The document discusses response spectra and methods for obtaining inelastic response spectra. It begins by showing a "comb-like" plot of a response spectrum with peaks at natural periods of a structure. It then provides two scenarios for obtaining an inelastic response spectrum: 1) Assuming a linear equivalent system and 2) Rotating the elastic response spectrum 45 degrees clockwise and using the resulting demand diagram. The document also discusses the improved ATC-40 procedure which matches equivalent damping ratios between capacity and demand diagrams.

1. 5.2 RESPONSE SPECTRUM
Comb of a music box:

2. Response spectrum
Comb SA 1
!
SA 2
T1 T2 Tn
h1 h2 hn
SA n
..
u
 Looks exactly like a “comb”

3. Response spectrum
Amplitude
Period
 This is not a “density
function”.

4. Simple approximation

5. Examples
 El Centro (NS)
 Kobe (NS)

6. “Umemura”
Spectrum?
a  T 2
    0 < T < T1
 2  2π 
  T 
S D =  v  T1 < T < T2
  2π 
 d T2 < T


Sh 1 .5
=
S 0.05 1 + 10h

7. Simple approximation
log S A = log SV + log 2π − log T
log S D = log SV − log 2π + log T

8. Tripartite plot?
Sv
JMA Kobe
Sd
Velocity (cm/s)
1 00
10
0
2 cm
/s
Sa
cm
00
10
10
2
/s cm
cm
0
10
10
0.1 1 10
Period (s)

9. Rotate it 45 degrees
clockwise?
Sv
JMA Kobe
Sd
Velocity (cm/s)
1 00
10
0
2 cm
/s
Sa
cm
00
10
10
2
/s cm
cm
0
10
10
0.1 1 10
Period (s)

10. Two scenarios for
obtaining inelastic
response spectrum

11. Assuming linear
equivalent system… Scenario #1
Equivalent
Equivalent period : h
Damping ratio
:

12. Again “Umemura”
Spectrum
a  T 2
    0 < T < T1
 2  2π 
  T 
S D =  v  T1 < T < T2
  2π 
 d T2 < T


Sh 1 .5
=
S 0.05 1 + 10h

13. Improved ATC-40 Procedure
Plot capacity and demand
diagrams in A-D format Demand diagram for µ = 2
Yielding branch of capacity
diagram intersects the demand
diagram for several μ Capacity diagram
At relevant intersection point, μ
from the two diagrams should
match (demand point).
 Interpolate between two μ
values or plot demand diagrams
at finer μ values if necessary
Demand point (µ = 4)

14. One more scenario for
obtaining nonlinear response
spectrum…
Scenario #2
µ
u plastic / u elastic =
2µ −1
u plastic / u elastic =
  1
2µ − 1
A
Acceleration
(inertia force)
a A’
Y
o Displacement
u plas t iic
uy u e la s t ic

15. Rotating original linear
spectrum 45 degrees
clockwise…
Sv
JMA Kobe
Sd
Velocity (cm/s)
1 00
10
0
2 cm
/s
Sa
cm
00
10
10
2
/s cm
cm
0
10
10
0.1 1 10
Period (s)

16. Inelastic response spectrum
(demand diagram) for this
case
Sa
A
Y
A’
T1
T2
Sd
T
µ=5 µ=5