Non-convergence of LRFR Iterative Shear Rating using General Procedure for Prestressed Concrete Members
This document describes a convergence issue encountered for an LRFR analysis of a post-tensioned multi-cell box (MCB) superstructure with the ‘General procedure – Appendix B5’ and ‘Consider Iterative Shear Rating’ control options selected. The BSSD ticket associated with this issue #5167.
Since the LRFD shear capacity is load-dependent for both factored moment and shear, the iterative shear rating procedure attempts to determine the total factored load for which the factored shear demand, 𝑉𝑢, is equal to the factored shear resistance, 𝑉𝑟.
The total load is composed of the dead load component (with subscript D) and the live load component (with subscript L). Let the factor 𝜆 be the live load multiplier. Then, the total shear and flexural demand can be expressed as follows:
𝑉𝑢 = 𝑉𝐷 + 𝜆∙𝑉𝐿
𝑀𝑢 = 𝑀𝐷 + 𝜆∙𝑀𝐿
Since the shear resistance is dependent on the shear and the flexural demands, it can be expressed as a function of the live load multiplier, i.e., 𝑉𝑟 = 𝑉𝑟(𝜆).
The iterative shear rating procedure attempts to find the specific value of the live load multiplier, 𝜆0 , such that:
𝑉𝑢(𝜆0) = 𝑉𝑟(𝜆0)
If the 𝜆0 value is found, then it becomes the rating factor, i.e., 𝑅𝐹 = 𝜆0.
The iterative process starts with the initial value of live load multiplier 𝜆 = 1, and then the value of 𝜆 is either decreased (Figure 1) or increased (Figure 2) depending on whether the initial shear resistance is smaller or larger than the initial shear demand.
In the first scenario where demand exceeds resistance initially, the live load multiplier is reduced during the iteration process causing reduction in total loads and an accompanying increase in resistance. Eventually demand and resistance become equal.
Similarly, in the second scenario where resistance exceeds demand initially, the live load multiplier is increased during the iteration process causing an increase in total loads and an accompanying decrease in resistance. Eventually demand and resistance become equal.
A problem can develop in the iteration process, however, whenever the sign of the dead load and live load moments are different. As the live load multiplier varies, the sign of the total moment demand can reverse. When the total moment demand reverses, the shear resistance can abruptly change due to implementation of the Modified Compression Field Theory (MCFT) in the AASHTO LRFD Bridge Design Specifications being predicated on the amount of prestressed and mild steel reinforcement located based on one-half the member depth, i.e., H/2.
To illustrate the issue, two specific load cases were evaluated for the controlling POI location in the BSSD‑5167 bridge model. The controlling shear load rating occurs in WEB 8 - Span 1 @ 88.30’ (0.8L pt). The controlling load case is ‘Min V with Concurrent M’. Two different vehicles were included in the analysis, 1) ‘P9 Split’ and 2) ‘P11 Split’. These are standard Caltrans permit vehicles and are very similar except the ‘P11 Split’ has two additional axles at the end. The load effects (moments and shears) caused by these two vehicles are very similar at the controlling POI location. For both permit vehicles, HL-93 live load was included in the adjacent lane.
The following tables show the effect of varying the LL load multiplier (l) on the load demands (Mu & Vu) and shear resistances (Vr). Table 1 is for the ‘P9 Split’ vehicle while Table 2 is for the ‘P11 Split’ vehicle.
Since the initial shear resistance exceeds the initial shear demand, the live load multiplier (l) is increased during the shear iteration process. As the live load multiplier increases, the magnitude of total negative moment demand decreases and, at a certain point, the total moment switches sign from negative to positive. For the ‘P9 Split’ live load case, the LL load multiplier when this sign reversal occurs is approximately 1.60, while for the ‘P11 Split’ live load vehicle, it is approximately 1.86.
This switching of sign of the total moment is not necessarily problematic except for post-tensioned concrete members having the prestress defined as a single force applied at a single C.G. height in combination with use of the ‘General procedure’ (or ‘General procedure - Appendix B5’) for the shear capacity computation method. For this situation, the total moment sign reversal usually results in a significant, abrupt change in shear resistance depending on whether the prestress C.G. is located above or below mid-height (H/2) of the member depth.
Furthermore, even the significant, abrupt change in shear resistance is not necessarily problematic except when the shear demand falls between the larger and smaller shear resistance values at the point of abrupt change in shear resistance. When this occurs, the shear iteration process gets caught in an oscillation pattern going back and forth between the change in sign of total moment where the shear resistance is discontinuous, and it fails to converge on a solution.
Figure 3 shows the situation for the ‘P11 Split’ vehicle in which the shear demand line intersects the shear resistance line between the larger and smaller shear resistance values at the point of discontinuity where the sign of total moment changes. For this vehicle, the program fails to converge on a solution. However, for the ‘P9 Split’ vehicle, the shear demand line does intersect with the shear resistance line just beyond the shear resistance point of discontinuity, and a solution is found.
When the shear iteration process fails to converge on a solution, BrR reports out the initial condition without any shear iteration being performed. This can potentially lead to very disparate rating factors being reported out for seemingly similar load cases at a given POI as shown below in Article ‘6A.4.2.1 Concrete Shear General’ spec check detailed output — the ‘P9 Split’ converged on an iterated solution while ‘P11 Split’ did not converge on an iterated solution.
The user needs to be aware of this potential issue and recognize it is not an error in the program, but rather an anomaly of the ‘General procedure’ shear capacity computation method.
The abrupt change in shear resistance is caused by whether the area of prestressing steel gets included in the computation for longitudinal strain, es (or, ex for Appendix B5) acting on the member cross-section. Aps can only be included for one sign of total moment because it can be located either above or below the C.G., not both. The area of prestressing steel not included for a particular sign of moment significantly reduces compressive strain in the member resulting in lower shear resistance.
Non-convergence of the shear iteration process usually occurs near contraflexure regions where the sign of moment can reverse and where the C.G. of post-tensioning transitions between the upper and lower regions of the member depth. Since total moment in a contraflexure region is typically much lower than the maximum moments, there is a high likelihood the member will not crack, i.e., Mu < Mcr. Because of this, it is recommended to select the ‘General Procedure’ for shear capacity along with the ‘Modify MCFT Theta’ control option.
When the ‘Modify MCFT Theta’ control option is used, the program will check if Mu < Mcr at the initial stage, and, if so, will compute the shear resistance using b = 4.8 and q = 29 degrees, and the issue of the program failing to converge on an iterative solution will be avoided. It is important to note that BrR only evaluates the ‘Modify MCFT Theta’ control option when the shear computation method is set to the ‘General Procedure’ (not the ‘General Procedure - Appendix B5’).