This page last updated on August 22, 1999

Linear Elastic Analysis of a Two-Story Frame

Learning Objectives:
After finishing this module, the student will be able to:

1. recognize the relationship between fundamental period of vibration and the time history pattern of building response.
2. explain the changes in a building's global and local responses due to the variation of peak ground acceleration and intensity of a ground motion acceleration.
3. list which parameters are considered global and local.
4. identify the variation of displacement and floor acceleration for different levels of the same buildings.
5. sketch a rough drawing of the base shear time history.
6. recognize the relationship between bending moment at the top and bottom of a column.
7. see that the axial load in a column may alternate between compression and tension.

To show the procedure and output of a time-history analysis we choose to analyze a small two-story one-bay frame.  The frame chosen is from Chopra's book, Example 10.5 on page 378.

To allow for modeling, we substituted values in for the mass and stiffness to arrive at a fundamental period of 1.0 seconds for the building.

The analysis was performed for two different earthquake ground motions: the 1940 El Centro and 1995 Kobe NS ground motion record.  These two ground motion have dramatic differences in peak ground acceleration and intensity.

The output from a time history analysis can be overwhelming in the magnitude of data.   We have tried to plot examples of some of the most important parameters.

Most engineers consider output parameters as being in one of two categories: global or local.  Global parameters include: base shear, floor displacement, floor acceleration, and building overturning moments.  These few parameters explain much of the overall behavior of the structure.  Local parameters include: bending moment, shear and/or axial load in an element, and displacement, strain, curvature and/or rotation of individual joints in the structure.  These local parameters define the specific demands on members for design of individual components.

The 2d beam elements used to model this analysis is commonly used by structural engineers.  Nodes are located at the centerline intersection of the beams and the columns in the frame.  These simple models are generally considered very effective in catching the correct value of the global parameters, but magnitudes of local parameters can be incorrect.  The appropriate means of modeling structures to capture the true local responses is still under debate.

Displacement Time History Response

The displacement of the first and second floors follows the same pattern but the second floor shows significantly higher displacements (as we should expect). 
            Click to see El Centro graph.
            Click to see Kobe EQ graph.

Note that the magnitude of the second floor displacement is significantly higher than the first floor.  Also the displacement due to the El Centro ground motion is less than half the Kobe EQ motion.

Acceleration Time History Response

We can also calculate the time-history response of the acceleration for each floor.   Note that the floor accelerations are significantly higher than the ground acceleration, indicating that the building frame is amplifying the acceleration.  
            Click to see El Centro graph.
            Click to see Kobe EQ graph.

Base Shear Time History Response

The base shear is the sum of the column shears at the base of the building.  The base shear follows the same pattern as the displacement time histories. 
            Click to see El Centro graph.
            Click to see Kobe EQ graph.

Column Axial Force Time History Response

For a local member we can see the response of the forces throughout the time history.   The axial force in the column fluctuates from tension to compression.  No gravity load was assumed on the column. 
            Click to see El Centro graph.
            Click to see Kobe EQ graph.

Column Bending Moment Time History