# Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models
## 1 Introduction
Another important application of the FE model updating is damage detection and structural health monitoring [25]. [PERSON] and [PERSON] [26] presented a way to locate damages in a structure based on the changes in Young modulus that indicate the loss of stiffness at the crack location. Thus, in the model updating process the Young modulus is chosen as the updating parameter. The method is based on the correlation of the reference FE model which depicts the undamaged structure with the test data (natural frequencies and mode shapes) obtained by measuring on the damage structure. [PERSON] et al. used a similar approach to identify bridge damage in their case study [27]. In the procedure they have proposed, the reference FE model is updated using a model that can reproduce the experimental modal data of the damaged state. These data were obtained by OMAX method (i.e., output-only data are used when updating the FE model). The crack location is determined by correction factors that express the change in bending and torsional stiffness of the model. This paper deal with procedures aimed at improving boundary conditions in the FE model. Conventional boundary conditions are usually defined as ideal. It is common practice that constraints in FE models are modelled as perfectly rigid, while in fact they have a certain degree of flexibility. Although this is acceptable in many cases, there are problems, such as vibration analysis, fatigue analysis, etc., in which it is necessary to obtain an accurate model response. The idealization of the boundary conditions can have a significant effect on the results and may cause them not to match the experimental test data. For this reason, an equivalence method is proposed to replace a conventional boundary condition or a certain part of the model. The equivalent boundary condition is represented by a bushing connector whose stiffness parameters can be determined either by a series of numerical static analyses or using the FE model updating. The updating method used in the paper is based on FE optimization analysis, in which the natural frequencies of the model are fitted to the frequencies determined by experimental modal analysis. The authors successfully used this approach to determine the material constants of the isotropic material [28], the homogenized laminate [29], and to determine the fiber orientation of the composite plate [30]. The use of the equivalence method allows to replace an ideal (perfectly rigid) boundary condition with a constraint of finite stiffness. The aim is to refine the model behavior in a given place and improve its response. The equivalent constraint can be modelled using a generalized spring element or a bushing connector. In this paper, the equivalence method is used to replace a part of the model. This requires determining stiffness at the point where the model is split. Two ways to determine equivalent stiffness coefficients are presented below. As will be shown, this approach provides a significant simplification of the FEmodel, while maintaining its dynamic behavior. The main benefit is then the shortening of the calculation time.