Stability assessment of a rubble-mound breakwater
Stability assessment of a rubble-mound breakwater under wave, wave run-down, and seismic loading
1. Introduction
Breakwaters are exposed to loading conditions that can vary considerably during their service life. In addition to the permanent weight of the structure, the analysis may need to consider direct wave pressures, wave run-up, changes in hydraulic conditions, and seismic action. Each condition may govern a different aspect of the design.
This example presents the analysis of a rubble-mound breakwater with a reinforced-concrete crown wall using DeepEX. Three loading stages are examined :
Direct wave loading on the seaward face of the crown wall;
Wave run-up along the seaward slope;
Seismic loading.
The assessment considers the global stability of the breakwater and its foundation, together with the local stability and structural response of the crown wall. The results demonstrate that the stage governing global stability does not necessarily govern the local crown-wall checks.
A. Analysis Description
The model represents a rubble-mound breakwater constructed over a frictional foundation layer. The breakwater comprises a quarried-rock core, filter and transition layers, and external armor layers. A reinforced-concrete crown wall is positioned near the seaward side of the crest.
The breakwater geometry is generated using the DeepEX Breakwater Wizard.
Breakwater Geometry
The main breakwater crest is located at elevation +8.0 m and has an overall width of 17.5 m. The main seaward slope is 2H:1V, while the harborside slope is 1H:1V.
A toe berm is provided at the base of the seaward slope. The toe berm is 2.0 m high and 4.0 m wide, with an outer slope of 1.5H:1V. The quarried-rock core is extended by 2.0 m at the toe.
The seabed elevation is −10.0 m, while the still-water level is located at elevation 0.0 m on both sides of the breakwater.
Table 1. Principal dimensions adopted in the breakwater model.
Parameter | Adopted value |
Main breakwater crest elevation | +8.0 m |
Overall crest width | 17.5 m |
Main seaward slope | 2H:1V |
Seaward front slope | 1.5H:1V |
Harbourside slope | 1H:1V |
Toe-berm height | 2.0 m |
Toe-berm width | 4.0 m |
Toe-berm slope | 1.5H:1V |
Core extension height at toe | 2.0 m |
Core extension length at toe | 2.0 m |
Seabed elevation | −10.0 m |
Still-water elevation | 0.0 m |
Breakwater Materials and Armor Layers
The main body of the breakwater consists of quarried rock. A 1.0 m thick rock filter layer separates the core from the external armor.
Two armor layers are included :
A 2.0 m thick concrete-block layer on the seaward side;
A 1.5 m thick riprap layer extending along the seaward and harborside slopes.
Table 2. Materials and armor layers adopted in the breakwater model.
Breakwater component | Material | Thickness |
Core | Quarried rock | Defined by breakwater geometry |
Filter layer | Rock | 1.0 m |
Primary seaward armor | Concrete blocks | 2.0 m |
Secondary armor | Riprap | 1.5 m |
The foundation and breakwater components are represented using frictional materials. The adopted unit weights and shear-strength parameters are presented in Table 3. Cohesion is neglected for all materials, and the analyses rely on the corresponding friction angles to represent their shear resistance.
Table 3. Soil and material strength parameters adopted in the analysis.
Material | γₜ (kN/m³) | c′ (kPa) | sᵤ (kPa) | φ′ (°) |
F | 19.62 | 0 | — | 32° |
TBR | 11 | 0 | — | 60° |
CBL | 23 | 0 | — | 50° |
RRP | 18 | 0 | — | 45° |
TROCKS | 18 | 0 | — | 60° |
QRK | 18 | 0 | — | 42° |
Crown-Wall Geometry
The crown wall extends from elevation +7.0 m to +12.0 m, giving a total height of 5.0 m. The wall has a 1.5 m thick stem, a 5.0 m wide base, and a 1.0 m thick base slab.
Table 4. Principal crown-wall dimensions.
Parameter | Adopted value |
Crown-wall top elevation | +12.0 m |
Crown-wall height | 5.0 m |
Stem thickness | 1.5 m |
Base width | 5.0 m |
Base-slab thickness | 1.0 m |
Displayed bending-moment capacity | 1,061.6 kN·m/m |
The crown wall is checked separately for bearing resistance, passive resistance, rotation, toe stability, basal stability, and structural bending. These local verifications are considered together with the global stability analysis of the complete breakwater.
B. Methodology
Loading Stages
The breakwater is evaluated under three distinct loading conditions.
Stage 1 – Direct Wave Loading
In the first stage, wave pressures are applied to the seaward face of the crown wall. The pressure distribution includes the horizontal wave action and uplift acting beneath the seaward portion of the wall base.
This stage is used to evaluate :
Wave-induced pressures on the crown wall;
Uplift beneath the wall base;
Crown-wall bending;
Local toe, passive-resistance, bearing, and rotational stability;
Global stability of the breakwater and foundation;
Wave overtopping.
Stage 2 – Wave Run-Up
The second stage represents wave run-up along the seaward slope of the breakwater. As waves travel up the slope, the water reaches an elevation above the still-water level and modifies the hydraulic loading acting on the structure.
This stage is analysed separately to assess the global stability of the breakwater and the corresponding local response of the crown wall under the adopted wave run-up condition.
Stage 3 – Seismic Loading
The third stage includes seismic action. Horizontal and vertical design accelerations are applied, and the resulting seismic earth pressures are calculated using the Mononobe–Okabe method.
This stage is used to assess the response of the breakwater and crown wall under earthquake loading without combining seismic action with the direct wave-pressure condition.
Wave-Analysis Parameters
Wave run-up is calculated using the TAW 2002a method, while wave impact is evaluated using the Pedersen 1986 approach. The seaward wave-pressure distribution is calculated using the Sainflou method, as modified by McConnell.
A significant wave height of 3.0 m and a peak period of 9.0 seconds are adopted. The incident wave is assumed to act normal to the breakwater.
Table 5. Principal wave parameters and calculation methods.
Wave parameter or option | Adopted value or method |
Design method | Wave run-up calculation |
Wave run-up method | TAW 2002a |
Run-up exceedance probability | 0.1% |
Wave-impact method | Pedersen 1986 |
Seaward pressure method | Sainflou, modified by McConnell |
Surface type | Concrete |
Armour type | Cubes/blocks |
Acceptable armour damage | 5% |
Wave run-down method | Thomson and Shuttler 1975 |
Significant wave height, Hₛ | 3.0 m |
Incident wave height at the toe | 3.0 m |
Local wave height | 3.0 m |
Wave inclination to the wall normal | 0° |
Peak period, Tₚ | 9.0 s |
Period associated with H₁⁄₃ | 8.0 s |
Average wave period | 7.0 s |
Local wave period | 6.0 s |
Storm duration | 6 h |
Flow-through adjustment factor | 0.7 |
Wave-reflection method | Postma 1989 |
Overtopping method | Automatic selection using EurOtop |
Uplift beneath the seaward wall base | Included |
Low-wave base-pressure factor | 0.5 |
Seismic Parameters
Horizontal and vertical design accelerations of 0.14g and 0.02g, respectively, are adopted. The crown wall is treated as flexible, with a user-defined structural response factor of R = 1.0.
The seismic earth pressures are calculated using the Mononobe–Okabe method and applied as external pressures. The Eurocode 8 Mononobe–Okabe pressure distribution is selected, and the seismic thrust is calculated to the bottom of the wall.
Table 6. Principal seismic-analysis parameters.
Seismic parameter or option | Adopted value |
Horizontal design acceleration | 0.14g |
Vertical design acceleration | 0.02g |
Wall behaviour | Flexible |
Structural response factor, R | 1.0 |
Seismic-pressure method | Mononobe–Okabe |
Pressure-distribution option | Eurocode 8 MO equation |
Water behaviour | Impervious |
Seismic thrust height | Calculated to the bottom of the wall |
Wall inertia | Not included |
Global Stability Analysis
Global stability is evaluated using the Morgenstern–Price method with circular slip surfaces. The search considers potential failure mechanisms passing through the breakwater body and the underlying foundation.
The global factor of safety is calculated independently for each loading stage. This verification is separate from the local crown-wall checks and should not be used as a substitute for them.
Crown-Wall Checks
The crown wall is evaluated for :
Bearing resistance beneath the base;
Passive resistance at the wall toe;
Rotational stability;
Required toe embedment;
Basal stability;
Structural bending capacity.
The direct wave-pressure stage also includes the applied wave-pressure distribution and uplift beneath the wall base.
2. Results and Discussion
The three loading stages produce different global and local responses. The seismic stage governs the global stability of the breakwater, whereas the direct wave-pressure stage governs the local crown-wall toe verification.
Principal Results
Table 7. Comparison of the principal stability results for the three loading stages.
Result | Direct wave pressure | Wave run-up | Seismic action |
Global factor of safety | 2.134 | 2.148 | 1.268 |
Rotational factor of safety | 2.237 | 9.114 | 92.297 |
Basal-stability factor of safety | 3.249 | 3.249 | 3.249 |
Displayed overtopping discharge | 0 | — | — |
¹ The direct wave-pressure output reports a negative passive/toe safety value of −9.906 and identifies the toe check as failed. This result should be treated as a failed local verification rather than as a physically meaningful negative factor of safety.
Direct Wave-Pressure Stage
The direct wave-pressure stage gives a global factor of safety of 2.134. The critical circular slip surface passes through the breakwater and extends into the foundation, indicating a comparatively high global stability margin for the analysed wave condition, subject to the project-specific acceptance criteria.
The local crown-wall results are less favourable. Although the rotational factor of safety is 2.237 and the basal-stability factor is 3.249, the passive-resistance and toe checks fail. DeepEX reports a minimum toe-safety value of −9.906, and the required toe depth corresponding to FS = 1.0 cannot be calculated.
This means that the satisfactory global factor of safety does not, by itself, establish the adequacy of the crown wall.
The local failure should be investigated by reviewing :
The magnitude and distribution of the wave pressures;
Uplift beneath the wall base;
The direction and side of wave application;
The passive resistance available in front of the wall;
The crown-wall base dimensions and toe geometry;
The interface properties between the concrete base and the underlying material.
The displayed wall-bending demand remains well below the reported moment capacity of 1,061.6 kN·m/m. The critical result is therefore associated with the local wall-foundation equilibrium rather than the structural bending resistance of the concrete section.
Wave Run-Down Stage
The wave run-down stage gives the lowest global factor of safety :
FSglobal = 2.148
This result is approximately 0.7% higher than the direct wave-pressure value of 2.134 and approximately 69% higher than the seismic result of 1.268. Wave run-up therefore does not govern the global stability of the analysed breakwater.
The local crown-wall checks are also satisfactory for this stage. The minimum toe factor of safety is 1.621, the rotational factor is 9.114, and the basal-stability factor is 3.249. The required toe embedment for FS = 1.0 is reported as 0.8 m.
Although this stage does not govern the current model, wave run-up remains an important design condition because it influences the hydraulic loading, maximum water elevation reached along the seaward slope, and potential overtopping response.
Seismic Stage
The seismic stage gives the lowest global factor of safety :
FSglobal = 1.268
This value is approximately 41% lower than the direct wave-pressure result and approximately 41% lower than the wave run-up result. The seismic stage therefore governs the global stability of the analysed breakwater.
The reduction in global stability reflects the additional destabilising effects introduced by the adopted horizontal and vertical accelerations, together with the seismic earth pressures calculated using the Mononobe–Okabe method.
Despite governing global stability, the seismic stage does not govern the local crown-wall checks. The minimum toe factor of safety is 13.201, while the rotational factor of safety is 92.297. The required toe embedment for FS = 1.0 is 0.5 m, and the basal-stability factor remains 3.249.
The global factor of safety of 1.268 should be compared with the acceptance criterion adopted for the relevant seismic design condition. If a higher minimum value is required, the breakwater geometry, slope configuration, material strengths, foundation resistance, or adopted seismic parameters may need to be reviewed.
Comparison of the Loading Conditions
The results demonstrate that different loading stages can govern different limit states.
The seismic stage controls the global stability assessment, producing the lowest global factor of safety of 1.268. The direct wave-pressure and wave run-up stages give similar global factors of safety of 2.134 and 2.148, respectively.
In contrast, the direct wave-pressure stage governs the local crown-wall verification because the passive-resistance and toe checks fail under the applied wave and uplift pressures. The wave run-up and seismic stages provide satisfactory local crown-wall results for the analysed conditions.
This distinction shows why global and local stability checks must be considered separately. A satisfactory global slip-surface result does not guarantee adequate resistance against sliding, rotation, bearing failure, or loss of toe resistance at the crown wall. Similarly, satisfactory local crown-wall checks do not confirm the global stability of the complete breakwater-foundation system.
3. Conclusions
This example evaluates a rubble-mound breakwater with a reinforced-concrete crown wall under direct wave loading, wave run-up, and seismic action.
The direct wave-pressure stage gives a global factor of safety of 2.134. However, the crown-wall passive-resistance and toe checks fail under the applied wave and uplift pressures. This local failure must be resolved even though the global stability result is comparatively high.
The wave run-up stage produces the highest global factor of safety, with FS = 2.148. The local crown-wall checks are also satisfactory, with a minimum toe factor of safety of 1.621.
The seismic stage governs global stability, producing a factor of safety of 1.268 under the adopted horizontal and vertical design accelerations of 0.14g and 0.02g. Although the local crown-wall checks show considerable stability margins, the global seismic result should be assessed against the acceptance criterion specified for the relevant seismic design condition.
The example demonstrates the importance of analysing several representative loading stages. Direct wave pressure, wave run-up, and seismic action influence the structure differently and may govern different limit states. A complete breakwater assessment should therefore consider both the global stability of the breakwater-foundation system and the local stability and structural capacity of the crown wall.
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