**7. Dynamic analysis and results**

For the purpose of the seismic evaluation of the bridge, dynamic analyses were conducted on the bridge model to determine the responses due to seismic actions represented by the uniform hazard spectrum and the selected sets of records. Elastic material properties of the model were assumed in the analyses. The dynamic analyses included both responsespectrum analyses and time-history analyses.

#### **Response-spectrum analyses**

272 Earthquake Engineering

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Spectral acceleration (g)

(b)

Spectral acceleration (g)

(a)

Since the seismic hazard based on the service life and the importance of the bridge corresponds to an annual probability of exceedance of 0.00027, it was necessary to scale the simulated accelerograms to be consistent with the uniform hazard spectrum (UHS) for a probability of exceedance of 0.00027 (Fig. 3). To determine the short-period hazard motions for the bridge, the simulated accelerograms for the M=6.0 event were scaled to have the same spectral values at the period of 0.2 s as that of the UHS for the bridge location. Similarly, the long-period hazard motions were obtained by scaling the simulated accelerograms for the

> M=6.0, R=50 km Design spectrum No.1 No.2 No.3 No.4

> M=7.0, R=100 km Design spectrum No.1 No.2 No.3 No.4

M=7.0 event to have the same spectral values as that of the UHS at the period of 2.0 s.

**Figure 14.** Design spectrum and scaled response spectra of simulated excitations; 5% damping

Trial time-history analyses showed that the largest responses of the bridge model are associated with the scaled accelerograms corresponding to the epicentral distances of R=50 km for the M=6.0 event and R=100 km for the M=7.0 event, and therefore, only these accelerograms were considered. The response spectra of the scaled short-period hazard accelerograms (R=50 km, M=6.0) and long-period hazard accelerograms (R=100 km, M=7.0) are shown in Figs. 14(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 Period (s)

0 0.5 1 1.5 2 2.5 3 3.5 4 Period (s)

(a) short-period hazard motions, (b) long-period hazard motions.

Response-spectrum analyses were performed for seismic actions represented by the uniform hazard spectrum. Separate response-spectrum analyses were carried out for the following two cases of seismic actions: (i) seismic actions in the longitudinal and vertical directions of the model; and (ii) seismic actions in the transverse and vertical directions. These two cases were considered appropriate since the longitudinal and the transverse modes are well separated, and the vertical modes are combined mainly with the longitudinal modes. The horizontal and the vertical actions were applied simultaneously at the bases of the piers. The horizontal seismic actions were represented by the horizontal uniform hazard spectrum (UHS) (Fig. 3), and the vertical actions were represented by a spectrum obtained by multiplying the horizontal UHS by 2/3. The factor of 2/3 is commonly used for defining vertical design spectra relative to horizontal spectra [9].

The analyses included the first 100 modes, which covered all natural periods above 0.02 s. A modal damping of 5% was used for all the modes. The response maxima at each joint of the models were computed by combining the modal responses using the complete quadratic combination (CQC) rule.

As required by the Canadian Highway Bridge Design Codes [24], the mass participation of the modes considered in the analysis is larger than 90% in each of the three principal directions of the model. Namely, the amounts of the mass participation of the longitudinal, transverse and vertical modes used in the analysis are 95.3%, 95.5% and 93.6% respectively.

#### **Time-history analyses**

Time-history analyses were conducted to determine the responses of the model subjected to the records of the selected sets. As in the response-spectrum analysis, simultaneous seismic excitations in the longitudinal and vertical directions, and in the transverse and vertical directions of the model were used in the time-history analysis. In each analysis, the seismic excitations consisted of a pair of scaled horizontal and vertical acceleration time histories applied at the bases of the piers.

The mode-superposition method was used in the time-history analysis. As in the responsespectrum analysis, the first 100 modes and modal damping of 5% for all the modes were considered in the time-history analysis. The response time histories were obtained at equal time interval of 0.005 s.

Dynamic Behaviour of the Confederation Bridge Under Seismic Loads 275

is for the design responses which were calculated by [7], 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 – short-period set (Fig. 10), the Saguenay records (Fig. 12), the Miramichi records (Fig. 13), and the simulated motions –

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

The 20% exceedance of the design responses by those from the UHS seismic actions in the longitudinal direction does not represent any concern regarding the seismic safety of the bridge. This is because of the following two reasons. First, conservative assumptions are involved in the design through the use of factored material strengths and specified safety factors, and therefore the actual *capacity* (i.e., resistance) of the bridge is substantially larger than the *demands* due to design loads. For example, considering only the resistance factors for concrete and reinforcing steel used in the design (i.e., φc=0.75 and φs=0.85, as specified in the Design Criteria [5]), the nominal flexural resistance of the bridge is about 20% larger than the design resistance. Other safety factors involved in the design, associated with the specified safety index [5], provide even larger resistances relative to the design resistance of the bridge. 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

short-period hazard set (Fig. 14(a)).

by that spectrum.

modes, i.e., periods longer than approximately 2.0 s (Fig. 3).
