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MECHANICS OF PROGRESSIVE COLLAPSE: WHAT DID AND DID NOT DOOM WORLD TRADE CENTER, AND WHAT CAN WE LEARN ?

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MECHANICS OF

PROGRESSIVE

COLLAPSE:

WHAT DID

AND DID NOT DOOM WORLD TRADE CENTER, AND WHAT CAN WE LEARN ?

ZDENĚK P. BAŽANT

Presented as a Mechanics Seminar at Georgia Tech,

Atlanta, on April 4 ,2007, and as a Civil Engineering

Seminar at Northwestern University, Evanston, IL,

on May 24, 2007

Jialiang Le

Mathieu Verdure

Yong Zhou

Frank R. Greening

David B. Benson

SPONSORS: Specifically none (except, indirectly, Murphy Chair funds, and general support for fracture mechanics

and size effects from NSF and ONR)

- Structural
- System
- framed
- tube

Previous Investigations

- Computer simulations and engrg. analysis at NIST — realistic,illuminating, meticulous but no study of progressive collapse.

- Mechanics theories of 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 — invalid
- 5. 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.

Initial Impact– only local damage, not overall

Tower designed for impact of Boeing 707-320 (max. takeoff weight is 15% less, fuel capacity 4% less than Boeing 767-200)

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 / 2p ≈ 40 cm

(about 40% of max.hurricane effect)

13% of columns

were severed

on impact, some

more deflected

a)

b)

c)

d)

e)

f)

I. Crush-Down Phase

II. Crush-Up Phase

- 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.)

Failure Scenario

Toppling

like

a tree?

Why Didn't the Upper Part Fall Like

a Tree, Pivoting About Base ?

FP

a)

c)

e)

MP

q

F1

m

h1

H1

d

x

MP

F1

b)

d)

f)

mg

F

Possible ?

(The horizontal reaction at pivot)

> 10.3× (Plastic shear capacity of a floor)

South tower

impacted

eccentrically

Plastic Shearing of Floor Caused by Tilting

(Mainly South Tower)

a

b

c

d

e

m

h

Elastically Calculated Overload

s

Dynamic elastic overload factor calculated for maximum deflection (loss of gravity potential

of mass m = strain energy)

x

The column response could not be elastic, but plastic-fracturing

- Overload due to step wave from impact! WRONG!

Can Plastic Deformation Dissipate the Kinetic Energy of Vertical

Impact of Upper Part?

q1

q2

q3

n = 3 to 4 plastic hinges per column line.

Combined rotation angle:

Dissipated energy:

Kinetic energy= released gravitational potential energy:

Collapse could not have taken much longer than a free fall

Only <12% of kinetic energy was dissipated by plasticity in

1st story, less in further stories

Plastic Buckling

F

P1

Case of single floor buckling

P1

MP

P1

u

q

L/2

L=2Lef

q

h

L

MP

Fc≥ Fs

…can propagate

dynamically

Fc < Fs

… cannot

P1

Yield limit

Yielding

F0

F0

Wf

Shanley

bifurcation

inevitable!

Plastic buckling

lh

Plastic buckling

LoadF

Elastic

Fc

Service load

Fs

Expanded scale

0

0

0

0.04h

0

0.5h

h

Axial Shortening u

- 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

Internal energy : φ(u) =

F(u')du'

Crushing of Columns of One Story

q1

q2

q3

u

ü = g – F(u) / m(z)

One-story equation of motion::

∫

0

Initial condition: v velocity of impacting block

K < Wc

Collapse arrest criterion: Kin. energy

F0

Crushing

ResistanceF(u)

Lumped Mass

Rehardening

λh

Wc

ΔFd

Wb

Crushing force,F

Dynamic Snapthrough

mg

ΔFa

Fc

Maxwell Line

u

0

uf

uc

u0

h

Floor displacement,u

Lower Fc for

multi-floor buckling!

b) Front decelerates

c) Collapse arrested

a) Front accelerates

λh

Real Crushing

Resistance F(z)

F0

F0

F0

λh

F(z)

ΔFd

W1= K

Fc

λh

ΔFd

W1 = W2

ΔFd

W1 = W2

Fc

mg

mg

Crushing force, F

mg

ΔFa

ΔFa

Fc

u

u

u

zc

0

0

0

h

h

h

v

v

v

v2 >v1

for Fc

v1

for Fc

λh

λh

λh

Deceleration

v1

Acceleration

v1

Deceleration

u

Deceleration

Floor velocity, v

Acceleration

u

u

0

0

0

h

h

h

Displacement

v

v2 > v1

v

v

1

1

v1

g-Fc/m

g-Fc/m

v1

v2 < v1

v1

t

t

0

0

0

tzc

tzc

time

Time t

h

h

Mean Energy Dissipation by Column Crushing, Fc, and

Compaction Ratio, λ, at Front of Progressive Collapse

Total potential = Πgravity - W

Internal energy (adiabatic) potential : W = ∫ F(z)dz

a) Single-story plastic buckling L = h

Floor n

n-1

n-2

n-3

n-4

Fpeak

λh

Wc

Wc

energy-

equivalent

snapthrough

= mean

crushing

force

Fc

Fc

b) Two-storyplastic buckling L = 2h

Crushing Force, F

Fpeak

2λh

Fc

Fc

c) Two-storyfracture buckling L = 2h

Fpeak

Fs

Service load

Fc

Fc

Distance from tower top, z

Fpeak = min (Fyielding, Fbuckling)

2 Phases of Crushing Front Propagation

Crush-Up

(Phase II of WTC

or Demolition)

Crush-Down (Phase I of WTC)

Mass shedding

Collapse front

Phase II

Collapse front

Δt

1D Continuum Model for Crushing Front Propagation

Rubble volume within perimeter

λ = compaction ratio =

h)

Tower volume

Crush-Down

.

a)

g)

m(z)v

Can 2 fronts propagate up and down simultaneously ?

– NO !

i)

Crush-Up

.

b)

ζ

m(z)g

m(y)y

C

z0

ζ

z

.

zΔt

C

z0

m(y)g

Fc’< Fcif slower

s0

Fc

Fc

B

Fc

Fc

than free fall

.

.

Fc

z

s =λs0

Phase 1

.

yΔt

μy2

downward

H

c)

d)

C

y

A

A

y0 = z0

η

y

e)

C

r0

B’

λz0

r = λr0

B’

B

λH

B

λ(H-z0)

B

Phase 1. Crush-Down

Phase 2. Crush-Up

Diff. Eqs. of Crushing Front Propagation

Front decelerates if Fc(z) > gm(z)

I. Crush-Down Phase:

z(t)

force

z0

Jetting air

Comminution

Buckling

Resisting force

Intact

Criterion of Arrest (deceleration): Fc(z) > gm(z)

z0

y(t)

II. Crush-Up Phase:

Compacted

Inverse: If functions z(t), m(z), l(z) are known, the specific energy dissipation in collapse, Fc(y), can be determined

Compaction ratio:

fraction of mass ejected outside perimeter

Resistance and Mass Variation along Height

Variation of resisting force due to column buckling, Fb, (MN)

Variation of mass density,m(z),

(106 kg/m)

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

Wf = 2.4 GNm

fall arrested

2

1.5

Tower Top Coordinate (m)

1

free

fall

0.5

phase 1

0

phase 2

λ= 0.18 , μ= 7.7E5 kg/m , z0 = 80 m , h = 3.7 m

Time (s)

(for no comminution, no air)

Collapse for Different Compaction Ratios

transition between

phases 1 and 2

Tower Top Coordinate (m)

free

fall

λ= 0.4

0.3

0.18

Wf = 0.5 GNm ,

μ= 7.7E5 kg/m ,

z0 = 80 m , h = 3.7 m

0

Time (s)

(for no comminution, no air)

for impact 2 floors below top

mg < F0,heated

5

(≈ 2.5 E7 GNm)

free

20

Tower Top Coordinate (m)

fall

55

phase 1

phase 2

λ= 0.18 , h = 3.7 m

μ= (6.66+2.08Z)E5 kg/m

Wf = (0.86 + 0.27Z)0.5 GNm

Time (s)

Collapse for Various Altitudes of Impact

(for no comminution, no air)

Crush-up or Demolition for Different Constant Energy Dissipations

Wf = 11 GNm

fall arrested

6

parabolicend

free

5

Tower Top Coordinate (m)

fall

4

3

2

0.5

λ= 0.18 , μ= 7.7E5 kg/m , z0 = 416 m , h = 3.7 m

Time (s)

asymptotically

(for no comminution, no air)

Resisting force as a fraction of total

Impacted Floor Number

Impacted Floor Number

Resisting Force /Total Fc

64

81

48

5

F

81

25

F

96

110

110

101

Fb

Fb

Crush-down ends

Crush-down ends

Fs

South Tower

Fs

North Tower

Fb

Fb

Fa

Fs

Fa

Fs

Fa

Fa

Time (s)

Time (s)

Impacted Floor Number

Impacted Floor Number

5

F

81

64

96

81

48

110

25

F

101

110

Crush-down ends

Crush-down ends

South Tower

North Tower

Time (s)

Time (s)

Resisting force / Falling mass weight

Fc / m(z)g

81

48

5 F

96

81

64

25

F

Fm

Fm

Fc

Fc

South Tower

North Tower

Time (s)

Time (s)

External resisting force and resisting force due to mass accretion

Resisting force FcandFm(MN)

Impacted Floor Number

Impacted Floor Number

Lay Criticism of Struct. Engrg. Consensus

Mass Centroid

Like a

Tree?

No !

No !

Ft

1) 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

~4º tilt due to asymmetry of damage

~25º (South Tower)

non-accelerated rotation about vertically moving mass centroid

2) Collapse was a free fall ! ? Therefore the

steel columns must have been destroyed

beforehand — by planted explosives?

Video Record of Collapse of WTC Towers

North Tower

South Tower

t

H1

c

Tilting Profile of WTC South Tower

2

t

m

e

1

s

Video

-recorded

(South

Tower)

Initial tilt

North

East

From crush-down

differential eq.

Free fall

First 20m of fall

South Tower

(Top part large falling mass)

Time (s)

Comparison to Video Recorded Motion

(comminution and air ejection are irrelevant for first 2 or 3 seconds)

Tower Top Coordinate (m)

From crush-down

differential eq.

Note

uncertainty range

Free fall

First 30m of fall

North Tower

Time (s)

Not fitted but predicted!

Video analyzed by Greening

Collapse motions and durations compared

Seismic and video records rule out the free fall!

H

From seismic data:

crush-down T ≈12.59s ± 0.5s

417 m

North Tower

with pulverization

Free fall

Impact of compacted rubble layer on rock base of bathtub

with expelling air

impeded by single-story buckling only

12.81s

12.62s

8.08s

12.29s

Seismic rumble

Most likely time

from seismic record

0 m

T

-20 m

Tower Top Coordinate (m)

Calculated crush-down duration vs. seismic record

North Tower

South Tower

with air ejection

& comminution

with air ejection

& comminution

Seismic error

Seismic error

Free fall

Free fall

Calculation

error

Crush-down ends

Calculation

error

Crush-down ends

with buckling only

with buckling only

Free fall

Free fall

Ground Velocity (m/s)

a

a

b

c

b

c

0

4

8

12

16

0

8

4

12

Time (s)

Time (s)

3) Pulverizing as much as 50% of concrete

to0.01 to 0.13 mm required explosives!

NO. — only 10% of kinetic energy sufficed.

- 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)

Comminution (Fragmentation and Pulverization) of Concrete Slabs

Schuhmann's law:

D

mass of particles < D

total

particle size

Energy dissipated

= kinetic energy

lossΔK

16 mm

0.12 mm

density of particle size

1

Impact on ground

Impact slab story

intermediate story

Cumulative Mass of Particles (M / Mt)

k

0.012 mm

= Dmin

1

0.16mm = Dmin

Particle Size (mm)

0.1

1

10

0.01

Kinetic Energy Loss ΔK due to Slab Impact

Momentum balance:

Fragments

Compacted layer

m

K

v1

Comminuted slabs

= msconcrete

Kinetic energy loss:

K

v2

Kinetic energy to pulverize concrete

slabs & core walls

K

Total:

K

K

(energy conservation)

Concrete

fragments

Gravitational energy loss

Air

Buckling

Fragment size of concreteat crush front

Maximum and Minimum

Fragment Size at Crush Front (mm)

Impacted Floor Number

Impacted Floor Number

101

110

64

25

48

5

F

110

F

96

81

81

Crush-down ends

Crush-down ends

Dmax

Dmax

Dmin

Dmin

South Tower

North Tower

Time (s)

Time (s)

Comminution energy / Kinetic energy of falling mass

5

96

81

48

F

F

64

101

110

110

81

25

Crush-down ends

North Tower

Crush-down ends

Time (s)

South Tower

Time (s)

Impacted Floor Number

Impacted Floor Number

Wf / К

Impacted Floor Number

Impacted Floor Number

81

48

5

F

110

81

64

F

101

110

96

25

Crush-down ends

Crush-down ends

North Tower

South Tower

Time (s)

Time (s)

Dust mass (< 0.1 mm) / Slab mass

Md / Ms

Energy Variation (GJ)

South Tower

North Tower

Loss of gravitational potential

Loss of gravitational potential

Ground impact

Ground impact

Comminution

energy

Comminution

energy

Time (s)

Time (s)

Loss of gravitational potential vs. comminution energy

h

4) Booms During Collapse!

—hence, planted explosives?

a

Air squeezed out

of 1 story in 0.07 s

If air escapes story-by-story, its mean velocity at base is

va = 461 mph (0.6 Mach), but

locally can reach speed of sound

Air Jets

200 m of concrete dust or fragments

(va< 49.2 m/s, Fa < 0.24 Fc, pa< 0.3 atm)

5) Dust cloud expanded too rapidly?

Expected.

1 story: 3.69 x 64 x 64 m air volume

North Tower Collapse in Sequence

- Note:
- Dust-laden air jetting out
- Moment of impact cannot be detected visually

Can we see the motion through the dust ?

Except that below dust cloud the tower

was NOT breaking,nothing can be learned !

Moment of ground impact cannot be seen, but from seismic

record: Collapse duration =12.59 s (± 0.5 s of rumble)

Note

jets

of

dust-

laden

air

6) Pulverized concrete dust(0.01 to 0.12 mm)

deposited as far as 200 m away? — Logical.

7) Lower dust cloud margin = crush front?

— air would have to escape through a rocket nozzle!

9) Red hot molten steel seen on video (steel cutting) — perhaps just red flames?

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.

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.

10) “Fracture wave” allegedly propagated in a material

pre-damaged, e.g., by explosives, led to free-fall collapse

—unrealistic hypothesis, because:

- 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.

- from WTC — little- from demolitions — much

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/m3

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

Weak Joints, Precast Members

Floor slab

Reinforcing Bar

Hotel New World

Singapore 1986

Generalization of Progressive Collapse

1) 1D Translational-Rotational

25th floor

--- "Ronan Point" type

Angular momentum and shear not negligible

Gas exploded

on 18th floor

2) 3D Compaction Front

Propagation

— will require finite strain simulation

Gravity-Driven Progressive Collapse Triggered by Earthquake

MAIN

RESULTS

- 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

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.)

Download 466.pdf & 405.pdf from Bažant’s website:

www.civil.northwestern.edu/people/bazant.html