5.2.1. Individual Events
The dynamic response of ‘Ponte delle Torri’ to the accelerograms listed in
Table 2 is simulated using the equivalent pier model introduced in the previous section. The masonry compressive strength,
, is set equal to
according to the experimental results collected in Reference [
31], whereas the tensile strength,
, is assumed equal to
. These values lead to a typical ratio
for masonry material, usually ranging between
and
. The characteristic length
is set equal to
. The material mechanical parameters are reported in
Table 4. Two-stage analyses are performed: first, the structure is subjected to self-weight, and then the ground acceleration is applied along the bridge out-of-plane direction. The structural response is monitored in terms of out-of-plane displacement of point A,
, located at the top side of the pier, as indicated in
Figure 8a. To be noted is that the displacement induced by the application of self-weight is not represented in the time histories of the displacement
reported in the following.
Figure 9a,
Figure 10a,
Figure 11a,
Figure 12a,
Figure 13a,
Figure 14a,
Figure 15a,
Figure 16a,
Figure 17a,
Figure 18a,
Figure 19a and
Figure 20a show the significant part of the response obtained by applying each accelerogram, starting from the pier undamaged configuration, with the aim of highlighting the effect of the onset of degrading mechanisms and better understanding the influence of accumulated damage during the seismic sequence. To this end, the responses evaluated assuming a linear elastic constitutive behavior for the material (blue lines in
Figure 9a,
Figure 10a,
Figure 11a,
Figure 12a,
Figure 13a,
Figure 14a,
Figure 15a,
Figure 16a,
Figure 17a,
Figure 18a,
Figure 19a and
Figure 20a) are compared with those obtained for the damaged structure considering the damage model described in
Section 4 (red lines in
Figure 9a,
Figure 10a,
Figure 11a,
Figure 12a,
Figure 13a,
Figure 14a,
Figure 15a,
Figure 16a,
Figure 17a,
Figure 18a,
Figure 19a and
Figure 20a).
Focusing on the elastic responses, various levels of dynamic amplification emerge (note that different ranges for y-axis are used in
Figure 9a,
Figure 10a,
Figure 11a,
Figure 12a,
Figure 13a,
Figure 14a,
Figure 15a,
Figure 16a,
Figure 17a,
Figure 18a,
Figure 19a and
Figure 20a). These can be interpreted on the basis of both the effects of
(see
Table 2) and displacement response spectrum (see
Figure 9c,
Figure 10c,
Figure 11c,
Figure 12c,
Figure 13c,
Figure 14c,
Figure 15c,
Figure 16c,
Figure 17c,
Figure 18c,
Figure 19c and
Figure 20c) of each earthquake. For instance, although the ground acceleration recorded at ALT station is characterized by low
intensity (
), this induces the largest response amplification, as the displacement spectrum shows high values in correspondence of the pier first natural period (indicated with dashed vertical line in
Figure 15c).
The lowest displacement amplifications are detected in cases of the FORC, FHC, and MMO records. In fact, although characterized by high values of the
(
,
, and
, respectively), their response spectra show low values (lower than
) in correspondence of the structural periods (see
Figure 9c,
Figure 16c,
Figure 20c). In general, it emerges that earthquakes with comparable response spectrum ordinates (consider also RM03-SPM and T1212-MNF-ACC-SPM
) lead to similar maximum displacements, regardless of their
values. This confirms that the ratio between the forcing and natural frequencies plays a significant role on the dynamic response amplification, and the overall behavior is influenced by several factors highly correlated with each other. Even more complex is the interpretation of the structural response when the nonlinear phenomena are accounted for: displacements can be reduced or amplified with respect to the elastic case, depending on the combination of various phenomena [
10,
41]. Indeed, the onset and evolution of damage modify the structure mechanical properties, making it more flexible; thus, the structural natural periods increase, as a consequence. Correspondingly, the ordinates of the displacement spectrum may either increase or decrease, depending on the frequency content of the accelerogram.
Schematic representation of the degrading mechanisms evolution during each record is shown in
Figure 9b,
Figure 10b,
Figure 11b,
Figure 12b,
Figure 13b,
Figure 14b,
Figure 15b,
Figure 16b,
Figure 17b,
Figure 18b,
Figure 19b and
Figure 20b by the time histories of the
defined in Equation (
9), whose initial value represents the damage state induced in the structure by self-weight.
The comparison of the maximum attained displacements in the elastic and nonlinear cases can, again, be better understood referring to the displacement spectra in
Figure 9c,
Figure 10c,
Figure 11c,
Figure 12c,
Figure 13c,
Figure 14c,
Figure 15c,
Figure 16c,
Figure 17c,
Figure 18c,
Figure 19c and
Figure 20c. Let us focus on the responses to some representative earthquakes taken as example. Under the ALT accelerogram, the pier response significantly involves the first mode of vibration, whose period (
) increases at the onset of damage, likely moving the structure towards the decreasing branch of the displacement spectrum. This causes the reduction of the dynamic amplification with respect to the elastic case. Conversely, increment of the maximum response emerges in case of T1212 record when damage is activated. As a matter of fact, the corresponding displacement response spectrum (
Figure 12c) shows increasing branches near the main periods of the structure. To better clarify, in
Figure 21a,b, the variation of the first two periods of the damaging structure (
), normalized with respect to the elastic ones (
), are plotted for ALT and T1212 earthquakes, respectively. These are derived from a step-by-step modal eigenvalues analysis, performed on the basis of the damaged stiffness matrix. Progressive increments of the natural periods emerge in the first part of the loading histories, and then these stabilize at constant values when the evolution of damage stopped. The final values of the damaged periods are also indicated with dashed blue vertical lines in the most significant part of the displacement response spectra shown in
Figure 21, further validating the reported considerations.
When the earthquake intensity induces moderate structural damage and/or the branches of the elastic response spectra do not show high gradients, no significant differences emerge between elastic and nonlinear response, as evident, for instance, in cases of SPM, ACC, FHC, and MMO. Obviously, the interpretation of the phenomenon is not straightforward, considering the continuous evolution of the degrading processes during the loading histories. Indeed, a succession of time intervals in which the response of the damaged structure is alternatively amplified or reduced with respect to the elastic one could occur.
Finally,
Figure 22 gives a direct comparison between the maximum values attained by the displacement for all analyzed events, where the blue and red bars refer to the elastic and damaged cases, respectively.
To further investigation,
Figure 23 shows the correlations, in terms of regression lines depicted in red, between the maximum Global Damage Index
evaluated for each event with (a) maximum response displacement
, (b)
, (c)
, and (d) Housner intensity
. It appears that the
quite well correlates with the Housner intensity, both indexes being evaluated as average quantities. Even more satisfying is the agreement obtained with
and
, as opposed to
, which shows the lowest level of correlation.
An interesting feature is highlighted in
Figure 23a: although the general trend is that the higher the maximum displacement, the higher the
, this is not always true, i.e., an increase of the maximum achieved displacement in some cases corresponds to a decrease of damage. For instance, the highest value of
is obtained for MNF, in spite of the highest value of
corresponding to the ALT record. This is due to the deformed configuration assumed by the structure during the loading history, that influences the areas where maximum tensile strains and, consequently, damage occur. Thus, the pier top displacement
could be not sufficient alone to fully interpret the structural damage state. Focusing on the responses to MNF and ALT accelerograms,
Figure 24a,b show the distribution of the damage variable
D on the pier amplified deformed configurations referring to points B and C in
Figure 13a and
Figure 15a, respectively. It appears that ALT mainly activates mode 1 deformed shape (
Figure 24b), whereas MNF seems to combine mode 1 and 2, inducing a more spread damage along the pier height (
Figure 24a). Moreover, similar intensities of the maximum local damage
D emerge, although these correspond to significantly different values of the pier top displacement, which is
and
for MNF and ALT, respectively.
The correlations shown in
Figure 23a–d are evaluated on the basis of the global damage index defined in Equation (
9). If other definitions were adopted, such as that referring to 95% fractile (
Figure 6), it is reasonable to assume low variations in correlation levels. Indeed, the comparison between the time histories of the
and
in
Figure 13b and
Figure 15b points out different damage intensities but a similar trend of evolution, as well as analogous ratios
/
.
In
Figure 25, other two correlations are shown: the first (
Figure 25a) relates
with maximum response displacement
, and the second (
Figure 25b) analyzes the relation between Housner intensity
and
. These are noteworthy in being characterized by high values of the correlation coefficients, unlike
and
quantities, that are almost unrelated.
Finally, some considerations are made on the single-pier model in comparison with the complete bridge model. The single-pier model neglects several effects, among which the horizontal bending of the “deck” and the torsion of the piers. These effects generally lead to a non-conservative solution. A further effect is related to the non-synchronous motion of the ground, which, however, is outside the scope of this study. The horizontal bending of the deck produces a collaboration between the squat, stiff piers, and the tall, flexible piers. This effect was taken into account by defining an equivalent pier, such as to match the frequency of the first mode of the complete and single-pier model. For the second mode, however, the results of the two models can be significantly different. In fact, it has been noted that, in the complete model, the second mode of the central piers appears in the third and fourth modes, whose periods are equal to 0.672 s and 0.550 s, respectively (see
Table 3). In the single-pier model, the second mode has a period of 0.285 s (
Figure 8), resulting in a spectral displacement less than that of the complete model for almost all the accelerograms analyzed (
Figure 9b,
Figure 10b,
Figure 11b,
Figure 12b,
Figure 13b,
Figure 14b,
Figure 15b,
Figure 16b,
Figure 17b,
Figure 18b,
Figure 19b and
Figure 20b). An evaluation of the torsional effects in the piers can be performed by means of the complete model by making a comparison between the first mode, which induces the maximum normalized out-of-plane displacement at the top of the central pier (point A) equal to 0.305 mm, and the second mode which induces the maximum torsional rotation at the top of the same pier, which corresponds to the maximum normalized displacements ±0.080 mm at the lateral ends of the pier, equal to 26% of the normalized displacement of the first mode.
5.2.2. Seismic Sequence
As mentioned before, the response of the equivalent pier is here analyzed considering the input motions of the previous section arranged in series. In detail, the seismic sequence is composed by the 12 records listed in
Table 2 in order to to reproduce the 20 historical events reported in
Table 1. Only the significant part of each event, determined on the basis of the results of
Section 5.2.1 in terms of onset of the degrading mechanism and maximum reached damage, was included in the sequence with the aim of reducing the overall time of analysis. Tails of zeroes, properly calibrated for every single accelerogram, were added to dampen the oscillations of each single event. The resulting ground acceleration time history is shown in
Figure 26a.
Figure 26b illustrates the result of the simulation through the comparison of the responses corresponding to a linear elastic (blue line) and damaged (red line) constitutive behavior.
Figure 26c contains the corresponding time history of the
. The coupled effect of frequency content of each signal and the accumulated damage during the loading history induces amplification or reduction of the displacements of the damaged structure with respect to the elastic case. To emphasize the influence of pre-existing degradation,
Figure 22 summarizes the maximum displacements achieved by the damaging structure when subjected to single seismic events (red bars) and those registered for the sequence of the same signals (yellow bars). It can be noted that displacement peaks might significantly vary, and some comments on this phenomenon are reported in the following, focusing on MFN and ALT events. For MFN record, the difference is not relevant since the damage caused by the individual accelerogram (
Figure 13b), i.e.,
, is slightly lower than that reached during the sequence (
Figure 26c), i.e.,
. Indeed, the structure approaches the MNF earthquake after passing through FORC and RM03 records, which induce a low degradation level, comparable to the initial state of the pier when subjected to the isolated MNF event. The same considerations do not hold for the ALT accelerogram, where different amplification responses are detected in case of seismic sequence and individual earthquake. This discrepancy is explained by the fact that, having experienced a degradation of its mechanical characteristics during the loading history, the pier shows altered dynamic properties.
The values of the maximum displacement experienced by the pier, shown in
Figure 22, are also reported in
Table 5, where the second column refers to the elastic response, while the third and fourth correspond to the damage case for individual event and seismic sequence. Moreover, in
Table 6, the percentage difference for each record is reported, comparing elastic versus damage displacement for individual event (second column) and the seismic sequence (third column). Lastly, in the fourth column, the percentage difference between damage displacement for the individual event versus the value obtained with the seismic sequence is shown. In a few cases, the elastic and damage responses for both cases (single event and sequence) are almost indistinguishable. However, considering the damage responses, displacements are lower or higher in the case of the seismic sequence if compared to the individual event. As described above for some relevant events, this is due to the variation of the mechanical properties caused by the damaging process. The pier subjected to the single events starts from an initial condition characterized by a low damage state induced by self-weight; for the sequence, its properties are affected by the further significant or irrelevant deterioration caused by the previous events. Thus, the variation of the structural stiffness causes the reduction or amplification of the seismic response.
Looking at the evolution of
in
Figure 26c, it appears that damage mainly grows during FORC, RM03, and MNF records and, then, assumes its constant maximum value
. The most relevant distributions of the damage variable
D on the pier undeformed configurations are shown in
Figure 27a–c, referring to the points indicated in
Figure 26c with red circle markers. Finally, an interesting feature should be noted: the
attains its maximum value, and a similar intensity, under the same seismic event (MNF), both in the case of sequence and individual events.