foundations: bearing capacity
DESCRIPTION
Foundations: Bearing CapacityTRANSCRIPT
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2 BEARING CAPACITY OF SHALLOW FOUNDATIONS
2.1 General characteristics
Shallow foundations transfer the loads to the ground at a level close to the surface. The side
friction between soil and foundation, and the shear resistance of the lateral soil, are neglected.
The lateral soil is seen as a surcharge acting at the level of the foundation base.
In most cases the ratio between the foundation width B and its depth D is less than 2. In any case
the minimum depth of the foundation base should be about 1 m.
The base of the foundation should be placed outside the zone of fluctuation of the water table.
For cohesive soils this reduces the possible heave/settlement induced by the wetting/drying
process of sensitive clays.
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The bearing capacity is not a characteristics of soils or of the foundation. In fact, it depends on
the interaction between the footing and the ground underlying it.
Depending on their dimensions, and on their major L and minor B sides, shallow foundation can
be subdivided into: spread or pad footings; strip footings; mat or slab or raft foundations.
Examples of spread footings (L B)
Scheme of a strip footing (L>>B)
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Schemes of mat foundations
The bearing capacity equations derived in the following refer to strip footings. They will be
corrected subsequently for the case of spread and mat foundations.
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2.2 Mechanisms of failure (De Beer, Vesic, 1958)
General failure (or failure by shearing)
I Active failure zone II Radial failure zone
III Passive failure zone (results from a model test on dense sand)
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Punching failure (or failure due to the volume decrease)
(results from a model test on loose sand)
The so called local failure mechanisms are intermediate between general and punching failure.
Only the general failure mechanism is considered here that applies to relatively dense soils. In
case of loose soils, where punching could occur, deep foundations should be considered instead
of footings.
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2.3 Terzaghis bearing capacity equation
This equation defines the limit value of a uniformly distributed
vertical load applied to the horizontal ground surface subjected to
a known lateral load q
The bearing capacity coefficients are functions
of the friction angle . Their expressions are worked out assuming:
1) Plane strain regime 2) Rigid-plastic material behaviour 3) Limit shear resistance of soil defined by Mohr-Coulomb yield condition 4) Principal stress direction at failure coinciding with vertical and horizontal directions
The three coefficients are separately evaluated on the basis of equilibrium and Mohr-Coulomb
condition. This implies the assumption that there is no mutual influence among them.
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2.4 Plane strain vs. plane stress regime
For linear elastic isotropic
materials:
[ ( )]
( )
Elastic plastic vs. rigid plastic behaviour
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Mohr-Coulomb criterion
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Apparent hardening in plane strain conditions
Element of elastic ideally
plastic material in plane
strain conditions
From the origin to point A the stress
path develops within the yield
surface: A = limit elastic state;
. From A to C the stress point moves
along the yield surface and depends on the plastic strain.
When C is reached, no further increase of could keep the stress point on the yield surface and, consequently, the element fails.
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Material element
Plane strain element
A C
A
C
III (out-of-plane stress)
The consequence of this is that the element exhibits an apparently hardening behaviour even in
the case of elastic perfectly plastic material.
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Mohr-Coulomb criterion in terms of principal stresses
Mohr-Coulomb criterion in plane strain regime assuming
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Plane strain Mohr-Coulomb criterion at the limit elastic state
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2.5 Derivation of the bearing capacity coefficients
Terzaghis equation can be applied in terms of total or of effective stresses, using undrained
or drained cohesion and friction angle.
In undrained conditions the total unit weight of
soil should be used.
In drained conditions, the dry and the submerged unit
weight of soil can be used, respectively, above and below the water table. If the water table level
is likely to change in time, it is safe to use the submerged weight
The bearing capacity coefficients are derived, in terms of the friction angle, evaluating the limit
load under particular conditions:
coincides with for coincides with for
depends on for
Water table
A=active zone
P=passive zone
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2.5.1 Bearing capacity coefficient
(conditions: )
Sign conventions for the stresses
Mohr circle for the passive zone
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Pole of the Mohr circle
- Consider a point of the Mohr circle that represents the stresses acting on a plane having known inclination,
e.g. point D corresponds to the stresses acting on
the ground surface.
- From point D draw a line parallel to the plane on which the stresses are applied (i.e. the horizontal line in this case).
- The line intersects the Mohr circle in a point P which is referred to as the Pole.
- If a line is drawn from the Pole, it intersects the Mohr circle in a point the represents the stresses acting on a plane parallel to that line.
Consequently, lines PA and PA are parallel to the planes along which failure occurs within the passive zone.
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Mohr circle for the active zone
Now we have to define the external limit of the passive zone, i.e. the position of point F.
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The position of point F is defined assuming that a radial equilibrium zone, bounded by a
logarithmic spiral, connects the active and passive zones.
The expression of qlim is obtained by imposing the rotational equilibrium of zone GEF with
respect to point G. To this purpose it is necessary to express the stresses acting on the planes
GE, GF and on the spiral arc EF.
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The stresses acting on plane GF are defined by the Mohr circle of the passive zone
The shear stress is
which, substituting V and H, becomes
The normal stress is
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Substituting one obtains
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The stresses acting on plane GE are defined by the Mohr circle of the active zone
The shear stress is
which, substituting V and H, becomes
The normal stress is
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Substituting one obtains
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It is not necessary to evaluate the stresses acting on the arch EF of the logarithmic spiral.
In fact, the angle between the normal to the tangent and the radius of the logarithmic spiral is
equal to . Considering that the spiral represents a failure line, Mohr-Coulomb relationship holds between
the normal and shear stresses acting on it: (note that the cohesion is not considered in evaluating Nq).
Consequently, at any point of the spiral the resultant of and is directed as the radius and does not affect the rotational equilibrium about the centre G of the spiral.
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The equation of equilibrium about point G reads
where
Substituting the expressions of one obtains
(note that if =0, )
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2.5.2 Bearing capacity coefficient (conditions: )
Mohr circle for the passive zone
The slope of the failure planes in the passive zone
coincides with that obtained for the case.
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Mohr circle for the active zone
The slope of the failure planes in the passive
zone coincides with that obtained for the case.
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The expression of is obtained by imposing the rotational equilibrium of zone GEF with respect to point G. To this purpose it is necessary to express the stresses along planes GE, GF
and along the spiral arch EF.
The normal stress on plane GF is
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The normal stress on plane EG is
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The rotational equilibrium of zone GEF about point G involves the contribution of the stresses
acting on planes EG and GF and on the spiral arc EF. They are referred to, respectively, as M(1),
M(2), M(3).
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Let evaluate first the contribution of the stresses and , acting on the logarithmic spiral EF , to the rotational equilibrium of zone GEF.
The resultant of stresses and coincides with the radius and does not affect the equilibrium. Consequently, only the shear stress =c affects the equilibrium of the arch.
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The above integral is solved considering that and introducing the
following expressions: and
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Hence, the rotational equilibrium of zone EFG reads
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2.5.3 Evaluation of in undrained conditions (=0)
Rotational equilibrium about point G
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2.5.4 Relationship between and (Thorme des tats correspondants, Albert Caquot, 1881-1976)
Surface (2) is obtained by translating surface (1) by a compressive stress . As a consequence, should correspond to determined under an increase of normal stress .
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(which corresponds to
)
Hence, =
( )
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2.5.5 Evaluation of in the case of inclined foundation and inclined load
Angles and are known while and have to be determined as a part of the solution.
Mohr circle for the active zone
triangle OAC:
triangle OCQ:
since
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Let now derive the expressions of the principal stresses I and II
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The expression of is obtained by imposing the rotational equilibrium of zone GEF about point G
Active zone: Passive zone:
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Rotational equilibrium about point G
where
Note that when the above expression of reduces to its standard form
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2.5.6 Evaluation of in the case of inclined foundation and inclined load
According to Caquot theorem, the expression of can be derived from that of by considering an applied normal stress . Since the normal stress is applied also on the foundation plane, the computed inclination of does not coincide with the inclination of the applied load Q. Consequently, the calculation should be based on an initial inclination
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Equilibrium in the direction tangent to the foundation plane
{
{
[
]
(
)
[
]
(
)
Considering that ,
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{
[
]
(
)
As previously observed, some attempts are necessary by changing the value of until the correct values of and, consequently, of are obtained.
If the applied load is normal to the inclined foundation plane, i.e. if , also vanishes and the expression of becomes
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The expression of in undrained conditions can be obtained assuming and adopting the same procedure previously described for evaluating in the presence of inclined foundation and load.
(Fig.1)
Passive zone GFI
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Active zone GEH
(Fig. 2)
(Fig. 3)
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Considering Figs. 2 and 3:
(
)
Consider now Figs. 1 and 2.
Points A and Q in Fig.2 represent the stresses acting, respectively, on lines EG and GH in Fig.1.
Due to the properties of Mohr circle, the angle between the lines AC and CQ in Fig.2 is twice
the angle between the lines EG and HG in Fig.1. Consequently:
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The angle in Fig. 1 is evaluated from the Mohr circle in Fig. 2
The angle is determined from the Mohr circle in Figs. 3,
;
(
)
and the angle is determined from Fig.1
(
)
(note that must be positive, i.e. )
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The expression of is obtained by imposing the rotational equilibrium of the radial zone GEF about point G.
where
;
;
Substitution of the expressions of and
; ; (
)
into that of leads to
[ (
) ]
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Substituting the expressions of and of ,
(
)
and considering that since c=1, the following expression for is arrived at
[
]
The implicit structure of the above equation requires an iterative solution process. A trial value
of is introduced into its right hand side term, thus obtaining a refined value of . This value is introduced again into the equation and the process continues until stabilizes. To choose the initial value of consider that and that . Hence, must fall within the interval:
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2.5.7 Bearing capacity coefficient (conditions: )
It can be shown that the previous solutions for and based on the described simplified procedure coincide with the correct solutions obtained with the Methods of Characteristics (see
e.g.: R. Hill, The Mathematical Theory of Plasticity, Oxford Press, 1950).
The same procedure cannot be applied to the case of (i.e. ) because the active zone underneath the foundation is not anymore characterized by straight failure lines.
This is due to the fact that the simultaneous presence of a surface load and of the soil self-weight leads to a rotation of the principal stresses which are not vertical and horizontal anymore.
As a result, the expression of obtained with the described simplified procedure overestimates the limit value of the load carried by the foundation.
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An approximated evaluation of can be obtained assuming straight failure lines in the active
zone, but considering their inclination as an unknown.
Having worked out the expression of in terms of , is determined by minimizing it with
respect to .
Mohr circle for the passive zone
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The Mohr circle of the active zone coincides with that for the passive zone if and are exchanged with each other.
The expression of the limit load in terms of is obtained by writing the equation of rotational equilibrium of the radial zone about point G.
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Finally, the approximated value of is reached by minimizing with respect to .
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Approximated expressions for Expression of and
(Vesic)
(Lundgren)
(Spangler) ( )
Bearing capacity coefficients
Nc Nq N 0 5.14 1.0 0
5 6.5 1.6 0.1 0.5
10 8.4 2.5 0.4 1.2
15 11.0 3.9 1.2 2.5
20 14.8 6.4 3.0 5.0
25 20.7 10.7 6.8 9.7
30 30.1 18.4 15.1 19.7
35 46.1 33.3 33.9 42.4
40 75.3 64.2 79.5 100.4
45 113.9 134.9 200.8 297.5
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2.6 Correction factors for Terzaghis bearing capacity equation
The original equation proposed by Terzaghi for a shallow strip footing subjected to a vertical
load
was corrected by Hansen in order to apply it also in other conditions
s = correction factors accounting for the foundation shape (rectangular or circular)
d = correction factors accounting for the foundation depth ( ) i = correction factors accounting for the inclination of load
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2.6.1 Shape factors
(in the following L and B denote, respectively,
the major and minor sides of the foundation)
Factors proposed by Hansen: Factors proposed by Vesic:
Vesic factor cannot be used when =0, in fact: In this case it is sufficient to substitute within the expression of :
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2.6.2 Depth factors
These factors account for the shear resistance of the lateral soil.
They should be used with care because their contribution could
vanish if the lateral soil is excavated after the completion of the
structure. This applies also to .
Factors proposed by Hansen: Factors proposed by Vesic:
If
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2.6.3 Load inclination factors
The vertical limit load in the presence of a known horizontal load
can be evaluated introducing these factors in Terzaghis equation. Note that this is different from computing the inclined limit load
by modifying the bearing capacity coefficients N as it was previously shown. Note also that the
horizontal load cannot exceed the limit value corresponding to the sliding failure of the
foundation on the underlying soil:
Case of a strip footing ( )
If
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Case of rectangular footing
If
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2.6.4 Factors accounting for the inclination of the foundation base
If
Note that the above coefficients permit computing the limit vertical load. This is different from
computing the limit load normal to the inclined foundation plane.
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2.6.5 Factors accounting for the inclination of the ground surface
If
These coefficients can be used only if is substantially smaller than . It is always advisable to perform also a stability analysis of the slope subjected to the overall
load of the structure. This is mandatory when .
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2.6.6 Factor accounting for the foundation settlement
Besides the fact that the evaluation of the settlements is always mandatory, a suitable factor of
safety FS (2.53) has to be adopted with respect to the bearing capacity also to avoid excessive settlements under working loads.
To avoid excessive settlements when dealing with compressible soils, reduced shear strength
parameters and could be adopted in the calculations of the bearing capacity.
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Alternately, Vesic proposed the use of the following reduction factors in the bearing capacity
equation.
If
The above reduction factors are disregarded if .
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2.7 Shallow foundations subjected to eccentric load
Under the simplifying assumption of linear pressure distribution between footing and underlying
soil, the following conditions could occur depending on the load eccentricity.
The literature does not provide simple equations for evaluating the bearing capacity of footing
subjected to non-uniform load. To circumvent this drawback the bearing capacity is evaluated
adopting an equivalent footing of reduced size and with constant pressure distribution.
(Note that L always represents the largest side of the equivalent foundation)
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T
N 0
In the general case, the footing is subjected to normal and shear forces and to bending moment.
To evaluate the bearing capacity in these
conditions it is necessary to determine
the N-M-T domain of the footing.
This problem will be discussed
during the exercise classes.
N
M
T
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2.8 Shallow foundations on layered soil
Consider first the case of a soil deposit that could be roughly subdivided into two layers.
If the shear resistance of layer (1) is smaller than
that of layer (2), either the parameters of soil (1) are
adopted for evaluating the bearing capacity or,
more advisably, the foundation plane is brought
down into layer (2).
If the shear strength of layer (1) is larger than that
of layer (2), the load pressure is spread on layer (2).
Then, the limit values of q and q are evaluated. The design of the foundation is based on the least
of the two bearing capacities.
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In the case of a sequence of layers having appreciably different shear strength characteristics,
the bearing capacity can be evaluated using one of the methods of slices used for slope stability
analysis.
For a chosen shape of the failure surface, the value of qlim is determined by imposing the global
equilibrium of all slices with respect to point C.
The minimum value of is found by a trial and error process by changing:
A common feature of these methods is that they introduce suitable assumptions on the
interaction forces between the slices so that they do not appear in the rotational equilibrium of
the sliding wedge of soil about point C.
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Fellenius method (1936)
This method is based on the assumption that the lateral forces and acting on each slice are parallel to the base of the slice.
The force Ni is determined through the equilibrium of the slice in the
direction normal to its base
Knowing the forces Ni and Ti, the equilibrium of the slices is imposed about point C obtaining
the value of that corresponds to failure. The process is repeated changing the shape of the failure surface until the minimum value of is reached.
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Bishops method (1955)
This method is based on the assumption that the lateral forces acting
on each slice are normal to the face of the slide (i.e. that they are
horizontal).
Consequently, the expressions of forces and are
Also in this case the lateral forces and do not appear in the global equilibrium of the assemblage of slices.
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2.9 Remarks on the pore pressure effects
To show the influence of pore pressure, consider a partially submerged slice having an
horizontal base.
= unit weight of soil without pore liquid = unit weight of soil with pores partially filled with water = unit weight of fully saturated soil
S = Degree of saturation =
; n = porosity =
The equation of (total stress) equilibrium of the slice in the vertical direction reads:
where is the normal effective stress and p is the pore pressure at the slice base.
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The pore pressure p consists of three components
= pore pressure due to the steady state seepage previous to the construction, this includes also the hydrostatic pore pressure
= pore pressure due to consolidation, i.e. to the variation of volume caused by the change of volumetric stresses
= pore pressure related to the change in volume due to the plastic dilation
The pore pressure is also referred to as excess pore pressure with respect to the steady state conditions . The evaluation of the three components of the pore pressure p is necessary for determining the
limit shear stress at the base of the slice that governs the stability problem,
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The component can be measured by means of piezometers before the beginning of construction. The remaining components must be calculated on the basis of the mechanical
parameters of soil and of the characteristics of the collapse mechanism.
Due to the uncertainties in evaluating and , quite often the stability analysis is divided in two independent stages:
- Long term analysis based on the effective stress parameters and that account only for assuming that the excess pore pressure has already dissipated;
- Short term analysis based on the undrained cohesion assuming that this parameter accounts for the effects of the initial excess pore pressure .
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2.10 Structural details of shallow footings
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2.11 Interaction of adjacent footings
Shallow foundations in seismic zone