大转动对称平面梁单元共旋列式研究  

Research on CR formulation of large rotational symmetric plane beam element

作　　者：邓继华 陈历强[1] 田仲初 谭平 DENG Ji-hua;CHEN Li-qiang;TIAN Zhong-chu;TAN Ping(School of Civil Engineering,Changsha University of Science and Technology,Changsha 410114,Hunan,China;Ministry of Education Key Laboratory of Safety Control of Bridge Engineering,Changsha University ofScience&Technology,Changsha 410114,Hunan,China;Earthquake Engineering Research&Test Center,Guangzhou University,Guangzhou 510405,Guangdong,China)

机构地区：[1]长沙理工大学土木工程学院,湖南长沙410114 [2]长沙理工大学桥梁工程安全控制教育部重点实验室,湖南长沙410114 [3]广州大学工程抗震研究中心,广东广州510405

出　　处：《长安大学学报（自然科学版）》2023年第4期60-67,共8页Journal of Chang’an University(Natural Science Edition)

基　　金：国家自然科学基金项目(52278142);湖南省自然科学基金项目(2023JJ30019);湖南省教育厅项目(21A0187);长沙理工大学青年教师成长计划项目(2019QJCZ059);长沙理工大学土木工程重点学科基金项目(17ZDXK03);广东省自然科学基金项目(2015A030310141);桥梁结构健康与安全国家重点实验室开放基金项目(BHSKL21-06-GF)。

摘　　要：针对已有研究中共旋法非线性平面梁单元存在切线刚度矩阵不对称的问题,以合理选择平面梁单元共旋坐标系的原点及坐标轴方向入手开展研究。首先,将单元共旋坐标系原点设在梁单元左右节点坐标的平均值处,轴则始终为单元左节点指向右节点的连线方向,该坐标系随节点的刚体转动和平动而运动,在扣除单元刚体位移的基础上计算出单元节点坐标和位移;其次,基于结构坐标系与共旋坐标系下虚功相等的原理再结合几何一致性原则,建立平面梁单元在大转动、小应变条件下具有对称性的切线刚度矩阵和节点抗力算法,结合能将荷载增量法与位移增量法统一于一体的非线性方程组求解方法开发相应计算程序,并对梁端受弯矩的悬臂梁和William肘式框架算例进行计算和对比。研究结果表明:对于前一个算例,将梁等分成20个单元,采用荷载增量法,分成10级均匀加载,得到的数值解与解析解吻合良好,即使在悬臂梁弯曲为1个圆时,两者的差别不到0.0001,完全可忽略不计;对于后一个算例,将框架每根杆件均匀划分成10个单元,在非线性方程组求解方法转换为位移增量法以后,不仅能顺利通过荷载-位移曲线的极值点,而且计算结果与William的试验结果基本一致。研究成果为平面梁结构的非线性分析提供了依据。According to the shortage of tangent stiffness matrix asymmetry of nonlinear plane beam element by using co-rotational method in former studies,researches were carried out to reasonably select the origin and coordinate axis direction of the plane beam element co-rotation coordinate system.Firstly,the origin was set at the average value point of the left and right element node coordinates,and the x-axis was always in the connecting direction from the left node to the right.The coordinate system moved with the rigid body rotation and translation of the node,and the element node coordinates and displacement were calculated on the basis of deducting the element rigid displacement.Secondly,based on the principle of equal virtual work in structural coordinate system and co-rotating coordinate system as well as the principle of geometric consistency,the symmetrical tangent stiffness matrix and the node resistance algorithm of plane beam elements under the condition of large rotation and small strain were established.Combined with the solution method of nonlinear equations which unifies the load increment method and displacement increment method,the corresponding calculation program was developed.The cantilever beam and William's toggle frame with bending moment at the beam end were calculated and compared.The results show that for the former example,the beam is divided into 20 elements,and load increment method is adopted by 10 uniform loading levels.The numerical solution obtained is in good agreement with the analytical solution and the difference between the two is less than 0.0001,which can be negligible even when the cantilever beam is bent into a circle.As for the latter example,each frame member is homogeneously divided into 10 elements.The extreme point of load-displacement curve can be successfully passed through after transforming the solution method of nonlinear equations to displacement increment method and the calculated value is basically consistent with William's test.The research results can provide a po

关 键 词：桥梁工程 平面梁单元 共旋法 几何非线性 对称切线刚度矩阵 虚功原理 

分 类 号：U441[建筑科学—桥梁与隧道工程]