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From: tadchem on 9 Dec 2009 17:46
On Dec 9, 4:33 pm, "Nasser M. Abbasi" <n...(a)12000.org> wrote:
> 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

Green's Functions

Tom Davidson
Richmond, VA
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