框架结构中矩形楼板动力响应求解的谱单元法

A spectral element method for modelling a rectangular plate vibration in frame structures

在进行轨道交通引起建筑结构振动响应计算时,为了提高高频响应的准确度,采用谱单元法(spectral element method,SEM)求解矩形楼板的动力响应,并针对既有谱单元法中板与梁(柱)单元自由度不相容问题,通过修正板的节点自由度,实现了板与梁、柱的耦合。将四边任意边界条件的矩形板分解为两个可以转化为类一维问题的单向板;基于Kirchhoff薄板的振动控制方程,对每一个单向板进行动力计算,并分析波的传播与横截面的振型。通过最小二乘法优化板的节点自由度,使之与梁、柱单元的自由度对应,实现谱单元法在梁板柱耦合结构中的应用。将谱单元法的计算结果与解析解和有限元法结果对比得出:在1~250 Hz内,谱单元法的计算结果比有限元准确,尤其在高频范围;谱单元模型节点数量较少,建模和计算效率高于有限元。

To improve the accuracy of the building vibration in the high frequency range,a modified spectral element method(SEM)was proposed to analyze the dynamic response of a rectangular plate and the plate coupled with beams and columns.First the plate was split into two sub-problems,each of which was solved as a quasi-one-dimensional problem using the spectral element method based on the governing equation of the Kirchhoff plate.Then the nodal dofs was modified to be in accord with the dofs of beams and columns by using the least-squares approach.After that the coupled’plate-beam-column’structure was solved entirely by the spectral element method.Some examples were given to validate the developed method and it shows that the spectral element method gives accurate result and is more computationally efficient than the finite element method.