7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 1/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 2/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 3/59
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,
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 4/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 5/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 6/59
- 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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 7/59
- 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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 8/59
-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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 9/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 10/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 11/59
The variation of the rest s f the term gives
Adding all the variations, we get the Euler equation as
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 12/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 13/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 14/59
-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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 15/59
-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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 16/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 17/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 18/59
-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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 19/59
-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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 20/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 21/59
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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 22/59
-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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 23/59
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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 24/59
-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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 25/59
-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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 26/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 27/59
FIG. 2
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 28/59
* Experimentai Points
FIG,3
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 29/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 30/59
FIG. 6
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 31/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 32/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 33/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 34/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 35/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 36/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 37/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 38/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 39/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 40/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 41/59
Distance Between Pickup and Conducting Surface ( inches)
FIG. 18 CAL I BRATION CURVE FOR PICKUP
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 42/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 43/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 44/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 45/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 46/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 47/59
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 48/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 49/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 50/59
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
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 51/59
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
http://slidepdf.com/reader/full/majumdar-s-1968 52/59
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
http://slidepdf.com/reader/full/majumdar-s-1968 53/59
. 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
http://slidepdf.com/reader/full/majumdar-s-1968 54/59
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
http://slidepdf.com/reader/full/majumdar-s-1968 55/59
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
http://slidepdf.com/reader/full/majumdar-s-1968 56/59
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
http://slidepdf.com/reader/full/majumdar-s-1968 57/59
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.
7/27/2019 Majumdar s 1968
http://slidepdf.com/reader/full/majumdar-s-1968 58/59
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
http://slidepdf.com/reader/full/majumdar-s-1968 59/59
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,
Top Related