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60 Chapter 3: Weight–Volume Relationships

Example 3.1 For a saturated soil, show that gsat
a

1w e b gw ba w 1e

Solution From Eqs. (3.19) and (3.20), gsat


1Gs  e 2gw 1e

(a)

and e
wGs or Gs


e w

(b)

Combining Eqs. (a) and (b) gives a gsat


e  e b gw w e 1w
a b gw ba w 1e 1e

Example 3.2 For a moist soil sample, the following are given. • • • •

Total volume: V
1.2 m3 Total mass: M
2350 kg Moisture content: w
8.6% Specific gravity of soil solids: Gs
2.71

Determine the following. a. b. c. d. e. f.

Moist density Dry density Void ratio Porosity Degree of saturation Volume of water in the soil sample

Solution Part a From Eq. (3.13), r


M 2350

1958.3 kg/m3 V 1.2

■

3.4 Various Unit-Weight Relationships

Part b From Eq. (3.14), rd


Ms M

V 11  w2V

2350
1803.3 kg/m3 8.6 b 11.22 a1  100

Part c From Eq. (3.22), rd
e


Gsrw 1e

12.712 110002 Gs rw  1
0.503 1
rd 1803.3

Part d From Eq. (3.7), n


0.503 e

0.335 1e 1  0.503

Part e From Eq. (3.18),

wGs
S
e

a

8.6 b 12.71 2 100
0.463
46.3% 0.503

Part f Volume of water:

Mw M  Ms

rw rw

M 1w
rw

M

2350 8.6 ¢ 1 100
0.186 m3 1000

2350  °

Alternate Solution Refer to Figure 3.7. Part a r


M 2350

1958.3 kg/m3 V 1.2

61

62 Chapter 3: Weight–Volume Relationships Volume (m3)

Mass (kg)

V
0.402 V
0.186

M
186.1

V
1.2

M
2350

Ms
2163.9

Vs
0.798

Water

Air

Solid

Figure 3.7

Part b M
1w

2350
2163.9 kg 8.6 1 100 Ms M 2350

rd

1803.3 kg/m3 8.6 V 11  w2V a1  b 11.22 100

Ms


Part c The volume of solids:

Ms 2163.9
0.798 m3
Gsrw 12.71 2 110002

The volume of voids: Vv
V  Vs
1.2  0.798
0.402 m3 Void ratio: e


Vv 0.402

0.503 Vs 0.798

Part d Porosity: n


Vv 0.402

0.335 V 1.2

Part e Vw Vv Mw 186.1 Volume of water: Vw


0.186 m3 rw 1000 S


3.4 Various Unit-Weight Relationships

63

Hence, S


0.186
0.463
46.3% 0.402

Part f From Part e, Vw
0.186 m3

■

Example 3.3 The following data are given for a soil: • Porosity: n
0.4 • Specific gravity of the soil solids: Gs
2.68 • Moisture content: w
12% Determine the mass of water to be added to 10 m3 of soil for full saturation. Solution Equation (3.27) can be rewritten in terms of density as


Gs w (1  n)(1 + w) Similarly, from Eq. (3.28)

sat
[(1  n)Gs + n] w Thus,


(2.68)(1000)(1  0.4)(1 + 0.12)
1800.96 kg/m3 sat
[(1  0.4)(2.68) + 0.4] (1000)
2008 kg/m3 Mass of water needed per cubic meter equals

sat 
2008  1800.96
207.04 kg So, total mass of water to be added equals 207.04  10
2070.4 kg

Example 3.4 A saturated soil has a dry unit weight of 103 lb/ft3. Its moisture content is 23%. Determine: a. Saturated unit weight, sat b. Specific gravity, Gs c. Void ratio, e

■

64 Chapter 3: Weight–Volume Relationships Solution Part a: Saturated Unit Weight From Eq. (3.12), gsat
gd 11  w2
1103 2 a 1 

23 b
126.69 lb/ft3
126.7lb/ft3 100

Part b: Specific Gravity, Gs From Eq. (3.16), gd


Gsgw 1e

Also from Eq. (3.20) for saturated soils, e
wGs. Thus, gd


Gsgw 1  wGs

So, 103


Gs 162.42

1  10.23 2 1Gs 2

or 103  23.69Gs
62.4Gs Gs
2.66 Part c: Void Ratio, e For saturated soils, e
wGs
(0.23)(2.66)
0.61

3.5

■

Relative Density The term relative density is commonly used to indicate the in situ denseness or looseness of granular soil. It is defined as Dr


emax  e emax  emin

(3.30)

where Dr
relative density, usually given as a percentage e
in situ void ratio of the soil emax
void ratio of the soil in the loosest state emin
void ratio of the soil in the densest state The values of Dr may vary from a minimum of 0% for very loose soil to a maximum of 100% for very dense soils. Soils engineers qualitatively describe the granular soil deposits according to their relative densities, as shown in Table 3.3. In-place soils seldom

Problems 69 1.0

Void ratio range, emax  emin

0.8

0.6 emax  emin
0.23 

0.06 D50

0.4

0.2

0.0 0.1

1.0 Mean grain size, D50 (mm)

10

Clean sands (FC
0 – 5%) Sands with fines (5 FC  15%) Sands with clay (15 FC  30%, PC
5 – 20%) Silty soils (30 FC  70%, PC
5 – 20%) Gravelly sands (FC 6%, PC
17 – 36%) Gravels

Figure 3.9 Plot of emax  emin versus the mean grain size (Cubrinovski and Ishihara, 2002)

Problems 3.1

For a given soil, show that

sat
d + n w 3.2 For a given soil, show that e b gw gsat
gd  a 1e 3.3 For a given soil, show that eSgw gd
11  e2w 3.4 A 0.4-m3 moist soil sample has the following: • Moist mass
711.2 kg • Dry mass
623.9 kg • Specific gravity of soil solids
2.68 Estimate: a. Moisture content b. Moist density

70 Chapter 3: Weight–Volume Relationships

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12 3.13

c. Dry density d. Void ratio e. Porosity In its natural state, a moist soil has a volume of 0.33 ft3 and weighs 39.93 lb. The oven-dry weight of the soil is 34.54 lb. If Gs
2.67, calculate the moisture content, moist unit weight, dry unit weight, void ratio, porosity, and degree of saturation. The moist weight of 0.2 ft3 of a soil is 23 lb. The moisture content and the specific gravity of the soil solids are determined in the laboratory to be 11% and 2.7, respectively. Calculate the following: a. Moist unit weight (lb/ft3) b. Dry unit weight (lb/ft3) c. Void ratio d. Porosity e. Degree of saturation (%) f. Volume occupied by water (ft3) The saturated unit weight of a soil is 19.8 kN/m3. The moisture content of the soil is 17.1%. Determine the following: a. Dry unit weight b. Specific gravity of soil solids c. Void ratio The unit weight of a soil is 95 lb/ft3. The moisture content of this soil is 19.2% when the degree of saturation is 60%. Determine: a. Void ratio b. Specific gravity of soil solids c. Saturated unit weight For a given soil, the following are given: Gs
2.67; moist unit weight,
112 lb/ft3; and moisture content, w
10.8%. Determine: a. Dry unit weight b. Void ratio c. Porosity d. Degree of saturation Refer to Problem 3.9. Determine the weight of water, in pounds, to be added per cubic foot of soil for: a. 80% degree of saturation b. 100% degree of saturation The moist density of a soil is 1680 kg/m3. Given w
18% and Gs
2.73, determine: a. Dry density b. Porosity c. Degree of saturation d. Mass of water, in kg/m3, to be added to reach full saturation The dry density of a soil is 1780 kg/m3. Given Gs
2.68, what would be the moisture content of the soil when saturated? The porosity of a soil is 0.35. Given Gs
2.69, calculate: a. Saturated unit weight (kN/m3) b. Moisture content when moist unit weight
17.5 kN/m3

Problems

71

3.14 A saturated soil has w
23% and Gs
2.62. Determine its saturated and dry densities in kg/m3. 3.15 A soil has e
0.75, w
21.5%, and Gs
2.71. Determine: a. Moist unit weight (lb/ft3) b. Dry unit weight (lb/ft3) c. Degree of saturation (%) 3.16 A soil has w
18.2%, Gs
2.67, and S
80%. Determine the moist and dry unit weights of the soil in lb/ft3. 3.17 The moist unit weight of a soil is 112.32 lb/ft3 at a moisture content of 10%. Given Gs
2.7, determine: a. e b. Saturated unit weight 3.18 The moist unit weights and degrees of saturation of a soil are given in the table.

3.19 3.20

3.21

3.22

3.23

3.24

(lb/ft3)

S (%)

105.73 112.67

50 75

Determine: a. e b. Gs Refer to Problem 3.18. Determine the weight of water, in lb, that will be in 2.5 ft3 of the soil when it is saturated. For a given sand, the maximum and minimum void ratios are 0.78 and 0.43, respectively. Given Gs
2.67, determine the dry unit weight of the soil in kN/m3 when the relative density is 65%. For a given sandy soil, emax
0.75, emin
0.46, and Gs
2.68. What will be the moist unit weight of compaction (kN/m3) in the field if Dr
78% and w
9%? For a given sandy soil, the maximum and minimum dry unit weights are 108 lb/ft3 and 92 lb/ft3, respectively. Given Gs
2.65, determine the moist unit weight of this soil when the relative density is 60% and the moisture content is 8%. The moisture content of a soil sample is 18.4%, and its dry unit weight is 100 lb/ft3. Assuming that the specific gravity of solids is 2.65, a. Calculate the degree of saturation. b. What is the maximum dry unit weight to which this soil can be compacted without change in its moisture content? A loose, uncompacted sand fill 6 ft in depth has a relative density of 40%. Laboratory tests indicated that the minimum and maximum void ratios of the sand are 0.46 and 0.90, respectively. The specific gravity of solids of the sand is 2.65. a. What is the dry unit weight of the sand? b. If the sand is compacted to a relative density of 75%, what is the decrease in thickness of the 6-ft fill?