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四边不同支承条件下矩形板的结构计算
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Structural Calculation of Rectangular Plates with Different Support Conditions of Four Edges
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    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.

    Abstract:

    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.

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YANG Chengyong, XU Qingtao, MA Wenhui?,HAN Xueguo.四边不同支承条件下矩形板的结构计算[J].湖南大学学报:自然科学版,2024,(1):101~111

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  • 在线发布日期: 2024-01-16
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