Rankine Active Force — Two-Layer Backfill with Water Table and Surcharge: 18-Env-A3 Geotechnical and Hydrogeological Engineering, December 2015, Q5

Question 5: Rankine Active Force — Two-Layer Backfill with Water Table and Surcharge (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.

Check: the source prints the surcharge as $q=15\ \text{kN/m}^3$, but a surcharge is a pressure (force per area), not a unit weight — kN/m³ is dimensionally impossible for a surface load. This is read as the standard convention $q=15\ \text{kN/m}^2=15\ \text{kPa}$.

Given.

Given data
Quantity	Symbol	Value
Total wall height	$H$	6 m
Depth to water table	$H_1$	2 m
Unit weight above WT	$\gamma_1$	16 kN/m³
Saturated unit weight below WT	$\gamma_2$	19 kN/m³
Friction angle above / below WT	$\phi_1,\phi_2$	32°, 36°
Surface surcharge	$q$	15 kPa

Find. (a) Rankine active force per unit length, $P_a$; (b) height of the resultant above the wall base.

Effective (Rankine) active-earth-pressure diagram in orange, hydrostatic pore-water pressure in blue — both plotted to scale against the wall face.

Approach. Compute each layer's own $K_a$ from its own $\phi$, apply it to the EFFECTIVE vertical stress at that depth (using the submerged unit weight $\gamma_2'=\gamma_2-\gamma_w$ below the water table), add the hydrostatic pore-water pressure separately, decompose the resulting diagram into elementary rectangles/triangles, then sum forces and take moments about the base.

Rankine coefficients. $$K_{a1}=\frac{1-\sin32^\circ}{1+\sin32^\circ}=\boxed{0.307},\qquad K_{a2}=\frac{1-\sin36^\circ}{1+\sin36^\circ}=\boxed{0.260}.$$
Effective vertical stress profile (submerged unit weight below WT, $\gamma_2'=19-9.81=9.19\ \text{kN/m}^3$): $$\sigma'_{v,\text{top}}=q=15.0,\quad \sigma'_{v,\text{WT}}=q+\gamma_1H_1=15+16(2)=47.0,$$ $$\sigma'_{v,\text{base}}=47.0+\gamma_2'(H-H_1)=47.0+9.19(4)=83.76\ \text{kPa}.$$
Lateral effective pressures — note the JUMP at the water table since $K_a$ changes even though $\sigma'_v$ is continuous: $$\sigma'_{h1,\text{top}}=K_{a1}(15.0)=4.61,\quad \sigma'_{h1,\text{bot}}=K_{a1}(47.0)=14.44\ \text{kPa (layer 1)},$$ $$\sigma'_{h2,\text{top}}=K_{a2}(47.0)=12.20,\quad \sigma'_{h2,\text{bot}}=K_{a2}(83.76)=21.75\ \text{kPa (layer 2)}.$$
Decompose into five elementary areas and sum. Layer 1 (surcharge rectangle + self-weight triangle), layer 2 (carried-over rectangle + incremental triangle), and the water-pressure triangle over the full 4 m submerged depth: $$F_1=9.22,\ F_2=9.83,\ F_3=48.81,\ F_4=19.09,\ F_w=78.48\ \text{kN/m},$$ $$P_a=F_1+F_2+F_3+F_4+F_w=\boxed{165.4\ \text{kN/m}}.$$
(b) Location of the resultant — take moments of each shape's own centroid height above the base ($y_1=5.00,\ y_2=4.667,\ y_3=2.00,\ y_4=y_w=1.333$ m): $$\bar y=\frac{F_1y_1+F_2y_2+F_3y_3+F_4y_4+F_wy_w}{P_a}=\frac{319.5}{165.4}=\boxed{1.93\ \text{m above the base}}.$$

Quantity	Value
$K_{a1}$ / $K_{a2}$	0.307 / 0.260
(a) Rankine active force, $P_a$	165.4 kN/m
(b) Height of resultant above base, $\bar y$	1.93 m

Concept & Theory

Topic. Rankine active earth pressure — layered soil with a water table and surcharge.

Key relations. $K_a=(1-\sin\phi)/(1+\sin\phi)$; $\sigma'_h=K_a\sigma'_v$; pore pressure $u=\gamma_w z_w$ added separately; $P=\sum(\text{areas})$, $\bar y=\sum(F_iy_i)/\sum F_i$.

Why this works. Rankine theory assumes a frictionless wall and semi-infinite, level backfill in plastic equilibrium, so the lateral EFFECTIVE stress at any depth is $K_a$ (using whichever layer's friction angle applies there) times the effective vertical stress at that depth; water pressure is purely hydrostatic and adds on top, independent of $K_a$.

Common pitfall. Using the total (not submerged) unit weight below the water table — double-counting the water's weight in both the effective-stress and pore-pressure terms — and forgetting the pressure diagram steps at a soil-layer interface even though vertical stress itself is continuous there.

Source. Craig & Knappett, Craig's Soil Mechanics, ch. on lateral earth pressure; Das, Principles of Geotechnical Engineering, ch. 13.

Question 5 of 6: Rankine Active Force — Two-Layer Backfill with Water Table and Surcharge

Question 5: Rankine Active Force — Two-Layer Backfill with Water Table and Surcharge (20 marks)