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# Page History

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1. Create a model (Model A) which contains the entire structure, including the column to be removed. Analyze this model to obtain the internal forces of the column which will be removed.
2. Create another model (Model B) in which the column is removed. Apply the column end forces, obtained during the analysis of Model A, to simulate the presence of the removed column.
3. Simulate the removal of the column by running a time-history analysis in which these equivalent column loads are reduced to zero over a short period of time. This is done by applying a ramp time function in which loads opposite to those of the equivalent column loads are scaled from zero to the full value. The duration of this event should match the time in which the column is removed.

Dead load may be applied through either of the following two approaches:

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• Use a single time-history load case to apply the dead load and the equivalent column load using one time function which gradually ramps these loads to their full values. The column removal load may then be applied through a separate time function which has a later arrival time.

This process enables observation of the dynamic effects which are associated with the removal of a structural member. Note that this is an idealized computational process which will not capture such effect as the impact of material collapsing onto the remaining structure.

The following are a number of points to consider.

First, some points on analysis.

1. In a response history analysis, the displacement-velocity-acceleration relationships are defined by the step-by-step integration method. The most common is the “constant average acceleration” method (also known as the “trapezoidal rule” or “Newmark beta=1/4 method). The relationships are not simply d(displ)/dt = veloc and d(veloc)/dt = accn. If you look at the velocities in the text file that you sent, you will see that the average values are OK, but they oscillate. This is probably because the text files are for a time step of 0.02 seconds, which is too long. I would expect the velocities to vary more smoothly for the shorter time step that you considered (0.001 sec).
2. The calculated accelerations can be very sensitive, and may oscillate wildly.
3. The amount of damping that you assumed may be much too large. You may have used the same Rayleigh damping properties that you would use for a dynamic earthquake analysis. If so, those properties will be based on the long period vibration modes for lateral motion. The periods for vertical vibration, when a column is suddenly removed, are much shorter, so those modes may be very heavily damped. If you use Rayleigh damping, the properties should be based on the dominant vertical periods of vibration with a removed column.
4. Incidentally, 5% damping is probably too large, for earthquake analysis as well as progressive collapse. For a tall building, a more reasonable value for earthquake analysis is 2%. For progressive collapse I suggest no more than 1% (based on vertical vibration periods), or even zero.

Second, some suggestions for running analyses.

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*Related Email:* * [5/6/2010 email on progressive collapse capabilities of SAP2000|\$1745561] *Related Incident:* * 23372: Modeling progressive collapse.

# References

• Powell, Graham

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