Guidelines for Inspection and Strength Evaluation of Suspension Bridge Parallel Wire Cables (2004)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

5-1 CONTENTS SECTION 5 ESTIMATION OF CABLE STRENGTH .................................................................................5-1 5.1 INTRODUCTION.......................................................................................................................................5-3 5.2 NOTATION .................................................................................................................................................5-3 5.3 ESTIMATED CABLE STRENGTH .........................................................................................................5-6 5.3.1 General.................................................................................................................................................5-6 5.3.2 Wire Groupings ..................................................................................................................................5-6 5.3.2.1 BROKEN WIRES IN EFFECTIVE DEVELOPMENT LENGTH .................................................5-6 5.3.2.2 REPAIRED WIRES IN EFFECTIVE DEVELOPMENT LENGTH ..............................................5-7 5.3.2.3 UNBROKEN WIRES IN EACH CORROSION STAGE ...............................................................5-7 5.3.2.4 CRACKED WIRES .........................................................................................................................5-8 5.3.2.4.1 Discrete Cracked Wires in Effective Development Length...........................................................5-8 5.3.2.4.2 Redevelopment of Cracked Wires That Fail..................................................................................5-9 5.3.2.5 EFFECTIVE NUMBER OF UNBROKEN WIRES ......................................................................5-10 5.3.2.6 FRACTION OF THE CABLE REPRESENTED BY EACH GROUP OF WIRES......................5-11 5.3.3 Strength of Unbroken Wires............................................................................................................5-11 5.3.3.1 SIMPLIFIED STRENGTH MODEL.............................................................................................5-13 5.3.3.1.1 Mean Tensile Strength of Uncracked Wires................................................................................5-13 5.3.3.1.2 Cable Strength Using the Simplified Model................................................................................5-14 5.3.3.2 BRITTLE-WIRE MODEL.............................................................................................................5-14 5.3.3.2.1 Compound Tensile Strength Distribution Curve .........................................................................5-14 5.3.3.2.2 Cable Force at Stress s .................................................................................................................5-15 5.3.3.2.3 Force in Cracked Wires That Break in Adjacent Panels...............................................................5-15 5.3.3.2.4 Strength of Unbroken Wires in the Cable....................................................................................5-16 5.3.3.3 LIMITED DUCTILITY MODEL ..................................................................................................5-16 5.3.3.3.1 Compound Ultimate Strain Distribution Curve ...........................................................................5-17 5.3.3.3.2 Force in Unbroken Wires at Strain e ...........................................................................................5-17 5.3.3.3.3 Force in Cracked Wires That Break in Adjacent Panels..............................................................5-17 5.3.3.3.4 Strength of Unbroken Wires in the Cable....................................................................................5-18 5.3.4 Redevelopment of Broken Wires.....................................................................................................5-18 5.3.5 Cable Strength ..................................................................................................................................5-19 5.4 FIGURES FOR SECTION 5....................................................................................................................5-21

5-2 FIGURES Figure 5.3.2.4.1-1. Fraction of discrete cracked wires in Stage k...............................................................5-21 Figure 5.3.2.4.2-1. Fraction of cracked wires in Stage k that are redeveloped...........................................5-22 Figure 5.3.2.6-1. Summary of calculations for number of wires in each group. .........................................5-23 Figure 5.3.3.1.2-1. Strength reduction factor, K..........................................................................................5-24

5-3 5.1 INTRODUCTION The Equations for estimating the current strength of the cable are presented in this section. Three strength models of increasing complexity are included, along with a graph that aids in the calculation of the simplest model. The section also includes information about separating the wires into groups that are in a similar state of deterioration, or that contain cracks or are broken. Grouping wires is required for applying the strength models. The strength is calculated at a specific inspected location along the cable, called the evaluated panel. The concept of effective development length, covered previously in Article 4.5.2, is crucial for counting the number of broken and cracked wires and calculating their contribution to cable strength. 5.2 NOTATION aw = nominal area of one wire used in lab analysis (5.3.3.1.2) (5.3.3.2.2) (5.3.3.3.2) (5.3.7.3.3) (5.3.4) Cd = fraction of 95% of the mean tensile strength of Group 2 wires that is developed in a broken wire at each cable band (5.3.4) Cdi = wire redevelopment factor for Panel i = Cd multiplied by the number of cable bands between Panel i and the evaluated panel (5.3.2.4.2) e = strain in the unbroken wires of the cable (5.3.3.3.1) F35(e) = fraction of cracked wires that are broken at strain e = Weibull cumulative distribution of ultimate strain of Group 5 (cracked) wires at strain e (5.3.3.3.3) F35(s) = fraction of cracked wires that are broken at stress s = Weibull cumulative distribution of tensile strength of Group 5 (cracked) wires at stress s (5.3.3.2.3) F3k(e) = Weibull cumulative distribution of ultimate strain of Group k wires (5.3.3.3.1) F3k(s) = Weibull cumulative distribution of tensile strength of Group k wires (5.3.3.2.1) (5.3.3.2.2) Fc(e) = compound cumulative distribution of ultimate strain (5.3.3.3.1) (5.3.3.3.2) Fc(s) = compound cumulative distribution of tensile strength (5.3.3.2.1) (5.3.3.2.2) i = number of a panel (5.3.2.1) (5.3.2.4.1) (5.3.2.4.2) (5.3.4) K = reduction factor (given in Figure 5.3.3.1.2-1 as a function of the coefficient of variation, σs/µs) (5.3.3.1.2) k = corrosion stage of wires (k = 1, 2, 3 and 4) (5.3.2.3) (5.3.2.4.1) k = corrosion stage of a group of wires (k = 2, 3, 4 and 5) (5.3.2.5) (5.3.2.6) (5.3.3.2.1) (5.3.3.3.1); (k=2,3,&4) (5.3.3.1.1) K = reduction factor (given in Figure 5.3.3.1.2-1 as a function of the coefficient of variation, σs/µs) (5.3.3.1.2) Le = number of panels in effective development length (5.3.2.1) (5.3.2.4.1) (5.3.2.4.2) (5.3.4)

5-4 max = maximum value of the expression inside the brackets (5.3.3.2.4) (5.3.3.3.4) N0k = number of unbroken Stage k wires in the evaluated panel (5.3.2.3) (5.3.2.4.1) (5.3.2.4.2) (5.3.2.5) N5 = number of discrete cracked wires in the effective development length (5.3.2.5) (5.3.3.1.1) (5.3.3.1.2) Nb = number of broken wires in the effective development length (5.3.2.1) (5.3.2.3) nb1 = number of broken wires in the evaluated panel (5.3.2.1) (5.3.4) nbi = number of broken wires in panel i (5.3.2.1) (5.3.4) Nc,k = total number of discrete cracked wires in the effective development length that are Stage k in the evaluated panel (5.3.2.4.1) (5.3.2.5) Ncr = effective number of redeveloped cracked wires in the effective development length (5.3.2.4.2) (5.3.3.2.3) (5.3.3.3.3) Ncr,k = effective number of broken cracked wires that are Stage k in the evaluated panel and can be redeveloped (5.3.2.4.2) Neff = effective number of unbroken wires in the evaluated panel (5.3.2.5) (5.3.2.6) (5.3.3.1.1) (5.3.3.1.2) (5.3.3.2.2) (5.3.3.3.2) Nk = number of Group k wires in the evaluated panel (5.3.2.5) (5.3.2.6) (5.3.3.1.1) Nr = number of broken wires that are repaired in the effective development length (5.3.2.2) (5.3.2.3) nr1 = number of broken wires that are repaired in the evaluated panel (i=1) (5.3.2.2) nri = number of broken wires in panel i that are repaired (5.3.2.2) (5.3.4) Nsk = number of Stage k wires in the evaluated panel (5.3.2.3) pi = number of cable bands between the evaluated panel and a wire break in panel i (5.3.4) pc,k = fraction of Stage k wires that are cracked (5.3.2.4.1) (5.3.2.4.2) pk = fraction of unbroken wires in the evaluated panel represented by Group k (5.3.2.6) (5.3.3.2.1) (5.3.3.3.1) puk = fraction of unbroken and uncracked wires in the cable represented by Group k (5.3.3.1.1) R = estimated cable strength (5.3.3.1.2) (5.3.5) Rb = cable strength attributable to broken wires in adjacent panels (5.3.4) (5.3.5) Ru = cable strength attributable to unbroken wires (5.3.3.2.4) (5.3.3.3.4) (5.3.5)

5-5 s = stress in unbroken wires of the cable (5.3.3.2.1) (5.3.3.2.2); stress in wires corresponding to the estimated cable strength calculated in Article 5.3.3 (5.3.4) s(e) = stress in wires determined from the average stress-strain curve for all wires at strain e (5.3.3.3.2) sd = redeveloped stress in the evaluated panel for a broken wire in an adjacent panel (5.3.4) Tcr(e) = maximum force in broken cracked wires that can be redeveloped in the evaluated panel at strain e (5.3.3.3.3) Tcr(s) = maximum force in broken cracked wires that can be redeveloped in the evaluated panel at stress s (5.3.3.2.3) Tu(e) = force in unbroken wires in the evaluated panel at strain e (5.3.3.3.2) Tu(s) = force in unbroken wires in the evaluated panel at stress s (5.3.3.2.2) µs = sample mean tensile strength of the combined groups of wires excluding cracked wires (5.3.3.1.1) µs2 = sample mean tensile strength of Group 2 wires (5.3.3.2.3) (5.3.3.3.3) (5.3.4) µsk = sample mean tensile strength of Group k wires (5.3.3.1.1) = sample standard deviation of tensile strength of the combined groups of wires excluding cracked wires (5.3.3.1.1) = sample standard deviation of the tensile strength of Group k wires (5.3.3.1.1) σsk σs

GUIDELINES COMMENTARY 5-6 5.3 ESTIMATED CABLE STRENGTH 5.3.1 General The strength of a cable at the evaluated panel is the sum of the strengths of wires in three categories: • all wires in the evaluated panel minus broken wires in that panel and nearby panels • wires that are cracked in nearby panels, affecting the strength of the same wires in the evaluated panel • wires that are broken in nearby panels, affecting the strength of the same wires in the evaluated panel Methods for evaluating each of the three categories are described below in Articles 5.3.3 and 5.3.4. 5.3.2 Wire Groupings The wires are assigned to groups that are numbered 2 to 5, corresponding to the corrosion stages they derive from. Stage 1 and Stage 2 wires are added together to form Group 2, because their properties are virtually identical. Stage 3 and Stage 4 wires become Group 3 and Group 4 respectively. All the discrete cracked wires are subtracted from their corresponding groups and added together to form Group 5. The number of discrete cracked wires in the effective development length is N5. Broken wires are treated separately. 5.3.2.1 BROKEN WIRES IN EFFECTIVE DEVELOPMENT LENGTH The total number of broken wires in the effective development length is Σ = = e i bib L nN 1 (5.3.2.1-1) where Nb = number of broken wires in the effective development length nbi = number of broken wires in panel i Le = number of panels in the effective development length (see Article 4.5.2) i = number of a panel If the evaluated panel is the only panel that has been C5.3.2.1 Wires that are broken in panels other than the panel being evaluated may affect the cable strength in the evaluated panel. The force that such a wire can sustain increases as the wire passes through one or more cable bands, until the wire is fully redeveloped. The increase in strength comes from the friction among the wires, caused by the clamping action of the cable bands and the cable wrapping.

GUIDELINES COMMENTARY 5-7 inspected, then it is assumed that all panels in the effective development length are alike, and that all values of nbi are equal to nb1, and that 1beb nLN ⋅ = (5.3.2.1-2) where nb1 = number of broken wires in the evaluated panel 5.3.2.2 REPAIRED WIRES IN EFFECTIVE DEVELOPMENT LENGTH The total number of repaired wires in the effective development length is Σ = = e i rir L nN 1 (5.3.2.2-1) where Nr = number of broken wires that are repaired in the effective development length nri = number of broken wires that are repaired in panel i If the evaluated panel is the only panel that has been inspected, then broken wires in that panel are the only ones to have been repaired, 1rr nN = (5.3.2.2-2) where nr1 = number of broken wires that are repaired in the evaluated panel (i=1) C5.3.2.2 If the evaluated panel is the only panel that has been inspected, numbering the order of the panels in the effective development length is not critical. If adjacent panels have been inspected, and if the methods specified in Article 5.3.4 and Appendix B are used to evaluate the cable, then the panels should be numbered according to the instructions in Appendix B, Article B.4.1. 5.3.2.3 UNBROKEN WIRES IN EACH CORROSION STAGE The number of unbroken wires in each stage of corrosion is determined by subtracting the unrepaired broken wires in the effective development length from Stage 4 wires in the evaluated panel, and when there are none remaining, from Stage 3 wires in the panel, as follows: when 4srb NNN ≤ − rbs NNNN +− = 404 (5.3.2.3-1) 303 sNN = (5.3.2.3-2)

GUIDELINES COMMENTARY 5-8 1202 ss NNN += (5.3.2.3-3) when 4srb NNN >− 004 =N (5.3.2.3-4) rbss NNNNN +−+= 4303 (5.3.2.3-5) 1202 ss NNN += (5.3.2.3-6) where N0k = number of unbroken Stage k wires in the evaluated panel Nsk = number of Stage k wires in the evaluated panel (Article 4.3.2) Nb = number of broken wires in the effective development length Nr = number of broken wires that are repaired in the effective development length k = corrosion stage of wires (k =1, 2, 3 and 4) 5.3.2.4 CRACKED WIRES The formulas in the following articles apply to situations in which only the evaluated panel has been inspected, and are based on the conservative assumption that all panels in the effective development length are in the same condition as the evaluated panel. Of all the cracked wires in the cable, these calculations are applied to discrete cracked wires only. A discrete cracked wire is a wire that is cracked in panel i but is not cracked in all the panels nearer than i to the evaluated panel. The effective number of discrete cracked wires that are assumed to be redeveloped in the evaluated panel due to friction at the cable bands is also required for the calculation. C5.3.2.4 If all panels in the effective development length have been inspected, the technique presented in Appendix B should be used. The technique is limited in its application because it is complex and the data are usually not available. It is recommended for severely deteriorated panels, in which case additional adjacent panels must be opened to obtain these data. 5.3.2.4.1 Discrete Cracked Wires in Effective Development Length The number of discrete wires in the effective development length is calculated separately for each corrosion stage. The number of discrete cracked wires in any stage, k, is given by the equation

GUIDELINES COMMENTARY 5-9 ( ) 1 1 ,,0, 1 − = Σ −⋅ ⋅ = iL i kckckkc e ppNN (5.3.2.4.1-1) where Nc,k = total number of discrete cracked wires in the effective development length that are Stage k in the evaluated panel N0k = number of unbroken Stage k wires in the evaluated panel pc,k = fraction of Stage k wires that are cracked i = number of a panel Le = number of panels in the effective development length k = corrosion stage of wires (k = 1,2,3 and 4) As stated above, this calculation is made separately for each stage. Usually, pc,1 and pc,2 will be zero; and pc,3 may be zero. The value of the expression ( ) 1 1 ,, 1 − = Σ −⋅ iL i kckc e pp (5.3.2.4.1-2) 5.3.2.4.2 Redevelopment of Cracked Wires That Fail Cracked wires that are assumed to fail as the cable stress is increased may redevelop part of their strength in the evaluated panel. Assuming all of the cracked wires are broken, redeveloped wires for each stage, k, are ( )Σ = − ⋅ −⋅ ⋅ = eL i di i k,ck,ckk,cr CppNN 1 1 0 1 (5.3.2.4.2-1) where Ncr,k = effective number of broken cracked wires that are Stage k in the evaluated panel and can be redeveloped Cdi = wire redevelopment factor for Panel i = in Equation 1 represents the fraction of discrete cracked wires in each stage in the effective development length. Values of this expression as a function of the fraction of cracked wires in each stage are shown graphically in Figure 5.3.2.4.1-1. The expression is called Nc,k/N0k.

GUIDELINES COMMENTARY 5-10 Cd multiplied by the number of cable bands between Panel i and the evaluated panel The total effective number of redeveloped wires is 432 ,cr,cr,crcr NNNN ++= (5.3.2.4.2-2) where Ncr = effective number of redeveloped cracked wires in the effective development length The value of the expression ( )Σ = − ⋅−⋅ eL i di i kckc Cpp 1 1 ,, 1 (5.3.2.4.2-3) 5.3.2.5 EFFECTIVE NUMBER OF UNBROKEN WIRES The effective number of unbroken wires in the cable is Σ = = 5 2k keff NN (5.3.2.5-1) in which kckk NNN ,0 −= (k = 2, 3, 4) (5.3.2.5-2) and Σ = = 4 2 ,5 k kcNN (5.3.2.5-3) where Neff = effective number of unbroken wires in the evaluated panel Nk = number of Group k wires in the evaluated panel N5 = number of discrete cracked wires in the effective development length in Equation 1 represents the effective fraction of cracked wires that are Stage k in the evaluated panel but are redeveloped because they are broken at stress, s. Figure 5.3.2.4.2-1 gives the effective fraction of cracked wires that will redevelop in the evaluated panel if they break, to be used in Equation 1. This expression is called Ncr,k/N0k. Each stage, k, is treated separately, and then combined using Equation 2.

GUIDELINES COMMENTARY 5-11 N0k = number of unbroken Stage k wires in the evaluated panel Nc,k = total number of discrete cracked wires in the effective development length that are Stage k in the evaluated panel k = corrosion stage of group of wires (k = 2, 3, 4 and 5) 5.3.2.6 FRACTION OF THE CABLE REPRESENTED BY EACH GROUP OF WIRES The unbroken wires in the cable are separated into four groups for the purpose of calculating cable strength. Each group has a different set of tensile strengths and/or ultimate strain properties. The values of Nk are used to calculate the fraction of the cable represented by each group of wires, k, eff k k N N p = (5.3.2.6-1) where pk = fraction of unbroken wires in the evaluated panel represented by Group k Nk = number of Group k wires in the evaluated panel Neff = effective number of unbroken wires in the evaluated panel k = corrosion stage of a group of wires (k =2,3,4 and 5) C5.3.2.6 5.3.3 Strength of Unbroken Wires The strength of the unbroken wires in the cable should be estimated using one of three strength models. The Limited Ductility Model is the most rigorous; the others are special cases of this model, using simplified assumptions. All three recommended models are further described in Appendix A. The choice of model depends on the extent of the deterioration found in the cable: • Use the Simplified Model for cables with no Stage 4 or cracked wires C5.3.3 The Limited Ductility and Brittle-Wire models are used to estimate the strength of a cable composed of many wires that are subjected to the same strain. The Simplified Model, which is based on the Brittle-Wire Model, subtracts all cracked and broken wires and uses a single distribution curve for the tensile strength of the remaining unbroken, uncracked wires. In the Limited Ductility Model, the cable is subjected to an incremental increase in strain The force in the cable at any strain is the sum of the forces in the individual wires at that strain. The wire forces vary in relation to the individual stress-strain diagrams. The calculation of the number of wires in each group is difficult to visualize and is summarized in Figure 5.3.2.6-1.

GUIDELINES COMMENTARY 5-12 • Use the Simplified Model (at the discretion of the owner or investigator) for cables in which cracked Stage 3 and Stage 4 wires account for up to 10% of the total wires. It is understood that the result in cable strength may be up to 10% less than the result using the Brittle Wire Model. • Use the Brittle-Wire Model if cracks are present in more than 10% of all the wires in the cable. • Use the Limited Ductility Model if the wires display unusual variations in tensile strength (sometimes due to varying carbon content), which are reflected in stress-strain curves that are also unusually varied. Individual wires share in carrying the cable tension until one reaches its ultimate strain, at which point it breaks and no longer resists any force. The total force in the cable is reduced accordingly by the previous force carried by the wire. Increasing the strain further causes the cable force to increase again, until the next wire breaks. The process continues until the cable force reaches a maximum value, after which wires break more rapidly than the force increases in the individual wires, resulting in decreased cable force with increased strain. The maximum cable force achieved is the cable strength. The Brittle-Wire Model is a special case of the Limited Ductility Model. In contrast to its parent model, it is assumed that all the wires are subjected to the same tensile stress at any given strain; thus, it is convenient to substitute an increasing stress in the calculation instead of an increasing strain. Any individual wire shares in carrying the tension in the cable until the stress in that wire exceeds its tensile strength and the wire fails, no longer participating in the cable force. As with the more general model, the cable force increases with increasing stress until a maximum value is reached, which is equivalent to the cable strength. The Limited Ductility Model requires determining the ultimate strain of each wire specimen and developing a full stress-strain diagram for each wire sample. The ultimate strain corresponds to the tensile strength of the wire (see Appendix A). It is also the strain at failure when there is no reduction of area, for instance from a crack. The percentage of elongation in a 10-inch gage length, determined in accordance with ASTM A370, cannot be used as the ultimate strain for the Limited Ductility Model, because it is measured only after necking down and does not include the elastic component of the strain. The Brittle-Wire Model requires knowing only the tensile strength of each specimen, which can be obtained by testing in accordance with ASTM A370. If a cable force vs. strain diagram is wanted by the investigator, then average stress-strain diagrams of the cable wires should be developed for each of the models. Several such diagrams are already needed for the Limited Ductility Model, whereas the Brittle-Wire Model requires only one.

GUIDELINES COMMENTARY 5-13 5.3.3.1 SIMPLIFIED STRENGTH MODEL The Simplified Model should be applied to cables that have very few cracked wires. The upper limit is no more than 10% of the total wire population.. The Brittle-Wire Model is used whenever this limit is exceeded. The Simplified Model is based on the Brittle-Wire Model; the estimated number of cracked wires (Group 5) and broken wires are omitted from the calculation, and the total number of wires in the cable is reduced accordingly. Although the strength may be underestimated by up to 20%, the Simplified Model is useful in locating the most severely deteriorated panel among those inspected. Then the more complex models can be applied to that panel for a more realistic strength estimate. C5.3.3.1 A single Weibull distribution that combines Groups 2, 3 and 4 is used in the Simplified Model. It combines the tensile strength distributions of the individual wire groups, with the relative size of each group taken into account. In order to minimize the computational effort required by the model, a factor is applied to the mean tensile strength, which is multiplied by the effective cable area. 5.3.3.1.1 Mean Tensile Strength of Uncracked Wires The fraction of the cable represented by Groups 2, 3 and 4 is combined with the sample mean values of minimum tensile strength of the representative specimens of each group to determine the sample mean tensile strength and standard deviation of the entire unbroken and uncracked wire population, using the equations Σ = ⋅ = 4 2 )( k skuks p µµ (5.3.3.1.1-1) ( ) 2224 2 ssksk k uks p µµσσ − += Σ = (5.3.3.1.1-2) in which 5NN N p eff k uk − = (5.3.3.1.1-3) where µs = sample mean tensile strength of the combined groups of wires, excluding cracked wires µsk = sample mean tensile strength of Group k wires σs = sample standard deviation of the tensile strength of the combined groups of wires, C5.3.3.1.1 The symbols µ and σ refer to the mean and standard deviation of a property of the entire population of wires in the cable. The mean and standard deviation used in cable strength models are determined from laboratory tests on a selection of wires removed from the cable during inspection and are called the sample mean and sample standard deviation, designated by µs and σs.        

GUIDELINES COMMENTARY 5-14 excluding cracked wires σsk = sample standard deviation of the tensile strength of Group k wires puk = fraction of unbroken and uncracked wires in the cable represented by Group k k = corrosion stage of a group of wires (k = 2, 3 and 4) Neff = effective number of unbroken wires in the evaluated panel N5 = number of discrete cracked wires in the effective development length Nk = number of Group k wires in the evaluated panel 5.3.3.1.2 Cable Strength Using the Simplified Model The cable strength is calculated from the equation KaNNR sweff ⋅⋅⋅− = µ)( 5 (5.3.3.1.2-1) in which where R = estimated cable strength aw = nominal area of one wire used in lab analysis C5.3.3.1.2 The strength reduction factor, K, is the brittle-wire strength of the combined groups of uncracked wires, divided by the product of mean tensile strength of the combined groups of uncracked wires and total area of uncracked wires. The derivation of K is given in Appendix A. 5.3.3.2 BRITTLE-WIRE MODEL The tensile strength of each test specimen is used to determine the minimum tensile strength of each wire sample. The minima are used to determine the sample means and standard deviations for each group of wires, which are combined to construct a compound tensile strength distribution curve. Whenever a cable force vs. strain diagram is required, the test laboratory determines the stress-strain curve up to the ultimate strain for at least one specimen from each sample wire. These curves are used to develop an average stress-strain curve for the entire cable. C5.3.3.2 The Brittle-Wire Model is recommended with few exceptions for determining the strength of the cable. The distribution of the tensile strength of the entire population of unbroken wires is a compound distribution curve developed from the distributions of the individual wire groups, with the relative size of each group taken into account. The Weibull distribution is used in the analysis, with the lower limit of tensile strength, s0, assumed to be zero (no wire can have a negative tensile strength). 5.3.3.2.1 Compound Tensile Strength Distribution Curve The fraction of the cable represented by each of the groups (calculated in Article 5.3.2.6) and the Weibull distribution curves for tensile strength of the specimens C5.3.3.2.1 The Weibull distribution is a Type 3 extreme value distribution function. The function extends from a minimum value, x0, to infinity. The cumulative distribution function is referred to as F3(x) in the K = reduction factor (given in Figure 5.3.3.1.2-1 as a function of the coefficient of variation, σs/µs)

GUIDELINES COMMENTARY 5-15 that represent each of the groups are combined to determine the compound distribution curve for the entire unbroken wire population. The equations used in the calculation are )(3)( 5 2 sFpsF k k kC ⋅= Σ = (5.3.3.2.1-1) where FC(s) = compound cumulative distribution of the tensile strength s = stress in unbroken wires of the cable pk = fraction of unbroken wires in the evaluated panel represented by Group k k = corrosion stage of a group of wires (k = 2, 3, 4 and 5) F3k(s) = Weibull cumulative distribution of tensile strength of Group k wires Guidelines. The equations for this function, as well as a method for determining the parameters of the function, are presented in Appendix A. The term x is the variable of the distribution and is replaced by s for tensile strength and by e for ultimate strain. 5.3.3.2.2 Cable Force at Stress s The total force in the unbroken wires at any value of cable stress, s, is given by ( )( )( )sFsaNsT Cweffu −⋅⋅⋅= 1)( (5.3.3.2.2-1) where Tu(s) = force in the unbroken wires in the evaluated panel at stress s Neff = effective number of unbroken wires in the evaluated panel aw = nominal area of one wire used in lab analysis C5.3.3.2.2 In Equation 1, the wires with tensile strength less than s are assumed to have zero force. This is accomplished mathematically, using the Survivor Function, (1- FC(s)). 5.3.3.2.3 Force in Cracked Wires That Break in Adjacent Panels The total force in the evaluated panel at cable stress, s, in wires that are cracked in other panels within the effective development length and that have a tensile strength less than that stress is given by ( ) )(395.0)( 52 sFaNsT swcrcr ⋅⋅ ⋅ = µ (5.3.3.2.3-1) where Tcr(s) = maximum force in the broken cracked wires that can be redeveloped in the evaluated panel at stress s

GUIDELINES COMMENTARY 5-16 F35(s) = fraction of cracked wires that are broken at stress s = Weibull cumulative distribution of tensile strength of Group 5 (cracked) wires at stress s Ncr = effective number of redeveloped cracked wires in the effective development length 2sµ = sample mean tensile strength of Group 2 wires 5.3.3.2.4 Strength of Unbroken Wires in the Cable The strength of the unbroken wires in the cable is given by ( ) ( )( )sTsTR cruu += max (5.3.3.2.4-1) where Ru = cable strength attributable to unbroken wires max = maximum value of the expression inside the brackets C5.3.3.2.4 ( ))()(max sTsTR cruu −= is determined by calculating the values of the expression for several values of s at suitably small increments (2 ksi) and seeking the maximum value. The entire calculation is best done on a computer spreadsheet program that incorporates the Weibull distribution functions. Alternatively, the expression can be evaluated for a single value of s, and an iterative program can then be used to determine the value of s that produces the maximum expression. Should a cable force vs. strain diagram be required, only the first technique provides the data to plot it. 5.3.3.3 LIMITED DUCTILITY MODEL The Limited Ductility Model requires the ultimate strain of each test specimen based on laboratory results, as well as a stress-strain curve up to the ultimate strain for each specimen tested. The data are used to determine the minimum value of the ultimate strain and the stress-strain curve of each sample wire. Whenever the stress-strain curves for the sample wires are essentially the same, an average curve for all the samples is constructed, and the method given below of estimating the cable strength is followed. Whenever the individual stress-strain curves for the sample wires vary from one another, the general form of the Limited Ductility Model, given in Appendix A, must be used. C5.3.3.3 In the Limited Ductility Model, the ultimate strain of the wires is used as the variable in the distribution functions. In the simple form of this model, presented in the articles below, the distribution of the ultimate strain for the entire population of unbroken wires is a compound distribution curve developed from the distributions of the individual wire groups, with the relative size of each group taken into account. The Weibull distribution is used in the analysis, with the lower limit of ultimate strain, e0, assumed to be zero (no wire can have a negative ultimate strain).

GUIDELINES COMMENTARY 5-17 5.3.3.3.1 Compound Ultimate Strain Distribution Curve The fraction of the cable represented by each of the groups, (calculated in Article 5.3.2.6), and the Weibull distribution curves for ultimate strain of the specimens that represent each of the groups, are combined to determine the compound distribution curve for the entire unbroken wire population. The equation used for the calculation is )(3)( 5 2 eFpeF k k kC ⋅= Σ = (5.3.3.3.1-1) where FC(e) = compound cumulative distribution of the ultimate strain e = strain in the unbroken wires of the cable pk = fraction of the unbroken wires in the evaluated panel represented by Group k k = corrosion stage of a group of wires (k = 2, 3, 4 and 5) F3k(e) = Weibull cumulative distribution of ultimate strain of Group k wires C5.3.3.3.1 The compound distribution curve is the sum of several independent Weibull distributions but is not itself a Weibull distribution. While there are no explicit equations for the distribution, it can be described by Equation 1. The compound distribution curve is used only when all the groups of wires have the same stress-strain curve and the cable force in unbroken wires is calculated by Equation 5.3.3.3.2-1. 5.3.3.3.2 Force in Unbroken Wires at Strain e If it can be shown that the average stress-strain curves for all the groups of wires are alike, then the following equation is used to estimate the force in unbroken wires in the cable at strain e: ( ) ( )( )( )eFesaNeT Cweffu −⋅ ⋅⋅= 1)( (5.3.3.3.2-1) where s(e) = stress in wires determined from the average stress-strain curve for all wires at strain e Tu(e) = force in unbroken wires in the evaluated panel in strain e Neff = effective number of unbroken wires in the evaluated panel aw = nominal area of one wire, used in lab analysis 5.3.3.3.3 Force in Cracked Wires That Break in Adjacent Panels

GUIDELINES COMMENTARY 5-18 The total force in the evaluated panel at cable strain, e, in wires that are cracked in other panels in the effective development length and that have an ultimate strain smaller than e is given by ( ) ( ) ( )eFaNeT swcrcr 52 395.0 ⋅⋅ ⋅ = µ (5.3.3.3.3-1) in which Tcr(e) = maximum force in broken cracked wires that can be redeveloped in the evaluated panel at strain e F35(e) = fraction of cracked wires that are broken at strain e = Weibull cumulative distribution of ultimate strain of Group 5 (cracked) wires at strain e Ncr = effective number of redeveloped cracked wires in the effective development length µs2 = mean sample tensile strength of Group 2 wires 5.3.3.3.4 Strength of Unbroken Wires in the Cable The strength of the unbroken wires in the cable is given by ( ) ( )( )eTeTR cruu += max (5.3.3.3.4-1) where max = maximum value of the expression inside the brackets Ru = cable strength attributable to unbroken wires C5.3.3.3.4 The value of the term ( ) ( )( )eTeT cru +max is determined by calculating the values of the expression for several values of e at suitably small increments (0.001 inch/inch) and seeking the maximum value. The entire calculation is best done on a computer spreadsheet program that incorporates the Weibull distribution functions. 5.3.4 Redevelopment of Broken Wires Wires that are broken in panels adjacent to the evaluated panel share in the cable tension because of the friction that develops at the intervening cable bands. If the tension in a wire exceeds the friction in the cable band adjacent to that panel, the wire will slip, but it will continue to carry a constant tension as the cable tension increases. The stress redeveloped in a broken wire is given by ( )2950 sdid .Cps µ⋅⋅⋅= , ssd ≤ (5.3.4-1) where sd = redeveloped stress in the evaluated panel C5.3.4 The stress in a broken wire that can be redeveloped is the number of cable bands between the break and the panel being evaluated multiplied by the stress redeveloped at each band. This redeveloped stress is, however, limited to the stress in the unbroken wires that corresponds to the cable strength calculated by the equations in Article 5.3.3. Generally, the stress in the unbroken wires will be more than 90% of the mean tensile strength of the Group 2 wires and the limitation can be disregarded, because the redeveloped stress with Le = 9 is not greater than 76% of the mean tensile strength of Group 2.

GUIDELINES COMMENTARY 5-19 for a broken wire in an adjacent panel pi = number of cable bands between the evaluated panel and a wire break in panel i i = number of a panel Cd = the fraction of 95% of the mean tensile strength of Group 2 wires that is developed in a broken wire at each cable band µs2 = sample mean tensile strength of Group 2 wires s = stress in the wires corresponding to the estimated cable strength calculated in Article 5.3.3 The contribution of the broken wires to the cable strength is given by ( ) dribi Le i iswb CnnpaR ⋅−⋅⋅⋅⋅ = Σ = )(95.0 2 2µ (5.3.4-2) where Rb = cable strength attributable to broken wires in adjacent panels aw = nominal area of one wire used in lab analysis Le = number of panels in effective development length nbi = number of broken wires in panel i nri = number of broken wires in panel i that are repaired If the evaluated panel is the only panel that has been inspected, then the contribution of the broken wires to the cable strength can be taken as ( ) ( )15.095.0 12 −⋅⋅⋅⋅⋅ = ebswb LnaR µ (5.3.4-3) where nb1 = number of broken wires in evaluated panel 5.3.5 Cable Strength The strength of the cable using either the Brittle-Wire Model or the Limited Ductility Model in the evaluated panel is given by

GUIDELINES COMMENTARY 5-20 bu RRR += (5.3.5-1) R = estimated cable strength Ru = cable strength attributable to unbroken wires Rb = cable strength attributable to broken wires in adjacent panels

5-21 5.4 FIGURES FOR SECTION 5 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 Fraction of wires in Group k that contain cracks, pc,k Fr ac tio n o f d is cr e te w ire s in G ro up k th a t a re c ra ck ed in e ffe c tiv e de ve lo pm en t l en gt h, N c, k/N 0k 5 9 7 3 panels = effective development length Figure 5.3.2.4.1-1. Fraction of discrete cracked wires in Stage k.

5-22 0.00 0.10 0.20 0.30 0.40 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 Percent of wires that contain cracks, pc,k Fr a c tio n o f c ra c ke d w ire s th at h av e fa ile d in e ff de v lg th th a t a re r e de ve lo pe d in P an el 1 , N cr ,k /(N 0k ) 5 9 7 3 panels = effective development length Figure 5.3.2.4.2-1. Fraction of cracked wires in Stage k that are redeveloped.

5-23 Figure 5.3.2.6-1. Summary of calculations for number of wires in each group. Type of wires Eq 5.3.2.6-1

5-24 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Coefficient of Variation (standard deviation / mean strength) K Figure 5.3.3.1.2-1. Strength reduction factor, K.