nite) to the maximum of the load does not necessarily modify the response。

Fig。 13 : FFT of input triangle-shape time histories (10 ms

duration), for several rise times

Role of the shear area

For complex cross-sections (like ship ones), the shear area might be difficult to calculate。 In order to study its influence on the response, several shear areas were considered (see Fig。 14)。 The fifth value corresponds to the shear factor of a rectangular cross-section and the first one was calculated with a specific tool from the real geometry of the amidship cross-section。

The shear force is proportional to the slope (in space) of bending moment :文献综述

Fig。 11 : Time history of the bending moment for the 10 ms and 15 ms load durations (same rise time 5 ms)

with m the mass of a section element and the total rotation of the cross-section。 Now, a reduction of the shear area tends to make the shear force decrease, and so the slope of bending moment in the same way。 Fi- nally, this tends to make the bending moment globally decrease (Fig。 14) for a given initial state。

In parallel, a reduction of the shear area makes the bending eigen frequencies increase, as expressed by the eigen circular frequency with transverse shear :

Fig。 12 : FFT of the triangular loads of 10 and 15 ms

durations

Influence of the load rate

For 10 ms duration and identical impulse of a triangle- shape time history, several values of the rise time were tested。 The time histories of the bending moment are not shown (see Fig。 9) because the rise time has very little effect。 Indeed, the Fig。 13 points out that every FFT are similar on a relatively wide frequency range, covering most of the frequencies contributing the more to the response (to 100 Hz approximately)。 Consequently, contrary to intuition, a very sharp rise (eventually    infi-

where the expression of c(n) depends on the boundary conditions。 Consequently, considering especially the frequencies lower than 2/T of a triangle-shape time history of load (see Fig。 2), the associated amplitudes are expected to be lower。 This would particularly affect the highest frequencies of this range。

The two previously described effects are opposite, what may explain the two observed tendencies concerning amplitudes on Fig。 14 : first, the maximum increases (for shear areas from 9。1 m2 to 5 m2), then decreases。 Moreover, the decreasing of the high frequencies con- tribution with decreasing shear area is quite clear。 For the lowest shear area, the effect is very significant, what prompts to have a value as representative as possible。 Furthermore, the points where the maximum occurs seem to move away (from the loaded zone) with de-

creasing  shear  factor。  But,  as  time  histories  are very

different for extreme shear factors (near 0 and near 1), the respective locations of the maximums are not really comparable。

Simulation of the Response of a Finite Beam of the Hull Girder Type (Homogeneous by Sec- tions)

Fig。 14 : Time history of the bending moment for several values of the shear factor at the element wherethe first peak is maximum

Influence of the boundary type (clamped and free)

The nature of the boundary conditions has an effect on both vibrations and wave propagations (when such waves meet these boundaries)。 However, in this last case, this depends on the distance between boundaries and the investigated point。 For example, for the re- sponses of Fig。 7 (clamped-free ends boundary condi- tions), this point is sufficiently far from both ends to present a first peak of bending moment (Gc group) non affected by the second (Gf group) or the reflection of the first group itself。