Abstract Within the context of response to slamming, a fast dy- namic approach was used to study the response of a hull girder beam to transverse impulsive load in detail, with consideration of both flexural wave propagation and transverse vibrations phenomena。 The load was ideal- ized but the influence of its main characteristic parame- ters on the response was examined。 In the same way, the complexity of the beam characteristics was progres- sively increased for more representativeness。 Finally, a comparison with a complete 3D model was performed。81337

Keywords Slamming; impulse; Timoshenko; beam; propagation; flexural waves; vibration。

Nomenclature

A : area of the cross-section

c : celerity of waves (phase velocity) E : Young modulus

G : shear modulus k : wave number

: shear factor (adjustment coefficient)

: mass density per unit volume

: circular frequency

Introduction

Slamming impacts lead to an increase of the dominant load effects, especially vertical bending moment and shear force。 Classification Societies and shipyards need to master these additional contributions, and then to agree upon a value of the dominant load effect (with corresponding probability) to consider for the dimen- sioning of the structure。

Some questions come up from observation of damages due to slamming or measurements at sea。 In   particular,

these questions are explained by the lack of knowledge about the inertial effects during the first times of the response, more precisely for durations not larger than the first period of the first vertical flexural vibration mode。 Some of these questions are : do fast dynamic effects such as propagation of possible flexural waves contribute to the response ? What are the phenomena explaining why the maximum bending moment could not occur at midship ?

The rigorous study of the response of a ship to a slam- ming impact requires a good estimation of such spatio- temporal impulsive loads on one hand and of the struc- tural response to these loads on the other hand。 The present paper is focused on this last aspect。

Firstly, beam finite elements models (based on Ti- moshenko formulation) have been used to study the response of the hull girder to transverse impulses, par- ticularly the influence of the load parameters (duration, impulse, rate of loading, etc。)。

Slamming loads are sufficiently short and energetic to excite modal response of global structures like hull girders (or even local structures like stiffeners)。 But other phenomena may occur as propagating flexural and shear waves, characterized by higher frequencies,  and so requiring sufficient input energy at these frequency levels。 These last ones are precised in the paper and linked to the characteristics of the load time history。

In reality, high frequency phenomena are damped the first。 Nevertheless, damping doesn’t affect very much the first periods。 So, it is worth taking an interest to this stage。

In the absolute, it would be interesting to get expected realistic results from complete 3D calculations (load and structural response)。 This is investigated in another study。 However, the more the simulation likes to be realistic, the more it is difficult to make allowances for the different phenomena occuring as they are combined。 The understanding often comes from the decoupling of the phenomena and the study of simplified models。 This was also the guideline of the work presented in this paper。

Theoretical Response of a Beam to a Transverse Shock

Propagation of harmonic flexural waves in a beam

We consider the complete Timoshenko theory of beams。 In determining the response of beams to sharp tran- sients, it is almost necessary to use this theory in con- trast to the Bernoulli-Euler theory, that predicts unreal- istic infinite phase velocities。 As a matter of fact, the Timoshenko theory adds the correction for rotary iner- tia, which alone enables to predict finite propagation velocities。 It also includes the effect of shear (additional contribution to slope of the cross-section due to shearing effects and making plane sections not remaining neces- sary plane), for better accordance with exact theory。