有水平摩阻的Winkler地基上四边简支矩形板弯曲问题的解析解

ANALYTIC SOLUTIONS TO THE BENDING PROBLEM OF RECTANGULAR THIN PLATES SIMPLY SUPPORTED ON FOUR SIDES ON A WINKLER FOUNDATION WITH HORIZONTAL FRICTION

  • 摘要: 假设板与地基之间的水平摩阻力正比于它们间水平位移差,根据受力平衡和本构关系建立了弹性地基上有水平摩阻矩形板的微分方程,并求解得到了四边简支板的级数形式解析解。以板中心作用集中荷载和方形均布荷载为例,分析水平摩阻对板挠度和弯矩的影响。结果表明:板与地基间的水平摩阻对矩形板的挠度和弯矩有削减作用;当地基板水平摩阻参数大于0.01时,地基水平摩阻力对板挠度和弯矩影响有可能超过2%,应予考虑;板尺寸和荷载尺寸会对水平摩阻的削减效果产生一定影响,需根据实际工况仔细考量;相对板长为5的方板中心处作用局部方形均布荷载时,水平摩阻的存在最大可使板最大挠度(即荷载中心点处挠度)下降约60%,板最大截面弯矩(即荷载中心点处弯矩)下降约70%。

     

    Abstract: Assuming that the horizontal friction between the plate and the foundation is proportional to the horizontal displacement difference between them, the differential equations for a rectangular plate resting on an elastic foundation with horizontal friction were established according to the force equilibrium and the constitutive relationship, and the analytical solution in series form was obtained for the four-side simply-supported plate. The effects of the horizontal friction on the deflection and bending moment of plates were analyzed by taking the concentrated load and the square uniform load acting at the center of the plate as examples. The results indicate that the horizontal friction between the plate and the foundation has a reduction effect on the deflection and bending moment of the rectangular plate. When the horizontal friction parameter is greater than 0.01, the influence of the horizontal friction on the deflection and bending moment may exceed 2% and should be considered. The size of the plate and the size of the load affect the reduction effect of the horizontal friction, which should be considered carefully according to the actual working conditions. When a square uniform load is applied at the center of a square plate with a relative length of 5, the horizontal friction can cause the maximum deflection of the plate (i.e., the deflection at the center of the load) to decrease by about 60 %, and the maximum section bending moment (i.e.,the bending moment at the center of the load) to decrease by about 70 %.

     

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