geosynthetic/soil studies using a geotechnical centrifuge

Geotextiles and Geomembranes 6 (1987) 133-156

Geosynthetic/Soil Studies Using a Geotechnical Centrifuge

Arthur E. Lord Jr

Geosynthetic Research Institute, Drexel University, Philadelphia, Pennsylvania 19104, USA

ABSTRACT

Presented herein is a brief introduction to the centrifuge as used in geotechnical engineering studies. The rationale for centrifuge studies and the basic background are discussed in the Introduction. More scientific detail about centrifuge studies is presented in the 'Physics of the centrifuge' section, including derivation of the radial stress distribution in the model. Some history and mention of previous uses is given in the section entitled 'History and uses of the geotechnical centrifuge'. Another section deals completely with scaling considerations. This is an important and difficult area and discussions of geotextile and geotextile/soil interactions are given. Previous centrifuge studies of reinforced slopes are detailed. The Drexel Geotechnical Centrifuge is described and a conclusion ends the body of the report and mention is made of future work planned for the Drexel system. The article casts a critical eye at certain aspects of the centrifuge problem. There appear to be some possible fundamental problems concerning the basics of centrifuge modeling that have not been adequately addressed theoretically or experimentally to date.

1 I N T R O D U C T I O N

The use of small-scale models to study physical phenomena is widespread in the engineering field. For example, wind tunnels are used extensively in aerodynamic studies and large hydraulic models of actual physical situations are of ten used. In the geotechnical field, however, the stress levels in a small laboratory model are not the same as the stress levels in the full scale

133 Geotextiles and Geomembranes 0266-1144/87l$03.50 O 1987, Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain

134 A r t h u r E. L o r d Jr

counter wff.

o r

Id tn f i co l model

L.-J

(at resf)

r 1

Cat rest}

Fig. 1. Schematic diagram of the centrifuge.

prototype. One possible remedy for this problem is the use of an artificial gravitational field, which makes the model appear heavier. The centrifuge provides a method for supplying an enhanced gravitational effect. 1 The increased gravitational field is supplied by the centripetal (radial) acceleration. (A common use of centrifuges is for separation of the components in liquid suspensions, e.g. separating the cream in milk and separating suspended particles in medical laboratory samples.)

Figure 1 is a schematic diagram of a centrifuge. The container, with the soil model, hangs downward when the centrifuge is at rest. As the centrifuge is rotated the model rotates upward and, at sufficiently high rotational speed, the model rotates in a horizontal plane. In this final position the centripetal acceleration is

V 2

a , - - r J (1) r

where v is the tangential velocity, r is the radius, and to is the angular velocity.

According to Newton's second law, the bottom wall of the container must push on the soil container in a direction toward the center with a total force F, = mar, where m is the mass of the soil. From Newton's third law, the soil mass pushes back on the container wall with an equal force mar which is the apparent weight. If the centripetal acceleration is ng (g = acceleration of gravity and n is an integer), then the soil is pushing outward with a force m ( n g ) = n ( m g ) , which is n times the zero velocity weight. Hence as far as self weight effects are concerned, the model corresponds to a prototype structure n times larger (in the radial direction) than the model. Therefore gravitational stresses in the model on the average are equivalent to those in a prototype n times as large. (It is assumed that strains and lateral stresses are equivalent in the model and prototype.) Thus prototype situations of

Geosynthetic/soil studies using a centrifuge 135

various sizes can be simulated by varying the rotational velocity of the centrifuge, with the same size model. The same size prototype can be simulated with different size models and different rotational speeds. This is called 'modeling of models' . Centrifuges capable of accelerations of 100 g or above have been used in the past.

It should be ment ioned that there are effects other than those of self weight which may be difficult or impossible to completely model in a centrifuge. However information as to the consequences of the lack of similarity between model and prototype can be obtained by the afore- ment ioned 'modeling of models' method. At any rate, in most cases, building full scale models is prohibitive from a cost and logistics standpoint and centrifuge modeling offers a very attractive (if not the only) experimental alternative to full scale experiments.

The combination of an experimental centrifuge effort with a theoretical numerical modeling approach would seem to be a viable approach to soil structure problems, including those involving geosynthetics.

2 PHYSICS OF THE C E N T R I F U G E

It is essential to know the radial stress distribution in a model undergoing centrifugation. Applying Newton's second law in the radial direction to a small e lement of the model rotating in the horizontal plane (see Fig. 2) we have 2

F 2 - El = d F = d m a , (2)

where

F = force

dm = mass of the element

ar = - - = radial acceleration r

Fur thermore ,

d m = t a d V = o d r A (3)

where

p = mass density

dV = volume of mass dm

dr = radial thickness of dm

A = area of drn

136 Arthur E. Lord Jr

tO

ro /r-A f,, II A

Fig. 2. Schematic drawing of a soil element in the rotating body, showing the radial forces.

and with the definition of stress,

dF do" - (4 )

A

we have (using eqns (2) and (3))

d F = p d r A a r

d F do" - - pa~dr (5)

A

The radial acceleration is given by eqn (1) yielding

do" = proJ 2 d r

Integrating from the free inner surface of the model (r = r0) where the stress is zero, we obtain, for a given to, the radial stress at the position ri

J'0~' do-= p~o ~ j'i~ rdr (6)

P°J2 (4 - ro 2) (6a) O'i = 2


D u e to the unfamil iar na ture of this general a rea to mos t pract ic ing geo techn ica l engineers , it might be o f help in placing actual SI units in eqn (6a) and de t e rmin ing some stress levels. W e shall use the pa rame te r s for a typical soil m o d e l in the centr i fuge:

O = 2000 kg/m 3 (2-times the density of water)

to = 27r(4/s) = 87r/s (240 rpm)

ri = 1.0m

r0 = 0 .9m

O " - -

2000 (87r) 2 ( 12 _ (0.9) 2 ) 2

~r = 1.2 x 105 N/m 2 (or 17.4 psi)

Us ing this in the express ion for vertical stress with dep th

tr = p g h (7)

1-2(105) = 2000(9.8)h

1.2(105 ) - h

2(0-98)(104 )

6 .12m = h

T h u s this stress co r re sponds to a dep th of 6.12 m in a real p ro to ty p e (1 g) soil s t ruc ture . T h e app rox ima te radial scaling factor n is easily seen to be (with ri - r0 = 0-1 m)

6.12 n - - - - 61.2

0.1

T h e scaling fac to r is also given by

V 2

n g = - - = to2r (8) F

which for o u r example comes ou t to be

n(9.8) = (8702(1)

(8~r) 2 n - - - - 64.4

9.8


The two values of n are slightly different due to the slight nonlinearity of eqn (6a).

If the model 's dimension is small with respect to the average radius of the centrifuge, then the radial stress distribution is approximately linear with depth from the free surface. This can be seen from eqn (6a).

2

_ - - r 0 (3" i

p¢o o'i = T [ ( r i + r0)(ri- r0)] (6b)

If ri - r0 = Ar < < r0 then eqn (6b) can be rewritten (ri ~ ro)

2

cri ~ p¢o 2roAr 2

cri = poj2 roAr (6c)

This gives the approximate linear dependence with depth. Equation (6a) also demonstrates the scale factor concept. Equation (6c) can be written

o-i = p(oJ2 ro)Ar = p a , A r (6d)

which agrees with the standard stress/depth relation (eqn (7)) o-~ = pgh if

Ar = h

and

a, = geff = ng

The deviation of the linear eqn (6c) from the true stress distribution eqn (6a) can be shown to be

Ar O't . . . . 1 + - - (9) O'linear 2r0

Table l shows this ratio as a function of Ar/2ro and also the percentage by which o-~ . . . . is below the true stress. It must be realized that if Ar represents the maximum depth of the model, then the average stress is only about 50% of this departure. The table shows that if the dimension of the model is 10% or less of the radius of the centrifuge, then the maximum deviation from the


TABLE 1 Deviation of True Stress from Linear Approximation

Ar/ ro 1 + Ar/2ro Linear model underestimation (%)

0.025 1.012 5 1-2 0-05 1.025 2.4 0.075 1.037 5 3.6 0.10 1.05 4.8 0.20 1.10 9 0.30 1.15 13 0.40 1.20 17 0.50 1.25 20

linear stress approximation is about 5% and the average deviation only about half of this. This is certainly acceptable and accounts for the fact that the 10% limit is often quoted in the centrifuge literature. (It seems that in some reported experiments the 10% limit is violated.)

While we are on the physics of the problem, it is interesting to calculate the angle, 0, from the vertical that the model makes as a function of rotational speed. The equations are worked out in the Appendix and only the result is given here. A transcendental equation results:

g tan 0 = to 2 (R + I sin 0) (10)

where R = length of the centrifuge arm, and l = distance from the arm pivot point to the center of mass of the model.

The results of a graphical solution of eqn (10) are shown in Fig. 3 for the case of R = 1 m and/ /R = 0.1 which is pertinent to many small centrifuges. Experimental verification ofeqn (10) is shown in Fig. 3, the data being taken on the Drexel one meter centrifuge. This result is quite important, for it shows that a centrifuge can only be used above a certain rotational speed if the model is to be rotating essentially in the horizontal plane (and hence have the radial force directed perpendicular to the bottom wall). Figure 3 gives this as about 90 rpm, which corresponds to a scale factor n of 22.

3 HI STORY AND USES OF THE GEOTECHNICAL CENTRIFUGE

This section will borrow heavily from the excellent review of Chaney and Fragaszy. ~ Phillips 3 in France (in 1869) made the first suggestion of the use of a centrifuge to study self-weight stresses in beams. Bucky 4 (USA) and Pokrovsky 5 (USSR) and Davidenkov 6 (USSR) worked on centrifuges in t h e


90

80

70

6 0

"~ 50 w

N 40 Z

3O

20

I0

0 0

/ / z / THEORY

, ~ . . . . EXPERIMENT

f

I i i i i i i i

I 2 3 4

FREQUENCY (revo/./sec)

60 180 2 4 0 120

RPM

Fig. 3. Angle of model (with respect to the vertical) versus the frequency of rotation. Comparison of theory and experiment.

early 1930s. A few projects were carried out subsequently in the U S A , 4"7'8

while in the USSR more than 50 centrifuges were built for model testing of dams, foundations, earth fill embankments, etc., in the period 1932-1980. 9

In the late 1960s Rowe at University of Manchester Institute of Science and Technology (UMIST) and Schofield at Cambridge University developed excellent geotechnical centrifuge programs. In the early 1970s Rowe built a large machine at Simon Engineering Laboratory at the University of Manchester and a 3.7 m (12 ft) radius machine was developed at Cambridge by Roscoe and James. The Cambridge project was taken over by Schofield on his return to Cambridge from Manchester in 1974. England is a most active center for geotechnical centrifuge studies and probably has produced more PhDs and hosted more sabbatical visitors than any other country in the world.

At present centrifuge activities are ongoing in many European countries and Japan. The USA had a late start in the use of the centrifuge on geotechnical problems. A small centrifuge was built at the University of California at Davis in 1972 under Chaney. Scott at the California Institute of Technology was active by 1975. Schmidt at Boeing in Washington State worked on cratering starting in 1976 and Ko at the University of Colorado began a program in 1978. An indication of the present activity in the USA is given in Table 2 (from Chaney and Fragaszyt). The very large centrifuges (8-10 m radius) at Moffet Field (previously


TABLE 2 List of US Centr ifuge Facilities

Institution Radius Acceleration Payload Capacity (ft a) (g) (lb b ) (g ton c)

Boeing C o m p a n y 4.6 600 122 66 Bureau of Mines at Mary land 3 high - - - - Univers i ty of Cal i fornia at Davis

Swing p la t form 3.25 Drum 2

Cal i fornia Inst i tute of Technology 4.2 Univers i ty of Co lo rado at Boulder 4.5 C o l u m b i a Univers i ty 1 Univers i ty of Florida

A 3-3 B 6.6

Univers i ty of Mary land 4.4 Missouri School of Mines at Rolla 3.5 New Mexico Engineer ing Research Insti tute 6 Pr ince ton Univers i ty 4-2 Queen's Univers i ty at Onta r io 1.1 Sandia Corpora t ion , CA-2 7

Swing 25 Fixed 25

20

175 - - 5 600 -- 900

50 - - - - 300 220 10 700 1 0.036

100 50 2.5 160 185 - - 200 1(~) 10

100 500 25 200 249 10 000 1 10 150 500 15 150 4 000 300 240 16 0O0 8(10

a 1 ft = 0.304 8 m. b l Ib = 0"453 6 kg. c I ton = 1.016 x 103 kg.

N A S A and run by the University of California at Davis) and Sandia are not operational at present as far as the author is aware. (The Davis effort has now been moved back to campus.) The Russian centrifuge at Baku is also extremely large and its status at present is unknown to the author.

Centrifuges have been employed in research on a large number of geotechnical-related problems:

• Beam-soi l interaction. 10 • Bearing capacity. 11-13 • Earthquake effects. 14 • Excavations. 15 • Flow simulation. 16 • Foundations. iv • Ice forces. 2 • Liquifaction. TM

• Phreatic surfaces observations.19


• Sand drains. 23 • Slurry walls. 24

• Slope stabil i ty--many projects. 25'26 • Subsidence. 27 • Tunnels. 28

4 SCALING CONSIDERATIONS

Scaling, or model-prototype similarity conditions are most important in centrifuge studies. The scaling of length and mass has already been discussed in Section 2. A brief (and quite incomplete) introduction to scaling will be a t tempted here with particular comments concerning the application of scaling to geosynthetics and the geosynthetic/soil interaction. Our t rea tment here relies heavily on Ref. 1 and the work o fHoekf l 9

Scaling relations can be determined in two ways:

• By evaluation of the differential equations governing the behavior, or • By dimensional analysis.

As an example of the first approach we consider the differential equation for one dimensional soil consolidation

/~u k(1 + e) 02/~/ - ( 1 1 )

0t ay 0z 2

where

u = pore water pressure e = void ratio k = coefficient of permeability y = unit weight of water a = coefficient of compressibility

z and t = the spatial and temporal variables

The scaling factor for z is 1/n yielding

Zp = nZm (12)

where the subscripts p and m refer to the prototype and model, respectively. The other quantities are scaled accordingly

U m = O[uU p (12a)

'Ym = ay~/p (12b)

tm = a t tp (12c)


km = akkp (12d)

am = a.ap (12e)

where o~ represents the scaling factor for the particular quantity and it is assumed that the void ratio need not be considered.

Equat ion (11) refers to the prototype. For the model and prototype to be similar in soil consolidation behavior, eqns (12) are put in eqn (11) and conditions placed on the o~s.

O(auu) _ akk(l +e) [O:(a,u) ] O(a,t) a, aa, y [ ~ J (13)

o r

Ou _ (Otkoqn____~2 ~ (k(l+___e) O~ (13a) Ot - ] - \ or, c% \ aT ] Oz 2

Comparing eqn (11) with eqn (13a) shows that for similarity of the model and prototype, the following relation must exist among the various scale factors

aka'n-----~2- 1 (14) OgaO( 3,

If the permeability, unit weight of the fluid and the compressibility of the soil are the same between model and prototype, i.e., ak = a, -- % = 1, then

1 (14a) O/t - - ?/2

Thus

1 tm= -~tp (15)

which says that the same process (e.g. consolidation of the model) happens nZ-times faster in the model than in the prototype. (Obviously with choices of the o~s other than one, tm 4= 1/n 2 tp.) This is easy to understand in physical terms. Consolidation is a diffusion-type process, in which the time for a process to occur depends on the pertinent distance squared. The model is only (1/n)-times as thick as the prototype, therefore (with the same k) the process takes only (1/n)2-times as long. In exactly the same manner one can easily show that a process such as Darcy flow which has a velocity associated with it will take (1/n)-times as long in the model as the prototype. Due to the

144 Ar thur E. Lord Jr

T A B L E 3 List of Scaling Relations

Full scale Centrifuge model Quantity (prototype) at ng

Linear dimension 1 l /n

Area 1 1/n 2 Volume 1 1/n 3 Time

Dynamic events 1 1/n Hydrodynamic events l 1 /n 2 Viscous flow 1 1

Velocity (distance/time) 1 1 Acceleration (distance/time 2) 1 n Mass 1 1/n 3 Force l l / n 2 E n e rgy l l / n 3 Stress (force/area) 1 l Strain (displacement/unit length) 1 l Density 1 l Frequency 1 n

great time savings involved, it would appear that flow and consolidation studies would be most popular in the centrifuge. In point-of-fact, they are not as yet. Therefore there may be either significant difficulties with the logistics of such measurements or with the scaling theory.

A set of scaling relations is given in Table 3 which is from Scott and Morgan. 3° (All entries in Table 3 are not obvious to the author, but this will possibly be addressed at a later date.)

In the second approach for determining the scaling relations, no differential eqn(s) is (are) available describing the process and therefore dimensional analysis must be used. A very readable treatment of this approach as applied to elastic problems is given in Hoek. 29 Buckingham's theory is used in which the displacement and stress at a point are written as a function of a pertinent number of dimensionless parameters. These parameters for the elastic problem are x, y, z, t, the Young's modulus of the material (Poisson's ratio is already dimensionless), density, g, applied stresses and forces and applied displacement, internal stresses and possibly other variables. For similarity between the model and the prototype as applied to the properties of the materials, the most pertinent result is (there may also be other relations to be satisfied 29)

Lp _ Ep Pm gm a (16)

Lm Em pp gp


where

L = the length E = modulus of elasticity (it could be another pertinent elastic constant but not

Poisson's ratio) g = acceleration ct -- the ratio between the resulting stress at an equivalent point in prototype

and model, i.e. O-p = Carm

For our purposes assume a = 1, and hence

L___i = E___2_p Pm gm (16a) Lm Em pp gp

Thus for modeling at 1 g, the scaling will have to be accomplished by changing the properties in the model such that

Lp _ Ep Pm

Lm Em pp (16b)

It is easy to see that if L p / L , = n and Ep = Em, then if the model material is n-times more dense, the self stresses are n-times larger. If the scale factor Lp/Lm < < 1, it may prove difficult to find materials with widely varying moduli and/or density. This points up the beauty of centrifuge modeling in that the scale factor can be achieved by changing the gravity field while using the same soil. More detail on scaling can be found in Hoek. 29 (m very often referenced book in the scaling area is Langhaar? ~)

Scaling considerations with regard to geotextiles and geotextile/soil interactions have been discussed by Blivet et al. 22 The present author has difficulty with some of the results of these workers. The limit equilibrium analysis of a geotextile reinforced slope will be given and then dimensional analysis applied to the resulting equations. Figure 4 shows a schematic diagram of a geotextile reinforced slope. A circular arc failure is assumed, and taking moments about the center of the circle, we can write the equil ibrium of moments in the conceptual form

EFo = 0 = mgR1--rlabRo--TR2 (17)

where

F0 = the sum of moments about point 0 m = the mass of the slipping (rotating) region r = the resisting shear stress on the failure arc (this is shear force per length of

arc) lab = arc length 'ab'

T = ultimate stress in the geotextile


b

Fig. 4. Schematic drawing of model for calculating limit equilibrium ofa geotextile reinforced slope.

Equat ion (17) is written for one unit thickness perpendicular to the plane of the drawing. Thus for dimensional analysis the one unit thickness is included.

For a purely frictional material (e.g. a granular soil) we can use

7i = Ni t a n 0 i (18)

ri = mig tan 01

Equat ion (17) becomes

(h ) mgR1 = ~ migtanOi liRo+ TR2 (17a)

i 1

Writing eqn (17a) in dimensional form, with a length scale factor of n,

(17b)

The first n (unparenthesized) on the right hand side comes from the fact that 7- is a force/length along the arc 'ab'. (Note that there would appear to be scaling problems with a c - ~b soil because the cohesion would not scale with mass. However , centrifuge slope stability studies are performed on cohesive soils with no mention of this problem, to the author's knowledge.) From eqn (17b):

I aT -- n2 (19)

i.e. scaling says that the tensile strength of the geotextile should be (l/n2) - times as large in the model as compared to the prototype. Blivet et al. 22 find that T should scale as 1/n. Regardless of which is correct, on physical


grounds both values must be wrong. If a much weaker geotextile is placed in the model, it will fail at much lower force than the designed tensile force which uses the full strength geotextile at the prototype size. This problem is undoubtedly associated with the fact that a geotextile (or geosynthetic in general) is not an inertia element, i.e. it doesn't rely on its weight (or imposed weight) for its strength.

The previous remarks apply to tensile failure of the geotextile. In the case of pullout failure, scaling theory will probably not give insight (or lack thereof). As long as the full friction can be developed (mobilized) between the geotextile and the soil, independent of thickness of the geotextile, then scaling will not be a factor.

A very important factor to keep in mind in all of this centrifuge work (and geosynthetic work, in general), is the effect of soil confinement on the properties of the geosynthetic. In this regard, there is very little experimental information available to date. For example, the tensile strength of a certain geotextile (like a needle punched nonwoven) could be very sensitive to soil confinement.

5 PREVIOUS REINFORCED SLOPE STUDIES USING THE CENTRIFUGE

There have been three reports of centrifuge studies involving reinforced slopes or walls as far as the author is aware. Two PhD theses were conducted at Manchester Institute of Science and Technology, 2°'21'33'34 both involving non-geosynthetic reinforcement. Multiple layer reinforcements were mild steel rods (1.5 mm diameter), aluminum strips (0.05 mm thick) and stainless steel strips (0.10 mm thick), their strength and roughness chosen so that the model could be brought to collapse either by slippage of strong strips or tensile failure of the rough, weak strips. The soil model of dry sand was of dimension 200 mm high by 400 mm long by 160 mm deep (of mass about 20 kg) and the average centrifuge radius was 1.5 m. (Note here Ar/ro ~- 160/ 1500 or 10-5%.) The strips were instrumented with many strain gages and two total stress transducers (of 10 mm diameter diaphragms) were also used. The strain gage data allowed the determination of the distribution of tension in the reinforcement layers. (A maximum value of tension along the strips was obtained in most cases.) The average lateral earth pressures acting on the strips were determined from the data. Combining the lateral earth pressure data with an estimate of the vertical earth pressure yielded a map of earth pressure coefficients. Much analysis was done concerning earth pressure coefficients and various methods of analyzing reinforced earth walls.


Blivet et a l . 22 have commented on the details of the large 5.5 m radius centrifuge (Nantes, France) devoted in large part to studies of geotextile reinforced walls and slopes. A dry sand model wall of dimensions 600 mm high x 800 mm long × 400 mm deep ( -300 kg) was strengthened by six layersof a nonwoven, spunbonded polypropylene of density 100 g/m E. The properties of the geotextile were:

failure stress = 6.6 kN/m failure strain = 21% initial modulus = 70 kN/m

The wall was instrumented with six horizontal displacement transducers (for the wall face), four surface settlement transducers (for the top) and four inclinometers with strain gages for internal displacements of the soil mass. The top of the wall could be surcharged with water. The wall was taken to 31 g at which value of acceleration the fabric was predicted to break. Breakage, however, did not occur. Results of horizontal displacement and surface settlement are given for equivalent wall heights of 3.6 m and 9 m. At 31 g there was overturning of the wall, i.e. a horizontal displacement of the top of the wall of 1% of its height with no displacement of the bottom of the wall. The results were said to be the first of a series of tests on the effect of geotextile reinforcement of walls and slopes.

6 THE D R E X E L GEOTECHNICAL CENTRIFUGE

Many of our centrifuge design concepts come from observations of the University of Kentucky's geotechnical centrifuge.32

Figures 5 and 6 are photographs of the Drexel centrifuge. It is built in a deep concrete-lined pit, which previously housed a low energy ( - 5 0 keV) particle accelerator. This serves as a very safe location in the case of a failure at a high speed of rotation. The centrifuge is belt-driven by a two horse- power motor. The radius of the arm is I m and with the bucket and counterweight shown is capable of a radial acceleration of about 60 g. The bucket can carry up to 445 N (100 lb), making this a 3g-ton machine. According to Table 2, this would be rated as a modest capacity centrifuge. The soil/geosynthetic models are constructed in aluminum boxes, with plexiglas end windows for viewing, of dimensions 11.5 cm x 15-3 cm x 25.4 cm (see Fig. 7). The electrical slip ring assembly shown has 24 contacts, which will connect displacement measuring probes, pore water cells and load cells from the model to the control room readouts. Strobe lighting, triggered magnetically by a Hall effect probe circuit, makes possible real

Fig. 5. Overhead photograph of the Drexel geotechnical centrifuge.

Fig. 6. Side photograph of the Drexel geotechnical centrifuge.


Fig. 7. Photograph of the model container on the Drexel geotechnicai centrifuge.

t ime observat ions on the model with a closed circuit TV system. These are seen in Figs 5 and 6. Reflecting balls or painted stripes placed in a regular array in the model , will allow general movement (and strains) to be mon i to red during rotation.

7 P R E L I M I N A R Y E X P E R I M E N T A L R E S U L T S

The Drexel centrifuge is still in its shake-down phase, and only preliminary exper imenta l results are available. We present comparison of the stability of two steep model sand slopes (of height 5.08 cm)---one unreinforced and one re inforced with a 'spider-netting'. 35 The spider-netting is a light-weight heat set, nonwoven staple filament polypropylene geotextile 'nailed' into the slope face, toe and top with finishing nails. Figure 8 shows the unreinforced slope after being rotated to an equivalent height of about 152.4 cm. It is seen that a definite slope failure has occurred. In the case of the reinforced slope (Fig. 9), it has maintained stability to an equivalent height of 335.3 cm, which is the maximum available at present on the centrifuge ( n g = 66 g).

Fig. 8. Model of reinforced sand slope of soft clay foundation after spinning to an equivalent height of 152.4 cm.

Fig. 9. Model of reinforced sand slope on soft clay foundation after spinning to an equivalent height of 335.3 cm.


8 CONCLUSION

This article has given a brief review of the geotechnical centrifuge together with descriptions of the Drexel centrifuge and a preliminary experiment. The centrifuge appears to be a viable modeling procedure to study soil/ geosynthetic problems at the large scale. It probably is the only full scale modeling approach available. The design aspect of geosynthetics, which is in its infancy, could very well use data from centrifuge related studies.

The author feels that there are fundamental aspects of centrifuge modeling still to be researched. For example, lateral stress effects are not well understood. Scaling has not been adequately addressed in many problems (e.g. clay soils, geosynthetic and geosynthetic/soil interaction). Placing instruments in small soil models could lead to unacceptable disturbance effects.

Some possible future work on the Drexel centrifuge includes:

• soft soil base/sand berms, with geotextile reinforcing elements. • spider netting soil-reinforcement. • forces developed in geomembrane liners on slope walls of landfills. • coastal erosion barriers. • fundamental centrifuge experiments (not project oriented).

ACKNOWLEDGEMENTS

The author would like to express his appreciation to the geotechnical engineering group at the University of Kentucky for their hospitality during a visit to their centrifuge. Dr Harry Stirling, a previous member of the Kentucky group, under whose guidance the Kentucky centrifuge was built, was most helpful in providing an introduction to the work of small centrifuges. Mr Darryl Greer and Mr Mike Ronayne, previous graduate students at Kentucky, were very helpful. Also thanks go out to Mr Tom Edmunds and Mr Rick Ackerman for design and construction help in the building of the centrifuge. As always, Dr Robert Koerner has been a stalwart colleague.

REFERENCES

1. Chaney, J. A. & Fragaszy, R. J., The Centrifuge as a Research Tool, Geotech. Test. Jnl, 7(4) (1984), pp. 182-7.

2. Clough, H. F., Wurst, P. L. & Vinson. T. S., Determination of Ice Forces with Centrifuge Models, Geotech. Test. Jnl, 9(2) (1986), pp. 49-60.


3. Phillips, Comptes Rend. Acad. Sci., Paris, Jan.-June 1869, p. 68. 4. Bucky, P. B., The Use of Models for the Study of Mining Problems, Tech. Publ.

425 AIMME, NY (1931), pp. 3-28. 5. Pokrovsky, G. I., Zeit. f. Tech. Physik, 14(4) (1933). 6. Davidenkov, N. N., A New Method of Using Models for the Study of

Equilibrium of Structures, Tech. Physics of the USSR, 3(1) (1936), pp. 131--6. 7. Clark, G. B., Geotechnical centrifuges for model studies and physical property

testing of rock and rock structures, Colorado School of Mines Quarterly, 74(4) (1981) (63 pp).

8. Panek, L. A., Design of Safe and Economic Structures, Trans. AIMME, 181 (1949), pp. 371-5.

9. Pokrovsky, G. I. & Fyodorov, I. S., Centrifuge Model Testing in the Construction Industry, Vol. I, Centrifuge Testing in the Mining Industry, Vol. II, Niedra Publishing House, Moscow (1969).

10. Leshchinsky, D., Frydman & Baker, R., Study of Beam-Soil Interaction Using Finite Element and Centrifuge Models, Canad. Geot. Jnl, 19(3) (1982), pp. 345-59.

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13. Kimura, T., Kusakabe, O. & Saitoh, K., Geotechnical Model Tests of Bearing Capacity Problems in a Centrifuge, Geotechnique, 35 (1985), pp. 33-45.

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17. Almeida, M. S. S., Davies, M. C. R. & Parry, R. G. H., Centrifuge Tests of Embankments on Strengthened and Unstrengthened Clay Foundations, Geotechnique, 35 (1985), pp. 425-41.

18. Arulanandan, M., Anandarajah, A. & Abghari, S. M., Centrifugal Modeling of Soil Liquifaction Susceptibility, J. Geotech. Engng, 109(3) (1982), pp. 281-300.

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A P P E N D I X

Calculation of the angle of the model (with respect to vertical) versus the rotational speed.

Refe r to Fig. A1, which is a schematic diagram of a model rotating at angular speed, oJ.

Newton ' s second law for the radial direction can be written

V 2

S, Fr = mar = m - - = mto2 r = T (A1) r

where

E F r = the sum of the forces in the radial direction = T (the tensile force in the connecting rod)

m = total mass of the model ar = radial acceleration of center-of-mass (c.m.) of model


an d

v = tangential velocity of c.m. r -- radius of c.m.

Us ing

r = R + l s i n O (A2)

w h e r e R and l are shown on Fig. A1.

.,~Ua 7" •

- R

r

I Y

N

mg

Fig. AI. Schematic diagram of the rotating model used to calculate the angle of model versus speed of rotation.

E q u a t i o n (A1) becomes

mtoE(R +/sin0) = T (A3)

N e w t o n ' s second law in the vertical direct ion (neglect ing any vert ical a cce l e r a t i on ) is

~Fy = 0 = N-mg (A4)

o r

N = m g


If there is to be no angular acceleration of the model about its center-of- mass, then there can be no moments about the c.m. Thus

tan0 - T (A5) N

i.e. the vector sum of T and N passes through the c.m. (the weight m g

already goes through the c.m.). Combining eqns (A3) and (A5)

NtanO = T

mg tan 0 = mtoZ(R +/sin O)

gtanO = toZ(R + lsinO) (lo)

geosynthetic/soil studies using a geotechnical centrifuge

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