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

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A-1 CONTENTS APPENDIX A MODELS FOR ESTIMATING CABLE STRENGTH ...........................................................A-1 A.1 INTRODUCTION ......................................................................................................................................A-2 A.2 NOTATION ................................................................................................................................................A-2 A.3 STRENGTH MODELS..............................................................................................................................A-4 A.3.1 Behavior of a Single Bridge Wire .....................................................................................................A-4 A.3.2 Strength Models for Wire Bundles...................................................................................................A-5 A.3.2.1 DUCTILE-WIRE MODEL .............................................................................................................A-5 A.3.2.2 BRITTLE-WIRE MODELS............................................................................................................A-5 A.3.2.2.1 Limited Ductility Model ...........................................................................................................A-6 A.3.2.2.2 Brittle-Wire Model (A Special Case) .......................................................................................A-6 A.3.2.3 BRITTLE-DUCTILE MODEL .......................................................................................................A-6 A.3.2.4 SIMPLIFIED MODEL....................................................................................................................A-6 A.4 STATISTICS FOR CABLE STRENGTH ANALYSIS ..........................................................................A-7 A.4.1 Extreme Value Distributions ............................................................................................................A-8 A.4.2 Weibull Distribution..........................................................................................................................A-8 A.5 A.5.1 Limited Ductility Model Equations................................................................................................A-10 A.5.1.1 DIFFERENT STRESS-STRAIN CURVES AVAILABLE FOR WIRES....................................A-11 A.5.1.2 ALL GROUPS OF WIRES HAVE THE SAME STRESS-STRAIN CURVE.............................A-12 A.5.2 Brittle-Wire Model Equations ........................................................................................................A-12 A.5.3 Simplified Model Equations............................................................................................................A-13 A.5.3.1 SINGLE DISTRIBUTION CURVE FOR TENSILE STRENGTH..............................................A-13 A.5.3.2 CABLE STRENGTH USING THE SIMPLIFIED MODEL ........................................................A-14 A.6 REFERENCES ..................................................................................................................... ................................................................................................................... ....................A-14 EQUATIONS FOR ESTIMATING CABLE STRENGTH USING THE WEIBULL DISTRIBUITON A-10

A-2 A.1 INTRODUCTION A parallel wire cable on a suspension bridge can be defined as a bundle of parallel structural elements in tension. A statistical approach to the problem of estimating the strength of a parallel element system is described in Rao [1] He presents two models based on different assumptions, either that the elements are ductile (they continue to carry load after reaching maximum capacity) or that they are brittle (they break when they reach maximum capacity). Engineers currently depend on several variations of these models to estimate the strength of the unbroken wires in a cable that has experienced deterioration. In practice, the models that include all the wires in a cable are more complex than Rao’s models would suggest. The series of tasks required for a proper assessment includes: • developing stress-strain diagrams for the wires • finding the minimum strength of a given length of wire • calculating distribution functions to describe how the wires vary • determining the effectiveness of the cable bands in redeveloping the strength of a wire broken at some distance from the point at which the strength is being determined • estimating the effect on the cable of deterioration in panels near the one being evaluated • estimating the strength of the cable in a given panel based on this data The following pages describe the basic models found in the literature for parallel systems and their application to estimating cable strength. A.2 NOTATION aw = nominal area of one wire, used in lab analysis (A.5.1.1) (A.5.1.2) (A.5.2) (A.5.3.2) e = specific value of strain (A.5.1.1) exp(x) = e (2.7183) to the power (x), the “exponential” of (x) (A.4.2) F3k(e) = Weibull cumulative distribution for ultimate strain of Group k wires (A.5.1.1) (A.5.1.2) F3k(s) = Weibull cumulative distribution for tensile strength of Group k wires (A.5.2.) f3X1(x) = Type III extreme value probability density distribution for the smallest values of random variable X (A.4.2.) F3X1(x) = Weibull extreme value cumulative distribution for the smallest values of random variable X (A.4.2) F3(s) = single Weibull distribution of the tensile strength representing all of Group 2 to 4 wires (without cracked wires), based on the sample mean and sample standard deviation of the combined groups.(A.5.3.2) F3(s ′) = single Weibull distribution of the tensile strength representing all the Group 2 to 4 wires, based on a mean tensile strength of 1.0 and a standard deviation of the combined groups, divided by µs(A.5.3.2) FC(e) = compound cumulative distribution of ultimate strain (A.5.1.2) FC(s) = compound cumulative distribution of tensile strength (A.5.2) k = corrosion stage of a group of wires (k = 2, 3, 4 and 5) (A.5.1.1) k = corrosion stage of a group of wires (k = 2, 3, and 4) (A.5.3.1) The text includes the results of tensile strength tests on corroded wires that were removed from three bridges, a description of the behavior of a single bridge wire, and the statistical equations required for evaluating the models. An additional source of information is the simulated ten-wire cable demonstration calculations that appear in the final report for NCHRP Project 10-57, on the accompanying CD.

A-3 km = subgroup of Stage k wires that follow stress-strain curve m (A.5.1.1) K = reduction factor from Chart 5.2.3.3.2-1 as a function of the coefficient of variation, σ s/µs. (A.5.3.2) max = maximum value of the expression inside the brackets (A.5.1.1) (A.5.1.2) (A.5.3.2) m = parameter of the Type III extreme value distribution for minimum values (A.4.2) m = number of a discrete stress-strain curve (A.5.1.1) M = number of different stress-strain curves considered (A.5.1.1) Neff = effective number of unbroken wires in the cable (A.5.1.1) (A.5.1.2) (A.5.2) Neff = effective number of unbroken wires and uncracked wires in the cable (A.5.3.2) pk = fraction of unbroken wires represented by Group k wires (A.5.1.2) (A.5.2) pk = fraction of unbroken wires and uncracked wires in the cable represented by Group k wires (A.5.3.1) pkm = fraction of cable represented by Subgroup km wires (A.5.1.1) Ru = cable strength attributable wires in the cable that are not broken (A.5.1.1) (A.5.1.2) (A.5.2) (A.5.3.2) s = stress in the unbroken wires of the cable (A.5.2) (A.5.3.2) s1 = s/µs (A.5.3.2) s(e) = stress in wires determined from average stress-strain curve for all wires (A.5.1.2) Sk(e) = survivor function, or that fraction of the wires in Group k that has an ultimate elongation greater than e (A.5.1.1) sm(e) = stress at strain e in wires that follow stress-strain curve m (A.5.1.1) Tkm = force in Subgroup km wires (A.5.1.1) Tu(s) = tensile force in unbroken wires in the cable at stress s (A.5.2) x = value of random variable X or X1 (A.4.2) x0 = minimum value of x for which the Type III distribution is valid (A..4.2) X1 = random variable representing the smallest values of property x of each wire (A.4.2) Γ( ) = Gamma function of the expression inside the brackets (A.4.2) µs = sample mean tensile strength of the combined groups of wires, excluding cracked wires (A.5.3.1) (A.5.3.2) µsk = sample mean tensile strength of Group k wires (A.5.3.1) µsx = sample mean of property x (A.2.3.1) (A.2.3.2) µsX1 = sample mean of the extreme value distribution of x (A.4.2) = parameter of the Type III extreme value distribution for minimum values (A.4.2) σ σ υ s = sample standard deviation of the tensile strength of the combined groups of wires, excluding cracked wires (A.5.2.1) sk = sample standard deviation of tensile strength of Group k wires (A.5.3.1) sx = sample standard deviation of property x (A.2.3.1) (A.2.3.2) sX1 = sample standard deviation of the extreme value distribution of x, (A.4.2) σ σ

A-4 A.3 STRENGTH MODELS A.3.1 Behavior of a Single Bridge Wire The modern method of manufacturing bridge wire is to cold draw a carbon steel rod through successively smaller dies until the specified diameter and tensile strength are reached. The process imparts strength to the wire, along with an elongated grain structure. Any bridge wire subjected to a tensile test stretches elastically to the proportional limit. It doesn’t exhibit a true yield point as the strain increases, but enters a strain-hardening range instead, immediately after the transition from elastic behavior. The stress continues to increase with the strain until the tensile strength is reached, at which point the wire necks down and fails, resulting in a reduction of area and a cup-and-cone fracture surface. There is no yield plateau as with milder steel materials. The strain at the tensile strength is the ultimate strain. Failure occurs almost immediately after the tensile strength and the ultimate strain are reached. Figure A.3.1-1. Typical stress-strain curve for bridge wire. [2] New wires have a tensile strength that varies little (the coefficient of variation is about 1.5% to 2%). Corroded wires that exhibit ductile fractures have a more variable tensile strength than new wires. Wires that are cracked to any degree 0 50 100 150 200 250 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 STRAIN (in/in) ST RE SS (k si) AVERAGE STRESS-STRAIN CURVE FOR 126 WIRES DATA FROM BEAR MNTN BRIDGE WIRE TESTS [ROEBLING, 1925] TENSILE STRENGTH 229 ksi ULTIMATE STRAIN 6.13 % NET DIAMETER OF WIRE 0.191 in NET STEEL AREA 0.02865 sq in STRESS BASED ON NET STEEL AREA OF WIRE PROPORTIONAL LIMIT TENSILE STRENGTH ULTIMATE STRAIN YIELD STRENGTH (0.2% OFFSET) The typical stress-strain curve for new bridge wire is shown in Figure A.3.1-1. The data were taken from Roebling [2] and represent the average results of tests on 126 wires from the Bear Mountain Bridge. Failure occurs almost immediately after the ultimate strain is reached. New, corroded and cracked wires all follow the same curve.

A-5 Table A.3.1-1 Mean tensile strength and coefficient of variation for wires from several suspension bridges Bridge Corrosion Stage Mean Tensile Strength of New Wire (ksi) Number of Samples Mean Tensile Strength (Fraction of New Wire) Coefficient of Variation New Wire 240 2 1.00 0.004 1-2 30 0.99 0.018 3 18 0.98 0.024 4 10 0.94 0.038 X Cracked 14 0.84 0.131 New Wire 231 0 1.00 N/A 1-2 19 1.00 0.089 3 12 0.97 0.119 4 7 0.89 0.085 Y Cracked 15 0.68 0.210 New Wire 236 20 1.00 0.020 1-2 29 0.98 0.031 3 29 0.95 0.038 4 33 0.94 0.041 Z Cracked 7 0.75 0.260 A.3.2 Strength Models for Wire Bundles Perry [3] describes 3 models that can be used to estimate the strength of a parallel wire cable. They are presented below, starting with the least conservative. One is for ductile wires, another is for brittle wires based on strain and the third, a subset of the second, is for brittle wires based on tensile strength. In studies of cables, which are bundles of filaments or wires, it is assumed that all the filaments are held together firmly at the ends and subjected to equal strain. Internal cable inspections of the Williamsburg Bridge [4] and Mid- Hudson Bridge [5] can be cited as evidence that the gap in a broken cable wire is equal to or slightly greater than the elastic elongation of the wire between cable bands under the dead load tension of the cable. The conclusion is that all the wires are effectively clamped at the cable bands, at least for normal working stresses. A.3.2.1 DUCTILE-WIRE MODEL In the Ductile-Wire Model, it is assumed that all of the wires in the cable share in the cable force until the entire cable breaks as a single unit. In other words, all of the wires are assumed to break simultaneously. For this to happen, the wires need not have equal strength, but they must be ductile (able to stretch under a constant load). Each wire elongates elastically and then plastically to the same degree as all of the other wires. The cable strength is the sum of the individual wire strengths, which is equal to the average wire strength multiplied by the number of wires in the cable. This model should not be applied to bridge cables, because there are always some cable wires that break before the entire cable does, and the efficiency of a cable of parallel wires has been found to vary between 94% and 96% [2, 6]. Efficiency is defined in these references as the actual breaking strength divided by the calculated strength of the Ductile-Wire Model. A.3.2.2 BRITTLE-WIRE MODELS have reduced and highly variable tensile strength. They exhibit brittle fracture surfaces with no reduction in area, often referred to as “square breaks.” The mean tensile strength and the coefficient of variation of wires removed from several suspension bridges are shown in Table A.3.1-1 for various stages of corrosion. Tensile strength in this table is based on the net steel area (the area not including the zinc coating).

A-6 The term brittle in the context of the Brittle-Wire Model is used to describe the behavior of an individual wire. It does not mean that the material in the wire is brittle, but that the wire fails suddenly when the strain or the stress in the wire reaches a certain level and no longer shares in the tensile force of the cable. Perry [3]describes two separate models under this name. The first is a general model that is referred to as the Limited Ductility Model in these Guidelines. The second is a specific case of the Limited Ductility Model, which is used by Perry and others to calculate cable strength under the general name, Brittle-Wire Model. It is also called the Brittle-Wire Model in these Guidelines. A.3.2.2.1 Limited Ductility Model In the Limited Ductility Model, a wire is assumed to fail suddenly when the strain in the wire reaches a certain level. Each wire in the cable can elongate only to its individual limit, which is called the ultimate strain of the wire. A specific wire that reaches this elongation will fail by definition. The limit is different for each wire, and so is the strength. The small amount of strain that occurs at reduced stress after the tensile strength is reached, as shown in Figure A 3.1-1, is ignored in the analysis. For any specific value of strain, it is assumed that each intact wire is subject to a tensile stress that corresponds to the strain in the stress-strain diagram for that wire. Whenever a wire breaks, the force carried by the wire is distributed to all of the unbroken wires in the cable in the same proportion as before the wire failed. The wires are assumed to break sequentially as each wire reaches its maximum elongation, and the cable strength is attained only after some of the wires break. To determine the cable strength, the cable strain is increased incrementally and then the number of wires that have reached their elongation limit and failed are calculated at each increment. For each subsequent calculation, the number of newly failed wires is subtracted from the number of still intact wires to determine the number of unbroken wires that remain. All of the wires in the cable are subjected to the same strain. The cable force is calculated as the sum of the forces in the intact wires at that strain. Wires will fail faster than the cable force can increase at some value of elongation. The maximum force attained is the cable strength. This technique uses a statistical method called “ordered statistics.” The strength can be estimated either by sorting the wires in order of ultimate elongation or by using the statistical distribution curve of this property. A.3.2.2.2 Brittle-Wire Model (A Special Case) In a special case of the Limited Ductility Model, it is assumed that all of the wires follow the same stress-strain diagram, and that the stress in all of the intact wires is the same at any specific value of strain. The model may be simplified by assuming that a wire fails suddenly when the stress in the wire reaches a certain level. In the Brittle-Wire Model that results, each wire in the cable can resist a stress only up to its specific limit, equivalent to its tensile strength. A specific wire that reaches this stress will fail by definition. The limit is different for each intact wire, which is assumed to carry the same tension as all of the other wires; hence, the model is also called the Load-Sharing Model. After a wire breaks, the force carried by the wire is distributed equally to all of the other unbroken wires in the cable. The wires are assumed to break sequentially as each wire reaches its tensile strength, and the cable strength is attained only after some of the wires have broken. Determining the cable strength requires increasing the cable stress in steps and calculating the number of wires that fail at each increment as they reach their tensile strength. The number of newly failed wires is subtracted from the number of previously intact wires to determine the number of unbroken wires. The cable force is calculated as the area of unbroken wires at a given level of stress multiplied by the wire stress. At some level of stress, wires will fail faster than the wire force can increase. The maximum force attained is the cable strength. This technique uses the same “ordered statistics” method as the Limited Ductility Model. The strength is estimated either by sorting the wires in order of tensile strength, or by using a statistical distribution curve of this property. A.3.2.3 BRITTLE-DUCTILE MODEL The Brittle-Ductile Model [7] takes into account the cable wires that fail at very low strains, which are subtracted from the total number of wires in the cable. The model assumes that the remaining wires in the cable are ductile. The strength calculation is then the same as for the Ductile Model, except that the number of wires is reduced. A.3.2.4 SIMPLIFIED MODEL

A-7 In a variation of the Brittle-Ductile Model, cracked and broken wires are subtracted from the total wires in the cable, and the Brittle-Wire Model is applied to the remaining intact wires. This alternative to the Brittle-Wire Model is called the Simplified Model in these Guidelines. A.4 STATISTICS FOR CABLE STRENGTH ANALYSIS Sample wires are removed from the cables and tested to provide the data required for estimation of cable strength. In the calculation of cable strength using sample statistics, data are used to develop statistical distribution curves that represent the spectrum of wire strengths or ultimate strains present in the cable. The curves are described below, before presenting their use in estimating cable strength. The equations that follow employ the results of physical tests performed on a selection of wires removed from an actual cable. The mean and standard deviations resulting from these tests are called sample means and sample standard deviations to differentiate them from the population means and population standard deviations that represent all of the wires in the cables. The terms µ (mean) and σ (standard deviation) apply to the entire population, while µs and σs apply to the samples. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 180 200 220 240 260 280 TENSILE STRENGTH, KSI f(s) F(s) = Fraction of cable weaker (1-F(s)) =Fraction of cable stronger FR A CT IO N O F CA BL E FR A CT IO N O F CA BL E Figure A.4-1. Typical probability density distribution curve 0.0 0.2 0.4 0.6 0.8 1.0 180 200 220 240 260 280 TENSILE STRENGTH, KSI FR A CT IO N O F CA BL E W EA K E R F(s) = Fraction of cable weaker (1-F(s)) =Fraction of cable stronger FR A CT IO N O F CA BL E W EA K E R Figure A.4-2. Typical cumulative probability curve Two types of distribution curves are discussed in the following text. The first is the probability density distribution curve shown in Figure A.4-1. This curve describes the fraction of the entire population that has a specific value of x. It is commonly known as the “Bell Curve” for a distribution that is Normal or Gaussian. The area under the curve is always unity, describing 100% of the population. The area under the curve to the left of a specific value of x is the fraction of th population that has a value less than or equal to that value of x. When this area is plotted against x, it becomes the cumulative probability curve (Figure A.4-2), and varies between 0 and 1. The area to the right of a specific value of x in Figure A.4-1 is the fraction of the population that has a value greater than the value of x.

A-8 The first of the distribution curves is represented by the expression f(x), while the second is represented by F(x). In these Guidelines, the expressions f3(x) and F3(x) are used to denote the recommended Weibull distribution. The value that is derived from these distributions is the fraction of the population that has a value of x (i.e., tensile strength or ultimate strain) greater than a specific value of x. It is also known as the Survivor Function and is given as (1-F (x)). It is equivalent to the area to the right of x on the probability density distribution. The term s is substituted for x for the distribution of tensile strength in these Guidelines, and e is substituted for x for ultimate strain. A.4.1 Extreme Value Distributions The distribution of the minimum tensile strengths of the wires must be obtained to determine the strength of the cable in any panel of length L. The extreme value distributions are very useful for this purpose. In the discussions and equations that follow, the general variable, X, represents either the tensile strength or the ultimate elongation in the strength models described previously. The derivation of the extreme value distributions is given in various references [1, 7-9] and will not be repeated here. The Type I and Type III extreme value distributions are useful in describing the minimum strength of wires. Type I (also known as the Gumbel) extends from -∞ to +∞, as does the Normal distribution. Type III, extending from a minimum value of x0 to a maximum of +∞, corresponds to a material with a lower limit of tensile strength or ultimate elongation. Rao [1]states that the parent function of this extreme value distribution is a Gamma distribution. The Gamma distribution is valued for x > x0, as is the Weibull distribution, and for very small standard deviations is virtually the same as a Normal distribution. The value of x0 is assumed to be zero, because the tensile strength and ultimate elongation of materials cannot be less than zero. The Type III extreme value distribution is a Weibull distribution, which is discussed in many references, among them Rao and Weibull. [1, 9] The parameters of the distribution are not implicit, but it is relatively simple to find a solution through trial-and-error using a spreadsheet. An alternative method is to use Weibull graph paper. [1] Perry [3]argues that the Weibull distribution is the only correct distribution in this instance. The Weibull distribution is used in these Guidelines because it allows limiting the minimum value of the variable, which is of particular importance for the Limited Ductility Model. Ultimate strains less than zero could result from the use of other distribution functions. A.4.2 Weibull Distribution The Type III extreme value distribution is the Weibull distribution. As stated above, it extends from a minimum value, x0, to +∞. This distribution is given in Rao [1] as ( ) − − − −= m X x xx xF 0 0exp13 1 υ (A.4.2-1) − − −⋅ − − ⋅ − = mm X x xx x xx x m xf 0 0 1 0 0 0 exp)(3 1 υυυ (A.4.2-2) where f3X1(x) = Type III extreme value probability density distribution for the smallest values of random variable X F3X1(x) = Weibull extreme value cumulative distribution for the smallest values of random variable X exp(x) = e (2.7183) to the power (x), the “exponential” of (x) x = a value of random variable X1 ¯

A-9 x0 = minimum value of x for which the Type III distribution is valid. It is taken as 0 in these Guidelines. X1 = random variable representing the smallest values of property x of each wire m = parameter of the Type III extreme value distribution for minimum values υ = parameter of the Type III extreme value distribution for minimum values The function F3X1(x) is the cumulative probability distribution. Its value at any value of x is the probable fraction of the entire population of wires for which the property represented by the general variable X1 is smaller than the specific value x. The probability density distribution f3X1(x) represents the fraction of the population for which one unit of property represented by X1 is equal to x. The unit of x is the basic unit of the property, e.g., ksi or inches/inch. The cumulative distribution is the integral of the density distribution between x0 and x. In these equations and the ones that follow, the term X1 represents the smallest values of X in a specific length of wire determined from tests on specimens cut from sample wires. The mode, mean and variance are given as ( ) m m m xx 1 00 1 mode − ⋅− += υ (A.4.2-3) ( ) + ⋅Γ− +== m xxX 11mean 001 υµ (A.4.2-4) ( ) +Γ − +Γ⋅ − == mm xX 1121variance 220 2 1 υσ (A.4.2-5) In these equations, x0 is the lower limit of the tensile strength or ultimate strain of the wire (usually taken as zero) and υ and m are parameters of the distribution. If the sample mean and standard deviation for the smallest values of x for each wire sample are given, µX1 becomes µsX1 and σ σX1 becomes sX1. ( )20122 111121 xmm sXsX − =− +Γ +Γ µσ σ (A.4.2-6) where µsX1 = sample mean of the extreme value distribution of X1 sX1 = sample standard deviation of the extreme value distribution of X1 Γ( ) = Gamma function of the expression inside the brackets The Weibull function is one of the functions included in the Microsoft Excel spreadsheet program, which makes it relatively simple to use once the parameters have been determined. The parameters and m are called beta and alpha, υ Eliminating the variable (υ - x0) in Equation 4 and Equation 5 results in the following equation, given by Castillo [10] that can be solved for m by successive approximation, illustrated in Figure A.4.2-1 (use the“Solver”function of a spreadsheet program such as Microsoft Excel for this purpose)

A-10 respectively, in Excel. The Weibull function in Excel uses x0 = 0. If another value is used, (x – x0) must be substituted for x, and (υ - x0) substituted for beta, in the Excel function. The recommended value of x0 in these Guidelines is zero. It can be argued that this value should be the dead load stress in the cable because the wires have been “tested” in service at this stress and the tensile strength cannot be smaller, but comparative calculations using the dead load stress for x0 in one case and zero in the other show that the difference in cable strength is very small, and Equations 1 to 6 can be simplified by omitting the term x0. STRENGTH TYPE 3 EXTREME VALUE DISTRIBUTION FOR MINIMUM VALUES MODELS WEIBULL DISTRIBUTION CALCULATION OF PARAMETERS FROM MEAN AND STANDARD DEVIATION WEIBULL USING EQUATIONS A.4.2-4 and A.4.2-6 PARAMETERS (ksi) THE MEAN AND STANDARD DISTRIBUTION OF THE TENSILE STRENGTH OF EACH GROUP OF WIRES DETERMINED FROM THE LABORATORY TESTS ARE USED TO DETERMINE THE PARAMETERS OF THE WEIBULL DISTRIBUTIONS. THE METHOD PRESENTED IN ARTICLE A.4.2 IS USED BELOW. THE VALUE OF x 0 IS TAKEN AS ZERO AND THIS TERM IS OMITTED IN THE EQUATIONS SHOWN IN THE CALCULATION BELOW Excel k = corrosion group 2 3 4 5, CRACKED parameter TENSILE STRENGTH DISTRIBUTION FOR EACH CORROSION GROUP mean tensile strength, µ s ksi 239.0 235.9 231.1 200.5 standard deviation, σ s ksi 4.3 5.7 8.7 26.3 CALCULATION OF WEIBULL PARAMETERS Eq. A.4.2-6: (1+2/m )/ 2(1+1/m ) = 1 - 2/ 2 m (assumed, then determined by solver) 70.6 52.4 33.4 9.1 alpha (Γ = GAMMA function) Γ(1+2/m ) 0.9844 0.9793 0.9688 0.9133 Γ(1+1/m ) 0.9920 0.9893 0.9836 0.9475 Γ (1+2/m )/Γ2(1+1/m ) 1.0003 1.0006 1.0014 1.0172 σ 2 18.490 32.490 75.690 691.690 µ 2 57121 55649 53407 40200 σ 2/µ 2 3.2E-04 5.8E-04 1.4E-03 1.7E-02 SOLVE FOR m USING SOLVER: Equation A.4.2-6 is solved for m by making the value of the expression Γ (1+2/m )/Γ2(1+1/m ) - 1 -σ2/µ2 equal to zero by varying m , using the "Solver" routine in Excel: (1+2/m )/ 2(1+1/m ) - 1 - 2/ 2 = 0 -4.7E-10 -8.1E-10 -4.9E-10 2.6E-10 CALCULATE : The value of υ υ is found by solving Equation A.4.2-4 for this variable and substituting the value of m found above: σ σ σ = µ µ µ /Γ Γ Γ Γ (1+1/m ) 240.9 238.4 235.0 211.6 beta Corrosion Group Figure A.4.2-1. Calculation of parameters of Weibull distribution by iteration. A.5 EQUATIONS FOR ESTIMATING CABLE STRENGTH USING THE WEIBULL DISTRIBUITON For both the Limited Ductility and Brittle-Wire Models, the Weibull distribution curve is used to estimate cable strength. The use of the expression (1-F3(x)) is equivalent to making the force equal to zero in wires that break as strain or stress is increased. A.5.1 Limited Ductility Model Equations Γ

A-11 A.5.1.1 DIFFERENT STRESS-STRAIN CURVES AVAILABLE FOR WIRES The separation of test samples into these subgroups would necessitate the removal of many more sample wires from the cable, and is not recommended. Thus, all wires in a specific group are assumed to follow the same distribution of the ultimate strain. At any value of strain, the fraction of unbroken wires in each subgroup of wires is represented by the survivor function, ( ) ( )( )eFeS kk 31−= (A.5.1.1-1) where Sk(e) = survivor function; or that fraction of the wires in Group k that has an ultimate elongation greater than e e = specific value of strain F3k(e) = Weibull cumulative distribution function for ultimate strain of Group k wires k = corrosion stage of a group of wires (k = 2, 3, 4 and 5) The total force in the wires of Subgroup km at strain e is ( )( )eFespaNT kmkmweffkm 31)( −⋅⋅⋅⋅ = (A.5.1.1-2) where Tkm = force in Subgroup km wires Neff = effective number of unbroken wires in the cable aw = nominal area of one wire, used in lab analysis pkm = fraction of cable represented by Subgroup km wires sm(e) = stress at strain e in wires that follow stress-strain curve m m = number of a discrete stress-strain curve km = subgroup of Stage k wires that follow stress-strain curve m The total cable force is the sum of the forces in the subgroups of wires, and the cable strength is the maximum force attained ( ) ( )( ) − ⋅⋅ ⋅⋅ = Σ Σ = = M m k kmkmweffu eFespaNR 1 5 2 31max (A.5.1.1-3)     The general form of the Limited Ductility Model is used whenever the stress-strain curves for wires vary significantly. This can occur when the carbon content of the wires varies from wire to wire because of poor manufacturing quality control or multiple suppliers. The stress-strain curves should be reduced to a limited number, M, of average curves, and each group of wires subdivided and assigned to the various M curves proportionally. This requires a map of the cable cross-section in each evaluated panel showing the distribution for each stress-strain curve. Additional samples must be taken to obtain the data to prepare these maps; a sampling pattern is suggested in the final report for NCHRP Project 10-57, on the accompanying CD. These additional samples may also be added to the random samples for the purpose of determining the tensile properties of the wires. It is recommended that a statistician be added to the investigation team when this procedure is followed.

A-12 where Ru = cable strength attributable to wires in the cable that are not broken M = number of different stress-strain curves considered This strength model requires M times as many calculations as when only a single average stress curve can be used for all wires, as in the following article. A.5.1.2 ALL GROUPS OF WIRES HAVE THE SAME STRESS-STRAIN CURVE Where all groups of wires have the same stress-strain curve, the term sm(e) becomes the same for all groups at any specific strain, e. The individual cumulative distributions may be combined into a single compound distribution ( not a Weibull distribution) and Equation A.5.1.1-3 simplifies to ( )( )( )eFesaNR Cweffu − ⋅⋅⋅= 1)(max (A.5.1.2-1) in which )(3)( 5 2 eFpeF k k kC ⋅= Σ = (A.5.1.2-2) where FC(e) = compound cumulative distribution of the ultimate strain pk = fraction of unbroken wires represented by Group k wires s(e) = stress in wires determined from average stress-strain curve for all wires A.5.2 Brittle-Wire Model Equations The compound tensile strength distribution is used in the Brittle-Wire Model (again, not a Weibull distribution). A single average stress-strain curve is used to represent the entire cable. The fraction of the cable that is essentially unbroken at any specific stress level s is given by the expression. ( )( )sFC− 1 (A.5.2-1) in which ( ) ( )sFpsF k k kC 3 5 2 ⋅= Σ = (A.5.2-2) where FC(s) = compound cumulative distribution of tensile strength

A-13 F3k(s) = Weibull cumulative distribution function for tensile stress of Group k wires s = stress in the unbroken wires of the cable The force in the cable is ( ) ( )( )[ ]sFsaNsT Cweffu −⋅⋅⋅= 1 (A.5.2-3) where Tu(s) = tensile force in unbroken wires in the cable at stress s and the cable strength is the maximum value that Tu(s) attains, ( )( )( )sFsmaxaNR Cweffu −⋅⋅⋅= 1 (A.5.2-4) This equation can be solved by trial-and-error using “Solver” in the Microsoft Excel spreadsheet program, as shown in Appendix C. If a cable force vs. strain diagram is required, the cable force must be calculated at selected increments of strain. A.5.3 Simplified Model Equations Using the simplified model requires: • Calculation of the effective number of wires, obtained by subtracting the unrepaired broken wires plus the estimated number of wires in Stages 3 and Stage 4 that contain cracks from the total number of wires in the cable (there will be no Group 5, because cracked wires are omitted from this calculation) • Calculation of the combined mean and standard deviation of the tensile strengths of the remaining wires that comprise Groups 2, 3, and 4, using Equation A-5.3.1-1 and Equation A-5.3.1-2) • Applying the Brittle-Wire Model to the wires, using the single distribution curve A.5.3.1 SINGLE DISTRIBUTION CURVE FOR TENSILE STRENGTH The fraction of the cable represented by Groups 2, 3 and 4 is combined with the sample mean and standard deviation values for the minimum tensile strength of the representative specimens of each group. The result is used to determine the sample mean tensile strength and standard deviation of the entire unbroken and uncracked wire population as follows: Σ = ⋅= 4 2 )( k skks p µµ (A.5.3.1-1) ( ) 2224 2 ssksk k ks p µµσσ − += Σ = (A.5.3.1-2) 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, excluding cracked wires       

A-14 σsk = sample standard deviation of the tensile strength of Group k wires pk = fraction of the unbroken and uncracked wires in the cable represented by Group k wires k = corrosion stage of a group of wires (k = 2, 3 and 4) A.5.3.2 CABLE STRENGTH USING THE SIMPLIFIED MODEL The estimated cable strength calculated according to the Brittle-Wire Model is given by Equation A-5.2-4. In the Simplified Model, which uses a single tensile strength distribution, the compound distribution FC(s) is replaced by the single Weibull distribution F3(s), resulting in ( )( )( )sFsmaxaNR weffu 31− ⋅⋅⋅= (A.5.3.2-1) where F3(s) = single Weibull distribution of the tensile strength representing all of the Group 2 to 4 wires (without cracked wires), based on the sample mean and sample standard deviation of the combined groups, calculated using equations A.5.3.1-1 and A.5.3.1-2 s = stress in the unbroken wires of the cable The sample mean tensile strength µs can be factored out of the expression in brackets in Equation A-5.3.2-2 to result in ( )( )( )sFsmaxaNR sweffu ′−⋅′⋅⋅⋅= 31µ (A.5.3.2-2) in which s' = s/µs F3(s') = single Weibull distribution of the tensile strength representing all the Group 2 to 4 wires, based on a mean tensile strength of 1.0 and a standard deviation of the combined groups, calculated using Equations A.5.3.1-1 and A.5.3.1-2, divided by µs Referring to Equation A.4.2-1, F3x1(x) does not change if the terms x, x0 and on the right-hand side of the equation are all divided by µs. The value of the distribution functions F3(s) and F3(s′) are identical for any specific value of s, if the distribution F3(s′) is based on µs′ = 1 and s′ = s/µσσ s. The value of the term K = max(s'·(1 - F3(s′))) can be determined as a function of the coefficient of variation, s/µ υ s. The calculation results in the curve of K vs. s/µσσ s shown in Figure 5.3.3.1.2-1, where K is the reduction factor to be applied in the equation KaNR sweffu ⋅⋅⋅= µ (A.5.3.2-3) where K = reduction factor from Figure 5.3.3.1.2-1 as a function of the coefficient of variation, σs/µs. A.6 REFERENCES 1. Rao, S.S., Reliability-Based Design. 1st ed, ed. R. Hauserman. 1992: McGraw-Hill, Inc. 569. 2. Roebling, J.A., Parallel Wire Cables for Suspension Bridges. 1925, John A. Roebling's Sons Company: Trenton, New Jersey. 3. Perry, R.J., Estimating Strength of Williamsburg Bridge Suspension Cables. The American Statistician, 1998. 52(3): p. No. 3.

A-15 4. Steinman et al., Williamsburg Bridge Cable Investigation Program: Final Report. 1988, New York State Department of Transportation & New York City Department of Transportation: New York, NY. 5. Modjeski and Masters, Mid-Hudson Bridge - Main Cable Investigation (Phase III). 1991, Modjeski and Masters, Inc.: Harrisburg, Pennsylvania. 6. Board of Engineers, The Bridge Over the Delaware River Connecting Philadelphia, PA and Camden, N.J.. 1927, Delaware River Bridge Joint Commission of the State of PA and NJ. 7. Matteo, J., G. Deodatis, and D.P. Billington, Safety Analysis of Suspension-Bridge Cables: Williamsburg Bridge. Journal of Structural Engineering, 1994. 120(11): p. 3197-3211. 8. Gumbel, E.J. and J. Lieblein, Statistical Theory of Extreme Values and Some Practical Applications. United State Department of Commerce - National Bureau of Standards, 1954. Series 33: p. 1 -51. 9. Weibull, W., A Statistical Distribution Function of Wide Applicability. Journal of Applied Mechanics, 1951. 73(18): p. 293-297. 10. Castillo, E., Extreme Value Theory in Engineering. 1988, San Diego, CA: Academic Press, Inc.