Determination of the stress state of a layer with a cylindrical cavity located on an elastic base and specified boundary conditions in the form of stresses Текст научной статьи по специальности «Физика»

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УДК 539.3

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

Kharkov, Ukraine DOI: 10.24411/2520-6990-2019-10610 DETERMINATION OF THE STRESS STATE OF A LAYER WITH A CYLINDRICAL CAVITY LOCATED ON AN ELASTIC BASE AND SPECIFIED BOUNDARY CONDITIONS IN THE FORM OF

STRESSES

The spatial problem of the theory of elasticity is solved for a layer with a longitudinal circular cylindrical cavity. The layer is located on an elastic base in the form of half-space. The layer and elastic base are homogeneous isotropic materials that are different from each other. At the cavity and at the upper boundary of the layer are specified stresses. Applying the generalized Fourier method with respect to the system of Lame equations and satisfying the boundary conditions, we obtain infinite systems of linear algebraic equations that are solved by the reduction method. As a result, displacements and stresses were obtained at various points of the elastic body. A numerical analysis of the stress-strain state was carried out for a concrete layer that lies on a clay base and has a longitudinal cylindrical cavity.

Keywords: cylindrical cavity in a layer, generalized Fourier method, Lame equation, elastic half-space

Introduction.

The problem of calculating problems for a layer with longitudinal cavities has existed for a long time. So in [1-3], on the basis of expansion in Fourier series, for a layer with a cylindrical cavity or inclusion, stationary problems of shear wave diffraction and determination of stresses are considered. Stationary problems of shear wave diffraction were also considered on the basis of the image method in [4]. In [5], using the method of superposition of general solutions, the problem for a layer with a circular hole was considered.

But for problems in a substantially spatial model or with several boundary surfaces, these methods cannot be applied. To solve such problems, it is necessary to use the generalized Fourier method [6].

Based on the generalized Fourier method, problems for a layer with a spherical hole are solved when the layer is stretched at infinity [7]. Problems for a space or half-space with cylindrical cavities and various boundary conditions were solved in [8-13]. The problems for a cylinder with cylindrical inclusions were solved in [14]. The problems for a layer with a cylindrical cavity in displacements, as well as for a

layer with a longitudinal cylindrical thick-walled pipe, were solved in [15, 16].

In this paper, we also propose an analytical-numerical solution to the problem posed on the basis of the generalized Fourier method.

Formulation of the problem.

In the elastic homogeneous layer is a longitudinal cylindrical cavity of radius R. The layer is ideally linked by its lower surface with a uniform elastic halfspace. The layer and half-space will be considered in the Cartesian coordinate system (x, y, z), the cavity will be considered in the cylindrical coordinate system (p, 9, z), combined with the coordinate system of the layer. The boundaries of the layer are located at a distance

y=h and y=

h.

It is necessary to find a solution to the Lame equation AUj + (l - 2a j )-1 WdivUj — 0,

where j=1 - corresponds to the layer, j=2 - half-space. At the upper boundary of the layer y=h and on the surface of the cavity P — R, the stresses are given:

FU (x, z)|y=h = F0 (x, z), FU (q, z)|p=r = FR (q, z)

where

Fh (x1, Z1) = ^(x) + ^y)e2 ) + ^yZe3 )

f0 (q, z ) = apr )e1(2) + xprq)42) + xpr h

(2)

(1)

(к )

famous features; ej , j = 1, 2, 3 - the unit vector of the Cartesian (k = 1) and cylindrical (k = 2) coordinate systems. At the boundary of the layer and half-space, the conjugation conditions are satisfied

U

У=-h

U

У=—h

(2)

3

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FU

y=-h

F2U 2

y=-h

(3)

FJUJ

where U\ — layer movements; U2 — displacement in half space;

2Gj [

aj ^ d ^ 1

EJ

J h n divUj +YnUJ+ 2(n x rotUJ)]; GJ ; ~ J

1 - 2 a j

2(1 + a j )

; a

E j - Poisson's ratio and elastic modulus of the layer (j = 1) or half-space (j = 2).

The specified functions will be considered as rapidly decreasing from the origin of coordinates along the z axis for the cylinder and along the z and x axes for the layer boundaries.

Solution method.

Choose the basic solutions to the Lamé equation for the Cartesian and cylindrical coordinate systems in the form [6] :

U± (x, y, --; X, p; a) = Njd V(Xz+px )±yy ;

N

Rk , m (p, z; X; a) = n{p ) Im (Xp)ei(Xz+mv); (4

Sk,m (p, V, z; X; a) = n[p) ^ (p; X) • e'(Xz+mv)] k = 1,2,3; f) = 1V ; N^ = 4(a- 1>f +1 v(y •); N3d) = irot^ •); N^ = 1V ;

( p )_ 1

N r =

X

X

5

V[p|) + 4(a- <V-#| ]; N(p )=' rot^ •);

X

N

sm (p; X) = (sign X)m Km (x|p); y = ^X2 + p2, -œ < X, p < œ,

—► —►

where Im (x), Km (x) - are the modified Bessel functions; Rj m ' S j m - are, respectively, the

- (-) - (+)

internal and external solutions to the Lamé equation for the cylinder; U^ , U^ - are the solutions to the

Lamé equation for the layer. We will present the solution to the problem in the form

- 3 œ œ -

ui = Z J Z Bk,m(X)"Sj,m(p,^z;X;a)dX+

j=1 -œ m=-œ

(5)

Z i i(^kl}(X,^ 4+^(*,y,z;X,Ia1 ) + H«(X,uk )(x,y,z;X,|i;a1 ))dpdX,

k=1 -œ-œ

3 œ œ/ , . , .

2 i i(Hk (Xi)'4(xy,z;X, 1;a2

k=1 - œ-œ

U

where

sk, m (p, V, z; X; a J ), 4+H*, y, z; X, a j ) and 4 ^(x, y, z; X, a j )

tf®(X, H ('»(x, H<2)(X, i)

solutions given by formulas (4), and the unknown functions (x, p), ^^ (x, p), jj^

Bj m (X) must be found from the boundary conditions (1) and the conjugation conditions (2), (3). To transition between coordinate systems, we use the theorems of basis solutions [16].

(6)

- the basic and

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To satisfy the boundary conditions at the upper boundary of the layer, the S» vectors in (5), using the

k, m

- (-)

transition formulas [16, formula 7], are rewritten in the Cartesian coordinate system through basic solutions U^

. From the obtained vectors, we find the stresses and equate (for y = h) to the given F^ (x, Z) one represented through the double Fourier integral. To satisfy the conjugation conditions on the flat contact surface of the layer

and the half-space (y= — h ) in displacements (2), for the right-hand side of (5) we apply the transition formulas - -(+)

from solutions Sk m to solutions U^ [16, formula 7]. Thus, we obtain three additional equations (one for

each projection) with unknowns HM-), HM-), H(2)(^, M-) and Bk m . Similarly, we write three additional equations for stresses (3).

Having obtained a system of nine infinite equations, we express the functions H( H(

4%, n) through Bk, m W.

s the form

64.a3 -y9 • e"3Y(h+h)-0(Y)

and h ^

The determinant A of this system has the form

a3- y - e

A = -

A6

where ^(y) - cumbersome function and as a result is omitted. The study ^(y) found that, with y>0, it has only positive values and is not zero. Since A> 0, this system of equations has a unique solution.

- (+)

To satisfy the boundary conditions on the cylinder p=R, using transition formulas from decisions U and - (—) -

U^ to decisions Rk m [16, formulas 8], right side (5) we rewrite in a cylindrical coordinate system through basic solutions

Rfr , s, . For the resulting vector, we find the voltage and equate to the given k, ^^ k, ^^

Fr Z), represented by the integral and the Fourier series. Instead H( ) and H( ) we sub-

stitute the previously expressed functions through Bk m . As a result, we obtain the totality of three infinite

systems of linear algebraic equations with respect to unknowns B^ m . These infinite systems have the properties of equations of the second kind and, as a consequence, the reduction method can be applied to them.

After determination B^ m we can find the values of unknowns Hj|)(^, M-), H

H( P-), which we previously expressed through B^ m . So all unknown expressions (5) and (6) will be found.

Numerical studies of stress.

There is a layer with a cylindrical cavity. The layer is perfectly adhered by the lower boundary to the elastic half-space.

The layer is concrete of class B20, Poisson's ratio a: = 0.16, elastic modulus E\ = 3250 kN/cm2. Half space - clay, a2 = 0.3, E2=10 kN/cm2.

The radius of the cylindrical cavity is R = 5cm. The distance from the center of the cylindrical cavity to the upper boundary of the layer is h = 10 cm., to the lower boundary of the layer h = 10 cm.

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At the upper boundary of the layer, stresses are set in the form

r2 L2,^2Y2 T(1)=T« =

(^(x,z) = -108 • (z2 + 102) 2 • (x2 + 102) 2, TjJi = Xy-! = 0, on the surface of the cyl

(R ) = X(R ) = X(R ) = 0

inder, the stresses are G,

Endless system was truncated to m=12. The integrals are calculated using Philon's quadrature formulas (for oscillating functions) and Simpson's ones (for functions without oscillations). The accuracy of the boundary conditions, for the specified values of m and given geometric parameters, is 10-3.

Figure 1 shows the stresses G ^ and G - along the z axis at x = 0, on various surfaces: y = h , y = R, y = -R and y = - h .

-40 -20 0 20

40

1.5 1

0.5 0

-0.5 -1 -1.5 -2 -2.5

0.5 0

-0.5 -1 -1.5

3 —- 1

"J ..............

4

N

2

/^1

-40

-20

20

40

____ ..•i"___c 1 / 54 —\- „4

■n. ■NM, "x 2 \ »

/1

Fig. 1. Stresses along the z axis in kN/cm2: a - G^ ; b - ctz ; 1 - aty = h, 2 - aty=R, 3 - aty = -R, 4 - aty = r h .

Stresses G^ (Fig. 1a) along the z axis decrease slowly, G z (Fig. 1b) - in proportion to the given Gp , quickly.

The maximum compressive stresses G^ and Gz arise at the upper boundary of the layer (Fig. 1, line 1).

In the upper part of the cavity (x=0, y = R), the compressive stresses G z are at r 6 > z > +6, and in the region z = 0 they take positive values, the stresses G^ are negative at any z.

In the lower part of the layer, the stresses G^ and G z are tensile (Fig. 1, lines 3, 4). As we approach the

lower boundary of the layer, the stresses G^ decrease (Fig. 1a), and G z increase (Fig. 1b). In fig. 2a shows the stresses on the surface of the cavity along the angle 9, at z = 0.

z

0

z

b

a

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1.5 1

0.5 0

-0.5 -1 -1.5

0 л/4 л/2 3л/4 л 5%/4 6л/4 7л/4 2л

N 2 . / "А

/ \

/ \ А 1 \

V / \

-40

-20

20

40

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2

4

■'О*.

3Ч \ • 5

x

b

Fig. 2. Stresses in kN/cm2: a - on the surface of the cylinder along the angle q; b - at the bottom of the layer

along the x axis; 1

аф- 2 -

a z, 3 -

ay, 4-

a x- * - a

z

Stresses Сф are maximum on the surface of the

cavity (Fig. 2a, line 1), the compressive values of which are in the upper part, tensile in the lower part

(ф=5л/4..7л/4). Stresses С z (Fig. 2a, line 2) have extreme values at the upper and lower points of the cavity (ф=л/2 и ф=3л/2).

Figure 2b shows the stresses at the lower boundary of the layer along the x axis, at z = 0.

At the lower boundary of the layer, tensile stresses

Сy arise under the cavity (Fig. 2b, line 3), and compressive stresses С y arise outside the cavity. The

maximum stresses С x and С z along the x axis

(Fig. 2b, respectively, lines 4, 5) arise outside the cylinder (at x = +8 cm), below the cylinder (at x = 0), the stresses are lower. Conclusions.

Based on the generalized Fourier method, a method is proposed for solving the spatial problem of the theory of elasticity for a layer with a lower face linked to an elastic half-space and weakened by a longitudinal cylindrical cavity. At the upper boundary of the layer and on the surface of the cavity, stresses are specified. The problem is reduced to an infinite system of linear algebraic equations that allows the application of the truncation method to it. Numerical studies of the stress state give reason to argue that its solution can be found with any accuracy by the proposed method, which is confirmed by the high accuracy of fulfilling the boundary conditions.

The solution method can be used in the design of road surfaces, building structures, as well as in mechanical engineering, provided that the problem is the same.

The presented stress state analysis can be used in the selection of geometric design parameters. References

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