板柱结构矩形弹性板弯曲精确解法

AN ACCURATE SOLUTION OF RECTANGULAR PLATE BENDING IN COLUMN SUPPORTED PLATE STRUCTURE

板柱结构是工程中经常采用的受力体系,但至今尚无一种精确解法分析板中内力分布。将板柱结构矩形板的弯曲划分为广义静定和广义超静定二类。对于前者利用静力平衡条件确定柱支反力后撤去柱支座,柱支条件下板的弯曲即转换为四边自由矩形板在原荷载和柱支反力共同作用下的弯曲。挠度表达式采用了新的通解形式,其变形曲线符合四边自由边界所限定的变形特征,并采用组合特解,即特解同时满足平衡微分方程,自由边界上剪力分布条件及自由角点上作用力条件,从而可以利用四边边界条件和柱支座处的位移条件直接求解。对于广义超静定弯曲需要利用叠加法求解。这种解法可以分析板柱结构在任意柱支条件下和任意荷载作用下板的弯曲。通过逆向分析验证法真实地说明了本解法具有很高的计算精度。

The column supported plate structure is a very common floor system. The accurate solution on the internal forces in the plate has not been attempted. The bending problem of a rectangular plate is assumed to be generalized statically determinate or generalized statically indeterminate. For the former, the column supports are replaced by the supporting reactions which are calculated using static equilibrium. The plate supported by columns is viewed as a plate with four free edges subjected to the original loads and the supporting reactions. The general solution of deflection which satisfies the free boundary conditions is used. Composite particular solutions are used that satisfy the equilibrium differential equation, the shear force conditions on the free edges and the concentrated force conditions at the points where columns are located. The internal forces of plate can be solved for directly using boundary conditions at four edges of the plate and the deformations of the column supports. For the generalized statically indeterminate case they are solved for using the superposition principle. The solution is effective for bending of the plate supported by columns at arbitrary locations under arbitrary loads and has the advantage of high accuracy that is proved by inverse analysis examples.