2.5. Response 2.5. Response SpectrumSpectrum
EARTHQUAKEEARTHQUAKE ENGINEERINGENGINEERING
2.5.1. Definition and Construction 2.5.1. Definition and Construction of Response Spectrumof Response Spectrum
2.5.2. Analysis Procedure 2.5.2. Analysis Procedure Using Response SpectrumUsing Response Spectrum
2.5.1. Definition and Construction of Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr2
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)
DefinitionDefinition
2.5.1. Definition and Construction of Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr3
ExampleExample
NorthBridge Earthquake
2.5.1. Definition and Construction of Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr4
Damping EffectsDamping Effects
2.5.1. Definition and Construction of Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr5
BenefitsBenefitsCharacterization of Ground Motions
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50
5
10
15
Response Spectrum
Period [sec]
Acc
eler
atio
n
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50
2
4
6
8
10
12
14
16
18
20
Response Spectrum
Period [sec]
Acc
eler
atio
n
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50
1000
2000
3000
4000
5000
6000
7000
8000
9000
Response Spectrum
Period [sec]
Acc
eler
atio
n
2.5.1. Definition and Construction of Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr6
BenefitsBenefitsEarthquake Resistance Design
Almost all Codes of Practice recommends the Response Spectrum as the Main Method for Earthquake Resistance Design
2.5.1. Definition and Construction of Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr7
Generation of Response SpectrumGeneration of Response SpectrumSelect 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
2.5.2. Analysis Procedure Using Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr8
SDOF Structures (Procedure)SDOF Structures (Procedure)Calculate the Structure dynamic parameters (K, M, , 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
The Maximum Base Shear
The Maximum Base Moment
2.5.2. Analysis Procedure Using Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr9
SDOF ExampleSDOF ExampleDetermine 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
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Response Spectrum
Period [sec]
Acc
eler
atio
n
2.5.2. Analysis Procedure Using Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr10
SDOF Example (Solution)SDOF Example (Solution) = 31.6278 rad /s T = 0.1988 s
From the ChartSa = 0.25 m/s
Umax = Sa/2 =0.00025 m =0.25 mmVmax = Sam =25.00 kN =2.50 tMmax = Vmax h = 100 kN.m =10.0 t.m
2.5.2. Analysis Procedure Using Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr11
MDOF Structures (Procedure)MDOF Structures (Procedure)
For each degree of freedom (Mode)
From the Response Spectrum Chart get the value of Sa corresponding to the mode i frequency i
The Max. Displacement at DOF number j
The Max. Story Force at Story j
The Max. Inter-Story Drift Story j
Calculate the Structure dynamic parameters (I and I) Corresponding to each DOF
Based on the Story Force, Story Shear, Moment and Base moment can be computed
Combine all modes by the SRSS formula
2.5. Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr12
MDOF ExampleMDOF ExampleInvestigate the inter-story drift and story shear for the given multistory building using the attached response spectrum chart
No of Stories = 2Story Height = 4 mStory Mass = 60 tStory Stiffness = 240000 kN/mDamping Ratio = 5%
2.5. Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr13
MDOF ExampleMDOF Example
0 0.5 1 1.5 2 2.5 3 3.5 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Period [sec]
Acc
eler
atio
n [g
]
= 39.09 T =0.1607 102.30 0.0614s
= -0.0679 -0.1098 -0.1098 0.0679
2.5. Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr14
MDOF Example (Solution)MDOF Example (Solution)
From the SpectrumSa1 = 0.45Sa2 = 0.315
Participation Factor = -10.66
-2.517
Displacements (Mode1 2 SRSS E-4u1 = 2.1 0.0832 2.13 mU2 = 3.44 -0.0514 3.45 m
2.5. Response Spectrum
Prof.Dr. Osman Shaalan Earthquake Engineering Dr. Tharwat Sakr15
MDOF Example (Solution)MDOF Example (Solution)
Forces (Mode1 2 SRSS)f1 = 19.54 5.22 20.23 kNf2 = 31.6 -3.23 31.77 kN
Inter-Story Drift (Mode1 2SRSS) E-41 = 2.1 0.0832 0.053%0
2 = 1.31 -0.134 0.033%0
Overturning MomentM = 335.05 kN.m