Cb Modification Factor 2

[12345abc6ttyui67]

Hi,

First foray into the AISC design codes, specifically 360-05 at the moment. I see that Cb can be calculated for different load patterns and different restraint conditions, and Cb is then carried through into determination of the allowable moment in Chapter F2.2 (if LTB governs).

Consider a simply supported beam, restrained against LTB / lateral flange displacement at mid span. Loads include a central point load, in combination with a uniformly distributed load along the entire length.

Cb for these cases are 1.67 (point load) and 1.3 (UDL). I realize Cb can conservatively be taken as 1.0, so in these cases it would also be conservative to use 1.3 (the lesser of 1.67 and 1.3 as appropriate in this case).

Just to check my thinking on this, I can either:
Conservatively use Cb = 1.3 as worst case for both load patterns.
Carry out two checks, first for the point load applied moment / allowable moment (using Cb = 1.67), combined with second check of UDL applied moment / allowable moment (using Cb = 1.3) and combining utilization factors and checking < 1.0.

Is there any 'quick' way to determine an 'average' Cb factor (based on the magnitude of point load to UDL for instance) which makes the most of the different Cb values, but doesn't require two checks?

 

[Lomarandil]

Sure, 1.3 or there are lots of rational ways to manipulate Cb as you describe. Cb itself isn't linear, but how it affects utilization is.

(assuming your beam is in the bucking controlled range where Cb applies, and that the sources of load act in the same direction)

----
The name is a long story -- just call me Lo.

 

[Hokie93]

A better approach is to draw the bending moment diagram and substitute the required values into Eq. (F1-1).

 

[12345abc6ttyui67]

Lo,

Do you have any guidance on manipulating Cb? I can get results to tie up correctly when doing two separate checks and combining utilisations at the end, but whenever I try and simplify things I start getting bogus answers.

Hokie93,

I would usually just draw the BMD, but I'm trying to create some logic which simplifies this step. For instance I'd like some 'reference tables' which allow Cb to be determined directly for different cases, like:
- If there is a central point load three times the UDL then Cb = XXX for the combined case
- If there are two point loads at 3rd points, total equal to the UDL, then Cb = XXX for the combined case

I'm actually quite surprised there are not more Cb values in the AISC Manual besides the dozen or so cases in Table 3.1. Essentially I am aiming to expand this table [EDIT: without having to draw out 100's of BMD's [/EDIT] to cover more cases where there are point loads, with UDLs, and with pinned or fixed supports. I've only just got stuck into this, so maybe I am missing something obvious, or maybe this data exists elsewhere I am unaware of?

 

[JAE]

I don't think you can separate the load types and derive separate Cb values. I've never heard of doing it that way at all.

Use a lower Cb value for conservatism - or calculate the Cb directly using the quarter point moments and maximum moment per AISC's formula for Cb as Hokie93 suggests.

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[RWW0002]

Here are a few other cases, but are based on a knowledge of the expected bending diagram shape and magnitude. This is from the ASIC webinar Steel Design After College

 

[Agent666]

As others have noted, you really need the final moment diagram as a starting point or use Cb = 1.0. There was a recent thread that showed that Cb < 1.0 can apply for cantilevers with no end restraint and an applied moment at the tip, so taking Cb = 1.0 as being conservative isn't always a sure thing if you have unusual restraint or loading situations.

You can't take the lowest value of the constituent load/moment diagrams like you want to, I can think of situations where the summation of loads is closer to the Cb = 1.0 case if loading cancels out to give a more constant moment gradient. Either work it out as intended or take Cb = 1.0, don't take the lowest value for the different types of loading you have, this can be non-conservative.

If the final moment diagram is the summation of the simple known load/moment relationships like the examples RWW0002 posted, then you can work out the moment for each of these constituent cases at the required locations and use superposition to construct a final moment diagram and then use the equation with this moment diagram. This is obviously pretty simple for your simply supported beam case. Simply calculate the moments at the relevant locations and apply the AISC equation.

Another option is working from first principles and undertaking a buckling analysis to calculate the Cb factor.

 

[12345abc6ttyui67]

RWW0002,

Thanks, these are useful!

Agent666,

Good point, I'd not considered that the final BMD may end up with a Cb less than either of the constituent load cases. Not sure why that hadn't registered, so thanks for pointing that out!

JAE,

I think I've been caught out because I'm also trying to verify the output from my analysis program vs. my hand calcs, so I've been taking simple cases (e.g. point load and restraint at mid span) to make sure Cb calculates correctly in the program, and then I've been trying to build up proof it also applies correctly for combined cases by trying to combine Cb values for individual cases.



Sounds like I'm heading off on a bit of a tangent so will just do the BMD for the combined cases and use F1-1.

 

[iv63]

Most of structural analysis softwares can calculate Cb automatically using the quarter point moments and maximum moment per AISC F1-1 (I know for sure STAAD and RISA can do it).
iv