The analysis of the strain- stress distribution of an elastic- plastic ...

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The analysis of the strain
-

stress distribution of an elastic
-

plastic body
during the process of its loading.

A.S. Zinov’ev

Saratov State Technical Universitiy

K.A. Omeluk

Saratov State Technical Universitiy


Defining of deflections and intensifications o
f constructions, which are
superficial, as a load is applied, is impossible without prerequisite about their
characteristics of deformation. In our days there is no unified model which can
represent difficult physico
-
mathematical characteristics of a botto
m soil. In
practical methods of computation of constructions on a deformable base some
models are widespread: such as Winkler conjecture, Elastic half
-
space
hypothesis, Vlasov model, Pasternak model. Let us study these models in turn.

Winkler conjecture
.

T
he base is represented as a set of independent
springs. A distribution of back pressure on the bottom of foundation can be
described by the following equation. The weak point of this method is that
ground surface, as it has been found experimentally, settl
es down not only in
places, where load has been applied to, but in neighboring places too.


Elastic half
-

space hypothesis.

It supposes using the model of elastic dissimilar
half
-
space to describe mechanical characteristics of a bottom soil. The main
argum
ent against the elastic hypothesis is that there is an
evidence of a great
number of permanent deformations in soil, these deformations almost don’t
work in tension. In our days analytical solutions of the problem of deformation
of constructions on a botto
m soil exist only for some subcases.



Vlasov model.

This model supposes that elastic foundation is a vertical
plate on an upper lineal edge of which a beam rests. Question is analyzed on
condition of flat
-

topped stressed state. The plate has a limited he
ight and bases
on incompressible base. This model provides an opportunity to take into account
the influence of tangent stresses in elastic base, which is absent in Winkler
model.

Pasternak model

says that an elastic base has two characteristics. The first

is
C1(a contraction coefficient) connects intensity of vertical earth back pressure
with setting of ground by Winkler dependency relation
1
ÑW



(2)

The
second characteristic is a shear coefficient, which is irrespective of C1. It h
as the
following dimensionality N/cm and it gives us an opportunity to quantify the
intensity of horizontal shear force
t


2
W
t C
x




(3).





The main disadvantage all models have in common is the fact that it’s
impossible t
o account for nonlineal character of deformation of environment
under the influence of external factors. And it is also impossible to account for
development of nonhomogenity of nonrigid characteristics of charged base,
when this nonhomogenity is caused by

external factors.



The model, which is described by Navier’s system of equations of
balance(written for augmentations of the
tensor component of a load
)can be used
as a solution of this problem. Characterization of equation of this model in
incrimination
s can give an opportunity to reduce nonlineal sum of deformation
of nonhomogeneous environment of base to a sequence lineal sums with
variable coefficients. Finally this model can be used for physically nonlineal
base, which is characterized with a nonhom
ogeneity. This nonhomogeneity
pervades all over the capacity size of deformed environment of the base and
shows as degradation of nonrigid and strength properties.




Picture

1

We will analyze physically nonlineal base, that is why we will use
exponential

relationship for describing the correspondence between stresses and
deformations of a base.




Picture

2

In figure 3 we can see the diagram of the vertical motion of the beam on
homogenous elastic
-

plastic layer of the base, when the different magnitudes

of
the monotonely rising load are distributed.

The more the load, the more the setting, at the same time the beam became more
incurved. It is connected with the fact, that, as we can see in figure 4, vertical
motions from the middle of the beam are faster
, than motions from edges.


Picture3

Picture 4

Let us compare results of computation for setting of an edge of the beam(which
cooperates with elastic
-
plastic base) with results of computation for setting of
elastic layer of base (fig. 5)



Picture 5

It’
s obvious that if exterior load is not more than 100 kPa the results of
calculations almost run in. Indeed, with the load being from 0 to 100 kPa, which
is equivalent to octahedral shear stress of from 0 to 150 kPa, force deformation
relationship
)
(
i
i
i
e




is close to direct, so the base works on scheme of linear
distorted ply within this interval of load.

With the increment of load up to q = 400 kPa ( = 417 kPa), the variation in
amounts of vertical motions of the edge of the beam makes approx
imately 26%.
If the load is q = 400 kPa, magnitude
Δ
W, calculated with elastic
-
plastic base for
the extreme point of the beam, is 16% higher than the comparable magnitude,
figured out in the case with cushion course.

So taking in account elastic
-

plastic characteristics of the base lead to
increasing of fl
exural deformation of a structural component and that results in
increasing of internal forces.

While load on a substantial construction increases, concentration of
vertical stresses under the edge of the beam decreases, nevertheless the
magnitude of affe
cting stresses in the process continues to increase.

Let us analyze the process of development of a plastic zone in the
homogeneous elastic
-

plastic base, which is charged by the load, which is
distributed regularly through the beam of finite rigidity.


A
t first, plastic zones appear under the edge of the beam and, while the
load increases up to 400 kPa, plastic zones are expanded into the depth of the
base under the centre of the beam, however zones of the highest intensity of
deformation are still concen
trated under the edge of the beam.

Let us analyze the so
-

called “incurvated element
-

base course” system in
condition of nonhomogeinity of rigid characteristics of the base course, when
there is “columnar” nonhomogeinity in the base. Let us assume, that
the area of
extra rigidity of the base is under the central portion of the beam. In this case,
zones of elastic deformation are concentrated under the edge of the beam (see
figure 6).



Picture 6

Picture 7

Let us analyze a design model, in which the area

of the lower rigid
characteristics is under the central portion of the beam. While the load is carried
over through the beam of the finite rigidity, zones of plastic deformations firstly
grow under the edge of the beam and secondly
-

under the central port
ion of it,
and the highest intensity of deformations is seen under the edge of the beam
during the process of loading(fig. 7)

In case that there is the double
-

layer base with occurrence from the surface of
the layer with lower nonrigid characteristics, th
en elastic zones will develop only
within this layer. In case, when the top layer of base is more rigid, then character
of development of zones of plastic deformations in the bottom course is similar
to development of plastic zones under the central portio
ns of the beam. The top
-
layer fulfills the role of the beam, which presses on the weak bottom course. In
the top layer plastic zones are concentrated under the edge of the beam.



Picture 8

Picture 9


Analyzing the results, in t
he conclusion I would like to say, that
depiction of deformations of the charged base has rather complicated appearance
and presence of some nonhomogeinity in the thickness of the base tangibly
affects the development of plastic zones. Thus, when we solve
the problem about
deformation of construction on an earth foundation, we should use models,
which can determine a strain
-
stress distribution in all points of a base and can
reckon in the influence of nonhomogeinity of rigid characteristics on process of
d
eformation.


List of references


1. Расчет конструкций на упругом основании/ М.И. Горбунов
-

Посадов,
Т.А.

Маликова, В. И. Соломин.


3
-
е изд, перераб. и доп.


М.:Стройиздат,

1984

679 с., ил.

2. Зиновьев А.С. Инкрементальная модель деформирования изгибаемого

элемента на нелинейном осн
овании с наведенной неоднородностью
свойств /

Диссертация на соискание ученой степени к.т.н.


Саратов, СГТУ.


2009.



120 с.