Question
How to use the Lack of Fit option during Construction Stage Analysis of a Cable-Stayed Bridge?
Answer
The Analysis Manual explains the calculation of LOF forces for truss & beam elements as follows.
Using the following models, we will demonstrate how to use the LOF option for a simple cable-stayed bridge.
Model A is a complete stage static analysis model used to calculate Unknown Load Factors (ULF) for the cables. Due to symmetry, there are only four unknown pretension forces. Each cable is assigned a 1 kN pretension load. In addition to the self-weight load case, these pretension loads are included in the load combination "ULF".
Let us determine the unknown load factors for the pretension forces.
Post analysis, go to Results tab > Bridge Specialization > Cable Control > Unknown Load Factor and click on the "Add New" button. Set the item name as "ULF" and click on "Table" to add the displacement constraints.
We assumed the following constraints to obtain the ULF, as displayed below.
1. Vertical deformation is limited to +/- 10mm at the nodes where the cable intersects the deck.
2. Horizontal deflection at the top of the pylon is limited to +/- 3mm.
We can input the above displacement constraints as inequalities in a tabular format. After entering the data in "mm", click "Close".
In the ULF detail window, set the Object Function Type as "Linear" and check ON the unknown pretension load cases. Since all the cables must be in tension, select the Sign of Unknowns as "Positive" and press the "Get Unknown Load Factors" button.
Click on the "Make Load Combination" button & name it "ULF Result" with a description, if any.
The load combination "ULF Result" will be generated as follows.
Model B is a forward construction stage analysis model. Here, we will use the above pretension forces as initial internal cable forces.
Switch on the "Lack-of-Fit Force Control" option in the Construction Stage Analysis Control data tab. Typically, this should be activated for cable elements only since the internal forces of the segments can't be adjusted on-site. However, for simplicity, all the elements are added to a structure group "All" and assigned for lack of fit force control.
After analysis, by using the same displacement constraints as in Model A, we will recalculate the unknown load factors for pretension loads at each construction stage for Model B.
The iterative analysis is done until the factors approach unity.
Below are the final pretension forces to be updated in Model B.
From Cable Control > Cable Force Tuning, we can see the stage-by-stage results due to cable pre-tensioning.
The sum of pretension and LOF is as follows. Note that the sum is positive, indicating tension in all the cables.
After obtaining the final pretension loads, any of these two procedures, as demonstrated in Model C or Model D, can be adopted to compute the final cable forces.
In the CS2 - Cable 1 stage for Model B, the truss forces for each cable are as follows. Even though the applied pretension in cable 1 is only 284.9 kN, the truss forces are 325.1 kN and 320.6 kN, respectively, as shown below.
This is because the internal force is further redistributed in the structure, resulting in a different final force in the element.
Once all the constraints are satisfied, we must turn off the lack-of-fit force control option and set the cable pretension force control as an external type. When the cable pretension force type is external, the force isn't redistributed in the system in the same stage. Hence, the applied load will equal the force in the cable element. Also, note that the truss forces at the and j ends of the cable are different. This difference is due to the self-weight of the cable. Ideally, the applied external pretension load should be the average of the maximum & minimum force in the truss. However, for simplicity, let's consider the truss force near the deck. So, the Pretension 1(ST) load should be changed to 320.6 kN. Similarly, Pretension 2, Pretension 3 & Pretension 4 values could be obtained from CS 4, CS 6 and CS 8 stages as shown below.
In Model C, the lack-of-fit option is turned off, and the cable pretension force control is changed to the external type.
In Model C, the pretension loads are modified as follows, based on the truss forces closer to the deck.
As expected, the vertical stage/step real displacements of the deck at every construction stage do not exceed 10 mm.
Another model, Model D, is also attached for reference, where the averaged truss forces are applied as cable pretension with the external option.
The results of Model D closely match with those of Model C.
These sample models demonstrate how the lack-of-fit option works for a simplified cable-stayed bridge.
Note:The lack of fit function is used when we are trying to determine the final cable forces. Once the forces are known, the lack-of-fit option will be disabled. Ensure cable pretension forces are set as “internal” when LOF is switched on. We do not apply the sum of pretension + LOF as the final external cable pretension loads; instead, the averaged truss forces from the construction stage model are applied as external cable pretension loads.