Hillel, pp. 341-382
Soil mechanics – a short introduction
Picture: Tennessee Department of Transportation
Picture: Ch. Salm, Terre AG
CE/ENVE 320 – Vadose Zone Hydrology/Soil Physics
Spring 2004
Copyright © Markus Tuller and Dani Or 2002-2004
Why mechanics of soils?
Like other solid materials (e.g. metals, rock), soils deform
when they are exposed to forces.
Force
Unlike many other materials in our
environment , soils show a wide range of
possible mechanical behavior which
influences considerably their use for …
Copyright© Markus Tuller and Dani Or2002-2004
Why mechanics of soils?
foundation of buildings…
…or agricultural production
Copyright© Markus Tuller and Dani Or2002-2004
Why mechanics of soils?
Understanding soil deformation behavior is crucial to:
• design slopes and retaining walls
• build tunnels in ‘soft’ rock
• assess hazards due to land slides
• prevent soil from compaction
• optimize soil management techniques
•…
Copyright© Markus Tuller and Dani Or2002-2004
Definition of stress and strain
The reaction of a solid body to a force F or a combination of forces
acting upon or within it can be characterized in terms of its relative
deformation or strain. The ratio of force to area where it acts is called
stress.
normal stress
s = Fn / A
shear stress
t = Fs / A
normal strain
e = dz / zo
shear strain
g = dh / zo
Note that compressive stresses and strains are positive and
counter-clockwise shear stresses and strains are positive.
Copyright© Markus Tuller and Dani Or2002-2004
Total vs. effective stresses
When a load is applied to soil, it is carried by the water in the pores as
well as the solid grains. The increase in pressure within the pore water
causes drainage (flow out of the soil), and the load is transferred to the
solid grains.
The rate of drainage depends on the permeability of the soil.
The strength and compressibility of the soil depend on the stresses within
the solid granular fabric. These are called effective stresses.
Copyright© Markus Tuller and Dani Or2002-2004
Stress in homogeneous soil
The total vertical stress acting on a soil element below the ground surface
is due to the weight of everything lying above: soil, water, and surface
loading.
In a homogeneous soil, the total
vertical stress sv on an element
with distance z from the surface is
determined by the weight of the
overlying soil and can be calculated
as:
s v  rb 1  Qm  g z
With rb the soil bulk density, Qm the gravimetric water content and g
the gravity constant. Typical values of rb are 1000 – 1800 kg m-3.
Copyright© Markus Tuller and Dani Or2002-2004
Stress in homogeneous soil
Any change in total vertical stress sv may also result in a change of total
horizontal stress sh on the same soil element. There is no simple
relationship between horizontal and vertical stress.
In a homogeneous soil, the total
horizontal stress sh on an element
with distance z from the surface can
be estimated as :
sh 

1 
sv
sv is the vertical stress and  soil Poisson’s ration. Typical values for
Poisson’s ratio are between 0.25 and 0.4. For practical purposes a ratio of
sh /sv = 0.5 provides a good first estimation.
Copyright© Markus Tuller and Dani Or2002-2004
Stress in a multi-layer soil
The total stress sv at depth z
is the sum of the weights of
soil in each layer above.
For example the total vertical
stress sv at a depth z in
layer 3 is
s v  r b1 1  Q m1  g d1
 r b 2 1  Q m 2  g d 2
 r b 3 1  Q m3  g z  d1  d 2 
where
rb1 , rb 2 , rb3
Qm1 , Qm 2 , Qm3
d1 , d 2 , d3
the bulk density of the layers 1 to 3
the gravimetric water content of the layers 1 to 3
the thickness of the layers 1 to 3
Copyright© Markus Tuller and Dani Or2002-2004
Stress in soil with a ‘wide’ surface load
The addition of a surface load will
increase the total stresses below
it. If the surcharge loading is
extensively wide, the increase in
vertical total stress below it may be
considered constant with depth
and equal to the magnitude of the
surcharge q.
The vertical total stress at depth z under a wide load q becomes then
s v  rb 1  Qm  g z  q
Copyright© Markus Tuller and Dani Or2002-2004
Stress in soil with a ‘narrow’ surface load
For narrow loads, e.g. stresses at the soil
surface under a strip footing or a wheel, the
induced total vertical stresses will decrease both
with depth and horizontal distance from the
center of the load.
In such cases, it is necessary to use a suitable
model to estimate the stress distribution in the
soil under the surface load (Boussinesq (1885),
Froehlich (1934) ). For a vertical load q,
homogeneously distributed over a circular area
of radius R, the vertical stress sv,q(z) in depth z
in the soil can be calculated as
R
q
z
s v ,q

2


 q
1

1

R
z



3 2



sv,q
Copyright© Markus Tuller and Dani Or2002-2004
Stress in soil with a ‘narrow’ surface load
The total vertical stress sv in the depth z due to a homogeneous surface
load q on a circular area and the overlying soil can therefore be
calculated as

2

s v  rb 1  Q m  g z  q 1  1  R z 


3 2



with rb the soil bulk density, Qm the gravimetric water content, g the
gravity constant, q the surface load and R the radius of the contact
area.
Copyright© Markus Tuller and Dani Or2002-2004
Uniaxial stress and strain – Hook’s law
Steel
wire
sa
Leonardo da Vinci’s (14521519) uniaxial tension test
Young’s modulus
E  ds a de a
Poisson's ratio
    de a de r
Young's modulus and Poisson's ratio are
measured directly in uniaxial compression or
extension tests, i.e. tests with constant (or
zero) stress on the horizontal surfaces.
Copyright© Markus Tuller and Dani Or2002-2004
Shear stress and strain
As the shear stress t’ increases materials distort (change shape). This
change in shape can be expressed as an angular shear strain g. The
shear modulus G' relates the change in shear stress dt’ to the change
in shear strain dg.
t
g
t
Shear modulus
G  dt  dg
Copyright© Markus Tuller and Dani Or2002-2004
Isotropic compression
As the isotropic stress s’ increases, materials compress (reduce
in volume). The bulk modulus K' relates the change in
volumetric strain dev=dV/V to the change in isotropic stress s’.
Bulk modulus
K '  ds de v
Copyright© Markus Tuller and Dani Or2002-2004
Stiffness of soil material
The relationship between a strain and stress is termed stiffness
OA: linear and recoverable
ABC: non-linear and irrecoverable
BCD: recoverable with hysteresis
DE: continuous shearing
The stress-strain curve of a soil has features which are characteristic for
different material behavior. Soils show elastic, plastic and viscous
deformation when exposed to stresses.
Copyright© Markus Tuller and Dani Or2002-2004
Elastic deformation
In linear-elastic behavior (OA) the stress-strain is a straight line and strains
are fully recovered on unloading, i.e. there is no hysteresis. The elastic
parameters are the gradients of the appropriate stress-strain curves and are
constant.
Young’s modulus
E  ds a de a  s a e a  const.
Poisson's ratio
    de a de r   e a e r  const.
Shear modulus
G  dt  dg  t  g  const.
Bulk modulus
K '  ds de v  s e v  const.
Copyright© Markus Tuller and Dani Or2002-2004
Typical values of elastic moduli E’ and ’
Typical E’
Unweathered overconsolidated clays
20 ~ 50 MPa
Boulder clay
10 ~ 20 MPa
Keuper Marl (unweathered)
Keuper Marl (moderately weathered)
Weathered overconsolidated clays
Organic alluvial clays and peats
Normally consolidated clays
Steel
30 ~ 150 MPa
3 ~ 10 MPa
0.1 ~ 0.6 MPa
0.2 ~ 4 MPa
205 MPa
Concrete
Soil
Rock
Steel
Concrete
>150 MPa
30 MPa
Typical ’
0.25-0.4
0.3
0.28
0.17
Copyright© Markus Tuller and Dani Or2002-2004
Relationships between elastic moduli
In bodies of isotropic elastic material the three stiffness moduli E',
K' and G' and Poisson’s ratio (') are related as:
G 
E'
21   
K'
E'
31  2 
Therefore the deformation behavior of an isotropic elastic material
can be described by only two material constants.
Copyright© Markus Tuller and Dani Or2002-2004
Plastic deformation
With increasing stress the material
behavior goes over from elastic to
plastic. This transition is called yield
(A). Plastic strains (AB) are not
recovered on unloading (BC).
Unloading (BC) and reloading (CD)
show a hysteresis. With increasing
strain (at constant stress) the material
eventually fails if brittle or flows if
ductile (E).
yield
Soils material behavior is often simplified
as elastic-perfectly plastic. During
perfectly plastic straining (AB), plastic
strains continue indefinitely at constant
stress. In a brittle perfectly plastic
material, the yield stress at point A this is
the same as the failure stress at a point
B.
Copyright© Markus Tuller and Dani Or2002-2004
Viscous deformation
Change in volume and shape of soils are generally time-dependent.
One way to capture this time-dependency is to model soil as a viscous
solids. For the case of simple shear for example, this means that the
shear stress t’ is proportional to the shear strain rate dg/dt. The
viscosity h relates the change in shear stress dt’ to the change in
shear strain rate dg/dt.
t
g
t
Viscosity
Linear viscosity
Shear
stress
.
.
h  dt  dg
h  t  g
Shear strain rate
Copyright© Markus Tuller and Dani Or2002-2004
Soil strength – precompression stress
The yield stress of a soil under
compression is called precompression
stress. It separates the stress ranges
where elastic and plastic deformation
can be expected.
If a soil is not loaded above the
precompression stress, deformation is
elastic (no irrecoverable deformation)
and rather small.
elastic
plastic
Precompression stress Pv
If a soil is loaded above the
precompression stress,
deformation is large and
plastic ( irrecoverable
deformation).
Copyright© Markus Tuller and Dani Or2002-2004
Soil strength – undrained shear
The maximum value of stress that may be sustained by a material is
termed strength.
The strength is independent of the normal stress since the response to
loading simple increases the pore water pressure and not the effective
stress.
The shear strength tf is a material parameter which is known as the
undrained shear strength su.
tf = (sa - sr) = constant
Copyright© Markus Tuller and Dani Or2002-2004
Soil strength – the angle of friction
The strength increases linearly with increasing normal stress and is zero
when the normal stress is zero.
t'f = s'n tanf'
f' is the angle of friction
In the Mohr-Coulomb criterion the material parameter is the angle of friction
f’ and materials which meet this criterion are known as frictional. In soils, the
Mohr-Coulomb criterion applies when the normal stress is an effective
normal stress.
Copyright© Markus Tuller and Dani Or2002-2004
Soil strength - cohesion
•..
The strength increases linearly with increasing normal stress and is positive
when the normal stress is zero.
t'f = c' + s'n tanf'
f' is the angle of friction
c' is the 'cohesion' intercept
In soils, the Mohr-Coulomb criterion applies when the normal stress is an
effective normal stress.
Copyright© Markus Tuller and Dani Or2002-2004
Typical values of f‘, c’ and su
Undrained shear strength
Hard soil
su > 150 kPa
Stiff soil
su = 75 ~ 150 kPa
Firm soil
su = 40 ~ 75 kPa
Soft soil
su = 20 ~ 40kPa
Very soft soil
su < 20 kPa
Drained shear strength
c´ (kPa)
f´ (deg)
Sands
0
30° - 45°
Clays
0 - 30 kPa
0 - 20°
Precompression stress Pv
soft
0-50 kPa
firm
50-150 kPa
stiff
> 150 kPa
Copyright© Markus Tuller and Dani Or2002-2004
Summary
The aim of this class was to
• introduce the concept of stress and strain in solid material,
especially soils.
• give you an idea what stiffness and strength of soil materials
are and
• provide you a short overview over the mechanics of one of the
most complex solid materials known.
Copyright© Markus Tuller and Dani Or2002-2004
Acknowledgment
I would like to acknowledge
• Prof. John Atkinson, City University, London, and Prof.
Sarah Springman, ETH, Zurich, who provided me with
many of the graphs I used for this presentation,
• Prof. Dani Or, UConn, Storrs, who gave me the
opportunity to talk about soil mechanics and
• you, the audience, for coming and staying with me.
Copyright© Markus Tuller and Dani Or2002-2004