**3. Result of Numerical Analysis**

*3.1. Method of Numerical Analysis*

The study assesses the effect of seismic excitations on the geodesic domes, which were designed in accordance with two different methods for creating these types of structures. As mentioned earlier, these methods have been described in detail in the papers [52–54]. The numerical program RFEM and the Time History method were used for the analysis. This method allows for the analysis of the structure in given time steps. The time step of numerical analysis was equal to 0.02 s. The load was applied in accordance with the standard rules [59], i.e., in two horizontal directions (X and Y) and in the vertical direction (Z). In addition, the seismic excitations were applied simultaneously in all direction (direction of X, Y and Z) to the rigid supports.

Table 1 shows the results for the analysed domes under seismic excitations. The analysed records had different durations and intensities. The dynamic (seismic) analysis was carried out on the basis of the numerical models presented in [52–54]. The abovementioned publications applied the rule that the most stressed strut in a given group was at a level of 90%. For the purposes of analysing the obtained results, eight numerical models were created, by different structural methods (method 1 or method 2). In addition, the seismic excitations (Ancona, Denizil, Friuli, and Kilini) were used. As a result, eight models were obtained, i.e., model I (method 1, Ancona), model II (method 2, Ancona), model III (method 1, Denizil), model IV (method 2, Denizil), model V (method 1, Friuli), model VI (method 2, Friuli), model VII (method 1, Kilini), and model VIII (method 2, Kilini).

#### *3.2. Displacements*

Based on the analysis, it can be seen that the maximum displacement values were recorded for the horizontal directions X and Y (for the Ancona and Denzil records in the Y direction, for the Friuli and Kilini records in the X direction). In the vertical direction, the displacements were much smaller (Table 2). For all four of the seismic records analysed, higher values were obtained for the dome designed according to method 1 than for the dome generated according to method 2.


**Table 2.** Results from numerical analysis.

In the case of structures shaped on the basis of method 1, the highest displacement values were obtained with the Ancona record (model I, 23.4 mm; Figure 5a). In the case of other records, the displacement values were smaller. In relation to Ancona, they accounted for 56% for Denizli (model III, 13.3 mm; Figure 5b), 17% for Friuli (model V, 3.9 mm; Figure 5c), and 8% for Kilini (model VII, −1.9 mm; Figure 5d). A similar tendency was noticeable with the domes created according to method 2. The Ancona forcing caused the highest displacement values (model II, −7.6 mm; Figure 5a), while for the remaining records, the maximum displacement values were 90% (model IV, Denizli, −6.9 mm; Figure 5b), 30% (model VI, Friuli, 2.3 mm; Figure 5c) and 14% (model VIII, Kilini, 1.1 mm; Figure 5d) of the values from the Ancon record, respectively.

**Figure 5.** Maximum displacements in numerical model records: (**a**) Ancona—models I and II, (**b**) Denizli—models III and IV, (**c**) Friuli—models V and VI, (**d**) Kilini—models VII and VIII.

There was also a noticeable difference in the size of the maximum displacements between the structures shaped according to methods 1 and 2. The greatest differences were obtained for the Ancona record, in which the dome shaped according to method 1 showed about three times greater maximum displacements (model I, 23.4 mm; Figure 5a) than the dome designed according to method 2 (model II, 7.6 mm; Figure 5b). In the case of the records of Denizli, Friuli, and Kilini, the differences were about two times greater for method 1 compared to method 2.

It should be added that the value of displacements for the analysed numerical models decreases with a decrease in the value of ground acceleration, which was recorded for individual earthquake records. The highest values were obtained for the Ancona record, i.e., the record with the greatest ground acceleration (3.740 m/s2).

It can also be seen that the domes were excited for approximately 1.5 to 4.0 s of the recording duration as a result of the imposed excitation. In the first stages of the excitation, the displacements had a negligible value, which was undoubtedly influenced by the characteristics of the given excitation (in the first seconds of recording, the acceleration values were low). However, after the occurrence of the maximum acceleration in a given record (Ancona, Denizli, Friuli, Kilini), the recorded displacements had a similar value to the maximum ones. This tendency was noticeable for all analysed records, regardless of the direction of the excitation. Even the decrease in acceleration did not cause any significant reduction in the displacements of the analysed domes. For both of the domes created according to methods 1 and 2, it can be seen that the moments of occurrence of the maximum values of displacement in the domes were similar to each other. They occurred at very similar time intervals. Only when recording Denizli (model III and model IV) did the maximum displacements appear at significant time intervals: for method 1 (model III) around 17 s and, for method 2 (model IV), approximately 5 s.

#### *3.3. Axial Forces*

Based on the numerical analysis, it can be concluded that the maximum axial forces occurring in the struts in both of the analysed methods of shaping geodetic strut domes (methods 1 and 2) were similar for each given excitation. Thus, there was no clear trend as in the case of displacements where, for the analysed models built according to method 1, the maximum values of displacement were always higher than for the models shaped according to method 2. It was noticed that for high-intensity records (Ancona and Denizli excitations—Figure 4a,b), higher values of axial forces were obtained for models I and III, where method 1 was used to generate the structure. However, in the case of records of lower intensity (Friuli and Kilini—Figure 4c,d), higher values were recorded for the dome designed according to method 2 (models VI and VIII). The difference in the maximum values of the axial forces of the domes generated according to methods 1 and 2, for the four records analysed, was from 4% to 32% and was largest (32%) was for the Ancona record.

The greatest forces were recorded with the Ancon excitation in model I (−69.54 kN— Figure 6a), which was designed according to method 1. In the other models, the values were lower and amounted to: (i) 47.01 kN (method 2, model II, Ancona—Figure 6a), (ii) 40.54 kN (method 1, model III, Denizli—Figure 6b), (iii) 37.63 kN (method 2, model IV, Denizli— Figure 6b), (iv) 13.31 kN (method 2, model VI, Friuli—Figure 6c), (v) 12.82 kN (method 1, model V, Friuli—Figure 6c), (vi) −6.76 kN (method 2, model VIII, Kilini—Figure 6d), and (vii) 5.56 kN (method 1, model VII, Kilini—Figure 6d). On the basis of the obtained results presented in Figure 6, it can be seen that the tensile and compression forces in the struts were of a similar value. In some models, almost identical values for the compressive and tensile forces were obtained.

It can also be seen that the axial forces for all analysed models (I–VIII) were recorded after the appearance of the maximum acceleration in the record. From that moment, they remained at relatively equal levels, similar to the displacements. Concentrating on Figure 6, it can also be seen that the maximum forces (for method 1 and 2 of strut dome design) appeared at a similar time. Thus, it can be concluded that the method of shaping the dome structure does not have a significant impact on the moment of the maximum axial forces in the structure appearance. Moreover, no clear tendency can be drawn as to how the length of the records and their intensity impacts the appearance of maximum values over time. As can be seen in Figure 6, the location of the maximum axial forces occurrence in time was very varied.
