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From page 99...
... 99 C H A P T E R 7 Illustrative Example Detailed Application of the Proposed Methodology To better explain the proposed methodology to estimate the cable strength, here a detailed example is presented, highlighting each single step of the procedure. All calculations were performed using MATLAB.
From page 100...
... 100 1) The number of columns of the table depends on the number of inspection locations and on the number of unit length wire segments, 𝑙𝑙𝑠𝑠, assumed representative of that given location.
From page 101...
... 101 Figure 37. Wire Corrosion Stage Data.
From page 102...
... 102 Figure 38. Wire Corrosion Stage Data.
From page 103...
... 103 Figure 39. Wire Corrosion Stage Data.
From page 104...
... 104 Table 7. Simulated Profiles Along the Length of All Wires on the Left Side of the Wedge, Sector 1 Ring # Corrosion stage/grade variation along the length of long wire 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
From page 105...
... 105 Table 8. Simulated Profiles Along the Length of All Wires on the Right Side of the Wedge, Sector 1 Ring # Corrosion stage/grade variation along the length of long wire Location 1 Location 2 Location 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 2 2 2 2 2 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 2 2 2 2 2 16 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 17 3 3 3 3 3 2 2 2 2 2 3 3 3 3 3 18 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 19 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 20 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 21 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 22 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 23 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 24 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 25 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 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 3 3 3 3 3 3 3 3 3 3 29 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 30 3 3 3 3 3 3 3 3 3 3 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 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 34 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 35 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 36 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 37 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 38 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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
From page 106...
... 106 Table 9. Simulated profiles along the length of all wires on the left side of the wedge.
From page 107...
... 107 Table 10. Simulated profiles along the length of all wires on the right side of the wedge.
From page 108...
... 108 Table 11. Simulated profiles along the length of all wires on the left side of the wedge.
From page 109...
... 109 Table 12. Simulated profiles along the length of all wires on the right side of the wedge.
From page 110...
... 110 Table 13. Simulated profiles along the length of all wires on the left side of the wedge.
From page 111...
... 111 Table 14. Simulated profiles along the length of all wires on the right side of the wedge.
From page 112...
... 112 Table 15. Simulated profiles along the length of all wires on the left side of the wedge.
From page 113...
... 113 Table 16. Simulated profiles along the length of all wires on the right side of the wedge.
From page 114...
... 114 Table 17. Simulated profiles along the length of all wires on the left side of the wedge.
From page 115...
... 115 Table 18. Simulated profiles along the length of all wires on the right side of the wedge.
From page 116...
... 116 Table 19. Simulated profiles along the length of all wires on the left side of the wedge.
From page 117...
... 117 Table 20. Simulated profiles along the length of all wires on the right side of the wedge.
From page 118...
... 118 Table 21. Simulated profiles along the length of all wires on the left side of the wedge.
From page 119...
... 119 Table 22. Simulated profiles along the length of all wires on the right side of the wedge.
From page 120...
... 120 Cumulative Distribution Functions for Different Corrosion Stage Wires 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 𝑙𝑙 𝑠𝑠 . As specified in this report, a sufficient number of wire specimens is recommended in order to have reliable CDF curves.
From page 121...
... 121 Table 23. Generate CDF Corrosion Stage συ Sorted συ Rank Empirical CDF (ksi)
From page 122...
... 122 Corrosion Stage συ Sorted συ Rank Empirical CDF (ksi)
From page 123...
... 123 Corrosion Stage συ Sorted συ Rank Empirical CDF (ksi)
From page 124...
... 124 Corrosion Stage συ Sorted συ Rank Empirical CDF (ksi)
From page 125...
... 125 Corrosion Stage συ Sorted συ Rank Empirical CDF (ksi)
From page 126...
... 126 Corrosion Stage συ Sorted συ Rank Empirical CDF (ksi)
From page 127...
... 127 Figure 40. Empirical CDF for Group 3 Wire Specimens An identical process has to be followed using the other corrosion stage wire specimens to obtain the CDFs corresponding to Corrosion Stage 1 (Group 1)
From page 128...
... 128 random numbers represent the values of the CDF for the wire segments in the model of the wire. To generate random numbers between 0 and 1, the function Rand can be used in Excel.
From page 129...
... 129 Repeating the same procedure for each single wire segment, we obtain the following values of the 𝜎𝜎𝑢𝑢 For Corrosion Stage 3: 1 2 3 4 5 6 7 8 9 10 0.1576 0.9706 0.9572 0.4854 0.8003 0.1419 0.4218 0.9157 0.7922 0.9595 226.9 252.7 251.1 236.1 243.2 226.6 234.0 248.3 243.0 251.2 For Corrosion Stage 2: 1 2 3 4 5 0.6557 0.0357 0.8491 0.9340 0.6787 238.7 215.0 246.1 251.1 238.9 At this point, the "estimated" variation of the wire strength for wire 28 on the left face of the wedged opening in sector 1 (and all the wires in the corresponding subsector) , with an "overall" classification as a Corrosion Stage 3 wire and modeled as: 3 – 3 – 3 – 3 – 3 – 2 – 2 – 2 – 2 – 2 – 3 – 3 – 3 – 3 – 3 will be represented as: Table 24.
From page 130...
... 130 Hence, in the final corrosion map of the cross-section, the wire 28 on the left face of the wedged opening in sector one and all the wires in the corresponding subsector will be categorized as Corrosion Stage 3 wires and their ultimate stress 𝜎𝜎𝑢𝑢,𝑙𝑙 = 215.0 ksi. This operation has to be done for each single wire exposed at the wedged openings and extended to every single wire in the corresponding subsectors.
From page 131...
... 131 • Corrosion Stage 1: 15 estimated broken wires • Corrosion Stage 2: 15 estimated broken wires • Corrosion Stage 3: 63 estimated broken wires • Corrosion Stage 4: 46 estimated broken wires. At this point, these broken wires have to be removed from the corrosion map obtained in the previous step.
From page 132...
... 132 Table 26. Estimated broken wires in the panels within the redevelopment length Corrosion Stage Group, k Proportion % Estimated broken wires in inspected panel Develop 25% strength Develop 50% strength Develop 75% strength 1 11.1% 15 (2 observed)
From page 133...
... 133 At the end of this process, the final corrosion map of the cable cross-section will still be comprised of 5941 wires but, for some of them, the ultimate stress 𝜎𝜎𝑢𝑢,𝑙𝑙 has been properly reduced to account for broken wires in the neighboring panels. At this point, the iterative procedure to estimate the ultimate strength of the entire cable can start.
From page 134...
... 134 The total number of effective wires in the cross-section, accounting for the broken wires in the neighboring panels, is: 5941 – 45 – 45 – 189 – 138 = 5524. At the end of this process, the final corrosion map of the cable cross-section will be comprised of 5524 wires but, contrary to Method 1, the ultimate stress 𝜎𝜎𝑢𝑢,𝑙𝑙 of these wires has remained unchanged.
From page 135...
... 135 8) The largest value of the cable force sustained by the long wires (the one corresponding to the (i-1)
From page 136...
... 136 Method 1 𝑅𝑅𝑢𝑢,𝑟𝑟𝑒𝑒𝑎𝑎𝑚𝑚 = 28,100 𝑘𝑘𝑖𝑖𝑘𝑘𝑘𝑘 Method 2 𝑅𝑅𝑢𝑢,𝑟𝑟𝑒𝑒𝑎𝑎𝑚𝑚 = 29,991 𝑘𝑘𝑖𝑖𝑘𝑘𝑘𝑘

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