ISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 11, May 2014

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1 ISSN: ISO 91:8 Certified Iteratioal Joural of Egieerig ad Iovative Techolog (IJEIT Volume, Issue 11, Ma 1 Aalsis of Rectagular Loaded Thi Plate usig the Classical Small eflectio Theor through Variatio Iteratio Method A. Karimpour 1, J. Vaseghi Amiri,.. Gaji 1,Civil ad Mechaical Egieerig epartmets, Babol Noushirvai Uiversit of Techolog, Babol, P. O. Box 8, Ira Abstract I the preset paper, the differetial equatio goverig o the flo over a thi plate (Bi-harmoic equatio has bee ivestigated aalticall b Variatioal Iteratio Method hich has ma applicatios i phsics ad egieerig. I this boudar value problem, satisfactio of boudar coditios has great importace ad depeds ol o the ature of the supports structures. Moreover, the results idicate that the metioed method ca be used for solvig differetial equatios goverig o this kid of problems ith differet boudar coditios i a situatio that there is o exact solutio. Idex Terms Variatioal Iteratio Method (VIM; Thi Plate; Boudar Value Problems (BVPs; Small eflectio Theor. I. INTROUCTION I egieerig problems, it is importat to ko the chage of deflectio ad stress i plates uder differet loads to desig slabs ad sstems perfectl. I this case, the classical small deflectio theor of thi plates i supplied problems is used to tackle this problem. Alog ith rapid developmet of scieces, ma differet methods ere proposed to solve various boudar value problems (BVP [1, ], such as homotop perturbatio method (HPM[-6] ad variatioal iteratio method (VIM [7 1].This paper is devoted to stud the behavior of rectagular elastic plate [1, 15 ad16] via VIM hich is sho i Fig. 1.This method gives rapid coverget successive approximatios ithout a restrictive assumptios or trasformatio that ma chage the phsical behavior of the problem. Mostl, there are restrictios for the exact aaltical solutio of plates ad there is o geeral exact solutio for differet tpes of boudar coditio. Numerical solutios are applied for complex boudar coditio ad plate shapes to overcome the metioed eakess. B usig VIM, e successfull fid the solutio of differetial equatio goverig o the rectagular plates hich has excellet agreemet ith classical solutios. II. BASIC IEA OF HE S VARIATIONAL ITERATION METHO To clarif the basic ideas of VIM, e cosider the folloig differetial equatio: Lu Nu gx, (1 Where L is a liear operator, N a oliear operator ad g x, a ihomogeeous term. Accordig to VIM, e ca express respectivel, the folloig correctio fuctioal i x ad directios, respectivel, as follos: x,, 1,,, u 1 x u x Lu Nu g d,,,,, u 1 x u x Lu x Nu x g x d Where 1 ad are Geeral Lagrage multiplier, hich ca be idetified optimall b the variatioal theor[17 18].The subscript idicates the th u ~ approximatio ad is a restricted variatio hich u~ meas that.cosequetl, the solutio is give b: u( x, lim ux, ( No, e ill appl the VIM to the folloig problems to illustrate the stregth of the method. ( ( Fig. 1. laterall loaded rectagular plate [1]. III. MATHEMATICAL MOELING OF THE PROBLEM The basic differetial equatio of lateral deflectio for thi plates i classical Small-eflectio theor is obtaied [1-16] 6

2 ISSN: ISO 91:8 Certified Iteratioal Joural of Egieerig ad Iovative Techolog (IJEIT Volume, Issue 11, Ma 1 as follos: 1 x, x, P( x, ( x, x x ( x, ( x, (5 x x ( x, Where is the middle surface of the plate or ( d ( x, P( x, P( x, the lateral plate displacemet, is lateral loads per (1 uit area ad is the plate bedig stiffess. I geeral, The Homogeeous Boudar Coditios (HBCs for rectagular Where is Lagrage multiplier for hich ca be plate for istace i x-directio [1, 15] are preseted as idetified optimall b the variatioal theor? Accordig to follos: VIM, the multiplier i directio i Eq. (1 ields the To HBCs for free edge: folloig statioar coditios: //// /// // (, ( 1, (, ( x x / (, (. (11 This i tur gives: x (6 ( For to HBCs for fixed edge: ( t 6 (1 Substitutig the value of the Lagrage multiplier, Eq. (1 ito the correctio fuctioal Eq. (1 gives the folloig iteratio relatio: x 1 x, x, (7 Ad for to HBCs for simple edge: ( x, ( x, ( x x d 6 ( x, Px (, (1 x (8 Where Ad the approximatio solutio is give is the Poisso ratio of plate material? u( x, lim ux, It should be oted that to HBCs must be satisfied at as. x x 1 x x, ad also the other to HBCs should be V. NUMERICAL EXAMPLES satisfied at 1,.I either case for aalsis of I this sectio, to examples are itroduced ad the VIM is plates, e must choose to HBCs for each edge of applied to get the results. The results are the compared ith rectagular plate hich depeds ol o the ature of the the ko exact solutios to esure the accurac of the plate support. method. A. EXAMPLE 1 IV. FORMULATION OF VIM Accordig to the VIM procedure, e ca costruct the correctio fuctioal for Eq. (5 as follo:,, x x 1 x (, (, ( d (, P(, (9 For the first case, a rectagular plate ith scatlig a a is cosidered ith costat plate flexural rigidit,. This plate has to edges hich simpl is supported ad the to edges fixed supported ad subjected to a lateral load as follos: x P( x, P si( a (1 The differetial equatio goverig o this case stud ad the related boudar coditios are preseted as follos: P x ( x, si( x x a (15 The e ill have: 7

3 (,, (, x ( a,, ( a, x a a ( x,, ( x, ISSN: ISO 91:8 Certified Iteratioal Joural of Egieerig ad Iovative Techolog (IJEIT a a ( x,, ( x, (16 The exact solutio of this problem is reported b Szilard [1]: sih( / sih( Pa ( x, {1 a a / cosh( / sih( / cosh( cosh( / sih( / a x }si( / cosh( / sih( / a (17 I order to compare the obtaied results ith the exact solutio, it is ecessar to fix the costat values. I this case, the scatlig of plate is devoted for comparig the VIM results ad the exact solutio. For a rectagular plate ith scatlig1 1, Eq. (1 ad Eq. (17 are chaged ito these forms: P( x, P si( x (18 Ad P ( x, {1 / cosh( / sih( / sih( ( sih( / cosh( / sih( / / cosh( / sih( / cosh( }si( x (19 The variatioal iteratio formula i this case is obtaied i the form of:,, x x 1 ( x, ( x, ( x x d 6 ( x, P si( x ( ( x, Begiig ith the iitial approximatio of the form of a polomial fuctio as: 1 i ( x, ( si( x Volume, Issue 11, Ma 1 si( x x (1 1(, [( ( 7 P I the same maer, the rest of the compoets of the iteratio formula, Eq. (, ca be determie respectivel. Usig the boudar coditios, Eq. (16, e obtai the values of four costats of 1,, ad as follos:.157p 1,.11117P, ( Table1 shos a good agreemet betee the VIM ad the exact solutio. The results sho that miimum errors occur o both sides i directio ad the error i the middle side is ver small ad uiform ith respect to the exact solutio hich shos accurate results for VIM. Table 1: The differece betee the exact solutio ad VIM x, / P solutios after four iteratios i example 5.1 at x=.5}. Exact[1] VIM Absolute error E E E- 1.97E E- 1.97E E E-.1651E-.9851E-1.7E-1.16E E E- 1.9E- 1.57E- 1.9E E E-.16E-.7E-1.7E-1.98E E E E E E E E-8.98E-8.7E-1 The - plots of the exact solutio ad the VIM results are plotted i Fig. hich idicates the accurac of VIM. Ad 1,, ad are uko costats to be determied ith four boudar coditios at directio. Usig the above iteratio formula, Eq. (, e ca directl obtai other compoets as: Fig.1: The Exact solutio of example 5.1 x, / P 8

4 x Fig.: The VIM solutio of example 5.1 B. EXAMPLE I aother case, a rectagular plate ith scatlig, / P ab is cosidered ith costat,, hich is the plate flexural rigidit, ad simpl supported at all edges ad subjected to a lateral load as follos: x P( x, P si( si( a b ( The correspodig differetial equatio ad boudar coditios of this case are: ( x, x x P x si( si( a b Ad the HBCs are defied as follos: (, (,, x ( a, ( a,, x ( x,, ( x, ( x, b, ( x, b This problem has exact solutio as follos [15]: ( x, P x si( si( 1 1 ( a b a b (5 (6 (7 No, e must devote scatlig of plate for comparig of result of VIM ad exact aalsis. For a rectagular plate ith scatlig 1 1, the Eq. ( ad Eq. (7 chage to these forms: P( x, P si( x si( ISSN: ISO 91:8 Certified Iteratioal Joural of Egieerig ad Iovative Techolog (IJEIT Volume, Issue 11, Ma 1 ( x, P si( x si( (9 Ad the variatioal iteratio formula i this case is obtaied i the form of: 1 x, x, (8 ( 6 ( x, ( x, ( x, x x P si( x si( d ( Like before, the iitial approximatio ( x, of accordig to the boudar coditios at x directio, e ca make oe i the form of a polomial as: 1 ( x, si( x (1 It is otable that the above formula is depedig o the order of equatio differetiatio. 1,, ad are uko costats to be determied ith boudar coditios at directio. B usig the above iteratio formula, Eq. (, e ca directl obtai other compoets as: si( x 1( x, [( P P P si( ( ] I the same maer, the rest of the compoets of the iteratio formula, Eq. (, ca be determied. Fiall, the coefficiets of 1,, ad are itroduced as follos: 1.86P.169P ( 9

5 ISSN: ISO 91:8 Certified Iteratioal Joural of Egieerig ad Iovative Techolog (IJEIT Volume, Issue 11, Ma 1 Table : The differece betee the exact solutio ad VIM x, / P at x=.5}. solutios after four iteratios i example 5. Y EXACT[15] VIM ABSOLUTE are i good agreemet ith the exact solutio. Error calculatios sho that the VIM method is orkig ver accurate ad oe ca use such methods i egieerig problems for the deflectio predictio of thi plates E E-.76 E-.8 E-.566 E-.8 E-.76 E E E- 1.5 E E E-.768E-.5E-.5669E-.E-.7586E E E-.798E-9 ERROR.E+ 8.18E E-7.51E-7.E-7.E-7.56E-7.71E-7.188E-7.6E-7.7E-9 The - plots of the exact solutio ad the VIM results are plotted i Fig. hich idicates the accurac of VIM. x Fig.1: The Exact solutio of example 5. x Fig.: The VIM solutio of example 5., / P, / P Table shos excellet agreemet ith the exact solutio. Results sho that the VIM ca be used for certai forms of plates i hich loadig ad the deflectio of plate have similar terms i oe directio. Therefore; oe ca itegrate it alog oe directio ad simplif the equatio. The importace of these terms is to satisf the boudar coditios alog the directio hich is ot itegrated. VI. CONCLUSIONS I this paper, the problem of thi plate ith the small deflectio theor is cosidered. To cases of rectagular ad square plates aalzed ad compared ith their o exact classical solutios. The variatioal iteratio method is used to obtai the solutio of plate deformatio uder differet loadig. Lagragia multipliers are obtaied ad used to costruct the VIM formulatio. The results obtaied b VIM REFERENCES [1] Waleed Al Haai, Luis Casasús, Numerical Algorithms, (1 (5 67. [] Liu. B, Appl. Math. Comput, 18( ( 7. [] Rafei. M, Gaji.. It. J. Nol. Sci. ad Num. Simu, 7( (6 1. [] Gaji., Sadighi. A, It. J. Nol. Sci. ad Num. Simu, 7( (6 11. [5] Gaji., Nurollahi M., MohseiLaguri E., Computers ad Mathematics ith Applicatios, 5(7-8 ( [6] He J. H, It. J. Mod. Phs. B, (18 ( [7] Seilam N. H., Khader M. M, Chaos Solito Fract, (1 (7 15. [8] Odibat Z. M., Momai S, It. J. Nol.Sci, 7 (1 (6 7. [9] Bildik N., Kouralp A., It. J. Nol.Sci, 7(1 (6 65. [1] He J.H, It. J. Nol.Mech, ( ( [11] He. J.H, Wu X.H, Chaos SolutioFract, 9 (1 (6 18. [1] He. J.H, It. J.Comput. Appl. Math, 7(1 (7. [1] Gaji.., Jaatabadi M., Mohsei LaguriE., Joural of Computatioal ad Applied Mathematics, Volume 7, Issue 1, 1 October 7, Pages 5-5. [1] Szilard. R, Theories ad applicatios of plate aalsis, Joh Wile & Sos Ic, (. [15] Ugural. A. C, Stresses i plates ad shells, McGra Hill,(1999. [16] Timosheko. S. P, Gere. J. M, Theor of elastic stabilit, McGra Hill,(1961. [17] M.Iokuti, et al., Geeral use of the Lagrage multiplier i oliear mathematical phsics, i :S.Nemat-Nassed (Ed, Variatioal Method i the Mechaics of Solids, Pergamo Press, 1978, pp [18] B.A.Filaso, The Method of Weighted Residuals ad Variatioal Priciples, Academic Press, 197. AUTHOR S PROFILE Ali Karimpour (First Author: He is from Ira ad he has Master of Sciece i civil egieerig (Structure Eg.. He is a scholar ma i Mazadara Uiversit i Ira ad orkig i a cosultat office as a Supervisio/Computatio egieer of structures ad teachig i some classes i some Uiversities. Also he is member of federatio of master builders ad Iraia Societ of Civil Egieerig. As ell as he has to liceses i to classes (Supervisio ad Computatio of buildigs. 1

6 ISSN: ISO 91:8 Certified Iteratioal Joural of Egieerig ad Iovative Techolog (IJEIT Volume, Issue 11, Ma 1 Javad Vaseghi Amiri: He is professor of Civil Egg epartmet ad head of Babol Noushirvai Uiversit of Techolog i IRAN. He has earl 5 papers published i various iteratioal ad atioal jourals ad cofereces. His ebsite is: avood omiri Gaji: He is famous professor of Mechaical Egg epartmet of Babol Noushirvai Uiversit of Techolog i IRAN. He has earl papers published i various iteratioal ad atioal jourals ad cofereces. Also he is Editor-i-chief of Iteratioal joural of oliear amic ad Eg. Sci. (IJNES, Editor of Iteratioal Joural of ifferetial Equatios ad Iteratioal joural of Noliear Sciece ad Numerical Simulatio (IJNSNS. His ebsite is: 11