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Bearing Capacity of Shallow Foundations - PDF

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									                   Bearing Capacity of Shallow Foundations

Foundations for structures are generally classified as "__________" and "____________".
Shallow is a relative term, and may be generally defined as foundations less than
approximately 3 m or less than the breadth of the footing.

Deep foundations generally refer to piled foundations, whereas shallow foundations
include pad foundations, raft foundations, and strip footings.

     B                             B

 Strip Footing                 Spread Footing

The performance and functional viability of a foundation depends on the interaction

between the structure which is supported and on the founding material. The behaviour of

the soil depends on the __________ and ___________ of the foundation, hence the

bearing capacity is not simply a function of the soil, but rather is also a function of the

specific foundation arrangement.

What we find from a practical perspective, is that _________________ generally limit the

capacity of a foundation, rather than the ______________________________ of the soil.

Hence the design procedure for foundations must include deformation considerations.

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There are then three main criteria which must be considered:

a. Adequate depth.

b. Limiting settlement.

         Typical limiting values for settlement are:
            SANDS: Max total settlement             40 mm for isolated footings
                                                    40 - 65 mm for rafts
                      Max differential settlement between adjacent columns
                                                    25 mm
          CLAYS: Max total settlement               65 mm for isolated footings
                                                    65 - 100 mm for rafts
                      Max differential settlement between adjacent columns
                                                    40 mm
c. Factor of safety against shear failure.

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Modes of Failure
There are three principal modes of shear failure:




General shear failure results in a clearly defined plastic yield slip surface beneath the
footing and spreads out one or both sides, eventually to the ground surface. Failure is
sudden and will often be accompanied by severe tilting. Generally associated with _____


Local shear failure results in considerable vertical displacement prior to the development
of noticeable shear planes. These shear planes do not generally extend to the soil surface,
but some adjacent bulging may be observed, but little tilting of the structure results.

Generally occurs in ____________________________ soils.

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Punching shear failure occurs in __________________ soils, and there is little or no
development of planes of shear failure in the underlying soil. Slip surfaces are generally
restricted to vertical planes adjacent to the footing, and the soil may be dragged down at
the surface in this region.

Definitions of Bearing Capacity

_____________________________, qu, is the maximum bearing pressure at which the
supporting ground is expected to fail in shear.


                              q                        GWT


There is a stress at the base of the footing due to the backfill and the footing mass

This can be approximated as:

                                  σ'o =

The net ultimate bearing capacity can be defined as:

                               qnf =

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_____________________________, qall is the ultimate bearing capacity divided by an
appropriate factor of safety.

Generally for bearing capacity of shallow foundations, the factor of safety is __________.
It has been found from experience that this is appropriate in order to limit deformations to
those allowable.

The applied stresses at the base of the footing due to the applied load is defined as:


The net (increased) applied stress at the footing base is defined as:


The Factor of safety for bearing capacity is then defined as


or in terms of allowable bearing stress, qall = q, thus


Solving for the allowable bearing stress we obtain


In the Canadian foundation Engineering Manual (CFEM) this expression is simplified to:


Which is conservative (less than the equation above by σ'o - σ'o/F, approx 0.67 σ'o)

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_______________________________, qp, is a conservative estimate of bearing capacity
for preliminary design purposes, taken from empirical data and local experience. Some
typical values are listed below.

Type of soil                                 qp (kPa)       Remarks
Cohesionless soil                                           Providing width B > 1m and
compact gravel or sand/gravel                     >600      groundwater level > B below
medium-dense gravel or sand/gravel            200 - 600     base of footing.
loose gravel or sand/gravel                       <200
compact sand                                      >300
medium-dense sand                             100 - 300
loose sand                                        <100
Cohesive soil                                               These soils are susceptible to
very stiff boulder clays; hard clays           300 - 600    long-term settlement.
stiff clays                                    150 - 300
firm clays                                      75 - 150
soft clays and silts                                 <75
very soft clays and silts                 not applicable

Ultimate Bearing Capacity of Shallow Foundations

As with most geotechnical applications which we've studied, the behaviour is first

idealized in the form a simplified model as the basis for a mathematical formulation.

This model and formulation are then modified by empirical corrections to provide a good

correlation between theoretical and observed behaviour.

The first case considered is that of an ________________________________________

of width B on a homogeneous, isotropic, weightless soil. A general shear failure is

assumed and the condition of limit equilibrium is analyzed for the soil which behaves as

and elastic-perfectly plastic material.

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The failure of the soil is divided into 3 zones as shown below:

         2)                                         , and


Slip is assumed to occur on the planes as shown.

If the footing is installed beneath the soil surface, the overburden pressure on the soil
adjacent to the footing is considered as a surcharge on the notional ground surface

         σo =                  (where D is the depth from the soil surface to footing base)

When failure is reached, the soil wedge beneath the footing (ABF) is displaced downward
developing a Rankine active state such that
                                            σ1 is _____________
                                            σ3 is _____________

The adjacent radial sections are then forced to rotate sideways and the principal stresses
rotate as well.

The passive wedges are forced upwards and away from the footing by the radial rotating
sections, and for these sections the stresses are:
                                                σ1 is _____________
                                                σ3 is _____________

Under undrained conditions the angle α in the figure is equal to:

                                                 α = 45 ± φ'/2

and φ = 0, thus the angles for the active and passive wedges are both equal to 45 and
hence the arc of the radial section is circular.

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For these conditions it has been determined that the bearing capacity may be calculated

         qu =

For drained conditions (φ > 0) the angles are as shown below, and as a consequence the
curve of the radial portion is not circular but rather is defined as a log spiral.

For these conditions it has been determined that the bearing capacity may be calculated

         qu =

where Nc and Nq are dimensionless factors dependent on φ'.

In order to account for the weight of the soil (so far it has been considered weightless)
another term must be added accounting for density. This was done by Terzaghi in 1943
for a strip footing of width B resulting in the equation:

         qu =

The parameters Nc, Nq, and Nγ are referred to as the Bearing Capacity Factors.

The values of these parameters can be calculated from the equations below, and are
tabulated on the following page.

          Nq = e π tan φ' tan2 (45 + φ' / 2)       Reissner 1924)

          Nc = cot φ' (Nq - 1)                     Prandtl (1921)

          Nγ = 1.8 (Nq - 1) tan φ'                 Hansen (1961) (Note: CFEM uses 1.5 vs. 1.8)

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Bearing Capacity Factors

   φ'      Nc           Nq     Nγ          φ'     Nc           Nq     Nγ    φ'         Nc     Nq      Nγ
    0     5.14         1.00   0.00         15    11.0         3.94   1.42   30        30.1   18.4    18.1
    1     5.38         1.09   0.00         16    11.6         4.34   1.72   31        32.7   20.6    21.2
    2     5.63         1.20   0.01         17    12.3         4.77   2.08   32        35.5   23.2    24.9
    3     5.90         1.31   0.03         18    13.1         5.26   2.49   33        38.6   26.1    29.3
    4     6.19         1.43   0.05         19    13.9         5.80   2.97   34        42.2   29.4    34.5
    5     6.49         1.57   0.09         20    14.8         6.40   3.54   35        46.1   33.3    40.7
    6     6.81         1.72   0.14         21    15.8         7.07   4.19   36        50.6   37.8    48.1
    7     7.16         1.88   0.19         22    16.9         7.82   4.96   37        55.6   42.9    56.9
    8     7.53         2.06   0.27         23    18.0         8.66   5.85   38        61.4   48.9    67.4
    9     7.92         2.25   0.36         24    19.3         9.60   6.89   39        67.9   56.0    80.1
   10     8.34         2.47   0.47         25    20.7         10.7   8.11   40        75.3   64.2    95.4
   11     8.80         2.71   0.60         26    22.3         11.9   9.53   41        83.9   73.9    114
   12     9.28         2.97   0.76         27    23.9         13.2   11.2   42        93.7   85.4    137
   13     9.81         3.26   0.94         28    25.8         14.7   13.1   43        105    99.0    165
   14     10.4         3.59   1.16         29    27.9         16.4   15.4   44        118    115     199
                                                                            45        134    135     241
Values of          Nc after Prandtl (1921)                                  46        152    159     294
                   Nq after Reissner (1924)                                 47        174    187     359
                   Nγ after Hansen (1961)                                   48        199    222     442
                                                                            49        230    265     548
                                                                            50        267    319     682

                                          Bearing Capacity Factors





                       0             10              20               30         40             50


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Modifications for different foundation configurations
The above calculations for bearing capacity were based upon an infinitely long strip

Modification to the calculation must be made for different shaped footings, and for
deeper foundations as the failure mode changes.

The bearing capacity equation becomes:

         qu =

Where the values of sc sq and sγ are shape factors, and dc dq and dγ are depth factors
calculated based on the values tabulated below:

      Footing Shape                   sc                       sq                   sγ
      Strip                          1.00                     1.00                1.00
      Rectangle              1 + (B/L) * (Nq/ Nc)        1 + (B/L) tanφ'      1 - 0.4 B/L
      Circle or square           1 + (Nq/Nc)                1 + tanφ'              0.6

     dc = 1 + 0.2 (Kp)1/2 (D/B)               and    dq = dγ = 1 + 0.1 (Kp) (D/B)

     The depth factors are generally quite small except for deeper footings

Soft Soils
Terzaghi recommended that the bearing capacity calculations be modified in soft
cohesive soils or low density cohesionless soils to account for the fact that the mode of
failure was different (local or punching vs general).

Recommended modifications are:

         Cohesive soils (modify c):

                   c=                               tanφ' =

         Cohesionless soil (modify φ')

                   CF =

                   where Dr = relative density                0 < Dr < 0.67

                   tanφ' = CF tanφ'measured

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Presence of the Water Table

In granular soils, the presence of water in the soil can substantially reduce the bearing
                                       Case 1      Case 2     Case 3      Case 4

                                                               GWT         D+B
Case 1:use γ' for the γDNq and ½BγNγ terms

                   Overburden        Beneath Footing

Case 2:for the γDNq = σ'Nq term calculate the effective stress at the depth of the footing
       σ' = σ-u = γD - γwhw, and
       for the ½BγN use γ'.

Case 3:use γ for the γDNq term, and
       use γ' for the ½BγNγ term.

Case 4:use γ for the γDNq and ½BγNγ terms.

In cohesive soils for short-term, end-of-construction conditions use:
       γ = γt and φ = 0        Nc = 5.14, Nq = 1, and Nγ = 0

         qu = 5.14c + γt D

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General Observations about Bearing Capacity

1. The cohesion term dominates in cohesive soils.

2. The depth term (γ D Nq) dominates in cohesionless soils. Only a small increase in D
   increases qu substantially.

3. The base width term (0.5 γ B Nγ) provides some increase in bearing capacity for both
   cohesive and cohesionless soils. In cases where B < 3 to 4 m this term could be
   neglected with little error.

4. No one would place a footing on the ground surface of a cohesionless soil mass.

5. It's highly unlikely that one would place a footing on a cohesionless soil with a
   Dr < 0.5. If the soil is loose, it would be compacted in some manner to a higher
   density prior to placing footings on it.

6. Where the soil beneath the footing is not homogeneous or is stratified, some judgment
   must be applied to determining the bearing capacity.

In practice,

For the short term :   we typically use φ = 0 so the bearing capacity equation simplifies

         qu =

For long-term performance, we usually use SPT blow counts and the charts.

• The allowable bearing pressure, qa, for a footing on sand can be estimated from the
  results of an SPT test by means of the relationship between the SPT index, N, and the
  footing width, as given in Fig. 10.1.

•   Values determined in this manner correspond to the case where the groundwater table
    is located deep below the footing foundation elevation.

•   If the water table rises to the foundation level, no more than half the pressure values
    indicated in Fig 10.1 should be used.

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•   The charts are based on SPT indices obtained from a depth where the effective
    overburden pressure is about 100 kPa (about 5m). Indices obtained from other depths
    must be adjusted before using the charts. Fig. 10.2 indicates a correction factor, CN,
    based on the effective overburden stress at the depth where the actual SPT was

•   The allowable bearing pressure determined from Figs. 10.1 and 10.2, are expected to
    produce settlements smaller than about 25 mm.


•   The SPT is subject to many errors which affect the reliability of the SPT index, N.

•   Correlation between the SPT index and the internal friction angle of sand is very poor.
    Consequently, the calculation of allowable bearing pressure from N values should be
    considered with caution.

•   Nevertheless, in Canadian foundation engineering practice, footings are frequently
    designed using the SPT results, which indicates that the complete borehole
    information, not just the N value, can provide a reliable basis for sound engineering

•   The SPT index is not appropriate for determining the bearing pressure in fine-grained
    cohesive soils.

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What about foundation failure beneath a waste pile on a soft foundation?

Let's examine how to estimate the maximum height of waste that may be placed on a soft

                        H                          Waste pile

                                                        Soft soil
                                      Failure Surface

•   Due to the inclined slope (solid line), calculation of the applied stress on the
    foundation soil is difficult and it does not lend itself to a simple bearing capacity

•   In addition, we must consider the shear strength of the material in the waste pile that
    is being failed (as in a slope stability problem).

•   However, if the material is similar to municipal waste, although it has some shear
    strength, it requires considerable displacement to mobilize the strength.

•   By the time strength of the waste pile is mobilized, the foundation may have already
    failed. Thus for simplicity sake, at this point neglect any strength in the waste

To initially overcome the geometry problem, we could conservatively estimate it as a
vertical face, as shown with the dotted line.

The allowable applied stress at the base of the soil would be equal to:

         qall =

Using the CFEM formulation for qall, considering short-term loading (φ = 0), and solving
for H, we obtain:


The factor of safety here would be less than the 2.5 to 3.0 used in typical foundation
design which is aimed at limiting displacements. An acceptable factor of safety would be
1.3 – 1.5.

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