Maximizing I Beam Strength

[ThirdString]

I am working with what is essentially a cast I beam. What I would like to do is to maximize strength and minimize weight for a given load. I only have general limitations on the dimensions. I started trying to solve the derivative of my moment of inertia equation, but it's been a bit too long. My reference books are not too much help either.
My question: does someone have an equation that I can use for this (and possible future projects) situation? Like I said, all dimensions can vary for my purpose.

 

[JAE]

Is this for a bridge beam?

The University of Nebraska developed a new kind of precast prestressed I bridge girder using studies to maximize the efficiency of the beam during initial release, construction loads, and design loads. Its called the NU girder.

Here's a link: NU Girder

 

[UcfSE]

You should try another book then, or use MathCAD or MATLAB to solve your diff-eq. There might even be a calculator or two out there with solvers that can handle what you have.

 

[ThirdString]

It's actually for a car jack stand. We do some testing that puts it into bending, which of course we're trying to minimize.

Now that I consider my specific problem a little more, I can only change the webbing. However, I am still interested in maximizing I and minimizing weight for future designs.

Thanks for the link, that was a great read.

 

[JStephen]

I don't think you'll find a cook-book approach to optimizing your particular beam.

If you apply a fairly rigid set of rules (say, a structural steel design code) to a beam section, you can then optimize the dimensions on a spreadsheet. Even this approach would get fairly involved, as you have to allow for the reductions for thin sections, allow for fillet radii, etc. But in the case of something like a jack stand, where it may have a ratchet mechanism on it, etc., you could only apply a steel code in an approximate sort of way, so I don't know that it would be worth the bother.

If you only look at section modulus and moment of inertia, without regard to buckling issues, then the optimum shape is to make the cross section enormous with paper-thin elements, which is obviously not a realistic design. If you hold the outside dimensions and disregard buckling issues, the optimum design is to make the web paper thin and put all the material into the flanges, which is once again an unrealistic design.

 

[ThirdString]

Thanks for the input JStephen. As I was running some numbers, I was getting the same results. So I'll throw buckling consideration into the mix and, with the outer dimensions i have, hopefully come up with something worthwhile!

 

[MichSt]

I don’t know how deflections play into your design, but they control over strength for most. You may want to consider a minimum moment of inertia to limit deflections within a reasonable limit.

 

[MikeHalloran]

You're probably optimizing the wrong equations.

The "i-beam" in a typical jackstand is a column with eccentric loading, not a beam.



Mike Halloran
Pembroke Pines, FL, USA

 

[ThirdString]

MichSt, deflections are the main concern. I suppose we do have a minimum MOI to work with, which is what I'm working with for the redesign. I'll have to remember to start with that for the new design.

Mike, I think you're right. Actually, in addition to material yielding, our stands must not deform in such a way to cause a 0.125" height change. I've found the equation for column displacement with an eccentric load, but that gives me horizontal displacement. What I need is the vertical displacement at the point of the load. I'll keep looking, I'll probably have to check out a structural engineering book I imagine.
Would it be reasonable to think I can get a good estimate for vertical displacement based on the before-and-after geometry calculated from the horizontal displacement?

 

[mitchelon]

I may be too late here, but if you need to calculate vertical displacement, here is what you should do. I take from your statements that we are talking about a steel section. Assuming buckling effects are insignificant, find the maximum stress on the cross section (e.g. P/A or P/A +- Pey/I) and obtain the strain from the material’s stress-strain diagram. This should give you a good approximation of the vertical deflection (“we ain’t making watches”). Unless you were doing this as a college problem and need to find the ideal section, the most efficient/economical section for this application should be one that is already available in the market.

 

[ThirdString]

We cast our own, so the specific size can be changed rather than purchased.

Thanks for the input, it's never to late! Well, the other answers pretty much helped me get what I needed to, but more info isn't a bad thing.