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The effect of cable corrosion on dynamic response of Stonecutters Cable-Stayed Bridge model

Publication: Canadian Journal of Civil Engineering
1 March 2023

Abstract

Cable corrosion is among one of the most common damage mechanisms in cable-stayed bridges. A finite element model of the long-span Stonecutters Cable-Stayed Bridge was built, and its dynamic behaviour was investigated considering localized groups of corroded cables. The reduction in cable cross-sectional area was used to simulate corrosion levels from 25% to 75% cross-sectional loss. The corroded cables were divided into two groups of four cables, arranged in symmetric and asymmetric distributions along both decks. Natural frequencies and mode shapes were compared with a reference case with no corrosion-damaged cables. A recorded wind load was applied on the deck to investigate the time-history response of the mid-span in lateral, vertical, and torsional directions. Frequency analysis was performed on the time-history response, and coupled motions at certain frequencies were observed for certain corrosion cases and were discussed.

1. Introduction

Cable-stayed bridges are widely used nowadays due to their lightweight structure, economic efficiency, and aesthetical appearance. However, these structures rely on high-strength cables, which transmit the loads from the deck to the towers, thus playing a critical role in supporting the entire bridge. Cable-stayed bridges in coastal areas are subjected to salty moist air, which penetrates inside the cable and reacts with the coating and high-strength steel wires after years of service (Bao 2014). The General Rafael Urdaneta Bridge in Venezuela opened in 1962 and had four cables rupturing due to deterioration in 1980. After the entire cable system was replaced with a new cable system in 1999, corrosion was still found in cables and sockets, which caused up to 30% tension loss in the cables (Sarcos-Portillo et al. 2003). The St. Nazaire Bridge in France built in 1974 also experienced severe corrosion of cables after several years of service, and similar cable corrosion was reported for the Köhlbrand Bridge located in Germany (Bao 2014). The Morandi Bridge located in Genoa, Italy, collapsed in 2018, with the suspicion of cable corrosion being the cause of the bridge failure (Pollock 2018).
Steel corrosion reduces the effective cross-sectional area, strength, and ductility of stay cables, thus affecting the dynamic characteristics and the stability of the entire stay cable-bridge system (Lonetti et al. 2011; Vikas et al. 2013). It is usual practice to conduct regular visual inspections to ensure the stay cables are in good condition or to employ non-destructive techniques, such as X-ray, acoustic emission, and magnetostrictive sensors, to detect corrosion progression without affecting the structure (Betti et al. 2014; Guo 2014; Zhang et al. 2021). The extent of corrosion in the cable section varies from wire to wire, while corrosion is usually more severe for the outer layer and mitigated along the radial direction of the cable section (Xu and Chen 2006). The ultimate stress of the affected cable wires has been found to decrease with corrosion (Barton et al. 2000; Fathali et al. 2020); however, the elastic modulus of the corroded wires remained the same (Li et al. 2012). Researchers have studied the effects that impact the corrosion rate and proposed time-dependent corrosion models (Bindschedler 2007; Liang 2008; Lu and He 2016). A FE nonlinear model, which considers the effects of local vibrations of the damaged cable-stayed systems and the large displacements in the girder and the pylons, was developed by Lonetti and Pascuzzo (2014) for determining the response of a cable-stayed bridge and a hybrid cable-stayed suspension bridge under the effect of a railway live load. The damage mechanism for the cables was modelled as a time-dependent cross-section reduction area based on the Continuum Damage Mechanics (CDM) theory. Ammendolea et al. (2020) investigated the effect of the sudden cable loss, for cable-supported arch bridges, by re-orienting the loads as opposite to the ones registered prior to cable loss and by using Kachanov's law for reducing the mechanical properties and initial stress when the cables fail. They concluded that there are discrepancies in the dynamic amplification factor estimated based on the standard analysis recommended by the Post-Tensioning Institute (PTI 2007) and the non-standard approach investigated, which could involve a considerable increase of stresses and deformations for higher transit speeds. A quantification of the dynamic amplification factors was attempted by Greco et al. (2013) for a cable-stayed bridge considering an accidental cable failure while under a moving load and concluded that these significantly depend on the speed of the moving load and on the mass schematization. The failure of the anchor stay cable was identified as the damage mechanism inducing a worse effect on the bridge response when considering the interaction between the girder and moving load.
The objective of this study was to investigate the dynamic behaviour under wind loading of Stonecutters Cable-Stayed Bridge when a localized group of stay cables is corroded. As the middle of the suspended span is more vulnerable during extreme wind events, corrosion damage of cables around this area was investigated. The analysis was conducted through a finite element model (FEM) of the twin-deck cable-stayed bridge to determine the impact of the location of the corroded cables and the effect of the level of damage on the dynamic structural response of the bridge. The degree of corrosion in the cables was accounted for through the effective cross-sectional area of the cable. Other material properties, such as the yield strength, were not changed, since the calculated maximum cable stresses due to the applied loads remained within the elastic region. In general, the design of cable-stayed bridges is conservative in terms of strength and number of cables, thus losing one single cable would not have a significant effect on the overall response of the bridge (Sabri 2012). However, no research is available for groups of several corroded cables, which could have a significant impact on the dynamic structural response of the bridge. Structural health monitoring data for Stonecutters Bridge is not available in the literature; therefore, a direct comparison of results was not performed. Moreover, bridge cable corrosion was not signalled for Stonecutters Bridge; thus, the current analysis represents a numerical investigation that aims at understanding the effect of several corroded cables on the overall change in the dynamic characteristics of the cable-stayed bridge.

2. Stay-cable bridge configuration and modelling

The twin-deck Stonecutters Bridge was built to connect the Nam Wan Kok and Tsing Yi islands, spanning the Rambler Channel in Hong Kong, with a total length of 1596 m and a main span of 1018 m (Kaji et al. 2010), as represented in Fig. 1. The bridge has a 53 m wide twin-box deck, which is connected by cross-girders that are spaced longitudinally at 18 m centre to centre. The cross-section of the bridge deck is shown in Fig. 2. The two independent towers are made of concrete to a height of up to +175 m and a composite of concrete and steel from level +175 to +293 m. The towers begin as an oval cross-section with 24 m across at the base and end up as a circular cross-section with 7 m diameter at level +175 m (Kite et al. 2007). The upper towers were designed to have a steel outer skin and inner steel anchor boxes for the stay cables. There are 224 parallel stay cables arranged as a semi-fan to support the deck at an interval of 18 m. The stay cables are prefabricated parallel wire strands and designed as a compact form, which can reduce the area exposed to wind. The cross-section of the stay cables comprised 7 mm galvanized wires with a tensile strength of 1770 MPa (Kaji et al. 2010). The lumped-mass FEM of the Stonecutters Bridge was built using ABAQUS software, as illustrated in Fig. 3, by determining the locations of the lumped masses that were derived from the geometry of gravity and shear centres of the real structure. The overall geometric details of the bridge were extracted from Falbe-Hansen et al. (2004) and Kaji and Grady (2010); however, some properties still needed to be determined. Because the cross-sectional shape of the steel and concrete decks is irregular, certain structural properties, tabulated in Table 1, were extracted with the help of the STAAD section analysis (Sabri 2012).
Fig. 1.
Fig. 1. Side view of the Stonecutters Cable-Stayed Bridge (reproduced from Kaji et al. 2010).
Fig. 2.
Fig. 2. Cross-section of the bridge deck (reproduced from Kaji et al. 2010).
Fig. 3.
Fig. 3. FE model of Stonecutters Cable-Stayed Bridge: (a) general view and (b) side view.
Table 1.
Table 1. Geometric properties of the deck (Sabri 2012).
The shear and mass centres, which are presented in Figs. 4a and 4b, lie in different locations due to the deck's unique shape. In the FEM, ABAQUS B31 element (2-node linear beam element in space) was used to simulate the twin-deck and the cross-girders connecting the decks, and the RB3D2 element (2-node 3D rigid beam), which serves as a stiff connection, was used to link the shear centre and mass centre together to ensure they move as one.
Fig. 4.
Fig. 4. Cross-section of the deck: (a) steel deck and (b) concrete deck (Sabri 2012).
The towers and piers were also simulated using B31 elements in ABAQUS. Stay cables are tension members that can only bear axial tensile forces but have no resistance to bending moments. Thus, the T3D3 element with three nodes and three degrees of freedom per node (3D 3-node quadratic truss element) was suitable to simulate the stay cables, which can only transfer axial loads and do not have bending and shear stiffnesses. Also, initial stay-cable pre-tensions were approximated based on the allowable stress at service load and the dimension of the longest stay cable reported by Kite et al. (2007), and a corresponding factored pre-tension was calculated and applied for other stay cables of different dimensions. The pre-tensions were adjusted such that the bridge deck displacement under dead load would match the displacement shown in the design drawing of the Stonecutters Bridge (Fig. 1). Finally, the stay-cable pre-tensions were re-adjusted such that the natural frequencies and vibration modes obtained would be similar to those reported by Hui et al. (2006) for the FE model of Stonecutters Bridge. Stay-cable pre-tensions in the range of 5.3 and 51.2 kN were calculated for each individual cable and were assigned to the T3D3 element to meet the requirements of real tendons and avoid numerical singularities and convergence problems when the elements are subjected to large-displacement analysis. The pre-tensions were re-estimated for each stay-cable cross-section reduction case and were updated through input files in the ABAQUS software before performing the investigation. The effect of the cable sag induced by its self-weight was considered by assigning the equivalent Young's modulus, as introduced by Ernst (1965) and shown in eq. 1, to each stay-cable truss element:
(1)
where is the equivalent truss element modulus of elasticity, Ecable is the actual cable modulus of elasticity, ρ is the cable material density, g is the acceleration due to gravity, Lx is the horizontal projected length of the cable, and σ is the cable's pre-stress. For the Stonecutters Bridge, the cables are made from a composition of 7 mm galvanized wires (tensile strength of 1770 MPa), totalling 168 mm in diameter, surrounded by a 10 mm thick HDPE coating (Kaji et al. 2010). To connect the towers and decks to the stay cables, multi-point constraints (MPCs) were employed, forcing the slave node to follow the movement of the degree of freedom of the master node. In this way, the stay cables can have an accurate response when coupled with the towers and decks, especially when performing a dynamic analysis to investigate vibration mode shapes.
Further details on deck, tower, and cable modelling and calibration of the Stonecutters Bridge model can be found in Sabri (2012) and Feng et al. (2016); the model was verified by comparing the numerical natural frequencies with the frequencies reported for the real bridge. Among the ten mode shapes resulting from the numerical modal analysis, the deck starts vibrating in symmetric horizontal (S-H-deck), vertical (S-V-deck), and torsional (S-T-deck) modes followed by the asymmetric horizontal (A-H-deck) and torsional (A-T-deck) modes of the deck, as can be noticed in Table 2. A particularity of this bridge model is the occurrence of the tower symmetric and asymmetric vibration modes (S-H-tower and A-H-tower) in the first five modes and the coupling between the tower and the deck vibration modes at higher frequencies (A-T-V-deck). The lowest natural frequency of 0.145 Hz corresponding to the first mode was the first horizontal motion of the bridge deck (S-H-deck), while the highest frequency was 0.581 Hz for the coupled deck–tower vibration mode. The natural frequencies were very close to those reported by Hui et al. (2006), who worked with the FEM model of the constructed Stonecutters Bridge; however, the mode shapes appear in a different order. This might be due to the fact that no dampers were considered in the current study between the deck and the towers.
Table 2.
Table 2. Natural frequency and vibration modes.

3. Cable corrosion scenarios and static deflection

Modal analysis was conducted for different case scenarios in which cables were damaged to corrosion levels of 25%, 50%, and 75% and were positioned at different locations along the length of the twin decks of the Stonecutters Bridge model. To identify the critical cases, the corroded cables were placed in symmetric and asymmetric positions with respect to the middle point of the deck. This type of localized cable corrosion simulation is preferred, as it is considered unlikely that undetected corrosion would appear on all cables simultaneously.
The remaining diameter of the corroded cable can be estimated from Faraday's law, assuming uniform corrosion (Andrade et al. 1993), as described by eq. 2:
(2)
where db (t) is the remaining diameter of the wire at time t (mm), icorr is the corrosion current density (μA/cm2), and t is the time during which corrosion has been occurring (years). For the Stonecutters Bridge model, the corrosion current density was assumed to be 100 µA/cm2 based on the recommendations formulated by Xie et al. (2015) for galvanized wires under 96 MPa tensile stress. Bridge stay cables have different diameters, depending on their location along the span. For a constant current density, cables with the smallest diameter would need less time to reach the effective cross-sectional area; therefore, the stay cables 1 to 8 near the middle of the span, which have a 182 mm diameter, were selected as the corrosion-affected cables in the current study. When substituting these parameters into eq. 1, the estimated time for 25%, 50%, and 75% corrosion-induced cross-sectional loss was around 20, 39, and 59 years, respectively. Therefore, for Stonecutters Bridge with a design life of 120 years (Kite et al. 2007), the assumed cable corrosion could happen during the bridge's service life.
The group of eight cables selected for corrosion simulation was classified into four main configurations, all concentrated around the middle of the span, as shown in Figs. 5a5d. The first main case considered four corroded cables for each deck, in symmetric arrangement with respect to the middle point of the deck (Fig. 5a). For the second main case, four corroded cables were considered on each deck; however, these were in asymmetric arrangement with respect to the mid-point of the span: four cables on the east side of the north deck and four cables on the west side of the south deck, as represented in Fig. 5b). The third main case had the eight corroded cables only on the north deck, arranged symmetrically on the west–east sides of the deck (Fig. 5c). Lastly, the fourth main case had the corroded cables placed on both decks, but all on the east side of the decks (Fig. 5d). Within each main cable corrosion case illustrated in Fig. 5, several subcases were also simulated by sliding the location of the corroded cables further away from the centre of the span to investigate the response affected by the distance of the corroded cables from the middle point of the span; thus, closest to the middle (MID), intermediately away from the middle (INT), and further away from the centre at the extreme exterior (EXT) subcases were investigated. Finally, within each subcase, different levels of corrosion were investigated: 25%, 50%, and 75% corrosion levels. These levels were selected according to previous studies, suggesting that an unnoticed 25% reduction is plausible (Bindschedler 2007; Yang 2018) and up to 30% tension reduction in cables due to corrosion was previously reported (Sarcos-Portillo et al. 2003); the 50% and 75% corrosion levels were investigated to determine the effect of critical cases when high corrosion could occur due to lack of monitoring. The three different damage percentages represent three different levels of cable corrosion: slight corrosion, medium corrosion, and severe corrosion, respectively. The corrosion level percentage denotes the percentage of cross-sectional area of the cable lost due to cable corrosion. The nomenclature used herein to distinguish the subcases is m-n-a, where m is the main case number, n is the subcase reference name, and a is the corrosion level represented by the percentage in area reduction in the affected cables. For example, subcase 3-MID-50 represents the main cable corrosion case 3 (both sides of the bridge with a single deck affected), subcase MID (cables closest to the mid-span affected), with a corrosion level of 50% in the affected cables.
Fig. 5.
Fig. 5. Four main configurations of corroded cables with respect to the middle point of the span: (a) main case 1, (b) main case 2, (c) main case 3, and (d) main case 4.
The Stonecutters Bridge has twin decks, with cables connected to the lateral edges of each deck; thus, cable corrosion could result in asymmetric static deflections of both decks under the effect of gravity. To easily compare the effect of the cable corrosion cases on the deck response under gravity load, Table 3 shows the maximum deflection and the corresponding location registered for each case, as well as the percentage increase in deflection with respect to the reference case, where no corrosion was applied to the cables. The maximum vertical deflection ranged from −7.04 × 10−1 and −7.48 × 10−1 m for the south deck, for cases 3-MID-75 and 4-MID-75, respectively. For the north deck, the maximum deflection ranged from −7.27 × 10−1 to −7.84 × 10−1 m for cases 1-MID-75 and 3-MID-75, respectively.
Table 3.
Table 3. Maximum static deflection (m) summary for all cases on both decks.
When the corroded cables are located at the same location on both decks (main cases 1 and 4), the same deflection occurred for both sides of the deck; thus, it can be concluded that symmetric distribution of affected cables can result in symmetric deck deflection. The deflections for main case 3 were relatively high values for the north deck and low for the south deck. Both decks deflected asymmetrically when the corrosion cable arrangement corresponded to main case 2. From Table 3, it can be noticed that the main span has the largest deflection for main case 4 on the south deck and for main case 3 on the north deck, when compared to other main cases for the north and south decks. The location of the largest deflection is not always on the side of the deck where the corroded cables are, as seen in subcase 4-MID-75, for which corroded cables were located on the east side, and the maximum deflection point was on the west side of the same deck.

4. Effect of cable corrosion on bridge's natural frequency

The natural frequency and mode shapes can effectively convey the dynamic characteristics of the bridge. For the Stonecutters Bridge model, the natural frequency analysis was conducted obtaining the first 30 modes of vibration, which contain the low and high orders of vibration modes. Numerical simulations investigated several corrosion levels of the stay cables, as well as the position of the corroded cables along the bridge deck. To further analyze the impact of corrosion damage of the stay cables, the frequency relative change with respect to the frequencies of the unaffected bridge, without cables corrosion, was computed. The percentage change in vibration frequencies for the four main cases at different levels of corrosion-induced damage is illustrated in Figs. 6a6d. It can be noted that most of the natural frequencies changed after introducing corrosion-induced damage in the cables, with some frequencies increasing and some of them decreasing; however, the relative change in frequencies associated to the second and sixth vibration modes, which are the first vertical and torsional vibrations of the decks, respectively, experienced the highest decrease. For the subcases 1-MID and 3-MID, the 75% corrosion level did not result in a significant relative change of the vertical and torsional frequencies, but it increased the frequency of the third mode, which is the second-order vertical vibration mode of the deck (Figs. 6a6c). As for the subcases 2-MID and 4-MID, which have an asymmetric distribution with respect to the middle line, the 75% corrosion level reduced the frequencies of the second and sixth modes, but it also increased the frequency of the third mode (Figs. 6b6d). Thus, the frequencies of the third and sixth modes, which are the second vertical and first torsional vibration modes, respectively, became closer to each other due to the critical corrosion condition, producing a dangerous trend for the deck to encounter a coupled vertical and torsional mode under certain aerodynamic loading conditions. This is considered a dangerous trend because, as noticed at Tacoma Bridge failure (Larsen 2000), the classical flutter can occur if the fundamental torsional and vertical modes have close frequencies. The effect on the frequency change induced by the 75% cable corrosion level case was investigated for cables located at different positions along the span of the bridge (MID, INT, and EXT), and the results for the four main cases are presented in Figs. 7a7d. For subcases 1-MID-75, 1-EXT-75, 3-MID-75, and 3-EXT-75, the 75% corrosion level resulted in an increase in frequencies for the second, third, and sixth modes; however, the subcases 1-INT-75 and 3-INT-75, for which the corroded cables are positioned starting from the third cable from the middle point, resulted in a significant decrease in frequency ratios for the second, third, and sixth modes, showing that the frequencies of the deck in the symmetric vertical and torsional vibrations are significantly affected (Figs. 7a7c). In the main cases 2 and 4, which have an asymmetric distribution with respect to the middle line, both subcases MID and INT (2-MID-75, 2-INT-75, 2-MID-75, and 2-INT-75) resulted in decreased frequencies for the second and sixth modes; the subcases 2-EXT-75 and 4-EXT-75 had limited influence on the increased ratio of the frequencies (Figs. 7b7d). As the corroded cables are located further from the mid-span, different cable distributions had less effect on changing the natural frequencies of the bridge model.
Fig. 6.
Fig. 6. Frequency relative change (%) for 25%, 50%, and 75% corrosion levels for subcase MID: (a) main case 1, (b) main case 2, (c) main case 3, and (d) main case 4.
Fig. 7.
Fig. 7. Frequency relative change (%) for 75% corrosion level, at MID, INT, and EXT locations: (a) main case 1, (b) main case 2, (c) main case 3, and (d) main case 4.

5. Bridge response to wind loading

To investigate the wind-induced response of the bridge decks, a wind load was applied to the bridge decks according to the Scanlan's mean wind load equations shown in eqs. 35 (Scanlan 1996), which represent the wind load in three directions, along-wind direction (drag), across-wind direction (lift), and overturning around the longitudinal deck direction:
(3)
(4)
(5)
where Dm, Lm, and Mm are the drag force, lift force, and torsional moment, respectively, ρ is the air density, is the mean wind velocity, B is the bridge deck width, L is the bridge length, α is the attack angle of the wind, and CD, CL, and CM are the drag, lift, and torsional moment coefficients, respectively.
Drag, lift, and torsional coefficients for 0° angle of attack were used, as reported from the wind tunnel experiment of Stonecutters Bridge (Larsen et al. 2012; Larsen and Larose 2015) as 0.07, −0.155, and −0.018, respectively. The coupled flutter aerodynamic instability and buffeting response indiced by the turbulent wind were not verified for the bridge model, as the current investigation focused on identifying the changes in the dynamic characteristics of the bridge, which could affect the response to wind loadings.
To determine the mean wind speed to be used in eqs. 35, a wind speed record collected through an anemometer installed on the Mann Parking Building at the University of Ottawa was used (Mehranfar 2014). As Stonecutters Bridge is located on an open channel and is exposed to high wind velocities, the measured wind data were increased five times to simulate the critical wind speed at the bridge site, reaching a maximum wind speed of 41 m/s among the 500 s of wind data and a mean wind speed of 20.75 m/s (Fig. 8). The modified maximum wind speed was close to the natural wind environment for an extreme wind storm in the Hong Kong area (Feng 2015).
Fig. 8.
Fig. 8. Mehranfar (2014)'s wind velocity record increased five times.
For the wind load analysis, 4-MID-75 was considered the most critical subcase, because besides inducing the highest static deflection, this subcase exhibited a decrease in frequencies for the second and sixth modes but a slight increase in the frequency of the third mode, a pattern which would bring the torsional and vertical modes closer, thus being more vulnerable to wind excitation. Figure 9b shows the time history of the deck mid-span vertical deflection for the subcase 4-MID-25, which presents periodic beat vibrations during the first 500 s of the applied wind load. The largest amplitude for case 4-MID-75 is 0.66 m at around 380 s (Fig. 9c), which is larger than 0.54 m during the last 50 s for case 4-MID-25 (Fig. 9b) and 0.33 m during the last 50 s for the reference case (Fig. 9a). Therefore, the vertical vibration amplitude increased with the corrosion level of the stay cables in subcase 4-MID.
Fig. 9.
Fig. 9. Mid-span displacement for subcase 4-MID: (a) reference case—vertical; (b) subcase 4-MID-25—vertical; (c) subcase 4-MID-75—vertical; (d) reference case—torsional; (e) subcase 4-MID-25—torsional; and (f) subcase 4-MID-75—torsional.
Figures 9d9f illustrate the torsional response changes for the reference case (no corrosion damage) and for 25% and 75% corrosion damage levels (subcases 4-MID-25 and 4-MID-75). When comparing these last two to the reference case, it is noticed that the deck's torsional response increases gradually for the subcase 4-MID-25 (Fig. 9e); however, the torsional vibration seems to have no specific pattern for subcase 4-MID-75 (Fig. 9f), in which the torsional amplitudes remain consistent during the 500 s of applied load. Furthermore, the amplitudes of the torsional response were very small, recording values of 0.0027, 0.0038, and 0.0022 rad for the reference, 4-MID-25, and 4-MID-75 cases, respectively.
Frequency analysis was performed to analyze the time-history response from the model by using the power spectrum density (PSD) to connect the power and the obtained frequencies by transforming the auto-correlation function of the time-history response. The peak frequency values of the vertical, lateral, and torsional responses can help identify the coupled movement between the vibration modes at the same frequency. Resonance of the structure could happen if the peak frequency of the response is the same or close to the natural frequency of the structure and therefore would enhance the amplitude of vibration. Figures 10a10c show the PSDs of the lateral, vertical, and torsional displacements of the mid-point of the deck for the reference case, for which the high spectral peaks are 0.146 and 0.446 Hz for the lateral response, 0.302 and 0.388 Hz for the vertical response, and 0.146, 0.292, and 0.446 Hz for the torsional response. The lateral and torsional motions are coupled at frequencies of 0.146 and 0.446 Hz when there is no corrosion in the cables. For subcase 4-MID-25 (Figs. 10d10f), the frequency peaks represented in the power spectrum for the lateral, vertical, and torsional responses are very close to the case unaltered by corrosion. However, a new peak appears for the vertical response at 0.45 Hz (Fig. 10e), which locks in with the 0.45 Hz peak present in the torsional response (Fig. 10f). Thus, the vertical and torsional motions show a tendency to couple at this frequency. Also, similarly to the reference case, the common peak of 0.146 Hz shows a coupling between the lateral and the torsional responses. For the subcase 4-MID-75 (Figs. 10g10i), the frequencies for all three responses increase slightly while still signalling a coupling between the lateral and torsional motions at 0.148 Hz (Figs. 10g10i) and between the vertical and torsional motions at 0.292 and 0.456 Hz (Figs. 10h10i).
Fig. 10.
Fig. 10. PSD results of mid-span for (a) reference case—lateral response; (b) reference case—vertical response; (c) reference case—torsional response; (d) 4-MID-25—lateral response; (e) 4-MID-25—vertical response; (f) 4-MID-25—torsional response; (g) 4-MID-75—lateral response; (h) 4-MID-75—vertical response; and (i) 4-MID-75—torsional response.
Applying the recorded wind load induces a coupled movement between lateral and torsional modes for the reference case, with frequencies of 0.146 and 0.446 Hz. When the corroded cables are distributed as 4-MID, one more peak frequency appears in the vertical deformation for all the cases due to less restraint provided by the cables. For all the cases, peak frequencies in all three directions for the cases with 25% corrosion are close to those for the reference case; however, there is a slight increase in the peak frequencies when the corrosion condition becomes worse. The distribution of corroded cables has less influence on creating more peak frequencies.
From the analysis of the results presented in Fig. 10, it is observed that corrosion in the stay cables induces coupled motions between the lateral and torsional and vertical and torsional wind-induced responses at 41 m/s, as illustrated in Fig. 11. For the lateral- and torsional-coupled motions shown in Fig. 11a, it is noticed that the common frequency is 0.146 Hz for most of the cases (except for 2-MID-75 and 4-MID-75, wherein a slightly different frequency of 0.148 Hz was recorded). The other less intense frequency peaks in the lateral motion followed the same pattern for all the cases. However, these were not close to the twisting vibration peak frequencies.
Fig. 11.
Fig. 11. Peak frequencies for wind-induced responses: (a) lateral and torsion and (b) vertical and torsion.
Figure 11b shows the frequency overlap between the vertical and torsional responses at 0.45 Hz for all cases with 25% corrosion level (1-MID-25, 2-MID-25, 3-MID-25, and 4-MID-25), 0.456 Hz for cases with 75% corrosion level (1-MID-75, 2-MID-75, and 4-MID-75), and 0.462 Hz for 3-MID-75, while no overlap was noticed for higher frequencies for the vertical and torsional responses. Thus, the increasing corrosion level leads to a coupling tendency between the lateral and torsional motions, which could compromise the structure if flutter instability is achieved. However, this was not registered for the dominant frequency peaks in vertical and torsional vibrations. The cable corrosion reduces the capacity to sustain the deck and increases the coupled responses of the deck when subjected to wind load.

6. Conclusions

The impact of corrosion on stay cables and the wind-induced response of the Stonecutters Cable-Stayed Bridge model were investigated for different levels of corrosion and cable location and distribution. It was noticed that the natural frequencies changed especially for the second (first vertical), third (second vertical), and sixth (torsional) modes; low corrosion levels (25% and 50%) reduced the deck's frequencies in the first vertical and torsional vibration modes, while the deck's frequencies in the second vertical mode increased for a corrosion level of 75%. For the cases with asymmetric distribution with respect to the centreline, the 75% corrosion level also registered a similar decrease for the second and sixth modes compared to the cases with symmetric distribution. As the corroded cables were located further from the mid-span, the distribution of corroded cables had less influence on changing the natural frequencies of the bridge. When wind loading was applied to the deck, it can be concluded that the corrosion levels of 25% and 50% had less influence on changing the dynamic wind-induced response of the bridge. The vertical vibration dominates the deck's response when the corrosion condition becomes critical (75%). In general, the cases with corrosion-damaged cables resulted in more coupled vibration modes of the deck under the effect of the wind loading. For all corrosion levels and cable distributions, the lateral and torsional responses coupled, and the frequency of coupled motions increased slightly with the increase in corrosion level. More significant for the onset conditions of flutter instability, however, is the tendency of coupling between the vertical and torsional responses. However, the frequencies were not dominant for the registered vertical and torsional responses; thus, the structural integrity of the bridge model was not affected. Localized damage for tower segments and for the deck girder will be further incorporated in the developed model, and response under wind and traffic loads will be investigated.

Acknowledgements

The research project presented in this paper was supported by the Natural Sciences and Engineering Research Council of Canada, Discovery Program. The authors would also like to thank Mr. A. Sabri and Mr. F. Feng who assisted in calibrating the FEM of the bridge.

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Information & Authors

Information

Published In

cover image Canadian Journal of Civil Engineering
Canadian Journal of Civil Engineering
Volume 50Number 4April 2023
Pages: 282 - 293

History

Received: 4 August 2021
Accepted: 9 August 2022
Accepted manuscript online: 19 October 2022
Version of record online: 1 March 2023

Key Words

  1. Stonecutters Cable-Stayed Bridge FEM
  2. cable corrosion
  3. wind-induced response
  4. long-span bridge frequencies

Authors

Affiliations

Yang Xiang
Edward Wong & Associates Inc., 441 Esna Park Drive, Unit 19, Markham, ON L3R 1H7, Canada
Author Contributions: Data curation, Formal analysis, Investigation, Methodology, Validation, Visualization, and Writing – original draft.
Étienne Noël
University of Ottawa, Civil Engineering Department, 161 Louis Pasteur, Ottawa, ON K1N 9K5, Canada
Author Contributions: Investigation, Validation, Visualization, Supervision, Writing – original draft, and Writing – review & editing.
University of Ottawa, Civil Engineering Department, 161 Louis Pasteur, Ottawa, ON K1N 9K5, Canada
Author Contributions: Conceptualization, Funding acquisition, Methodology, Resources, Supervision, Writing – original draft, and Writing – review & editing.
University of Ottawa, Civil Engineering Department, 161 Louis Pasteur, Ottawa, ON K1N 9K5, Canada
Author Contributions: Conceptualization, Funding acquisition, Investigation, Resources, Software, Supervision, Writing – original draft, and Writing – review & editing.
Beatriz Martin-Pérez served as an Associate Editor at the time of manuscript review and acceptance; peer review and editorial decisions regarding this manuscript were handled by another Editorial Board Member.

Author Contributions:

Conceptualization: ED, BM-P
Data curation: YX
Formal analysis: YX
Funding acquisition: ED, BM-P
Investigation: YX, ÉN, BM-P
Methodology: YX, ED
Project administration:
Resources: ED, BM-P
Software: BM-P
Supervision: ED, BM-P
Validation: YX, ÉN
Visualization: YX, ÉN
Writing – original draft: YX, ÉN, ED, BM-P
Writing – review & editing: ÉN, ED, BM-P

Competing Interests

Data repository can be made available upon request. The authors and co-authors have no competing interests related to the current project.

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