Cantilever retaining wall

Design of cantilever retaining wall.

Problem: Design a cantilever retaining wall using input data.

1. Draw cross section of retaining wall showing reinforcement


2. Draw longitudinal section showing curtailment.

Cantilever retaining wall.

Design steps:

1. Height of Cantilever wall above ground level.


2. Density of soil in kN/m³.


3. Angle of internal friction in degree.


4. Safe bearing capacity of soil in kN/m².


5. Coefficient of friction as 0.5.


6. Strength of concrete in M20


7. Strength of steek in Fe-415

Design input data for cantilever retaining wall.

Dimensions of retaining wall.
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 +D_f

Thickness of base slab.

\[T = \frac{H}{{12}}\]

Adopt Thickness of base slab, T_Base = Thickness of stem, T_Stem

Height of stem, H_Stem = H-T_Base

Width of base slab, W_Base =0.5H to 0.6H

Adopt base slab width, W_Base

Adopt toe slab width, W_Toe = ( 1 / 3 ) . W_Base

W-top_stem = Top width of stem should be 200mm to 300mm.

Dimensions of retaining wall.

Maximum bending moment.

\[M = {K_a}\left( {\frac{{w \times {h^3}}}{6}} \right)\]

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 of stem.

\[{S_{stem - main}} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]

Calculate the distribution steel for stem.

\[D_{st} = 0.12 \times {A_{c/s}}\]

Calculate distribution spacing of stem.

\[{S_{stem - distribution}} = \frac{{As{t_{provided}}}}{{{D_{st}}}} \times 1000\]

Forces acting on retaining wall.

Pressure distribution at the base is computed by taking moments of all forces about heel point a

Table: Stability calculation

• K_a = Active earth pressure constant.
• H_Stem = Height of stem slab.
• T_Base = Thickness of base slab.
• W_Base = Width of base slab.
• W_Heel = Width of heel slab.
• W_Toe = Width of toe slab.
• W-top_stem = Width of stem slab.
• W_triangle = Width of triangle of stem slab.
• ρ_Soil = Density of soil.
• ρ_concrete = Density of concrete.

Serial No.	Types of loads	Vertical force (kN)	Distance from a (m)	Moment about a (m)
1.	Rectanglur portion of stem slab	W11 = HStem . W-topstem . ρconcrete ( ↓ )	a11 = ( W-topstem . 0.5 ) + WHeel	M11 = W11 . a1 ( ↶ )
2.	Triangular portion of stem slab	W12 = HStem . Wtriangle . ρconcrete ( ↓ )	a12 = ( Wtriangle / 3 ) + W-topstem + WHeel	M12 = W12 . a12 ( ↶ )
3.	Base slab	W2 = TBase . WBase . ρconcrete ( ↓ )	a2 = WBase . 0.5	M2 = W2 . a2 ( ↶ )
3.	Weight of soil	W3 = HStem . WHeel . ρsoil ( ↓ )	a3 = ( WHeel . 0.5 )	M3 = W3 . a3 ( ↶ )
4.	Moment of earth pressure ( ← )	-	-	M4 = 1 /6 .( Ka . ρsoil . HStem3 )
5.	Total	ΣW = W1 + W2 + W3	-	ΣM = M1 + M2 + M3 + M4

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.

\[{\sigma _{_{\max }}} = \frac{{\sum W }}{b}\left( {1 + \frac{{6e}}{b}} \right)\]

\[{\sigma _{_{\min }}} = \frac{{\sum W }}{b}\left( {1 - \frac{{6e}}{b}} \right)\]

soil pressure distribution of retaining wall

We need to check for overturning and sliding

\[{F_1} = \frac{{0.9 \times \sum {{M_s}} }}{{\sum {{M_o}} }} > 1.4\]

ΣM_s = Stabilizing moment

ΣM_o = Overturning moment

\[{F_2} = \frac{{0.9 \times \mu \sum W }}{{\sum {{P_H}} }} > 1.4\]

ΣP_H = Total horizontal pressure.

ΣW = Total weight of retaining wall.

Forces acting on Heel slab of retaining wall.

The maximum bending moment on the heel slab is calculated by taking moments of all the forces about the points b.

• σ_a = Upward soil pressure at a.
• σ_b = Upward soil pressure at b.

Table: Moments in heel slab

Serial No.	Types of loads	Magnitude of Load ( kN )	Distance from b ( m )	Moment about b ( m )
1.	Earth pressure ( ↓ )	W1 = HStem . WHeel . ρsoil	a1 = ( WHeel . 0.5 )	M1 = W1 . a1 ( ↷ +ve )
2.	Heel slab ( ↓ )	W2 = TBase . WHeel . ρConcrete	a2 = ( WHeel . 0.5 )	M2 = W2 . a2 ( ↷ +ve )
3.	Reactangular soil pressure ( ↑ )	U1 = σa . WHeel	a3 = ( WHeel . 0.5 )	MU1 = U1 . a3 ( ↶ -ve )
4.	Triangular soil pressure ( ↑ )	U2 = 1 / 2 . ( σb - σa ). WHeel	a4 = ( WHeel . 2 ) / 3	MU2 = U1 . a4 ( ↶ -ve )
5.	Total	-	-	ΣM = M1 + M2 - ( MU1 + MU2 )

Maximum service bending moment.

\[\sum {moment} = M\]

Factored moments.

\[{M_u} = 1.5 \times M\]

calulate Area of steel A_st.

\[{M_u} = \left( {0.87 \times {f_y} \times {A_{st}} \times d} \right)\left( {1 - \frac{{{A_{st}} \times {f_y}}}{{b \times d \times {f_{ck}}}}} \right)\]

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.

\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]

Forces acting on Toe slab of retaining wall.

The maximum bending moment on the toe slab is calculated by taking moments of all the forces about the points c.

• D_f = Depth of foundation.
• σ_a = Upward soil pressure at c.
• σ_b = Upward soil pressure at d.

Table: Moments in toe slab

Serial No.	Types of loads	Magnitude of Load ( kN )	Distance from c ( m )	Moment about c ( m )
1.	Earth pressure over toe slab ( ↓ )	W1 = ( Df-TBase ). WToe . ρsoil	a1 = ( WToe . 0.5 )	M1 = W1 . a1 ( ↶ -ve )
2.	Self weight of toe slab ( ↓ )	W2 = TBase . WToe . ρConcrete	a2 = ( WHeel . 0.5 )	M2 = W2 . a2 ( ↶ -ve )
3.	Reactangular soil pressure ( ↑ )	U1 = σc . WToe	a3 = ( WToe . 0.5 )	MU1 = U1 . a3 ( ↷ +ve )
4.	Triangular soil pressure ( ↑ )	U2 = 1 / 2 . ( σc - σd ). WToe	a4 = ( WToe . 2 ) / 3	MU2 = U1 . a4 ( ↷ +ve )
4.	Total	-	-	ΣM = MU1 + MU2 - ( M1 + M2 )

Maximum service bending moment.

\[\sum {moment} = M\]

Factored moments.

\[{M_u} = 1.5 \times M\]

calulate Area of steel A_st.

\[{M_u} = \left( {0.87 \times {f_y} \times {A_{st}} \times d} \right)\left( {1 - \frac{{{A_{st}} \times {f_y}}}{{b \times d \times {f_{ck}}}}} \right)\]

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.

\[{S_v} = \frac{{As{t_{provided}}}}{{As{t_{required}}}} \times 1000\]

p_p= 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

\[F.S = \left( {\frac{{\mu w + {p_p}}}{P}} \right) > 1.4\]

A_{st-shearkey, Minimum percentage of reinforcement in shear key = 0.3%.}

\[{A_{st - shearkey}} = 0.003 \times b \times D\]

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, V_u

\[Vu = 1.5 \times V\]

Nominal shear stress, τ_v

\[{\tau _v} = \frac{V}{{b \times d}}\]

Check for permissible shear stress, A_st-shearkey

\[\left( {\frac{{100{A_{st - shearkey}}}}{{bd}}} \right)\]

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.

\[{\tau _c} < {\tau _v}\]

Cross section of cantilever retaining wall.

Reinforcement detailing of retaining wall.

Longitudinal section of cantilever retaining wall.

Sectional side view of cantilever retaining wall.