isope-i-03-311_similitude and stability of rubble unequal in weight and shaped

8/12/2019 ISOPE-I-03-311_Similitude and Stability of Rubble Unequal in Weight and Shaped

http://slidepdf.com/reader/full/isope-i-03-311similitude-and-stability-of-rubble-unequal-in-weight-and-shaped 1/6

Similitude and Stability of Rubble Unequal in Weight and Shaped

Ryuji KOHPort Hydraulic Research Center, General Directorate of Railways, Ports and Airports Construction,

Macunkoy, Ankara, Turkey

ABSTRACT

A Froude model usually assimilates prototype, neglecting othersimilarities. However similitude should be also considered as to rubblemound. Prototype rubble is not completely reproduced by modelcrushed stone. Some indices are introduced to elucidate weightdistribution and shape of rubbles to quantify characteristics of

prototype and model. Theoretical granular model reveals stabilityconvergence, alleviating inequality of weight and shape.

KEY WORDS: similitude of rubble, weight distribution, shapeindices, stability convergence

INTRODUCTION

Hydraulic experiment is ordinarily carried out with a Froude model.Prototype is reduced to model with similar shape. Prototype rubble isreplaced by model crushed stone. Rubbles are commonly prescribed ina range of weight as a structural material. This range is converted toweight range of crushed stone, multiplying a cube of scale. Here is noapprehension on similarity of weight distribution and shape, althoughthey are unquestionably dissimilar. Model experiment easily distortsresult with dissimilar weight distribution and shape of crushed stone.This apprehension motivates to quantify characteristics of rubble withsome indices. Granular model evaluates the effect of unequal weightand shape on stability.

BACKGROUND

Stone (rubble) should be sound, durable and hard, sound enough not to

fracture or disintegrate, and angular (SPM,1984). Besides SPM requiresthat for a quarry stone armor unit, the greatest dimension should be nogreater than three times the least dimension to avoid placing slab-shaped stone.

The above is the sole reference to quantity. Rubbles are inevitablyirregular and different from each other in their weight and shape. It isnecessary to quantify characteristics of rubble mound for similitudeanalysis, however difficult it seems to be. The author introduces someindices for weight distribution and shape, and evaluates stability ofunequal rubbles.

OVERALL INDICES

(1) Angle of repose and void ratio

Rubble mound consists of numerous rubbles varying within a range oweight and shape.

Angle of repose and void ratio are reasonably adoptable as overaindices of rubble mound as aggregate, regarding stability against wave.

Angle of repose is the maximum stabilizing angle of circular consurface of stacked rubbles. For granular materials such as rubbles, iregarded as the same as internal friction anglei , where µi= tan expresses coefficient of internal friction against shear failure.

Void ratio e is given as e = VV /VS , where VV and VS are volume ovoid and soil (rubble) , respectively, and ordinarily manifested adecimal.

Void is sometimes expressed with analogous “ porosity n ” which idefined as n = VV / ( VS + VV )×100%, related to e as n = e /(1+e×100% and usually indicated in percent (%). Void ratio e should bstrictly distinguished from porosity n.

Model ought to have the same angle of repose and void ratio a prototype. This is a necessary condition, although it may not be sufficient condition. It is easy to measure angle of repose and calculatvoid ratio in model. For prototype, void ratio is necessary to designand angle of repose should be measured through in-situ experimensimply stacking rubbles to form circular cone.

(2) Previous measurements of and e

1) Angle of repose

Internal friction anglei is adopted as 30° for sand and 40° for back-firubble, for designing port structures ( JPHA, 1989). For model quarrstone, friction coefficient µ= tan somewhat varies with placinmethod and medium ( Hudson, 1959) (Table 1).

Proceedings of The Thirteenth (2003) International Offshore and Polar Engineering Conference

Honolulu, Hawaii, USA, May 25 – 30, 2003

Copyright © 2003 by The International Society of Offshore and Polar Engineers

ISBN 1 –880653-60 – 5 (Set); ISSN 1098 –6189 (Set)

761



Table 1. Angle of reposePlacing Medium µ 0 cotdumped

dostacked

do

waterair

waterair

0.98 - 1.130.79 – 0.90

1.09 – 1.260.97 – 1.22

44.4 – 48.538.3 – 42.047.5 – 51.644.1 – 50.7

0.88 – 1.021.10 – 1.270.79 – 0.920.82 – 1.03

average water 1.06 – 1.20 46.7 – 50.2 0.83 – 0.94

It is expected that can be measured with sufficient accuracy, for fluctuates within 10 % of its range under a fixed condition. Here cot means stable gradient in still water. Accordingly the difference betweencot and cot implies surplus stability ( : angle of rubble moundslope).

As expected, augments with strength, surface roughness, angularity,cube, compaction and widely ranging weight distribution of rubbles(JGS, 1982). This suggests that is a function of weight distributionand shape of rubbles.

Mogami (1968 and 1970) proposed an empirical formula. +=+= C U k ek log),1/(sin ..…………………………….(1)

where k and are constants affected by material and weightdistribution, and material, respectively (UC: uniformity coefficient ofweight distribution). Eq (1) may be advantageous, since it does notexplicitly comprise shape effect which will implicitly affect k and .

2) Void ratio e

The three dimensional arrangements of uniform spheres have e valuesas 0.91 (simple cube), 0.65 (cubical tetrahedron), 0.43 (tetragonalspheroid) and 0.35 (pyramid and tetrahedron)( Deresiewicz, 1958).

Sands have e values as 0.67 (carefully compacted uniform sand),0.64~0.70 (coarse sand), 0.70~0.92 (medium sand), 0.79~0.96 (finesand) and 0.67~1.27 (coastal sand) (Homma, 1954).

Rock- fill materials, ranging from rock to silt for impermeability,naturally have e values widely dispersing from 0.26 to 1.50, dependingon size (weight) distribution and roller compaction, deduced from rock-

fill dams in Japan (JGS, 1982).Unit weight 1.8 tf/m3 (sand and rubble) is commonly adopted fordesigning port structures and converted to e = 0.47, which seems to bemost reasonable.

Hudson (1959) picked up quarry groups A to G, each with nearly thesame weight, shape and specific weight and measured e ( Table 2).Void ratio is small in near sphere A and rises in flat G, where a, b and care largest, medial and least latera of quarry. This measured e seems to

be slightly high, for tested quarry stones are mostly uniform.

Table 2. Void ratioQuarry A B C D E F G

a/c 1.5 1.6 1.6 1.7 2.2 2.6 3.3

b/c 1.2 1.3 1.3 1.4 1.5 1.5 2.5c 0.64 0.69 0.67 0.75 0.72 0.85 0.89

CONSTITUENT

From preceding review, overall indices and e can be decomposed intoconstituent indices as weight distribution and shape.

(1) Weight distribution

In soil mechanics, weight (grain size) distribution is conventionallexpressed with cumulative curve, integrating weight distribution curv(Figure 1 and Table 3).

Table 3. Weight distribution and cumulative curveLine Weight distribution Cumulative curve

I function Unit functionII Uniform Linear functionIII Error function Relaxation unit function

IV Prevalent heavy rubbles Concentrated heavy rubblesV Prevalent light rubbles Concentrated light rubbles

1) Typical distributions

Line I corresponds to uniform blocks such as artificial concrete blocksLine II is completely uniform distribution. Line III reflects carefullselected rubbles with nearly the same weight or poorly fabricateconcrete blocks. Line IV contains much heavy rubbles executed by conscientious contractor. Line V consists of much light rubbleconstructed by a vulgar contractor.

Figure 1. Weight distribution and cumulative curve

762



2) Actual distribution

Prototype and model rubbles draw irregular distributions considerablydifferent from these typical curves.

Soil mechanics adopts some indices to characterize actual distribution.Uniformity coefficient UC and coefficient of curvature UC

/ are given asfollows:

60102

30/

1060 /,/ D D DU D DU C C == ……………………………(2)

where Dn is grain size corresponding to passing n% soil of total weight(JGS, 1982).

UC signifies range of grain size distribution or non-uniformity. UC/

indicates smoothness of cumulative curve which shows stepdistribution unless smooth. Quality of grain size distribution is judgedfor general use with UC and UC

/ (JGS, 1982) (Table 4).

Table 4. Grain size distribution bad

good Uniform grain size Step distributionUC 10

1 < UC’ C U

UC < 10 UC 10UC’ < 1 or UC’ >

C U

However additional study will be needed to clarify favorable conditionof UC and UC/ for rubble mound, considering Figure 1.

Grain size, easily obtained by sieving, is peerless expression for soilsuch as silt, sand and gravel, because it is impossible to weigh each

particle. For a rubble, weight is contrarily the only precise quantity, fordimension cannot be unitarily defined.

Weight distribution of rubbles can be manifested as cumulative curvewith weight W (abscissa) and passing weight in % (ordinate) both inCartesian scale. Wn or weight of a rubble with passing n% of totalweight is converted into a sphere with a diameter Dn.

3 /6 S nn W D = ………...…………………………………………(3)

where S is unit weight of rubble.Then UC and UC

/ can be rewritten as

36010

230

/31060 /,/ W W W U W W U C C == …………………………..(4)

,defying difference of S and shape.

It is supposedly reasonable to regard rubble as sphere except flatrubble.

3) Weight distribution indices

For further study, the following indices are recommended:Median weight W50, Equality factor IE = W25/W75 <1,

Median factor IM = W50/(W25+W75)<1 ……………………………(5)

Median weight is doubtlessly meaningful. Equality factor is small indispersive rubbles and tends to 1 in uniform rubbles. Median factor is0.5 in uniform distribution (Line II in Figure 1), and ranges between 0.5and 1 in prevalent heavy rubbles( Line IV) and between 0 and 0.5 in

prevalent light rubbles (Line V) (Table 5). These indices are newlyintroduced from a practical viewpoint. They also suggest weight rangesof each quarter (25%) of total rubbles.

Table 5. Weight distribution indicesLine W50 IE IM

W25 W75 0.5 1.0 0.5 1.0I = = = =II > < > < =III >> =IV > < < > <

V > < < <

Eq. 5

(2) Shape indices

1) Previous expression

Irregular shape of rubble can be stereographically delineated to tracthree-dimensional coordinates of surface S (x, y, z). However it iunproductive work for practical use.

For granular material, shape has been measured with indices such aroundness, angularity and ratios of three latera (Mogami, 1969). Thesindices can be adopted when they are related to mechanical strength ogranular material.

For instance, it is known that flat or slender granule is weak an porous.

Roundness and angularity seem to be impractical indices because ointricate measurement.

2) Model crushed stone

Setting a side conceptional view, ten pieces were taken from modecrushed stones. Beyond expectation, their shapes can be roughlapproximated as polyhedrons, disregarding minor irregularities. Theare classified into 1 tetrahedron (trigonal pyramid), 2 pentahedron(quadrangular pyramid or truncated trigonal pyramid), 4 hexahedron(irregular tetragonal prism), 2 heptahedrons (deformed gabled house oirregular pentagonal prism) and 1 octahedron (irregular pentagona

prism with gabled top face).

Predominance of hexahedron is reasonable and favorable, as massivcube-like stones are selected and supposed to be most stable.Practically octahedron has most faces, seeing additional small facehardly affect shape and void ratio in polyhedron with more faces tha8.

Polyhedron is an excellent and expressive index among others fomodel crushed stone, still it necessitates some supplementary study tquantify shape.

3) Prototype rubble

Rubble approximates to a spheroid rather than a polyhedron, roundinoff corners by colliding impact.

The following indices are introduced in relation to and e:

Isotropy factor II = c/a<1, Flatness factor IF = ab/c2 >1, Slendernesfactor IS = a2/bc >1, Prism factor IP = V/abc <…………………………..………………………………………….(6)where V is volume of rubble. IP = 1 for rectangular prism and = R/6 0.52 0.5 for spheroid. In-situ measurement is necessary to obtaithese shape indices.

763



STABILITY

Stability depends on weight distribution, shape and arrangement ofrubbles. Setting aside intricate analysis, stability is evaluated regardingunequal weight and shape.

(1) Stability factor S

Here stability is defined as energy E necessary to slide rubble up toinstability.

==

+=+=

=

l sW W

d sW E

S

A

A

)2/(,)/1(

)2//()cossin1()2//()sin(cos

/

2/

/

µ µ µ

µ

…………………………………………………………………..….(7)where µA: apparent friction coefficient, W/: weight of rubble in

water , s: sliding distance, : slope angle of objective rubble, l: orbitalradius of sliding rubble and µ: friction coefficient of rubble (Koh, 2001)(Figures 2 and 3).

Figure 2. Sheltering by 2 footing rubbles

Figure 3. Critical stability

Here centrifugal force ml2 is neglected, as angular velocity is smal

Ratio (<1) of friction forceµA W/ to wave force P is primary index ostability.

I AS I

S S A A

fH d ab gH f

abc g P W

/)1/(}{

/}3/4)/1({//

µ

µ µ

=

==

where : density of water,S: density of rubble( = S/g, g: gravitationaacceleration), f: wave pressure factor, d( =4c/3): average depth orubble in wave direction z, a. b.c: radius in x.y..z axes and HI: incidenwave height.

E is turned into non-dimensional expression asr s H d f r s E I AS ///)1/(/Pr / == µ

Stability factor S is here introduced.r sr d f H r E S A I S ///}/)1//{(Pr)/( == µ ……….….....(8)

where r is radius of standard rubble.Putting r = 1, d and s become non-dimensional.

f sd S A / µ = ………………………………………..……………..(9)

can evaluate the effect of inequality.

(2) Unequal radius a

Rubble is categorized into ranges W of 2~4t, 4~6t and 6~8t, whiccorrespond to equivalent radius ranges a of 0.88~1.10, 0.93~1.06 an0.95~1.04 as a ratio to median radius. Thus it suffices to consider a =0.9 and 1.1.

022 /8/coth,2/)1(, L H kh f K f f f f I E K RS K S =+=+= ………...…..(10

where K R : reflection coefficient, k=2R/L, L: shallow water wavlength, L0: deep water wave length, and suffixes S and K signify statiand kinetic pressure, which act on whole surface area R and exposesurface area ER, respectively ( E: exposed ratio). f is exemplified foH = 2m, T = 6s and h = 5m, since it varies with wave condition.

1) a = 1.0

For reference, uniform rubbles are considered with mound slope cot1.5 (Figure 2 ).

c = a = r = 1 , ,sin3,3,1,2 / ======== j AAl AG BGCA BC AB

=U=cot-11.5=33.69°, V=4R/3, A=R, d=V/A=4/3.

Sheltered area AS between 2 intersections P1 and P2 of circles A and Bis calculated from simultaneous equations.

B : 122=+ y x , A : 1)()1( 22

=+ j y x

Then exposed ratioV E is given, subtracting AS from total cross-sectionaarea R as

/1 S E A= …………………………………………..………….(11)

l, s,µA , f and S are successively obtained (Table 6).

2) a = 0.9

In order to evaluate the effect of unequal weight ( diameter ), it iassumed that a rubble with diameter a ( W 1 , sphere A in Figure 2slides on footing standard rubbles (a =1 ).

.)2(1)1(,1 222+=+===+= aaa BG ABl AGa AB

Putting a = 0.9,

764



231 9.0,3/9.04),616.1/cos3(cos

,616.1)29.0(9.0

×=×=

==+=

AV

l

and other values are likewise calculated.

Table 6. Stability factorSpheroidShape Sphere

Slender Flat

Eq.

A 0.9 1.0 1.1 1.5 2.0 1.2 1.4B 0.9 1.0 1 .1 0.82 0.71 0.69 0.51

a/b 1.0 1.0 1.0 1.83 2.82 1.72 2.74II 1.0 1.0 1.0 0.547 0.355 0.581 0.365 6IS 1.0 1.0 1.0 3.35 8.01 1.72 2.74 6IF 1.0 1.0 1.0 1.83 2.82 2.96 7.51 60 26.90 33.69 38.71 3.093 27.92 3.594 37.27 7

/2 - 1.101 0.983 0.895 1.031 1.084 0.944 0.920 7I 1.616 1.732 1.847 1.680 1.631 1.665 1.686 7

µA 1.064 1.046 1.029 1.054 1.061 1.038 1.034 7s 1.779 1.703 1.653 1.732 1.768 1.572 1.551 7d 1.200 1.333 1.467 1.093 0.947 1.600 1.867 8

E 0.482 0.637 0.700 0.517 0.466 0.642 0.675 11,14f 0.645 0.660 0.666 0.648 0.644 0.661 0.664 10S 3.522 3.598 3.747 3.079 2.758 3.950 4.509 9

+ 0.9 1.0 1.1 0.82 0.71 1.2 1.4 20

- 1.087 1.0 0.946 1.044 1.079 0.915 0.895 20K 0.979 1.0 1.041 0.856 0.767 1.098 1.253 20

3) a = 1.1

Similarly to a = 0.9,

231 1.1,3/1.14),847.1/cos3(cos

,847.1)21.1(1.1

×=×=

==+=

AV

l

(3) Unequal shape

1) Slender spheroid a = 1.5

To abstract the effect of unequal shape, here are examined slenderspheroid and flat spheroid, which have the same weight as standardsphere. Slender and flat spheroids imply 3b>a>b=c and 3b>a=c>b,respectively.

Major axis is supposed to be horizontal and perpendicular to wavedirection, minimizing potential energy and resistance against wave

force ( Figure 4).

A point of contact P ( x ,y ) and AG = l are decided on 1-1 sectionwhich contains A, B, G and P, solving simultaneous equations with thesame tangent, differentiating them ( Figure 4).

Figure 4. Slender spheroid

)(/)1(//,1/)(/)1(:,1: 22222222 yl a xb y xdxdybl ya x A y x B ===+=+ …

……………………………………………………………….……..(12)Slender spheroid is supposed to have the same weight as standasphere.

Sheltered area AS is obtained from simultaneous equations.

1/)(/)1(:,1: 222222=+=+ b j ya x A y x B ……..….…..…….(13

Exposed ratio E is given asab AS E /1= …………..…………….………….………..…….(14

For a = 1.5, 3/43/4 2abV == and b = 0.82.

sin),/cos3(cos,cos3 /1/ l j AAl I A ====

l, j, s, µA , f and S are calculated with Eqs.( 12 ) to ( 14 ) in Table 6.

2) Slender spheroid a = 2.0

b=0.71, from V=4R/3=4Rab2 .Eqs.( 12 ) to ( 14 ) give all values in the same way as a = 1.5

( Table 6 ).

3) Flat spheroid a = 1.2

Flat spheroid is most stable when minor axis is vertical ( Figure 5).

Figure 5. Flat spheroid

A point of contact P ( x, y ) is on 2-2 section together with A, A/ and D

754.11441.1

,441.1cos3

2222/

//

=+=+

===

ID I A

D A I A

P and AA/ = j are solved with simultaneous equations and thedifferentials.

==

=+=+

)1(/)754.1(//

1/)(/)754.1(:,1:

22

222222

ya xb y xdxdy

b j ya x A y x B……..…….…(15)

2 intersections P1( x1 , y1 ) and P2( x2 , y2 ) are gained fromsimultaneous equations.

1/)(/)1(:,1: 222222=+=+ b j ya x A y x B ………..….……..(16)

Sheltered area AS consists of A1 for sheltered circle B and A2 fosheltered ellipse A.

765



With arguments )/(tan 111

1 x y= and )/(tan 22

12 x y

= ,

2/)2sin(,,2/ 11221 ==+= A A A AS ……….…………(17)

To calculate A2, ellipse A is transformed into circle with radius a,multiplying b by a/b. 1 and 2 are measured clockwise from someradial direction.

2/)2sin(, 212 == ab A …………………………….…..(18)

Exposed ratio E has the same expression as slender spheroid.

ab AS E /1= ………………………….……...………………..(19)

l is variable for flat spheroid.

22/1 j I A AI l +== for initial condition = and 12 += bl for final

condition =/2. However l can be averaged as 2/)( 21 l l l += , for l

differs little with .

For a = 1.2, 3/43/4 2baV == and b = 0.69.

All values are derived in the same manner as slender spheroid.

4) Flat spheroid a = 1.4

b = 0.51 and calculation is the same as a = 1.2.

(4) Stability convergence

It is natural to presume that stability S is proportional to average depthd. In order to prove this presumption, relative depth K + and relativestability K are listed in Table 6, where d 1 and S1 are d and S forstandard sphere a = 1, respectively.

It is clear that K deviates less than K +. Then relative resistance K is

introduced as

11

1

)//()/(,/

,/,

f s f s K d d

K S S K K K K

A A µ µ =

===

++ ………………..………….……….(20)

and also attached in Table 6.

Thus K alleviates deviation of K + , diminishing µA and s and

augmenting f, when d increases, and vice-versa.

This counteraction tends to converge stability within narrower rangethan the deviation of K + affected by unequal weight and shape.

“Stability convergence” is expected to make over-sensitive solicitudeunnecessary, concerning unequal weight and shape of rubbles.

CONCLUSION

Hydraulic model should maintain similitude not only on hydraulics butalso on rubbles.

Prototype rubbles are replaced by model crushed stones with dissimilarshapes. Rubble and crushed stone can be regarded approximately as

polyhedron and spheroid, respectively.

As overall indices, angle of repose and void ratio e of rubble moundshould the same, keeping similar weight distribution and shape betweenrubble and crushed stone.

Median weight W50, equality factor IE = W25 / W75 <1 and mediafactor IM = W50 / ( W25 + W75 ) <1 are recommended as weighdistribution indices of rubbles.

Isotropy factor II = c / a<1, flatness factor IF = ab / c2>1, slendernesfactor IS = a2 / bc >1 and prism factor IP = V / abc<1 are introduced ashape indices.

Stability is defined as energy necessary to slide a rubble up tinstability. Stability factor S =µAsd / f can evaluate the effect of weighand shape of rubble on stability.

Relative stability K = S / S1is a product of relative depth of a rubble i

wave direction K + = d / d1 and relative resistance K = (µAsd / f )

(µAsd / f )1 as += K K K , where suffix 1 means corresponding value

for standard sphere r=1. K counteracts K + and stability tends t

converge, alleviating deviation of unequal weight and shape.

“Stability convergence” is expected to turn over-sensitive solicitudunnecessary concerning unequal weight and shape of rubbles.

ACKNOWLEDGEMENT

The author expresses his gratitude for assistance of Geo. Eng. Mr. AziÜNAL from Port Hydraulic Research Center in Ankara during th

preparation this paper.

REFERENCES

Deresievicz, H. ( 1958 ) : Mechanics of granular matter, Advan. AppMech. , Acad. Press Inc. Vol. 5, pp. 233 ~ 306

Homma, M. ( 1954 ) : Hydraulics – hydrodynamics for engineers, p233, Maruzen Publ.*

Hudson, R. Y. (1959 ) : Laboratory investigation of rubble – moun breakwaters, J. WW3., Proc. ASCE, pp. 93 ~ 121

Japanese Geotechnical Society ( JGS, 1982 ) : Handbook for somechanics, 1505p.*

Japan Port and Harbor Association ( JPHA, 1989 ) : Technical standarfor port structures and explanation 469p.*

Koh R. ( 2001 ) : Theoretical analysis on wave transmission, reflectioand stability of rubble moun breakwater with a granular model, Proc11th ISOPE Conference, Vol. pp. 645 ~ 650

Mogami, T. (1968 ) : On angle of internal friction and k of gravel, Soand Foundation, Vol. 16, No. 11*

Mogami, T. ( 1969 ) : Soil mechanics, pp. 905 ~ 908, Gihodo Publ*

Mogami, T. ( 1970 ) : Construction material of dam – especially sheastrenght of material, Large Dam, No. 54*

Shore Protection Manual ( SPM, 1984 ), Vol. , 2nd ed. P. 6 ~ 97, DepArmy, US Army Corp. Eng., Washington DC

* : in Japanese

766

isope-i-03-311_similitude and stability of rubble unequal in weight and shaped

Documents