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