Civil Engineering Reference
In-Depth Information
one-dimensional Riera-Model can only be seen as a rough approximation although
this approximation should give a result on the safe side. In the examinations after
September 11, 2001 the mathematical models were refined, in particular to take into
account the time dependent spatial distribution on the target. In addition to modified
Riera-Models complex computer simulations such as those used in the automobile
industry to analyze crashes were used. While the complex analyses are verified by
crash tests in the automobile industry such analyses are almost impossible to realize
in the case of an airplane crash and have therefore not taken place.
Within the frame of the examinations to the consequences of an impact of large
commercial aircraft, an enhanced mathematical model for the realistic compilation
of the existing conditions was also applied by the author. In this model, the aircraft
is broken down into discrete masses which are spatial distributed according to the
geometry. In particular the masses are assumed to be distributed along the fuselage
and main wings. The individual masses are connected by springs that are able to
transmit axial forces, bending moments and shear forces using a special non-linear
approach to the spring characteristics.
Figure
14.3
shows the basic layout of the mathematical model. The airplane is
treated as a plane model in the x-y plane. The third axis can be neglected for the
given problem. A row of nodes with assigned masses
M
i
are assumed in the flight
direction in the axis of the fuselage (x-axis). These are connected via axial force
springs
R
i
which describe the force-deformation relationship during the deforma-
tion. At the same time the masses are connected by special elements that describe
the flexural behaviour of the fuselage in the x-y-axis. The two main wings are
coupled to the fuselage as rigid elements at a certain angle to the longitudinal axis.
Again different rows of individual masses with axial force springs are attached to
these in the direction of flight. The jet engines are also represented by a group of
springs and masses which are connected to the wings through special coupling
springs.
The axial force springs used in the simulation have in principal a characteristic
curve as shown in Fig.
14.4
. Starting with an initial stiffness
k
0
a maximum force
R
B
can be carried in relation to the displacement
u.
This corresponds to the bursting
load of the fuselage or the wing element. Beginning at a value
u
V
a hardening with a
parabolic gradient takes place until a specified force
R
C
is reached at
u
C
. The
unloading and reloading takes place with a stiffness of
k
E
. These parameters must
be determined for each section
i
of the aircraft model. Once deformation of the
normal force spring reaches the value
u
C
, the masses are considered to be rigidly
coupled. At this point in time a balancing of momentum of the two concerned
masses take place by calculating a new resulting speed.
The method of calculation consists of an integration in time of a coupled spring-
mass system. In principle it is based on the finite element method. All masses are
assigned a starting speed as an initial condition. This corresponds to the approach
speed being considered. Upon impact with a rigid target this speed is gradually
decreased though the effect of the springs. To determine the impact load-time
function the forces in the respectively first facing spring is evaluated (cf. Fig.
14.3
)
and the spring force is plotted over time. Specifically these are the springs in the
