LRFD Metal Loss and Service-Life Strength Reduction Factors for Metal-Reinforced Systems (2011)

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99 List of Symbols A constant in Equation (23) Ac cross sectional area of reinforcement at the end of service Ac 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 Fy statistical variable for Fy Fult ultimate strength of steel Fult 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) A P P E N D I X G List of Symbols and Summary of Equations

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 ΔS 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 β reliability index βT target reliability index φ resistance factor γi load factor for the ith load source as used in Equation (6) γQ load factor as used in Equations (21) and (22) λAc bias of remaining cross section defined as the ratio of measured (actual) to nominal (computed) values λFy biasofyieldstressdefinedas theratioofmeasured(actual) to nominal (computed) values λR resistance bias defined as the ratio of measured (actual) to nominal (computed) values λQ load bias defined as the ratio of measured (actual) to nominal (computed) values ρ resistivity of fill material σH, σv horizontal and vertical stress, respectively, at depth of interest in the reinforced zone ΔσH supplemental factored horizontal pressure due to external surcharges σs standard deviation of steel corrosion rate as used in Equation (18) σz 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= ( )1 100

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 for plain steel elements 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 ρmin greater than 1,000 Ω-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 where C is the time for zinc depletion (C = t1 + , which is computed 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 μm/yr for a given service life, tf , and initial thickness of zinc coating, zi, is computed as Load and Resistance Factor Design (LRFD) LRFD is a reliability-based design method by which loads and resistances are factored such that γ φi ni nQ R≤∑ ( )6 X yr t yr z yr f iμ μ μμm m m m ( ) = × − − −( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟12 2 30 4 ⎟ yr ( )5 C t z r t r i z z = + − ×( )⎛ ⎝⎜ ⎞ ⎠⎟1 1 12 X t yrs C yrs r yr f sμ μ m m( ) = ( )− ( )( ) × ( )4 X yr t fμ μ m m( ) = ×80 30 8. ( ) if then m m t z X yr tf i> ⎛⎝⎜ ⎞⎠⎟ ( ) = ⎛ ⎝⎜ ⎞ ⎠⎟ ×25 50 1 54. μ μ f i f i yr z t z X 0 65 1 54 2 25 . . ( )− × ( ) ≤ ⎛⎝⎜ ⎞⎠⎟ μm if then μm( ) = 0 2( ) 101

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: Chapter 2—Research Approach Yield Limit State Reinforcement loads are computed based on the horizontal stress carried by the reinforce- ments computed as The maximum reinforcement tension per unit width of wall is computed from σH based on the vertical spacing of the reinforcements as Equations (8) and (9) describe the demand placed on the reinforcements, the capacity is the yield resistance of the reinforcements computed as for strip-type reinforcements and for steel grid-type reinforcements For galvanized reinforcements For plain steel reinforcements 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 λR y c y c F A F A =   ( )15 ΔS r ts f= × ×2 14( ) C yrs z r r i z z = + − ×( ) 2 2 13 1 2 ( )b Δ Δ S r t C C t S C t s f f f = × × −( ) < = ≥ 2 0 13 For For a( ) A n D D D S S D c i = × × = − < π   2 1 4 12( ) ,Δ Δfor and 0 for ΔS Di≥ A bE E S S S S S S c c c = = −( ) < ≥Δ Δ Δfor and 0 for, ( )11 R F A S y c H = ( )10 T SH Vmax ( )= σ 9 σ σ σH v HK= + Δ ( )8 g R Q R Q R Qi R n Qi ni, ( )( ) = − = − >∑λ λ 0 7 102

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: 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: 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: 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: For each value of X the bias of the remaining cross section (strip-type reinforcements) is com- puted as: 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 λAc. The bias of the remaining tensile strength was then computed as: λ λ λR Ac Fy= × ( )20 λAc f S X S t C = − ×( ) − × × −( )[ ] 2 2 12 19( ) P X X t z r r f r F X t zf i z z s s z z s f> ′[ ] = ( ) − ′( ) −, , , , ,σ σ 1 i z z r r dr( )( )( )∞∫ 0 18( ) Model II m m : (X t years year designμ μ( ) = −( ) ×10 56 17b) Model I m m : (X t years year designμ μ( ) = −( ) ×10 28 17a) CR ≈ −1 400 160 75, ( ).ρ Q Qn Q= × λ , and based on the LRFD equation [Equation 6], R R Q R n R Q n = × = × ×λ λ γφ 103

Closed-form Solutions for Reliability Index. For a specific limit state and a single load source, the reliability index (β) and the resistance factor (φ) can be related using the following formula (Allen et al., 2005), which assumes that the load and resistance bias both have normal distributions: In the case of lognormal distributions for load and resistance bias: 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 β versus φR, 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 The resistance bias is computed as follows: 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: D D X X D D X D i i i ∗ = − ∗ ∗ < ∗ = ∗ ≥ 2 2 2 0 26 for 2 for 2 ( ) A D c  = ∗π 2 4 25( ) λR ult c F A T =   nominal ( )24 X side A yr side t yr μ μm m⎛⎝⎜ ⎞⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ ( )0 8 23. ( ) β γ λ φ λ = +( ) +( )⎡⎣⎢ ⎤ ⎦⎥ + ln ln Q R R Q Q RCOV COV COV 1 1 1 2 2 Q RCOV2 21 22( ) +( )[ ] ( ) β γ φ λ λ γ φ λ = ⎛ ⎝⎜ ⎞ ⎠⎟ − ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ + Q R R Q R Q R RCOV CO 2 VQ Qλ( )2 21( ) 104

The following equation is recommended for computing nominal sacrificial steel requirements for galvanized reinforcements in marginal quality fills: X side t yrs yr side f μ μm m⎛⎝⎜ ⎞⎠⎟ = −( ) × ⎛ ⎝⎜ ⎞ ⎠⎟10 28 2( 9) High Quality Fill: m m X side yr side μ μ⎛⎝⎜ ⎞⎠⎟ = 13 × ( )t yr ( )28 Good Quality Fill: m m X side yr side μ μ⎛⎝⎜ ⎞⎠⎟ = 80 × ( )t yr0 8 27. ( ) 105