**Answer:** Progressive collapse may be modeled using SAP2000 or ETABS in the sense that certain members may be removed to stud=
y the effect on the structure. Progressive collapse and dynamic effects may=
be evaluated using time-hist=
ory analysis as follows:

- 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.
- Create another model (Model B) in which the column is removed. Apply th= e column end forces, obtained during the analysis of Model A, to simulate t= he presence of the removed column.
- Simulate the removal of the column by running a time-history analysis i= n which these equivalent column loads are reduced to zero over a short peri= od 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:=

- Apply the dead load together with the equivalent column load in a nonli= near-static load case. Then the time-= history load case, in which the column is removed using a time function, sh= ould start at the end of this nonlinear-static load case.

- Use a single time-history load case to apply the dead load and the equi= valent column load using one time function which gradually ramps these load= s 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 associ= ated with the removal of a structural member. Note that this is an idealize= d 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.

- In a response history analysis, the displacement-velocity-acceleration = relationships are defined by the step-by-step integration method. The most = common is the =E2=80=9Cconstant average acceleration=E2=80=9D method (also = known as the =E2=80=9Ctrapezoidal rule=E2=80=9D or =E2=80=9CNewmark beta=3D= 1/4 method). The relationships are not simply d(displ)/dt =3D veloc and d(v= eloc)/dt =3D 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, wh= ich is too long. I would expect the velocities to vary more smoothly for th= e shorter time step that you considered (0.001 sec).
- The calculated accelerations can be very sensitive, and may oscillate w= ildly.
- The amount of damping that you assumed may be much too large. You may h= ave used the same Rayleigh damping properties that you would use for a dyna= mic earthquake analysis. If so, those properties will be based on the long = period vibration modes for lateral motion. The periods for vertical vibrati= on, 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 colu= mn.
- Incidentally, 5% damping is probably too large, for earthquake analysis= as well as progressive collapse. For a tall building, a more reasonable va= lue for earthquake analysis is 2%. For progressive collapse I suggest no mo= re than 1% (based on vertical vibration periods), or even zero.

Second, some suggestions for running analyses.

- Before doing any analysis, be clear on the purpose. For many structures= , the purpose of the analysis is to show that if one column is suddenly rem= oved, there is at most a small amount of damage. This is relatively easy, a= nd may require only linear static analysis. For some structures actual coll= apse may be a concern. In this case the analysis is much more complex. Also= , the calculated response is likely to be so sensitive to the modeling assu= mptions that the analysis may be little more than an academic exercise. The= attached paper by Graham Powell can provide some guidance.
- Before running a dynamic analysis, start with a static analysis (remove= the column, analyze for static gravity loads plus static loads equal to th= e column forces (upwards load), then add static downwards load equal to two= times the column forces). Use this to get an upper bound estimate of the v= ertical deflection (the factor of two is the dynamic amplification factor f= or an undamped elastic structure when the column is suddenly removed), and = also to see if there is any substantial inelastic behavior.
- If a dynamic analysis is needed, it is probably a good idea to start wi= th an elastic analysis. For zero damping the results should be close to tho= se for a static analysis with an impact factor of 2.0. Be sure to use a tim= e step that is short enough to capture the vertical vibrations, which are l= ikely to have short periods. The analysis results may be sensitive to the a= ssumed distribution of vertical masses, and to the assumed damping. Since t= here is only one substantial displacement cycle when a column is suddenly r= emoved, it can be argued that there is less effective damping than for eart= hquake response, with many cycles. For damping, be very careful with the be= ta-K part of Rayleigh damping. It may be wise to specify only alpha-M dampi= ng, or to assume zero damping.
- If an elastic analysis indicates that there may be significant inelasti= c behavior, run an inelastic static analysis. If the amount of inelastic be= havior is small, this should run OK. Among other things, look at the energy= balance (of the type considered in the attached paper).
- If an inelastic dynamic analysis is still needed, then run one, but be = aware that the calculated response is likely to be very sensitive to modeli= ng assumptions such as strengths, strain hardening ratios, damping, mass di= stribution, and whether the catenary effect is considered.

Powell, Graham. Collapse Analysis Made Easy (More Or Less)=

Smilowitz, R. Analytical= Tools for Progressive-Collapse Analysis. Weidlinger Associates.

Agnew, E., Marjanishvili, S. (2006). Dynamic Analysis Procedure for Prog= ressive Collapse.

*Structure Magazine*, 24-27.