73%.

Note!

The square-root-simplification may in certain cases not be satisfying. One may find values for m and a directly from the laboratory curves by picking a couple of M-σ’-pairs and combine to find the two unknown parameters.

11

5.4 Clays (a=0) For the normally consolidated clays, a linear idealization is used.

The stiffness expression becomes:

'')01(

σσσσ mmM

aa =

⋅=

−

,

and by including the definition of modulus, M=dσ’/dε one gets:

[ ] ''

'

'0

0

'ln1''

11''1

''

σσ

σ

σ

σσσ

ε

σσε

σεσ

md

m

dm

d

mddM

==⇒

=⇒

⋅==

∫

,

and the expression for strain in NC-clay becomes

''ln1

0σσε

m= .

This matches very well with the ε-logp’- representation by using a linear curve in the log-diagram.

''log

0σσε C= .

The relation between m and C is:

3.210ln1 CCm

== .

Modulus numbers for NC-clays varies in the range 10-30, and has a strong relation to the water content w of the soil. This is shown in the following figure.

Figure 8 Modulus number m for clay versus water content w. From Janbu, 1970.

12

6 A few examples 6.1 Wide surface load – primary consolidation settlement

The case is simply set as in the figure, with OC-clay on top, typical for a desiccated (dry crust) clay that can be characterized by constant modulus.

The NC-clay has a modulus number of 17, and is simply expected to have a linear modulus vs effectives stress relation. The effective stress profile is drawn, and the strain is evaluated at 4 levels. Settlement from OC-clay: 1.5 cm, from NC-clay: 21 cm.

Figure 9 Simple analysis of surface load on OC-clay above NC-clay

13

6.2 Ground water withdrawal in sand.

14

6.3 Example from Dunn, Anderson and Kiefer The example from this book regards the effect of ground water drawdown from surface level to depth 15.5m. The soil strata are 25m sand over 9m NC-clay. The clay data are as given in Ch. 3.1, and in the following a spreadsheet-based calculation of the stresses, strains and settlement is shown.

Figure 10 Example stress profiles with data from Dunn, Anderson and Kiefer.

(a) (b) (c)

Figure 11 Strain profiles and surface settlement with modulus numbers in sand msand=150, 250 and 500 in (a), (b) and (c) respectively.

Total Sand Clay Final settlement [m]

0.322

0.105

0.217

Total Sand Clay Final settlement [m]

0.280

0.063

0.217

Total Sand Clay Final settlement [m]

0.248

0.031

0.217

Stresses [kPa] before drawdown vs depth [m]

0

5

10

15

20

25

30

35

40

0 200 400 600 800

sigtot_0

sigef_0

porepressure eff.stress

Stresses [kPa] afterdrawdown vs depth [m]

0

5

10

15

20

25

30

35

40

0 200 400 600 800

sigtot_finalsigef_final

porepressureeffective stress

Change in eff.stress [kPa]vs depth [m]

0

5

10

15

20

25

30

35

40

0 50 100 150

dsigef

0

5

10

15

20

25

30

35

40

0 0.02 0.04

Strain[] vs depth[m]

Dept

h

0

5

10

15

20

25

30

35

40

0 0.02 0.04

Strain[] vs depth[m]

Dept

h

0

5

10

15

20

25

30

35

40

0 0.02 0.04

Strain[] vs depth[m]

Dept

h

15

The example simply shows that the strain field in sand is significant if the sand is loose and saturated, with msand typically in the lower range. In (a), 30% of the settlement would come from the sand. The more realistic values for the example would perhaps be msand of 250 and more as in (b) and (c).

Note that the settlement due to strain in the clay layer equals exactly the value in the example in the book. A coincidence maybe…

7 Loading and unloading Loading and unloading of soil is very common. In clays we know that this gives the effect of preconsolidation, as well as swelling when the load is taken away. For sands, the situation is similar, and some examples are shown below.

Figure 12 Oedometer test on sand with loading-reloading cycles. Work by Tom Schantz, Stuttgart, and Lars

Grande, Trondheim.

0

0.005

0.01

0.015

0.02

0.025

0.03

0 50 100 150 200 250Stress (kPa)

Stra

in

0

10000

20000

30000

40000

50000

60000

0 50 100 150 200 250

Stress (kPa)

Mod

ulus

(kPa

)

16

a)

b)

c)

Figure 13 The first load cycle from the test. Tom Schantz/Lars Grande. a) Oedometer curve b) Modulus curves c) Theoretical fit to the virgin curve d) The reloaded oedometer curve in ε-logp’-diagram.

(d)

The test is performed on sand in the laboratories at the Technical University of Stuttgart. It shows that the sand repeatedly comes back to a “virgin” condition. It may not show very clear on the oedometer curve in Figure 12a), but the development of stiffness in Figure 12b) reveals this fact.

The interpretation of these curves in Janbu’s modulus concept would be a modulus number of m=180 and an stress exponent a=0.43 so that the theoretical modulus-curve is

First load cycle

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 10 20 30 40 50 60

Stress (kPa)

Stra

inFirst load cycle

0

10000

20000

30000

40000

50000

60000

70000

80000

0 10 20 30 40 50 60

Stress (kPa)

M (k

Pa)

Comparison with theoretical fit

0

10000

20000

30000

40000

50000

60000

70000

80000

0 10 20 30 40 50 60

Stress (kPa)

M (k

Pa)

First load cycle

0.008

0.0085

0.009

0.0095

0.01

0.0105

0.011

0.0115

0.012

0.0125

1 10 100

Stress (kPa)

Stra

in

17

)43.01()1(

100'100180' −−

⋅=

⋅=

σσσσ

a

aamM .

The first load cycle from the test is “blown up”, and studied separately. It has been loaded to a level of 27kPa before unloading to 4kPa and reloaded to 50kPa.

The modulus show a very stiff unloading, and a less stiff reloading which falls down to a virgin behaviour at stresses above pc’. A fit between the laboratory curve and the theoretical curve is shown.

Finally, a plot of the reloading curve (from 4kPa to 50kPa) is shown in ε-logp’, merely to show the problem that such a presentation in this case hides the vital information of the preconsolidation pressure level.

Janbu’s early work also show some loading-reloading curves on a low to medium dense Klaebu sand.

Modulus number m=200, a=0.5.

Figure 14 Loading-reloading curves on sand. Janbu (1970) and (1985).