11
CHAPTER 4. BEARING CAPACITYIn fig. 4.1 is shown a strip footing,
which is a shallow foundation supporting a load-bearing wall. When
establishing the area A of contact between the foundation and the
soil, two fundamental requirements must be satisfied:
to ensure safety against the risk of shear failure of the
supporting soil (fig. 4.1 a),
to limit the settlement s of the foundation to values allowable
for the structure and for its normal exploitation (fig. 4.1 b).
a.
b.
Fig. 4.1The problem of bearing capacity, this chapter is dealing
with, refers to the first of the two above outlined
requirements.
Bearing capacity represents the ability of a soil to carry a
load.
The allowable bearing capacity is defined as the maximum
pressure which may be applied to the soil such that the two
fundamental requirements are satisfied.
The ultimate bearing capacity is defined as the least pressure
which would cause shear failure of the supporting soil immediately
below and adjacent to a foundation.
As shown in the chapter 7 (ar fi cap1), the problem of ultimate
bearing capacity is a special case of limiting or plastic
equilibrium in a soil mass.
In the following paragraphs, the particular problem of the
ultimate bearing capacity of shallow foundations will be
considered.
4.1 Failure modes
Present knowledge concerning the way in which failure of the
soil supporting shallow foundations takes place is based on
analysis of both causes of accidents in which various structures
lost stability and interpretation of experimental data. The
experiments were conducted, in general, at small scale in
installations allowing to visualize the trajects followed by soil
particles during the process of gradual loading until the failure
condition was reached.
On that basis, three main modes of failure were recognized,
depending, in essence, on the ground conditions.
a. general shear failure
Continuous failure surfaces develop between the edges of the
footing and the ground surface (fig. 4.2 a). As the pressure is
increased towards the value of the ultimate bearing capacity pf,
the state of plastic equilibrium is reached initially in the soil
around the edges of the footing then gradually spreads downwards
and outwards. Ultimately, the state of plastic equilibrium is fully
developed throughout the soil above the failure surfaces. Heave of
the ground surface occur on both sides of the footing, although the
final slip movement would occur only on one side, accompanied by
tilting of the footing, as shown in fig. 4.1 a. The load-settlement
diagram, which accompanies this mode of failure, shown in the
diagram a in fig. 4.3, puts into evidence clearly the values of the
ultimate bearing capacity pf for which deformations increase
indefinitely. The transition from the initial, quasi-linear, part
of the diagram and the point corresponding to pf is a short
one.
Fig. 4.2
The general shear failure (sometimes named complete shear
failure) is typical for soils of low compressibility (dense sands,
stiff clays) and for rocks.
b. local shear failure
In this mode of failure, there is significant compression of the
soil under the footing and only partial development of the state of
plastic equilibrium. The failure surfaces do not reach the ground
surface and tilting of the foundation is unlikely to occur. The
load-settlement diagram (b in the fig. 11.3) shows that the
ultimate bearing capacity is not clearly defined and is
characterized by the occurrence of relatively large settlements.
This mode of failure is associated with soils of medium to high
compressibility, (non-cohesive soils of medium relative density,
cohesive soils of medium consistency).
c. punching shear failure
This mode of failure occurs when there is compression of the
soil under the footing, accompanied by shearing in the vertical
direction around the edges of the footing. As the pressure is
increased, the foundation penetrates into the soil like a piston.
There is no heave of the ground surface away from the edges and no
tilting of the footing. The load-settlement diagram (c in fig. 4.3)
shows that large settlements are also characteristics to this mode
of failure and the ultimate bearing capacity, like in the case b,
is not well defined. Punching shear failure, is associated with
soils of very high compressibility such as loose sands and soft
clays.
Fig. 4.3
In cases of local shear and punching shear failures, the
ultimate bearing capacity should be defined based on a deformation
criterion. Available experimental data show that settlements of
shallow foundations corresponding to a failure load are of the
order of (3%...7%) B for clay soils and of (5%...15 %) B for sands
where B is the width of the foundation. Hence, a settlement of 10%
B could be adopted as a deformation criterion for any soil
condition in order to define pf (fig. 4.4). It follows also that
plate load tests on compressible soils should be conducted to
settlements equal to at least 0.25 B, to be able to define the
ultimate load from the load-settlement diagram.
Fig. 4.4Besides the nature of the soil, the mode of failure
depends also on other factors such as:
the depth of the foundation; punching shear failure will occur
in a soil of low compressibility, for instance dense sands, if the
foundation is located at considerable depth (deep foundation);
the kind of loading; a dense sand subjected to cyclic loading
will exhibit punching shear failure;
the rhythm of loading; a saturated, normally consolidated clay,
exhibits a general shear failure under a sudden loading, when no
volume change takes place, and a punching shear failure when the
rhythm of applying the load is slow and after each load stage the
time required for the consolidation of the soil is provided.
4.2 General hypothesis adopted for computing the ultimate
bearing capacity
For the computation of the ultimate bearing capacity pf the
following hypothesis are adopted:
a continuous failure surface characteristic for the general
shear failure mode (fig. 4.5);
Fig. 4.5
the failure condition is fulfilled in each point of the failure
surface;
the shear strength of the soil between the level of the
foundation and the ground surface (part CD of the failure surface)
is neglected;
the friction between the soil above the level of the foundation
and the lateral face of the foundation (EB) is neglected;
the friction between the soil located above and below the
foundation level (on the line BC) is neglected;
the friction between the base of the foundation (AB) and the
soil to which it c.. in contact, is neglected.
With these hypothesis, the soil located above the foundation
level is replaced by a surcharge q = D, where D is the foundation
depth.
4.3 Ultimate bearing capacity in the case of a failure surface
made by two planes
The two failure planes (fig. 4.6) have the inclinations in
respect to the horizontal of and , corresponding to the development
in the mass of soil under the footing of two Rankine zones on both
sides of a imaginary, fictitious, perfectly smooth (frictionless)
wall BD, namely the active zone on the left of the wall and the
passive zone on the right of the wall.
Fig. 4.6
Computing pf is based on expressing the active earth thrust Pa
behind a vertical wall BD limited by an horizontal ground surface,
on which a surcharge pf is applied, and the passive resistance Pp
in front of the same wall, limited by an horizontal ground surface
on which a surcharge q = D is applied (fig. 4.7).
Fig. 4.7
(4.1 a)
(4.1 b)
To find pf, the condition Pa = Pp is written, considering
that:
(4.2)
The expression (11.2) can be put into the form:
(4.3)
where , named bearing capacity factors, are depending on the
angle of internal friction , and have the following
expressions:
(4.4)
4.4 Ultimate bearing capacity in the case of a curved failure
surface
The problem is solved in three phases, corresponding to the
following conditions:
a. cohesionless, weightless soil (
b. frictionless, weightless soil ()
c. soil with weight ()
a. In the case of a soil without cohesion and weight, a suitable
failure mechanism for a strip footing is shown in fig. 4.8. The
footing, of width B and infinite length, carries a uniform pressure
on the surface of a mass of homogeneous, isotropic soil. When the
pressure becomes equal to the ultimate bearing capacity pf the
footing will be pushed downwards into the soil mass, producing a
state of plastic equilibrium, in the form of an active Rankine
zone, below the footing, the angles ABC and BAC being (). The
downward movement of the wedge ABC forces the adjoining soil
sideways, producing outward lateral forces on both sides of the
wedge. Passive Rankine zones ADE and BGF develop on both sides of
the wedge ABC, the angles DEA and GFB being (). The transition
between the downward movement of the wedge ABC and the lateral
movement of the wedges ADE and BGF takes place through zones of
radial shear ACD and BCG. In his solution, Prandtl admits that the
surfaces CD and CG are logarithmic spirals, to which BC and ED, or
AC and FG, are tangential. The equation of the spiral is where is
the angle between the initial radius ro and the one corresponding
to a point on the spiral; is the angle made by the radius with the
normal in any point of the spiral. A state of plastic equilibrium
exists above the surface EDCGF, the remainder of the soil mass
being in a state of elastic equilibrium.
Fig. 4.8
To find pf, first the equilibrium of the wedges ABC and BDE, as
equilibrium of forces on vertical direction, will be considered.
Then, the equilibrium of the transition zone BCD, as equilibrium of
moments toward the point B, will be written.
On the conjugated failure planes AC and CB are acting the
reactions RI, making an angle with the normal (fig. 4.9 a).
The equation of projection of forces on the vertical
direction:
(4.5)
On the conjugated failure planes BD and DE are acting the
reactions RIII, making an angle with the normal (fig. 11.9 b).The
equation of projection of forces on the vertical direction:
(4.6)
The equilibrium of the transition zone II (fig. 4.9 c) is
expressed in terms of the moment around the point B.
Fig. 4.9
The arc of the spiral CD belongs to the failure surface,
therefore the reaction RII makes an angle with the normal to the
arc. Hence, the direction of RII coincides with the direction of
the radius and RII produces no moment in respect to B. The moment
equation becomes:
But r1 = ro
(4.7)
By writing:
equation (4.7) becomes:
pf = q Nq
(4.8)
From (11.8) follows that, in the case of a cohesionless and
weightless material, there is a bearing capacity only if there is a
surcharge q.
To consider the effect of the cohesion, a normal stress equal to
c cot is added to the normal stresses p and q. The equation (11.8)
becomes:
(4.9)
By writing
equation (11.9) becomes:
pf = q Nq + c Nc
Nq and Nc are bearing capacity factors depending on .
An additional term should be added to equation (4.10) to take
into account the self-weight of the soil. Experimental observations
showed that a wedge of soil remaining in elastic state, with faces
making an angle with the horizontal, is developed below the
foundation and moves downwards together the foundation, tending to
produce the lateral movement of the soil along the failure surfaces
CDE and CFG (fig. 4.10). The passive resistance of the soil mass
above the failure surfaces is mobilized. The problem consists on
computing the passive resistance force Pp of a mass of soil (),
limited by a horizontal ground surface, behind a wall BC with
inclination and height H = .
Fig. 4.10
The failure surface CDE is made of the line DE, corresponding to
the passive Rankine zone BDE, and by the arc of logarithmic spiral
CD.
The passive resistance force Pp can be expressed:
(4.11)
The equilibrium of the elastic wedge ABC:
(4.12)
The ultimate bearing capacity is:
(4.13)
The following notation was used:
Terzaghi assumed that and obtained the value of the passive
resistance force in the hypothesis of a curved failure surface.
Adding the additional term bringing the effect of the self-weight
of the soil, the expression of the ultimate bearing capacity pf
becomes:
(4.14)
Relations of the kind of (4.14) were established by Terzaghi and
other authors. Most of them differ only with respect of the third
component, introducing the influence of the self-weight of the
soil. These relations are theoretically incorrect for a plastic
material since they are superposing terms corresponding to
different failure figures such as those represented in fig. 4.8 and
4.10. However, the error implied is considered to be on the safe
side and is accepted in engineering practice.
4.4 Ultimate bearing capacity in the case of a purely cohesive
soil
This is a particular case of the problem previously considered.
The failure mechanism shown in fig. 11.8 is transformed, when , in
the one shown in fig. 4.11.
Equation (11.10) becomes:
(4.15)
(For , Nq = 1)
Fig. 4.11
One defines as netto ultimate bearing capacity the difference
between the critical pressure in the geological pressure at the
level of the foundation base:
(4.16)
The problem is to find the bearing capacity factor Nc for this
case (.
An approach similar to the one used for the case ( is
adopted:
Equilibrium of forces acting on the prism I (fig. 4.12 a)
(4.17)
The normal stress acting on the faces AC and BC:
(4.18)
Equilibrium of forces acting on the prism III (fig. 4.12 b)
(4.19)
Fig. 4.12
The normal stress acting on the faces BD and DE:
(4.20)
pf is obtained by writing the condition that the moment of all
forces acting on the failure prism, in respect to the point B, is
zero. Normal pressures acting on the circular are CD having the
direction of the radius, do not give moment toward B.
(4.21)
But AC = BC = BD = DE = r
Relation (4.21) becomes:
;
(4.22)
(4.23)
(4.24)
Skempton has shown that, in fact, the netto ultimate bearing
capacity increases with the depth D of the foundation until a depth
D = 5B (fig. 4.13), reaching a limit value 9 for Nc.
Fig. 4.13
For rectangular foundations B x L, for which , Skempton proposed
the relation:
(4.25)
PAGE 104
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