This document discusses time history analysis and response spectrum analysis of a 3-story steel frame structure. It describes modeling the structure in SAP2000, applying loads and base excitation from the 1940 El Centro earthquake. Time history analysis was conducted to determine the structure's response over time. Response spectrum analysis provided the peak response for different vibration periods. The results show time history and response spectrum curves, along with modal properties like periods, frequencies, and participation ratios. The analyses help understand the structure's dynamic behavior under earthquake loads.

1. Time history analysis and Response spectra
By: Anaya Bhobe and Abhishek Jain

2. Contents
Introduction
Time History Analysis
Response Spectrum
Methodology
 Model
 Steps followed for Time History Analysis
 Steps followed for Response Spectrum
 Result
Conclusions

3. Introduction
 We know that Calamity such as Earthquake causes
a big wave of deaths, resulting in a huge loss of life
and property.
 Now-a-days, people started to give an emphasis on
the Earthquake Resistant Structures. In the similar
direction, this term paper is a small work for
determining the response of a three story frame
under normal conditions.
This term paper deals with the dynamic response of
a structure.
 This term paper includes the modelling of a three
story frame and the results of Dynamic Analysis.

4. Time History Analysis
 Provide response of structure during and
after the application of load.
 To calculate time history, we must solve the
equation of motion for the respective system.
 Time History can be calculated by Linear
Finite Element Analysis Software but as the
number of degrees increases, the system
become more complex and we will require to
analysis our system by Non Linear Finite
Element Analysis Software.
 Time History is greatly affected by the
Damping, Stiffness and Mass of the system.

5. Time History For SDOF
For Damped System:-
Equation of motion for SDOF:-
mü(t) + cu̇(t) + ku(t) = -müg(t)
u(t) = ρe-ξwnt ( 1 – cos(wt-ϴ))
u(t) = U1 ( 1 – cos(wt- ϴ))
The above function is the time dependent,
where U1 is the amplitude of the system.
For Undamped System:-
Equation of motion for SDOF:-
mü(t) + ku(t) = -müg(t)
u(t) = U2 ( 1 – cos (wt))
The above function is the time dependent,
where U2 is the amplitude of the system.

6. Response spectrum
 It helps in understanding the structural behavior response of the model under different loading
conditions.
The response spectrum provides a convenient means to summarize the peak response of all
possible linear SDF systems to a particular component of ground motion.
A plot of the peak value of a response quantity as a function of the natural vibration period Tn of
the system, or a related parameter such as circular frequency ωn or cyclic frequency fn, is called
the Response Spectrum Analysis.
 It is also affected by Damping and Stiffness.
 A response spectrum is used for deciding the exact amount of damping that a structure is
required.

7. Methodology
 After Modelling, Loads have been assigned.
 IBC (International Building Code-2006) have
been used to model the structure.
 Then Structure is analyzed individually for the
time history and the response spectrum.
 The results i.e. the joint acceleration, joint
displacements, modal periods, its frequencies
and the modal participation ratios are
tabulated in a systematic manner to provide
the required data for further designing of the
structure.
About the Model:-
 Designed by SAP 2000 Program.
 Loads including Dead Load which is equal to
the self weight of the structure
 Base Excitation Data is taken from 1940 EI
Centro Earthquake Acceleration.
 Damping Ratio takes as 5%.
 Structure gets deflected horizontally.

8. Model used in SAP2000
1. The model is created in SAP2000 for a simple three- storied frame structure. The structure is
firmly fixed at its base. The following dimensions and properties are assigned to the structure.
Period of the structure - 0.2seconds.
Material - Steel
Steel size - 0.5*0.5 ft.
Floor to floor height - 10ft
Damping - 5%
Length, X - 20ft
Breadth, Y - 20ft

9. The model as designed in SAP2000

10. Steps followed in SAP for time history calculation
1. After the frame structure has been
modelled and assigned with all its
properties and loads, the time history
function has to be defined.
2. The accelerometer readings as obtained
from the El Centero N-S acceleration are
defined as shown.
3. The Time History FUNC2 is then applied to
the structure which gives us the time
history response of the structure.

11. Steps followed for calculation of response spectrum
1. After the structure has been analyzed, we
can select the response spectrum curves
from the display menu in the top bar of the
menu bar.
2. We need to select the joint in our case
Joint 5 to perform the response spectrum
analysis at that point.
3. Also we need to specify the damping
constants like 0%, 3%, 10%, 20% and 50%.
4. Other parameters like the direction, axes,
etc can also be defined to obtain a good
response spectrum curve.

12. Results obtained
Response Spectrum(left) and Time history analysis(right) as obtained from the three stories framed structure modelled in
SAP2000

13. Tabular Data as obtained after the analysis
Modal periods, frequencies of Mode 1 to 12 and the modal participation ratios of the three axes are displayed.

14. Conclusion
1. The time history and response spectrum are thus obtained from the SAP 2000 software.
2. Similarly the time history and response spectrum can be obtained for the individual joints of
the modelled structure.
3. The mode shapes and frequencies as obtained from the time history analysis help us in
understanding the behavior of the structure over time, thus providing us valuable
information needed to take precautionary measures.
4. Also the response spectrum contributes for an effective design as well by helping us
understand the dynamic behavior of the structure when the base excitation is applied, thus
helping us in building an effective design of the structure.

15. Thank you!! Any questions?