Why does the lower c-phi layer add no effective active soil thrust?

Its pressure Ka*sigma - 2c sqrt(Ka) is negative at top and base (tension)

What effect does uplift under the base have on stability checks?

It reduces net vertical load and resisting moment, lowering FS

It increases the net vertical force

It only affects the horizontal force

How is base sliding resistance computed here?

Sum V tan(delta) + c_a B, with delta = 2/3 phi and c_a = 2/3 c

What does e = 0.526 m < B/6 = 0.833 m tell us?

Resultant is in the middle third, so no tension at the base

2080 Baisakh · 12 marks

Stability of gravity wall, two layers

Short answer

FS against overturning = 4.87, sliding = 1.79; eccentricity 0.53 m (< B/6), maximum base pressure ≈ 88 kN/m²; bearing capacity is adequate; the wall is stable.

A concrete gravity retaining wall of height 6 m retains two layers of horizontal backfill soil. The upper layer is 3 m deep cohesionless soil of unit weight $20\ \text{kN/m}^3$ and angle of internal friction $32^\circ$ and the bottom layer is c- $\phi$ soil of unit weight $22\ \text{kN/m}^3$ , cohesion $30\ \text{kN/m}^2$ and angle of internal friction $27^\circ$ . There is water table at the depth of 2 m from the ground surface. The back of the retaining wall is smooth and vertical with the top width 1.2 m and bottom width 5 m. Check the stability of the retaining wall. Take unit weight of concrete as $25\ \text{kN/m}^3$ .

Assumptions: smooth vertical back; the sloping face is on the front (toe) side, so the wall is a rectangle $1.2\times6$ m plus a triangle $3.8\times6$ m; no embedment or passive resistance; water in the backfill acts at the wall back and uplift acts under the base. $\gamma_w = 9.81$ kN/m $^3$ , and the unit weights given are taken as saturated below the water table.

1. Active pressure (Rankine)

K_{a1} = \tan^2(45^\circ-16^\circ) = 0.3073,\quad K_{a2} = \tan^2(45^\circ-13.5^\circ) = 0.3755,\quad 2c\sqrt{K_{a2}} = 36.77

Upper layer, 0-2 m (dry/moist): $p$ at 2 m $= 0.3073\times20\times2 = 12.29$ kN/m $^2$ .
Upper layer, 2-3 m (submerged, $\gamma'=10.19$ ): $\sigma_v'=40+10.19 = 50.19$ , $p = 15.42$ kN/m $^2$ .
Lower layer (c- $\phi$ ): top $p = 0.3755\times50.19-36.77 = -17.92$ ; base $\sigma_v' = 86.76$ , $p = -4.19$ . Both are negative, so the c- $\phi$ soil gives no effective active pressure (tension zone neglected).
Water pressure: $0$ at 2 m depth, $9.81\times4 = 39.24$ kN/m $^2$ at the base.

Part	Force (kN/m)	Arm above base (m)	Moment
Triangle 0-2 m	12.29	4.667	57.35
Rectangle 2-3 m	12.29	3.500	43.02
Triangle 2-3 m	1.57	3.333	5.22
Water 0.5 x 39.24 x 4	78.48	1.333	104.64
Total $\sum H$	104.63		210.23

2. Wall weight and uplift

Part	Force (kN/m)	Arm from toe (m)
Rectangle 1.2 x 6 x 25	180.0	4.400
Triangle 0.5 x 3.8 x 6 x 25	285.0	2.533
Uplift (taken uniform, $39.24\times5$ )	-196.2	2.500
$\sum V$ (net)	268.8

$M_r = 180\times4.4+285\times2.533 - 196.2\times2.5 = 1023.5$ kN m/m.

3. Checks

Overturning: $FS_{ot} = 1023.5/210.2 = 4.87$ (>2). Safe. (Without uplift it would be 7.20.)
Sliding: base friction $\delta = \tfrac23\times27^\circ = 18^\circ$ , adhesion $c_a = \tfrac23\times30 = 20$ kN/m $^2$ :

FS_s = \frac{\sum V\tan18^\circ + c_aB}{\sum H} = \frac{268.8\times0.3249 + 20\times5}{104.6} = 1.79\ (>1.5)

Safe.

Eccentricity and base pressure: $\bar x = (1023.5-210.2)/268.8 = 3.026$ m, $e = |2.5-3.026| = 0.526$ m $< B/6 = 0.833$ m, so no tension. The resultant lies toward the heel:

q_{max} = 87.7,\ q_{min} = 19.9\ \text{kN/m}^2\ (\text{effective, after uplift})

Bearing capacity (Vesic factors for $\phi=27^\circ$ : $N_c=23.94$ , $N_\gamma=14.47$ ; $B' = 5-2e = 3.95$ m, surface footing, $\gamma'=12.19$ ):

q_u = cN_c + \tfrac12\gamma' B' N_\gamma = 30\times23.94 + 0.5\times12.19\times3.95\times14.47 = 1067\ \text{kN/m}^2

$FS_b = 1067/q_{max}$ is well above 3. Safe.

Answer: $FS_{ot} = 4.87$ , $FS_{sliding} = 1.79$ , $e = 0.53$ m $< B/6$ , $q_{max} \approx 88$ kN/m $^2$ with a large bearing margin. The wall is stable (assuming the stated adhesion and uplift conditions).

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All Foundation Engineering chapter 3 questions →

Stability of gravity wall, two layers

1. Active pressure (Rankine)

2. Wall weight and uplift

3. Checks

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