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Design code: ACI 318-05
Calculate factored shear force at bottom of stem
V_{u} = 1.6*(g K_{a} H^{2}/2+q K_{a} H) [6.1]
Where 1.6 is load factor, g is unit weight of soil, K_{a} is active lateral earth coefficient, h is height of earth, q is surcharge.
Calculate shear strength of stem
fV_{c}=0.75*(2Öf_{c}’) b d [6.2]
Where 0.75 is strength reduction factor, f_{c}’ is compressive strength of concrete, b is one foot width of wall, d is effective depth of stem and is equal to thickness of stem minus 2” cover and half bar size.
Compare shear force with shear strength, design shear reinforcement when necessary.
If fV_{c}³ V_{u} no shear reinforcement is required
If fV_{c}< V_{u} increase thickness of stem or design shear reinforcement
Calculate factored moment at base of stem
M_{u}=1.6*(g K_{a} H^{3}/6+q K_{a} H^{2}/2) [6.3]
Design flexural reinforcement for stem
Reinforcement ratio:
[6.4]
Where
R=M_{u}/(0.9bd^{2}), m =F_{y}/(0.85f_{c}’), F_{y} is yield strength of steel.
The required reinforcement, A_{s} = rbd should be within maximum reinforcement.
The required minimum reinforcement is the smaller of
A_{s,min}=(3Öf_{c}’/F_{y})
or (4/3) A_{s}. if A_{s} is
less than A_{s,min}
(ACI 10.5)
The minimum total vertical reinforcement ratio for wall (both faces)
is
0.0012 for deformed bars #5 or smaller or 0.0015 for other bars
(ACI 14.3.2)
One of common mistake in retaining wall design is neglecting or
inadequate horizontal reinforcement. When retaining wall gets
too long, the wall will crack due to shrinkage of concrete. Vertical
control joints and horizontal reinforcement are normally used to
control cracks in the stems. The
spacing of control joist depends on the amount of horizontal
reinforcement. Larger
spacing requires heavier reinforcement. The
reinforcement ratio recommended by Concrete Reinforcing Steel
Institute (CRSI) is shown below.
Design horizontal reinforcement to avoid shrinkage cracks.
Figure 1: Joint spacing related to steel for shrinkage.
(Reproduced from CRSI handbook)
The minimum total horizontal reinforcement ratio for wall (both faces) is
0.002 for deformed bars #5 or smaller or 0.0025 for others.
Determine minimum width of expansion joints.
In some case, when temperature change is large and the retaining wall has to be water tied, expansion joist are used. The width of expansion joint depends on temperature change and the length between joints. Without consider the contribution of horizontal reinforcement, the width of expansion joints can be calculated as
D=1.5*(0.0000055*T*L) [6.6]
Where 0.0000055 is coefficient of expansion of concrete per degree F, T is maximum range of temperature difference, L is the length of wall between expansion joints, 1.5 is factor of safety.
Forces that apply to the heel
are weight of soil, footing, surcharge, and footing bearing
pressure. Weight
of soil, footing, and surcharge are downward forces. Footing
bearing pressure is upward forces. Sometime,
footing bearing pressure are neglected to be conservative.
Otherwise, factored footing pressures are calculated as follows:
1. The center of the total weight from the edge of toe is
X_{u} = (1.2*M_{R}-1.6M_{o})/(1.2W) [6.7]
Where W is total weight of retaining wall including stem, footing, earth and surcharge.
2. The eccentricity, e_{u} = B/2-X_{u}
3. If e_{u} £ B/6, the maximum and minimum footing pressure is calculated as
Q_{max} = 1.2 (W/B)[1+6 e_{u} /B] [6.8]
Q_{max} = 1.2 (W/B)[1-6 e_{u} /B] [6.9]
Where, Q_{max}, Q_{min} are maximum and minimum factored footing pressure, B is the width of footing.
The factored footing pressure at any point in the footing is calculated as
Q = Q_{min} + (Q_{max}-Q_{min})*(B-L)/B
Where B is the width of footing, L is the distance from toe
If e_{u} > B/6, the maximum footing pressure is calculated as
Q_{max} = (1.2 W)(2)/(3 X_{u}) [6.10]
The length of bearing area is
L_{b} = 3*X_{u}
The footing pressure at any point in the bearing zone is
Q = Q_{max}*(L_{b}-L)/L_{b} [6.11]
L is the distance from toe
The critical section of shear in the heel is taken at the face of stem instead of at one-effective depth from the stem because it does not produce compression to the stem according to ACI code.
Calculated factored shear force at face of stem
V_{u} = 1.2*(W_{e} +W_{hl}+W_{q})-R [6.12]
Where 1.2 is load factor, W_{e} is weight of earth, W_{hl} is weight of heel, W_{q} is weight of surcharge, and R is resultant of factored bearing pressure.
Calculated shear strength of stem.
fV_{c}=0.75*(2Öf_{c}’) b d [6.13]
Where 0.75 is strength reduction factor, f_{c}’ is compressive strength of concrete, b is one foot width of wall, d is effective depth of stem and is equal to thickness of stem minus 2” cover and half bar size.
Compare shear force with shear strength, if fV_{c}< V_{u}, increase thickness of stem.
The critical section of moment is at the face of stem. The heel reinforcement is calculated as follows:
Calculate factored moment at face of toe
M_{u}=1.2*(W_{e}+W_{hl}+W_{q})*C/2-R*X_{r} [6.14]
Where C is the length of heel, X_{r} is the distance from R to face of stem.
Design flexural reinforcement for heel
Reinforcement ratio:
Where
R=M_{u}/(0.9bd^{2}), m =F_{y}/(0.85f_{c}’), F_{y} is yield strength of steel.
The required reinforcement, A_{s} = rbd should be within maximum reinforcement.
The required minimum reinforcement is the smaller of
A_{s,min}=(3Öf_{c}’/F_{y})
or 1.33 A_{s} if
As is less than A_{s,min}
(ACI 10.5)
Reinforcement ratio: 0.002 for grade 40, 50 deformed bars, 0.0018 for grade 60 deformed bars.
The forces that apply to the bottom of toe is footing bearing pressure. In a normal situation, the length of toe is shorter than that of heel. The maximum shear force is less than of heel. The depth of footing for heel is usually enough for toe. It is also a normal practice to bend the dowel bars at the bottom of stem for toe reinforcement. It is normally sufficient for toe reinforcement. In some situation, when toe is extra long, then, it will be necessary to check shear strength and design reinforcement for toe.
Calculate factored shear at one-effective depth from face of stem
If e_{u} £ B/6, the factored footing pressure at one-effective depth from face of stem is
Q = Q_{min} + (Q_{max}-Q_{min})*(B-L_{c})/B [6.15]
If e_{u} > B/6, the factored footing pressure at one-effective depth from face of stem is
Q = Q_{max}*(L_{b}-L_{c})/L_{b} [6.16]
Where L_{c} is the distance from edge of toe to one effective depth from front face of stem.
The factored shear force at the critical section is
V_{u} = (Q + Q_{max})*L_{c}/2-W_{c} [6.17]
Where L_{c} is weight of concrete and soil above toe.
Calculate shear strength of toe
The shear strength of the concrete is
fV_{c}=0.75*(2Öf_{c}’) b d
Calculate factored moment at the front face of stem
If e_{u} £ B/6, the factored footing pressure at one-effective depth from face of stem is
Q = Q_{min} + (Q_{max}-Q_{min})*(B-L_{d})/B
If e_{u} > B/6, the factored footing pressure at one-effective depth from face of stem is
Q = Q_{max}*(L_{b}-L_{d})/L_{b}
Where L_{d} is the distance from edge of toe to front face of stem.
The factored moment at the critical section is
M_{u}=R*X_{r}-W_{t}*L_{d}/2 [6.18]
Where X_{r} is the distance from the resultant force to the front face of stem, W_{t} is weight of concrete and soil above toe.
Design flexural reinforcement
reinforcement ratio:
Where
R=M_{u}/(0.9bd^{2}), m =F_{y}/(0.85f_{c}’), F_{y} is yield strength of steel.
The required reinforcement, A_{s} = rbd should be within maximum reinforcement.
The required minimum reinforcement is the smaller of
A_{s,min}=(3Öf_{c}’/F_{y})
or 1.33 A_{s} if
As is less than A_{s,min}
(ACI 10.5)