АНАЛИТИЧЕСКИЙ РАСЧЕТ КОМБИНИРОВАННЫХ ФУНДАМЕНТНЫХ ПЛИТ ПРИ ДЕЙСТВИИ НА НИХ АНТИСИММЕТРИЧНЫХ НАГРУЗОК Текст научной статьи по специальности «Строительство и архитектура»

DOI:10.22337/2587-9618-2023-19-4-83-94

ANALYTICAL ANALYSIS OF COMBINED FOUNDATION PLATES, SUBJECTED TO AN ACTION OF ANTISYMMETRIC LOADS

Elena B. Koreneva Valery R. Grosman 2

1 Moscow Higher Combined-Arms Command Academy, Moscow, RUSSIA 2 Academy ofWater Transport, the branch ofRussian University ofTransport (MIIT), Moscow, RUSSIA

Abstract: The combined foundation plates, having circular form and consisting of several parts which have different laws of thickness variation, are under study. The constructions under examination are subjected to an action of antisymmetric load distributed along circumferences according to the laws cos 9 or sin 9 . Similar problems occur in the cases of seismic and wind loads. The interaction with elastic foundation is taking into account. The conditions of the construction's different parts conjugation are fulfilled. In this work for solution of such problems the analytical approach is used for the first time. The method of compensating loads (MCL) is applied. The solutions are obtained in closed form in terms of Bessel functions. As an example the foundation plate, consisting of two parts, is considered in detail. The inner part of the slab under study has the variable thickness, the outer one has the constant thickness.

Keywords: combined plates, antisymmetric loads, Bessel functions

АНАЛИТИЧЕСКИЙ РАСЧЕТ КОМБИНИРОВАННЫХ ФУНДАМЕНТНЫХ ПЛИТ ПРИ ДЕЙСТВИИ НА НИХ АНТИСИММЕТРИЧНЫХ НАГРУЗОК

Е.Б. Коренева В.Р. Гросман 2

1 Московское высшее общевойсковое орденов Жукова, Ленина и Октябрьской Революции Краснознаменное училище, г. Москва, РОССИЯ 2 Академия водного транспорта, филиал Российского университета транспорта (МИИТ), г. Москва, РОССИЯ

Аннотация: Рассматриваются комбинированные фундаментные плиты, представляющие собой конструкции, имеющие в плане круговую форму и состоящие из нескольких участков с различными законами изменения толщины. Изучается действие на подобные конструкции антисимметричных нагрузок, распределенных по окружностям по законам cos 9 или sin 9 . Подобные задачи возникают при расчетах на действие ветровых и сейсмических нагрузок. Учитывается взаимодействие плиты с упругим основанием. Выполняются условия сопряжения отдельных участков. Впервые для решения подобных задач используется аналитическая методика. Используется метод компенсирующих нагрузок. Получены решения в замкнутом виде, выраженные в функциях Бесселя. В качестве примера изучается комбинированная плита, состоящая из двух участков. Внутренняя часть этой конструкции имеет переменную толщину, внешняя часть - постоянную толщину.

Ключевые слова: комбинированные плиты,антисимме^ичные нагрузки, функции Бесселя

1. INTRODUCTION

The constructions with piecewise thickness occur among modern structures and buildings. Among them foundation slabs of circular form and bottoms of cylindrical reservoirs are to be mentioned. The combined foundation plates

subjected to an action of symmetric loads were studied in the works [1] and [2]. The inner parts of these plates had the variable thickness, the outer ones - the constant thickness. The influence of upper parts of structures and interaction with the elastic subgrade were examined. For the first time for analysis

of similar constructions in [1] and [2] analytical method was applied. The solutions were obtained in closed forms in terms of Bessel functions.

Analytical methods for plates and shells analysis, in particular, connected with the use of special functions, occur in literature. The monographies [3] and [4] can be named. The book [5] is devoted to calculation problems of orthotropic and isotropic plates of variable thickness of different forms; the plates under study are subjected to an action of complicated loads. In the present time the modern software allows to investigate various structures and building in detail. The numerical methods, in particular, the finite elements method, are widely utilized. The buckling problems of orthotropic plates with various boundary conditions were considered in [6]. In the work [7] 3D free vibration problems of cross-ply laminated plates were under study. Vibrations of rectangular orthotropic and isotropic plates of linear thickness were investigated in [8].

Numerical analysis on experimental research on buckling of closed shallow conical shells under external pressure was given in [9]. The work [10] concerns free vibration analysis of a blade of variable thickness with arbitrary boundary conditions. The optimization problems of rectangular plates subjected to ther-mo-mechanical loads we examined in [11]. Comparative assessment of finite element modelling techniques for wind turbine rotors blades was given in [12]. Nonlinear primary resonance analysis of nanoshells was stated in [13]. Vibration problems of layered plates were considered in[14]. Statics, vibration and stability problems of rectangular plates with various boundary conditions were studied in the work [15]; the approximate analytical equation decomposition method was used.

The publications [16], [17], [18], [19] concerns the computation problems of combined plates with piecewise variable thickness, the exact analytical solutions in terms of special functions were obtained.

The present work considers the combined circular foundation plate, consisting of two parts. The plate's inner part has the variable thickness, the outer one has the constant thickness. The plate is subjected to an action of wind and seismic loads, which are considered as antisymmetric ones which vary according to the laws cos 0 or sin 0. The action of the construction's upper part on the foundation slab is analyzed. The properties of the elastic subgrade are described by Winkler's model.

2. THE ANALISYS OF THE INNER PART OF THE COMBINED PLATE

The inner part of the combined plate subjected to an action of antisymmetric loads is under study. The flexural rigidity of this part when x0 < x < x1 is varying according to the law

D = D0 x4. (1)

The corresponding law of thickness variation is:

h = h0x^ . (2)

The antisymmetric distortion of the construction under study is caused by contour or lateral loads, varying according to the laws:

Q(0) = Qi cos 0,

M (0) = M1 cos 0, (3)

q(r, 0) = q(r ) cos 0,

where Q(G) and M(9) are accordingly intensities of a contour transverse force and a contour bending moment.

It is assumed that the properties of the elastic subgrade are described by Winkler's model. The differential equation, describing the antisymmetric bending of the circular plate of variable thickness, resting on an elastic Winkler's basis, is:

nv2V2 + dD L d^w , 2±a _ the solutions (6) are to be determined. Their

L>\ v w + ^ |2 3 + x ^2 propertiesaredescribedin[5].

2 , 2 For determination of the fundamental functions

-A. (tw + A wl+ d-D J + (4) the Cauchy functions Z{ (x0; x)(i = 1,2,3,4) are

2 dx x3 J dx2 I dx2 ___

x

at first to be obtained.

+ a| 1 dw_1 w 11 - (q(x) -cw)r4 For calculation the properties of Wandermond's

V x dx x2 J] ' determinant are used. As a result we get:

where D is the cylindrical rigidity, c - modulus xo [a^T/ -a a -1

r Z1(xo;x) = 7T~2 i Ha1+ 1)xo 1 x 1 ' of subgrade, x = —, r0 is the constant. 2\a3 - a1 j "

ro

Inourcasethecylindricalrigidityvariesaccord- + (tt1 -1)x0'1 x a1 1 J-—[(a3 + 1)x0a3 x ing to the law (1). Then the equation (4) comes a3

tothefollowingform: _ ^3 x-03 -1 J.

a1 2

,•"«3 v«3 -1 .

d4w 10 d3w (17 + 4a) d2w Z (x . x) _ , x02 . i(a32 ~ 3a1)x-«1 x«1 -1 --1----------¿2\'x0>*j~ 2 T\ 1 X0 x

dx4 x dx3 x2 dx2 A«2 -«J1

3(3 - 4g) dw 3(3 - 4g) b4 qr4 (5) (a2 ~ 3«1) -«1 -«1-1 (o^, -----1----w H--j w —-, x '

x3 dx x4 x4 D0 a1 a3

4 y x "a3 xa1_1 i V 1 3a3/ x -«3 x-a3-1 I.

cr 0 0 '1

Homogeneous differential equation, correspond- Z3 (x0; x) - / —^ i -——v" ' x_tt1 _1 +

where P4 = -j^. a3 J (7)

x0 {_ x0a1 (6 + «1 )x-a1 -1

ingto(5),istheequationoftheEulerclass[20]. ^^ - )[ «1

By means of substitutions z = ez and w = ue~2 x~Qai(6-a1) x/3 (6 + a3)

the equation (5) can be reduced to the equation + " x + " x

with constant coefficients. 1 3

As a result we get the following expression for x^™3 (6 -a3) _a3_11 the deflections of the plate:

«3

w - x 1

^xa1 + A2xa2 + A3xa3 + A4xa4 ]cos 9, (6) Z( (xo; x) = x04 Up^ _1

■ +

a1

x «1 x-a1 -1 x-a1 xa3 -1 x 03 x-03 -1 Jv 0 0 0

where a1 2 3 4 -±y 2(2-ct)± V2a2 -p4 . It should be marked that aj = -a2, a3 = -a4. a1 a3 a3 J

In the case when the value of P4 is sufficiently As it was mentioned above for the case of suffi-

large, the roots of the characteristic equation are. , , 04 , ,,

. ,, , . ciently large p the roots a1234 are mutually

mutually complex conjugate. y ° 1,2,3,4 y

First the interaction of the inner part of the complex conjugate. Then introducing the nota-

foundation slab and the upper part of the con- tions

struction under study will be considered. For

this aim the fundamental functions wi(xx) for a12 =£±Si, a34 = -s±Si

we write the general solution of the homogeneous differential equation, corresponding to (5), in the following form:

(8)

w(x) - xE [B1 sin(8 In x) + B2cos(8 In x)] + + x_E [B3 sin(8 In x) + B4cos(8 In x)].

Introducing the notation

sin(8 In x) = a(x), cos(ôInx) = y(x).

We cite below one of Cauchy functions as an example:

*0 / -E £ Z4 (x0; x) = 4s§(s2° +§2 j (X0EXE X

X I- [Y( X° )5 + ea( x° )]y(x) + [y( x° )e- 5a( x°)] x (9) x a( x)} - x° x"E {[- y( x°)5 + ea( x°)]y (x) -- [ey( x°) + 5a( x°)]a( x)}}.

Further, using (7) and taking into account the properties of fundamental functions, we get the following expressions:

w1 (x°; x) = Z1 (x°; x) + Z 3 (x0; x) -

x°

3 Z 4 (x°; x);

X,

0

G

w2 (xo ; x) = z2 (xo ; x)--z3 (xo ; x) +

Xn

+

(Q + 2)

Xo

Z 4 (x0; x);

^3(xo;x)=

i

+-

D( Xo)

l_dD( xo)

D( xo) dx

W4(Xo;x)=■

i

(io)

It is assumed that the influence of the construction's upper part is transmitted on the foundation slab inner part as moments MA cos 9 and forces Qa cos 0, distributed along circumference with the radius xA, where xq ^ x^ ^ xi. As a

result we can write the following expression for the deflection:

Wnner = [W0 W1(x0 ix) + r0 w2 (x0 Îx) "

- M0r02 w3 (x0 ;x) - Q0r0W4 (x0 ;x) +

+ MArQ2 W3 (xA ; x) + QArQ W4 (xA ; x)]cos 9 -

= [wi (xo ;x)+MAr0 w3 ( xa ;x)+ + QArl W4 (xa ; x)]cos 9 - [wi (xo ; x) + + wII( xA ; x)]cos 9,

where w0, , M0, Q0 are accordingly the deflection, the angle, the moment, the force when x — xq •

When for computation we assume that the influence of the upper part is transmitted on the foundation inner part as the load q(x) cos G,

distributed over surface of the ring xa ^ x < xB . The result can be obtained by integrating in (11) the term containing QA.

3. THE COMPUTATION OF THE OUTER PART OF THE PLATE. THE BASIC SOLUTION

We go to the consideration of the outer part of the ring plate, subjected to an action of antisymmetric load. The thickness of the foundation slab part under study is constant. The general solution of the differential equation describing the antisymmetric bending of a circular plate resting on an elastic Winkler's subgrade is:

D( Xo)

Z 4 (xo; x)-

w = (AlUi ( x) + A2V1 ( x) + 4/1 (x) + + A4 g1 (x)) cos 9,

(12)

where u1, v1, f1, g1 are Bessel functions [4],

[21]; ^^^ A3 , A4

D

coefficients; x =

l

l = 4

4

Method of compensating loads (MCL) [1], [2], [4] is used for the receiving of the solution. The basic and the compensating solutions are to be determined.

First the basic solution will be obtained. When the circular plate is loaded by a concentrated force applied at the centre, the solution has the following form:

w-■

Pl2 4 D

fo( x).

(13)

qal

w =-x

4D

: J f0 i^a2 + x2 - lax cos(0 - 9) ^jcos QdQ.

(14)

Here the symbol ' denotes that when n = 0 the 1

coefficient — should be introduced. The expression (15) is valid when a< x. When a> x in the right part of the expression (15) the terms a and x must be replaced. Assuming

Z0 = H((1) (ViV«2 + x2 - 2axcos(0 - 9)

we get the following expressions fulfilling the integration and separating the real and the imaginary parts: when x <a

(0)

W = Wj =

naql3 2D

k (x)/1 (a)- V1 (x )g!(a)]x (16)

The solution (13) mentioned above is the fundamental influence function when P = 1. Integrating this solution, we can obtain the basic solutions for consideration of several problems. Let us assume that this part of the plate is subjected to the action of the load qcos 9 , distributed along the circumference with the reduced radius a.

Further, the principle of addition of the effects will be used.

For the determination of the deflection of the point with the coordinates x and 9 the following expression will be used:

x cos 6,

when x > a

(0)

W = Wjj =

naql 2D

u (a)/1 (x)- V1 (a)g! (x)]x (17) x cos 9.

Using the Wronskian of the Bessel equation it can be shown that the received solutions satisfy to the conjugation conditions when x = a . The problem of infinite plate subjected to an action of the load x) cos 9, where x) is the given function, is considered for the obtaining of the basic solution.

When the construction under study is subjected to an action of the load q(^) distributed along the circumference with the reduced radius a, we will present the load in the following form:

For this integral calculation the formula of the cylindrical functions addition is used:

Z0 ^a2 + x2 - 2axcos(9~9) )= 1

= 2 E'Jn (a)Zrn( x) cos n(0 - 9)-

n=0

(15)

q = a1 cos 9 + b1 sin 9

and using the formulae (16) and (17) we get the

solutions:

when a< x

w = X 2 D

x [a1 \u1 (a) f (x) - v1 (ag (x)]sin 9 + + bi k (a)fi (x) - vi (a)g! (x)\cos 9},

(18)

when a > x

w = -

%al 2 D

q = x(a1 cos 6 + b1 sin 6)

or

are:

4~ [Z"Jn (z)] = z"Jn-i(.zX

az

0- [znH®( z)]= znH(Hi( z). az

J xu0 (x)Jx = ~x= \u1 (x) + v1 (x)],

i xv0 (x)Jx = —-j= \u1 (x) - v1 (x)] J V2

j xf0(x)dx = -j= [fi(x) + gl(x)], j xgo(x)dx = L/i(x) - gi(x)l

(24)

x [a1 [u1 (x) fi (a) - v1 (x)g1 (a)]sin 9 + (19) + ¿1 [u1 {x)f1 (a)" V1 (x )g1 (a)]cos 9}-

For the basic solution receiving for the load q - F(x, 0) the formulae (18) and (19) should

be used. The load can be represented in the form:

(20)

q = x1 (a1 cos 0 + b1 sin 0). (21)

The integrating for these cases is highly simple. The formulae for Bessel functions derivation

(22)

Let us assume z = x*Ji. Fulfilling integration and separating real and imaginary parts, we get the following expression:

(23)

The solution for the case of the load varying according to the law (20) and distributed over a surface of the ring a1 < x < a 2 is to be received. For this aim the expressions (16) and (17) are to be integrated. In the mentioned expressions the load q must be replaced by the

elementary load q0adz - q0ada. It should be

14

marked that the multiplier —, which is equal to 1

—, appears.

c

Further we integrate the received solutions within the limits a = a1 and a = a 2 . When x < a1 <a 2 the solution has the following form:

w = 2c~T2 ^2 ^2(a2 ) + g2(a2)]--«i2 l^K) + g 2 (a1)]}u1 (x) + (25) + {a2I/2O2)-g2(a2)]_a3 x X [/2M - g 2 (a1)])V2 (x))cOS 9;

when x > a 2 > a1

w = 2^2 ^2 [u2(a2) + v2(a2)] " -a2 [u2^1) + MuMMx) + (26)

+ {a2 [u2 (a2) - v2 (a2)] -a2 x x [u2 ) - v2 ) ]}g1(x))cos

when a1 < x <a 2

xux{x) + a2 [/2(a2) - g2(a2>]x x Vl (x) - a2 [112 (a1) + V2 (a1)^,/1 (x) -

V2I

-a? [m2 (a1) - v2 (a1)]g1 ( x)- 2 x — Icos 9.

n '

(27)

dx form:

from the basic solution in the following

w0 = A1 sin 9 + B1 cos

dw0 dx

= C1 sinq + D1 cosq>.

These formulae give the expressions for the deflections of the foundation plate.

4. THE OUTER PART OF THE PLATE. THE COMPENSATING SOLUTION

The compensating solution should be determined for the plate's outer part investigation. This solution satisfies to boundary conditions and with the basic solutions satisfies to the resolving differential equation of the problem. For the foundation slab under study, which is subjected to an action of antisymmetric loads, the case of free edges is more actual. Below, however, another cases of boundary conditions will be considered.

The sought solution can be represented as the result of an action of two compensating loads q1 and q2, which are applied along the concentric circumferences with the reduced radius a^ and a 2. Let us note the reduced radius of the outer contour as P .

a) First it is assumed that the outer boundary of the plate, subjected to an action of antisymmetric load, is clamped. For example it is possible for the plate when x = p connected with the upper part of the construction which is represented by the shell of the rotation and subjected to an action of seismic or wind loads. Let us present the deflection w0 and the angle

dwr.

The compensating loads q1 and q2, which with the sum of the basic solution are satisfying to the case of the clamped boundary, are to be determined.

As a result we get the following system of equations:

«1 Jqi(9)/0(Va2 +P2 -2a^cos(0-9))/0 +

0

+ a2jq2(0)/0(Va2 +p2 -2a2p005(8-9)+ (30)

4 D

+—wo = 0,

a1 f/0 (A2 +P2 - 2a1p cos(9-9) jd/9 + o

+ ^2(0)^/0(Va2 + ß2 -2a2pcos(0-9)]dQ+ (31)

4D w

l2 dx

= 0.

The given above integral equations (30) and (31) express the conditions of equality of the values of deflections and angles to zero at the contour.

Further using the formulae of Bessel functions addition the integration is fulfilled [21]. The functions q1 and q2 are to be expanded into trigonometrical series. As a result we get a system of algebraic equations with respect to the coefficients of these series. Here this system is not cited.

The compensating solution can be written in the following form when x < a :

wk - (a1 sin 9 + b1 cos ^>)u1 (x) + + (c1 sin 9 + d1 cos ^)v1 ( x).

(32)

(28) The coefficients of the expression (32) can be determined from the following equations, using (30) and (31):

WP) + CiVi(P) + Ai = 0, | aiUi' (P) + CiVi' (P) + Cil = 0;! bA(P) + ^iVi(P) + Bi = 0, | b1u1' (p) + d1vl' (p) + D1l = 0.J

Solving these systems, we obtain: Cilvi(P) - AiVi' (P)

(33)

(34)

a = ■

h =~

C =

di =

Vi(PK (P) - «i(P)Vj' (P)

D1/v1(P) - B^' (P) v,(ß)«i' (P) - «i(P)Vi' (P)

Ci/ui (P) - M1' (P) Vi(P)V (P) - Mi(p)vi' (P)'

Di/ui(P) - Biui' (P) Vi(P)ui' (P) - Ui(P)Vi' (P) ■

wlr =-

i

ufM )(ß) = -Ui(p)-VÎM)(ß) = Vj(ß) -

ß

i-a

i V (p) ~ Vi(P)

v (ß)-i «i(p>

p

fi(M }(ß) = - /iO) -gi(M }(ß)=gi(ß) -

i-a

ß i-a

P

gi' (ß)gi(P) fi' (P)fi(P)

' (43) ■ (44)

The following compensating solution is obtained:

i

(35) wk = u

(36) x №M'

(37) + [Li iß Vi

(38) \im i ■

«i(P)«iM )(ß) " Vi(ß)Vi(M )(ß)

[[Mi ß Vi(P) + A(u( m )(P)] sin 9 +

(45)

We receive introducing (35)-(38) into expression (32):

* V1(P)u/(P)-U1(PK(P) x ({lA V (P) - A1V1' (P)]sin 9 + + [D1/V1 (P) - B1V1' (P)]cos 9}^ (x) - (39)

- {[C1/u1 (P) - A1u1' (P)] sin 9 + + [D1/u1 (P) - B1u1' (P)] cos 9}v1 (x)).

b) The case when the outer boundary is simply supported is under study. The deflections are represented in the form (28). The radial bending moments on the contour are represented in the form:

Mr - M1 sin 9 + L1 cos 9. (40) The following notations are introduced:

"{[Mi ß Ui(P) + A(v(m )(P)] sin 9 + + [L( ßUi(P) + B(v(m)(P)]cos9Wx) 1.

c) The case of free contour of the fundament plate is under consideration. It is known that the bending moment and the reduced transverse force are equal to zero at the contour. The boundary bending moment is defined by the expression (40). The boundary reduced transverse force will be written in the form:

Q - = N1 sin 9 + 01 cos 9. (46)

UU

The following notation, taking into account (40), and (46) are introduced:

up](ß) = -Ui' (P) - (i -a) ^

Vi'(ß) -

Vi(ß)

P

Viß](ß) = Vi' (P) - (i-a)^ ß2

ui' (P) -

«i(P)'

(41)

(42)

fi[ß](ß) = - fi' (P) " (i-a) ^

giö](ß) = gi' (P) - (i

ß2

gi' (P) " fi' (P)

P J

gi(ß) P J

fi(P) "

P _

(47) ' (48)

(49)

(50)

As a result we get the compensating load:

/2_i_

^ =" £^(P)vi - ](P) - u|M ](P>F ](P) X X {([/^1u|M ](P) - M1u\Q](^)]sin 9 + + [lOy u|M] (p) - V|e] (P)] cos 9)U1 (x) - (51)

- ([/N 1 u|M ] (p) - M1v|e] (p)]siw 9 + + [/O1v|M ](P) - Z1v|e](P)]cos 9>1( x)\

The solution for the outer part, according to the MCL, is represented by the sum of the basic and the compensating solutions [22], [23]:

w - w0 + wk. (52)

The solution of the problem under study can be presented as the sum of the expressions (11) and (52) for the study of bend of the whole combined plate.

5. THE CONCLUSIONS

The work receives the exact solution of the antisymmetric deformation of the combined plate which inner part has variable thickness and the outer one - the constant thickness. The solution is obtained in closed form in terms of Bessel functions. Method of compensating loads (MCL) is used. An influence of the construction's upper part is taken into account. The received results can be used for the analysis of the combined plates subjected to an action of seismic and wind loads.

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11. Khakpour Komarsofla M. et al. Optimization of Three-Dimensional up to Yield Bending Behaviour Using the Full Layer-Wise Theory for FGM Rectangular Plate Subjected to Thermo-Mechanical Loads. // Compos. Struct. https://doi.org/10.1016/j.compstruct.2020.11 3172.

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13. Sarafraz A., Sahmani S. and Aghdam M.M. Nonlinear Primary Resonance Analysis of Nanoshells Including Vibrational Mode Interaction Based on the Surface Elasticity Theory. // Applied Mathematics and Mechanics (English Edition). 2020. https://doi.org/10.1007/10483-020-2564-5.

14. Saira Javed, F.H.H. Al Mukahal and S.B.A. El Sayed. Geometrical Influence on the Vibration of Layered Plates. Hindawi. // Shock and Vibration. V. 2021. Article ID 8843358. Pp. 1-17. https://doi.org/10.1155/2021/8843358.

15. Koreneva E.B., Grosman V.R. Equation Decomposition Method for Solving of Problems of Statics, Vibration and Stability of Thin-Walled Constructions. // International Journal for Computational Civil and Structural Engineering. 2020. 16(2). Pp. 6370.

16. Koreneva E.B., Grosman V.R. Analitich-eskij Raschet Kombinirovannyh Kon-struktsij |Analytical Computation of Combined Constructions]. // Stroitelnaya Me-hanika i Raschet Sooruzhenij. 2020. 2. Pp. 28-32 (in Russian).

17. Koreneva E.B., Grosman V.R. The Problems of Computation of Combined Plates with Piece-Wise Variable Thickness. Solution in Orthogonal Polynomials. // International Journal for Computational Civil and Structural Engineering. 2020. 16(2). Pp. 3034.

18. Koreneva E.B. Analysis of Combined Disks with Piecewise Thickness. // International Journal for Computational Civil and Structural Engineering. 2021. 17(2). Pp. 1221.

19. Koreneva E.B. Izgib Kombinirovannyh Plastin Kysochno-Peremennoj Tolschiny [The Flexure of Combined Plates with Piece-Wise Thickness]. // Stroitelnaya Me-hanika i Raschet Sooruzhenij. 2021. 2. Pp. 10-18 (in Russian).

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23. Koreneva E.B. Metod Kompensirujuschih Nagruzok dlya Reshenija Zadach o Tsikli-cheski-Simmetrichnom Izgibe Anizotropn-yh Plastin, Kontaktirujuschih s Uprugim Osnovanijem [The Method of Compensating Loads for Solving of Problems of Cyclic Symmetrical Flexure of Anisotropic Plates, Resting on an Elastic Subgrade]. // Stroitelnaya Mehanika Inzhenernyh Kon-struktsij i Sooruzhenij. 2021. 2. Pp. 99-111 (in Russian).

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analysis of cross-ply laminated plates with one pair of opposite edges simply supported. // Composite Structures. - 2005. - 69. -Pp. 77-87.

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tropic and orthotopic rectangular plates with linearly varying thickness by discrete singular convolution method. // Applied Mathematical Modelling. - 2009. - 33. - Pp. 3825-3835.

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10. Li C., Cheng H. Free vibration analysis of a rotating varying-thickness-twisted blade with arbitrary boundary conditions. // Journal of Sound and Vibration. -https://doi.org/10.1016/]'. - jsv. - 2020. -11579/

11. Khakpour Komarsofla M. et al. Optimization of three-dimensional up to yield bending behaviour using the full layer-wise theory for FGM rectangular plate subjected to thermo-mechanical loads. // Compos. Struct. - 2020. -https://doi.org/10.1016/j.compstruct. -2020. - 113172.

12. VanSkike W.P., Hale R.D. Comparative assessment of finite element modelling techniques for wind turbine rotors blades. // American Institute of Aeronautics and Astronautics. - Downloaded by University of Texas at Austin on January 8. - 2020 // http://arc.aiaa.org. - Pp. 1-18.

13. Sarafraz A., Sahmani S. and Aghdam M.M. Nonlinear primary resonance analysis of nanoshells including vibrational mode interaction based on the surface elasticity theory. // Applied Mathematics and Mechanics (English Edition). - 2020. -https://doi.org/10.1007/10483-020-2564-5.

14. Saira Javed, F.H.H. Al Mukahal and S.B.A. El Sayed. Geometrical influence on the vibration of layered plates. // Hindawi. -Shock and Vibration. - V. 2021. - Article ID - 8843358. - Pp. 1-17. https://doi.org/10.1155/2021/8843358.

15. Koreneva E.B., Grosman V.R. Equation decomposition method for solving of problems of statics, vibration and stability of thin-walled constructions. // International Journal for Computational Civil and Structural Engineering. - 2020. - Vol. 16. - Issue 2. - Pp. 63-70.

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17. Koreneva E.B., Grosman V.R. The problems of computation of combined plates with piece-wise variable thickness. Solution in orthogonal polynomials. // International Journal for Computational Civil and Structural Engineering. - 2020. - Vol. 16. - Issue 2. - Pp. 30-34.

18. Koreneva E.B. Analysis of combined disks with piecewise thickness. // International Journal for Computational Civil and Structural Engineering. - 2021. - Vol. 17. - Issue 2. - Pp. 12-21.

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Решения в функциях Лежандра. // Строительная механика и расчет сооружений. - 2021. -№ 2.-С. 10-18.

Elena B. Koreneva, Professor, Dr.Sc., Moscow Higher Combined-Arms Command Academy, 2, Golovacheva str., Moscow, 109380, Russia. Phone: +7(499)175-82-45, E-mail: elena.koreneva2010@yandex.ru

Valery R. Grosman, Engineer, Academy of Water Transport, 2, Novodanilovskaya nab., corp.1, Moscow, 117105, Russia. Phone: +7(499)618-52-56, E-mail: elena.koreneva2010@yandex.ru

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23. Коренева Е.Б. Метод компенсирующих

нагрузок для решения задач о цикличе--

пластин, контактирующих с упругим основанием. // Строительная механика инженерных конструкций и сооружений. -2021. -№ 2.-С. 99-111.

Коренева Елена Борисовна,профессор, доктор технических наук, Московское высшее общевойсковое орденов Жукова, Ленина и Октябрьской Революции Краснознаменное училище, 109380, Россия, г. Москва, ул. Головачева, д.2, тел.: +7(499)175-82-45, E-mail: elena.koreneva2010@yandex.ru.

Гросман Валерий Романович, инженер, Академия водного транспорта,117105, Россия, г. Москва, Новоданиловская наб., д.2, корп.1, тел.: +7(499)618-52-56, E-mail: elena.koreneva2010@yandex.ru.