I .rAObl 979 FOflEIGN TECI’*IOLOGY DIV WRIGHNPAflERSON AFO OHIO F/S 20/11SEKAVIOØ Of A MILDLY SLOPING CYLINDRICAL PANEL UNDER THE EFFECT—ETC(u)OCT 77 A S VOLMIR, A F DANILENKO

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FTD—ID(RS)T—~857—77

FOREIGN TECHNOLOGY DIVISION

~~~~~~ BEHAVIOR OF A MILDLY SLOPING CYLINDRICAL PANELUNDER THE EFFECT OF WIND GUSTS

çj.u ..r. by

A. S. Vol’mir, A. F. Danh lenko

Approved for public release;distribution unlimited.

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FTD —ID(RS)T—1857—77US Sift S 1if~PRAN

~~~~~~~~~~~ EDITED TR ANSLATION

FTD—ID(RS)T—l857—77 19 October 1977

MICROFICHE NR:

BEHAVIOR OF A MILDLY SLOPING CYLINDR ICAL PANELUNDER THE EFFECT OF WIND GUSTS

By: A. S. Vol’inir, A. P. Danilenko

English pages: 14• Source: Stroitel ’naya Nekhani.ka I Rascbet

Sooruzheniy , No. 14 , 1971, pp. 66—68.

Country of origin : USSRTranslated by: Robert 0. HillRequester : FTD/PHEApproved for public release; distributionunlimited.

THIS TRANSLATION IS A RENDITION OF THE ORIGI.HAL FOREIGN TEXT WITHOUT ANY ANALYTI CAL OREDITORIAL COMMENT. STATEMENTS OR THEORIES PREPARED BY:ADVOCATEDOR IMPLIEDARE THOSE OF THE SOURCEAND DO NOT NECESSARILY REFLECT THE POSITION TRANSLATION DIVISIONOR OPINION OF THE FOREIGN TECHNOLOGY DI. FOREIGN TECHNOLOGY DIVISIONVISION. WP .APB , OHIO.

FTD — ID ( RS ) T— 18 57—77 Date 19 Oct19 77

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U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM

Block Italic Transliteration Block Italic TransliterationA a A a A , a P p P p R , r5 6 5 6 B, b C c C c S , S

B B B . V , v T T T m T, tr r r , G ,g Y y y y U , u

ii 8 D, d F, fE e 5 ’ Ye , ye; E, e* X x X x Kh , kh

4W Zh , zh Lt u Li i, Ts, ts3~~ 3 , Z, z Ch , chkIM I f u 1, 1 W w LI! iu Sh, shP1

~‘ R a7 Y , y [14 u~ 111 iq Shch , shchK R X x K, k

/1 ii A L, 1 H bi M ~ Y, yM M M , m b b b ~H N N , n 3~~ .g , E, e

o o 0 o 0, o Hi ~o Yu , yuf l n t i n P, p H i Ya,ya

*~~~~ initially , after vowels , and after b, b ; e elsewhere .When written as ë in Russian , transliterate as y~ or §.

RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS

Russian English Russian English Russian English

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BEHAVIOR OF A MILDLY SLOPINGCYLINDRICAL PANEL UNDER THEEFFECT OF WIND GUSTS

A.S. Vol’mir , A .F. Danilenko (Moscow )

The investigation of the forced oscillations of elements of

structures under the effect of a nulsating wind pressure is of

interest in connection with the possibility of their fatigue

failure . Examined in works [1, 2, 3] and a number of others isthe behavior of flexible one—dimensional structures under a wind

load treated as a stationary Gaussian random process. Using a

similar anproach , we find the average squares of the normal

movements and stresses of a two—dimensional structure , which is

what a shell is.

Let us take a mildly sloping cylindrical panel with lenRth

a and width b , the edges of which are hinged supported and freely

shifted in the t lan . Let apply to the curvilinear edges a static

compressive load p uniformly distributed over the width. The

equations of motion of such a panel have the form [14]

v’ v’ ~~~~~ — ~~~ !.1~1. V’ VI . I •~R ox1 1’Ox’ ~~ + E ~-i~ii~’ (21

where •1z.,.’) is the normal movement ; ø(x,y. fl — stress func—

tion~ D — cylindrical rigidity; E — elastic modulus ; p — density;R and h — radius of curvature and thickness of the panel; mean

value of the random time function q(t) = 0 and the spectral den—

sitv J 3 ] (s) ~~~p ,C.V.) ’S, (a~; w — frequency; p0 — air density;

C0 — aerodynamic coefficient ; V0

- average wind speed;

S, (.)—2.’L! ~iV. (1 +~~~.—sp e c t r a 1 density of oulsations of wind speed [5];

.—•~•-~•—.

— -~~- : : :~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~

- -

a2 — dispersion of wind speed; ~~.LI V, — dimensionless fre—

quency ; L — scale of turbulence.

Let app rox imate t he normal movemen t in the form of a decom-position by forms of the free sustained oscillations

w(x .y.i)~ i~~~I_a (1)sin (msxIa)sLn (nJIl,/b). (8)

From equat ions ( 2 ) and (3) we find the stres s funct ion

•(x , y~ t)=Ea’ 3 2 R~~ ~~m’(m’ + fl~ M)~~ ~~ (0 $in (m ~ x/a~ siii (n ~ #/b) —p b’/2 . (4)

where X—m/b.

Substituting (3 ) and (14 ) into equation (1) and integrating

by the Buvnov—Galerkin method , we obtain , by introducing formally

• the dissioative terms , the equation of force oscillations of the

panel in the form [6]

Imn + 2P ~~~ ~ ma 1 ,., + cu~~ !.a Q~, (1) (m. n 1 .3. .. .), (5)

where ~~~ is the damping factor ; the generalized forces

Q,,(A)~~ I6q(1)/3’p hm n (m, n~~~I ,3..•.); wm~~= (*th/ab)(p ,~,,—p~)°5 — thefrequency of the m and nth form of oscillations taking the com-

pressive load into account ; c — the propagation velocity of the

longitudinal waves in the material of the panel; parameter of the

upper cr it ical stress ~.nrn — ~‘ (& 1-n’).1)’ (12).’ (1 _~‘) r 1 +k’).’~,4I~Q,’+.s’).’)r’

— curvature parameter; p — coefficient of lateraldeformation ; the parameter of the static compressive load

m’/E M.

Using the expressions for the generalized forces from (5 ) ,we find the relative spectral density of the generalized forces

3Q~,. Q1,(~) (I6/z’pk)’ts.(w)/iIIrJ (i,~ , 1,r~~ I3 , ...).

Following [6 ] , we determine the mean nroducts of the gener—alized coordinates and generalized velocities

L1~~

~i=— .’ --

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~— —- -~~--—. --..----- -— — •

!~7I~, = -&,j,, (X ~1 / V,1 + X,, ,‘ Yg, + A);

— ~~~ (V. I L)’ (Z11 / V11 + Z~,.I Yir —A).

-

~,-L~-- X~1 ~ j j [5 ~~~j —

~~ r + ~~, (~ ~~ —~~~ -

Ye, c;j~, [(~

, —~,f + (4 ~~, ~~J7} [(~

, + i )2 + (2D1, ;~,)2J ;

z1, ~~~~~ [s~~,+ ~~~~~~~~~~~~~~ ~,)1; A [(~~,+i) (~,+ l)1 ’;

z111,. = (16p, aC.L’/*’t e h V.)’ (j7Ir)4 (i ,J ~ I.r~~~13 . . . .);

- - Xf r , Yi,. Zi, are obtained from X 11, Y~j . Z~1 by replacement of the sub-

scripts i by 1 , j by r and conversely .

The expressions obtained for the mean products make it poss—

Ible to find jointly from (3) and (14 ) the mean square of the

movements1 1,

w’ (x. F,,b, sin i s!.! sin sin sin

and the normal stress es ac ting along the x ax is in the ex tremefibers

~~ (x (~~ \2 ‘fir a ’ (1’ + i)’ )” )

— k )‘ ~~~~

£ ~ a ’ / [ 2 (1 — ii’) (i’ + j ’)~’) ’ J

11~ ~ ‘

~~~~

1.)&~:;).s) —

~~~ .~~~~~~~~~ J i;;~;; sin ~~~~~ sin sin

and according to the Rice formula — the frequency of inter—

section of the assigned level of movement or stress.

As an example let us investigate In the first approximation

the rat io of the stan dard of stress to the average stres swrit t en for the greatest stressed point of the panel

(~t~’

7z _12 6 ’~ 1 ~~ ‘)~ j~~~

’+ i~ ,i • + ~~~~~~

With an increase in frequency this ratio raoidly grows from zero

un to its maximum , and then it slowly approaches the value equal

to ~~ (~ r’,~

_2aIv.. Assum ing a 0 ,l V0 , we obtain (the

region of low frequencies,Where the solution in a linear

_ 3 A

—- - ~~~~

—•—- - —- .‘---.-~.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ...,—-- -.-- ..

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ . — _ - --~- —-~-

formulat ion is Incorrect , is excluded ) [(~~)‘5 i&5 J 1 i o ~ s. where

H . ~ is the number of the standards (usually 2—3 ).

Thus the pulsating stresses are comparable to the static

stresses and therefore must be considered in calculations for

strength and fatigue .

Undoubtedly , this problem requires a more complete study

on the basis of the nonlinear theory . The following approaches

to the solution to the problem of the nonlinear oscillations of

plates and shells caused by a gusty wind are possible : 1) the

atrnroach used in [7] allows solving the nonlinear problem by

means of the spectral method; 2) the approach proposed in [8]

nermits in a number of cases using the Fokker-Planck—Kolmogorov

equation.

Bibliography

I. M. •. 6e~ * I s~I u~~~~~.aN~HalI pIc~0? NI~? N 6a~ ri a. ~.IcisIIe Iel9S. ~CIPCUI~aIIIN

~rr1i~’ ?~~~ 7~~r t. ~~~~~~~~~~ .Jouvnal of Ui. Struct ural Div Islor , 1157. *3.3. ~~. H. L i t t L .bury . Dy~~~Ic •zclt. t len of HrIaI str uctur.. by wind turbulenc.. .Maicon

R.vIew~ • 1915. * 1 .4 A• C. b oA bUI p. VcToIqy~Pocn A. opuapyeiiblx cHc~eu. H.yx.., 1967..5 H• L B a a a a . el ,o. H. 3. fl ~ a y c a sp. Typ6y. leaTnoc m s c,o6oaitol a,agoc$epe. rupo-

oa o Tl I. CTHacTI ie~ ae uetoau I clp oHTeAbii oA -M exam~ e. roccTpoaasxaT, Ills.7. ~~~. A. • 141p.m. 0 ~~~~~~~~ IsasIsaI u npauoyroamuo aA.cuniu. 115 saUAIcnsIN

L .l1•~~~~~~~~iI’~? •riillUU IlSiS ~j ~;11,1•, ynpyl as ft$ISNSI api CJl7~~~~~~I s.i~ 7Ss.&.

.H,.sci iu. *u CCCP.. Cops. 4Ms.auusii N N5 iCf~QiUN~~ . 915. *5.

L.A

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