Construct an ADR spectrum for ductility ratio of 1, 1.5, 2, 4 and 6 for a ground motion having peak ground acceleration, peak ground velocity and peak ground displacement as 1 g, 90 cm/sec and 60 cm, respectively. Make use of Newmark–Hall amplification factors for 5% damping and 84.1 percentile.

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Accepted Answer

The inelastic response spectra for this ground motion for ductility ratio of 1.5, 2, 4 and 6 were developed in Example 13.5. An alternative format was shown in Example 13.6. The same data was used to develop the ADRS. The reduction factor R depends upon the acceleration or velocity-sensitive regions given by Equation (14.31). The spectral displacements are computed using Equation (14.32). Typical calculations for generating inelastic ADRS are shown in Table 14.8. Similar calculations can be done for other ductility ratios. The double lines in the table show change in reduction factor region.
The inelastic ADRS is shown in Figure 14.22.

R = 1 for [latex]T < T_{A}[/latex] (14.31a)

[latex]R=(2\mu -1)^{y/2}[/latex] for [latex]T_{A}< T

[latex]R=(2\mu-1)^{0.5} ext{for} T_B\lt T\lt T_C^{\prime}[/latex] (14.31c)

[latex]R=(T/T_C)\mu ext{for} T_C^{\prime}\lt T\lt T_C[/latex] (14.31d)

[latex]R=\mu ext{for} T\gt T_C [/latex] (14.31e)

[latex]\gamma =\frac{\log\left\lgroup\frac{T}{T_A} ight group }{\log \left\lgroup\frac{T_B}{T_A} ight group } [/latex] (14.31f)

[latex]S_{ ext{d inelastic}} =µ S_{ d inelastic}/R[/latex] for [latex] T

[latex]S_{ ext{d inelastic}} =S_{ d inelastic}[/latex] for [latex] T >T_{C}[/latex] (14.32b)

Table 14.8 calculations for R, [latex]S_{a}[/latex] and [latex]S_{d}[/latex] coordinates

µ = 1			µ = 1.5			µ = 2
[latex]S_{a}[/latex]/g	T(sec)	[latex]S_{d}[/latex] (cm)	[latex]R_{y}[/latex]	[latex]S_{a}[/latex]/g	[latex]S_{d}[/latex] (cm)	[latex]R_{y}[/latex]	[latex]S_{a}[/latex]/g	[latex]S_{d}[/latex] (cm)
1	0.01	0.002	1	1	0	1	1	0
1	0.03	0.022	1	1	0.03	1	1	0.04
2.71	0.125	1.053	1.41	1.92	1.12	1.73	1.56	1.22
2.71	0.2	2.696	1.41	1.92	2.86	1.73	1.56	3.11
2.71	0.25	4.213	1.41	1.92	4.47	1.73	1.56	4.86
2.71	0.3	6.067	1.41	1.92	6.43	1.73	1.56	7.01
2.71	0.35	8.258	1.41	1.92	8.76	1.73	1.56	9.54
2.71	0.4	10.785	1.41	1.92	11.44	1.73	1.56	12.45
2.71	0.45	13.65	1.41	1.92	14.48	1.8	1.51	15.17
2.71	0.5	16.852	1.50	1.81	16.85	2.00	1.36	16.85
2.3	0.6	20.596	1.50	1.53	20.60	2.00	1.15	20.6
1.95	0.7	23.767	1.50	1.30	23.77	2.00	0.98	23.77
1.77	0.8	28.178	1.50	1.18	28.18	2.00	0.89	28.18
1.6	0.9	32.237	1.50	1.07	32.24	2.00	0.80	32.24
1.5	1	37.311	1.50	1.00	37.31	2.00	0.75	37.31
1	1.5	55.967	1.50	0.67	55.97	2.00	0.5	55.97
0.7	2	69.648	1.50	0.47	69.65	2.00	0.35	69.65
0.55	2.5	85.505	1.50	0.37	85.51	2.00	0.28	85.51
0.45	3	100.741	1.50	0.30	100.74	2.00	0.23	100.74

Question 14.4: Construct an ADR spectrum for ductility ratio of 1, 1.5, 2, ...

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