Multi-scale modeling (M-S model)

The multi-scale model (M-S model) of the Memorial Bridge is created through a stepwise procedure, where the groups of members that were initially modeled with the shell elements in the SH-model are replaced with a single beam element to develop an efficient M-S model. This stepwise procedure for the M-S model development started with the least critical to the most critical groups of members.

After the dimensional reduction of each group, a comprehensive comparison is made between the structural responses of the M-S model to the SH-model and B-model and ultimately to the field-collected data. The optimum location for the interface point of different dimensions is determined through minimizing the difference between the structural response of the M-S model to the other FE models and the field-collected data. The error estimations are performed using the second term of the energy norm expressed through the following equation (36):

Eq.(4.26)

where J2 _{is the stress jump for bending and shear force, p is defined as the polynomial interpolation} order, h and t are the dimension and the thickness of the shell element respectively (37). In the following, the procedure and the criteria for the selection of the members including braces, floor beams, diagonals and, chords, for dimensional change are discussed.

4.5.4.1Braces

Initial evaluation of the SH-model indicated that the braces are one of the least stressed group of the bridge’s members, which make them the leading candidates for the dimensional reduction. The braces of the Memorial Bridge are connected through the bolted joints to the web

2 (1 ) 24 error h t J dy Ep

n

d

ò

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of the gusset-less connection. The developed out-of-plane constraint equations are applied at the bolted area to couple the braces to the gusset-less connections shown in Figure 4-9a. The results of the SH-model show that the axial performance of the braces does not generate significant stress concentration at the connected gusset-less connection as it dissipates around the bolted area.

4.5.4.2Floor Beams

Floor beams connecting the eastern and the western trusses of the bridge at the bottom chords are the highest stressed members due to the traffic loads. The initial results of the SH-model show that the floor beams can be replaced by a single beam element when they are coupled to the shell element with an appropriate interface location. In this model, the floor beams are coupled to the edge of the stiffeners at the bottom chord representing the bolted connection between the floor beam and the stiffener shown in Figure 4-9b. This modeling approach provides a planar coupling to the floor beams along the stiffeners. There are also skewed beams tying the floor beams to the bottom gusset-less connections.

The procedure for reducing the dimension of the skewed beams is similar to the braces as they have significant axial performance. However, due to the high transferred load from the floor beam to the gusset-less connection, the concentrated stresses may require a larger area of coupling at the gusset- less connection compared to the actual bolted area. This larger area allows for stress concentration dissipation while it increases the time of analysis by increasing the number of DOFs involved in the constraint equations.

4.5.4.3Diagonals

The diagonals connect the gusset-less connections at the top chord to the bottom chord, as shown in Figures 4-9a and 4-9b, respectively, through the bolted joints and have a dominant axial

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performance. The initial analysis of the SH-model showed that in the multiscale modeling of the diagonals with the beam elements, the interface point location is less influential on the global performance of the model. The coupling between the different dimensions is performed at the bolted end of the diagonals.

4.5.4.4Top and bottom chord

Top and bottom chord plate girders at the Memorial Bridge are uniformly connected to the gusset-less connections, as shown in Figures 4-9a and 4-9b. The initial structural analysis results of the SH-model showed that these members are the high-stressed regions that require careful considerations for dimensional reduction. The error estimation procedure in Eq. (4.26) is applied to find the appropriate location for the interface point. The results of the estimated error show that the interface location for the top and bottom chord may not be identical. This difference originates from the unequal performance of the chord members as well as the connected members to them.

The complex geometry of the gusset-less connection also requires more consideration in selecting the interface location (Saint Venant principle). The appropriate position of the interface point is determined through minimizing the estimated error in Eq. (4.26) between the structural response of the M-S model as compared with the SH-model and the field-collected data. In the developed M-S model, the interface location at the bottom chord is located at a distance equal to three times the depth of the cross-section from the connection center point. The interface distance can be reduced to two times the depth of the cross-section for the top chord connection to ensure that the interface point will not conflict with the flange curvature. The defined interface location depends on the geometric properties and the structural performance of the component as well as the connected members that may require either planar or out-of-plane coupling conditions. In the

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less complex bridges requiring the planar coupling condition, the distance for the interface point can be considerably decreased, which reduces the number of higher dimension elements (e.g., shell element) required for an appropriate M-S model.

Shown in Figure 4-9a and Figure 4-9b, are the coupling conditions of the members modeled with beam elements and coupled to the gusset-less connections modeled with shell elements for the top and bottom chord, respectively. Similar efforts are performed for the gusset-less connections at the tower. The finalized multiscale model is shown in Figure 4-10. In the next section, all four developed models are compared to the field- collected data for model verification purposes and to highlight the advantage and disadvantage of each developed model. In addition, the comparisons are quantified using statistical postprocessing.

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