## FDEs

Moderator: Mike Everman

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### calculate the sag of a SS W-beam (re: FDEs)

Simply Supported ...

(c.t.)

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### calculate the sag of a SS W-beam (re: FDEs)

NOTE:

The last file I posted has been modified since I first put it up!

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### calculate the sag of a SS W-beam

Hi Forrest,

It's literally been decades since I have had to do calculus, but I suspect that the M term that you provide is not quite right for a beam supported at both ends -- I would expect it to be symmetrical about l/2.

If this observation is correct, I have to confess that I only found this by building a SAGE notebook that is parallel to your calculations then plotting the sag. If it is incorrect, then, in the words of Emily Litella, "Never mind".
Attachments
untitled.GIF

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### the Litella function

... umm ...
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### calculate the sag of a SS W-beam (re: FDEs)

OK, Pezman, I might have made a mistake. Let's see.

Also,

as x -> 0, M(0) = 0 and
as x -> L, M(L) =0

as they should.

I stand behind what I have posted earlier.

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### Beam sag agreement

I agree that you were correct in the first place and that my "correction" was incorrect -- hence my posting of the "Litella" function.

Time permitting, I'll try to maintain a separate, parallel thread that builds a SAGE notebook illustrating your presentation. Fun for me, maybe instructive for others, and clearly I could benefit from some practice to resurrect my rusty applied math skills.

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### a little 'calculus action' (re: FDEs)

Having done the M(x) derivation, I realized there is now not
much to solving the differential equation. If you know calculus,

If you do not, well, why not?

In this day and age everything you desire to learn is 'out there' on the
Internet. I learned calculus in high school. Every technical book I
read now, uses it in some fashion. I guess what I am trying to say is,

"It's not a waste of time!"

(c.t.)
Last edited by WebPilot on Sat Feb 09, 2008 4:11 am, edited 1 time in total.

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### some 'algebraic manipulation' (re: FDEs)

With this installment I have derived the deflection equation.

(c.t.)

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### almost 2 1/2" in a 60' beam (re: FDEs)

you do not need to know 'calculus' nor how the diffEQ was solved
analytically. The analytic solution will be used as the 'right answer'
and will be used to visualize how 'close' the numerical answer
approaches it.

I have chosen to use a spreadsheet only because a lot of people
have this software, and are somewhat familiar with how to use it.

There is no need to learn another piece of software in order to
learn the basics of fdm. There is enough to learn already.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Here, I have placed the deflection equation into a spreadsheet. I am
assuming the reader can do this.

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### rearranged deflection eq. for spreadsheet use (re: FDEs)

(c.t.)

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### a few more words on x/L (re: FDEs)

Maybe you understand this; maybe you do not. It's for the
'udonots' I am writing.

Dimensionless parameter, x/L:

Let's say that you wanted to know the deflection at a point, 20',
from either end (it's a symmetrical deflection, remember). The
beam is 60' long, so

x/L = 20 feet / 60 feet = 0.333

The units, feet, cancel out so you are left with a unitless, pure number.

x/L is a dimensionless parameter.

Also note from the spreadsheet plot (mine and yours, I hope), that
the maximum 'sag' occurs at station, 0.5 or at a point 1/2 the length
of the beam.

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A lot of the input so far is just 'text'' save for:

- the little red triangles in column A are just

- in rows B5:8 I used Excel's insert - name - define capablity.
Instead of using 'cell pointers' I can now use names (e.g.
W, I, E, and L) in formulae

- in F4 I have defined X as 96 (no units in the cell, do it
in the next one)

- in F6 I have determined the deflection at this value for X.
The formulation is shown in the cell edit window.

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### For this, Excel seems like a lot of work ...

I like the presentation so far, but I would just offer the opinion that Sage has Excel soundly beat for these kinds of problems. I was already able to do all this stuff in Excel and yet found the combined effort of learning Sage and then using it to implement the problem to be far easier than implementing anything comparable in Excel.

With a free online Sage notebook accounts, everyone with a browser has Sage (if they want it). The results are nicer looking and the notebook artifacts can be integrated, e.g. into papers (for example, your equations can be emitted as Tex, which can then be pasted into OpenOffice).

So, imho (and in the spirit of learning), anyone brave enough to learn these equations is more than brave enough to learn Sage.
https://www.sagenb.org/home/pub/1678/

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### what is a derivative? (re: FDEs)

Pez,

For the 2nd time you have felt compelled to tell me, et al, of the