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
What I learned from Ken Lee as a role model
Friction Angles for Sand, Gravel and Rockfill
Why this topic? • Ken’s interest – my interest. • Data is available that has not been thoroughly evaluated in previous studies.
50 years of previous studies
When you dig into something, dig deep. Keep an open mind about what is useful and practical.
Thanks to Youngjin Park – Binod Tiwari – Chris Meehan –
Virginia Tech Virginia Tech Virginia Tech
Sathi Sathialingam – Yoshi Moriwaki –
WCC (MWDSC) WCC (MWDSC)
Harry Seed – Clarence Chan – Dean Marachi – Ed Becker –
Berkeley RTF Berkeley RTF Berkeley RTF Berkeley RTF
Outline
• 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)
• Review of the basics • Compilations of data • Evaluation of effects • Improving estimates of φ
Components of shearing resistance (Lee and Seed, 1967)
Greater expansion, higher
φ (Lee and Seed, 1967)
Dr = 100%
6
Dr = 38%
6
σ3 = 2,048 psf 3
σ / 1
3
5
σ / 1
σ
o i t a r s s e r t s l a p i c n i r P
σ3 = 21,500 psf
4
3
2
1
o i t a r s s e r t s l a p i c n i r P
σ3 = 2,048 psf
4
σ3 = 26,000 psf
3
2
1 0
5
%15 n i a r 10 t s e 5 c i r t e m 0 u l o V -5
10
15
20
σ3 = 21,500 psf
5 10 15 Axial strain - %
Secant values of
0 %15 n i a 10 r t s e 5 c i r t e m 0 u l o V -5
σ3 = 2,048 psf
0
Curved failure envelopes
5
σ
20
5
10
15
20
σ3 = 2,048 psf σ3 = 26,000 psf
0
5 10 15 Axial strain - %
20
φ
φ is defined by a single test
(Marachi et al., 1969)
Variation of
φsecant with confining pressure
σ φ = φ0 − ∆φlog10 3 pa
Marachi et al. (1969)
Compilations of test results
WCC, working for MWD on the Diamond Valley Reservoir, doubled the size Leps’s compilation (from 109 tests to 226 tests)
Leps (1970) – 109 tests 70.0
φ = 55 degrees - 7 degrees x Log10(σN in psi) 60.0
) 50.0 e e r g e d ( 40.0 e l g n A n 30.0 o i t c i r F 20.0
10.0
0.0 1
10
100
1,000
10,000
Normal stress on slip surface (psi)
Woodward-Clyde working on the Diamond Valley Reservoir For the Metropolitan Water District of Southern California – 226 tests
Leps’s comments on his compilation of φ-values for rockfill
70.0
φ = 57 degrees - 7 degrees x Log10(σN in psi)
65.0
Virtues:
60.0
) 55.0 e e r g e 50.0 d ( e l g 45.0 n a n o 40.0 i t c i r F 35.0
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.”
30.0
25.0
20.0 1
10
100
1,000
10,000
Normal stress on slip surface (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
Effect of relative density
First conclusions:
(Becker et al., 1972) σ3 = 30 psi, Dmax = 0.5 in.
Material
Dr = 50%
Dr = 85%
Pyramid Dam argillite
φ = 48o
φ = 51o
Venato sandstone
φ = 38o
φ = 41o
Oroville amphibolite
φ = 43o
φ = 48o
• 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.
φ increases as Dr increases
Effect of gradation 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 C u > 4
►
26 tests on sands with C u > 6
►
30 tests on sands with C u < 6
U.S. Standard Sieve Opening (in)
100
1 3/4
1/2 3/8
1/4 #4
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.
U.S. Standard Sieve Number #10
#20
#40
Effect of gradation (Becker et al. (1972)
#60 #100 #140 #200
90
Oroville gradation
80 ) % ( t h g i e w y b r e n i f t n e c r e P
"Pyramid gradation" -- 2-inch max particle size Cu = 7.2, C c = 1.3 USCS classification = GW
70 60
Cu = 35 Pyramid gradation Cu = 7
50
"Oroville gradation" -- 2-inch max particle size C u = 35, C c = 4.2 USCS classification = GP
40 30 20 10 0 100
10
1 Grain Size (mm)
Grain size curves from Becker, Chan and Seed (1972)
0.1
0.01
Higher Cu, higher φ
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
Particle strength
Categories suggested by Leps (1970) • Weak – qu = 500 psi to 2,500 psi
For example:
• Average – qu = 2,500 psi to 10,000 psi
Gravel with Cu = 35, rather than GP for the Oroville Dam material.
• Strong – qu = 10,000 psi to 30,000 psi
Materials tested by Marachi et al. (1969) and Becker et al. (1972) qu = 28,000 psi
Strong
Pyramid
qu = 15,500 psi
Strong
Readily broken with a hammer
Average
Easily broken with a hammer
Very weak
Particles disintegrated upon saturation
Venato sandstone Colorado sandstone
qu = 5,000 psi
qu < 500 psi
Strong particles
Very difficult to break with a hammer
Oroville amphibolite
argillite
Particle strength
Average strength particles
Very weak particles
Becker, Chan and Seed (1972)
Tests to measure strengths of rockfills Ratio of particle size to specimen size Investigation
Dmax Specimen size
• Holtz and Gibbs (1956) and Leslie (1963) showed that particles larger than 1/5 th to 1/10th of the specimen diameter tend to interfere, resulting in measured strengths that are too high.
Zeller and Wullimann (1957) – Göschenanalp Dam
4 in. 20 in. diameter
Marsal (1967, 1970) – El Infernillo Dam & Mica Dam
7 in. 44 in. diameter
• These findings have led to the widely accepted rule:
Marachi et al. (1969) – Oroville Dam & Pyramid Dam triaxial
6 in. 36 in. diameter
Becker et al. (1972) – Oroville Dam & Pyramid Dam plane strain
4 in.
Varadarajan et al. (2003) – Ranjit Sagar Dam & Purulia Dam
3 in. 15 in. diameter
Specimen diameter should be
≥
6 x Dmax
24 in. by 54 in. rectangular
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 Material Pyramid Dam argillite
Dmax = 0.47 in
Dmax = 2 in
Dmax = 6 in Dr = 100%
φ=
51o
φ=
48o
φ=
Crushed basalt
φ = 50o
φ = 48o
φ = 49o
Venato sandstone
φ = 41o
φ = 41o
φ = 40o
Oroville amphibolite
φ = 48o
φ = 46o
φ = 48o
φ decreased as Dmax increased (or stayed the same)
Dr = 71%
Dr = 44%
48o
Becker et al., (1972)
Effect of particle size – modeled gradations – quarried rockfill Varadarjan et al. (2003)
Effect of particle size – modeled gradations - alluvial rockfill Varadarjan et al. (2003) 55.0
Dmax
50.0 ) e e r g e d 45.0 ( e l g n A n 40.0 o i t c i r F
35.0
3.2 in 2.0 in
1.0 in Alluvial rockfill - G ravel, C u = 138 USCS classification = GW Relative density = 87%
30.0 1.00
10.00
100.00
(σ3/pa)
φ decreased as Dmax increased, for σ3 > 4 atmospheres
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
Effect of moisture condition
• At the same void ratio, material with angular particles has higher φ than material with rounded particles (Chen, 1948, and Vallerga, et al., 1957).
•
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
• 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 (Lee, Seed and Dunlop, 1967)
Review of Leps’s comments
Sand
σ3
Oven dry
Saturated
Antioch
14 psi
φ = 48o
φ = 41o
85 psi
φ = 40o
φ = 31o
14 psi
φ = 44o
φ = 41o
85 psi
φ = 38o
φ = 37o
14 psi
φ = 31o
φ = 31o
(Minus No. 4)
Antioch (No. 50 to No. 100)
Sacramento River (No. 50 to No. 100)
Monterey (No. 20 to No. 30)
Ottawa (No. 20 to No. 30)
Based on information available now, we can say: 1. Relative density – 125 tests are available to evaluate the important effect of relative density.
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.
2. Gradation – its effect has been isolated through tests with the same particles, same D r , same σ3 (tests on Oroville material at Pyramid gradation compared to tests on the Oroville material at the Oroville gradation).
These tests show, for example, that increasing Dr from 40% to 100% results in an increase in φ of about 10 degrees at confining pressure of 15 psi (Sacramento River sand, Lee and Seed, 1967).
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.
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.
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 D r = 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).
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.
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.
The same compactive effort likely resulted in the same, or nearly the same, relative density for the two materials .
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.
σ φ = φ0 − ∆φlog10 N pa 2.
Separate data into these three groups – Gravel with Cu > 4, Sand with Cu > 6, and Sand with Cu < 6
2.
Use a simple equation for the variation of φ0 and with relative density.
φ0 = A + B (Dr ) ∆φ = C + D ( Dr ) σ φ = A + B (Dr ) − C + D ( Dr ) log10 N pa
55.00
Dr
50.00
) s 45.00 e e r g e d ( 40.00
Include the effect of relative density on φ0 and ∆φ
3.
Example – Pyramid Dam material, Dmax = 0.5 in.
φ
Dr = 80%
∆φ
80% 56°
10°
5%
7°
Dr = 5%
35.00
30.00 1.00
φ0
10.00
50°
100.00
σN/pa (log scale)
∆φ
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.
σ φ = A + B (Dr ) − C + D (Dr ) log10 N pa
σ φ = A + B (Dr ) − C + D (Dr ) log10 N pa 60.0
Gravel with Cu > 4: A = 44 degrees B = 10 degrees C = 7 degrees D = 2 degrees
A = 39 degrees B = 10 degrees C = 3 degrees D = 2 degrees
50.0
) 40.0 e e r g e d (
Standard deviation = 3.1 degrees
50.0
) 40.0 e e r g e d (
Standard deviation = 3.2 degrees
d 30.0 e t a l u c l a C20.0
69 tests
60.0
Sand with C u > 6:
d 30.0 e t a l u c l a C20.0
26 tests
10.0
10.0
0.0
0.0 0.0
10.0
20.0
30.0
40.0
50.0
60.0
0.0
10.0
20.0
Measured (degree)
σ φ = A + B (Dr ) − C + D (Dr ) log10 N pa A = 34 degrees B = 10 degrees C = 3 degrees D = 2 degrees
40.0
50.0
50.0
) 40.0 e e r g e d (
Standard deviation = 3.2 degrees
d 30.0 e t a l u c l a C20.0
30 tests
10.0
0.0 0.0
10.0
20.0
30.0
40.0
50.0
Measured (degree)
50
60.0
Material type
A
B
C
D
Standard Deviation
Gravel with Cu > 4
44
10
7
2
3.1°
Sand with Cu > 6
39
10
3
2
3.2°
Sand with Cu < 6
34
10
3
2
3.2°
50 Distribution of measured values of φ in WCC data (226 data points)
45
Distribution of measured values of φ in WCC data (226 data points)
45
40
40 Normal distribution
35 y 30 c n e u 25 q e r F20
35
Lognormal distribution
y 30 c n e u 25 q e r F20
Measured values
15
15
10
10
5
5
0
Measured values
0 30.0
32.5
35.0
60.0
σ φ = A + B (Dr ) − C + D (Dr ) log10 N pa
60.0
Sand with C u < 6:
30.0
Measured (degree)
37.5
40.0
42.5
45.0
47.5
50.0
φ (degrees)
52.5
55.0
57.5
60.0
62.5
65.0
30.0
32.5
35.0
37.5
40.0
42.5
45.0
47.5
50.0
φ (degrees)
52.5
55.0
57.5
60.0
62.5
65.0
50
50 Distribution of measured values of φ in WCC data (226 data points)
45
Distribution of measured values of φ in WCC data (226 data points)
45
40
40 Normal distribution
35 y 30 c n e u 25 q e r F20
35
Lognormal distribution
y 30 c n e u 25 q e r F20
Measured values
15
15
10
10
5
5
0
Measured values
0 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
30.0
32.5
35.0
37.5
40.0
42.5
45.0
φ (degrees)
47.5
50.0
52.5
55.0
57.5
60.0
62.5
65.0
φ (degrees)
Example
Significance of standard deviation Sand with Cu > 6 Probability that the actual value could be smaller than the best estimate value
A = 39, B = 10, C = 3, D = 2
Number of standard deviations below best estimate
Normal distribution
Lognormal distribution
1
16%
16%
φ = (39 + 10(0.75)) – (3 + 2(0.75))(log(4,000/2,116) φ = 46.5 – 4.5(0.28) = 45 degrees
2
2%
2%
Probability of
3
0.1%
<0.1%
Relative density = 75%
σN = 4,000 psf
(pa = 2,116 psf)
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
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:
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.
• Pressure
• Particle strength
• Relative density
• Particle shape
• Gradation (Cu)
• Moisture
• Grain size (sand, gravel, rockfill)
σ φ = A + B (Dr ) − C + D (Dr ) log10 N pa
Summary
Summary
In addition to estimates of φ, the correlation equations give the standard deviations for the estimated values of φ.
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.
The standard deviations are the same for all three cases – about 3 degrees.
Knowledge of the standard deviation makes it possible to determine the reliability of the estimated value of φ.
