Question 14.7: Construct an ADR spectrum for ductility ratio of 1, 2, 3, 4,...
Construct an ADR spectrum for ductility ratio of 1, 2, 3, 4, 5 and 6 for a ground motion having peak ground acceleration, peak ground velocity and peak ground displacement as 1g, 90 cm/sec and 60 cm, respectively. For elastic response spectrum, make use of Newmark–Hall amplification factors for 5% damping and 84.1 percentile. For constant ductility spectra, make use of reduction factors as specified in the Appendix B, Eurocode 8-Part 1.
Plot the elastic spectral acceleration-spectral displacement curve for 5% damping using Equation (14.26) along with radial lines for T = 0.5, 1, 1.5 and 2 sec as in previous examples. The reduction factors for spectral accelerations specified in Eurocode 8 are as follows:
R = (µ − 1)T/T_{C} + 1 for T < T_{C} (14.33a)
R = µ for T >T_{C} (14.33a)
S_{di}=\frac{T_{I}^{2}}{4\pi ^{2}} S_{ai}g (14.26)
The spectral displacements are obtained using Equation (14.32). Typical calculations for generating inelastic ADRS are shown in Table 14.11. The double lines in the table show change in reduction factor R region. Similar calculations can be done for other ductility ratios.
The constant ductility ADRS is shown in Figure 14.25.
S_{\text{d inelastic}} =µ S_{\text{d inelastic}}/R for T < T_{C} (14.32a)
S_{\text{d inelastic}} =S_{\text{d inelastic}} for T > T_{C} (14.32b)
Table 14.11 Calculations for r, S_{a} and S_{d} coordinates
| µ = 1 (1 g) | µ = 2 | µ = 3 | ||||||
| S_{a}/g | T | S_{d}(m) | R | S_{a}/g | S_{d}(m) | R | S_{a}/g | S_{d}(m) |
| 1 | 0.01 | 0.0000 | 1.0200 | 0.9804 | 0.0000 | 1.0400 | 0.9615 | 0.0001 |
| 1 | 0.03 | 0.0002 | 1.0600 | 0.9434 | 0.0004 | 1.1200 | 0.8929 | 0.0006 |
| 2.71 | 0.125 | 0.0105 | 1.2500 | 2.1680 | 0.0169 | 1.5000 | 1.8067 | 0.0211 |
| 2.71 | 0.2 | 0.0270 | 1.4000 | 1.9357 | 0.0385 | 1.8000 | 1.5056 | 0.0449 |
| 2.71 | 0.3 | 0.0607 | 1.6000 | 1.6938 | 0.0758 | 2.2000 | 1.2318 | 0.0827 |
| 2.71 | 0.4 | 0.1079 | 1.8000 | 1.5056 | 0.1198 | 2.6000 | 1.0423 | 0.1244 |
| 2.71 | 0.5 | 0.1685 | 2.0000 | 1.3550 | 0.1685 | 3.000 | 0.9033 | 0.1685 |
| 2.33 | 0.6 | 0.2086 | 2.0000 | 1.1650 | 0.2086 | 3.0000 | 0.7767 | 0.2086 |
| 1.83 | 0.8 | 0.2913 | 2.0000 | 0.9150 | 0.2913 | 3.0000 | 0.6100 | 0.2913 |
| 1.5 | 1 | 0.3731 | 2.0000 | 0.7500 | 0.3731 | 3.0000 | 0.5000 | 0.3731 |
| 1 | 1.5 | 0.5597 | 2.0000 | 0.5000 | 0.5597 | 3.0000 | 0.3333 | 0.5597 |
| 0.93 | 1.6 | 0.5922 | 2.0000 | 0.4650 | 0.5922 | 3.0000 | 0.3100 | 0.5922 |
| 0.81 | 1.8 | 0.6528 | 2.0000 | 0.4050 | 0.6528 | 3.0000 | 0.2700 | 0.6528 |
| 0.71 | 2 | 0.7064 | 2.0000 | 0.3550 | 0.7064 | 3.0000 | 0.2367 | 0.7064 |
| 0.55 | 2.5 | 0.8551 | 2.0000 | 0.2750 | 0.8551 | 3.0000 | 0.1833 | 0.8551 |
| 0.45 | 3 | 1.0074 | 2.0000 | 0.2250 | 1.0074 | 3.0000 | 0.1500 | 1.0074 |
