Plane Shear Beam Element Based on CR Total Deformation Nonlinear Beam-column Theory

DOI：

 作者 单位 曾国良1，2，颜东煌1，陈常松1?覮，董道福3 （1. 长沙理工大学 土木工程学院，湖南 长沙 410114； 2. 湖南联智科技股份有限公司，湖南 长沙 410019； 3. 江西省长大桥隧研究设计院有限公司，江西 南昌 330031）

基于CR全量非线性计算模式，结合考虑梁元剪切效应的Timoshenko梁理论，采用梁柱理论（稳定函数格式）建立精细高效的综合考虑单元轴力、弯矩、剪力及其耦合效应的平面梁元分析理论. 采用梁柱理论，结合CR坐标法，完善高精度目标下的平面梁元几何非线性问题快速求解方法；以二阶梁柱理论为基础，推导考虑剪切效应的转角位移方程具体格式，获得刚度系数的精确稳定函数表达式，在保证数值精度及计算过程稳定性的前提下，将上述稳定函数幂级数化且提出对应的截断准则，随后推导了剪切变形影响下考虑梁元弓弦效应的单元节点计算表达式；编写程序对本文提出的方法与经典算例进行对比检验，并结合实际工程进行分析. 研究结果表明：对一般的多跨连续梁结构，利用本文计算理论能获得与经典算例一致精度的计算结果；对发生大位移的类如肘型刚架结构，利用本文理论可获得其考虑剪切效应的荷载-位移全过程变形曲线，其跃越屈曲第一及第二临界点亦与经典文献符合；对不同主梁形式的特大跨度斜拉桥结构，利用本文理论亦可得到合理的整体变形特征，同时明显提高分析效率.

Based on the CR full nonlinear calculation model and the Timoshenko beam theory，and considering the beam element shear effect，the beam-column theory（stability function format） is adopted to establish a fine and efficient plane beam element analysis theory，which considers the axial force，bending moment，shear and their coupling effect of the element. The beam-column theory and the CR coordinate method are used to improve the fast solving method for the geometric nonlinear problem of plane beam element under the high precision target. On the basis of the two order beam-column theory，the concrete form of the angular displacement equation for the shear effect is derived，and the exact stability function expression of the stiffness coefficient is obtained. The power series of the above stability function is transformed and the corresponding truncation criterion is put forward on the premise of ensuring the numerical accuracy and the stability of the calculation process. Subsequently，the formula for calculating the node force of the element considering the effect of shear deformation on the bowstring effect is derived，and the program is compiled and compared with classic examples. Moreover，it is analyzed with the actual engineering. The results show that，for the general multi-span continuous beam structure，the calculation results of the classical calculation can be obtained by the calculation theory proposed in this paper. For the class of large displacement，such as the elbow rigid frame structure，the full process deformation curve of the load-displacement considering the shear effect can be obtained by this theory，and the first and two critical points of leap buckling are also consistent with the classical literature. For the long-span cable-stayed bridge with different main girder sections，reasonable overall deformation characteristics can be obtained by using this theory，and the efficiency of analysis can also be significantly improved.