In the article "strut-and-tie modeling for pier caps", we have discussed the definition of strut-and-tie analysis and how to construct a strut-and-tie model using the example of pier cap. After creating the geometry of a strut-and-tie model, the next step usually is calculating dead and live loads from the superstructure. This article discusses how to determine the boundary loads for a pier cap with a superstructure that has irregular geometries.
What is Lever Rule?
Applying the lever rule allows us to consider the deck to be simply supported between girder lines in the transverse direction. We can then use a line girder analysis to determine the tributary length for loads on each supported superstructure span, and use different live load placements that maximize these reaction force (shear and moment) from the deck at the girders following the AASHTO rules for lane width, vehicle placement, and multiple presence provisions.
Is Lever Rule Always Able to Obtain Superstructure Dead and Live Load?
However, there are cases where lever rule cannot be applied. Figure 1 shows a bridge with a significant skew, and with a girder line that stops at the top of a pier, and with cross frames that are oriented at a different angle than the piers, and with different span lengths. Using lever rule in this case would not be accurate to determine what the superstructure reaction should be that is applied to the pier cap. Therefore, we included the superstructure model and the piers in the same analysis model and used the superstructure model to determine what the bearing loads should be. For this analysis to be accurate, all the live load forces for each of the different cases need to be concurrent with one another, and we cannot use a live load envelope. This article shows the process of determining the concurrent live loads using midas Civil to find the load cases that maximizes the shear and moment reactions at the pier cap.
Figure 2 shows the zoomed view of the region highlighted in blue box from figure 1. What we are trying to do is to figure out what would be the live load from the superstructure that maximizes the vertical shear and negative moment at the element that is highlighted by the blue box. We can define that as an equivalent static load case so that we can be sure its loads are concurrent. In this example, we are interested in obtaining the maximum HL-93 shear and negative moment in the element 30053 part i, shown in figure 3.
To determine which live load maximizes the loading conditions at element 30053i, the following steps can be followed. As shown in figure 4, in midas Civil post-processor, we can select all the individual load patterns, and select element of interest (30053i). After requesting the list of forces at the element 30053i, we can visually determine which of the live load pattern that maximizes the forces at element 30053i, as shown in figure 5.
After determining the load pattern that maximizes the loading conditions at element 30053i (load pattern: C2_w/Dbl Trk_min), we can visualize this load pattern and save it as an equivalent static load. To visualize this load pattern: Results -> Moving Load -> Moving Tracer, would show us the position of the vehicle that contribute to the load which maximizes the shear and vertical moments at the element of interest, as shown in figure 6.
To define this load pattern as an equivalent static load case for strut-and-tie analysis, follow the highlighted blue boxes in figure 6 and click "Write Min/Max Load to File", we can get a text-based file (.mct file) shown in figure7.
After saving this .mct file, you can go to Tools - MCT Command Shell -> Open to open the above saved .mct file and hit "run". This would create a static load case shown in figure 8, and by adding this equivalent static live load case, we can get the concurrent loads for our strut-and-tie model.
Daniel Baxter is a bridge department manager in Michael Baker's Minneapolis office, where he also serves as a project manager and senior bridge engineer. He has been involved in both the rehabilitation, design, and load rating of many complex bridges such as the Third Avenue Bridge, Winona Bridge, High Bridge, Franklin Avenue Bridge, Lafayette Bridge and Hastings Bridge, all over the Mississippi River. He also recently led the analysis of 18 existing pier caps with sectional and strut-and-tie methods and helped develop procedures for the design of 38 new pier caps using the strut-and-tie method. He also serves as an NHI-certified instructor for multiple NHI bridge design and load rating training courses, including Strut-and-Tie Modeling for Concrete Structures. He is highly experienced in complex finite element analysis, has given eight presentations at the International Bridge Conference, and has published several papers in peer-reviewed journals. He graduated with a B.A. in Physics from Carleton College in 2001 and received a B.S. in Civil Engineering and an M.S. in structural engineering in December 2003 from Washington University in St. Louis. He is a licensed professional engineer in Minnesota and a licensed structural engineer in the state of Illinois.