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There are several types of shallow foundations by djh75337

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									TYPES OF SHALLOW FOOTINGS



                                       bc
                                 - column width

                                                Critical
           Critical section                     section
            for bending                        for shear
                                                           1
                                                                   2
                  de                                           150mm (min)
  - effective depth
  determined from                                              75mm
  bending and shear                                            (min) cover to
                                   B                           earth face
                             Bmin = bc + 300 mm
                       B is determined by satisfying
                       serviceability and ultimate limit states.

                                     (a) Isolated footing




 (b) Isolated footing and floor
               slab
(c) Combined footing               (d) Mat foundation




(e) Waffle mat                                          (f) Tendons layout in a post-tensioned mat

               bw
          - wall width



                    0.5 bw (min)


                               bw (min)


            2bw


(g) Strip footing


                                                         (h) A strip footing under construction
(i) Location of footings near slopes




(j) Location of footings on s slope
                                          (k) Stepped footings
                 Figure 4.1 Shallow footings and approximate initial dimensions.




There are several types of shallow foundations. Some are shown in Figure 4.1. A spread or
individual forting, also called isolated footing, transfers individual column loads directly to the
soil (Figures 4.1a,b). If a single spread footing interferes with another spread footing, the two can
be combined to form a combined footing (Figure 4.1c). When there is a large number of spread
footings required for a structure it may he more economical to use a mat or raft foundation. If the
overburden pressure of a soil removed to embed a raft foundation is equal to the stress imposed
by the structure, the raft is termed fully compensated or floating raft (Figure 4.1d). If the
overburden pressure is a fraction of the imposed stress, the raft is partially compensated. Some
rafts are thickened in the form of a waffle as shown in Figure 4.1e. Post-tensioned mat or floor
slab are often located just below the surface (Figure 4.1f).     Raft foundations are also integrated
with pile foundations. These are called pile-raft foundations. Walls are normally supported
using a strip foundation (Figures 4.1g,h). A strip foundation is one in which the width (B) to
length (L) ratio is small (B/L >> 10). On sloping ground, a footing can rest on top of the slope,
on the slope or can be stepped as illustrated in Figures 4.1i,j,k.


THEORETICAL BEARING CAPACITY BASED ON LIMIT EQUILIBRIUM METHOD
       The theoretical bearing capacity equations in common use in design practice were
obtained using the limit equilibrium method. Prandtl (1920) showed theoretically that a wedge
of material is trapped below a rigid plate when it is subjected to concentric loads. Terzaghi
(1943) applied Prandtl's theory to a strip footing (length is much longer than its width) with the
assumption that the soil is a semi-infinite, homogenous, isotropic, weightless rigid-plastic
material. Based on Prantl’s theory, failure of the footing occurs by a wedge of soil below the
footing pushing its way downward into the soil. The failure mechanism postulated by Terzaghi
(1943) is shown in Figure 4.2a.
                            Df                                 F

                 Considered by
                 Meyerhof




                           Df




                                 Df




                                      Figure 4.2 Failure mechanisms


       The zone ABC is a rigid wedge of soil trapped beneath the footing. Zone ABD is fan like
with radial slip lines stopping on a logarithm spiral slip line. The other, zone ADE, consists of
                                            φ′          φ′
slip lines oriented at angles of 45 +          and 45 −    to the horizontal and vertical planes,
                                            2           2
respectively. The zone ADE is called the Rankine passive zone. The surfaces, AB and AD, are
frictional sliding surfaces and their actions are similar to rough walls being pushed into the soil.
The pressure exerted is called passive earth pressure. The shape of the surface BD depends on
            γB
the ratio        where γ is the unit weight of the soil and σzo is the overburden pressure. For a
            σ zo
                    γB
weightless soil,         → 0, and BD is a logarithmic spiral; for γB = 0, BD is a circle; for γB ≠ 0 ,
                    σ zo
BD is between a circle and a logarithm spiral. For clean, coarse-grained soils, BD is always a
circle (Vesic, 1973). For dense coarse-grained soils or heavily over-consolidated fine-grained
soils, the failure planes are expected to reach the ground surface and the failure mode is called
general shear failure. For loose coarse-grained soils or lightly over-consolidated fine-grained
soils, the failure planes, if they are developed, are expected to lie within the soil layer below the
base of the footing and extend laterally. The failure mode is called local shear. Another type of
shear failure is possible. For very loose soil, the failure surfaces may be confined to the surfaces
of the rigid wedge. This type of failure is termed punching shear.


Each of these collapse mechanism depends on the relative density of the soil and the depth of
embedment (Figure 4.3).




Figure 4.3 Collapse mechanism based on relative density and embedment depth (Source: Vesic,
1973)


In saturated fine-grained soils, the settlement to mobilize general shear failures under a surface
footing is 3% to 7% of the foundation width and about 15% for deeply embedded footings. For
coarse-grained soils, the corresponding settlements are 5% to 15% for surface footings and 25%
for deeply embedded footings.




We will consider the theoretical bearing capacity equations for short-term (undrained condition)
and long-term (drained condition) separately.        A total stress analysis (TSA) utilizing the
undrained shear strength, su, is employed for short-term loading of fine-grained soils. An
effective stress analysis (ESA) utilizing the effective friction angle, φ′, is employed for long-term
and short-term loading of fine-grained and coarse-grained soils. All the equations are based on
the following assumptions.
1. The embedment depth is not greater than the width of the footing (Df < B)
2. General shear failure occurs
3. The shear strength of the soil above the footing base is negligible (Terzaghi’s and Hansen’s
assumption).
4. The soil above the footing base can be replaced by a surcharge stress (= γDf).
5. The soil is a rigid-plastic, homogenous medium. No displacement occurs prior to collapse.
The appropriate friction angle to use for an ESA is the peak friction angle,φ′p. However, φ′p is
not a fundamental soil parameter. Its magnitude depends on the ability of the soil to dilate which
in turn depends on the relative density and the normal effective stress.
         We will consider the theoretical net ultimate bearing capacity (qult) of shallow footings
based on limit equilibrium proposed by Meyerhof (1963), Vesic (1973) and Hansen (1970).
Terzagi’s equation is not given because it has been generally superseded in engineering practice
by Meyerhof (1963), Vesic (1973) and Hansen (1970) methods. The author has modified some
of these equations to separate total stress analysis from effective stress analysis.


Meyerhof’s (1963) equations
Vertical load

TSA: q ult = 5.14s u s c d c + γD f

ESA: q ult = γD f (N q − 1)s q d q + 0.5γB′N γ s γ d γ

Inclined load

TSA: q ult = 5.14s u d c i c + γD f

ESA: q ult = γD f (N q − 1)d q i q w q + 0.5γB′N γ d γ i γ w γ

where

         π tan φ′                  φ′
                                        ) ; N γ = (N q − 1) tan(1.4φ′ ) and the other factors are given in Table 4.1.
                                    p
Nq = e          p
                    tan 2 (45o +                                    p
                                   2
Vesic’s (1973) equations
Vertical load

TSA: q ult = 5.14s u s c d c + γD f

ESA: q ult = γD f (N q − 1)s q d q w q + 0.5γB′N γ s γ d γ w γ

         π tan φ′                  φ′
                                        ) ; N γ = 2(N q + 1) tan φ′ and the other factors are given in Table 4.1.
                                    p
Nq = e          p
                    tan 2 (45o +                                  p
                                   2

Hansen’s (1970) equations
All loads


TSA: qult = 5.14 su (1 + sc + dc - ic - bc - gc)
ESA: qult = γ Df (Nq - 1) sq dq iq bq gq wq + 0.5 γ B′ Nγ sγ dγ bγ gγ wγ
where

         π tan φ′                  φ′
                                        ) ; N γ = 1.5(N q − 1) tan φ′ and the other factors are given in Table 4.1
                                    p
Nq = e          p
                    tan 2 (45o +                                    p
                                   2

The factors, Nq and Nγ, are bearing capacity factors that are functions of φ′p; sc, sq and sγ are
shape factors, dc, dq, dγ are embedment depth factors, ic, iq, iγ are load inclination factors, bc, bq,
bγ are base inclination (base tilt) factors, gc, gq, gγ are ground inclination factors, wq and wγ are
ground water factors. Various equations have been proposed for the various factors as given in
Table 1.            The bearing capacity equations apply for a single resultant load with normal Vn and
horizontal components HB parallel to the width, B (the short side) and horizontal components HL
parallel to the length, L (the long side). When investigating potential failure along the short side,
use HL and for failure along the long side, use HB.
Table 4.1 Geometric factors for use in theoretical bearing capacity equations

Shape factors
                                                                   Hansen
           Meyerhof                               Vesic
   sc                B′                                B′              B′
           1 + 0.2                                1+               0.2    ; θ = 0 ; vertical loads only
                     L′                                L′              L′
                                                                          B′
                                                                   0.2i cB ; for inclined loads, short side
                                                                          L′
                                                                   failure
                                                                             L′
                                                                   0.2i cB      ; for inclined loads, long side
                                                                             B′
                                                                   failure
   sq                     B′                           B′              B′
           1 + 0.1K p                             1+      tan φ′   1+     sin φ′ ; θ = 0 ; vertical loads
                          L′                           L′              L′
                                                               p                p


                               φ′                                           B′
           K p = tan 2 (45 +
                                p
                                    ); φ′ > 10o                    only 1 +    i qB sin φ′ ; for inclined
                                                                            L′
                                        p                                                p
                               2
                                                                   loads, short side failure
                                                                         L′
                                                                   1+       i qL sin φ′ ; for inclined loads,
                                                                         B′
                                                                                      p

                                                                   long side failure
   sγ                     B′                                  B′             B′
           1 + 0.1K p                             = 1 − 0.4        1 + 0.4      φ′ ; θ = 0 ; vertical loads
                          L′                                  L′             L′
                                                                                 p


                               φ′                                  only
                                    ); φ′ > 10o
                                p
           K p = tan 2 (45 +            p                                    B′i γB
                               2                                   1 − 0.4          ; for inclined loads, short
                                                                             L′i γL
                                                                   side failure, sγ > 0.6 should be used
                                                                             L′i γL
                                                                   1 − 0.4          ; for inclined loads, long
                                                                             B′i γB
                                                                   side failure, sγ > 0.6 should be used
Depth factors
                                                                                            Hansen
             Meyerhof                                     Vesic
    dc                 Df                                           Df                          Df
             1 + 0.2                                      1 + 0.4      ; D ≤ B′ ; see       0.4      ; D ≤ B′ ; see note 1
                       B′                                           B′    f
                                                                                                 B′     f

                                                          note 1                                       D
                                                                           Df               0.4 tan −1 f ; Df > B′ ; see note 1
                                                          1 + 0.4 tan −1        ;                      B′
                                                                           B′
                                                          Df > B′ ; see note 1
    dq                      Df                                                                                   2 Df
             1 + 0.1 K p       ; φ′ > 10o
                             ′ p                                                    D f 1+2 tan φ′ (1-sin φ′ ) B′ ; D ≤ B′
                                                                                                   p         p
                            B                             1+2 tan φ′ (1-sin φ′ ) 2
                                                                                    B′
                                                                   p         p
                                                                                                                           D
                                                                                          1+2 tan φ′ (1-sin φ′ ) 2 tan -1 ( f )
                                                                                                                           B′
                                                                                                   p         p
                                                          1+2 tan φ′ (1-sin φ′ ) 2 tan -1
                                                                   p         p
                                                                                          ;D>B   ′
                                                          ; see note 1                       see note 1
    dγ                      Df                            1;   D f < B′                     1;   D f < B′
             1 + 0.1 K p       ; φ′ > 10o
                            B′
                                  p

Notes: note 1 – set all depth factors to 1 if the shear strength of the soil above the base is small
Load inclination factors
                                                                      Hansen
              Meyerhof
    ic                    2
              ⎛ θo ⎞                                                                     Hi
              ⎜1 − ⎟ ; load inclined in direction of                  0.5(1 − 1 −               ) ; see note 2
              ⎝ 90 ⎠                                                                   Vn A′s u
              footing width                                           Q is the applied vertical load with Vn as the normal
                                                                      load, Hi is the horizontal load, the subscript i
                    ⎡ ⎛    α ⎞       ⎤                                denotes direction.
              cos θ ⎢1− ⎜1− a ⎟ sin θ⎥ ; load inclined in                          Q
                    ⎣ ⎝ π+ 2 ⎠       ⎦                                                      Vn
              direction of footing length                                              +θ
                                                       2 1
              αa = adhesion factor that is usually      to                    HB
                                                       3 2
              for short-term loading


                                                                                       HL         L
                                                                             B


    iq                    2
                                                                                 Hi 5
              ⎛ θo ⎞                                                  (1 − 0.5      ) ; 45o ≤ φ′ ≤ 300 ;i q > 0 ; see note
              ⎜1 − ⎟ ; load inclined in direction of                             Vn
                                                                                               p

              ⎝ 90 ⎠
                                                                      3
              footing width                                                                       5
                                                                      ⎛           ηo             ⎞
                                                                      ⎜    (0.7 −     )H i       ⎟
                    ⎛    sin θ      ⎞                                             450
                                                                      ⎜1 −                       ⎟ ; for η = 90
                                                                                                          o     0
              cos θ ⎜1 −            ⎟ ; load inclined in
                    ⎜ sin φ′        ⎟                                 ⎜          Vn              ⎟
                    ⎝         p     ⎠                                 ⎜                          ⎟
              direction of footing length                             ⎝                          ⎠

    iγ                    2
                                                                                 Hi 5
              ⎛ θo ⎞                                                  (1 − 0.7      ) ; 45o ≤ φ′ ≤ 300 ;i γ > 0 ; see note
              ⎜1 − ⎟ ; load inclined in direction of                             Vn
                                                                                               p

              ⎝ 90 ⎠
                                                                      2
              footing width                                                                       5
                                                                      ⎛        ηo                ⎞
                    ⎛    sin θ      ⎞                                 ⎜ (0.7 −     )H i          ⎟
                                                                      ⎜1 −     450               ⎟ ; for η = 90
                                                                                                          o     0
              cos θ ⎜1 −            ⎟ ; load inclined in
                    ⎜ sin φ′        ⎟                                 ⎜       Vn                 ⎟
                    ⎝         p     ⎠                                 ⎜                          ⎟
              direction of footing length                             ⎝                          ⎠


Notes: note 2– i c ,i q , i γ = i cB ,i qB , i γB for the horizontal force along the width, Hi = HB, (failure along the long
side, HB dominant) or i c , i q , i γ = i cL , i qL , i γL for the horizontal force along the long side, Hi = HL, (failure along
short side, HL dominant)
Base and ground inclination factors (see Figure )
           Hansen
   bc       ηo
                 ; β < φ′ ; ηo + βo < 90o
                        p
            147
   bq       exp( −2η tan φ′ ); η i s in radians
                          p

            β < φ′ ; ηo + βo < 90o
                 p
   bγ       exp( −2.7 η tan φ′ ); η i s in radians
                             p

            β < φ′ ; ηo + βo < 90o
                 p
   gc        βo
                 ;; β < φ′ ; ηo + βo < 90o
                         p
            147
   gq       (1 − 0.5 tan β]5 ; β < φ′ ; ηo + βo < 90o
                                    p
   gγ       (1 − 0.5 tan β]5 ; β < φ′ ; ηo + βo < 90o
                                    p




                    Vn
                                        Df                    +β



                                  B/2
         Hi              B/2
         +η
Figure 4.4 Footing on a slope: definition of terms in Hansen’s equation

The effective width B′ and effective length, L′, giving an effective area, A′ = B′ L′ must be used
in the theoretical bearing capacity equations. When the location of the resultant load (load
center) is not coincident with the centroid (center of area) of the footing, the footing dimension is
theoretically adjusted to align the load center with the centroid. Some possible cases are shown
in Figure 4.5.
                 Vn                                                         Vn
                                                Vn
                                               eB
                                          eL                                   My


                                                                          Mx
                        L                                                           L
                                                    L              B
       B                        B

       B′ = B, L′ = L           B′ = B - 2eB, L′ = L - 2eL              Mx         M
       A′ = A = BL              A′ = B′L′                          eB =      ;e L = Y
                                                                         Vn         Vn
                                                                   B′ = B - 2eB, L′ = L - 2eL
                                                                   A′ = B′L′
             Q                                                Vn
                 +θ                                      eB
                                                    HB
                                                         eL


            HL                                                L
                        L
                                           B
   B

                                          B′ = B - 2eB, L′ = L - 2eL
                                          A′ = B′L′
                                          Use both the inclination factors and the
                                          effective width in the equations

Figure 4.5 Some possible cases of eccentric loads


You have to make some adjustment to the theoretical bearing capacity equations for ground
water condition. The term γDf in the bearing capacity equations for an ESA refers to the vertical
stress of the soil above the base of the foundation. The last term γB refers to the vertical stress of
a soil mass of thickness, B, below the base of the footing. You need to check which one of three
ground water situations is applicable to your project.
Situation 1: Ground water level at a level B below the base of the footing. No correction is

required, i.e., wq = wγ = 1
Situation 2:       Ground water level within a depth B below the base of the footing

Df < z < (B + Df ) (Figure 4.3a).


wq = 1, w γ =
                ( z-Df ) +    γ′       D   z
                                   (1 + f − )
                  B          γ sat     B B


                                                                                            z
                                          Df                                                         Df
                                                z
                                                                                           (Df –z)



                             B                                                 B




       (a) Ground water within a depth B below base          (b) Ground water within embedment depth


                              Figure 4.3 Ground water effects below base of footing.

Situation 3: Ground water level within the embedment depth 0 ≤ z ≤ Df (Figure 4.3b)

       z    γ′       z             γ′
wq =      +      (1 − ) and w γ =
       D f γ sat     Df           γ sat



DESIGN OF SHALLOW FOOTINGS


It is customary for practicing engineers to make an estimate of the footing size and then check
that it satisfies serviceability and ultimate limit states. Typical initial dimensions for isolated and
strip footings are shown in Figure Remember that these dimensions are only initial guidelines.


The procedures to design a shallow rectangle footing using ASD are as follows:


   1. Select a footing size


   2. Determine settlement using a load factor of 1.
   3. Check that settlement is less than or equal to the serviceability limit state. If settlement
       exceeds serviceability limit state, increase the footing size and recalculate. If settlement
       is much less (> 10% ) than the serviceability limit state, then reduce footing size and
       recalculate.


   4. Calculate ultimate bearing capacity (qult) using the bearing capacity equations if lab tests
       results are available or from empirical equations using field test data. For fine-grained
       soils, calculate the ultimate bearing capacity for both short-term (TSA) and long-term
       (ESA) conditions. For free draining coarse-grained soils, calculate the ultimate bearing
       capacity for long-term (ESA) conditions, which is also applicable to short-term
       conditions.


                                                                   q ult
   5. Calculate the allowable bearing capacity using q a =               + γDf
                                                                   FS


                                    My             Mx
   6. If the eccentricities, eB =         ; eL =      , where x and y are the axes parallel to the width
                                     P             P
                                                                       6e B       6e
       and length respectively, are not zero then calculate Γ =             or Γ = L
                                                                        B          L


   7. Calculate B′ = B − 2eB and L′ = L − 2eL


   8. Calculate    ( Pa )max = P(1 + Γ)   where Pa is the applied load and Γ is the higher value

       calculated in item 6.


   9. Check that q a B′L′ ≥ ( Pa )max


The performance or resistance factors to determine the bearing capacity of shallow footings
based on LFRD are shown in Table 7X.A
Table 7 X. A Performance factors for bearing capacity calculations using LRFD
                                                                         Resistance
                                                                         Factor, ωR
                     ESA: Coarse-grained and fine –grained soils
                          •   φ′cs from lab tests                        0.95*
                          •   φ′p from lab tests                         0.80

                          •   SPT                                        0.45

                          •   CPT                                        0.55

                          •   Plate load test                            0.55

                     TSA: Fine-grained soils
                          •   su from lab tests                          0.60
                          •   Shear vane                                 0.60
                          •   CPT                                        0.50

                          •   Plate load test                            0.55

                     Rock                                                0.6

                * Author’s recommendation


The procedures to design a shallow rectangle footing using LRFD are as follows:


1.      Select a footing size.
2.      Determine settlement using a load factor of 1.
3.      Check that settlement is less than or equal to the serviceability limit state. If settlement
        exceeds serviceability limit state, increase the footing size and recalculate. If settlement
        is much less (> 10% ) than the serviceability limit state, then reduce footing size and
        recalculate.


4.      Determine the maximum factored load (Pu) by combining loads according to IBC (2000)
        or other codes.


5.      Calculate ultimate bearing capacity (qult) using the bearing capacity equations if lab tests
        results are available or from empirical equations using field test data. For fine-grained
        soils, calculate the ultimate bearing capacity for both short-term (TSA) and long-term
        (ESA) conditions. For free draining coarse-grained soils, calculate the ultimate bearing
        capacity for long-term (ESA) conditions, which is also applicable to short-term
        conditions.
                                      My            Mx
6.      If the eccentricities, eB =        ; eL =      , where x and y are the axes parallel to the width
                                      P             P
        and length respectively, are not zero then calculate B′ = B − 2eB and L′ = L − 2eL


                           Pu
7.      Calculate q u =        ( if eB = 0, then B = B′; if eL = 0, then L = L′)
                          B′L′


8.      Calculate the gross resistance


     R = ψR qult + ρi γ Df


     where γ Df is the effective soil pressure above the footing base, ρi is the dead load factor for
     earth pressure (usually 0.9).
9.      Check that R ≥ q u
Suggested changes if serviceability and ultimate limit states are not satisfied after first trial.
Case 1: Settlement too large
Actions: (1) Increase size of footing. This will reduce the applied stress.
         (2) Consider a mat foundation if isolated footing becomes too large with the potential for
stress overlaps. The mat foundation will spread the load over a larger area than isolated footings
thereby reducing the applied stress.
          (3) Consider soil improvement such as dynamic compaction (coarse-grained soils) and
soil reinforcement by, for example, geotextiles (see Chapter 11).
          (4) Consider deep foundations.
Case 2: Ultimate bearing capacity too low.
Actions: Same as above

								
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