sm 05 phase relationship
DESCRIPTION
Soil MechanicsBasicsTRANSCRIPT
Soil Mechanics ICE‐222CE 222
Chapter 2: Phase relationship of soils
1
Three phases of soil 2
Soil element Three phases
Air VW = 0
in natural state of the soil element
ume
= V
. = W
Air
Water Vw
VaVv
VWw
Wa = 0To
tal v
olu
Tota
l wt.
Solids V
Vw
V
V
W
wW
To Solids VsVsWs
Weight‐Volume relationships 3
Weight Volume
Air Va
V
Wa = 0
Water Vw
Vv
VWw
W VW
Solids VsVsWs
Weight relationships 4
The common terms used for weight relationships
are moisture content, wmoisture content, w
and unit weight γunit weight, γ.
Moisture content, w 5
Weight
Moisture content is also known as water content, w, is expressed either in terms of weight or mass.
W
AirWa = 0
Weight%100×=
s
w
WWw
WaterWwW
%100×=s
w
MMw
SolidsWs
A small w indicates a dry soil, while a large w indicates a wet one.
Usual values: 3 to 70%
Moisture content values greater than 100% are found in soft soils below ground water table.
Unit weight, γ 6
Geotechnical engineers often need to know the unit weight, γ:
W=γ
V=γ
Two variations of unit weight are commonly used, the dry unit weight, γd, and the unit weight of water, γw:
Ws wW
Vs
d =γw
ww V=γ
Normally we use γ = 9 81 kN/m3 = 62 4 lb/ft3 for fresh water andNormally we use γw = 9.81 kN/m = 62.4 lb/ft for fresh water, and γw = 10.1 kN/m3 = 64.0 lb/ft3 for sea water
Th i i h f il b l h d bl (GWT) iThe unit weight of soil below the ground water table (GWT) is called saturated unit weight, γsat.
7
Volume relationships 8
The common terms used for volume relationships
are void ratio, e
andand porosity, n
andanddegree of saturation, S
Void ratio, e 9
Void ratio, e, is defined as the ratio of the volume of voids to the volume of solids.
Air Va
Volume
s
vVVe =
Water Vw
Vv
V
s
Densely packed soils have low void ratios
Solids V Vs
Vhave low void ratios.
Typical values in the field Solids Vs syprange from 0.1 to 2.5.
Porosity, n 10
Porosity, n, is defined as the ratio of the volume of voids to the total volume.
Air Va
Volume
VVn v=
Water Vw
Vv
V
V
Typical values in the field f
Solids V Vs
Vrange from 0.09 to 0.7.
Solids Vs s
Degree of Saturation, S 11
Degree of saturation, S, is the percentage of the voids filled with water.
V
Air VaWa = 0
Weight Volume%100×=v
w
VVS
h l f
Water Vw
Vv
VWw
W
S has max. value of 100% when all of voids are filled with water. Such
Solids V Vs
V
Ws
Wsoil are called saturated soils.
Solids Vs sWs
S values above GWT are usually 5 to 100%.
S = 0 is found in very arid areas.
e – n relationship 12
nVV
VV
VVevv
vv =====n
VV
VVVVVV
evvvs −=
−=
−=
−==
11
eVV
VV
VV s
v
s
v
vve
VV
VVVVVV
n
s
vs
s
vss
vs
vv+
=+
=+
=+
==11
ne = en =n
e−
=1 e
n+
=1
γ – γd relation13
WWW ws ⎥⎤
⎢⎡
+
Moisture content, w
VWW
W
VWW
VW ss
sws
⎥⎦
⎢⎣
+=
+==γ
D it i ht
( ) ( ) ( )wwVW
VwW
dss +=+=
+= 111 γ
Dry unit weight , γd
VV
( )1( )wd += 1γγ
γ( )wd +
=1γγ
Specific gravity of solids, Gs14
The specific gravity of any material is the ratio of its density to that of water.
In case of soils, we compute it for the solid phase only, and express the results as the specific gravity of solids, Gs:
ssss V
WVWGγγ
==wsw V γγ
This is quite different from the specific gravity of the entire soil hi h ld i l d lid d i h f dmass, which would include solid, water, and air. Therefore, do not
make the common mistake of computing Gs as γ/γw.
For most of soils, Gs is from 2.60 to 2.80.
Specific gravity of solids, Gs15
Typical values of e, w and γd16
γ‐e‐w relationship 17
ws
ss V
WGγ
= wsss VGW γ=Vs = 1
wss GW γ=wsγ
s
wWWw = sw wWW = wsw wGW γ=
Weight
Air
Volumes
ww
WVγ
=Air
Water Vw = wGs
Vv = e
Ww = wGsγw
wγ
sws wGwG==
γ
S lid
w s
V = 1+eW
swγ
wGV =Solids Vs = 1Ws = Gsγwsw wGV =
γ‐e‐w relationship 18
VW
=γeWW ws
++
=1
γe
wGG wsws++
=1
γγγ
( )eGw ws+
+=
11 γγ ( ) e
Gw
ws+
=+ 11
γγe
G wsd +=
1γγ
Weight
Air
Volumee+1
G wsγ Air
Water Vw = wGs
Vv = e
Ww = wGsγw
ews
d +=
1γγ
lid
w s
V = 1+eW
1−= wsGe γSolids Vs = 1Ws = Gsγw
1d
eγ
γ‐e‐w relationship 19
wVS wGS s wGe s= wGSe =v
wV
S =e
S s=S
e = swGSe =
Weight
Air
Volume
wGSe = Air
Water Vw = wGs
Vv = e
Ww = wGsγw
swGSe =
lid
w s
V = 1+eW
SwGe s=
Solids Vs = 1Ws = GsγwS
γ‐e‐w relationship 20
swGSe =As we know
d W WW ws + wGG wsws + γγandVW
=γeWW ws
++
=1
γe
wGG wsws++
=1
γγγ
Weight
Air
Volume
eeSG wws
++
=1
γγγAir
Water Vw = wGs
Vv = e
Ww = wGsγw
( )eSG ws +=1
γγ
lid
w s
V = 1+eW
e+1γ
Solids Vs = 1Ws = Gsγw
γ‐e‐w relationship (fully saturated) 21
wss GW γ= swGSe = At fully saturated conditionS = 1
wsw wGW γ= ww eSW γ= ww eW γ=
WW +Weight
V = V = e
VolumeV
WW swsat
+=γ
WaterVv = Vw = e
V = 1+e
Ww = eγw
eeG wws
sat ++
=1
γγγ
Solids V = 1
V = 1+e
W( )eG wssat
+=
γγ Solids Vs 1Ws = Gsγwesat +1
γ
γ‐n‐w relationship 22
sW
If V = 1, n = Vv/V => Vv = n V = Vv + Vs => 1 = n + Vs => Vs = 1–n
( )ws
ss V
WGγ
= wsss VGW γ= ( )nGW wss −= 1γ
Weight
Air
Volumesw wWW =
Air
Water
Vv = n
Ww = wGsγw(1–n))1( nwGW wsw −= γ
S lidV = 1
Solids Vs = 1-nWs = Gsγw(1–n )
γ‐n‐w relationship 23
( ) ( )1
11 nwGnGV
WWVW wswsws −+−
=+
==γγγ
( )( )wnG ws +−= 11γγ
Weight
Air
Volume
Air
Water Vw = wGs
Vv = n
Ww = wGsγw(1–n)
Solids V 1V = 1
W G (1 ) Solids Vs = 1-nWs = Gsγw(1–n )
γ‐n‐w relationship 24
( ) ( )nGnGVW
wswss
d −=−
== 111 γγγ
( )nG wsd −= 1γγ
Weight
Air
Volume
Air
Water
Vv = n
Ww = wGsγw(1–n)
Solids V 1V = 1
W G (1 ) Solids Vs = 1-nWs = Gsγw(1–n )
γ‐n‐w relationship (fully saturated) 25
As we know that, S = 1 at saturation swGe = swG
nn
=−1
( )nGW wss −= 1γ ( ) sGnwn −= 1
Weight Volume ( ) sww GnwW −= 1γ
Water
Vw = wGs
Vv = nWw = nγw
nW γ=
S lid
w s
V = 1
ww nW γ=
Solids Vs = 1-nWs = Gsγw(1–n )
γ‐n‐w relationship (fully saturated) 26
( )1
1 wwswssat
nnGV
WWVW γγγ +−
=+
==
( )[ ] wssat nnG γγ +−= 1
Weight Volume
Water
Vw = wGs
Vv = nWw = nγw
S lid
w s
V = 1Solids Vs = 1-nWs = Gsγw(1–n )
γ‐n‐w relationship (fully saturated) 27
( )nGn
WWw ww
sat −==
1γγ
( )sat Ge
nGnw =−
=1( )nGW wss 1γ ( ) ss GnG 1
Weight Volume
Water
Vw = wGs
Vv = nWw = nγw
S lid
w s
V = 1Solids Vs = 1-nWs = Gsγw(1–n )
Summary 28
Relative density 29
Relative density (Dr) is commonly used to indicate the in‐situ denseness or looseness of granular soils.
ee
minmax
max
eeeeDr −
−=
Relative density 30
eG ws
d +=
1γγ 1−=
d
wsGeγγ 1
(min)max −=
d
wsGeγ
γ
1min −= wsGeγ
γ
(max)dγ
S b tit ti d i t ti f DSubstituting e, emin, and emax into equation of Dr
11
minmax
max
eeeeDr −
−= ⎥
⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−=
−
−=
d
d
dd
ddddrD
γγ
γγγγγγ (max)
(min)(max)
(min)(min)
11dd γγ (max)(min)