An Exact Solution for the Deflection of a Clamped Rectangular Plate under Uniform Load

Transcription

1 Applied Matheatical Science, Vol. 1, 007, no. 3, An Exact Solution for the Deflection of a Claped Rectangular Plate under Unifor Load C.E. İrak and İ. Gerdeeli Itanbul Technical Univerity Faculty of Mechanical Engineering Mech. Eng. Dept., Guuuyu TR-337, Itanbul, Turkey irak@itu.edu.tr Abtract Rectangular plate under unifor load, x = ±a, y = ±b, are conidered. An exact olution i preented in which each ter of the erie i trigonoetric and hyperbolic, and identically atifie the boundary condition on all four edge. The olution ha three ter in which the firt ter correpond to the cae of a trip and the other two ter denote the effect of the edge. The ethod ued to obtain the olution i iple and traightforward. In order to illutrate the ethod the nuerical value of the deflection are calculated and copared with thoe of the previou paper. It i found that there i a reaonable agreeent between the reult of the and thoe of thi paper. Matheatic Subject Claification: 7K0, A3, 35L0 Keyword: Rectangular plate; claped plate; deflection; unifor load 1 Introduction Thin plate are coon tructural eleent eployed in any engineering application and are ubject to a wide variety of excitation,

2 130 C.E. İrak and İ. Gerdeeli including acoutic excitation. The proble of a rectangular plate claped on four ide and carrying a uniforly ditributed load i of great iportance and any paper have been devoted to thi ubject. Many author have calculated the deflection of uniforly loaded rectangular plate with claped edge uing different ethod. Soe of the are approxiate ethod. The accuracy of the analytic olution copiled and developed in the literature varie: thoe for iply upported plate are exact, other approxiate. For fixed rectangular thin plate no accurate reult appear to be available. Approxiate olution were alo uggeted, but thee reulted in a notable lo of accuracy [1]. Many author have calculated the deflection of rectangular plate with variou upport uing different ethod [-9]. Soe of the are approxiate ethod. Two ain ethod of approach have been ued for obtaining the olution of the axiu deflection for fixed thin rectangular plate under unifor load; thee are the double coine erie and the uperpoition ethod a a generalization of Hencky' olution [10]. The proble of the uniforly loaded rectangular plate with fixed at all edge ha been olved by Hencky and independently by Boobnoff. Boobnoff ade exact calculation for everal apect ratio of the plate while Hencky ade refined calculation only for the cae of a quare plat [3]. Hutchinon ha ued the olution fro which wa preented in [11] and tabulated deflection for uniforly loaded rectangular plate. Obtaining the nuerical value of deflection for a rectangular plate ay be difficult. A ingle coine erie for rectangular fixed plate have been preented in [1, 13]. Thi paper analye the deflection of a rectangular fixed thin plate under uniforly ditributed load. In thi paper, a coprehenive ethod i preented for the nuerical olution of the fixed rectangular plate proble under uniforly ditributed load and boundary condition. A olution of the governing equation in ter of trigonoetric and hyperbolic function i given. The ethod i baed upon the claical coine erie expanion and found to be eaier and ore effective. By uing thi ethod, a rectangular plate having four edge fixed and ubjected to uniforly ditributed loading ha been odeled. For an iotropic plate with a Poion coefficientν = 0. 3, table giving the value of the deflection, each with a different value of the apect ratiob / a are preented. A nuerical ethod for dealing uniforly noral loaded rectangular plate i copared to the iilar nuerical technique ued in [5,9] and the reult have been copared with thoe in the literature. The reult how reaonable agreeent with the other available reult, but with a uch ipler and a ore practical approach.

3 Exact olution 131 Deflection of laterally loaded rectangular plate According to the claic theory of plate bending, a all deflection i defined a all copared with the plate thickne. The governing differential equation for iotropic hoogenou thin plate i DW ( xy, ) = pxy (, ) (.1) ( ) where W i the all deflection of the plate idurface, ( ) denote the bi-haronic operator and i Laplacian operator, p i intenity of lateral preure on the plate and D i the contant flexural rigidity ter of the aterial propertie, Young odulu of the aterial E and Poion ratio ν (taken equal to 0.3), and the plate thickne, h, 3 Eh D = (.) 1(1 ν ) The proble of a rectangular plate fixed at four ide with a unifor load i conidered. Taking the origin of coordinate at the center of the plate and x- and y-axe parallel, to the ide a and b of the plate, and it i uppoed throughout the paper that a b. The boundary condition for (.1) are obtained by requiring the olution to atify two precribed condition at each boundary point [1]. The boundary condition are cobination of W W = 0 and = 0 (.3) n along all edge. When a boundary point i fixed, W and W / n are taken to be zero there. A direct erie olution wa not obtained for the boundary condition of fixed edge. The olution to the hoogenou biharonic equation (.1), the known olution for the iply upported plate with the unifor loading giving the deflection function for the trip cae i cobined with that for a olution of deflection function which how the effect due to the edge W = W + W e. A ethod i for efficiently deterining a very large nuber of ter in the erie. The erie i given by π y co pb y (, ) 1 b π x x π x DW xy = E coh inh + + G b = 1,3... cohα b a b π x co a πy πy + H yinh btanhαcoh = 1,3... bcohα a a, (.) where α = π a/bi deflection coefficient. It i een that W ( x, y) given by Eq.(.) vanihe in the iddle of the plate and atifie Eq.(1). The firt ter give the deflection function for the trip cae in b y b. The

4 13 C.E. İrak and İ. Gerdeeli other how the effect due to the edge. The coefficient E, G and H in Eq.(.) ut be choen that the boundary condition (.3) are atified. Thee coefficient are to be deterined fro the condition that the lope at the boundarie i zero. Subtuting Eq.(.) in Eq.(.3), the following yte of linear equation for deterining the coefficient E, G and H are obtained. co b y E coh G inh 1 ( α + α) =, (.5) = 1,3,... π y cohα b α + tanh α π y ( 1) π Eα tanhα + G co H 1,3,... a = b = 1,3... b coh β = 1,3... πy πy yinh btanh β coh = 0, (.6) a a ( 1) π x x x Ecoh π G inh π + acohα b a b π x co a β H tanh β + 0 = = 1,3... b coh β, (.7) where α = π a/bi deflection coefficient and β = πa / b i oent coefficient. Thu, taking W = 0 we obtain tanh ( 1) E + G α = π π (.8) where G i found by expreing the lope at y = ± b in a coine erie of the for π b tanhα π E tanhα + G + G b a ( 1) π + H = 0, (.9) = 1,3,.. π π ba + b a Siilarly, for zero lope at x = a we find H i obtained in a erie of the for

5 Exact olution tanh α E G π ab ab ab + ( 1) + = 1,3... b a b a b a a β + H tanhα + 0 b = (.10) coh β After the uual procedure of deterining the coefficient of E, G and H a Fourier erie, the equation : E G tanh α ( 1) + = b αa Eα tanhα + G tanhα + a b H 3 3 = 1, π π, (.11) 3 α ( 1) + = 0, (.1) π b + a b( 1) b + / E G ( 1) tanh β = 1,3,.. a b = 1,3... a b a 8α α H tanh α +. coh α 1+ π where α = π a/bi deflection coefficient and β = πa / b i oent coefficient. a (.13) The nuerical calculation of the coefficient coplete the olution of Eq.(.1). Value of the deflection at the center point ay be deterined fro the known relation in the theory of plate. Thee equation are olved nuerically by neglecting all ter higher than a given order, which reult in a yte of iultaneou equation. The forula i valid for ot coonly ued etal aterial that have Poiion' ratio around 0.3. In

6 13 C.E. İrak and İ. Gerdeeli fact, the Poion' ratio ha a very liited effect on the diplaceent and the above calculation norally give a very good approxiation for ot practical cae. The increaing uage of flat plate in the contruction of panel in uch teel tructure a bridge and deck ha called attention to the need for ore inforation on the behavior of rectangular plate with uniforly ditributed load. The center deflection of rectangular plate with fixed at four edge and ubject to the action of uniforly ditributed load i an iportant proble that ha received coniderable attention becaue of it technical iportance. The value of the deflection at the point x = 0, y = 0 i very iportant and it i given in the following for pb E H tanh β W (0,0) = 1 D + = 1,3... cohα = 1,3... coh β (.1) Fro the boundary condition (.3), G = 0 for the center deflection of the rectangular plate. It i clearly een that there i no contribution of the coefficient G on the deflection. The erie in thi expreion converge very rapidly, and ufficient accuracy i obtained by taking only the firt ter. The expanion of the center deflection function of the rectangular plate i in a erie for and the econd ter of the erie i negligible and that by taking only the firt ter the forula for deflection i obtained. When the apect ratio goe to infinity, the center deflection tend to qb / D. Thi reult i in good agreeent with [5,9]. 3 Nuerical exaple and reult The proble to be conidered i a uniforly loaded rectangular plate with it edge fixed a in Figure 1. Thi proble wa olved uing a variety of nuerical ethod and copared to the exact olution.

7 Exact olution 135 Figure 1. A rectangular plate uniforly loaded and fixed at four edge. The nuerical value of the deflection can be calculated if the coefficient E, G and H are known for all value ofb / a and can be deterined by Eq.(.11), (.1) and (.13). The nuerical value of coefficient are obtained by a et of linear equation a given in [3]. Accuracy of the nuerical value of the depend on the nuber of linear equation. Coputation of the coefficient E, G and H were alo ade for b / a equal to 1.0, 1., 1., 1.6, 1.8,.0 and. Uing the value of the coefficient E, G and H, we proceed to find the axiu deflection in the plate under conideration. The deflection at the center i then found fro Eq.(.1) and ay be reduced to the for pb W xy = 0 = α, (.15) = 0 D In the calculation for the coefficientα, the value of Poion ratio ha been taken a ν = Fro Eq.(.15), Table 1 of coefficient for variou value of b / a of the ide of the plate wa copiled. Thi table aggre eentially with that copiled by Henkcy baed on hi ethod of olving Eq.(.1).

8 136 C.E. İrak and İ. Gerdeeli Table 1. Deflection for claped rectangular plate with unifor load ( ν = 0. 3, b a) b / a w(0,0) Evan Taylor & Govindjee pb / D pb / D pb / D It i oberved that the reult of the preented tudy are in good agreeent with other. The reult how that about twice the accuracy can be obtained. In coparion to the other ethod, preented ethod ha the advantage that the erie repreentation atifie the partial differential equation exactly. However, it i clearly een that the ethod in thi paper i iple and traightforward. The reult preented in thi paper can be copared with thoe of the previou paper. Concluion A ethod for treating uniforly loading on thin rectangular plate ha been preented. By uing thi ethod, a rectangular plate having four edge fixed and ubjected to uniforly ditributed loading ha been odeled, the reult of calculation for axiu deflection for everal ratio of the ide of the plate have been coputed, and the reult have been copared with thoe in the literature. The reult how cloe agreeent with other analyi ethod. However, it i clearly een that the ethod in thi paper i iple and traightforward. Acknowledgent The author expre their incere gratitude to Prof. M.Ein Erdoğan for guidance in thi work and would like to thank Profeor E. Minchev for hi tiulating correpondence. Reference [1] D. Wang and A.I. El-Sheikh, Large-deflection atheatical analyi of rectangular plate, Journal of Engineering Mechanic,131 (005)

9 Exact olution 137 [] H. Hencky, Der Spannungzutand in Rechteckigen Platten, Ph.D Thei, Dartadt, R. Oldenburg, Munich, Berlin, [3] I.A. Wojtazak, The calculation of axiu deflection, oent, and hear for uniforly loaded rectangular plate with claped edge, J. Appl. Mechanic, Tran. ASME, (1937), A173-A176., [] S.P. Tiohenko, Bending of rectangular plate with claped edge, Proceeding of the Fifth International Congre of Applied Mechanic, Maachuett Intitute of Technology, Cabridge, (1938), 0-3. [5] T.H. Evan, Table of oent and deflection for a rectangular plate fixed on all edge and carrying a uniforly ditributed load, J. Appl. Mechanic, Tran. ASME, 6 (1939), A7-A10. [6] D. Young, Analyi of claped rectangular plate, Journal of Applied Mechanic, Tran. ASME, 7 (190), A139-A1. [7] J.R. Hutchinon, On the bending of rectangular plate with two oppoite edge iply upported, J. Appl. Mechanic, Tran. ASME, 59 (199), [8] C.M. Wang, Y.C.Wang and J.N.Reddy, Proble and reedy for the Ritz ethod in deterining tre reultant of corner upported rectangular plate, Coputer and Structure, 80 (00), [9] R.L. Taylor and S. Govindjee, Solution of claped rectangular plate proble, Counication in Nuerical Method in Engineering, 0 (00), [10] R.Szilard, Theory and analyi of plate, Prentice-Hall, Englewood Cliff, New Jerey, 197. [11] S.P. Tiohenko and S. Woinowky-Krieger, Theory of Plate and Shell (nd edn.), McGraw-Hill, New York, [1] C.E. İrak and I. Gerdeeli, A nuerical ethod for claped thin rectangular plate carrying a uniforly ditributed load, Int. J. Appl. Mech. No.3 (007). (accepted for publication) [13] C.E.Irak and I. Gerdeeli, The proble of iotropic rectangular plate with four claped edge, Indian Acadey of Science SADHANA (007) (accepted for publication) [1] G. Burge and E. Mahajerin A nuerical ethod for laterally loaded thin plate, Coputer Method in Applied Mechanic and Engineering 9 (1985) Received: February 3, 007