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99 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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100 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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101 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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102 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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103 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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104 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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105 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