Retaining Wall Technical Guidance
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Transcript of Retaining Wall Technical Guidance
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Retaining Wall Technical Guidance
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.
Comprehensive step by step calculations for retaining wall analysis are provided below, or click:
Example Problems for Retaining Wall Analysis
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 , andtoward 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:
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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 behindthe 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/ft
2)
Where,
Ps = 1 K H2
kN/m2
(lb/ft2
) earth pressure due to soil2
Pw = 1 wH2 kN/m
2 (lb/ft
2) earth pressure due to pore water
2 Pq = qKH kN/m
2 (lb/ft
2) earth pressure due to surcharge (i.e. building, vehicle load)
Pe = 3 K hH2 kN/m
2 (lb/ft
2) earth pressure due to earthquakes
8
and,
Ps = lateral earth pressure due to soil
Pw = lateral earth pressure due to pore waterP
q = lateral earth pressure due to surcharge (i.e. building, vehicle load)
Pe = lateral earth pressure due to earthquakes
K = K A, K P or K o lateral earth pressure coefficient
K A = (1 - sin ) coefficient for active conditions
(1 + sin )
K P = (1 + sin ) coefficient for passive conditions
(1 - sin )
K o = 1 - sin coefficient for at-rest conditions
K h = 3 K earthquake coefficient
4 = effective unit weight of soil medium kN/m2(lb/ft
2)
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w = 9.1 kN/m2
(62.4 lb/ft2) = unit weight of water
= angle of internal friction degrees
H = height of retaining wall m (ft)
q = surcharge on soil, if any kN/m2(lb/ft
2)
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 K subH2 + 1/2 wH
2 (lb/ft
2)
sub = submerged soil unit weight (lb/ft3)
= sat - w
sat = saturated soil unit weight (lb/ft3)
w = unit weight of water (lb/ft3)= 62.4 lb/ft
3
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 1 KH2 kN/m
2 (lb/ft
2)
2 sin cos
where,
K = K A or K P lateral earth pressure coefficient;
K A = active, K P = passive (see above)
K A = sin2 ( + ) cos
sin (sin - )[1 + SQRT[(sin ( + ) sin ( - ))/(sin ( - ) sin ( + ))]]2
K P = cos2
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[1 - SQRT[(sin sin ( - ))/(cos )]]2
= effective unit weight of soil medium kN/m2(lb/ft
2)
= angle of internal friction degrees
H = height of retaining wall m (ft)
= 2/3 = angle of wall friction degrees
= angle of wall face from horizontal (90 degrees for vertical wall) degrees
= 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 thiswebsite.
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 therebycausing 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.
(R SL/R H) > F.S.
R SL = Resistance to sliding= ( Wi + R V)tan + cAB when a key is not used
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= ( Wi + R V)tan + cAB + PP when a key is used
R H = R cos = horizontal component of resultant lateral earth pressure (kN/m
2) (lb/ft
2)
R V = R sin = vertical component of resultant lateral earth pressure (kN/m2) (lb/ft
2)
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
Wi = summation of weights (see this link ), that includes:
o weight of footing
o weight of wall
o weight of soil directly above the entire width of the footing
soil
= effective unit weight of soil medium kN/m3(lb/ft
3)
concrete = unit weight of concrete = 23.6 kN/m3 (
150 lb/ft3)
A = area of soil or concrete unit (see this link ) m2(ft
2)
= angle of internal friction (deg)
= external friction angle (deg)
= (2/3)
cA = adhesion (kN/m2)(lb/ft
2)for concrete on soil only
= c, for c = (23.9 kN/m2) (500 lb/ft
2) or less
= 0.75c, for c = (47.9 kN/m2) (1000 lb/ft
2)
= 0.5c, for c = (95.8 kN/m2) (2000 lb/ft
2)
= 0.33c, for c = (191.5 kN/m2) (4000 lb/ft
2)
c = cohesion (kN/m2) (lb/ft
2)
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 retainingwalls 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
( Wixi + R VxV)/(R Hy) > F.S.
where:
Wixi = summation of moments about the retaining wall toe. (see this link ), that includes:
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o weight of footing
o weight of wall
o weight of soil directly above the entire width of the footing
o distance between toe of wall and centroid of specific weight
Wi = A 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)
R V = R sin
= vertical component of resultant lateral earth pressure (kN/m2) (lb/ft
2)
xV = distance from toe of the retaining wall system to the centroid of
the resultant vertical earth pressure (R V) in the x-axis (horizontal) direction (ft)
(see this link )
R H = R cos
= horizontal component of resultant lateral earth pressure (kN/m2) (lb/ft
2)
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)
soil = effective unit weight of soil medium kN/m3(lb/ft
3)
concrete = unit weight of concrete = 23.6 kN/m3 (
150 lb/ft3)
A = area of soil or concrete unit (see this link ) m2(ft
2)
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 buildingfoundations. Technical guidance for these analyses can be found on this website under the following headings:
Bearing Capacity
Settlement Analysis
Example Problems for Retaining Wall 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 hasa soil unit weight, , of 21.2 kN/m
3 (135 lb/ft
3), and an angle of internal friction, , of 36 degrees. There
will be vehicle surcharges of 14.4 kN/m2 (300 lb/ft
2). The retaining wall will be constructed for passive
conditions.
Given
unit weight of soil backfill, = 21.2 kN/m3 (135 lbs/ft
3) *see typical values
vehicular surcharge, q = 14.4 kN/m2 (300 lbs/ft
2) *from wall use determination
angle of Internal Friction, = 36 degrees *see typical values
wall height, H = 2.1 m (7 ft)
passive case (wall moves toward retained soil)
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Solution
Soil parameters, and , 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/ft
2)
coefficient for passive conditions
K = K P = (1 + sin ) = (1 + sin 36) = 3.85
(1 - sin ) (1 - sin 36)
lateral earth pressure due to soil
Ps = 1 K H2
2
= 1 3.85(21.2 kN/m3 m
2 = 180.0 kN/m metric
2
= 1 3.85(135 lb/ft3
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 (K H) at the bottom of the wall. The
resultant soil pressure, area of the triangle = 0.5K H2, 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 wH2 = 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/m
2
(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 K hH2
8 K
h = 3 K = 3 (3.85) = 2.89 earthquake coefficient
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4 4
Pe = 3 K hH2
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(K hH2, 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
unit weight of soil backfill, = 21.2 kN/m3 (135 lbs/ft
3) *see typical values
vehicular surcharge, q = 14.4 kN/m2 (300 lbs/ft
2) *from wall use determination
angle of Internal Friction, = 36 degrees *see typical values
= ()2/3 = 24 degrees
c = 23.9 kN/m2 (500 lb/ft
2) = 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)
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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
Solution
F.S. = 2.0 for passive earth pressure conditions.
(R SL/R H) > F.S.
R SL = Resistance to sliding
= ( Wi + R V)tan + cAB when a key is not used
= ( Wi + R V)tan + cAB + PP when a key is used
R H = R cos
= (398 kN/m)cos 24 = 364 kN/m metric
= (27,990 lb/ft)cos 24 = 25,570 lb/ft standard
R V = R sin
= (398 kN/m)sin 24 = 162 kN/m metric
= (27,990 lb/ft)sin 24 = 11,385 lb/ft standard
Wi = summation of weights (see this link ) for a depiction
W1 = soil(width of soil block above footing)(height of soil block above footing)= 21.2 kN/m
3(1.68 m)(1.83 m) = 65.1 kN/m metric
= 135 lbs/ft3(5.5 ft)(6 ft) = 4455 lb/ft standard
W2 = concrete(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/ft
3(0.83 ft)(6 ft) = 750 lb/ft standard
W3 = concrete(width of footing)(height of footing)= 23.6 kN/m
3(2.13 m)(0.30 m) = 15.1 kN/m metric
= 150 lbs/ft3(7 ft)(1 ft) = 1050 lb/ft standard
Wi = W1 = W2 = W3 = 91.1 kN/m (6,255 lb/ft)
cA = c for c = (23.9 kN/m2) (500 lb/ft
2) or less
= 23.9 kN/m2 (500 lb/ft
2)
B = 2.13 m (7 ft)
F.S. = R SL/R H = (214 lb/ft)/(364 kN/m) = 0.6 metric
F.S. = R SL/R H = (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
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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, = 21.2 kN/m
3
(135 lbs/ft
3
) *see typical values vehicular surcharge, q = 14.4 kN/m2 (300 lbs/ft
2) *from wall use determination
angle of Internal Friction, = 36 degrees *see typical values
= ()2/3 = 24 degrees
c = 23.9 kN/m2 (500 lb/ft
2) = 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
( Wixi + R VxV)/(R Hy) > F.S.
Wixi = summation of the moments (see this link ) for a depiction
W1 = soil(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 = concrete(width of wall)(height of wall above footing) = 23.6 kN/m
3(0.253 m)(1.83 m) = 10.9 kN/m metric
= 150 lbs/ft3(0.83 ft)(6 ft) = 750 lb/ft standard
W3 = concrete(width of footing)(height of footing) = 23.6 kN/m
3(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
Wixi = 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
R V = R sin
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= (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
R H = R cos = (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. = ( Wixi + R VxV)/(R Hy)
= 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)