再论文克勒地基板极限承载力的弹塑性解

为了完善文克勒地基板极限承载力的理论基础，针对圆形荷载作用下的文克勒地基上无限大板达到其极限承载力状态时的破坏性状，即板顶出现环状裂缝，以板切向弯曲曲率是否达到其弹性极限弯曲曲率为准则将板划分为2个区域，环内屈服区釆用刚塑性假设，环外弹性区采用线弹性假设，联立方程求解，推演得到了文克勒地基板上作用圆形均布或刚性承载板荷载的极限承载力问题的解析解。分析了材料泊松比、残余弯矩比、荷载圆半径、荷载类型等因素对地基板的极限承载力、屈服区和环裂区范围的影响，并将分析结果与刚塑性解、其他弹塑性解进行对比。最后，归纳给出了2种荷载形式下的地基板极限承载力近似计算式。研究结果表明：板材料的泊松比对屈服区半径、环状裂缝半径及极限承载力影响不大；当荷载圆半径较小时，环状裂缝发生在屈服区，当荷载圆半径较大时，环状裂缝出现在弹性区；板残余弯矩比对环裂半径的影响随荷载圆半径的变化而变化；随着板残余弯矩比的减小，屈服区边界内缩，板极限承载力减小；在荷载圆半径相同时，刚性承载板荷载的环裂半径比圆形均布荷载的环裂半径略大，相应的极限承载力也稍大，最大偏差约为30%；在常见荷载圆半径范围内，刚塑性解的极限承载力比所提方法弹塑性解要大，偏差随荷载圆半径的增大而减小。

Re-discussion on Elastic-plastic Solution of the Ultimate Bearing Capacity of a Plate on a Winkler Foundation

To improve the theoretical basis of the ultimate bearing capacity of a Winkler foundation plate, this study focused on the failure behavior of an infinite plate on a Winkler foundation under circular load when it reaches its ultimate bearing capacity (i.e., when circumferential cracks appear on the top of the plate). The plate was divided into two regions (yield and elastic) based on whether the circumferential bending curvature of the plate exceeded its elastic limit bending curvature. The rigid-plastic and linear-elastic assumptions were used inside and outside the yield region, respectively. Then, an analytical solution of the ultimate bearing capacity of the plate resting on a Winkler foundation with a circular uniform distributed load and a rigid bearing plate load was derived. The influence of material Poisson's ratio, residual bending moment ratio, load radius, and load type on the ultimate bearing capacity, yield region radius, and circumferential crack radius of the foundation slab were analyzed, and the results were compared with those of the rigid-plastic and other elastic-plastic solutions. Finally, an approximate formula for calculating the ultimate bearing capacity of a plate resting on a Winkler foundation in two load forms was given. The results show that the Poisson's ratio of the plate material has little effect on the yield region radius, circumferential crack radius, and ultimate bearing capacity. When the load radius is either short or long, the circumferential crack occurs in either the yield or elastic region, respectively. The influence of the residual bending moment ratio on the radius of the circumferential crack changes with the load radius. As the residual bending moment ratio decreases, the yield region radius and the ultimate bearing capacity of plate decrease. When the load radius is the same, the circumferential crack radius of the plate subjected to the rigid bearing plate load is slightly longer than that of the plate subjected to the circular uniform load, and the corresponding ultimate bearing capacity is also greater, where the maximum deviation is approximately 30%. In the range of common load circle radii, the ultimate bearing capacity of the rigid-plastic solution is greater than that of the solution proposed in this study, and the deviation decreases with an increasing load circle radius.