Figure 2 Long-term recovery of the fundamental frequency of the
buildings (ANX, THU).
a, Co-seismic frequency computed over the years, showing slow
dynamics after the important earthquakes in 2005, 2008 and 2011.b, Zoom-in on the recovery after the 2011 event, showing the
conditioning cycles (i.e., R3a, R3b,
R3c). a, b, Each symbol corresponds to a single
earthquake, and the colour scale is related to the PTA. The larger
symbols were used to monitor the backbone curve. c, Log–linear
recovery of the normalized frequency variation (∆f/ff =
(f-ff)/ff , where ff =
maximum final frequency), which indicates the slopes computed from the
log–linear method applied to the periods shown in a andb .
Relaxation models applied to long-term structural recovery
Firstly, from the relaxation model24 shown in Figure
3a, the a and G parameters were computed as proxies for the
elasticity. The parameter a , which is inversely proportional to
the pre-seismic elastic modulus, increased sharply during the
post-Tohoku recovery. Parameter G, which is directly proportional to the
co-seismic elasticity, decreased. This confirms the increase in the
softening in both of these buildings. Secondly, the ratio
τmax/τmin computed from the relaxation
function in Figure 3b22 increased from
~6 to ~23 in the ANX building, and from
~9 to ~18 for THU. This ratio denotes
the different time-scale mechanisms that act in the time–logarithmic
segment of the recovery and characterizes the diversity of the crack
sizes14. From this we can infer that after the 2011
event the variety of the cracks in the ANX and THU structures was
quadrupled and doubled, respectively. Additionally, the gradual
reduction in τmax/τmin during the
re-loading cycles confirms the progressive crack-closing process during
the aftershocks inferred from the recovery slopes. In addition, after
the Tohoku earthquake we observe clear changes in the maximum frequency
variation (i.e., ∆f/f) for both models, which increased from
~3% to ~12% for ANX and from
~13% to ~33% for THU (Fig. 3a, b),
which is consistent with the modulus softening and then the global
change in the structural states.
A complete signature of the recovery process is given by the relaxation
spectrum23 shown in Figure 3c. We detect mechanisms
over five orders of magnitude in time, from t ~0.1 to
~1200 seconds; i.e., extreme values that are not
revealed by the previous models. The spectrum bandwidth represents the
range of the dominant relaxation times, and as the ratio
τmax/τmin, this serves as a hint of the
diversity of the crack sizes that are closed over the time of the
recovery. These data suggest that a large variety of crack sizes was
activated following the 2008 event (i.e., R2). The post-Tohoku spectrum
does not indicate new types of cracks; nevertheless, the maximum
spectrum amplitude is ~3.5-fold that observed in the
periods prior to Tohoku. This implies that the crack density increased
around 3.5-fold after 2011 for both of these structures. It can also be
noted that the spectra of the THU building are approximately 3-fold
those of the ANX building, showing the different levels of damage
between the buildings even before 2011. At the same time, conditioning
effects might have been significant in the recovery process for the ANX
building, which activated mechanisms with relaxation times in the order
of 101-102 seconds. In contrast,
during the recovery of the THU structure, the conditioning cycles just
contributed to the activation of inner small mechanisms, as shown by the
narrow left-shifted spectra R3a,b,c (Fig. 3c, right).
Thus, theoretical models applied to earthquake data from real buildings
fit the recovery of the fundamental frequency after earthquakes. These
data indicate that non-linear elastic processes within the structural
bond system might explain the transitory and permanent variations of
structural dynamic responses to seismic events. In particular, the
relaxation parameters reveal the internal material changes that are
related to cracking and stiffness degradation; i.e., in relation to the
structural health and safety of a building.