majumdar s 1968

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RUCKLING OF THIN ANNULAR PLATES

DUE TO RADIAL COMPRESSIVE LOADING

Thesis by

Saurindranath Majumdar

En Partial Fulfillment of the Requirements

For the Degree of

Aeronautical .Engineer

California Institute of Technology

Pasadena, California

1968

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ACKNOWLEDGMENT

The author wishes to ex pre ss his s ince re appreciation to

(1) Dr, Ern est E. Sech ler , who suggested the experiments , for his

help and encouragement , (2) Dr. C. D. Babcock fo r his advice and

comments , (3) M. Je ss ey , C, Hemphil l and R. Luntz for thei r help

in set t ing up the e xper iments , (4) Mrs. Betty wood fo r h e r excel lent

drawings and Mrs. El izabeth Fox fo r bearing with my handwrit ing

and typing the manuscript so skilZfullyo

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ABSTRACT

Buckling of c ir cu la r annular plates with the outer edge

clamped and the inne r edge f r e e loaded with a uniform ra di al corn-

pre ssi ve forc e applied a t the outside edge has been studied both

theoretically and experimentally. A differential equation of equi-

libriu m of the buckled plate ha s been developed fo r any gen era l

deflection pa tte rn and solutions corre sponding to the buckled fo rm

w ( r )Cos

n8

have been sought. The differential equation hasn

been solve d exactly for n = 0 and n = 1 and approximately fo r

hig he r va lue s of n a s well a s for n = 0 and 1. The solutions in-

dicate that , fo r sm al l ra t ios of inner to outer radius , the pla tes

buckle into a radia l ly symmetr ic buckling mode, but for the ra t io

of inn er to outer radiu s exceeding a ce rt ai n minimum value the

minim um buckling load corr es po nd s to buckling modes with

waves along the circum feren ce, the number of which depends

on the part icular rat io of the inner and outer radii . Tes t s we re

ca r r i ed out using th in a luminium pla tes and the re su l ts agreed

reasonably well with the theoret ica l predic t ions,

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TABLEOFCONTENTS

PART- TITLE

LlST OF SYMBOLS

I INTRODUCTION

PZ THEORY

Assumptions

Der ivat ion of Cove rning Equations

Approximate Solution

U-II TEST RESULTS

Tes t Se r i e s A

Tes t Se r i e s B

9V DISCUSSION OF EXPERIMENTAL RESULTS

V CONCLUSION

V I REFERENCES

VII APPENDIX ADerivatio n of buckling te m pe ra tu re

Effect of el as ti ci ty of ring on prebucklingsere s s distr ibution

The eff ect of the twisting of the s te e l rin gon the buckling load of the plate

APPENDIX B

Imperfe ct plate

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LIST OF SYMBOLS

a = Outer radius

b = Inner radius

h = Thickness of plate

N = Radia l com press ive force at the outer edge0

No= Cri t ica l rad ia l compress ive force a t the outer edge

Gr

E = Modulus of e la st ic it y of aluminium

E = Modulus of ela st ic it y of s te elS 2

= Stif fne ss of plate =Eh"

1

v = Poisson ' s ra t io , a ssumed equal to 1 3

Nr= Ra d ia l s t r e s s r e su l t a n t

Ne= Circumferent ia l s t ress resu l tan t

Nre= Shear s t r e s s re su ltant

ee=

In-plane radia l and c i rcumferen t ia l s t ra in pr io r tobuckling

e ' e ' = In-plane st ra in during bucklingr ' B

U = Stra in energy

V = Potential energy

U s V = In-plane radial and circu mfer ential displacementperturbations

w = Tr ans ver se d isp lacement per turbat ion

T = Tempera ture r i se above ambient

e c= Theo retical buckling tem per atu re

Tc= Experimentally observed buckling tem per atu re

(Is' a~

= Coefficients sf th er m al expansion of ste el and aluminium

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- 1 -INTRODUCTION

It i s well known ( l ) * hat (assuming a radially symmetric buck-

ling mo de) the rad ial buckling load No of a ci rc ul ar plate with a holec r

a t the center can be expressed a s

where k is a nu me ric al fa ct or , the magnitude of which depends on the

b/a ratio, The value of k for vari ou s b/a ra tio s for a clamped outside

edge and fre e inner edge i s shown in f igure 1 . It is seen that k

r eaches a minimum f or b/a = 0.2 and fo r ra t ios la rg er than 0.2, k

in cr ea se s rapidl y without bound, Th is m ay be explained by noting

that fo r b/a approaching unity, the com pre sse d ring with the ou ter

boundary clamp ed behaves Pike a long co m pr es se d rec tan gu lar plate

clamped along a long side and fr e e along the other. Such a plate will

buckle into ma ny waves. Thus i t could reasona bly be expected that

in the cas e of a narr ow ring , sev era l waves will be form ed along the

circu mfe renc e and the values of k obtained by assumi ng a symmet r i c

buckling mode will have high er value s .The stabi lity of a thin annular plate under uniform c om pre ssi ve

for ces applied at both edges was tre ate d by BPsson (4) and Schubert (5)

fo r several, boundary conditions, but in the se investiga tions, the de-

f lec t ion surface was assumed to be radia l ly symm etr ic , Yamaki (2)

took into account the pos sibili ty of w aves in the c i rcumferen t ia l d i rec -

tion of the buckled plate, but hi s calculations showed that for the ca se

* Numbers in parentheses indicate referen ce n umbers a t the end.

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- 2 -with the out er edge clamped and the inner edge fre e, the minimum

buckling load s ti l l correspon ded to a radially sy mm etr ic buckling

mode,

The purpose of thi s re p or t was to study the buckling m o d e of

a ci rc ul ar iso tropic plate with a concentric hole loaded radially a t

the outer edge which was clam ped and having the edge of the hole free .

The effe ct of the b/a ra tio on the buckling load was sought and the

possibility of ant isymmetr ic modes of buckling was investigated. A

shor t series of sim ple te st s were carried out to check the validity of

the theory.

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-3 -THEORY

Assumptions

(i) The usual Kirchhoff' s hypothesis rega rdin g the varia tion

of displacement and s tr e s se s through the thickness of the plate a r e made.

(ii) The displacement perturbations a r e assum ed to be s ma ll

so that the in-plane Lam4 st re ss es p rio r to buckling do not undergo

any appreciable amount of change during buckling.

(iii) The system i s assum ed to be perfect .

Derivation of Governing Equation

The in- lane equation of equilibrium is

The Lam6 solution for the plane s tr es s case is

The s t ra ins p r io r to buckling a r e

Pe r = E(Nr-vNe) (4a)

Pe g =- No-vNr)

h E(4b

The st ra in ene rgy due to contractio n of the midplane before buckling

is given by

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During buckling, the in-plane s tr e ss e s do fur th er work and the in -

plane st ra in s after bending may be written a s

Assuming that the forc es Nr, Ne rem ain con stant during bending, the

s tr a in energ y due to additional contraction of the middle plane i s

The st ra in energy due to bending is -2B a w- - - - -

P Brae 2 aerdedr

Th eref ore , the total stra in energy of the plate is

The fi r s t integra l of eq. (7) can be writt en with the help of eq. (2) a s

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aNeSince = 0, by using the boundary conditions, the above becomes

which i s equal to the work done by the ex ter nal force . Thus

where W = work done by ex ter nal fo rc e

U' = work produced by the in-plane $t re ss es due to bending.. . Total potential ene rgy of the s ys te m , V, is given by

The d iffe ren tial equation of equilibrium ma y be obtained fro m eq. (9)

by making the potential energ y of the sy ste m have a stationa ry value

The variatio n of the f i r s t two t e r m s g ive

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The variation of the rest s f the term gives

Adding all the variations, we get the Euler equation as

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and the boundary conditions

Since in the presen t c as e the outer edge is clamped and the inn er edge

f r e e

and

We a ls o have

fi= a t r = aa r

(14)

w = O a t r = a (15)

Substituting fo r Nr nd N8 rom eq. (3) into eq. (1 1) and defining

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we get

T ry w = A Cos n8 as a solution.n

Substituting this into (1 7 ) and cancell ing Cos nB we get

Making a substitution of z =6 ivesb

Eq. (19 ) together with the boundary conditions (1 2)- (1 5) properly

transformed poses an eigenvalue problem. When n = 0, the result

i s radially sy mmetric buckling and eq. (19) reduces to

dAn

where $ =-z

A solution of eq. (20) satisfying boundary condition (13) i s

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-9-

The im posi tion of boundary conditions (12) and (141, properly trans -

for me d, yield the following equation

A plot of the res ult obtained i s given in Timoshenko 's "Theor y of

Ela stic Stability" and i s reproduced he re on fig. 2.

n = 1

Eq. (19) reduces to

Substituting

A = @z andn

%'= +dz

this reduces to

This could fu rth er be reduced to

where c i s a n a rb i t ra ry constant and .b p Z - 4

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-1 0-

The general solution of eq. (25) i s

9 = 1. [A J ( z ) + B J (z) CSz P -P - 1 , P (26

where

and i s called the Lornmel function, With the help of eq. (23 ), eq. (26)

can be written as

The boundary conditions (12)-(E5) reduce to

Satisfying these boundary conditions leads to the following equations

For a nontrivial solution of A and E3, we must have

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A plot of the result is given on fig , 2 .

Approximate Solution (Energy Method)

As s u m e a deflection pattern

w = An(r) Cos n 9

Putting this expression in eq, ( 9 ) and integrating over 8 , we get

for n;F 0

and

for n = 0

where ( )' denotes differentiation with respect to r .

For the purpose of calculation the function An was chosen to be

1t sat isfie d a ll the displacement boundary conditions. From f ig , 2

it is seen that the difference between the exact and the approximate

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solutions for n = 0 and n = 1 is not too large, especially for large b/a

ratio. Hence, in carrying out the approximate solutions fog n greater

than 1, the same form f o r An was chosen.

Putting this value of An in eqs. (30) nd (31) nd substituting

for Nr and No rom eq. (3) nd minimising V with respect to wo, we

for n # 0

and

From eq. (16)

awhere k = A (7 - 1)

b

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-13-

The value of k a s a function of b/a h as been plotted fo r n = 0,

1 , 2 , 3 , 4, 5, 6 a n d l o o n f i g . 3 , The e xa ct so lu t i o n s f o rn = Oan d

n = 1 a r e plotted with the approxim ate ones in fig. 2 and it i s seen

that they ag re e reasonably well fo r high b/a ratio. It is evident fr om

fig. 3 that the tr aje cto ry of the minimum buckling load i ncr eas es

with b/a ra ti o without bound. Th is could be explained by the follow-

ing analogy. It is known that a long rectangular plate with one of its

long edges fixed and the o the r fr ee has a buckling load given by

where d is the width of the plate and k' i s a constant.

In the case of a circular plate when b/a approaches unity,

the compressed ring behaves like a plate, as descri bed above.

Therefore , if we redefine a constant kt such that

k'DNe (a) =-r (a-b)'

then k' should appro ach a finite limit a s b/a a ppr oac hes unity.

A plot of k' against b/a shown in fig. 4 indicates that the kt o r r e s -

ponding to minimum buckling load remains finite as b/a approaches

unity. The value of k' for an infinitely long rectangular plate is

shown in f ig , 4.

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-14-

TEST RESULTS

To check the validity of the theo ry, some simple te s ts were

carr ie d out using 0.041" thick aluminium plate s clamped between

two 0 . 5 " thick steel r ings by means of 12- 1/2"$ high tensile stee l

bolts as shown in fig. 5. The inner dia me ter of the st ee l ring s was

8'' and the outer diameter 10". The following siz ed holes were used

in the plates:

b- b/a0' 0

0.5! ' 0,125

2 ! ' 0 . 5

2.5 I s 0.625

The loading was accom plished by heating the whole a sse mb ly , so

that due to different coefficients of expansions, the steel rings put

a uniformly distributed radial comp ressive load on the plate. The

effect of the elasticity of the rings on the s tr e s s distribution in the

pla tes and on the assumption of clamped edge condition is dis-

cussed in Appendix A, The tes ts were done in two parts .

Tes t Ser ies A

The assembly was placed inside a Missimers environment

chamber in which the temperature could be held at any value for any

length of time. Str ain gauges we re attached to t h e two fa ces of the

plate and a pair of opposing gauges R 1 and R were connected to two2

legs of a Wheatstone bridge a s shown in fig. 7. The reading of the

voltmeter, amplified by a factor of 10 , w as directly proportional to

the difference between the strains experienced by R 1 nd R Z and

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gave a m e a su re of the bending of the plate a s shown in Appendix B,

St ra in gauges were connected both in the circ um fere ntia l and

in the rad ial direct ion for the plates with b = 0" , 0.5 and 2' I, and

only in the circum ferentia l directio n fo r the plate with b = 2.5". SR4

Baldwin gauges we r e us ed fo r the plat es w ith b = 0" and b = 0.5" and

tem pera tu re compensated microm easurement fo il gauges w ere used

fo r the plates with lar ge r holes. Copper -Constantine thermocouples

wer e so ldered onto a bra ss washer and seve ra l of these br as s wa shers

we re at tached to the plate a s well a s to the s tee l r ings. After each

inc rem ent of te m pe ra tu re, the ass em bly was allowed to soak heat

fo r about 45 minutes to one hou r, o r until the tem per atu re indicated

by the different thermocouples w er e the sam e. This tempera ture

was checked with the reading of a ther m om et er placed inside the

chamber , The maximum er ro r in measur ing the tempera ture was

+ OF. Derivation of the cri tic al tem per atu re Bc a t which buckling-

occurs is given in Appendix A.

Tes ts w ere car r ie d out to meas ure the d i f ference in the co-

-6efficients aE expansion of aluminium and ste el. A value of 6.6 x 10

p e r OF was used for (aA-%) f o r the purpo se of ca lculation s.

Due to the presen ce of initial imperfectio ns the plat es s ta rt ed

to bend fro m the beginning of loading an d a Southwell type plot was

used to determine the buckling tem per atu re Tc f the perfect plates.

The reli abil ity of th e Southwell plot h as been proven fo r a solid

plate in Appendix B. In plotting the Southwell plot, points ne ar the

or ig in have been ignored s ince for sma l l T the percentage e r r o r in

measur ing T is larg e; als o points corresponding to tem per atu res that

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a r e comparable with Bc have been ignored, since the Southwell plot

does not hold for such large values of T. See Figs . 8-15 inc.

T e s t S e r i e s B

An attem pt was made to m ea su re the nu mber of waves along

the c irc um fer en ce of the plate s during buckling by mea ns of a n in-

ductance pickup. The plate clamped by the rin gs was placed on a

tu rn table; the pickup was attached by mea ns of an a r m to a g radu ated

optical bench and could be ra is ed o r lowe red by mea ns of a turning

knob a s shown in fig. 16. The asse mb ly was heat ed by me ans of a

PO00 watt quar tz iodine photographic lam p. The ge neral tes t setup

i s shown in fig. 17. The la mp was connected in se ri es with a rheostat

so that the cu rre nt through it could be controlled. The pickup was

fixed a t a given height fro m the plate and the c ur re nt through the lam p

was increas ed in steps. At ev ery ste p, sufficient t ime was allowed

to let the assem bly r eac h an equilibrium state and then the plate was

rot ate d by turning the tu rn table a nd the output of the pickup was

plotted di rec tly on a n X -Y plotter . Fig , 18 gives the calibration

curve for the pickup and the effect of tem per atu re on i t . I t i s se en

that the effect of tem pera ture on the calibration curve i s sm all and

for the purpose of mea surin g the number of waves around the ci r -

cumferenc e i t was adequate. It was ass um ed that , though the tem pe r-

atu re distr ibution in the plate was n onuniform, i t would only defo rm

the shap es of the waves around the c irc um fer enc e and would not

change the number, The tes t res ult s from the different sized plates

a r e g iven on f igures 19 to 23. A thermocouple attached to the plate

gave a n ave rag e value of the te m pe ra tu re of the plate.

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-1 7 -DISCUSSION O F EXPERIMENTAL RESULTS

The theo ret ic al buckling te mp eratu re Bc f o r each cas e has

been indicated together with the expe rimentally obse rved buckling

tempera ture Tc in fig. 8 to fig. 15. The valwsof k computed from

the experim ental ly observed buckling tem pe rat ure have been plotted

in f ig. 3 . The difference between the th eore t ical and the experim en-

tal ly observed values of k a r e of the ord er of 1 0% . This discrepancy

is mainly due to the inaccuracy in me asuri ng the tem pera ture and

al so due to the s l ight tem pera ture variat ion in the plate and the steel

rings which could not be avoided, Another possible so urc e of e r r o r

lay in the fac t th at , though the voltage output fro m the st ra in gauges

was amplified 10 t ime s , its magnitude was ve ry sm all and thus

sma l l e r r o r in measur ing the vol tage resul ted in considerable re la -

t iv e e r r o r .A

dir ec t displacement me asure me nt of the plates under

loading would be mo r e d esi rab le, but setting up displace men t me as -uring devices inside the furnac e was inconvenient because of lac k of

space a s well as giving tempe rature problem s, By using two strain

gauges on the two fa ce s of the p late s a s two leg s of a Wheatstone

bridg e, the effect of temp era tur e on the voltage output was min im ise d

because both s t r a in gauges were heated to a lm ost the same tem pera -

tur e and any effect of tem pera ture on the re sis tan ce of the s tr ai n

gauges was balanced out.

In t e s t s e r i e s B, the cal ibrat ion curve for voltage vs . dis tance

va rie d s l ightly with tem pera ture and so the dis tance given on f igs . 19

to 23 a r e not ve ry accura te . But, since the purpose of the te st s was

to m ea su re the number of waves around the c ircum feren ce only, the

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effect of te m pe ra tu re on the calibr ation curve did not affect the

resul t s . The plots in figs. 19 and 20 fo r b/a = 0 and 0,125 show

the buckle pattern to be radially sym me tric , a s expected. In

fig , 21, f or the cas e of b/a = 0.5 the buckling mode i s se en to be

radial ly symmetr ic , i. e. , n = 0, According to the app roximate

an aly sis , whose resu lts a r e shown in fig. 3 , the buckling mode

should be n = 1 though the c urve corresponding to n = 0 l ie s closely

above it . In fig. 2 where the exact solutions a r e drawn, the cu rves

f o r n = 0 and n = 1 inters ect a t b/a = 0.5. The fac t that the ex pe r-

imental re su lt showed n = 0 could be explained by observin g tha t

the plate had more initial imperfection in the n = 0 mode than i n

the n = 1 mode, which ag re es with the physical intuition that im -

perfections with longer wavelengths a r e m or e probable than with

sho rte r wavelengths and that axisymrnetr ic imperfections a r e m ost

predominant. F o r the ca se with b/a = 8.625 (fig. 22) the plate

f i r s t s t a r t s to def lec t in the n = 0 mode, but with r is e of tem pe r-

atu re i t goes into the n = 2 mode. According to fig. 3 , the cur ves

fo r n = 2 and n = 3 a lmost in te rse c t a t b /a = .625. F o r t h e % c a se

with b/a=

0.75 (fig. 231, the plate fi r s t s ta rt s deflecting in the

axisy mm etric mode, but with r is e of tem pe rat ure goes into the

n = 5 mode, which ag re es with the theoretica l wave num ber given

in the figure. The se te st s concllusively prove tha t, fo r clamped/* '

outside edge and fr e e inne plate buckles with waves

ference for b/a exceeding a cer ta in minimum

value close to 0.5.

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-19-

CONCLUSION

(1 The energy method using just one t e r m gives a reasonable

approximation to the tr ue buckling load, parti cula rly for lar ge b/a

rat ios.

( 2 ) F o r sm all values of b/a rat io the radial ly sym me tric mode

gives the lowest buckling load. As b/a i s in cr ea se d, a n unsyrnmet-

r i c mode with waves i n the circum ferential direct ion gives a lower

buckling load than the sy m m et ri c mode and the number of waves

in cre ase s with an increasing b/a rat io.

( 3 ) The theo retic al buckling load in cr ea se s beyond bound a s b/a

approaches unity, but i t mu st be r em em ber ed that as b/a approaches

unity, the thickness h beco me s of the sa m e o rd e r of magnitude a s

(a- b), and hence the theor y which is based on the assu mption that h

i s ve ry sm all compared to the other dimensions of the plate is no

longer valid.

(4 It would be interes ting to me as ur e the initial imp erfection

of the plate and do a Fou r ier analys is on it and experim entally

m e as u re how the coefficient of e ach component grow s with the load.

(5) Fo r lar ge b/a rat io s, the buckling cur ves for different n 's

a r e crowded together. Thus fo r any given b/a rat io in this ran ge,

the plate will probably s ta rt to buckle i n that mode in which the

maximum ini t ial imperfection i s present . To check whether the

plate bifurcates into another mode a t higher lo ad, a full nonlinear

analysis ha s to be c arr ied out.

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-20-

REFERENCES

1, Timoshenko, S. Theory of Elastic Stability, McGraw Hill,

New York (1936).

2 . Yamaki, N. : "Buckling of a Thin Annular P la te Under Uniform

C om pr es si on s' , Jou rna l of Applied Mech anics, A.S. M. E .,

Vol. 25, p. 267 (1958).

3. Timoshenko, S . : Strength of Mate rials, P a r t 11, D. Van Nostrand

Comphny, h c . , New York (1930)..4, Olsson, R. @ran: "Uber axia lsym me trisch e h ic k u n g &inner

Mreisr ingplat tenl\ hgenieur-Archiv, Bd. $ (1937), S. 449.

5, Schubert , 8 . : "Die Beullast dUnner Kre isrin gpla tten, d i e am

Aussen- enrnd Innenrand g le ich m~ ssi ge nDruck erfalhren",

Zeitschrift fGr angewandte Mathematik und Mechanik,

Bd 25/27.

6. Dean, W , Pi.: "The Ela stic Stability of an Annular Pla te" ,

Proceedings of the Royal Society of London, England, S e ri e s

A, Vol, PO6 (19241, p. 268.

7. Meissner , E, "her das knicken kreisringf%miger Scheiben" ,

Schew eizerche Bauzeitung, Bd. 101 (1 9331, s. 87.

8. Timoshenko, S. : n 'Theory s f Pla t es and Shel lss , McGraw-Hill

Book Company, hc. , New York and London, 1940.

9 . Luke, YeL. h t e g r a l s of B ess el Functions, McGraw Hill Book

Company, Hnc .10. Jahnke -Emde: Tab les of Fun ction s, Ver lag m d Druck von

B. C. Teubner in Leipzig and Berlin.

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FIG. 2

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* Experimentai Points

FIG,3

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FIG. 6

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Distance Between Pickup and Conducting Surface ( inches)

FIG. 18 CAL I BRATION CURVE FOR PICKUP

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APPENDIX A

Derivation of Buckling Temperature

Alternatively

Since displacements are radially sy-mmetric

e = -u

du and e = -dr ' 8 r

hE dhEand y =-etting P =- a -Vl - v

The in -plane equilibrium equation i s

Substituting for Nr and No n terms of u and T

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The solution i s

Case I. Plate without hole

and u at r = a i s aBTa, neglecting the elastic deformation of the rings.

*@ u = aBTr

Putting the value of u in equation (32)

From eq. (1)

Case 11. Plate with hole

Xn this case the boundary conditions are

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Solving for A and I3 and substituting for these in the solution for u

2 2asa (1 -v)+aAb ( l t v ) 2 2

u = T r - (1tv)a b T

2 (aA-a*);.

a2(1-v)+b2(l+v) a ( ~ - v ) t b l t v )

Substituting this value of u in eq. (32)

From eq. ( 1 )

Effect of elasticity s f ring on prebuckling stress distribution

In fig, 24, let p be the pressure on the plate applied by the

ring, Then the pres sure on the ring = f$

, Radial displacement of ring

h a2

= a T a + % =

s

hen the boundary condition (35a) should be replaced by

and the other equation is as before

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Solving

and

2 32 2

phb a ( l t v )a b (aA-ae) ( l tv )T

B = - 2 (37b)

b ' ~ ~ t ( a ' ( 1v)+bZ(l v ) } a (1 - v ) + b t ( l + v )

Using the condition that Nr = -ph at r = a

where A and B are given above.

Solving for p

This value of p substituted irm eq, ( 37 ) gives the value of A and B in

terms of T,

7/27/2019 Majumdar s 1968


2 2 3 2(aA-os)(a -b )( l t v) ha b T

2 2 Es 2 2-b )+bl tF{a (l-v)-i-b ( ~ - v ) + b~ t v ) }

The second ter m in the expression fo r A and I3 give the cor rec t ion

due to ela sti cit y of th e ring. The magnitude of the cor re cti on h as

been calculated using the following data

-6 0a s = 6 . 7 x 1 0 per F, ~ = 1 3 . 3 r l 0 - ~ ~ e r O ~

For a = 4 " , b = O W

% e r r o r in A fo r neglecting elas ticity of r ing

B = 0 fo r both cases.

-6A = 8 . 8 ~ 0 T t . 1 6 9 x l om6T

% e r r o r in A fo r neglecting ela sti cit y of rin g

7/27/2019 Majumdar s 1968


. O/ o er ror in B for neglecting elasticity of ring

The e r r o r s involved a r e seen to be small,

The effect of the twisting of the s teel ring on We buckling load of

the plate

Considering the ring acted on by the twisting moment Mt p e r

unit length as shown in fig. 25, i t can be shown (3) that the moment-

rotation cha racteristic is given by

Mt = L0

E bt3s

where L=-12a log (1 + t /a)

Consider a solid circu lar plate loaded radially and clamped by elastic

support with moment-rotation c ha rac ter isti c given by

Mt = Le

The equation of equilib rium i s

N0 2

Let = a and or = u, the above re duces to

The solution of this equation is

where @ =-w and S is the ~ o r n r n e lunction.du 1,1

7/27/2019 Majumdar s 1968


Since # i s f in ite a t u = 0, B1 = C1 0. The other boundary condition

i s

F o r the specimens used in the te st

( t )d r

The solution is

F o r a p erfectly clamped plate

dw= - L -r

r= a

Thus fo r the specimens used the assumption of per fectly clamped

r= a

edge condition is justified and can reas onab ly be extended to pla tes

with hole s ,

7/27/2019 Majumdar s 1968


The pair of radial strain gauges used in tes t s er i e s A measures

the difference in the radia l st ra in s and a pa ir of circu mfe renti al

s t ra in gauges measure r the difference in the cixcumfe ential s trains

between the top and the bottom sur fac es of the plate.

dLw. . e r (top) - E (bottom)= t-rIt dw

ee (top) - E @(bottom) - -dr

Thus the pair of radial st rai n gauges me asu re the rad ial curvature

of the plate and the p ai r of the circumfe rential gauges measure the

circumferential curvature of the plate,

Imperfec t plate

Consider a solid circu lar plate with radially sylnme tric initial

imperfection wQ(r). The equation of equilibrium is (for small w and w )0

7/27/2019 Majumdar s 1968


Since the deflections a re radial ly sym metric

1 d d l d No 1 d d(w+w*--d r [ r ~ f ~ Z i r % i } ]T T F ' d r )

Since there is no la te ra l load C = 0.1

N0

Letting = @ and substituting hr = u, where a2 =

L e t J l ( o a ) = O h a v e r o o t s a l , aZ, .....dw

00

ana J (- U) =n l

- b J (a r )a n l n

n = l n = l

dwowhere #o =-rThe equation reduces to

with boundary conditions,+= 0 at u = 0 and @ = 0 a t u = aa.

The Green's function fo r this o perato r is

J, ( e ) ~ ,M )J l ( a ) fo r & ' ~ u 4 a a

J1(aa) IUsing the Green's function the solution of the above differential equa-

tion can be writ ten as

7/27/2019 Majumdar s 1968


For small load No, this can be written as

Taking only the first two terms of the series

- 3 . 8(using al -a 7.01

and a2 =-a

Coefficients of bl and b2 together with the first order correction are

tabulated on the following page for r = a/2.

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N Coefficient of bl Coefficient of bZ0-o 0th order Correction 0th order CorrectionCr

N

O <.C 1 . 0 , the relationhus for

is quite accurate,

Differentiating eq. (39) it can be shown that

-Yz(ar)J1"'1

dJ1b,r)where ~ i ( a ~ r )

dr

defining

7/27/2019 Majumdar s 1968


the above reduces to

and as before for small loading

N0and al so for- < 1 . 0

Nocr

is quite accurate.

Thus from eqs, (38a) , (38b) , and (40) and (41 we conclude that a

Southwell type plot may be used with the test data to find the buck-

ling load,

majumdar s 1968

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