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

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.