四边不同支承条件下矩形板的结构计算  

Structural Calculation of Rectangular Plates with Different Support Conditions of Four Edges

作　　者：杨成永[1] 许清滔 马文辉 韩薛果 YANG Chengyong;XU Qingtao;MA Wenhui;HAN Xueguo(School of Civil Engineering,Beijing Jiaotong University,Beijing 100044,China;Anhui Transport Consulting&Design Research Institute Co.,Ltd.,Hefei 230088,China;Beijing Urban Rail Transit Construction Engineering Co.,Ltd.,Beijing 100176,China)

机构地区：[1]北京交通大学土木建筑工程学院,北京100044 [2]安徽省交通规划设计研究总院股份有限公司,安徽合肥230088 [3]北京城建轨道交通建设工程有限公司,北京100176

出　　处：《湖南大学学报（自然科学版）》2024年第1期101-111,共11页Journal of Hunan University:Natural Sciences

基　　金：国家自然科学基金高铁联合基金重点支持项目(U1934210)。

摘　　要：采用带补充项的傅立叶级数作为挠度函数,针对四边不同支承矩形薄板,推导了确定待定系数的方程组,给出可处理简支边、固支边和自由边任意组合条件下统一的结构计算公式.探讨了集中荷载作用处弯矩级数解不收敛的处理办法,以及双向板简化为单向板需要达到的长宽比问题.结果表明,集中荷载作用处的弯矩,可采用挠度值按中心差分公式进行计算,差分步长可取10 mm.对边支承对边自由板及一边固支三边自由板,可视作单向板.当四边支承板的长宽比达到2∶1、2.5∶1及4.5∶1时,可分别简化为两端固支、一端简支一端固支及两端简支单向板.三边支承一边自由板长宽比达到1∶1及2∶1时,可分别简化为两端固支(及一端简支一端固支)及两端简支单向板;长宽比达到6∶1时,可简化为悬臂单向板.两邻边支承两邻边自由板若要简化为悬臂单向板,在两支承边为固支时,长宽比需要达到2∶1;在支承边为一边简支一边固支时,长宽比要达到1.5∶1.For a rectangular thin plate with different support conditions of four edges,a double Fourier series with additional terms was taken as the deflection function,and linear algebraic equations for solving the undetermined coefficients were derived.A unified structural calculation formula was obtained for the plate with any combination of simply supported,clamped,and free edges.The non-convergence of a series solution of bending moment at a point where a concentrated load is applied was discussed.In addition,the aspect ratio problem that needs to be achieved when simplifying a two-way plate is also discussed.The results show that the bending moment at the point can be calculated through the finite difference method using the deflection value,where a differential step length of 10 mm was selected.A plate with two opposite edges supported and two other edges free,and a plate with one edge clamped and the other three edges free,can be taken as a one-way plate.A plate with all four edges supported can be treated as a one-way plate with two ends clamped,one end simply supported and the other end clamped,or two ends simply supported if its aspect ratio reaches 2∶1,2.5∶1,or 4.5∶1,respectively.A plate with three edges supported and the last edge free can be seen as a one-way plate with two ends clamped(and one end simply supported and the other end clamped),or two ends simply supported if its aspect ratio gets 1∶1 or 2∶1,separately;this type of plate can also be simplified as a cantilevered one-way plate at an aspect ratio of no less than 6∶1.A plate with two adjacent edges supported and the other two edges free can be taken as a cantilevered one-way plate if its aspect ratio reaches 2∶1 when the two supported edges are all clamped,or if the aspect ratio reaches1.5∶1 when one edge is clamped and an adjacent edge is simply supported.

关 键 词：矩形薄板 结构分析 支承条件 级数解 收敛 单向板 

分 类 号：U411[交通运输工程—道路与铁道工程]