WEEK 8

Soil Behaviour at Small Strains: Part 1

11. Strain levels and soil behaviour

Soil’s shear stress-strain relationships

exhibit many typologies. For example,

some of them are ductile with

continuing strain-hardening, and

some of them are brittle with

significant post-peak strain-softening.

The objective of Week 8-13 is to

understand what sort of soil

behaviour we should expect from

a given soil at a given condition, and

consider what impact the observed

features have on engineering problems.

In doing so, it is convenient to look at

soil behaviour at different strain levels.

This will allow us to focus on

γ

τ

Small strain

Large strainτ

Shear

This will allow us to focus on

stiffness at small strains, yield

characteristics at medium strains

and strength at large strains.

This view is applicable in principle for

compression behaviour that we have

studied last week. The only difference

is that we normally do not invoke a

notion of ‘strength’ in compression.

1

Medium strain

p′

p′

vε

Compression

12. Small-strain stiffness and non-linearity

12-1. Definitions of soil stiffness

- Tangent stiffness (Gtan, Etan , etc.)

- Secant stiffness (Gsec, Esec , etc.)

- Initial (elastic) stiffness (G0, E0 , etc.)

(Equivalent to tangent stiffness at

very small strains)

Upon unloading and reloading,

elastic stiffness is normally

observed (but not necessarily

identical to the initial stiffness).

Normally, soils’ stiffness is largest

at very small strains, exhibiting

gradual degradation as the strain

becomes larger (due to plastic

straining).

How small is small? There is no

formal definition or consensus on

γ

τ

tanG

secG0G

secG

0G

Unloading

&

reloading

formal definition or consensus on

“small strain”, but when we say

small strains, usually we talk about

strains smaller than order of 10-4

(imagine, 1 µm over 10 mm).

12-2. Some history: Background to recognition of small-strain stiffness

Importance of the stiffness non-linearity at small strains started to be recognised mainly

after the 1970s. This development had two technical factors in its background;

sophistication in laboratory tools and the advent of personal computers. New laboratory

tools allowed resolving ever smaller strains with higher accuracy. The computer allowed

non-linear numerical analyses, which provided a way to utilise the new laboratory findings

on small-strain stiffness for practical problems. Without PCs, prediction needs to be based

on analytical solutions, which normally exist for very simple, linearly elastic stress-strain

relationships. So in many senses, general recognition of the stiffness non-linearity at small

strains coincided with the turning point of soil mechanics from the classical era to the

modern.

2

γlogUp to order of 10-4

(0.01% strain)

12-3. Testing techniques for measuring small-strain stiffness

(i) Laboratory: Static tests

Triaxial apparatus with local instrumentation

is most commonly used for both research

and practice. Hollow cylinder apparatus and

plane strain apparatus are also used, but

mainly for research purposes. Here we

limit the scope to triaxial apparatus.

However, the principle itself of local

instrumentation is same in any apparatus.

Global instrumentation is erroneous due to

- Bedding errors

- Non-parallel ends

- Load cell and system compliance

Local instrumentation is capable of avoiding

these errors, providing more accurate

strain and hence stiffness measurement.

LVDTs

Suction capLoad cell

Bender elementsystem(also in other side of soil specimen)

Mid-height PWPtransducer

Soil specimen

Tie rod

Perspex wall Drainage

Ram

Porous stone

(Global)displacementtransducer

Radial belt

Ram pressure chamberfilled with oil

Bearing

To oil/air interfaceor CRS-pump

Example of triaxial apparatus withWhy not abolish all global instrumentation

and just use local one then? It is easier said

than done; local transducers are expensive

and requires expertise in handling.

3

Example of triaxial apparatus with

local instrumentation (Nishimura, 2006)

Specimen

externald

0H

Load cell

internald

0H ′

0

externalrnalaxial_exte

H

d=ε

0

internalrnalaxial_inte

H

d

′=ε

From external (global) instrumentation:

rnalaxial_exte

axialexternal

ε

σ

∆

′∆=′E

From internal (local) instrumentation:

rnalaxial_inte

axialinternal

ε

σ

∆

′∆=′E

Examples of local transducers

These devices have very high resolutions in displacement measurement. Consider how

high the resolution needs to be to measure, say, Young’s modulus for strain of 10-5

(0.001%)?

Local Displacement Transducer

(LDT; Goto et al., 1991)

Axial displacement transducer using

inclinometer (Burland & Symes, 1982)

4

Linear Variable Differential Transformer

(LVDT) for axial displacement

(Cuccovillo&Coop, 1997)

LVDT for radial displacement

(Drawing provided by

Prof. Matthew Coop)

Example of measurements

Note how different the magnitudes of stiffness are when measured externally and internally.

This is a typical result; you can find numerous similar comparisons in literature for sands,

silts, soft clays, etc. However, the error involved in global measurement of strains is more

significant for stiffer soils. The same problems of bedding and system compliance are

Triaxial compression on soft mudstone (Goto et al., 1991)

significant for stiffer soils. The same problems of bedding and system compliance are

encountered in oedometer tests too.

5

Another example: Lightly over-consolidated North Sea Clay

(Jardine et al., 1984)

(ii) Laboratory: Dynamic tests

Most of the dynamic tests are based on elastic or visco-elastic wave theory. The magnitude

of strain is associated to the magnitude of oscillation amplitude. The strain levels involved

are normally very small (<10-5), in many cases small enough to regard the obtained

stiffness as the initial elastic stiffness.

One dimensional wave equation is

where G is the shear modulus, m the viscosity

and r the mass density of soil. If the viscosity

is disregarded,

Where is the shear wave velocity.

Bender element tests:

tx

u

x

uG

t

u

∂∂

∂+

∂

∂=

∂

∂2

3

2

2

2

2

ρ

µ

ρ

x

)(xu

Case of one-dimensional shear wave2

22

2

2

x

uV

t

us∂

∂=

∂

∂

ρ/GVs =

hv

Bender elements

SoilSpecimen

A bender element is made up of piezo-ceramic

semiconductors. It generates shear waves when

energised, and conversely, it sends electric signals

when receiving shear waves. So by installing

a couple of them as transmitter and receiver,

and measuring the travel time between a given

distance, Vs and then G can be calculated.

A caution is required; soil stiffness

is anisotropic (the topic of next

week), and you need to know

which shear modulus you are

measuring; Gvh Ghv or Ghh?

6

v (or z)

h (or r)

hh

-0.5 0 0.5 1 1.5 2

Time [mSec]

-100

-50

0

50

100

Am

plit

ud

e o

f sig

nals

in a

rbitra

ry u

nits

First arrivalt = 0.514 mSec

InputOutput

TE4: After consolidationf = 9 kHz, vh-direction

Beginningof signal

Example of London Clay (Nishimura, 2006)

Resonant Column test

In contrast to bender element tests, in which typically a pulse wave is transmitted to monitor

its velocity, a sample is put in steady state oscillations in resonant column tests. By

gradually changing the input frequency at a constant input force (or torque) amplitude, the

frequency at which the oscillation becomes maximum is sought (i.e. the resonance

frequency is sought). From the resonance frequency, the sample’s stiffness is obtained.

If the oscillation is compression – extension, E is obtained (E or E’?)

If the oscillation is cyclic torsional, G is obtained.

The resonant column apparatus is

normally purpose-built, unlike

auxiliary tools such as bender elements.

This poses some inconveniences.

However, it has a big advantage; by

changing the input force, the oscillation

amplitude (hence strain amplitude) can

be changed. This is a useful feature for

estabilishing G – γ curves over a wider

strain range.

Various types of resonant column

Active Active

Active

Passive

F F

F

Ka

Ca

Ca

(a) Fixed-free (b) Fixed-base-spring –top (c) Free-free

Ka

7

Shear modulus measured in crag and Tertiary soils

(LC: London Clay, TC: Thanet Sand; Hight et al., 1997)

(iii) Field

Shear wave velocity measurement: Cross-hole and down-hole methods

The principle of these field methods is same as that of bender element tests. A receiver

(and transmitter in down-hole methods) is placed inside a borehole, or if the soil is soft, it

may be installed in a penetration cone (seismic cone penetration test; SCPT).

These method measures shear wave

velocity, which is a body wave. There

are also techniques which use

surface wave (Reighley wave).

Making waves above a seismic cone

8

Cross-hole measurement

(Hight et al., 1997)

Down-hole measurement

(Hight et al., 1997)

Example of comparison between different method:

30

20

10

0

De

pth

be

low

GL

[m

]0 100 200

Gvh [MPa]

-10

0

10

20

Ele

va

tio

n [

m O

D]

Down-hole (BH407, North)*

Down-hole (BH407, East)*

Resonant column (rot. core)

Bender element

Resonant column(range for blocks)

*Shear wave was transmittedfrom two sides of borehole

Biii

Bii

Bi

B1

C

FinallyR

In old days, the stiffness moduli measured in dynamic and static tests used to be

considered two fundamentally different things due to the strain-rate effects, because the

dynamic moduli were always far larger than the static ones. After it was found that the

static moduli had been underestimated by global measurement, the agreement of the

moduli between dynamic and static tests has been seen (Tatsuoka & Shibuya, 1991).

One problem solved?

9

Shear modulus Gvh of natural London Clay measured

by different laboratory and field methods (Nishimura, 2006)

50

40

-20

A3Lithological unit:

12-4. Importance of small-strain stiffness non-linearity: Case studies

(i) Excavation: Simpson et al. (1979)

One of the early examples of geotechnical non-linear finite element analysis is on

construction of an underground car park in front of the Palace of Westminster in the 1970s.

To avoid affecting the historic building,

the ground deformation caused by

the excavation needed to be predicted

with high accuracy.

A Class A prediction had been given

by elastic analysis by Ward and

Burland (1973). The problem was

revisited by Simpson et al. (1979)

by non-linear analysis.

Palace of Westminster with Big Ben Clock Tower

10

Cross-section

(Continued; Simpson et al., 1979)

The non-linear analysis was capable of

simulating the observed ground movements

with good accuracy.

An interesting episode is that the linear elastic

and non-linear analyses predicted the tower’s

leaning towards opposite directions.

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Modelling of stress-strain relationships

Predicted ground movements

(ii) Shallow foundation: Jardine et al. (1995)

Experiments at Bothkennar site, Scotland

Loading on a 2.4m x 2.4m footing on soft silty clay.

Analysis with a non-linear model predicted better the observed settlement than with linear

elasticity. The elastic analysis predicts that the influence of the footing settlements reaches

very far. In reality, it does not, as the non-linear analysis indicates.

Testing pad

12

Testing pad

Predicting and observed settlements

D

r

rδcδ

(iii) Shallow - deep foundation: Izumi et al. (1997)

Rainbow Bridge, Tokyo

(Construction work: 1987-1993)

140,000 tf anchorages built on Tertiary

Mudstone

(©Google 2011)

13

Cross-sections

(Continued: Izumi et al., 1997)

Proper consideration of stress-strain non-linearity at small strains led to significant

improvement in settlement prediction.

Note how conventional testing methods

underestimating the small-strain stiffness

led to over-estimation of the settlement.

3-D FEM mesh

14

Non-linear stiffness Settlement: Predictions and observations

Simulation cases

(iv) Tunnelling: Addenbrooke et al. (1997)

Jubilee Line Extension Project, London

Prediction of settlement troughs with non-linear numerical models

Jubilee Line (Grey-coloured)Cross-section

15

Model L4&J4: Non-linear models

fitted to locally instrumented triaxial

extension tests

Stiffness non-linearity from experiments and models

(Continued; Addenbrooke et al., 1997)

Linear elasticity is useless in predicting the settlement trough, which is deeper and

narrower than linear elasticity predicts.

However, even the non-linear stress-strain models do not do a perfect job. Research is

going on to see any other factor is being missed, such as anisotropy and the influence of

loading histories.

Settlement trough

Tunnel

excavated2D FEM mesh

16

Settlement at the ground surface due to excavation of first (west-bound) tunnel

References

Addenbrooke, T.I., Potts, D.M. and Puzrin, A.M. (1997) “The influence of pre-failure soil

stiffness on the numerical analysis of tunnel construction,” Geotechnique 47(3) 693-712.

Burland, J.B. and Hancock, R.J.R. (1977) “Underground car park at the House of

Commons, London: Geotechnical aspects” The Structural Engineer,” The Journal of The

Institution of Structural Engineers 87-100.

Burland, J.B. and Symes, M. (1982) “A simple axial displacement gauge for use in the

triaxial apparatus,” Geotechnique 32(1) 62-65.

Cuccovillo, T. and Coop, M.R. (1997) “The measurement of local axial strains in triaxial

tests using LVDTs,” Geotechnique 47(1) 167-171.

Goto, S., Tatuoka, F., Shibuya, S. Kim, Y.-S. and Sato, T. (1991) “A simple gauge for local

small strain measurements in the laboratory,” Soils and Foundations 31 136-180.

Hight, D.W., Bennell, J.D., Chana, B., Davis, P.D., Jardine, R.J. and Porovic, E. (1997)

“Wave velocity and stiffness measurements of the Crag and Lower London Tertiaries at

Sizewell,” Geotechnique 47(3) 451-474.

Izumi, K., Ogihara, M., and Kameya, H. (1997) “Displacement of bridge foundations on

sedimentary softrock; a case study on small strain stiffness,” Geotechnique 47(3) 619-

632.

Jardine,R.J., Symes, M.J., and Burland, J.B. (1984) “The measurement of soil stiffness in

the triaxial apparatus,” Geotechnique 34(3) 323-340.

Jardine, R J, Lehane, B M, Smith, P,R and Gildea, P A (1995) “Vertical loading

experiments on rigid pad foundations at Bothkennar,” Geotechnique 45(4) 573-599.

Nishimura, S. (2006) “Laboratory study on anisotropy of natural London Clay,” PhD Thesis, Nishimura, S. (2006) “Laboratory study on anisotropy of natural London Clay,” PhD Thesis,

Imperial College London.

Simpson, B., O’Riordan, N.J. and Croft, O.D. (1979) “A computer model for the analysis of

ground movements in London clay,” Goetechnique 29(2) 149-175.

Tatsuoka, F. and Shibuya, S. (1991) “Deformation characteristics of soil and rocks from

field and laboratory tests,” the 9th Asian Regional Conference on Soil Mechanics and

Foundation Engineering, Vol.1, 101-170.

Ward, W. H. & Burland, J. B. (1973). The use of ground strain measurements in civil

engineering. Phil. Trans. R. Sot. A274 421-428.

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