# Special Finite Elements with Adaptive Strain Field on the Example of One-Dimensional Elements
## 3 General Theoretical Assumptions of One-Dimensional Multi-Area Elements
\right]\cdot\mathbf{J}\cdot d\xi=\left[\begin{array}{cc} \frac{2(EA)_{k}}{L}\frac{(\chi_{(1+1}\cdot\eta_{i+1}-\chi_{l}\cdot m_{l})^{2}} {(\xi_{l+1}-\xi_{l})}&-\frac{2(EA)_{k}}{L}\frac{(\chi_{(1+1}\cdot\eta_{i+1}- \chi_{l}\cdot m_{l})^{2}}{(\xi_{l+1}-\xi_{l})}&\\ -\frac{2(EA)_{k}}{L}\frac{(\chi_{(1+1}\cdot\eta_{i+1}-\chi_{l}\cdot m_  
the calculation of the multi-layered subsoil, where each layer is characterized by different thickness and strain modulus [29]. One of the examples where subsidence values have been calculated is the analysis of a layered subsoil under a building wall. In the presented case the calculations are limited to one-dimensional elastic model of subsoil [30]. In the computational test, the subsidence values for the multi-layered subsoil under the wall of a 5-storey building were compared, as shown in Figure 8. For calculation a reinforced concrete wall measuring 9.6 m \(\times\) 14.0 m \(\times\) 0.15 m, loaded by the floor ceilings and dead load has been taken. Calculations were made in three variants, using different calculation methods. In the variant I the calculations were carried out according to hypothesis of the elastic subsoil parameters ([PERSON]'s hypothesis) and the formulas given in [30]. The problem of modeling the multi-layered subsoil using the hypothesis of the elastic subsoil is to determine the stiffness of the layer (as a spring). The solution is to use exactly the [PERSON]'s hypothesis, according to which the subsidence of the elastic subsoil \(s\) is proportional to the acting load \(q\), as in the following Expression (12):  
\[q=\ k_{z}\cdot s \tag{12}\]  
The value of flexibility coefficient \(k_{z}\) for homogeneous soil to the depth \(z\)[30] can be determined according to the formula:  
\[k_{z}=\frac{E_{0}}{\omega\cdot B\cdot(1-v^{2})} \tag{13}\]  
On the other hand, in case of the multi-layered subsoil, the coefficient \(k_{z}\) is the sum of the flexibility coefficients for the individual layers of the subsoil. For a single layer \(i\), this coefficient is determined from the following expression:  
\[k_{z}^{i}=\frac{E_{0}^{i}}{\Delta\omega_{i}\cdot B\cdot(1-v^{2})}\text{and} \quad\Delta\omega_{i}=\omega_{i}-\omega_{i-1}, \tag{14}\]  
Figure 8: Parameters of multi-layered subsoil calculation model. where:  
\(q\)--acting load,  
\(E_{0}\)--strain modulus of the subsoil,  
\(B\)--the width of the loaded area (foundation),  
\(\
u\)--coefficient of the lateral expansion of the ground,  
\(\omega_{i}\) (\(\Delta\omega_{i}\))--influence coefficient, depending on the shape of the loaded area (foundation) determined according to the corresponding tables and nomograms [30]. In the variant II, calculations have been made using a two-dimensional FEM model in order to compare and control the results. For plane elements, that were implemented in ground subsoil modelling, a transformation of the strain modulus has been applied in order to use the flexibility of the subsoil as in [PERSON]'s hypothesis:  
\[E^{i}=E^{i}_{0}.\frac{h_{i}}{\Delta\omega_{i}} \tag{15}\]  
where:  
\(E^{i}_{0}\)--strain modulus for particular layer of the ground,  
\(h_{i}\)--the thickness of the particular layer,  
\(\Delta\omega_{i}\)--influence coefficient, depending on the shape of the loaded area (foundation) determined according to the corresponding tables and nomograms [30]. In the variant III, calculations have been performed using special one-dimensional finite elements integrated in sub-areas, presented in the paper and implemented in the authors' analysis system \"ORCAN\" [23]. As a results of the calculations, the following subsidence values have been obtained. Variant I: [PERSON]'s hypothesis:  
\[s_{1}=