Generation of RotDnn Response Spectra

To have a better understanding of the RotDnn measure proposed by Boore (2010) and comprehend the differences between the two response spectra definitions, RotD50 and RotD100, it is worth to look at an example. For a given pair of ground motion records, the two orthogonal as-recorded time series can be combined into a resultant time series at a particular azimuth (or rotation angle, 𝜃), as shown in Figure 3.13, for which the corresponding response spectrum can be generated. The response spectrum can be defined as the maximum of the absolute amplitude of an oscillator response, which can occur in any direction.

For one oscillator with a corresponding period of vibration, the response time series

𝒂_𝑹𝑶𝑻(𝒕; 𝜽) can be computed for a range of azimuths at a rotation angle (𝜃) increment following

Equation 3.1 (Boore, 2010):

𝒂_𝑹𝑶𝑻(𝒕; 𝜽) = 𝒂_𝟏(𝒕)𝒄𝒐𝒔(𝜽) + 𝒂_𝟐(𝒕)𝒔𝒊𝒏(𝜽) Equation 3.1

Where 𝑎₁(𝑡) and 𝑎₂(𝑡) are the original pair of horizontal acceleration time series, which correspond to H1 and H2 in Figure 3.13. This procedure can be repeated for all possible oscillators and all nonredundant angles (0° to 180°) at a chosen angle increment.

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Figure 3.13: Combination of a pair of orthogonal as-recorded acceleration time series for a particular rotation angle and the corresponding response spectrum for the resultant time series.

In this example, using the ground motion data from the 2011 Christchurch earthquake (Kaipoi North School Station), the angle increment is equal to 30° (Figure 3.14), meaning that for each period of vibration, a resultant time series was computed at 6 rotation angles (𝜃). From this, the peak response (acceleration, velocity, and displacement) for each period at each azimuth was identified. The peak responses for three fundamental periods (1s, 2s, and 3s), in terms of relative displacements, are summarized in Table 3.2.

41 It can be observed that for a single period the response varies depending on the orientation, and the maximum response does not occur at the same orientation for all oscillators’ periods. Sorting the displacement response from lowest to highest at each period (right side of Table 3.2), allows the identification of the percentile (nnth) that defines the measure of ground motion for that oscillator’s period. As introduced by Boore (2010), the minimum response, or RotD00, corresponds to the lowest values of response (pink), the median response, or RotD50, to the values at the 50th percentile, and the maximum response (blue), RotD100, to the maximum values (red).

Table 3.2: Displacement response for three oscillators with periods 1, 2, and 3 seconds for the seven chosen azimuths at 30° increments.

Having identified the RotDnn responses at the 00th, 50th, and 100th percentiles for the three oscillator periods of 1s, 2s, and 3s, the displacement response spectrum for this particular pair of ground motions for the 2011 Christchurch earthquake can be plotted. Figure 3.15 shows the possible range of displacement response for these three fundamental periods under a particular pair of ground motion records, from minimum (RotD00) to maximum (RotD100).

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Figure 3.15: RotDnn displacement response spectra at 00th, 50th, and 100th percentiles for oscillators with fundamental periods of 1s, 2s, and 3s under the 2011 Christchurch earthquake record at the Kaipoi

North School station.

Following the procedure previously described, the displacement response spectrum for a particular pair of components can be generated at smaller rotation angle increments and for a wider range of fundamental periods. With the consideration of sufficient rotation angles, the RotD50 response (median) would approach the mean response. Figure 3.16 shows a comparison of the spectral displacements for the different definitions obtained for periods at increments of 0.02 seconds and 10 different azimuths (10° rotation angle increments). As expected, the response given by the RotD100 definition is above the rest, but at some discrete periods the response given by the other three curves (RotD50 spectrum and the two individual signals’ spectra) is close to the maximum. For example, at a period of 3 seconds, it is clear that the response is very different between the four, but at a period of 3.8 seconds the difference is not as obvious. From this, it can be inferred that for a structure with a fundamental period of 3.8 seconds there would not be a significant difference if any of the four definitions were used in its design. However, this is not true for a structure with a fundamental period of 3 seconds, for which this decision would impact the design capacity of said structure.

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Figure 3.16: Displacement response spectra for the two original orthogonal components, and for the two Rotnn definitions, RotD50 and RotD100.

The two definitions, RotD50 and RotD100, discussed here are the definitions adopted by U.S. codes. Nonetheless, it is worth noting that the RotDnn measure develop by Boore (2010) has the characteristic of being able to define any percentile of the response beyond just the median and maximum. As mentioned in Chapter 2 – Section 2.1, ASCE 7 uses conversion factors to generate RotD100 spectra based on RotD50, while AASHTO currently uses geometric mean response spectra, but is expected to move to the RotD50 spectra definition for the next edition of the code. As previously noted, this is not a deliberate choice of endorsing the RotD50 definition. It is rather a default position that is most similar to the current AASHTO code. As research is conducted for MDOF bridge systems, alternatively recommendations may be proposed.

The aforementioned procedure was employed to generate the displacement response spectra, both RotD50 and RotD100, for a suite of 65 pairs of unscaled real acceleration records. For each pair of ground motion records, the spectra was generated solving the equation of motion with the Duhamel integral for a period range of 1 to 5s at seven different damping ratios, ranging

44 from 5% (𝜇 = 1) to 17.37% (𝜇 = 8) as characterized by the damping ductility relationship for bridge structures (Priestley et al., 2007).

RoD50 displacement response spectra

RoD100 displacement response spectra

Figure 3.17: Displacement Response Spectra for the suite 65 pairs of ground motion records.

0 25 50 75 100 125 150 175 200 225 250 275 0 1 2 3 4 5 Displacem ent [cm ] Period [s] 0 25 50 75 100 125 150 175 200 225 250 275 0 1 2 3 4 5 Displacem ent [cm ] Period [s]

45 In sum, a total of 910 response spectra curves were used in this study. This was done using a MATLAB code generated for this research, which can be found in Appendix C. The median and maximum elastic displacement response spectra for all 65 ground motion records are shown in Figure 3.17. From Figure 3.17 it can be observed that the suite of 65 ground motion records chosen for this analyses cover a wide range of spectral displacements across the period range. As expected, the response given by the RotD50 definition is, in all cases, lower than the response given by the RotD100 definition. The assumptions and decisions taken in the SDOF design approach are explained in detail in the next section in this chapter.

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