Determination of stress state for a layer with a longitudinal cylindrical thick-walled tube under given mixed conditions on boundary surfaces Текст научной статьи по специальности «Математика»

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объединяющем разработку, транспортирование и укладку грунта.

Литература:

1. Тилинин Ю.И. Абразия берега и развитие рекреационной прибрежной территории финского залива в г. Сестрорецке Ленинградской области/ Тилинин Ю.И, Козлов, О.В. Рулёва К.С.// Современное строительство и архитектура. - 2018 - № 4(12) - С. 725-729. - Екатеринбург, 2018

2. Хазанович К. К. Геологические памятники Ленинградской области: [очерк-путеводитель] / К.К. Хазанович. - Л.: Лениздат, 1982. - 78 с.:

3. Тилинин Ю.И., Хорошенькая Е.В. Глубинное укрепление обводненных песков при намыве территории //Сборник научных трудов участников

межвузовской научно-практической конференции (6 апреля 2018 г.) / Под редакцией кандидата технических наук Галушко М.М. - СПб: ВИ(ИТ) ВА МТО2018. - 299 с.

4. Anton Gajdo, Vladimir Verstov and Antonina Judina Comparative Efficiency Investigation of Various Types of Dynamic Influences on the Dipped Pile -World Applied Sciences Journal 23 (6): 817-822, 2013 ISSN 1818-4952 © IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.23.06.13109.

5.Юдина А. Ф. Технологические процессы в строительстве / А. Ф. Юдина, В. В. Верстов, Г. М. Бадьин. - М.: Издательский центр «Академия», 2013. - 304 с.

UDC 539.3

Miroshnikov Vitaly Yuryevich PhD, Associate Professor Kharkiv National University of Construction and Architecture

Kharkov, Ukraine DOI: 10.24411/2520-6990-2019-10710 DETERMINATION OF STRESS STATE FOR A LAYER WITH A LONGITUDINAL CYLINDRICAL THICK-WALLED TUBE UNDER GIVEN MIXED CONDITIONS ON BOUNDARY SURFACES

A substantially spatial problem of elasticity theory for a layer with a longitudinal circular cylindrical tube solved in it is solved. The layer and tube are rigidly fixed together. It is necessary to study the stress-strain state of the elastic bodies of both the layer and tube.

On the lower boundary of the layer, displacements are given; on the upper boundary of the layer and the inner surface of the tube, stresses; on the boundary of the layer and tube, conjugation conditions. The solution to the spatial problem of the theory of elasticity is obtained using the generalized Fourier method in relation to the system of Lame's equations in the cylindrical coordinates associated with the tube, and the Cartesian coordinates associated with the boundaries of the layer. By satisfying both the boundary and conjugation conditions, we obtain infinite systems of linear algebraic equations that are solved by the truncation method. As a result, we obtain displacements and stresses at different points of both the elastic layer and elastic tube.

A numerical analysis of the stress-strain state of the elastic body of the layer and tube is carried out. Graphs of the normal stresses on the inner and outer surfaces of the tube are presented.

Keywords: thick-walled tube in the layer, Lame's equation, generalized Fourier method.

Introduction

When designing tunnels, underground facilities and protective screens, there is a need to determine the stress-strain state in such structures. To achieve this, it is necessary to have a calculation method that matches the calculation scheme and allows getting the result with the required accuracy.

There are papers for the layer with a transverse circular cavity or inclusion [1 - 3]. However, the methods used in them can not be applied to the layer with a longitudinal cavity or inclusion.

Papers [4-5] consider the stationary problems of wave diffraction for the layer with a longitudinal cylindrical cavity or inclusion, with the problems based both on the Fourier decomposition method and the image method.

This paper uses an analytical-numerical approach, and is based on the generalized Fourier method [6]. On the basis of this method, also solved are the problems for a half-space with a cylindrical cavity or inclusion [9-11], as well as the one for a cylinder with cylindrical inclusions [12].

Formulation of the Problem

In an elastic homogeneous layer, parallel to its boundaries, an infinite circular cylindrical thick-walled tube with an external radius R\ and an interior one R2 is located.

The tube will be considered in the cylindrical coordinate system (p, 9, z), and the layer, in the Cartesian coordinate system (x, y, z), which is equally oriented and associated with the system of coordinates of the tube. The upper boundary of the layer is located at the

distance y = h, the lower one, at the distance y= — h . It is necessary to find a solution to the Lamé equation

AUj + (l - 2aj )-1 WdivtJj = 0

where aj is Poisson's coefficient for the layer (j = 1) or the tube (j = 2).

On the lower boundary of the layer, the displacements Ui (x, Z) y=-~ = U~ (X, Z) are given; on the upper boundary of the layer, the stresses FU1 (x, Z )| y=h = FUh0 (x, Z ) ; on the inner

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surface

h

of

stresses

the tube, the FU2 (9, Z )p=r = FR (9, Z); on the boundary

of the tube and layer, the conjugation conditions

U1 * )|p=R1 = Û 2 * )|p=r

FÛ1 (9 *)p=R = FU2 (9 *)p=r, ,

(1) (2)

where FUj = 2 • Gj •

CT

j

a

n • divÛv +— Û j + — (n x rotUv)

1 - 2 •CT j j an j 2V j 7

is the stress op-

erator; q,, Gj, Ûj are the elastic constants and displacements of the layer (j = 1) or the tube (j = 2);

Û0 (w, * )=xWgi(1)+CT<h )g21) + xWg<1) ,

(w, *)= Û^ + ûflf + ui~)e<1), (3)

Fr0 (9,* )=CTp> e(2) + tM?22> + Tp?

are known functions; e(), (j = 1,2,3) are the unit vectors of the Cartesian (k = 1) and cylindrical (k = 2) coordinate systems.

All known vectors and functions will be considered as fast falling to zero at great distances from the origin of the coordinate z for the tube and the coordinates x and z for the boundaries of the layer. Solving the Problem

Choose the basic solutions to the Lamé equation for the specified coordinate systems in form [6]

u± (w, y, *; X, = N^é (X*+pw>±Yy, ; Rk m (p, 9, *; X) = \ (Xp>'(X*+m9); Sk m (p, 9, *; X) = Nkf) [(sign Xf Km (X|p)-é < *+m9) j k = 1,2,3;

Nd = 1V ; N2d ) = 4 (CT- 1)?« +1 V(y •) ; N(d ) = i rOt(é<1) •); N« = 1V ;

XXX X X

(4)

N.

(p ) = 1 X

V

a

p^

. ap.

Y

+ 4 (ct- 1)fv-e3(2) —

dz

; N3<P )= L rot (42 )•);

where Im (j), Km (j) are the modified Bessel functions; Rfc m, S^ m , k=1, 2, 3 are, respectively, the

■ m w, Km(X) are the modified Bessel functions; Rk m internal and external solutions to the Lamé equation for the cylinder; u^

( ) U(+) are the solutions to the Lamé

equation for the layer.

The solution to the problem will be presented in the form

30

CHEMICAL SCIENCES / «€©LL©qyMm-J©yrMaL»#22î4â),2@19

3 œ œ

Ui = Z J Z Bk m (à)- 4 ,m (p, 9, z; À)dÀ +

k=1-œ m=-œ 3 œ œ,

Z J j(-Hk(à,p)- Uk[+)(x,y, z;à,p) + Hk(à,p)- Uk[ )(x,y,z;À,p))dpdÀ

+ Z J J(Hk<

k=1-œ —œ

3 œ œ

U2 = ZJ Z Ak,m (à)-Rk,m (P, 9 z; à)+ Ak,m (à)-Sk ,m (P, 9 z;À)dÀ

k=1-œ m=-œ

(5)

(6)

where, Sk,m (P, 9 z;À), Rk,m fo 9 z; à)

uk+)(x , y, z; à, p) and uk ^ (x, y, z; à, p) are the basic solutions given by formulas (4), and the unknown functions Hk (À, p), H k (À, p,),

Bk,m (à) , Ak,m (à) and Ak,m (à) must be found from boundary conditions (3) and conjugation conditions (1) and (2).

For the transition in basic solutions between coordinate systems, we use the formulas [14].

To fulfill the boundary conditions at the boundaries of the layer, we rewrite the basic solutions S^ m in (5), using transition [14, formulas 7], in the Cartesian

coordinate system through the basic solutions u

k

(for y = h) and u(for y= - h ). We equate the

re-

~ 0 i \ sulting vectors, fory= _ h , to the given U~ (X, z j,

and for y= _ h , we find the stresses and equate them to Fh (x, z j. We give the vectors U~ (x, Z j and

Fh (x, zj in advance through the double Fourier integrals.

The resulting system of 6 equations has a determinant

32 - G3 -y5 - ch x - x2 +(3 - 4a)-ch2 x+(1 - 2a)

\2

4

À

where x = y(h + h ), G is the shear modulus.

The square brackets of this determinant coincide with known results [15].

From the obtained equations, we find the functions

hk (à, p) and Hk (À, p) through bk m (à) .

To take into account conjugation conditions (1), we decompose, in (5), the basic solutions u(±) by means [14, formulas 8], turning them into the solutions

Rk m. We then equate p=R1 therein. This will fulfill

condition (1).

To take into account conjugation conditions (2),

we find the vectors FU1 and FU 2 from solutions

(5) and (6), decompose the basic solutions u(±) therein by means of [14, formulas 8], turning them into the solutions Rk m, and equate p = R1. This will fulfill condition (2).

These two conditions give 6 equations, conjugating all the unknowns in equations (5) and (6).

To take into account the boundary conditions on the inner surface of the tube, we apply the stress operator to the right-hand side of (6), and equate (for p=R2)

to the specified FR (9, z) given by the integral and

Fourier series.

From the resulting system of equations, we exclude the previously found functions hk (à, p) and

H k (À, p) through bk m (à) . Having gotten rid of

the series m and integrals À, we obtain a collection of nine infinite systems of linear algebraic equations for

identifying the unknowns Ak m (à) , ~ ot(à) and Bk,m (À).

For the obtained infinite systems of equations, we will apply the truncation method. The numerical studies show that the determinant of the truncated system does not turn into zero for any m, for 0<m<10, and, consequently, this system of equations has a unique solution. Having solved this system of equations, we will

find the unknowns, Ak m (à) , ak ot(à) and

Bk,m (À).

We substitute the functions b, (à) obtained

k ,m V /

from the infinite system of equations into the expressions for hk (à, p) and Hk (À, p). This will determine all unknown problems.

Numerical Studies of the Stressed State A B30 grade concrete tube is located in a homogeneous isotropic clay layer in parallel with its surfaces. Layer: Poisson's coefficient a1 = 0.3, the elastic modulus E1=10 kN / cm2. Tube: Poisson's coefficient a2 = 0.16, the elastic modulus E2= 3250 kN / cm2. The outer

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tube radius R = 70 cm, the internal one R2 = 60 cm.

The layer thickness h + h = 340 cm. The distance from the upper boundary of the layer to the tube center h = 170 cm.

With the weight of the processing equipment taken into account, on the upper boundary of the layer, the stresses

4h)(x, z )= oV2

108 ■(z2 +102

r2 ■(

x2 +102

M-M-

TyX — T yyZ — 0 are given; on the lower boundary

the displacements

0 are given. On the in-

of the layer, (~ ) = TT (~ )

4 )

Uh = U

x ^ y - z ner surface of the tube, there are no stresses = 0.

O(p) _ T(p) _ T(p) ^p ^pp Lpz

-50

-25

0

25

50

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

\ 3

✓

X X V N V ✓ ✓ ✓ ✓

- - - .V" ✓

A finite system of equations of order m = 8 was solved. The accuracy of the fulfillment of the boundary conditions for the indicated values of geometric parameters was equal to 10-4.

Fig. 1 shows the normal stresses along the z axis at the upper point of the tube on the outer and inner surfaces.

The greatest stresses are C^ (Fig. 1, line 2), which reach the maximum values at z = 0: on the outer surface of the tube, the compression C^ = -0.5 kN /

cm2; on the inner surface of the tube, the tension C^

= +0.564 kN / cm2. It should be noted that the stresses on the tube surface, along the z axis, fall very slowly (compared to the specified function at the boundary of the layer).

-50

-25

25

50

0.6 0.5 0.4 0.3 0.2 0.1 0

y * * * \ 2 'x N N X

* * * t N X X

3

b

Fig. 1. Stresses on the surfaces of the tube along the z axis, at x = 0 (in kN / cm2): a - on the outer surface (y -

+ R1); b - on the inner surface (y = + R2); 1

Op ; 2

Op; 3

O,

Fig. 2 shows the stresses on the tube surface (along the radii R1 and R2) in the plane z = 0.

0.4 0.2 0 -0.2 -0.4 -0.6

0 %/4 %/2 3%/4 % 5%/4 6%/4 7%/4 2%

/ \ » 1 /

\ \ \ 1 1 ' / / \ \ \ / / > ...

v.. 1 ••. » I \ A /' / /

t t t \ 3/ / 1 / 2 \ \ X / * *

I \ * / /

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6

0 %/4 %/2 3%/4 % 5%/4 6%/4 7%/4 2%

/ 1 \ \ \

1 1 1 1 \ V- 2 * / / X \ \

1 1 .. « 3 / / ...... \ *

....... 1 1 T 1 < .....1. 1 I 1 \ \ \

/ / * * \ \ \ 1 / / / % \ \ X

X. / (

b

Fig. 2. Stresses on the surface of the tube along the radii R1 and R2, at z = 0 (in kN / cm2): a - on the outer surface; B - on the inner surface; 1 - Op ; 2 - Op ; 3 - Oz

Along the radii, the stresses vary from compression to tension and vice versa. Thus, on the outer surface of the tube (Fig. 2a) at the upper and lower points, there is compression, to the right and left, tension. At the inner surface of the tube (Fig. 2b), in the upper and lower points, there is tension, on the left and right, compression. In addition, the stresses in absolute value on

the inner surface of the tube are higher than on the outer one.

Along the radius R1 in the elastic body of the layer, the stresses are very small (in comparison with those in the elastic body of the tube), which is the result of the difference in the layer and tube materials.

Conclusions

0

z

z

a

a

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CHEMICAL SCIENCES /

On the basis of the generalized Fourier method, the problem for the layer with a longitudinal cylindrical thick-walled tube and different boundary conditions at the boundaries of the layer and tube is calculated.

The proposed analytical-numerical calculation method allows us, with the given accuracy, to determine the stress-strain state of the elastic body, taking into account its infinite boundaries and conjugation conditions for the layer and tube.

The numerical study of the stress-strain state of the concrete tube, which is in a layer of clay under the action of loading on the surface of the layer, proves that the greatest stresses arise on its inner surface. In addition, in comparison with the given function, there is a very slow decrease in the stresses along the z axis.

The numerical studies of the algebraic system make it possible to state that its solution can be found with any degree of accuracy by the method of reduction. This is confirmed by the high accuracy of the implementation of boundary conditions.

References

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