PROBLEMS OF STABILITY
O.V. Inozemtseva
Saratov State Technical University
Equilibrium instability stability heterogeneity elasticity plasticity
The problems of static equilibrium, durability, stability and instability of
structures, which interact with non-elastic foundations, are quite urgent
nowadays. This issue is devoted to the problems of stability and represents the
results of the scientific research that concerns non-elastic bases for these
structures.
Such a research must include stress-strain analysis both of the structural
elements and the soil foundation. Let’s consider uni-layered and multi-layered
foundations with different deformative properties and soil foundations,
subjected to industrial (artificial) or natural moisturing, resulting in the decrease
of mechanical properties. The results of this investigation are patterned in
numerous graphical figures, illustrations. This issue deals with heterogeneous
elastic foundation, composed of one and two layers and obeying the
requirements of Vlasov-Leontiev foundational model. The structure, which
interacts with these foundations, is high enough to experience instability under
the action of forces. I should notice that this instability doesn’t represent
buckling of the structure. I consider that case when this structure experiences
lateral deviation, which induces yielding (vertical displacements) of the
foundation, so we should mean the problem of stability both of the construction
and its foundation. Employing the bifurcation approach and variational
principles I get the critical value of the load, applied to the construction. And
this is the main problem for me as a constructor. Besides, this enables me to
consider the behavior of the system structure-foundation, after changing its
initial equilibrium configuration. Using special system of equations it is possible
to consider non-elastic heterogeneous foundations with induced heterogeneity.
The environmental influence in this case includes effects of underground waters
and other liquids or gases, changes of temperature-humidity relationship, and so
on. Modulus of deformation is reduced with the increase of soil humidity and
depends on temperature level of the environment. And being the main
characteristic of foundation strength, decreasing modulus of deformation affects
the behavior of the construction on the soil foundation.
It is known, that the degree and character of heterogeneity of properties of
structural materials and soil basis may be various. The reasons, due to which this
heterogeneity occurred in the soil base, are various too. Presence of
heterogeneity of mechanical properties influences strength, stiffness, stability
and durability of structural materials, structural elements and their bases. The
theory of induced heterogeneity, which is widely employed in the research,
deals with heterogeneity as a process. This means that it has its origin and is
developing under influence of external factors, which induce it, during the
certain time period. Thus, we are concerned about such a question: in what way
can we take this process in account without knowing its main characteristics and
without having an opportunity to reconsider the free-body diagram of the
structure every time the heterogeneity changes the foundation properties? And I
should notice that the solution must be obtained with an acceptable accuracy,
because it is used in the designing of the structure and choosing the materials
that can make this structure able to resist deformations, caused by this
foundation heterogeneity.
It is possible to consider the induced heterogeneity as a process, when a
complex mathematical model of the system structure - foundation is built. Such
model generally has three kinds of non-linearity, including physical nonlinearity of structural material and non-linear character of external influences.
Such model is based upon fundamental concepts of mechanics of deformable
solids and the phenomenological approach. Numerical methods, such as
Boundary element method (BEM), Finite difference method (FDM), Finite
element method (FEM), provide such complex mathematical models. I got my
solutions with employing Finite difference method, which made me able to build
a finite difference model of the soil foundation that was under consideration in
this issue.
The
phenomenological
approach
allows
constructing
a
formal
mathematical model, which, even in the absence of full clarity in interpretation
of the physical part of complex physical and chemical processes in material,
adequately reflects these processes and may be exact enough in the given range
of process’ parameters. The model of induced heterogeneity should be referred
to this class of models. The principle of virtual work for a solid body with
induced heterogeneity lays in the basis of this model and allows an interpretation
of governing equations as a system of differential equations with quite clear
physical sense. One should note that variational principles are widely used in the
cases, when the principle of virtual work is reduced to the potential energy
minimum principle: among all varieties of acceptable states the state of
equilibrium is characterized by minimum of potential energy. In particular, this
principle is widely applied in the linear and nonlinear theory of elasticity and
plasticity.
Today on the territory of Russia about 900 cities have a high level of
underground waters. The total area of saturated built up territories in the country
is equal to 8000 km2 . Lately in the city of Saratov level of subsoil waters have
been rising intensively from the depth of 9-12 meters up to 2.5-5 meters on
some platforms, and this water level growth caused deformation of buildings
and constructions.
Thus, the research of stress-strain state and estimation of serviceability of
the structures, which interact with the basis and are under aggressive natural or
technogenous influences, are very topical.
Nowadays we can find many papers, significant number of monographies,
which are devoted to the problem of stress-strain state of the non-linear
foundations. The problem of stability and instability of structures, including
buckling of columns, is one of the complex problems in structural mechanics,
and there are a lot of issues dealing with such a problem. But on the other hand
there are almost no publications, which connect the above-mentioned problems
together, and no investigation is carried on in this field of mechanics. But this
task remains an urgent one and requires scientific development: it is known that
designing of high structures is widely spread both in industrial and housing
construction.
Consider a model of a structure that is high enough to experience
instability and is located on a heterogeneous foundation (Fig. 1).
Fig. 1
The foundation is assumed to be a linearly elastic one, composed of two
layers with different deformation properties (Fig. 2).
Fig. 2
Employing the variation method, proposed by V.Z. Vlasov and theory of
elastic foundation, we get the model of bilayered elastic foundation, obeying the
following equations:

k0 W0  k0 W1  2t0 W   t0 W1  q,

 k0 W0  (k0  k1 )W1  2t0 W   2(t0  t1 )W1  0.
(1)
Here k0, k1 are the foundation modules, characterizing the ability of the
upper and the lower layers, respectively, to resist compressive loads; t0, t1 are
the foundation modules, characterizing the ability of the upper and the lower
layers, respectively, to resist shear loading.
Consider the above-mentioned structure, subjected to a small lateral
deviation. Equilibrium equations in this case depend on the type of the
structure’s support (Fig. 3, 4, 5).
We can distinguish a flexible supporting platform or an absolutely rigid
plate. The former construction provides a flexible transmitting of the pressure to
the foundation. Assuming that the structure is supported by a latter construction
(an absolutely rigid plate), we apply a kinematical loading to it.
Fig. 3
Fig. 4
Fig. 5
Consider a number of numerical model examples, illustrating the
bifurcation approach to determination of the stability of high structures. For the
first example we take a model of a high structure on the unilayered foundation,
following the requirements of the Vlasov-Leontiev model of foundation. The
numerical results were obtained for the following parameters: E0=104 kPa;
E1=E2; ν0=0,25; ν0=ν1; H=10 m; a=0,5 m; h0=2,7 m.
Owing to the absolute rigidity of the supporting plate, transmitting the
pressure to the foundation, we apply a kinematical loading to the foundation
(Fig. 5). This means that the structure is subjected to a small lateral deviation.
Equilibrium equations in this case are:
P  (Ql  Qr ) 
k0 S y
(W0 r  W0l )  2ak0W0l  0,
2a
k0 S y
k
P  a(Ql  Qr )  0 I y (W0 r  W0l ) 
(W0 r  W0l )  0.
2a
2
(2)
Here indexes “l” and “r” characterize the indicated values as those,
occurring under left or right edges of the plate, respectively; S y and Iy are the
first and the second moments of the plate about y-axis (section modulus and
moment of inertia); Δ – eccentricity increment. Concentration of the reactive
stresses around left and right edges of the plate is denoted by concentrated
forces Ql and Qr , respectively.
The second equation (Eq. 2) can be converted to a homogeneous with
respect to Δ:
4ta2
P 
H
k0
2k0 a3

  0.
2t
3H
(3)
The critical value of the load can be obtained as follows:
Pcr 
4ta2
H
k 0 2k 0 a 3

.
2t
3H
(4)
Reconsider the above-mentioned structure, subjected to the off-center load
with the initial eccentricity: Δ0=Hsin(φ0) (Fig. 4). The relationship between an
angular deviation of the structure φ and the value of the increasing load P is:
P(
4ta2
H
k0
2k a 3
sin 
cos   0 cos 2  )
.
2t
3H
sin(   0 )
(5)
In the P- φ (load -deviation) diagram for that case, when φ0=0, the point
of significance is the critical point of bifurcation of the 2 nd type (for Pcr=221,5
kN). In this point the trivial (zero) solution, corresponding to the initial straight
equilibrium configuration, intersects the curve, corresponding to the deviated
(disturbed) equilibrium equation, when  0  0 . In that case, when an initial
angular deviation is applied  0  0 , the critical point of bifurcation of the 2nd
type is converted to a limit point (for Pcr=221,5 kN). The foregoing is depicted
in Fig. 6.
Employing linearization to Eq.7 we get the solution of acceptable
accuracy for the stable portion of the corresponding curve. Notice, that as P
approaches Pcr, the maximum value of deviation increases infinitely without
bound (Fig. 6).
Fig. 7
Fig. 7 gives a graphical representation of the linearized equations. The
curves represent the increase of vertical displacements (yielding) under left and
right edges of the foundation plate. The corresponding increments of vertical
displacements under the plate and beyond its limits are patterned in Fig. 8.
Fig. 8
Consider a structure, located on the heterogeneous bilayered foundation
(Fig. 4). Bifurcation load in this case can be found as follows:
4 0 a 2 2k *a 3
Pcr 

;
0 H
3H
k *  k0 (1 
k0
k0  k1 )
(6)
Here k* is the foundation modulus, characterizing its ability to resist
compressive loads.
4 0 a 2
The equation member
0 H
characterizes the influence of stress
concentration, denoted as Ql an Qr, around both edges of the supporting plate.
Here:
l 20 l111
l11l 20
0
0
 0  l  1 ;   3t 0 l1 1 (! 0 1 );
l 2 2
l1 l 2
0
1
l i0  k 0  k1  2i2 (t 0  t1 ); l i1  k 0  i2 t 0 ;
i  r 2  (1) i r 4  s 4 r 
3k 0 t 0  k 0 t1  k1t 0
k 0 k1
; s4
.
t 0 (3t 0  4t1 )
t 0 (3t 0  4t1 )
(7)
Assume that increase of the humidity of the lower layer induces decrease
of its deformation modulus E1.
Fig. 9
Experimental data, obtained for black clay, gives the relationship between
the deformation modulus E1 and soil humidity C. This relationship is of
hyperbolical type (Fig. 9): E1(C)=C/(bC-d). Here, suggesting that the humidity
level differs from 25.8 % to 32.8 % (the corresponding values of stresses are in
the range 0.1 to 0.2 MPa, respectively), the coefficients b, d are taken as
follows: b=2,0687; d=-50,725.
Fig. 10
This enables us to obtain the relationship between the value of the critical
load and the humidity of the lower foundation layer. The foregoing is depicted
in Fig.10 for the following parameters: E0=104 kPa; E1(C)=C/(bC-d); ν0=0,25;
ν1=ν0; H=10 m; a=0,5 m; h0=h1=1,35 m.
The increase of the humidity level of the lower foundation layer induces
the decrease of the critical value of the load (Fig. 10). The corresponding
diagram shows its 7% reduction.
Иноземцева О.В. Расчет на устойчивость сооружений на неоднородном
нелинейно деформируемом основании: Дис. канд. тех. наук. Саратов, 2006. 144 с.