Large deflection analysis of rhombic sandwich plates placed on elastic foundation

Transcription

1 Indian Journal of Engineering & Materials Sciences Vol. 5, February 008, pp. 7-3 Large deflection analysis of rombic sandic plates placed on elastic foundation Gora Cand Cell a*, Subrata Mondal b & Goutam Bairagi c a Mecanical Engineering Department, Jalpaiguri Government Engineering College, Jalpaiguri 735 0, India b Birla Institute of Tecnology, Kolkata , India c Civil Engineering Departmet, Jalpaiguri Government Engineering College, Jalpaiguri 735 0, India Received Marc 007; accepted 6 December 007 Tis paper presents non-linear analysis of rombic sandic plates placed on elastic foundation under uniform load. Banerjee s ypotesis involving a ne form of energy expression in te total potential energy of te system as been employed. As a consequence te differential equation is decoupled keeping intact its non-linear caracter. Te aim of te present study is to analyze te non-linear beaviour of rombic sandic plates placed on elastic foundation for different ske angles. Te results ave been obtained bot for movable and immovable edges from a single cubic equation. Numerical results (central deflection versus load) ave been computed and compared it knon results (K=0) for square sandic plates only. Te corresponding linear analysis (K=0) is also presented. Te results tus obtained are in good agreement it tose repoted in te literature. Non-linear analysis of tin elastic plates resting on elastic foundation is interesting to design engineers for teir ide applications in te practical field. Cien and Cen successfully carried out te analysis on te effect of initial stresses of nonlinear vibration of laminated plates on an elastic foundation. Nonlinear partial differential equations based on Mindlin plate teory are derived for nonlinear vibration of laminated plates. Civalek 3 as utilized armonic differential quadrature-finite differences coupled approaces for geometrically nonlinear static and dynamic analyses of rectangular plates on elastic foundation. Winkler-Pasternak foundation model as been considered. Tis analysis is attractive for te practical field. Sladek et al. reported mesless local boundary integral equation metod for simply supported and clamped plates resting on elastic foundation. Simply supported and clamped tin elastic plates resting on a to parameter foundation ere analyzed. Te governing partial differential equation of fourt order for a plate is decomposed into to coupled partial differential equations of second order. One of tem is Poisson s equation and te oter is Helmoltz s equation. In te literature on nonlinear analysis of elastic plates te folloing to metods are attractive: *For correspondence ( gccell@rediffmail.com ) In te generalized differential quadrature metod te global Lagrange interpolation polynomial is used 5, viz.: N x xk g j ( x) =, j =,,..., N k=, k j x x j k ereas, singular convolution metod te folloing integral equations aving a singular convolution is defined 6 as F ( t) = ( T * η )( t) = T ( t x) η( x) dx ere, T(t-x) is a singular kernel. Te discrete singular convolution algoritm can be realized by using many approximation kernels. It is ell knon tat a good number of structural design utilizes sandic type construction in te fabrication of major structural components. A ig strengt-to-eigt ratio is acieved by combining a relatively tick ligt-eigt core it to tin-ig strengt faces. Te problem of large deflection of sandic plates as been investigated by several researcers, among ic orks of Reissner 7, Wang 8, Hoff 9 and Eringen 0 need special mention. Reissner 7 presented an exact analysis of finite deflection of sandic plates. Wang 8 gave a general teory of large deflections of sandic plates and sells. Hoff 9 and

2 8 CHELL et al.: RHOMBIC SANDWICH PLATES Eringen 0 eac developed a teory of bending and buckling of sandic plates. All tese investigations are, oever, confined to rectangular sandic plates under mecanical loading only. Kamiya employed Berger s ell knon tecnique to solve nonlinear problems of sandic plates using a ne set of governing equations it a correction factor. Tis ork as been restricted to a particular plate geometry. Dutta and Banerjee offered a simple approac to investigate nonlinear static as ell as dynamic beaviours of sandic plates. Later, Ray et al. quite elegantly investigated in te nonlinear termal beaviour of sandic plates. Dumir 3 and Dumir & Baskar orked on nonlinear analysis of rectangular plates on elastic foundations. Civalek 5 presented geometrically nonlinear analysis of sells. Some more interesting orks on sandic structure ave been reported 6-9 ere nonlinear analyses ave been carried out elegantly and are attractive to design engineers. Literature on large deflection analysis of elastic ske sandic plates demands special attention because of teir ide applications in space industry. Plates placed on elastic foundation are also attractive to design engineers for teir ide applications. Civalek and Yavas 6 ave carried out analysis on large deflection static analysis of rectangular plates. Non-linear analysis of sandic plates on elastic foundation is rarely reported. Te aim of te present study is to use a set of uncoupled differential equations in oblique coordinates to analyze nonlinear beaviours of rombic sandic plates placed on elastic foundation under uniform loading using Banerjee s Hypotesis. To obtain te central deflection ( 0 /) versus load (q 0 a /E ) Galerkin tecnique as been used. Numerical results tus obtained for different ske angles ave been plotted in graps. Results for linear and nonlinear analysis ave been compared it te results obtained by Cakraborty 0 and Dutta & Banerjee for square sandic plates (K=0) only. Te numerical results of te non-linear beaviours for different ske angles of rombic sandic plates placed on elastic foundation are believed to be ne. Governing Equation We consider a rombic sandic-plate (Fig.) it an isotropic core as ell as isotropic upper and loer faces of identical tickness; ile te faces respond to te bending and membrane action of te plate, te core is assumed to transfer only sear deformation. Moreover, te tickness of upper and loer faces is sufficiently tin in comparison it core tickness (>>t) to ignore a variation of sear in te tickness direction of te faces. Under mecanical loading te governing equations for sandic plates on elastic foundation in rectangular Cartesian coordinate are given as: m Et G EtI.. ν..( ν ) G.( ν ) x y Etλ G( ν ) x y.... x x y y xy x y Fig. (a) Sandic plate and (b) Ske sandic plate

3 CHELL et al.: RHOMBIC SANDWICH PLATES 9 ere, I m = q G. k = 0 () G x ν y () For ske angle θ, (Fig. b) e get te folloing transformation: = sec θ sinθ. x xy y sec θ sin θ. = 3 3 x xy xy ( sin θ ) xy y Putting te above transformations in Eqs () and (), e get te folloing set of differential equations: Et ( ν ) sec θ sin θ.. x x y y m EtI sec θ sinθ G( ν ) x xy G tan θ ν.sec θ sin θ. y y x xy Etλ secθ sinθ y G( ν ) x y sec θ sin θ. y x xy y secθ sinθ sec θ x y x sin tan θ θ x y y y y sec sin sec sin θ θ θ θ xy y x y q G sec θ sin θ. y G x xy k=0 y I m = Analysis secθ sinθ x Let us assume = y πx πy 0 sin sin a a (3) ν () y...(5) πx πy q = q0 sin sin (6) a a Tis form of clearly satisfies te required simply supported edge conditions. It is to be noted tat to determine te desired solution, as been cosen in te form of double sine series ensuring convergence of te solution. Moreover, putting tis form of in te given differential equation, e get te error function πx πy ( x, y) because = 0 sin sin is not te a a exact solution of te differential equation. We are no required to minimize tis error by using te ell knon Galerkin s Tecnique. Tis tecnique gives ( x, y) dxdy = 0

4 0 CHELL et al.: RHOMBIC SANDWICH PLATES Tus, te solver of te differential equation is Galerkin s Tecnique involving te evaluation of te above double integrals, ic is simple. If e no integrate Eq. () over te entire plain area of te plate e get π = I m { sec θ ( sin 0 θ ) ν } (7) 8 a It is to be noted tat for movable edge conditions I m = 0 Putting Eqs (5), (6) and (7) in Eq. (3), e get te error function ( x, y), Galerkin s Tecnique requires (, y) dxdy = 0 x (8) Evaluating te integrals in (8) e get te folloing form of cubic equation, determining te central deflection 0 ; A B C = 0 (9) π Et 6 ere, A = { sec θ ( sin θ) ν}. 6( ν ) Ga { sec θ tan θ sec θ sin θ sec θ ν sec θ} 6 Eλπ t 3 sec sec tan θ θ θ ( ν ) Ga tan θ sin θ sin θ sec θ sec θ tan θ θ θ θ θ θ tan.sin ) sec sec sin π t { sec θ ( sin θ) ν}. 6( ν ) sec θ tan θ ν λπ t { sec θ 6sec θ 3( ν ) { } ( ) } 3tan θ 3tan θsin θ 6sin θ 3 t k B = π sec θ( sin θ) a 3 ( ν ) E and π Etsec θ q0a C = ( ν ) Ga E Numerical Results Tables -3 and corresponding graps (Figs. -0) sos te results for te central deflection parameter for different load function of a rombic sandic plates on elastic foundation under uniformly q0a E Table Linear analysis (K = 0) For θ = 0 θ = 30 θ = 5 θ = 60 0 (knon) [0] Table Non-linear analysis (immovable edge conditions K = 0) q For θ = 0 For θ = 30 For θ = 5 For θ = 60 0a E 0 0 (knon) []

5 CHELL et al.: RHOMBIC SANDWICH PLATES q0a E 0 Table 3 Non-linear analysis (movable edge conditions K = 0) For θ = 0 For θ = 30 For θ = 5 For θ = 60 0 (knon) [] Fig. Linear analysis Fig. Nonlinear analysis (movable edge conditions K=0, K=0.5, K=.0) Fig. 3 Nonlinear analysis (immovable edge conditions K=0, K=0.5, K=.0) distributed load for different values of K it te folloing dimensions: E = N/m t = m ν = 0.3 G = N/m λ = ν = m a = 0.5 m Fig. 5 Nonlinear analysis (immovable edge conditions K=) Conclusions Te proposed differential equations are uncoupled and tus simple. From te same Cubic Eq. (9) for 0, te results of immovable and movable edge conditions can be obtained. Numerical results (bot linear and non-linear) for rombic sandic plates on elastic foundation are obtained for different

6 CHELL et al.: RHOMBIC SANDWICH PLATES Fig. 6 Nonlinear analysis (immovable edge conditions K=5) Fig. 9 Nonlinear analysis (movable edge conditions K=5) Fig. 7 Nonlinear analysis (immovable edge conditions K=8) Fig. 8 Nonlinear analysis (movable edge conditions K=) ske angles and presented tose in tables. For square sandic plate te results are compared it tose of knon results 0, for K=0 and are in good agreement. Te results for te oter ske angles for different values of K are believed to be completely ne. Te Fig. 0 Nonlinear analysis (movable edge conditions K=8) numerical results presented in different tables for different ske angles and different values of K, offer an interesting observation. As θ increases, i.e., as te plate tends toards rombic sape, te deflection 0 decreases, i.e., stress decreases. Tis is quite expected because it te increase of ske angles, te plate offers more rigid structure. And also as te value of K increases, ic means elastic force 0 increases, te deflection decreases, as elastic force alays acts in upard direction, tereby opposing te deflection. Acknoledgement We are grateful to Dr B Banerjee, Head of te Department of Matematics, Jalpaiguri Government Engineering College (Retd.), Jalpaiguri, West Bengal, for is valuable suggestions.

7 CHELL et al.: RHOMBIC SANDWICH PLATES 3 Nomenclature E = Young s modulus. q,q 0 = uniform load ν = Poisson s Ratio a = size of plate. t = face tickness. = core tickness. G = sear modulus. u,v, = displacement in x, y, z directions respectively. λ = constant depending on te Poisson s ratio of te plate materials. I m = first invariant of average face strain k = Winkler foundation parameter K = ka = stiffness parameters of Winkler foundation θ 3 E = Ske angle References Dutta S & Banerjee B, Int J Nonlinear Mec, 6(3) (99) Cien Rean-Der & Cen Cun-Seng, Tin-Walled Struct, (8) (006) Civalek O, J Sound Vib, 9(-5) (006) Sladek J, Sladek V & Mang H A, Comput Metod Applied Mec Eng, 9(5-5) (00) Ng C H W, Zao Y B & Wei G W, Comput Metod Appl Mec Eng, 93 (00) Civalek O & Yavas A, Int J Sci & Tecnol, () (006) Reissner E, J Aeronaut Sci, 5 (98) Wang C T, Principle and application of complementary energy metod for tin omogeneous and sandic plates and sells it finite deflections, NACA Tecnical Note 60 (95) Hoff N J, Bending and buckling of rectangular sandic plates, NACA Tecnical Note 5 (950) Eringen A C, Bending and buckling of rectangular sandic plates, Proc st U.S. Nat Cong Applied Mecanics, ASME Ne York, (95) Kamiya N, AIAA J, () (976) Ray A, Dutta S & Banerjee B, Meccanica, 8 (993) Dumir P C, Acta Mec, 7 (988) 33-. Dumir P C & Baskar A, Comput Metod Appl Mec Eng, 67 (988) -. 5 Civalek O, Int J Pressure Vessels Piping, 8(6) (005) Siau Le-Cung & Kuo Si-Yao, J Eng Mec, 30(0) (00) Kare R K, Rode V, Garg A K & Jon S P, J Sandic Struct Mater, 7() (005) Mania Radosla, J Compos Struct, 69() (005) Acary G G S & Kapuria S, J Acta Mec, 8(-) (006) Cakraborty A, An efficient layer-ise finite element model for static, vibration and buckling analysis of composites and sandic laminates it inter laminar imperfections, P.D. Tesis, Indian Institute of Tecnology, Karagpur, India, 00.