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.

Failure mechanisms and derivation of equations

Bearing capacity • 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)]

Upper and lower bound solutions

Failure mechanisms and derivation of equations
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.

Semi-circular slip mechanism

Failure mechanisms and derivation of equations 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
= [(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.

Circular arc slip mechanism

Failure mechanisms and derivation of equations Consider a slip surface which is an arc in cross section, centred 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
= [ (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
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

Bearing capacity of shallow foundations

Bearing capacity
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
g = unit weight of the soil removed.

Bearing capacity equation (undrained)

Bearing capacity of shallow foundations 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

Bearing capacity equation (drained)

Bearing capacity of shallow foundations
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.

Bearing capacity factors

Bearing capacity equation (drained)
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')

Shape factors

Bearing capacity equation (drained)
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
B = breadth, L = length

Depth factors

Bearing capacity equation (drained)
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

Factor of safety

Bearing capacity of shallow foundations
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.

Presumed bearing values

Bearing capacity
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

Bearing capacity of piles

Bearing capacity 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.

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 mobilised 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 mobilised 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.

Driven piles in non-cohesive soil

Bearing capacity of piles
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. Judgement is therefore left to the experience of the engineer.

Ultimate pile capacity

Driven piles in non-cohesive soil
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.

Standard penetration test

Driven piles in non-cohesive soil The standard penetration test is a simple in-situ test in which the N-value is the mumber 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.

Meyerhof (1976) produced correlations between base and frictional resistances and N-values. It is recommended that N-values first be normalised with respect to effective overburden stress:

Normalised 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

Cone penetration test

Driven piles in non-cohesive soil
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

Bored piles in non-cohesive soil

Bearing capacity of piles
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.

Driven piles in cohesive soil

Bearing capacity of piles
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 porewater. 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.

Bored piles in cohesive soil

Bearing capacity of piles
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 mobilised 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 mobilisation of full end-bearing capacity by large-diameter piles requires much larger displacements than are required to mobilise full skin-friction, and therefore safety factors of 2.5 to 3.0 may be required to avoid excessive settlement at working load.

Carrying capacity of piles in layered soil

Bearing capacity of piles
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, Meyerhof'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. Effects of groundwater

Bearing capacity of piles
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 centimetres 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 minimised if the concrete has a high cement/aggregate ratio and is well compacted during placement.