The Expected Sliding Distance Model of Meandering Damage for Caisson Breakwater

In the past decades, many references have studied the mechanics of meandering damage for breakwater. However, up to the present time, no model has been proposed to compute the consequence of this kind of damage quantitatively. So, this paper tries to solve this problem by combining the expected sliding distance technique and the diffraction model. 1 Preface The meandering damage of breakwater is one of horizontal damages, which usually occurs near the tip of breakwater and appears like a snake along the axial line of breakwater. Takahashi (2000) discussed the causes of this phenomenon and pointed out that the periodic variation of wave height in front of breakwater due to the interaction of wave diffraction and wave reflection is the internal reason for it. Takahashi also listed some other influencing factors, such as different longitudinal sectional geometries of breakwater and different incident wave heights along the wave crest line. Fig-1 is an example of wave height distribution in front of breakwater (Hitachi, ). From Fig-1, it is seen that wave height shows a regular and weakening fluctuation. At the position away from the breakwater tip, the wave height tends to double of incoming wave height. The maximum wave height turns up near the breakwater tip, which is the main reason of the head area of breakwater failing easily. Expected sliding computation distance model was proposed by Shimosako and Takahashi (1998). They called it “deformation-based reliability design method”, which integrated the Monte Carlo technique and the stochastic behaviors of design factors. Goda (2000) introduced the economic optimization model into this model and put forward the conception of optimal wave height. Goda (2001) discussed the effect of extreme wave statistics on expected sliding distance and suggested a new spread parameter to characterize the extreme distribution functions. Kim and Takayama (2003) adopted the Doubly-Truncated Normal Distribution to describe the uncertainty of design factors and discussed the effect of the seed of random variable on the expected sliding distance in the computation process. Kim ea al. (2005) used the model to compute sliding distance due to typhoon Tokage considering caisson tilting. Figure1 The variation of wave height along a breakwater 2 The Expected Sliding Distance Model for Meandering Damage Generally, when waves are incident to a breakwater, a standing wave system is formed in its front. If the breakwater reflects the incident waves completely, the standing wave height at the front wall becomes twice the incident height. However, in actual sea condition, the standing wave height along the axial line of breakwater undulates owing to the two-dimensional effect induced by the interaction of wave diffraction and reflection. So, before establishing the meandering damage model, it is necessary to obtain wave height in the front and at the rear of breakwater. In this paper, the diffraction model proposed by Penney and Price (1952) is used. For the sake of simplicity, only the situation of semi-infinite caisson breakwater of perfect wave reflection is considered in this paper. For other cases, such as island breakwater, the computation process is the same. In the classical paper by Penney and Price (1952), the distribution function of wave height around breakwater ) , ( y x F was presented as (based on the small-amplitude wave theory): ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + + = ∫ ∫ ∞ − ∞ − − − − ξ ξ π π ' 2 2 ) 2 / ( ) 2 / ( 2 1 ) , ( du e e du e e i y x F u i iky u i iky (1) Where, ξ , ' ξ and r are defined by: ) ( 4 y r L − ± = ξ , ) ( 4 ' y r L + ± = ξ , 2 2 y x r + = (2) L is the wave length and k is the wave number. The signs of ξ and ' ξ are taken depend on the quadrant in which the solution is being applied, see Fig-2. As ) , ( y x F is complex, it contains both wave amplitude and phase information. The above case is for the situation of normal wave incidence on a semi-infinite breakwater and for the case of obliquely incident wave on a semi-infinite breakwater, Shore Protection Manual (1977) presents the diffraction diagrams for many wave directions. After obtaining the wave height in the front and at the rear of a semi-infinite caisson breakwater, the consequence induced by meandering damage can be figure out by the expected sliding distance model. Fig-3 is the whole computation flow: