Estrategia creativa

To assess the reliability of the girder, statistical parameters were assigned to both the load (fatigue code truck weight) and resistance (stud connector strength). A procedure from the International Institute of Welding (Hobbacher, 2005) was used to assign a stud fatigue resistance curve to each connector in the model. These curves each have a slope of m = 3, and a vertical shift dictated by the failure distribution observed in n = 33 stud failures in the Waterloo beam testing program. The IIW procedure accounts for the uncertainty in estimating the mean and standard deviation of log(C) due to the small sample size, giving values associated with a two-sided confidence interval of p = 75% of each. In Equation 6-10, i represents the random variable corresponding to the logarithm of the y-intercept, C, for the S-N fatigue curve of a given stud. This is equal to the mean log(C) value of the sample data subtract the product of k and the standard deviation of log(C). The factor k is given in Equation 6-11, and may be used to simplify the Equation 6-10

into Equation 6-12. Note that the inverse normal distribution function shown will give a random variate with a mean of zero and a standard deviation of one.

Equation 6-10

i= −x k s 

1

t

n

k

n



−

=

+ 

Substituting p = 0.75 and n = 33 gives:

Equation 6-12

i=12.33− 

0.500

Equation 6-12 gives i, or log(C), for a given stud by substituting the actual mean (log(C) = 12.42) and standard deviation (0.424) of the sample data into Equation 6-10, along with k from Equation 6-11 for p = 0.75 and n = 33. The method in Equation 6-10 through Equation 6-12 contrasts with the simplified approach of ignoring the uncertainty in the estimation of the mean and variance, which is given in Equation 6-13 and Equation 6-14.

Equation 6-13

i= − x

 s

Equation 6-14

i=12.42− 

0.424

By comparing Equation 6-12 and Equation 6-14 it can be noted that the effective mean is 0.7% lower, and the effective standard deviation is 18% higher with the more conservative IIW procedure. Figure 6-11 shows both the IIW curve and the simplified approach, plotted next to the test failures and the current CSA S6 design SN curve.

The probabilistic nature of the effective fatigue code truck weight was previously described in Section 6.1, and will not be further discussed. Details for the full MCS procedure may also be found there. The results of the analysis are now presented.

Figure 6-10: Mean SN (solid) and -2 standard deviation SN (dashed) used in analysis.

Results of three simulations (trials) of the probabilistic analysis are shown in Figure 6-11, two of which have been truncated based on the previously established failure definition. The deterministic simulation from Figure 6-9 is also shown for comparison purposes. Several observations can be made: the probabilistic results show a more gradual failure, with the initial stud failure happening earlier and with greater numbers of cycles between points of interest two, three, and four. In addition, the average number of cycles until failure in the probabilistic simulations is much lower on average than that of the deterministic simulation. No variability in stud strengths in the deterministic case represents the ideal situation for a long-lasting shear connection but does not provide the same length of warning time once the connection begins to fail. Evidently, the variability in stud strengths leads to early individual stud failures, which cause increased stress levels in the connection and shift the “unzipping” process to an earlier time frame.

In Figure 6-12, the failure patterns are shown for the same three simulations compared to the deterministic failure pattern. Studs remaining are shown on the y-axis, with stud locations on the x-axis; the graph should be read from top to bottom, as time passes, and studs fail. The deterministic pattern shows that after the stud with the highest stress fails, the closest neighbouring stud will fail next because it has taken over some shear from the adjacent, failed connector. The shear connection failure occurs from both outsides towards the centre of the bridge. However, in the probabilistic (stochastic) failure patterns it can be noted that initial and subsequent failures occur at random locations. Eventually an unzipping pattern similar to the deterministic pattern emerges, however, when a critical number of studs have failed in one area. All simulations shown in Figure 6-11 and Figure 6-12 were completed assuming no post-failure stiffness in the studs.

10 100 1000

1.0E+05 1.0E+06 1.0E+07

S tre ss Ran ge (M P a) = V ·Q ·s / ( I·A ) Cycles (#)

Figure 6-11: Deflection increase and # cycles, including sample simulation failures.

Figure 6-12: Comparison between deterministic and stochastic failure patterns.

To determine the reliability of the system, the probability distribution of the number of cycles until failure, N, must be estimated. A lognormal distribution was used to fit the data in a procedure that will now be demonstrated for 100 probabilistic simulations with studs having no post-failure stiffness. Three of the 100 simulations were already shown in Figure 6-11 and Figure 6-12. The factor of safety (FS) for each simulation is obtained by dividing N by the design life of 87.6 million cycles. The average N of all simulations was 199.8 million cycles (FS = 2.28) with a standard deviation of 37.1 million cycles (0.424·FS). On average, the first individual connector failure occurred after 23.2 million cycles.

Probability paper plots showing the goodness of fit for three distribution types are given in Figure 6-13. The upper left plot is for the normal distribution, and it shows a reasonable goodness of fit with a coefficient of determination (R2_{) value of 95.5%. Equation 6-3 gives a β value of 2.99, or 3.02 when using the mean and}

standard deviation from the normal distribution probability paper plot. A normal distribution is generally not thought to fit fatigue data as well as a lognormal distribution, however, and will not be used further.

A lognormal distribution gives a better goodness of fit with R2 _{= 98.7% (upper right plot in Figure 6-13). The}

respectively, for the 100 simulations, which is close to the values estimated using the probability paper plot (λ = 0.808·ln(FS), ζ = 0.185·ln(FS)). Equation 6-4 yields a β value of 4.50, or 4.37 when using the mean and standard deviation from the probability paper plot.

RELIABILITY NORMAL: β = 2.99 - 3.02

LOGNORMAL: β = 4.37 – 4.50 LOGNORMAL: β = 5.07*

*left side only

Figure 6-13: Probability paper plots for normal, lognormal, and modified lognormal distributions. The lower left plot in Figure 6-13 shows a lognormal probability paper plot with only the left half of the data used to draw the line of best fit. This is done because the pattern of the failures falling below the mean is more important than those falling above the mean (a violation of the design fatigue life will always be from the weaker half of the failure data). It was thought that this method would be particularly useful when considering the helpful effects of the endurance limit; considering only the left half of the data ensures that simulation results from “runouts” (or very long-lasting simulation results) will not increase the level of scatter estimated by the probability paper plot. This avoids the problem of runouts causing an illogical downward shift of the design S-N curve (e.g. the 95% or 97.7% survival probability curve) because their negative effect on the scatter outweighs their positive effect on the mean. No endurance limit was used for the 100 probabilistic simulations being considered but the standard deviation estimated by the left side only probability paper plot was still lower at 0.154·ln(FS). A reliability value of 5.07 resulted from the use of this method, and the probability plot goodness of fit fell in between the normal and lognormal values with R2 ₌

The probability density function for the 100 simulations is shown in Figure 6-14, along with the lognormal distribution using the estimated population parameters. The results from the simulations presented so far indicate that current code practice for the design of shear studs is very conservative as it relates to the simple span geometry used in the analysis. Not only is the reliability of the system estimated at close to β = 4.5, but this analysis has assumed no consideration of the endurance limit, no post-failure stiffness of the studs, and neglects the redundancy effects of having multiple girders.

The level of reliability that should be present in the full shear connection could be a matter of debate but should probably be between β = 3 and 4 based on Figure 2-13 from the CSA S6 Commentary (2014). This considers the fact that degradation of the shear connection could result in the collapse of the girder if enough studs are cracked and a ULS event occurs. This depends on the post-failure stiffness assumed for each stud and the amount of over-design with respect to the critical moment; the bridge under consideration probably would not fail under ULS conditions even if all studs were assumed to have failed, yet maintained 50% of their initial stiffness as was present at minimum in the testing. However, not only could degradation result in collapse or catastrophic failure, but the connection is largely uninspectable. The only times that inspections can be made are if decks are being replaced, which may start to happen less as design lives for decks increase through the practice of waterproofing and the use of stainless-steel reinforcing. One aspect of the shear connection failure that would shift the target reliability index lower is the gradual nature of the failure and the accompanied increase in deflection, which may provide some warning.

Figure 6-14: Probability density function of girder shear connection failures. 0% 5% 10% 15% 20% 25% P roba bi li ty # Cycles (N) Design Life N = 87,600,000 Cycles

One hundred simulation was chosen as the number of trials necessary for this analysis based on Figure 6-15, where the reliability index is shown as the number of simulations is performed. There are diminishing returns for performing more than one hundred simulations.

Figure 6-15: Reliability index with the number of Monte Carlo Simulations performed.