General Bearing Capacity

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General Bearing Capacity

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  • Foundation EngineeringFoundation Engineering

    Lecture #04Lecture #04

    The Bearing Capacity of Soils

    -Terzaghis Ultimate Bearing Capacity-Terzaghis Ultimate Bearing Capacity-Meyerhofs Method

    -Brinch Hansen Method-Vesics Method

    Luis A. Prieto-Portar, 2009

  • These wheat storage silos in Canada are an example of the bearing capacity failure of a large foundation. This failure alerted engineers to the mechanism of how surface loads may

    exceeded the shear strength of the soils beneath the foundation.

  • Some foundations in congested cities must consider how they transfer loads onto adjacent utilities and the adjacent buildings foundations, in addition to the underlying soil.

    This is a typical excavation to repair a utility in a New York City street.

  • Torontos Queen and Yonge Street intersection.

  • A shallow foundation must:1. be safe against an overall shear failure in the soil that supports it. 2. cannot experience excessive displacement (in other words, settlement).

    The definitions of bearing capacity are,qo is the actual contact pressure of the soil at the footings invert;qult is the load per unit area of the foundation at which the shear failure in soil occurs and is called the ultimate bearing capacity of the foundation; andqall is the load per unit area of the foundation that is supported without an unsafe movement of the soil, and is called the allowable bearing capacity.

    CONCENTRIC LOAD ( NO ECCENTRICITY)CONCENTRIC LOAD ( NO ECCENTRICITY)

    B X LB

    q

    Q

    y

    xFigure 1.

  • A continuous footing resting on the surface of a dense sand or a stiff cohesive soil is shown in Figure 2a with a width of B. If a load is gradually applied to the footing, its settlement will increase. When the load per unit area equals qult a sudden failure in the soil supporting the foundation will take place, with the failure surface in the soil extending to the ground surface. This type of sudden failure is called a general shear failure.

    If the foundation rests on send or clayey soil of medium compaction (Figure 2b), an increase of load on the foundation will increase the settlement and the failure surface will gradually extend outward from the foundation (as shown by the solid line). When the load per unit area on the foundation equals qult, the foundation movement will be like sudden jerks. A considerable movement of the foundation is required for the failure surface in soil to extend to the ground surface (as shown by the broken lines). The load per unit area at which extend to the ground surface (as shown by the broken lines). The load per unit area at which this happens is the ultimate bearing capacity qult. Beyond this point, an increase of the load will be accompanied by a large increase of footings settlement. The load per unit area of the footing qult, is referred to as the first failure load (Vesic 1963). Note that the peak value of qis not realized in this type of failure, which is called the local shear failure in soil.

    If the foundation is supported by a fairly loose soil, the load-settlement plot will be like the one in Figure 2c. In this case, the failure surface in soil will not extend to the ground surface. Past the value qult, the load-to-settlement plot will be steep and practically linear. This type of failure is called the punching shear failure.

  • S e t t l e m e n t

    L o a d / u n i t a r e a , q

    q u

    B L o a d / u n i t a r e a , q

    q u

    q u ( 1 )

    B

    ( a ) F a i l u r ei n s o i ls u r f a c e

    Dense sands and stiff cohesive soils.

    S e t t l e m e n t

    L o a d / u n i t a r e a , q

    q uq u

    S u r f a c ef o o t i n g

    B

    ( c )s u r f a c e F a i l u r e

    ( b ) F a i l u r es u r f a c e

    Figure 2. The nature of bearing capacity failures in soils: (a) the general shear failure; (b) the local shear failure; and (c) the punching shear failure. (Redrawn from Vesic, 1973).

    Soils of medium density or stiffness.

    Loose and soft soils.

  • R e l a t i v e d e n s i t y D0 . 6

    5 0 0

    1 0 0

    2

    a

    n

    d

    1

    u

    B

    8 0 7 0

    9 0

    2 0 0

    3 0 0

    4 0 0

    Buq

    1

    P u n c h i n g S h e a r

    0 . 3

    7 0 06 0 0

    0 . 2 0 . 4

    L o c a l S h e a r

    0 . 5

    G e n e r a l S h e a r

    r

    0 . 7 0 . 8

    2

    u

    (

    1

    )

    6 0

    S m a l l s i g n s i n d i c a t ef i r s t f a i l u r e l o a d

    8 i n . C i r c u l a r p l a t e6 i n . C i r c u l a r p l a t e4 i n . C i r c u l a r p l a t e2 i n . C i r c u l a r p l a t e2 X 1 2 i n . R e c t a n g u l a r p l a t e( r e d u c e d b y 0 . 6 )

    D r y u n i t w e i g h t ( l b / f t )8 5

    2 0

    1 0

    3 0

    21

    4 0

    5 0

    12 B

    9 0

    u ( 1 )q

    3

    9 5

    a

    n

    d

    B

    7 0 B1

    L e g e n d

    Figure 3. Variation of qu(1) / B and qu / B for circular and rectangular plates on the surface of sand. (Adapted from Vesic, 1963).

  • Based on experimental results from Vesic (1963), a relation for the mode of bearing capacity failure of foundations can be proposed (Figure 4), where

    Dr is the relative density in sand,Df is the depth of the footing measured from the ground surface,B is the width and L is the length of the footing (Note: L is always greater than B).

    0 . 4R e l a t i v e d e n s i t y D

    P u n c h i n g S h e a r f a i l u r e

    1

    00 . 2

    r

    G e n e r a l S h e a rL o c a l S h e a rf a i l u r e f a i l u r e

    0 . 6 0 . 8 1 . 0

    B5

    4 D f

    3

    D

    f

    /

    B

    f a i l u r e

    2

    f a i l u r e f a i l u r e

    Figure 4. Modes of foundation failure in sand (after Vesic, 1963).

  • Figure 5. Range of settlement of circular and rectangular plates at ultimate loads for Df / B = 0 in sand (after Vesic, 1973).

  • Terzaghis Ultimate Bearing Capacity Theory:

    Using an equilibrium analysis, Karl Terzaghi expressed in 1943 the ultimate bearing capacity qu of a particular soil to be of the form,

    (for strip footings, such as wall foundations)

    (for square footings, typical of interior columns)

    (for circular footings, such as towers, chimneys)

    = c + + 0.5 B

    = 1.3c + + 0.4 B

    u c q

    u c q

    q N q N N

    q N q N N

    (for circular footings, such as towers, chimneys)

    where, q = Df is the removed pressure from the soil to place the footing, Nc, N, and Nqare the soil-bearing capacity factors, dimensionless terms, whose values relate to the angle of internal friction . These values can be calculated when is known or they can be looked up in Terzaghis Bearing Capacity Factor Table.

    Terzaghis bearing capacity equations have been modified to take into account the effects of the foundation shape (B/L), depth (Df) and the load inclination (i). This will be discussed further in the next section for the General Bearing Capacity equation.

    = c + + 0.3 B u c qq N q N N

  • (1970)

    (1951, 1963)

  • The factor of safety FS against a bearing capacity failure is thus defined as,

    where qall is the gross allowable load-bearing capacity and qnet is the net ultimate bearing capacity.

    The factor of safety is chosen according the function of the structure, but never less than 3 in all cases.

    The net ultimate bearing capacity is defined as the ultimate pressure per unit area of

    =ult

    allqqFS

    The net ultimate bearing capacity is defined as the ultimate pressure per unit area of the footing that can be supported by the soil in excess of the pressure caused by the surrounding soil at the foundation level.

    Therefore,

    A footing will obviously not settle at all if the footing is placed at a depth where the weight of the soil removed is equal to the weight of the columns load plus the footings weight.

    = = net ult ult fq q q q D

  • Choosing the Bearing Capacity Factors.

    Based on laboratory and field studies of bearing capacities, the basic failure surface in soil suggested by Terzaghi, has been modified by Meyerhof, Hansen and Vesic. The relation for Nq was presented by Reissner (1924),

    whereas the equation for Nc was originally derived by Prandtl in 1921 to explain the behavior of hot steel being extruded through dies,

    2 tan tan tan 45 e e

    2

    = + =

    q PN Kpi pi

    Nc = (Nq 1) cot

    and finally, Caquot and Kerisel (1953) and Vesic (1973) proposed a relation for N , latermodified by Coduto as,

    2( 1) tan '1 0.4sin(4 ')

    qNN

    +

    =

    +

  • A) Terzaghis Bearing Capacity Factors.

    2 0 75 2

    22 45 21

    5 7 0 0

    2 1

    =

    +

    = = = >

    +

    ' '( . / )tan

    q '

    q' 'c c '

    '

    eNcos ( / )

    NN . for or N for

    tan

    ( N )tan

    pi pi pi pi

    2 1

    1 0 4 4+

    =

    +

    '

    q'

    ( N )tanN

    . sin( )

  • B) Meyerhofs Bearing Capacity Factors.

    C) Brinch Hansens Bearing Capacity Factors.

    2 45 2

    1

    1 1 4

    'tan 'q

    '

    c q

    '

    q

    N e tan ( / )N ( N )cotN ( N )tan( . )

    pi pi pi pi

    = +

    =

    =

    2 45 2

    1

    1 5 1

    'tan 'q

    '

    c q

    '

    q

    N e tan ( / )N ( N )cotN . ( N )tan

    pi pi pi pi

    = +

    =

    =

  • ANGLE OF FRICTION TERZAGHI MEYERHOF HANSEN (DEGREES) Nc Nq N Nc Nq N Nc Nq N

    0 5.70 1.00 0.00 5.1