Originally posted by captainnotsoobvious
reply to post by LightSpeedDriver
Another Truther that can't defend his beliefs...
Sorry, you're right, my attitude of having to have facts before I believe something, of not simply spouting lies because I want my point to "win"... of being critical of everyone and everything, not just the stuff I don't like...
Yes, that's what's wrong with the world... yes, I will fail in life because of my high standards..
nailed it..
First, let us review the basic argument . After a drop through at least the height h of one story heated by fire (stage 3 in Fig. 2 top), the mass of the upper part of each tower has lost enormous gravitational energy, equal to m0gh. Because the energy dissipation by buckling of the hot columns must have been negligible by comparison, most of this energy must have been converted into kinetic energy K = m0v2/2 of the upper part of tower, moving at velocity v.
Calculation of energy Wc dissipated by the crushing of all columns of the underlying (cold and intact) story showed that, approximately, the kinetic energy of impact K > 8.4 Wc ). It is well known that, in inelastic buckling, the deformation must localize into inelastic hinges. To obtain an upper bound on Wc, the local buckling of flanges and webs, as well as possible steel fracture, was neglected (which means that the ratio K/Wc was at least 8.4).
When the subsequent stories are getting crushed, the loss m0gh of gravitational energy per story exceeds Wc exceeds 8.4 by an ever increasing margin, and so the velocity v of the upper part must increase from one story to the next. This is the basic characteristic of progressive collapse, well known from many previous disasters with causes other than fire (internal or external explosions, earthquake, lapses in quality control)
Merely to get convinced of the inevitability of gravity driven progressive collapse, further analysis is, for a structural engineer, superfluous. Further analysis is nevertheless needed to dispel false myths, and also to acquire full understanding that would allow assessing the danger of progressive collapse in other situations.
...
steel will not just crush down upon itself
Variation of Mass and Buckling Resistance along the Height Near the top, the specific mass (mass per unit height) μ = 1.02 × 106 kg/m. In view of proportionality to the cross section area of columns, μ = 1.05 × 106 kg/m at the impact level (81st floor) of South Tower.
Although precise data on μ(z) are unavailable, it appears sufficient to use the approximation μ(z) = k0ek2z + k1 (where k0, k1, k2 = constants), with a smooth transition at the 81st floor to a linear variation all the way down. The condition that R H 0 μ(z)dz be equal to the total mass of tower (known to be roughly 500,000 tons) gives μ = 1.46 × 106 kg/m at the base. There are various local complexities whose possible effects were estimated in calculations (e.g., the fact that 16 of 47 core columns at the bottom were much more massive than the rest).
However, they appeared to have no appreciable effect on the overall response, particularly on the diagram of z(t) and the collapse duration. The total energy dissipation per unit height, which represents the resisting force Fc, consists not only of energy Fb dissipated by the inelastic hinges formed during column buckling, but also of energy Fs required for comminuting concrete floor slabs, energy Fa required for expelling air from the tower, and energy Fe required for ejecting particles and fragments.
Based on Fig. 5 and Eq. 8 of Baˇzant and Zhou (2002), we have, for three-hinge column buckling: Fb = Z uf 0 F(u)du h , F(u) = XN i=1 2[Mai(i) +Mbi(0i)] Li sin i , i = arccos 1 − u Li (6) where F = axial force resultant of all the columns in the story; u = vertical relative displacement between column ends, uf = final u-value; 0i = 2i; i, 0i = hinge rotations at the ends and middle of column i, which are functions of u; Mai,Mbi = bending moments in inelastic hinges at the ends and middle of column i, as functions of i or 0i; and Li = initial clear length of columns i. For plate-type four-hinge buckling, similar simple expressions apply.
Although some core columns were rectangular, their plastic bending moments Mp were nearly proportional to the column cross section areas because, in the weak buckling direction, most core columns had the same width as the perimeter columns. Thus the curve F(u) corresponding to perfect plasticity (Mai = Mbi = Mpi) is not difficult to estimate from the weight of all the columns in a story.
However, three effects doubtless intervened to reduce F(u): 1) multi-story buckling of some columns; 2) softening due to local plastic flange buckling, and 3) fracture of steel in inelastic hinges (the last two likely occurred only at large buckling deflections for which F(u) is small).
The available data are insufficient to make an accurate estimate of these effects, and even the data on the flange thicknesses in the perimeter and core columns of all the stories are missing.
So we simply apply to Fb an empirical correction factor ( 1) which is reasonably expected to lie within the range (0.5, 0.8) for normal structural steel (yield limit 250 MPa), but in the range (0.1, 0.3) for the high-strength steel (yield limit 690 MPa) which was used for perimeter columns in the lower stories.
The high-strength steel has a much lower ductility, which must have caused fractures (with a drop of axial force to zero) very early during buckling, and must have been the cause of formation of large multistory fragments seen to fall from the lower part of tower. Consequently,the energy dissipated (which is equal to the area under the load-displacement curve of column) was probably about the same for high- and normal-strength columns.
As the website 911research.wtc7.net pointed out, "Bazant must be a super-genius to understand how two skyscrapers could crush themselves to rubble, a newly observed behaviour for steel structures, and write a paper about it in just two days."
Previous analysis of progressive collapse showed that gravity alone suffices to explain the overall collapse of the World Trade Center towers.
However, it has not been checked whether the allegations of controlled demolition by planted explosives have any scientific merit.
The present analysis proves that they do not. The video record available for the first few seconds of collapse agrees with the motion history calculated from the differential equation of progressive collapse but disproves the free fall hypothesis (on which the aforementioned allegations rest).
Although, due to absence of experimental crushing data for the lightweight concrete used, the theory of comminution cannot predict the size range of pulverized concrete particles, it is shown that the observed size range (0.01 mm – 0.1 mm) is fully consistent with this theory and is achievable by collapse driven gravity alone, and that only about 7% of the total gravitational energy converted to kinetic energy of impacts would have sufficed to pulverize all the concrete slabs and core walls (while at least 158 tons of TNT per tower, installed into many small holes drilled into each concrete floor slab and core wall, would have been needed to produce the same degree of pulverization).
The collapse, in which two phases—crush-down followed by crush-up—must be distinguished, is described in each phase by a nonlinear second-order differential equation for the propagation of the crushing front of a compacted block of accreting mass. Expressions for consistent energy potentials are formulated and an exact analytical solution of a special case is given.
The governing differential equations [5] are:
They were derived by continuum homogenization of the energy dissipation per story [5]; t = time, z = vertical (Lagrangian) coordinate = distance of the current crushing front from the initial position of the tower top; the superior dots denote time derivatives; m(z) = cumulative mass of the tower above level z; Fc = resisting force = energy dissipation per unit height,
where Wd = energy dissipation per story due to buckling; Fb = energy per unit height consumed for buckling of steel columns; Fs = energy per unit height consumed by fragmenting (or comminuting) concrete floor slabs and core walls; Fa = energy of expelling air (laden with dust), per unit height (in [5], Fs and Fa were neglected).
Here V0 = initial volume of the tower, V1 volume of the rubble on the ground into which the whole tower mass has been compacted; / (z) = effective compaction ratio = (1/h)× the thickness of the layer of debris to with each story is compacted; kout = fraction of mass that is ejected outside the tower perimeter before it receives significant downward acceleration, and Wd(z) = total energy dissipation up to level z (for the idealized special case of = Fc = out = 0 and constant μ = dm/dz, Eq. (2) reduces to the differential equation (zz˙)˙ = gz, .
Eq. (2) may be rewritten where Fm = force required to accelerate to velocity z˙ the stationary mass accreting at the crushing front, and ¯μ = d[m(1−)]/dz = non-ejected part of the accreting mass per unit height. This force causes a greater difference from free fall than do forces Fb, Fs and Fa combined.
Some critics believe that the bottom of the advancing dust cloud seen in the video represented
the crushing front. However, this belief cannot be correct because the compressed air exiting
the tower is free to expand in all directions, including the downward direction, which causes
the dust front to move ahead of the crushing front (the only way to prevent the air from jetting
out in all directions would be to shape the exit from each floor as a diverging nozzle of a rocket,
which was obviously not the case).
The main point to note is that the curve identified from the video record grossly disagrees with the free fall curve, for each tower.
The belief that the towers collapsed at the rate of free fall has been the main argument of the critics claiming controlled demolition by planted explosives.
The video record alone suffices to prove this argument false. For the South Tower, the difference between the free fall curve and the curve calculated from Eq. (2) is less pronounced than it is for the North Tower.
The reason is that the initial upper falling mass of the South Tower is nearly twice that of the North Tower, causing the resisting force to be initially a much smaller fraction of the falling weight.
Calculations show that the duration of the entire crush-down phase exceeds the free fall duration by 57.6% for the North Tower, and by 41.9% for the South Tower (Fig. 6b). This is a significant difference, which can be checked against seismic records registered at Lamont-Doherty Earth Observatory of Columbia University [23], shown in Fig. 6b. The first tremor, which is weak (and is marked as a), is assumed to represent t = 0, i.e., the moment of impact of the upper part of tower onto the lower part (a correction of 0.07 s is made for the delay due to the travel time of the sound wave along the steel columns to the ground). The sudden displacement increase at instant b (Fig. 6b) is attributed to ejected large structure segments that hit the ground outside the tower perimeter. Their travel durations,9.74 s for the North Tower and 8.4 s for the South Tower, are much shorter than the crushdown
because there is no mass accretion, although they must be a bit longer than a free fall in
vacuum because of air drag (which is, however, relatively small for massive pieces). The free
fall times for ejected mass are 8.61 s and 7.91 s for the North and South towers, respectively
(they are not the same as the free fall times shown in the Fig. 6b, because the ejected mass is
hitting the ground, while the free fall time shown in Fig. 6b corresponds to the upper falling
mass hitting the top of debris pile which is above the ground level).
Engineer Zdeněk P. Bažant is best known as a world leader in scaling research in solid mechanics . His research focuses on the effect of structure size on structural strength as it relates to the failure behavior of the structure. He also has made outstanding advances in structural stability , fracture mechanics, the micromechanics of material damage , concrete creep , and probabilistic mechanics
Born in Prague to a geotechnical engineering professor and a sociology Ph.D., and the grandson of a professor of structural mechanics and former university president, Bažant was the winner of the 1955 Mathematical Olympics.
He studied civil engineering at Czech Technical University (ČVUT), where he was first in his class. He was awarded the C.E. degree with the highest distinction in 1960. While working as a bridge engineer for the state consulting firm Dopravoprojekt in Prague, he studied for his Ph.D. in structural mechanics at the Czech Academy of Sciences, which he received in 1963.
In his dissertation on concrete creep theory, he developed a new method to analyze fracturing and cracking in concrete structures.He went on to earn a postgraduate diploma in theoretical physics from Charles University in 1966 and attained docent (Associate Professor) habilitation in concrete structures from ČVUT in 1967.
He is an ISI highly cited researcher in Engineering, which places him among the 250 most cited authors in all engineering fields worldwide. He was elected to the National Academy of Engineering in 1996, the National Academy of Sciences in 2002, Fellow of the American Academy of Arts and Sciences in 2008, and is a registered Structural Engineer in the state of Illinois. He has supervised 60 Ph.D.s in addition to receiving six honorary doctorates of his own (ČVUT 1991, TU Karlsruhe 1997, CU Boulder 2000, Politecnico di Milano 2001, INSA Lyon 2004, and TU Vienna 2005).
Originally posted by plube
reply to post by Drunkenparrot
is not an aceptable critique to dismiss empirical research.
Bazant must be a super-genius to understand how two skyscrapers could crush themselves to rubble, a newly observed behaviour for steel structures, and write a paper about it in just two days
now you will go here and say such things....but read it...and check the sources...and you might learn a few things.....i have check the sources...and have gone to the JEM read the papers....and have read through all the things you just cut and pasted so many times....WHY....BECAUSE it is in my field of work.
Dr. Bazant attempts to explain the balance of energies at a point in time immediately after
collapse initiation. He states that,
" To arrest the fall, the kinetic energy of the upper part, which is equal to the potential
energy release for a fall through the height of at least two floors, would have to be
absorbed by the plastic hinge rotations of one buckle, i.e., Wg/Wp would have to be less
than 1. Rather, Wg /Wp ?= 8.4 (3) if the energy dissipated by the columns of the critical
heated floor is neglected."
In other words, the energy available to progress the collapse was 8.4 times greater than
the energy required to progress the collapse.
The first error which Dr. Bazant has made is his assumption that all of the available
energy would be utilised exclusively in the destruction of the uppermost storey of the
lower section. This is physically impossible under any and all circumstances.
Nor is this deformation of the columns in the upper and lower sections limited to their
elastic range. It must be noted that the columns in the upper section could not deliver
a greater force than they themselves were able to transmit. In a situation where the
columns in the upper section were asked to deliver loads at magnitudes sufficient to cause
plastic deformation of columns in the lower section, then they themselves would
simultaneously suffer plastic deformation at levels proportionate to their ability and
applied loads.
WTC Technical Information Repository
Attn: Stephen Cauffman,
NIST, 100 Bureau Dr., Stop 8611,
Gaithersburg, Md. 20899-8610.
Dear Sirs:
I have examined the documents¹ you provided on your theory of the collapse of WTC 7 due to fires by way of thermal expansion. It is apparent that you have spent a great deal of time, effort, money and thought on this project.
However, like Ptolemy’s Theory of Epicycles, you begin with a faulty and unproven assumption. It is also the least likely assumption based on the evidence. Therefore, although your computer modeling may be intricate, your results are completely speculative and have no connection with the reality of what happened to that building. You are simply “adding epicycles” to a theory based on a false premise.
Your theory essentially rests on two physical observations:
There were office fires in WTC 7 that burned for some hours.
The building completely collapsed.
Observation 1 is not in dispute, except as to the location, extent, and effect of the fires. You never observed these fires from inside the building, and you have no actual measurements of the thermal expansion and deformation of the structural steel beams whatever. You never examined any of the steel.
Observation 2 runs contrary to 100 years of experience with the behavior of steel-framed buildings that have caught on fire. Every one of them was subjected to thermal expansion, but never before has there been such a collapse. To now postulate that a collapse did occur due to office fires is the height of scientific recklessness.