Cantilever Retaining Wall
res: res/sdd-unit-1-comp-1.pdf
Design a cantilever retaining wall to retain an earth embankment of \(3m\) high above ground level. The density of soil is \(17kN/m^2\) and its angle of repose is \(30^\circ\). The horizontal surcharge. The SBC is equal to
1. Depth of Foundation¶
\(q_a\) = SBC (Safe Bearing Capacity of Soil) = \(200kN/m^2\)
\(\gamma_s\) = Density (Unit weight of soil) = \(17kN/m^3\)
\(\phi\) = Angle of internal friction = \(30^\circ\)
= (200/17) × [(1-sin30°)/(1+sin30°)]²
= (200/17) × [(1-0.5)/(1+0.5)]²
= (200/17) × [0.5/1.5]²
= (200/17) × (1/3)²
= (200/17) × (1/9)
= 1.3m
If retaining 3m of soil, depth is 1.3m → H = 4.3
2. Selecting Preliminary Dimensions¶
You draw this diagram after the calculation of step 2, I've just mentioned it here for initial visualization.
Base slab thickness¶
= H/12
Width of footing¶
Heel width¶
Toe width¶
Diagram should be drawn at this point
Step 3: Calculating forces and moments¶
| S.No. | Parts | Vertical Load (kN) | Distance from heel (m) | Moment (kNm) |
|---|---|---|---|---|
| 1 | W₁ Soil on heel | 1.5 × 4.3 × 17 = 109.65 | 1.5/2 = 0.75 | 82.23 |
| 2 | W₂ Stem | 0.36 × 4.3 × 25 = 38.7 | 1.5 + 0.36/2 = 1.68 | 65.01 |
| 3 | W₃ Footing | 0.36 × 3 × 25 = 27 | 3/2 = 1.5 | 40.5 |
| ΣW = 175.35 | ΣM = 187.74 |
W₁ = Soil on heel (the dirt sitting on right side)
- Formula: (Heel width) × (total H) × (soil density)
- = 1.5 × 4.3 × 17 = 109.65 kN
- Distance from heel = 1.5/2 = 0.75m (halfway across heel)
- Moment = 109.65 × 0.75 = 82.23 kNm
W₂ = Stem (the vertical concrete wall)
- Formula: (stem thickness) × (stem height) × 25
- Why 25? Because concrete ALWAYS weighs 25 kN/m³ (memorize this!)
- = 0.36 × 4.3 × 25 = 38.7 kN
- Distance from heel = 1.5 + (0.36/2) = 1.68m
- Moment = 38.7 × 1.68 = 65.01 kNm
W₃ = Base slab (horizontal bottom piece)
- Formula: (base thickness) × (total width B) × 25
- = 0.36 × 3 × 25 = 27 kN
- Distance from heel = 3/2 = 1.5m (middle of base)
- Moment = 27 × 1.5 = 40.5 kNm
Step 4: Calulating Earth Pressure¶
Step 4 in the design process for a Cantilever Wall is Calculating the Earth Pressure. This step involves determining the forces exerted by the retained soil on the wall, which are critical for subsequent stability checks and reinforcement design.
The calculation is broken down into two main parts:
1. Earth Pressure Coefficients¶
This involves calculating both the Active Earth Pressure Coefficient (\(K_a\)) and the Passive Earth Pressure Coefficient (\(K_p\)).
Active Earth Pressure Coefficient (\(K_a\))¶
The coefficient of active earth pressure (\(K_a\)) is calculated using the following general formula, which accounts for the internal friction angle (\(\phi\)) of the soil and the angle of backfill inclination (\(\delta\)):
This formula is applied when the backfill surface is inclined, as seen in Example 1.2, where the backfill inclination angle (\(\delta\)) is \(15^\circ\).
If the backfill is horizontal (i.e., \(\delta = 0\)), the formula simplifies, or a simplified version is used. For the specific case where \(\phi = 30^\circ\) and the backfill is horizontal (implied by Example 1.1), the value of \(K_a\) is \(0.333\). In one example, with an inclined backfill (\(\delta=15^\circ\) and \(\phi=30^\circ\)), the calculated \(K_a\) is \(0.374\).
Passive Earth Pressure Coefficient (\(K_p\))¶
The coefficient of passive earth pressure (\(K_p\)) is calculated using a simplified formula involving the internal friction angle (\(\phi\)):
In examples where \(\phi = 30^\circ\), this formula yields a \(K_p\) value of \(3\).
2. Force due to Active Earth Pressure (\(P_a\))¶
Once the Active Earth Pressure Coefficient (\(K_a\)) is determined, the total Force due to active earth pressure (\(P_a\)), which acts horizontally against the wall, is calculated:
Where:
- \(K_a\) is the Active Earth Pressure Coefficient.
- \(\gamma_e\) (or \(\gamma_s\) in Step 1) is the unit weight of the soil.
- \(H\) is the overall depth of the wall (or the height of the vertical section of the earth retained).
In the case where the backfill is inclined, the total active force (\(P_a\)) acts at an angle, and it is decomposed into horizontal and vertical components:
- Horizontal component: \(P_a \cos \delta\)
- Vertical component: \(P_a \sin \delta\)
The total active force (\(P_a\)) calculated is essential for proceeding to Step 5, where the stability of the wall against overturning and sliding is checked.
Analogy: Calculating the earth pressure in a retaining wall design is like determining the strength of the current against a boat. You first calculate the coefficients (\(K_a\) and \(K_p\)) which represent the properties of the water (soil)—how easily it flows or resists movement. Then, you use those coefficients to calculate the total force (\(P_a\)) exerted by the moving water (active pressure) against the boat's hull (the wall), and finally, you determine the individual directions (horizontal and vertical components) of that force to see if the boat will tip over or slide sideways.