Determination of the stress state of the medium in the form of a layer having a cylindrical inclusion and conditions for smooth contact with half-space Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

/ TECHNICAL SCIENCE

УДК 539.3

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

Kharkov, Ukraine DOI: 10.24411/2520-6990-2019-10642 DETERMINATION OF THE STRESS STATE OF THE MEDIUM IN THE FORM OF A LAYER HAVING A CYLINDRICAL INCLUSION AND CONDITIONS FOR SMOOTH CONTACT WITH

HALF-SPACE

The solution of the spatial problem of the theory of elasticity for a composite medium in the form of a layer with a longitudinal circular cylindrical inclusion and a half-space associated with the layer by normal displacement and normal stress at zero tangential stresses is proposed. Layer, inclusion and half-space are homogeneous isotropic materials, distinct from each other. Using the generalized Fourier method, which allowed satisfying the boundary and conjugation conditions to obtain infinite systems of linear algebraic equations. The latter are solved by the reduction method. As a result, displacements and stresses were obtained. A numerical analysis of the stress state was carried out for a plastic layer reinforced with a steel rod and smooth contact conditions associated with the half-space of rubber.

Keywords: cylindrical inclusion in a layer, generalized Fourier method, Lame equation, conjugation conditions.

Introduction.

When designing composite joints in mechanical engineering, it becomes necessary to determine the stress with a given accuracy. This can be achieved using analytical or numerical-analytical methods of calculation. One of the effective numerical-analytical methods for solving spatial problems with several boundary surfaces is the generalized Fourier method [1].

Based on the generalized Fourier method, problems are solved for a layer with a spherical hole, when the layer is stretched at infinity [2], problems for space or half-space with cylindrical cavities and various boundary conditions [3-8], for a cylinder with cylindrical inclusions [9 - 11], for a layer with a cylindrical cavity in displacements [12], for a layer with a longitudinal cylindrical thick-walled pipe [13], as well as a layer with a cavity on an elastic base [14].

In this paper, we also propose a numerical and analytical solution to the problem based on the generalized Fourier method.

Formulation of the problem.

In a homogeneous elastic layer, parallel to its boundaries, there is a circular cylindrical inclusion of radius R. The lower surface of the layer has smooth contact with a homogeneous elastic half-space. The layer and half-space will be considered in the Cartesian coordinate system (x, y, z), the inclusion in the cylindrical (p, 9, z), combined with the Cartesian coordinate system. The boundaries of the layer are located at a distance y=h and y= — h , the boundary of the half-space

at a distance y= — h .

It is necessary to find a solution to the Lame equation AUj + (l — 2a ] y1 VdivUj = 0, where j = 1 -corresponds to the layer, j = 2 - to half-space, j = 3 - to

inclusion. At the upper boundary of the layer y=h, displacements U1 (x, z )| y=h = U0 (x, z ) or stresses

F1U1 (x, z ) y=h = Fh (x, z ) are given, Where

U0 (x, z ) = U«ef1) + U^« ,

F0 (x, z ) = + +

(1)

are known functions

; г (k >, ; = 1

J

2, 3 - are the

unit vectors of the Cartesian (k = 1) and cylindrical (k = 2) coordinate systems. On the boundary of the layer and half-space, the conditions of smooth contact are satisfied

Uy]\ ~ = Uy i ~

-77(2)1

G

(1)

'У y=-h

= g(2)

У y=-h

(2) (3)

rM

yx\y=-t

= r(2), ~ =r(l), ~ =r(2), ~ = 0 (4)

У \y=-h y* \y=-h y* \y=-h V '

at the boundary of the layer and inclusion, the conjugation conditions are satisfied

U1 ((Pi z}p=R = U3 ((p, z}p=R , (5) FU1 (P z^R = FU3 (P z^R , (6)

where U^), ) - normal displacements and stresses in the layer (j = 1) or in half-space (j = 2);

TECHNICAL SCIENCE / «C®LL®qyMm-J®yrMaL»#2@î44),2@19

FU

1 - 2 o,

_ _ d A ^ = 2G ([-—-j— n div U +—U + — (n x ro/U)] ered as fast falling to zero at great distances from the cn 2 origin of the coordinate z for the tube and th

nates x and z for the boundaries of the layer.

All known vectors and functions will be consid-l as fast falling to zero at great distances from the origin of the coordinate z for the tube and the coordi-

Ei r ; G, =-; O,, E - elastic constants of a

j 2(1 + ) J J layer (j = 1), half-space (j = 2) or inclusion (j = 3).

Solving the Problem.

Choose the basic solutions to the Lamé equation for the specified coordinate systems in form [1]:

U±(x, y, z;A, ,;a) = N¡d ^ ; RKm (p,ç, z; A;a) = Nkp ) Im (Ap)e'(Az+mÇ ; SKm (p, ç, z; A; a) = Nkp ) [^ (p; A) • } k = 1,2,3;

(7)

Nid)= 1V ; N 2) = 4 (a-l)e2l)+1 V(y •) ; N3d > = j- rot(e^) ;

N( p ) = - v ; N 2p ' =1 1 A A

v|p—| + 4(a- 1)V-ef^

dp

dz

; Np)= Í rot(^3(2)•);

A

smp;A)=(signA)mKm\\p); y = yj\ + /2, /<œ,

where Im (x), Km (x) - are modified Bessel external solutions of the Lame equation for the cylin-

functions; Rm, - respectively, internal and

der; U

(-) u(+)

k

- solving the Lame equation for a

layer.

The solution to the problem is presented in the

form

3 » »

Ui =ËÎÊ (A) Skmm (p,ç, z;A;a )dA +

k=1 _»m=-»

(8)

¿ j j(/«(A, ,)• Uf(x, y, z; A, )+/~«(A, ,)• U«(x, y, z;A, )]d,A:

k=1 -»-»

U2 =

¿ j j(/k2)(A,,)-Uk+)(x,y,z; A,))d,dA, (9)

k=1 —CO—CO

3 » »

U3 =E j Ê A,m (A)-Rk ,m (PÇ z;A;^3 ^

k=1 -»m=-»

(10)

r(+)

where 4 m Rm (p,v, z;A) ,

(x,y,z;A,/i;a]) h u(f](x,y,z;A,/i;a}) -are the basic solutions given by formulas (7), and the unknown functions H^(A, /), H^(A, /),

Hk)(\ /), Bkm(A) and AkM (a) must be found

Uk . If displacements are specified at the boundary of the layer, then the resulting vector is equated (for y = h) to the given U^ (x, z), if the stresses are specified, then from the obtained vector we find the stresses and

equate (for y = h) to the specified Fh (x,

z ). We pre-

from the boundary conditions (1) and the conjugation liminarily rePresent the vectors Uh (x,z) and

Fh° (x, z ) through the double Fourier integral. So

conditions (2) - (6).

For the transition in basic solutions from one coordinate system to another, we use the formulas [13]. To satisfy the boundary conditions on the upper

surface of the layer, the vectors Sk m in (8), using the

transition formulas [13, formula 7], are rewritten in the Cartesian coordinate system through the basis solutions

we

get three equations (one for each projection).

To satisfy the conjugation conditions on the flat contact surface of the layer and the half-space (2), for the right-hand side of (8) we apply the transition formulas from the solutions Sk m of the cylinder to the

<<C®ILL®qUQUM~J®U©MaL>>#20qq),2(0]9 / TECHNICAL SCIENCE

solutions [13, formula 7]. We equate the resulting

vector with expression (9) for ey and y = _ h .

We

similarly fulfill condition (3) by finding the stresses for (8) and (9). To fulfill conditions (4), we find from (8) and (9) the voltage t^X ~, t^X ~ and equate them

yx \y=-h yz \y=-h

to zero.

Thus, we obtain nine equations (three of the boundary conditions and six of the conditions of the conjugation of the layer with half-space) with unknowns hf (à,h), H(1 (à, h) , H(l)(x,n) and

Bkm W.

From this system of equations we express the functions H®(A,|), Hf (A,|) and H®(A,l)

in terms of Bk ,m (A).

The study of the determinant showed that it has only positive values and does not vanish, therefore, the system of equations has a unique solution.

To satisfy the conditions of conjugation of the layer and inclusion, we write three equations for displacements (5). Moreover, to express Ux (p, z)

>\p=R-

it

is necessary to use the formula for the transition from

,m . Applying

solutions and U^ ) to solutions R

the stress operator to the obtained expression, we can write down three more equations for stresses (6). Eliminating from these six equations the previously found

hP(ah) and d) through Bkm(a), and

also freeing ourselves from series in m and integrals over X, we obtain an infinite system of linear algebraic

-40 -20 0 20

0,0

equations of the second kind for finding the unknowns Bkm (A) and Ak ,m (A).

A study of the determinant of this system of equations showed that its value is not equal to zero for any X.

Infinite systems of equations are solved by the reduction method.

After defining Bk ,m (A) and Ak ,m (A), we can find the values of the unknowns H^(1(A,I), H P(A,fl), Hk 2(A,i) , which we previously expressed through

B m(A). So all unknown expressions (8) - (10) will be found.

Numerical Studies of the Stressed State. The layer is plastic, Poisson's ratio a: = 0.38, elastic modulus Ei= 1700 MPa. Half space is rubber, a2 = 0.47, E2 = 8 MPa. Inclusion - steel a2 = 0.25, E2=2 105 MPa.

The radius of the cylindrical inclusion R = 10mm. The distance from the inclusion center to the upper and

lower boundary of the layer is h =h = 20 mm.

The stresses in the form

af(x, z) = —108 \z2 +102)—2-(x2 +102)—2, = 0.

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 10-3.

Figure 1 shows the stresses on the upper (Fig. 1a) and on the lower (Fig. 1b) surface of the layer along the x axis, at z = 0, in MPa.

40

-40 -20

0

20

40

-0,2 -0,4 -0,6 -0,8 -1,0

x

0,40 0,30 0,20 0,10 0,00 -0,10

a b

Fig. 1. Stresses along the x axis: a - on the upper surface of the layer;

..... N¿1 .............

M

2 \ \ № // £/

Y-. 7

2

-A-

"y1

x

b - on the lower surface of the layer; 1 - Oy, 2 -Ox, 3 -

For given voltages Oy (Fig. 1a, line 1), stresses slower than stresses Oy both along the x axis and along

, ~ , the z axis.

cx and O on the upper surface of the layer (Fig. 1a, On the lower surface of the layer (Fig. 16) , stresses lines 2, 3) at the point x = z = 0 have maximum values

ox and cr2 acquire positive values, which slowly de-

that are less than given O,,, but they decrease much , z ^ ,

y crease along the x and z axes.

Stresses < and < , in the middle of the isthmus

x z y

between the inclusion and the upper boundary of the

TECHNICAL SCIENCE / «€©LL©MyM-J©yrM&L>>ffi2§(ffl]q201]9 layer (Fig. 2a), tend to decrease, but at the boundary surfaces the stresses are maximum.

10 12 14 16

18 20

0,0 -0,2 -0,4 -0,6 -0,8 -1,0

3

... \

2 4

> \ \ •.

1 -4—' \ ^___ \

3 2 1 0 -1 -2 -3

0 л/4 л/2 3л/4 л 5л/4 6л/4 7 л/4 2

3

> 2

1 Л

»»f

•

b

Fig. 2. Stresses in MPa: a - on the isthmus between the inclusion and the upper boundary of the layer;

b - on the inclusion surface, along the angle ф; 1 -

ay, 2-

a ^ 3 - a :

On the inclusion surface, the maximum negative stresses cr2 (Fig. 2b) are in the upper part of the inclusion (ф = л / 2), positive - in the lower part (ф = 3 л / 2). Moreover, due to the difference in the characteristics of the material, the stresses cr2 on the inclusion surface are higher than those specified on the surface of the layer.

Conclusions.

A method is proposed for solving the spatial problem of the theory of elasticity for a layer coupled by smooth contact with an elastic half-space and strengthened by a longitudinal cylindrical inclusion. 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 composite compounds in mechanical engineering, with similar conditions for the formulation of the problem.

References

1. Николаев А. Г., Проценко В. С. Обобщенный метод Фурье в пространственных задачах теории упругости. - Харьков: Нац. аэрокосм. университет им. Н.Е. Жуковского «ХАИ», 2011. - 344 с.

2. Проценко В. С., Николаев А. Г. Пространственная задача Кирша. -Математические методы анализа динамических систем. - 1982. - Вып. 6. - С. 3 - 11.

3. Проценко В. С., Попова Н.А. Вторая основная краевая задача теории упругости для полупространства с круговой цилиндрической полостью. -Доповщ НАН Украши. - 2004. - № 12. - С. 52-58.

4. Николаев А. Г., Орлов Е. М. Решение первой осесимметричной термоупругой краевой задачи для трансверсально-изотропного полупространства со сфероидальной полостью. - Проблеми обчислю-вально! мехашки i мщносп конструкцш. - 2012. -Вип.20. - С. 253-259.

5. Мiрошнiков В. Ю. Перша основна задача теори пружносп в просторi з N паралельними кру-говими цилшдричними порожнинами. - Проблемы машиностроения. - 2017. - Т. 20, № 4. - С. 45-52.

6. Miroshnikov V. Yu. First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities. - Journal of Mechanical Engineering. - 2018. - Vol. 21, №. 2. - P. 12-18. https://doi.org/10.15407/pmach2018.02.012

7. Protsenko V., Miroshnikov V. Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned. - Eastern-European Journal of Enterprise Technologies. Applied mechanics. -2018. - Vol 4, № 7 (94). - P. 43 - 50. https://doi.org/10.15587/1729-4061.2018.139567

8. Miroshnikov V. YU. Evaluation of the stressstrain state of half-space with cylindrical cavities. -Вюник Дшпровського ушверситету. Серiя: Ме-хашка. - 2018. - Vol. 26, № 5. - P. 109 - 118.

9. Николаев А.Г., Танчик Е.А. Распределение напряжений в ячейке однонаправленного композиционного материала, образованного четырьмя цилиндрическими волокнами. - Вюник Одеського нацюнального ушверситету. Математика. Ме-хашка. - 2013 - Т.18 - Вип. 4(20). - С. 101-111.

10. Nikolaev A. G., Tanchik E. A. Stresses in an elastic cylinder with cylindrical cavities forming a hexagonal structure. - Journal of Applied Mechanics and Technical Physics. - №57 (6). - 2016. - P. 1141-1149. https://doi.org/10.1134/S0021894416060237

11. Nikolaev A. G., Tanchik E. A. Model of the Stress State of a Unidirectional Composite with Cylindrical Fibers Forming a Tetragonal Structure. - Mechanics of Composite Materials. - №52 (2). - 2016. -P. 177-188. https://doi.org/10.1007/s11029-016-9571-6

12. Miroshnikov V. Yu. The study of the second main problem of the theory of elasticity for a layer with a cylindrical cavity. - Strength of Materials and Theory of Structures. - №102. - 2019. - P. 77-90. https://doi.org/10.32347/2410-2547.2019.102.77-90

13. Miroshnikov V. Investigation of the Stress Strain State of the Layer with a Longitudinal Cylindrical Thick-Walled Tube and the Displacements Given at

У

a

<<Ш11ШетиМ~^®и©Ма1>#Ш44)),2Ш9 / TECHNICAL SCIENCE

the Boundaries of the Layer. - Journal of Mechanical Engineering. - 2019. - Vol. 22, N 2. - P. 4452. https://doi.org/10.15407/pmach2019.02.044

14. Miroshnikov V. Yu. Determination of the stress state of a layer with a cylindrical cavity located УДК 621.314

Kazanskiy S.

Ph.D., Associate Professor of electric networks and systems,

Mossakovskiy V. TF of electric networks and systems National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»

DOI: 10.24411/2520-6990-2019-10643 COMPARATIVE ANALYSIS OF STATE STANDARDS FOR LOADING CAPACITY OF THE OIL-

IMMERSED POWER TRANSFORMERS

on an elastic base and specified boundary conditions in the form of stresses. Colloquium-journal.- №18 (42). -2019. - P.50-55 DOI: 10.24411/2520-6990-201910610

Abstract

It was stated the necessity of increasing the operational reliability of oil-immersed power transformers in electric networks. It was analysed the state standards which provides guidance for loading capacity of power transformers. It is pointed on some differences in the criteria of determining the actual thermal mode, in particular, the winding temperature insulation and the thermal life-time.

Key words: oil-immersed power transformer, loading capacity, winding temperature insulation.

Oil-immersed power transformer is one of the most important elements of electric power transmission systems. There is a significant shortage of transforming power in electrical networks, and this impedes the connection of new consumers and hinders the networks development. Therefore, increasing the load capacity of oil-immersed power transformers is an application task. Performing this task, the requirements for safe and reliable operation must be strictly observed [1].

Table 1 presents the results of the analysis of statistical information on 536 failures and failures of oil-

Statistical analysis of damages of step-down

1996 -

immersed power transformers of different voltage classes, which occurred in the period 1996-2010 in 58 utility companies in 21 countries [2].

Table 2 presents the results of a failure analysis of 200 lower power transformers from 100 to 500 kV that occurred during 2000 and 2010 in 32 countries [3]. The results summaries indicate that problems with winding and insulation account for between 36% and 50% of the total number of causes, with thermal effects accounting for 16% of failures. Therefore, one of the most common causes of failures of oil-immersed power transformers is damage to the winding insulation due to thermal overload.

Table 1

transformers with voltage over 100 kV during 2010 [2]

Failure cause Percentage Failure cause Percentage

HV winding 19,4 Electrical screen 0,56

MV winding 5,6 HV bushings 13,99

LV winding 5,6 MV bushings 2,8

Tapping winding 3,36 LV bushings 0,37

HV lead exit 3,17 Core & magnetic circuit 2,43

MV lead exit 1,68 Flux shunts 0,37

LV lead exit 1,12 Tank 0,75

Phase to phase isolation 0,75 Cooling unit 1,12

Winding to ground isolation 1,31 Tap changer 31,16

Winding to winding isolation 0,37 Current transformer 0,37

Table 2

Statistical analysis of damages of 200 step-down transformers with voltage from 100 to 500 kV during _2000 - 2010 [3]_

Failure cause Percentage Failure cause Percentage

Tap changer 30,0 HV winding 6,5

Cooling unit 1,0 MV winding 5,0

Tank 1,0 LV winding 11,5

Core and magnetic circuit 4,0 Tapping winding 2,0

LV bushings 0,5 HV lead exit 3,5

MV bushings 5,0 MV lead exit 3,5

HV bushings 12,0 MV lead exit 2,5