Appendix B
Quantile IDA Curves for
Single-Spring Systems
This appendix presents quantile (16th, 50th and 84th percentile) incremental
dynamic analysis (IDA) curves from focused analytical studies on individual
spring single-degree-of-freedom (SDOF) systems. These systems consist of
spring types 1 through 8, with characteristics described in Chapter 3. This
collection of curves is intended to present the range of results for
representative short (T=0.5s), moderate (T=1.0s), and long (T=2.0s) period
systems, both with and without cyclic degradation. In the figures, the
vertical axis is the intensity measure Sa(T,5%), which is not normalized, and
the horizontal axis is the maximum story drift ratio, .max, in radians. IDA
curves with cyclic degradation (black lines) are shown along with IDA
curves without cyclic degradation (grey lines). Differences between the
black and grey lines in the plots indicate the effect of cyclic degradation
given the characteristics of the particular spring and period of vibration.
B-2 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-1 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 0.5s
Figure B-2 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 1.0s.
Figure B-3 Quantile IDA curves plotted versus Sa(T,5%) for Spring 1a and Spring 1b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-3
Figure B-4 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 0.5s.
Figure B-5 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 1.0s.
Figure B-6 Quantile IDA curves plotted versus Sa(T,5%) for Spring 2a and Spring 2b with a period of T = 2.0s.
B-4 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-7 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 0.5s.
Figure B-8 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 1.0s.
Figure B-9 Quantile IDA curves plotted versus Sa(T,5%) for Spring 3a and Spring 3b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-5
Figure B-10 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 0.5s.
Figure B-11 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 1.0s.
Figure B-12 Quantile IDA curves plotted versus Sa(T,5%) for Spring 4a and Spring 4b with a period of T = 2.0s.
B-6 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-13 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 0.5s.
Figure B-14 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 1.0s.
Figure B-15 Quantile IDA curves plotted versus Sa(T,5%) for Spring 5a and Spring 5b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-7
Figure B-16 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 0.5s.
Figure B-17 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 1.0s.
Figure B-18 Quantile IDA curves plotted versus Sa(T,5%) for Spring 6a and Spring 6b with a period of T = 2.0s.
B-8 B: Quantile IDA Curves for Single-Spring Systems ATC-62
Figure B-19 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 0.5s.
Figure B-20 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 1.0s.
Figure B-21 Quantile IDA curves plotted versus Sa(T,5%) for Spring 7a and Spring 7b with a period of T = 2.0s.
ATC-62 B: Quantile IDA Curves for Single-Spring Systems B-9
Figure B-22 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 0.5s.
Figure B-23 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 1.0s.
Figure B-24 Quantile IDA curves plotted versus Sa(T,5%) for Spring 8a and Spring 8b with a period of T = 2.0s.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-1
versus Normalized Intensity Measures
Appendix C
Median IDA Curves for
Multi-Spring Systems versus
Normalized Intensity Measures
This appendix contains normalized plots of median incremental dynamic
analysis (IDA) curves from focused analytical studies on multi-spring singledegree-
of-freedom (SDOF) systems. All systems are composed of two
springs representing a primary lateral-force-resisting system and a secondary
gravity system with the characteristics described in Chapter 3. Multi-spring
systems carry a designation of “NxJa+1a” or “NxJa+1b” where “N” is the
peak strength multiplier (N = 1, 2, 3, 5, or 9), “J” is the lateral-force-resisting
spring number (J = 2, 3, 4, 5, 6, or 7), and 1a or 1b is the gravity system
identifier. In all figures, the vertical axis is the normalized intensity measure
R = Sa(T1,5%)/Say(T1,5%), and the horizontal axis is the maximum story drift
ratio, .max, in radians. The period of vibration for each system is indicated in
parentheses.
C.1 Visualization Tool
Given the large volume of analytical data, customized algorithms were
developed for post-processing, statistical analysis, and visualization of
results. The accompanying CD includes an electronic visualization tool that
was developed to view results of multi-spring studies. The tool is a
Microsoft Excel based application with a user-interface that accesses a
database of all available multi-spring data. By selecting a desired spring
combination (“NxJa+1a” or “NxJa+1b”), stiffness level (stiff or flexible), and
intensity measure (normalized or non-normalized), users can view the
resulting quantile (median, 16th, and 84th percentile) IDA curves for the
combination of interest.
C-2 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a (1.87s)
2x2a (1.32s)
3x2a (1.08s)
5x2a (0.84s)
9x2a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2b (1.87s)
2x2b (1.32s)
3x2b (1.08s)
5x2b (0.84s)
9x2b (0.62s)
Figure C-1 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a and Nx2b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2b+1a (1.53s)
2x2b+1a (1.18s)
3x2b+1a (1.00s)
5x2b+1a (0.80s)
9x2b+1a (0.61s)
Figure C-2 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1a and Nx2b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2a+1b (1.53s)
2x2a+1b (1.18s)
3x2a+1b (1.00s)
5x2a+1b (0.80s)
9x2a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10 S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
2b+1b (1.53s)
2x2b+1b (1.18s)
3x2b+1b (1.00s)
5x2b+1b (0.80s)
9x2b+1b (0.61s)
Figure C-3 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1b and Nx2b+1b with mass M=8.87ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-3
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a (1.87s)
2x3a (1.32s)
3x3a (1.08s)
5x3a (0.84s)
9x3a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3b (1.87s)
2x3b (1.32s)
3x3b (1.08s)
5x3b (0.84s)
9x3b (0.62s)
Figure C-4 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a and Nx3b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a+1a (1.53s)
2x3a+1a (1.18s)
3x3a+1a (1.00s)
5x3a+1a (0.80s)
9x3a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3b+1a (1.53s)
2x3b+1a (1.18s)
3x3b+1a (1.00s)
5x3b+1a (0.80s)
9x3b+1a (0.61s)
Figure C-5 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1a and Nx3b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3a+1b (1.53s)
2x3a+1b (1.18s)
3x3a+1b (1.00s)
5x3a+1b (0.80s)
9x3a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
3b+1b (1.53s)
2x3b+1b (1.18s)
3x3b+1b (1.00s)
5x3b+1b (0.80s)
9x3b+1b (0.61s)
Figure C-6 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1b and Nx3b+1b with mass M=8.87ton.
C-4 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4a (1.18s)
2x4a (0.84s)
3x4a (0.68s)
5x4a (0.53s)
9x4a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4b (1.18s)
2x4b (0.84s)
3x4b (0.68s)
5x4b (0.53s)
9x4b (0.39s)
Figure C-7 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a and Nx4b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4a+1a (1.08s)
2x4a+1a (0.80s)
3x4a+1a (0.66s)
5x4a+1a (0.52s)
9x4a+1a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4b+1a (1.08s)
2x4b+1a (0.80s)
3x4b+1a (0.66s)
5x4b+1a (0.52s)
9x4b+1a (0.39s)
Figure C-8 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1a and Nx4b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4a+1b (1.08s)
2x4a+1b (0.80s)
3x4a+1b (0.66s)
5x4a+1b (0.52s)
9x4a+1b (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
4b+1b (1.08s)
2x4b+1b (0.80s)
3x4b+1b (0.66s)
5x4b+1b (0.52s)
9x4b+1b (0.39s)
Figure C-9 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1b and Nx4b+1b with mass M=8.87ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-5
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5a (1.02s)
2x5a (0.72s)
3x5a (0.59s)
5x5a (0.46s)
9x5a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5b (1.02s)
2x5b (0.72s)
3x5b (0.59s)
5x5b (0.46s)
9x5b (0.34s)
Figure C-10 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a and Nx5b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5a+1a (0.95s)
2x5a+1a (0.70s)
3x5a+1a (0.58s)
5x5a+1a (0.45s)
9x5a+1a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5b+1a (0.95s)
2x5b+1a (0.70s)
3x5b+1a (0.58s)
5x5b+1a (0.45s)
9x5b+1a (0.34s)
Figure C-11 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1a and Nx5b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5a+1b (0.95s)
2x5a+1b (0.70s)
3x5a+1b (0.58s)
5x5a+1b (0.45s)
9x5a+1b (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
5b+1b (0.95s)
2x5b+1b (0.70s)
3x5b+1b (0.58s)
5x5b+1b (0.45s)
9x5b+1b (0.34s)
Figure C-12 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1b and Nx5b+1b with mass M=8.87ton.
C-6 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6a (1.87s)
2x6a (1.32s)
3x6a (1.08s)
5x6a (0.84s)
9x6a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6b (1.87s)
2x6b (1.32s)
3x6b (1.08s)
5x6b (0.84s)
9x6b (0.62s)
Figure C-13 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a and Nx6b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6a+1a (1.53s)
2x6a+1a (1.18s)
3x6a+1a (1.00s)
5x6a+1a (0.80s)
9x6a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6b+1a (1.53s)
2x6b+1a (1.18s)
3x6b+1a (1.00s)
5x6b+1a (0.80s)
9x6b+1a (0.61s)
Figure C-14 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1a and Nx6b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6a+1b (1.53s)
2x6a+1b (1.18s)
3x6a+1b (1.00s)
5x6a+1b (0.80s)
9x6a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
6b+1b (1.53s)
2x6b+1b (1.18s)
3x6b+1b (1.00s)
5x6b+1b (0.80s)
9x6b+1b (0.61s)
Figure C-15 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1b and Nx6b+1b with mass M=8.87ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-7
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7a (1.87s)
2x7a (1.32s)
3x7a (1.08s)
5x7a (0.84s)
9x7a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7b (1.87s)
2x7b (1.32s)
3x7b (1.08s)
5x7b (0.84s)
9x7b (0.62s)
Figure C-16 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a and Nx7b with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7a+1a (1.53s)
2x7a+1a (1.18s)
3x7a+1a (1.00s)
5x7a+1a (0.80s)
9x7a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7b+1a (1.53s)
2x7b+1a (1.18s)
3x7b+1a (1.00s)
5x7b+1a (0.80s)
9x7b+1a (0.61s)
Figure C-17 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1a and Nx7b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7a+1b (1.53s)
2x7a+1b (1.18s)
3x7a+1b (1.00s)
5x7a+1b (0.80s)
9x7a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=8.87
7b+1b (1.53s)
2x7b+1b (1.18s)
3x7b+1b (1.00s)
5x7b+1b (0.80s)
9x7b+1b (0.61s)
Figure C-18 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1b and Nx7b+1b with mass M=8.87ton.
C-8 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2a (3.74s)
2x2a (2.65s)
3x2a (2.16s)
5x2a (1.67s)
9x2a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2b (3.74s)
2x2b (2.65s)
3x2b (2.16s)
5x2b (1.67s)
9x2b (1.25s)
Figure C-19. Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a and Nx2b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2a+1a (3.05s)
2x2a+1a (2.37s)
3x2a+1a (2.00s)
5x2a+1a (1.60s)
9x2a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2b+1a (3.05s)
2x2b+1a (2.37s)
3x2b+1a (2.00s)
5x2b+1a (1.60s)
9x2b+1a (1.21s)
Figure C-20 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1a and Nx2b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2a+1b (3.05s)
2x2a+1b (2.37s)
3x2a+1b (2.00s)
5x2a+1b (1.60s)
9x2a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
2b+1b (3.05s)
2x2b+1b (2.37s)
3x2b+1b (2.00s)
5x2b+1b (1.60s)
9x2b+1b (1.21s)
Figure C-21 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx2a+1b and Nx2b+1b with mass M=35.46ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-9
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3a (3.74s)
2x3a (2.65s)
3x3a (2.16s)
5x3a (1.67s)
9x3a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3b (3.74s)
2x3b (2.65s)
3x3b (2.16s)
5x3b (1.67s)
9x3b (1.25s)
Figure C-22 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a and Nx3b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3a+1a (3.05s)
2x3a+1a (2.37s)
3x3a+1a (2.00s)
5x3a+1a (1.60s)
9x3a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3b+1a (3.05s)
2x3b+1a (2.37s)
3x3b+1a (2.00s)
5x3b+1a (1.60s)
9x3b+1a (1.21s)
Figure C-23 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1a and Nx3b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3a+1b (3.05s)
2x3a+1b (2.37s)
3x3a+1b (2.00s)
5x3a+1b (1.60s)
9x3a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
1
2
3
4
5
6
7
8
9
10
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
3b+1b (3.05s)
2x3b+1b (2.37s)
3x3b+1b (2.00s)
5x3b+1b (1.60s)
9x3b+1b (1.21s)
Figure C-24 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx3a+1b and Nx3b+1b with mass M=35.46ton.
C-10 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
14
16
18
20
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4a (2.37s)
2x4a (1.67s)
3x4a (1.37s)
5x4a (1.06s)
9x4a (0.79s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
14
16
18
20
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4b (2.37s)
2x4b (1.67s)
3x4b (1.37s)
5x4b (1.06s)
9x4b (0.79s)
Figure C-25 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a and Nx4b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4a+1a (2.16s)
2x4a+1a (1.60s)
3x4a+1a (1.32s)
5x4a+1a (1.04s)
9x4a+1a (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4b+1a (2.16s)
2x4b+1a (1.60s)
3x4b+1a (1.32s)
5x4b+1a (1.04s)
9x4b+1a (0.78s)
Figure C-26 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1a and Nx4b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4a+1b (2.16s)
2x4a+1b (1.60s)
3x4a+1b (1.32s)
5x4a+1b (1.04s)
9x4a+1b (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
30
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
4b+1b (2.16s)
2x4b+1b (1.60s)
3x4b+1b (1.32s)
5x4b+1b (1.04s)
9x4b+1b (0.78s)
Figure C-27 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx4a+1b and Nx4b+1b with mass M=35.46ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-11
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5a (2.04s)
2x5a (1.45s)
3x5a (1.18s)
5x5a (0.91s)
9x5a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5b (2.04s)
2x5b (1.45s)
3x5b (1.18s)
5x5b (0.91s)
9x5b (0.68s)
Figure C-28 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a and Nx5b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5a+1a (1.91s)
2x5a+1a (1.39s)
3x5a+1a (1.15s)
5x5a+1a (0.90s)
9x5a+1a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5b+1a (1.91s)
2x5b+1a (1.39s)
3x5b+1a (1.15s)
5x5b+1a (0.90s)
9x5b+1a (0.68s)
Figure C-29 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1a and Nx5b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5a+1b (1.91s)
2x5a+1b (1.39s)
3x5a+1b (1.15s)
5x5a+1b (0.90s)
9x5a+1b (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
20
25
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
5b+1b (1.91s)
2x5b+1b (1.39s)
3x5b+1b (1.15s)
5x5b+1b (0.90s)
9x5b+1b (0.68s)
Figure C-30 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx5a+1b and Nx5b+1b with mass M=35.46ton.
C-12 C: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6a (3.74s)
2x6a (2.65s)
3x6a (2.16s)
5x6a (1.67s)
9x6a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6b (3.74s)
2x6b (2.65s)
3x6b (2.16s)
5x6b (1.67s)
9x6b (1.25s)
Figure C-31 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a and Nx6b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6a+1a (3.05s)
2x6a+1a (2.37s)
3x6a+1a (2.00s)
5x6a+1a (1.60s)
9x6a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6b+1a (3.05s)
2x6b+1a (2.37s)
3x6b+1a (2.00s)
5x6b+1a (1.60s)
9x6b+1a (1.21s)
Figure C-32 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1a and Nx6b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6a+1b (3.05s)
2x6a+1b (2.37s)
3x6a+1b (2.00s)
5x6a+1b (1.60s)
9x6a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
5
10
15
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
6b+1b (3.05s)
2x6b+1b (2.37s)
3x6b+1b (2.00s)
5x6b+1b (1.60s)
9x6b+1b (1.21s)
Figure C-33 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx6a+1b and Nx6b+1b with mass M=35.46ton.
FEMA P440A C: Median IDA Curves for Multi-Spring Systems C-13
versus Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7a (3.74s)
2x7a (2.65s)
3x7a (2.16s)
5x7a (1.67s)
9x7a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7b (3.74s)
2x7b (2.65s)
3x7b (2.16s)
5x7b (1.67s)
9x7b (1.25s)
Figure C-34 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a and Nx7b with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7a+1a (3.05s)
2x7a+1a (2.37s)
3x7a+1a (2.00s)
5x7a+1a (1.60s)
9x7a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7b+1a (3.05s)
2x7b+1a (2.37s)
3x7b+1a (2.00s)
5x7b+1a (1.60s)
9x7b+1a (1.21s)
Figure C-35 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1a and Nx7b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7a+1b (3.05s)
2x7a+1b (2.37s)
3x7a+1b (2.00s)
5x7a+1b (1.60s)
9x7a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
2
4
6
8
10
12
S
a
(T
1
,5%) / S
a,y
(T
1
,5%)
.
max
M=35.5
7b+1b (3.05s)
2x7b+1b (2.37s)
3x7b+1b (2.00s)
5x7b+1b (1.60s)
9x7b+1b (1.21s)
Figure C-36 Median IDA curves plotted versus the normalized intensity measure R = Sa(T1,5%)/Say(T1,5%) for
systems Nx7a+1b and Nx7b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-1
versus Non-Normalized Intensity Measures
Appendix D
Median IDA Curves for
Multi-Spring Systems versus
Non-Normalized Intensity
Measures
This appendix contains non-normalized plots of median incremental dynamic
analysis (IDA) curves from focused analytical studies on multi-spring singledegree-
of-freedom (SDOF) systems. All systems are composed of two
springs representing a primary lateral-force-resisting system and a secondary
gravity system with the characteristics described in Chapter 3. Multi-spring
systems carry a designation of “NxJa+1a” or “NxJa+1b” where “N” is the
peak strength multiplier (N = 1, 2, 3, 5, or 9), “J” is the lateral-force-resisting
spring number (J = 2, 3, 4, 5, 6, or 7), and 1a or 1b is the gravity system
identifier. In the figures, the vertical axis is one of two ground motion
intensities IM = Sa(1s,5%) or Sa(2s,5%), which are not normalized, and the
horizontal axis is the maximum story drift ratio, .max, in radians. The period
of vibration for each system is indicated in parentheses.
D.1 Visualization Tool
Given the large volume of analytical data, customized algorithms were
developed for post-processing, statistical analysis, and visualization of
results. The accompanying CD includes an electronic visualization tool that
was developed to view results of multi-spring studies. The tool is a
Microsoft Excel based application with a user-interface that accesses a
database of all available multi-spring data. By selecting a desired spring
combination (“NxJa+1a” or “NxJa+1b”), stiffness level (stiff or flexible), and
intensity measure (normalized or non-normalized), users can view the
resulting quantile (median, 16th, and 84th percentile) IDA curves for the
combination of interest.
D-2 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(1s,5%) (g)
.
max
M=8.87
2a (1.87s)
2x2a (1.32s)
3x2a (1.08s)
5x2a (0.84s)
9x2a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(1s,5%) (g)
.
max
M=8.87
2b (1.87s)
2x2b (1.32s)
3x2b (1.08s)
5x2b (0.84s)
9x2b (0.62s)
Figure D-1 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a and Nx2b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
2a+1a (1.53s)
2x2a+1a (1.18s)
3x2a+1a (1.00s)
5x2a+1a (0.80s)
9x2a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
2b+1a (1.53s)
2x2b+1a (1.18s)
3x2b+1a (1.00s)
5x2b+1a (0.80s)
9x2b+1a (0.61s)
Figure D-2 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a+1a and
Nx2b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
2a+1b (1.53s)
2x2a+1b (1.18s)
3x2a+1b (1.00s)
5x2a+1b (0.80s)
9x2a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 S
a
(1s,5%) (g)
.
max
M=8.87
2b+1b (1.53s)
2x2b+1b (1.18s)
3x2b+1b (1.00s)
5x2b+1b (0.80s)
9x2b+1b (0.61s)
Figure D-3 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx2a+1b and
Nx2b+1b with mass M=8.87ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-3
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3a (1.87s)
2x3a (1.32s)
3x3a (1.08s)
5x3a (0.84s)
9x3a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3b (1.87s)
2x3b (1.32s)
3x3b (1.08s)
5x3b (0.84s)
9x3b (0.62s)
Figure D-4 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a and Nx3b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3a+1a (1.53s)
2x3a+1a (1.18s)
3x3a+1a (1.00s)
5x3a+1a (0.80s)
9x3a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3b+1a (1.53s)
2x3b+1a (1.18s)
3x3b+1a (1.00s)
5x3b+1a (0.80s)
9x3b+1a (0.61s)
Figure D-5 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a+1a and
Nx3b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3a+1b (1.53s)
2x3a+1b (1.18s)
3x3a+1b (1.00s)
5x3a+1b (0.80s)
9x3a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
3b+1b (1.53s)
2x3b+1b (1.18s)
3x3b+1b (1.00s)
5x3b+1b (0.80s)
9x3b+1b (0.61s)
Figure D-6 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx3a+1b and
Nx3b+1b with mass M=8.87ton.
D-4 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
4a (1.18s)
2x4a (0.84s)
3x4a (0.68s)
5x4a (0.53s)
9x4a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
4b (1.18s)
2x4b (0.84s)
3x4b (0.68s)
5x4b (0.53s)
9x4b (0.39s)
Figure D-7 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a and Nx4b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4a+1a (1.08s)
2x4a+1a (0.80s)
3x4a+1a (0.66s)
5x4a+1a (0.52s)
9x4a+1a (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4b+1a (1.08s)
2x4b+1a (0.80s)
3x4b+1a (0.66s)
5x4b+1a (0.52s)
9x4b+1a (0.39s)
Figure D-8 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a+1a and
Nx4b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4a+1b (1.08s)
2x4a+1b (0.80s)
3x4a+1b (0.66s)
5x4a+1b (0.52s)
9x4a+1b (0.39s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
4b+1b (1.08s)
2x4b+1b (0.80s)
3x4b+1b (0.66s)
5x4b+1b (0.52s)
9x4b+1b (0.39s)
Figure D-9 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx4a+1b and
Nx4b+1b with mass M=8.87ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-5
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
5a (1.02s)
2x5a (0.72s)
3x5a (0.59s)
5x5a (0.46s)
9x5a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
5b (1.02s)
2x5b (0.72s)
3x5b (0.59s)
5x5b (0.46s)
9x5b (0.34s)
Figure D-10 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a and Nx5b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5a+1a (0.95s)
2x5a+1a (0.70s)
3x5a+1a (0.58s)
5x5a+1a (0.45s)
9x5a+1a (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5b+1a (0.95s)
2x5b+1a (0.70s)
3x5b+1a (0.58s)
5x5b+1a (0.45s)
9x5b+1a (0.34s)
Figure D-11 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a+1a and
Nx5b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5a+1b (0.95s)
2x5a+1b (0.70s)
3x5a+1b (0.58s)
5x5a+1b (0.45s)
9x5a+1b (0.34s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
S
a
(1s,5%) (g)
.
max
M=8.87
5b+1b (0.95s)
2x5b+1b (0.70s)
3x5b+1b (0.58s)
5x5b+1b (0.45s)
9x5b+1b (0.34s)
Figure D-12 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx5a+1b and
Nx5b+1b with mass M=8.87ton.
D-6 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6a (1.87s)
2x6a (1.32s)
3x6a (1.08s)
5x6a (0.84s)
9x6a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6b (1.87s)
2x6b (1.32s)
3x6b (1.08s)
5x6b (0.84s)
9x6b (0.62s)
Figure D-13 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a and Nx6b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6a+1a (1.53s)
2x6a+1a (1.18s)
3x6a+1a (1.00s)
5x6a+1a (0.80s)
9x6a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6b+1a (1.53s)
2x6b+1a (1.18s)
3x6b+1a (1.00s)
5x6b+1a (0.80s)
9x6b+1a (0.61s)
Figure D-14 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a+1a and
Nx6b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6a+1b (1.53s)
2x6a+1b (1.18s)
3x6a+1b (1.00s)
5x6a+1b (0.80s)
9x6a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S
a
(1s,5%) (g)
.
max
M=8.87
6b+1b (1.53s)
2x6b+1b (1.18s)
3x6b+1b (1.00s)
5x6b+1b (0.80s)
9x6b+1b (0.61s)
Figure D-15 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx6a+1b and
Nx6b+1b with mass M=8.87ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-7
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S
a
(1s,5%) (g)
.
max
M=8.87
7a (1.87s)
2x7a (1.32s)
3x7a (1.08s)
5x7a (0.84s)
9x7a (0.62s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
S
a
(1s,5%) (g)
.
max
M=8.87
7b (1.87s)
2x7b (1.32s)
3x7b (1.08s)
5x7b (0.84s)
9x7b (0.62s)
Figure D-16 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a and Nx7b
with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7a+1a (1.53s)
2x7a+1a (1.18s)
3x7a+1a (1.00s)
5x7a+1a (0.80s)
9x7a+1a (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7b+1a (1.53s)
2x7b+1a (1.18s)
3x7b+1a (1.00s)
5x7b+1a (0.80s)
9x7b+1a (0.61s)
Figure D-17 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a+1a and
Nx7b+1a with mass M=8.87ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7a+1b (1.53s)
2x7a+1b (1.18s)
3x7a+1b (1.00s)
5x7a+1b (0.80s)
9x7a+1b (0.61s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
a
(1s,5%) (g)
.
max
M=8.87
7b+1b (1.53s)
2x7b+1b (1.18s)
3x7b+1b (1.00s)
5x7b+1b (0.80s)
9x7b+1b (0.61s)
Figure D-18 Median IDA curves plotted versus the intensity measure Sa(1s,5%) for systems Nx7a+1b and
Nx7b+1b with mass M=8.87ton.
D-8 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
2a (3.74s)
2x2a (2.65s)
3x2a (2.16s)
5x2a (1.67s)
9x2a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
2b (3.74s)
2x2b (2.65s)
3x2b (2.16s)
5x2b (1.67s)
9x2b (1.25s)
Figure D-19 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a and Nx2b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2a+1a (3.05s)
2x2a+1a (2.37s)
3x2a+1a (2.00s)
5x2a+1a (1.60s)
9x2a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2b+1a (3.05s)
2x2b+1a (2.37s)
3x2b+1a (2.00s)
5x2b+1a (1.60s)
9x2b+1a (1.21s)
Figure D-20 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a+1a and
Nx2b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2a+1b (3.05s)
2x2a+1b (2.37s)
3x2a+1b (2.00s)
5x2a+1b (1.60s)
9x2a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
2b+1b (3.05s)
2x2b+1b (2.37s)
3x2b+1b (2.00s)
5x2b+1b (1.60s)
9x2b+1b (1.21s)
Figure D-21 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx2a+1b and
Nx2b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-9
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
3a (3.74s)
2x3a (2.65s)
3x3a (2.16s)
5x3a (1.67s)
9x3a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
3b (3.74s)
2x3b (2.65s)
3x3b (2.16s)
5x3b (1.67s)
9x3b (1.25s)
Figure D-22 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a and Nx3b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3a+1a (3.05s)
2x3a+1a (2.37s)
3x3a+1a (2.00s)
5x3a+1a (1.60s)
9x3a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3b+1a (3.05s)
2x3b+1a (2.37s)
3x3b+1a (2.00s)
5x3b+1a (1.60s)
9x3b+1a (1.21s)
Figure D-23 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a+1a and
Nx3b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3a+1b (3.05s)
2x3a+1b (2.37s)
3x3a+1b (2.00s)
5x3a+1b (1.60s)
9x3a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
3b+1b (3.05s)
2x3b+1b (2.37s)
3x3b+1b (2.00s)
5x3b+1b (1.60s)
9x3b+1b (1.21s)
Figure D-24 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx3a+1b and
Nx3b+1b with mass M=35.46ton.
D-10 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
4a (2.37s)
2x4a (1.67s)
3x4a (1.37s)
5x4a (1.06s)
9x4a (0.79s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
4b (2.37s)
2x4b (1.67s)
3x4b (1.37s)
5x4b (1.06s)
9x4b (0.79s)
Figure D-25 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx4a and Nx4b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4a+1a (2.16s)
2x4a+1a (1.60s)
3x4a+1a (1.32s)
5x4a+1a (1.04s)
9x4a+1a (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4b+1a (2.16s)
2x4b+1a (1.60s)
3x4b+1a (1.32s)
5x4b+1a (1.04s)
9x4b+1a (0.78s)
Figure D-26 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for (systems Nx4a+1a and
Nx4b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4a+1b (2.16s)
2x4a+1b (1.60s)
3x4a+1b (1.32s)
5x4a+1b (1.04s)
9x4a+1b (0.78s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
4b+1b (2.16s)
2x4b+1b (1.60s)
3x4b+1b (1.32s)
5x4b+1b (1.04s)
9x4b+1b (0.78s)
Figure D-27 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx4a+1b and
Nx4b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-11
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
5a (2.04s)
2x5a (1.45s)
3x5a (1.18s)
5x5a (0.91s)
9x5a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
S
a
(2s,5%) (g)
.
max
M=35.5
5b (2.04s)
2x5b (1.45s)
3x5b (1.18s)
5x5b (0.91s)
9x5b (0.68s)
Figure D-28 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a and Nx5b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5a+1a (1.91s)
2x5a+1a (1.39s)
3x5a+1a (1.15s)
5x5a+1a (0.90s)
9x5a+1a (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5b+1a (1.91s)
2x5b+1a (1.39s)
3x5b+1a (1.15s)
5x5b+1a (0.90s)
9x5b+1a (0.68s)
Figure D-29 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a+1a and
Nx5b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5a+1b (1.91s)
2x5a+1b (1.39s)
3x5a+1b (1.15s)
5x5a+1b (0.90s)
9x5a+1b (0.68s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
5b+1b (1.91s)
2x5b+1b (1.39s)
3x5b+1b (1.15s)
5x5b+1b (0.90s)
9x5b+1b (0.68s)
Figure D-30 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx5a+1b and
Nx5b+1b with mass M=35.46ton.
D-12 D: Median IDA Curves for Multi-Spring Systems FEMA P440A
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6a (3.74s)
2x6a (2.65s)
3x6a (2.16s)
5x6a (1.67s)
9x6a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6b (3.74s)
2x6b (2.65s)
3x6b (2.16s)
5x6b (1.67s)
9x6b (1.25s)
Figure D-31 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a and Nx6b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6a+1a (3.05s)
2x6a+1a (2.37s)
3x6a+1a (2.00s)
5x6a+1a (1.60s)
9x6a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6b+1a (3.05s)
2x6b+1a (2.37s)
3x6b+1a (2.00s)
5x6b+1a (1.60s)
9x6b+1a (1.21s)
Figure D-32 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a+1a and
Nx6b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6a+1b (3.05s)
2x6a+1b (2.37s)
3x6a+1b (2.00s)
5x6a+1b (1.60s)
9x6a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S
a
(2s,5%) (g)
.
max
M=35.5
6b+1b (3.05s)
2x6b+1b (2.37s)
3x6b+1b (2.00s)
5x6b+1b (1.60s)
9x6b+1b (1.21s)
Figure D-33 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx6a+1b and
Nx6b+1b with mass M=35.46ton.
FEMA P440A D: Median IDA Curves for Multi-Spring Systems D-13
versus Non-Normalized Intensity Measures
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
7a (3.74s)
2x7a (2.65s)
3x7a (2.16s)
5x7a (1.67s)
9x7a (1.25s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
S
a
(2s,5%) (g)
.
max
M=35.5
7b (3.74s)
2x7b (2.65s)
3x7b (2.16s)
5x7b (1.67s)
9x7b (1.25s)
Figure D-34 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a and Nx7b
with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7a+1a (3.05s)
2x7a+1a (2.37s)
3x7a+1a (2.00s)
5x7a+1a (1.60s)
9x7a+1a (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7b+1a (3.05s)
2x7b+1a (2.37s)
3x7b+1a (2.00s)
5x7b+1a (1.60s)
9x7b+1a (1.21s)
Figure D-35 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a+1a and
Nx7b+1a with mass M=35.46ton.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7a+1b (3.05s)
2x7a+1b (2.37s)
3x7a+1b (2.00s)
5x7a+1b (1.60s)
9x7a+1b (1.21s)
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S
a
(2s,5%) (g)
.
max
M=35.5
7b+1b (3.05s)
2x7b+1b (2.37s)
3x7b+1b (2.00s)
5x7b+1b (1.60s)
9x7b+1b (1.21s)
Figure D-36 Median IDA curves plotted versus the intensity measure Sa(2s,5%) for systems Nx7a+1b and
Nx7b+1b with mass M=35.46ton.
FEMA P440A E: Uncertainty, Fragility, and Probability E-1
Appendix E
Uncertainty, Fragility, and
Probability
The concepts presented in this report are compatible with current
probabilistic trends in performance-based seismic design. A probabilistic
context allows explicit consideration of the variability and uncertainty
associated with each of the contributing parameters. An important concept in
probabilistic procedures is the development and use of fragility curves.
Use of fragility curves, and explicit consideration of uncertainty in
performance assessment, is described in ATC-58 Guidelines for Seismic
Performance Assessment of Buildings (ATC, 2007). Fragility curves are also
used to determine the margin of safety against collapse in FEMA P695
Quantification of Building Seismic Performance Factors (FEMA, 2009).
This appendix explains the conversion of incremental dynamic analysis
(IDA) results into fragilities, and presents equations that could be used to
calculate annual probabilities for collapse, or any other limit state of interest.
E.1 Conversion of IDA Results to Fragilities
Incremental dynamic analysis results can be readily converted to fragilities.
Figure E-1 shows an example of IDA results for a single structure subjected
to a suite of ground motions of varying intensities.
Figure E-1 IDA results for a single structure subjected to a suite of ground
motions of varying intensities.
E-2 E: Uncertainty, Fragility, and Probability FEMA P440A
In this illustration, sidesway collapse is the governing mechanism, and
collapse prediction is based on dynamic instability or excessive lateral
displacements. Using collapse data obtained from IDA results, a collapse
fragility can be defined through a cumulative distribution function (CDF),
which relates the ground motion intensity to the probability of collapse
(Ibarra et al., 2002). Studies have shown that this cumulative distribution
function can be assumed to be lognormally distributed. Figure E-2 shows an
example of a cumulative distribution plot obtained by fitting a lognormal
distribution to the collapse data from Figure E-1.
Figure E-2 Cumulative distribution plot obtained by fitting a lognormal
distribution to collapse data from IDA results.
Lognormal distributions are defined by a median value and a dispersion
parameter. The median collapse capacity, Sa50% , indicates a ground motion
intensity that has a 50% chance of producing collapse in the system. It also
indicates the point at which half of the ground motions will produce collapse
at higher intensities, and half will produce collapse at lower intensities. For
each mode of collapse, the record-to-record dispersion can be estimated as:
84% 16% ( ) ( )
2
a a
RTR
. ln S ln S
.
. (E-1)
Figure E-3 provides conceptual collapse fragility curves showing the
probability of collapse due to loss of vertical-load-carrying capacity (LVCC)
or lateral dynamic instability (LDI). These are events are mutually exclusive,
meaning that either one or the other can occur, but both events cannot occur
at the same time.
FEMA P440A E: Uncertainty, Fragility, and Probability E-3
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
S a (T 1 )
P( C | Sa )
combined
vertical collapse
lateral collapse
Figure E-3 Conceptual collapse fragility curves for sidesway (lateral)
collapse, vertical collapse, and a combination of both.
The total probability of collapse due to either mode can then be represented
(Cornell et al., 2005) as:
P(C | sa) = P(CLDI | sa) + P(CLVCC | NCLDI ,sa).P(NCLDI | sa (E-2)
where:
P(CLDI | sa) is the probability of collapse due to lateral dynamic instability at
a ground motion intensity level sa ,
P(CLVCC | NCLDI ,sa) is the probability of collapse due to loss of vertical-loadcarrying
capacity, given that collapse due to lateral dynamic instability has
not occurred at intensity sa , and
P(NCLDI |sa) is the probability of no collapse due to lateral dynamic
instability at a ground motion intensity level sa.
Since P(NCLDI |sa) is equal to 1– P(CLDI | sa), then Equation E-2 can also be
written as:
P(C | sa) = P(CLVCC | NCLDI , sa)+P(CLDI | sa) - P(CLVCC | NCLDI sa).P(CLDI | sa) (E-3)
E.2 Calculation of Annualized Probability
The results of an incremental dynamic analysis expressed as a cumulative
distribution function can be used in combination with a seismic hazard curve
to generate mean annual frequencies (MAF) for collapse (or for other limit
states of interest). This process is the integration of the limit state CDF (e.g.,
E-4 E: Uncertainty, Fragility, and Probability FEMA P440A
fragility representing the probability of collapse as a function of spectral
acceleration) with respect to the probability of occurrence of the intensity
measure (e.g., hazard curve representing the annual probability of exceeding
a full range of spectral accelerations). The mean annual frequency of
collapse, col . , or other limit state of interest, can be approximated (Cornell,
2002) as:
. .exp 1 2 2
col Sa C 2 RTR . .. . .. k . ..
. .
(E-4)
where . . Sa C . . is the mean annual probability of the median spectral
acceleration associated with collapse. The parameter k is the slope of the
hazard curve, and can be calculated as:
(10/ 50)
(2/50)
(2/50) (2/50)
(10/ 50) (10 / 50)
ln
1.65
ln ln
aT
aT
S
S
aT aT
aT aT
H
H
k
S S
S S
. .
.. ..
. . . .
. . . .
.. .. .. ..
. . . .
(E-5)
FEMA P440A F: Example Application F-1
Appendix F
Example Application
This appendix presents an example application of a simplified nonlinear
dynamic analysis procedure. The concept originated during the conduct of
focused analytical studies comparing force-displacement capacity boundaries
to incremental dynamic analysis results. In this procedure, a nonlinear static
analysis is used to generate an idealized force-deformation curve (i.e., static
pushover curve). The resulting curve is then used as a force-displacement
capacity boundary to constrain the hysteretic behavior of an equivalent
single-degree-of-freedom (SDOF) oscillator. This SDOF oscillator is then
subjected to incremental dynamic analysis.
The steps for conducting a simplified nonlinear dynamic analysis are
outlined in the following section, and illustrated using an example building.
Alternative retrofit strategies are evaluated using the same procedure. Use of
the procedure to develop probabilistic estimates of performance for use in
making design decisions is also illustrated.
F.1 Simplified Nonlinear Dynamic Analysis Procedure
The concept of a simplified nonlinear dynamic analysis procedure includes
the following steps:
. Develop an analytical model of the system.
Models can be developed in accordance with prevailing practice for
seismic evaluation, design, and rehabilitation of buildings described in
ASCE/SEI Standard 41-06 Seismic Rehabilitation of Existing Buildings
(ASCE, 2006b). Component properties should be based on forcedisplacement
capacity boundaries, rather than cyclic envelopes.
. Perform a nonlinear static pushover analysis.
Subject the model to a conventional pushover analysis in accordance
with prevailing practice. Lateral load increments and resulting
displacements are recorded to generate an idealized force-deformation
curve.
F-2 F: Example Application FEMA P440A
. Conduct an incremental dynamic analysis of the system based on an
equivalent SDOF model.
The idealized force-deformation curve is, in effect, a system forcedisplacement
capacity boundary that can be used to constrain a hysteretic
model of an equivalent SDOF oscillator. This SDOF oscillator is then
subjected to incremental dynamic analysis to check for lateral dynamic
instability and other limit states of interest. Alternatively, approximate
incremental dynamic analysis can be accomplished using the idealized
force-deformation curve and the Static Pushover 2 Incremental Dynamic
Analysis open source software tool, SPO2IDA (Vamvatsikos and
Cornell, 2006).
. Determine probabilities associated with limit states of interest.
Results from incremental dynamic analysis can be used to obtain
response statistics associated with limit states of interest in addition to
lateral dynamic instability. SPO2IDA can also be used to obtain median,
16th, and 84th percentile IDA curves relating displacements to intensity.
Using the fragility relationships described in Appendix E in conjunction
with a site hazard curve, this information can be converted into annual
probabilities of exceedance for each limit state. Probabilistic information
in this form can be used to make enhanced decisions based on risk and
uncertainty, rather than on discrete threshold values of acceptance.
F.2 Example Building
The example building is a five-story reinforced concrete frame residential
structure with interior unreinforced masonry infill partitions in the upper
stories, and a soft/weak first-story. An exterior elevation of the building is
shown in Figure F-1 and first floor plan is shown in Figure F-2. Reinforced
concrete columns in each orthogonal direction provide lateral resistance to
seismic forces. As indicated in Figure F-2, the first story includes a mixture
of components with column-like proportions and components that are more
like slender shear walls.
This building is a prototypical example of a soft/weak story structure.
Concentration of inelastic deformations in the first story presents an obvious
potential story collapse mechanism.
F.3 Structural Analysis Model
To investigate the potential for collapse in this structure, it is reasonable to
assume that the response can be represented by a SDOF model. The first
story column components are classified for modeling purposes in accordance
FEMA P440A F: Example Application F-3
with ASCE/SEI 41-06. Most of the columns in this example are classified as
shear-controlled or flexure-shear controlled. Wall-like column components
are shear-controlled along the strong axis of the member.
Figure F-1 Example building exterior elevation.
Figure F-2 Example building first floor plan.
Modeling parameters for the column components can be characterized by the
conceptual force-displacement relationship (“backbone”) specified in
ASCE/SEI 41-06. The modeling parameters selected for the components in
this example are taken from Chapter 6 of ASCE/SEI 41-06, and depicted in
Figure F-3. In both cases, the residual strength, c, is taken as zero.
F-4 F: Example Application FEMA P440A
The column components are assembled into a model of the structural system
as shown in Figure F-4. Inelastic response is assumed to occur
predominantly in the first story. First-story columns are taken as fixed at the
base on a rigid foundation. The stiffness of the column components are
based on elastic properties in flexure and shear. The model includes soil
flexibility, allowing for rigid body rotation due to the response of the
structure above. Soil stiffness parameters are taken from Chapter 4 of
ASCE/SEI 41-06, assuming a relatively soft soil site (site Class E).
n V
.total columns y .
b . 0.01
a . 0
A c . 0
B,C
D,E
n V
total . columns y .
b . 0.01
a . 0
A c . 0
B,C
D,E
(a)
n V
walls
h
.
e . 0.01
d . 0.0075
A c . 0
B,C
D,E
n V
walls
h
.
e . 0.01
d . 0.0075
A c . 0
B,C
D,E
(b)
Figure F-3 Force-displacement modeling parameters for: (a) column
components; and (b) wall-like column components.
When developing a SDOF representation of a system, it is important to
account for foundation rotation in assessing column distortions. The
resulting SDOF model represents the relationship between the total first floor
drift, including contributions from the foundation, sys fdn cols . .. .. , and the
applied inertial loads, V .
FEMA P440A F: Example Application F-5
fdn .
col . Column
distortion
Foundation
rotation
1st story mechanism-all
inelasticity in columns
Rigid foundation on elastic supports
Effective
height of
inertial
forces, h* hcm
h1st
W W
sys . fdn col . 1st . . . .. h
cm col 1st fdn cm . .. h .. h
V V
eff K.
fdn .
col . Column
distortion
Foundation
rotation
1st story mechanism-all
inelasticity in columns
Rigid foundation on elastic supports
Effective
height of
inertial
forces, h* hcm
h1st
W W
. . sys fdn col 1st . . . .. h
cm col 1st fdn cm . .. h .. h
V V
eff K. eff K.
(a.) (b)
Figure F-4 Structural analysis model showing: (a) assumptions; and (b)
distortions.
F.4 Nonlinear Static Pushover Analysis
The analytical model is subjected to a conventional pushover analysis.
Results are shown in Figure F-5.
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
V
sys .
LVCC .
LDI .
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
V
sys . 0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
V
sys .
LVCC .
LDI .
Figure F-5 Pushover curve from nonlinear static analysis.
F-6 F: Example Application FEMA P440A
F.5 Evaluation of Limit States of Interest
Collapse in real structures can be caused by sidesway collapse (lateral
dynamic instability) or by loss of vertical-load-carrying capacity. In this
example, the following two limit states are defined (both are shown in Figure
F-5):
LVCC . the total system rotation at which loss of vertical-load-carrying
capacity occurs (i.e., when first story columns fail due to shear
distortion). In this example, the critical column distortion for loss of
vertical-load-carrying capacity is taken as 1% inelastic rotation,
which occurs when the total system rotation sys . = 1.2%.
LDI . the total system rotation at which lateral dynamic instability occurs
(i.e., when first story columns lose all lateral-force-resisting
capacity). In this example, this is taken to occur when the total
system rotation sys . = 4.0%.
The target displacement for a given intensity is estimated using the
Coefficient Method:
2
0 1 2 4 2
e
t a
. C C C S T g
.
.
Uniform hazard spectra for the example site are shown in Figure F-6 and
Figure F-7.
Figure F-6 Uniform hazard spectrum for intensity corresponding to 10%
chance of exceedance in 50 years (from USGS).
FEMA P440A F: Example Application F-7
Figure F-7 Uniform hazard spectrum for intensities corresponding to 2%
chance of exceedance in 50 years (from USGS).
For an intensity corresponding to a 10% chance of exceedance in 50 years
(i.e., 475-year return period), and a period T = 0.3s:
Sa .1.36g
The strength of the model is:
Fy . 0.45g
which results in:
aT 3.0
y
R S g
F
. . for the 10%/50 year hazard level.
The coefficients are:
C0 . first mode participation factor = 1.0,
1 2
1 1 1.3
e
C R
aT
.
. . . ,
where a = 50, and
2
2
1 1 1 1.04
800 e
C R
T
. . .
. . .. .. .
. .
.
This results in a target displacement of:
F-8 F: Example Application FEMA P440A
1.6 t .
. inches, or .sys .1.6/100 .1.6%.
This is greater than the acceptable limit for loss of vertical-load-carrying
capacity ( LVCC . ) taken as sys . = 1.2%.
To check for lateral dynamic instability, the proposed equation for Rdi is:
3
3
a
c e r u r
di e
y c y
R b T F T
. F
. . . . . . . .
. .. .. . . .. .. . . . . . .
where Te is the effective fundamental period of vibration of the structure, .y
,
.c
, .r
, and .u
are displacements corresponding to the yield strength, Fy,
capping strength, Fc, residual strength, Fr, and ultimate deformation capacity
at the end of the residual strength plateau. Determination of these parameters
requires a multi-linear idealization of the pushover curve, as shown in Figure
F-8.
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
Idealization A
Idealization B
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0.00% 1.00% 2.00% 3.00% 4.00%
System rotation
Base shear (%g)
Idealization A
Idealization B
Figure F-8 Pushover curve from nonlinear static analysis and two idealized
system force-displacement capacity boundaries.
Parameters a and b are functions given by:
1 exp( ) e a . . .dT
2
1 . ..
.
. ..
.
. .
c
r
F
b F
FEMA P440A F: Example Application F-9
and parameter d is a constant equal to 4 for the example building (assuming
the presence of stiffness degradation).
Using the above expressions along with parameters from Idealization ‘A’ in
Figure F-8, results in:
Rdi . 2.6
which is less that the calculated value of R = 3.0 for intensities corresponding
to the 10%/50 year hazard level. The parameter R is a ratio equal to the
strength necessary to keep a system elastic for a given intensity, divided by
the yield strength of the system. Higher values of R imply lower values of
system yield strength. Values of R that exceed Rdi mean that the structure
does not meet the minimum strength necessary to avoid lateral dynamic
instability at this hazard level.
At higher intensities (e.g., 2%/50 year hazard level) the calculated value of R
would be even higher (R = 5.3 >> Rdi = 2.6), illustrating how the comparison
between system strength and the limit on lateral dynamic instability would
change for a different hazard level.
In summary, the example structure does not meet acceptability criteria for
loss of vertical-load-carrying capacity and lateral dynamic instability at the
10%/50 year hazard level. Thus, a nonlinear response-history analysis must
be performed.
F.6 Incremental Dynamic Analysis
The resulting force-displacement relationship from the pushover curve can be
used to generate a force-displacement capacity boundary for the system. An
incremental dynamic analysis (IDA) can then be applied to a SDOF oscillator
constrained by the resulting force-displacement capacity boundary.
Performing an incremental dynamic analysis will allow determination of the
ground motion intensity at which various limit state deformations occur.
For the example building, an approximate incremental dynamic analysis is
performed using the open source software tool, SPO2IDA. Use of SPO2IDA
along with Idealization ‘A’ in Figure F-8 results in the median, 16th, and 84th
percentile IDA curves shown in Figure F-9. The figure also includes the
estimate of Rdi for lateral dynamic instability.
F-10 F: Example Application FEMA P440A
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.0% 1.0% 2.0% 3.0% 4.0% 5.0%
System rotation
Spectral acceleration
IDA-50% IDA-84% IDA-16% Capacity Boundary
Loss of vertical
carrying capacity
(LVCC)
Lateral dynamic
instability (LDI)
Rdi eqn.
Figure F-9 Results of approximate incremental dynamic analysis using
SPO2IDA.
F.7 Determination of Probabilities Associated with
Limit States of Interest
From Figure F-9, median values of intensity causing loss of vertical-loadcarrying
capacity (LVCC) or lateral dynamic instability (LDI) in the example
building can be obtained. Using the expressions in Appendix E, the
dispersion and mean annual frequencies (MAF) associated with these limit
states can be determined. The resulting data is presented Table F-1.
Table F-1 Mean Annual Frequencies for Collapse Limit States
Limit state/collapse mode
Sa50
MAF Sa50 .. MAF
collapse
Loss of vertical load
carrying capability 0.92 0.0050 0.20 0.0060
Lateral dynamic instability 1.26 0.0025 0.32 0.0040
LVCC or LDI 0.88 0.0060 0.17 0.0069
For the example building, fragilities associated with loss of vertical-loadcarrying
capacity (LVCC) or lateral dynamic instability (LDI) are derived
from the median values of spectral acceleration and dispersions in Table F-1,
as illustrated in Figure F-10.
FEMA P440A F: Example Application F-11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00 1.00 2.00 3.00
Spectral accel., S aT
Probability of collapse, P(C|Sa