Back to Soil Mechanics
Based on part of the GeotechniCAL reference package
by Prof. John Atkinson ,
City University, London

Compression and swelling

The relationship between volume change and effective stress is called compression and swelling. (Consolidation and compaction are different.) The volume of soil grains remains constant, so change in volume is due to change in volume of water.

Compression and swelling results from drained loading and the pore pressure remains constant. If saturated soil is loaded undrained there will be no volume change.


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Mechanisms of compression

Compression of soil is due to a number of mechanisms:
rearrangement of grains
fracture and rearrangement of grains
distortion or bending of grains

On unloading, grains will not unfracture or un-rearrange, so volume change on unloading and reloading (swelling and recompression) will be much less than volume change on first loading (compression).

In compression, soil behaviour is:



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Common cases of compression and swelling

In practice, the state of stress in the ground will be complex. These are simple theories for two special cases.

Isotropic:
Equal stress in all directions. Applicable to triaxial test before shearing.

p' = (s'a + 2s'r) / 3
= mean stress
ev = DV / Vo
= volumetric strain
 

One-dimensional:
Horizontal strains are zero. Applicable to oedometer test and in the ground below wide foundations, embankments and excavations.

s'z = vertical stress
ev = DV / Vo
= DH / Ho
= De / (1+eo)
= volumetric strain
 


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Isotropic compression and swelling

Isotropic compression and swelling is applied at the start of a triaxial test.

p' = (s'a + 2s'r) / 3
= mean stress
V = Vo - DVw
= volume
ev = DV / Vo = Dv / vo
= volumetric strain
v = V / Vs
= specific volume
 

As the mean stress p' is raised and lowered there are volumetric strains and the specific volume changes.

p'o = initial mean stress
vo = initial specific volume

Note the paths of compression, swelling and re-loading.
 
 


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Equations

For isotropic compression and swelling there are simple relationships
between specific volume v and (the natural logarithm of) the mean stress p'.

First loading
normal compression line
OAD on the graph
v = N - l ln p'

Unloading and reloading
swelling line
BC on the graph
v = vk - k ln p'

N, l and k are soil parameters.
vk and p'y locate the particular swelling line.

p'y is referred to as the yield stress.

If the current stress and the history of loading/unloading are known, the current specific volume can be calculated.
 
 


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Bulk modulus

Isotropic compression can be represented by a bulk modulus K' or by the slope of the normal compression line l (or the slope of a swelling line k): these are related.

K' = dp' / dev
v = N - l ln p'
dv / v = - l dp' / vp'

Hence K' = vp' / l (or vp' / k )

Bulk modulus K' depends on v and p'. Both of these will change during compression or swelling and so K' is not a soil constant.
 
 


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Typical values for isotropic compression parameters

The compression and swelling parameters l and k are soil properties and the values depend on the nature of the soil.
Typical values wL Ip l
very high plasticity clay 80 50 0.29
high plasticity clay 60 34 0.20
intermediate plasticity clay 42 23 0.14
low plasticity clay 30 12 0.07
quartz sand     0.15
carbonate sand     0.34

For clays l » Ip / 170.
k / l is relatively large (e.g. 0.25 - 0.35) because clay particles can bend and distort.

For sands l is relatively large due to particles crushing (but states only reach NCL at high pressure).
k / l is relatively small (e.g. 0.1) because sand particles crush and rearrange during first compression.
 
 


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Overconsolidation

If the current state of soil is on the normal compression line it is said to be normally consolidated. If the soil is unloaded it becomes overconsolidated.

(Soil cannot usually be at a state outside the normal compression line unless it is bonded or structured).

At a state A the overconsolidation ratio is
Rp = p'y / p'a
(on NCL Rp = 1.0 and soil is normally consolidated).

Note: p'y is the point of intersection of the swelling line through A and the NCL. This is usually close to the maximum past stress.
 
 


Back to Isotropic compression and swelling

State

The current state of a soil is described by the stress p', the specific volume v and the overconsolidation ratio Rp (for a complete description the shear stress q' is required).

The state at A is given by any two of
va , p'a , Rp = p'y / p'a

All states with the same Rp fall on the lines parallel with the NCL.

ln Rp = ln ( p'y / p'a )
= ln p'y - ln p'a

Many features of soil behaviour, especially shear modulus and peak strength, increase with increasing overconsolidation.
 


Back to Isotropic compression: state

Change of state

Loading and unloading
(relevant to all soils)
Change of state A to B can only be achieved by normal compression along CD followed by swelling along DB. Note that the yield stress corresponding to B is larger than the yield stress corresponding to A.

Vibration or compaction
(relevant to sands)
or creep
(relevant to clays)
Change of state can occur directly from A to B. Note that the yield stress corresponding to B is larger than the yield stress corresponding to A.

 


Back to Isotropic compression: state

Critical state

There is a critical overconsolidation ratio which separates states in which the soil will either compress or dilate during shear. This corresponds to the critical state line CSL. Look at the possible specific volumes (v) that can occur at a mean effective stress p'.

wet side of critical
(W on the graph)
vw > vc at stress p'
water content ww is larger than critical wc
· loose
· normally consolidated
or lightly overconsolidated
· compress during drained shear

dry side of critical
(D on the graph)
vd < vc at stress p'
water content wd is smaller than critical wc
· dense
· heavily overconsolidated
· dilate during drained shear
 
 


Back to Isotropic compression: state

Normalising parameters

Normalising parameters change the current state to a normalised state so that all states with the same overconsolidation ratio have the same value.

Equivalent specific volume
vl = va + l ln p'a
Equivalent pressure
ln p'e = ( N - va ) / l
Critical pressure
ln p'c = ( G - va ) / l

If A is on the wet side of critical
ve > G
p'a / p'c > 1

If A is on the dry side of critical
ve < G
p'a / p'c < 1
 
 


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One-dimensional compression and swelling

One-dimensional loading is applied in an oedometer and occurs in the ground beneath wide foundations, embankments or excavations.

s'z = vertical effective stress
H = height or thickness

vertical strain = volumetric strain
ev = DH / Ho = De / (1+eo)
where Ho, eo and s'o are initial values.

As the vertical stress s'z is raised and lowered the top of the sample settles or heaves, or the layer contracts or expands.

Note that the compression-swelling-recompression curve is similar to that for isotropic compression, but the axes used are (s'z, e) rather than (p', v).
 
 


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Equations

For one-dimensional compression and swelling there are simple relationships between the void ratio and the logarithm of the vertical effective stress s'z.

First loading:
normal compression line (NCL)
OAD on the graph
e = eN - Cc log s'z

Unloading and reloading:
swelling-recompression line (SRL)
BC on the graph
e = ek - Cs log s'z

· eN, Cc and Cs are soil parameters
· ek and s'y locate a particular swelling line

If the current stress s'o and the history of loading and unloading are known, the current void ratio can be calculated. e.g.
eo = eN - Cc log s'y + Cs (s'y - s'o )

 


Back to One-dimensional compression: equations

One-dimensional modulus and compressibility

The one-dimensional stiffness modulus is the slope of the stress/strain curve:

M' = Ds'z / Dev or
E'o = Ds'z / Dez (since eh = 0)

The reciprocal of stiffness is compressibility. The one-dimensional coefficient of compressibility is the slope of the strain/stress curve:

mv = De / (Ds'z (1+e))
= 1 / E'o

E'o and mv apply for the normal compression line and for swelling and recompression lines, and depend on the current state, on the history and on the increment of loading, so they are not soil constants.
Since mv varies with s'z, its value is often quoted for s'z = 100kPa.
 
 


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Overconsolidation

If the current state of soil is on the normal compression line it is said to be normally consolidated. If the soil is unloaded it becomes overconsolidated.

Soil cannot usually be at a state outside the normal compression line unless it is bonded or structured.

At a state A the overconsolidation ratio is
Ro = s'y / s'a
(on NCL Ro = 1.0 and soil is normally consolidated).

Note: s'y is the point of intersection of the swelling line through A and the NCL. This is usually, but not always, close to the maximum past stress (see change of state).
 
 


Back to One-dimensional compression and swelling

Horizontal stress in one-dimensional loading

During one-dimensional loading and unloading the horizontal effective stress s'h will change since the condition of zero horizontal strain (eh = 0) is imposed.

The ratio Ko = s'h / s'z is known as the coefficient of earth pressure at rest.

Ko depends on
· the type of soil
· the overconsolidation ratio (Ro)
· the loading or unloading cycle

Approximations
normally consolidated soils:
Konc » 1 - sinf'c
overconsolidated soils:
Ko » Konc ÖRo
 
 


Back to One-dimensional compression and swelling

State

The current state of a soil is described by the stress s', the void ratio e and the overconsolidation ratio Ro (for a complete description the shear stress t' is required).

The state at A is given by any two of
ea , s'a , Ro = s'y / s'a

All states with the same Ro fall on the lines parallel with the NCL.

log Ro = log ( s'y / s'a )
= log s'y - log s'a

Many features of soil behaviour, especially shear modulus and peak strength, increase with increasing overconsolidation.
 
 


Back to One-dimensional compression: state

Change of state

Loading and unloading
(relevant to all soils)
Change of state A to B can only be achieved by normal compression along CD followed by swelling along DB. Note that the yield stress corresponding to B is larger than the yield stress corresponding to A.

Vibration or compaction
(relevant to sands)
or creep:
(relevant to clays)
Change of state can occur directly from A to B. Note that the yield stress corresponding to B is larger than the yield stress corresponding to A.
 
 


Back to One-dimensional compression: state

Critical state

There is a critical overconsolidation ratio which separates states in which the soil will either compress or dilate during shear. This corresponds to the critical state line CSL. Look at the possible voids ratios (e) that can occur at an effective stress s'a.

wet side of critical
(W on the graph)
ew > ec at stress s'
water content ww is larger than critical wc
· loose
· normally consolidated
or lightly overconsolidated
· compress during drained shear

dry side of critical
(D on the graph)
ed < ec at stress s'
water content wd is smaller than critical wc
· dense
· heavily overconsolidated
· dilate during drained shear
 
 


Back to One-dimensional compression: state

Normalising parameters

Normalising parameters change the current state to a normalised state so that all states with the same overconsolidation ratio have the same value.

Equivalent void ratio
el = ea + Cc log s'a
Equivalent stress
log s'e = ( eN - ea ) / Cc
Critical stress
log s'c = ( eG - ea ) / Cc

If A is on the wet side of critical
el > eG
s'a / s'c > 1

If A is on the dry side of critical
el < eG
s'a / s'c < 1
 
 


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Wet and dry states

Soils whose states lie on the normal compression line (NCL) are normally consolidated. There is a critical overconsolidation ratio that corresponds with the critical state line (CSL).

A lightly overconsolidated soil has a state which lies above the CSL.
A heavily overconsolidated soil has a state which lies below the CSL.

States lying above the CSL are said to be on the wet side of critical.
States lying below the CSL are said to be on the dry side of critical.

In the diagrams: va > vb, and yet since the stress at B is greater, state B is on the wet side of critical, while state A is on the dry side of critical.

 


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Wet and dry states

Soils whose states lie on the normal compression line (NCL) are normally consolidated. There is a critical overconsolidation ratio that corresponds with the critical state line (CSL).

A lightly overconsolidated soil has a state which lies above the CSL.
A heavily overconsolidated soil has a state which lies below the CSL.

States lying above the CSL are said to be on the wet side of critical.
States lying below the CSL are said to be on the dry side of critical.

In the diagrams: va > vb, and yet since the stress at B is greater, state B is on the wet side of critical, while state A is on the dry side of critical.
 
 


Back to Wet and dry states

State parameters

A measure of the initial state of a soil are the distances it lies at from the CSL, in terms of either volume or stress. These distances are expressed as state parameters:

Stress state parameter
Ss = pa' / pc'
ln Ss = ln pa' - ln pc'
Volume state parameter
Sv = va - vc

The state parameters are related:
Sv = ln Ss
Normally consolidated state:
Sv = l ln Ss = 0

States on the wet side of critical:
Sv and ln Ss are positive
States on the dry side of critical:
Sv and ln Ss are negative

 

Produced by Dr. Leslie Davison, University of the West of England, Bristol, 2001