Month: September 2011

World Trade Center

We will never forget the terrorist attack that took place in New York City on September 11th, 2001.

The events were tragic and at the same time remarkable in their physical aspects.

Scientists and engineers simulated jet colliding with World Trade Center. Their simulation could be used to understand which elements in the building’s structural core were affected.Most of the structural damage from such aircraft collisions was caused by the mass of the fluid on the craft.  The mystery of the WTC collapse is to investigate the mechanics of the impact and collapse events. This involves following the energy transfer processes from the initial shock of the aircraft collisions to the final collapse. There are 3 basic principles of mechanics that are invoked:

  • conservation of energy
  • conservation of linear momentum
  • principle of virtual work
The conservation of energy says that the initial kinetic energy (E_k ) is converted into the energy dissipated by plastic deformation and fracture: the airframe (E_p ), the external column ($latex E_e $), the floor (E_f ) and the core structure (E_c ).
E_k = E_p + E_e + E_f + E_c

  • E_k is the kinetic energy of the airplane;
  • E_p is the energy dissipated by the crushing and breakup of the aircraft;
  • E_e is the energy required to cut through the exterior columns;
  • E_f is the energy dissipated by the floors;
  • E_c is the energy absorbed by the core column destruction.

Some energy is lost by friction and is converted into the elastic vibration of the building.

The main geometric dimensions of a Boeing 767-200ER are

Length: lf = 48.51m
Wing span: lw = 47.57m Fuselagediameter: D=5.03m
Max. take -off mass: M = 179,330kg

Given that the maximum take-off mass of the airplane is 179,330 kg, that the airplane wasn’t full of passengers (only 65 of 216 max. capacity), the mass of the airplane is estimated to be

M = 127 tons

It’s assumed that impact velocities were 240 m/s and 192 m/s for the South and North Tower respectively. Hence, the initial kinetic energy of the airplane is:

E South=1/2 MV^2 =3658MJ

The average estimated impact velocity of the airplane hitting the North Tower was V0 = 192 m /s. The corresponding kinetic energy was

ENorth = 2540MJ

The points of impact were centered at floor 96 for WTC 1 and floor 81 for WTC 2. For the general case of n floors collapsing we define a collapsing mass M_c :

M_c = nm_f

where m_f is the mass of one WTC floor.

Plastic energy is dissipated thought two mechanisms:

1) the first mechanism is plastic deformation through the tensile tearing;

2) the second mechanism is associated with the momentum transfer.

The law of conservation of momentum states that:

m1 v1 = m2 v2

We will use this law for the inelastic collision where the colliding masses essentially merge into a single mass that continues to descend. For the case of 1 floor collapsing onto another identical floor:

m1 =mf

m2 =2mf

Thus:  mf v1 = 2mf v2

-> v2 = 1⁄2 v1

The momentum transfer between the airframe and the 1st barrier of external columns was responsible for most of the energy dissipated in this phase: E_e =26MJ.

Loss of kinetic energy to plow the airframe through the model structure was E_f = 1221MJ for North Tower and E_f = 1040MJ for the South Tower.

A source of inaccuracy in the determination of energy dissipated in failing the aircraft is presented by the impact orientation.  The lower bound on the energy expanded to distressing the aircraft was E_p = 962MJ .

The energy to be dissipated by the core structure is E_c =141MJ for the North Tower and E_c = 1630MJ for the South.