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

April 12, 2012 17

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%

April 12, 2012 19

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

April 12, 2012 22

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