BUCKLING OF CIRCULAR PLATES WITH AN INTERNAL ELASTIC RING SUPPORT AND OUTER EDGE RESTRAINED AGAINST TRANSLATION

Transcription

1 oural of Egieerig Sciece ad echology Vol. 7, No School of Egieerig, aylor s Uiversity BUKLNG OF RULAR LAES WH AN NERNAL ELAS RNG SUOR AND OUER EDGE RESRANED AGANS RANSLAON LOKAVARAU BHASKARA RAO, *, HELLALLA KAMESWARA RAO School of Mechaical & Buildig Scieces, V Uiversity, heai ampus, Vadalur-Kelamaam Road, heai-6000, amiladu, dia Departmet of Mechaical Egieerig, Sreeidhi stitute of Sciece & echology, Ghataesar, Hyderaad-500, dia *orrespodig Author: hasarau_0@yahoo.com Astract his paper deals with exact solutio for elastic uclig of circular plates restig o iteral elastic rig support ad elastically restraied agaist traslatio at the outer edge. he classical plate theory is used i derivig the goverig differetial equatio for a circular plate restig o iteral rig support ad liearly restraied at the outer edge ad sujected to i-plae compressive load. From the preset study it is foud that there exists a uclig mode switchig from axi-symmetric to asymmetric as we vary the radius of iteral elastic rig support. this case, the uclig mode is ot axisymmetric as is geerally expected. he plate ucles i a asymmetric mode as the radius of the iteral rig support ecomes smaller. he cross-over rig support radius varies from 0.55 to times the plate radius, depedig upo the stiffess of the traslatioal sprig support at the outer edge. Based o this data, the optimum radius of the iteral rig support ca e determied. Extesive data is preseted i taular form so that sigificat coclusios ca e arrived at o the ifluece of traslatioal restraits, oisso s ratio, ad oudary coditios o the uclig ehavior of isotropic circular plates. he umerical results otaied from the preset study are foud to e i close agreemet with the previously pulished data. Keywords: ircular plate, Buclig, Elastic rig support, Flexile edge, Mode switchig.. troductio he predictio of uclig of structural memers restraied laterally is a importat topic i the desig of various egieerig compoets. particular, the 9

2 9 Loavarapu B. Rao ad hellapilla K. Rao Nomeclatures No-dimesioal rig support radius parameter D Flexural rigidity of a material, N.mm E oug s modulus of a material, N/mm h hicess of a plate, mm No-dimesioal uclig load parameter K raslatioal sprig stiffess at outer edge, N/mm K raslatioal sprig stiffess of iteral elastic rig, N/mm N Uiform i - plae compressive load, N/mm R Radius of a plate, mm No-dimesioal traslatioal flexiility parameter at outer edge No-dimesioal traslatioal flexiility parameter of iteral elastic rig v oisso s ratio circular plates with a iteral elastic rig supports fid applicatios i aeroautical istrumet moutig ases for space vehicles, rocet lauchig pads, aircrafts ad aval vessels istrumet moutig ases. Based o the Kirchhoff s theory, the elastic uclig of thi circular plates has ee extesively studied y may authors after the pioeerig wor pulished y Brya []. Sice the, there have ee extesive studies o the suject coverig various aspects such as differet materials, oudary ad loadig coditios [, ]. However, these sources oly cosidered axisymmetric cases, which may ot lead to the correct uclig load. troducig a iteral elastic rig supports may icrease the elastic uclig capacity of i-plae loaded circular plates sigificatly. Laura et al. [] ivestigated the elastic uclig prolem of the aforesaid type of circular plates, who modeled the plate usig the classical thi plate theory. their study oly axisymmetric modes are cosidered. Kuuasseril ad Swamidas [5] are proaly the first to cosider elastic rig supports. hey formulated the equatios i geeral, ut preseted oly the case of circular plate with a free edge. Although the circular symmetry of the prolem allows for its sigificat simplificatio, may difficulties very ofte arise due to complexity ad ucertaity of oudary coditios. his ucertaity could e due to practical egieerig applicatios where the edge of the plate does ot fall ito the classical oudary coditios. t is a accepted fact that the coditio o a periphery ofte teds to e i etwee the classical oudary coditios free, clamped ad simply supported ad may correspod more closely to some form of elastic restraits, i.e., rotatioal ad traslatioal restraits [6-]. a recet study, Wag ad Wag [6] showed that whe the rig support has a small radius, the uclig mode taes the asymmetric form. But they have studied oly the circular plate with elastically restraied edge agaist rotatio. he purpose of the preset wor is to complete the results of the uclig of circular plates with a iteral elastic rig support ad elastically restraied edge agaist traslatio y icludig the asymmetric modes, thus correctly determiig the uclig loads. oural of Egieerig Sciece ad echology ue 0, Vol. 7

3 Buclig of ircular lates with a teral Elastic Rig Support agaist raslatio 95. Defiitio of the rolem osider a thi circular plate of radius R, uiform thicess h, oug s modulus E, flexural rigidity D ad oisso s ratio v. t is sujected to a uiform i-plae load, N alog its periphery, i.e., alog the outer edge at r o-dimesioal radius of outer edge ad it is supported y a iteral elastic rig support at r o-dimesioal radius of support, as show i Fig.. he prolem at had is to determie the elastic critical uclig load of a circular plate with a iteral elastic rig support ad elastically restraied edge agaist traslatio. Fig.. Buclig of a ircular late with a teral Elastic Rig Support ad Restraied Edge agaist raslatio.. Formulatio of the rolem he plate is elastically restraied edge agaist traslatio at the edge of radius R ad supported o a iteral elastic rig of smaller radius R as show i Fig.. he plate uder cosideratio is sujected to a uiformly distriuted i-plae load N, alog the outer periphery, i.e, at r ad supported o elastic rig support at r, as show i Fig.. Let suscript deote the outer regio r ad the suscript deote ier regio 0 r. Here, all legths are ormalized y R. Usig the classical thi plate theory Kirchhoff theory, the goverig fourth order differetial equatio for elastic uclig of a aular plate may e expressed i polar coordiates r, θ as D w N w 0 where w is the lateral displacemet, N is the uiform compressive load at outer edge. After ormalize legths y the radius of the plate R, Eq. ca e writte as D w w 0 w 0 where the Laplacia operator, r r r r θ where r is the radial distace ormalized y R. D Eh / is the flexural rigidity, w w/ R, is ormalized trasverse displacemet of the plate. R N / D is o-dimesioal load parameter. Suppose there are odal diameters. polar coordiates r, θ set oural of Egieerig Sciece ad echology ue 0, Vol. 7

4 96 Loavarapu B. Rao ad hellapilla K. Rao w r, θ u r cos θ 5 osiderig the oudess at the origi, the geeral solutio [] for the two regios is u log r r r r r r u r 5 r 6r 7 Sustitutig Eqs. 6 ad 7 ito Eq. 5 yields the followig log r w r, θ r r r cos θ r w r, θ [ 5 r 6r ] cos θ 9 where top form of the Eq. 9 is used for 0 symmetric ad the ottom form is used for 0 asymmetric,,,,, 5 ad 6 are costats,. ad. are the Bessel fuctios of the first ad secod ids of order, respectively. he oudary coditios at outer regio of the circular plate is give y the followig expressios u ' r 0 0 Vr r Ku r V r he radial Kirchhoff shear at outer edge is defied as follows D r [ u ''' r u '' r u ' r u ] R r 6 Equatios ad yields the followig '' r u '' r u ' r u r u r [ u ] ' where KR D Apart from the elastically restraied edge agaist traslatio, there is a iteral elastic rig support costrait ad the cotiuity requiremets of slope ad curvature at the support, i.e., r u u 5 u ' u ' 6 u '' u '' 7 u '' ' u ' '' u where K R / D is the ormalized sprig costat, K of the traslatioal sprig. Boudary coditios at the outer edge at r Equatios, 0 ad yields the followig [ ] 0 9 oural of Egieerig Sciece ad echology ue 0, Vol. 7

5 Buclig of ircular lates with a teral Elastic Rig Support agaist raslatio 97 oural of Egieerig Sciece ad echology ue 0, Vol. 7 [ ] { } 0 0 where ; ; ; ; ; ; otiuity requiremets at the support, i.e., r Equatios, 9 ad 5-6 yield the followig 0 log [ ] where ; ; ; ; ; ; he top form of Eqs. 9, 0 ad - are used for 0 axisymmetric uclig ad the ottom form is used for 0 asymmetric uclig.. Solutio For the give values of, v,,, ad the aove set of equatios gives a exact characteristic equatio o-trivial solutio of the coefficiets,,,, 5 ad 6. For o-trivial solutio, the determiat of [] 6x6 must vaish. he prolem is solved usig Mathematica, computer software with symolic capailities.

6 9 Loavarapu B. Rao ad hellapilla K. Rao 5. Results ad Discussio he code developed i Mathematica is used to determie the uclig load parameter for ay rage of traslatioal costraits. he fidigs are preseted i oth taular ad graphical form. he uclig loads are calculated for various iteral rig support radii, ad traslatioal sprig stiffess parameter. he uclig load parameters for axisymmetric ad asymmetric modes for various values of traslatioal stiffess parameters of elastic rig support, y eepig traslatioal sprig stiffess parameter, costat, is preseted i ale. Figure shows the variatio of uclig load parameter, versus elastic rig support radius parameter, for traslatioal sprig stiffess parameter of a elastic rig support 00 y eepig traslatioal sprig stiffess parameter costat 000. t is oserved from Fig. that the 0 axisymmetric solutio weaves with asymmetric solutio. Whe 0, the utethered plate ucles axisymmetrically with.057. Whe icreased eyod , the asymmetric mode gives the correct lower uclig load. his persists util where the 0 mode agai determies the uclig load. t is oserved from the Fig. that the uclig is govered y the axisymmetric mode 0, whe 0. 5 ad he optimal locatio of a iteral elastic rig support opt is 0.56 ad the correspodig uclig load parameter opt is For this case, a iteral elastic rig support, whe placed at a optimal positio, ca icrease the uclig load capacity y %. Fig.. Buclig Load arameter versus teral Elastic Rig Support Radius for Various Values of 00 whe 000. Figure shows the variatio of uclig load parameter, with respect to the iteral elastic rig support radius, for traslatioal sprig stiffess parameter of a elastic rig support 0 6 y eepig traslatioal sprig stiffess parameter costat 000. t is oserved from the Fig., that for a give value of y eepig costat, the curve is composed of two segmets. his is due to the switchig of uclig modes. For a smaller iteral elastic rig support radius, the plate ucles i a asymmetric mode i.e.,. this segmet as show y dotted lies i Fig. the uclig load decreases as oural of Egieerig Sciece ad echology ue 0, Vol. 7

7 Buclig of ircular lates with a teral Elastic Rig Support agaist raslatio 99 decreases i value. For larger iteral elastic rig support radius, the plate ucles i a axisymmetric mode i.e., 0. this segmet as show y cotiuous lies i Fig. the uclig load icreases as decreases up to a pea poit correspods to maximum uclig load ad thereafter decrease as decreases i value as show i Fig.. he cross over radius is t is oserved from the Fig. that the asymmetric uclig mode domiates small elastic rig support radius parameter situatio, i.e., Whe the 0 axisymmetric mode gives the correct lower uclig load. he optimal locatio of a iteral elastic rig support opt is 0.90 ad the correspodig uclig load parameter opt is Similarly for this case, a iteral elastic rig support, whe placed at a optimal positio, ca icrease the uclig load capacity y 0%. ale. Buclig for Axisymmetric ad Asymmetric Modes Load arameters for Differet Values of raslatioal Stiffess of Elastic Rig arameter whe 000 ad v t is oserved that the effect of the supportig elastic rig is aset if the rig radius is oe, ecause at the outer fixed edge domiates. Fig.. Buclig Load arameter versus teral Elastic Rig Support Radius for various values of 0 6 whe 000. oural of Egieerig Sciece ad echology ue 0, Vol. 7

8 00 Loavarapu B. Rao ad hellapilla K. Rao Similarly the effect of the iteral elastic rig support is egligile for 0. he ifluece of stiffess parameter of elastic rig support is ot very much o uclig load at lower values of. Ad also it is oserved that the ifluece of stiffess parameter of elastic rig support is more o uclig load at higher values of. he uclig loads icreases with icrease i value of. he results of this id were scarce i the literature. However, the results are compared for the followig cases: i Whe, i.e., whe the support is rigid, the uclig is govered y the asymmetric mode whe 0.5 ad this is i well agreemet with the results of Wag ad Wag [6]. Whe, i.e., rigid rig support, the optimum locatio is at a radius of 0.66, with a uclig load of that is i well agreemet with the results of Wag ad Wag [6]. ii Whe ad, i.e., whe the support is rigid ad simply supported edge at the oudary, the uclig load parameters are compared with those otaied y Wag et al. [9] as show i ale ad the results are i good agreemet. ale. ompariso of Buclig Load arameter, with Wag et al. [9] for Simply Supported Edge for ad v 0.. Wag et al. [9] reset oclusios he paper itroduced a Mathematica code for calculatio of uclig load. he uclig prolem of circular plates with a elastically restraied edge agaist traslatio has ee solved aalytically. Also the uclig loads are give for various traslatioal sprig stiffess parameter at the edges that simulate the traslatioal restraits where represets a simply supported edge to free edge where. t has ee oserved that the uclig mode switches from a asymmetric mode to a axisymmetric mode at a particular iteral rig radius parameter. his cross over radius depeds o the traslatioal sprig stiffess parameters ad. he crossover radius varies from 0.55 for 00 to for 0 6 y eepig wo-dimesioal plots are draw for a wide rage of traslatioal costraits. this paper the characteristic equatios are exact; therefore the results ca e calculated to ay accuracy. hese exact solutios ca e used to chec umerical or approximate results. ompariso of studies demostrates the accuracy ad staility of the preset wor. he taulated uclig results are useful to desigers i viratio cotrol, structural desig ad related idustrial applicatios. oural of Egieerig Sciece ad echology ue 0, Vol. 7

9 Buclig of ircular lates with a teral Elastic Rig Support agaist raslatio 0 Refereces. Brya, G.H., 9. O the staility of a plae plate uder thrust i its ow plae with applicatio to the uclig of the side of a ship. roceedigs of the Lodo Mathematical Society,, Wolowisy,.H Buclig of the circular plate emedded i elastic sprigs, a applicatio to geophysics. ommuity o ure ad Applied Mathematics, 5, Brush, D.O.; ad Almroth, B.O Buclig of ars, plates ad shells. McGraw-Hill, New or.. Laura,.A.A.; Gutierrez, R.H.; Sazi, H..; ad Elvira, G Buclig of circular, solid ad aular plates with a itermediate circular support. Ocea Egieerig, 77, Kuuasseril, V.X.; ad Swamidas, A.S Viratios of cotiuous circular plates. teratioal oural of Solids ad Structures, 06, Wag,..; ad Wag,.M. 00. Buclig of circular plates with a iteral rig support ad elastically restraied edges. hi-walled Structures, 99, Kim,.S.; ad Diciso, S.M., 990. he flexural viratio of the isotropic ad polar orthotropic aular ad circular plates with elastically restraied peripheries. oural of Soud ad Viratio,, amai, N., 95. Buclig of a thi aular plate uder uiform compressio. oural of Applied Mechaics, 5, Wag,.M.; ad u Myit Aug, 005. Buclig of circular Midli plates with a iteral rig support ad elastically restraied edge. oural of Egieerig Mechaics,, oural of Egieerig Sciece ad echology ue 0, Vol. 7