THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS Текст научной статьи по специальности «Математика»

PHYSICS AND MATHEMATICS

THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS

Candidate of Physical and Mathematical Sciences, Zelensky A. G.,

Ukraine, Dnipro, Associate Professor of the Department of Building Mechanics and Materials Resistance of the State Higher Educational Institution "Pridneprovsk State Academy of Civil Engineering and Architecture "

DOI: https://doi.org/10.31435/rsglobal_ws/30112019/6764

ARTICLE INFO

Received: 20 September 2019 Accepted: 11 November 2019 Published: 30 November 2019

KEYWORDS

mathematical theory of transversal-isotropic shallow shells of arbitrary thickness, Legendre poly-nomials, method of successive approximations.

ABSTRACT

The method of sequential approximations (MSA) in mathematical theory (MT) of transversal-isotropic shallow shells of arbitrary thickness is developed. MT takes into account all components of stress-strain state (SSS). SSS and boundary conditions are considered to be functions of three varia-bles. Three-dimensional problems are reduced to two-dimensional decompositions of all the compo-nents of the SSS into series in the transverse coordinate using Legendre polynomials and using the Reisner variational principle. The boundary conditions for stresses on the front surfaces of the shell are fulfilled precisely. Previous studies have shown the high efficiency of this MT. The boundary-value problem for a shallow shell is reduced to sequences of two boundary-value problems for the respective plates. One sequence describes symmetric deformation relative to the median plane, and the other sequence is skew symmetric. MSA makes it easier to find a common solution of differential equations (DE) for shallow shells. Highly accurate results for SSS are already in the first approxi-mation. MSA can be used when solving problems for shallow shells by other theories.

Citation: Zelensky A. G. (2019) The Method of Successive Approximations in the Mathematical The-Ory of Shallow Shells of Arbitrary Thickness. World Science. 11(51), Vol.1. doi: 10.31435/rsglobal_ws/30112019/6764

Copyright: © 2019 Zelensky A. G. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

1. Introduction.

Problem solving for shells and plates is performed on the basis of classical and refining theories, using equations of three-dimensional elastic theory and on the basis of variants of mathematical theory. Classical and clarifying theories are based on various physico-geometric assumptions [1-3, 8, 12, 15, 17, 20, 22, 23]. The limits of using these theories for different classes of problems require further research. The most common in practical use are the theories of the Tymoshenko-Reisner type [20, 22, 23] and their various modifications [3, 8, 12, 17]. Clarifying theories include theories based on specific deformation models [11]. The main drawback of all the clarifying theories is the inability to increase the accuracy of the solution of the problems within these theories.

The use of three-dimensional elasticity theory in the analytical solution of boundary value problems for plates and shells [6, 14] is too much of a problem for mathematical physics, since all the components of the SSS and boundary conditions are functions of three coordinates. At the same time, three-dimensional SSS occurs in thick plates and shells, in the field of local, discontinuous and non-smooth loads, under the action of other SSS concentrators. And so there is an urgent need to develop

and construct theories that take into account all the components of the SSS and boundary effects as a functions of the three variables. And so that these theories can be used to analytically solve boundary value problems. with the required accuracy. These qualities are satisfied by the variants of MT, which are based on a mathematical approach in the image of the components of the SSS with infinite rows in transverse coordinates. These theories are devoid of physico-geometric assumptions. Different mathematical series are used: tensor [9], power [13], using the Lezhran-dra polynomials [4, 5, 7, 10, 16, 18, 19, 24]. Three-dimensional problems are reduced to two-dimensional by different methods: operating [4, 5, 24], variational [7, 10, 16, 18, 19], others [15]. The MT variants have different accuracy depending on the approach of reducing three-dimensional problems to two-dimensional ones and the method of representing the SSS in the form of mathematical series.

In this article, MSA is developed in solving boundary value problems for transversely isotropic shallow shells of arbitrary thickness based on the MT variant [25-28]. Shells can be subjected to arbitrary transverse loads. All SSS components that are functions of three variables are taken into account. The MT is based on the representation of the SSS components in the form of infinite rows with a transverse coordinate using Legendre polynomials. The transverse normal and tangent stresses are approximated by taking into account the three-dimensional DE equilibrium theory of elasticity such that the boundary conditions in the stresses on the face surfaces are satisfied exactly. Three-dimensional problems for shells are reduced to two-dimensional problems based on the Reisner variational principle [21]. This method of constructing the MT variant showed efficiency and high accuracy [25, 26]. As the number of additives in the mathematical series increases, the order of the systems of equations and the complexity of solving them increases, but the accuracy of the solution increases. The MSA makes it possible to reduce the complex boundary-value problem for the shell to simpler boundary-value problems for the corresponding plates with symmetric and oblique deformation relative to the median plane.

2. Problem statement.

We study the transversal isotropic shallow shell of constant arbitrary thickness h h in a rectangular coordinate system x,y,z . The surface of the isotropy coincides with the median surface. The axes x, y belong to the plan of the shell, and the axis z is perpendicular to the plane of the plan of the shell and is directed in the direction of the convexity (up) (- h /2 < z < h /2). On the upper and

lower surfaces of the shell there is a static transverse load qx(x,y) and q2(x,y) directed downwards. All SSS components are functions of three coordinates. Boundary conditions on the front surfaces:

^z(z = h/2) = -qj(x,y) ; ^(z = -h/2) = q2(x,y) ; ^(z = ±h/2) = ^(z = ±h/2) = 0 (1)

The transverse loads on the upper and lower surfaces are depicted as the sum of two additions: oblique symmetric q /2 and symmetric p /2 loads relative to the middle surface:

°z(z = ±h / 2) = (+q(x> y) - p( x y))/2, p(x y) = qi(x y) - q2(x y), q( x y) = qi(x> y) + q2(x y).

The boundary conditions on the side surface may be different.

The displacement components are represented by the Fourier-Legendre series in the coordinate z :

U(x,y,z) = £Pk(2z/h)uk(x,y), (U,V;uk,vk) ; W(x,y,z) = £Pk-i(2z/h)wk(x,y), (2) k=0 k=1 where Pk(2z/ h) is Legendre polynomials; uk,vk, wk - sought components in displacements.

If in (2) in tangential displacements we take into account terms with indices k = 0,1,2,...,n, then we call this approximation K0-n. If we take into account additives with k = 0,1,2,3 indexes, this is an approximation of K0123 or K0-3.

Since the shell is of arbitrary thickness, tangential displacements are taken into account in the shear deformations of yyz, yyz [1] (in the theory of thin shells they are neglected):

£x =dU / dx + k1W ; =dV / dy + k2W ; ez =dW / &; yy =dU / dy + dV / cx; /xz =dW / dx + 5U / dz - k[U, (x, y; U ^ V; k[ ^ k'2), (kt = 1/R; k't = kt, i = 1,2), where R1, R2 is the principal radii of curvature of the middle surface of the shell. Clarifying additives in the expressions for the transverse angular deformations contain k{, k'2.

Here are general structural formulas for stress components [25], which derive from the DE system of the spatial theory of elasticity and the Reisner variational equation:

°xz(x> z) = Z Pi fxi; ^y z(x z) = Z Pi *yi; & z(x> y>z) = Z Pisz,;

i=0 i=0 i=0 ^

(x y>z) = Z Pisxi ,

x ^ ; sxi ^ syi); &yx

(x y,z) = Z Pity xi, i=0 i=0

where txi ...,tyxi -functions that depend on the displacements of uk(x,y),vk(x,y), wk(x,y) and

mechanical-geometric parameters (MGP).

3. Displacements, stresses and boundary conditions in the K0-n approximation

3.1. Components of displacements and stresses in the shell. The displacement components are determined according to (2):

n n

U(x,y,z) = ZPi(2z/h)Uk(x,y), (U,V;Uk,Vk); W(x,y,z) = £Pk-X(2z/h)wk(x,y), (4) k=0 k=1 The stress components according to (3):

n+1 n+1 n+2

&xz(xy, z) = Z p, lxi; °yz (x y, z) = Z Pityi; &z (x y, z) = Z p,sZ, ;

i=0 i=0 i=0 n+2 n+2 n (5)

^x(xy,z) = ZPisxi; Vy(xy,z) = ZPisyi; &xy(xy,z) = ZPityxi■ i=0 i=0 i=0

The transverse normal and tangent stresses satisfy exactly the conditions (1).

For the approximations K0-3 and K0-5, the functions are given in [26].

3.2. Boundary conditions. The boundary conditions are obtained from the Reisner variation equation:

n h

J {Z TTT-¡7((sxj lx +tyxj ly - xsj )$UJ + (tyxj lx + syjly - y*J ">Svj ) + (s) /=0 (2J + 1)

(S) J , (6)

n-1 h

+ Z (txj lx + tyj ly - zSj )S wj+i)}ds = 0

j=o (2 J +1) j y y J J

In (6) lx, ly - is the cosines of the angles between the normal vector to the lateral surface and the coordinate axes; S - contour of the shell; xsj(x,y),ysj(x,y), zsj(x,y) - members in mathematical series

of the image of the external loading Xv (x, y, z), Yv (x, y, z), Zv (x, y, z) by Legendre polynomials:

h/2

xj (x, y) = (2j +1)( J Xv (x, y, z)Pj (2 z / h)dz)/h,(xj ^ ySj; Xv ^ Yr; j = 0,1,.., n);

-h/2 h, (7)

z s j (x, y) = (2j +1)( J Xv (x, y, z)Pj (2 z / h)dz)/h,(j = 0,1,..., n -1),

-h/2

where Zn must balance the transverse load on the upper and lower surfaces of the shell.

Equations (6) and (7) yield different boundary conditions. Here are some of them.

1) Boundary conditions in displacements. Only the displacement components Up (x, y, z), Vp (x,y, z), Wp (x,y, z) are known on the side surface T of the shell. Boundary conditions:

Uj(x,y) = Ujp(x,y); Vj(x,y) = Vjr(x,y), (j = 0,1,...,n); wj(x,y) = wjp(x,y),(j = 1,...,n); x,y e S,

where

Ujp (x,y) = j1 \Ur (x,y,z)Pj (2z/h)dz, (Ujp ^ v^ ,Up ^ Vp),(j = 0X„., n));

z 2-1 (8)

Wjp(x,y) = j- JWp(x,y,z)Pj-1 (2z/h)dz, (j = 1,...,n).

z

2) Boundary conditions in stresses. Only the external load Xv (x, y, z), Yv (x, y, z), Zv (x, y, z) is specified on the side surface. Then we have the following boundary conditions:

oo

oo

oo

sxj (x> y) lx + tyxj (x> y) ly = xsj (x> y); tyx j (x> y) lx + syj (x> y) ly = y.sj (x> y)>

(j = 0,1,..., n); t xj (X, y) lx + tyj (X, y) ly = zsj(x, y), 0' = 0,1,..., « -1); x, y g

3) The boundary conditions for the freely fixed at the edges of the shells:

Vj(x = 0,y) = Vj(x = a,y) = 0, (j = 0,1,...,«); Wj(x = 0,y) = Wj(x = a,y) = 0, (j = 1,...,n);

sXj (X = 0,y) = sXJ (x = a,y) = 0, ( j = 0,1,..., n); j J (10) Uj (x,y = 0) = Uj (x,y = b) = 0, (J = 0,1,..., n); Wj (x,y = 0) = Wj (x,y = b) = 0, (J = 1,..., n);

Syj ( x, y = 0) = Syj ( x, y = b) = 0, (j = 0,1,...,«).

4) Boundary conditions for rigidly secured shells:

Uj ( x = 0, y) = Uj ( x = a, y) = 0, Vj ( x, y = 0) = Vj ( x, y = b) = 0, ( j = 0,1,..., n); j j J J (11)

Wj (x = 0,y) = Wj (x = a,y) = Wj (x,y = 0) = Wj (x,y = b) = 0, (j = 1,..., n).

In the approximations K01, K0-3, K0-5, to obtain displacements, stresses and boundary conditions, it is necessary to put n = 1; n = 3; n = 5 in (4) - (11), respectively. 4. The method of successive approximations

4.1. The K0-3 approximation. The system of equilibrium DE has the 22nd order:

Dj,1U0 + Dj,2V0 + Dj,3U1 + Dj,4V1 + D j,5U2 + Dj,6V2 + Dj,7U3 + + Dj 8V3 + Dj,9W1 + Dj,10 W2 + Dj,11W3 = Djpq y\ (j = 1 2 11),

where

d2 d2 d2

D1,1 =nn—J + ru2—r + Vl*0> D1,2 = 7121 T-Z- , D1,3 = k1l1x1 , D1,4 = 0 :

dx2 dy2 dxdy

d2 d2 ' d d

D1,5 = 7131T"2 + k1l1x2> D1,6 = 7131^^ D1,7 = k1l1x3 , D1,8 = 0 , D1,9 = k1w1^~, D1,10 = 7151 ,

dx2 dxdy dx dx

d dp d2 d2 ' -D1,11 = k1w3 — , D1 pq =7u0W- , D2,2 =7112—J + 7111 + k2l1y0 , D2,3 = 0 , D2,4 = k2l1y1 , dx dx dx dy

d2 d2 ' d

D2,5 = 7131 ^ - , D2,6 = 7131 + k2l1y2 , D2,7 = 0 , D2,8 = k2l1y3 , D2,9 = k2w1T~ , dxdy dy2 dy

d d dp d2 d2 ' D2,10 =7151T- , D2,11 = k2w3^T~, D2pq = 7u0; D3,3 + ß112 + ^113 + k1l2x1/5

d^ dy dy dx dy

d2 d2 d2 D3,4 = ß121 T-Z- , D3,5 = k1 k3u2 , D3,6 = 0 , D3,7 = ß131 + ß133 + k1l2x3 / 5 , D3,8 = ß1

^ ~ , D3,5 k1k3u2 , D3,6 0 , D3,7 = ß131 _ 9 1 ß133 1 "ll2x3 ' 5 , D3,8 ß131 ~ ~ ,

dxdy dx2 dxdy

d d d dq

D3,9 = ß151^T , D3,10 = k3w2^T , D3,11 = ß161 , D3pq = ßu1^~ ,

dx dx dx dx

d2 d2 ' d2

D4,4 =ß11^T-T + + ß113 + k2l2y1/5 , D4,5 = 0 , D4,6 = k2k3u2 , D4,7 = ß131 , (13)

' 111 ^s 2 ' 113 2 2y 1 ' 4,5 ' 4,6 2 3M 4,/ / 131

dx 2 dy 2 dxdy

d2 ' d d d dq

D4,8 =ß13^~2 + ß133 + k22l2y3/5 , D4,9 = ß151 ^ , D4,10 = k4w2T^ , D4,11 = ß161 ^ , D4 pq = ßu1^ ;

d2 d2 d2 D5,5 = 7331 —T + 7332 —T + 7333 + (k1)2k5u2, D5,6 = 734^1^" , D5,7 = k1 k5u3 , D5,8 = 0 ,

, dx2 dy2 , dxd^

D = k — D =7 — D = k — D =7 ^p

d5,9 = , d5,10 = 7351 ^ , D5,11 = Ä5w3 ^ , D5pq =7u2 ~ ,

dx ox dx dx

d 2 d 2 I d

D6,6 = 7332 ^ + 7331 T-2 + 7333 + (k2)2k6v2 , D6,7 = 0 , D6,8 = k2k5u3 , D6,9 = k6w1 ^ ,

, dx2 dy2 , , dy

D6,10 =/351 , D6,11 = k6w3 ^ , D6 pq =7u 2 , D7,7 =0331 + $332 + 0333 + (k\)2 k7u3 , 8y 8y 8y 8x 8y

8 8 8 8 8q

D7,8 = 0341 - - , D7,9 = 0351^T , D7,10 = k7w2^T , D7,11 = 0361^T , D7pq = 0u3^T ;

8x8y Sr 8x 8^ 8x

82 82 * 2 8 8 8

D8,8 = 0332~7 + 0331~7 + 0333 + (k2) k8v3 , D8,9 = 0351^~ , D8,10 = k8w2^~ , D8,11 = 0361 TT" ,

, 8x2 8/ , 8y ' 8y ' 8y

8q 2 2

D8 pq = 0u3^ , D9,9 = 055\^ + r1w1, D9,10 = r1w2 , D9,11 = 056\^ + r1w3 , D9 pq = k9 pp + 0w1q ,

2 2 D10,10 =/551^ + 7552 + r2w2 , D10,11 = r2w3 , D10 pq = k10qq + 7w2p , D11,11 =0661^ + 0662 + r3w3 ,

Dn pq = 0w3q+kn pp.

It is shown that the differential matrix of the DE (12) system is symmetric (Dj = Dji). In (13):

1 1 h

7111 = h(d0- — d10e20) , 7112 = hG , 7121 = h(G + d0V- — d10e20) , 7131 =— d10e22 ,

7131 = - ~h d10e22 , k1w1 = (d0k1v - ^ d10e20 + ~T h11)h , 7151 =-hd10 q22 , 10 10 h 10

k1w3 = (T h13 - "T^ d10e22)h, 7u0 = 3h d10 ; k2w1 = (d0k2v - ^ d10e20 + "t h11)h , h 10 20 10 h

k2w3 = ("V" h13 - ^ d10e22)h ; 0111 = h (d0d10 e31) , 0112 = 1 hG , 0113 =- 2 l1x1 ,

h 10 3 70 3 h

h 3 19 ' h 2 2

0121 =- (G + d0V- — d10e31) , k3u2 =— G , 0131 =- — d10e33 , 0133 =-T l1x3 , 0151 = -T h11 ,

3 70 15 70 h h

k3w2 = h d0k1v + h22 - ^ hd10q32 , 0161 =- 2 h13 -hd10q33 , 0u1 = ^ hd10 ;

3 5 70 h 70 21

k4w2 = h d0k2v + h22 - hd10q32 , 7331 = \ (d0 + ^ d10e22) , 7332 = 1 hG , 7333 =-~6 l2x2 , 3 5 70 5 7 5 5h

k5u2 = k6v2 =--2 Gh , 7341 = h (G + d0V + 1 d10e22) , k5u3 = 3 G , k5w1 = hk12d10e20 - ^ Gh ,

7351 = 1(hd10q22 - 6 h22) , k5w3 = hd10e22 + hk1vd0 + ^ k1hG , 7u2 = ^hd10 5 7 h 35 5 15 10

k6w1 = hk12d10e20 - ^ hG , k6w3 = 1(hd10e22 + k2vhd0 + 2 k2hG ) ,

0331 = h (d0 + 1 d10e33) , 0332 = 1hG , 0333 = - l3x3 , k7u3 = k8v3 = - G h ,

7 15 7 7h 70

0341 = h (G + d0V + -1 d10 e33) , 0351 =--6 h31 , k7w2 = ^^ hd10 q32 - 3k1h22) ,

7 15 7h 35 3

0361 = hd10 q33 - 6 h33 ) , k8w2 = ^ (1 hd10 q32 - 3k2h22) , 0u3 = hd10, 0551 =-h11,

7 15 h 35 3 18

k2 k ^ r1w1 = h(kvd0 - T2 d10e2(i) , r1w2 =-TT hd10 q22 , 0551 =-h11, 0561 =-h13 , r1w3 =- "T2 hd10e22 ,

dWe2o) , rlw2 =--¡2 hd10q22 , 055l = -hH , 056l = -hl3 , r1w3 =-"I® hdl° ' 0Wl =-1' r551 =-1 h22 ' 7552 =-1922 , r2w2 = h(1 -vd 0 - ^ d™ ^32) ,

k 1 2k 1 2 3

-2(e22 + hd10933) , -10q = --2hd10 ' 7w2 , 0661 =-—hG , 0662 =-—>

r3w3 = h (kvd0 + k22 d10e22) , Pw3 =-3 , k11p = kT2 hd10 ; h11 = 14 Gh , h13 G'h , h22 = 7 Gh ,

5 7 7 10 15 15 6

7 14

h31 = 7 G h, ^33 = hn, (x ^ y; k1 ^ k2); q21 = -7d30 k12, q22 = , q23 = 2d 30 k12,

5 hd 20

66 _ 22 . _7

q32 = -11d30 k12 , q33 = — , e20 =-7d30; e22 = 2d30; e31 = -11d30; e33 = ~ d30; e2 p =-";

hd20 3 2

22 2

e3q =-—; d0 = E/(1 -v ),d10 = Ev'/(E'(1 -v)), d20 = (1 - 2dwv')/E', ¿30 = dw /d20;;

kXv = k1 + k2v, k2v = k2 + k1v, kv = k^v + k2k2v, where E,E',v,v',G,G' is the mechanical parameters of the transversely isotropic material.

The DE system (12) is not divided into two systems that describe independently symmetric and oblique deformation. This indicates the interdependence of symmetric and oblique deformation of the shells. For plates, DE systems are separated.

To obtain the MSA equations, we transfer all the additions of the left-hand sides of equations (12) containing the curvatures of the shell to the right-hand side. We will have the following system in the i = 1,2,... approximation:

where

LjA) + LJ,2 v0° + + LjAvil) + LjA} + LJA ) + Ljj + + LjV) + Lj 9w« + Lj>10w« + L]Xwf = Li-* (x,y), (j - 1,2,...,11),

, _ d2 d2 _ d2 T _T L1,1 -Y111 ~T + Y112~T, L1,2 -Y121 - - , M,3 - L1,4 - 0 ,

dx dy exey

q2 Q2 ^

L1,5 -f131~^, L1,6 -^131 - - , L1,7 - A,8 - L1,9 - 0, L1,10 - Y151 T- , L1,11 - 0 ,

dx2 dxdy dx

L1-q -Yu0p,x - (k1 hx0u0 + k1 l1x1u1 + k1 hx2u2 + k1 hx3u3 + k1w1wl,x + k1w3w3,x ^ ;

_ d2 d2 _ _ _ d2 _ d2 L2,2 - Y112~T + Y111~2 , L2,3 - L2,4 - 0 , L2,5 - ^131 ~ - , L2,6 - 7131

(14)

dx2 dy2 ' , , , dxdy ' , dy2 '

d

L2,7 - L2,8 - L2,9 - 0 , L2,10 - ^151 T- , L2,11 - 0,

dy

L2pi} -Yu0p,y - (k2 l1 y0v0 + k2 l1 y1v1 + k2 l1 y2v2 + k2 l1 y3v3 + k2w1w1,y + k2w3w3,y ) );

d2 d2 d2 d2 L3,3 - ß111 + ß112~2 + ßm , L3,4 - ß121 a - , L3,5 - L3,6 - 0 , L3,7 - ß131 "TT + ß133 ,

dx dy dxdy dx

d2 d d

L3,8 = ß131 ^ - , L3,9 - ß151 T- , L3,10 - 0, L3,11 - ß161 T" , dxdy dx dx

L3-l) -ßu1l,x - (k1hx1u1 + k1l2x1u1/5 + k1 k3u2u2 + k1l2x3u3/5 + k3w2w2,x -1) ;

d2 d2 d2 d2 L4 4 = ßn2 —2 + ß111 T"Y + ß113 , L4,5 - 0, L4,6 - 0 , L4,7 = ß131 _ _ , L4,8 = ß131 T~J + ß133 ,

-9 / 111 _ 9 / 11^ 4,5 7 4,6 7 4,/ / 131 Ä Ä 7 4,8 / 131 _ ^

dx2 dy ^ dxdy dyx

d d L4,9 - ß151 , L4,10 - 0 , L4,11 - ß161^T ,

dy dy

L4pq -ßu1),y - (k2^1 y1v1 + k2^2y1v1/5 + k2 k3u2v2 + k2hy3v3/5 + k4w2w2,y )(''-1) ;

_ d2 d2 _ d2 L5,5 - ^331 + Y332 ~T + Y333 , L5,6 - ^34^ - , dx2 dy 2 dxdy

_ _ _ _ d _

L5,7 - L5,8 - L5,9 - 0, L5,10 - ^351 T" , L5,11 - 0 ,

dx

L5 pq =7u2P,x - (—1 l1 x2u0 + —1 — 3u 2u1 + (—1 f -5 u 2u2 + -1-5u 3u3 + -5 w1w1,x + -5 w3w3,x f 1 ;

>2 ö2

L6,6 = T332 ~T + Y331~T + 7333 , L6,l = L6,8 = L6,9 = 0 , L6,10 = 7351 ^T , L6,11 = 0 , öx öy Oy

L(6~Pl =7u2P,y - (—2 l1 y2v0 + — 2 —3u 2v1 + (—2 f —5u 2v2 + — 2 —5u 3v3 + —6w1w1,y + —6w3w3,y t'^ ö2 ö2 „ ö „ ö

Ll,l = 0331 ^ 2 + 0332 ^ 2 + 0333 , Ll,8 = 0341 - - , L7,9 = 0351 - , L7,10 = 0 Ll,11 = 0361 -ö ; Llpq = 0u3q,x p i—,1 l1x3 u0 + —,112x3 u1/5 + —\—5 u3 u2 + (—1 Y —lu 3 u3 + —l w2w2,x f ''1

- 9 / 332 _ 9 / 33^ 7,8 / 341 ^ ^ ? 7,9 / 3J1 /-.

8x 8y2 , 8x8y , 8x

8 8x

82 82 8 8 L8,8 = 0332 ~2 + 0331~T + 0333 , L8,9 = 0351^~ , L8,10 = 0, L8,11 = 0361^T , 8x 8y 8y 8y

4pl1q =0u3q,y - (k211 y3V0 + k212 y 3V1/5 + k2 k5u 3V2 + (k2 ? k8v3 V3 + k8v 2 w2,y f1 ;

2 2 t—7 2 2

L9,9 = 055\^ , L9,10 = 0 , L9,11 = 0561^ , L\0,\0 = 755\^ + 7553 , L10,11 = 0, L\\,U = 066\^ + 0663 , L(9-1q = k9 pp + 0w11 - (k1w1U0,x + k2 w1V0, y + k5 w1U2, x + k6 w1V2, y + r1w1 W1 + r1w 2 W2 + r1w3 W3 ;

=7w2 p + k10 qq - (k3w2U1,x + k4 w 2V1,y + k7 w2U3,x + k8 w 2V3, y + r1w2 W1 + r2 w 2 W2 + r2 w3W3t~11 ; LUpq = k11 pp + 0w31 - (k1w3 U0,x + k2 w3 V0, y + k5 w3 U2,x + k6 w3V2, y + r1w3 w1 + r2 w3 w2 + r3w3 .

In the zero approximation (i = 0), the system of equations (12) has the following form:

t u (0) + j V (0) + j u (0) + j V (0) + j u (0) + j V (0) + j u (0) + Lj,1U0 + Lj,2V0 + Lj,3U1 + Lj,4V1 + Lj,5U2 + Lj,6 V2 + Lj,7 U3 +

+ Lj^0 + Lj,9w10) + Lj^w™ + Lj-w^ = Ljpq(x,y), (j = 1,2,...,11),

where

L1 pq =Yu0p,x ; L2pq = Yu0p,y ; L3pq = 0u1q,x ; L4pq =0u1q,y ; L5pq =7u2p,x ; L6pq =Tu2p,y ; L7pq = 0u3q,x ; L8pq = 0u3q,y ; L9pq = k9pqp + 0w1q ; L10pq = Yw2p + k10qq ; L11 pq = k11 pp + 0w3q .

Equations 1, 2, 5, 6, 10 of systems (14) and (15) (10th order) describe symmetric deformation of the corresponding plates, and 3, 4, 7-9, 11 - skew symmetric deformation (12th order).

4.2. Approximation K013. In approximation K013, the DE (16th order) system consists of the first-fourth, seventh-ninth, and eleventh equations (12). In addition, you need to put p(x, y) = 0 and

consider only the functions of u0, v0,u1, v19 u3, v3, w1, w3, q . System (12) for i = 1,2,— then looks like:

Lh1uf + Lj,2 vlp + Lj 3u1i) + Lj^i) + Lj^uf +

(16)

+ LJV) + LJ9w1(i) + Lj-w^) = x,y), (j = 1,2,3,4,7,8,9,11),

where

_ 82 82 _ 82 _ _ _ _ _ L1,1 = 7111 ~T + 7112~T, L1,2 = 7121 ~ - , L1,3 = 0 , L1,4 = 0, L1,7 = 0, L1,8 = 0 , L1,9 = 0, 8x 8y 8x8y

L1,11 = 0 , ¿CI =-(k1 l1x0U0 + k1 l1x1u1 + k1 l1x3U3 + k1w1w1,x + k1w3w3,x t-1 ; 82 82

L2,2 = 7112~T + 7111 ~T , L2,3 = 0, L2,4 = 0 , L2,7 = 0 , L2,8 = 0, L2,9 = 0 , L2,11 = 0,

, 8x 8y2 , , , , , ,

L{2-)q =-(k2 l1 y0V0 + k2 l1 y1v1 + k2 l1 y3V3 + k2w1w1,y + k2w3w3,y ;

82 82 82 82 82

L3,3 = 0111~w + 0112~2 + 0113 , L3,4 = 0121 - - , L3,7 = 0131~2 + 0133 , L3,8 = 0131 2 8y 2 2

8 j _ a 8 T(i-1)

'1518x' L3,11 =01618x'4 pq

„ 9 ' f11'2 _ 9 ' f113! "3,4 1-121 ~ ~ 5 "3,1 f131 - 9 l" 133 ^ "3,8 1-131 ~ ~ '

öx2 öy2 öxöy öx öxöy

ö ö

L3,9 = 0151 -T-, L3,11 = 0161 -T-, L3~p1q = 0u1q,x - (—\l1x1u1 + —1l2 x1u1 /5 + —1l2 x 3u3 > ^t''1 \

d2 d2 d2 d2 d

L4,4 = ß\\2~T + ß111 7"7 + ßH3 ' L4,7 = ß131 a - , L4,8 = ß131 77 + ß133 , L4,9 = ß151 7" dx dy dxdy dy dy

L4.11 =#61 ^ , = #^q,y - (k'2k yV + k2h y1v1/5 + 3V3/5)0-1);

a2 a2 a a

L7,7 = #331 "T + #332 77 + #333 , L7,8 = #341 - - , L7,9 = #351 T" '

ax ay axay ax

a

L7.11 = #361 -, L(7-1) = #„3^,x - (k1 /1x3 "0 + k1 /2x3 "1/5 + (k1)2k7„3 ^-1);

a2 a2 a a

L8,8 = #332 77 + #331 77 + #333 , L8,9 = #351 7" , L8,11 = #361 7" ,

' ax2 ay2 ' ay ' ay

4-q = #u3q,y - (k'2/1 y3 Vo + k'2/2y3V1/5 + (k2 )2 k8v3 V3 )(i-1) ;

L9,9 = #551V2 , L9,11 =#561V2 , = #w1q - (k1w1u0,x + k2 w1v0,y + r1w1 w1 + r1w3 w3)( -1);

L11,11 =#661V 2 +#663 , =#w3q - (k1w3 u0,x + k2 w3 v0, y + r1w3 w1 + r3w3 w3)0 -1) •

In the zero approximation (i = 0) the system of DE (j = 1,2,3,4,7,8,9,11) is as follows:

T „(0) + T v(0) + r ,/0) . (0) , , „(0) . (0) , , w(0) , r ,.,(0) _ t .A nTl

Lj,1u0 + Lj,2V0 + Lj,3u1 + Lj,4V1 + Lj,7u3 + Lj,8V3 + Lj,9w1 + Lj,11w3 = Ljpq (x,У), (i/)

where L1 pq = L2pq 0 ; L3pq = #u1q,x ; L4pq =#u1q,y ; L7pq = #u3q,x ; L8pq = #u3q,y ; L9pq = #w1q •

Equations 1, 2 of systems (16) and (17) describe the symmetric deformation of the plates, and 3, 4, 7-9, 11 - skew symmetry. Similarly, DE systems are obtained for other approximations.

The systems DE (15), (17) coincide for the corresponding plates., And (14) and (16) are structurally different only in the right parts. Therefore, each of the systems (14)-(17) can be divided into systems that separately describe the vortex boundary effect, the internal SSS, and the potential boundary effect. Methods of transformation and decoupling of such systems are given in [26].

In MSA, at each approximation, the general solutions must satisfy the same set boundary conditions. In the method of perturbations of geometric parameters [27] in a null approximation by a small parameter, the general solutions must satisfy the given boundary conditions, and in subsequent approximations the corresponding homogeneous boundary conditions.

4.3. Numerical results. The effectiveness of MSA was investigated in a boundary value problem for transversally isotropic shallow shells, freely fixed on the lateral surface (10). The transverse skew symmetric load q(x,y) = qmn sin(mnx/a)sin(n^y /b) (qmn - const) (DE (16) and (17) systems are

considered). The following MGP were accepted: G'/ G = 0,1; E'/E = 1; a = b; v' = v = 0,3; m = n = 1; k[ * 0; k'2 * 0; R1 = R2; R1 /a = 10; 20; 40; h/a = 1/3; 1/5; 1/10. Numerical results show that in the zero approximation of the difference between the SSS components and the results obtained by the direct solution of the DE equilibrium system, for crx (x, y, z)/ q(x, y) is less than 3.9%, for W(x, y, z) E /(q(x, y) h) - less than 1.1%. In the first approximation for the difference does not exceed 1%. This indicates a high convergence of MSA.

5. Conclusions.

1) The method of sequential approximations in MT of transversely isotropic shallow shells of arbitrary thickness is developed. In the zero approximation of MSA, the systems of equations for shells coincide with the equations for the corresponding plates. In the following approximations, the left parts are the same and coincide with the equations for the plates, and the right parts of the equations depend on the curvatures and components in the displacement components of the previous approximations.

2) By this method, the boundary value problem for the shell is reduced to a sequence of boundary value problems for the corresponding plates with symmetric and oblique deformation. Then inhomogeneous high-order DE can be reduced to low-order equations. MSA makes it easier to find a common solution for shallow shells.

3) Numerous studies have shown a high convergence of results.

4) MSA can be used to solve problems for shallow shells based on other theories.

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