Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 285

8.4.4. Using stochastic equations

After these basic reminders, it seems timely to say a few words about stochastic
equations, as they represent an important investigative tool as far as the calculation
of randomly stimulated structures is concerned. Although engineers seldom use
them, for simple cases, they can provide us with quite useful solutions for
approximate methods and the interpretation of digital simulations.

Consider a not necessarily linear N dof (degree of freedom) structure, stimulated
by a stationary Gaussian random noise (“no memory” random process). Its response
is a “one memory step” ( Markov ) 2N dimension vectorial process (the dofs and
their first derivatives). Such a process is completely characterized, as we have
already said, by its second order joint probability or, which is equivalent, by its
transition probability p 2 (y, t / y 0 , t 0 ), with t > t 0 , and y represents a 2N-component
vector.

We can show that such probability verifies a so-called Fokker-Planck equation,
with partial derivatives according to y and t, the terms of which can be explained
from those of the motion equations of the structure.

Let us take as a simple example a 1-dof dampened oscillator, with a stiffness
derived from a potential:

2 2

dx/dt 2İȦ dx/dt + d U x / dx = f t [8.21]

Initially the oscillator is at rest. We will take y 0 = 0 and t 0 = 0. Furthermore, we
will only be interested in its set response. Then the transition probability does not
depend on t time and it will be written as q (x, v), with v = dx/dt. The source process
F(t) is a S 0 level random noise (its correlation function will be S 0 G(W), with G(W) =
delta functional).

The Fokker-Planck equation confirmed by q (x, v) is:

22

ª º

v q/ x wwww / 2İȦv + dU/dx q S / 2 w w q/ v = 0 [8.22]

¬ ¼

0

the solution to which is: