19710004581

8/12/2019 19710004581

http://slidepdf.com/reader/full/19710004581 1/34

THEORETICAL A N D EXPERIMENTAL

VIBRATION AND BUCKLING RESULTS

FOR BLUNT TRUNCATED CONICAL SHELLS

WITH RING-SUPPORTED EDGES

by Sidney C . Dixon, Robert Miserentino,und M . Lutrelle Hudson

Lungley Research CenterHumpton, Vu 3365

N AT I O N A L A E R ON A U TI CS A N D S PA C E A D M I N I S T R A T I O N WA S H I N G T O ND. C DECEMBER 1 9 7

8/12/2019 19710004581


1. Report No. 2. Governmen t Accss ion No.

NASA TN D-7003 I4. Title and Subtitle

THEORETICAL AND EXPERIMENTAL VIBRATION ANDBUCKLING RESULTS FOR BLUNT TRUNCATED CONICALSHELLS WITH RING-SUPPORTED EDGES

7. Author(s)

Sidney C. Dixon, Robe rt Misere nt ino , and M. L at re l l e Hudson

9. Performin g Organization Name and Address

NASA Langley Research Center

Hampton , Va. 23365

12. Sponsoring Agency Name and Address

Nationa l Aero nau t ics and Space Admin is t ra t ion

Washin gton, D.C. 20546

15. Supplementary Notes

.. ~

16. Abstract

TECH LIBRARY KA FB,NM

I l illI1I 3. Recipient s Catalog No.

5. RewJrt DateDecember 1970

6. Performinq Orqanization Code-I

8. Performing Organizat ion Report No.

L-7061

1.0. Work Unit No.

124- 08- 20-04

11. Contract or Grant No .

13. Type of Report and Period Covered

Technical Note

14. Sponsoring Agency Code

The v ib ra t ion and buck l ing cha rac te r i s t i c s of r ing-suppor ted con ica l she l l s have

been inves t iga ted theore t i ca l ly and exper imenta l ly. Theore t i ca l r es u l t s ind ica te tha t r ings

des igned to p rov ide edge r es t ra in t be tween s im ple and c lamped suppo r t on the bas i s of

buckl ing calcul at ion s do not provide the equ ivalent sup por t on the ba si s of vib rat io n calcu

l a t io n s f o r m o d e s h a v in g f ew c i r c u m f e r e n t i a l w a v e s. T h e n a t u r a l f r e q u e n c i e s f o r t h e s e

modes were c ons iderab ly be low the min imum f requen c ies fo r sh e l l s wi th s imply suppor te d

o r c l a m p ed e d g e s . F o r s u c h c a s e s , s tu d i e s i n d ic a t e t h a t su b s t a n t ia l i n c r e a s e s i n r i n g s i z e

a nd m a s s a r e r e q u i r e d t o e ff e c t a s ign i f i can t inc rea se i n the min imum f requency of r in g-

s u p p or t e d s h e l l s .

Exper imenta l buckl ing and v ib ra t ion resu l t s we re ob ta ined fo r four b lunt t runca ted

c o n ic a l s h e l l s e s s e n t i a l l y c l a m pe d a t t h e s m a l l e n d a nd r i n g - s u p p o r t e d a t t h e l a rg e e n d .

Buckl ing was induced by ae rodynam ic load ing in wind- tunne l t e s t s . Theor e t i ca l r es u l t s

f o r t he s a m e s h e l l s are in good qua l i t a t ive agr eem ent with the exper imen ta l r es u l t s , bu t

the quan t i t a t ive agree men t is only f a i r.

19. Security Classif. of this report) 20. Security Classif. of this page) 21. No. of Pages 22. Price*

Unclass i f i ed Unc lass i f i ed 32 3.00~ ~~~

For sale b y t h e N a t i o n a l Te c h n i c a l I n f o r m a t i o n Se r v ic e , S p r i n g f i e l d , Vi rg i n i a 2 2 1 5 1

8/12/2019 19710004581


-

THEORETICAL AND EXPERIMENTAL VIBRATION

AND BUCKLING RESULTS FOR BLUNT TRUNCATED CONICAL SHELLS

WITH RING-SUPPORTED EDGES

By Sidney C. Dixon, Robert Miserentino,

and M. Latrelle Hudson

Langley Resea rch C enter

SUMMARY

The vibrat ion and buckling char acte rist i cs of r ing-supported conical shell s have

been invest igated theoret ica l ly and experimental ly. Theoret ica l res ult s indicate that

rings designed to provide edge rest ra in t between sim ple and clamped sup port on thebasis of buckling calc ulat ion s do not provide the equival ent su ppo rt on the basis of vibra

t ion calculat ions for modes having few circumferential waves. The natural frequencies

for these modes we re considerably below the minimum frequencie s fo r sh ells with simp ly

supported or c lamped edges . Fo r such cases, s tudies indica te tha t subs tant ia l incr ease s

in r ing size and m ass a r e r equ i r ed to e ffec t a significant increase in the minimum f requency of ring-supported shells.

Experim ental buckling and vibrat ion resu lts w ere obtained fro m a pre l iminary

investigation of fo ur blunt truncated conical shel ls esse ntia lly clamped at the sm al l end

and r ing-supported a t the lar ge end. Buckling was induced by aerodynamic loading in

wind-tunnel tests . Theoret ic al res ult s for the shel ls are in good qualitativ e agreem ent

with the experimental re sul ts , but the quanti tat ive agreement is only fair. The ra ther

larg e differen ces between theo ret ical and expe rimental buckling res ult s are attr ibuted to

the l arge permanent deformations in the s hells that resulte d fr om buffet ing as the s ta r t ing

shock wave moved down the wind tunnel.

INTRODUCTION

Ring st iffeners are often used t o provide edge su ppor t fo r sh ell s of revolutio n in

such aer ospa ce applications as proposed planetary entry vehicles and rocket nozzles.

(See refs . 1 and 2.) Since a r ing-supported shel l may be subjected to both static and

dynami c loading, a thorough knowledge of the static and dynamic cha ra ct er is ti cs of suc h

shell-ring configurations is needed. Recently, the re has been considerable theoret ical

re se ar ch effort on the buckling of rin g-suppo rted s he ll s (e.g., refs. 1 and 3 to 9), but only

. . . . .

8/12/2019 19710004581


l imited theoretical re sul ts on the free-vibration char act eris t ic s of such shells ar e avail

able (e.g., refs. 2 and 10 t o 13); exper imenta l resul t s a re even more l imited ( refs . 6, 12,

and 13).

In the p re sen t investigat ion, the buckling and vibratio n of blunt ring-supported

conical shell s are considered both theoretically and experimentally. Theoretical vibra

t ion results are obtained by us e of the appro xima te theory of r ef ere nc e 7, which tr ea ts

the vibration, buckling, and flu tte r of trunca ted orthot ropic co nical shel ls with genera lized

elastic edge support, such as the restra int imposed by end stiffening rings. The analy sis,

which uses a Donnell-type shell theory and a membrane pre s t res s s ta te , accounts for the

sti ffne ss and in er ti a cha rac te ri st ic s of the end ring s in the boundary conditions by use of

expressions presented in reference 10. Calculations are pre sen ted which show the dif

fere nce between the vibration chara cte ris tic s of ring-supp orted she lls and shell s with

cla ssi cal edge suppor ts. Additional calculations ar e presen ted which indicate the effect

of end -ring s i ze on the vibrati on char act eri st ic s of a lightweight entry-vehicle

configuration.

Experimental buckling and vibration data we re obtained fro m a preliminary investi

gation of fo ur blunt cone models essent iall y clamped at the s ma ll end and ring-supported

at the lar ge end. The sti ffn ess of the end rings for these models w a s very close to the

minimum stiffn ess require d t o provide edge re st rai nt between simp le and clamped

suppo rt on the ba si s of t heo ret ica l buckling calcu latio ns. Shell buckling was induced by

aerodynamic loading at a Mach number of 3 in the Langley 9- by 6-foot the rma l st ru ct ure s

tunnel; the vibration data were obtained prio r t o the wind-tunnel te st s. The experimental

resu lts a re presented and compared with the theoretical predict ions in an appendix.

SYMBOLS

The units for the physical quantities defined in thi s paper a r e given both in the

U.S. C ustom ary Units and in the International System of Uni ts (SI). Appendix A presents

fac tor s relati ng these two sy st em s of units.

A r cross-s ectio nal ar ea of end ring

A nondimensional ar ea para met er, = ErAr/R1B

B extensional stiffn ess

E Young's modulus of iso tro pic shel l

2

8/12/2019 19710004581


E r Young's modulu s of end ri ng

f frequency

h wall thickness of isotropic shell

hC thi ckn ess of co re of sandwich sh el l

hf thic kness of fa ce sh eet of sandwich sh el l

I moment of in er ti a

Iy,Iz,Iyz centroidal moments and product of ine rti a of ring c ro ss secti on fo rconical coordinates

Ic,It,Ict; centroidal moments and product of ine rti a of rin g cro ss sec tion forcylindrical coordinates

fy7fz,fYz nondimensional i nert ia parame ters, f = :Ar

J torsion constant for r ing cro ss section

J nondimensional torsion-constant parameter, J = J/R?Ar

n number of circumferential waves

P uniform lat eral external pre ssur e

R17R2 rad ii of conical fru st um at small and large end, respectively

r cross-s ectio nal rad ius of tubular end rin g

t wall thickne ss of end ring

v7w circu mferen tial and normal displacements of middle s ur fac e ofconical frustum

X t ransf ormed coordinate, x = In y y1 1

3

8/12/2019 19710004581


x1 = l (Yz/Yl)

Y,z orthogonal conical coordinates (see fig. 7)

y 1 J2dis tance f rom ver tex

tosm al l and la rg e end of cone, respectively

2 0 ecce ntric ity of r ing centroidal axis measured f r om ins ide she l l sur face ,

posit ive for internal r ing

CY se mi ve rt ex angle of cone

shell densi ty per unit are a

P Poisson's rat io of isotropic shell

P r Poiss on's ra ti o of end ring

c 9 t orthogonal cylindrical coordinates (see fig. 1)

P density of r ing ma te ri al

i s nondimensional m as s par amete r, p = pAr/YR1

Subscript :

max maximum

RESULTS

Analytical re su lt s were obtained fro m the approximate analy sis of r efere nce 7. Inthat analysis, linea r Donnell-type equations that utilize the membrane s t re s s state arederived, and the effects of in-plane ine rti as and pr es tr es s deforma tions ar e neglected.

The resulting equations ar e solved by an assum ed displacement method. In the presentinvestigation , up to 40 er ms were used in the computat ions to insur e converged resu lts .

Computations wer e a ssum ed to be converged when the r es ul ts f or N + 4 er m s differed by

less than 1 percent f rom the res ul t s for N t e r m s .

Calculations ar e presented for two conical she lls previously studied in refere nce 11

by us e of the anal ysi s of refe ren ce 10, a rigorous nume rical analy sis based on Novozhilov

she ll theory. Th es e calculations ar e compared with the resu lts of refe rence 11 t o assess

4

8/12/2019 19710004581


the effects of th e assum ptions in the app roximate analysis (ref. 7) used in the present

investigation. Additional calculatio ns are pres ente d to show the difference between vibra

t ion charact erist ics of r ing-supported sh ell s and shells with classical edge suppo rts and

to indicate the effect of r in g si ze on the vibration ch ara cte ris tic s of rin g-suppor ted coni

cal she lls . Exp erim enta l buckling and vibration data obtained fr om a prelimina ry invest i

gation of blunt cone models es sen tia lly clamped at the sm all end and r ing-supported atthe lar ge end are given in appendix B. Shell buckling w a s induced by aerodynamic loading

at a Mach number of 3 in the Langley 9 by 6-foot th erm al str uc tu re s tunnel; the vibration

da ta wer e obtained pr io r to the wind-tunnel tests.

Comp arison of Re su lt s F ro m Donne11 and Novozhilov Th eo ri es

The vibrat ion ch ara cte ris t ic s of two des igns for a proposed planetary entry capsule

were s tudied in re ference 11. The configurations are designated s hel ls A and B herein

and are shown, along with pertin ent st ruc tur al dat a, in figu re 1 and table 1 . The shells

are supported at the lar ge end by an aluminum tubular ring. The sect ion pro per tie s ofthe r ing for s he l l B include both those of the tubular ri ng and an aluminum channel ring

ins ert ed into the end of the sandwich cor e. Both she ll configurations are 120' a = 6trun cate d cones of aluminum honeycomb sandwich cons truc tion . F o r the purpos es of

analysis , the sandwich c ore laye r is assumed t o provide negligible extensional stif fnes s

while providing sufficient t ran sve rse she ar and normal st i ffness to validate the thin-shell

hypothesis of nondeformable no rm al s. The sm al l end of she ll A was assum ed to be

clamped. A payload wa s as sum ed to be attached to sh ell B by an I-section payload rin g

at the sm al l end of the she ll. Calculations fo r shel l B considere d th e finite stif fnes s of

the payload rin g. In addition, the payload m as s was uniformly distributed throughout the

payload r ing by inc reas ing the r in g density in such a manner that the r ing m as s equaled

the total ma ss of the payload and attachment. In ord er to account for the m as s of a heat-

sh ield laye r, a sur fac e de nsity of 0.70 Ibm/ft2 (3.42 kg/m2) was added to the sur fa ce

density of the s tr uc tu ra l sandwich to obtain the total density of the s hel l wall fo r both

shel l s .

In f igure 2, the n atur al frequencies of shel ls A and B determined from the theoryof the p res ent investigation (solid lines) are compared with the re su lt s of r efe ren ce 11

(circles); the resul ts of re fere nce 11 we re calculated from the theory of refe renc e 10.

In view of t he lim itati ons of t he Donne11 the ory , res ul ts fr om the p re se nt inves tigation

are presented only fo r n 2 2 . The res ult s in f igure 2(a) are f o r a shel l clamped at thesm al l end and r ing-supported at the lar ge end (shell A), where as the res ult s i n f igu re 2(b)

are fo r a shel l with r in gs on both ends (shell B). As can be se en in f igure 2, the frequen

cies obtained fr om the pr esen t (modal) analysis a r e in good ag reement with the frequen

cies obtained from the m or e r i goro us (numerical) analysis of refe renc e 10, which isbased on Novozhilov sh el l theory. The differenc es are fro m about 1 to 7 percent for

5

8/12/2019 19710004581


shel l A (clamped/ring support) and fr om about 5 to 1 2 percent for s hell B (r ing/r ing

support). Thus, the theory of ref ere nce 7 appe ars to be adequate fo r s tud ies of the

vibra t ion charac ter i s t ics of blunt ring-supported isot ropic shell s.

Ef fe ct s of Si ze of End Rings

In most s tudi es of ring-suppo rted conical she lls , the end ring s have been designed

on the basis of buckling con sid era tio ns. (See, fo r e xample, ref. 11.) The buckling char

ac t e r i s t i c s of r ing-supported shells are indicated in ske tch (a). As can be se en , buckling

of sh el ls with end rin gs ma y oc cur in ei th er of two modes: If the r ings are not very stiff,the sh ell buckles alm ost inextensionally into ver y few circu mfer entia l waves (usually 2or 3 at loads which vary considerab ly with rin g stiffne ss, and which can be s ev er al or de rs

of magnitude s m al le r than the buckling loads fo r clamped sh el ls (ref. 4). If the r ing is

sufficiently stiff, the s hel l buckles extensionally into a higher number of cir cumf eren tial

waves a t loads brac keted by the buckling loads fo r sim ply suppo rted and clamped edges;the se loads do not var y significantly with ring stiff ness. Use of the s ma ll es t (lightest)

ri ng require d to su pp re ss the inextensional for m of buckling generally re su lt s in minimum

total ma ss of the s hel l and ring for a she ll designed to withstand buckling. (See ref. 9.)

Thi s design philosophy was us ed in the design of th e end ring s fo r she ll s A and B. To

dete rmin e the effectiveness of these ring s on the basis of vi brati on calcu latio ns, the

natural frequencies for the s am e shel ls with various class ical boundary conditions we re

E x t e n s i o n a l b u c k l i n g

~ 2

R in g s t i f f n e s s

Sketch (a)

6

8/12/2019 19710004581


obtained for comparison with the frequencies for the ring-supported shel ls. The res ul ts

are presented in figure 3 . The solid curves a r e for r ing-snpported shells , the short-

dash cur ves a r e for shel ls with both edges simply supported, and the long-dash curvesa r e for shel ls with both edges clamped. As can be seen in the f igure, the resu lts fo rclamped and supported edges indic ate the minimum frequency occu rs fo r n = 4. For

n > 4 , the resu lt s for ring-supported shel ls ar e bracketed by the resu lt s for clamped andsimply supported shells. However, fo r n < 4 , the result s for r ing-supported shells fallwel l below the re su lt s for clamped or simply supported she lls and indicate the minimumfrequency occurs at n = 2 . Thus, a rin g designed to provide edge res trai nt between

simpl y supported and clamped on the ba si s of buckling calculations does not provide theequivalent support on the basi s of vib ration calculations fo r sma ll values of n, and rin gstiffne ss must be considered in calculations for th ese modes.

The mode shapes for shel l B (ring/ring support) and for a clamped shell ar e indi

cated in figure 4. The shel l with ring support vibrates in essentially an inextensional

mode for n = 2, with the maximum amplitude occurring at the l ar ge end of the cone(fig. 4(a)). Eit her of the edge re st ra in ts v = 0 o r w = 0 , which apply to both simpleand clamped support , su pp res se s the inextensional mode of vibration. Fo r higher values

of n, the inextensional mode does not occu r fo r the ring-supported sh el l, and the mode

shapes approach those fo r the fully clamped shell. (See fig s. 4(d) to 4(f) . ) Similar trends

fo r a conical shel l clamped on the sm all end and ring-supported on the l arge end ar e

presented in reference 2 .

The ve ry low frequ enc ies t hat occ ur fo r low values of n may re sul t in the unde

sirable dynamic response of, for example, an entry vehicle in the launch configuration

subjected to the launch vibration environment. Calculations were therefo re made todeterm ine the effect of in cre ase s in the siz e of the base ring on the frequency resul ts fo r

shel l A. Frequency res ul ts for n = 2 and n = 4 for vari ous values of r/R2 ar eshown in figure 5. In ord er t o investigate the effects of both the finite-coupled ela sti c

res tr ain t of the ring and the ine rti a loading of the vibrating rin g mas s, calculations wer e

made for rings with a density p corresponding to aluminum (solid curv es) and al sowith p = 0 (dashed curves). The rat io of the ring cross-se ction radius to ring thickness

w a s kept constan t at r/t = 34.4, the value corresponding to the original base ring.

The re su lt s of figu re 5 indicate that the effect of the i ne rt ia l for ce of the vibrat ing

r ing ma ss was sma l l fo r r p 2 = 0.0294, the s iz e of the or igina l base ring. A s the rat ior/R2 inc reas es, the effect of rin g ma ss dec rea se s for n = 4 but inc reas es for n = 2 .The minimum frequency for the ring-supported shel l occurs in the n = 2 mode when

< 0.11 and occ urs in the n = 4 mode when 22 2 0.11; further inc reas es in r ing sizeR2 R2do not significantly alter thi s minimum frequency. Fo r 22 > 0.088, the minimum

R2

7

8/12/2019 19710004581


frequency for the r ing-supported shell is brack eted by the minimum frequencies fo r the

she lls with simp le and clamped supports.

The total ma s s of the rin g fo r r/R2 = 0.088 is over eight tim es the ma ss of the

r ing for r/R2 = 0.0294. Thus , inc rea sin g the minimum frequency of ring-supported

shell s by increasing the si ze of the base r ing can re sul t in a signif icant in creas e in base-r ing mass . The va r ia t ion of t he mode shapes w ith r - 2 is shown in figu re 6 . For

r/R2 = 0.0294 and r/R2 = 0.0555, the she ll vibra tes in ess entia lly an inextensional mode

with n = 2 . For r/R2 = 0.1670, the vibra tion mode ha s become extensional but still

differs somewhat fro m the mode for the shell with clamped edges.

The theoret ical res ult s presented in f igures 3 and 5 indicate that rin gs designed to

provide edge re st ra in t between simply suppor ted and clamped on the basi s of buckling

calculations do not provide th e equivalent supp ort on the b as is of vib ration calcul ations.

Resu lts of the preli mina ry e xperimental investigation describ ed in appendix B tend to

verify these tre nds . Appendix B also shows that theoret ical res ult s for the she l l s a r ein good qualitative agree ment with the experimental data but that the quantitative agr ee

ment is only fair. The differences between theo retic al and experimental buckling resul ts

a r e at tr ibuted to the l arge permanent deformations in the she lls that res ulted from buf

feting as the st ar ti ng sh ock wave moved down the wind tunnel.

CONCLUDING REMARKS

The vibration and buckling cha rac ter ist ics of ring-supported conical she lls have

been investigated theoretically and experimentally. Theo retic al re su lt s indicate thatrin gs designed to provide edge rest rai nt between simple and clamped support on the

ba si s of buckling calculat ions do not provide the equivalent su ppo rt on the basi s of vib ra

tion calculations for modes having few circumfe rentia l waves. The natural frequencies

for these modes we re considerably below the minimum frequen cies for s hel ls with simple

and clamped support edges. For such case s, the pre sent s tudy indicates that substantial

increases in r ing s ize and ma ss ar e requi red to e ffec t a significant incr eas e in the mini

mum frequency of ring-supp orted she lls .

Experimental buckling and vibration data wer e obtained fr om a preliminary invest i

gation of four b h n t truncated conical shel ls essentially clamped at the s mal l end and ring-supported at the la rg e end. Buckling w a s induced by aerod ynam ic loading in wind-tunnel

tes ts . Theore t ica l resu l t s for the she l l s are in good qual itati ve agre em ent with the exper

imental data , but the quantitative agreement is only fair. The rathe r la rge differences

between theoretic al and experimental buckling r es ul ts are attr ibuted to the large

8

8/12/2019 19710004581


permanent deformations in the shel ls that resulted from buffeting as the starting shock

wave moved down the wind tunnel.

Langley Researc h Center,

National Aeronautics and Space Administration,

Hampton, Va., October 16, 1970.

9

8/12/2019 19710004581


APPENDIX A

CONVERSION O F U.S. CUSTOMARY UNITS TO SI UNITS

The International System of Units (SI) was adopted by the Eleventh General Conference on Weights and Measu res, Paris, Octo ber 1960. (See ref. 14.) Conversion fac tors

fo r the units used herein are given in the following table:

Physical quantity U .S. Customaryfac tor SI unitUnit

* * ) - _.~

A r e a . . . . . . . . . . . . . . in2 0.6452 x 10-3 meters2 (m2)

Length. . . . . . . . . . . . . in. 0.02 54 meters (m)

Mass/area . . . . . . . . . . lbm/ft2 4.882 kilogr am s/m eter 2 (kg/m2:M a s s density . . . . . . . . . lbm/in3 2.768 X l o 4 kilograms/meter3 (kg/m3:

Moment of inertia. . . . . . . in4 0.4162 x 10-6 meters4 (m4)

Pres sur e . . . . . . . . . . . {ps;/;bf/in2 6.895 X 103 newtons/meter2 (N/m2)

4.788 X 10 newtons/meter2 (N/m2)

Young's and shear moduli . . lbf/in2 6.895 X lo3 newtons/mete r 2 (N/m2)

Temperature . . . . . . . . . O Fl° +14:g.67 (kelvins (K)

*Multiply value given in U.S. Custo mary Un its by conv ers ion fact or to obtain

equivalent value in SI Units.

**Prefixes to indica te mu lti ple s of units are as follows:

Prefix Multiple 1-

gigs (GI 109

kilo (k) 103

centi (c) 10-2

mill i (m) 10-3

10

8/12/2019 19710004581


APPENDIX B

PRELIMINARY EXPERIMENTAL INVESTIGATION

Since experimental dat a for vibration or buckling of ring-suppo rted sh el ls are very

limited, a preliminary experimental investigation w a s conducted to provide additionalexperimental data and trends.

Models

The models were blunt truncated conical shells with a ra di us ra ti o R2/R1 of

2.544. Two models had a semivertex angle a of 60° and nominal lar ge ra di us thicknes s rat io R2/h of 132. F o r the other two models, a = 70° and R2/h = 118. Each

model w a s machined from a solid piece of magnesium. Construction det ail s, ma te ri al

pro pe rti es, and dimensions of the models a r e given in figure 7. A solid spherical nose

cap w a s rigidly attached to a mounting sting in suc h a manner t hat the sm al l end of eachmodel w a s essent ially clamped. The lar ge end w a s supported by an inte gral end ring;pr op ert ie s of the end ri ng s are given in fig ure 7. The st if fne ss of the end ri ngs fo r the

models w a s very close to the minimum stiffness requi red to provide re str ain t between

simply supported and clamped at the la rg e end of th e she ll on the ba si s of th eo ret ic al

buckling calculations.

Te st Technique

Vibration tests. - The models were sting mounted in the wind-tunnel t est sectio n

(fig. 8) and wer e vibrated pri or to the wind-tunnel tes ts . The models wer e excited withan air-jet sha ker, which is describ ed in refe renc e 15. The frequency w a s var ied between5 and 1000 Hz until an inc reased response indicated a natural frequency. The force

applied w a s then increased, and a contact accelerometer w a s moved manually about theshe ll outer surfac e until the positions of the node lines we re located. The frequency w a sdetermined by an electronic counter, which sampled the accelerometer output.

Buckling tests .- Buckling w a s induced by aerodynamic loading; the buckling tests

w e r e conducted in the Langley 9- by 6-foot thermal structures tunnel, which is a Mach 3blowdown facility exhausting to the atmosphe re. The tunnel stagnat ion-pr essure operating

range is from 50 t o 200 ps ia (380 to 1380 kN/m2). A complete descri ption of the facili tyis given in refer ence 16. The models were subjected to severe buffeting as the start ing

shock wave moved down the wind tunnel durin g tunnel sta rt up . The model s we re su pporte dby re tra cta ble suppor t vanes during tunnel sta rt up to minimize the effect of the buffeting.

Figure 8 is a photograph of the test section pri or to a tes t ; it shows the model i n place,

the model support vanes partially ret ract ed, the sting, and the sting support. The tunnel

11

8/12/2019 19710004581


APPENDIX B - Continued

was started at the minimum dynamic pre ss u re of 1480 lbf/ft2 (71 kN/m2) with the sting-

mounted model supported from behind by the eight retractable supp ort vanes. Once the

flow was established, the vanes were retra cted 30 inche s (76 cm), and the dynamic pre s

su re was then incre ased unti l buckles wer e seen. The stagnation temperature for alltests was 200' F (366 K .

Eac h of the fou r models was tested once; the longest test lasted about 26 seconds.

Recorded data con sis ted of motion pictu res taken at 400 frames pe r second, base pres

su re s , tunnel pres sur es , model tempera ture , and axial force . The externa l pre ssu re p

actin g on the s he ll was a ss um ed to be uniform and adequately given by the magnitude of

the total axial for ce divided by the total frontal area of the model. Resu lts pres ented in

re fe rence 1 7 indicate that the pre ss ur e on a 120° cone va ri es smoothly along the cone

meridian; maximum variat ion from the aver age pr es su re w as of the or de r of 10 percent .

Use of the r es ul ts of ref ere nce 17 in conjunction with the mea sur ed base pr es su re s gave

essent ial ly the sam e va lues for the externa l pressu re as obtained fr om the axial-force

measurements .

Results

Vibrat ion r esu lts .- The meas ured resonant frequencies for the recorded modes

(for one half-wave in the longitudinal dire ction ) are given in table 2. The minimum f requency occurs at n = 3 for the 120° cones and at n = 2 fo r the 140' cone s. The ove r

all t rends are s imi l a r t o t he t r ends p re sented in r e f e rences 18 and 19 fo r clamped/free

shell s . The large r end r ing (r ing 2) generally increased the frequencies only sl ightly

and actually dec rea sed the frequen cy for the 120° cone at n = 2.

Buckling results .- The mea sur ed buckling pr es su re and circumfer entia l wave num

ber n are given in tab le 3 fo r each shell . All shells suffered permanent deformation

(fig. 9) due to buffeting as the st ar ti ng shock wave moved down the wind tunnel durin g

tunnel st ar tu p. The 120° cone with rin g 1 was s o damaged that t he end ring was perma

nently deformed, and the model failed immediately af te r the supporti ng vanes wer e

re t rac ted , at a pr es su re lower than those for the other thre e models ( table 3 . The other

thr ee shells were visibly deformed into an n = 8 mode (fig. 9) by the eight model sup

por t vanes, but the end ri ng s we re essent ially undeformed. The se large prebuckling

def orm ati ons undoubtedly affect ed th e buckling load and mod es. Even though the qual ity

of the data was ad verse ly affected by the tes t technique, the d ata are presented because

of th e uniquene ss of th e tests, which simulate d the type of loading on plane tary en try

vehicles.

The shell deformations wer e mos t sev ere for the s hel ls with the smal ler cone

angle (als o sma lle r she ll thickness, fig. 7) and the sm al le r end r ings. The shells with

the smal ler end r ings ( r ing 1) appea red to buckle initially into an n = 4 mode; the sh el ls

12

8/12/2019 19710004581


APPENDIX B Continued

with the la rg er rin gs (ring 2) buckled at higher press ure s in to an n = 7 mode. All shells

ultimately collapsed into an n = 3 mode as dynamic press ure was increased; a photo

gr ap h of a collapsed model is shown in figure 10.

Com pari son of The ory and Exp erim ent

Vibrat ion results . Theoretical and experimental vibrat ion res ult s a r e shown in

figure 11 i n t e r m s of the shell frequencies and wave number n. The curv es repr esen t

theoret ical res ult s for various class ical boundary conditions, a clamped edge at t he sma l l

end and either a clamped, simply supported, or f r ee edge at the lar ge end; the symbols

are for clamped/ring support ; and the solid symbols repr esent experimental data. At

n = 2 or 3 the theoret ical effects of the r ings a r e sl ight , and the shells vibrate essential ly

as a clamped/f ree shel l. As n inc rea ses , the effect of the rin gs becomes mor e pro

nounced, and for n > 7, the frequencies a re bracketed by the frequencies for clamped/

supported and clamped/clamped edges. The theo retic al effect of the incre ase in rin g

s i ze is slight.

The theory predicts the co rr ec t value of n for the minimum frequency fo r each

shell , and the theoret ical and experimental resul ts ar e in good qualitat ive agreem ent for

the ent ire range of n considered . However, the quantitative agre emen t is only f a i r.

Excluding n = 2 , the l arge st difference between theor et ical and experimental re sul ts

was 26 percent; experimental frequencies were general ly les s than the res ult s predicted

by theory. The thebret ical and experimental vibrat ion res ult s re veal that the end rings

did not provide edge re st ra in t between simply supported and clamped for n < 7 . In fac t ,

at n = 2 , the experimental data differ from the theoret ical frequencies for clamped or

supported edges by a fa ct or of 4 fo r the 120' cone and by nea rly a n or de r of ma gnitudef or the 140' cone .

Buckling results .- Theoretical and experimental buckling results are shown in fig

u r e 1 2 where buckling pr ess ure is shown as a function of c irc um fer ent ial wave number

n. In the theoret ical calculations, each shell w a s subjected to la te ra l ex terna l pre ssu re ,and static equilibrium w a s maintained by an axial load applied at the s ma ll end of t he cone,

as indicated by the sketch in the figu re. The so lid curves represent theore t ica l resu l t s

for various classical boundary conditions, clamped at the sm all end and ei ther clamped,

simply supported, o r f r e e at the large end; the symbols ar e for s hells clamped at the

sm all end and r ing-supported at the lar ge end; experimen tal data a r e indicated by thesolid symbols. The re su lt s for s hell s with classic al edge support indicate for each she l l

a single minimum value of p re ss ur e, which occu rs at n = 8, 7 (120°, 140°) fo r sh el ls

with either clamped or simply supported edges at the large end and at n = 5, 4 (120°,

140') f o r sh el ls with a f re e edge at the large end. However, the res ult s for r ing support

indicate two relative minimums, one at n = 3 and one at n = 7 . The lowest theoret ical

13

8/12/2019 19710004581


8/12/2019 19710004581


REFERENCES

1. Cohen, Ger ald A.: Th e Effect of Ed ge Co ns tr ai nt on th e Buckling of Sandwich andRing-Stiffened 120 Deg ree Conical Shel ls Subjected to Ex ter nal Pr e s s u r e . NASA

CR-795, 1967.

2 . Newton, R. A.: Free Vi bra tio ns of Rocket Nozzles . AIAA J. , vol. 4, no. 7, July 1966,pp. 1303-1305.

3. Almroth, B. 0.; and Bushne ll, D.: Com pute r An aly sis of Vario us She lls of Revolution.

AIM J . , vo l. 6 , no. 10, Oc t. 1968, pp. 1848-1855.

4. Cohen, Ger ald A.: Buckling of Axially Co mp res sed C yli nd ric al She lls With Ring-

Stiffened Ed ge s. AIAA J . , vol. 4, no. 10, Oct. 1966, pp. 1859-1862.

5. Bushnell, David: Buckling of S phe ric al Sh ell s Ring-Supported at the Edge s. AIAA J . ,

vol. 5, no. 11, Nov. 1967, pp. 2041-2046.

6. Wang, Leon Ru-Liang: Effe cts of Edge Rest rai nt on the Stability of S phe ric al Caps .AIAA J . , vol. 4, no. 4, Apr. 1966, pp. 718-719.

7. Dixon, Sidney C.; and Hudson, M. Latrel le: Fl ut te r, Vib ratio n, and Buckling ofTruncated Orthot ropic Conical Shells With Generalized Elas tic Edge Res tra int .

NASA TN D-5759, 1970.

8. Bushnell , David: Inextensi onal Buckling of Sp her ica l She lls With Edg e Ri ngs .

AIAA J . , vol. 6, no. 2, Feb. 1968, pp. 361-364.

9. Dixon, Sidney C.; and Carine , John B.: Pre lim ina ry Design Pro ced ure for End Rings

of Isotr opic Conical Shel ls Loaded by Exter nal Pr es su re . NASA TN D-5980, 1970.10. Cohen, Ger ald A.: Compute r Analysi s of As ym me tri c Free Vib rat ion s of Ring-

Stiffened Orth otr op ic Shel ls of Revolution. AIAA J . , vol. 3 , no. 12, Dec. 1965,

pp. 2305-2312.

11. Cohen, Gerald A.; Fo st er , Richard M.; and Scha fer, Ev er et t M.: Ana lysis of Conceptual

Designs fo r the Voyager Entr y Capsule. Cont ract No NAN-5554-1, Philco-FordCorp. (Available as NASA CR-66580.)

12. Sewall, John L.; and Ca th er in es , Donne11 S.: Analytical Vibration Study of a Ring-

Stiffened Conical Shell and Com pari son With Expe rime nt. NASA T N D-5663, 1970.

13. Stee ves, Ea r l C.; Durling, B ar ba ra J.; and Walton, William C., Jr. : A Method for

Computing the Response of a Gene ral Axisymmetr ic Shell With an Attached

Asymme tric Stru cture . AIAA Stru ctura l Dynamics and Aeroelastic ity SpecialistConference and the ASME/AIAA 10th Str uct ure s, Str uct ura l Dynamic s, and

Ma te ri al s Conference, Ap r. 1969, pp. 302-328.

15

8/12/2019 19710004581


8/12/2019 19710004581


TABLE 1.- MODEL PROPERTIES

(a) Shel l A.

Shell prope rties:hf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.016 in. (0.041 cm )

h , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.53Oin. (1.35cm). . . . . . . . . . . . . . . . . . . . . . . 1.469 Ibm/ft2 (7.172 kg/m2)E . . . . . . . . . . . . . . . . . . . . . 10.5 X 106 lbf/in2 (72.4 GN/m2)p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 0 . 3 2

Ring properties:r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.65 in. (6.73 cm )t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.077 in. (0.196 cm )A, . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28 in2 (8.26 cm2 )

I < , FI , J / 2 . . . . . . . . . . . . . . . . . . . . . . .4 .4 5 in4 (185.2 cm4)I<[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .z o . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.70 in. (6.86 cm )E, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E

p r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1p . . . . . . . . . . . . . . . . . . . . . . . . 0.1 Ibm/in3 (2768.0 kg/m3)

(b) Shell B.

Shell properties:hf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.020 in. (0.051 cm )hc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.75 in. . (1.91 cm)y . . . . . . . . . . . . . . . . . . . . . . . 1.714 Ibm/ft2 (8.368 kg/m2)

E . . . . . . . . . . . . . . . . . . . . . 10.5 X l o 6 lbf/in2 (72.4 GN/m2)I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.32

Ring properties:Small end:

A, . . . . . . . . . . . . . . . . . . . . . . . . . . 0.648 in2 (4.18 cm2)15 . . . . . . . . . . . . . . . . . . . . . . . . . . 0.517 in4 (21.5 cm4)15 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.058 in4 (44.0 cm4 )I<< . . . . . . . . . . . . . . . . . . . . . . . . . -0.45 in4 (-18.7 cm4 )J . . . . . . . . . . . . . . . . . 0.1045 X 10-3 in4 (4.35 X 10-3 cm4)

z o . . . . . . . . . . . . . . . . . . . . . . . . . . 0.805 in. (2.04 cm)

E, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E

r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1p . . . . . . . . . . . . . . . . . . . . . . 151.7 Ibm/in3 (4.2 Gg/m3)

Large end:

Ar . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8in2 (11.6cm2)

IC . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 in4 (412.0 cm4)IF, . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 in4 (482.8 cm4)ICF, . . . . . . . . . . . . . . . . . . . . . . . . . -1.25 in4 (-52.0 cm4)J . . . . . . . . . . . . . . . . . . . . . . . . . 18. 45i n4 (767.9 cm4)z o . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.01 in. (7.64 cm )E, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E

p r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 1

p . . . . . . . . . . . . . . . . . . . . . . 0.1 lbm/in3 (2768.0 kg/m3)

17

8/12/2019 19710004581


TABLE 2.- EXPERIMENTAL FREQUENCIES O F

n

23

45

6

7

RING-SUPPORTED SHELLS

f , Hz

120° cone 140' cone

Ring 1 Ring 2I

Ring 1 Ring 2

271 261 14 1 146

249 258 24 7 253

302 324 343 3 70576 478 512

608

8 60 882 838 865

TABLE 3.- EXPERIMENTAL BUCKLING LOADS FOR

RING-SUPPORTED SHELLS

3 19.1 131.7

m;3 23.5 162.0

3 a14.9 '102.7

120 7 3 18.9 130.3

ashell end ri ng damaged during tunnel startu p.

18

8/12/2019 19710004581


7 in .+

6 8 . 6 cm)

J

t u r e I

9 0 cm)( 2 2 9 n .

(a) Shell A.

S a n a w i c h a r i e I I s t r u c t u r e7

(b) Shell B.

Figure 1 . - Idealized configurations of des ign s fo r proposed entry ve hi cle .

19

8/12/2019 19710004581


T h e o r y

- P r e s e n t

O R e f . I O

L I 1 1 I

0 2 4 6 8

n

(a) Shell A .

0 2 4 6 8

n

(b) Shell B.

Figure 2.- F.ree-vibrat ion characterist ics of conical sh ell s with

ring-sup ported edges. (See f ig. 1 fo r shel l geometry.)

20

8/12/2019 19710004581


- -

IO0

f , H z 50

0

I O 0

f , H z 5 0

0

-C I ampe d / r i n g

- S u p p o r t e d / s u p p o r t e d

- - C l a m p e d / c l a m p e d

1 1 J

2 46 8

n

(a) Shell A.

\

.

E d g e r e s t r a i n t

- - - S u p p o r t e d / s u p p o r t e d

C I a m p e d / c I a m p e d

02 4 6 8

n

(b) Shell B.

Figure 3 . - Effects of edge support on free -vib ratio n ch ara cte ris tic s of

conical she lls . (See fig. 1 for shell geometry.)

21

8/12/2019 19710004581


8/12/2019 19710004581


- -

- - A . .I .o e -

w

Wmax

- I . o0

- 1 . 0 . L0

0

C I ampe d/c I a m p e d s u p p o r t

R i n g / r i n g s u p p o r t1

. 5 I .oX-

X I

(d) n = 5 .

~

5 I .oX

X I

e) n = 6 .

5 I . oX

f) n = 8 .

Figure 4.- Concluded.

23

8/12/2019 19710004581


8/12/2019 19710004581


w

0W m a x

- I .o

W

Wma x

W

w m a x

-1.0

- I a m p e d / c I a m p e d s u p p o r tL C I a m pe d / r i n g s u p p o r t

5

X

X I

(a) - 0.0294.

X I

(b) L = .0555.R2

O \

I . o

I

I . o~ -

0 . 5X

X I

(c) L= .1670.R2

Figure 6 . - Variation of vibration mode shapes with

base- r ing s ize for shel l A . n = 2.

25

8/12/2019 19710004581


M a t e r i a I P r o p e r t i e s1

I

5 . 4 x I O 6 4 4 . I I O.OC4 1 I 7 7 2 3 5

I

M o d e l D i m e n s i o n s

a R I R h

i n . cm i n . cm i n . c m

6 0 3 . 9 3 9 . 9 8 I O 2 5 . 4 . 0 7 6 , 1 9 3

7 0 ° 3 . Y 3 9 . 9 8 I O 2 5 . 4 I , 0 8 5 . 2 1 6

I Y I YZ J P Zo/R2

- . . _ _0 0 0 5 3 . 0 0 1 0 8 , 4 8 8 9 , 0 1 5 54 2 9 0 - 0 0 3 8 2 . 0 0 0 3 9

60 , 2 . 5 3 5 2 . 0 0 5 7 4 - . _ . 0 0 0 6 6 , 0 0 0 9 7 . O O I 0 5 . 6 1 0 0 . 0 1 5 7

70 I . 3 7 1 9 - 0 0 3 4 7 .0 0 0 6 3 . O O I 3 0 ,001 12 . 4 2 3 8 . 0 1 6 8

7 0 ° 2 . 4 5 0 2 . 0 0 4 8 9 . 0 0 0 9 8 . O O I06 00 I08 . 5 1 3 1 O 170,.... ~. . .. - ~

Figure 7 .- Construction details of test models.

8/12/2019 19710004581


L-68- 9859.1Figure 8.- Model in wind-tunnel tes t section.

8/12/2019 19710004581


(a) Ring 1, a = 120'. (b) Ring 2 ,a

= 120°.

L- 70-4786c )Ring 1, a = 140'. (d) Ring 2, a = 140°.

F igu re 9.- Initial deformations in test models induced by

buffeting during tunnel startup.

28

8/12/2019 19710004581


L- 69- 7 2

Figure 10.- Te st model aft er collapse into n = 3 mode.

29

8/12/2019 19710004581


1600-

I 2 0 0

f , HZ 8 0 0

400

-1200

f , Hz 8 0 0

-400

T h e o r y E x p e r i m e n t

C l a s s i c a l s u p p o r t @ R i n g I

0 C l a m p e d / r i n g I. R i n g 2

17 C l a m p e d / r i n g 2

C l a m P e d / c l a m p e d

C I a m p e d / s u p p o r t e d e

h e Be1 1 I I

n

(a) 1 2 0 ~cone.

C I a m p e d / s u p p o r t e d

O C I

amped/free------ _ _ 1 1

n(b) 14 ° cone.

Figure 11 . Comparison of theoretical and experimental vibration results f o r blunt

conical shel ls clamped at sma ll end and ring-supported at lar ge end.

30

8/12/2019 19710004581


200 T h e o r y

I C l a s s i c a l s u p p o r t

I 5 0

p , p s i 100

50

0

0 C l a m p e d / r i n g I

C l a m p e d / r i n g 2

E x p e r i men t


\ \

4 8 12

n

(a) 120O cone.

2 0 0

8 0 0

P , k N / m 2

40 0

3

Figure 12.- Comparison of theoretical and experimental buckling results for blunt

conical sh ell s clamped at small end and ring-supported at large end.

31

8/12/2019 19710004581


C I a m p e d/c a m p e

20c

I 5 0

P P s i I O 0

5 0

0

T h e o r y

C l a s s i c a l s u p p o r t

C I a m p e d / r i n g I


E x p e r m e n t

e c m p e d / r i n g I

C a m p e d / r i n g 2

C I a m p e d/c a m p e

11 I

4 8 12

n

(b) 140’ cone.

Figure 12.- Concluded.

I200

8

P k N / m 2

400

0

19710004581

Documents