Performing structural analysis of the cable-stayed Owensboro Bridge was quite a challenge because:

• The usual large number of loading conditions had to be applied to a complex structural model of the cable-stayed structure. Loadings included:
  – Dead load
  – Live load
  – Wind load (with its additional complexities for a cable-supported structure)
  – Seismic load
  – Thermal effects
  – Dynamic effects.
• Temporary construction stage conditions had to be studied because the bridge is cable-stayed and will be constructed by cantilevering deck sections from the towers. We analyzed more than 100 construction stages for dead load, live load and wind.
• Accidental cable loss and cable replacement, two distinct conditions, had to be analyzed.
• Four critical construction stages that were to be tested in the wind tunnel were analyzed for mode shapes and natural frequencies.

The analysis was further complicated by the fact that cable-stayed bridges are nonlinear structures.

Linear vs. Nonlinear Structural Analysis

When doing structural analysis we normally assume that displacements are small. Small displacements do not affect the stiffness characteristics of the structure and load is directly proportional to displacement. The structure is said to be linear.

In cable-stayed structures, however, displacements are large. Large displacements do affect the stiffness characteristics of the structure and load is not proportional to displacement. The structure is said to be nonlinear.

Nonlinear analysis requires an iterative approach that adjusts the geometry and the stiffness matrix after every analysis cycle until the results of two successive iterations are within a specified tolerance. Theoretically, nonlinear computer modeling and analysis is only slightly more difficult than linear analysis, requiring additional input procedures and additional members. In practice, however, nonlinear analysis requires a lot more debugging and tweaking of the model than does linear analysis, taking up considerably more computer and design time. We used LARSA to run the analysis, a program that has linear, nonlinear, static and dynamic capabilities.

Keeping Analysis Manageable

With early planning we kept the analysis manageable, which, in turn, kept costs down for computing and engineering time. We used some simplifications that reduced the number of nonlinear runs. For example, we did nonlinear analysis for dead load, but for live load we did linear analysis and applied a nonlinear magnification factor. (The magnification factor was arrived at by comparing linear and nonlinear analysis runs for specific load conditions.) To obtain the live load forces we generated influence lines and plotted the forces (moments, shears and axial forces) for each bridge element interactively on GDS CADD.

We further reduced the number of nonlinear runs by performing calculations to determine that the design is governed by cable loss cases rather than cable replacement cases. Since each condition alone would have required one nonlinear run for each cable (on one side of the bridge), these calculations saved 48 runs.

Computer Model

We developed one basic computer model and then adjusted it as needed for each analysis run. The 3-dimensional model extended from expansion joint to expansion joint, a total length of 1,031 meters. Some of the more relevant characteristics of the model are listed below:

• The deck was made up of edge girders and equivalent transverse beams at cable points.
• Horizontal cross bracings between edge girders simulated the deck transverse rigidity.
• Prestress force was input for the cable elements.
• Each cable was modeled with four elements for the nonlinear analysis.
• For the linear and dynamic analyses, each cable was modeled as a single element from top to bottom due to stability requirements.
• Joints were located along edge girders at cable points, field splices and midpoints between cables.
• Each tower was modeled in 3-dimensions.
• At the top of the tower, a single vertical member with horizontal diagonal members connecting the anchor points was modeled.
• The approaches were modeled in 2-dimensions. The superstructure was treated as a single line element to reduce the size of the stiffness matrix.

Figure 1: Mode Shapes

Analyzing Mode Shapes

With long-span bridges being sensitive to wind, dynamic analysis must be implemented to ensure their stability under low and high wind speeds. The analysis provides the period and frequency of vibration of the structure at different wind speeds. These data are then used in wind tunnel testing and seismic analysis.

A separate 3-D computer model was developed for the dynamic analysis. It was adjusted for proper mass distribution, maintaining the total mass and mass moment of inertia of the system.

The dynamic analysis was run for 25 modes. (See Figure 1 for examples of mode shapes.) This covered vertical bending, lateral bending and torsional rotations of the structure. Periods and frequencies were extracted for each mode and the deformation shapes were plotted on CADD.

Next, we ran the seismic analysis to evaluate seismic induced forces in transverse and longitudinal directions . We used response spectra from AASHTO for Type I Soil and the frequencies obtained from the dynamic analysis, combining the frequencies by the complete quadratic combination method.

Conclusion

Structural analysis of the Owensboro Bridge was a major undertaking, even with the measures taken to reduce the effort. Planning ahead and anticipating the volume of analysis to be performed was critical to the success of the project.