131
Sample Problems
Problem 6.1
The cross section of cantilever retaining wall is show below. The angle of external friction between concrete and
base soil is 20, the adhesion is 100 lb/ft2, and cohesion is 200 lb/ft2. The density of concrete is 150 lb/ft3. The factor
of safety against overturning, factor of safety against sliding, maximum stress at the toe, and minimum stress at the
heel are most nearly
Factor of Safety
against Overturning
(A)
(B)
(C)
(D)
132
2.5
3.2
3.8
4
Factor of Safety
against Sliding
1.9
2.3
2.5
3
Maximum Stress
at the Toe (lb/ft2)
3,000
4,200
5,000
6,000
Minimum Stress
at the Heel (lb/ft2)
500
700
1,000
2,000
Solution
The following procedure can be used to analyze the retaining wall with given dimensions and soil parameters.
a) Factor of safety against overturning
Step 1. Calculate the Rankine active earth coefficient with horizontal backfill and vertical wall. Use Equation No.
65, page 123. Press ENTER.
Display
Keys
Description
I?
0 R/S
Angle of sloping backfill with respect to horizontal
A?
30 R/S
Angle of internal friction of soil
K = 0.3333
--
Rankine active pressure coefficient
Step 2. Compute the resisting moment about the toe. Use Equation No. 66, page 124. Press ENTER.
Display
Keys
Description
W?
2 R/S
Stem thickness at the top (ft)
Y?
20 R/S
Height of stem (ft)
C?
150 R/S
Density of concrete (lb/ft3)
B?
13 R/S
Length of base (ft)
E?
8 R/S
Length of heel (ft)
U?
3 R/S
Stem thickness at the bottom (ft)
T?
2 R/S
Thickness of base footing (ft)
H?
0 R/S
Vertical distance from top of stem to backfill (ft)
S?
120 R/S
Z?
0 R/S
I? 0.0000
Density of backfill (lb/ft3)
Surcharge load (lb/ft2)
R/S
Angle of sloping backfill with respect to horizontal
G?
4 R/S
Thickness of backfill from mudline to top of toe (ft)
D?
115 R/S
K? 0.3333
R=
227,070.0000
R/S
--
Density of soil at the toe (lb/ft3)
Rankine active pressure coefficient
Resisting moment about the toe (ft-lb)
Step 3. Calculate the overturning moment. Use Equation No. 67, page 126. Press ENTER. Press R/S every time
you are prompted. You’ll get O = 70,986.6667 (ft-lb).
Step 4. Compute the factor of safety against overturning. Use Equation No. 68, page 126. Press ENTER. Press R/S
every time you are prompted. You’ll get L = 3.1988.
133
b) Factor of safety against sliding
Step 1. Calculate the total weight. Use Equation No. 69, page 127. Press ENTER. Press R/S every time you are
prompted. You’ll get V = 31,520.0000 lb/ft.
Step 2. Compute the factor of safety against sliding. Use Equation No. 70, page 128. Press ENTER.
Display
V?
31,520.0000
N?
B? 13.0000
Keys
R/S
20 R/S
R/S
F?
100 R/S
P?
*
Description
Total weight (lb/ft)
Angle of external friction between soil and concrete
Length of base (ft)
Adhesion of soil at the base (lb/ft2)
Rankine passive coefficient of soil at the base
P? 2.7698
R/S
D? 115.0000
R/S
Density of soil at the toe (lb/ft3)
G? 4.0000
R/S
Thickness of backfill from mudline to top of toe (ft)
T? 2.0000
R/S
Thickness of base footing (ft)
X?
200 R/S
Cohesion of soil at the base (lb/ft2)
S? 120.0000
R/S
Density of backfill (lb/ft3)
Y? 20.0000
R/S
Height of stem (ft)
H? 0.0000
R/S
Vertical distance from top of stem to backfill (ft)
I? 0.0000
R/S
Angle of sloping backfill with respect to horizontal
E? 8.0000
R/S
Length of heel (ft)
K? 0.3333
R/S
Rankine active earth pressure coefficient
Z? 0.0000
R/S
Surcharge load (lb/ft2)
M = 2.3244
--
Factor of safety against sliding
* Keys for calculating Rankine passive earth pressure coefficient of soil at the base using RPN:
45 ENTER 28 ENTER 2 ÷ +
45 ENTER 28 ENTER 2 ÷ +
(HP 35s)
(HP 33s)
28 

K p  tan 2  45  
2

c) Maximum stress at the toe
Step 1. Calculate the eccentricity. Use Equation No. 71, page 129. Press ENTER. Press R/S every time you are
prompted. You’ll get J = 1.5481 for the eccentricity (ft). B/6 (13 ft/6 = 2.167 ft) is greater than the
eccentricity.
134
Step 2. Compute the maximum stress at the toe. Use Equation No. 72, page 130. Press ENTER. Press R/S every
time you are prompted. You’ll get Q = 4,157.0414 for maximum stress at the toe (lb/ft2).
d) Minimum stress at the heel
Calculate the minimum stress at the heel. Use Equation No. 73, page 131. Press ENTER. Press R/S every time you
are prompted. You’ll get Q = 692.1893 for minimum stress at the heel (lb/ft2).
The answer is (B).
Notes: If the stem is not tapered, the stem thickness at the bottom (U) is 2 feet. In case there is no backfill soil at the toe, (G)
and (D) are zero.
Problem 6.2
A reinforced concrete retaining wall is holding a sandy soil with an inclined backfill of 15° from horizontal. The
angle of external friction between concrete and base soil is 20°, adhesion is 100 lb/ft2, and cohesion is 200 lb/ft2.
The density of concrete is 150 lb/ft3. Ignore the passive resistance of soil below the mudline. The factor of safety
against overturning and sliding are most nearly
Factor of Safety
against Overturning
(A)
(B)
(C)
(D)
2.8
3.1
3.5
4
Factor of Safety
against Sliding
1.1
1.5
2
2.5
135
Solution
The factor of safety against overturning and sliding can be obtained in the following steps:
a) Factor of safety against overturning
Step 1. Calculate the Rankine active earth coefficient with sloping backfill and vertical wall. Use Equation No. 65,
page 123. Press ENTER.
Display
Keys
Description
I?
15 R/S
Angle of sloping backfill with respect to horizontal
A?
30 R/S
Angle of internal friction of soil
K = 0.3729
--
Rankine active pressure coefficient
Step 2. Compute the resisting moment about the toe. Use Equation No. 66, page 124. Press ENTER.
Display
Keys
Description
W?
2 R/S
Stem thickness at the top (ft)
Y?
20 R/S
Height of stem (ft)
C?
150 R/S
Density of concrete (lb/ft3)
B?
13 R/S
Length of base (ft)
E?
8 R/S
Length of heel (ft)
U?
3 R/S
Stem thickness at the bottom (ft)
T?
2 R/S
Thickness of base footing (ft)
H?
0 R/S
Vertical distance from top of stem to backfill (ft)
S?
120 R/S
Z?
0 R/S
I? 15.0000
Density of backfill (lb/ft3)
Surcharge load (lb/ft2)
R/S
Angle of sloping backfill with respect to horizontal
G?
0 R/S
Thickness of backfill from mudline to top of toe (ft)
(ignored)
D?
115 R/S
K? 0.3729
R=
280,670.1544
R/S
--
Density of soil at the toe (lb/ft3)
Rankine active pressure coefficient
Resisting moment about the toe (ft-lb)
Step 3. Calculate the overturning moment. Use Equation No. 67, page 126. Press ENTER. Press R/S every time
you are prompted. You’ll get O = 101,398.1318 ft-lb.
Step 4. Determine the factor of safety against overturning. Use Equation No. 68, page 126. Press ENTER. Press
R/S every time you are prompted. You’ll get L = 2.7680.
136
b) Factor of safety against sliding
Step 1. Calculate the total weight. Use Equation No. 69, page 127. Press ENTER. Press R/S every time you are
prompted. You’ll get V = 35,004.9195 (lb/ft).
Step 2. Compute the factor of safety against sliding. Use Equation No. 70, page 128. Press ENTER.
Display
V?
35,004.9195
N?
B? 13.0000
Keys
R/S
20 R/S
R/S
F?
100 R/S
P?
0 R/S
Description
Total weight (lb/ft)
Angle of external friction between soil and concrete
Length of base (ft)
Adhesion of soil at the base (lb/ft2)
Rankine passive coefficient of soil at the base and toe
(ignored)
D? 115.0000
R/S
Density of soil at the toe (lb/ft3)
G? 0.0000
R/S
Thickness of backfill from mudline to top of toe (ft)
(ignored)
T? 2.0000
R/S
Thickness of base footing (ft)
X?
200 R/S
Cohesion of soil at the base (lb/ft2)
S? 120.0000
R/S
Density of backfill (lb/ft3)
Y? 20.0000
R/S
Height of stem (ft)
H? 0.0000
R/S
Vertical distance from top of stem to backfill (ft)
I? 15.0000
R/S
Angle of sloping backfill with respect to horizontal
E? 8.0000
R/S
Length of heel (ft)
K? 0.3729
R/S
Rankine active pressure coefficient of backfill
Z? 0.0000
R/S
Surcharge load (lb/ft2, kN/m2)
--
Factor of safety against sliding
M = 1.1144
The answer is (A).
Notes: If the stem is not tapered, stem thickness at the bottom (U) is 2 feet. If no backfill soil at the toe, (G) and (D) are zero.
137
Problem 6.3
A reinforced concrete retaining wall is supporting a sandy soil backfill with surcharge. The angle of external friction
between concrete and base soil is 20°, adhesion is 300 lb/ft2, and cohesion is 100 lb/ft2. The density of concrete is
150 lb/ft3. Include the passive resistance of soil below the mudline. The factor of safety against overturning and
sliding are most nearly
Factor of Safety
against Overturning
(A)
(B)
(C)
(D)
Factor of Safety
against Sliding
2.5
3
4
5
1.0
1.5
2.5
2.7
Solution
The calculations involving factor of safety against overturning and sliding are as follows:
a) Factor of safety against overturning
Step 1. Determine the Rankine active earth coefficient with sloping backfill and vertical wall. Use Equation No. 65,
page 123. Press ENTER.
Display
Keys
Description
I?
0 R/S
Angle of sloping backfill with respect to horizontal
A?
30 R/S
Angle of internal friction of soil
138
Display
K = 0.3333
Keys
--
Description
Rankine active pressure coefficient
Step 2. Calculate the resisting moment about the toe. Use Equation No. 66, page 124. Press ENTER.
Display
Keys
Description
W?
2 R/S
Stem thickness at the top (ft)
Y?
18 R/S
Height of stem (ft)
C?
150 R/S
Density of concrete (lb/ft3)
B?
16 R/S
Length of base (ft)
E?
9 R/S
Length of heel (ft)
U?
3 R/S
Stem thickness at the bottom (ft)
T?
2 R/S
Thickness of base footing (ft)
H?
0 R/S
Vertical distance from top of stem to backfill (ft)
S?
115 R/S
Density of backfill (lb/ft3)
Z?
200 R/S
Surcharge load (lb/ft2)
I? 0.0000
R/S
Angle of sloping backfill with respect to horizontal
G?
4 R/S
Thickness of backfill from mudline to top of toe (ft)
D?
110 R/S
K? 0.3333
R=
315,565.0000
R/S
--
Density of soil at the toe (lb/ft3)
Rankine active pressure coefficient
Resisting moment about the toe (ft-lb)
Step 3. Compute the overturning moment. Use Equation No. 67, page 126. Press ENTER. Press R/S every time you
are prompted. You’ll get O = 64,444.4444 ft-lb.
Step 4. Calculate the factor of safety against overturning. Use Equation No. 68, page 126. Press ENTER. Press R/S
every time you are prompted. You’ll get L = 4.8967.
b) Factor of safety against sliding
Step 1. Calculate the total weight. Use Equation No. 69, page 127. Press ENTER. Press R/S every time you are
prompted. You’ll get V = 33,740.0000 (lb/ft).
Step 2. Compute the factor of safety against sliding. Use Equation No. 70, page 128. Press ENTER.
Display
V?
33,740.0000
N?
Keys
R/S
20 R/S
Description
Total weight (lb/ft)
Angle of external friction between soil and concrete
139
Display
B? 16.0000
Keys
R/S
F?
300 R/S
P?
*
Description
Length of base (ft)
Adhesion of soil at the base (lb/ft2)
Rankine passive coefficient of soil at the base and toe
P? 2.7698
R/S
D? 110.0000
R/S
Density of soil at the toe (lb/ft3)
G? 4.0000
R/S
Thickness of backfill from mudline to top of toe (ft)
T? 2.0000
R/S
Thickness of base footing (ft)
X?
100 R/S
Cohesion of soil at the base (lb/ft2)
S? 115.0000
R/S
Density of backfill (lb/ft3)
Y? 18.0000
R/S
Height of stem (ft)
H? 0.0000
R/S
Vertical distance from top of stem to backfill (ft)
I? 0.0000
R/S
Angle of sloping backfill with respect to horizontal
E? 9.0000
R/S
Length of heel (ft)
K? 0.3333
R/S
Rankine active pressure coefficient of backfill
Z? 200.0000
R/S
Surcharge load (lb/ft2, kN/m2)
M = 2.7291
--
Factor of safety against sliding
* Keys for calculating Rankine passive earth pressure coefficient of soil at the base using RPN:
45 ENTER 28 ENTER 2 ÷ +
45 ENTER 28 ENTER 2 ÷ +
The answer is (D).
140
(HP 35s)
(HP 33s)