Derechos civiles

The procedure to calculate the flange lateral stress from the 2D grid analysis for the Erected Fit detailing method has been specified in literature [9] and [7]. A brief summary of the procedure is provided here. Displacements corresponding to the concrete dead load from the 2D GA are used to calculate forces in cross-frame members. These forces are then resolved into vertical and lateral components at the connection point of the cross-frame and the girder. The flange is assumed simply supported or fixed ended between connections adjacent to the connection at which lateral force is obtained. Using the lateral bending moment at the location of lateral load of this idealized beam model, the lateral stress is calculated using flexural formula. In order to obtain fl from 3D FEM analysis, the mean value of the longitudinal stress at the two edges of the top flange is subtracted for the longitudinal stress at one of the edges of the top flange.

Flange lateral bending stresses obtained from different methods of analysis are compared for Girder 8 of Bridge A in Figure 3.7 and for Girder 4 of Bridge B in Figure 3.8 for the Erected Fit detailing method at the TDL stage. It can be noticed in both Figure 3.7and Figure 3.8 that fl is almost zero for the traditional 2D grid analysis that does not include warping term in modeling the torsional stiffness of girders. More appropriate values of fl are obtained by modeling the torsional stiffness of the girder correctly, i.e., taking into account the warping torsional stiffness. This warping torsional stiffness is incorporated into the improved 2-D grid analysis. Increase in the torsional stiffness of the girder by incorporating the warping stiffness makes the girder stiffer. The stiffness attracts more force and therefore flange lateral bending stresses increase. The effect is

more pronounced in fl than in the vertical deflections since small movement can have large effect in stress.

In the 2D grid analysis two assumptions can be made for the segment of girder between three consecutive cross-frames for the calculation of the lateral moment as explained in NCHRP 725 [9]. Assuming a simply supported (s-s) boundary condition for the segment gives more value of the lateral moment and thereby conservatively estimates

fl, whereas assuming a fix-fix boundary condition for the segment gives un-conservative estimates of fl. The boundary condition is somewhere between fix-fix and s-s in reality. However, such boundary condition is difficult to model. Results of this study indicate that the average of fl values obtained based on the two assumption constitutes an acceptable approach, which is in agreement with the recommendations of NCHRP 725 [9].

It can be concluded that the improved 2-D grid analysis with an average value of

Figure 3.7: Comparison of flange lateral bending stress calculated by different analysis method in Girder 8 of Bridge A

Figure 3.8: Comparison of flange lateral bending stress calculated by different analysis methods in Girder 4 of Bridge B

3.2.5 Cross-frame Forces

Cross-frame forces can be obtained from the 2D GA by multiplying displacements at connections of a cross-frame to girders to the axial stiffness of members in the cross-frame. Detail of this approach is given in NCHRP 725 [9]. Cross-frame forces from the 3D FEM analysis can be directly obtained from forces in link elements used for modeling cross-frame members. Comparison of cross-frame forces obtained from different methods of analysis is done and is shown for the top chord of cross-frames in bay 1 of Bridge A in Figure 3.9 and for the top chord of cross-frames in bay 4 of Bridge B in Figure 3.10 for the Erected Fit detailing method at the TDL stage.

It can be observed that the difference between cross-frame forces obtained from different methods of analysis is significant. Comparison also indicates that cross-frame forces are highest for the improved 2D grid analysis and lowest for the 3D FEM analysis in case of Bridge A. The improved 2D-grid analysis significantly over-estimates cross- frame forces compared to the 3D FEM analysis. Cross-frame forces evaluated from the traditional 2D-grid analysis are essentially zero, due to the gross underestimation of the girder torsional stiffness in the traditional 2D-grid methods for Bridge A. The difference in cross-frame forces for Bridge B is not very significant. The results of a broad range of analyses on the different bridges demonstrate that the improved 2-D grid analysis is sufficient to calculate the cross-frame forces. The results from the improved 2D-grid analysis are generally accurate and conservative compared to results from 3D FEM analysis.

Figure 3.9: Comparison of cross-frame forces calculated by different analysis method for Bridge A

Figure 3.10: Comparison of cross-frame forces calculated by different analysis method for Bridge B