FOUNDATION PLATES (outline)

FOUNDATION PLATES (outline)

to Lecture no.3

1. Introductory comments

Horizontal elastic plate resting on an elastic subsoil is to be solved, so its settlements w(x,y), but first of all - internal forces for material design (STR). Most of design cases meet complicated situations which need use of numerical methods, like finite elements or finite differences – applied to both the slab itself and the subsoil, sometimes also to a superstructure. Such situations definitely should be focused on:

• plates of complex shapes and varied loads, corresponding to building shapes and complex plan of foundations,

• plates of varying thickness, increased under columns or walls, plates with opening, foundation grids a.o.,

• complex geological structure of the subsoil including local weak-soil lenses, leaned strata, geological discontinuities,

• pile-plate foundations in which there is a significant contribution of rigid plates in load transmission to the subsoil,

• foundation plates situated on mining areas, karst and loess soils etc.

Sophisticated nonlinear elasto-plastic soil models are seldom used in design practice with soil-foundation interaction analysis, because load values q [kPa] are rather small under vast foundation plates; moreover, plates can reduce local unevenness of such loadings.

There is a standard approach focused on the simplest linear elastic model of the subsoil under a plate which seems to be the Winkler model. There are no reported evidences that the Winkler hypothesis can contribute to a real danger due to underestimated internal forces and finally lead to a structural failure; of course, the role of appropriately selected value of the subsoil coefficient C [MN/m³] and the used partial safety factors is of primary importance.

From the physical viewpoint, the model of elastic half-space or finite thickness layers is less controversial then the Winkler model consisting of separate springs; the more significant advantages are taking into account possible inhomogeneities of the subsoil and spreading of settlements outside a loaded place (foundation) known as “the neighbour’s effect”. On the other side, the model of the elastic half-space overestimates the role of deeply situated regions in the subsoil (usually of worse geological recognition); this maybe caused by stiffness of the subsoil, which usually increases with depth, as well as vertical/horizontal orthotropic properties of the subsoil. In addition, if one imposes on this a very subjective model of ground layering interpreted by a geologist or a geotechnical engineer, i.e. subdivision into strata assumed as homogeneous - the elastic half-space model loses some of its advantages.

2. Numerical methods

Design engineers dispose of large number of commercial computer codes addressed to soil-foundation interaction modelling; no doubt that they are more universal, well-tested, more reliable and user friendly then most own author’s codes.

Note that some popular commercial codes often say about the elastic half-space model but in fact they base on the simplest Winkler model using a specially selected “effective value” of the subsoil coefficient kz [kN/m³], so C [kN/m³] – see the material about subsoil homogenization on this WWW.

Numerical modelling is generally out of the scope of the courses GHB003321/CEB007361.

3. Some very approximate methods

The ideas presented hereafter can find only a tentative application, as a first approximation.

3.1. Rigid foundations:

a) for a centrally loaded rigid foundation, the response of the Winkler subsoil is constant, because uniform settlements cause the same reaction in each (separate)

„Winkler spring”;

a careful analysis is however recommended if the Winkler model is acceptable for the considered real case !

b) for a centrally loaded rigid foundation the response of the elastic half-space under a beam of the width B is concentrated under the outer sections;

roughly speaking, such a redistribution of contact stresses under a beam is on the level of ±25% on 4 quarters of the beam width B.

By analogy, for a centrally loaded rigid foundation, the response of the elastic half- space under a square footing is concentrated under the corners, like ±50% on 8 of 18 square subsections; details are discussed elsewhere (Example no.4 in the Lecture no.1).

The truth can be situated somewhere in between, so an envelope of the two cases a) and b) is recommended for a safe and economic estimation of internal forces.

3.2. Regular system of columns of (almost) the same vertical loads:

The directions x and y are separated.

For this purpose, two projections can be visualized and next two 1D “corresponding beams” are being solved which have the following stiffness:

• EI = ELh³/12 . . . resulting in a beam reaction rB(x) [kN/m]

• EI = EBh³/12 . . . resulting in a beam reaction rL(y) [kN/m].

To find these reactions under the corresponding beams the loadings Pij are also projected along the axes:

• in the L-direction P^Bi = Σ P^ij (sum up due to j),

• in the B-direction PLj = Σ Pij (sum up due to i).

Estimation of the 2D response can be taken as r(x.y) [kPa]

, = ∙

∑ ∑

For rectangular plates and regular systems of columns the method can lead to relatively useful results.

Exercise:

The above algorithm of separation of directions can also be applied to „evaluation” of Pij loadings on the top surface of the plate (where the forces are known by definition); the estimation errors are as follows:

1100x1650/3520 = 516 ≈ 500, 1120x1650/3520 = 525 ≈ 550, 1300x1650/3520 = 609 ≈ 600, 1100x1870/3520 = 584 ≈ 600, 1120x1870/3520 = 595 ≈ 570, 1300x1870/3520 = 691 ≈ 700.

q = qśr

q = qśr (1±0,25)

1,5 1,0 1,0 1,5 1,0 0,50 0,50 1,0 1,0 0,50 0,50 1,0 1,5 1,0 1,0 1,5

500 550 600

600 570 700

1100 1120 1300

1650

1870

x y

4. Analytical methods

Situation is much easier for thin plates than for thick ones. because the hypothesis on planar deformation of cross-sections can be used; moreover, foundation plates are in practice relatively thin.

For the elastic half-space assumed as the subsoil model, there is only one situation which can be solved analytically with no great problems (more or less…) – a cylindrical plate which is centrally loaded by a vertical concentrated force P (rotational symmetry); generally, a numerical support is unavoidable.

The simple Winkler model gives, of course, wider possibilities¹.

Let a horizontal plate rest on the Winkler subsoil, thickness

of the p of the plate is constant and the plate stiffness is defined as s:

=

_ν

∙

^[kNm2/m] = [kNm]

being the equivalent of EI for beams, x,y – plane of horizontal coordinates.

The constitutive relation for the Winkler model is as simple as possible:

, = ∙ ,

^[kPa].

Kirchhoff generalized the Euler-Bernoulli method for thin beams and obtained for plates the following partial differential equation for the settlement function w(x,y):

⋅ _{+ 2 ∙ + ! + ∙ , = "}

_#

_,

(1) where qo denotes an outer vertical loading applied to the top-surface of the considered plate; the homogeneous case happens if qo disappears, as for beams.

On vertical surface-elements, both in the plate interior and on an external boundary, there are considered 3 moments M and 2 shearing forces Q; the axial force N (of secondary importance, if happens) is ignored, as for the Euler-Bernoulli beams. In particular, on the boundary surface-elements of a rectangular plate LxB

(x,y oriented) or on its boundary which is composed of such x,y oriented surface-elements, the following generalized Euler-Bernoulli relations are true.

Bending moment Mx (on unit length) around the y axis in a cross-section perpendicular to x:

$ = %

^{) /}_/

& ∙ ' (& = − ∙ + ν _{∙ !}

Bending moment My (on unit length) around the x axis in a cross-section perpendicular to y:

$ = %

^{) /}_/

& ∙ ' (& = − ∙ + ν _{∙ !}

Twisting moment Mxy = Myx (on unit length) in a cross-section perpendicular to x or y:

$

-

= %

^{) /}_/

& ∙ . (& = − 1 − ν _{∙ ∙}

Shearing force in a cross-section x (on unit length, y-oriented):

0 =

¹²

+

¹³

= − ∙ + !

Shearing force in a cross-section y (on unit length, x-oriented):

0 =

¹⁴

+

¹³

= − ∙ + !

As an exercise corresponding to the Bleich method for beams considered in the Lecture no.2, an analogous version (outline) of the Bleich method adopted for plates can be formulated which is almost the same.

1. The idea of the Bleich method for beams focuses on the use of the fundamental solution of the first kind, so on a vertical force P applied to a doubly-infinite beam.

1 The list of referenced text-books is very long and very useful; in particular, it should be recommended for testing any author’s own codes (benchmarks).

(0,0) y

qo(x,y)

P

x

Hereafter, a finite plate – not a finite beam – is extended to infinite plate in both directions x,y and again a vertical concentrated force P is applied to the origin (0,0). For such a situation, the fundamental solution of the first kind is known as the Green function, very popular term in theory of all linear theories where the superposition principle is assumed (such is the Boussinesq solution in particular).

2. The solution is of course axially symmetrical, therefore another version (1) will be used, expressed in cylindrical coordinates (ρ,ϕ); but the solution is independent of the angle, so w(ρ,ϕ) = w(ρ) = const(ϕ) and for qo =0, i.e. with the exception of one point (0,0), it has the following form (homogeneous equation):

⋅ ₅

₆₇⁶

₊

₇

_∙

₆₇⁶

₈ 9 + ∙ 9 = 0 (2)

This exchange of ∂ to d makes a great difference, because solving of ordinary differential equations is much easier than partial different equations; however, easier does not mean easy. Indeed, (2) has functional coefficients, not constants. An effective tool for such problems was already mentioned in Example no.2 in the Lecture 2 – this is a polynomial series expansion; w(ρ)=Σci⋅ρⁱ is substituted to (2) and this way an infinite sequence of ci coefficients can be found (and the series itself is convergent).

3. As previously for beams, the following dimensionless coordinate is introduced ξ = ρ/LW ,

where this time

;

_<

= =

^>_? [m]. The solution of (2) belongs to a class of special functions and it is known as the Kelvin function Kei(ξ), expressed as an infinite polynomial series; its plot, which can be found in tables, is in fact close to the solution e^-ξ⋅(cosξ+sinξ) for beams.

Coming back to the Cartesian coordinates:

. =

_B∙?∙^A

C

∙ DEF ξ =

_B∙?∙^A

C

∙ DEF G

^{H I ) I}

C

J

(3) where P ist applied at (xo,yo), not necessary at (0,0).

4. By integration of the Green function (3), so as usually dP = qo(xo,yo)dxodyo and so on, any distributed load qo

can be taken into account (inhomogeneous equation) but all the time this is the case of the infinite plate.

5. Let

B

denote a boundary of real finite plate. We focus on a free (unloaded) boundary, without loss of generality. The boundary is composed of elements which are parallel either to x or y; outer load is in form of qo or Pi.

6. The boundary collocation method.

This is a kind of computer-oriented discretization: boundary conditions will be satisfied only at a finite (large) number of discrete points Ai, i=1,2,…,n >>1.

Thus the number of virtual forces equals 5n and a system of 5n x 5n algebraic equations will be formulated that has its equivalent 4x4 for beams. Clearly, the boundary conditions will not be met between the points Ai but the function (3) is very regular and deviations can be small - if the number of Ai points is “big enough” and they are placed in a rational way.

If, for example, there is no qo loading but 4x6=24 concentrated forces Pi and there are 30 points Aj distinguished as the collocation points, so the superposition of 24+5x30 = 174 contributions from concentrated forces must be calculated.

Conclusion: this is the approximate solution – the differential equation is fully satisfied, but boundary conditions only at selected points.

Final conclusion:

At each point A of the free boundary, both 3 moments M and 2 for- ces Q described above, yielding from (3), must be reduced to zero.

Therefore, 5 virtual Bleich forces Ti have to be introduced, situated outside the real finite plate, and the solution of the type (3) can be applied to each of them.

However, the number of such points A is infinite, even dense along the boundary

B

. Thus the reducing virtual forces TAi , in infinite number, will also have to be distributed in a continuous dense way, so apparently on 5 lines surrounding the real plate. Finding of such 5 functions is not an easy task, so the method has to be simplified for practical applications.

A

TA1

TA3

TA5

TA4

TA2

qo(x,y)

x y

B

specific feature of this version of the Bleich method has only some topological reasons – for beams there are only two ends taken as the boundary, for plates this is a continuous line with infinite number of points;

therefore, 1D versus 2D are qualitatively v.different cases in numerical modelling.