Structural full-field responses reconstruction by the SVD and pseudo-inverse operator-estimated force with two-degree multi-scale models

Abstract

The measured structural response currently is in a deficient state for structural health monitoring (SHM). To obtain complete structural responses, a good solution is first estimating the external input, then applying the estimated input to the finite element model (FEM) of a structure, and calculating the complete structural responses by the FEM. The key content is estimating the unknown structural input from the limited measurements, which indicates that an under-determined differential equation should be solved. In the previous literature, the dynamic equilibrium equation is usually converted into the state space and solved as a discrete-time linear system. In this paper, the external input time history is estimated from the difference of the state space vectors at different moments combining with the singular value decomposition. The complete structural responses are then obtained by applying the estimated load vectors to the FEM. Adopting a single degree of freedom system, the robustness of the approach is theoretically deduced. The modeling error can merely change the estimated external force, while the reconstructed structural responses retain accuracy. To improve computational efficiency, the two-degree multi-scale FEM is proposed. The first model is a simplified model of the original structure, only adopting interior elements such as the beam elements. This model is to quickly estimate the external inputs and to reconstruct the displacement responses. The second model is a FEM adopting both interior and superior elements such as the beam and shell elements, and it can reconstruct both the displacement and detailed strain–stress responses. Finally, numerical simulations of a complex bridge FEM are carried out demonstrating that the proposed approach has high accuracy under different loading conditions, including the vehicle load, impact load, and random load. An in-field experiment has also validated the proposed framework applies to the quasi-static plus dynamic loading condition.