PLASTIC BUCKLING OF SSSS THIN RECTANGULAR PLATES SUBJECTED TO UNIAXIAL COMPRESSION USING TAYLOR-MACLAURIN SHAPE FUNCTION

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1 I. J. Sruc. & Civil Egg. es. 013 U G Eziefula e al., 013 esearch Paper ISSN Vol., No., Noveer IJSCE. All ighs eserved PLASTIC BUCKLING OF SSSS THIN ECTANGULA PLATES SUBJECTED TO UNIAXIAL COMPESSION USING TAYLO-MACLAUIN SHAPE FUNCTION D O Owuka 1, O M Iearugule 1 ad U G Eziefula * *Correspodig auhor:u G Eziefula, uchechi.eziefula@yahoo.co I his paper, a soluio for he plasic ucklig of a hi recagular isoropic plae wih four siply suppored edges uder uifor i-plae copressio is preseed. The plasic ucklig equaio was derived usig a deforaio heory of plasiciy ad a work priciple. The plae aalysis was carried ou hrough a heoreical forulaio ased o Taylor-Maclauri series ad applicaio of eergy ehod. The approxiae shape fucio for he plae oudary codiios usig he Taylor-Maclauri series was rucaed a he fifh er. The shape fucio was susiued io he plasic ucklig equaio ad he criical plasic ucklig load was oaied. The plae ucklig coefficie was deeried for aspec raios wihi he rage of 0.1 ad 1.0 a icrees of 0.1. The resuls were copared wih soluios fro previous sudies ad he average perceage differece was 0.091%. This differece deosraes ha he Taylor- Maclauri series shape fucio is a very good approxiaio of he exac values for he displacee fucio of he defored SSSS plae. Keywords: Criical ucklig load, Deforaio plasiciy heory, Displacee fucio, Iplae copressio, Taylor-Maclauri series, Thi plae INTODUCTION Based o he sress-srai relaioship, ucklig of plaes ay e classified as elasic ucklig or plasic ucklig. Elasic ucklig is ased o Hooke s law where i is assued ha he proporioal lii of he plae aerial is greaer ha he ucklig sress. I ay pracical cases, however, ucklig ay occur i he plasic rage. The acual ucklig load i he plasic rage is always lower ha he ucklig load i he elasic rage. Hece, i is ecessary o carry ou plasic ucklig aalysis 1 Depare of Civil Egieerig, Federal Uiversiy of Techology, Owerri, Nigeria. School of Egieerig Techology, Io Sae Polyechic, Uuagwo, Nigeria 168

2 I. J. Sruc. & Civil Egg. es. 013 U G Eziefula e al., 013 usig plasiciy heories so as o deerie he accurae ucklig load whe ucklig occurs i he plasic rage. The wo cooly used plasiciy heories i plae ucklig are he deforaio heory pioeered y Ilyushi (197) ad he flow (or icreeal) heory developed y Hadela ad Prager (198). The deforaio heory of plasiciy is aheaically less cosise i copariso wih he flow heory of plasiciy. However, os researchers accep ha he plasic ucklig loads y deforaio heory are always i eer agreee wih experieal resuls ad ha hey have lower uerical values ha hose oaied fro he flow heory. This is he well-kow paradox of plae plasic ucklig, ad a uiversally acceped soluio of he plasic ucklig paradox has o ye ee preseed (Pride ad Heierl, 199; Iskaso ad Pifko, 1969; Becque, 010). I fidig soluios o plae ucklig proles for oh he elasic ad plasic rages, he use of Fourier series or rigooeric series i esiaig he shape fucio of he defored plae exiss i lieraure. Irrespecive of he plasiciy heory used, soe researchers used he uerical approach while ohers used he equiliriu ad eergy approaches i fidig soluios o plasic ucklig of plaes. Sudies y researchers such as Sowell (198), Iyegar (1988), She (1990) ad Wag e al. (00) ivolved he use of rigooeric series. The use of Taylor s series i solvig plae ucklig proles has araced very lile aeio. To he es of he researchers kowledge, he Taylor s series has o ee used i he eergy approach for aalyzig he plasic ucklig of SSSS plaes. Therefore, he ai of his sudy is o use he Taylor-Maclauri series o solve he plasic ucklig prole of a hi recagular isoropic plae wih four siply suppored edges sujeced o uiaxial i-plae copressive loads. The prole defiiio is illusraed i Figure 1. The goverig equaio derived i he aalysis is ased o he deforaio heory of plasiciy usig Sowell s approach. Figure 1: SSSS Thi ecagular Plae uder Uiaxial I-plae Loadig a x N x SSSS PLATE N x y 169

3 I. J. Sruc. & Civil Egg. es. 013 U G Eziefula e al., 013 MATHEMATICAL FOMULATION Sowell (198) expressed he differeial equaio of equiliriu for he plasic ucklig of a hi, fla, recagular plae uder uifor copressio i he x-direcio as: 1 3E w w w Nx w 0 Es x x y y D x...(1) Where E is he age odulus, E s is he seca odulus, N x is he ucklig load, D is he plasic flexural rigidiy of he plae ad w is he displacee i he z-axis. Trasforig he x y coordiae syse o Q coordiae syse, we have x ; a y Q I should e oed ha ad Q are diesioless paraeers. Eziefula (013) applied a echique ased o Iearugule e al. (013) where Equaio (1) was rasfored usig he priciple of coservaio of work i a saic coiuu. Eziefula (013) ade N x he sujec of forula ad oaied N x 1 1 H 1 3 E H H H H Dp H Q p Es p Q Q 1 1 H H Q where...() p = a/...(3) 3 Es D 9...() w = AH...(5) p is he aspec raio, is he plae hickess, a ad are he legh ad widh of he plae respecively, H is he plae ucklig coefficie ad A is apliude of he shape fucio. Iearugule (01) expaded he shape fucio usig Taylor-Maclauri series ad oaied w w x, y 0. 0 x x. y y ( F ) x F ( ) y!!...(6) where F () (x 0 ) is he h parial derivaive of he fucio wih respec o x ad F () (y 0 ) is he h parial derivaive of he fucio w ad respec o y.! ad! are he facorials of ad respecively while x 0 ad y 0 are he pois of origi. He rucaed he ifiie power series a = = ad go 0 w J K. Q...(7) where J K F F 0 0 a! 0!...(8a)...(8) The oudary codiios for a SSSS plae are w 0 0; w (9) 170

4 I. J. Sruc. & Civil Egg. es. 013 U G Eziefula e al., 013 w 1 0; w (10) w Q 0 0; w Q (11) w Q 1 0; w Q (1) Susiuig Equaios (9) ad (11) io Equaio (7) gave J 0 = J = 0; K 0 = K = 0 Susiuig Equaio (10) io Equaio (7) ad solvig he resulig wo equaios siulaeously gave J 1 = J ; J 3 = J Siilarly, susiuig Equaio (1) io Equaio (7) ad solvig he resulig wo equaios siulaeously gave K 1 = K ; K 3 = K Susiuig he values of J 0, J 1, J, J 3, J, K 0, K 1, K, K 3 ad K io Equaio (7) gave 3 3 w J J J K Q K Q K Q 3 3 J K Q Q Q...(13) Fro Equaios (5), (7) ad (13), we have A J K...(1) 3 3 H Q Q Q...(15) Parial derivaives of Equaio (15) wih respec o, Q or oh ad Q gave H H Q Q Q 3 3 H H Q Q Q Q (16)...(17) H H 1 Q Q Q 3 3 Q Q Q...(18) H H 1 Q Q 3 Q 3...(19) Expadig ad iegraig Equaios (16), (17), (18), ad (19) parially wih respec o ad Q i a closed doai respecively resuled i 1 1 H H Q (0) 1 1 H H Q Q...(1) 1 1 H H Q Q...() 1 1 H H Q (3) Susiuig he values i Equaios (0), (1), () ad (3) io Equaio () gave N x D E p p Es () The plasic ucklig load ay e expressed as 171

5 I. J. Sruc. & Civil Egg. es. 013 U G Eziefula e al., 013 D N x H...(5) where E H p p Es...(6) ESULTS AND DISCUSSION The resuls fro his sudy gave he equaio of criical plasic ucklig load as D E N x, C p p E s...(7) Fro Iyegar (1988), he exac soluio for he plasic ucklig of a SSSS plae usig Sowell s approach is N D 1 3 E E x, C s p p...(8) I Equaio (8), ad are he ucklig odes. For he firs ode of ucklig, = 1. Also, sice we are ieresed i fidig he lowes value of N x a which he plae uckles, us e equal o oe (Iyegar, 1988). Hece, Equaio (8) ay e siplified o N D E p E x, C s p...(9) The facor, E /E s is equal o oe i elasic ucklig u is value is always less ha uiy i plasic ucklig. I his paper, he uerical value of E /E s is ake o e equal o 0.9. Tale 1 shows he values of H fro his prese sudy ad Iyegar (1988) for differe aspec raios usig E /E s = 0.9. Tale 1: Values of H for Plasic Bucklig of Uiaxially Copressed SSSS Thi ecagular Plae p = a/ H fro Prese Sudy H fro Iyegar (1988) Perceage Differece

6 I. J. Sruc. & Civil Egg. es. 013 U G Eziefula e al., 013 Fro Tale 1, he highes perceage differece is 0.175% for p = 0.1, while he lowes perceage differece is 0.066% for p = 1.0. The average perceage differece ewee he soluio fro his prese sudy ad Iyegar s soluio is 0.091%. Iyegar s soluio is a exac soluio oaied fro rigooeric series while he soluio fro he prese sudy is a approxiae soluio ased o Taylor-Maclauri series. The soluio fro he prese sudy is a upper oud soluio. I ca e oserved ha he closeess of he wo soluios iproves as he aspec raio icreases fro 0.1 o 1. CONCLUSION I his sudy, plasic ucklig aalysis of a hi, fla, recagular, isoropic SSSS plae was carried ou usig Sowell s plasiciy heory ad Taylor-Maclauri series shape fucio. A work echique was applied o deerie he plae ucklig coefficie for differe aspec raios of he plae. The resuls showed ha he soluio is a very close approxiaio of he exac soluio. Therefore, he Taylor-Maclauri series is adequae for approxiaig he defored shape of he SSSS plae i he plasic ucklig aalysis. EFEENCES 1. Becque J (010), Ielasic Plae Bucklig, ASCE Joural of Egieerig Mechaics, Vol. 136, No. 9, pp Eziefula U G (013), Plasic Bucklig Aalysis of Thi ecagular Isoropic Plaes, M.Eg Thesis Suied o Posgraduae School, Federal Uiversiy of Techology, Owerri. 3. Hadela G H ad Prager W (198), Plasic Bucklig of ecagular Plaes uder Edge Thruss, NACA Techical epor, No Iearugule O M (01), Applicaio of a Direc Variaioal Priciple i Elasic Sailiy Aalysis of Thi ecagular Fla Plaes, Ph.D Disseraio, Federal Uiversiy of Techology, Owerri. 5. Iearugule O M, Eu L O ad Ezeh J C (013), Direc Iegraio ad Work Priciple as New Approach i Bedig Aalyses of Isoropic ecagular Plaes. The Ieraioal Joural of Egieerig ad Sciece, Vol., No. 3, pp Ilyushi A A (197), The Elaso-plasic Sailiy of Plaes, NACA Techical Meoradu, No Isakso G ad Pifko A (1969), A Fiie Elee Mehod for he Plasic Bucklig Aalysis of Plaes. AIAA Joural, Vol. 7, No. 10, pp Iyegar N G (1988), Srucural Sailiy of Colus ad Plaes, Ellis Horwood, Chicheser. 9. Pride A ad Heierl G J (199), Plasic Bucklig of Siply Suppored Copressed Plaes, NACA Techical Noe, No She H S (1990), Elaso-plasic Aalysis for he Bucklig ad Posucklig of ecagular Plaes uder Uiaxial Copressio. Applied Maheaics ad Mechaics, Eglish Ediio, Vol. 11, No. 10, pp Sowell E Z (198), A Uified Theory of he 173

7 I. J. Sruc. & Civil Egg. es. 013 U G Eziefula e al., 013 Plasic Bucklig of Colus ad Plaes, NACA Techical epor, No Wag C M, Che Y ad Xiag Y (00), Plasic Buckig of ecagular Plaes Sujeced o Ierediae ad Ed I-plae Loads, Ieraioal Joural of Solids ad Srucures, Vol. 1, pp