Torsional Warping Constant, Cw 1

[psuengr1]

Can anyone tell me how to calculate the warping constant for a composite plate girder? The formula for the girder itself is from AASHTO C6.9.4.1.3-1. However, how do I account for the transformed slab?

 

[KootK]

However, how do I account for the transformed slab?

One could make an estimate of this but I suspect that it's the wrong approach for your situation. You're doing a lateral torsional buckling (LTB) check, right? Assuming that to the be the case, you've got a few popular approaches to choose from, in order of decreasing capacity:

1) Convince yourself that the slab provides continuous torsional bracing to the beam and LTB is not possible.

2) Assume that the connection to the slab forces LTB to occur about a point of rotation located at the underside of the slab. This is called restrained axis buckling in the literature and yields a pretty high capacity. It's a fair bit of work to do the first time however.

3) Assume that your slab provides only lateral flange restraint but that LTB will occur about the shear center of the girder. This is your "normal" LTB check between points of lateral restraint.

While I do have some ideas about how to calculate Cw based on a transformed section including your slab, I'll hold off on that until you confirm that you still want to head in that direction.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[psuengr1]

Thank you for your response! Actually, what I am trying to do is model a bridge superstructure in STAAD for a long term composite section (3n) for my bridge barrier (DL2) loading. In speaking with Bentley, STAAD cannot handle a transformed section based on 3n and can only look at a short term composite section (fully composite) so I cannot include the deck as plates in my model. Therefore, they suggested that I manually input the transformed section properties for my girders based on 3n. My bridge deck is severely skewed so I am trying to calculate an equivalent torsional moment of inertia as recommended by NCHRP report 725 Sect. 3.2.2 but I need the value of Cw for my composite section in order to do so.

 

[KootK]

You're very welcome. I'm not a bridge guy and, to be honest, I don't fully understand what you're trying to do here. To me, it still sounds as though this composite Cw thing is the wrong way to go. That said, were I to attempt to calculate Cw for a composite section including a steel plate girder and a concrete slab, here's what I'd do:

1) Construct a transformed section (TS) ignoring the presence of the web. Keep your concrete width the real width and perform the transformation using the thickness instead.

2) Vertically locate the shear center of the TS. Do this by finding the combined centroid of the section with one important difference: replace the areas in the calculation with the lateral moments of inertia of each flange.

3) Calculate the lateral moment of inertia of the bottom flange and multiply it by the square of the distance from the flange centroid to the TS shear center. in^6

4) Calculate the lateral moment of inertia of the transformed top flange and multiply it by the square of the distance from the transformed flange centroid to the TS shear center. in^6.

5) Add #3 and #4 to get an estimate of Cw.

Note that this procedure gives no account of any cracking or long term effects in the concrete.


I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[Hokie93]

You might consider transforming the concrete slab to steel (using 3n) and then using the warping constant (C_w) equation in Roark's Formulas for Stress and Strain for an unequal-flanged wide-flange beam. In the 6th edition of the text, the equation is in Table 21, Case No. 7.

 

[KootK]

@Hokie: in that procedure, one would ignore the steel top flange contribution entirely, right? And just use the transformed concrete on its own? If so, I agree, that should provide a simple and fairly accurate lower bound estimate of Cw. Presumably the concrete "flange" is much wider than the steel flange.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[Hokie93]

@KootK: Yes, that is what I had in mind.

 

[psuengr1]

Thanks for your suggestions. I am still having difficulty. Please see the attached pdf. Typically, when calculating your standard strong axis and weak axis moments of inertia for a transformed section, you divide the effective slab width by the modular ratio and calculate everything accordingly. The transformed section that is used is as shown in Case 1 of the attached. I have calculated my torsional moment of inertia accordingly. However, for my last term in the equation where I am including the J term for the slab, is it correct to take the width as the slab thickness of 8" and the thickness as the transformed slab width of 3.87" as I have done? The other issue is depending on whether you transform the slab width or the slab thickness you can get substantially different result. I am having a tough time figuring out what the correct procedure would be to correctly estimate the torsion stiffness of the composite beam.

 

[KootK]

@KootK: Yes, that is what I had in mind.

Yeah, that would be... much easier really.

The transformed section that is used is as shown in Case 1 of the attached.

In my opinion Case 1 is not applicable to either the J or the Cw calculation. It should be case two for Cw with the following properties:

1) Transformed Concrete Width = physical concrete width.

2) Transformed Concrete Vertical Centroid Location = vertical location of centroid for physical concrete slab

For Cw, the thing that matters most is the lateral moment of inertia of the transformed slab/flange. And that's all about width as the width is the h in bh^2/12.

However, for my last term in the equation where I am including the J term for the slab, is it correct to take the width as the slab thickness of 8" and the thickness as the transformed slab width of 3.87" as I have done?

While it's surely possible to come up with a transformed section for J, I doubt that it would be worth the trouble. It would be a different transformed section than you're using for Cw. I think that you'd be better off just adding in a bt^3/3 term to your total using the actual concrete dimensions and shear modulus. Moreover, you may want to reconsider using the J value of the concrete at all. If concrete cracks in torsion, it's torsional stiffness drops upwards of ten fold.

We've been talking about Cw but are you trying to come up with one transformed section for use with strong axis bending, weak axis bending, Cw, and J? If so, I'm not sure that's even possible.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.