Closed and approximate analytical solutions for rectangular Mindlin plates
Closed and approximate analytical solutions for rectangular Mindlin plates
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矩形 Mindlin 板的闭合近似解析解
DOI：
10.1007/bf01182359
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发表时间：
2001
期刊：
影响因子：
2.7
通讯作者：
V. Naumenko
中科院分区：
文献类型：
--
作者：
K. Naumenko;J. Altenbach;H. Altenbach;V. Naumenko
根据Nádai-Lévy和Vlasov-Kantorovich方法，讨论了矩形板Mindlin板方程的封闭解和近似解析解。对于弹性、均质和各向同性板，二维边值问题的三个未知量被表示为依赖于单个坐标的函数的乘积的级数。对平面内坐标方向之一的函数，对特殊边界条件的控制偏微分方程和对一般情况的虚位移原理，导出一组常微分方程。这些方程的解析解提供了取决于其他面内坐标的函数的表达式。对于其中一个坐标方向上的简支边板和另一个坐标方向上的任意齐次边界条件，Nádai-Lévy方法提供了一个封闭的或精确的解，因为位移和应力合成的无穷级数可以被截断以获得任何所需的精度。在一般情况下，非简支边的迭代Vlasov-Kantorovich方法产生一个近似的解析解。这两种方法都是不敏感的厚度减少方面的准确性和代表的边界层的解决方案的指数函数。矩形板与各种类型的边界条件的应用。
SummaryBasing on the Nádai-Lévy and the Vlasov-Kantorovich methods closed and approximate analytical solutions of Mindlin's plate equations in the case of rectangular plates are discussed. For elastic, homogeneous and isotropic plates three unknowns of the governing two-dimensional boundary value problem are formulated as series of products of functions depending on a single coordinate. Specifying the functions for one of the in-plane coordinate directions the governing partial differential equations for a special type of boundary conditions and the principle of virtual displacements for the general case yield a set of ordinary differential equations. The analytical solution of these equations provides expressions for functions depending on the other in-plane coordinate. For plates with simply supported edges for one of the coordinate directions and for arbitrary homogeneous boundary conditions for the other one the Nádai-Lévy method provides a closed or exact solution in the sense that the infinite series for displacements and stress resultants can be truncated to obtain any desired accuracy. In the general case of nonsimply supported edges the iterative Vlasov-Kantorovich method yields an approximate analytical solution. Both methods are nonsensitive to a reduction of the thickness with respect to accuracy and represent the boundary layer solutions in terms of exponential functions. Applications to rectangular plates with various types of boundary conditions are presented.