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

1. 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

2. Collaborators: 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)

3. Structural • System • framed • tube

4. 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.

5. 1Review of ElementaryMechanics of Collapse

6. 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)

7. 13% of columns were severed on impact, some more deflected

8. 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

9. Toppling like a tree?

10. 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)

11. South tower impacted eccentrically

12. Plastic Shearing of Floor Caused by Tilting (Mainly South Tower) a b c d e

13. 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!

14. 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

15. 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

16. 2Gravity-Driven Propagation of Crushing Front in Progressive Collapse

17. 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

18. 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!

19. 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

20. 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)

21. 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

22. 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

23. 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

24. 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)

25. Energy Potential at Variable Mass Crush-Down Crush-Up Note: Solution by quadratures is possible for constant average properties, no comminution, no air ejection

26. 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)

27. 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)

28. 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)

29. 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)

30. 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)

31. 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

32. 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

33. 3 Critics Outside Structural Engineering Community:Why Are They Wrong?

34. 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

35. 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

36. t  H1 c Tilting Profile of WTC South Tower 2 t m e 1 s Video -recorded (South Tower) Initial tilt North East

37. 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

38. 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

39. 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)

40. 3) Pulverizing as much as 50% of concrete to0.01 to 0.13 mm required explosives! NO. — only 10% of kinetic energy sufficed. How much explosivewould 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)

41. 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

42. 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

43. 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)

44. 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

45. 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

46. 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

47. 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

48. 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 !

49. 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

50. 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?