In this section, the procedure used for estimating the damping ratio is presented, as well as an analysis of the values obtained.

　

The hysteretic damping ratio estimation can be performed, by means of the loading-unloading cycles, using the following expression⁴:

where:

- η=Ediss/2πEsto

- Ediss is the energy dissipated in one cycle.

- Esto is the energy stored in one cycle.

　

The equivalent viscous damping ratio is given by:

The dissipated energy, Ediss , is the area related to one hysteresis cycle (area A, in Figure 15), while the energy stored in one cycle, Esto, is the area of the triangle whose sides b and h are indicated in this figure. Therefore, the equivalent viscous damping ratio, for each test, can be estimated through the following equation:

It has been developed a system for processing the data from the tests (total of 1425) and, consequently, obtaining the damping ratio values. This system consists of the signal visualisation, the use of filters for conditioning the signals⁵ , the separation of the signals and, finally, the damping ratio evaluation.

　

By applying the procedure described on the latter section, damping ratio values were estimated for all test cases. The parameters varied in the tests were: dynamic displacement amplitude, exciting frequency, axial loading and internal pressure. The analysis of the results lead to the following conclusions:

- the damping ratio of the riser specimen decreases as the displacement amplitude increases. For a better understanding about how the displacement amplitude affects the damping ratio, Figure 16 shows a plot of variation of the resultant of applied forces versus imposed displacement, for two cases of dynamic amplitude. It can be seen in this figure that the first stage strongly influences the dissipated energy. Given that the variation of the force applied in this stage is very similar for the two cases, the areas related to the dissipated energy are proportional to the dynamic displacement amplitude. Nevertheless, since the stiffnesses of the two cases are very similar, the area related to the stored energy varies with the square of the dynamic amplitude ratio. Thus, the equivalent viscous damping ratio tends to decrease as the displacement amplitude decreases;

- the damping ratio of the riser specimen tends to increase as the exciting frequency increases;

- the damping ratio of the riser specimen tends to decrease as the axial loading increases;

- the damping ratio of the riser specimen tends to increase as the internal pressure increases.

The damping ratio estimations presented a large variation band, and the maximum value obtained was approximately eleven times greater than the minimum one. Nevertheless, the values of this band were expected according to the technical literature^6,7.

In order to verify if the specimen behaviour has been altered after the set of tests, the first test case (no pressure and no axial loading) was repeated. By comparing the damping ratio values obtained through the initial and the final tests, no significant variation was detected.

　

Different types of tests were carried out, through a complex experimental setup, aiming to study the influence of some parameters (internal pressure and axial loading, especially)on the bending stiffness and the hysteretic damping ratio of a flexible riser specimen.

The results indicate that the bending stiffness of the riser specimen is not affected neither by the internal pressure nor by the axial loading application.

With respect to the damping ratio, a large band of values was obtained. It has been shown that these values are influenced by the parameters varied during the tests(displacement amplitude, exciting frequency, internal pressure and axial loading).