## 3 Validation of the Proposed Semi-Analytical Solution
### Validation of the Proposed Semi-Analytical Solution Regarding Experimental Results  
The innovative steel seismic energy dissipation device (Figure 5) consists of one lower plate anchored into the structure foundations, several vertical components with circular cross-section variable across their length which are fixed into the lower plate, and one active plate which is indirectly connected to the isolated structure. During an earthquake, the active plate moves, and, after annealing the gaps, bends the vertical components which then absorb seismic energy, entering the inelastic range of the material. Figure 4: A flow chart of the semi-analytical algorithm. Within the innovative project, the experimental testing of the vertical components with different geometric characteristics was realized, and force-displacement relationships were defined [26]. The tested vertical components are cantilever beams with linearly variable cross-section. The height of the conical body of the vertical components is 190 mm. Two combinations of the cross-sections diameter of the conical body were analyzed: VC1--bottom diameter 32 mm and top diameter 25.6 mm; VC2--bottom diameter 20 mm and top diameter 12 mm. At the top of the vertical components, a slave cylinder \(\Omega\) 24 \(\times\) 60 mm was introduced in order to accept the external load and transfer it to the conical body minimizing stress concentrations in the zone of force application. The prototypes of vertical components were fixed into the testing equipment via nut and thread on the base of the vertical component (Figure 6). The vertical components were made of steel C45 [31], which had been chosen as suitable for the aspects of ductility, toughness, and fatigue resistance. The mechanical properties of the steel material are presented in subsequent text. A specific test platform was created for the experiment, consisting of two basic segments: rigid base structure, which the vertical components were fixed to, and a laterally mobile structure connected to a horizontal hydraulic actuator for displacement application. The laterally mobile structure is constructed in a way to not constrain the slave cylinder at the top of the vertical components. The initial position of the vertical component and characteristic deformed configuration during the test are shown in Figure 7.  
During the experimental testing of the vertical components, the displacement was applied in a controlled so-called \"time mode\" with an increment of 1 mm/s. The displacement in the direction of the horizontal actuator piston was monitored using an electronic displacement transducer (LVDT), while simultaneously monitoring the force increase via an electronic dynamometer. For recording signals, an adequate multi-channel measuring-acquisition system connected to a personal computer was used, where rate sampling was  
Figure 5: The geometry and components of the innovative steel seismic energy dissipation device. Figure 6: The geometry of the tested vertical components of the energy dissipation device. With the aim of defining a mathematical model and an application of the semi-analytical solution for the vertical components, some idealizations of the problem are introduced. It is assumed that the vertical component is clamped into a rigid base structure of the test equipment. Therefore, in the contact between the nut and rigid base structure of the test equipment, there is no rotation of the nut, but the upper surface of the nut is free. Considering this, in the mathematical model, it is adopted that the vertical component is clamped at the level of the middle plane of the nut. The zone of the load application is in the middle plane of the slave cylinder at the top of the vertical component. The upper part of the slave cylinder from the load application zone toward the free end does not significantly affect the stress-strain state of the vertical component; therefore, that part is neglected.