Buckling Analysis of Polar Orthotropic Circular and Annular Plates of Uniform and Linearly Varying Thickness with Different Edge Conditions

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1 Joural of Sold Mechacs Vol., No. (00 pp Bucklg Aalyss of Polar Orthotropc Crcular ad Aular Plates of Uform ad Learly Varyg Thckess wth Dfferet Edge Codtos F. Farhata,*, A. Golshah Mechacal Egeerg Faculty, Islamc Azad Uversty - Brach of Khomeshahr, Boulvard Mazare, Khomeshahr, Isfaha, Ira Ira Arcraft Maufacturg (HESA, Moallem Hghway, Isfaha, Ira Receved 8 May 00; accepted 0 July 00 ABSTRACT Ths paper vestgates symmetrcal bucklg of orthotropc crcular ad aular plates of cotuous varable thckess. Uform compresso loadg s appled at the plate outer boudary. Thckess vares learly alog radal drecto. Ier edge s free, whle outer edge has dfferet boudary codtos: clamped, smply ad elastcally restrat agast rotato. The optmzed Rtz method s appled for bucklg aalyss. I ths method, a polyomal fucto that s based o statc deformato of orthotropc crcular plates bedg s used. Also, by employg a expoetal parameter deformato fucto, egevalue s mmzed respect to ths parameter. The advatage of ths procedure s smplcty, comparso wth other methods, whle whole algorthm for soluto ca be coded for computer programmg. The effects of varato of radus, thckess, dfferet boudary codtos, rato of radal Youg modulus to crcumferetal oe, ad rato of outer radus to er oe aular plates o bucklg load factor are vestgated. The obtaed results show that plate wth detcal thckess, creasg of outer radus decreases the bucklg load factor. Moreover, crease of thckess of the plates results crease of bucklg load factor. 00 IAU, Arak Brach. All rghts reserved. Keywords: Bucklg; Orthotropc Plates; Crcular ad aular plates; Dfferet edge codtos INTRODUCTION O RTHOTROPIC crcular ad aular plates are always used by mechacal, cvl, aerospace ad structural egeers ad desgers. Some applcatos of these systems are pressure vessel valve, reforced crcular plates by radal ad crcumferece supporter, composte plates, cylder head cover, bulkhead plates submares, separated plates arcraft, optcal leses, ad acoustc trasducers rockets. Woosky [], for the frst tme, studed the problem of elastc stablty of orthotropc crcular plates. He troduced umerc results by usg Bessel fucto for bucklg of plates. Mek et al. [] studed o varato of thckess o bucklg of orthotropc rectagular plate. Laura et al. [3] foud crtcal load of bucklg for sotropc aular plate wth costat thckess by optmzed Raylegh-Rtz method. Imposed boudary codto of plate for ether er edge or outer edge was uder dfferet supports. Caco [4] studed o bucklg of crcular ad aular sotropc plate wth varable thckess that was used as a part of submare. Ths plate was cosdered wth free support o er edge ad clamped support ad resstat of rotato for outer edge. The thckess of plate s expoetal fucto of ts radus. He aalyzed the plate by optmzed Raylegh-Rtz method. Bremec et al. [5] also troduced oe optmzed rate of varato of thckess for bucklg of sotropc plates whch both er edge ad outer edge was uder costat * Correspodg author. E-mal address: farhata@aukhsh.ac.r (F. Farhata. 00 IAU, Arak Brach. All rghts reserved.

2 F. Farhata ad A. Golshah 57 The work doe by -plae radal force s gve as: radal loadd ad thckess was vared radal drecto. Bucklgg fucto was lear fasho ad solved by umerc method m uder smply ad clamped support. Guterrez et al. [6] cosdered bucklg ad vbrato of the sotropc plate p wth varable thckesss o elastc support usg Raylegh-Rtz method, ad obtaed acceptable results. Lag et al. [7] foud atural frequeces of oe orthotropc crcularr ad aular plate wth varable thckess usg u Raylegh-Rtz method that agreed wth the result of fte elemet method. I ths paper, bucklg of orthotropc crcular ad aular platess wth lear varato of thckess uder costat compressve radal loadg s studed. The boudary codto off aular plates are F-C plates (free er, ad clamped outer edge or clamped crcular sold oes, F-S plates (free er, smply outer edge or smply supported sold crcular oes, ad aular plates wth free er support ad resstat elastc agast rotato outer edge. Crcular plates cota plates wth clamped, smply ad elastc resstat agast rotato boudary codto. Solvg bucklg dfferetal d equato of orthotropc crcular or aular plate wth varable thckess s mpossble by aalytcal method m ad t should be solved usg umercal or eergy method. For ths reaso, optmzed Raylegh- of Rtz method s used. The results of ths method s more precsely tha Raylegh-Rtz method. For optmzato Raylegh-Rtz method, oe expoetal parameter approxmate fucto s cosdered. Egevalues (bucklg load factor that s obtaed, are mmzed accordg to ths expoetal parameter. Furthermore, the comparso betwee results of ths method ad the results of ANSYS commercal fte elemet package s doe. The effects of thckess varato, v boudary codtos, Youg module rato radus ad crcumferece axs, varato of rato of er radus to outer oe o bucklg load factor are cosdered. THEORY. Basc formulato f of the problem The formulato of the problem s derved uder the followg assumptos:. The plate s the state of plae stress.. The stress-stra relatoshp follows orthotropc materal. 3. The plate s th. Therefore, the Krchhoff assumptos are corporated. 4. The thckess s vared the drecto of radus of the plate. Cosder a crcular aular plate wth varable thckess h(r, a, b er ad outer radus, respectvely as show Fg.. For bucklg aalyss, the -plae dsplacemet u ad v may be eglected ad oly out-of-plae deformato w s cosdered. The goverg eergy fuctoal ca be gve by: J=U+VV ( where U ss the stored stra eergy per ut volume that the polar coordate system for plae stress s gve as follows [8]: U = òòò v ( r r + + r r r d r d dz ( V = òò A é æ wö N æ wö ù ê r ç + N çè r ç ø çè êë ø ú r d r d úû (3 Fg. Schematc vew of aular plate wth varable thckess (cetrally thcker crcular plate. 00 IAU, Arak Brach

3 58 Bucklg Aalyss of Polar Orthotropc Crcular ad Aular Plates I the preset study, the axal radal symmetry s assumed. Therefore, plate s depedet of azmuthal varable: a æ wö V Nr = ò ç r dr çè r (4 ø b For orthotropc aular plate, V s derved as follows [9] é ùæ ö ò (5 - ë û a + - N0 a - b w V = r - ú r dr + h b a a b ê ç r úçè r ø Also, usg the stress-stra relato for orthotropc materal, substtutg to Eq. (, oe may obta: é z æd wö æd wöædwö ædwö ù U = E r E d d d r ç E r r z òòò + ç + - ç v r dr ç dr çèdr ø çèrdr ê è ø è ø (6 ø ë ú û As show Fg the thckess of plate s cotuously varyg h(r, wth tegratg alog the thckess, the stored stra eergy ca be expressed by the followg equato: where b é ædwö ædwöædwö æ dwö ù U = Dr ( r ç r dr ò + + ç ç dr ç dr çèrdr ø çèrdr (7 è ø è ø ø a ê ë ú û 3 Eh r ( r E D( r Dr ( r =, = = ( - E D ( r r r r (8 I above equato, Dr ( r ad D ( r deote crcumferetal ad radal bedg stffess of the plate, respectvely. I ths study, aular ad sold crcular plates wth cotuously varyg thckess are cosdered. The varato of thckess alog the radus drecto ca be expressed as follows: æ r ö æ ö hr ( = h o + ç çèa ø çè ø (9 where h o ad a represet the thckess the cetre (er radus for aular plate ad the outer radus of plate, respectvely. I order to cosder the fluece of o uform thckess o bucklg load factor, two values are assged for the parameter, =0, for uform ad learly varyg thckess, respectvely. s o-dmesoal geometrc parameter that may be postve (cetrally ther crcular plate or egatve (cetrally thcker crcular plate ad s defed as follows: For crcular sold plates: ha = - (0 h o For crcular aular plates: hb- ho = ( h ( r o b 00 IAU, Arak Brach

4 F. Farhata ad A. Golshah 59 I above equato, r b s the rato of er radus to the outer oe (b/a. I the sake of coveece, the varables Eqs. (7 ad (5 are trasformed to dmesoless oe, so the total potetal eergy fuctoal s as follows: ì é ù a æd Wö æd Wöæ dw ö æ dw ö é r ùæ b dwö ü - JW ( = ïgr ( R RdR ï + D ò í ý ç 0 dr ç ç dr ç RdR çrdr r ê r b R úç dr ( ï êè ø è øè ø è ø ú - è ø ï b ïî ë û ë û ïþ that, Eh r gr = + R D = R= a 3 3 r o ( (, 0, ( - r (3 where λ s the bucklg load factor that s related to bucklg load as follows: N a 0 = (4 D 0. Optmzed Rtz method Cosderg the fact that Raylegh-Rtz method s a upper boud method, the determed egevalue s more tha real oe. Therefore, f oe ca optmze t somehow, the results wll be closer to real oe. I geeral, the fucto whch troduces the ukow quatty, s a lear combato of shape modes, as follows: N f ( x = å c ( x (5 = where c s the ukow costats. Accordg to the dea of optmzato, by performg the Optmzed Raylegh- Rtz method, t s qute coveet to approxmate f(x by meas of a summato: N f ( x = å c ( x, k (6 = Note that k above equato s the expoetal optmzato parameter. Regardg to the Eq. (6, out-plae dsplacemet W(R s gve as follows: N WR ( = å cw( Rk, (7 = By mmzg potetal eergy fuctoal Raylegh-Rtz method: J = 0, =,..., N c (8 Total umber of N lear homogeous algebrac equatos are geerated that the ukows are costats c. It forms the egevalue problem that the egevlaues are the values of bucklg load parameter. The o-trval codto leads to a trascedetal equato whose lowest root s the desred bucklg load factor [4]. Sce / k = 0, by requrg, oe s able to optmze the fudametal egevalue. 00 IAU, Arak Brach

5 60 Bucklg Aalyss of Polar Orthotropc Crcular ad Aular Plates.3 Bucklg of aular plate Regardg to the Eq. (7, w (R,k s defed as follows [9] w R a R br R k + - ( = ( + + (9 Ukow costats a ad b are determed by applyg boudary codtos..3. Clamped outer edge (F-C plate For a clamped outer edge, ra or R, the goverg boudary codtos are: ì w ( = 0 ï í dw ï ( = 0 ïî dr (0 By substtuto Eq. (9 to eq. (0, the a ad b values are determed as follows: -- k a =, b = + - k + -k (.3. Smply supported outer edge (S-F plate For a smply supported outer edge, the out-plae dsplacemet must satsfy the followg codtos: ì w ( = 0 ï í d w dw ï ( + ( 0 = ïî dr dr ( Cosderg the boudary codto outer edge, the a ad b values are determed: ( + ( a = ( k( k + + k( 3 k b = ( k( k + + (3.3.3 Elastcally restraed agast rotato For the case of elastcally restraed agast rotato: ì w ( = 0 ï í dw é d w dw ( gr ( ( ( dr dr ù =- + ê dr ú ïî ë û (4 where a ad b values are determed as follows: 00 IAU, Arak Brach

6 F. Farhata ad A. Golshah 6 Q -L S -Q a =-, b =- S -L S -L (5 I above equato, S, Q, ad L are defed as follows: [ ] [ ] [ ] S = ( k+ - + g( R (- + + k+ L = ( + + g( R ( Q = ( - + g( R (- + + (6 where = ak / D0 represets the dmesoless flexblty coeffcet. Usg the Eq. (8, total umber of N lear homogeous algebrac equatos s geerated. The o-trval codto leads to set zero the determat of coeffcets matrx, as follows: A- B = 0 (7 where A j ad B j are determed as follows: éæ d w öæd w ö d d d d j w w j dw w ù æ öæ ö æ öæ ö j ædw öæ w ö j Aj = g( R R dr r ò b dr dr dr R d R R d R R dr dr êçè ø èç ø çè øçè ø ç è øçè øú ç è øçè dr ë û ø (8 é d r ù æ b dw ö æ w ö - j Bj = R - R dr + -r ê rb b R ú ç dr ç è dr ø ë û (9 I the above equato, for the case of smply supported ad clamped outer edge, =0 ad =, respectvely..4 Bucklg of crcular sold plate Regardg to Eq. (7, for crcular sold plate w (R s defed as follows [9] w R a R br R k + ( - ( = ( + + (30 I order to obta a ad b as costats, the followg boudary codtos must be satsfed:.4. Clamped outer edge, (C-F plate k -- b =, a = + - k + -k (3.4. Smply supported outer edge, (S-F plate ( + ( a = ( k( k + + k( 5 4 k b = ( k( k + + (3 00 IAU, Arak Brach

7 6 Bucklg Aalyss of Polar Orthotropc Crcular ad Aular Plates.4.3 Elastcally restraed agast rotato Q -L S -Q a =-, b =- S -L S -L where S = ( k+ - é g( R K( 3 k ù êë úû L = (+ - é g( R K( ù êë úû Q = ( - é + g( R K( ù êë úû (33 (34 where A j ad B j are determed as follows: éæ d w öæd w ö d d d d ( j w w j dw w ù æ öæ ö æ öæ ö j æd w( öæ w ö j Aj = g( R R dr 0 ò dr dr dr RdR RdR RdR RdR êçè øè ç ø çè øçè ø ç è øçè øú ç è øçè RdR ë û ø (35 æ d dw ö æ w ö - j Bj = R ò R dr 0 ç èdr ç øè ç dr ø (36 I the above equato, lke before, for the case of smply supported 0 ad clamped outer edge. 3 NUMERICAL RESULTS Ths secto presets a umber of umercal examples that shows the good performace of the proposed method, whch was mplemeted Mathematca 5. computer program. The results of the developed optmzed Rtz method are compared wth some other results whch were obtaed from FE method that are developed ANSYS 0 commercal package. All calculato has bee performed for E r 0000 Mpa, υ ө 0.3, a= m, h o 0.08 m. It was revealed that coverget bucklg load factor s obtaed wth 4-term seres. Table ( shows ths covergece for cetrally thcker aular orthotropc plate. I presetg results, the dmesoless bucklg load factor s used. Value of ths factor s obtaed for plates of uform ad learly cotuously thckess. 3. Aular plate Table depcts the fluece of parameters,, r b o the bucklg load factor. It s observed that wth creasg the amout of β, bucklg load factor s creased too. Oe observes that orthotropc plate ( > has more stffess agast bucklg occaso comparso to sotropc oe. Furthermore, creasg parameter toward postve values makes the bucklg load factor to crease. Regardg to Table, some values of r b (r b > decreases the bucklg load factor λ, meawhle some other values makes t decrease. As t was expected, show Table 3, clamped boudary codto represets the hghest value of factor whle the smply supported oe shows the lowest. The fluece of parameters,, r b s the same as clamped case. Tables 4-6 depct the fluece of rotatoal costat o bucklg load factor for dfferet value of. It s revealed that whe 0, the boudary codto s closer to smply supported case ad the loadg factor shows lower tha whe the outer edge s clamped ad stff agast rotato. Table Covergece study for aular orthotropc plate r b m IAU, Arak Brach

8 F. Farhata ad A. Golshah 63 Table Bucklg load factor varato wth smply supported outer edge r b Table 3 Bucklg load factor varato wth clamped outer edge r b IAU, Arak Brach

9 64 Bucklg Aalyss of Polar Orthotropc Crcular ad Aular Plates Table 4 Bucklg load factor varato wth edge elastcally restraed (= r b Table 5 Load factor varato wth edge elastcally restraed ( = r b Crcular sold plate Fgs. ad 3 llustrate the varato of buckg load factor wth respect to boudary codto case, orthotropc ad geometry parameters ad. As show, plate wth <0 (cetrally thcker sold plate, h a /h o < has lower bucklg load factor tha plate wth >0 (cetrally ther crcular plate, h a /h o >. Bucklg factor patter for all boudary codtos are smlar. 00 IAU, Arak Brach

10 F. Farhata ad A. Golshah 65 Table 6 Load factor varato wth edge elastcally restraed ( = r b Fg. Varato of bucklg load factor wth respect to orthotropc parameter ad o-dmesoal geometrc parameter wth clamped outer edge. Fg. 3 Varato of bucklg load factor wth respect to orthotropc parameter ad o-dmesoal geometrc parameter wth smply supported outer edge. 00 IAU, Arak Brach

11 66 Bucklg Aalyss of Polar Orthotropc Crcular ad Aular Plates Fg. 4 Varato of bucklg load factor wth respect to orthotropcc parameter for dfferet boudary codtos.. Table 7 Comparso of results for bucklg load factor Optmzed Rtz method (I ad ANSYS (II I II I 9.87 II I.56 II Fg 4 llustrates the fluece of rotatoal costat o bucklg load factor for dfferet values of. For dfferet boudary codtos, the obtaed result s the same as the aular plate. I fact, for smply supported outer edge, the evaluated bucklg load factor s lower tha clamped ad elastcally restraed agast rotato outer edge. As show Table 7, the results obtaed from preset method compare very well wth obtaed from ANSYS commercal package. Ths cofrms the accuracy of Optmzed Rtz method bucklg aalyzg of crcular plates. 4 CONCLUSIONS The bucklg aalyss of orthotropc crcular aular ad sold plates uder uform radal compresso loadg wth uform ad learly varyg thckess was preseted. The er edge s free whle the outer edge s uder dfferet types of classcal boudary codtos ad also wth edges elastcally restraed agast rotato. Ths s mplemeted by optmzed Rtz method. I ths method, a optmzato expoetal parameter s utlzed that bucklg load factor was mmzed wth respect to t. Followg aree some of the cocludg remarks: I. Icreasg orthotropc parameter ad radus rato (er to outer radus,, creases the resstace of plate agast bucklg pheomea. II. Plate wth clamped boudary codto exhbts the hgher value of the bucklg parameter, whle the smply supported case shows lowest. Also, plate wth edges elastcallyy restraed agast rotato has value of betwee two boudary codto cases. III. Cetrally ther crcular plate has hgher bucklg parameter tha cetrally thcker oe. REFERENCES [] Woowsk-Kreg ger S., 958, Bucklg stablty of crcular plates wth crcular cyldrcal Aeolotropy, Igeeur- Archv 6: IAU, Arak Brach

12 F. Farhata ad A. Golshah 67 [] Mek T., Huybrechts S., Galey J., 999, The effect of varyg thckess o the bucklg of orthotropc plate, Joural of Composte Materals 33: [3] Laura P.A.A, Guterrez R.H., Saz H.C., Elvra G., 000, Bucklg of crcular, sold ad aular plates wth a termedate crcular support, Ocea Egeerg 7: [4] Caco P.M., Reyes J.A., 003, Bucklg of crcular aular plates of cotuously varable thckess used as teral bulkheads submersbles, Ocea egeerg 30: [5] Bremec B., Kosel F., 006, Thckess optmzato of crcular aular plates at bucklg, Th-Walled Structures 3: [6] Guterrez R.H., Romall E., Laura P.A.A., 996, Vbrato ad elastc stablty of th crcular plates wth varable profle, Joural of Soud ad Vbrato 95: [7] Lag B., Zhag Sh., Che D., 007, Natural frequeces of crcular aular plates wth varable thckess by a ew method, Joural of Pressure Vessels ad Ppg 84: [8] Tmosheko S.P., Gere J.M., 96, Theory of Elastc Stablty, Secod Edto, McGraw-Hll, New York. [9] Gupta U.S., Lal R., Verma C.P., 986, Bucklg ad vbratos of polar orthotropc aular plates of varable thckess, Joural of Soud ad Vbrato 04: IAU, Arak Brach