confused with AISC code about section modulus 1

[Duckinline]

The definition of Sc in F10-7 of Specification for Structural Steel Building (2005) is not clear to me. The note on page 16.01-60 says: "Sc=elastic section modulus to the toe in compression relative to the axis of bending. For bending about one of the geometric axes of an equal-leg angle with no lateral-torsion restraint, Sc shall be 0.8 of the geometric axis section modulus". My problem is a L3x2x3/16 single angle beam with lateral bracing subjected to uniform loading. In my book, Sc is equal to the section modulus of the extreme fiber in compression about the X-axis passing through the centroid of the whole section. However, my colleague thinks Sc should be the section modulus of the small segment (rectangular shape) of the leg in compression, regarding to the horizontal axis of the small segment. Obviously, my own Sc is larger than that of my co-work. Which interpretation is right?

 

[akastud]

If the toe is in compression (ever) then your colleague is correct.

akastud

 

[Duckinline]

Thank you, akastud. Is the Sc the section modulus of the whole vertical rectangular web, or of just the small segment below the N.A.?

 

[JAE]

Sc is for the whole section, referred to the compression side of the section.

If you remember, S = I/y where I is the moment of inertia of the entire cross section about the applicable axis.

y is the distance from the NA to the point on the section you are dealing with. So if you have a leg tip in compression, y is the distance from the NA to that tip. If the other side is in compression, then y is the distance to that point.

For your L3x2x3/16, the properties are:

Ix = 0.847 in4
Sx1 = 0.414 in3 (this is referred to the tip of the 3" leg
Sx2 = 0.89 in3 (this is referred to the heel of the 3" leg

So if your 3" leg tip is in compression, you would have Sc = 0.414. If the heel is in compression, Sc = 0.89.

Based on your description in your original post, the use of just a portion of the cross-section to calculate a section modulus (per your colleague) is incorrect.

 

[Duckinline]

JAE, I have the same thoughts as you have. But, I cannot convince my colleague, because per my interpretation the allowable bending stress of a L3-1/2X4X1/4 section, in which the 4" vertical leg's tip in compression, is larger than the yield stress. That is not making sense. The calculation is as following.
L3-1/2x4x1/4 section. The N.A.distance from flange (3-1/2" wide) is 1.14". The Sx=1.01 for the whole section. If Sc=Sx in F10-7, the allowable stress before local buckling is 57ksi for 50ksi yielding steel. Do this make sense?

 

[hokie66]

You are obviously very confused. JAE has given you an example of how to calculate the elastic section modulus. But the section modulus does not determine the allowable bending stress. The section you quoted reduces the design section modulus on the compressive side by a factor of 0.8, but it says nothing about allowable stress.

 

[csd72]

The difference between the behaviour of a W shape and that of an angle is that the angle will go into biaxial bending around its major and minor axis(w and z).

This bending would result in both a horizontal and a vertical component of bending/deflection.

When you have continuous lateral restraint this negates the lateral component and therefore you can treat it as bending around the x-x axis only.

For laterally unrestrained angles, I believe that the 0.8 factor allows for the additional stress from the lateral bending component.

I agree with JAE.

 

[miecz]

The Commentary to section F10 describes this in depth.

 

[Duckinline]

Given a L3-1/2x4x1/4 angle with lateral restraint to prevent torsion, to find out the nominal flexural strength Mn, F10-7 applies and gives Mn=Fy*Sc*(2.43-1.72*(b/t)*Sqrt(Fy/E)). Fy=50ksi, E=2.9e7psi, b=4",t=1/4". If Sc is interpreted as the section modulus of whole section at the compressed tip, Sc=1.01. Mn=65 k-in. Then, the Fbx=0.9*Mn/Sx, provided Sx=Sc, Fbx=57.9Ksi >Fy. Does this conclusion make sense? I have to admit this is a very interesting topic. Thank you all guys, and all comments are welcomed.

 

[JAE]

Duckinline,

Your value of 65 in-kips you calculated is correct. This is the value of Mn at which the leg will theoretically buckle locally.

I believe that the moment capacity of your angle is, per AISC, the minimum of yielding, lateral torsional buckling, and leg local buckling.

Since you state that your angle is fully braced, LTB doesn't apply.

So the yield moment is (per equation F10-1):

Mn = 1.5My = 1.5(Sx)(Fy) = 1.5(1.01)50 = 75.75 in-kips

So the lower value is the local leg buckling mode = 65 in-kips per your calculation above.

Therefore:
[φ]Mn = 0.9(65) = 58.5 in-kips

 

[Lion06]

duckline-
I think part of the problem is that you are using the elastic section modulus (Sx) to calc your Fbx, but a portion of the vertical leg is yielding in compression before it reaches your 0.9Mn moment. The actual yield moment is 50.5 ki-in. This isn't a yielding capacity as defined in F10-1, but it is the moment at which the section will begin to yield. F10-1 is when the entire section has yielded.