1. EARTHQUAKE
ENGINEERING
2.5. Response Spectrum
2.5.1. Definition and Construction
of Response Spectrum
2.5.2. Analysis Procedure
Using Response Spectrum

2. 2.5.1. Definition and Construction of Response Spectrum
Definition
Response Spectrum is a Plot of the Maximum Peak
Response of Single Degree of Freedom System as a
Function of its Natural Period (Frequency) For a
Specific Dynamic Force (or Earthquake)
Prof.Dr. Osman Shaalan
2 Earthquake Engineering Dr. Tharwat Sakr

3. 2.5.1. Definition and Construction of Response Spectrum
Example
NorthBridge
Earthquake
3 Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr

4. 2.5.1. Definition and Construction of Response Spectrum
Damping Effects
4 Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr

5. 2.5.1. Definition and Construction of Response Spectrum
Benefits
Characterization of Ground Motions
Response Spectrum
15
20
9000
18
8000
16
7000
14
10
6000
Acceleration
Acceleration
12
5000
10
4000
8
5
3000
6
2000
4
1000
2
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Period [sec]
Prof.Dr. Osman Shaalan
5 Earthquake Engineering Dr. Tharwat Sakr

6. 2.5.1. Definition and Construction of Response Spectrum
Benefits
Earthquake Resistance Design
Almost all Codes of
Practice recommends
the Response Spectrum
as the Main Method for
Earthquake Resistance
Design
Prof.Dr. Osman Shaalan
6 Earthquake Engineering Dr. Tharwat Sakr

7. 2.5.1. Definition and Construction of Response Spectrum
Generation of Response Spectrum
Select the Required excitation & Damping Ration
Choose SDOF System
Change the Period of the System in the practical range
For each Period make a Time History and Get the
Maximum Required Response (Displacement – Velocity –
Acceleration)
Plot The Response Vs Period
REFER TO THE MATLAB
SEGMENT ATTACHED
Prof.Dr. Osman Shaalan
7 Earthquake Engineering Dr. Tharwat Sakr

8. 2.5.2. Analysis Procedure Using Response Spectrum
SDOF Structures (Procedure)
Calculate the Structure dynamic
parameters (K, M, ω, T)
Select the Appropriate Value of Damping Ratio
From the Response Spectrum Chart get the Spectral Value
of the required response (Sa, Sv or Sd) Commonly Sa
The Maximum Displacement umax = S a / ω 2
The Maximum Base Shear Vmax = S d × K = ( S a / g ) × W
The Maximum Base Moment M max = Vmax × h
Prof.Dr. Osman Shaalan
8 Earthquake Engineering Dr. Tharwat Sakr

9. 2.5.2. Analysis Procedure Using Response Spectrum
SDOF Example
Determine The Maximum Displacement, Base Shear
and Overturning moment of the following Frame for 5%
damping (Response Spectrum Chart is Given)
Stiffness= 100000 kN.m2 - Mass = 100 t - Height = 4 m
Response Spectrum
4.5
4
3.5
3
Acceleration
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6
Period [sec]
Prof.Dr. Osman Shaalan
9 Earthquake Engineering Dr. Tharwat Sakr

10. 2.5.2. Analysis Procedure Using Response Spectrum
SDOF Example (Solution)
ω = 31.6278 rad /s
T = 0.1988 s
From the Chart
Sa = 0.25 m/s
Umax = Sa/ω 2 =0.00025 m =0.25 mm
Vmax = Sa×m =25.00 kN =2.50 t
Mmax = Vmax ×h = 100 kN.m =10.0 t.m
Prof.Dr. Osman Shaalan
10 Earthquake Engineering Dr. Tharwat Sakr

11. 2.5.2. Analysis Procedure Using Response Spectrum
MDOF Structures (Procedure)
Calculate the Structure dynamic parameters (φ I and ω I)
Corresponding to each DOF
For each degree of freedom (Mode)
From the Response Spectrum Chart get the value of Sa
corresponding to the mode i frequency ω i
The Max. Displacement at DOF number u ji = Γiφ ji S ai / ωi
2
j
The Max. Inter-Story Drift Story j ∆ ji = Γi (φ ji − φ( j −1) i ) S ai / ωi
2
The Max. Story Force at Story j F ji = Γi m jφ ji S ai
Based on the Story Force, Story Shear,
Moment and Base moment can be computed N
Combine all modes by the SRSS formula
R= ∑ Ri2
i =1
Prof.Dr.
11 Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr

12. 2.5. Response Spectrum
MDOF Example
Investigate the inter-story drift and story shear for the
given multistory building using the attached response
spectrum chart
No of Stories =2
Story Height =4m
Story Mass = 60 t
Story Stiffness = 240000 kN/m
Damping Ratio = 5%
Prof.Dr. Osman Shaalan
12 Earthquake Engineering Dr. Tharwat Sakr

13. 2.5. Response Spectrum
MDOF Example
0.5
0.45
0.4
0.35
Acceleration [g]
0.3
0.25
0.2
0.15
0.1
0.05
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Period [sec]
Prof.Dr. Osman Shaalan
13 Earthquake Engineering Dr. Tharwat Sakr

14. 2.5. Response Spectrum
MDOF Example (Solution)
ω = 39.09 T =0.1607
102.30 0.0614s
φ= -0.0679 -0.1098
-0.1098 0.0679
From the Spectrum Participation Factor
Sa1 = 0.45 Γ = -10.66
Sa2 = 0.315 -2.517
Displacements (Mode1 2 SRSS E-4
u1 = 2.1 0.0832 2.13 m
U2 = 3.44 -0.0514 3.45 m
Prof.Dr. Osman Shaalan
14 Earthquake Engineering Dr. Tharwat Sakr

15. 2.5. Response Spectrum
MDOF Example (Solution)
Inter-Story Drift (Mode1 2
SRSS) E-4
∆1 = 2.1 0.0832 0.053%0
∆2 = 1.31 -0.134 0.033%0
Forces (Mode1 2 SRSS)
f1 = 19.54 5.22 20.23 kN
f2 = 31.6 -3.23 31.77 kN
Overturning Moment
M = 335.05 kN.m
Prof.Dr. Osman Shaalan
15 Earthquake Engineering Dr. Tharwat Sakr