The response quantities obtained from both the response-spectrum analysis and the timehistory¬†analysis included bending moments, shear forces, axial forces, and displacements. A¬†detailed review of the response results showed that the observations from the shear forces¬†and the axial forces were the same as those from the bending moments. Given this, only the¬†bending moments and the displacements were used for the evaluation of the seismic¬†performance of the bridge. However only the results for bending moments are shown there,¬†the results for deflections can be found in .
For simplicity in discussing the results, the simultaneous excitations in the longitudinal and¬†vertical directions are referred to as excitations in the longitudinal direction (or longitudinal¬†excitations), and those in the transverse and vertical directions are referred to as excitations¬†in the transverse direction (or transverse excitations). This is the case for both the responsespectrum¬†and the time-history analyses.
To assist in understanding the results from the analyses, it is useful to describe the¬†convention for the moments, as used in this study. In reference to the coordinate system¬†shown in Fig. 5, longitudinal moments in the bridge girder are those that act about the Yaxis,¬†and transverse moments are those that act about the Z-axis. For the piers, the moments¬†that result from longitudinal excitations and act about the Y-axis are referred to as “moments¬†in the longitudinal direction”, and those that result from transverse excitations and act about¬†the X-axis are referred to as “moments in the transverse direction”.
The moments at the joints of the model resulting from the response-spectrum analysis¬†represent the maximum absolute values and by definition are positive. The time-history¬†analysis provided a comprehensive set of results for each excitation motion. Time histories¬†and maximum positive and negative values for the moments and displacements were¬†obtained for the joints of the model. Moment and displacement envelopes for both the¬†girder and the piers were determined using the largest absolute values of the computed¬†(positive and negative) maxima for each of the selected sets of ground motions.
The comparisons of bending moments are shown in Figs. 15 and 16. Figure 15(a) shows the¬†envelopes of the longitudinal moments in the bridge girder for seismic actions in the¬†longitudinal direction, and Fig. 15(b) shows the envelopes of the transverse moments for¬†seismic actions in the transverse direction. The moment envelopes are plotted using the¬†corresponding values at selected sections along the bridge girder. Similarly, Figs. 16(a) and¬†16(b) present the moment envelopes for pier P31 for excitations in the longitudinal and¬†transverse directions respectively. The moment envelopes for the other piers are similar to¬†those for pier P31, and they are not shown here. The designation “Design” in Figs. 15 and 16 is for the design responses which were calculated by , and “UHS” is for the responses due¬†to seismic actions represented by the uniform hazard spectrum. Furthermore, the¬†designations “World-wide”, “Saguenay”, “Miramichi”, and “Simulated” are respectively for¬†the responses due to the selected world-wide records ‚Äď short-period set (Fig. 10), the¬†Saguenay records (Fig. 12), the Miramichi records (Fig. 13), and the simulated motions ‚Äď¬†short-period hazard set (Fig. 14(a)).
For the purpose of clarity, the results from the response-spectrum analysis (i.e., the “Design”¬†and the “UHS” results) are discussed first. It can be seen from Fig. 15(a) that for the seismic¬†actions in the longitudinal direction, the UHS envelope of the moments in the bridge girder¬†is somewhat higher than the design envelope. Also, the values of the UHS envelope for the¬†pier (Fig. 16(a)) resulting from the longitudinal seismic actions are larger than those of the¬†design envelope in the upper 25 m of the pier. The largest differences are approximately¬†20%. These observations for the longitudinal seismic actions were expected because the¬†periods of the predominant longitudinal and vertical modes of the bridge are shorter than¬†1.5 s, i.e., these are within the range in which the uniform hazard spectrum is higher than¬†the design spectrum (Fig. 3). For seismic actions in the transverse direction, the UHS¬†envelopes of the moments in the bridge girder and in the pier (Figs. 15(b) and 16(b),¬†respectively) are all smaller than the design values. This is because the uniform hazard¬†spectrum is lower than the design spectrum for the periods of the predominant transverse¬†modes, i.e., periods longer than approximately 2.0 s (Fig. 3).
The second reason is related to the conservatism of the response resulting from the uniform¬†hazard spectrum. By definition, the uniform hazard spectrum at the bridge location¬†represents the envelope of the spectral contributions of all possible earthquakes in the¬†surrounding area that affect the seismic hazard at the location. This implies that the seismic¬†response resulting from the uniform hazard spectrum represents the envelope of the¬†response contributions from earthquakes with different magnitudes and at different ¬†distances from the bridge location, assuming that all the earthquakes occur at the same time.
Obviously, the response from such combined earthquake actions is much larger than the¬†responses from each of the earthquakes considered separately. These considerations clearly¬†show that the response-spectrum analysis using the uniform hazard spectrum provides¬†significantly larger responses than those from expected seismic ground motions represented¬†by that spectrum.
In regard to the response results obtained from the time-history analysis of the model ¬†subjected to the selected sets of excitations, it can be seen in Figs. 15 and 16 that the¬†maximum moments are all smaller than the design responses for both the longitudinal and¬†transverse excitations. This was expected based on the spectral characteristics of the¬†excitation motions. As described earlier, the response spectra of the excitation motions used¬†in the analysis (i.e., the World-wide short-period set, the Saguenay set, the Miramichi set,¬†and the simulated short-period set) are all lower than the design spectrum for periods¬†longer than approximately 0.5 s (Figs. 10, 12-14), i.e., within the period range of the¬†longitudinal and transverse modes that produce almost the entire response. The¬†contributions of the modes with periods below 0.5 s, where the spectra of the excitation¬†motions exceed the design spectrum, are very small.