Acceleration Time Histories at Tops of Foundation Footing, Tower, and Downhole
Fig.
3 shows equiscaled plots of acceleration time histories for all horizontal and vertical channels at the top locations of footings at Piers W2 and E2, as well as at the tower location. They are considered the horizontal and vertical input excitation of the SAS structural system supported by the tops of foundation footings. It is not surprising that the amplitudes of accelerations (longitudinal, transverse, and vertical) are greater for those footing top locations that are in deeper alluvial geotechnical media (Pier E2 and the tower location) as compared to those at Pier W2, which is on rock (Channels 4, 5, and 6). Also, the strong shaking duration of the Pier W2 channels (4, 5, and 6) are visually smaller in their respective orientations than those of Pier E2 and the tower location. Simply stated and as expected, motions at the rock location are amplified at alluvium locations.
However, to compute the strong shaking duration of input horizontal motions, the recorded accelerations (recorded by Channels 23 and 24) at the borehole bottom (rock at elevation is 59.5 m) are used. Fig.
4(a) is developed to display acceleration time history plots of horizontal downhole data (of Channels 23 and 24). Fig.
4(b) shows the assessment of strong ground shaking duration a using summed squared horizontal accelerations method (
Trifunac and Brady 1975;
Boore and Thompson 2014). The 5%–95% method is used to define the strong shaking duration as applied in the figure in which both horizontal longitudinal and transverse Channels 23 and 24 at the borehole bottom indicate 13.25 s as an estimate of the duration of strong shaking input motions to the SAS. Later in the paper, this figure is referred to in discussions of the elongated duration of shaking at the superstructure of the SAS.
Amplitude Spectra and Spectral Ratios (Longitudinal, Transverse, and Vertical)
First, amplitude spectra of longitudinal, transverse, and vertical accelerations recorded by the channels of accelerometers (Table
1) are computed. Second, spectral ratios of the amplitude spectra of accelerations are computed. Longitudinal spectral ratios are computed with respect to (w.r.t) channel (ch)23 (the longitudinal downhole channel at elevation
at rock). Similarly, transverse spectral ratios are computed w.r.t ch24 (the transverse downhole channel) and vertical spectral ratios are computed w.r.t ch22 (the vertical downhole channel). However, using acceleration data did not yield the expected low frequency (
) amplitudes in the spectra or the ratios. Therefore, in this paper, only the amplitude spectra of displacements and their spectral ratios are presented.
In Fig.
8, both the amplitude spectra [Figs.
8(a–c)] of longitudinal displacements and the spectral ratios in Fig.
8(d) (with respect to the amplitude spectrum of downhole ch23) allow identification by peak picking of the lowest frequency at
and others at 0.71, 0.83, and 2.08 Hz. This first modal frequency at
is clear only in these plots of spectral ratios of amplitude spectra of displacements and not in those computed from accelerations. The frequency at 0.71 Hz most likely appears due to coupling with modes other than longitudinal. It is also noted that both the amplitude spectra and spectral ratios are repeatable for all channels in each frame.
In Fig.
9(a) computed amplitude spectra of transverse displacements and (b) their spectral ratios that ents allow the identification of frequencies as
, 0.77, 1.78, and 3.13 Hz. Again, both the amplitude spectra and spectral ratios are repeatable for all channels. Similarly, in Figs.
9(c–e), computed amplitude spectra of vertical displacements of deck and cable locations, and Figs.
9(f–h), spectral ratios of the amplitude spectra, allow the identification of frequencies at
and
.
Displacement Time Histories and Amplitude Spectra for Cables
Vertical acceleration and displacement time histories of cable channels (20,17, 60, and 54) are plotted in Fig.
10. Their amplitude spectra are included and are consistent with the amplitude spectral and spectral ratio peak for the first mode at
for the deck (Fig.
10). This is not surprising because cables are connected to the deck with hangers, which ensures that in the vertical direction, cable and deck motions have to be compatible both in amplitude and frequency.
In Figs.
10(a–c), time histories of previously discussed cable-specific oriented horizontal [longitudinal (CL) and transverse (CT)] displacements are plotted. C1L and C1T refer to the 180-m-long (590.55-ft) side span cable, and C2L and C2T refer to the 385-m (1,261.10-ft) main span cable [Fig.
10(a)]. Orientation angles of the cables w.r.t the longitudinal axis of the bridge are shown in Fig.
2(a). Fig.
10(a) displays that the amplitudes of longitudinal displacements (C1L and C2L) are comparable in amplitudes and are therefore plotted in one frame. Hence, in Fig.
10(d), it follows that their amplitude spectra also have comparable amplitudes to fit within the plotting frame but have peaks at slightly different frequencies (
and 0.8 Hz).
On the other hand, the amplitudes of cable transverse displacement time histories for C1T (side span) and C2T (main span) are significantly (
) different. Hence, Fig.
10(b) hosts the C1T time histories and Fig.
10(c) hosts the C2T time histories. Therefore, it follows that amplitude spectra of transverse displacements C1T and C2T are plotted separately in Figs.
10(e and f), respectively.
In the time history plots [Figs.
10(a and b)], repetitious beating is observed and the corresponding amplitude spectra have distinct spectral peaks. The same is not repeated for the cable transverse time histories [Fig.
10(c)] of the main span (C2T) locations, as defined by the displacement time histories of Channels 61 and 55. Consequently, amplitude spectra of displacement of Channels 61 and 55 display multiple peaks within the 0–2 Hz frequency band. This may be interpreted to be due to the longer lengths of main span cables and the interaction of the same with the tower and deck. The frequencies identified by peak picking are summarized in Fig.
14 later in the paper. The beating effect is discussed in more detail in a separate section also later in the paper.
System Identification and First Modal Shapes
Subspace State Space System Identification (N4SID), coded in Matlab (
Mathworks 2020), is used to identify modal shapes, frequencies (f), and critical damping percentages (
). Background information for N4SID is provided in Van Overschee and De Moor (
1994) and Ljung (
1999) and are not repeated herein. Displacement data was used for N4SID computation. However, using all available data in all orientations could not be used in one execution of N4SID because of the large size of the number of equations to be solved. Hence, N4SID is applied using several orientations and structural-component-specific displacement subset data to successfully estimate the prestated modal shapes, frequencies (f), and critical damping percentages (
).
In Fig.
11(a), the extracted first translational and vertical mode shapes for only the deck are shown. The identified transverse first mode frequency (period),
is 0.36 Hz (2.79 s), and the critical damping percentage is 2.11%, both of which are provided in the figure. The identified vertical first mode frequency (period),
, is 0.24 Hz (4.14 s), and the critical damping percentage is 2.64%, both of which are provided in the figure. It is noted that the deck first mode shapes in the vertical direction for the side span and main span are in the opposite sense vertically. This is expected for this bridge because there is no physical restraint of the deck at the tower location; however, the hanger-cable-tower pull action for the main span forces the side span to displace in the opposite sense vertically. There is no other explanation.
Similarly, in Fig.
11(b), the extracted first longitudinal and transverse mode shapes for the tower only are shown. The identified longitudinal first mode frequency (period)
is 0.47 Hz (2.14 s) and the critical damping percentage is 3.14%, both of which are provided in the figure. The identified transverse first mode frequency (period)
is 0.36 Hz (2.79 s) and the critical damping percentage is 4.83%, both of which are provided in the figure.
It is noted that the frequencies limited to the first mode as identified by N4SID method compare well with those identified by peak picking using spectral ratios. On the other hand, the damping percentages computed by the N4SID method are difficult to confirm except for that by the well-known logarithmic decrement method (
Chopra 2001). An example using the logarithmic decrement method is provided later in the paper. All identified dynamic characteristics (frequencies and damping) are summarized in Fig.
14 later in the paper.
Beating Effect
Beating effects have been observed in response data of many buildings and have been studied in depth by Boroschek and Mahin (
1991) resulting in a formula for computing beating periods of buildings when torsional and translational natural periods of vibration are close (or closely coupled)
where
and
= translational and torsional periods, respectively;
= beating period; and
= beating frequency. The beating period (
) is twice the inverse of beating frequency (
) (
). Beating effect is mainly observed when the damping in the structural system is small. It occurs when repetitively stored potential energy, produced during coupled translational and torsional deformations, turns into repetitive vibrational energy.
To the best knowledge of the author, there are only a few publications that refer to beating effects of long-span bridges that are discussed or studied using earthquake response data. The presence of beating effects are observed in the earthquake response data recorded by the seismic monitoring array of the Cape Girardeau, Missouri, bridge (
Çelebi 2006a) and the Carquinez, California, suspension bridge data set, recorded during the
South Napa earthquake of August 24, 2014 (
Çelebi et al. 2019). However, in the Golden Gate Bridge earthquake response study using data from three different events, a beating effect was not observed (
Çelebi 2012). In addition, within the last three decades, including the study by Boroschek and Mahin (
1991), there are a limited number of studies related to some tall buildings experiencing beating phenomena, as observed from recorded earthquake response data, including: (1) a 13-story building during the 1989 Loma Prieta, California, earthquake (
Çelebi and Liu 1998); (2) the 20-story Atwood Building in Anchorage, Alaska, during the
Tazlina Glacier earthquake in April 2005, along with other earthquakes (
Çelebi 2006b); (3) the tallest California building, the 73-story Wilshire Grand of Los Angeles, during the
Ridgecrest, California, earthquake of July 5, 2019 (
Çelebi et al. 2020); and (4) the 61-story (the tallest building in San Francisco) Salesforce Tower during the January 4, 2018,
Berkeley earthquake (
Çelebi et al. 2019). A detailed study to quantify the effect of beating in lengthening the vibrational duration, and therefore the vibrational energy, of tall buildings is presented in a recent paper (
Çelebi 2018). Finally, it should be stated that the subject of beating behavior induced by seismic or strong winds is, unfortunately, not included in structural dynamic textbooks either.
Why beating is important is displayed in Figs.
12(a and b), in which several cycles of beating in extends the duration of strong shaking in the translational (ch51) and vertical accelerations (ch50) at the
location of the main span of the deck. As done in Fig.
4, 5%–95% normalized summed squared acceleration method (
Trifunac and Brady 1975) yields the estimated structural strong shaking duration as
and significantly longer than the
for the ground motion (Fig.
4). Hence, (1) it is therefore deduced that going from
to
implies an elongation of strong shaking duration of the structure, at least for a significant percentage of the length of the time difference between 60 and 13.25 s; and (2) the additional step-like vibrational energy (as represented by the normalized summed squared acceleration) is added several times, and on a regular and repetitious basis, due to beating. Note that the observed length of each beating cycle time (or beating period) differs for the vertical as compared to transverse acceleration time history; both, however, exhibit the steps in normalized energy as represented by the normalized summed square of acceleration. From this plot (Fig.
12), the beating periods in the translational and vertical direction of the main span deck are extracted to be
and
, respectively. As a result, the repetitious exchange between vibrational and potential energy that causes the elongated response may result in low-cycle fatigue of the elements of the suspension bridge (e.g., the cables, hangers, deck and tower). It is important to mention that the data record length is important when computing summed squared acceleration time history. The reason for this is that the record length of 149 s in the data set used in this study is based on a preset start recording trigger and an end recording acceleration threshold (as opposed to continuous recording and selecting the record length on demand). This means that it is likely that the beating continues beyond the stop recording threshold (149 s in this case), possibly elongating the shaking even more than the estimated
. Finally, it was stated earlier that important requisite conditions for beating effects to materialize are a low critical damping percentage of the structure and its components (tower, deck, and cables) and closely coupled modes.
The computed critical damping percentages of components using the N4SID system identification method are all
(varying between 2.11% and 4.83%). Another way to assess the low critical damping percentage is by the well-known textbook logarithmic decrement method (
Chopra 2001) as demonstrated in Fig.
12(c), which results in 1.43% damping and which easily allows for the identification of the repetitious beating period as 15.5 s. The low-amplitude shaking data set used in this study implies only elastic response.
About Apparent Frequency
The fact that frequencies and critical damping percentages are identified for the components (tower, deck, and cables) of the SAS does not allow for direct comparison with those characteristics as computed for the total SAS structural system in its global coordinates (as has been done by
Nader et al. 2002). Hence, an exploration of making use of an apparent frequency computation is appropriate to estimate global coordinate–based frequencies for the SAS, using those identified for the components (tower, deck, and cables). This is being attempted even though the frequencies of the components are identified from low-amplitude shaking data. The Nader et al. (
2002) frequencies (periods) are computed for the much larger–amplitude SEE event, and for longitudinal, transverse, and vertical orientations of the SAS, respectively, as 0.263 Hz (3.80 s), 2.75 Hz (3.64 s), and 0.222 Hz (4.50 s).
The apparent frequency (
) approach is widely known and used in soil-structure analyses of buildings as:
where
and
are the frequencies of the structure and soil, respectively (
Jacobsen and Ayre 1958;
Luco et al. 1986;
Trifunac et al. 2001). This relationship is described by Trifunac et al. (
2001) as a combination rule whereby apparent frequency,
, (period,
) is shorter (longer) than the shortest frequencies (longest periods) of interacting parts.
From the aforementioned relationship, the apparent frequency (
) and period (
) are computed directly as
However, in the experience from the Golden Gate Bridge study (
Çelebi 2012), as also may be applicable for the majority of long-span suspension bridges, when, for example, only the deck (
) and tower (
) are considered, and when the difference between them is large enough, the apparent frequency is identical to the lower of the two, which always is that of the deck. For the Golden Gate Bridge, the apparent frequency (
) of 0.13 Hz was computed from the interacting tower’s longitudinal (
) and the deck’s vertical (0.13 Hz) motions, and is the same as the deck’s vertical motion (0.13 Hz) [as computed by
].
Similarly, in the case of the Carquinez Suspension Bridge study (
Çelebi et al. 2019), the tower longitudinal frequency (0.39 Hz) is significantly higher than that of the deck vertical frequency (0.17 Hz). The resulting apparent frequency,
(at
) is very close to
(at 0.17 Hz).
Applying Eq. (
2) to the SAS and using deck vertical frequency (period) as 0.25 Hz (4.0 s), and tower longitudinal frequency (period) as 0.3 Hz (3.33 s), apparent frequency (period)
(
) is obtained as 0.19 Hz (5.20 s). Similarly, using tower longitudinal frequency (period) as 0.3 Hz (3.33 s), and tower transverse frequency (period) as 0.33 Hz (3.0 s), apparent frequency (period)
(
) is obtained as 0.22 Hz (4.48 s). Table
2 summarizes the apparent frequencies for these three bridges (the SAS, Golden Gate, and Carquinez). Fig.
13 displays the variation of apparent periods against both the main span only and the total length of the deck. There is surprisingly almost a linear variation even though the SAS has only one tower and one side span. Naturally, more empirical data is required to make a claim of reliable correlation.