elastic stability of annular thin plates with one free edge

Hindawi Publishing CorporationJournal of StructuresVolume 2013 Article ID 389148 9 pageshttpdxdoiorg1011552013389148

Review ArticleElastic Stability of Annular Thin Plates with One Free Edge

Nagarjuna Jillella and John Peddieson

Department ofMechanical Engineering Tennessee Technological University 115West 10th Street Box 5014 Cookeville TN 38505 USA

Correspondence should be addressed to John Peddieson jpeddiesontntechedu

Received 21 April 2013 Accepted 20 June 2013

Academic Editor Lucio Nobile

Copyright copy 2013 N Jillella and J Peddieson This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The elastic stability of annular thin plates having one free edge and subjected to axisymmetric radial edge loads at the other edgeis investigated The supported edge is allowed to be either simply supported or clamped against axial (transverse) deflection Bothcompression buckling and tension buckling (wrinkling) are investigated To insure accuracy twomethods of solving the appropriateeigenvalue problems are used and found to yield essentially identical results A selection of these results for both compression andtension buckling is presented graphically and used to illustrate interesting aspects of the solutions

1 Introduction

This paper deals with the elastic stability of an annular thinplate subjected to axisymmetric in-plane edge loads Thecase of equal compressive loads applied at both boundarieswas dealt with definitively by Yamaki [1] In this casethe in-plane radial and circumferential stress resultants areuniform and equal producing a closed-form transcendentalequation from which the buckling loads can be deduced Ifthe in-plane boundary loads (either tensile or compressive)are unequal the in-plane stress resultants are variable andunequal creating a more complicated situation in whichnumerical work is normally required Various aspects ofthis general problem have been investigated by Timoshenkoand Gere [2] Mansfield [3] Majumdar [4] Yu and Zhang[5] Machinek and Troger [6] Coman and Haughton [7 8]Coman and Bassom [9 10] Noh et al [11] and Jillella andPeddieson [12] A particularly interesting facet of this class ofproblems is that radial tensile loads can produce compressivecircumferential stress resultants over a portion of the platethat lead to radial wrinkling (tension buckling) It should benoted that wrinkling analysis has often been carried out usingtension field theory (see for instance Coman [13] for a recentdiscussion) Since that approach is not used in this work thevast pertinent literature will not be reviewed herein

The present paper reports an investigation of the elasticstability of an annular thin plate having one load-free edge

The other edge is subjected to uniform radial tension orcompression and is either simply supported or clampedagainst axial deflection This investigation both overlapssome findings of the papers referenced above (as discussedsubsequently) and adds significant new information

The remainder of the paper is organized as followsFirst the pertinent governing equations are reviewed Nexta selection of results for both compression and tensionbuckling is presented and discussed Finally a summary ofthe work and a recapitulation of important conclusions aregiven

2 Governing Equations

In this section the governing equations are reviewed forthe bending of thin elastic homogeneous isotropic annularplates of uniform thickness undergoing small deflections andsubjected to axisymmetric in-plane loads (see for instanceTimoshenko and Gere [2] for more details) Consider anannular plate having respective inner and outer radii 119903

119894and

119903119900and thickness ℎ The plate is assumed to be linearly elastic

with Youngrsquos modulus 119864 and Poissonrsquos ratio 120592 It is convenientto describe the problem in terms of the respective radialazimuthal (circumferential) and axial cylindrical polar coor-dinates 119903 120579 and 119911 with the plane 119911 = 0 being coincident

2 Journal of Structures

with the platersquos middle surface and the 119911 axis being the axisof symmetry The appropriate differential equation is

119863nabla4119908 minus (119873

119903120597119903119903119908 +119873

120579(

120597119903119908

119903

+

120597120579120579119908

1199032)) = 0 (1)

where 119908 is the axial (transverse) displacement

nabla2119908 = 120597

119903119903119908 +

120597119903119908

119903

+

120597120579120579119908

1199032

(2)

is the two dimensional Laplacian operator

119863 =

119864ℎ3

12 (1 minus 1205922)

(3)

is the plate modulus 119873119903is the radial normal force resultant

and 119873120579is the circumferential normal force resultant In all

equations appearing in the present work partial derivativeoperators are understood to operate only on the immediatelyfollowing symbol The normal force resultants are foundby solving the uncoupled plane stress problem associatedwith axisymmetric radial loading at the inner and outerboundaries (see for instance [10]) to get

119873119903=

119873119894((119903119900119903)2minus 1) + 119873

119900((119903119900119903119894)2minus (119903119900119903)2)

(119903119900119903119894)2minus 1

119873120579=

minus119873119894((119903119900119903)2+ 1) + 119873

119900((119903119900119903119894)2+ (119903119900119903)2)

(119903119900119903119894)2minus 1

(4)

where119873119894and119873

119900are the respective inner and outer boundary

tensile force resultants Support conditions to be employedsubsequently are

clamped 119908 = 0 120597119903119908 = 0

simply supported 119908 = 0 119872119903= 0

free 119872119903= 0 119881

119903= 0

(5)

where

119872119903= 119863(120597

119903119903119908 + 120592(

120597119903119908

119903

+

120597120579120579119908

1199032)) (6)

is the radial bending moment resultant and

119881119903= minus 119863(120597

119903119903119903119908 +

120597119903119903119908

119903

minus

120597119903119908

1199032minus

(3 minus 120592) 120597120579120579119908

1199033

+

(2 minus 120592) 120597119903120579120579119908

1199032) + 119873

119903120597119903119908

(7)

is the radial Kirchhoff shear force resultantFor the purpose of numerical analysis it is convenient to

convert the equations to dimensionless forms Towards this

end it is helpful to define dimensionless quantities (denotedby superposed asterisks) through the equations

119903 = 119903119900119903lowast 119903

119894= 119903119900119903lowast

119894 119908 = ℎ119908

lowast 119872

119903=

119863ℎ119872lowast

119903

1199032

119900

119872120579=

119863ℎ119872lowast

120579

1199032

119900

119881119903=

119863ℎ119881lowast

119903

1199033

119900

119873119903= 119873119888119873lowast

119903 119873

120579= 119873119888119873lowast

120579

119873119894= minus119873119888119899lowast

119894 119873

119900= minus119873119888119899lowast

119900

(8)

where119873119888is a characteristic in-plane load When (8) are sub-

stituted into the appropriate equations discussed previouslyand for simplicity the asterisks are dropped from the results(with the understanding that all subsequent equations except(14) are in terms of dimensionless quantities) the results are

nabla4119908 minus 120582

2(119873119903120597119903119903119908 +119873

120579(

120597119903119908

119903

+

120597120579120579119908

1199032)) = 0 (9)

119873119903= minus

119899119894(11199032minus 1) + 119899

119900(11199032

119894minus 11199032)

11199032

119894minus 1

(10)

119873120579=

119899119894(11199032+ 1) minus 119899

119900(11199032

119894+ 11199032)

11199032

119894minus 1

(11)

119872119903= 120597119903119903119908 + 120592(

120597119903119908

119903

+

120597120579120579119908

1199032) (12)

119881119903= minus (120597

119903119903119903119908 +

120597119903119903119908

119903

minus

120597119903119908

1199032minus

(3 minus 120592) 120597120579120579119908

1199033

+

(2 minus 120592) 120597119903120579120579119908

1199032) + 1205822119873119903120597119903119908

(13)

where

1205822=

1198731198881199032

119900

119863

(14)

defines an in-plane load dimensionless parameter which canbe thought of as a dimensionless buckling load

Substituting

119908 =

infin

sum

119899=0

119908119899cos (119899120579) (15)

into (9) leads to the sequence of one dimensional problems

1199081015840101584010158401015840

119899+

2119908101584010158401015840

119899

119903

minus (

21198992+ 1

1199032+ 1205822119873119903)11990810158401015840

119899

+

((21198992+ 1) 119903

2minus 1205822119873120579)1199081015840

119899

119903

+

1198992((1198992minus 4) 119903

2+ 1205822119873120579)119908119899

1199032

= 0

(16)

Journal of Structures 3

with a prime denoting differentiation with respect to 119903 In asimilar manner substituting (15) into (12) and (13) yields thecorresponding Fourier cosine series coefficients

119872119903119899= 11990810158401015840

119899+ 120592(

1199081015840

119899

119903

minus

1198992119908119899

1199032) (17)

119881119903119899= minus (119908

101584010158401015840

119899+

11990810158401015840

119899

119903

minus (

1

1199032+ 1198992(2 minus 120592) + 120582

2119873119903)1199081015840

119899

+

1198992(3 minus 120592)119908119899

1199033)

(18)

which when substituted into (5) produce the boundaryconditions

clamped 119908119899= 0 119908

1015840

119899= 0 (19)

simply supported 119908119899= 0 119872

119903119899= 0 (20)

free 119872119903119899= 0 119881

119903119899= 0 (21)

Equation (16) and an appropriate combination of (19)ndash(21) constitute an eigenvalue problem for the dimensionlessbuckling parameter 120582 which must be solved numerically forthe configurations to be considered below To insure accuracytwo independent methods of doing this were employed Thefirst was a variant of the compound matrix method usedby Coman and Haughton [7 8] and the second was theimperfection method used by Jillella and Peddieson [12]The interested reader is referred to these publications andthe corresponding references cited therein for the details ofthe two approaches In all cases the predictions of the twoprocedureswere essentially identical A representative sampleof these predictions is presented in the next two sections

3 Results for Load-Free Inner Edge

Majumdar [4] considered the problem of compression buck-ling of a freeclamped annular plate having a uniform radialcompressive loading at the outer edge and no radial loading atthe inner edge (load-free inner edge) This is a generalizationof the problem discussed by Timoshenko and Gere [2] whodealt with only axisymmetric buckling Majumdar [4] foundthat the cases of 119899 = 0 (axisymmetric) and 119899 = 1 (onecircumferential node) produced exact closed-form transcen-dental equations in terms of Bessel functions from whichthe buckling loads could be determined with the formeryielding results in agreementwith those obtained byMeissnerquoted in [2] For 119899 ge 2 no exact closed-form transcendentalequations could be found and estimates of the buckling loadswere made using the RayleighRitz method Machinek andTroger [6] in preparation for a postbuckling study reportedlimited results based on numerical solutions of the exacteigenvalue problem for 119899 ge 2 Coman and Bassom [10]in preparation for an analytical study employing asymptoticmethods recently reproduced these numerical results Noother work on the load-free inner edge configuration appearsto have been published For this reason it was decided

to investigate a set of configurations involving an annularplate with a load-free inner edge These have either uniformtension or compression acting on either a clamped or simplysupported outer edge

In the present notation a radial loading involving a freeinner edge and uniform tension or compression at the outeredge corresponds to

119899119894= 0 119899

119900= plusmn1 (22)

(with the top sign indicating compression and the bottomsign tension) which according to (10) and (11) produces

119873119903= ∓

11199032

119894minus 11199032

11199032

119894minus 1

119873120579= ∓

11199032

119894+ 11199032

11199032

119894minus 1

(23)

Here the quantity119873119888stands for the magnitude of the dimen-

sional outer edge stress resultant It can be seen from (23)that both stress resultants have the same sign for all 119903 Thusradial compression produces circumferential compressionand radial tension produces circumferential tension In thisconfiguration therefore tension buckling is impossible andresults will be presented for compression buckling only

Predictions relevant to compression buckling arereported in Figures 1 2 3 and 4 for freeclamped supportconditions Figure 1 shows curves of buckling load versusinner radius associated with the first eleven circumferentialbuckling modes for 120592 = 13 the value thought to be usedby Majumdar [4] A value of 120592 is not explicitly stated in [4]but 120592 = 13 is used to obtain the results reported in [2] towhich comparisons are made in [4] The composite curvecreated by combining the lowest portions of all individualcurves contained in Figure 1 (and subsequent similar figures)indicates the lowest buckling load for any given value ofthe dimensionless inner radius (radius ratio) In additionfor plates exhibiting initial imperfections each individualcurve characterizes the buckling behavior associated witha particular circumferential imperfection pattern (see forinstance [2] for a detailed discussion) Thus for exampleit is expected that the buckling loads associated with anaxisymmetric imperfection pattern would be determinedfrom the curve labeled 119899 = 0 in Figure 1 even though theyare not always the lowest possible buckling loads

The curves corresponding to 119899 = 0 and 1 depictedin Figure 1 can be directly compared with Figure 1 of [4]and agree well with the accuracy of visual inspection It isinteresting to note that for 119899 = 0 the trend of a small decreasein the value of the buckling load for the smaller inner radii iscaptured by the numericalmethod RayleighRitz predictionsof buckling loads for 0 le 119899 le 6 and 119899 = 10 are presentedin Figure 2 of [4] The curve for 119899 = 0 fails to capturethe decrease in 1205822 discussed above In addition the energymethod predicts the lowest buckling load at 119903

119894= 05 to

correspond to 119899 = 1 rather than to 119899 = 0 as indicated bythe exact solution This effect is also captured by the currentmethodology as illustrated by Figure 1


100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0

Figure 1 Buckling load versus inner radius (freeclamped 119899119894= 0

119899119900= 1 120592 = 13)

As stated earlier the composite curve formed by com-bining the lowest portions of curves for each circumferentialbucklingmode (value of 119899) indicates the lowest buckling loadIt is therefore possible that the composite curve associatedwith the RayleighRitz method could be an accurate repre-sentation of the buckling behavior even if some individualcomponents of the composite curve are inaccurate Thisproposition was tested by comparing the lowest bucklingloads for selected inner radii predicted by both the presentnumerical method and Figure 2 of [4] (to the accuracy ofvisual inspection) It was found that the maximum errorsassociated with the RayleighRitz method are in the 10ndash15range Thus the RayleighRitz composite curve is reasonablyaccurate

It is often asserted in the literature that compressionbuckling loads are insensitive to the value of Poissonrsquos ratioIt is of interest to use the results presented herein to testthis assertion Figures 2 3 and 4 (and other similar setsof predictions to be presented subsequently) depict thecomposite curves of minimum buckling load versus innerradius for 120592 = 01 03 and 05 The difference betweenthe highest and lowest buckling load values for these threePoissonrsquos ratios at a given inner radius can be expressed as apercentage 119875 of the lowest value For this set of predictions 119875was found to be in the 8ndash37 range

Figure 3 can be directly compared with both Figure 4of [6] and Figure 2(a) of [10] To the accuracy of visualinspection agreement between the former and the latter twoappears to be excellent thus providing additional validationof the two independent numerical procedures used hereinFigures 1 2 3 and 4 all exhibit a rapid increase in the numberof circumferential nodes exhibited by the buckling mode asthe inner radius increases This behavior (here associatedwith compression buckling) will also be observed in one ofthe tension buckling configurations to be discussed below

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 03)

Results for compression buckling of the freesimplysupported configuration are reported in Figures 5 6 and7 It is clear that the lowest buckling load corresponds toaxisymmetric buckling for all values of Poissonrsquos ratio Thedifference produced by the change from a clamped to simplysupported edge is striking Here 119875 is in the 2ndash30 range

To be consistent with the results presented in [4] pre-dictions were reported above only for cases satisfying theinequality 1205822 le 100 This fact accounts for the differencebetween the number of circumferential modes for whichinformation was presented in Figures 1ndash4 and the number forwhich it was presented in Figures 5ndash7


100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 05)

4 Results for Load-Free Outer Edge

Yu and Zhang [5] considered the problem of tension bucklingof a simply supportedfree annular plate having a uniformradial tensile loading at the inner edge and no radial loadingat the outer edge (load-free outer edge) Yu and Zhang [5]estimated the buckling loads by using the Galerkin methodComan and Haughton [8] in preparation for an analyticalstudy presented limited results based on numerical solutionsof the exact eigenvalue problem for this configuration Nohet al [11] reproduced some of these results using the finiteelement method No other work on the load-free outer edgeconfiguration appears to have been published It was there-fore decided to investigate a set of configurations involvingan annular plate with a load-free outer edgeThese have eitheruniform radial tension or compression acting on either aclamped or simply supported inner edge

In the present notation a radial loading involving a freeouter edge and uniform tension or compression at the inneredge corresponds to

119899119894= plusmn1 119899

119900= 0 (24)


119873119903= ∓

11199032minus 1

11199032

119894minus 1

119873120579= plusmn

11199032+ 1

11199032

119894minus 1

(25)

Here 119873119888stands for the magnitude of the dimensional inner

edge stress resultant It can be seen from (25) that thestress resultants have opposite signs for all 119903 Thus radialtension produces circumferential compression and radial

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0

Figure 5 Buckling load versus inner radius (freesimply supported119899119894= 0 119899

119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 03)

compression produces circumferential tension In this con-figuration therefore both tension and compression bucklingare possible and results for both will be presented

Figures 8 9 and 10 show curves of buckling load versusinner radius for several circumferential tension bucklingmodes associated with the simply supportedfree configu-ration In this set (and subsequent similar sets) of figuresresults will be presented only for circumferential bucklingmodes exhibiting buckling loads in the range 1205822 le 200 tobe consistent with [5] As before this will produce differentmaximum and minimum values of 119899 in different figures Inparticular no buckling was observed in this range for 119899 = 0


100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 05)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10

Figure 8 Buckling load versus inner radius (simply supportedfree119899119894= minus1 119899

119900= 0 120592 = 01)

or 119899 = 1 thus accounting for the absence of the associatedcurves from Figures 8ndash10

Figure 9 can be directly compared with the Galerkinpredictions presented in Figure 3(a) of [5] for 2 le 119899 le6 Quantitative comparisons of the results with those readfrom Figure 3(a) of [5] (to the accuracy of visual inspection)reveal that maximum errors associated with the Galerkinmethod are in the 20ndash25 range Figure 9 can also becompared directly with Figures 3 and 4 of [8] To the accuracyof visual inspection agreement seems excellent providingfurther validation of the present numerical approaches

Figures 8ndash10 illustrate the influence of Poissonrsquos ratioon buckling loads with 119875 being in the 58ndash67 range

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 03)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 2

34 5 6 7 8 9 10

Figure 10 Buckling load versus inner radius (simply supportedfree 119899

119894= minus1 119899

119900= 0 120592 = 05)

These figures also exhibit the rapid increase in bucklingmodenodes with increasing inner radius mentioned earlier Herethis phenomenon is associated with tension buckling whilepreviously it was associated with compression buckling Aprogram of analytical work begun by Coman and Haughton[8] (using Rayleighrsquos quotient) and continued by Coman andBassom [9] (using asymptotic methods) reveals that a valueof the inner radius exists for each value of 119899 beyond whichtension buckling does not occur

Results for tension buckling of the clampedfree con-figuration are reported in Figures 11 12 and 13 Whilethe qualitative influence of Poissonrsquos ratio is slight 119875 isin the 37ndash42 range Comparison of Figures 8ndash10 with


200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 6 7 8 9 10

Figure 11 Buckling load versus inner radius (clampedfree 119899119894= minus1

119899119900= 0 120592 = 01)

0 01 02 03 04 05 06 07 08 09

ri

n = 2 34 5 6 7 8 9

10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 03)

Figures 11ndash13 suggests that the change from a simply sup-ported to a clamped inner edge does not produce a significantchange in the nature of the tension buckling behavior (insharp contrast to the compression buckling cases discussedearlier) In particular the rapid increase in buckling modenodes with increasing inner radius appears in both It wouldbe interesting to apply the asymptotic analysis of [9] todetermine whether there are limiting inner radii above whichtension buckling will not occur in this configuration as wellIt would also be interesting to investigate the possibility of aunified analytical approach to the rapid node increase phe-nomenon which appears for both tension and compressionbuckling

0 01 02 03 04 05 06 07 08 09

ri

n = 2

3 45 6

7 8 9 10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 05)

As discussed in detail by Jillella and Peddieson [12] theuse of a tension field model for wrinkling (tension buckling)analysis does not allow either the dependence on axialsupport conditions or the wrinkling pattern to be predictedExamples were given in [12] in which the axial supportdependence was significant Figures 8ndash13 show that thechange from a simply supported to a clamped inner edge doesnot change the qualitative nature of the wrinkling behaviorfor a free outer edge but does (as expected) significantly raisethe buckling load values Figures 8ndash13 also associate a uniquevalue of 119899 with each inner radius for which buckling occurs(except in cases in which two curves cross) This in turnquantitatively defines the wrinkle pattern

Results for compression buckling of the simply sup-portedfree configuration are reported in Figure 14 Since thelowest buckling load corresponds to axisymmetric bucklingfor all values of Poissonrsquos ratio one value has been selectedas representative for graphical presentation Additional com-puted results (not shown) reveal that 119875 is in the 34ndash44range

Results for compression buckling of the clampedfreeconfiguration are reported in Figure 15 Again the lowestbuckling load corresponds to axisymmetric buckling for allvalues of Poissonrsquos ratio and one representative value hasbeen chosen for graphical depictionThe results of additionalsimulations (not shown) indicate that 119875 is in the 8ndash14range In contrast to the case of tension buckling the changefrom a simply supported to a clamped inner edge producesa significant change in the buckling load versus inner radiusbehavior

5 Conclusion

The foregoing discussed the elastic stability of thin elastichomogeneous isotropic annular plates of uniform thickness


0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0

Figure 15 Buckling load versus inner radius (clampedfree 119899119894= 1

119899119900= 0 120592 = 01)

Two independent numerical approaches were used to solvethe eigenvalue problem associated with buckling analysisPredictions were obtained for several configurations involv-ing a load-free inner or outer edge with axisymmetric radialloading at the opposite edge exhibiting either compressionor tension buckling Two of these were used to verify thenumerical approach by comparison with previous publishedresults and several involved new results Some importantconclusions are as follows

First some of the results reported herein exhibit signifi-cant sensitivities of thin plate buckling loads to Poissonrsquos ratioOf the configurations investigated the largest sensitivity wasobserved for simply supportedfree tension buckling while

the smallest was observed for clampedfree compressionbuckling however no definite pattern is obvious

Second standard elastic stability methodology provides aunified approach to both tension and compression bucklingof thin plates In particular when applied to tension buckling(wrinkling) this approach can make certain quantitativepredictions of which tension field theories are incapable Themost important of these are the wrinkle pattern (bucklingmode shape) and the effect of axial support conditions Thepresent work provides examples of such predictions Thepapers by Jillella and Peddieson [12] and Coman [13] containfurther discussion of the relative merits of tension field andthin plate models for the analysis of flat sheet wrinkling

Third the plate thickness enters the dimensionless equa-tions employed herein only through the dimensionless buck-ling load1205822 (which depends on the platemodulus119863which inturn depends on the thickness ℎ) Thus no difficulties arisefor small thicknesses when using the approaches employed inthe present work Such difficulties are often cited as reasonsfor preferring tension field models to plate or shell modelsfor the analysis of thin sheet wrinkling Many commercialfinite element codes are based on shear deformation plate andshell models which of course sometimes exhibit difficultiesin dealing with small thicknessesThese difficulties howeverare generic and not specific to the prediction of wrinklingphenomena

References

[1] N Yamaki ldquoBuckling of a thin annular plate under uniformcompressionrdquo Journal of Applied Mechanics vol 25 pp 267ndash273 1958

[2] S Timoshenko and J GereTheory of Elastic Stability McGraw-Hill New York NY USA 2nd edition 1960

[3] EHMansfield ldquoOn the buckling of an annular platerdquoQuarterlyJournal of Mechanics and Applied Mathematics vol 13 no 1 pp16ndash23 1960

[4] S Majumdar ldquoBuckling of a thin annular plate under uniformcompressionrdquo AIAA Journal vol 9 no 9 pp 1701ndash1707 1971

[5] T X Yu and L C Zhang ldquoThe elastic wrinkling of an annularplate under uniform tension on its inner edgerdquo InternationalJournal of Mechanical Sciences vol 28 no 11 pp 729ndash737 1986

[6] A Machinek and H Troger ldquoPost-buckling of elastic annularplates at multiple eigenvaluesrdquo Dynamics and Stability of Sys-tems vol 3 pp 79ndash98 1988

[7] C D Coman and D M Haughton ldquoLocalized wrinkling insta-bilities in radially stretched annular thin filmsrdquoActaMechanicavol 185 no 3-4 pp 179ndash200 2006

[8] C D Coman and D M Haughton ldquoOn some approximatemethods for the tensile instabilities of thin annular platesrdquoJournal of Engineering Mathematics vol 56 no 1 pp 79ndash992006

[9] C D Coman and A P Bassom ldquoSingular behaviour in ageneralized boundary eigenvalue problem for annular plates intensionrdquoQuarterly Journal ofMechanics andAppliedMathemat-ics vol 60 no 3 pp 319ndash336 2007

[10] C D Coman and A P Bassom ldquoOn a class of bucklingproblems in a singularly perturbed domainrdquo Quarterly JournalofMechanics and AppliedMathematics vol 62 no 1 pp 89ndash1032009


[11] S Noh M Abdalla and W Faris ldquoA study of annular platebuckling problem with tension loaded at inner edgerdquo AnaleleUniversitatii Eftimie Murgu vol 17 pp 97ndash103 2010

[12] N Jillella and J Peddieson ldquoModeling of wrinkling of thincircular sheetsrdquo International Journal of Non-Linear Mechanicsvol 47 no 1 pp 85ndash91 2012

[13] C D Coman ldquoOn the applicability of tension field theory toa wrinkling instability problemrdquo Acta Mechanica vol 190 no1ndash4 pp 57ndash72 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



with the platersquos middle surface and the 119911 axis being the axisof symmetry The appropriate differential equation is

119863nabla4119908 minus (119873

119903120597119903119903119908 +119873

120579(

120597119903119908

119903

+

120597120579120579119908

1199032)) = 0 (1)

where 119908 is the axial (transverse) displacement

nabla2119908 = 120597

119903119903119908 +

120597119903119908

119903

+

120597120579120579119908

1199032

(2)

is the two dimensional Laplacian operator

119863 =

119864ℎ3

12 (1 minus 1205922)

(3)

is the plate modulus 119873119903is the radial normal force resultant

and 119873120579is the circumferential normal force resultant In all

equations appearing in the present work partial derivativeoperators are understood to operate only on the immediatelyfollowing symbol The normal force resultants are foundby solving the uncoupled plane stress problem associatedwith axisymmetric radial loading at the inner and outerboundaries (see for instance [10]) to get

119873119903=

119873119894((119903119900119903)2minus 1) + 119873

119900((119903119900119903119894)2minus (119903119900119903)2)

(119903119900119903119894)2minus 1

119873120579=

minus119873119894((119903119900119903)2+ 1) + 119873

119900((119903119900119903119894)2+ (119903119900119903)2)

(119903119900119903119894)2minus 1

(4)

where119873119894and119873

119900are the respective inner and outer boundary

tensile force resultants Support conditions to be employedsubsequently are

clamped 119908 = 0 120597119903119908 = 0

simply supported 119908 = 0 119872119903= 0

free 119872119903= 0 119881

119903= 0

(5)

where

119872119903= 119863(120597

119903119903119908 + 120592(

120597119903119908

119903

+

120597120579120579119908

1199032)) (6)

is the radial bending moment resultant and

119881119903= minus 119863(120597

119903119903119903119908 +

120597119903119903119908

119903

minus

120597119903119908

1199032minus

(3 minus 120592) 120597120579120579119908

1199033

+

(2 minus 120592) 120597119903120579120579119908

1199032) + 119873

119903120597119903119908

(7)

is the radial Kirchhoff shear force resultantFor the purpose of numerical analysis it is convenient to

convert the equations to dimensionless forms Towards this

end it is helpful to define dimensionless quantities (denotedby superposed asterisks) through the equations

119903 = 119903119900119903lowast 119903

119894= 119903119900119903lowast

119894 119908 = ℎ119908

lowast 119872

119903=

119863ℎ119872lowast

119903

1199032

119900

119872120579=

119863ℎ119872lowast

120579

1199032

119900

119881119903=

119863ℎ119881lowast

119903

1199033

119900

119873119903= 119873119888119873lowast

119903 119873

120579= 119873119888119873lowast

120579

119873119894= minus119873119888119899lowast

119894 119873

119900= minus119873119888119899lowast

119900

(8)

where119873119888is a characteristic in-plane load When (8) are sub-

stituted into the appropriate equations discussed previouslyand for simplicity the asterisks are dropped from the results(with the understanding that all subsequent equations except(14) are in terms of dimensionless quantities) the results are

nabla4119908 minus 120582

2(119873119903120597119903119903119908 +119873

120579(

120597119903119908

119903

+

120597120579120579119908

1199032)) = 0 (9)

119873119903= minus

119899119894(11199032minus 1) + 119899

119900(11199032

119894minus 11199032)

11199032

119894minus 1

(10)

119873120579=

119899119894(11199032+ 1) minus 119899

119900(11199032

119894+ 11199032)

11199032

119894minus 1

(11)

119872119903= 120597119903119903119908 + 120592(

120597119903119908

119903

+

120597120579120579119908

1199032) (12)

119881119903= minus (120597

119903119903119903119908 +

120597119903119903119908

119903

minus

120597119903119908

1199032minus

(3 minus 120592) 120597120579120579119908

1199033

+

(2 minus 120592) 120597119903120579120579119908

1199032) + 1205822119873119903120597119903119908

(13)

where

1205822=

1198731198881199032

119900

119863

(14)

defines an in-plane load dimensionless parameter which canbe thought of as a dimensionless buckling load

Substituting

119908 =

infin

sum

119899=0

119908119899cos (119899120579) (15)

into (9) leads to the sequence of one dimensional problems

1199081015840101584010158401015840

119899+

2119908101584010158401015840

119899

119903

minus (

21198992+ 1

1199032+ 1205822119873119903)11990810158401015840

119899

+

((21198992+ 1) 119903

2minus 1205822119873120579)1199081015840

119899

119903

+

1198992((1198992minus 4) 119903

2+ 1205822119873120579)119908119899

1199032

= 0

(16)



119872119903119899= 11990810158401015840

119899+ 120592(

1199081015840

119899

119903

minus

1198992119908119899

1199032) (17)

119881119903119899= minus (119908

101584010158401015840

119899+

11990810158401015840

119899

119903

minus (

1

1199032+ 1198992(2 minus 120592) + 120582

2119873119903)1199081015840

119899

+

1198992(3 minus 120592)119908119899

1199033)

(18)


clamped 119908119899= 0 119908

1015840

119899= 0 (19)


119903119899= 0 (20)

free 119872119903119899= 0 119881

119903119899= 0 (21)






119899119894= 0 119899

119900= plusmn1 (22)


119873119903= ∓

11199032

119894minus 11199032

11199032

119894minus 1

119873120579= ∓

11199032

119894+ 11199032

11199032

119894minus 1

(23)





119894= 05 to



100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 13)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 03)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 05)




119899119894= plusmn1 119899

119900= 0 (24)


119873119903= ∓

11199032minus 1

11199032

119894minus 1

119873120579= plusmn

11199032+ 1

11199032

119894minus 1

(25)



100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 03)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 05)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 01)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 03)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 2

34 5 6 7 8 9 10


119894= minus1 119899

119900= 0 120592 = 05)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 6 7 8 9 10


119899119900= 0 120592 = 01)

0 01 02 03 04 05 06 07 08 09

ri

n = 2 34 5 6 7 8 9

10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 03)


0 01 02 03 04 05 06 07 08 09

ri

n = 2

3 45 6

7 8 9 10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 05)




5 Conclusion



0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 01)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119872119903119899= 11990810158401015840

119899+ 120592(

1199081015840

119899

119903

minus

1198992119908119899

1199032) (17)

119881119903119899= minus (119908

101584010158401015840

119899+

11990810158401015840

119899

119903

minus (

1

1199032+ 1198992(2 minus 120592) + 120582

2119873119903)1199081015840

119899

+

1198992(3 minus 120592)119908119899

1199033)

(18)


clamped 119908119899= 0 119908

1015840

119899= 0 (19)


119903119899= 0 (20)

free 119872119903119899= 0 119881

119903119899= 0 (21)






119899119894= 0 119899

119900= plusmn1 (22)


119873119903= ∓

11199032

119894minus 11199032

11199032

119894minus 1

119873120579= ∓

11199032

119894+ 11199032

11199032

119894minus 1

(23)





119894= 05 to



100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 13)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 03)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 05)




119899119894= plusmn1 119899

119900= 0 (24)


119873119903= ∓

11199032minus 1

11199032

119894minus 1

119873120579= plusmn

11199032+ 1

11199032

119894minus 1

(25)



100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 03)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 05)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 01)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 03)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 2

34 5 6 7 8 9 10


119894= minus1 119899

119900= 0 120592 = 05)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 6 7 8 9 10


119899119900= 0 120592 = 01)

0 01 02 03 04 05 06 07 08 09

ri

n = 2 34 5 6 7 8 9

10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 03)


0 01 02 03 04 05 06 07 08 09

ri

n = 2

3 45 6

7 8 9 10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 05)




5 Conclusion



0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 01)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 13)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 03)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 05)




119899119894= plusmn1 119899

119900= 0 (24)


119873119903= ∓

11199032minus 1

11199032

119894minus 1

119873120579= plusmn

11199032+ 1

11199032

119894minus 1

(25)



100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 03)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 05)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 01)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 03)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 2

34 5 6 7 8 9 10


119894= minus1 119899

119900= 0 120592 = 05)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 6 7 8 9 10


119899119900= 0 120592 = 01)

0 01 02 03 04 05 06 07 08 09

ri

n = 2 34 5 6 7 8 9

10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 03)


0 01 02 03 04 05 06 07 08 09

ri

n = 2

3 45 6

7 8 9 10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 05)




5 Conclusion



0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 01)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5 6 7 8 9 10

n = 0


119899119900= 1 120592 = 05)




119899119894= plusmn1 119899

119900= 0 (24)


119873119903= ∓

11199032minus 1

11199032

119894minus 1

119873120579= plusmn

11199032+ 1

11199032

119894minus 1

(25)



100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 01)

100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 03)




100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 05)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 01)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 03)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 2

34 5 6 7 8 9 10


119894= minus1 119899

119900= 0 120592 = 05)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 6 7 8 9 10


119899119900= 0 120592 = 01)

0 01 02 03 04 05 06 07 08 09

ri

n = 2 34 5 6 7 8 9

10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 03)


0 01 02 03 04 05 06 07 08 09

ri

n = 2

3 45 6

7 8 9 10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 05)




5 Conclusion



0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 01)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100

80

60

40

20

0

1205822

0 01 02 03 04 05 06 07 08 09

ri

1

2

3

4

5

6

n = 0


119900= 1 120592 = 05)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 01)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 67

8 9

10


119900= 0 120592 = 03)

200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 2

34 5 6 7 8 9 10


119894= minus1 119899

119900= 0 120592 = 05)




200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 6 7 8 9 10


119899119900= 0 120592 = 01)

0 01 02 03 04 05 06 07 08 09

ri

n = 2 34 5 6 7 8 9

10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 03)


0 01 02 03 04 05 06 07 08 09

ri

n = 2

3 45 6

7 8 9 10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 05)




5 Conclusion



0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 01)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










200

175

150

125

100

75

50

25

00 01 02 03 04 05 06 07 08 09

1205822

ri

n = 23

4 5 6 7 8 9 10


119899119900= 0 120592 = 01)

0 01 02 03 04 05 06 07 08 09

ri

n = 2 34 5 6 7 8 9

10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 03)


0 01 02 03 04 05 06 07 08 09

ri

n = 2

3 45 6

7 8 9 10

1205822

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 05)




5 Conclusion



0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 01)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 0

12

200

180

160

140

120

100

80

60

40

20

0


119894= 1 119899

119900= 0 120592 = 03)

0 01 02 03 04 05 06 07 08 09

1205822

ri

n = 01

200

180

160

140

120

100

80

60

40

20

0


119899119900= 0 120592 = 01)






References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









elastic stability of annular thin plates with one free edge

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