Deterministic MLA Simulation

To assess the reliability of the shear connection as a whole, and not just the reliability of a single connector, information about the progressive failure of studs must be obtained. Since the studs that undergo the highest stress range with the passage of a fatigue code truck are located at the ends of the girder, these studs can be expected to fail first. A prediction can even be made as to how many cycles it will take for this to happen, using fatigue results from beam tests completed and described in Chapters 3 and 4, or using the current design rules. However, it cannot be known what will happen after the first connector fails without the use of the FE program, informing how the shear stud forces will redistribute in the full girder and how long subsequent critical studs will remain intact before also experiencing fatigue failure.

The deterministic MLA simulation is a single simulation case where critical studs (those with the highest stress levels) fail after the expected number of cycles and are reprogrammed with a reduced stiffness. Two cases of reduced stiffness were considered: 50% reduction in stiffness, and 100% reduction in stiffness (complete failure of the studs). The latter was used for conservatism, assuming that fatigue failure completely disengaged a stud in transferring any longitudinal shear force. The 50% reduced stiffness case was meant to reflect the observations made during the beam testing at the University of Waterloo (UW), where studs were found to fail in a crack pattern that allowed them to transmit shear even after a large fatigue crack was present, or even propagated completely from one side of the stud to the other. Cracks

0% 20% 40% 60% 80% 100% 120% 0 5,000 10,000 15,000 20,000 25,000 30,000 D eg re e o f In te ra c ti o n ( % )

were found to move into the top flange of the steel section, resulting in a wedge capable of transmitting shear through mechanical bearing. On average, studs were found to lose only about 20% of their stiffness, long term, after a fatigue failure. A reduced stiffness value, termed the “post-failure stiffness” (PFS), of 50% was chosen as a conservative approximation (see Figure 6-8). In both cases, the conservative assumption is made that stiffness loss occurs immediately; in the tests, the studs were actually found to lose stiffness gradually as the fatigue cracks propagated. Results of this analysis will be shown for the 100% stud post- failure stiffness reduction assumption until near the end of this chapter, when the probabilistic simulation for code calibration is presented.

Figure 6-8: Post-failure long-term stiffness (PFS) reduction of CS and PS studs.

The fatigue endurance limit concept presents a challenge when conducting failure simulations. The current fatigue code provisions state that any shear stud experiencing a stress range of under 24 MPa with the passage of a fatigue code truck sustains no damage, and will not fail regardless of the number of loading cycles. As discussed previously, almost all highway bridges designed in Canada, including the bridge under discussion, are required to have studs designed at this endurance limit due to the high number of loading cycles expected. Furthermore, the design stress may be 24 MPa, but the actual maximum FE connector stress for this bridge is only 20 MPa, as shown in Figure 6-4. To avoid the complexities introduced by the endurance limit, the decision was made to conservatively ignore this limit, and assume the S-N curve to maintain a slope of m = 3 past the endurance limit. This is consistent with the assumption made in the probabilistic simulation, since statistically an endurance limit is arguably fictitious if the truck weight statistical distribution does not have a maximum value. The S-N curve used in the deterministic MLA simulation was the mean curve obtained from the UW beam test results shown in Figure 4-23.

Since the simulation is meant to capture the results of multiple stud fatigue failures, an overall failure definition for the girder shear connection was necessary. One obvious failure point is the violation of the ULS strength requirement; if the number of studs remaining on the girder was less than the number required to resist the factored moment on the girder, this simulation should certainly be terminated. For the example girder, this corresponds to a degree of shear connection of approximately 50%, or 78 studs. However, a more conservative approach was desired. Another approach could be to define failure when the bridge girder reaches a degree of shear connection of 100%, or 154 studs remaining. This would indicate that the margin between the studs required for strength and fatigue was depleted. This approach would be intuitive, but would still result in about 270 stud failures, or 64% of the studs initially installed on the bridge.

An investigation into the deflection of the example girder as studs failed progressively revealed that a sharp increase in deflection occurred after approximately 204 studs failed. This can be observed in Figure 6-9, where the deflection increase is plotted on the y-axis, and the number of studs failed (top), as well as the number of cycles (bottom) is shown on each x-axis. Points 1 through 4 shown in each plot correspond to points of interest during the progressive failure of the shear connection. Point 1 represents the undamaged girder, Point 2 represents the point of increase in deflection, Point 3 represents the point where a 100% degree shear connection remains, and Point 4 represents the point where only a 50% degree of shear connection remains (and Mr approximately equals Mf).

The definition of failure for the shear connection was conservatively chosen as Point 2, the start of the sharp increase in deflection when plotted with the number of cycles (N). A threshold of 15% deflection increase was chosen as the start of the sharper increase. For the example girder, this corresponds to a deflection increase from an initial 22.3 mm to approximately 25.6 mm, an increase of 3.3 mm. This is a conservative failure definition, and a convenient choice since the main output of the simulation, N, is not sensitive between Points 2 and 3 (and even Point 4 for the deterministic case, but this will not hold true probabilistically). The insensitivity to Points 2 through 4 is shown by the steep incline in the curve in Figure 6-11 (bottom). With this failure definition, Figure 6-9 (bottom) reveals that the deterministic MLA simulation results in a fatigue life of approximately 350 million cycles for the girder, or 4 times the number of cycles expected. This is reassuring, but there can be no indication of the reliability of this system without considering the probabilistic nature of the input variables. In the following section the probabilistic parameters are discussed and a probabilistic MLA simulation is performed.

Figure 6-9: Deflection increase plotted with # studs failed (top) and # cycles (bottom).