非理想边界拱的面内失稳模式与屈曲荷载

康厚军+易壮鹏+曾有艺

摘要：将拱结构中既非固结也非铰支的非理想边界考虑为沿不同方向的具有一定刚度的弹性约束，利用变形几何关系和能量变分原理推导了拱的非线性平衡方程.以圆弧拱为例建立了径向均布荷载下外荷载与结构内力、径向位移之间的关系，通过定义拱的深浅参数和临界约束刚度进行分析并得到了跳跃屈曲、分岔屈曲等的发生条件及存在区间.本文方法所得屈曲路径和屈曲荷载与有限元法所得结论吻合良好，且用数值法分析了不同约束刚度时的屈曲路径和临界荷载.结果表明，临界深浅参数和临界约束刚度对圆弧拱的屈曲模式及屈曲临界荷载影响显著.

关键词：屈曲;圆弧拱;非理想边界;分岔屈曲;变分原理

中图分类号：O343.9 文献标识码：A

Planar Buckling Mode and Critical Load

for Arch Structure with Non-ideal Boundary

KANG Hou-jun1， YI Zhuang-peng2， ZENG You-yi2

（1. College of Civil Engineering， Hunan Univ，Changsha，Hunan410082， China;

2.School of Civil Engineering and Architecture， Changsha Univ of Science and Technology，Changsha，Hunan410114， China）

Abstract： The non-ideal boundary conditions （neither fixed nor hinged） of arch structure were considered as an elastic constraint with certain stiffness in different directions， and the nonlinear equilibrium equation was determined by using deformation geometric relation and energy variation principle. A circular arch under radial uniform load was taken as an example to establish the relationships between the external load and the internal force， and the radial displacement. By defining the shallowness and critical constraint stiffness， the snap-through buckling and bifurcation buckling were studied and the occurrence condition and distribution range were investigated. The buckling path and critical buckling load in the proposed method were in good agreement with the results from the finite element method. And the numerical method was used to study the buckling path and critical buckling load for different stiffness of elastic constraint. The results show that the critical shallowness and the critical constraint stiffness play a fundamental role in the buckling mode and critical buckling load for circular arch.

Key words： buckling; circular arch; non-ideal boundary;bifurcation buckling; variation principle

拱结构[1]在土木、机械和航天航空等领域应用广泛.拱作为一种基本结构构件具有优良的受力特性，其力学特性受到国内外学者[2-3]广泛关注.如周期激励下内共振[4-6]时的分岔和混沌特性，冲击荷载作用下弹性浅拱的跳跃屈曲[7]等.静力方面，近年来，Pi等[8-10]采用解析法与有限元法对各种荷载与边界条件下拱结构的非线性屈曲特性进行了深入系统的研究.卫星等[11]探讨了多种参数对拱结构考虑2阶效应时弹性屈曲特性的影响.程鹏和童根树[12]综述了径向均布荷载下圆弧拱的面内屈曲特性.郭彦林等[13]提出了压弯圆弧拱平面内稳定承载力的设计建议公式.

这些文献侧重于研究边界为理想固结或铰支时拱的力学性能.结构的复杂分析在很多情况下需考虑复杂边界，如：大跨系杆拱桥中系杆将两端连起来，系杆与竖向弹性支座、地基的作用可抽象为轴向、径向弹性约束;机械拱臂或曲臂与相邻结构弹性连接，共同受力，可将其考虑为弹性约束;弹性地基上的拱型结构在外荷载作用下边界考虑为弹性更加合理.本文以圆弧拱为例，将非理想边界考虑为径向、轴向弹性约束，通过能量变分原理[14]建立非线性平衡方程，得到外荷载与结构内力、位移的关系，并分析屈曲模式与结构重要参数的关系.

1基本方程与稳定分析

1.1变形几何关系

图1（a）所示圆弧坐标下的圆弧拱，径向均布荷载q增至一定程度时会发生分岔屈曲（图1（b））或跳跃屈曲（图1（c）），2Θ为展开角，R为半径，θ为角坐标，kvi， kwi（i=±Θ）分别为两端径向、轴向弹性约束刚度.屈曲前变形呈现非线性，求解屈曲荷载及变形时需考虑屈曲前非线性的影响.图1（a）中拱上任意一点P总的轴向应变εP=εm+εb，其中薄膜应变εm和弯曲应变εb分别为：