I trust this answers your questions:
First let's calculate the mass of concrete on each WTC 1 floor as follows:
Core floor area = 862 m^2
Out-of-core (Office space) floor areas:
2 long one-way slabs = 1,225 m^2
2 short one-way slabs = 486 m^2
4 two-way slabs = 1,137 m^2
Total out-of-core area = 2848 m^2
The floors in the core areas were made of normal weight concrete, density 1760 kg/m^3
The floors in the office areas were made of lightweight concrete, density 1500 kg/m^3
Volume of 5-inch normal weight concrete per floor = 109.5 m^3
Weight of normal weight concrete per floor = 193 tonnes
Volume of 4-inch thick lightweight concrete per floor = 289.4 m^3
Weight of lightweight concrete per floor = 434 tonnes
Total weight of concrete on one floor of WTC 1 = 627 tonnes
Let's now assume that the concrete on the 95th floor of WTC 1 was impacted by the mass of the 15-storey block of floors above the aircraft impact zone. In order to determine the energetics of this collapse we note that the drop distance was 3.7 meters and with the relation v = Sqrt(2gh) we find the impact velocity, vi, was 8.52 m/s. Then, using Bazant’s value for the mass of the upper section of WTC 1, M15 = 5.8 x 10^7 kg, the kinetic energy of the falling mass at the moment of impact is given by:
E = ½ M15 vi2 = 0.5 x 5.8 x 10^7 x (8.52)^2 Joules = 2.1 x 10^9 Joules
Thus we see that the first major energy transfer in the collapse of WTC 1 occurred when 2.1 gigajoules of kinetic energy was delivered to the 627 tonnes of concrete on the first impacted (~95th) floor. We now consider how concrete would behave under this degree of impact loading.
In order to compare the behavior of different materials under impact loading and other damaging events such as explosions, we need to consider the mass specific energy input, Ei, imparted to the material, defined as the energy input per unit mass. Thus for the particular case of the WTC 1 collapse we determine Ei (with the gram as the unit of mass) as follows:
Ei = (2.1 x 10^9) / (0.627 x 10^9) J/g = 3.36 J/g
Thus we conclude that the mass specific energy input of the first impact of the upper section of WTC 1 on the layer of concrete on the 95th floor was 3.36 joules per gram. This level of energy input will now be evaluated by comparison to published data on the energetics of a wide range of impact phenomena in order to establish:
· Is a ~ 3 J/g impact potentially damaging to concrete? And, if so, what degree of
concrete pulverization is to be expected from such an impact?
The impact strength of concrete and other brittle materials such as rocks, minerals, glass and ceramics, has traditionally been determined by a number of techniques:
· The Drop Hammer: See for example: B. P. Hughes, “Concrete Subjected to High Rates of Loading in Compression.” Magazine of Concrete Research 24, 25, (1972) and P. H. Bischoff et al. “Impact Behavior of Plain Concrete Loaded in Uniaxial Compression.” Journal of Engineering Mechanics 121, 685, (1995)
· The Ballistic Pendulum: See for example: H. Green, “Impact Strength of Concrete.” Proceedings of the Inst. of Civil Engineers 28, 383, (1964) and B. P. Hughes et al. “The Impact Strength of Concrete Using Green’s Ballistic Pendulum.” Proceedings of the Inst. of Civil Engineers 41, 731, (1968).
· The Split Hopkinson Bar: See for example: B. Lundberg “A Split Hopkinson Bar Study of Energy Absorption in Dynamic Rock Fragmentation.” International Journal of Rock Mechanics and Mineral Science 13, 187, (1976) and C. A. Ross et al. “Split Hopkinson Pressure Bar Tests on Concrete and Mortar in Tension and Compression.” ACI Materials Journal 86, 475, (1989).
All these techniques involve a projectile (hammer) of known mass and kinetic energy striking a fixed target (concrete or rock sample). The effects of the impacts are usually monitored by the recoil behavior of the system and other experimental aids such as strain gauges and high-speed photography. See for example: S. Mindess, “A Preliminary Study of the Fracture of Concrete Beams Under Impact Loading Using High Speed Photography.” Cement and Concrete Research 15, 474, (1985).
A survey of the literature quoted above shows that most researchers used hammers or equivalent projectiles with masses in the range 3 to 350 kg dropped from heights in the range 0.2 to 3.5 meters. The actual combinations of hammer mass and drop height were such that impact kinetic energies in the range 10 to 2000 joules were investigated. In addition, because samples with weights from 20 grams to 70 kilograms were impacted, data and impact behavior for mass specific energy inputs between 0.02 and 0.9 J/g are available in the above references. These data show that significant fracturing of concrete occurs at impact energies above 0.1 J/g.
The fact that concrete is observed to fragment from impact energies ~ 0.1 J/g is consistent with the known properties of brittle materials. Thus the total elastic strain energy, Us, that may be stored in a material of mass M, up to the point of the initiation of fragmentation is given by the relation:
Us = sy^2 M / 2(rho) E
sy is the yield stress
rho is the density
E is Young’s Modulus
(See for example: T. Waza et al. in “Laboratory Simulation of Planetesimal Collision 2. Ejecta Velocity Distribution” Journal of Geophysical Research 90(B2), 1995, (1985).)
Substituting appropriate values of sy (40 MPa), rho (1500 kg/m^3) and E (10 GPa) for the lightweight concrete used in WTC 1 (See Appendix A of the FEMA WTC Report: “Overview of Fire Protection in Buildings”) we find that the mass specific elastic strain energy of concrete, or the energy per unit mass is:
Us / M = 53 J/kg » 0.05 J/g
This energy is half of the previously noted experimental impact energy (0.1 J/g) required for significant fragmentation of concrete. Thus it is theoretically predicted and experimentally verified that when an impact energy in excess of about 0.05 J/g is supplied to lightweight concrete, it ceases to behave elastically and undergoes brittle fracture.
In order to further quantify impact fragmentation of concrete we need to consider its fracture energy, Gf , defined as the energy needed to create a unit area of fracture surface. For typical, normal strength, concrete Gf is ~ 100 Joules/ m^2. (See, for example A. Hillerborg. “Results of Three Comparative Test Series for Determining the Fracture Energy Gf of Concrete” Materiaux et Constructions (Materials and Structures) Vol 18, No. 107, 407, (1985), or F.H. Wittmann et. al “Probabilistic Aspects of Fracture Energy of Concrete” Materials and Structures 27, 99, (1994).)
Because a single particle crushed into smaller particles exhibits a larger surface area, we need to multiply the fracture energy of 100 Joules/m^2 by the total surface area of the crushed particles to determine the minimum energy required to produce the crushed particles. It is a minimum energy because we are neglecting any possible kinetic energy of the crushed particles in cases where particles are violently ejected from the original sample by the impact. This neglect of the recoil energy of the fragments is valid in cases where the impacted material is reasonably well confined to a compartment or sealed container during the impact.
By way of an example, consider a 10 cm x 10 cm x 10 cm cube of concrete weighing 2 kg being shattered by an impact into one thousand 1 cm x 1 cm x 1 cm cubes. The original cube had six faces, each with an area of 100 cm2, so that the initial surface area was 600 cm2. After impact, the total surface area has increased to 6,000 cm^2. Thus 5400 cm^2, or 0.54 m^2, of new surface has been created. Since the fracture energy of concrete is 100 J/m2, 100 x 0.54, or 54 joules of energy were required to fragment the sample.
An analysis of the data presented in the studies by Hughes, Mindess and Bischoff noted above shows that the impact energies actually employed in their studies of concrete utilized about 1000 joules to fragment a 10 cm block of concrete into 1 cm, or smaller, cubes, in which case the fragmentation process actually used only about 5.4 % of the available energy. This suggests that impact fragmentation is not a very efficient process. Nonetheless, in order to apply these concepts to the pulverization of concrete in the WTC collapse we first need to consider “real-world” particle size distributions of impact fragmented concrete and determine how fragmentation efficiency varies with the mass specific energy input.
The conversion of large blocks of a brittle material into smaller fragments by crushing, grinding, hammering, drilling, or explosive blast, - processes collectively referred to as comminution - has been extensively studied because of its importance in mining and quarrying. (See for example: Proceedings of the First International Symposium on Rock Fragmentation by Blasting, Luleå, Sweden (1983); D. A. Shockley et al. in “Fragmentation of Rock Under Dynamic Loads” Int. J. Rock Mechanics 11, 303, (1974); M. Kabo et al. in “Impact and Comminution Processes in Soft and Hard Rock” Rock Mechanics 9, 213 (1977).)
The impact fragmentation of brittle bodies by energetic collisions has also been of considerable interest to astronomers and physicists as an aid to understanding the formation and evolution of asteroids and comets. Thus a large number of reports have been published in which a variety of targets, generally in the 100 - 2000 gram size range, have been impacted by high speed projectiles to investigate the fragmentation of the target. While hard rocks such as granite have been extensively studied, softer targets, frequently assembled with cement or mortar, have also been investigated. (See for example: D. R. Davis et al. in “On Collisional Disruption: Experimental Results and Scaling Laws” Icarus 83, 156 (1990); T. Waza et al. in “Laboratory Simulation of Planetesimal Collision 2. Ejecta Velocity Distribution” Journal of Geophysical Research 90(B2), 1995 (1985).)
Through these and related studies, a considerable body of data is available on the comminution of brittle materials - from granite and basalt, through intermediate strength materials such as limestone, concrete and shale, to soft minerals such as calcite and gypsum - thereby covering a wide range of hardness and fracture toughness. What has emerged from these studies is that the fragmentation of brittle materials by fast dynamic loading using explosions and/or collisions results in a range of fragment sizes that follow a universal power law distribution. (See for example: A. Carpinteri in “One, Two, and Three-Dimensional Universal Laws for Fragmentation due to Impact and Explosion” Journal of Applied Mechanics 69, 855 (2002); F. Ouchterlony in “The Swebrec Function: Linking Fragmentation by Blasting and Crushing” Mining Technology (Trans. Inst. Min. Metall. A) 114, A29, (2005).)
In his classic early paper on comminution theory, R. J. Charles showed that only two independent parameters, a size modulus and an energy factor, are sufficient to characterize most particle size distributions. (See R. J. Charles in “Energy-Size Reduction Relationships in Comminution.” Trans. AIME 208, 80, (1957)).
Thus, a commonly used size distribution function, first proposed by R. Schuhmann, is usually expressed by:
M (<d) / Mt = (d / dmax)^k
d is the diameter of a specified fraction of all the particles
dmax is a size modulus denoting the largest particles in the distribution
M (<d) is the total (cumulative) mass of fragments of size smaller than d
Mt is the total mass of all the particles from an impacted solid target
k is a numerical constant related to the energy imparted to the material
A log-log plot of Schuhmann’s function yields a straight line of slope k. Values of k in the range 0.4 - 0.6 are appropriate for concrete-like materials undergoing hard impact.
It should be noted that the Schuhmann function is not particularly accurate, (or even meaningful!), as M (<d) / Mt approaches 1.0. However, this in no way detracts from the usefulness of this formalism because a number of investigators have shown that for a mass-specific kinetic energy input ~ 3 J/g, the ratio of the largest fragment mass to the total mass of the target is about 0.1. (See for example: A. Fujiwara et al. “Destruction of Basaltic Bodies by High-Velocity Impact.” Icarus 31, 277 (1977); D. R. Davis et al. “On Collisional Disruption: Experimental Results and Scaling Laws.” Icarus 83, 156 (1990).).
The Schuhmann function is also not very accurate as M (<d) / Mt approaches zero. Inspection of particle size data for materials fragmented by impact shows that the Schuhmann function over-estimates the amount of “fines” in a sample – the “fines” in the present context corresponding to particles smaller than about 10 microns. As a result, most “real-world” particle size distributions show a distribution that falls significantly below the Schuhmann value of M (<d) / Mt for particles smaller than 10 microns. For this reason a lower limit of 10 microns will be used in the following discussion.
The results of the comminution theory presented above and applied to the pulverization of WTC concrete, where the initial mass-specific kinetic energy of the upper section of WTC 1 was 3.36 J/g, indicate that the first experimental point on a Schuhmann function plot of crushed concrete would be for M (<d) / Mt equal to 0.1. We also note that the initial size metric of WTC concrete is the shortest characteristic length of the material as used in the buildings, which is about 10 cm. Thus, on the basis of first order comminution theory, the largest WTC concrete fragments are predicted to be approximately 1 cm in diameter.
A well-known approach to further classifying the size distributions of crushed materials is through a series of parameters t10, t20, t50, t100, etc, which represent the cumulative weight-percent of particles passing a sieve size of d/10, d/20, d/50, d/100, etc, where d is the original size of the particles. Furthermore, t10 is traditionally selected as a convenient single measure of the fineness of a crushed sample. (See O. Genç et al. in “Single Particle Impact Breakage Characterization of Materials by Drop Weight Testing.” Physicochemical Problems in Mineral Processing 38, 241, (2004).)
From a survey of Genç et al’s particle size data on the comminution of concrete-like materials such as cement clinker and limestone we have been able to develop a series of Schuhmann plots representing the particle size distribution of these materials fragmented over a range of impact energies from 1 to 20 J/g. For an impact energy of 3.4 J/g on concrete-like materials we find that t10, the cumulative weight-percent of particles passing a sieve size of 10 % of the material’s original characteristic length, is approximately the same for all samples studied and may be taken to be 32 ± 1 %. This value of t10 corresponds to a plot with a value of 0.5 for the Schuhmann parameter k.
It may also be shown, (See Genç et al’s paper), that t10 is a function of the mass specific impact energy, Ei:
t10 = A [ 1 - exp(-bEi) ]
A is a constant that represents the maximum value of t10 for a material
b is a constant for a given material expressed in units of grams per joule.
Data reported by L. M. Tavares in his paper “Optimum Routes for Particle Breakage by Impact.”, in Powder Technology 142, 81, (2004), show that the constant A is equal to 65 % for concrete-like materials. Since we also know that t10 is 32 % for a mass specific impact energy, Ei, of 3.36 J/g, we conclude that the constant b is 0.2 g/J for these materials.
With these values for the parameters A and b it is possible to construct Schuhmann plots for a range of values of Ei and thereby determine the expected particle size distribution for a given impact energy. In practice it is more convenient to use an inverse procedure in which the Schuhmann parameter k is varied in 0.1 increments between zero and one, and a t10 value determined for each value of k. The derived t10 values may then be used to calculate the associated value of the mass specific impact energy, Ei. These values are presented in Table 1 together with the associated values of Em, the theoretical minimum fracture energy defined as:
Em = (1 x 10-3) Gf (J/m2) x Sigma [Sav(m2/kg)] J/g
Gf is the previously defined energy needed to create a unit area of fracture surface
and is equal 100 Joules/ m^2 for concrete.
Sav is the average mass specific surface area, equal to 4000/dav(microns) m^2, where dav(microns) is the average particle size in a specified mass range - the contributions from each size range being summed over the entire range down to 10 mm.
(N.B. The derivation of the formula for Sav is described in F. Greening’s article “Energy Transfer in the WTC Collapse Events of September 11th 2001”.)
Substituting for Gf and Sav in the above equation leads to the simple relation:
Em = 400/ dav(microns) J/g
Also included in Table 1 are values of the percentage of the impact energy utilized to fragment the material; the data show that this is consistently below 15 %.
Table 1: Calculated Impact Energies, Ei, and Fracture Energies, Em,
for Schuhman Parameters Between 0.9 and 0.2
Schuhmann Parameter, k t10 (%) Ei (J/g) Em (J/g) Em / Ei (´100) (%)
0.9 12.6 1.06 0.059 5.6
0.8 15.8 1.38 0.087 6.3
0.7 20.0 1.82 0.136 7.5
0.6 25.1 2.41 0.223 9.3
0.5 31.6 3.35 0.373 11.1
0.4 39.8 4.68 0.622 13.3
0.3 50.1 7.28 0.998 13.7
0.2 63.1 17.45 1.455 8.3
Table 1 shows that t10, (the cumulative weight-percent of particles passing a sieve size of 10 % of the material’s original size), increases as k decreases and the impact energy, Ei, increases. The same trend holds true for t20, t50, t100, etc, meaning that a larger percentage of material is crushed into finer particles as the impact energy increases, as expected. However, these and similar data also show that once the impact energy exceeds about 10 J/g or three times the initial (minimum) value for the WTC collapse, the energy consumed in fracturing the concrete, Em, remains essentially constant at about 1.6 J/g.
This observation suggests that the fracturing process saturates as more and more impact energy is supplied to the material because the probability of fracturing a very small particle by impact is much lower that the probability of fracturing a large piece of the same material. Thus crushed concrete tends towards a limiting, constant, size distribution at impact energies greater than ~ 10 J/g. For an initial 10 cm cube of concrete this limiting size distribution is (approximately):
10 cm – 1 cm 1 cm – 1 mm 1 mm – 100 mm 100 mm – 10 mm Less than 10 mm
30 % 20 % 15 % 10 % 25 %
A study of the growth of the kinetic energy of the upper section of WTC 1 as the Tower collapsed shows that the mass specific impact energy of the first four collisions increased from 3.4 J/g (1st impact), to 6.4 J/g (2nd impact), to 8.7 J/g (3rd impact), to 11.7 J/g (4th impact) - See Greening’s “Energy Transfer in the WTC Collapse Events of September 11th 2001” and subsequent Addendum. Hence, by the 4th impact, the energy supplied to the concrete was sufficient to cause it to fragment to the limiting size distribution noted above. At this point, and for all subsequent impacts, the energy consumed in pulverizing the WTC 1 concrete was essentially constant and progressively less than 15 % of the available impact kinetic energy.
Thus we conclude that 50 % of the WTC 1 concrete was pulverized to particles less than 1 mm in diameter, (and 30 % was smaller than 100 microns). For all impacts of the upper section of WTC 1, less than 15 % of the available impact kinetic energy was dissipated in pulverizing the concrete.