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7 Audit Task 'c'. Estimation of a deterioration model
The objective of this task is to consider whether the assessed performance of the main cables over the longer term is reasonable. The NCHRP Guidelines were used as the basis for this assessment and the audit has considered not only whether the Guidelines have been followed but also whether the Guidelines are strictly applicable to the Forth Road Bridge main cables.
The strength degradation calculations also follow the general procedures of the NCHRP, but they have been carried out to a more rigorous level. They have also, at FETA's request, been developed for time periods of fifteen years and longer into the future. It should be noted that the NCHRP guidelines stipulate that the method should only be used to project 10% of the current age into the future (which is only four years for the Forth Road Bridge). It should also be noted that there has never been any calibration of the calculation method, and while a ten year look ahead may produce plausible results for a century old bridge, we consider that the results must be used with extreme caution for a forty year-old bridge. The uncertainty in the results is considerably greater than for the calculated strengths for the bridge in the as-inspected condition.
In recognition of this, Weidlinger departed from the basic NCHRP method in an effort to produce meaningful results. A great deal of thought went into the methodology, which we will outline below:
In Weidlinger's latest calculations it is assumed that the cables began corroding immediately after construction (also not in accordance with NCHRP, but more reasonable).
Based on the conditions found during the inspection, Weidlinger had developed a lower bound model which assumed that the cable can be divided into 3 "environmental zones", the top segment, bottom segment and side segments (both sides taken as one segment). The corrosion rates in these zones vary. This in itself is plausible and has been observed in studies of some of the Honshu-Shikoku Bridges (See Structural Engineering International 3/2000).
Using the wire populations in these zones, Weidlinger modelled a hypothetical cable comprised of the three different corrosion zones, with the more severe corrosion occurring at the outer part of the cable and deeper wires having less advanced corrosion. This expedient is used to aid in calculations only and isn't meant to represent the real situation. After projecting the wire deterioration for each zone, the results are then combined to project the strength of the real cable.
The projected corrosion rate of individual wires is not based on repeated inspections of actual bridges (there have not been repeated inspections that can be used to develop such data). Corrosion rates are actually based on laboratory tests being conducted in the US as part of a study commissioned by the Triborough Bridge and Tunnel Authority. This study is headed by Weidlinger with Ammann & Whitney as a named sub-consultant. In these tests, individual wires are being subjected to various corrosive environments and monitored for the advancement of corrosion; the wires are held under varying tensions. It has been observed to date that wires progress from Stage 1 to 2 to 3 to 4 over approximately 6 month intervals for each step. To date (over 2 years of the current testing according to Weidlinger) these tests have not produced any cracked wires. It is intended to continue testing for another 8 years.
From these test data, Weidlinger correlates the corrosion rate inside the cable by a simple proportionality. For example, a Stage 3 wire in the cable got that way after 40 years, while the laboratory wire got to Stage 3 in 1 year. The real case wire therefore corroded at one fortieth the rate of the laboratory wire.
Using the above-described linear correlation, Weidlinger's model predicts how many wires will go from one existing stage to the next in a given time period
During the performance of the audit calculations, it was noted that if one assumes a linear rate of corrosion advancing from one stage to the next, the so-called deterioration rate of the cable becomes independent of the test data and is simply the corrosion stage divided by the number of the years the bridge has been corroding (in this case, the age of the bridge).
Weidlinger's assumptions extend to the rate of cracking and breaking wires, i.e. Weidlinger has assumed a Stage 4 wire becomes a cracked wire (called a Stage 5 wire for convenience in the current calculations) based on the same correlation between the laboratory's 6 months and the actual bridge wire's age. This is done even though the laboratory tests have not produced cracked wires in that time period (no cracks have been produced to date in the current tests nor in earlier tests performed at Columbia University).
The assumption that only the outer six rings of wires are cracked is based on test specimens taken only from the outer 11 wires of the cable. No samples were removed from deeper in. Weidlinger have advised that deeper wires cannot be repaired thus none were removed.
At the time of writing this report, Weidlinger indicated that they were continuing to refine their calculations. It appears they have only performed the analysis for panel 100S-100N of the east cable, so far (albeit that this is the worst panel discovered). Weidlinger did not indicate when their report would be finalised. The outcome of these calculations was a 6.7% strength loss in 2004 as noted in section 6.5.
We have performed independent calculations of the future degradation of the cable using Weidlinger's assumptions as well as alternative assumptions regarding the length of time a Stage 4 wire becomes cracked (Category 5), then broken (Category 6).
Using the same assumptions and calculation method as Weidlinger (with some approximations), we have independently computed results similar to Weidlinger's. These were advanced to produce the following plots for comparison to illustrate the sensitivity of the calculations to the assumed linear progression from one stage of corrosion to the next. The Weidlinger model assumes all wires advance to the next stage after 6 months. As explained earlier, there is no laboratory data to support the assumption that Stage 4 wires begin to crack after 6 months nor that the next stage of deterioration (the cracked wires begin to break) occurs after another 6 month interval.
We have examined the effect of these final two stages taking a period of T years under the laboratory condition to crack ( i.e. Stage 4 wires become Category 5 wires) and that Category 5 wires begin to break T years later (they become Category 6 wires). Values of T of 0.5 (Weidlinger's assumption) to 1.0 and are plotted in Figure 1, but, it should be noted, they have been based on the consensual 8% loss of strength model at 2004, not the 6.7% model. The purpose of this is to demonstrate the variance in the predictive models. Cable strength units are kilonewtons. 4
Figure 1 - Cable Strength Deterioration Estimates
An additional plot is included to highlight the range of predicted loss of strength models. These are shown against notional factors of safety of 2.0 and 1.9. It can be seen that the point in time when the safety factor for the main cable falls below these values for the BSALL live loading is between 2013 to 2018.
Plot of uncertainty of acceptable cable capacity
Figure 2 - Range of strength loss models
The results of the above calculations illustrate that the projected strength loss is very sensitive to the rate of deterioration of Stage 4 wires and that the degradation models should not be viewed as definitive, especially for projections beyond the 10% value given in the NCHRP Guidelines. However, it confirms that without intervention there will be a significant loss of strength in the longer term.
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