圆拱结构平面外稳定分析方法

为研究圆拱结构的平面外稳定问题，根据其与圆曲梁变形的相关性，首先考虑二阶弯矩效应建立了圆曲梁平衡方程，结合圆曲梁几何方程和物理方程，推导了考虑大位移的自由扭转圆曲梁的挠度控制方程和扭转角控制方程，分别给出了圆曲梁的挠度和扭转角的解析解一般格式及相应简化格式，同时获得了圆曲梁变形和内力表达式；在此基础上，提出了圆拱结构平面外分岔失稳和极值点失稳的分析方法；计算了4种圆拱结构的平面外分岔失稳临界荷载系数及失稳模态，并与文献模型作对比分析；根据4种圆拱结构的荷载-位移曲线进行了极值点失稳分析. 研究结果表明:采用本文模型计算的两端铰支圆拱平面外分岔失稳临界荷载系数与文献模型结果相差为0，而且可以得到工程中常用但鲜有人研究的跨中单铰拱及两端插支拱的平面外分岔失稳临界荷载；各类圆拱在面内均布径向荷载作用下的平面外分岔失稳模态均为单波对称；圆拱径向荷载的存在不改变圆拱面外荷载位移曲线的线性特征，却降低了面外抗弯刚度，当径向荷载达到某一值时，面外抗弯刚度为0，则发生面外失稳. 

Out-of-Plane Stability Analysis Method for Circular Arch Structures

In order to analyze the out-of-plane stability behavior of circular arch structures, according to the deformation correlation of circular arch with circular curved beam, the equilibrium equations of circular curved beam were firstly established by taking secondary moment effect into account. Combined with the geometric equation and physical equation of circular curved beam, the deflection control equation and torsion angle control equation of circular curved beam with free torsion were derived by considering large displacement. Both the general formats and corresponding simplified formats of analytical solution for circular curved beam deflection and torsion angle were obtained. Meanwhile, the expressions for deformation and internal force of circular curved beam were also derived. On this basis, the methods to analyze the out-of-plane bifurcation instability and extreme point instability of circular arch structure were presented. Critical load coefficients and their instability modes of four kinds of circular arch structure were calculated when out-of-plane bifurcation instability occurs, and the calculation results between the proposed model and the models from the literature were discussed; their load-displacement curves were calculated, and the extreme point instability of circular arch structures was analyzed. The results show that the critical load coefficient of the circular arch with two ends simply supported can be calculated by this model and has no difference with other models. In addition, this model is also useful to calculate the critical load coefficient of that with single hinge at mid-span or that with two ends inserted supported which are rarely seen in research. The out-of-plane bifurcation instability modes of all kinds of circular arch under in-plane uniformly distributed radial load are in the form of single symmetric wave. The radial load does not change the linear character of the out-of-plane load vs. displacement curve, but reduces the out-of-plane flexural rigidity. When the radial load reaches a certain value, the out-of-plane flexural rigidity becomes 0, and then the out-of-plane instability occurs.