Reinforced concrete design of retaining wall
Contents
Design thickness of stem
Design code: ACI 318-05
-
Calculate
factored shear force at bottom of stem
Vu = 1.6*(g
Ka H2/2+q
Ka H)
[6.1]
Where
1.6 is load factor, g
is unit weight of soil, Ka is active
lateral earth coefficient, h is height of earth, q is surcharge.
-
Calculate
shear strength of stem
fVc=0.75*(2Öfc’) b d
[6.2]
Where 0.75 is strength reduction factor, fc’
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
fVc³
Vu no shear
reinforcement is required
If
fVc<
Vu
increase thickness of stem or design shear reinforcement
Design vertical reinforcement of stem
-
Calculate
factored moment at base of stem
Mu=1.6*(g
Ka H3/6+q
Ka H2/2)
[6.3]
-
Design
flexural reinforcement for stem
Reinforcement ratio:
[6.4]
Where
R=Mu/(0.9bd2),
m =Fy/(0.85fc’), Fy is yield strength of steel.
The required reinforcement,
As = rbd
should be within maximum reinforcement.
The required minimum reinforcement is the smaller of
As,min=(3Öfc’/Fy)
or (4/3) As. if As is less than As,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:
Calculated factored footing pressure
1.
The center of the total weight from the edge of toe is
Xu
= (1.2*MR-1.6Mo)/(1.2W)
[6.7]
Where W is total weight of retaining wall including stem,
footing, earth and surcharge.
2.
The eccentricity, eu = B/2-Xu
3.
If eu £
B/6, the maximum and minimum footing pressure is calculated as
Qmax =
1.2 (W/B)[1+6 eu /B] [6.8]
Qmax =
1.2 (W/B)[1-6 eu /B] [6.9]
Where, Qmax, Qmin
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 = Qmin +
(Qmax-Qmin)*(B-L)/B
Where B is the width of footing, L is the distance from toe
If eu > B/6,
the maximum footing pressure is calculated as
Qmax =
(1.2 W)(2)/(3 Xu) [6.10]
The length of bearing area is
Lb =
3*Xu
The footing pressure at any point in the bearing zone is
Q = Qmax*(Lb-L)/Lb
[6.11]
L is the distance from toe
Design thickness of footing
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
Vu = 1.2*(We
+Whl+Wq)-R
[6.12]
Where
1.2 is load factor,
We is weight of earth, Whl is weight of heel, Wq is weight of surcharge, and R
is resultant of factored bearing pressure.
-
Calculated
shear strength of stem.
fVc=0.75*(2Öfc’) b d
[6.13]
Where 0.75 is strength reduction factor, fc’
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 fVc<
Vu, increase thickness of stem.
Design heel reinforcement in transverse direction
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
Mu=1.2*(We+Whl+Wq)*C/2-R*Xr
[6.14]
Where C is the length of heel,
Xr is the distance from R to face of stem.
-
Design
flexural reinforcement for heel
Reinforcement ratio:
Where
R=Mu/(0.9bd2),
m =Fy/(0.85fc’), Fy is yield strength of steel.
The required reinforcement,
As = rbd
should be within maximum reinforcement.
The required minimum reinforcement is the smaller of
As,min=(3Öfc’/Fy)
or 1.33 As if As is less than As,min
(ACI 10.5)
Design longitudinal reinforcement for shrinkage and temperature
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.
Design thickness of toe
-
Calculate
factored shear at one-effective depth from face of stem
If eu £
B/6, the factored footing pressure at one-effective depth from face of stem is
Q = Qmin +
(Qmax-Qmin)*(B-Lc)/B
[6.15]
If eu > B/6,
the factored footing pressure at one-effective depth from face of stem is
Q = Qmax*(Lb-Lc)/Lb
[6.16]
Where Lc 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
Vu = (Q +
Qmax)*Lc/2-Wc
[6.17]
Where Lc is weight of concrete and soil above toe.
-
Calculate
shear strength of toe
The shear strength of the concrete is
fVc=0.75*(2Öfc’) b d
Design reinforcement for toe
-
Calculate
factored moment at the front face of stem
If eu £
B/6, the factored footing pressure at one-effective depth from face of stem is
Q = Qmin +
(Qmax-Qmin)*(B-Ld)/B
If eu > B/6,
the factored footing pressure at one-effective depth from face of stem is
Q = Qmax*(Lb-Ld)/Lb
Where Ld is the distance from edge of toe to front face of
stem.
The factored moment at the critical section is
Mu=R*Xr-Wt*Ld/2
[6.18]
Where Xr is the distance from the resultant force to the
front face of stem, Wt is weight of concrete and soil above toe.
-
Design
flexural reinforcement
reinforcement ratio:
Where
R=Mu/(0.9bd2),
m =Fy/(0.85fc’), Fy is yield strength of steel.
The required reinforcement,
As = rbd
should be within maximum reinforcement.
The required minimum reinforcement is the smaller of
As,min=(3Öfc’/Fy)
or 1.33 As if As is less than As,min
(ACI 10.5)
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