MOHAMMAD H. KARGARNOVIN, Professor Mechanical Engineering Department, Sharif University of Technology Azadi Ave., P.O.Box , Tehran, Iran

Transcription

1 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 Buckling of Generll Orthotropic Rectngulr Simpl Supported Edgewise Plte under Compound In-Plne Linerl Bending- Compressive Loding, Using the Ritz Method MOHAMMAD H. KARGARNOVIN, Professor Mechnicl Engineering Deprtment, Shrif Universit of Technolog Azdi Ave., P.O.Box , Tehrn, Irn nd AHMAD MAMANDI, Ph.D. Student Aerospce Engineering Deprtment, Fcult of Mechnicl Engineering, Science nd Reserch Brnch, Islmic Azd Universit P.O.Box , Tehrn, Irn Astrct: - In this stud the uckling nlsis of n orthotropic rectngulr plte with simpl supported edges sustining linerl in-plne ending (extensive-compressive compound loding on its two opposite edges using the Ritz method ssuming one prmeter for deflection solution is crried out to determine the iticl uckling lod for plte-loding sseml configurtion. Using powerfull energ method nd deriving expressions for internl energ nd totl work done the externl in-plne compound loding nd derivtive of sutrction of them s totl potentil energ function for this sstem under investigtion with respect to ssumed prmeter gives out us minimum iticl in-plne uckling lod fctor such s considered this derivtive to e zero which it s the principle of minimum potentil energ. Ke-Words: - Orthotropic Rectngulr Plte, Simpl Supported, Compressive-Bending Loding, the Principle of Minimum Potentil Energ, the Ritz Method, Approximtion Method, Criticl Buckling Lod 1 Introduction According to development nd vst usge of pltes s one of mostl structurl components in oth of ordinr nd dvnced constructions in gret rnge of mechnicl sstems like s utomoiles frme odies, ship decks nd in erospce structures for instnce in wing oxes components, ris, wing stiffeners, dividers in irplne fuselge nd missiles nd rockets interior elements nd in oil, gs nd petrochemicl industries equipments for exmple in vessels, storge tnks s impingement or ffle pltes nd mn more etc. the expecting of sfe opertion nd immunit mong their opertion is ver significnt for us. One of the commonl mchine elements filure during opertion which occurs is uckling. The im of the uckling nlsis is to estimte the mximum lod tht structure cn support efore it ecomes elsticll unstle or efore it collpses. The oserved uckling lod depends on how uckling is defined. We cn define tht the uckling lod is defined s tht lod t which the first uckle ecomes visile to the unided ee, or sudden deformtion of slender memers or thin wlled structures under compressive loding (or n comintion of in-plne-lterl loding due to sudden relese of internl energ in the form of ending deformtion. In this pper, the proper pproximte-nlticl nlsis is crried out to exmine the elstic uckling ehvior of n orthotropic rectngulr plte with respect to the geometric prmeters. This will enle designer to predict the elstic uckling lod of plte quickl, resonl, ccurtel, economicll nd relil. It is ssumed tht the pltes re of constnt thickness nd re free from imperfections. Models with thin prts tend to uckle under xil loding. Buckling cn e defined s the sudden deformtion tht occurs when the stored memrne (xil energ is converted into ending energ with no chnge in the externll pplied lods. Mthemticll, when uckling occurs, the totl stiffness mtrix ecomes singulr [1]. The linerized uckling pproch, used here, solves liner sstem of equtions to estimte the iticl uckling lod fctors nd the ssocited uckling shpes. ISBN: pge ISSN:

2 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 Buckling nlsis clcultes the smllest (iticl loding required to uckle structure. In fct ech uckling lod is ssocited with uckling mode. Designers re usull interested in the uckling lod corresponding to the lowest mode ecuse it is ssocited with the lowest iticl lod. A more vigorous pproch to stud the ehvior of models t nd eond uckling level requires the use of nonliner design nlsis like s Finite Element Method codes. Slender structures nd structures with slender prts loded in the xil direction uckle under reltivel smll xil lods. Such structures m fil in uckling while their stresses re fr elow iticl levels. For such structures, the uckling lod ecomes iticl design fctor. Stock structures, on the other hnd, require lrge lods to uckle, therefore uckling nlsis is usull not required. The prolem of uckling of pltes under vrious shpes nd oundr conditions hs een studied in some fmous clssic nd cdemic reference text ooks [-3] nd some of numerous literture works [-7]. Bhrt Kln nd Bhskr [8] hve investigted the uckling of rectngulr orthotropic pltes sujected to non-uniform compressive lods nd the hve presented n nlticl solution for their stud. Bo, Jing nd Roerts [9] used the oth of Finite Element nd nlticl solution for ending nd uckling of flt, rectngulr, orthotropic thin pltes considering pltes with ll edges simpl supported, two edges simpl supported nd two edges clmped, nd ll edges clmped. Fn nd Tong [1] studied the ehvior of uckling model of simpl supported orthotropic rectngulr plte under ixil lods on the sis of the nlticl solution. The lw of uckling models in reltion to lods nd generl solution of differentil eqution for uckling displcement function of orthotropic rectngulr thin plte to solve the stilit prolem of rectngulr plte with ritrr oundries hs een presented in their work. Spillers nd Lev [11] studied the Optiml design for plte uckling in generl solution. Also Lev nd Sokolinsk [1-1] studied the Rleigh-Ritz optiml design of orthotropic pltes for uckling in some tpes of loding. Wng et l. [15], Hslch et l. [16] nd Stein [17] extended theoreticl nlsis for the post-uckling ehviour of simpl-supported orthotropic rectngulr pltes sujected to comined in-plne ixil compression nd edge sher perturtion technique. Yng et l. [18], Hrik et l. [19] nd Jirng et l. [] studied stilit of uckled orthotropic rectngulr pltes. Timoshenko et l. [1] hs studied the uckling of n isotropic (customr engineering mterils rectngulr plte with vrious comined loding nd oundr conditions in pssed centur sed on development nd extension of Boonov [] nd Johnson et l. [3] works. However there isn t n developed investigtion nd stud for n orthotropic rectngulr simpl supported plte with such s tpe of loding in literture works up to now. The sic im of this presented reserch is to stud nd formulte the effect of plte geometr spect rtio to determine iticl vlue of uckling lod fctor for n orthotropic rectngulr simpl supported plte using n pproximte-nlticl solution for first iticl mode predominntl. The Ritz Method [] The so-clled Ritz method is convenient procedure for determining solutions the principle of minimum potentil energ. The essence of this pproch is desied for the cse of elstic ending of pltes s follows. First choose solution for the deflection w in the form of series contining undetermined prmeters mn ( m, n 1,,.... The deflection so selected must stisf the geometric oundr conditions. The sttic oundr conditions need not to e fulfilled. Clerl, proper choice of the deflection expression is importnt to ensure good ccurc for the finl solution. Thus, it is desirle to ssume n expression for w which is nerl identicl with true ent surfce of the plte. Next, emploing the selected solution, determine the potentil energ Π in terms of mn. (This demonstrtes tht the mn ' s govern the vrition of the potentil energ. In order tht the potentil energ e minimum t equilirium: Π Π,..., 11 mn (1 The foregoing represents sstem of lgeric equtions which re solved to ield the prmeters mn. Introducing these vlues into the ssumed expression for deflection, one otins the solution for given prolem. In generl, mn includes onl finite numer of prmeters nd the finl results re therefore onl pproximte. Of course, if the ssumed w should hppen to e the exct one, the solution will then e exct. Advntges of the Ritz pproch lie in the reltive ese with which mixed edge conditions cn e hndled. This method is mong the simplest for ISBN: pge 3 ISSN:

3 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 solving plte nd shell deflections mens of hnd clcultion. The ppliction of the strin-energ techniques in the tretment of ending, stretching, s well s uckling prolems of pltes nd shells, will e discussed through out the mn text ooks nd litertures from pst centur until now []. 3 Prolem Formultion We consider n orthotropic rectngulr simpl supported plte like s Fig. 1 in which its edges in x nd x sustin linerl loding in its midplne of plte. Proper eqution for expression of this tpe of loding is [1] (1 ( Nx N C Fig. 1 An orthotropic rectngulr plte with simpl supported edges, sustining compound linerl ending-compressive loding In which N is intensit of compressive lod t edge nd C is numericl multiplier. B chnging in C vlue, we cn otin some different cses. For exmple if C, there is uniform compressive loding nd for cse C, there is pure ending stte. If C <, we get compound compressionending cse such s hs een shown in our prolem figure tpicll. If C >, similrl we hve comintion of extensive-ending cse [1]. 3.1 Governing Equtions We cn consider doule trigonometric Fourier series for uckled plte displcement field like s [] x nπ wx (, mn sin sin (3 m 1 n 1 Clerl, the foregoing stisfies the oundr conditions t the simpl supported plte edges. For clculting of iticl compressive lod N, we use energ method. Therefore for interior ending energ due to ove displcement field, we cn write following expression [] 1 w w w w U ( ( D x + D + D x x w + Gx ( dxd x Where x ( te te te x, x tg Dx D, Dx, Gx ( The expressions for Dx, D, Dxnd G x represent the flexurl rigidities nd the torsionl rigidit of n generl orthotropic plte, respectivel in principl directions x,. B inserting Eq. (3 into this eqution we otin U D nπ x nπ + Dx ( ( sin sin 1 x nπ ( sin sin mn x m 1 n 1 nπ x nπ + D ( sin( sin( nπ x nπ + Gx ( ( cos( cos( dxd And oserving from orthogonlit reltions tht L ( i i iπx i πx sin sin dx L L L ( i i ( j j jπx j πx cos cos dx L L L ( j j L We otin 1 m mn U mnπ Dx ( + D ( ( x m 1 n 1 n m n + D ( + G ( ( x (6 (7 (8 Finll the strin energ ssocited with the ending of the uckled plte is given ISBN: pge ISSN:

4 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 m mn n U D + H + D 8 Where mnπ x ( ( ( m 1 n 1 (9 H D + G (1 x x The totl work done these externl in-plne lods during the uckling of the plte is [] 1 w w w w W N + N + N dxd x( ( x A x x (11 For this prolem N x (1 C N, N Nx (1 More simplifiction ields 1 C w W N (1 ( dxd x A N C x nπ (1 ( cos( sin( mn m 1 n 1 A N x nπ ( cos( sin( mn dxd m 1 n 1 A N C x nπ ( cos( sin( mn A m 1 n 1 Using some useful reltions s elow dxd dxd (13 ( i j iπ jπ sin sin dx ( i j, i± j even 8ij ( i j, i± j odd π ( i j (1 We cn conclude little lgeric simplifiction N W ( mn 8 m 1 n 1 8 mn mi ( mn m 1 n 1 π n 1 i ( n i NC ni (15 In this reltion, i numer vlue must e such tht n± ire lws odd numers. B equling totl work done with internl energ relted to ending we otin iticl lod for N s elow N m mn n π mn Dx ( + H( + D ( m 1 n 1 C 3 mnmini ( mn ( mn m 1 n 1 m 1 n 1 π n 1 i ( n i In generl form (16 N f( C,,, D, D, H (17 x In order to scertin the iticl vlue of the N, the coefficient mn is to e found such tht N is minimum. This, it cn e demonstrted (Sec. [], is equivlent to requiring tht Π e minimum. Upon ppliction of Eq. (1, thus we hve Π Π U W, mn We otin liner sstem of equtions s (18 m mn n π mn Dx ( + H( + D ( C 16 mini N ( mn ( mn π i ( n i (19 We dd ll of equtions with the sme determined m. It mens we consider eqution elow for plte displcement insted of Eq. (3 [1] x nπ wx (, sin mn sin ( n 1 It mens we will tke one prmeter of the series solution s mn coefficient. It is in ccordnce with this ssumption tht the uckled plte in long x xis hs een divided to m hlf-wves. This ssumption is logicl, ecuse the sstem of Eq. (19 cn e divide into some groups in which ech of them hs presied vlue of m,i.e. it shows uckling shpe of plte which is in the form of some hlf-wves with nodl lines prllel to the direction of xis. If one determinnt of them is equl to zero then generl determinnt of whole of sstem of Eq. (19 is equl to zero vlue. We cn ssume one hlf-wve mong of two nodl lines s plte with simpl supported edgewise tht ehve like s one hlf-wve uckled plte. B considering m 1 in Eq. (19 nd ppling reltion s elow ISBN: pge 5 ISSN:

5 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 N t (1 It ields 1 n 1 n n π Dx( + H( + D( t π C π 16 N ( ( And.. ni 1i 1n 1n π i ( n i 1 n n n π Dx + H( + D( t C ni π (1 8 1i 1 n C i ( n i Finll t 1i π i ( n i ( (3 n n t C 1 n Dx + H( + D( (1 π ( ni 8C These clcultion hs een crried out on the whole i s in the form n± i to e odd numer. These equtions re liner homogenous sstem of equtions with respect to 11, 1,... nd the hve een stisfied Equling 11, 1, to e zero vlues tht re in ccordnce with flt (unreformed plte shpe. For hving non-zero vlue of 11, 1, the determinnt of these equtions must e zero. In this w we cn find n eqution for clculting of vlues of compressive stresses. This clcultion cn e done using successive pproximtion method. We egin considering first eqution of sstem of Eq. ( nd we ssume tht ll of prmeters re zero except of 11 prmeter. Then we hve smller thn uniforml compressive stresses in comprison. Eq. ( gives out us expected result. In the cse C, Eq. (6 is dpted to expression for iticl stress of n orthotropic plte which is compressed uniforml. For otining of second pproximtion it must e considered two equtions of sstem of Eq. ( hving 11, 1. So we hve t C 11 Dx + H( + D( (1 π t 1 8C π 9 And t 11 (7 8C + π 9 t C Dx + H( + D(. (1 1 π (8 For hving non-zero solution, determinnt must e zero, so it is + c det c d + In which Dx + H( + D( t C (1 π t c 8C π 9 t C d Dx + H( + D( (1 π At lst we cn otin s elow (9 (3 t C Dx + H( + D( (1 (5 π + + ( d (( d ( d c (31 Conclusion is π D ( ( x H D t( C + + (6 So if we consider three equtions of sstem of Eq. (19 nd ssuming C, we will tke Estimting of first pproximtion is onl for smll vlues of C in the cse which ending stresses re ISBN: pge 6 ISSN:

6 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 3t + ( + ( 11 1 Dx H D 9π 3t 11 + Dx + H( + D( 1 9π (3 96t 13 5π 96t Dx + H( + D( 13 5π B introducing some more sustitutive vriles 3 e 9π t f Dx + H( + D( 96t g 5π 3 9 h Dx + H( + D( Similrl determinnt must e equl to zero e det e f g g h (33 (3 Then we cn otin once more gin clcultion of third pproximtion which hs good ccurc for pure ending cse s f h g + he (35 Oserving from Eq. (31 nd Eq. (35, Generl form of for generl orthotropic rectngulr simpl supported plte under ssumed loding ecomes 3. Cse stud nd results To tke some results for more deliertion of presented formultion, t this step we consider n orthotropic rectngulr simpl supported edges plte with following dt nd specifictions [] D D, D 3 D, D. D, G.5D x x x And so for plte geometr we consider 1( m, for instnce. Now we determine the iticl uckling lod nd stress of simpl supported edges plte under the ction of compound ending-compressive loding. In tle 1, Vlues of k used in Eq. (36 for our cse stud with given specifictions for m 1 hs een presented. Fig., shows vrition of the uckling lod fctor k with respect to the spect rtio r for vlue of m 1, C 1 nd C. The vrition of the uckling lod fctor k s function of spect rtio r for m 1,, 3, nd C /3,1,1.5 re plotted in Fig. 3 through Fig. 5. Oviousl, for specified m nd C, the mgnitude of k depends upon r onl. Referring to the Fig. 3, the mgnitude of N nd the numer of hlf-wves m for n vlue of the spect rtio r cn redil e found. In the cse of r 1.5, for instnce, from Fig.3, k nd m. The corresponding iticl lod is π D N under which the plte will uckle into two hlf-wves in the direction of the loding. It is lso oserved from Fig. 3 tht n orthotropic plte m times s long s it is wide will uckle in m hlf-sine wves. Thus, long plte ( with simpl supported edges under n xil compression tends to uckle into numer of squre cells of side dimension ; its iticl lod for ll prcticl purposes is given Eq. (31 []. π k f( D,, x D H (36 t We cn otin results for m, 3,... lso. To otin desired oject we must use Eq. (19. Once more gin, such similr w gives out us other uckling modes [1]. ISBN: pge 7 ISSN:

7 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 Tle 1, Vlues of k used in Eq. (36 for cse stud with given specifictions for m 1 C / / / / Fig.. Vrition of the uckling lod fctor k with spect rtio r for vlue of m 1, C 1 nd C Fig.. Vrition of the uckling lod fctor k with spect rtio r for vlues of m 1,, 3, nd C 1 Fig. 3. Vrition of the uckling lod fctor k with spect rtio r for vlues of m 1,, 3, nd C 3 Fig. 5. Vrition of the uckling lod fctor k with 3 spect rtio r for vlues of m 1,, 3, nd C ISBN: pge 8 ISSN:

8 Proceedings of the 3rd IASME / WSEAS Interntionl Conference on CONTINUUM MECHANICS (CM'8 Conclusion In this stud n pproximte-nlticl solution for investigtion of uckling of n orthotropic rectngulr simpl supported plte in in-plne comined extensive (ending-compressive loding stte to otin iticl uckling lod fctor hs een presented. The predominnt dvntge of this formultion is using of simpl energ method directl to derive properl formultion. It hs een shown tht inesing of r vlue, the uckling lod fctor curve ehve for ech uckling mode s we hoped similrl in nlog with isotropic rectngulr simpl supported pltes [-3] nd [1]. References: [1] Chndruptl T.R., Blegundu A.D., Introduction to Finite Element, nd ed., Printice-Hll, [] Ugurl A.C., Stresses in Pltes nd Shells, nd ed., WCB/McGrw-Hill, [3] Timoshenko S., Woinowsk-Krieger S., Theor of Pltes nd Shell, nd ed., McGrw-Hill, [] Tung T.K., Surdens J., Buckling of rectngulr orthotropic pltes under ixil loding, Journl of Composite Mterils, Vol.1, No., 1987, pp [5] Wng, J. T.S., Biggers S.B., Dickson J.N., Buckling of composite pltes with free edge in edgewise ending nd compression, AIAA journl, Vol., No.3, 198, pp [6] Frshd M., Ahmdi G., Perturtion solution to the prolems of generl orthotropic rectngulr pltes, Irnin Journl of Science nd Technolog, Vol.1, No., 1971, pp [7] Whitne J.M., Leiss A.W., Biggers S.B., Dickson, J.N., Anlsis of simpl supported lminted nisotropic rectngulr plte, AIAA journl, Vol.8, No.1, 197, pp [8] Bhrt Kln J., Bhskr K., An nlticl prmetric stud on uckling of non-uniforml compressed orthotropic rectngulr pltes, Journl of Composite Structures, Vol.8, 8, pp [9] Bo G., Jing W., Roerts J.C., Anltic nd finite element solutions for ending nd uckling of orthotropic rectngulr pltes, Interntionl Journl of Solids nd Structures, Vol.3, No.1, 1997, pp [1] Fn Y., Tong X., Buckling model nlsis of simpl supported orthotropic rectngulr pltes under ixil lods, Fuhe Cilio Xueo/Act Mterile Composite Sinic, Vol.1, No.1, 1993, pp [11] Spillers W.R., Lev R., Optiml design for plte uckling, Journl of structurl engineering, Vol.116, No.3, 199, pp [1] Lev R., Rleigh-Ritz optiml design of orthotropic pltes for uckling, Structurl Engineering nd Mechnics, Vol., No.5, 1996, pp [13] Lev R., Sokolinsk V., Preuckling optiml design of orthotropic vrile thickness pltes for inplne loding, Journl of Structurl Optimiztion, Vol.9, No., 1995, pp [1] Lev R., Sokolinsk V., Optiml design of pltes for sher uckling, Journl of Computers nd Structures, Vol.63, No., 1997, pp [15] Wng H., Ou M., Wng T., Post-uckling ehviour of orthotropic rectngulr pltes, Journl of Composite Structures, Vol.1, No.1, 1991, pp [16] Hslch Jr., Henr W., Post-uckling stilit of orthotropic, liner elstic, rectngulr pltes under ixil lods, Interntionl Journl of Mechnicl Sciences, Vol.8, No.11, 1986, pp [17] Stein M., Postuckling of long orthotropic pltes under comined loding, AIAA journl, Vol.3, No.8, 1985, pp [18] Yng D.S., Lio Y., Hung Y., Stilit nlsis for orthotropic thin pltes, Gongcheng Lixue/Engineering Mechnics, Vol.19, No.3,, pp [19] Hrik I.E., Blkrishnn N., Stilit of orthotropic rectngulr pltes, Applied Mthemticl Modeling, Vol.18, No.7, 199, pp. -. [] Jirng F., Jinqio Y., Exct solutions of uckling for simpl supported thick lmintes, Composite Structure, Vol., No.1, 1993, pp [1] Timoshenko S., Gere J.M., Theor of Elstic Stilit, nd ed., McGrw-Hill, [] Boonov J., Theor of Structures of Ships, 1 st ed., Vol., st. Petersurg, 191. [3] Johnson J.H., Noel R.G., Buckling of n rectngulr simpl supported plte, Journl of Aeronutics Science, Vol., 1953, p535. ISBN: pge 9 ISSN: