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APPENDIX G
List of Symbols and Summary of Equations
List of Symbols
A constant in Equation (23)
Ac cross sectional area of reinforcement at the end of service
A c statistical variable for Ac
b width of strip-type reinforcement
C time in years for zinc depletion from galvanized reinforcements
COVQ coefficient of variation for load bias
COVR coefficient of variation for resistance bias
CR corrosion rate used in Equation (16)
Di initial diameter of bars/wires
D diameter of bar or wire corrected for corrosion loss
Ec strip thickness corrected for corrosion loss
fz(rz) pdf representing zinc corrosion rates, rz in Equation (18)
Fs cumulative density function representing steel corrosion rates in Equation (18)
Fy yield strength of steel
F y statistical variable for Fy
Fult ultimate strength of steel
F ult statistical variable for Fult
g random variable representing safety margin
K coefficient of lateral earth pressure
k constant in Equation (1)
n exponent for Equation (1), or number of longitudinal wires in Equation (12)
pf probability of occurrence (e.g., probability that yield stress will be exceeded before
the end of intended service life)
P[X1 X2] probability of X1 given X2 in Equation (18)
Q random variable representing "measured or actual" load
Qni nominal (i.e., computed) loads from sources that may include earth loads, sur-
charge loads, impact loads or live loads
Qn nominal load from single source
r0 the lowest rate of zinc corrosion for which base steel will be consumed within tf and
is equal to zi/tf as used in Equation (18)

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rs mean steel corrosion rate
rz1 mean of the initial rate of zinc corrosion, i.e. until reaching t1
rz2 mean corrosion rate of zinc subsequent to t1
rz mean zinc corrosion rate [used in Equation (18)]
R random variable representing "measured or actual" resistance
Rn nominal (i.e., computed) resistance
S initial thickness of strip-type reinforcements
S loss of thickness due to corrosion
SH horizontal spacing of reinforcements
SV vertical spacing of reinforcements
t time (years) in Equation (1)
t1 time for which rz1 prevails, usually taken as 2 to 3 years
tf service life in years
tdesign design life used in Equations (17a) and (17b)
Tmax maximum reinforcement tension at a given level per unit width of wall
Tnominal nominal tension/prestress applied to rock bolts during installation and used in Eq. (24)
Trem remaining tensile strength
x loss of thickness per side or loss of radius as used in Equation (1)
X loss of steel
X given amount of steel loss used in Equation (18)
zi initial thickness of zinc coating for galvanized reinforcements
reliability index
T target reliability index
resistance factor
i load factor for the ith load source as used in Equation (6)
Q load factor as used in Equations (21) and (22)
Ac bias of remaining cross section defined as the ratio of measured (actual) to nominal
(computed) values
Fy bias of yield stress defined as the ratio of measured (actual) to nominal (computed) values
R resistance bias defined as the ratio of measured (actual) to nominal (computed) values
Q load bias defined as the ratio of measured (actual) to nominal (computed) values
resistivity of fill material
H, v horizontal and vertical stress, respectively, at depth of interest in the reinforced zone
H supplemental factored horizontal pressure due to external surcharges
s standard deviation of steel corrosion rate as used in Equation (18)
z standard deviation of zinc corrosion rate as used in Equation (18)
Summary of Equations
Chapter 1--Background
Durability and Performance Issues for Earth Reinforcements
Romanoff (1957) proposed the following power law to predict rates of corrosion of buried
metal elements:
x = kt n (1)

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Darbin et al. (1988) and Elias (1990) proposed equations, having the same form as Equation (1),
to estimate steel loss for plain steel and galvanized elements, respectively. These models are devel-
oped using measurements of corrosion from elements buried in fill representative of MSE construc-
tion. The following models apply to galvanized and plain steel reinforcements, respectively:
for galvanized elements
1.54
z m 0.65
if tf > i then X ( m ) = 50 × t f ( yr ) - 2 × z i ( m )
25 yr
1.54 (2)
z
if tf i then X ( m ) = 0
25
for plain steel elements
m 0.8
X ( m ) = 80 ×tf (3)
yr
For Equation (2) loss of base steel occurs subsequent to depletion of the zinc coating, and zi is
the initial zinc thickness. Equation (2) is applicable to the range of fill conditions representative of
MSE wall construction that exhibit min greater than 1,000 -cm. Data reviewed for Equation (3)
are based on the NBS data set for plain steel and include a wider range of fill conditions.
Although corrosion rates for both galvanized and plain steel clearly vary exponentially with
respect to time, a number of models (including the AASHTO model) approximate loss of steel
using linear extrapolation for the purpose of design. Calibration of LRFD resistance factors for
galvanized reinforcements assumes that the steel cross section is not consumed before the zinc
coating, which serves as the sacrificial anode protecting the base steel. Since the zinc layers do
not contribute to the tensile strength of the reinforcements, strength loss is also delayed until the
zinc is consumed, and loss of steel section is described according to Equation (4). In general the
thickness of steel, X, consumed per side over the design life, tf, may be computed as
m
X ( m ) = ( t f ( yrs ) - C ( yrs )) × rs (4)
yr
( zi - rz1 × t1 )
where C is the time for zinc depletion (C = t1 + C = t1 + , which is computed
rz 2
based on the initial zinc thickness, zi, the initial corrosion rate for zinc, rz1, the subsequent zinc
corrosion rate, rz2, and the duration for which rz1 prevails (t1 - usually taken as 2 to 3 years). The
corrosion rate of the base steel subsequent to zinc depletion is rs.
Equation (5) is based on Equation (4) but uses the AASHTO model parameters where the
steel loss per side (X) in m/yr for a given service life, tf , and initial thickness of zinc coating, zi,
is computed as
m ( zi - 30 m )
X ( m ) = 12 × t f - 2 yr - yr (5)
yr m
4
yr
Load and Resistance Factor Design (LRFD)
LRFD is a reliability-based design method by which loads and resistances are factored such that
iQni Rn (6)

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Load and resistance factors are applied such that the associated probability of the load exceed-
ing the resistance is low. The limit state equation corresponding to Equation (6) is:
g ( R, Q ) = R - Qi = R Rn - Qi Qni > 0 (7)
Chapter 2--Research Approach
Yield Limit State
Reinforcement loads are computed based on the horizontal stress carried by the reinforce-
ments computed as
H = K v + H (8)
The maximum reinforcement tension per unit width of wall is computed from H based on
the vertical spacing of the reinforcements as
Tmax = H SV (9)
Equations (8) and (9) describe the demand placed on the reinforcements, the capacity is the
yield resistance of the reinforcements computed as
Fy Ac
R= (10)
SH
for strip-type reinforcements
Ac = bEc
Ec = ( S - S ) for S < S , and 0 for S S (11)
and for steel grid-type reinforcements
D2
Ac = n × × (12)
4
D = D1 - S for S < Di , and 0 for S Di
For galvanized reinforcements
S = 2 × rs × ( t f - C ) For C < t f
(13a)
S = 0 For C t f
( zi - 2 × rz1 )
C = 2 yrs + (13b)
rz 2
For plain steel reinforcements
S = 2 × rs × tf (14)
Using the statistics and observed distribution for measurements of corrosion rate, the bias of
the remaining strength is computed and used as input for the reliability-based calibration of
resistance factor. The bias is computed as
F
y Ac
R = (15)
Fy Ac

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Resistance Factor Calibration
The Monte Carlo analysis for calibration of resistance factor computes values for the limit state
function, g = R - Q, considering the uncertainty of R and Q, and renders the probability that
g <0. The variables R and Q can be related to nominal value as follows:
Q = Qn × Q , and based on the LRFD equation [Equ
uation 6],
R × Q × Qn
R = R × Rn =
Chapter 3--Findings and Applications
Trends
A power law was regressed to achieve the "best fit" with the data describing the relationship
between corrosion rates and fill resistivity rendering the following equation, which is limited to
galvanized reinforcements that are less than 20 years old:
CR 1, 400-0.75 (16)
Metal Loss Models and Reliability
Two different metal loss models for computing nominal sacrificial steel requirements with
respect to marginal quality fills are studied to illustrate how this impacts the reliability of service
life estimates. The first model (Model I) is from Jackura et al. (1987) for "neutral" fill and the
second model (Model II) is a similar form, but with double the corrosion rate for steel as follows:
m
Model I : X ( m ) = ( t design - 10 ) years × 28 (17a
a)
year
m
Model II : X ( m ) = ( t design - 10 ) years × 56 (17
7 b)
year
Verification of Monte Carlo Analysis
Sagues Formulation. Equation (18) was proposed by Sagues et al. (2000) to compute the prob-
ability that loss of base steel, X, from galvanized reinforcements exceeds a given threshold, X as:
P [ X > X t f , z i , rz , z , rs , s ] = f z ( rz )(1 - Fs (( X ) ( t f - z i rz ))) drz (18)
r0
For each value of X the bias of the remaining cross section (strip-type reinforcements) is com-
puted as:
(S - 2 × X )
Ac = (19)
[S - 2 × 12 × (tf - C )]
wherein the AASHTO metal loss model, Equation (5), is used in the denominator to compute
nominal remaining cross section. A mean and standard deviation were determined from the dis-
tribution of the computed bias to describe the variation of Ac. The bias of the remaining tensile
strength was then computed as:
R = Ac × Fy (20)

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Closed-form Solutions for Reliability Index. For a specific limit state and a single load source, the
reliability index () and the resistance factor () can be related using the following formula (Allen
et al., 2005), which assumes that the load and resistance bias both have normal distributions:
Q
R - Q
R
= (21)
2
COVR Q R + (COVQ Q )
2
R
In the case of lognormal distributions for load and resistance bias:
ln Q R (1 + COVQ 2
) (1 + COVR2 )
= R Q (22)
ln [(1 + COVQ )(1 + COVR )]
2 2
For a given load factor, and known load and resistance statistics, Equations (21) and (22) are
satisfied for selected values of resistance factor, rendering related pairs of reliability indices and
resistance factors. From the computed pairs of versus R, resistance factors can be selected cor-
responding to the targeted level of reliability.
Type II--Condition Assessment
Rock Bolts
Metal loss of exposed portions of the reinforcement behind the anchor plate, or other areas,
may be expressed using the Romanoff equation as
m m 0.8
X = A t ( yr )
side
(23)
yr side
The resistance bias is computed as follows:
A
Fult
R = c
(24)
Tnominal
D 2
Ac = (25)
4
D2 = Di - 2 X for 2X < Di
(26)
D2 = 0 for 2X Di
Chapter 4--Conclusions
and Recommendations
Recommended Resistance Factors for LRFD
The following equations are recommended to estimate nominal sacrificial steel requirements
for plain steel reinforcements (i.e., not galvanized) for good and high quality fills:

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m m
Good Quality Fill: X = 80 × t 0.8 ( yr ) (27)
side yr side
m m
High Quality Fill: X = 13 × t ( yr ) (28)
side yr side
The following equation is recommended for computing nominal sacrificial steel requirements
for galvanized reinforcements in marginal quality fills:
m m
X = ( t f - 10 yrs ) × 28
side
(29)
yr side