1、PDF外文：http:/
2、 第 1 页 中文 2300 字 Analytical solution nonrectangular plate with in-plane Variable stiffness Tian-chong YU， Guo-jun NIE， Zheng ZHONG， Fu-yun CHU (School of Aerospace Engineering and Applied Mechanics, Tongji University,Shanghai 200092, P.R. China) Abstract: Th
3、e bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate var
4、ies in the plane following a power form， and Poisson s ratio is constant A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation， and an analyt
5、ical solution is obtained for a thin rectangular plate subjected to the distributed loads The validity of the present solution is shown by comparing the present results with those of the classical solution The influence of in-plane variable stiffness on the deflection and bending moment is studied b
6、y numerical examples The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness. Keywords: in-plane variable stiffness ,power form， Levy-type solution， rectangular plate Chinese Library Classification 0343 2010 Mathematics Subje
7、ct Classification 74B05 &nb
8、sp; 第 2 页 1 Introduction The term” variable stiffness” implies that the stiffness parameters vary spatially throughout. The structure1 Functionally graded materials (FGMs) are inhomogeneous composites， in which. the mechanica
9、l properties vary smoothly with the position to meet the predetermined functional. performance The structures composed of the FGMs are of variable stiffness There are extensive literatures on the bending， vibration， and fracture of the FGM structures2-9 The deformation of a functionally graded
10、 beam was studied by the direct approach10 An efficient and simply refined theory was presented for the buckling analysis of functionally graded plates by Thai and Choi11 Jodaei et a112 dealt with the three dimensional analysis of functionally graded annular plates using the state space based
11、differential quadrature method(SSDQM) Wen and A1iabadi13investigated functionally graded plates under static and dynamic loads by the local integral equation method(LIEM) There are also some works on the FGM shells and cylinders14-17 However， most of the studies on the FGMs deal with material
12、stiffness varying along the thickness direction The studies on the plates with in plane variable stiffness are quite few Shang 18 studied the rectangular plates with bidirectional linear stiffness with two opposite edges simply supported and the other two edges arbitrarily supported under the distri
13、buted loads Yang 19 investigated the structural analysis of the plates with unidirectionally varying flexural rigidity by the Galerkin line method Liu et a1.20 analyzed the free vibration of a Functionally graded isotropic rectangular plate with in plane material in homogeneity using the Levy-type s
14、o1ution Uymaz et a1 21 Considered the functionally graded plates with properties Varying in an in plane direction based on a five degree of-freedom shear deformable plate theory with different boundary conditions In this paper， the Levy-type solution22-23is presented for the bending of a thin
15、rectangular plate with in plane variable stiffness under the distributed load 2 Basic equations Consider a thin rectangular plate of length A and with B with in-plane Variable stiffness， as shown in Fig.1. Introduce a Cartesian coordinate system 0 XYZ such that 0 X A， 0 Y B We assume. T
16、hat the flexural rigidity of the plate D=D(X,Y)is a function of X and Y. The governing equation of the plate with in plane Variable stiffness can be obtained as Where W is the transverse displacement， V is Poisson s ratio， and Q is the normal pressure on The plate &
17、nbsp; 第 3 页 It is assumed t
18、hat the flexural rigidity of the plate varies only along the Y-direction according to the following power form: Where Y and P are two material parameters describing the in homogeneity of D, Do is
19、the flexural rigidity at， Y=0, and Db is the flexural rigidity at Y=b. In this case, Eq. (1) can be reduced to 3 Solution The rectangular plate is assumed to be simply supported along two opposite edges parallel to the Y-direction. To solve the governing equation with the prescribed boundary conditi
20、ons, a generalized Levy-type approach is employed as where Ym(y) is an unknown function to be determined. Substituting Eq. (4) into Eq. (3) yields the following differential equation:
