【摘要】 <正> 一.前言 波兰科学院通讯院士W.诺瓦茨基教授不久以前在北京所作的关于他在弹性薄板方面的工作的介绍报告,引起我们的兴趣与注意。在讨论的过程中,本文著者曾试从有限变换式(fnite transform)的角度上来说明诺瓦茨基教授从物理观点导出的一个普遍的方法。 下面我们将讨论边缘为夹住的或自由支承的弹性圆及圆环形薄板在任意横向载荷分布下的弯曲问题,说明有限汉克尔变换式(fnite Hankel transform)的应用。更多还原

【Abstract】 The present paper has the purpose of illustrating the usefulness of finite transform method to certain problems in the theory of elastic thin plates and shells. This method supplements from the mathematical side the idea of Prof. W. Nowacki in his treatment of problems of this type^[1].In particular, the finite transforms of a function f（x） over the interval 0≤x≤1 are defined aswhere I_m represents Bessel function of the first kind and of m-th order, ξ_m,i denoting the ;-th root of the equationThen it may be proved that^[2]Using this method, we can treat the problem of a circular plate under arbitrary lateral load with clamped edge or simply supported edge.The problem of a circular clamped plate of flexural rigidity D and radius a under arbitrary lateral load p（r, θ） reduces to a boundary-value problem of the following type: to find a dimensionless deflection function W（x, θ） satisfying the differential equationand the boundary conditions at x= 1where, θ denoting the deflection,We expand q（x, θ） into series of the form and write out W（x, θ） also in the formWe have boundary-value problems of the type:Applying the method of finite Hankel transform, we find thatand are constants to be determined. By inversion, we obtainObviously the condition X_m implies is satisfied and to satisfy the conditionThen, we can express the dimensionless deflection W（x, θ） in the form By specifying the load, many problems of interest can be solved, including those considered by H. Reissner and others.By the same procedure, the problem ot a clamped circular plate under tension and arbitrary lateral load, and that of a simply-supported circular plate can be solved without difficulty.While considering the problem of a ring-shaped plate under arbitrary lateral load, we use a different type of finite Hankel transform. If f（x） is defined in the interval 1≤x ≤λ -（λ> 1）, the finite Hankel transforms are given by the formulas:Y_m denoting Bessel function of the second kind and of m-th order, （m,i） being the i-th root of the equationThe application of this type of finite Hankel transform to ring-shaped plate under arbitrary lateral load is analogous to the case of circular plate, in that some of the boundary conditions are used after the inversion.更多还原