Field Verified Model Example

The case of a bow string arch bridge with a very slender tie and a large arch rib which serves as the primary bending element will be examined (Kulicki, 1988). This particular bridge exhibited relatively large and long lasting vibrations under the passage of isolated trucks. The resultant problems were two-fold. The global vertical displacement field of the tied-arch spans was excessive and, due to the public’s loss of confidence in the bridge, intolerable. Secondly, out- of-plane distortions of the floorbeams, which were related to the global displacement, had resulted in ever-increasing fatigue cracking in the webs of the welded floorbeams at locations

Figure 4-10 – Circles showing locations of Typical Fatigue Cracks on Welded Floorbeams

A combined field and analytic approach was taken as it was clear that field verified computer models would provide the most confident basis for evaluating proposed retrofits on the bridge, which had a history of prior problems. Since the observed problems were dynamic in nature, investigations began with field studies using accelerometers and strain gages. Analytic studies utilized extensive finite element time history dynamic modeling and sub-structuring. The finite element models used to study the global dynamics were three-dimensional assemblages of plate bending, beam and truss elements as shown in Figure 4-11.

Figure 4-11 - Isometric View of Three-Dimensional Finite Element Model

Figure 4-12 is a comparison of analytically-obtained displacements compared to field-estimated displacements for two test trucks moving northbound and two test trucks moving southbound. Field data on displacements were obtained by estimating vertical displacement by sighting from a pier cap to a bolt pattern five panels away using a transit. While this type of measurement was quick and convenient, more accurate measurements could be taken with LIDAR, lasers and sometimes string potentiometers. Despite the expedient nature of the measurements taken in this case they were sufficient to produce confidence in the global model.

Figure 4-12 - Two Trucks – Unit V (Three-Dimensional Model)

Measured frequencies and mode shapes under imposed or ambient excitation provide a more comprehensive comparison assessment of the accuracy of a computer model because they combine the effects of stiffness and mass. Additionally, while the frequencies represent an integrated response, the mode shapes can sometimes point to local deficiencies or errors in a model. A comparison of field-measured and calculated frequencies and mode shapes for the arch bridge under consideration are shown in Figure 4-13. In this figure, the closed circles represent measured normalized amplitudes at the discrete accelerometer locations, while the continuous solid lines represent calculated mode shapes. Note that the mode numbers indicate that not all of the theoretical modes are significantly excited by the controlled truck traffic used in the field studies.

The measurement of strains was made under vehicles of known weight and under random traffic can be particularly useful where behavior of an individual member or detail is being investigated as may often be the case in fatigue or cracking investigations.

In the current case strain measurements were compared to analytic results obtained using a substructure shown in Figure 4-14, which contains 670 plate bending elements, 541 beam elements, 14 truss elements and 156 boundary elements interconnecting 592 joints, resulting in 5,690 degrees of freedom. For a given instant of the time-history obtained using the global model shown in Figure 4-11, the displacement field surrounding the substructure model was applied, which was then allowed to take the deformation pattern and, hence strains required to accommodate the global displacements. Figure 4-15 shows a comparison of field-measured and calculated strain histories from the substructure model at four locations in unwelded gaps between plates. These gaps were cope size, i.e. about an inch in length.

Figure 4-15 - Field vs. Calculated Strain Time Histories

The comparison of field and analytic results indicates that the computer modeling accurately represents the physical situation. This step provided a confident base onto which various retrofits could be superimposed.

Often investigations such as those described above involving a field verified computer model are important not only in understanding the problem at hand, but also in evaluating potential retrofits. In the case at hand, the effect of two promising retrofit concepts on the dynamic response under vehicular traffic were studied using the three-dimensional finite element models. A typical comparison of the two concepts and the original response is shown in Figure 4-16. This comparison shows the time history of quarter point displacements under truck passage calculated for the original configuration and hypothetical retrofits based either on adding inclined hangers or conversion of the tie into a stiffening truss. The results of these studies indicate that while both were quite effective in reducing the unacceptable vertical displacement, the stiffening truss retrofit concept was more effective in reducing the crack-inducing distortion of the welded floorbeams.

Figure 4-16 - Quarter Point Displacements for Unmodified and Retrofitted Structure

In this case, the implemented solution involved the addition of a stiffening truss under traffic using the original tie girder as the bottom chord, and the addition of numerous gap closing plates and angles. Field tests after retrofit indicated that the reduction in displacement was almost exactly as predicted in the computer simulations.

4.5 RAMIFICATIONS OF INCORRECT BOUNDARY CONDITIONS

Boundary conditions seem to be a recurring problem in FEA. Consider the structure shown in

Figure 4-20. This was a relatively straightforward bridge designed by grid analysis. The designer had a good model for this structure, except that the rotational degree of freedom corresponding to the global "X" axis was fixed instead of being released at all of the bearings. This did not allow the diaphragms at the piers and abutments to respond correctly to the imposed loadings and deformations, and also had the effect of producing artificially stiff ends on the girders by virtue of vector resolution between global and local systems. The effect of this condition on the reactions obtained at the abutments and piers was dramatic.

For brevity, Table 4-3 shows only results for the far abutment. It can be seen that the reactions are quite non-uniform. with a very substantial uplift reported at the acute angle. Note the moments at the pinned ends of the girders in the moment diagram reflecting the incorrect reactions shown in Figure 4-17. Also shown in Table 4-3 are the correct reactions determined when the structure was modeled as a grid with proper boundary conditions at the supports using a generic computer program for matrix structural analysis. In this case, a positive reaction is found at all bearings, and a significantly different moment diagram also resulted as shown in Figure 4-17.

investigated utilizing a relatively complete three-dimensional finite element analysis. The deck slab, girders and cross-frames were modeled in their proper relative positions. The comparison of reactions obtained with the grid and FEA models was excellent.

Table 4-3 - Live Load Reaction

FAR ABUTMENT SUPPORT REACTIONS GIRDER NO. “GRID” CORRECT SUPPORT CONDITION DESIGNER’S INCORRECT REACTIONS

“GRID” INCORRECT SUPPORT CONDITIONS USING DESIGNER’S

ASSUMPTIONS

(VERTICAL – k) (VERTICAL – k) (VERTICAL – k) (MOMENT X k-ft)

1 66.68 223.61 223.24 811.24 2 64.36 47.07 47.14 1231.35 3 64.93 86.29 85.49 1229.66 4 66.62 117.94 117.94 1237.95 5 69.46 -81.63 -81.51 813.37

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References

Anger, Georg, Ten-division Influence Lines for Continuous Beams, F. Ungar Publishing Company, 1956.

Hambly, Edmund C., Bridge Deck Behavior, John Wiley and Sons, Inc., New York, NY, 1976. Kulicki, J.M., Mertz, D.R., and Murphy, R.E., Dynamic response and subsequent retrofit of a

tied-arch bridge, 13th IABSE Congress Report, 1988.

Table of Contents 5.1 DEAD LOADS ... 5-2 5.1.1 Modeled Components ... 5-2 5.1.2 Non-Modeled Components ... 5-2 5.2 LIVE LOADS ... 5-3 5.2.1 AASHTO LRFD Requirements ... 5-3 5.2.2 Additional Requirements ... 5-3 5.2.3 Modeling Vehicle Live Loads ... 5-3 5.2.4 From Wheels to Loads ... 5-5 5.2.5 Live Load Optimization ... 5-5 5.2.6 Wind Loads ... 5-9 5.2.7 Centrifugal and Braking Forces ... 5-10 5.2.8 Thermal Loads ... 5-10 5.2.9 Verifying Live Loads ... 5-11 5.3 PRESTRESSING LOADS ... 5-12 5.4 NONLINEAR LOADS ... 5-12 APPENDIX 5A ... 5-13 APPENDIX 5B ... 5-14 References ... 5-15

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