by SanderO » Sat Feb 11, 2012 3:42 pm
Based on the cross sectional areas of the columns at the 83rd floor the percentage of each column's share of the total support can be deduced. The 24 perimeter columns were 67.5% of the core's axial strength and the 23 central columns accounted for 32.5% of the core's total strength.
4.62% 2.81% 3.40% 2.48% 2.21% 3.35% 2.81% 4.62%
1.57% 1.75% 1.63% 1.57% 1.57% 1.63% 1.75% 1.57%
1.81% 1.81% 1.12% 0.47% 0.39% 1.07% 1.93% 1.66%
1.57% 2.16% 1.20% 0.94% 0.93% 1.65% 1.67%
1.57% 1.85% 1.64% 1.11% 1.04% 1.65% 1.65% 1.57%
4.74% 2.92% 3.86% 3.48% 2.06% 3.53% 2.92% 4.74%
Row 800 had only 7 columns
If all of one outside row of the core was take out...say 501-508 about 26% of the core's total strength would be gone in one fell swoop. All the floors connected to that side of the core would collapse - 3 of them since each column carried three floors.
The axial loads supported by row 500 would then be redistributed to the remaining columns. Assuming that the 500 loads were equally redistributed to all remaining core columns (I doubt it would be equally distributed) each core column would see 26/41 or an increase share of the total load of .56%.
Here's the new percentages in row 600 after 500 is gone:
2.13% 2.31% 2.19% 2.13% 2.13% 2.19% 2.31% 2.13%
Here is were the FOS comes into the picture. The redistributed loads to row 600 have increased by about 1/3. If the FOS was 1.65 it has not been reduced to about 1.33
Look at row 700 with the .56% of the additional load:
2.36% 2.37% 1.68% 1.03% 0.95% 1.63% 2.49% 2.22%
It appears that columns 704 and 705 are now seeing more than 2x their design load. they would fail with an initial FOS of 2 let alone an initial FOS of 1.65.
And the above two rows are for equal load redistribution. If the load redistribution was more on the columns directly under the hat truss, then 8 columns would not carry redistributed loads...602, 603, 606, 607, 902, 903, 906, 907. The hat truss had 8 trusses, 4 in each axis... 2 over the perimeter and two of the center columns. If those 8 columns saw no redistributed loads then the redistribution would fall to 33 columns... 26% of the lost capacity again equally redistributed to the remaining 33 columns or about an additional .8% of the total original core axial load. In this scenario if the original FOS was 1.65 the following columns would fail 601, 604, 605, 608,801 805, 901, 904, 905, 908. When this happens there would be another rapid redistribution to the remaining columns which would all buckle except maybe 1001 and 1008.
The same scenario would apply if it was row 1000 taken out and the tower's structure above would lose the entire core, the facade columns would then buckle as well or the floors at and above the core destruction would collapse and the facade would lose bracing.
FOS was the key to how long the core could stand and how few columns could support the entire axial load.
It should be noted that the typical FOS for a steel frame is 1.42 and it appears that the FOS for the twins was a low as 1.65 but no more than 2.0 depending on the total mass of the towers. And of course the FOS was not uniform for each column and not the same at each level. There was some variation.