基于应变梯度理论和逆有限元法的Timoshenko梁形状传感建模  

Shape sensing modeling of Timoshenko beam based on the strain gradient theory and iFEM method

作　　者：赵飞飞 郭彦浩 保宏 王伟 张峰 Feifei Zhao;Yanhao Guo;Hong Bao;Wei Wang;and Feng Zhang(School of Electronic Engineering,Xidian University,Xi’an 710071,China;School of Mechanics,Civil Engineering and Architecture,Northwestern Polytechnical University,Xi’an 710072,China)

机构地区：[1]不详

出　　处：《Acta Mechanica Sinica》2023年第12期71-85,共15页力学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant Nos.51775401,and 51675398)。

摘　　要：大型结构的几何非线性变形严重威胁着结构体系的安全.因此,对在役结构变形进行实时监测具有重要意义.然而,目前基于线弹性理论提出的逆有限元法(iFEM)并不适用于非线性变形,为此,提出了一种非线性逆有限元法来建立结构形状感知模型.首先给出了应变梯度Timoshenko梁模型的运动学和动力学变量,并建立了其几何非线性变形控制方程.然后,推导出了转角函数的解析解,并建立了非线性形状感知模型.其中,采用等几何分析(IGA)方法构造插值形状函数.由于位移函数以旋转形式表示,所以可以有效地避免“剪切锁定”问题.随后,利用离散表面应变测量值推导出了实际转角函数,并建立了转角自由度在直角坐标系与曲线坐标系之间转换关系.最后,以一悬臂梁为例,将重构位移与理论位移进行了比较.数值结果表明,不论是集中荷载还是分布荷载,所提的非线性逆有限元法均具有良好的重构性能,重构误差都小于2.5%.The geometrically nonlinear deformation of the large scale structure seriously endangers the structural system safety.Thus,it is of great significance to real-time monitor the structural deformation in service.However,the current inverse finite element method(iFEM),which is presented based on the linear elastic theory,is not suitable for nonlinear deformation.This paper proposes a nonlinear iFEM for establishing the shape sensing model.Initially,the kinematics and kinetics of the strain-gradient Timoshenko beam model are presented,and the governing equations for geometrically nonlinear behavior are formulated.Then,the analytical solution of the rotation function is presented and the nonlinear shape sensing model is established.Therewith,isogeometric analysis(IGA)approach is employed to construct the interpolation shape functions.Due to displacement functions expressed in terms of rotations,the“shear locking”problem can be effectively avoided.Subsequently,the experimental rotation functions are deduced using discrete surface strain measurements and the rotation transformation is established between Cartesian and curvilinear coordinate systems.Finally,a cantilevered beam is used as a case study to compare the reconstructed with theoretical displacements.The numerical results demonstrate the excellent performance of the proposed formulation,where the reconstructed errors are less than 2.5%for both concentrated and distributed loads.

关 键 词：TIMOSHENKO梁 应变梯度理论 逆有限元法 转角自由度 直角坐标系 形状函数 曲线坐标系 等几何分析 

分 类 号：TP212.9[自动化与计算机技术—检测技术与自动化装置]