static and dynamic loads on the first row of single layer

i

Static and dynamic loads on the first row

of interlocking, single layer armour units

Master thesis

M.A. van de Koppel

May 2012

Delft University of Technology

Faculty of Civil Engineering & Geosciences

Section Hydraulic Engineering

Delta Marine Consultants

Department Coastal Engineering

Static and dynamic loads on the first row

of interlocking, single layer, armour units

Master thesis

M.A. van de Koppel

This research is performed for the partial fulfilment of requirements for the Master of Science degree

at the University of Technology, Delft, The Netherlands

Graduation Committee:

Prof.dr.ir. W.S.J. Uijttewaal Delft University of Technology

Ir. J. van den Bos Delft University of Technology

Ir. H.J. Verhagen Delft University of Technology

Ir. M. Muilwijk Delta Marine Consultants / BAM Infraconsult

List of trademarks in this report:

Delta Marine Consultants (DMC) is a registered trade name of BAM Infraconsult bv, The

Netherlands

Xbloc is a registered trademark of Delta Marine Consultants (DMC), The Netherlands

Xbase is a registered trademark of Delta Marine Consultants (DMC), The Netherlands

The use of trademarks in any publication of Delft University of Technology does not imply any

endorsement or disapproval of this product by the University.

Delft University of Technology

Faculty of Civil Engineering & Geosciences

Section Hydraulic Engineering

Delta Marine Consultants

Department Coastal Engineering

Static and dynamic loads on the first row of interlocking, single layer, armour units

iii

Preface

This thesis contains the results of a study on the static and dynamic loads on the first (bottom) row of

interlocking, single layer armour units. This study was performed in order to fulfil the requirements

for the degree of Master of Science in Civil Engineering at Delft University of Technology.

This research was supervised and evaluated by a graduation committee consisting of the following

persons: Prof. Dr. Ir. W.S.J. Uijttewaal, Ir. J. van den Bos Ir. H.J. Verhagen of Delft University of

Technology and Ir. M. Muilwijk of Delta Marine Consultants / BAM Infraconsult. I would like to thank

the members of this graduation committee for their very useful advice and their supervision on this

research.

During this research physical model test were performed in the wave flume of Delta Marine

Consultants which is a trademark of BAM Infraconsult. I would like to thank Delta Marine Consultants

for the support of my research and I would like to thank the personnel of Delta Marine Consultants

for their advice and the good working environment they created.

I would like to thank Greta and Joost for their editorial contribution to this thesis. And last but not

least I thank my parents for their support throughout my study and also my girlfriend, Leonie, for her

editorial contribution and her support.

I wish the reader of this thesis as much pleasure and interest as I had conducting this research.

Michael van de Koppel

Gouda, May 2012


iv

Abstract

Interlocking, single layer concrete armour units are placed in a specific grid depending on the type of

armour unit. Within this grid, armour units are placed in horizontal rows. The number of horizontal

rows of single layer armour units on a breakwater is limited to 20. This limit is proposed in order to

prevent major settlements, which might affect the interlocking of the armour units. The limit on the

number of rows is based on experience from prototypes and is not yet confirmed in a systematic

study. Then number of rows also might have an effect on the load on the first (bottom) row of armour

units, which affects the structural integrity of the armour units. The load on the first row of armour

units is however unknown. The research presented in this thesis is a study on the load on the first

(bottom) row of concrete armour units placed on a breakwater.

The objective of this research was to study the magnitude of the occurring load and the relevant

processes influencing this load. The total load on the first row of armour units can be decomposed in

a number of loads from which the static and the dynamic load were further studied in this research.

Based on literature two hypotheses were developed; one regarding the static load and one regarding

the dynamic load.

Experiments

The static load and the dynamic load were studied in separate experiments. The static load was

studied in an experiment in which this static load on the first row of (366 gram Xbloc) armour units

was continuously measured as function of the number of rows applied on the slope of a model

breakwater (up to 20 rows) with a slope of 37 degrees (3V:4H). This experiment, which was repeated

15 times, resulted in a graph that displays the measured static load on the first row of armour units as

function of the number of rows applied on the slope of the model breakwater. A second static

experiment was executed in order to study the individual force balance. The critical angle at which an

armour unit is just stable was determined. From this data the individual force balance could be

determined.

The dynamic load was studied in a physical model test in a wave flume. The dynamic load was

measured during tests with regular waves of 20% to 100% of the maximum wave height

corresponding to the used (62 gram Xbloc) armour unit and a wave period corresponding to an

Iribarren number of 3, 4 and 5 for all of the described wave heights. Several parameters were varied

resulting in a total of 10 tests, each with a unique combination of parameters which were repeated

for at least three times. The varied parameters are: the number of rows applied on the breakwater,

the relative packing density (RPD) of the armour layer, the permeability of the core, the smoothness

of the underlayer and the slope of the breakwater. These cases (different tests) were compared to a

reference case. The reference case is a breakwater model with a slope of 3:4, a total of 20 rows with a

RPD of 100% and a normal permeable core and a normal smoothness of the underlayer. The

parameters were changed one at the time, relative to the reference case.

Results and conclusions static load

Based on a theoretical analysis of the individual force balance of an armour unit a linear relation

between the number of rows applied on a breakwater and the static load was expected. The results

from the static experiment showed however a relation between the static load on the first row of

armour units and the number of applied rows which was not linear but flattened with an increasing

number of rows and approached a constant static load of 1,1 times the unit weight per armour unit

(see figure).


Theoretical and measured static load on the first row of armour units

The static load on the first row of armour units originates from the along slope, individual force

balance of the armour units. The force balance of a single unit consist of the along slope weight

component of the unit and the friction of the unit with the under layer. An imbalance of this

individual force balance results in a load on the lower positioned armour units. The transfer of these

loads on lower positioned armour units add up and result eventually in the load on the first row of

armour units. The relation between the number of rows and the static load approaches and

equilibrium, which can be explained by the difference between the individual force balances and thus

the different transfer of forces to the under layer by the various armour units.

From the study on the critical angle of the slope at which an armour unit is just stable it was learned

that a certain number of armour units are positioned in a very stable position. These armour units are

able to transfer high forces to the under layer and as a result, transfer less force to the lower

positioned armour units. The influence of these units on the static load on the first row of armour

units was modelled in a one‐dimensional model which was able to reproduce the shape of the

measured static load on the first row of armour units as function of the number of rows applied on

the slope of a breakwater. These very stable positioned armour units are thus responsible for the

observed flattening of the relation between the static load on the first row of armour units and the

number of rows applied on the slope of a breakwater.

The influence of an external load, imposed on a certain row, on the static load on the first row was

studied based on this model and the measured static load. It appeared from this study that an

external load imposed on a higher row had less influence on the static load on the first row of armour

units. An external load imposed on row 10 or higher added less than 10% of this external load to the

static load on the first row of armour units. From this analysis it was concluded that loads imposed on

row 10 or higher do not have a significant influence on the load on the first row of armour units.

Results and conclusions dynamic load

The measurements of the dynamic load showed two clear phenomena. The first phenomenon was a

periodic behaviour of the dynamic load with a certain peak‐to‐peak amplitude. This peak‐to‐peak

amplitude was related to the downwash. The dynamic load appeared to be a harmonic load with the

same period as the waves imposed on the model. The dynamic load was the result of the flow of

water along the armour layer. The maximum dynamic load on the first row of armour units occurred

simultaneous with the maximum downwash. The following relation between the downwash velocity

and the peak‐to‐peak amplitude of the dynamic load was found: 2

peak to peak

,max

A =a downwash

downwash

U

U

Based on a theoretical analysis it was concluded that the downwash is the main mechanism

influencing the dynamic load on the first row of armour units. The influence of porous flow from the


core through the armour layer, which destabilises the armour units and might induce a higher load on

the first row of armour units was only studied in a theoretical analysis. The influence of this

mechanism was found to be in the order of 13%, much smaller than the remaining load which is

induced by the downwash.

The following parameters were of influence on the peak‐to‐peak amplitude of the load on the first

row of armour units; the permeability of the core, the smoothness of the under layer and the slope of

the breakwater (a flatter slope results in smaller peak‐to‐peak amplitudes). The number of rows

applied on the breakwater had no (clear) influence on the load on the first row of armour units for the

tested number of rows (15 to 25). Based on the static model it was determined that external loads

imposed on row ten or higher have a limited influence on the load on the first row of armour units.

The dynamic load on an armour unit can be regarded as an external load and, based on this

statement, it can be concluded that dynamic load imposed on row ten or higher have a limited effect

on the load on the first row of armour units. The influence of the initial packing density on the peak‐

to‐peak amplitude of load on the first row of armour units was found to be unclear for the packing

density measured over the entire slope.

The second observed phenomenon was the increment of the wave averaged load (equilibrium load)

on the first row of armour units during the test. During the tests the harmonic load oscillated around

an equilibrium line which showed a positive trend. The measured load after testing was significantly

higher than the measured load at the beginning of the tests. The following relationship was found

between the relative wave height and the increment of the load on the first row of armour units: 2

=3,1 0,80equilibrium

g x max max

F H H

F N H H

This trend was independent of the number of rows and the RPD, although the RPD of the first ten

rows might have had some influence. The parameters which were found to have an influence on the

peak‐to‐peak amplitude also had an influence on the increase of the equilibrium load. A smooth

under layer and an impermeable core resulted in a larger increase of the equilibrium load during the

tests compared to the reference case. Furthermore, a flatter slope also resulted in a larger increase of

the equilibrium load compared to the reference case with a steeper slope.

Furthermore, a relation between the armour unit stability (as described in the Van der Meer formula)

and the increase of the equilibrium load was even more accurate compared to the relation between

the wave height and the dynamic load. Based on this relation it can be concluded that the increase of

the equilibrium load is related to the (hydraulic) armour unit stability while the peak‐to‐peak

amplitudes are related to the occurring flow.

The total load on the first row of armour units is composed of the static load and the dynamic load:

peak to peak( ) ( , ) A + =

2

static y equilibrium total

g x g x g x g x

F N F H No waves F

F N F N F N F N

The objective of this research is to determine the governing loads on the first row of armour units and

to establish a quantitative relation between the influencing parameters and this load. Based on this

research, it was concluded that the static load combined with the increase of the equilibrium load are

the governing loads on the first row of armour units. The static load is a function of the number of

rows and reaches a constant value around ten rows. The increase of the equilibrium load depends on

the wave action (wave height) but did not show a dependency on the number of rows applied or the

total packing density. The dynamic load oscillating around the equilibrium load with certain peak‐to‐

peak amplitude is mainly the result of the downwash but is as result of the much larger increase of

the equilibrium load during the wave action, of minor importance.


vii

Table of Contents

Preface ..................................................................................................................................................... iii Abstract ................................................................................................................................................... iv Table of Contents ................................................................................................................................... vii List of Figures ......................................................................................................................................... viii List of Tables ............................................................................................................................................. x List of Symbols ......................................................................................................................................... xi 1 Introduction .................................................................................................................................... 1

1.1 Objective ................................................................................................................................. 2 1.2 Layout of the report ................................................................................................................ 2

2 Literature study ............................................................................................................................... 3 2.1 Static load ............................................................................................................................... 5 2.2 Dynamic load .......................................................................................................................... 7

3 Hypothesis .................................................................................................................................... 19 3.1 Research approach ................................................................................................................ 20

4 Experiments .................................................................................................................................. 21 4.1 Static experiment .................................................................................................................. 21 4.2 Hydraulic experiment ............................................................................................................ 23

5 Results and analysis of static test ................................................................................................. 35 5.1 Model .................................................................................................................................... 37 5.2 Influence of an external load on an arbitrary row ................................................................ 43

6 Results and analysis of hydraulic tests .......................................................................................... 44 6.1 Peak‐to‐peak amplitude of the load on the first row of armour units ................................. 44 6.2 Trend of the equilibrium load ............................................................................................... 57

7 Discussion ..................................................................................................................................... 73 7.1 Static load on the first row of armour units .......................................................................... 73 7.2 Dynamic load on the first row of armour units ..................................................................... 76 7.3 The total load on the first row .............................................................................................. 81

8 Conclusion and recommendations ............................................................................................... 83 8.1 Conclusion ............................................................................................................................. 83 8.2 Applicability and limitations of the study ............................................................................. 86 8.3 Recommendations ................................................................................................................ 87

References .............................................................................................................................................. 88 Appendices ............................................................................................................................................. 90


viii

List of Figures

Figure 2‐1 Stress in an armour unit during a wave cycle [BURCHARTH, 1993] ............................................ 4 Figure 2‐2 Type of loads as function of the location relative to the still water level [BURCHARTH, 1993] . 4 Figure 2‐3 Force balance armour unit ...................................................................................................... 5 Figure 2‐4 Breaker types as a typed by the Iribarren number (ξ) BATTJES (1974) [SCHIERECK, 2004] .......... 8 Figure 2‐5 Notational permeability coefficients as defined by Van der Meer (1988) [U.S. ARMY CORPS OF

ENGINEERS, 2002]........................................................................................................................................ 9 Figure 2‐6 Wave‐induced set‐up (from HOLTHUIJSEN (2007)) ................................................................... 10 Figure 2‐7 Definition of run‐up as local maximum in elevation (from U.S. ARMY CORPS OF ENGINEERS,

2002) ....................................................................................................................................................... 10 Figure 2‐8 Influence of permeability on the run‐up and internal water level (from Burcharth (1993) via

U.S. ARMY CORPS OF ENGINEERS, 2002) ........................................................................................................ 11 Figure 2‐9 Visualised flow pattern of a breaking wave approaching the maximum level of run‐up (from

TSAI, 1997) ............................................................................................................................................... 12 Figure 2‐10 Visualised flow pattern of a breaking wave approaching the maximum level of run‐down

(from TSAI, 1997) ..................................................................................................................................... 12 Figure 2‐11 Visualised flow patter of a non‐breaking wave approaching the maximum level of run‐up

(from TSAI, 1997) ..................................................................................................................................... 12 Figure 2‐12 Visualised flow patter of a non‐breaking wave approaching the maximum level of run‐

down (from TSAI, 1997) ........................................................................................................................... 12 Figure 2‐13 Record of wave run‐up, velocities of water flow at the toe and pressure at the toe for a

breaking wave (from TSAI, 1997) ............................................................................................................. 13 Figure 2‐14 Flow pattern inside a breakwater (from Barends et. al. (1983) via ABBOTT AND PRICE (1994))

................................................................................................................................................................ 14 Figure 2‐15 Contribution of each term of the Forchheimer equation to the total gradient [VAN GENT,

M.R.A. 1995] ........................................................................................................................................... 15 Figure 2‐16 Dynamic forces on an armour stone [HALD, 1998] .............................................................. 16 Figure 2‐17 Drag and inertia coefficients relative to the KC number TØRUM, 1994 [HALD, 1998] ........... 17 Figure 2‐18 Rise and subsidence of the phreatic level due to in and outflow of water ......................... 17 Figure 2‐19 Flow around a armour unit.................................................................................................. 18 Figure 4‐1 Conceptual design of static experiment ................................................................................ 21 Figure 4‐2 Sketch of the critical slope experiment ................................................................................. 23 Figure 4‐3 Side and frontal view of breakwater model with measuring frame ...................................... 24 Figure 4‐4 Cross section of along slope frame profile and U‐profile at location of one of the wheels .. 24 Figure 4‐5 Longitudinal view of the wave flume [VAN ZWICHT, 2009] ...................................................... 25 Figure 4‐6 Frontal and side view of a Xblox single layer armour unit .................................................... 27 Figure 4‐7 Cross‐section of the breakwater model with under structure .............................................. 29 Figure 4‐8 Sketch of cross‐section with 15, 20 and 25 rows .................................................................. 30 Figure 4‐9 Cross‐section of a breakwater with 20 rows for a slope of 2:3 and a slope of 3:4 (striped) . 30 Figure 4‐10 Positions of wave gauges .................................................................................................... 34 Figure 5‐1 Measurements of load on the first row during static tests ................................................... 35 Figure 5‐2 Measurements including curve fit ......................................................................................... 36 Figure 5‐3 Measurements of the static load including measurements of MUILWIJK (2011) (in gray) ..... 37 Figure 5‐4 Force balance armour unit .................................................................................................... 38 Figure 5‐5 Critical angle of Xbloc armour units on a rock under layer ................................................... 39 Figure 5‐6 Diagonal transfer of forces between armour units ............................................................... 40 Figure 5‐7 Vertical transfer of forces between armour units ................................................................. 40 Figure 5‐8 Static model based on measured values ............................................................................... 41 Figure 5‐9 Measured data and static model ........................................................................................... 42


Figure 6‐1 Measured load during a flume test ....................................................................................... 44 Figure 6‐2 Maximum run‐down, maximum water level at wave gauge, time; 1:31:42 ......................... 45 Figure 6‐3 Wave front passes upwards through still water level, time; 1:31:58 .................................... 45 Figure 6‐4 Maximum run‐down, maximum water level at wave gauge, time; 1:32:15 ......................... 45 Figure 6‐5 Wave front passes downwards through still water level, time; 1:32:35 ............................... 45 Figure 6‐6 Points of interest compared to the water level at the toe .................................................... 46 Figure 6‐7 Hydraulic test 3(3), subtest 0,6 Hmax 0,874 Hz ....................................................................... 46 Figure 6‐8 Boxplot wave height versus peak‐to‐peak amplitude of the load ......................................... 47 Figure 6‐9 Curve fit wave height versus peak‐to‐peak amplitude of the load ....................................... 48 Figure 6‐10 Theoretical run‐up versus the measured run‐up ................................................................ 51 Figure 6‐11 Curve fit measured run‐up versus peak‐to‐peak amplitude of the load ............................. 52 Figure 6‐12 Downwash velocity from energy conservation ................................................................... 53 Figure 6‐13 Movement of the wave front over the breakwater slope ................................................... 54 Figure 6‐14 Comparison downwash velocity formulae .......................................................................... 55 Figure 6‐15 Curve fit calculated downwash velocity versus peak‐to‐peak amplitude of the load ......... 56 Figure 6‐16 Measured load on the first row of armour units during a flume test ................................. 58 Figure 6‐17 Type 1 .................................................................................................................................. 59 Figure 6‐18 Boxplot of relative increase of the equilibrium load versus the wave height. .................... 63 Figure 6‐19 Equilibrium load versus the wave height ............................................................................ 64 Figure 6‐20 Curve fits equilibrium line versus wave height ................................................................... 65 Figure 6‐21 Equilibrium load on the first row of armour units relative to the armour unit stability ..... 66 Figure 6‐22 Packing density versus wave height .................................................................................... 68 Figure 6‐23 Boxplot of the development of the packing density versus the wave height ..................... 68 Figure 6‐24 Equilibrium load versus the packing density ....................................................................... 69 Figure 6‐25 Equilibrium load of the load on the first row versus the relative subsidence ..................... 70 Figure 6‐26 Peak‐to‐peak amplitude of the load on the first row versus the equilibrium load ............. 71 Figure 7‐1 Theoretical and measured static load on the first row of armour units ............................... 74 Figure 7‐2 Layout of flume test model ................................................................................................... 77 Figure 7‐3 Basic form of increase of the equilibrium load ..................................................................... 79


x

List of Tables

Table 2‐1 Types and origins of loads on armour units [BURCHARTH, 1993] ................................................ 3 Table 2‐2 Run‐up parameters according to [VAN DER MEER AND STAM, 1992]........................................... 11 Table 4‐1 Length of armour layer and crest height ................................................................................ 28 Table 4‐2 Parameters varied in the test program .................................................................................. 32 Table 4‐3 Wave program regular waves ................................................................................................. 33 Table 5‐1 Analysis of the influence of an external load.......................................................................... 43 Table 5‐2 Analysis of the increase of the static load after the addition of an extra row ....................... 43 Table 6‐1 Curve fits of individual test, relation between wave height and amplitude .......................... 49 Table 6‐2 Run‐up parameters according to [VAN DER MEER AND STAM, 1992]........................................... 51 Table 6‐3 Curve fits of individual test, relation between run‐up and amplitude ................................... 53 Table 6‐4 Curve fits of individual test, relation between downwash velocity and amplitude ............... 56 Table 6‐5 Overview decrease of peak‐to‐peak amplitude during a subtest .......................................... 57 Table 6‐6 Types of trend curves ............................................................................................................. 59 Table 6‐7 Occurrence of trend curve types ............................................................................................ 60 Table 6‐8 Number of waves corresponding to the “half‐life” time ........................................................ 61 Table 6‐9 Averaged measured load on the first row of armour units after testing ............................... 62 Table 6‐10 Curve fits of individual test, relation between wave height and equilibrium load .............. 64 Table 6‐11 Curve fits of individual test, relation between armour unit stability and equilibrium load . 67 Table 6‐12 Curve fits of the peak‐to‐peak amplitude versus the wave height and the equilibrium load

versus the wave height ........................................................................................................................... 71 Table 6‐13 Results of spectrum analysis wit WaveLab and calculation of expected equilibrium load .. 72 Table 6‐14 Measured increase of the equilibrium load ......................................................................... 72


xi

List of Symbols

a 1. Direction coefficient of the squared term of a curve fit [‐]

2. Coefficient in the run‐up formula of VAN DER MEER AND STAM [‐]

3. Dimensional porous friction coefficient [s∙m‐1]

ab Wave amplitude at the bottom [m]

A 1. Surface perpendicular to the direction of the flow [m3]

2. Coefficient in the run‐up formula of Battjes [‐]

Apeak‐to‐peak Peak‐to‐peak amplitude on the first row [F]

b 1. Direction coefficient of the slope term of a curve fit [‐]


3. Dimensional porous friction coefficient [s∙m‐1]

c Coefficient in the run‐up formula of VAN DER MEER AND STAM [‐]

C Coefficient in the run‐up formula of Battjes [‐]

Cf Friction coefficient between the wall and the flow [‐]

CD Drag coefficient [‐]

CI Inertia coefficient [‐]

CL Lift coefficient [‐]

d 1. Depth (distance between surface elevation and bottom) [m]


dstone Protuberant height of under layer stone [m]

dx Horizontal centre to centre distance between armour units [m]

dy Vertical (along slope) centre to centre distance between armour units [m]

D Grain diameter [m]

Dparticle Diameter of a stone particle [m]

Dn50 Median nominal diameter [m]

DXbloc Unit height Xbloc [m]

E Expected value [‐]

EJONSWAP Variance density of the JONSWAP spectrum [m2∙s]

f Coefficient of friction [‐]

f1(ξ) Factor between average downwash velocity and peak downwash velocity [‐]

ffreq Frequency [s‐1]

ffreq,peak Peak frequency [s‐1]

FD Drag force [N]

Fend Load on the first row of armour units at the end of a (sub)test [N]

Fequilibrium Dynamic component of the load on the first row of armour units [N]

Ff Friction force between armour unit and under layer [N]

Fg Weight of armour unit [N]

Fh Along slope component of armour unit weight [N]

FI Inertia force [N]

FL Lift force [N]

Fn Normal force armour unit [N]

Fr Froude number [‐]

Fr(n) Total resulting force of single armour unit along slope at row n [N]

Fres,unit Resulting force of single armour unit along the slope based on the

force balance of a single unit [N]

FS Seepage force [N]

Fstatic Static component of the load on the first row of armour units [N]

Fstart Load on the first row of armour units at the start of a subtest [N]


g Gravitational acceleration [m∙s2]

H Wave height [m]

H1/3 Mean of the highest one‐third of waves: Significant wave height

Measured from a wave record [m]

Hmax Maximum wave height (approximation: Hmax≈2Hs) [m]

Hs Significant wave height [m]

imax Pressure gradient at the bottom [‐]

I ith element (of a dataset) [‐]

I Hydraulic gradient [‐]

k Wave number (2π/L) [m−1]

K Permeability coefficient [m∙s‐1]

KD Damage coefficient in the Hudson formula [‐]

KC Keulegan Carpenter number [‐]

l Characteristic length (model or prototype) [m]

L Wave length [m]

Larmour layer Along slope length of the armour layer [m]

Ldownwash Downwash length (along slope length between Ru and Rd) [m]

Lseepage Seepage length [m]

Lx Distance between the centers of the armour unit’s at the far left and

the far right of a row [m]

Ly Distance between the centers of an armour unit on the first row

And the centre of an armour unit on the highest row [m]

N Number of waves [‐]

N1/2 Number of row at which the half of the eventual growth

of the static load is reached [‐]

Nx Number of units on a horizontal row [‐]

Ny Number of horizontal rows [‐]

n Porosity of the porous medium [‐]

n exceedence percentage [‐]

P Permeability in the Van der Meer formula [‐]

r Residual [‐]

R Coefficient of determination [‐]

Rd Run‐down [m]

Re Reynolds number [‐]

RPD Relative packing density [‐]

Ru Run‐up level [m]

Rucore Run‐up level at the core [m]

Ruhunt Run‐up level calculated with the Hunt formula [m]

Run Run‐up level only exceeded by n % of the waves [m]

Rumax Maximum calculated level of run‐up [m]

s Wave steepness [‐]

S Damage level in the Van der Meer formula [‐]

t Time [s]

t1/2 Half‐life of trend growth curve [s]

T Wave period [s]

Tm Mean wave period [s]

U Velocity [m∙s‐1]

Ub Velocity of the orbital motion at the edge of the boundary layer [m∙s‐1]

Uhunt Downwash velocity calculated based on the Hunt formula [m∙s‐1]

Up Porous flow velocity [m∙s‐1]

Udownwash,average Average downwash velocity [m∙s‐1]

Udownwash,max Maximum downwash velocity [m∙s‐1]


‐ xiii ‐

V Volume [m3]

Vi Volume under the crest of the wave [m3]

Vinternal Internal water volume (core) [m3]

VRu Run‐up volume [m3]

VXbloc Volume of a Xbloc armour unit [m3]

W85 Rock weight that is exceeded by 15% of the rocks in the under layer [kg]



WXbloc Weight of a Xbloc armour unit [kg]

x Value on the horizontal axis of a graph (used for curve fits) [‐]

X Random dataset [‐]

y Value on the vertical axis of a graph (used for curve fits) [‐]

α Slope of breakwater [°]

αcrit Critical slope angle of armour unit [°]

αf Dimensionless permeability coefficient Forchheimer equation [‐]

α JONSWAP Scaling parameter JONSWAP spectrum [‐]

β Angle of incidence of waves with the breakwater [‐]

βf Dimensionless friction coefficient Forchheimer equation [‐]

γ Coefficient representing the effect of the added mass [‐]

γr Rough slope reduction coefficient in run‐up formula of Battjes [‐]

γb Berm reduction coefficient in run‐up formula of Battjes [‐]

γh Wave distribution coefficient in run‐up formula of Battjes [‐]

γβ Angle of incidence coefficient in run‐up formula of Battjes [‐]

γJONSWAP Scaling parameter JONSWAP spectrum [‐]

∆ Relative density (ρs‐ ρw)/ ρw [‐]

ξ0 Iribarren number, surf similarity parameter (additive 0 is for deep water) [‐] λ Scale factor [‐]

ρs Density of stone (or concrete in in the case of concrete armour units) [kg∙m‐3]

ρw Density of water [kg∙m‐3]

σ Standard deviation [‐]

σJONSWAP Scaling parameter JONSWAP spectrum [‐]

τ Mean life‐time of trend growth curve [s]

τw Shear stress under a wave [N∙m‐2]

ω Angular frequency of a wave (2π/T) [rad∙s‐1]

ν Kinematic viscosity [m2∙s‐1]


‐ 1 ‐

1 Introduction

Rubble mound breakwaters are used in many places around the world were activities such as

navigation need protection from high waves. Rubble mound breakwaters can be armoured by rocks

or concrete armour units. The state of art regarding concrete armour units are interlocking, single

layer, armour units. These armour units are placed on a granular core and under layer(s) and cover

the complete slope of the breakwater from the toe of the structure at the sea bottom to the crest of

the breakwater.

Figure 1‐1 Placement of Xbloc single layer armour units (source xbloc.com)

Interlocking, single layer concrete armour units are placed in a grid which depends on the type of

armour unit. Within this grid, armour units are placed in horizontal rows. The number of horizontal

rows of single layer armour units on a breakwater is limited to 20. This limit is proposed in order to

prevent major settlements, which might affect the interlocking of the armour units. The limit on the

maximum number of rows is based on experience from prototypes and is not yet confirmed in any

systematic study. Then number of rows also might have an effect on the load on the first (bottom)

row of armour units, which affects the structural integrity of the armour units. The load on the first

row of armour units is however unknown. The research presented in this thesis is a study on the load

on the first (bottom) row of concrete armour units placed on a breakwater.

In some cases the combination of the dimensions of the armour units (determined based on the

occurring wave height) and the maximum number of rows is not enough to cover the whole slope.

This problem is usually solved by applying larger armour units in order to cover the whole slope. The

relative strength of these larger units is less than the relative strength of the smaller units. When

applying larger units the strength of the armour units could be normative just as the settlement

criterion.

In order to make a judgement about the strength of the first row of armour units the load on this row

must be known. The load on the first row of interlocking, single layer armour units was studied in this

study. The total load is composed of a static load of the above laying armour units and a dynamic load

of the waves which is partially transferred to the lowest row of armour units. It is unknown which

processes are governing for the load on the lowest row of armour units as well as their quantitative

contribution to the total load.

Introduction

‐ 2 ‐

1.1 Objective

The objective of this study was to determine the governing loads on the lowest row of armour units

and to qualify and quantify the relevant processes that induce this load.

The loads were studied in physical model tests in order to extend the present knowledge.

1.2 Layout of the report

Chapter 2 gives an overview of the known literature introducing the problem and giving a further

explanation of the relevant processes. A number of gaps in the existing literature were identified and

those relevant for this research were analysed. Based on the literature study, two hypotheses were

formulated which are stated in chapter 3.

These hypotheses were tested which physical model tests described in chapter 4. This chapter

describes two experiments; one experiment for the study of the static load and one experiment for

the study of the dynamic load. The results of these experiments and an analysis of these results are

presented in chapter 5 and 0. The results and analyses of these results described in these chapters are

discussed in chapter 7. Based on this discussion these hypotheses were evaluated, conclusions are

presented and recommendations for further research are made in chapter 8.


‐ 3 ‐

2 Literature study

Research was performed on the structural integrity of (slender) concrete armour units. This research

was triggered by the failure of the breakwater at the port of Sines (Portugal) in 1978 and other cases

of breakage of slender concrete armour units in that period. The breakwater at the port of Sines was

armoured with 40 tons Dolos armour units [MAGOON, et.al. 1994]. One of the most well‐known

research programs on the structural integrity of concrete armour units is the research by BURCHARTH

(1993) which focused on the stresses in Dolos armour units. BURCHARTH (1993) divided the number of

loads in a number of types including a type for the static loads and the dynamic loads (table 2‐1).

These two types of loads are the loads considered in this research. The research of BURCHARTH (1993)

is further described in the rest of this section. Paragraph 2.1 gives a further description of the static

load and the parameters influencing this load and paragraph 2.2 describes the dynamic load and the

processes and parameters influencing this load.

Table 2‐1 Types and origins of loads on armour units [BURCHARTH, 1993]

Types of Loads Origin of loads

Static Weight of units

Prestressing of units due to wedge effect and

arching caused by movement under dynamic loads

Dynamic Pulsating:

Gradually varying wave forces

Earthquake

Impact:

Collision between units when rocking or rolling,

collision with under layer or other structural parts

Missiles of broken units

Collision during handling, transport and placing

High‐frequency wave slamming

Abrasion Impacts of sand, shingle etc. in suspension

Thermal Temperature differences during the hardening

process after casting

Freeze – thaw

Chemical Alkali‐silica and sulphate reactions, etc.

Corrosion of steel reinforcement

The research of BURCHARTH (1993) focussed on the static, pulsating and impact loads. The static loads

are loads which are present without the presence of wave action and are the result of the weight of

the armour units and possible prestressing of armour units. The pulsating load is the load due to the

gradually varying flow and earthquakes. Earthquakes were not further considered in the research of

BURCHARTH (1993). The impact loads are peak loads of a short duration and are the result of collisions

between armour units (rocking) and impact forces from waves (wave slamming). It was not possible

to directly study the breakage of armour units due to various loads on a reduced scale since the

relative strength of a model unit is much too high compared to the strength of a full‐scale armour

unit. In the research of BURCHARTH (1993) the stresses in the armour units were measured and not the

actual loads on the armour units which cause these stresses.

Literature study

‐ 4 ‐

The stress in an armour unit during a wave cycle is visualised in figure 2‐1:

Figure 2‐1 Stress in an armour unit during a wave cycle [BURCHARTH, 1993]

BURCHARTH (1993) analysed the stress in armour units around the waterline (in the active zone) and

concluded that beneath the still water level the static and pulsating stresses are dominant while

above still water level the impact stresses have a significant influence (see figure 2‐2).

Figure 2‐2 Type of loads as function of the location relative to the still water level [BURCHARTH, 1993]

Based on the research of BURCHARTH (1993) it was concluded that the static and pulsating load are of

primary importance for the load on the first (bottom) row of armour units, since these loads have a

governing influence on armour units placed under the still water line. Furthermore, the impact forces

due to wave slamming are directed inwards and were supposed to have less effect on the along slope

load on the first row of armour units. The stress in the armour units was the primary interest in the

research of BURCHARTH (1993). The determination of stresses in armour units is a straightforward

approach that resulted in directly usable results if one is interested in a certain type of armour unit.

Observations regarding the relative influence of the various stresses are also valid for (bulky) armour

units of another type. However, the magnitude of the stress in these armour units is different

compared to the stress in slender Dolos armour units. In order to obtain results independent of the

type of armour unit a load based approach was followed in order to determine the (average) external

load on the armour units. The actual stress in the armour units is outside the scope of this research

and should be determined based on the armour unit characteristics in subsequent research.

The total load on the first row of armour units is, in line with the research of BURCHARTH (1993),

decomposed in a static load and a dynamic load. The static load is described in paragraph 2.1 and the

dynamic load is the subject of paragraph 2.2.


‐ 5 ‐

2.1 Static load

The static load is defined as the load on the first row of armour units resulting from the higher

positioned rows of armour units during conditions without waves. The static load on an armour unit is

the result of the static instability of higher positioned armour units. Each single armour unit has its

own static force balance. This is visualised in figure 5‐4. If the force balance of a single armour unit is

not in balance by itself, a residual force on the lower placed armour units compensates this imbalance

(Frn).

Figure 2‐3 Force balance armour unit

The individual force balance is the balance between the gravitational force (Fg), the normal force (Fn)

and the friction force (Ff) on an armour unit. The weight vector is the main loading vector influencing

the static balance of a single armour unit. This vector has a vertical direction and is determined by the

weight of the armour unit (W) and the gravitational acceleration (g) which results in the gravitational

force (Fg). The gravitational force (Fg) of an armour unit can be decomposed in an along slope

component (Fh) and a component perpendicular to the slope (Fn) which called the normal force. The

size of these components in relation to each other is determined by the angle of the slope on which

the armour unit is placed on:

sin( )

cos( )

h g

n g

F F

F F

(2.1)

The force balance in along slope direction is of main interest for this research since the forces in along

slope direction contribute to the resulting static load on the first row of armour units. The force

balance in along slope direction consists of the weight vector parallel to the slope (Fh) and the friction

between the armour unit and the under layer (Ff). This friction force is determined by the normal

force between the armour unit and the under layer (the weight vector component perpendicular to

the slope (Fn)) multiplied by a friction coefficient:

cos( )f n gF f F f F (2.2)

Literature study

‐ 6 ‐

The along slope force balance for an individual armour unit is a balance between the along slope

weight component (Fh) and the friction force between the under layer and the armour unit (Ff). The

armour unit is in static balance when the friction force is equal to the gravitational force component

parallel to the slope:

sin( ) cos( )h n

g g

F F

F f F

(2.3)

Residual forces between the armour units occur when the along slope forces are not in balance. The

total force balance of an armour unit is now influenced by external forces (a residual force on the unit

from (a) higher positioned unit(s) and a residual force of the considered unit on (a) lower positioned

unit(s)). This results in an extended along slope force balance which is a balance between the along

slope weight component (Fh), a possible residual load of the above positioned armour unit (Fr(n+1)), the

friction force between the under layer and the armour unit (Ff) and the residual force of the lower

positioned armour unit (Fr(n)):

( 1) ( )

( 1) ( )sin( ) cos( )

h r n f r n

g r n g r n

F F F F

F F f F F

(2.4)

Based on this force balance it was concluded that if the single units are in balance by themselves, the

residual forces between the armour units will be zero and the static load on the first row of armour

units will also be zero. If the armour units are not in balance, this will result in residual forces between

the armour units and eventually in a static load on the first row of armour units. The friction between

the armour units is a function of the characteristics of the material of the armour units and the under

layer. If the friction coefficient is the same for all armour units then the slope is the only variable

parameter. The residual force between the armour units can then be determined based on the slope

at which the armour unit are positioned. A positive residual force between the armour units leads to a

linear increasing load on the first row of armour units as function of the number of rows applied on

the slope of the breakwater.

MUILWIJK (2011) did experiments regarding the static load on the first row of armour units. From this

research followed that a number of armour units are not stable by itself on breakwater with a slope of

3:4 which resulted in a residual force between the armour units. Furthermore, MUILWIJK (2011) found

that the number of rows which has an influence on the load on the lowest row of armour units is not

the same as the total number of rows. MUILWIJK (2011) also investigated the influence of the packing

density on the static load and found that an increase of the packing density results in an increase of

the static load on the first row of armour units.


‐ 7 ‐

2.2 Dynamic load

The dynamic load is defined as the load on the bottom row of armour units during conditions with

wave attack minus the static load. The dynamic loads as defined by BURCHARTH (1993) consist of a

pulsating load and an impact load. The pulsating load consists of flow induced forces and forces due

to earthquakes. The latter are not further considered in this research. The impact forces are

composed of collision forces between armour units (due to rocking) and wave slamming.

These loads vary during the wave cycle of a wave attacking a breakwater. The cycle of a wave on a

breakwater can be decomposed into the following phases:

Incoming wave

In the first phase the crest of the wave is travelling from the toe of the structure to the wave breaking

zone (active zone). The wave height and length are transformed on the slope of the breakwater. The

armour units are loaded by the orbital velocities in this phase (pulsating load).

Breaking and run‐up

The second phase is during the breaking and run‐up of the wave. This phase is reached when the ratio

between wave height and water depth of a wave travelling upslope on a breakwater approaches the

breaking limit. During the breaking process energy is converted into turbulent motions and the

remaining energy is converted to potential energy which can be seen as run‐up of the wave on (and

in) the slope. In this phase armour units are loaded by impact loads and pulsating loads which occur

mostly in the active (breaking) zone.

Reflecting wave (reflection and run‐down)

When the highest run‐up is reached the elevated water level starts to descend rapidly inducing a

downwash of water along and in the slope and outflow of water from the under layer and core

through the armour layer, which results in the reflection of the wave.

The wave‐induced water motions on and in a breakwater are described in separate sections beginning

with the transformation of waves on the foreshore and the slope of a breakwater, followed by

processes as wave breaking and run‐up and completed by the reflection of waves from the

breakwater.

2.2.1 Dynamic processes

2.2.1.1 Incoming wave

Waves propagating on a slope are transformed both in shape and direction. The direction of wave

propagation is influenced by refraction and diffraction. The transformation of the shape of the wave

(height, length) is caused by shoaling, dissipation, additional growth due to the wind, wave‐current

interactions and wave‐wave interactions [U.S. ARMY CORPS OF ENGINEERS, 2002].

The bottom velocities under an incoming wave are mainly parallel to the slope. Directly under the

crest the velocities at the bottom are in the direction of wave propagation and reverse 180 degrees at

the passage of the trough. The velocities are driven by a pressure gradient at the bottom [SCHIERECK,

2004]:

2cosh( )max

k Hi

k d

(2.5)

In which k is the wave number and H the wave height. The velocities at the bottom impose a shear

stress on the slope which causes a load on the armour units.

Literature study

‐ 8 ‐

This shear stress can be related to the velocities by the following expression:

21 with :U and: sin

2 sinhb

w w f b b b b

aC U a U U t

k d

(2.6)

In which τw is the shear stress under a wave, ρw is the density of water, cf is the friction coefficient

between the wall (or armour layer surface) and the flow, Ub is the velocity of the orbital motion at the

edge of the boundary layer, ω is the angular frequency of the wave, ab the wave amplitude at the

bottom, k the wave number and d the water depth. The shear stress in the expression is related to

the orbital motion at the edge of the boundary layer (ub). The shear of the water mass with the slope

dissipates energy in the form of friction.

2.2.1.2 Breaking wave and run‐up along the slope

As waves propagate further along the slope, the wave height increases until the waves become

unstable. A waves becomes unstable because the wave is either too steep or the water depth too

small. As result of this instability the wave breaks. The limit of the non‐breaking wave is described by

Miche (1944) from [SCHIERECK, 2004]:

20,142 tanhbH L d

L

(2.7)

The breaking process is characterised by the dimensionless Iribarren number or surf similarity

parameter Battjes (1974) from [SCHIERECK, 2004] :

In which α is the slope of the breakwater and L is the wave length with the additive 0 which indicates

deep water. The Iribarren number is the ratio between the wave steepness and the steepness of the

slope (of the breakwater). The Iribarren number can be used to characterize the type of wave

breaking. The possible breaking types are visualised in figure 2‐4. The Iribarren numbers lower than

2,5 to 3 indicate a collapsing type of wave breaking while higher Iribarren numbers indicate a surging

type of wave breaking. A surging wave surges up and down the slope while less energy is dissipated

compared to the collapsing breaker types. Due to the steep slopes at which interlocking armour units

are applied the most observed breaker type during this research was the surging breaker type.

Figure 2‐4 Breaker types as a typed by the Iribarren number (ξ) BATTJES (1974) [SCHIERECK, 2004]

The stability of the armour units in breaking waves is described by the (rewritten) Hudson formula

[SCHIERECK, 2004]:

0

tan

/H L

(2.8)


‐ 9 ‐

In this formula; Hs is the significant wave height, ∆ the relative density of the concrete ((ρs‐ ρw)/ ρw), D

the diameter of the considered grain particle (or unit height of the armour unit) and KD the damage

coefficient for the Hudson formula. The damage coefficient KD contains the degree of allowable

damage at which the breakwater is considered as failed. The Hudson formula has a number of

limitations regarding the parameters which are not directly taken into account such as the wave

period, the permeability of the breakwater and the number of waves. The Van der Meer formula

[SCHIERECK, 2004] does incorporate these parameters:

50 0,2

0,18 0.5

50 0,2

0,13

(plunging breakers, )

6,2

(surging breakers, )

1,0 cot

sn transition

sn transition

P

HD

SP

N

HD

SP

N

(2.10)

In which P is the permeability parameter according to Van der Meer which should be determined

based on figure 2‐5. S is the damage level as defined by Van der Meer which roughly varies between 1

and 10 and N the number of waves for which the stability of the structure is considered. In practice

equilibrium is reached for 6500 waves.

Figure 2‐5 Notational permeability coefficients as defined by Van der Meer (1988) [U.S. ARMY CORPS OF ENGINEERS,

2002]

The difference between surging breaker types and non‐surging breaker types (collapsing, plunging

and spilling breaker types) is mentioned earlier and this separation is also found in the Van der Meer

formula which gives two expressions: one for surging and one for plunging breaker types. The

applicability of the two expressions is determined by the Iribarren number related to the transition

Iribarren number:

1

0,31 0,56,2 tan Ptransition P

(2.11)

For Iribarren numbers smaller than the transition Iribarren number the first expression for plunging

breakers should be used, while for Iribarren numbers larger than the transition Iribarren number the

second expression for surging breakers should be used.

The transformation of the wave height and the breaking process affects the transport of momentum

of the waves [HOLTHUIJSEN, 2007]. This horizontal variation in the transport of wave‐induced

momentum (radiation stress) causes a change in the opposing force on the water mass. This radiation

stress gradient is balanced by a gradient in the hydrostatic pressure. The gradient in the hydrostatic

pressure corresponds to a tilted (mean) water surface which causes a higher water level at the

3

50

cotsD

n

HK

D

(2.9)

Literature study

‐ 10 ‐

breakwater. This increase of water level due to wave breaking on a slope is usually referred to as set‐

up in literature.

Figure 2‐6 Wave‐induced set‐up (from HOLTHUIJSEN (2007))

The run‐up is the maximum surface elevation above the still‐water level. It is composed of the wave

induced set‐up as described above and the time varying variation around this time averaged set‐up.

The variations of this set‐up create an oscillating water level around the time averaged (mean) set‐up

which is called swash. The level of the (mean) wave induced set‐up and the maximum level of the

swash is referred to as run‐up.

Figure 2‐7 Definition of run‐up as local maximum in elevation (from U.S. ARMY CORPS OF ENGINEERS, 2002)

There are several formulae available for the determination of maximum run‐up (Ru). Run‐up formulae

for impermeable slopes are generally based on the formula suggested by Hunt (1959) [HUGHES, 2004]:

2

tan2,3

Ru

H H

T

(2.12)

The Hunt formula has a coefficient of 2,3 which is a dimensional coefficient in ft‐1/2∙s‐1. The Hunt

formula can be formulated in a dimensionless expression when rewritten using the deep‐water

Iribarren number:

01,0Ru

H (2.13)

The Hunt formula was modified by Battjes in 1974 in order to derive the (n) percentage run‐up level

[U.S. ARMY CORPS OF ENGINEERS, 2002];

nr b h

s

RuA C

H (2.14)

In which Run is the run‐up level only exceeded by n % of the waves, A and C are coefficients

dependent on ξ and n. The reduction factors; γr, γb, γh and γβ are applied in order to correct for the

influence of the surface roughness (γr), the influence of a berm (γb), the influence of wave conditions


‐ 11 ‐

of which the wave height distribution deviates from the Rayleigh distribution (γh) and the influence of

the angle of incidence β of the waves (γβ).

Now‐a‐days run‐up formulae are of this form. In this research rubble mount breakwaters armoured

with concrete single layer armour units were used. These have a rough surface and are relatively

permeable. Xbloc armour units were used for the model tests performed in this research. Delta

Marine Consultants, the patent holder of Xbloc, recommends the run‐up formula of VAN DER MEER AND

STAM (1992) for the calculation of run‐up on a breakwater armoured with Xbloc armour units. The run‐

up formula of VAN DER MEER AND STAM (1992) is given in the following expression:

for 1,5

for 1,5

n

cn

Rua

HsRu

bHs

(2.15)

The relative run‐up of structures with a permeable core (P=0,4) is limited to a maximum value [VAN

DER MEER AND STAM, 1992]:

nRud

H (2.16)

The, b, c and d parameters are presented in table 6‐2:

Table 2‐2 Run‐up parameters according to [VAN DER MEER AND STAM, 1992]

Run‐up level n(%) a b c d

0,1 1,12 1,34 0,55 2,58

1 1,01 1,24 0,48 2,15

2 0,96 1,17 0,46 1,97

5 0,86 1,05 0,44 1,68

10 0,77 0,94 0,42 1,45

Significant 0,72 0,88 0,41 1,35

Mean 0,47 0,6 0,34 0,82

The run‐up on breakwaters with a permeable core is smaller compared to the run‐up on breakwaters

with an impermeable core. A permeable slope will absorb part of the wave action by an inflow of

water. This causes a rise of the internal water level (phreatic line). The inflow of water into the

structure reduces the flow velocities along the slope and reduces the run‐up on the slope. A structure

with a higher permeability has more inflow of water, a higher internal set‐up and a lower level of run‐

up on the slope. The inflow of water and the internal set‐up are illustrated in figure 2‐8.

Figure 2‐8 Influence of permeability on the run‐up and internal water level (from Burcharth (1993) via [U.S.

ARMY CORPS OF ENGINEERS, 2002])

Literature study

‐ 12 ‐

2.2.1.3 Reflecting wave: downwash along the slope

When a wave has reached its highest level of elevation on the slope of a breakwater the flow reverses

and the elevated water mass flows down along the slope till the elevation level of the trough is

reached. This downward directed flow on the slope of a breakwater is called downwash. The reversal

of the flow creates a reflection of the wave by the structure.

The downwash of the water mass along a breakwater slope depends on the highest level of surface

elevation: the level of run‐up. Depending on the wave characteristics of the incoming wave and the

characteristics of the breakwater, this downwash can induce a flow extending to the toe of the

breakwater [TSAI, 1997]. The breaker type is influencing the character of the downwash flow. A

breaking wave is more likely to induce a downwash that extends to the toe of the breakwater

compared with a non‐breaking wave. Whether the downwash reaches the toe or not also depends on

the water level above the toe.

Figure 2‐9 Visualised flow pattern of a breaking wave

approaching the maximum level of run‐up (from TSAI,

1997)

Figure 2‐10 Visualised flow pattern of a breaking

wave approaching the maximum level of run‐down

(from TSAI, 1997)

Figure 2‐9 and figure 2‐10 show the flow patterns of a breaking wave on a breakwater slope near the

moment of maximum run‐up and the moment of maximum run‐down. It can be observed that the

flow at the toe, and all positions at a higher level along the slope, is influenced by the downwash.

Figure 2‐11 Visualised flow patter of a non‐breaking

wave approaching the maximum level of run‐up

(from TSAI, 1997)

Figure 2‐12 Visualised flow patter of a non‐breaking

wave approaching the maximum level of run‐down

(from TSAI, 1997)


‐ 13 ‐

Figure 2‐11 and figure 2‐12 show the flow patterns of a breaking wave on a breakwater slope near the

moment of maximum run‐up and the moment of maximum run‐down of a non‐breaking wave. The

flow characteristic of this wave may be similar to the water particle motions of a periodic standing

wave due to the interaction of the incoming wave and the reflected wave. Besides, it can be seen that

the flow caused by the downwash does not reach the toe and that the direction of the flow is more

perpendicular to the slope, directed upwards during run‐down and downwards during run‐up.

The above observations can be related to the magnitude of the wave period. A wave with a larger

wave period is more likely to extend the influence of the downwash along the whole slope up to the

toe of the slope while a wave with a smaller period is influencing a smaller part of the slope.

In the same experiment executed by TSAI (1997) the run‐up, velocity and pressure at the toe were

measured in a continuous record (figure 2‐13).

Figure 2‐13 Record of wave run‐up, velocities of water flow and pressure at the toe for a breaking wave (from

TSAI, 1997)

From figure 2‐13 it can be observed that the record of the flow velocity and pressure at the toe has

the same periodicity as the measured surface elevation. However, the periodic curve of the flow

velocity at the toe and the pressure at the toe is shifted with respect to the surface elevation curve.

The maximum velocity at the toe roughly occurs at the maximum gradient of the surface elevation.

Flow inside a breakwater

During the period of downwash, the water level on the slope of the breakwater decreases and

becomes lower than the water level inside the breakwater. This creates a hydraulic gradient which

induces an outflow of water from the core, through the armour layer.

The surface along the slope through which the inflow takes place is larger than the surface along the

slope through which the outflow takes place. Furthermore, the average path of inflow is shorter than

the path of outflow. The combination of these two processes is that more water flows into the

structure than flows out of the structure during a wave cycle. As a consequence of this imbalance, the

average water level inside the breakwater (phreatic line) is elevated above the average still water

level on the outside of the breakwater (the side of the breakwater loaded by waves) [ABBOTT AND PRICE,

1994].

Literature study

‐ 14 ‐

The gradients of the phreatic line give rise to flows inside the breakwater (see figure 2‐14).

Figure 2‐14 Flow pattern inside a breakwater (from Barends et. al. (1983) via ABBOTT AND PRICE (1994))

Porous flow in a granular medium can be described by various equations. The simplest and oldest

equation for stationary flow in through porous media is the law of Darcy (1856) [VERRUIT AND VAN

BAARS, 2005]:

pU K I (2.17)

This formula can be rewritten as:

p

p

UI a U

K (2.18)

The law of Darcy gives a relation between the hydraulic gradient (I), the permeability coefficient of

the soil (K) and the filter velocity (Up) (which is depth‐averaged is this form of the Darcy law). The law

of Darcy is valid for laminar flow without convective inertia forces. The law of Darcy was extended by

Forchheimer in 1901 with a quadratic (turbulence) term [VAN GENT, 1995]:

p p pI a U b U U (2.19)

The permeability coefficient in the Forchheimer equation is represented by a, which is equal to K‐1.

Van Gent (1995) gives the following expression for a, original derived by Kozeny (1927):

23 2

1f

na

n g D

(2.20)

In this expression ν is the kinematic viscosity, D is the grain diameter, n is the porosity of the porous

medium and αf is a dimensionless coefficient to be determined empirically. Van Gent (1995) gives the

following expression for (b) original derived by Ergun (1952):

3

1 1f

nb

n g D

(2.21)

In this expression βf is a dimensionless coefficient which has to be determined empirically. The first

term of the Forchheimer equation represents the laminar contribution, while the second term

represents the contribution of turbulence. The Forchheimer equation is used for turbulent flow

through a porous medium and for flow in transition between laminar and turbulent flow [BURCHARTH

AND ANDERSEN, 1995].


‐ 15 ‐

In order to use the Forchheimer equation for un‐stationary flow Kochina (1952) added an extra time

dependent term to the original equation [VAN GENT, 1995]:

p

p p p

UI a U b U U c

t

(2.22)

In this expression c is a dimensionless coefficient which includes the effect of the added mass. Van

Gent (1991) gives the following expression for c [VAN GENT, 1995]:

11

1 A

nc nc

n g n g

(2.23)

This expression contains the parameter γ which is a dimensionless coefficient representing the effect

of the added mass. Van Gent (1995) executed an experiment in order to investigate the contributions

of the different terms. These measurements were performed in an oscillating wave tunnel [VAN GENT,

1995]. The results of these experiments are presented in figure 2‐15.

Figure 2‐15 Contribution of each term of the Forchheimer equation to the total gradient [VAN GENT, M.R.A.

1995]

Based on this figure it can be concluded that the contribution of the b‐term is the most important and

that the contribution of the c‐term is only of significant importance at the zero crossing of the signal

(reversion of the flow).

The basic equation for flow inside a breakwater is the Forchheimer equation. Various methods are

developed that determine the flow inside a breakwater and the motion of water outside of the

breakwater in relation to each other. Two methods are mentioned; the volume exchange model

which couples the outside motion of water to the internal motion of water by the conservation of

(water) volume and the exchange of water volume through the surface of the breakwater [JUMELET,

2010] and the ComFLOW model which is a Volume‐of‐Fluid model that solves the incompressible

Navier‐Stokes equations [WENNEKER, 2010].

Literature study

‐ 16 ‐

2.2.2 Dynamic load on a single armour unit

A single armour unit is loaded by both static loads and dynamic wave induced (hydraulic) loads. The

hydraulic force balance consists of a number of terms. The forces acting on a single armour unit (in

this case an armour stone) are schematised in figure 2‐16:

Figure 2‐16 Dynamic forces on an armour stone [HALD, 1998]

Various forces are displayed in figure 2‐16. They are the result of the flow around the armour unit and

the gravitational force. Since the flow around an armour unit positioned on a breakwater slope is not

stationary, the size and direction of these forces vary in time. The left side of the figure schematises

the situation during downwash of water along the breakwater slope. Since in this situation the

dynamic forces act in the same direction as the along slope static load on the first row of armour

units, this situation is of primary interest.

The figure depicts a number of forces being:

The gravitational force (Fg);

The lift force (FL);

The drag force (FD);

The inertia force (FI);

The seepage force (FS);

The lift, drag and inertia forces are all a function of the downwash velocity (Vdownwash) (V, in figure

2‐16). These forces can be described with the classical, simplified Morrison equations [HALD, 1998].

The simplified (the drag related term without the inertia related term) Morrison equations are given

since BURCHARTH (1993) established that the drag term dominates the inertia term.

D D w downwash downwash

L L w downwash downwash

downwashI M w

F C A v v

F C A v v

dvF C V

dt

(2.24)

In these equations ρw is the density of water, A is the surface of the considered body perpendicular to

the flow, Vdownwash is the downwash velocity, V the volume of the stone and CD, CL, CI, are coefficients

depending on the shape of the armour unit, the Reynolds number (Re) and the Keulegan‐Carpenter

number (KC). The Keulegan Carpenter number represents the ratio between the influence of

turbulence and inertia and reads:

V TKC

n D

(2.25)

in which V is a characteristic velocity, T is the wave period, n is the porosity and D is the particle size.


‐ 17 ‐

The Morrison equations are derived for a body in undisturbed flow. In this case the armour unit

(body) is sheltered by other units which disturb the flow. Despite this deviation from the original

circumstances, the Morrison equations are widely used. Research was performed in order to establish

the drag, lift and inertia coefficients for stones. Figure 2‐17 gives the results of the research of TØRUM,

1994 who related the coefficients to the KC number.

Figure 2‐17 Drag and inertia coefficients relative to the KC number TØRUM, 1994 [HALD, 1998]

The seepage velocity is the velocity of the flow from the core of the breakwater through the under

layer and through the armour layer. This seepage flow imposes a force on an armour unit (during

outflow) in the upward direction, perpendicular to the slope and in downward direction, parallel to

the slope. The seepage flow is the result of the porous flow in the core of the breakwater which is

caused by of the internal water gradient.

Figure 2‐18 Rise and lowering of the phreatic level due to in and outflow of water

In figure 2‐18 the phreatic level of the water in the core of a breakwater is sketched. The water flows

into the core of the breakwater during the period of uprush of water till the maximum run‐up is

reached. When the maximum run‐up is reached the internal phreatic level is also at its maximum and

the inflow of water reverses resulting in an outflow of water. The driving mechanism behind the

outflow of water is thus the hydraulic gradient of the elevated phreatic level.

As stated before, the hydraulic gradient can be described with the Forchheimer equation:

f p p pI a v b v v (2.26)

The porous flow at every point in the core of the breakwater can be calculated based on the water

level inside the core and the Forchheimer equation [MANSARD AND FUNKE, 1980]. The seepage flow at

the interface between the under layer and the armour layer is of primary interest since at this

interface the actual loads on the armour units occur.

Literature study

‐ 18 ‐

In figure 2‐19 a schematisation of all relevant occurring flows is given:

Figure 2‐19 Flow around a armour unit

In order to determine the force on an armour unit in downward, along slope direction, the downwash

velocity and the seepage velocity should be determined. The drag and inertia force of the armour unit

can be calculated based on the Morrison equations, the calculated downwash velocity and the along

slope component of the force resulting from the seepage flow.

The force imposed on the armour unit in upward direction, perpendicular to the slope is composed of

two components. One component is a lift component which can be determined with the Morrison

equation with as input the downrush velocity. The second component is the force resulting from the

seepage flow in perpendicular direction to the slope which can be determined with the Forchheimer

equation.

These two components result in the total force in upward direction, perpendicular to the slope. This

total force reduces the normal force what results in a reduced friction force. The reduced friction

force may increase the load on the lower laying armour units.


‐ 19 ‐

3 Hypothesis

Based on the literature study in the previous chapter, it was concluded that the load on the first

(bottom) row of armour units is composed of various loads, being the static load, the dynamic load,

the abrasion load, the thermal load and the chemical load. In this research the static load and the

dynamic load were further investigated. The influence of earthquakes, arching and wedge effects and

rocking were left outside the scope of this research. Based on the division between static load and

dynamic load in literature, two hypotheses were tested in this research; one regarding the static load

and one regarding the dynamic load.

Based on the extended along slope force balance (equation 2.4) it was reasoned that if the single

armour units are not in balance by themselves they impose a residual load on the lower positioned

armour units. The residual force on this lower positioned armour unit can be transferred to the under

layer by this unit or can be transferred further along the slope to lower positioned armour units. This

creates an armour layer of armour units each with its own force balance and with residual forces

between them. From the top row downward to the first row the residual forces add up and impose a

load on the first row of armour units.

If all individual force balances would be identical to each other than all residual forces originating

from the individual force balances would have the same magnitude. The positive residual forces

between the armour units lead to a linear increasing load on the first row of armour units as function

of the number of rows applied on the slope of the breakwater. However, experiments performed by

MUILWIJK (2011) shows that the relative increase of the static load with the number of rows, decreases

with the number of rows applied on a breakwater. These statements are summarised in the following

hypothesis for the static load:

The static load on the first row of armour units is the result of the individual residual forces (resulting

force from individual force balance) per armour unit which add up to a total resulting force on the first

row of armour units. The increasing rate of the static load, decreases with the number of rows applied

on the breakwater slope which is caused by a number of armour units which can bear (a part of) the

residual forces of armour units placed on higher rows.

The governing dynamic (hydraulic) load is the dynamic load in downward, along slope direction with

the largest magnitude. The dynamic load during the time period of the incoming and breaking wave is

mainly directed in upward along slope direction or in a direction perpendicular to the slope. During

the time period of the reflecting wave a downwash of water flows in downward, along slope

direction, in the same direction of the considered static load on the first row of armour units.

Furthermore, an outflow of water from the core destabilises the armour units beneath the water line.

It was assumed that this situation is governing for the (maximum) dynamic load on the first row of

armour units, which results in the following hypothesis regarding the dynamic load:

The combination of downwash on and through the armour layer, and the outflow of water from the

inside of the breakwater through the armour layer, is determining for the maximum loading of the

first row of armour units. The transfer of the dynamic forces imposed on an individual unit correlates

to the transfer of residual forces as described in the static load hypothesis.

Hypothesis

‐ 20 ‐

3.1 Research approach

In order to study the load on the first row of armour units and to test the hypotheses, the load on the

first row of armour units had to be determined. This load was examined by physical model tests.

Physical model test are still widely used within the field of coastal engineering and are still the basis of

many research programs. The static load and the dynamic load were measured in different

experiments.

The static load experiments performed in this research were of the same nature as the experiments

executed by MUILWIJK (2011). However in this research a different size of model unit was used and the

number of tests was increased in order to improve the general validity of the conclusion drawn from

the experiments of MUILWIJK (2011) and the experiments performed in this research.

The dynamic load was examined with a physical model in a wave flume. The hypothesis described a

significant influence of the downwash velocity on the dynamic load. Various parameters which

influence the downwash velocity were varied and studied in this experiment. Furthermore the

influence of the seepage flow was studied in this experiment.


‐ 21 ‐

4 Experiments

The load on the first row of armour units was investigated with an experiment with only static load

and with an experiment with dynamic loads. The static experiment is described in paragraph 4.1 and

the hydraulic experiment is described in paragraph 4.2.

4.1 Static experiment

A static experiment was developed and executed in order to determine the load on the first row of

armour units in static conditions (without the influence of waves). The experiment was designed for

dry conditions thus also the influence of buoyancy of the armour units was outside the scope of this

experiment. The load on the first rows of armour units was investigated based on the number of rows

applied on the slope of a breakwater.

The concept of this experiment was to measure the load on the first row of armour units with a

movable beam which was connected to force gauges which measures the load imposed on this beam.

This concept is sketched in figure 4‐1.

Figure 4‐1 Conceptual design of static experiment

The movable beam was made as a T‐shaped retaining structure which was able to move in along

slope direction due to the wheels which under the beam. The beam was supported in along slope

direction by suspension cables connected to two force gauges (one on each side of the slope). These

force gauges were of the type MG‐200W, had a range of 0‐1000 N and a resolution of 1 N. This

construction enabled the measurement of forces imposed on the beam in along slope direction.

Forces were imposed on this beam by the armour units which were placed directly against this beam

(and were also supported by the under layer). The first row of armour units placed on the beam was

placed with a regular orientation (the reader is referred to appendix A for photographs of this

placement). The load imposed by this first row on the beam was read from the force gauges and

noted after the complete placement of the first row. Subsequently, a second row was placed on the

slope and the total force imposed by the two installed rows on the beam in along slope direction was

measured and noted. This routine was repeated until a total of 20 rows were placed on the slope.

In this way the load on the beam as function of the number of rows applied on the slope was

measured. The objective of this research was to measure the load on the first rows of armour units.

However, in this experiment instead of the load on the first row of armour units, the load imposed by

the first row of armour units on a movable beam was measured. In this experiment the beam is

Experiments

‐ 22 ‐

“replacing” the first row of armour units. The load imposed by the first row of armour units is in

reality thus the load imposed by the first row of armour units above the lowest row of armour units

(row 0) on this lowest row of armour units. In practice, this lowest row can consist of an actual row of

armour units (Xbloc) or a special armour unit designed for the placement on the lowest row. (Xbase).

The model armour units used in this experiment were Xbloc single layer armour units (see figure 4‐6)

with a weight of 366 gram and a unit height of 7,9 cm. The armour units were placed in a staggered

grid corresponding to a packing density of 1,20 / DXbloc2 (the reader is referred to paragraph 4.2.2.3 for

more information about the packing density). This packing density resulted in a horizontal centre‐to‐

centre distance between the armour units of 10,43 cm (1,32 times the unit height) and a vertical

centre‐to‐centre distance between the armour units of 4,98 cm (0,63 times the unit height). Twenty

armour units were placed on a horizontal row which gave a total row width of nineteen times the

centre‐to‐centre distance between the armour units plus one time the unit height of an armour unit

(205,9 cm). The test slope used for this experiment was wider than this row width. The excess width

was filled up. The along slope length of the armour layer with 20 rows was calculated in the same way

what resulted in an along slope length of 102,5 cm.

The armour units were placed on an under layer which was placed on a wooden slope of 36,87

degrees (3:4). The under layer was made of stones with a standard grading which had a W50 of 36,6

gram (the requirements for under layers are described in paragraph 4.2.2.4). The reader is referred to

appendix A for the weight distribution of the stones used for this under layer.

A total of 15 tests were done with a varying relative packing density. The relative packing density

(RPD) is defined as the requested packing density divided by the applied packing density:

1 1 =

x y x y

x y

N N d dRPD

L L

(4.1)

In which Nx is the number of armour units in a horizontal row, Ny is the number of horizontal rows

applied on the slope, dx is the horizontal centre‐to‐centre distance between armour units, dy is the

along slope centre‐to‐centre distance between armour units, Lx is the actual measured distance

between the centre of the armour unit at the far left side of a horizontal row and the centre of the

armour unit at the far right side of the same horizontal row and , Ly is the actual measured along slope

distance between the centre of the armour unit at the lowest row and the centre of the armour unit

at highest row.

An armour layer placed accordingly the previous described centre‐to‐centre distances has a RPD of

100%. The RPD was varied in this research in order to investigate the influence of the packing density

on the static load. Five tests were done with a RPD lower than 98%, five tests were done with a RPD

of 98% to 102% and five tests were done with a RPD higher than 102%.

4.1.1 Critical angle

In addition to the described static experiment, a second static experiment was designed in order to

investigate the individual force balance of an armour unit. The force balance of the individual units

was researched in order to determine the friction between an armour unit and the under layer.

Knowledge about the friction between an armour unit and the under layer was necessary in order to

determine the theoretical load in the first row of armour units. This theoretical load was compared to

measured load in the previous described experiment.

The friction between an armour unit and the under layer was studied by determining the critical angle

of an armour unit on a slope of under layer material. The critical angle is the angle of the slope at


‐ 23 ‐

which an armour unit is just stable. The friction between the armour unit and the under layer is equal

to the along slope component of the gravitational force of an armour unit in this case. The along slope

component of the gravitational force of an armour unit is a function of the slope and can be easily

calculated. If the critical angle and the weight of an armour unit are known then it is possible to

calculate the friction between the armour unit and the slope.

The critical angel was investigated by placing the armour units on a slope of under layer material that

could be tilted so that the angle of the slope increases:

Figure 4‐2 Sketch of the critical slope experiment

The slope was gradually increased until the armour units started to move. The angle at which the

armour units started to move was noted as the critical angle.

This test was done with Xbloc armour units with a weight of 119 gram and a unit height of 5,4 cm. The

stones used in the applied under layer had a W50 of 11,9 gram (the reader is referred paragraph

4.2.2.4 for more information about the selection of material for an under layer). The test was

executed 200 times so a dataset of 200 data points was collected.

4.2 Hydraulic experiment

A hydraulic experiment was designed and executed in order to determine the dynamic (hydraulic)

load on the first row of armour units caused by the wave action on the slope of a breakwater. The

static experiment which determined the static load on the first row of armour units consisted of a

movable beam which supported the armour layer. This beam was supported by suspension cables

which were connected to two force gauges which measured the load imposed on the beam by the

armour layer. This concept was also used for the hydraulic experiments.

The armour units used in the hydraulic experiment were also supported by a movable beam

connected to a force sensor. Instead of using suspension cables for the support of the movable beam,

a stiff frame was used for the connection between the movable beam and the force sensor. This was

done because the dynamic loading of the breakwater induced rapid changes of the tensile forces in

the connection between the movable beam and the force sensor. Therefore, a stiff connection was

chosen to enable the direct transfer of changes in tensile forces to the force sensor.

Experiments

‐ 24 ‐

Figure 4‐3 Side and frontal view of breakwater model with measuring frame

The frame placed on the breakwater is visualised in figure 4‐3. The left part of figure 4‐3 displays the

frontal view of the frame on the breakwater with some armour units placed on it and the right part of

figure 4‐3 displays a side view of the frame placed on the breakwater. The frame transferred the

forces imposed on the lowest horizontal beam of the frame to the force sensor which was placed on a

horizontal beam. This horizontal beam was placed on top of the flume. The frame consisted of

aluminium L‐profiles. Because the tension forces occurring in the profiles in along slope direction

were limited, the dimensions of the profile could be rather small. The profiles in along slope direction

had a width of 2 cm.

The frame was able to move freely in along slope direction since any obstruction would influence the

outcome of the measurements. In order to achieve a free moving frame in along slope direction the

frame was fitted with wheels (at both sides two wheels, both at the top of the model breakwater and

at the toe of the prototype breakwater). The wheels moved in a U‐profile which was positioned on

the under layer and directly against the glass (side) of the wave flume (figure 4‐4). The width of the u‐

profile was 20 mm, the height was 15 mm and the thickness of the walls and base was 2 mm. The

wheels which fit in the U‐profile had a width of 14 mm, a diameter of 14 mm and a total height of 17

mm. Two types of profiles were used for the frame. The profile in along slope direction was a L‐profile

with a height and a width of 20 mm. The horizontal profiles were also L‐profiles but had a height of

10mm and a width of 20 mm (figure 4‐4).

Figure 4‐4 Cross section of along slope frame profile and U‐profile at location of one of the wheels

In the described hydraulic experiment, the load imposed by the first row of armour units on the frame

was measured instead of the load on the first row of armour units. Statements made about the load

on the first row of armour units refer to this measured load which is in reality the load on the armour


‐ 25 ‐

units (Xbase or a row of Xbloc armour units depending on the actual design) which is “replaced” by

the frame. When statements are made referring to the load on the first row of armour units as

function of loads developed in the first five rows, then the load between these five rows and the

frame or the (not applied in the hydraulic tests) row of Xbases is meant.

4.2.1 Wave Flume

The hydraulic tests were executed in the wave flume of Delta Marine Consultants in Utrecht, The

Netherlands. This flume has a length of 25 m, a width of 0,6 m and a height of 1,05 m. The maximum

water level is 0,7 m and the maximum wave height that can be generated is 0,3 m. The waves are

generated with a piston wave generator (of Edinburgh Designs) which can generate regular and

irregular waves. The reflected waves are compensated by the wave generator.

Figure 4‐5 Longitudinal view of the wave flume [VAN ZWICHT, 2009]

4.2.2 Breakwater Model

4.2.2.1 Scaling laws

The scaling of a prototype breakwater with related wave conditions to a model breakwater with

related wave conditions should be done in such a way that the characteristics of the prototype agree

with the characteristics of the model. If the scaling of the prototype to the model is incorrect then the

results obtained from the physical model test become inaccurate and therefore unreliable. The scale

(λ) is determined based on geometric, dynamic and kinematic similarity [FROSTICK et.al., 2011].

Geometric similarity is obtained as all corresponding linear dimensions between the model and the

prototype are scaled according to the same scale factor (λL):

mL

p

l

l (4.2)

Kinematic similarity is obtained when all time‐depended processes in the model have the same

constant relation to the time‐depended processes in the prototype. Kinematic similarity thus means a

similarity of motion (of the fluid particles):

mt

p

t

t (4.3)

Dynamic similarity is obtained if there is a similarity between the occurring forces in the model and

the occurring forces in the prototype. The scale factor between all vectorial forces of the prototype

and the model should be equal:

mF

p

F

F (4.4)

These conditions can be fulfilled with the similitude of the Froude or Reynolds number in combination

with geometric similarity [HUGHES, 1993].

Experiments

‐ 26 ‐

The Froude number is defined as the ratio of a characteristic velocity to a gravitational wave velocity.

The Froude number of the model should be the same as the Froude number of the prototype:

p m

U U

g l g l

(4.5)

The Reynolds number is defined as the ratio of inertial forces to viscous forces. The Reynolds number

of the model should be the same as the Reynolds number of the of the prototype:

/ /w wp m

U l U l

(4.6)

This condition is impossible to fulfill for models at a reduced scale [HUGHES, 1993]. HUGHES (1993)

states that the flow conditions throughout the primary armour layer must remain turbulent. The best

approach to prevent the viscous forces to become too large relative to the inertial forces is to use a

model scale as large as possible.

The physical model tests were executed with short waves which are dominated by gravitational forces

(relative to the viscosity). Therefore Froude scaling (which corresponds to linear geometric scaling)

was used which means that the viscosity, elasticity and surface tension forces could not be scaled

correctly causing scale effects. According to HOLTHUIJSEN (2007) surface tension affects waves with

wave periods shorter than 0,25 seconds. These waves were not applied in this research and the

influence of surface tension was therefore not further considered.

As stated before, the viscous forces could not be scaled correctly when using the Froude criterion. However, as long as the flow velocities stay high enough to keep the flow in the primary armour layer

turbulent the Reynolds criterion is reasonably satisfied. The flow velocities in both layers are relatively

high due to the high permeability of the primary and secondary armour layer which results in

relatively high Reynolds numbers. In most cases laminar flow is not a problem in the primary and

secondary armour layer [VAN ZWICHT, 2009]. There is however a possibility of non‐turbulent flow in the

core. In order to minimise the scaling effect of viscous forces a minimum Reynolds number for the

flow in the core is specified. The flow in the core of the model is turbulent for Reynolds numbers

higher than 1∙102 [HUGHES, 1993].

4.2.2.2 Foreshore

Normally a foreshore is applied in order to simulate the bathymetry in front of the prototype to be

modelled. The model used in this research was not related to an actual prototype and was aimed to

be a model of a “general rubble mound breakwater” although such a breakwater does not exist in

reality. In this research a flat foreshore was used which means that the waves generated by the wave

generator were not transformed by this foreshore. The generated waves were comparable to the

waves near the structure. However in practice the waves are transformed by the foreshore and the

results and expressions found in this research should be related to the waves near the structure.

4.2.2.3 Armour units

The physical model was armoured with Xbloc single layer armour units. Various sizes of Xbloc model

units were available which ranges from an unit height of 2,9 cm to 7,8 cm. The width of the flume was

reduced by the frame leaving a width of 55 cm for a row of armour units. The armour units should be

placed at a horizontal centre‐to‐centre distance (dx) of 1,32 times the unit height (DXbloc) from each

other [TEN OEVER, 2011]. The width of the armour layer can be determined based on the number of

armour units at a horizontal row and the size of these units. The width of the armour layer should be


‐ 27 ‐

smaller than 55 cm. The application of a larger unit reduces the number of units at a horizontal row.

In this research Xbloc model units with a unit height of 4,3 cm and a weight of 61,7 gram were used

what corresponds to a total number of ten armour units at a horizontal row. The thickness of the

armour layer is 0,97 times the unit height of the armour unit for Xbloc armour units. The unit height

of 43 mm resulted in an armour layer thickness of 42 mm.

Figure 4‐6 Frontal and side view of a Xbloc single layer armour unit

Xbloc armour units are placed in a staggered grid with packing density (number of armour units per

m2) of 1,20 / DXbloc2. This packing density was converted into centre‐to‐centre distances (dx and dy) for

the armour units on the grid. A packing density of 1.20 / DXbloc2 corresponds to a horizontal centre‐to‐

centre distance of 1,32 times DXbloc and a horizontal centre‐to‐centre distance of 0,63 DXbloc [TEN OEVER,

2011].

An armour unit height of 4,3 cm and a total number of 10 units at a horizontal row corresponds to a

horizontal centre‐to‐centre distance of 1,30 times DXbloc and an along slope centre‐to‐centre distance

of 0,64 times DXbloc which is within margins.

The significant wave height corresponding to the determined model armour unit was determined

with the following formula [DELTA MARINE CONSULTANTS, 2011]:

3 with 32,77

sxbloc Xbloc xbloc

HV D V

(4.7)

The applied armour units had a density of 2279 kg m‐3 and the water in the wave flume has a density

of 1000 kg m‐3. This results in a corresponding significant wave height of 11 cm.

4.2.2.4 Under layer

As under layer Delta Marine Consultants recommends the use of standard gradings as listed in the

Xbloc design table (appendix B) and specified in the Rock Manual, CIRIA C683 (2007). As alternative

non‐standard grading’s can be used if they fulfill the following requirements [TEN OEVER, 2011]:

15

Xboc 50 Xbloc

85

1W W

15

1 1W W W

11 9

1W u W

7

Xbloc

Xbloc

(4.8)

Experiments

‐ 28 ‐

In which WXbloc is the weight of the Xbloc, W85 is the rock weight that is exceeded by 15% of the rocks

in the under layer, W50 is the rock weight that is exceeded by 50% of the rocks in the under layer and

W15 is the rock weight that is exceeded by 85% of the rocks in the under layer.

Normally, a model is scaled according to the model scale and the prototype parameters. In this case

there was no real prototype and therefore no strict model scale. The model was linked to a fictional

prototype in order to determine the prototype under layer dimensions and scale them back to the

model scale. As prototype armour unit a armour unit with a unit height of 1,96 m was chosen which

sets the model scale to 45,75. Xbloc armour units with a unit height of 1,96 m are placed on an under

layer with a standard grading of 300 to 1000 kg (Xbloc design table, appendix B). This grading was

constructed on scale with a mixture of sieve gradings. The standard grading of 300 to 1000 kg on a

scale of 45,75 was made with a mixture of 45 % of the sieve grading 11,2 to 16 mm and 55% of the

sieve grading of 16 to 22,4 mm. One is referred to appendix B for more information of the applied

grading relative to the standard grading. The Dn50 of the stones used for the under layer was 13,3 mm.

The minimum thickness of the under layer was calculated from the thickness described in the Xbloc

design table (1,2 m), (appendix B) and the scale factor of 45,75. This gave a under layer thickness of

28,5 mm.

4.2.2.5 Slope

The conventional slope for Xbloc armour units is a slope of 3:4 (36,9 degrees). This slope was the

standard slope in this research. In order to investigate the effect of the slope steepness a flatter slope

of 2V:3H (33,7 degrees) was tested as well.

4.2.2.6 Crest height and width

One of the objectives of the physical model research was to investigate the effect of a varying number

of rows. Three different numbers of rows were incorporated in the test program; 15, 20 and 25 rows.

The numbers of rows correspond to a length of the armour layer and crest height. The length of the

armour layer was calculated with the following expression:

armourlayer xbloc y yL D N d (4.9)

The length of the armour layer and the crest height corresponding to the number of rows applied on

the slope can be found in the following table:

Table 4‐1 Length of armour layer and crest height

Number of rows (ny) Larmour layer Crest height (slope 3:4) Crest height (slope 2:3)

15 42,8 cm 29,9 cm 27,9 cm

20 56,6 cm 38,1 cm 35,6 cm

25 70,3 46,4 cm 43,4 cm

The minimum crest width is 2,28 times Dxbloc which was 98 mm with the chosen model units.

4.2.2.7 Water level

The water level was chosen in such a way that the overtopping during the tests was limited. The run‐

up was calculated with the run‐up formula of VAN DER MEER AND STAM (1992), for every combination of

the number of rows, wave height and period included in the wave program (described later). It

appeared that with a water level of 12 cm above the bottom level of the toe overtopping only occurs

for tests with wave heights of 0,6 Hmax and higher.


‐ 29 ‐

The largest wave height of the wave program (paragraph 4.2.3.1) is the maximum wave height which

is about equal to two times the significant wave height [HOLTHUIJSEN, 2007]. A significant wave height

of 11 cm thus gives a maximum wave height of 22 cm. Depth induced wave breaking occurs for waves

with wave heights which are higher than 0,88 times the local water depth according to the criterion of

Miche (equation 2.7) [SCHIERECK, 2001]. This means that for wave heights of 22 cm a minimum water

depth of 25 cm is needed.

In order to maintain the water level of 12 cm from the bottom of the toe and to have enough water

depth in front of the breakwater for the highest waves a compromise was found in raising the

breakwater with an extended structure underneath the actual breakwater. The slope of the designed

breakwater was extended until a vertical height of 43 cm was added to the breakwater. The surface of

the structure underneath the Xbloc armoured model breakwater was protected with gabions:

Figure 4‐7 Cross‐section of the breakwater model with under structure

The under structure could had two possible effects on the load of the armour units. The first effect

could be that the absence of a toe structure and a seafloor at the lowest row of armour units had as

consequence that the downwash flow was not hampered by a toe structure or the seabed. The

downwash flow would normally be bend in “seaward” direction while in this design the flow was able

to remain parallel to the slope what might induce higher load on the armour units. The second effect

could be that the outflow of water from the core occurs at a larger surface, resulting in lower

velocities (but the same total volume of outflow). This effect may lower the load on the armour units.

The first effect imposed a possible higher load on the armour units and thus a higher load on the first

row of armour units. Measurements influenced by this effect were thus conservative. The lower

outflow velocities results in a lower load on the armour units which is unintended. An impermeable

layer at the horizontal level of the lowest row of armour layer was applied in order to compensate for

this effect and to obtain conservative results. The effect of this layer was investigated by removing

this layer and comparing the results to the results of a test without this impermeable layer.

4.2.2.8 Cross‐section

Based on information of the armour layer, under layer, slope, number of rows and crest height it was

possible to determine the dimensions of the breakwater and design the cross‐section of the

breakwater. In this research the number of rows was varied and a breakwater with 15, 20 and 25

rows and a slope of 3:4, was designed. Furthermore, the slope was varied so a fourth design of a

breakwater with 20 rows and a slope of 2:3 was designed.

Experiments

‐ 30 ‐

The basic idea regarding the breakwaters with a 3:4 slope was to design a 3:4 breakwater with 25

rows and corresponding crest height. The width of the crest was for this breakwater equal to the

minimum value of a crest with Xbloc armour units (2,28 times Dxbloc). Based on this initial design the

breakwaters with 20 and 15 rows were designed by removing the top of the breakwater design with

25 rows.

Figure 4‐8 Sketch of cross‐section with 15, 20 and 25 rows

Since the lower part of the breakwater remained the same, the width of the breakwater at water

level also remained the same and the crest width increased for breakwaters with 15 and 20 rows. For

a more detailed design the reader is referred to appendix B.

The breakwater with a 2:3 slope was a modified version of the breakwater with a slope of 3:4 and 20

rows. This design was made by adjusting only the seaward slope of the breakwater. The seaward

slope was rotated from a slope of 3:4 to a slope of 2:3 at the point where the water level intersects

with the centre of the armour layer. In this way the number of submerged armour units in a situation

without waves is the same for a breakwater with a slope of 3:4 and a breakwater with a slope of 2:3.

Figure 4‐9 Cross‐section of a breakwater with 20 rows for a slope of 2:3 and a slope of 3:4 (striped)


‐ 31 ‐

4.2.2.9 Core

A geometrical scaled core may induce problems with the similarity of the flow in the core of the

breakwater. A geometrical scaled core leads to small diameters of the material used in the core and

this could lead to a too low permeability of the core and a resulting flow which is not similar to the

flow in the core of the prototype. BURCHARTH, et.al (1999) developed an empirical formula for the

wave induced pressure gradient in the core which, in combination with the Forchheimer equation,

can be used to estimate the pore velocities in the core. These velocities can be used to determine the

characteristic Froude number of the prototype. This Froude number should be the same in the model.

Based on the method proposed by BURCHARTH, et.al (1999), it is possible to determine the core

diameters of the model in such a way that the characteristic pore velocity corresponding to Froude

number of the model equals the characteristic pore velocity corresponding to Froude number of the

prototype. For further reading the reader is referred to [BURCHARTH, et.al, 1999].

The Dn50 of the material used in the core of the prototype breakwater was determined by applying the

filter rules of Terzaghi for the interface between the core and the under layer. Following this

approach a Dn50 of 41 cm was determined for the material used in the core of the prototype

breakwater. Based on the method described by BURCHARTH, et.al (1999) a characteristic pore velocity

for the prototype of 0,2374 m∙s‐1 was found which corresponds to a characteristic pore velocity of the

model obtained from Froude scaling of 0,03517 m∙s‐1. A Dn50 of 0,90 cm of the material used in the

core was determined corresponding to a characteristic pore velocity of 0,03518 m∙s‐1. This is well in

line with the requested characteristic pore velocity obtained from Froude scaling.

As input parameters for the Forchheimer equation (equation 2.19) an αf of zero was used and a βf of

3,6 was used. The porosity of both the prototype as the model were assumed to be 0,4.

As stated before, laminar flow should not occur in the core. The Reynolds number in the core was

calculated and has a value of 8,22∙103. The calculated Reynolds number is above the requested

minimal Reynolds number of 1∙102 [HUGHES, 1993].

Based on the determined Dn50 of the required material used as core, a standard grading was selected

which fulfils this condition. The standard grading of 60 – 300 kg fulfils this condition (scaled to

prototype scale based on the just determined Dn50 for the core material). This grading was constructed

with a mixture of 20 % of the fraction of 5,6 to 8 mm, 40 % of the fraction of 8 to 11,2 mm and 40 %

of the fraction of 11,2 to 6 mm. The reader is referred to appendix B for the graph of the applied and

required grading

4.2.3 Test program

In the previous paragraphs some statements were made about the parameters which were varied in

the test program. First of all the number of rows applied on the breakwater was varied during the test

program. By varying the number of rows influence on the dynamic load on the first row of armour

units could be investigated. A higher number of rows means that the waves imposed forces on a

larger number of rows (and thus armour units). The expectation was that the measured load for a

larger number of rows would be higher. Tests were executed with 15, 20 and 25 rows.

The second parameter that was varied in the test program was the packing density. The packing

density influences the flow through the armour layer (a higher packing density reduces the

permeability of the armour layer) and thus influences the load on the armour units. A high packing

density was expected to have a decreasing effect on the flow velocities resulting in a lower load on

Experiments

‐ 32 ‐

the first row of armour units. Relative packing densities of 97%, 100% and 104% were tested in this

research (the reader is referred to paragraph 4.1 for the definition of the relative packing density).

The third parameter that was varied was the permeability of the core. The permeability of the core

influences the run‐up on the slope and the volume of water that flows into and out of the core during

a wave cycle. A breakwater with a normal permeable core was compared with a breakwater with an

impermeable core. In this way the influence of the outflow from the core through the armour layer on

the stability of the armour units and thus on the load on the first row, was studied. The same wave

height will induce higher levels of run‐up at a breakwater with an impermeable core which will result

in higher downwash velocities. In order to study the effect of the outflow from the core, tests with

the same downwash velocities had to be compared. Furthermore, the test with an impermeable core

might give some insight into the load on the first row of armour units for tests with the same wave

heights.

The fourth parameter that was varied was the smoothness of the under layer. A smooth

(impermeable) under layer was expected to transfer more load to the first row of armour units

compared to the test with an impermeable core but with a normal under layer. This test was also

used to measure the static load on the first row with armour units placed on a breakwater with a

smooth underlayer.

The last parameter that was varied was the slope of the breakwater. The conventional slope for

breakwaters with Xbloc armour units is 3:4. In this program also a flatter slope of 2:3 was tested.

Interlocking armour units are less stable on a flatter slope so the expectation was that the relatively

instable armour units transfer more load to the first row of armour units.

In order to limit the number of experiments a base case was defined; being a breakwater with a

permeable core, a slope of 3:4, 20 rows, a normal under layer and a packing density of 100%. The

permeability of the core, the number of rows, the slope, the smoothness of the under layer and the

packing density were only varied in reference to this basis case. This means that not every parameter

was tested against every other parameter, thereby reducing the number of tests.

Table 4‐2 Parameters varied in the test program

Parameter Variation

Number of Rows 15 / 20 /25 rows

Packing density 97%, 100% and 104%

Core Permeable / impermeable core

Under layer Normal / Smooth and impermeable

Slope 3:4 and 2:3

Each test was executed at least three times. For the full test program and photographs of the

constructed breakwater the reader is referred to appendix B.

4.2.3.1 Wave program

The main part of the hydraulic tests was done with regular waves. This was done because in this

research the main objective was to investigate the effect of the relevant processes which could be

better identified with regular waves. Regular waves in a wave flume are reflected by the structure

what may result in a standing wave pattern. The wave lengths were determined in such a way that

standing patterns did occur.


‐ 33 ‐

Five wave heights were tested and each of these wave heights was combined with three periods

which correspond to a Iribarren number of 3, 4 and 5:

Table 4‐3 Wave program regular waves

H = 0,2 Hmax H = 0,4 Hmax H = 0,6 Hmax H = 0,8 Hmax H = 1,0 Hmax

H (cm) 4,4 H (cm) 8,8 H (cm) 13,2 H (cm) 17,6 H (cm) 22

T(s) 0,67 T(s) 0,94 T(s) 1,14 T(s) 1,33 T(s) 1,46

ffreq (Hz) 1,50 ffreq (Hz) 1,06 ffreq (Hz) 0,874 ffreq (Hz) 0,749 ffreq (Hz) 0,687

L (m) 0,70 L (m) 1,38 L (m) 2,03 L (m) 2,75 L (m) 3,25

S (‐) 0,06 S (‐) 0,06 S (‐) 0,06 S (‐) 0,06 S (‐) 0,07

ξ (‐) 2,98 ξ (‐) 2,97 ξ (‐) 2,94 ξ (‐) 2,96 ξ (‐) 2,88

T(s) 0,94 T(s) 1,33 T(s) 1,60 T(s) 2,00 T(s) 2,29


L (m) 1,38 L (m) 2,75 L (m) 3,90 L (m) 5,87 L (m) 7,41

S (‐) 0,03 S (‐) 0,03 S (‐) 0,03 S (‐) 0,03 S (‐) 0,03

ξ (‐) 4,21 ξ (‐) 4,19 ξ (‐) 4,08 ξ (‐) 4,33 ξ (‐) 4,35

T(s) 1,14 T(s) 1,78 T(s) 2,00 T(s) 2,67 T(s) 2,67


L (m) 2,03 L (m) 4,74 L (m) 5,87 L (m) 9,57 L (m) 9,57

S (‐) 0,02 S (‐) 0,02 S (‐) 0,02 S (‐) 0,02 S (‐) 0,02

ξ (‐) 5,10 ξ (‐) 5,51 ξ (‐) 5,00 ξ (‐) 5,53 ξ (‐) 4,95

The wave program was executed for every test beginning at the upper left side of the table (0,2 Hmax,

with a frequency of 1,50 Hz) and ending at the lower right side of the table (1,0 Hmax, 0,375 Hz).

The base test was also executed with a wave program of irregular waves. Irregular waves can be

characterised by a wave spectrum. The irregular waves in this research were generated according to a

JONSWAP spectrum:

42/1

2

5

442 52

f ffreq freq freqpeak

freqpeak

f

fe

JONSWAP freq JONSWAP freq JONSWAPE f g f e

(4.10)

With EJONSWAP is the variance density, ffreq the frequency , ffreq,peak the peak frequency (0,687 Hz) which

was obtained from the peak period corresponding to the significant wave height and a breaker type

of 4, g is the gravitational acceleration, αJONSWAP is a scaling parameter which is based on the Person‐

Moskowitz spectrum and is 0,0081 in the applied spectrum, γJONSWAP a scaling parameter which is 3,3

in the standard JONSWAP spectrum and σJONSWAP which is also a scaling parameter which is for the

standard JONSWAP spectrum 0,07 if ffreq ≤ ffreq,peak and 0,09 for ffreq > ffreq,peak.

For further reading about the JONSWAP spectrum the reader is referred to [HOLTHUIJSEN, 2007].

The significant wave heights were obtained from the applied spectra and were equal to the design

wave height corresponding to the applied armour units (Hs=11 cm). A total number of 2000 waves

were imposed on the structure during the tests with irregular waves corresponding with a storm

duration of 5,5 hours.

Experiments

‐ 34 ‐

4.2.4 Measurements

The main objective of the hydraulic tests was the measurement of the load on the first row of armour

units. This load was measured by connecting the frame which supports the lowest row of armour

units to a LSH load cell with a capacity of 0 to 100 kg and a combined error of 0,035 % of the rated

output. The reader is referred to appendix B for more information about the load cell.

The load cell was connected to an amplifier which provided a current to the load cell and fed a signal

to the computer which was read using the program DASYLab 11.0 [DASYLAB, 2009]. This program was

used to filter the noise from the signal by filtering the frequencies. Initially, all frequencies were

measured but, after filtering, the number of measured frequencies was reduced so that only the wave

induced effects were measured. The measured signal was converted from voltage to force by

multiplying the signal with a factor 4,94 which was determined by calibration using known weights.

The waves were measured by wave gauges at three places in the flume. A group of four wave gauges

was placed at a distance of 5,35 m from the wave generator side of the wave flume. The distance

between the wave gauges was respectively 0,3 m, 0,4 m and 0,2 m. A second group of wave gauges

was placed closer to the structure at a distance of 15,6 m from the wave generator and 5,5 m from

the structure (for the wave gauge closest to the structure

Figure 4‐10 Positions of wave gauges

The signal of the described wave gauges was not directly linked (in time) to the measured data of the

load cell. The signal of the described wave gauges was used in order to evaluate the wave heights and

periods after the completion of a test. The signal of an eighth wave gauge was directly coupled to the

measurements made with de load cell. This wave gauge was placed at the toe of the model

breakwater. The signal of this wave gauge was not used to evaluate the wave height but only to

evaluate the wave cycle in relation to the measured load on the first row of armour units.

The run‐up and run‐down along the slope was visually read (and noted) from a ruler positioned on the

glass of the wave flume at the interface of the armour layer and the under layer.

The settlement of the top row of armour units was measured by hand. The settlements were

measured after three subtests with the same wave height were executed. The measured settlements

were a total value for all tests with the same wave height. The subsidence of the armour units was

measured relative to a rod which was placed at a fixed location which corresponds to a RPD of 100%.

Finally, all subtests were filmed by two cameras which were positioned at the side of the breakwater

and at the top of the flume. Photographs were taken before and after each subtest from the same top

position which gives a view of the armour layer perpendicular to the surface of the armour layer. The

reader is referred to appendix B for photographs of the video cameras and photo camera positions

and other photographs of the hydraulic model.


‐ 35 ‐

5 Results and analysis of static test

In the previous chapter two experiments are described. One experiment was designed and executed

in order to study the static load on the first row of armour units and one experiment which was

designed and executed in order to study the dynamic load on the first row of armour units. In this

chapter the results of the experiment concerning the static load and the analysis of these results are

presented.

The experiment was done by measuring the load on the first row of armour units in a physical model.

This is described in the previous chapter. Furthermore, measurements of the individual critical angle

of a Xbloc armour unit (the angle at which an armour unit is still just stable) on a slope of under layer

material were executed in order to study the force balance of a single armour unit.

The measurements of the critical angles are used for a model which can simulate the behaviour of the

rows of armour units and was made to interpret the measured load on the first row.

Figure 5‐1 visualises the results of the measurements of the static load on the first row of armour

units as a function of the number of rows applied on the slope. The load is presented in a

dimensionless form, by dividing it by the gravitational force of a single armour unit multiplied by the

number of units in a row. In this way the average load on a single armour unit at the first row

expressed in the number of armour unit weights is obtained.

Figure 5‐1 Measurements of load on the first row during static tests

The legend at the right of the figure displays the relative packing density (RPD) at the end of the static

test. The tests are grouped in three groups; a group with a low packing density (lower than 99%,

marked green in the figure), a group with a normal packing density (between 99% and 101%, marked

orange in the figure) and a group with a high packing density (higher than 101%, marked red in the

figure). It can be observed that in general (on average) a higher packing density is related to a higher

static load on the first row of armour units.

The load on the first row of armour units increases linear with the number of rows for about the first

five rows. For slopes with more than five rows the increasing rate of the load on the first row of

armour units (Fstatic) decreases and approaches a constant value, in the order of 1,1 times the unit

Results and analysis of static tests

‐ 36 ‐

weight of an armour unit per unit. There is however some spread in the results. A breakwater with 20

rows imposes on average a static load of 1,1 times the unit weight on the first row of armour units

while the measurements varies between plus 18% and minus 30% of the average load on the first row

of armour units.

The standard deviation (σ) can be calculated as well as the variation from the dataset of measured

values with the following formula [DEKKING et al., 2005]:

2 2( ) [ ] ( [ ])Var X E X E X (5.1)

With this expression a standard deviation of 0,16 was found. Based on this standard deviation and the

schematisation of the measured end values, it can be stated that there is a probability of 95,6% that

an random test would have an end value of the static load between 0,78 and 1,42 times the unit

weight of an armour unit per unit.

Figure 5‐2 Measurements including curve fit

The static measurements can be represented by a curve fit which shows the earlier mentioned shape

of the relation between the number of rows and the measured load on the first row. The requested

relation should approach a constant value over time and should have an initial steep slope what

decreases and approaches a constant value over time. This relation was found in the following

expression for exponential decreasing growth:

1/2

ln(2)

( )

yN

N

static y end end startF N F F F e

(5.2)

This relation describes the load on the first row as a function of the number of rows on a breakwater

slope. The first row corresponds to startF which is 0,38 times the unit weight per unit. There is some

scatter of startF since this value is very dependent on the placement of the first row. A photograph of

the placement of this first row can be found in appendix A and this placement is comparable to the

placement in practice.

The load on the first row increases with the total number of rows placed on the breakwater (Ny). At

row N1/2 (which is 4,2) the half of the eventual constant value (Feind) (which is 1,11) is reached and this


‐ 37 ‐

parameter is used in the expression as the “half‐life value”. These values are applied which gives the

following expression:

4,62 4,62( )

1,12 0.76 1,12 0.76y yN N

static y

g x

F Ne e

F N

(5.3)

Similar tests were executed by MUILWIJK (2011) who used Xbloc model units with a unit height of 18,2

cm and a unit weight of 4,8 kg. MUILWIJK (2011) executed three tests with a low relative packing

density of about 95% and two tests with a high RPD of more than 100%. The results of MUILWIJK (2011)

are added to the measurements made in this research and visualised in Figure 5‐3.

Figure 5‐3 Measurements of the static load including measurements of MUILWIJK (2011) (in grey)

The results of MUILWIJK (2011) are similar to the results obtained in this study. The results of MUILWIJK

(2011) are curves with the same form and to a lesser extent the same values of the load on the first

row as function of the number of rows applied on the slope. The influence of the RPD as described by

MUILWIJK (2011) was also found in this research. However, the influence of the relative packing density

is more prominent in the research done by MUILWIJK (2011).

The initial static load on the first row was also measured during the hydraulic tests. The results of

these measurements are given in appendix C. These measurements of the static load on the first row

are in line with the above presented results.

5.1 Model

A simple one‐dimensional model was developed in order to gain insight in the transfer of load to the

under layer and the transfer of forces between the armour units. This eventually leads to the load on

the first row of armour units. The basis of this model is the individual force balance of an armour unit

and the interaction between the other armour units and the under layer.


‐ 38 ‐

The individual force balance is visualised in figure 5‐4:

Figure 5‐4 Force balance armour unit

The unit weight (Fg) of an armour unit can be decomposed in an along slope component (Fh) and a

component perpendicular on the slope (Fn):

sin( )

cos( )

h g

n g

F F

F F

(5.4)

The weight component perpendicular on the slope equals the normal force and is of importance for

the friction between the under layer and the armour unit:

cos( )f n gF f F f F (5.5)

The load on the first row of armour units is a load in the along slope direction. Therefore, only the

force balance in along slope direction is investigated. The along slope force balance is a balance

between the along slope weight component (Fh), a possible residual load of the above positioned

armour unit (Fr(n+1)), the friction force between the under layer and the armour unit (Ff) and the

residual force of the lower positioned armour unit (Fr(n)) (see figure 5‐4). In formula:

( 1) ( )

( 1) ( )sin( ) cos( )

h r n n r n

g r n g r n

F F F F

F F f F F

(5.6)

In this balance the interaction between the under layer and the armour units is simply schematised as

friction. This friction is a factor times the normal force (friction coefficient (f) times the normal force

(Fn)). The friction coefficient depends on the material used for the under layer (rock, gravel) and

armour units (concrete) and is the same for all units of a single breakwater. The friction coefficient is

investigated by determining the slope angle at which an armour unit starts to move. This is

investigated by placing Xbloc model units on a tillable rock slope and then gradually increase the

angle of the slope. At a certain (critical) angle the armour unit will start moving since the forces are

out of balance.


‐ 39 ‐

Prior to this point the armour unit is just stable and it is possible to calculate the friction coefficient

from the force balance:

cos( ) sin( )

sin( )

cos( )

f h

g crit g crit

crit

crit

F F

f F F

f

(5.7)

The reader is referred to the previous chapter for further reading about this experiment set up. The

angles at which a armour unit starts to move are visualised by the histogram in figure 5‐5:

Figure 5‐5 Critical angle of Xbloc armour units on a rock under layer

This diagram shows that there is not one single critical angle at which all armour units starts to move.

Some armour units start to move at fairly low angles while other units remain stable for angles of 50

degrees and more. The majority of the units start rolling down (instead of sliding down) the

breakwater slope when they start to move. This indicates that the interaction between the armour

unit and the under layer is more complicated than only friction between the armour unit and the

under layer. The surface of the under layer is not flat and also the armour units have protuberances

(legs). It is therefore possible that an armour unit “hooks” with a leg behind one of the protuberances

of the under layer. Armour units positioned in this way start to move at very large angles and since a

force between the armour unit and the under layer can be transferred by pressure instead of friction,

are able to transfer very high forces to the under layer.

Although sliding of the armour units is not the governing failing mechanism, the above proposed

approach based on friction coefficients was used as basis for a model describing the static load as a

function of the number of rows. This schematisation was made since there is no test data available

making a distinction between failure by rolling or sliding. Furthermore, the calculation of the friction

coefficient is just an intermediate step in order to arrive at the individual balance or imbalance of a

armour unit. The individual balance of a single armour unit reads;

,res unit h fF F F (5.8)

Fres,unit is the resulting force of a single armour unit without the influence of lower positioned of higher

positioned armour units. Due to the variable friction coefficients, calculated from the measurements

of the critical angles, Fres,unit can be either negative or positive. A positive resulting force means that

the armour unit will impose a load on the lower laying armour units, a negative resulting force means

that the armour unit is able to bear (part of) a possible resulting force of the above positioned armour


‐ 40 ‐

units. Since tension between the armour units is not possible this resulting bearing capacity is only

capitalised when the above positioned armour unit has a positive resulting force (is out of balance

due to its own force balance or due to the residual forces of the armour units positioned above this

unit).

The above described mechanisms were modelled in a simple one dimensional (line) model. In reality a

breakwater is covered with armour units placed on a certain grid (a staggered grid in case of Xbloc

armour units).

Figure 5‐6 Diagonal transfer of forces between armour units

A single (Xbloc) armour unit has direct contact with two lower positioned units and two higher

positioned armour units. Forces (pressure) occur between the armour unit of consideration and the

surrounding four units. The distribution of the imbalance of a single unit between the lower laying

armour units is unknown. This distribution should be studied in order to study the exact load on a

single armour unit on the first row since this distribution could have a significant effect on the load of

a particular single armour unit at the first row. This research is however focused on the row‐averaged

load on a single unit at the first row. For a row‐averaged load on a single unit on the first row it is not

important how the residual force of an armour unit is distributed over the two under laying armour

units. It is therefore possible to model the transfer of loads between units of a certain row to the

units on the lower laying row along straight vertical lines.

Figure 5‐7 Vertical transfer of forces between armour units

The distribution of forces between the individual armour units is the basis of the presented model.

The measured critical angles were converted into a residual force of a single armour unit (Fres,unit).

There was no single critical angle measured but a number of critical angles so the transformation of


‐ 41 ‐

the critical angles in residual forces again gives a dataset of residual forces and not one single residual

force for each armour unit. This dataset was used as actual input for the static model. For each row a

single value was randomly draw from the residual force dataset. The residual force of the highest row

was then evaluated. On this row there was no residual force of higher positioned armour units so the

individual residual force (Fres,unit) equals the total residual force of this row (Fr(n)). If the total residual

force ended up negative then the residual force was round of to zero since it is not possible to have

tension forces between armour units. The second highest row of armour units has as input its own

individual residual force (Fres,unit) which is randomly drawn from the individual residual force dataset

and the total residual force of the highest row (in previous calculation Fr(n) but in this calculation Fr(n+1)

since a lower row is considered). Those two add up to the total residual force which was round of to

zero when negative. The round of residual force of this row is the input for the third highest row and

so on. In formula:

( ) , ( 1)

( ) ( )if 0 then 0

r n res unit r n

r n r n

F F F

F F

(5.9)

The reader is referred to appendix E for further reading and the applied MATLAB code.

If this model is executed with as input the individual residual forces calculated from the measured

critical angles displayed in figure 5‐5 the following curve for the static load on the first row as function

of the number of rows on the slope results from this model:

Figure 5‐8 Static model based on measured values

As stated before the individual residual force was calculated from the measured critical angles of the

model armour units. In appendix E the start of rolling is analysed. Rolling of an armour unit occurs

when the working line of the gravity force creates a momentum around the point of rotation and this

rotation is in “seaward” direction. This occurs for angles larger than 45 degrees. In the schematised

situation the rolling of an armour unit over a perturbation of the under layer was analysed. From the

calculation following on this schematisation it can be derived that the angle at which armour units

supported by a perturbation of the armour layer start to roll is about 49 degrees. The units which

were still stable at 49 degrees or higher are armour units which are supported by a large perturbation

of the under layer. It is not possible to derive the individual residual force from the measured critical


‐ 42 ‐

angles of these units since this critical angle only represents an imbalance in momentum and not in

along slope forces. The perturbation can transfer much higher forces to the under layer than what is

possible with only the transfer of forces by friction.

The approach of the calculation of the individual residual force based on friction is abandoned for the

armour units which were still stable at angles higher than 49 degrees. These armour units were

modelled as armour units with a very high bearing capacity for residual forces (relative to the other

residual forces). This results in the following curve for the static load on the first row as function of

the number of rows on the slope:

Figure 5‐9 Measured data and static model

It can be observed that the correspondence of the static model with the measured values is very

good, and well enough to represent the measured values with this curve. The basis of the model is the

distribution of the individual residual forces and the bearing capacity of the units supported by

perturbations of the under layer which appeared to be responsible for decreasing growth of the load

on the first row with an increasing number of rows and the approach of a constant value at a large

number of rows (about ten rows or higher).

The shape of the curve obtained from the static model is well in line with the measurements of the

static load. However, the absolute values obtained from the static model are lower than the

measured values. The outcomes of the static model are multiplied with a factor 6,1 in order to obtain

the curve displayed in figure 5‐9. The lower results of the static model are probably the result of the

measurement of the individual residual load. The individual residual loads were measured with units

which were not surrounded by other units. From this situation an individual residual force was

obtained and in the model this residual force was first transferred to the under layer and the surplus

was transferred to the lower positioned units. This approach gives priority to the transfer of force to

the under layer. The armour units are placed on top of each other and will transfer a large part of

their along slope gravitational force to the under laying armour unit whether the theoretical friction

capacity is reached or not. The residual load is then a result of the support of the armour unit by the

under laying armour units and the under layer. Detailed research has to be performed on the support

of the under layer versus the support of the under laying armour units in order to achieve a complete

force balance and thereby a correct model. The packing density may be used as a starting point of this

research since the packing density can be linked to the relative contact of an armour unit between the

under layer and the surrounding armour units.


‐ 43 ‐

5.2 Influence of an external load on an arbitrary row

The developed static model describes the transfer of forces via lower positioned armour units to the

first row of armour units. In the above described static model these forces are the individual residual

force of the armour units. With this model it is also possible to impose an external load on a certain

row by adding this force to the individual residual force of the armour units on this row. By imposing a

load on an arbitrary row of a breakwater it is possible to calculate the transfer of this load to the

lower laying rows and the under layer. Part of the original imposed load is thus transferred to the

under layer by the under lying rows and just a part of the original imposed load on a certain row will

eventually reach the first row of armour units. An analysis was done on the percentage of the

imposed load which adds to the load on the first row of armour units as function of the row on which

the load is imposed. Table 5‐1 gives the results of this analysis. A load on, for instance, row 5 ads only

29% of this load to the original load on the first row of armour units.

Table 5‐1 Analysis of the influence of an external load

Input at row 1 2 3 4 5 6 7 8 9 10

Output at first row 0,76% 0,58% 0,47% 0,36% 0,29% 0,22% 0,20% 0,15% 0,12% 0,11%

Input at row 11 12 13 14 15 16 17 18 19 20

Output at first row 0,10% 0,08% 0,05% 0,04% 0,04% 0,04% 0,03% 0,02% 0,02% 0,01%

This conclusion can be verified by observing the results given in table 5‐2 that gives the measured

load on the first row as function of the number of rows in a modified way. In this table the increase of

the load on the first row between the addition of the last row and the addition of the second last row

relative to the load imposed on the first row by a single row of armour units is given. In this way a

similar table as table 5‐1 was obtained whereby in this case the along slope load of the top row is

treated as an external load. Based on table 5‐2 it can also be concluded that the influence of an

external load decreases with the number at which this load is imposed.

Table 5‐2 Analysis of the increase of the static load after the addition of an extra row

Number of applied rows 1 2 3 4 5 6 7 8 9 10

Output at first row 100% 48% 29% 22% 24% 14% 12% 14% 7% 5%

Number of applied rows 11 12 13 14 15 16 17 18 19 20

Output at first row 7% 9% 2% 3% 3% 5% 2% 2% 1% 3%

Results and analysis of hydraulic tests

‐ 44 ‐

6 Results and analysis of hydraulic tests

The results of the dynamic test program described in chapter 4 and an analysis on these results are

given in this chapter. The most important measurements obtained from this test program are the

measurements of the load on the first row of armour units. A dataset of load measurements during

one of the tests is visualised in figure 6‐1. For all other tests the reader is referred to appendix D

Figure 6‐1 Measured load during a flume test

This plot visualises two major phenomena during this (and all other) test:

1. the load imposed on the measurement frame shows a periodic behaviour with a certain

peak‐to‐peak amplitude.

2. this periodic load has no constant equilibrium line (wave‐averaged dynamic load) but instead

this “equilibrium” line shows a positive trend during a subtest until a new equilibrium is

reached at the end of a subtest.

Both phenomena are described in this chapter. Paragraph 6.1 focuses on the measurements of the

peak‐to‐peak amplitude of the load on the first row of armour units while paragraph 6.2 focuses on

the trend of the equilibrium line of the load on the first row of armour units. The equilibrium line of

the load on the first row of armour units is called “equilibrium load” in this thesis.

6.1 Peak‐to‐peak amplitude of the load on the first row of armour

units

The analysis of the periodic behaviour of the load on the first row of armour units was focussed on

the shape of the periodic behaviour and on the magnitude of this behaviour. In paragraph 6.1.1 an

analysis of the shape of the periodic behaviour of the load on the first row and the correlation

between the measured load and the measured water level at the toe is presented. Paragraph 6.1.2

gives the results of an analysis of the correlation between the magnitude of the peak‐to‐peak

amplitude of the load on the first row of armour units and various test parameters. In paragraph 6.1.3

a short analysis of some special cases of decreasing peak‐to‐peak amplitudes during the subtests is

presented.


‐ 45 ‐

6.1.1 Shape and period of incoming waves

The shape and wave period of the incoming waves and peak‐to‐peak amplitude of the load was

studied by analysing the videos of two tests. These videos were analysed frame by frame and the

analysis of these videos was linked to the measured data of the water level at the toe and peak‐to‐

peak amplitude of the load. This analysis was performed in order to determine during which phase of

the wave (uprush, downwash) the load on the first row of armour units is at its maximum. The

occurring processes during this maximum are of significant importance for the dynamic load. The

analysis of the wave cycle leads to the period of time during which the relevant processes occur.

Several points of interest like the moment of maximum run‐up, maximum run‐down, maximum water

level at the toe and minimum water level at the toe can be determined with an accuracy of 2 ms by

analysing the videos of a subtest. The videos of the tests have a constant frame rate of 28 frames per

second, from which it is possible to calculate the exact time in milliseconds since each frame has a

“duration” of 2 ms. Part of the video analysis is displayed in this paragraph. The reader is referred to

appendix F for further reading.

Figure 6‐2 Maximum run‐down, maximum water

level at wave gauge, time; 1:31:42

Figure 6‐3 Wave front passes upwards through still

water level, time; 1:31:58

Figure 6‐4 Maximum run‐down, maximum water

level at wave gauge, time; 1:32:15

Figure 6‐5 Wave front passes downwards through still

water level, time; 1:32:35

Figure 6‐2 to figure 6‐5 are screenshots of the video made of the hydraulic test of the model

breakwater with 20 rows during the subtest with a wave height of 0,6 Hmax and a wave period of 0,874

Hz. During this test the moment of maximum run‐down equals the moment of maximum water level

at the toe and the moment of maximum run‐up equals the moment of minimal water level at the toe.

This behaviour is however not general occurring and depends on the test characteristics (slope, wave

height, wave length). The time differences between the points of interest like maximum run‐up and

maximum run‐down were analysed. The results are visualised in figure 6‐6:


‐ 46 ‐

Figure 6‐6 Points of interest compared to the water level at the toe

In figure 6‐6 the points of interest are linked to the movement of the water level at the toe. In this

particular case the maximum run‐up occurs at the same time as the minimum water level at the toe

and the maximum run‐down occurs at the same time as the maximum water level at the toe. The

time difference between maximum run‐up and maximum run‐down is equal to 50% of the wave

period (T). As stated before the maximum run‐up and the minimum water level are only coupled in

this test and deviations of this coupling were observed in other tests (appendix F). The time difference

relative to the wave period between the passing of the wave front through the still water level in

upward direction, the maximum level of run‐up, the passing of the wave front through the still water

level in downward direction and the maximum level of run‐down is however independent on the test

characteristics and has a time difference of about 25% of the wave period in between the points of

interest. The time between the maximum run‐up and maximum run‐down is about 50% of the wave

period. The results of this video analysis can be coupled to the measured data which is visualised in

figure 6‐7.

Figure 6‐7 Hydraulic test 3(3), subtest 0,6 Hmax 0,874 Hz


‐ 47 ‐

figure 6‐7 Figure 6‐7 the measured load data is displayed as well as the data from the wave gauge.

The points of maximum run‐up and maximum run‐down are marked on the plot of the wave gauge

data. The vertical striped lines have an intersection with the measured load curve at the points in time

of maximal run‐up or run‐down. Between these vertical lines the flow at the slope is either in upward

motion (uprush) or in downward motion (downwash). From

figure 6‐7 and the analysed subtest described in appendix F it can be observed that the positive peaks

of the peak‐to‐peak amplitude of the load on the first row occur during the downwash. Since these

maxima occur during the downwash further analysis was performed on the peak‐to‐peak amplitude

related to characteristics of this downwash (run‐up, run‐down and occurring velocities).

6.1.2 Magnitude of peak‐to‐peak amplitude of the load on the first row

The peak‐to‐peak amplitudes were collected by plotting the original test data and were manually read

from the created graph. The amplitudes were collected in a dataset (appendix D) which contains

various other parameters like the wave height per test, the measured run‐up, relative packing density

et cetera. From this dataset various plots can be made. A study on relations between the peak‐to‐

peak amplitude and the wave height, run‐up and calculated velocity on the slope is presented in this

paragraph.

6.1.2.1 Relation between peak‐to‐peak amplitude and wave height

First of all the relation between the wave height and the peak‐to‐peak amplitude was studied. The

peak‐to peak amplitude (relative to the weight of the used model units) was plotted against the wave

height (relative to the, with the model unit corresponding, maximum wave height (Hmax≈2∙Hs)

[HOLTHUIJSEN, 2007]) which is visualised in the boxplot of figure 6‐7. Based on this figure it can be

observed that the spread of the peak‐to‐peak amplitudes at a specific wave height increases with

increasing wave height.

Figure 6‐8 Boxplot wave height versus peak‐to‐peak amplitude of the load

In figure 6‐9 the wave height is again plotted against the peak‐to‐peak amplitude but in this figure

two curve fits are added which are determined with a least‐square method which is available as

standard Matlab routine [MATLAB, 2009]. The first curve fit is a fit for all tests (ignoring the different

test characteristics of the tests). This second curve fit is a fit of the tests with a varying initial packing


‐ 48 ‐

density and varying number of rows but only of tests with the same slope (3:4) and permeability of

the core (“full” permeability). This distinction was made since variations of the slope and

(im)permeability of the core were found to have a significant influence on the peak‐to‐peak

amplitude (see table 6‐1). The second curve fit represents the relation between the wave height and

the peak‐to‐peak amplitude for the basis case.

Figure 6‐9 Curve fit wave height versus peak‐to‐peak amplitude of the load

The equation of the curve fit of all tests is

2

peak to peakA =0,27 0,020max max

H H

H H

2 with a norm

of residuals of 2,6774 (which is a measure for the accuracy of the curve fit according the least square

method, a lower norm of residuals corresponds to a more accurate fit). The norm of residuals is the

squared root of the sum of all squared deviations (residuals) between the curve fit and the points in

the dataset [BETZLER, 2003] :

2

1

i

Norm of residuals

f x

n

ii

i i

r

r y

(6.1)

The norm of residuals is used as measure in order to compare the different curve fits to each other. If

the curve fit (f(xi)) at point xi is equal to the measured value (yi) the residual will be zero. If this holds

for all data points then the norm of residuals will approaches zero and a perfect data fit is achieved. If

the curve fit is a better fit on the presented data the norm of residuals will thus be closer to zero. The

R‐squared value is also often used as measure of the goodness of a curve fit. The norm of residuals is

related to the R‐squared value:

22

1

2

1(Norm of residuals)

Total variation1

( )n

ii

n

iiR

r

y y

(6.2)

The curve fit of the data from of all tests except the tests with an impermeable core and a flatter

slope equals

2


H H

H H

and has a norm of residuals of 2,4834. In

practice only the last equation can be used in order to calculate the peak‐to‐peak amplitude based on


‐ 49 ‐

the known wave height since this equation is based on a general applied breakwater design with an

permeable core and the often applied slope of 3:4. The first equation contains data from all tests and

only confirms the shape of the second equation. The second equation should thus be used for future

calculations (for breakwaters with a slope of 3:4 and a permeable core).

Beside these curve fits a curve fit was made for every test series. The curve fits for the individual tests

has the following equation but with varying coefficients:

These fits are presented by their coefficients corresponding to equation 6.3 in

table 6‐1.

Table 6‐1 Curve fits of individual test, relation between wave height and amplitude

Test series Coefficient a Coefficient b Norm of residuals

Hydraulic test 1 (15 rows) +0,23 ‐0,04 0,245

Hydraulic test 1_2 (15 rows) +0,26 ‐0,04 0,162

Hydraulic test 2 (20 rows, RPD =1) +0,25 +0,01 0,428

Hydraulic test 3 (20 rows RPD =1.04) +0,20 ‐0,03 0,286


Hydraulic test 6 (20 rows, continuous) +0,49 ‐0,05 1,049

Hydraulic test 7 (impermeable core) +0,11 +0,02 0,166

Hydraulic test 8 (impermeable core,

smooth under layer)

+0,06 +0,03 0,219

Hydraulic test 9 (25 rows) +0,07 +0,01 0,131

Hydraulic test 10 (slope 2:3) +0,21 +0,03 0,613

Based on figure 6‐9 and table 6‐1,it was concluded that there is a quadratic relation between the

wave height and the peak‐to‐peak amplitude of the load on the first row. The a‐coefficient is about

one order larger than the b‐coefficient what gives support to the significance of the quadratic term.

It can be observed that a large number of the b‐coefficients have a negative value. No negative peak‐

to‐peak amplitudes were measured (and simply do not exist) and the negative b‐coefficients are only

the result of the least‐square curve fit and have no physical meaning. Although the form of the peak‐

to‐peak amplitude equation is clear, the influence of the different parameters is less clear.

The a‐coefficient is a good measure for the steepness of the curve since the quadratic term is found

to be the predominant term in equation 6.3. A higher a‐coefficient means a steeper curve and higher

amplitudes at the same wave height compared to curves with a lower a‐coefficient.

The a‐coefficients of the tests were compared to each other. It was found that the a‐coefficients of

the tests with 15 rows, 20 rows with an initial RPD of 100% and 104% and the test with a slope of 2:3,

are comparable to each other. The a‐coefficients of the tests with 20 rows, and an initial RPD of 97%

and the test with 20 rows but with a continuous execution, are higher and the a‐coefficients of the

tests with a (smooth) impermeable core and the a‐coefficients of the test with 25 rows are lower.

Based on the limited difference between the a‐coefficients of the tests with a normal initial RPD and

the test with a high initial RPD, it was expected that there would be a limited difference between

those test and the test with a low initial RPD. This was however not observed here. The a‐coefficient

of the tests with a high initial RPD is about 20% lower than the a‐coefficient of the test with a normal

initial RPD while the a‐coefficient of the test with a low initial RPD is about 250% higher than the a‐

coefficient of the test with a normal RPD.

2

peak to peakA = max max

H Ha b

H H (6.3)


‐ 50 ‐

A limited difference was also expected between the a‐coefficients of the test with 20 rows and the

test with 20 rows and a continuous execution, since these tests have the same layout (number of

rows, packing density). The only difference between those two tests is the way the tests were

executed. The test with 20 rows is executed subtest by subtest with some time between the subtests

in order to make pictures. The test with 20 rows and a continuous execution is executed directly after

each other without much time in between. The difference between the a‐coefficients of both tests is

however considerable with the test with 20 rows and a continuous execution having an a‐coefficient

which is 200% larger than the a‐coefficient of the test with 20 rows. The spread between the

measured data points of the test with 20 rows and a continuous execution is however much larger

than the spread of the data points of the test with 20 rows. The results of the test with 20 rows are

therefore considered to be more reliable.

The a‐coefficients of the tests with a (smooth) impermeable core and the test with 25 rows are lower

than the a‐coefficients of the tests with 15 rows, 20 rows with a RPD of 100 and 104% and the test

with a slope of 2:3. A deviation of the a‐coefficients of the curve fits of the test with an impermeable

core and the test with an impermeable core and a smooth under layer was expected since both tests

have an impermeable core. The permeability of the core has, as stated before, an influence on the

absorption of wave energy by the breakwater. A breakwater with an impermeable core absorbs less

wave energy which leads to the expectation that a more impermeable core would lead to higher

peak‐to‐peak amplitudes of the load on the first row. The opposite was however observed.

The third test that has a lower a‐coefficient is the test with 25 rows. Since the test with 15 rows and

the test with 20 rows have a similar a‐coefficient, a similar a‐coefficient for the test with 25 rows was

expected on beforehand. The a‐coefficient of the test with 25 rows is however only 27% of the a‐

coefficient of the test with 20 rows.

6.1.2.2 Relation between peak‐to‐peak amplitude and measured run‐up

During the tests the run‐up was measured by reading the run‐up level from a tape measure which

was positioned parallel to the slope at the interface between the armour layer and the water. The

run‐up incorporates the effect of the wave height and wave period (by the breaker parameter) and is

also influenced by the layout of the breakwater (slope, roughness of the armour layer and

permeability of the breakwater). The run‐up represents the potential energy just before the run‐down

and the flow of water from the core through the armour layer. The run‐up is a parameter which

characterises the driving force of those two phenomena and is a possible good parameter to relate

the peak‐to‐peak amplitudes too.

The measured run‐up is plotted against the theoretical run‐up calculated according the formula of

VAN DER MEER AND STAM (1992) in order to verify whether this measurements are in line with the

theoretical run‐up.


‐ 51 ‐

This plot is presented in figure 6‐10.

Figure 6‐10 Theoretical run‐up versus the measured run‐up

Figure 6‐10 shows a reasonable correlation between the measured run‐up and the calculated

theoretical run‐up. The run‐up formula of VAN DER MEER AND STAM (1992) is given in the following

expression:

for 1,5

for 1,5

n

cn

Rua

Hs

Rub

Hs

(6.4)

All test can be characterised by a surf similarity parameter (ξ) larger than 1,5 so for the experiments

done in this research the second expression is valid. Since the tested structure is a structure with a

permeable core (P=0,4) the relative run‐up is limited to a maximum value [VAN DER MEER AND STAM,

1992]:

nRud

H (6.5)

The a, b, c and d parameters are presented in table 6‐2:

Table 6‐2 Run‐up parameters according to [VAN DER MEER AND STAM, 1992]

Run‐up level n(%) a b c d

0,1 1,12 1,34 0,55 2,58

1 1,01 1,24 0,48 2,15

2 0,96 1,17 0,46 1,97

5 0,86 1,05 0,44 1,68

10 0,77 0,94 0,42 1,45

Significant 0,72 0,88 0,41 1,35

Mean 0,47 0,6 0,34 0,82

Figure 6‐10 shows, besides the measured data, two curves for d=2,58 and d=2,15. Most of the

measured run‐up levels are lower than the theoretical run‐up levels calculated with a d‐value of 2,58

since a d‐value of 2,15 corresponds to a more average run‐up level.


‐ 52 ‐

Figure 6‐11 shows a plot of the peak‐to‐peak amplitude of the load on the first row against the

measured run‐up. The measured run‐up is presented in a dimensionless form by dividing the

measured run‐up by the theoretical maximum run‐up. This theoretical maximum run‐up is calculated

with the run‐up expression of VAN DER MEER, J.W. AND STAM. C.J.M. (1992) WIth a conservative d‐value

of 2,58 and the maximum occurring wave height (Hmax).

Figure 6‐11 Curve fit measured run‐up versus peak‐to‐peak amplitude of the load

A clear difference can be observed between the curve fits of figure 6‐11 and of figure 6‐9. Both the

curve fit for all test as the curve fit for of all tests except the tests with an impermeable core and a

flatter slope show a linear behaviour. This is caused by the group of data points of the test with 25

rows which have high levels run‐up but very low corresponding peak‐to‐peak amplitudes (relative to

the expected amplitude). The equation of the curve fit of all tests is 2


Ru Ru

Ru Ru

2 with a norm of residuals of 2,495. The curve fit of the

data from of all tests except the tests with an impermeable core and a flatter slope equals 2


Ru Ru

Ru Ru

and has a norm of residuals of 2.2405. If the data of

the test with 25 rows would be neglected then the curve fit for all test would have a quadratic

character like the curve fits of the peak‐to‐peak amplitude as function of the wave height.

Although the shape of the curve fits is not an improvement of the curve fits of the peak‐to‐peak

amplitude as function of the wave height, there is an improvement regarding to the norm of residuals

which is lower for both the curve fits. The curve fits of all individual tests are given by the following

equation but have varying coefficients:

2

peak to peakA =max max

Ru Rua b

Ru Ru

(6.6)


‐ 53 ‐

These fits are presented by their coefficients corresponding to equation 6.6 in table 6‐1. Test 6 is not

included in this table since the run‐up was not measured during test 6.

Table 6‐3 Curve fits of individual test, relation between run‐up and amplitude




Hydraulic test 2 (20 rows, RPD =1) +0,23 ‐0,07 0,435





smooth under layer)

+0,02 +0,08 0,254


Hydraulic test 10 (slope 2:3) ‐0,07 +0,30 0,644

It can be observed that a large number of the b‐coefficients have a negative value. No negative peak‐

to‐peak amplitudes were measured (and simply do not exist) and the negative b‐coefficients are only

the result of the least‐square curve fit and have no physical meaning. The curve fit of test 10 deviates

from the other curve fits by the linear character of this curve fit. It can be observed in the table that

the norm of residuals of this curve fit is relatively high which indicate a poor fit. The linear character

of the curve fits is therefore not further investigated since it is the result of an irregular dataset.

6.1.2.3 Relation between peak‐to‐peak amplitude and downwash velocity

One of the hypotheses of this research is that the downwash is responsible for a significant part of the

governing load on the first row of armour units. Therefore, the relation between the downwash

velocity and the peak‐to‐peak amplitude was analysed. The downwash velocity was however not

measured so the downwash velocity was calculated from the other, known parameters.

In general, there are two types of expressions for the downwash velocities. The first, and widely used

expression, is based on the energy conservation principle. A single water particle at the top of a wave

just before breaking (figure 6‐12) is considered. The vertical distance between this particle and the

still water level is approximately equal to the eventual run‐up (Ru). Following an energy conservation

approach for the free fall of this particle till the still water level (approximating the vertical distance to

be equal to the run‐up) leads to the well‐known expression for free falling objects:

2U g Ru (6.7)

Figure 6‐12 Downwash velocity from energy conservation

The second approach is based on the actual movement of the wave front. The wave front moves with

a periodic movement on the slope of a breakwater (figure 6‐13). As stated in the literature study and

confirmed in paragraph 6.1.1, the period of time between maximum run‐up and maximum run‐down

is about 50 % of the wave period (T). During this time the wave front travels the along slope distance

between the maximum run‐ level and the maximum run‐down level (equation (6.8)).


‐ 54 ‐

; avarage

1/2

downwash

Ru RdU

T

(6.8)

Figure 6‐13 Movement of the wave front over the breakwater slope

This approach was followed in ABBOTT AND PRICE (1994) what gives an expression for the average

downwash velocity based on the Hunt formula:

1

122

/ sin( ) 2 1

/2 cos( )hunt

hunt

RuU g H

T

(6.9)

Furthermore, ABBOTT AND PRICE (1994) stated that the maximum velocity is a factor times the average

downwash velocity:

;1

;

( )downwash max

downwash hunt

Uf

U (6.10)

Research was performed on the size of 1( )f by Führböter and Witte (1989) and Battjes and Roos

(1975) [ABBOTT AND PRICE, 1994]. This research was however performed for moderate slopes, flatter

than 1:3 which is much flatter than the used breakwater slope in the model tests. Investigations in

this research were therefore based on the average downwash velocity:

L

0,5 sin( ) 0,5downwash

downwash

Ru RdU

T T

(6.11)


‐ 55 ‐

Both formulae are visualised in figure 6‐14:

Figure 6‐14 Comparison downwash velocity formulae

Figure 6‐14 shows that both downwash formulae give results in the same order and can therefore in

principle (at least for the analysis of the data of this model tests) each be used for the calculation of

the downwash velocity. The equation 6.11 is however much more related to the occurring water

motion at the breakwater slope. This expression incorporates the run‐up which is related to the wave

height and breakwater parameters such as the slope, roughness of the armour layer and permeability

of the breakwater as described earlier. Furthermore, the wave period is clearly involved in this

expression. For these reasons equation 6.11 was used in this research to determine the downwash

velocities.

The measured peak‐to‐peak amplitudes were plotted against the downwash velocity calculated with

equation 6.11 (with as input the measured run‐up and run‐down) divided by the maximum downwash

velocity. This maximum downwash velocity was calculated with equation 6.11. The maximum

downwash length was determined by calculating the maximum run‐up corresponding to the

maximum wave height with the run‐up expression (equation 6.4) of paragraph 6.1.2.2 (with a d‐value

of 2,58) and adding the maximum run‐down to this (which is assumed to be half of the calculated run‐

up). The used wave period was the wave period corresponding to the maximum wave steepness of a

wave with the maximum wave height. In this way a Udownwash,max of 0,98 m/s was obtained.


‐ 56 ‐

The plot is presented in figure 6‐15:

Figure 6‐15 Curve fit calculated downwash velocity versus peak‐to‐peak amplitude of the load

For this plot two curves were fitted as well. The equation of the curve fit of all tests is 2

peak to peak

,max ,max

A =0,31 0,053downwash downwash

downwash downwash

U U

U U

2 with a norm of residuals of 2,360. The

curve fit of the data from of all tests except the tests with an impermeable core and a flatter slope

equals 2

peak to peak

,max ,max

A =0,48 0,020downwash downwash

downwash downwash

U U

U U

and has a norm of residuals of 2,136.

These curve fits are even more accurate than the curve fits of last paragraph and represent the best

correlation between the measured peak‐to‐peak amplitude of the load on the first row and another

single parameter found in this research.

Based on the presented relation between the downwash velocity and the peak‐to‐peak amplitude

and the relation between the run‐up and the wave parameters it is possible to calculate the peak‐to‐

peak amplitude from the wave parameters.

Table 6‐4 gives the coefficient of the curve fits of the individual test corresponding to the general

curve fit:

2

peak to peak

,max ,max

A = downwash downwash

downwash downwash

U Ua b

U U

(6.12)

Table 6‐4 Curve fits of individual test, relation between downwash velocity and amplitude





‐ 57 ‐


Hydraulic test 3 (20 rows RPD =1.04) +0,34 +0,01 0,340




smooth under layer)

+0,06 +0,06 0,254


Hydraulic test 10 (slope 2:3) +0,35 +0,01 0,511

It can be observed that a number of the b‐coefficients have a negative value. No negative peak‐to‐

peak amplitudes were measured (and simply do not exist) and the negative b‐coefficients are only the

result of the least‐square curve fit and have no physical meaning. The positive b‐coefficients are an

order smaller than the a‐coefficients and can therefore be neglected except for test 7 and 8.

6.1.3 Decrease of peak‐to‐peak amplitude during test

The majority of tests (92%) have a constant peak‐to‐peak amplitude during the whole subtest.

However, for a few subtest the peak‐to‐peak amplitude decreases during the subtest. In table 6‐5 an

overview is given of the whole test program. The empty fields indicate that no decrease of peak‐to‐

peak amplitude was observed during the test while the none‐empty fields indicate that a decrease of

the peak‐to‐peak amplitude was observed. The numbers in the table represent the number of

subtests during which a decrease of amplitude was observed and the total number of subtest of this

wave height and period within the total test. For instance 3:4 on the row of test 2, column 0,6 Hmax,

0,87 Hz means that during 3 of the 4 subtests with the wave parameters of this columns belonging to

test 2, a decrease of the peak‐to‐peak amplitude was observed.

Table 6‐5 Overview decrease of peak‐to‐peak amplitude during a subtest

Wave

parameters

Wave height

0,2 Hmax

Period [Hz]

1,5 1,1 0,87

Wave height

0,4 Hmax

Period [Hz]

1,1 0,75 0,56

Wave height

0,6 Hmax

Period [Hz]

0,87 0,62 0,50

Wave height

0,8 Hmax

Period [Hz]

0,75 0,50 0,38

Wave height

1,2 Hmax

Period [Hz]

0,69 0,44 0,38

Number of

test

1

1_2 1:3

2 3:4 1:4 2:4 3:4 2:4

3 1:4

4 1:4 4:4 1:4 3:4 1:4 2:4 1:4

6 1:3 1:3

7 3:3 1:3

8 1:3 1:3

9 1:4 2:4 2:4 2:4

10 1:3 1:3 1:3 1:3

Total: 6 11 5 10 6 5

From the table it can be observed that most cases with decreasing peak‐to‐peak amplitude during the

test occur during a subtest with a larger wave height that the previous subtest. Furthermore, it was

observed from the photographs before and after the subtest that the most subsidence of the armour

units occurred during the same test whereby a decrease of the amplitude was observed. Based on

these observations it can be concluded that a decrease of the amplitude during a subtest occurs

during subtests with a just increased wave height and a relatively large subsidence of the armour

units.

6.2 Trend of the equilibrium load

In the beginning of this chapter was stated that the periodic behaviour of the measured load is not a

periodic movement around a constant equilibrium line but a periodic movement around an


‐ 58 ‐

“equilibrium” line (equilibrium load) which shows a trend. This trend is further analysed in this

paragraph.

Various trends of the equilibrium load were observed in the plots of the measured load on the first

row during the subtests (figure 6‐16). The subtests displayed in figure 6‐16 show a positive trend (in

general). A test consists of 15 subtests whereof five times three subtests with the same wave height

(see table 4‐3Table 4‐3 Wave program regular waves). The first three subsequent subtests had the

same wave height after which the wave height was increased and the following three subsequent

subtests were executed with this increased wave height. The plot of figure 6‐1 shows that the first of

these three subsequent subtests with the same wave height has the most positive trend what is in

line with the other test results which can be found in appendix D.

Figure 6‐16 Measured load on the first row of armour units during a flume test

6.2.1 Shape of the trend of the equilibrium load

In figure 6‐16 a typical form of the positive trend was observed. The positive trend of the equilibrium

load is initially steep and follows a curve of decreasing slope. After a certain period of time the slope

of this curve decreases to zero and a maximum for this subtest is reached.


‐ 59 ‐

Figure 6‐17 Type 1

The plot of figure 6‐17 gives an illustration of this curve, obtained from a random other test. This

curve, named “type 1”, was observed in a large number of other tests. This curve type was however

not the only observed trend curve type. Flat trend curves and other forms of curves can also be

observed in figure 6‐16. By analysing all test data an inventory of all occurring trend curve types was

made. The occurring trend curve types were schematised and collected in table 6‐6. In appendix F an

example is given for each mentioned type of trend curve.

Table 6‐6 Types of trend curves

A1 A2 A3 A4

B5 B6

B7

B8

C9 D10

A total of ten different trend curve types are identified. These curve types can be grouped in four

groups; A, B, C and D.

Group A contains type 1 to type 4, which have similar curve characteristics comparable with the

previous description of type 1. The difference between type 1 to type 4 lies in the first part of the


‐ 60 ‐

curve. Type 1 is the simplest curve type and shows a positive trend right from the beginning of the

test. Type 2 to type 4 shows this positive trend too, but this positive trend starts after a certain time

from the beginning of the subtest. Type 2 shows an oscillating measurement of the equilibrium load

before the definitive trend; type 3 shows a negative dip before the positive trend and the definitive

positive trend curve of type 4 starts after a positive bulge and a negative dip. The curves of the

equilibrium load of types 1 to 4 all approach a constant value which is named Fend. The curves start at

Fstart and follow a curve of exponential decreasing growth which approaches Fend:

1/2( ) with: =ln(2)

startt t

equilibrium end end start

tF t F F F e

(6.13)

The form of the curve described with this equation depends on the “mean lifetime” (τ) which depends

on the “half‐life” t1/2. This “half‐life” represents the point in time when the half of the eventual

growth is reached.

Group B consists of four curve types which are characterised as a “flat” type of curve. The subtests

characterised as type 5 are flat and have an end value which is similar to the start value of the test

without clear minima or maxima during the subtest. The subtests characterised as type 6, type7 and

type 8 also have an end value that is comparable to the start value but these types do have (various)

maxima or minima during the subtest.

Group C resembles the curve types which characterise subtests with a lower end value compared to

the start value of the subtest. This group contains just one type, type 9 being equal to type 1 but

vertically mirrored. The equilibrium load of the type 9 tests can also be approached by equation 6.13

for the equilibrium load of type 1.

Group D, resembles a collection of curves which characterise subtests with a positive trend of the

equilibrium load. These subtests can however not be characterised by the same curve as the types in

group A since these subtest do not show an exponential decreasing growth of the equilibrium load.

The only type in this group represent subtests with a linear increase of the equilibrium load during the

subtest.

All subtests were studies and for every subtest a characterising type was established. Based on the

analysis of the occurrence of the trend curve types a pattern, it was observed between the wave

height and the occurring trend type. The number of times that a certain trend type occurs is displayed

in table 6‐7 and visualised in figure F17 to figure F21 of appendix F.

Table 6‐7 Occurrence of trend curve types

Group Type No of occurrence

A (exponential decreasing

growth)

1 223

262 2 15

3 16

4 8

B (flat)

5 99

139 6 23

7 7

8 10

C (mirrored exponential decreasing growth)

9 33 33

D (linear growth)

10 33 33


‐ 61 ‐

This table shows that type 1 (exponential decreasing growth) is the most frequently occurring wave

type followed by type 5 (flat). From the figures in appendix F it can be observed that the subtests with

the lowest wave height (H=0,2 Hmax) are dominated by the curve types of group B while the subtests

with a higher wave height are mostly characterised by curve types of group A. If the difference

between first parts of the curve types is not taken into account, then for 56% of the subtests the

trend of the equilibrium load can be characterised by a trend curve with an exponential decreasing

growth and even 63% if also negative growth is defined as growth. The majority of the other subtests

(30%) was characterised as “flat”.

The flat test can be characterised by a simple relation since the equilibrium load was given by a

constant number. The majority of the subtests during which a trend of the equilibrium load was

observed (90%) can be described by the earlier mentioned equation for exponential decreasing

growth:

1/2( ) with: =ln(2)

startt t


tF t F F F e

(6.14)

For each subtest the trend of the equilibrium load was approached by a plot of this equation. An

example of a subtest to which a plot of the above equation was added is given in figure 6‐1. The

parameters of this curve fit were determined by hand and are displayed in appendix D. The “half‐life”

value of each test was used to determine the number of waves corresponding to the “half‐life” time.

The number of waves corresponding to the “half‐life” time is given in table 6‐8

Table 6‐8 Number of waves corresponding to the “half‐life” time

Group Type Average al

tests [‐] 0,2 Hmax [‐] 0,4 Hmax [‐] 0,6 Hmax [‐] 0,8 Hmax [‐] 1,0 Hmax [‐]

A (exponential

decreasing growth)

1 15.2 33 20 31 4 19 10 8 9 11 15 19 7 17 12 8

2 9.9 25 13 12 9 6

3 12.8 33 8 8 4 13 13 14 17 15 4 12

4 12.5 14 3 9 10 24 15

C (mirrored exponential decreasing growth)

9 9.1 16 10 6 6 1 4 14 11 10 7

Average [‐]: 14.0 27 17 21 6 11 7 8 8 13 14 14 5 15 12 7

The point in time (or corresponding number of waves) at which a certain percentage (n) of the

eventual growth of the equilibrium load can be determined from the “half‐life” time by the following

expression:

1/2

ln(0,5)

ln( )nt t

n (6.15)

By determining the point in time at which on average a certain percentage of the eventual increase is

realised it is also possible to determine the corresponding number of waves which on average

correspond to this growth. On average the half of the eventual growth was realised after 14 waves.

With the given relationship it is possible to determine the average number of waves between the

start of the subtest and the point in time at which 90% of the eventual growth is reached, which was

92 waves.

6.2.2 Magnitude of the increase or decrease of the equilibrium load

An increase or decrease of the equilibrium load is defined as the difference between the load on the

first row of armour units measured before the beginning of a subtest and the load on the first row of

armour units measured after the execution of this subtest. In most cases (63%) the load measured

after a subtest was higher than at the beginning of the subtest. In other cases the measured load


‐ 62 ‐

before and after the subtest was comparable (30%) and even in a few cases lower (7%) than at the

beginning of the subtest. At the end of the test all measured loads were increased significantly.

In this paragraph the results of an analysis on the increase of the equilibrium line of the load on the

first row of armour units (equilibrium load) in relation to the end measurements of the load after

testing with each other are presented. Furthermore, the results of an analysis on the relation

between the equilibrium load and the wave height, packing density and amplitude of the load on the

first row is presented.

6.2.2.1 End measurements of the load on the first row after testing

The average measured load on the first row of armour units (excluding the initial static load) at the

end of the test is given in table 6‐9.

Table 6‐9 Averaged measured load on the first row of armour units after testing

Tests Load on first row after

test [N]

Load on first row after

test [F/Funit]

Initial (static) load

[F/Funit]

Hydraulic test 1 “15 rows” 12,8 2,12 1,30


Hydraulic test 3 “high packing density” 9,27 1,53 1,15

Hydraulic test 4 “low packing density” 12,6 2,08 0,85

Hydraulic test 6 “20 rows” 15,5 2,56 1,22 (dry)

Hydraulic test 7 “impermeable core” 47,9 7,92 1,38

Hydraulic test 8 “flat, impermeable

under layer”

37,8 6,25 1,65


Hydraulic test 10 “flatter slope” 23,3 3,85 0.92

The end values of the load measurements were compared to each other in order to determine

possible influences of the varied test parameters.

The test with 15 rows, the test with 20 rows, the test with 20 rows but with a continuous execution

and the test with 25 rows were compared to each other (which were tests with a varying number of

rows but which had the rest of the test parameters in common). It was not possible to derive a

relation between the number of rows and the measured end values of these tests. The end values of

the test with 20 rows and the test with 20 rows but with a continuous execution were comparable

which was in line with the expectations, because the only difference between those tests was the

method of execution. The end value of the test with 15 rows was 18% lower. Based on this value it

was expected that the end value of the test with 25 rows would be higher than the end values of the

test with 20 rows and the test with 20 rows but with a continuous execution. The end value of the

test with 25 rows was however lower than the end value of the test with 20 rows and the test with 20

rows but with a continuous execution. This observation is comparable to the observations made

about these tests regarding the measured amplitude of the load on the first row.

The tests with varying packing density showed no clear relationship between the end value and the

varying packing density. The test with the highest packing density had the lowest end value, the test

with a normal packing density had the highest end value and the test with the lowest packing density

had an end value which lies in between the two other end values.

The test with an impermeable core and the test with impermeable core and a smooth under layer

resulted however in a much higher measured end values compared to the test with a permeable core.

This is in line with the amplitude analysis which also showed a deviation in amplitude for these tests.

The impermeable core reflected more of the wave energy, thereby increasing the instability of the

armour units which translates into higher loads of armour units on each other. The end value of the


‐ 63 ‐

test with an impermeable core and a smooth under layer was lower than the end value of the test

with an impermeable core.

The end value of the test with a flatter 2:3 slope appeared to be higher than the test with the same

amount of rows (20 rows) and a 3:4 slope.

6.2.2.2 Relation between the equilibrium load and the wave height

Besides the end values, which are the end results of the various tests also the development of the

increase of the equilibrium load till this end value was studied. All subtests with the same wave height

were binned. The total increase or decrease of the equilibrium loads during the three binned subtests

with the same wave height, relative to the end value are plotted in a boxplot in figure 6‐18

Figure 6‐18 Boxplot of the relative increase of the equilibrium load versus the wave height.

From this figure it can be read that the average equilibrium load did not increase or decrease during

test with a wave height of 0,2 Hmax. For wave heights higher than 0,2 Hmax the equilibrium load starts

(on average) to increase to 10% of the end value during the subtests with a wave height of 0,4 Hmax,

30% of the end value during the subtests with a wave height of 0,6 Hmax, 65% of the end value during

the subtests with a wave height of 0,8 Hmax and of course 100% during the subtests with a wave height

of 1,0 Hmax.

There is no linear relationship between the increase equilibrium load of the wave height and the

measured load on the first row of armour units. An increase of the wave height created a higher

increase than was expected based on a linear relationship. The relation between the wave height and

the relative equilibrium load seems to be of a quadratic.

In order to determine a relationship between the wave height imposed on the structure and the

equilibrium load, the equilibrium load was directly plotted against the wave height in figure 6‐19,

without balancing the equilibrium load against the measured end value as was done in figure 6‐18.


‐ 64 ‐

This approach was followed because the end value was not known on beforehand and a relation

based on this end value would be of limited applicability.

Figure 6‐19 Equilibrium load versus the wave height

Figure 6‐19 contains the data of all tests but not the data of the tests with an impermeable core,

because the much steeper relation between the wave height and equilibrium load of the test with an

impermeable core and the test with an impermeable core and a smooth under layer would disturb

the picture and curve fit. Of course the test with an impermeable core and the test with an

impermeable core and a smooth under layer are also relevant and a plot including the data of these

tests can be found in appendix F. Furthermore, the equations of the curve fits are given in table 6‐10.

However, since the considered cases in the test with an impermeable core and the test with an

impermeable core and a smooth under layer are rather uncommon the data is not used for the

general curve fit. The curve fit of all tests, but without the tests with an impermeable core, is: 2


g x max max

F H H

F N H H

(6.16)

This expression has a norm of residuals with the measured dataset of 6,086. Table 6‐10 gives the

coefficients of the individual curve fits which are of the form:

Table 6‐10 Curve fits of individual test, relation between wave height and equilibrium load





Hydraulic test 4 (20 rows RPD =0.97) +2,3 ‐0,30 0.954

Hydraulic test 6 (20 rows, continuous) +2,3 ‐0,08 1.848

Hydraulic test 7 (impermeable core) +13 ‐0,74 2,358


smooth under layer)

+0,12 +6,1 2,570


Hydraulic test 10 (slope 2:3) +4,6 ‐0,74 1,334

2

peak to peakA = max max

H Ha b

H H (6.17)


‐ 65 ‐

In these curve fits the negative b‐coefficients does have a meaning. A decrease of the equilibrium load

was measured for a number of tests which results in the negative b‐coefficients. It can also be

observed that the a‐coefficients are larger than the b‐coefficients confirming the quadratic character

of the relation. The curve fit of test 8 seems to be a deviation from the other curve fits. However, in

figure 2‐1 can be observed that the difference between the curve fits of test 7 and test 8 is

considerable but that both curve fits are in the same range.

Notable is the similarity of the equations of these curve fits compared to the curve fits which describe

the relation between the peak‐to‐peak amplitude and the wave height. In subparagraph 6.2.2.5 the

relation between the amplitudes and the increase of the equilibrium load was further analysed.

Based on these curve fits it is also possible to analyse the effects of the varying test parameters. This

was however more difficult compared to the similar analysis of paragraph 6.1.2.1 because in this case

the b‐coefficients cannot be disregarded since they have a significant influence on the shape of the

curve fit. As a consequence it is not possible to simply compare a‐coefficients with each other. The

curve fits were therefore plotted in a new graph:

Figure 6‐20 Curve fits equilibrium line versus wave height

Based on this graph, it was found that the curve fits of the test with 15 rows, 20 rows and a RDP of

100%, 20 rows and a RPD of 97%, 20 rows and a continuous execution and the test with 25 rows were

reasonable comparable to each other. Test 3 (high in initial packing density) appeared to have a

flatter curve fit which means that the expected increase of the load on the first row is lower. The test

with the flat slope (test 10) corresponded to the curve fit with a steeper curve.


‐ 66 ‐

6.2.2.3 Relation between the equilibrium load and the armour stability

It is clear that there is a relation between the wave height imposed on the structure en the increase

of the equilibrium load. The curve fits presented in figure 6‐20 for the test with an impermeable core

and for the test with a smooth under layer and an impermeable core deviated from the main group of

curve fits. In order to incorporate the effect of the permeability of the core, the slope of the armour

unit and the breaker parameter (which contains the wave period) the increase of the equilibrium load

in relation to the stability of the armour unit was studied. A slightly rewritten version of the original

Van der Meer formula is used as expression for the armour unit stability VAN DER MEER (1988)

[SCHIERECK, 2001]:

50 0,2

0,18 0.5

50 0,2

0,13

(plunging breakers,

6,2

(surging breakers, )

1,0 cot

sn transition

sn transition

P

HD

SP

N

HD

SP

N

(6.18)

In the case of the model breakwater the following values for the following parameters were used:

permeability (P) = 0.4 (0.01 for the impermeable core);

damage level (S) = 2 (only minor damage);

number of waves (N) = 7500 (corresponds to the equilibrium damage level);

The equilibrium load was plotted as function of the stability parameter given in expression 6.15 what

gives the following figure:

Figure 6‐21 Equilibrium load on the first row of armour units relative to the armour unit stability


‐ 67 ‐

The figure shows a reasonable relation between the stability of an armour unit (calculated based on

the Van der Meer formula) and the equilibrium line. A curve fit is added to the plot for all tests except

the tests with an impermeable core, as was done in figure 6‐19:

2

50 50 =96,2 8,58equilibrium

n n

g x

FD D

F N

(6.19)

This curve fit has a norm of residuals of 5,72 which is slightly better than the curve fit based on the

relation between the equilibrium load and the wave height. The results of the test with an

impermeable core are much more in line with the other results because the impermeability is

included in the armour unit stability. Table 6‐11 gives the coefficients of the curve fits of the individual

test corresponding to the general curve fit:

2

50 50 =equilibrium

n n

g x

Fa D b D

F N

(6.20)

Table 6‐11 Curve fits of individual test, relation between armour unit stability and equilibrium load






Hydraulic test 6 (20 rows, continuous) +61,2 ‐0,5 1,843

Hydraulic test 7 (impermeable core) +72,0 ‐0,1 2,549


smooth under layer)

+4,0 +14,6 2,449


Hydraulic test 10 (slope 2:3) +120 ‐4,9 1,387

In these curve fits the negative b‐coefficients does have a meaning. A decrease of the equilibrium load

was measured for a number of tests which results in the negative b‐coefficients. It can also be

observed that the a‐coefficients are much larger than the b‐coefficients confirming the quadratic

character of the relation.

6.2.2.4 Influence of packing density

A significant subsidence of the armour units was observed during the hydraulic tests. This subsidence

was measured through the subsidence of the top row of armour units although this was initially not

part of the test program. The measured subsidence was transformed in an increase of the packing

density since the packing density is a parameter which was already part of the test program and is

independent of breakwater dimensions, as the length of the slope and the size of the armour unit.

Since the subsidence varied with the wave height, also the packing density varied with the wave

height. The packing density is plotted against the wave height in figure 6‐22.


‐ 68 ‐

Figure 6‐22 Packing density versus wave height

The subsidence increased with an increasing wave height. The curves did not show a gradual

subsidence with an increasing wave height but instead a moderate increase at wave heights smaller

than 80% of the maximum wave height followed by a severe increase of the subsidence for the higher

wave heights. This can also be observed in the boxplot of figure 6‐23. A curve fit on the data of figure

6‐22 has the following expression: 2

=0,035 1,0max

HRPD

H

(6.21)

Figure 6‐23 Boxplot of the development of the packing density versus the wave height


‐ 69 ‐

This boxplot shows the initial gradual increase of the packing density followed by a more severe

increase. The increase of the load on the first row of armour units has to be the result of wave

induced modifications of the breakwater. Since the packing density is the only measured parameter

which can give an indication of possible modification of the armour layer the relation between this

wave‐induced modification of the armour layer (subsidence of the rows of armour units) and the

increase of the load on the first row of armour units was studied.

The equilibrium load of the load on the first row of armour units versus the packing density is plotted

in figure 6‐24.

Figure 6‐24 Equilibrium load versus the packing density

The plotted curves in figure 6‐24 show a very chaotic pattern. The curves are initially very steep and

flatten toward the top of the curve. This could be expected from figure 6‐22 which shows a dominant

increase of the packing density at the high wave heights while from figure 6‐18 it can be observed

that the increase of the equilibrium load relative to the wave height is much steeper.

It is however hard to deduce a reliable relationship between the equilibrium load of the load on the

first row and the packing density. In figure 6‐24 three curve fits are added. The green line is a fit on

the test data of tests with a low initial packing density, the blue line is a fit on the test data of tests

with an initial packing density of about 100% and the cyan coloured line is a fit on the test data of the

tests with a high initial packing density. The green line of the tests with an initial low packing density

shows that during these tests the packing density remained reasonably constant till an increase of the

equilibrium load of about one unit weight. During the further increase of the equilibrium load a strong

increase of the packing density was observed. This “bend” in the curve fit was also found in some

cases of the test with a normal initial packing density of about 100%. The bend in this curve fit was

however situated at higher values of the equilibrium load which means that the initial packing density

remained the same until higher wave heights and thus increase of the equilibrium load was reached.

The curve fit of the test with a high initial packing density showed a more unclear picture. In general

there was not a clear bend which could be identified.


‐ 70 ‐

Based on figure 6‐24, it was not possible to deduce an equation which incorporated the effect of the

initial packing density on the development of the packing density during the hydraulic tests and the

value of the equilibrium load of the load on the first row.

A theoretical relation between the equilibrium load and the packing density can however be obtained

if a combination is made of equation 6.21, which gives a relation between the packing density and the

wave height, and equation 6.16, which gives a relation between the wave height and the equilibrium

load of the load on the first row. These equations were combined resulting in the following equation: 2

1 1 =3,1 0,8

0,035 0,035

equilibrium

unit

F RPD RPD

F

(6.22)

A plot of the relative subsidence (relative to the end value of the subsidence) versus the equilibrium

load gives a far smoother picture:

Figure 6‐25 Equilibrium load of the load on the first row versus the relative subsidence

Based on this figure, it can be concluded that there is a correlation between the (relative) subsidence

and the increase of the equilibrium load. Since this plot was based on the relative subsidence it was

not possible to incorporate the effect of this correlation in a predicting equation for the equilibrium

load on the first row of armour units.

6.2.2.5 Relation between the equilibrium load and the peak‐to‐peak amplitude

In this section an analysis on the relation between the equilibrium load and the amplitude of the load

on the first row is presented. Since both the peak‐to‐peak amplitude and the equilibrium load can be

related to the wave height, it was also possible to relate the peak‐to‐peak amplitude to the

equilibrium load. If a relation whereby the equilibrium load increases after an increase of the peak‐to‐

peak amplitude would exist then it can be concluded that a higher peak‐to‐peak amplitude triggered

the increase of the equilibrium load.


‐ 71 ‐

A comparison was made between the curve fits of the peak‐to‐peak amplitude in relation to the wave

height and the relation between the equilibrium load and the wave height. The coefficients of the two

groups of curve fits are given in table 6‐12:

Table 6‐12 Curve fits of the peak‐to‐peak amplitude versus the wave height and the equilibrium load versus

the wave height

Test Curve fit amplitude Curve fit equilibrium load

Hydraulic test 1 (15 rows) a: 0,23 b: ‐0,037 a: 3,0 b: ‐1,1

Hydraulic test 2 (20 rows, RPD =1) a: 0,25 b: 0,014 a: 3,0 b: ‐0,43

Hydraulic test 3 (20 rows RPD =1,04) a: 0,20 b: ‐0,033 a: 2,5 b: ‐1,2

Hydraulic test 4 (20 rows RPD =0,97) a: 0,63 b: ‐0,036 a: 2,3 b: ‐0,30

Hydraulic test 6 (20 rows) a: 0,49 b: ‐0,053 a: 2,29 b: ‐0,076

Hydraulic test 7 (impermeable core) a: 0,11 b: 0,023 a: 12,7 b: ‐0,74

Hydraulic test 8 (impermeable core, smooth under layer) a: 0,061 b: 0,031 a: 0,12 b: 6,1

Hydraulic test 9 (25 rows) a: 0,065 b: 0,012 a: 3,8 b: ‐1,57

Hydraulic test 10 (slope 2:3) a: 0,21 b: 0,032 a; 4,6 b: ‐0,74

In both groups of curve fits was observed that a number of curve fits were similar to each other

(which are marked orange in the table), a number of curve fits was observed which lies underneath

the “average” group (marked green) and a number of curve fits was observed which lies above the

“average” group (marked red). For a good relation between the peak‐to‐peak amplitude and the

increase of the equilibrium load the curve fit of a subtest should be in the same colour‐group. This

was however not observed.

This observation can also be made based on the plot of figure 6‐26. No relation between the

amplitude of and the equilibrium load could be derived from this plot.

Figure 6‐26 Peak‐to‐peak amplitude of the load on the first row versus the equilibrium load


‐ 72 ‐

6.2.3 Trend of the equilibrium load during test with irregular waves

A total of three tests were executed with irregular waves. The spectra of these tests were analysed

and based on the expression for the trend of the equilibrium load as function of the wave height

(expression 6.16) and the number of waves, a calculation of the end value of the equilibrium load was

made. This calculation can be found in appendix F. The results of this calculation are summarised in

table 6‐13:

Table 6‐13 Results of spectrum analysis wit WaveLab and calculation of expected equilibrium load

Hydraulic test 5(1) Hydraulic test 5(2) Hydraulic test 5(3)

Hs (measured) 11,2 11,17 11,14 [cm]

Hmax (measured) 20,22 20,19 20 [cm]

Hmax ( based on Xbloc) 21,28 21,28 21,28 [cm]

Number of waves 2790 2819 2709 [‐]

Exceedence probability of 92 waves (90 % of end value)

3,30% 3,26% 3,40% [‐]

Corresponding (H/Hs)2

(Wavelab) 1,66 1,655 1,635 [‐]

H(prob:90 waves) 14,43 14,37 14,24 [cm]

Increase of Fequilibrium 0,79 0,79 0,77 [F/(Fg∙Nx)]

Increase of Fequilibrium 4,81 4,76 4,65 N

The theoretical expected increase of the equilibrium load with the applied spectrum was about 0,8 of

the weight of an armour unit. This result was compared to the actual measured increase of the

equilibrium during these tests. The measured increase of the equilibrium load during the test with

irregular waves is given in table 6‐14.

Table 6‐14 Measured increase of the equilibrium load


Measured increased of Fequilibrium 0,88 0,80 0,74 [F/(Fg∙Nx)]

Measured increase of Fequilibrium 5,3 4,85 4,48 N

Based on the comparison of the measured increase of the equilibrium load with the theoretical

expected increase of the equilibrium load, it could be concluded that the resemblance between the

measured and calculated increase of the equilibrium load is reasonable. However, in this analysis the

wave record of the entire test was used as input for the calculation while in figure D‐5 can be

observed that the most of the eventual increase of the equilibrium load was reached after 1000 s (850

waves). The calculation was also made with this reduced number of waves. This calculation gives a

lower expected increase of the equilibrium load of 0,44 times the unit weight while at this point in

time the measured equilibrium load is 0,73 times the unit weight.


‐ 73 ‐

7 Discussion

The load on the first row of armour units was decomposed in a static load and in a dynamic load. Both

types of loads were studied in experiments and the results of these experiments were subsequently

given and analysed in chapter 5 (static load) and chapter 6 (dynamic load). These analyses are

discussed in this chapter. Paragraph 7.1 discusses the static load, paragraph 7.2 discusses the dynamic

load and the combined, total load is discussed in paragraph 7.3.

7.1 Static load on the first row of armour units

The along slope force balance of an armour unit on the slope of a breakwater was analysed and this

resulted in the following equations:

sin( )

cos( )

sin( ) cos( ) 0

h g

n g

g g

F F

F F

F f F

(7.1)

This balance consists of an along slope component of the weight of the armour unit (Fg) and the

counteracting friction force which is a function of the weight component perpendicular to the slope

and the friction coefficient. The friction coefficient between a Xbloc armour unit and the rock under

layer depends on the materials used (concrete and rock) and the size of the rocks. Various friction

parameters for concrete on rock or gravel were found in literature like f = 0,6 [D’ANGREMOND AND VAN

ROODE, 2001] and f = 0,564 ‐ 0,679 (from STÜCKRATH (1996) which was obtained from Coastal

Engineering Manual table VI‐5‐62 [U.S. ARMY CORPS OF ENGINEERS, 2002]). If a friction coefficient of 0,6

and the common breakwater slope for Xbloc armour units of 36,87 degrees (3V:4H) is applied then

the following along slope force balance is obtained, which results in a positive residual force :

sin( ) cos( )

sin(36,87) 0,6 cos(36,87) 0,12

res

g

res

g

Ff

F

F

F

(7.2)

Based on this simple calculation, a linear relation between the number of rows and the increasing

load on a single armour unit at the first (bottom) row of armour units was expected with a linear

increase of 0,12*Fg per row. The theoretical load on a single armour unit at the first row of a

breakwater with 20 rows in total is thus 2,4 times the weight of this armour unit.

In practice, this theory was found to be incomplete. A physical model test was executed in order to

obtain the average load on a single unit at the first row of armour units as function of the number of

rows applied on the slope of the breakwater. The results of these tests are visualised in figure 5‐1 and

are schematised in figure 7‐1. This figure also shows the relation between the theoretical load and the

measured load.

Discussion

‐ 74 ‐

Figure 7‐1 Theoretical and measured static load on the first row of armour units

This visualisation shows that for the first number of rows (about five rows) the measurements are in

line with the theoretical determined relation between the number of rows and the load on the first

row (the slope of 0,12 was found for the static load between row 1 and row 5). For a higher number

of rows applied on the breakwater slope (Ny), the load on the first row approaches a constant value in

the order of 1,1 times the weight of an armour unit per unit (with a standard deviation of 0,16). The

measurements can be described by the following expression:

4,62

( )1,12 0.76

yN

static y

g x

F Ne

F N

(7.3)

The maximum number of rows applied in the tests executed in this research is 20. Based on the

theoretical approach, it was expected that the measured load on a single armour unit at the first row

of armour units would be in the order of 2,4 times the weight of this armour unit. The measured load

was however only 1,1 times the weight of an armour unit. The measured start value (along slope

static load of a single row) is 0,38 times the unit weight. This start value depends on the position of

the armour units on the first row relative to the under layer and the toe. The armour units on the first

row were placed according to the valid standard.

The deviation of the results from the theoretical load on the first row indicate that the transfer of

forces between the armour units and the under layer does not solely occur by friction as this would

lead to much higher loads then measured.

An experiment was performed in order to investigate the critical angle at which an armour unit starts

to move. This experiment was executed in order to derive the friction coefficient. From this

experiment we collected a dataset of critical angles that was transformed into a dataset of friction

coefficients. Based on equation 7.1, it was expected that the angle at which an armour unit is just

stable is about 31 degrees. The measured critical angles are however much higher with an average

critical angle of 38 degrees. This also indicates that armour units are still stable when the maximum

friction capacity is reached. Furthermore, it was observed that a significant number of units started to

roll down the slope when they got out of balance. These observations lead to the conclusion that a

second phenomenon is influencing the stability of a single armour unit on a breakwater slope.

Considering the contact between the armour units and the under layer it can be observed that the

under layer is not smooth but contains irregularities and perturbing stones. The armour units, which

are not smooth either but consist of pointy legs and (less pointy) noses, can be placed in these

irregularities or against a perturbing stone. In this way a direct point of contact is created through

which large forces can be transferred from the armour unit to the under layer by means of pressure.

The critical slope experiment demonstrated a very high critical angle of more than 49 degrees for a

number of units. These units are stable due to friction but, since friction alone cannot explain such


‐ 75 ‐

high critical angles, it was concluded that it also due to the contact of the armour units with

perturbing stones and other irregularities. These very stable armour units can transfer large forces to

the under layer.

During the hydraulic test program three tests were executed on a smooth wooden under layer. The

friction coefficient between wood and concrete is similar to the friction coefficient used for concrete

on rock. The measured static load just after the construction of the model was therefore expected to

follow the theoretical linear relationship of the number of loads and the static load. This means that

the measured static load on a smooth under layer should be much higher than the measured static

load for the tests with concrete armour units on a “rock” under layer. This is indeed the case with an

average static load on the first row of 1,65 times the weight of an armour unit per unit (appendix E).

These results support the statement that the friction is not the only mechanism influencing the static

load, but that the smoothness of the under layer also has an effect on the static load.

A model was constructed in order to investigate whether these very stable armour units, which are

able to cope with high forces, are responsible for the levelling of the relation between the number of

rows and the load on the first row of armour units. Our model included the behaviour of the armour

units as described before (it used the dataset of the critical angles as an input and models the units

with a critical angle of more than 49 as very stable armour units with a very high capacity for the

transfer of forces to the under layer, see paragraph 5.1 ). This model is able to produce the shape of

the relationship between the load on the first row and the number of rows applied on the

breakwater. And this is very well in line with the found relationship. It can thus be concluded that the

varying stability of the armour units and the presence of very stable armour units in particular are

responsible for the levelling of the relationship of the static load and the number of rows and

eventually for the constant end value for a higher number of rows.

The outcome values of the model initially were very low and were calibrated with a multiplication

factor of 6,1. The measured critical values apparently resulted in friction force capacities which are [?]

not fully utilized in practice. In practice, armour units are positioned against lower positioned armour

units and will impose a load on these units whether the friction capacity is used or not. The armour

units in this study were modelled with a preference for friction and only the resulting force after

subtraction of the full friction capacity was imposed on the lower positioned armour units in this

model. This preference for friction instead of a more equal distribution of force between the lower

positioned armour units and the under layer is the most plausible cause of the lower outcomes of the

model (without the multiplication factor). The actual values of the static load have however no

influence on the shape of the relation between the number of rows and the static load on the first

row. This relationship still holds and the model can thus perfectly be used for the explanation of the

effect of the presence of very stable armour unit in an armour layer on the static load on the first row

of armour units.

The results found in the static tests show high dispersion. The static load has a standard deviation of

0,16 [units[. The spread in the results of the static load can be explained by the stochastic character of

the transfer of loads to the under laying armour units or to the under layer. Based on the model it can

be determined that for an infinite number of columns the spread of the average static load on an

armour unit will reduce to zero. However, since the number of units in a horizontal row used in this

research was limited to 20, dispersion of the measured static load was found.

With the static load model the influence of an external load on a certain row was studied. From this

study it appeared that the influence of a certain external load on the load on the first row decreases

as the row number at which the external load is imposed increases. External loads which are imposed

on row 10 or higher add less than 10% of this external load to the load on the first row of armour

Discussion

‐ 76 ‐

units. This implicates that only loads imposed on row one to ten have a significant influence on the

load on the first row.

The presented relationship of the static load on the first row and the number of rows applied on the

breakwater gives an average load on an armour unit at the first row. In order to derive the stresses in

this armour unit, which are required for a structural integrity calculation, the load on a single armour

unit should be determined. This can be done by measuring individual loads or via an extension of the

model to a two dimensional model which models the transfer of force of one unit to two lower

positioned armour units and, in this way, through the entire armour layer and eventually to the first

row of armour units.

7.2 Dynamic load on the first row of armour units

The dynamic load was examined with a physical model in a wave flume. The results of this experiment

show two major phenomena:

the measured dynamic load shows a periodic behaviour with a certain peak‐to‐peak

amplitude;

the wave‐averaged dynamic load (the equilibrium load) is not constant but increases during

the sub tests until an equilibrium in reached at the end of a subtest;

The first phenomenon is discussed in paragraph 7.2.1 and the second phenomenon is discussed in

paragraph 7.2.2.

7.2.1 Amplitude of the load on the first row of armour units

The correlation between the motion of water on the breakwater slope and the load on the first row of

armour units was studied in chapter 0. From this analysis, described in paragraph 6.1, it can be

concluded that the motion of water on the breakwater slope has a direct relation with the peak‐to‐

peak amplitude of the load on the first row of armour units. The measured load shows a periodic

behaviour with the same period as the incoming waves and positive maxima during the downwash of

water over and through the armour layer. The relation between the measured amplitudes and the

wave height, run‐up and downwash velocity was studied and the relation between the peak‐to‐peak

amplitude and the downwash velocity was found to be the most accurate. This relation is of the form: 2

peak to peak

,max ,max

A =a downwash downwash

downwash downwash

U Ub

U U

(7.4)

The values of coefficients in this expression for a breakwater with a permeable core and a slope of 3:4

are 0,48 for the a‐coefficient and ‐0,02 for the b‐coefficient. Since b is much smaller than a, the

expression can be reduced to: 2

peak to peak

,max

A =a downwash

downwash

U

U

(7.5)

This is a pure quadratic relation between the downwash velocity and the load on the first row of

armour units. This relation is in line with the Morrison equation which states that the force imposed

by the flow on a body (only taking into account the drag and thus disregarding the inertia force) is:

1

2w dF C A U U (7.6)

According to the Morrison equation the force imposed by the flow has a quadratic relation with the

flow velocity which is in line with the presented expression for the peak‐to‐peak amplitude of the load

on the first row of armour units ( peak to peakA ). This relation between the downwash velocity and the

peak‐to‐peak amplitude was expected based on the know literature and was found in this research.


‐ 77 ‐

A considerable spread was found in the results of the dynamic tests. Regarding the peak‐to‐peak

amplitude it can be stated that the dynamic load can be treated as an external load on each armour

unit. The transfer of dynamic forces through the armour layer is therefore comparable to the transfer

of static forces through the armour layer. In the previous paragraph it was concluded that the

stochastic character of the distribution of forces between the lower positioned armour units and the

under layer is influencing the (spread of the) static load. An external load is influenced by the same

mechanism. The transfer of dynamic load, imposed on armour units, through the armour layer onto

the first row of armour units is thus also a stochastic process which leading to a dispersion of the

found results. A second origin of the spread of the results is the influence of the measurement frame.

An analysis of the influence of the frame on the results was presented in appendix D. Based on this

analysis it can be concluded that the spread related to the influence of the frame might be up to 20%

of the measured peak‐to‐peak amplitude.

The relation between the downwash velocity and the peak‐to‐peak amplitude is clear but the

influence of the breakwater layout (number of rows, packing density et cetera) is however less clear.

Two tests with a total number of 15 rows (Ny) were executed. The difference between those tests was

that during the first tests with 15 rows a horizontal impermeable layer at toe level was applied in

order to prevent the internal flow of water from the actual breakwater model to the lower part of the

construction which was not protected by armour units and was only constructed in order to make

higher wave heights possible (figure 7‐2), while this was not the case during the second test with 15

rows.

Gabio

ns

Figure 7‐2 Layout of flume test model

This difference in layout was not observed in the measurements of the two tests so there was

concluded that the eventual flow between the base of the with armour units protected structure and

the rest of the structure has no influence on the peak‐to‐peak amplitude. From this it was also

concluded that the subsoil is of no influence on the peak‐to‐peak amplitude of the load on the first


Discussion

‐ 78 ‐

7.2.1.1 Influence of varied parameters

Several parameters were varied in the hydraulic experiment. The varied parameters were:

The initial relative packing density (RPD);

The number of rows (Ny);

The permeability of the core;

The smoothness of the under layer (in combination with an impermeable core);

The slope of the breakwater;

Looking at the variations made in the initial RPD the outcomes were not evident. The results show

that a higher RPD of 104% leads to relative small deviation (‐20%) of the peak‐to‐peak amplitudes

measured during the tests with a RPD of 100%. The measurements of the peak‐to‐peak amplitude of

the tests with a low RPD of 97% show however a much higher peak‐to‐peak amplitude (+250%). The

comparison of these tests did not provide a relationship between the RPD and the peak‐to‐peak

amplitude. It could be possible that there is a very curved relation between the peak‐to‐peak

amplitude and the relative packing density. This is however not in line with the relation between the

downwash velocity and the peak‐to‐peak amplitude that only explains an influence of the RPD by the

hydraulic drag of the armour layer. It is therefore, based on this research, not possible to make a clear

statement about the influence of the initial RPD on the peak‐to‐peak amplitude of the load on the

first row of armour units.

Also the variation of the number of rows has no clear, direct influence on the peak‐to‐peak amplitude

of the load on the first row. The results of tests with 15 rows and 20 rows are very well comparable.

The peak‐to‐peak amplitude measured during the test with 25 rows is however much lower. During

this test only the highest waves were able to reach the highest rows which may cause this deviation.

This means that during the subtest with lower wave heights the highest rows where just dead weight.

This may have dampened the peak‐to‐peak amplitude of the load on the first row.

Two tests were executed with an impermeable core and one of these tests also had a smooth under

layer. It was expected that the permeability of the core would reflect more of the wave energy with as

a result a higher downwash velocity along the slope and thus a higher load on the first row.

Furthermore, a smooth under layer was supposed to decrease the contact between the under layer

and the armour units, creating a higher (static) instability of the armour units and thus resulting in a

larger peak‐to‐peak amplitude. This was not directly observed from the analysis of paragraph 6.1.

However, when looking at the analysis of the dataset of the load on the first row in combination with

the videos a very strong increase of the “equilibrium” line was observed (compared to the test with a

normal permeable core) followed by the (in the amplitude dataset included) small amplitude. At the

beginning of these tests the armour layer settled very fast (within a few waves) which was the result

of the impermeable core. During this quick settlement a solid new equilibrium was reached that

apparently does not facilitate the transfer of loads through the armour layer until the lowest row of

armour units. This is in agreement with the observation described in paragraph 6.1.3 that describes

the decrease of the amplitude during a number of tests with a strong increase of the equilibrium load.

These extreme settlements made it impossible to measure the peak‐to‐peak amplitude correctly and

it is therefore recommended to investigate the influence of the core permeability with a test program

including less permeable cores and not complete impermeable cores.

The last variation of the layout that was tested was the variation of the slope. All tests except the last

test were performed on a breakwater with a slope of 3:4 (36,9 degrees) while the last test was

performed on a breakwater with a slope of 2:3 (33,7 degrees). A flatter slope leads to less interlocking

and a higher static stability of the armour units and theoretically to a better support of the armour


‐ 79 ‐

units by the under layer. This should lead to more transfer of forces to the under layer and thus a

lower load on the first row of armour units. The observed difference between the peak‐to‐peak

amplitude on the first row of the test with a slope of 36,9 degrees and the tests with a slope of 33,7

degrees is small (‐16%) but confirms this theory.

At last a theoretical approximation of the dynamic (peak‐to‐peak amplitude) load is made in appendix

F. This calculation shows that the influence of the flow from the breakwater core through the armour

layer has a limited influence on the eventual dynamic load in the order of 13% of the total dynamic

load on the first row of armour units. Furthermore, this calculation shows that the load per row of

armour units is only a number of times smaller than the total measured dynamic load. It can thus be

concluded that only a limited amount of rows transfer their dynamic load to the first row of armour

units.

7.2.2 Trend of the equilibrium load on the first row of armour units

The previous paragraph discussed the peak‐to‐peak amplitude of the load on the first row of armour

units around an equilibrium line. However, this equilibrium line (equilibrium load) appeared to be not

constant and increased during most of the subtests (in 63% of the cases). This means that the

measured value before the beginning of the subtest is lower than the measured value after the

completion of the test. In most of the remaining cases (30%) there was no trend of the equilibrium

line observed.

The increase of the equilibrium load has in most cases the same basic shape which is called “Type 1”:

Figure 7‐3 Basic shape of increase of the equilibrium load

Typical for this type is the decreasing growth which approaches a constant end value. The fact that a

constant end value is approached is a general feature which holds for most tests (of course not for the

“flat” tests where nothing happened with the equilibrium load at all). From this observation it can be

concluded that there is a maximum end value of the load on the first row of armour units depending

on the wave related input (wave height, period). The shape of this basic shape can be described with

the following equation which contains the begin value, end value and the “half‐life” time of the

examined subtest:

1/2( ) with: =ln(2)

startt t


tF t F F F e

(7.7)

An analysis of the “half‐life” times of the subtest showed that on average 90% of the eventual

increase is reached after 92 waves. Based on the observation that a maximum end value is reached

during a subtest, it can be concluded that due to the wave action on the breakwater slope, the

breakwater is transformed, with as a result that a new configuration was reached corresponding to a

new equilibrium load on the first row of armour units.

The equilibrium load increases with a quadratic relation with the wave height. This relation is

comparable to the relation between the peak‐to‐peak amplitude and the wave height. During waves

of 0,2 Hmax there is no significant increase of the equilibrium load. The increase of the equilibrium load

Discussion

‐ 80 ‐

begins at a wave height of about 0,4 Hmax and this growth increases with the wave height. This can be

described by the following relation between the wave height and the equilibrium load of the load on

the first row: 2


g x max max

F H H

F N H H

(7.8)

There is a relation between the hydraulic stability of an armour unit and the equilibrium load on the

first row of armour units. This was investigated by comparing the armour unit stability, calculated

with the Van der Meer formula, to the measured increase of the equilibrium load which appeared to

be related to each other. The Van der Meer formula includes the permeability of the core and this

thus resulted in a relation that includes the results of the test with an impermeable core. A higher

level of instability of the armour units leads to a higher load on the first row of armour units.

7.2.2.1 Influence of varied parameters

Several parameters were varied in the hydraulic experiment. The varied parameters were:

The initial relative packing density (RPD);

The number of rows (Ny);

The permeability of the core;

The smoothness of the under layer (in combination with an impermeable core);

The slope of the breakwater;

The end values, measured at the end of the test, were compared with each other. Regarding the end

values, it can be observed that a large number of tests have a similar end value of 1,5 to 2,6 times the

initial static load on the first row. Three tests were higher than this range of end values. The test with

a flatter slope has on average an end value of 3,85 and the tests with a (smooth) impermeable under

layer have end values in the range of 6 to 8 times the initial static load on the first row.

Based on these outcomes, it was concluded that there is no significant influence of the initial packing

density or the number of rows. There is a significant influence of the slope of the breakwater: a flatter

slope gives higher end values, and the permeability of the core: a more impermeable slope gives

higher end values. The significant influence of the permeability of the core is in line with the

expectations since an impermeable core reflects the waves which lead to higher (hydraulic)

instabilities of the armour units.

The test with the smooth, impermeable under layer leads to even more instability of the armour units

and therefore even higher equilibrium loads were expected for this test. The equilibrium load

measured during this test appeared to be lower than the equilibrium load of the test with only an

impermeable core. A possible reason for this deviation may be that during the test with only an

impermeable core (with a plastic sheet between the core and the under layer) the under layer itself

was instable. This was concluded from the observation that during the test the under layer settled as

well as the armour units. The higher end values of the test with a flatter slope are in line with the

increase instability of the armour units (less interlocking due to a flatter slope)

Equation 7.8 in combination with equation 7.7 which gives the equilibrium load dependent of the

elapsed time (number of waves) should make it possible to make a first estimation of the load on the

first row of armour units based on a known wave climate. A calculation was made on order to make

an estimation of the increase of the equilibrium load during a test with irregular waves, based on this

method. The determined theoretical increase of the equilibrium load showed some resemblance to

the measured equilibrium load. A calculation based on the full wave record showed good


‐ 81 ‐

correspondence with the measured final equilibrium load. However, calculations of intermediate

equilibrium loads based on the first half of the wave record showed a significant deviation from the

measured corresponding equilibrium load.

Finally the relation between the development of RPD and the development of the equilibrium load

was determined. No significant relation between the development of the RPD and the equilibrium

load was found. However, there was found that the packing density increased with increasing wave

height and therefore with an increasing equilibrium load.

7.3 The total load on the first row

The total load on the first row of armour units is composed of the static load and the dynamic load. In

order to calculate the total load on the first row of armour units the dynamic load has to be added to

the static load:

peak to peak( ) ( , ) A + =

2

static y equilibrium total

g x g x g x g x

F N F H No waves F

F N F N F N F N (7.9)

The static load was measured in various tests and these measured static loads are in agreement with

each other. The equilibrium load was measured during hydraulic tests with wave‐induces dynamic

loads. These measurements show a correlation with the wave height (or downwash velocity for the

peak‐to‐peak amplitudes) but less correlation with the variation of parameters such as the number of

rows applied on the slope (Ny) and the initial relative packing density.

The analysis of the static tests showed that an external load imposed on row ten or higher has a

limited influence on the eventual static load on the first row of armour units. The influence of row ten

or higher on the initial static load is very limited, furthermore, the dynamic loading of armour units at

row ten or higher can be treated as an external load on this row which thus has a limited influence on

the measured load on the first row. Therefore, it makes sense that the influence of Ny on the peak‐to‐

peak amplitude an the equilibrium load, for a larger number of rows then ten, is limited.

This reasoning can be extended in order to explain the varying measurements of the equilibrium load

and peak‐to‐peak amplitude related to the RPD. If only the first ten rows have a significant influence

on the measured load on the first row of armour units then only the RPD of the first ten rows is of

importance for this load on the first row of armour units. In the executed test program slopes with 15

to 25 rows were tested and the RPD’s were measured for the entire slope. Only the first ten rows

seem to be of importance so only the RPD of the first ten rows had an influence on the load on the

first row. The RPD of the first ten rows should be in line with the RPD of the entire slope but this is not

necessarily the case and since the RPD of the first ten rows was not measured this cannot be studied.

It is therefore still possible that the RPD (of the first ten rows) has an influence on the load on the first


The static model demonstrated that the presence of very stable armour units (regarding to forces in

downward, along slope direction) is responsible for the stabilisation of the static load for breakwaters

with ten rows or more. These very stable units are randomly distributed over the entire armour layer.

It was observed that for hydraulic tests with a significant increase the armour units tend to move

(settle) downwards along the slope. When the very stable units (regarding to the downward, along

slope force but not necessarily regarding to the hydraulic stability) start to move, they will be

repositioned. The repositioned unit can end up in a comparable, very stable position but since only

the minority of the armour units is positioned in this position, it is likely to end up in a less stable

position. The wave‐induced movement of armour units (which can be moderate and is only in a

minority of the cases rocking) will thus on average induce a slight motion of the armour units

Discussion

‐ 82 ‐

(settlement) which decreases on average the number of very stable armour units en thus leads

positive to a trend of the equilibrium load.

On average, the number of very stable armour units decreases, which leads to a positive trend of the

equilibrium load but this does not exclude the possibility of an increase of the number of very stable

armour units due to wave‐induce loading of the breakwater. On average, the majority of armour units

is not a very stable positioned armour unit so the wave‐induced loading of the breakwater will

eventually result in a decrease of the number of very stable armour units and therefore in a positive

trend of the equilibrium load. However, during time limited tests it may be possible that the majority

of the mobilised armour units end up in a more stable position than at the beginning of the test. This

leads to a (temporary) decrease of the equilibrium load. This effect was observed in the form of

negative dips before the eventual increase of the equilibrium load during tests characterised as type

A2 to A4, B‐6 to B8 and C9. This mechanism is a stochastic process which influences the equilibrium

load. The spread of the equilibrium loads found in the results of the hydraulic experiments can be

explained by this mechanism.


‐ 83 ‐

8 Conclusion and recommendations

In this chapter the hypotheses stated in chapter 3 are evaluated and some extra conclusions are

drawn in addition to the statements made in last chapter. Also some recommendations for further

research are given.

8.1 Conclusion

The total load was decomposed in a static load and a dynamic load. A hypothesis was developed for

each of these loads and both loads were studied in two different experiments.

8.1.1 Static load

The static load was studied in an experiment which resulted in a graph of the average static load on

an armour unit at the first row as a function of the number of rows applied on the breakwater. The

static load increases linear with the number of rows applied on the breakwater for 1 to 5 rows. The

static load starts approaching a constant value for breakwaters with 6 to 10 rows. A constant static

load is reached for breakwaters with 10 or more rows. This is visualised in figure 7‐1.

The approach of a constant value of the relation between the static load and the number of rows

applied on the breakwater can be described by a one‐dimensional model. This model is able to

reproduce the shape of the found relation based on measured critical angles of the individual armour

units (the slope at which an armour unit is just stable). These critical angles were transformed to

friction force between the armour units and the underlayer that was used to determine the individual

along slope force balance of an armour unit. A number of armour units were found to be stable at

very high angles (larger than 49 degrees). These armour units are very stable on conventional slopes

and are thus able to transfer relatively large forces to the under layer.

The measured values could not exactly be reproduced using the model (without a multiplication

factor). This is due to the unknown distribution of force between the under layer and the lower

positioned armour units (the distribution of force to the under layer by friction is favoured in this

model). However, since the shape of the reproduced relation is the same as the measured relation, it

can be concluded that the varying stability of the armour units and in particular the presence of very

stable armour units, are responsible for the approach of a constant static load for a breakwater slope

on which more than 10 rows are applied.

Based on the static model and the measurements was concluded that external loads on armour unit

at row 10 or higher, have a limited influence on the load on the first row of armour units.

The hypothesis regarding to the static load was:

The static load on the first row of armour units is the result of the individual residual forces (resulting

force from individual force balance) per armour unit which add up to a total resulting force on the first

row of armour units. The increasing rate of the static load, decreases with the number of rows applied

on the breakwater slope which is caused by a number of armour units which can bear (a part of) the

residual forces of armour units placed on higher rows.

Conclusion and recommendations

‐ 84 ‐

This hypothesis is found to be true and covers a large part of the results regarding the static load

determined in this research. The individual residual forces are of significant importance to the

eventual load on the first row of armour units. The approach of a constant value of the static load for

a higher number of rows was found in this research (or simply said the levelling of the relation

between the static force and the number of applied rows). This levelling is indeed the result of

armour units which can bear (a part of) the residual forces of armour units placed on higher rows. It

was also found (based on the investigation made with the static model) that a number of very stable

armour units are responsible for the complete levelling of relation between the static force and the

number of rows.

8.1.2 Dynamic load

The dynamic load was examined with a physical model in a wave flume. The results of this experiment

show two major phenomena:

the measured dynamic load shows a periodic behaviour with a certain peak‐to‐peak

amplitude;

the wave‐averaged dynamic load (the equilibrium load) is not constant but increases during

the sub tests until an equilibrium in reached at the end of a subtest;

The peaks of the periodic behaviour of the dynamic load occur simultaneously with the downwash of

water along the slope of the breakwater. Furthermore, a relation of the downwash velocity and the

peak‐to‐peak amplitude of the dynamic load was found.

The measured peak‐to‐peak amplitudes are related to the calculated downwash velocities. The

maximum force occurs at the point in time when the downwash velocity is maximal. The downwash is

therefore marked as the major component that influences the peak‐to‐peak load on the first row of

armour units. The importance of the outflow of water through the armour layer is, based on the test

measurements, less clear. The peak‐to‐peak amplitudes during tests with an impermeable core where

lower than test with a permeable core but since the equilibrium load during these tests increased

significantly (which occurs in a number of cases parallel to the decrease of the peak‐to‐peak

amplitude), these measurements are less reliable. Furthermore, no influence of the impermeable

layer at toe level was found and the theoretical calculation of the dynamic load showed a contribution

of only 13% to the total dynamic load which supports the statement that the outflow of water

through the armour layer is of less importance compared to the dynamic load induced by the

downwash. The periodic behaviour of the dynamic load is related to the hypothesis regarding the

dynamic load:

The combination of downwash on and through the armour layer, and the outflow of water from the

inside of the breakwater through the armour layer, is determining for the maximum loading of the

first row of armour units. The transfer of the dynamic forces imposed on an individual unit correlates

to the transfer of residual forces as described in the static load hypothesis.

This hypothesis relates to the measured peak‐to‐peak amplitudes and did not foresee a trend of the

equilibrium load. The hypothesis covers therefore just a part of the studied phenomena. The dynamic

load is composed of an equilibrium load which is the dynamic load averaged over the wave period

and a harmonic peak‐to‐peak amplitude around this equilibrium load. The hypothesis is corect with

respect to the importance of the downwash for the harmonic behaviour of the dynamic load. The

influence of the flow from the core on the dynamic load is found to be limited based on a theoretical

analyses but was not confirmed with experiments.


‐ 85 ‐

The following parameters are of influence on the peak‐to‐peak amplitude of the load on the first row

of armour units; the permeability of the core, the smoothness of the under layer and the slope of the

breakwater (a flatter slope results in smaller peak‐to‐peak amplitudes). The number of rows applied

on the breakwater has no (clear) influence on the load on the first row of armour units for the tested

number of rows (15 to 25). Based on the static model it was determined that external loads imposed

on row 10 or higher have a limited influence on the load on the first row of armour units. The dynamic

load on an armour unit can be seen as an external load and based on this statement it can be

concluded that dynamic load imposed on row 10 or higher have a limited effect on the load on the

first row of armour units. For a smaller number of rows (10 or smaller) the dynamic load will be lower

but this was not tested in this study. The influence of the initial packing density on the load on the

first row of armour units was found to be unclear for the packing density measured over the entire

slope. There may be parallels between the packing density of the first ten rows and the measured

loads. For future research it is recommended to investigate the individual settling of armour units and

to investigate the relation between the settling of all individual units and the increase of the

equilibrium load.

Besides the peak‐to‐peak amplitudes which were foreseen in the hypothesis, a trend of the

equilibrium load was observed which is more significant than these peak‐to‐peak amplitudes. This

trend is independent of the number of rows and the initial RPD, although RPD of the first ten rows

might have some influence. The total RPD increased as the equilibrium load increased but no clear

relation was found between the increase of the RPD and the increase of the equilibrium load.

The parameters which were found to have an influence on the peak‐to‐peak amplitude also have an

influence on the increase of the equilibrium load. A smooth under layer and an impermeable core

resulted in a larger increase of the equilibrium load during the tests compared to the reference case

with 20 rows and a normal under layer and core. Furthermore, a flatter slope also resulted in a

greater increment of the equilibrium load compared to the base case with a steeper slope.

A relationship of the wave height and the equilibrium load was found. This relation has a quadratic

character and shows a much dispersion. Furthermore, a relationship of the armour unit stability (as

described in the Van der Meer formula) and the increase of the equilibrium load was found which still

shows large dispersion, but less than the relationship of the wave height and the equilibrium load.

Based on this relation it can be concluded that the increment of the equilibrium load is related to the

(hydraulic) armour unit stability while the peak‐to‐peak amplitudes are related to the occurring flow.

The objective of this research was to determine the governing loads on the first row of armour units

and to establish a quantitative relation between the influencing parameters and the load on the first

row of armour units. Based on this research, it was concluded that the static load combined with the

increase of the equilibrium load are the governing loads on the first row of armour units. The static

load is a function of the number of rows and reaches a constant value around ten rows. The increase

of the equilibrium load depends on the wave action (wave height) but did not show a dependency on

the number of rows applied or the total initial packing density. The dynamic load oscillating around

the equilibrium load with certain peak‐to‐peak amplitude is mainly the result of the downwash but is

as result of the much larger increase of the equilibrium load during the wave action, of minor

importance.

Referring to the background of the problem and to the above described results, it was concluded that

the number of rows applied on the breakwater (for 15 rows or more), has no significant influence on

the average load on the first row of armour units. When considering the average load, it can thus be

concluded that it is more favourable to apply an extra row then applying a larger armour unit in order

to cover the whole slope.

Conclusion and recommendations

‐ 86 ‐

8.2 Applicability and limitations of the study

This study investigated the load on the first row of armour units during static and dynamic conditions.

Understanding was gained of the fundamental processes which influence the static load and the

dynamic load on the first row of armour units. It was found that the static load is very much

dependent on the individual along slope force balance of the individual armour units. The variation of

these force balances leads to the approach of a constant value of the static load when a large number

of rows (more than 10) are applied on the slope of the breakwater.

The dynamic load appeared to be a periodic load around a (wave‐averaged) equilibrium load which

shows a positive trend that approaches a constant value over time. The periodic behaviour is related

to the downwash velocity and the positive trend of the equilibrium load is related to the armour unit

stability, as defined in the Van der Meer formula, and can be related to the wave height.

These results improve the knowledge on this subject and give insight into the background of the

observed phenomena. However, it is not possible to make design calculations based on the outcomes

of this research. First of all, the average load on an armour unit at the first row was studied in this

research and not the peak load on a single armour unit. In order to make a design calculation of the

structural integrity of the armour units, the peak load on an armour unit should be known in order to

derive the occurring stresses in the armour unit. An important next step for future research is the

determination of the peak loads on, and peak stresses in, the armour units of the first row. Several

recommendations regarding the study of the force distribution between the under layer and the

lower positioned armour units and possible measurement methods of the peak load per unit are

stated in the following paragraph.

Secondly, the spread observed in both the static results as the dynamic results is substatial. This

spread is the result of the stochastic character of the described processes. In general, designs are

based on a conservative estimation of the known forces. A design based on a conservative estimation

of the static and the dynamic load would result in an uneconomic design and is therefore not advised.

Finally, the results found in this research are obtained from test with regular waves while in practice

irregular waves are imposed on breakwaters. Some agreement was found between the results of

tests executed with irregular waves and the wave height‐dynamic load relations found in the results

of the tests executed with regular waves. However, no absolute agreement was found and the

number of tests executed with irregular waves was limited. For future research it is advised to study

the dynamic load during test executed with irregular tests while simultaneously expanding the

number of tests.

The mentioned first and last points of consideration already impose many challenges for future

researchers. These subjects should be studied in order to achieve full benefit of this research.

However, this research already clarifies a number of fundamental processes such as the cause of the

constant value of the static load for more than 10 rows applied on the slope of a breakwater and the

relation between the downwash and the periodic dynamic load. Based on this new gained knowledge,

future research can be performed.


‐ 87 ‐

8.3 Recommendations

A number of recommendations for further research can be made improving this research or for future

research. n this research it was concluded that the influence of more than ten rows on the load of the

first row is relatively low. Further hydraulic research could be done with only ten rows of amour units

so the influence of for instance the relative packing density on the part of the breakwater were the

influence actual occurs can be determined. The subsidence of every row (or even every single armour

unit) should be measured and not only the subsidence of the top row since this may vary and may

have an influence.

In this research the water level was kept at a constant level and thus at the same distance from the

toe. This distance might have an influence since the maximum velocities occur around the water line.

A variation of the water level may thus have an influence on the measured load on the first row. It is

recommended to investigate the influence of the distance between the toe and the water level in

further research.

In practice the waves imposed on a breakwater are irregular waves. A short analysis of the effect of

irregular waves on the load on the first row of armour unit was performed in this research but the

results of this analysis were not completely clear and more tests with irregular waves should be

performed in order to obtain a more valuable analysis. The irregular wave spectra imposed on the

breakwater during these test had all the same significant wave height and peak period. In order to

study all possible irregular waves much more tests with varying significant wave heights and peak

periods should be performed.

A number of numerical models are available which model (a part of) the behaviour of a rubble mound

breakwater. For further understanding of the subject and verification of the obtained results a

numerical model study is recommended.

All these recommendations concern the improvement of the present research. However in order to

have a value for the design process of a breakwater it has to be known whether armour units fail

because of the load on the armour units or not.

The load on an armour unit should be translated into stresses which occur in this armour unit in order

to determine whether a load on an armour unit will lead to problems. This can be done by applying

the full load on an unfavourable spot of the armour unit but more realistic is an approach whereby

the contact points between the armour units are identified and whereby the calculated load is

imposed on these spots.

The maximum load on a specific armour unit has to be known in order to determine the stress in this

armour unit. In this research only an average load for the entire row was identified. The static load on

a specific armour unit can be determined by an improved static model. In order to improve this model

the model should be a two dimensional model which models the distribution of forces between an

armour unit and the two under laying armour units and the under layer. The distribution of the

residual force of one unit to the two underlying units and the support of a single layer by underlying

units and under layer (distribution of force between under layer and underlying units) has to be

known in order to model this correctly.

The (peak) load per unit should also be determined during hydraulic tests. This can be done by

measuring the load on a armour unit with one pressure sensor per armour unit. With this method

peak loads can be identified and with this method there is no influence of the measurement frame.


‐ 88 ‐

References

ABBOTT, M.B. AND PRICE, W.A. (1994) Coastal, Estuarial and Harbour Engineers’ Reference Book.

Chapman & Hall, London, p. 349‐438

BATTJES, J.A. (1974) Computation of Set‐Up, Long shore Currents, Run‐Up and Overtopping due to

wind‐generated waves Dissertation, Delft University of Technology, Delft, The Netherlands

BETZLER, K. (2003) Fitting in Matlab. Short Lecture Notes, Universität Osnabrück, Osnabrück, Germany

BURCHARTH, H.F. (1993) Structural Integrity and Hydraulic Stability of Dolos Armour layers. Series paper

9 Hydraulics & Coastal Engineering Laboratory, Aalborg University, Aalborg, Denmark

BURCHARTH, H.F. AND ANDERSEN, O.H. (1995) On the one‐dimensional steady and unsteady porous flow

equations Journal of Coastal Engineering Vol. 24 p 233‐257

BURHART, H.F., ZHOU, L. AND TROCH, P. (1999) Scaling of core material in rubble mound breakwater

model tests. Proceedings of COPEDEC V, Cape Town, South Africa

CIRIA. CUR, CETMEF (2007) The Rock Manual. The use of rock in hydraulic engineering (2nd edition)

C683, CIRIA, London, United Kingdom

D’ANGREMOND, K. AND VAN ROODE, F.C. (2001) Breakwaters and closure dams. Delft University Press,

Delft, The Netherlands

DEKKING, F.M. AND KRAAIKAMP, C. AND LOPUHAÄ, H.P. AND MEESTER, L.E. (2005) A Modern Introduction to

Probability and Statics: understanding why and how. Springer‐Verslag, London, United Kingdom, p.

89‐99

DELTA MARINE CONSULTANTS (2011) Guidelines for Xbloc Concept Designs. Brochure Delta Marine

consultants (available at xbloc.com), Gouda, The Netherlands

FROSTICK, L.E. AND MCLELLAND, S.J. AND MERCER, T.G. (EDS) (2011) User Guide to Physical Modelling and

Experimentation: Experience of the HUDRALAB Network, CRC Press/Balkema, Leiden, The Netherlands

HALD, T. (1998) Wave Induced Loading and Stability of Rubble Mound Breakwaters. Dissertation,

Hydraulics & Coastal Engineering Laboratory, Aalborg University, Aalborg, Denmark

HOLTHUIJSEN, L.H. (2007) Waves in Oceanic and Coastal Waters. Cambridge University Press,

Cambridge, united Kingdom

HUGHES, S.A. (1993) Physical Models and Laboratory Techniques in Coastal Engineering, World

Scientific Publishing, Singapore

HUGHES, S.A. (1994) Estimation of wave run‐up on smooth, impermeable slopes using the wave

momentum flux parameter. Proceedings 24th ICCE, Kobe, Japan, p. 1085‐1104

JUMELET, H. D. (2010). The influence of core permeability on armour layer stability. Msc. thesis, Delft

University of Technology, Delft, The Netherlands

References

‐ 89 ‐

MAGOON, O.T., WEGGEL, J.R., BAIRD, W.F., EDGE, B.L., WHALIN, R.W., DAVIDSON, D.D., MANSARD, E. (1994)

Rehabilitation of the West Breakwater –Port of Sines, Portugal, proceedings 24th ICCE, Kobe, Japan, p.

3608‐3613

MANSARD, E.P.D. AND FUNKE, E.R. (1980) The Measurement of Incident and Reflected Spectra Using a

Least Squares Method, proceedings 17th ICCE, Hamburg, Germany, p. 154‐172

MATLAB version 7.9.3 (2009), computer software, The MathWorks Inc., Natick, Massachusetts, United

States of America

DASYLAB version 11.0 (2009), computer software, Measurement Computing, Norton, Massachusetts,

United States of America

MUILWIJK, M.P. (2011) Xbloc model test report: forces on the first row, Report Delta Marine

consultants, Gouda, The Netherlands

SCHIERECK, G.J. (2001) Introduction to Bed, Bank and shore protection. Delft University Press, Delft, The

Netherlands

TEN OEVER, E. (2011) Specifications for the application of Xbloc. Report Delta Marine consultants,

Gouda, The Netherlands

TSAI, C.P. (1997) Downrush flow from waves on sloping seawalls. Journal of Ocean Engineering Vol.

25 p 295‐308

U.S. ARMY CORPS OF ENGINEERS, (2002) Coastal Engineering Manual. U.S. Army corps of Engineers,

Washington, D.C., United States of America, chapter 2‐3, chapter 6‐5,

VAN DER MEER, J.W. AND STAM. C.J.M. (1992) Wave run‐up on smooth and rock slopes. ASCE, Journal of

WPC and OE, Vol. 188, No5, New York, United States of America, pp.534‐550

VAN GENT, M.R.A. (1995) Wave Interaction with Permeable Coastal Structures. Dissertation, Delft University of Technology, Delft, The Netherlands

VAN ZWICHT, B.N.M. (2009) Manual hydraulic model testing, Report No 968000 Xbloc, Delta Marine

consultants, Gouda, The Netherlands

VERRUIT, A. VAN BAARS, S. (2005) Grondmechanica. Delft University Press, Delft, The Netherlands

WAVELAB version 3.04 (2008), computer software, Hydraulics & Coastal Engineering Laboratory,

Aalborg University, Aalborg, Denmark

WENNEKER, I., WELLENS, P. AND GERVELAS, R. (2010) Volume‐of‐Fluid model ComFLOW simulations of wave

impacts on a dike, proceedings 32th ICCE, Shanghai, China, p. 1‐12

‐ 93 ‐

Appendices

Table of contents

A. Design of static tests ..................................................................................................................... 94 A.1 Photographs of static test ..................................................................................................... 94 A.2 Model information ................................................................................................................ 95 A.3 Photographs of critical angles test ........................................................................................ 96

B. Design of hydraulic test ................................................................................................................ 97 B.1 Xbloc design table ................................................................................................................. 97

B.1.1 Under layer ................................................................................................................... 98 B.1.2 Core .............................................................................................................................. 99

B.2 Cross‐section ....................................................................................................................... 100 B.3 Photographs of Model ........................................................................................................ 101 B.4 Test program ....................................................................................................................... 108 B.5 Load cell .............................................................................................................................. 109

C. Results of static tests .................................................................................................................. 110 D. Results of hydraulic tests ............................................................................................................ 112

D.1.1 Wave analysis ............................................................................................................. 112 D.2 Measurements .................................................................................................................... 113 D.3 Influence of measurement frame ....................................................................................... 127 D.4 Influence of the water level variations during wave attack ................................................ 127

E. Analysis of static tests ................................................................................................................. 129 E.1 Critical angle regarding tot rolling....................................................................................... 129 E.2 Measurements of static load in test prior to hydraulic tests .............................................. 130 E.3 Static model ........................................................................................................................ 130

F. Analysis of hydraulic tests ........................................................................................................... 133 F.1 Wave analysis hydraulic test 3(3) hydraulic test 2(2) ......................................................... 133 F.2 Analysis on trend of the “equilibrium” line ......................................................................... 142

F.2.1 Trend of the equilibrium load ......................................................................................... 145 F.2.2 Trend of the equilibrium load during tests with irregular waves ................................... 147

F.3 Theoretical approximation of dynamic load ....................................................................... 149

‐ 94 ‐

A. Design of static tests

This chapter gives further information on the design of the static tests. Paragraph A.1 gives a number

of photographs of the static test, in paragraph A.2 the weight distribution of the under layer is given

and paragraph A.3 gives a number of photographs of the critical slope test.

A.1 Photographs of static test

Figure A‐1 Static test (the wooden beam can move along slope and the load imposed on this beam

is measured with the load sensor which can be seen on the picture and a load sensor on the other

side)

Figure A‐2 Frontal view of static test

‐ 95 ‐

Figure A‐3 Wheel under movable beam

Figure A‐4 Placement of first row

A.2 Model information

One single stone fraction was used for the construction of the underlayer. Figure A‐1 gives the weight

histogram of the under layer material which was composed by weighing a total of 50 stones by hand.

Figure A‐5 Weight histogram of under layer material

‐ 96 ‐

A.3 Photographs of critical angles test

Figure A‐6 Overview of critical slope tests

Figure A‐7 Rear view of critical slope test

Figure A‐8 Random placement by hand (leg down)

‐ 97 ‐

B. Design of hydraulic test

This chapter gives further information on the design of the static tests. Paragraph B.1 gives the design

table for Xbloc single layer armour units, in paragraph B.1.1 and paragraph B.1.2 the weight

distribution of the under layer and the core is given, paragraph B.2 gives a drawing of the cross‐

section of the breakwater model, paragraph B.3 gives a number of photographs of the hydraulic

experiment, paragraph B.4 gives the test program and paragraph B.5 gives an information sheet of

the used load cell.

B.1 Xbloc design table

Figure B‐1 Xbloc design table [DELTA MARINE CONSULTANTS, 2011]

‐ 98 ‐

B.1.1 Under layer

The under layer and the core were composed according to the requirement as described in the main

report. Figure B‐2 and Figure B‐3 displays the required grading of the under layer and the core and the

applied under layer material and core material.

Figure B‐2 Grading applied and requested grading of under layer

Used stone density: 2659 kg∙m‐3

available stone gradings [mm]

available stone gradings [gramm]

Corresponding prototype stone grading’s (scale 1: 45,75)

sieve curve 60‐300kg (model scale)Cumulative / percentage of total

2 4 mm 0.01 0.10 gramm 1.20 9.63 kg 0 % 0 %

4 5.6 mm 0.10 0.28 gramm 9.63 26.41 kg 0 % 0 %

5.6 8 mm 0.28 0.80 gramm 26.41 77.01 kg 0 % 0 %

8 11.2 mm 0.80 2.21 gramm 77.01 211.31 kg 0 % 0 %

11.2 16 mm 2.21 6.43 gramm 211.31 616.05 kg 45 % 45 %

16 22.4 mm 6.43 17.65 gramm 616.05 1690.45 kg 100 % 55 %

22.4 31.5 mm 17.65 49.09 gramm 1690.45 4700.98 kg 100 0

31.5 > mm 49.09 > gramm 4700.98 > kg 0

‐ 99 ‐

B.1.2 Core

Figure B‐3 Grading applied and requested grading of core

Used stone density: 2659 kg∙m‐3

available stone gradings [mm]

available stone gradings [gramm]

Corresponding prototype stone grading’s (scale 1: 45,75)

sieve curve 60‐300kg (model scale)Cumulative / percentage of total

2 4 mm 0.01 0.10 gramm 1.20 9.63 kg 0 % 0 %

4 5.6 mm 0.10 0.28 gramm 9.63 26.41 kg 0 % 0 %

5.6 8 mm 0.28 0.80 gramm 26.41 77.01 kg 20 % 20 %

8 11.2 mm 0.80 2.21 gramm 77.01 211.31 kg 60 % 40 %

11.2 16 mm 2.21 6.43 gramm 211.31 616.05 kg 100 % 40 %

16 22.4 mm 6.43 17.65 gramm 616.05 1690.45 kg 100 % 0 %

22.4 31.5 mm 17.65 49.09 gramm 1690.45 4700.98 kg

31.5 > mm 49.09 > gramm 4700.98 > kg

‐ 100 ‐

B.2 Cross‐section

Figure B‐4 Cross‐section of breakwater with a slope of 3:4 and 15, 20 and 25 rows (units in mm)

‐ 101 ‐

B.3 Photographs of Model

Figure B‐5 Overview of model

Figure B‐6 Gabions (below water level minus 12 cm)

‐ 102 ‐

Figure B‐7 Armour layer of Xbloc model units

Figure B‐8 Toe

‐ 103 ‐

Figure B‐9 Measurement of Ly (the wooden rod is placed at the 100 % RPD position)

Figure B‐10 Load sensor

‐ 104 ‐

Figure B‐11 Wave gauge (positioned at toe level)

Figure B‐12 Signal amplifiers (left wave gauge, right load sensor) and computer with measurement

software DASYLAB (2008)

‐ 105 ‐

Figure B‐13 Camera positions

Figure B‐14 Wave gauges near the wave paddle

‐ 106 ‐

Figure B‐15 Wave gauges near the structure

Figure B‐16 Wave paddle

‐ 107 ‐

Figure B‐17 Plastic sheet in order to make the core impermeable

Figure B‐18 Smooth, impermeable under layer (wood)

Figure B‐19 Placement of first row on the frame

‐ 108 ‐

Figure B‐20 Gabion with Xbloc units (used in order to measure the influence of the frame)

B.4 Test program

Table B‐1 Hydraulic test program

‐ 109 ‐

B.5 Load cell

Figure B‐21 Detailed information of used load cell

‐ 110 ‐

C. Results of static tests

The static load on the first row of armour units has been measured by force gauges and read manually

after a complete row was added to the breakwater slope. The measurements were divided by the

weight of a single armour unit and the number of units at a row in order to determine the load on a

single armour unit at the first row relative to its own weight (Table C‐1). The last column of this table

gives the relative packing density which was calculated from the measured distance between the

centre of the lowest row of armour units and the highest row of armour units (Ly)

Table C‐1 Results of static test (the presented load is relative to the unit weight)

These measured static loads were plotted relative to the number of rows applied on the breakwaters

slope which gives Figure C‐1:

‐ 111 ‐

Figure C‐1 Results of static tests

‐ 112 ‐

D. Results of hydraulic tests

D.1.1 Wave analysis

The incoming waves were measured with four wave gauges at the beginning of the wave flume and

three wave gauges near the structure (see figure 4‐10). The wave data of the regular test was

analysed with the program WAVELAB (2008). This analysis gives the measured wave height and periods

of the incoming waves. Since al test were executed with a wave series of regular waves with the same

settings, only the results of the subtests of hydraulic test 2(3) (20 rows, RPD of 100%) are visualised in

Table D‐1.

Table D‐1 Results wave analysis regular waves

Table D‐2 Requested wave height and period Test Wave height (H) Wave period (T)

H = 0,2 Hmax T = 0,874 Hz 44 mm 1,14 s

H = 0,2 Hmax T = 1,06 Hz 44 mm 0,94 s

H = 0,2 Hmax T = 1,50 Hz 44 mm 0,67 s

H = 0,4 Hmax T = 0,562 Hz 88 mm 1,78 s

H = 0,4 Hmax T = 0,749 Hz 88 mm 1,34 s

H = 0,4 Hmax T = 1,06 Hz 88 mm 0,94 s

H = 0,6 Hmax T = 0,500 Hz 132 mm 2,00 s

H = 0,6 Hmax T = 0,624 Hz 132 mm 1,60 s

H = 0,6 Hmax T = 0,874Hz 132 mm 1,14 s

H = 0,8 Hmax T = 0,375 Hz 176 mm 2,67 s

H = 0,8 Hmax T = 0,500 Hz 176 mm 2,00 s

H = 0,8 Hmax T = 0,749 Hz 176 mm 1,34 s

H = 1,0 Hmax T = 0,375 Hz 220 mm 2,67 s

H = 1,0 Hmax T = 0,437 Hz 220 mm 2,29 s

H = 1,0 Hmax T = 0,687 Hz 220 mm 1,46 s

The results of the measured wave height and period with the requested (set) wave height and period

show a very good resemblance with the measured and requested wave period. The measured wave

height deviates slightly from the requested wave height. This deviation is acceptable.

‐ 113 ‐

D.2 Measurements

Besides the wave data also the load on the first row of armour units (or in this case the measuring

frame) was measured with a load censor. This gave a measurement of the force measured by this

sensor over time. During the execution of the test program a constant increase of the measured load

over time was observed even when the load senor was unloaded. This trend was identified at the

beginning of each subtest and the measured data was corrected for this trend. For each of the

executed tests one plot of the measured load on the first row relative to time is visualised in Figure

D‐1 to Figure D‐11:

Figure D‐1 Measured load record of hydraulic test 1


‐ 114 ‐



‐ 115 ‐



‐ 116 ‐

Figure D‐7 Measured load record of hydraulic test 6_2 (directly after test 6 without rebuilding the

slope)


‐ 117 ‐



‐ 118 ‐


‐ 119 ‐

In order to compare the results visualised in the above figures much of the information which is

contained in these graphs was obtained from the corresponding dataset or manually read from the

graphs and noted in Table D‐3 to Table D‐17. The information obtained from the graphs includes:

the peak‐to‐peak amplitude (of the load) at the beginning of the subtest (A0)

the start time of the load measurements compared to the beginning of recording (tstart)

the end time of the load measurements compared to the beginning of recording (tend)

the measured force at the beginning of the subtest (F0)

the measured force at the end of the subtest (Fend)

the read “half‐life” time of the increase of the equilibrium load (thalf)

the determined type of the occurring trend of the equilibrium load (type)

the total increase of the equilibrium load during this subtest, read from the graphs (∆measured)

the eventual peak‐to‐peak amplitude of the load (A∞)

the time between the initial peak‐to‐peak amplitude and the eventual amplitude (◦T)

the equilibrium load at the end of the subtest relative to the maximum measured equilibrium

load (the equilibrium load after the last subtest)

the increase of the equilibrium load during the subtest relative to the maximum measured

equilibrium load (the equilibrium load after the last subtest)

the total increase of the equilibrium load during this subtest, calculated with a MATLAB script

(∆calculated)

Table D‐3 Results of hydraulic tests: 0,2 Hmax 1,50 Hz

‐ 120 ‐



‐ 121 ‐



‐ 122 ‐



‐ 123 ‐


Table D‐11 Results of hydraulic tests: 0,6 Hmax 0500 Hz

‐ 124 ‐



‐ 125 ‐



‐ 126 ‐



‐ 127 ‐

D.3 Influence of measurement frame

In order to measure the forces on the first row of armour units a measurement frame was built to

support the first row of armour units. This measurement frame was however also loaded by wave

action. In order to have an indication of the influence of the measurement frame on the eventual

measured amplitudes, the load on the frame without armour units on top of the frame was measured

during a full test program. In order to simulate the same flow conditions as during normal test a

gabion filled with Xbloc armour units was placed on the breakwater slope. This gabion was not

supported by the measurement frame. The results of these measurements are given in Table D‐18:

Table D‐18 Measured influence of the measurement frame Measured peak‐to‐peak

amplitude [N] 0.04 0.06 0.06 0.11 0.09 0.14 0.65 0.33 0.42 1.82 0.98 0.95 2.49 1.31 1.34

Measured Influence

measurement frame [N] 0.00 0.03 0.05 0.08 0.08 0.04 0.14 0.13 0.11 0.38 0.26 0.24 0.85 0.53 0.48

In this table it can be observed that the influence of the measurement frame is not unambiguous. The

measured amplitude of the frame is relatively large for the lowest wave heights but stabilises at a

level of 20% of the total peak‐to‐peak amplitude. It is therefore not possible to simply subtract the

influence of the measurement frame of the total peak‐to‐peak amplitude and therefor the influence

of the measurement frame results in an uncertainty of about 20% of the peak‐to‐peak amplitude.

D.4 Influence of the water level variations during wave attack

The Influence of the water level variations during wave attack was studied in order to determine

whether part of the measured amplitudes could be related to the variation of the buoyancy of the

armour units.

The force balance of a single armour unit is:

sin( ) cos( )

sin(36,87) 0,6 cos(36,87) 0,12

res

g

res

g

Ff

F

F

F

(D.1)

If an armour unit is submerged the effective weight is reduced (the submerged weight has to used).

Furthermore the friction coefficient reduces since the friction between wet surfaces is less compared

to the friction between dry surfaces. In Table D‐19 the both the wet and dry friction coefficients are

given:

Table D‐19 Friction coefficients concrete1 Materials Friction coefficient

Cement concrete on dry gravel 0,50 – 0,60

Cement concrete on dry rock 0,60 – 0,70

Cement concrete on wet rock 0,50

The submerged weight can be calculated by multiplying the weight of the armour unit by the

following factor:

1 http://www.supercivilcd.com/FRICTION.htm

‐ 128 ‐

; g;buoyancy 10001 1 0,561

2279

g submerged g water unit

g g g concrete unit

F F F g Vol

F F F g Vol

(D.2)

This factor was applied on the weight of an armour unit and entered in equation (D.1) what gives the

following force balance for a submerged armour unit:

0,561 sin( ) 0,561 cos( )

0,561 sin(36,87) 0,5 0,561 cos(36,87) 0,112

res

g

res

g

Ff

F

F

F

(D.3)

The difference between the outcome of equation D.3 and the outcome of equation D.1 is very

dependent on the wet friction coefficient. Based on the spread of the friction coefficients presented

in Table D‐19 and in earlier mentioned references (chapter 7) no reliable conclusions could be drawn

from the calculated difference between the residual force of a dry unit compared to the residual force

of a submerged unit. Possible influences of the water level variation (on the peak‐to‐peak amplitude)

are therefore not further considered in this research.

‐ 129 ‐

E. Analysis of static tests

E.1 Critical angle regarding tot rolling

The critical angle regarding rolling of an armour unit was investigated in order to determine a

conservative angle at which movement by rolling will take place instead of sliding.

Figure E‐1 Critical angle

During the tests which were executed in order to establish the critical angle of a Xbloc armour unit

two types of failure where observed. One type of failure is sliding of the armour unit which can be

linked to the reached friction capacity between the armour unit and the under layer. The second type

of failure is rolling of an armour unit which can be linked to a momentum imbalance of an armour

unit. The angle at which this occurs was further studied by a schematisation displayed in Figure E‐1.

The initiation of rolling was schematised as rolling of an armour unit over a protuberant stone of the

under layer with a certain protuberant height (dstone). The armour unit start rolling around point RC

and therefore the moments around this point were evaluated. The armour unit will start rolling when

the working line of the weight of an armour unit (Fg) is at the left side of point RC. Whether this occurs

or not is a function of the slope of the breakwater and the protuberant height of the under layer

stone and can be calculated with use of the line segments X1 and X2. When X1 is larger than X2, the

armour unit will start rolling.

With the help of some simple geometric relations the following relations for X1 and X2 were obtained:

‐ 130 ‐

21 cos(180 45 )

22 sin( )stone

DX

X H

(E.1)

The protuberant height of the under layer stone was however unknown. The Dn50 of the stones used

for the under layer in this test was 13.3 mm which is about one third of the unit height of the used

armour unit. The protuberant height of the under layer stone is assumed to be 25% of the Dn50 which

equals one twelfth of the unit height of the armour unit. If one enters this value in the above given

equation on arrives at a critical angle regarding tot rolling of an armour unit of 49 degrees.

E.2 Measurements of static load in test prior to hydraulic tests

The initial static load on the first row was also measured during the hydraulic tests. The results of

these measurements are given in Table E‐1

Table E‐1 Measurements of static load prior to hydraulic testing Test Initial load "dry" Initial load "wet" Slope (∆F/∆W)

RPD 100% 1,42

RPD 100% 1,34

RPD 100% 1,15

RPD 103% 1,07

RPD 103% 0,92

RPD 97% 0,85

RPD 97% 0,84

RPD 100% 1,09 0,042

RPD 100% 0,93

RPD 100% 0,9

RPD 100% 1,13

RPD 100% 1,15 ∆ <0.05

RPD 100% 1,28 ‐0.014

RPD 100% 1,38

RPD 100% 1,45 0.0562

RPD 100% 1,22

RPD 100% “Smooth UL” 1,865 ‐0.0088

RPD 100% “Smooth UL” 2,01 ∆ <0.05

RPD 100% “Smoot UL” 1,06 ∆ <0.05

RPD 100% 25 rows 0,64

RPD 100% 25 rows 0,86

RPD 100% 25 rows 0,77

RPD 100% 25 rows 0,99

RPD 100% slope 2:3 1,06

RPD 100% slope 2:3 0.66

RPD 100% slope 2:3 0.77

The measurments presented in this table are in line with the measurments made during the static

test. This table also shows the effect of a smooth underlayer which gives much higher static loads

then the other tests.

E.3 Static model

The load on the first row of armour units was calculated by simple one dimensional model. The

working of this model is described in the main report. The MATLAB code of this model is given in

Table E‐2.

‐ 131 ‐

Table E‐2 MATLAB‐code static model

function [A] = Staticmodel(vargin)

% Read dataset residual forces (calcualted from critical angles, total number of items in the dataset =

200)

A= xlsread('ResidualForces','G7:G206');

% Number of iterations

n = 10000;

% Draw random value from residual forces dataset

X1 = A(ceil(length(A)*rand(1,n)));

...

...

...

X20 = A(ceil(length(A)*rand(1,n)))

% Calculation of force transfer between rows for n times

for i = 1: n

% Calculation of force on the first row of a slope of 20 rows

% Calculation of total residual force row 20

Fres20 = X20(i);

if Fres20<0;

Fres20 = 0;

End

%Calculation of total residual force row 19

Fres19 = X19(i)+Fres20;

if Fres19<0;

Fres19 = 0;

end

...

...

...


if Fres1<0;

Fres1 = 0;

end

F20(i,:)=Fres1;

% Calculation of force on the first row of a slope of 19 rows

%Calculation of total residual force row 19 (which is now the highest row)

Fres19 = X19(i);

if Fres19<0;

Fres19 = 0;

end

...

...

...


if Fres1<0;

Fres1 = 0;

end

F19(i,:)=Fres1;

% this is repeated for a slope of 18, 17, 16 etc rows

...

‐ 132 ‐

...

...



Fres1 = X1(i);

if Fres1<0;

Fres1 = 0;

end

F1(i,:)=Fres1;

end

%Calculation of mean values per row

F1=mean(F1;

...

...

...

F20=mean(F20);

%Por results

F=[F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 ];

plot(F)

‐ 133 ‐

F. Analysis of hydraulic tests

F.1 Wave analysis hydraulic test 3(3) hydraulic test 2(2)

An analysis of the waves imposed on the breakwater model in relation to the measured load during

hydraulic test 3(3), subtest 0,6 Hmax 0,874 Hz was performed. The video of this test is analysed frame

by frame for the time interval 1:28 to 1:33 [min:sec] and compared to the measured water level at the

wave gauge and measured load on the first row.

For the interval 1:28 to 1:33 the video is analysed frame by frame in order to determine at which

point in time the highest and lowest level of run‐up occurs and how these points corresponds in time

to the total wave period (T). Furthermore the points in time when the wave front passes the still

water level and the points in time of a water level minimum or maximum at the wave gauge (toe)

were determined. Figure F‐1 shows the screenshots of the video with occurring points of interests.

1:28:16 1:28:32

Wave gauge max, Run‐down max Upwards through still water level

1:28:47 1:28:49

Run‐up max Wave gauge min

1:29:10 1:29:23

Downwards through still water level Wave gauge max

‐ 134 ‐

1:29:25 1;29:43

Run‐down max Upwards through still water level

1:30:00 1:30:02

Wave gauge min Run‐up max

1:30:18 1:30:29

Downwards through still water level Wave gauge max

1:30:31 1:30:43

Run‐down max Upwards through still water level

1:31:08 1:31:13

Wave gauge min Run‐up max

‐ 135 ‐

1:31:29 1:31:42

Downwards through still water level Run‐down max, Wave gauge max

1:31:58 1:32:15

Upwards through still water level Wave gauge min, Run‐up max

1:32:35

Downwards through still water level

Figure F‐1 Screenshots from side video (hydraulic test 3(3), subtest 0,6 Hmax 0,874 Hz)

‐ 136 ‐

The points of interest and times of occurring are collected in Table F‐1.

Table F‐1 Point of interest Time [m:s:ms] Period based on

time difference

Time compared

to t0 (1:28:16)

Compared to

average wave

period

Point at the wave

period when pint of

interest occurs

Maximum water

level at wave

gauge

1:28:16, 1:29:23

1:30:29, 1:31:42

67ms, 66ms,

73ms

t0, t0+67ms

t0+133ms,

t0+206ms

0, 0.98

1,95 3,02

0,99

Maximum level

of run‐down

1:28:16, 1:29:25

1:30:31, 1:31:42

69ms, 66ms,

71ms

t0, t0+69ms

t0+135ms,

t0+206ms

0, 1,01

1,98 3,02

1,00

Wavefront

passing through

still water level

(up)

1:28:32, 1;29:43

1:30:43, 1:31:58

71ms, 60ms,

75ms

t0+16ms, t0+87ms

t0+147ms,

t0+222ms

0,23 1.28

2,16 3,26

0,23

Minimum water

level at wave

gauge

1:28:49, 1:30:00

1:31:08, 1:32:15

71ms, 68ms,

67ms

t0+33ms,

t0+104ms

t0+172ms,

t0+239ms

0,48 1,52

2,52 3,5

0,51

Maximum level

of run‐up

1:28:47, 1:30:02

1:31:13, 1:32:15

75ms, 71ms

62ms

t0+31ms,

t0+106ms

t0+177ms,

t0+239ms

0,45 1,55

2,60 3,50

0,54

Wavefront

passing through

still water level

(down)

1:29:10, 1:30:18

1:31:29, 1:32:25

68ms, 71ms,

56ms

t0+54ms,

t0+122ms

t0+193ms,

t0+249ms

0,79 1,79

2,83 3,65

0,77

The average wave period was determined based on the time difference between subsequent points

of interests (for instance the time between subsiding maximum water levels at the wave gauge).

Following this approach an average wave period of 68,2 milliseconds (1,14 s) was obtained which

correspond well to the wave period of the incoming wave of 0,874 Hz (1.44 s) ). By analysing the time

difference between the points of interest and t0 (1:28:16) the average point in time when the points

of interests occur relative to the wave period was determined (Figure F‐2).

Figure F‐2 Occurring of points of interest relative to the wave period (T)

The occurrence of maximum run‐up and run‐down is visualised by a plot of the measured data in

Figure F‐3.

‐ 137 ‐

Figure F‐3 Hydraulic test 3(3), subtest 0,6 Hmax 0,874 Hz

Figure F‐3 shows that the positive peaks of the oscillating load on the first row took place between

the maximum run‐up of the wave on the slope and the maximum run‐down of the wave.

The described method is repeated for hydraulic test 2(2), subtest 0,6 Hmax 0,500 Hz on the time

interval of 1:02 to 1:10 min:s in order to obtain some general applicability of the above made

statements. The avarage period of the this test is 119,9 ms (2,0 s) which has a good correspondence

with the wave period of of the incoming wave of 2,0 s.

1:02:21 1:02:52

Run‐up max Downwards through still water level

1:03:21 1:03:27

‐ 138 ‐

Wave gauge min Run‐down max

1:03:56 1:04:36

Upwards through still water level, Wave gauge max Run‐up max

1:04:50 1:05:23

Downwards through still water level Wave gauge min

1:05:28 1:06:02

Run‐down max Upwards through still water level, Wave gauge

max

1:06:24 1:06:51

Run‐up max Downwards through still water level

‐ 139 ‐

1:07:23 1:07:24


1:07:53 1:08:01

Upwards through still water level Wave gauge max

1:08:22 1:08:53

Rup max Downwards through still water level

1:09:20 1:09:27


1:09:54 1:09:58

‐ 140 ‐

Upwards through still water level Wave gauge max

Figure F‐4 Screenshots from side video (hydraulic test 2(2), subtest 0,6 Hmax 0,500 Hz)

Table F‐2 Wave state Time [m:s:ms] Period based

on time

difference

Time compared to

t0 (1:28:16)

Time compared to

average wave period

Time between t0 and wave state [in

times per period]

Maximum water

level at wave

gauge

1:03:56, 1:06:02

1:08:01, 1:09:58

127ms, 119ms,

117ms

t0, t0+127ms

t0+246ms, t0+363ms

0, 1.06

2.05, 3,03

1.04

Maximum level

of run‐down

1:03:27, 1:05:28

1:07:24, 1:09:27

121ms, 116ms,

123ms

t0‐29, t0+92ms

t0+208ms, t0+331ms

‐0.24, 0.77

1.73, 2.76

0.76

Wavefront

passing through

still water level

(up)

1:03:56 1:06:02,

1:07:53 1:09:54

126ms, 111ms,

121ms,

t0, t0+126ms

t0+237ms, t0+358ms

0, 1.05

1.98, 2.98

1.00

Minimum water

level at wave

gauge

1:03:21, 1:05:23

1:07:23, 1:09:20

122ms, 120ms,

117ms

t0‐35ms, t0+87ms

t0+207ms, t0+324ms

‐0.29, 0.73

1.73, 2.7

0.72

Maximum level

of run‐up

1:02:21, 1:03:36

1:06:24, 1:08:22

135ms, 108ms

118ms

t0‐95ms, t0+40ms

t0+148ms, t0+266ms

‐0.79, 0.33

1.23, 2.22

0.25

Wave front

passing through

still water level

(down)

1:02:52, 1:04:50

1:06:51, 1:08:53

118ms, 121ms,

122ms

t0‐64ms, t0+54ms

t0+175ms, t0+297ms

‐0.53, 0.45

1.46, 2.48

0.47

Figure F‐5 Occurring of points of interest relative to the wave period (T)

‐ 141 ‐

Figure F‐6 Hydraulic test 2(2), subtest 0,6 Hmax 0,500 Hz

Table F‐2 in combination with Figure F‐5 and Figure F‐6 show that the time difference between the

point in time with the highest run‐up and the point in time with the maximum run‐down is equal to

the half of the wave period. This is the same as observed in the analysis of hydraulic test 3(3), subtest

0,6 Hmax 0,874 Hz. The positive peaks of the oscillating load on the first row also take place between

the maximum run‐up of the wave on the slope and the maximum run‐down of the wave. A slight

phase shift was however observed. The occurrence of water level peaks at the wave gauge (toe) is

however not comparable to the previous analysis and is dependent on the specific test characteristics.

‐ 142 ‐

F.2 Analysis on trend of the “equilibrium” line

Various types of trend curves of the equilibrium load were distinguished. The observed trend types

are displayed in Table F‐3:

Table F‐3 Types of trend curves

A1

A2

A3 A4

B5 B6

B7

B8

C9

D10

This table gives a schematic representation of the occurring trend of the equilibrium load. In Figure

F‐7 to Figure 16 examples of the mentioned curve types are displayed:

Figure F‐7 Type 1 Figure F‐8 Type 2


‐ 143 ‐


Figure F‐13 Type 7 Figure F‐14 type 8

Figure F‐15 type 9 Figure F‐16 Type 10

The occurrence of the trend curve types during the different subtests was investigated. No relation

between the test parameters and the occurrence of a certain curve type was found. A relation was

however observed between the occurrence of trend curve type 1 and 5 and the wave height. This is

visualised in Figure F‐17 to Figure 21 which are plots of the occurrence of each wave type during a

certain wave height:

‐ 144 ‐

Figure F‐17 Occurrence of trend types H=0,2 Hmax



‐ 145 ‐



It can be observed in these plots that the subtests with a wave height of 0,2 Hmax were mostly of type

5 (flat). The subtests with a wave height of 0,4 Hmax could still be characterised as type 5 (flat) for a

significant amount of trend curve types but also for large number as of trend curve types of type 1 (of

exponential decreasing growth). The subtests with a higher wave height are dominated by trend

curve types of type 1. Recalling the statement that the increase of the equilibrium load is larger at

higher wave heights can be imagined that the flat type of curve trends (type 5) are less dominant

during tests with a higher wave height.

F.2.1 Trend of the equilibrium load

The analysis of the relation between the equilibrium load and the wave height and packing density

was done neglecting test 7 (impermeable core) and 8 (smooth, impermeable underlayer) since the

data of these tests disturb the curve fit on the other data. During Test 7 and 8 a large increase of the

equilibrium load took place due to the impermeability of the core. The plots including the data of test

7 and 8 are displayed in Figure F‐22 to Figure F‐24

‐ 146 ‐

Figure F‐22 Equilibrium load versus the wave height

Figure F‐23 All tests packing density versus wave height

‐ 147 ‐

Figure F‐24 All tests equilibrium load versus the packing density

F.2.2 Trend of the equilibrium load during tests with irregular waves

A total of three tests were executed with irregular waves. The spectra of these tests were analysed

and based on the relation for the trend of the equilibrium load relative to the wave height and the

number of waves, a calculation of the estimated end value of the equilibrium load was made.

The wave data was collected with wave gauges placed in the wave flume. This data was analysed with

the program WAVELab (2008). With this program a reflection analysis was carried out which is based

on a least square analysis whereby the incident and reflected spectra are resolved from the measured

spectra in three separate points (three separate wave gauges) [MANSARD AND FUNKE, 1980]. The result

of this analysis was the measured incoming significant wave height, maximum wave height, number

of waves and wave height distributions of the test with irregular waves. The wave height distributions

of hydraulic test 5‐1 to hydraulic test 5‐3 are visualised in Figure F‐25 to Figure F‐27:

Figure F‐25 Wave height distribution Hydraulic test (5‐1)

‐ 148 ‐



The with WAVELab determined significant wave height, maximum wave height and total number of

incoming waves are given in Table F‐4

Table F‐4 Results wave analysis wave lab


Hs (measured) 11.2 11.17 11.14 [cm]

Hmax (measured) 20.22 20.19 20 [cm]

Hmax ( based on Xbloc) 21.28 21.28 21.28 [cm]

Nr of waves 2790 2819 2709 [‐]

In paragraph 6.2.1 was derived that 90% of the final increase of the equilibrium load is reached after

92 waves. The wave height which is reached for at least 92 times during the wave test is thus of

intrest for future calculations. Instead of counting the number of waves this number of waves was

converted in a exceedence probability by dividing the 92 waves by the total number of incoming

waves:

Table F‐5 Exceedence probability corresponding with 92 waves


Exceedence probability of 92

waves (90 % of end value) 3.30% 3.26% 3.40% [‐]

‐ 149 ‐

These exceedence probabilities can be used to derive the corresponding wave height in Figure F‐25

to Figure F‐27:

Table F‐6


Corresponding (H/Hs)^2 1.66 1.655 1.635 [‐]

H(prob:90 waves) 14.43 14.37 14.24 [cm]

The derived wave heights can be used to calculate the increase of the equilibrium load with the

equation derived in paragraph 6.2.2.2:

2


unit max max

F H H

F H H

(F.1)

Since a total number of 92 waves corresponds to 90% of the, with this expression, calculated

equilibrium load the calculated value has to be multiplied by 0,9:

Table F‐7


Increase of Fequilibrium 0.79 0.79 0.77 [F/Fg]

Increase of Fequilibrium 4.81 4.76 4.65 N

The theoretical expected increase of the equilibrium load with the applied spectrum is thus about 0,8

of the weight of an armour unit.

F.3 Theoretical approximation of dynamic load

In order to verify whether the measured dynamic load was in line with the theory presented in this

report a straight forward approximation of the dynamic load based on certain measured parameters

and the wave input parameters used in the test program was made. This calculation of the dynamic

load and comparison with the measured dynamic load only refers to the measured peak‐to‐peak

amplitude and is independent of the measured increase of the dynamic load.

As stated before the drag force and inertia force and lift force can be determined with the Morrison

equation with as input the downwash velocity. The downwash velocity was calculated with the

following expression [ABBOTT AND PRICE, 1994]:

; avarage0,5

downwash

Ru RdU

T

(F.2)

This expression uses the maximum level of run‐up and run‐down (measured along the slope) and the

wave period. This equation gives the average downwash velocity which is lower than the maximum

downwash velocity. The peak of the downwash velocity is given by the following expression BATTJES

AND ROOS (1975) [ABBOTT AND PRICE, 1994]:

; max

1

; avarage

downwash

downwash

Uf

U (F.3)

Some research have been done in order to determine the value of f1(ξ). These researches were

executed on smooth slopes with a maximum slope of 1:3. The determined values for f1(ξ) found by

BATTJES AND ROOS (1975) are therefore only useful as first approximation. The values for f1(ξ) of

‐ 150 ‐

Irribaren numbers of 3, 4 and 5 are prospectively 3,6, 2,7 and 2 BATTJES AND ROOS (1975) from [ABBOTT

AND PRICE, 1994].

The run‐up was determined with the run‐up expression of VAN DER MEER AND STAM (1992) (equation

2.15). The run‐up were run‐down are however also measured during the execution of the hydraulic

tests. In this approximation of the dynamic load these measured values are used.

Based on the measurements of the run‐up and run‐down, the expression for the downwash velocity

and the Morrison equation it is possible to theoretically approximate the forces induced by the

downwash along the slope. In order to approximate the force imposed on a armour unit by the

seepage flow the seepage flow had to be determined.

In order to do this in a correct way the seepage flow should be calculated with the pressures in the

breakwater and the Forchheimer equation for all points along the interface of the under layer and the

armour layer during the complete wave period. This can be done with a program like ComFlow

[WENNEKER, 2010]. However in this first approximation a more simple approach was followed. This

approach is based on the Volume Exchange model described by JUMELET, 2010 and others.

Figure F‐28 Phreatic level in a breakwater

In the Volume Exchange model the internal water volume is schematised as a triangular shape (which

is in reality a more curved shape) which can be described by the following expression:

2int

1 1

2ernal coreV n Ru

I

(F.4)

In this equation the run‐up at the core an own parameter (Rucore) since the run‐up at the core is not

the same as the run‐up at the top of the armour layer. The outflow of water occurs during about half

of the wave period. The average seepage velocity can thus be determined by dividing the internal

water volume by the half of the wave period and the surface of the breakwater slope through which

the seepage takes place.

This approach was for this first approximation even further schematised. The internal volume is

sketched in Figure F‐28 and was calculated without determining the hydraulic gradient. The distance

between point 2 and the intersection with the armour layer on a horizontal line was determined from

visual observations of the videos made during the hydraulic test and is found to be smaller than two

times the wave height (about 1,4 times the wave height). In order to be conservative in this first

approach a length of two times the wave height was used. Furthermore the following input

parameters were used:

‐ 151 ‐

Table F‐8 Input parameters theoretical approximation dynamic load

Drag coefficient 0.35 [‐]

Lift coefficient 0.2 [‐]

Inertia coefficient 0.2 [‐]

ρw 1000 kg/m3

Crossesction

surface 0.001356 m2

f1(ξ) 2 2.7

Slope (α) 0.931596 rad

Lseepage 2H [m]

porosity (n) 0.4 [‐]

Table F‐9 gives a number of relevant test input parameters such as the wave height’s used in the test

program and the corresponding wave periods. Furthermore the measured run‐up and along slope

distance between the maximum level of run up and the maximum level of run‐down is given:

Table F‐9 Wave input parameters and measured run‐up and run‐down (averaged)

H [m] 0.044 0.044 0.044 0.088 0.088 0.088 0.132 0.132 0.132 0.176 0.176 0.176 0.22 0.22 0.22

T [s] 0.67 0.94 1.14 0.94 1.33 1.78 1.14 1.60 2.00 1.33 2.00 2.67 1.46 2.29 2.67

Rumeasured

[m] 0.061 0.065 0.067 0.112 0.149 0.167 0.196 0.256 0.285 0.293 0.360 0.383 0.379 0.398 0.412

Rudmeasured

[m] 0.077 0.094 0.103 0.159 0.234 0.294 0.288 0.404 0.453 0.439 0.585 0.549 0.554 0.716 0.583

With this parameters presented in Table F‐8 and the earlier stated expression for the downwash

velocity and the Morrison equations it is possible to calculate the downwash velocity and the velocity

induced forces on the armour unit:

Table F‐10 Calculation of downwash and downwash induced forces on a armour unit

Udownwash; avarage [m/s] 0.23 0.20 0.18 0.34 0.35 0.33 0.50 0.50 0.45 0.66 0.58 0.41 0.76 0.63 0.44

Udownwash; peak [m/s] 0.46 0.54 0.65 0.68 0.94 1.19 1.01 1.36 1.63 1.32 1.58 1.48 1.52 1.69 1.57

Fdrag [N] 0.22 0.26 0.31 0.32 0.45 0.56 0.48 0.65 0.77 0.62 0.75 0.70 0.72 0.80 0.75

Flift [N] 0.13 0.15 0.18 0.18 0.26 0.32 0.27 0.37 0.44 0.36 0.43 0.40 0.41 0.46 0.43

The above described approach was applied in order to determine the internal volume. This volume

flows out of the breakwater during a half wave period through a surface which was approximated as

the surface between the still water level and the maximum level of run‐down. The seepage velocity

was determined by dividing the internal velocity by the time of outflow and the outflow surface. With

this seepage velocity the force on the armour unit was determined.

Table F‐11 Calculation of seepage flow and seepage induced forces on a armour unit

Vinternal [M*M] 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.04 0.05 0.04

Useepage;perpendiculair 0.13 0.09 0.07 0.18 0.13 0.10 0.22 0.16 0.13 0.25 0.17 0.13 0.29 0.19 0.16

Useepage; along slope 0.17 0.12 0.10 0.24 0.17 0.13 0.30 0.21 0.17 0.34 0.23 0.17 0.39 0.25 0.21

Fdrag, seepage [N] 0.08 0.06 0.05 0.11 0.08 0.06 0.14 0.10 0.08 0.16 0.11 0.08 0.19 0.12 0.10

Fuplift, seepage [N] 0.06 0.04 0.04 0.09 0.06 0.05 0.11 0.08 0.06 0.12 0.08 0.06 0.14 0.09 0.08

‐ 152 ‐

The determined forces are forces in along slope direction and forces in upward direction,

perpendicular to the slope. These last forces reduce the normal force between the armour unit and

the under layer and leads in this way to a higher along slope force. The upward directed forces,

perpendicular to the slope were multiplied with the friction force and the in this way caused

reduction of the friction force was added to the along slope dynamic force:

Table F‐12 Total downwash induced dynamic force and total seepage induce dynamic force

Fdue downwash 0.29 0.34 0.41 0.43 0.60 0.76 0.64 0.87 1.04 0.84 1.01 0.94 0.97 1.08 1.00

Fdue. seepage 0.12 0.08 0.07 0.17 0.12 0.09 0.21 0.15 0.12 0.23 0.16 0.12 0.27 0.17 0.15

The total dynamic force was obtained by adding the seepage induced dynamic force tot the

downwash induced dynamic force. In the second row the total force relative to the unit weight is

given:

Table F‐13 Total theoretical dynamic force

Falong slope total 0.41 0.43 0.48 0.60 0.72 0.84 0.85 1.01 1.16 1.07 1.16 1.06 1.24 1.25 1.15

Falong slope total /Fg 0.07 0.07 0.08 0.10 0.12 0.14 0.14 0.17 0.19 0.18 0.19 0.18 0.21 0.21 0.19

In Table F‐14 an comparison between the measured peak‐to‐peak amplitude of the dynamic load and

the theoretical derived is made. Add the first row the measured peak‐to‐peak amplitude is given and

on the second row of the table the number of times that the measured peak‐to‐peak amplitude

transcends the approximated dynamic load is given. This number is equal to the number of rows

which apparently influence the dynamic load on the first row.

Table F‐14 Comparison between measured peak‐to‐peak dynamic forces and calculated forces

Measured amplitude 0.04 0.06 0.06 0.11 0.09 0.14 0.65 0.33 0.42 1.82 0.98 0.95 2.49 1.31 1.34

No of rows 1 1 1 1 1 1 5 2 2 10 5 5 12 6 7

Based on these results it can be concluded that the seepage flow has a minor share in the total

dynamic load on the first row of armour units. Furthermore it can be concluded that since the total

measured load on the first row of armour units is just a few times larger than the dynamic load

imposed on the first row by a single armour unit, only a limited number of rows have a share in the

total dynamic load on the first row of armour units.

static and dynamic loads on the first row of single layer

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