SME
Soil Mechanics · Problem type
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.”
Chart
σ′ₕ(1.5) ≈ 90.31
Shaded: ∫ from 0 to 1.5 of σ′ₕ(z) dz = 67.74
This is an example. Type your own values to start a new problem.
Kₚ
=
(1
+
sin(
φ′
°))
÷
(1
−
sin(
φ′
°))
σ′ₚ (kPa)
=
Kₚ
×
γ (kN/m³)
×
D (m)
Pₚ (kN/m)
=
½
×
σ′ₚ (kPa)
×
D (m)
z̄
Height of the thrust
calculated · m
Formulas
Kₚ
=
1 + sin(φ′)
1 − sin(φ′)
3.2546
=
1 + sin(32°)
1 − sin(32°)
σ′ₚ
=
Kₚ
×
γ
×
D
90.3148 = 3.2546 × 18.5 × 1.5
Pₚ
=
1
2
×
σ′ₚ
×
D
67.7361
=
1
2
×
90.3148
×
1.5
z̄
=
D
3
0.5
=
1.5
3
Assumptions
•
Rankine: a smooth vertical face and level ground in front of the wall.
•
The wall pushes into the soil; this takes far more movement than the active state.
•
Designers often count only part of Pₚ, with a factor of safety of 2 or more.
Step-by-step
WE KNOW
φ′ = 32°, γ = 18.5 kN/m³, D = 1.5 m
FIND
passive coefficient (Kₚ), passive pressure at the base (σ′ₚ), passive thrust (Pₚ), height of the thrust (z̄)
Step 1 · Find passive coefficient (Kₚ)
Use Kₚ = (1 + sin(φ′)) ÷ (1 − sin(φ′))
Rankine’s
passive
wedge:
tan²(45°
+
φ′
2
)
=
1 + sin φ′
1 − sin φ′
.
Kₚ
=
1 + sin(32°)
1 − sin(32°)
Kₚ
=
1 + 0.5299
1 − 0.5299
Kₚ
=
1.5299
0.4701
Kₚ = 3.254588
Step 2 · Find passive pressure at the base (σ′ₚ)
Use σ′ₚ = Kₚ × γ × D
The vertical stress γz at the base, times the coefficient, pushes sideways on the wall.
σ′ₚ = 3.254588 × 18.5 × 1.5
σ′ₚ = 60.21 × 1.5
σ′ₚ = 90.3148 kPa
Step 3 · Find passive thrust (Pₚ)
Use Pₚ = ½ × σ′ₚ × D
The pressure grows from 0 at the top to its base value: the thrust is the triangle’s area.
Pₚ
=
1
2
×
90.3148
×
1.5
Pₚ = 45.1574 × 1.5
Pₚ = 67.7361 kN/m
Step 4 · Find height of the thrust (z̄)
Use z̄ = D ÷ 3
The thrust acts at the centroid of the triangle, one third of the height above the base.
z̄
=
1.5
3
z̄
=
0.5
m
Check
Put the numbers back in. Both sides should match.
3.254588
=
1 + sin(32°)
1 − sin(32°)
✓
90.3148 = 3.254588 × 18.5 × 1.5 ✓
67.7361
=
1
2
×
90.3148
×
1.5
✓
0.5
=
1.5
3
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Lateral earth pressure
Main lesson
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