Chapter 3 Influence of frequency and girder mass on seismic response of skewed bridges
The seismic vulnerability of skewed bridges had been observed in many past earthquakes. Researchers have found that the in-plane rotation of the girders was one of the main reasons for the vulnerability of these types of bridges. To date, not many experimental works have been done on this topic, especially those including pounding between adjacent structures. In this study, shake table tests were performed on bridge-abutment systems with different skew angles. Bridge segment with straight (0°), 30°, and 45° decks were considered alternately. For each case, pounding between the bridge deck and abutments is allowed and prevented to occur at will. Simulated earthquakes were considered as the ground motions. The bridge segment and abutments experience spatially uniform ground motions, i.e. the ground motion at the base of the bridge and abutments are identical. The abutments and bridge are assumed to have fixed base conditions. The results obtained for skewed bridges having the same mass as the straight bridge will be compared to those with bridges having the same longitudinal frequency.
Bridge prototype and 1:100 scale model
A 1:100 scale bridge model was constructed based on the Newmarket Viaduct Replacement Bridge located in Auckland, New Zealand. The bridge is 100 m long, with a pier-to-pier distance of 50 m. The compressive strength of the concrete is assumed to be 50 MPa. Table 3.1 shows a summary of the dimension and dynamic properties of the prototype bridge.
The prototype was scaled down using principles of similitude adopted by Makris (2014), Chen et al. (2017), and Qin et al. (2013). The fundamental theory was based on Buckingham’s π theorem (Buckingham, 1914). The scale factors adopted were shown in Table 3.2.
Based on the scale factors, the dimensions of the straight bridge model were calculated, as shown in Table 3.3. Polyvinylchloride (PVC) was used to construct the bridge girder and piers. Throughout all the tests, the model was kept elastic to allow for repeatability of subsequent tests.
The dimensions of the bridge girders were selected to allow for assumption of rigid-body motion. In total, five bridges were constructed: a straight bridge, two 30° bridges, and two 45° bridges. The size of the bridge piers was kept the same for all bridges. For each skew angle, one of the bridges had the same longitudinal frequency as the straight bridge. The mass of the girders was adjusted accordingly to achieve this. The cases of skewed bridges with the same frequencies as the straight are hereafter denoted as ‘F’, e.g. for the 30° skewed bridge when pounding was considered, 30° (F) (Pounding). The other bridges for each skew angle were designed to have the same girder mass. This means that the longitudinal frequencies of the bridges will differ. The cases of skewed bridges with the same mass as the straight are hereafter denoted as ‘M’, e.g. for the 30° skewed bridge when pounding was considered, 30° (M) (Pounding). A summary of the cases considered were shown in Table 3.4. The values in italic were kept the same.
The bridge segment and abutments were fixed on a uniaxial shake table with a payload of 10 kN. For the skewed bridges, the abutments were adjusted to be parallel to the face of the bridges. The bridge and abutments experience spatially uniform excitations. In the cases where pounding was considered, the abutments were fixed at 1 mm apart from the bridge segment. The gap size was scaled down according to a thermal expansion gap of the prototype of 10 cm. When pounding was not considered, they were spaced sufficiently apart so that the bridge segment did not come in contact with the abutments. The set-up of the bridge-abutment model considering pounding is shown in Figure 3.1.
In order to measure the pounding forces at the bridge-abutment interface, two pounding and measuring heads were constructed using PVC. The pounding heads were attached to the ends of the bridge, whereas the measuring heads were attached to the abutments. Two pounding and measuring heads were used on each side of the bridge to measure the pounding force at the acute and obtuse corners of the skewed bridges. The impact force was measured at each measuring head using a strain gauge attached to the back of a piece of steel. Figure 3.2 shows the pounding and measuring heads used in the experiments.
The bending moments developed near the top and base of the piers were calculated using measurements from strain gauges attached to the piers. The displacements of the bridge girders relative to the abutments were measured using laser transducers. Due to the direction 23 of excitation, there were no movements in the transverse direction for the straight bridge. The transverse displacements were only measured for the skewed bridges.
Ten earthquake excitations were stochastically simulated based on the design spectra for shallow soil (Class C) ground conditions as specified in NZS 1170.5 (Standards New Zealand, 2004). The numerical approach used for simulating the ground motions has been explicated in Chouw and Hao (2005).
The response spectra of three of the unscaled simulated ground motions and the corresponding NZS design spectra were plotted in Figure 3.3.
Bridge design specifications
Many current bridge design specifications have not considered the effects of pounding. This oversimplification could be one of the reasons for the occurrence of girder unseating. In this study, the recommendations specified in the NZTA bridge manual (New Zealand Transport Agency, 2016) will be used as an example to evaluate the adequacy of design standards when designing the seat lengths of skewed bridges.
NZTA bridge manual
The required seat length, SL, for a straight bridge recommended by the NZTA bridge manual is given in Clause 5.5.2 (d) as:
where D is the relative movement between span and support.
For short-span skewed bridges, the NZTA bridge manual recommends that the required seat length in both the longitudinal and transverse directions be increased by up to 25% of that of the straight bridge. This requirement implies that the expected relative displacements of the skewed bridge are only up to 25% larger than that of the straight bridge. The actual relative displacements obtained from the experiments will be compared with the NZTA recommendation.
Results and discussions
Bending moment developed in bridge piers
Figure 3.4 shows the maximum bending moment (BM) of the straight, 30°, and 45° bridges with and without considering the effects of pounding. Except for the 30° (F) case, the bending moments of all bridges were always larger when pounding was not considered. A ‘capping’ behaviour was observed when pounding occurred; the abutments restricted the movements of the girders. In the 30° (F) case, the bending moments with and without pounding were similar. Comparing the (M) and (F) cases, the bending moments of the skewed bridges tend to be smaller in the latter case.
The average maximum bending moments near the top of the piers were shown in Table 3.5. As expected, the largest bending moment was observed for the straight bridge without considering pounding. This is partly because the orientation of the piers of the straight bridge meant that it would receive the largest excitation in the bending direction, e.g. if the excitation has an amplitude of “1”, the piers will experience “1” excitation. However, with the skewed bridges (skewed piers), the piers would only be subjected to “1” × cos(θ) in the direction of bending of the piers, where θ is the skew angle as illustrated in Figure 3.5. However, as the direction of the excitation is only related to the translational responses of the bridge, this trend was not seen with the responses of the skewed bridges. This is because the responses of the skewed bridges also consist of the rotational responses in addition to the translational responses.
Interestingly, with the larger skew angle of 45°, the bending moments were larger than that of the 30°. This is likely due to the larger movements of the girder of the 45° bridge in the longitudinal direction, as discussed in Section 3.4.2. For both skewed bridges, the case where they had the same mass as the straight bridge (M) had the larger response independent to the occurrence of pounding.
Relative displacement of bridge girders
The maximum displacements (Rd) of the girders of the straight, 30°, and 45° bridges relative to the abutments in the longitudinal direction were shown in Figure 3.6. The relative displacements were measured at the centre of the girder of the bridges. The displacements show almost an identical trend as the bending moments (Figure 3.4). This proves that the bending moments developed in the bridge piers are very closely related to the longitudinal displacements of the girders. However, the transverse displacements may not have the same correlation with the bending moments.
The average maximum displacements and ratio of that of the NP to P case were calculated and shown in Table 3.6. Similar trend between the bending moments and longitudinal displacements was also seen with the ratio of the responses in the NP to P cases. This trend is probably due to the smaller movement of the 30° bridge in the NP case compared to the 45° bridge. Smaller movement of the girder in the NP case means that when pounding occurs, the magnitude of collisions will likely also be smaller. Hence, the trend that was observed in Table 3.5 and Table 3.6. From Table 3.5, the maximum bending moments of the piers of the straight, 30° (F), 30° (M), 45° (F), and 45° (M) bridges in the pounding cases were on average 0.38, 1, 0.82, 0.67, and 0.57 times that of the no pounding case, respectively. With the relative longitudinal displacements (Table 3.6), they were 0.27, 1.06, 1.06, 0.62, and 0.59 times, respectively.
The transverse displacements of the skewed bridges were plotted in Figure 3.7. The displacements of the bridges in all cases were larger when pounding was not considered. This was very likely due to the movement restrictions provided by the abutments when pounding occurred. The transverse displacements in the (M) case were larger than in the (F) for both the 30° and 45° skewed bridges. This amplification was most significant in the 30° bridge without considering pounding.
The average maximum transverse displacements of the skewed bridges with and without considering pounding are shown in Table 3.7. Pounding reduced the average maximum transverse displacement by up to 29%, likely due to the restrictions provided by the abutments. The results also show that the displacements in the (M) case were significantly larger than those in the (F) case. Hence, the subsequent large-scale tests (Chapters 4 to 8) considered skewed bridges with identical mass rather than frequencies. The larger amplification was seen with the 30° bridge, where the transverse displacements without considering pounding were increased from 2.2 mm in the (F) case to 3.2 mm in the (M) case (amplification of approximately 1.45 times) and with pounding from 1.8 mm in the (F) case to 2.4 mm in the (M) case (amplification of approximately 1.32 times).
To evaluate the girder unseating potential of the skewed bridges based on the NZTA recommendations, the average maximum relative displacements of each bridge in the longitudinal and transverse (applicable only to skewed bridges) were normalised against the longitudinal displacement of the straight bridge for both with or without pounding cases. For example, to assess the longitudinal displacement of the 30° (M) bridge in the no pounding case, the average maximum displacement was normalised against the longitudinal displacement of the straight bridge in the no pounding case, i.e.
The normalised displacements were given in Table 3.8. As the NZTA recommendations specified that the relative displacements of skewed bridges in both the longitudinal and transverse displacements were amplified by 25% of that of the straight bridge. As the straight bridge has no transverse displacements, the displacements of the skewed bridges were designed based on the longitudinal displacement of the straight bridge. This recommendation implies that the relative displacements of the girders of skewed bridges were expected to only be at most 25% larger than that of the straight. As shown in Table 3.8, when pounding was not considered, the seat length recommended by the NZTA bridge manual was adequate.
Since pounding is difficult to avoid during strong seismic events due to the typically small thermal expansion gap, the relative displacements of the girders could potentially be underestimated when pounding in fact does occur, as can be clearly seen from the results in Table 3.8. The values in bold were larger than 1.25, meaning that the NZTA requirements underestimated the displacements in those cases. All of the displacements of the skewed bridges in the transverse direction were severely underestimated. In the worst case i.e. 45° (M), Transverse (P), the average displacement was 2.6 times larger than that of the straight bridge in the longitudinal direction, approximately twice the 1.25 times amplification specified in the NZTA bridge manual. In the longitudinal direction, although in most cases, the NZTA requirements was sufficient to accommodate for the relative movements of the girders, when pounding was considered, there was still potential for the girder of skewed bridges to unseat, e.g. in this case the 45° (M), Longitudinal (P) case was underestimated.
In-plane rotations of girders
Figure 3.8 shows the maximum in-plane rotations of the girders of the skewed bridges. For the 30° bridge, the rotations were larger when pounding was not considered, whereas for the 45° bridge, the opposite was observed. This means that pounding potentially has a stronger effect on the in-plane rotations of the girders as the skew angle increases. The rotations were larger when the skewed bridges had the same mass as the straight bridge, compared to when they had the same fundamental frequencies.
Although the 30° bridge had smaller skew angle compared to the 45° bridge, the latter had smaller in-plane rotations in both cases with and without considering pounding. For example, in the 30° (F) case without considering pounding, the girder had an average maximum rotation of 0.105°, but for the same case, when the skew angle was 45°, the average maximum rotation of the girder was only 0.057°, about 0.46 times smaller than that of the 30° bridge. A past study conducted by Catacoli et al. (2014) reported that the torsional responses do not necessarily arise just from a bridge being skewed, and that only happened when pounding occurred. The results in Table 3.9 showed that for the cases considered, the in-plane rotations of the girders can not only be induced when pounding did not occur, they can in fact, be even larger than those induced when pounding was present. This highlights the need for skew angles to be considered in more detail when designing skewed bridges.
This chapter investigates the effects of two dimensions of skewed bridges on their seismic response through shake table tests: mass and frequency. By keeping the dimensions of the piers identical, the mass of the bridges need to be altered to obtain the same translational frequencies between the straight and skewed bridges, and vice versa. Bridge-abutment models with straight (0°), 30°, and 45° skew angles were considered. Two bridges were considered for each skew angle – skewed bridges that had the same translational frequency as the straight bridge (F), and those that had the same mass as the straight (M). It was revealed that:
The bending moments developed in the bridge piers were directly proportional to the relative displacement of the bridge girder in the longitudinal direction than the transverse displacement and in-plane rotations of the girder.
The bending moments and longitudinal displacements of the 30° (F) bridge were similar with or without considering pounding, as the bending moments generated in the cases without considering pounding was similar to those of the pounding case. In all the other cases, the responses tend to be larger when pounding was not considered. This is because of the restrictions to the girder movements provided by the abutments when pounding was considered. The transverse displacements of the girder of the skewed bridges, in both the (M) and (F) cases, were larger when pounding was not considered compared to when it was present.
The in-plane rotations of the girders of the 30° bridge were smaller when pounding was not considered compared to when pounding occurred, in both the (M) and (F) cases, but for the 45° bridge, pounding caused larger rotations instead.
Skewed bridges with the same mass as the straight bridge counterpart tend to have larger responses compared to where they had the same frequencies, due to the larger in-plane rotations induced.
The seat length of skewed bridges recommended by the NZTA bridge manual could be underestimated, especially in the transverse direction when pounding occurs. In the longitudinal direction, only the displacement of the 45° (M) bridge was underestimated.
Table of Contents
List of publications
List of Figures
List of Tables
Symbols and notations
Chapter 1 Introduction
1.1 Motivation and scope
Chapter 2 Literature review
2.1 Seismic vulnerability of skewed bridges
2.2 Influence of pounding on seismic response of bridges
2.3 Effect of supporting soil on seismic response of bridges
2.4 Experimental studies
2.5 Current design specifications
Chapter 3 Influence of frequency and girder mass on seismic response of skewed bridges
3.1 Bridge prototype and 1:100 scale model
3.2 Ground motions
3.3 Bridge design specifications
3.4 Results and discussions
Chapter 4 Influence of pounding and skew angle on seismic response of bridges
4.1 Bridge prototype and model
4.2 Experimental set-up
4.3 Ground motions
4.4 Girder seat length requirements
4.5 Effect of bridge-abutment pounding and skew angle
Chapter 5 Effect of skew angle and supporting soil on seismic response of bridges
5.1 Bridge model and experimental set-up
5.2 Effect of supporting soil and skew angle on bridge response
Chapter 6 Seismic performance of skewed bridges with simultaneous effects of pounding and supporting soil
6.1 Large-scale model and experimental setup
6.2 Effect of pounding and supporting soil simultaneously on bridge response
Chapter 7 Influence of ground motion characteristics on seismic response of skewed bridges
7.1 Ground motions
7.2 Results and discussion
Chapter 8 Estimation of response of skewed bridges considering pounding and supporting soil
8.1 Results and discussions
Chapter 9 Conclusions and recommendations for future research
9.2 Recommendations for future studies
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Influence of pounding and supporting soil on the seismic response of skewed bridges