Rankine active earth pressure on a layered c–φ backfill: 16-Civ-A4 Geotechnical Materials and Analysis, May 2015, Q5

Question 5: Rankine active earth pressure on a layered c–φ backfill (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Two cohesive-frictional backfill layers, a uniform surcharge and a water table 1 m into the lower layer.

Given data
Item	Layer 1 (0–3 m)	Layer 2 (3–5 m)
Unit weight $\gamma$ (moist)	18.0 kN/m³	18.5 kN/m³
Saturated unit weight	—	20.0 kN/m³
Friction angle $\phi'$	30°	25°
Cohesion $c'$	20 kPa	25 kPa
Surcharge $q$	20 kPa (over the full backfill surface)
Water table	depth 4 m; $\gamma_w = 9.81$ kN/m³

Find. (i) the active earth-pressure distribution with depth, and (ii) the active thrust per metre of wall once the tension zone has cracked.

Approach. Build the vertical effective-stress profile (surcharge + soil, buoyant below the water table), apply the Rankine active relation $\sigma_a' = K_a\sigma_v' - 2c'\sqrt{K_a}$ layer by layer, add hydrostatic water pressure below the water table, then integrate only the positive pressure once the tensile (cracked) zone is discounted.

Figure 4 — Rankine active pressure envelope. The upper backfill is in tension (cracked, discounted); positive earth pressure develops in the lower layer, with hydrostatic water pressure added below the water table.

Active coefficients. $$K_{a1}=\tan^2\!\left(45^\circ-\tfrac{30^\circ}{2}\right)=0.333,\qquad K_{a2}=\tan^2\!\left(45^\circ-\tfrac{25^\circ}{2}\right)=0.406.$$
Vertical effective stress profile. With the surcharge included and buoyant weight below the water table: $$\sigma_v'(0)=20,\quad \sigma_v'(3)=20+18(3)=74,\quad \sigma_v'(4)=74+18.5(1)=92.5,$$ $$\sigma_v'(5)=92.5+(20-9.81)(1)=102.7\ \text{kPa}.$$
Active earth pressure, upper layer. With $2c_1'\sqrt{K_{a1}} = 2(20)(0.577)=23.1\text{ kPa}$: $$\sigma_a'(0)=0.333(20)-23.1=-16.4\ \text{kPa (tension)},\qquad \sigma_a'(3^-)=0.333(74)-23.1=+1.6\ \text{kPa}.$$ The pressure is negative down to the tension-crack depth where $\sigma_v' = 2c_1'/\sqrt{K_{a1}}=69.3$ kPa: $$z_{c1}=\frac{69.3-20}{18}=\boxed{2.74\ \text{m}}.$$
Active earth pressure, lower layer. With $2c_2'\sqrt{K_{a2}} = 2(25)(0.637)=31.9\text{ kPa}$: $$\sigma_a'(3^+)=0.406(74)-31.9=-1.8,\quad \sigma_a'(4)=0.406(92.5)-31.9=+5.7,\quad \sigma_a'(5)=0.406(102.7)-31.9=+9.8\ \text{kPa}.$$ The lower layer is in tension until $z_{c2}=3.0+\dfrac{2c_2'/\sqrt{K_{a2}}-74}{18.5}=3.24\ \text{m}$.
Water pressure. Hydrostatic below the water table: $$u(4)=0,\qquad u(5)=9.81(1)=9.8\ \text{kPa}.$$
Active force after cracking (part ii). Once the tensile zones crack they carry no load, so the thrust is the area of the positive pressure blocks plus the water thrust: $$P_{a1}=\tfrac12(1.6)(3-2.74)=0.2,\quad P_{a2}=\tfrac12(5.7)(4-3.24)+\tfrac12(5.7+9.8)(1)=9.9,$$ $$P_w=\tfrac12(9.8)(1)=4.9\ \text{kN/m}.$$ $$\boxed{P_a = P_{a1}+P_{a2}+P_w \approx 15.0\ \text{kN/m}.}$$

Check: the cohesion given for the backfill (20–25 kPa) is large relative to the 5 m wall height, so most of the retained soil is in tension and contributes nothing once cracked — the active thrust is modest and is dominated by the water pressure. This is the intended teaching point (cohesive backfill sheds most of its active load through tension cracks), not an error. It assumes the open cracks stay dry; if a crack fills with water, an additional hydrostatic thrust over the cracked depth must be added, which would substantially increase $P_a$. Because the two layers do not share the same $c'$, the diagram legitimately shows a small positive sliver at the base of layer 1 followed by renewed tension at the top of layer 2.

Question 5 — results
Quantity	Value
$K_{a1}$ / $K_{a2}$	0.333 / 0.406
Tension-crack depth (layer 1 / layer 2)	2.74 m / 3.24 m
Active earth pressure at base (z = 5 m)	9.8 kPa
Water pressure at base (z = 5 m)	9.8 kPa
Active thrust after cracking, $P_a$	≈ 15.0 kN/m

Topic. Rankine active earth pressure on a stratified cohesive-frictional backfill with surcharge and groundwater.

Key relations. $K_a=\tan^2(45^\circ-\phi'/2)$; $\sigma_a'=K_a\sigma_v'-2c'\sqrt{K_a}$; tension-crack depth $z_c$ where $\sigma_a'=0$; total lateral pressure $=\sigma_a'+u$; thrust $=$ area of the pressure diagram.

Why this works. Cohesion lets a soil stand in tension near the surface, which the Rankine formula shows as a negative active pressure. Real backfill cannot sustain tension, so it cracks; after cracking only the positive pressure and the water pressure push on the wall. Working in effective stress and adding water pressure separately keeps the buoyancy and the hydrostatic thrust correct.

Common pitfall. Integrating the negative (tension) part into the thrust, forgetting to switch to buoyant unit weight below the water table, or omitting the separate water-pressure triangle. Also, applying a single $2c'\sqrt{K_a}$ across the layer boundary instead of recomputing it for each layer’s $c'$ and $K_a$.

Source. Das & Sobhan, Principles of Geotechnical Engineering, Ch. 13 (lateral earth pressure; Rankine active with $c'$–$\phi'$ soil, tension cracks).

Question 5 of 6: Rankine active earth pressure on a layered c–φ backfill

Question 5: Rankine active earth pressure on a layered c–φ backfill (20 marks)