# Wave Impact Pressures on Stepped Revetments

^{*}

## Abstract

**:**

_{m}

_{0}/S

_{h}< 3.5 and a constant slope of 1:2 are analyzed with respect to (1) the probability distribution of the impacts, (2) the time evolution of impacts including a classification of load cases, and (3) a special distribution of the position of the maximum impact. The validity of the approved log-normal probability distribution for the largest wave impacts is experimentally verified for stepped revetments. The wave impact properties for stepped revetments are compared with those of vertical seawalls, showing that their impact rising times are within the same range. The impact duration for stepped revetments is shorter and decreases with increasing step height. Maximum horizontal wave impact loads are about two times larger than the corresponding maximum vertical wave impact loads. Horizontal and vertical impact loads increase with a decreasing step height. Data are compared with findings from literature for stepped revetments and vertical walls. A prediction formula is provided to calculate the maximum horizontal wave impact at stepped revetments along its vertical axis.

## 1. Introduction

## 2. Previous Studies on Wave Impacts on Stepped Revetments

_{h}= 0.026 m were analyzed. A recurved seawall was incorporated at the crest of both sloping structures. Reference [8] analyzed the wave loads on stepped structures in a specific range of Iribarren numbers of 2.8 < ξ < 6.3 and step ratios in a range of 9.0 < H

_{m}

_{0}/S

_{h}< 11.0 with H

_{m}

_{0}as zeroth moment wave height. Similarly, Reference [8] remarks the importance of the short duration shock pressures (impacts) resulting from the rapid compression of an air pocket trapped between the front of a breaking wave and the wall. For vertical walls the authors in Reference [6] note that ‘the shock pressure exerted by a breaking wave is due to the violent simultaneous retardation of a certain limited mass of water that is brought to rest by the action of a thin cushion of air, which in the process becomes compressed by the advancing wave front’ [9]. The position of the highest measured impacts was dependent on the initial still water level (SWL). According to the analysis of impact distributions, the maximum impacts at different wall elevations rarely occur simultaneously. This finding is particularly valid in the case of a non-vertical wall, such as the stepped wall studied here, since some wave energy is dissipated through turbulence. In some cases, a negative impact duration was measured, which is interpreted as a characteristic of turbulence and air entrainment occurring at the base of each seawall step. Finally, Reference [8] summarizes a discussion about the importance of shock pressures for the actual design of a stepped seawall. According to the discussion, pressures of such short duration should not be used for establishing the design load case. Rather, it is recommended to consider the smaller surge pressures with a longer duration to determine the critical dynamic load.

_{h}= 0.015 m), focusing on wave run-up and wave overtopping. The vertical wave impact was measured on a single step with a sampling rate of only 100 Hz, which is considered as rather inappropriate for measuring rapid, i.e., almost instantaneous, wave impact pressures. As such, analyzed data represent merely averaged maximum impact pressures P

_{max}of six test repetitions (with a standard deviation of STD~0.1 P

_{max}) to depict the inherent loading bias within each experiment.

## 3. Experimental Set-Up, Test Conditions and Procedures

_{1/3,max}= 0.42 m with T

_{max}= 2.0 s) were generated with a piston-type wave maker. Two model set-ups, constructed from plywood, with varying step heights (large steps: S

_{h}= 0.3 m, small steps: S

_{h}= 0.05 m) are placed in a 0.7 m wide sub section over a horizontal flume bottom at a distance of L

_{F}= 81.6 m from the wave paddle (Figure 1). The relative flume length with respect to the tested wave length L

_{p}is 10 < L

_{F}/L

_{p}< 36.

_{sample.}= 50 Hz. Three of the sensors are positioned at a distance larger than two wave lengths L from the toe of the revetment, to determine incident wave conditions, as calculated by a reflection analysis. One sensor is placed at the toe of the stepped revetment and another in the shallow water region of the still water level. Pressure impacts on the stepped revetment are recorded by seven pressure transducers, which are placed along the horizontally (f

_{sample.}= 2.4 kHz) and vertically (f

_{sample.}= 19.2 kHz) orientated step fronts (Figure 1). An impression of the set-ups is given in Figure 2 for the analyzed step heights of 0.05 m (a) and 0.3 m (b). In order to capture a profile of wave induced loads on an 1:2 inclined stepped revetment, sensor locations are varied in relative water depths —6.0 < z/H

_{m}

_{0}< 2.0, relative to the still water level. The pressure sensors (ATM.1ST/N fabricated by sts-sensors) have a range from zero to 150 mbar and a non-conformity of ±0.1% from full scale. The sensors are connected by a serial interface connection (RS232) to the data acquisition and provide an output signal from zero to 10 V. The configuration of the probes allows a local (over a single step) and a more global interpretation (for the whole revetment).

_{m}

_{0}/L

_{p}< 0.04 (with H

_{m}

_{0}: spectral wave height, L

_{p}: wave length calculated from the peak period T

_{p}measured at the gauge array in a distance of x = 8.6 m (1 L

_{p}< x < 3.8 L

_{p}) from the toe of the revetment) and Iribarren numbers of 2.5 < ξ < 4.9. Three different water levels h

_{s}with intermediate water depths (0.13 < h

_{s}/L

_{p}< 0.49) are tested. Additionally, the corresponding freeboard height R

_{c}and the number of waves N in each test are given. A total of 13 tests are conducted for steps with a height of S

_{h}= 0.05 m (1.1 < H

_{m}

_{0}/S

_{h}< 2.8) and 10 tests with a step height of S

_{h}= 0.3 m (0.2 < H

_{m}

_{0}/S

_{h}< 0.6).

_{p}.

_{max}. The quantity of a finite number N of waves that cause N individual impacts p within a single test is defined as P. If the quantity P is sorted in a descending order, the maximum recorded impact P

_{max}is defined as max{P} or P

_{(i=1)}, as given in Equations (1) and (2).

_{max}is very important for the design of a structure, it presents significant scatter when formulating predictions for practical design purposes. Therefore, the collected pressure magnitudes will rather be described with a probability of exceedance (e.g., 2% of all incident waves reveals the probability of exceedance impact pressure P

_{98%}, Equation (3)), which offers more representative and reliable predictions.

_{99.6%}is calculated on the basis of the number of waves (N) given in Table 1. The impact with a certain probability of exceedance (P

_{99.6%}in this case) is calculated finally by the mean of the n highest impact events according to Equation (4).

## 4. Results and Discussion

_{h}= 1.7) and Figure 3i–l over a revetment with large steps (H/S

_{h}= 0.3). For the two stepped configurations, the still water level SLW is at the level of a step edge.

_{h}= 1.7), the general flow processes of wave breaking, wave run-up and wave run down sequence is almost in analogy with the processes on a smooth slope. The main differences are: (1) The impact of the breaking wave causes a splash-up at the vertical step front (f) during the wave run-up process, (2) the aeration during the wave run-up is more intense during the run-up process caused by larger friction induced by the steps while interacting with the incoming waves, and (3) higher turbulence and continuous air intrusion (g). Yet, in contrast, the wave run-down induces air into the water body and is retarded compared to the smooth revetment (h). For large step heights (H/S

_{h}= 0.3), the flow processes differ significantly from a normal wave run-up as shown in (a–h). The incident breaking wave interacts with the reflected backflow of a previous wave at the first step edge (i). The interaction leads to an impact on this first step edge, an upwards-directed flow and a wave breaking in very shallow waters over the step edge (j). Then, the wave is propagating as spilling breaker over the horizontal step front (k) until it impacts and resonates with the second vertical step front (l). The impact induces a second splash up.

#### 4.1. Probability Distribution of Impact Pressures

_{m}

_{0}= 0.084 m and H

_{m}

_{0}= 0.089 m (for test 103 and 209, respectively) and a corresponding Iribarren number of ξ = 2.9 (2.8 for test 209). Figure 4a provides measurements of wave impacts on slopes with a small step height (H

_{m}

_{0}/S

_{h}= 1.68, h

_{s}= 1.1 m, sensor P1). A total of 876 individual wave impacts have been recorded with pressure sensor P1 with a maximum recorded pressure impact of P

_{max}= 1.26 kPa. The wave impacts are presented with a certain probability of exceedance. The figure presents an idealized normal-distribution, shown as a dashed line, as reference. As is evident from the presented figure, the wave impacts on this stepped revetment follow a log-normal distribution. Over and under predictions from the idealized normal-distribution given with the dashed line appear for impacts larger than P

_{95%}and smaller than P

_{10%}.

_{m}

_{0}/S

_{h}= 0.3, h

_{s}= 0.921 m, sensor P2) for the same hydraulic boundary conditions, as in Figure 4a. The maximum pressure impact P

_{max}of 1114 individual impacts is 1.68 kPa, which is about 35% larger when compared to the small steps. As for Figure 3c, slight deviations from the ideal log-normal distribution are detected for impacts smaller than P

_{10%}, while impacts larger than P

_{90%}deviate significantly from this trend. These deviations indicate that the influence of the wave-interaction with large steps differs for high and low impact scenarios. The apparent difference in the wave breaking and wave run-up at these large steps compared to slopes with and without smaller steps (Figure 3) is a significant transformation of the incident wave which breaks over the very shallow horizontal step front. Hence, the log-normal distribution of the induced pressures of incident wave heights is disturbed and transformed over large steps directly at the structure, and in analogy, the wave impact distribution is affected. The increase in the wave impacts for large steps can be explained by the fact that the vertical step fronts fall completely dry during the run-down process, whereas for small step heights, there is always a thin water flow remaining from the previous run-down. This thin water layer (often aerated) reduces the wave impact of the next breaking wave.

_{max}are based on pulsating loads and larger impacts on impacting loads. Reference [15] found that often only a part of all impacts show a good agreement with the log-normal distribution according to the stochastic pattern of the pressures in the impact area. This observation is confirmed for the present experimental configuration. In contrast, when re-considering only the highest impacts during one wave impact event on the area, it turns out that measured data points follow an almost linear relation in a log-normal plot. Hence, Figure 4c gives a detailed log-normal probability distribution of only the 18 largest impacts (>P

_{98%}) of test 209, confirming evaluations stemming from Reference [15] to be also valid for stepped revetments.

#### 4.2. Temporal Evolution and Load Cases of Impact Pressures

_{q}. Partially breaking waves induce peaks in a range of 1.0 < P

_{max}/P

_{q}< 2.5. Pulsating impacts due to standing waves occur when the impact peak and its subsequent quasi-static peak P

_{q}are in the same order of magnitude. Figure 5 sketches a parameter definition describing both a pressure impact event (a), and an exemplary time series of the maximum impact event during test 204 for horizontal oriented pressure sensors P1, P2, and P3 (b). A typical time evolution with an impact and subsequent quasi-static peak is observed. The highest impact is recorded by sensor P2, which is located near the still water level SWL. The oscillation of the signal and negative pressures, as also observed by Reference [8], are interpreted as characteristically mimicking physical processes of turbulence and air entrainment occurring at the base of each revetment step. With increasing distance to the SWL, the impact amplitude decreases (P1 and P3). While sensor P2 recorded a violently impacting load, P1 was impacted by a slightly breaking wave and P3 experienced a pulsating impact.

_{m}

_{0}/S

_{h}= 1.7 (small steps) and subfigures (g–l) to step ratios of H

_{m}

_{0}/S

_{h}= 0.3 (large steps). Each subfigure displays the measured impact event of a certain probability of exceedance (P

_{max}to P

_{10%}) over the relative time. The time t is normalized by the wave period T

_{m–}

_{1,0}. Time step t/T = 0 is set at the time of maximum impact. Each peak impact (based on the raw data) is given proportionally to the maximum measured peak P

_{max}of the entire test duration.

_{max}is reached by 5% of all impact events (P

_{95%}) (d), whereas in the case of the large steps (h), it is only 0.4% (P

_{99.6%}) of all impact events.

_{m}

_{0}< S

_{h}) demonstrate a behavioral function like vertical walls in reflecting incoming wave energy. Functional processes are mimicked in two distinct phases: A clear initial impact followed by a quasi-static peak P

_{q}. For small steps (H

_{m}

_{0}> S

_{h}) the impact peak is also clearly visible, but the subsequent quasi-static peak P

_{q}is not as prominent as for large steps or vertical walls. These differences are caused due to different flow principles. At large steps, the water level in front of the step front is constantly rising after the initial impact, thus inducing the quasi-static load. At small steps, which dissipate energy from the up-rushing wave tongue, a highly aerated flow emerges after the initial impact (Figure 3f,g) and the absolute depth under the wave crest is evidently lower. This leads in general to smaller quasi-static peaks. For the specific case presented in Figure 6, impacting loads are observed in the range of about 5% of all impacts at the small steps and 2% of all impacts at the large steps. The initial violent impact dissipates more energy than a pulsating load induced by wave run-up. As a result, it can thus be deduced that the overall energy dissipation at small step heights is larger, i.e., more effective. Furthermore, the impact-mitigating effect due to the practically permanent existence of a thin water layer on top of the individual small steps of the revetment during the wave run-down phase stemming from the preceding wave (Figure 3) plays a dominant role in impact reduction.

_{r}and impact pressure duration t

_{d}, as defined in Figure 7 and depicts the correlation between the dimensionless impact duration t

_{d}/t

_{r}and the dimensionless impact rising time t

_{r}/T

_{m}

_{–1,0}. The data are grouped in terms of the step ratios H

_{m}

_{0}/S

_{h}, the direction of the measured wave impact (horizontal or vertical with respect to gravity) and the position of the impact (above or below the still water level SWL). As a meaningful reference, Figure 7 presents the empirical projections for rising times and durations of impacts at vertical walls provided by Reference [16]. Impacting load cases (P

_{max}> 2.5 P

_{q}) are characterized by a short relative rising time (0.02 < t

_{r}/T

_{m–}

_{10}< 0.2), while pulsating loads are distinguished by their longer peak durations (t

_{r}/T

_{m–}

_{10}> 0.2). The dominance of a peak is characterized by the relative impact duration t

_{d}/t

_{r}. The highest impact loads (e.g., P

_{max}or P

_{99.6%}) correspond to short relative impact durations t

_{d}/t

_{r}in combination with a short relative rising time t

_{r}/T

_{m–}

_{10}. The larger the relative impact duration t

_{d}/t

_{r}, the more critical the impact gets in terms of causing damage or instability of the revetment.

_{r}/T

_{m–}

_{10}> 0.2. Horizontal impacts (filled marker) or vertical impacts (empty marker) have comparable impact rising times and impact durations. Below the SWL, no impacting load case was detected for either small or large steps. The recorded loads have a pure hydrostatic nature induced due to the water level changes induced by wave run-up and run-down over the pressure sensors. Stronger impacts are mitigated by means of a sufficient water layer (“cushion”) effectively sheltering the submerged steps’ fronts from more violent impacts. The stepped shape of the slope retards the wave run-down leading to a permanent water cover of the revetment below the SWL.

_{m}

_{0}/S

_{h}< 0.6 and filled black diamonds for small steps 1.0 < H

_{m}

_{0}/S

_{h}< 3.5) ranges from very short impacts (t

_{r}/T

_{m–}

_{10}< 0.05) up to long load cases (t

_{r}/T

_{m–}

_{10}> 0.3). The latter ones represent a full run-up and run-down phase inducing pulsating loads. While the impacts at small steps show an increase in the peak duration for decreasing relative peak rising times up to t

_{d}= 4t

_{r}, the maximum duration of t

_{d}= 2t

_{r}is observed for large steps. This finding can be explained with the fact that impacts at large steps are often caused by breaking waves that directly hit the step front, whereas for small steps, the step fronts are covered by thin water layers. Rising times and peak durations of vertical impacts above SWL (empty blue squares for large steps 0.2 < H

_{m}

_{0}/S

_{h}< 0.6) and empty black diamonds for small steps 1.0 < H

_{m}

_{0}/S

_{h}< 3.5) scatter significantly and do not follow a clear, yet visual trend. The formation and progression of an aerated water layer over the horizontal step front influences the recorded impact significantly and explains the scattering in the data. Pulsating loads can be explained by the water layer thickness during the wave run-up and run-down. Impacting conditions occur only very close to the SWL. The impact rising times for vertical walls and stepped revetments are in the same range. Reference [16] found a minimum peak duration of t

_{d}= 2t

_{r}for vertical walls, whereas the minima for stepped revetments is t

_{d}= t

_{r}. These differences may be due to the differences in the sampling frequency of the impact pressure sensors (0.6 to 1.0 kHz at [16] and 19.2 kHz in the present study). Lines of best fit for the different load cases are calculated according to Equation (5) with corresponding regression coefficients a, b and c given in Table 2.

#### 4.3. Spatial Distribution of Impact Pressures

_{j}, the maximum impact P

_{j,max}is normalized by the maximum pressure impact of all compared sensors (P

_{max}= max{P

_{j,max}}). For this case, the absolute maximum was recorded by sensor P2 of the revetment with the small steps (compare Figure 1).

_{m}

_{0}/S

_{h}= 0.3), while the maximum horizontal impact is about four times lower than at the small steps (Test 103, H

_{m}

_{0}/S

_{h}= 1.7). Additionally, data from Reference [8] are shown. These data represent very small step heights (H

_{m}

_{0}/S

_{h}= 11) and indicate a maximum horizontal wave impact of about 1.5 times larger than for the steps with a step ratio of H

_{m}

_{0}/S

_{h}= 1.7. The smaller the step height in relation to the wave height, the higher the recorded horizontal loads become. The reduced impact for an increasing step height can be ascribed to the delayed run-down of the previous wave impact (compare Figure 3a,e). This run-down causes a constant water layer on the revetment, which buffers the wave impact. Moreover, the plunging wave breaking at a 1:2 slope with small steps (H

_{m}

_{0}/S

_{h}≥ 1.7) triggers impacting wave loads whereas the transformation of the breaking wave by large dominant step edges (H

_{m}

_{0}/S

_{h}= 0.3) leads to the formation of spilling wave breaking over the horizontal step front and triggers, thereby pulsating wave impacts.

_{max,vertical}= 0.5 P

_{max,horizontal}). Note that, in the context of the onshore-orientated wave propagation and wave breaking process, this finding is reasonable for design and constructional purposes in any practical applications of stepped revetments. The vertical impact for the large steps is negligible as it represents only the hydrostatic pressure induced by the overflow of the incident wave. Additionally, data from Reference [10] are given. The relative step height of H

_{m}

_{0}/S

_{h}= 7.5 is comparable to the data set provided by Reference [8] for horizontal wave impacts. The maximum presented impact is about two times larger than for the steps with a step ratio of H

_{m}

_{0}/S

_{h}= 1.7. An increase in the vertical impact loads is observed for decreasing step heights and is explained analogous to the horizontal wave impact.

_{99.6%}/(ρgH

_{m}

_{0}) on the abscissa and the relative position to the SWL (z/H

_{m}

_{0}) on the ordinate. The probability wave impact P

_{99.6%}is selected following the approach of Reference [11] to allow a comparison with the underlying data for wave impacts on vertical walls. For both examined step heights, the maximum pressure impact is close to the SWL. The maximum pressure decreases significantly within a range of ± z/H

_{m}

_{0}around the SWL. Within a range of ±2 z/H

_{m}

_{0}, the normalized pressure impact becomes low. The P

_{99.6%}wave loads measured over the stepped revetment tend to be about 50% smaller than those measured at a vertical wall. On the contrary, the impacts on the steps around the SWL exhibit pressures peak impacts which are comparable to those measured on vertical walls by Reference [11] (data points of single impacts not given in this figure scatter significantly along the abscissa). A dual-sided envelope curve (Equation (6)) with coefficients a, b and c according to Table 3 representing the best fit of data (coefficient of determination: R

^{2}= 0.62, STD = 0.189) is given to describe the correlation of the vertical distribution z/H

_{m}

_{0}of horizontal impacts 0.01 < P

_{99.6%}/(ρgH

_{m}

_{0}) ≤ 3.6.

_{m}

_{0}for revetments with large (0.2 < H

_{m}

_{0}/S

_{h}< 0.6, Figure 10a–d) and small (1.0 < H

_{m}

_{0}/S

_{h}< 3.5, Figure 10e–h) steps are shown. Each row represents data with a certain probability of exceedance, as described in Section 4.1. For comparison, the reference line for the horizontal impact forces [11] and the trend line according to Equation (6) are given. The pressure distribution predicted for P

_{99.6%}values by Equation (6) also shows a good agreement for P

_{max}values, if the step height is larger than the wave height (a). P

_{max}values for revetments with step heights smaller than the wave height (e) show higher impacts, which scatter in a range of ±2 z/H

_{m}

_{0}. The higher and more variable distributed impacts for small steps compared to large steps can again be ascribed to the influence of higher aeration [8,9]. With increasing probability of exceedance ((c,d) for large steps and (g–h) for small steps) the peaks around the SWL become less prominent, although the peaks are still visible. This trend is reasonable as the statistical distribution of the individual wave heights in a wave spectrum follows a Rayleigh distribution and the mean wave height of all waves, exceeding a certain threshold, decreases significantly with increasing probability of exceedance. Additionally, the shape of the front of a breaking wave governs the subsequent wave impact as the wave’s asymmetry has a secondary importance on the resulting impact. The overall distribution over the water depth becomes more homogeneous with the decreasing probability of exceedance for reasons discussed in Section 4.2.

## 5. Laboratory and Scale Effects

_{m}

_{0}, T

_{p}), minimized by an analysis of the incident waves very close to the structure.

## 6. Conclusions

_{m}

_{0}/S

_{h}< 3.5.

_{m}

_{0}< S

_{h}) show similar load cases compared to vertical walls. For small steps (H

_{m}

_{0}> S

_{h}) the impact peak is also clearly visible but the subsequent quasi-static peak P

_{q}is not as prominent as for large steps or vertical walls. Therefore the recommendation of Reference [7] to calculate the forces on stepped structures with the same method as for vertical walls, which only holds true for large steps.

_{r}/T

_{m–}

_{10}> 0.2. No impacting load case was detected at stepped revetments with large or small step height below the SWL. Real impacts are buffered by a water layer protecting the steps from violent impacts. Impacting conditions occur only very close to the SWL. Compared to impacts on vertical walls, the impact rising times are in the same range. The minima for stepped revetments is t

_{d}= t

_{r}. As the importance of aeration in the run-up to the wave impact is identified as a future study, it should focus on this effect in analogy to Reference [15].

_{m}

_{0}> S

_{h}) is located slightly below the SWL with an amplitude of about 50% of the maximum horizontal impact. The vertical impact for the large steps (H

_{m}

_{0}< S

_{h}) is negligible as it represents only the hydrostatic pressure induced by the overflow of the incident wave. An increase in the vertical impact loads is seen for decreasing step heights analogous to the horizontal wave impact. The maximum impact decreases significantly within a range of ± z/H

_{m}

_{0}, mainly in the range of ±2 z/H

_{m}

_{0}. The highest (P

_{99.6%}) wave loads measured over the stepped revetment tend to be about 50% smaller than those measured at a vertical wall. On the contrary, the impacts on the steps around the SWL showed impacts comparable to those measured on a vertical wall. Higher and more variable distributed impacts for small steps (H

_{m}

_{0}> S

_{h}) compared to large steps can be explained by the influence of higher aeration. With increasing probability of exceedance, the peaks around the SWL are less significant.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Model set-up of pressure sensors and position of the SWL for two 1:2 inclined stepped revetments. (

**a**) Cross-section of the flume set-up, (

**b**) detail of the large step configuration (S

_{h}= 0.3 m), and (

**c**) detail of the small step configuration (S

_{h}= 0.05 m).

**Figure 3.**Demonstration of flow processes for relative time-steps in the run-up process on revetments with smooth surface (

**a**–

**d**), small step heights (

**e**–

**h**), and large step heights (

**i**–

**l**).

**Figure 4.**Recorded pressure impacts and log-normal probability distributions of the maximum impact pressures for (

**a**) small step heights (test 103, pressure sensor P1, H

_{m}

_{0}/S

_{h}= 1.7) and (

**b**) large step heights (test 209, pressure sensor P2, H

_{m}

_{0}/S

_{h}= 0.3). (

**c**) Gives a detail of the log-normal probability distribution of the 18 largest impacts in test 209.

**Figure 5.**(

**a**) Parameter definition describing a pressure impact event and (

**b**) exemplary time series of the maximum impact event during test 204 for pressure sensors P1, P2 and P3.

**Figure 6.**Comparison of pressure events with a certain probability of exceedance between small steps ((

**a**–

**f**), H

_{m}

_{0}/S

_{h}= 1.7, test 103) and large steps ((

**g**–

**l**), H

_{m}

_{0}/S

_{h}= 0.3, test 209).

**Figure 7.**Correlation of the dimensionless impact duration t

_{d}/t

_{r}and the dimensionless impact rising time t

_{r}/T

_{m–}

_{1,0}at stepped revetments compared to findings for vertical walls. Given trend lines following Equation (5) represent upper envelopes of the findings. Straight horizontal lines represent the corresponding lower envelope line.

**Figure 8.**Horizontal (

**a**) and vertical (

**b**) relative maximum pressure impact distribution over large (dashed, Test 209, H

_{m}

_{0}= 0.089 m, and ξ = 2.8) and small (solid, Test 103, H

_{m}

_{0}= 0.084 m, and ξ = 2.9) stepped revetments.

**Figure 9.**Normalized pressure impact relative to the SWL (z = 0) for a 1:2 inclined stepped revetment.

**Figure 10.**Normalized horizontal pressure impacts with varying probability of exceedance and corresponding relative water depth z/H

_{m}

_{0}for large (0.2 < H

_{m}

_{0}/S

_{h}< 0.6) and small (1.0 < H

_{m}

_{0}/S

_{h}< 3.5) stepped revetments including theoretical laws for stepped revetments according to Equation (6) and for vertical walls according to Reference [11].

Test | S_{h} | R_{c} | H_{m}_{0} | T_{p} | h_{s} | N | ξ | H_{m}_{0}/L_{p} | H_{m}_{0}/h_{s} | H_{m}_{0}/S_{h} |
---|---|---|---|---|---|---|---|---|---|---|

# | (m) | (m) | (m) | (s) | (m) | (-) | (-) | (-) | (-) | (-) |

101 | 0.05 | 0.121 | 0.056 | 1.43 | 1.100 | 1298 | 3.7 | 0.018 | 0.051 | 1.12 |

102 | 0.063 | 1.20 | 1.100 | 427 | 3.0 | 0.028 | 0.057 | 1.26 | ||

103 | 0.084 | 1.38 | 1.100 | 1261 | 2.9 | 0.029 | 0.076 | 1.68 | ||

104 | 0.082 | 1.38 | 1.100 | 1256 | 3.0 | 0.028 | 0.074 | 1.63 | ||

105 | 0.082 | 1.38 | 1.100 | 1261 | 3.0 | 0.028 | 0.075 | 1.64 | ||

106 | 0.084 | 1.37 | 1.100 | 167 | 2.9 | 0.029 | 0.076 | 1.67 | ||

107 | 0.088 | 2.11 | 1.100 | 1422 | 4.2 | 0.015 | 0.080 | 1.76 | ||

108 | 0.114 | 2.20 | 1.100 | 171 | 3.8 | 0.019 | 0.104 | 2.28 | ||

109 | 0.119 | 2.81 | 1.100 | 179 | 4.6 | 0.016 | 0.108 | 2.38 | ||

110 | 0.143 | 2.26 | 1.100 | 162 | 3.5 | 0.023 | 0.130 | 2.86 | ||

111 | 0.211 | 0.085 | 2.09 | 1.010 | 1410 | 4.2 | 0.016 | 0.084 | 1.71 | |

112 | 0.085 | 2.07 | 1.010 | 1434 | 4.1 | 0.016 | 0.084 | 1.70 | ||

113 | 0.085 | 2.08 | 1.010 | 1371 | 4.1 | 0.016 | 0.084 | 1.70 | ||

201 | 0.30 | 0.121 | 0.111 | 1.37 | 1.100 | 1413 | 2.6 | 0.038 | 0.101 | 0.37 |

202 | 0.129 | 3.18 | 1.100 | 307 | 4.9 | 0.016 | 0.117 | 0.43 | ||

203 | 0.116 | 1.38 | 1.100 | 1428 | 2.5 | 0.040 | 0.106 | 0.39 | ||

204 | 0.167 | 2.08 | 1.100 | 278 | 3.0 | 0.030 | 0.151 | 0.56 | ||

205 | 0.211 | 0.064 | 1.36 | 1.010 | 1586 | 3.3 | 0.023 | 0.064 | 0.21 | |

206 | 0.091 | 1.34 | 1.010 | 1390 | 2.8 | 0.033 | 0.090 | 0.30 | ||

207 | 0.166 | 2.01 | 1.010 | 1515 | 2.9 | 0.032 | 0.164 | 0.55 | ||

208 | 0.300 | 0.064 | 1.41 | 0.921 | 1339 | 3.4 | 0.021 | 0.069 | 0.21 | |

209 | 0.089 | 1.38 | 0.921 | 1294 | 2.8 | 0.031 | 0.097 | 0.30 | ||

210 | 0.170 | 2.11 | 0.921 | 1468 | 2.9 | 0.032 | 0.184 | 0.57 |

**Table 2.**Coefficients a, b and c for Equation (5) and corresponding minimal impact duration t

_{d,min}.

Geometry | Impact Direction | a | b | c | t_{d,min} |
---|---|---|---|---|---|

Vertical walls [16] | horizontal | 2 | 8 | −18 | 2.0 t_{r} |

Stepped revetments | horizontal | 1 | 8 | −12 | 1.0 t_{r} |

vertical | 3 | 8 | −18 | 1.5 t_{r} |

z (SWL at z = 0) | a | b | c |
---|---|---|---|

z ≥ 0 | −1171.64 | 745.72 | −2831.66 |

z < 0 | 4.97 | −2.87 | −6.15 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kerpen, N.B.; Schoonees, T.; Schlurmann, T.
Wave Impact Pressures on Stepped Revetments. *J. Mar. Sci. Eng.* **2018**, *6*, 156.
https://doi.org/10.3390/jmse6040156

**AMA Style**

Kerpen NB, Schoonees T, Schlurmann T.
Wave Impact Pressures on Stepped Revetments. *Journal of Marine Science and Engineering*. 2018; 6(4):156.
https://doi.org/10.3390/jmse6040156

**Chicago/Turabian Style**

Kerpen, Nils B., Talia Schoonees, and Torsten Schlurmann.
2018. "Wave Impact Pressures on Stepped Revetments" *Journal of Marine Science and Engineering* 6, no. 4: 156.
https://doi.org/10.3390/jmse6040156