AD-756 103

LARGE DEFLECTIONS OF PLATES WITH NON- LINEAR ELASTICITY AND HOLLOW SHELLS WITH HINGE-SUPPORTED EDGES

M. S. Kornishin, et al

Foreign Technology Division Wright-Patterson Air Force Base, Ohio

26 January 19'/3

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FOREIGN TECHNOLOGY DIVISION

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LARGE DEFLECTIONS OF PLATES WITH NONLINEAR ELASTICITY AND HOLLOW SHELLS WITH HINGE-SUPPORTED EDGES

M. S,

by

Kornishin, H. N. StolyaroVj and N. I. Dedov

rr^'! D D.^

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4 Raproduced by

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Foreign Technology Division Air Force Systems Comn>aiid U, S. Air Force

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LARGE DEFLECTIONS OP PLATES WITH NONLINEAR ELASTICITY AND HOLLOW SHELLS WITH HINGE-SUPPORTED EDGES

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Translation I AU TKOKisi (FhM mm», mUmt. Mtiat. laal naaMj ""■"

M, S. Komlshln, N, N. Stolyarov, N, I. Dedov %■ »%romr OATK

1971 •«. COWTKACT OH •RANT NO.

b. rnoJBCT NO. 782

Tfl. TOTAL NO- Or »A«U

9 tft. NO. Of MKP1

3 a«. omoiNATOira *■*ORT NUMBCRISI

FTD-HT-23-l8l'4-72

10. OltTRISUTION STATCMKNT

•ft. OTHtR RCFORT NO!*) (Atf oUit numbtm Aat mmy I Mtrnpott)

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II. SUPPLEMCHTARV NOTKS

II. AftlTRACT

ta- IRONBORINC MILITARY ACTIVITY

Foreign Technology Dividion Wright-Patterson AFB, Ohio

In-this-work, method of finite differences is used to solve problems dealing with large deflections of rectangular plates l«-pian with nonlinear elasticity and hollow shells with hinge-supported edges, which are under the effect of external normal pressure uniformly distributed along the entire surface or along the central rectangular area. The curvature parameter values at which a crack in a cylindrical panel occurs are established. Critical loads and their corresponding deflections havt;,he«ri found for panels with different curvature parameters.

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EDITED TRANSLATION PTD-HT-23-]8lit-72

LARGE DEFLECTIONS OF PLATES WITH NONLINEAR ELAS- TICITY AND HOLLOW SHELLS WITH HINGE-SUPPORTED EDGES

Bv; M. S. Kornishin, N. M. Stolyarov, and N. 1. Dedov

English pages: 9

Source: AN SSSR. Kasanskiy Fiziko-Tekhnicheskly. Trudy Seminara pc Teorli Obolochek. - ,, No. 2, 1971, PP ^9-58.

Requester. FTD/PHE

Translated by: TSgt Victor Mesenzeff

Approved for public release; distribution unlimited.

TMI TtAMtLATIOM IS A RINOITidM OP TK1 ORIOI. MAL FORUM Tin fflTI<OUT AMY ANALYTICAL 09 iMTORIAL eOMMINT. STATIMiNTS OR THIORiKS APVOCATIOORUtfLIROAReTNOIiOPTHI IOURCI ANOOO NOT MSCf ISAilLV RIPLICT THB POSITIOH OR OMMON OP TNi PORIION TiCMNOLOOY OC VIIION.

PR!PARIO RYi

TRANSLATIOM DIVIUOM PORRIOM TSCMMOLOSY OIVUION WP.AP», ONIO.

FTD-HT- 23-1814-72 p^^ m 1^,

U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM

Block Italic Transliteration Block Italic Transliteration A a A a A, a P p p p R, r E 6 B 6 B, b c c C c S, s B B B t V, v T T T m T, t r r r t G, g y y y y U, u fl a n d D, d o * <P <P F, f E « E t Ye, ye; E, e* X X X X Kh, kh Ä w M OK Zh, zh u u u H Ts, ts 3 a 3 3 Z, z H H H V Ch, ch H M H U I, i m m Ul Ul Sh, sh a ft n ä i. y ui m m «/ Shch, shch K K K K K, k "b i> L l n

Jl Jl n A L, 1 bi u u ht Y, y M M M M M, m b b b b i

H K H H N, n 3 3 9 3 E, e 0 0 0 0 0, 0 10 10 10. 10 Yu, yu n n n n P. P H * X H Ya, ya

♦ ye initially, after vowels, and after t,, B; £ elsewhere. When written as S in Russian, transliterate as yS or e. The use of diacritical marks is preferred, but such marks may be omitted when expediency dictates.

FTD-HT-23-1814-72

r

All figures, graphs, tables, equations, etc.

merged into this translation were extracted

from the best quality copy available.

FTD-HT-23-lbliW2 ü /

LARGE DEFLECTIONS OF PLATES WITH NONLINEAR ELASTICITY AND HOLLOW SHELLS WITH HINGE-SUPPORTED EDGES

M. S. Kornlshln, N. N. Stolyarov, and N. I. Dedov

In this work, method of finite differences is used to solve

problems dealing with large deflections of rectangular plates in

plan with nonlinear elasticity and hollow shells with hinge-

supported edges, which are under the effect of external normal

pressure uniformly distributed along the entire surface or along

the central rectangular area. The curvature parameter values

at which a crack in a cylindrical panel occurs are established.

Critical loads and their corresponding deflections have been

found for panels with different curvature parameters.

1. Principal dependences. Let us consider a hollow shell

which is rectangular In plan with sides 2a, 2b and thickness h.

We will place the origin of the coordinates in the center of the

shell and direct axes x, y parallel to the sides of the plan.

We introduce the following designations: <? - stress

function; ü, v, w, - corresponding displacements of a point cf

the middle surface along axes x, y, z; k, , kp - shell curvatures:

FTD-HT-23-18U-72

mkm^^j_^^m

written here in total derivatives;

* ^»r^nra-j^ JtffZ&i*-. s-^.^

■5? /■'»''W-] ' /^.3.T.^)-/';"'»/.(*>"0. (1,)

-Tii01»^"] •'■ A».'*/.^-^«^^-0- (12)

^[^^-^•/.W-o. (I3)

The conditions of articulation (z ■ 0; x * 0)

2* J-ii-...0 — condition of symn^try of vibration shapes,

Boundary equations when :■ }.l^

d r it't i r rfx L/:/* 7ii>J 0 " absence of shearing force,

'"•£ - absence of bending moment,

8" 07» '■''-*. 0— absence of torque.

Boundary conditions when x ■■/♦:

FTD-HT-23-1315-72

• ■ ■ -^

Assuming that 6^ =o, from relationships (1) we obtain

(2)

jw -..'•' :*::'■,>■)

Q

Here and subbequently 1, 2 - index transposition symbol.

The expressions for stresses and moments

-CM

7^.

P'u~ j Öu?^' ('^-V- n'^l r'-h:i

after certain transformations with the use of relationships (2)

will assume the form

Fk f.*. ~ T,~-rj;t(t,yt£}+*ltl, ([X. ^Tii^^:,

K Eh En ^/7^J^+/^W^^' ^,' 'V^'-^'v,

(3)

where

Eh

(4)

\*^' l^I^t

FTD-HT-23-1814-72

Coefficients P, Qj, R, ..., X, Y, Z represent Integrals

[Translator's note; equation (5) is not indicated - assumed to

be one of the following]

flfti i!'. iSji

■O.Sn •Lih -ZSr.

ojk am &*

■li.:h -ftTA -O.S\

OS*. . ilSn S>.x

'OJh -9M -Mb

where

1 (f-ßKt-v) ^" nv^T^T" (I-JIJ(I~J)

Having expressed T-,1, Tp0, T12 in terms of the stress

function ^,

'» " %* » '// " rri > '/£ ~ 'xy . 'u 7yy

from relationships (3) we obtain

£'=iT(**yV~ftl'6T''~JUAfv-)' Eh i

It!- £k v/y £/,

(6)

^^MHta^^^a^^l «HI

Using the well known nonlinear equations of equilibrium

and the condition of compatibility of deformations of the

theory of hollow shells, and introducing dimensionless variables

at . y « - ^ tt

= f' y^f' Ar-r' ^^r- ^pj- K^u

T frt* K, _ fin I)1 M „ firJ1 M _ Ell t

we' will obtain a system of two nonlinear differencial equations

(7)

= A' Ur/y - X'4 ^X tt^y ^ 7;^yy "/ ^^ ^.U "/^ ^^.y ^

+Xt&HUtZX+2XANUtXv+6M;J,y .

Presented below are solution results of the geometrically and

physically nonlinear problems of bending a square plate and a

square cylindrical panel using external normal pressure with

boundary conditions of hinged support, which can be presented as

with x-t\

with y=±/

• -^- '---

Conditions (8V- indicate the equality to zero on the con-

tour of normai and -angential stresses, and also the deflection

and moment,

2. Method for solving nonlinear problems using the elec-

tronic digital computer and the calculation results. To resolve

the problem, we will use a method of finite differences. Having

selected a rectangular net 10 * 10, we substitute the initial

system of differential equations (7) "with a system of differ-

ence equations, approximating the biharmonic operator and all

the derivatives with the symmetric difference equations with —2 — an error on the order of 0(h ) where h - net spacing. The de-

rivatives of function r in boundary conditions we approximate

with an error on the order of 0(h ].

In accordance with the boundary conditions the contour

values of functions ^* = 0 , (pK=o , and the values of functions u\'i • QH » beyond the contour, are expressed In terms of the

intracontour values according to formulas [2]:

with ac:-±l

with ii=±1

4„. ^.JL .. ~-2.Ui- -*if +r-u' ■*■-■■.-■ J-~}ä'\:

Reproduced from best available copy.

%^3^r°.=-^

Due to the symmetry of deformational state of the shell

relative to axes x and y the number of difference equations is

reduced to 50. The obtained system of 50 nonlinear algebraic

equations was solved using a method of total iteration, simi-

larly to those described in monograph [2].

mmmm

Part of the calculatJon reoults with u = 0.3, A = 0,

18-10 for ^.'f. =<. /. i-'-J~^ is presented in the table and

on figures. The calculations were carried out for the following

loads: load of constant intensity P, distributed throughout the

surface s, = 4ab; with Intensity Pp - along the cylindrical

area s- = ^ab; with intensity P^ along area s, = 0.36ab; with

intensity ?u along area s^ = O.O^ab. We used ehe following

designations in the figures:

1

h i -

where /,- - load intensity parameter, f- - parameter of total

3oad, ;.*. - deflection parameter at the center.

r: '. 3i | f J ^ i 55 : 6^ 72 : c -

''='--»

£9.2 57.-r 133,1 •?£.: 2^5.5 2':•

-■■; i.:5 1.75 ? 2 2.2; 2.25 I.'-

.-''■ !-E-5

*0 .^f 60.2 CI.7 SB. 2 5-T.C -t-.'

3.75 T.75 5.5 6.5 7.5 '.

r-- -\ f! - 79.8 IIS.J ItV.2 ??2.? 256.5

.,! __ 2.C 2,0 2.0 2,25 2.25

tf.r 75.9 7-J.] Vc..5 V,'.,7

}.5 4.5 5.7L- 6.5 7.5

*,% - 13 16 18 21 2,'t

(,'. - If: 21 22 2r: 27

The table shows the obtained, with the consideration of

(?~ *'}'''']" J and without the consideration of {}"-■) of physical

nonlinearity, parameter values for upper (c\ I and lower (i*)

critical loads and their corresponding deflection parameters in

Reproduceci Iron. best avai ̂

able cop£.

FTD-HT-23-1814-72

center (V/, KT*) for a square cylindrical panel uniformly loaded

along the entire surface in the plan with different curvature

parameters K? and with ratio ,V^=o,oo5 • Given here also are the

values expressed in per cent for relations

*r KM

It is evident from the table that physical nonlinearity

lowers the critical loads significantly. With an increase in

parameter Kp the effect of physical nonlinearity noticeably

Increases, moreover, its effect on fl" is somewhat greater than

on p1 .. At the same time, for the critical states the corre-

sponding deflection parameters a^., and ^ r^ are very close

or coincide.

For a twice nonlinear case (fcf). Fig. 1 shows relationships

ft (k'i) for a number of values Kp and k/& =0,0053 and Pig. 2 shows

relationships £nC-^J for a square panel with Kp = 40 and A/^ = 0,002

with loading areas which differ with respect to size.

Here - parameter of the total load per area.

Fig. 1

FTD-HT-23-I814-72

■A-^n^M^iMrii

obvious algebraic transfornations v;e get:

(26)

If we introduce still more designations to certain equations

(•»./») (1. -^r/,) 1 (?./1./»).

iVf») (Tf^-V») I (.V^'?*4».

(C.A)-- (vc,./*).

(27)

then eouation (26) is rewritten in the form of

/..•/.(0) i-/.♦•<?,(0) /»'(«.•/.) i /':(*-?») \ PHC'/*). (28)

By performing algebraic calculations fo^ equation (232) and the

second trio of vectors B2J Yz, 012 s we get equation

^•/.(0) + /..:-9,(0) /,>,•/,) I />M*2 ?.) f^^-A). (29)

Here the numbers £„,, l^2, and vectors ap, Sp, 5- are determined

by the same formulas as numbers K-, , l-,p and vectors a.., b,, C,,

by, of course substituting index 1 in these formulas for the

indicated quantities and index 2 for vectors a, B, y.

From Pig. 2 it is evident that the localization of the

load on a smaller area leads to a considerable increase in

maximum deflection and to a decrease in the value of the upper

critical load. Thus, with a load distributed along the central

area Sp equalling one fourth of the total panel area (curve p, )

deflections *.; In the precritical state having Increased almost

rhree times, while cracking load 2* has decreased by approxi-

mately two times, as compared to the corresponding values for

a panel uniformly loaded along the entire surface (curve ft ).

With a load of ^ the crack virtually disappears.

Figure 3 shows relationships ß^i with various ratios

h/ß fcr a square panel ( A =1, ^ = (0) uniformly loaded along the

e?itire surface. It is evident that with an increase of this

ratio the effect of physical nonlinearity on deflections and

critical loads increases significantly.

Fig. 3.

BIBLIOGRAPHY

1. Key^epep r. He^iiiefitraa yßxaniiKa. :.!., I!.'!, 1961.

2. KoorüT-;:!.; M.C. KenrHeimue 33.i;3>i;i Teop;;;^ nnacr;iH ;) nor.ovy.x

cSozoneK 1: vevo^:-; HX peL:oH;in. .'.!.,"i-layi:«'', 1954.

5. G'^ipoK:;;! '.\.'i,, raKoesa M.C. !!3r;;3 rraciiüi «a KaTer;;a;.i:i, ':r

«i:iiK!i,jr;eroch jiDMKe.'iHOv.v 3a;;oH.v ynpyrocT;i. Cö.'MjCiior.ors::;:^

i;o reop;:;" iiäac?;;;; ,1 o5o.io<iOK", ',; ■>, !:3.;.r;ri', 19Ö6.

FTD ■HT-?3-l8l^- . 0