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The stress-strein state from its own weight in ground base with trapezoidal cutout

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In order to determine the deformation of the base, the shrinkage of the structures erected on them, as well as to study the strength of the bases, it is necessary to determine the stresses in the soil massif. The stresses in the soil massif from the action of the structure are superimposed on the existing stresses in it from the action of its own weight, taking into account the excavation during the development of the pit. Therefore, the paper considers the stress-strain state of the ground base with a trapezoidal cutout under its own weight. It is shown that the effect of the own weight of the ground base with a trapezoidal cutout can be replaced by the fictitious action of the ball tensor of forced deformations and external load. At the same time, the previously known methods of replacing the own weight of the soil base by the action of fictitious surface forces, existing in the presence of rigid restrictions, are special cases of the solutions obtained in this work. In modern conditions, in relation to the total amount of accumulated professional knowledge, the volume of active information resources is increasing, and construction practice is constantly enriched with new experimental and theoretically sound accurate knowledge.
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MATEC Web of Conferences 193, 03047 (2018) https://doi.org/10.1051/matecconf/201819303047
ESCI 2018
The stress-strein state from its own weight in
ground base with trapezoidal cutout
Elephan Agakhanov1,*, Murad Agakhanov2 and Edward Batmanov1
1 Dagestan State Technical University, Imam Shamil Ave. 70, 367015, Makhachkala, Russia
2Moscow State University of Civil Engineering, Yaroslavskoye shosse, 26, 129337, Moscow, Russia
Abstract. In order to determine the deformation of the base, the shrinkage
of the structures erected on them, as well as to study the strength of the
bases, it is necessary to determine the stresses in the soil massif. The
stresses in the soil massif from the action of the structure are superimposed
on the existing stresses in it from the action of its own weight, taking into
account the excavation during the development of the pit. Therefore, the
paper considers the stress-strain state of the ground base with a trapezoidal
cutout under its own weight. It is shown that the effect of the own weight
of the ground base with a trapezoidal cutout can be replaced by the
fictitious action of the ball tensor of forced deformations and external load.
At the same time, the previously known methods of replacing the own
weight of the soil base by the action of fictitious surface forces, existing in
the presence of rigid restrictions, are special cases of the solutions obtained
in this work. In modern conditions, in relation to the total amount of
accumulated professional knowledge, the volume of active information
resources is increasing, and construction practice is constantly enriched
with new experimental and theoretically sound accurate knowledge.
1 Introduction
In order to assess the deformation of the base, the shrinkage of the structures erected on
them, as well as the study of the strength of the bases, it is necessary to determine the
stresses from the action of the own weight of the soil [1-11].
On the available in the soil massif stress from the action of its own weight (initial
stress), stress arising from the action of the structure is superimposed. The initial stress in
the general case is determined by gravity (own weight of the soil) and change of these
forces in the process of forming the soil massif, tectonic and seismic impacts, and other
factors.
2 State of the problem
In this article, the conditions are established that allow us to represent the effect of the own
weight of the soil in the form of the sum of the impact of surface forces and forced
* Corresponding author: muradak@mail.ru
2
MATEC Web of Conferences 193, 03047 (2018) https://doi.org/10.1051/matecconf/201819303047
ESCI 2018
deformations. Since the feasibility of each action is limited by the possibilities of modeling
techniques, the conditions presented are important for experimental problem solving.
During the period of site infrastructure works the initial stress state of the soil massif
may also change, particularly as a result of dredging in the development of excavation,
dewatering, ramming, or rolling the soil, etc. In this case one should speak about a modified
initial stress state of the base, which interacts further with the stresses from the building.
3 Method of research
Let us consider the stress-strain state of the ground base with a trapezoidal cutout, which is
under the influence of its own weight, i.e. under the action of volume forces
0
x
F
,
0
y
F
,
z
Fg
, (1)
where
constg
is the volume weight of the soil.
Fig. 1. Ground base with trapezoidal cutout under its own weight.
The system of equations to describe the stress-strain state in the area of the base by the
action of its own weight looks like [12]
     
, ,,
20
i
i jj j ij i
F
UU S G
gg g
  
, (2)
   
 
 
,,
2 0,5 0
i j j i ij j
G U U Sn
gg g
d

 

, (3)
     
 
 
,,
2 0,5
ij i j j i ij
G UU S
g gg g
sd

 

, (4)
where
() ()
ii
SE
gg
ns
or in case ν ≠ 0,5
() ()
,
12
jj
SU
gg
n
n
. (5)
It makes sense to obtain the possibility of stress-strain state from the given forces of its
own weight in the form of the stress-strain state caused by the resultant effects of surface
loads Pi and forced deformations ξ.
3
MATEC Web of Conferences 193, 03047 (2018) https://doi.org/10.1051/matecconf/201819303047
ESCI 2018
0
x
F
0
y
F
z
Fg
constg
     
, ,,
20
i
i jj j ij i
F
UU S G
gg g
  
   
 
 
,,
2 0,5 0
i j j i ij j
G U U Sn
gg g
d

 

     
 
 
,,
2 0,5
ij i j j i ij
G UU S
g gg g
sd

 

() ()
ii
SE
gg
ns
() ()
,
12
jj
SU
gg
n
n
The stress-strain state in the area of the base under the action of the load distributed
along the boundary surface and directed normally to this surface and forced deformations is
described by a system of equations [13, 14]
     
,, ,
, ,,
20
PP P
i jj j ij i
UU S
xx x
 
, (6)
   
 
 
,, ,
,,
2 0,5
PP P
i j j i ij j i
G U U S nP
xx x
d

 

, (7)
     
 
 
, ,, ,
,,
2 0,5
P PP P
ij i j j i ij
GU U S
x xx x
sd

 

, (8)
where
( ,) ( ,)PP
ii
SE
xx
nsx 
or in case ν ≠ 0,5
( ,) ( ,)
,
1
12 12
PP
jj
SU
xx
nn
x
nn
 

. (9)
For identical equality of displacement
 
 
,P
ii
UU
x
g
the following conditions must be
met:
 
 
,
,,
22
Pi
ii
F
SS
G
xg
 
, (10)
 
 
,
22
P
i ii
GS n GS n P
xg
 
. (11)
Taking into account the correlations (5) and (9), these conditions take the form
,
12
ii
EFx
n 
; (12)
0
12
ii
EnPx
n
, (13)
If
 
 
,P
ii
UU
x
g
, then from (4) and (8) we get
 
 
 
 
,,
2
PP
ij ij ij
GS S
xx
gg
ss d



. (14)
Taking into account the correlations (5) and (9) we get
 
 
,
12
P
ij ij ij
E
x
g
ss d x
n
. (15)
We will rewrite the expression (13) taking into account that
ii
P Pn
0
12
EPx
n
, (16)
4
MATEC Web of Conferences 193, 03047 (2018) https://doi.org/10.1051/matecconf/201819303047
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Comparing the expression (16) differentiated by i with (12) we will obtain
i
PF
i
. (17)
The expression (15), taking into account (16), takes the following form
 
 
,P
ij ij ij
P
x
g
ss d 
. (18)
According to the expression for volume forces (1) of (12) and (17) we have
12 z
E
n
xg
 
, (19)
Pzg
. (20)
Consequently, the determination of the stress-strain state from the given volume forces
(the own weight of the soil) is reduced to the determination of the stress -strain state from
the forced deformations of the ball type (temperature effect) (19) and the external surface
load (20), distributed respectively according to linear and hydrostatic laws (Fig. 2). More
precisely, the displacements in these problems coincide identically, and the stresses are
related according to the correlation (18).
Fig. 2. The ground base with trapezoidal cutout under action of replacing loads.
The methods of experimental determination of the stress-strain state under the action of
surface loads and forced deformations, to which the initial problem is reduced, are
methodically developed with completeness sufficient to obtain effective solutions.
4 Analysis of results
Some special cases.
1. Base with flat surface. The surface load then according to (20) is zero
0P
, hence
the stress
 
0
p
ij
s
and expression (18) take the form
 
   
ij ij ij ij ij
Pz
xx
g
ssdsdg
. (21)
5
MATEC Web of Conferences 193, 03047 (2018) https://doi.org/10.1051/matecconf/201819303047
ESCI 2018
i
PF
i
 
 
,P
ij ij ij
P
x
g
ss d 
12 z
E
n
xg
 
Pzg
0P
 
0
p
ij
s
 
   
ij ij ij ij ij
Pz
xx
g
ssdsdg
Under compression conditions of deformation of the base should be assumed
   
0
xy
xx
ee 
. (22)
from the conditions of equivalence of the directions of ox and oy
   
xy
xx
ss
(23)
Obviously, that from the forced deformations
x
, defined by expression (19), stresses
 
z
x
s
are equal to zero, i.e.
 
0
z
x
s
(24)
Then taking into account (24) from expression (21) we have
 
z
z
g
sg 
(25)
Substituting the known correlation between the deformations
 
x
x
e
and stresses
 
x
x
s
       
1()
x x yz
E
x x xx
e s ns s x

 

(26)
with expressions (19), (22), (23) and (24), after trivial transformations, we have
   
12
1
xy
z
xx
n
ss g
n
. (27)
Taking into account (27) from expression (21) we have
   
1
xy
z
gg
n
ss g
n
 
. (28)
Then for the coefficient of lateral pressure of the soil we obtain
 
 
 
 
1
y
x
zz
g
g
gg
s
sn
hn
ss

. (29)
The obtained expressions (25), (28) and (29) for stresses
 
z
g
s
,
 
x
g
s
,
 
y
g
s
, and the
lateral pressure coefficient of the soil coincide with the known [4]. The difference of signs
in the expressions (25) and (28) is due to the fact that in soil mechanics the reverse rule of
signs is accepted, and the equations in this work are written following the rules of signs
accepted in the theory of elasticity.
2. We take the Poisson's ratio equal to 0.5. According to (19) the forced
deformations are then
0x
, hence the stress and correlation (18) takes the form [15]
6
MATEC Web of Conferences 193, 03047 (2018) https://doi.org/10.1051/matecconf/201819303047
ESCI 2018
 
P
ij ij ij
P
g
ssd 
. (30)
provided a flat surface
 
ij ij ij
Pz
g
s d dg
. (31)
In this case, all three normal stresses are equal, that is, the lateral pressure coefficient is
equal to one and there is a uniform (hydrostatic) compression. This solution also coincides
with the known [4]. If the Poisson's ratio in the expressions (28) and (29) is replaced by 0.5,
then together with (25) they represent the same solution.
3. Let the forced deformation not cause stress, i.e.
 
0
ij
x
s
. Then the solutions (30) and
(31) used for the case 2 are acceptable for all values of the Poisson's ratio.
5 Conclusions
The conditions allow us to present the effect of the own weight of the soil in the form of the
sum of the effects: the impact of surface forces and forced deformation. It is shown that the
solutions of the problems by the presented method coincide with the previously known
solutions.
References
1. G.E. Agakhanov, Scientific review 12, 733-736 (2014)
2. G.E. Agakhanov, V.B. Melekhin, Scientific review 4, 90-93 (2016)
3. E.C. Agakhanov, M.K. Agakhanov, International Journal For Computational Civil And
Structural Engineering 2, 24-28 (2014)
4. V.A. Florin, Fundamentals of Soil Mechanics (Gosstroiizdat, 1959)
5. E.K. Agakhanov, M.K. Agakhanov, MATEC Web Conf. 86, 01012 (2016)
6. E.K. Agakhanov, M.K. Agakhanov, Higher education proceedings. North-Caucasus
Region. Engineering sciences 1, 25 (2005)
7. G.E. Agakhanov, Scientific review, 12 (2014)
8. G.E. Agakhanov, Bulletin of Dagestan State Engineering University, 1 (2015)
9. E.K. Agakhanov, M.K. Agakhanov, Izvestiya Vuzov. Severo-Kavkazskiy region.
Tekhnicheskie nauki, 1 (2005)
10. E.C. Agakhanov, M.K. Agakhanov, Industrial and Civil Construction 11, 40-44 (2015)
11. V.I. Andreev, A.S. Avershyev, International Journal for Computational Civil and
Structural Engineering 9, 3 (2013)
12. E.K. Agakhanov, Bulletin of Dagestan State Technical University. Technical science 2,
39-45 (2013)
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3, 8-15 (2015)
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15. E.K. Agakhanov, M.K. Agakhanov, MGSU Bulletin 3, 140-143 (2010)
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Objective . Determination of the stress state of a ground base with a trapezoidal cutoff from the action of own weight, according to the conditions of equivalence of effects, is reduced to determining the stress state from the external surface load distributed according to the hydrostatic law. Methods . The problem of determining the stresses in the structure foundations at any degree of areas development of a plastic strain of the soil has a strict mathematical formulation, and similarity criteria can be obtained using a simpler construct of similarity theory. The simulation is performed by using similarity criteria based on which the model is executed, the loading conditions are determined, and the transition from the values measured on the model to the corresponding values of the full-scale structure is carried out. Similarity criteria can be obtained either with the help of similarity theory or with the help of dimensional analysis. An even greater effect of increasing the self-weight of a model made of transparent optically sensitive material can be achieved using the immersion method in conjunction with the centrifugal modeling method. If necessary, the stresses in the model area are fixed using the "freeze" method. Result . Using the equations system of the mixed problem of the elasticity and plasticity theory, and the scale method, similarity criteria are established for modeling stresses in the foundations of buildings and structures. Limitations on the choice of similarity multipliers for loose soils, the possibility of using the method of centrifugal modeling, as well as features of modeling connected soils are noted. Conclusion . A necessary condition for the similarity of the stress states of loose homogeneous bases in nature and the model is the equality of the similarity multipliers of the geometric scale and the force factor.
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Severo-Kavkazskiy region
  • E K Agakhanov
  • M K Agakhanov
  • Izvestiya Vuzov
E.K. Agakhanov, M.K. Agakhanov, Izvestiya Vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki, 1 (2005)
  • G E Agakhanov
G.E. Agakhanov, Scientific review 12, 733-736 (2014)
  • G E Agakhanov
  • V B Melekhin
G.E. Agakhanov, V.B. Melekhin, Scientific review 4, 90-93 (2016)
  • E C Agakhanov
  • M K Agakhanov
E.C. Agakhanov, M.K. Agakhanov, International Journal For Computational Civil And Structural Engineering 2, 24-28 (2014)
Higher education proceedings
  • E K Agakhanov
  • M K Agakhanov
E.K. Agakhanov, M.K. Agakhanov, Higher education proceedings. North-Caucasus Region. Engineering sciences 1, 25 (2005)
  • E C Agakhanov
  • M K Agakhanov
E.C. Agakhanov, M.K. Agakhanov, Industrial and Civil Construction 11, 40-44 (2015)
  • V I Andreev
  • A S Avershyev
V.I. Andreev, A.S. Avershyev, International Journal for Computational Civil and Structural Engineering 9, 3 (2013)
  • E K Agakhanov
  • M K Agakhanov
E.K. Agakhanov, M.K. Agakhanov, MGSU Bulletin 3, 140-143 (2010)