Weight-Volume Relationships

and Plasticity
Weight-Volume Relationships
- relationships between
a.Unit weight
b.Void ratio
c.Porosity
d.Moisture content
e.Specific gravity
 Volume and Weight Relationships
V=𝑉! + 𝑉" = 𝑉! + 𝑉# + 𝑉$
e=void ratio
Where :
𝑉"
V=total volume of the soil 𝑒=
sample 𝑉!
n=porosity
𝑉! =volume of soil soil solids
𝑉" =volume of voids 𝑉"
𝑛=
𝑉
𝑉# =volume of water S=degree of saturation
𝑉$ =volume of air
𝑉#
S=
𝑉"
𝑊 = 𝑊# + 𝑊! w=moisture content or water
Where: content
W = total weight of soil sample 𝑊#
𝑊# =weight of water 𝑤=
𝑊!
𝑊! =weight of soil
𝛾 = 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 of soil per volume
𝑊
𝛾=
𝑉
#"
) )! +)" )! ,+ #! )! ,+-
𝛾= *
= *
= *
= *

𝛾. = 𝑑𝑟𝑦 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡

𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑠𝑜𝑖𝑙
𝑚
𝜌=
𝑉
𝜌. = 𝑑𝑟𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦
𝑚/
𝜌. =
𝑉
 𝑊/ = 𝐺/ 𝛾-
𝑊- = 𝑤𝑊/ = 𝑤𝐺/ 𝛾-
52 Chapter 3: Weight–Volume Relationships
𝐺/ = speciEic gravity of soil solids
) )! +)" 0! 1" +-0! 1" ,+- 0! 1"
𝛾=* = * = ,+2
= ,+2
Va

𝑊/ 𝐺/ 𝛾- Total Total W V
𝛾. = = weight volume V

𝑉 1+𝑒 !W !V

Ws Vs
𝑊- 𝑤𝐺/ 𝛾-
𝑉- = = + 𝑤𝐺/
𝛾- 𝛾- (a) (b)
𝑉- 𝑤𝐺/
𝑆= = Air Water Solid

𝑉3 𝑒 Figure 3.1 (a) Soil element in natural state; (b) three phases of the soil element

𝑆𝑒 = 𝑤𝐺/ Porosity (n) is defined as the ratio of the volume of voids to the total volume, or

Vv
n! (3.4)
V
 Vs ! 1 Vs ! 1 56 Chapter 3: Weight–Volume Relationships
Vs ! 1
Weight Volume

𝑊 𝑊! + 𝑊# Figure
𝐺! 𝛾Figure3.33.3Figure
# + 𝑒𝛾# 𝐺! + 𝑒3.3𝛾#
𝛾!$% = = = Saturated soil = element
Saturated with volume
𝑉 𝑉 Saturated
1 + 𝑒 soil 1 + 𝑒soil
element element
with volume with
W volume
! eg V !V !e
SolidSolid Solid
ter Water 𝑒 =of𝑤𝐺
soil solids
of! soil equal to solids
of equal
solids soil one
to oneequal to one

11 "
11 w2G
" w2Gsr11
wsrw w 2 Gsrw
Density ! r !
DensityDensity
!r!!r!
" (3.21)
(3.21)
W V!1"e
1 "1 e" e (3.21)
1"e
GsrGwsrw Ws ! Gsg Vs ! 1
DryDry
density ! r!
density d !
rd ! Gsrw (3.22)
(3.22)
Dry density !1 "
1 " erd e! (3.22)
1"e
1Gs1G
"s e2r
" e2r
w w
turated density
Saturated density
! r! !sat !
sat r 1Gs " e 2 rw (3.23)
(3.23)
1 "
Saturated density ! rsat ! 1 e
" e (3.23)
3 3
1"e
water
r ! 1000! 1000
kg/m kg/m
. . Water Solid
3
yay
e ofbewater
derived
derived 1000
by kg/m
referring
by!referring to. the
to the soilsoil element
element shown
shown in Figure
in Figure 3.4,3.4,
l soil
21) maysolids
solids is equal
isbeequal
derived toand
to 1by 1referring
and
the the volume of voids
to theofsoil
volume is equal
element
voids to to
shown
is equal e. in
e. Figure 3.4,
me of soil solids is equal to 1 and the volume of voids is equal to e. 11 " w
Density ! r !
 volume equal to one, as shown in Figure 3.5. From Eq. (
ght of water (Ww) can then
Wsso
!Vbe expressed
Gsg! 11 as follows:
2 The weight of soil solids (W
nn. (3.24)
1, then Vv is equal to
W n, G sgw 1"
w 11 "
" n2 s) and
TheW relationship
s ! wW
s among
! wG unit
g weight,
11 " n 2porosity,
! G g 11 " n2 and moisture content can
(3.25)
(3.26) be developed in a Vv
d ! be expressed
! n
w s s w
ater (Ww) cangthen as11wG
follows:
" gn211 " ns2 w (3.25) (3.24)
!
V
WW!
manner !wWG g! w1 that spresented
V wssimilars s to w in the preceding section. Consider a soil that has a total
y unit weight equals volume equal to one, as shown in Figure 3.5. From Eq. (3.24)(3.4), Weight
s ! Gsgw 11 " n2
Wequals
he dry unit weight Volume
unit weight equals Ww ! wWs ! wGsgw 11 " n2 (3.25)
W W G
G gg 11
11 "" n n
2 2 Vv
s ! wGsgw 11!
s w
dry unit weight gW !
gw ! wW
d equals
s
d !W !
s
!
s w "! Gn2gG 11
g " 11n "
2 nn 2! (3.25) (3.26)
(3.26)
s # Ww 11 V
s ws w
g! V V
! Gsgw 11 " n2 11 # w2 (3.27) V !n
weight equals W V Gsgw 11 " n2
moist unit weight equals s
unit weight equals gd ! Weight ! ! Gsgw 11 " n2 Volume W ! (3.26)
Gsg (1 # n)
V 1
shows a soil W sample
s GWthat
g
ssw # Wwis
11 " saturated
n2 and has V ! 1. According to this figure,
gd ! g!!W # W ! Gs!
s V w
gw 11
Gs" gwn11 2 11"#n2 w2 (3.26)
(3.27) V!1
ist unit weightV gequals
! 1 ! Gsgw 11 " n 2 11 # w 2 (3.27)
Ws # Ww 11V " n2Gsgw # ngw
reg3.6 ! a soil sample
sat shows ! that is saturated and has V ! ! 1.3 11 " n2 Gto
According thisn4figure,
s # gw (3.28)
weight equalsV W s # W w 1
shows a soilWsample g ! that is !
saturatedGsgwand 11 "has n2V 11! # 1. V !n
w2According to
Ws ! G
this (3.27)
sg (1 # n)
figure,
Vs ! 1 # n
s # Ww 11V" n 2 Gsgw # ngw
gsat ! ofW
ure content !
aVssaturated
# W soil sample can !be3 11expressed
" n 2 Gs # as n 4 gw (3.28)
W ! G!w
sg"(1
1
#gn) 11 " n2 11 # w2
g
W !# W
3.6 shows as soil sample 11
that Gisn 2G
w sgw #and
ssaturated ngwhas V ! 1. According to this (3.27)
figure,
g !
wV! ! 3 11 " n 2 G # n 4 g (3.28)
moisture
sat content
V of a saturated
Ww soil sample
ng
1 w can be expressed n as s w
w ! !
11 " n2G # ng ! V!1 (3.29) Air Water Solid
s a soil W # W g
gsat sample
s thatwis saturated and
s whas V ! 1.
! W!W sw 11 "ng n2
w
gwGs w ! n According
11 3"11 n2G
" n2G to this figure, (3.28)
s s # n4gw
ure content of aVsaturated
w!
Ws
!soil sample
11 " n12 gwGs
can! be expressed as
11 " n 2 Gs
(3.29)
Ws # Ww 11 " n2Gsgw # ngw
!isture content ! of a saturated
Ww ngw !
soil sample can3 11
be" n2Gn s # as
expressed n4gw (3.28)
Example
A moist soil has these values: V=7.08x10-3 m3, m=13.95 kg, w=9.8% and Gs=2.66
Determine the following:
a. Density
b. Dry density
c. Void ratio
d. Porosity
e. Degree of saturation (%)
f. Volume occupied by water
In the natural state, a moist soil has a volume of 0.30 m3 and weighs 5500 N. The oven dry
weight of the soil is 4911 N. If 𝐺! = 2.74, calculate the following:
a. moisture content
b. moist unit weight
c. dry unit weight
d. void ratio
e. Porosity
f. degree of saturation
A representative soil specimen collected from the field weighs 1.8 kN and has a volume of
0.1 m3. The moisture content as determined in the laboratory is 1.2%. Given Gs=2.71,
determine the following:
a. Moist unit weight
b. Dry unit weight
c. Void ratio
d. Porosity
e. Degree of saturation
A saturated soil has a dry unit weight of 16.2 kN/m3. Its moisture content is 20%. Determine:
a. Saturated unit weight
b. Specific gravity
c. Void ratio
The following data are given for a soil: porosity = 0.45, specific gravity of the soil solids=2.68,
and moisture content = 10%. Determine the mass of water to be added to 10 m3 of soil for
full saturation.