Coulomb theory includes wall friction and a possible inclined wall back face. For a vertical wall with horizontal backfill, it reduces to a wall-friction correction.
Determine the lateral pressure of soil at rest 6 m below the ground surface. The soil has $\phi=30^\circ$, dry unit weight $17.5$ kN/m3, and a deep water table.
Problem 131: Total Lateral Pressure with Water Table
Calculate total lateral pressure at 6 m depth. $K_0=0.40$, water table is 2 m below ground surface, and the extracted solution uses 18.8 kN/m3 above the water table and effective unit weight 9.4 kN/m3 below it.
$$\sigma_h'=K_0[2(18.8)+4(9.4)]$$
$$\sigma_h'=0.40(75.2)=30.08\text{ kPa}$$
$$u=9.81(4)=39.24\text{ kPa}$$
$$\sigma_h=30.08+39.24=69.32\text{ kPa}$$
Answer: Total lateral pressure is 69.32 kPa.
Problem 132: At-Rest Pressure with Water Table at Surface
For sand with water table at the ground surface, compute total at-rest lateral pressure at 5 m. Saturated unit weight is 20.5 kN/m3 and $K_0=0.507$.
A 5.4 m retaining wall supports horizontal cohesionless backfill with $\gamma=17.30$ kN/m3 and $\phi=36^\circ$. Determine at-rest force per unit wall length.
$$K_0=1-\sin36^\circ=0.41$$
$$P_0=\frac{1}{2}K_0\gamma H^2$$
$$P_0=\frac{1}{2}(0.41)(17.3)(5.4)^2$$
$$P_0\approx103.4\text{ kN/m}$$
Answer: $P_0\approx103.4$ kN/m.
Problem 134: At-Rest Force with Surcharge
A 6 m vertical wall supports horizontal cohesionless backfill with $\gamma=16$ kN/m3, $\phi=32^\circ$, and uniform surcharge $q=15$ kPa. Determine at-rest lateral force per unit wall length.
A 6 m wall is prevented from moving and supports horizontal backfill as shown in the PDF. The extracted solution uses $\phi=32^\circ$ and $\gamma=16$ kN/m3. Compute lateral force per unit wall length.
$$K_0=1-\sin32^\circ=0.47$$
$$P_0=\frac{1}{2}(0.47)(16)(6)^2$$
$$P_0=135.36\text{ kN/m}$$
Answer: $P_0=135.36$ kN/m.
Problem 136: Friction Angle from Active Coefficient
Using Rankine theory for horizontal backfill, find $\phi$ if $K_a=0.50$.
$$0.50=\frac{1-\sin\phi}{1+\sin\phi}$$
$$0.50+0.50\sin\phi=1-\sin\phi$$
$$\sin\phi=\frac{1}{3}$$
$$\phi=19.5^\circ$$
Answer: $\phi=19.5^\circ$.
Problem 137: Rankine Active Force
A 6 m retaining wall supports horizontal backfill as shown. The extracted solution uses $K_a=0.307$, consistent with $\phi=32^\circ$, and $\gamma=16$ kN/m3. Compute active force.
A vertical 6 m wall retains horizontal backfill with $e=0.60$, $G_s=2.60$, $w=24\%$, and $\phi=25^\circ$. Compute active force if the water table is at the ground surface.
A frictionless vertical wall retaining horizontal backfill is 8 m high. Soil has $\gamma=17.17$ kN/m3, $\phi=20^\circ$, and $c=13$ kPa. Compute maximum tensile crack depth.
A 6 m frictionless wall supports soil with $\gamma=17.4$ kN/m3, $c=14$ kPa, and $\phi=26^\circ$. Compute Rankine active force before tensile cracks occur.
A 5.5 m vertical frictionless wall supports soft backfill with $\gamma=15.5$ kN/m3, undrained friction angle $\phi=0^\circ$, and $c=16.6$ kPa. Determine lateral force after tensile cracks occur.
Problem 144: Rankine Active Force for Inclined Backfill
A 9 m wall supports inclined cohesionless backfill making $\alpha=15^\circ$ with the horizontal. The unit weight is 18 kN/m3 and $\phi=30^\circ$. Compute total active force using Rankine theory.
Problem 145: Tensile Crack in Inclined Cohesive Backfill
A frictionless wall supports inclined backfill at $\alpha=5^\circ$ with the horizontal. The soil has $c=10$ kPa, $\gamma=16.5$ kN/m3, and $\phi=30^\circ$. Find the tensile crack depth.
Problem 146: Coulomb Active Force with Wall Friction
A 6 m vertical retaining wall has cohesionless horizontal backfill with $\gamma=15.5$ kN/m3, $\phi=30^\circ$, and wall-soil coefficient of friction $\mu_w=0.268$. Compute active static force using Coulomb theory.
Three states of lateral earth pressure depend on wall movement. The at-rest condition applies when no lateral strain occurs (e.g., rigid basement walls). Jaky's formula for normally consolidated soil:
$$K_0 = 1 - \sin\phi'$$
When a wall moves away from the soil, stress decreases to the active state (minimum lateral pressure). When a wall is pushed into the soil, stress increases to the passive state (maximum lateral pressure). Rankine's coefficients for a level cohesionless backfill:
For cohesive backfill ($c$-$\phi$ soil), the active horizontal stress is $\sigma_h = K_a\gamma z - 2c\sqrt{K_a}$, producing a tension zone near the surface to depth $z_c = 2c/(\gamma\sqrt{K_a})$.
Problem: At-Rest Pressure on Rigid Basement Wall
A rigid basement wall retains 3.5 m of backfill. The soil has $\phi=30°$ and $\gamma=18$ kN/m³ and is normally consolidated. Find the at-rest horizontal stress at the base of the wall, total at-rest force per unit length, and its point of application.
$$K_0 = 1 - \sin 30° = 1 - 0.5 = 0.5$$
$$\sigma_{h,\text{base}} = K_0\gamma H = 0.5(18)(3.5) = 31.5\text{ kPa}$$
Total force (triangular pressure diagram):
$$P_0 = \frac{1}{2}\sigma_{h,\text{base}} H = \frac{1}{2}(31.5)(3.5) = 55.1\text{ kN/m}$$
Acts at $H/3$ from the base:
$$\bar{y} = \frac{H}{3} = \frac{3.5}{3} = 1.17\text{ m from base}$$
Answer: $\sigma_{h}=31.5$ kPa at base, $P_0=55.1$ kN/m at 1.17 m above base.
Problem: Rankine Active Force with Uniform Surcharge
A 5.0 m retaining wall has a level backfill with $\phi=32°$, $\gamma=17.5$ kN/m³, and $c=0$. A uniform surcharge $q=20$ kPa acts on the surface. Find total active force per unit length and the location of its resultant from the base.
Surcharge adds a uniform pressure block; self-weight adds a triangular block:
$$P_{a,q} = K_a q H = 0.3073(20)(5) = 30.73\text{ kN/m at }H/2=2.5\text{ m from base}$$
$$P_{a,\gamma} = \frac{1}{2}K_a\gamma H^2 = \frac{1}{2}(0.3073)(17.5)(25) = 67.22\text{ kN/m at }H/3=1.67\text{ m from base}$$
$$P_a = 30.73 + 67.22 = 97.95\text{ kN/m}$$
Moment about base: $30.73(2.5) + 67.22(1.67) = 76.83 + 112.26 = 189.09$ kN·m/m.
$$\bar{y} = \frac{189.09}{97.95} = 1.93\text{ m from base}$$
Answer: $P_a=97.95$ kN/m acting 1.93 m above the base.
Problem: Rankine Active Force with Water Table in Backfill
A 5.0 m retaining wall has a backfill with the water table at 2.0 m below the surface. Above the water table: $\gamma_{dry}=16$ kN/m³. Below: $\gamma_{sat}=19$ kN/m³. Soil has $\phi=30°$, $c=0$. Compute total lateral force (effective soil pressure + water pressure) and its resultant height above the base.
Moment about base $= 10.67(4.33)+32.01(1.5)+13.76(1.0)+44.15(1.0)=152.2$ kN·m/m.
$$\bar{y} = \frac{152.2}{100.6} = 1.51\text{ m from base}$$
Answer: Total lateral force $= 100.6$ kN/m acting 1.51 m above the base.
Problem: Retaining Wall — Factors of Safety
A gravity retaining wall with height $H=5$ m has a net active force $P_a=65$ kN/m acting at 1.80 m above the base. The total weight of wall plus retained soil is $W=200$ kN/m acting 1.00 m from the toe. The base width is $B=2.8$ m and base friction angle $\delta_b=28°$. Passive resistance is neglected. Find FS against overturning about the toe and FS against sliding.
Answer: $FS_{OT}=1.71$ and $FS_{\text{sliding}}=1.64$. Both exceed 1.5 — wall is adequate.
Problem: Active Force on Cohesive Backfill with Tension Crack
A vertical wall retains a 6.0 m high cohesive backfill with $c=25$ kPa, $\phi=20°$, and $\gamma=18$ kN/m³. Find: (a) the tension crack depth $z_c$, (b) net active force assuming the crack forms but remains dry, and (c) net active force if the tension crack fills with water.
