This is the initial condition. Load is applied to the upper portion of the truss.
The initial deflection due to load was about 7 inches (I applied a very heavy load). Next, a heat flux is applied to the lower truss chord to simulate a fire below.
Over time, the truss sags. After heating to about 1,300°F, the truss sags to 25 inches and pulls in the perimeter wall by 1.2 inches.
Close-up of the wall at maximum deflection.
I have now started working on using this model in a manner of testing the hypothesis of dropping a truss(s) attached to the perimeter wall onto another truss that is attached to a perimeter wall, to see if it will cause a progressive collapse.
The other FEA I alluded to on the debate both sides forum was one done by Newtons Bits (found here and here).
I created a column with dimensions 14” x 14” x .25” and 37 feet long. E=9,700,000 psi and an applied load at the top and 6 kips in the middle.
The maximum deflection was 1.58 inches.
The minimum load to initiate instability for a E=9,700,000 psi column was around 40 kips, applied in middle.
Progression of the instability (I fused together several steps into one picture)2
Here are the results of adding a horizontal force to the existing model.
First, I added a 6 kips horizontal “push-in” force. The maximum inward bowing of the column increased from 1.2” (w/o horizontal force) to 1.75”
I then decided to see what would happen if I increased the load. I applied 50 kips.
Here was the corresponding chart for inward bowing. Notice initially the bowing increased to a maximum of about 2.5”, then as the truss continued to heat up, the expansion coefficient (as a function of temperature) caused the column to be pushed back a little. However, something happened at around time 52. There was a “jump” of some sort and the perimeter column continued to bow inward as time and heat increased.
Here was the reason for the “jump”. As the truss continued to heat up and push back against the perimeter column, the truss eventually deformed out of the x-y plane.
This could very well explain what was happening in the Usmani model. Prior to around 1,850 seconds, the column and truss behaved as predicted. However, at around 1,850 seconds, several data points were taken in close proximity.
From 1,850 seconds onward, the rate of perimeter inward bowing increased faster than the truss sag. There is the possibility that Usmani used a horizontal force to induce inward bowing of the perimeter column.3
I put together the Usmani model to see if there was any truth to his conclusions. In short, the answer is no.
The three red trusses in the middle are the ones heated up to simulate a two story fire.
I used all 3D shell elements to reduce the analysis time. Usmani used 2D shell elements for the trusses and a beam element for the perimeter wall. On the three heated floors, I used concrete and a 8 kN/m load. On the other floors, I used the equivalent of 8 kN/m. Usmani did not provide the properties for the steel or concrete so I used what was in NIST NCSTAR 1-6D.
I used the same constraints for the perimeter column and truss ends and the temperature profile as given in Usmani figure 9. The image below is after heating for 3,600 seconds. It isn’t surprising that all we get is expansion and pushing on the perimeter column and sag in the truss.
In the next image, I had to manipulate the forces in order to get the necessary buckling. First, I applied a horizontal force around 150 kips and an out of plane force of 3 kips to cause buckling in the z direction. The model resisted the buckling as shown below.
You can see the end of the truss going through the perimeter column on the left. In reality, there would be eight locations where the bridging truss would prevent the z direction buckling.
Usmani was analyzing in 2D and hence they should not get any out of plane buckling, assuming they modeled everything correctly. Even in my model, I wasn’t getting any out of plane action since forces were in the y direction and everything was perfectly symmetrical. I had to induce the out of plane buckling.
Here is a better view of the discrepancy in the shell model when I added in the horizontal and out of plane forces. It really isn't disconnected, but as the truss chord deforms, it penetrates into the perimeter column since it doesn’t know the geometry is there. I didn't set the flag to check the model for self interaction or parts coming into contact with other parts. If I did, you would not see this penetration and the chord would bend around the perimeter column. It would take more computer time and wouldn’t change the overall result.
My point was that in order to obtain the results of Usmani, one would need to apply a non-existent horizontal force and ensure the truss deforms out of the z-plane. This is something that should not happen in a 2D analysis.
Instead of fixing and redo the shell model, I went ahead and created the three fire floors using solid elements. They are what Usmani claims to have modeled in his 2D shell and beam analysis. This is about as accurate as one can get without actually building a full scale model.
I applied the 8kN/m load to the floor and raised the temperature of the truss from 20 C to 600 C using the 2 floor profile given in Usmani over 3,600 seconds. Note, the displacements as compared to the shell model (17” vs. 25” vertical).
Here is a chart of the previous model. Note the lateral displacement of the wall initially dips below the x-axis (pushed out), then goes positive (pushed in). I applied quite a bit of load just to get about 5 inches. The NIST considers this pull-in force the reason for the column being pulled in.
The reason I could not get higher than 5 inches pull in deflection is when I continue to add horizontal force, the perimeter column collapses into itself. This is something that cannot be observed in Usmani’s 2D shell and beam model.
