A 3 m high retaining wall supports sandy soil of unit weight 18.5 kN m-3. The angle of shearing resistance (φ) is 30° and the surface of soil is horizontal. The magnitude of the active thrust (in kN m-1) and its acting point from the top (in m) are
GATE AG - 2016,
Answer: C
The active thrust is 27.75 kN per metre, acting 2 m from the top.
By Rankine's theory, for a level backfill of cohesionless soil:
\[ \begin{aligned} K_{a} &= \frac{1 - \sin 30^{\circ}}{1 + \sin 30^{\circ}} = \frac{1}{3} \\ P_{a} &= \frac{1}{2}K_{a}\gamma H^{2} \\ &= \frac{1}{2} \times \frac{1}{3} \times 18.5 \times 3^{2} = 27.75 \end{aligned} \]
The pressure increases linearly with depth, so the triangle's resultant acts at \(H/3 = 1\) m above the base, which is 2 m below the top.
The options with 1 m measure the point from the base instead of the top. 249.75 is \(\frac{1}{2} \times 18.5 \times 3^{2} \times 3\), using \(1/K_{a} = 3\), the passive coefficient, instead of \(K_{a}\).
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