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Analysis of disturbing influence of traffic load on soil body
Janat Musayev,1 Algazy Zhauyt,2
1
Department of Transport engineering and technologies, Kazakh Academy of Transport and Communications
named after M.Tynyshpayev, Almaty 050012, Kazakhstan
2
Department of Applied Mechanics and Basics of Machine Design, Kazakh National Technical University
named after K.I.Satpaev, Almaty 050013, Kazakhstan
Correspondence should be addressed to Janat Musayev; mussaev1975@mail.ru
Stress waves propagate in soil in case of earthquake and man-made effects (traffic flow, buried explosions,
shield-driven pipes and tunnels, etc.). The wave point-sources are those located at the distances equal to more
than two-three waves lengths, that significantly simplifies solving of a problem of these waves strength
evaluation. Distribution of stress and displacement by the stress waves propagation in elastic medium is a
complex pattern. The stress distribution in propagating waves depends on a type and form of source, conditions
of the source contact with medium and properties of mediums in the vicinity of the source. The point-sources
and their combinations are selected in such a way to model an influence of machines and processes on soil body
in case of shield-driven pipes (tunnels).
Key words: waves * propagation * equations * concentrated * tunnels
1.
Introduction
In confined environment the wave pattern becomes
more difficult due to the boundary reflection of waves. For
evaluation of different factors influence on the nature of
waves propagation it is convenient to divide the waves
propagation problem into several stages.
At the first stage tasks of waves propagation in infinite
medium from the point-sources of the following different
types are reviewed:
- concentrated force,
- double force without moment (two concentrated
forces acting contrariwise along one line and applied at
small distance 2h from each other),
- two double forces without moment (double couple
without moment) acting at right angles and in a single
plane,
- three double forces without moment acting in three
orthogonally related directions (center of expansion),
- double forces with moment (couple of forces),
- combination of two couples of forces with sum of
moments equal to zero,
- uniform pressures in alveole at a section in length d,
- tangential stresses applied at a section of alveole
contour in length d,
- tangential stresses applied to alveole contour in length
d.
Solution of tasks enables evaluation of types of wave
sources influence on a type and parameters of waves
propagating over a distance.
At the second stage tasks of wave propagation from the
point-sources in semi-infinite elastic medium are reviewed.
At this stage tasks of vibration of semi-infinite elastic
medium surface from different point-sources of waves at
various depths with account of waves reflected from free
surface are solved. The solutions make it possible to
evaluate a dynamic impact on the environment in case of
earthquake, as well as the dynamic impact of different
devices and machines used for underground works. This
article deals with a task of waves propagation from the
point-sources of different types in the infinite space. In the
performance of the tasks the Fourier integral transform and
generalized functions are used[1,3,4].
2. Materials and Methods
For dynamic tasks both infinite and semi-infinite
spaces and for bounded areas we shall use differential
equations of motion
 U j ,ii     U i ,ij   U j  f j ,
i, j  1,2,3 .
(1)
Supposing that in the generalized functions outside
area  occupied by the elastic area the displacement and
stress functions are equal to zero[1]. Then equation (1) may
be in a form:
 U j ,ii     U i ,ij   U j 
 

  U j cosn, xi  s  ,i 
    U i cosn, xi  s , j 
 
  ji s cosn, xi  s    U k  t 0
 t    U
t  T  
 
j
t T
   t    U 
  U k
t 0
k
t T
 t  T ,
(2)
 ij - Kronecker symbol,  s - delta function at
the area boundary,  t ,  t  - delta function and its
Where
U 
time derivative.
 
ij
j s
and
 
ij
s
  and
- jump of U j
functions upon outside passage through the
boundary of area  . Since outside of this area these
functions are equal to zero, symbols U j and  ij
 
 
 2  12   22   32
wave velocity
present values of these functions at the area boundary.
U  , U  , U  , U 
Functions
j t 0
j t T
j t 0
j t T
represent initial and final conditions, i.e. displacements
and velocities of medium points at t  0 , t  T .
In the expressions below and above the summation
over repeated indices is performed.
Let us introduce the notations:
 

X j   U j cosn, xi  s  ,i 
    U i cosn, xi  s , j 
 
k
t T
j
(3)
interval (0, Т), and is equal to zero outside of this area and
interval.
Since all functions represented in the finite functions
are equal to zero outside the area, consequently jumps of
functions at the area boundary are boundary conditions.
Depending on the assigned tasks a part of these functions
is set, the other is defined in the course of solution[11].
Let apply the Fourier transform for the differential
equation system solution. Let us multiply the left and right
expi  k x k     and
integrate with respect of four variables:  x1 , x 2 , x3 ,  , in
member of the equation by
other words let apply the Fourier transform to system of
equations (2). Here  1 ,  2 ,  3 means parameters of the
 - frequency.
For description of the Fourier functions
let in symbol

the
transform are
Fourier
ij

ij
,U j , F j 
,U j , F j . The following properties of
used
upon
U i
~
  i  j U i or in tonsorial
x j
~
~
U i , j   i  j U i and Ui    2 U i .
- ratio of P-
to S-wave velocity
. Having used the agreed notations and the
Fourier transform let represent the system of equations (2)
in a form:
~
Xk
~
~
2
    U k    1  k  lU l 

2
2
2



(4)

~
X k includes not only
the force impacts on medium but also kinematic ones.
Solving of algebraic equations system (4) may be writtendown in the following form:
Expression (3) contains information on mass load,
impacts on the medium boundary and initial conditions. If
the medium motion is considered at the finite interval (0,
Т) of time, the final conditions of displacements and
velocities functions of the medium points at time Т are
included, which represent unknown quantities; in the used
method the solutions fulfill a role of “integration
constants”. F j coincides with f j in area  and in
Fourier transform space-wise,


  2

c1 
In the right member of equation (4)
  ji s cosn, xi  s    U k  t 0
 t    U j t T t  T    U k t 0 
  t    U
 t  T   F
 
 
c2 
  2


and
integration:
notations
Let divide the left and right members of system of
equations (2) by  and introduce the notations:





~
~
 2  2   2  2 X j   2  1  j k X k
~
Uk 
 c12  2   2  2  2   2


(5)
For detection of displacements it is necessary to perform
an inverse Fourier transform:
U j  x1 , x2 , x3 ,  
1

4 2
~
~
X j   2  1  j k X k


 2  2   2  2 

 c12  2   2  2  2   2 
W
 e  xk k    dW
(6)
where W means a space of variables
1 ,  2 ,  3 ,   and
dW  d1 , d 2 , d 3 , d .
For the infinite space symbols
~
X j and X j have only
~
a generalized load F j and F j .
3. Results and Discussion
Let consider the method of solution of the tasks of
waves propagation in the infinite elastic medium from the
point-sources with application of the Fourier transform and
generalized functions from the point-sources using the
example of a task of the concentrated force F t effect
for comparison with famous Love solution[2].
Upon effect of the concentrated force at the origin of
coordinates and in х3 – direction expressions (3) take the
form:

X l  0 , X 2  0 , X 3  F t  x3 
(7)
and correspondingly the Fourier transform
~
~
X 3  F   .
Under such conditions the displacement components
take the form:
U 1 x1 , x2 , x3 ,  
~
1 3 X 3
 2 1

2
2 
2
4   c1 W    2  2  2   2 
 e   xk k    dW ,
U 2 x1 , x2 , x3 ,  

e

cos
r

,
F
t

4   r c12  c1 
U  0 ,
Ur 
U 
~
 2 3 X 3
 1
4 2  c12 W  2   2  2  2   2 
2
  xk  k   


1
4   c12
2
 




U  r ,  / 2 
U r r ,0 
(8)
Let us introduce the notation:
r  x12  x22  x32 .
Considering radiation conditions and using asymptotic
development[3] of the Fourier integrals (8) and neglecting
components representing fluctuation of near-field we will
get
U1 
 2r

4   r x1x3

  0
c 22
.
c12
(11)
For majority of rocks ratio (11) at Poisson coefficient
  0,25 is approximately equal to three.
The achieved method is used upon getting of functions
describing wave propagation from different sources. Wave
radiation patterns from such sources are provided below
with the required clarifications. Propagation of stress
waves from the point-sources of different types acting in
the infinite elastic medium. The graphs below have been
obtained with use of MATLAB program complex[5-10].
1
3.1. Waves propagation from concentrated force

cos
r

,
F
t

4   c12 r  c1 
u  0 ,
ur 
1 
r 1 
r 
  2 F  t    2 F  t   ,
 c1  c1  c2  c2 
 2r
U2 

4   r x 2 x3
1
u 
1 
r 1 
r 
  2 F  t    2 F  t   ,
 c1  c1  c2  c2 
U3 

 .

longitudinal motion directed along the force line
~  2  2   2  2   2  1  32
 X 3  2   2 2  2   2 
W
 e   xk k    dW .

sin 
r
F  t 
2
4   r c2  c2
(10)
An interesting fact shall be pointed out: over all
distances from the point of force cross motions
perpendicular to the force line    / 2 exceed the
dW ,
U 3  x1 , x 2 , x3 ,  
In consequence of symmetry of displacements and
stresses about axis х3 it is possible to get rather convenient
expressions, if a spherical coordinate system going through
the coordinates center is applied and  shall be defined as
an angle between the radial coordinate and positive axis
х3

sin 
r
F  t 
2
4   c2 r  c2
(12)

 .

 2r

4   r x32
1
1 
r  1 
r
  2 F  t    2 F  t 
 c1  c1  c 2  c 2

 .

(9)
We point out that it is possible to get more complete
transforms of integrals (8). In such case there are solutions
fully coinciding with Love solutions[2] for the concentrated
force in the infinite space. This article deals with
propagation of stress waves from the point-sources of
different types in the infinite space based on the
asymptotic development of Fourier integrals.
а – P waves
FIGURE 1: Diagrams
concentrated force.
b – S waves
of
waves
propagation
from
3.2.
Propagation of waves from combination of two
forces (double force without moment)
ur 
2h cos2  
r


,
F
t

4   c13 r  c1 
u  0 ,
u 
(13)
2 h sin cos 
r 
F   t   .
3
4   c2 r
 c2 
а – P waves
b – S waves
FIGURE 4: Diagrams of waves
combination of two double forces.
propagation
from
3.4. Propagation of waves from combination of three
couples of forces without moment directed in parallel with
three orthogonal axes
ur 
а – P waves
FIGURE 2: Diagrams of
combination of two forces.
u  0 ,
b – S waves
waves
propagation

2h
r


,
F
t

4   c13 r  c1 
from
(15)
u  0 .
а – P waves
FIGURE 3: Clarification to diagram of waves propagation
from combination of two forces without moment.
FIGURE 5: Diagrams of waves propagation
combination of three couples of forces.
from
3.3.
Propagation of waves from combination of two
double forces (two double force without moment)
3.5. Propagation of waves from two couples of forces
ur 
2h sin 2  
r
F   t   ,
3
4   c1 r  c1 
u  0 ,
u 
2 h sin  cos 
r 


F
t

 c  .
4   c23 r
2 

ur  0 ,
(14)
u  

2h sin
r 



F
t

4   c23 r  c2 
u  0.
,
(16)
а – S waves
FIGURE 6: Diagrams of waves propagation from two
couples of forces.
3.6.
Propagation of waves from combination of two
couples of forces with sum of moments equal to zero
2 h sin cos sin 2  
r


ur 
F
t

 c 
4   c13 r
1 


,
a-P waves


2 h sin cos2   sin 2 
r 


u 
F
t

 c 
4   c23 r
2 

2 h sin cos sin  cos 
r 


u 
F
t

 c 
4   c23 r
2 

b- S waves
FIGURE 8: Diagrams of propagation of waves created by
uniform pressure.
3.8. Propagation of waves created by tangential stress at
alveole contour in length d
ur  0 ,
(17)
u 

 a2d
r
   t  
4   c 2 r  c1 
,
(19)
u  0
а – P waves
b – S waves
FIGURE 7: Diagrams of waves propagation from
combination of two couples of forces with sum of
moments equal to zero.
3.7. Propagation of waves created by uniform pressure in
alveoleat a sections in length d
a - S waves
 
c2
 a2d 
r
1  2 22 cos2   p   t  
4   c1 r 
c1
  c1 
u  0 ,
FIGURE 9: Diagrams of propagation of waves created by
tangential stress at alveole contour in length d.
ur 
 a 2 d sin  cos 
r 
u 
p   t   .
2   c2 r
 c2 
(18)
3.9. Propagation of waves created by tangential stress
applied to alveole contour at section in length d
The forms of waves propagating from different
sources comply with the law of force variation and are
derivatives of functions describing variation of forces
and stresses.
2  a d cos 
r
p   t  
2
4   c1 r
 c1 
u  0
ur 
u  0 u 
2  a d sin
4   c22 r
(20)

r 
  t   .
 c2 
Conflict of Interests
The authors declare that there is no conflict of
interests regarding the publication of this paper.
References
[1]
а – P-waves
b – S-waves
FIGURE 10: Propagation of waves created by tangential
stress applied to alveole contour.
Remark: Derivative with respect to all expressions
above between the brackets is marked with a prime.
4. Conclusions
Solutions of stress waves propagation in elastic
medium from different sources have been provided.
The point-sources and their combinations have been
selected in such a way to model the influence of
machines and processes on soil body in case of shielddriven pipes (tunnels).
P-waves and S-waves propagate from each source
of waves.
The waves propagate from concentrated force, the
forms of these waves comply with the law of force
variation. Wave amplitude decreases as
1
due to
r
radiation in space.
Upon propagation of waves from double force
without moment the wave forms represent a derivative
of functions describing the law of force variation. The
wave amplitude also decreases as
space.
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due to radiation in
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