MECHANICS OFPROGRESSIVE COLLAPSE: WHAT DID AND DID NOT DOOM WORLD TRADE CENTER, AND WHAT CAN WE LEARN ?
StructuralSystem
- framed tube
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:
11Review of ElementaryReview of ElementaryMechanics of CollapseMechanics of Collapse
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)
13% of columns were severed on impact, somemore deflected
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)
T opplinglike a tree?
(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)
South tower impacted eccentrically
Plastic Shearing of Floor Caused by Tilting(Mainly South Tower)
a b c d e
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
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:
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!
22Gravity-Driven Gravity-Driven
Propagation of Crushing Propagation of Crushing Front in Progressive Front in Progressive
CollapseCollapse
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
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!
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
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
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
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
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
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
Energy Potential at Variable Mass
Crush-Down
Crush-Up
Note:Solution by quadratures is possible for constant average properties, no comminution, no air ejection
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)
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)
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)
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)
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
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
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
33 Critics Outside Critics Outside
Structural Engineering Structural Engineering Community:Community:
Why Are They Wrong?Why Are They Wrong?
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
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?
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
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
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
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
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.
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
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
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
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
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
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)
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
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
Moment of ground impact cannot be seen, but from seismic record: Collapse duration = 12.59 s (± 0.5 s of rumble)
Notejetsofdust-ladenair
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.
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.
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.
44How the findings can be How the findings can be
exploited by tracking exploited by tracking demolitions demolitions
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
Collapse of 2000 Commonwealth Avenue in Boston under construction, 1971(4 people killed)The collapse was initiated by slab punching)
Murrah Federal Building in Oklahoma City, 1995(168 killed)
Ronan Point Collapse
U.K. 1968
Reinforcing Bar
Floor slab
Weak Joints, Precast Members
Hotel New World
Singapore 1986
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
Gravity-Driven Progressive Collapse Triggered by Earthquake
• 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
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