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Center forGeotechnical Practice and Research
200 Patton HallBlacksburg, VA 24061
Virginia Polytechnic InstituteAnd State University
The Charles E. Via, Jr.Department of Civil Engineering
CENTER FORGEOTECHNICAL PRACTICE AND RESEARCH
Retaining Wall Stability 2.05Workbook Documentation
by
J. Michael Duncan
andBingzhi Yang
Report of a study performed by the Virginia Tech Center forGeotechnical Practice and Research
May 2002
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Virginia Polytechnic Institute
And State University
The Charles E. Via, Jr.
Department of
Civil and Environmental Engineering
CENTER FOR
GEOTECHNICAL PRACTICE AND RESEARCH
Retaining Wall Stability 2.05
Workbook Documentation
by
J. Michael Duncan
and
Bingzhi Yang
Center for Geotechnical Practice and Research
200 Patton Hall, Virginia Tech
Blacksburg, Virginia 24061-0105
May 2002
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Contents
Page
Introduction 3
Features 3
Suggestions for use 3
Methods of Analysis and Assumptions Employed 8
Earth Pressure Loads 8
Sliding Resistance 10
Bearing Pressures 10
Distribution of bearing pressure for bearing capacity 10
Distribution of bearing pressure for footing thickness 12Bearing Capacity of Cohesive Soil Foundations 12
Bearing Capacity of Granular Soil Foundations 13
Bearing Capacity Factor of Safety 13
Stem Thickness and Footing Thickness 13
Figure 1 Retaining wall stability computation sheet 4
Figure 2 Bearing capacity computation sheet for granular soils 5
Figure 3 Bearing capacity computation sheet for cohesive soils 6
Figure 4 Suggested sequence for use of retaining wall stability workbook 7
Figure 5 Freebody and earth pressure loads used in evaluating stability 9
Figure 6 - Bearing pressures for calculating bearing capacity and footing thickness 11
Appendix A: List of Symbols 15
Appendix B: Equations 17
Appendix C: References 20
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Introduction
This report provides documentation for an Excel workbook named Retaining WallStability 2.05, which was developed by the Center for Geotechnical Practice and Research
(CGPR) in the Department of Civil and Environmental Engineering at Virginia Tech. The
first version of the workbook was developed for Virginia Tech classes by Mike Duncan and
Robert Mokwa in 1998. The current version, which involves several new features andrefinements, was developed in 2002 by Mike Duncan and Bingzhi Yang, with support from
the CGPR.
The following sections of the report describe the features of the workbook, suggest how itcan be used efficiently, and describe the methods of analysis and assumptions it employs.
The symbols and equations used in the spreadsheet are listed in the appendices.
Features
The Retaining Wall Stability 2.05 Workbook includes 3 worksheets:
1. Retaining Wall Stability Computation Sheet 2.05 (see Figure 1).
2. Bearing Capacity Computation Sheet for Granular Soils 1.00 (See Figure 2).
3. Bearing Capacity Computation Sheet for Cohesive Soils 1.00 (See Figure 3).
As can be seen at the bottom of Figure 1, the retaining wall stability spreadsheet computesthe factor of safety against sliding, the position of the resultant on the base, and the factor of
safety against overturning around the toe of the wall. It also computes stem thickness and
footing thickness based on structural requirements (these are advisory, not intended forstructural design), and the volume of concrete per foot of wall, which provides an
approximate indication of wall cost.
As can be seen at the left side of Figures 2 and 3, the bearing capacity worksheets use data
transferred from the wall stability worksheet to compute bearing capacity factors of safety forretaining walls founded on granular or cohesive materials. As shown at the right sides of
Figures 2 and 3, these bearing capacity worksheets can also be used, with values input directly
into the sheets, to calculate bearing capacity factors of safety for other footings subjected to
eccentric and inclined loads.
Suggestions for Use
Because the spreadsheets show computed results as quickly as data is entered, they can be
used efficiently to determine wall dimensions that satisfy requirements with regard to safety
against sliding, bearing capacity and overturning. A procedure for using the spreadsheets is
shown in Figure 4. Initial estimates of wall dimensions are entered, along with values of soilproperties and surcharge loads. If the computed factors of safety are found not to be
acceptable, the dimensions of the wall can be adjusted to achieve stability. Possible
modifications include changing the width of the base (B), changing the position of the stemon the base (bt), changing stem and footing thickness, adding a key, or changing the depth of
the key (Dk).
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Figure 1 Retaining wall stability computation sheet
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Figure 2 Bearing capacity computation sheet for granular soils
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Figure 3 Bearing capacity computation sheet for cohesive soils
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Input wall dimensions, soil
properties, and structural
properties
Check stem thickness andfooting thickness
If not OK, change stemand/or footing
thickness
Check factor of safety for
sliding and overturning
If not OK, change base
width, position of the
stem on base, or add akey
Check factor of safety for
bearing capacity offoundation
If not OK, change base
width or position ofstem on base
Check volume of concrete Is there a more efficientshape for the wall crosssection?
Figure 4 Suggested sequence for use of retaining wall stability workbook
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Generally, the least costly wall will be one which has all factors of safety close to the
minimum required. The volume of concrete per foot of wall provides an approximate measure
of wall cost. In computing the volume of concrete in the key, it is assumed that the width ofthe key is equal to the width of the stem at the bottom of the stem (t 1).
Methods of Analysis and Assumptions Employed
Earth Pressure Loads.
The earth pressures that result from the weight of the backfill and surcharge load on the
surface of the backfill act on the vertical plane extending from the heel of the wall to thesurface of the backfill. The freebody used to calculate the forces on the base of the footing,
and the factors of safety with respect to sliding and bearing capacity, includes the reinforced
concrete retaining wall and the trapezoidal zone of backfill above the rear portion of the
footing (behind the stem), as shown in Figure 5.
The earth pressure due to the weight of the backfill is defined using the equivalent fluid
pressure method. The earth pressure due to the weight of the soil increases linearly with
depth, according to the relationship
zeqhb = (1)
where hb = horizontal earth pressure due to weight of backfill (psf), eq = unit weight of the
equivalent fluid, which would exert the same lateral pressure as the backfill (the units ofeqare pcf, or pounds per cubic foot), and z= depth below the surface of the backfill (ft). The
position of the resultant force (Eh2) due to hb is specified by the user in terms of y/H, where y= height of Eh2 above the base of the footing, and H = height of the vertical plane from the
heel of the wall to the surface of the backfill.
The earth pressure due to the surcharge does not vary with depth. Its magnitude is
sshs qK= (2)
where hs = horizontal earth pressure due to surcharge (psf), Ks = surcharge pressure
coefficient (dimensionless), and qs = surcharge pressure on the surface of the backfill (psf).
The resultant force (Eh1) due to hs is assumed to act at 0.5H above the heel of the wall.
The spreadsheet provides an option to indicate whether the surcharge acts on the backfillover the portion of the footing behind the stem, or only on the backfill behind the footing.
The factor of safety against sliding is smaller if it is assumed that the surcharge pressure will
not act above the footing.
The spreadsheet provides an option for specifying shear load as well as normal load on the
vertical plane through the heel of the wall. The magnitude of the shear load is
2
2
1HKE bfvv = (3)
where Ev = vertical shear load (lb/ft),Kv = vertical shear load coefficient (dimensionless), bf= unit weight of backfill (pcf), and H= height of the vertical plane through heel of the wall
(ft).
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W
W
W
E
E
E
P
T
N
h1
h2v
bf
s
p
fWk
H
Wff
(a) Freebody used in evaluating stability
Eh1H H
Eh2
(b) Earth pressures due to surcharge (Eh1) and weight of backfill (Eh2)
Figure 5 Freebody and earth pressure loads used in evaluating stability
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Sliding Resistance.
In computing the factor of safety against sliding, it is assumed that the concrete footing is
cast on a layer of granular material. Factor of safety against sliding is computed using thefollowing equation:
T
PNFp
s+= (4)
whereFs is factor of safety against sliding (dimensionless), = coefficient of friction betweenthe wall footing and the granular material beneath (dimensionless),N= resultant normal force
on the base due to weight of wall plus the weight of backfill and the vertical earth pressure
loads (lb/ft), Pp = passive earth pressure force on the front of the key (lb/ft), and T =mobilized horizontal shear load on the bottom of the footing (lb/ft).
Resistance to sliding due to passive earth pressure on the front of the footing is ignored,
on the basis that the soil above the bottom of the footing may be weak and compressible as a
result of freeze/thaw or poor compaction. However, if there is a key beneath the footing (if
Dk > 0), passive resistance on the key is included in the resistance to sliding. The passivepressure resisting sliding is calculated using the Rankine earth pressure theory. The passive
pressure force on the front of the key is:
+++= )2
(45tan)2
(45tan2 2
DD
pkp ZcDP (5)
whereDk= depth of key below bottom of footing (ft), c = cohesion intercept of soil in front of
key (psf), and = friction angle of soil in front of key (degrees). Zp is the depth from the
ground surface in front of the wall to mid-height of the key (ft), expressed as2
k
fp
DDZ += ,
whereDf = depth from ground surface in front of wall to the bottom of the footing (ft), whichis the same depth as the top of the key.
Bearing Pressures.
Different distributions of bearing pressure are used to compute the bearing capacity of the
foundation and to compute the minimum thickness of the footing, as shown in Figure 6.
Distribution of bearing pressure for bearing capacity. To account for the effect ofeccentric loading on the footing, a reduced effective footing width is used, as suggested by
Meyerhof (1953). The effective footing width used in computing bearing pressure is 2x,
wherex = distance from the toe of the footing to the point of application of the normal load N.
For a footing width of 2x, the footing is centrally loaded, and the bearing pressure is uniform.The bearing pressure for a reduced footing width of 2x is given by:
x
Nq
2= (6)
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N
B
q
x
Effective width = 2x
x
(a) Bearing pressure for bearing capacity calculations
Nx
B
q
Nx
B
2x
max
qminqmax
B-x
e=62
Bx
B e=
62
Bx
B>
(b) Bearing pressures for footing thickness calculations
Figure 6 - Bearing pressures for calculating bearing capacity and footing thickness
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Using an effective footing width smaller than the actual width is conservative, because a
portion of the footing is neglected, and it simplifies the calculations because the bearing
pressure is uniform.
Distribution of bearing pressure for determining the footing thickness. Non-uniform
bearing pressure (Figure 6b) is adopted for the purpose of computing the minimum footing
thickness, in keeping with conventional structural engineering practice. The maximum and
minimum bearing pressures are given by:
+=
63
26
)6
1(
max Beif
x
N
Beif
B
e
B
N
q (7)
=
60
6)
61(
min B
eif
Beif
B
e
B
N
q (8)
where xB
=2
e is the eccentricity of the normal force, which is the distance between the
normal force and the center of the footing, andB = the width of the footing.
Bearing Capacity of Cohesive Soil Foundations.
The bearing capacity of cohesive soil is computed using the simplified equations
developed by Brinch Hansen (1957) for= 0 soils. Brinch Hansens equation are expressedas follows:
fucult DSNq += (9)
where qult= ultimate bearing capacity (psf),Nc = bearing capacity factor (dimensionless), Su =
undrained shear strength of soil (which has = 0), = unit weight of soil (pcf), andDf= depthfrom ground surface in front of wall to the bottom of the footing (ft). For the sake of
conservatism, the value of used in computing qult is the smaller of the unit weights of the
backfill and foundation.
The value ofNc in Eq (9) is expressed as follows:
)3.11()
2
2.01(5
N
T
x
DN
f
c += (10)
where 2x = effective footing width (ft) as discussed above, x = distance from edge of the
footing to the point of application of the normal load (ft), T= shear load on the footing (lb/ft),
andN= normal load on the footing (lb/ft).
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Bearing Capacity of Granular Soil Foundations.
The bearing capacity of granular soil is computed using the simplified equationsdeveloped by Meyerhof (1956) forc = 0 soils. Meyerhofs bearing capacity formula can be
expressed as follows:
+= )21)(
2(
10
22000 21
60
N
T
x
DCC
xNq
f
wwult (11)
where qult = ultimate bearing capacity (psf),N60 = average value of Standard Penetration Test
blow count, corrected to 60% of the theoretical hammer energy, within a depth equal to 3x
below the bottom of the footing, x = distance from the toe of the footing to the point ofapplication of the normal load N (ft), 2x = effective footing width (ft) as discussed above
under bearing pressures, T= shear load on the footing (lb/ft), and N = normal load on the
footing (lb/ft).
The dimensionless factors Cw1
and Cw2
adjust for the position of the water table. Theirvalues depend on the position of the water table with respect to the ground surface and the
bottom of the footing, and are determined as follows:
Cw1 = 0.5 for water table at bottom of footing or higher,
Cw1 = 1.0 for water table 2x below bottom of footing or deeper,
Cw1 = varies linearly with position of the water table between bottom of footing and depth2x below bottom of footing,
Cw2 = 0.5 for water table at ground surface,
Cw2 = 1.0 for water table at bottom of footing or deeper,
Cw2 = varies linearly with position of the water table between ground surface and bottomof footing,
Bearing Capacity Factor of Safety.
The factor of safety against bearing capacity failure is calculated as:
qqF ultbc /= (12)
whereFbc = bearing capacity factor of safety (dimensionless), qult = ultimate bearing capacity
(psf), and q = bearing pressure (psf).
Stem Thickness and Footing Thickness.
The stem thickness (at the bottom of the stem) and the footing thickness are calculated
based on considerations of shear and moment capacity, calculated in accordance with theBuilding Code Requirements for Structural Concrete (ACI, 1999). Resistance factors of 0.85
for shear and 0.9 for moment are used in calculating the stem and footing thickness. Cover of
three inches is used for both stem and footing.
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The required thickness for shear capacity is computed using the following equation:
'2 cn fbdV = (13)
Where Vn = Nominal shear capacity (lb/ft)
b = Section width = 12 in/ft
d= Effective section depth (in)fc= 28-day compressive strength of concrete (lb/in
2)
The required thickness for moment is computed using the following equations:
)2
(a
dfAM ysn = (14)
(15)ysc fAabf ='85.0
max =bd
As (16)
Where
Mn = Nominal moment capacity (lbft/ft)d = Effective section depth (in)fc= 28-day compressive strength of concrete (lb/in
2)
fy = Yield strength of reinforcing steel (lb/in2)
As= Area of steel reinforcement (in2)
a = Depth of equivalent rectangular compressive stress block (in)
b = Section width = 12in/ft
= Reinforcement ratio = steel area divided by concrete areamax = Maximum allowable reinforcement ratio = 0.75bb = Value of corresponding to balanced capacity of concrete and steel
The desired value of/b is input by the user. The maximum value of /b allowed is0.75. In practice, walls are usually designed using values of/b ranging from 0.35 to 0.50(MacGregor, 1992).
This workbook has been developed for use by geotechnical engineers to evaluate wallstability and safety with respect to bearing capacity and sliding. The values of stem and
footing thickness shown in the spreadsheet are advisory. They are intended to provide
guidance with regard to reasonable wall dimensions rather than structural design. Therefore,the computed values of stem and footing thickness are not inserted automatically as wall
dimensions, leaving the user in full control of all wall dimensions.
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Appendix A: Symbols
a = Depth of equivalent rectangular compressive stress block (in)
As= Area of reinforcement (in2)
b = Section width = 12 in/ftbh = Width of heel (ft)
B = Width of the footing (ft)
c = cohesion intercept of soil in front of key (psf)
Cw1 and Cw2 = water table correction factors (dimensionless)
d= Effective section depth (in)
Df = depth from ground surface in front of wall to the bottom of the footing (ft)
Dk= depth of key below bottom of footing (ft)
e = eccentricity of normal force from the center of the footing (ft)Ev = vertical shear load (lb/ft)
Eh = Horizontal earth pressure (lb/ft)
Eh1 = Horizontal earth pressure caused by surcharge (lb/ft)
Eh2 = Horizontal earth pressure caused by the weight of backfill (lb/ft)
fc= 28-day compressive strength of concrete (lb/in2)
fy = Yield strength of reinforcing steel (lb/in2)
Fbc = bearing capacity factor of safety (dimensionless)
Fom = factor of safety against overturning (dimensionless)Fs = factor of safety against sliding (dimensionless)
hw = Height of the stem (ft)
H= height of the vertical plane from the heel of the wall to the surface of the backfill (ft)
Ks = surcharge pressure coefficient (dimensionless)
Kv = vertical shear load coefficient (dimensionless)
Mn = Nominal moment capacity (lbft/ft)
N= normal load on the footing (lb/ft)
N60 = average value of Standard Penetration Test blow count within 2x below the bottom ofthe footing
Nc = bearing capacity factor (dimensionless)
Pp= passive earth pressure force (lb/ft)
q = bearing pressure (psf)
qs = surcharge pressure (psf)
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qult= ultimate bearing capacity (psf)
Su = undrained shear strength of soil (which has = 0)
t1 = thickness of stem at the bottom (ft)
t2 = thickness of stem at the top (ft)
T= shear load on the footing (lb/ft)
Vn = nominal shear capacity (lb/ft)
Wbf= Weight of backfill (lb/ft)
Wf= Weight of footing (lb/ft)Wff= Weight of soil in front of the stem and above the footing above bt (lb/ft)
Ws = Weight of stem (lb/ft)
x = distance from edge of the footing to the point of application of the normal load (ft)
2x = effective footing width (ft)
x1 = horizontal distance from the weight center of the stem to the toe (ft)
x2 = horizontal distance from the weight center of backfill to the toe (ft)
y = height ofEh2 above the base of the footing (ft)
z= depth below the surface of the backfill (ft)
Zp = depth from the ground surface in front of the wall to mid-height of the key (ft),
= friction angle of soil in front of key (degrees)
= unit weight of soil (pcf)
bf= unit weight of backfill (pcf)
eq = unit weight of the equivalent fluid, which would exert the same lateral pressure as thebackfill (pcf)
= coefficient of friction between the wall footing and the granular material beneath thefooting (dimensionless)
= Reinforcement ratio (dimensionless)b = Reinforcement ratio corresponding to balanced failure (dimensionless)max = Specified maximum reinforcement ratio (dimensionless)
hb = horizontal earth pressure due to weight of backfill (psf)
hs = horizontal earth pressure due to surcharge pressure (psf)
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Appendix B: Equations
Forces and dimensions:
1tbBb th =
sseqh KHqHE +=25.0
ssh KHqE =1 2
25.0 HE eqh =
25.0 HKE bfVv =
bfhwbf bdHhW )(5.0 +=
150= BdWf (150pcf)
150)(5.0 21 += ws httW (150pcf)
1501 = kk DtW
)( dDbW ftff =
)(3
22
21
2
221
2
11
tt
ttttBx t +
++=
)(3
)2(2
dHh
dHhbBx
w
w
h +
+=
shffksfbfv qbWWWWWEN ++++++= with surcharge over footing
ffksfbfv WWWWWEN +++++= without surcharge over footing
hET=
N
yHHqKWbBWWxWxWxBEx
eqssfftfksbfV )5.05.0(2/2/22
312 ++++++=
without surcharge over the footing
N
bBbqyHHqKWbBWWxWxWxBEx
hhseqssfftfksbfV )2/()5.05.0(2/2/22
312 +++++++=
with surcharge over the footing
6/)(2/)( 32 dHdHqKM eqss += (Moment at the bottom of the stem)
sseq KqdHdHV )()(5.02 += (Shear force at the bottom of the stem)
xB
e =
2
)6
1(maxB
e
B
Nq += if
6
Be
x
Nq
3
2max = if
6
Be >
)6
1(minB
e
B
Nq = if
6
Be
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x
Nq
2=
Passive pressure on the key:
{ })2/45(tan)2/45tan(2 2 +++= pkp ZcDP
2/kfp DDZ +=
Sliding through granular soil:
T
PNF
p
s
+=
Overturning:
yHHqK
BEbWxWxWxWBWF
eqss
vtffkbfsf
om 22
321
5.05.0
2/2/
+
+++++= without surcharge over footing
yHHqK
bBbqBEbWxWxWxWBWF
eqss
hhsvtffkbfsf
om 22
321
5.05.0
)2/(2/2/
+
++++++= with surcharge over
footing
Bearing capacity for cohesive soil foundations
fucult DSNq += , is the smaller of the unit weights of the backfill and foundation
)3.11)(2
2.01(5N
T
x
DN
f
c +=
qqF ultbc /=
Bearing capacity for granular soil foundations
)21)(2
(10
)2(2000 21
N
T
x
DCC
xNq
f
ww
SPT
ult +=
qqF ultbc /=
Water Table Cw1 Cw2
At the ground surface 0.5 0.5
At Base of the Footing 0.5 1.02x below footing 1.0 1.0
For intermediate water table depths, interpolate linearly between these values.
Shear capacity and moment capacity'2 cn fbdV =
)2
(a
dfAM ysn =
ysc fAabf ='85.0
max =bd
As
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)000,87
000,87(
'85.0 1
yy
c
bff
f
+=
=1 0.85 when f 4000' psic
=
1000
'05.005. c
f1 when 4000 psifc 8000'<
= 0.65 when f 8000'> psic
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Appendix C: References:
1. ACI(1999), Building Code Requirements for Structural Concrete(ACI 318-99),
American Concrete Institute.
2. Hansen, J. B. and Hansen, B. (1957), Foundations of structures (a) General
Subjects and Foundations other than piled foundations, General Report, 4th
ICSMFE,London, Vol II, pp 441-447.
3. MacGregor, James G. (1992), Reinforced Concrete: Mechanics and Design, Prentice
Hall, Englewood Cliffs, NJ4. Meyerhof, G. G. (1953), The Bearing Capacity of Foundations under Eccentric and
Inclined Loads. Third International Conference on Soil Mechanics and Foundation
Engineering. Zurich. Proceedings, Vol. 1, pp. 440-445.