Thermal Bending of Circular Plates for Non-axisymmetrical Problems

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1 Copyrigt 2 Tec Science Press SL, vol.4, no.2, pp.5-2, 2 Termal Bending of Circular Plates for Non-axisymmetrical Problems Dong Zengzu Peng Weiong and Li Suncai Abstract: Due to te complexity of termal elastic problems, analytic solutions ave be obtained only for some axisymmetrical problems and simply problems. Using te Green function, te boundary integral formula and natural boundary integral equation for te boundary value problems of biarmonic equation is obtained. Ten based on bending solutions to circular plates subjected to te non-axisymmetrical load, by te Fourier series and convolution formulae, te bending solutions under non-axisymmetrical termal conditions are gained. Te formulas for te solutions ave ig convergence velocity and computational accuracy, and te calculating process is simpler. Examples sow te discussed metods are effective. Keywords: termal bending problems, circular plate, boundary integral formula, natural boundary integral equation, Introduction Termal elastic problems are important one of solid mecanics. Due to te complexity of termal elastic problems, analytic solutions ave been obtained only for some axisymmetrical problems and simply problems. For general loads and general boundary conditions, te numerical computation is te main metod. For bending problems of solid circular plates, Fu Bao-lian adopted te reciprocal teorem and took te solution of te clamped circular plate as te basic solution to discuss some bending solutions under axis-symmetrical loads. Wang An-wen introduced te point source function to discuss te non-symmetrical bending problems under te concentrated force; Yu De-ao discussed bending problems of plates wit te natural boundary element metod. Using te above metod, Li Sun-cai discussed te bending problems of solid circular plates] and bending deflections for annular infinite plates under te boundary loads. On te basis of te same metod, expanding te boundary slope into Fourier series, and using several convolution formulae, te boundary integral formula and natural boundary integral equation for te,

2 6 Copyrigt 2 Tec Science Press SL, vol.4, no.2, pp.5-2, 2 boundary value problems of termal bending of Circular Plates are obtained. Te formula for te solutions as ig convergence velocity and computational accuracy, and te calculating process is simple. Examples sow te discussed metods are effective. 2 Boundary integral formula and natural boundary integral equation Te differential equation of elastic plate bending problems is: 2 u = q(r,θ) D = f (r,θ) () Were, is te Laplacian operator, u is te deflection of te plate, q is te surface density of external loads,d is te bending rigidity of te plate, is te plate in a circle domain. For convenient, suppose te circle is a unit circle. Using te Green formula of te bending problems for tin plates, we get: (u 2 v v 2 u)dp = Γ (u u v v v + n n n u v n u)ds + v f dp (2) Were dp = dxdy, Γ is te edge of te circular plate. Suppose u = u(p) satisfying te biarmonic equation, and letv = G(p, p ), wic is te Green function of te biarmonic equation in, and ten te Poisson integral equation of te bending problem of te plate can be found u(p) = Γ [ ] n G(p, p )u (p ) G(p, p )u n (p ) ds + G(p, p ) f (p )dp, p, (3) Were p = (x,y), p = (x,y ), u n = u n Γ, dp = dx dy, is te Laplacian related to p. Te Green function outer te unit circular domain can be obtained from te basic solution of te biarmonic equation G(p, p ) = 6π { [r 2 + r 2 2rr cos(θ θ ) ] ln r2 + r 2 2rr cos(θ θ } ) + r 2 r 2 2rr cos(θ θ ) + ( r2 )( r 2 ) (4)

3 Termal Bending of Circular Plates 7 Were, P and P represent te polar coordinate (r,θ) and (r,θ ) respectively. Tus G r = = ( r 2 ) 2 4π [ + r 2 2r cos(θ θ )] n G r = = ( r2 ) 2 [ r cos(θ θ )] 2π [ + r 2 2r cos(θ θ )] 2 Hence, te Poisson integral formula of te bending circular plates ( r2 ) 2 [ r cos(θ θ )] 2π [ + r 2 2r cos(θ θ )] 2 u ( r 2 ) 2 (θ) 4π [ + r 2 2r cos(θ θ )] u n(θ) + Were, is te convolution wit regard to θ, u (θ), u n (θ) denotes te deflection and slope at te edge. For te supported edge, u =, te above equation will be educed to ( r 2 ) 2 4π [ + r 2 2r cos(θ)] u n(θ) (5) Suppose M is te differential boundary operator in te polar coordinate system, te bending moment Mu Mu = [µ u + ( µ) 2 ] r 2 u = M r Γ D Were, µ is Poisson ratio. Let te boundary operator acts on Eq. (5), and use te limit formula of generalized function, te natural boundary integral equation of te bending problems [Li (22)] can be obtained as Mu = M + µ u n (θ) 2π sin 2 ( θ 2 ) u n(θ) (6) (7)

4 8 Copyrigt 2 Tec Science Press SL, vol.4, no.2, pp.5-2, 2 3 Termal elastic equation and boundary conditions Te termal elastic equation is 2 u = q (r,θ) D = f (r,θ) Were q is te surface distribution density of te equivalent load. Suppose is te tickness of te plate, E is elastic modulus, α is te termal expansion coefficient. D is te bending rigidity. In general, suppose te termal linear distribution along te plate tickness: q = µ M T, f (r,θ) = ( µ)d M α( + µ) T = T (r,θ) WereT (r, θ) is te termal distribution function on te surface of te plate, 2 M T = αe T (r,θ)zdz = αe2 2 T (r,θ) 2 Te equivalent boundary condition of te clamped bending plate is u Γ =, u n Γ =.Te equivalent boundary condition of te simply bending plate is u Γ =, Mu = - M T D( µ) αt (,θ)(+µ) = on Γ If in te plate tere is no eat source, T (r,θ) =, q =, for te simply plate, u (θ) =, equation (5) and (6), will be reduced to ( r 2 ) 2 4π [ + r 2 2r cos(θ θ )] u n(θ) (8) αt (,θ)( + µ) = + µ R 2 u n (θ) 2πR 2 sin 2 ( θ 2 ) u n(θ) (9) For te clamped plate, if in te plate tere is no eat source, T (r,θ) =, q =, tere are no deflections. Te following are te plates on te ead sources.

5 Termal Bending of Circular Plates 9 Example Suppose T (r,θ) = + r f (r,θ) = - M T D( µ) α( + µ) = r For te clamped plate on te eat source, from eq. (5) 2π 2π = α + µ) 2π Te numerical computation is according to te axisymmetrical solution α + µ) 2 ( ( ) r3 r 2 + ) 9 G(r,θ;r,θ )dr dθ For te simply plate on te eat source, firstly using te following equation to get u n Mu 2π MG(r,θ;r,θ ) = = +µ R 2 u n (θ) 2πR 2 sin 2 ( θ 2 ) u n(θ) r 2 ) 2 4π(+r 2 2r cos(θ θ )) () αt (,θ)(+µ) u n (θ) = 5α 3 + α +µ 3 = ( + µ)u n (θ) Ten using te convolution formula r 2 2π( + r 2 2r cosθ) coskθ = rk coskθ

6 Copyrigt 2 Tec Science Press SL, vol.4, no.2, pp.5-2, 2 We can get ( r 2 ) 2 4π[+r 2 2r cos(θ θ )] u n(θ) + 2π = 5α( r2 ) 6 α +µ) ( ( r3 r 2) + 9) Example 2 Suppose te center of T (r,θ) is ( 2,) T (r,θ) = r 2 + ( ) 2 2r 2 2 cosθ f (r,θ) = ( µ)d M T = α( + µ) T (r,θ) = α( + µ) 2 4r 2 4cosθ + Figure : Deflections of te clamped circular plate Figure 2: Deflections of te clamped plate For te clamped plate on te eat source, from eq. (5) 2π

7 Termal Bending of Circular Plates For te simply plate on te eat source, firstly using eq. () to get u n, suppose u n (θ) = m= b m cosmθ + m= b m sinmθ Te left of Eq. () is expanded to series α(+µ) T (,θ) + 2π 2 r 2 ) 2 r 4π(+r 2 2r cos(θ θ )) 4r 2 4cosθ + dr dθ = a m cosmθ + a m sinmθ m= m= Substituting te above equation into Eq. () wic is an integral wit a strongly singular Poisson kernel, and using te convolution formula, we can get b k = Ten, + µ + 2k a kb k = + µ + 2k a k ( r 2 ) 2 4π[+r 2 2r cos(θ)] ( r b k coskθ 2 ) 2 4π[+r k= 2 2r cos(θ)] b k sinkθ k= + 2π = a k ( r 2 )r k 2(+µ+2k) coskθ + a k ( r2 )r k 2π 2(+µ+2k) sinkθ + k= k= Suppose µ=.3, D=, αá/=, te deflections of te circular plates from example 2 are following 4 Conclusions Based on Green function metod, te boundary integral formula and natural boundary integral equation wit strongly singular kernel are educed for te biarmonic equation of te termal bending problem of te plate supported at te boundary. Te convolution formulae are utilized to get te solutions of deflection and slope directly for simple problems. As to complex problems, te Fourier series is be used

8 2 Copyrigt 2 Tec Science Press SL, vol.4, no.2, pp.5-2, 2 to get te solutions wic ave nicer convergence velocity and computational accuracy, and te calculating process is simpler. For te oter complicated load, it can be solved wit te similar metod or by te superposition wit te solutions of above examples. Acknowledgement: Tis work is supported by grants of National Basic Researc Program of Cina, No. 27CB294 and yout National Researc foundation of Cina, No References Candrasekara K. (2): Teory of plates, Hyderabad, Universities Press (India) Linited, Clesc A. (962): Teorie der Elasticitaet fester Koerper. Fu Bao-liang (2): Te New Bending Teorem of te Tin Plates on Reciprocal Metod. Beijing: Science press, Gelfand, I. M., Silov, G. E. (964): Generalized Functions, Academic Press, New York. Li Sun-cai (22): Dong Zeng-zu., Xie Wei-ong. Application of natural boundary element metod to te bending problem of te elastic tin plate. J. Xuzou Normal University, 2(4): 2-5. Wang An-wen (2): Solution to Asymmetric Bending of Circular Plates under Single Load by Using Point-Source Function. Acta Mecanic Sinica. 24(3), Timosengko, S.P., Woinowsky-Krieger, S. (959): Teory of Plates and Sells. 2nd ed., New York, McGraw-Hill. Yu De-ao (22): Natural Boundary Integral Metod and Its Applications. Kluwer Academic Publisers/Science Press. Zu Jia-lin. (984): Te boundary integral equation metod for solving Diricelt problem of plane biarmonic equation, J. Comp. Mat., Vol.6, No.3,