Problem: Design a counterfort retaining wall using input data.
- Draw cross section of retaining wall showing reinforcement
- Draw longitudinal section showing curtailment.
Design steps:
- Height of counterfort wall above ground level.
- Density of soil in kN/m3.
- Angle of internal friction in degree.
- Safe bearing capacity of soil in kN/m2.
- Coefficient of friction as 0.5.
- Spacing of counterforts.
- Strength of concrete in M20.
- Strength of steek in Fe-415.
Minimum depth of foundation
\[{D_f} = \frac{p}{w}{\left( {\frac{{1 - \sin \phi }}{{1 + \sin \phi }}} \right)^2}\]
Overall depth of wall, H = Height of Cantilever wall above ground level +Df
Thickness of base slab.
\[T = 2 \times L \times H\]
Adopt Thickness of base slab, T = Thickness of stem
Height of stem, h = H-T
Width of base slab, b = 0.6 * H to 0.7 * H
Adopt base width, b
Calculate toe projection
\[{T_w} = \frac{B}{4}\]
Top width of stem should be 200mm to 300mm.
Maximum horizontal pressure on stem.
\[P = {K_a} \times \gamma \times {H_{stem}}\]
Maximum bending moment.
\[M = \frac{{P \times {L^2}}}{{12}}\]
Factored bending moment.
\[M_{u} = 1.5 \times M\]
Calculate the required effective depth.
\[d = \sqrt {\frac{{{M_u}}}{{0.138 \times {f_{ck}} \times b}}}\]
Calculate the area of steel for stem.
\[{M_u} = \left( {0.87 \times {f_y} \times {A_{st}} \times d} \right)\left( {1 - \frac{{{A_{st}} \times {f_y}}}{{b \times d \times {f_{ck}}}}} \right)\]
Calculate main reinforcement spacing.
\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
Calculate the distribution steel for stem.
\[D_{st} = 0.12 \times {A_{c/s}}\]
Calculate distribution reinforcement spacing.
\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
Pressure distribution at the base is computed by taking moments of all forces about heel point a
Table: Stability calculation
| Serial No. | Loads | Magnitude of Load (kN) | Distance from a (m) | Moment about a (m) |
|---|---|---|---|---|
| 1. | Weight of stem slab | |||
| 2. | Weight of base slab | |||
| 3. | Weight of soil | |||
| 4. | Moment of earth pressure | |||
| 5. | Total |
Distance of point of application of resultant from end a.
\[z = \frac{{\sum M }}{{\sum W }}\]
Eccentricity.
\[e = z - \frac{b}{2}\]
\[e < \frac{b}{6}\]
Minimum and maximum pressures at the base.
\[{\sigma _{_{\max }}} = \frac{{\sum W }}{b}\left( {1 + \frac{{6e}}{b}} \right)\]
\[{\sigma _{_{\min }}} = \frac{{\sum W }}{b}\left( {1 - \frac{{6e}}{b}} \right)\]
We need to check for overturning and sliding
\[{F_1} = \frac{{0.9 \times \sum {{M_s}} }}{{\sum {{M_o}} }} > 1.4\]
ΣMs = Stabilizing moment
ΣMo = Overturning moment
\[{F_2} = \frac{{0.9 \times \mu \sum W }}{{\sum {{P_H}} }} > 1.4\]
ΣPH = Total horizontal pressure.
ΣW = Total weight of retaining wall.
The maximum bending moment on the heel slab is calculated by taking moments of all the forces about the points b.
Table: Moments in heel slab
| Serial No. | Loads | Magnitude of Load ( kN ) | Distance from b ( m ) | Moment about b ( m ) |
|---|---|---|---|---|
| 1. | Weight of earth pressure | |||
| 2. | Weight of heel slab | |||
| 3. | Upward soil pressure | |||
| 4. | Total |
\[\sum {moment} = M\]
Factored moments.\[{M_u} = 1.5 \times M\]
calulate Area of steel Ast.\[{M_u} = \left( {0.87 \times {f_y} \times {A_{st}} \times d} \right)\left( {1 - \frac{{{A_{st}} \times {f_y}}}{{b \times d \times {f_{ck}}}}} \right)\]
Calculate main bar reinforcement.\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
Calculate distribution reinforcement.\[As{t_d} = 0.12\% \times {A_{c/s}}\]
Calculate spacing of reinforcement.
\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
The maximum bending moment on the toe slab is calculated by taking moments of all the forces about the points c.
Table: Moments in toe slab
| Serial No. | Loads | Magnitude of Load ( kN ) | Distance from c ( m ) | Moment about c ( m ) |
|---|---|---|---|---|
| 1. | Weight of earth pressure over toe slab | |||
| 2. | Weight of toe slab | |||
| 3. | Upward soil pressure | |||
| 4. | Total |
\[\sum {moment} = M\]
Factored moments.\[{M_u} = 1.5 \times M\]
calulate Area of steel Ast.\[{M_u} = \left( {0.87 \times {f_y} \times {A_{st}} \times d} \right)\left( {1 - \frac{{{A_{st}} \times {f_y}}}{{b \times d \times {f_{ck}}}}} \right)\]
Calculate main bar reinforcement.\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
Calculate distribution reinforcement.\[As{t_d} = 0.12\% \times {A_{c/s}}\]
Calculate spacing of reinforcement.
\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
pp= intensity of passive earth pressure developed just in front of the shear key.
p = soil pressure just in front of the shear key.
Total horizontal earth pressure.\[{P_p} = {K_p}P\]
\[{K_p} = \left( {\frac{{1 + \sin \phi }}{{1 - \sin \phi }}} \right)\]
a = Depth of the shear key.
\[{P_p} = {p_p}a\]
Factor of safety against sliding\[F.S = \left( {\frac{{\mu w + {p_p}}}{P}} \right) > 1.5\]
Minimum percentage of reinforcement in shear key = 0.3%.\[{A_{st}} = 0.003 \times b \times D\]
Calculate spacing of reinforcement.
\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
Networking shear force, V
\[V = \left( {1.5\sum{P - \mu W} } \right)\]
Factored shear force, Vu
\[Vu = 1.5 \times V\]
Nominal shear stress, τv
\[{\tau _v} = \frac{V}{{b \times d}}\]
Check for permissible shear stress
\[\left( {\frac{{100{A_{st}}}}{{bd}}} \right)\]
From the table:19 of IS:456-2000, permissible shear stress should be within limit.
\[{\tau _c} > {\tau _v}\]
Redesign if, permissible shear stress is more than nominal shear stress.
\[{\tau _c} < {\tau _v}\]
Thickness at the top of retaining wall.
Thickness of counterfort.
Maximum working moment in counterfort.
\[M = {K_a} \times \left( {\frac{{w \times {h^3}}}{6}} \right) \times L\]
Factored moments.
\[{M_u} = 1.5 \times M\]
Maximum bending moment of counterfort and calculate area of steel.
\[{M_u} = \left( {0.87 \times {f_y} \times {A_{st}} \times d} \right)\left( {1 - \frac{{{A_{st}} \times {f_y}}}{{b \times d \times {f_{ck}}}}} \right)\]
Calulate minimum area of steel Amin as per IS 456 : 2000.
\[As{t_{\min }} = \left( {\frac{{0.85 \times b \times d}}{{{f_y}}}} \right)\]
Calculate number of bars by choosing maximum area out of two formula.Nb is the number of bars required for counterfort.
\[{N_b} = \frac{{As{t_{required}}}}{{As{t_{provided}}}} = \frac{{As{t_{required}}}}{{\left( {\frac{{\pi \times {d^2}}}{4}} \right)}}\]
Coming soon.
Pstem = Pressure intensity at stem.
Total working load = Pstem . ( Counterfort spacing - Thickness of counterfort)
Factored load = 1.5 . Total working load
Ast1 = Factored load / ( 0.87 . fy)
Ast2 = 0.12% . b. D
Calculate spacing of reinforcement.
\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
Provide spacing
Pheel = Total Pressure intensity at heel.
Total working load = Pheel . ( Counterfort spacing - Thickness of counterfort)
Factored load = 1.5 . Total working load
Ast1 = Factored load / ( 0.87 . fy)
Ast2 = 0.12% . b. D
Calculate spacing of reinforcement in 2 legged. Multiple two times.
\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]
Provide spacing
Cross section of cantilever retaining wall.
Longitudinal section of cantilever retaining wall.
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