摘要：
为了找到一种针对任意荷载作用下任意拱轴线弹性支撑拱的非线性稳定性研究方法,对集中荷载作用下任意轴线两端竖向弹性支撑浅拱的面内屈曲特性开展研究,推导了量纲一化的非线性平衡方程,并通过算例分析了屈曲路径和临界荷载的分布特点,并将其结果与有限元解进行了对比验证.推导过程中采用具有相同弹性支撑梁的屈曲模态作为形函数,不截断地展开拱轴线、外部荷载和结构位移,得到基本平衡状态以及极值点屈曲、分岔屈曲的平衡方程;建立了外荷载、结构位移与结构内力之间的对应关系,进而得到2种屈曲形式的平衡路径和临界荷载;分析了弹性刚度参数对2种屈曲条件下平衡路径与极限荷载分布规律的影响.研究结果表明:采用给出的方法计算的结果与有限元解吻合良好,可以追踪结构发生屈曲的全过程;极值点屈曲和分岔屈曲同时存在,当弹性支撑参数由对称变为不对称时,极值点屈曲路径在特定位置分成基本路径和独立的分离路径,某些位置的分岔屈曲路径变成极值点屈曲路径,并伴随相应临界荷载点的出现和消失;临界荷载仅在量纲一的弹性约束参数较小时随之发生变化,当约束刚度增至一定程度时临界荷载不再随约束刚度的变化而改变.推导的集中荷载下任意拱轴线形竖向弹性支撑浅拱面内屈曲求解公式,可为最终实现任意荷载下任意轴线弹性支撑拱非线性稳定性的解析求解提供参考.%To find a research method that can solve the nonlinear stability of elastically supported arch with arbitrary arch axis under arbitrary load,the buckling characteristics of a planar shallow arch with vertical elastic supports at both ends of an arbitrary axis under a concentrated load were investigated in this research.The nonlinear equilibrium equations of dimensional normalization were derived and the distribution characteristics of buckling paths and critical loads were studied by an example.The analytical results were compared and validated with the finite element analysis results.The buckling modes of the beam,which has the same elastic supports as shallow arch,were applied as shape functions to expand the arch axis,external load and structural deformation without any truncations.Then the equilibrium equations of basic equilibrium states,primary buckling and bifurcated buckling were obtained.The corresponding relationship among the external load,the structural displacement and internal force of structure was established,and the equilibrium paths and critical loads of the primary and bifurcated buckling of shallow arches were further achieved.The effects of elastic stiffness parameters on equilibrium paths and ultimate load under two kinds of buckling conditions were analyzed.The results show that these two results match well with each other,and the presented method can trace the whole buckling process of shallow arch structure.The primary buckling and bifurcated buckling exist simultaneously.When elastic supports change from symmetry to asymmetry,the primary equilibrium paths split into basic paths and independent separated paths at specific locations.Some bifurcated equilibrium paths transfer to primary one with the appearance and disappearance of corresponding critical load points.The critical loads are only sensitive to the smaller elastic constraint parameters.When constraint stiffness increases to a certain extent,the critical load will no longer change with the constraint stiffness.The solving equations for planar buckling of vertically supported shallow arch with elastic supports and arbitrary arch axis under a concentrated load deduced in this paper can eventually provide a reference to realize the analytical solution of nonlinear stability of shallow arch with elastic supports and arbitrary arch axis under arbitrary loads.
