AMERICAN
SOCIETY
OF
CIVIL
ENGINEERS
Founded
November
5, 1852
TRANSACTIONS
Paper
No. 3213
LABORATORY lNVESTIGATION
OF
RUBBLE-MOUND BREAKWATERS
Robert
Y.
Hudson,l
F.
ASeE
With
Discussion
by
Messrs
Jose
Reis d
William H. Booth,
Jr.;
F~ancis
B
Sli:h~a~valho
and
Daniel Vera-Cruz;
Lillevang; Thorndike
Savill
~
J . erd'
LReland
B.
Jones; Omar J.
,
r.,
an obert
Y.
Hudson.
SYNOPSIS
This
paper
reports
on a
lab
t . .
States
Army
Engineer
waterway~r~
ory
~nvesbgati?n
co~ducted
at
the
United
to
determine
criteria
for
the
deSignX~~~lment
Stat~on,
VlCksburg,
Mississippi,
waters.
Small-scale
breakwater
section~,:strucbon
of
rubble-mound
break-
wave
flume
119
ft
long, 5
ft
wide
4
ft
d
re
hand-con.structed
m a
concrete
generated
waves
to
determine
the
stab·l·t
ee
f
P
,
and
subjected
to
mechanically
All
1 Y 0
the
armor
units
genera
stability
equation
has
b'
.
experimental
program
and
correlate
eten
denved
and
is
being
used
to
guide the
i~portant
unknown
functions
in
the
gen~~!~~~:~\\
From
t~e
test
data
obtained
mmed
for
se
l
ected
breakwater
andtest
-w
ll.y.equabon
have
been
deter-
stability
formula
has
been
obtained.
ave
condlbons,
and
a new
breakwater
In
conjunction
with
the
stabilit
t t
each
breakwater
section
and
wav:
coe~.~~
wave
run-up
data
are
obtained
for
obtained
that
enable
the
thickness
an~
1
lon
t.ested.
Also,
measur
'
ements
are
dlfferent
types
of
armor
units
to
be
d t
por
.o
slty
of
cover
layers
composed
of
Th
e
ermmed
e new
stability
formula
and
the
.'
provided
essential
information
f
e.xpenmental
data
obtained
so
far
have
mound
breakwaters
with
protect~:e
a;O~~~rl~ved
method
of
designing
rubble-
and
tetrapod
armor
units
Test
.
yers
composed
of
quarry-stone
data
for
other
special
shapes
~f
m
p;ogress
(1959)
to
obtain
experimental
cas
-concrete
armor
units
(cubes,
tetra-
Note. Published essentially as printed
here
.
t~e
Waterways and Harbors Division
as
P
'd.
m
September,
1959,
in the Journal
of
glven a
re
those
in
effect
when
the
p~per
ro~.ee
mgs
Paper
2171.
Positions and titles
Transactions.
or
lscussion
was
approved for publication in
1 Hydr. Engr., Chi
of
Wave
Act' S t
Sta., Vicksburg, Miss.'
lOn
ec.,
U.
S.
Army Engr. Waterways Experiment
610
BREAKWATERS
611
hedrons,
and
tribars)
should
increase
considerably
the
accuracy
of
rubble-
mound
breakwater
design.
lNTRODUCTION
Small-scale
tests
of
rubble
- mound
breakwaters
have
been
in
progress
at
the
U.
S.
Army
Engineer
Waterways
Experiment
Station,
Vicksburg,
Missis-
sippi,
almost
continuous
ly
since
1942.
During
the
period
from
1942
to
1950,
variouS
phases
of
rubble-mound
breakwater
construction
were
investigated
for
the
Bureau
of
Yards
and
Docks,
Department
of
the
Navy.
The
most
im-
portant
findings
of
that
investigation
concerned
the
accuracy
of
Iribarren
' s
formula
(1), (2), (3).2
It
was
concluded
(4)
that
the
Iribarren
formula
can
be
used
for
the
design
of
rubble-mound
breakwaters
only
if
experimental
coef-
ficients,
of
the
kind
developed
during
the
investigation
conducted
for
the
Bureau
of
Yards
and
Docks,
are
available
for
the
complete
range
of
variables
encountered
in
the
design
of
full-
scale
structures
.
In
1951, a
comprehensive
investigation
of
rubble-mound
breakwaters
(5)
was
begun
at
the
Waterways
Experiment
Station
for
the
Office,
Chief
of
Engineers,
U.
S.
Army.
This
investigation
(in
progress
in
1959)
is
similar
to
the
study
conducted
for
the
Bureau
of
Yards
and
Docks
except
that
it
is
larger
in
scope;
it
includes
the
necessary
range
of
important
variables
that
affect
the
stability
of
rubble-mound
breakwaters.
To
insur
e
optimum
designs
for
breakwaters,
design
engineers
should
have
accurate
information
concerning
the
required
weight
for
the
individual
armor
units
in
the
protective
cover
layer,
along
the
length
of
the
structure,
as
a
function of;
(a)
shape
of
unit,
(b)
specific
weight
of
unit,
(c)
specific
weight
of
water
in
which
the
structure
will
be
situated,
(d)
beach
slope
seaward
of
the
breakwater,
(e)
dimensions
of
waves
at
the
location
of
the
proposed
struc-
ture,
(f)
seaside
slope
of
breakwater,
(g)
porosity
of
protective
cover
layer,
(h)
thickness
of
cover
layer,
and
(i)
porosity
and
thickness
of
underlayers
on
which
the
armor
units
are
to
be
placed.
In addition,
design
engineers
should
be
able
to
determine
quantitatively;
(a)
the
height
of
breakwater
above
still-
water
level
necessary
to
prevent
excessive
overtopping
by
wave
run-up,
(b)
the
depths
below
still-water
level
to
which
the
cover
layer
should
extend,
(c)
the
amount
of
damage
that
will
be
inflicted
on
a
breakwater
section
not
de-
signed
for
overtopping
when
waves
higher
than
the
selected
design
wave
occur,
and
(d)
the
best
design
of
back
slopes
for
preventing
failure
when
overtopping
of
the
breakwater
is
permitted.
Information
should
also
be
availab
le
for
de-
signing
the
seaward
end,
or
head,
of
the
breakwater.
The
test
program
under
discussion
includes
tests
to
provide
the
design
data
and
quantitative
infor.mation
that
has
been
outlined.
However,
tests
de-
scribed
in
this
paper
are
concerned,
for
the
most
part,
with
the
types
of
rubble-mound
breakwaters
in
which
that
part
of
the
breakwater
section
sub-
jected
to
the
most
intense
wave
action
is
composed
of a
pile
of
quarry
stone
armor
units
placed
pell-mell,
and
those
in
which
the
protective
cover
layers
are
composed
of two
layers
of
cast
-
concrete
armor
units
placed
pell-mell
over
one
or
two
quarry-stone
underlayers.
2 Numerals in
parentheses-thus,
(I)-refer
to corresponding items
in
the Bibli-
ography-see
Appendix
1.
612
BREAKWATERS
After
the
comprehensive
investigation
was
begun,
it
was
found
that
the
Iribarren
formula
has
limitations
that
render
it
unsatisfactory
for
use
in
correlating
stability
data
from
tests
of
small-scale
rubble-mound
breakwaters.
Thus,
it
was
necessary
to
reanalyze
the
phenomenon
that
results
when waves
attack
a
rubble-mound
breakwater
in
order
to
develop
a
more
general
sta-
bility
equation.
This
paper
describes
the
apparatus
and
testing
te
c
hniques
used
in the
laboratory
investigation
,
explains
why
it
was
cons
i
dered
necessary
to
abandon
the
.
use
of
Iribarren
' s
formula
in
correlating
test
data,
and
presents
the
deri
-
vation
of a
more
general
stability
equation
that,
with
the
experimental
data
obtained
to
date,
was
used
to
develop
a
simple
formula
for
the
weight
of
armor
units
necessary
to
insure
the
stability
of
rubble-mound
breakwaters.
Informa
-
tion
concerning
wave
run-up,
and
the
thickness
and
porosity
of
cover
layer
materials
is
also
presented.
For
this
paper,
a
rubble
-
mound
breakwater
is
considered
to
be
one con-
structed
with
a
core
of
quarry-run
stones,
sand,
slag,
or
other
suitable
ma-
terials,
protected
from
wave
action
by
one
or
more
stone
underlayers
and a
cover
layer
of
relatively
large,
selected
quarry
stones
or
specially-shaped
concrete
armor
units.
Notat
i
on.-The
letter
symbols
adopted
for
use
in
th
is
paper
are
defined
where
they
first
appear,
in
the
illustrations
or
in
the
text,
and
are
arranged
alphabetically,
for
convenience
of
reference,
in
Appendix II.
DISCUSSION
OF
IRIBARREN'S FORMULA
Iribarren
's
original
formula
for
the
weight
of a
rmor
units
in
rubble-mound
breakwaters,
in
its
general
form,
revised
(6)
to
make
it
dimensionally
homo-
geneous,
and
·
retaining
the
coefficient
of
friction
as
a
variable,
reduces
to
W =
r
K'
Y
r
J.l3
H3
.....
(1)
in
which W r
is
the
weight
of
individual
armor
units,
Y r
is
the
specific
weight
of
the
armor
units
,
Sr
is
the
specific
gravity
of
the
armor
units
relative
to
the
water
in
which
the
breakwater
is
situated
(Sr = Y r / Y w), '
/1
is
the
effective
coefficient
of
friction
between
armor
units,
H
is
the
hei
g
ht
of
wave
attacking
the
breakwater,
CI
is
the
angle,
measured
from
the
horizontal,
of
the
exposed
breakwater
slope,
and
K'
is
an
experimentally
determined
coefficient.
The
accuracy
of
this
-
formula
was
discussed
by
Hudson
and
Jackson
(4), and
Hudson
(6)
in
1953. At
that
time
it
was
concluded
that
the
Iribarren
formula
could
be
used
to
correlate
the
test
data,
and
that
it
could
be
made
sufficiently
accurate
for
use
in
designing
full-scale
rubble-mound
breakwaters,
if
suffi-
cient
test
data
were
available
to
evaluate
the
experimental
coefficient
(K').
After
the
comprehensive
testing
program
was
begun
,
and
shortly
after
the
conclusions
concerning
the
adequacy
of
Iribarren
' s
formula
were
published
,
preparations
were
initiated
for
tests
to
determin
e
the
stability
of
armor
units
as
a
function
of
armor-unit
shape
.
These
included
a
study
to
establish
the
values
of
the
friction
coefficient
(/1)
that
should
be
used
for
the
variOUS
shapes
of
armor
units
in
the
experimentaldet
e
rminationofK'
in
Iribarren
' s
formula.
The
first
armor
units
of
special
shape
for
which
friction
coefficients
were
BREAKWATERS
613
measured
were
cubes
and
tetrapods.
Tetrapod
is
the
name
of a
patented
armor
unit
of
special
shape
that
was
developed
at
the
Laboratoire
Dauphinois
d'Hydraulique
Ets
.
Neyrpic,
Grenob
le,
France
(7).
The
tests
showed
that
the
friction
coefficient
in
Iribarren
' s
formula,
as
measured
by
the
tangent
of
the
angle
of
repose
(cP)
,
varied
appreciably
with
the
shape
of
armor
unit
and
the
method
of
placing
these
units
in
the
cover
layer.
These
results
led
to
the
realization
that
the
experimental
coefficient
(K')
in
Iribarren's
formula
could
not
be
determined
accurately
from
small-scale
breakwater
stability
tests
unless
accurate
comparative
values
of
the
friction
coefficient
could
be
ob-
tained
for
the
different
shapes
of
armor
units.
This
realization
was
made
more
acute
by
the
fact
that
Iribarren's
force
diagram,
from
which
his
basic
stability
equation
was
derived,
is
predicated
on
the
assumption
that
the
fric-
tion
between
armor
units,
specifically
that
component
of
the
friction
force
parallel
to
the
breakwater
slope,
is
the
primary
force
that
resists
the
forces
of
wave
action
and
determines
the
stability
of
the
armor
units.
Results. of
coefficient-of
-
friction
determinations
for
three
sizes
of
quarry
stones,
and
for
concrete
cubes
and
tetrapods
are
shown
in
Table
1.
Fig.
1
shows
the
shapes
of
these
armor
units
. About
seventy
repeat
tests
of
the
TABLE I.-FRICTION COEFFICIENTS
OF
ARMOR
UNITS
Method of
Quarry Stone
Concrete
Concrete
No.
Measurement
Cubes
Tetrapods
Wr
= 0.10 lb
Wr
= 0.30 lb
Wr
= 0.62
lb
Wr = 0.80 lb
Wr
= 0.211b
(1)
(2)
(3)
(4)
(5)
(6)
C2
Dumped
in
water
1.02
0.98
1.13
1.20
1.10
ffi
Dumped
in
air
0.79
0.90
0.87
1.34
---
Stacked
in
water.
1.09
1.19
1.26
1.36
1.78
0
Stacked
in
air
0.97
1.12
1.22
1.75
---
Avg
(all meth-
ods)
0.
97
1.05
1.12
1.41
---
Avg
(CD
and@
1.06
1.
09
1.20
1.28
1.44
O.30-lb,
quarry-stone
armor
units
were
conducted
to
determine
the
range
of
J.l
for
units
of
this
type.
It
was
found
that
/1
varied
from
a low of
0.78
to
a
high of 1.28,
with
an
average
value
of 0.98.
Thus,
/1
varies
not only witlr
armor-unit
shape
and
method
of plaCing,
but
it
also
varies
considerably
from
test
to
test
for
the
same
armor
unit.
The
curves
of
Fig.
2
were
prepared
using
the
modified
Iribarren
formula
(Eq. 1),
and
show
the
effects
of
varia-
tions
in
the
measured
value
of
/1
on
the
computed
values
of
K'.
Because
W
is
directly
proportional
to
K',
variations
in
!1
have
the
same
effect
on
com:
puted
values
of
Wr
as
they
do
on
K'.
It
can
be
seen
that
for
steep
breakwater
slopes,
small
variations
in
the
measured
value
of
/1
cause
large
variations
in
the
computed
values
of
K'
and
W
r
.
This
becomes
more
significant
when
it
is
recalled
that
the
use
of -
concrete
armor
units
of
special
shape
is
more
apt
to
be
economically
feasible
only
for
the
steeper
breakwater
slopes
.
614
BREAKWATERS
FIG
.
l.-TYPES
OF
ARMOR UNITS
FOR
WHICH
FRICTION
COEFFICIENTS
WERE
DETERMINED
I.O~
-
\~./
--
. -
--
..
./
:,../'/,
f-
___
----_
'i"
r
,0
//,'i"~'--:'qo'
./"
---
/
..
././
'i"
o~o
I.·j
/
/
..
Ii
/ j /
/'
-
/1//
/ /
/
EXAMP
L.e:
ASSUME
p.
=
1.0
AS
S
UM
E
J1
=
1.2
-
('/
/ / / /
ACTUAL
Jl
=
1.2
ACT
UA
L
J1
=
1.0
COT
a
"i.
CHAN
G E
COT
a
%
CHANGE
IN
Wr
IN
Wr
-
-
1.25
7.
1.2
5
360
-
I I I I
1.50
65
1.50
,.2
2,00
41
2.
00
6.
-
I ; I
3.00
18
3.
00
22
I
4.00
14
4.0
0
17
I I I I
COT a
FIG
. 2.
-VARIATIO
NS
OF
K'
WITH
11
IN
THE
MODIFIED
IRIBARRE
N
FORMULA
BREAKWATERS 615
Bas
ed
on
the
results
of
the
tests
to de
termine
friction
coefficients,
corre-
lation
of
test
data
by
the
use
of
Iribarren
's
formula
was
abandoned,
and
a new
stability
equation,
similar
to
the
Iribarren
formula
but
capable
of
more
ge
n-
eral
application,
was
derived
.
ANALYTICAL BASES
OF
STABILITY EQUATION
When
short-period
wind
waves
impinge
on
a
pervious
rubble-mound
breakwater,
the
resulting
int
erp
lay
of
forces
de
ve
l
oped
by
the
wave-induced
water
motion
and
the
resisting
action
of
the
armor
units
in
the
cover
layer
is
extremely
complex,
and
attempts
to
describe
the
phenomenon
quantitatively
by
rigorous
theoretical
analyses
have
not,
as
yet,
been
successful.
Waves
at
a
breakwater
may
break
completely,
projecting
a
jet
of
water
approximately
perpendicular
to
the
slope,
break
partially
with
apoorly
defined
jet,
or
estab
-
lish
an
oscillatory
motion
of
the
water
particles
along
the
breakwater
slope
similar
· to
the
motion
of a
clapotis
at
a
vertical
wall.
Characteristics
of
the
motion
of
water
particles
when
short
-
period
w
ind
waves
encounter
a
rubble
-
mound
breakwater
are
determined
by
the
wave
steepness
(H
i A),
the
relative
depth
(d
/ A),
the
relative
height
(H
/
d)
,
the
depth
of
water
at
the
toe
of
the
breakwater
slope
(d),
the
ang
le of
the
beach
slope
seaward
of
the
breakwater
(a),
angle
of
seas
id
e
slope
of
the
breakwater
with
the
horizontal
(a),
the
angle of
obliquity
of
the
a
tt
ack
ing
waves
(j3),
and
the
shape
.,
thickness,
and
porOSity of
the
cover
layer
and
underla
yer
materials
(.11,
r,
and
P,
respec
-
tively) .
Th
e
ability
of an
armor
unit
in
the
cover
la
yer
to
resist
the
forces
ca
used
by wave
action
is
determin
ed
by
the·
buoyant
weight
of
the
armor
unit
(W~),
the
position
of
the
unit
relative
to
the
still-water
l
evel
(z),
the
angle
of
seaside
slope
(a), .
the
he
i
ght
of
breakwater
crown
above
still-water
level
(h),
the
width of
breakwater
crown
(m),
the
shape
of
unit
(.11),
porOSity
of
the
armor
units
in
place
(P),
thickness
of
the
cover
layer
(r),
the
porosities
and
thick-
nesses
of
the
underl
ayers,
and
the
method
of
placing
the
breakwater
material,
especially
the
armor
units
in
the
cover
layer
(dumped
pell
-
mell,
placed
in
some
orderly
manner
to
obtain
wedging
action,
or
stacked
without wedging
action).
Short-period
wind
waves
incident
on
a
rubble
-m
ound
breakwater
develop
dynamic
forces
that
tend
to
lift
and
roll
the
armor
units
from
the
breakwater
slope
.
These
forces
consist
of a
drag
force
1 2
Yw
2
Fd
=
2"
Cd
ka 1 g V
............
. .
..
. (2)
and
an
inertia
force
F = C k 1
3
Y
w
av
( )
m
mv
gat
· ·
··
·······
·····
·
3
in
which Cd
is
a
drag
coefficient,
C
m
is
a
virtual-mass
coefficient,
1
is
a
characteristic
linear
dimension
of
the
unit
such
that
the
projected
area
of
the
unit
perpendicular
to
the
ve
l
ocity
is
ka
1
2
,
and
the
volume
of
the
unit
is
kv 1
3
,
Y
w
is
the
specific
weight of
the
water
in
which
the
breakwater
is
to
be
situ
-
ated,
g
is
acceleration
due to
grav
ity
,
and
.V
is
the
velocity
of
the
water
flowing
around
or
impinging
on
the
armor
units
in
the
cover
layer.
Because
of
the
difficulties
inherent
in
an
attempt
to
evaluate
the
separate
sets
of coeffiCients,
616
BREAKWATERS
Cd
ka
and
C
m
kv,
that
would
involve
either
direct
measurement
or
a
d~rived
expression
of
the
acceleration
(oY / 0 t)
in
terms
of
the
wave
charactenshcs,
and
in
order
to
simplify
the
force
equation
used
to
correlate
test
data,
the
effects
of
acceleration
are
combined
with
the
drag
force.
The
resulting
equa
-
tion
is
F = C
12
Y
w
y2
q q g
...
..
...
(4)
1
oy
in
which
C
q
,
the
total
coefficient,
is a
function
of
the
terms
y2
at'
Cd
ka
'
and
C
m
kv'
..
.
The
velocity
of
the
water
jet
resultmg
from
a
breakmg
wave
(Y
b
) is
equal
to
the
particle
velocity
at
the
wave
crest
that,
at
the
instant
of
breaking,
is
equal
to
the
celerity
of
the
wave
form.
Thus,
for
shallow-water
waves,
as
d/A
-O,
..........
.
.....
...
(5)
Also
at
breaking,
Hb = k db,
in
which
k = f(H/ A).
Therefore
,
by
substitution
,
Y
b
2
=
~
Hb
...
..
. .
....
.
..
....
.
(6)
Substituting
this
value
of
velocity
in
Eq
. 4,
the
expression
for
th
e
force
exerted
on
an
armor
unit
by
a
breaking
wave,
in
terms
of
wave
height,
is
_ 2 Y
w
( )
F q - C
q
1 k Hb . . . . . . . . . . . . . . .
..
7
For
breakwaters
constructed
by
dumping
or
by
placing
armor
units
es
-
sentially
pell-mell
,
the
forces
resisting
displacement
are
the
buoyant
weight
of
the
individual
units
and
the
friction
between
units
.
Except
for
isolated
in-
stances
in
which
wedging
action
is
involved,
friction
between
armor
units
can
be
neglected,
and
the
principal
resisting
force
for
pell-mell-constructed
cover
layers
can
be
assumed
to
be
W~
= kv
l\Yr
- Y
w
)
.................
(8)
in
which
Y
r
is
the
specific
weight
of
the
armor
units
.
For
inCipient
instability
of
armor
units
in
a
rubble-mound
breakwater,
or
fill
slope,
subjected
to
breaking
waves,
W~
= F
q
,
or
kv
I\Y
r
- Y
w
}
= C
q
12
Y;
Hb
..
.
..
.
...
..
. (9)
Letting
Sr
= Yr/ Y
w
'
and
substituting
in
Eq.
9.
ky
l(Sr
_
1)
= C
q
:
b
or
Hb k
(ky)
1
(Sr
-
1)
=--cq
. . . . . .
... ...
.. ..
(10)
The
weight
of
an
armor
unit
in
air
is
Wr
=
ky
1
3
Y
r'
or
1 =
(ky
W;r)
1/ 3
........
.
.....
. . . . (11)
BREAKWATERS
617
Substituting
this
value
of
1
in
Eq.
10,
Y
1/ 3 H
r b
k
(ky)2
/3
.............
(12)
in
which
k (kyi/
3
C
q
The
forces
that
tend
to
displace
armor
units
from
breakwater
slopes
when
the
waves
do
not
break,
or
break
only
partially,
are
not
the
same
as
those
forces
that
result
from
breaking
waves,
nor
do
they
act
in
the
same
directions.
However
the
order
of
magnitude
of
the
nonbreaking
wave
forces,
and
the
~ffects
df
these
forces
on
the
stability
of
rubble-mound
breakwaters,
should
be
approximately
the
same
as
those
caused
by
breaking
waves
.
It
is
believed,
therefor:e
,
that
Eq. 12
adequate
ly
represents,
at
least
in
the
first
approxima-
tion
the
major
forces
of
both
breaking
and
nonbreaking
waves.
Thus,
for
both
typ~s
of
short-period
wave
mot
ion
s
on
rubble-mound
breakwaters,
and
intro-
ducing
those
variables
that
were
not
included
in
the
derivation
of
Eq.
12,
the
most
general
equation
used
in
this
investigation
to
guide
the
testing
program
and
correlate
test
data
is
_
('
Cd'
em'
ka' kv'
~2
:~,
H/
A,
d/ >")
y
r
l/
3
H
-,--=-:"--:;-",
= f
H/
d, d,
a,
P,
r,
h, m, z,
{3,
and
(
Sr
- 1) w
r
1
/
3
the
method
of
placing
armor
units
..
:'
.(13)
In
Eq.
13,
Cd
and
C
m
are
functions
of
,1
and
the
Reynolds
number
(R),
and
ka
and
kv
are
functions
of ,1.
The
term
~2
~
~,
that
is
a
form
of
Iversen's
modulus
for
accelerated
motion
(8),
is
omitted
from
the
list
of
variables
tested
in
this
investigation
because
of
the
difficulty
of
obtaining
accurate
veloc
it
y-
time
histories
of
the
flow
around
individual
armor
units.
.
In
the
first
phase
of
this
testing
program,
the
upper
portion
of
the
small-
scale
breakwaters
was
constructed
of
rocks
Simulating
quarry
stones,
all
pieces
of
which
were
of
nearly
the
same
weight
,
specific
weight,
and
shape.
In
addition
the
crown
width
of
the
breakwater
test
sections
was
standardized
at
three
ti~es
the
average
diameter
of
the
armor
units;
the
angle
9f
obliquity
of
the
test
waves
was
0°
,
and
the
cover
layer
was
extended
to
a
depth
below
still-water
level
sufficient
to
insure
that
the
stability
of
the
structure
would
not
be
influenced
by
the
stones
used
in
the
lower
portion
of
the
test
section.
For
those
tests
in
which
the
no-damage
criterion
was
used
in
the
selection
of
design
-
wave
heights,
the
c
rown
heights
above
still-water
level
were
sufficient
to
prevent
overtopping
by
the
test
waves.
For
those
tests
in
which
the
wave
heights
used
were
greater
than
the
previously
selected
design-wave
heights,
the
crown
heights
above
still-water
level,
and
the
depths
to
which
the
cover
l
ayers
extended
below
still-water
level,
were
equal
to
the
previously
selected
design-wave
heights.
For
all
tests
,
the
water
depth
between
the
wave
gener-
ator
and
the
breakwater
was
constant,
and
was
sufficient
to
prevent
the
ratio
H/ d
from
influencing
the
action
of
waves
on
the
structure.
For
the
tests
con
-
ducted
,
the
variation
in
Reynolds
number
was
comparatively
small.
Tests
in
618
BREAKWATERS
a
larger
wave
flume
at
the
laboratory
of
the
Beach
Erosion
Board,
Washing
-
ton, D.
C.,
are
being
conducted
to
determine
the
effects
of
this
variable
on
the
stability
of
armor
units
in
rubble-mound
breakwaters.
When
damage
is
allowed
to
occur
to
the
breakwater
(by
use
of wave
heights
greater
than
the
design-wave
height),
the
geometry
of
the
structure,
the
mo-
tion
of
the
water
particles,
and
the
resulting
forces
on
the
breakwater
differ
from
those
resulting
from
tests
in
which
the
no-damage
criterion
is
used.
Thus,
a
damage
parameter,
D,
defined
as
the
percentage
of
armor
units
dis-
placed
from
the
cover
layer
by wave
action,
is
included
as
a
prime
variable
.
For
the
breakwater
sections
investigated
in
the
first
phase
of
the
testing
program,
in
which
the
armor
units
were
rocks
simulating
rounded
and
smooth
quarry
stones
placed
pell-mell
1/3
H
'Yr
( )
1/3
=
f(CI,
HIA,
dlA,
and
D)
..
.
......
.(14)
,Sr
- 1
Wr
In
the
second
phase
of
the
testing
program
the
armor
units
used
were
patterned
after
the
tetrapod,
and
the
rubble
mound
was
protected
by two
or
more
layers
of
armor
units
placed
over
one
or
two
quarry-stone
underlayers.
For
these
tests
'Y
1/ 3 H
(
r)
1/3
=
f(CI,
HI
A,
dlA,
r)
Sr
- 1
Wr
..........
..
(15)
The
dimensionless
parameter
on
the
left
side
of
Eqs.
13
through
15
is
desig-
nated
the
stability
number
(N
s
)
for
rubble-mound
breakwaters.
EXPERIMENTAL
EQUIPMENT
AND
PROCEDURE
Test
Apparatus
.-Th
e
breakwater
stability
tests
are
conducted
in
a
con-
crete
flume
5
ft
wide, 4
ft
deep,
and
119
ft
long,
equipped
with
a
plunger-type
wave
generator
. Wave
heights
are
measured
with
a
parallel
-
rod-type
wave
gage,
and
recorded
on
a
direct-writing
oscillograph.
The
wave-
h
eight
meas-
uring
apparatus
consists
of
the
wave
gage
(two
liB-in.
stainless
steel,
paral-
lel
rods
1. 2
ft
long,
spaced
2 in.
apart),
a
balancing
circuit,
a
Brush
universal
analyzer,
and
a
magnetic
oscillograph
.
Cross-section
measurements
of
the
small-scale
breakwaters
are
obtained
with
a
sounding
rod
equipped
with
a
circular
spirit
level
for
plumbing
, a
scale
graduated
in
thousandths
of a foot,
and
a
ball-and-socke
t foot
that
facilitates
adjustment
to
the
irregular
surface
of
the
breakwaters.
The
foot
is
Circular,
and
for
each
test
the
diameter
of
the
foot
is
equal
to
one-half
the
average
diameter
of
the
armor
units
.
Ty
pes
of
Tests
Conducted.-Two
primary
types
of
stability
tests
are
being
conducted
in
this
investigation.
First,
design-wave
heights
are
determined
for
breakwater
sections
of
sufficient
height
to
prevent
overtoppin
g by
the
test
waves
.
Design
- wave
height
is
defined
as
the
maximum
wave
height,
measured
at
the
location
of a
proposed
breakwater
before
it
is
constructed,
that
will
not
damage
the
cover
layer.
The
removal
of
as
much
as
1% of
the
total
number
of
armor
units
in
the
cover
layer
is
considered·
to
be
"no
damage
."
The
second
type
of
tests
being
conducted
is
concerned
with
determination
of
safety
factors
for
breakwater
sections
deSigned on
the
basis
of
the
criteria
BREAKWATERS
619
established
from
results
of
the
no-damage
and
no-overtopping
tests.
For
the
safety-factor
tests,
breakwater
sections
are
constructed
in
the
wave
flume
in
accordance
with
the
results
of
the
nO-damage
and
no-overtopping
tests,
and
the
amount
of
damage,
as
determined
by
the
percentage
of
armor
units
re-
moved
from
the
cover
layer,
is
obtained
as
a
function
of wave
height.
Wave
heights
greater
than
the
previously
selected
design
- wave
height
for
the
no-
damage
and
no-overtopping
criteria
are
used
in
these
tests.
HARBORSIDE
ARMOR
UNITS
(W
r
)
CORE
MATERI
A L
(
a)
HARBORS
I
DE
UNDE
R
LAYERS
CORE
MATER
I
AL
(b)
HARBORSIDE
~
,,'/.
,;O~~~~~~~~~
CORE
MATERIAL
0';:"
STON
ES
'"
W=O
. 1
0TO
0.20
LB
(c
)
SEASIDE
_---'v"-_
SW L
SEASIDE:.
_--,Y,--_
SWL
SEASIDE
FIG.
3.
-ELEMENTS
OF
BREAKWATER
SECTIONS
TESTED
In
addition
to
the
two
previously
mentioned
types
of
tests,
special
tests
are
conducted
from
time
to
time
to
determine
optimum
designs
for
specific
breakwaters
. In
these
tests,
design-wave
heights
may
be
determined
for
con-
ditions
other
than
no-
damage
and
no-overtopping.
Breakwater
Sections
Test
e
d.
-Rubble-mound
breakwaters
of
the
types
shown
schematically
in
Figs.
3
(a)
and
3
(b)
have
been
used
in
most
of
the
sta-