THE SIMPLEST LINEAR MODELS OF THE DEFORMABLE SUBSOIL to Lecture no.2
1. Behaviour of the ground subsoil
Loaded soil medium behaves in an elastic-plastic way, i.e. after removing the load, it only partially returns to its pre-load condition. This phenomenon is quite difficult to simple physical modelling, because there is a long list of factors causing such material behaviour:
irreversible mutual displacements of grains, crushing and breaking of particles, spatial structures of the soil skeleton, dependence on the history of previous loads, changes in porosity - not to mention a serious complication for media saturated with water or partly with water and partly with pore air. Complex computer programs are required for effective numerical treatment of these phenomena in a sample scale, and even much more so for solving practical situations and expert tasks. Moreover, careful modelling of soil behaviour and use of good software do not solve problems, because the greatest attention should concern determining adequate numerical values of parameters for calculations. This situation definitely applies to the Third Geotechnical Category (GC 3, EC7-1).
GC 2 is not so sharp in requirements, and just GC 2 should be seen as a background to further considerations.
Simplifications of design situations are as follows:
1) building objects are basically typical, they are not very complicated in shape nor particularly responsible structures;
2) loads are monotonically increasing during construction or are constant (dead loads), with a small percentage of variable loads; therefore, problems with un-loading and re-loading are not very significant;
3) the period in which loads increase (construction-rising time) is "large enough" to dissipate excess pore pressure; this is, of course, relative - in relation to the filtration capabilities of the subsoil;
a several-month increase in the load of a saturated silt can be a "relatively short" period, while a several-day increase in the load of a coarse sand - is a quite sufficient period to complete consolidation of the soil layer;
4) the considered loads are "relatively small" – in relation to the load capacity of the ground.
This last point needs to be explained.
The basic foundation dimensions (e.g. BxL) are determined by the EC7-1 GEO bearing capacity:
𝑉 < 𝑅 =𝑅 𝛾 which concerns a very unlikely design situation1.
1Has anyone seen reinforced concrete of the weight not 25 kN/m3 but 25x1,35 = 33,75 kN/m3 ?
Clearly, it may also be the effect of construction inaccuracy, i.e. the volume of the element, but not of the order of 35%.
The safety margin on the right is exactly R = 1.4; for simplification assume that it is 2. On the left, the safety margin is between 1.35 and 1.50 (with the predominance of the former) - suppose it is also
2. In relation to the “typical” situation - if the characteristic values are considered as “typical” - degree of the bearing capacity achievement can be estimated on the level of 50% or less,
= Vk /Rk = 1/(22) = 1/2. In fact, this coefficient = 50% is probably even overestimated, because the characteristic value, according due to EC7-1, is already defined as "a cautious estimation".
Fig. shows how the settlements w increase as increases,
i.e. the load Vk increases with a constant bearing capacity Rk. In the range of 4050% of the load-bearing capacity, the nonlinearity of the load-settlement curve is not yet large and the straight (secant) line is a good approximation thereof, especially for non-cohesive soils (black line), maybe worse for cohesive soils (red line).
w
4050%
100%
2. Global models
This term defines the so-called global analogues, i.e. mechanical systems that globally, macroscopically, behave somewhat similar to a foundation placed on a deformable subsoil.
2.1. The Winkler model
Introducing the Cartesian system of coordinates x,xB on the horizontal plane (x along L, xB along B for considered rectangular foundations), the following relation is assumed:
𝑤(𝑥, 𝑥 ) = ( , ) or equivalently 𝑞(𝑥, 𝑥 ) = 𝐶 ∙ 𝑤(𝑥, 𝑥 ) (1a) q = loading of the subsoil boundary at the point (x,xB), kPa,
w = settlement of the subsoil boundary at the point (x,xB), m.
For a foundation beam L >> B, the loading is usually studied per 1m of length independently of the beam width B, so 𝑟(𝑥)=∫−𝐵/2+𝐵/2𝑞(𝑥, 𝑥𝐵)𝑑𝑥𝐵 , kN/m.
In addition, foundation beams are very rigid in transversal cross-sections, so the settlement of the beam practically does not depend on xB, w(x,xB) = w(x);
double-sided integrating of (1a) over the width B of the beam results in:
𝑤(𝑥) = ( )∙ or 𝑟(𝑥) = 𝐵 ∙ 𝐶 ∙ 𝑤(𝑥) (1b) The fundamental shortcoming of the Winkler model should be strongly emphasized - the Winkler model does not provide any settlement of the area next to the loaded foundation (w = 0 if and only if q = 0), which does not always correspond to reality. The constant C can be determined from measurements of w for the applied q, i.e.
C = q/w. Another method, an inverse analysis, is used in the design project.
2.2. Conditions for using the Winkler model
The shortcoming of the Winkler model is not so much significant if there is only a shallow soil layer (or layers) under foundation and an undeformable bedrock is still deeper, so the thickness of the subsoil H is “small” – in relation to the foundation width B.
Clarifying the meaning of the term "small" is not consistent, but usually it is assumed that H < 1,01,5B.
This can be easily justified.
Examples of surface settlement curves w(x,xB) are shown as solid lines in Fig.:
blue lines for a “great” H, red ones for a “small” H.
The global analogue is a system of identical separa- ted springs on a stable, undeformable bedrock;
therefore, only directly loaded springs settle down.
The model has one elastic constant C [kN/m3].
Settlements are mainly determined by vertical stresses z on which the vertical strains z
depend (linearly). Settlement w(x,xB) is an integral of z in the depth range from 0 to H, recall also settlement calculations on narrow strips in the Design Project in Soil Mechanics.
In other words, the integral of the function z
over the depth 0H (area under the plot) is a graphical visualization of the settlement w(x,xB) of a homogeneous layer at (x,xB).
As can be seen in Fig., outside the loaded place, the hatched area is small if H is small (red, on the right), and the settlements are large, if H is large (blue, on the left).
z
z
H
H
The above plots in Fig. are only provisional and contain one more simplifying assumption:
the stress diagrams z are assumed the same regardless of the thickness of the H layer and are adopted as for the elastic half-space, excluding the section below the depth H (deletions). More accurate numerical calcula- tions show that this is not always the case; for "small" H values, the stress z increases in the lower part of the layer.
2.3. Improved Winkler models
The Pasternak model
Regardless of the above two concepts, it is necessary to introduce a second elastic constant, that is why this approach is called a two-parameter model.
Focusing on the simplest case of a beam in L direction, equilibrium equation for the Pasternak shearing layer in a segment B x g x dx (g denotes thickness of the layer) is as follows:
𝑟(𝑥)𝑑𝑥 = 𝑟 (𝑥)𝑑𝑥 + 𝜏(𝑥) ∙ 𝑔 ∙ 𝐵 − 𝜏(𝑥) +𝑑𝜏(𝑥)
𝑑𝑥 ∙ 𝑔 ∙ 𝐵 thus
𝑟(𝑥) = 𝐵 ∙ 𝐶 ∙ 𝑤(𝑥) − 𝐵 ∙ 𝐷 ∙ ( ) (2) The Hooke law for the Pasternak layer which works only in shearing is used: = G =Gdw/dx
(G = the Kirchhoff modulus at shearing) and a new constant is introduced D = gG [kN/m].
Of course, G and g do not exist physically and the constant D will be determined in a different way.
There is r(x)=0 outside the beam, so (2) turns into a simply differential equation which describes the shape of the deformed soil surface next to the foundation for x L. The solution is:
𝑤(𝑥) = 𝑤 ∙ 𝑒𝑥𝑝{−𝛼 ∙ (𝑥 − 𝐿)} (3)
where wL stands for the settlement of the right end of the beam and 𝛼 = [m-1].
There is therefore the effect of "the neighbour influence".
For the value of C already found, the constant D cannot be determined directly, but it can be determined using the so-called inverse analysis. To do this, it suffices to measure settlements at several points outside the foundation and evaluate a constant so that the curve best fits these points (Data Fitting, least squares method).
It is worth analyzing what happens with the subsoil reaction under an infinitely rigid centrally loaded beam.
Equation (2) is true at every internal point along the beam, so for 0 < x < L, as usually for differential equations, because derivatives are defined in open intervals; there maybe various conditions at the ends. In this open interval, of course, there is w(x) = const, so simply rs(x) = BCw(x), which looks like the Winkler model. Outside the beam, in the Pasternak shearing layer, there is =Gdw /dx, which allows the calculation of the shearing force Q =gB = -BDwLexp {- (x-L)} along the shearing layer. This value also makes sense in the limit passage for x > L, x L, so this limit force must be transferred to the beam under its edge x = L, because action equals reaction.
The basic disadvantage of the Winkler model (mutual indepen- dence of all springs, no settlement outside the foundation) can be easily removed if a thin elastic layer is laid on the springs: it could be a flexible membrane in tension or a layer created from rigid
"bricks" connected by elastic joints and working under shear (so as assumed Pasternak).
dx r(x) = load on the upper surface of the Pasternak layer,
i.e. under the foundation (or zero outside it), rs(x) = load on the lower surface of the Pasternak layer,
i.e. from the Winkler springs, rs(x) = BCw(x), r(x)
rs(x)
(x)
(x)+d/dxdx P
Conclusion:
under a rigid foundation beam resting on the Pasternak subsoil, there is a constant reaction rs = const in the interior but there are also two concentrated forces Q = BDwL acting at the beam ends x = 0, x = L.
This means that along the open interval (0,L), the subsoil reaction equals rs = (P-2Q)/L = const which is, however, less than for the standard Winkler model, r = P/L. In particular, the beam settlement on the Pasternak subsoil is less than on the Winkler subsoil, though both the springs and the loading are the same. It cannot be surprising, since the beam is like being „suspended” by the Pasternak layer (or by a membrane in tension).
The most important practical conclusion, however, is the observed redistribution of the subsoil reaction under a rigid beam, being typical for subsoil models, which take into account the "influence of the neighbour" or
settlement outside the loaded foundation. Such the redistribution increases the maximum moment that occurs in the considered beam. The presented redistribution in the form of two Q forces at the beam ends is a prototype for similar analyses in the case of elastic half-space, which is discussed in detail in Example 4 in Lecture 1 and later on in this material.
Similar study is possible in the transversal cross-section for 0 xB B (for 1m of the beam length) but full modelling of a (x,xB) rectangular footing in w 2D gets complicated.
The Kerr model
Such the discontinuity or „submerging” can be observed in practice and its modelling is not easy, even using the finite element approach.
3. Local models
This group includes all solutions of material continuum theory, so modelling takes place locally in elementary volumes dxdxBdz, generally in elasto-plastic formulation.
3.1. Elastic half-space
Vertical force dP[kN] concentrated at a point (xo,xBo) causes a surface settlement wo(x,xB) of the horizontal boundary and the solution found by Boussinesq is as follows:
𝑤 (𝑥, 𝑥 ) =
∙ ∙ (4) where Es = Eo/(1-2) , and 𝜌 = (𝑥 − 𝑥 ) + (𝑥 − 𝑥 ) denotes the horizontal distance in cylindrical
coordinates.
By making use of the superposition law, the case of vertical loads q(xo,xBo) distributed on the area D can be found by integrating (4) over D for dP = q(xo,xBo)dxodxBo; in particular, for D = BxL one gets the co-called Steinbrenner integrals, known as the method of corner points.
So:
𝑤(𝑥, 𝑥 ) =
∙ ∙ ∬ ,
( )
𝑑𝑥 𝑑𝑥 (5)
Equation (5) corresponds to (1a), bonding settlements w and loadings q.
It is clear here, what is the main problem with the applications of the elastic half-space model:
P
It is a three-parameter model that can be described as Pasternak+Winkler, so there are 3 elasticity constants C1, D, C2. All the features of the Pasternak model are preserved and additionally there is a discontinuity of settlements at the edge of the foundation - the settlements of the foundation are larger than the ones of the adjacent soil (“submerging” of the foundation);
the discontinuity can be used to determine the value of C2 for the upper springs.
If solving the beams and the plates on elastic subsoil (Lecture no.3), there is a need to expressed q as depending on w but (5) defines the reverse – that is w which is expressed as depending on q , moreover depending in a complex (integral) form. Reversal of the expression (1a) is trivial but it is not so with (5).
Over half a century ago, Gorbunov-Posadov proposed a method of reversing the relation (5) and even published ready charts for internal forces for beams and rectangular plates - see elsewhere on this WWW.
At present, these charts cannot compete with modern numerical methods of mechanics, but the idea itself is general and worth recalling.
In simplification, Gorbunov-Posadov used double power series assuming that:
w(x,xB) = aij(x)i(xB)j , q(xo,xBo) = bij(xo)i(xBo)j , [(x-xo)2+(xB-xBo)]-1/2 = cij(x-xo)i(xB-xBo)j.
The cij coefficients are known as determined by the expansion of 1/, the amn coefficients are assumed as known; finally, the bpq coefficients are expressed using the cij and amn coefficients;
this means the approximate expression of the function q by making use of the function w.
To do this, substitute the three series into the appropriate places in equation (5), multiply the series under the integral, double-integrate term by term, and finally compare all coefficients at (x)i(xB)j on both sides of (5) . The approximate nature of the solution results from taking into account only a finite number of terms in the double series, which are always infinite2.
3.2. The Sadovsky contact problem
For a centrally loaded B-wide beam3, a certain integral equation of the theory of elasticity has to be solved where the following mixed boundary conditions are assumed on the founding level:
under the beam: = 0 (smooth contact) and w = const (rigid, no rotation), -B/2 < xB < +B/2,
on the outside of the beam: = 0 and z = 0 (free part of the surface, without load).
The obtained solution for vertical contact stress under the beam:
(𝑥 ) = ∙ ∙
/
The subsoil reaction is strongly concentrated near the ends of the transversal cross-section (Fig.), even such that z(B/2)=+. The tendency to redistribute the subsoil reaction towards the ends of the rigid foundation is confirmed, as in Example 4 in Lecture 1 and in the Pasternak model; this finding increases bending moments for the foundation, if compared to uniform subsoil response.
3.3. Horizontal elastic layers
The situation becomes more complex in a case of one finite elastic layer, H meters thick, where an undeformable bedrock happens at the depth H < + below the founding level; in addition to the boundary conditions on the founding plane (e.g. as in point 3.2), some conditions at the depth H must also be formulated;
first of all, this is zero settlement on the bottom of this undeformable bedrock, i.e. w(H) = 0.
If the bottom of the layer is assumed to be " perfectly smooth" (not very realistic), then both components of shearing stresses should vanish (H) = 0; but if it is assumed to be "perfectly rough", then both components of the horizontal displacement vanish u(H) = 0, v(H) = 0.
2Probably, the number of considered terms was significantly reduced in the Gorbunov-Posadov solutions obtained 50-60 years ago.
3There exist also similar but nonsymmetrical solutions for eccentric loads of a rigid beam.
It is the search for contact vertical stress r = z under a rigid, smooth punch mounted on the elastic half-space. Analytical solutions exist only for two-dimensional cases - an infinitely long B-wide beam in a plane state of displacements and for a circular punch (in cylindrical coordinates); they both have a similar form.
EI=+ P
-B/2 +B/2
z
+ +
In case of a multilayered subsoil, the discussed boundary conditions apply to the highest and the lowest layers;
additionally, appropriate continuity conditions are introduced at all interfaces between the layers. There are simple and effective methods to solve such problems of elasticity, but they require a computer support: to a lesser extent (Fourier series expansions for each layer, see ZEM_SIN) or to a greater extent (FEM).
4. Bilateral vs. unilateral bonds
Finally, a few remarks on the possible detachment of the foundation from the subsoil.
In the general case, at each point (x,xB) under foundation, the settlement of the subsoil w(x,xB) should be distinguished from the settlement of the foundation itself y(x,xB). With the vertical axes downwards, w and y are both usually positive, but not always equal. The case y > w is not possible, because it would mean
"interpenetration" of the foundation and the ground, at most the relation y = w is possible. However, the situation y < w can happen, when the foundation breaks away from the ground, a gap is formed at the bond. At the contact between foundation and subsoil, any tension forces can appear, even for cohesive soils. This is because foundations are constructed on a previously prepared layer of lean concrete and technically it is not possible (nor rational) to join these two concrete layers. A situation is possible when w = 0 but y < 0, i.e. locally a part of the foundation rises above the original ground level and breaks away from the subsoil, creating a certain local gap between them. Such a situation, at least in relation to permanent loads, should be avoided in the design of foundations whose role is to transfer loads to the ground, i.e. to work in full contact and interaction with the subsoil; this should be a signal to redesign the foundation, if necessary.
Summary:
bilateral bonds - this is the case when the contact of the foundation with the subsoil is permanent, it fulfills the condition y = w for both compression and tension; for unilateral bonds (red line) the condition y = w occurs only for compression; for example, if w = 0 and y < 0, a gap appears between the foundation and the subsoil;
there is no difference between the two bonds, when the foundation is pushed into the subsoil.
y gap
0
w
