Foundations: Bearing Capacity

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2 BEARING CAPACITY OF SHALLOW FOUNDATIONS

2.1 General characteristics

Shallow foundations transfer the loads to the ground at a level close to the surface. The side

friction between soil and foundation, and the shear resistance of the lateral soil, are neglected.

The lateral soil is seen as a surcharge acting at the level of the foundation base.

In most cases the ratio between the foundation width B and its depth D is less than 2. In any case

the minimum depth of the foundation base should be about 1 m.

The base of the foundation should be placed outside the zone of fluctuation of the water table.

For cohesive soils this reduces the possible heave/settlement induced by the wetting/drying

process of sensitive clays.

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The bearing capacity is not a characteristics of soils or of the foundation. In fact, it depends on

the interaction between the footing and the ground underlying it.

Depending on their dimensions, and on their major L and minor B sides, shallow foundation can

be subdivided into: spread or pad footings; strip footings; mat or slab or raft foundations.

Examples of spread footings (L B)

Scheme of a strip footing (L>>B)

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Schemes of mat foundations

The bearing capacity equations derived in the following refer to strip footings. They will be

corrected subsequently for the case of spread and mat foundations.

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2.2 Mechanisms of failure (De Beer, Vesic, 1958)

General failure (or failure by shearing)

I Active failure zone II Radial failure zone

III Passive failure zone (results from a model test on dense sand)

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Punching failure (or failure due to the volume decrease)

(results from a model test on loose sand)

The so called local failure mechanisms are intermediate between general and punching failure.

Only the general failure mechanism is considered here that applies to relatively dense soils. In

case of loose soils, where punching could occur, deep foundations should be considered instead

of footings.

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2.3 Terzaghis bearing capacity equation

This equation defines the limit value of a uniformly distributed

vertical load applied to the horizontal ground surface subjected to

a known lateral load q

The bearing capacity coefficients are functions

of the friction angle . Their expressions are worked out assuming:

1) Plane strain regime 2) Rigid-plastic material behaviour 3) Limit shear resistance of soil defined by Mohr-Coulomb yield condition 4) Principal stress direction at failure coinciding with vertical and horizontal directions

The three coefficients are separately evaluated on the basis of equilibrium and Mohr-Coulomb

condition. This implies the assumption that there is no mutual influence among them.

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2.4 Plane strain vs. plane stress regime

For linear elastic isotropic

materials:

[ ( )]

( )

Elastic plastic vs. rigid plastic behaviour

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Mohr-Coulomb criterion

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Apparent hardening in plane strain conditions

Element of elastic ideally

plastic material in plane

strain conditions

From the origin to point A the stress

path develops within the yield

surface: A = limit elastic state;

. From A to C the stress point moves

along the yield surface and depends on the plastic strain.

When C is reached, no further increase of could keep the stress point on the yield surface and, consequently, the element fails.

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Material element

Plane strain element

A C

A

C

III (out-of-plane stress)

The consequence of this is that the element exhibits an apparently hardening behaviour even in

the case of elastic perfectly plastic material.

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Mohr-Coulomb criterion in terms of principal stresses

Mohr-Coulomb criterion in plane strain regime assuming

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Plane strain Mohr-Coulomb criterion at the limit elastic state

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2.5 Derivation of the bearing capacity coefficients

Terzaghis equation can be applied in terms of total or of effective stresses, using undrained

or drained cohesion and friction angle.

In undrained conditions the total unit weight of

soil should be used.

In drained conditions, the dry and the submerged unit

weight of soil can be used, respectively, above and below the water table. If the water table level

is likely to change in time, it is safe to use the submerged weight

The bearing capacity coefficients are derived, in terms of the friction angle, evaluating the limit

load under particular conditions:

coincides with for coincides with for

depends on for

Water table

A=active zone

P=passive zone

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2.5.1 Bearing capacity coefficient

(conditions: )

Sign conventions for the stresses

Mohr circle for the passive zone

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Pole of the Mohr circle

- Consider a point of the Mohr circle that represents the stresses acting on a plane having known inclination,

e.g. point D corresponds to the stresses acting on

the ground surface.

- From point D draw a line parallel to the plane on which the stresses are applied (i.e. the horizontal line in this case).

- The line intersects the Mohr circle in a point P which is referred to as the Pole.

- If a line is drawn from the Pole, it intersects the Mohr circle in a point the represents the stresses acting on a plane parallel to that line.

Consequently, lines PA and PA are parallel to the planes along which failure occurs within the passive zone.

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Mohr circle for the active zone

Now we have to define the external limit of the passive zone, i.e. the position of point F.

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The position of point F is defined assuming that a radial equilibrium zone, bounded by a

logarithmic spiral, connects the active and passive zones.

The expression of qlim is obtained by imposing the rotational equilibrium of zone GEF with

respect to point G. To this purpose it is necessary to express the stresses acting on the planes

GE, GF and on the spiral arc EF.

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The stresses acting on plane GF are defined by the Mohr circle of the passive zone

The shear stress is

which, substituting V and H, becomes

The normal stress is

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Substituting one obtains

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The stresses acting on plane GE are defined by the Mohr circle of the active zone

The shear stress is

which, substituting V and H, becomes

The normal stress is

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Substituting one obtains

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It is not necessary to evaluate the stresses acting on the arch EF of the logarithmic spiral.

In fact, the angle between the normal to the tangent and the radius of the logarithmic spiral is

equal to . Considering that the spiral represents a failure line, Mohr-Coulomb relationship holds between

the normal and shear stresses acting on it: (note that the cohesion is not considered in evaluating Nq).

Consequently, at any point of the spiral the resultant of and is directed as the radius and does not affect the rotational equilibrium about the centre G of the spiral.

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The equation of equilibrium about point G reads

where

Substituting the expressions of one obtains

(note that if =0, )

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2.5.2 Bearing capacity coefficient (conditions: )


The slope of the failure planes in the passive zone

coincides with that obtained for the case.

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The slope of the failure planes in the passive

zone coincides with that obtained for the case.

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The expression of is obtained by imposing the rotational equilibrium of zone GEF with respect to point G. To this purpose it is necessary to express the stresses along planes GE, GF

and along the spiral arch EF.

The normal stress on plane GF is

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The normal stress on plane EG is

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The rotational equilibrium of zone GEF about point G involves the contribution of the stresses

acting on planes EG and GF and on the spiral arc EF. They are referred to, respectively, as M(1),

M(2), M(3).

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Let evaluate first the contribution of the stresses and , acting on the logarithmic spiral EF , to the rotational equilibrium of zone GEF.

The resultant of stresses and coincides with the radius and does not affect the equilibrium. Consequently, only the shear stress =c affects the equilibrium of the arch.

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The above integral is solved considering that and introducing the

following expressions: and

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Hence, the rotational equilibrium of zone EFG reads

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2.5.3 Evaluation of in undrained conditions (=0)

Rotational equilibrium about point G

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2.5.4 Relationship between and (Thorme des tats correspondants, Albert Caquot, 1881-1976)

Surface (2) is obtained by translating surface (1) by a compressive stress . As a consequence, should correspond to determined under an increase of normal stress .

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(which corresponds to

)

Hence, =

( )

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2.5.5 Evaluation of in the case of inclined foundation and inclined load

Angles and are known while and have to be determined as a part of the solution.


triangle OAC:

triangle OCQ:

since

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Let now derive the expressions of the principal stresses I and II

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The expression of is obtained by imposing the rotational equilibrium of zone GEF about point G

Active zone: Passive zone:

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Rotational equilibrium about point G

where

Note that when the above expression of reduces to its standard form

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2.5.6 Evaluation of in the case of inclined foundation and inclined load

According to Caquot theorem, the expression of can be derived from that of by considering an applied normal stress . Since the normal stress is applied also on the foundation plane, the computed inclination of does not coincide with the inclination of the applied load Q. Consequently, the calculation should be based on an initial inclination

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Equilibrium in the direction tangent to the foundation plane

{

{

[

]

(

)

[

]

(

)

Considering that ,

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{

[

]

(

)

As previously observed, some attempts are necessary by changing the value of until the correct values of and, consequently, of are obtained.

If the applied load is normal to the inclined foundation plane, i.e. if , also vanishes and the expression of becomes

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The expression of in undrained conditions can be obtained assuming and adopting the same procedure previously described for evaluating in the presence of inclined foundation and load.

(Fig.1)

Passive zone GFI

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Active zone GEH

(Fig. 2)

(Fig. 3)

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Considering Figs. 2 and 3:

(

)

Consider now Figs. 1 and 2.

Points A and Q in Fig.2 represent the stresses acting, respectively, on lines EG and GH in Fig.1.

Due to the properties of Mohr circle, the angle between the lines AC and CQ in Fig.2 is twice

the angle between the lines EG and HG in Fig.1. Consequently:

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The angle in Fig. 1 is evaluated from the Mohr circle in Fig. 2

The angle is determined from the Mohr circle in Figs. 3,

;

(

)

and the angle is determined from Fig.1

(

)

(note that must be positive, i.e. )

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The expression of is obtained by imposing the rotational equilibrium of the radial zone GEF about point G.

where

;

;

Substitution of the expressions of and

; ; (

)

into that of leads to

[ (

) ]

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Substituting the expressions of and of ,

(

)

and considering that since c=1, the following expression for is arrived at

[

]

The implicit structure of the above equation requires an iterative solution process. A trial value

of is introduced into its right hand side term, thus obtaining a refined value of . This value is introduced again into the equation and the process continues until stabilizes. To choose the initial value of consider that and that . Hence, must fall within the interval:

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2.5.7 Bearing capacity coefficient (conditions: )

It can be shown that the previous solutions for and based on the described simplified procedure coincide with the correct solutions obtained with the Methods of Characteristics (see

e.g.: R. Hill, The Mathematical Theory of Plasticity, Oxford Press, 1950).

The same procedure cannot be applied to the case of (i.e. ) because the active zone underneath the foundation is not anymore characterized by straight failure lines.

This is due to the fact that the simultaneous presence of a surface load and of the soil self-weight leads to a rotation of the principal stresses which are not vertical and horizontal anymore.

As a result, the expression of obtained with the described simplified procedure overestimates the limit value of the load carried by the foundation.

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An approximated evaluation of can be obtained assuming straight failure lines in the active

zone, but considering their inclination as an unknown.

Having worked out the expression of in terms of , is determined by minimizing it with

respect to .


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The Mohr circle of the active zone coincides with that for the passive zone if and are exchanged with each other.

The expression of the limit load in terms of is obtained by writing the equation of rotational equilibrium of the radial zone about point G.

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Finally, the approximated value of is reached by minimizing with respect to .

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Approximated expressions for Expression of and

(Vesic)

(Lundgren)

(Spangler) ( )

Bearing capacity coefficients

Nc Nq N 0 5.14 1.0 0

5 6.5 1.6 0.1 0.5

10 8.4 2.5 0.4 1.2

15 11.0 3.9 1.2 2.5

20 14.8 6.4 3.0 5.0

25 20.7 10.7 6.8 9.7

30 30.1 18.4 15.1 19.7

35 46.1 33.3 33.9 42.4

40 75.3 64.2 79.5 100.4

45 113.9 134.9 200.8 297.5

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2.6 Correction factors for Terzaghis bearing capacity equation

The original equation proposed by Terzaghi for a shallow strip footing subjected to a vertical

load

was corrected by Hansen in order to apply it also in other conditions

s = correction factors accounting for the foundation shape (rectangular or circular)

d = correction factors accounting for the foundation depth ( ) i = correction factors accounting for the inclination of load

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2.6.1 Shape factors

(in the following L and B denote, respectively,

the major and minor sides of the foundation)

Factors proposed by Hansen: Factors proposed by Vesic:

Vesic factor cannot be used when =0, in fact: In this case it is sufficient to substitute within the expression of :

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2.6.2 Depth factors

These factors account for the shear resistance of the lateral soil.

They should be used with care because their contribution could

vanish if the lateral soil is excavated after the completion of the

structure. This applies also to .

Factors proposed by Hansen: Factors proposed by Vesic:

If

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2.6.3 Load inclination factors

The vertical limit load in the presence of a known horizontal load

can be evaluated introducing these factors in Terzaghis equation. Note that this is different from computing the inclined limit load

by modifying the bearing capacity coefficients N as it was previously shown. Note also that the

horizontal load cannot exceed the limit value corresponding to the sliding failure of the

foundation on the underlying soil:

Case of a strip footing ( )

If

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Case of rectangular footing

If

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2.6.4 Factors accounting for the inclination of the foundation base

If

Note that the above coefficients permit computing the limit vertical load. This is different from

computing the limit load normal to the inclined foundation plane.

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2.6.5 Factors accounting for the inclination of the ground surface

If

These coefficients can be used only if is substantially smaller than . It is always advisable to perform also a stability analysis of the slope subjected to the overall

load of the structure. This is mandatory when .

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2.6.6 Factor accounting for the foundation settlement

Besides the fact that the evaluation of the settlements is always mandatory, a suitable factor of

safety FS (2.53) has to be adopted with respect to the bearing capacity also to avoid excessive settlements under working loads.

To avoid excessive settlements when dealing with compressible soils, reduced shear strength

parameters and could be adopted in the calculations of the bearing capacity.

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Alternately, Vesic proposed the use of the following reduction factors in the bearing capacity

equation.

If

The above reduction factors are disregarded if .

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2.7 Shallow foundations subjected to eccentric load

Under the simplifying assumption of linear pressure distribution between footing and underlying

soil, the following conditions could occur depending on the load eccentricity.

The literature does not provide simple equations for evaluating the bearing capacity of footing

subjected to non-uniform load. To circumvent this drawback the bearing capacity is evaluated

adopting an equivalent footing of reduced size and with constant pressure distribution.

(Note that L always represents the largest side of the equivalent foundation)

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T

N 0

In the general case, the footing is subjected to normal and shear forces and to bending moment.

To evaluate the bearing capacity in these

conditions it is necessary to determine

the N-M-T domain of the footing.

This problem will be discussed

during the exercise classes.

N

M

T

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2.8 Shallow foundations on layered soil

Consider first the case of a soil deposit that could be roughly subdivided into two layers.

If the shear resistance of layer (1) is smaller than

that of layer (2), either the parameters of soil (1) are

adopted for evaluating the bearing capacity or,

more advisably, the foundation plane is brought

down into layer (2).

If the shear strength of layer (1) is larger than that

of layer (2), the load pressure is spread on layer (2).

Then, the limit values of q and q are evaluated. The design of the foundation is based on the least

of the two bearing capacities.

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In the case of a sequence of layers having appreciably different shear strength characteristics,

the bearing capacity can be evaluated using one of the methods of slices used for slope stability

analysis.

For a chosen shape of the failure surface, the value of qlim is determined by imposing the global

equilibrium of all slices with respect to point C.

The minimum value of is found by a trial and error process by changing:

A common feature of these methods is that they introduce suitable assumptions on the

interaction forces between the slices so that they do not appear in the rotational equilibrium of

the sliding wedge of soil about point C.

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Fellenius method (1936)

This method is based on the assumption that the lateral forces and acting on each slice are parallel to the base of the slice.

The force Ni is determined through the equilibrium of the slice in the

direction normal to its base

Knowing the forces Ni and Ti, the equilibrium of the slices is imposed about point C obtaining

the value of that corresponds to failure. The process is repeated changing the shape of the failure surface until the minimum value of is reached.

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Bishops method (1955)

This method is based on the assumption that the lateral forces acting

on each slice are normal to the face of the slide (i.e. that they are

horizontal).

Consequently, the expressions of forces and are

Also in this case the lateral forces and do not appear in the global equilibrium of the assemblage of slices.

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2.9 Remarks on the pore pressure effects

To show the influence of pore pressure, consider a partially submerged slice having an

horizontal base.

= unit weight of soil without pore liquid = unit weight of soil with pores partially filled with water = unit weight of fully saturated soil

S = Degree of saturation =

; n = porosity =

The equation of (total stress) equilibrium of the slice in the vertical direction reads:

where is the normal effective stress and p is the pore pressure at the slice base.

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The pore pressure p consists of three components

= pore pressure due to the steady state seepage previous to the construction, this includes also the hydrostatic pore pressure

= pore pressure due to consolidation, i.e. to the variation of volume caused by the change of volumetric stresses

= pore pressure related to the change in volume due to the plastic dilation

The pore pressure is also referred to as excess pore pressure with respect to the steady state conditions . The evaluation of the three components of the pore pressure p is necessary for determining the

limit shear stress at the base of the slice that governs the stability problem,

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The component can be measured by means of piezometers before the beginning of construction. The remaining components must be calculated on the basis of the mechanical

parameters of soil and of the characteristics of the collapse mechanism.

Due to the uncertainties in evaluating and , quite often the stability analysis is divided in two independent stages:

- Long term analysis based on the effective stress parameters and that account only for assuming that the excess pore pressure has already dissipated;

- Short term analysis based on the undrained cohesion assuming that this parameter accounts for the effects of the initial excess pore pressure .

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2.10 Structural details of shallow footings

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2.11 Interaction of adjacent footings

Shallow foundations in seismic zone

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Foundations: Bearing Capacity