Geometric Shell Stability Theory

Table of contents :
Front Cover
TABLE OF CONTENTS
U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM
INTRODUCTION
1. STRICTLY CONVEX SHELLS DURING SUPERCRITICAL DEFORMATIONS.
2. LOSS OF STABILITY OF STRICTLY CONVEX SHELLS.
3. CYLINDRICAL SHELLS DURING SUPER CRITICAL DEFORMATIONS.
SUPPLMENT 1 SOME QUESTIONS OF DYNAMICS.
SUPPLEMENT 2 ISOMETRIC TRANSFORMATIONS OF CYLINDRICAL SURFACES.

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FTD-ID(RS)T-2219-78 %

I

FOREIGN TECHNOLOGY DIVISION

GEOMETRIC SHELL STABILITY THEORY by /f. y. Pogorelov

Approved for public release; distribution unlimited.

I

FTD- ID(RS)T-2219-78

UNEDITED MACHINE TRANSLATION FTD-ID(RS)T-2219-78 MICROFICHE NR:

23*February 1979

ylf£)- 7f' d'COOjLji

GEOMETRIC SHELL STABILITY THEORY By:

A. V. Pogorelov

English pages: Source:

459

Geometricheskaya Teoriya Ustoychivosti Obolochek, Izd-vo "Nauka", Moscow, 1966, pages 1-296

Country of origin: USSR This document is a machine translation. Requester: FTD/PHE iLppr-nu.-M-i

for

public

r p l i - 3 . a n ; --------------------------------------- -----

distribution unlimited.

THIS TRANSLATION IS A RENDITION OF THE ORIGINAL FOREIGN TEXT WITHOUT ANY ANALYTICAL OR EDITORIAL COMMENT. STATEMENTS OR THEORIES ADVOCATED OR IMPLIED ARE THOSE OF THE SOURCE ANDDO NOT NECESSARILY REFLECT THE POSITION OR OPINION OF THE FOREIGN TECHNOLOGY DlVISION.

FTD-

ID(RSJt_-2219-78

PREPARED BY: TRANSLATION DIVISION FOREIGN TECHNOLOGY DIVISION WP-AFB, OHIO.

Date 23 Feb 19

79

TABLE OF CONTENTS U. S. Board on Geographic Names Transliteration System ................................................ Introduction ...........................................

1

Chapter One. Strictly Convex Shells During Supercritical Deformations .............................

16

Chapter Two. Loss of Stability of Strictly Convex Shells ..........................................

146

Chapter Three. Cylindrical Shells During Supercritical Deformations .............................

258

Supplement I.

Some Questions of Dynamics ................

380

Supplement II. Isometric Transformations of Cylindrical Surfaces ...................................

426

U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM Block A a b 6 B B A A E e hi w 3 3 M H Pi H H K

Italic A a 5 6 B $ i r n d E 4 Hi OK 3 3 H U R a K K

J\ /I

n

A

L ,

1

u

bi

hi

w

I'i it

M

M

M,

m

to

b

b

k

H

H

H

N,

n

3

3

3

9

0 o

0

0

o .

0

KJ K3

!0

to

n

n

n

P, P

R

A

r

H

r

n

Transliteration A, a B, b v, V G. g D, d Ye , ye; E, e* Zh , zh z. z I, i Y, y K, k

Block P P C c T T y y 0 $ X X u u H H 111 UJ EJ A b h

fl

fl

y

y

Transliteration R, r S, s T, t U, u

0 X u

» •

■
*

,

•

•

m

o,= 4 •103 kg/ca* relative deformations are less than

o,/£ = 2 •10-J.

Page 12.

This means that any elastic deforaation of steel shell is accompanied by relative change in the metrics of its median surface less than 2*10'3. Therefore, if t m s deformation leads to considerable changes in the fora of shell, then it is alaost geometric bending.

Purther, the noraal coaditions of the attachaent of the edge of shell guarantee geoaetric the nondeforaability of its nedian surface

DOC * 78221901

PAliB

in the class of regular surfaces. Therefore the bending, which correspond to supercritical deroraations, belong to a broader class of piecewise-regular surfaces. Tars Beans that the surface, which reproduces the fora of shell during supercritical deforaation, aust have fin/edges. On the surface of shell, these fin/edges are saoot hed.

The nearness of the supercritical deforaation of shell to certain of its isoaetric conversion with special feature/peculiarities along lines {fin/edges) creates the specific specific character in the energy distribution of deforaation according to the surface of snell. Namely, it noticeably is concentrated in the vicinity of fin/edges. The considerations, based on variation principle, sake it possible to refine the fora of the deforaed shell near fin/edges and to deteraine strain energy in the vicinity of fin/edges depending on the geoaetric values, which relate to fin/edge. As a result functional (i (strain energy) proves to be deternined on isoaetric transforaations of aedian surface, which reproduce the fora of shell during supercritical deforaation. So we coae to the following variation principle A.

Considerable supercritical deforaation elastic-and shells under the action of the given load is close to that fora of the isoaetric “t • '- • ‘ * ' * conversion of initial surface, w m c h coaaunicates steady-state value

DOC = 78221901

PAGE

d 7

to the functional W = U p - A? .

Page 13.

This functional is determined during piecewise-regular isonetric transformations F of aediaa surface of shell. Its tern/conponent/addend

Uf

(strain energy) is determined by the

following formula:

V Sere

a*, mad

A£2

V

- main changes normal of curvatures with

transition from the initial fora of shell F to the isometric transformation P; 2 a - angle uetneen the tangential planes of surface F along fia/edge (fin/edges) and

k,

y; p - radius of curvature carved y; kt

- normal surface curvatures F in the direction, perpendicular

to fin/edge y ,

k - normal surface curvature F in the appropriate

direction; 6 - thickness of saeli, E - modulus of elasticity, and y - Poisson ratio. Constant c-0.19. integration in first term is ’......... /V/ fulfilled by surface area F, and ia remaining two .

is not exceeded time/temporary strength of materials

o. (Ve we

consider that the tensile strength is elastic limit.). The . i.

e

•

I

«
. for exaaple, after taking the convex hull of surface P. If we surface ‘

•* - -

.

.

-e~

•

F with attached edge y indicated it allow/assuaed nontrivial

DOC *

78221901

PAGB

* *

jL X .

isometric transforeation in class regular of surfaces, then the closed surface O', obviously, would allov/assuae isometric • •-

* • *

.

.

.

•

.

«-

transformation ia the class of convex surfaces. But this is impossible in view of the theorem about the unique determination for such surfaces..Affirmation xs proved.

As noted above, surface F, being rigid/inflexible in one class •

•

•

.

1

•

• .

„

«

.....

of surfaces, can be bent xn a broader class. In particular, regular, attached along the edge strictly convex surface is nonbendable in the class of regular surfaces, but besdable in the class of piecevise-regular surfaces. Of this, us convinces an example of mirror bulge. Here isometric transformation is connected with the disturbance/breakdown of regularity (by formation of fin/edge) along certain curved, that limits convex region on surface..In connection with this let us examine the following question. The how most coamon/general/total isometric transformation of regular, attached on edge, over strictly convex surface in the class of piecevise-regular •

•

*•

•

•

surfaces, if the disturbance/breakdovn of regularity you do solve along assigned/prescribed curve jr, which limits the convex region G on surface of F? »• -*• •

••

*

•

G* - part of surface F, arranged/located out of region G and •

t

•

• •

•

*•

** '

-

*

*

*

-

i- - —

—

- —

—

adjoining the edge. First of all, we confirm that during any isometric transformation of surface of F with the

DOC = 7 8 2 2 1 9 0 1

PAGE

M

.

m

, ,

,

attachaent of edge of surface gives initial conditions for the Cauchy problea indicated. The inalterability of region G* will entail the •

••

•

4

— *

♦

.

.

.

..

.

-. —

.

.. . .

inalterability of its edge y. Thus, during the isoaetric transforaation of surface of P is deformed only part G, aoreover y an edge of region G - reaains fired. Let during the isoaetric • •* • • ■ • • •

T its

transforaation P into

•

»

•

-

- » 1 ► *

0

part G transfer/con vert in

If surface G is directed by convexity to the saae side, as G, •

•

• •

*

. . . . .

W v *■

-

•—

i i ii— i i kit

•

..

«-

then F will be convex surface, it is not difficult to conclude that in this case it aust coincide with F. For this, it suffices to use the reasoning, with the aid of which is establish/installed unique • • •

. . . . .

. . .

-

—

~

V..

.

.

.

.

deteraination F in the class of regular surfaces. Thus, if surface F •

•

•

.

* •.

*e

••

* .*

•

*

.

e

allow/assuses nontrivial isoaetric transforaation, then it is •

a

•• *

•

*

*

*

*

’■ •

•

•• •

•

■• •

necessary to count that G is directed by convexity in the other . . -«e

e

. .

n .i

*

.

.

. ^

.

.

. . .

I*.

a.

. . .

direction. In this case, surfaces G and G, having overall edge y .

JS %•/

PAGE

DOC * 78221901

c o a p o M tha closed con v a x surface (Pig. 2) . let us designate it Surfacs

allow/assunes xsoaetric aappiag onto itsalf. This • 1* ....... representation consists of conpartson to each point P of region 6 of •

•

•

•

• •

«

'*

the corresponding according to rsosetry point of region G and each point P of region ti of the corresponding on isonetry point of region G.

In view of the unigue datecarnation of the closed convexs surface, the constructed isonecrtc representation of surface

a, onto

itself sust be reduced to notion or to notion and nirror reflection. 1 *■“ * • • • • * - * • ••— • Since points carved y during isoaetric representation reaaia fixed, .

•

V .

v

«•

» * « • • • * . .

.

»

.

.

.

.

*.

the natter is reduced to the nirror reflection of surface

4*

relative to a certain plane. Curve r. being fixed, sust lie/rest at t

.

•«

I- * •

*

»

i

•e

•

•

•

•

this plane. Thus, we coae to following conclusion.

-

•

DOC

78221901

PAGE

?’*'

m

Pig. 2.

Page 23.

Isoaetric transforaatioa of strictly convex regular surface, I

*

f

' a

attached on edge, in the class of piecewise-regwlar surfaces with ••

«

-

%e

e-

-ee • •

•

•

•

•

•

•

•

•

••

-

•

-

•

»•

disturbance of regularity only by lengthwise curve y, which U n i t s • M *• • -' * the convex region G, is possible oaly if curved y flat/plane, and in •

A . . . .

-

.

-

|

.

•—

.

.

.

•

•

■ / . . . .

—

- - w

«

•

*•

.

.

.

• —

this case it is reduced to the airror reflection of region G in plane by curve 7 *.

Let us turn now to the supercritical deforaationa of elastic «

•

i f *

. ,

«

*

„

.

..

— a

• a.

*— .

•

*•«

shells. First of all, let us explain the concept of supercritical defornation. By supercritical deforaatioa we sill understand defor nation, with which the shell experience/tests considerable changes in exterior fora. Sues defornations appear usually as a result of the loss of stability of the shell whea the effective load

DOC * 78221901

PAG*

** 2 ( r

reaches critical value, heace aad aaae - supercritical deforaations. • .•

•

•'

•

••

«

•

*.

•

« «

.

.

I

The materials froa which are manufactured the shells, as a rale, *

'•

•

•'

•

•

*

a- • » < % » «

—

4

do not allow/assume considerable iaternal strains. For example, for steel with tensile strength •

*•

a

%r

of vicinity by curve y is coaputed in it strain energy (Pig. 3). Proa •

"h*

...•»

.

..

*

*

.

•

i-

.

. . . . .

a

a a

,

.

.

.

demonstrative considerations about the deformation of shell in the '

'*

k

'

a.

.

.

. .

.

...

vicinity of fin/edge, ue consist that the strain energy of the chosen

DOC = 78221901

PA£>£ i

cell/eleaent it is consisted in essence froa energy of curvature in the plane, perpendicular to txu/edye and energy of expansion-coapressioa in the direction of fin/edge.

Let the section/cut of surface of P by the plane, perpendicular to fin/edge at point P, in coordinates r, z be assigned by the equation t = z(r).

a/

Let us designate through u and V shifts of the points of surface P during its deforeation in P: u - on the principal noraal, and T - on binoraal curved

y

at point P. Then, if the tangential planes of

surface P by leagthvise curve j

fora snail angle, then change in the

noraal curvature upon transfer froa surface of P and P in the direction, perpendicular to fin/edge, it sill be it is equal A,i ~ v", shere the differentiation is conducted according to to the variable

PAG*

DOC = 78221901

*4 3

Pig. 3.

Page 27.

If now a change of the noraal curvature in the sane direction upon —

■.......................

-V

transfer fron the initial fora of ft to P is designated Ah# then the total change in the noraal curvature during deforaation P in P will be equal Aj* = r" 4 Ah. Energy of curvature 0 " , the choaea cell/eleaent is deteraiaed froa •

,«.*

»

•«

. •

***

•

* • •

•* — •

»-•*■•••

•

.

•

the foraula . . . .

. . .

,

.

.

.

.

.

=-§-{ J V + A*)» o» and */ - noraal surface ............ .. J* ‘ curvature P also in the direction, perpendicular y, but froa the side of internal half-neighborhood ai11 l l __( - ^ _ = 0 . a,’>+ a«>= — 1. ffl,

©2

1

Por obtaining the second approach/approxiaaticn, let us substitute obtained values of a ti*) and

in 1 and Q we solve the systea .(2) =0 , —©,— 11— ©—2 +1 P ( 'a'l1”.= + op)- — 1 .1?)

Analogously are located the subseguent appr oach/approxiaat ions,

Page 39.

By the described aethod, being liaited to the second

DOC * 7 8 2 2 1 9 0 2

PAUh

VS^ h

approach/approximation, is obtained the value y ^ I2

max11»'|~ I.

Por the appropriate functions u, V, it will be

max |u j~0,5.

* § He focus attention

on these values because with their aid are determined the constants c( and c( * in formulas for a d u i u i voltage/stresses

(p. 2 ).

In connection with the study of the problee concerning elasto-plastic supercritical deforaations, we will now propose another sethod of solving tne variational problem for functional J. This solution, approximated actually, will be based on the demonstrative representations of the character of the deformation of the shell near fin/edge, with study of which is connected our variational problem.

In Pig. 3 to the right (page 26) they are depicted the section/cut of the deformed shell oy the plane, perpendicular to A /

fin/edge, and section/cut oy the sane plane of surface P, which approaches the form of shell. Mew the variables u, V, which we now use, are the respectively standardized/noraalized radial displacement of the point of surface If during its deformation in P and the standardized/noraalized angle or rotation to tangent. The standardization of angle is carried out in such a way that its value at point A(s=v0 ) is equal to -I.

DOC = 7 8 2 2 1 9 0 2

PAGE

V? t \

On the basis of the representation of the local character of the deformation of shell in the zone of powerful bending, it is logical to assume that after point 8 , where v=0 , value V it remains small and in the differential linkage of the variables u, V term v*/2 can be disregarded. Then communication/connection will take the fora u' 4-t»= 0 .

Purther, it is obvious, that the maximum of the bending of the deformed shell must be reached in immediate proximity of point A. Hence it follows that near point A value V ,

which is determining the

value of bending, changes oareiy, and logical to consider V* of constant in certain vicinity point A.

Page U 0 .

Taking into account two considerations indicated, let us search for the minimum of functional J on many functions u, V, which satisfy the conditions . ^

1: Flpn s < o 2. ripH s > o

-.1

v + -5- = 0,

v' — const.

« - t - v = 0.

Key: with.

Here

0

- parameter, which is subject to variation. The minimum of

functional J with given one # will be known function from JmtH = J(o). T o t

0

:

determining value J„, we minimize this function on a: Jo = min J(o).

p im a

DOC * 78221902

y f

y

Let us find function J(w). Set/assuaiog with s$# ,

l

v = j = const,

after integration we will obtain v := -j" “4“ const.

Since v--1 with s*0# then

The parameter X Bakes

siaple

sense. Specifically, ,

at which V turns into zero*

such this

X*w. Thus, with sfw we

value

s9

have

From the equation «'+ " + -£ = 0 ve find function u(s) with s$«: « = — - ^ ( S - O)* — ^ - ( s — o)» 4 - const.

Integration constant is detersined by boundary condition u(0)=0,

and

it is equal to v/3. Thus, vita s{« it will be u—-

^ -(a — af -

^r(* —

+ 5.

Page hi.

The values of functions u, V at the end • of cut (0, »)

are

respectively equal to v/3 and 0 they are initial values for the varied functions u, V on the renaming part of the seni-axis (a , «) .

DOC * 78221902

PAt>E

Since with s>« by h y p o t h W U «'+ « = 0 , that functional J can be presented in the fors

a

y=J(*'’+«5)d*+O/(*'*+«?)rf* 0

function u(s), that realizes tne ainiaue of functional on seai-axis (#,

it satisfies the equation of Euler tt|v

u = 0.

Its general solution, wnich disappears at infinity, allow/assunes the representation u = c{e°'s 4- c7ea>‘.

where u t and w 2 - roots of the characteristic equation u« + 1 = 0 with negative real part, i.e., «D,= --p-(l--0 . wj= - 7 = 5 (I+ The constants c, and c, are deterained by the conditions of the coupling of functions u, V with s*w. He have u (a) = c,eu’n +

V(O) — — U'(0) =

== j ,

— Cj«,**•" — CjWj*

= 0.

Hence O U ),

3/2