The static stability of stones on mild slopes under wave attack is investigated by this research. First it was checked whether there are any numerical models that could be used to describe the stability of mild slopes. As such, there is no existing model at this moment that can accurately model the stability of mild slopes. To model this correctly, the model has to be able to model plunging and spilling waves, include bed sediment transport and to model the velocity and acceleration due to a breaking wave over multiple layers in depth. The existing models do comply mostly with one of these specifications. To get accurate results, the stability should be modelled with all of the specifications and not one model complied with this. Therefore, it was decided to investigate the stability of stones on mild slopes under wave attack with physical modelling tests at the research institute Deltares. The physical tests were executed with a constant slope angle of tan α = 1:10 and a constant nominal median stone diameter. The wave steepness (that determines the type of breaking for a constant slope angle), the significant wave height, the number of waves and the layer thickness are varied in between the tests to determine the influences of each parameter on the stability. Of each test an erosion profile is made by photogrammetry from which the profile-related damage parameters can be determined. In the bed strips with coloured stones are laid down to follow the transport of the stones. According to these erosion profiles three damage parameters were determined. The damage level, S, as used by Van der Meer (1988), the erosion depth, de, and the damage depth, E3. The damage depth, E3, is used to describe the damage for mild slopes. The range of the damage depth for start of damage to failure is from 0.5 to 2.3. These values are only applicable for a layer thickness of 2.5dn50 and a slope angle of tan α = 1:10. The development of the damage depth is investigated for multiple parameters. The breaker type gives a change in development of the damage depth. For a wave condition with plunging breakers the damage depth development is higher per increasing wave height than for a wave condition that includes both plunging and spilling breakers. For the transition zone from plunging to spilling breakers the damage depth increase is very small if the wave height increases. The damage depth is influenced by the number of waves, because after about 11,000 waves the development of the damage depth is still linearly increasing with the number of waves. The damage depth was not stabilized towards a maximum value after this number of waves. It can be concluded that the transport of the stones is mainly all below the still water level. This also followed from the damage zones. These indicated that the total area of the damage is in between the -2Hs and SWL. The maximum area location is around -0.8Hs w.r.t. the SWL. The transport is mostly upslope, which is attributed to the mechanism whereby wave forces are strong in upslope direction. The location where the most stones are picked up depend on the breaker type, the wave height and the layer thickness. Plunging breakers cause damage more downslope than spilling breakers. Plunging breakers cause more local damage and have a much smaller runup than the spilling breakers. Per increasing wave height, the location of damage is shifted more downslope due to the increase in breaker depth.

The static stability of stones on mild slopes under wave attack is investigated in this research. The first part of the research is focused on reproducing the physical scale model tests regarding profile change of Kramer (2016) numerically with the model XBeach-G. The erosion profiles modelled with the bed-load transport formulas of Nielsen (2006) and Van Rijn (2007) in XBeach-G do not match the erosion profiles of the profile change experiments of Kramer (2016). The bed-load transport formulas of Nielsen (2006) and Van Rijn (2007) are not able to model the sediment transport in XBeach-G accurately. Furthermore, XBeach-G cannot determine the velocity and acceleration near the bed, because the model solves the flow due to currents and waves for a single layer. Therefore, it can be concluded that XBeach-G should not be used to describe static stability of stones on mild slopes under wave attack. For the application of dynamically stable structures (which is not investigated in this research), XBeach-G functions satisfactorily (Postma, 2016). For further research, a model that solves the hydrodynamics for multiple layers should be applied. In this way, the hydrodynamics near the bed can be used to describe the static stability of stones. The aim of the second part of the research is to develop a design method that describes the static stability of stones on mild slopes under wave attack. The basis of this design method is the initiation of motion of a stone and the hydrodynamic forces that initiate this movement. The hydrodynamic forces and corresponding mobility parameters are determined with the velocity and the acceleration near the bottom. Using Bubble Image Velocimetry (BIV), the velocity and the acceleration are derived from the videos of the BIV experiments of Kramer (2016) with regular waves breaking on a slope. It is found from the results of the BIV analysis that the effective, adapted Shields parameter θ’McCall can be used to describe movements of stones on mild slopes under wave attack. This mobility parameter has been determined with the bed shear stress of McCall (2015), which added an inertia term to include the influence of accelerations. For initiation of motion of stones, it appears that the stability parameter θcr could be a value of 0.024 (in case no slope correction factor has been applied). To substantiate a design method that describes the static stability of stones on mild slopes under wave attack, the value of 0.024 could be used to define a threshold for initiation of motion of stones. More experiments need to be executed to optimize this value of the stability parameter. Moreover, a statistical value for the stability parameter could be used (like θcr,1%) to describe the static stability of stones by means of a certain number of stones that are allowed to move for a certain number of waves.

Until now, an extensive design method for mild slopes has not been available. The aim of this thesis is to understand the stability of rock on mild slopes under wave attack for impermeable cores in order to optimize designs. Physical model tests have been executed to study this stability for a 1:8 slope. Mossinkoff (2019) executed physical model tests for a 1:10 slope and a re-analysis of these tests is included in this thesis as well. Damage caused by entrained rocks is quantified by damage parameters using stereophotogrammetry and coloured rocks in strips. In this study, the influence of several hydraulic and structural parameters on damage parameters has been investigated for mild slopes. A positive correlation has been found between the significant wave height and the damage parameters. Besides this, based on the analysis it can be concluded that the wave steepness and damage parameters are negatively correlated. An increase in layer thickness of the rocks does not seem to increase the stability of rock on mild slopes. However, the slope angle does have an effect on the stability as a milder slope is associated with less damage. Another conclusion is that more damage continues to be observed even after 15 000 waves. Based on the results of the physical model tests, a design formula is developed for mild slopes to be able to increase efficiency in designs of coastal structures for these mild slopes. This study also provides evidence that rocks on mild slopes have different characteristics of damage and damage development compared to steep slopes. The largest share of entrained rocks transport in upward direction and rocks on mild slopes seem to be more mobile compared to steep slopes. This suggested that it might be more efficient to study the moment when the filter layer or the core becomes visible instead of the static stability of rock within the armour layer itself.

Since the behaviour of damage formation on steep slopes varies from mild slopes, a comprehensive definition of damage needs to be established for different situations. Knowledge about the transition of rock stability between steep and mild slopes is lacking. This study presents a reliable and efficient design method. The goal of this study is to describe the stability of rock slopes (1:6 - 1:10). Insight into governing processes and quantified damage characteristics are gained by analyzing wave attack in the transition zone of mild and steep slopes. The method is based on physical model test series with deep water wave conditions and perpendicular wave attack, which were performed to understand the stability of slope 1:6. The slopes were measured with stereophotogrammetry and processed in a 3D model with Agisoft Metashape software. Before wave attack, an initial profile is retrieved which can be compared with cumulative damaged profiles. Automatically detectable Ground Control Points are used as reference to assign scaling information and location recognition. The quantity and location of normative damage on mild slopes are identified for six damage parameters, based on different relations regarding erosion area and depth. The tests results have led to a number of key findings. Damage formations are developed by dominant downward (bar profile) or upward transport (berm profile). In general, a thicker layer and a milder slope give more upward transport. Besides the change in transportation direction, no significant differences in stability numbers were observed for various layer thicknesses with similar wave characteristics. For an increasing slope angle, the influence on stability significantly decreases between various wave steepnesses, due to the impact of a higher amount of more damaging plunging waves. The definition of initial, intermediate and failure damage limits are specified and the test profiles are matched per limit. Initial and intermediate damage are found at a constant value of _3_.3 for a varying layer thickness and slope angle based on the definition and matching profile analysis. Failure of the slope is defined by the event in which the impermeable layer or filter layer becomes visible, which occurs for a layer thickness of 2.53=50 at a damage level of _3_.3 = 1.5. Research at 1:6 slope shows that a higher failure limit is observed for layer thickness 53=50. Due to the lack of test results of other slopes (1:8 - 1:10), specific values cannot yet be indicated for the range 1:6 - 1:10. The lack of testing is also due to limits of the flume and wave machine. In order to remain at least 13=50 over the slope, an acceptable damage limit for a design with a layer thickness of 2.53=50 is found at _3_.3 = 0.9. For a layer thickness of 53=50, an acceptable value of _3_.3 = 1.5 holds a remaining layer of 2.53=50 due to the increase of bed mobility for higher damage numbers. The design formula of mild slopes is based on _3_.3 because this gives the lowest bias error, variability and measurement errors. The design formula for mild slopes is used to estimate stability, nominal rock diameter and acceptable damage for conditions within the applicability range of performed tests. This new method for mild slope design indicates that a higher allowable stability can be used compared to the previously used (extrapolated) method of Van der Meer (1988). A displacement is not necessarily linked to erosion due to alternating upward and downward transport on mild slopes. It can be discussed to what extent a rock, after displacement, contributes to the stability of the design. This can be investigated by the ratio of mobility and erosion over the entire slope. This study has shown that the damage parameters, based on displacements, are larger than the width-averaged erosion parameters. This appears to be a useful ratio for distinguishing mild slopes from steep slopes. In the latter these parameters are considered equal. Requirements for mild slope stability can also be imposed on the basis of the ratio between mobility and erosion as described in this research.