April 12, 2012
Vermelding onderdeel organisatie
1
Computation of armour layersa comparison of the classical method, the PIANC method and a probabilistic method
ct5308 Breakwaters and Closure Dams
H.J. Verhagen
Faculty of Civil Engineering and GeosciencesSection Hydraulic Engineering
April 12, 2012 2
Three methods of computation
• The Classical Method• The method of the partial coefficients (PIANC)• A full probabilistic method
April 12, 2012 3
Classical computation
31 1 1ln(1 ) ln(1 0.2) 4.5 10 22550L
f pt
−=− − =− − = ⋅ =
0.40.1
0.36.7 1.0ss design odm
n
H N sd N− −⎛ ⎞
= +⎜ ⎟Δ ⎝ ⎠
April 12, 2012 4
Scheveningen case
Hss = 8.64 mNod = 0.5Δ= 1.75 (ρ=2800 kg/m3)N= 4000 wavessm = 5.6 %
dn = 3.28 m W = 38ton
Hudson gives (KD = 5. slope 1:1.5)dn = 2.5 m and W = 45 ton
0.40.1
0.36.7 1.0ss design odm
n
H N sd N− −⎛ ⎞
= +⎜ ⎟Δ ⎝ ⎠
April 12, 2012 5
Depth limitation in Scheveningen
Waterdepth at Scheveningen is 6 m below m.s.l.This is 9.5 m below Design Water Level
Using γ = 0.5, this makes that Hss can never be more than 4.75 m
In that case, the result is:Van der Meer W = 6.3 tonHudson W = 7.5 ton
April 12, 2012 6
But, there is an increased occurrence
exceedance every 0.6 years
April 12, 2012 7
This implies...
During lifetime (5 years) 50/0.6 = 85 storms
Thus 85 times in lifetime “nearly” damage
In total thus 85* 400 = 34000 waves
Including this in Van der Meer gives 24 tons(but this is outside the range of vdMeer)
April 12, 2012 8
The real design in Scheveningen
• Van der Meer was not available• Hudson underestimates because of the fact that
the number of waves are not included• Hudson with deep water waves overestimates• Model tests were performed for Scheveningen• This resulted in a block weight of 25 ton blocks
with a density of 2400 kg/m3
April 12, 2012 9
use of partial safety coefficients
• PIANC committee nr 12 (1992)Analysis of Rubble Mound Breakwaters
• Design should be based on probabilistic considerations• Level 2 and 3 were considered too difficult• So, a level 1 approach is adopted (i.e. use of partial safety
coefficients
1/3( cot )n D sZ A D K Hα= Δ −
3 cotDH KD
α=Δ
April 12, 2012 10
definition of coefficients
,design load load
i i chariX Xγ= •
,resisti chardesign
i resisti
XX
γ=
April 12, 2012 11
Extended Z-function
( )1/ 3,* cot 0ch n ch D H ch
A
AZ D K Hα γγ
= Δ − ≥
1/ 3,
cot
cot 0n chch chD H ch
A Dn
DAZ K H
α
α γγ γ γ γΔ
⎡ ⎤Δ= − ≥⎢ ⎥
⎣ ⎦
April 12, 2012 12
values for γH 3ˆ1 1' ˆ
ˆˆ
TT Hpf s k Pfs sTHsH FHsTfs
H kP NH
βγ σ⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟+ −
⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦= + +
T required service timePf target probability of failure in required service timeσFHs normalised standard deviation for FHsHT estimate of Hs once per T yearsH3T estimate of Hs once per 3T yearsHTpf estimate of Hs corresponding to a return period of TpfTpf return period corresponding to a probability Pf that HTp
will be exceeded during service life time T:( )
11/1 1
TPf fT P
−⎡ ⎤= − −⎢ ⎥⎣ ⎦
April 12, 2012 13
Elements in the equation
3ˆ1 1' ˆ
ˆˆ
TT Hpf s k Pfs sTHsH FHsTfs
H kP NH
βγ σ⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟+ −
⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦= + +
correction formeasurement errorsshort term variability
correction for “life time”correction for statistical uncertainty
April 12, 2012 14
PIANC method
1 exps LN t
sstL
HQα
γβ
⎧ ⎫⎡ ⎤⎛ ⎞−⎪ ⎪= − −⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
1
lnln 1 exp tLss
s L
QHN T
α
γ β⎡ ⎤⎧ ⎫⎛ ⎞⎪ ⎪= + − −⎢ ⎥⎨ ⎬⎜ ⎟
⎪ ⎪⎢ ⎥⎝ ⎠⎩ ⎭⎣ ⎦
April 12, 2012 15
PIANC, determination of Hss
α 1.24 β 1.17 γ 1.22 Ns 87.3
8.158.819.06
7.718.398.64
HsstL
Hss3tL
Hsstpf
Hss for t =tL (50,100)Hss for t =3tL (150,300)Hss for t =t20% (225,450)
life time 100
years
life time 50 years
April 12, 2012 16
Values for σparameters Method of determination Typical
value for σ’
Wave height Significant wave height offshore Hss nearshore determined from offshore Hss taking into account typical nearshore effects (refrac-tion, shoaling, breaking)
Accelerometer buoy, pressure cell, vertical radar Horizontal radar Hindcast, numerical model Hindcast, SMB method Visual observation (Global wave statistics Numerical models Manual calculation
0.05 – 0.1 0.15 0.1 – 0.2 0.15-0.2 0.2 0.1-0.2 0.15-0.35
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Safety coefficient
31 1 0.05'
tLpfss
ftLssss L
t H k Pss HH QtL f
HQ P N
βγ σ⎛ ⎞⎛ ⎞⎜ ⎟+ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= + +
8.391 1 38 0.27.718.64 0.050.2 1.13
7.71 0.2 1746ssHγ⎛ ⎞⎛ ⎞+ − ⋅⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠= + + =⋅
April 12, 2012 18
Parts in the safety coefficient for load
base example
use σ’ = 0.35 use N = 10 storms
use σ’ = 0.35 and N = 10
basic safety coefficient
100% 87% 99% 84%
measurement and short term errors
0% 13% 0% 13%
statistical uncertainty
0% 0% 1% 3%
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The partial safety coefficient for strength (γA)
( )* 1 lnA fk Pαγ = − •
kα coefficient fitted from probabilistic computations
Pf target probability of failure in the required service lifetime of the structure
April 12, 2012 20
equations for Cubes and Tetrapods
0.40.1
0.36.7 1.0s odom
n
H N sD N
−⎛ ⎞= +⎜ ⎟Δ ⎝ ⎠
0.40.1
0.36.7 1.0 0.5s omovom
n
H N sD N
−⎛ ⎞= + −⎜ ⎟Δ ⎝ ⎠
0.50.2
0.253.75 0.85s odom
n
H N sD N
−⎛ ⎞= +⎜ ⎟Δ ⎝ ⎠
0.50.2
0.253.75 0.85 0.5s omovom
n
H N sD N
−⎛ ⎞= + −⎜ ⎟Δ ⎝ ⎠
Tetrapods
Cubes
April 12, 2012 21
values for partial safety coefficients
Formula Condition kα kβ Hudson 0.036 151 Van der Meer Plunging
Surging 0.027 0.031
38 38
Van der Meer Tetrapods 1:1.5 0.026 38 Van der Meer Cubes 1:1.5 0.026 38 Van der Meer Accropods 0.015 33 Van der Meer low crested rock 0.035 42 Van der Meer rock toe berm 0.087 100 Van der Meer run-up ξ <1.5
run-up ξ >1.5 0.036 0.018
44 36
ks = 0.05
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Cubes-equation, including safety coefficients
0.40.1
0.30
1 6.7 1.0 tLodm n Hss ss
z
N s d HN
γγ
−⎛ ⎞+ Δ ≥⎜ ⎟
⎝ ⎠
April 12, 2012 23
Application for Scheveningen
0.40.1
0.30
1 6.7 1.0 tLodm n Hss ss
z
N s d HN
γγ
−⎛ ⎞+ Δ ≥⎜ ⎟
⎝ ⎠
Nod = 1N = 1500Δ- 1.75s = 2.5 %
dn = 2.07W = 25 tons
Remark:In this equation Hss = 7.71(i.e. the 1/50 wave)
April 12, 2012 24
Comparison with Classical Method
8.391 1 38 0.27.718.64 0.050.2 1.13
7.71 0.2 1746ssHγ⎛ ⎞⎛ ⎞+ − ⋅⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠= + + =⋅
= 1.12
7.71 * 1.12 = 8.63 m
Hss in classical method was 8.71
April 12, 2012 25
Probabilistic approach• Use VaP• In VaP Weibull is possible, but• parm 1 = u = β + γ = 2.39• parm 2 = k = α = 1.24• parm 3 = ε = γ = 1.22
• VaP computes probability per event, not per year• So multiply final result with storms/year (87)• Target probability of failure is thus:
51 1 5.07 10225 87.5fP −= = ⋅
April 12, 2012 26
Cube equation in VaP0.4
0.10.30 1.0 tLod
m n ssNG A s d HN
−⎛ ⎞= + Δ −⎜ ⎟⎝ ⎠
April 12, 2012 27
The statistical uncertainty in VaP
0.40.1
0.30 1.0 tLodm n ss
NZ A s d MHN
−⎛ ⎞= + Δ −⎜ ⎟⎝ ⎠
M has a mean Mmean = 1 and a standard deviation σ
April 12, 2012 28
determination of the standard deviation
' MM
ss designHσσ−
=
. , 2000M z x acc to Godaσ σ σ=
12 2
1.31 2
1.0 ( )
exp
z
a y c
Na a a N
σ
−
⎡ ⎤+ −⎣ ⎦=
⎡ ⎤= ⎣ ⎦N = number of stormsy = reduced variate
April 12, 2012 29
example
VaP normal calculation: dn = 2.40 m = 37.5 tonincluding uncertainty:
σx = 6.85 (follows from dataset)σz = 0.024σM = 0.02 dn = 2.42 m = 37.5 ton
Dataset with only 100 storms:σM = 0.175 dn = 2.70 m = 55 ton
April 12, 2012 30
Shallow water Wave height is limited by waterdepth.
Waterlevel HvH is Gumbel distributed:
1 exp exp surgehQ
γβ
⎡ − ⎤⎛ ⎞= − − −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
γ(u) is intercept (2.3)α is slope (3.289)α =1/ β = 3.289
VaP uses u and α
April 12, 2012 31
cube equation for shallow water
( )0.4
0.10.30 1.0od
m n br surge depthNG A s d h hN
γ−⎛ ⎞= + Δ − +⎜ ⎟⎝ ⎠
required probability of failure:Pf = 1/225 = 0.0044
(because statistic is already based on yearly storms)
April 12, 2012 32
Result of VaP calculation for Shallow water (γbr = 0.55)
5.0
15.0
25.0
35.0
45.0
0.000 0.002 0.004 0.006 0.008 0.010 0.012probability of failure
bloc
k w
eigh
t (to
n)
sigma=.1 sigma=.2target sigma=.05
Plot of require block asfunction of failure prob.for different values of
the standard dev. of γbr
April 12, 2012 33
Scheveningen