Slope stability formulae for rubble mound structures are usually developed for head-on conditions. Often, the effects of oblique waves are neglected, mainly because it is assumed that for oblique wave attack, the reduction in damage compared to perpendicular wave attack is insignificant. When the incident waves are oblique, the required armour size can be reduced compared to the perpendicular wave attack case. Therefore, it is important to consider the wave obliquity influence on slope stability formulae as a reduction factor. One of the most recent formulae for estimating the stability of rock-armoured slopes, referred to as Etemad-Shahidi et al. (2020), was proposed for perpendicular wave attack. The aim of this study is to develop a suitable wave obliquity reduction factor for the above-mentioned stability formula. To achieve this, first, laboratory experiment datasets from existing reliable studies were selected and analysed. Then, previously suggested reduction factors were evaluated and a suitable reduction factor for the mentioned stability formula were suggested. The suggested reduction factor includes the effect of wave obliquity and directional spreading explicitly. It is shown that the stability prediction is improved by using the wave obliquity reduction factor.

Stability formulae for armour layers of rubble mound breakwaters are generally developed for perpendicular wave attack and do not include effects of oblique waves. Waves usually attack breakwater obliquely as the sea wave is three dimensional. Several studies have been performed to investigate the effect of wave angle (beta) on the armor stability. Galland (1994), Yu et al. (2002), Wolters and Van Gent (2010) and van Gent (2014) performed laboratory experiments to consider effects of oblique waves on the stability of armour layers. They performed tests with long-crested and/or short-crested waves on rock and concrete armours. The aim of this study is to find an appropriate and compatible reduction factor for EBV stability formulae.

Toe design is an important task for coastal engineers as it ensures the stability of the main armor layer and prevents scour in front of the armor slope. Several laboratory experiments have been conducted to investigate the toe stability using different testing approaches, i.e. damage due to a single test condition and cumulative damage due to a number of conditions. In addition, the methods of measuring and reporting damage to the toe are not the same as some researchers have counted only stones that were washed away from the toe; while others counted all the stones that have been displaced. Several formulas with different levels of success have been developed based on these studies. However, the scatter in the results is large and effects of some parameters are disregarded. The aims of this study are (a) to briefly review the abovementioned differences and exiting formulas, and (b) to develop a and physically sound formula for common design conditions, which considers the effect of all governing parameters. To achieve this, first a comprehensive data base from existing reliable studies was collected. Then nondimensional parameters which capture effects of governing parameters such as wave height, wave period, water depth, toe depth, toe width and foreshore slopes were deployed to develop a stability formula using physical reasoning and regression analysis. The new formula outperforms existing formulae for toe stability. The coefficient of variation of the formula was also determined to be used for probabilistic design applications. Finally, some design hints are provided for practitioners.

Estimation of the required armor size is a major task in the design of coastal structures under wave loading such as breakwaters and revetments. Several semi-empirical formulas have been developed for this purpose. However, these formulas are often either limited to certain water depth conditions or do not incorporate the permeability of the structure in an appropriate way. The main objectives of this study are to (a) develop a unified physically sound formula for the estimation of the required rock size in all relevant water depth conditions and (b) to relate the effects of the permeability of the structure directly to physical parameters. To achieve these, first a comprehensive data base of deep and shallow water experiments within the design conditions was built. Then physical reasoning along with a robust data mining approach, i.e. M5 model tree, were used to develop formulas for armor stability. In the stability formula, wave characteristics such as the significant wave height and spectral energy mean period (T_m-1,0) are invoked while the permeability is incorporated using the ratio between the size of the core material and the armor stones. Accuracy metrics such as discrepancy ratio and scatter index indicated high performance of the model in different conditions. Finally, a probabilistic formula and some guidelines are provided for practicing engineers.