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4 STRIP FOOTINGS

4.1 Strip footing on Winkler soil

(the distributed pressure on the soil is p/b)

Differential equation of the elastic curve:

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The distributed load p on Winkler soil is expressed as

Neglecting the distributed load q the governing differential equation becomes,

which introducing the following constant

becomes

The integral of the above homogeneous differential equation is

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4.1.1 Analytical solution for a load applied to footings of unlimited length

Consider a strip footing of unlimited length subjected to a concentrated load P

Since the deflection of the beam vanishes for (i.e. ( ) ), the expression of the elastic curve reduces to

Because of symmetry, for we have , hence

Consequently, the expressions of y and y are

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In order to reach the expression of the integration constant C3 we can follows two ways.

a) The first one consists in imposing the equilibrium of the beam in the vertical direction

b) Another way is to take into account the symmetric distribution of the internal shear force T

about the point where the external load P is applied, i.e. for

Considering that , for x=0 we have

To apply the above equation it is necessary to

express the third derivative of y(x).

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Which, for x=0, becomes

Substitution of this equation into leads to

where

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Substituting C3 into the expression of the elastic curve we reach the following equation which

holds for positive values of x

Hence, the solution for of the strip footing subjected to a load P is

where

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4.1.2 Analytical solution for a moment applied to footings of unlimited length

Consider a strip footing of unlimited length subjected to a concentrated moment W

In this case the equation of the elastic curve

fulfills the following conditions

Consequently, the governing equation reads

Also in this case two ways can be followed to reach the expression of the integration constant C3

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a) The first one consists in imposing the rotational equilibrium of the beam

b) Another way is to take into account the symmetric distribution of the internal moment M

about the point where the external load P is applied, i.e. for

Introducing this condition into the equation

expressing the bending moment

we have

To apply this equation it is necessary to express the second derivative of the governing equation.

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Which, for x=0, becomes

Substitution of this equation into leads to

where

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Substituting C3 into the expression of the elastic curve we reach the following equation that

holds for positive values of x

Hence, the solution for of the strip footing subjected to a moment W is

where

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4.1.3 Solution for a strip footing of limited length

If the strip footing has a relatively small length L (i.e. ) its deformation can be

neglected and the foundation is equivalent to a spread footing.

In the case of a large distance L between the applied loads and the edges of the strip footing (i.e. ) the problem can be analysed adopting the solutions for foundation of unlimited

length.

This condition, however, seldom occurs in practice.

In the remaining cases it is necessary to work out the solution for the beam of limited length.

To this purpose, the superposition of suitable loading conditions on the beam of unlimited length

is adopted.

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Let consider a beam of limited length subjected to a given loading condition

and let apply the same loads to a beam of unlimited length. This beam is also subjected to

external loads and moments (PA, WA, PB, WB) applied to points A and B that correspond to the

edges of the actual foundation of limited length.

In order to evaluate the loads applied to the end points of the unlimited beam it is sufficient to

enforce that at these points the internal shear force and bending moment vanish.

This leads also to the solution of the actual beam. In fact, in the portion AB, the same differential

equation governs both beams. In addition, the same boundary conditions apply. Hence, the

solutions of both problems coincide.

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The following equations enforce that the internal forces (TA, MA, etc.) at point A and B of the

unlimited beam vanish.

,

, etc. represent the internal forces in the unlimited beam due to the applied loads.

( ) represents the internal shear force on the right of point A of the unlimited beam due

to a unit load applied load in A; etc.

The solution of the above system leads to the values of PA, MA, etc.

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4.1.4 Solution based on the principle of minimum potential energy

For a conservative systems, among all its cinematically admissible displacement fields the one

corresponding to a stable equilibrium condition minimizes the total potential energy.

(I) admissible displacement field

(II) non-admissible displacement field

(III) displacement field that minimizes the total potential energy

The solution is found by minimizing the total potential energy expressed in terms of the admissible displacement fields ui (i=1,2, , )

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In the present context, the total potential energy has three contributions: elastic strain energy of the beam B; elastic strain energy of Winkler soil W; work potential of the applied loads WP. Note that the external loads do not vary with the deflection of the beam.

Rayleigh Ritz method is adopted to express the potential energy as a function of the admissible

displacement fields. In particular, these fields are expressed through series of functions that

fulfil the displacement boundary conditions and the strain compatibility relationships.

These functions depend on unknown parameters which will be evaluated by minimizing .

For sake of simplicity let divide the applied loads into their symmetric and non-symmetric parts.

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Symmetric loading condition

Taking into account the symmetry of the vertical

displacements with respect to the origin, they are

expressed as follows

where is the displacements of the beam ends: . This definition of implies than n must represent odd numbers 1, 3, 5, 7,

As a consequence, the expression of ( ) involves N+1 unknowns: N coefficients (n=1,3,5) and the displacement . The N+1 equations necessary to evaluate these unknowns consist of the equilibrium of the beam

in the y direction and of N equation expressing the minimum of the total potential energy.

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Equilibrium in the vertical direction.

(p is the pressure on

Winkler soil)

Substitution of the above expression into that of ( )

leads to

Considering that

the equation of equilibrium in the vertical direction becomes

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Non-symmetric loading condition

In this case the vertical displacements are expressed

as follows

where is the displacement of the ends of the beam: . This definition of implies than m represents even numbers 0, 2, 4, 6,

As a consequence, the expression of ( ) involves M+1 unknowns: M coefficients (m=0, 2, 4) and the displacement . The M+1 equations necessary to evaluate these unknowns consist of the rotational equilibrium

of the beam and of M equation expressing the minimum of the total potential energy.

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Equilibrium to rotation about the origin of axes.

Substitution of the above expression into that of y(x)

leads to

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Let now apply the principle of minimum total potential energy .

The expression of the deformed beam is

where if i is even , while if i is odd

The contributions to the potential energy are

The N+M equations are obtained by imposing that

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4.1.5 Finite difference solution

This is a discrete method in which the unknowns are represented by the displacements of a limited number of nodes.

The discretization involves the governing equation

where q is the distributed applied load and p is the reaction of Winkler soil.

It is necessary to express the derivatives of ( ) in terms of the nodal displacements.

First derivative:

Forward derivative: ; Backward derivative:

Central derivative:

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Second derivative:

Finite difference approximation:

Third derivative:

Finite difference approximation:

Forth derivative:

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Substitution of the expression of the forth derivative into the equation of the elastic curve

leads to

where the pressure pi of Winkler soil is expressed as

and the applied forces Qi are reduced to distributed loads over the distance x

The solving system of linear equation is obtained by writing the equation of the elastic curve for

all nodes of the beam.

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A drawback of the finite difference method concerns the imposition of the boundary conditions.

In the present case they correspond to vanishing internal shear force and moment at the ends of

the beam.

Consider the end node i of the beam,

Before solving the system it is necessary to eliminate the displacements of nodes i+1 and i+2.

To this purpose let impose the boundary conditions: and

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It should be observed that in the case of strip footings on Winkler soil the finite difference

solution does not present marked computational differences with respect to that based on the

finite element method.

In both cases, in fact, the behaviour of the beam is represented by a banded stiffness matrix K.

In the finite element context, the coefficients representing the stiffness of Winkler springs are

added to the terms of the main diagonal of the stiffness matrix of the structure.

This introduces constraints that eliminate the rigid movements of the beam (its stiffness matrix

would be singular otherwise) and allows for the solution of the system of linear equations.

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4.2 Strip footing on elastic half space

The finite difference method can be used for representing the beam behavior also in the case of

elastic half space. However this method is limited to relatively simple structures and presents

difficulties when applied to the geometrically complex cases.

Since for both finite difference and finite element methods the structure is represented by its

stiffness matrix, the same solution procedure can be applied in both cases when analyzing a

structure on elastic half space.

The finite difference stiffness matrix has only one degree of freedom per node (y displacement),

while the finite element one involves, in general, also the x displacement and the rotation of the nodes.

In the finite element cases, the stiffness coefficients of Winkler springs are added to the main

diagonal terms of the structural stiffness matrix that correspond to the y displacements.

The stiffness coefficients of the foundations on elastic half-space are obtained through

Boussinesq solution as follows.

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The pressure exerted by the beam on the half space is seen as a series of distributed loads on

known areas

(side view)

(plan view)

The settlement of point j is expressed as

where the coefficients are evaluated according to Boussinesq solution.

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. .

.

. . . . . . . . fpi yj

fk

The relationship between pressures and settlement can be cast in the following matrix form

where is the matrix of coefficients, deriving from Boussinesq solution, that represents the deformability of the half space; is the nodal force vector corresponding to the pressures and

vector collects the nodal displacements of the foundation.

Note that matrix is fully populated and, in general, non-symmetric. Note also that the same relationship can be worked out for any structure resting on elastic half space.

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Let be the diagonal matrix which contains only the terms of the main diagonal of matrix and that coincides with the flexibility matrix of Winkler soil.

[

]; [

]

The relationship between nodal displacements u of the structure and loads acting on it is,

where the banded stiffness matrix of the structure, being unconstrained, is singular and where are the nodal forces due to the soil pressures .

Note that, in the finite element case, the vector of nodal displacement contains additional degrees of freedom with respect to the displacements of the foundations.

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The stiffness matrix of the half-space is,

;

hence, the solving system could be obtained by overlapping the two stiffness matrices.

( )

It has to be considered, however, that matrix is fully populated and, in general, non-symmetric. Consequently also is so. This would prevent its implementation in standard finite element codes which consider the stiffness matrix as symmetric and banded.

To overcome this drawback, the same procedure already outlined for the analysis of statically

undermined structures on spread footings can be adopted.

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First, the stiffness matrix corresponding to the flexibility matrix is evaluated,

[

]

then the following iterative process is performed.

a) The problem is solved overlapping matrices and (this corresponds to the analysis with Winkler soil) obtaining the nodal forces exchanged between structure and soil.

( )

b) These forces are used to evaluate the settlements of the foundations on the half-space

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c) The structural problem is solved again, without introducing the half-space stiffness matrix,

imposing to the structure the displacement of the foundations as nodal constraints,

and the forces are re-calculated.

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

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

markedly different from that of the structure (e.g. a very stiff structure on a very deformable

soil). 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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4.3 Strip footing on Pasternak and on Repnikow soils

Pasternak soil is an obsolete model that adds to Winkler springs a layer S reacting only to shear.

This model and the subsequent one proposed by Repnikov are mentioned here only for

providing some historical hints.

Pasternak soil (Petr Leontevic Pasternak, 1930)

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The equation of equilibrium of layer S is

Assuming , the equilibrium equation becomes

The stress strain relationship of layer S reads

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Substitution of the stress strain relationship into the equilibrium equation leads to

Introducing the above equation, and that of Winkler soil , into the equation of the

elastic curve of the beam,

The governing equation for Pasternak soil is finally arrived at

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Repnikov soil (L. N. Repnikow, 1967)

This model consists of the superposition of Winkler springs and elastic half-space. It aims at

limiting the drawbacks of the two original modes:

a) settlements developing also at a large distance from the foundation in the case of elastic half-

space;

b) settlements developing only below the foundation in the case of Winkler soil.

However, the actual difficulties in predicting the foundation settlements depends on the non-

linear behaviour of soils rather than on the linear model adopted in the calculations.

Since analytical solutions applicable in engineering practice for strip footings on elastic half

space are not available, also Repnikow soil requires the use of the numerical solution procedure

previously described.

Winkler, Parternak and Repnikow models present the same drawback: the constants K (Winkler

springs) and G (Pasternak layer S) do not represent physical properties of soils and,

consequently, cannot be determined by tests on soil samples.

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4.4 Additional types of continuous footings

Grid foundations: they consist of combined footings formed by intersecting continuous strip footings. In most cases the load of the structure are applied at their intersection points.

They are adopted to limit the differential settlements of the structure.

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Mat or slab or raft foundations: these are continuous footings having a slab like shape. They can have depressions or openings. In most cases they are chosen to reduce the differential

settlements. They could be economically convenient with respect to spread and strip footings

when the total area of the isolated foundations exceeds about 50% of the total area of the

building.

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Cellular mats can be used also to host facilities, e.g. garages or technical plants.

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4.5 Compensated foundations

These are foundations in which the stress relief due to excavation is approximately balanced by

the applied stress due to the foundation. They are usually adopted in very soft soils when the use

of deep foundation would not be convenient.

A compensated foundation normally comprises a deep basement and a rigid slab.

Main problems of compensated foundations are:

- differential settlements - stability of the excavation walls - stability of the bottom of excavation - effect of the pore water pressure

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Log scale

.

= unloading

. . (1) (2) (3)

(1) Initial in situ condition

(2) End of swelling due to unloading

(3) Construction of building

Mechanism governing the settlements of a compensated foundation.