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NCHRP 12-79 was a big research project, and it took a big research team to deliver it. The lead organization of the research team was the Georgia Institute of Technology, with Professor Don White as the Principal Investigator. HDR was a major subconsultant, performing most of the 1D and 2D analysis modeling. We also had the Markosky Engineering Group, Tensor Engineering, High Steel Structures, and Professor Ted Galambos on the team. What I really want you to notice about this team is that it represents a broad cross-section of the steel bridge industry. We had academic researchers, practicing design engineers, steel detailers, steel fabricators, and steel erectors, all contributing their knowledge, insights, and experience to the research.

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So, what we did we want to learn? Well, we asked three basic questions: First, when are simplified 1D or 2D analysis methods sufficient… that is sufficient at predicting constructed geometry and evaluating strength and stability of steel girder bridges during all stages of erection? Second is the converse question: When are more refined 3D methods necessary? Finally, while answering these first two questions, we also wanted to learn more about the behavior of steel girder bridges during erection, deck placement, and under final loads.

We asked these questions for a reason… we wanted to provide some guidance to the industry. Specifically we wanted to provide recommendations on the appropriate level of analysis for bridges based on their geometric and construction complexity. We also wanted to provide guidelines for erection procedure plans and the associated erection engineering analysis.

And, we wanted these recommendations to be quantitative, not qualitative or subjective. To that end, the recommendations of NCHRP 12-79 are based on analyses of over 70 existing and parametric I-girder and tub-girder bridges.

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You might be asking yourself “What’s the big deal here? What’s so special about curved or skewed steel girder bridges? What makes them different from straight, non-skewed bridges?”

Curved and skewed steel girder bridges behave in ways that are fundamentally different than straight, non-skewed bridges. Much of this is due to the torsion that occurs in these structures.

WALK THROUGH BULLETS.

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There are a number of different methods available to analyze curved and skewed steel girder bridges. For the purposes of our studies, we grouped the various analysis methods into three broad categories, which we called the three “Levels of Analysis.” In the order of increasing levels of refinement and complexity, these are Approximate, or 1D methods, 2D methods, and 3D methods. Each of the over 70 bridges we studied as part of the 12-79 research project were analyzed using all three levels of analysis. We’ll discuss each level of analysis in a little more detail on the next few slides.

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For each of the over 70 bridges we studied, we followed a fairly methodical analytical process. First, we defined the bridges. In the case of existing bridges, their plans provided that definition. But most of the bridges we studied were what we called “parametric” bridges, bridges configured with a certain set of geometric parameters we wanted to evaluate. For the parametric bridges we first performed a design, determining flange sizes, web depths, cross-frame member sizes, etc. Each bridge was designed to meet all appropriate AASHTO LRFD design requirements, so that each bridge we studied represents a realistic design. Once we had the design completed, we developed a conceptual erection sequence for each bridge, and we identified several critical stages in the erection sequence. Then we analyzed each of those erection stages… we performed approximate analysis, 2D analysis, and 3D analysis of each critical stage of erection. Then we compared all the analysis results and evaluated the accuracy of the simpler 1D and 2D methods vs. the benchmark 3D analysis results.

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The nonlinear 3D ABAQUS models were quite sophisticated, as you can see from the graphic on this slide.

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As you might imagine, comparing the structural response for various interim and final stages of erection, for a variety of response categories, for a variety of structural elements, at a number of different locations in the bridge, for multiple different analysis methods, involves a LOT of numbers. Comparing the predictions of each of the various analysis methods and understanding the results was a major task for the grad students at Georgia Tech. They used graphs to help illustrate some of these comparisons, and they also performed complex statistical analyses, developing what they called the “error index” for each response category to make recommendations on level of analyses.

We would compare stresses, deflections, cross frame forces. The 3D analysis was considered the “benchmark” and we would measure the difference of the 1D and 2D predictions vs. the 3D predictions.

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Most current 2D software packages model girders using line elements which neglect warping. As you know, there are two main elements in the torsional response of a beam: St. Venant, or pure torsion, and Warping Torsion.

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The catch is that in open-section members such as I-shaped girders, the main source of torsional resistance is warping, also called Flange Lateral Bending. The girders in curved or skewed steel I-girder bridges are subject to torsion, and omitting consideration of warping can dramatically underpredict the torsional stiffness of the girders, and that can potentially lead to inaccuracies in the analysis.

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Here is the proposed improvement to the modeling of torsional stiffness of I-girders, using an equivalent torsion constant, Jeq. This can be accomplished using some simple formulas which approximate the warping stiffness of an I-shaped girder. These formulas were developed prior to the 12-79 research project, but are being recommended in the 725 report. The first formula is for the case of warping fixed at both ends, for a segment of a girder within a span. The second formula is for the case of warping fixed at one end and free at the other end, for the segment at the end of a girder.

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If you have a bridge with truss-type cross-frames, you have to find a way to model the stiffness of that truss structure in a single line elements. There are a number of ways to do this. Two of the more common methods currently used are illustrated here… we called them the Shear Stiffness Method and the Flexure Stiffness method. In each case, you would build a separate model of the truss-type cross frame, apply a unit force effect, and use the resulting displacement to calculate an equivalent moment of inertia for the single line element used to model the cross frame. Regardless of which method is used, key stiffness parameters are omitted, again leading to the potential for inaccuracies in the analysis.

So, what can we do about this?

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In the NCHRP 725 report, we proposed two different methods for modeling the stiffness of a truss-type cross-frame using a single line element. The first method is to develop a custom finite element that includes consideration of all six key degrees of freedom in the cross-frame. For each degree of freedom, you can write stiffness parameters, as shown on this slide.

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You can proceed to do this for all six degrees of freedom, the axial, shear, and moment (or end rotation) degrees of freedom.

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The result is the stiffness matrix at the top of this slide. The eight highlighted terms represent the independent terms in the matrix. The rest of them are repetitive. In other words, we need eight stiffness coefficients to fully capture the cross-frame behavior.

The lower left matrix is the stiffness matrix for an equivalent Euler-Bernoulli beam; i.e., the matrix that we normally use for representing a cross-frames using a single line element. You can see that in the Euler-Bernoulli matrix there are a lot of zero terms, stiffness parameters inherently neglected in this simplified approach. Also note that the previously mentioned Shear Stiffness Method and Flexure Stiffness method each capture only some of the stiffness parameters, not all of them.

Finally, I’d like you keep an eye on the term in the full stiffness matrix in the red box. READ TERM

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You can see that same term here (READ TERM). This slide shows the programming of that full stiffness matrix in the custom finite element module of the FEM program LARSA. LARSA offers a module where users can program their own custom finite element, which is what one of the grad students working on the 12-79 research project, Andres Sanchez, did here. It was a relatively simple exercise… I think Andres said he developed this custom element in about half a day.

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Now, you may not want to have to program custom finite elements every time you design a bridge, so we also proposed an alternate approach. This alternate approach uses a shear-deformable (Timoshenko) beam element. First you calculate an equivalent moment of inertia from a pure bending model, then you calculate an equivalent shear area from a pure shear model. This is similar to the current Flexure Stiffness Approach mentioned earlier, but by using a shear-deformable element and considering the shear stiffness you end up with a better, more complete representation of the stiffness of the cross frame.

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These figures are taken from the 725 Report and illustrate this method. You can see it is relatively straightforward.

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So, how much better can 2D analysis be if you do a better job of modeling I-girder torsional stiffness and cross-frame stiffness? Lets look at some case studies…

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So with two improvements, we can realize a dramatic improvement in 2D analysis methods. But our intent is not for designers to be manually adjusting torsion constants in models or writing custom finite elements for cross-frames as part of production bridge design. Instead, our hope is that the software vendors will implement these improvements in 2D bridge design software programs.

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Next I’d like to talk about the Analysis Methods Matrix, perhaps the most significant result of the 12-79 research.

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The analysis methods matrix is a scorecard, or report card, addressing each of the analysis methods we studied. As mentioned earlier in this presentation, we analyzed over 70 bridges, comparing the results of 1D and 2D analysis methods to the results of 3D analysis methods. We graded the 1D and 2D methods based on how accurately their results matched the results of the 3D methods. We evaluated the normalized mean error and provided each method with a letter grade score. A grade of A indicates that the 1D or 2D method had a normalized mean error of 6% or less compared to the 3D method. A B indicates that the 1D or 2D method had a normalized mean error of 7 to 12%, and so on. We broke down the grades based on different categories of structural response and different categories of bridge geometry.

Here is the result. This is the analysis methods matrix for I-girder bridges. We also provided a similar matrix for tub-girder bridges.

You can see we broke down the grades by structural response, providing categories for major axis bending stress, vertical displacements, cross frame forces, flange lateral bending, and girder layover at bearings.

You can also see that within each response category, we further broke down the grades by bridge geometry. You can see that there are six geometry categories. First, there are two categories for curved girder bridges with radial supports, indicated by the letter C. The two categories reflect various ranges of what we called the Connectivity Index, I-sub C. The Connectivity Index is based on the radius of curvature and the number of cross frames per span. Next there are three categories, labeled with the letter S, for straight skewed bridges. The three categories reflect various ranges of what we called the Skew Index, I-sub S. The Skew index is based on not only the severity of the skew, but also the ratio of the bridge’s width versus span length. Finally, there is a category labeled C and S, for bridges which are both curved and skewed.

You can also see that in each case, the table provides both a Worst Case score, and the Mode of Scores. The Worst Case score is the score of the bridge in that category

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for which the worst accuracy was recorded. The Mode of Scores is the most frequently reported score from all bridges in that category. Seeing both the Worst Case Score and the Mode of Scores gives some feel for the range of scores in that category.

The Analysis Methods Matrix is a powerful tool for helping you choose an appropriate level of analysis for your particular bridge.

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Now, you might be thinking “Hmmm, I wonder how these scores would change if those 2D analysis improvements like the Equivalent Torsion Constant and the improved cross-frame modeling techniques were incorporated into 2D modeling?” Good question! Here’s the answer. We performed a study where we evaluated the accuracy of “improved” 2D models vs. the 3D benchmark models. We used the Equivalent Torsion Constant and the improved cross-frame modeling techniques. As you can see, the scores for the Improved 2D models are a lot better than the scores for the Traditional 2D models, mostly As and Bs, with a few Cs. No more Ds and Fs.

We’re hoping that the industry will adopt these improvements, and we’re encouraging several software vendors to pick up these changes in their programs.

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The goals of the Appendix B guidelines are three-fold.

First, to ensure that construction plans, methods of analysis, and other calculations are generally sufficient for predicting the constructed geometry of a bridge, in order to facilitate fit-up.

Next, to ensure adequate strength and stability during all stages of erection.

Finally, to achieve better consistency in erection plans and calculations.

We recognized that every bridge is different, and a relatively simple bridge may only need a few pages of plans and calculations to document its erection procedures, while a more complicated bridge may require extensive plans and a significant amount of relatively complex analysis. We tried to provide a comprehensive set of guidelines, organized somewhat like a shopping list, such that items that clearly don’t apply can be ignored, but all items are listed so that no critical items slip through the cracks.

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The recommended guidelines include specification language, commentary, and checklists. These recommendations have since been discussed and fine-tuned by the AASHTO/NSBA Steel Bridge Collaboration and are scheduled to appear in the upcoming 2nd Edition of their Steel Bridge Erection Guide Specification, S10.1.

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So, in today’s presentation, we reviewed the key findings of NCHRP Research Project 12-79, including recommended improvements to I-girder torsional models and cross frame models for 2D analysis, a new method for calculating flange lateral bending stresses in straight skewed bridges, improvements for calculating tub girder internal framing forces, and discussions of lack of fit effects and cross frame detailing methods.

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