The New Self-Anchored Suspension Bridge of the San Francisco Bay Bridge System: A Preliminary Study of Its Response and Behavior during a Small Earthquake

The overall SAS structural system is depicted using the schematics in Fig. 2. A concise summary of the structural system is provided in Strong Motion Center (2023). Accordingly, the SAS is the bridge structure between Pier W2 and Pier E2 (including the tower and foundations), as seen in Figs. 2(a and b), and the main deck comprises dual orthotropic steel box girders each having a width of 27.88 m (91.47 ft) The box girders are not attached to the tower. The westbound (W) and eastbound (E) girders are connected by steel crossbeams and are suspended with hangers from cables, which, in turn, are hung over a saddle system and supported at the top of a single steel tower at an elevation of 160 m (524.93 ft). The self-anchored suspension describes the cables that are anchored at Pier E2 and looped around Pier W2 through saddles (e.g., at top of the tower). In other words, there are no anchorages, as in the vast majority of suspension bridges (e.g., the two side-by-side suspension bridges between San Francisco and Yerba Buena Island, which are west of the SAS, or the Golden Gate Bridge connecting San Francisco and Northern California).

The aforementioned structural attributes are depicted in vertical and plan views in Fig. 2, which will later be reviewed for the details of the instrumentation.

It is noted herein that additional general in-plan orientations of the structure, assigned as longitudinal (L) along the length of the deck and transverse (T) perpendicular to the length of the deck, are added in the modified Fig. 2(a). These are proclaimed herein as the global coordinates of the overall SAS structure. As indicated in the figure, the total length of the deck (and therefore the SAS Bridge) is 622.5 m (2,042 ft).

The tower is a unique design [Fig. 2(b)]. There are four steel shafts that are interconnected with steel shear links. In addition, the shafts are rigidly connected at the saddle at the top of the tower, which supports and traffics the cables. The shear links are designed to experience inelastic behavior and therefore have been cyclic tested in laboratory. Detailed description of these tests are provided by McDaniel et al. (2003). The overall design of the SAS Bridge is detailed in Nader et al. (2002, 2014), and Penzien et al. (2003).

In addition, both Piers W2 and E2 comprise a concrete cap beam that is supported by concrete columns. The columns of Pier W2 are tie-downed to the foundation. The Pier E2 concrete cap beam is connected to the box girders (of the deck) with shear keys and bearings.

The foundation of the bridge is best described as such: (1) at Pier W2 as concrete footing on top of 2.5 m cast-in-drilled-hole (CIDH) concrete piles at the westbound side only; (2) at the tower as concrete footing on top of a total of 13 70-m-long piles (2.5-m-diameter concrete piles with steel casing and 2.2-m-diameter CIDH piles, all rock socketed at an elevation of $-59.5m$); (3) at Pier E2 as each footing cap (below the westbound and eastbound decks) rests on 8 105.5-m-long, 2.5-m-diameter cast-in-steel-shell (CISS) piles; (4) the W and E footing caps are connected by a 5.7-m-deep concrete beam; and (5) all piles are socketed to rock.

The vertical elevation in Fig. 2(b) also shows an approximate rock line and an approximate grade. The rock is described as sedimentary and what is above it as alluvium (Strong Motion Center 2023). Nader et al. (2014) describe the rock as Franciscan Formation, and the alluvium, comprising sandy clays and sand, as Alameda Formation.

Details of seismic hazard evaluations of SAS are best described in Nader et al. (2002, 2014). In summary, a safety evaluation earthquake (SEE) and a functional evaluation earthquake (FEE) have been defined. They report that the SEE was defined by a probabilistic seismic hazard analysis with a 1,500-year return period. Design for a functional evaluation earthquake is expected to result in the SAS providing immediate full service with essentially elastic performance (but with minimal damage or minor inelastic response). On the other hand, with a SEE design, the SAS is expected to be functional even after it sustains repairable damage that can be repaired within a few days. Further details are found in Nader et al. (2002, 2014). However, the zero-period accelerations (ZPA) for the design events have not been identified in these references.

Nader et al. (2002) further report that (1) the design of the SAS is governed by the SEE, and (2) the analyses using this criterion computed longitudinal, transverse, and vertical fundamental periods (frequencies) of the SAS as 3.80 s (0.263 Hz), 3.64 s (2.75 Hz) and 4.50 s (0.222 Hz).

It is important to restate that in this paper, recorded data from sensors deployed at components of the SAS are used. Therefore, the fundamental periods (frequencies) reported by Nader et al. (2002) in the global longitudinal, transverse, and vertical coordinates of the bridge are not comparable with those dynamic response characteristics identified using the data from the small 2019 earthquake.

Fig. 2, modified from the originals in Strong Motion Center (2023) depicts locations and orientations of most of the seismic monitoring sensors (accelerometers, tiltmeters, and displacement sensors), the north (N) and reference north (Nref) as well as the orientations of global coordinates (L and T). For ease in following the analyses of records from the many sensors, Table 1 provides the locations, orientations, and sensor types. It is noted herein that the data from tilt sensors and displacement sensors are not available. Horizontal cable sensors are oriented according to cable-specific longitudinal and translational orientations C1L and C1T for the side span and C2L and C2T for the main span. It is noted that Channels 2 and 3 malfunctioned and Channels 56, 57, and 58 were not active.

The one event during which the vast majority of accelerometers recorded the response of the SAS Bridge is the October 14, 2019, $M_{w}4.6$ Pleasant Hill earthquake [$22:33:42.810$ pacific daylight time (PDT), coordinates 37.9380° N and 122.0570° W, and 14.0 km depth that occurred at an epicentral distance of 29.9 km from the SAS, whose coordinates are 37.8152° N and 122.3589° W] (Strong Motion Center 2023). Table 1 summarizes the channel numbering system according to location and orientation.

It is important to note herein that the recorded response data served by CSMIP via Strong Motion Center (2023) are both (1) raw (unprocessed accelerations); and (2) processed [acceleration (a), velocity (v), and displacement (d)] versions. Processing is executed by CSMIP. Details of processing filters are always included in the headers of each avd data for each channel. In this study, only processed a, v, and d data sets are used as imported Strong Motion Center (2023). Further discussion of data processing is outside the scope of this study.

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.

For all structural histories, the figures display the strong shaking duration which, as expected, is longer than that of the input motion strong shaking duration. And, as discussed earlier and displayed in Figs. 4(a and b), particularly in the displacement time histories of numerous channels (Figs. 5–7), there is strong evidence of a repetitious beating effect that will be discussed in detail later in the paper.

Upfront and at the expense of being repetitious, it is important to note that sensors are deployed at the three main structural components of SAS (deck, tower, and cables) along with the foundation. Therefore, the frequencies identified from the data set will belong to the structural component and not the overall SAS with the global coordinate system.

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 $-59.5m$ 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 ($<0.5\mathrm{Hz}$) 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 $\sim 0.30\mathrm{Hz}$ and others at 0.71, 0.83, and 2.08 Hz. This first modal frequency at $\sim 0.30\mathrm{Hz}$ 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 $\sim 0.33$, 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 $\sim 0.25$ and $\sim 0.68–0.90\mathrm{Hz}$.

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 $\sim 0.25\mathrm{Hz}$ 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 ($\sim 0.7$ 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 ($\sim 6\mathrm{times}$) 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.

Subspace State Space System Identification (N4SID), coded in Matlab (Mathworks 2020), is used to identify modal shapes, frequencies (f), and critical damping percentages ($\xi$). 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 ($\xi$).

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), $f_1(T_1)$ 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), $f_1(T_1)$, 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) $f_1(T_1)$ 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) $f_1(T_1)$ 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 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 $T_1$ and $T_2$ = translational and torsional periods, respectively; $T_b$ = beating period; and $f_b$ = beating frequency. The beating period ($T_b$) is twice the inverse of beating frequency ($f_b$) ($T_b=2{/f}_b=2/|f_1{-f}_2|$). 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 $M_{w}6.0$ 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 $M_{L}4.96$ 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 $M_{w}7.1$ 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, $M_{w}4.4$ 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 $1/3$ 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 $\sim 60s$ and significantly longer than the $\sim 13.25s$ for the ground motion (Fig. 4). Hence, (1) it is therefore deduced that going from $\sim 13.25$ to $\sim 60s$ 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 $\sim 14.7$ and $\sim 10.6s$, 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 $\sim 60s$. 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 $<5\%$ (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.

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 ($f_a$) approach is widely known and used in soil-structure analyses of buildings as: $1{/f}_a^2=1/f_1^2+1/f_2^2$ where $f_1$ and $f_2$ 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, $f_a$, (period, $T_a$) is shorter (longer) than the shortest frequencies (longest periods) of interacting parts.

From the aforementioned relationship, the apparent frequency ($f_a$) and period ($T_a$) 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 ($f_{\mathrm{deck}}$) and tower ($f_{\mathrm{tower}}$) 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 ($f_a$) of 0.13 Hz was computed from the interacting tower’s longitudinal ($\sim 0.7\mathrm{Hz}$) 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 $f_a=\surd ((f_1^2·f_2^2)/(f_1^2+f_2^2)$].

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, $f_a$ (at $\sim 0.17\mathrm{Hz}$) is very close to $f_{\mathrm{deck}}$ (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) $f_a$ ($T_a$) 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) $f_a$ ($T_a$) 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.