SME
Soil Mechanics
Lateral earth pressure
Use this for “A 6 m wall holds back a dry sand with φ′ = 32° and γ = 18.5 kN/m³. Find Kₐ, the active thrust and where it acts.”
Chart
σ′ₕ(6) ≈ 34.11
Shaded: ∫ from 0 to 6 of σ′ₕ(z) dz = 102.3
This is an example. Type your own values to start a new problem.
Kₐ
=
(1
−
sin(
φ′
°))
÷
(1
+
sin(
φ′
°))
σ′ₐ (kPa)
=
Kₐ
×
γ (kN/m³)
×
H (m)
Pₐ (kN/m)
=
½
×
σ′ₐ (kPa)
×
H (m)
z̄
Height of the thrust
calculated · m
Formulas
Kₐ
=
1 − sin(φ′)
1 + sin(φ′)
0.3073
=
1 − sin(32°)
1 + sin(32°)
σ′ₐ
=
Kₐ
×
γ
×
H
34.1057 = 0.3073 × 18.5 × 6
Pₐ
=
1
2
×
σ′ₐ
×
H
102.3171
=
1
2
×
34.1057
×
6
z̄
=
H
3
2
=
6
3
Assumptions
•
Rankine: a smooth vertical wall and a level, dry, cohesionless backfill.
•
The wall moves away from the soil enough to reach the active state.
•
Pₐ is per metre of wall and acts at H ÷ 3, the centroid of the triangle.
Step-by-step
WE KNOW
φ′ = 32°, γ = 18.5 kN/m³, H = 6 m
FIND
active coefficient (Kₐ), pressure at the base (σ′ₐ), active thrust (Pₐ), height of the thrust (z̄)
Step 1 · Find active coefficient (Kₐ)
Use Kₐ = (1 − sin(φ′)) ÷ (1 + sin(φ′))
Rankine’s
active
wedge:
tan²(45°
−
φ′
2
)
=
1 − sin φ′
1 + sin φ′
.
Kₐ
=
1 − sin(32°)
1 + sin(32°)
Kₐ
=
1 − 0.5299
1 + 0.5299
Kₐ
=
0.4701
1.5299
Kₐ = 0.307259
Step 2 · Find pressure at the base (σ′ₐ)
Use σ′ₐ = Kₐ × γ × H
The vertical stress γz at the base, times the coefficient, pushes sideways on the wall.
σ′ₐ = 0.307259 × 18.5 × 6
σ′ₐ = 5.684 × 6
σ′ₐ = 34.1057 kPa
Step 3 · Find active thrust (Pₐ)
Use Pₐ = ½ × σ′ₐ × H
The pressure grows from 0 at the top to its base value: the thrust is the triangle’s area.
Pₐ
=
1
2
×
34.1057
×
6
Pₐ = 17.05 × 6
Pₐ = 102.3171 kN/m
Step 4 · Find height of the thrust (z̄)
Use z̄ = H ÷ 3
The thrust acts at the centroid of the triangle, one third of the height above the base.
z̄
=
6
3
z̄
=
2
m
Check
Put the numbers back in. Both sides should match.
0.307259
=
1 − sin(32°)
1 + sin(32°)
✓
34.1057 = 0.307259 × 18.5 × 6 ✓
102.3171
=
1
2
×
34.1057
×
6
✓
2
=
6
3
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Passive resistance
Use this for “A wall’s toe is buried 1.5 m in sand with φ′ = 32° and γ = 18.5 kN/m³. Find Kₚ and the passive resistance in front of it.”
At-rest pressure
Use this for “A basement wall 6 m high holds back sand with φ′ = 32°, OCR = 2 and γ = 18.5 kN/m³. Find K₀ and the at-rest thrust.”
Active thrust with the backfill under water
Use this for “A 6 m wall holds back saturated sand (φ′ = 32°, γₛₐₜ = 20 kN/m³) with water up to its top. Find the total thrust.”