FDEs

[WebPilot]

Simply Supported ...

(c.t.)

[WebPilot]

NOTE:

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

[pezman]

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".

[pezman]

... umm ...

[WebPilot]

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.

[pezman]

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.

[WebPilot]

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

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.)

[WebPilot]

With this installment I have derived the deflection equation.

(c.t.)

[WebPilot]

Before I return to the thread, I want to reassure the reader that
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.

[WebPilot]

(c.t.)

[WebPilot]

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.

[WebPilot]

[WebPilot]

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

- the little red triangles in column A are just
comments (notes to myself).

- 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.

[pezman]

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/

[WebPilot]

Pez,

For the 2nd time you have felt compelled to tell me, et al, of the
advantages of SAGE. Ok, that's your opinion.

I catch your 'drift'.

For the record, SAGE, would not be my choice for a replacement.

So far, I have done nothing difficult at all. The reader doesn't need
to be able to derive the deflection equation - he/she can look it up
in a book.

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

what is a derivative? ...

(c.t.)