MECHANICS OF PROGRESSIVE COLLAPSE - trentglobal.edu.sg · Previous Investigations • Computer...
Transcript of MECHANICS OF PROGRESSIVE COLLAPSE - trentglobal.edu.sg · Previous Investigations • Computer...
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MECHANICS OFPROGRESSIVE COLLAPSE: WHAT DID AND DID NOT DOOM WORLD TRADE CENTER, AND WHAT CAN WE LEARN ?
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StructuralSystem
- framed tube
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Previous Investigations• Computer simulations and engrg. analysis at NIST — realistic,
illuminating, meticulous but no study of progressive collapse.
• Northwestern (9/13/2001) — still valid • E Kausel (9/24/2001) — good, but limited to no dissipation
3. GC Clifton (2001) — “Pancaking” theory: Floors collapsed first, an empty framed tube later? — impossible 4. GP Cherepanov (2006) — “fracture wave“ hypothesis — invalid5. AS Usmani, D Grierson, T Wierzbicki…special fin.el. simulations
• Lay Critics: Fletzer, Jones, Elleyn, Griffin, Henshall, Morgan, Ross, Ferran, Asprey, Beck, Bouvet, etc.
Movie “Loose Change” (Charlie Sheen), etc.
• Mechanics theories of collapse:
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11Review of ElementaryReview of ElementaryMechanics of CollapseMechanics of Collapse
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Momentum of Boeing 767 ≈ 180 tons × 550 km/h
Momentum of equivalent mass of the interacting upper half of the tower ≈ 250, 000 tons × v0
Initial velocity of upper half:
v0 ≈ 0.7 km/h (0.4 mph)
Assuming first vibration period T1 = 10 s:
Maximum Deflection = v0T / 2π ≈ 40 cm
Initial Impact – only local damage, not overallTower designed for impact of Boeing 707-320 (max. takeoff weight is 15% less, fuel capacity 4% less than Boeing 767-200)
(about 40% of max.hurricane effect)
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13% of columns were severed on impact, somemore deflected
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Failure Scenario• 60% of 60 columns of impacted face (16% of
287 overall) were severed, more damaged.• Stress redistribution higher column loads.⇒• Insulation stripped steel temperatures ⇒ up to 600oC→yield strength down -20% at
300oC,-85% at 300oC, creep for > 450oC. 4. Differential thermal expansion +
viscoplasticity floor trusses sag, pull ⇒perimeter columns inward (bowing of columns = buckling imperfection).
5. Collapse trigger: Viscoplastic buckling of hot columns (multi-floor) → upper part of tower falls down by at least one floor height.
• The kinetic energy of upper part can be neither elastically resisted nor plastically absorbed by the lower part of tower ⇒ progressive collapse (buckling + connections
sheared.)
I. Crush-Down Phase II. Crush-Up Phase
a) b) c) d) e) f)
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T opplinglike a tree?
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(The horizontal reaction at pivot) > 10.3× (Plastic shear capacity of a floor)
δ
Possible ?
mg F
mgF8
3max =
1H
mxθ⋅⋅
H1
m
x
θ
MPF1
MPF1
h1
FP
Why Didn't the Upper Part Fall Like a Tree, Pivoting About Base ?
a)
b)
c)
d)
e)
f)
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South tower impacted eccentrically
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Plastic Shearing of Floor Caused by Tilting(Mainly South Tower)
a b c d e
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m
h Dynamic elastic overload factor calculated for
maximum deflection (loss of gravity potential of mass m = strain energy)
a) Overload due to step wave from impact! WRONG!
⇒ The column response could not be elastic, but plastic-fracturing
Elastically Calculated Overload
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θ1 θ2
θ3
Can Plastic Deformation Dissipate the Kinetic Energy of Vertical Impact of Upper Part?
Only <12% of kinetic energy was dissipated by plasticity in 1st story, less in further stories
⇒Collapse could not have taken much longer than a free fall
n = 3 to 4 plastic hinges per column line.
Combined rotation angle:
Dissipated energy:
Kinetic energy = released gravitational potential energy:
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Plastic Buckling
Fc ≥ Fs
…can propagate dynamically
Fc < Fs
… cannot
hL=2Lef
P1 P1
θu
LL/2θ
P1MP
MPP1
Plastic buckling
Wf
Fc FsService load
Loa
d F
Axial Shortening u
00 0.5h h
Yield limit
λh
F0
00 0.04h
F0
Elas
tic
Yielding
Plastic buckling
Expanded scale
Case of single floor buckling
F
Shanleybifurcationinevitable!
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22Gravity-Driven Gravity-Driven
Propagation of Crushing Propagation of Crushing Front in Progressive Front in Progressive
CollapseCollapse
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Two Possible Approaches to Global Continuum Analysis
• Stiffness Approach homogenized elasto-plastic strain-softening continuum — must be NONLOCAL, with characteristic length = story height … COMPLEX !
• Energy Approach – non-softening continuum equivalent to snap-through*
— avoids irrelevant noise …SIMPLER !________________________
* analogous to crack band theory, or to van der Waals theory of gas dynamics, with Maxwell line
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mg
F0
Fc
0
CrushingResistance F(u)
Wcλh
ΔFd
ΔFa
h
Crushing of Columns of One Story
Floor displacement, u
Cru
shin
g fo
rce,
F
ucu0 uf
ü = g – F(u) / m(z)
K < Wc
Internal energy : φ(u) =
∫
Wb
b
bMaxwell Line
Dynamic Snapthrough θ1 θ2
θ3
Collapse arrest criterion: Kin. energy
One-story equation of motion::
Reh
arde
ning
Initial condition: v v velocity of impacting block
Lumped Mass
Lower Fc formulti-floor buckling!
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tzctzc
v1
v2 > v1vg-Fc/m
1
h
a) Front accelerates
h0
F0
Fcmg
F(z)
h
F0
mg
Fc
v1
Cru
shin
g fo
rce,
Fb) Front decelerates c) Collapse arrested
v
v2 < v1
time
Flo
or v
eloc
ity,
v
u
h
for Fc v1
v
u
u
g-Fc/m1
v
u
v2 >v1v
h
v1
for Fc
0
0
0
00 0
hu
v
0
v1
v1
W1 = K
mg
F0
zc
Fc
0
Real CrushingResistance F(z)
W1 = W2
u
λhΔFd
ΔFa
W1 = W2ΔFd
ΔFa
λh ΔFd
Deceleration
Acceleration
DecelerationAcceleration
Deceleration
λh
λhλhλh
Displacement
t tTime t
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h h
Fc
a) Single-story plastic buckling L = h
Fc
Fc
Floor n n-1 n-2 n-3 n-4
Wc Wc
Fpeak
Fc
Fpeak
Fc
Fs Service load
Fc
Fpeak
b) Two-story plastic buckling L = 2h
c) Two-story fracture buckling L = 2h
Fpeak = min (Fyielding, Fbuckling)
Internal energy (adiabatic) potential : W = ∫ F(z)dz
Compaction Ratio, λ, at Front of Progressive Collapse
λh
2λh
Cru
shin
g F
orce
, F
Distance from tower top, z
Total potential = Πgravity - W
Mean Energy Dissipation by Column Crushing, Fc, and
energy-equivalentsnapthrough = mean crushingforce
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Mass shedding
Phase II
Collapse front
Crush-Down (Phase I of WTC)
Crush-Up (Phase II of WTC or Demolition)
Collapse front
2 Phases of Crushing Front Propagation
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1D Continuum Model for Crushing Front Propagation1D Continuum Model for Crushing Front Propagation
C
A
z0
s0
z
H
B
B
y0 = z0C y
B
CB’
y η
ζ
r0 B’
B
z0C
Phase 1. Crush-Down Phase 2. Crush-Up
Fc
Fc’< Fc if slowerthan free fallPhase 1
downwardz&
Δt
m(z)g
FcFc Fc
Fc
m(y)g
a)
b)
c)d)
e)
g)Crush-Down
Crush-Up
h)
i)
Can 2 fronts propagate up and down
simultaneously ? – NO !
s = λs0
λ(H-z0)
A
r = λr0 λz0
λH
λ = compaction ratio = Rubble volume within perimeterTower volume
zΔt.
m(z)v.
m(y)y.
yΔt.
μy2.z.
ζ
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Diff. Eqs. of Crushing Front PropagationI. Crush-Down Phase:
II. Crush-Up Phase:
fraction of mass ejected outside perimeter
Inverse: If functions z(t), m(z), λ(z) are known, the specific energy dissipation in collapse, Fc(y), can be determined
Front decelerates if Fc(z) > gm(z)
z(t)
y(t)
Intact
Compacted
Compaction ratio:
z0
z0
Criterion of Arrest (deceleration): Fc(z) > gm(z)
Buckling Comminution Jetting airResisting force
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0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0
0
1 0 0
2 0 0
3 0 0
4 0 0
Variation of resisting force due to column buckling, Fb, (MN)
1 1 . 2 1 . 4 1 . 6
0
1 0 0
2 0 0
3 0 0
4 0 0
Variation of mass density, m(z),(106 kg/m)
Resistance and Mass Variation along Height
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Energy Potential at Variable Mass
Crush-Down
Crush-Up
Note:Solution by quadratures is possible for constant average properties, no comminution, no air ejection
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Collapse for Different Constant Energy Dissipations
Time (s)
Tow
er T
op C
oord
inat
e (m
)
Wf = 2.4 GNm
2
1.5
10.5
0
free
phase 1
phase 2
fall
λ= 0.18 , μ= 7.7E5 kg/m , z0 = 80 m , h = 3.7 m
fall arrested
(for no comminution, no air)
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Collapse for Different Compaction RatiosT
ower
Top
Coo
rdin
ate
(m)
Time (s)
λ= 0.4 0.30.18
0
transition between phases 1 and 2
Wf = 0.5 GNm , μ= 7.7E5 kg/m , z0 = 80 m , h = 3.7 m
freefall
(for no comminution, no air)
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Collapse for Various Altitudes of Impact
for impact 2 floors below top
5
20
55
Time (s)
Tow
er T
op C
oord
inat
e (m
)
(≈ 2.5 E7 GNm)
mg < F0,heated
freefall
phase 1phase 2
λ= 0.18 , h = 3.7 mμ= (6.66+2.08Z)E5 kg/mWf = (0.86 + 0.27Z)0.5 GNm
(for no comminution, no air)
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Crush-up or Demolition for Different Constant Energy Dissipations
Time (s)
Tow
er T
op C
oord
inat
e (m
) Wf = 11 GNm
65432
0.5
parabolic endfree
fall
λ= 0.18 , μ= 7.7E5 kg/m , z0 = 416 m , h = 3.7 m
fall arrested
asymptotically
(for no comminution, no air)
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Resisting force as a fraction of totalR
esis
ting
For
ce /T
otal
Fc
0 4 8 1 2
0 %
2 5 %
5 0 %
7 5 %
1 0 0 %
0 4 8 1 2
0 %
2 5 %
5 0 %
7 5 %
1 0 0 %
FbFb
Fs
Fa
Fs
Fa
Fb
Fs
Fa
Fb
Fs
Fa
96 81 48 5 F 110 81 64 25 F 101
Time (s) Time (s)
Impacted Floor Number Impacted Floor Number
North Tower South Tower
Crush-down ends
Crush-down ends
110
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Fc / m
(z)g
Resisting force / Falling mass weight
0 4 8 1 2
0 . 1
1
1 0
1 0 0
0 4 8 1 2
0 . 1
1
1 0
1 0 096 81 48 5 F 110 81 64 25 F 101 110
Time (s) Time (s)
Impacted Floor Number Impacted Floor Number
North Tower South Tower
Crush-down ends
Crush-down ends
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External resisting force and resisting force due to mass accretion
Res
istin
g fo
rce
Fc a
nd F
m (M
N) Impacted Floor Number Impacted Floor Number
Time (s)0 4 8 1 2
0
1 2 5 0
2 5 0 0
Fm
Fc
North Tower
96 81 48 5 F
Time (s)0 4 8 1 2
0
1 2 5 0
2 5 0 0
Fm
Fc
South Tower
81 64 25 F
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33 Critics Outside Critics Outside
Structural Engineering Structural Engineering Community:Community:
Why Are They Wrong?Why Are They Wrong?
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Lay Criticism of Struct. Engrg. Consensus1) Primitive Thoughts:
Euler's Pcr too high Buckling possibility denied Plastic squash load too high, etc. Initial tilt indicates toppling like a tree? — So explosives must been used !
Shanley bifurcation
No ! — horizontal reaction is unsustainable
No !No !
Like a Tree?
~4º tilt due to asymmetry of damage
~25º (South Tower)non-accelerated rotation about vertically moving mass centroid
Mass Centroid
Ft
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South TowerNorth Tower
Video Record of Collapse of WTC Towers
2) Collapse was a free fall ! ? Therefore the steel columns must have been destroyed beforehand — by planted explosives?
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Tilting Profile of WTC South Tower
East
)cos1(2
1 θ−−∆=∆H
tC
North
∆1
∆2
θe∆m
∆t
θs Video-recorded(South Tower)
Initial tilt
H1
∆t
∆c
θ 2
H1
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Comparison to Video Recorded Motion(comminution and air ejection are irrelevant for first 2 or 3
seconds)
Not fitted but predicted! Video analyzed by Greening
0 1 2 3
3 8 0
4 0 0
4 2 0
Tow
er T
op C
oord
inat
e (m
)
First 30m of fall
North Tower
Free fall
From crush-down differential eq.
Time (s)
0 1 2
4 0 0
4 1 0
4 2 0
South Tower(Top part − large falling mass)
First 20m of fall
From crush-down differential eq.
Time (s)
Free fallNote uncertainty range
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417 mH
T
8.08s 12.29s 12.62s
12.81s
Free fall
impeded by single-story buckling only
with pulverization
with expelling air
Most likely time from seismic record
From seismic data: crush-down T ≈ 12.59s ± 0.5s
-20 m0 m
Seismic rumble
Impact of compacted rubble layer on rock base of bathtub
Seismic and video records rule out the free fall!
North Tower
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Calculated crush-down duration vs. seismic record
Tow
er T
op C
oord
inat
e (m
)
Seismic error
a bc
0 4 8 12Time (s)
0
1 5 0
3 0 0
4 5 0
Free fall
with air ejection & comminution
Crush-down ends
with buckling only
South Tower
Calculationerror
0 4 8 12 16
a
bc
Seismic error
Time (s)
Calculationerror
0
1 5 0
3 0 0
4 5 0
North Towerwith air ejection & comminution
Free fall
Crush-down ends
with buckling only
Gro
und
Vel
ocity
(µ
m/s
)
Free fall Free fall
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How much explosive would be needed to pulverize 73,000 tons of lightweight concrete of one tower to particles of sizes 0.01— 0.1mm ?
• 237 tons of TNT per tower, put into small drilled holes (the energy required is 95,000 MJ; 30 J per m2 of particle surface,
and 4 MJ per kg of TNT, assuming 10% efficiency at best).
(similar to previous estimate by Frank Greening, 2007)
3) Pulverizing as much as 50% of concrete to 0.01 to 0.13 mm required explosives! NO. — only 10% of kinetic energy sufficed.
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Comminution (Fragmentation and Pulverization) of Concrete Slabs
kt DDMDM )/()( max=Schuhmann's law:
Dtotal particle sizemass of particles < D
)(d
)( 3)(
min
DMD
DGDWK
D
D
ff ∫==∆
ρ
Energy dissipated = kinetic energy loss ΔK
density of particle size
Cum
ulat
ive
Mas
s of
Par
ticle
s (M
/ M
t)
1k
0.16mm = Dmin
Impa
ct sla
b stor
y
interm
ediat
e stor
y
Impa
ct on g
round
0.012 mm = Dmin
0.01 0.1 1 10
10.12 mm
Particle Size (mm)
16 mm
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Kinetic Energy Loss ΔK due to Slab ImpactMomentum balance:
∑+=i iivmmvmv 21
Fragments
2max for (all )iv v i∆ =Kinetic energy loss:
2 21 2
1 1 ( )
2 2 imv m m vγ ∆ = − + ∑
2 2 [1 / ( )]
s
s
mz
h m m z
γ∆ =+
(energy conservation) total b aU W W∆ = ∆ = ∆ + ∆ + ∆Total:Concrete fragments
BucklingGravitational energy loss
m
v1
v2
Compacted layer
Comminuted slabs
Kinetic energy to pulverize concrete slabs & core walls
= ms concrete
Air
A
K
K
K K
K
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Fragment size of concrete at crush front
Max
imum
an
d M
inim
um
Fra
gmen
t Siz
e at
Cru
sh F
ront
(m
m)
0 4 8 1 2
0 . 0 0 1
0 . 0 1
0 . 1
1
1 0
Time (s) Time (s)
North Tower
Dmin
Dmax
96 81 48 5 F 110
Impacted Floor Number81 64 25 F 101 110
Impacted Floor Number
0 4 8 1 2
0 . 0 0 1
0 . 0 1
0 . 1
1
1 0
Dmin
Dmax
South Tower
Crush-down ends Crush-down
ends
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Wf /
КComminution energy / Kinetic energy of
falling mass
0 4 8 1 2
0 . 1 %
1 %
1 0 %
1 0 0 %
Crush-down ends
Time (s)
North Tower
96 81 48 5 F 110
0 4 8 1 2
0 . 1 %
1 %
1 0 %
1 0 0 %
Crush-down ends
Time (s)
South Tower
81 64 25 F 110101Impacted Floor Number Impacted Floor Number
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Dust mass (< 0.1 mm) / Slab massM
d / M
s
0 4 8 1 2
0
0 . 5
1
0 4 8 1 2
0
0 . 5
1
Time (s) Time (s)
96 81 48 5 F 110 81 64 25 F 101110Impacted Floor Number Impacted Floor Number
Crush-down ends
Crush-down ends
North Tower South Tower
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Loss of gravitational potential vs. comminution energy
0 4 8 1 2
0
5 0 0
1 0 0 0
0 4 8 1 2
0
5 0 0
1 0 0 0
Ene
rgy
Var
iatio
n (G
J)
Comminution energy
Ground impact Ground impact
Comminution energy
Loss of gravitational potential
Loss of gravitational potential
North Tower South Tower
Time (s) Time (s)
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4) Booms During Collapse! —hence, planted explosives?
If air escapes story-by-story, its mean velocity at base is va = 461 mph (0.6 Mach), butlocally can reach speed of sound
5) Dust cloud expanded too rapidly? Expected.
(va < 49.2 m/s, Fa < 0.24 Fc, ∆ pa < 0.3 atm)
1 story: 3.69 x 64 x 64 m air volume
200 m of concrete dust or fragments
Air Jets
Air squeezed outof 1 story in 0.07 s
a
h
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North Tower Collapse in Sequence
Can we see the motion through the dust ?Can we see the motion through the dust ?Except that below dust c loud the tower Except that below dust c loud the tower was NOT breaking,was NOT breaking, nothing can be learned nothing can be learned !!
Note:• Dust-laden air jetting out• Moment of impact cannot be detected visually
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Moment of ground impact cannot be seen, but from seismic record: Collapse duration = 12.59 s (± 0.5 s of rumble)
Notejetsofdust-ladenair
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9) Red hot molten steel seen on video (steel cutting) — perhaps just red flames?
7) Lower dust cloud margin = crush front? — air would have to escape through a rocket nozzle!
6) Pulverized concrete dust (0.01 to 0.12 mm) deposited as far as 200 m away? — Logical.
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8) Temperature of steel not high enough to lower yield strength fy of structural steel, to cause creep buckling?
fy reduced by 20% at 300ºC, by 85% at 600ºC (NIST). Creep begins above 450ºC. Steel temperature up to 600ºC confirmed by annealing studies at NIST.
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10) “Fracture wave” allegedly propagated in a material
A uniform state on the verge of material failure cannot exist in a stable manner, because of localization instability. Wave propagation analysis would have to be nonlocal, but wasn't “Fracture wave” cannot deliver energy sufficient for comminution.
pre-damaged, e.g., by explosives, led to free-fall collapse — unrealistic hypothesis, because:
9) Thermite cutter charges planted? — evidenced by residues of S, Cu, Zi found in dust? But these must have come from gypsum wallboard, electrical wiring, galvanized sheet steel, etc.
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44How the findings can be How the findings can be
exploited by tracking exploited by tracking demolitions demolitions
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Proposal: In demolitions, measure and compare energy dissipation per kg of structure.
Use: 1) High-Speed Camera 2) Real-time radio-monitored accelerometers: Note: Top part of WTC dissipated 33 kJ/m 3
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Collapse of 2000 Commonwealth Avenue in Boston under construction, 1971(4 people killed)The collapse was initiated by slab punching)
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Murrah Federal Building in Oklahoma City, 1995(168 killed)
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Ronan Point Collapse
U.K. 1968
Reinforcing Bar
Floor slab
Weak Joints, Precast Members
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Hotel New World
Singapore 1986
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Generalization of Progressive Collapse
1) 1D Translational-Rotational--- "Ronan Point" typeAngular momentum and shear not negligible
2) 3D Compaction Front Propagation
Gas explodedon 18th floor
— will require finite strain simulation
25th floor
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Gravity-Driven Progressive Collapse Triggered by Earthquake
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• All WTC observations are explained.
• All lay criticisms are refuted.
Download 466.pdf & 405.pdf from Bazant’s website: www.civil.northwestern.edu/people/bazant.html
MAIN RESULTS
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References• Bažant, Z.P. (2001). “Why did the
World Trade Center collapse?” SIAM News (Society for Industrial and Applied Mathematics) Vol. 34, No. 8 (October), pp. 1 and 3 (submitted Sept. 13, 2001) (download 404.pdf).
• Bažant, Z.P., and Verdure, M. (2007). “Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions.” J. of Engrg. Mechanics ASCE 133, pp. 308—319 (download 466.pdf).
• Bažant, Z.P., and Zhou, Y. (2002). “Why did the World Trade Center collapse?—Simple analysis.” J. of Engrg. Mechanics ASCE 128 (No. 1), 2--6; with Addendum, March (No. 3), 369—370 (submitted Sept. 13, 2001, revised Oct. 5, 2001) (download 405.pdf).
• Kausel, E. (2001). “Inferno at the World Trade Center”, Tech Talk (Sept. 23), M.I.T., Cambridge.
• NIST (2005). Final Report on the Collapse of the World Trade Center Towers. S. Shyam Sunder, Lead Investigator. NIST (National Institute of Standards and Technology), Gaithersburg, MD (248 pgs.)
: www.civil.northwestern.edu/people/bazant.html