多分布荷载下拱结构面内屈曲近似解析

Approximate closed-form solution for in-plane buckling of arches subjected to multi-pattern distributed load

经典拱桥稳定理论基于单一分布荷载,而现代拱桥的荷载已发展成为由桥面水平均布荷载与主拱圈弧长均布荷载组成的多分布荷载,拱桥稳定理论亟需适应拱桥工程的发展。针对多分布荷载下拱结构面内线性屈曲没有解析解的现状,提出一种高精度近似解析方法用以解决该问题。基于拱结构线性屈曲后拱轴无伸缩的基本假定与包含全部非线性项的应变表达式,推演直角坐标系下任意拱结构线性屈曲应变表达式。在该屈曲应变基础上,基于虚功原理与泰勒近似展开得到多分布荷载下拱结构虚功的近似表达。根据Ritz法与拱结构屈曲波形函数的基本假定,推演多分布荷载下拱结构面内线性屈曲荷载的近似解析。与有限元法结果的对比表明,提出的基本假定满足工程需求,多分布荷载下拱结构面内线性屈曲时满足拱轴无伸缩基本假定与屈曲波形函数基本假定。推演的多分布荷载下拱结构面内线性屈曲荷载的近似解析,在大范围矢跨比与荷载比参数下具有较高的精度,两铰拱与无铰拱近似解析的最大相对误差为3.24%与8.52%,在矢跨比小于1/4的拱桥工程参数范围内,两铰拱近似解相对误差约为0.35%,无铰拱近似解析相对误差少于0.67%。

The basis of the classical arch bridge stability theory is the single distributed load,while the load of modern arch bridges has developed into a multi-pattern distributed load consisting of the horizontal uniformly distributed load along the span from the deck system and the uniformly distributed load along the arc from the main arch ring.The arch bridge stability theory needs to be adapted to the development of arch bridge engineering urgently.Because of the lack of analytical solution for the in-plane elastic buckling of arches subjected to multi-pattern distributed load,an approximate closed-form solution was proposed to solve this problem.Based on the assumption that no membrane strain occurs in the buckling arch and the strain-displacement expression with all nonlinear terms,the linear strain-displacement expression of buckling arch was derived.According to the obtained strain-displacement expression,an approximate closed-form solution of virtual work of arch subjected to multi-pattern distributed load was derived.The approximate closed-form solutions for the in-plane elastic buckling of arches subjected to multi-pattern distributed load were derived based on the Ritz method and the buckling mode functions of arch.Comparisons against the results of the finite element method demonstrated that the basic assumptions proposed in this paper meet the engineering requirements,and the basic assumption that no membrane strain occurs in the buckling arch and the buckling mode functions can be satisfied.The approximate closed-form solutions for the in-plane elastic buckling of arches subjected to multi-pattern distributed load have high accuracy,and the maximum relative errors of pin-end arch and fixed-end arch are 3.24%and 8.52%in the large range of the rise-to-span ratio and the load ratio parameters.The relative error is about 0.35%for pin-end arches and is less than 0.67%for fixed-end arches when the rise-to-span ratio is less than 1/4.