A study of the collapse of the world trade center.

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

5. 1 Review of Elementary Mechanics of Collapse

6. Momentum of Boeing 767 ≈ 180 tons × 550 km/h Momentum of equivalent mass of the interacting upper half of the tower ≈ 250, 000 tons × v 0 Initial velocity of upper half: v 0 ≈ 0.7 km/h (0.4 mph) Assuming first vibration period T 1 = 10 s: Maximum Deflection = v 0 T / 2  ≈ 40 cm 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) (about 40% of max.hurricane effect)

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

9. Toppling like a tree?

10. (The horizontal reaction at pivot) > 10.3 × (Plastic shear capacity of a floor)  Possible ? mg F H 1 m x  M P F 1 M P F 1 h 1 Why Didn't the Upper Part Fall Like a Tree, Pivoting About Base ? a) b) c) d) e) f) F P

11. South tower impacted eccentrically

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

14. Can Plastic Deformation Dissipate the Kinetic Energy of Vertical Impact of Upper Part? Only <12% of kinetic energy was dissipated by plasticity in 1 st 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:  1  2  3

15. Plastic Buckling F c ≥ F s … can propagate dynamically F c < F s … cannot h L=2L ef P 1 P 1  u L L/2  P 1 M P M P P 1 Plastic buckling W f F c F s Service load Load F Axial Shortening u 0 0 0.5 h h Yield limit  h F 0 0 0 0.04 h F 0 Elastic Yielding Plastic buckling Expanded scale Case of single floor buckling F Shanley bifurcation inevitable!

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

18. mg F 0 F c 0 Crushing Resistance F(u) W c u λ h Δ F d Δ F a h Crushing of Columns of One Story Floor displacement, u Crushing force, F u c u 0 u f ü = g – F ( u ) / m ( z ) K < W c Internal energy : φ (u) = F ( u' )d u' u 0 ∫ W b   Maxwell Line Dynamic Snapthrough Collapse arrest criterion: Kin. energy One-story equation of motion: : Rehardening Initial condition: v  velocity of impacting block Lumped Mass Lower F c for multi-floor buckling!  1  2  3

19. t z c t z c v 1 v 2 > v 1 v g-F c /m 1 h a) Front accelerates h 0 F 0 F c mg F(z) h F 0 mg F c v 1 Crushing force, F b) Front decelerates c) Collapse arrested v v 2 < v 1 time Floor velocity, v u h for F c v 1 v u u g-F c /m 1 v u v 2 >v 1 v h v 1 for F c 0 0 0 0 0 0 h u v 0 v 1 v 1 W 1 = K mg F 0 z c F c 0 Real Crushing Resistance F(z) W 1 = W 2 u λ h Δ F d Δ F a W 1 = W 2 Δ F d Δ F a λ h Δ F d Deceleration Acceleration Deceleration Acceleration Deceleration λ h λ h λ h λ h Displacement t t Time t

20. F c a) Single-story plastic buckling L = h F c F c Floor n n-1 n-2 n-3 n-4 W c W c F peak F c F peak F c F s Service load F c F peak b) Two-story plastic buckling L = 2h c) Two-story fracture buckling L = 2h F peak = min ( F yielding , F buckling ) Internal energy (adiabatic) potential : W = ∫ F ( z )d z Compaction Ratio, λ , at Front of Progressive Collapse λ h 2 λ h Crushing Force, F Distance from tower top, z Total potential = Π gravity - W Mean Energy Dissipation by Column Crushing, F c , and energy- equivalent snapthrough = mean crushing force h h

21. 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

22. 1D Continuum Model for Crushing Front Propagation C A z 0 s 0 z H B B y 0 = z 0 C y B C B’ y η ζ r 0 B’ B z 0 C Phase 1. Crush-Down Phase 2. Crush-Up F c F c ’< F c if slower than free fall Phase 1 downward Δ t m (z)g F c F c F c F c 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 = λ s 0 λ ( H-z 0 ) A r = λ r 0 λ z 0 λ H λ = compaction ratio = Rubble volume within perimeter Tower volume z Δ t . m ( z ) v . m ( y ) y . y Δ t . μ y 2 . z . ζ

23. Diff. Eqs. of Crushing Front Propagation I. Crush-Down Phase: II. Crush-Up Phase: fraction of mass ejected outside perimeter Inverse: If functions z ( t ), m ( z ),  ( z ) are known, the specific energy dissipation in collapse, F c ( y ), can be determined z ( t ) y ( t ) Intact Compacted Compaction ratio: z 0 z 0 Criterion of Arrest (deceleration): F c ( z ) > gm ( z ) Buckling Comminution Jetting air Resisting force force Front decelerates if F c ( z ) > gm ( z )

24. Variation of resisting force due to column buckling, F b, (MN) Variation of mass density, m(z), (10 6 kg/m) Resistance and Mass Variation along Height

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 (for no comminution, no air) Time (s) Tower Top Coordinate (m) W f = 2.4 GNm 2 1.5 1 0.5 0 free phase 1 phase 2 fall λ = 0.18 , μ = 7.7E5 kg/m , z 0 = 80 m , h = 3.7 m fall arrested

27. Collapse for Different Compaction Ratios (for no comminution, no air) Tower Top Coordinate (m) Time (s) λ = 0.4 0.3 0.18 0 transition between phases 1 and 2 W f = 0.5 GNm , μ = 7.7E5 kg/m , z 0 = 80 m , h = 3.7 m free fall

28. Collapse for Various Altitudes of Impact (for no comminution, no air) for impact 2 floors below top 5 20 55 Time (s) Tower Top Coordinate (m) (≈ 2.5 E7 GNm) mg < F 0, heated free fall phase 1 phase 2 λ = 0.18 , h = 3.7 m μ = (6.66+2.08Z)E5 kg/m W f = (0.86 + 0.27Z)0.5 GNm

29. Crush-up or Demolition for Different Constant Energy Dissipations asymptotically (for no comminution, no air) Time (s) Tower Top Coordinate (m) W f = 11 GNm 6 5 4 3 2 0.5 parabolic end free fall λ = 0.18 , μ = 7.7E5 kg/m , z 0 = 416 m , h = 3.7 m fall arrested

30. Resisting force as a fraction of total Resisting Force /Total F c F b F b F s F a F s F a F b F s F a F b F s F a 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

31. F c / m(z)g Resisting force / Falling mass weight 96 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

32. External resisting force and resisting force due to mass accretion Resisting force F c and F m (MN) Impacted Floor Number Impacted Floor Number Time (s) F m F c North Tower 96 81 48 5 F Time (s) F m F c South Tower 81 64 25 F

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

35. South Tower North 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?

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

37. 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 Tower Top Coordinate (m) First 30m of fall North Tower Free fall From crush-down differential eq. Time (s) Note uncertainty range South Tower (Top part   large falling mass) First 20m of fall From crush-down differential eq. Time (s) Free fall

38. Collapse motions and durations compared 417 m H 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 ≈ 12.59 s ± 0.5s -20 m 0 m Seismic rumble Impact of compacted rubble layer on rock base of bathtub Seismic and video records rule out the free fall! North Tower

39. Calculated crush-down duration vs. seismic record Tower Top Coordinate (m) Seismic error a b c 0 4 8 12 Time (s) Free fall with air ejection & comminution Crush-down ends with buckling only South Tower Calculation error 0 4 8 12 16 a b c Seismic error Time (s) Calculation error North Tower with air ejection & comminution Free fall Crush-down ends with buckling only Ground Velocity (  m/s) Free fall Free fall

41. Comminution (Fragmentation and Pulverization) of Concrete Slabs Schuhmann's law: D total particle size mass of particles < D Energy dissipated = kinetic energy loss Δ K density of particle size Cumulative Mass of Particles ( M / M t ) 1 k 0.16mm = D min Impact slab story intermediate story Impact on ground 0.012 mm = D min 0.01 0.1 1 10 1 0.12 mm Particle Size (mm) 16 mm

42. Kinetic Energy Loss Δ K due to Slab Impact Momentum balance: Fragments Kinetic energy loss: (energy conservation) Total: Concrete fragments Buckling Gravitational energy loss m v 1 v 2 Compacted layer Comminuted slabs Kinetic energy to pulverize concrete slabs & core walls = m s concrete Air  K K K K K

43. Fragment size of concrete at crush front Maximum and Minimum Fragment Size at Crush Front (mm) Time (s) Time (s) North Tower D min D max 96 81 48 5 F 110 Impacted Floor Number 81 64 25 F 101 110 Impacted Floor Number D min D max South Tower Crush-down ends Crush-down ends

44. W f / К Comminution energy / Kinetic energy of falling mass Impacted Floor Number Impacted Floor Number Crush-down ends Time (s) North Tower 96 81 48 5 F 110 Crush-down ends Time (s) South Tower 81 64 25 F 110 101

45. Dust mass (< 0.1 mm) / Slab mass M d / M s Time (s) Time (s) 96 81 48 5 F 110 81 64 25 F 101 110 Impacted Floor Number Impacted Floor Number Crush-down ends Crush-down ends North Tower South Tower

46. Loss of gravitational potential vs. comminution energy Energy Variation (GJ) Comminution energy Ground impact Ground impact Comminution energy Loss of gravitational potential Loss of gravitational potential North Tower South Tower Time (s) Time (s)

47. 4) Booms During Collapse! — hence, planted explosives? If air escapes story-by-story, its mean velocity at base is v a = 461 mph (0.6 Mach) , but locally can reach speed of sound 5) Dust cloud expanded too rapidly? Expected. ( v a < 49.2 m/s, F a < 0.24 F c ,  p a < 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 out of 1 story in 0.07 s a h

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. 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.

53. 4 How the findings can be exploited by tracking demolitions - from WTC — little - from demolitions — much

54. 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

55. Collapse of 2000 Commonwealth Avenue in Boston under construction, 1971 (4 people killed) The collapse was initiated by slab punching)

56. Murrah Federal Building in Oklahoma City, 1995 (168 killed)

57. Ronan Point Collapse U.K. 1968 Reinforcing Bar Floor slab Weak Joints, Precast Members

58. Hotel New World Singapore 1986

59. Generalization of Progressive Collapse 1) 1D Translational-Rotational --- &quot;Ronan Point&quot; type Angular momentum and shear not negligible 2) 3D Compaction Front Propagation Gas exploded on 18 th floor — will require finite strain simulation 25th floor

60. Gravity - Driven Progressive Collapse Triggered by Earthquake