Time history analysis and Response spectra

By: Anaya Bhobe and Abhishek Jain

ContentsIntroductionTime History AnalysisResponse SpectrumMethodology Model Steps followed for Time History Analysis Steps followed for Response Spectrum Result Conclusions

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.

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.

Time History For SDOF For Damped System:-

Equation of motion for SDOF:-

mü(t) + c (t) + ku(t) = -müu̇ g(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.

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.

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.

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

The model as designed in SAP2000

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.

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.

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

SAP2000

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

Conclusion1. 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.

Thank you!! Any questions?