static and dynamic loads on the first row of single layer
TRANSCRIPT
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
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).
Static and dynamic loads on the first row of interlocking, single layer, armour units
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
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 [‐]
2. Coefficient in the run‐up formula of VAN DER MEER AND STAM [‐]
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]
2. Coefficient in the run‐up formula of VAN DER MEER AND STAM [‐]
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]
Static and dynamic loads on the first row of interlocking, single layer, armour units
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]
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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]
W50 Rock weight that is exceeded by 50% of the rocks in the under layer [kg]
W15 Rock weight that is exceeded by 85% 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]
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, 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)
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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)
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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].
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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)
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
ffreq (Hz) 1,06 ffreq (Hz) 0,75 ffreq (Hz) 0,624 ffreq (Hz) 0,500 ffreq (Hz) 0,437
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
ffreq (Hz) 0,87 ffreq (Hz) 0,56 ffreq (Hz) 0,500 ffreq (Hz) 0,375 ffreq (Hz) 0,375
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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Results and analysis of static tests
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Results and analysis of static tests
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Results and analysis of static tests
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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:
Results and analysis of static tests
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Results and analysis of static tests
‐ 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
peak to peakA =0,33 0,052max max
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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 4 (20 rows RPD =0.97) +0,63 ‐0,04 0,953
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)
Results and analysis of static tests
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Results and analysis of static tests
‐ 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
peak to peakA =0,04 0,20max max
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
peak to peakA =0,11 0,19max max
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)
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Test series Coefficient a Coefficient b Norm of residuals
Hydraulic test 1 (15 rows) +0,78 ‐0,18 0,194
Hydraulic test 1_2 (15 rows) +0,65 ‐0,13 0,193
Hydraulic test 2 (20 rows, RPD =1) +0,23 ‐0,07 0,435
Hydraulic test 3 (20 rows RPD =1.04) +0,40 ‐0,05 0,265
Hydraulic test 4 (20 rows RPD =0.97) +0,66 ‐0,12 1,071
Hydraulic test 7 (impermeable core) +0,07 +0,07 0,182
Hydraulic test 8 (impermeable core,
smooth under layer)
+0,02 +0,08 0,254
Hydraulic test 9 (25 rows) +0,05 +0,04 0,131
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)).
Results and analysis of static tests
‐ 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)
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Results and analysis of static tests
‐ 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
Test series Coefficient a Coefficient b Norm of residuals
Hydraulic test 1 (15 rows) +0,19 +0,08 0,417
Hydraulic test 1_2 (15 rows) +0,36 ‐0,02 0,309
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 57 ‐
Hydraulic test 2 (20 rows, RPD =1) +1,05 ‐0,60 0,987
Hydraulic test 3 (20 rows RPD =1.04) +0,34 +0,01 0,340
Hydraulic test 4 (20 rows RPD =0.97) +0,80 ‐0,24 0,957
Hydraulic test 7 (impermeable core) +0,04 +0,10 0,275
Hydraulic test 8 (impermeable core,
smooth under layer)
+0,06 +0,06 0,254
Hydraulic test 9 (25 rows) +0,14 +0,02 0,287
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
Results and analysis of static tests
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Results and analysis of static tests
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
equilibrium end end start
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
Results and analysis of static tests
‐ 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 2 “20 rows” 15,7 2,60 1,00
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 9 “25 rows” 14,0 2,31 1,09
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Results and analysis of static tests
‐ 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
=3,1 0,80equilibrium
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
Test series Coefficient a Coefficient b Norm of residuals
Hydraulic test 1 (15 rows) +3,0 ‐1,1 1,337
Hydraulic test 2 (20 rows, RPD =1) +3,0 ‐0,43 2,400
Hydraulic test 3 (20 rows RPD =1.04) +2,5 ‐1,2 1,308
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
Hydraulic test 8 (impermeable core,
smooth under layer)
+0,12 +6,1 2,570
Hydraulic test 9 (25 rows) +3,8 ‐1,57 1,498
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)
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Results and analysis of static tests
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Test series Coefficient a Coefficient b Norm of residuals
Hydraulic test 1 (15 rows) +75,8 ‐5,8 1,368
Hydraulic test 2 (20 rows, RPD =1) +78,2 ‐2,9 2,404
Hydraulic test 3 (20 rows RPD =1.04) +62,4 ‐5,5 0,962
Hydraulic test 4 (20 rows RPD =0.97) +61,6 ‐2,1 0,995
Hydraulic test 6 (20 rows, continuous) +61,2 ‐0,5 1,843
Hydraulic test 7 (impermeable core) +72,0 ‐0,1 2,549
Hydraulic test 8 (impermeable core,
smooth under layer)
+4,0 +14,6 2,449
Hydraulic test 9 (25 rows) +95,1 ‐8,3 1,625
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.
Results and analysis of static tests
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Results and analysis of static tests
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Results and analysis of static tests
‐ 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
Hydraulic test 5(1) Hydraulic test 5(2) Hydraulic test 5(3)
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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
row of armour units.
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
equilibrium end end start
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
=3,1 0,80equilibrium
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
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
row of armour units.
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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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.
Static and dynamic loads on the first row of interlocking, single layer, armour units
‐ 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
Figure D‐2 Measured load record of hydraulic test 2
‐ 114 ‐
Figure D‐3 Measured load record of hydraulic test 3
Figure D‐4 Measured load record of hydraulic test 4
‐ 115 ‐
Figure D‐5 Measured load record of hydraulic test 5
Figure D‐6 Measured load record of hydraulic test 6
‐ 116 ‐
Figure D‐7 Measured load record of hydraulic test 6_2 (directly after test 6 without rebuilding the
slope)
Figure D‐8 Measured load record of hydraulic test 7
‐ 117 ‐
Figure D‐9 Measured load record of hydraulic test 8
Figure D‐10 Measured load record of hydraulic test 9
‐ 118 ‐
Figure D‐11 Measured load record of hydraulic test 10
‐ 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 ‐
Table D‐4 Results of hydraulic tests: 0,2 Hmax 1,06 Hz
Table D‐5 Results of hydraulic tests: 0,2 Hmax 0,874 Hz
‐ 121 ‐
Table D‐6 Results of hydraulic tests: 0,4 Hmax 1,06 Hz
Table D‐7 Results of hydraulic tests: 0,4 Hmax 0,749 Hz
‐ 122 ‐
Table D‐8 Results of hydraulic tests: 0,4 Hmax 0,562 Hz
Table D‐9 Results of hydraulic tests: 0,6 Hmax 0,874 Hz
‐ 123 ‐
Table D‐10 Results of hydraulic tests: 0,6 Hmax 0,624 Hz
Table D‐11 Results of hydraulic tests: 0,6 Hmax 0500 Hz
‐ 124 ‐
Table D‐12 Results of hydraulic tests: 0,8 Hmax 0,749 Hz
Table D‐13 Results of hydraulic tests: 0,8 Hmax 0,500 Hz
‐ 125 ‐
Table D‐14 Results of hydraulic tests: 0,8 Hmax 0,375 Hz
Table D‐15 Results of hydraulic tests: 1,0 Hmax 0,687 Hz
‐ 126 ‐
Table D‐16 Results of hydraulic tests: 1,0 Hmax 0,437 Hz
Table D‐17 Results of hydraulic tests: 1,0 Hmax 0,375 Hz
‐ 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
...
...
...
Fres1 = X1(i)+Fres2;
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
...
...
...
Fres1 = X1(i)+Fres2;
if Fres1<0;
Fres1 = 0;
end
F19(i,:)=Fres1;
% this is repeated for a slope of 18, 17, 16 etc rows
...
‐ 132 ‐
...
...
%Calculation of total residual force row 1 (which is now the highest row)
%Calculation of total residual force row 1 (which is now the highest row)
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
Wave gauge min Run‐down max
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
Wave gauge min Run‐down max
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
Figure F‐9 Type 3 Figure F‐10 Type 4
‐ 143 ‐
Figure F‐11 Type 5 Figure F‐12 Type 6
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
Figure F‐18 Occurrence of trend types H=0,4 Hmax
Figure F‐19 Occurrence of trend types H=0,6 Hmax
‐ 145 ‐
Figure F‐20 Occurrence of trend types H=0,8 Hmax
Figure F‐21 Occurrence of trend types H=1,0 Hmax
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 ‐
Figure F‐26 Wave height distribution Hydraulic test (5‐2)
Figure F‐27 Wave height distribution Hydraulic test (5‐3)
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
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]
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
Hydraulic test 5(1) Hydraulic test 5(2) Hydraulic test 5(3)
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
Hydraulic test 5(1) Hydraulic test 5(2) Hydraulic test 5(3)
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
=3,1 0,80equilibrium
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
Hydraulic test 5(1) Hydraulic test 5(2) Hydraulic test 5(3)
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.