As another example the behavior of a three storey building subjected to dynamical loading of time varying frequency content is presented. The Tabas (Iran – 0.68g) earthquake was chosen for this particular analysis and the method of wavelets was applied for the construction of the corresponding time-frequency spectrum. The analysis demonstrates the response of the structure to a short of “sliding” resonance between the first eigen frequency of the structure and the primary frequency of the excitation, that both are shifted as time progresses. The structure due to plastification and the excitation due to a time – frequency variation that often occurs in strong ground motion excitations.
The original Tabas accelerogram is presented in Figure 12 accompanied by its corresponding time – frequency spectrum (Figure 13).
From the time-frequency spectrum one can identify the three peaks presented in Table 0. Since Max2 and Max3 have similar values and arise in relatively close time interval they can be substituted by an “equivalent” spectral frequency presented in the last row.
The 3D structure studied in the present example is depicted in Figure 14. A convenient set of geometrical/mechanical properties and hysteretic parameters was used in order to achieve an eigen frequency variation for the primary mode which would start from a value of 8.15Hz for a duration of
11.1 sec approaching 5.55 Hz around 13 sec. A dynamic analysis was performed using the Tabas accelerogram as excitation in the x direction The structure’s response is presented in Figure 15, where it can be observed that after t = 13.5 sec the top floor displacement increases rapidly, leading the structure finally to failure at t = 14.4 sec.
By careful examination of the results it turns out that most of the critical sections of the structural elements yield and fail within the time interval of 11 to 14.4 sec. Similar evidence is provided by examining the variation of the primary frequency of the building, that remains steady at approximately 8.15Hz for the first 12.5 sec, decreasing rapidly in steps due to the successive degradation phenomena to the value of 5.55Hz at around 13.5 sec (Figure 15).
Figure 15. a) Top floor displacement time history – b) Frequency Variation.
In order to verify whether the failure is due to the previously described pattern of successive resonance a non linear analysis is performed on the previous structure after filtering out the second highest spectral frequency of the excitation (of approximately 5.552Hz), i. e by considering an excitation without the time – frequency variation. For this case although the structure suffers a severe damage does not collapse as it remains steady at a top storey displacement of the order of 50 mm (Figure 16).
Figure 16. Top story displacement time history after filtering the accelerogram.
By examining the status of elements and its evolution in time a different development in the failure mechanism is observed mainly for t>13 sec, which is due to the absence of the filtered frequency.
A new computer program, “Plastique”, for the inelastic analysis of structures is presented. A macro modeling approach is implemented, combined with a spread plasticity and a yield penetration model to account for the inelastic phenomena on structural response. By introducing a smooth Bouc Wen type hysteretic model a close to reality simulation of R/C cyclic response is achieved. An iterative procedure is used in order to solve the equilibrium equations. The program proves to be a versatile tool offering different analysis options. In comparison to other similar nonlinear analysis programs, “Plastique” is capable of performing a full 3D inelastic analysis, using the same model for every plan orientation of the seismic excitation.
Inelastic analysis reveals a more realistic response of the structure, demanding though a more accurate description of the structural properties and their variations. Seismic hazard analysis becomes even more crucial, as compared to an elastic analysis, while time-frequency distribution of the frequency content of an earthquake, as demonstrated in the second example, can be decisive on the fate of the structure.