Bending Analysis of Isotropic Rectangular Plate with All Edges Clamped: Variational Symbolic Solution

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1 Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): Scholarlink Research Institute Journals, 0 (ISSN: -706) jeteas.scholarlinkresearch.org Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): 86-85(ISSN: -706) Bending nalsis of Isotropic Rectangular Plate with ll Edges Clamped: Variational Smbolic Solution Osadebe N. N and ginam C Department of Civil Engineering Universit of Nigeria, Nsukka Corresponding uthor: ginam C bstract n alternative variational approach is developed for the bending analsis of thin isotropic clamped plate. The method b-passes the tedious and rigorous solution of plate s differential equation of equilibrium involved in the classical and numerical methods. The method is a modification of Ritz variational approach. It is based on total potetial energ. B the fomulation, the deformation surface of the clamped plate with uniforml distributed load is approimated to be the sum of products of constructed polnomals in the and aes. The constructed polnomials satisf the plate s geometrical boundar conditions in addition to being interdependent and continuous. The sum of product of the constructed polnomial is substituted into plate s differential equations and then solved through minimization principle. Consequentl, the deflections and the bending of the thin isotropic clamped plate are obtained in analtical form thus enabling the evaluation of these quantities at an arbitrar point on the plate with uniforml distributed load at various plate aspect ratios of.0 to.0. The solution is done for first, second, third and four terms- polnomials representing st, nd, rd and th approimations respetivel. The variational procedure elucidates the ease and convergence of the results. Kewords: clamped rectangular plate; deflections; moments; uniform load; variational method INTRODUCTION Plates are plane two dimensional structural elements used etensivel in Mechanical, eronautical and Civil Engineering to bear heav loads due to it weight and econom. Ventsel and Kraunamather (00) classified plates into thin, membrane and thick. Plates ma be either isotropic or orthotropic. The modelled plate of the research is steel with the poisson ratio, (ϑ), of 0.0. The deflection of isotropic thin rectangular plates clamped at four edges and under the action of uniforml distributed load has received considerable attention due to its technical importance (Imrak & Gerdemeli, 006) thus, the essence of this research. There are man classical solutions for isotropic linear elastic thin plate. Timoshenko & Woinowsk-Kreiger (959) developed classical solutions for thin plate. pproimate solutions have also been suggested b notable researchers, but there is notabl loss of accurac (Wang & El-Sheik, 005). Various methods have been developed in the evaluation of rectangular plates with various boundar conditions: Henck (9), Timoshenko (98), Evans (99), Young (90), Hutchinson (99), Wang et al (00), Talor & Govindgee (00). These are generall accepted as approimate methods. Popular methods commonl used in evaluating the maimum deflections for the clamped thin plates are classical and numerical methods (Finite Element Method- FEM, Finite Difference Method-FDM, and Finite Strip Method- 86 FSM). For the classical solution b Timoshenko (959), two main approaches have been developed for obtaining the solution of maimum deflection for clamped thin rectangular plates under uniform load. These are double cosine series (Szilard,97) and the superposition method as a generalization of Henck s solutionenck s solution is well known to have accelerated convergence but possess risk with regard to overflow/underflow problem in evaluation of trignometric functionsutchinson (99) etended the works of Timonshenko (959) and evaluated defletions for the uniforml loaded rectangular plate. However, determination of the numerical values of maimum deflections for a rectangular plate is cumbersome and rigorous (Imrak & Gerdemeli, 007). Bending analsis of rectangular plates is proposed using a combination of basic functions and finite difference energ techniques (De, 00).The analsis of rectangular plate subjected to a uniforml distributed load with both ends fied was done b Wojtaszak (97) and Evans (99). Evans (97) developed and proposed design tables for deflections and moments for plate fied on all edges and subjected to uniforml distributed load. Toda, smbolic manipulation sstems have become popular in engineeing analsis (Betzer, 990). The method is capable of manipulating both numbers and smbols. The computer-aided algebraic computations can considerabl reduce tedious and rigorous analtical calculations and at the same time improve the results. In this research, it is shown how the smbolic

2 Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): 86-85(ISSN: -706) computer manipulation approach (Quick Basic) can be emploed using the well-established variational method of Ritz to solve the Bending problem of clamped isotropic rectangular plates with or without elastic foundations. The variational results obtained with this procedure will be compared with those of classical methods. FORMULTION OF PLTE EQUTION The general equation of plate is formulated using total potential Energ principles. Total potential Energ consists of strain Energ of deformation U and potential Energ of Eternal work, we, assuming the element of the structure under the transverse load remains elastic and is under adiabatic condition. ssuming Hooks low is strictl obeed, the Strain Energ of the plate is () du Where = normal stress along the -ais = normal stress along the -ais = shear stress along the - plane., and are the respective strains on,, aes and - plane. But z w w. ( a) z w w. ( b) ( ) z w. ( c) w z w z ( a) ( b) w z ( c) Where E = modulus of Elasticit = Poisson Ratio The Strain Energ U can be written in terms of curvature b substituting the respective values of stresses and Strains of equations (a-c) and (a-c) into equation () and simplifing to obtain z w w w w w U dz. ] dd _ () Integrating the first term of Equation over the entire thickness of the surface from h to h and simplifing, we obtain D w w w w w U. dd _ (5) Where D = Fletural Rigidit = E h ( ) In the present contet, the plate is acted upon b uniforml distributed transverse load. Therefore, the eternal work W e = q w (, ) d d. (6 ) Therefore total potential Energ = U W e t (7 a ) = D w w w w w. q dd _ (7 b) METHODOLOGY The Direct variational method adopted here is Ritz which is based on minimum total potential Energ. Recall that the total potential Energ of plate from equation 7b is D w w w. w w qw (, ) _ (7 b ) Where W (,) is the plates deformation surface which is being approimated in this stud as a variable Separable polnomial as C ( ) ( ) C ( ) ( ) C ( ). ( )... C ( ). ( ) (8) n n n where,,..., and,,... are constructed co ordinate functionsin aes respectivel. n n ( ) is derviable from ( ) b replacing b and a b b. Equation (8) could be simplified further b putting h ( ). ( ), h ( ). ( ) h ( ). ( )... h ( ). ( ) (9) n n n Substituting equation () into equation (8), the deformamtion surface of the plate could now be written as W (, ) C h C h C h C h. W (, ) HC 0 Where H = [h h h... h n ] C = [C C C... C n ]. The functions of H polnomial of equation (0) must satisf the kinematic boundar conditions and are linearl independent and continuous. These functions 87

3 Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): 86-85(ISSN: -706) of equation (0) are subsequentl substituted into the total potential energ equation of (7b) above and on simplifing, becomes: D T T T T C H C C. C T T T T T C H C ( ) C H C C H q dd () T D C ( ) C () T W h e r e H d d T H d d H d d T H d d T B H. d d For the equilibrium condition of the plate under the transverse loading to be maintained the total potential Energ will be minimum. ie 0, i,,, () C i D ( ) ( ) T C q B a C q C C q ( ) ( ) C q D b C q w h ere q q b ( c ) b b and B b b i j ( d) Figure ll round clamped thin rectangular plate subjected to uniforml distributed Load q. First pproimation (one- term of h polnomial) For this approimation, the deformation function representing the deflection surface of the plate is (using onl the first term of equation (0) W (, ) = Where C = unknown coefficient h = ( ) ( ) C h - (6) ( ) Where (6 ) a a a a ( ) (6 ) b b b b b Section B - B, Having determined the unknown coefficients, C, C C, and C of equation (b), the coefficients are substituted into equation (0) to obtain the deformation surface of the plate in analtical form. Subsequentl the deflection and moments on an arbitrar point on the plate can be obtained using the following equations. W (, ) H C (5 a ) w w M D (5 b) w w M D (5 c) w M D. (5 d ) NLYSIS OF THE PLTE ND RESULTS Considering the isotropic thin rectangular plate with all sides clamped as shown in figure () below: Therefore, the deflection surface will be approimated b the equation: (, ). (7) W C a a a b b b ssuming a square plate, (a=b) the equation (7) is substituted into equation (7b), the results is integrated over the entire surface area of plate via equation (). The integrand is minimized and solved via equation, to obtain the unknown coefficient C =.0. The determined coefficient is substituted into equation (6) to obtain the deformation surface of the plate in analtical form as: W(, ).0 (7 a) a a a b b b The deflection at an point on the plate (, ) is determined b substituting the values for an arbitrar point (, ) into equation (7a). Moments at 88

4 Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): 86-85(ISSN: -706) similar point(, ) are obtained b putting equation (7a) into respective equations (5b-c). Similarl, the results of the deflections and moments are determined b substituting the values of (, ) at an arbitrar point for various plate aspect ratios using equations (5a-d). The results are presented respectivel on tables and. The accurac of the results obtainable from this first approimation is poor although the response pattern of the deformation surface is good. For maimum deflection values displaed in table for plate aspect ratios of.0 c b.0 ; the errors range a from nil for c =.0 to about.8% for c =.0. s would be epected, the deflection is evaluated with higher accurac than the moment field M, M, M which, in turn is evaluated to greater level of accurac than the shear-force field Q, Q, q, q. This is because of the fact that the stress couples and the transverse Shear forces are proportional to the second and third derivatives of the displacement function respectivel. Thus for plates having aspect ratio.0 c.0, the errors in maimum bending moment (M ) at a, b Table range from.6% (c =.6) to 0.08% at (c=.). For maimum bending moment along long span (M ), the errors range form 9.8% (at c =.0) to 58.5% (at c =.0). The edge (negative) moments have errors ranging from 0.5% (at c =.50) to 6.77% (at c =.0). Second pproimation (- terms of h polnomial) In the second approimation, the deformation function describing the surface of the plate will be represented b terms of h polnomial of equation (0) W(, ) C h C h. (9) where h ( ). ( ) h ( ). ( ) b u t ( ) ( ) ; ( ) ( ) Therefore, W(, ) C. a a a b b b C a a a b b b. (0) Similarl, equation (0) is substituted into equation (7b), integrated via equation () and the integrand minimized and solved via equations () to obtain the coefficients C =5.57, and C =-.0. Therefore, in analtical form, the deflection surface of the plate for -term polnomial becomes, 89 W(, ) a a a b b b.0. (0 a) a a a b b b The respective deflections at the centre of the plate and moments at the centre of span and edges are obtained for various plate aspect ratios b using equations (5a-d). The results are similarl presented on tables,,,, and 5. The two- term polnomial solution ields better accurate results than preceding one term polnomial. From table, it is observed that for plate aspect ratios.0 c.0, the errors in determined deflections range from 0.05% (at c =.0) to.5% (at c =.5). Evidentl, the evaluated results of bending moments (M, M ) and transverse Shear forces corresponding to.0 c.0 (Table and ) are not as accurate as the displacement counterpart, although there is highl improve results over the corresponding one-term polnomial. s shown in tables and, and for aspect ratios considered (.0 c.0) the errors of Short span moment (M ) range form Nil (at c =.9) to.99% (at.0) while that of long span moment range form 0.98% (c =.5) to 9.9% (at =.0). There is observed improvement in the accurac of the results. The edge negative moments, errors range form.0% (c =.0) to 9.0% (c =.0) and Nil (c =.) to 0.% (at c =.0) for short span (M ) and long span (M ) respectivel. Third pproimation (-terms of h polnomial) The deformation surface of the plate for the third approimation will be represented b the -term polnomial of equation (0) as: W(, ) Ch C h Ch. () where h ( ) ( ) h h ( ). ( ). W (, ) C.. a a a a b b a a a b C C 5. 5 () a b b a a a a a a a a In the similar wa of section (.), the unknown coefficients of C, C, and C are evaluated to be 9.76, -.6, and -.6 respectivel. The deformation surface of the pate in analtical form becomes: (, ) W a a a a b b a a a b () b b b a a a a a a a a Similarl, the respective deflections and moments are evaluated and the results presented on tables,,,, and 5.

5 Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): 86-85(ISSN: -706) Higher accurate results are obtained with the term polnomial. For deflection, the maimum errors range from Nil (at c =.0, through c =.0) to.5% (at c =.0). The improvement in the accurac of results in equall observed with the moment field (M, M ). s depicted in table, the errors of moment range from 0.76% (at c =.0) to.6% (at c =.0). s evidence in table, the long span moments have errors ranging from.60% (c =.0) to.66% (at c =.0). For edge moments (M, M ), errors of short span solutions (M ) range from.56% (c =.0) to maimum of.69% (c =.60). While long span (M ) range from 0.5% (c =.0) to 0.% (c =.0). Fourth pproimation (four terms of h polnomials) For the fourth approimation using four terms of h- polnomial, the functional representing the deformation surface of the plate is written as: - W(, ) Ch C h C h C h. () where h h ( ) ( ) ( ). ( ) h h ( ). ( ) ( ). ( ) Finall, the four terms polnomial of equation () is substituted into potential energ equation (0), integrated, minimized, and solved to obtain coefficients. Similar analses are done for deflections and moments as in the proceeding sections to obtain deflections at centre and, mid span and edge moments. The results are shown on tables, and. s indicated in the -term polnomial, the th pproimation solutions do not show marked difference from the rd thus indicating full convergence was obtained with -term h-polnomial for deflections, and all the moments fields M, M, M, M thus elucidating the validit and reliabilit of the variational procedure. Table -Maimum deflection coefficients ( ) of isotropic all round clamped rectangular plate under uniforml distributed load for various plate aspect ratios (ϑ=0.0). Span ratio = (b/a) Classical method Deflection(W ma) = qa : at = a/, = b/ D Present Stud term of h terms of h terms of h terms of h (0.05%) 0.000(0.05%) 0.006(Nil) 0.006(Nil) (6%) 0.005(.67%) (Nil) (Nil) (5.8%) (.7%) 0.007(Nil) 0.007(Nil) (5.76%) 0.009(.57%) (Nil) (Nil) (6.8%) (0.97%) (-0.97%) (-0.97%) (6.8%) 0.000(.5%) 0.008(-0.9%) 0.008(-0.9%) (7.8%) 0.00(0.%) 0.008(-0.87%) (8.8%) 0.008(Nil) 0.007(-0.%) 0.007(-0.%) (9.80%) 0.00(-0.%) 0.000(-0.%) 0.000(-.0%) (.%) 0.008(-0.0%) 0.007(-0.80%) 0.007(-0.80%) (.8%) 0.005(-.8%) (-.57%) (-.57%) Table -Maimum short span moments coefficients (β) in all round clamped isotropic rectangular plates under uniform load for various plate aspect ratios ( =0.0). Short span moment (M ma) = qa : at = a/, = b/ Classical method Present Stud term of h terms of h terms of h terms of h (9.05%) 0.06(.99%) 0.05(-.60%) 0.05(-.60%) (0.08%) 0.096(.%) 0.06(-0.76%) 0.055(-0.76%) (7.7%) 0.06(9.0%) 0.09(-.67%) 0.098(-0.%) (6.5%) 0.09(6.7%) 0.00(-.%) 0.05(-.67%) (6.5%) 0.068(5.%) 0.0(-%) 0.0(-.58%) (6.%) 0.08(.80%) 0.058(-.7%) 0.05(-.08%) (6.0%) 0.09(.5%) 0.07(-.6%) (6.5%) 0.000(.0%) 0.080(-.06%) 0.087(-.8%) (6.8%) 0.005(.0%) 0.087(-.9%) 0.055(-.7%) (7.9%) 0.007(nil) 0.09(-.9%) 0.058(-.0%) (8.5%) 0.009(-0.7%) 0.09(-.6%) 0.095(-.%).The values in the bracket indicate the % deviation of the present stud from the classical solution 850

6 Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): 86-85(ISSN: -706) Table -Maimum long span moments coefficients (β ) in all round clamped rectangular plates under uniform load for various plate aspect ratios ( =0.0). Long span moment (M ma) = qa ; at = a/, = b/ Classical method Present Stud term of h terms of h terms of h terms of h (9.8%) 0.00(.90%) 0.05(-.60%) 0.05(-.60%) (.8%) 0.00(.90%) 0.06(-.6%) 0.00(-.76%) (.%) 0.05(.07%) 0.0(-.65%) 0.06(-0.88%) (0.8%) 0.06(.80%) 0.0(-.6%) 0.0(-.95%) (.9%) 0.0(0.9%) 0.00(-.77%) 0.098(-6.60%) (7.99%) 0.00(-0.98%) 0.09(-5.%) 0.089(-6.90%) (.8%) 0.086(-.6%) 0.08(-6.%) (7.5%) 0.076(-5.75%) 0.069(-7.%) 0.07(-.9%) (50.57%) 0.06(-7.7%) 0.058(-9.9%) 0.0(-8.96%) (55.5%) 0.05(-9.9%) 0.07(-0.9%) 0.00(-.%) (58.5%) (-.66%) 0.09(-.0%) Table -Maimum short span edge moments coefficients ( ) in all round clamped isotropic rectangular plates under uniform load for various plate aspect ratios ( =0.0). Short span moment (M ma) = qa : at = a/, = b/ Classical method Present Stud term of h terms of h terms of h terms of h (-7.5%) -0.05(-9.0%) -0.05(.56%) (.7%) (-.7%) -0.09(-5.%) (.89%) (.99%) (-6.77%) (-.%) (.9%) -0.06(.%) (-5.68%) (-9.90%) (.7%) (.80%) (-.89%) (-7.99%) -0.07(.7%) (6.0%) (-0.0%) (-6.7%) (.5%) (0.0%) (.9%) (-5.7%) (.69%) (.88%) (-.5%) (.50%) (-0.%) (5.9%) (-.8%) -0.08(.%) -0.08(.%) (7.78%) (-.8%) (.(%) -0.09(-.7%) (9.%) (-.0%) -0.08(.8%) -0.08(.5%.The values in the bracket indicate the % deviation of the present stud from the classical solution. " Table 5-Maimum long span edge moments coefficients ( ) in all round clamped isotropic rectangular plates under uniform load for various plate aspect ratios ( =0.0). Classical method Long span moment (M ma) = " " qa at = a/, = b/ : Present Stud term of h terms of h terms of h terms of h (-7.5%) -0.05(.90%) -0.05(.56%) (.6%) (-.%) (-.97%) -0.05(-.%) (0.9%) (5.05%) (.0%) (0.5%) ((-0.7%) (-.97%) (.07%) (-0.8%) (.%) (-6.9%) (nil) (-.06%) (.06%) (-.%) (-.%) (-.95%) (-0.88%) (-5.%) (-.65%) (-.5%) (-9.7% (-.0%) -0.05(-.7%) (-5.78%) (-5.90%) (-5.95%) -0.05(-6.8%) (-.7%) (-57.09%) (-7.88%) -0.05(-8.5%) (-.85%) (-60.%) -0.05(-0%) -0.05(-0.%) -0.05(-0.5%). The values in the bracket indicate the % deviation of the present stud from the classical solution. 85

7 Journal of Emerging Trends in Engineering and pplied Sciences (JETES) (5): 86-85(ISSN: -706) CONCLUSION The well-known Ritz mathematical and variational method was developed and successfull applied in the analsis of uniforml loaded clamped isotropic rectangular plate. The tedious, time-consuming and rigorous computations have been circumvented using Quick Basic programming language. The smbolic variational approach appears simple, acceptable and understable b an Civil/Structural Engineer.The results obtained with the present stud compare favourable with the classical solution (Timonshenko and Woinowsk-Kreer - 959). In the course of this stud, several methods of analses especiall the numerical methods were reviewed etensivl. The most widel accepted classical method though acknowledge as satisfactor for most engineering problems are usuall ver difficult and rigorous. The modelled Ritz variational method formulated using total potential energ principle circumvents the rigorous procedure inherent in the classical solution through minization principle. The results of this procedure are obtained in analtical form thus, enabling the determination of deflection and moments at an arbitrar point of the plate unlike numberical methods that give results onl at the nodal point. lso the results of moments at the clamped position compare favourabl with the classical and numerical solution. This developed model can convenientl applied to thin rectangular of variable thickness and is therefore recommended for certain problems on plate analsis that do not have eact solution or their eact solutions are too complicated to be obtained analticall. REFERENCES Beltzer,. I., (990): Engineering nalsis via Smbolic Computation- a breakthrough, pplied Mechanical Revolution Vol. : pp 9-7 De, S.S. (00): Semi-Numerical nalsis of Rectangular Plates in Bending, Computers and Structures Vol. Issue 56 pp Hutchinson, J. R., (99): On the Bending of Rectangular Plates with two opposite edges simpl supported, Journal of pplied Mechanics Trans. SME 59: pp Kaneda, K, Takamine, T. Chinen, Y. and Iraha, S. (00): n naltical stud and pplication of Earth Pressures loaded rectangular plates in the condition of two adjacent sides fied and the other sides free, Journal of structural Engineering, Vol. 50: pp Mbakogu, F. C., and Pavlovic, M. N. (000): Bending of Clamped orthotropic rectangular Plates: variational smbolic Solution, Computers & structures, Vol.77: pp.7-8. Szilard, S. (97): Theor and nalsis of Plate, Prentice Hall, Eaglewood, New Jerse. Szilard, R. (00): Theories and pplication of Plate nalsis: Classical Numerical and Engineering Methods; John Wile and Sons, New York. Talor, R. L. and Govindjee, S. (00): Solution of Clamped Rectangular Plate Problems, Commnications in Numerical methods in Engineering, 0: pp Timoshenko, S.P. (98): Bending of Rectangular Plates with Clamped edges, Proceedings of 5 th International pplied Mechanics, MIT-US, pp. 0-. Timonsheko, S.P. and Woinowsk-Kreiger, S. (959): Theor of Plates and Shells nd Ed., McGraw- Hill, New York. Wang, D. and Sheik,. (005): Large Deflections, Moments and Shears for uniforml Loaded Rectangular Plate with Camped Edge, J. pplied Mechanics Trans. SME, pp Wang, C, M., Wang, Y. C. and Redd, J. N. (00): Problems and Remed for the Ritz method in determining stress resultant of corner supported rectangular plates, Computers & Structures, Vol. 80: pp 5-5. Wojtaszak, I.. (97): The calculation of maimum deflections, moments and shears for uniforml loaded plate for clamped edge, Journal of pplied Mechanics, Trans. SME 7, pp. 9- Imrak, C. E. and Gerdemeli, I. (007): n Eact Solution for the Deflection of Clamped Rectangular Plate under Uniform Load, pplied Mathematical Sciences, Vol. no. pp 9-7. Imrak, C. E. and Gerdemeli, I. (007): The Problem of Isotropic Rectangular plate with four Clamped edges, Indian cadem of Science SDHN Vol. part, pp