# Tensile Bending Stresses in Mortar-Grouted Riprap Revetments Due to Wave Loading

^{*}

## Abstract

**:**

_{x,max}/(ρ

_{w}gH

_{m}

_{0}) are given for four configurations of MGRRs that are of great practical relevance.

## 1. Introduction

#### 1.1. Motivation

_{90/250}(coarse particles of diameter 90–250 mm according to standard DIN EN 13383-1 [3]), LMB

_{5/40}or LMB

_{10/60}(light mass with stone mass of 5–40 kg and 10–60 kg according to DIN EN 13383-1, respectively). They can be constructed either by filling the entire pore volume of the riprap with mortar (fully grouted) or by partially filling the pore volume of the riprap with mortar while at the same time ensuring a mortar bond between the individual stones in the uppermost stone layer (partially grouted). Hence a fully grouted MGRR has an impermeable top layer with a rough surface, whereas a partially grouted MGRR has a rough, porous and permeable top layer. Figure 1 depicts the cross-sections of MGRRs and gives an impression of a partially and a fully grouted top layer.

^{2}) can be chosen using these technical standards and guidelines.

- Flexible revetments: Limited interaction between individual elements of the revetment results in the individual elements resisting wave loading mainly by their weight while the whole revetment can deform by redistribution of the single elements and thereby easily adapt to subsoil settlement. Non-grouted (loose) riprap forms a flexible revetment, see for example [8].
- Revetments with some flexibility: Revetments made of individual elements with high interaction or monolithic revetments that due to their creep behavior can adapt to subsoil settlement to a certain degree while remaining coherent. This type of revetment can resist higher wave loads than fully flexible revetments given the same size of the individual elements or the same revetment thickness, but especially asphalt is subject to fatigue and abrasion [9]. Asphalt revetments or riprap grouted with asphalt are examples of revetments with some flexibility.
- Revetments with no flexibility: These monolithic revetments or revetments made of individual elements with high interaction are too stiff to follow subsoil settlement. It is therefore possible that cavities beneath the revetment are present, resulting in a non-continuous bedding of the revetment. This type of revetment can also resist higher wave loads than fully flexible revetments given the same size of the individual elements or the same revetment thickness, but it is subject to breaking and crack formation. Mortar-grouted riprap revetments are revetments with no flexibility [10].

- For very permeable revetments such as loose riprap the pressure beneath the revetment almost instantaneously adapts to the pressure on the revetment. In case of small amounts of grouting mortar, MGRRs also represent this type of revetment.
- With decreasing permeability of the top layer the load on the top layer is increasingly dominated by the pressure difference on and beneath the top layer. For pattern placed revetments the leakage length, which takes into account the permeability and thickness of both the filter layer and top layer of the revetment, is a measure for the loading of the top layer [15]. In case of high amounts of grouting mortar, MGRRs can qualitatively represent this type of revetment. However, the concept of the leakage length assumes flow in the filter layer parallel to the embankment and can therefore not be applied for MGRRs that are in most cases placed on a geotextile filter layer, because the thickness of a geotextile filter is in the order of a few centimeters and does not allow any significant flow in its plane.
- In the case of a non-permeable top layer, the pressure beneath the revetment adapts to the pressure on the revetment only for time periods of several hours or longer (tides, storms), but not for the time periods of individual waves [9]. This is the case for fully grouted MGRRs and for example for asphalt revetments.

#### 1.2. Modelling Tensile Stresses for Crack Formation Assessment

_{max}and width B (see Figure 2), which are found in hydraulic experiments, see for example Oumeraci et al. [19]. The wave impact magnitude is determined with the following equation [16]:

_{p}is a dimensionless wave impact coefficient, ρ

_{w}the density of water, g the acceleration due to gravity and H

_{s}the significant wave height. Major influences on the wave impact coefficient k

_{p}are structural parameters of the revetment, namely roughness and porosity, as well as the revetment slope angle α [20]. Table 1 gives an overview of the different parameters used for each type of revetment.

#### 1.3. Objective and Outline

## 2. Materials and Methods

#### 2.1. Plate on an Elastic Foundation Model

_{yy}is the bending stiffness of the plate, w its deflection, k

_{s}the modulus of subgrade reaction and q the external load. The x-axis is defined in the longitudinal direction of the plate.

_{y}is calculated from the deflection as follows:

^{®}, Natick, Massachusetts, United States). The algorithm gabamp.m (Gauss algorithm with column pivoting for asymmetric band matrices) of Dankert and Dankert [23] was used to solve the resulting system of equations.

#### 2.2. Full Scale Hydraulic Model Tests

_{t}= 0.40 m. Before the first test phase, both revetments were grouted with 80 L/m

^{2}mortar. The revetments were grouted by hand by experienced contractors that have been grouting MGRRs for decades. Assuming a porosity of 0.45 before grouting (riprap dumped in dry conditions, medium density according to [5]) this resulted in permeable revetments after grouting with a porosity of n = 0.25 (d

_{t}= 0.40 m) and n = 0.32 (d

_{t}= 0.60 m), respectively.

^{2}, confirming the assumption of a porosity of 0.45 before grouting, which results in an available pore volume of V

_{p}= 180 L/m

^{2}(V

_{p}= n × d

_{t}× 1000 L/m

^{3}= 0.45 × 0.40 m × 1000 L/m

^{3}) before grouting.

^{2}, confirming that the lower part of the cross-section of the top layer still exhibited a free pore volume and was only partially grouted. The potential free pore volume before any grouting of the top layer took place (before test phase one) was 270 L/m

^{2}(0.45 × 0.60 m × 1000 L/m

^{3}), thus a pore volume of 90 L/m

^{2}was still available in the lower part of the top layer in test phase two. The overall volumetric porosity of the revetment equaled 0.16, while it was at the same time impermeable for any flow perpendicular to the embankment.

_{10,H50mm}= 2.86 × 10

^{−3}m/s, thickness 8 mm, characteristic opening size 0.1 mm), which was directly placed on the sand embankment. The toe structure of the revetment below the lowest wave run-down was made of concrete blocks in order to support the revetment. By this means, an efficient construction of the revetment was ensured, covering only the relevant section of the slope with grouted riprap material. The crest of the embankment was constructed of concrete blocks behind which any overtopping water was collected in a basin and discharged back into the flume during the tests.

_{5/40}with a median weight of G

_{50}= 23.5 kg and a narrow grading with d

_{85}/d

_{15}= 1.5. The grouting mortar can either be produced by adding additives to the mortar or by mixing it in a colloidal mixer with high velocity shear action [24] to ensure an appropriate consistency of the fresh grout for a distribution of mortar inside the pore space of the riprap that results in a “sufficient” permeability and bonding of the individual stones. The latter method was chosen for grouting the riprap for the experiments in the GWK. The mortar was tested and fulfilled all requirements stated by the code of practice “Use of Cementitious and Bituminous Materials for Grouting Armourstone on Waterways” [5].

_{m}

_{0}= 0.38–0.94 m and wave periods from T

_{m−}

_{1,0}= 2.7–9.0 s were generated which resulted in surf similarity parameters ξ

_{m−}

_{1,0}= 1.55–4.64. In this way a wide range of wave loading conditions and breaker types were covered. The still water level in all experiments was set to 4 m.

_{m−}

_{1,0}, the fictitious wave length in deep water according to EurOtop [27] was calculated:

#### 2.3. Boundary Conditions PEF

#### 2.4. Plate on an Elastic Foundation Model

#### 2.4.1. Bending Stiffness

- The materials cannot shift against each other at their interfaces (no slippage).
- Since the behavior of a mortar-grouted riprap is to be described up to the first crack formation, a linear relationship between stress and strain is assumed in the uncracked state according to Hooke’s law.

_{RR}and E

_{M}are the moduli of elasticity of riprap (index RR) and mortar (index M), respectively, A is the area of each component in the cross-section and $\overline{z}$ is the distance of the center of gravity of each surface from the center of the ideal total cross-section, which corresponds to the center of gravity of the cross-section under the assumptions made. The bending stiffness of the top layer is therefore calculated as follows:

_{RR}and a

_{M}are the area shares of the components in the total cross-section, d

_{t}is the top layer thickness, b the cross-section width and μ

_{RR}and μ

_{M}Poisson’s ratio of riprap and mortar, respectively. The stress in the respective components can be calculated as a function of the distance from the center of the ideal cross-section according to:

_{M}= 0.06 (coefficient of variation σ’ = 0.10) for the mortar and μ

_{RR}= 0.23 (coefficient of variation σ’ = 0.11) for the riprap.

#### 2.4.2. Modulus of Subgrade Reaction

_{s}, which must be determined in physical tests for the respective soil. Peters [15] points out that there are no measurements of the modulus of subgrade reaction for soils beneath revetments. For the calculations carried out here, the modulus of subgrade reaction is therefore obtained from literature. As a non-exhaustive summary, Table 5 shows the range of possible values of moduli of subgrade reaction and shows which values were used in other publications to calculate the stresses in the cross-section of a revetment using a PEF model.

^{3}for the modulus of subgrade reaction for soils beneath revetments are realistic because, in contrast to soils beneath other structures like building foundations or roads, the soil beneath a revetment is often not treated and compacted as elaborately. Especially in the case of permeable revetments, the external loads additionally cause the pore water pressures in the soil beneath the revetment to oscillate and thus loosen rather than compact the grain structure of the soil, which further reduces the modulus of subgrade reaction [15,34]. Richwien [34] points out that when designing revetments with a PEF model, the soil properties that may change during a storm event must be considered by adjusting the modulus of subgrade reaction accordingly. The calculations for MGRRs are performed with k

_{s}= 10 MN/m

^{3}and k

_{s}= 50 MN/m

^{3}. These values are intended to represent sandy soils in different degrees of compaction during cyclic wave loading.

#### 2.4.3. Parameter Variations

## 3. Results

#### 3.1. Paramter Combination 1

_{x,max}/(ρ

_{w}gH

_{m}

_{0}) at the edge of the cross-section (the maximum bending stresses either occur at the top or bottom edge of the cross-section) of the partially grouted MGRRs over the surf similarity parameter ξ

_{m-}

_{1,0}for parameter combination 1, i.e., k

_{s}= 50 MN/m

^{3}. The relative stress is calculated by dividing the stress by a fictitious hydrostatic pressure at wave height H

_{m}

_{0}. The magnitude of the bending stresses depends in particular on the mean wave height H

_{m}

_{0}and the way the waves break on the revetment or are reflected at the revetment, which is taken into account by depicting the bending stresses as a function of the surf similarity parameter ξ

_{m-}

_{1.0}. In Figure 8 and all following figures depicting the stresses, no distinction is made between the maximum stress at the upper edge or bottom edge of the top layer.

_{x,qst}resulting from the quasi-static component of the wave load and the impact component σ

_{x,imp}resulting from wave impacts:

_{1}and c

_{2}are empirical factors. Equation (10) gives meaningful limit values for quasi-static loads, as f(ξ

_{m−}

_{1,0})→0 for ξ

_{m−}

_{1,0}→0, i.e., for spilling breakers the stresses tend to zero. For ξ

_{m−}

_{1,0}→∞, f(ξ

_{m−}

_{1,0}) tends to a limit value that is dependent on the revetment characteristics. As the maximum relative stresses for ξ

_{m−}

_{1.0}> 2.5 mostly occur during wave run-down, this limit value is particularly dependent on the time it takes for the pressure beneath the top layer to adapt to the pressure on the top layer, which is dependent on the porosity and permeability of the top layer.

_{m−}

_{1,0})→0 for ξ

_{m−}

_{1,0}→0 and f(ξ

_{m−}

_{1,0})→0 for ξ

_{m}

_{−}

_{1,0}→∞, as the number of impact loads tends to zero for spilling breakers as well as for reflected waves. For this reason, the Technical Advisory Committee on Flood Defence [9] as well as Alcérreca-Huerta and Oumeraci [11] represent the magnitude of wave impacts by a Rayleigh distribution. The sum of quasi-static load and impact load (Equation (9)) tends to the quasi-static load (f

_{total}(ξ

_{m}

_{−}

_{1,0}) → f

_{quasi-static}(ξ

_{m}

_{−}

_{1,0})) for ξ

_{m}

_{−}

_{1,0}→ ∞. The quasi-static component of the relative bending stress can be understood as the lower envelope function and the sum of the quasi-static component and the component due to the impact load as the upper envelope function. The empirical parameters c

_{1}–c

_{4}in Equations (10) and (11) are given for each MGRR configuration in Table 7. In Figure 8 and all following figures depicting the relative bending stress, the solid lines show the validity range of the functions for which pressure measurement data from the physical model tests in the GWK is available. The dashed lines show the values of the function for smaller surf similarity parameters where no pressure measurements are available.

_{m}

_{−}

_{1.0}≤ 2.5 in Figure 8 the relative bending tensile stress and its scattering increase due to the higher probability of occurrence of wave impacts and due to the increase in their magnitude. For larger surf similarity parameters (ξ

_{m}

_{−}

_{1.0}> 2.5), the relative bending stress seems to reach a limit value. The non-breaking waves occurring in this range of surf similarity parameters therefore do not seem to be able to generate larger pressure differences as a result of wave rundown, since the top layers are permeable and thus the pressure beneath the top layers can adapt quickly to the pressure on the top layers. Given the same load, the relative bending stresses are lower for the 0.60 m thick revetment, because of its higher bending stiffness. This results in lower stresses in the 0.60 m thick revetment for almost all wave loads measured in the GWK, see also Figure 9.

_{m}

_{−}

_{1.0}≈ 2.4, compared to other values for similar surf similarity parameters, are due to a particularly high wave impact load. This highlights the inherent natural variability of the magnitude of the wave impact load, which becomes particularly apparent when considering the maximum values of the bending tensile stresses. Figure 9 shows the calculated maximum relative tensile bending stress σ

_{x}/(ρ

_{W}gH

_{m}

_{0}) at the edge of the cross-section of the fully grouted MGRR and regrouted MGRR as a function of the surf similarity parameter ξ

_{m}

_{−}

_{1,0}for parameter combination 1.

_{m}

_{−}

_{1.0}≤ 2.5) than in the case of permeable top layers (cf. Figure 8).

_{m−}

_{1,0}≈ 1.9 again highlights the inherent natural variability of the magnitude of the wave impact load.

_{m−}

_{1.0}≤ 2.5 is small in the case of the regrouted top layer (d

_{t}= 0.60 m). The pressure changes more quickly beneath the top layer in the case of the regrouted top layer compared to the fully grouted top layer for the same wave conditions. The peak pressure beneath the regrouted revetment in the instant of maximum stresses during wave impact was about 20% of the peak pressure on the revetment in that same instant. As a consequence, the pressure difference is reduced. This also decreases the variation in the relative bending stresses for ξ

_{m−}

_{1.0}≤ 2.5 as the pressure gradient along the top layer decreases in comparison to the fully grouted MGRR. Another reason for lower variation of the relative bending stress for the regrouted top layer is the fact that the variation of the peak pressure on the revetment was smaller than for the fully grouted MGRR, which is most probably caused by the natural variability of the wave impact load in combination with slightly different surface characteristics of the top layers.

_{regrouted}= 0.16 remained.

_{m−}

_{1.0}> 50 m (corresponds approximately to surf similarity parameters ξ

_{m−}

_{1.0}> 3 in Figure 9), greater wave run-down heights were observed for both impermeable cover layers. Because the PTs did not cover the entire length of the revetment (the most seaward PT on the top edge of the top layer has a coordinate of z = 2.28 m), they did not resolve the complete load figure during wave run-down for such long wavelengths. Therefore, for the impermeable MGRRs and wavelengths L

_{m−}

_{1.0}> 50 m, the PTs do not provide reliable boundary conditions for the model. The calculated relative bending stresses are therefore not meaningful. The validity of the given functions and empirical parameter (see Table 7) is therefore limited to wavelengths L

_{m−}

_{1,0}< 50.

#### 3.2. Parameter Combination 2

_{x}/(ρ

_{W}gH

_{m}

_{0}) at the edge of the cross-section (a) of the partially grouted MGRRs and (b) of the fully and regrouted MGRRs over the surf similarity parameter ξ

_{m-}

_{1,0}for parameter combination 2. Again, functions for the upper envelope of the results are defined.

_{s}= 10 MN/m

^{3}) leads to greater deflections and thus to higher relative bending stresses for the same external load and bending stiffness of the revetments. The relative bending stresses are therefore increased in comparison to parameter combination 1. The same remarks regarding the validity range of the equations apply as for parameter combination 1. Table 7 gives an overview of the empirical coefficients used for the equations.

## 4. Discussion

_{s}= 50 MN/m

^{3}, then the tensile bending stresses in all experiments in the GWK were σ

_{x}< 0.1 N/mm

^{2}for the partially grouted MGRRs and σ

_{x}< 0.3 N/mm

^{2}for the fully grouted and regrouted MGRRs. These stresses are lower than the average values of the tensile strength of colloidal mortar (β

_{t}≈ 0.96 N/mm

^{2}) as well as the adhesive tensile strength (β

_{t,adh}≈ 0.55 N/mm

^{2}) and adhesive bending tensile strength (β

_{bt,adh}≈ 1.44 N/mm

^{2}) between riprap and colloidal mortar as determined by the Institute of Building Materials Research at RWTH Aachen University [28,31]. The laboratory conditions in the GWK facilitated quality assurance of the application of the mortar as well as of the grouting mortar itself, supporting the assumption that the same strengths and adhesive strengths were present in the GWK experiments. The results of the PEF model therefore suggest that no cracks over the whole width of the revetments would occur and none were observed in the GWK experiments. Therefore, the observations are in line with the results of the PEF model.

## 5. Conclusions

_{x,max}/(ρ

_{w}gH

_{m}

_{0}) are given for four configurations of MGRRs that are of great practical relevance. The procedure described herein can be used for further boundary conditions, e.g., for determining the equations of the enveloping functions of the 98% quantile of the relative bending stresses.

_{m−}

_{1,0}> 2.5 the load resulting from the wave run-down must also be considered. A detailed analysis and subsequent parametrization of the pressure distributions along the top layer due to wave loading will be presented in a follow-up paper.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Introduction

#### Appendix A.2. Materials, Methods and Results: Riprap

**Figure A1.**(

**a**) Stone (specimen 3) of group 1 and (

**b**) detail of stone surface. (

**c**) Stone (specimen 6) of group 2 and (

**d**) detail of its surface. Stone group 1 has a coarser grain size and darker minerals than stone group 2 (after [31]).

^{3}with a coefficient of variation of 4%. The porosity was 0.3% with a coefficient of variation of 24%.

**Figure A2.**Testing the static modulus of elasticity on a stone under compression. (

**a**) Specimen in test facility (Instron 5587), (

**b**) detail showing the strain gauges and (

**c**) stone after failure (after [31]).

**Table A1.**Static modulus of elasticity of revetment stones under compression and tension as well as compressive strength, Poisson’s ratio and tensile strength (after [31]).

Compression | Tension | |||||||
---|---|---|---|---|---|---|---|---|

Specimen No. | Group | Compressive Strength β_{D} | Modulus of Elasticity E_{C} | Poisson’s Ratio µ | Speci-men No. | Group | Tensile Strength β_{Z} | Modulus of Elasticity E_{T} |

(N/mm^{2}) | (N/mm^{2}) | (-) | (N/mm^{2}) | (N/mm^{2}) | ||||

3 | 1 | 268.3 | 75,385 | 0.218 | 3 | 1 | 13.4 | 64,861 |

4 | 1 | 231.2 | 60,951 | 0.223 | 4 | 1 | 10.3 | 40,394 |

7 | 1 | 143.4 | 67,475 | 0.200 | 7 | 1 | 8.6 | 52,698 |

6 | 2 | 257.0 | 63,331 | 0.247 | 1 | 2 | 8.5 | 77,009 |

9 | 2 | 171.4 | 76,771 | 0.268 | 2 | 2 | 11.7 | 69,932 |

10 | 2 | 141.6 | 68,686 | 0.213 | 8 | 2 | 11.6 | 53,792 |

9 | 2 | 16.2 | 74,870 | |||||

Mean | 202.2 | 68,767 | 0.228 | Mean | 11.47 | 61,936 | ||

Coefficient of variation | 0.28 | 0.09 | 0.11 | Coefficient of variation | 0.24 | 0.22 |

#### Appendix A.3. Materials, Methods and Results: Grouting Mortar

**Figure A3.**Testing the static modulus of elasticity on grouting mortar under compression. (

**a**) Specimen in test facility (Form+Test Prüfsysteme, ALPHA 4-3000) and (

**b**) detail showing the measuring chain and inductive displacement transducers (after [31]).

**Table A2.**Static modulus of elasticity of grouting mortar under compression as well as compressive strength and Poisson’s ratio (after [31]).

Compression | Tension | |||||
---|---|---|---|---|---|---|

Specimen No. | Compressive Strength β_{D} | Modulus of Elasticity E_{C} | Poisson’s Ratio µ | Specimen No. | Tensile Strength β_{Z} | Modulus of Elasticity E_{T} |

(N/mm^{2}) | (N/mm^{2}) | (-) | (N/mm^{2}) | (N/mm^{2}) | ||

1 | 35.6 | 22,532 | 0.061 | 1 | 1.1 | 18,621 |

2 | 35.4 | 23,229 | 0.054 | 2 | 1.0 | 17,766 |

3 | 33.1 | 22,962 | 0.067 | 3 | 1.0 | 21,329 |

4 | 37.8 | 24,287 | 0.066 | 4 | 0.7 | 20,085 |

5 | 34.5 | 23,591 | 0.061 | 5 | 1.0 | 19,802 |

6 | 31.6 | 23,020 | 0.052 | |||

Mean | 34.7 | 23,270 | 0.060 | Mean | 0.96 | 19,520 |

Coefficient of variation | 0.06 | 0.03 | 0.10 | Coefficient of variation | 0.16 | 0.07 |

#### Appendix A.4. Discussion

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**Figure 1.**View of (

**a**) a partially grouted top layer and (

**b**) a fully grouted top layer (Photos: Kreyenschulte, 2015) as well as schematic representations of the cross-section of (

**c**) partially grouted MGRRs and (

**d**) fully grouted MGRRs (modified after [4]).

**Figure 2.**Plate on an elastic foundation (PEF) and wave impact load, idealized as triangular load (after [16]).

**Figure 3.**Partially grouted MGRRs (v

_{g}= 80 L/m

^{2}) in test phase one (Photos: Kreyenschulte and Kühling, 2017).

**Figure 4.**(

**a**) Schematic longitudinal cross-section of the GWK showing the wave paddle at the left, the instrumentation along the flume and the revetment on the right. (

**b**) Top view of the revetment section with the pressure transducers on and beneath the top layer in red and black, respectively.

**Figure 6.**(

**a**) Top view and (

**b**) close up of the placement of the steel channel for the PTs in the top layer. (

**c**) Demolition of the 0.40 m thick revetment on the south side before test phase two in order to construct a new revetment. The steel channel is visible in the lower right part of the picture. Notice also the partial grouting of test phase one, recognizable by portions of free pore space and grout (Photos: Kreyenschulte, 2017).

**Figure 7.**Model of the MGRRs as a plate on an elastic foundation with boundary conditions and model parameters (after [28]).

**Figure 8.**Maximum relative tensile bending stress σ

_{x,max}/(ρ

_{w}gH

_{m}

_{0}) dependent on the surf similarity parameter ξ

_{m−}

_{1,0}with enveloping functions (permeable, partially grouted MGRRs, parameter combination 1). The quasi-static component of the relative bending stress from Equation (9) is shown as a dotted line. The dashed line for illustration shows the enveloping functions for surf similarity parameters for which no experimental data is available (after [28]).

**Figure 9.**Maximum relative tensile bending stress σ

_{x,max}/(ρ

_{w}gH

_{m}

_{0}) dependent on the surf similarity parameter ξ

_{m−}

_{1,0}with enveloping functions (impermeable, regrouted and fully grouted MGRR, parameter combination 1). The quasi-static component of the relative bending stress from Equation (9) is shown as a dotted line. The dashed line for illustration shows the enveloping functions for surf similarity parameters for which no experimental data is available (after [28]).

**Figure 10.**Maximum relative tensile bending stress σ

_{x,max}/(ρ

_{w}gH

_{m}

_{0}) dependent on the surf similarity parameter ξ

_{m−}

_{1,0}with enveloping functions (parameter combination 2) for (

**a**) permeable, partially grouted MGRR and (

**b**) impermeable, regrouted and fully grouted MGRR (after [28]).

**Table 1.**Parametrization of the wave load in the design of different kinds of monolithic or coherent revetments with a PEF model.

Revetment | k_{p} | Width B | Slope Angle Tan (α) | Reference |
---|---|---|---|---|

Asphalt | approx. 2–6 * | H_{s} | 1:3–1:8 | [9,16] |

Polyurethane bonded revetment | 4 | H_{s} | 1:3 | [17,21] |

Riprap fully grouted with asphalt | approx. 2–6 * | H_{s} | 1:3–1:8 | [18] |

Partially Grouted | Fully Grouted | |||
---|---|---|---|---|

Section | North | South | North | South |

Configuration Nr. | 1 | 2 | 3 | 4 |

Top layer thickness d_{t} (m) | 0.6 | 0.4 | 0.6 | 0.4 |

Amount of mortar v_{g} (L/m^{2}) | 80 | 80 | 80 + 100 | 180 |

Estimated porosity n before grouting (-) | 0.45 | 0.45 | 0.45 | 0.45 |

Porosity n after grouting (-) | 0.32 | 0.25 | 0.16 | 0 |

Pore volume V_{p} after grouting (L/m^{2}) | 190 | 100 | 90 | 0 |

**Table 3.**Static modulus of elasticity E

_{stat,}

_{33}of the components of MGRRs, dependent on the load (after [31]).

Component and Load | Static Modulus of Elasticity E_{stat,}_{33} (N/mm^{2}) | ||||
---|---|---|---|---|---|

Minimum | Mean | Maximum | Standard Deviation | Coefficient of Variation (–) | |

Riprap (Compression) | 60,951 | 68,767 | 76,771 | 6327 | 0.09 |

Riprap (Tension) | 40,394 | 61,936 | 77,009 | 13,433 | 0.22 |

Mortar (Compression) | 22,532 | 23,270 | 24,287 | 606 | 0.03 |

Mortar (Tension) | 17,766 | 19,520 | 21,329 | 1374 | 0.07 |

Top Layer Thickness d_{t} (m) | Description | Static Modulus of Elasticity E_{stat,}_{33} (N/mm^{2}) | Area Share of Components in Cross-Section (–) | Bending Stiffness
$\overline{\mathit{E}\mathit{I}}\text{}$(MNm^{2})
| ||
---|---|---|---|---|---|---|

Riprap | Mortar | a_{RR} | a_{M} | |||

0.40 | Fully grouted | 60,000 | 20,000 | 0.55 | 0.45 | 224.00 |

Partially grouted, v_{g} = 80 L/m^{2} | 0.20 | 197.33 | ||||

0.60 | Regrouted, v_{g} = 80 + 100 L/m^{2} | 60,000 | 20,000 | 0.55 | 0.29 | 698.40 |

Partially grouted, v_{g} = 80 L/m^{2} | 0.13 | 642.00 |

Soil | k_{s} (MN/m^{3}) | Description | Reference |
---|---|---|---|

“Sandy soil“ | 10–20 | Recommendation for soils beneath revetments, wave impact load | [15] |

“Dense sand“ | 100 | Beneath asphalt revetments, wave impact load | [32] |

64 | |||

“Sand“ | 40–100 * | Beneath asphalt revetments | [9] |

35–60 | Beneath asphalt revetments, quasi-static load | [33] | |

50–90 | Beneath asphalt revetments, wave impact load |

^{2}for medium to well compacted sand, proctor density >95%.

Top Layer Thickness d_{t} (m) | Description | Bending Stiffness
$\text{}\overline{\mathit{E}\mathit{I}}\text{}$(MNn^{2})
| Modulus of Subgrade Reaction k_{s} (MN/m^{3}) | Denomination |
---|---|---|---|---|

0.40 | Fully grouted | 224.00 | 50 | Parameter combination 1 |

Partially grouted, v_{g} = 80 L/m^{2} | 197.33 | |||

0.60 | Regrouted, v_{g} = 80 + 100 L/m^{2} | 698.40 | ||

Partially grouted, v_{g} = 80 L/m^{2} | 642.00 | |||

0.40 | Fully grouted | 224.00 | 10 | Parameter combination 2 |

Partially grouted, v_{g} = 80 L/m^{2} | 197.33 | |||

0.60 | Regrouted, v_{g} = 80 + 100 L/m^{2} | 698.40 | ||

Partially grouted, v_{g} = 80 L/m^{2} | 642.00 |

**Table 7.**Empirical coefficients of the enveloping functions of the maximum relative bending stress for each parameter combination and top layer configuration.

d_{t} (m) | Description | c_{1} | c_{2} | c_{3} | c_{4} | Restriction of Validity * | |
---|---|---|---|---|---|---|---|

0.40 | Fully Grouted | 20 | 0.2 | 30 | 1.8 | L_{m}_{−}_{1,0} <50 m | Parameter combination 1 (k _{s} = 50 MN/m^{3}) |

Partially grouted, v_{g} = 80 L/m^{2} | 7 | 0.4 | 14 | 1.5 | - | ||

0.60 | Regrouted, v_{g} = 80 + 100 L/m^{2} | 20 | 0.2 | 10 | 1.8 | L_{m}_{−}_{1,0} <50 m | |

Partially grouted, v_{g} = 80 L/m^{2} | 5 | 0.4 | 8 | 1.5 | - | ||

0.40 | Fully grouted | 40 | 0.2 | 50 | 1.8 | L_{m}_{−}_{1,0} <50 m | Parameter combination 2 (k _{s} = 10 MN/m^{3}) |

Partially grouted, v_{g} = 80 L/m^{2} | 12 | 0.4 | 20 | 1.5 | - | ||

0.60 | Regrouted, v_{g} = 80 + 100 L/m^{2} | 30 | 0.25 | 18 | 1.8 | L_{m}_{−}_{1,0} <50 m | |

Partially grouted, v_{g} = 80 L/m^{2} | 10 | 0.4 | 14 | 1.5 | - |

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## Share and Cite

**MDPI and ACS Style**

Kreyenschulte, M.; Schüttrumpf, H.
Tensile Bending Stresses in Mortar-Grouted Riprap Revetments Due to Wave Loading. *J. Mar. Sci. Eng.* **2020**, *8*, 913.
https://doi.org/10.3390/jmse8110913

**AMA Style**

Kreyenschulte M, Schüttrumpf H.
Tensile Bending Stresses in Mortar-Grouted Riprap Revetments Due to Wave Loading. *Journal of Marine Science and Engineering*. 2020; 8(11):913.
https://doi.org/10.3390/jmse8110913

**Chicago/Turabian Style**

Kreyenschulte, Moritz, and Holger Schüttrumpf.
2020. "Tensile Bending Stresses in Mortar-Grouted Riprap Revetments Due to Wave Loading" *Journal of Marine Science and Engineering* 8, no. 11: 913.
https://doi.org/10.3390/jmse8110913