11 Bearing Capacity

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    CHAPTER 11. BEARING CAPACITY

    In fig. 11.1 is shown a stripfooting, which is a shallow foundation supporting a load-bearing wall. When establishing the area A of contact between the foundation andthe soil, two fundamental requirements must be satisfied:

    - to ensure safety against the risk of shear failure of the supporting soil (fig. 11.1a),

    - to limit the settlement s of the foundation to values allowable for the structureand for its normal exploitation (fig. 11.1 b).

    a. b.Fig. 11.1

    The problem ofbearingcapacity, this chapter is dealing with, refers to the first ofthe two above outlined requirements.

    Bearingcapacityrepresents the ability of a soil to carry a load.The allowablebearingcapacityis defined as the maximumpressure which may be

    applied to the soil such that the two fundamental requirements are satisfied.

    The ultimatebearingcapacityis defined as the leastpressure which would causeshear failure of the supporting soil immediately below and adjacent to a foundation.

    As shown in the chapter 7, the problem ofultimatebearingcapacity is a specialcase of limiting or plastic equilibrium in a soil mass.

    In the following paragraphs, the particular problem of the ultimatebearingcapacityof shallow foundations will be considered.

    11.1 Failure modes

    Present knowledge concerning the way in which failure of the soil supporting shallowfoundations takes place is based on analysis of both causes of accidents in whichvarious structures lost stability and interpretation of experimental data. Theexperiments were conducted, in general, at small scale in installations allowing tovisualize the trajects followed by soil particles during the process of gradual loadinguntil the failure condition was reached.

    On that basis, three main modes of failure were recognized, depending, in essence,on the ground conditions.

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    a. general shear failure

    Continuous failure surfaces develop between the edges of the footing and the groundsurface (fig. 11.2 a). As the pressure is increased towards the value of the ultimatebearing capacity pf, the state of plastic equilibrium is reached initially in the soilaround the edges of the footing then gradually spreads downwards and outwards.Ultimately, the state of plastic equilibrium is fully developed throughout the soil abovethe 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 bytilting of the footing, as shown in fig. 11.1 a. The load-settlement diagram, whichaccompanies this mode of failure, shown in the diagram a in fig. 11.3, puts intoevidence clearly the values of the ultimate bearing capacity pf for which deformationsincrease indefinitely. The transition from the initial, quasi-linear, part of the diagramand the point corresponding to pf is a short one.

    Fig. 11.2

    The general shear failure (sometimes named complete shear failure) is typical forsoils 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 footingand only partial development of the state of plastic equilibrium. The failure surfacesdo 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 capacityis not clearly defined and is characterized by the occurrence of relatively largesettlements. This mode of failure is associated with soils of medium to high

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    compressibility, (non-cohesive soils of medium relative density, cohesive soils ofmedium 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. Asthe pressure is increased, the foundation penetrates into the soil like a piston. Thereis no heave of the ground surface away from the edges and no tilting of the footing.The load-settlement diagram (c in fig. 11.3) shows that large settlements are alsocharacteristics to this mode of failure and the ultimate bearing capacity, like in thecase b, is not well defined. Punching shear failure, is associated with soils of veryhigh compressibility such as loose sands and soft clays.

    Fig. 11.3

    In cases of local shearand punching shear failures, the ultimate bearing capacityshould be defined based on a deformation criterion. Available experimental datashow that settlements of shallow foundations corresponding to a failure load are ofthe order of (3%...7%) B for clay soils and of (5%...15 %) B for sands where B is thewidth of the foundation. Hence, a settlement of 10% B could be adopted as adeformation criterion for any soil condition in order to define pf (fig. 11.4). It followsalso that plate load tests on compressible soils should be conducted to settlementsequal to at least 0.25 B, to be able to define the ultimate load from the load-settlement diagram.

    Fig. 11.4

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    Besides the nature of the soil, the mode of failure depends also on other factors suchas:

    - the depth of the foundation; punching shear failure will occur in a soil of lowcompressibility, for instance dense sands, if the foundation is located atconsiderable depth (deep foundation);

    - the kind of loading; a dense sand subjected to cyclic loading will exhibitpunching 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, andapunching shear failure when the rhythm of applying the load is slow and aftereach load stage the time required for the consolidation of the soil is provided.

    11.2 General hypothesis adopted for computing the ultimate bearingcapacity

    For the computation of the ultimate bearing capacity pf the following hypothesis areadopted:

    - a continuous failure surface characteristic for the general shear failure mode(fig. 11.5);

    Fig. 11.5

    - the failure condition ctanf += 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 lateralface of the foundation (EB) is neglected;

    - the friction between the soil located above and below the foundation level (onthe line BC) is neglected;

    - the friction between the base of the foundation (AB) and the soil to which it c.. incontact, is neglected.

    With these hypothesis, the soil located above the foundation level is replaced by asurcharge q = D, where D is the foundation depth.

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    11.3 Ultimate bearing capacity in the case of a failure surface made bytwo planes

    The two failure planes (fig. 11.6) have the inclinations in respect to the horizontal of

    )

    2

    45( o

    + and )

    2

    45( o

    , corresponding to the development in the mass of soil under

    the footing of two Rankine zones on both sides of a imaginary, fictitious, perfectlysmooth (frictionless) wall BD, namely the active zone on the left of the wall and thepassive zone on the right of the wall.

    Fig. 11.6

    Computing pf is based on expressing the active earth thrust Pa behind a vertical wallBD limited by an horizontal ground surface, on which a surcharge pf is applied, andthe passive resistance Pp in front of the same wall, limited by an horizontal groundsurface on which a surcharge q = D is applied (fig. 11.7).

    Fig. 11.7

    aafa2

    a KHc2KHpKH2

    1P += (11.1 a)

    ppp2

    p KHc2KHqKH2

    1P ++= (11.1 b)

    To find pf, the condition Pa = Pp is written, considering that:

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    po KB)

    245(tanBH =

    +=

    pa K

    1K =

    ppppp

    f

    p

    Kc2KqKH2

    1

    K

    1c2

    K

    p

    K

    1H

    2

    1++=+

    [ ] [ ]

    )KK(c2Kq)KK(B2

    1

    KKKc2KqKKKB21

    K

    Kc2H

    2

    1KKc2KqKH

    2

    1p

    21

    p23

    p2p

    21

    p25

    p

    ppp2pp

    2pp

    p

    ppp

    2p

    2pf

    +++=

    =+++=

    =+++=

    (11.2)

    The expression (11.2) can be put into the form:

    ++= NB2

    1NcNqp cqf (11.3)

    where N,N,N cq , named bearing capacity factors, are depending on the angle of

    internal friction , and have the following expressions:

    )KK(N);KK(2N;KN 21

    p25

    p21

    p25

    pc2pq =+==

    (11.4)

    11.4 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 ( )0;0c;0 ==

    b. frictionless, weightless soil ( 0;0;0c == )

    c. soil with weight ( 0 )

    a. In the case of a soil without cohesion and weight, a suitable failure mechanism fora strip footing is shown in fig. 11.8. The footing, of width B and infinite length, carriesa uniform pressure on the surface of a mass of homogeneous, isotropic soil. Whenthe pressure becomes equal to the ultimate bearing capacity pf the footing will bepushed downwards into the soil mass, producing a state of plastic equilibrium, in theform of an active Rankine zone, below the footing, the angles ABC and BAC being (

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    245o

    + ). The downward movement of the w

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