FLEXURAL ANALYSIS OF SIMPLY SUPPORTED RECTANGULAR MINDLIN PLATES UNDER BISINUSOIDAL TRANSVERSE LOAD

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1 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om FLEXURAL ANALYSIS OF SIMPLY SUPPORTE RECTANGULAR MINLIN PLATES UNER BISINUSOIAL TRANSVERSE LOA Clifford Ugouku Noji, Benjamin Okudili Mama, Hyginus Nanko Ona and Carles Cinua Ike epartment of Civil Engineering University of Nigeria, Nsukka, Enugu State, Nigeria epartment of Civil Engineering Enugu State University of Siene & Tenology, Enugu State, Nigeria ABSTRACT Tis study onsidered te solution of te system of oupled partial differential euations of euilirium of retangular Mindlin plates sujeted to isinusoidal transverse load over te entire plate domain. Te metod of undetermined parameters as used. Solutions ere assumed in terms of unknon parameters for te unknon displaements in te governing euation tat apriori satisfied all te simply supported oundary onditions on te plate edges. Te unknon parameters ere otained y solving a system of algerai euations. Te in plane displaement, transverse displaement, inplane stresses and transverse stresses ere otained from te determined parameters using te euations of Mindlin plate teory. Comparison of te results of tis study it te literature results sos agreement it te previous ork of Reddy and Pagano. It as also found tat for tin plates, te results agree remarkaly ell it lassial plate teory. Keyords: mindlin plate teory, lassial plate teory, metod of undetermined parameters, isinusoidal transverse load.. INTROUCTION Plate prolems are enountered freuently in engineering appliations like ridge deks, floor slas, retaining alls, raft foundations, airraft and spaeraft panels. In general, plates are lassified into tree major ategories - tin plates, moderately tik plates and tik plates - depending upon te ratio of tikness to te governing in plane dimension of te plate. Tey an e furter ategorized ased on te nature of te deformation as small-deformation plates or large deformation plates; and te material properties as anisotropi or isotropi, omogeneous or non-omogeneous [, ]. Te plate prolem in general, is a tree dimensional elastiity prolem tat is solved y solving te differential euations of elastiity teory []. Solutions are impossile even in most simple prolems of plate geometry and loading. Hene, simplifiations are made, yielding te various teories or models of plates. Kiroff developed te first omplete teory of plate flexure. Based on Bernoulli s ypotesis, for tin eams, Kiroff derived te same partial differential euation of euilirium for plate ending as otained y Navier, oever using a different energy metod [, 5, 6]. Kiroff plate teory also alled te lassial plate teory (CPT) or te Kiroff - Love plate teory is a linear infinitesimal teory. Te assumptions of te CPT are [7]: a) points on te plate lying initially on a normal to te plate middle surfae remain on te normal to te middle surfae even after ending deformation. ) te normal stresses in te transverse diretion an e negleted i.e. zz 0, xz yz 0 ) te middle surfae of te plate is free of deformation, and remains neutral during ending. Flexure and streting i are te to dominant eaviours of plates are unoupled in te CPT, ut an e analysed togeter y linear addition [8]. Tis is a merit of te CPT. Anoter merit of te CPT is tat te resulting governing euation i is a fourt order partial differential euation is linear and as only one unknon; te transverse displaement (x, y). espite te merits of te CPT, it as te folloing limitations: a) CPT is appliale to tin plates it small defletions ere te sear stress an e disregarded. For moderately tik plates, sear stress ontriution must e aounted for and te CPT eomes inadeuate. Also, for large defletions, inplane stresses need to e inorporated yielding non-linear euations ) Solutions for transverse sear stresses are inonsistent it te teory of elastiity results ) CPT presents ompliations in te appliation to finite element analysis and oter omputer appliations [].. Resear aim and ojetives Te resear aim is te analysis of simply supported retangular Mindlin plates under isinusoidal transverse loads. Te ojetives are: a) to employ te metod of undetermined parameters to solve te governing partial differential euations of retangular Mindlin plates it simply supported edges, and under isinusoidal transverse loads. 80

2 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om ) to otain maximum values of inplane displaements and stresses xx, xz, xy for various ratios of plate tikness to least inplane dimension for suare Mindlin plates.. THEORETICAL FRAMEWORK. Governing euations of te Mindlin plate teory Te governing partial differential euations of euilirium of retangular Mindlin plates under transverse distriuted loads (x, y) are te folloing system of tree oupled partial differential euations in terms of tree xy and ( xy, ). [9] unknon variales (x, y), (, ) x x y x y y () x y Mxx (7) x y y x Myy (8) y x. RESULTS. Solution to te governing euations for Bisinusoidal transverse loads A retangular Mindlin plate lying on te domain 0 x a, 0 y, z as son in Figure- is onsidered in tis study. x x y 0 x x y xy x () y y x y 0 y x xy y () Figure-. Retangular Mindlin plate. ere (x, y, z = 0) = (x, y) is te transverse defletion of te plate, x and y are te rotations aout te x and y axes of lines normal to te middle plane efore deformation, is te Poisson s ratio of te plate material, is te plate flexural rigidity, is te searing stiffness of te plate, x, and y are te in-plane Cartesian oordinate variales, z is te transverse Cartesian oordinate variale and is te Laplaian in to dimensions, given y x y Te plate flexural rigidity, is given y: E (5) ( ) () Te plate is sujeted to te ation of isinusoidal transverse load (x, y) given generally y te Fourier doule Sine series: mx ny ( x, y) (9) ere are te Fourier oeffiients of te load distriution. By te Fourier series teory, a mx ny ( x, y) dxdy a (0) 00 Te sket of te distriution of te isinusoidal load on te plate domain is son in Figure-. ere E is te Young s modulus of elastiity, is te plate tikness Te searing stiffness of te plate is given y: = kg (6) ere k is te sear orretion fator, and G is te sear modulus. Te internal ending fores are given y: 8

3 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om mx ny ( x, y) x () mx ny ( x, y) A os sin () y mx ny ( x, y) B sin os (5) Figure-. istriution of isinusoidal load on te plate domain. Te prolem is ten to solve te system of Euations (-) for te ase of distriuted transverse load given y Euation (9) for simply supported edges x = 0, x = a, and y = 0, y =. Te oundary onditions are: Mxx ( x 0, y ) 0 () y( x 0, y ) 0 () ( x 0, y) 0 () Mxx ( x a, y ) 0 () y( x a, y ) 0 (5) ( x a, y) 0 (6) Myy ( x, y 0) 0 (7) ( xy, 0) 0 (8) x ( x, y 0) 0 (9) Myy ( x, y ) 0 (0) x( x, y ) 0 () ( x, y ) 0 () Te defletion funtions tat satisfy all te oundary onditions Euations (-) at te edges x = 0, x = a, y = 0, y = are given in terms of unknon generalized displaement parameters, and te sape funtions, as: ere, A, and B are te unknon generalized displaement parameters tat are sougt in te metod of undetermined parameters and m, and n are integers. m =,,, n =,,, Applying te metod of undetermined parameters, Euation (-5) and (9) are sustituted into Euations (-) to otain onditions for te unknon generalized parameters to e solutions to te prolem. Tus, m n m n mx ny mx ny A os sin x mx ny B sin os y mx ny (6) mx ny mx ny A os sin A os sin x m n y m n mx ny mx ny B sin os A os sin x y m n m n x m x n y 0 (7) mx ny mx ny B sin os B sin os y m n x m n mx ny mx ny A os sin B sin os x y m n m n y m x n y 0 (8) 8

4 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om Simplifiation of Euations (6-8) yield: m n m n mx ny A B mx ny (9) m n A a m n m mx ny B os sin 0 (0) a n m B a m n n mx ny A sin os 0 () From Euations (9-), e otain te system of tree algerai euations: m n m n A B m m n m n A B 0 a n m n n m A B 0 a a Solving Euations () (), e otain: A m n a m n a m a m n a () () () (5) (6) B n m n a (7) Te transverse defletions are ten otained as follos: m n (8) mx ny ( x, y) m n m n a ( x, y) mx ny mx ny a m n m n m n m n K (9) ( x, y) ( x, y) ( x, y) (0) ere K ( x, y) ( x, y) mx ny () m n m n a () m n m n mx ny a K (x, y) is te Kiroff tin plate solution for defletion of plates under isinusoidal load and (x, y) is te additional defletion due to te effet of sear deformation. Also, te rotations are: x m mx ny os sin a () m n m n a 8

5 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om y n mx ny sin os () m n m n a. Bending moments Using Euations (7) and (8), te ending moments are found to e: M xx m n m m mx ny a a m n a r a (50) For suare Mindlin plates, r =, at te enter, x = a/, y = / and te enter defletion is: ( ) k (5) (5) (5) n n sin m x sin n y m n M xx m n m n mx ny m n a (5) (6) 0 6( ) k 0 k( ) For k = 5/6, ( 0.) (5) (55) (56) M yy m n n m mx ny a m n a (7) 0 For 7 (57) x y. Case of load ( x, y) 0 For tis ase, m =, n = and x y ( x, y) 0 m n a (8) (58) (59) (60) ( x, y) 0 sin x sin y sin x sin y 0 (9) m n ( r ) ( r ) ere r is te plate aspet ratio 0 0 K (6) 0 0 s (6) 8

6 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om (6) Te results of enter defletion of Mindlin plate and Kiroff plate for values of / ranging from 0 to are son in Tale-. Tale-. Center defletion of Mindlin plate and Kiroff plate under isinusoidal load x y 0. 0 K 0 s 0 m % ifferene Alternative ay of presentation of results Te defletion at te plate enter is: 0 ( ) (6) E ( ) k 0 ( ) ( ) E ( ) k (65) 0 E E E 56 E (7) (7) (7) 0 ( ) E k 0 ( ) E k For E E For, E (66) (67) (68) (69) (70) (7) E For E E E u x (75) (76) (77) (78) 85

7 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om u a 0 a 0 0 u a a r 0 0 ( a) u a ( ) 0 u a ( r ) r r 0 ( ) u ( r ) a E 0 6( ) r u E ( r ) 0 6( ) r u E ( r ) For 0., r = (79) (80) (8) (8) (8) (8) (85) u E 0 u 0.00 E 0 u.0 0 E (86) (87) (88) Te maximum stresses at z = 0, and z are alulated using Ez x y xx x y Ez y x yy y x Ez x y xy ( ) y x E zx G x x x ( ) x (89) (90) (9) (9) Te results for te displaements u,, and te maximum stresses xx, xz, xy at z = 0, z for te present study are presented in Tale togeter it te results for te same prolem of tik plates under isinusoidal load solved y Pagano and Reddy. 86

8 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om Tale-. Inplane displaement ux 0, y, z, transverse displaement x y z in-plane normal stress a,, and transverse sear stress 0,, 0 a 0 Plate teory used xx x y z in isotropi suare plate sujeted to isinusoidal load. Plate model used u ( 0 /E),, 0, zx x y z ( 0 /E) xx xy ( 0 ) zx ( 0 ) Mindlin (present study) FSPT Reddy [7] HSPT Pagano [8]) Elastiity teory Kiroff CPT Mindlin (present study) FSPT Reddy [7] HSPT Pagano [8] Elastiity teory Kiroff CPT ISCUSSIONS In tis ork, te prolem of flexural analysis of retangular moderately tik plates modeled using te Mindlin plate teory, under transverse isinusoidal load distriution on te entire plate domain (0 x a, 0 y ) as een solved for te ase of simply supported edges (x = 0, x = a, y = 0, y = ). Te metod of undetermined parameters as used to solve te system of tree oupled partial differential euations in terms of te tree unknon displaements (x, y), x( xy, ) and y( xy, ). Trigonometri asis (oordinate) funtions tat satisfy apriori all te oundary onditions of te simply supported edges ere used in terms of undetermined parameters, A and B. Te asis (oordinate) funtions ere onstruted to satisfy all te oundary onditions on te simply supported plate edges, and te funtions represented y Euations () - (5) eame te trial funtions of te prolem for i te solutions for te unknon parameters, A and B ere sougt y sustitution of te euations into te governing PE of te Mindlin plate in order to find te onditions for i te unknon parameters ould yield a solution. A system of tree algerai euations ere ten otained as Euations () - (). Solution of te system of algerai euations yielded te unknon displaement parameters as Euations (5) - (7). Te trial displaement fields or displaement funtions ere ten ompletely determined as Euations (9), () and () for te general ase of aritrary distriuted load. Te solution as ten applied to te partiular ase of isinusoidal transverse load to otain te expression for te enter defletion as Euation (55) or Euation (57) for te ase k and te en te sear orretion fator, 5 6 Poisson s ratio, 0.0. Similarly, te expression for te inplane displaement u at z as otained as Euation (85). Normal and sear stresses ere alulated using Euations (89) - (9) and teir maximum values ere son taulated in Tale- for te ratios of a Te variation of te enter a 0. and 0.5. defletion of te plate it te plate tikness to governing span ratio (/a) for suare plates as omputed and son in Tale ; i also displays orresponding results for te lassial Kiroff-Love plate teory. Tale sos tat for i falls itin te a 0 ategorization of tin plate, te relative differene in te results of te Mindlin plate teory of te present ork and te lassial Kiroff plate teory is a negligile 0.%. For te Mindlin plate enter defletion a 0 exeeds te enter defletion predited y te lassial tin plate teory of Kiroff y 5.6%. For, a te enter defletion of te Mindlin plate is greater tan tat of te Kiroff plate y 6.66%; illustrating te signifiant role of sear deformation as te ratio /a inreases and te plate eomes tiker. Tale- also ompares te results of te present study it earlier studies done y Reddy, and Pagano. Te results displayed in Tale- sos tat te present ork agrees it te results otained previously y Reddy o used a Higer Order sear deformation plate teory (HSPT) and y Pagano o used te metod of te teory of elastiity. 5. CONCLUSIONS Te onlusions of tis study are as follos: a) te metod of undetermined parameters an e suessfully applied to te flexure analysis of moderately tik plates it simply supported edges, under isinusoidal transverse loads, provided te asis funtions of te trial displaement fields are made to 87

9 VOL., NO. 5, AUGUST 08 ISSN ARPN Journal of Engineering and Applied Sienes Asian Resear Pulising Netork (ARPN). All rigts reserved..arpnjournals.om satisfy all te oundary onditions at te simply supported edges. ) te metod simplifies te omputation of te oundary value prolem of solving a system of tree oupled partial differential euations to an algerai prolem of simultaneous euations in terms of te undetermined displaement parameters. ) te lassial Kiroff plate teory i disregards te ontriutions of sear deformation under predits te enter defletion en te tikness-governing span ratio (/a) exeeds /0; and for su ases is not a suitale teory for moderately tik plates. [6] Onarte E. 0. Strutural Analysis it te Finite Element Metod. Springer, Neterlands. [7] Reddy J.N. 00. Meanis of Laminated and Composite Plates and Sell Teory and Analysis, nd Edition. CRC Press, Boa Raton F.L. [8] Pagano N.J Exat Solutions for Bi-diretional Composite and Sandi Plates. Journal of Composite Materials. : 0-. [9] Mindlin R Influene of Rotary Inertia and Sear on Flexural Motions of Isotropi, Elasti Plates. ASME. Journal of Applied Meanis. 8: -8. d) te results for te transverse defletion at te enter of te plate agree remarkaly ell it te results of te same prolem solved y Reddy using te Higer order sear deformation plate teory and y Pagano using teory of elastiity. e) for, te transverse defletion of te plate a 0 enter for Mindlin plate teory eome signifiantly similar to te results otained y Kiroff plate teory. REFERENCES [] Gugal Y.M. and Sayyad A.S. 00. Free Viration of Tik Ortotropi Plates using Trigonometri Sear eformation Teory. Latin Amerian Journal of Solids and Strutures. 8: 9-. [] Gugal Y.M. and Paar M.. 0. Bukling and Viration of Plates y Hyperoli Sear eformation Teory. Journal of Aerospae Engineering and Tenology. (): -. [] Cantaraiit P., Kongtong P., Sompornjaroensuk Y., Viootjak J. 05. Note on te Matematial evelopment of Plate Teories. Advaned Studies in Teoretial Pysis. 9(): 7-55, Hikari Ltd. [] Kumar S.K. 0. Analysis of Composite Plates using Element Free Galerkin Metod. Master of Tenology (Strutural Engineering) Tesis, National Institute of Tenology Bourkela. [5] Hosseini S.B. 06. Implementation and Analysis of Error Estimates for p MITC Finite Elements of Reissner-Mindlin Plates. Master of Siene in Tenology Tesis, Sool of Engineering Aalto University. 88

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