Question 17.6: Construct a Newmark–Hall design spectrum for a maximum groun......
Construct a Newmark–Hall design spectrum for a maximum ground acceleration equal to that of Northridge earthquake 0.308g and for concrete buildings.
(a) Determine maximum ground motion parameters
Maximum ground acceleration | \ddot{u}_{g}({t})|=0.308\,g
Maximum ground velocity = 1219.2 × 0.308
|\dot{u}_{g}(t)\ |_{\mathrm{max}}=375.5~\mathrm{mm/s}
Maximum ground displacement = 914.4 × 0.308
|U_{g}(t)\ |_{\mathrm{max}}= 281.63~\mathrm{mm}
(b) Determine the amplified response parameters for p = 0.05
From table amplified S_{p a}=2.6\times0.308~g
= 0.801g m/s²
Amplified S_{p v}=1.9\times375.5
= 713.45 mm/s
Amplified S_{p d}=s d=1.4\times281.63
= 394.28 mm
(c) Construction of design spectrum
(i) Draw the maximum ground motion polygon using \left|\ddot{u}_{g}(t)\right|_{\mathrm{max}}, |\dot{u}_{g}(t)\,|_{\mathrm{max}},|\,u_{g}(t)\,|_{\mathrm{max}}.
(ii) Draw the amplified displacement S_{d} bound parallel to the maximum ground displacement.
(iii) Draw the amplified velocity S_{v} parallel to the maximum ground velocity line. It intersects displacement bound at T_{1} = 3.5 s.
(iv) Draw the amplified acceleration S_{pa} bound to maximum ground acceleration. It intersects velocity bound at 0.55 = T_{2}. Extend the amplified acceleration bound downward left to the point corresponding to T_{3} = 0.17.
(v) Draw the amplified acceleration bound linearly from the point corresponding to T_{3} = 0.17 so that it intersects a line at T_{4} = 0.033 s with maximum ground acceleration.
This spectrum is shown in Fig. 17.20.
Researchers have developed procedures to construct the design spectra. From the ground motion parameters the recommended values for firm ground are T_{a} = 1/33; T_{b} = 1/8; T_{e} = 10; T_{f} = 33 ~s. The amplification factors for α_{a}, α_{v}, α_{d} for S_{pa}, S_{pv}, S_{d} were developed for two different non-accedence probabilities 50% and 84.1% as given in Table 17.3. Newmark–Hall elastic spectrum construction is shown in Fig. 17.21.
| Table 17.3 Amplification factors | ||||||
| Damping (%) |
Median 50% | One sigma 84.1% | ||||
| α_{a} | α_{v} | α_{d} | α_{a} | α_{v} | α_{d} | |
| 1 | 3.21 | 2.31 | 1.82 | 4.38 | 3.18 | 2.73 |
| 2 | 2.7 | 2.03 | 1.63 | 3.66 | 2.92 | 2.42 |
| 5 | 2.12 | 1.65 | 1.39 | 2.71 | 2.3 | 2.01 |
| 10 | 1.64 | 1.37 | 1.2 | 1.99 | 1.84 | 1.69 |
| 20 | 1.17 | 1.08 | 1.01 | 1.26 | 1.37 | 1.38 |
| ρ | 3.21– | 2.31– | 1.82– | 4.38– | 3.38– | 2.73– |
| 0.368 ln ρ | 0.41 ln ρ | 0.27 ln ρ | 0.04 ln ρ | 0.67 ln ρ | 0.45 ln ρ | |
