Heiwa:
Consider each WTC tower as being divided (mathematically) into upper and lower sections such that an upper mass Mu acts on a lower mass Ml and the mass of the tower Mt is thus [Mu + Ml]. In a normal, undamaged, tower the load acting on Ml is Mu .g. Then, for example, if we consider the upper mass to be say 50,000 tonnes, the downward-acting force on the top of the lower section is 50,000,000 kg x 9.8 m/s^2 ~ 500 MN. In the Twin Towers this load was shared more or less equally between the core and the perimeter columns.
Let us suppose that, under the compressive load of the upper section, the lower section of a tower acts like a giant spring and obeys Hooke’s Law. This means that the downward displacement, d, of the lower section due to the static weight of the upper section is proportional to the applied compressive force, Fu. This leads to the familiar result:
Fu = Mu. g = k.d,
where k is the stiffness, also called the spring constant, of the lower section of the tower.
Now since Young’s modulus, E, equals stress, s, divided by strain, the downward displacement of the top of the lower section of a tower due to static loading by mass Mu, is given by,
d = L. Mu .g /AE
where L is the length of the lower section and A is the effective cross sectional area of the structural steel.
Representative values of L and A would be 300 meters and 5 m^2, respectively, while E for structural steel is typically ~ 200 GPa. It follows that d is ~ 15 cm and since k is equal to Mu. g / d we have k = 3 GN/m. The elastic energy stored by this compression is (500 x 0.15) MJ or 75 MJ. Now since structural steel has an elastic strain energy capacity of 50 J/kg, the building can handle the weight of the upper section because the resulting strain energy is taken up by a large mass of steel below it. There was, in fact, at least 15,000,000 kg of structural steel available to support a 50,000,000 kg upper section, so we have 75 MJ of elastic strain energy per 15 x 10^6 kg or 5 J/kg which is well below the 50 J/kg elastic capacity of the structural steel.
Now consider what happens if there is a failure of a significant number of columns supporting a 50,000,000 kg upper section of a tower. Prior to such a failure we had a stable building in which the downward-acting force on the lower section was countered by an equal and opposite reaction force on the upper section. If the perimeter wall columns should suddenly fail at the interface between our upper and lower sections, the wall will unload some or all of the reaction force so that the downward-acting force now exceeds the reaction force acting on the upper section. This creates a net accelerating force on the upper section of the tower.
Data reported in NCSTAR 1-6D show that the total perimeter column load at the 83rd floor of WTC 2 was about 250 MN. It follows that after unloading by the perimeter columns at or near this floor, the upper section will move downwards under the action of this force with an acceleration, a, given by:
a = Force /Mass ~ 250 MN/50,000,000 = 5 m/s^2 ~ ½ g
Assuming that the upper section drops one story height or 3.7 meters, the stiffness, k, of the “spring” that held up the lower section of the tower has been reduced from 3 GN/m to 68 MN/m, (250 MN/3.7 m). We also note that the work done by the upper section in collapsing one floor, Wc, is given by:
Wc = Force x Distance = 250 MN x 3.7 m = 925 MJ
Now, since the loss in potential energy is Mu. g. h, we see that the amount of kinetic energy gained by the upper section after falling 3.7 meters is:
K.E. = (Mu. g. h) - Wc = (1813 – 925) MJ = 888 MJ
It follows from the equation K.E. = ½ Mu. v^2, that the upper section would be moving with a velocity v ~ 6 m/s after this 3.7 meter drop. It is interesting to note that Wc is identical to the quantity I have previously called E1 and the calculated value of 925 MJ given above is in good agreement with the range of values proposed for E1.
The real-world situation in the twin towers was, however, a little more complex than the simplified model described above because collapse initiation was actually caused by tipping of the upper section. Thus, considering the case of WTC 2, floor truss sagging and/or failures on the east side of this tower between the 80th and the 84th floors affected the lateral bracing between the core and the exterior of the building and led to an inward bowing of the east perimeter wall as seen in Figure 6-21 of NCSTAR 1-6. This bowing caused a measurable tipping of the upper section of WTC 2 that, according to NIST, had already lowered the east side of the upper section of the building by 30 cm about 15 minutes before total collapse started. The associated loss of potential energy by the upper section at this point in time is easily calculated to be about 75 MJ; this energy was converted into strain energy in the east perimeter wall columns because it created shear stresses in the upper and lower splices of these columns.
Just prior to collapse, the inward bowing of the east wall of WTC 2 was well over 50 cm so that the affected perimeter columns eventually failed in shear at the splice bolts. This created a net downward force on the east side of the upper section of WTC 2 that allowed it to tip 3.3 degrees before striking the floor below. However, at this point in time, ¾ of the perimeter wall at or near the 83rd floor - a wall that supported 50 % of the mass of the upper section of the building - would have been destroyed.
The kinetic energy imparted to the upper section of WTC 2 by this tipping motion was actually rotational kinetic energy, (RKE), and is given (approximately) by the formula:
RKE = 1/6 Mu. h^2. {d(theta)/dt}^2
where h is the height of the upper section and d(theta)/dt is the angular velocity in rads/s. Substituting appropriate values into this equation we find that the rotational kinetic energy imparted to the upper block of WTC 2 during the first second of collapse was at least 350 MJ. This would be more than sufficient to shear off many perimeter columns surrounding the floors immediately below the impact zone and so on all the way down the tower.
