Nominal moment strength & plastic moment capacity

[rmc1980]

Hi,
How is that for a rectangular cross-section bent about the minor axis, the nominal moment strength is equal to plastic moment capacity??
Isn't nominal moment strength = stress x b x d^2/6
plastic moment capacity = stress x b x d^2/4

 

[csd72]

rmc,

This is an international forum. Which country and code are you using?

Also how do you come to your original conclusion.

regards

csd

 

[rmc1980]

This is a sentence that I came across and wasn't able to understand while reading the beam base plate design in the Steel Design text book by Segui

 

[271828]

Post-1989 AISC, the nominal strength has been based on the plastic moment.

 

[rmc1980]

I am sorry but I didn't quite clearly understand what 271828 said..Is there any explanation for my doubt based on mechanics of rectangular section?
Is it because d<<<b that when the section bends about the minor axis there is not much difference between its nominal moment capacity and plastic moment capacity??

 

[rb1957]

i'd say that the plastic bending form factor for a rectangle is 1.5, as you've applied.

possibly, reading your last post, if d <<< b then the nominal bending moment is so low, that 1.5x is still very low, so why bother ...

 

[271828]

It doesn't have anything to do with b vs d.

In AISC steel design post-1989, in almost all cases (bars, circles, I-shapes, etc.), the nominal flexural strength is the plastic moment. The excepts are usually sections for which stability controls--such as local buckling of a beam flange, lateral-torsional buckling of an I-shape bent about its strong axis, etc.

 

[UcfSE]

I think you are getting confused with the terminology. The nominal moment strength is whatever we choose; we define it to be what we want, feel safe using, calculated or arrived at empirically. So in this case we're saying that the highest load the rectangular section bent about its minor axis can take is equal to the plastic moment. That is being defined as the nominal moment. Perhaps this definition of nominal moment has changed since you last investigated this particular bit of engineering.

 

[Lion06]

rmc1980-
The reason that a rectangular section has a nominal moment capacity = stress*b*d^2/4 is that there are no buckling concerns for this type of shape loaded in this manner. Because of this, you can achieve the full plastic moment capacity of the beam before any buckling would occur.
That being said, you would use the plastic section modulus instead of the elastic section modulus.
The plastic section modulus for a rectangular shape is bd^2/4 as opposed to bd^2/6 for the elastic section modulus.
This is the same reason that you can use 0.75Fy as an allowable bending stress instead of 0.6Fy or 0.66Fy in the green book.

 

[rb1957]

i don't know your specific structural business as well as i know airplane structure, but i would have thought that bending the weak axis of a rectangle (with d<<<b) would be susceptable to local buckling, and that the maximum compression stresses would be limited by fcy (often less than fty) but then you carry a factor of safety (by limiting your allowable to .75fy), but i'd query using a higher allowable for plastic bending; possibly i'd query how the relatively large deformation required to generate these stresses would be allowed by the strcuture around the item in question.

just my 2c worth ...

 

[Lion06]

rb1957-
The difference between LRFD and ASD is that you wouldn't consider the deflections that would cause the "ultimate" or plastic moment capacity of the beam. You check deflections with the service loads. At ultimate strength, you are concerned with strength not serviceability. Serviceability is based on service loads.
Also, bending about the weak axis REDUCES the suceptibility to buckling, locally or otherwise. Buckling is resisted by, among other things, the stiffness of the member in the axis perpindicular to loading. If that axis is stronger than the axis of bending (as is the case when bending a rectangular section about its weak axis) then buckling is not an issue. Buckling is an issue when the section can't support the plastic moment capacity before it will buckle, that is when you must consider it.

 

[JAE]

StructuralEIT - I think what rb1957 was asking about was not Lateral Torsional Buckling (LTB) but Local Flange Buckling, which does become an issue if the tube width/thickness ratio gets large. (i.e. chapters B and K in the AISC specs)

 

[Lion06]

JAE-
I would agree with that condition, but I just read rectangle, not tube.

 

[271828]

rmc, I think you need to better define what you have. Is it a solid rectangular bar or a tube as JAE suggests?

By b & d, are you referring to one of those as the thickness of a tube?

Do you have a light-gage tube with b&d as the side dimensions? In that case, you'd be right that the b/t ratio of the longer side would probably be controlled by local buckling. Segui probably wouldn't cover that--need to go to a cold formed steel textbook like the one by Yu. There's all kinds of crazy post-buckling stuff to deal with in that case.

 

[apsix]

271828
Original question states that it is a rectangle, the formulas indicate that it is solid.

rmc1980
The 'nominal moment strength' is also equal to the plastic moment capacity for bending about the major axis if there is enough restraint to prevent Lateral Torsional Buckling (although some Design Codes may stipulate otherwise).
As said above this is only true for LRFD, not ASD

 

[SuperStressed]

I am not certain about my response, however my understanding of structural mechanics has led me to the following...

I believe that you need to refer back to first principles, as opposed to relying upon the predefined formulae as stated above.

For any section, the calculation for moment capacity is based upon the section modulus, be it plastic or elastic. However, for a symmetrical section, the plastic neutral axis and elastic neutral axis are located at the same point; straight through the centroid of area (assuming the rectangle consists of a single material). So as Force(Compression)=Force(tension)=M/z, where z is normally elastic but in this plastic neutral axis, then your moment calculations are directly related to plastic analysis.

 

[apsix]

SuperStressed

With respect there appears to be a couple of errors in your post.
First, M/z = extreme fibre stress (not force).
Secondly, when calculating stress due to bending of a ductile section you can not choose either elastic or plastic z. Bending will be elastic until yielding first occurs. Further increase in the moment will then result in first partial plastic and finally fully plastic bending. Throughout this plastic stage the maximum stress will be the yield stress.

 

[BlastResistant]

rmc –

You are correct on all accounts: yes, the nominal moment capacity is equal to plastic moment capacity. As you state the nominal moment strength Mn, is equal to Fy x S, where S is the elastic section modulus, b x d^2/6. And, the plastic moment strength, Mp is equal to Fy x Z, where Z is the plastic section modulus, bxd^2/4.

If you look closely at the equations for S and Z, you will see that S=(2/3)Z. Per AISC equation F1-1, for solid rectangular bars bent about the minor axis, the maximum value allowed for Mn is Mp = 1.5 x My. Hence, if My = Fy x S, then Mn = 1.5 x Fy x S = 1.5 x Fy x (2/3)Z = Fy x Z = Mp. Therefore, for flat rectangular bars the maximum value for Mn is Mp.

For compact sections, you need to check Z/S with respect to the axis of bending, which for a flat rectangular bar bent about the weak axis would be is 1.5. This is the factor you multiply My by to get the equation Mp = 1.5 My, which as described above equals Mn.

This is for LRFD, which I believe Segui’s text is based on.

Hope this helps!

By the way, just to clarify previous posts which might be a little confusing, for rectangular bars LTB is the applicable limit state for major axis bending. Local buckling is not a limit state for either major or minor axis bending.

 

[vonlueke]

BlastResistant: The so-called nominal flexural strength, Mn, is Fy*Z as stated in your second and third paragraphs, not Fy*S as stated in your first paragraph. And this Mn value also applies to bending about the major axis (if the member has adequate lateral bracing), not just bending about the minor axis. This was pointed out by apsix. And this Mn value also applies to ASD (in the current AISC code), not just LRFD. Last, lateral-torsional buckling is a limit state for major axis bending of rectangular sections, not "the" limit state as stated in your last paragraph.

rmc1980: The term nominal flexural strength is somewhat semantic. What might be clearer is the actual allowable bending stress on your solid rectangular cross section. According to the current AISC specification (AISC 2005), or AISC Steel Construction Manual (SCM), 13 ed., 2006, if the strength limit state governs, then the allowable bending stress is 0.90*Fy (for bending about the major or minor axis) for allowable strength design (ASD). Or for limit states design (LSD), called load and resistance factor design (LRFD) in USA, the equivalent, unfactored allowable bending stress (corresponding to unfactored M*c/I) for solid rectangular sections is 0.964*Fy if 100% of the applied load is dead load. These allowable stresses are significantly higher than the 0.75*Fy cited above by StructuralEIT. By the way, the above publication changed the name "allowable stress design (ASD)" to "allowable strength design (ASD)."

In other words, for solid rectangular sections in bending about their major or minor axis, subjected to unfactored (service) loads, the above publication states that the factor of safety against extreme fiber yielding is 1.11 for ASD; or only 1.037 for LRFD if all load is dead load.

 

[ThingsLearner]

As I was reading through this threat, I could find the correct deffinitions of elastic (the whole section remains elastic with triangular stress distribution) and plastic (the whole croos-section is under Sigma_p) bending moment capacities.

The only thing that I want to add from structural mechanics point of view when I am not talking about any code is this:

Pay attention that when you say Mp=stress*b*d^2/4 you are assuming "elastic perfectly-plastic material behavior".

All of these results are coming from basic equilibrium of your cross section. (nothing more complicated or confusing).