Why is the total active thrust horizontal here?

Because the water table is present

Why does gamma_sat = 21 not enter the main solution?

The water table is below the base of the wall, so the backfill is above it

How does the uniform surcharge q enter the thrust?

As a rectangle of pressure Ka q over H with arm H/2

It only affects vertical forces

2079 Bhadra · 9 marks

Active thrust with surcharge and water table

Short answer

Rankine Ka = 0.333; total active thrust is 152 kN/m horizontal (perpendicular to the back), applied 2.33 m above the base.

A 6 m high retaining wall having a vertical back has horizontal cohesion-less backfill having $\gamma = 17\ \text{kN/m}^3$ , $\phi = 30^\circ$ and carrying a uniform surcharge of $25\ \text{kN/m}^2$ . It has a water table at a depth of 3.5 m from the base of the retaining wall (as printed). Determine the magnitude and direction of the total active thrust. Take $\gamma_{sat} = 21\ \text{kN/m}^3$ .

Reading of the data: the water table is stated "at 3.5 m from the base of the retaining wall (as printed)", i.e. it lies below the base of the 6 m wall. The backfill within the wall height is then above the water table and $\gamma = 17$ kN/m $^3$ is used; $\gamma_{sat}$ does not enter. (An alternative reading, with the water table 3.5 m below the top, is given at the end.)

Smooth vertical back and horizontal backfill, so Rankine applies and the thrust is horizontal.

K_a = \tan^2(45^\circ - 15^\circ) = 0.3333

Pressures (kN/m $^2$ )

Top: $K_a q = 0.3333\times25 = 8.33$
Base: $K_a(q + \gamma H) = 0.3333\times(25 + 17\times6) = 42.33$

Thrust

Part	Force (kN/m)	Arm above base (m)
Surcharge (rectangle) $K_a q H = 0.3333\times25\times6$	50.00	3.00
Soil (triangle) $\tfrac12 K_a\gamma H^2$	102.00	2.00
Total	152.00	2.33

Answer: total active thrust $P_a = 152$ kN/m, horizontal (perpendicular to the wall back), acting 2.33 m above the base.

Alternative reading (water table 3.5 m below the top, $\gamma_{sat}=21$ , $\gamma'=11.19$ ): pressures are 8.33 (top), 28.17 (at 3.5 m), 37.49 (base, effective) plus water 0 to 24.53. The total thrust is then 176.6 kN/m (soil 145.9 + water 30.7), acting 2.12 m above the base.

✓ Answer checked by Claude Sonnet 5.5

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