Bearing Capacity
BEARING CAPACITY EQUATIONS IN DESIGNING SHALLOW FOUNDATIONS1.1
BEARING CAPACITY EQUATIONS:1.1.1 Terzaghis bearing Capacity
TheoryTerzaghi (1943) was the first to present a comprehensive
theory for evaluating the ultimate bearing capacity of shallow
foundations with rough base.According to this theory a foundation
is shallow if its depth Df is less than or equal to its width.
Later investigators, however, have suggetsed that foundation with
Df equal to 3 to 4 times their width may be defined as shallow
foundations.
Fig.1.1: Typical Shallow Foundation
Fig.1.3: Terzaghis concept of Footing with five distinct failure
zones in foundation soil
Ultimate bearing capacity,
If the ground is subjected to additional surcharge load q,
then
. Net ultimate bearing capacity, Or,
Safe or allowable bearing capacity,
Here, F = Factor of safety (usually2-3)c = cohesion of soil, =
unit weight of soil, D = Depth of foundation
q = Surcharge at the ground level, B = Width of foundationNc,
Nq, N = Terzaghis bearing capacity factors depend on soil friction
angle .Where:
Nc = cot(Nq 1) Nq = (e 2(3/4-/2)tan ) / 2 cos2(45+/2)N = 1/2 tan
( Kpr /cos2 -1)
Kpr = passive pressure coefficient.Table 1: Bearing capacity
factors for different
NcNqNgN'cN'qN'g
05.71.00.05.71.00.0
57.31.60.56.71.40.2
109.62.71.28.01.90.5
1512.94.42.59.72.70.9
2017.77.45.011.83.91.7
2525.112.79.714.85.63.2
3037.222.519.719.08.35.7
3452.636.535.023.711.79.0
3557.841.442.425.212.610.1
4095.781.3100.434.920.518.8
45172.3173.3297.551.235.137.7
48258.3287.9780.166.850.560.4
50347.6415.11153.281.365.687.1
Effect of shape of Foundation
The following are the corrections for circular, square and
rectangular footings.Circular footing,
Square footing,
Rectangular footing,
Table 2: Summary of Shape factors for different shapes of
footing
Shapescsqs
Strip111
Square1.310.8
Round1.310.6
Rectangle
1
The equation for bearing capacity explained above is applicable
for soil experiencing general shear failure. If a soil is
relatively loose and soft, it fails in local shear failure. Such a
failure is accounted in bearing capacity equation by reducing the
magnitudes of strength parameters c and as follows.
Table summarizes the bearing capacity factors to be used under
different situations. If is less than 360 and more than 280, it is
not sure whether the failure is of general or local shear type. In
such situations, linear interpolation can be made and the region is
called mixed zone.
Table 3: Bearing capacity factors in zones of local, mixed and
general shear conditions.
Local Shear FailureMixed ZoneGeneral Shear Failure
< 28o28o < < 36o > 36o
Nc1, Nq1, N1Ncm, Nqm, NmNc, Nq, N
1.1.2 General Bearing Capacity EquationIt is evident that
Tergaghis equation is only valid for the case of general shear
failure because no soil compression is allowed before the failure
occurs.
Meyerhof, Hansen, and Vesic further extended Terzaghis bearing
capacity equation to account for footing shape (si), footing
embedment depth (d1), load inclination or eccentricity (ii),
sloping ground (gi), and tilted base (bi). Chen reevaluated N
factors in Terzaghis equation using the limit analysis method.
These efforts resulted in significant extensions of Terzaghis
bearing capacity equation. The general form of the bearing capacity
equation can be expressed as:qu = c.Nc. Sc. dc. ic + q.Nq. Sq. dq.
iq + 0.5.BN. S. d. i
Equations are available for shape factors (sc, sq, s), depth
factors (dc, dq, d) and load inclination factors (ic, iq, i). The
effects of these factors are to reduce the bearing capacity.Table
4: Bearing capacity factors for general bearing capacity
equation
Note: Nc and Nq are same for all four methods; subscripts
identify author for M = Meyerhof; H = Hansen; V = Vesic; C =
Chen.
Vesic suggested that a flat reduction of might be too
conservative in the case of local and
punching shear failure. He proposed the following equation for a
reduction factor varying with
relative density Dr:
Bearing Capacity from Standard Penetration Test (SPT)The SPT is
widely used to obtain the bearing capacity of soils directly
.Meyerhof (1956, 1974) published equation for computing the
allowable bearing capacity for a 25 mm (1 inch) settlement.
IN FPS SYSTEM:
qall = but (1+0.33)1.33 and B 4.0 ft.qall = but (1+0.33)1.33 and
B > 4.0 ft.
Where: qall = Allowable bearing pressure in ksf, for H = 1inch
settlement.D = Depth of foundation (ft)
B = Width of foundation (ft).IN SI UNIT:
qall = but (1+0.33)1.33 and B 1.2 m.qall = but (1+0.33)1.33 and
B > 1.2 m.Where: qall = Allowable bearing pressure in kpa, for H
= 25 mm settlement.
D = Depth of foundation (m)
B = Width of foundation (m).The Standard blow count can be
computed from the measured N as follows:
= CN.N.
EMBED Equation.3
EMBED Equation.3 Where: CN=and
= Effective overburden pressure in (kpa).Hammer for = 1.14
(normally)Rod length correction = 1.00 when rod length >10.0m =
0.95 when rod length 6-10m
= 0.85 when rod length 4-6m
Sampler correction = 1.00 without liner.Borehole diameter
correction= 1.00 for 60 mm-120 mm = 1.05 for 150 mm.The allowable
soil pressure for any settlement Hj is =. Where Ho = 25 mm or 1
inch and Hj = settlement that can be tolerated in mm or inch.Parry
(1977) proposed computing the allowable bearing capacity of
cohesion less soils as:
qa = 30N55 (kpa) for DBWhere, N55 is the average SPT value at a
depth about 0.75B below the proposed base of the footing. The
allowable bearing pressure qa is computed for settlement checking
as qa = (kpa) for a Ho = 20 mmAngle of internal friction can be
calculated by using SPT value as:
Here, is the effective overburden pressure at the location of
the average N55 count.
N/avg. is an average value of the SPT blow counts, which is
determined within the range of depths
from footing base to 1.5B below the footing. In very fine or
silty saturated sand, the measured SPT blow count (N) is corrected
for submergence effect as follows:
BEARING CAPACITY FROM PLATE LOAD TESTFor clay soil qult is
independent of footing size, giving qult,foundation = qult, load
test and for (c-) soil qult,foundation = M+ N where M includes the
Nc and Nq terms and N is the term. Practically for sand use the
following relation qult,foundation = qult, load test.The use of
this equation is not recommended unless the is not much more atan
about 3.0.Housels method for bearing capacity from plate load
testHousels (1929) and Williams (1929) both gave an equation for
using at least two plate load tests to obtain an allowable load Ps
for some settlement as Ps = Aq1 + pq2 (kpa or ksf)Where A = area of
plate used for the load test, m2 or ft2.P = perimeter of the load
test plate, m or ft.
q1= bearig pressure of interior zone of plateq2= edge shear of
plate31.5.2 Layered SystemsWestergaard [70], Burmister [21-23],
Sowers and Vesic [62], Poulos and Davis [55], and Perloff
[54] discussed the solutions to stress distributions for layered
soil strata. The reality of interlayer
shear is very complicated due to in situ nonlinearity and
material inhomogeneity [37,54]. Either
zero (frictionless) or with perfect fixity is assumed for the
interlayer shear to obtain possible
FIGURE 31.9 Pressure bulbs based on the Bossinesq equation for
square and long footings. (After NAVFAC 7.01,
1986].)
solutions. The Westergaard method assumed that the soil being
loaded is constrained by closed
spaced horizontal layers that prevent horizontal displacement
[52]. Figures 31.10 through 31.12 by the Westergaard method can be
used for calculating vertical stresses in soils consisting of
alternative layers of soft (loose) and stiff (dense) materials.
31.5.3 Simplified Method (2:1 Method)
Assuming a loaded area increasing systemically with depth, a
commonly used approach for computing the stress distribution
beneath a square or rectangle footing is to use the 2:1 slope
method as shown in Fig. below. Sometimes a 60 distribution angle
(1.73to1 slope) may be assumed.The pressure increase .q at a depth
z beneath the loaded area due to base load P is
FIGURE 31.10 Vertical stress contours for square and strip
footings [Westerqaard Case].(After NAVFAC 7.01, 1986.) Where
symbols are referred to Figure 31.14. The solutions by this method
compare very well with those of more theoretical equations from
depth z from B to about 4B but should not be used for depth z from
0 to B [14]. A comparison between the approximate distribution of
stress calculated by a theoretical method and the 2:1 method is
illustrated in Figure 31.15.
31.6.2 Settlement of Shallow Foundations on Sand
SPT Method
DAppolonio et al. [28] developed the following equation to
estimate settlements of footings on
sand using SPT data:
Where 0 and 1 are settlement influence factors dependent on
footing geometry, depth of embedment, and depth to the relative
incompressible layer (Figure 31.17), p is average applied pressure
under service load and M is modulus of compressibility. The
correlation between M and average SPT blow count is given in Figure
31.18. Barker et al. [9] discussed the commonly used procedure for
estimating settlement of footing on sand using SPT blow count
developed by Terzaghi and Peck [64,65] and Bazaraa [10].
FIGURE 31.18 Correlation between modulus of compressibility and
average value SPT blow count. (After DAppolonia
et al [28].)1.6 FACTOR OF SAFETYIt is the factor of ignorance
about the soil under consideration. It depends on many factors such
as,
1. Type of soil
2. Method of exploration
3. Level of Uncertainty in Soil Strength
4. Importance of structure and consequences of failure
5. Likelihood of design load occurrence, etc.
Assume a factor of safety F = 3, unless otherwise specified for
bearing capacity problems. Table 7.5 provides the details of
factors of safety to be used under different circumstances.
Table 7.5 Typical factors of safety for bearing capacity
calculation in different situations
7.12 Presumptive Safe Bearing Capacity
It is the bearing capacity that can be presumed in the absence
of data based on visual identification at the site. National
Building Code of India (1983) lists the values of presumptive SBC
in kPa for different soils as presented below.
A : Rocks
Sl NoDescriptionSBC (kPa)
1Rocks (hard) without laminations and defects. For e.g. granite,
trap & diorite3240
2Laminated Rocks. For e.g. Sand stone and Lime stone in sound
condition1620
3Residual deposits of shattered and broken bed rocks and hard
shale cemented material880
4Soft Rock440
B : Cohesionless Soils
Sl NoDescriptionSBC (kPa)
1Gravel, sand and gravel, compact and offering resistance to
penetration when excavated by tools440
2Coarse sand, compact and dry440
3Medium sand, compact and dry245
4Fine sand, silt (dry lumps easily pulverized by fingers)150
5Loose gravel or sand gravel mixture, Loose coarse to medium
sand, dry245
6Fine sand, loose and dry100
C : Cohesive Soils
Sl NoDescriptionSBC (kPa)
1Soft shale, hard or stiff clay in deep bed, dry440
2Medium clay readily indented with a thumb nail245
3Moist clay and sand clay mixture which can be indented with
strong thumb pressure150
4Soft clay indented with moderate thumb pressure100
5Very soft clay which can be penetrated several centimeters with
the thumb50
6Black cotton soil or other shrinkable or expansive clay in dry
condition (50 % saturation)130 - 160
Note :
1. Use d for all cases without water. Use sat for calculations
with water. If simply density is mentioned use accordingly.
2. Fill all the available data with proper units.
3. Write down the required formula
4. If the given soil is sand, c = 0REFERENCES:1. Joseph E.
Bowles, Foundation Analysis and Design, Fifth Edition, McGraw Hill,
New York, pp.213-343.2. Braja M. Das, Principles of Foundation
Engineering, Sixth Edition, India edition,
pp.81-146._1394151183.unknown
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