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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。
In order to find out a nonlinear stability research method of arbitrary arch elastic support arch under arbitrary load, the in-plane buckling characteristics of shallow arch with vertical elastic support at any axis under concentrated load are studied, A non-linear equilibrium equation is established. The distribution characteristics of buckling paths and critical loads are analyzed by examples. The results are contrasted with the finite element solutions. In the process of derivation, the buckling mode with the same elastic support beam is taken as the shape function, and the arch axis, external load and structural displacement are unbroken, and the equilibrium equations of the basic equilibrium state and the buckling and bifurcation of the extremum point are obtained. Load, displacement of structure and internal force of structure, and then obtained two kinds of buckling forms of the equilibrium path and the critical load; analyzed the elastic stiffness parameters of the two kinds of buckling conditions under the equilibrium path and the ultimate load distribution law. The results show that the calculated results agree well with the finite element method, and the whole process of structural buckling can be tracked. The extreme point buckling and bifurcation buckles exist simultaneously. When the elastic support parameters change from symmetrical to asymmetric , The extreme point buckling path is divided into the basic path and independent separation path at a certain position, the bifurcation buckling path in some locations becomes the extreme point buckling path, accompanied by the emergence and disappearance of the corresponding critical load point; the critical load is only in the amount When the elastic constraint parameter of a class I is smaller, the critical load will no longer change with the change of the restraint stiffness when the restraint stiffness increases to a certain extent. The formulas for solving the buckling of a shallow vertical elastic arch with arbitrary arch under concentrated load can be provided for the ultimate solution of the nonlinear stability analysis of an arch with arbitrary axis under any load.