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Example S.1 Cantilever retaining wall with horizontal backfill, no surcharge
Example S.2 Cantilever retaining wall with horizontal backfill, surcharge, and key
Important Notes:
1. Lateral earth pressure is normally calculated based on Rankine or Coulomb’s theories. Lateral earth pressure is assumed distributed triangularly. The location of resultant is at 1/3 of height. If there is surcharge, lateral earth pressure from surcharge is distributed uniformly. The resultant is at ½ of height. The lateral earth pressure is calculated at the edge of heel.
2. The rotating point for overturning is normally assumed at bottom of toe. The height of soil used to calculate lateral earth pressure should be from top of earth to the bottom of footing.
3. Elements that resisting overturning are weight of stem, weight of footing, weight of soil above footing. If there is a surcharge, the weight of surcharge can also be considered.
4. The factor of safety against overturning is resisting moment divided by overturning moment. Acceptable factor of safety is between 1.5 to 2.
5. The driving force that causes retaining wall to slide is the lateral earth pressure from soil and surcharge. The forces that resist sliding are passive pressure at toe, the friction at the base of the footing; and the passive pressure against the key if used.
6. The factor of safety against sliding is the total resisting force divided by total driving force. Acceptable factor of safety is between 1.5 to 2.
The factor of safety against overturning is calculated as
1. The overturning moment is calculated as
Where g is unit weight of soil, K_{a} is active pressure coefficient, and H is the height from top of earth to bottom of footing, q is surcharge.
2. The resisting moment is calculated as
where Ws,Wf,We,Wk,Wq are weight of stem, footing, earth, key and surcharge, X_{s},X_{f},X_{e},X_{k},X_{q} are distance from the center of stem, footing, earth, key, and surcharge to the rotation point at toe.
3. The factor of safety against over turning is calculated as
The bearing pressure is calculated as follows
1. The center of the total weight from the edge of toe is
Where W is total weight of retaining wall including stem, footing, earth and surcharge.
2. The eccentricity, e = B/2-X
3. If e £ B/6, the maximum and minimum footing pressure is calculated as
Where, Q_{max}, Q_{min} are maximum and minimum footing pressure, B is the width of footing.
4. If e > B/6, Q_{min} is zero,
5. Q_{max} should be less than allowable soil bearing capacity of footing soil.
1. The driving force for sliding is calculated as
2. The friction resisting force at the base of footing is calculated as
where m is friction coefficient between concrete and soil. m is often taken as tan (2/3 f). f is internal friction of the soil.
3. The passive resistance at the toe of retaining wall is calculated as
Where Kp is passive earth pressure coefficient, h is the height from top of soil to bottom of footing at toe. If a key is used to help resist sliding, h is the height from top of soil to the bottom of the key.
4. The factor of safety is calculated as
Given:
Height of stem: 10 ft
Thickness of stem: 1 ft
Thickness of footing: 1 ft
Width of footing: B = 6 ft
Length of heel: 4 ft
Length of toe: 1 ft
Height of soil above heel: 10 ft
Height of soil above toe: 1 ft
Unit weight of backfill soil: g =
115 pcf,
Internal friction angle of soil: 30 degree,
Friction coefficient between concrete and soil:
0.5
Allowable soil bearing capacity for footing soil:
3000 psf
Requirement: Check stability against
overturning and sliding, and soil bearing capacity.
1. Check overturning stability
Rankine 's active earth coefficient: K_{a} =
tan (45-f/2)2=0.333
Height from top of backfill soil to bottom of
footing: H = 10 +1 = 11 ft
Consider one-foot width of soil
Overturning moment: Mo=g K_{a} H^{3}/6
= 8504 ft-lb
Calculate Resisting moment:
Weight of stem: W_{s}=150x10x1=1500
lbs
Distance from center of stem to edge of toe: X_{s}=1.5
ft
Weight of footing: W_{f}=150x6x1=900
lbs
Distance from
center of footing to edge of toe: X_{f}=3 ft
Weight of earth: W_{e}=115x10x4=4600
lbs
Distance from center of earth to edge of toe: X_{e}=4
ft
Resisting moment: M_{R} =
1500x1.5+900x3+4600x4 = 23350 ft-lb
Factor of safety: FS = M_{R}/M_{o} =
23350/8504 = 2.75 >1.5 O.K.
2. Check soil bearing capacity:
Total weight of retaining wall: W=1500+900+4600 =
7000 lbs
Distance from center of resultant to toe,
X = (M_{R}-M_{o})/W =
(23350-8504)/7000 = 2.12 ft
Eccentricity: e = 6/2 – 2.12= 0.88 ft <
1/6 width of footing, 1 ft
Maximum and minimum footing pressure:
<
3000 psf O.K.
3. Check sliding stability
Driving force: P_{h} =
=g K_{a} H^{2}/2
= 2319 lbs
Friction resistance at bottom of footing: F_{r} = m W
= 0.5x7000 = 3500 lbs
Rankine's passive earth coefficient: K_{p} =
tan (45+f/2)2=3
Height from top of backfill soil to toe: h = 2 ft
Passive resistance at toe: P_{p} =
=g K_{p} H^{2}/2
= 115x3x22 =690
lbs
Factor of safety against sliding:
FS= (F_{r}+P_{p})/P_{h} =
(3500+690)/2319 = 1.81 >1.5 O.K.
Given:
Height of stem: 10 ft
Thickness of stem: 1 ft
Thickness of footing: 1 ft
Width of footing: B = 6 ft
Length of heel: 4 ft
Length of toe: 1 ft
Height of soil above heel: 10 ft
Height of soil above toe: 1 ft
Depth of key: 1 ft
Unit weight of backfill soil: g =115
pcf
Internal friction angle of soil: f =
28 degree
Friction coefficient between concrete and soil: m =
0.5
Allowable soil bearing capacity for footing soil:
4000 psf
Surcharge above backfill soil: q = 200 psf
Requirement: Check stability against
overturning and sliding, and check soil bearing capacity.
1. Check overturning stability
Active earth coefficient: K_{a} =
tan (45-f/2)2=0.361
Height from top of backfill soil to bottom of
footing: H = 11 ft
Consider one foot width of soil:
Overturning moment: M_{o}=g K_{a} H^{3}/6
= 13580 ft-lb
Calculate Resisting moment:
Weight of stem: W_{s}=150x10x1=1500
lbs
Distance from center of stem to edge of toe: X_{s}=1.5
ft
Weight of footing: W_{f}=150x6x1=900
lbs
Distance from
center of footing to edge of toe: X_{f}=3 ft
Weight of earth: W_{e}=115x10x4=4600
lbs
Distance from center of earth to edge of toe: X_{e}=4
ft
Weight of key: W_{k} =
150x1x1=150 lbs
Distance from center of key to edge of toe: X_{k}=2
ft
Weight of surcharge: W_{q} =
200 x 4 = 800 lbs
Distance from center of surcharge to edge of toe: X_{q} =
4 ft
Resisting moment: M_{R} =
1500x1.5+900x3+4600x4+150x2+800x4 = 26850 ft-lb
Factor of safety: FS = M_{R}/M_{o} =
268500/13580 = 1.98 >1.5 O.K.
2. Check soil bearing capacity:
Total weight of retaining wall:
W=1500+900+4600+150+800 = 7950 lbs
Distance from center of resultant to toe: X =
(26850-13580)/7950=1.67 ft
Eccentricity: e = 6/2-1.67 = 1.33 ft > 1/6
width of footing, 1 ft
Maximum footing
pressure:
<
4000 psf O.K.
3. Check sliding stability
Driving force: P_{h} =
=g K_{a} H^{2}/2
+ q K_{a} H=
115x0.333x112+ 200x0.333x11 = 3306 lbs
Friction resistance at bottom of footing: F_{r} = m W
= 0.5x7950 = 3975 lbs
Rankine's passive earth coefficient: K_{p} =
tan (45+f/2)2=2.77
Height from top of backfill soil to bottom of key: h
= 3 ft
Passive resistance at toe: P_{p} =
=g K_{p} H^{2}/2
= 115x2.77x32 =1433
lbs
Factor of safety against sliding:
FS= (F_{r}+P_{p})/P_{h} =
(3975+1433)/3306 = 1.64 >1.5 O.K.
Given:
Height of stem: 10 ft
Thickness of stem: 1 ft
Thickness of footing: 1 ft
Width of footing: B = 6 ft
Length of heel: 4 ft
Length of toe: 1 ft
Height of soil above heel at retaining wall: 10
ft
Height of soil above toe: 1 ft
Unit weight of backfill soil: g =115
pcf
Slope of backfill soil: b =
10 degree
Internal friction angle of soil: f =
30 degree
Friction coefficient between concrete and soil: m = 0.5
Allowable soil bearing capacity for footing soil:
3000 psf
Requirement: Check stability against
overturning and sliding, and check soil bearing capacity.
1. Check overturning stability
Rankine's active earth coefficient:
Height from top of backfill soil to bottom of
footing: H = 10+1+4 tan b =
11.71 ft
Consider one foot width of wall,
Overturning moment: M_{o}=g K_{a} H^{3}/6
= 115x0.35*11.713/6 = 10740 ft-lb
Calculate Resisting moment:
Weight of stem: W_{s}=150x10x1=1500
lbs
Distance from center of stem to edge of toe: X_{s}=1.5
ft
Weight of footing: W_{f}=150x6x1=900
lbs
Distance from
center of footing to edge of toe: X_{f}=3 ft
Divide the earth above heel to two portions: a
rectangular portion and a triangular portion
Weight of rectangular portion of earth: W_{e1}=115x10x4=4600
lbs
Distance from center of rectangular portion of earth
to edge of toe: X_{e1}=4 ft
Weight of triangular portion of earth: W_{e2}=115x(4x4tanb/2)=162
lbs
Distance from center of earth of triangular to edge
of toe: X_{e2}=1+1+4x2/3 = 4.67 ft
Resisting moment: M_{R} =
1500x1.5+900x3+4600x4+162x4.67 = 24110 ft-lb
Factor of safety: FS = M_{R}/M_{o} =
24110/10740 = 2.24 >1.5 O.K.
2. Check soil bearing capacity:
Total weight of retaining wall: W=1500+900+4600+162 =
7162 lbs
Distance from center of resultant to toe: X =
(24110-10740)/7162=1.87 ft
Eccentricity: e = 6/2-1.13 = 1.13 ft > 1/6
width of footing, 1 ft
Maximum footing
pressure:
<
3000 psf O.K.
3. Check sliding stability
Driving force: P_{h} =
=g K_{a} H^{2}/2
= 115x0.35x1.732/2 =2754 lbs
Friction resistance at bottom of footing: F_{r} = m W
= 0.5x7162 = 3581 lbs
Rankine's passive earth coefficient:
Height from top of backfill soil to toe: h = 2 ft
Passive resistance at toe: P_{p} =
=g K_{p} H^{2}/2
= 115x2.78x22 =638
lbs
Factor of safety against sliding:
FS= (F_{r}+P_{p})/P_{h} = (3581+638)/2754 = 1.53 >1.5 O.K.