Duncan Friction Angle of Sand Stone11

Friction Angles for Sand, Gravel and Rockfill

by J. Michael Duncan

Notes of a lecture presented at the Kenneth L. Lee Memorial Seminar

Long Beach, California April 28, 2004

1

Friction Angles for Sand, Gravel and Rockfill

What I learned from Ken Lee as a role model

When you dig into something, dig deep.

Keep an open mind about what is useful and practical.

Why this topic?

• Ken’s interest – my interest.

• Data is available that has not been thoroughly evaluated in previous studies.

Thanks toYoungjin Park – Virginia TechBinod Tiwari – Virginia TechChris Meehan – Virginia Tech

Sathi Sathialingam – WCC (MWDSC)Yoshi Moriwaki – WCC (MWDSC)

Harry Seed – Berkeley RTF Clarence Chan – Berkeley RTFDean Marachi – Berkeley RTFEd Becker – Berkeley RTF

50 years of previous studies• Ken Lee and various co-authors – Seed, Dunlop, Singh,

Farhoomand

• Vallerga

• Zeller and Wullimann

• Marsal

• Tom Leps

• WCC for MWDSC on Diamond Valley Dam

• Marachi et al.

• Becker et al.

• 35 others, through Varadarajan et al. (2003)

Outline

• Review of the basics

• Compilations of data

• Evaluation of effects

• Improving estimates of φ

2

Components of shearing resistance (Lee and Seed, 1967)

0 5 10 15 201

2

3

4

5

6

Prin

cipa

l stre

ss ra

tio -

σ 1/σ

3

-5

0

5

10

15

Volu

met

rice

stra

in -

%

0 5 10 15 20Axial strain - %

σ3 = 21,500 psf

σ3 = 21,500 psf

0 5 10 15 201

2

3

4

5

6

Prin

cipa

l stre

ss ra

tio -

σ 1/σ

3

-5

0

5

10

15

Vol

umet

rice

stra

in -

%

0 5 10 15 20Axial strain - %

σ3 = 26,000 psf

σ3 = 26,000 psf

Dr = 100% Dr = 38%

σ3 = 2,048 psf

σ3 = 2,048 psf

σ3 = 2,048 psf

σ3 = 2,048 psf

Greater expansion, higher φ (Lee and Seed, 1967)

Curved failure envelopes

(Marachi et al., 1969)

Secant values of φ

φ is defined by a single test

Marachi et al. (1969)

Variation of φsecant with confining pressure

σφ = φ − ∆φ

3

0 10a

logp

Compilations of test results

3

Leps (1970) – 109 tests

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

1 10 100 1,000 10,000

Normal stress on slip surface (psi)

Fric

tion

Ang

le (d

egre

e)

φ = 55 degrees - 7 degrees x Log10(σN in psi)

WCC, working for MWD on the Diamond Valley Reservoir,doubled the size Leps’s compilation (from 109 tests to 226 tests)

Woodward-Clyde working on the Diamond Valley Reservoir For the Metropolitan Water District of Southern California – 226 tests

20.0

25.0

30.0

35.0

40.0

45.0

50.0

55.0

60.0

65.0

70.0

1 10 100 1,000 10,000

Normal stress on slip surface (psi)

Fric

tion

angl

e (d

egre

e)

φ = 57 degrees - 7 degrees x Log10(σN in psi)

Leps’s comments on his compilationof φ-values for rockfill

Virtues:

1. “Presents a good overall perspective of the relation of friction angles to normal pressure.”

2. “Illustrates the dearth of information at normal pressures below 10 psi.”

Leps’s comments (continued)Shortcomings:

1. “Only roughly indicates the effect of relative density.”

2. “Only roughly indicates the effect of gradation.”

3. “Only vaguely suggests the effect of particle strength.”

4. “Gives no clue as to the influence of particle shape.”

5. “Offers no evaluation of influence of degree of saturation of the rock particles.”

These factors can be evaluated by examining data now available

4

Effect of relative density

φ = 48oφ = 43oOroville amphibolite

φ = 41oφ = 38oVenatosandstone

φ = 51oφ = 48oPyramid Dam argillite

Dr = 85%Dr = 50%Material

φ increases as Dr increases

(Becker et al., 1972) σ3 = 30 psi, Dmax = 0.5 in.First conclusions:

• Relative density is the most important single factor governing friction angles of granular materials.

• It is essential to know relative densities in order to isolate the effects of the other factors on Leps’s list.

Data is available for 125 tests on materials with particle sizes up to 6 inches, where the relative densities of the test specimens are known:

► 69 tests on gravels with Cu > 4

► 26 tests on sands with Cu > 6

► 30 tests on sands with Cu < 6

The Unified Soil Classification System is not a good guide with respect to the influence of gradation on friction angles.

The classification “GP” is usually said to indicate uniformly graded or gap-graded material. This is not correct.

A gravel that is neither uniform nor gap-graded may classify as GP.

Effect of gradation

Grain size curves from Becker, Chan and Seed (1972)

0

10

20

30

40

50

60

70

80

90

100

0.010.1110100Grain Size (mm)

Perc

ent f

iner

by

wei

ght (

%)

#10 #20 #40 #60 #100 #140 #200#4

U.S. Standard Sieve Number

1/43/81/23/41

U.S. Standard Sieve Opening (in)

"Pyramid gradation" -- 2-inch max particle sizeCu = 7.2, Cc = 1.3USCS classification = GW

"Oroville gradation" -- 2-inch max particle sizeCu = 35, Cc = 4.2USCS classification = GP

Oroville gradationCu = 35

Pyramid gradationCu = 7

Higher Cu, higher φ

Effect of gradation (Becker et al. (1972)

5

Better than USCS classification (GW, GP):

(1) Describe as Gravel or Sand, based on the percent passing the #4 sieve,

(2) With the value of Cu

For example:

Gravel with Cu = 35, rather than GP for the Oroville Dam material.

Particle strength

Categories suggested by Leps (1970)

• Weak – qu = 500 psi to 2,500 psi

• Average – qu = 2,500 psi to 10,000 psi

• Strong – qu = 10,000 psi to 30,000 psi

Materials tested by Marachi et al. (1969) and Becker et al. (1972)

Particles disintegrated upon saturation

Very weakqu < 500 psiColorado sandstone

Easily broken with a hammer

Averagequ = 5,000 psiVenatosandstone

Readily broken with a hammer

Strongqu = 15,500 psi

Pyramidargillite

Very difficult to break with a hammer

Strongqu = 28,000 psi

Oroville amphibolite

Strong particles

Average strength particles

Becker, Chan and Seed (1972)

Particle strength

Very weak particles

Ratio of particle size to specimen size

• Holtz and Gibbs (1956) and Leslie (1963) showed that particles larger than 1/5th to 1/10th of the specimen diameter tend to interfere, resulting in measured strengths that are too high.

• These findings have led to the widely accepted rule:

Specimen diameter should be ≥ 6 x Dmax

Tests to measure strengths of rockfills

15 in. diameter3 in.Varadarajan et al. (2003)– Ranjit Sagar Dam & Purulia Dam

24 in. by 54 in. rectangular

36 in. diameter

44 in. diameter

20 in. diameter

Specimen size

4 in.Becker et al. (1972) – Oroville Dam & Pyramid Dam plane strain

6 in.Marachi et al. (1969) – Oroville Dam & Pyramid Dam triaxial

7 in.Marsal (1967, 1970) – El Infernillo Dam & Mica Dam

4 in.Zeller and Wullimann (1957) –Göschenanalp Dam

DmaxInvestigation

6

Scalping and modeling to remove particles that are too large for test equipment

Modeled grain size curves for Pyramid Dam material (Marachi et al., 1969)

Dr = 100% Dr = 71% Dr = 44%

Becker et al., (1972)

Effect of particle size (Becker et al., 1972)Dr = 85%, σ3 = 30 psi, modeled gradations

φ = 48oφ = 46oφ = 48oOroville amphibolite

φ = 40oφ = 41oφ = 41oVenatosandstone

φ = 49oφ = 48oφ = 50oCrushed basalt

φ = 48oφ = 48oφ = 51oPyramid Dam argillite

Dmax = 6 inDmax = 2 inDmax = 0.47 inMaterial

φ decreased as Dmax increased (or stayed the same)

Dr = 100% Dr = 71% Dr = 44%

Becker et al., (1972)

7

Effect of particle size – modeled gradations – quarried rockfillVaradarjan et al. (2003)

φ decreased as Dmax increased, for σ3 > 4 atmospheres

30.0

35.0

40.0

45.0

50.0

55.0

1.00 10.00 100.00

(σ3/pa)

Fric

tion

Ang

le (d

egre

e)

Dmax

3.2 in

2.0 in

1.0 in

Alluvial rockfill - Gravel, Cu = 138USCS classification = GWRelative density = 87%

Effect of particle size – modeled gradations - alluvial rockfillVaradarjan et al. (2003)

φ increased as Dmax increased

Göschenanalp Dam material – scalped gradations(Zeller and Wullimann, 1957)

Effect of particle size (Zeller and Wullimann, 1957), scalped gradations

Effect of grain shape

• At the same void ratio, material with angular particles has higher φ than material with rounded particles (Chen, 1948, and Vallerga, et al., 1957).

• At the same compactive effort, material with angular particles has very nearly the same φ as material with rounded particles – only one degree higher (Vallerga, et al., 1957).

Effect of moisture condition

• For most sands, moisture reduces φ.

• Lee, Seed and Dunlop (1967) found that the value of φ for a saturated specimen of Antioch sand was 9.6 degrees less than for an oven-dry sample.

• For Ottawa sand, values of φ for saturated and oven dry specimens were the same.

• Conclusion – always test in a moist condition

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Effect of moisture (Lee, Seed and Dunlop, 1967)

φ = 31oφ = 31o14 psiOttawa(No. 20 to No. 30)

φ = 37oφ = 38o85 psiMonterey(No. 20 to No. 30)

φ = 41oφ = 44o14 psiSacramento

River(No. 50 to No. 100)

φ = 31oφ = 40o85 psiAntioch(No. 50 to No. 100)

φ = 41oφ = 48o14 psiAntioch(Minus No. 4)

SaturatedOven dryσ3Sand

Review of Leps’s comments

Leps noted that the compilation he had assembled:

1. Only roughly indicated the effect of relative density.

2. Only roughly indicated the effect of gradation.3. Only vaguely suggested the effect of particle

strength.4. Gave no clue as to the influence of particle

shape.5. Offered no evaluation of influence of degree of

saturation of the rock particles.

Based on information available now, we can say:

1. Relative density – 125 tests are available to evaluate the important effect of relative density.

These tests show, for example, that increasing Dr from 40% to 100% results in an increase in φof about 10 degrees at confining pressure of 15 psi (Sacramento River sand, Lee and Seed, 1967).

At a confining pressure of 180 psi, the increase increasing Dr from 40% to 100% results in an increase in φ of about 5 degrees.

2. Gradation – its effect has been isolated through tests with the same particles, same Dr, same σ3(tests on Oroville material at Pyramid gradation compared to tests on the Oroville material at the Oroville gradation).These tests show that an increase in the coefficient of uniformity (Cu) from 7 to 35 results in an increase in φ of 3 degrees at a confining pressure of 30 psi, and 2 degrees at a confining pressure of 650 psi.Cu is a good indicator of friction angle, but the USCS classification “GP” is misleading, because it applies to materials that are neither uniform nor gap-graded.

3. Particle strength – its effect has been isolated through tests at the same relative density on materials with strong, average, and very weak particles.The effect of particle strength is due to the fact that materials with strong particles suffer less breakage, and dilate more during shear.

3. (continued) At σ3 = 30 psi, and Dr = 70%, the measured friction angles for materials with strong particles were 10 degrees higher than for materials with very weak particles, and 5 degrees higher than for material with average strength particles.At high pressures, the effect of particle strength is less (only about 3 degrees at σ3 = 650 psi).

9

4. Particle shape – tests by Vallerga et al. (1957) showed that, at the same compactive effort, material with angular particles had φ only one degree higher than material with rounded particles.

The same compactive effort likely resulted in the same, or nearly the same, relative density for the two materials .

5. Degree of saturation – the tests by Lee, Seed and Dunlop (1967) showed that moisture reduces the friction angles for materials with particles that contain micro-cracks.

Moisture has no effect on friction angles for materials with completely sound particles that contain no cracks, like Ottawa sand.

Improving estimates of φ:1. Represent the variation of φ with confining pressure

using the same type of equation as used by Leps(1970) and WCC.

2. Include the effect of relative density onφ0 and ∆φ

3. Separate data into these three groups –Gravel with Cu > 4, Sand with Cu > 6, and Sand with Cu < 6

σφ = φ − ∆φ

N

0 10a

logp

Example – Pyramid Dam material, Dmax = 0.5 in.

30.00

35.00

40.00

45.00

50.00

55.00

1.00 10.00 100.00

σN/pa (log scale)

φ (d

egre

es) Dr = 80%

Dr = 5%

7°50°5%

10°56°80%

∆φφ0Dr

2. Use a simple equation for the variation of φ0 and ∆φwith relative density.

( )

( )

( ) ( )

φ = +

∆φ = +

σ φ = + − +

0 r

r

Nr r 10

a

A B D

C D D

A B D C D D logp

3. Determine best-fit values of A, B, C, and D by trial and error using the data available for tests on gravels and sands where relative density is known.

Use the standard deviation of the computed value of φ as a measure of best fit.

10

Gravel with Cu > 4:

A = 44 degreesB = 10 degreesC = 7 degreesD = 2 degrees

Standard deviation= 3.1 degrees

69 tests

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Measured φ (degree)

Cal

cula

ted

φ (d

egre

e)

( ) ( ) σ φ = + − +

N

r r 10a

A B D C D D logp

Sand with Cu > 6:



26 tests

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0


Cal

cula

ted

φ (d

egre

e)

( ) ( ) σ φ = + − +

N

r r 10a

A B D C D D logp

Sand with Cu < 6:



30 tests

( ) ( ) σ φ = + − +

N

r r 10a

A B D C D D logp

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0


Cal

cula

ted

φ (d

egre

e)

3.2°

3.2°

3.1°

Standard Deviation

231034Sand withCu < 6

231039Sand withCu > 6

271044Gravel with Cu > 4

DCBAMaterial type

( ) ( ) σ φ = + − +

Nr r 10

a

A B D C D D logp

0

5

10

15

20

25

30

35

40

45

50

30.0 32.5 35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0 62.5 65.0

φ (degrees)

Freq

uenc

y

Distribution of measured values of φ in WCC data(226 data points)

Measured values

Normaldistribution

0

5

10

15

20

25

30

35

40

45

50

30.0 32.5 35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0 62.5 65.0

φ (degrees)

Freq

uenc

y


Lognormaldistribution

Measured values

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0

5

10

15

20

25

30

35

40

45

50

30.0 32.5 35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0 62.5 65.0

φ (degrees)

Freq

uenc

y


Measured values

Normaldistribution

0

5

10

15

20

25

30

35

40

45

50

30.0 32.5 35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0 62.5 65.0

φ (degrees)

Freq

uenc

y


Lognormaldistribution

Measured values

Significance of standard deviation

Lognormal distributionNormal distribution

<0.1%0.1%3

2%2%2

16%16%1

Probability that the actual value could be smaller than the best estimate value

Number of standard deviations below best estimate

ExampleSand with Cu > 6A = 39, B = 10, C = 3, D = 2Relative density = 75%σN = 4,000 psf (pa = 2,116 psf)

φ = (39 + 10(0.75)) – (3 + 2(0.75))(log(4,000/2,116)φ = 46.5 – 4.5(0.28) = 45 degrees

Probability of φ less than 42 degrees is 16%Probability of φ less than 39 degrees is 2%Probability of φ less than 36 degrees is ≤ 0.1%

Summary

Information available from tests on sands gravels and rockfills provides a basis for determining the effects of several factors on friction angles through “all other things equal” comparisons:

• Pressure• Relative density• Gradation (Cu)• Grain size (sand,

gravel, rockfill)

• Particle strength• Particle shape• Moisture

Summary

Using the results of 125 tests for which relative density is known, it is possible to make estimates of φ that reflect the effects of pressure, relative density, gradation, and grain size.

( ) ( ) σ φ = + − +

Nr r 10

a

A B D C D D logp

12

Summary

In addition to estimates of φ, the correlation equations give the standard deviations for the estimated values of φ.

The standard deviations are the same for all three cases – about 3 degrees.

Summary

The correlation equations make it possible to account in a logical way for the effects of pressure, relative density, grain size, and gradation on the value of φ.

Knowledge of the standard deviation makes it possible to determine the reliability of the estimated value of φ.

Duncan Friction Angle of Sand Stone11

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