|
Comprehensive step by step calculations for retaining wall analysis are
provided below, or click:
On this page, you will find an abundance of information relating to:
- Retaining wall and lateral earth pressure variables,
- Rankine
analysis,
- Coulomb analysis,
- Graphical methods,
-
Log spiral theory,
-
Sliding, and
- Overturning
Retaining Wall Variables
Magnitude of stress or earth pressure acting on a retaining wall depends on:
- height of wall,
- unit weight of retained soil,
- pore water pressure,
- strength of soil (angle of internal friction),
- amount and direction of wall movement, and
- other stresses such as earthquakes and surcharges.
Lateral Earth Pressure Variables
Lateral earth pressures are analyzed for either "Active," "Passive" or "At-Rest" conditions.
Active conditions exist when the retaining wall moves away from the soil it retains.
Passive conditions exist when the retaining wall moves toward the soil it retains.
At-Rest conditions exist when the wall is not moving away or toward the soil it retains.
Conditions for active, passive and at-rest pressures are usually
determined by the structural engineer. Basically, at-rest pressures exist
when the top of the wall is fixed from movement. Active and passive
pressures are assumed when the top of the wall moves at least 1/10 of 1% of
height of wall in the direction away from , and toward the soil it retains,
respectively. Some theorize that at-rest pressures develop over time, when a
retaining wall is constructed for the active case.
Retaining Wall Analysis Methods
Lateral earth pressures are typically analyzed, as presented below, from one of the
following methods:
- Rankine Analysis
- Coulomb Method
- Log Spiral Theory
After determining lateral earth pressures, retaining wall analysis and
design also includes:
- Sliding
- Overturning
- Bearing capacity and settlement
- Structural design of wall
Rankine Analysis
Basically, lateral earth pressures are derived from the summation of all
individual pressure (stress)
areas behind the retaining wall. These pressure areas are triangular in shape with the base of the
triangle at the base of the wall for the soil component and pore water
component. Pressure areas for surcharges are rectangular in shape, and earthquake
pressures are usually analyzed with a nearly 'upside-down' triangle. See the
RANKINE ANALYSIS link for an
excellent presentation of determining lateral earth pressures using the
Rankine Analysis.
For the Rankine analysis, assumptions include:
- horizontal backfill
- vertical wall with respect to the retaining soil
- smooth wall (no friction)
Resultant Lateral Earth Pressure, R
The resultant lateral earth pressure, R, is the summation of all
individual lateral earth pressure components.
R = Ps + Pw
+ Pq + Pe kN/m2 (lb/ft2)
Where,
Ps = 1 KgH2 kN/m2 (lb/ft2) earth pressure due to soil
2
Pw =
1 gwH2 kN/m2 (lb/ft2) earth pressure due to pore water
2
Pq = qKH
kN/m2 (lb/ft2) earth pressure due to surcharge
(i.e. building, vehicle load)
Pe = 3
KhgH2 kN/m2 (lb/ft2)
earth pressure due to earthquakes
8
and,
Ps = lateral earth pressure due to soil
Pw =
lateral earth pressure due to pore water
Pq = lateral earth pressure due to surcharge
(i.e. building, vehicle load)
Pe = lateral earth pressure due to earthquakes
K = KA, KP or Ko
lateral earth pressure coefficient
- KA = (1 - sin
f) coefficient for active conditions
(1 + sin
f)
- KP = (1 + sin
f) coefficient for passive conditions
(1 - sin
f)
- Ko = 1 - sin
f coefficient for at-rest conditions
Kh = 3 K
earthquake coefficient
4
g = effective
unit weight of soil medium kN/m2
(lb/ft2)
gw = 9.1 kN/m2 (62.4 lb/ft2)
= unit weight of water
f =
angle of internal friction
degrees
H = height of retaining wall
m (ft)
q = surcharge on soil, if any kN/m2
(lb/ft2)
water table
Engineering judgment should allow for some pore water pressure behind a retaining wall due
to stormwater or other water source. For a water table behind the wall, why would you analyze a
partially submerged backfill? You could reasonably expect for almost every situation that a
partially submerged backfill will become fully inundated during the life of the wall. The
following lateral earth pressure equation is for a water table at the top of the wall. This
equation is composed of a soil component plus a pore water component. Add the above surcharge
and earthquake components if necessary.
P = 1/2 KgsubH2 + 1/2
gwH2 (lb/ft2)
gsub = submerged soil unit weight (lb/ft3)
=
gsat -
gw
gsat = saturated
soil unit weight
(lb/ft3)
gw = unit weight of water (lb/ft3)
= 62.4 lb/ft3
See the following link for an excellent presentation of determining
lateral earth pressures using the Rankine Analysis
RANKINE ANALYSIS
Coulomb Method
The Coulomb Method:
- Allows for friction between the retaining wall and soil
- May be used for non-vertical walls
- Allows for non-horizontal backfill
(inclined), but must be planar
- Backfill must be cohesionless for inclined backfill
- Assumes a planar slip surface, similar to Rankine
- Is used for Active and Passive (see above) conditions only
- Assumes a homogeneous backfill
- Any surcharge must be uniform and cover entire surface of driving wedge
P = 1 g 1
KH2 kN/m2 (lb/ft2)
2
sin q cos
d
where,
K = KA or KP
lateral earth pressure coefficient;
KA = active, KP = passive
(see above)
- KA =
sin2 (q +
f) cos
d
sin
q (sin
q - d)[1 + SQRT[(sin (f +
d) sin (f -
b))/(sin (q
- d) sin (q
+ b))]]2
- KP =
cos2
f
[1 - SQRT[(sin
f sin (f -
b))/(cos
b)]]2
g = effective
unit weight of soil medium kN/m2
(lb/ft2)
f =
angle of internal friction
degrees
H = height of retaining wall
m (ft)
d = 2f/3 = angle of wall friction degrees
q = angle of wall face from horizontal (90 degrees for
vertical wall) degrees
b = angle of backfill (0
degrees for horizontal backfill)
degrees
Graphical Methods
Graphical methods are more in-depth than the Rankine or Coulomb Analysis.
Until some examples are presented on this website, look for more information in the following
downloadable publication:
NAVFAC 7.02 -
Foundations and Earth Structures. This publication has a graphical solution for lateral
earth pressure analysis. Other publications with
Coulomb solutions may be found in the
publications
section of this website.
Log Spiral Theory
Since a planar slip surface, as assumed for both Rankine and Coulomb Methods, is reasonable
for active earth pressure conditions, this assumption may yield unreasonable results for
passive earth pressure conditions. The Log Spiral Method assumes a curved slip surface, and
therefore should be used for all passive earth pressure conditions.
Horizontal backfill is required for this method. If backfill is not horizontal, then it may
be reasonable to use engineering judgment and include the sloping portion of the backfill as
a surcharge.
Geotechnical Info .Com does not currently have procedures and examples for the Log Spiral
Method. Please check the retaining wall
publications
section of this website for additional resources that may have information on the Log Spiral
Method.
Sliding
Sliding failure is a result of excessive lateral earth pressures with
relation to retaining wall resistance thereby causing the retaining wall
system to move away (slide) from the soil it retains.
See a depiction for calculating the factor of safety for retaining wall
sliding from the following link:
SLIDING ANALYSIS
The following factors of safety (F.S.) are typically used for analyzing sliding:
F.S. = 1.5 for active earth pressure conditions.
F.S. = 2.0 for passive earth pressure conditions.
(RSL/RH) > F.S.
RSL = Resistance to sliding
= (SWi + RV)tan
d + cAB when a
key is not used
= (SWi + RV)tan
d + cAB + PP
when a key is used
RH = R cos d
=
horizontal component of resultant lateral earth pressure (kN/m2)
(lb/ft2)
RV = R sin d
=
vertical component of resultant lateral earth pressure (kN/m2)
(lb/ft2)
R = Ps + Pw
+ Pq + Pe (see Rankine
Analysis above)
PP = Ps
(use Rankine where K is passive)
= Soil pressure exerted on
key using passive earth pressures
SWi = summation of weights (see this
link), that includes:
- weight of footing
- weight of wall
- weight of soil directly above the entire width of the footing
gsoil = effective
unit weight of soil medium kN/m3
(lb/ft3)
gconcrete = unit
weight of concrete = 23.6 kN/m3 (150 lb/ft3)
A = area of soil or concrete unit (see this
link) m2
(ft2)
f =
angle of internal friction
(deg)
d =
external friction angle
(deg)
= (2/3)f
cA = adhesion (kN/m2)(lb/ft2)for
concrete on soil only
= c,
for c = (23.9 kN/m2) (500 lb/ft2) or less
= 0.75c,
for c = (47.9 kN/m2) (1000 lb/ft2)
= 0.5c,
for c = (95.8 kN/m2) (2000 lb/ft2)
= 0.33c,
for c = (191.5 kN/m2) (4000 lb/ft2)
c =
cohesion (kN/m2) (lb/ft2)
B = footing width (m) (ft)
See a depiction for calculating the factor of safety for retaining wall
sliding from the following link:
SLIDING ANALYSIS
Overturning
Overturning failure is a result of excessive lateral earth pressures with
relation to retaining wall resistance thereby causing the retaining wall
system to topple or rotate (overturn). Sliding governs the design of retaining walls most of the time,
especially for walls less than 8 feet in height. However, overturning must
be analyzed.
See a depiction for calculating the factor of safety for retaining wall
overturning from the following link:
OVERTURNING ANALYSIS
Factor of safety (F.S.) is typically 1.5 when analyzing overturning
(SWixi
+ RVxV)/(RHy) > F.S.
where:
SWixi
= summation of moments about the retaining wall toe. (see this
link), that includes:
- weight of footing
- weight of wall
- weight of soil directly above the entire width of the footing
- distance between toe of wall and centroid of specific weight
Wi = Ag = weight of individual soil or
concrete component (see this link)
(kN) (lb)
xi = distance from toe of the retaining wall system
to the centroid of
each individual weight in the
x-axis direction (horizontal) (m) (ft)
RV = R sin d
=
vertical component of resultant lateral earth pressure (kN/m2)
(lb/ft2)
xV = distance from toe of the retaining wall system
to the centroid of
the resultant vertical earth
pressure (RV) in the x-axis (horizontal) direction (ft)
(see this
link)
RH = R cos d
=
horizontal component of resultant lateral earth pressure (kN/m2)
(lb/ft2)
y = distance from the bottom of the retaining wall
to the
resultant
earth pressure location in the y-axis (vertical)
direction (m) (ft)
R = Ps + Pw
+ Pq + Pe (see Rankine
Analysis above)
gsoil = effective
unit weight of soil medium kN/m3
(lb/ft3)
gconcrete = unit
weight of concrete = 23.6 kN/m3 (150 lb/ft3)
A = area of soil or concrete unit (see this
link) m2
(ft2)
See a depiction for calculating the factor of safety for retaining wall
sliding from the following link:
OVERTURNING ANALYSIS
Bearing Capacity and Settlement
Bearing capacity and settlement for wall foundations can be determined in the same manner
as building foundations. Technical guidance for these analyses can be found on this website
under the following headings:
Bearing Capacity
Settlement Analysis
Example #1: Using the Rankine analysis,
determine the individual lateral earth pressures, and resultant lateral
earth pressure on a 2.1 m (7 ft) rigid concrete retaining wall. The free draining gravel
backfill has a soil unit weight,
g, of 21.2 kN/m3 (135 lb/ft3), and an
angle of internal friction,
f, of 36 degrees. There will be vehicle
surcharges of 14.4 kN/m2 (300 lb/ft2). The retaining
wall will be constructed for passive conditions.
Given
- unit weight of soil backfill,
g = 21.2 kN/m3 (135 lbs/ft3) *see typical
g values
- vehicular surcharge, q = 14.4 kN/m2 (300 lbs/ft2) *from
wall use determination
- angle of Internal Friction,
f = 36 degrees *see typical
f values
- wall height, H = 2.1 m (7 ft)
- passive case (wall moves toward retained soil)
Solution
Soil parameters,
g and
f, are determined from laboratory testing. Engineering soil properties
from a known granular material source is sometimes used. Some engineers use
conservative soil parameters based on the soil classification without
laboratory testing. It is good practice to avoid cohesive soils, and use
gravel type materials for retaining wall backfill.
From the Rankine Analysis equation provided above, the resultant (total)
pressure exerted on a retaining wall is:
R = Ps + Pw
+ Pq + Pe kN/m2 (lb/ft2)
coefficient for passive conditions
K = KP = (1 + sin
f) =
(1 + sin 36) =
3.85
(1 - sin
f)
(1 - sin 36)
lateral earth pressure due to soil
Ps = 1 KgH2
2
= 1 3.85(21.2 kN/m3)(2.1 m)2 = 180.0 kN/m
metric
2
= 1 3.85(135 lb/ft3)(7
ft)2 = 12,734 lb/ft
standard
2
The soil pressure component is triangular behind the retaining wall. This
means that the theoretical lateral earth pressure due to soil is minimum
(zero) at the top of the wall, and maximum (KgH) at the bottom of the wall. The resultant soil pressure,
area of the triangle = 0.5KgH2, acts at the bottom 1/3 of the wall (i.e.
centroid of the triangle). In this case, the resultant location is H/3, or
0.7 m (2.3 ft) from the bottom of the wall.
lateral earth pressure due to pore water pressure
Pw = 1 gwH2 = 0
because backfill is above water table
2
The pore water pressure component is also triangular, similar to the soil
component. The resultant location is H/3 from the bottom of the wall.
lateral earth pressure due to surcharge
Pq = qKH
= 14.4 kN/m2 (3.85)(2.1 m) = 116.4 kN/m
metric
= 300 lb/ft2 (3.85)(7 ft) = 8085 lb/ft
standard
The surcharge pressure component is rectangular behind the retaining
wall. This means that the theoretical lateral earth pressure due to the
surcharge (qK) is the same at both the top of the wall, and bottom of the
wall. The resultant surcharge pressure, area of the rectangle = HqK, acts in
the middle of the wall (i.e. centroid of the rectangle). In this case, the
resultant location is H/2, or 1.05 m (3.5 ft) from the bottom of the wall.
lateral earth pressure due to earthquakes
Pe = 3
KhgH2
8
Kh = 3 K =
3 (3.85) = 2.89
earthquake coefficient
4
4
Pe = 3
KhgH2
8
= 3 (2.89)(21.2 kN/m3)(2.1 m)2 = 101.3 kN/m
metric
8
= 3 (2.89)(135 lb/ft3)(7 ft)2 = 7169 lb/ft
standard
8
The earthquake pressure component is nearly an upside down triangle behind
the retaining wall. The resultant earthquake pressure, area of the triangle
= 3/8(Kh)gH2, acts at the upper 1/3 of the wall (i.e.
centroid of the triangle). In this case, the resultant location is H/3, or
0.7 m (2.3 ft) from the top of the wall.
resultant lateral earth pressure
R = Ps + Pw
+ Pq + Pe
R = 180.0 kN/m + 0 + 116.4 kN/m + 101.3 kN/m = 398 kN/m
metric
R = 12,734 lb/ft + 0 + 8085 lb/ft + 7169 lb/ft = 27,990 lb/ft
standard
The position of the resultant pressure, y, is determined by taking the
moments of each individual pressure about the base of the wall:
R(y) = Ps(H/3) + Pw(H/3)
+ Pq(H/2) + Pe(2H/3)
y = 180.0kN/m(0.33(2.1m)) + 0 + 116.4kN/m(0.5(2.1m)) +
101.3kN/m(0.67(2.1m))
398 kN/m
=
0.98 m from bottom of wall
metric
y = 12,734lb/ft(0.33(7ft)) + 0 + 8085lb/ft(0.5(7ft)) +
7169lb/ft(0.67(7ft))
27,990 lb/ft
= 3.2 ft from bottom of wall
standard
Conclusion
The resultant pressure behind the retaining wall is 398 kN/m (28 kips/ft)
at a distance of 0.98 m (3.2 ft) from the bottom of the wall.
***********************************
Example #2: Using the results from the Rankine
analysis in example problem #1, determine the factor of safety for the
concrete retaining wall to resist sliding due to lateral earth pressures exerted on
the wall. The wall foundation is on soils with a cohesion of 23.9 kN/m2 (500 lb/ft2).
The retaining wall is not threatened by earthquakes, so
omit the dynamic component. The retaining wall dimensions are provided
below.
Given
Solution
F.S. = 2.0 for passive earth pressure conditions.
(RSL/RH) > F.S.
RSL = Resistance to sliding
= (SWi + RV)tan
d + cAB when a
key is not used
= (SWi + RV)tan
d + cAB + PP
when a key is used
RH = R cos d
= (398 kN/m)cos
24 = 364 kN/m
metric
= (27,990 lb/ft)cos 24 = 25,570 lb/ft
standard
RV = R sin d
= (398 kN/m)sin 24 = 162 kN/m
metric
= (27,990 lb/ft)sin 24 = 11,385 lb/ft
standard
SWi = summation of weights (see this
link) for a depiction
W1 =
gsoil(width of
soil block above footing)(height of soil block above footing)
= 21.2 kN/m3(1.68 m)(1.83 m) = 65.1 kN/m metric
= 135 lbs/ft3(5.5 ft)(6 ft) = 4455 lb/ft
standard
W2 =
gconcrete(width
of wall)(height of wall above footing)
= 23.6 kN/m3(0.253 m)(1.83 m) =
10.9 kN/m metric
= 150 lbs/ft3(0.83 ft)(6 ft) = 750 lb/ft
standard
W3 =
gconcrete(width
of footing)(height of footing)
= 23.6 kN/m3(2.13 m)(0.30 m) = 15.1 kN/m metric
= 150 lbs/ft3(7 ft)(1 ft) = 1050 lb/ft
standard
SWi =
W1 =
W2 =
W3 = 91.1 kN/m (6,255 lb/ft)
cA = c for c = (23.9 kN/m2) (500 lb/ft2) or less
= 23.9 kN/m2 (500 lb/ft2)
B = 2.13 m (7 ft)
F.S. = RSL/RH = (214 lb/ft)/(364 kN/m) =
0.6
metric
F.S. = RSL/RH = (14,824 lb/ft)/(25,570 lb/ft) = 0.6
standard
Conclusion
The factor of safety with relation to retaining wall sliding is 0.6. This
factor of safety is unacceptable. In order to increase the F.S., we can
design a number of combinations including adding a key beneath the footing,
increasing the footing width, and using tie-backs. Also, note that soil
above the footing in front of the wall was not accounted for in this
problem. Depending on the footing depth, this soil aids in the sliding
resistance.
***********************************
Example #3: Using the results from the Rankine
analysis in example problems #1 and #2, determine the factor of safety for the
concrete retaining wall to resist overturning due to lateral earth pressures exerted on
the wall. The retaining wall dimensions are provided
below.
Given
- unit weight of soil backfill,
g = 21.2 kN/m3 (135 lbs/ft3) *see typical
g values
- vehicular surcharge, q = 14.4 kN/m2 (300 lbs/ft2) *from
wall use determination
- angle of Internal Friction,
f = 36 degrees *see typical
f values
-
d = (f)2/3 = 24 degrees
- c = 23.9 kN/m2 (500 lb/ft2) =
cohesion
- wall height, H = 2.1 m (7 ft)
- wall thickness, h = 0.30 m (1 ft)
- footing thickness, t = 0.30 m (1 ft)
- footing width, B = 2.1 m (7 ft)
- distance from the footing edge (toe) to face of wall in front of wall, 0.46 m
(1.5 ft)
- R = 398 kN/m (27,990 lb/ft) from example problem #1
- y = 0.98 m (3.2 ft) from example problem #1
Solution
Factor of safety (F.S.) is typically 1.5 when analyzing overturning
(SWixi
+ RVxV)/(RHy) > F.S.
SWixi = summation of the
moments (see this
link) for a depiction
W1 =
gsoil(width of
soil block above footing)(height of soil block above footing)
= 21.2 kN/m3(1.68 m)(1.83 m) = 65.1 kN/m metric
= 135 lbs/ft3(5.5 ft)(6 ft) = 4455 lb/ft
standard
W2 =
gconcrete(width
of wall)(height of wall above footing)
= 23.6 kN/m3(0.253 m)(1.83 m) =
10.9 kN/m metric
= 150 lbs/ft3(0.83 ft)(6 ft) = 750 lb/ft
standard
W3 =
gconcrete(width
of footing)(height of footing)
= 23.6 kN/m3(2.13 m)(0.30 m) = 15.1 kN/m metric
= 150 lbs/ft3(7 ft)(1 ft) = 1050 lb/ft
standard
x1 = (width of footing in front of wall) + (width of
wall) + (1/2 of width of soil block above footing)
= 0.457 m + 0.253 m + 0.5(1.676 m) = 1.55 m metric
= 1.5 ft + 0.83 ft + 0.5(5.5 ft) = 5.1 ft
standard
x2 = (width of footing in front of wall) + (1/2 of
wall width)
= 0.457 m + 0.5(0.253 m) = 0.583 m metric
= 1.5 ft + 0.5(0.83 ft) = 1.9 ft
standard
x3 = (1/2 width of footing)
= 0.5(2.13 m) = 1.07 m metric
= 0.5(7 ft) = 3.5 ft
standard
SWixi
=
W1x1
+
W2x2
+
W3x3
= (65.1 kN/m)(1.55 m) + (10.9 kN/m)(0.583 m) + (15.1 kN/m)(1.07 m) = 123.4 kN metric
= (4455 lb/ft)(5.1 ft) + (750 lb/ft)(1.9 ft) + (1050 lb/ft)(3.5 ft) = 27,821
lb standard
RV = R sin d
= (398 kN/m)sin 24 = 162 kN/m
metric
= (27,990 lb/ft)sin 24 = 11,385 lb/ft
standard
xV = (width of footing in front of wall) + (width of
wall)
= 0.457 m + 0.253 m = 0.71 m metric
= 1.5 ft + 0.83 ft = 2.3 ft
standard
RH = R cos d
= (398 kN/m)cos
24 = 364 kN/m
metric
= (27,990 lb/ft)cos 24 = 25,570 lb/ft
standard
y = 0.98 m (3.2 ft)
F.S. = (SWixi
+ RVxV)/(RHy)
= 123.4 kN + (162 kN/m)(0.71
m) = 0.7 metric
(364 kN/m)(0.98 m)
= 27,821 lb + (11,385
lb/ft)(2.3 ft) = 0.7 standard
(25,570
lb/ft)(3.2 ft)
Conclusion
The factor of safety with relation to retaining wall overturning is 0.7.
This factor of safety is unacceptable. In order to increase the F.S., we can
design a number of combinations including moving the wall further from the
footing toe, increasing the footing width, decreasing the wall height, and
using tie-backs. Also, note that soil above the footing in front of the wall
was not accounted for in this problem. Depending on the footing depth, this
soil aids in the overturning resistance.
***********************************
Please look at the information and related sources for Retaining Walls
in the
publications or
software links.
Or, post a question in the Geotechnical Forum.
You are encouraged to provide any additional information or evaluation concerning the
content of Geotechnical Info .Com. Comments can be sent
here.
Tell a friend! about
Geotechnical Info .Com
|
