# Estimation of Structural Deformed Configuration for Bridges Using Multi-Response Measurement Data
## 1 Introduction
(2018) was used as a basic method in various studies for estimating displacement [PERSON] et al. (2018); [PERSON] et al. (2018); [PERSON] et al. (2018); [PERSON] et al. (2018); [PERSON] et al. (2018). To overcome the limitations of conventional displacement sensors, various studies have been conducted to indirectly estimate the displacement using other response measurement data. However, these studies only focused on the estimation of the displacement at a certain point. The structural deformed shape (SDS), including the global displacement of a structure, is more useful for evaluating structural conditions compared to the displacement at a certain point because it can be used as a displacement load to evaluate the internal force of the structure through inverse analysis. Recently, as Global Navigation Satellite Systems (GNSSs), which can compensate for the shortcomings of conventional displacement sensors, have been developed, studies on estimating SDS using displacement data have been conducted [PERSON] et al. (2018); [PERSON] et al. (2018). [PERSON] et al. estimated the SDS based on the shape superposition and least-squares method in the same way as in previous studies, but a structural shape was applied as a shape function instead of a mode shape [PERSON] et al. (2018); [PERSON] et al. (2018). Although the mode shape derived by frequency analysis is the dynamic response including mass and stiffness properties of the structure, mode shape does not represent deformed shape by loading. Alternatively, the structural shape based on the structural stiffness is derived by static analysis applying a unit load to each node of the FEM model. The effectiveness of the estimation algorithm was verified using an FEM model (beam, truss, and beam + truss). Although the method showed good performance with a sufficient number of measured data points, application of the SDS estimation method for on-site structures has been quite limited because collectingsufficient displacement data measured from a GNSS can be quite expensive. Thus, the development of an affordable SDS estimation method that can reduce the required number of GNSS is essential for the cost-effective field application of the SDS estimation technique. Therefore, in this study, SDS estimation method using only displacement data developed by [PERSON] et al. [20] was extended to use multi-response (displacement, slope, and strain) data in combination. The addition of slope and strain data can reduce the required number of GNSS and cost for estimating rational SDS, because slope and strain sensors are generally cheaper than GNSS. An estimation algorithm was developed based on shape superposition with various combined response data (displacement, slope, and strain) and the least-squares method, and the structural shape was used as a shape function. The proposed SDS estimation method was verified using an FEM model (beam, truss). Additionally, three important issues that may affect the estimation accuracy were also reviewed during the verification process: first, comparison of shape function type (mode shape vs. structural shape); second, the placement method of the multi-response sensor; and third, the effectiveness of using multi-response data for SDS estimation. Comparisons of methodology between this study and the literature are listed in Table 1. The effectiveness of the estimation algorithm was verified by comparing it with the existing method presented in [20]. As a result of the comparison, the proposed method can reduce the number of displacement data required to estimate rational SDS by using additional slope and strain data. It is expected that cost-effective structural health monitoring (SHM) can be established using the proposed estimation method.