Hydraulic Loading & Strength

- overview -

Krystian Pilarczyk

krystian.pilarczyk@gmail.com

www.enwinfo.nl, (english, downloads)

Stability criteria

bop

s

cr

cosF =

DH

k

h1 = K

-0.2

hr

3/2op

s

cr

F =

DH

with maximum 8.0 = DHs

cr

L / H

tan =

opsop

g2u

K

KK 0.035 = D2cr

s

hT

sin

sin - 1 = K

2

s

k

h = K

s

-0.2

h

Current (general) (Pilarczyk)

Waves

Block revetments

b = 0.5 for rip rapb = ½ to 2/3 for blocks and gabions

Breaker index

or

bop

s

cr

cosF =

DH

1.05

Current attack

• Principles

• Velocity distribution

• Turbulence

• Design formulas

• Uncertainties

RSCv

2/13/21SR

nv

Chezy

Manning

C = h1/6/n ;

Strickler: n = ks1/6/25

C= 25(h/ks)1/6

R~h

For granular (non-cohesive) banks, fluvial erosion is modelled as for sediment transport

(with = bank angle):

Fluid Lift (FL)

Fluid Drag (FD)

Friction ()

Particle Weight (W)

Downslope componentof particle weight (Wd)

Normal component of particle weight

(Wn)

Steve Darby

Stability criteria revetments rock - current attack

• (a) K = f G, or

• ½CDρwU²πD²/4=fπD³/6(ρs-ρw)g, providing:

• U²/(2g Δ D) = (2/3)f/CD = φAssuming θ = 42o(for rock), f = tan 42o = 0.90, and CD= 1.0, one obtains: U²/(2g Δ D) = φ = 0.60.

(b) The moment with respect to the turning point S gives the equation: F b = G a, or (½CF ρw U²πD²/4)b=(πD³/6(ρs-ρw)g)a, providing:

U²/(2g Δ D) = (2/3)(a/b)/CF = φAssuming a = b, CF = 1.0, then φ= 0.67. CF is a combination of coefficients for drag and lift forces.

Isbash (1935)

Ub2/(2gD) = φ

Ub = bottom velocityD usually = D50

Dn50 = 0.84 D50

φ = stability factor:φ = 0.7 for exposed stones

and 1.4 for embedded stones

22

tan

tan1cos

sin

sin1

KsSlope factor

Ub2/(2gD Ks) = φ

or

Ub2/(2g φ Ks) =

D

Comparison formulas

Because in Vietnam design codes follow the Chinese codes (which is not correct):

Beginning of movement of granular materials

)()( *

2*

ec

cr

cws

crcr Rf

gD

u

gD

CgUgRIu w //*

)()( *

2*

ec

cr

cws

crcr Rf

gD

u

gD

CgUgRIu w //*

Shields Diagram (D*=d50(g/2)^1/3

us

k

h = K

-0.2

hr

k

h1 = K

-0.2

hr

or

In Pilarczyk formula:

Velocity distribution/profiles

Under ice cover

Comparable with flow in pipes

RSCv Chezy

2/13/21SR

nv Manning

R(hydraulic radius) ~ h (average

depth);

S = energy slope ~ bed slope

C = h1/6/n ;

Strickler: n = ks1/6/25

C= 25(h/ks)1/6

Mean velocity in a stream/rivers (uniform flow):

C = 18 log (12 h/kr)

Inter-relationships:C and n = resistance parameters/functions

Velocity profiles Log law vs. power law. The 1/6 power law is a standard profile, commonly applied to river flow.

Family of curves representing log-law profiles for varying roughnesses. All have the same mean velocity. The red curve uses the roughness parameter z0 = 1/10 of the depth, while the blue line corresponds to z0 = 10-6 times the depth (i.e. a very smooth bottom). The magenta line (z0 = 0.0005 times the depth) corresponds approximately to a 1/6 power law, while the blue line corresponds approximately to a 1/12 power law

A standard USGS practice allows hydrographers to estimate the average flow velocity by either measuring the velocity at 0.6 times the depth or by averaging the velocities measured at 0.2 and 0.8 times the depth.

Standard USGS practice. The red and blue lines give the velocities computed using the 0.6 depth and 0.2/0.8 depth methods, as a fraction of the true mean velocity. The resulting velocity is close to 1.00 over a wide range of bottom roughnesses. The arrows for the 1/6 and 1/12 power laws show the bottom roughness that give log profiles most closely matching these power laws. The 1/6 power law corresponds to typical river flow

and the 1/12 power law might apply in a man-made channel.

Secondary flow. Top: pattern of cross-channel velocity. Bottom: Down-channel velocity contours. The depth of maximum velocity (dashed line) is below the surface, and deepens closer to the side.

How secondary flows change the vertical profile of velocity

http://www.rsnz.org/publish/nzjmfr/1997/15.pdf

Velocity profile

Measurements in UK rivers:

Ub = 0.74 to 0.9 Udepth-average

Ub measured at 10%of the water depth

above the bed

???????

U/Uave=(x+1)(y/h)x ; x = 1/6 developed profile, x = 1/10 non-developed profile

For y =0.4h, U=Uave

y/h = 0.1

x = 1/6, U/Uave= 0.80

x = 1/10, U/Uave= 0.87

U/Usurf=(y/h)x

U/Uave=(x+1)(y/h)x ; x = 1/6 developed profile, x = 1/10 non-developed profile

For y =0.4h, U=Uave

y/h = 0.1

x = 1/6, U/Uave= 1.167 (0.1)0.167 = 1.167 0.681 = 0.80

x = 1/10, U/Uave= 1.1 (0.1)0.1 = 1.1 0.794 = 0.87

y/h = 0.4

x = 1/6, U/Uave= 1.167 (0.4)0.167 = 1.167 0.858= 1

x = 1/10, U/Uave= 1.1 (0.4)0.1 = 1.1 0.912 = 1

0

0,5

1

1,5

2

0 1 2 3 4 5

v(y) [m/s]de

pth

[m]

Velocity Distribution (log)

0

11 ln

yv y V gdS

d

1 lnyd

- =

At what elevation does the velocity equal the average velocity?

For channels wider than 10d

0.4k » Von Kármán constant

V = average velocityd = channel depth

1y de

= 0.368d

0.4d

0.8d

0.2d

V

Designation of Design Flow Velocity Bangladesh

General approach to current attack

- logarithmic velocity profile (Chezy),

sh k

h

gg

C 121log

2

18

22

22

50

6log75.5

12log

18

D

h

k

h

gDg

Ucr

scr

Strickler’s resistance formula for developed velocity profile,

3/13/1

1322

625

ssh k

h

k

h

g

- non-developed profile (Neill, 1967, Pilarczyk, 1995),

2.02.0

13232

ks

h

k

h

sh

StricklerC = 25(h/kr)1/6

C = 18 log (12 h/ks)~ 18 log (1+12h /ks)(for small h/ks)

crhcD

gU

2/2

assuming ks = 2 D50.

crhcD

gU

2/2

- logarithmic velocity profile (Chezy),

sh k

h

gg

C 121log

2

18

22

22

u

g d

C

gd

u

Cc

n

cn

c

c 5050

2

2

dn50 = Dc

Schiereck 2001,

Introduction to Bed, Bank and Shore Protection

C = 18 log (12 h/ks) ~ 18 log (1+12 h/ks)

(for small h/ks)

C K

u* K = Dn

2cs

c

2 2v

50

Often written as

Kv=1 to 1.6; usually 1.2 (turbulence)

Ks = slope factor

crc g

C

D

gU )2

(2/ 22

2.02

5.2)(

h

D

Dg

U c

cws

w

Neill (1967)

Kh = 33/Λh

k

h12log

2 = K

s

2h

crhcD

gU

2/2

- logarithmic velocity profile (Chezy),

sh k

h

gg

C 121log

2

18

22

22

Kh=33/Λh = 33/C2/2g = 33 2g/C2= 33 2g/[18 log (12h/ks)]2

= 2/[log(12h/ks)]2

C = 18 log (12 h/kr)

g2u

K

KK 0.035 = D2cr

s

hT

k

h = K

s

-0.2

h

1k

h12log

2 = K

s

2h

k

h12log

2 = K

s

2h

Kh = 33/Λh

k

h1 = K

s

-0.2

h

Fully-developed profile

Non-developed profile

ks=2D

ks=D

Kh=(h/ks)-1/3

C K

u* K = dn

2cs

c

2 2v

50

g u = d or =

d g

u or d g u2cc

c 27.07.122.1

Isbash, 1930

du

ffdg

u

dgcc

ws

cc

**

2* Re

u

g d

C

gd

u

Cc

n

cn

c

c 5050

2

2

d V Mn 3 3 /

In Schiereck 2001

Shields, 1936

Practical relation;

Including velocity and slope factor

For = 1.33*10-6 m2/s and = 2650 kg/m3, values of the grain size in mm are indicated on the graph.

Shields Diagram (D*=d50(g/2)^1/3

In Schiereck 2001

General Stability Formula for current attack (Pilarczyk)

g2u

K

KK 0.035 = D2cr

s

hT

sin

sin - 1 = K

2

s

k

h = K

s

-0.2

h

k

h1 = K

s

-0.2

hor

• Shields parameter ψ:• riprap, small bags 0.035• placed blocks, large geobags 0.05 • blockmats 0.05 to 0.07

(0.07- if cabled and/or washed-in)

• gabions 0.07 • geomattresses 0.07

Stability parameter Φ: continuous protection edgesRiprap and placed blocks, 1.0 1.5Sand-filled unitsBlock mats, gabions, washed-in blocks, 0.75 1.0 (1.2)Concrete-filled geobags, and geomattresseson proper subgrade (no soil deformation)

Turbulence factor KT:

• Normal turbulence: abutment walls of rivers: KT 1.0

• Increased turbulence:• river bends: KT 1.5• downstream of stilling basins: KT 1.5

• Heavy turbulence• hydraulic jumps: KT 2.0• strong local disturbances: KT 2.0• sharp bends: KT 2.0 (to

2.5)

• Load due to water (screw) jet: KT 3.0 (to 4.0)

(KT = kt2 in Rock Manual ‘07)

(kt U)2

39

Flow attack• pressure fluctuations

/ turbulence• drag forces• lift forces

load

strength

u

gD

Design formula

u

gD

K

K Ks

T h

2

0 035.

Example:• Mattress on a bank of a straight channel, 2 m deep: u

gD3

Rock specifications

Weight gradings and size relations for the standard light and heavy grading classes

1 3

1 0/

,na

WD

g

1 2

1 24/

,sa

WD

g

0 806,n sD D

Ds=equivalent sphere diameter

sD

)

1 3

5050

/

na

WD

g

50

50

0 84,nD

D

Comparison with Maynord USACE:

• Pilarczyk’s formula can be transformed into structure similar to Maynord formula, namely:

5.25.225.1

50 2

035.0

hgK

UhC

hgK

Uh

KD

s

P

scr

Tn

Maynord:

5.2

1

30

hgK

UhCD MD30 = 0.70 D50 (approximately)

CEM2002

Expressed as u =f(d30, h/d30)

CM = Sf Cs Cv CT

(useful for wide grading)

5.2

1

30

hgK

UhCD M

• CM = Sf Cs Cv CT

• Sf = safety factor; 1.1 to 1.5• Cs = 0.3 for angular rock and 0.375 for rounded rock• Cv = velocity distribution coefficient• Cv = 1.0 for straight channels• Cv = 1.25 downstream conrete channels;at the end of

dikes• Cv = 1.283 -0.2log(R/w) for outside bends (1 for R/w > 26)• CT = blanket thickness coef., typically = 1

K1 = side slope correction factor (less conservative than Ks);K1 = -0.672+1.492ctgα-0.449 (ctgα)2+0.045(ctgα)3

sin

sin-1 = K

2

s

Utoe/Uavg= 1.75 – 0.5 log (R/w)Example USACE guide:

hmax/have= 2.07 – 0.19 log (R/w -2)Bend scour/depth

Maynord: For ctgα=3, U=3.5m/s, h=10m, Sf=1.5,Cs=0.3, Cv=1.25, CT=1 and K1=0.97:

• D30 = 0.23 m and D50 = 0.33 m ( Dn50= 0.28m)• Applying Ks = 0.9• D30 = 0.25 m and D50 = 0.36 m ( Dn50= 0.30m)

Pilarczyk:

g2u

K

KK 0.035 = D2cr

s

hT

KT=1.4, Kh =(D/h)0.2= 0.5, Φ=1, Ks=0.9, ψc=0.03 to 0.035:

Dn50= 0.35m for ψc=0.03

Dn50= 0.30m for ψc=0.035

50

50

0 84,nD

D

Escarameia and May

g

UcD b

Tn 2

2

50

ct = 12.3 r – 0.20

riprap (valid for r 0.05):

r = turbulence intensity defined at 10% of the water depth above the bedTypical turbulence levels

SituationTurbulence level

Qualitative Turbulence

intensity (r)

Straight river or channel reaches

normal (low)

0.12

Edges of revetments in straight reaches

normal (high)

0.20

Bridge piers, caissons and spur dikes; transitions

medium to high

0.35 – 0.50

Downstream of hydraulic structures

very high 0.60

0

0,05

0,1

0,15

0,2

0,25

0,3

0 0,2 0,4 0,6 0,8

Ud/(gh)0.5D

n50

/h

Pilarczyk, not fully developed profile

Pilarczyk, fully developed profile

Escarameia and May

Maynord

Figure 2 Increased turbulence (kt = 1.5, r = 0.2 and Cv = 1.25)

Figure 1 Normal turbulence (kt = 1.0, r = 0.12 and Cv = 1.0)

0

0,05

0,1

0,15

0,2

0,25

0,3

0 0,2 0,4 0,6 0,8

Ud/(gh)0.5

Dn5

0/h

Pilarczyk, not fully developed profile

Pilarczyk, fully developed profile

Escarameia and May

Maynord

Figure 1 Normal turbulence (kt = 1.0, r = 0.12 and Cv = 1.0)

Figure 2 Increased turbulence (kt = 1.5, r = 0.2 and Cv = 1.25)

RM 2007

Figrs. 5.93 and 5.94

Hoffmans 2006

• r0 = 0.10 - 0.15 normal turbulence, rivers

• 0.15 - 0.20 non-uniform flow with increased turbulence as below stilling basins, outer bends in rivers

• 0.20 - 0.25 high turbulence as below hydraulic jumps, local disturbances, sharp outer bends

• 0.25 - 0.30 jet impact, hydraulic jump

s

n Kg

UrD

16.0

200

( )gd

Ur

Δ7.0=Ψ

200

gD

Ur

2007.0

s

n Kg

UrD

17.084.0

200

50

50

0 84,nD

D

Uo = depth-average velocity

ro = turbulence related factor

C

gc

U

uc

U

kr

ave0

0

*0

00 ===

where kave is depth-averaged turbulent kinetic energy

C is the Chézy coefficient.with co (≈1.2) a coefficient

For hydraulically smooth conditions, that is for C = 75 m½/s, r0 ≈

0.05 and for hydraulically rough conditions, i.e. C = 25 m½/s, r0 ≈

0.15.

Hoffmans 2006 (cont.)

The depth-averaged relative turbulence intensity (ro)

Soil types

Possible soil gradings and uncertainties

Hydraulic and coastal structures in international perspective; Pilarczyk, K.W.

TPG001.pdf (1016.1 KB) http://repository.tudelft.nl/view/hydro/uuid%3A9c8267d4-cdb6-4c53-8045-15afa6294e04/

http://www.slideshare.net/Pilarczykhttp://www.slideshare.net/Pilarczyk/1-geosyntheticsampgeosystems-pilarczyk-pres-final

; http://www.slideshare.net/Pilarczyk/geosyntheticsampgeosystems-in-coastal-engineering-pilarczyk2009

; http://www.isbnlib.com/author/Krystian_Pilarczyk

www.enwinfo.nl, (english, downloads, design revetments)