Risk-Based Inspection and Strength Evaluation of Suspension Bridge Main Cable Systems (2023)

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53 C H A P T E R 4 Development of the Guidelines- Probabilistic Approach for Strength Evaluation As an improvement on the methods used in NCHRP Report 534, the Random Field Method is adopted to calculate the strength of a suspension bridge main cable. In this section, the method is described in detail, and compared with those used in NCHRP Report 534. It should be noted that the Random Field Method replaces the multiple methods contained in NCHRP Report 534. Estimation of Strength Discussion of the Random Field Method The process of accounting for the variation of wire strength along the length of a wire within a panel is the main difference between the proposed Random Field methodology and the current NCHRP NCHRP Report 534. With the current NCHRP 534 methodology: • Each wire is inspected at multiple locations along the length of the panel and assigned a corrosion stage based on the observed conditions; the wire is then reassigned the highest stage (i.e. worst corrosion) recorded in any of the panel segments. • The tensile strength of each test specimen is used to determine the minimum tensile strength of a wire sample. • Tensile strengths of specimens are grouped according to the corrosion stage of the wire sample (Groups 1 through 4), and “cracked” wires (i.e. wires determined to have pre- existing cracks) are separated into their own group (Group 5). • The tensile strength distribution curve of each group is based on the Weibull distribution (Type 3 EVD). The Random Field methodology departs from the current method in the following ways: • Each wire is inspected at multiple locations along the length of the panel and assigned a corrosion stage based on the observed conditions; wires of unit length are assigned the actual corrosion stage recorded within that panel segment. • The tensile strength of each test specimen is treated independent from the wire sample. • Tensile strengths of specimens, including “cracked” wires, are grouped according to their observed corrosion stage (cracked wires are no longer treated as a separate group). • The tensile strength distribution curve is based on the estimated empirical Cumulative Distribution Function (CDF) directly.

54 The RT feels that the Random Field methodology is a significant improvement as it considers a large amount of corrosion stage data that is currently recorded, but not utilized, by the NCHRP 534 methodology. It is anticipated that in some cases the Random Field methodology will yield cable strength results similar to those of NCHRP 534. However, in other cases, the differences can be as high as 20% or more, with the Random Field methodology providing the more rigorous results. A thorough description of the Random Field methodology is contained in the sections that follow. To provide the reader with a better understanding of how to implement the proposed methodology, a worked example using the Random Field methodology is provided in Chapter 7 of this report. Mapping of Corrosion Stage Variation At each inspection location, the corrosion stage of every wire on the 16 faces of the eight wedged opening is identified and recorded, following the procedures outlined in the proposed Guidelines. This inspection data is then used to build a model of the corrosion stage variation along the entire effective development length 𝐿𝐿𝑒𝑒 for each exposed wire (and for the wires in the corresponding half-sector). A table is useful for preparing this information, similar to the one in Table 3, which shows the corrosion stage for all wires on one face of a wedged opening at that specific location. (Note that a complete table showing the variation along the length 𝐿𝐿𝑒𝑒 of the corrosion stage for all the wires exposed on the 16 faces of the eight wedged position at three inspection locations per panel is included in the worked example in Chapter 7 of this report.) The number in each cell represents the corrosion stage of a wire (and of the wires in the corresponding subsector) To create these tables, the following process is suggested: 1) The number of columns of the table depends on the number of inspection locations in the panel and on the number of wire segments of unit length, 𝑙𝑙𝑠𝑠 , assumed representative of that given location. In the example shown in Figure 24, there are three inspection locations and 5 × 𝑙𝑙𝑠𝑠 wire segments inspected at each location, so the total number of cells in a row (e.g. the total number of columns in the table) will be 3 × 5 = 15. The sequence of the 15 numbers will indicate the variation of the corrosion stage of a given wire along the effective length 𝐿𝐿𝑒𝑒 of the panel. 2) To fill the cells for each single wire inspected, each unit length is assigned the corrosion stage observed within that inspected segment of the panel. For example, assuming there are three panel segments and Stage 3 is observed at Location 1, Stage 2 at Location 2 and Stage 3 again at Location 3, a sequence of 15 numbers representative of the corrosion stage variation of the wire on one face of the wedge line is shown as (note that this corresponds to Wire 28 in Table 3): 3 – 3 – 3 – 3 – 3 – 2 – 2 – 2 – 2 – 2 – 3 – 3 – 3 – 3 – 3 3) This operation is repeated for every individual wire exposed within the wedge lines (both faces) and extended to the wires in the corresponding subsectors. 4) To produce the corrosion map, this specific wire (and all the wires in the corresponding subsector) will be considered as a Stage 3 wire (based on the worst corrosion, i.e. highest stage, within the inspected length of the panel). At the end of this operation, all wires in the cross-section of the cable will have a model of the corrosion stage variation along the effective length 𝐿𝐿𝑒𝑒 of the panel and a corrosion map of the cable cross-section can be presented graphically, as shown in Table 3.

55 Table 3. Corrosion Stage Variation Along Observed Wire Ring # Corrosion Stage Variation along Wire (Wedge 1, left side) Location 1 Location 2 Location 3 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 5 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 6 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 7 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 8 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 11 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 12 3 3 3 3 3 2 2 2 2 2 3 3 3 3 3 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 14 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 15 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 17 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 18 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 19 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 21 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 22 3 3 3 3 3 2 2 2 2 2 3 3 3 3 3 23 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 24 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 25 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 26 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 27 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 28 3 3 3 3 3 2 2 2 2 2 3 3 3 3 3 29 3 3 3 3 3 2 2 2 2 2 3 3 3 3 3 30 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 31 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 32 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 33 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 34 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 35 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 36 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 37 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 38 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 39 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 40 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 41 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 42 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 43 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 44 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 45 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 46 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

56 Figure 24. Corrosion Map of Cable (with Broken Wire Locations) [Guidelines Figure B.3.3-2] Cumulative Distribution Functions for Different Corrosion Stage Wires Different CDFs can be considered for best fit to the empirical one, including log normal, Weibull and beta. However, it is the recommendation of the RT to use the estimated empirical cumulative probability distribution functions directly.

57 To build the empirical CDF of the wire strength for the different corrosion stages, it is necessary to have the experimental results of ultimate tension tests on wire specimens of length 𝑙𝑙𝑠𝑠. As specified in the section on sampling/wire splicing of the Guidelines, a sufficient number of wire specimens is recommended in order to have reliable CDF curves. For the sake of discussion, the procedure to obtain empirical CDFs is described for Stage 3 wires but an identical procedure is followed for the wires in Stages 1, 2 and 4. To obtain the corresponding empirical CDF, the ultimate stress values 𝜎𝜎𝑢𝑢 , must be arranged from the smallest to the largest. The values are ranked in ascending order, starting at one and continuing to the maximum number of specimens in that stage. Note that the rank value also indicates the number of wire specimens that have an ultimate stress 𝜎𝜎𝑢𝑢 equal to or less than the value of the 𝜎𝜎𝑢𝑢 corresponding to the ranked specimen. For example, for the wire specimen having a rank value of 10, this means that there are 10 wires that have an ultimate stress 𝜎𝜎𝑢𝑢 that is less than or equal to that specimen. This is important in computing the probability that a wire segment has the ultimate stress 𝜎𝜎𝑢𝑢 equal or less than a certain value. Figure 25. Empirical CDF for Corrosion Stage 3 Wire Specimens Once the ultimate stresses have been sorted and ranked, then the value of the corresponding CDF can be easily obtained by dividing, for each single wire, the corresponding rank by the total number of wires tested plus one. (Note: the additional 1 is added so that for the last specimen, the probability is slightly less than 1). A sample CDF for Group 3 wires developed using this process is shown in Figure 25. An identical process is then used to obtain the CDFs corresponding to Corrosion Stage 1, Corrosion Stage 2 and Corrosion Stage 4 wires.

58 Once the CDFs are available, they can be used to obtain different random realizations of the spatial variation of the wire strength along the entire effective length of the panel. Simulating The Variation Of The Wire Strength Along The Effective Panel Length This process for determining the variation in corrosion stage along the wire outlined in the section on CDFs for different corrosion stage wires has to be repeated for each wire in the exposed 16 faces at the wedged openings (note that a full table of results is presented in the worked example in Chapter 7). The value of the ultimate stress (𝜎𝜎𝑢𝑢) of each wire determined in this way is assumed to be representative of the ultimate stress of the wires in the corresponding subsector (as described in the section of the proposed Guidelines on synthesis of information from field and laboratory). As an example, refer to wire 28 in Table 3. It was classified as a Stage 3 wire and was modeled as a combination of 10 wire segments of corrosion stage 3 and 5 segments of corrosion stage 2, as follows: 3 – 3 – 3 – 3 – 3 – 2 – 2 – 2 – 2 – 2 – 3 – 3 – 3 – 3 – 3 Random numbers are then generated to represent the values of the CDF for the wire segments in the model of the wire (the function Rand can be used in Excel to generate random numbers between 0 and 1). For example, 10 random numbers uniformly distributed in the interval [0,1] are generated for the Stage 3 wire units; and five random numbers uniformly distributed in the interval [0,1] are generated for the Stage 2 wire units. For each generated random number, representing the value of the CDF for a given wire unit, it is possible to obtain the value of the corresponding 𝜎𝜎𝑢𝑢 using the CDF tables developed in the section on cumulative distribution functions for different corrosion stage wires. For example, if the random number generated for one unit of Stage 3 wire is 0.1576, then this value of the CDF corresponds to a 𝜎𝜎𝑢𝑢 of 226.9 ksi; if the value of the CDF for the second unit length is 0.9706, then the corresponding 𝜎𝜎𝑢𝑢 is 252.7 (see Figure 26). These values of 𝜎𝜎𝑢𝑢 are obtained from the table of the values of the CDF as follows: • If the random value of the CDF is equal to one of the values of the CDF in the table (i.e. identical to the 𝜎𝜎𝑢𝑢 determined from testing of an individual specimen), then the corresponding value of the 𝜎𝜎𝑢𝑢 is used directly. • If the random value of the CDF is not one of the values in the table, but it is in between two values, linear interpolation is used between these two values in the table of the CDF to determine the value of the 𝜎𝜎𝑢𝑢.

59 Figure 26. How to Obtain the Values of σ_u from the CDF The procedure is repeated for each single wire segment to obtain values of the 𝜎𝜎𝑢𝑢. For example, considering Wire 28 on the left face of Wedge Line 1 (and all the wires in the corresponding subsector), with an “overall” classification of Stage 3 wire and corrosion variation along its length modeled as: 3 – 3 – 3 – 3 – 3 – 2 – 2 – 2 – 2 – 2 – 3 – 3 – 3 – 3 – 3 The “estimated” variation of the wire strength is shown in Table 4. Due to the fact that a wire will break at its weakest point (“weakest link” assumption), the ultimate stress of this representation of Wire 28 will be 𝜎𝜎𝑢𝑢,𝑙𝑙 = 215.0 ksi. Hence, in the final corrosion map of the cross-section, Wire 28 on the left face of Wedge Line 1, and all the wires in the corresponding subsector, will be categorized as Stage 3 wires with an ultimate stress 𝜎𝜎𝑢𝑢,𝑙𝑙 = 215.0 ksi. This operation is repeated for each single wire exposed at the wedged openings and extended to every single wire in the corresponding subsectors. At the end of this operation, the final corrosion map of the cross-section will be obtained; each of the wires will be categorized in one of the four corrosion stages (the worst along the inspected length) and will have a value of the ultimate stress 𝜎𝜎𝑢𝑢,𝑙𝑙 that accounts for the spatial variation of the wire strength along the effective length 𝐿𝐿𝑒𝑒 of the panel.

60 Table 4. Ultimate Stresses Determined from CDF Segment Corrosion Stage 𝝈𝝈𝒖𝒖 ksi 3 226.9 3 252.7 3 251.1 3 236.1 3 243.2 2 238.7 2 215.0 (Minimum) 2 246.1 2 251.1 2 238.9 3 226.6 3 234.0 3 248.3 3 243.0 3 251.2 For example, for a cable comprised of 6080 wires (206 Stage 1; 2559 Stage 2; 2827 Stage 3 and 488 Stage 4), then at the end of this operation the final corrosion map used to calculate the ultimate strength will consist of: • 206 Corrosion Stage 1 wires with associated 206 values of 𝜎𝜎𝑢𝑢,𝑙𝑙 • 2559 Corrosion Stage 2 wires with associated 2559 values of 𝜎𝜎𝑢𝑢,𝑙𝑙 • 2827 Corrosion Stage 3 wires with associated 2827 values of 𝜎𝜎𝑢𝑢,𝑙𝑙 • 488 Corrosion Stage 4 wires with associated 488 values of 𝜎𝜎𝑢𝑢,𝑙𝑙 Broken Wires in Effective Development Length As described in the section on number of broken wires (3.2.2 of the proposed Guidelines), there are two kinds of broken wires. The first kind consists of the observed broken wires, whose exact locations are known from the corrosion map (Figure 22) and can be therefore removed. The second kind includes estimated broken wires within the panel under consideration and an equivalent number of broken wires representing the effect of broken wires from adjacent panels within the redevelopment length. Two methods were evaluated by the RT for the calculation of the redevelopment of broken wires (note that cracked wires are no longer treated as a separate group, and therefore the calculations for cracked wires currently included in NCHRP 534 have been eliminated): • Method 1 increases the force in the broken wire as the distance from the break increases along the effective development length (i.e. due to friction at the cable bands). This method provides a rational basis for calculation as it realistically models the behavior of the wires in the cable. • Method 2 reduces the number of broken wires as the distance from the break increases along the effective development length. This method, which is similar to that of NCHRP 534, is somewhat simpler than Method 1 from a calculation standpoint but is counterintuitive as it does not provide a realistic model of the behavior of the wires in the cable.

61 Based on the example provided in Chapter 7 using data from a typical suspension bridge, Method 1 provides an approximately 7% lower value of the calculated cable strength. Upon further evaluation of these two methods by the research team, it was decided to carry forward Method 2 for use in the Guidelines. This method is similar to that of NCHRP 534 and helps maintain continuity between the old and new Guidelines. Both methods utilize the same redevelopment coefficient, Cdi, calculated using Article 3.4.2 of the proposed Guidelines. Assuming that adjacent panels have broken wires in equal numbers as those in the inspected panel, then the strength reduction of these broken wires in the adjacent panels must be accounted for in the estimation of the cable strength in the inspected panel. The associated redevelopment factors 𝐶𝐶𝑑𝑑𝑑𝑑 , 𝑖𝑖 =1, 2, …, 7, are listed in Table 5. To help visualize the redevelopment of the strength of broken wires in adjacent panels, refer to Figure 27, where a schematic representation of the cable is presented. For the example shown in the figure, Le = 7 so the redevelopment length affects three panels (4 cable bands) on each side of the inspected panel. Table 5. Redevelopment Factors, Cdi Panel i 1 2 & 3 4 & 5 6 & 7 Cdi 0 0.25 0.50 0.75 Figure 27. Redevelopment Length for the Inspected Panel (Panel 1) Method 1 The total number of broken wires in an inspected panel (observed broken and estimated broken), 𝑁𝑁𝑏𝑏1, can be determined using the methods in Article 3.2.2 of the proposed Guidelines. The next step is to remove the observed broken wires and estimated broken wires at random from the corresponding corrosion stage groups in the same proportion as for the observed broken wires. The location of observed broken wires is determined from the inspection data forms (Figure 2.2.5.1.3-1 of the proposed Guidelines) and the corrosion stage is determined from the corrosion map using the procedure described in Article 4.2 of the proposed Guidelines. An example corrosion map showing the corrosion stages and location of broken wires is shown in Figure 22. The number of observed broken wires can be determined for each corrosion stage, and must sum to give the total number of observed broken wires in the evaluated panel, 𝑁𝑁𝑜𝑜𝑏𝑏1: 𝑁𝑁𝑜𝑜𝑏𝑏1 = ∑ (𝑁𝑁𝑜𝑜𝑏𝑏1)𝑠𝑠𝑠𝑠4𝑠𝑠𝑠𝑠=1 (2) Cable Panel 1 Panel 2 Panel 3 Panel 4 Panel 5 Panel 6 Panel 7 𝑁𝑁𝑏𝑏7 𝑁𝑁𝑏𝑏6 𝑁𝑁𝑏𝑏5 𝑁𝑁𝑏𝑏4 𝑁𝑁𝑏𝑏3 𝑁𝑁𝑏𝑏2 𝑁𝑁𝑏𝑏1 Cable bands Panel under investigation

62 The total number of broken wires in the evaluated panel, 𝑁𝑁𝑏𝑏1, consists of both observed broken wires and estimated broken wires. The number of broken wires in each corrosion stage is proportioned from the observed broken wires from the same corrosion stage: (𝑁𝑁𝑏𝑏1)𝑠𝑠𝑠𝑠 = (𝑁𝑁𝑜𝑜𝑜𝑜1)𝑠𝑠𝑠𝑠 𝑁𝑁𝑜𝑜𝑜𝑜1 ∙ 𝑁𝑁𝑏𝑏1, rounded to next highest integer (3) 𝑁𝑁𝑏𝑏1 indicates the total number of broken wires (observed broken and estimated broken) in the evaluated panel in each of the four corrosion stages. 𝑁𝑁𝑏𝑏1 = ∑ (𝑁𝑁𝑏𝑏1)𝑠𝑠𝑠𝑠4𝑠𝑠𝑠𝑠=1 (4) where (𝑁𝑁𝑏𝑏1)𝑠𝑠𝑠𝑠 = number of Stage k broken wires in the evaluated panel If 𝑁𝑁𝑇𝑇𝑇𝑇𝑇𝑇 is the total number of wires in the cable, then the total number of effective (unbroken) wires at the inspected panels is: 𝑁𝑁𝑒𝑒𝑒𝑒𝑒𝑒 = 𝑁𝑁𝑇𝑇𝑇𝑇𝑇𝑇 − 𝑁𝑁𝑏𝑏1 (5) Using a uniform probability density function, select at random (𝑁𝑁𝑏𝑏2 + 𝑁𝑁𝑏𝑏3) wires in the cross-section under investigation (keeping the same proportions of corrosion stages as the broken wires) and multiply the ultimate strength of these wires by 𝐶𝐶𝑑𝑑2,3 = 0.25. Again, using a uniform probability density function, select at random (𝑁𝑁𝑏𝑏4 + 𝑁𝑁𝑏𝑏5) wires in the cross- section under investigation (keeping the same proportions of corrosion stages as the broken wires) and multiply the ultimate strength of these wires by 𝐶𝐶𝑑𝑑4,5 = 0.5. Finally, using a uniform probability density function, select at random (𝑁𝑁𝑏𝑏6 + 𝑁𝑁𝑏𝑏7) wires in the cross-section under investigation (keeping the same proportions of corrosion stages as the broken wires) and multiply the ultimate strength of these wires by 𝐶𝐶𝑑𝑑6,7 = 0.75. At this point, a new map of the ultimate strength of the wires in the cross-section of the inspected panel is available, which differs from the one obtained at the end of the section on Cable Inspection of this Chapter. In this new map, (𝑁𝑁𝑏𝑏2 + 𝑁𝑁𝑏𝑏3) wires will have their ultimate strength reduced by 75%, (𝑁𝑁𝑏𝑏4 + 𝑁𝑁𝑏𝑏5) wires will have their ultimate strength reduced by 50%, and (𝑁𝑁𝑏𝑏6 + 𝑁𝑁𝑏𝑏7) wires will have their ultimate strength reduced by 25%, while the remaining wires: 𝑁𝑁𝑟𝑟𝑒𝑒𝑟𝑟 = 𝑁𝑁𝑇𝑇𝑇𝑇𝑇𝑇 − 𝑁𝑁𝑏𝑏1 − (𝑁𝑁𝑏𝑏2 + 𝑁𝑁𝑏𝑏3) − (𝑁𝑁𝑏𝑏4 + 𝑁𝑁𝑏𝑏5) − (𝑁𝑁𝑏𝑏6 + 𝑁𝑁𝑏𝑏7) (6) will keep the same ultimate strength as previously obtained in Chapter 4 of the proposed Guidelines. Method 2 An alternative way to account for broken wires in adjacent panels is to obtain an equivalent number of broken wires within the panel under consideration and then proceed to remove them from the cross- section in question. This approach is similar to the one given in the NCHRP 534. The redevelopment length Le and redevelopment factor Cdi can be estimated by Article 3.4.2 of the proposed Guidelines. Then, the equivalent number of broken wires within the panel under consideration, 𝑁𝑁𝑒𝑒𝑏𝑏, accounting for the redevelopment length in broken wires in neighboring panels, can be estimated by: 𝑁𝑁𝑒𝑒𝑏𝑏 = 𝑁𝑁𝑏𝑏1 + ∑ 𝑁𝑁𝑏𝑏𝑑𝑑 ∗ (1 − 𝐶𝐶𝑑𝑑𝑑𝑑) 𝐿𝐿𝑒𝑒 𝑑𝑑=2 (7)

63 where 𝑁𝑁𝑏𝑏𝑑𝑑 is the number of broken wires in the 𝑖𝑖 −th panel, 𝑁𝑁𝑏𝑏1 is the number of broken wires in the evaluated panel; and 𝐶𝐶𝑑𝑑𝑑𝑑 is the redevelopment factor in the 𝑖𝑖 −th panel. It is then suggested to remove all the estimated equivalent broken wires on the basis of the same proportion as the observed broken wires. The number of broken wires from corrosion stage can be determined as follows: (𝑁𝑁𝑏𝑏1)𝑠𝑠𝑠𝑠 = (𝑁𝑁𝑜𝑜𝑜𝑜1)𝑠𝑠𝑠𝑠 𝑁𝑁𝑜𝑜𝑜𝑜1 ∙ 𝑁𝑁𝑏𝑏1, rounded to next highest integer (8) This means that for one realization of the overall cable strength, the number of broken wires calculated above for each corrosion stage will be identified randomly on the corrosion map as broken wires, and subsequently removed from the cross-section in the evaluation of its overall cable strength. Hence, the effective number of wires in the cross-section under investigation will be: 𝑁𝑁𝑒𝑒𝑒𝑒𝑒𝑒 = 𝑁𝑁𝑇𝑇𝑇𝑇𝑇𝑇 − 𝑁𝑁𝑒𝑒𝑏𝑏 (9) Methods of Determining Cable Strength Once the final map of the ultimate strength of the “effective” wires is finalized using either Method 1 or Method 2 described in the previous section, the process of estimating the overall strength of the cable can start. The proposed methodology will be an iterative process (similar to the Brittle Wire Model in NCHRP 534) where, at each load step, the load carried by each wire is checked against the wire’s ultimate strength. If such a load exceeds the wire’s strength, the wire is removed and its load redistributed equally to the other wires in the cross-section. This process will stop when the remaining wires will not be able to carry the applied load and this load will represent the cable’s overall strength. By repeating this process for a large number of realizations of the cross-sectional map of the wire strength, it will be possible to obtain a probability distribution of the cable’s overall strength. To calculate one random realization of the overall cable strength, the strength map of the cross- section is established by randomly generating and recording the ultimate strength of every “effective” long wire within the cross-section. The number of “effective” wires is obtained following one of the two methods described in the section on estimation of strength. The overall cable strength Ru can then be determined by an iterative incremental loading process that, at each step, identifies newly failed wires, removes them, and redistributes their loads equally to the surviving wires. The iterative process can be executed in Excel (using subroutines) or in MATLAB. Once the final map of the ultimate strength of the “effective” wires is finalized using either Method 1 or Method 2 described in the section on estimation of strength, the process of estimating the overall strength is as follows (note that the description that follows uses the stress in the wire but it is equally applicable to a force based approach): 1) Select an initial estimate of the force, 𝐹𝐹0, carried by the cable under service load conditions.

64 2) Divide this force by the number of effective wires in the cross-section to find the force carried by a single wire, 𝐹𝐹0,𝑑𝑑: 𝐹𝐹0,𝑑𝑑 = 𝐹𝐹0 𝑁𝑁𝑒𝑒𝑒𝑒𝑒𝑒 (10) 3) Divide 𝐹𝐹0,𝑑𝑑 by the area 𝐴𝐴𝑤𝑤 of a single wire (0.02895 in2) to obtain an average stress 𝜎𝜎𝑎𝑎𝑎𝑎 : 𝜎𝜎𝑎𝑎𝑎𝑎 = 𝐹𝐹0,𝑖𝑖 𝐴𝐴𝑤𝑤 (11) 4) Compare the value of 𝜎𝜎𝑎𝑎𝑎𝑎 with all the values in the map of long wire ultimate strengths 𝜎𝜎𝑢𝑢,𝑙𝑙 as obtained in Article 4.2.3 of the proposed Guidelines. If there are values of the wire ultimate strengths 𝜎𝜎𝑢𝑢,𝑙𝑙 that are less than the value of 𝜎𝜎𝑎𝑎𝑎𝑎, then the corresponding wires are eliminated and a new number of effective wires, 𝑁𝑁𝑒𝑒𝑒𝑒𝑒𝑒′ is obtained. 5) Repeat Step 2 to Step 4 until the newly computed 𝜎𝜎𝑎𝑎𝑎𝑎 is smaller than any ultimate strength 𝜎𝜎𝑢𝑢,𝑙𝑙 of the remaining effective wires. 6) Increase the cable force by an arbitrary small increment (Δ𝐹𝐹) and consider the new cable force (for example, let Δ𝐹𝐹 = 100 kips): 𝐹𝐹1 = 𝐹𝐹0 + Δ𝐹𝐹 (12) 7) Repeat Steps 2 through 6 until, at the i-th iteration, the corresponding 𝜎𝜎𝑎𝑎𝑎𝑎 will be greater than the ultimate strength of all the remaining wires. This would mean that all the wires are broken. 8) The largest value of the cable force sustained by the wires (the one corresponding to the (i- 1)-th iteration, 𝐹𝐹(𝑑𝑑−1)) will represent an estimate of the overall cable strength for that given realization: 𝑅𝑅𝑢𝑢 = 𝐹𝐹(𝑑𝑑−1) (13) Monte Carlo Simulations for the Estimation of the Overall Cable Strength. The procedure described in the previous section provides one value of the ultimate strength of the cable 𝑅𝑅𝑢𝑢 based on the specific realization of the spatial variation of the wire strength. To obtain reliable statistics of the overall cable strength, this procedure must be repeated a large number of times (for the example in Chapter 3, the procedure was repeated 10,000 times using a subroutine created in MATLAB). Hence, the distribution of the overall cable strength, with its mean and standard deviation, is obtained based on a population of 10,000 values. This iterative process can be executed in Excel (using subroutines) or in MATLAB. For a new repetition, the CDF’s determined in Articles 4.2.1 and 4.2.2 of the proposed Guidelines remain unchanged. Starting from Article 4.2.3, a new map of the ultimate stress 𝜎𝜎𝑢𝑢,𝑙𝑙 is obtained, following the same process described in Articles 4.2.3 and 4.2.4 and a new value of 𝑅𝑅𝑢𝑢 is derived through the process described in Article 4.2.5. The COV is continuously monitored during the Monte

65 Carlo simulations and the process stops when the COV drops below a pre-specified value (e.g. 2%). For a flowchart of the calculation procedure, see Figure 29. For the specific example contained in Chapter 3 of this report, and using Method 1, the distribution of cable strength is as shown in Figure 28. Taking the mean of all the calculated 𝑅 values results in a cable strength of: 𝑅 , 28,100 𝑘𝑖𝑝 Figure 28. Distribution of Cable Strength, Ru For comparison purposes, the calculated cable strength determined per NCHRP 534 is 27,800 kips (1% lower) using the Simplified Strength Model or 31,800 kips (13% higher) using the Brittle Wire Model. Results using the BTC method were also published for this particular bridge in the report “BTC Method for Evaluation of Remaining Strength and Service Life of Bridge Cables – Final Report NYSDOT Project C-07-11” (Mahmoud, 2011). For this panel, the calculated cable strength was 28,261 kips (0.6% higher). The proposed Random Field methodology provides the cable strength in terms of a probability distribution function. Thus, the cable strength is estimated in probabilistic terms incorporating error information rather than a single numerical value. Predicted Factor of Safety The cable capacity of the evaluated panel (panel 𝑖) is the mean value computed using the procedure described in Article 4.2.1 of the proposed Guidelines,𝑅 .

66 The FOS is determined by dividing the calculated capacity by the demand forces, Fi , in the evaluated panel (the calculation of cable force is not covered in this report). ui i i RFOS F = (14) Figure 29. Flow Chart for Calculation Procedure [Guidelines Figure 4.1-1]

67 Comparison of Proposed Random Field Method with NCHRP 534 To allow for a meaningful comparison of the proposed Random Field Method with respect to the current NCHRP 534 report, Table 6 represents a summary of pertinent changes broken down by article number (both in the current NCHRP 534 report and within this report). As well, a flow chart is provided in Figure 30 outlining the entire inspection and strength evaluation process, with changes from the current NCHRP NCHRP Report 534 method highlighted in red (and denoted “New”). Note that all Article references in the flow chart are based on the proposed Guidelines. As discussed previously in this Chapter, the primary difference between the proposed Random Field methodology and the current NCHRP NCHRP Report 534 is the process of accounting for the variation of wire strength along the length of a wire within a panel. The RT feels that the Random Field methodology is a significant improvement as it considers a large amount of corrosion stage data that is currently recorded, but not utilized, by the NCHRP 534 methodology. The proposed changes to NCHRP 534 are primarily limited to the cable strength calculation. There are no changes to the cable opening work performed by contractors in the field work. There are also no changes to the inspection procedures performed by investigators in the field. The Random Field methodology departs from the current method in the following ways: • Each wire is inspected at multiple locations along the length of the panel and assigned a corrosion stage based on the observed conditions; wires of unit length are assigned the actual corrosion stage recorded within that panel segment. • The tensile strength of each test specimen is treated independent from the wire sample. • Tensile strengths of specimens, including “cracked” wires, are grouped according to their observed corrosion stage (cracked wires are no longer treated as a separate group). • The tensile strength distribution curve is based on the estimated empirical CDF directly. It should be noted that the change in the way cracked wires are treated means that fractographic examination of wires is no longer required under the proposed Random Field Method. Because of this, it will not be possible to calculate the cable strength using both the NCHRP 534 approach and the Random Field approach. It is recommended that all wire specimens be retained after testing to allow for fractographic examination in the event that investigators wish to perform a back-to-back comparison between the two methods. Table 6. Comparison between NCHRP 534 and Proposed Guidelines NCHRP NCHRP Report 534 Article Guidelines Article Description NCHRP 534 Methodology Random Field Methodology 2.2.5.1 2.2.4.1 First Internal Inspection Minimum 3 panels/cable Minimum 4 panels/cable.

68 NCHRP NCHRP Report 534 Article Guidelines Article Description NCHRP 534 Methodology Random Field Methodology 2.2.5.2, 2.2.5.3 and 2.2.5.4 2.2.4.1 Second Internal Inspection • 4 panels/cable (if Stage 1 or 2) • 6 panels/cable (if Stage 3 or superficial Stage 4) • 16%-20% of all panels/cable (if Stage 4 >3 wires deep) • 100% panels/cable (if >10% Stage 4 or broken wires) Same as above. 2.2.5.5 Ch. 6 (of Final Report) Acoustic Monitoring Recommend installation of acoustic monitoring system when Stage 3 wires (or worse) are found in cable Discussion of current state of the art NDE/SHM methods provided. 2.3.1.2.6 2.2.3.3 Inspection QA plan More than one inspector should make observations Added recommendation for independent review for verification purposes. 2.4.3.1 2.2.5.1.3 Recording of corrosion stages Use four corrosion stages (photographs in Figure 1.4.2.2-1) No change. 2.4.3.1 2.2.5.1.3 Recording of corrosion stages At least three segments along panel Wire condition is recorded at 3, 4 or 5 segments (depending on panel length).

69 NCHRP NCHRP Report 534 Article Guidelines Article Description NCHRP 534 Methodology Random Field Methodology 2.4.3.1 2.2.5.1.3 and 3.2.1 Recording of corrosion stages Only the highest stage found along the length of the wire is used in the analysis of cable capacity (C2.4.3.1) in accordance with the "weakest link" theory Corrosion stage recorded in each segment is utilized in calculations in accordance with the Random Field theory. 2.4.3.1 2.2.5.1.3 and 3.2.1 Cross-sectional map of wire stages Wires are "reassigned" the highest stage found along the length of the wire in each panel Mapping is still based on highest corrosion stage of wire in any segment in panel, but wires are not "reassigned" as this data is subsequently used for the Random Field calculations. 2.4.3.2 3.2.2 Broken Wires No change No change. 2.4.3.5.1 2.2.6.1.1 Number of Samples Number of samples is shown in Table 2.4.3.5.1- 1 Changed to a target number of specimens rather than a number of sample wires. Note that total target number is now 320 specimens, which has reduced considerably from the 1085 recommended under NCHRP 534 (assuming 9-Stage 2, 10-Stage 3 and 10- Stage 4 specimens per sample wire). 2.4.3.5.2 2.2.6.1.1 Sample Location Specified locations of samples are provided A basic assumption of the Random Field Method is that samples can be taken from any panel.

70 NCHRP NCHRP Report 534 Article Guidelines Article Description NCHRP 534 Methodology Random Field Methodology 2.4.3.5.2.c 2.2.6.1.1 Percentages of Stage 3 and Stage 4 wires Specific percentages are provided The statistics for determining the number of specimens in each group is currently based on only one bridge where data is available. The RT reviewed new test data from two additional bridges to verify the information and benchmark the new methodology. 2.4.3.5.2.c 2.2.6.1.1 Random sampling "Samples should be selected at random in each inspected panel, using tables of random sample locations prepared in advance for several different groupings of Stage 3 or Stage 4 wires" A basic assumption of the Random Field Method is that samples can be taken from any panel. Samples are selected based on visual classification of corrosion stage, not at random. 2.4.3.5.3 3.1.1.1 Number of Specimens in Each Sample When the corrosion stage varies along the length of a sample wire, the specimens to be tested for strength shall be cut from the worst areas of the wire (C2.4.3.5.3) All specimens from a sample wire are tested (assuming the target value is not exceeded). 2.4.3.5.3 2.2.6.1.3 Lengths of Samples Sample length is provided (though this appears to be based on a bridge with a 40 ft. panel length) The sample length is as long as practicable. It is not always possible to extract 16 to 20 foot samples when panel length is short (many older bridges are < 25 ft).

71 NCHRP NCHRP Report 534 Article Guidelines Article Description NCHRP 534 Methodology Random Field Methodology 3.2.1 3.1.1.2, 2.2.5.1.3 Specimen Preparation "All of the specimens from a given sample should be at the same stage of corrosion, but it is understood that this is not always possible" Variation of corrosion stage along the length of the wire is explicitly accounted for in the calculations. 3.2.1 3.1.1.2 Specimen Preparation "Before sample wires are cut into specimens of suitable length for testing, they should be inspected and assigned to the appropriate corrosion stage" Properties of specimens are assigned to the appropriate group based on the recorded corrosion stage (not that of the sample wire). 3.2.1 3.1.1.2 Specimen Preparation "If possible, NDE testing to locate pre-existing cracks should be performed on individual wires before they are cut, so that the worst cracks can be arranged to appear near the center of the specimen" No reliable means are available to identify pre-existing cracks prior to tensile testing. 3.2.4 3.1.1.4 Fractographic Examination of Suspect Wires Pre-existing cracks are identified so that properties of cracked wires are determined separately Pre-existing cracks are identified after testing but the properties of these wires are included in their corrosion group. 3.2.5 3.1.1.2 Examination of Fracture Surface for Pre-existing Cracks Cracked wires are treated as a separate group in estimating cable strength Cracked wires are not treated as a separate group; they are included with the wire group according to corrosion stage.

72 NCHRP NCHRP Report 534 Article Guidelines Article Description NCHRP 534 Methodology Random Field Methodology 4.4.1 4.2.2 Wire Properties - Groups Groups 1 - 4 and "cracked" wires in Group 5 Group 5 is eliminated as cracked wires are grouped according to their respective corrosion stage. 4.5.2 3.4.1 Effective Development Length and Redevelopment Coefficient No guidance for special areas, such as anchorages or unsupported backstays (with no cable bands) No change. 5.3.2 4.2.2 Wire Groupings All the discrete cracked wires are subtracted from their corresponding groups and added together to form Group 5 Group 5 has been eliminated - cracked wires are grouped by the wire's corrosion stage. 5.3.2.3 4.2.4 Unbroken Wires in each corrosion stage Broken wires within the effective development length are subtracted from the total Although an alternate method was initially studied by the RT, it was decided to follow the current NCHRP approach of subtracting broken wires. No change. 5.3.2.4 4.2.4 Cracked wires Cracked wires are redeveloped Cracked wires are no longer treated separately. They are included within Group 3 and/or Group 4. 5.3.2.5 4.2.4 Effective Number of Unbroken Wires Unbroken wires includes cracked wires that are redeveloped Cracked wires are no longer treated separately. They are included within Group 3 and/or Group 4.

73 NCHRP NCHRP Report 534 Article Guidelines Article Description NCHRP 534 Methodology Random Field Methodology 5.3.3 4.2 Strength of Unbroken Wires Weakest Link theory Random Field Theory. 5.3.3.1, 5.3.3.2 and 5.3.3.3 4.2 Simplified Strength Model, Brittle Wire Model and Limited Ductility Model Multiple models for strength calculation Cable strength is calculated using an iterative procedure whereby the load is increased incrementally in the cable (similar in concept to the Limited Ductility Model). The sections in the Guidelines follow the same organization as NCHRP Report 534 and are intended to provide a thorough explanation of the inspection procedures and methodology used during an actual main cable inspection. In order to help engineers who may already be familiar with NCHRP Report 534, the following flowchart highlights the important changes that have been incorporated in the Guidelines.

74

75 Figure 30. Inspection and Strength Evaluation Procedure Flowchart [Guidelines Figure 1.2.3-1]

76 Predicted Remaining Life The calculation of remaining life for a given main cable of a suspension bridge requires information not only on the rate of deterioration that controls the cable strength calculation to the present time, but how that deterioration rate will be affected by any maintenance or preservation actions. Without quality information in these areas, it will not be possible to develop an accurate estimate of remaining life. To be able to collect such data would require in-service monitoring of the environmental conditions within the interior of a main cable, which is not currently feasible based on presently available technology (outside of a laboratory). Estimating the remaining life of a main cable is important to owners as it can be used to drive preservation, rehabilitation, and replacement planning for the cable system, and possibly, the bridge as a whole. Ideally, a remaining life calculation would account for the deterioration rate from the opening of the bridge to the date of evaluation, as well as the anticipated conditions the bridge will experience in the future that affect the cable health. The current guidance on cable inspection and strength evaluation, NCHRP NCHRP Report 534, was published in 2004. Before that, there was a lack of consistent guidance for determining the condition of a given cable. Even after the publication of the report, some time elapsed before the methods of the report were generally adopted. Inspections of main cables are on average an infrequent occurrence (i.e. every 10 years or more), with the exception being those cables that have been found to be in a significantly deteriorated state, which have been inspected more frequently (i.e. between 5 and 10 years). For most cables, subsequent inspections often involve opening new panels that had not previously been opened, and only a few panels are opened multiple times. Considering the above, and the significant differences in environment and design and construction details across the current inventory of suspension bridges that may impact the deterioration rate of the cables, the total amount of data available from which to determine deterioration rates is quite small. The ideal set of data would contain many years’ worth of inspection and strength calculation data from the same locations along a given cable from which a regression analysis could be used to fit a theoretical deterioration curve. The reality is that only 3 or 4 data points are typically available for a given location along a cable, and any deterioration rate derived from such a small data set would be prone to gross errors, and likely would not provide useful information for an owner. For cables that have experienced significant deterioration, it is likely that an owner would be implementing preservation or rehabilitation actions such as dehumidification, to try to slow the rate of deterioration and extend the life of the cable. The success of any such action would depend on the details of the implementation, and the specific configuration of a given cable system. There is very little data available on the long-term impact of such actions on which to base an estimate on the degree to which the remaining life is extended. The use of cable monitoring systems, such as an acoustic emissions based system, can provide useful qualitative data regarding the deterioration rate of cables as indicated by the frequency of wire breaks. However, there are limitations to the accuracy of such methods so it is not possible to make a direct correlation between the cable strength and the rate of detected wire breaks. Other nondestructive evaluation (NDE) and SHM systems are available but none provide the quantitative data necessary to determine the remaining life of a main cable (a summary of currently available NDE/SHM techniques is provided in Article 6.0 of the this report).

77 The pairing of a quality cable investigation and strength evaluation program performed at a regular frequency, alongside a cable monitoring system, would appear to be the best option for an owner looking to establish the strength vs. time relationship for the cables on their bridge. The development of more generic relationships that might apply across the bridge inventory does not appear practical due to the limited information, and the relatively limited extent of applicability of the information that is available.