Solution of Problems in the Theory of Shells by Fdm

8/19/2019 Solution of Problems in the Theory of Shells by Fdm

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or another asymptotic repre senta tion has been used, Investi gators have also obtained some

a p p r o x i m a t e s o l u t i o n s o f s h e l l - t h e o r y p r o b l e m s i n a n a l y t i c f o r m b y d i s r e g a r d i n g t h e s m a ll

terms in the fundamen tal equations.

The approaches to the solution of geome tric ally nonline ar proble ms in the theory of

plates and shells that were used at that time are describ ed c hiefly in the monog raphs of

P. F. Pap kov ich [98], V. I. Feo dos' ev [122], and A. S. Vol 'mir [20], A numb er of nonl ine ar

p r o b l e m s f o r f l e x i b l e p l a t e s a n d s h a l l o w s h e l l s w e r e s o l v e d b y m e a n s o f d i v e r s e v a r i a t i o n a l

approaches, in partic ular by constructin g series expansions for one of two resolving func-

tions: by P. F. Papkovic h's method, whe n the function used is the defle ction and the stress

functio n is found by integrat ing the equations of compat ibility; or else by the method of

K. Z. Gallmov [27], in whic h we construct a represe ntati on for ~the stress func tion and the

d e f l e c t i o n i s d e t e r m i n e d b y i n t e g r a t i n g t he e q u a t i o n o f e q u i l ib r i u m . B e y o n d t h i s t h e p r o b l e m

i s s ol v e d by u s i n g t h e B u b n o v - G a l e r k i n p r o c e d u re . A n o t h e r a p p r o a c h u s e d f or s o l v i n g n o n -

linear problem s was one based on the use of the Bubno v-Gal erkin method for the entire system

o f eq u a t io n s , T h e p o s s i b i l i t y o f s o l v i n g v a r i o u s s h e l l - t h e o r y p r o b l e m s b y a n a l y t i c m e t h o d s

has also been disc usse d in [5, 56, 74, 75, 102, 120, 76, 129, 130, 86],

The use of these approaches for solving linear and nonlinea r proble ms in the theory of

plates and shells made it possible, on the one hand, to obtain sol utions in analytic form;

the advantages of such solutions lie primarily in the fact that they are general and closed.

However, on the other hand, analytic methods were succes sful only in construc ting so lutions

for simple problems; plat es and shells of uncom plica ted shape for well- defin ed types of load-

i ng . T h us , i n t h e p r e c o m p u t e r a g e t h e p o s s i b i l i t i e s o f o b t a i n i n g s o l u t i o n s f o r p r o b l e m s

in shell theory were severely limited.

T h e i n v e n t i o n o f c o m p u t e r s a n d t h e ir w i d e s p r e a d i n t r o d u c t i o n i n t o t h e w o r k o f p r a c t i c a l

c a l c u l a t i o n m a d e i t p o s s i b l e t o u s e t h e m e t h o d s o f n u m e r i c a l a n a l y s i s f o r s o l v i n g p r o b l e m s

in the the ory of shells and thus to expand subs tantia lly the class of solvabl e shell-t heory

problems. As early as the begin ning of the 1960s, A. A. Doro dnit syn [62] wrote that high-

speed electronic comp uters have expan ded by a factor of tens and even hundred s of thousands

the computa tional pos sibil ities open to us even today, and in the near futu re the produ ctivi ty

of computers will be increased by several more orders of magnit ude. This has created a com-

pletel y new quali tative situation. The 'precomputer' age was character ized by a crisis of

computing equipment, but during the period fo llowing the appeara nce of computers the old

m e t h o d s o f n u m e r i c a l a n a l y s i s h a v e b e e n e x h a u s t e d v e r y r a p i d l y , a n d t o d a y w e h a v e a ' c r is i s

o f m e t h o d s ' . T h e r e f o r e a t t he p r e s e nt t i m e t h e d e v e l o p m e n t o f a c c u r a t e n u m e r i c a l m e t h o d s

in all branches of the exact sciences is an extremely importa nt task of theoret icians.

As V. I. Feodos 'ev remarks in his book [125], toda y's computer s have such capabiliti es

that whe n they are used, quanti ty becomes quality. The compute r becomes a means for studying

phenomena, and we are faced wit h a new branch of mathem atics -- computer analysi s.

L. I. Sedov, in his report at the Fourth All-Unio n Congress on Theor etica l and Applied

M e c h a n i c s , r e m a r k e d t h a t a c h a r ac t e r i s t i c f e a t u r e o f t o d a y ' s m e c h a n i c s i s t h e w i d e s p r e a d u s e

o f c o m p u te r s . F u r t h e r m o r e , t o g e t h e r w i t h t h e d e v e l o p m e n t o f e x p e r i m e n t a l m e t h o d s o f i n -

vest igat ion for solving the problems that arise, the main trend is the develo pment of meth ods

of numeric al solution using hlgh-s peed computers [117].

I n a r e p o r t d e l i v e r e d a t t h e A l l - U n i o n C o n f e r e n c e d e d i c a t e d t o t h e m e m o r y o f Y u , A .

Shlmanskll, V. V. Novo zhilo v discussed in detail the question of the appro ach to the solutio n

o f t h e p r o b l e m a s a w h o l e: f r o m t h e f o r m u l a t i o n o f t h e p r o b l e m t o t h e u s e ( or i n t e r p r et a t i o n )

of the result s obtained. The ability to solve a probl em in a simple manne r represents a

profoun d under stand ing of the problem [91]. To do this, we must conside r prob lems for cruder

models, whic h will e nable us to unders tand the pheno menon as a whole and to give some esti-

mat e of the res ults that can be expected. In R. W. Han~ning's boo k [128] the mot to for the

engineer is The purpose of calcula tions is not to find numbe rs but to gain underst anding ;

on the other hand, V. V. Novoz hilov proposes replacing this mott o with the following: The

purpose of calculatio ns is not only to find number s but also to gain understan ding [91 , 92].

Actually, in solving a probl em by means of computers, we must cons ider the certain chain

c o n s i s t i n g of t h e f o l l o w i n g l i nk s : t h e m a t h e m a t i c a l m o d e l , t h e m e t h o d , t h e a l g o r i t h m , t h e

program, the computation, and the analysis of the results; we must also take account of the

connect ions between all these links. Special attentio n is drawn to this by A. A, Samarski i

in his discu ssion of a comput ationa l experiment [115]. There fore in constr ucting the math e-

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matica l model of a problem, we must provi de an opportunity for efficientl y realizing it,

taking account of the choice of the method the algorithm, and the formulat ion of the pro-

gram, and it is no less important to have feedback between all these links and the mathe-

matical model. Only if we combine all these features can we obtain an effective solution

of the problem. All of this is fully applicable to the solution of problem s in the theory

of shells.

In analyzing the various approa ches to the numerical solution of problems in the theory

of plates and shells,* as noted earlier, w e shall discuss linear and geome trically nonlinear

problems of shell theory. The characteristics of the solution of linear problems in shell

theory by numerical methods using computers wi ll be analyzed by considering two major classes

of problems: shells of revolution of arbitrary design wit h axially symmetric and asymmetric

loading, and shells of complicated shape, The problems relating to the first of these classes

are described by ordinary differential equations. The second class of problems leads to the

solution of partial differenti al equation s for regions of complicated shape, The solution

of nonlinear problems in shell theory by numerica l method s is considered for shel ls of revolu-

tion under axially symmetric deformation and for some shells of different shape which are

described by two-dlmensi onal nonlinear equations.

2. Linear Problems. Shells of Revoluti on of Arbitra ry Design with Thickness Varyin g

along the Generator, under Ax ia l yS ym me tr ic an d Asymmetric Loading, This class of problems

is distingui shed by the fact that under axiall y symmetr$c types of loading it can be described

immediately by boundary-value problems for systems of ordinary differential equations with

variabl e coefficients, and in the case of loadings which are not axially symmetric this class

admits of the separation of variables, us ing expans ions of the loading and of the desired

functions in Fourier series in the circular coordinate [90, 40]. Therefore the method s for

solving this class of problems are applicable also to other classes of problems in shell theory

which are described by ordinar y differe ntial equations. To solve the class of problems dis-

cussed here, the following metho ds ha ve been used,

The Finite-Difference Method (FDM ). One of the methods used to solve boundary-value

problems relatin g to the deforma tion of shells of revolution of arbit rary design is the FDM,

by means of which the so lut ion of one-dimensional boundary-value problem can be reduced to

the solution of a system of linear algebraic equations. Since shell-theory problems involve

local and boundary effects of the stressed state, the system obtained may prove difficult

to justify. For the solution of such systems the method of difference facto rization is used

[114, 28]. The computational difficulties in connection with this method arise out of the

need to preserve the infor mation of the forward course, which for large systems may occ upy

a large part of the computer,s operational memory [14 3]. Such an approach also requires a

certain degree of inven tiven ess in the approxim ation of the boundary conditions, especially

if they are expressed i n terms of the leading derivatives.

By using the MFD, V. N. Bulgakov [16] solved a number of proble ms for toroidal shells

acted upon by axially symmetric distributed, contour, and centrifugal forces, Ya. M.

Grigorenko [38] used the FDM to ob ta in solutions for conical shells of variable rigidity

under ant isymmetric loads such that as the aperture angle of the cone decreased, the eigen-

values of the matri x of the system of equations became substantially different, leading to

instability of the computation, and in those cases the method of differe nce factorizat ion

was used to solve the proble m [ 114, 28]. In the monog raph of V. V. Boloti n and Yu. N. Novich-

kov [15] the followi ng approac h was used for solving problems in the theory of plates and

shells in addition to the usual approaches: the boundary-value problem is reduced to finite-

difference equations with constant coefficients; the theory of finite-dif ference equations

is used to solve them [29], and solution s which are exact in this sense are constructed at

a discrete set of points.

The Finite-Element Method (FE M). The first work on the solution of problems for shells

of revolut ion under axially symmetric loads by using the FEM was done by non-Soviet authors

[64]. They considered shells of revolut ion of a specified shape and proposed various finite-

element constructions for their calculation. For the solution of problems relating to the

stressed-de formed state of shells of revolution, other numerical methods, whic h will be

considered below, apparent ly proved more effective, The finite-ele ment method is more widely

used for solving two-dim ensional problem s in the theory of shells.

*Here we consi der chiefly the work of Soviet authors, since the results that can be achieved

by using numerical methods depend substantial ly on the computers used.

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However, we should draw attenti on here to some studies in whic h the FEM was used to

construc t computing complexes for solving problems relating to shell-t ype systems, Such a

c o m p u t i n g c o m p l e x f o r c a l c u l a t i o n s r e l a t i n g t o sh e l l s o f r e v o l u t i o n u n d e r a x i a l l y s y m m e t r i c

loads was developed by L. A. Rozin and L. B. Grinze [Iii], who used an algori thm based on

the metho d of sepa rating the operato rs of the equations of shell theory and realize d it in

the form of a unive rsal program. The algor ithm is in fact the soluti on of a separated

proble m using the FEM [105]. In the studies by V. A. Postnov and V. S, Korne ev [104], the

FEM was used to work out an algo rithm and set up a computing comp lex for solving proble ms

for shells of revolution. Here the finite element chosen was a conical element. B. Ya.

Kantor and V. M. Mitk evic h proposed an approa ch to the solution of problem s relating to the

stressed--deformed state of shell-t ype stru ctures of revo lution wit h a branched meridian.

To solve the problem, the entire stru cture is subdivided into segm ents and the finite-ele -

ment proced ure is applied to these. Such an appro ach was realized in the comput ing complex

of [71].

The Integ ral-E quatio n Method. I. A. Birger and his pupils [12, 13, 65] considered a

c l a s s o f p r o b l e m s c o n c e r n i n g t h e s t r e s s e d - d e f o r m e d s t a t e of s h e l l s o f r e v o l u t i o n a c t e d u p o n

by axial ly symmetric and antisyn~netric loads. The problem can be descri bed by means of

m o d i f i e d b o u n d a r y - v a l u e a n d n o r m a l i n t e g r a l e q u a t i o n s o f t h e F r e d h o l m a n d V o l t e r r a t y pe s ,

w h i c h s i m p l i f i e s t h ei r c o n s t r u c t i o n f r o m th e o r i g i n a l d i f f e r e n t i a l r e l a t i o ns . T h e i n t e g ra l

e q u a t i o n s s o o b t a i n e d c a n b e s o l v ed b y i t e r a t i v e n u m e r i c a l m e t ho d s . F o r p o o r l y d e f i n e d

p r o b l e m s t h e i n t e g r a l - e q u a t i o n m e t h o d i s u s e d i n c o n j u n c t i o n w i t h t h e o r t h o g o n a l - f a c t o r i z a -

tion method [63].

Meth ods of Reduc tion to Cauchy Problems. By virtu e of the linear ity of the class of

p r o b i e m s o n t h e d e f o r m a t i o n o f s h e l l s o f r e v o l u t i o n w h i c h w e a r e c o n s i d e r i n g h e r e , s o l u t i o n s

for them can he found by the usual method of reducing a bound ary-v alue pr oblem to a number

of Cauchy problems [8] each of which can be solved by one of the known numeric al methods

~Runge-Kutta, Adams--Starmer, and others). However , in such cases, as alrea dy mentioned,

owing to the boun dary and local effects occu rring in thin shells, this approa ch may be un-

suitable, since the comp utati on becomes unstab le [18, 31, 8]. In the cases indicate d the

probl em becomes rigid [119]. To overcom e these difficul ties, in vestig ators have worked out

a n u m b e r o f m e t h o d s b y w h i c h t h e n u m e r i c a l s o l u t i o n o f t h e b o u n d a r y - v a l u e p r o b l e m i s r e d u c e d

to a stable computatio n process. Such method s include: the metho d of facto rlzati on in dif-

ferenti al form, proposed by I. M. Gel land and O. V, Loku tsiev skil [28], the metho d of con-

t i n u o u s o r t h o g o n a l i z a t i o n d e v i s e d b y A . A . A b r a m o v [ 1 ] , a n d t h e m e t h o d o f d i s c r e t e o r t h o g o n a l -

ization devised by S. K. Godun ov [30].

W e s h a l l n o w c on s i d e r t h e p r o p e r t i e s a n d p o s s i b i l i t i e s o f u s i n g t h e s e m e t h o d s f o r s o l v -

ing problem s in shell theory. The method of diffe renti al factor lzati on [28, 8] consists in

a n e q u i v a l e n t r e p l a c e m e n t o f a b o u n d a r y - v a l u e p r o b l e m s f o r a s y s t e m o f l i n e a r d i f f e r e n t i a l

e q u a t i o n s w i t h a n u m b e r o f C a u c h y p r o b l e m s f or s y s t e m s o f n o n l l n e a r d i f f e r e n t i a l e q u a t i o n s ,

w h i c h a r e t h e n s o lv e d n um e r i c a l l y . T h e s e m e t h o d s y i e l d s t a b l e p r o c e s s e s f o r t he f o r w ar d a n d

reverse course. On the basis of physic al representatio ns, an analogo us approach to the solu-

t i o n of b o u n d a r y - v a l u e p r o b l e m s f o r s h e l l s o f r e v o l u t i o n w i t h a x i a l l y s y m m e t r i c l o a d i n g h a s

been used in a mono grap h by V. S. Chuvikovskli, O. M. Palii, and V. E. Spiro [131]. Some

a s p e c t s o f t h e d i f f e r e n t i a l - f a c t o r i z a t i o n m e t h o d i n t h e s o l u t i o n o f p r o b l e m s i n s t r u c t u r a l

m e c h a n i c s h a v e b e e n d e s c r i b e d i n a n a r t i c l e b y V. L . B i d e r m a n [ 9 ]. I n u s i n g t h i s m e t h o d , w e

m a y e n c o u n t e r d i f f i c u l t i e s c a u s e d b y t h e un b o u n d e d i n c r e a s e i n t h e e l e me n t s o f t h e f a c t o r i z a -

t i o n m a t r i c e s a n d t h e f a c t o r i z a t i o n v e c t o r [ I i , I 0] .

The method of continuous orth ogona lizat lon [i, 87] is actually based on the continu ous

o r t h o g o n a l i z a t i o n o f t h e s o l u t i o n v e c t o r s o f C a u c h y p r o b l e m s a s t h e a r g u m e n t v a r i e s ; i t m a k e s

it possible to carry over any number of bounda ry conditions. In this method, un like the pre-

ceding one, there is no increas e in the matrices, but the right sides of the equations are

m u c h m o r e c o m p l i c at e d , a n d t h e r e f o r e i t s u s e r e q u i r e s a f a r l a r g e r a m o u n t o f c o m p u t a t i o n .

Some features of the use of this method in the solut ion of shell-th eory problem s were first

consider ed by Ya. M. Grlgore nko in [42, 39].

T h e m e t h o d o f d i s c r e t e o r t h o g o n a l i z a t i o n [ 3 0 ] m a k e s i t p o s s i b l e t o o b t a i n a s t a b l e c o m -

p u t a t i o n p r o c e s s b y o r t h o g o n a l i z i n g t h e s o l u t i o n v e c t o r s o f C a u c h y p r o b l e m s a t f i n i t e n u m b e r

of points in the interval of vari atio n of the argument. The effect iveness and high accurac y

of the method wer e also noted in the book by R. Bellman and R, Kalaba [6 ], Today the method

of discr ete orthogon aliza tion is widel y used for solving problem s in shell theory. The fea-

t u r e s of t h e u s e o f t h i s m e t h o d i n s h e l l - t h e o r y p r o b l e m s a r e d i s c u s s e d i n a m o n o g r a p h b y

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Ya. M. Grigore nko [40]. Grigor enko also shows that the amount of stored informat ion can be

s u b s t a n t i a l l y r e d u c e d b y a sl i g h t m o d i f i c a t i o n o f th e a l g o r i t h m. T h e e f f e c t i v e n e s s o f u s i n g

t h i s m e t h o d i n s h e l l - t h e o r y p r o b l e m s i s n o t e d i n m o n o g r a p h s b y A. V . K a r m i sh i n , V . A, L y a s -

kovets, V. I. Myachen kov, and A. N. Frolo v [73] and by V. L. Bider man [ii]. Algor ithms for

c a l c u l a t i n g v a r i o u s s h e l l - t y p e s y s t e m s b y G o d u n o v ' s m e t h o d h a v e b e e n r e a l i z e d i n c o m p u t i n g

complexe s and progra m packages [40 , 73, 43, 88].

S o m e o t h er a p p r o a c h e s t o t h e n u m e r i c a l s o l u t i o n o f p r o bl e m s f o r s h e l l s o f r e v o l u t i o n

u n d e r a x i a l l y s y m m e t r i c a n d n o n - a x i a l l y - s y m m e t r i c l o a d s h a ve b e e n d e s c r i b e d i n [ 1 3 5 - 1 42 ] .

T h e p r o p o s e d n u m e r i c a l m e t h o d s a r e u s u a l l y d e s i g n e d f o r u s e w i t h c o m p u t e r s t h a t h av e l a r g e

o p e r a t i o n a l m e m o r i e s .

3. Linear Problems. Shells of Complic ated Shape. In order to chara cteri ze the use

a n d d e v e l o p m e n t o f n u m e r i c a l m e t h o d s i n p r o b l e m s f o r s h e l l s o f c o m p l i c a t e d s h a pe , i t is

d e s i r a b l e t o a n a l y z e t h e e x i s t i n g s c h e m e s f o r s o l v i n g c e r t a i n c l a s s e s o f p r o b l e m s , w i t h d u e

r e g a r d f o r t h e i r u s e i n c a l c u l a t i o n s f o r s h e l l - t y p e s t r uc t u r e s . T h e r e f o r e w e s h a l l c o n s i d e r

b e l o w o n l y t h o s e a p p r o a c h e s t o th e n u m e r i c a l s o l u t i o n o f p r o b l e m s w h i c h h a v e b e e n e f f e c t i v e l y

used for solving c ertain class es of probl ems in shell theory.

T h e F i n i t e - D i f f e r e n c e M e t h o d . T h e F D M h a s b e e n w i d e l y u s e d f o r t h e c a l c u l a t i o n o f

shallow shells in the works of A. A. Nazar ov and his pupils [89]. These studie s discuss

s o m e m e t h o d s f o r a p p r o x i m a t i n g t h e d i f f e r e n t i a l r e l a t i o n s b y d i f f e r e n c e r e l a t i o n s f o r v a r i o u s

c a s e s o f d e f o r m a t i o n o f s h e l ls w h o s e c o n t o u r s a r e f i xe d i n d i f f e r e n t w a y s , i n c l u d i n g t h e

case of a free boundar y and that of an elasticall y built -in boundary. The accur acy of the

metho d is explaine d by the use of examples. The FDM is applied to the calcu lation of shallow

s h e l ls a n d c o v e r i n g s w i t h d o u b l e c u r v a t ur e .

V. I. Gulyaev and E. A. Gotsuly ak [58, 33] have made effectiv e use of the FDM for the

c a l c u l a t i o n o f v a r i o u s s h e l l - t y p e s y s t e ms . I n i n v e s t i g a t i n g s h e l l s o f c o m p l i c a t e d s h a p e,

i t p r o v e d t o b e d e s i r a b l e t o u s e t h e F D M i n c o n j u n c t i o n w i t h t h e a p p a r a t u s o f t e n s o r a n a l y s i s ,

so that it was possi ble to desc ribe in general form the geometry of the deform ed surface.

S u c h an a p p r o a c h i s c o m b i n e d w i t h t h e p o s s i b i l i t y o f a r b i t r a r y c h o i c e o f t h e a r r a n g e m e n t a n d

d e n s i t y o f t h e d i f f e r e n c e n e t w o r k a n d t h e a b i l i t y t o t a k e a c c ou n t o f i t s v a r i a t i o n i n t he

p r o c e s s o f d e f o r m a t i o n . T h e t e n s o r r e p r e s e n t a t i o n o f t he g e o m e t r i c a n d m e c h a n i c a l c h a r a c t e r -

i s t i c s o f t h e s h e l ls u n d e r i n v e s t i g a t i o n i n a n a r b i t r a r y c u r v i l i n e a r c o o r d i n a t e s y s t e m c o n -

s i d e r a b l y e x p a n d s t h e r a n g e o f p r o b l e m s w i t h o u t i n c r e a s i n g t h e a m o u n t o f i n p u t i n f o r m a t i o n

r e q u i r ed . A n a t t e m p t t o c o n s t r u c t f i n i t e - d i f f e r e n c e e q u a t i o n s i n e x p l i c i t f o r m l e a d s to

c u m b e r s o m e e x p r e s s i o n s w h i c h a r e h a r d t o r e p r e s e n t i n an a l y t i c f o r m . S i n c e t h e r e l a t i o n s o f

the theory of shells are linear comb inatio ns of stresses, strains, and displacement s, it was

f o u n d c o n v e n i e n t t o s et u p a p r o g r a m f o r f o r m u l a t i n g e q u a t i o n s f r o m s u b p r o g r a m s w h i c h r e a l i z e

t h e c o r r e s p o n d i n g a n a l y t i c t r a n s f o r m a t i o n s a n d r e p e a t t h e f u n d a m e n t a l s t e p s of t h e d e r i v a t i o n

o f th e r e s o l v i n g e q u a t i o n s o f s h e ll - t h e o r y . B y u s i n g t h e a p p r o a c h s o w o r k e d o u t , i n v e s t i g a t o r s

have solved a number of problems in the deter minat ion of the stressed state of spiral shells,

w a v e - s h a p e d s h e l l s w h o s e m i d d l e s u r f a c e v a r i e s a c c o r d i n g t o a s i n u s o i d a l l a w i n t wo c o o r d i n a t e

directions, and spherical shells with nonco axial skew cuts [58, 57].

The Metho d of Calcul ating Skew Shell Systems. I. F, Obrazt sov and his associ ates [93,

95, 94] worked out an effect ive a pproac h to the calc ulati on of thin-wal led shell systems of

skew type, whi ch are wide ly used in aviat ion tec hnolo gy and other areas. In the general case

t h e y c on s i d e r a c o n i c a l s h e l l o f a r b i t r a r y a r r a n g e m e n t , w h i c h i s a u n i v e r s a l c a l c u l a t i o n m o d e l

for a skew thin-wal led stru cture. On the basis of Lagran ge's varia tiona l principle, they

c o n s t r u c t a g e n e r al m e t h o d f o r c a l c u l a t i n g a c o n i c a l s h e l l o f a r b i t r a r y a r r a n g e m e n t , b a s e d

o n a s p e c i a l c h o i c e o f b a s i s f u n c t i o n s a n d o n r e d u c i n g a t w o - d i m e n s i o n a l b o u n d a r y - v a l u e p r o b l e m

to a one-d imen siona l one by the Vlasov--Kantorovich method. In the case of rigid equations ,

t he u s e o f t he m e t h o d o f d i s c r e t e o r t h o g o n a l i z a t i o n [ 3 0 ] i s p r o p o s ed . T h e a p p r o a c h s o w o r k e d

o u t is s u c c e s s f u l l y b e i n g u s ed i n t h e c a l c u l a t i o n o f v a r i o u s s t r u c t u r a l e l e m e n t s o f m o d e r n

aircraft.

T h e M e t h o d o f P a r a m e t r i z i n g t h e M i d d l e S u r f a c e o f a S h e ll b y M e a n s o f a F i c t i t i o u s D e -

formation. V. N. Paim ushi n [97, 96] has worked out an appro ch to the soluti on of shell-

t h e o r y p r o b l e m s f o r n o n c a n o n i c a l r e g i on s . I n c o n s i d e r i n g s u c h p r o b l e m s, w e e n c o u n t e r d i f f i -

c u l t i e s i n v o l v e d i n s a t i s f y i n g t h e b o u n d a r y c o n d i t i o n s w h e n d i f f e r e n t a p p r o x i m a t e m e t h o d s

are used for finding the solution. To overco me these difficulties, we must reduce such

problems to classic al form. In the above -ment ione d studies Paimushin~ by constr ucting a

param etriz atio n for the midd le surface of the shell, reduces the probl em to a classica l

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one~ which can be solved by known numeri cal methods. He has worked out methods for param-

etrizing the middl e surface for shells in which that surface has a complic ated shape, shells

w h i c h h a v e c o m p l i c a t e d s h a p e s i n t h e p la n e , a n d i n ge n e r a l c a s e , s h e l l s w i t h a r b i t r a r y g e o m -

etry, whic h are based on the constructi on of a mappi ng of some canonic al reg ion onto the

m i d d l e s u r f a c e b y m e a n s o f a f i c t i t i o u s d e f o r m a t i o n o f t h a t s u r fa c e . T h e t w o - d i m e n s i o n a l

proble m is then reduce d by some method to a one-d imensi onal one, whi ch is solved by the

method of finite sums. The numeri cal metho d of calculat ing shells with complicat ed geom-

etry has been reali zed in the form of a computing comple x [97] which has been used for

solving a number of problems for elements of shell-t ype structu res of complicated shape.

The Finite -Elemen t Method. The FEM has been widel y used in problem s of the theory of

plates and shells. By mean s of the FEM a continuo us proble m is replace d with a probl em that

h a s a f i n i t e n u m b e r o f p a r am e t e r s . T h e s u b d i v i s i o n of t h e r e g i o n i n t o f i n i t e e l e m en t s m a k e s

i t p o s s i b l e t o co n s i d e r s h e l l s o f f a i r l y c o m p li c a t e d c o n f i g u r a t i o n . I n s u ch a n a p p r o x i m a t i o n

as a rule, we do not know that some relations of the theory of shells are satisfied, and as

a result, the state of each element and of the entire system is described onl y approximately.

We then use some varia tion al principle to solve the problem. Therefo re the FEM is both a

n e t w o r k m e t h o d a n d a v a r i a t i o n a l m e t h o d [ 10 5] . I n c o m p a r i s o n w i t h t h e c l a s s i c a l v a r i a t i o n a l

methods, the FEM is more alg orith mic and leads to a system of algebraic e quation s with spar-

sely filled band matric es of limited width. However, the use of the FEM for calcul ating

t h i n s h el l s i n v o l v e s d i f f i c u l t i e s i n e n s u r i n g t h a t t h e d e f o r m a t i o n s o f a d j a c e n t e l e m e n t s a r e

compati ble and in approx imati ng smooth surfaces by distort ed elements. A great deal of at-

tention is therefore devoted to approa ches that mak e it possib le to overc ome these difficulties ,

The real izati on of the algor ithms cons tructed on the basis of the FEM depends to a large extent

on the compute rs used. The FEM is wide ly used for solving shell- theor y problem s in the studies

carried out by V. A. Postnov and his pup ils [105, 107, 106] and by L. A. Ro zin and his pupils

[105, 109, ii0], in which the FEM is used to constru ct com puting co mplexes and to solve a

number of complicat ed problems in shell-theory. Some aspects of the use of the FEM in pro-

blems in the theory of plates and shells have b een discussed in the works of A. G. Ugod chik

and his associates [121]. The use of the FEM for calcu latin g shells with double curvatu re

has been discus sed in the mono grap h of A. V. Aleksandr ov, B, Ya. Lashch enikov, N, N. Sha posh-

nikov, and V. A. Smirnov [ 3] . Compli cated and isoparametri c finite elements have been used,

so that it has been possible to combine ele ments not only at nodal points but at interme diate

p o i n t s a s w e l l . T h e F E M h a s b e e n f u r t h e r d e v e l o p e d f or t h e c a l c u l a t i o n o f c o m p l i c a t e d s t r u c -

tures in the metho d of supere lements [103], in which the entire str ucture is separated into

individual parts or substructures. This method, unli ke the FEM, has certain theoret ical

feat ures of its own.

Methods of Reducin g Problem s to One-Di mensi onal Ones. For solving some classes of two-

dimen sional prob lems in the theory of shells, inve stigato rs have used the metho d of integral

relations wor ked out by A. A. Dorodn itsyn and his asso ciates [61 , 62, 7], whic h is in a

certai n sense a generaliz ation of the Vlasov--Kantorovich meth od and of the metho d of strai ght

lines [ 19, 72, 7, 8]. The process of solving the probl em by the meth od of integral relations

consists of two stages. In the first stage the origina l system of partial diffe renti al equa-

t i o n s w i t h g i v e n b o u n d a r y c o n d i t i o n s o r r e g u l a r i t y c o n d i t i o n s i s a p p r o x i m a t e d b y a s y s t e m o f

o r d i n a r y d i f f e r e n t i a l e q u a t i o n s o f h i g h o r d e r w i t h t h e C o r r e s p o n d i n g b o u n d a r y c o n d i t i o n s .

I n t h e s e c o n d s t a ge , t h e s o l u t i o n o f t h e o n e - d i m e n s i o n a l b o u n d a r y - v a l u e p r o b l e m s o o b t a i n e d

is carried out by using some numeric al method. This metho d was used by Ya. M, Grigorenko ,

A. T. Vasilenko, and E. I. Bespalo va to solve a whole class of problem s for closed and open

shells with variab le param eters in one directio n [40, 44, 43],

I n a d d i t i o n t o t h e m e t h o d o f i n t e g r a l r e l a t i o n s , a n o t h e r m e t h o d u s e d f o r s o l v i n g t w o -

d i m e n s i o n a l p r o b l e m s i n s h e l l t h e o r y i s t h e m e t h o d o f s t r a i g h t l i n es , i n w h i c h t h e t r a n s i -

t i o n f r o m t h e t w o - d i m e n s i o n a l t o t h e o n e - d i m e n s i o n a l p r o b l e m i s c a r r i e d o u t b y m e a n s o f a

f i n i t e - d i f f e r e n c e a p p r o x i m a t i o n w i t h r e s p e c t to o n e v a r i a b l e , T h e r e s ul t i n g b o u n d a r y - v a l u e

p r o b l e m f or a s y s t e m o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s c a n b e s o l v e d n u m e r i c a l ly , T h i s a p -

proac h was used by Ya. M. Gri gorenk o and A, I. Shinkar' [45, 43] to calculate nonc ircu lar-

c y l i n d e r s h e l l s a n d s h e l l s o f r e v o l u t i o n w i t h v a r i a b l e p a r a m e t e r s i n t w o d i r e c t i o n s ,

N. P. Fleishma n and Ya. G. Savula [113, 112] calculate d shells of complicate d shape by

t h e m e t h o d o f r e d u c i n g a t w o - d i m e n s i o n a l b o u n d a r y - v a l u e p r o b l e m t o a o n e - d i m e n s i o n a l o ne ,

b a s e d o n t h e d i s c r e t i z a t i o n o f o n l y t h e d e s i r e d c o m p o n e n t s o f t h e d i s p l a c e m e n t v e c t o r , w i t h -

o u t s u b d i v i d i n g t h e r e g i o n i n t o f i n i t e e l e m e n ts , T h i s e l i m i n a t e s t h e v e r y s e r i o u s e r r o r

introduce d into the calculat ion results by the represen tati on of the shell in the form of a

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collect ion of elements of specified shape. This appro ach has been used for solving problems

i n t he c a l c u l a t i o n o f p i p e s w i t h a n a r b i t r a r y c u r v i l i n e a r a x i s,

T h e V a r i a t i o n - D i f f e r e n c e M e t h o d ( VD M) . T h e V D M i s w i d e l y u s e d fo r s o l vi n g t w o - d i m e n -

sional problems in shell -the ory [134, 52]; it is based on a vari atio nal formul ation of the

p r o b l e m a n d t h e r e d u c t i o n o f t h i s f o r m u l a t i o n b y m e a n s o f a d i f f e r e n c e a p p r o x i m a t i o n t o a

v a r i a t i o n a l p r o b l e m f o r d i s c r e t e v a l u e s o f t h e d e s i r e d f u n c t i o n s a t t h e n o d e s o f t h e n e t w o r k

r e g io n . T h e o r i g i n a l f u n c t i o n a l i s d i s c r e t i z e d a n d r e p l a c e d w i t h a s u m w h o s e m i n i m u m c o n d i -

tion leads to the solution of a system of algebr aic equatio ns for the nodal values. In com-

p a r i s o n w i t h t h e n e t w o r k m e t h o d , t h e V D M c o n s i d e r a b l y s i m p l i f i e s t h e r e a l i z a t i o n o f t h e b o u n d -

a r y c o n d it i o n s , a n d l e ad s t o a w e l l - d e t e r m i n e d s y m m e t r i c m a t r i x o f c o e f f i c i e n t s o f t h e sy s -

tem of algebraic equations. The use of the VDM enabl es us to avoid a number of diffi culti es

r e s u l t i n g f r o m t h e u s e o f t h e F E M , i n p a r t i c u l a r t h e c u m b e r s o m e n u m e r i c a l i n t e g r a t i on , t h e

r e q u i r e m e n t o f m a k i n g t h e a d j a c e n t e l e m e n t s c o m p a t i b l e , a n d t h e i s o l a t i o n o f t he d i s p l a c e m e n t s

of the shell as a rigid en tity [121]. The VDM has been widely used for the calcu latio n of

shells of complicat ed shape in the works of A, L. Sinyavski i [118] and A, G, Ugodc hikov and

h i s a s s o c i a t e s [ 12 1] . S o m e c o m p a r a t i v e e s t i m a t e s o f t h e f i n i t e - e l e m e n t m e t h o d a n d t h e v a r i a -

t i o n - d i f f e r e n c e m e t h o d a r e g i v e n i n V. G . K o r n e e v ' s s t u d y [ 78 ],

T h e M e t h o d o f R - F u n c t i o n s . I n r e c e n t y e a r s i n c r e a s i n g a t t e n t i o n h a s b e e n g i v e n in th e

c a l c u l a t i o n o f p l at e s a n d s h e l l s o f c o m p l i c a t e d c o n f i g u r a t i o n t o t h e so - c a l l e d m e t h o d o f R -

functions, or the structur al method, pro posed by V. L. Rvache v [108, i01], The essence of

t h e m e t h o d c o n s i s t s i n t h e f o l l o w in g . T h e p i c t u r e o f t h e d e s i r e d f i e l d d e p e n d s n ot o n l y o n

t h e p h y s i c a l l a w s t a k e n i n t o a c c o u n t b y t h e f u n d am e n t a l d i f f e r e n t i a l e q u a t i o n s o f t h e c o r r e -

s p o n d i n g p r o b l e m b u t a l s o o n th e s h a p e o f t h e r e g i o n a n d i t s b o u n d a r y . T h e e x i s t e n c e o f

g e o m e t r i c i n f o r m a t i o n i n p r o b l e m s i n v o l v i n g t h e c a l c u l a t i o n o f f i e l d s c r e a t e s s p e c i f i c d i f ~

ficulties arisin g out of the need to take account of this informa tion at the analytic level.

T h e m e t h o d o f R - f u n c t i o n s m a k e s i t p o s s i b l e t o i n c l u d e t h e g e o m e t r i c i n f o r m a t i o n i n t h e s t r u c -

tural formula in such a way as to satisf y the boun dary c onditions of the problem, Then we

c a n u s e s o m e v a r i a t i o n a l p r o c e d u r e w i t h a n u m e r i c a i r e a l i z a t i o n , T h e m e t h o d o f R - f u n c t i o n s

h a s b e e n u s e d f o r s o l v i n g p r o b l e m s r e l a t i n g t o t h e f l e x u r e o f p l a t e s w i t h c o m p l i c a t e d s h a p e s

[ 1 0 1 ]

4 . N o n l i n e a r P r o b l e m s. S h e l l s o f R e v o l u t i o n u n d e r A x i a l l y S y m m e t r i c D e f o r m a t i o n . I n

o r d e r t o a n a l y z e t h e n u m e r i c a l a p p r o a c h e s t o t h e s o l u t i o n o f n o n l i n e a r o n e - d i m e n s i o n a l p r o -

b l e m s i n s h e ll t h e o r y , w e s h a l l c o n f i n e o u r s e l v e s t o c o n s i d e r i n g s h e l l s o f r e v o l u t i o n , w h i c h

admit of axial ly syrmnetric defo rmat ion b ecause of special f eatures of the defo rmat ion p rocess,

thin walls, and other factor s [35].

T h e F i n i t e - D i f f e r e n c e M e t h o d . O n e o f t he m e t h o d s w i d e l y u s e d i n p r o b l e m s i n t h e t h e o r y

o f f l e x i b l e p l a t e s a n d s h e l ls i s t h e f i n i t e - d i f f e r e n c e m e t h o d , i n w h i c h d i f f e r e n t i a l r e l a -

t i o n s a r e a p p r o x i m a t e d b y t h e i r f i n i t e - d i f f e r e n c e a n a l o g o u s, a n d a n o n l i n e a r b o u n d a r y - v a l u e

p r o b l e m i s r e d u c e d t o th a t o f s o l v in g a s y s t e m o f n o n l i n e a r a l g e b r a i c o r t r a n s c e n d e n t a l e q u a -

tions.

M. S. Korn ishi n [79] inv estiga ted in detail the feature s of the appl icat ion of the FDM

t o t h e s o l u t i o n o f n o n l i n e a r p r o b l e m s c o n c e r n i n g c i r c u l a r p l a t e s a n d s h a l l o w s h e l l s o f r e v o l u -

t io n. Q u e s t i o n s o f t h e u s e o f f i n i t e - d i f f e r e n c e m e t h o d s w i t h h i g h a c c u r a c y h a v e a l s o b e e n

c o n s i d e re d . F o r t h e s o l u t i o n o f t h e s y s t e m s o f n o n l i n e a r a l g e b r a i c e q u a t i o n s t h e r e h as b e e n

p r o p o s e d a n a p p r o a c h t o t h e c h o i c e o f t h e i n i t i a l a p p r o x i m a t i o n b y e x t r a p o l a t i o n f o r o th e r

v a l u e s o f t h e p a r a m e t e r s , w h i c h p r e v e n t s t h e a c c u m u l a t i o n o f e = r o r s a s t h e p a r a m e t e r v a r i e s ,

In the studies of V. I. Feodos'e v [123, 25, 126] the FDM was used for solving a problem

c o n c e r n i n g t h e d e f o r m a t i o n o f a s p h e r i c a l s h e l l u n d e r u n i f o r m p r e s s ur e . T h e s o l u t i o n o f a

probl em by the FDM was obtain ed for the entire interval of var iati on of the load, in cluding

u p p e r a n d l o w e r c r i t ic a l v a l u e s , f o r d i f f e r e n t v a l u e s o f t h e r e l a t i v e d e f l e c t i o n . F o r f ix e d

valu es of the geome try and the load, the FDM yields solutions for the nonlin ear and linear

parts of the shell, whi ch are joined at the common boun dary [25], In [126] the method of

d i f f e r e n c e f a c t o r i z a t i o n i s u s e d f o r c a l c u l a t i n g a n o n s h a l l o w s p h e r i c a l s h el l , T h e s o l u t i o n

found by the FDM is taken as an exact solut ion and is used as a standard for compari son wit h

the results of solutions obtain ed b y other method s [123].

T h e M e t h o d o f R e d u c i n g a N o n l i n e a r B o u n d a r y - V a l u e P r o b l e m t o a S y s te m o f N o n l i n e a r E q u a -

tions and a Cauchy Problem. A number of proble ms in the theory of flexi ble shells and plates

c a n be d e s c r i b e d b y m e a n s o f n o n l i n e a r s y s t e m s o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s w i t h a p -

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propr iate bound ary conditions. These include, in particular, pro blems relating to the axially

s y m m e t r i c d e f o r m a t i o n o f c i r c u l a r p l a t e s a nd s h e l l s of r e v o lu t i o n , T h e m e t h o d o f r e d u c t i o n

i s b a se d o n t h e e q u i v a l e nt r e p l a c e m e n t o f t h e i n i t i a l b o u n d a r y - v a l u e p r o b l e m w i t h a s y s t e m o f

n o n l i n e a r a l g e b r a i c o r t r a n s c e n d e n t a l e q u a t i o n s a n d a C a u c h y p r o b l e m w i t h a n i n i t i a l v a l u e

which is the solution of th t system of equations. This yields a system of nonline ar equa-

t i o ns w h i c h i s n o t g i v e n in e x p l i c i t f or m bu t i s d e t e r m i n e d a l g o r i t h m i c a l l y . C o n s e q u e n t l y

the solution of this system is found by means of the discre te analog of Newt on's method,

w h e r e t h e d e r i v a t i v e s i n t h e J a c o b i a n a r e a p p r o x i m a t e d b y d i f f e r e n c e r a t i o s a n d a r e c a l c u l a t e d

from the solutions of Cauchy prob lems [133, 17, 53],

T h e r e d u c t i o n m e t h o d m a k e s i t p o s s l b l e t o f i n d a s o l u t i o n of a n o n l i n e a r b o u n d a r y - v a l u e

problem both in the subcritic al and in the superc ritica l region. The diffi cultie s in the

numeric al realiz ation of this metho d are due to the choice of the initial approx imatio n and

the rigidi ty of the Cauchy problems for the cases of strong boundary and local effects. Using

the reduc tion method, N. V. Valishv ili [17] solve many probl ems c oncer ning the stressed--de-

formed state of spherical and conical shells with axia lly symmetr ic loading, over their entire

region of deformation, including the limiting values.

The Step Metho d for Solving a Nonlin ear Boundar y-Val ue Problem. V. I. Feodos' ev [123,

124] proposed an approach to the solution of nonlin ear probl ems in shell the ory whic h is

based on the consi derati on of defo rmat ion as a time-variant process. Time is regarde d as a

p a r a m e t e r d e f i n i n g t h e d e v e l o p m e n t o f t he d e f o r m a t i o n s. T h e s t a ti c e q u a t i o n s of e q u i l i b r i u m

a r e r e p l a c e d b y eq u a t i o n s o f m o t i o n c o n t a i n i n g i n e r t i a l t e r m s. T h e ot h e r r e l a t i o n s r e m a i n

u n c h a n g e d , a l t h o u g h t h e f u n c t i o n s a p p e a r i n g i n t he m a r e n o w a l s o d e p e n d e n t o n t im e , I t is

assumed that the forces are given functio ns of time. In the case of static loading it is as-

sumed that the vari ation of the forces is proportion al to time. By mean s of the Bubnov--

G a l e r k i n m e t h o d , t h e p r o b l e m i s r e d u c e d t o a C a u c h y p r o b l e m f o r t h e c o e f f i c i e n t s o f t h e d e -

sired expansions of the displace ments, and these coeffi cients depend o n a time parameter.

T h e r e s u l t i n g C a u c h y p r o b l e m c a n b e i n t e g r a t e d by a s t e p w i s e n u m e r i c a l m e t h o d . O n t h e b a s i s

of this method a spherical cupola can be calculated over its entire re gion of deformation.

T h e L i n e a r i z a t i o n M e t h o d . T h e l i n e a r i z a t i o n m e t h o d o f s o l vi n g n o n l i n e a r b o u n d a r y - v a l u e

problem s [133, 53] is an analog of Newton's method for solving a system of nonlin ear e qua-

tions in the functio nal space of vector function s of the desired solutions of the nonlinear

bounda ry-va lue problem. In this method we carry out the lineariz ation of a system of dif-

f e r e n t i a l e q u a t i o n s a n d b o u n d a r y c o n d i t i o n s a n d c o n s t r u c t a n i t e r a t i v e p r o c e s s f o r s o l v i n g

t he n o n l i n e a r b o u n d a r y - v a l u e p r o b l e m , f o r w h i c h w e a r e g i v e n t h e i n i t i a l a p p r o x i m a t i o n , a n d

a t e a c h s te p w e s o l v e a l i n e a r b o u n d a r y - v a l u e p r ob l e m . I n s h e l l - t h e o r y p r o b l e m s a l in e a r

b o u n d a r y - v a l u e p r o b l e m c a n b e s o l v e d b y t h e m e t h o d o f d i s c r e t e o r t h o g o n a l i z a t i o n [ 30 ],

T h e p r e v i o u s l y m e n t i o n e d m e t h o d o f r e d u c i n g a n on l i n e a r b o u n d a r y - v a l u e p r o b l e m t o a s y s -

tem of nonlinear e quations and a Cauchy problem, as noted in the mono grap hs by A. V. Karmishi n

et el. [73], ... has a serious shortc oming due to the prope rties of probl ems for shell-type

structures. As a result of this, it is applic able to a fairly nar row class of problems, de-

fined by the characteris tic len gth of the shell, and consequ ently cann ot be used for automa t-

i n g th e c a l c u l a t i o n s of s h e l l - t y p e s t r u c t u r e s . T h e r e a s o n f o r th i s i s a p p a r e n t l y t h a t i n

t h e s o l u t i o n o f C a u c h y p r o b l e m s , o w i n g t o t h e i r n o n p h y s i c a l n a t u r e , t h e r e m a y a r i s e d i f -

f i c u l t i e s c a u s e d b y t h e r i g i d i t y o f t h e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s , T o s o l v e r i g i d

C a u c h y p r o bl e m s , w e m u s t u s e s p e c i a l m e t h o d s w h i c h e n t a i l a d d i t i o n a l c o m p u t a t i o n a l d i f f i c u l -

ties. In these cases it is found

th t

f o r t h e s o l u t i o n o f n o n l i n e a r b o u n d a r y - v a l u e p r o b l e m s

it is more effectiv e to use ite rative proce sses s uch that at each step we solve a linear

b o u n d a r y - v a l u e p r o b l e m w h i c h a d m i t s o f a s o l u t i o n b y s o m e s t a b l e n u m e r i c a l m e t h o d o f f a c t o r -

ization, when in its real izat ion the process of solving the Cauchy probl ems in bounded in-

t e r v a l s o f i n t e g r a t i o n is s t a b l e [ 7 3 , 3 6, 5 3 ]. T h e a b o v e - m e n t i o n e d p r o c e s s of l i n e a r i z a t i o n

w i t h t h e u s e o f t h e d i s c r e t e - o r t h o g o n a l i z a t i o n m e t h o d i s o n e m e t h o d o f t h i s k i n d. U s i n g s u c h

an approach, V. I. Myac henko v and A. I. Frolov [73, 36, 88] solved a number of nonli near

problem s for shells of revol utio n in the subcritical region,

Ya. M. Grl gorenk o and N. N. Kryukov [48, 53] gave a gener aliza tion of this approac h to

t h e c a se o f s o l v i n g n o n l i n e a r p r o b l e m s f o r s h e l l s o f r e v o l u t i o n o v e r t h e e n t i r e r e g i o n o f

d e f o r m a t i o n , i n c l u d i n g t h e l i m i t i n g v a l u e s o f t h e l oa d . T h i s a p p r o a c h i s b a s e d o n r e g a rd i n g

t h e l oa d a s c o m p o n e n t s o f t h e d e s i r e d v e c t o r f u n c t i o n , w h i c h d e p e n d s o n t h e c h a r a c t e r i s t i c

parameter, whe re the paramet er selecte d is the

m ximum

valu e of the deflection. This trans-

f o r m a t i o n e n a b l e s u s t o o b t a i n a u n i q u e l y d e f i n e d p r o c e s s f o r c o n s t r u c t i n g t h e r e l a t i o n b e -

tween loading and deflection.


9/18

T h e M e t h o d o f P a r a m e t r i c C o n t i n u a t i o n o f t h e S o l u t i o n, I n s o l v i n g n o n l i n e a r b o u n d a r y -

v a l u e p r o b l e m s f o r s y s t e m s o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s b y t h e m e t h o d o f r e d u c t i o n t o

n o n l i n e a r e q u a t i o n s a n d a C a u c h y p r o b l e m o r b y t he m e t h o d o f l i n e a r i z a t i o n, w e s o m e t i m e s

e n c o u n t e r d i f f i c u l t i e s c a u s e d b y t h e c h o i c e o f t h e i n i t i a l a p p r o x i m a t i o n t o e n s u r e c o n v e r -

g e n c e of t h e i t er a t i v e p r o c e s s. O n e w a y o f o v e r c o m i n g t h e s e d i f f i c u l t i e s i s t o u s e t h e m e t h -

od of param etric c onti nuat ion [133, 53]. In this method, instead of the origin al nonline ar

b o u n d a r y - v a l u e p r o b l e m, w e c o n s i d e r a b o u n d a r y - v a l u e p r o b l e m c o n t a i n i n g a p a r a m e t e r , s o t h a t

a t i t s i n i t i a l v a l u e t h e s o l u t i o n o f t h e p r o b l e m i s k n o w n a n d a t t h e m a x i m u m v a l u e o f t h e

p a r a m e t e r t h e b o u n d a r y - v a l u e p r o b l e m i s e q u i v a l e n t t o t h e o r i g i n a l o n e , T h u s, t h e d e s i r e d

vecto r fun ction can be regard ed as a funct ion of the paramete r, and this enables us to for-

m u l a t e t h e C a u c h y p r o b l e m f o r t h e d e s i r e d v e c t o r f u n c t i o n o n t h e b a s i s o f t h e p a r am e t e r .

T h i s i s a c h i e v e d b y d i f f e r e n t i a t i n g w i t h r e s p e c t t o th e p a r a m e t e r t h e s ys t e m o f e q u a t i o n s

w e h a v e o b t a i n e d. T h e s o l u t i o n o f t h e C a u c h y p r o b l e m c a n b e f o u n d d i r e c t l y b y i n t e g r a t i ng

t h e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s b y s o m e n u m e r i c a l m e t h o d , o r e l s e w e c a n u s e a d i s c r e t e

m e t h o d f o r i t s s o l u t io n , s o l v i n g t h e n o n l i n e a r b o u n d a r y - v a l u e p r o b l e m a t e a c h s te p , f o r e x a m p l e

by the linea rizat ion method. In this connec tion I. I. Voro vich [21] believ es

th t

t h e m e t h o d

o f p a r a m e t r i c c o n t i n u a t i o n c a n b e t r e a t ed a s a v a r i a n t o f t h e s t e p m e t h o d p r o p o s e d b y V . I .

Feodo s'ev [123, 124]. If the load is used as the parameter, then in solving the prob lem we

f i n d t h e v a l u e o f t h e d e s i r e d v e c t o r f u n c t i o n f o r s o m e r a n g e o f t h i s p ar a m e t e r ,

I. I. Vor ovi ch and V. F. Zipalova [21] showed that the metho d of parame tric conti nuati on

e n a b l e s u s t o o b t a i n b y c o m p u t e r t h e n u m e r i c a l v a l u e o f a n y f u n c ti o n a l o f t h e s o l u t io n w h e n

t h e p r o b l e m i s s o l v e d b y th e d i r e c t m e t h o d i n h i g h a p p r o x i m a t i o n s . T h e m e t h o d i s i l l u s t r a t e d

b y u s i n g t h e s o l u t i o n o f a p r o b l e m c o n c e r n i n g t h e a c t i o n o f a u n i f o r m p r e s s u r e a n d a co n -

centrated load on a spherical cupola. In [22, 23] I, I. Voro vich and N, I, Minako va, using

t h e m e t h o d o f p a r a m e t r i c c o n t i n u a t i o n o f t h e s o l u t i o n a n d t h e B u b n o v - G a l e r k i n p r o c e d u r e ,

o b t a i n e d t h e s o l u t i o n o f a p r o b l e m c o n c e r n i n g t h e d e f o r m a t i o n o f a n o n s h a l o w s p h e r i c a l s h e ll ,

T h e y e s t a b l i s h e d t h e l i m it s o f a p p l i c a b i l i t y o f t h e r e s u l t s o f t h e c a l c u l a t i o n s p e r f o r m e d f o r

shallow shells.

T h e m e t h o d o f p a r a m e t r i c c o n t i n u a t i o n w a s u s e d i n c o m b i n a t i o n w i t h t h e m e t h o d o f l i n e a r -

i z a t i o n a n d d i s c r e t e o r t h o g o n a l i z a t i o n b y A . N. F r o l o v a n d T . I. K h o d t s e v a t o i n v e s t i g a t e

the superc ritica l defo rmati on of shells of revol ution [127]. The devel opmen t of this ap-

proac h is given in a study by S. A. Kabr its and V. F. Terent'ev [68]. Prob lems c oncerni ng

the defo rmat ions of toroidal and spheri cal s hells have been solved by this meth od [127, 68],

E. I. Grig olyu k and V. I. Shala shi in [37] propos ed special forms of the proce ss of con-

t i n u o u s a n d d i s c r e t e p a r a m e t r i c c o n t i n u a t i o n o f t h e s o l u t i o n w h i c h h a v e s o m e a d v a n t a g e s i n

their algorithms. Using this approach, V. I. Shala shili n [132] solved a probl em on the de-

f o r m a t i o n o f p ar t o f a c l o s e d c i r c u l a r - t o r o i d a l s h e l l a c te d u p o n b y a u n i f o r m p r e s s u r e a n d

demon stra ted the proc ess of format ion of a depr essio n at the top of the torus. Ya. M, Grigor-

enko and N. N. Kry ukov [47, 53] used the metho d of para metri c contin uatio n of the solution to

s o l v e a p r o b l e m c o n c e r n i n g t h e d e f o r m a t i o n o f a n a n n u l a r p l a t e d a c t e d u p o n b y a t ra n s v e r s e

c o n t o u r l o a d a n d i n v e s t i g a t e d t h e p r o c e s s o f s o l v i ng t h e p r o b l e m a s t h e p a r a m e t e r v a r i e s .

A. V. Koro vaits ev [81, 82], using a direct transi tion to the Cauchy prob lem and applying the

p a r a m e t r i c - c o n t i n u a t i o n m e t h o d [ 1 33 ] , o b t a i n e d t h e so l u t i o n o f a p r o b l e m c o n c e r n i n g t h e d e -

f o r m a t i o n o f n o n s h a l l o w s p h e r i c a l s h e ll s . T h e q u e s t i o n o f th e f o r m u l a t i o n o f t h e b o u n d a r y

conditi ons at the top of the shell was investigated. It should be noted

th t

the metho d of

p a r a m e t r i c c o n t i n u a t i o n o f t h e s o l u t i o n r e q u i r e s a l a r g e a m o u n t o f c o m p u ta t i o n , s i n c e a n o n -

linear bounda ry-va lue probl em is solved at each step [133, 53].

N u m e r i c a l M e t h o d s B a s e d o n V a r i a t i o n a l P r i n c i p l e s , F o r t he s o l u t i o n of n o n l i n e a r s h e l l -

t h e o r y p r ob l e m s , v a r i a t i o n a l m e t h o d s a r e a l s o u s e d i n a d d i t i o n t o t he m e t h o d s l i s t e d a b o v e ,

M. S. Kornish in [79] and B. Ya, Kantor [69] made extensiv e use of Ritz's metho d for solvi ng

this class of problems. In the real izat ion of this method, a proced ure reduc ing the origina l

v a r i a t i o n a l p r o b l e m t o a s y s t e m o f n o n l i n e a r a l g e b r a i c e q u a t i o n s i s u s e d; t h e a l g e b r a i c s y s -

tem is then solved numerically .

U s i n g t h e R i t z m e t h o d a n d t h e r e p r e s e n t a t i o n o f t h e d e s i r e d f u n c t i o n s i n d i s p l a c e m e n t s ,

M . S. K o r n i s h i n o b t a i n e d s o l u t i o n s f o r a n u m b e r o f p r o b l e m s c o n c e r n i n g p l a t e s a n d s h a l l o w

shells [79]. B. Ya. Kantor, solving nonlin ear prob lems in the theory of shells of vari able

t h i c k ne s s , u s e d t h e v a r i a t i o n a l e q u a t i o n s o f m i x e d t y p e p r o p o s e d b y N, A. A l u m y a ~ [ 4 ] . H e

s o l v e d p r o b l e m s c o n c e r n i n g t h e a x i a l l y s y m m e t r i c d e f o r m a t i o n o f c o n ic a l a n d s p h e r i c a l c u p o l a s

a c t e d u p o n b y f o r c e s a n d t e m p e ra t u r e s . O n e o f th e f u n d a m e n t a l c o n d i t i o n s f o r t he e f f e c t i v e

9


10/18

use of this approach is the automati on of the computing process. This approach has been

further developed in the variati onal-se gmental method, in which a variational method is

used for individual segments, after whic h the joining cond ition is realized [70].

Some questions of improving the effectiveness of the numerical realiz ation of varia -

tional method s and the Bubnov-Ga lerkin method in solving nonlinear problems in the theory

of plates and shells are discussed in I. V. Svirskii's monograp h [116].

We should also mentio n at this point the surve y articles by i. I, Vorovich and N. I.

Minakova [24] and by E. I. Grigolyuk and V. I. Mamai [35], which are devote d to the use

of numerical method s in investigatin g nonlinear deformations of spherical shells.

5. Nonlinear Problems. Shells of Variou s Shapes. The solution of two-dimensi onal

nonlinear problems in the theory of plates and shells entails severe computational difficu l-

ties, and therefore even fewer such classes of problems in shell theory than of linear

problems have been brought to numerical realization. Here we shall discuss only a few ap-

proaches to the numerical solution of nonlinear problems in shell theory whic h have been

most widely used for sp ecific calculations.

The Finite-Difference Method. The finite-difference method is widely used for solving

two-dimensional nonlinear boundary-va lue problems in the theory of flexible shells and plates.

One of the first to investiga te the properties of the use of this method in solving nonlinear

problems on the deforma tion of flexible plates and shells is M. S. Kornishin [79, 80]. He

obtained numerical results for vario us cases of plate and shell deformatlon~ He considered

questions concerning the realiza tion of the FDM at boundary points and gave an esti mate of the

the effectiveness of vario us met ho ds and of the error in the solutions obtained. He used

computers for solving the systems of nonlinear algebraic equations,

A. S. Vol'mir and A. Yu. Birkgan [14] , using the FDM, obtain ed soluti ons of a number

of problems on the deforma tion of flexible plates and shells. The problems were solved on

computers on the basis of the method of successive approximations. The investigators achieved

the necessa ry accuracy, which was greater than the accuracy of the solutions obtained by varia-

tional approaches in cases with a small number of parameters.

The solution of nonlinear problems for shells of complicated shape in the works of V. I.

Gulyaev and E. A. Gotsul yak [59, 34, 26] was carried out by the FDM, using computers, Initial-

ly the vector d ifferentia l relations are replaced with their vector finit e-diffe rence analogs,

and then, by project ion in a local basis, the transition to scalar relations is made, Such

an approach enables us to eliminate completely the error caused by rigid displace ments in ap-

proximating the covariant derivativ e of the vecto r function, whic h leads to a much faster

convergence of the solutions of shell-theor y problems, In additi on to this, we must also note

some other important advant ages of the proposed scheme= the valid ity of the differe nce rela-

tions at the break-poin ts of t h e middle surface of the shell along the network lines, the ab-

sence of any nodes outside the contours, ~nter al ~ at the free boundary, and the reduction

of the order of diffe rentia tion of the functions describing the geometry of the middl e sur-

face of the shell. The resulting system of nonlinear difference equations can be solved by

the parametric-co ntlnua tion method [60 , i33], usin g Newton's method for a fixed value of the

parameter. The method has been used for solving problems concerning tube shells with variable

configurati on of their cross sectio n and a curvilinear center llne, and also those concernin g

toroidal shells made up of skew cylinders.

The Finite-E lement Method. The use of the FEM in solving nonline ar boundary -value prob-

lems in shell theory leads to a nonlinear system of algebraic or transcendental equati ons

and its num eric al solution. To solve such systems, V. A, Postno v and his pupil s [105-107]

propo se using vario us successive-a pproximat ion procedures, In particular, for calculating

flexible composite shells (cylindrical and toroidal), the solution of the nonline ar system

of algebraic equation s is constructed by the method of iteractions, using the idea of a step

process of loading.

Studies carried out at Gorky Univer sity under the direction of A. G, Ugodchik ov [121]

dealt with a number of problems concerning the stressed state of thick-walled shells, u sing

variou s finite-elemen t schemes in a geometrically nonlinear formulation~. The studies analyzed

the stressed--deformed state of articulat ed struct ures of the type of shells with c ut-in tubes

and with variou s kinds of Inflows, from the standpoints of the theory of thin shells and

the mecha nics of a continuous medium.

89


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Vo Vo Kabanov and L. P. Zhel eznov [66] have develop ed an algorit hm for investigat ing

t h e F E M u s e d t o o b t a i n t h e n o n l i n e a r d e f o r m a t i o n o f c i r c u l a r - c y l i n d e r s h e l l s u n d e r l o a d s

w h i c h a r e n o t a x i a l l y s y mm e t r i c. T h e f i n i t e e l e m e n t us e d i s a c u r v i l i n e a r q u a d r i l at e r a l .

T h e s y s t e m o f n o n l i n e a r e q u a t i o n s i s s o l v ed b y t h e m e t h o d o f i t e r a c t i o n s i n c o m b i n a t i o n w i t h

the step method wit h respect to the loading. The algorithm is reali zed on a comput ing complex.

Me tho ds of Reduc tion to One-D imens ional Problems. In the works of V. V. Petro v [I00,

9 9] , n o n l i n e a r b o u n d a r y - v a l u e p r o b l e m s i n t h e t h e o r y o f p l a t e s a n d s h e l l s a r e s o l v e d b y t h e

m e t h o d o f s u c c e s s i v e l o a di n g s . I n t h is m e t h o d t h e e n t i r e p r o c e s s c o n s i s t s of s u c c e s s i v e

stages, at each of which we sol ve a linear boundary- value proble m for a small part of the

load, taking account of the internal str esses and defo rmati ons known from the preced ing stage.

S u c h a n a p p r o a c h e n a b l e s u s t o l i n e a r i z e t h e o r i g i n a l n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s a n d

r e d u c e t h e p r o b l e m t o t h e s u c c e s s i v e s o l u t i o n o f l i n e a r p r o b l e m s . A t e a c h s t a t e o f l o a d in g ,

t h e p r o b l e m i s d e s c r i b e d b y a n e w c a l c u l a t i o n s y s t e m , I n t h i s m e t h o d , w h e n w e l i n e a r i z e t h e

n o n l i n e a r o p e r a t o r, w e a c t u a l l y u s e t h e F r e c h e t d e r i v at i v e . S o m e m a t h e m a t i c a l J u s t i f i c a t i o n

for this kind of iterative meth od is given in L. Collatz s book [77]~ The metho d of succes~

sive loadings and t he Vlasov--Kantorovich metho d ha ve been used in combi natio n for solving a

n u m b e r o f t w o - d i m e n s i o n a l n o n l i n e a r p r o b l e m s r e l a t i n g t o p l a t e s a n d s h e l l s,

T h i s c o m b i n a t i o n o f m e t h o d s i s w i d e l y u s e d i n t h e w o r k s o f V . A . K r y s k o [ 8 3 ] f o r c a l -

c u l a t i n g f l e x i b l e s h e l l s w i t h r e c t a n g u l a r c r o s s s e c t i o n i n t he p l a n v i e w , T h e p r oc e s s y i e l d s

a t e a c h a p p r o x i m a t i o n a s y s t e m o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s, w h i c h i s s o l v e d b y t h e

m e t h o d o f a u x i l i a r y f u n c t i o n s .

Ya. M. Grigo renko and N. N. Kryek ov [49-51, 53], in order to solve two-di mension al non-

l i n e a r p r o b l e m s i n s h e l l t h e o ry , u s e d a n e f f e c t i v e a p p r o a c h b a s e d o n r e d u c i n g t h e t w o - d i m e n -

s i o n a l p r o b l e m t o a o n e - d i m e n s i o n a l o n e b y t h e m e t h o d o f s t r a i g h t l i n e s a n d s o l v i n g t h e l a tt e r

p r o b l e m b y t h e l i n e a r i z a t i o n m e t h o d i n c o m b i n a t i o n w i t h t h e m e t h o d o f d i s c r e t e o r t h o g o n a l i z a -

tion. This approach has been used for solvin g a number of proble ms relating to the defor ma-

t i o n o f c i r c u l a r p l a t e s a n d s h e l l s o f r e v o l u t i o n w i t h v a r i a b l e t h i c k n e s s i n t w o c o o r d i n a t e

d i r e c t i o n s u n d e r a l o a d w h i c h i s n o t a x i a l l y s y m m e t r i c [ 4 9 -5 1 ] ,

Ya. M. Grigor enko and A. M. Timonin [54, 55] propos ed an appro ach to the solution of

t w o - d i m e n s i o n a l p r o b l e m s c o n c e r n i n g t h e d e f o r m a t i o n o f f l e x i b l e s h e l l s of r e v o l u t i o n w i t h

v a r i a b l e p a r a m e t e r s a l o n g t h e g e n e r a t o r u n d e r l o a d s w h i c h a r e n o t a x i a l l y s y m me t r i c . F o r

this class of nonlin ear problems, using Fourier series in the circula r coordinate, it is pos-

sible to separat e the variable s, as in the case of a hinge -supp orted flexib le plate [120],

and reduce the problem to a one-di mensi onal one, retain ing a finite number of terms of the

e x p a n s i o n . T h e s o l u t i o n o f t h e p r o b l e m so o b t a i n e d i s f o u n d b y t he m e t h o d o f s i m p l e i t e r a -

tion, whi ch makes it poss ible to separate the entire system of equations into subsystems of

mini mal ordera Stepwi se loading [55] is used for satisf actory converg ence of the process,

L i n e a r b o u n d a r y - v a l u e p r o b l e m s a r e s o l v e d b y a s t a b l e n u m e r i c a l m e t h o d [ 3 0 ]. S o l u t i o n s f o r

s o m e s h e l l s o f r e v o l u t i o n h a s b e e n o b t a i n ed .

V. V. Kabanov used the follow ing appr oach [67, 66] for inve stigati ng the nonlinear de-

f o r m a t i o n of c i r c u l a r - c y l i n d e r s h e l ls . T h e d e s i r e d s o l u t i o n o f t h e o r i g i n a l b o u n d a r y - v a l u e

probl em for a system of partial dif ferent ial equatio ns is sought in the form of a series in

t r i g o n o m e t r i c f u n c t i o n s. U s i n g t h e B u b n o v - G a l e r k i n p r o c e d ur e , t h e t w o - d i m e n s i o n a l p r o b l e m

is reduce d to a one-d imensi onal one. For each harmonic, we obtain a related s ystem of ordinary

d i f f e r e n t i a l e q u a t i o n s . T h i s s y s t e m i s s o l v e d b y a s te p m e t h o d w i t h r e s p e c t t o l o a d i ng , T h e

n o n l i n e a r b o u n d a r y - v a l u e p r o b l e m i s s o l v e d b y

the metho

of linearization, and the linear

p r o b l e m i s s o l ve d b y t he m e t h o d o f d i f f e r e n c e f a c t o r i z a t i o n . U s i n g t h e f o r e g o i n g a p p r o a c h ,

the stressed--deformed state of cylind rical shells has been found in cases with loadings

w h i c h a r e n o t a x i a l l y s y m m e tr i c .

6. Conclusion. Summing up our analysi s of some appro aches to the numeri cal solution

o f l i n e a r a n d n o n l i n e a r b o u n d a r y - v a l u e p r o b l e m s i n t h e t h e or y o f s h e l ls , w e n o t e t h e f o l l o w -

ing funda mental features.

I n t h e n u m e r i c a l s o l u t i o n o f l i n e a r p r o b l e m s i n s h e l l t h e or y , w e u s e e s s e n t i a l l y m e t h o d s

w h i c h h a v e a l r e a d y b e e n t r i e d an d p ro v e n . I n t he c a s e o f o n e - d i m e n s i o n a l n o n l i n e a r p r o b l e m s ,

especi ally if the solut ion is found in the entire region of deformation, including the limit

point s and bran ch points [24, 35], the search for the most e ffectiv e meth ods of soluti on is

s t i l l c o n t i n u i n g. F o r t w o - d i m e n s i o n a l n o n l i n e a r p r o b l e m s, t h e r e ar e a f e w s t u d i e s i n w h i c h

s u p e r c r i t i c a l d e f o r m a t i o n i s i n v e s t i g a t e d ~

891


12/18

N u m e r i c a l m e t h o d s a r e w i d e l y u s e d i n t h e s o l u t i o n o f l i n e ar p r o b l e m s a n d n o n l i n e a r

problem s in the subcriti cal region.

One characteris tic of the numerical so lution of problems is that in each of the propos ed

approa ches there is a tendency to cover a certain class of problems in shell theory and make

effecti ve use of all the possib iliti es of the numer ical solutio n of the problem s of that

class. That is to say, the need to solve individual problems leads the investigator to

s e a r c h f o r a c l a s s of p r o b l e m s w h i c h i n c l u d e t h e p r o b l e m s o r i g i n a l l y e n c o u n t e r e d . H e r e w e

need not as yet raise the quest ion of automa ting the entire process oY individu al stages of

the solution of problems on computers, wh ich require s muc h more eff ort and time. As examples

o f s u c h a p p r o ac h e s , w e m a y c i t e t h e s o l u t i o n s o f c e r t a i n c l a s s e s o f s h e l l - t h e o r y p r o b l e m s

mentioned in [iii, 71, 2].

T h e ne x t s t e p i n t h e n u m e r i c a l s o l u t i o n o f s h e l l - t h e o r y p r o b l e m s m a y b e r e g a r d e d a s b e i n g

t h e c r e a t i o n o f p a r t l y o r t o t a l l y a u t o m a t e d c o m p u t i n g c o m p l e x e s f o r s o l v i n g c e r t a i n c l a s s e s

of problems. As a rule, a totally automated computi ng compl ex can be worked out for a more

restrict ed class of problems. One of its main adv antag es is that it can be used by unskil led

personnel. For broader classes of problems, the computi ng complex consists of a standard and

a n o n s t a n d a r d p a r t. T h e s t an d a r d p a r t o f t he c o m p l e x i s f u l l y a u to m a t e d , w h i l e t h e n o n s t a n d -

ard part is set up for each subclass of the given class of problems ac cording to the minim al

i n i t i a l i n fo r m a t i o n. S u c h a p p r o a c h e s t o t he s o l u t i o n o f s h e l l - t h e o r y p r o b l e m s h a ve b e e n r e a l -

ized in [41, 73, 104].

F u r t h e r p r o g r e s s i n t h e f i e ld o f a l g o r i t h m i z a t i o n , p r o g r a m m i n g , a n d c o m p u t e r i m p r o v e m e n t

will lead to the creat ion of progr am packages for solving appl ied proble ms in broad fields,

i n c l u d i n g p r o b le m s i n t h e t h e o r y o f s h e ll s . T h e p a c k a g e s o f a p p l i e d p r o g r a m s a r e w o r k e d o u t

o n t h e b a s i s o f t h e m o d u l a r p r i n c i p l e o f p r o g r a m i n g , i . e. , r e p r e s e n t i n g t h e e n t i r e p r o b l e m

a s t h e s u m t ot a l of r e l a t i v e l y e l e m e n t a r y p r o b l e m s w h i c h a r e i n d e p e n d e n t i n s o m e s en s e. C o n -

s e q u e n t l y t he e n t ir e p a c k a g e c o n s i s ts o f i nd i v i d u a l b ut i n t e r c o n n e c t e d p r o g r a m m o d u l e s . I n

shell -theo ry problems, pa ckage s of applied prog rams have been formulated in [88, 46, 59].

In the reali zati on of individual programs, computing complexes, and packages of applied

p r o g r a m s f o r t h e c o m p u t e r i z e d n u m e r i c a l s o l u t i o n o f s h e l l - t h e o r y p r o b l e m s , a s i g n i f i c a n t r o l e

is played by the followin g factors. The effect ivenes s of const ructin g an appro ach to the

s o l u t i o n of t h e p r o b l e m d e p e n d s t o a l a r g e e x t e n t o n t h e r a t i o n a l c o r r e s p o n d e n c e b e t w e e n t h e

class of problem s considered and the metho d chosen for its numer ical solution. Sometime s a

failure to take account of some featur es of the class of problem s in questio n can have a

s e r i o u s i m pa c t o n t he e f f e c t i v e n e s s o f t h e a p p r o a c h u s e d . I n t h e p r a c t i c a l a p p l i c a t i o n o f

m e t h o d s o f n u m e r i c a l a n a l y s i s t o t he c a l c u l a t i o n o f s h e l l - t y p e s t r u c t u r e s a m a j o r r o l e i s

p l a y e d b y t h e p h y s i c a l i n t e r p r e t a t i o n o f t h e re s u l t s o b t a i n e d , w h i c h i s v e r y i m p o r t a n t i n

e s t i m a t i n g t h e r e l i a b i l i t y o f t h e n u m e r i c a l s o l u t i o n o f t h e p r o b l em . T o g e t h e r w i t h m a n y

o t h e r i n d u c t i v e m e t h od s , t h e e v a l u a t i o n o f t h e r e s u l t s f r o m t h e p h y s i c a l p o i n t o f v i e w s o m e -

times makes it possib le to avoid the oretical errors. In the elabor ation of approac hes to

t h e n u m e r i c a l s o l u t i o n o f s h e l l - t h e o r y p r o b l e m s t h e r e n a t u r a l l y a r i s e s t h e q u e s t i o n of s o m e

optim izatio n of each stage of the solutio n of the problem, However, in the broad sense, an

o p t i m a l o r r a t i o n a l a p p r o a c h m u s t e m b r a c e t h e e n t i r e c a l c u l a t i o n p r o c e s s o f t h e s o l u t i o n o f

a given class of problems in shell theory, takiBg ac count of all its stag es [85].

C o m p u t e r s c a n b e e f f e c t i v e l y u t i l i z e d t o s o l v e s h e l l - t h e o r y p r o b l e m s o n l y w h e n t h e y a r e

t a k e n i n a r a t i o n a l c o m b i n a t i o n w i t h n u m e r i c a l m e t h o d s a n d m a t h e m a t i c a l m o d e l s o f t h e c l a s s e s

o f p r o b l e m s u n d e r c o n s i d e r a t i o n.

i.

2

3.

5.

L I T E R A T U R E C I T E D

A . A. A b r a m o v , T r a n s f e r of b o u n d a r y c o n d i t i o n s f o r s y s t e m s o f o r d i n a r y d i f f e r e n t i a l

equations (variant of the facto rizat ion metho d), Zh. Vychisl. Mat. Fiz., No. 3, No, 3,

542-545 (1961).

Ya. M. Grigorenk o (editor), Algor ithms and Programs for Solving Problems in the Mecha nics

of a Defor mable Solid [in Russian], N auko va Dum~a, Kiev (1976).

A. V. Aleksandrov, B. Ya. Lashchenik ov, N. N. Shaposhnikov, and V. A, Smirnov, Me thods

of Calculatin g Rod Systems, Plates, and Shells by the Use of Compute rs [in Russian],

Stroiizdat, Moscow (1976), Part 2.

N. A. Alumyag, A varia tiona l formula for the inves tigat ion of thin-walled elastic shells

in the pos t-c rit ica l stag e, Prikl. Mat. Me kh., 1__4, No. 2, 197-2 02 (1950).

S. A. Ambartsum yan, Theor y of Aniso tropic Shells [in Russian], Nauka, M osc ow (1961),

892


13/18

6 . R . E . B e l l m a n a n d R . E . K a l a b a , Q u a s i l i n e a r i z a t i o n a n d N o n l i n e a r B o u n d a r y - V a l u e P r o b l e m s

[ R u s s i a n t r a n s l a t io n ] , M i r , M o s c o w ( 1 96 8 ).

7 . O . M . B e l o t s e r k o v s k i i a n d P . I . C h u s h k i n , " A n u m e r i c a l m e t h o d f o r i n t e g r a l r e l a t i o n s , "

Z h . V y c h i s l . M a t . F i z . , N o . , N o . 5 , 7 3 1 - 7 3 9 ( 1 9 6 2 ) .

8 . I . S . B e r e z i n a n d N . P . Z h i d k o v , M e t h o d s o f C a l c u l a t i o n [ in R u s s i a n ] , F i z m a t g i z , M o s c o w

(1962), Vol. 2.

9 . V . L. B i d e r m a n , " U s e o f t h e m e t h o d o f f a c t o r i z a t i o n f o r t h e n u m e r i c a l s o l u t i o n o f p r o b l e m s

i n s t r u c t u r a l m e c h a n i c s , " I n z h . Z h. M e k h . T v e r d . T e l a , N o , 5 , 6 2 - 6 6 ( 1 9 6 7 ).

I 0. V . L . B i d e r m a n , " S o m e c o m p u t a t i o n a l m e t h o d s f o r s o l v i n g p r o b l e m s i n s t r u c t u r a l m e c h a n i c s

w h i c h h a v e b e e n r e d u c e d t o o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , " R a s c h e t y P r o c h n o s t ' , N o . 1 7 ,

8 - 3 6 ( 1 9 7 6 ) .

I I, V . L . B i d e r m a n , M e c h a n i c s o f T h i n - W a l l e d S t r u c t u r e s [ i n R u s s i a n ] , M a s h i n o s t r o e n i e ,

M o s c o w ( 1 9 7 7 ) .

1 2. I . A . B i r g e r , S o m e M a t h e m a t i c a l M e t h o d s f o r t h e S o l u t i o n o f E n g i n e e r i n g P r o b l e m s [ i n

R u s s i a n ] , O b o r o n g i z , M o s c o w ( 1 9 5 6 ) .

1 3. I . A . B i r g e r , C i r c u l a r P l a t e s a n d S h e l l s o f R e v o l u t i o n [ i n R u s s i a n ] , O 5 o r o n g i z , M o s c o w

(1968).

1 4. Y u . A. B i r k g r a n a n d A . S. V o l ' m i r , " I n v e s t i g a t i o n o f l a r g e d e f l e c t i o n s o f a r e c t a n g u l a r

p l a t e b y m e a n s o f d i g i t a l c o m p u t e r s , " I zv . A k a d . N a u k S S S R , M e k h a n i k a i M a s h i n o s t r o e n i e ,

No. 2, 100- 106 (1959).

1 5. V . V . B o l o t i n a n d Y u . N. N o v i c h k o v , M e c h a n i c s o f M u l t i l a y e r S t r u c t u r e s [ i n R u s s i a n ] ,

M a s h i n o s t r o e n i e , M o s c o w ( 1 98 0 ) .

1 6 . V . N . B u l g a k o v , S t a t i c s o f T o r o i d a l S h e l l s [ i n R u s s i a n ] , I z d . A k a d , N a u k U k r . S S R ( 1 9 6 2 ) .

1 7. N . V . V a l i s h v i l i , C o m p u t e r M e t h o d s f o r C a l c u l a t i n g S h e l l s of R e v o l u t i o n [ i n R u s s i a n ] ,

M a s h i n o s t r o e n i e , M o s c o w ( 1 97 6 ).

1 8. V . S. V l a d i m i r o v , " A n a p p r o x i m a t e s o l u t i o n of a b o u n d a r y - v a l u e p r o b l e m f o r a s e c o n d-

o r d e r d i f f e r e n t i a l e q u a t i o n , " P r i k l . M a t . M e k h . , 1 9 , N o , 3, 3 1 5 - 3 2 4 ( 1 9 5 5 ) .

1 9. V . Z. V l a s o v , G e n e r a l T h e o r y of S h e l l s a n d I t s A p p l i c a t i o n s i n T e c h n o l o g y . S e l e c t e d

S t u d i e s [ i n R u s s i a n ] , I z d . A k a d . N a u k S S S R , M o s c o w ( 1 9 6 2 ) , V o l . i .

2 0. A . S . V o l ' m i r , F l e x i b l e P l a t e s a n d S h e l l s [ i n R u s s i a n ] , G o s t e k h i z d a t , M o s c o w ( 1 9 5 6 ).

2 1. I . I . V o r o v i c h a n d V . F , Z i p a l ov a , " O n t h e s o l u t i o n o f n o n l i n e a r b o u n d a r y - v a l u e p r o b l e m s

i n t h e t h e o r y o f e l a s t i c i t y b y t h e m e t h o d o f p a s s a g e t o a C a u c h y p r o b l e m , " P r i k l , M a t .

Mekh ., 29, No. 2, 121- 128 (1965).

2 2. I . I . V o r o v i c h a n d N . I . M i n a k o v a , " S t a b i l i t y o f a n o n s h a l l o w s p h e r i c a l c u p o l a , " i b i d , ,

32, No. 2, 332 -33 8 (1968).

2 3 . I . I . V o r o v i c h a n d N . I . M i n a k o v a , " I n v e s t i g a t i o n o f t h e s t a b i l i t y o f a n o n s h a l l o w

s p h e r i c a l c u p o l a i n h i g h a p p r o x i m a t i o n s , " I zv . A k a d . N a u k S S S R , Me k h . T v e r d . T e l a, N o .

2, 121 -128 (1969).

2 4 . I . I . V o r o v i c h a n d N . I. M i n a k o v a , " T h e p r o b l e m o f s t a b i l i t y a n d n u m e r i c a l m e t h o d s i n

t h e t h e o r y o f s p h e r i c a l s h e l l s , " i n: A c h i e v e m e n t s o f S c i e n c e a n d T e c h n o l o g y . M e c h a n i c s

o f D e f o r m a b l e S o l i d s [ i n R u s s i a n ] , V I N I T I A k a d o N a u k S S S R ( 1 9 7 3 ) , V o l , 7 , p p . 5 - 8 6 ,

2 5. A . G . G a b r i l ' y a n t s a n d V . I. F e o d o s ' e v , " A x i a l l y s y m m e t r i c f o r m s of e q u i l i b r i u m o f a n

e l a s t i c s p h e r i c a l s h e l l s a c t e d u p o n b y a u n i f o r m l y d i s t r i b u t e d p r e s s u r e , " P r i k l , M a t ,

Mek h., 2_~5, No. 6, 109 1- 11 01 (1961),

2 6. V . V . G a i d a i c h u k , E . A . G o t s u l y a k , a n d V . I. G u l y a e v , " B r a n c h i n g o f t h e s o l u t i o n s o f

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