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# Allowable bending stress of rectangular plate

## Allowable bending stress of rectangular plate

(OP)
I'm using the 9th edition of the AISC steel manual and am getting myself all confused.  I have a situation where I'm trying to use a steel .25" x 2.5" steel plate to stiffen an aluminum "beam" in bending.  I'm orienting the plate such that I'm loading it in bending along the strong axis, where my Mom of I = .325in^4.  I can only laterally brace this piece at 12" intervals.  I suspect that when this thing is bent about the strong axis that my compression edge(not sure I can call it a flange as I'm basically just analyzing a prismatic rectangle) is going to buckle out of plane.  I'm confused as to what my allowable bending stress would be for 36ksi plate steel.

How do I look at this in regards to Table B5.1?  Do I even need to?  It seems to me that I need to look at the "Unstiffened elements simply supported along one edge......." which gives me a limiting width-to-thickness ratio of 76/Fy^.5.  Is that right?  If so, then what do I do with that then in Chapter F?  There is no axial load carried by this piece.....only bending.

### RE: Allowable bending stress of rectangular plate

As you've noticed, the 9th edition doesn't do a good job of addressing strong-axis bending on a rectangular plate.  Use the 13th edition, where specification F11 does deal with this case, including lateral torsionla buckling.

(OP)

### RE: Allowable bending stress of rectangular plate

remember too, dissimiliar materials ... "rule of mixtures"

### RE: Allowable bending stress of rectangular plate

PostFrameSE

The manual tables are not there, but all of the spec is and that is all you need.

### RE: Allowable bending stress of rectangular plate

Dissimilar metals corrosion and expansion/contraction - plus the extreme difficulty of 100% joining of the reinforcing plate (in steel) to the (more flexible!) aluminum base structure?

I don't recommend this configuration at all.  Better to use a thicker Al plate.  Then weld the Al-Al.

### RE: Allowable bending stress of rectangular plate

Be careful when designing welded aluminum sections.

Depending on the type of weld (longitudinal or transverse) the allowable stresses of a welded section can be significantly reduced.  See the Aluminum Design Manual.

### RE: Allowable bending stress of rectangular plate

PostFrameSE - simply go to this link:

http://www.aisc.org/content.aspx?id=2884

...and download the FREE AISC specification (Specification for Structural Steel Buildings (ANSI/AISC 360-05)
) which is inside the 13th Edition.

### RE: Allowable bending stress of rectangular plate

(OP)
I appreciate all of your responses.  I'm dealing with a 2.5" x .25" thick member bent about the strong axis.  Given 36ksi steel, a Cb=1.0 and Lb=12", could somebody confirm that my calcs are correct, in that my allowable bending stress would be 29,246psi and my moment capacity is 7,604in-lbs?  If that's right, I'm comfortable with this methodology.

Thanks

### RE: Allowable bending stress of rectangular plate

Lb*d/t^2 = 480 < 1.9 E/Fy
Z= bt^2/4 = 0.3906 in^3
FyZx = 14.0625 k-in

I am getting an Mn based on equation F11-2 19.0791 k-in.

### RE: Allowable bending stress of rectangular plate

(OP)
Thanks slickdeals,

If I break out my formula more it would look like this:
Mn = [1.52 - .274(480)*36/29,000]*My

That becomes 1.3567*My

My I believe should be 36*.26 = 9.36in-k since Sy=.25*2.5^2/6 = .26

Did you use Mp rather than My???

Therefore, I think Mn = 1.3567*9.36 = 12.69in-k.

My allowable moment capacity = 12.69/1.67 = 7.6in-k.

Assuming I'm wrong and you're right, where did I go wrong?

Thanks

### RE: Allowable bending stress of rectangular plate

Ref. Theory of Elastic Stability by Timoshenko and Gere

For pure bending of a narrow rectangular section about its major axis:

Mcr = π√(E*Iy*G*J)/L

E = 29,000,000 psi
G = 12,000,000 psi
Iy = b*h3/12 = 0.003255 in^4
J = b*h3/3 = 0.01302 in^4
L = 12"

So Mcr = 31,800"# > My = 9375"#

Plastic moment, Mp = 14060"#

For an unbraced length of 12", lateral buckling does not appear to govern, so you can use the allowable stress specified by your code.  I would not expect it to be as high as 29,246 psi, however.

BA

### RE: Allowable bending stress of rectangular plate

PostFrameSE:
Yes, you are right. It appears My should be used and not Mp because you are not reaching Mp.

### RE: Allowable bending stress of rectangular plate

(OP)
Hopefully this is my last question.  If we can agree that the yield moment is around 7.6in-k (including the 1.67 reduction factor), doesn't it follow that the allowable fiber bending stress is = to the M / S?  If S is = .26in^3, then the fiber bending stress has to 29.2ksi right?  Am I missing something BA??

Thanks all.

### RE: Allowable bending stress of rectangular plate

Your nominal moment, Mn, is higher than the yield moment, My, but not all the way to the plastic moment, Mp.

So, in theory and in line with the factor of safety, your stress diagram will not be linear.  The extreme fiber stress will be 21.6 ksi (Fy/1.67).  As you come down the member from the top fiber, the stress will remain 21.6 ksi, until you reach a point where it does start to decrease, until you have zero stress at the neutral axis.

In actuality, the extreme fiber stress is 29.2 ksi.  Since this is below yield, the stress diagram will remain linear.

### RE: Allowable bending stress of rectangular plate

PostFrameSE,

The moment, 'My' at first yield is:

My = Fy*S = 36,000*0.2604 = 9.375"k

I don't use the AISC code and I don't use ASD, so I can't remember for certain, but as I recall, the maximum allowable stress you can use is Fy/1.67 = 21.65"k which corresponds to an allowable moment of 5.6"k.  This would suggest that one of us is missing something.

BA

### RE: Allowable bending stress of rectangular plate

am kinda lazy to look for it.. where can i find how they come up with 0.75 for base plate

baseplate ASD
Fb = 0.75 Fy

### RE: Allowable bending stress of rectangular plate

westy, that is in the compact shape direction, he is bending in the other direction, the non-compact direction.

Michael.
Timing has a lot to do with the outcome of a rain dance.

### RE: Allowable bending stress of rectangular plate

BA, the AISC equation is checking for lateral torsional buckling, which does control.

If it did not control, you'd be allowed to go to Mp, not just My, in which case the allowable moment would be 50% higher than the values you listed.

### RE: Allowable bending stress of rectangular plate

nutte,

The original question in this thread was in respect to allowable stresses.  Under no circumstances would you be able to go to Mp if you are using ASD.  The maximum allowable stress using ASD is Fy/1.67 as I understand it.

As for the AISC equation, I can't comment because I don't have it, but the equation I provided earlier was for lateral torsional buckling of a narrow rectangular section according to Timoshenko and Gere.  Are you saying it is wrong?

BA

### RE: Allowable bending stress of rectangular plate

Under ASD, you are allowed to go to Mp.  In the old ASD, for a wide flange, this was done using 0.66 Fy for the allowable, being a 10% increase over the normal 0.6 Fy, correlating to a general ratio of Zx/Sx for wide flange shapes equal to 1.10.

Under the current AISC 13th edition ASD, you're allowed to go to Fy Zx / 1.67 for your moment, if LTB doesn't control.

In this case, using AISC's LTB equations, you get a nominal moment higher than My, but less than Mp.

I'm not as familiar with the Timoshenko/Gere equations, so I can't comment on them.  Regardless, the AISC equations do say that LTB controls.

### RE: Allowable bending stress of rectangular plate

nutte,

I wasn't aware of the change to ASD, but I admit it makes sense.

I cannot believe there is such a huge difference in results between AISC and Timoshenko for lateral torsional buckling.  Could you post the AISC equation for comparison?

BA

### RE: Allowable bending stress of rectangular plate

nutte,
I'm still trying to understand the difference between your reference and mine and, for that matter between your code and mine.  The following is another reference on lateral torsional buckling which appears to tie in with Timoshenko and Gere:

http://www3.imperial.ac.uk/portal/pls/portallive/docs/1/2075917.PDF

BA

### RE: Allowable bending stress of rectangular plate

BA:

The Timoshenko equation you posted is the same as Equation F11-4 in the AISC specification.  Equation F11-3, the one that applies to our specific case, must be an for an inelastic buckling region.

Timoshenko:  Mcr = pi*√(E*Iy*G*J)/L

F11-4:  Fcr = 1.9 E Cb / (Lb*d/t^2)

We can manipulate Timoshenko's equation as such:

Iy = d*t^3/12
G = E/2.6
J = d*t^3/3

Mcr = pi*E^0.5*(E/2.6)^0.5*(d*t^3/12)^0.5*(d*t^3/3)^0.5/L

Mcr = 1.948*E*d*t^3/6/L

Multiply both top and bottom by d, resulting in:

Mcr = 1.948*E*d^2*t^3/6/L/d

Replace d*t^2/6 on top with Sx:

Mcr = 1.948*E*t^2*Sx/(L*d)

Remove Sx, to get Fcr, and pull t^2 to the bottom.

Fcr = 1.948*E/(L*d/t^2)

For Cb=1, this is AISC's equation, with 1.948 rounded to 1.9.

### RE: Allowable bending stress of rectangular plate

nutte,

Yes, thanks for that.  I guess we are in the inelastic buckling range with this problem.  Tricky little devil, isn't it?

BA

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