Experimental Methods in Earthquake Engineering - Dynamic Behavior of Concrete and Seismic Engineering - page 172

a linked to the load 1 r , which is a function of the 1

depends on the acceleration

d . Generally it implies an iterative resolution process. Such an
approach is used by the PSD method [SHI 91], even if it is not easy to implement in
so far as the structure itself is involved in the iterations. The implementation can be
considered as a modified Newton-Raphson method converging in a linear way at
best. Actually a sub-relaxation coefficient is introduced into the successive iterations
to force convergence in a monotonous way and without any parasitic discharge. The
stability of this scheme will be discussed later.

1

displacement

Nevertheless, it is possible to introduce an implicit D - Newmark scheme without
resorting to iterations. To do this we use an operator splitting (OS) method ([COM
97], [NAK 90]). This maintains the stability of the scheme, as it is implicit for the
elastic part of the response, but requires no iteration whatsoever, as it remains
explicit as far as the non-linear part is concerned. The OS method is based on the
following approximation of the reaction force

r : 1

n 1 n 1 n 1 n 1 I n 1 n 1

r ( d ) r ( d ) Kd d ( )

[5.4]

where K is a stiffness matrix which has to be higher than (or equal to) the tangent
K stiffness matrix of the system. That condition ensures the unconditional
stability of the scheme. In this case, the time step ' can be any time step, and thus
it is chosen according to the experiment carried out and not to verify any condition
of stability. The D-OS scheme appears as the natural extension to the implicit case of
the explicit scheme of the central finite differences.

As structures tend to soften, K can be chosen as the elastic stiffness matrix
K or the initial stiffness matrix of the structure. The experimental installation
obviously allows us to obtain a very good approximation of this matrix.

Eventually, we should note that the Newton-Raphson method of the iterative
scheme also uses a K matrix with the same properties in terms of stability as those
of the D -OS scheme. If the matrix is chosen properly, then the scheme will be
unconditionally stable as well.

5.2.2. Implementation at ELSA

The checking system implemented at ELSA was quite original compared with
previous analog implementations. As this system is entirely digital, it can easily be