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3 SETTLEMENTS OF SPREAD FOOTINGS

3.1 Physical causes of settlements

Immediate settlements caused by the deviatoric and volumetric deformation of the soil skeleton which, in turn, depends on the increase of effective stresses due to the applied loads.

Increase of the effective stresses due to the reduction of the pore pressure induced, e.g., by the lowering of the water table level due to drainage systems or to hydrogeological causes.

Consolidation of clay, i.e. time dependent transfer of stresses from the liquid to the solid phases of soil.

Swelling clays: These are clays with a high Activity A, i.e. ration between the Plasticity Index and the percent in weight of clay particles (size less than 2 m). Non-active clays A1.25

Active clays undergo large volume increase (swelling), when wetted (e.g. by rainfalls), and

large volume reduction (shrinkage), when dried.

This can produce cyclic seasonal heave/settlement of footings if their base is too close to the

ground surface.

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Heave due to frost: In cold areas, if shallow foundations rest on saturated soils, appreciable heaves can be caused by freezing of the pore water. The settlement/heave caused by swelling

soils and by frost cannot be prevented. In these cases it is necessary to place the footings

below the level at which these phenomena take place, or to use deep foundations.

Subsidence: It consists of settlement of large areas due to, e.g., to the lowering of the water table level due to hydrogeological causes; to pumping of water from drainage wells; to

extraction of natural gas or oil from deep rocks, etc. Subsidence is usually associated to local

differential settlements that could damage the buildings.

Liquefaction of saturated sand deposits: This depends on the temporary loss of shear strength of loose sand due to a rise of excess pore water pressure during cyclic loading, e.g. caused by

seismic events.

Seismic densification of dry or non-saturated sand: This is due to the densification of loose sands above the groundwater level due to ground vibrations.

Here only the settlements due to the applied loads will be considered on the bases of:

a) Winkler soil model; b) Bussinesq elastic half-space. The evaluation of settlements due to

consolidation will be discussed during the exercise classes.

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3.2 Winkler soil (E. Winkler, 1867)

The soil is seen as equivalent to a liquid which applies to the base of the footing a pressure

linearly dependent on the local settlement. In other words, the foundation is treated as the hull of

boat immerged in water. Winkler soil can be also seen as a continuous bed of independent

springs. This soil model was originally proposed for the analysis of rails.

A linear relationship is assumed between the local pressure q exerted by the foundation on the

soil (that is equal to the subgrade reaction of Winkler soil) and the settlement ,

where the Winkler constant k, having dimension of force/volume, represents the unit weight of

the equivalent liquid.

It is not possible to determine the parameter k through laboratory tests on soil sample because

the weight of the equivalent liquid does not represent a physical property of a solid material.

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The only way to estimate k is based on the results of in situ plate load tests.

Even in the linear range the load settlement curve depends on the size of the plate.

This, as already observed, is due to the fact that Winkler soil cannot model the behaviour of

continuous solid, even in the elastic case.

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The settlement , or the parameter , is obtained from the linear part of the test and is scaled to the dimensions of the actual foundation through empirical relationships.

S0 = settlement of plate

S = settlement of footing

B0 = size of plate

B = size of footing

k0 = Winkler constant for the plate

k = Winkler constant for the footing

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In the case of a rigid footing, a linearly distributed interaction pressure exists between

foundation and Winkler soil.

In the design practice, this linear distribution is assumed also when evaluating the settlement on

Boussinesq half-space.

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3.3 Elastic half-space (Joseph Valentin Boussinesq, 18421929)

In 1885 Boussinesq proposed the solution for the stress

and strain distribution within an isotropic linear elastic

half space subjected to a point load normal to its surface.

The solution provides also the expression for the

displacement of points of the half space surface.

The solution can be applied also in the case of a known load p(x,y) distributed on an area A,

and to the settlement of a rigid footing:

In the case of a circular footing with radius R:

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Note that for Winkler soil the settlement depends solely on the applied pressure q, while in the

case of elastic half space it depends also on the size of the foundation.

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3.4 Contact pressure between soil and footing

The actual contact pressure between foundation and soil depends on the characteristics of soil

and on the stiffness of the foundation. Consider the case of a rigid spread footing:

For saturated clay (undrained conditions), the contact pressure is similar to that predicted by the

elastic half space solution.

For relatively shallow foundations on granular soil, the contact pressure tends to decrease in the

vicinity of the footing edges.

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Contact pressures for Winkler soil

Contact pressures for elastic half space

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The analysis of settlements of shallow foundations on elastic half space is customarily carried

out neglecting the actual distribution of the soil/foundation contact pressure.

A constant or linear distribution of the applied load is assumed in most cases, hence neglecting

the influence of the foundation stiffness.

Note that, in the case of a rigid footing, the above distributions are correct for Winkler soil

while they are not theoretically for the elastic half space.

The distribution of the contact pressure, however, has an appreciable influence only on the state

of stress within the footing. In particular, Winkler soil could underestimate the bending moment

for stiff slab foundations, e.g. if the applied load is uniformly distributed on the foundation, the

bending moment vanishes for Winkler soil.

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Example:

Bending moment at the center of the footing

Difference %

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3.5 Settlement of footings on elastic half space

Let consider the footing as equivalent to a uniformly

distributed load q over a rectangular area.

The settlements of points of the loaded area can be

evaluated by means of the following equation, where

the coefficient depends on the position of the point within the area.

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The analytical expression of the corner settlement is as follows

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The solution for the homogeneous half space can be also used in the case of layered soils

assuming that the stress state is not affected by the change of the elastic properties with depth.

a) The stress distribution is evaluated below a corner of the foundation

subjected to the uniformly distributed load q

b) The distribution of vertical strain is evaluated along the axis z:

c) The settlement is calculated by integrating the vertical strains:

The integration depth H should reach possible soft layers and should fulfill the requirements:

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[ ( )]

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Settlement of non-corner points

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Settlements due to bending moment

Settlement of point M:

Settlement of point C:

If :

If :

Rotation:

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Settlements of non-corner point P due eccentric load on A

The loading conditions (a) and (b) can be expressed as the superposition of the following cases

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Rotation of nearby footings

The rotation B of foundation B due to the load q on foundation A can be easily evaluated on the basis of the settlements of pints P1 and P2

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3.6 Settlements of statically determined structures

In this case the settlements do not influence the forces that the structure exerts on its foundations.

Consequently the reaction forces can be evaluated directly, i.e. by solving the structural problem without considering the foundation settlements.

Then, the settlements are determined on the basis of the reactions .

Note that, in the case of Winkler soil, the coefficients vanish for and that, in the case of half space, the matrix of coefficients is fully populated and not necessarily symmetric.

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In order to evaluate the parameter k, necessary for determining the coefficients in the case of Winkler soil, the following scheme can be adopted,

which, being , leads to

( )

In this way the direct coefficient for Winkler soil coincide with those of the half space, that in turn depend on the size of the foundation. However the indirect coefficients are missing.

Consequently the total settlements in the two cases could never coincide.

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3.7 Settlements of statically indeterminate, or redundant, structures

In this case the settlements influence the forces that the

structure exerts on its foundations. Hence, the reactions depend on the settlements .

In the case of Winkler soil it is easy to add the stiffness

of each foundation (represented by a spring) to the stiffness

matrix of the structure. In fact, only some diagonal terms have

to be added to it.

Hence, the structural problem can be easily solved considering only the direct settlement of each

footing. This leads to both the reaction forces and to the settlements .

In the case of half space, the stiffness matrix of soil cannot be added to that of the structure

because it is fully populated and, in general, non-symmetric.

In this case and iterative process can be adopted for solving the linear problem of structure-

foundation interaction without using the complete stiffness matrix.

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a) The reactions of the foundations are determined by solving the structural problem adopting only the coefficients expressing the direct settlements (i.e. the Winkler soil scheme)

( ) ( ) ( )

b) The settlements of the foundations on the half space are evaluated considering both direct and indirect coefficients

c) The structural problem is solved again, and the reactions are re-calculated, imposing to the foundations the settlements evaluated at step (b)

( ) ( ) ( )

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Steps (b) and (c) are repeated until no appreciable changes of settlements and reactions occur.

This iterative procedure could not converge when the global stiffness of the foundations is

markedly different from that of the structure. In this case, a suitable provision could consist in

averaging at step (c) the settlements obtained from two subsequent iterations

( )

where

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3.8 Tolerable settlements of foundations

The maximum value of tolerable differential settlements strongly depends on the characteristics

of the structure, of its components (walls, pavements, technical plants, etc.), of its materials, of

its age, etc. Some approximated values are listed below.

Allowable deflection according to Polshin and Tokar (1957):

for for

Categories of potential damage (Bjerrum, 1963) Safe limit for flexible brick walls ( ) 1/150 Danger of structural damage to most buildings 1/150

Cracking of panels and brick walls 1/150

Visible tilting of high buildings 1/250

First cracking of panel walls 1/300

Safe limit for no cracking in buildings 1/500

Danger to structural frames with diagonals 1/600

Damages to machinery of industrial plants 1/750