Singly Symmetric Reinforced Steel Beam - Plastic Moment vs Yield Moment

[MelBWasHere]

I have an (E) 50ksi wide flange beam that will be reinforced with a 50ksi 1"x12" cover plate stitch welded to the bottom flange. The beam is in positive bending (top flange in compression, bottom reinforced flange in tension).

I want to check the web slenderness. According to attached AISC 360-16 Table B4.1b Case 16, I need to calculate the plastic bending moment (M[sub]p[/sub]) and the moment at yielding of the extreme fiber (M[sub]y[/sub]). "Of the extreme fiber" is ambiguous to me, but it suggests that there is technically an M[sub]yt[/sub] & M[sub]yc[/sub], in this case moment at yielding of the extreme tension fiber (bottom flange/cover plate) and moment at yielding of the extreme compression fiber (top flange).

I naturally thought to use M[sub]yt[/sub] since that's where the coverplate is, but figured it would be more conservative to use M[sub]yc[/sub] since that is the lesser of the two (also valid when considering M[sub]y[/sub] = F[sub]y[/sub]S[sub]x[/sub], where S[sub]x[/sub] = the minimum elastic section modulus taken about the x-axis, per the index).

However, while calculating, I realized M[sub]yt[/sub] > M[sub]p[/sub], which confused me since the plastic moment should be larger than the yield moment. This inequality suggests that permanent deformation occurs before yielding, which doesn't make sense to me. I understand why the elastic section modulus at the bottom is larger than that at the top. Can anyone explain what Myt > Mp physically means?
I thought at first it had to do with the plastic section modulus, but it remains the same, regardless of whether we're looking at the compression flange or the reinforced tension flange. I also thought maybe Mp = FuZx, but AISC defines it as FyZx.

Values:
M[sub]yc[/sub] = 50ksi*S[sub]xc[/sub] = 1028.621 kip-ft
M[sub]yt[/sub] = 50ksi*S[sub]xt[/sub] = 1867.696 kip-ft
M[sub]p[/sub] = 50ksi*Z[sub]x[/sub] = 1389.11 kip-ft
S[sub]xc[/sub] = 246.869 in[sup]3[/sup]
S[sub]xt[/sub] = 448.247 in[sup]3[/sup]
Zx = 333.387 in[sup]3[/sup]

 

[driftLimiter]

I suppose this could happen if your section was soo unsymmetric that the entire compression region is at the yield stress, and the extreme tension fiber hasn't even reached yield stress.

Generally I think of Mp as being both the compression and tension region of the cross section at uniform yield stress, but I don't normally deal with highly unsymmetric shapes.

 

[Celt83]

What are the wide flange dimensions?

How did you calculate S and Z?

Also make sure you are checking against section F4 in ANSI/AISC 360 since the new shape is singly symmetric.

 

[MelBWasHere]

See attached images. S & Z come from SkyCiv. I've been using F4 for my reinforcement calcs, but I figured I'd check web slenderness to see if I could use F5 instead. However, this issue does come up when calculating the web plastification factor. I'm sure the capacities are fine, but I wanted to address the specific occurrence of the yield moment being larger than the plastic moment, since I've never seen that before.

 

[271828]

I naturally thought to use Myt since that's where the coverplate is, but figured it would be more conservative to use Myc since that is the lesser of the two

My is the min of Myc and Myt.

Can anyone explain what Myt > Mp physically means?

It doesn't mean much. With singly symmetrical members, the min of Myc and Myt is real, but the other is fictitious.

The following is the stress distribution at Myt. All this calculation knows is there's a linear stress distribution (My/I), Fy at the bottom, and zero stress at the centroid. It doesn't care that the stress exceeds Fy for most of the compression zone.

 

[271828]

You're generally on the right track. Follow Table B4.1 Case 16. With My = min(Myc,Myt) the method should work out ok and you'll be able to determine the web slenderness. Then you'll be heading over to F4 (compact or noncompact web) or F5 (slender web) for the Mn calcs.

 

[Celt83]

That cover plate size drives the plastic neutral axis into the bottom flange, this makes the moment arm for the tension region very small. May also be influenced by the fact that this wide flange shape is slender in compression, the PNA position with the cover plate essentially puts 90%+ of the section into compression.