Approximate Solutions of the Finite Deflection Equations for Axially Compressed Thin Cylindrical Shell

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APPB0XIMÂ3ÎS SOLUTION Of THS f IHITE DSfLLCTIOH EQUATIONS fOH AXIALLY C0MPHSS5ED THIM OYLimsiOM* SHKLLS

DISSSRTATION

Submitted in p a r t i a l fu lfilm e n t o f the requirem ents fo r the degree o f Doctor o f Philosophy a t the P o ly tech n ic I n s t i tu t e of Brooklyn by

Joseph KeGQpner August 1949

Appr Head of Ssroartment

ProQuest Number: 27594654

All rights reserved INFORMATION TO ALL USERS The q u a lity of this re p ro d u c tio n is d e p e n d e n t u p o n the q u a lity of the co p y su b m itte d . In the unlikely e v e n t that the a u th o r did not send a c o m p le te m a n u scrip t and there are missing p a g e s, these will be n o te d . Also, if m a te ria l had to be re m o v e d , a n o te will in d ic a te the d e le tio n .

uest P roQ uest 27594654 Published by ProQuest LLO (2019). C o p y rig h t of the Dissertation is held by the A uthor. All rights reserved. This work is p ro te cte d a g a in s t u n a u th o rize d co p yin g under Title 17, United States C o d e M icroform Edition © ProQuest LLO. ProQuest LLO. 789 East Eisenhower Parkway P.Q. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346

Approved by the Guidance Committee:

M ajor:

A pplied Mechanics

H. J . k o ff ( (aalrm an) P ro fe s so r o f A ero n au tical E ngineering

W.nor:

M athematics

è. D. W hitford A sso c ia te P ro fe s so r o f M athematics

Minor:

A ero n au tica l E ngineering

P ro fe s s o r and^Mbad o f Department of A ero n au tical E ngineering and A pplied Mechanics

M. J jL '^ is s n e r P ro fe s s o r o f Aerodynamics and Aero s tr u c tu r e s

To C.B.X.

BIOGRAPHICiL SKETCH The a u th o r was horn in Brooklyn on A p ril 25, 1923*

He re c e iv e d

h i s prim ary and secondary e d u catio n s a t P u b lic School 99 T echnical High School.

and Brooklyn

A fte r coBç>letion of the c o lle g e p re p a ra to ry

course a t th e l a t t e r , he e n ro lle d a t th e P o ly tech n ic I n s t i t u t e of Brooklyn from which he re c e iv e d the degree o f B achelor o f A ero n au tical E n g ineering l a 1943.

H is t h e s i s

was e n t i t l e d "Experim ental and

E elajcation Methods A pplied to th e S tre s s A n aly sis o f a P l a t Sheet and S trin g e r C om bination.”

While earn in g h i s d eg ree, he was employed as

an a s s i s t a n t in th e A ir c r a f t S tru c tu re s L aboratory. In 1944 th e a u th o r re c e iv e d an appointm ent as a e ro n a u tic a l eng in eer w ith th e N a tio n a l Advisory Conaalttee f o r A eronautics a t L a n ^ e y F ie ld , V irg in ia .

Here he jo in e d th e s t a f f of th e S tru c tu re s S esearch D iv isio n

w ith which he rem ained f o r th re e y e a rs . D e sirin g to f u r th e r h i s ed u catio n , the au th o r In 1947 re tu rn e d to th e P o ly tec h n ic as a re se a rc h fe llo w and earned th e degree o f M aster o f A e ro n au tic al îb ig in eerin g th e same y e a r.

The t i t l e of h i s th e s is

was "D iffu sio n o f T e n sile S trin g e r Loads in R einforced P l a t P an els w ith C u t-O u ts."

A fte r re c e iv in g th e M aster’ s degree, he continued h is g rad u ate

study in o rd e r to q u a lify f o r th e D octorate in A pplied Mechanics.

The

r e s e a rc h d e sc rib e d in th e p r e s e n t paper was i n i t i a t e d in th e P a ll o f 1948 and was p a r t of a p r o je c t sponsored t y th e O ffice o f Naval R esearch . The w r ite r i s a u th o r o r co -au th o r of the fo llo w in g p u b lic a tio n s : H off, H .J ., Levy, R obert S ., and Kemgmer, Joseph:

Numerical

P ro ced u res f o r th e C a lc u la tio n of th e S tre s s e s in Monocoques. I - D iffu s io n of T e n sile S trin g e r Loads in R einforced P a n e ls.

NACA TN Ho. 934, 1944. H off, N .J ., and Hempner, Joseph.* Numerical P rocedures f o r th e C a lc u la tio n o f the S tre s s e s in Monocoques. I I - D iffu sio n o f T e n sile S trin g e r Loads in R einforced P l a t P an els w ith Cut-O uts. HACA TN Ko. 950, 1944. Kenç)ner, Joseph*

A p p lic a tio n o f a Numerical P rocedure to S tre s s

A n aly sis o f S trin g e r-R e in fo rc e d P an els.

HACA Wartime R eport

L-11, 1945. Dub erg , John E. , and Kempner, Joseph*

S tre s s A n aly sis by R ecurrence

Formula o f R einforced C irc u la r C y lin d ers Under L a te ra l Loads. NAGA TN No. 1219. 1947. Kempner, Joseph, and Duberg, John:

C harts fo r S tre s s A n aly sis

o f R ein fo rced C irc u la r C ylinders %cier L a te ra l Loads.

NAGA

TH No. 1310, 1947. Kenpner, Joseph*

R ecurrence Formulas and D if f e r e n tia l E quations

f o r S tr e s s A n aly sis o f Cambered Box Beams.

HACA TH No. 1466, 1947.

Boley, &*uno A., Renpner, Joseph, and Mayers, J .:

A Numerical

Approach to th e I n s t a b i l i t y Problem o f Monocoque C y lin d ers. Subm itted to th e NAGA, June, 1948.

jm m m Æ D m m The a u th o r w ishes to ex p re ss h ia a p p re c ia tio n to Dr. N .J. Hoff f o r su g g estin g th e p r e s e n t th e s is to p ic and f o r h is c o n sta n t encour­ agement d u rin g th e course o f th e re se a rc h . To M essrs. J .P . Ohawla, D.W. Booth, and R.A.V. P an d alai and to C ^ t . Sadet t i n Cunturkun, the a u th o r owes h i s unbounded a p p re c ia tio n , f o r w ith o u t th e cap ab le a s s is ta n c e of th ese men t h i s d i s s e r t a ti o n Could n o t have been consummated. Since th e in v e s tig a tio n re p o rte d on in t h i s th e s is r e p re s e n ts one phase o f work on a c o n tra c t w ith the O ffice of Naval R esearch, th e a u th o r e35>resses h is thanks f o r the monetary a id extended by t h i s o rg a n iz a tio n .

ABSTRACT The known c o E p a tib ility and e q u ilib riu m eq u atio n s f o r the d e te rm in a tio n o f the s tr e s s e s in and d isp lacem en ts of th in -w a lle d is o tr o p ic c y lin d r ic a l s h e lls undergoing f i n i t e d e f le c tio n s a re redeveloped.

The

e q u a tio n s o b tain ed a re a c c u ra te to w ith in second-order sm all q u a n titie s . T b ro u ^ th e a p p lic a tio n o f a method o f su ccessiv e «^proxim ations, th ese nonr-linear p a r t i a l d i f f e r e n t i a l eq u a tio n s a re reduced to a s e t o f f iv e n o n - lin e a r a lg e b ra ic e q u atio n s f o r the d ete rm in a tio n o f th e sq>plled a x ia l com pressive s t r e s s c o n s is te n t w ith the main ta in ence o f f i n i t e b u c k lin g c o n fig u ra tio n s .

To f a c i l i t a t e num erical work th e f iv e eq u atio n s

a re approxim ated by th re e s im ila r eq u atio n s w ith which the b u lk o f the nu m erical com putations i s concerned.

The th re e -e q u a tio n s o lu tio n y ie ld s

a minimum a p p lie d a x ia l com pressive s tr e s s equal to 0.149 E t/R a s courp a red to 0.605 E t/R o f the c l a s s i c a l l i n e a r th e o ry .

The s t r e s s obtained

co rresp o n d s to a wave shape o f the inward diamond p a tte r n type having a maximum wave a n p litu d e 8.93 tim es the sk in th ic k n e ss and to a r a t i o of c irc u m fe re n tia l to a x ia l wave le n g th equal to 0.436.

TABLE (37 OûHTENTB

I n tr o d u c tio n .....................

1

Symbols..........................................

3

S tatem ent o f Problem and IM derlying Assum ptions......................

5

Development o f D if f e r e n tia l E q u a tio n s

6

.....................

^ p r o x im a te S o lu tio n o f E q u ilib riu m and C o m p atib ility E q uations f o r A x ia lly Compressed S h e lls ...............................10 D e s c rip tio n o f N um erical S o lu tio n .......................

27

R e s u lts and D isc u ssio n

31

Ref e r e n o e s .

.....................

F i g u r e s . .......................

............

34 36

m m oD ucT im The d e te rm in a tio n o f th e f a l l i n g s tr e s s o f on a x ia lly coaipressed c i r c u l a r th in -w a lle d c y lin d e r ap p ears to have been fo rm u lated f i r s t a s a f i n i t e d e f le c tio n problem by L.H. D onnell in 1934 (re fe re n c e l ) . U sing second o rd e r expressicm s f o r the median su rface s t r a i n s , Donnell a t t e s t e d to re s o lv e th e problem f r a a energy c o n s id e ra tio n s .

However,

th e a n a ly s is was o v e r-s isg ^ llfi# d and hence le a d to an u n s a tis f a c to r y s o lu tio n (re fe re n c e 2 ). I n 1941 Th* von Ksirmsa, and H.S. T sien (re fe re n c e 3) p re se n te d second o rd e r co n ç> atib ility and e q u ilib riu m eq u atio n s f o r th e aforem entioned p ro b l^ .

I t was shown in re fe re n c e 4 , and In the p re s e n t t e x t , th a t

t h e i r e q u ilib riu m eq u atio n was incom plete.

However, th ese men d id n o t

u se th e e q u ilib riu m e q u atio n b u t in s te a d used the s tr a i n energy ^ p r o a c h . The r e s u l t s p re se n te d in re fe re n c e 3 in d ic a te d th a t a q y lin d w could be m ain tain ed In e q u ilib riu m under a x ia l "reduced s tr e s s e s "

cf R a s

low a s 0.195 as compared to 0.&05 o f the c la s s i c a l l i n e a r theory* The work o f

and Tsien has been r e fin e d and extended r e c e n tly

by s e v e ra l in v e s tig a to r s (re fe re n c e s 2, 0, and 6 ).

For the most p a r t ,

th e s e men have accep ted Harman and T sie n ’ s energy ex p ressio n s and assumed d e f le c tio n functicm and co n sequently, a r riv e d a t the same minimum reduced s t r e s s mentioned above. In the p r e s e n t p ap er second o rd e r c o m p a tib ility and e q u ilib riu m e q u a tio n s a re p re s e n te d f o r th e a n a ly s is o f # e f i n i t e b u ck lin g ch arac­ t e r i s t i c s o f th in -w a lle d , i s o tr o p ic , c i r c u l a r c y lin d e rs s tr e s s e d w ith in th e e l a s t i c l i m i t .

These eq u a tio n s a re a p p lie d to th e in v e s tig a tio n o f

th e bu ck lin g c o n fig u ra tio n s o f a x ia lly c o # r e s s e d c y lin d e rs whose le n g th

- 1 —

to diam eter r a t i o l a la r g e r than th r e e - q u a r te r s .

%%e method of suocesslv©

ap p roxim ations e»(ployed i a th e approxim ate s o lu tio n of the d e fin in g eq u a tio n s le a d s to a d e f le c tio n fu n c tio n # iich . in c lu d e s th e fu n c tio n assumed in r e f e r ­ ence 3 . The a n a ly s is p re se n te d h e r e in d i f f e r s from th a t o f

and T sien

and o th e rs ( re f e r e n c e s 2, 5, and 6) sin ce i n the p r e s e n t method the e q u ilib riu m e q u a tio n i s d e a l t w ith d i r e c t ly and «qjproximately s a tis fie d # w hereas the p re v io u s I n v e s tig a to r s have re p la c e d t h i s e q u a tio n w ith e q u iv a le n t energy eatpressi CQ C

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