SME
Soil Mechanics · Problem type
Field density by the sand cone
Use this for “1,620 g of 1.5 g/cm³ sand fills the test hole, and the soil dug out of it weighs 2,070 g at w = 10%. Find the dry unit weight.”
Diagram
e
=
Vᵥ
Vₛ
=
370
710
=
0.521;
S
=
Vw
Vᵥ
=
0.509.
Mₛ
=
M
1 + w
=
2,070
1.1
=
1,880
g;
Mw
=
188
g.
The solids fill 710 of the 1,080 cm³.
This is an example. Type your own values to start a new problem.
V (cm³)
=
msand (g)
÷
ρsand (g/cm³)
ρ (g/cm³)
=
M (g)
÷
V (cm³)
ρd (g/cm³)
=
ρ (g/cm³)
÷
(1
+
w
%)
γd
Dry unit weight
calculated · kN/m³
Gₛ
Specific gravity of solids
example
e
Void ratio
calculated
Formulas
V
=
msand
ρsand
1,080
=
1,620
1.5
ρ
=
M
V
1.9167
=
2,070
1,080
ρd
=
ρ
1 +
w
÷ 100
1.7424
=
1.9167
1 + 10 ÷ 100
γd = ρd × 9.81
17.0932 = 1.7424 × 9.81
1
+
e
=
Gₛ
×
1
ρd
1
+
0.5209
=
2.65
×
1
1.7424
Assumptions
•
The sand used is the sand in the hole: the sand left in the cone is already taken off.
•
All the soil dug out of the hole is weighed; w comes from a sample of it, oven dried.
•
ρw = 1 g/cm³, so Gₛ ÷ ρd is 1 + e; 1 g/cm³ weighs 9.81 kN/m³.
Step-by-step
WE KNOW
msand = 1,620 g, ρsand = 1.5 g/cm³, M = 2,070 g, w = 10%, Gₛ = 2.65
FIND
hole volume (V), moist density (ρ), dry density (ρd), dry unit weight (γd), void ratio (e)
Step 1 · Find hole volume (V)
Use V = msand ÷ ρsand
The sand’s mass over its density is the volume it fills: the hole’s.
V
=
1,620
1.5
V
=
1,080
cm³
Step 2 · Find moist density (ρ)
Use ρ = M ÷ V
The soil’s mass over the hole’s volume is its moist density.
ρ
=
2,070
1,080
ρ
=
1.9167
g/cm³
Step 3 · Find dry density (ρd)
Use ρd = ρ ÷ (1 + w ÷ 100)
Taking
the
water
out
leaves
the
solids:
divide
by
1
+
w
.
ρd
=
1.9167
1 + 10 ÷ 100
ρd
=
1.9167
1 + 0.1
ρd
=
1.9167
1.1
ρd = 1.742424 g/cm³
Step 4 · Find dry unit weight (γd)
Use γd = ρd × 9.81
Each
1
g/cm³
(1000
kg/
m
3
)
weighs
9.81
kN/
m
3
.
γd = 1.742424 × 9.81
γd
=
17.0932
kN/
m
3
Step 5 · Find void ratio (e)
Use 1 + e = Gₛ × 1 ÷ ρd
ρw
=
1
g/cm³:
the
solids
alone
would
be
Gₛ
g/cm³;
spread
over
1
+
e
they
make
ρd.
e
=
Gₛ
×
1
ρd
−
1
e
=
2.65
×
1
1.742424
−
1
e
=
1.5209
−
1
e
=
0.5209
Check
Put the numbers back in. Both sides should match.
1,080
=
1,620
1.5
✓
1.9167
=
2,070
1,080
✓
1.742424
=
1.9167
1 + 10 ÷ 100
✓
17.0932 = 1.742424 × 9.81 ✓
1
+
0.5209
=
2.65
×
1
1.742424
✓
Topic check
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