Coastal and Hydraulics Laboratory

Rubble Mound Stability

Presenter: Jeffrey A. Melby, Ph.D.

Email: melbyj@wes.army.mil

WHAT IS A RUBBLE MOUND?

RUBBLE MOUND STABILITY

ContentsINTRODUCTION

Structure TypesFunctional vs Structural DesignExamplesArmor Selection Criteria

HYDRAULIC STABILITYPhysical Model StudiesInitiation of Armor MovementImportant ParametersDamage DeverlopmentStability Equations

Based on CEM Chapter VI-5-2 (f)

STRUCTURE TYPES

BREAKWATERSMultilayer

+ Traditional trapezoidal section+ Composite+ Low-Crested+ Submerged

Uniform+ Reef

JETTIESGROINSREVETMENTS

FUNCTIONAL VS STRUCTURAL DESIGN

FUNCTIONAL DESIGNPlan and Profile GeometryWave Transmission - Overtopping, Flow ThroughWave DiffractionWave Reflection

STRUCTURAL DESIGNFilter Sizing for Stability and Filtering Armor StabilityGeotechnical Settlement and Slip Circle FailureToe StabilityArmor Unit SizingArmor Layer GeometryArmor Unit Structural CapacityCap or Wavewall Stability and Structural CapCrest and Backside Stability

OCEAN CITY INLET JETTIES, MARYLAND

· Constructed in ‘30's using W= 5.4 t, 1V:2H· Repaired in ‘50's with W = 8.1 t, 1V:3H· Extended in 1985 with W = 13.5 t, H=5.5 m, KD=2.7

NOTE EBB TIDAL SHOAL

VENTURA, CALIFORNIA

HUMBOLDT JETTIES, CA

LUARCA, SPAIN

LLANOS, SPAIN

ARMOR SELECTION CRITERIA

Consider purpose of armorHydraulic stabilityStructural capacity, materialsEngineering performance vs cost

Volumetric efficiencyEase of construction

ARMOR LAYER LAYOUT & OPTIMIZATION

· N = Number of armor units A = Surface area on slope· δ = Packing density coefficient V = Volume of individual armor unit· W = Armor unit weight γ = Armor specific weight· n = Number of thicknesses kΔ = Layer coefficient· r = Total armor layer thickness VT = Total volume for N units· P = Armor layer porosity

23

13

1100

.T n n

N W PV where V and nkA

V N V r nk D where D V

δ δγ

−

Δ

Δ

⎛ ⎞= = = −⎜ ⎟⎝ ⎠

= = =

ARMOR LAYER LAYOUT AND OPTIMIZATION

VOLUME OPTIMIZATION

• N/(δA) = V(-2/3)

• VT/(δA) = V1/3

• So the total number of units can decrease much faster than the total volume increases as the armor size increases

• If equipment is constant, then it may be more economical to go with larger units ( increase volume)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 2 4 6 8 10 12

Volume

NumberVolume

N/(δA) = V(-2/3)

VT/(δA) = V1/3

HYDRAULIC STABILITY

· PHYSICAL MODEL STUDIES

· INITIATION OF ARMOR MOVEMENT

· ARMOR STABILITY DERIVATION

· STABILITY PARAMETERS

· DAMAGE DEVELOPMENT

· STABILITY EQUATIONS

PHYSICAL MODEL STUDIES

FROUDE AND GEOMETRIC SIMILITUDE

INITIATION OF ARMOR MOVEMENT

INITIATION OF ARMOR MOVEMENT

2 2

3

2

( )

D L

G

w n

s w n

n

n

s

F x Fwave forcerestoring force F

D vg D

vg DH if v gHD

N

ρρ ρ

=

=−

=Δ

= =Δ

=

PRIMARY SEA STATE STABILITY PARAMETERS

· Hs, H1/10 = Wave height (at toe)· Tm = Mean wave period· Lom = Deep water wave length for mean period· Lm = Local wave length for mean period· som= Hs/Lom = Deep water wave steepness· Nw or Nz = Number of waves at mean period· Hmo = 4(mo)1/2 = Spectral significant wave height · Tp = Peak wave period· Lop = Deep water wave length for peak period· h = Water depth at toe· ρw = Mass density of water

SECONDARY SEA STATE STABILITY PARAMETERS

· WAVE ASSYMMETRY· SHAPE OF WAVE SPECTRUM· WAVE GROUPING· WAVE INCIDENT ANGLE

STRUCTURALSTABILITY PARAMETERS

· α = Structure slope from horizontal· ρs = Mass density of armor units· Rock grading· Mass and shape of armor· Dn50 = (M50/ρs)1/3 = Nominal stone diameter (cube)· Packing density, layer thickness· P = Hudson’s porosity· P = van der Meer’s notional permeability of entire structure· S = Ae / D2

n50 = Normalized eroded area or damage· Armor placement

COMBINED PARAMETERS

· SR = ρs/ρw - 1 = Relative mass density· Ns = Hs/Dn50 = Stability number· Ns* = Ns sp

-1/3 = Ahrens’ stability number

· ξom = tan /(som)1/2 = Surf similarity parameter· SR = Sa - 1 = Relative mass density· Ns = Hs/Dn50 = Stability number

SURF SIMILARITY PARAMETER

DAMAGE PROGRESSION

Redondo Beach, California, 1988

DAMAGE DEFINITIONS

· COUNTING METHOD• Appropriate for small amounts of damage (CAU’s)• ‘No movement’ is lower limit• Armor units rocking (important for dolosse)• Individual units displaced• D = number displaced / total number in active region• Nod = number displaced / total number in strip Dn wide

· CAU: concrete armor unit

DAMAGE DEFINITIONS

· PROFILING OR DISPLACED AREA METHOD• Eroded Volume: Hudson, Jackson, D%, active region• Eroded Area: Broderick and Ahrens, S = Ae/D2

n50

• 0.6 < S/D% < 1.25• If S/D% = 0.8, then D = 5% corresponds to 0 < S < 4• Note that S determined from average profile can be very

different from average S of several profiles

DAMAGE CLASSIFICATION

· STONE• INITIAL DAMAGE: “no damage” value in 1984 SPM...D = 0-

5% displacement by volume or S = 0 - 2 by profiles• INTERMEDIATE DAMAGE: S = 2 - 12• FAILURE: Underlayer exposed through a hole at least Dn50

in diameter, D > 20%, S = 8-20

SOUTHWEST PASS, LA

DAMAGE CLASSIFICATION

· CONCRETE ARMOR UNITS• INITIAL DAMAGE: Core-Loc and Accropode D = 0-1%,

Dolosse D=0-2%, All shapes Nod = 0• INTERMEDIATE DAMAGE: Core-Loc and Accropode D = 1-5%• FAILURE: Core-Loc and Accropode D> 10%,

Cube Nod = 2, Dolosse D > 15%Tetrapod Nod =1.5, Accropode Nod = 0.5

LESSER ANTILLES

DAMAGE PROGRESSION

· Melby And Kobayashi 1998

DAMAGE PROGRESSION

ERODED AREA PREDICTION

Standard Deviation shows cross-shore variation

DAMAGE PROFILES

DAMAGE DEVELOPMENT

DAMAGE PROGRESSION

SHAPE OF ERODED PROFILE

Melby & Kobayashi’s OBSERVATIONS

· Equations verified for wide range of wave heights, wave periods, water depths, and stone gradation

· Damage does not progress to equilibrium· Armor gradation did not produce a measurable

difference in damage progression or eroded profile· Significant alongshore varability of damage · Damage initiation varied more than advanced damage· Spectral or time series parameters produce similar

results

Melby & Kobayashi’s LIMITATIONS

· Only single breakwater and beach slopes tested but conservative for most conditions

· Equations are limited to trunk section with head-on waves

· Multilayer trapezoidal cross section· Damage initiation not specifically investigated· Breaking waves only but should be conservative for

non-breaking conditions· Does not account for breaking stones

ARMOR STABILITY EQUATIONS

• MULTI-LAYER STONE• MULTILAYER LOW-CRESTED STONE• MULTILAYER SUBMERGED STONE• STONE REEF BREAKWATERS• BACKSIDE STABILITY• CONRETE ARMOR UNITS

CROSS SECTION

HUDSON EQUATION

· W = W50 = Median weight of armor unit· γa = Specific weight of armor material· H1/10 = Average of highest 10% of wave heights· KD = Tabulated empirical stability coefficient· Sa = Specific gravity of armor material· α = Seaside angle of armor slope from horizontal

1/ 3

5031/10

3

( cot )

( 1) cot

Dn

a

D

HNs KD

HWK Sa

α

γα

= =Δ

=−

STABILITY COEFFICIENTS

RIPRAP

· Riprap can also be designed using Hudson equation using the median weight W50, H1/10, and KRR = 2.2 for breaking waves and KRR = 2.5 for non-breaking waves (Ahrens 1981b)

· Typical stone size distribution 0.125 W50 < W < 4.0 W50

· Melby and Kobayashi observed that riprap armor deteriorates at the same rate as uniform armor for similar median weights because the larger stones hold the matrix together

STONE PLACEMENT

BUFFALO, NEW YORKKEYED AND FITTED

UMPQUA, OREGONSPECIAL PLACEMENT

GALVESTON BREAKWATER, TEXAS

STONE PLACEMENT

UNKNOWN SITERANDOM PLACEMENT

HUDSON EQUATION FOR BREAKING WAVES

· Probably the most extensive data set

· Many structure and armor types

· Breaking waves means depth-limited waves

· KD is lowest stability measurement in lab using severely breaking regular waves

CARVER’s MODIFICATIONS TO THE HUDSON EQUATION

Hudson Stability Coefficient vs Relative Depth

VAN DER MEER EQUATIONS

Hs

ΔDn501.0P 0.13 S

Nw

0.2cotα ξP

om Surging waves: ξm > ξmc

Hs

ΔDn506.2P 0.18 S

Nz

0.2ξ 0.5

om Plunging waves: ξm < ξmc

ξom [6.2P 0.31 tanα ]1

(P 0.5)

VAN DER MEER’S PARAMETERS

• Hs = Significant wave height at toe• Δ = Sa - 1 = Relative mass density• Nw = Number of waves at mean period• P = Notional permeability of entire structure• S = Ae / D2

n50 = Normalized eroded area / damage• ξ om = tan α/(Hs/Lom)1/2 = Surf similarity parameter• Lom = gTm

2/2π

VAN DER MEER’S PERMEABILITY

VAN DER MEER EQUATIONSBREAKING WAVES

H2%

ΔDn501.4P 0.13 S

Nw

0.2cotα ξP

om Surging waves: ξm > ξmc

ξom [6.2P 0.31 tanα ]1

(P 0.5)

Hs = H2%

H2%

ΔDn501.4(6.2)P 0.18 S

Nz

0.2ξ 0.5

om Plunging waves: ξm < ξmc

VAN DER MEER’S LIMITATIONS

· Limited to deep to intermediate depth· Breaking wave equations are based on 8 tests using

only spilling breakers - not conservative for most applications

· Damage is for constant wave conditions · Equations are for a trunk section or revetment with

head-on waves· Equations are limited to uniform armor stone

MULTI-LAYER OVERTOPPED STRUCTURES

Rc/h a*10^4 b Hs/Lp0.29 0.07 1.66 <0.03 850.39 0.18 1.58 <0.03 700.57 0.09 1.92 <0.03 200.38 0.59 1.07 >0.03 45

· Nod = Number of displaced units· Na = Total number of units· Ns = Hs / Dn50· sp = Hs / Lp = Local wave steepness

POWELL AND ALLSOP (1985)

Nod

Naa exp[b s 1/3

P Ns] or Nss 1/3

p

bln 1

aNod

Na

MULTI-LAYER OVERTOPPED STRUCTURES

· Replace Dn50 by fi (Dn50) in original equations· Rc = Freeboard· sop = Hs / Lop = Deep water wave steepness

VAN DER MEER 1991 - ARMOR REDUCTION COEF.

fi 1.25 4.8Rc

Hs

sop

sπ

1

0 <Rc

Hs

sop

sπ< 0.052

MULTI-LAYER SUBMERGED STRUCTURES

· h = Water depth at toe

· h’c = Crest height over sea bed

· Ns* = Ns(sp)-1/3 = Spectral stability number

VAN DER MEER, 1991

h c

h(2.1 0.1S) exp( 0.14Ns )

REEF BREAKWATERS

At = Initial cross-sectional area of structure

VAN DER MEER, 1990 AFTER AHRENS, 1987

h cAt

exp(aNs )h c

a 0.28 0.045At

(h c)20.034

h c

h6×10 9 A 2

t

D 4n50

BACKSIDE STABILITY

JENSEN, 1984

ROUNDHEAD STABILITY

· Heads and bends usually sustain damage at a much lower wave height than the trunk

· Causes include reduced support from neighboring armor and high overflow wave velocities and wave refraction

ROUNDHEAD STABILITY

· Use equation of Carver and Heimbaugh (1989) for stone and dolos

· Can also use Hudson equation· Increase stability by increasing roundhead diameter· Roundhead slopes are often flatter· Trunk and head units must interlock· Toe detail is very important

YAQUINA JETTIES, NEWPORT, OREGON

· Original structures 1 km long in ~1900

· Structures extended to 1.5 km in 30's - 40's then to 2.1 km in 60's. W = 18.5 t, S=2.58, Hb = 8.2 m, 1V:2H, KD=10

Outer 500 ft repaired twice: W = 18 t, S=2.66, Hb = 6 m, KD = 3.7Then W = 29 t, Hb = 8.5 m,KD=6.3

Coastal and Hydraulics Laboratory

Stability Conclusions

· Stability of stone breakwaters predicted by many equations• Use appropriate equation• Use conservative design assumptions because

equations are based on idealized laboratory conditions

· Least expensive option may not be one with least material

· Lobby for aesthetics

VATIA STONE