Bearing Capacity of Soil
Bearing Capacity
The ultimate load which a foundation can support
may be calculated using bearing capacity theory. For preliminary design,
presumed bearing values can be used to indicate the pressures which would
normally result in an adequate factor of safety. Alternatively, there is a range
of empirical methods based on in situ test results.
The ultimate bearing capacity (qf) is the value of bearing
stress which causes a sudden catastrophic settlement of the foundation (due to
shear failure).
The allowable bearing capacity (qa) is the maximum bearing
stress that can be applied to the foundation such that it is safe against
instability due to shear failure and the maximum tolerable settlement is
not exceeded. The allowable bearing capacity is normally calculated from the
ultimate bearing capacity using a factor of safety (Fs).
When excavating for a foundation, the stress at founding level is relieved by
the removal of the weight of soil. The net bearing pressure
(qn) is the increase in stress on the soil.
qn = q -
qo
qo = g D
where D is
the founding depth and g is the unit weight of the soil
removed.
- A relatively undeformed wedge of soil below the foundation forms an active
Rankine zone with angles (45º + f'/2).
- The wedge pushes soil outwards, causing passive Rankine zones to form with
angles (45º - f'/2).
- The transition zones take the form of log spiral fans.
For purely
cohesive soils (f = 0) the transition zones become
circular for which Prandtl had shown in 1920 that the solution is
qf = (2 + p) su =
5.14 su This equation is based on a weightless soil.
Therefore if the soil is non-cohesive (c=0) the bearing capacity depends on the
surcharge qo. For a footing founded at depth D below the surface, the
surcharge qo = gD. Normally for a shallow
foundation (D<B), the shear strength of the soil between the surface and the
founding depth D is neglected.
radius of the fan r = r0 .exp[q.tanf'].
q is the fan angle in radians (between 0 and p/2)
f' is the angle of
friction of the soil
ro = B/[2 cos(45+f'/2)]
The bearing
capacity of a soil can be investigated using the limit theorems of ideal
rigid-perfectly-plastic materials.
The ultimate load capacity of a footing can be estimated by assuming a
failure mechanism and then applying the laws of statics to that mechanism. As
the mechanisms considered in an upper bound solution are progressively
refined, the calculated collapse load decreases.
As more stress regions are considered in a lower bound solution, the
calculated collapse load increases.
Therefore, by progressive refinement of the upper and lower bound solutions,
the exact solution can be approached. For example, Terzaghi's mechanism gives
the exact solution for a strip footing.
 Suppose the
mechanism is assumed to have a semi-circular slip surface. In this case, failure
will cause a rotation about point O. Any surcharge qo will resist
rotation, so the net pressure (q - qo) is used. Using the equations
of statics:
- Moment causing rotation
- = load x lever arm
- = [(q - qo) x B] x [½B]
- Moment resisting rotation
- = shear strength x length of arc x lever arm
- = [s] x [p.B] x [B]
- At failure these are equal:
- (q - qo ) x B x ½B = s x p.B x B
- Net pressure (q - qo ) at failure
- = 2 p x shear strength of the soil
- This is an upper-bound solution.
 Consider a
slip surface which is an arc in cross section, centered above one edge of the
base. Failure will cause a rotation about point O. Any surcharge qo
will resist rotation so the net pressure (q - qo) is used. Using the
equations of statics:
- Moment causing rotation
- = load x lever arm
- = [ (q - qo) x B ] x [B/2]
- Moment resisting rotation
- = shear strength x length of arc x lever arm
- = [s] x [2a R] x [R]
- At failure these are equal:
- (q - qo) x B x B/2 = s x 2 a R x R
- Since R = B / sin a :
- (q - qo ) = s x 4a /(sin a)²
- The worst case is when
- tana=2a at a = 1.1656 rad = 66.8 deg
- The net pressure (q - qo) at failure
- = 5.52 x shear strength of soil
The ultimate bearing capacity of a foundation is
calculated from an equation that incorporates appropriate soil parameters (e.g.
shear strength, unit weight) and details about the size, shape and founding
depth of the footing. Terzaghi (1943) stated the ultimate bearing capacity of a
strip footing as a three-term expression incorporating the bearing capacity
factors: Nc, Nq and Ng, which are related to the angle of friction (f´).
qf =c.Nc
+qo.Nq
+ ½g.B
.Ng
For drained loading, calculations are in terms of effective stresses;
f´ is > 0 and N c, Nq and
Ng are all > 0.
For undrained loading,
calculations are in terms of total stresses; the undrained shear strength
(su); Nq = 1.0 and Ng = 0
c = apparent cohesion intercept
qo =
g . D (i.e. density x depth)
D =
founding depth
B = breadth of foundation
g = unit weight of the soil removed.
- Skempton's equation is widely used for undrained clay soils:
- qf = su .Ncu + qo
- where Ncu = Skempton's bearing capacity factor, which
can be obtained from a chart or by using the following expression:
- Ncu = Nc.sc.dc
- where sc is a shape factor and dc is a depth
factor.
- Nq = 1, Ng = 0,
Nc = 5.14
- sc = 1 + 0.2 (B/L) for B<=L
- dc = 1+ Ö(0.053 D/B ) for D/B < 4
Terzaghi (1943) stated the bearing capacity of a
foundation as a three-term expression incorporating the bearing capacity
factors
Nc, Nq and Ng.
He proposed the following equation for the
ultimate bearing capacity of a long strip footing:
qf =c.Nc
+qo.Nq
+ ½g.B
.Ng This equation is applicable
only for shallow footings carrying vertical non-eccentric loading.
For
rectangular and circular foundations, shape factors are introduced.
qf = c .Nc .sc + qo
.Nq .sq + ½ g .B .Ng .sg Other factors can be used
to accommodate depth, inclination of loading, eccentricity of loading,
inclination of base and ground. Depth is only significant if it exceeds the
breadth.
The bearing capacity
factors relate to the drained angle of friction (f').
The c.Nc term is the contribution from soil shear strength, the
qo.Nq term is the contribution from the surcharge pressure
above the founding level, the ½.B.g.Ng term
is the contribution from the self weight of the soil. Terzaghi's analysis was
based on an active wedge with angles f' rather than
(45+f'/2), and his bearing capacity factors are in
error, particularly for low values of f'. Commonly used
values for Nq and Nc are derived from the Prandtl-Reissner
expression giving
Exact values for Ng are not directly
obtainable; values have been proposed by Brinch Hansen (1968), which are widely
used in Europe, and also by Meyerhof (1963), which have been adopted in North
America.
- Brinch Hansen:
- Ng = 1.8 (Nq - 1) tanf'
- Meyerhof:
- Ng = (Nq - 1) tan(1.4 f')
Terzaghi presented modified
versions of his bearing capacity equation for shapes of foundation other than a
long strip, and these have since been expressed as shape factors. Brinch
Hansen and Vesic (1963) have suggested shape factors which depend on f'. However, modified versions of the Terzaghi factors are
usually considered sufficiently accurate for most purposes.
|
|
sc |
sq |
sg |
| square |
1.3 |
1.2 |
0.8 |
| circle |
1.3 |
1.2 |
0.6 |
| rectangle (B<L) |
1+ 0.2(B/L) |
1+ 0.2(B/L) |
1 - 0.4(B/L) | B = breadth, L = length
It is usual to assume an
increase in bearing capacity when the depth (D) of a foundation is greater than
the breadth (B). The general bearing capacity equation can be modified by the
inclusion of depth factors.
- qf = c.Nc.dc +
qo.Nq.dq + ½ B.gNg.dg
- for D>B:
- dc = 1 + 0.4 arctan(D/B)
- dq = 1 + 2 tan(f'(1-sinf')² arctan(B/D)
- dg = 1.0
- for D=<B:
- dc = 1 + 0.4(D/B)
- dq = 1 + 2 tan(f'(1-sinf')² (B/D)
- dg = 1.0
A factor of safety
Fs is used to calculate the allowable bearing capacity qa
from the ultimate bearing pressure qf. The value of Fs is
usually taken to be 2.5 - 3.0.
 
The factor of safety should be applied only to the
increase in stress, i.e. the net bearing pressure qn. Calculating
qa from qf only satisfies the criterion of safety against
shear failure. However, a value for Fs of 2.5 - 3.0 is sufficiently
high to empirically limit settlement. It is for this reason that the factors of
safety used in foundation design are higher than in other areas of geotechnical
design. (For slopes, the factor of safety would typically be 1.3 - 1.4).
Experience has shown that the settlement of a typical foundation on soft clay
is likely to be acceptable if a factor of 2.5 is used. Settlements on stiff clay
may be quite large even though ultimate bearing capacity is relatively high, and
so it may be appropriate to use a factor nearer 3.0.
For preliminary design purposes, BS 8004 gives
presumed bearing values which are the pressures which would normally result in
an adequate factor of safety against shear failure for particular soil types,
but without consideration of settlement.
| Category |
Types of rocks and soils |
Presumed bearing value |
| Non-cohesive soils |
Dense gravel or dense sand and gravel |
>600 kN/m² |
| |
Medium dense gravel,
or medium dense sand and gravel |
<200 to 600 kN/m² |
| |
Loose gravel, or loose sand and gravel |
<200 kN/m² |
| |
Compact sand |
>300 kN/m² |
| |
Medium dense sand |
100 to 300 kN/m² |
| |
Loose sand |
<100 kN/m² depends on
degree of
looseness |
| Cohesive soils |
Very stiff bolder clays & hard clays |
300 to 600 kN/m² |
| |
Stiff clays |
150 to 300 kN/m² |
| |
Firm clay |
75 to 150 kN/m² |
| |
Soft clays and silts |
< 75 kN/m² |
| |
Very soft clay |
Not applicable |
| Peat |
|
Not applicable |
| Made ground |
|
Not applicable |
Presumed bearing values for Keuper
Marl
| Weathering |
Zone |
Description |
Presumed bearing value |
| Fully weathered |
IVb |
Matrix only |
as cohesive soil |
| Partially weathered |
IVa |
Matrix with occasional pellets less than 3mm |
125 to 250 kN/m² |
| III |
Matrix with lithorelitics up to 25mm |
250 to 500 kN/m² |
| II |
Angular blocks of unweathered marl with virtually no matrix |
500 to 750 kN/m² |
| Unweathered |
1 |
Mudstone (often not fissured) |
750 to 1000 kN/m² |
 The ultimate
bearing capacity of a pile used in design may be one three values:
the
maximum load Qmax, at which further penetration occurs without
the load increasing;
a calculated value Qf given by the
sum of the end-bearing and shaft resistances;
or the load at which a
settlement of 0.1 diameter occurs (when Qmax is not clear).
For large-diameter piles, settlement can be large, therefore a safety factor
of 2-2.5 is usually used on the working load.
- A pile loaded axially will carry the load:
- partly by shear stresses (ts)
generated along the shaft of the pile and
- partly by normal stresses (qb) generated at the base.
- The ultimate capacity Qf of a pile is equal to the base
capacity Qb plus the shaft capacity Qs.
- Qf = Qb +
Qs = Ab . qb + S(As . ts)
- where Ab is the area of the base and As is the
surface area of the shaft within a soil layer.
Full shaft capacity is mobilized at much smaller displacements than those
related to full base resistance. This is important when determining the
settlement response of a pile. The same overall bearing capacity may be achieved
with a variety of combinations of pile diameter and length. However, a long
slender pile may be shown to be more efficient than a short stubby pile. Longer
piles generate a larger proportion of their full capacity by skin friction and
so their full capacity can be mobilized at much lower settlements.
The proportions of capacity contributed by skin friction and end bearing do
not just depend on the geometry of the pile. The type of construction and the
sequence of soil layers are important factors.
Driving a pile has different effects on the soil
surrounding it depending on the relative density of the soil. In loose soils,
the soil is compacted, forming a depression in the ground around the pile. In
dense soils, any further compaction is small, and the soil is displaced upward
causing ground heave. In loose soils, driving is preferable to boring since
compaction increases the end-bearing capacity.
In non-cohesive soils, skin friction is low because a low friction 'shell'
forms around the pile. Tapered piles overcome this problem since the soil is
recompacted on each blow and this gap cannot develop.
Pile capacity can be calculated using soil properties obtained from
standard penetration tests or cone penetration tests. The ultimate
load must then be divided by a factor of safety to obtain a working load. This
factor of safety depends on the maximum tolerable settlement, which in turn
depends on both the pile diameter and soil compressibility. For example, a
safety factor of 2.5 will usually ensure a pile of diameter less than 600mm in a
non-cohesive soil will not settle by more than 15mm.
Although the method of installing a pile has a significant effect on failure
load, there are no reliable calculation methods available for quantifying any
effect. Judgment is therefore left to the experience of the engineer.
- The ultimate carrying capacity of a pile is:
- Qf = Qb + Qs
- The base resistance, Qb can be found from Terzaghi's
equation for bearing capacity,
- qf = 1.3 c Nc + qo Nq + 0.4
g B Ng
- The 0.4 g B Ng term may be ignored,
since the diameter is considerably less than the depth of the pile.
- The 1.3 c Nc term is zero, since the soil is non-cohesive.
- The net unit base resistance is therefore
- qnf =
qf - qo =
qo (Nq
-1)
- and the net total base resistance is
- Qb = qo (Nq -1)
Ab
- The ultimate unit skin friction (shaft) resistance can be found
from
- qs = Ks .s'v
.tand
- where s'v = average vertical
effective stress in a given layer
- d = angle of wall friction, based on pile
material and f´
- Ks = earth pressure coefficient
- Therefore, the total skin friction resistance is given by the sum
of the layer resistances:
- Qs = S(Ks .s'v .tand
.As)
- The self-weight of the pile may be ignored, since the weight of the
concrete is almost equal to the weight of the soil displaced.
- Therefore, the ultimate pile capacity is:
- Qf = Ab qo Nq
+ S(Ks .s'v
.tand .As)
Values of Ks and d can be related to
the angle of internal friction (f´) using the following
table according to Broms.
| Material |
d |
Ks |
| low density |
high density |
| steel |
20° |
0.5 |
1.0 |
| concrete |
3/4 f´ |
1.0 |
2.0 |
| timber |
2/3 f´ |
1.5 |
4.0 |
It must be noted that, like much of pile design, this is
an empirical relationship. Also, from empirical methods it is clear that
Qs and Qb both reach peak values somewhere at a depth
between 10 and 20 diameters.
It is usually assumed that skin friction never exceeds
110 kN/m² and base resistance will not exceed 11000 kN/m².
 The standard
penetration test is a simple in-situ test in which the N-value is the number of
blows taken to drive a 50mm diameter bar 300mm into the base of a bore hole.
Schmertmann (1975) has correlated N-values obtained from SPT tests against
effective overburden stress as shown in the figure.
The effective overburden stress = the weight of material
above the base of the borehole - the wight of water
e.g. depth of soil = 5m, depth of water = 4m, unit weight of soil
= 20kN/m³, s'v = 5m x 20kN/m³ - 4m x
9.81kN/m³ » 60 kN/m²
Once a value for f´ has been estimated, bearing
capacity factors can be determined and used in the usual way.
Mayerhof (1976) produced correlations between base and frictional resistances
and N-values. It is recommended that N-values first be normalized
with respect to effective overburden stress:
Normalized N = Nmeasured x 0.77
log(1920/s´v)
| Pile type |
Soil type |
Ultimate base resistance
qb (kPa) |
Ultimate shaft resistance
qs (kPa) |
| Driven |
Gravelly sand
Sand |
40(L/d) N
but < 400 N |
2 Navg |
| |
Sandy silt
Silt |
20(L/d) N
but < 300 N |
|
| Bored |
Gravel and sands |
13(L/d) N
but < 300 N |
Navg |
| |
Sandy silt
Silt |
13(L/d) N
but < 300 N |
|
L = embedded length
d = shaft diameter
Navg = average value along shaft
End-bearing
resistance
The end-bearing capacity of the pile is assumed to be equal
to the unit cone resistance (qc). However, due to normally occurring
variations in measured cone resistance, Van der Veen's averaging method is used:
qb = average cone resistance calculated over a depth
equal to three pile diameters above to one pile diameter below the base level
of the pile. Shaft resistance
The skin friction can also
be calculated from the cone penetration test from values of local side friction
or from the cone resistance value using an empirical relationship:
At a
given depth, qs = Sp. qc
where Sp = a coefficient dependent on the type of pile
| Type of pile |
Sp |
Solid timber )
Pre-cast concrete )
Solid steel driven ) |
0.005 - 0.012 |
| Open-ended steel |
0.003 - 0.008 |
The design process for bored piles in
granular soils is essentially the same as that for driven piles. It must be
assumed that boring loosens the soil and therefore, however dense the soil, the
value of the angle of friction used for calculating Nq values for end
bearing and d values for skin friction must be those
assumed for loose soil. However, if rotary drilling is carried out under a
bentonite slurry f' can be taken as that for the
undisturbed soil.
Driving piles into clays alters the
physical characteristics of the soil. In soft clays, driving piles results in an
increase in pore water pressure, causing a reduction in effective stress;.a
degree of ground heave also occurs. As the pore water pressure dissipates with
time and the ground subsides, the effective stress in the soil will increase.
The increase in s'v leads to an increase in
the bearing capacity of the pile with time. In most cases, 75% of the ultimate
bearing capacity is achieved within 30 days of driving.
For piles driven into stiff clays, a little consolidation takes place, the
soil cracks and is heaved up. Lateral vibration of the shaft from each blow of
the hammer forms an enlarged hole, which can then fill with groundwater or
extruded pore water. This, and 'strain softening', which occurs due to the large
strains in the clay as the pile is advanced, lead to a considerable reduction in
skin friction compared with the undisturbed shear strength (su) of
the clay. To account for this in design calculations an adhesion factor, a, is introduced. Values of a can be
found from empirical data previously recorded. A maximum value (for stiff clays)
of 0.45 is recommended.
The ultimate bearing capacity Qf of a driven pile in cohesive soil
can be calculated from:
Qf = Qb + Qs
where the skin friction term is a summation of layer resistances
Qs = S( a
.su(avg) .As)
and the end bearing term is
Qb = su .Nc
.Ab
Nc = 9.0 for clays and silty clays.
Following research into bored
cast-in-place piles in London clay, calculation of the ultimate bearing capacity
for bored piles can be done the same way as for driven piles. The adhesion
factor should be taken as 0.45. It is thought that only half the undisturbed
shear strength is mobilized by the pile due to the combined effect of swelling,
and hence softening, of the clay in the walls of the borehole. Softening results
from seepage of water from fissures in the clay and from the un-set concrete,
and also from 'work softening' during the boring operation.
The mobilization of full end-bearing capacity by large-diameter piles
requires much larger displacements than are required to mobilize full
skin-friction, and therefore safety factors of 2.5 to 3.0 may be required to
avoid excessive settlement at working load.
When a pile extends through a number
of different layers of soil with different properties, these have to be taken
into account when calculating the ultimate carrying capacity of the pile. The
skin friction capacity is calculated by simply summing the amounts of resistance
each layer exerts on the pile. The end bearing capacity is calculated just in
the layer where the pile toe terminates. If the pile toe terminates in a layer
of dense sand or stiff clay overlying a layer of soft clay or loose sand there
is a danger of it punching through to the weaker layer. To account for this,
Mayerhof's equation is used.
The base resistance at the pile toe is
qp = q2 +
(q1 -q2)H / 10B but £ q1
where B is the diameter of the pile, H is the thickness between the base of
the pile and the top of the weaker layer, q2 is the ultimate base
resistance in the weak layer, q1 is the ultimate base resistance in
the strong layer.
The presence and movement of
groundwater affects the carrying capacity of piles, the processes of
construction and sometimes the durability of piles in service.
Effect on bearing capacity
In cohesive soils, the permeability is
so low that any movement of water is very slow. They do not suffer any reduction
in bearing capacity in the presence of groundwater.
In granular soils, the
position of the water table is important. Effective stresses in saturated sands
can be as much as 50% lower than in dry sand; this affects both the end-bearing
and skin-friction capacity of the pile.
Effects on construction
When a concrete cast-in-place pile is
being installed and the bottom of the borehole is below the water table, and
there is water in the borehole, a 'tremie' is used.
 With its lower end lowered to the bottom of the borehole, the
tremmie is filled with concrete and then slowly raised, allowing concrete to
flow from the bottom. As the tremie is raised during the concreting it must be
kept below the surface of the concrete in the pile. Before the tremie is
withdrawn completely sufficient concrete should be placed to displace all the
free water and watery cement. If a tremie is not used and more than a few
centimeters of water lie in the bottom of the borehole, separation of the
concrete can take place within the pile, leading to a significant reduction in
capacity.
A problem can also arise when boring takes place through clays. Site
investigations may show that a pile should terminate in a layer of clay.
However, due to natural variations in bed levels, there is a risk of boring
extending into underlying strata. Unlike the clay, the underlying beds may be
permeable and will probably be under a considerable head of water. The 'tapping'
of such aquifers can be the cause of difficulties during construction.
Effects on piles in service
The presence of groundwater may lead
to corrosion or deterioration of the pile's fabric.
In the case of steel
piles, a mixture of water and air in the soil provides conditions in which
oxidation corrosion of steel can occur; the presence of normally occurring salts
in groundwater may accelerate the process.
In the case of concrete
piles, the presence of salts such as sulphates or chlorides can result in
corrosion of reinforcement, with possible consequential bursting of the
concrete. Therefore, adequate cover must be provided to the reinforcement, or
the reinforcement itself must be protected in some way. Sulphate attack on the
cement compounds in concrete may lead to the expansion and subsequent cracking.
Corrosion problems are minimized if the concrete has a high cement/aggregate
ratio and is well compacted during placement.
Source : http://environment.uwe.ac.uk
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