【正文】 n Gale kin line method (in Chinese). Journal of Wuhan, Institute of Chemical Techs logy, 18(1), 5760 (1996) [20] Liu, D. Y., Wang, C. Y., and Chen, W. Q. Free vibration of FGM plates with inplane material in homogeneity. Composite Struct,92, 10471051 (2022) [21] Uymaz, B., Aided, M., and File, S. Vibration analyses of FGM plates with inplane material in homogeneity by Ritz method. Composite Structures, 94, 13981405 (2022) [22] Badeshi, M. and Saudi, A. R. Levytype solution for buckling analysis of thick functionally graded rectangular plates based on the higherorder shear deformation plate theory. Applied Mathematical Modeling, 34, 365 于 3673 (2022) [23] Thai, H. T. and Kim, S. E. Levytype solution for buckling analysis of orthotropic plates based on two variable refined plate theories. Crocoites Structures, 93, 17381746 (2022) 第 11 頁 對在同一平面上矩形板可變剛度的解 析 天創(chuàng)宇，聶國軍，鄭鐘，傅潤珠 （學院航天工程與應(yīng)用力學，同濟大學，上海 202292， ） 摘要： 對 平面可變剛度矩形薄板彎曲問題進行了研究。 然而，大多數(shù)功能梯度材料的研究是與材料剛度沿變厚 度方向變化有關(guān)的。 （ 4）代入式 （ 3） 式可以得到 以下差分方程： 上述方程的解決方案由兩部分組成，即，齊次微分方程的一般解 YMO沒有齊次微分方程的特解 YMO。 （ ii）若彎曲剛性 D0和 Db滿足 D0Db,的板的撓度與剛度參數(shù)對 P 0與 P 0（參見圖 2和圖 5）。將所得到的結(jié)果 可以用來評估板的 平 面內(nèi) 可 變剛度的各種近似的理論和數(shù)值模型的有效性和準確性。 剩下的兩個邊（的邊界條件為： Y =0和 Y= b）的由下式給出： 簡支的邊界條件（ S） 邊界條件（ C） 自由邊界條件 8（ F） 第 17 頁 我們認為，支持簡單的條件式。 0XYZ直角坐標系 0≤ X≤ A， 0≤ Y≤ B 我們認為。梯度復(fù)合材料（功能梯度材料）在功能上有非均勻復(fù)合材料的力學性能，這種平滑變化的位置，以滿足預(yù)定的功能。 關(guān)鍵詞： 平面變剛度，電源形式，征費型的解決方案，矩形板 第 12 頁 1 引言 “變剛度”意味著整個結(jié)構(gòu)具有的剛度參數(shù)在空間上 [1] 的變化。 2 基本方程 考慮一個薄的矩形板的長度 A 和寬度 B 的平面變剛度，如圖 1 所示。 在式（ 4）和（ 17），的矩形板的橫向位移，可以表示為 邊界條件 這是假設(shè)的兩個相對的邊平行于 y 方向的簡支和其他兩個邊有任意邊界條件，如免費，簡單的支持，或鉗制條件。上的偏轉(zhuǎn)和 彎曲力矩的可變參數(shù)的影響進行了研究，通過數(shù)值例子。這是因為平均彎曲剛性板具有 p 0大于板帶 p 0。 因此，可以表示為方程（ 5）的解。對于板在平面上的可變剛度的研究相當少。 第 1 頁 Analytical solution nonrectangular plate with inplane Variable stiffness Tianchong YU， Guojun NIE， Zheng ZHONG， Fuyun CHU (School of Aerospace Engineering and Applied Mechanics, Tongji University,Shanghai 202292, . China) Abstract: The bending problem of a thin rectangular plate with inplane variable stiffness is studied. The basic equation is formulated for the twooppositeedge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form， and Poisson’ s ratio is constant． A fourthorder partial differential equation with variable coefficients is derived by assuming a Levytype form for the transverse displacement. The governing equation can be transformed into a Whittaker equation， and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads． The validity of the present solution is shown by paring the present results with those of the classical solution． The influence of inplane variable stiffness on the deflection and bending moment is studied by numerical examples． The analytical solution presented here is useful in the design of rectangular plates with inplane variable stiffness. Keywords: inplane variable stiffness ,power form， Levytype solution， rectangular plate Chinese Library Classification 0343 2022 Mathematics Subject Classification 74B05 第 2 頁 1 Introduction The term” variable stiffness” implies that the stiffness parameters vary spatially throughout. The structure[1] ． Functionally graded materials (FGMs) are inhomogeneous posites， in which. the mechanical properties vary smoothly with the position to meet the predetermined functional. performance． The structures posed of the FGMs are of variable stiffness． There are extensive literatures on the bending， vibration， and fracture of the FGM structures[29]． The deformation of a functionally graded beam was studied by the direct approach[10]． An efficient and simply refined theory was presented for the buckling analysis of functionally graded plates by Thai and Choi[11]． Jodaei et a1[12] dealt with the three— dimensional analysis of functionally graded annular plates using the state— space based differential quadrature method(SSDQM) ． Wen and A1iabadi[13]investigated functionally graded plates under static and