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From: Nasser M. Abbasi on 9 Dec 2009 16:33
The Euler beam equation is 4th order ODE given by

E I y''''[x]= load

For a beam, of some length L, and load is defined as intensity load, i.e.
force per unit length in same direction as y.

http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_equation

But what if the load on the beam is an applied external couple M as some
point? say at m=L/3.

How would one add this as a "load" ? If I were to simulate it as a point
force P going up and another point force P going down separated by some
delta from each others such that they result in the applied couple acting at
the point m, then this would work, and would give the correct solution for
y(x), but the problem now is that I introduced an artificial shear force
bump at the point m, where it did not exist before.

i.e. Given this


couple here, say clock wise
-----------------------o-------------------------------------------


I change the above to the following


P
^
|
+-----------------+---+-------------------------
|
v
P

Where I make sure that 2*P*delta=M where delta is the half the distance
between the 2 forces above.

Now I can write the ODE as

E I y''''[x] = P*dirac(x-m-delta)-P*dirac(x-m+delta)

Where dirac is the dirac delta function.

and now can solve it.

But the shear force, which comes from y'''(x) now has that extra bump which
comes from these 2 phantom forces P which were added which is not correct.

I can solve this problem easily by starting from the moment diagram and
integrate it twice to get y(x), but wanted to see how to do it starting from
the original Euler ODE equation.

Any other ideas on how to approach this?

thanks,
--Nasser


From: eric gisse on 9 Dec 2009 17:07
Nasser M. Abbasi wrote:

[...]

> I can solve this problem easily by starting from the moment diagram and
> integrate it twice to get y(x), but wanted to see how to do it starting
> from the original Euler ODE equation.

Why can't you just integrate the ODE directly?

>
> Any other ideas on how to approach this?
>
> thanks,
> --Nasser

From: Nasser M. Abbasi on 9 Dec 2009 17:42

"eric gisse" <jowr.pi.nospam(a)gmail.com> wrote in message
news:hfp757$2tp$2(a)news.eternal-september.org...
> Nasser M. Abbasi wrote:
>
> [...]
>
>> I can solve this problem easily by starting from the moment diagram and
>> integrate it twice to get y(x), but wanted to see how to do it starting
>> from the original Euler ODE equation.
>
> Why can't you just integrate the ODE directly?
>
>>
>> Any other ideas on how to approach this?
>>
>> thanks,
>> --Nasser
>

The problem is how to set the load (the RHS of the ode, i.e. the forcing
function).

Euler ODE as given, is defined for forcing function which is either a point
load or a distributed load, not a couple.

As I said, if I replace the couple by an equivalent 2 point forces, this
introduces an internal shearing force bump which is not present in the
original problem.

This is an academic exercise, as the problem can be easily solved by other
means. I just was wanted to see how solve it from the 4th order ODE itself,
but to do this, I need to figure how to set up the load right.

--Nasser


From: eric gisse on 9 Dec 2009 18:37
Nasser M. Abbasi wrote:

>
> "eric gisse" <jowr.pi.nospam(a)gmail.com> wrote in message
> news:hfp757$2tp$2(a)news.eternal-september.org...
>> Nasser M. Abbasi wrote:
>>
>> [...]
>>
>>> I can solve this problem easily by starting from the moment diagram and
>>> integrate it twice to get y(x), but wanted to see how to do it starting
>>> from the original Euler ODE equation.
>>
>> Why can't you just integrate the ODE directly?
>>
>>>
>>> Any other ideas on how to approach this?
>>>
>>> thanks,
>>> --Nasser
>>
>
> The problem is how to set the load (the RHS of the ode, i.e. the forcing
> function).
>
> Euler ODE as given, is defined for forcing function which is either a
> point load or a distributed load, not a couple.

Why is this a problem?

Why does the ODE handle a point load or a load defined over the entire beam
but *not* a load on two points? It is like saying the heat equation can
handle sources/sinks at one point or a continuum of points, but not two.

>
> As I said, if I replace the couple by an equivalent 2 point forces, this
> introduces an internal shearing force bump which is not present in the
> original problem.

As would any non-constant load. Different loads on different points would
impart shearing forces regardless.

>
> This is an academic exercise, as the problem can be easily solved by other
> means. I just was wanted to see how solve it from the 4th order ODE
> itself, but to do this, I need to figure how to set up the load right.
>
> --Nasser

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