@_summary Of Rubble Mound Breakwater Design Equations

08a Summary of Rubble Mound Breakwater Design Equations B crown/cap

ocean side

bay/harbor side

crest armor layer, W R

DHW SWL

hc

first underlayer

α

hb

αb

h

second underlayer t

toe

core/base

Bt

bedding and/or filter

Design Concept/ Procedure 1. Specify Design Condition Æ design wave (H1/3, Hmax, To, Lo, depth, water elevation, overtopping, breaking, purpose of structure, etc.) 2. Set breakwater dimensions Æ h, hc, R, ht, B, α, αb 3. Determine armor unit size/ type and underlayer requirements 4. Develop toe structure and filter or bedding layer 5. Analyze foundation settlement, bearing capacity and stability 6. Adjust parameters and repeat as necessary Design Wave 1. Usually H1/3, but may be H1/10 to reduce repair costs (Pacific NW) (USACE recommends H1/10) 2. The depth limited breaking wave should be calculated and compared with the unbroken storm wave height, and the lesser of the two chosen as the design wave. (Breaking occurs in water in front of structure) 3. Use Hb/hb ~ 0.6 to 1.1 4. For variable water depth, design in segments Set BW Dimensions (controlled by height & slope) 1. Design elevation (peak crown elevation) = DHW + set-up + run-up + freeboard 2. Run-up (CEM VI-5, pp. 3-19) based on surf zone parameter at the structure tan α ξm = , Hs Lm α ≡ breakwater face slope,

Hs ≡ significant wave height at toe of structure Lm ≡ corresponding wave length at toe of structure a. Example (CEM equation VI-5-13, see p. VI-5-17 for coefficients) for ξm ≤ 1.5 R u ,i % HS = Aξ m

for 1.5 < ξm ≤ (D/B)1/C

R u ,i % H S = Bξ Cm

for ξm > (D/B)1/C

R u ,i % HS = D

b. Reduced Run-up R uR H S = (R u ,i % H S )γ surface γ berm γ shallow γ wave roughness

3. Relative Freeboard

R *m =

R Hs

water

angle

sm 2π

4. Overtopping Discharge (CEM VI-5, pp. 19-33) Q (gH s Tm ) = a exp(− b R *m γ ) (Table VI-5-8) example: Owen model: 5. Armor "Stone" Weight (CEM VI-6, pp. 48-105) example: rock armor, non-overtopping, 2 layers (Table VI-5-22): γ a H3 3 K D (SG − 1) cot α

W= 1/ 3

W 6. Armor thickness t = nk ∆    γa 

, n = number of layers (at least 2) 1/ 3

W 7. Crest width (minimum n = 3): B = 3k ∆    γa  8. Number of armor units per surface area

P  γ a  Na  = nk ∆ 1 −   A  100  W 

1/ 3

W 9. Underlayer thickness t = nk ∆    γa  10. Underlayer Guidance:

First Underlayer (directly under the armor units) minimum two stone thick (n = 2) (1) under layer unit weight = W/10 • if cover layer and first underlayer are both stone

2/3

•

if the first underlayer is stone and the cover layer is concrete armor units with KD ≤ 10 (2) under layer unit weight = W/15 when the cover layer is of armor units with KD > 10 Second Underlayer - n = 2 thick, W/20 Layer Primary Armor Layer First Underlayer Second Under Layer Base/ Core Material

Weight Ratio W/1 W/10 W/200 W/4000

Equivalent Diameter Ratio 1 2.15 2.7 2.7

11. Underlayer Volume (from geometry) for any layer: V A ≈ t (a + 2c ) L a

c = h 2 + (h cot α )

2

α

T

t

c h H

B

b

= h 1 + cot 2 α b = B − 2 T sin α = A + 2(H cot α − T sin α ) a = b − 2h cot α

= A + 2(H − h ) cot α − 2 T sin α = A + 2T(cot α − csc α )

12. Core Material Guidance

D15 <5 d 85 • Under dynamic load (i.e. wave forces), more restrictive rules apply: D50 W ≤ 2.5 to 3 , which gives ≤ 15 to 25 (assumes W ∝ D3) d 50 wbase 13. Filter Design Criteria D (breakwater design) • To prevent material from leaching out: 15 < 4 to 5 d 85 d85 = dia. exceeded by the coarsest 15% of the base mat'l D15 = dia. exceeded by the coarsest 85% of the filter mat'l D (embankment design) • To prevent pore pressure build-up: 15 > 4 to 5 d15 D • To maintain filter layer internal stability: 60 < 10 (i.e. well sorted material is D10 D preferred). Poorly sorted material is not suitable for filters Æ 60 ≥ 20 D10 •

For sorted material (e.g. quarry stones) under static (calm) load :

14. Toe Design 1/ 3

W a. D50 =    γs 

(assumes cubes)

b. Toe Stone Weight (iterative process to satisfy stone size and toe depth requirements) W=

γS H 3

N s3 (SG − 1)

3

,

various options for calculating a minimum Ns3 i.

Table VI-5-45 for regular waves,

ii. Table VI-5-48 (primarily for composite breakwater)  (1 − K )2 h t  1− K  h maximum of N s = 1.3 1 / 3  b + 1.8 exp − 1.5  K1 / 3 H s   K  Hs  or Ns = 1.8, where K =

2kh b sin 2 kBb sinh 2kh b

c. Toe Berm Width (Bt) should be the maximum of Bt = 2H or Bt = 0.4h, and at least 3 stones wide. d. Toe height should be at least 2 stones. 15. Scour Depth

ds = f (ξ ) = 0.5 to 1.0 , with 1.0 at ξ ~ 2.7 H

Settlement & Bearing Capacity:

1. Evaluate the ultimate bearing capacity pressure qu 2. Determine a reasonable factor of safety (FS) based on available subsurface surface information, variability of the soil, soil layering and strengths, type and importance of the structure and past experience. FS will typically be between 2 and 4. (marine applications 1.5-2.5) 3. Evaluate allowable bearing capacity qa by dividing qu by FS; i.e., qa = qu /FS 4. Perform settlement analysis when possible and adjust the bearing pressure (i.e. breakwater design) until settlements are within tolerable limits. The resulting design bearing pressure qd may be less than qa . Settlement analysis is particularly needed when compressible layers are present beneath the depth of the zone of a potential bearing failure. Settlement analysis must be performed on important structures and those sensitive to settlement. Ultimate Bearing Capacity For saturated, submerged soils strip foundations: q u = q c + q q + q γ = cN c + qN q + 0.5γ′BN γ circular foundations

q u = q c + q q + q γ = 1.3cN c + qN q + 0.3γ′BN γ

square foundations

q u = q c + q q + q γ = 1.3cN c + qN q + 0.4γ′BN γ

qc, qq, qγ = load contributions from cohesion, soil weight and surcharge Nc, Nq, Nγ = bearing capacity factors for cohesion, soil weight and surcharge obtain from equations or tables (see notes) based on friction angle c = cohesion strength of soil q = soil weight, q = γ′D

γ ' = effective bulk density of soil ( γ′ = γ − γ w =

G −1 γw ) 1+ e

B = width of the foundation D = the depth of penetration of the foundation qa =

qu FS

n Ι  Settlement in Sand: ρ = C1C 2 ∆σ ∑  z  ∆zi i =1  E i

Settlement in Clay ρ = ρ/i + ρc + ρs Primary Consolidation Settlement (ρc) Normally consolidated: ρc =

 σ '+ ∆σ v  Cc H  log 0 1 + e0  σ0 ' 

Cc =

Over-consolidated:

ρc = −

− ∆e log[(σ0 '+ ∆σ v ') σ0 ']

 σ '+ ∆σ v  CR H  log 0 1 + e0  σ0 ' 

typically CR is 10-20% of Cc •

For thick clay, divide into n multiple layers & use appropriate equation for each layer (shear stress is computed at center of sub-layers) n

ρc = ∑ ρci i =1

•

Consider depth to 2B for square foundation (BxB) or 4B for strip foundations (BxL), B is the width (below this depth, the load has dissipated and is zero)

•

most marine soils are overconsolidated - sedimentation increases the surcharge on the soil, but subsequent erosion removes much of the load

Secondary Consolidation Settlement (ρs) C H t  ρS =  α  log  F   tP   1 + e0 

where: tF = time at which magnitude of secondary compression required tP = time corresponding to end of primary consolidation

Note: C'α= [Cα/(1+eP)] found by Mesri to have correlation with water content, can replace Cα/(1+e0) with C'α in above equation for ρS (see Das Fig. 8.19) Uniform Cα/Cc ratio (USACE) Inorganic 0.025 - 0.065 Clay 0.025 - 0.085 Silt 0.030 - 0.075 Peat 0.030 - 0.085