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Stability of shore protection structures

Extensive treatments on the stability of shore protection structures under the influence of waves and currents can be found in the Rock Manual [1], the Coastal Engineering Manual [2] and the Toe structures management manual [3]. This article provides a short introduction to the topic by summarising a few important notions. The thumb rules mentioned in the article are rough estimates that are not intended for optimizing the design of shore protection structures.

## Introduction

Shore protection structures are generally built to protect shorelines from ongoing erosion or to shelter areas from strong wave action. Usually they are not meant to provide protection against flooding. They influence shoreline morphology in several ways; this influence is described in the articles Seawalls and revetments, Human causes of coastal erosion and Detached breakwaters. Shore protection structures exist in two different kinds: modular and monolithic. Monolithic structures are inflexible, usually made of concrete, cemented or interlocking stone, steel or wood. Modular structures are flexible and consist in most cases of piled-up heavy elements, that can – to some extent – adjust their position relative to each other under the action of waves. The elements can be rock clasts or concrete units of different forms. These structures are usually called 'rubble mound'. Another type of flexible structures used for shore protection are sand-filled geocontainers; most popular are so-called geotubes. These structures are dealt with in the article Sand-filled geosystems in coastal engineering. In this article we focus on rubble-mound structures. The main advantages of these structures, compared to monolithic structures, are: (1) flexibility - the potential to accommodate (small) changes in seabed/beach level; (2) potential to dissipate wave energy, thus reducing wave loads on the structure, and also reducing the tendency for scour; (3) the (relatively) low costs of construction, maintenance and adaptation.

Failure of a shore protection structure means: damage that result in structure performance and functionality below the minimum anticipated design[2]. A structure usually fails when a large number of elements (10-20% in a cross-section, depending on the slope) is displaced by wave action or if the structure collapses after being undermined. Three crucial design criteria therefore pertain to:

1. the slope of the structure,
2. the weight of the elements and
3. the scour at the toe.

Other important design criteria refer to the form of the elements, the permeability (packing density) of the structure and the gradation (size range) of the rock elements.

The other design question is: under which conditions is failure of the structure allowed? This question is related to the planned lifetime (design life) of the structure. Occurrence of the design conditions should have low probability; the recurrence period should be larger than the design life. Available records of water levels, currents and waves are often not long enough to determine the design conditions corresponding to the desired recurrence period. This issue is discussed in the section #Determination of the design conditions.

## Design of shore protection structures

Shore protection structures can be built on the beach or offshore in the surf zone. In the first case the structure is essentially a revetment protecting the backshore, dune foot or cliff base. In the second case the structure is a breakwater, aligned approximately parallel to the shoreline. The breakwater aims to reduce the wave attack on the beach or to protect fairways and harbors from high waves.

It is assumed that design conditions correspond to a situation with high waves and a strong wind- and wave-driven set-up of the mean water level. The water level may exceed the crest of the breakwater but should not exceed the upper edge of the revetment. In both cases the toe of the structure is subjected to the scouring action of waves and currents.

Fig. 1. Left panel: Offshore rubble-mound breakwater. Right panel: Rubble-mound revetment protecting the backshore.

Typical designs for rubble-mound shore protection structures are shown in Fig. 1. The designs consist essentially of a sediment core (the backshore, in case of a revetment), a geotextile, a granular underlayer, a double armor layer and a toe structure. The functions of these components are:

• The geotextile prevents erosion of the core by blocking the passage of sediment particles; however, it allows the passage of porewater. Care must be taken to ensure that the filter cloth is not damaged during installation.
• The granular underlayer protects the geotextile against puncturing by the overlying armor elements. It softens hydraulic pressure gradients and reduces lift forces on the armor elements (recommended layer thickness > 60 cm, grainsize 10-20 cm[2]).
• The armor elements protect the structure against damage by wave action.
• The toe prevents sliding of the armor elements along the slope of the structure and prevents undermining of the structure due to scouring.

The crest of the breakwater in Fig. 1 emerges above sea level. Breakwaters with the crest below sea level (so-called low-crested breakwaters) are also often applied, instead of emergent breakwaters. The distance between crest level and sea level (the so-called freeboard $R_c$) is an important design parameter for low-crested breakwaters.

## Failure modes

Several processes can cause failure of the shore protection structure; important processes are shortly described below.

• Hydraulic instability of the armor layer. This instability, due to wave loading, is discussed in the section #Armor layer stability.
• Scour (erosion) at the toe of the construction. Toe scour due to waves and currents is discussed in the section #Toe stability and scour protection. Failure of the toe construction can cause sliding down of the armor layer.
• Erosion of the rear side (lee side) of an offshore rubble mound breakwater (Fig. 2). The armor layer at the rear side must be designed to withstand erosion by overtopping waves [4]. The erosional power of overtopping can be reduced by the incorporation of a crown element[1], see Fig. 3.
• Erosion at the top of a rubble mound protection revetment, see Fig. 4. The height of the revetment should be determined such that wave run-up under design conditions does not produce significant erosion of the backshore. For wave run-up estimates see: Wave run-up.
 Fig. 2. Erosion of the rear side of the breakwater due to overtopping waves. Fig. 3. Breakwater with crown element. Fig. 4. Overtopping of revetment by wave run-up.

• Instability of the seabed under the structure due to soil liquefaction, causing the structure to collapse, see Fig. 5. A loosely packed saturated sandy seabed can liquify under heavy load; the risk of liquefaction is greatest with well-sorted round sand particles. A seabed with a high content of clay particles (> 40%) can also liquify when loaded and subjected to strong pressure fluctuations (by wave action, seismic events)[5]. In both cases the sediment skeleton is destroyed by excess pore pressure. The susceptibility of a soil to liquefaction can be checked through Standard Penetration Tests or Cone Penetration Tests. Mitigating measures consist of soil consolidation (e.g., vibro compaction) and/or enhancing soil drainage.
• Washout of fine sediments from the core of the structure. Porewater flow due to a hydraulic pressure gradient inside the core can entrain fine sediment particles through the voids between the larger particles. In the absence of a filter, fine particles are washed out of the core, giving rise to the so-called piping phenomenon. The voids inside the core gradually increase until the structure collapses. The presence of a geotextile filter prevents the escape of sediment particles from the core. (A geotextile filter is the simplest solution, a granular filter may do as well.) The openings of the filter should be small enough to prevent the passage of sediment particles. However, the openings must be large enough to evacuate excess porewater from the core.
• Slip surface failure inside the core, see Fig. 6. Slip surface failures develop when the effective shear stress (mediated by interparticle contacts) on a slip plane exceeds a critical limit. In saturated sediment bodies, the limit value decreases with increasing porewater pressure. Pore pressure builds up when fluctuations in the external pressure (by wave action, seismicity) cannot be leveled fast enough by porewater flow. This is the case if the fraction of fine sediment is high, causing low permeability of the sedimentary body. The permeability can be increased by the installation of drains. Coarse sand, gravel or quarry run are recommended as core material for rubble mound breakwaters. The permeability of the geotextile must also be enough to prevent the development of excess pore pressures. A safe assumption is that the permeability of the geotextile must be at least 10 times that of the permeability of the fill material it is filtering, see Sand-filled geosystems in coastal engineering.
 Fig. 5. Collapse of the armor layer due to seabed liquefaction under the toe of the breakwater. Fig. 6. Slip failure of the core of the breakwater that may occur if the core is water saturated, insufficiently drained and subjected to cyclic loading.

## Armor layer stability

### Emergent structures

It is assumed that the armor layer has been put in place properly, i.e., well packed such that there are many contact points between the armor elements. The functionality of the structure will be maintained during its intended lifetime if the armor elements stay largely in place with only small adjustments of their positions. This should be the case even under the heaviest wave attack considered in the design. The most crucial design parameters are the slope of the structure and the weight of the elements. In many cases it is recommended to optimize the design by means of hydraulic model tests, in order to achieve the best functionality at the lowest costs. This is especially true for situations with a complex geometrical setting. In the following some empirical rules are presented that allow a rough initial estimate of the design parameters and corresponding construction costs.

We assume that stability is related to the significant wave height $H_s$ at the toe of the structure under design conditions. Then it is usual to define the stability number

$N=\Large\frac{H_s}{\Delta \, D}\normalsize , \qquad (1)$

where $D$ is the average diameter of the armor elements and $\Delta=(\rho_r/\rho_w)-1$ is the relative density of the armor elements. For rock $\Delta \approx 1.65$ and for concrete $\Delta \approx 1.4$. The critical maximum value of the stability parameter $N_s$ for which the armor layer is stable, $N \le N_s$, depends on several parameters:

• The slope $\alpha$ of the structure (the slope angle should not exceed the friction angle of about 35 degrees, i.e. $\cot \alpha \ge 1.5$; in earthquake-prone areas the critical slope angle is smaller);
• The type of filter/underlayer;
• The form of the armor elements;
• The gradation of the armor elements (thumb rule: the size of the 20% smallest elements should not be smaller than half the size of the 20% largest elements);
• The packing/porosity of the structure;
• The acceptable damage $S_d$ during the lifetime of the structure;
• The number of waves $n$ corresponding to design conditions (related to storm duration; often a maximum number $n$=7500 is assumed);
• The period and steepness of the design waves;
• The mode of wave breaking (plunging or surging).

Several empirical formulas have been established for $N_s$, generally based on experiments in hydraulic models. One of the most simple and often used formulas is due to Hudson (1959)[6]

$N_s = (K_D \cot \alpha)^{1/3} , \qquad (2)$

where $\cot \alpha$ is the inverse average slope (width/height) of the structure. The factor $K_D$, the stability coefficient, depends on the parameters listed above (except the slope). The formula of Hudson was derived from experiments with non-breaking waves (the water depth at the toe of the structure being much larger than the design significant wave height). Combination of (1) and (2) gives an estimate of the required minimum weight $W = g \rho_r D^3$ of cubic armor elements,

$W \approx \Large\frac{\rho_r g H^3}{\Delta^3 K_D \cot \alpha}\normalsize . \qquad (3)$

A value $K_D=1$ is recommended for structures with a geotextile filter, in case no damage is allowed[1]. In this case the safe diameter of the armor elements for a structure with inverse slope $\cot \alpha = 2$ equals approximately half the design significant wave height. A higher value $K_D=3-4$ can be taken in case of a granular filter [7], implying that a granular filter may save costs.

Formulas for $N_s$ have been derived that take into account some other factors in addition to the slope. Van Gent et al. (2004)[8] considered the acceptable damage $S_d$ and the number of waves $n$ occurring during the design storm. For a structure with geotextile filter under breaking design waves he proposed the formula

$N_s=1.75 \, (\cot \alpha)^{1/2} \, n^{-1/10} \, S_d^{1/5} . \qquad (4)$

The damage factor is defined as $S_d=A_e/D^2$, where $A_e$ is the cross-sectional area of the structure eroded by wave action. (Damage can also expressed as the percentage $N_d$ of displaced elements; the relationship in case of a double armor layer is $N_d \approx (1.2 – 1.6) \, S_d$.) For a structure with inverse slope $\cot \alpha = 2$, $S_d=4 , \; n=7500$, the safe diameter of the armor elements is again close to half the design significant wave height. Herera et al. (2017) [7] conducted lab experiments with $n=1000$ breaking waves on a rubble-mound breakwater with inverse slope $\cot \alpha = 1.5$. They found for the stability parameter the relationship $N_s = 1.57 \, S_d^{1/6}$. This is about 40% larger than the corresponding estimate Eq. (4) and thus suggests that Eq. (4) overestimates damage of the armor elements. Other formulas for the stability parameter $N_s$ can be found in the Rock Manual[1] and the Coastal Engineering Manual[2].

Fig. 7. Examples of concrete armor units.

Special armor elements have been designed that provide better stability than rock clasts, see Fig. 7. Experimentally determined stability numbers for randomly placed interlocking armor elements are in the range $N_s = 2 - 3$. Interlocking concrete armor elements are susceptible to breakage[2]. Breakage of armor elements compromises the stability of the armor layer.

### Submerged structures

Submerged structures transmit part of the incident waves. The transmission coefficient $C_t=H_{transmitted}/H_{incident}$ depends on the ratio $r$ of freeboard $R_c$ (water column above breakwater crest) and incident significant wave height, $r=R_c/H_s$. The transmission is large, $C_t\gt 0.8$, if $r\gt 1$. For smaller values of $r$ the transmission decreases to $C_t \approx 0.5$ for $r=0$ [1].

The stability criterion for submerged breakwaters is a bit less severe than for emergent breakwaters, as far as the front armor layer is concerned. However, the crest and the rear side of the breakwater are more vulnerable if the freeboard is smaller than the design wave height ($r\lt 1$) [9]. Damage at the rear side of the breakwater strongly affects the integrity of the whole structure, see Fig. 2. The size of the armor elements at the front side, the crest and the rear side should be similar in case of frequently overtopped breakwaters.

## Toe stability and scour protection

The toe is a crucial component of a structure: it supports the armor layer and it prevents undermining by wave- and current-induced erosion. Different toe designs are possible; here we focus on toe structures consisting of loose elements, see Fig. 1. In some designs the toe is protected by sheet piles; these constructions will not be considered here. The seabed near the toe is susceptible to erosion. Therefore, the toe is often located below the seabed at the expected scour depth (right panel Fig. 1). If the seabed is not erodible, or if scour protection measures are applied, the toe can be situated on the seabed (left panel Fig. 1).

The toe is placed on a geofilter that prevents washout of fine sediment from below the toe. The geofilter is protected by an underlayer of medium-sized stones on which larger toe elements are placed. The stability of the toe elements is ruled by considerations similar as for the armor layer. The stability number for the toe elements is larger (up to factor 2) than for the armor elements if (under design conditions) the height of the toe (relative to the seabed) is much lower than the water depth [1]. In that case the toe elements can be smaller than the armor elements. This has several advantages: less scour around the toe, less risk of soil liquefaction and lower costs. However, if the strongest wave attack occurs at low water levels, similar sizes are required for the toe and armor elements. The risk of soil liquefaction under the toe (Fig. 4) has to be investigated by laboratory tests [3]. Soil compaction and/or draining measures may be needed for soils with a high fraction of fine sediments.

Fig. 8. Beach scour in front of a revetment; Delta Flume test.

Structures that disturb the natural flow induce a modification of the seabed morphology. Erosion dominates in the vicinity of these structures as a consequence of increased local energy dissipation of waves and currents. Erosion is greatest at the toe of reflective vertical structures. The scour depth is smaller for structures with a more gentle slope and a greater porosity. As a rule of thumb, the maximum scour depth $S_m$ at vertical structures can be as large as the significant wave height $H_s$ under design conditions. The scour depth is an increasing function of the ratio water depth / wavelength. Beach lowering occurs in front of structures protecting the backshore when waves collapse on the structure under storm conditions [10], see Fig. 8. Natural restoration of the beach level can occur under calm conditions in case of sufficient sand supply. In other cases, the option of artificial nourishment can be considered, see the articles Artificial nourishment and Shore nourishment.

Fig. 9. Principle of the falling apron.

Obliquely incident currents generate a longshore current (see Shallow-water wave theory) that can strongly enhance erosion at the toe of the structure. This is particularly relevant for offshore breakwaters, where it is necessary to protect the seabed in front of the toe. This can be done with a stone cover, provided the stone size has been adjusted to the Shields criterion for critical shear stress, see Sand transport. An alternative is a so-called 'falling apron', a row of wide-graded quarry stone stacked at the edge of the toe, which falls into a developing scour hole and prevents further erosion, see Fig. 9. A similar idea is a design in which the seaward face of the breakwater is extended with a berm that can adjust to local changes in the seabed level.

Currents can be very strong at the extremities of offshore shore-parallel breakwaters, especially for long breakwaters (length >> distance to the shore); these currents are driven by water set-up landward of the breakwater. Hence, particular attention is needed for scour protection of the seabed at the extremities of a long offshore breakwater. The use of a flexible mattress for protecting the seabed can be considered in case of very strong scour.

Shore protection structures always require maintenance, in particular the recharge of scour protection layers. An important point of attention for the design is therefore to ensure that the work is easily accessible for maintenance.

## Determination of the design conditions

Design conditions are the most destructive hydrodynamic conditions that the structure must withstand. These conditions must have a low probability of occurrence during the design lifetime. The structure may suffer some damage during design conditions, but this damage should not compromise the intended functionality of the structure. Conditions that occur more frequently should not produce damage, or the damage must be so small that the total accumulated damage from a number of events does not compromise the functionality of the structure.

The ideal situation is the availability of: (1) a record of field observations which is much longer than the design lifetime, covering all hydrodynamic conditions relevant to the design, (2) a hydraulic or numerical model capable to simulate the impact of the observed hydrodynamic conditions on the design for the entire duration of the observation record. With these tools the design can be optimized without the need to specify design conditions. However, this ideal situation hardly occurs in practice. A possible solution consists in constructing a dataset as intended under (1) from historical meteorological records, using a numerical hydrodynamic model.

The modelling step (2) is generally very demanding in time and costs and the modelling state of the art is not sufficiently advanced to yield accurate results. In practice, structures are tested for a selected set of hydrodynamic conditions that represent extreme events with a low probability of occurrence during the design lifetime (based on acceptable risk agreed between contractor and client). The extreme hydrodynamic conditions are derived from a statistical analysis of available data. The most important design parameters are:

• The significant wave height $H_s$ at the toe of the structure (the 2% highest wave height $H_{2\%}$ is sometimes preferred);
• The wave period $T$;
• The wave incidence direction;
• The extreme water level;
• The storm duration.

For offshore breakwaters in deep water, the extreme significant wave height and storm duration are usually the most critical design parameters. If available historical records of these parameters are too short for extracting directly the design values, they can be obtained from extreme value analysis. For the significant wave height this can be done, for example, by fitting significant wave heights above a certain threshold level to a Weibull distribution. The scour depth at the toe of the structure depends in particular on wave height and wave direction; frequent periods of energetic waves are generally more relevant than exceptional extreme conditions. For estimating the scour depth, knowledge is required of the correlation between wave height and wave direction during such periods.

For breakwaters and revetments in shallow water the wave height is limited by depth-induced breaking. Deep-water wave data therefore have to be transformed to shallow water conditions (as explained in Shallow-water wave theory). The most appropriate design parameter in this case is the 2% wave height $H_{2\%}$ rather than the significant wave height $H_s$[1]. In deep water $H_{2\%}=1.4 \, H_s$ according to the Rayleigh distribution, but in shallow water this relationship is modified, see Statistical description of wave parameters. The extreme water level is another important design parameter for structures in shallow water and on the beach: it determines the water depth above the toe of the structure and it influences the height of the waves breaking on the structure, the run-up and possible overtopping of the revetment. Extreme water levels are related to storm events and therefore correlate with extreme wave heights, but they are also influenced by the astronomical tide that is uncorrelated with wave height. The most critical combination of extreme water level and extreme wave height that may occur can be found by modelling the local hydrodynamic conditions with input from historical meteorological records and by constructing in this way a dataset of sufficient length.

## References

1. CIRIA/CUR/CETMEF 2007. The Rock Manual. The use of rock in hydraulic engineering (2nd ed.). C683. London: CIRIA
2. USACE, 2012. Coastal engineering manual. Report No 110-2-1100. Washington DC: US Army Corps of Engineers https://www.publications.usace.army.mil/USACE-Publications/Engineer-Manuals/u43544q/636F617374616C20656E67696E656572696E67206D616E75616C/
3. Bradbury, A., Rogers, J. and Thomas, D. 2012. Toe structures management manual. Environment Agency https://www.gov.uk/government/publications/toe-structures-management-manual
4. Van der Meer, J.W. and Veldman, J.J. 1992. Singular points at berm breakwaters: scale effects, rear, round head and longshore transport. Coastal Eng. 17: 153-171
5. Chávez, V., Mendoza, E., Silva, R., Silva, A. and Losada, M.A. 2017. An experimental method to verify the failure of coastal structures by wave induced liquefaction of clayey soils. Coastal Engineering 123: 1–10
6. Hudson, R.Y. 1959. Laboratory investigation of rubble-mound breakwaters. J. Waterw. Harb. Div. ASCE 85: 93–121.
7. Herrera, M.P., Gomez-Martín, M.E. and Medina, J.R. 2017. Hydraulic stability of rock armors in breaking wave conditions. Coastal Engineering 127: 55–67
8. Van Gent, M.R.A., Smale, A.J. and Kuiper, C. 2004. Stability of rock slopes with shallow foreshore. In: J.A. Melby, ed., Proceedings 4th Int. Coastal Structures Conf., Portland, 2003. ASCE, Reston, VA
9. Burger, G. 1995. Stability of low-crested breakwaters: stability of front, crest and rear. Influence of shape and gradation. Report H1878/H2415, WL|Delft Hydraulics
10. Sutherland, J., Brampton, A.H., Obhrai, C., Dunn, S. and Whitehouse, R.J.S. 2007. Understanding the lowering of beaches in front of coastal defence structures, stage 2. Joint Defra/Environment Agency Flood and Coastal Erosion Risk Management R&D Programme, R&D Technical Report FD1927/TR. London: Defra