# Special Finite Elements with Adaptive Strain Field on the Example of One-Dimensional Elements
## 2 Research Problem  
As an object of analysis, damaged posttensioned concrete girder KBOS-24 [22] has been chosen. In this case, cracks were observed between the column and the top chord of girder, which has been shown in Figure 2b. Analyzing the problem, it was found that the valid girder model can and should take into account hinge connections at both ends of column. This means at the places of connection with rigid top and bottom chord. In Figure 2b, an extreme case of concrete discontinuity between the column and the top chord of the girder has been shown. The same damages (but much smaller) have been also noticed on the other columns of the girder KBOS-24 as well. According to the research, these cracks are mainly caused by the technology of realization. It should be mentioned that such a types of girders were produced and assembled in Poland in the 60's of the last century, when the Polish economy was still recovering from the war damage. Potential calculation models have been illustrated in Figure 3. First model, presented in Figure 3a, is a typical 3D model made in professional computer program. The second, shown in Figure 3b, is the computational model that was made using the \"ORCAN\" Structure Analysis System (Version 0.98, Bialystok University of Technology, Bialystok, Poland) [23]. The former model, apart from a lot advantages, also has disadvantages like quite problematic geometry or difficulties in finding the internal forces (shear or normal). The latter model is simplified but based on calculations made during the design process of posttensioned girders. In this model the columns were modeled using multi-area double-hinged linear elements. The theoretical assumptions of multi-area elements have been described in the next chapter. Figure 2: Posttensioned concrete girder KBOS-24: (**a**) View of erected roof construction; (**b**) Damage of the column; (**c**) View (span) of whole girder. In case of construction damage shown in Figure 2b, analysis of the crack propagation can be done in many different methods. The most obvious is the rearrangement of the mesh discretization in the calculation, computational model. This method of analysis is effective but computationally expensive. It also requires complex calculation algorithms regarding nonlinear analysis. Of course, nonlinear analysis can be modeled using the Gauss integration points in the finite element. However, in relation to the rearrangement of the mesh discretization method, this solution can be less precise. This is caused by the necessity of approximation for results obtained in the Gauss points, e.g., transmission of strain values to the nodes of finite element. In the performed analysis, the authors decided to use one-dimensional multi-area elements as girder columns, shown in Figure 3b. This is justified as follows:  
* The girder column can be modeled using single one-dimensional double-hinge element, because it is proved by the practical aspects concerning erection of the concrete posttensioned girders; lack of concrete continuity in the area of column connections with girder chords was observed very often. * In the procedure of the integration in sub-areas (described in the next section), the width of crack can be easily adjusted, and the localization of crack occurrence can be chosen as well. This second aspect is especially relevant in transient processes, where the location of damage is unknown in advance and depends on the state of structure response. * There is an opportunity whereby elastic elements with significant differences of stiffness characteristics can be connected in series and described by a single finite element. The adequacy of the proposed method has been checked exactly on the example of the girder column. For this purpose, the single column was separated and the tension force P was applied, as shown in Figure 4. The value of the displacement in form of extension of such a rod (bar element) was calculated.