Assuming harmonic waves solutions in the governing equations of an infinite Timoshenko beam brings the dispersion equation c(k) (Graff, 1975), and the c() relation by using the identity =kc subsequently。 This result is presented in Fig。 1。 It shows that the propaga- tion of such waves is dispersive : a harmonic wave of frequency can propagate only at a specific velocity c()。 Another important result is the existence of an upper  limit  celerity  clim,  contained  between  the   two

nal pulse will propagate with its own inpidual velocity (see Fig。 1)。

By definition, a shock expresses a load discontinuity。 In the slamming case, it is due to the sudden contact with water。 These loads can look like strong impulses and are characterized by high amplitude levels and short global durations (lower than 0。5 s)。 In the frequency domain, this implies non negligible energy at high frequencies。

Response to this kind of load can be characterized  by the propagation of internal force waves。 In practice, these ones may dissipate quickly due to 3D effects, structural, material and radiation dampings。 Damping doesn’t affect the first periods of a signal much。 But it is referred to a standing wave。 So, a further  analysis should be devoted to the effects of damping on a single harmonic propagating wave (which for the circumstance could be seen as the superposition of many standing waves)。 Meanwhile, no damping was considered in this study。

Simulation of the Response of a Semi-infinite Homogeneous Beam to a Transverse Shock

Calculations have been performed with LS-DYNA, a general finite elements software, based on an explicit time scheme (finite difference method), suited for    fast

cL   (celerity  of  longitudinal waves)

dynamic phenomena, like wave propagations,    because

(celerity of pure shear waves) :

of their discontinuous nature。

Main characteristics are : cross-section area = 9。1 m2, moment of inertia of cross-section = 1000 m4, material : steel (= 7800 kg/m3, E = 210 GPa)。

Due to discretization, the response frequencies are lim- ited。 In bending for example, the maximum    frequency

It  is  linked  to  the    frequency

reachable is the maximum of natural bending frequency

which is characteristic of a length of shear wave close to the beam thickness。 Beyond lim, the motion is charac- terized by pure shear waves propagating with the celer- ity cT。

of a single element (free-free ends) over the entire model。 According to the Euler-Bernoulli theory, this frequency expresses :

Fig。 1 : Celerity of flexural waves with shearing and rotary inertia (a), with rotary inertia alone (b)

(Guyader, 2002)

Response to a transverse pulse

Suppose we consider a pulse shape at a given time t=t0 to be a Fourier superposition of harmonic waves。 Then, as time advances, each Fourier component of the   origi-

where L designates the length of the element here。 For the present model, this leads to 24000 Hz approxi- mately。 Rigorously, taking into account of shear and rotary inertia effects make this value decrease (Guyader, 2002), as presented subsequently for shear。 In this last case, the maximum frequency should rather be around 220 Hz for a shear factor equal to 0。063 (value calcu- lated with a specific tool for a characteristic amidship cross-section geometry of a passenger ship) and 860 Hz for a unit shear factor by comparison。

The first test was used to verify if the phenomena previ- ously described were really accounted for。 A semi- infinite homogeneous elastic beam was loaded by a transverse external pressure applied uniformly on a certain length from the end point。 The time  history shape was a triangle of duration T=10 ms, which FFT is called back in Fig。 2。 It has to be emphasized that most of the energy is contained in the frequency range from 0 to 2/T。