Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Chapter 4
Solutions to Exercises
Exercise 4.1
Inclination of failure plane to direction of minor principal stress:
!=
"
29°
+ 45° =
+ 45° = 59.5° 2
2
⇒ Inclination with respect to direction of major principal stress:
= 90° – 59.5° = 30.5°
Plot Mohr circle:
Effective normal stress on failure plane = 295 kPa
Shear stress on failure plane = 161 kPa
Maximum shear stress = 184 kPa
Published by Wiley
ISBN-13: 978-0470673393
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Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Exercise 4.2
(a) From page 99:
(σ1 – σ3) = (σ1 + σ3) sin φ' + 2c' cos φ'
Let σ1 be denoted by the axial stress, σa and σ3 be denoted by the radial stress,
σr .
σa = (σa + σr) sin φ' + σr + 2c' cos φ'
σa = σa sin φ' + σr sin φ' + σr + 37.6
σa – σa sin φ' = σr sin φ' + σr + 37.6
σa (1 – sin φ') = σr (1 + sin φ') + 37.6
!r =
! a (1! sin " ') ! 37.6 250(1! sin 20°) ! 37.6
= 95 kPa =
(1+ sin 20°)
(1+ sin " ')
(b)
(σ1 – σ3) = (σ1 + σ3) sin φ' + 2c' cos φ'
(σ1 – 250) = (σ1 + 250) sin 20° + 40 cos 20°
(σ1 – 250) – σ1 sin 20° = 123.1
σ1 (1 – sin 20°) = 373.1
σ1 = 567 kPa
Published by Wiley
ISBN-13: 978-0470673393
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Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Exercise 4.3
160
140
Test 2
2
Shear stress (kN/m )
120
Test 1
100
80
60
40
20
0
0
50
100
150
200
250
300
350
400
450
500
550
-2
Horizontal displacement (x10 mm)
Peak (Test 2):
τf (peak) = 150 = σ tan φ'
⇒ tan φ' = 150/200; φ' = 37°
Ultimate (Test 1):
τf (ultimate) = 116 = σ tan φ'
⇒ tan φ' = 116/200; φ' = 30°
Published by Wiley
ISBN-13: 978-0470673393
http://www.wiley.com/go/smith/soil
Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Exercise 4.4
Calculate s1 values for each test:
Test 1: σ3 + (σ1 – σ3) = 200 + 200 = 400 kPa
Test 2: σ3 + (σ1 – σ3) = 400 + 370 = 770 kPa
Test 3: σ3 + (σ1 – σ3) = 600 + 540 = 1140 kPa
Plot Mohr circles:
600.0
Shear stress (kPa)
500.0
400.0
300.0
200.0
100.0
0.0
0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
Normal stress (kPa)
From plot, c' = 10 kPa; φ' = 17°
Published by Wiley
ISBN-13: 978-0470673393
http://www.wiley.com/go/smith/soil
Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Exercise 4.5
Calculate s1 values for each test:
Test 1: σ3 + (σ1 – σ3) = 200 + 140 – 50 = 290 kPa
Test 2: σ3 + (σ1 – σ3) = 400 + 256 – 128 = 528 kPa
Test 3: σ3 + (σ1 – σ3) = 600 + 380 – 200 = 780 kPa
Plot Mohr circles:
600.0
Shear stress (kPa)
500.0
400.0
300.0
200.0
100.0
0.0
0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
Normal stress (kPa)
From plot, c' = 0; φ' = 19°
Published by Wiley
ISBN-13: 978-0470673393
http://www.wiley.com/go/smith/soil
Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Exercise 4.6
B=
Δu a 240
= 0.6
=
Δσ 3 400
Deviator stress against strain:
1400
1200
Deviator stress (kPa)
1000
800
600
400
200
0
0
2
4
6
8
10
12
14
16
18
20
Strain (%)
Pore water pressure against strain:
350
300
Pore water pressure (kPa)
250
200
150
100
50
0
0
2
4
6
8
10
12
14
16
18
20
Strain (%)
Published by Wiley
ISBN-13: 978-0470673393
http://www.wiley.com/go/smith/soil
Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Pore pressure coefficient A against strain:
Strain
(%)
0
1
2
3
4
5
7.5
10.0
12.5
15.0
Deviator
stress (kPa)
0
240
460
640
840
950
1100
1150
1170
1150
u (kPa)
240
285
300
270
200
160
110
75
55
50
Δ ud
45
60
30
-40
-80
-130
-165
-185
-190
A
0.188
0.130
0.047
-0.048
-0.084
-0.118
-0.143
-0.158
-0.165
A= A
B
0.313
0.217
0.078
-0.079
-0.140
-0.197
-0.239
-0.264
-0.275
0.400
0.300
Pore pressure coefficient A
0.200
0.100
0.000
0
2
4
6
8
10
12
14
16
18
20
-0.100
-0.200
-0.300
-0.400
Strain (%)
Published by Wiley
ISBN-13: 978-0470673393
http://www.wiley.com/go/smith/soil
Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Exercise 4.7
Normal load
(kN)
0.2
0.4
0.6
Normal stress (kPa)
-3
= 0.2 / 3.6 × 10 = 55.6
111.1
166.7
Shear force at failure (N)
= 33.4 × 8.4 = 280.6
400.0
526.7
Shear stress (kPa)
= 280.6 / 3.6 = 77.9
111.1
146.3
70
60
Shearing force (N)
50
40
30
20
10
0
0
2
4
6
8
10
12
14
16
18
Strain (%)
Published by Wiley
ISBN-13: 978-0470673393
http://www.wiley.com/go/smith/soil
20
Smith’s Elements of Soil Mechanics. 9th Edition
Ian Smith
Shear stress (kPa)
160.0
140.0
120.0
100.0
80.0
60.0
40.0
20.0
0.0
0.0
50.0
100.0
150.0
200.0
Normal stress (kPa)
From plot, c' = 43 kPa; φ' = 32°
Published by Wiley
ISBN-13: 978-0470673393
http://www.wiley.com/go/smith/soil