考虑地基水平摩阻的Winkler地基上板模型解析

Model and Solution of a Plate on Winkler Foundation With Horizontal Friction

在Winkler地基模型基础上,采用板与地基之间的水平摩阻应力正比于板与地基接触点间水平位移假设,推演了包含板弯曲和轴向拉压效应的非线性地基板微分方程;应用摄动法分析竖向集中力作用下,板与地基水平摩阻应力对板内力(弯曲力、轴向力和剪应力)、位移(挠度、水平位移)的影响规律,实证了板与地基水平摩阻应力对板剪应力的非线性影响可以忽略不计,从而得到了轴对称条件下计入地基水平摩阻的Winkler地基上薄板的线性微分方程.随后通过汉克尔变换得到了轴对称无限大板的解析解,以及包含四项复宗量贝塞尔函数的轴对称圆形板的解析解,给出了板内力、位移的计算式;最后,通过算例分析了板与地基间水平摩阻状况对板挠度、截面弯矩的影响规律.结果表明:地基板挠度与弯矩随着板与地基间水平摩阻的增大而减少;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;无限大板作用圆形均布荷载时,水平摩阻的存在最大可使板最大挠度(即荷载圆中心点处挠度)下降约50%,板最大截面弯矩(即荷载圆中心点处弯矩)下降约70%.

Based on the Winkler foundation model,we adopted the assumption that the horizontal friction stress between the plate and foundation is proportional to the horizontal displacement between the contact points of the plate and foundation.Further we derived the differential equations of the nonlinear foundation plate considering the effects of plate bending and axial tension and compression;The regression method is applied to analyze the influence of the horizontal friction stress between the plate and foundation on the internal force(bending force,axial force and shear stress)and displacement(deflection and horizontal displacement)of the plate under the action of vertical concentrated force,and it is proved that the nonlinear effect of the horizontal frictional stress between the plate and the foundation on the plate shear stress can be neglected,so as to obtain the linear differential equation of the thin plate on Winkler foundation under the axisymmetric condition taking into account the horizontal frictional resistance of the foundation.Subsequently,the analytical solutions of the axisymmetric infinite plate and the axisymmetric circular plate containing the four-term complex zoned Bessel function are obtained by the Hankel transform,and the computational equations of the internal forces and displacements of the plate are provided.Finally,the influence of the horizontal friction between the plate and the foundation on the plate deflection and the section bending moment is analyzed through arithmetic examples.The results show that,the deflection and bending moment of the foundation plate decrease with the increase of the horizontal friction between the plate and the foundation.When the horizontal friction parameter between the plate and the foundation is greater than 0.01,the influence of the horizontal friction resistance between the plate and the foundation on the deflection and bending moment may exceed 2%and should be considered.When an infinite plate is loaded with a circular uniform load,the horizontal friction can cause the maximum deflection of the plate(i.e.the deflection of the plate at the center of the load circle)to decrease by about 50%and the maximum section bending moment of the plate(i.e.the section bending moment of the plate at the center of the load circle)to decrease by about 70%.