1. Computation of armour layersa comparison of the classical method, the PIANC method and aprobabilistic methodct5308 Breakwaters and Closure DamsH.J. VerhagenApril 12, 2012 1Faculty of Civil Engineering and GeosciencesVermelding onderdeel organisatieSection Hydraulic Engineering
2. Three methods of computation• The Classical Method• The method of the partial coefficients (PIANC)• A full probabilistic methodApril 12, 2012 2
4. Scheveningen caseHss = 8.64 m Hss−design ⎛ Nod 0.4 ⎞ −0.1 = ⎜ 6.7 0.3 +1.0 ⎟ smNod = 0.5 Δdn ⎝ N ⎠Δ= 1.75 (ρ=2800 kg/m3)N= 4000 wavessm = 5.6 % dn = 3.28 m W = 38tonHudson gives (KD = 5. slope 1:1.5)dn = 2.5 m and W = 45 tonApril 12, 2012 4
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 ton Hudson W = 7.5 tonApril 12, 2012 5
6. But, there is an increased occurrence exceedance every 0.6 yearsApril 12, 2012 6
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 7
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/m3April 12, 2012 8
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 H = 3 K D cot α ΔD Z = A ΔDn (KD cot α ) 1/ 3 − HsApril 12, 2012 9
10. definition of coefficients design Xi =γ i load • X i,char load resist X i ,char Xidesign = γiresistApril 12, 2012 10
11. Extended Z-function 1/ 3 Ach Δ ch Dn,ch ⎡ cot α ⎤ Z= ⎢KD ⎥ − γ H H ch ≥ 0 γ A γ Δ γ Dn ⎣ γ cot α ⎦ A Δ ch Dn,ch ( K D cot α ) 1/ 3 Z= − γ H H ch ≥ 0 γ* AApril 12, 2012 11
12. values for γH ⎡ ⎛ H 3T ⎞ ˆ ⎤ ˆ T pf Hs ⎢1+⎜ s −1⎟ kβ Pf ⎥ ks γH = + σ FHs ⎢ ⎜ H s ⎟ ⎣ ⎝ ˆT ⎠ ⎥ ⎦ + ˆT Hs Pf NT 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: −1 ( TPf = ⎡1 − 1 − Pf ) 1/ T ⎤ ⎢ ⎣ ⎥ ⎦ April 12, 2012 12
13. Elements in the equation ⎡ ⎛ H 3T ⎞ ˆ ⎤ ˆ T pf Hs ⎢1+⎜ s −1⎟ kβ Pf ⎥ ks γH = + σ FHs ⎢ ⎜ H s ⎟ ⎣ ⎝ ˆT ⎠ ⎥ ⎦ + ˆT Hs Pf N correction for measurement errors short term variability correction for “life time” correction for statistical uncertaintyApril 12, 2012 13
15. PIANC, determination of Hss life time life time 50 years 100 yearsHss for t =tL (50,100) HsstL 7.71 8.15Hss for t =3tL (150,300) Hss3tL 8.39 8.81Hss for t =t20% (225,450) Hsstpf 8.64 9.06 α 1.24 β 1.17 γ 1.22 Ns 87.3April 12, 2012 15
17. Safety coefficient t pf ⎛ ⎛ H sstL ⎞ 3 ⎞ H ⎜1+ ⎜ tL −1⎟ kβ Pf ⎟ 0.05 γH = ss +σ QL ⎜ ⎜ H ⎝ ⎝ ss ⎟ ⎠ ⎟ ⎠ + ss QtL Pf N ⎛ ⎛ 8.39 ⎞ ⎞ 8.64 ⎜1+⎜ −1⎟38⋅0.2 ⎟ 0.05 γ Hss = + 0.2 ⎝ ⎝ 7.71 ⎠ ⎠ + =1.13 7.71 0.2 ⋅1746April 12, 2012 17
18. Parts in the safety coefficient for load base use σ’ = 0.35 use N = 10 use σ’ = 0.35 example storms and N = 10basic safety 100% 87% 99% 84%coefficientmeasureme 0% 13% 0% 13%nt and shortterm errorsstatistical 0% 0% 1% 3%uncertainty April 12, 2012 18
19. The partial safety coefficient forstrength (γA) γ A =1 − * ( kα • ln Pf ) kα coefficient fitted from probabilistic computations Pf target probability of failure in the required service lifetime of the structureApril 12, 2012 19
20. equations for Cubes and Tetrapods Cubes Hs ⎛ Nomov 0.4 ⎞ −0.1 Hs ⎛ N 0.4 ⎞ −0.1 = ⎜ 6.7 0.3 + 1.0 ⎟ som − 0.5 = ⎜ 6.7 od +1.0 ⎟ som ΔDn ⎝ N ⎠ ΔDn ⎝ N 0.3 ⎠ Tetrapods Hs ⎛ 0.5 Nod ⎞ −0.2 Hs ⎛ 0.5 Nomov ⎞ −0.2 = ⎜ 3.75 0.25 + 0.85 ⎟ som = ⎜ 3.75 0.25 + 0.85 ⎟ som − 0.5 ΔDn ⎝ N ⎠ ΔDn ⎝ N ⎠April 12, 2012 20
21. values for partial safety coefficientsFormula Condition kα kβHudson 0.036 151Van der Meer Plunging 0.027 38 Surging 0.031 38Van der Meer Tetrapods 1:1.5 0.026 38Van der Meer Cubes 1:1.5 0.026 38Van der Meer Accropods 0.015 33Van der Meer low crested rock 0.035 42Van der Meer rock toe berm 0.087 100Van der Meer run-up ξ <1.5 0.036 44 run-up ξ >1.5 0.018 36 ks = 0.05April 12, 2012 21
22. Cubes-equation, including safetycoefficients 1⎛ N 0.4 ⎞ −0.1 ⎜ 6.7 od +1.0 ⎟ sm Δdn ≥ γ Hss Hss tL γz ⎝ N 0.30 ⎠April 12, 2012 22
23. Application for Scheveningen 1⎛ 0.4 Nod ⎞ −0.1 ⎜ 6.7 0.30 +1.0 ⎟ sm Δdn ≥ γ Hss Hss tL γz ⎝ N ⎠ Nod = 1 N = 1500 Remark: Δ- 1.75 In this equation Hss = 7.71 s = 2.5 % (i.e. the 1/50 wave) dn = 2.07 W = 25 tonsApril 12, 2012 23
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: 1 1 Pf = = 5.07 ⋅10−5 225 87.5April 12, 2012 25
26. Cube equation in VaP ⎛ N 0.4 ⎞ −0.1 G =⎜ A od 0.30 +1.0 ⎟ sm Δdn − Hss tL ⎝ N ⎠April 12, 2012 26
27. The statistical uncertainty in VaP ⎛ Nod 0.4 ⎞ −0.1 Z = ⎜ A 0.30 + 1.0 ⎟ sm Δdn − MHss tL ⎝ N ⎠ M has a mean Mmean = 1 and a standard deviation σApril 12, 2012 27
28. determination of the standarddeviation σM σ M = H ss − design σ M = σ zσ x acc. to Goda , 2000 1 ⎡1.0 + a ( y − c) ⎤ 2 2 σz = ⎣ ⎦ N N = number of storms a = a1 exp ⎡ a2 N ⎣ −1.3 ⎤ ⎦ y = reduced variateApril 12, 2012 28
29. example VaP normal calculation: dn = 2.40 m = 37.5 ton including uncertainty: σx = 6.85 (follows from dataset) σz = 0.024 σM = 0.02 dn = 2.42 m = 37.5 ton Dataset with only 100 storms: σM = 0.175 dn = 2.70 m = 55 tonApril 12, 2012 29
30. Shallow water Wave height is limited by waterdepth. Waterlevel HvH is Gumbel distributed: ⎡ ⎛ h −γ ⎞⎤ Q =1 − exp ⎢ − exp ⎜ − surge ⎟⎥ ⎣ ⎝ β ⎠⎦ γ(u) is intercept (2.3) α is slope (3.289) α =1/ β = 3.289 VaP uses u and αApril 12, 2012 30
31. cube equation for shallow water ⎛ Nod 0.4 ⎞ −0.1 G = ⎜ A 0.30 + 1.0 ⎟ sm Δdn − γ br ( hsurge + hdepth ) ⎝ N ⎠ required probability of failure: Pf = 1/225 = 0.0044 (because statistic is already based on yearly storms)April 12, 2012 31
32. Result of VaP calculation for Shallow water (γbr = 0.55) 45.0 block weight (ton) 35.0 25.0 15.0 5.0Plot of require block as 0.000 0.002 0.004 0.006 0.008 0.010 0.012function of failure prob. probability of failure for different values of sigma=.1 sigma=.2the standard dev. of γbr target sigma=.05 April 12, 2012 32
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