## Nominal moment strength & plastic moment capacity

## Nominal moment strength & plastic moment capacity

(OP)

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

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

## RE: Nominal moment strength & plastic moment capacity

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

Also how do you come to your original conclusion.

regards

csd

## RE: Nominal moment strength & plastic moment capacity

## RE: Nominal moment strength & plastic moment capacity

## RE: Nominal moment strength & plastic moment capacity

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??

## RE: Nominal moment strength & plastic moment capacity

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 ...

## RE: Nominal moment strength & plastic moment capacity

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.

## RE: Nominal moment strength & plastic moment capacity

## RE: Nominal moment strength & plastic moment capacity

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.

## RE: Nominal moment strength & plastic moment capacity

just my 2c worth ...

## RE: Nominal moment strength & plastic moment capacity

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.

## RE: Nominal moment strength & plastic moment capacity

## RE: Nominal moment strength & plastic moment capacity

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

## RE: Nominal moment strength & plastic moment capacity

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.

## RE: Nominal moment strength & plastic moment capacity

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

## RE: Nominal moment strength & plastic moment capacity

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.

## RE: Nominal moment strength & plastic moment capacity

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.

## RE: Nominal moment strength & plastic moment capacity

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.

## RE: Nominal moment strength & plastic moment capacity

alimit 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 majororminor 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.

## RE: Nominal moment strength & plastic moment capacity

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).